Strategic Approach in Multi-Criteria Decision Making: A Practical Guide for Complex Scenarios (International Series in Operations Research & Management Science, 351) [2nd ed. 2024] 3031444523, 9783031444524

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International Series in Operations Research & Management Science

Nolberto Munier

Strategic Approach in Multi-Criteria Decision Making A Practical Guide for Complex Scenarios Second Edition

International Series in Operations Research & Management Science Founding Editor Frederick S. Hillier, Stanford University, Stanford, CA, USA

Volume 351 Series Editor Camille C. Price, Department of Computer Science, Stephen F. Austin State University, Nacogdoches, TX, USA Associate Editor Joe Zhu, Business School, Worcester Polytechnic Institute, Worcester, MA, USA Editorial Board Members Emanuele Borgonovo, Department of Decision Sciences, Bocconi University, Milan, Italy Barry L. Nelson, Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, IL, USA Bruce W. Patty, Veritec Solutions, Mill Valley, CA, USA Michael Pinedo, Stern School of Business, New York University, New York, NY, USA Robert J. Vanderbei, Princeton University, Princeton, NJ, USA

The book series International Series in Operations Research and Management Science encompasses the various areas of operations research and management science. Both theoretical and applied books are included. It describes current advances anywhere in the world that are at the cutting edge of the field. The series is aimed especially at researchers, advanced graduate students, and sophisticated practitioners. The series features three types of books: • Advanced expository books that extend and unify our understanding of particular areas. • Research monographs that make substantial contributions to knowledge. • Handbooks that define the new state of the art in particular areas. Each handbook will be edited by a leading authority in the area who will organize a team of experts on various aspects of the topic to write individual chapters. A handbook may emphasize expository surveys or completely new advances (either research or applications) or a combination of both. The series emphasizes the following four areas: Mathematical Programming : Including linear programming, integer programming, nonlinear programming, interior point methods, game theory, network optimization models, combinatorics, equilibrium programming, complementarity theory, multiobjective optimization, dynamic programming, stochastic programming, complexity theory, etc. Applied Probability: Including queuing theory, simulation, renewal theory, Brownian motion and diffusion processes, decision analysis, Markov decision processes, reliability theory, forecasting, other stochastic processes motivated by applications, etc. Production and Operations Management: Including inventory theory, production scheduling, capacity planning, facility location, supply chain management, distribution systems, materials requirements planning, just-in-time systems, flexible manufacturing systems, design of production lines, logistical planning, strategic issues, etc. Applications of Operations Research and Management Science: Including telecommunications, health care, capital budgeting and finance, economics, marketing, public policy, military operations research, humanitarian relief and disaster mitigation, service operations, transportation systems, etc. This book series is indexed in Scopus.

Nolberto Munier

Strategic Approach in Multi-Criteria Decision Making A Practical Guide for Complex Scenarios Second Edition

Nolberto Munier INGENIO, Polytechnic University of Valencia Valencia, Spain

ISSN 0884-8289 ISSN 2214-7934 (electronic) International Series in Operations Research & Management Science ISBN 978-3-031-44452-4 ISBN 978-3-031-44453-1 (eBook) https://doi.org/10.1007/978-3-031-44453-1 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2019, 2024 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

What do Elon Musk, Albert Einstein, Nicola Tesla and Madame Curie have in common? They all show that it is OK to be unconventional. Separateness had helped the innovators be independent thinkers, freeing them to break the rules and ignore the assumptions that constraint others. —Melissa A. Schilling

Preface and Road Map

Book Structure The book is divided into four parts, as follows: Part 1: Theory and Analysis of MCDM Problems: History of MCDM and how it is performed (Chaps. 1 and 2) Part 2: Theory and Analysis of MCDM Problems: What can be done by using the MCDM process? (Chaps. 3–5) Part 3: Theory and Analysis of MCDM Problems: Proposes SIMUS as a strategic procedure to tackle real-world scenarios (Chaps. 6–11) Part 4: Practice of Problem Solving Using MCDM: Support for Practitioners (Chaps. 12–14)

vii

Introduction

The decision-making process is a human activity in which the human being, as the decision maker, can hardly escape from the influence of multiple circumstances that, in the end, give shape to what will become the winning decision. With the aim of reaching this winning decision, Multicriteria Decision Making (MCDM in short) has become one of the most important and fastest growing subfields of Operations Research and Management Science. It started in the Second World War with the contribution of Kantorovich and continued with the modern MCDM, under the influences of the Utility Theory in a first instance and Multiple Objective Mathematical Programming as a second stream of influence. There is plenty of literature that examines and analyzes the MCDM timeline as a discipline. Although there is a clear advance of the MCDM field with the incorporation of new methods, the discipline that analyzes MCDM processes has evolved in a way that might indicate that the roots of its own existence have been forgotten, by not considering some critical aspects that are key in the correct interpretation of a scenario, and regardless which method is used. The author of this book observed that under the comfortable “umbrella” of continuity, there is an incessant number of MCDM methods that are not restricted by any kind of normative or protocol to guide them, nor to assure the quality of the assessment. As a result of the above, the author wants to put in evidence an old claim of many scholars in MCDM who are worried about the issue that, for a same problem, MCDM methods deliver different results, anomaly that is known as the “DecisionMaking Paradox.” Although, this only represents a technical problem that, of course deserves attention, it arises other elemental questions like “Are all existing MCDM methods valid to solve all kind of problems?” However, there is no procedure in the MCDM field to guide the DM in this quandary. One wonders whether it is possible to decommission or discontinue this current pattern, by proposing as an alternative a structured decision process more in line with actual requirements of MCDM scenarios and especially with reality, some sort of a framework within which the MCDM techniques have to be applied. Our motivation for this book derives from this point, in the sense that current MCDM methods seem ix

x

Introduction

to be more concerned with the mathematics behind their application than from an accurate application of the MCDM principia as a discipline. Therefore, the objective of this book is twofold: First, to highlight the need for a larger debate on some critical issues regarding the application of MCDM methods, and as a second objective it aims at providing the reader a strategic, practical, and structured guide to deal with multiple and complex scenarios. To address these significant issues, the book proposes an innovative procedure. Keeping in mind the previous objectives, this author will not go further in this direction of current MCDM process. Instead, our line of reasoning focuses on the immobility that affects the discipline regarding some structural aspects. Consequently, this author believes that it is necessary to innovate, to look for new ways to solve the same problems differently and more efficiently. In this respect, just as almost everyone could agree with this introductory paragraph, it is highly probable that all scholars of MCDM would also agree with the fact that, at present, the essence of these methods is governed by mathematical procedures and subjective assumptions, being the latter something inherent to the human being, and by ignoring many aspects that are present in a scenario. The author will come back to this critical point later on. Instead, our main concern is rooted in the origins of the structural pillars that give coverage to one assumption that, in many cases, is taken for granted. It refers to the supposed rational reasoning of the Decision Maker (DM) in the process of defining criteria and weighting them in order to select alternatives or scenarios, which later can provoke other problems such as the above mentioned. If the MCDM process is supposed to provide support to the DM and avoiding subjectivity, then it is worth to provide the DM with techniques or methods that keep away, as far as possible, the possibility of introducing value judgments that may not represent reality. It is not said that the MCDM process is based only on good data and on a mathematical procedure, and then leaving the DM in a secondary role. This book considers the DM as the most important element of the MCDM process and aims to put him/her where is most needed, in time and in form. In the author’s opinion, the DM is most needed at the end of the process, examining results, analyzing consequences that the best alternative must generate, and providing stakeholders with a wide spectrum of different possibilities, and especially risks, that may jeopardize the best selection due to subjective assumptions and uncertain data. Two main concepts have been explained so far, but not enough highlighted: It was said that this book is mainly strategic and that MCDM process had to provide techniques to the DM. Based on these concepts, a question arises: Is it supposed that the DM must provide useful and strategic information to stakeholders? The answer to this obvious question is one of the principal contributions of this book and probably its largest strategic value. This work delivers a free software for modelling complex scenarios which incorporates an innovative sensitivity analysis (SA) that to the author’s knowledge has never been developed. The output of this SA gives strategic answers to the decision-making process. Related to these complex scenarios, this book does not provide the reader with a procedure, as the typical manual that explains, step by step, what have to be done to

Introduction

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solve a problem; instead, it describes what should be done to tackle some classical problems, and to consider reality which is, in essence, a complex scenario. This is the main and perhaps more important pillar of the book, and it is performed by building a frame of reference for the description of the problem and its alternatives, as objective as possible. The book acknowledges that decision-making is to a large extent a subjective affaire. It recognizes that it is very necessary because the different mathematical methods are only tools and not designed to give solutions, but to support and help the DM, who can use this tool to make a rational and documented decision. The word “documented” is key, since acting on results the DM has a solid base on which base his decisions. This is why we think that the DM’s crucial role is at the end of the process, not at the beginning, as is normal nowadays. Linked to this, it cannot be lost in sight the fact that a problem is surrounded by a series of elements and collateral elements that the DM cannot leave aside when modelling reality, as well as the reality of the problem. Consequently, we want to bring up again the need to address scenarios in their full complexity, considering the circumstances and the collateral elements it is composed of. This book, with practical foundations, provides the reader with a template as a guidance to reflect, as much as possible, complex scenarios, and this is one point that we understand is missing in the present-day MCDM modelling. As a bottom line, this book highlights the idea of a systemic representation of the problem if the aim is to keep it as close to the reality as possible. As it was mentioned above, obtaining different solutions to the same problem is an uneasiness for many MCDM scholars since its reasons have not been thoroughly explained or understood, and let alone solved, although it is revealed on many occasions. Sometimes, the different solutions are due to considering different types of weights for criteria or by ignoring interrelations between criteria, or by parceling out a problem. This book could be classified as a practical guide because certain concepts or situations are explained in a simple way and those explanations are accompanied by numerous examples to be able to support them, but in addition it aims to work with a new paradigm in MCDM. The book is structured in three clearly different parts. The first one is devoted to exploring through the history and development of the discipline and the way it is performed nowadays. It specifically involves Chaps. 1 and 2. Included in this part, the book highlights those drawbacks and problems that scholars have identified in the different MCDM methods and techniques. As indicated above, the motivation to raise this aspect is to provoke the necessary debate on the validity of the theoretical pillars that sustain the discipline, considering the generalized absence of representing reality. The second part of the book includes Chaps. 3–5 and with the intention of answering an important question: What should be done to assure a quality MCDM process? The purpose is to offer a theoretical response to the drawbacks identified in the first part. Finally, the third part encompasses Chaps. 6–12 and is devoted to introduce and explain in a simple language and using graphic aids the Linear Programming concept

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Introduction

and the SIMUS method, based on Linear Programming, as the new toolkit that is suggested to deal with MCDM process. In Chap. 8, it is analyzed and wholly exemplified a new procedure for sensitivity analysis, which is always of the utmost importance in decision-making. As in most parts of the book, the explained procedure is innovative and based on sound mathematical principles. It provides examples that sustain what was said above about the kind of information that stakeholders need. Chapter 9 is devoted to Group Decision-Making using SIMUS. An actual and complex example is provided together with a simulation of debate amongst members of the group. The system is based on a progressive analysis of the scenario by sequentially addressing each objective, considering potential changes and examining their applicability or not, measured by quantified values. Chapter 10 tackles a very important aspect; it is related to selecting the best strategy and using the very well-known SWOT (Strength, Weakness, Opportunities, and Threats) technique. It is exemplified by a complex and actual scenario, and the result quantitatively selects the best strategy, and so doing it is a step forward, since SWOT finishes by determining the SWOT matrix of strategies, but not selecting the best one. Chapter 11 analyzes the reasons for lack of agreement amongst results from different methods and proposes the use of a proxy method which would allow to determine the closest solution to the proxy. Chapter 12 is some sort of tutorial that simulates receiving 101 different queries and responding them. Chapter 13 addresses best practices in MCDM. Chapter 14 details and solves real-life complex scenarios in different fields. Finally, in the appendix, the theory of linear programming is explained in tabular format for easy comprehension. It is completed with a very important issue, since it demonstrates through eight different examples that SIMUS is not subject to rank reversal.

Prologue to this Second Edition

Springer Nature, considering the interest for this book’s first edition (2019), decided to launch a second edition revised and considerably expanded. The first edition is oriented to the theory of MCDM, by analyzing, discussing in depth, and illustrating every component of this discipline, aiming for the practitioner to have a clear understanding, and be able to tackle many different problems or scenarios, examinations of results, and their analysis. In other words, it gives the knowledge of what to do. This second edition is in reality a tutorial that explains how to proceed. It starts with Chap. 12, which is a novelty in technical books, since it simulates receiving questions and consultations from students, practitioners, and MCDM professors, about many different aspects of this discipline. There are about 100 questions. This second edition simulates receiving questions of the most diverse type and answers them in three stages: (1) Explanation of the meaning and scope of the question, (2) Real-world application example, and (3) Discussion. In this way, the requester acquires a complete knowledge. Not all queries receive the same treatment, mainly due to its relative importance, and may be a couple of lines, while others take many pages, with tables and graphics and screen capture of results. This author is transferring in this chapter 30+ years’ experience in studying and solving MCDM problems and situations. There is no doubt that people, mainly students and practitioners, have many questions when they are in need of solving a problem using MCDM. This is the purpose of Chap. 12; the 100 questions are not only answered, but also explained, and in many cases, illustrated with the modelling and solving of complex real-life scenarios. The questions involve all aspects of MCDM. Chapter 13 is devoted to modelling, the most important steps in MCDM, while Chap. 14 heavily draws from more than 60 papers written by this author and addresses and enlarges capital points on the themes treated in the first 11 chapters of the book. xiii

Contents

Part I 1

2

Theory and Analysis of MCDM Problems: History of MCDM and How It Is Performed

Multi-criteria Decision-Making, Evolution, and Characteristics . . . 1.1 History and Evolution of Multi-criteria Decision-Making Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Some Background Information on Decision-Making . . 1.2 Introduction to Most Common and Used Heuristic Methods . . . 1.3 The Decision-Making Paradox . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Which Is the Best MCDM Method? . . . . . . . . . . . . . . . . . . . . 1.5 Considering and Modelling Reality . . . . . . . . . . . . . . . . . . . . 1.6 Is It Possible to Represent Reality Faithfully? . . . . . . . . . . . . . 1.7 Conclusion of This Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Initial Decision Matrix and Its Relation with Modelling a Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Basic Components of the Initial MCDM Decision Matrix . . . . 2.1.1 Stakeholders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Decision-Maker or Group of DMs . . . . . . . . . . . . . . 2.1.3 Objective/s That the Scenario Must Attain . . . . . . . . 2.1.4 Scenario/s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.5 Alternatives, Projects or Options . . . . . . . . . . . . . . . 2.1.6 Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.6.1 Areas Included in Criteria . . . . . . . . . . . 2.1.6.2 Capacity of Criteria to Evaluate Alternatives . . . . . . . . . . . . . . . . . . . . . 2.1.6.3 Actions for Criteria . . . . . . . . . . . . . . . . 2.1.6.4 Resources and Restrictions for Criteria . . 2.1.7 Performance Values . . . . . . . . . . . . . . . . . . . . . . . . 2.1.8 Decision Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.9 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 3 7 8 10 11 11 13 13 15 15 15 16 16 16 16 17 17 21 23 23 24 24 24 xv

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2.2 2.3

2.4 2.5

2.6 2.7

Routines to Perform with Data . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rank Reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Possible Causes for RR . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Brief Information on Rank Reversal in Different MCDM Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2.1 Rank Reversal in AHP . . . . . . . . . . . . . . 2.3.2.2 Rank Reversal in TOPSIS . . . . . . . . . . . 2.3.2.3 Rank Reversal in PROMETHEE . . . . . . 2.3.2.4 Rank Reversal in ELECTRE . . . . . . . . . 2.3.2.5 Rank Reversal in SAW . . . . . . . . . . . . . The Uncertain Best Solution . . . . . . . . . . . . . . . . . . . . . . . . . Characteristics of Components of the Initial Decision Matrix . . 2.5.1 The MCDM Process as a System . . . . . . . . . . . . . . . 2.5.2 Alternatives Relationships . . . . . . . . . . . . . . . . . . . . 2.5.3 Alternatives Heavily Related: A Case—Selecting Proposals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.4 Including and Excluding Alternatives—Conditions by a Third Party . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.4.1 Actual Cases . . . . . . . . . . . . . . . . . . . . . 2.5.5 Forced Alternatives—An Actual Case: Fulfillment of Previous Commitments . . . . . . . . . . . . . . . . . . . . 2.5.6 Criteria Selection . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.7 Resources—An Actual Case: Oil Refinery . . . . . . . . 2.5.8 Criteria Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.9 Annual Budget Restriction—An Actual Case: Five Yrs Development Plan . . . . . . . . . . . . . . . . . . . . . . 2.5.10 Criteria Correlation . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.11 Risk: A Fundamental Criterion . . . . . . . . . . . . . . . . 2.5.12 Examining Differences in Results for the Same Problem Between Assumed Weights and Weights from Entropy: Case Study—Electrical Transmission Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.13 Working with a Variety of Performance Values—An Actual Case: Environmental Indicators . . . . . . . . . . . 2.5.14 The “Z” Method for Determining Some Performance Values for Qualitative Criteria . . . . . . . . . . . . . . . . . 2.5.15 The Z Matrix—CASE STUDY: Determining Risk Performance Values for Inputting in Risk Criteria . . . 2.5.16 Need to Work with Performance Values Derived from another Data Table . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.17 Conditioning the Decision Matrix to Obtain a Specified Number of Results . . . . . . . . . . . . . . . . . . . . . . . . . Additional Conditions Required for Methods . . . . . . . . . . . . . Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 The Two Types of Sensibility Analysis . . . . . . . . . .

24 25 26 28 30 30 31 31 31 31 32 32 32 33 35 36 36 37 37 38 38 38 39 40

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2.7.2

A Critical Analysis of the Way Sensitivity Analysis Is Performed Nowadays . . . . . . . . . . . . . . . . . . . . . 2.8 Conclusion of This Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part II 3

4

5

54 57 58

Theory and Analysis of MCDM Problems: What Can Be Done by Using the MCDM Process?

How to Shape Multiple Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Developing the Best Strategy: Case Study—Selecting Projects for Agribusiness Activities in Different Scenarios . . . . . . . . . 3.3 Solving the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Conclusion of This Chapter . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. .

63 63

. . . .

65 72 72 72

The Decision-Maker, a Vital Component of the Decision-Making Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Decision-Maker (DM) Functions—Interpretation of Reality . . . 4.1.1 First Level: Building the Initial Decision Matrix . . . . 4.1.2 Second Level: Selecting a Method to Use . . . . . . . . . 4.1.3 Third Level: Following the Process . . . . . . . . . . . . . 4.1.4 Fourth Level: Examining the Result . . . . . . . . . . . . . 4.1.5 Synergy Between the DM and the Method . . . . . . . . 4.2 Conclusion of This Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73 73 73 76 78 78 79 79 80

Design of a Decision-Making Method Reality-Wise: How Should it Be Done? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Interpreting Reality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Areas Where Reality Is Not in General Interpreted . . . 5.2.1.1 Scenarios . . . . . . . . . . . . . . . . . . . . . . . 5.2.1.2 Alternatives . . . . . . . . . . . . . . . . . . . . . 5.2.1.3 Criteria . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1.4 Performance Values . . . . . . . . . . . . . . . . 5.2.1.5 Results Delivered by MCDM Methods . . 5.3 Check List for Aspects to Be Normally Considered When Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Working Template for Modeling a Scenario in MCDM and for Selecting a Method to Solve it . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Conclusion of This Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81 81 85 86 86 87 88 88 89 89 91 91 98

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Part III 6

7

8

Theory and Analysis of MCDM Problems: Proposes SIMUS as a Strategic Procedure to Tackle Real-World Scenarios

Linear Programming Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Basic Mathematical Background . . . . . . . . . . . . . . . . . . . . . . 6.2 The Initial Decision Matrix (IDM) . . . . . . . . . . . . . . . . . . . . . 6.3 Solving the LP Problem Graphically: Case Study—Power Plant Based on Solar Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 The Two Sides of a Coin . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Description of the Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Graphical Explanation of Correlation . . . . . . . . . . . . . . . . . . . 6.7 Is Rank Reversal Present in Linear Programming? . . . . . . . . . . 6.8 Conclusion of This Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

101 101 103

The SIMUS Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Background Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 How SIMUS Works—Case Study: Power Plant Based on Solar Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Normalization by SIMUS . . . . . . . . . . . . . . . . . . . . 7.3 SIMUS Application Example: Case Study—Power Plant Based on Solar Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Special Circumstances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Ties in Scores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Need to Use Formulae for Performance Factors . . . . 7.4.3 Errors in the Decision Matrix . . . . . . . . . . . . . . . . . 7.4.4 Dealing with Non-linear Criteria . . . . . . . . . . . . . . . 7.5 Is SIMUS Affected by Rank Reversal? . . . . . . . . . . . . . . . . . . 7.6 Testing SIMUS in Rank Reversal . . . . . . . . . . . . . . . . . . . . . . 7.6.1 Case 1: Investment in Renewable Sources of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.2 Case 2: Rehabilitation of Abandoned Urban Land . . . 7.6.3 Case 3: Determining Sustainable Indicators . . . . . . . 7.7 Solving Multi Scenarios Simultaneously . . . . . . . . . . . . . . . . . 7.7.1 Analysis of Global Solution—What to Produce and where? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7.2 What Projects Go into Each Scenario . . . . . . . . . . . . 7.8 Conclusion of this Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

117 118

Sensitivity Analysis by SIMUS: The IOSA Procedure . . . . . . . . . . 8.1 Background Information . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Example - Agroindustry for Export . . . . . . . . . . . 8.2 Data that the DM Must Input in IOSA . . . . . . . . . . . . . . . . . 8.3 DM Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

104 107 108 109 115 115 116

120 126 127 131 131 132 135 136 138 138 139 142 143 148 148 150 152 152 155 155 157 160 162

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8.4 8.5

Sequence for Sensitivity Analysis by SIMUS/IOSA . . . . . . . . . Report to Stakeholders: Type of Concerns and Questions Expressed by the Stakeholders Relative to this Production Problem and DM Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Conclusion of this Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

164

Group Decision-Making Case Study: Highway Construction . . . . . 9.1 Background Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Construction of the Decision Matrix: A Case – Construction of a Highway in China . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Loading Data into SIMUS . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Step-by-Step Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Detailed Analysis by the Group . . . . . . . . . . . . . . . . . . . . . . . 9.5.1 First Objective (Minimize Construction Cost) . . . . . . 9.5.2 Second Objective (Minimize Maintenance Cost) . . . . 9.5.3 Third Objective (Minimize Delays in Transit) . . . . . . 9.5.4 Four Objective (Maximize Safety) . . . . . . . . . . . . . . 9.5.5 Fifth Objective (Maximize Lighting) . . . . . . . . . . . . 9.5.6 Six Objectives (Minimizes Breaking Connectivity Between Areas Due to the Highway) . . . . . . . . . . . . 9.5.7 Seventh Objective (Minimize Construction Time) . . . 9.5.8 Eighth Objective (Environmental Impacts) . . . . . . . . 9.5.9 Ninth Objective (Minimize Traffic Noise) . . . . . . . . 9.6 Conclusion of this Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

169 169

9

10

11

SIMUS Applied to Quantify SWOT Strategies . . . . . . . . . . . . . . . . 10.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Application Example: Case Study—Strategy for Fabricating Electric Cars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Construction of the Numerical SWOT Matrix . . . . . . . . . . . . . 10.4.1 Market and Government . . . . . . . . . . . . . . . . . . . . . 10.5 Preparing an Excel Matrix with Data . . . . . . . . . . . . . . . . . . . 10.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7 Conclusion of This Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analysis of Lack of Agreement Between MCDM Methods Related to the Solution of a Problem: Proposing a Methodology for Comparing Methods to a Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Objective of this Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Causes for Discrepancies on Results . . . . . . . . . . . . . . . . . . . . 11.3 Subjective Preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.1 Subjective Weights . . . . . . . . . . . . . . . . . . . . . . . . .

164 167 167

170 172 174 174 174 176 177 178 178 179 180 180 181 183 184 185 185 186 189 190 190 194 196 198 198

199 199 201 201 201

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11.3.2 Objective Weights . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.3 Inconsistencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.4 Evaluating Results . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.5 The Proxy Approach . . . . . . . . . . . . . . . . . . . . . . . . 11.3.6 Selecting a MCDM Method . . . . . . . . . . . . . . . . . . 11.3.7 The DM Role . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.8 What MCDM Method Can Be Chosen as a Proxy? . . 11.3.9 Measuring Similitude Between Rankings . . . . . . . . . 11.3.10 Example as How Rankings Can Be Compared . . . . . 11.4 Conclusion of this Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part IV 12

13

202 203 204 204 205 207 207 209 211 213 214

Practice of Problem Solving Using MCDM: Support for Practitioners

Support and Guidance to Practitioners by Simulation of Questions Formulated by Readers and Detailed Answers and Examples . . . . . 12.1 Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Sequence of the MCDM Process . . . . . . . . . . . . . . . . . . . . . . 12.3 Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Resources and Limits for Criteria . . . . . . . . . . . . . . . . . . . . . . 12.5 Performance Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7 Group Decision Making . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.8 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.9 SWOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.10 Sensitivity Analysis (SA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.11 Role of the Decision-Maker (DM) . . . . . . . . . . . . . . . . . . . . .

219 224 230 267 281 282 289 290 291 299 300 302

Best Practices: Modelling and Sensitivity Analysis in MCDM . . . . . 13.1 The SIMUS Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.1 Composite Indexes . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.2 Macro Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.3 Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.4 Alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.5 Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.6 Performance Values . . . . . . . . . . . . . . . . . . . . . . . . 13.2.7 Attributes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.8 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.9 Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.10 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Modelling and the Role of Stakeholders . . . . . . . . . . . . . . . . . 13.4 Areas Where SIMUS Has Been Used . . . . . . . . . . . . . . . . . . . 13.5 Comments and Advices in Black, Examples in Italics . . . . . . .

305 305 306 306 308 308 309 309 310 310 311 311 311 312 312 314

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13.6 13.7

Recommendations to Practitioners . . . . . . . . . . . . . . . . . . . . . Complex and Complicated Scenarios . . . . . . . . . . . . . . . . . . . 13.7.1 Background Information . . . . . . . . . . . . . . . . . . . . . 13.7.2 Aspects to Consider by the DM . . . . . . . . . . . . . . . . 13.7.2.1 Data Acquisition . . . . . . . . . . . . . . . . . . 13.7.2.2 Criteria Selection . . . . . . . . . . . . . . . . . . 13.7.2.3 Weights . . . . . . . . . . . . . . . . . . . . . . . . 13.7.2.4 Criteria Units . . . . . . . . . . . . . . . . . . . . 13.7.2.5 Criteria Types . . . . . . . . . . . . . . . . . . . . 13.7.2.6 Cardinal Data . . . . . . . . . . . . . . . . . . . . 13.7.2.7 Selecting a Method . . . . . . . . . . . . . . . . 13.7.2.8 Working with a Method . . . . . . . . . . . . . 13.7.2.9 Difficulties than May Be Encountered in Interpreting a Solution . . . . . . . . . . . . 13.7.2.10 Role of Sensitivity Analysis . . . . . . . . . . 13.8 Using SIMUS for Decision-Making . . . . . . . . . . . . . . . . . . . . 13.9 Analyzing Variations in Criteria Limits (RHS) . . . . . . . . . . . . 13.10 Analyzing Variations in Alternatives Scores . . . . . . . . . . . . . . 13.11 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Some Complex and Uncommon Cases Solved by SIMUS . . . . . . . . 14.1 Case Study: Simultaneous Multiple Contractors’ Selection for a Large Construction Project . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1.1 Background Information . . . . . . . . . . . . . . . . . . . . . 14.1.2 The Case: Construction of a Large Power Plant . . . . 14.1.3 Conclusion of this Case . . . . . . . . . . . . . . . . . . . . . 14.2 Case Study: Quantitative Evaluation of Government Policies Regarding Penetration of Advanced Technologies . . . . . . . . . . 14.2.1 Background Information . . . . . . . . . . . . . . . . . . . . . 14.2.2 Process Structure . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.3 The Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.4 Analysis of Different Policies . . . . . . . . . . . . . . . . . 14.2.5 Conclusion of This Case . . . . . . . . . . . . . . . . . . . . . 14.3 Case Study: Selecting Hydroelectric Projects in Central Asia . . 14.3.1 Background Information . . . . . . . . . . . . . . . . . . . . . 14.3.2 Conclusion of This Case . . . . . . . . . . . . . . . . . . . . . 14.4 Case Study: Community Infrastructure Upgrading for Villages in Ghana . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.1 Background Information . . . . . . . . . . . . . . . . . . . . . 14.4.2 Areas and Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.2.1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . 14.4.3 Conclusion for This Case . . . . . . . . . . . . . . . . . . . . 14.5 Case Study: Urban Development Study for the Extended Urban Zone of Guadalajara, According to Sustainability Indicators, Mexico . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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317 317 318 320 320 320 323 323 325 325 325 326 326 328 329 333 335 337 337 339 339 339 341 350 350 350 351 352 360 361 361 361 364 364 365 365 368 369

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14.5.1

Background Information . . . . . . . . . . . . . . . . . . . . . 14.5.1.1 Projects . . . . . . . . . . . . . . . . . . . . . . . . 14.5.1.2 Criteria . . . . . . . . . . . . . . . . . . . . . . . . . 14.5.1.3 Project by Municipalities Considering . . . 14.5.1.4 Projects that Are Shared for more than One Municipality . . . . . . . . . . . . . . . . . . . . . 14.5.1.5 Maximum Amounts Available for Municipality Considering . . . . . . . . . . . . 14.5.1.6 Result . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5.2 Conclusion of This Case . . . . . . . . . . . . . . . . . . . . . 14.6 Case Study: Selection of the Best Route Between an Airport and the City Downtown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6.1 Background Information . . . . . . . . . . . . . . . . . . . . . 14.6.2 The Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6.3 Conclusion of This Case . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Simplex Algorithm: Its Analysis—Progressive Partial Solutions . . Demonstration of Absence of Rank Reversal in SIMUS . . . . . . . . . . . . Solving a Problem with SIMUS Software . . . . . . . . . . . . . . . . Adding an Exact Copy of an Existing Project . . . . . . . . . . . . . Adding Project 6 “Worse” than Others . . . . . . . . . . . . . . . . . . Adding New Project x7 Keeping Project x6 and with x3 = x6 = x7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adding a New Project Identical to Other and Simultaneously Adding Another Considered the Best . . . . . . . . . . . . . . . . . . . Deleting Project from the Original . . . . . . . . . . . . . . . . . . . . . Summary of Scenarios and Results . . . . . . . . . . . . . . . . . . . . .

370 371 372 373 373 374 374 375 376 376 377 380 380 381 381 386 387 388 389 390 391 391 392

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397

Part I

Theory and Analysis of MCDM Problems: History of MCDM and How It Is Performed

Chapter 1

Multi-criteria Decision-Making, Evolution, and Characteristics

Abstract This chapter deals with the commencement, history, and evolution of multi-criteria decision-making process. It gives the reader a bird’s-eye glance of the birth, development, and present-day status of this discipline, which is nowadays taught in most technical universities around the world. Its purpose is to make the reader aware of why it was conceived and developed and what can a practitioner expect from it. It mentions the main actors and their roles, as well as enumerates the factors or aspects that need to be considered.

1.1 1.1.1

History and Evolution of Multi-criteria Decision-Making Methods Some Background Information on Decision-Making

Decision-making is as old as civilization; however, we can identify with the name the reason for the beginning of the scientific procedure. Benjamin Franklin devised in the eighteenth century a system of using a list with two entries and then assigning weights, the latter still largely used today. He wrote, in a letter to Joseph Priestly, on September 1772: My way is to divide half a sheet of paper by a line into two columns; writing over the one Pro and over the other Con. Then during three- or four-days’ consideration, I put down under the different heads short hints of the different motives, that at different time occur to me, for or against the measure. When I have thus got them altogether in one view, I endeavor to estimate their respective weights; and where I find two, one on each side, that seem equal, I strike them both out. If I judge some two reasons con equal to some three reasons pro, I strike out five; and thus proceeding, I find where the balance lies; and if after a day or two of further consideration, nothing new that is of importance occurs on either side, I come to a determination accordingly.

It is interesting that this scientist used the word “balance,” which precisely is what MCDM heuristic methods do; that is, instead of optimizing results, which most of

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 N. Munier, Strategic Approach in Multi-Criteria Decision Making, International Series in Operations Research & Management Science 351, https://doi.org/10.1007/978-3-031-44453-1_1

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1 Multi-criteria Decision-Making, Evolution, and Characteristics

the time is impossible, they aim at finding a consensus, agreement or compromise among all the intervening parties in a project selection. The series of issues conducted with the actual methods is as follows: Thurstone introduced the pair-wise comparison system for measurements, which he referred to as the law of comparative judgment (Wikipedia). This is a fundamental procedure and is used for most decision-making methods. In its origins projects were selected using measurements or indicators such as the net present value (NPV), that is, discounting the after-tax cash flows by the weighted average cost of capital method, which possibly started in the early 1900s. Another metric is the internal rate of return (IRR), which is the interest rate at which the net present value of all the cash flows (both positive and negative) from a project or investment equals zero, which probably originated at the same time as the NPV metric. Another standard is the benefit–cost ratio (B/C), that is, the relationship between the net present value of the benefits cash flow and the negative cash flow, both at a certain discount rate. There are more indicators, but these are the most significant. For instance, another very important is the payback period (PB), which expresses the time elapsed until the investment is recovered. All of them led to an economic and financial analysis and were the only methods used to select projects by comparing their respective indicators; the higher, the better. However, projects involve and are dependent on many more factors than money. Most important are social issues such as people’s welfare, disposable income, public health, and education, as well as environment, resources, sustainability, externalities (i.e., goods and actions that do not have a market value), etc. An example of the last one is the erosion produced by logging when reforesting does not follow. Erosion means loss of fertile soil, and it can lead not only to the destruction of the natural capital of a country but also to catastrophic consequences when heavy rains loosen rocks and produce large quantities of mud, which can destroy populations along their way because there are no barriers to stop it. This is an aspect normally not considered when analyzing forestry projects; the same for air, soil, and water contamination, mineral depletion, etc. It goes without saying that no social problems or environmental issues were considered in the early times. Probably the most significant scientific approach to decision-making took place during the WWII. At that time, the most important objective for the Russian government was to win the war against the Germans that had invaded their territory. Russia is a country of unaccountable natural wealth in people, science, fuels, and minerals, as well as utilities like transportation and electricity generation and in manufacturing (especially war equipment, at that moment), food production, etc., all of them vital for the war effort. However, there were priorities since some resources were more important than others, and then it was crucial, since time was of the essence, to determine which of the resources should be developed with the highest potential with this objective in mind.

1.1

History and Evolution of Multi-criteria Decision-Making Methods

5

For that reason, the Russian government commissioned Leonid Kantorovich, a multifaceted Russian engineer, mathematician, and economist, to determine or select the optimal mix of resources and utilities that should be developed to maximize the war effort. His work gave birth to linear programming (LP), an algebraic procedure that could do the job, and it did. After the war in 1975, he and Tjalling Koopmans, an American economist of Dutch origin, were awarded the Nobel Prize in Economics for “His theories on the optimal assignment of scarce resources.” Among his books are Mathematical Methods for Organization and Production (1939), Contribution to the Theory of Optimal Allocation of Resources (1959), and Optimal Solution in Economy (1972). The system worked well and most probably contributed to a large extent to the Russian Army advance and victory over the German Army by providing food, clothing, and weapons, but it was exceedingly complex and time-consuming in a time when computers did not exist. Both scientists paved the road for the birth of multi-criteria decision-making (MCDM), where a set of alternatives are subject to compliance with a set of criteria. The better an alternative matches the criteria requirements, the better. For this reason, MCDM can be defined as the process of selecting one of several courses of action, alternatives or options, which must simultaneously satisfy many different conflicting and even contradictory criteria. In 1948, the American physicist and mathematician, George Dantzig, created an algorithm called Simplex, which turned feasible the solution of complex LP problems in selecting the best and optimal solution. In the early l990s, the American Dan Fylstra, a pioneer in the early software products, with a main role in the development of the legendary VISICALC, the first spreadsheet program, developed the software for the Simplex algorithm, which is since 1991 an add-in of Excel and then available to everybody with the Microsoft Office loaded in his/her computer. Probably in the mid-twentieth century, researchers and practitioners working on project selection and without a doubt also motivated by social organizations such as the United Nations, the World Bank, and many others began to realize that the system based on the abovementioned financial technologies completely ignored social aspects such as people relocation and effects on population due to new projects, as well as people health related to them, and that no significance was given to their contamination to air, soil, and water, because they neither considered aspects such as sustainability and externalities, such as, deforestation and erosion, nor depletion of aquifers. They realized that LP, which at that time (and still at present) was heavily used, could be advantageously utilized for selecting projects considering social, military, supply, storage, capacities, environmental issues, etc. That is, because of the effect and benefits that LP and Dantzig algorithm was having in thousands of large industries such as oil refineries, food production, transportation networks, and personal assignment, researchers started to see that now they had a tool that could address projects that were subject to many more objectives than only economics. Therefore, LP had a boost in popularity. However, LP has a serious drawback because it can work with as many criteria as wished, maximizing and minimizing

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Multi-criteria Decision-Making, Evolution, and Characteristics

them, but it can manage only one objective at a time; however, projects normally must deal with different objectives at the same time, which can also be contradictory. However, LP looks for complete objective data and results, ignoring the subjective component of the decision-making process, and as a consequence, the position and expertise of the DM were minimized, to say the least. In this respect, Buchanan et al. (1998) opine that “The role of the decision-maker is downplayed and the method is depicted as an objective entity.” For this reason, researchers tried to develop heuristic1 methods where optimality was not the goal but to find a compromising solution and thus satisfying stakeholders, society, and the environment and not only choosing the best alternative but also generating a ranking of alternatives. The reader may be wondering why to also select other alternatives in descending importance once the best alternative is found. The reason lies in the fact that alternatives chosen are a consequence of data inputted and a mathematical calculus. However, sometimes, the best alternative is not the most appropriate bearing in mind factors that cannot be considered in the modelling, because they are exogenous to the scenario, most probably uncertain, and very difficult to model. For instance, if a company is manufacturing a series of products that sells nationally and internationally, there are factors such as government policies, competition, and international demand that are exogenous to production and marketing. In this context it could be that the best-selected alternative is very sensitive to variations of some exogenous variables, such as demand, and then its variation within certain limits can cause that best alternative is no longer the best, while the second best alternative, for instance, could not be affected. This is the reason why a MCDM problem, once a solution is obtained, must be evaluated considering the sensitivity point of view, using what is called “Sensitivity Analysis.” This is not new since also in the early days of using financial indicators, a sensitivity analysis also had to be performed, for instance, to determine how an increase in bank interest rates could influence a selected project; however, nowadays it is considered much more complicated because of the different factors that intervene. Here, we need to introduce a fundamental component of the MCDM process, the decision-maker (DM), who oversees stakeholders’ wishes and demands in designing and preparing a mathematical method with those requirements (called the decision matrix). He is also responsible for selecting the criteria, choosing the method to be used to solve it, and interpreting and analyzing the results based on his expertise, examining the exogenous factors that can affect the best alternative, and then making his/her recommendations to stakeholders. Naturally, these results and 1

In computer science, artificial intelligence, and mathematical optimization, a heuristic (from Greek εὑρίσκω “I find, discover”) is a technique designed for solving a problem more quickly when classic methods are too slow, or for finding an approximate solution when classic methods fail to find any exact solution. This is achieved by trading optimality, completeness, accuracy, or precision for speed. In a way, it can be considered a shortcut (Wikipedia).

1.2

Introduction to Most Common and Used Heuristic Methods

7

recommendations must rest on solid grounds, and the DM should be prepared to answer the questions that without a doubt will formulate stakeholders.

1.2

Introduction to Most Common and Used Heuristic Methods

In this book, the following expressions are used: Problem: General name for a set of projects, alternatives, options, and strategies that an entity wants to pursue. The entity may be a company, a building developer, the government, the military, health and educational institutions, software companies, municipal offices, etc., that is, whoever is interested in building or manufacturing something tangible (bridges, buildings, cars, computers, software, etc.) or intangible (improve disposable income, ameliorate people living conditions, reduce crime, etc.). Scenario: It is the generic denomination for a certain problem; however, it includes all the factors that are related to it. Model: It refers to building or modelling the initial data in a decision matrix, which is a mathematical method. Method: It addresses the different MCDM processes, routines, and techniques used to solve the method. The first heuristic method for decision-making was developed in Europe in 1965 by Bernard Roy and called ELECTRE. It is the French acronym for “ELimination Et Choix Traduisant la Realité,” or in English, “Elimination and Choice Expressing Reality.” It is widely known as a product of the French school. It belongs to the outranking category methods in decision-making. By outranking, it is understood that there is a strong enough argument to support a conclusion that a is at least as good as b and not a strong argument to the contrary (Belton & Stewart, 2002). There are several versions aiming at different objectives. The PROMETHEE method, “Preference Ranking Organization Method for Enrichment Evaluation,” also originated in Europe, was introduced by Brans and Mareschal (1986). It also adopted the outranking procedure. It has an interesting visual feature (GAIA), which is a geometrical analysis for interactive aid. Fuzzy sets and fuzzy logic were introduced by Zadeh (1965), an engineer, computer scientist, and mathematician. This theory was further applied to decision-making; however, this technique, which essentially tries to reduce uncertainty, is not a MCDM method, and therefore it does not solve this kind of problem; however, it has been widely used in conjunction with many different MCDM methods. This author believes that even considering the importance of using fuzzy logic in MCDM, it is being misused because very often the procedure is inputted with low, medium, and high values that come from subjective appreciations.

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Multi-criteria Decision-Making, Evolution, and Characteristics

To this respect, use fuzzy logic in railways performance analysis, but they input the technique with data of low and high values obtained using SIMUS and, consequently, mathematically correct since they are based on reliable data. Oprocovic (1980) developed VIKOR, an acronym in Serbian for “VIseKriterijumska Optimizacija I Kompromisno Resenje,” in English, “Multicriteria Optimization and Compromise Solution.” It also can work with fuzzy logic. Hwang and Yoon (1981) developed “The Technique for Order of Preference by Similarity to Ideal Solution”—TOPSIS. It assumes that there are both an ideal positive and an ideal negative solution and tries to find a result that has the shortest geometric distance to the ideal positive and the longest to the negative. The “Analytical Hierarchy Process,” AHP, introduced by Saaty (1980), is based on the additive concept established by earlier methods such as MAUT. AHP generates weights for criteria and for alternatives, the first fundamental to compute the second. For weights extraction, the method uses the eigenvalue method based on pair-wise comparisons between criteria. Values in the decision matrix represent DM preferences. Fuzzy logic is employed in AHP and in other methods; however, most scholars, including Saaty, do not sustain this procedure since AHP is fuzzy by itself. Munier (2011) developed SIMUS “Sequential Interactive Method for Urban Systems.” The method is grounded on LP and selects alternatives based on opportunity costs, which are derived from the data. One feature is that it produces two solutions for the same problem from the same data using two different procedures; however, both solutions give the same ranking. It is important to mention that not always a DM builds the decision matrix; in some scenarios, because of their complexity, there is a group of DMs that after an agreement produce the necessary data.

1.3

The Decision-Making Paradox

Since the development and implementation of methods in the 1960s to solve multicriteria problems, researchers have been puzzled by the fact that different mathematical approaches to solve this kind of scenario produce different results. Probably, nowadays it can be asserted that this is due to the fact that decision-making is in a large extent a subjective problem and that explains that starting from the same initial matrix, using mathematical tools, and looking for the same objective results are different. As far as this author’s knowledge, there is not at present a rule or procedure that can be applied to solve this quandary and maybe never will. Just to clarify this issue, assume a certain scenario and proceed by: • Building the decision matrix and using it as the starting point for different methods,

1.3

The Decision-Making Paradox

9

• Selecting the objective to achieve, • Assuming that all methods follow mathematical principles, In so doing it is logical to expect that results or rankings must be very close from one method to another; however, in checking the results, most probably one will see that there are different solutions or rankings, even for different versions of the same method. This has been called “The decision-making paradox” (Triantaphyllou, 2000). Many researchers and practitioners have commented and detected this paradox; however, not a uniform response is recorded. This circumstance appears weird because if all methods start with the same data, use mathematical procedures, and aim at the same objective, why results are different? According to this author’s opinion, the reason, among others, is that each method has a dosage of subjectivity related with DM preferences, and not all DMs are on the same issue. Consequently, for the same problem, a DM in AHP, for instance, gets weights for criteria that are a product of his preferences, while weights for a DM working with PROMETHEE are extracted by other means, for example, using Monte Carlo, Delphi, or any other method. In addition, the same DM in AHP quantifies his preferences using a ratio scale (the Saaty Fundamental Scale) (1980), while the DM in PROMETHEE uses his own judgment for thresholds and preference functions, which are based on statistics. Another DM working with ELECTRE IV employs no weights, and one more DM in TOPSIS draws on a geometric distance of his preference. There are also methods that generate their own weights based on data and, therefore, objective. After examining hundreds of published works on a myriad of different scenarios and using various MCDM methods, and naturally, assuming that mathematics in several methods is correct, this author believes that discrepancies are produced in three main areas, namely, modelling, DM subjectivity at several levels, and wrong use of criteria weights. Since it is apparent that differences emerge because of subjective judgment, it appears rational to think that a solution to this problem could be using methods without subjectivity, for instance, not employing assumed weights for criteria. However, this might be unrealistic since most probably in a certain problem the significance or importance of one criterion normally is perceived as different from another, and this is important. This book proposes solving MCDM scenarios without weights; however, it does not mean not considering the criteria’s relative importance. Far from it, the difference is that weights should be employed only when the DM reckons that they are necessary in some criteria, and this is done when a first tentative weightless solution has been reached, that is, at the end of the mathematical process, not at the beginning as is performed nowadays. The assignment of these weights will be then based on solid grounds and in the same conditions for all methods and when the DM, according to his experience, believes that the importance of a certain criterion has not been considered. Once the correction is done, the method is run again, and results are compared. It could very

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1 Multi-criteria Decision-Making, Evolution, and Characteristics

well be that this second run yields the same result, thus meaning that the weighted criterion is not as significant as thought. This is some sort of trial-and-error process that can be repeated as many times as necessary, and at the end the DM may be assured that his corrections are correct. The difference is radical, since with this procedure the DM is acting and putting his experience to work on unbiased results. This is also the principle proposed in this book for group decisions (Chap. 9), with the difference that it is done at the end of each partial run, that is, during the process and not at its very end.

1.4

Which Is the Best MCDM Method?

There is consensus among researchers that there is no method better than another is, and consequently all of them are considered good and adequate for selection. In addition, it is also understood that there is not a universal method that can deal with any type and size of the problem; unfortunately, not many practitioners are apparently aware of this fact and use the same method for very different problems. For instance, subjective problems normally employ the pair-wise comparison method based on DM preferences. This procedure is good for trivial problems and for those where the decision consequences directly affect the DM or his company. As an example, assume purchasing a house; if the selected property does not fulfill the DM expectations, it is he who will suffer because of that, and if a company is selecting personnel, a good or a wrong decision will affect the company and nobody else. However, there are projects whose consequences, good or bad, will affect the proper existence of the company pursuing it, and perhaps the life of hundreds or even thousands of people, and also hurting the environment. An example could be selecting a place to install a plant for assembly cars; in these cases, it is obvious that the DM preferences are no longer valid or even considered. Is it possible to establish a level of confidence when using a method? No, it is impossible to know; however, it is frequent to hear that an author often says that he/she applied a certain method successfully. What he should say is that he reached a result that satisfies him or his group, but no more than that. His assertion is clearly inaccurate because if he does not know what the best solution for a real problem is, he cannot say that his method attained it since he cannot compare it with something whose value he does not know! Some researchers suggest comparing results for one problem solved by method A, with the same problem solved by methods B and C. Apparently it would be a good approach; however, it appears difficult because, most probably, the three methods will yield different results. Naturally, if results from the three methods coincide, there is a high chance that they are “correct.” Nevertheless, one needs to be careful in extracting conclusions when comparing results in this case. For instance, if the same problem is solved using AHP, which furnish criteria weights, and they are then applied to other methods such as PROMETHEE and TOPSIS (known as hybrid PROMETHHE or Hybrid TOPSIS), it could be that the

1.6

Is It Possible to Represent Reality Faithfully?

11

three results coincide. However, one must be aware that in this case the results may be misleading because the last two methods use partial values derived from the first. In addition, since AHP weights derive from DM’s preferences, they are subjective, and then they transfer that subjectivity to the last two methods. Chapter 7 proposes a method that it is believed can help in determining the methods that draw near the ideal. Since this ideal or real solution is unknown, this is only an approximation of reality.

1.5

Considering and Modelling Reality

In the examination of papers that solved various scenarios using different methods, this author found a noticeable and perturbing fact: Most of them do not take into account the whole scenario and ignore some actual factors or demands; that is, in their modelling they do not consider many real aspects that shape a scenario. Consequently, results by ignoring important details may not be as reliable as expected. For instance, in any project, criteria are normally interrelated; that is, one may have some sort of relationship, direct or indirect, with another; however, they are customarily considered independent. As a typical example, contemplate the purchase of a car by selecting one among several methods of a maker. Say that these alternatives or options are subject to three criteria, namely cost, fuel consumption, and comfort. It is obvious that if the purchaser is taking only into consideration the price tag, he does not realize that the other two criteria also have an influence on costs. Therefore, the selection must consider the three criteria together. Modelling is probably the most important step in MCDM scenarios, since if it is incomplete, as often happens when examining most published papers, the algorithm, whatever it may be, delivers a solution that does not represent reality.

1.6

Is It Possible to Represent Reality Faithfully?

What is in fact “reality” in a MCDM context? It would be first necessary to define reality. The Oxford Dictionary defines reality as the state of things as they actually exist, as opposed to an idealistic or notional idea of them. In the MCDM concept, if it is planned to build a route between A and B, obviously it does not exist yet; however, there are tangible aspects that do exist, such as conditions of the terrain between A and B, rivers to be crossed, forests to be avoided, workforce to be hired, and contractors to be selected. These are very real aspects and that can be precisely measured. However, there is also another kind of reality that certainly exist but that cannot be accurately measured, such as people’s opinion about the problems, inconveniences that will cause the route if it divides their city in two, or noise. It is necessary

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1 Multi-criteria Decision-Making, Evolution, and Characteristics

to remark that this does not mean people’s opinion about the technical economic or technical feasibility of the undertaking: it only reflects how it interferes with their lives. Both types of realities are objective; they truly exist; however, their treatment and especially their attributes, or values according to each alternative, must be obtained by different means. Both are expressed by numbers; however, the first type is objective and reliable, while the second one is subjective and uncertain. These two clear aspects are fundamental for the alternatives to be evaluated according to quantitative criteria (as in the first case) and qualitative criteria (as in the second case), and both simultaneously. At this point it is advisable to clarify the two meanings of the term “Objective” in MCDM. When it is used to establish a goal, an aim, or an ambition, it is a noun and refers explicitly to that. For instance, the objective may be to maximize the benefit, minimize the cost, or minimize contamination, and it is indefinite; that is, there are no values and no units. In its second meaning it is an adjective and denotes the condition of being objective, that is, according to the Oxford Dictionary, “(of a person or their judgement) not influenced by personal feelings or opinions in considering and representing facts,” that takes things as they are, or at their face value. From this point of view, this book supports the notion that reality exists and can be measured exactly or approximately. Why this disquisition about reality? Because some scholars and some methods argue that reality does not physically exist and that it only is in the mind of the people. Buchanan et al. (1998) state, “The first assumption is that there is a reality to describe. This metaphysical premise, founded on seventeenth-century Western scientific thought, essentially states that there is one reality shared by all. A commonly shared reality, then, is fundamentally objective; i.e., it is made up of elements originating independently from us.” From the researchers’ point of view, reality is what a scenario looks like, what it is subject to, and its facts and conditions; however, probably it is not possible to represent it faithfully and perhaps never will. For instance, assume two alternatives, such as a tunnel and a bridge for crossing a river. In addition to those typical technical factors such as cost, investment amortization, and expected traffic and revenues, there are other aspects to consider, namely: Social unrest from people removed from their homes to make room for the tunnel approaches, or cost of periodic painting the bridge with antioxidants, or provisions to be taken in case of serious car accidents in the tunnel, or complementary works for both, as well as benefits for people that both undertakings will bring, risks associated with each project, selection of contractors considering experience in each type of work, etc. Consequently, a method for selecting projects must consider as many aspects of the scenario as possible; otherwise, MCDM is just an arithmetic exercise. At the present time, the incomplete data may get results that are a consequence of Deciding by omission, that is, following the reasoning that Since I can’t consider this fact because my modelling or my MCDM method does not allow it, thus, I forget about it.

References

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It is not ignorance; probably most practitioners know about this, but there is no attempt to remediate the situation. This author does believe that reality, if not in its totality, can be replicated in modelling, and this is the main objective of this book. In successive sections, this reality is analyzed and exemplified. Is this enough? Probably not, but we are using the tools and techniques available nowadays; may be in the future new tools will attain this objective, but that future and the new tools it can bring is of course unpredictable.

1.7

Conclusion of This Chapter

This first chapter starts at giving the reader a bird’s-eye view of the evolution of MCDM by following a historical line of its development. Its next purpose is an update on the state of the art of this discipline with very brief comments on the most used methods. It continues by examining some problems such as decision-making paradox. The Next section addresses some common questions that most practitioners pose. The balance of the chapter aims at introducing a concept that is the leitmotiv of this book and related to the lack of considering reality that permeates actual MCDM practice.

References2 Belton, V., & Stewart, T. J. (2002). Implementation of MCDA: Practical issues and insights. In Multiple criteria decision analysis. Springer. Brans, J. V., & Mareschal, B. (1986). How to select and how to rank projects: The PROMETHEE method. European Journal of Operational Research, 24(2), 228–238. Buchanan, J., Hening, E., & Hening, M. (1998). Objectivity and subjectivity in the decision-making process. Annals of Operations Research, 80(1998), 333–334. Dantzig, G. (1948). Linear Programming and extensions. R-366-PR-Corporation. Hwang, C., & Yoon, K. (1981). Multiple attribute decision making: Methods and applications, A State- of - the - Art Survey. Springer-Verlag. Kantorovich, L. (1939). Mathematical methods of organizing and planning production. Management Science, 6(4) July, 1960, 366–422. Munier, N. (2011). A Tesis Doctoral - Procedimiento fundamentado en la Programación Lineal para la selección de alternativas en proyectos de naturaleza compleja y con objetivos múltiples. Universidad Politécnica de Valencia, España. Oprocovic, S. (1980). VIseKriterijumska Optimizacija I Kompromisno Resenje (Multicriteria Optimization and Compromise Solution). Science Watch, April 2009.

2

These references correspond to the author mentioned in the text. However, there are also publications that are not mentioned in the text but that have been added for the reader to access more information about this chapter; they are identified with (*).

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Roy, B. (1965). Classement et choix en présence de points de vue multiples (la méthode ELECTRE). Revue française d’informatique et de recherche opérationnelle, 2(8), 57–75. Saaty, T. (1980). Multicriteria decision making - The analytic hierarchy process. McGraw-Hill. Triantaphyllou, E. (2000). Multi-criteria decision making: A comparative study (p. 320). Kluwer Academic Publishers (now Springer). ISBN 0-7923-6607-7. Zadeh, L. (1965). Fuzzy sets. Information and Control, 8, 338–353.

Chapter 2

The Initial Decision Matrix and Its Relation with Modelling a Scenario

Abstract This chapter is mainly devoted to a critical task: modelling a scenario. It addresses two main aspects: a) Elements of the initial decision matrix. b) How to method a scenario. Naturally, it is impossible in the second aspect to deal with the innumerable cases that correspond to a myriad of different projects and scenarios. The chapter aims at providing as much information as possible and as being a guide for the practitioner. It condenses conclusions from the examination of many cases proposed by researchers and practitioners around the world and using different methods and procedures. From here, the author extracted critical aspects that should be considered. All these points lead to the formulation of a sound and realistic modelling that replicates a scenario as closely as possible.

2.1

Basic Components of the Initial MCDM Decision Matrix

For a given problem or scenario the components of the MCDM process are:

2.1.1

Stakeholders

The people who manage projects and who are responsible for the best selection. Theirs is the final responsibility for a decision.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 N. Munier, Strategic Approach in Multi-Criteria Decision Making, International Series in Operations Research & Management Science 351, https://doi.org/10.1007/978-3-031-44453-1_2

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2.1.2

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The Initial Decision Matrix and Its Relation with Modelling a Scenario

Decision-Maker or Group of DMs

The practitioner/s, expert/s, or individual/s that, based on demands from stakeholders, deliver results and formulate recommendations for them to decide.

2.1.3

Objective/s That the Scenario Must Attain

They are established by stakeholders; objectives are in general indefinite since no value is attached to them.

2.1.4

Scenario/s

The set of different alternatives, projects, or options. It also applies to places or locations where all alternatives and criteria may apply jointly or separately.

2.1.5

Alternatives, Projects or Options

These are feasible undertakings, as well as opportunities and strategies, which must be determined to attain the objective/s. These are generally jointly selected between the stakeholders and company departments (engineering, accounting, financial, human resources, environment, etc.), which normally give information about the ability, experience, and financial conditions the company has to undertake such kind of projects. For instance, a company manufacturing electric bikes may consider expanding by fabricating small electric cars, electric batteries, or electric forklifts; however, it needs to examine if it has the capacities and conditions for these developments. Consequently, the stakeholders establish the goals and the means to reach them while the company’s departments develop the data and analysis about the feasibility of each one. This feasibility is not only related to the economy and financial point of view, but it also considers government, social, environmental, and sustainability aspects. Later, once the high management has decided on what to do, company departments must examine each project and produce the corresponding data.

2.1

Basic Components of the Initial MCDM Decision Matrix

2.1.6

17

Criteria

These are the evaluators of alternatives. They are determined according to the nature and characteristics of the alternatives and, therefore, are different for diverse projects. Criteria must be selected by the stakeholders and the DM in such a way that they contemplate what each stakeholder wants. For instance, a financial representative will probably be interested in maximizing returns and minimizing payback periods for each project. Hence, a criterion for IRR or NPV and other addressing investment recovery time must be implemented. The environmental representative will be interested in minimizing noxious emissions and considering allowable limits. The accounting representative will certainly ask for minimizing working capital needed for each project, and the human resources representative will ask for maximizing the quality of personnel that have to be contracted, etc., and do not forget to include criteria for risks in their different categories, as for personnel, delays, safety, equipment, etc.

2.1.6.1

Areas Included in Criteria

There is a common question between practitioners: Which areas of activity must be included in selecting criteria? Well, obviously, those related to the alternatives to be evaluated, but as mentioned, a scenario may have “peripherals”; that is, there could be activities that in some way or another influence a project. For instance, in a scenario for determining the location of a plant to produce photovoltaic panels (PV), it is very important to have access to easy shipping around the world and for which the maritime transport is crucial. Say for instance that the potential locations are three different ports in north Europe. Therefore, port facilities, especially for handling containers, is essential, and it is then a sine qua non condition for a place to be considered, along with many other requisites. There are however other aspects not normally perceived and not related whatsoever to the PV load that may have a decisive influence in the selection, and it is winter weather. Why? Because it could very well be that a port might become inoperative because blocked by ice in wintertime. To help the DM consider these types of circumstances, the following list, which is of course incomplete, intends to serve as a checking list for him “to check what could be directly or indirectly related with the formulation of the initial decision matrix.” Normally it is not for the DM to consider these facts since they should be incorporated by the respective department when planning the work and estimating costs; however, it is the responsibility of the DM that aspects related to them be present in the initial decision matrix by the respective criteria. Therefore, appropriate criteria must be added in each case. Areas: Agriculture—Consider criteria to:

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The Initial Decision Matrix and Its Relation with Modelling a Scenario

• Maximize soil fertility, that is, adequacy for certain crops. • Maximize weather (e.g., Abundance of rains, minimized droughts, or adequate temperature and sunlight). • Maximize water quality (salty, with undesirable chemicals, etc.). • Maximize water availability (i.e., where the water comes from: rains, wells, irrigation channels managed by a public entity, etc.). • Minimize plagues (types of plagues and cost of pesticides, etc.). • Minimize occurrences of hails and floods when crops are in different areas or various scenarios. City rehabilitation—Consider criteria to: • • • •

Maximize feasibility for new undertakings according to city bylaws. Maximize legal conditions (for instance, plots without any encumbrances). Maximize benefits for city citizens. Maximize job generation. Civil Construction—Consider criteria to:

• Maximize job generation. • Minimize projects at high altitudes since they result in the reduction of working hours and efficiency due to the altitude, which can also affect equipment efficiency and concrete work. • Minimize special conditions posed by weather. For instance, in projects in hot weather, consider a criterion related to the need for cooling concrete before pouring. • Maximize data on geological and hydrological conditions, that is, a criterion that qualifies these conditions for each project. • Maximize data as in long tunnelling, taking into consideration testing of samples obtained by drilling for each type of project. • Minimize hazards such as flooding and fall of rocks. • Maximize search for underground utilities (e.g., if a deep excavation is needed, it is necessary to verify if there are water, storm, and sewage trunks, electric wires, and telecommunications optic fiber). • Minimize time for repeated tasks. If repeated actions are present, there is a possibility of savings in cost (e.g., in high-rises, flooring may have a decreasing cost using a learning curve). • Minimize risk for personnel accidents. It means estimating risk values for each alternative. • Minimize accessibility to job sites. • Minimize the necessity to make repairs for transportation. In remote places transportation of equipment may be hindered by not prepared harbor facilities (for instance, low-capacity cranes), as well as inadequate roads and bridges to stand heavy weights, etc. • In roads, minimize roads with potential problems. (For instance, in an actual case a road had to be rerouted because the original one was projected over a sacred

2.1

Basic Components of the Initial MCDM Decision Matrix

19

aboriginal cemetery). Different projects may show very different conditions regarding this aspect. Alternative roads exist for each project site. Economy—Consider criteria to: • • • •

Maximize expected values for project for IRR and NPV. Minimize payback periods per project. Minimize working capital needed in each project. Maximize residual value per project. Education—Consider criteria to:

• • • • •

Maximize the number of students. Minimize the number of students per classroom. Maximize the number of computers. Minimize absenteeism. Maximize teacher’s preparation. Energy—Consider criteria to:

• Different aspects that influence selection such as: For hydroelectric plants: River flow statistics, type of turbines, maintenance costs, efficiency, energy transmission line, etc. In other power plants: Type of equipment, unsatisfied demand, contamination (in air, soil, and water) • Maximize job generation by each alternative. • Maximize people’s opinion (For instance, there may be people against the construction of nuclear power plants nearby). • Minimize air, water, and soil contamination. Environmental—Consider criteria to: • Minimize the effect on wildlife affected by projects. • Minimize projects affecting preserved and protected areas, as well as swamps, original forest, national parks, etc., that may be disturbed by different projects. • Maximize limits for water extraction from aquifers, keeping maximum extraction below the replenishment rate. • Minimize noise produced by aircraft when approaching to land at an urban airport and when taking off. • Minimize soil contamination due to airplanes de-icing in airports. Financing—Consider criteria to: • Minimize interest rate from bank loans for each project. • Maximize return on investment. Garbage disposal—Consider criteria that relate to: • Construction of landfills: Maximize studies on soil conditions and especially permeability is considered for different alternative sites.

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The Initial Decision Matrix and Its Relation with Modelling a Scenario

• Construction of domestic garbage incinerators: Maximize people’s opinion (generally negative), noxious emissions, short- and long-distance metal depositions on soil, etc., are recorded in criteria for different incinerator locations. Health—Consider criteria to: • • • •

Maximize eradication of infectious diseases. Maximize hygiene in hospitals. Minimize newborn death. Maximize care for new mothers. Human resources—Consider criteria to:

• • • • •

Maximize the experience of candidates. Maximize the number of articles published. Maximize academic degree. Maximize positions held. Minimize training. Industrial—Consider criteria to:

• • • • • • • • •

Minimize how a new product affects the present operation of the company. Minimize stock of raw for each project. Minimize production time. Minimize rejection of faulty pieces. Minimize personnel training. Minimize externalities that each product may cause (noise, odors, vibration, etc.). Maximize delivery on time of raw materials and subproducts and components. Minimize electricity consumption for each product. Maximize quality control standards for each product. Infrastructure—Consider criteria to:

• Maximize scope (Km, m2, patients in hospitals, etc.) for each different project, for instance, sewerage, paving, and social programs. • Maximize construction of schools and health centers. Marketing—Consider criteria to: • • • • • •

Minimizing competition effect on own product. Maximize market penetration for each product. Maximize total quality management (TQM) for each product. Maximize estimated market penetration. Maximize estimated demand. Minimize time to fulfill demands. Politics—Consider criteria to:

• Minimize/maximize government-announced import duties for raw materials. • Minimize regulations that may affect products differently. • Maximize/minimize the effect of government policies.

2.1

Basic Components of the Initial MCDM Decision Matrix

21

Safety—Consider criteria to: • Maximize safety procedures and risk for personnel and for each project. • When selecting potential main contractors, maximize their records on safety in similar jobs. • Maximize probabilistic values for different for increasing street safety. • Minimize probabilistic values showing a reduction in car accidents on highways. Social—Consider criteria to: • Maximize/Minimize estimated performance values for different policies. • Maximize aspiration level for some projects (for instance, number of patients treated per year). • Minimize cost for plans to get kids out of drugs. Software design—Consider criteria to: • • • •

Minimize the estimated time for completing each different program. Minimize the estimated cost of each program. Minimize man-hours needed. Maximize applications. Sustainability—Consider criteria to:

• Minimize values of erosion produced by different projects. • Minimized values for accounting externalities. Telecommunications—Consider criteria to: • • • • •

Broadband selection. Maximize the number of equipment to be connected. Maximize range. Minimize cost. Minimize maintenance cost. Urban infrastructure—Consider criteria to:

• Urban highways: Be sure that there are criteria to minimize noise and that contemplate remediation work. • Maximize that the opinion of people affected by the undertaking is recorded. • Maximize registration on people’s opinions about the esthetics of an elevated road in comparison with a ground-level road.

2.1.6.2

Capacity of Criteria to Evaluate Alternatives

Since the main role of criteria is for alternative evaluation, it appears that it is essential to have a means to gauge this capacity. A criterion is a vector (row or column) whose components are the performance values, that is, cardinals that indicate the contribution of each alternative to that criterion objective. If

22

2

The Initial Decision Matrix and Its Relation with Modelling a Scenario j=1

Concentration of H2SO4 in μg/m3 Probability of each value

11.10 0.14

j=2

j=3

j= 4

4.21 0.05

1.23 0.01

50.20 0.64

j=5

1.99 0.03

j= 6

Sum of values

10.36 0.13

79.09

Constant K = -1/ln (6) = -0.558111

Entropy Ej1 Ej2 Ej3 Ej4 Ej5 Ej6 -0.28 - 0.16 - 0.06 - 0.29 - 0.09 - 0.27

Sum of Ej ∑ =1 ln( ) -1.14

Entropy Criterion Ci ∑ =1 ln( )

Quantity of information

0.64

0.36

Fig. 2.1 Different H2 SO4 concentrations for different processes when there is high discrimination

performance values in a vector are very similar to each other, the criterion capacity is probably zero because it is impossible to use it for evaluation, for there is almost the same value for each alternative; that is, capacity is a function of discrimination between criterion values; then the larger the dispersion, the better. This capacity can be mathematically measured using Shannon’s (1948) theorem, introducing the entropy concept, which paves the way for determining criteria weights that objectively measure their respective capacity. Entropy (S) essentially means “loss,” therefore, the higher the entropy, the lower the information content, because it means that some information is not recoverable, or unread, or lost, like noise in a message. For this reason, to determine the criteria relative importance, it is used what is called Information (I), which expression is: I = 1-S. If a criterion shows S = 1, then its information content I = 1-1 = 0. Consequently, high discrimination corresponds to low entropy and better information. Conversely, high entropy corresponds to low discrimination and worse information. This is illustrated in Fig. 2.1, where there are a series of performance values for vector conforming criterion “Ci” for minimizing H2SO4 concentration for six different industrial processes. As seen, there is an appreciable discrimination for criterion Ci (Ei). Its entropy is 0.64, and the quantity of information that it delivers is 0.36. This may be considered a measure or weight of the criterion significance. When this procedure is applied to all criteria, it is then possible to rank criteria according to their significance; the higher the information content or weight, the better. It is important to take into account that these weights may be used for alternative evaluation since they are built based on dispersion or discrimination of data. Consequently, they are totally objective and then hold constant irrelevant of the person doing the analysis. Assume now that the same criterion has values as shown in Fig. 2.2. Observe that because of the close values of concentration for different processes, the entropy is high, 0.97, and the quantity of information is 0.03, that is, practically negligible. Therefore, this criterion is irrelevant for alternative evaluation and can be eliminated. This is a method that can be employed to reduce the number of criteria in a certain scenario.

2.1

Basic Components of the Initial MCDM Decision Matrix j=1

Concentration of H2SO4 in μg/m3 Probability of each value

9.01 0.164

j=2

j=3

8.90 0.162

7.96 0.145

j= 4

8.96 0.163

23 j=5

j=6

Sum of values

10.05 0.183

10.01 0.182

54.89

Constant K = -1/ln (6) = -0.558111

Entropy Ej1 Ej2 Ej3 Ej4 Ej5 Ej6 -0.29 -0.29 - 0.28 - 0.29 - 0.31 - 0.32

Sum of Ej ∑ =1 ln( )

Entropy Criterion C i K ∑ =1 ln( )

Quantity of information

0.998

0.002

- 1.79

Fig. 2.2 Different H2SO4 concentrations for different processes when there is low discrimination

Weights determined by entropy can be used in any MCDM method, except AHP and ANP, which determine weights on their own. Nevertheless, most actual methods continue using criteria weights derived subjectively (from AHP). Probably, they are good to gauge criteria relative importance, even when this ranking can change depending on the analyst, but they are not suitable for evaluating alternatives. Another point to consider is that weights derived from entropy are not subject to any bias from the DM. Unfortunately, it is not possible to state the same for weights derived from DM’s preferences.

2.1.6.3

Actions for Criteria

They establish goals for criteria, maximizing and minimizing or equalling them to their resources and restrictions. For instance, a criterion may call for maximizing the use of stocked raw material, another may call for minimizing risk, and another may call for equating available funds. It utilizes mathematical symbols to indicate these actions.

2.1.6.4

Resources and Restrictions for Criteria

Used for specifying the availability of resources to which the criteria are related. There can also be restrictions to limit the scope of criteria. For instance, in an oil refinery, there are limits for storing gasoline because capacity in tanks is limited, and also normally, there are different storage tanks with different capacities for each other product; this has to be inputted to the method; otherwise, the method may assume that capacity in infinite.

24

2.1.7

2

The Initial Decision Matrix and Its Relation with Modelling a Scenario

Performance Values

These cardinals indicate the participation of each alternative in satisfying each criterion. These values may be real (quantitative criteria) or uncertain (qualitative criteria), and mixed in any proportion; that is, there can be any blend of maximization, minimization, and equating criteria.

2.1.8

Decision Matrix

All the above elements are arranged in a matrix, which may have any size, where alternatives are in columns (in some methods, in rows), criteria are in rows (in some methods in columns), and in their intersection the performance values. It also indicates for each criterion its resources and/or restrictions, as well as its action.

2.1.9

Methods

When the decision matrix is built, it is in most methods the starting point, and using a mathematical procedure to solve it. That is, solving means employing algorithms to select the alternative that best complies with the criteria. The importance of each alternative is gauged through a numerical value (score), the highest the better (for any action), and followed by a series of other alternatives with lower values (ranking). Some methods produce only the best alternatives; however, most methods yield both, the best alternative and the ranking. The set of scores or ranking is the result.

2.2

Routines to Perform with Data

The components of the initial decision matrix have been briefly analyzed. Once the decision matrix is complete, there are two very important and unavoidable procedures or routines to follow. The first one is “Normalization,” done just before running the data in the selected method; without it no process can give significant and valid results because to compare quantities all of them must have the same units. The second one is “Sensitivity Analysis,” done just after the end of the process and when results are obtained. Without it, no results can be considered reliable, and information given to stakeholders may be misleading. The next section comments about them.

2.2

Routines to Perform with Data

2.2.1

25

Normalization

When selecting criteria in a MCDM scenario, they are normally chosen by the DM to address different aspects in different fields, related with the alternatives, and in line with stakeholders’ wishes. Thus, it is possible to have economics criteria with performance values (in any currency and per unit). There could be environmental criteria articulated in different units such as particles suspension (in mg/m3) and SO2 concentration (in ppb) (parts per billion), while social criteria may be shown in percent or in births/100,000 inhabitants, or in reduction of crime (number/year). Consequently, for the MCDM process to work with this data, it is necessary to convert all data to the same units. This is called “Normalization.” Normalization may be done using different procedures. Most known are: a) The sum of performance values in a row as: aij
=

aij n 1 aij

8j

where: aij = original value aij* = normalized value That is: Find the sum of all performance values in a row and then divide each by this sum. b) Largest value in a row. aij 8j aij
= max aij That is: Find the largest performance value in a row and then divide each value by the largest.

c) Euclidean formula aij
=

√

aij n ðaijÞ2 1

8j

That is: Compute the formula in the denominator and then divide each performance value by it. d) Maximum/minimum ratio aij - min aij aij
= max aij - min aij 8j That is, compute the difference between (aij) and (min aij) and divide this result by the difference between (max aij) and (min aij). In general, all normalization procedures must give the same results except for the last one (d). This computation differs from the first three since it incorporates a subtracting symbol as the minimum performance value. This is advantageous in MCDM because the first three procedures produce some concentration of values (less discrimination), while the last one favors discrimination.

26

2

2.3

The Initial Decision Matrix and Its Relation with Modelling a Scenario

Rank Reversal

Given a Multi-criteria Decision-Making (MCDM) scenario, say with four projects A-B-C-D, subject to several criteria and solved by any method, the result indicates precedence of some projects over others and this preference/equality constitutes a ranking. For instance, in a case the ranking—after using any decision-making method— could be: B≽A≽D≽C. The symbol “≽” means is preferred or equal to, or precede to; therefore, B is preferred to A, which is preferred to D, which is preferred to C. Rank reversal (RR) is a phenomenon producing changes in a ranking, materialized by results alteration or even reversal in the order of preferences when alternatives are added or deleted. It was discovered by Belton and Gear (1983) on the Analytic Hierarchy Process (AHP) (Saaty, 1987). Rank reversal is considered undesirable since it shows weakness in the method used for decision-making by suggesting instability of the solution to changes in the alternatives. Because of this, some authors propose that a comparison between different methods to determine the most appropriate and reliable—something that has not been achieved yet—could be made by considering robustness and strength, which is, keeping ranking stability when the original system of projects is modified by changing the number of projects. Wang Taha and Elhag (2006) and Maleki and Zahir (2013) performed an exhaustive analysis of RR occurrence in different methods. Experience shows that several actions may alter a ranking as follows: 1. 2. 3. 4.

Adding a worse project Adding a better project Adding a project which is a near copy or is identical to another Deleting a project

If a new project “E,” worse than any in the ranking, is added, sometimes this addition causes irregularities. Common sense and intuition say that if “E” is worse than all others, it should go to the end of the ranking and not alter the ordering. Conversely, if “E” is better than all the others, then it should go to the top of the ranking and, again, not alter its original order. Therefore, both cases are not producing RR but placing the new project in some position or intercalating it into the ranking. For instance, considering the abovementioned ranking, it could appear as E≽B≽A≽D≽C if “E” is the best of them all, or B≽A≽D≽C≽E if it is the worst, or B≽A≽ E≽D≽C if it is better than “D” and “C.” Observe that in the two first cases, the addition of “E” does not alter the existing precedence, and that in the third case, the ranking is also preserved since it only incorporates “E” preference regarding “D” and “C,” and then respecting their original order. If “E” is identical to any other of the original set, its inclusion should not produce RR and, consequently, without any influence in the ranking. This is what common sense says; however, reality shows differently.

2.3

Rank Reversal

27

Literature about RR asserts that if a worse project is introduced, no changes should be produced in the ranking. However, how is a project defined worse or better? This is a fundamental issue not often addressed. According to some authors, it depends on how it complies with all criteria in maximizing and minimizing. That is, if an alternative vector has the maximum values for all maximization criteria and the lowest for minimization criteria, it is apparently the one that best complies with them, although this circumstance is rarely, if even, achieved. There are always alternatives that may have the highest performance values for some maximization criteria, but not for all of them, as well as not producing the minimum values for all minimization criteria. It is also possible, and it is very frequent, that two or more alternatives have the same values for a certain criterion. There is no doubt about the necessity of determining the causes for RR, and diverse theories were developed to explain it. Analysis and discussions have been going on for years and with different explanations, and certainly, this book does offer none; instead, it proposes to analyze the structure of a new entering alternative vector. From this author’s point of view, this is the hub of the question because on what basis is it possible to assert that a project vector is worse or better than others? Each new project or alternative vector has two main components: a) The value of its cost or benefit (Cj) b) Its performance values for the set of criteria (aij) Consider an initial decision-making matrix for a scenario that has projects in columns and criteria in rows, which is the standard structure in mathematical equations, as criteria are. For criterion “i3” the performance value (a34), that is, the performance value on the third row or criterion (3) and the fourth column or project (4)), may be better than any other performance value for this row, while for criterion “i2” it could be the opposite. Regarding cost and benefits, it is immediate to see which project is the best, but it is not the same for performance values. That is, checking if the new project has a lesser cost or a larger benefit than others is not enough; the performance values also play a very important, if not a larger, role than (Cj). However, comparing the influence of its (aij) is a more complicated issue and not seen immediately, because it is not the numerical value that matters but the vectors interaction. Therefore, this author sustains that it is impossible just by observing the vector for a new project to decide if it is worse or better than others. If a vector is introduced into an existing system, the result may indicate if it produces RR or not. If, as analyzed at the beginning, it locates in any position of the ranking without modification of the existing preferences, it is probably safe to affirm that it does not produce RR. However, if the existing arrangement from the initial projects is altered, then most possibly it is safe to affirm that it produces alterations. Wang et al. go further in this analysis by establishing that a reliable and stable method for decision-making should not produce RR when it is subject to these three different tests:

28

2

The Initial Decision Matrix and Its Relation with Modelling a Scenario

Test number 1: “An effective MCDM method should not change the indication of the best project when a nonoptimal project is replaced by another worse project (given that the relative importance of each decision criterion remains unchanged.” Test number 2: “The rankings of projects by an effective MCDM method should follow the transitivity property.” Test number 3: “For the same decision problem and when using the same MCDM method, after combining the rankings of the smaller problems that a MCDM problem is decomposed into, the new overall ranking of the projects be identical to the original overall ranking of the un-decomposed problem.” Other researchers believe that the most difficult situation appears when two projects have very close performances (or are near copies), or when they are identical. See Saaty (1987) and Belton and Gear (1983). Cascales and Lamata (2012) even assert that “It is well known that when the projects are very close the order between them can depend on the method used on their evaluation.” For a maximization criterion, the new project may have a performance value that is worse than all of the others in that criterion, or better, or in between. Consequently, by asserting that a new project vector is worse than those existents, all performances in all criteria, as well as the corresponding (Cj), must be worse than the balance, which in reality is possible but uncommon. Some authors (Wang & Triantaphyllou, 2006) try to analyze this issue by using random numbers in a simulation, which certainly may respond to realism for a new project vector. This author’s opinion is that there could be situations where the existence of better performances can lead to an alteration of the ranking—but not to a RR— because there should not be alteration in the existent preferences as was commented above, and then it cannot be considered an irregularity but a right consequence of its characteristics. Most methods produce RR; however, there are some such as ANP that is not affected (as claimed), while AHP is affected. There are many theories about the reasons for RR, especially for those methods that work with weighted sum of priorities. The SIMUS method, analyzed in Chap. 7, does not produce RR, and the reason is explicitly explained in Sect. 7.8.

2.3.1

Possible Causes for RR

Why RR appears? Nobody really knows for sure; however, there is consensus that different aspects can produce it. Many scholars argue that the rank reversal phenomenon is unpreventable when some MCDM method is applied. Some believe that this phenomenon seems to be an inherent problem related to criteria, which are measured on different units (Shing et al., 2013). According to the literature, most all known methods produce RR, and several hypotheses have been elaborated to explain this issue without a valid explanation about its origins since potential causes differ with methods, which suggests that the

2.3

Rank Reversal

29

phenomenon is attached to a particular approach. RR has been detected and studied by many researchers, and since the impossibility to name all of them there are mentioned here only a reduced number of researchers such as Wang Taha & Elhag (2006) and Cascales & Lamata (2012) for addressing “Technique for Order Preference by Similarity to an Ideal Solution” (TOPSIS); Wang & Triantaphyllou (2006) for ELECTRE “Élimination Et Choix Traduisant la Réalité” (ELECTRE); Mareschal et al. (2008) and Verly and De Smet (2013) for “Preference Ranking Organization Method for Enrichment Evaluations” (PROMETHEE-GAIA); Belton & Gear, 1983 and Saaty & Sagir, 2009 for “Analytical Hierarchy Process” (AHP); and Wang & Luo (2009) for “Simple Additive Weighting” (SAW). However, “linear programming” (LP) developed by Kantorovich (1939) and made available to practical situations by the “Simplex Method” (Dantzig, 1948), which was the first method to treat MCDM problems, is apparently immune to RR. See also (Wang & Luo, 2009). To examine RR, a ranking given for a certain problem is analyzed. Say for instance that for five projects identified as P1, P2, P3, P4, P5, the ranking is as follows: P5≽ P3 ≽ P1 ≽ P4 ≽ P2. This ranking is called “Original,” where the “best” solution is obviously P5. Usually project P5 is called “optimal” and the others are called “nonoptimal.” This author believes that these definitions are a misnomer or inappropriate; optimal, at least in MCDM parlance, is identified as a result that cannot be improved, or Pareto1 efficient. That is, Pareto optimality is when no “solution can be better off without making another solution being worse off”, but using heuristic methods, it is unknown in most situations what the optimal solution is, except using mono-objective LP that can guaranty that a solution is optimal. In addition, there could be no optimal solutions in a case where contradicted objectives exist, such as maximizing benefits and at the same time minimizing costs; it is evident that both notions are conflicting, and then it is difficult if not impossible to find a solution optimizing simultaneously both objectives. Since it is not known which optimal solution is (if it exists), it is not convenient in heuristic methods to qualify results as optimal or suboptimal. This is the reason why all MCDM heuristic methods aim at a feasible but compromising solution, that is, a solution satisfying all parties. However, in order to agree with other authors, these two terms are also used in this book. Suppose this ranking: P5≽ P3 ≽ P1 ≽ P4 ≽ P2. Here, the transitivity principle establishes that if P3≽ P1, and P1≽ P4, then P3≽ P4. Assume P4 is deleted; in this case, after the method is solved by a certain method, the result is P5 ≽ P2 ≽P3 ≽ P1. Is this rationally, correct? No, it is not. Why? Because now P2 precedes P3 and P1 and there is no reason for that since P2 should have kept its position at the end of the ranking. It could also have been P2 ≽ P3 ≽ P1≽ P5. As can be seen, the result is reversed. That is, RR can convert a project

1

Vilfredo Pareto. Italian engineer and economist.

30

2 The Initial Decision Matrix and Its Relation with Modelling a Scenario

from being the best to the worst, and this is a serious issue and the reason why researchers are investigating this matter. From the literature, it appears that the main causes that can trigger RR are: • Adding a new project that is worse than an optimal or nonoptimal project. • Deleting a project. • When two projects have the same or close scores. For instance, the scores could be P5 = 0.236 ≈ P2 = 0.240. • Adding a copy of a project. For instance, when introducing a new project that is a copy of an existing project (see Belton and Gear, cited). • Decomposing the problem into pairs of projects. When finding the precedence and combined, the transitivity condition is not satisfied. As an example: If P3 > P1 and P1 > P4, then P3 > P4. If in a pair-wise comparison between P3 and P4, its result is P3 > P4, then transitivity is respected; however, if the result in the last comparison is P3 < P4, transitivity is not satisfied. Triantaphyllou (2001) postulates that ranking irregularities may occur when decomposing a decision problem into a set of smaller problems—each one defined by a pair of alternatives—and subject to the same criteria as the original problem. This disaggregation or decomposition is illustrated in Sect. 2.5.1.

2.3.2

Brief Information on Rank Reversal in Different MCDM Methods

This Section gives only a glimpse of the RR problems in several MCDM methods. There is abundant bibliography for the reader to consult on each method and by various different authors.

2.3.2.1

Rank Reversal in AHP

RR was detected by Belton and Gear (cited) while examining the analytic hierarchy process (AHP), and some efforts have been made to correct this problem that invalidates any method or at least makes it less reliable. AHP has been the most studied and criticized method, perhaps because it is the most popular and was the starting point for analysis of the RR problem that extended to other methods; there are different opinions regarding why this is happening in this method. According to Cascales and Lamata (cited), RR is apparently produced by normalization when a new activity is added. It is attributable to a change of the denominator of the Euclidean formula used.

2.3

Rank Reversal

31

Other opinions are based on the fact that AHP original method uses an additive scheme or “Simple Additive Weighting” (SAW), which considers dependency between projects and criteria for which it is called the “Relative Method.” After suggestions from several researchers, its creator agreed to use a multiplicative scheme or “Simple Multiplicative Weighting” (SMW) originating the “Absolute Method,” without that dependency, which apparently solved the problem. Other authors think that RR in AHP is produced by scaling the eigenvectors because the criteria are not independent. Wang, Y.M. and Elhag (cited) also point out that RR is produced by a change in local priorities before and after an alternative is added or deleted and propose an approach to keep priorities unchanged to avoid RR.

2.3.2.2

Rank Reversal in TOPSIS

TOPSIS method also presents RR. Some researchers, such as Cascales and Lamata (cited), present a new modification to the Hwang and Yoon algorithm to solve the problem.

2.3.2.3

Rank Reversal in PROMETHEE

Eppe & De Smet, 2015 analyze RR in PROMETHEE II and derive the exact conditions for their occurrence. See also Verly et al. (cited).

2.3.2.4

Rank Reversal in ELECTRE

According to Zopounidis and Pardalos (2010), RR in this method comes from a) the existence of discriminating thresholds and the values that should have been assigned to them and b) the fact that such a comparison is conditioned by the way the actions are compared to remaining actions.

2.3.2.5

Rank Reversal in SAW

According to Shing et al. (cited), this method is also affected when an alternative is added or dropped.

32

2.4

2

The Initial Decision Matrix and Its Relation with Modelling a Scenario

The Uncertain Best Solution

Another problem not directly related to RR that makes the scenario more complex is the fact that the same problem solved by different methods often gets different results (Wang & Triantaphyllou, 2006); consequently, the DM may wonder what method is best for his scenario, because one method may reach a solution without RR, but unfortunately, it is unknown if that solution is the best because nobody knows the answer. Consequently, if a new activity, worse than the best, is added, some methods may register a RR while others do not. As said, the problem is that since it is not known which is the best solution, the DM can accept a method without RR, but he does not have a hint about the reliability of the solution. Unfortunately, there is no way to know it since there is consensus regarding that there is not a method better than another.

2.5

Characteristics of Components of the Initial Decision Matrix

Section 2.1 enumerates the different components of a MCDM; it now proceeds to examine each one in detail. Section 2.1 enumerated the different components of a MCDM, and this Section proceeds to examine each one in detail.

2.5.1

The MCDM Process as a System

A MCDM process is a system composed of objectives, scenarios, alternatives, criteria, actions, resources, and performance values, and consequently all of them are usually connected. Examining hundreds of projects from different authors and in different fields, it is evident that there is always some kind of relationship between criteria. Regarding this issue, Sect. 1.5 describes a very common situation. It is then important, when dealing with a scenario for selecting the best alternative among many, and all of them subject to criteria, to investigate, even summarily, if criteria are independent or related between them by direct or indirect links. For some complicated scenarios, some methods choose to split it into parts and solve each separately. As a procedure for analysis, it is fine since it allows examining each part in depth, but not appropriate for decision-making. There is even a very well-known system in Project Management called WBS (Work Breakdown Structure) that uses this decomposition to analyze components of complicated subsystems. However, it is not correct to analyze a subsystem without

2.5

Characteristics of Components of the Initial Decision Matrix

33

Fig. 2.3 Scheme of alternatives and criteria

considering the influence that other subsystems or parts of another subsystem may have on it. An example is proposed (Figs. 2.3 and 2.4), following a procedure described in Triantaphyllou (2001). Suppose that there are three alternatives, A-B-C, subject to seven criteria, a-b-c-de-f-g. See the initial matrix in Table 2.1. Each alternative contributes with its performance value to the criteria, although it is not mandatory that an alternative must contribute to all criteria. Suppose that the problem is partitioned and pairs are compared, that is: A compared to B, A compared to C, and B compared to C. Figure 2.3 shows dependencies. Solving this problem by pairs, using the SAW method, the following rankings are obtained: Pair A/B, scores are: A = 0.36 and B = 0.28 then A > B. Pair A/C, scores are: A = 0.36 and C = 0.07 then A > C. Pair B/C, scores are: B = 0.28 and C = 0.07 then B > C. Consequently, the ranking is A > B > C. If the problem is solved simultaneously and using the same method, the three alternatives are ranked A > C > B. Consequently, results are different when taking pairs of alternatives, in comparison to working with all alternatives simultaneously.

2.5.2

Alternatives Relationships

Alternatives or projects may be independent, meaning that each one is not related to another, and they can be executed independently.

34

2

The Initial Decision Matrix and Its Relation with Modelling a Scenario

Fig. 2.4 Z matrix applied to examining the rail option

For instance, three independent projects in a portfolio are a new high school building, a program for keeping kids out of the streets, and building a new wastewater treatment plant. In some cases, it often happens that projects are linked for a specific purpose. For instance, it could be that for programming reasons, project A must precede project B, as in the following example. Assume three independent projects for infrastructure work in North Ave. Project A: Build storm rainwater drains. Project B: Install wiring for LED lighting. Project C: Paving. The three projects are independent; they can be executed with various deadlines. However, common sense and engineering planning indicate that Project A > Project B > Project C. The mathematical symbol “>” in this case indicates precedence. Then, if the DM has a problem of this type, he must be able to model this precedence. This is a real and frequent occurrence but not always considered. And, sometimes

2.5

Characteristics of Components of the Initial Decision Matrix

35

Table 2.1 Initial decision matrix

Alternatives Criteria

A

B

C

Weights

a

0.36

0.14

0.18

0.03

b

0.13

0.41

0.19

0.09

c

0.24

0.08

0.51

0.15

d

0.48

0.28

0.24

0.23

e

0.34

0.28

0.32

0.31

f

0.50

0.25

0.27

0.12

g

0.30

0.30

0.50

0.07

happens that a pavement barely 3 months old is broken, to allow for some work underground, when it should have been foreseen.

2.5.3

Alternatives Heavily Related: A Case—Selecting Proposals

Suppose that a large construction project for building an oil refinery is calling for bids to select a main contractor; normally a set of preselected construction firms will submit proposals. The owner very often finds that two or more companies submit together, each one on its own specialty, but complementing. This is called a “joint venture,” and then the two or three firms must be considered as one, but of course with different performance values for their services and with different costs. The modelling must take this into account, even if other companies are bidding on their own.

36

2

The Initial Decision Matrix and Its Relation with Modelling a Scenario

2.5.4

Including and Excluding Alternatives—Conditions by a Third Party

2.5.4.1

Actual Cases

a) City connectivity. b) Conditions on quantity and quality of river water. Assume a portfolio of projects where all of them are feasible. The DM must require information regarding if the different projects are inclusive or exclusive, that is, if the execution of one project precludes or not the execution of other projects. As an example, a scenario calls for the construction of a road and passenger link between two areas, separated by a firth. The alternatives are: A. Build a bridge. B. Build a tunnel. C. Establish a ferry service. It appears at first sight that once an alternative is chosen, the other two are discarded, or perhaps not. That is, alternatives may be including or excluding, since it could be stated that if the tunnel is built the other two options are not considered, or perhaps the scenario demands that “If the option bridge is selected, then the ferry option might also be considered.” The DM needs to put these two conditions in the model. How can he do that? By using mathematical conditional sentences, as for instance the “IF. . .. . .then . . . .” expression. Of course, the DM must be sure that the MCDM method he selects can manage mathematical expressions of this type. In another actual case, consider a very common scenario, which is the origin of multiple conflicts, and that very often requires Court participation, even at the international level. A national government decides to build a series of undertakings along a river that traverses several provinces. These undertakings are: a) using water for irrigation in three locations and with different crops, b) building a small hydropower plant by taking advantage of the river slope in a section of the river, and c) diverting water through a channel to supply drinking water to a city. The DM examines these undertakings along the river and talks with people that will be positively or negatively affected by these works. In the last city, at the mouth of the river where it discharges into the sea, he receives this warning from the City Hall Major: You can do whatever you want upstream, but make sure that we do not receive salty and contaminated water from agriculture and from industries located upstream or make certain that our fish industry will be not damaged if you build a dam, that will be a barrier for trout’s spawning, or

2.5

Characteristics of Components of the Initial Decision Matrix

37

Be sure that the amount of water we will receive after your projects upstream, will not produce water shortages for our irrigation undertakings. These comments warn the DM on aspects that he had not thought about, one of them being the level of salinity, at present negligible, that will have the water in the mouth of the river because of irrigation projects upstream. Consequently, this fact must also be modelled, establishing a criterion to limit salinity at a certain percentage.

2.5.5

Forced Alternatives—An Actual Case: Fulfillment of Previous Commitments

A large city (about 3,000,000) was considering working with its ten satellite cities to design a plan for infrastructure, including sewerage, housing, roads, and education. In this case, there were about 14 alternatives or projects and as many as 30 criteria. A MCDM method indicated that, considering the budget available for a 5 yrs plan, the following projects should be funded: Projects: 3-5-6-9-11. When the DM submitted this result to the City Major, he said, “Where is project 2?” The DM replied that it has not been selected since it has a very low priority. The Major stated that said project must be included since it was one of his campaign promises. Therefore, the DM had to modify the modelling to consider that project 2 must be in the final list, irrelevant to what the mathematical method says.

2.5.6

Criteria Selection

A solution normally depends on certain criteria while the others are irrelevant. For instance, in the car example, common sense says that cost, speed, and fuel consumption are without a doubt important, while a criterion such as style may not be so important and significant for car selection. Consequently, to determine the influence on the output by changing a criterion importance, a MCDM must be able to identify which are the criteria that define the output, and this is again not related with weights. Only when we know the defining criteria, it is possible to investigate their effects. At the present time, and as per this author’s knowledge, the only procedure that can deliver this information is linear programming.

38

2.5.7

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The Initial Decision Matrix and Its Relation with Modelling a Scenario

Resources—An Actual Case: Oil Refinery

Many methods do not consider resources and their availability; however, resources, whatever they might be, are not limitless; therefore, limitations in capital, duration, workforce, machinery, storage space, etc., are ignored. For the same token, sometimes they are not paying attention to established limits for air, water, and land contamination produced by projects. Let us see an example. In the oil refinery business, it is necessary to determine, probably daily, the production of gasoline, diesel oil, kerosene, etc., which of course depends on market demand. However, this demand is not constant, and then refineries need to determine how much of their final products must be stored in tanks to compensate for market fluctuations. Storage is expensive, bulky, and limited; therefore, it is essential to know how many cubic meters of storage capacity the refinery has for each one of its different types of final products, and data must be added to the initial decisionmaking in a storage capacity criterion. Another fundamental data needed relates to the daily capacity of transportation for delivering the products to the market, and their frequency, again in a transportation criterion. By reading published papers, it appears that only about two MCDM methods, out of probably two dozen, consider this limitation in resources.

2.5.8

Criteria Range

A criterion cannot be increased or reduced infinitely. Each one has a range from a minimum to a maximum value. When any of these limits are exceeded, for instance, the daily production exceeds capacity, the storage and the transportation criteria are no longer relevant, and another criterion such as demand must be considered. Its interpretation in a MCDM scenario is that these criteria will no longer have an effect on the best alternative. For this reason, the MCDM method must indicate the permissible range of variation of each one.

2.5.9

Annual Budget Restriction—An Actual Case: Five Yrs Development Plan

Normally, governments, companies, institutions, etc., elaborate a five-year plan for developing different types of projects. There is a budget for each project and also an annual budget that corresponds to the sum of disbursements for all projects in that year and based on different percentage rates of completion for each one. There could also be undertakings underway from past years that have to be finished and are also probably in different stages of completion.

2.5

Characteristics of Components of the Initial Decision Matrix

39

There are projects underway that must be completed, as well as new projects that start at different times, and that do not finish at the same time. In addition, each project has its own duration in months or years and an estimated percentage of completion during each year. The scenario is complex since a selection of projects must be done to use the maximum amount of available funding, not exceeding annual budgets and in compliance with all criteria related to social, environmental, and risk issues. Naturally, if there are enough funds and time, obviously, all projects will be selected. However, more often than not, funds are scarce and then the problem consists in selecting the projects and their ranking that, considering the funding, can comply with all restrictions from the economic, financial, and timing points of view. It is then possible to generate a set of criteria, one for each year during the planning horizon, and where the aij are articulated as percentages of completion, and all of them calling for maximization and using the “≤” operator, since expenses cannot be greater than funds available. Obviously, each criterion must be limited by the available annual funding, and also, as usual, there are criteria related to benefits, costs, environment, public opinion, etc. If the total funds available for the whole portfolio are less than the summation of the annual budgets, obviously, some projects are not going to be selected; a MCDM method should identify the best combination of projects that makes the maximum use of scarce funds.

2.5.10

Criteria Correlation

What is it? Sometimes, in addition to simple relationships between criteria, as in the trivial case of purchasing a car commented in Sect. 1.5, there are correlated criteria. In correlated criteria, the increase or decrease in one criterion corresponds to a proportional increase or decrease in another. Also, an increase in one criterion may correspond to a decrease in another. That is, both criteria move either in the same direction or in opposite directions. This is expressed by the correlation factor where a correlation of (+1) means a perfect direct or positive correlation, while a value of (-1) denotes a perfect inverse correlation. Zero correlation reveals that no relationship exists between the two variables. Obviously, the higher the correlation, which is close to +1 or - 1, the better the fit; however, it is necessary to bear in mind that correlation, whatever its values, does not necessarily mean a cause-and-effect relationship because there could be other factors that can be influencing. Where can we find it? In many scenarios, have a look at this example. A hydroelectric powerhouse is designed to generate a certain amount of kWh per year; however, there is a strong correlation between annual generation and river flow, which in turn is heavily correlated with snowfalls, rain, or both. The modelling

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2 The Initial Decision Matrix and Its Relation with Modelling a Scenario

must consider these correlations, and not only that, it must allow the analysis of different hydrological conditions related to the alternative selected. Correlation between criteria and their effect can be easily visualized. See Sect. 6.6.

2.5.11

Risk: A Fundamental Criterion

Risk can be defined as: (the probability of occurrence of an event) times (the impact or consequences that this event can generate). Unfortunately, risk is not always considered in MCDM; however, there is risk—perhaps due to different events—with no exceptions, in all projects. We can then say that there is a risk by not considering risk. Consequently, criteria calling for minimization of different kind of risks are a must in an IDM (Initial Decision Matrix). Most probably, these criteria are qualitative because they consist of results derived from probability values and not on exact figures. There are no universal units for risk; however, its definition is that the figure indicates the damage, of any kind, produced by a certain risk. The same type of risk-generating event, such as an earthquake, may produce heavy losses in human lives and in money in case that a highway collapse, as happened with the Cypress Street Viaduct in 1989 in Northern California, which killed 42 people. The same earthquake affecting another area can cause similar devastating effects and losses of human lives but also very dangerous consequences, as happened in 2011 in Fukushima, Japan, which destroyed a power nuclear plant and provoked the release of a radioactive plume. Therefore, units for measuring a risk may be variable; however, all of them are related to losses. And this is something that the DM needs to consider. When several projects are examined for selection, it is impossible to establish in which part of the undertaking a situation may create a risk. Risk conditions are everywhere, from the management’s decision to go ahead with a project up to its conclusion and sometimes even further. In MCDM, risk may be present in many scenarios; however, the most important are: • Personnel risk. This should be contemplated in all projects, but mostly in heavy civil and mechanical works (high-rises construction, tunnelling, underwater work, bridges, etc.). This requires the DM to contemplate criteria devoted to additional cost for personnel training, safety people on site, safety measures, control, etc. In some countries, it is mandatory for all personnel, in any position, to take and approve a training course, which in the United States is called OHSA (Occupational Health and Safety Act). Some projects require more intensive preparation than others and different training durations. It naturally involves a cost and, in addition, paying the employee his/her normal hourly rate, including during the course.

2.5

Characteristics of Components of the Initial Decision Matrix

41

• Risk by suppliers’ delays. This is another very important item since a delay in delivering equipment may cause over costs because personnel and machinery are idling and waiting for equipment and materials to arrive. Criteria addressing this issue must consider, for instance, in selecting equipment from overseas, delays in delivering from the manufacturer, and potential delays in transporting it to the job site. Of course, it cannot be known if equipment may or may not be delayed; however, a criterion can be established to take account of the difficulties that logistics pose for every project. In some large projects such as hydro-projects, it is normal that qualified personnel from the owner periodically visit the manufacturers’ overseas premises to verify quality and that the progress of work goes according to schedule. At the same time, it is necessary to consider delays that can take place during transportation to port and during the ocean trip. This naturally implies provisions for additional funds and perhaps more time than scheduled, information that must be incorporated in the IDM. • Accessibility risk. One common issue is to detail for each project, in a criterion such as “Access,” the conditions regarding access to the site for personnel, equipment, and supplies; some projects are so inaccessible that the construction site can only be reached by helicopter. This author learned about a metallurgical project in a mountainous area, where lack of consideration of this aspect provoked a delay of nearly 3 weeks in receiving heavy equipment by truck, because even when there was a road, the bridges were not designed to stand heavy loads and needed to be reinforced. Obviously, this risk was not contemplated, as it should, during the planning stage; however, it is the DM’s obligation to take it into account because these are real circumstances and they are aspects that could very well impact selection. It could also be that for the same project there were two different seaports where equipment should be unloaded and loaded into trucks. Here, there are then two ports alternatives to ponder for the same project; that is, it is a decision nested inside one of the alternatives. • Local conditions risk. In some countries, it is known that whatever permits a company may have with the local or central government, there are also local chieftains that demand that the contractor receive their “blessing”; this must be taken seriously. This author bears witness to a case where the company did not pay attention to these courtesies. The result was that on a certain day all local workers abandoned the place because a chieftain put a curse on them if they continue working for the company. Some regions are maintaining fights with other regions for reasons such as territory dominance or, more frequently, for political motivations. This can paralyze the work and even may have serious consequences at a personal level. As an actual case, some years ago, a company was projecting the construction of an oil pipeline along 1600 Km, and there were several routes to choose from. Once the MCDM study was performed and selected the best route, the company

42

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The Initial Decision Matrix and Its Relation with Modelling a Scenario

rejected it and instead decided for one of the least preferred; the reason was sabotage. This threat was initially known by the company; however, it was not considered in structuring the IDM. • People opposition. This fact is normally very serious. There are many actual examples involving large companies that did not evaluate the risk posed by people that oppose a project. For instance, in 1991, a huge hydroelectric project in Western Canada was halted by the Supreme Court, who ruled for the native people who were complaining about the effect of water extraction from a lake that was home to trout, their staple, and that would impact on their way of life. The ruling provoked the closing of the project after the company invested hundreds of millions of dollars. The native people’s argument and resistance were well known even before the work commenced, but the company did not evaluate it as a serious risk. Something similar occurred in South Argentina when a company began work for an open pit gold mine, and even when they did know of the people’s opposition, they did not evaluate the risk. The result was that the same people forced the company to abandon operations. In another case, the risk involved lack of evaluation even at the international level, when a country opposed the construction of a paper mill factory by a neighbor country at the shores of the river they shared, under the argument that the plant would contaminate the water with consequences for people living downstream of the river. • Weather. There could be a risk here. For instance, in the mentioned hydro-plant in Western Canada, special precautions were taken to prevent snow avalanches from destroying a construction camp. The risk was so serious that the Canadian Army authorized the use of a cannon to shell the snow and ice in a nearby mountain before it reached the conditions for an avalanche. In this case, the risk was correctly evaluated and measures taken. In high-altitude work that takes place at about 4000 meters above sea level (MASL), it is necessary to consider the fact that workers are subject to very harsh conditions, which can lower their performance. Two mining projects, for instance, one at 3800 MASL is at a disadvantage with another project at say 800 MASL. This must be contemplated in structuring the IDM. • Health conditions. Some places, especially in Africa, are subject to diverse diseases provoked by mosquitoes, contaminated water, or excessive heat and heavy rains. There is a certain risk that the DM must consider for each project according to data available for such places. There is also an additional cost because the personnel must be medically tested and immunized. He must also remember that in isolated sites there must be a clinic staffed by a physician and/or perhaps by a qualified nurse.

2.5

Characteristics of Components of the Initial Decision Matrix

2.5.12

43

Examining Differences in Results for the Same Problem Between Assumed Weights and Weights from Entropy: Case Study—Electrical Transmission Line

The following example refers to an overhead transmission line between two places separated by hilly country. Three potential routes are evaluated by ten criteria considering economic, environment, people’s opinion, terrain, and risks (see Table 2.2). This scenario is solved using TOPSIS (Hwang & Yoon, 1981). Options are in columns, while criteria are in rows. Notice that there are minimization and maximization actions. Assume that the DMs have elicited criteria weights, as seen at left in Table 2.3 in column “Elicited weights from DM”; notice the high values (underlined) assigned to “Costs” (0.21) and “Distance” (0.14), which seems reasonable, since cost is indeed a very important factor and distance is paramount considering this kind of undertaking. When solved, the ranking shows that the best option is the project labelled “Suquia River Valley” (score: 0.53), followed by option “Suquia Valley/Mountains” (score: 0.52), and in the last place, the option “Trevel Mountains” (score: 0.49). Observe that the scores are very close and then, the DM may wonder if it is legitimate and safe for him to choose the first option, when there is a slight difference in scores with the second one. What would happen if instead of using criteria weights elicited from preferences, these weights were obtained by entropy, that is, objective weights as explained in Sect. 2.1.6.2? These weights, shown at the left of Table 2.4, come from a separate computation and are not included in TOPSIS; they are only provided to illustrate their use; however, their calculation follows the example posted in Sect. 2.1.6.2. Table 2.2 Initial data Criteria Total distance Total cost People’s opinion Slope Savings in trans.towers Crossing scenic area Ecosystem protection Swamps crossed Earthquake risk Forestry

Suquia River Valley 427 1389 40 80 34

Trevel Mountains 395 1519 60 40 24

Suquia Valley/ Mountains 419 1418 30 40 30

Action MIN MIM MAX MIN MAX

Sum of rows 1241 4326 130 160 88

27

84

23

MIN

134

27

67

23

MAX

117

19 4 5

2 11 10

10 7 7

MIN MIN MIN

31 22 22

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The Initial Decision Matrix and Its Relation with Modelling a Scenario

Table 2.3 Selection of route for transmission line using elicited weights by DMs

Elicited weights by DM

Suquia Trevel Suquia River Mountains Valley / Criteria Valley Mountains

Distance 0.14 0.21 Cost 0.1 People's opinion 0.09 Slope 0.07 Sav. trans. towers 0.1 Crossing sce.areas 0.11 Ecosyst.protection 0.02 Swamps crossed 0.07 Earthquake risk 0.09 Forestry Dist. rom each project to ideal (R+) Dist. from each project to ideal (R-) Result

0.048 0.067 0.031 0.045 0.027 0.02 0.022 0.012 0.013 0.02 0.0480 0.0540 0.53

0.045 0.074 0.046 0.023 0.019 0.063 0.063 0.001 0.035 0.041 0.0560 0.0540 0.49

0.047 0.069 0.023 0.023 0.023 0.024 0.017 0.006 0.022 0.029 0.0490 0.0550 0.52

(A+) (A-) Positive ideal Negative ideal solution solution MIN MIN MAX MIN MAX MIN MAX MIN MIN MIN

0.045 0.067 0.046 0.023 0.027 0.017 0.063 0.001 0.013 0.021

MAX MAX MIN MAX MIN MAX MIN MAX MAX MAX

0.048 0.074 0.023 0.045 0.019 0.063 0.022 0.012 0.035 0.041

Ranking: Suquia River Valley - Suquia Valley/Mountains - Trevel Mountains

Table 2.4 Selection of route for transmission line using elicited weights from entropy

Elicited weights by entropy

Criteria

Distance 0.001 Cost 0.001 0.052 People's opinion 0.075 Slope 0.013 Sav. trans. towers 0.230 Crossing sce. areas 0.154 Ecosyst.protection 0.327 Swamps crossed 0.099 Earthquake risk 0.099 Forestry Dist. from each project to ideal (R+) Dist. from each project to ideal (R-) Result

Suquia Trevel Suquia River MountainsValley / Valley Mountains 0.000 0.000 0.000 0.000 0.000 0.000 0.016 0.024 0.012 0.037 0.019 0.019 0.005 0.003 0.004 0.046 0.144 0.039 0.036 0.088 0.030 0.200 0.021 0.105 0.018 0.049 0.031 0.011 0.023 0.060 0.1880 0.1100 0.1040 0.1030 0.1900 0.1440 0.36 0.63 0.58

(A+) (A-) Positive ideal Negative ideal solution solution MIN MIN MAX MIN MAX MIN MAX MIN MIN MIN

0.000 0.000 0.024 0.019 0.005 0.039 0.088 0.021 0.018 0.011

MAX MAX MIN MAX MIN MAX MIN MAX MAX MAX

0.000 0.000 0.012 0.038 0.004 0.144 0.030 0.200 0.094 0.023

2.5

Characteristics of Components of the Initial Decision Matrix

45

When solved, the ranking shows that the best option is now the “Trevel Mountain,” project, and then reversing the result shown in Table 2.3. Notice that there is also a large discrimination between scores. Table 2.3 showed that the most important criteria with preferences for elicited weights were “Cost” (0.21) and “Distance” (0.14). Look now at their values (underlined) with entropy-derived weights in Table 2.4. For the same two criteria it shows: “Cost” (0.001) and “Distance” (0.001). Consequently, evaluating alternatives with these two criteria would be irrelevant since they do not have any significance from their information content. As a matter of fact, they can be eliminated. However, how does one know that entropy-derived weights are more reliable than elicited weights? Look at Table 2.2; for criterion “Cost,” there is only an 8.5% difference between extreme values, while for “Distance,” this difference is 7.5%. These values show that these two criteria are not appropriate to make alternative evaluation because their performance values are very close; in other words, they have very low evaluation capacity. In Table 2.4, the criterion “Swamps crossed” has the largest weight of 0.327, followed by “Crossing scenic areas”; look at Table 2.2; the difference between extreme values in “Swamps cross” criterion shows a difference of 89%, which shows that this criterion is appropriate to evaluate alternatives. For the same token, the difference between extreme values in “Crossing scenic areas” criterion is 73%. An analysis on weights significance is now performed using data from Table 2.3. Suppose that environmentalists make a strong objection since “Cost” and “Distance” have much higher influence because of their assumed weights than all weights corresponding to environmental issues. Consequently, the DM and stakeholders decide to lower the cost weight from 0.21 to 0.17, and then increase weights for “Crossing scenic area” (to 0.11), “Ecosystem protection” (to 0.12), and “Forestry” (to 0.11). By running the software, the result shows (albeit not displayed here) that the ranking has not changed, although the weight for “Cost” has been reduced in (0.210.17) / 0.21 = 0.19 or 19%, which is significant. This result proves that criterion “Cost” has no relevance, as already mentioned, to qualify alternatives since its computed entropy-derived weight (as indicated in Table 2.4) is 0.001 and, therefore, negligible. Naturally, varying a criterion weight does not change the relationships between performance values; it simply changes the trade-off values between criteria. The capacity that said vector has for evaluation is completely ignored, as Greco (2006) states when referring to the actual system of elicited weights “But, this kind of analyses has rather a theoretical interest than a practical one.” It is then understandable how an alternative that has the highest performance in a criterion that calls for maximization, which in addition has the largest weight, could not be chosen because that criterion may not have the capacity to discriminate between alternatives, even if it is deemed the most important.

46

2.5.13

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The Initial Decision Matrix and Its Relation with Modelling a Scenario

Working with a Variety of Performance Values—An Actual Case: Environmental Indicators

Normally, all performance values, that is, those that indicate the contribution of each alternative to each criterion, are positive. However, some projects may require using simultaneously positive, negative, and absolute values for a given problem. Therefore, it is necessary to contemplate them all in the IDM. Suppose that the objective is to select 25 environmental indicators in a country out of, say, 250 original indicators; those indicators are subject to a set of criteria, not all of them applicable to all indicators. The relationship between indicators and criteria may come from correlation, statistics, experts’ opinion, etc. However, a certain indicator may be positively correlated with a criterion, meaning that both increase and decrease at the same time, but there could also be negative correlation where there is an inverse relationship between both, and in this case, the correlation is negative; hence, the negative value must be inputted in the decision matrix. However, there is another aspect to bear in mind; normally, the procedure for determining the importance of each indicator calls for adding up all values for each one, but in so doing, the negative values would be deducted from the addition, when they must be added, and thus it is necessary to work with absolute values. This condition must also be indicated in the IDM. It is also necessary for the method to work with mathematical notation for large numbers, that is, using the mathematical notation. For instance, 150,000,000 can be expressed as 150 × 106, and the method must be able to convert those numbers to a decimal format.

2.5.14

The “Z” Method for Determining Some Performance Values for Qualitative Criteria

In general, the decision-making process may be thought of as a set consisting of information, figures, statistics, facts, and numbers. Sometimes, these components or elements of the set have an individual value per se; however, in many others they are interrelated. It can be that an element, if materialized, influences some elements, which in turn impinge on others in some sort of domino effect. Consequently, it appears that considering elements as isolated entities, per usual, does not portray reality, and as such it may produce unreliable results. This is different from correlation since this is a series of activities, one of them receiving information from the precedent, working with it, and then transferring its results to the next in line. A common example is noise inherent to an airport operation. Noise is important by itself, but what about its effects on areas close to or near the airport? It can be very upsetting for people living in the vicinity because of the high noise levels, day and

2.5

Characteristics of Components of the Initial Decision Matrix

47

night, and in fact, there are stiff regulations in place that specify procedures to be followed during aircraft takeoff and landing. Arriving and departure flight paths are designed to avoid flying in populated areas, as well as the replacing noisy old equipment. This single factor, noise, increments with frequency and number of flights, and then its reduction or increment has a very important economic effect, as well as on human health and the environment. In this case, it is necessary to detect, analyze, and evaluate these derived effects and impacts, and this can be quantified using the “Z method” (Munier, 2015). As an example, consider the following scenario: City Hall plans to build an urban light transit system and has three alternatives: a) Utilize an abandoned railway track that was used by freight trains. b) Use it as a dedicated road for express buses. c) Convert it into an urban avenue for car traffic. The three of them have advantages and disadvantages, some of them of qualitative nature such as noise and vibration from trains and noise and fumes contamination by buses and cars. Therefore, it is necessary to select one of the three options, subject to a set of criteria. Figure 2.4 shows how they are related when the light transit option is examined. The main purpose of this analysis is to generate performance values for the initial matrix that reflect the final or accumulative result of the series in linked activities. Components are: “Actions” vector. It enumerates activities due to project operations. “Effects” matrix. Reflects and quantifies outcomes of actions. “Receptors” matrix. Links and quantifies outcomes on recipients, that is, people, wildlife, and financial aspects from the effects matrix. “Consequences” matrix. It links receptors with consequences or impacts on them from the receptor matrix. “Response” matrix. It registers mitigation or rehabilitation measures that can be taken to decrease consequences from the consequences matrix. As can be seen, the action vector output is an input for the effects matrix; its output is in turn an input to the receptor matrix and so on. The action vector and the four matrices are called “The Z-matrix system.” Train operation, “Action,” generates Air contamination, Noise and Vibration effects. One of them, “Noise” for instance, has been measured in 78 dB, which is quite high. Receptors for this level of noise are Neighbors and Wildlife. If for neighbors, 1.781 people are affected, as shown in the receptor’s matrix. “Consequences Matrix” refers to people complaining due to noise, potential structural damage to buildings because of trains passing very closely to them, and light contamination since night illumination invades nearby properties. The assessment for people’s criticism and protests is valued at 5897 Euros due to the number of complaints. Response from the City Hall is the installation of sound barriers and

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2 The Initial Decision Matrix and Its Relation with Modelling a Scenario

some type of dampers to reduce vibration, as well as dimmed lighting, at a cost of 194,810 Euros. Therefore, it can be perceived how a single action, a train passing, produces a chain reaction or a series of impacts that are linked. These are the performance values that should be inputted into the decision matrix in the respective criteria and for the train alternative. As can be observed, there could be many Z-paths considering different effects, receptors, consequences, and responses. In this elemental case, the result is that the project cost could increase by 194,810 Euros and was probably not foreseen. Using this tool, it is then possible to have more realistic values for the decisionmaking analysis and even add more evaluation criteria as a consequence of its use. It can also be designed to portray and quantify serial risks for a developer, to analyze the convenience, or to cancel the project.

2.5.15

The Z Matrix—CASE STUDY: Determining Risk Performance Values for Inputting in Risk Criteria

This case refers to a study of a nuclear power station located on a seashore and in an area that registers high earthquake activity and has been designed based on the Fukushima accident, that is, the reasons for failure in that nuclear power plant are compared here, and a series of options are developed to avoid replication. Therefore, the objective of this case is to develop appropriate performance values to input them in the corresponding criteria calling for minimization of risks in the IDM. These options could be, for instance, different ways to increase safety in reactor rod control. In Fig. 2.5, navigate through the framework along the shaded areas and large arrows. This procedure determines those performance values by considering direct, indirect, and induced effects, from an initial action (1) potential risks), which in this case is an earthquake and/or tsunami, and determines its effects that are logged in the effects matrix (2). This data is the input for a consequences matrix (3) that registers how effects influence people and installations. Its output goes to a consequences matrix (4) that enumerates plans for mitigation. Finally, results are the input of a response matrix (5) where the value of each mitigation measure is assessed, and this being for a certain action, in this case, “Earthquake in a Richter scale” performance value. This analysis can help in determining, thinking backward, those potential risks that could be decreased, for instance, by improving people’s capacity through training. It is believed that this procedure can be applied to any of those MCDM methods that do not determine performance values using the DM’s preferences. The case goes as follows: Suppose that in this region there is a threat of earthquake activity, with an expected probability of 0.6. It can produce different effects such as flooding,

Fig. 2.5 Z matrix for determining total impact in a coastal nuclear plant

2.5 Characteristics of Components of the Initial Decision Matrix 49

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2 The Initial Decision Matrix and Its Relation with Modelling a Scenario

which is assumed as 0.4, with external interruption of electric power of 0.2, with probability of affecting the containment structure of 0.02 and with a capacity of affecting cooling pumps of 0.5 (see Fig. 2.5). At the same time, or independently, consider that a tsunami produced by the earthquake has a probability of occurrence of 0.3, and that there is 0.2 conditional probability of waves reaching a maximum height of say 11 meters. Naturally, this last value varies according to the location and the height over the sea level where the plant is built. According to some researchers, there is a linear log relationship between the magnitude of an earthquake and a tsunami height. Therefore, the model could incorporate a formula to automatically compute the expected tsunami height corresponding to a quake activity as measured by seismographs. To make clear this example, let us analyze one effect, the “Containment structure failure.” It has a conditional probability of 0.02, and this value is outputted to matrix 3 (dashed line). Here, for a consequence such as “Plums of radioactive isotopes” (first row in matrix 3, with a value of damage of 0.06), the impact due to failure in the containment is 0.06 × 0.02. For the second consequence, “Size of population affected,” the impact is assumed to be 0.3. Consequently, the impact due to failure in the containment will be 0.3 × 0.02. The same procedure repeats for the last three consequences, and their impacts are also indicated. Presume that the failure in the containment could produce another risk related to “Operations failure” and “Control rod failure” with a 0.003 probability each. Observe also that “External electrical interruption” in matrix 2 affects all receptors (black dashed line) with different impacts shown in black values. These impacts output to the containment column and are added to the former values due to containment failure. Therefore, the whole row indicates the impacts produced by both the containment failure and the electrical interruption. Consequently, for each receptor values are added, and their results are outputted to matrix 4. Choosing, for instance, “People evacuation plan,” it is affected by connections (the latter comes from “Cooling pumps failure,” dashed line). This mitigation issue has an estimated effectiveness of 0.7, meaning that it is believed that about 70% of the people leaving at a certain radius of the plant can be evacuated in a certain number of hours. The value outputted from matrix 4 is: (0.06 × 0.003 + 0.06 × 0.02) × (1-0.7) × 0.35 = 0.000145. 0.3 × 0.003 + 0.3 × 0.02 × (1-0.42) × 0.58 × 0.35 = 0.004002 0.2 × 0.5 × (1-0.7) × 0.35 = 0.00105 Total: (0.000145 + 0.004002 + 0.00105) = 0.011457 or 1.14%. This value is registered in the results solid black row at the top of the last matrix (5). As a practical example let us suppose that a population to be evacuated amounts to say 38,562 people, then 38,562 × 0.0114 = 440 people do not have immediate relocation and must wait for further transfer.

2.5

Characteristics of Components of the Initial Decision Matrix

51

This computation must be done for every potential risks, effects, consequences, and mitigation. The result will display a complete picture of a scenario. This framework allows for: 1-. Attention to whatever aspect a determined complex plant (nuclear or any other) could require. 2-. It requires a thorough analysis and discussion of the whole undertaking. 3-. By examining the results, it is possible to detect which is the area that must be improved. In the example posted and solved, it is obvious that the areas that need a careful reassessment are related to “Cooling pumps” (2) (green line), together with “Containment structure failure” (blue line) and involving “Operations and Control rods failure” (red line). It appears to indicate that more strict protocols, safety devices, and training are necessary to improve the system.

2.5.16

Need to Work with Performance Values Derived from another Data Table

Sometimes quantitative performance values cannot be placed directly into the initial matrix because they are not single values but the result of several inputs that are linked by mathematical formulas. Changing any of the original components (for instance, cost, demands, environmental limit, workforce, etc.) will change the result to be inputted, as often happens in practice. To consider this circumstance, it is sometimes convenient to have the performance values in the initial matrix automatically computed by a formula whose inputs come from another table. This is a very helpful procedure for a quick analysis of the results of different policies, even before using a MCDM method, and it is easily done by inserting the corresponding formula in the initial matrix. This is illustrated with an example in Chap. 7, Sect. 7.4.2.

2.5.17

Conditioning the Decision Matrix to Obtain a Specified Number of Results

In some projects, the original data has many alternatives, maybe in the hundreds. It is the classic case of a list of indicators of any type, each one containing information. For instance, in social issues one indicator of a series could inform about the percentage of students not finishing third-level education. In this case, the DM needs to reduce this initial number; otherwise control would be almost impossible. There are statistical procedures to do this, and the DM must establish which is the final number of projects or alternatives he wishes.

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However, in addition, there is another requirement, in the sense that this much reduced number must capture as much information as possible contained in the initial data. This is necessary because normally, an indicator does not represent an isolated circumstance, since many indicators are related, and often one influencing another. For instance, in this case, desertion from university may be provoked by different factors, such as schooling costs, lodging costs, lack of work, marriage, better opportunities, lack of interest, need to economically help the family, etc. Consequently, if this indicator is shortlisted, it should contain as much information as possible from other related indicators. An example can be seen in Munier (2011a, b), where this condition is met by using entropy.

2.6

Additional Conditions Required for Methods

It has been emphasized that MCDM methods should contemplate real conditions such as the abovementioned, which may appear in any scenario. In addition, the different methods must be able to: • Work not only with a DM but also with a group of DMs. • Work on different scenarios. • Often, the result of a MCDM method shows ties between two or more alternatives. This is a serious problem since the DM may find himself back at square one. The method must give some indication to the DM as to how to proceed in that circumstance. • Most MCDM methods suffer from rank reversal as examined in Sect. 2.3. Although the reason for the problem is not yet clearly understood, it is necessary to find means to correct or eliminate it. This is a pending matter.

2.7

Sensitivity Analysis

Objective No decision-making is complete without performing a sensitivity analysis; it is a fundamental and mandatory step in MCDM. The main reason is that inputted data may be uncertain, and therefore, it is necessary to find out if a solution holds when this uncertain data varies. Hence, sensitivity analysis, after problem-solving, can effectively contribute to making accurate decisions (Alinezhad & Amini, 2011). Another reason is that exogenous factors (which the DM or the company have no control on), such as government policies, competition, weather, exchange rate, and many others, may influence the performance of the selected project when it is in

2.7

Sensitivity Analysis

53

operation. If these factors can be identified and evaluated based on their historical performance, it is possible to assess the potential risk that they pose to the selected project. Normally, the DM may believe that the best alternative found by any heuristic method is the best. Yes, it could be the best, considering that it has been obtained using a mathematical procedure and reasonable assumptions; however, it could not be the best from the operative perspective, and from this point of view, the DM must use his expertise, know-how, and common sense in examining it and by accepting or not, what the method suggests. It is paramount to analyze the stability of the solution found when some known parameters change their values either incrementally or decrementally. This is also known as determining “robustness.” To this respect, sensitivity analysis is used to determine, in most cases, how variations in criteria weights modify the ranking, and then a solution is considered stable if the ranking holds for different variations; however, another important issue must be contemplated, and it is how the first alternative will perform regarding variations exogenous parameters.

2.7.1

The Two Types of Sensibility Analysis

There are three areas where sensitivity analysis is performed, but both with the same purpose: a) Evaluate changes in the best alternative and ranking due to variations of criteria. b) Evaluate changes in the best alternative and ranking due to variations in performance factors. c) Evaluate changes in the ranking when the objective values are varied. The first one has already been addressed, and now it is necessary to examine what happens when certain performance values are changed either individually or jointly. A choice is to change one or several performance values at the same time, run the software again, and get the results. If the original ranking does not change, it means that it is strong despite the variation of a particular or several performance values. In the third case, the analysis consists of changing one of several coefficients of the objective function and seeing how it affects the ranking. The same procedure as in the b) case may be applied; however, some methods, such as linear programming, can determine it automatically. Normally, only the first case is usually considered. Sensitivity analysis is normally included in most methods; there is abundant material on this issue and some recommended publications are Felli and Hazen (1997) and Jansen et al. (1997). How to measure strength, stability or robustness? By observing if there are variations in the ranking when the inputs change in small amounts. Assume, for instance, that the best alternative is A2, and the ranking is A2≽A4≽A3≽A1≽A6≽A5; the symbol “≽” means “Preferred to.” Choose the criterion with the highest weight, for instance C3, and make a small change of its

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weight, say in 2%; run the software again and observe what happens with the best alternative. If it has changed and now the ranking is, for instance, A4≽A2≽A3≽A1≽A6≽A5, it means that the former alternative A2 is too sensitive to small variations of C3, and consequently, it ceases to be the best alternative and probably should be replaced by A4. Conversely, if A2 holds for large variations of C3, say up to 20%, evidently A2 will remain being the best alternative during this variation interval and this is a proof of its strength. Observe that when A2 is displaced by A4 to a second best, the other relations hold, that is, A3 > A1 > A6 > A5. This can be considered strength of the ranking; that is, the ordering must be maintained in the balance of the ranking. Suppose now that the DM decides to add an alternative A7 that is the best for all of them (this can only be assumed—but not guaranteed—when comparing the performance values in the A7 vector with all others). If the method is run again, the result could probably be A7≽A4≽A2≽A3≽A1≽A6≽A5, where A7 is now the best alternative. Observe that other than this change all the other preferences hold. It is often seen, especially in projects with many alternatives and criteria, that no variation is allowed whatsoever for the most important criterion. In that case, the criterion is critical, and this is evidence that the DM can use to justify even rejecting the best alternative and choosing the next one.

2.7.2

A Critical Analysis of the Way Sensitivity Analysis Is Performed Nowadays

Suppose there are three alternatives Alt.1, Alt.2, and Alt.3 subject to a set of criteria A-B-C-D-E-F. When solved, their ranking is Alt.1≽Alt.2≽Alt.3, and thus, Alt. 1 is preferred. Most methods perform sensitivity analysis by changing the weight of a selected criterion and observing at what value the ranking changes. In fact, it constitutes just a sensitivity analysis for weights (although more appropriate for trade-offs), but not for the output, since weights or trade-offs cannot be used to evaluate it. Assume that the DM chooses criterion D (that refers to international prices for raw materials) because it has the largest weight and starts increasing it. This increment can be made at purpose, as in this case to analyze the output response, or by estimating potential future variations of this criterion such as price fluctuations, which is a parameter on which the DM or the company has no control. In this example, Alt.1 is selected as the best with a score of 8.1, followed by Alt.2 with a score of 4.9 and by Alt. 3 with score 2.3 (see Table 2.5). When the DM starts increasing the weight for criterion D, in a 0.1 interval, he observes that the score for Alt.1 decreased to 7.4. More increase of criterion D-weight produces additional decrease in Alt.1; at the same time, both scores for Alt.2 and Alt.3 increase. When the weight increases three intervals, that is, 0.3, the scores of Alt.1 and Alt.2

2.7

Sensitivity Analysis

55

Table 2.5 Variation of alternative scores as a function of changes in criterion “D” Alternative scores Alt. 1 8.1 Alt.2 4.98 Alt. 3 2.3

7.4 5.31 3.11

6.7 5.72 3.92

6 6.13 4.73

5.3 6.54 5.54

4.6 6.95 6.35

3.9 7.36 7.16

3.2 7.77 7.97

2.5 8.18 8.78

10 9 8.1

8

7.4

Scores

7

6.7

6 5

5.72

5.31

4.98

5.3

4.6

3.9

3.92

3

8.78 8.18

5.54

4.73

4

7.36 7.16

6.95 6.35

6.54

6 6.13

7.97 7.77

3.2

3.11

2.5

2.3

2

Alt. 1

1

Alt.2

Alt. 3

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Intervals for C1

Fig. 2.6 Plotting alternatives scores against a criterion variation

coincide, and further increases produce even higher scores for Alt.2 and Alt. 3 and decreases for Alt. A1. Table 2.5 shows these scores for the three alternatives, while Fig. 2.6 displays how the three alternatives interact. Consequently, at a D-weight increment of 0.3, Alt.1 loses its first position since now Alt. 2 has a higher score. Continuing increasing weight of D provokes that an increment of 0.65 Alt. 3 gets a higher value than Alt. 2 and becomes the best. Thus, this is how the DM reports to the stakeholders that the ranking changes when the weight of criterion D reaches a certain value. In all truth it does not appear to be very relevant; probably they would instead need another type of information relating the performance of the selected alternative when subject to variation of some parameters; that is, they need to know how strong the alternative is, in real-world situations. Just for clarification, assume that a company has plans for producing several cosmetic products and that one of them has been selected by the MCDM method as the most promising. In this regard, the stakeholders surely will demand to know the stability of this selection when it is subject to external factors beyond the company’s control, such as: • Competition prices. • Market share. • People’s confidence, etc.

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In the real world, the three of them may change independently, for instance, price of competition decreasing by 5%, market share decreasing by 2.5%, and people confidence increasing by 3%. The stakeholders need to know quantitatively what could happen with their star product, if at some point in time the three parameters vary as shown. This is the kind of information that the company needs, not when a product is supplanted by another because a single criterion change. In addition, which are the drawbacks of this actual procedure in most MCDM processes? 1. Criteria independence. Let us consider that criterion D can vary while keeping the other five constant. This is a mechanism known as “Ceteris paribus” (meaning “other things being equal”) and employed in Economics, but that is not realistic in this context because all criteria are related; therefore, they cannot be disaggregated. 2. Same weight for two or more criteria. It is usual for several criteria to get the same weight, and in this situation, one wonders which of them is chosen by the DM. This put the DM in a quandary since all of them would have the same effect on the ranking, according to this procedure. 3. All criteria vary proportionally. It is assumed that the variations of all criteria are proportional to variations of criterion D. Actually this assumption does not resist any analysis because, for instance, if price increases, it does not necessarily mean that demand will decrease in the same proportion, let alone in a linear proportion, as the very well-known Law of Supply and Demand demonstrates. 4. Criteria capacity. The DM uses a criterion with no influence whatsoever in the output variation because its only merit is that it has the maximum value among all criteria but does not have any relationship with the alternatives. 5. Several criteria. Normally, there are several criteria that affect an output, not only one, and all of them must be considered. Therefore, it is assumed that only one criterion is responsible for the best alternative selection when there could be many and affecting it differently. Even if these many were considered, the question lies in determining which are the criteria that may affect the performance of the first alternative selected. To clarify this issue, assume a case for evaluating different products from a manufacturing company. The analysis shows a ranking such as Product D ≽ Product A ≽ Product B ≽ Product C, and that criteria weights are as follows: C1 (Cost): 0.25. C2 (Environmental damage): 0.12.

2.8

Conclusion of This Chapter

57

C3 (People satisfaction): 0.35. C4 (Demand): 0.15. C5 (Import duties): 0.13. All of these weights are constant; therefore, they are the same whatever the selection reached. The most important criterion, as per its highest value, is C3 = 0.35. However, it does not mean that it is significant for product D; it could be negligible. It could happen that D, because of its characteristics, is very sensible to demand (criterion C4), yet is one of the least important factors. If demand for this product is variable—as usually is—perhaps it is not convenient to select product D, but product A, which is less sensible to demand variations. This is purely a DM’s job in analyzing the consequences of selecting the first product, and this is where his expertise and knowledge can be used. However, there is little he can do if he does not know the influence of C4 in the first selected product. Notice that this analysis is not related to changing the ranking by varying weights of criteria; it deals with determining which is the most important criterion—not necessarily in numerical value—but according to its influence in the future on the first product. Taking into account all the above, it is this author’s opinion that the information that sensitivity analysis delivers to stakeholders, as it is performed nowadays, is rather irrelevant because it only tells when the ranking changes, which could be important, provided that this process was correctly done, that is, without subjectivity. This reasoning does not apply to all criteria weights because there are weights that can be used effectively; they are called Objective weights.

2.8

Conclusion of This Chapter

To model a scenario is not an easy task; as mentioned and exemplified in previous sections, said model must reflect reality as much as possible, and it involves not working with pre-conceived ideas or by cutting corners with approximations. Unfortunately, it is very common that parctitiuonerd for different reasons, try to assimilate reality to a mathematical model when it should be exactly the opposite. Reasoning such as, We decide not to consider dependency between criteria, because we think it is not necessary, is a poor argument to justify the unjustifiable, since it is very rare that criteria are not related. Limiting the number of criteria due to workload, or because of computing time, or seeing as cost of performing pair-wise comparisons, is irresponsible and biased, since the modelling is subordinated to a method’s characteristics or limitations, or to budget. Ignoring in some scenarios that alternatives are time sensitive is disregarding reality, because our world is rarely static, but dynamic. Not considering resources available for criteria is naive because it is assuming that they are limitless.

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However, the most important aspect is not realizing that whatever the project, it is designed to serve people, and this is true for a new road, a new cosmetic product, a new course in a university, the launching of a new airline, or the construction of a hydro dam. Unfortunately, this is overlooked in many cases when considering economic, commercial, technical, social, and environmental points of view. As experience shows, even when a project is designed to address directly or indirectly the people’s needs, population is seldom consulted, and this is a serious matter. It is unacceptable that the DM, irrelevant of his/her knowledge, background, and preparation, establishes preferences of his/her own on matters that affect a large quantity of people or even override what public opinion says. It is still even weirder to assume that this person possesses an encyclopedic experience in diverse fields such as agriculture, environment, economics, and engineering, which could be the case for different criteria in a project and used for evaluation. The DM can never match the information that people affected by the project can produce. This has nothing to do with technical information, of course, but with the project consequences on people living in the project area. They know, better than anybody, to what extent the project will benefit or harm them. This is the information that must be inputted in the decision matrix. To illustrate this point, take, for instance, a highway project through a city, which will allow for faster commuting to downtown; there may be people that benefit from it, but also people that do not, and will even be damaged by this project, for instance, by cutting access to schools, shopping centers, hospitals, etc. Therefore, people must be consulted and their synthesized results must be inputted into a decision table. This way, the method receives information from all parts that will be affected by the project in one way or another. This can be done by consulting people, to learn their opinions, through polls and surveys. Sometimes it is embarrassing for city planners to design a project, and before its implementation, to be told by people about things that they had not considered.

References2 Alinezhad, A., & Amini, A. (2011). Sensitivity analysis on TOPSIS technique: The result of change in the weights of one attribute on the final ranking of alternatives. Journal of Optimization in Industrial Engineering, 7(2011), 23–28. Belton, V., & Gear, T. (1983). On a shortcoming of Saaty’s method of analytic hierarchies. Omega, 11, 228–230. Cascales, M.-T., & Lamata, M.-T. (2012, September). On rank reversal and TOPSIS method. Mathematical and Computer Modelling, 56(5–6), 123–132. Dantzig, G. (1948). Linear Programming and extensions. R-366-PR-Corporation.

2

These references correspond to authors mentioned in the text. However, there are also publications than are not mentioned in the text but that have been added for the reader to access more information about this Chapter; they are identified with (*).

References

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Eppe, S., & De Smet, Y. (2015). On the influence of altering de action set on PROMETHEE’s II relative ranks. CoDE-SMG. Technical Report Series Technical Report No. TR/SMG/2015-002 March 2015 – Université Libre de Bruxelles. Felli, J, & Hazen, G. (1997). Sensitivity analysis and the expected value of perfect information. Retrieved February 05, 2018, from http://citeseerx.ist.psu.edu/viewdoc/download? doi=10.1.1.39.3737&rep=rep1&type=pdf Greco, S. (2006). Multiple criteria decision analysis: State of the art surveys. Springer. Hwang, C., & Yoon, K. (1981). Multiple attribute decision making: Methods and applications, A State- of - the - Art Survey. Springer-Verlag. Jansen, B., De Jong, J., Roos, C., & Terlaky, T. (1997, August 16). Sensitivity analysis in Linear Programming: Just be careful!! European Journal of Operational Research, 101(1), 15–28. Kantorovich, L. (1939). Mathematical methods of organizing and planning production. Management Science, 6(4) July, 1960, 366–422. Maleki, H., & Zahir, S. (2013, May–August). A comprehensive literature review of the rank reversal phenomenon in the analytic hierarchy process. Journal of Multi-criteria Decision Analysis, 20(3–4), 141–155. Mareschal, B., De Smet, Y., Nemery, P. (2008). Rank reversal in the PROMETHEE II method: Some new results. Conference: Industrial Engineering and Engineering Management, 2008. IEEM 2008. IEEE International Conference. Munier, N. (2011a). Methodology to select a set of urban sustainability indicators to measure the state of the city, and performance assessment. Ecological Indicators, 11(5), 1020–1026. https:// doi.org/10.1016/j.ecolind.2011.01.006 Munier, N. (2011b). A Tesis Doctoral - Procedimiento fundamentado en la Programación Lineal para la selección de alternativas en proyectos de naturaleza compleja y con objetivos múltiples. Universidad Politécnica de Valencia, España. Munier, N. (2015). The Z matrix applied to risk determination. Internal report- INGENIOUniversidad Politécnica de Valencia, España. Saaty, T. (1987). Rank generation, preservation, and reversal in the analytic hierarchy decision process. Decision Sciences, 18(1987), 157–177. Saaty, T., & Sagir, M. (2009). An essay on rank preservation and reversal. Mathematical and Computer Modelling, 46(5–6), 930–941. Shannon, C. (1948, July, October). A mathematical theory of communication. The Bell Systems Technical Journal, 27, 379.423–623.656. Shing, Y., Lee, S., Ghun, S., & Chung, D. (2013). A critical review of popular multi-criteria decision-making methodologies. Issues in Information Systems, 14(1), 358–365. Triantaphyllou, E. (2001). Two New Cases of Rank Reversals when the AHP and Some of its Additive Variants are Used that do not Occur with the Multiplicative AHP. Journal of MultiCriteria Decision Analysis, 10, 11–25. Verly, C., & De Smet, Y. (2013). Some results about rank reversal instances in the PROMETHEE methods. International Journal of Multicriteria Decision Making, 3(3), 2013. Wang Taha, M., & Elhag, M. (2006). An approach to avoiding rank reversal in AHP. Decision Support Systems, 42(2006), 1474–1480. Wang, X., & Triantaphyllou, E. (2006). Ranking irregularities when evaluating alternatives by using some multi-criteria decision analysis methods. Handbook of Industrial and System Engineering. CRC Press – Taylor & Francis Group. Wang, Y., & Luo, Y. (2009). On rank reversal in decision analysis. Mathematical and Computer Modelling, 49(5–6), 1221–1229. Zopounidis, C., & Pardalos, P. (2010). Handbook of Multicriteria Analysis. Springer.

Part II

Theory and Analysis of MCDM Problems: What Can Be Done by Using the MCDM Process?

Chapter 3

How to Shape Multiple Scenarios

Abstract In a portfolio with different projects, and where a project may participate simultaneously in various scenarios, this chapter addresses the issue of assigning projects to each one, which usually have different demands and characteristics. These scenarios may involve, for instance, different plots of land situated in different places or countries, and where different kinds of undertakings for land use are contemplated. That is, its objective is to consider simultaneously all possible scenarios (different plots), undertakings (different land uses), and projects (different plans) and to determine the best projects for each scenario. In addition, it can also find which are the more significant or important scenarios and their restrictions. The objective of this chapter is to pose this type of multiple scenarios problem, and whose resolution is described in Chap. 7, Sect. 7.7.

3.1

Introduction

Most problems in MCDM refer to the selection of an alternative from a given set and subject to a group of evaluation criteria. Normally, in the set of alternatives all of them are of the same type (for instance, select the best place to install a car assembly plant, where all alternatives are locations), or selection for purchasing a house (all alternatives are houses), etc. However, in many cases alternatives belong to different classes and apply to different areas, and this constitutes a complex scenario, although realistic. Consider that the City Hall of a city has a five-year plan in four areas / departments (I—Infrastructure, P.H. —Public Health, E—Environment, and S—Safety on the streets). Each one of these areas has their own portfolio of projects and plans, as for instance: I—Infrastructure: (1) Construction of storm sewerage in the northeast area of the city, (2) paving 900 m of North Avenue, (3) purchasing equipment for street city cleaning, and (4) replacing high-pressure sodium street lights with LED street lights in 45 streets. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 N. Munier, Strategic Approach in Multi-Criteria Decision Making, International Series in Operations Research & Management Science 351, https://doi.org/10.1007/978-3-031-44453-1_3

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3 How to Shape Multiple Scenarios

P.H.—Public Health: (1) Build a new hospital in downtown, (2) build an outpatient center for HIV/AIDS, and (3) install and maintain a walk-in clinic in a peripheral part of town. E—Environment: (1) Increase green space from 7m2/inhabitant to 12 m2/inhabitant and (2) develop a plan for domestic garbage collection to improve recycling. S—Safety: (1) Purchase 75 patrol cars for the police force and (2) fight street crime. As can be seen, there are different quantities and different costs for projects within each department, and each one has its own budget. Naturally, no one is willing to relinquish its share of the City Hall budget to another, so consequently, every department budget must be respected, and then the objective is to select the best projects within each department and within its funds. All alternatives are subject to a set of criteria; however, criteria for evaluating sewerage and paving will probably be different from a set of criteria evaluating health undertakings and different from other criteria set for evaluating environment projects or safety. It does not mean that all criteria are different from one set of projects to another; most probably, funding, financing, manpower, and transportation will be common for all of them, as well as the 5-years’ time for completion. Therefore, all projects in the four departments will be competing for funding and financing not only between the different sets but also within a department, and it also can happen that a project may belong to different departments. In addition, the following situations may arise: a) There is not enough funding for all projects in the prescribed period, meaning that some of them will not be selected. It is possible that more than one department has a portfolio of projects where execution costs exceed the budget. b) Normally, it is necessary to establish as a condition that each department budget will be used at its maximum and for its own projects. If this last condition is not established, it could very well be that some departments will have many projects that get financing, while other departments may have a reduced number of projects or even none. c) There could be other conditions and restrictions that make the problem still more complicated. d) Projects may be inclusive or exclusive. This example portrays a single scenario, the city, with different projects and in different areas. However, in other cases, projects for each area are not defined because a project can be developed in different places. This type of situation, which is very common when alternatives are within a portfolio and where some alternatives may take place, even repeating themselves in different areas, is normally addressed as “multiple scenarios”; the objective of this analysis is to find the best mix of projects using the best conditions in each area. In this way, projects are assigned to areas where they can be developed at full potential.

3.2

Developing the Best Strategy: Case Study—Selecting Projects. . .

65

Fig. 3.1 Scheme for agribusiness

3.2

Developing the Best Strategy: Case Study—Selecting Projects for Agribusiness Activities in Different Scenarios

Let us illustrate a multiple scenario by using an actual example of an agribusiness multinational (the company) involving crops, livestock, and fruit production, with several large plots of land on different countries. In each plot there are different undertakings or projects that can be executed depending on both project and plot characteristics. Figure 3.1 shows diverse agribusiness activities. Out of them, this company deals with: Agriculture, with four different crops (wheat, corn, soybean, and rice). Livestock, with beef and pork meat.

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3 How to Shape Multiple Scenarios

Fruit production, with orange groves, including a potential orange juice concentrate plant. There are several potential locations where it is possible to develop different projects or products, and considering that not all of them can be built in all locations due to restrictions of climate, water, risk, pests, etc. Then, there are several strategies that involve finding the best undertakings for each location. The company contemplates the following potential undertakings: 1. Cultivate crops, all of them for export; consequently, the project (whatever the crop selected) must include the construction of grain storage or silos; that is, there is a conditioning here since grains cannot be cultivated without storage facilities, and storage is unnecessary if there are no grains; therefore, this conditioning must be imputed in modelling. 2. Livestock farming, including cattle and pigs. 3. Development of orange groves. Oranges will be used on the site to produce frozen orange concentrate by the same company. Consequently, the orange concentrate plant must be included as another project, and this is another condition that must be inputted in the modelling. The orange concentration plant will be built only if the orange groove project is selected. However, the opposite does not follow the same conditioning, since oranges can be sold to a local, and existing, juice concentrate plant. As mentioned, not all the plots have the same characteristics related to the type and quality of soil, weather, water availability, yields/ha, inherent risks, pests or plagues, etc. In addition, different crops require different amounts of water, and they are subject to different market demands and prices, and due to varied types of soil, they have different yields. Considering all of these conditions, the company developed four strategies (see Fig. 3.2), which, based on its experience, defined the products that best match local conditions. Strategy 1: It will take place in their smallest (700 ha) but best plot (the flagship of the company). • It is a fertile land, with quality water and fairly good weather. • The plan is to cultivate wheat, corn, and soybean and livestock production for fine beef and pork meat to be sold alive directly to the meatpacking industry; therefore, they do not need cold storage units. • The plot is not suitable for rice and orange groves because it is rather a cold country. • There is no need for silos since grain is trucked after harvest to a nearby harbor for export. • They need to determine the most important ranking of these five projects (wheat, corn, soybean, beef meat, and pork meat); that is, which is the most convenient undertaking. Strategy 2: It pertains to a larger plot (753 ha) with the following characteristics:

3.2

Developing the Best Strategy: Case Study—Selecting Projects. . .

67

Scenario 2 Potential undertakings Agriculture: Proj. A2, A3, A4 Scenario 3 Scenario 1 Livestock production: Proj. A6, A7 Potential undertakings Potential undertakings Orange groves: Proj. A8, A9 Agriculture: Proj. A1,A2, A4 Agriculture: Proj. A1, A2, A3 Livestock production: Proj. A6, A7 Livestock production: Proj. A6, A7 753 ha 820 ha 700 ha Scenario 4 Potential undertakings Agriculture: Proj. A1, A3 Livestock production: Proj. A6, A7 Orange groves: Proj. A8, A9

850ha

Fig. 3.2 Plots in different regions or scenarios, feasible undertakings, and proposed projects

• It is not a good soil for cultivating wheat but fine for corn, soybean, and rice. • The land is good for animal farming and suitable for orange groves, because it is sunny country. • The estimated IRR (internal rate of return) is very high for both crops and livestock, but most especially for the orange groves. • Water is available but not as abundant as in Strategy 1. • Risk is minimal because there is abundant information on the area, as well as multiple similar and profitable undertakings within a 150-Km radius. • There is no need for silos since grain is trucked after harvest to a nearby harbor for export. • As a negative aspect the area is known for pests, which, although can be treated, involves a risk and higher expenses. Strategy 3: It pertains to another large plot (820 ha) with the following characteristics: • Adequate for wheat, corn, and rice, but not for soybean and orange groove. • Enjoys good and reliable weather. • Water is not abundant but enough for irrigation, although slightly salty. If necessary, water needs can be supplemented from wells. • Silos for grain storage are needed. • Land is suitable for livestock. • It is not good for orange groves because the area is not warm and sunny enough. Strategy 4: It pertains to the largest plot of land (850 ha) with the following characteristics:

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• Good conditions for wheat and soybean but not for corn and rice. • There is no need for silos since grain is trucked after harvest to a nearby harbor for export. • Enjoys excellent and reliable weather. • Adequate for livestock farming. • It is an excellent climate for orange groves. • Its major disadvantage is the existence of pests of different types, and consequently entails a risk, • Its IRR (internal rate of return) is relatively low in crops but very high in livestock and oranges, although, unfortunately, the plot is not too large for orchards, and then the processing plant, which must have a large capacity due to economies of scales, must outsource oranges from neighboring groves. Therefore, the company is faced with determining for each plot which are the best undertakings to maximize profits. Table 3.1 shows the available data, where: • Production capacity: A function of the plot size and statistics about yield/ha from current information and international studies from the World Bank and FAO (Food and Agriculture Organization), United Nations. • Gross Benefit: Estimated considering yield/ha and international prices for each crop. The same procedure applies to livestock. Land and sales taxes and inflation were also estimated, as well as foreseeable increases in wages and salaries due to cost of living in each area. • Estimated IRR: Obtained by the Financial Department of the company, based on projection over a 30-year period of the financial statements (Balance sheet, Net Earnings, Interest to be paid to Banks due to loans, Working Capital, etc.). • Water consumption: Estimated based on the need for crops (considering that demand for water is different for each crop), live stocks, and orange groves, founded on local statistics for each undertaking. • Weather: Forecasting based on statistics from the Meteorological Services for each area and considering the last 15 years, as well as forecasts based on current estimates due to global warming. • Risk: Appraisal based on world fluctuation of prices for commodities and minimum level of ranges. • Pests or plagues: Obtained from the Ministry of Agriculture in each area, as well as from the World Bank and the OECD (Organisation for Economic Co-operation and Development). • Frozen concentrate: The company owns three juice concentrate plants; therefore, it has enough information about their costs, operating capital, and investment needed. There are nine projects as follows: A1—Wheat A2—Corn A3—Soybean

Scenarios Scenario or plan 1 Prod. capacity /ha (C1) (tons) Gross benefit/ha (C2) (€) Estimated IRR (C3) (%) Scenario or plan 2 Prod. capacity /ha (C1) (tons) Gross benefit/ha (C2) (€) Estimated IRR (C3) (%) Water consump. (C4) (lt/ha) Weather (C5) (1 to 5 scale, the higher the better) Scenario or plan 3 Prod. capacity /ha (C1) (tons) Gross benefit/ha (C2) (€) Estimated IRR (C3) (%) Water consump. (C4) (lt/ha) Risk (C6) (%) Plagues (C7) (%) Scenario or plan 4 Prod. capacity /ha (C1) (tons)

2301

86904 0.049 12852 0.061 0.012

88127 0.067 15631 0.058 0.01

88958 0.077 11230 4

85478 0.061 13697 4

3821

1952

3401

1956

89157 0.089

85937 0.058

89321 0.063

3327

2037

3542

2198

Soybean (A3)

Corn (A2)

Wheat (A1)

Table 3.1 Data for each scenario

91304 0.093 25630 0.062 0.03

1025

91452 0.082 25842 3

2023

Rice (A4)

75

50

70

75

Silos f/grain storage (A5)

9E+06

106932 0.091 567000 0.08

789500

98630 0.098 567000

2E+06

101237 0.059

2E+06

Beef meat (A6)

287000

91236 0.085 423000 0.07

289100

98631 0.085 423000

123000

89752 0.061

358000

Pork meat (A7)

1457000

100157 0.102 800000 5

2100000

Orange juice plantation (A8)

97

140

Frozen concentrate (A9)

Developing the Best Strategy: Case Study—Selecting Projects. . . (continued)

MAX MAX MIN MIN MIN Actions MAX

Actions MAX

MAX MAX MIN MAX

MAX MAX Actions MAX

Actions MAX

3.2 69

Scenarios Gross benefit/ha (C2) (€) Estimated IRR (C3) (%) Water consump.(C4) (lt/ha) Weather (C5) (1 to 5 scale, the higher the better) Risk (C6) (%) Plagues (C7) (%)

Table 3.1 (continued)

0.07 0.01

Wheat (A1) 85962 0.065 15631 4

Corn (A2)

0.03 0.011

Soybean (A3) 89627 0.071 12785 5

Rice (A4)

Silos f/grain storage (A5)

0.07 0.015

Beef meat (A6) 102457 0.091 567000 4 0.08 0.021

Pork meat (A7) 89631 0.085 423000 5 0.09 0.08

Orange juice plantation (A8) 1023789 0.09 800000 5

Frozen concentrate (A9)

MIN MIN

MAX MAX MIN MAX

70 3 How to Shape Multiple Scenarios

3.2

Developing the Best Strategy: Case Study—Selecting Projects. . .

71

A4—Rice A5—Silos construction A6—Beef meat A7—Pork meat A8—Orange groves A9—Orange concentrates plant Subject to the following criteria and actions, with units indicated in Table 3.1. C1—Maximize production capacity. C2—Maximize gross benefit. C3—Maximize internal rate of return (IRR). C4—Minimize water consumption. C5—Maximize fine weather. That is, an area is gauged according to the quality of its weather. C6—Minimize risk, for instance, drought, hail, etc. C7—Minimize the use of pest control, that is, the less pest control needed, the better. Notice that: Scenarios. The four distinctive scenarios have their own criteria with characteristics as follows: Scenario 1 is subject only to three criteria (production, gross benefit, and estimated IRR), Scenario 2 is subject to five criteria (production, gross benefit, estimated IRR, water consumption, and weather), Scenario 3 is subject to six criteria (production, gross benefit, estimated IRR, water consumption, risk, and plagues), Scenario 4 is subject to seven criteria (production, gross benefit, estimated IRR, water consumption, weather, risk, and plagues). Plague values are not the same for all scenarios. Performance values show how each undertaking contributes to each criterion, expressed by its action. Dependencies: The four strategies include dependencies and conditions. From the point of view of crops, the construction and especially the size of the silos depends on the respective selection. For instance, it must be inputted into the IDM that project A5 (Construction of silos) will only be built if at least one crop project is selected in scenario 3. This is expressed mathematically by Project A3 = Project A5. The symbol “=” indicates that the score for A3 must be the same as the score for A5. There is also dependency on the concentrate plant (Project A9), which construction will only be contemplated if the orange groves project (Project A8) is chosen, then Project A8 ≥ Project A9. The symbol “≥” means that there exists technical reasons or preference regarding subordination of A9 to A8 since the orange groves get preference to the concentration plant, or in other words, producing oranges is a

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more interesting business for the company than building a concentrate plant, since it can be built elsewhere, or even subcontracted.

3.3

Solving the Problem

To solve this problem, a MCDM method should take into account: 1. That all four scenarios must be considered simultaneously. 2. The economic importance of each scenario is evaluated, for instance, by the parameters “gross profit” and “IRR.” 3. That the problem will be examined through sensitivity analysis. 4. Since criteria do not have the same units of measure, normalization is necessary. A sequence for a MCDM to deal with complex scenarios was proposed and its solution will be given in Chap. 7, Sect. 7.7.

3.4

Conclusion of This Chapter

The aim of this chapter was to show how a complicated problem involving different scenarios, or areas with different projects, each one with special conditions such as precedence between projects, as happens in actual scenarios, can be modelled in MCDM.

References1 *Montibeller, G., Gummer, H., & Tumidei. D. (2007). Combining scenario planning and multicriteria decision analysis in practice- Kingston Business School. Kingston University, London, England, UK Working Paper LSEOR 07.92 ISBN: 978-0-85328-047-7. *Munier, N. (2011). A strategy for using multicriteria analysis in decision-making – A guide for simple and complex environmental projects. Springer. *Shortridge, J., & Guikema, S. D. (2016). Scenario discovery with multiple criteria: An evaluation of the robust decision-making framework for climate change adaptation. Risk Analysis, 36(12), 2298–2312. *Zolfani, S., Maknoon, R., & Zavadskas, E. (2016). Multi attribute decision making (MADM) based scenario. International Journal of Property Management, 20(1), 101–111.

1

Publications identified with (*) are not mentioned in the text but have been added for the reader to access more information related to this Chapter.

Chapter 4

The Decision-Maker, a Vital Component of the Decision-Making Process

Abstract Decisions are taken by human beings. It does not matter how many different methods are available for this activity and how accurate they seem to be; they are simply tools to organize and process information and to support the DM. Once the processing of data is finished, they provide results but not definite valuable conclusions. It is the decision-maker who analyzes this information and decides and advices the stakeholders based on it. This chapter is devoted to examining the functions of this important entity that has a great responsibility, and that must provide realistic and well-founded conclusions for the Board of Directors to make the final decision.

4.1

Decision-Maker (DM) Functions—Interpretation of Reality

The DM operates at three consecutive levels as follows:

4.1.1

First Level: Building the Initial Decision Matrix

It deals with information to input into the mathematical method and has several steps. First step: Normally, the DM learns from the Board of Directors or stakeholders which are the objectives of certain high-level policies that usually include different projects that are in the pipeline. It is his obligation to require information about the technical and economic feasibility of each one, as well as the social and environmental consequences of each project. Second step: Once the DM knows the projects involved in a certain plan, he must interact with stakeholders to learn what must be taken into account from their perspective, and most probably there will be different opinions and needs from © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 N. Munier, Strategic Approach in Multi-Criteria Decision Making, International Series in Operations Research & Management Science 351, https://doi.org/10.1007/978-3-031-44453-1_4

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different company departments. He is not to judge them but to have their concerns imputed in the modelling. As an example, if the company in the dry food business decides to enter in a line of frozen new products, there will be different requirements; the accounting manager could ask those benefit/cost ratios be considered, while the packing manager may be concerned about the cost of necessary equipment, and the production manager aiming at determining if the structure and space of the existing plant will be able to cope with actual work in progress in addition to the new one (whatever it could be). There will be inquires to be included in the analysis, for instance from the marketing department, to find out if the present company distribution network will be capable of handling the new product, while the purchasing department will be concerned with on-time logistics, considering that some inputs are perishable, or that they have to be imported from overseas, in which circumstance the average maritime transport time may be a critical factor. This analysis with interested parties may last weeks, but it is an essential step; otherwise, the DM will be working in the dark, but it is he who must direct the inquiries in order to get the answers he needs. Third step: Assuming that the DM receives all pertinent data from responsible sources, his next task is structuring a system that contemplates all requisites. That is, the DM has to build a mathematical method that incorporates all the data and requirements from all departments. This is a critical step because it is the raw material for selecting the criteria that will be used for evaluating the different projects or alternatives. Is there any limit to the quantity of criteria to consider? No, it shouldn’t be any, because if for whatever reasons the DM decides not to consider, say working capital for each project, he is probably not considering what the accounting manager demanded. Criteria must then obey stakeholders’ and managers’ requirements and, in addition, there could be more criteria generated by the DM himself. Decision-making carries a great responsibility, and if the DM believes that there are some things that escape his expertise, he must propose to work with a group decision-making. What could be the aspects that he does not know? Many, because criteria normally involve different fields such as marketing, purchasing, production, and financing, and thus it is not expected that the DM be knowledgeable in all of them. Once the DM has built the initial matrix, he must discuss it with all interested parties to make sure that nothing has been left. Maybe the need for some criteria is not clearly seen, for instance, those related with people, especially with people that could be benefitted or hurt by a project. Naturally, this aspect does not mean asking people about technical issues but about how they perceive that a project affects them. As an actual case, a chocolate plant already established in a city planned to increase operations. Everything seemed fine and in place; however, a neighbor who was told by a relative about the project requested to meet the project committee and asked how the plant would eliminate the chocolate smell, which albeit agreeable, may become very disturbing in the neighborhood.

4.1

Decision-Maker (DM) Functions—Interpretation of Reality

75

Not surprisingly, nobody has thought about it. This is the kind of input the DM must extract from the people that will be affected by a project, and once the response is known, input it to the initial decision matrix. This case indicates that the DM, as an expert, must suggest taking certain measures that the company management did not think or did not know about. In this case, probably a poll or survey should be conducted to know what people think about it. In another actual example, an urban developer had bought a plot of urban land and made plans to build houses. When everything was ready, the DM realized something that nobody noticed, or if noticed, it did not deserve attention. The plot had been classified by the City Hall as an industrial place, and then bylaws did not allow for human habitation. Had the DM overlooked this fact, processed the method, and got results, he would have realized that it was a futile exercise. Fourth step. In this step, the DM must check if he has all the relevant information. For instance, he needs to have the values with which each alternative contributes to each criterion, that is, the performance values. These can be quantitative and reliable as in costs, quantities, manpower, equipment output, etc., produced by the company departments, suppliers, vendors, etc. There can also be qualitative and uncertain, as in weather, people’s opinion, government measures, transport delays, etc. Naturally, these are more difficult to compute, and in some methods, they are expressed by DM preferences or are based on polls and surveys, market studies, or on statistics. An example of the latter is found on projects aiming at building a hydroelectric power plant, where the flow of water, for instance, depends on many circumstances, and then it must be based on hydrographical statistics in a certain extended period. The same may apply to energy produced and energy consumption. In these cases, it could be valuable for the DM to consider a minimum and a maximum flow and feed the method with both values by using two criteria, with the same data but with different actions and thresholds. As mentioned, all criteria are linked in some way or another to resources. These can be funds, capacities, personnel, equipment, land size, water availability, etc. These resources are also limits of validity for criteria. For instance, an environmental criterion will have, instead of resource, a maximum limit of contamination of, say, NOx; therefore, this criterion will call for minimizing the spewing of this noxious gas from the different alternatives or their combination. Another criterion may refer to water to dwellings and will call for maximizing the daily quantity of water available for person up to a maximum of, say, 260 liters/day-person. In several scenarios, this quantity may be different from one to another depending on the characteristics of the area. Now that the DM has determined the criteria that will go into the decision matrix, he must analyze each pair to determine if they are independent or if they are somehow related. This is important because some MCDM methods only consider independent criteria. Therefore, with this examination he has a hint about which method to use. Some scholars and practitioners claim that a MCDM must be as inexpensive as possible. Nobody can deny the logic of this; however, nowadays projects are

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normally very complex, and just to pretend to make a selection using an easy-tounderstand and inexpensive method is rather utopian. In this aspect, this book is not pointing out a method as being the best. It aims at detailing the potential existing conditions to give the DM the means for him to decide the most fitting his problem.

4.1.2

Second Level: Selecting a Method to Use

It deals with the selection of the method to process the information in the decision matrix. There are probably more than two dozen methods for MCDM, generally based on three platforms: weighted addition, outranking, and distance to an ideal point. There are hundreds of articles published in reputable journals about the mathematics of each method, and consequently that is not examined in this book. One of the most common questions formulated by practitioners in a scientific forum such as ResearchGate refers to which the best method in MCDM is. The problem of selecting an appropriate method is a hard task and one for which the DM must take into account the following aspects: 1. Knowing a method. It does not mean that the practitioner needs to have a deep knowledge of the mathematics involved. Fortunately, that is not necessary; however, he must be aware of the characteristics of the methods and its limitations and how to work with them. This is similar to driving a car; it is not necessary to know how the engine works, but it is fundamental to be aware of the capabilities, as well as strong and weak points of the vehicle. 2. Type of problem, that is, a simple or a complex one, considering what was stated in Sect. 4.1.1. 3. Size of the problem by the number of alternatives and criteria. Not all methods can handle big data. 4. Relation between alternatives. Not all methods can handle precedence. In some projects there are multiple preferences between alternatives. In this case they must be inputted into the decision matrix. For instance, for whatever reasons alternative A1 must precede alternatives A4, A9, and A12. This is easily done in SIMUS using a table ad hoc. See example in Sect. 3.2. 5. Dependency or independency of criteria. Not all methods can handle dependency between criteria and most especially correlation. As an example, AHP cannot work with dependency between criteria; however, the analytic network process (ANP), which is the more general form of AHP, can. 6. If there is correlation, find the method that can handle it. 7. Some practitioners and researchers recommend using the most affordable method, money wise. It is understood that this procedure is not advisable; the DM will get a result but most probably biased because not all aspects present in a real scenario are inputted.

4.1

Decision-Maker (DM) Functions—Interpretation of Reality

77

There is consensus between practitioners about three aspects: a) No method is better than another. This assert comes from considering that all methods are mathematical formulations, and that they are correct. The differences come from the subjective component inherent to decision-making and how each method handles it. The main problem is that subjectivity is intrinsic to each DM, and therefore, the same problem, solved by the same method and using the same data, may give different results, if solved by two or more DMs, and this is frankly not admissible. Some researchers recommend using two or three different methods to solve the same problem and then compare results. However, this procedure practically brings the DM back to square one, because he has then to decide which the best method is. In addition, this recommendation does not consider software costs and the necessity for the DM to study and master each one, which is not very realistic. In Chap. 11 it is proposed a method that far for being perfect can help in this endeavor. b) There is no guarantee that a method will find the best solution (provided that it exists). This assertion is absolute and without discussion; nobody knows which the best solution is, and consequently, there is no benchmark to compare results. It is also evident that if the best solution is known beforehand, no MCDM would be needed. c) Each MCDM method was designed with a purpose, and for this reason there is not a method that can be applied to any possible scenario. This is due to the fact that all methods have structural limitations, and some are able to treat some kind of problems but not able to tackle another. From this point of view, it appears that to treat personal and corporate scenarios, such as for instance selecting personnel, the analytic hierarchy process (AHP) is the most adequate. The reason is because its structure allows for the DM to express his preferences on something that he knows very well, and that in addition, and probably the most important issue, the result of his/her decision will exclusively benefit (or hurt) his or his company and nobody else. If the scenario is complex, involving perhaps thousands of people, investing thousands of millions of dollars, and changing perhaps forever the way people live and work, it is a completely different scenario. This is a very common undertaking in highway construction, large factories, oil refinery plants, urban planning, high-speed trains, etc. As an example, the gigantic Three Gorges Dam in China provoked the relocation of between 1.2 and 2 million people (see Yardley 2007). Another similar effect took place when the Itaipú Hydroelectric Plant was built on the Paraná River between Brazil and Paraguay (see Mina 2011). The lake behind the dam flooded extensive populated areas, whose inhabitants had to be monetarily compensated; however, it created a social phenomenon when Brazilian people in the flooded area were not given enough monetary compensation for them to buy new land in Brazil and migrated to neighbor Paraguay. In these cases, it does not make sense to work with a DM or even with a group of DMs preferences, because it is impossible for him or them to appraise the magnitude

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4 The Decision-Maker, a Vital Component of the Decision-Making Process

of many problems that the undertaking may cause to people. For this reason, it is necessary to utilize more elaborate MCDM methods.

4.1.3

Third Level: Following the Process

Generally, once data is inputted, and by pressing the start key, the process starts and stops only when producing the final result. This is fine; however, it is more convenient if the DM may participate during the process, maybe simply as an observer or perhaps changing some values. This can be done in some methods.

4.1.4

Fourth Level: Examining the Result

How good is a result? Very difficult to know; it is only the outcome of a mathematical process, and thus, from this point of view most probably it is good. However, to determine how “good” or “bad” a result is, it is necessary a thorough appraisal by the DM. Any method yields the best alternative and a ranking for others, both from the mathematical point of view. It is important here to consider how the DM feels about the best alternative. Since he was in charge of studying the whole data, his feeling is important albeit of course not decisive; it is just an element of the evaluation. Obviously, the best alternative is the one that best meets the criteria terms. However, since some qualitative data is uncertain it is necessary to determine how potential variations in those uncertainties affect the stability of the result, that is, how the result holds considering these variations. This is called “Strength” of the solution and has been discussed in Sect. 2.7.1. It could very well be that the best alternative is very sensitive to variations of a criterion such as international price for the product manufactured by a company. Knowing this fact, it is then important to study which are the potential variations— probably from statistics—for other parameters that can affect this price and find out if the best alternative holds its position for a range of those variations. This is the very essence of sensitivity analysis. It is obvious that if the best alternative is sensitive to small variations of international price, it is not the best option, and probably the DM should select the next one in the ranking. The DM must ask himself what stakeholders need to know and try to answer those coming questions. There is no doubt that the stakeholders may be interested in knowing which their best flagship product for exports is; however, they are perhaps more interested in other issues related to the profits or losses of the company when other parameters change and that affect their star product. Then, most probably the DM will be questioned about what will happen if some parameters, exogenous to the company, change and affect their best product. These parameters could be local

4.2

Conclusion of This Chapter

79

market penetration, competition prices, potential increase in raw material prices, review of labor agreements, etc. The DM must be prepared to answer these questions. This is examined and exemplified in Sect. 8.4.

4.1.5

Synergy Between the DM and the Method

All of the analyzed aspects put on evidence that MCDM is not an exact science even using scientific tools. It is a mix of DM knowledge, experience, and common sense as well as comprehension of heuristic modelling capacities and scope. However, even furnished with the best data, the method is not a mathematical equation where values are imputed, and results obtained and accepted without discussion because are supported by theorems, as it would be if the DM were given the task to calculate the area of a right-angled triangle with a base of 5 cm and a height of 12 cm. In MCDM the method only supports the DM in the treatment of thousands of possible combinations of projects, suggesting a solution that distillates a ranking of numbers, and sometimes very confusing as in the case of ties. The method has to be able to help the DM in giving numerical data that he can translate into an educated reason for accepting or rejecting alternative D instead of alternative A, even when the mathematics of the method suggests it. It is necessary to remember that the task of the DM is to interpret, analyze, and consent or discard the results from the mathematical method, which is simply a tool; the reasoning is always left to the DM.

4.2

Conclusion of This Chapter

This chapter examined in some detail the function of the DM by breaking his labor into four levels, from the very beginning to the very end. It is believed that this description portrays the different tasks assigned to the DM for him to deliver his decision and, most important, with arguments to support what he says, not because he feels or prefers in a certain way. Some scholars think that the DM decisions must be guided by intuition, emotions, and moods; this could be true for trivial problems where the consequences of his/her decision will fall on him/her but it is not certainly applicable to serious problems.

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References1 *Encyclopaedia of Management. (n.d.). Multi-criteria decision making. http://www. referenceforbusiness.com/management/Mar-No/Multiple-Criteria-Decision-Making.html Mina, T. (2011). The Itaipú Dam – Impacts of construction. http://large.stanford.edu/courses/2011/ ph240/mina2/ *Roles in the Buying Multi-Criteria Decision Process. (n.d.). http://www.blackbeltselling.co.uk/ members/wp-content/uploads/additional-resources/lesson-6/6-5%20Decision_making_roles% 20_%20Detailed%20V1_1.pdf Yardley, J. (2007). Chinese dam projects criticized for their human costs. https://www.nytimes. com/2007/11/19/world/asia/19dam.html

1

Publications identified with (*) are not mentioned in the text but have been added for the reader to access more information related to this Chapter.

Chapter 5

Design of a Decision-Making Method Reality-Wise: How Should it Be Done?

Abstract Modelling, which is the preparation of a mathematical model, normally the IDM, is the core of decision-making, and results depend on how it is built. This chapter analyzes what aspects are normally missing.

5.1

Modelling

This is a complex task that demands not only a profound knowledge of the scenario under study but also skills to replicate it as close as possible. There is no norm, regulation, or directive as to how to model a scenario. Naturally, each case is different but there is a principle that should always be applied. It is: Reality is not to be ignored; it means designing a model reflecting the scenario as close as possible. For instance, in the very well-known elemental example of purchasing a car, there could be several models from the same maker or from different makers, and the buyer normally bases his/her selection on price, trying to minimize it. This is perfectly reasonable; however, is the purchaser sure that the price is the most important and only aspect to take into account? Not that he is wrong, but perhaps what he does not consider is that the purchasing price is not an isolated figure. Initial car prices are also associated with maximum speed, structural safety, style, power, comfort, etc., and all of them are linked with price. In addition, during operation, there will be expenses that add to the initial price, for instance, fuel consumption, repairs, and insurance; consequently, the buyer must be aware of how these elements are combined time-wise to define the true price of the vehicle. It is also common to find cars with very similar prices; however, the difference may be in their operation, comfort, or style. Thus, for a certain similar price car A may have high fuel consumption and high comfort, while car B may have lower fuel consumption but without ancillaries. In other words, all attributes are normally linked. The existent interrelationships between criteria, attributes, or factors are more the norm than the exception, but usually, it is not considered. Therefore, this is one of the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 N. Munier, Strategic Approach in Multi-Criteria Decision Making, International Series in Operations Research & Management Science 351, https://doi.org/10.1007/978-3-031-44453-1_5

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reasons for disagreement between methods, when one method takes into account something while another overlooks it. Another aspect that may produce differences—and it may depend on the DM—is the number of criteria in the model. Small or trivial projects normally may have less than ten criteria; however, complex projects may have hundreds. Some methods are not prepared to handle large quantities of criteria and alternatives; consequently, the DM must select the method that best copes with them and does not limit the number of criteria based on economic considerations because the workload, as it happens in methods using pair-wise comparisons, is labor-intensive. From this point of view, methods such as PROMETHEE, ELECTRE, TOPSIS, SIMUS, VIKOR, and others are better prepared for this task. There is another critical factor: generally, not all criteria have the same importance usually measured as a weight. The problem lies in how to assign weights to each criterion, especially if there are many. There are several ways for determining weights, some subjective and some objective; consequently, if two MCDM methods use different type of weights, the results will be different. As a conclusion there are customary aspects that are ignored in modeling, for instance: a) Too much flexibility in deciding not to introduce real-world aspects such as: 1-. Ignoring interrelationships between criteria and even the existence of correlation. That is, all criteria are thought as independent, which could be a gross error. 2-. Not considering technical precedence between alternatives. If they exist, but are not taken into account, this may lead to absurd results. For instance, the result may indicate that alternative D is the best, followed by B, and then D > B; that is, D is more important or preferred to B; however, the result is ignoring the fact that to start D, alternative B must be completed, and then it should be B > D. The problem is that most methods do not contemplate adding this restriction. 3-. Not using resources. Criteria are always dependent on resources, whatever their action and in different aspects. Thus, one criterion may call for equalizing available funds, another for maximizing stock consumption, and a third for minimizing noise. Consequently, and since resources are scarce and limited, there must be a cardinal value for each one, that is funds or budget in the first case, quantities in stock in the second, or a cardinal limit for noise, normally in decibels, established by competent authority.

5.1

Modelling

83

Not considering these aspects produces unrealistic results. Again, some methods incorporate resources in their modelling, but not all of them. Consequently, this is another factor that contributes to obtaining different results, not only subjectivity, but a lack of appropriate consideration of reality, or because the method structure does not allow its introduction. 4-. Not using all the necessary criteria. In some methods such as AHP, a maximum number of criteria less than 10 is recommended. Whatever the reason supporting this action, this is equivalent to ignoring reality and consequently developing a model which is not representative. 5-. Not considering that some results must be in binary format. For instance, in an industrial location problem, it has to be decided if a plant will be located in location A or B or C. Obtaining decimal scores for each location makes no sense, because it is not possible to install say 67% of the plant in location B and 33% in location A. Binary results are needed where “1” identifies a location and “0” precludes others. b) Decision-maker subjectivity at several levels. 1-. Preferences have no relationship with reality. In AHP and ANP, the DM develops the initial matrix where values are established by his own preferences and according to a scale; in so doing, he presumes that transitivity between his values holds. That is, if he establishes that project A > project B and project B > project C, he expects that project A > project C. Mathematically this is correct; however, if he does not get this transitivity within an error of 10%, he modifies his preferences in order to be within that margin. That is, he is changing his former preference values in a universe of his own, which in most cases is disconnected with reality, where in general there is no transitivity. This is one reason why different DMs modelling the same scenario get different results. This pseudo reality is also reckoned by Buchanan et al. (1998) when they express that “Thus, a subjective description of reality is someone’s account of his own perception of that reality. With this type of description, the observer is trying to convey the way reality appears to him.” In other methods such as PROMETHEE, the DM establishes acceptance thresholds and assumes certain transfer functions according to his experience and knowledge. It appears that even when there is subjectivity, it is framed by historical and statistical tools. In ELECTRE The method partitions the set of solutions into two areas. One of them is called “nucleus,” which clusters feasible and most favorable alternatives, and the second one, which involves the least favored. The method is obviously interested in the first

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case, i.e., those in the nucleus or “kernel,”1 since any alternative off that kernel is outranked by an alternative pertaining to the kernel. ELECTRE belongs to the outranking methods family, because it establishes a binary relationship between alternatives regarding all criteria. It states that “a” outranks “b” and is expressed as “aSb” if the number of favorable criteria favors “a” and if there are no strong oppositions toward “a.” A “concordance matrix” is built, comparing paired alternatives, where outranking exists if there is a strong supremacy in criteria, and there is another “discordance matrix,” which opposes the former in the sense that it opposes the supremacy of one alternative over another. Naturally, it comes up immediately as a question about which value one can consider greater enough to justify a supremacy of one alternative over another, which leads to the threshold concept. In TOPSIS, the DM selects the distance formula for the ideal solution, out of several, against a subjective decision that may alter results. c) Wrong use of criteria weights. Some methods use the so-called criteria weights for evaluating criteria. These weights are not even good for determining criteria relative importance, because they are trade-offs, not weights; therefore, no clarification is needed. They are not good for alternatives evaluation. The reason is because subjective criteria weights are normally determined by preferences, a procedure that does not consider the ability or capacity of a criterion for evaluation. This subject was discussed in Sect. 2.1.6.2. Shannon (1948) paved the way for determining objective criteria weights without human intervention and entirely based on the initial data. Consequently, if a method uses objective weights, most probably the result will be different from the result produced by a method using subjective weights (see example in Sect. 2.5.12, Tables 2.3 and 2.4). Considering all the abovementioned factors, it appears that discrepancy in results is not a paradox, but the logical consequence of personal appreciations and judgments, or in other words, subjectivity. At first thought, it would appear that starting from an elaborated decision matrix generated by a DM as examined in Sect. 4.1.1 a solution could be to eliminate preferences in the process, especially on criteria weights, and then theoretically at least, all methods should give the same results. However, this would be neither real nor practical; decision-making is to a large extent a subjective discipline; one cannot ignore that the result given by a method is only a result obtained by a mathematical procedure, which serves as a guide and reference for the DM to make decisions. With this concept in mind, this book proposes transferring DM subjective appreciations from the top to the bottom of the process and not using criteria weights, i.e., taking advantage of the DM’s knowledge and expertise, and applying them to

1

Kernel. Subset of the elements of a set in which a function is transformed in an identity element in other set.

5.2

Interpreting Reality

85

results obtained free of subjectivity. It is then possible that different methods yield very similar rankings, and then each one may be examined and even modified with a critical eye. Milan Janic (2002) present a case for a project solved by three different methods and each one using three different kinds of weights: (a) not using weights, (b) working with weights obtained by simulation, or (c) weights derived from entropy. As a conclusion, the authors state that “When the same MCDM method used weights for criteria obtained from different procedures then, depending on the procedure, either the same or different results emerged. This implies that the weights of the criteria and not the MCDM method, should be considered more carefully when dealing with this and similar MCDM problems.” The proposal of using weightless criteria methods is not new, but what is proposed here is that when the weightless results are known, they must be examined by the DM, and if according to his opinion, a certain criterion deserves to be weighted, he must do so, and then run the software again. If the new result shows no difference with the original, it may indicate that said criterion did not deserve to be weighted, since its contribution after weighting did not change. The DM even may assign weights to all criteria, but now working on reliable results obtained when all criteria have the same importance; that is, he is weighting on solid bases and not on preferences which most of the times he cannot justify. However, he can compute weights derived from entropy and assign them to criteria and run the software again. In this way, he will have the certainty of the real significance of each criterion. The suggestion to put the DM experience and knowledge at the end of the process, to correct aspects that he believes can be improved, can be better understood using an analogy. Engineers and designers test scale models of cars and airplanes called mock-ups, made of wood or clay, in a wind tunnel. When the model is subject to strong winds, they can extract conclusions about how the model behaves under certain conditions, and from there, how the real thing will behave. If the mock-up performs as expected or at least within a certain range, it could indicate that the initial values were correct. If not, it is possible to modify some of them; in these engineering cases the “right” solution is never known, the same as in MCDM problems. Then, similarly, the DM by examining how the best solution reacts to external parameters can make the necessary adjustments. But again, this is done when there is a mathematical solution, not at the beginning of the process.

5.2

Interpreting Reality

The core of this book addressed the course of action followed by most MCDM methods. It is not a criticism of any of them; it only intends to point out the perceived unawareness of different aspects of reality. In this author’s opinion, this is a serious issue that somehow replicates what happened in the 1950s when projects evaluation

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5 Design of a Decision-Making Method Reality-Wise: How Should it Be Done?

only considered economic and financial issues, and then ignored reality posed by social, environmental, and sustainability factors. At present, it is believed that the actual process must be improved by trying to replicate reality, even partially, since total replication does not seem feasible in the near future and with the tools that we have nowadays.

5.2.1

Areas Where Reality Is Not in General Interpreted

For this analysis the following areas are examined: 1. 2. 3. 4. 5. 6.

Scenarios Alternatives Criteria Performance values Results Sensitivity analysis

5.2.1.1

Scenarios

This is probably the most important area because if a scenario is not modelled as precisely as possible, results are debatable; a scenario may be a simple or a complex entity, but in both cases inherent, concealed factors need to be considered. From this point of view, it appears that at present many factors are ignored, probably because not all the ramifications are examined, and this can be checked by analyzing published papers on MCDM. If an experienced and qualified DM examines at random, papers addressing scenarios solved by diverse MCDM methods, most probably he will be puzzled and surprised to notice that most of these papers do not reflect factors or issues that are there for everybody to see. He probably will ask: ‘Didn’t the author realize that he should not be using this method for this scenario’, or ‘What would happen if this issue was inputted?’, or ‘Why this issue was not contemplated? or ‘Obviously the author took such aspect for granted, when experience shows that it is incorrect’

The next question will probably be: Was this done because the issue was not important, or since the method employed could not handle it, or for the simple reason that he wanted to simplify a complex problem?

To clarify this issue, assume a real and frequent scenario that calls for selecting the best mode of transportation for merchandise, with alternatives such as single light and heavy trucks, bi-trains, rail, river, and air options, obviously within a certain scope of load and distance. What is sometimes missing in scenarios is that they do not include all possible and feasible intermodals with trucks and trains, or a

5.2

Interpreting Reality

87

combination of trailer on flatcar (TOFC), that is, trains carrying trucks, or roll-on and roll-off vessels, etc. That is, normally only a single transportation mode is considered. This examination is not directly related to the MCDM process; however, it is fundamental for defining the scope of the analysis and especially for foreseeing aspects that may be appraised when selecting criteria. For instance, in the road transportation scenario, it is not enough to bear in mind costs and duration of the trip; it is necessary to assume that some failure may occur in the system, for instance something unfathomable such as the break of the refrigeration system in a truck transporting perishable goods. In these cases, unforeseeable threats such as weather, heavy traffic, machine failure, and road accident must be examined, as well as their consequences such as deteriorated products, heavy fines, and loss of contract. Consequently, control measures as well as potential mitigation measures needed to be contemplated. This is naturally related to risk, something that normally is not even thought about. Perhaps some practitioners may think that this is not related whatsoever to MCDM; however, it is, because from here the DM will decide the criteria that have to be chosen for alternatives selection. In this example, for alternative “Using long-distance refrigerated trucks,” versus “Railroad,” criteria will be, of course, linked to total cost for each option, as well as travel time and unloading costs. But also, there should be criteria related to cost of control measures and mitigation measures, and all of this is associated with a cardinal value for risk (probability of occurrence times magnitude of potential loss). In the case of a railroad line, additional costs must be added for loading the merchandise in the railway yard (unless there is a spur line in the sender premises), as well as unloading, and perhaps transferring the load to a truck at destination. In this circumstance, trucks have a clear advantage over railroads. As can be seen, the DM needs to discuss this sort of concealed factors to allow him to build the IDM.

5.2.1.2

Alternatives

Not many studies contemplate existing relationships between alternatives or projects—for most, they are just alternatives. However, there are cases where their characteristics must be included in the scenario. It has already been mentioned in Sect. 2.5.2 that alternatives may be interrelated or not, inclusive or exclusive, and this is something rarely, if ever, considered. For instance, it is not very often that a MCDM process takes into account that if project A is selected, then all other projects are automatically excluded. This calls for a binary result as was mentioned in Sect. 5.1. This condition must be included in the IDM; however, this author never found a project treating this aspect.

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5.2.1.3

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Design of a Decision-Making Method Reality-Wise: How Should it Be Done?

Criteria

It is here, where reality is possibly more infringed. For instance, it is very often recommended in AHP not to exceed more than ten criteria; without challenging this popular assertion, which apparently obeys psychological experiments, it is obvious that it constitutes a blatant violation of a rational MCDM process (since in so doing, the DM is avoiding incorporating criteria that may have a large influence in the result, and therefore disregarding reality). Another omission, already commented, is not considering resources and their limits. Once the IDM is built, some methods make a partition of the problem in its components analyzing each part separately. This was examined and exemplified in Sect. 2.5.1, where it was demonstrated that results differ when the system is solved partially and as a whole; however, this is a common practice. This is not a breach of reality but a violation of the principles that direct how a system works. The pair-wise procedure when applied to subjective criteria may in some cases be biased, since a DM is not qualified to represent or interpret by his own preferences the opinion of thousands or even millions of people on a certain issue. The way to proceed is asking for the opinion of that people, not on technical issues of course, but on how a project will benefit or hurt them, as was commented in Sect. 4.1.1. The evidence is so overwhelming that this could be considered axiomatic. The issue is still more debatable when, against all logic, the DM is forced to take decisions related to very different fields, from public health to engineering and from environment to financial issues. Again, this is not advisable, but it is done.

5.2.1.4

Performance Values

In quantitative criteria, they are normally reliable because they are homologated amounts from suppliers and vendors of equipment, services, and suppliers. They also come from the technical departments of the company, and consequently are obtained by a thorough process. Most MCDM methods use these reliable values and match reality; however, the AHP and ANP methods are generated by DM preferences, and consequently, most of the time, do not represent reality, but the DM’s personal vision of reality. In addition, the preferences by the DM are expected to reach transitivity, a fact that normally does not happen in the real world. Regarding performance values for qualitative criteria, especially on aspects involving population, these must be obtained by sampling, and then condensing population preferences in percentages or crisp values. It is obvious that obtaining these values from the DM preferences is something far away from reality.

5.3

Check List for Aspects to Be Normally Considered When Modelling

5.2.1.5

89

Results Delivered by MCDM Methods

Obviously, they may be biased considering all violations enumerated in Sect. 5.2 and having a relative value. It is common to read in the literature that using a certain method in a problem produces a successful result. Obviously, this may be true or not, but it is impossible to assert it since the “true” result, if it exists, is unknown, and consequently, no comparison is possible.

5.3

Check List for Aspects to Be Normally Considered When Modelling

The objective of this section is to propose a means for the decision-maker (DM) to keep in mind all the necessary aspects to be considered for preparing the decision matrix, to reflect as close as possible the existent conditions in a scenario. It is just an organized reminder that is complemented with comments and examples (in italics) to clarify the meaning of each term. The check list is a template or form to be completed by the user, with the possibility to mark in appropriated boxes which are the aspects that are included in his/her study. In many of them, the template offers comments, suggestions, and hints, with the purpose of making clear the meaning and scope of each term. A few include questions that the DM could query himself and then find an answer. Once completed, this may constitute a valuable document, as it is the whole source for the decision matrix. It could also be an effective and important record for the DM support and defend his selection in front of stakeholders, and if in the future, it is necessary to conceive a certain solution, they already know how to achieve the selection of a best alternative and ranking. The following list is in reality an extension, as well as a condensation, of the aspects required in modelling a scenario. It does not pretend to be complete, and its only purpose is to help the DM when considering which elements must be included and aim at replicating reality, as much as possible. All MCDM methods start with the information provided by the decision matrix, but depend largely on the characteristic of each scenario, and these rarely replicate another, even a similar one. Consequently, there is not a standard decision matrix, and very rarely assumptions used in one scenario can be applied to another. Therefore, each scenario is unique, and as such must be addressed. Literature shows that this fact is not very often considered, and it is usual to read that an author used a certain procedure and method for a scenario, for instance selecting personnel to be hired for office work, while another author employs the same method to solve a scenario for selection of projects in a river basin. The first one is a simple scenario where candidates are subject to very well-known demands.

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Design of a Decision-Making Method Reality-Wise: How Should it Be Done?

The second one involves a series of very different requirements, works with different kinds of undertakings, poses very difficult environmental and political questions, may affect thousands of people, may use hundreds of criteria, and very often faces demands which are very difficult to model. Then the DM needs to ask himself if the method he is thinking about, or that he knows better, or using MCDM software his company owns, is appropriate for his scenario. But which is the motive by which a method cannot tackle any scenario? The same reason that a light truck with a weight limit of less than 3 tons cannot handle a 15 tons weight, as a heavy truck can. Each one is designed and has components according to their intended use. Something similar happens in decision-making; the DM must use the appropriate method. The following list enumerates conditions for a MCDM method and process. A method must comply with these conditions: 1. Possibility to work with different scenarios simultaneously, as happens in some settings 2. Work with many different objectives, quantitative and qualitative, in any mix, as is practically the norm in most cases 3. Allow for group decision-making 4. Offer a reasonable computing cost 5. Consider in a portfolio of projects, different starts and finishes, annual percentages of completion, compliance with annual budget, etc. 6. It must have a decision table reflecting as faithfully as possible, a scenario which includes project precedence, standing alone, or complementary. 7. Be able to work with positive or negative values for alternatives performance, as well as with integer and decimal values. 8. Not be limited by the number of alternatives and criteria. Not complying with this condition normally indicates that the alternatives are improperly evaluated, since many important aspects are ignored. 9. Capable of working with quantitative and qualitative criteria in any mix, 10. Able to solve scenarios where criteria and alternatives are either independent or related. 11. Have criteria limits because resources are always limited in real situations. 12. Give the same result, irrelevant of who is the DM doing the analysis. 13. Whatever the method used, results must be coincident, or at least very similar. 14. No subject to rank reversal in any condition. 15. In case of ties in alternatives scores, be able to give information and support for the DM to choose between them. 16. Be able to indicate and document which are the most important criteria, related to the best alternative chosen and ranked. This is paramount for performing a sensitivity analysis. 17. Provide information about range of validity of a criterion when it is made to vary to analyze output.

5.5

Conclusion of This Chapter

91

18. Have strong sensitivity analysis capabilities, by supplying quantitative information of effects in the output due to changes in the input. It must give the DM enough quantitative data for him to sustain his decision to stakeholders.

5.4

Working Template for Modeling a Scenario in MCDM and for Selecting a Method to Solve it

This section presents the template in Table 5.1. No method is recommended since it is a DM’s choice after considering the requirements for a scenario. The template is no more than a check list for the DM to consult and to realize if he is missing aspects in his modelling. It is for his judgment to incorporate them or not. However, the template will indicate how close to reality his modelling is and allow him to decide if what is missing is or is not significant. The template has been divided into different components, designed to guide the DM, starting with the definition and types of objectives, and ending with sensitivity analysis. In general terms, it has been exemplified for easier comprehension, and in some cases, hints are given.

5.5

Conclusion of This Chapter

This chapter deals with modelling and its main purpose is to call the attention of the DM to what is not being done at present in MCDM processes. To this end, it analyzes different procedures related to too much flexibility, DM subjectivity, and wrong use of criteria weights; it is not meant to be a criticism but a way to highlight on what should be done. All of these aspects lead to interpreting reality and from this point out of view scenarios, alternatives, criteria, and performance values; that is, the components of a method were examined. The chapter finishes with a condensation of these points materialized in a template or check list, intended to help the DM in keeping track of all aspects that must be taken into account.

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Design of a Decision-Making Method Reality-Wise: How Should it Be Done?

Table 5.1 Template for modelling a scenario Template for modelling a scenario in MCDM and for selecting a method to solve it Please mark your response Objectives Number of objectives

One objective

Several objectives

Fixed objectives

Contradic tory objectives

Feasible or unfeasible objectives

Scenarios

One For instance, minimize cost

Several For instance, maximize benefits, minimize environmental contamination, minimize costs or maximize transportation

Related Scenario A is somehow related to scenario B. Example: Project A is an orange plantation in one location, while an orange concentrate plant B (project B), that process oranges, is located in another location

Independen t Example: A manufactur ing plant in some place and another different plant in another place

Simultaneous Some alternatives can be in some places but not in all of them, while other alternatives may be in other different places

Alternatives, Projects, or Options Alternatives types

Relationship between alternatives

Discrete or infinite

Somehow related Example: Project A calls for placing sewerage pipes deep in a road. Project B involves paving the same road Obviously A must precede B

Independent

Inclusive Several alternatives are chosen (ranking)

Need for equal scores Example: Calling for proposals, there could be joint ventures. Then, alternatives with equal scores are needed, and this condition must be inputted in the IDM

Exclusive (if alternative A is selected, alternative D cannot be selected). For instance, in scenarios for industrial location in different places; it is one or the other

Complementary If alternativ e A is selected, alternativ e D has to also be selected. Example: In remote locations the constructi on of a hydro dam, necessitat es installatio n of lodging facilities for workers

Exposed to external factors Such as variations in offer and demand

Do nothing Status quo

Some alternatives are already underway Example: When some alternatives in a portfolio are already under execution; very common in medium term plans

Imposed When for whatever reasons an alternative may be required to be in the ranking. Example: A pledge to execute a project made during an electoral campaign

Time phased Alternatives take place at different times, that is, they do not start and finish at the same time

(continued)

5.5

Conclusion of This Chapter

93

Table 5.1 (continued) Are all alternatives feasible?

For sure

If result shows equal scores (ties)

Do you know how to break the tie?

Y N

Some are in doubt

If unfeasible, just discard them

Dubious feasibility. Insist that more studies be made

By DM opinion

By sensitivity analysis

By using max/min normalization Not guaranteed

Reasonable preparation cost

Matrix preparation Initial matrix modelling cost

Low preparation cost

Moderate

Labor intensive

Computing time

Low

Extensive

Very extensive

High preparation cost

Criteria

Criteria identification

Normalization of criteria performance values

Alternatives are subject to criteria, but the latter have to respond to alternatives evaluation needs. Example: Criteria for purchasing equipment are not the same that criteria for selecting personnel

Since criteria normally have different units, normalization is necessary for them to be compared

Different normalization procedures: a) Dividing each performance values by their sum in the row aij*=∑

=1

b) Dividing each performance value by the maximum value in the row aij*=

Y

N

Do not forget to specify criteria units. There could be any mix

Hint: If you were using one of the first three normalization methods, and if your result shows a tie between two or more alternatives scores, run again the method that you are using and change to max-min normalization procedure. The first three procedures tend to little discrimination, while the last provides more discrimination

Are the results different? No if using a), b) or c) but a little different in magnitude if using d) (In SIMUS). It may be different for other methods

max

c) Using Euclidean formula aij*=

√ ∑ =1 (

)^2

d) Using max-min formula aij*=

max

−min −min

Being aij*= The normalized performance value Different fields for criteria

Externalities

Compensation

Engineering

These are side effects produced by commercial or industrial activities that do not have a market value.

Financing

Economics

Is it important in this context? Yes, very important, because it demands complementary works which execution have to be added to the cost of a project. More than once, a project has been halted by environment authorities Examples: Need of reforestation in logging operations, installing sound barriers in highways and factories

Social

Environment

Sustainability

This is a very common feature and not generally considered. There are externalities in a logging operation, because it produces erosion. Also, in mining operations (minerals, oil and gas) because it produces depletion of natural resources. Noise is an externality because it affects people health and wildlife

In many large projects such as hydroelectric schemes, sometimes people have to be relocated since their land will be flooded by the lake that will be formed behind the dam. Entrepreneurs or the government are legally bound to provide not only a new dwelling but also monetary compensations. These expenses could be very high, especially when there are hundreds of people to be relocated

Are there externalities in this scenario? If Yes, have they been contemplated?

Is there in your project a situation like this?

Y

N

Y N

(continued)

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Table 5.1 (continued)

Ancillary equipment

In projects such as the construction of large bridges or projects that use large amounts of concrete, there are ancillary cost projects that are not related with the main project, but without them, its construction is not possible. In the first case there could be giant cranes that have to be purchased and erected over huge foundations. Of course, their costs of mounting, operating and dismounting these ancillary structures have to be added to the whole project

Is there in your project a similar situation?

Clusters

In some scenarios, for example in a municipal five-year plan, there could be several clusters not related between them, for instance Infrastructure, which may involve various projects such as sewerage construction or improving garbage collection. Other cluster may be Social, including construction of schools or hospitals. Another could be Environment, with projects to increase green spaces, reducing NOx concentration, and promote garbage recycling. Most of them have the budget as a common factor, which is normally assigned in different amounts to each cluster. Naturally, the sum of partial budgets must be less or equal to the whole budget

If this is your case, the modelling must consider these clusters and respective projects

As many as necessary Do not limit the number of criteria due to workload or because the method you chose cannot handle large number of them. If that is the case look for another method

Limited to a certain number This could affect the accuracy of the results since many aspects are not considered or dismissed

By similitude to other scenarios Not a good deal, because normally there are not two identical scenarios or projects

Criteria weights. How are they elicited?

By DM preferences These are subjective weights

By group preferences These are subjective weights

By standard deviation These are objective weights

Criteria feasibility

For instance, a criterion that calls for unreasonable benefit, such as 35 % of Internal Rate of Return (IRR)

Check if company resources comply with each criterion threshold

How many criteria?

Criteria duality

Y

N

Are they time related? That is, do they have different starting and finishing dates?

Y

Y

If Yes, how do you model this aspect?

N

N

N

Y Have you consid ered the costs for this apprai sal?

From entropy These are objective weights

No weights From Some methods do not statistics use weights, but they These are consider criteria objective relative importance weights by other means and using inputted data In a call for bids check that the Are resources shared? bidder fulfills the request for May be there are enough tenders, for instance, number of resources, but some of them are or will be used in other projects own equipment, manpower, etc. under way or planned, and then there could be conflicts and overlapping a) Checked b) Not checked

a) Checked b) Not checked

Y

N

From similar projects These are doubtful weights. Not recommended

Check if the call allows that two or more bidders bid together for the same job (joint venture). Very common in practice a) Checked b) Not checked

Sometimes a criterion must be subject to maximum and minimum values at the same time. For instance, in urban water distribution schemes, there is a minimum threshold of daily amount of water per person fixed by the World Health Organization, and a maximum fixed by the water distribution entity, to avoid waste. In this case, the performance values of said criterion are duplicated in two rows, one for maximum, and the other for minimum, and thus constituting two different criteria but with the same performance values. Consequently, the operator in the first case is’ ≥’ and ‘≤’ in the second case. In Chapter 9, Section 9.2 there is a real example of criteria duality.

Which is the data source for quantitative criteria

Technical departments

Compensatory and non-compensatory methods

The MCDM method that you plan to use belongs to the

Y Suppliers N

category of compensatory or noncompensatory?

Data source for qualitative criteria

International Y Organization:

Y Data collected N

C

N C

If compensatory. get advice about its characteristics to learn if it is adequate to solve your problem

DM estimates

Y

World Bank, United N Nations, OECD, etc.,

N

Y

Y Governmen t

Research N

N

Compens atory methods:

Non-compensatory methods: ELECTRE,

Be aware that in projects which criteria are considered nonindependent (most projects,

AHP, ANP, SAW, TOPSIS.

PROMETHEE, SIMUS

because criteria normally are related directly or indirectly) compensatory methods are generally not acceptable

Similarity with other projects

Surveys and polls

Effect on other areas

Unknown

(continued)

5.5

Conclusion of This Chapter

95

Table 5.1 (continued) Yes

Need for using fuzzy data?

Relationships between criteria

Type of criteria

Some linked

Quantitative Mostly contain reliable performance factors such as: - Production /h of equipment -Firm purchase price for materials or services -Number of people- to operate equipment- Contamination produced by a certain installation - Efficiency - Number of students - Type of dwelling - Water consumption - Sewage produced by a new dwelling complex

Criteria meaning

Criteria weights

Budget

Some correlated a) Large correlation ρ > 0.70 b) Moderate correlation 0.45< ρ x2.

106 Fig. 6.1 Polygon formed by criterion A (maximize)

6

Linear Programming Fundamentals

PV

Polygon for criterion A

PS

0 A

Fig. 6.2 Polygon formed by criterion A (maximize) and criterion D (minimize)

PV

Polygon for criterion A and criterion D combined PS

0 D

A

This description aims at explaining why an LP scenario can work with many different alternatives, that is, many dimensions, and with many criteria. In linear programming it is normally established that the scores must be positive, and for that reason, only the first quadrant is considered. In some cases, however, it could be desirable not to establish this limitation: in the latter case, the program can be instructed to display scores irrelevant of their sign. Since x = 0.56 and x2 = 0.41, the objective function Z will be equal to the sum of the product of x1 times its cost, plus the product of x22 times its cost. That is: Z = 0.56 × 0.72 + 0.41 × 0.68 = 0.68. Any other combination for this expression will give a higher cost. In a scenario with three alternatives, that is, in three dimensions, each criterion is not represented by a line but by a plane, and thus the inequality is represented by a volume—called a polyhedron—and in the case of maximization, extends from

6.4

The Two Sides of a Coin

107

Fig. 6.3 Polygon formed by all criteria

PV

Z Polygon 0.41

0

a (optimal solution)

PS

0.56 D

Z C

B A

below the plane, to the planes corresponding to three coordinate’s axes. The objective function Z will then be a plane. In a scenario with more than three alternatives, which is n-dimensions, each criterion is represented by a hyperplane and the inequality is represented by an n-dimensional figure—called a polytope—and in the case of maximization, it extends from below the hyperplanes, to the hyperplanes corresponding to the n-dimensions coordinates axes. The objective function Z will then be a hyperplane. The algebraic development of the simplex method will find the same result. It starts from an initial condition of no solution, where Z = 0 (i.e., in the origin of coordinates); each alternative is compared simultaneously with all others, and the alternative with the highest opportunity cost is selected. Since the dimensions are fixed and established by the number of alternatives, the entering alternative must be compensated with a leaving alternative. The latter is determined using the ratios between the criteria independent terms and values of the vector corresponding to the entering alternative. It is here where the relative importance of each criterion is considered, since its importance is used to identify the alternative that must leave, and thus, keeping constant the dimensional space.

6.4

The Two Sides of a Coin

Every LP problem solved by the simplex algorithm produces two optimal results. It appears to be a contradiction, because the two optimal results are far from each other. An analogy can help in their understanding. A coin has two faces: obverse and reverse, or more colloquially head and tails. They are different, but are an

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indissoluble part of the coin, and even when they have different characters (or information), their value is the same. This is what happens with an LP problem that produces two different solutions, with the same value, but with different information, as the coin. One solution is called the “primal problem’ and the other is the “dual problem,” which is the “reverse” of the primal. The primal problem is generated when the method starts with the IDM. Its solution produces: a) Scores for each alternative in decimal, integer, or binary format according to the instruction given by the DM b) A ranking of alternatives The problem solved graphically in Sect. 6.3 and providing scores for PS and PV and its ranking is an example of primal. The dual problem result is generated at the same time as the primal and works with the transpose of the IDM. Its solution produces: c) Shadow prices, which are marginal values for each criterion d) Information about the allowed variation for each criterion A shadow price value is related simultaneously to both objectives and criteria. Thus, for a certain criterion its shadow price informs how much the respective objective increases or decreases (depending on its sign) each time that the criterion RHS is increased in one unit. For an explanation of the meaning of the last concept, see Sect. 8.1.1. Even when these two problems (primal and dual) give different results, they are completely related and complement each other. The objective function Z has a quantitative value in the primal problem which coincides with the Z quantitative value in the dual. This duality of LP problems has great importance, and it will be illustrated in Chaps. 7 and 8.

6.5

Description of the Method

SIMUS is grounded on the fact that both objective functions and criteria have the same mathematical linear structure, and due to that they are interchangeable. This property is the core of the SIMUS method, which operates according to this sequence: 1. 2. 3. 4. 5. 6.

Choosing a criterion Deleting it from the decision matrix Using it as objective function Getting an optimal result (if it exists) Saving these results in a matrix (called efficient results matrix) or ERM Returning the used criterion to the matrix

6.6

Graphical Explanation of Correlation

109

7. Choosing another criterion and repeating the process, until all criteria (or only those selected by the DM) have been used as objective functions. It can be seen that when complete, the ERM becomes a Pareto efficient matrix, and indisputably optimal. From then on, the ERM is the starting point for two different procedures; these two approaches create two distinctive solutions, which coincide not in their relative scores but in their rankings. That is, it is equivalent to solving a problem with two different methods, and both delivering the same results. Naturally, the method does not produce an optimal solution as LP does, since it has a heuristic component, and then, it offers a compromise solution, as other MCDM heuristic procedures. In addition, because it is possible to have as many objective functions as criteria, it produces a final table with “shadow prices,” or criteria marginal values, with a set of marginal values for each objective, and permits extracting valuable information about selected projects and breaking ties in the scores of two or more projects. SIMUS demands a strong interaction with the DM along the calculation process, and more intensively when analyzing results. Its strength resides in this close association between the method and the DM because the method only suggests a result, as a ranking, but more importantly, it gives the DM, by examination of the marginal values for criteria, the vision for him to foresee the potential threats that can exist in the future, for each one of the alternatives of the ranking. Naturally, it also allows a sensitivity analysis by using not the criteria weights, but the marginal values, permitting the DM to know how sensible a solution is, when certain parameters change. Using these marginal values is central for developing an efficient sensitivity analysis.

6.6

Graphical Explanation of Correlation

This section is connected to Sect. 2.5.10 where the criteria correlation issue was addressed; it illustrates in a graphic manner how correlation may affect the result. Following this demonstration, we will explain how to incorporate criteria correlation in the IDM. For this explanation an example is proposed consisting in two different routes for highways between points N and M, where one of which must be selected. These routes or alternatives are represented as x1 following a river valley route, labelled “the valley road,” and x2 following a high mountain route labelled “the mountain road.” See in Table 6.2 the IDM with the two alternatives subject to six criteria A, B, C, D, E, and F. The valley road, with many curves, is considerably longer with an average maximum speed of 95 km/hr. It is not subject to harsh weather, but may have flooding areas, although not very frequent. The mountain road is designed as a high-altitude mountain route with many viaducts and tunnels which make it almost linear, and thus allowing a high speed of 130 km/hr.

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Linear Programming Fundamentals

Table 6.2 Initial decision matrix for road selection Safety Speed Rock falling risk Landslide risk Flood risk Road control

x1 Valley highway 60 95 0.08 0.25 0.18 0.04

x2 Mountain highway 80 130 0.23 0.03 0.02 0.25

MAX MAX MIN MIN MIN MAX

x2

Fig. 6.4 Criteria A and B considered independently Z

B A 9

Z

0 1.5

B

A

x1

In this graph, which demonstrates influence of correlation, only two criteria are considered to gain clarity in the illustration. They are “A,” “safety” (in solid red) and “B,” “speed” (in solid blue) both calling for maximization. Figure 6.4 represents this case graphically following the procedure described and illustrated in Sect. 6.3. These criteria are linear inequalities and its equations can be represented by straight lines. These lines intersect and define a polygon that is shown shaded in the graphic, which is the common space for both criteria, and contains all the infinite solutions of this system. The objective function Z (in dashed black line) that calls for fatalities’ minimization is obtained from statistics of fatalities per 1,000,000 car, bus, and truck travellers, as follows: Minimize Z = 7.2 x1 + 3.3 x2. It is tangential to the vertex of the polygon that corresponds to the optimal solution, identified with v. This vertex defines the values of the two coordinates or alternative scores, which are x1 = 1.5 and x2 = 9. For this solution, the objective function is Z = 7.2 × 1.5 + 3.3 × 9 = 40.5 (fatalities per 1 million people). Observe that x2 > x1 or that the mountain alternative is preferred. This is the optimal solution when both criteria are considered independent in the IDM. Criteria can vary in two ways, as follows:

6.6

Graphical Explanation of Correlation

111 x2

Fig. 6.5 Criterion A varies to A’ and criterion B remains constant

B A A’

Z

6

0

4

B Z

A’

A x1

a) By changing their allowable limits (RHS), either by incrementing or decrementing them. The variation will be represented by a new criterion line parallel to the former. b) By changing the performance value (aij), just for one alternative or for both. The change will be represented by a new criterion line with a different slope regarding the former. In this example, it is assumed that the performance value for alternative x2, high mountain, and criterion A, safety, is changed. Figure 6.5 represents this situation by varying criterion A. After the change the new A line will be the red dashed A’. Observe that A and A’ are not parallel, due to the change of the aij for the high mountain alternative; check also that the line for criterion B remains constant. A new polygon is now formed, and the objective line Z identifies the new optimal point, which in turn determines the two new coordinates or scores for x1 = 4 and x2 = 6, with Z = 7.2 × 4 + 3.3 × 6 = 48.6. Again, x2 >x1. Up to now, correlation has not been pondered. Let us see what happens when it is considered. If a correlation exists between criteria A and B, it can be computed using the Pearson correlation coefficient; say for instance that it is ρ = 0.86. This means that weather conditions, such as snowfalls or heavy rains on the area affecting safety on the road or criterion A, will impact speed, criterion B, since common sense indicates that lower safety conditions mean lower speed. Assume that a performance value for A is varied for the high mountain project due to heavy snowfall, which is frequent in wintertime; consequently, its new line in now A”A” as illustrated in Fig. 6.6. Due to the change in A and because of the correlation between A and B, the latter also changes, and then the new line is portrayed as a dashed B′ (in blue), Consequently, a new polygon is formed and a new optimal vertex detected by the objective

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Fig. 6.6 Criteria A and B are correlated

Linear Programming Fundamentals

x2 x2 A

B

A’’

B’

Z

3.4

0

5

B

Z

x1 B’

A’’

A

function (which has not changed) determines the new scores for safety and speed as x1 = 5 and x2 = 3.4, with Z = 7.2 × 5 + 3.3 × 3.4 = 47.2. Observe that now the ranking is reversed since x1>x2; that is, the valley road is preferred to the mountain road. This indicates that a very thorough analysis must be done to estimate the frequency and severity of snowfalls, as well as heavy rains. The first can even produce a temporary closing of the mountain road, as it happens in different parts of the world, and last for several days. The second, albeit not analyzed here, will probably have the same effects, and may cause the same problems due to mud avalanches. Consequently, if this is a frequent phenomenon, the DM may consider that maybe the mountain road is not the best alternative. Observe that if the objective function Z had an initial value of 40.5 per 1,000,000 travellers, and if safety is lowered independently from speed, the percentage of fatalities increases to 48.6 per 1,000,000 travellers, since the same high speed holds whatever the weather. However, when a change in safety is considered, which also affects speed of 0.86 and then decreases it, fatalities expressed by Z reach a lower value of 47.2 per 1,000,000 travellers. There is no doubt that considering correlation results follows reality. There is also the possibility of changing the objective function, varying for whatever reason, its coefficients. In this case, Z will have a different slope, which most probably will produce other selection. In addition, it must be noted that the DM should be able to change the limits of criteria (RHS), the performance values of one or all alternatives, and the coefficients of the functional, all at the same time.

6.6

Graphical Explanation of Correlation

113

Table 6.3 Pearson correlation coefficient (ρ) between criteria

Criteria Safety Speed Rocks falling risk Landslides risk Flood risk Road control

A B C D E F

Criteria A B Safety Speed 1 0.86 1

C Rocks falling risk 0.07 0.05 1

D Landslides risk 0.56 0.12 0.59

E Flood risk 0.80 0.02 0.40

F Road control 0.76 0.70 0.03

1

0.61

0.21

1

0.07 1

Analytical Procedure It was possible to develop this example graphically because there were only two alternatives or dimensions. But, with problems involving many alternatives, it is not feasible, but it is often done, to use the simplex algorithm and modify the IDM. To proceed follow these steps: 1. Detect pairs of criteria that show correlation, which is simply performed using the Excel formula for Pearson correlation. Table 6.3 shows the correlation matrix for this example and the six criteria. 2. The DM must decide which level of correlation is important to him. For instance, he may decide that it is ρ ≥ 0.75. Consequently, only three pairs of criteria are chosen, and shown with their correlation coefficients in bold. The selected pairs are: Safety and speed (A/B) Flood risk and safety (E/A) Road control and safety (F/A)

` 3. Add those pairs (denoted bolded), to the initial decision matrix (IDM) after the normal criteria, as shown in Fig. 6.7. 4. Now, at the intersection of project x1 column and the row corresponding to pair flood risk and safety (cell D43), place the conditional formula indicated in Fig. 6.7. 5. Another formula at the intersection of project x2 column and the row corresponding to the pair safety and speed (cell E42). 6. Place another formula at the intersection of project x2 column and the row corresponding to the pair road control and safety (cell E44). In this way, all the respective values will be accredited to the corresponding cells in the original IDM, altering it, as a function of correlation.

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Linear Programming Fundamentals

Fig. 6.7 Modified IDM by adding correlated pairs of criteria

Here, it is assumed that there is an increase of 12% in safety procedures for the mountain highway, an increase of 15% in safety procedures for the valley highway, a decrease of 20% risk in road control for mountain highway. Safety and Speed (A/B) The new performance factor for speed in mountain highway will be 130-(80*1.12-80) *0.86 = 122. Consequently, there has been a reduction in speed from 130 km/hr to 122 km/hr, because of more controls.1 Flood Risk and Safety (E/A) The new performance factor for safety for valley highway will be 60-(0.25*1.15-0.25) × 0.8 = 60. Consequently, there have been no impacts in safety for the valley highway due to floods. Road Control and Safety (F/A) The new performance factor for safety in mountain highway will be 25-(60*1.20-60) *0.76 = 41. Consequently, a decrease in road control produces a decrease in safety from 60 to 41. 7. Run SIMUS, and once a result is obtained, make several test runs, using different values for changes to see if the initial selection of alternatives changes. The procedure explained is a sensitivity analysis aimed at changes in either the RHS and/or the performance values, considering correlated criteria. Just by changing one value in a formula, which is used again and again, it is possible to have a very accurate panorama of what may happen when the values change. It is perhaps necessary to explain that the set of correlated pairs and formulas are added to the IDM for reference only, since they do not participate in the computation, despite that their results do. In a situation like this example, the DM only limits to put in the corresponding formulas the variations that he considers may take place, either individually or jointly; the IDM will be automatically updated. Then running the software, the result

1

To avoid confusion the operator (*) used in computing is also used in the text, in lieu of operator (x).

6.8

Conclusion of This Chapter

115

will indicate if there are changes in the ranking because of variations in the correlated pairs. Even when this procedure was developed to be used in SIMUS, it appears that there is no inconvenience to employ it in other MCDM methods.

6.7

Is Rank Reversal Present in Linear Programming?

Research by this author in many published papers about MCDM failed to find references about rank reversal (RR) appearing in problems solved by LP. Whatever the method, it is evident that RR produces a loss of robustness that makes a method unreliable. From the mathematical point of view RR violates the invariance principle of utility theory (Cascales & Lamata, 2012) which establishes that a DM should not be affected by the way projects are presented (see also Cox, 2016). Whatever the method used, it is necessary to remember that they employ mathematical models that organize the available data and by means of some algorithms or norms suggest a solution. Keeping in mind, the only purpose of these methods is to support the DM judgment. Consequently, RR could be a serious issue for him because experience shows that a problem using the IDM suffers several modifications, and then it is necessary to examine different options, deleting or adding projects to examine their effect in the result. Section 7.6 presents three cases with increasing levels of complexity, and then the SIMUS method is tested for RR. It is necessary to consider that these tests involve more severe requirements than normally encountered in the literature for other methods, since here, in addition to the normal requirements of deletion and addition of one project or alternative, it deals with two and three alternatives deleted simultaneously, as well as deleting the “best” project and even adding a project better than the “best.” Case 1 includes 25 different tests. Case 2 involves 32 tests, and Case 3 examines 3 tests. There are then 60 tests on different problems, where no RR was found. Naturally, this is not enough to guarantee that the SIMUS method is 100% free of RR, but it obviously provides a very good basis for confidence.

6.8

Conclusion of This Chapter

Linear programming is a complex issue, and this book is not the place to develop LP theory. However, just to give the reader an idea of how it works—by the way, in a completely different manner as other MCDM methods—this chapter illustrates by means of a graphic example how LP operates. Naturally, nobody solves a complicated scenario using the graphic method because it is also workable up to three

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Linear Programming Fundamentals

alternatives, but it is useful to understand the interactive process followed by the simplex algorithm when it seeks iteratively for better solutions, as well as the reasons why the method admits a large quantity of alternatives and criteria. Perhaps the more important point is the fact that the reader can positively see how the different criteria are treated simultaneously and then replicate reality. A very important concept is also introduced related to the primal and dual problems in every LP problem, using the simplex method. This concept is central in LP and is largely used in this book, especially in sensitivity analysis. The chapter finishes with an introduction about rank reversal in LP and its importance in decision-making and paving the way for a full analysis in the next chapter, by proposing three large and complex examples that were solved by SIMUS, subject to many different changes but not producing rank reversal.

References2 Cascales, M. T., & Lamata, M. T. (2012, September). On rank reversal and TOPSIS method. Mathematical and Computer Modelling, 56(5–6), 123–132. Cox, A. (2016). Good decision-making guide for public bodies. Retrieved May 6, 2018, from http:// www.arthurcox.com/wp-content/uploads/2017/10/Good-Decision-Making-Guides-Collection1-7-booklet.pdf Dantzig, G. (1948). Linear Programming and extensions. United States Air Force. Fylstra, D. (2018). (Solver). Retrieved May 05, 2018, from https://www.solver.com/ Kantorovich, L. (1939). The best uses of economic resources. Munier, N. (2011). A Tesis Doctoral – Procedimiento fundamentado en la Programación Lineal para la selección de alternativas en proyectos de naturaleza compleja y con objetivos múltiplesUniversidad Politécnica de Valencia, España.

2

These references correspond to authors mentioned in the text.

Chapter 7

The SIMUS Method

Abstract This chapter aims at explaining the SIMUS method, trying to show without formulas how it works. Its purpose is to illustrate the DM about its principles and characteristics for him or her to understand and apply it without going into complex mathematical demonstrations. That is, one thing is to understand a method, to know how to use it, and to know how to get the most from it, and another is to be knowledgeable about its mathematical intricacies. SIMUS is a hybrid method based on linear programming, weighted sum, and outranking methods. If the reader is interested or perhaps rather curious about how LP works, in the Appendix there is a detailed and accessible explanation. Since SIMUS is also grounded on the two abovementioned techniques it produces two results but with the same ranking. It is the equivalent of solving a problem with two distinctive methods and getting coincident rankings. Naturally, it does not mean that SIMUS delivers the “true” solution, if it exists, but these two similar outputs offer a good deal of reliability. Although SIMUS is a heuristic method, the compromising solution obtained is based on the Pareto efficient matrix. An application example illustrates how to load the data into the SIMUS software and shows its operation. The chapter continues explaining how to incorporate specific and real-world issues in the model and ends examining why both LP and SIMUS do not produce rank reversal.

On SIMUS method: Method developer: Nolberto Munier (2011). Software developer: Nolberto Munier (2011). Proprietary rights: Faculty of Economics, Universidad Nacional de Córdoba, Argentina. SIMUS System Administrators: Dr. Catalina Alberto and Magister Claudia Carignano (Argentina)– Tel: (54) 351437300 ext. 48,650. Availability: Free. Support: Technical questions about structuring the Initial Decision Matrix and mathematical explanation of the algorithm contact: Nolberto Munier (Canada): [email protected]. Tel: (1) 6137707123. (continued) © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 N. Munier, Strategic Approach in Multi-Criteria Decision Making, International Series in Operations Research & Management Science 351, https://doi.org/10.1007/978-3-031-44453-1_7

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The SIMUS Method

Catalina Alberto- email: [email protected] Claudia Carignano- email: [email protected]

7.1

Background Information

As commented in Sect. 1.2, the SIMUS method uses the simplex algorithm for LP. From the very beginning this algorithm, considered one of the most significant created in the twentieth century, had very important applications especially related to oil refineries and other large undertakings. Nowadays, after nearly 70 years it continues being used by thousands of companies around the world; one of its many advantages is that it offers a solution that is optimal. However, it had and still has two large disadvantages: (a) it works only with one objective and (b) accepts only quantitative criteria. This is possibly the main reason why many different methods appeared, which take into consideration that in realworld scenarios there are usually more than one objective, as well as a mix of quantitative and qualitative criteria, and then, they cannot be solved by LP. These methods are heuristic and do not offer optimal responses, because they are looking for a compromise solution while satisfying the DM or a group of DMs. Aiming at remediating these LP drawbacks, the SIMUS method was developed, which makes possible working with as many objectives as necessary, even in the hundreds, and also permits qualitative criteria. It does not offer an optimal solution, but as in other methods, a compromise one. However, this compromise solution is based on a Pareto efficient matrix, and therefore, the solutions obtained using weighted sum and outranking are grounded on optimal data. In addition, as per this author’s experience after examining methods and solutions in hundreds of different kinds of projects, it seems that practitioners are not generally interested in optimal solutions, even if it could be attained, but in a feasible and convenient one. This lack of interest could be attributed to the fact that optimality is an unfamiliar concept and not very well understood, and for that reason, they are more inclined to accept methods that are more transparent and that reflects their assumptions. However, if heuristic algorithms have some advantages, their main disadvantage, and a serious one, is the uncertainty produced by too many subjective decisions and hypotheses. Then, it transpires that a good method should try to take the best of both approaches. SIMUS has no need for weights, although they can be incorporated if the DM wants to use them for both criteria and alternatives. Due to the lack of subjectivity in preparing the initial decision matrix (IDM), as well as its capacity to model a wide range of real-world situations, it is perceived as the method that best mimics reality, although, of course, imperfectly.

7.1

Background Information

119

SIMUS core is grounded on the principle that both criteria and objectives have the same mathematical structure, and then, criteria are used alternatively as objectives or as criteria, since they constitute the same element. The only difference is that objectives are in general indefinite (as maximize wellbeing, minimize poverty, etc.), while criteria have a very definite purpose and goal; for instance, maximize the project return, as close as possible to a certain value, which is in general the internal rate of return (IRR) or the net present value (NPV) normally established by the board of directors of the company or entrepreneur, or minimize air contamination, or maximize manpower use, etc. This double use of criteria allows SIMUS to inform about a very important fact: it indicates which criterion, among many, contributes to the selection of an alternative and provides a numerical value for each one. This permits us to perform a sensitivity analysis that gives valuable quantitative information to stakeholders, especially linked with the performance of the best alternative when subject to external influences. For instance, in a scenario where it is necessary to select the most efficient mode of bulk transportation for grain using either (or both) rail or trucks, between production zones and ports for export, SIMUS allows us to investigate, based on quantitative values, how the variation of exogenous factors such as competition, speed, safety, weather, facilities for loading and unloading, etc., influences the alternative selected, and even compute different associated risks. Despite its advantages, SIMUS is no more complicated than other methods, and perhaps less, and certainly a DM does not need to understand its algebra, same as in other methods. Application is made easy by using SIMUS software (Lliso & Munier, 2014), and the model normally designed in Excel can electronically be inputted to the software. SIMUS is completed with IOSA (input-output sensitivity analysis) that automatically loads data from SIMUS and delivers a graphical and quantitative result. Both SIMUS and IOSA software are available free and without limitations on the Web at: http://decisionmaking.esy.es/es/SIMUS. Normally, the IDM is built in an Excel spreadsheet although this is not mandatory since data can be loaded directly in the main SIMUS screen. The first method allows for the electronic transfer of the information to SIMUS software. Whatever the method for building the IDM, SIMUS uses an Excel Add-in, called “Solver,” which is accessed through the “Data” tab, and located at the top right of the screen. The Solver is in all Excel spreadsheet; however, if it does not show up in the computer screen, it must be retrieved following this sequence: Excel Options, Add-ins, Manage Excel Add-ins, Go, and then check the Solver box. IOSA allows the DM and stakeholders to analyze how future potential changes related to external and unpredictable factors might affect the performance of the best alternative, making possible to compute and quantify risks associated with these changes. SIMUS/IOSA does not use weights, and consequently, results are totally objective. It does not mean that it ignores that there are differences in the relative importance between criteria. It considers them, although in a different and more exact manner than existent methods, since it is based on a mathematical ratio, and

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7

Fig. 7.1 Basic initial decision matrix

The SIMUS Method

A

Alternatives B C

Criterion 1

42

18

25

Criterion 2

36

9

47

Criterion 3

4

11

15

Criterion 4

15

13

13

Criterion 5

76

49

72

Criterion 6

8

5

9

----------

-

-

-

----------

-

-

-

51

56

48

Criterion 10

inherent in the algebra of the simplex algorithm (See Sect. 7.5 and especially Appendix A.1). Each scenario is saved in a SIMUS library and is retrievable at any time, as well as deleted or modified. Results are also automatically saved in an Excel file called “Projects.” IOSA is thoroughly explained in Chap. 8. The software has a large tutorial with many examples. Sensitivity analysis in IOSA is performed using shadow prices or marginal values for each criterion, which are automatically computed. Thus, it is possible to determine quantitatively and without subjectivity how much each objective changes due to a unit variation of a criterion. SIMUS can handle complex problems incorporating aspects whose absence in present methods has been extensively discussed in previous chapters. It also can work with as many alternatives and criteria as deemed necessary, without any limit, and with any mix of quantitative and qualitative criteria.

7.2

How SIMUS Works—Case Study: Power Plant Based on Solar Radiation

It starts with the IDM as other methods, (Fig. 7.1) where alternatives are in columns while criteria are in rows. However, it has some aggregates that allow the DM to represent mathematically real-world aspects, as exemplified in Fig. 7.2. The IDM represents a set of linear inequalities as was analyzed in Sect. 6.1. This initial matrix is completed with actions (Max—Min—Equal), operators (≤. ≥, =), and resources or independent terms (RHS). Performance values are obtained as described in Sect. 6.1. Notice that the objective function does not have coefficients. Their coefficients will sequentially be the performance values of criteria.

7.2

How SIMUS Works—Case Study: Power Plant Based on Solar Radiation

121

Alternatives

Action Max

Objective function Criterion 1

A

B

C

--42

--18

--25

Operator ≤

RHS 12.36

Min

Criterion 2

36

9

47

≥

4.28

Max

Criterion 3

4

11

15

≤

452.90

Min

Criterion 4

15

13

13

≥

6.27

Min

Criterion 5

76

49

72

≥

10.09

Max

Criterion 6

8

5

9

≤

1 23.50

----------

-

-

-

------

----------

-

-

-

------

51

56

48

10.70

Min

Criterion 10

Fig. 7.2 Initial decision matrix utilized in LP Alternatives Action

A

B

C

Max

Objective function

---

---

---

Operator

RHS

Max

Criterion 1

42

18

25

≤

12.36

Min

Criterion 2

36

9

47

≥

4.28

Max

Criterion 3

4

11

15

≤

452.9 0

Min

Criterion 4

15

13

13

≥

6.27

Min

Criterion 5

76

49

72

≥

10.09

Max

Criterion 6

8

5

9

≤

1 23.50

----------

-

-

-

------

----------

-

-

-

------

51

56

48

Min

Criterion 10

≥

10.70

Fig. 7.3 The first criterion becomes objective function, and removed from the IDM

Figure 7.3 shows that the first criterion becomes an objective function, and then its performance values are removed from the IDM and attached to the objective function. The objective function is then maximized, according to the action of the first criterion. The software is run and an optimal solution is attained (if it exists) for that objective. After that, the coefficients are returned as performance vales to the IDM (Fig. 7.4). The second criterion values are now removed and used for coefficients of the objective function. The objective function is then minimized, according to the action of the second criterion (Fig. 7.5).

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7

The SIMUS Method

Alternatives Action

A

B

C

18 ---

25 ---

Operator ≤

Max Max

Objective function Criterion 1

42 ---

RHS 12.36

Min

Criterion 2

36

9

47

≥

4.28

Max

Criterion 3

4

11

15

≤

452.90

Min

Criterion 4

15

13

13

≥

6.27

Min

Criterion 5

76

49

72

≥

10.09

Max

Criterion 6 ----------

8 -

5 -

9 -

≤

123.50 ------

Min

Criterion 10

≥

10.70

----------

-

-

-

51

56

48

------

Fig. 7.4 Coefficients from the objective function are returned to the IDM as performance values Alternatives Action

A

B

C

--18

--25

Operator ≤

RHS 12.36

Min Max

Objective function Criterion 1

--42

Min

Criterion 2

36

9

47

≥

4.28

Max

Criterion 3

4

11

15

≤

452.90

Min

Criterion 4

15

13

13

≥

6.27

Min

Criterion 5

76

49

72

≥

10.09

Max

Criterion 6

8

5

9

----------

-

-

-

---------Min

Criterion 10

-

-

-

51

56

48

----------≤

123.50

Fig. 7.5 The second criterion becomes objective function, and is removed from the IDM

The procedure continues until all criteria performance values are used as coefficients of the objective function. After each solution is obtained, the alternative scores are stored into a new matrix with alternatives in columns and objectives “Z” in rows (Table 7.1). Thus, in the first iteration, when using the first criterion as objective function, its result is score 15.44 for alternative A and no scores for alternatives B and C. In the second iteration the result shows a score of 0.48 for alternative B and no scores for A and C. In the third iteration, the scores are 0.69 for B and 0.07 for C with no score for A and so on.

7.2

How SIMUS Works—Case Study: Power Plant Based on Solar Radiation

Table 7.1 Efficient results matrix (ERM) Objective Z 1 Objective Z 2 Objective Z 3 Objective Z 4 Objective Z 5 Objective Z 6 Objective Z 10 Table 7.2 Normalized efficient results matrix (ERM) Objective Z 1 Objective Z 2 Objective Z 3 Objective Z 4 Objective Z 5 Objective Z 6 Objective Z 10

Alternatives A 15.44

B 0.48 0.69 0.14 0.48

B 1 0.9 1 1

0.04

C

0.07

0.49 0.46

0.02

Alternatives A 1

123

C

0.1

1 0.95

When the process described above is complete, this new matrix, called ERM (efficient results matrix), becomes a Pareto efficient matrix since all results are optimum. It is possible, and very common, that there are scores for one, two, or three alternatives to the same objective function, and as well, there could be that there are no scores for any of them. In this case, it means that there is no optimal solution for that objective function. The geometrical interpretation, see Fig. 6.3 in Chap. 6, is that there is no tangent point between the polygon and the objective function. Possibly, there is a solution, but it is not efficient and, for that reason, there are no scores. As mentioned, using an objective function does not guarantee that all the alternatives will get a score. It simply means that if the alternative left out is introduced into the solution, it will produce a decrease of the benefit or an increase in the cost, or whatever the objective function might be. Once the ERM is complete, it must be normalized (see Table 7.2). Notice that this matrix does not produce changes in the scores, just normalize them. 1. For Z1 there is only one solution and that is alternative A with a score of 1. 2. For Z10 there are scores of 0.04 for alternative A and 0.96 for alternative C. 3. There could be objectives with scores for three alternatives, although there is none in this example. 4. There are objectives that do not give any value, and thus do not select any alternative, meaning that there is no optimal solution for that objective.

124 Table 7.3 Solution and ranking using the first procedure

7 Alternatives A 1

Objective Z1 Objective Z2 Objective Z3 Objective Z4 Objective Z5 Objective Z6 Objective Z10 0.05 Sum of column (SC) Participation factor (PF) Normalized part. factor (NPF) Final result (SC × NPF)

The SIMUS Method

B

C

1 1 0.67 1

1.05 2 0.29 0.30

0.33

3.67 4 0.57 2.10

1 0.95 2.28 3 0.43 0.98

5. Not all alternatives participate in an equal number of criteria. Alternative A contributes to objectives Z1 and Z10. Alternative B contributes to objectives Z2, Z3, Z4, and Z5, and alternative C contributes to objectives Z3, Z6, and Z10. The normalized efficient results matrix is the starting point to compute scores for alternatives; SIMUS follows two different procedures: a) Analyzing each column, and producing a first solution b) Analyzing each row and producing a second solution First solution Obtained by adding up all scores in each column. This produces a total value (SC) (Table 7.3). ERM ranking: alternative B > alternative C > alternative A. However, an alternative is more valuable considering how many objectives are satisfied. The ideal would be to fulfill all of them. For this reason, for each alternative the number of objectives that it satisfies is computed, and this number is called “participation factor” (PF), which is normalized, and then generating a “normalized participation factor” (NPF), which performs as a “weight” for each alternative. Finally, this weight is multiplied for the sum (SC × NPF), and thus, the result is a weighted sum, giving the score for each alternative. Second Solution This second procedure examines for each row which alternative is dominant and finds its differences between scores with the other alternatives. For instance, it can be seen from Table 7.3 that in the first row alternative A has only a value for objective Z1. Consequently, it outranks alternatives B and C by 10 = 1, each. For objective Z2 alternative B has the only value. Consequently, it outranks alternatives A and C by 1-0 = 1, each. For objective Z3 alternative B has the only value. Consequently, it outranks alternatives A and C by 1-0 = 1, each.

7.2

How SIMUS Works—Case Study: Power Plant Based on Solar Radiation

125

Table 7.4 Project dominance matrix (PDM)

Subordinate alternatives (in columns) Dominant alternatives (in rows)

A

A B C Sum of columns (Subordinate alternatives)

B

C

1+0.05

1

1+1+0.67+ 1=3.67 0.33+1+0. 90=2.23

1+0.95=1. 95

3.67 + 2.23 = 5.90

1.05 + 1.95 = 2.95

PDM Ranking:

1+1+0.34+ 1=3.34 `

Sum of rows (Dominant alternatives) 1.05+ 1 =2.05 3.67 + 3.34 = 7.01 2.23 + 1.95 = 4.18

Differences Net (Dominant Subordinate) values 2.05 – 5.90

–3.85

7.01 – 2.95

4.06

4.18 –4.34

–0.16

1 +3.34 = 4.34

Alternative B > Alternative C > Alternative A

For objective Z4 alternative B outranks by 0.67-0 = 0.67 alternative A, and by 0.67-0.33 = 0.34 alternative C. Also, alternative C outranks alternative A by 0.330 = 0.33. For objective Z5 alternative B has the only value. Consequently, it outranks alternatives A and C by 1-0 = 1, each. For objective Z6 alternative C outranks by 1-0 = 1 alternatives A and B. For objective Z10 alternative C outranks by 0.95-0.05 = 0.90 alternative A, and by 0.95-0 = 0.95 alternative B, while alternative A outranks alternative B by 0.05 + 0 = 0.05. The result of these differences is stored in a new matrix called “project dominant matrix” (PDM) which is a square matrix formed by alternatives in columns and in rows (Table 7.4). These differences are added up for each alternative. Summation of row values gives the total outranking value for each dominant alternative, while summation of columns values gives the total value for each subordinate alternative. The difference between dominant and subordinated values for the same alternative provides its score (as seen in the last column of the PDM matrix), and then it forms the ranking. The PDM shows that this ranking is alternative B > alternative C > alternative A. Subordinate alternatives (in columns). PDM ranking: alternative B > alternative C > alternative A. Observe that both procedures assign different scores to alternatives, which is logical, because they come from different methods. However, starting from the same

126

7

The SIMUS Method

data (the normalized ERM matrix), and following two different procedures (weighted sum in the first, and outranking in the second), the ranking is identical. This is a distinctive feature of SIMUS. This result shows that by neither using weights nor subjectivities, two different procedures reach the same result.

7.2.1

Normalization by SIMUS

Now a question arises: Which normalization system to use? The result should always be the same regardless of the normalization method used. For instance, look at this example in Table 7.5 depicting a MCDM problem with 7 schemes or alternatives, and 15 criteria. SIMUS works with as many criteria performing as objective function, as wished (in this case the DM chose only the first 8 criteria, since he considered that the other 7 criteria were not relevant to perform as objectives). However, for computation purposes the total 15 are used. In the present case, which is considered as reference, normalization is done by dividing each performance value by the sum of all performance values in a row. Figure 7.6 shows the result. Using the same procedure but applying other normalization methods, SIMUS produced the same ranking as shown in Table 7.6. As can be seen, rankings and even alternatives scores coincide in the four normalization methods, at least in SIMUS.

Table 7.5 Initial matrix with seven schemes and 15 criteria C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15

Scheme 1 0.15 0.20 0.10 0.36 0.16 0.20 0.19 0.42 0.20 0.25 0.34 0.20 0.20 0.20 1.00

Scheme 2 0.19 0.22 0.11 0.25 0.17 0.24 0.19 0.03 0.20 0.12 0.33 0.11 0.16 0.22 1.00

Scheme 3 0.15 0.25 0.08 0.40 0.16 0.29 0.19 0.16 0.21 0.24 0.33 0.18 0.31 0.22 1.00

Scheme 4 0.18 0.20 0.09 0.21 0.19 0.22 0.19 0.17 0.20 0.23 0.34 0.20 0.22 0.22 1.00

Scheme 5 0.21 0.16 0.07 0.22 0.50 0.32 0.19 0.42 0.26 0.23 0.16 0.43 0.10 0.23 1.00

Scheme 6 0.18 0.16 0.11 0.19 0.19 0.30 0.19 0.16 0.19 0.23 0.34 0.17 0.08 0.20 1.00

Scheme 7 0.19 0.20 0.09 0.18 0.20 0.35 0.20 0.42 0.19 0.14 0.17 0.30 0.19 0.21 1.00

7.3

SIMUS Application Example: Case Study—Power Plant Based on Solar Radiation

127

Fig. 7.6 Capture of SIMUS final screen showing ERM and PDM scores and rankings Table 7.6 Scores and rankings from SIMUS using different normalization methods Normalization method used Sum of values in a row Largest value in a row Euclidean formula Max / min

7.3

Scheme score Scheme ranking Scheme score Scheme ranking Scheme score Scheme ranking Scheme score Scheme ranking

0.26 3 0.26 3 0.26 3 0.26 3

0 1 0 1 0 1 0 1

0.60 7 0.60 7 060 7 0.60 7

0.02 4 0.02 4 0.02 4 0.02 4

0.02 5 0.02 5 0.02 5 0.02 5

0.02 6 0.02 6 0.02 6 0.02 6

0 2 0 2 0 2 0 2

SIMUS Application Example: Case Study—Power Plant Based on Solar Radiation

The same problem used for illustration in Sect. 7.2 will be solved by SIMUS in order to exemplify how the software works. Only the elemental functions are commented; the software has a comprehensive tutorial with 12 real-world examples fully developed. 1. The DM opens SIMUS first screen as shown in Fig. 7.7 and starts by identifying a project by its name. In this example it is “Power” written in the “New project name” box.

128

7

The SIMUS Method

Fig. 7.7 First screen in SIMUS introducing basic data for power example

Fig. 7.8 Main screen with data loaded—observe the RHS values

2. He selects the language. There are two options: English and Spanish; in this case, he selected “English.” If he instead chooses “Spanish,” then he must press the “ES” key. 3. The DM fills boxes “Number of criteria” and “Number of alternatives” with the respective values. In this case, data was imported from Excel and electronically uploaded in SIMUS, by pressing the “Import/Export” key. The DM presses the “Generate” key (if he inputted the number of criteria and alternatives). If he imported data the IDM appears automatically. Figure 7.8 shows the IDM in the main screen. Data values from the IDM are seen at left, loaded electronically from Excel, or manually. The DM establishes the action for each criterion, indicates the corresponding operator, and finally writes in the “RHS values” column the figures for resources, limits, or thresholds. If the DM wants to use criteria weights, he can input them in the “Weight” column. If the DM wants to also work with alternatives weights, he can input them in the “Project weight” row. Both weights may be inputted independently or jointly. Right up under the columns “Project” the DM can input the alternatives precedence (not used in this example), for instance, 3 > 2 (alternative 3 precedes

7.3

SIMUS Application Example: Case Study—Power Plant Based on Solar Radiation

129

Fig. 7.9 Selecting a normalization method

Fig. 7.10 SIMUS requests information regarding how the DM wants to proceed

alternative 2), and then mark it. The DM can choose to work with all criteria as objective or only with some of them; his selection is inputted in the “Choose targets” box (target and criteria have the same meaning). By default, SIMUS considers all criteria, that is, 7 in this example. Down, in the “Results row” the DM may specify the format in which he wishes the results appear, that is, in decimal, integer, or binary construct. 4. By striking the “Save matrix” key this project is saved in the library. By pressing the “Normalize & Validate” key, the computation process is ready to initiate and the following screen appears (Figure 7.9). 5. Here the DM is requested to make a selection about the normalization system wanted, out of four different options. In this example, he chose the Euclidean formula. Pressing one of the options the next screen appears (Fig. 7.10). The result, considering the first objective, can be seen in the “Function per criteria” column, in this case showing that objective Z1 = 12.11 and that according to

130

7

The SIMUS Method

Fig. 7.11 Final screen with ERM solution and shadow prices for each objective

this objective, only alternative A is chosen with a score of 15.44, as shown in the ERM matrix. 6. The DM is asked if he wants to work step by step (which is the way used in groups decision-making), or if he wishes the process to continue up to the end without interruption. In this latter case, he pressed “Automatic”; the process continues until it reaches the final screen (Fig. 7.11). This screen has been partitioned in ERM (Fig. 7.11) and PDM (7.12) for better reading, although it is unique in the computer screen. The whole screen shows the results from the primal problem at left (in blue), that is, scores and ranking, and at the same time the results from the dual problem, i.e., the shadow prices at right (in green). The screen is divided into three parts. Blue, Green, and Brown. Blue refers to the primal solution from one method. Green refers to the dual solution. Brown refers to the primal solution from the other method. Figure 7.11 portrays components for the first solution, i.e.: • The ERM matrix or Pareto efficient matrix where all scores are optimal (in blue) • The normalized ERM matrix (in blue) • The final scores for all alternatives (in the solid blue row)

7.4

Special Circumstances

131

Fig. 7.12 Final screen with PDM solution and allowable intervals for criteria variation

• The ranking (in red) • The shadow prices for each objective (in green) Figure 7.12 portrays components for the second solution, i.e.: • • • •

The PDF matrix (in brown). The final scores for all alternatives (in the solid brown column). The ranking (in red). The table depicting the variability ranges for RHS, for both increments and decrements (in green). The larger the interval, the more the robustness of a criterion.

As can be seen, scores are different using the two different procedures, which is expected, since mathematical operations are different—sum and multiplication in ERM and sum and differences in PDM—however, the rankings are identical.

7.4

Special Circumstances

In many cases, there are special circumstances in the modelling and in the results. They are:

7.4.1

Ties in Scores

Very often two or more alternative scores coincide, or are very close. In that situation, what should one do?

132

7

The SIMUS Method

SIMUS offers several options to deal with this problem: 1. Run the software again but instruct the method to produce a result in binary numbers that are “1” “go” or “0” “not go.” Since in this case the method works with an integer programming algorithm, it follows a more rigorous procedure (looking only for integers scores) than when working with linear programming, and then identifying the best alternative. 2. As seen, for each problem SIMUS generates different scores according to ERM or PDM procedures. It is possible to use these differences to analyze the discrimination between the values of two or more alternatives in conflict, by checking the different scores. If, for instance, the ERM scores for the first two selected alternatives are A = 2.39 and D = 2.38, by checking the scores in PDM for these two alternatives may show different scores say A = 5.8 and D = 7.91. It can be seen that alternative D is better positioned than alternative A, and then D should be selected. 3. Examining the shadow prices and analyzing the corresponding values for the two alternatives in conflict, the DM will probably find that they are different. Then the DM can check the corresponding criteria for those shadow prices and examine their influence in each one of the alternatives in conflict. If say, criterion for alternative D affects the result in a lesser degree than criterion for alternative A, then from the point of view of sensitivity of the solution to their variation, D is a better alternative.

7.4.2

Need to Use Formulae for Performance Factors

Case Study: Housing Development Assume the following case: A City Hall is calling for tender to build utility networks in different areas, called Green Valley, Altavista, and Saint Paul. Each bidder may submit proposals for building an individual network or for all of them. The DM is interested in networks with the minimum cost of construction, and for this reason, he needs to make a decision about what networks are most profitable. Data is given by City Hall. The three areas already have water supply from a nearby water treatment plant; however, it has reached full capacity, which creates water shortages especially in summertime. Another water treatment plant is under construction to serve several areas of the city, including these three districts. Regarding sewage none of the three areas is linked to the city sewerage network and utilize cesspools, a system the city wants to eliminate. Concerning gas service, only one of the districts is connected to the city gas network. Criteria to evaluate these projects are: • Yearly water volume to supply • Yearly sewage volume to be generated • Yearly gas volume to supply

7.4

Special Circumstances B 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

C

133 D

E

F

Green Valley Water Sewage Gas Population 1200 Density 5.24 Number of dwellings 229 Dwellings w/o services 26 Avg. water consumption 160 Avg. sewage production Avg. gas consumption Water cost 0.01 Sewerage cost Gas cost Distance to trunks 2.5 Distance to trunks Distance to trunks

229 26

229 26

G

H

I

J

Saint Paul Altavista Water Sewage Water Sewage Gas 908 356 255 35 160

216

255 35

1450 7.44 195 100 160

216

195 100 216

0.02 0.01 0.02

0.01 0.02

0.02

0.16 1.6 4

0.8 5.2

3.5

K

4.2

Units People People/dwelling 195 Quantity 100 (%) L/person L/person 0.02 m³/person $/L $/L 0.16 $/m³ Km Km 4 Km

Fig. 7.13 Basic data which is subject to change

• • • • • • • • • •

Water payment Sewerage payment Gas payment Water infrastructure cost Sewerage infrastructure cost Gas infrastructure cost Water connection cost to dwellings Sewerage connection cost to dwellings Gas connection cost to dwellings Total investment in each project Data needed:

• • • • • • • • • • •

Population of each area. Density per dwelling. Number of dwellings in each area. Dwellings without services. Average consumption of drinking water per person. Average production of sewage per person. Average gas consumption per person. Water, sewerage, and gas costs. Distance from the closest trunks for water, sewage, and gas to each area. There is a budget for this project which cannot be surpassed. Sewerage works will only be executed when piping for water supply is complete.

This information is depicted in Fig. 7.13; however, normally data is changed frequently during the analysis of work to be done (before the decision-making process), such as new estimates, new market studies, and construction materials price.

134

7 Cell D29 = $D$8*(D9/100) *$D$7*D10*365

D

Water 29 30 31 32 33 34 35 36 37 38 39 40 41

E

F

Green Valley Sewage Gas

J

K

Water

Saint Paul Sewage

Gas

84680000 22619415

182208

185595 491962

114318000 10585.000 846800

452388

2286360

364.42

Gas payment

48000

1694 36200

54600

Sewage infrastructure cost

23000 50400

58000

33800

Gas infrastructure cost

52212

64000 63240

75112

Sewage connecting cost to dwellings Gas connecting cost to dwellings Total cost

I

2277.600

Sewage payment

Water connecting cost to dwellings

H

Altavista Water Sewage

24598080

Yearly gas volume to supply

Water infrastructure cost

(1) Cell O29 = Sum (D29:K29) (2)

18559520

Yearly water volume to supply Yearly sewage volume to generate

Water payment

G

100212

75112

48360 81345

661810 695610

99440

The SIMUS Method

81345

64350 71360

122350

563550 627550

L

M

LHS values 33,798,770 161,535,495 118.03 1,214,603 3,230,710 19 240,111 284,598 714 270,498 385,682 6,284 1,872,979

N

O

RHS values

≤ ≤ 161,535,495 ≤ 12,863 ≥ 1,214,603 ≥ 3,230,710 ≥ 2,058 ≥ 107,200 ≥ 163,000 ≥ 97,800 ≥ 163,812 ≥ 220,807 ≤ 1,225,360 = 1,872,979

Fig. 7.14 Introducing algebraic expressions in the initial matrix

The sources of this data are: • Population data for of each area, from City Hall data bank • Density, as well as number of existing dwellings without service, from City Hall • Average water consumption from international organizations such as the World Health Organization (WHO) • Sewage production as well as gas consumption from statistics • Prices for these services from the respective suppliers • Distances for installing piping that connects trunks with areas, from City Hall’s Engineering Department. These are the values the user can work with for each utility. For instance, it is essential to know the volume of water needed in each area, for the entrepreneur’s engineering department to compute the characteristics of piping to be installed. This is illustrated for Green Valley Area taking data from Fig. 7.13 as follows: • • • • •

Number of dwellings, from cell D8 = 229 units Percentage of dwellings without this service: from cell D9 = 26% Average density, from cell D7 = 5.24 people/dwelling Average water consumption, from cell D10 = 160 l/person-day Number of days in a year: 365

Then, the total annual water consumption calculation for Green Valley is performed using formula (1) that must be written in cell D29, as shown in Fig. 7.14. In this case, the total volume of water to be supplied to the three districts is equivalent to the sum of their demands. To perform this operation the user inputs formula (2)—an Excel function—on cell O29 of vector RHS. When the D29 cell formula is solved, result is: 229 × 0.26 × 5.24 × 160 × 365 = 18,220,800 m3/year. When the O29 cell formula is solved, result is: 18,220,800 + 18,559,520 + 84,680,000 = 121,460,320. See Fig. 7.15.

7.4

Special Circumstances

135 D

Water 29 30 31 32 33 34 35 36 37 38 39 40 41 42

E

F

Green Valley Sewage Gas

G

H

Altavista Water Sewage

I

J

K

Water

Saint Paul Sewage

Gas

Yearly water volume to supply 18220800 18559520 84680000 Yearly sewage volume to generate 24598080 22619415 114318000 Yearly gas volume to supply 2278 10585 Water payment 182208 185595 846800 Sewage payment 491962 452388 2286360 Gas payment 364 1694 Water infrastructure cost 48000 36200 23000 Sewage infrastructure cost 54600 50400 58000 Gas infrastructure cost 33800 64000 Water connecting cost to dwellings 52212 63240 48360 Sewage connecting cost to dwellings 75112 81345 64350 Gas connecting cost to dwellings 661810 563550 . Total cost 100212 75112 695610 99440 81345 71360 122350 627550

L

M

N

O

LHS 121460320 161535495 118 1214603 3230710 19 240111 284598 714 270498 385682 6284 1872979

≤ ≤ ≤ ≥ ≥ ≥ ≥ ≥ ≥ ≥ ≥ ≤ =

RHS 121460320 161535495 12863 1214603 3230710 2058 107200 163000 97800 163812 220807 1225360 1872979

Fig. 7.15 Detecting errors

The advantage in using formulas is that changes may be done in the original data of Fig. 7.13 and be automatically registered in the IDM in Fig. 7.14. Running the software, it allows for easily examining how the change in some performance values modifies the results. It could be the case for instance, if for whatever reasons, the entrepreneur decides to decrease the population that will live in some location while increasing in another.

7.4.3

Errors in the Decision Matrix

Performing the above case using SIMUS indicates (albeit not shown) that no feasible solution could be found. Naturally, the entrepreneur would be interested in finding why it is not feasible; he discovers the reason by comparing vis-à-vis on each criterion quantities in columns LHS (which are the values reached by each criterion after the computation) and RHS (which are the values of resources available). By doing this examination, the reason is clear: there are two criteria that do not comply with the respective operator. This is indicated in Fig. 7.15 in two dashed boxes. It can be clearly seen that criterion “Gas infrastructure cost” in row 37 in the LHS column has a value (714) that is not greater than the corresponding value in column RHS (97,800) as required by the “≥” operator, and therefore, this restriction is not satisfied leading to a non-feasible solution. Similar happens for row 34. For “Gas payment.” However, the DM also notices that something is amiss, in the sense that the symbol for row 40 for criterion “Gas connecting cost to dwellings” is “≤” when it should be “≥,” as also used for water and sewage, since what the DM wants is to be sure that the cost will be at a minimum 1,225,360 Euros, because this is the budget established by City Hall. In this case, results are meaningless because there is no feasible solution. Data is corrected and SIMUS ran again. Table 7.7 shows the final results in SIMUS screen.

136

7

The SIMUS Method

Table 7.7 Capture of SIMUS final screen showing result after correction Green Valley Water Sewage 1.21 0 2 0 0.15 0

Gas 0.11 1 0.08

Altavista Water Sewage 0.79 0 1 0 0.08 0

Up to:

Up to:

Up to:

100 0.62 0.16 34

200 0.44 0.17 34

400 0.28 0.3 38

Saint Paul Water Sewage 1 2 1 2 0.08 0.15

Gas Sum of column (SC) 1.89 Participation factor (PF) 2 Normalized participa0.15 tion factor (NPF) Final result (SC × NPF) 0.19 0 0.01 0.06 0 0.08 0.31 0.29 ERM ranking sewage (Saint Paul)—Gas (Saint Paul)—Water (Green Valley)—Water (Saint Paul)—Water (Altavista)—Gas (Green Valley)

Units Unit cost (A) Unit benefit (A) Labor (A)

Unit benefit (A)

Unit cost (A)

Labor (A)

0.35

0.7 0.62

0.6

0.3

0.5

0.44

0.4 0.3

Unit cost (A)

0.25

Polynomial

0.15

0.3

0.2 0.28

0.16

0.17

0.1

0.2

y = 0.01x2 - 0.21x + 0.82

0.1

0 100

200

400

Unit benefit (A) Polynomial y = 0.06x2 - 0.17x + 0.27

0.05 0 100

200

400

39 38 37 36 35 34 33 32 31

38

Labor (A) Polynomial 34

34 y = 2x2 - 6x + 38

100

200

400

Fig. 7.16 Linearization of quadratic criteria for product A

It can be seen that networks in the Saint Paul area are the best choice for the entrepreneur with the execution of the three networks. It is followed by the Green Valley area; however, the sewerage network is not chosen. The same with sewerage in the Alta Vista area.

7.4.4

Dealing with Non-linear Criteria

Performance values are not always linear. For instance, cost per unit is not uniform normally, since it is not the same to purchase 20 units than 100, 300, or 1000; i.e., the unit cost is a function of quantities purchased. In these cases, selecting one product over other depends on the final quantities ordered. Consequently, this aspect needs to be considered since there is no linear relationship but a non-linear. As an example, assume two products A and B with three different prices each quantities-wise. They are subject to three criteria that are unit cost, benefit, and labor. The three of them are non-linear. Figure 7.16 illustrates this case for product A and Fig. 7.17 for product B.

7.4

Special Circumstances

137

Units Unit cost (B) Unit benefit (B) Labor (B)

Up to: 100 16.8 8.6 29

Unit cost (B) 20 16.8

16.47 12.61

10 7.28 5

Unit cost (B) Polynomial

y = –0.57x2 – 2.48x + 19.85

0 100

150

Up to: 350 7.28 16.47 31

Unit benefit (B)

20 15

Up to: 150 12.61 10.59 29

350

15 10

10.59 8.6

5 0 100

150

Labor (B)

31.5 31 30.5 Unit 30 benefit (B) 29.5 Polynomial 29 28.5 y = 1.945x2 – 28 3.845x + 10.5 27.5

31 Labor (B) 29

y = 1x2 – 3x + 31 100

350

Polynomial

29

150

350

Fig. 7.17 Linearization of quadratic criteria for product B Efficient Results Matrix (ERM) Normalized A1 A2 A3 B1 B2 Z1-Unit cost Z2-Unit benefit Z3-Labor Sum of Column (SC) Participation Factor (PF) Norm. Participation Factor (NPF) Final Result (SC x NPF)

B3

1.00 1.00 1 0.33

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0.33

0

0

0

0

0

ERM Ranking

A1 - A2 - A3 - B1 - B2 - B3

Fig. 7.18 Capture of SIMUS final screen for ERM

Observe that each product has different unit prices in accordance with quantities purchased and that these are different for both products. The relation between prices and quantities is represented by curves instead of straight lines. The function equation is shown in each case, and it is quadratic. Each curve can be replaced by many straight lines (in this case three), each one representing a price according to a quantity. Consequently, the whole scenario can be represented in the IDM subject to the three criteria. The IDM will be then formed by six columns, three for product A and three for product B, and subject to three criteria. Solved by SIMUS, the results from ERM and PDM are shown in Fig. 7.18. Also, verify that the same solution has been reached for both SIMUS procedures. The large differences in scores between the best option and the others are also reflected in the second procedure (brown solid column). Product A up to 100 units appeared to be the best option, since obviously the cost, benefit, and labor advantages for smaller quantities are not convenient.

138

7.5

7

The SIMUS Method

Is SIMUS Affected by Rank Reversal?

As explained in Sect. 2.3, many methods suffer RR and the purpose of this section is to show that SIMUS is not affected by it. The reason for this is the rigid mathematical structure followed in the simplex algorithm which is repeatedly used for each new objective. As explained in Sect. 6.3 the simplex algorithm starts at the origin of coordinates and iteratively advances vertex by vertex always improving the value of the functional “Z” which is expressed as: Z = nj = 1 ðαjxj βjyjÞ Maximize=Minimize where: x and y: Variables or projects α and β: Solution or scores for projects j: Number of projects from 1 to “n” The simplex works using a tableau that has all the data of the problem and ordered in a certain manner. In each iteration, the algorithm selects the best project to enter in a new solution by comparing the contribution of all projects (Cj) in improving the solution from the last iteration (Zj), that is, (Cj-Zj), and obviously chooses that with the greatest difference. Once this selection is made, the simplex determines the project that must be eliminated from the solution using a minimum ratio between (Bi / aij) considering the column of the selected project. Consequently, if in an existing problem a new project vector, worse than all the others, is added to the system, it will be never considered. For the same token, if a new project is added and is better than another, it will be selected by the same algebraic mechanism. Any book on LP explains this mechanism which is in reality straightforward and simple, and for this reason, it is not explained here. For a good explanation of the simplex tableau and the mathematical procedure followed by simplex, see in Appendix references Kothari (2009), page 83.

7.6

Testing SIMUS in Rank Reversal

In order to examine SIMUS performance, three scenarios of different complexity are analyzed and tested with different modifications of the original projects’ arrangement. For each scenario, the test starts by determining a ranking which becomes the “Original.” All modifications are performed, and their rankings compared with this original. The following situations are considered: a) b) c) d)

Adding a new project vector worse than the optimal or any suboptimal Adding a new project vector better than the optimal or any suboptimal Deleting a project vector which is worse than the optimal or any suboptimal Pair-wise comparison of projects by decomposing the original scenario, by one pair at a time, and subject to the same constraints, and verifying transitivity

7.6

Testing SIMUS in Rank Reversal

139

e) Comparing the ranking between each pair with the original f) Adding a new project vector with very close values or even identical to another existent project vector, without deleting the original

7.6.1

Case 1: Investment in Renewable Sources of Energy

The case refers to a European entrepreneur contemplating the construction of a solar park (or solar power plant) by using two different technologies, either independently or working in parallel. These technologies are (a) installing solar dishes to take advantage of heat generated by the sun using thermo-electrical equipment and (b) using photovoltaic plates to exploit the photoelectric principle, that is, the direct transformation of solar energy into electricity. Between both technologies, there are four different technical configurations. Table 7.8 shows the IDM with the four criteria, namely internal rate of return (IRR), net present value (NPV), payback period (PBP), and project horizon. SIMUS is applied, following the changes commented above and repeating each with different combinations, totaling 26 tests. Table 7.9 condenses the results for this scenario. The table is organized with the different modifications in rows. The first row indicates the original ranking obtained by SIMUS, that is, 4-1-3, by its two solutions. The first column codes modifications by IDs. The second column identifies modifications. The third column indicates the ranking obtained applying the modification on its left, and reveals the result from the ERM matrix when this matrix is examined following the first procedure in SIMUS. The fourth column indicates the ranking obtained applying the modification on its left, and reveals the result of the PDM matrix when the ERM is examined following the second procedure in SIMUS. Both ERM and PDM rankings must coincide or show minimal differences. The fifth column indicates if the original ranking is preserved. The three last columns are based on Triantaphyllou’s (2001) proposal of three principles to measure robustness, in the sense that in each situation and whatever it may be, its ranking must coincide with the original. These three principles are: Table 7.8 Initial data for Case 1—Investment in renewable sources of energy

Criteria IRR (%) NPV (euros) PBP (years) Project horizon (years)

Projects P1 P2 P3 Projects contribution to criteria 5.7 5.9 6 6200 6050 4800 4.2 4.2 2.8 5.7 5.9 6

P4 6.1 3800 3.1 6.01

Action MAX MAX MIN MAX

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Table 7.9 Test for Case 1—Investment in renewable sources of energy

I.D. [ NB] Original (4 alternatives) 4-1-3 Deleting a single project [NB] [D] DELETING P1 4-3-2 [AA] DELETING P2 1/3/4 [R] DELETING P3 4-1-2 [Q] DELETING P4 1-3 Deleting pairs of projects [Z] DELETING P1 and P2 3-4 [Y] DELETING P1 and P3 4-2 [H] DELETING P1 and P4 3-2 [P] DELETING P2 and P3 4-1 [AB] DELETING P2 and P4 1-3 [I] DELETING P3 and P4 1-2 Comparing pairs of projects [W] COMPARING P1 and P2 1-2 [G] COMPARING P1 and P3 1-3 [E] COMPARING P1 and P4 4-1 [X] COMPARING P2 and P3 3-2 [N] COMPARING P2 and P4 4-2 [V] COMPARING P3 and P4 3-4 Equal values for a pair of projects [AC] P3 EQUAL P2 4-1 [AD] P4 EQUAL P3 1-3 [AE] P1 EQUAL P3 4-1-2 Adding a new project [A] Adding P5P4 5-3-4 [A] Comparing pairs of projects [L] Comparing P1 and P5 1-5 [M] Comparing P2 and P5 2-5 [J] Comparing P4 and P5 4-5 [K] P5>P4 Comparing P3 and P5 5-3-1

4-1-3

Determining robustness using Triantaphyllou's three criteria No change Transitivity Transitivity in best between between Original best smaller smaller ranking project problems problems & preservation composed ORIGINAL

4-2-3 OK 1/3/4 Not '>' but ' ' OK 4-1-2 instead OK 1-3 Changes since P4 was deleted 3-4 4-2 3-2 4-1 1-3 1/2

Transitivity (1) between original values

1/2 1-3 4-1 3-2 4-2 3-4

OK Transitivity (2) Transitivity (3)

4-1 1-3 4-1-2

OK

OK OK OK OK OK

OK

OK OK OK OK OK

Transitivity (4) Comment (2) Comment (3) OK Transitivity (5)

OK OK OK OK OK

OK OK OK

OK OK

4-1-3 OK OK 5-3-4 Changes since P5 > P4 1-5 2-5 4-5 5-3-1

Comment (1)

OK

OK

OK OK

OK

Transitivity (1) If P3>P4 [Z],and P4>P2[Y] then P3>P2 as in the small problem [H] and in the original (2) If P1>P3 [G] and P3>P2 [X], then P1>P2 as in both the small problem [W] and the original (3) If P4>P1 [E] and P1>P2 [W], then P4>P2 as in both the small problem [N] and in the original I(4) If P3>P4[V] and P4>P2 [N], then P3>P2 in both the small problem [X] and in the original (5) If P4>P1[AC] and P1>P3 [AD], then P4>P3 in both the small problem [AE] and in the original Comments ( 1) It is OK because it refers when P5>P4 (2) Original does not change as we use a non-optimal alternative and replaced it by a worse one

7.6

Testing SIMUS in Rank Reversal

141

1-. Whatever the modifications, the best project must be kept. It is considered in this book that this principle must be accepted after studying the modification, because when inputting a new alternative which is better than the best, it is expected that it will replace the original best. 2-. The initial ranking must be decomposed into pairs of alternatives, or small projects that is, without the others, and subject to the same criteria. Their rankings must preserve transitivity between pairs. That is if A≽B and B≽C, then A≽C. 3-. When all small projects are grouped, and their rankings compared with the initial or composed ranking, there must be coincidence in precedence. That is, if A≽B when analyzing only a pair of projects, this ranking must agree with the same pair in the composed ranking. As seen in n7.9 these three principles are preserved in all cases. For instance, if deleting P3 [ID “R”], the resulting ranking is 4-1-2 (a dash sign (-) is used instead of “≽,” to simplify writing, but the meaning is the same). If this ranking is compared to the original ranking, which is 4-1-3 it can be seen that both are equivalent. It is true that in comparing the pair, Project 2 popped up instead of Project 3; however, the ranking is preserved since Project 3 was deleted as required. In addition, even if the comparison were not exactly in the same order, what is important is the preference, not its position. That is, if an original ranking was for instance 3-5-7-9-4-6 and the requirement is to delete Project 5, and if in the decomposition the result is something like 3-4-6, it can be seen that preference is preserved even when 5 is not present. However, if a ranking was for instance 3-9-7, it does not preserve the original because in the original ranking, Project 9 does not precede Project 7. Observe that in this “R” ID, the preference is maintained since 4 precedes 1 as in the original. Analysis of this Case The following aspects have been considered: 1) 2) 3) 4) 5)

Adding a new worse activity ID [A] to the original problem Adding a new better activity ID [C] to an original problem with four activities Deleting a single project from the original, IDs [D-AA-R-Q] Deleting pairs of projects from the original, IDs [Z-Y-H-P-AB-I] Comparing pairs of projects from original and verifying transitivity, IDs [W-G-EX-N-V] 6) Comparing pairs of projects when new project is added, IDs [L-M-J-K] 7) Giving the same values to pairs of projects from the original, IDs [AC-AD-AE]. It can be seen that in all cases, the original ranking has been preserved, except, when a new project, better than the best, is inputted, as in ID [C]. • Original ranking preservation. Ranking is preserved in all cases, except in ID [X], [W] which is reversed when comparing small problems because of the necessity to keep transitivity as explained in (4). • Change in the best project. The best project (P4) is being kept as seen in IDs [D], [R]. In ID [AA], there is no clear preference by P4 because there is a tie with

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projects P1 and P3. In ID [Q] there is no preference for P4 because it has been deleted; however, notice how the ranking is kept. When deleting pairs of projects, the P4 preference is maintained in ID [Y] but not in ID [H] because P4 is deleted. P4 preference is not maintained when deleting simultaneously P1 and P2 ID [Z]; however, transitivity is kept as described in (1). This alteration in small problems related to composed ranking is explained because of the need to keep transitivity. • Transitivity between smaller problems. Please read comments. • Transitivity between smaller problems and composed ranking: Please read comments.

7.6.2

Case 2: Rehabilitation of Abandoned Urban Land

This case refers to an American port city, where containerization resulted in maritime wharves and railroad yards falling into disuse. City Hall wants them to be rehabilitated. Table 7.10 shows initial data with seven potential schemes and 14 criteria. C1: Transportation, C2: Job generation, C3: Environmental impact, C4: Financial feasibility, C5: Aesthetics, C6: Soil permeability, C7: Water demand, C8: Energy demand, C9: Sewerage demand, C10: Municipal infrastructure, C11: Link to subway network, C12: Green space recovery, C13: Business activity, C14: City Council Opinion. In this case, the following requirements were examined:

Table 7.10 Initial Data for Case 2—Rehabilitation of abandoned urban land

Criteria C1 transportation C2 job generation C3 environmental impact C4 financial feasibility C5 aesthetics C6 soil permeability C7 water demand C8 energy demand C9 sewerage demand C10 municipal infrastructure C11 link to subway network C12 green space recovered C13 business activity C14 City Council opinion

Different schemes proposed for land use P1 P2 P3 P4 P5 P6 Projects or schemes contribution to criteria (aij) 0.153 0.185 0.153 0.183 0.207 0.183 0.196 0.218 0.250 0.201 0.164 0.156 0.098 0.113 0.079 0.094 0.065 0.113 0.356 0.251 0.401 0.207 0.219 0.190 0.185 0.185 0.185 0.190 0.182 0.185 0.423 0.025 0.164 0.174 0.423 0.164 0.196 0.196 0.211 0.196 0.256 0.190 0.251 0.120 0.240 0.230 0.225 0.319 0.196 0.196 0.211 0.101 0.102 0.101 0.164 0.170 0.164 0.185 0.404 0.194 0.338 0.196 0.201 0.199

0.330 0.180 0.307 0.289

0.330 0.180 0.307 0.289

0.341 0.196 0.223 0.222

0.164 0.429 0.101 0.321

0.341 0.168 0.079 0.299

P7 0.190 0.200 0.088 0.175 0.200 0.418 0.185 0.142 0.101 0.203

Action MAX MAX MIN MAX MAX MAX MIN MIN MIN MIN

0.167 0.302 0.190 0.351

MAX MAX MAX MAX

7.6

1. 2. 3. 4. 5. 6. 7.

Testing SIMUS in Rank Reversal

143

Adding a new project, IDs [NS], [NA]. Comparing projects from the new scenario after adding, IDs [NAI], [NAJ]. Assuming identical values for two projects, IDs [NK], [NAL]. Deleting a single projects IDs [NC], [NH], [NQ]. Deleting pairs of projects, IDs [NE], [NF], [NG]. Replacing pairs of projects, IDs [NR], [NH]. Comparing pairs of projects IDs [NJ], [NK], [NP], [NV], [UN], [NL], [NI],[NW], [NAB], [NAD], [NAC], [NAF], [NAA],[NAG]. Table 7.11 shows results on 30 tests.

• Original ranking preservation: Kept in all cases. • Change in the best project: Kept in all cases (Project 5), except when deleting the best project IDs [NG], [NH]. • Transitivity between small projects: Satisfied where there are enough values to compare, since three IDs are needed. • Transitivity between small projects and composed ranking: IDs [NJ], [NK], [NI]. See comments about the others. Comments (1) Notice coincidence of rankings between this pair (5-3) and pair (5–7). Because Projects P3 and P7 have the same scores (thus, a tie). (2) The three pairs do not yield any solution because each pair cannot comply with all criteria. Subordinate Projects P2, P6, and P4 have the same score. Consequently, if P5 ⊀ P2 it cannot precede P6 and/or P4. Transitivity condition: If P3 > P2 and P5 > P3, then P5 > P2 as in the original. (3) Subordinate Projects P2, P6, and P4 have the same score. Consequently, if P5 ⊀ P2 it cannot precede P6 and/or P4. (4) No precedence since both projects have the same score.

7.6.3

Case 3: Determining Sustainable Indicators

In this case, there are 17 indicators that involve eight areas, such as economic growth, social capital, society, health, sustainability, environment, natural resources, and education. These indicators are subject to 16 criteria also grouped into education, social, health, sustainability, environment, economy, and OECD (Organisation for Economic Co-operation and Development) framework. This table has been obtained from expert’s opinions quantitatively linking indicators and criteria, as well as using correlation analysis. Notice Table 7.12 has been partitioned because its extension makes viewing and reading difficult.

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Table 7.11 Test for Case 2—Rehabilitation of abandoned urban land

Original ranking Test I.D. Tests ERM Ranking PDM Ranking preservation NT Original problem (6 Alternatives) 5-3-6-4 5-3-6-4 ORIGINAL Add Adding to original a project worse than others NS ADDING P2< P5 5-3-6-4 5-3-6-4 OK 5-3-6-2/4 OK NA ADDING P3 Alt. 3. This is one of the ways that SIMUS shows a solution when two or more alternatives tie.

9.6

Conclusion of this Chapter

SIMUS is applied here to solve a decision-making problem where decisions derive from a collaborative effort or group, with a DM as a leader, or with many DMs. The example separates the problem into two very distinctive areas, the Technical Department and the Decision-Making Group. The first one supplies the information needed that is the alternatives to consider and cardinal data. The second arranges this data and obtains a preferred solution. Naturally, one of them cannot work without the other and they complement. The decision-making process is a subjective issue because the final decision is not what a method or formula indicates, but attained by people who make suggestions, analyze results and finally make a decision. SIMUS is simply a tool that allows information to be organized, that applies sound and proven mathematical concepts and that in so doing, helps the decision-making process. Decision-making in many methods needs weights for criteria, is subject to personal preferences, or assumes levels for determining when an alternative outranks another. This provokes that the same problem with the same data delivers different results. SIMUS does not need any of those subjective concepts, and because of that, it is independent of whoever makes the calculations, unless, as in this example, different people take different approaches, but once these approaches are approved the model does not depend on the analyst’s preferences. As appreciated in the above recreation the Group works on a first ranking obtained from the initial matrix, with values as reliable as possible from various

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Group Decision-Making Case Study: Highway Construction

company departments including Engineering, Economics, Financial, Accounting, Human Resources, Environment, and Safety. Some values are quantitative, while other qualitative numbers are produced by different statistical procedures, including fuzzy logic, but not for personal preferences. In the process each objective is examined independently by all members of the Group: if there are no objections from any member, it is approved and then the analysis is centered on the next objective. If in this new one objective there are objections, then they are considered either from one member or several members. The software is run again and the partial result due to these changes is analyzed. This process involves two measurements: The first one is quantitative because it is the value of the objective function Z. The second is subjective because, even if it is based on shadow prices the comparisons between benefits and hindrances are made by the Group, although helped by the quantitative significance of these marginal values. The final result reflects the opinion of every member of the Group. It is believed that this example illustrates the power of this new method, which provides reliable results considering that the same solution is reached by two different procedures, and that calls for a strong DMG participation.

References1 *Indrani, B., & Saaty, T. (1993, February–March). Group decision making using the analytic hierarchy process. Mathematical and Computer Modelling, 17(4–5), 101–109. *Munier, N. (2011b) A strategy for using multicriteria analysis in decision-making: A guide for simple and complex environmental projects. Springer.

1

Publications that are not mentioned in the text but that have been added for the reader to access more information about this chapter; they are identified with (*).

Chapter 10

SIMUS Applied to Quantify SWOT Strategies

Abstract This Chapter describes a methodology aimed at selecting a strategy based on SWOT (Strength, Weakness, Opportunities, and Threats) model (Humphrey, Swot analysis for management consulting. SRI Alumni Newsletter. SRI International, 1970). Once the SWOT matrix is established, it is converted to a numerical SWOT matrix, and from there, the methodology uses Linear Programming (LP) to make the selection and to establish a ranking. The procedure quantitatively selects the best strategy.

10.1

Background

Using a mathematical procedure for identifying best strategies is not new; diverse authors such as Alptekin (2013), Chang and Hwang (2006) David (2009) (QSPM), Dyson (2004), Hashemi (2011), and others have already used mathematical methods to determine the “best” strategy. There are Multi Criteria Decision Making tools such as TOPSIS (Technique for Order Preference by Similarity to Ideal Situation) (Hwang & Yoon, 1981), AHP (Analytical Hierarchy Process), (Saaty, 1980), and others to generate scores for each strategy and making possible an educated selection. The procedure described in this Chapter is, as per this author’s knowledge, the first to apply LP with several objectives to find the best strategy, where the objectives are the different strategies, that is, combinations of external factors (Opportunities and Threats) and internal factors (Strength and Weakness), for one or several projects, as in a portfolio. However, most methods end by determining a SWOT matrix where the best properties of each strategy are described, and then it is up to the management to decide which of these strategies is “the best.” The method explained in this Chapter works with SIMUS, and it advances beyond the SWOT matrix by not only identifying the best strategy but also quantifying it. Consequently, SIMUS starts from the SWOT matrix and makes the selection.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 N. Munier, Strategic Approach in Multi-Criteria Decision Making, International Series in Operations Research & Management Science 351, https://doi.org/10.1007/978-3-031-44453-1_10

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SIMUS Applied to Quantify SWOT Strategies

Procedure

Following SWOT procedures and given a project or a set of projects, the first step consists of building a matrix that links for each project the two internal issues (Strength and Weakness) (in columns), and the two external issues (Opportunities and Threats) (in rows). At the intersection of columns and rows, and a strategy is identified. Second step is developing the strategies that express the modus operating for each pair. Third step consists of assigning numerical values to each pair derived from taking into account how opportunities and threats quantitatively relate to the company's strengths and weaknesses. This is done by considering how an opportunity influences the company objective when paired with the company's strength, and determining how a new project based on it, influences other current production of the company, or it if can use idle resources. From here, the financial influence of this new project is computed by determining a control parameter, such as the Internal Rate of Return (IRR), the Net Present Value (NPV), or any other factor. Similar procedure applies for comparing the same opportunity with company’s weakness, which allows for determining the risk of developing the project. Finally, both the IRR and the Risk are compared and a net value is obtained. This procedure is repeated for each opportunity and each threat. Fourth step is the application of SIMUS to help choose the best strategy. To set up the mathematical model, strategies are placed in columns (SO, WO, ST, WT), and the n-external factors on rows (Opportunities, Threats). Table 10.1 follows the traditional SWOT format with Opportunities and their corresponding advantages for the company, as well as disadvantages or Threats. The same course of action for Strength and Weakness, this data is detailed in Table 10.2. Table 10.3 shows the derivation of respective values for Table 10.2. LP demands that each criterion must obey an action such as maximization, minimization, or equalization according to the nature of the criterion. Thus, a criterion related to opportunities will have a maximization action, since an entrepreneur wishes to have a link between this factor, and Strength, as robust as possible. Conversely, a criterion like Threats will probably have a minimization action, since the entrepreneur wishes to have a link between this factor and Weakness, as feeble as possible. However, because each criterion is formed by two opposite strategies, it is impossible to use the same action for both, since one wants to maximize a strategy that combines strength and opportunities but cannot at the same time maximize a strategy that combines weakness with threats. For this reason, it is necessary to use different signs for these values. SWOT analysis is mostly based on subjective or not deterministic values such as the evaluation of an existing opportunity considering the strength of the firm or among others the evaluation of an existing threat and a known internal weakness. This method uses generally probabilistic values for linking these concepts, which,

10.2

Procedure

187

Table 10.1 Information gathered by the company Opportunities O1 Statistics show an increase in sales of small cars O2 Government has announced a 50% increase in duties for importing cars O3 Gasoline prices are increasing O4 There are subsidies for developing renewable resources O5 Banks are offering better rates for purchasing cars using renewable energy sources Threats T1 Government approved a 3.8% increase in electric bills T2 Statistics show that fatal accidents are more frequent in motorcycles and small cars which raises insurance cost T3 There is a well-established company importing Swiss electric cars whose prices are 13% below ours T4 The number of years for an owner to renew his car is increasing T5 There are not enough electric car charging stations Strengths S1 Our car is small and light because of its carbon fiber body S2 It has an efficient battery S3 It has a reinforced body S4 Low electricity consumption Weaknesses W1 We have a limited budget W2 Lack of expertise in electric cars W3 Our manufacturing cost is too high W4 Our prototype has not been tested in harsh winter conditions

however, are the outcome of a thorough analysis based on known, although not certain, facts. That is, the firm can estimate the probable link between an opportunity and its strengths and weaknesses using correlation analysis and estimates market penetration of its product by using marketing tools such as “Quality Function Deployment” (QFD) (Akao, 1997). However, they are not fixed or deterministic values but well-documented estimates. As a bottom line, the firm gets a matrix with comparisons between external and internal factors, and quantitatively evaluated. However, as important as this step is, it is only a description of strategies related to factors and expresses the pros and cons of each pair of internal and external factors. It is a good summary for the next step, which consists of quantitatively measuring the relationship between each pair. One decision-making method to solve this uncertainty is SIMUS.

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Table 10.2 SWOT matrix with strategies Strengths (S) S1—Light weight S3—Strong structure S4—Low electricity consumption Opportunities (O) O1—Sale of small cars shows a positive trend O2—Government intention to increase in 50%, duties for importing electric cars

SO Strategy (Recognition of potential) There is potential here since both opportunities match our strength, as the trend shows an increase in sales for small cars as our product is. In addition, if the government measure materializes, we will have a competitive advantage over imported vehicles from Switzerland. Rest to see the correlation existent among these factors, that is, between values of this trend and our sales regarding conventional engines, in order to get if both follow the same performance. However, we must also analyze the impact that this new production could have on our current manufacturing regarding labor and equipment. Make a simulation of the cash flow.

Threats (T) T1—Due to the agreement between government and electric suppliers there will be a 3.8% increase in electricity bills T2—Statistics show that in serious street accidents, motorbikes, and small cars rate surpasses that of normal size cars T3—Imported cars are 13% cheaper than our model

ST Strategy (Accepting risk) As in other aspects, we have to address each threat separately. First, we have to continue with our studies to use a fuel cell installed in the car, thus generating its own electric power. This way the cost of electricity is irrelevant for us as long as hydrogen can be purchased at today’s prices. Then this last aspect should be carefully analyzed. The cell should be imported because it is not manufactured in the country; however, its

Weaknesses (W) W1—Limited budget W2—Lack of expertise in this type of undertaking W3—High manufacturing cost WO Strategy (Accepting a confrontation) If we compare existing opportunities with our weaknesses, it is a confrontation, however, with appropriate measures we can win it. Therefore, our objective would be to first get investment money by attracting investment capital from private sources or from the stock market. In order to do that, we need to have our numbers about production cost, which is probably our greatest weakness. The other weaknesses, such as lack of expertise, can be solved by an aggressive campaign to train our people and perhaps hiring a consultant in this type of manufacture. It will obviously influence our limited budget but it could be recovered because there will be fewer rejects and possible car recalls. WT Strategy (Reducing weakness) We cannot decrease the threats but we can decrease the damage it can do, caused by our weaknesses, by attracting investment capital from private sources. Another measure that we can explore is the decreasing of manufacturing costs by at least 10%, by using robots in the assembly line. And a third measure could be reducing the price gap between imported cars and (continued)

10.3

Application Example: Case Study—Strategy for Fabricating Electric Cars

189

Table 10.2 (continued) import is not affected by the probable increase in import duties. If we can get the fuel cell, then our vehicle will be perhaps 4.5% more expensive than imported cars but this will cancel the 3.8% increase in the electric bill. In addition, we will have a technological advantage over other firms. Regarding accidents, we do not have to improve our product since it is robust enough as per its unique honeycomb design and that has been extensively tested, but nevertheless, we need to advertise this fact. Obviously, there is risk in this strategy, and the company needs to establish the maximum risk it is willing to accept. If the estimated risk is larger than the former, we should not pursue this project.

10.3

ours, by lowering our profit margin.

Application Example: Case Study—Strategy for Fabricating Electric Cars

The following example illustrates the method. A company, assembling small cars equipped with conventional fuel engines, is considering the fabrication of the same car but driven by electric motors. The firm believes that there is potential for this type of vehicle. The complete list of internal and external factors gathered by the company is detailed in Table 10.1. Out of this list, the company makes a short list of its more significant strengths and weaknesses, opportunities and threats existing, as indicated in Table 10.2, as well as comments about their interaction, and thus defining strategies. The strategies are: SO: Promote the concept that small electric cars are the solution for urban traffic, as well as pointing at no air contamination when compared to fuel-powered cars. WO: Get financing for this project, as well as hire an expert for this new production. ST: Think of production of electric cars fed with imported hydrogen cells. WT: Get financing, as well as decrease manufacturing costs, by using advanced robotic technology.

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SIMUS Applied to Quantify SWOT Strategies

Construction of the Numerical SWOT Matrix

Table 10.3 shows the derivation of values for the construction of the numerical SWOT matrix for this example.

10.4.1

Market and Government

Involves rows with external factors and their impact. First column: Opportunities and Threats are detected. There could be many different opportunities and threats, and this column groups those considered significant by the company, but of course, all of them must be analyzed. There must be documentation regarding each concept materialized as statistics, official government announcements, competition advertisements and prices, political scenarios, technical advances, etc. Second column: Related to how these measures impact society and markets. This is important because they may influence company projects. Factors can be expressed as volume or demand for the product under consideration, environmental regulations, economy, social issues, etc. Information must come from official publications, reliable sources, international prices, statistics, etc. The value found here for this opportunity is 1468 cars. Third column: Associated to the extent to which these external factors are related or affect company business. There could be positive, as well as, negative correlations. For instance, if there is an increase in fuel prices, most probably it will have a negative impact regarding demand of any product of service using this type of fuel. How intense is this relationship can be known by correlation analysis (albeit not necessarily in a cause–effect relationship). If this increase affects the company product under analysis, it will be necessary to compute a quantitative value expressing this influence. In this case it is found to be 0.9. That is, the increasing sales trend have a correlation of 0.9 regarding company sales, at least considering actual production of gas engine vehicles. Fourth column: Share of the market (0.19). This has been ascertained from statistics of past sales and also using different marketing tools such as the Quality Function Deployment (QFD) method. Fifth column: If the company considers these opportunities and threats, how will they affect the company’s current production or the operation of the plant? In other words, how will the new production interfere with current and foreseen production? It is related to personnel, budgets, equipment, expertise, etc. For instance, a new production might produce delays, poorer quality, or smaller production of other products from the same plant for which there are previous compromises with clients. Or, it could very well be that the firm have idle capacity which can be utilized. In this case, there is no interference or damage to current production because the firm has idle capacity. Therefore, the company can expect a production of 251 cars which is

10.4

Construction of the Numerical SWOT Matrix

191

Table 10.3 Derivation of values for construction of a numerical SWOT matrix (comments are in green italics) `

Our company Strengths 1 2 3 4 5 Measure of How does How much What would benefits to this could be our be our society opportunity participation production credited match our in the market? considering Opportunities to this strength? our current detected opportunity (Strategic plans? fit) Con-sider Explain Percentage in interference Explain method Explain nature of with actual decimals opportunity production used reasons for this and correlation financing

Statistics show that there was an increase of sales of small cars in the last 10 years period [See report with trend]

Examining

sales trend for small cars the regression coefficient that is, the average year increase, is: 1468 units

Our car is small and light because its carbon fiber body.

Share of the market: 0.19

[From Marketing Department is believed analysis and that there is conclusions a positive from the correlation Quality with people Function acceptance Deployment estimated tool (QFD), at: 0.9 considering competition as well] Therefore, it

[No interference since we have idle capacity] Impact: 1468 x 0.9 x 0.19 =251 [Considering our on-going and future production of other products already approved]

6 How much is the expected IRR? Make sure in establishing an acceptable minimum value Make sure that impact of this new project on current production is considered Estimated IRR: 0.068

Weaknesses 7 Our weakness or negative aspects regarding this opportunity How can we reduce this internal weakness?

8

Net balance

Financial analysis 9 10

Acceptable risk

Conclusion

Compute Risk

Because our limited budget, the [This figure construction of a plant to comes by build the car mathematical will simulation jeopardize introducing car our other production on plans. The our financial estimated statements, risk could and analyzing be: the result, taking into Probability account the of this whole happening: portfolio of 0.35 projects. In this way the [Considering new project personal and impact on our equipment resources is capacity] contemplated] Impact on cash flow: 0.16 [As per cash flow statement]

Risk = 0.35x0.16 = 0.056 [No Estimated Government This will There is no Share Because our interference IRR: 0.066 indeed has doubt that estimated in limited announced increase 11 this 20% since we budget, the its intention % the price measure [This figure construction have idle to increase of those will benefit [Consultations cap-city] of a plant to comes by 50% import imported our mathematical build the car with car duties for vehicles in company Impact: will simulation dealers electric cars the local perspectives suggest that 1307 x 0.9 x introducing jeopardize although no market, 0.20: our other car and it is we can date has been inducing 235 units production on plans. believed increase our established decreasing that the our financial Therefore, share of the because the demand [Considering statements the risk correlation market but new duty is estimated our on-going and analyzing could be: will hold in only a little, affecting as an plans and the result, 0.9 because we commercial average in future Probability deeming the are a new relationships 1307 units of this whole maker that production of with other happening: portfolio of people do not other procountries [Data from 0.35 projects. In ducts know yet] car this way the already [Information dealers] [Taken into new project approved] from impact on our account personal and resources is newspapers] contemplated] equipment capacity]

Net percentage on invested capital

Our tolerance to risk is 0.07

0.012 is a small percentage on earnings, however, bearing in mind that it corresponds to the first From the three years risk point of operation, of view the and that it project is includes equipment accepted considering amortization It is this opportunity acceptable

= 0.068 – Then, the 0.056 = computed 0.012 risk is below our accepted risk 0.056 < 0.07

Net earnings on invested capital = 0.066 – 0.056 = 0.010

Our tolerance to risk is 0.07

0.01 is a small percentage on earnings, Then, the however, computed considering risk is that it below our corresponds accepted to the first three years risk of operation, 0.056 < and that it 0.07 includes equipment From the amortization, risk point of view the It is acceptable project is accepted considering this opportunity

(continued)

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Table 10.3 (continued) Impact in cash flow: 0.16 [As per cash flow statement] Risk = 0.35x0.16 = 0.056

Our company

Market 1

Threats detected Explain nature of threat

Government announced an increase of 3.8 % on electric bills for domestic consumers [This measure will be enforced starting January 1st]

2 Measure of threats to society credited to this threat Explain method used

The new tax aims at the development of renewable sources of energy. We have used a proportional

correction factor and estimate that the annual rate of sales will decrease in less than 1.8%, that is to 1468 – 0.018 x 1468 = 1441

Strengths 3 4 5 How is How What the much would correlatio could be be our n between our product this threat participat ion and our ion in the conside strength? market? ring our current Explain Percenta plans? ge in reasons for this decimals correlatio n

It is believed that there will still be the same correlatio n of 0.9

6 How much is the expected IRR?

Make sure in establishing an acceptable minimum value

Financial analysis Weaknesses 7 Our weakness or negative aspects regarding this probability

8

9

10

Compute Risk

Make sure that impact of this new project on current production is considered Estimated Share Because our [No IRR: 0.069 limited estimate interd in: budget, the ference [This figure construction 0.22 since of a plant to we have comes by mathematic build the car [We idle have an capacit al will simulation jeopardize our advantag y] e here introducing other plans. because Impact: car Therefore, the average 1441 x production risk could be: consump 0.9 x on our Probability of tion of 0.22 = financial present statements this 285 and happening models is of 9.73 analyzing 0.35 the result [Considering kWh/100 km, taking into personal and while account the equipment our car, whole capacity] because portfolio of its projects. In Impact on reduced this way the cash flow: new project 0.16 weight, uses only impact on [As per cash 7.74 our flow resources statement] kW/100k is m] contemplat Risk = ed] 0.35x0.16 = 0.056

AccepNet balance table risk

Net percent on investe d capital = 0.069 – 0.056 = 0.013

Conclusion

Our tolerance to risk is 0.07

0.013 is a small percentage on earnings, however, Then, the since it compute corresponds d risk is to the first below three years our of operation, accepted and that it risk includes equipment 0.056 < amortization 0.07 It is From the acceptable risk point of view the project is accepted consider ing this opportun ity

(continued)

10.4

Construction of the Numerical SWOT Matrix

193

Table 10.3 (continued) Government and public statistics show that motorbikes and small cars are subject to more frequent accidents that those for medium size and large cars [Municipal statistics and report are included]

Cost [From our Financial and Engineering Departments]

There is indeed concern amongst buyers about safety as expressed by car dealers, however, economy, parking space and cost, normally prevail and it is assumed that the effect would be negligible, therefore the original annual average of 1468 units is held

Imported cars are about 13 % cheaper than our model and this is viewed as a threat because competitio n

Our Share of vehicle the has a little market: advantage 0.23 on this issue because its special honeycomb design. Therefore, it is believed that the correla2tio n will increase to 0.92

Car dealers estimate that even with the advantage s of our model there is a high inverse correlatio n between costs and demand, of about 65%

Impact: 1468 x 0.92 x 0.23 =309

Because our Estimated IRR: 0.069 limited budget, the [This figure construction comes by of a plant to mathematic build the car will al simulation jeopardize our introducing other plans. Therefore, the car production risk could be: on our Probability of financial statements this happening: and 0.35 analyzing [From the result considering personal and equipment the whole portfolio of capacity] projects. In this way the new project impact on Impact in our cash flow: resources 0.18 is [Considering contemplat personal and ed] equipment capacity]

From the Impact: Estimated cost 1468 x IRR: 0.037 point of 0.65 x [This figure view our 0.15 comes by share of =143 mathematic the al market simulation could introducing drop to car 0.15 production on our financial statements and analyzing the result considering the whole portfolio of projects. In this way the new project impact on our resources is contemplat ed]

Risk = 0.35x0.18 = 0.063 Because our limited budget, the construction of a plant to build the car will jeopardize our other plans. Therefore, the risk could be: 0.28 [Considering personal and equipment capacity] Impact in cash flow: 0.30 [From cash flow statement] Risk = 0.28x0.30 = 0.084

Net earning s on investe d capital

Our tolerance to risk is 0.07

Then, the compute 0.069 – d risk is below 0.063: = 0.003 our accepted risk 0.056 < 0.07

0.003 is a negligible percentage on earnings It is acceptable with reservations

From the risk point of view the project is accepted based on this opportun ity

Net earning s on investe d capital 0.084 0.047= 0.037

Our tolerance to risk is 0.07 Then, the compute Un d risk is acceptable higher than accepted risk

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the value arrived by multiplying market reaction, by company sales correlation, and by share of said market. Sixth column: This is related to company finances. That is, even if the new product may look attractive from the economic point of view, it is necessary to compute its return considering its impact on the whole company; this is done by computing the IRR. It can be determined by using projected financial statements when introducing the new parameters such as the amount of working capital for the new production, bank loans, tax deferrals, and labor availability. This computation produces a return that may or may not be acceptable to the company. In this case, it is 6.8% which shows the return the company can expect by combining the new product with the current production. Seventh column. It is related to company weaknesses; the computation materializes by estimating the risk of this new production. Risk is the product of the probability that something can happen and the impact if it indeed happens. Probability can be computed in different ways, for instance, by considering personal and equipment actual load, against forecasted, experience, capital shortage, suppliers delays, etc., and impact can be calculated from the balance sheet and cash flow statements. When the forecasted economic and financial results of the firm including other products are analyzed by parameters such as the IRR and net earnings, it may be found that there is a gain or a loss. In other words, it means that a certain measure of the impact can be obtained. Then a risk can be computed and a sensitivity analysis performed regarding the influence of variations in prices, efficiency, volumes, demand, etc. on this risk. Since the probability of jeopardizing current production is 0.35 and the impact is 0.16, the risk is the product of these values and is equal to 0.056. Eighth column: This column compares benefits, expressed as IRR and costs expressed as impact on cash flow. Then, in this case, it is 0.068–0.056 = 0.012. Nineth column: In this column, the company tolerance to risk is compared with the computed risk. If the tolerance risk is greater than the computed risk, then the project can be accepted at least from each particular opportunity or threat and rejected if computed risk is greater than the accepted risk. In this case, tolerance to risk is 0.07, and computed risk is 0.056, consequently, 0.07–0.056 = tolerance is greater (0.014), which makes it acceptable. Tenth column: indicates the conclusion for each particular opportunity or threat. This analysis must be made for each external factor associated with the joint result from internal factors, and of course, for each project.

10.5

Preparing an Excel Matrix with Data

Values found in the analysis of the numerical SWOT matrix in the seventh column are loaded into an Excel decision matrix and the results of their computing are shown in Table 10.4, whose only purpose is to check all data and make the necessary

10.5

Preparing an Excel Matrix with Data

195

Table 10.4 Decision matrix for this project

Opportunities (O) • Sale of small cars shows a positive trend • Government intention to increase 50% import duties for electric cars Threats (T) • Due to an agreement between government and electric suppliers there will be a 3.8% increase in electricity bills • Statistics show that in serious street accidents motorbikes and small cars rate surpass that of normal size cars

Strengths (S) • Light weight • Strong structure • Few imported parts SO Strategy Use our main advantage; the weight of the car as well as its local fabrication to increase our participation in the market

Weaknesses (W) • Limited budget • Lack of expertise in this type of undertaking WO Strategy Attracting investment capital from private sources

ST Strategy Continue with our studies to use a fuel cell installed in the car, which generates its own electric power The cell should be imported because is not manufactured in the country and its import is not affected by the probable increase in import duties Advertisement pointing to the robustness of our vehicle because of its honeycomb design

WT Strategy Attracting investment capital from private sources Decrease manufacturing costs by at least 10%, by using robots in the assembly line. Lessen the price gap between imported cars and ours by reducing our profit margin

corrections before their input into the SIMUS decision method. Observe that columns correspond to strategies while rows indicate opportunities and threats. Once this information is inputted into SIMUS, the method yields the following final results for selection. The capture of the computer screen has been split for better visualization in two, corresponding to ERM and PDM matrices, respectively, showing its two rankings (Figs. 10.1 and 10.2). In Fig. 10.1, the solid blue row corresponds to ERM and shows that the most favorable strategy is Strategy ST with a score of 0.63, followed by Strategy SO with a score of 0.20. In last place is strategy WT with a score of 0.09. These are three very well-differentiated values (showing large discrimination, which is convenient). Ranking is: ST - SO - WT - WO. The second split part of the screen corresponds to PDM and shows in the solid brown column that the most favorable strategy is Strategy ST with a score of 5, followed by Strategy SO with a score of 0 and Strategy WT with a score of -1, and finally Strategy WO with -4. Ranking is: ST - SO - WT - WO.

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Efficient Results Matrix (ERM) Normalized SO WO ST WT Objective Z1 (Opp.1) Objective Z2 (Opp. 2) 1.00 Objective Z3 (Threat 1) 0.57 0.43 Objective Z4 (Threat 2) 1.00 Objective Z5 (Threat 3) Sum of Column (SC) 1.00 1.57 0.43 Participation Factor (PF) 1 2 1 Norm. Participation Factor (NPF) 0.20 0.40 0.20 Final Result (SC x NPF) 0.20 0 0.63 0.09

ERM Ranking

Shadow prices Z2 Z3 Z4 1.17

Z1 Opport. 1 Opport. 2 Threat 1 Threat 2 Threat 3

Z5

2.17 0.50 0.50

ST - SO - WT - WO

Fig. 10.1 Screen capture showing ERM solution Project Dominance Matrix (PDM) Subordinated strategies Dominant strategies SO WO ST WT Column sum of subordinated strategies

SO

WO 1

0 2 1 3

PDM Ranking

2 1 4

ST 1 0 0 1

Row sum of subordinated strategies

WT 1 0 2

3 0 6 2

Net dominance 0 –4 5 –1

3

ST - SO - WT - WO

Fig. 10.2 Screen capture showing PDM solution

10.6

Discussion

Analyzing the selected strategies, it can be seen that: • Both procedures coincide and discriminate, which confirms selection, being Strategy ST “Strength and Threats” the best, and Strategy SO “Strength and Opportunities” the second best. • ST which has a score of 0.63, is well differentiated from the second strategy SO (0.20) and from strategy WT (0.09), casts no doubt which is in the first position. This large discrimination, which also can be visualized in the PDM score is an important issue, since if two scores are similar, adoption of the best strategy becomes controversial and may render the whole analysis useless. • Examining strategy ST in a normalized ERM matrix, notice that the two most important objectives are Z3 “3.8% increase in electric bills,” with a score of 0.57, and Z4 “Accidents with small cars,” with a score of 1.00. Objective Z3 has two shadow prices, corresponding to Threat 2 “More accidents for small vehicles” with a value of 0.50, and to Threat 3 “Cost” with a value of 0.50. Consequently, an increment in the RHS value for Threat 2 will affect objective Z3 in 0.50 units.

10.6

Discussion

197

It means that if the rate of accidents increments, it will have a direct consequence on Z3, since there will be fewer small vehicles on the streets, which translates into savings in electric bills. Apparently, this does not affect the company; however, it also means a decrease in the sales of locally made electric vehicles. However, that is only mathematics, and apparently indicates an illogical relationship, which does not give much intelligence to the company. On the other hand, they reveal that safety has paramount importance, and then, knowing this fact, the company may use this information to improve it, by adding airbags in its cars, or by reinforcing its structure with carbon fibers or by installing low-cost side impact bars, or by a collapsible steering wheel. That is, the value of this information is that it enlightens the company about something that it probably did not realize before. It is not the numbers that are important, but what they mean; in this case the company receives information that can debilitate the threat and fortify its position in the market. For the same token, an increment for the RHS value for Threat 3, will increase objective Z3 to 0.50 units in. It means that if the price of imported electric cars increments (perhaps as a consequence of a continuous depreciation of the local currency), it has a direct consequence on Z3, since there will be fewer imported small vehicles on the streets. This fact may convert a threat into an opportunity for the company, since the proportion of locally made electric cars will probably increase. Consequently, this information may lead the company to make more deeper studies about the potential evolution of the local currency, based perhaps on statistics. If objective Z4 “More accidents for small cars” is analyzed, it can be seen that it has only one shadow price, that belongs to Threat 1 “3.8% increase in electric bills.” Consequently, a unit increment in this price will have a very strong effect on accidents by small cars on the streets, since there are fewer vehicles on the streets. Again, this is mathematics, but what the company may surmise from this is that there is a close relationship between accidents and electricity prices, and then, it reinforces the findings of the above paragraph in the sense that for the company is essential to challenge the threat of accidents with better safety measures. Observe that the second-best strategy, SO, is also composed of Strength, and this supports the feeling that Strength is the best asset the company has. In fact, when the company affirms that its car is light because of its fiber carbon body, it is also saying that it is safe since this is one of the most resistant materials in the market with a high tensile strength, as well as high rigidity, has resistance to high temperatures, not flammable, and because of these characteristics, it is used in many industries, especially in aerospace. This could then be a good selling strategy.

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Conclusion of This Chapter

This Chapter addresses SWOT, the very well-known procedure for analyzing strategies. However, the SWOT matrix does not identify which is the best strategy, so SIMUS is applied to define it. A real case is proposed that includes SIMUS methodology to develop quantitative values for the competing strategies by means of an extensive analysis, which is condensed in tabular format. Then, data from it is used as the IDM and SIMUS are applied. The two results coincide in selecting the best strategy.

References1 Akao, Y. (1997). QFD past, present, and future. In International Symposium on QFD’97 – Linköping. Alptekin, N. (2013, Winter). Integration of SWOT analysis and TOPSIS method in strategic decision-making process. The Macrotheme Review, 2(7). Chang, H., & Hwang, W. (2006). Application of a quantification SWOT analytical method. Mathematical and Computer Modelling, 43, 158–169. David, M., David, F., & Davis, F. (2009). The quantitative strategic planning matrix applied to a retail computer store. The Coastal Business Journal, 8, 1. Dyson, R. (2004). Strategic development and SWOT analysis at the University of Warwick. European Journal of Operations Research, 152(3), 631–640. Hashemi, N. F., Mazdeh, M. M., Razeghi, A., & Rahimian, A. (2011). Formulating and choosing strategies using SWOT analysis and QSPM matrix: A case study of Hamadan Glass Company. In Proceedings of the 41st International Conference on Computers and Industrial Engineering. *Humphrey, A. (1970 (2005)). Swot analysis for management consulting. SRI Alumni Newsletter. SRI International Hwang, C., & Yoon, K. (1981). Multiple attribute decision making: Methods and applications, A state-of-the-art survey. Springer. *Munier, N. (2011). A strategy for using multicriteria analysis in decision-making – A guide for simple and complex environmental projects. Springer. Saaty, T. (1980). Multicriteria decision making - The analytic hierarchy process. McGraw-Hill.

1

Publications than are not mentioned in the text but that have been added for the reader to access more information about this Chapter; they are identified with (*).

Chapter 11

Analysis of Lack of Agreement Between MCDM Methods Related to the Solution of a Problem: Proposing a Methodology for Comparing Methods to a Reference

Abstract It is a proven fact that at present, there is no course of action that can evaluate or validate the reliability of the solution reached by a MCDM method, because the “true” solution is not known, and it is impossible to make a comparison to assess the efficiency of a result found. This Chapter presents a procedure that can help in this endeavor. It proposes to use a proxy of the true solution, to test a result of any MCDM method; this proxy solution must be the consequence of a more faithful model to replicate as much as possible real-world conditions, as well as the absence of subjectivity in criteria weighting, and the result achieved by an indisputable mathematical procedure. For this purpose, this book suggests using the SIMUS method that fulfils these conditions. In so doing, a problem is solved by this method and its result is used as a benchmark to determine the closeness to this result by other methods. To measure closeness to the proxy, it is suggested to use the Kendall Tau Rank Correlation Coefficient (Kendall, Biometrika 30(1–2):81–89, 1938).

11.1

Objective of this Section

In MCDM, there are three questions that have been not answered yet: (a) What method is most adequate to solve a problem? (b) How to select the most appropriate? (c) How to know if the solution found is correct or at least closer to the “true” one? This Section aims at examining the different reasons to explain questions. Some scholars argue that a true solution does not exist, while others assume that it does; TOPSIS (Hwang & Yoon, 1981), one of the most popular methods goes even further by assuming that there are both a positive ideal and a negative ideal solution. In addition, mono-objective Linear Programming (Kantorovich, 1939), always finds this optimum or ideal solution, provided that it exists. These last four words appear to be a contradiction and need an explanation.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 N. Munier, Strategic Approach in Multi-Criteria Decision Making, International Series in Operations Research & Management Science 351, https://doi.org/10.1007/978-3-031-44453-1_11

199

200

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Analysis of Lack of Agreement Between MCDM Methods Related to the. . .

In LP, it is always a good practice to quote a warning to a potential solution adding “if it exists.” Its meaning refers to a concept that can be perfectly understood by examining graphic results for different objectives, in a two or three-dimensional problem. The possibility of no existence is then due to a geometrical problem, for instance, the best vertex is a polygon (see Chap. 6) formed by some of its lines or criteria in the infinitum. In this book, this author considers that the ideal or true solution exists, conformed by a conjunction of physical, geological, and nature-wise factors. The mere existence of human race could be a case; because out of the many different evolutionary forms nature chose one, and that becomes a reality. It is like Nature found the best combination of multiple factors, to produce a humanoid. The procedure suggested here does not aim at a validation because that is considered impossible in multi-objective decision-making problems, but in generating a final scenario that could be similar to the real one. That scenario is created by a proxy method such as SIMUS. This procedure appears rational since normally the true results are unknown, led by a sort of circumstances that no human being cannot contemplate, let alone evaluate. For instance, in deciding to construct hydroelectric dams, there are many technical, economic, environmental, and social aspects that are and should be considered. However, there are others such as earthquakes, ice melting regimes, wildlife reactions to their territory being invaded, or people's attitudes to being displaced from their homes and lands, that nobody can predict. In addition, MCDM problems are sometimes so complex that no one can expect an optimal solution; for this reason, methods are not looking for an optimal solution from mathematical computations, but for a compromise solution (Compromise Programming), Zeleny (1974), Yu (1973), Cochrane and Zeleny (1973). The only thing that can be done is to construct a model as close as the real world, with existent conditions, and without any human interference on data, other than establishing the alternatives and the criteria they are subject to. A review of the literature shows that there are not too many publications addressing the issue of developing a method that could allow results to be evaluated, and as per this author's knowledge, it has never been suggested the use of a proxy to appraise results. This discussion is organized as follows: Section 11.2 analyses the causes for discrepancies and their effects on results. Section 11.3 examines aspects to be considered when selecting a MCDM method; Section 11.4 refers to selecting a benchmark method.

11.3

Subjective Preferences

11.2

201

Causes for Discrepancies on Results

Since the development and implementation in the 1960s, of methods to solve multicriteria decision problems, researchers have been puzzled by the fact that different mathematical approaches to solving the same problem produce different results (Triantaphyllou and Mann 1989, Mulliner et al. 2016, Ceballos et al. 2016). Nowadays, there is agreement that this is due to the fact that decision-making is to a large extent a subjective activity, which explains that starting from the same initial matrix, using mathematical tools and aiming at the same objective, the results can be diverse.

11.3

Subjective Preferences

Subjective preferences and opinions are constrained by “bounded rationality,” as Simon (1957) defined them, and later Tversky and Kahnemann (1974) produced their seminal work on predictable biases and mistakes; some methods accept that a DM is able to make decisions, and guessing that he is rational, which should be defined first what this word means in the MCDM context. This dubious rationality can explain why two DMs (or even the same DM after a time), may produce different preferences on the same problem and consequently, establishing different weights for criteria.

11.3.1

Subjective Weights

In this book, the author believes in part by the reason commented, that the pair-wise comparison of criteria and the extraction of weights, should not be utilized in a MCDM scenario, considering that a criterion may change its value. Some methods do not use weights, Zardari et al. (2015), and this book agrees with this procedure. Naturally, this does not mean that criteria of relative importance must be ignored; quite the opposite. A method must avoid criteria weights, but at the same time it must be mathematically built for criteria significance to be considered, as SIMUS does, using data from the IDM and from partial results obtained from the iterative computation process to select an alternative. Regarding subjective weights, and even considering DM’s honesty, goodwill, experience, and knowledge, it is necessary to admit that his conclusions may be highly debatable, especially when he cannot give a convincing explanation for his preferences, which translate into weights. In addition, except in very special cases, there is no way to evaluate his decisions, as well as his competence, especially when they affect a large amount of people, who can express it themselves.

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Analysis of Lack of Agreement Between MCDM Methods Related to the. . .

As an example, in some scenarios where the projected alternatives involve relocating maybe hundreds or even thousands of people, as happened in the Three Gorges Dam project in China, it is inconceivable that a DM will solely rely on preferences, to make a decision that affects people, in lieu of considering people opinions and preferences; a procedure that, violates the Arrow Impossibility Theorem (Arrow, 1951). Decision-making is too serious a matter to be left to a person or even to a group’s personal preferences. Since most of the MCDM methods use the AHP’s, Saaty’s (2008) first stage for determining criteria weights, it is understandable that results are usually different and depend on who makes the evaluation, even if the mathematical procedure is considered correct. Of course, there is also subjectivity in other methods when the DM has to select thresholds and make assumptions that usually do not have any mathematical justification. Considering all these aspects, it really would be a miracle that different methods coincide in their results. Moshkovich et al. (2012) assert that it is difficult to select an appropriate multiple criteria ranking method because “criterion weightings and scale transformations for criterion values produced significant differences in the ranking of alternatives when two different methods are used for the aggregation of the preferential information.” Stewart (1996) also maintains that MCDM tools are used, especially in a group setting, to provide a methodology for selecting a course of action by a procedure that is perceived to be fair and just to all interests, concepts with which this author agrees. However, is it right to assume that a DM decision is fair for all parties involved? Reliable and objective weights can be extracted by employing different techniques, and to this respect, Lieferink et al. (2014), express that whether criteria weights are elicited with the same or with different techniques, it does influence them and consequently the results. Accordingly, there are many reasons that make MCDM methods yield different rankings for the same problem. Naturally, a very frequent question arises: Which of the different methods gives the most accurate solution? Until today, this question remains unanswered, and chances are that it never will.

11.3.2

Objective Weights

Objective criteria weights can be extracted using different procedures. In general, they are based on data from the IDM. Probably any method can use objective criteria weights derived from entropy, following Shannon’s (1948) concept, or ratio weights PSI, Petković, or using a statistic such as variance. What is the main difference between subjective and objective weights? The difference lies in their intended use, since subjective weights for criteria are utilized just to find out the relative importance of criteria, while objective weights are good for also determining such relative importance, but, more importantly, to reveal the capacity of each criterion to evaluate alternatives, which is based in the discrimination of performance values in each criterion. There is no doubt that subjective

11.3

Subjective Preferences

203

weights are not designed to qualify criteria for evaluating alternatives, while objective weights are designed considering that purpose. If for whatever reason, a method requires using weights, it can use objective weights derived from the original data, instead of subjective and arbitrary weights derived from pair-wise comparison, expert’s opinions, or Delphi. There is overwhelming evidence that criteria weights are the main responsible factor for discrepancies; therefore, eliminating them seems a rational procedure.

11.3.3

Inconsistencies

Zanakis et al. (1998) among others, also pointed out those MCDM methods might produce different rankings when applied to the same problem, under the same conditions. According to these authors, this inconsistency occurs because (i) each method uses different weighting calculations; (ii) the algorithms differ in their approach to selecting the best solution; and (iii) some algorithms introduce additional parameters that affect the solution. Moreover, this situation can be intensified by differences in weighting extraction among different decision-makers, even with similar preferences. They also found that ranking differences derive from the process of weighting the criteria, and are more intensive in scenarios with many alternatives, and besides using the same weighting vector the ranking order may vary depending on the method used. For this reason, it is conceivable that an ideal or proxy solution obtained by a method without preferences and assumptions may be used to contrast results; and this is the core of the procedure proposed here. The author of this book also sustains that in reality, the DM preferences belong to his own personal scenario that is in general different from the real scenario that he tries to model, since it attempts to reduce his inconsistence that is, making his reasoning transitive, when most probably reality is intransitive. Transitivity is a main issue here because weights extraction is based on looking at transitivity in the DM preferences and as Fishburn (1991) reports “Transitivity is obviously a great practical convenience and a nice thing to have for mathematical purposes, but long ago this author ceased to understand why it should be a cornerstone of normative decision theory.” The use of subjective weights for criteria has, of course, a direct influence on results, and as Wang and Triantaphyllou (2008) express, this problem relates to rank reversal, when they state “Irregularities in the ranking of alternatives occur when the MCDM method does not meet the following requirements: (i) maintaining the indication of the best alternative even when one of the alternatives is replaced by another worse alternative and the weightings determined for the criteria remain the same; (ii) obeying the property of transitivity for the final ranking of alternatives; (iii) providing the same ranking as for the original problem when the decision problem is divided into parts.”

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11.3.4

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Analysis of Lack of Agreement Between MCDM Methods Related to the. . .

Evaluating Results

Evaluation of results should be a fundamental step in any MCDM process (Lieferink et al., 2014). Different researchers such as Wallenius et al. (2007) describe a method that can “validate” results through progressive articulation of preferences, and it can be proved that the procedures converge in the sense of termination, after a finite number of iterations. Stewart (1996) states that it can work in some instances, but not in others, and since most MCDM methods work with preferences, a method should provide a convergence of preferences after a few interactions, although by no means a mathematical convergence may prove the practical validity of a procedure. In addition, a procedure must not introduce biases, which is one of the arguments that hold for researchers against the DM preferences, and that is shared by this author because examination of publications shows that many times this happens. For instance, authors of an article published in a known journal said that they have decided not to consider correlation in criteria, even when existing in their scenario. Too bad that also the reviewers found this assertion legitimate!

11.3.5

The Proxy Approach

Regarding the proxy solution, it is believed that it can be best understood by taking as examples other fields in science. Naturally, developing such a proxy solution may be as difficult as knowing the true solution—however, it is not impossible. An analogy is found in the pharmaceutical industry when developing a new drug; it cannot be tested on human beings and then examined results; it is tested on mice, assuming they are a good proxy for human beings. In MCDM, it is impossible to say that the proxy is close to the true solution, because it is unknown, but most probably it will be close to it by replicating reality as much as possible, that is, recreating the same conditions existent in a given scenario. However, there is a difficulty here, because literature shows that MCDM methods do not model reality as they should, perhaps due to technical difficulties or by ignoring certain actual facts. Consequently, it would be necessary to have two kinds of proxies: (a) The one that replicates the decision-making matrix, acknowledging that most probably it does not take into account several existing facts. (b) The proxy that modelled the initial decision-making trying to replicate, as much as possible, those aspects that were not considered by the method tested. Most probably, the results from the proxy will be different and also give two different kinds of information: In (a), the proxy can be used to measure how close the result from the tested method is. In this case—where there is flexibility due to no consideration of actual

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facts—the comparison may prompt the DM to a revision of data and maybe changing some values. In (b), the result of the proxy may illustrate to the DM, how the aspects that were not considered influence the final result, and it may involve a drastic modification of the tested method. As an example, if the scenario includes relationships between criteria or between alternatives, but they are ignored, it means excessive flexibility in formulating the true nature of a problem, which is the case (a). If these aspects are indeed considered, the approach will belong to case (b). It is proposed here a procedure for case (a) that uses a proxy of the “true” solution as a benchmark; it does not guarantee solving this quandary with 100% accuracy, but at least gives a rational and quantified response by qualifying different methods solving the same problem. That is, the procedure does not compare one method against other or others; it compares each method against the proxy solution. The proxy must be a method without any preferences, that is, it must process the data as it is furnished by reliable sources, without any alteration by the DM, and of course, using the same initial matrix employed by other methods. In simple cases, if external factors are not inputted into the scenario (that is personal preferences), it is highly likely that the proxy will be similar to the real solution, and solving it, it could be that it is near the “best” solution, if it exists. It is a proven fact that at present, there is no course of action that can evaluate or validate the reliability of the solution reached by a MCDM method, because the “true” solution is not known, and then it is impossible to make a comparison to assess the efficiency of a result found. This chapter presents a procedure that can help in this endeavor. It proposes to use a proxy of the true solution to test a result of any MCDM method; this proxy solution must be the consequence of a more faithful modelling, to replicate as much as possible the real-world conditions, as well as the absence of subjectivity in criteria weighting, and with the result achieved by an indisputable mathematical procedure. For this purpose, this book suggests using the SIMUS method that fulfils these conditions. In so doing, a problem is solved by this method and its result is used as a benchmark to determine the closeness to this result by other methods. This analysis is performed using the: The Kendall Tau Rank Correlation Coefficient to check the closeness of the tested ranking to the proxy. Thus, when comparing with the proxy and for the same scenario more than one method solution, that one with the highest correlation, and then closest to the proxy values, can be appraised as the method giving the “best” solution.

11.3.6

Selecting a MCDM Method

This issue is the source of innumerable questions by practitioners. Normally it is recommended to apply a method according to the characteristics of the problem, which is a number of alternatives and criteria, workload involved, ability to

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introduce real-world aspects, etc. It sounds logical to use the easiest methods; however, it is not recommended because most probably these methods do not have the capacity to properly represent a scenario. Sometimes, and perhaps in matter of costs in acquiring a new software (usually expensive), or possibly because it is necessary to learn it, many practitioners treat different kinds of problems with the same method. That is, it is assumed consciously or unconsciously that a certain method is able to solve any MCDM problem, whether it is simple or complex; however, as has been posted by different researchers, this is not the case—at least with the methods available as of today. It also appears that this subject is not properly addressed in university courses on MCDM, where the rudiments of the techniques are taught. As experience shows, another feature is that sometimes practitioners do not consider certain aspects or characteristics of a problem, either by trying to simplify the modelling or by not being aware of actual existing conditions in the scenario. Selecting a certain method is normally related to this aspect, because using a method that does not have the capacity to solve intricate problems, will produce debatable results which will be certainly different when the chosen one takes into account these facts. A very common example is using methods that do not pay attention to existing relationships between alternatives. Naturally, a very frequent question arises: Which of the different methods gives the most accurate solution? Until today this question remains without a response, and chances are that it never will have one. A good way to help this decision is to analyze which are the features of the methods that best fit the modelling. For instance, there is an advantage in using methods that allow interrupting the process at will for the DM to examine how it evolves and make corrections if necessary (see Chap. 4, Sect. 4.1.3). Once the method delivers a final solution, then the full capacity and ability of the DM applies—correcting, modifying, and even rejecting the whole result or questioning the ranking and making modifications if considered necessary—and this is closely related with the kind of information that the method provides with its solution. A method that only delivers a ranking and its score is of little use if it does not allow an appropriate sensitivity analysis to be performed. Some methods call for partitioning the whole problem into subsets or clusters and then studying each one in particular. This author believes that the system is right for studying the problem in-depth, but it is erroneous to solve each subset independently and then its results added up to those of other subsets. As pointed out in DTLR (2001), the purpose (of disaggregation) is to serve as an aid to thinking and analysis but not advisable for taking a decision. Another element to consider for selecting MCDM methods is if the problem needs to be addressed by compensatory or non-compensatory methods. The first one compensates the disadvantage in one criterion with an advantage in another, while in the second, criteria stand on their own. If stakeholders express their different demands for criteria—for instance, one can ask for minimum cost while another for minimum contamination—it is not realistic to believe that the first will gladly accept an increase to benefit the second.

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This author believes that compensatory methods are not realistic, and that it would be better to work with non-compensatory methods.

11.3.7

The DM Role

At first sight, it appears that the proxy can be developed without the DM intervention. This is definitely not so, since the proxy needs the DM as in any other method, but within a different sequence. That is, as is normally done nowadays, the DM modifies the scenario with his preferences at the very beginning of the process and thus alters it. What he must do is translate his/her experience to the end of the computation process, and then work on the results. This position is also supported by Buchanan et al. (1998) when stating that “. . .alternatively, but less common the DM can be presented with a set of solutions from which the most preferred is chosen a posteriori, the ADBASE system of Steuer used in the context of multiple objectives, Linear Programming is an example of this latter category” (by “category” it refers to a priory and a posteriori articulation of preferences). Thus, the DM needs to model the scenario, which is a simplified representation of reality, providing a complete and realistic representation of the decision environment by incorporating all the elements required to characterize the essence of the problem under study (Massachusetts Institute of Technology, MIT). That is, establishing the necessary criteria according to the type of problem and alternatives to be evaluated, and in consultation with stakeholders, the DM, based on his experience and knowhow has the elements he needs for modelling. He can decide to ignore or not—based on his views—which factors of the real world must be incorporated and which left out, that is to say, he manages the flexibility of his approach. As Ishizaka and Nemery (2013) state, “The modelling effort generally defines the richness of the output.” Once this is done, he must select the most appropriate method to solve the problem. It is also for the DM to consider, when establishing criteria, if they can be analyzed by a sensitivity analysis regarding their potential changes. Keney and Raffia (1993) provided a theoretically sound integration of the uncertainty associated with future consequences. It is the DM who must convince the stakeholders that a certain alternative is the best and why, not because the method suggests it, but for the reason that, in addition to this, the selection is backed up for his experience.

11.3.8

What MCDM Method Can Be Chosen as a Proxy?

Any MCDM method can be used as a benchmark, as long as it does not introduce subjectivities. Because of the last characteristic as per the following reasons, SIMUS is proposed here, although there are also other methods that can also do the job:

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(a) It does not use weights, which when subjective, are the main culprit for discrepancies (Stanujkic, 2014). However, the procedure reckons that not all criteria have the same importance and this is key in its iteration process. This fact, that in most MCDM is taken into consideration by using subjective weights, is also contemplated in SIMUS, but the algorithm it uses, developed by Dantzig and named “Simplex,” determines in each iteration, the relative importance of each criterion by means of a ratio test which selects the alternative that must be out of a precedent iteration solution (see Appendix A.1). In the next iteration, the alternative entering the solution is selected using the costs of opportunity. That is, the method works with data from the initial matrix and uses well-established economic concepts. (b) It does not take any view regarding acceptance or rejection thresholds when comparing alternatives. (c) It can work with any normalization procedure without changing the final results. It agrees with Jahan and Edwards (2015) who postulates that it is a basic rule that when normalizing identical data with different units or scales, the same results are obtained (see Sect. 2.2.1). (d) It is a not compensatory method (see Sect. 11.3.6); therefore, criteria importance can vary independently without affecting others. This is not to be confused with relationships between criteria, such as correlation, aspects the method can also handle (see Sect. 2.5.10). (e) It is based on proven and sound mathematical principles that have been uncontested since their development in the 1950s, and most importantly, producing a Pareto efficient matrix, which mean that the two coincidental solutions start from optimal values (see Sect. 6.5). (f) It starts in the same manner as other methods, that is, building the IDM. (g) It works with a powerful software (Lliso, 2014), which is free and may be uploaded from the Web. (h) The method rests heavily on the know-how, opinion, and experience of the DM; however, it does not require DM preferences but demands his expertise once the solution is known (Sect. 8.3). (i) It has the unique property of automatically producing two identical rankings, for the same problem, and using two different mathematical procedures, both starting from a Pareto efficient matrix, where consequently, all the values are optimal (see Sect. 7.2). (j) It allows a positive analysis of sensitivity, more realistic than that performed in actual methods, since it works on variations of criteria thresholds, not on weights (see Sect. 8.3). Due to all of these conditions, we believe that SIMUS is the adequate MCDM method to act as a proxy. A logical and immediate question arises: If it is accepted that SIMUS results are the closest to the “true” one, why is it not used for all problems instead of trying to solve them through different methods? There are many reasons:

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(a) A method selection is a prerogative of the DM; he may feel more comfortable working with weights that reflect his preferences as in AHP and ANP (Saaty, 1996), and/or applying them to other methods. (b) He may believe that alternatives should be pair-compared and considering preferences, indifference as in PROMETHEE (Brans & Vincke, 1985) or veto thresholds as in ELECTRE (Roy, 1991). (c) He may like the idea of reducing the distance to an ideal point as in TOPSIS and VIKOR (Oprocovic, 1980), and using the most convenient, out of several. (d) Because he already possesses software of some method, is familiar with it, and reasonably wants to use it, and does not see any reason to change. (e) Because certain trivial problems where the personal opinion and preferences are indeed very important, and especially where the consequences of the selection will affect the DM or his company, such as selecting a university, purchasing a car, contracting personnel, or choosing an apartment for rent, he can use SIMUS, but it appears that a method such as AHP or SAW (MacCrimon, 1968) is perhaps more appropriate.

11.3.9

Measuring Similitude Between Rankings

Assume that a problem has been solved by a certain method (XXX), which produced a best alternative and a ranking, and that the same problem is solved by SIMUS, so the DM has the two rankings, using SIMUS as a benchmark. Even if the comparison is mathematical, the DM must exert his ability, knowledge, and know-how to interpret it. He must not blindly accept what the two results show, because he must detect the reason for differences; as a consequence, he may accept, reject or modify the XXX result and introduce weights for certain criteria, if he believes that some criteria should be considered more significant than others. To measure the strength of the relationship between XXX and SIMUS the Kendall Tau Rank Correlation Coefficient (τ) is employed, assigning the proxy τ = 1, because it correlates to itself, as XXX is correlated against the proxy. Assume for instance that the τ between the proxy and XXX method is 0.65 (moderate correlation), which indicates that the latter is short 0.35 from the former; however, it follows to a certain extent the ups and downs of the proxy. The DM must study this difference to investigate why it is produced. After examining XXXs and SIMUS’ decision matrices, he realizes for instance the differences are: 1. In part due to the fact that XXX did not take into account a condition imposed by stakeholders, for instance, that a certain project of the portfolio is already under execution, and then it must be incorporated as an alternative, under the condition that whatever the result, it must be chosen. Consequently, his XXX initial matrix does not reflect reality and should be corrected (this is the “b” kind explained in Sect. 11.3.5).

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2. Maybe the difference can be attributed to the subjectivity in determining criteria weights. The DM can perhaps review his weights and run his XXX method again (this is the “a” kind explained in Sect. 11.3.5). 3. The XXX method did not take into account, for instance, that there exists a strong correlation between criteria C5 and C9. 4. The XXX method is not using resources and restrictions, say for contamination; as a consequence, the selected alternative surpasses the maximum level of contamination allowed. 5. The XXX method did not consider that more than one alternative may have the same score; because the scenario acknowledges that there are joint ventures, for instance, selecting vendors. 6. The XXX method assigned a large weight to a criterion whose performance values are very similar, and then it is not very useful for alternative selection. In other words, a high weight has been assigned to a criterion that is irrelevant (see Sect. 2.1.6.2). 7. Subjective weights were assigned to a qualitative criterion calling for people’s opinions about each project, when they were not even consulted, while for the same criterion SIMUS considered the data emerging from surveys and polls that included people that will be affected by the project (see Sect. 2.5.4). 8. The XXX method did not take into account that the selection must be dynamic because in a portfolio of projects, for example, not all of them start or finish at the same time, and that each one must be performed in a yearly percentage established by the engineering and financing departments of the company (see Sects. 2.5.9 and 2.5.12). 9. The XXX method did not contemplate that different scenarios can take place at the same time and that an alternative may be a part of only one scenario or in several (See Chap. 3). 10. The XXX did not consider that a criterion, with the same values, can have two different calls, one for maximizing and other for minimizing, in two different rows (see Table 5.1 on “Criteria duality”). 11. The XXX did not consider that two or more scores could tie (see Sect. 8.5). 12. The XXX gave several scores. Comparing with the proxy, it appears that the problem called for only one solution, that is, a binary result, which the XXX ignored (see Sect. 6.4). 13. Engineering established that because of scheduling or maybe financial reasons, alternative C must precede alternative E, thus C getting a highest score than E or C > E. However, the XXX shows that E > C, which naturally invalids the result. The DM finds the reason for this reversal, in the sense that this precedence was not registered in the XXX decision-matrix (see Sects. 2.5.2, 5.3, 7.3, and 7.6 Case 1). Of course, there are more aspects that the DM could consider to analyze because of the discrepancies with the proxy, but these are very common. In reality, the differences between the two methods normally indicate that something has not been taken into account which influences the final result. Of course, the DM can choose to

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Table 11.1 Scores for SIMUS and TOPSIS and their ranks SIMUS A B C D

Scores 0.18 0.36 0.21 0.30

SIMUS Rank 4 1 3 2

TOPSIS A B C D

Scores 0.10 0.25 0.19 0.32

TOPSIS Rank 4 2 3 1

ignore the differences, or try to include them in XXX, and then run the method again with the added issue. The biggest advantage of LP is that when a result is obtained for each objective, it is optimal, and as such, it is the true result (mathematically) and cannot be improved. In this aspect, the Simplex algorithm, used by LP, acts like an algebraic formula where whatever the realistic values imputed, the result will be correct, even if it is unknown. Since SIMUS can consider either individually or simultaneously all of the commented situations and even more, it is then reasonable to consider it as the proxy, since as far as this author's knowledge, there is no other method with those characteristics.

11.3.10

Example as How Rankings Can Be Compared

A comparison in MCDM context means analyzing the differences in ordering between rankings. If from the proxy method, the ranking is B-D-A-E and it is the same ranking for XXX we say that there is a perfect correlation or τ = 1. Example: Given a problem with four alternatives A-B-C-D, subject to a series of criteria, assume for instance that the scores for SIMUS are A = 0.18, B = 0.36, C = 0.21, and D = 0.30 and that scores from TOPSIS are: A = 0.10, B = 0.25, C = 0.19, and D = 0.32. The best value for SIMUS is B, which receives a 1, followed by the second that is D, with a 2, the third is C with 3 and the fourth is A with a 4. Similarly, for TOPSIS, it will be: D = 1, B = 2, C = 3, and A = 4. Table 11.1 condenses these scores. Figure 11.1 shows their respective graphs. Notice that both broken lines follow the same direction; that is, both decrease at the same time, then increase together, and finally decrease jointly. If both broken lines were exactly parallel, then the correlation coefficient would be 1, which is perfect correlation. Calculation of τ: It proceeds to work only with the TOPSIS ranking column (T), for determining concordance and discordance values. See Table 11.2. For the first row (4) count how many numbers are underneath it.

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Scores

Fig. 11.1 Graphic showing the two methods’ scores

4.5 4 3.5 3 2.5 2 1.5 1 0.5 0

4 3

A

2

2

1

1

B

C

D

Projects SIMUS

TOPSIS

Table 11.2 Computation of concordance and discordance values Ranking SIMUS 4 1 3 2

Ranking TOPSIS 4 2 3 1 Sum of values

Concordance Counting to lower (C) 3 1 1 0 5

Discordance Counting to higher (D) 0 1 0 0 1

Since below (4) there are 3 values lesser than it, the number 3 is placed in correspondence with this number (4) ranking at right in column (C). For the second row, there is only one number below it. Put 1 at its right. For the third row, there is only one number below. Put 1 this value at the tight. For the fourth row, there are no numbers below it. Put 0. For the discordance, do the same in ascending order. For the first row (4), there is no number on top of this. Place 0 at its right in column (D). For the second row (2), there is only one number larger than it. Place 1 at its right. For the third and fourth rows, there are no numbers larger than 3 and 1. Put 0 at their right. Add up values in concordance and in discordance columns. Concordance (C) = 5 Discordance (D) = 1 Apply now Kendall τ formula (1): τ=

ðC - DÞ ð 5 - 1Þ = = 0:66 ð C þ D Þ ð 5 þ 1Þ

ð1Þ

This means that the result from TOPSIS approximates SIMUS in 66%. By examining Fig. 11.1 notice that the slope of decreasing and increasing values is different for both methods; SIMUS had steeper slopes than TOPSIS. That is, even

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when the two methods have the same directions, the decrement and increment values are different. In this elemental example, it appears that the rate of increases and decreases in SIMUS is greater than in TOPSIS. This means that the SIMUS solution has more discrimination between scores, when compared with those of TOPSIS. This is also a good indication since the greater the discrimination the better, because low discrimination is related to very close values between alternatives and even ties, which makes it difficult for the DM to make a selection. It can then be concluded that the TOPSIS solution is close to the proxy. If the same problem is also solved by other methods, say for instance ELECTRE (Roy, 1991) and PROMETHEE (Brans & Vincke, 1985), their proximity to the proxy helps to indicate the best method for that particular problem. The reason by which the proposed procedure uses Kendall Tau instead of the more popular Pearson Correlation Coefficient (Pearson, 1895), is that both do not measure the same thing. Pearson measures the linear trend while Kendall measures a monotonic trend and even if the relation is not lineal; it also indicates the ordinal association between two quantities. Since Kendall relates concordances and discordances on positions, it characterizes the probability that two variables are in the same order, against the probability that they are in a different order, which is important in this study. Considering these differences between Pearson and Kendall, their correlation values for the same problem are different: Pearson is normally higher than Kendall Tau. The reason for this proposal is that two variables may vary in the same direction, but generally in different amounts or in different slopes. It can be seen in Fig. 11.1 that both SIMUS and TOPSIS vary in the same direction, however, in different proportions.

11.4

Conclusion of this Chapter

This chapter examines the different reasons that cause two different MCDM methods to give different rankings. An analysis has been done for each of those and the conclusion is that the differences are produced by subjectivities. Considering this fact, it is proposed not to use subjective weights. To compare results, it is proposed to use a proxy MCDM that gives results utilizing only the data furnished in the IDM; that is, with no interference from the DM. The SIMUS method is proposed to act as a proxy because of its complete absence of subjectivity in preparing the initial data and processing. Thus, it is also suggested that SIMUS be used as a benchmark to compare its results with the other methods and measure the proximity between certain methods and the proxy, using the Kendall Tau Rank Correlation Coefficient.

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Part IV

Practice of Problem Solving Using MCDM: Support for Practitioners

Chapter 12

Support and Guidance to Practitioners by Simulation of Questions Formulated by Readers and Detailed Answers and Examples A Hundred One Questions a Practitioner May Ask on Different MCDM Aspects, Difficulties Encountered in Solving Complex Scenarios, Detailed Responses, Hints, Explanations, Examples, References Abstract This chapter incorporates innovation in science and technical books, at least in the MCDM area, since it simulates readers addressing the author with questions of varied nature related to situations that may be present in decisionmaking when modelling a problem. There are 101 questions and doubts distributed over eleven main topics, covering the whole spectrum of the decision-making scene. Each question, concern, or interest is thoroughly addressed normally in three parts. First, explaining it, second, posing an example, and third, by an analysis. When corresponding, references to the theoretical part of the book, that is, to the eleven precedent chapters, are suggested. Thus, the reader has a complete theoretical and practical explanation of his/her question that blends two aspects: an instruction manual and personal guidance or tuition. The chapter contains 12 graphics and 22 tables, which is a measure of the depth the different questions are answered. It has an only purpose: to answer people’s concerns, for instance, How can I model this problem? Which is the best procedure for. . . ..? When to use Entropy and how to calculate it? Is there an example where I can see how to proceed?

Introduction The first section of this book intended to establish the foundations, bases, and conditions for MCDM projects, in a sort of theoretical approach, addressing what must be done. This second section has a working, hands-on approach, aiming at how to proceed to get results and, thus, aims at facilitating the work of decision makers (DMs), professors, practitioners, and students when they must solve intricate problems.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 N. Munier, Strategic Approach in Multi-Criteria Decision Making, International Series in Operations Research & Management Science 351, https://doi.org/10.1007/978-3-031-44453-1_12

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There are myriads of different problems or scenarios in all aspects of anthropogenic activity, thus involving economics, engineering, health, environment, sustainability, manufacture, medicine, energy, agriculture, civil construction, metal working, government policies, education, etc., and very rarely similar, even if two refer to the same issue, for instance, designing a road. Therefore, each problem is unique, unlike from another, and even if the difference is small, most probably the results will not be the same because of the dissimilar characteristics and demands. The enormous variety of projects, plans, strategies, schemes, and policies, added to stakeholders’ expectations and uncertainty of some data, makes MCDM a very difficult task, and each one a distinctive endeavor, and its variety and the special characteristics of each scenario, constitutes the main feature of this activity. This segment aims at being a guide for participants and as remote tutor by simulating receiving questions from practitioners and responding to them by explaining its scope and importance and how to model a particular problem in the Initial Decision Matrix for its further processing by a MCDM method. It can be resumed as How can I manage my scenario? To make things more complicated, each one of the four elements that conform the initial matrix, that is, alternatives, criteria, data, and resources, have many variations and interrelationships that make it impossible to solve the problem manually and, hence, the necessity to use mathematical methods, which give a solution depending on assumptions that are normally different in each one of more than about 200 existent methods, the number that increases steadily. Results from all MCDM methods are numbers or scores that quantify each alternative, the higher, the better, in both maximization and minimization, and a ranking is generated when ordered from highest to lowest. This ranking is the tool we use to select the best alternative, or the second best, or the third best, and so forth. Even if an alternative is chosen by a MCDM method to be the best, probably it is, due to the data and the mathematic procedures involved; however, there are also other factors that must be considered, for instance, the stability of this solution to changes in criteria, that may suggest to select the second or the third best; therefore, normally, the solution given by a method is not straight forward, let alone definitive. It is here where the DM judgment is most relevant, by analyzing and examining the solution, asking others for more information, searching, and finding the exogenous facts that are not in the Initial Decision Matrix, which might have a strong influence on the result. When the DM has considered all these factors, many of them unknown but that can be studied through statistics, he is in a condition to submit a recommendation to the owner of the project and stakeholders. It is necessary to bear in mind that the DM must possess documented reasons to justify his selection of an alternative due to the fact that the mathematical result is only a guide since the decision is taken by the DM. The aim of this chapter is to answer questions “On demand.” Example:

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221

Question: I have heard that criteria weights derived from entropy are objective and recommended, but what is entropy and why is it better? Explanation: What is entropy? Entropy is a concept derived from Thermodynamics that roughly indicates “disorder.” For instance, superheated steam has millions of “particles” moving in any direction; this means that it is very difficult to know where a certain particle is at a certain moment, that is, there is complete uncertainty on its position. In this condition, it is said that there is a very high entropy; in other words, entropy is a concept linked with uncertainty. As an example, in communications, speaking over the phone, a person at the other end of the line may have difficulty in understanding because there is a lot of noise, or the sentences are garbled, or words are missing, or the message is corrupted, or a poor understanding of the language. In this case, the meaning and content of the message are very uncertain, and similarly to Thermodynamics, it is said that it has high entropy. The maximum value for entropy (S) is “1” (complete uncertainty) and “0” (certainty). Its difference from 1 (D) indicates the “quantity of information” or capacity of a criterion for evaluation. Thus, for a value of entropy S = 0.85, the information quantity is D = 1 0.85 = 0.15. In 1948, Claude Shannon discovered this similitude between messages or information received through any mean with the thermodynamic concept of entropy; this discovery had a tremendous impact because it was the foundation of Information Theory. In MCDM we can apply this concept in determining the real weight or significance of a criterion, considering that the set of its performance values also transmit a message. Consequently, its importance depends on the quantity of information it can deliver, and in turn, it is grounded on the dispersion or discrimination of its values. To understand this concept, think that if the values in a criterion are very similar, they are unable to evaluate alternatives since these evaluations will be very close. When we compute the entropy for each criterion, it gives its real significance, which is invariant whoever the DM is. What is the advantage in using entropy, or speaking properly, its complement? Because it is an objective weight developed based on the data contained in each criterion. An elemental example: Suppose that we have to select among 4 different movies and have knowledge that they have been appraised by art critics in a scale 1 to 10 as: 2-2-1-2 (Table 12.1). It is very difficult to extract a conclusion because all values are very close, and the information quantity or capacity for evaluation is very low or perhaps null. Different is the case if the values are 3-8-4-7. In this case, the quantity of information and the corresponding capacity evaluation is much higher. Therefore,

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Table 12.1 Initial decision matrix

Press 1 561,896 3,200 2,536

Alternatives or options Press 2 Press 3 562,397 561,821 4,000 3,700 4,219 3,018

Press 4 563,601 3,700 4,621

Sum 2,249,715 14,600 14,394

Table 12.2 Computation of probabilities Probabilities for each performance 0.250 0.219 0.176

0.250 0.274 0.293

0.250 0.253 0.210

0.251 0.253 0.321

discrimination is fundamental, and this is what value “D” indicates. The larger the discrimination, the less the uncertainty. Example of calculation of entropy and quality of information: Assume you have four printing presses subject to three criteria. 1. Compute for each criterion their sum (column at right of Table 12.1). Notice that criterion C1 has four very close values; therefore, it is expected that its D will be zero or nearby. 2. Normalize these values by dividing each performance value by the sum of all values in its criteria and save them in a new matrix. Thus, for the first performance value: 561,896/2,249,715 = 0.250 (Table 12.2). 3. Compute the value of constant K = - 1/ln(n) (being “n” the number of alternatives). = -1/ln (4); K = - 0.721. 4. Multiply each normalized value by its natural logarithm. Example for the first number (0.250) will be: 0.250 ln (0.250) = -0.346. Table 12.3. 5. Find the sum of each row and multiply it by K. This is the entropy (Ei) of the criterion, but we want its complement Di, which is the quantity of information. 6. Compute all Ds and find their sum.

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Table 12.3 Computation of criteria weights

Computation of criteria weights Ei= ∑

ln

Ei x K

pj ln pj –0.346 –0.333 –0.306

–0.347 –0.355 –0.360

Criteria Difference entropy (Ei) D)

–0.346 –0.348 –0.328

–0.347 –0.348 –0.365

1.386 1.383 1.358

1.000 0.998 0.980 Σ Ei

Criteria weight (wi)

1 - (Ei x K)

D / Σ Ei

0.000 0.002 0.020 0.023

0.000 0.100 0.900

7. Normalize each value using this total. Thus, the criterion with the highest normalized Di is the most significant. These values are the weights used for the criteria (Table 12.3). As confirmation of the observation in Table 12.3, notice the value of D1 for the first criterion. In is zero, that is, it does not convey any information, as presumed. This tutorial is the core of Segment II; it formulates in plain language 99 potential questions and situations that appear in most types of projects and scenarios. Each one is defined, commented on, and solutions are proposed to answer it. Most questions are illustrated with real-life cases and examples, the decision matrix is built, with tables and figures complementing the picture, and the problem is finally solved. It also provides hints for the reader. All the 99 questions or consultations are distributed in numbered sections, for instance, related to Criteria (Sect. 12.3), with 36 questions, to Sensitivity analysis (Sect. 12.10), with 6 questions, Scenarios (Sect. 12.3), with 4 questions, Results (Sect. 12.8) with 12 questions, etc., and thus, if the reader has a doubt on “criteria,” all questions regarding that sector are under that label. The reader must go to the Table of Contents, Chap. 12, identify the area of interest, for instance, “Criteria,” and within it, find if there is a question similar to his/her. Please, consider that: Questions: Are in bold Brief explanation of the subject questioned: In black Examples and solutions: In italics Consider that all answers are linked to using the SIMUS method; it is not known if these answers can be found with other methods

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12.1

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Scenarios

Question 12.1.1: How to Select Contractors? This is a common activity in MCDM scenarios; the different contractors are the options or alternatives in columns in the Initial Decision Matrix, and the conditions they have to comply are the criteria in rows. These conditions are, for instance: The number of projects in the last 15 years and their amounts, the number of engineers, architects, etc., the number of contracts finished on time, the number of times the contractor was taken to court, the number of own equipment the constructor will have in this site or sites, etc. This is the normal procedure; however, especially in large projects, the matrix is a little more complicated. Example: Selecting simultaneously contractors for each construction area in a large project In a large hydroelectric construction project, the promoter, a national government agency, wants to select main contractors at the international level, simultaneously to subcontractors for all trades. Therefore, it sends invitations to selected firms worldwide asking for bids. There is neither a limit on the number of companies that can participate nor in the formation of joint ventures. The promoter is asking for bids for 12 different activities identified by their codes. CC—Construction companies (Main contractors) LC—Land clearing E&F—Excavation and foundations ME—Mechanical and Electrical equipment CON—Concrete suppliers STR—Steel truss and roofing Si—Siding FLOO—Flooring IE—Industrial Electric EINS—Building Electric IP—Industrial plumbers IPAIN—Industrial painters The selection of all contractors is based on 80 criteria. The number and type of criteria are different for each trade, and joint ventures are accepted. Question 12.1.2: What Are Complex Scenarios, and How to Manage Them? Normally, one deals with only one scenario, for instance, select a location among several sites to install an industry, but this case refers to various scenarios and where the same alternative may be simultaneously in different ones. An example can be found in Chap. 3, Sect. 3.2. Question 12.1.3: Complex Scenarios There is not a clear definition of what is a complex scenario because of the variety of projects, situations, characteristics, etc. When one considers that alternatives,

12.1

Scenarios

225

criteria, and resources in the hundreds are related among them, it is possible to understand the complexity of relationships and, most especially, that any of them may influence others. It is very seldom that we can optimize this set of different elements, and in reality, there is consensus that there is no optimal solution in multicriteria. In truth, we cannot optimize costs and at the same time optimize benefits since they are opposite. Therefore, MCDM tries to achieve a compromise solution, that is, a solution that satisfies all parties. Elemental and trivial problems can generally be represented as a linear hierarchy, that is, a frame where decisions and precedence run top-down, without transversal or horizontal relationships among objective, criteria, and alternatives. This is called the “Military hierarchy” used by the military since the dawn of civilization, the same as pharaohs in Egypt and Mesopotamia, and also pursued by feudalism in the Middle Ages. Orders and instructions came from the top and lower echelons obey, without the possibility of contributing with their own ideas and solutions. This is the way industrial organizations worked in the nineteenth century until well advanced twentieth century, when selecting alternatives consisted in a costbenefit analysis and not taking into account anything else. Thus, it was a relatively easy process. Everything changed by the 1950s, when enterprises and decision-makers realized that new aspects, other than benefit and cost, should be considered in the decision process, promoted by social and environmental pressure. These aspects were, among others, social issues, like limiting the maximum number of working hours for workers in a day, health benefits, better working conditions, measures to be implemented to avoid work accidents, etc., as well as demands for environmental groups, supported by international agencies like United Nations, regarding energy conservation, care about water consumption, prohibition to dump industrial liquid wastes into rivers, etc. These aspects required hiring more people and creating new departments in the company related to social issues and environment and, most important, to decision power. Obviously, the old lineal structure collapsed; now, there were people not in the first level but high enough to convey and discuss their ideas and even decisions to the upper echelons in a sort of feedback mechanism. Some of them even had the power to stop work if there were serious violations. Consequently, the original lineal structure was no longer appropriate; large companies realized that and started to change from the rigid lineal structure to more complex industrial schemes, which lead to the creation of complex networks where many things are interconnected. Due to these reasons, MCDM techniques that appeared in the 1950s addressed the decision problem with many alternatives subject to many criteria, linked to the traditional areas of economy, finances, and engineering, but now incorporating government, risk, health, contamination and pollution, sustainability, transportation, supply, informatics, etc. In many countries there are associations between government, university, and industries to work together in developing new products, and of course, this complicated a little more the decision-making process.

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At present, there are perhaps more than 200 MCDM methods that have been developed with the network scheme in mind, although there is one method that still considers that the lineal structure is correct and can be applied to complex scenarios. A very complex project is ITER, which is the building of the first large fusion reactor, where many nations and hundreds of engineers, physicists, chemists, etc., participate. All this extense discourse was meant to illustrate what a complex project is, since there is not, as said before, an agreed definition, but of course, most projects have only a certain degree of complexity. Common complex scenarios involve large engineering projects, mainly in selecting between diverse options, like: • Circular economy, related to society, economics, and environment. • Construction of wind and photovoltaic (PV) farms for energy generation, depending on weather, sun, and technology. • Crime reduction, with strong links to human rights. • Domestic and industrial wastes, associated with economics and environment. • Energy generation using hydro, thermal, nuclear, wind, sea waves, PV, etc., with strong links to demand, industrial grow, and society. • Farm equipment, connected with economies of scale. • Government policies, normally concomitant with society and economics. • Housing development with different styles and sizes, market dependent, which complicates the construction of large undertakings. • Industrial location, correlated to many different aspects of society, environment, and transportation. • Large irrigation undertakings, fundamentally linked with the efficient use of land, water availability, weather, etc. • Multinational assets and plans, like sewerage networks associated with public health. • Off-shore wind farms, intimately connected to navigation, existing subaquatic utilities, etc. • People relocation, an extremely important and sensitive issue from social and economics points of view. • Poverty reduction, one of the largest problems in all societies, and interrelated with public health, economics, and hunger. • Power equipment, linked with many different technical, economics, and environmental aspects. • Railways, one of the main ways the world moves its production as well as people. • River contamination, one of the biggest problems in oil-producing countries and very related to the mere existence of natural life. • Road construction, always expanding, because of population growth and commerce. • Supply chain, fundamental in our present economy, and depending on weather, roads, demand, transportation, and time.

12.1

Scenarios

227 Scheme of Dependency

Alternatives

A1

A2

A3

1

4 Criteria importance to evaluate alternatives

computed by entropy Criteria selection depend on alternatives

Criterion Criterion Criterion Criterion

C1 C2 C3 ----

Cm

Alternatives selection do not depend on trade-offs values a11 a21 a31 ---am1

a12 a22 a32 --am2

a13 a23 2

a33 --amn

Entropy S1 Entropy S2 Entropy S3 --Entropy Sn

3

Fig. 12.1 Sketching multiple relationships at different levels

• Sustainability at city, regional, and national level, linked to our own future and thus with multiple connections. • Transportation modes involving railways, barges, maritime, and trunks. • Urban highways, linked to oil and cement industries as well as economics and society. • Urban issues, a fast-increasing subject linked to a myriad activity. SIMUS works with complex scenarios and is especially adaptable because its algebraic structure uses inequations instead of equations, thus providing a large latitude that allows solving complex scenarios. The drawing in Fig. 12.1 is a scheme that describes relationships between different elements. Here, the reader can see that alternatives are the first to be selected, followed by the set of criteria chosen and the weights (if used). Observe that alternatives cannot be evaluated by criteria trade-offs, but for criteria weights obtained by entropy or statistics. Example: Two cities are separated by a river, and both constitute a metropolitan area of about 2 million people. Because of that, they share budgets for improvement from national sources (Fig. 12.2). City A has three main districts and city B has two. Each district has its own scenarios, with a total of 15 for the two cities, and the problem consists in determining which of the 15 scenarios must be developed. For city A District 1: Three projects as follows: • Construction of a community center • Enlarging the gas network • Improving streets condition (Infrastructure)

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Fig. 12.2 Network for two cities

District 2: Three projects as follows: • Housing development • People safety on streets • Infrastructure (water) District 3; Two projects as follows: • House development • Infrastructure (water) For city B District 1: Three projects as follows: • House development • Urban roads • Construction of a park and botanic garden District 2: Four projects as follows: • Improvement of the business center • House development • Urban streets • Park

12.1

Scenarios

229

– Notice that there is interrelationship between districts 1 and 2 of city A (dashed line in brown) because they are geographically very close districts and share many utilities. – House development in districts 2 and 3 of city A share house development; however, they are exclusive; that is, if one is developed, the other is not chosen (blue dot arrow). – District 3 in city A is related to district 1 in city B (double solid black arrow). – For scheduling reasons, infrastructure for district 3, city A must precede infrastructure for district 2, also in city A (solid black arrow). – Observe that there is an exclusive restriction between two sub-alternatives in district 3, city A. This is a fictitious example of a network but was designed to illustrate some of the relationships that may link criteria and sub-criteria, showing precedence and exclusivity. A cluster of alternatives, and how an alternative may be in different places. A real example of a very complex problem is an oil refinery. The industries involve 5 areas, that is, oil extraction, transportation to refinery, oil refining, storage, and transportation of final products to the market. The complication lies in the sequence or precedence between processes, that is, extraction → transportation to refining plant → refining → storage → transportation to market. On top of that, there is a matter of holding a balance between oil extraction, refining, and transportation to market, where storage in tanks acts as a buffer between oil products from refining and the market. As seen, this is a very complex problem that can be solved by MCDM using linear programming. Question 12.1.4: What Is Modelling? Modelling in MCDM is a mathematical representation of the scenario. Normally, in industry, a very usual procedure is the construction in wood of a physical model to scale (called “mock-up”) of a car or an airplane, subject to certain conditions, generated by adequate means (for instance, a wind tunnel), which replicates the physical working environment of the device, and that allows scientists to extract conclusions that are further applied to the real contraption. In MCDM there is not a material device but a problem, or more generally a scenario, and thus we use mathematical methods. The equivalent to a “mock-up” is a matrix or mathematical method where all characteristics of the problem are included. Considering all the data, a MCDM method is applied that gives us an idea of the “behavior” of the problem. Based on this result, it allows the DM to make corrections, not to reality, but to the initial matrix, adding or deleting alternatives or criteria and adjusting, if possible, existing performance values based on analysis, research, or experience. Modelling can thus be defined as the construction of this matrix, incorporating, as much as possible, all characteristics of the problem.

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In the opinion of this author, this is the greatest failure in present-day MCDM practice because most of them are unable to model reality. To make things worse, by using subjective weights, these methods modify reality by altering data.

12.2

Sequence of the MCDM Process

Question 12.2.1: How Is the Sequence in a MCDM Process? Specifically, Which Are the Steps and in Which Order? The sequence of solving a problem in MCDM is always the same. It starts with the definition of alternatives, criteria, performance factors, and resources that allows to build an Initial Decision Matrix where all this information is loaded. In SIMUS, alternatives or options are always in columns and criteria in rows, following the format of mathematical equations and inequations. The most important factor in all methods is to build an Initial Decision Matrix that replicates as close as possible all the characteristics of the scenario. SIMUS also incorporates sensitivity analysis (using IOSA add-in) that automatically feeds from SIMUS results. Example: Flowchart for MCDM process and sensitivity analysis using SIMUS/ IOSA software. It illustrates the whole process, from the decision matrix to reporting to stakeholders. (a) It starts with a finite number of pre-determined alternatives, usually through technical studies, which support the DM selection. See the green box in Fig. 12.3.

Fig. 12.3 Sequence followed by SIMUS/IOSA

12.2

Sequence of the MCDM Process

231

(b) It follows the DM criteria selection, considering all features in the scenario, and finishes by establishing the performance values, that is, the amount with which each alternative contributes to each criterion. Quantitative values normally come from technical departments, from vendors and suppliers, and from data like distances, prices, workforce, etc., while qualitative values are supplied by surveys, simulation, or statistics, and also as crisp values from fuzzy calculations. The DM also needs to specify each criterion for the respective action, that is, maximize, minimize, or equalize, and establish the type of results he wants, for instance, in decimal, integer, or binary scores. He must also include the quantity of available resources, percentages of allowed contamination, etc., for each criterion. This is the Initial Decision Matrix with which SIMUS works. (c) Processing this data, SIMUS shows in its primal screen the scores for each objective, which are optimal (see black table at right). At the same time, SIMUS delivers the dual screen (in green) with the values of the marginal utilities for each objective. These values are used later in sensitivity analysis. (d) Then, the IOSA screen, in pink, takes over and computes the total utility curve for each objective (in black at left) according to the DM requirements. From here, the DM analyzes the results, changes what he considers necessary, run the software again with the changes, and, when satisfied, prepares the final report for the stakeholder (green box). Question 12.2.2: Which Is the Best Method to Solve My Problem? Researchers agree that there is not a best method to solve a specific problem, let alone any kind of problem. Consequently, the DM must choose the method that better fulfills the scenario requirements and avoid subjective assumptions affecting initial data. Perhaps the most important aspects to consider once a method has been chosen is its ability to solve the problem, and it depends on many factors, such as the capacity to modelling a scenario that replicates the problem as much as possible. Unfortunately, most MCDM methods are unable to do it. Suggestion. This author has prepared an interactive template to help the reader to select a MCDM method according to the problem characteristics. It is a matrix with 10 different MCDM methods labels in columns, from the easiest to the most complex. In rows are 54 characteristics of projects. The reader can select which of these characteristics are present in his/her problem. The last column of the matrix shows the number of MCDM methods that satisfy each characteristic. As an example, assume that the DM problem has these characteristics. Characteristics # 1—Is a simple scenario. 18—Independent criteria. 20—There is clustering. 46—Necessity to evaluate criteria relative importance.

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That is, a total of 4. When this data is inputted, the matrix will inform the DM about: – The number of methods that comply with each requirement. For instance: SAW complies with only 3. AHP complies with 4 (that is, with all of them). PROMETHEE complies with only 2, and so on. – The last row shows for each method how much they are short in satisfying or fulfilling these 4 requirements. Obviously, the less this difference, the better, and thus the most adequate method for this problem is AHP, which gets 4–4 = 0, because it complies with the 4 requirements, while TOPSIS is short in 1 and ELECTREE is short in 2. This table can be found in Sect. 4.14 in Munier—Mathematical modelling of decisions problems—Using the SIMUS method for complex scenarios—Springer, where there is a detailed explanation and application examples. Question 12.2.3: Is It Possible to Get an Optimum in a MCDM Process? Normally not, because very often there are opposite criteria or objectives, such as maximizing a benefit and at the same time minimizing a cost. However, you can get an optimum when using linear programming (LP). This is the only MCDM method that can deliver optimal solutions in single-objective problems, provided that they exist. However, it has two serious drawbacks related to its scope of application, since it works with only one single objective, and with quantitative criteria, a situation that is not usual in practice, except in large undertakings like oil refineries, steel plants, paper mills, etc. Nevertheless, SIMUS, based on LP, overcomes these two drawbacks but does not produce optimal results, same as all heuristic MCDM problems. It does not use any type of weights and is able, because of its algebraic structure, to model efficiently most scenarios, including any mix of quantitative and qualitative (objective and subjective) criteria. Question 12.2.4: How Do I Know If a Criterion Is Satisfied? Quantitative criteria have low and high limits that express their scope of acceptance, like the maximum amount of funds available, the maximum number of hectares accessible, the minimum quantity of water per hour, the percentage of contamination, etc. SIMUS shows in its result in what proportion that goal or resource has been satisfied or used after obtaining the best result. It is the same for qualitative criteria, for instance, establishing a minimum necessary number of people required to accept a project or the maximum percentage expected as a return on an investment. The software also provides the same information as in quantitative criteria.

12.2

Sequence of the MCDM Process

233

Question 12.2.5: How Do I Know that My Scenario Is Not Feasible? A feasible solution is one that satisfies all criteria. Unfortunately, most MCDM methods assume that there is always a solution for a problem, and this may not be true in some cases. It can happen because the combination of equations and inequations, that is, criteria, do not always converge in the sense of forming a feasible solutions space. As an example, the WHO (World Health Organization), established that the minimum amount of water per person and per day is about 140 liters, and usually there is also a maximum amount of water per person-day, to avoid water wasting. Therefore, if in a project it is specified that as a maximum, water supply will be 85 liters, but with a minimum of 100, obviously this causes infeasibility since there is no solution, for there is not a value that can satisfy both requirements simultaneously. Of course, there are also other aspects that produce a no solution, as, for instance, errors in data and in the mathematical symbolism. Look at Table 12.4; it informs the reader that there is not a feasible solution. Observe that in the fourth row, in column “Requirements,” it is required to build at least 35 type 2 houses. This requirement may obey, perhaps, to market analysis. Column “Results from computation” indicates, in the same row, that the minimum number of units of type 2 houses must be 31. Consequently, it is clearly seen that the criterion is not conformed; therefore, the project is unfeasible. The value of 31 houses was obtained because the software found that some resources could be better used in building more houses of other types. Observe that in all the other criteria the relation between requirements and values from the software solution is maintained. It can be perceived the importance that all alternatives and criteria be analyzed simultaneously, something that only a method can do, because it has the capacity of recording resources, comparing and balancing them in all criteria. Unfortunately, all MCDM methods, except linear programming, Goal programming, and SIMUS, ignore this fact of capital importance by treating each criterion separately and then adding up results. Question 12.2.6: How to Instruct the Method to Rank a Certain Project as the Best? Sometimes, due to political or scheduling reasons, a project must be in the ranking and in the first position, irrelevant of whether the method selects it or not. Example: Selecting sites in a hydroelectric project but incorporating in the final solution a project which is under construction. In this actual case, a study was performed to select the best places for hydropower schemes in Nepal, on the Himalayas. There were about 16 locations preselected; however, project #3 was under execution and, consequently, it should be in the final list. This strong restriction was incorporated into the initial matrix by placing a 1 at the intersection corresponding to column 3 and a criterion called “Mandatory

55

Message from the model:

Number of dwellings to be built by hectare

1.19 33.12

52 5.52 5.20 1,573

65

1.36 37.80

80 6.30 6.15 1,796

95

1 1 1 1 1 1

160

Plan for type 3 house 26,500

80 2,211

3,800 368 341 105,000

4,588

7,402 30 30 31 31 10 10 71 71 1,442,149 1,442,149

Result from computation

30 31 10 71 'Solver has not found a feasible solution'

0.98 27.30

3 S ewage production baths & kitchens (m /day-house) Electric energy consumption (kWh/day-house)

Total cost (Euros) 1,442,149

47 4.55 4.00 1,297

Minimum floor space (m /house) Maximum density (persons/house) Minimum density (persons/house) Water consumption (liters/day-house)

2

Maximum floor space area (m /house)

1 1 1 1

1 1

122

Plan for type 2 house 22,000

Urban development

≤ ≤

≥ ≤ ≥ ≤

≤

≤ ≥ ≤ ≥ ≤ ≥ ≤ ≥ ≤ ≤ ≥

90 3,000

3,800 400 340 105,000

5,000

7,700 30 40 35 48 5 10 50 95 1,500,000 1,250,000

Total number of houses

m3/day kWh/day

2

m persons persons liters/day

2

m

2

m dwellings dwellings dwellings dwellings dwellings dwellings dwellings dwellings Euros/ha Euros/ha

Units

Requirements Action

12

2

1 1 1 1

69 1 1

Area of each lot (m2) Minimum number of units of type 1 houses Maximum number of units of type 1 houses Minimum number of units of type 2 houses Maximum number of units of type 2 houses Minimum number of units of type 3 houses Maximum number of units of type 3 houses Minimum number of dwellings Maximum number of dwellings Maximum total investment (Euros/ha) Minimum total investment (Euros/ha)

Plan for type 1 house Investment per dwelling (Euros): 16,800

Table 12.4 Unfeasibility

234 Support and Guidance to Practitioners by Simulation of. . .

12.2

Sequence of the MCDM Process

235

selection,” using as a limit (RHS) the number 1 and the mathematical symbol “=”. The criterion will then be: 1 = 1. In this way, the software was instructed to consider that only this project must precede any other project in the final ranking. Of course, the project is still subject, as the other 15, to the same set of criteria, but for the “Mandatory selection” criterion it is the only one. See this case in Sect. 14.3.1. Question 12.2.7: How to Address Urban Planning in Peri-Urban Areas? This is a typical case in urban planning, when large undertakings like construction of sewerage, water, roads, etc., are shared between several cities and subject to the same set of criteria. The aim here is to determine which undertakings will be developed in such a way that each city uses his allotted share of funding, considering that each one must receive a reasonable and equitable distribution of projects. Example: Urban planning in a city and its satellite cities. This project refers to a large metropolitan area involving 27 peripheral cities, subject to 24 criteria, with many projects that must be shared only by some of them, while various others are significant for a single city. The objective of this exercise is to determine which is the set of road or thoroughfare projects that most benefit the city and suburbs. Criteria are divided into three clusters. The first one refers to the existent relationship between projects and municipalities. That is, municipality H needs projects B and C, while municipality M need projects D and F, and perhaps also C. The second one identifies projects with municipalities. That is, Municipality P is related to projects H and J. The third one is linked to the amount of funds for each municipality. The DM establishes the minimum number of projects for each municipality. To make sure that the result is correct, the DM examines the results comparing columns LHS (results from computer) and RHS (goals to achieve). They reveal that all criteria are satisfied since the computation values are higher, and sometimes equal, as the requirements. In this case it is convenient to use a binary matrix (i.e., using only 0s and 1s) combined with a quantitative row for criterion “Cost.” See this project in Sect. 14.5. 1. There is no problem in incorporating more quantitative and qualitative criteria. For instance, an example for the latter could be the level of approval or disapproval by the people in each municipality for the construction of a domestic waste incinerator to serve several municipalities. Surely, none of them will be happy to have the plant in their territory. Question 12.2.8: Can I Identify the Indicators that Have Real Influence in a City for Improving Quality of Life? Yes, you can. It is first necessary to define quality of life and what it entails. Normally, it involves many different areas, from health to transportation, municipal services, environment, cultural issues, etc.

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An example is proposed, obviously largely reduced, to illustrate how to proceed. Example: There is a large city (Central city), surrounded by 14 smaller cities in its outskirts. The case aims at determining a final set of 20 indicators to apply for control of city advance regarding liveability. Initially there are 76 initial indicators, in 27 clusters or areas, subject to 30 criteria. The areas for indicators are: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.

Environmental education Strengthening education Citizen participation Alternate transportation Urban fabric Regulations Investments in science and technology Dwellings Irregular settlements Decentralization Vertical dwellings Cultural issues Tourism Constant control of public transportation Private investment Program evaluation Space quality per dwelling Communication at government level Organization manuals Relationship between university and government Government timing Continuity Development plans Green spaces Protected areas lost Investment on environmental services Economic dwellings

Criteria relate to four clusters: • • • •

Sustainability goal (14 criteria) General selection criteria (5 criteria) Areas participation (8 criteria) Environment Canada Framework (3 criteria)

The initial matrix, Table 12.5, has been reduced to only 10 indicators to facilitate readability; however, the partial result shown in the solid blue row was obtained considering the 76 indicators and the 30 criteria. Due to the large number of indicators, the DM wants a final set of only 20.

Units:

INDICATORS:

ASPECTS:

1. - Sustainability goals Inter-generational equity Intra-generational equity Minimal impact on the nat environ. Living off the interest of renew. res. Minimal use of non-renew. res. Long-term economic development Individual well-being Enforcement Decentralization Citizen's participation and control Government participation Household economics Local economic conditions Government efficiency 2. - General selection criteria Representative Understandable by potential users Comp. w/ind. develop.n other sites Cost effective Attractive to the media 3. - Areas participation Environment Social - Economics Infrastructure Municipal Administration Social Economics Citizen Participation Municipal organization 4.- Environment Canada framework Condition Stress Response

CRITERIA

INDICATORS:

ASPECTS:

1 1 1

1 1

Citizen participation Qty. of proposals received

0

Strengthening

Qty. of courses given

1

1 1

1

1 1

1 1 1 1 1 1

1

1 1

1 1 1 1 1 1

1 1 1 1 1 1

number

Qty. of proposals received

number

Citizen participation

Strengthening

Qty. of courses given

Use of bikes

Alternate transpor.

Invest. in sci. & technology

0

Percentage to research activities

Invest. in sci. & technology

1

1

0

Qty. of areas in compliance

1

1

1 1 1

1 1 1 1 1

1 1 1 1

%

Qty. of areas in compliance

1

Qty. of paperwork reassigned

1

1

1

1 1

1 1 1 1 1

1 1 1 1

number of files

Qty. of paperwork reassigned

0

Qty. of dwellings built

Vertical dwellings

1

1 1 1

1

1 1 1 1 1

1 1 1 1

number

Qty. of dwellings built

Vertical dwellings

Culture

0

Food system

Culture

1

1

1 1

1 1 1 1 1 1 1 1

%

Food system

Private investment

Private investment

1

1

1

1 1 1 1 1

1 1 1 1 1 1

%

0

0

Public Percentage of transportation private control investment

Transportation

1

1

1 1

1 1 1 1 1

1 1 1 1 1 1

hours/employee

Public Percentage of transportation private control investment

Transportation

LHS

2 1 1 1 2 2 2 1 3 3 2 76

> > > = 0 0 1

2 3 5 10 3

Req. 2 1 4 2 4 1 2 7 1 2 4 6 1 6

> > > > > > > >

> > > > >

> > > > > > > > > > > > > >

RHS

1 0 0 1 0 0 0 1

0 0 1 2 1

Comp. 2 2 2 2 2 1 1 0 1 1 1 1 2 2

Choose at least:

3 indicators related with condition 4 indicators related with stress 3 indicators related with response Choose 16 aspects

2 indicators related with environment issues 1 indicators related with social-economics issues 1 indicators related with infrastructure issues 1 ind. related w/ municipal administration issues 3 indicators related with social issues 2 indicators related with economics issues 2 ind.s related with citizens' participation issues 1 ind.s related to municipal organization issues

6 representative indicators 5 indicators being clearly understood by citizens 10 indicators able to be compared with standards 10 indicators being cost effective 10 indicators attractive to the media

2 inter-generational indicators 1 intra-generational indicators 4 indicators showing minimal impact 2 ind. w/ living off the inter. of renew. res. 2 ind. w/ minimal use of non-renew. res. 1 ind. related with long term econ. develop. 2 indicators related with individual well-being 7 indicators related with enforcement 1 indicator related with decentralization 2 ind. related with citizen's particip. and control 4 indicators related with govern. participation 2 indicators related with household economics 1 indicator related with local econ. conditions 6 indicators related with government efficiency

Sequence of the MCDM Process

0

Use of bikes

Alternate transpor.

1

1

1 1

1

1 1 1 1

1 1 1 1 1 1

1 1 1 1 1 1

%

Percentage to research activities

1 1 1 1 1 1

1 1 1 1 1 1

bikes/100 pers.

Table 12.5 Initial Decision Matrix (Reduced)

12.2 237

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Observe the box above the table; it is used for the DM to indicate the number of indicators he wants. Nevertheless, there is a lower limit because if the number of indicators is too low, say for instance 5, the software will indicate that this system is unfeasible because the small number of indicators cannot simultaneously satisfy the 30 criteria. The upper limit is, in this case, 76. In this way, the software reduces the original number of 76 indicators to 20, and this is usually a necessary step for the DM to be able to control progress with a more manageable smaller number of indicators. In this example it is interesting to notice that: 1. The matrix is fully composed by 1s and 0s, that is, it is a binary matrix. Its objective is to show in rows inclusive indicators, which number is variable. For instance, criterion (1) “Intergenerational equity” indicates that at least two of those identified indicators have to incorporate this criterion. This restriction can be seen in column RHS, in red. Observe that criterion (8) “Enforcement” is required to be at least in 7 indicators. This difference shows that the DM thought, in selecting the RHS values, that this criterion is by far more important than the former and then should be present in more indicators. This is done to make sure that all criteria are considered in quantities defined by the DM and in accordance with their importance. 2. Criteria are grouped into four clusters, each one with a different general purpose. For instance, the first cluster addresses criteria involving social, environment, law, economics, and government. The second cluster groups hard restrictions, for instance, that the selected criteria must be representative, understandable to users, cost-effective, etc. The third cluster entails areas that participate. This is important to ensure that all fundamental areas are considered. The fourth cluster involves a decision-making framework that is a copy of the framework developed by the Organisation for Economic Co-operation and Development (OECD); it is composed of three criteria. The first analyzes the actual condition of the system, the second refers to the stress produced by the actual conditions, and the third addresses how society responds to that stress. 3. Going to the right end of this matrix, there are tough requirements in column RHS (the last one), establishing how many indicators must comply with each criterion. Notice how the LHS column, that is, results from the computation, matches the requirements from RHS; this guarantees that the result is optimal. 4. At the top of the figure, observe a box with the number 20. This box is used by the DM to specify the number of final indicators he/she wants, and of course, it can be changed as often as he/she wishes. This will instruct the software to reduce the original number of indicators to the desired final set. This reduction is normally done in many studies using statistical tools such as Principal Components Analysis (PCA); however, as shown, it can also be performed using LP.

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5. Observe in the last row the value 1 for each criterion, which sum is 76. Its purpose is to make sure that the software considers all 76 indicators. 6. Notice that the result, depicted in the solid blue line, is in binary format. In the hypothetical case, although not in this problem, where the DM needs only one alternative selected, he will put the value 1 in the RHS or Requirements. This case could be, for instance, selecting only one location out of, say, three to install an industrial plant. Obviously, in this case the plant cannot be partitioned in several locations.

Question 12.2.9: How to Input Acceptance of Joint Ventures in a Call for Bids? To register this request in the initial matrix, consider that those bidders that submit together must reach the same score. Say there is a supplier S1 that offers steam turbines and another supplier S2 that provides the electric generator. Both companies join efforts transitorily to win this bid, and with shared responsibility for the equipment, which is very convenient for the bidder. Instruct the software by putting a “1” below S1 and another “1” below S2′; put the symbol “=” and a “2” below RHS. In this way, the software will consider that S1 and S2 are inclusive and that both must be sleeted together. Contractor 1 1

Contractor 3 1

Action Equal

RHS 2

Question 12.2.10: In a Portfolio of Projects Where Not All Projects Start at the Same Time, How to Select the Best Time for Execution of Each One? Most probably these projects have different durations in years. 1—Establish the rate of completion of each project in each year. 2—Add a criterion for each year of construction for the whole period. Example: Regional planning. Determining schedule for bridges upgrading (see Table 12.6). It refers to a lacustrine area with abundant small rivers and creeks that empty into a river. The zone is home to small villages and hamlets that are connected by 10 bridges. In the capital city of the area, the City Hall decided to upgrade all these bridges over a 10-year period. The Engineering Department determined the structural condition of each bridge, as well as which works should be done. Bridge 1 was deemed too old to be repaired, and then is not considered in this study. The Engineering Department established three periods: • Immediate repair • Repairs in a 1- to 5-year span • Repairs in the 5- to 10-year span The nine bridges or alternatives were subject to the following criteria:

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Table 12.6 Capture from SIMUS result screen

• • • • • •

Replacement cost Repair cost Economics Load criterion Urgency of upgrade depending on structural conditions Different financing sources (regional, provincial, and national)

The objective was to determine in which of the three established periods each bridge will be repaired during the 10-year window. This problem was solved using SIMUS, and the final result is in Table 12.6. Look at the last row in solid blue. The results indicate for each bridge the best time for their repair. The results show the periods where each bridge must be intervened as follows: Repair in: Bridge 2: 1–5-year period Bridge 3: 1–5-year period Bridge 4: 1–5-year period Bridge 5: Immediate Bridge 6: 1–5-year period Bridge 7: Immediate Bridge 8: Immediate Bridge 9: 1–5-year period Bridge 10: 6–10-year period Question 12.2.11: When I Have Ties Between Alternatives, How to Choose the Best One If Both Have the Same or Very Close Score? Ways of breaking ties between alternatives.

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When the final result shows that two or more alternatives have the same scores, that is, a tie, it puts the DM in a difficult position since he/she must decide which is the best alternative, that is, he is back to “square one.” As far as this author’s knowledge, there is no procedure to break the tie, albeit perhaps using a different normalization method, like min-max, because this procedure normally delivers results with larger discriminations. In SIMUS, there are several methods to break this tie such as: (a) First method: SIMUS produces two results or solutions for the same problem, and both coincide in selecting the best alternative and ranking; however, the scores values are different. This difference can be used to break the tie because, normally, the second result (based on outranking) shows a difference between the scores in conflict in the first result (based on weighted sum). However, it could also be that the tie repeats too in the second solution; then another method can be applied as follows: (b) Second method: Run the SIMUS software again, but now ask for the result to be expressed in integer or in binary format. Because the Solver (which is the software that resolves the simplex linear programming algorithm) will now use a more stringent algorithm (Gomory), most probably there will be a difference between the two results in conflict, either by a different integer or by one of them identified as 1 while the other identified as 0. Choose the identified with 1. If this does not solve the tie, try another procedure like this: (c) Third method: It works with the marginal values of the criteria. The criteria that conform to the alternatives in conflict will be different. To break the tie, choose the criterion that has the maximum value, if maximizing, and see to which alternative it belongs. This way, the tie can be broken. Question 12.2.12: If I Add or Delete an Alternative, How Can I Avoid Rank Reversal (RR)? It is not known yet why this problem, detected in the 1980s, appears, and there are several theories regarding why it happens in some MCDM methods, but none of them are satisfactory, and it seems that the reasons are particular for each method. There is only a couple of methods where this phenomenon is absent, like linear programming, SIMUS, and SPOTIS. Researching in the literature, it seems that this condition is related to involving other alternatives when a new one is added or deleted. In the three abovementioned methods, these changes are done to a reference that is not linked with existing alternatives. LP does not produce RR, and this is due to its algebraic structure; it is an iterative procedure, and thus, at each iteration, the selection of the alternative to be inputted is done by computing opportunity costs for each one. Therefore, if one alternative is deleted, it is now inexistent for the algorithm, and then, it will only work with the remaining alternatives; consequently, the original ranking is not altered. If a new alternative is added, the algorithm will consider it, again, using opportunity costs. If its opportunity cost is better than those of all other alternatives,

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it will be selected. If it is worse than all the other alternatives, it will not be selected. It can be seen that according to the value of the new alternative (that is, its performance values), it can be the best, the worse, or to have an intermediate position. The important thing is that whatever its characteristics and alternatives being deleted or added, there must be no influence on the relative position of the other alternatives. SIMUS, as mentioned, works on the opportunity economics concept, while SPOTIS chooses arbitrary best and worst ideals, far enough of the respective lows and highs of the problem, and thus, all alternatives are compared using these constant values. Question 12.2.13: I Have Detected Dependency Between Alternatives, How Should I Consider Them? If you detect dependencies, they must be inputted into the decision matrix. Example: Dependencies between alternatives or projects are inputted into the initial matrix just by creating a criterion establishing through the “>” mathematical symbol the relation the DM wants. If for whatever reasons alternative A must precede alternative B, add a criterion that specifies that A > B, or if the opposite, put B > A. It is more convenient to use the (Project–Symbol–Project) dedicated table located upright on the second SIMUS screen. The DM can create as many precedence as needed, and even from a single alternative to many others. The total number of precedence is equal to the number of alternatives. For instance, if project 1 must have the same score as project 3, as in a joint venture, put 1 = 3. If project 2 must precede project 4 and project 7, because, for instance, to start project 4 and 7 it is necessary to have project 2 complete, then put 2 > 4, and 2 > 7. You can establish as many precedence as necessary; however, many of them may render the problem unfeasible because it would become very restricted. Question 12.2.14: I Need to Use Weights If for whatever reasons you want to use weights either for criteria or for alternatives, even when SIMUS does not utilize them, follow this procedure: For criteria, multiply the corresponding criterion weight by each one of the performance values of that criterion. Put these results in the Initial Decision Matrix. For alternatives, put the weights in the last row labelled “Project weight” in the second SIMUS screen. SIMUS will automatically consider them. Question 12.2.15: I Need to Use Negative Performance Values Most performance values are positive, but they can also be negative. For instance, there are cases where the contribution of alternatives to criteria is negative. An example is determining the location of a domestic landfill. In a criterion all performance values may be zero, positive, or negative, but you cannot mix positive and negative values in the same criterion. When dealing with negative values, take care of the corresponding action because if the criterion calls for maximization, the software will look for the least

12.2

Sequence of the MCDM Process

243

negative value. If the criterion calls for minimization, it will look for the most negative. Question 12.2.16: Is There a Limit for the Number of Criteria to Use? In general, MCDM methods try to find the best solution, and for that, the number of objectives or criteria is paramount for having the problem properly modelled; consequently, the number of criteria depends on the problem, and there is not a rule or a number limiting them. Some MCDM methods try to limit the number of criteria and sub-criteria based on a psychological theory related to human capacity and because of the heavy workload involved in weighting them. In the opinion of this author, both reasonings are very debatable and without any support. The DM must consider as many criteria as necessary to properly evaluate alternatives. Since in SIMUS, objectives and criteria are equivalent, the method can use all criteria as objectives, whatever their number up to 100 criteria. However, sometimes, the DM may consider that some criteria are not important enough to work as objectives; in that case, he can select as objectives only the criteria he is interested in. Question 12.2.17: I Need Only One Alternative to Be Selected Sometimes, the DM does not want to have a ranking but only one alternative selected. This is the case, for instance, in location analysis for installing a new industry, where only one location is needed. However, in general, it is more convenient to have a ranking of locations from the best to the worse. The reason is that occasionally, the best location has drawbacks that were not considered, and then another alternative must be selected as the best. Example: A firm is looking for a suitable location where it is fundamental to the existence of a harbor for exporting its products. It may select one based on aspects such as land cost, suitable manpower, export taxes, government facilities like subsidies, etc. Suppose that a MCDM method indicates a port such as Montreal, as the best, among others like New York and Baltimore. However, the DM, in his pot result analysis, observes that an aspect that is fundamental for export wasn’t considered: The port of Montreal is inactive for four months of the year because it is blocked by ice, something that was not taken into account when building the decision matrix. In this circumstance the DM probably will opt for the second-best port in the ranking. Or it could be, as often happens, that a sensitivity analysis shows that the best alternative is very vulnerable to the demand of a product, which is unstable. This finding may force the DM to choose another alternative of the ranking that is not subject to small variations of some criteria. Anyway, for only one alternative selected, create a criterion with “1s” in all columns corresponding to alternatives. Use the symbol “=” and put a “1” in the RHS. In this way, the software is instructed to select only one alternative.

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Question 12.2.18: I Want to Make Sure that All Alternatives or Projects in the Scenario Are Considered for Selection: Is This Possible? You cannot do this in most MCDM methods because this request has to be modelled, and most, if not all, methods do not allow it. Solution in SIMUS: Create a criterion in your initial matrix and put “1s.” in the cells corresponding to all alternatives. If, for example, you have, say, 26 alternatives, put “26” in in the RHS cell, or the second term of the inequation, and use the symbol “=.” Due to this criterion, the method is instructed to comply with it; therefore, in the solution there must be 26 scores. However, more often than not, this may not happen because there are alternatives with “0s” as scores. This is possibly due to the fact that not all alternatives can comply with the satisfaction of all criteria, even at a minimum degree, and thus, not considered. Question 12.2.19: If the Scenario Has Exclusive and Inclusive Alternatives, How Can I Model Them? Inclusive alternatives mean that they are independent one of the other, and thus, they can be together in a scenario. For instance, there could be a city program to build schools, pave roads, improve safety on the streets, prevent flooding, etc. All of them participate in a 5-year plan to be selected because none alternative excludes others. When for instance, two alternatives like “vacationing in a mountain resort” and “vacationing in a beach resort,” at the same time gap, are exclusive, because it is one or the other, since they cannot be selected together in the allotted period of time. Question 12.2.20: I Have a Set of Criteria, but Not Alternatives; How Can I Generate an Alternative Based on Those Criteria? Thus, you have a set of criteria and from them you want to determine the best alternative that satisfies them all, but you do not have alternatives; you have to “fabricate” them. Example: Design a road from an airport to city downtown. This case refers to the selection of the most direct route between a city airport, located on the outskirts of the city, and downtown. At present, there exists in-between a disordered network of streets with different speeds and directions, avenues, roads, and junctions, identified by dash lines, as seen in Fig. 12.4. There is not a dedicated road to link both sites; it has to be created. It shows the different thoroughfares that can be utilized to drive between the two points, and there are many different combinations that can be used. Therefore, the problem consists in combining the thoroughfares, junctions, roundabouts, and bridges in such a way as to “fabricate” the best route, subject to criteria requirements. The problem is a little more complicated because the future highway must also serve some established sites (commercial centers) between the airport and downtown; therefore, access to and exit from them is mandatory. These commercial centers are identified as Junctions 1, 2, 3, 4, 5, 6, and 7 (see Fig. 12.4). Most junctions can be accessed from more than one road; for instance, junction 6 can be accessed either via junction 5 or from junction 4. The latter can be accessed

12.2

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Airport *

Junction 2 and Junction 1 are mutually exclusive Junction 2

Junction 3 Roundabout

Junction 1 Junct. 1 and Junct.4 (Alt. 1) are inclusive Junct. 1 and Junct. 4 (Alt. 2) are exclusive

Alt. 1 and Alt. 2 are mutually exclusive Junct. 4 alt.1 Junct. 4 alt.2 Road to be upgraded River Downtown Bridge

Junction 5

Railw ay Underpass Junction 6 Junct. 4 (Alt. 1) and Junction 6 are inclusive Underpass and Junction 7 are inclusive Junction 7 Junct. 6 and Junct. 7 are inclusive Solid red= Selected route Dashed black:= Potential thoroughfares * bing.com/images

Fig. 12.4 Creating a route for a highway construction

either from junction 1, or junction 3 or from a roundabout that connects with the airport and with junction 2. Junction 4 presents another characteristic since it offers two alternative roads to downtown, a shorter, through a bridge on a river, and a larger path via junction 6. It is obvious that junctions 1 and 2 are exclusive since the highway cannot reach both simultaneously, while junction 4 is inclusive since it allows for the existence of two different alternatives, that is, the shorter, via the bridge over the river, and the larger route, via the underpass at the railway tracks. These conditions can be modelled using binary constraints for both inclusive and exclusive alternatives. When these restrictions are established as criteria, the problem consists of finding the most direct route between the airport and downtown. In this case, it is the red solid line. Question 12.2.21: I Have a Portfolio of Alternatives to Be Developed in a Six-Year Period: The Selected Alternatives Cannot Start at the Same Time, Due to Budgetary and Planning Reasons, but Distributed Along a Specified Time It can be said that this is the normal procedure in real scenarios, very common when an entrepreneur or the government develops a series of projects, for instance, Paving 23 streets, building 5 underpasses in city roads traversed by railway tracks, building

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high-rises of different heights and in different places, construction 6 km of cycle lanes, etc. Each of these projects takes many months and even years to complete and starts at different times due to company schedule, based on funds availability as well as manpower. Therefore, for a certain plan we need to assume that time is a component that must be considered. As an illustration, a developer has plots of land in diverse sites in a city. He also has some projects underway, say, high-rises of different heights and architecture, and they must be considered jointly with the new ones because the developer has a total annual budget and a limited number of workers, machinery, bank loans, etc. Consequently, he must select the high-rises to build, taking into account that: • The buildings underway will finish at different years. • Not all the new undertakings will start and finish in the same years; this is decided by the planning and scheduling department of the company and based on funds, manpower equipment, etc., availability since these resources are limited. • In each year every building will advance at its own rate, which is different from the precedent year and also unlike to the next year. • There is an annual budget that must be complied with and that cannot be surpassed. This is an actual scenario and, as can be seen, quite complex. Example: Table 12.7 illustrates this problem. Notice that projects 1, 2, and 4 start in 2015 and finish along three different years; Project 5 starts in 2018 and will finish in 2020, while project 6 starts in 2016 and finishes in 2018. This is what really happens in the real world. In addition, the performance values for each alternative indicates, in percentages, the rate of progress in each of the six-year period, and because of that, the same criterion takes different values each year. This information also comes from the Planning and Scheduling Department of the company. Consequently, there are different starts and finishing dates and percentages of completion. Furthermore, there is a six-year budget to meet and also an annual budget. The problem is then to select those projects that comply with all criteria. This is indeed a tough scenario; however, it can be solved. The trick is to consider, besides the normal criteria, six criteria, each of every year, and then load the different percentages of completion. Another criterion is to establish a limited budget for each year, and verify that the summation of the six years adds up to the total budget. Naturally, previously, it is necessary to consider at the initial period that the annual budget varies because of inflation and interest rates. The following example illustrates it. In this example, there is a sub-matrix with only one criterion, but can be many, and then another sub-matrix corresponding to years from 2015 to 2020.

2029

2028

2027

2026

2025

2024

Criteria Total construction costs in 000’s Euros for each project Other criteria (social, environment, quality, etc.) Xxxxx Yyyyy Zzzzz Criteria related to construction time

5420

Project 2 6589

Project 3 22,157

Project 4

0.3 × 14520 = 4356 0.25 × 14520 = 3630 0.25 × 14520 = 3630 0.20 × 14520 = 2904

0.3 × 5420 = 1626 0.45 × 5420 = 2439 0.25 × 5420 = 1355 0.70 × 6589 = 4612.3 0.3 × 6589 = 1976.7

0.15 × 22157 = 3323.55 0.35 × 22157 = 7754.95 0.30 × 22157 = 6647.10 0.15 × 22157 = 3323.55 0.05 × 22157 = 1107.85

Investment for project/year (×000) × percentage of completion

14,250

Data Project 1

Table 12.7 Portfolio of projects to be developed in a six-year period

0.55 × 14780 = 8129 0.25 × 1478 = 3695 0.2 × 14780 = 2956

14,780

Project 5

0.15 × 18932 = 2839.8 0.80 × 18962 = 15145.6 0.05 × 18932 = 946.6

18,992

Project 6

Total

82,388

2956

6779.55

19915.45

26777.7

16663.55

Amounts that must be available in each year (×000) 9305.55

82,388

Symbol RHS values

EQUAL =

Action

12.2 Sequence of the MCDM Process 247

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Observe that the annual budget is depicted for each year. This is obtained as the sum product between each project’s annual percentage and its budget. Notice that for each project, the sum of the performance values—in percentages—adds up 1. In this case, the total amount is 82,388 Euros, seen in the RHS column, coincides with the summation of the annual expenditures. In this condition, it is natural that all projects will be selected. However, if the annual expenses for each year surpass the assumed budget, there will be some projects that will not be selected.

Question 12.2.22: The Engineering Department Tells Me that There Is Precedence Between Alternatives: How Should I Proceed? In many projects there is indeed precedence between alternatives, sometimes due to sharing resources with other alternatives or by scheduling, probably de most common, or by physical dependency. Use the little table right up in SIMUS second screen. You can put as many precedence as the number of alternatives, that is, if there are, say, 12 alternatives, you can establish 12 preferences, even repeating alternatives. For instance, 1 > 4, 1 > 6, 1 > 10, etc., and thus, establishing that alternative A precedes alternatives 4, 6, and 10. However, these are hard restrictions and too many may turn the project unfeasible. Question 12.2.23: The Scenario Involves Projects that Are Shared by Various Entities: How Can I Introduce this Restriction in the Method? This case is better understood using an example. Example: Assume a project for water supply involving different methodologies for six cities; however, not any project may be applied to all cities. The different projects are: building an aqueduct from a river, drilling water wells, building a wastewater treatment plant, and water recycling. Assume that your cities are: A, B, C, D, E, and F (see Table 12.8). You can build your initial matrix as usual, that is, you put the different cities in columns (since these are the unknown, not the projects), and in rows the criteria they are subject to, like the population of each city, water consumption per person, water consumption from industries, water consumption for the agricultural belt of each city, etc. Using binary criteria, you use them for the different projects. Now, you have to instruct the software by specifying to what cities it applies. Say for instance, that the aqueduct project is designed to supply water to cities A, C, and F, for they are close by but cannot serve cities B, D, and E because they are far away from the aqueduct route. If in the matrix you have in columns the three cities that are in the plan, and since you want to instruct the method that cities A, C, and F share the facility, put “1s.” In the corresponding cells on row “Aqueduct,” use the symbol “=” and the value 3 in RHS. In this way the method will know that only the three cities must have the same project, none others.

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Table 12.8 Projects to be shared by several entities Aqueduct

City A 1

City B

City C 1

City D

City E

City F 1

Symbol =

RHS 3

Table 12.9 Selection of only one project Criterion

Bremen 1

Toulouse 1

Barcelona 1

Symbol =

RHS 1

Question 12.2.24: I Need to Select Only One Alternative Among Several Example: Say you have three options but only one should be selected; this is a very common scenario, for instance, in selecting a location for installing an industrial plant. Assume that the potential locations are Bremen, Toulouse, and Barcelona. You have three locations, but you need only one. Procedure: Place 1s. in all of them, use the symbol “=”, and put 1 in RSH. This way the method is instructed to consider only one alternative out of 3 (Table 12.9). Of course, you have to add the technical, environmental, economic, social, risk, etc. criteria you consider necessary using integers, percentages, or crisp numbers from a fuzzy analysis. However, you may wonder if you can mix the binary row or rows with the normal criteria set. Yes, you can, but do not forget to normalize. SIMUS does this automatically, and you can select which normalization method you prefer. Question 12.2.25: I Need to Reduce the Number of Options, Alternatives, or Indicators This is a common problem, especially when dealing with indexes or indicators that normally are numerous, perhaps more than one hundred. For instance, to build a composite indicator to monitor a country’s growth, all areas of human activity must participate, that is, society, economics, environment, natural resources, GDP, crime, education, and industry. Since usually these metrics must be updated yearly, it is an expensive, cumbersome, and time-consuming task; therefore, there is interest in having a set of manageable indicators, say for instance, 20 or 25, that best represent the original set. This is a task that can be done using SIMUS. It normally starts with the analysis by experts of the existing relationships between indicators and certain criteria, gauged, for instance, by using a 1 to 10 and -1 to -10 scale, the largest the value, the greater the direct or inverse relationship with the criteria. For instance, if one indicator is a forested area, it is strongly related to GDP because logging generates jobs for forestry laborers and truck drivers, provides jobs to sawmills and furniture manufacturers, etc. This is done by experts. For instance (Table 12.10), expert Alice finds that for her, the relationship between indicator ID “1” is very strongly related to criterion (6). Because of that she gives it a value of 9. Expert Dan comparison is very close since

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he assigns a value of 8. Expert Michelle reckons that there is a relationship but not so strong, and thus, she assigns a 7. The average is then 8. However, when the indicator is compared with environmental issues, normally the relationship is negative because, as in the case of logging, the more its intensity, the greater the damage to the environment due to erosion. For this reason, these evaluation values are absolute, and they are the base for selecting the best indicators. How do we indicate that we want only a fraction of all indicators? By using binary notation and the symbol “≤,” or “=,” meaning that the total number of indicators must be, say, less than, or that must be exactly a certain value. Notice that the last row of one of the tables has 1’s corresponding to each indicator and the symbol “=,” meaning that the DM wants 8 indicators selected out of the total of 19. If we ask for a set with less than 25 indicators, we use the symbol “≤” and the value 25. In the last row, with the label “Indicators selected,” that are exactly 8 indicators 1-4-7-8-11-12-13-15. Observe that the software also delivers information (al bottom right) about the percentage of indicators for each one of the three areas, that is, Human health, Natural resources, and Transportation.

Question 12.2.26: How to Reduce the Number of Options, Alternatives, or Indicators but Keeping in the Final Set the Maximum Amount of Information from the Initial Set? Procedure: In SIMUS the DM can indicate the number of the final set of alternatives you want. However, he needs to capture as much information as possible from the initial set. To do this, the DM uses Shannon Entropy Theorem, which calls for computing the entropy of each element, which measures the quantity of information contained in data. The values of the differences D = (1 - entropy (S)) can be used as the weight for each indicator and can be put in the corresponding box at the bottom of the initial matrix. See Fig. 7.8, in Chap. 7, row “Project weight.” Is it also possible to link this procedure with the reduction of indicators. See Question 12.2.25. Question 12.2.27: How Do I Know if Resources Are Enough to Get the Best Solution? If They Are Not, How Can I Identify Which Is the Resource that Is the Most Critical and How Much Should It Be Increased? Resources have limits as well as environmental percentages. The ideal is to have the resource used in full, usually, a shortage of a resource affects others resources and the alternatives. In the final solution the DM may determine to what extent a resource was used. For instance, in a thermal power plant, contamination due to emissions of noxious gases is a big concern. In analyzing the selection of alternatives to generate electricity, it is possible to determine the amount of, say, SOx that is spewed.

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Table 12.10 Reducing the number of indicators m Selecting environmental indicators Indicators proposed AREA

Human health

Natural resources

Transportation

A B C D E F G H I J K ASPECTS I.D. WEIGHT 8 6 4 9 3 4 7 7 6 2 8 Indicator I.D. 1 2 3 4 5 6 7 8 9 10 11 Normalized weight 0.066 0.050 0.033 0.074 0.025 0.033 0.058 0.058 0.050 0.017 0.066

L M N O P Q R S Sum of weights 9 9 5 8 5 5 7 9 121 12 13 14 15 16 17 18 19 0.074 0.074 0.041 0.066 0.041 0.041 0.058 0.074

Experts' estimates of each indicator relevance (The larger the better) CRITERIA LINKED W/ TECH. SELECTION

Alice

1) Relevant to environmental sustainability 2) Scientifically credible and tested 3) Indicator is agreed by researchers 4) Relevant at different scales 5) Links to policy priorities 6) Relationship to human activity

10 9 9 7 9 9

7 7 8 6 5 8

10 8 7 5 8 8

10 9 9 7 9 6

8 4 7 3 7 6

6 4 8 2 6 5

7 4 2 0 7 0

10 9 7 5 7 7

10 9 9 7 9 9

10 9 9 7 9 9

* * * * * *

10 9 8 7 9 8

10 9 9 7 8 9

10 9 9 7 10 9

* * * * * *

7 7 8 4 7 7

7 7 8 4 7 7

7 7 8 6 5 8

7 7 7 5 7 7

1) Relevant to environmental sustainability 2) Scientifically credible and tested 3) Indicator is agreed by researchers 4) Relevant at different scales 5) Links to policy priorities 6) Relationship to human activity

9 8 8 7 8 8

* * * * * *

9 8 7 6 8 9

9 7 8 6 8 8

9 7 8 6 6 8

8 8 4 6 9 7

9 7 8 8 7 8

7 7 7 5 7 7

7 7 7 5 7 7

Daniel 7 6 8 5 7 6

9 8 7 7 9 8

* * * * * *

8 8 7 6 6 9

10 9 9 7 10 9

9 8 7 6 6 8

* * * * * *

7 8 8 6 5 8

8 8 7 5 5 8

7 7 7 5 7 7

1) Relevant to environmental sustainability 2) Scientifically credible and tested 3) Indicator is agreed by researchers 4) Relevant at different scales 5) Links to policy priorities 6) Relationship to human activity

7 7 7 5 7 7

7 8 7 3 5 6

7 7 7 5 7 7

9 8 8 3 7 7

9 8 8 7 8 8

9 6 7 6 6 8

9 8 8 6 8 7

9 7 8 6 8 7

8 7 8 6 6 7

Michelle 8 8 8 6 7 7

8 8 8 5 6 8

8 6 7 6 5 8

8 7 8 6 5 8

* * * * * *

(*)

7 7 8 9 * 7 7 8 8 * 8 8 8 8 * 6 6 5 5 * 5 5 6 6 * 8 8 8 8 * No estimate available for this indicator

Results Action from computation Requirements

Average values

1) Relevant to environmental sustainability 2) Scientifically credible and tested 3) Indicator is agreed by researchers 4) Relevant at different scales 5) Links to policy priorities 6) Relationship to human activity Forcing all indicators to be considered

8.67 8.33 8.00 6.67 7.67 8.00

7.00 7.50 7.50 4.50 5.00 7.00

8.67 7.67 7.00 5.33 7.67 8.00

9.33 8.00 8.33 5.33 8.00 7.00

8.67 6.33 7.67 5.33 7.00 7.33

7.67 6.00 6.33 4.67 7.00 6.67

8.33 6.33 6.00 4.67 7.33 5.00

8.67 7.67 7.33 5.33 7.33 7.00

8.33 7.67 8.00 6.00 7.33 7.67

8.33 7.67 8.33 6.00 7.67 7.33

8.00 7.50 7.50 6.50 7.00 8.00

5.67 5.33 5.33 4.33 4.67 5.33

9.00 8.50 8.00 6.50 7.00 9.00

1

1

1

1

1

1

1

1

1

1

1

1

1

CRITERIA LINKED WITH AREAS

9.33 8.67 8.67 6.33 8.67 8.67

9.00 8.00 7.50 5.50 6.00 8.00

7.50 7.50 8.00 4.50 6.50 7.50

7.33 7.00 7.67 5.33 5.67 7.67

7.67 7.33 7.67 5.67 5.00 8.00

7.00 7.00 7.00 5.00 7.00 7.00

66.67 59.67 58.00 44.83 55.00 57.33

≥ ≥ ≥ ≥ ≥ ≥

5.67 5.33 5.33 4.33 4.67 5.00

1

1

1

1

1

1

8

=

8

2.00

≥

2.00

Indicate here number of indicators required

(Minimum percentage of indicators per area) 7) Human health 1 8) Natural resources use 9) Transportation

1

1

1

1

1 1

1

1

1

1

1 1

1

1

1

1

1

1

Indicator I.D.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

Indicators selected Objective function 0.537

1

0

0

1

0

0

1

1

0

0

1

1

1

0

1

0

0

0

0

4.00

≥

4.00

2.00

≥

2.00

8

See here number of indicators produced Percentage obtained of indicators for HUMAN HEALTH area Percentage obtained of indicators for NATURAL RESOURCES area Percentage obtained of indicators for TRANSPORTATION area

0.25 0.50 0.25

Table 8.4 Data from experts - Decision table and indicators selected

If it is below the upper limit, it is OK, and although it would be better to have a level of contamination as far as possible from the maximum limit, examining the solution may give a very good idea about the plant’s efficiency. Example. A farmer has a plot of land of 120 has, where he can grow wheat, barley, and soybean. These crops are subject to six criteria as shown in Table 12.11. Notice that regarding the size of land—that is, criteria 5 and 6—their performance values are “1s.” This is the way to instruct the MCDM method that the land is a resource that has to be considered by the three crops and not only by one or two. Why this condition?

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Table 12.11 Initial data LHS data

Criteria Exploitation cost per hectare Gross benefits per hectare Water consumption per hectare Local consumption in tons Maximum available land in hectare Minimum size of economically feasible land

LHS computed values

Operator

RHS values

Projects or alternatives Wheat Barley Soybean 13,584 14,908 16,214

Actions MIN

≥

89,321

85,937

89,157

MAX

≤

102,369,51

17,630

13,697

11,230

MAX

≤

1,302,567

417

362

189

MAX

≤

56,327

1

1

1

MAX

≤

120

1

1

1

MAX

≤

101

Because otherwise, the method may dedicate the whole land to only one crop, and if the farmer wants to diversify his production, to be covered in case the price of the most promising crop goes down, this condition is convenient. The farmer wants to know the answer to his question: What crop to cultivate and how many hectares for each one? When solved, SIMUS informs that the most important crop is soybean, with a score of 4.12, followed by wheat with 0. 34, and barley with 0.01. It also shows that soybean complies will all criteria except water consumption. However, to his surprise, the farmer finds that the maximum land he can use is 116 has, and he wonders why he cannot use in full his 120 has. The reason is that he has not enough water for his 120 has and with these types of crops. How does he know that this is the reason? Because to develop 120 hectares the farmer will need a minimum—and considthe crop with lowest water consumption (soybean)—of ering 120 × 11,230 = 1,347,600 liters of water, and he has only 1,302,567 liters. From here, the farmer can choose to increase water availability by drilling a water well; however, he must analyze if its costs compensate to have four hectares idled. This case illustrates how the DM can use his/her knowledge and expertise by interpreting results that come from certain data and without any subjectivity. A detailed explanation and discussion of this case can be accessed by pressing the “Help SIMUS” key in SIMUS first screen, and under “Crop selection.”

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Question 12.2.28: How Can I Keep Stocks of Material Under Control in Manufacturing? Benefits, costs, production, as well as keeping an adequate and economic stock of materials are paramount aspects in manufacturing, and they can be addressed using MCDM methods. The last one normally has a direct influence in costs. Example: Assume a company that manufactures five models of robot vacuums. Most methods share a large number of parts, and the manufacturer wants to know which is the robot model that maximizes his benefits, a typical MCDM scenario. Related to this, there are different strategies that the manufacturer can choose. For instance, most probably he establishes a lower limit of parts to consume that allows continuing production until restocking comes, since the manufacturer does not want to stop his production lines because there are no more parts. Establishing a lower limit that can be easily determined normally based on supplier production time allows the production to continue until the parts or materials arrive. At the same time, the manufacturer establishes an upper limit due to many reasons, like lack of space in his warehouse, insurance cost, maintenance cost, financial reasons, etc. Therefore, in his initial matrix the manufacturer needs to put these two limits. For the lower limit, and being determined its value, it may use a criterion “Minimum limit of reorder level,” and apply the “≥” symbol, meaning “Use as much as possible but above a certain minimum level of stock.” For the upper limit, once determined, he must establish another criterion, “Maximum limit of storage,” identical to the former, but with the “≤” symbol, that is, “Use as much as possible but below a certain maximum level of stock.” With these two instructions the software is instructed to keep the consumption between these two limits. It is the system used by supermarkets; every time a client buys an item, it is automatically deducted from the existence on the shelves using its barcode. When the lower level is reached, an automatic order is sent to the supplier for restocking. Question 12.2.29: How to Learn if the Different Objectives Have Been Satisfied and in What Extent? MCDM problems work with different alternatives, subject to a set of criteria that in reality are objectives. The solution gives a compromise result, that is, it is not optimal and, in most cases, is of interest to learn how the selected alternative satisfies each objective. Therefore, in most methods, it is difficult to learn in which extent the objectives are satisfied; consequently, we get a “blind” result. As far as this author’s knowledge, there is only one method that gives this information: SIMUS, which, at the end of the process, informs for each objective how close or far was the result when compared with the goal established. To understand this, it is necessary to remember that SIMUS uses the linear programming algorithm (simplex) for every criterion when they are sequentially removed from the decision matrix and used as objective functions (Z).

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Example: Suppose a criterion C5 that calls for maximizing production to no more than 500 units of a product because this is the maximum demand. This is the goal or value for the right-hand side of the inequation (RHS). When this production criterion is used as an objective function (Z5), assume that the value in the left-hand side of the inequation, or computer result, is LHS =485 units. It means that this criterion or maximum production is satisfied in 97%. If criterion C3 calls for minimizing water consumption but at no less than 19 liters/unit, then RHS = 19. When C3 is used as an objective function (Z3), it delivers LHS = 22.5 liters per unit, meaning that it exceeds the minimum by 18%. Question 12.2.30: I Have to Prepare a Complete Overhaul and Refurbishing Plan for a Fleet of Diesel-Electric Locomotives, Subject to Many Criteria: How Do I Select, Considering State of Each Equipment, Usage, and Age, When Each One Must Be Scheduled for Repair? In reality, this is not a typical scenario to be solved by MCDM; however, taking into account that each locomotive taken individually is a project, and that all locomotives are subject to the same set of criteria, it is believed that it is adequate to be solved by MCDM methods. We have to start with an inventory of the fleet of locomotives, with a complete history of their state since the last servicing. It is assumed that units are of different age, models, thousands of km run, electricity consumption, etc., and based on their history, as well as performance, engineers are able to make estimates regarding the need or frequency of service. Example: This is a similar problem but refers to refurbishing 9 road bridges in the Ottawa Region in Canada, involving immediate, high, medium (1–5 years span), and low priorities (5–10 years span). See question 12.2.10 for a detailed explanation of this project. Criteria refer to costs for immediate (high, medium, and low) repair, as well as for 1–5- and 5–10-year spans, as well as costs for studies on work to be done, and actual heavy-haul capacity, compared with design capacity. Financial criteria, replacement costs, economic and time criteria were also included, as well as origin and timing for different financial sources. Question 12.2.31: I Have Been Ordered to Include as the Best an Alternative that Was Not Selected by a MCDM Method: How Can I Proceed? This is not indeed a correct procedure from your boss, but unfortunately it is quite common. Example: A Major for a city has been elected, and now in office, he must comply with the pledge made during his electoral campaign, to develop a children park in a certain part of the city. This project is identified as alternative “D,” and there are seven alternatives or projects in total in his four-year plan, labelled A, B, C, D, E, F, and G. He informs the DM that his alternative must come first in a list of selected undertakings. The DM needs to instruct the software about this demand, and this is done in SIMUS establishing, in the special little table at right of SIMUS second screen, this precedence as follows: D > A, D > B, D > C, etc., that is, D preceding everything.

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Question 12.2.32: How Are Risks Considered? This is a big question and, more often than not, ignored. In formulating projects there are risks everywhere, from the birth of the idea until the end of the project. There are risks regarding the feasibility of an undertaking, especially data from the sources in new projects, for instance, estimating the return, risk because of the weather, or for the characteristics of the soil, as in the case of boring a tunnel. No less important are personal risks, machinery malfunction, legal, economic, environmental risks, etc. There are risks by omission or not forecasting what can happen. The issue is really complex. Risks must be considered in any project. First, it is necessary to determine which are the potential risks, their nature and consequences, availability of historical data, and their probability of occurrence. Study the scenario of the project; in a construction, think that are personal risks due to the absence of security measures like harnesses, not use of adequate hats and boots, gloves, etc. In all projects evaluate risks derived of lack of compliance of environmental regulations and limits. In projects like selection of industrial location, analyze, for instance, history of strikes in each site or country that can paralyze a work. In chemical plant projects, ruminate about what can go wrong regarding the release of noxious gases that can hurt people and the environment. All of these risks can generally be considered as criteria in a decision matrix and once risks have been identified for a certain scenario, for each type of alternative, give them a performance value based on probabilities of occurrence. You can even consider minimum and maximum probabilities for each event. Think about the consequences of each type of risk. Example In an actual scenario, a large project was designed to transport oil from an Asian country to the Black Sea by constructing an oil pipeline. Three different routes were envisioned with distances of 1560 km, 1297 km, and 1527 km, respectively. Each route had a variety of topographical features including earthquakes in one of them, as well as social unrest in another, and a third with political problems leading to sabotage of existing infrastructure or stealing gasoline through drilling of pipelines. Therefore, a risk criterion can be added with the probabilistic values for this type of attacks that can damage the pipeline. This was an actual project where a MCDM study was done, and that selected one of the three routes. However, the owner did not accept these results because the risk due to sabotage was not duly considered, and chose another route. Obviously, the risk factor was not properly appraised in the MCDM study. The best way to proceed is to create a criterion for each type of risk that can affect one, two, three, or more projects with possibly different magnitudes. Even in long distances, all projects can be sectorized, each sector reporting different risk according to its characteristics. In this case each project can be considered a cluster involving different sectors, but all participating together in the alternatives evaluation.

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Question 12.2.33: How to Select the MCDM Method to Solve My Problem? Background Information This is probably the most frequent question. It is not easy to answer it due to the large number of existing MCDM methods, and because there is not a method that can solve all kinds of problems. “Method” refers to the different procedures, algorithms, and techniques existent in the market to solve MCDM scenarios. Just to mention the most known are AHP, ANP, ELECTRE, Goal Programming, PROMETHE, SAW, SIMUS, TOPSIS, VIKOR, WASPAS, etc., and based on different principles. Therefore, it is a legitimate question to ask oneself which method to use and a dilemma to select the most appropriate. Primarily bear this in mind: Not all methods can solve whichever problem you may have, consequently, do not jump at a method because you have its software, or because you know it, or since it is easy, or by considering that a friend used it, or because it is the most popular. Bear in mind that not all methods are suitable for all scenarios, simply because scenarios are not alike; each one has a distinctive feature that demands different actions. As an analogy consider a driver’s license; it allows for driving a car, but not for driving a bus, because the latter has more requirements. The same with scenarios; you can use a method that is good for selecting which restaurant you can go for dinner, but not for determining the best way to collect industrial waste. Unfortunately, many practitioners do not think this difference is important. In many ways, the selection of a MCDM method depends on the scenario. But what is a scenario? It is the core of the problem and defines which is the undertaking or goal and the candidate alternatives, projects, or options to attain it. For instance, a scenario could be the construction of an electric power generation plant in a certain site, which can be accomplished by using different alternatives such as steam turbines, gas turbines, diesel engines, hydro turbines, etc., and then it is necessary to find the best option, or even a mix of them. You cannot use a method where criteria must be independent, because normally it does not exist in the real world, like determinations of potential paths to build a highway. You cannot use a method based on a rigid lineal hierarchical structure, which is good for trivial problems like selecting a movie, but not for determining the best place for building hydroelectric dams. In the trivial example, the consequences, good or bad of a decision fall on a person which is fully responsible for the selection. In the second case, we are talking about very costly projects which consequences affect the lives of perhaps millions of people, and with enormous effects on industry, and which are conformed by an intricate network of activities. In addition, remember that most possible a DM will be required to answer questions from the stakeholders, which response depends on the capacity of the method employed. This is often seen not only at the corporate level but also in the academic thesis, where the stakeholders or the tribunal can embarrass the speaker with questions which answers he doesn’t know, but that should know. Remember that these people are interested in results, not on the method used; usually for them it is irrelevant and probably they have never heard about them, but for sure, they will

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be concerned about the procedure the DM followed to reach a decision in order to make sure that all important aspects linked to the scenario have been considered. Consider the very real scenario where the DM must make his presentation in front of shareholders of his company and recommending, say, equipment A to generate electric power out of three A, B, and C options. Assume that a stakeholder may ask the DM if he has considered in his selection the rates of air contamination produced for the three types of equipment, and where the project stands relative to the minimum and maximum limits for contamination, to see if the company can eventually enlarge its plants to generate more energy. The DM may say: “Naturally, I considered air contamination generated by each of our projects, in reality it is one of the evaluation criteria, which of course is minimized.” The stakeholder may ask again: “Yes, I am aware of that, but that is not what I asked; my question is related to where we stand relative to the minimum and maximum limits for contamination in order to see if we can eventually enlarge our plants to generate more energy. For example, do we have room for increasing steam production in project A and still be below the maximum contamination limit established by environmental agencies, and if we do, what is the margin?” If the DM says, “No, I didn’t consider it because the method I used does not allow criteria limits.” What do you think the reaction of the stakeholder could be? Probably the DM will be fired for incompetence, because it is evident that he is not doing his job as should. Also remember that when exposing his results almost certainly somebody will ask, “What if we do this or that. . .? Or how will affect the stability of the result you found to variations in say demand, or competition or prices? In other words, how strong of robust is the solution, including the ranking, (output) to variations of inputs?” Of course, this is “Sensitivity analyses, and then the DM must be sure that the method he chose allows for a rational examination and provides meaningful answers. If a practitioner develops his project without an analysis about which is the appropriate method, he most probably will be performing no more than a number crunching process and getting poor results, because his method has not considered all the characteristics of the scenario. If a colleague says, for instance, “I used the xxx method and I got very good results,” may be the practitioner can ask him how does he know that the results are as good as he says. All MCDM methods select an alternative and provide a ranking, but it does not mean that these results are the best. Maybe a short and poor answer would be “Because I used a MCDM method.” In reality what the DM did was to use a MCDM method that gave a result, but there is no proof that it is the best. It was designed to give a solution, no more than that. Question 12.2.34: Which Is the Best Method? Then, which is the method that provides with certainty the right solution?

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It does not exist, except when using linear programming that if there is an optimum solution it will find it. Why then don’t we use linear programming? Because it works with a single objective and with only quantitative criteria, and these conditions are generally not present in real-life scenarios; however, it has been for about 75 years and still is the method of choice when there is only one objective, as one can find in oil refineries, paper plants, steel production, airlines, etc. Since it is impossible to realize if a result is good or bad, common sense says that the practitioner must consider a method that fits his needs, Appraisal at First Glance If the DM has a small problem and, after studying it, decides to have 3 projects and 9 criteria, he can do a first and rapid appraisal of what could possibly be the results once the decision matrix has been built. If this matrix has alternatives in columns and criteria in rows (or their inverse), observe the values of each alternative along the whole range of each criterion. If some criterion calls for maximization look at the project that has the highest value (see Table 12.12). If the criterion calls for minimization look at the project with the minimum value, and in each case identify it with a square for project A, an oval for project B, and a rhomboid for project C. When done with all criteria, count the number of squares for project A, the number of ovals for project B, and the number of rhomboids for project C. However, this procedure gives only a hint, and not very relevant. What is important is the discrimination or quantitative differences between projects and for the same criterion. This assertion is grounded on Shannon’s entropy and largely corroborated in practice, since it is the base of Information Theory. In this example notice that in criterion C1, that calls for maximization, project B has a large difference with the other projects; this is what interests you. Write linguistically this difference at right. The same happens for criterion C2. That also calls for maximization. Criterion C3 calls for minimization; the lowest value pertains to project A, although the difference with other projects is small. Continue examining each criterion the same way. Observing Table 12.12 with ordinal values at right it is evident that project B has three large differences in three criteria. For project A notice that there is only one criterion with a large difference and two with small differences. For project C there are three small differences. As a conclusion, it appears that the order in the ranking is: Project B > Project A > Project C. Why? Because B offers the largest values, according to the maximization action, the same for A, which offers the minimum values and calls for minimization, and finally C that also complies with the maximization and minimization actions. If this a good solution? We don’t know.

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Table 12.12 An example

A

Projects B

C

22 1.26 0.27 42 0.05 16.8 46.7 2.5 12.9

35 2.19 0.3 51 0.05 22.8 46.2 3 11.7

26 1.23 0.31 39 0.03 23.1 47.8 3 10.9

Criteria C1 C2 C3 C4 C5 C6 C7 C8 C9

Differences between projects A B C Action MAX MAX MIN MAX MIN MIN MAX MIN MIN

Large diff. Large diff. Small diff.

Large diff. Small diff.

Large diff. Small diff. Small diff. Small diff.

Nevertheless, we can develop this scenario using a MCDM method, and then compare results. Using the SIMUS method which result is seen in Fig. 12.5 verifies this ranking. Again, it does not mean that this is the best solution for this scenario; we have simply compared two methods and got the same result. In fact, three different methods were compared because SIMUS automatically produces two solutions using two different procedure and both must coincide. Figure 12.5 shows SIMUS printout for this problem, and the coincidence of two rankings can be observed (in red), even when scores are different. Chances are that when the DM solves the system mathematically using a method, he may get an approximate ranking from the first and the second best, compared with yours. If not, just examine the problem and ask why the mathematical method does not follow your appraisals. You can get interesting answers. Needless to say, even in case of perfect coincidence, as could be the case, it proves nothing, but at least gives the possibility to compare the result of the method with another method, in this case, your examination and appraisal. Question 12.2.35: Which Are the Steps for Modelling? They are: 1. Designing the mathematical model Build your mathematical model taking into account all the information you have, and in addition, examine your scenario looking for information you should have, bearing in mind that your mathematical model must consider reality as much as possible. It is not for the DM to supply the data, since it must respond to the scenario; the DM must attempt to replicate it, and not the opposite, that is, the scenario adjusting to your needs. Therefore, remember that there should be no limits regarding the maximum number of criteria. Some MCDM methods are very restrictive about this because

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12

Efficient Results Matrix (ERM) Normalized Project A Project B Project C C1 C2 C3 C4 C5 C6 C7 C8 C9

1.00 1.00 0.59 0.91

0.41 0.09

1.00 1.00 1.00 1.00 0.21

0.79

Sum of Column (SC) Participation Factor (PF) Norm. Participation Factor (NPF)

2.50 4 0.44

3.71 5 0.56

2.79 3 0.33

Final Result (SC x NPF)

1.11

2.06

0.93

SIMUS First solution: Project B > Project A > Project C Project Dominance Matrix (PDM) Subordinated projects Dominant proj.

Proyecto 1

Project 1 Project 2

3.2

Project 3

2.8

Column sum of subordinated projects

6.0

Proyecto 2

Proyecto 3

Row sum of dominant projects

2.0

2.5

4.5

3.5

6.7

2.6 4.6

5.4

Net dominance

–1.5 2.1 –0.6

6.0

SIMUS Second solution: Project B > Project A > Project C

Fig. 12.5 SIMUS solution

of the heavy load work, but limiting the number of criteria most probably will lead to an unreliable result. Modelling is the most important and most difficult task in a MCDM process. It does not matter what MCDM method the DM employs to solve a problem, as long as the scenario is properly built and represented in a mathematical model or Initial Decision Matrix, and of course, knowing if the chosen method is up to the job. In order to build the mathematical model replicating reality as close as possible, the DM needs to consider the aspects that are detailed in the following sections and that apply to the scenario. Once he thinks that his model incorporates all available information and demands from reality, he will be in position to select the method to be used to solve this scenario. How? By checking the characteristics of each method, for which there is abundant literature in the Web. Sometimes, the problem is so complex that the DM may need to partition it into two very well-defined and independent areas, and the result of the first partition is applied as an input to the second. Be aware that this partition is not the same as that recommended by some MCDM methods to solve a problem. In general, a problem or scenario is a system, and as such, the DM cannot study the different components, find the best result, and then add them up. This is not allowed, because normally, there are

12.2

Sequence of the MCDM Process

261

interrelationships between the different components. The whole may not be equal to the sum of the parts. What suggested above about partitioning is a different scenario, since the DM is considering two distinct systems, that is, between segment A-B and between segment B-C, with different characteristics, and without connections between them. That is, the selection in the first section is not related to the selection in the second. The DM can solve the first segment with the criteria he considers to apply, and when done, he solves the second segment, using the selected options or even the same, but with different or perhaps equal criteria. In this way, the results will show the best option between A and B and the best option between B and C. Most methods are not designed to deal with all types of problems and especially with complexity. If the scenario is complex, the DM must disregard simple methods because they do not have the capacity to solve these problems. 2. Find the nature of the problem Is the scenario complex or trivial? But what is a complex scenario? See also Question 12.1.2. It is a scenario that has not only many alternatives and criteria but also many interrelations between them. In addition, a complex scenario may have several projects and several sub-scenarios and precedence and restrictions. For instance, in studying a river basin, there are perhaps hundreds of criteria and dozens of alternatives of projects of diverse nature, like irrigation, electric generation, cattle raising, tourism, industry, etc., as well as restrictions of different kinds. The Rhine, Danube, Mississippi, Orinoco, and Yang Tse river basins are clear examples of complex problems when one analyses the best uses of their waters. One of the most usual restrictions is that water for downstream projects must have certain values and limits regarding quality, for instance, related to chemical, salt content, and sewage contamination coming from upstream undertakings, which means that upstream projects are forced to consider these restrictions, and thus, they must be inputted in the modelling. The no compliance of these restrictions has produced more than one international conflict; for instance, the construction of a paper mill in Fray Bentos, in Uruguay, on the shore of the Uruguay River that divides Uruguay and Argentina. Argentina complained about contamination produced by the plant that the river transported downstream to the La Plata River. This produced the two countries asking for an International Court of Justice verdict. An excellent example of a complex undertaking can be read in a work performed by the MIT on the Colorado River in Argentina in 1973. See Jared Cohon “Multi objective Programming and Planning”—Academic Press Inc. This example shows that there are no shortcuts when we must solve a complex problem by using methods that are not capable to manage this scenario. This example, which is a classic in the MCDM process literature, was solved by linear programming and simulation long before the appearance of the heuristic methods we know today.

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Complexity can be appreciated when comparing this scenario with simple ones such as those involving personal decisions (such as purchasing a car) or a corporate scenario (such as hiring people or selecting contractors). • How many objectives are? There could be opposite objectives like maximizing benefits while minimizing costs. • Are these objectives achievable? • How reliable is the data? This applies for both, quantitative and qualitative data; for instance, values for machinery performance, do they have certificates from the manufacturer as well as tests results, or they are accepted without verification? • What type of normalization is to be used? In some MCDM methods, the selection of the normalization procedure may have effects on the results. 3. Alternatives or options or projects Ask yourself the following questions: • Are there precedence between alternatives? • Are all legal documents in order? • Are alternatives linked in such a way that two or more alternatives must have identical scores (such as in joint ventures)? Specifically, if contractor A is selected, contractor D must also be selected. • Are there alternatives under execution? Example: Alternatives C and F are projects that have already started. At the time of analysis, C has a completion of 45% while F is 19%, and both are incorporated on a portfolio, together with other projects to be selected, and then competing for resources. These projects must be in the final ranking and, probably, in the first place if they are a priority. • Is there a need to group activities according to areas and sub-areas? Example: In a problem to select environmental indicators, the DM must consider different recipients of pollution, like soil, air, water, etc. • For soil, for instance, maybe there is a need to contemplate several kinds of contaminations, especially in projects like oil extraction. • Is there any suggestion or imposition regarding that a certain alternative must be selected? For instance, for political reasons project D must be executed, irrelevant if it is or not selected by the method. • Are there excluding alternatives? Example: If project C (building a bridge to cross a river) is selected, Project A (building a tunnel below the river) cannot be in the ranking. It is one or the other. • Do alternatives allow for decimal scores or they need integer scores? Example: Only one alternative must be selected, and there is no ranking, which implies working with binary results. • Does the scenario deal with alternatives that can take place in different places?

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Sequence of the MCDM Process

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Example: Projects A and C can be executed in sites 3 and 4; however, project C can also be implemented in site 5. • Are two or more alternatives with the same or very similar performance values? • What precision is the DM willing to accept in the alternative scores? 4. Criteria • Is there a limit to the number of criteria? No, the DM can create as many as he needs • Is it possible for a mix of maximization, minimization, and equalization actions? Yes. As an example, the model may have a criterion calling for maximizing benefits, another requiring minimizing environmental damage, and another imposing equalization of capital to invest. • Are criteria all quantitative or all qualitative (even with linguistic variables), or there is a mix of both? Normally, there is a mix. • Are weights for criteria needed? If so, how are they obtained? Yes, they are needed, because otherwise it is assumed that all criteria have the same importance, which is not realistic. They may be obtained by subjective appreciations (not recommended) or from objective entropy (recommended). • What are the types of criteria? Quantitative or qualitative? A mix? Quantitative refers to certain values, like distances, prices, environmental limits, etc. Qualitative refers to uncertain values like population approval of a projects, demands, financial returns expected, etc. Is there correlation between criteria? If there is, the method selected must be able to cope with it. See Question 12.3.2. Yes, there could be a correlation between criteria, and if it exists, it must be considered in MCDM. • Are there limits for each criterion in plus and in minus in performance values? No, there is no limit. You can have them in any mix. • Are clusters needed? Sometimes, yes. For instance, in city planning there could be several criteria clusters, normally corresponding to different departments of the City Hall. Thus, for project or alternative “Storm water collector” you may have a number of independent projects and criteria like “Maximum budget for NW-SE storm water collector,” “Maximum length of NW-SE stormwater collector,” “Underground temporary stormwater storage tank,” and finally “Maximum budget for storm water collectors,” the last allows for assigning a budget to this cluster.

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• Is there a need for using mathematical formulas? There can be mathematical formulas for both, performance values and resources (RHS). The first is very useful to make many tests on different performance values, using an auxiliary table or matrix that is related by a simple algebraic expression to a corresponding criterion in the decision matrix. In assigning different values in the auxiliary matrix, you can use a formula that reflects the new values in the decision matrix. Executing the software, the DM can obtain results or rankings considering as many values as wished. See detailed example in Question 12.5.1. There are also correlation formulas to be used between criteria. See detailed example in Sect. 6.6. For resources, you can use formulas expressing dependency of a resource regarding another. See example in Question 12.4.2. • According to this problem, is it convenient to incorporate as criteria some frameworks? For instance, some well-known frameworks like that from the OECD (Organization for Economic Co-operation and Development) makes explicit the existing interactions between different issues; this framework relates pressure, stress, and response. That is, the pressure of an action on the environment, the stress that it causes, and the results after applying improvement or remediation measures. See application in Table 12.5, in Question 12.2.8. Performance values • • • • • • •

Are performance values expressed in cardinals or in linguistics? Maybe there is a need to build a scale to convert linguistic values into cardinals, How much confidence offers the data? Is it necessary to use fuzzy numbers? Is there a need to express performances by mathematical formulas? Is it necessary to input negative performance values? If in the result there are two or more alternatives with the same scores. How to proceed? See Question 12.2.11. Sensitivity analysis

• The DM must talk with stakeholders to learn about what they will demand to know when he submits his conclusions. • He must find the method to answer stakeholders’ questions. Sensitivity analysis is a must; therefore, the DM must know how to perform it. Question 12.2.36: In What Types of Scenarios the SIMUS Method Can Be Used? SIMUS can be used in simple, complex, and very complex scenarios, but it is not recommended for trivial ones. Due to its algebraic structure using inequations instead of equations, it can handle a large variety of problems involving multiple scenarios and allows for modelling most characteristics present in real-life problems.

12.2

Sequence of the MCDM Process

265

It has the capacity to accept up to 200 alternatives and up to 100 criteria, and thus, well over the range of typical MCDM scenarios in practice. It also has the capacity to determine which are the binding criteria, that is, those that has a direct influence on the best solution. An important aspect is that its solution also informs in what percentage each criterion or objective has been satisfied. In addition, each run solves simultaneously two problems: The Primal, which delivers the scores of the alternatives and their ranking, and the Dual, which shows the marginal utility of each criterion, that is used in sensitivity analysis, as well as producing the total utility curve for each criterion/ objective. The fact that it does not use any kind of weights is a big advantage in producing sound results obtained without human participation. Nevertheless, SIMUS dynamically generates criteria weights in each iteration, used to eliminate dominated alternatives. The method heavily relies on the DM since, once a solid, mathematically true result is obtained in each run, he gets a robust platform to examine, research, analyze, and modify values, criteria, or alternatives, and even reject the result, according to his experience, know-how, and reasoning. SIMUS follows the bottom-up approach, that is, data may be modified by the DM post result, as opposite the top-down approach where initial data is altered and changed according to the DM preferences at the initiation of the process. SIMUS is probably the easiest MCDM method to work since the only thing a DM must do is to build the usual Initial Decision Matrix, with the aggregate of two or three mathematical symbols. Where can SIMUS be used? Table 12.13 gives the reader a glimpse of the type of scenarios managed by SIMUS. Bolded italics identify the variety of fields or areas, followed by a summary description of different projects including simple and highly complex scenarios. Question 12.2.37: Is It Permissible to Divide a Problem for Easy Resolution? No, it is incorrect, because MCDM is a system with all elements interconnected; therefore, selecting a certain pair of alternatives, making an evaluation, and then adding this evaluation to that for another pair of alternatives violates the system structure. Partitions are great for analysis but not for MCDM. It is very easy to show that if we have three alternatives A, B, and C subject to a set of criteria and solve the system using, for instance, the Weighted Sum Method, we will get different rankings if we solve it by pairs than if we solve it simultaneously.

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Table 12.13 Sample list of projects SIMUS addressed Bulgaria—(railways): (a) Performance measures for urban, regional, and international trains (b) Alternatives evaluation for containers transportation (c) Determining performance in ten countries from a Black Sea to Baltic Sea line (d) Planning passenger traffic when existing uncertainties Canada (a) (Assets management). Plan for bridges renewal (b) (Environment). Determination of environmental indicators at the country level (c) (Domestic waste). Best policy for collection of domestic waste (d) (Public health). Plan to improve health service during COVID 19 (e) (Electric energy.). Prefeasibility to prioritize sources of renewable energy Cuba—(industry and metal work) (a) (Tooling). Optimization of high-speed milling of thin plates (b) (Renewable energy). Wind energy in four locations Ghana-(Urban planning). Provision of basic infrastructure to five cities Indonesia (a) Modal transportation). Selecting the transportation mode for imported goods (b) (Electric Energy). Waste to energy Iran: (Sustainability). Classifying sustainability in Mashhad urban region Italy:-(Society). Pondering best site for installing domestic waste incinerators Maroc—(Urban planning). Removing and locating people in new sites México—(Metropolitan planning). (a) Study of Guadalajara urban and satellites cities, involving environment, economics, and infrastructure. (b) Selection of urban indicators in Leon city. Nigeria—(River) (a) Determination of the largest concentration of pollutants in the Delta of the Niger River (b) Determination of sediments, benthic and macro-level for water quality in Niger River Palestine—(International help)—Selection of projects to be sent to financial agencies of donor countries Portugal—(Mining) (a) Feasible alternatives for landscaping in transition processes post-mining (b) Example in pyrite Iberic Belt. Spain (a) (Irrigation). Selection of renewable energy sources for agriculture projects (b) (Air conditioning). Determination of the most important thermodynamic parameters in laminar condensers (c) (Road transportation). Fuel selection for 18-wheelers (d) (Urban planning). Run-down areas rehabilitation (e) (Assets). Planning and scheduling of regional road assets South Africa—(Infrastructure). Selecting the best route for construction of a high-voltage line. Sri Lanka—(Public health)—Framework to improve service quality during COVID 19 pandemic. UK (Wales) (Metropolitan planning). Identification of urban and peri-urban indicators.

12.3

12.3

Criteria

267

Criteria

Question 12.3.1: If I Want to Use Weights in MCDM Methods that Do Not Use Them, How Can I Proceed? In most MCDM methods you can use weights for criteria derived from entropy, statistics, MonteCarlo, AHP, SWARA, etc. Normally, the software asks for them, and there is no difficulty in placing for each criterion the corresponding weight. There are very few MCDM methods that don’t use pre-determined weights. It does not mean that they consider that all criteria have the same importance, but that the respective significance is computed based on data imputed, and thus, without human intervention. The most known methods are CRITIC, based on statistics, and SIMUS, based on linear programming. As said, SIMUS does not utilize any weights; however, if the DM wants, he can input his for both alternatives and criteria. If you want to work with weights in LP and SIMUS, they can be inputted for both, criteria and alternatives, separately or at the same time. For alternatives, just put your weights in the single row at the bottom of SIMUS second screen. The software will consider them. For criteria, you must make a prior operation that consists of multiplying each criterion weight by its corresponding performance factor. Put the corresponding results in the Initial Decision Matrix. Question 12.3.2: I Have Detected Correlation Between Criteria. Should I Consider It? If the Answer Is “Yes,” How Can I Input the Correlation Factor in the Method? There are not too many scenarios where criteria are independent, since in most cases they are linked in one way or another. In most MCDM methods this is not an inconvenient and the problem can be solved as any other. However, in some cases, there could be correlation, that is, the variation of one criterion may influence the values of the other. In these cases, this correlation must be inputted in the Initial Decision Matrix. Example: In this scenario there are two different potential routes to build a highway between two cities A and B. The first road is a mountainous path at high altitude, which uses tunnels and viaducts, allowing the highest speed, and also subject to heavy snowfalls in winter. The second road follows a river valley at low altitude and prone to flooding. The two routes are subject to a set of six criteria, some of them correlated. Figure 12.6 illustrates it. Solution: x1 = 5 and x2 = 3.4, and thus, Z = 28.6. In the following example, in a different scenario, the Pearson correlation coefficient between criteria shows, for example, that the most correlated criteria pairs are: “Safety” and “Speed” (ρ = - 0.86), consequently, are inversely correlated, since safety decreases as speed increases.

268 Fig. 12.6 Criteria A and B are correlated—When A varies to A’, B also varies to B’ Original alternative (A) also participates in the final solution.

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Support and Guidance to Practitioners by Simulation of. . .

X2 B A

A’ B’

Z

3.4

0

5

B

Z

A’ B’

X1

“Flood risk” and “Safety” (ρ = - 0.92), the same, they are inversely correlated, since the higher the flood risk the lower the safety. “Road Control” and “Safety” (ρ = 0.79). They are positively correlated, since to a greater control corresponds an increase in safety. These correlations can be modelled by using the “IF” function in Excel, as shown in Fig. 12.7. It shows for the mountain highway that when safety increases 12%, then speed decreases from 130 to 122 km/h, according to this formula: 130 - (80 × 1.12+ 80) 0.86 = 122 km/h. To solve it analytically, build the Initial Decision Matrix as shown. What have we learnt from this result? This indicates that if statistics show that heavy snowfalls are frequent and strong enough to close the highway, it is possible that the maximum speed of 130 km/h, cannot be maintained for long periods, and that could make that the Valley Highway becomes the favorite, instead of the Mountain Highway. Question 12.3.3: I Have Heard that Entropy Derived Weights Are Objective and Recommended, but What Is Entropy and Why It Is Better? What is entropy? Entropy is a concept derived from Thermodynamics that broadly indicates disorder. For instance, superheated steam has millions of “particles” moving in any direction. This means that it is very difficult or impossible, to know where a certain particle is at a certain moment, that is, there is complete uncertainty on its position. In this condition, it is said that there is a very high entropy; in other words, entropy is a concept linked with disorder and uncertainty. When somebody speaks over the phone, sometimes the person at the other end of the line has difficulty in understanding, because there is a lot of noise, or the

12.3

Criteria

269

Fig. 12.7 Introducing IF(. . .. . . .), conditional function

sentences are garbled, or words are missing, or the message is corrupted, or the language understanding is very poor. In this case, the meaning and content of the message are very uncertain, and similarly to thermodynamics, it is said that there is high entropy. Entropy maximum value is “1” (complete uncertainty), and “0” (certainty). Shannon, in 1948, discovered this similitude of messages with Thermodynamics, and derived the formula that allows us to determine entropy in a message, or in our case, in a criterion. In reality, we are not interested in the entropy (S) value, but in its difference to “1,” that is, (D) = (1-S), which is called “Quantity of Information,” a very important concept that paved the way to Information Theory. What is the advantage in using entropy, or speaking properly, its complement D? Because it is an objective weight, developed based on the data contained in each criterion. Entropy implies also concentration of values, while quantity of information (D) measures the dispersion or discrimination of values, and this property is paramount for evaluating alternatives. Elemental example: Suppose that we have to select among 4 different movies and have knowledge that they have been appraised by art critics in a scale 1 to 10 as: 2-2-1-2 (Table 12.14). It is very difficult to extract a conclusion, because all values are very close, that is, practically no differences between movies. Different is the case if the values are: 3-8-4-7. In this case, the discrimination of qualifications is very clear and gives the best values to movie number 2. Therefore, discrimination is fundamental, and this is what value D indicates. The larger the discrimination, the less the uncertainty. Example of calculation of entropy and quality of information: Assume you have four equipment subject to three criteria.

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Table 12.14 Initial decision matrix

Press 1 561,896 3,200 2,536

Alternatives or options Press 2 Press 3 562,397 561,821 4,000 3,700 4,219 3,018

Press 4 563,601 3,700 4,621

Sum 2,249,715 14,600 14,394

Table 12.15 Computation of probabilities Probabilities for each performance 0.250

0.250

0.250

0.251

0.219

0.274

0.253

0.253

0.176

0.293

0.210

0.321

1. Compute for each criterion their sum, seen in the column at right. 2. In each criterion, normalize its performance values by dividing each one by their sum and save them in a new matrix. Thus, for the first performance value: 561,896/2,249,715 = 0.250 (Table 12.15). 3. Compute the value of constant K = - 1/ln(n) (being “n” the number of alternatives). = -1/ln (4); K = - 0.721. 4. Multiply each normalized value by its natural logarithm. Example for the first number (0.250) will be: 0.250 ln (0.250) = -0.346. Table 12.16 5. Find the sum of each row and multiply it by K. This is the entropy of the criterion, but we want its complement “D” to 1, which is the quantity of information. These are the weights you can use for criteria. The most important criterion is the third one, because it has the largest D, or quantity of information. Question 12.3.4: I Heard that Criteria Weights Can Be Assessed Using Shadow Prices? What Are They? When a scenario is solved using linear programming (LP) utilizing the simplex algorithm, two problems are solved automatically and simultaneously by the

12.3

Criteria

271

Table 12.16 Computation of criteria weights Criteria

-0.346

-0.347

Sum -1.386

Entropy D =1 - (Ei × K) K = -1/ln (4) = 0.7212 S = Sum × K 0.999

-0.348

-0.348

-1.383

-0.328

-0.365

-1.358

Ei = Ei = ∑ ej × K ej = pj × ln(pj) -0.347 0.346 -0.355 0.333 -0.360 0.306

Difference (D)

0.001

Quantity of information (I) D normalized 0.04

0.997

0.003

0.12

0.979

0.021

0.84

D/∑i D

∑DI = 0.025

algorithm. The first is called “Primal” that gives the optimal solution if it exists. The second is called “Dual” and displays the “Shadow prices” or “Marginal values” for each criterion or objective. The shadow price is a value that indicates how much a solution changes when the corresponding criterion is increased/decreased in one unit. It is the marginal value or marginal utility of each criterion. Therefore, it gives its real importance. In SIMUS it is used in Sensitivity Analysis, in the add-in called IOSA. Shadow prices are useful because they allow building the total utility curves for each objective. They also put a value to externalities, which is something without a market price, like noise, depletion of natural resources, or damages to the environment. Also, in the market, they may indicate how much a person is willing to pay to get a certain good. IOSA uses them in sensitivity analysis, instead of weights, and thus working with realistic values. Example: Shadow prices are used in an agroindustry project for export. See Sect. 8.1.1. in Chap. 8. Question 12.3.5: How Can I Get Objective Criteria Weight Using PSI? Answer: See paper from Attri, “Application of preference selection index method for decision making over the design stage of production system life cycle.” Question 12.3.6: How Can I Get Objective Criteria Weights Using SWARA? Answer: See paper from Vescovic and Stevic, “Evaluation of the railway management method by using a new integrated method DELPHI-SWARA-MABAC.”

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Table 12.17 Example of clusters Areas intervening in composite indicators EC. GROWTH Indicator I.D. 1 Capital and Indicators labor factors Objective areas

2 Trust

Experts'evaluation on relationship between indicators and objectives

Objectives

Education 1-Enhance all levels of education Social 2-Minimize gender differences at work Social 3. Maximize income/capita Social 4-Maximnize people sufficiency Health 5. Reduce child deaths Health 6-Improve maternal health Health 7-Minimice infectious diseases Sustainability 8-Guaranty sustainability Environment 9-Reduce global warming Environment 10. Reduce acid rain Economy 11. Improve productivity Economy 12. Diversify the economy Economy 13 Increase exports OECD framework 14. Pressure OECD framework 15. State OECD framework 16. Response

SOCIAL CAPITAL 3 4 5 6 Groups and Social Collective networking Diffusion inclusion actions

5

7 7 –7 6

7

–9

7 6

7 9 7

7

8

7 –9

7 –8 9 6

–6

8 7

8 4 7 7 –8

8

6

–6 –4 8 8 9 9 8 7

8 8

9 7

Question 12.3.7: How Can I Work with Alternatives Clusters? A cluster in MCDM can be defined as a number of similar things grouped together under a generic name. For instance, under the label “Environment” we can have aspects such as “Forested area,” “Replanted area after logging,” “Aquifers,” “Erosion,” etc., all of them subject to a set of criteria. In a criterion, “State of the environment in year 2023,” we put numbers at its intersections with the corresponding columns under the Environment label, like thousands of hectares in forested areas, hundreds of hectares replanted, state of the aquifers, and areas affected by erosion. As another example, Table 12.17 shows a cluster for “Social capital” in a project to determine a composite indicator. There are also clusters in criteria for Social, Health, Environment, Economy and OECD (Organisation for Economic Co-operation and Development) framework, not shown here due to readability. Observe that for each indicator there are positive and negative values between an indicator and a criterion, corresponding to degrees of relationships, according to expert’s estimates. If positive, a high value indicates a direct high relationship. If negative, they show a negative relationship, that is, the indicator has an opposite relationship. For instance, for indicator “Social inclusion,” there is a high positive relationship (7), with criterion “Education,” and a very high negative relationship (- 9), with criterion “Gender differences.”

12.3

Criteria

273

Question 12.3.8: Is It Correct to Use Pair-Wise Comparison for Determining Criteria Importance? If Not, Why? Pair-wise comparison of criteria is not performed in SIMUS; however, because of its importance in other methods, a short explanation follows to answer this question. Pair-wise comparison is a great tool, but not for MCDM, and at least in AHP, considers two criteria, and arbitrarily determines that one is more important than the other, and also gives a value of that importance; all of this, based on intuition. According to the Cambridge Dictionary, intuition is: An ability to understand or know something immediately based on your feelings rather than facts For this author, this procedure is incorrect because intuition can be used on personal matters, like I intuit that it is a nice place, but not for serious scenarios. Needless to say, there is no mathematical or common sense support for this procedure. In other methods, like PROMETHEE, the pair-wise comparison between alternatives is performed considering real values for each alternative, for example, distances in two different routes between two points A and B. In this case, the pair-wise comparison makes sense. In addition, in AHP, the importance of a criterion over another is arbitrary, because it depends on who is the DM; if DM 1 establishes a value of preference between criteria C5 and C6, it could happen that DM 2 may have a different value, and thus, there can be as many values as DMs. Even considering that the DM is correct in his/her appreciation, it is done independently of the alternatives they are supposed to qualify. If he says, for instance, that he always prefers quality over price, he needs to refer them to the alternatives; otherwise, it is irrelevant because it is not always possible to generalize. It could very well be that considering alternatives A, B, and C, this preference may make sense comparing A and B, but different when comparing A and C, and thus, the DM cannot be sure that the same preference apply, or perhaps be reversed. Example: Consider a simple example such as selecting a restaurant, out of three, for dinner. Table 12.18. The DM says that for him, quality is more important than price and, thus, in comparing restaurant A with B, he will select A, and thus, he prefers to sacrifice price (i.e., paying more for a meal) to gain in quality (i.e., getting a better meal). Consequently, there is a trade-off between quality and price. Now he compares A and C, and says that the difference in price is large and the difference in quality is small. Consequently, he may change his preference and choose to sacrifice a little quality but win in price. Therefore, it is incorrect to establish a preference that applies to everything, let alone without considering alternatives.

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Table 12.18 Initial decision matrix Criteria Quality Price

Restaurants A

B

C

9 45

7 38

8 39

Question 12.3.9: In a Set of Alternatives, Subject to a Set of Criteria, Is it Correct to Analyze Alternatives One by One or in Pairs, Compute Some Value, and Then Add Them Up? If Not Why? It is incorrect, since a scenario is a system. It can be partitioned for a deeper analysis in more detail, but each part cannot be solved independently and results added. Avery simple example demonstrates this. Consider three projects A, B, and C. Choose A and B and calculate the best alternative. Do the same for A and C and for B and C, and add results. Now consider the three of them jointly and find a result. Most probably, the latter is different from the former. Remember that the whole not always is equal to the sum of the parts. Why is this? Because when considering A and B, the DM does not take into account, as very often happens, that B is or may be related to C. Question 12.3.10: Is There Any Way to Know If a Problem Is Unfeasible? Yes, there is, at least in SIMUS. Normally, it happens when there are errors in the symbols for criteria. It could also be that there is incongruence. Consider for a certain criterion that LHS (the computed result for a criterion), must be lower than RHS the wished goal a criterion). If LHS = 56 and RHS = 48, for a relation type “≤” or maximization, meaning that LHS must be as a maximum lower or equal to RHS, there is obviously an incongruence, since the symbol is not respected. In this case, the problem is unfeasible since the criterion is not satisfied. Question 12.3.11: Is It Necessary to Work with Weights for Criteria? It is not mandatory but most probably necessary. If they are not used, it is assumed that all criteria have the same relative importance, and in general, this is not realistic. There is normally a logical scale of values between criteria, when, for instance, “cost” is considered more important than say “company image.” What is however important is the manner in which these weights are determined; subjective weights, as those based on preferences, are not really appropriate for alternatives selection or for sensitivity analysis, because the final result is dependent on DM preferences. On the contrary, objective weights, as those from entropy, ratio analysis, statistics, etc., may be very useful, since they are independent of the DM feelings, and in addition, invariant, since they do not change whoever the DM may be.

12.3

Criteria

275

Table 12.19 Selection roads Weighted decision matrix Estimated weights Criteria 0.14 Distance 0.21 Cost 0.1 People's opinion 0.09 Slope 0.07 Sav. trans. towers 0.1 Crossing sce. areas 0.11 Ecosys. protection 0.02 Swamps crossed 0.07 Earthquake risk 0.09 Forestry Dist. from each project and ideal (R+) Dist. from each project and ideal (R–) Result

Suquia Trevel Suquia River Mountains Valley / Valley Mountains 0.048 0.045 0.047 MIN 0.067 0.074 0.069 MIN MAX 0.031 0.046 0.023 0.045 0.023 0.023 MIN 0.027 0.019 0.023 MAX 0.02 0.063 0.024 MIN MAX 0.022 0.063 0.017 MIN 0.012 0.001 0.006 0.013 0.035 0.022 MIN MIN 0.02 0.041 0.029 0.0480 0.0560 0.0490 0.0540 0.0540 0.0550 0.53 0.49 0.52

(A+) ideal solution 0.045 0.067 0.046 0.023 0.027 0.017 0.063 0.001 0.013 0.021

MAX MAX MIN MAX MIN MAX MIN MAX MAX MAX

(A–) ideal solution 0.048 0.074 0.023 0.045 0.019 0.063 0.022 0.012 0.035 0.041

Ranking: Suquia River Valley - Suquia Valley/Mountains - Trevel Mountains

However, since criteria have probably different relative significance, we must someway determine them. In this respect, objective weights derived from the problem data can be used, like those based on standard deviation of criteria data or using entropy. There are methods like CRITIC that generate these weights, and others, like linear programming and SIMUS, that internally determine criteria relative importance by other means, but based on data. An important feature of LP and SIMUS is that weights are not constant, for the whole set of weights is computed in each iteration in accordance with the new alternative selected. Question 12.3.12: Is There Any Proof of the Disadvantage of Using Subjective Weights? Look at this scenario solved by SIMUS to select a road. Table 12.19 models a road problem with three alternatives and ten criteria. Subjective weights were assigned to criteria and shown on the “Estimated weights” column as 0.14 and 0.21, respectively, and being the highest weights of the set. Notice that both criteria have very similar performance values in each one. Assuming that this data is correct, one realizes that these two criteria have very poor performance values for alternative evaluation. When this problem is solved by SIMUS, the result shows that these two criteria do not participate in the alternatives selection; they are irrelevant, something that is also corroborated by computing their entropy, which can be seen in Table 12.20.

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Table 12.20 Problem solved by SIMUS Weighted decision matrix Weights from entropy Criteria Distance 0.001 0.001 Cost 0.052 People's opinion 0.075 Slope 0.013 Sav. trans. towers 0.230 Crossing sce. areas 0.154 Ecosys. protection 0.327 Swamps crossed 0.099 Earthquake risk 0.099 Forestry Dist. from each project and ideal (R+) Dist. from each project and ideal (R–) Result

Suquia Trevel Suquia River Mountains Valley / Valley Mountains 0.000 0.000 0.000 MIN 0.000 0.000 0.000 MIN MAX 0.016 0.024 0.012 0.037 0.019 0.019 MIN 0.005 0.003 0.004 MAX 0.046 0.144 0.039 MIN MAX 0.036 0.088 0.030 0.200 0.021 0.105 MIN 0.018 0.049 0.031 MIN MIN 0.011 0.023 0.060 0.1880 0.1100 0.1040 0.1030 0.1900 0.1440 0.36 0.63 0.58

(A+) ideal solution 0.000 0.000 0.024 0.019 0.005 0.039 0.088 0.021 0.018 0.011

MAX MAX MIN MAX MIN MAX MIN MAX MAX MAX

(A–) ideal solution 0.000 0.000 0.012 0.038 0.004 0.144 0.030 0.200 0.094 0.023

Ranking: Trevel Mountains - Suquia Valley / Mountains -Suquia River Valley

If criteria weights are computed by entropy, they are: Distance: 0.001 and Cost = 0.001, very far from the subjective weights assigned (Distance = 0.14 and Cost =0.21). Therefore, both SIMUS and Entropy coincide. Remember that SIMUS is not related to entropy; therefore, they are two independent assessors.

Question 12.3.13: Why Subjective Weights Are Not Appropriate for Alternatives Selection The main purpose of criteria weights is to ponder criteria relative importance for alternatives evaluation. However, subjective weights as those obtained by the DM preferences do not reflect reality, but what a DM intuits. Even if the weights are obtained by a group of DMs, and although chances are that they truly represent criteria importance, they are always subjective, and another DM or another DM’s group can suggest different values, and of course, they might be subject to the DM’s vested interests. Therefore, the influence that they have, generally on the performance values, is debatable. The best weights are objective, and generally they can be produced by the analysis of the initial data; therefore, given the same initial conditions, they are constant.

12.3

Criteria

277

Do All Methods Use Weights? Probably all methods do use weights; the issue is how to determine and apply them rationally. In addition, when the variation of weights is used for sensitivity analysis, it is assumed that an increase of say 1 unit does normally have the same influence on all performance values; however, increasing 1 unit in a performance of 10 does have not the same effect on a performance value of 100. Question 12.3.14: I Have Read that One of the Advantages of Linear Programming and the SIMUS Methods Is that They Do Not Use Weights: Does It Mean that All Criteria Have the Same Importance? Correct, they do not use subjective or objective weights for criteria, but it does not mean that they do not take into account criteria relative importance, for both methods use another concept to define criteria significance. As a matter of fact, their consideration is capital in the algebra of LP, since calculated criteria importance is constantly updated in each of the process’ iterations, which may be in the hundreds. This can be understood since LP is an iterative process, that is, each step is based on the precedent step or solution, and in each one the whole set of computed criteria importance is recalculated. The concept of weight in LP is not aimed at quantifying criteria relative importance, but in ranking them relative to their capacity to get rid of a dominated alternative. In LP the Simplex algorithm starts adding extra alternatives (or auxiliary alternatives), and the first solution, based on them, is equal to “0.” In order to improve it, one of the real alternatives is added; consequently, an extra alternative must be eliminated to keep constant the size of the dimensional space, contingent to the number of initial real alternatives. Determining which of the auxiliary alternatives must be eliminated depends on the ratio between the values of criteria and the performance values of the alternative selected to enter. It can be seen then that this ratio or “weight” is paramount in the LP algorithm. This can be clearly seen in Section A.1 of the Appendix. Question 12.3.15: How LP or SIMUS Select Alternatives? Each iteration yields a global result; to improve it, an alternative is chosen either for increasing benefits or for reducing costs. For this selection, each alternative is individually examined through the economics concept of “Cost of opportunity,” that is, determine how much you lose if not selecting a certain alternative. In other words, “cost of opportunity” is the cost of the foregone opportunity. Obviously, you select the alternative that produces the lowest loss. It is calculated using the formula (Zj - Cj) for both, benefits and costs, where Zj is the objective function for alternative “j” and Cj is the benefit/cost with which an alternative “j” contributes. Consequently, when maximizing a benefit, it is chosen the alternative with the most negative difference, or the one with the most positive difference when minimizing costs. See the Simplex Tableau in Section A.1 in the Appendix to better understand this procedure.

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Table 12.21 Criteria and restrictions Criteria C1 Benefits (€) C2 Raw material (kg) C3 Manpower (number) C4 Environmental damage (ppm) C5 Equipment (units)

Product A 5000 100 200 300 1

Product B 4000 160 100 100 3

Action Max Max Max Min Max

Limits 6000 350 260 100 3

When a new alternative is selected, its Zj = 0, and if its contribution to benefit is say “5,” the difference is—(0 - 5) = 5, and thus, it is preferable to another alternative, which contribution is say 2.5. Same reasoning, but opposite when minimizing costs. This independent analysis is the reason why there is no rank reversal in LP. Question 12.3.16: Do Criteria and Restrictions Have the Same Meaning? Many people use indistinctly the word criteria and restrictions; however, they are different. Criteria establish the conditions to which alternatives are subject, while restrictions may indicate the scope of criteria. For instance, in a criterion related to NOx contamination produced by different equipment (Alternatives), the criterion is composed of the set of performance values corresponding to each equipment. A restriction is used to establish a limit to this contamination, normally expressing that whatever the selection is done, the contamination cannot surpass that limit. Therefore, restrictions establish a limit or threshold. Table 12.21 illustrates the concept of limits to resources, but these restrictions can also express allowable limits of contamination, as well as lower and upper limits. In the last case, the criterion is restricted to oscillate between two extreme values. Example: A company manufactures two products A and B (Table 12.21) using as a maximum 260 workers as shown in criterion C3 (Manpower), calling calls for maximization, that is, to manufacture as much as possible. Therefore, the solution must indicate the quantity of A and B to manufacture, considering all criteria and their restrictions. In this case, manpower cannot exceed 260 persons. The reader may be puzzled to see that there is a limit for benefits. The reason is that there is a certain amount of funds, and the company has estimated that the maximum benefit they can obtain using them is 6.000 Euros. Therefore, this restriction is due to scarcity of funds. Criterion C4 strangely says that the damage must be at least a minimum, when it should be zero. The reason for this apparent no sense is that it is not realistic, in this case, to manufacture something without producing some damage to the environment, for instance, the fumes of the diesel engine used to generate electricity for the process. This can be seen everyday. For instance, while eating bananas, there is an unavoidable waste, that cannot be reduced: Its skin or peel, and remember that even breathing produces air contamination: CO2.

12.3

Criteria

279

Question 12.3.17: Is It Possible to Get the Same Result for a Given Problem by Using Different MCDM Methods, and the Same Criteria Weights? Yes, it is possible, but not guaranteed. Even when the DM uses the same weights and the same factors in each method, there is no guarantee because each method has different subjective approaches. There is very interesting paper by Janic and Reggiani, where three different methods (SAW, TOPSIS, and AHP) are used to run the same problem in three different scenarios, such as: (a) Using the same weights. (b) Using weights derived from simulation. (c) Using for SAW and TOPSIS the entropy methods, while AHP uses its own, based on preferences. Comparing the rankings of the three methods they show, for SAW and TOPSIS: 1. They coincide with the selection of the best alternative in the three scenarios. 2. The two rankings disagree for the three scenarios. 3. AHP also coincides with the best selection, but gets a different ranking. Question 12.3.18: What Is a Trade-Off? A trade-off is an interchange of values. Diminishing the quantity of something gets an increase in another. For instance, when in a pair-wise comparison we say that we prefer A over B, it means that we are willing to sacrifice some attribute of B in order to get some gain in A. It can also be seen as a balance, a compromise. Question 12.3.19: What Are Externalities? Externalities are actions that do not have a market value. For instance, forest exploitation has a diverse market value given by the unit price of timber harvested and processed in its different formats (beams, panels, logs, etc.) and producing benefits. However, the erosion caused by logging does not have a market value, or on other words, it is not considered a loss, when in reality it is; erosion is then an externality. Others are noise, air, water, and soil pollution, fish depletion, etc. Externalities are usually ignored with computing the Gross Domestic Product (GDP), when it should be deducted, as benefits are added. These are very important aspects in projects selection and in MCDM, and therefore, must be considered as criteria. For instance, for each logging project it can be assessed the erosion it can cause. These are the performance values for a criterion called “erosion.” This can be done using most MCDM methods, where erosion is just another criterion that needs to be minimized. When the problem is solved by LP or SIMUS, the dual problem gives a value for such a criterion by means of the shadow price. Therefore, many scholars agree that this shadow price can be interpreted as a market value for this externality, since it may represent how much somebody is willing to pay to get something.

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Another typical externality is produced when an industrial plant, say a pulp mill, is built on the shores of a river. The effluent of the plant, even treated, is normally discharged into the river. However, the remaining contamination can produce serious damage to the fishing industries downstream, because the fish may be too sensible to this pollution, and then may be dying or leaving the area. This type of effects produced by industrial plants or by other activities, such as agriculture upstream, is the origin of many international conflicts and have halted more than some projects, so, better to keep an eye on it. Question 12.3.20: I Have Minimum and Maximum Limits for the Same Criterion: How Can I Proceed? In other words, in my problem there is not a limit, but an interval, for instance, in my project I have a minimum of water for the process, but, at the same time, water is scarce, and I need an upper limit of usage. How can I indicate this?

This is a very common requirement in most industries, and it has an easy solution using what is called a “Dual criterion.” It consists in configuring the initial decision by the addition of two rows for the same criterion. In both, the performance values are the same, and what changes is the mathematical symbol “≥,” denoting minimization, or lower limit, and “≤” denoting maximization for the upper limit, and putting in the resource’s columns the corresponding minimums and maximums values. It could also be that there is one set of performance values for minimization and another different set for maximization, that is, different performance values for the same resource; however, the resource limits will be the same as in the former case. Most probably the two results, that is, corresponding to no change and change of performance values, will be different. Example An urban development project aims at building a new housing complex involving three different types of dwellings, according to their size. Data for alternatives and criteria is in Table 12.22. Observe that the number of units (houses, in 3 types) for each type of dwellings, as well as minimum and maximum number of dwellings, investment and floors space, are dual criteria. Results from computation are in column “Results for computation,” while target values are in column “Requirements.” Therefore, it is possible to see how in each criterion results meet requirements or targets. Type 1 houses: Minimum number of units: The result from computation is 30, the same as the target, Maximum number of units: 30 houses, against the target of 40. This seems to indicate that 40 houses are too many for this type, considering the other criteria restrictions. Type 2 houses: Minimum number of units: 35, same as the target, Maximum number of units: 35, far below the target of 48. This seems to indicate that 48 units are too many for this type, considering the other criteria restrictions. Type 3 houses: Minimum number of units: 7, against 5 as the target.

12.4

Resources and Limits for Criteria

281

Table 12.22 Using the same criterion with maximums and minimums

Investment per dwelling (Euros):

Plan for Plan for Plan for type 1 house type 2 house type 3 house 16,800 22,000 26,500

Requirements Action

Results from computation

Units

7,480 30 30 35 35 7 7 72 72 1,462,813 1,462,813

m2 dwellings dwellings dwellings dwellings dwellings dwellings dwellings dwellings Euros/ha Euros/ha

Area of each lot (m 2 ) Minimum number of units of type 1 houses Maximum number of units of type 1 houses Minimum number of units of type 2 houses Maximum number of units of type 2 houses Minimum number of units of type 3 houses Maximum number of units of type 3 houses Minimum number of dwellings Maximum number of dwellings Maximum total investment (Euros/ha) Minimum total investment (Euros/ha)

69 1 1

1 1 1 1

1 1 1 1

1 1 1 1 1 1

Maximum floor space area (m2 /house)

55

65

95

4,602

m2

≤

5,000

Minimum floor space (m2 /house) Maximum density (persons/house) Minimum density (persons/house) Water consumption (liters/day-house)

47 4.55 4.00 1,297

52 5.52 5.20 1,573

80 6.30 6.15 1,796

3,800 375 346 106,757

m2 persons persons liters/day

≥ ≤ ≥ ≤

3,800 400 340 115,000

Sewage production baths & kitchens (m³/day-house) Electric energy consumption (kWh/day-house) Total cost (Euros) 1,462,813 Number of dwellings to be built by hectare

0.98 27.30

1.19 33.12

1.36 37.80

81 2,248

m 3 /day kWh/day

≤ ≤

90 3,000

30

35

7

72

122

160

1 1

≤ 7,700 ≥ 30 ≤ 40 ≥ 35 ≤ 48 ≥ 5 ≤ 10 ≥ 50 ≤ 95 ≤ 1,500,000 ≥ 1,250,000

Total number of houses

Maximum number of units: 7, below the maximum 10 as the target. In the first case this could indicate that the estimated target (5) is too low, and in the second case the target is also too low. Consequently, studying these three cases, it appears that Type 3 is the most convenient size considering all criteria. These results allow making multiple analyses to reduce the total cost of 1,462,813 Euros, studying diverse possibilities, for instance, floor spare in Types 1 and Type 2. If the first, it is reduced from 47 m2 to 45 m2, and the second from 52 m2 to 49 m2, the total cost will be 1,362,000 Euros, that is, a saving of 82,826 Euros.

12.4 Resources and Limits for Criteria Question 12.4.1: How a Criterion May Have Two Different and Opposite Limits: May I Work with Both at the Same Time? Yes, you can, the idea is to create a range, and letting the algorithm of the MCDM method to select the best alternative. As a consequence, in the solution, that criterion will have an intermediate value between minimum and maximum. See Question 12.3.19. Procedure: Just use two identical criteria, that is, with the same performance values, in two rows. Minimize one criterion by using the “≥” symbol that establishes

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a minimum value, and maximize the other by using the “≤” symbol, which establishes the upper limit. Question 12.4.2: I Have a Criterion Resource or RHS that Is a Function of Other Criteria: How Can I Model That? Just put the formula that relates both in the corresponding RHS with reference to another RHS. For instance, if criterion C8 is related to criterion C12, just state, for instance, that RHS8 = 3.25*RHS12. Therefore, each time you alter C12 you will also alter C8. Question 12.4.3: What Are the Characteristics of Resources? Resources can apply to everything that measures and defines the scope of criteria. They can be quantities, percentages, or limits of contamination, normally expressed in different units. They are a fundamental part of MCDM, since there in nothing unlimited in our world, and it applies to maximum/minimum available investment, ($) minimum/maximum cost, minimum acceptable percentage of return for an investment or IRR (%), maximum allowable environment limits (ppm), minimum/ maximum manpower (number), maximum/minimum volumes (m3), higher/lower pressure (kg/cm2), quantity of raw materials (ton), simple numbers (2, 6, 0.23. . . .), algebraic formulas, etc. These are the elements that define, together with the performance values, the form of an inequation, that is, line, plane, or hyperplane, which determine the geometrical figure that contains all the feasible solutions of a problem. These are also known as thresholds. In some cases, the amount or quantity of a resource depends on the outcome of another. As an example, the oil industry works as some sort of assembly line, in the sense that the output of a stage or area is input for another. For instance, crude oil production is the input to the oil refinery that has a limited refining capacity. Diverse products from the refinery, like gasoline, fuel oil, kerosene, etc., are its output that go to a resource such as storage tanks, also with limited capacity, then loaded in trucks, another resource with certain capacity, and finally go to the tanks of a gas station to satisfy demand.

12.5

Performance Factors

Question 12.5.1: How Can I Use Formulas for Performance Factors? This is not a common problem, but a DM may have the necessity of inputting formulas in the decision matrix instead of simple numbers. Other than using the “IF. . . ..” formula for correlation, sometimes, especially if performing an analysis of alternatives prior to their selection, there is a need of examining how certain criteria vary and results change when changing numerical values.

12.5

Performance Factors

283

Table 12.23 Initial decision matrix data in excel (data matrix is not shown) Cell D29 = $D$8*(D9/100) *$D$7*D10*365 D

Water 29 Yearly water volume to supply 30 Yearly sewage volume to generate 31 Yearly gas volume to supply 32 Water payment 33 Sewage payment 34 Gas payment 35 Water infrastructure cost 36 Sewage infrastructure cost 37 Gas infrastructure cost 38 Water connecting cost to dwellings 39 Sewage connecting cost to dwellings 40 Gas connecting cost to dwellings 41 . Total cost

182208

48000

52212

100212

E

F

Green Valley Sewage Gas

G

H

Altavista Water Sewage

I

Water

J

Saint Paul Sewage

K

Gas

18559520 84680000 24598080 22619415 114318000 2277.600 10585.000 185595 846800 491962 452388 2286360 364.42 1694 36200 23000 54600 50400 58000 33800 64000 63240 48360 75112 81345 64350 661810 563550 75112 695610 99440 81345 71360 122350 627550

L

M

N

LHS values 33,798,770 ≤ 161,535,495 ≤ 118.03 ≤ 1,214,603 ≥ 3,230,710 ≥ 19 ≥ 240,111 ≥ 284,598 ≥ 714 ≥ 270,498 ≥ 385,682 ≥ 6,284 ≤ 1,872,979 =

O

RHS values 161,535,495 12,863 1,214,603 3,230,710 2,058 107,200 163,000 97,800 163,812 220,807 1,225,360 1,872,979

Example: An entrepreneur is thinking of building dwellings in three different areas. Each one involves three infrastructural facilities, namely, provision of water, sewage, and gas, and for two of them, water and sewage. All of these eight projects are subject to a set of criteria. However, because of uncertainty, the developer wants to solve the problem using different values for these eight undertakings. For that purpose, he builds a matrix with all data, called “Data matrix,” and prepares an “Initial Decision Matrix” where all values are referenced to the data matrix. Considering that values in the latter may be a combination of several values taken from the data matrix, he put formulas in the initial matrix as seen in Table 12.23. Data matrix as well as a complete explanation of this problem can be seen in Chap. 7, Section 7.4.2, Table 7.13. For instance, to compute the yearly water volume to supply Green Valley site (cell D 29 in the Table 12.4), he writes Cell D29 = $D$8*(D9/100) *$D$7*D10*365. That is, the amount of water to be supplied depends on these criteria values of the data matrix, where: Cell D8: “Number of dwellings” Cell D9: “Dwellings without services” in percentage Cell D7: “Density” Cell D10: “Average water consumption” It can be seen that using these two matrices, the DM can change whatever he wants in the data matrix and be immediately reflected in the Initial Decision Matrix. Running the software, the results will reflect these changes. It is clear that the developer can make this type of simulation as many times as whished. The results

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Table 12.24 Serial risk in the Z matrix

in each run will indicate the total amount of water, sewage, and gas needed in each site. Question 12.5.2: How Can I Get Qualitative Performance Factors? Qualitative values may come from different sources, such as experts’ opinions, Delphi, statistics, Monte Carlo, polls, surveys, and samples. They can also come as crisp values from fuzzy logic and intuitionistic analysis. Many projects refer to undertakings that in one way or the other will involve thousands of people. In this case, it is necessary to perform surveys in the area/s to be affected by the project, and ask people what they think about the project, as well as their opinion about benefits or harm it will bring. This is very important because, often, DMs back in their offices, sometimes thousands of kilometers away, don’t know the area, and don’t have an idea of these problems, which are very real and known for people living in those areas. It is normally amazing the wealth of information that a survey can provide. This is highly qualitative data and can be statistically consolidated in percentages for each different project, using one criterion for benefits and another for losses. Some MCDM methods consider that the DM may decide for people and thus use his own estimates, which of course is incorrect, since nobody can vote or give his/her opinion on behalf of others. By the way, the Arrow Impossibility Theorem, addresses this same issue. Question 12.5.3: What Are Serial Risks? Sometimes, a risk may provoke another, which in turn generates another, and so forth. That is, these are serial risks than can act as a domino. Therefore, it is of interest to determine how an initial risk generates others, and when the intermediate risks have diverse significance. Table 12.24 details the interaction of one vector and

12.5

Performance Factors

285

four matrices, called “Z Matrix,” that simulates a case in a coastal nuclear plant where the initial risk is a tsunami and divided into five are as: 1. 2. 3. 4. 5.

Potential risks vector Effects matrix Consequences matrix Mitigation matrix Total impact results matrix As seen, the result is the determination of the total impact. This example can be examined in more detail in Chap. 2, Section 2.5.15.

Question 12.5.4: How Can I Compute Performance Factors Considering the Impacts of One Over Another? This is not an easy task because it is hard to evaluate the impact numerically, and it has not received too much attention from researchers. This author has developed a method, called “Z Matrix,” that can help because it allows in determining the accumulative effects and produces a single number that can be inputted into the Initial Decision Matrix. Example: Analysis of a new railway service. It refers to a new City Hall project to take advantage of abandoned railway tracks, in order to build a Light Railway System for urban transportation. Figure 12.8 proposes an example and shows how it works. See also Section 2.5. 15 in Chap. 2. It starts with a vector with the origin or cause of the issue, that is, the trains’ traffic. This vector will produce diverse effects such as contamination, noise, and vibration. We choose to analyze “Noise.” The “Effects” matrix shows that trains will generate noise with an intensity of 78 dB, which is quite high. This noise will affect receptors in the “Receptors matrix,” for instance, to neighbors and affecting 1781persons. In addition, and related to that people, there will be consequences, which are detailed in the “Consequences matrix” and with a cost of 5897 Euros because people complain. This leads to the necessity of constructing a sound barrier, which amount to 194,810 Euros as seen in the “Response Matrix.” Therefore, this is the value that should be inputted in the initial matrix for a criterion such as noise. The same procedure should be applied to effects such as contamination and vibration, and to receptor wildlife, as well as consequences to people complaints and light contamination. Question 12.5.5: How Can I Work with Non-lineal Performance Factors? In many cases the performance values for activities are considered lineal when in reality they are not. For example, when alternatives have unit costs as a criterion, it could very well be that these values are not real, since costs and demand for instance, vary in a non-linear relationship. Therefore, it cannot be assumed that cost is a constant.

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Fig. 12.8 The Z procedure applied to a new railways service Table 12.25 Initial Decision Matrix for product A

Units Unit cost (A) Unit benefit (A) Labor (A)

Up to: 100 0.62 0.16 34

Up to: 200 0.44 0.17 34

Up to: 400 0.28 0.3 38

Example: Assume two products, A and B, which prices vary as a function of the quantity purchased, which is the norm in every transaction. That is, normally prices per unit of product decrease when the demand increases. This relationship is not lineal. See Table 12.25 for product A. Up to 100 units the unit price is 0.62 Euros. From 100 to 200 units the unit price is 0.44 Euros, and from 200 to 400 units the unit price is 0.28 Euros.

12.5

Performance Factors

287

Consequently, the unit benefit increases from one quantity to another, as well as labor, while cost decreases, all in a nonlinear mode, represented by the respective curves. These curves may be divided into linear segments, and thus, each one is considered an option in MCDM. This example is detailed in Fig. 7.16 for product A and in Fig. 7.17 for product B, in Sect. 7.4.4, Chap. 7. Observe that each product has been decomposed in lineal intervals where LP can be applied. This problem can be solved by SIMUS, and the result is shown as the blue solid lines in Fig. 7.18. It can be seen that product B is the best. Question 12.5.6: Is It Possible to Have some Performance Factors Equal to Zero? Yes, it is possible and very usual. Example: To select between three different sources of electric energy. Assume that the scenario contemplates a power plant burning coal, a second power plant burning gas, and a third using wind power. Also presume that one of the criteria refers to air pollution. The coal-burning plant will most probably have a value of contamination, say in NOx, which is imputed in the decision matrix as the corresponding performance factor. The natural gas plant will undoubtedly have a value much lower than the first. However, the wind power plant will have a performance factor equal to zero, because, obviously, it does not produce that type of air contamination. Nevertheless, this last may have other environmental defects that are absent from the two first, for instance, “noise” or “birds killing.” As a bottom line a “0” in the decision matrix means that a certain alternative does not contribute to a criterion. Question 12.5.7: How Can I Indicate in the Decision Matrix that There Are Different Kinds of Contamination or Pollution? Create a criterion for each type of pollution generated by the different alternatives, in air, soil, and water, that is, one criterion displaying the contribution of all alternatives due to CO2, another criterion for Particulates, another for SO4, and so forth, And, obviously, you need to instruct the software about the high and the low limits for each one. If one criterion calls for lower and higher limits, there are two rows with the same performance values, but with different symbols: “≤” for higher limit and “≥” for the lower limit. Of course, all polluters will probably have different units; don’t worry, normalization will take care of that. Table 12.26 shows several gases and their values in corresponding units. There must be no restrictions regarding the number of criteria addressing contamination; use as many as you need. Remember that each criterion must have a limit (RHS), which is established internationally or locally, for maximum amount of pollution accepted for each one. Missing to declare a contaminant may induce environmental agencies or even credit institutions to reject the project.

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Table 12.26 Most common pollutants found in MCDM projects

Pollutant Nitrogen gases Family of Sulfur gases Carbon monoxide

Criteria labels: contamination levels, minimum and maximum NOx SOx

CO Hydrocarbons

Various gases

Smog Particulate

Dust Lead

Pb

Most common origin Cars, electric energy, generation plants Electric energy generation plants burning coal with a high content of Sulfur. Example: Oil refineries Idling cars Gasoline and gas oil operated vehicles Farms producing methane gas from animals’ digestive tract Mainly from vehicles exhaust combined with solar light Electric energy generation from plants burning coal industries Construction works. Atmospheric action Car burning leaded gasoline

Restrictions or RHS Values in maximum and in minimum for different contaminants and in various units Values are only for demonstration 0.05 0.7

5.2 2.89

Low/high 45

75 0.026

But Why Should I Put a Low Limit for Contamination: Is it Not Supposed that the Low Limit Should Be Zero? Yes, in theory, but not in practice, because when a project is in operation there are always contamination, as little as it can be; for instance, there are always emissions from a power plant. It can be greatly reduced by different means, but it is practically impossible to eliminate it in a 100%. Remember that the mere fact of breathing generates CO2 . Sometimes it appears that there is no contamination. For instance, a power plant, located on the shore of a lake or river, equipped with gas-fired boilers, takes water from the lake and uses it to cool condensers, returning hot water to the lake. There is some contamination here, because the hot water may affect the aquatic life and help the growth of algae. Question 12.5.8: Where Performance Factors Come From? Normally performance factors are obtained from the different departments of the promoter, namely engineering, accounting, financing, economics, etc., as well as from consolidated budgets from suppliers, or from private agencies and government bodies, or from environmental and international agencies.

12.6

Normalization

289

This information normally is quantitative and/or qualitative, and in any mix. Quantitative data is supported from studies and computations from each department, consequently, it is considered reliable. Qualitative data may also come from these sources as is the case for expected Internal Rate of Return (IRR), Net Present value (NPV), Working capital, etc., computed by the respective departments in a company. It also may be acquired from external sources, like surveys, consultations, authorized opinions, etc. To clarify this point, in activities that can affect communities, like the construction of a domestic waste incinerator, or the approval to install a factory in an urban area, etc., the values for a criterion like “Acceptance by interested parties,” are normally taken from surveys. For instance, a mining company tried to build a gold extraction and concentration plant in a site, with one large village close. The project will greatly benefit the impoverished area; however, when people were consulted, there was an overwhelming reject for the project, since it could contaminate aquifers that the village used for their fresh water supply. Other sources of qualitative data may come from statistical trends of demands of products to be manufactured by a company,

12.6

Normalization

Question 12.6.1: What Are the Different Methods I Can Use for Normalization? There are several. 1. One of them computes the sum of all performance values of a criterion and then divides each one for this sum. 2. Other divides each performance values by the largest value in the criterion. 3. The Euclidean method divides each performance value by the square root of the sum of performance values squared. 4. Another popular method is the max/min, where the numerator is the difference between each performance value and the minimum performance value in the row, and this difference is divided by the difference between the maximum and the minimum performance values. This method has the advantage of producing larger discrimination than the other three methods. This author suggests that it can be conveniently used when there are ties in the final result of a MCDM project. There are more methods for normalization. Figure 12.9 shows the 4 methods offered by SIMUS, any of them can be selected by the DM. Selection of normalization methods by user. Pressing the question mark the user can see the algebraic formula corresponding to each method. Most methods give different solutions according to the formula selected, not in SIMUS, all methods, except the Minimum/Maximum deliver the same solution.

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Fig. 12.9 Drop-down menu of normalization methods offered by SIMUS

See for more information Chap. 2, Section 2.2.1, and Chap. 7, Section 7.2.1. Question 12.6.2: I Have Heard that Different Normalizations Produced Different Results, Even for the Same Method True, and there are different opinions for the reasons. Some scholars claim that not all normalization methods give the same result; however, there is neither evidence nor explanation for that. SIMUS gives the same results using any of the four mentioned procedures. See Table 7.6 in Sect. 7.2.1.

12.7

Group Decision Making

Question 12.7.1: I Need to Make Decisions by a Group of Decision-Makers: How Can I Unify Their Responses and Decide Which Is the Best? There are several methods corresponding to MCDM methods. In SIMUS, it starts with a result obtained for instance by a DM, and with a quantitative optimum value of the objective function (Zi). The objective value Zi plays a fundamental role in this area, because for each modification it shows a quantitative value. If several DMs agree to review the result, this is done objective by objective, or step by step, and observing the first result, that is, optimizing the first objective; the DMs may agree or not with it. If some of them disagree, every one may suggest the changes that should take place, and these are done in the initial decision-making. All changes from different DMs are considered simultaneously. The software is run again and a new optimal value for the objective function (Z1 corrected) is obtained. If Z1 corrected > Z1, in case of maximization, it means that the changes are beneficial, since they produced a better result. The new values are then imputed to the Initial Decision Matrix, and Z1 corrected is taken as the yardstick for reference. If the objective is minimization, this result (Z1corrected > Z1), indicates that the changes were not adequate, and the original matrix holds. This procedure is repeated until all objectives have been tested and the final ranking belongs to the last approved changes. In this way, any new objective Zn corrected is inspected regarding the objective Zn, and at the end, the result responds to a complete agreement between the DMs,

12.8

Results

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because it is not based on personal preferences, but in quantitative values for Z. See Chap. 9 for a real example and detailed explanation. Question 12.7.2: Consolidation of Results from the DMs in a Decision-Making Group It is explained in question 12.7.1 that the consolidation of opinions does not depend on the DMs but on a quantitative result that reflected the opinions of all DMs. See Chap. 9 for a real example and detailed explanation.

12.8

Results

Question 12.8.1: How Results from Different Methods on the Same Problem Can Be Compared? Comparing different rankings can be easily done using correlation. However, having a strong correlation does not ensure that they represent a “true” result. Example: Comparing results between different methods using the Kendall correlation coefficient. Assume you have the rankings from methods XXX and YYY from the same problem and want to compare them. You can compute the concordance and the discordance between both, and then apply the Kendall Tau Correlation Coefficient. This is shown in Table 12.27 by comparing the rankings from SIMUS and TOPSIS, of course, for the same problem. Add up values in concordance and in discordance columns. Concordance (C) = 5 Discordance (D) = 1 Apply now the Kendall τ formula. τ = 0:66 This means that the result from TOPSIS approximates SIMUS in 66%, but no more than that.

Table 12.27 Calculation of Kendall Tau correlation coefficient Ranking SIMUS 4 1 3 2

Ranking TOPSIS 4 2 3 1 Sum of valúes

Concordance Counting to lower (C) 3 1 1 0 5

Discordance Counting to higher (D) 0 1 0 0 1

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Question 12.8.2: I Need a Result with Quantities, Not with Decimals or in Binary Format: How Can I Get Them? For that, it is necessary to configure the Initial Decision Matrix using a mix of binary and nominal values, and you must indicate, using “1s.” the alternatives that must be considered. Example: It illustrates the case where it was necessary to distribute loans from the World Bank, with a total fixed amount, between 5 cities in Ghana, Africa. Each city had its own characteristics and needs and also different repayment capacity for the loan. The initial matrix is shown in Table 12.28, together with the result. The result in hectares for each urban area is shown in the solid blue row. Observe the high percentage of land use or efficiency according to this result. See Sect. 14.4 for a more detailed information. Question 12.8.3: Is There Any Way to Detect Errors in the Initial Decision Matrix? Yes, in SIMUS at least, they are easily detected either because the solution (LHS), does not agree with the goal established (RHS), or because one or more symbols may be wrong. Namely, putting “≤” for minimization, or “≥” for maximization. Example: Assume for instance a manufacturing plant that have parts and raw material stored, and both needing a certain time for replenishment. Therefore, the production manager has fixed for each item a certain minimum quantity that, when reached, produces a replenishment order to the supplier. Normally, this minimum quantity is subject to the replenishment time. The larger it is, the greater the minimum quantity for ordering. Consequently, the manager establishes a lower limit of existence in storage, say 1200 units of a certain part. But, at the same time he determines that the maximum amount in storage, driven by storage capacity, must be 945. Obviously, this is incongruent, and then the project is unfeasible. In another example, involving a house development, as shown in Table 12.29, it could be that a criterion was assigned the symbol “≥” or “Greater than,” in lieu of the corresponding “≤” or “Lower than,” in a maximization problem. Observe values in dashed boxes. It is evident that 19 is not greater than 2058 and that 714 is not bigger than 97,800, and it is not common to see these large differences between LHS and RHS. It appears that there was an error in putting the symbols. Change them and run the software again. In other cases, SIMUS detects an unfeasible solution. In this instance, the software highlights the corresponding objective and also leaves a written message. Question 12.8.4: May I Select Projects When There Are Not Enough Resources for All of Them? Yes, SIMUS will determine priority in using scarce resources and assigns a “0” score for projects for which there are not enough resources. In some cases, it indicates the percentage that can be reached in those projects with the available funds.

12.8

Results

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Table 12.28 Initial Decision Matrix

Question 12.8.5: How Many Results Formats May I Choose From? In SIMUS you can choose between decimal, binary, or integer. The selection can be done in the last row of SIMUS second screen, as shown in Fig. 12.10 (oval). Question 12.8.6: Is It Correct to Announce Success When Applying a Certain Method? If Not, Why? No, it is incorrect, because we never now which is the true result. When somebody expresses that a problem was successfully solved, it is only a presumption, since what the practitioner can only acknowledge is that a MCDM method performed as expected, following its algorithm.

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Table 12.29 Errors in symbols D

Water

E

F

Green Valley Sewage Gas

G

H

Altavista Water Sewage

I

J

K

Water

Saint Paul Sewage

Gas

29 Yearly water volume to supply 18220800 18559520 84680000 30 Yearly sewage volume to generate 24598080 22619415 114318000 31 Yearly gas volume to supply 2278 10585 32 Water payment 182208 185595 846800 33 Sewage payment 491962 452388 2286360 34 Gas payment 364 1694 35 Water infrastructure cost 48000 36200 23000 36 Sewage infrastructure cost 54600 50400 58000 37 Gas infrastructure cost 33800 64000 38 Water connecting cost to dwellings 52212 63240 48360 39 Sewage connecting cost to dwellings 75112 81345 64350 40 Gas connecting cost to dwellings 661810 563550 41 . Total cost 100212 75112 695610 99440 81345 71360 122350 627550 42

L

M

N

O

LHS 121460320 161535495 118 1214603 3230710 19 240111 284598 714 270498 385682 6284 1872979

≤ ≤ ≤ ≥ ≥ ≥ ≥ ≥ ≥ ≥ ≥ ≤ =

RHS 121460320 161535495 12863 1214603 3230710 2058 107200 163000 97800 163812 220807 1225360 1872979

Fig. 12.10 Loading alternative weights

It found the solution of the problem for the data inputted, but it does not mean that it reached the solution of the problem. The reason is that most of the times the DM cannot input all the characteristics of the scenario, because the MCDM used does not allow it. Therefore, what the DM models is only a simplification of the real scenario, and thus, the result corresponds to that simplification, not to the real problem. Normally, MCDM methods, except linear programming and SIMUS, are not able to model aspects like resources, precedence, inclusivity and exclusivity, comparisons, negative performance factors, various scenarios, time, etc. Question 12.8.7: Robustness of a Solution: What Does It Mean? This is a fundamental question, and thus, it is answered in some detail.

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Robustness, also known as “Strength,” refers to the best alternative holding its position when there are changes in the importance of a criterion or criteria that determined it. These changes, either elicited by the DM (to check strength) or due to exogenous factors (like demand variations), are on criteria that explain the best alternative selection (Binding criteria). Assume that one of those criteria is demand; its variation can affect the best selection and forcing to relinquish its position. What we want to know is how much the demand can increase or decrease without altering the selected alternative in the ranking. If a criterion can change to a certain extent or range without altering the ranking, then the selected best alternative is said to be strong. If the criterion can change only in a very limited range, the best alternative is said to be weak. If the criterion is fixed (does not have any range), it is critical, and the best alternative extremely sensible, because it ceases to be the best at the slightest change in demand. In this circumstance, the DM, depending on the type of criteria and probabilities for changing, probably will opt for the second-best alternative. If the range of the criterion is wide, in both, increasing and decreasing, it can change within that range, and it will not affect the best alternative. For instance, if the criterion corresponds to cost and has relatively wide range, it means that fluctuation in cost within those limits will not affect the best alternative position. If there are, say, 9 criteria, not all of them are important, only those that for a certain objective are responsible for defining the best alternative. These criteria are called “Binding criteria,” and the ones we are interested in to determine the strength of a solution. How to identify the binding criteria? SIMUS indicates them in the shadow price table. See Table 14.4 in Sect. 14.2.3. Fine, but how do I know how much a criterion can change? Good, then, we have two inquiries here: 1. How to identify the binding criteria? These are those that have a shadow price attached. (Fig. 12.11, green table at right). See also this concept explained in Sect. 7.3, Fig. 7.11. 2. How do we know how much each one can change? Identifying binding criteria SIMUS determines them by linking the primal and dual problems (see Fig. 12.11). Observe that: (a) In ERM, the best alternative is “Product B” with a score of 1.21. It satisfies objectives Benefits (Z1), Raw materials (Z2), and Equipment (Z5), (b) Go to Shadow Prices box, enter the respective Z numbers, and for each one you will find the shadow prices for the corresponding criteria. For instance, for Z1

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Efficient Results Matrix (ERM) Normalized Resume of shadow prices considering different Product A Product B objectives (Zi) in columns and criteria (Ci) in rows Z1 Benefits 0.59 0.41 Z2 Raw materials 0.40 0.60 Criteria Z1 Z2 Z3 Z4 Z5 Z3 Manpower 1.00 C1 Benefits 0.01 0.04 0.001 Z4 Env. Danage C2 Materials Z5 Equipment 1.00 C3 Manpower 22 C4 Env. Danage Sum of Column (SC) 1.99 2.01 C5 Equipment 600 31.82 Participation Factor (PF) 3 3 Norm. Participation Factor (NPF) 0.60 0.60 RHS allowable limits. Final Result (SC x NPF) 1.21 1.19 Objective Increase Decrease function (Z) values Z1 Z1 per objective RHS 5 3.80 1.70 7520 RHS 3 340 160 177.27 Z2 Z2 240 RHS 5 1.50 1.80 0 RHS 1 1520 2000 4.50 Z3 Z3 RHS 1 9000 4333 Z5 Z5 RHS 1 3333 2000

Fig. 12.11 Snapshot of SIMUS last screen

they are Manpower (C3) and Equipment (C5). These are the binding criteria; consequently, the other three are irrelevant. Finding how much a criterion can change (c) Go to the small green table within an oval, enter with Z1 and find the corresponding criteria ranges in plus and minus that they can be increased or decreased without altering the best selection (Product B). Notice that it has two columns: The first at the left indicates maximum limits for incrementing a criterion, while the right one indicates minimum limits. Between these two limits a criterion may change without altering the position of the activity selected. Thus, objective Z1 (Benefits). Enter with objective Z1 at the top of the table and observe that there are two criteria C5 (or RHS 5) and C3 (or RHS 3). The range for C5 is 1.50 for increases and 1.80 for decreases, that is, there is a range of (1.80–1.5) = 0.30, which is not bad, and indicating that criterion C5 (Equipment) has a modest variation, and thus, from this point of view, alternative B is safe. For criterion C3 the range is (9000 – 4.333 = 4,667,180) which is excellent, and thus, alternative B has no problem with these two criteria. Objective Z2 (Raw materials). It is linked with criteria C1 and C5. The RHS allowable indicates that for Z2, C5 has a range of 1.80 for decreasing and 1.50 for increasing, that is, a difference of 0.30, which is a moderate range. C1 has 2000 - 1500 = 500, which is a very good ranger. Objective Z3 (Manpower). It can vary from 4333 to 9000, with a good range of 9000–4333 = 4667 Objective Z5 (Equipment)

12.8

Results

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Alternatives (From stakeholders) (1) A

Apply MCDM method (4)

B Best alternatives and ranking Identifying Binding criteria (3)

B >A (5)

Criteria defined based on alternatives (2) Sensitivity analysis (6)

Fig. 12.12 Scheme of process

It is only related to criterion C1. Entering with Z5, we see that the range is 3333 - 2000 = 1333, again, an excellent range. From this analysis we can conclude that the position of best alternative B is very strong. See Fig. 12.12 for the scheme of the process. Question 12.8.8: Can a Result Be Validated? No, because we don’t have any yardstick to compare to. Some authors suggest that solving a problem by different methods, getting a ranking and then, through correlation, determining closeness between them, assume than the two nearest in ranking are which better represent reality. There is not mathematical support for this assumption; therefore, from this author’s point of view, that process does not indicate the best solution. In his opinion, a true comparison would be to contrast the result from each method to some trend or variation of the real problem. Question 12.8.9: Can I Be Certain that a Solution Is Correct? There is no reason to believe that it is incorrect, since it was produced following a mathematical algorithm, in theory. The result is probably correct according to the data inputted in the decision matrix, and the assumptions made, but it does not mean that the problem is solved correctly. This is the quid of the question, for if we use arbitrary weights, or work on hypothesis that don’t have any mathematical support, as in assuming that the weights the DM has in mind agree with reality, and many others, it will be naïve to think that the solution corresponds to reality. It could be an approximation, but no more than that. To get a correct answer, we need certain or reliable data and a procedure that is strictly based on mathematical considerations, and not on personal opinions based on intuitions. The opinion of the DM is very important and should considered, but in form and in time, and certainly not altering arbitrarily the values given by the

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developers of a project; the opinion, the know-how, and the experience of the DM must be taken into account, but after a solution, based on reliable values is obtained. Question12.8.10: If I Solved the Same Problem by Different Methods, Is There Any Way to Know Which Is the Best? This question has been answered in 12.8.9. This author has a theory based on the fact that, since criteria are loaded with alternatives values, and alternatives depends on criteria discrimination, there is a mutual interaction, which is a very well-known subject in mathematics. It is believed that we should compare the trend of prevailing alternatives from reality, with the ranking of alternatives from a MCDM method. Tests have shown that this hypothesis might be correct, but until now, it is only a theory. Question12.8.11: Why Are Differences in the Rankings Obtained from Various Methods and for the Same Problem? This is another old question and still unsolved. If algorithms are sound, if we use the same data, with the same criteria weights, and aiming at the same objective, results should be, if not exactly the same, at least similar. In this author’s opinion, it is due to several causes, for instance: • Some MCDM algorithms are based on different principles and assumptions with no mathematical support and common sense, for instance, the DM deciding that a criterion is the best and another the worse, using only his opinion. • Criteria weights applied in different forms. • Some methods consider resources, but not most of them; therefore, they work on the wrong concept that they are inexhaustible or unlimited. • Different degree of the DM participation, for instance, in AHP it is very strong and arbitrary, in PROMETHEE also has a strong participation in selecting thresholds and transfer functions, however, there is a sense of reasoning and know-how, in TOPSIS the DM decides type of distance to use, in Best and Worse Method, the DM selects the best and the worse criteria, and so forth. Considering these causes, there would be a miracle if results coincide. Question 12.8.12: Since We Do Not Have a Yardstick to Compare a Result with, Could We Use Some Sort of a Proxy Measure? Yes, in engineering problems we input data in formulas, and we are certain that the result will be correct, according to data entered. If this data come from trials, experiments or measuring quantities, the result may be used to calibrate a corresponding analytical development, or hypothesis, but unfortunately there is nothing like that in MCDM. Example: Car traffic between two cities depends on the distance between them, the population of both cities, and other known factors, in a formula called “Gravity Method.” Consequently, it appears easy to compute the flow between two cities using this formula. However, the distance is raised to the power of “α,” which is an unknown

12.9

SWOT

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value. To use this method in predicting traffic, we should know α. It can be done by computing the total traffic in a certain period using traffic counters. Knowing the true traffic, we can determine α in our formula. Unfortunately, we cannot use this procedure in MCDM problems because we do not have a formula, and nothing to compare to. Consequently, our results are assumed to be correct, but it is only an assumption. For this reason, this author has thought in using some type of proxy yardstick, and proposed use the SIMUS method for that. Why SIMUS? Because, as in real-word scenarios there is not human intervention nor weights. It can be explained as this: In a problem such as site location for an industry, hundreds of factors intervene and interact, without any human participation. For instance, demand of energy depends on population, or the price per KW/h may be fixed by the market, or even by the weather or by the existence of deep-water port may be ideal to build a port which is related with maritime tariffs. The existence of raw materials is also something that does not depend on human action; however, it relates to the extraction and transportation to the plant, and so on. That is to say, there is a lot of factors that are “hidden” and linked by nature, and of course, without any type of preference or weights. This is exactly what happens in SIMUS; with its large capacity to manage a very large number of criteria or characteristics (100), without weights of any case, devoid of any assumption, the method works only with existent data and thus could replicate, even approximatively, what naturally happens in a site.

12.9

SWOT

Question 12.9.1: How to Get Strategies Selected in SWOT Using MCDM Methods? SWOT (Strength—Weakness—Opportunities—Threats), is a very well-known procedure for identifying a series of strategies, but it does not select which of them is the best. That is, using SWOT, one only gets a set of strategies subject to a set of criteria, but applying MCDM it is possible to identify the most preferred. To see a complete example of SWOT listing strategies, and SIMUS selecting the best one for manufacturing electric cars, see Chap. 10.

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Sensitivity Analysis (SA)

Question 12.10.1: How Is SA Performed at Present? Conventionally, it is based in increasing the weight of the criterion that has the maximum weight and observing if the best alternative holds. This procedure is erroneous for the following: • The highest weight does not guarantee that the criterion chosen is the best appropriate. This is only intuitive. • Only one criterion is selected when all criteria that are responsible for the best solution or binding criteria should be considered. • Increasing or decreasing the weight of a criterion geometrically produces a parallel displacement to the right, if maximizing, or to the left if minimizing. • The weight, in reality a trade-off, does not have any significance to evaluate alternatives. Consequently, SA is being performed using a faulty procedure. In performing SA with SIMUS, all binding criteria are varied simultaneously, some increasing and others decreasing. Every time that the RHS of the criterion is added in one unit, the corresponding objective augments in a constant amount equal to the shadow price of said criterion, and the same happens when the RHS is decreased in one unit. That is, the objective escalates lineally as long as the criterion is expanded within its limits. Same for decreasing. Question 12.10.2: How Can I Compute or Estimate the Potential Risk of a Selected Alternative? Risk is a very important aspect in MCDM, not only because data can be uncertain, but also for its ubiquity and consequences in every aspect of projects. There could be risks related to: – – – –

Delays from suppliers of products, components, and machinery. Work stoppage, due to strikes and discontent. Not reliable data. Quality of procedures. For instance, in concrete slabs, the level of intensity of concrete vibration when it is poured, is a risk that can compromise slab strength. – Personnel risk, due to movement of heavy equipment on site. – Weather, that can delay work, produce flooding, or dangerous snow avalanches, etc. – Financial risks, etc. However, there are another type of risks related with exogenous factors, and that neither the DM nor the stakeholders can control. For instance, stocks oscillations, demands, competition, performance, etc. For example, a company manufacturing two cosmetic products for export, has selected one of them, using MCDM. Once this decision is made, it is convenient to detect which are the risks involved, as for instance, variation in international demand, price fluctuations, supply delays, etc. With statistics over several years, it

12.10

Sensitivity Analysis (SA)

301

is possible to determine the trend of each one of these exogenous factors, and use this information to calculate the potential risk for this product. Therefore, it is feasible to determine the effect or influence that each one of these exogenous factors will have in the products to be exported. To do this the DM needs the performance curve for the objective that conform the solution (See Sect. 8.1.1 in Chap. 8). Then, he has to select the exogenous factors than can alter the solution (that is, the factors such as weather, demand, international prices, etc., on which he has no control). Once this is done using statistics, he may run the software again and determine the utility curves. Comparing the results of these curves to the original one, the risk can be computed. Risk is the product of the probability of occurrence of an event times the impact it can produce. He already knows the probability because statistics, and thus, he can compute the impact by finding the difference between Z–Z1 values. By multiplying the probability of occurrence by this difference or impact, he will get the risk due to an exogenous factor. Naturally, this exercise can be performed as many times as necessary and considering different exogenous factors. Question 12.10.3: Is It Possible to Have an Illustration that Shows the Variation of the Objective Due to Variations in the Criteria? Yes, IOSA, a SIMUS add-in, generates total decreasing and increasing utility curves. See Sect. 8.1.1 in Chap. 8 for a detailed explanation. Question 12.10.4: How Can Relevant or Binding Criteria Be Identified for Sensitivity Analysis? In SIMUS they are identified by the dual problem. See Sect. 7.3 in Chap. 7, as well as Question 12.8.7. Question 12.10.5: Is it Correct to Use Only One Criterion for Sensitivity Analysis? If Not, Why? No, it is not, because Systems Theory does not allow to vary only one component and keeping the others frozen. This is called (OAT), “One at the time.” This procedure comes from the concept known as “Ceteris Paribus” in Economics. The Dictionary (Merriam Webster), translates ceteris paribus as If all other relevant things, factors, or elements remain unaltered. The concept is considered very debatable by many economists, who consider that all criteria must be treated jointly. This is called (AAT), “All at the time.” Question 12.10.6: How Sensitivity Analysis Should Be Performed? It is a matter under discussion, but certainly not in the way it is done now. In SIMUS, the selected criteria are evaluated according to their shadow prices, that is, mathematically. Thus, increasing or decreasing a criterion RHS one unit, it translates into an increase or decrease of the objective equivalent to the corresponding shadow price that may impact on the best alternative.

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12.11

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Support and Guidance to Practitioners by Simulation of. . .

Role of the Decision-Maker (DM)

Question 12.11.1: If We Have MCDM Software, Why Do We Need a DM? There is consensus on this. MCDM problems are not solved with the application of formulas. To solve a scenario we need an organizer, an inquisitive person, a rational human being with capacity to analyze, with the ability for research and formulate questions, with experience and know-how, and with common sense. This is a Decision-Maker (DM). There are many MCDM methods to solve different kinds of problems and following algorithms assumed to be sound and based on solid mathematical principles, axioms, and theorems. However, they do not decide, they only deliver a result on which the DM can work. It is he or she who decide after their analysis. Question 12.11.2: Can a MCDM Result Override the DM Decision? No, a result from a MCDM method is simply the consequence of completing a matrix with numbers, and have them processed by an algorithm. It is only a base for the DM to work, supported by a result grounded on reliable data and solid assumptions. It does not matter whatsoever elaborated and complete a method could be; it does not have the capacity of analysis, the experience, knowhow, an information the DM possesses. The only time that a formula overrides the DM occurs in AHP, where the DM must review his/her estimates because a formula says so. Obviously, this is not reasonable. Question 12.11.3: Why Do We Need a Group of Decision Makers? Because if there is much uncertainty, working with a group of DMs dilutes in some extent that uncertainty, by allowing discussions and the interchange of information and expertise. This is beneficial if the values produced by each member of the group are based on research, analysis, experience and know-how, and not on intuitions, or preferences. In reality, each member of the group must justify his/her preference with reasons and common sense. However, the drawback of the procedure lies on how to consolidate the different opinions and estimates. SIMUS solves this problem examining each criterion sequentially and working iteratively, that is, getting a solution in each iteration, and trying to improve it in the next iteration. Say that after the first result, a solution is obtained. All DMs are free to comment on it and propose whatever measure, or changing values to improve it. Everybody may express openly his/her comments and suggestions, there is no discussion on this. All suggestions, even contradictory, are incorporated to the initial matrix and SIMUS run again to get the results for the second iteration. This result is optimal, whatever the data inputted, and gives a solution with a crisp value (Z), and it does not matter if the criterion analyzed calls for maximization or minimization; the final value is always positive.

12.11

Role of the Decision-Maker (DM)

303

If a criterion calls to maximizing benefits, and the Zi value is greater than the Zi-1 value of the last iteration, it shows that the suggestions of all DMs improved the solution, and then, adopted. If a criterion calls for minimizing costs, and the Zi value is lower than the Zi-1 value of the last iteration, it shows that the suggestions of the DMs improved the solution and then, adopted. If the reverse occurs, no change is made. Therefore, no subjectivities are present; the process just compare two optimal real values. An advantage of the procedure is that during the entire process, from the beginning to the end, changes and results keep logged and can be reviewed at any time in the future. An example and detailed analysis can be seen in Chap. 9. Question 12.11.4: Can a DM Establish Preferences Between Criteria When a Scenario May Affect Many People? No, because nobody can vote or decide for another one. This is also scientifically addressed in. Arrows Impossibility Theorem. It is one of the greatest criticisms from many researchers to the use of the pair-wise comparison procedure. Question 12.11.5: Why a Final Selection by the DM May Be Different from Another DM on the Same Problem? Normally, decisions from DMs are based on intuitions, and thus, they can produce different values from different individuals even for the same problem. Question 12.11.6: Once Solved a Problem, Is There Any Way to Determine Which Was the Most Important Criterion? Yes, each time that SIMUS uses a criterion as objective function (Z), it shows a value for Z. Comparing the Z’s values for all criteria, the most important and relevant criterion is that with the highest Z value. It is easily seen because the last screen displays a table with these values (see Table 12.30).

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Table 12.30 Values (in oval), for objective functions (Zi)

Chapter 13

Best Practices: Modelling and Sensitivity Analysis in MCDM

Abstract Considering the questions posed by hundreds of practitioners in a scientific forum such as Research Gate (RG) over several years, it appears that there exists confusion about the different elements of a MCDM scenario, as well as the scope of them, and how to approach some real-life situations. This section tries to clarify them. Needless to say, it neither contemplates all possible problems nor clarifies all doubts, but gives solutions to which are considered the most common and frequent qualms formulated by students, practitioners, and professors in this area. It shows, using numerous examples, how to model in MCDM real problems into the initial decision-matrix, and thus, replicating the scenario under study. In so doing, it fills a void in the published literature where it is noticeable that the scenario is not replicated in its whole dimension, which may lead to arguable results. The chapter tries to give answers to some questions, and mainly to point out certain features which are common to most projects and how to incorporate them in the modelling. No formulas are used. Instead, there are real-life examples, common sense, and reasoning, portraying actual scenarios, reflecting real issues, and solved either by Linear Programming (LP) or by SIMUS, a multi-criteria hybrid method, which can deal with most real features. These methods are not perfect, far from it, and perhaps there are better MCDM approaches, however, so far, to his author knowledge, no other method exits able to solve these problems, other than the two mentioned.

13.1

The SIMUS Method

MCDM method is applied to solve all examples in this book, SIMUS (Sequential Interactive Modelling for Urban Systems), was developed by Munier (2011a, b), with software in Visual Basic by Lliso and Munier (2014). It is based on Linear Programming, and thus, totally different from other methods. Its algebraic structure, unlike any other, makes it very apt to model real projects to a © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 N. Munier, Strategic Approach in Multi-Criteria Decision Making, International Series in Operations Research & Management Science 351, https://doi.org/10.1007/978-3-031-44453-1_13

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much higher degree than other MCDM structures. It does not work with lines (or equations) but with areas (inequations). This is the reason it performs very well in complex problems. Its main characteristics are: 1. It does not need weights of any kind, however, it takes into account the different significance between criteria, using an automatic ratio procedure, based on data, to quantify criteria. 2. In a single run, it builds matrix ERM (Efficient Results Matrix) with Pareto efficient scores for each objective. Taking this matrix as a starting point, SIMUS yields two identical solutions, grounded on two different MCDM methods, and thus, they control each other. 3. SIMUS is probably the easiest MCDM method to work with. The DM takes the data and builds the matrix. No pair-wise comparisons are needed, no weights, nothing to guess, and no assumptions to apply. However, regarding weights, the DM opinions, expertise, and knowledge are taken into account when the final result is reached. In this way, the DM works on a mathematical obtained platform where he can make as many changes as wished as well as input his/her preferences. SIMUS also allows working in groups (see Chap. 9), where all opinions are considered, inputted, and evaluated for each criterion. At the end, all criteria have been analyzed and pondered all suggestions, modifications and changes from the members of the group. 4. It informs the user if the problem is unfeasible, i.e., when not all criteria are satisfied. 5. Very fast. A 12 × 10 matrix may take less than 3 min to solve in a laptop. 6. It is not a commercial software, and therefore, free for everybody to use at its full capacity (200 alternatives and 100 criteria). 7. Has an extensive tutorial with 12 solved real cases.

13.2

Definitions

Scenario: It is the problem under study and formed by alternatives, criteria, performance values, resources, and restrictions. Most problems deal with single scenarios, but others address multiple scenarios.

13.2.1

Composite Indexes

Sometimes, a scenario consists in analyzing a set of indicators, pertaining to many different fields, to obtain a single number out of many statistics that groups them all, i.e., building a “composite index.” This is a common approach in economic and social studies.

13.2

Definitions

307

Indicators for HC Life expectancy Education GINI Index

Indicators for Technology INPUTS Internet penetration Telephone penetration Technology Electricity consumption OUTPUT Tertiary science and eng Patents Scientific papers Mean years of schooling Literacy rate TAI index Goods & Services EXTERNALITIES & PROD. FACTOR

Indicators for Society Employment pop. ratio Population health Education Income /capita Housing Premature deaths Green space /capita Gender equality Indicator for Country Progress

AT IN TK

SC

Economic Growth (GDP) NC

Indicators for Social Capital Trust Networks Collective action Diffusion Social inclusion

Indicators for Environment Hydro electricity Air contamination Tourist management Fisheries Wildlife

EXTERNALITIES Indicators for NC Forest depletion Water resources Erosion Oil, Coal and Gas Mineral depletion

Influence of Technology in Country Progress

Fig. 13.1 Scheme of network for a composite indicator

Example: Assume that the practitioner wants to analyze how a country's progress develops over the years. This is a complex issue, and intervenes factors such as the Gross Domestic Product (GDP), Quality of Life, Gini Index, Natural Resources, Health, Environment, Technical issues, Social Capital, Influence of technology, and Gender equality. Each one of them is also composed of different subindexes. The starting point is the schematic network depicted in Fig. 13.1. Main fields are shown as rectangles, like Human Capital (HC), Technology in both inputs and outputs, national or imported, Society, Externalities, Environment, and Natural Capital (NC), that in general feed nodes like Social Capital (SC) and Economic Growth (GDP). At the end, all converge to the Indicator for Country Progress oval. Construction of this network is the fundamental step, and it allows for constructing the Initial Decision Matrix with indicators in columns, which are yearly updated, with positive and negative values, and thus, each year it is possible, by comparison to the precedent year, to register an advance or backtracking, in the respective area. The set of indicators is subject to a set of criteria in a row. Regarding indicators, it is necessary to consider that a positive increment from year (n-i) to year (n) does not necessarily mean progress; an increase in indicator “Crime rate” is obviously negative for the country, and so it must be considered. Similarly, a negative difference does not always mean backtracking. If it refers, for instance, to a decline in newborn deaths, this is very positive for society. Mainly in the environment area, indicators detailing aspects such as erosion due to logging, or depletion of aquifers, must be considered, because they are negative aspects for a country, and consequently, must be taken into account, something that unfortunately is not done in present-days.

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This project necessitates a large number of people, especially experts in each discipline and analysts to interpret results. At least, as has been exposed here, it is believed that it can be applied to a single country and not for country comparisons, since what may be very important for a country, may be not significant for others. One example is Canada, where forest industry is very important, while oil is fundamental for Saudi Arabia.

13.2.2

Macro Planning

It is often necessary to design a whole plan for a city, a region, or a country, to be executed over several years. It is divided into different most important areas like Departments (in the USA, for instance, Department of the Environment), or ministries (for example, Interior Ministry). At a different scale, this subdivision also applies to provinces, states, and cities. For each of these large divisions, a budget is assigned, not only for their staff functioning, but mainly for projects to be developed by each one, in education, public works, health, tourism, etc. Normally, funds are not enough for all plans, and thus, MCDM methods are conveniently applied to select the projects that can be executed with the available funds.

13.2.3

Strategies

In MCDM methods there are two main strategies to follow, once the decision matrix is built. The “Top-Down approach,” and the “Bottom-Up approach.” In the first one, the decision matrix (top of the process), is modified by the DM, using subjective weights for criteria, something inexistent in real life. Because of that, the result (bottom of the process) reflects the opinion of the DM, even when there is no guarantee that it applies to real life. In addition, the result depends on a certain DM decision, and it could be that another DM may produce a different result. In the second approach, data is not modified in any way by the DM, and thus, the (bottom) is assumed mathematically correct, if the data is reliable. However, at this stage, the DM has a strong participation since he/she may modify data, but supported by solid reasons; thus, he can add or delete alternatives and criteria and even reject the result. Consequently, there is a bottom-up evaluation of results, that incorporates the know-how, expertise, research, and experience of the DM. Observe that in the top-down approach, there is normally no research, no expertise, and no reasoning in determining weights, but normally personal intuitions and feelings, in determining weights.

13.2

Definitions

309

As said, in the bottom-up approach, the DM has a solid platform (result at bottom), obtained by real data, and can make corrections based not only on his/her expertise, but also on researching about aspects that were not considered in the Initial Decision Matrix, as exogenous issues like demand, performance (from sources that have had similar problems, for instance, equipment performance, inconvenient, failures, etc.), but especially, because the results may identify sensitive criteria that where unknown at the time of building the decision matrix. For example, at the beginning, it is ignored which of the criteria may have the largest influence on the alternative selected, something that is known with the results. In the bottom-up approach, consequently, the DM is able to consult with stakeholders on a respective criterion, and perhaps modify it. For example, a manufacturing company plans to purchase foundry equipment and has three brands to choose from: A, B, and C. With all data collected and using an MCDM method the best equipment seems to be C. The DM studied the result and did some research about the reliability of the C equipment. He got good referrals, however, one of the consulted colleagues in the industry said that equipment C is good equipment but not for long hours of continuous service, because its frame heats up very much after working continuously for about 6 or 7 hours, and it must be stopped for about 1 hour to cool it down. This is a serious drawback since the company plans to woks in two shifts 8 hours each, and thus, with this equipment perhaps there could be problems in the second shift, and not being able to achieve the intended production of pieces. This fact was not known at the initial stage of the process and was only detected when the DM did some research, based on results. Most defenders of the top-down procedure declare that the preference of the DM is fundamental, and this is true, but when applied to solid data and guided by reasoning, not by intuition.

13.2.4

Alternatives

The different projects or options, normally finite, that pertain to a scenario.

13.2.5

Criteria

Represent most of the aspects that alternatives are subject to. In practically all MCDM methods criteria are equivalent to linear equations. In Linear Programming (LP) and SIMUS, criteria are expressed as linear inequalities, i.e., equations, with a Left Hand Side (LHS), grouping the set of performance values, a Right-Hand Side (RHS), with the values/limits for resources, and both sides separated by mathematical symbols, such as: “≤” (Less or equal than), “≥” (Greater or equal than), “=”

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(Equal to), and according respectively the action called by each criterion; in the first case “Maximization,” in the second “Minimization,” and in the third “Equalization.” Maximization translates in getting the maximum value, but less than the maximum allowable, for instance, funds. This is expressed by “≤.” Minimization translates in getting the minimum value, but greater than a minimum, for instance, low-level existence of parts in the warehouse. This is expressed by “≥.” Equalization translates in matching an exact quantity, i.e., no more or less, for instance, equipment. This is expressed by “=.” One especial feature of SIMUS is its ability to provide two solutions for the same problem, both coincident in the rankings, even when they are obtained by two different procedures. This aspect, in addition to reinforcing the reliability of the method, is very useful in other aspects, for instance, in breaking alternative ties, i.e., when two or more alternatives have the same score. The set of inequalities is the heart of Linear Programming and SIMUS, and they can model most real-life situations. A clarification: Some authors identify criteria as “Restrictions”; they are not. Restrictions are the RHS values to restrict criteria scope. In other words, criteria are restricted by limits.

13.2.6

Performance Values

They are the contributions, positive or negative, of each alternative to each criterion. For a certain criterion, there could be values from each alternative or only for some of them. They can be expressed in decimal, integer, or in a binary format. An Initial Decision Matrix can be completely binary, i.e., only “1s” and “0s”; they have a especial meaning and are very useful to represent some scenario features as seen in this book. In an Initial Decision Matrix, there could be a combination of submatrices, that is, sub-matrices with normal performance values, as well as binary sub-matrices. Both are examined in this book. See for instance Table 14.11, Sect. 14.6, Chap. 14. Performance values can also consist of algebraic formulas, as well as values expressed in mathematical notation for large numbers. Sometimes, uncertain values are processed by “Fuzzy logic,” which works with probabilities and delivers crisp values.

13.2.7

Attributes

These are the characteristics of the performance values of a criterion. For instance, a criterion attribute is the dispersion or discrimination between its values, or their quantitative or qualitative character. Do not confuse criteria with attributes, for the latter are the characteristics inherent to the values within a criterion, which is thus,

13.2

Definitions

311

an “envelope” of values with certain attributes, for instance, dispersion or discrimination.

13.2.8

Results

They give the mathematical solution of a MCDM problem and can be expressed in decimals, integers, or binary format. The last one is very useful when the result must indicate integer results, where the “1s” means selected alternatives and the “0s” those not selected. The results expressed in decimals establish a ranking of alternatives, the higher the better, irrelevant if the objective calls for maximization or minimization.

13.2.9

Weights

LP and SIMUS do not use weights, but if the DM wishes them, they can be inputted for both, criteria and alternatives, separately or at the same time. For criteria, a previous operation is needed, by multiplying the corresponding weight by each value of the criterion. For the second, just put the weights in the boxes at the bottom of the table. See Fig. 7.8 (row # 9), Sect.7.3, Chap. 7.

13.2.10

Sensitivity Analysis

No MCDM process is complete without a Sensitivity Analysis (SA). Its importance lies in the fact that values for criteria are often uncertain, and consequently, it is necessary to examine how the solution or ranking (Output) changes, when the criteria values (Inputs), are increased or decreased. In SIMUS, a criterion's importance is measured by its marginal value or shadow price (see Sect. 6.4, Chap. 6), which indicates how much an objective changes due to one increase or decrease of a criterion. A result may be dependent on one criterion or on several. In this last case, all criteria changes must be considered at the same time. SIMUS allows for both, considering only one criterion (not recommended), or for all of them jointly (highly recommended). Present-day methods contemplate the change of only one criterion, arbitrarily chosen, and assume that the other criteria are not affected, which is not considered correct. In addition, present-day methods choose the criteria to be changed assuming, without any mathematical foundation, that the best criterion to consider is that with the highest weight.

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SIMUS quantifies the relative importance of criteria using data from the Initial Decision Matrix, and determines scientifically the weight of each criterion based on its marginal value. This value is called “Shadow price” and is automatically computed for each objective. This allows, by using a SIMUS add-in called IOSA (Input-Output Sensitivity Analysis), to draw the graphic that indicates how the curve that represents an objective change, due to variations in the input. See Figs. 8.4, 8.5, and 8.6, and Sect. 8.1.1, Chap. 8. This curve, in turn, permits determining quantitatively, the different risks when the best alternative is affected simultaneously by different exogenous factors or parameters that the promoter cannot control. For instance, demand variation, delays from suppliers, weather, etc. As can be appreciated, LP and SIMUS methodology sharply depart from the principles used in present-day methods, and this is the reason by which the former can represent and solve very complex problems, and even, in some of them, give optimal results (using LP), while in others, yielding satisfactory results (using SIMUS).

13.3

Modelling and the Role of Stakeholders

According to “stakeholder/s” is an individual or any group who can affect or is affected by the achievement of an organization’s objectives. Therefore, in MCDM stakeholders are the Directors, Division Managers, and Shareholders who have an interest in the project, and who normally have answers to the questions a DM may formulate. For the main objective of a project, the Board of Directors articulates policies and establishes goals. Managers know what they need, and shareholders may express their support or disapproval of a project. There is also a large segment affected by the project, and it is the people that will be the recipients of the benefits, for instance, new jobs, higher capacitation, improvements in their lives, etc., but also they will be the receivers of the negative aspects, like unwanted displacement from their sites of residence, as in the case of building large hydro-power plants, odors, when building a landfill, as well as their properties losing value, or potential contamination from a nuclear plant accident.

13.4

Areas Where SIMUS Has Been Used

Just to give the reader an idea of SIMUS applications, the following forcefully incomplete list, illustrates about the fields where the method has been used. Agriculture:

13.4

Areas Where SIMUS Has Been Used

313

(1) Selection of crops. (2) Determination of best energy sources for irrigation purposes. Air transportation: Airport expansion. City assets: Plan for upgrading and construction of road bridges in a lacustrine area, in a 10-year plan. City planning: Rehabilitation of abandoned areas. Construction: Road construction and GIS. Country development: Determination of country growth composite indicators. Economics: (1) Analysis of economic growth and sustainable development. (2) Identification of factors affecting investments. Energy: (1) Determination of renewable energy sources that will be used in energy transition. (2) Determining better sites for wind and PV farms. (3) Wind turbines farms. (4) Photovoltaic farms. Environment: Environmental Improvement Considering Aichi’s Targets. Fuzzy logic: SIMUS application in railways. Health: Improving service quality during COVID-19 Pandemic. Highways: (1) Selection of routes using Geographical Information Systems (GIS). (2) Using existing road infrastructure to build a highway between an airport and the city downtown. International aid: Analysis for economic aid to countries in need. Landscaping: Post-mining transitional landscaping. Metal working: (1) Milling (2) High-speed milling Mining: (a) Transition for post-mining operations (b) Mining waste MCDM: Proxy results to reflect reality. People relocation: From rundown neighborhoods to new areas. Project location: Determining the best location for the construction of domestic waste incinerators. Public Health: Determining geographical locations in urban areas for health centers. Railways: Analysis for urban railway passengers’ services.

314

(1) (2) (3) (4) (5)

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Best Practices: Modelling and Sensitivity Analysis in MCDM

Urban Regional National International Freight scenarios

Region planning: Determining indicators for a large metropolitan area as well as satellite cities. Risk: Project alternatives considering risk. River environmental condition: Determination of more contaminated areas in a large river. Solar energy Strategies: Strategies selection from SWOT analysis. Supporting stakeholders in decision-making Sustainability: (a) (b) (c) (d)

Classifying sustainability for an urban region. Growth vs. sustainable development. Determining sustainable indicators to measure the state of a city. Sustainable urban mobility plan (SUMP). Thermodynamic: Determination of thermodynamical parameters in condensers. Transportation:

(1) Mode selection. (2) Fuel selection. Urban access: Urban sustainability and mobility (SUMP). Urban and peri-urban: Determination of urban and peri-urban indicators. Urban policies: Policies to reduce urban risk for pedestrians. Urban mobility (SUMP). Urban sewerage networks Urban rehabilitation: Several cities. Waste: Selection of domestic waste collection. Water distribution: Maintenance of certain sectors of water trunks. Water: (1) Recycling policies (2) Waste to water energy Wind energy

13.5

Comments and Advices in Black, Examples in Italics

• In all scenarios the DM needs: Alternatives, Criteria, Performance values, and Resources.

13.5

Comments and Advices in Black, Examples in Italics

315

In which order are they determined? Common sense indicates that everything starts with the alternatives or options, which the DM receives from the owner of a project or from stakeholders. From then on, the DM needs to contact all interested parties in order to learn about their requirements and wishes. There will surely be contradictions among different stakeholders, but it is not the duty of the DM to manage differences; at this stage he must be interested in collecting reliable information from specialists and experts. He must consider them all, at the same level, i.e., all are equally important, and he is not to judge the significance of one criterion over another. With this information, the DM may prepare his/her battery of criteria. Once completed, he must consult again the interested parties to make sure that everybody’s concern has been addressed. From there, he may be able to build the Initial Decision Matrix. • If you are a DM, make sure that you fully understand the problem. If not, do not be afraid to ask questions to the respective departments, stakeholders, and sources. Do not assume that you have a complete and thorough understanding of the problem because possibly you are ignoring many facts. Remember that the sources will not hide information, but they do not know what you need, and then perhaps do not consider it necessary to tell you everything. • In some projects, there could be conflicts of interest that normally have a very large influence on the project. It happens for instance, when there are several parties involved in a scenario, and when a party can adopt measures, like proposing a project, that affects another party. Example: A large study performed by the Massachusetts Institute of Technology (MIT), for the best uses of the Colorado River in Argentina, involved five provinces in the river basin, from its headwaters high in the Andes Range to its discharge into the Atlantic Ocean. One of the provinces proposed a project to use water for irrigation; this directly affected the other downstream provinces due to salinization, as well as contamination due to pesticides and fertilizers. This fact needed to be considered since there was a conflict of interests. • Consult stakeholders to learn what they expect from your work. This is important because later, when you have the solution, you must be able to answer stakeholders’ questions. • Stakeholders normally are not aware of the existence of MCDM, and then, it is the DM responsibility to insist that they contemplate all situations, for example, risk. It may occur when a project is operating, something that was not anticipated but may occur. For instance, personnel being hit by a forklift going blackguards and without a sound warning. Accidents may happen in any project. There are sad examples of lack of vision of the project planners and DMs, such as the disaster in a chemical plant in Bhopal, India, in a dam in Italy, in a mining operation, in Guyana, in the Challenger accident in the United States, etc.

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• Make sure that data is as accurate as possible. Look for inconsistency, for instance, the DM can find that in three quotations for similar equipment and from three different suppliers, two of them have close prices, but the third one is about 10 times less than the other two. There is an inconsistency here; possibly a zero was accidentally dropped for the third offer. Do not be afraid if there are many criteria, because if so, it is because they are representing the scenario more accurately. Do not limit the number of criteria because of the work load; if you do, the result most probably will be biased, and you may get embarrassed when a stakeholder may ask why you did not consider a certain fact. • Verify with the respective sources if there are relationships between alternatives or if they are inclusive or exclusive. If there are dependencies get a full explanation of them, and ask for clarifications about the reasons for these dependencies. • When the DM delivers his recommendations, he must be sure of his results, and certain that they are consistent and well founded. Responding that some values are from his own preferences is not acceptable, and he may have an embarrassing moment. Personal preferences do not have room in serious MCDM process. The DM must work with facts, and use his experience for building a model, and for analyzing outcomes from the method result. • Weighting criteria subjectively is not a good practice. If the DM believes that they are necessary for whatever reasons, he may use objective weights. There are several options for working with objective weights. There is no doubt that the opinion of the DM is very valuable, but in time and form; certainly not altering data. It is much better to modify a reliable result and change data, adding or deleting criteria when there is a solid foundation, which is given by the MCDM method obtained without human participation. • Determine which criteria are objective and which are subjective. For the latter, think of a method to compute their values more accurately, for instance, surveys, using fuzzy, gray numbers, etc. • Large projects, in general, will affect a lot of people over years. Do not make the mistake of deciding for them. Conduct surveys and polls in all projects that affect people. • Use statistics to gauge answers, and input the resulting values in the Initial Decision Matrix. • If the subjective criteria will affect a large number of people make sure to conduct a survey to know their opinion, not about the technicalities of each problem, but on what extent each project will benefit or not their lives. This is extremely important; many large and very large projects around the world have been halted because of people disagreement, which led to Court orders to stop work, whatever its advance. Remember that you, as DM, cannot decide by the people.

13.7

13.6

Complex and Complicated Scenarios

317

Recommendations to Practitioners

• Once the set of criteria is designed, examine it for interrelationships and correlation. • Determine an input resource and its limits, both lower and higher. • If the performance values are prices from different countries, make sure to convert them to the same monetary unit. • Make sure that quantities and prices are linear. • In case of equipment from overseas, including in the initial matrix the necessary time from the trip from the factory to the harbor, ocean trip, waiting time on the wharves, trip from the port of arrival to the job site, delays in customs, state of roads, bridges, etc. as well as port facilities for unloading. The distance and time factors may have a heavy weight in decision-making. • If criteria deal with environmental issues, remember that there is always a lower limit of contamination; normally it cannot be zero, because whatever the anthropogenic activity, it will always alter the environment, even the act of breathing. • Select a MCDM method that better models the scenario and data. • Keep away from compensatory methods. • Perform a sensitivity analysis. Do not make it by changing criteria weights, because they are meaningless for this purpose.

13.7

Complex and Complicated Scenarios

This section examines some uncommon and complex scenarios that illustrate the ability of LP to solve them. Normally, they are related to the need to consider a scenario as a whole and where peripheral considerations intervene, but they are linked to the main objective. It is examined here the components for modelling a scenario to be solved by Multi-Criteria Decision-Making (MCDM) methods. From the technical literature, it is apparent that not too much attention to detail is, in general, placed in modelling an actual scenario, or maybe it is considered too complex to be represented by a mathematical method, and these two conditions may coexist. There is no doubt that MCDM is a subjective activity, but it is also true that efforts should be made to replicate reality as close as possible. From this point of view, this author considers that it is not enough to have a portfolio of projects or alternatives and a set of criteria to evaluate them. Reality cannot be even remotely replicated if conditions of both, alternatives and criteria are not thoroughly examined, and perhaps linked, i.e., alternatives among them, criteria amid them, as well as between alternatives and criteria. This aspect is analyzed here proposing some ideas to improve reality representation, and in so doing get more reliable final results. It also gets into sensitivity analysis, not in the conventional way, as it is normally done, that is, investigating

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how variations of the weight in a certain parameter affect the ranking. Here, SA is performed scientifically, determining the binding criteria and their allowed range of variation, and considering all of them simultaneously. In so doing it makes the Decision-Maker (DM) fully responsible for the alternative or project chosen, since it is he/her who take the final decision, not a mathematical method, which is only a tool, but that gives valid arguments to proceed.

13.7.1

Background Information

MCDM is a complex activity that aims at selecting projects or alternatives that are subject to many criteria used for their evaluation. It has two main constituents, and one of them supporting the other; they are: Decision-Maker He is the person who studies a specific scenario to determine the best way to develop it (the scenario could be, for instance, the installation of an industrial plant for which there are several suitable locations), with only one to be selected. In this example, the DM has information on locations to be evaluated, gathers data, selects criteria, analyzes numerical data (cardinal values), and inputs everything in an appropriate mathematical model, which contains all the information. When completed, he examines results, and may agree, reject, or modify them, and takes the final decision by himself or by reaching a compromise agreement when there are several DMs. He must get a final selection of projects among many, based on a numerical score for each project that the MCDM method produces. These scores are arranged in descending order, the higher the better, to form a ranking. Mathematical Method The second constituent is a variety of different mathematical methods and algorithms based on different principles and theories, whose only purpose is to order and process the information inputted, and deliver a potential solution, suited to be analyzed by the DM. This solution is no more than a sort of a suggestion to the DM, with the advantage that it is the result of various mathematical procedures. However, it does not consider many different aspects of reality because they are difficult or impossible to model or quantify. MCDM is not like a mathematical formula where if data is inputted, the result will be exact. This is not possible in MCDM because there is not such a formula, nor is possible to develop one to get the right results, simply because we do not know all intervening variables. Selecting an adequate method is essential, and although it is generally recognized that there is not a method better than others, it must have the capacity to deal with the scenario. A hunter may kill a small bird with a sling and using small stones, but he cannot kill an elephant with the same weapons and ammunition. Something similar happens in decision-making, there are certain problems where personal preferences and opinions are paramount, such as deciding what

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319

car to buy, or where to go on summer vacations, and these situations can be easily solved. But for projects as every day found in industry, commerce, environment, social life, politics, government, finance, public health, education, military, etc., there is no room for preferences, predilections, or opinions; the DM must work with reliable data. Consequently, it is not acceptable to select a location, amongst several, for developing a large project like building for instance a steel mill, investing hundreds of millions of Euros, modifying the landscape forever, and shaping the future life of thousands of people, based on preferences and opinions. The choice must be grounded on reliable and documented information and using a mathematical method able to handle hundreds or thousands of criteria and alternatives, with the ability to produce new results fast, when new data is inputted or eliminated, and without spending hundreds of man-hours to modify an initial scheme. Does such a method exist nowadays? Possibly not and most probably never will, because scenarios may be extremely complex. This condition places the DM in the position of selecting a method that he/she believes fits better his/her scenario. There are complicated and complex scenarios; these two words seem synonymous but they are not. Complicated refers to the links between diverse elements that work along definite patterns. For instance, a watch is a complicated mechanism, formed by different parts that are perfectly defined, that have an objective and with a performance that can be predictable. Some methods are adequate for complicated scenarios with many relationships and dependencies but which outcome is predictable. As an example, a person goes to a GM car dealer to purchase a car. There are different models with different characteristics such as color, performance, safety, and fuel consumption. The person must take a decision but he knows that finally he will leave the place with a car, by following a certain decision process. This is a complicated process, and it follows a predictable pattern. Complex refers to the links between diverse elements but which relations change continuously. For instance, a river basin is complex because it links many rivers as well as different sources, and performance is unpredictable because conditions change continuously, as for instance, the amount of rain, snowfalls, electricity demand, etc. As a complex scenario, assume a portfolio of different alternative construction undertakings, all of them large, expensive, and with may be two dozen contractors each, all related, and with precedence and simultaneity links, as well as annual budgets to adhere to and different completion periods. There are many unpredictable factors that may interfere with the process like stormy weather, supplier’s delays, concrete quality, inflation, efficiency, and accidents. Because of this, it is difficult to predict when a project will finish and at what cost. This is a complex scenario. Therefore, the DM must have a good knowledge of the different existing methods for MCDM and the abilities and capacities of each one, as well as the degree of difficulty, and especially the amount of labor and time involved in preparing the mathematical model. This is his first task; he can do the same exercise using two or

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three different methods, but more often than not their results, expressed as alternative rankings, will not coincide (see Triantaphyllou et al., 1998, for a good introduction).

13.7.2

Aspects to Consider by the DM

13.7.2.1

Data Acquisition

It is necessary to get reliable data, which may be known (for instance, technical values for equipment, that come from the manufacturer), as well as incomplete data (for example, missing values in a statistical series for construction accidents), or simply inexistent values (such as in the case of R&D projects, where results of intermediate actions may be unknown or just hinted, because this is the first time an experiment is performed), or data may be corrupted, or very difficult and costly to get. Assume as an example that the DM needs to produce a short set of indicators to measure the environment. As initial data he has 350 environmental indicators and since it would be very difficult and time-consuming to deal with all indicators, forcefully he must reduce their number to a manageable figure, say to a final set of 20 or 25. Of course, he will be losing a big deal of data. The selected set must provide thus, as much information as possible; in this case, his initial battery of 350 indicators has a lot of information content in each one of them. Statistics provide several means for doing this reduction, however, there is a caveat; the resulting set of final indicators must concentrate the maximum amount of information contained in the original indicators. Can it be done? Yes, it is possible using for instance a statistical technique known as Principal Components Analysis (CPA). See Question 12.2.8. in Chap. 12.

13.7.2.2

Criteria Selection

Criteria are used to evaluate a set of alternatives; therefore, it is necessary to choose, out of many possible criteria, those who pertain to the alternatives under study. Criteria type and nature depend on them, since it is not the same to select criteria to maximize stock market gains, than to choose criteria for minimizing cost in an agricultural project, or to maximize labor employed, etc. Assume that the DM has a portfolio of four projects from which he must select the best, and further, to rank them, i.e., which project will be second, third, and fourth. How does he evaluate them, or still perhaps more importantly, what criteria to use? How does he know what criteria are needed? The answer could be based on experience or similarity with other scenarios, but most probably, by expertise and capacity to envisage the whole scenario, and by determining and judging what aspects are involved, and their importance.

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There is no rule for selecting criteria, however, other than common sense, all aspects related to the project must be contemplated. Evidently, it is impossible to consider all feasible criteria for a project; for this reason, it is enumerated here some cases and illustrated with examples. Possibly the best way to develop criteria is by brainstorming, from where a good deal of ideas and procedures can be obtained, giving way to new criteria. For instance, if a scenario is related to an airport expansion, naturally, the DM will look for incrementing in some way the airport activity, and then, it can be said that his main objective is to make the airport more profitable. Thus, one objective could be incrementing passengers traffic and decreasing operating costs, as well as maximizing benefits. In the old days there was only one objective, and was related with economics, but nowadays, public opinion and stringent regulations on the environment and safety, force to include other objectives not directly related with the airport operation. Therefore, criteria must address not only the economic aspect but also noise, job creation, infrastructure needed, etc. Is there any limit to the number of criteria used? It all depends on the method used. Some of them require extensive and costly elaboration regarding the criteria selected; others can work with any number of them without too much difficult. However, in all methods, it is necessary to assign performance values for each alternative and for each criterion, and this can be cumbersome, expensive, timeconsuming, and prone to errors. Let us illustrate this subject with some examples: In irrigation projects, there may be different alternatives such as building a dam to storage water in a concrete or an earth dam, or by other options. There may be conditions that affect some types of dams but not another, and these constraints must be considered. As an example, in tropical zones the quality of concrete poured may be affected by temperature, while an earth dam is not affected, therefore, temperature weather could be added as a criterion. There are criteria that apply to all projects of this type, for instance, to quantify for each project, the degree of damage produced by flooding and destruction of a nearby forest, when the reservoir filling reaches its designed water levels. Risk in its different meanings must always be taken into account, and then it is necessary to add as many criteria as needed, since there may be financial, political, environmental, accidental, geological, etc. risks. Population acceptance is another criterion to be taken into account and it is very important, as well as if people removal if necessary. In some undertakings, people relocation is a big and complex project by itself. Are some criteria strongly linked or correlated with others? For instance, a criterion addressing sales is probably linked with a criterion that deals with quality and with another focusing on competition. This type of correlation must be examined because it could be that a variation of the values in one of them affects the values of another. See Question 12.3.2 in Chap. 12. What about legal aspects? It is paramount to check if the diverse options are free of encumbrances, litigation, illegal occupancy, popular complains due to contamination, etc., For instance, in a location analysis, an already selected urban location with free title and without encumbrances for a chocolate producing plant, had to be

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rejected, because it was discovered that there could be neighborhood complains about chocolate smell, nice, but upsetting people when it is permanent. This must be given consideration during the planning stage. As another and actual example, a very large hydro project in Western Canada was ordered by the Supreme Court of Canada to hold construction, after spending hundreds of millions of dollars, because the company did not pay attention to complains of native people living in the area. That is, since the company knew about this issue even before commencing the work, it should have foreseen potential damage to the project with the corresponding criterion addressing it. What about noise, transportation needs, qualified workers, water availability, energy cost, political unrest, etc.? As an actual example for the last, a huge oil pipeline in Eastern Europe was built not following the MCDM selected route, but another more expensive, among several, due to sabotage risk, from potential actions of some political groups. Thus, a criterion calling for minimizing political perturbation along the different routes should have been considered. What about complementation with existing undertakings? For instance, a City Hall planning to develop large plots of abandoned wharves competing for different undertakings, needs to consider access, and especially a close connection with existing city transportation systems. This situation is very common all around the world where large railway yards and maritime and river wharves located on prime land, and no longer in operation, require to be rehabilitated and put to good use. There are other restrictions. For instance, a large city in South America had ambitious plans to utilize a large plot in downtown area which is considered World Heritage by the United Nations. Plans had to be changed because they were not approved by the UN since the high-rises would alter the skyline of the preserved place. The UN threatened to withdraw support—which is very important—if the city persisted in its plans, and it certainly did not. In the Yucatan Peninsula, there was a plan to link two cities with two routes. Unfortunately, the selected option had to be abandoned because the planners did not realize that it would pass nearby a native cemetery, considered a sacred place. If, for instance, there are plans to build an oil refinery, it is necessary to take into account, in addition to the distillation columns, the construction of oil storage tanks or ancillary structures for handling and storing the different final products. Therefore, restrictions considering tank capacities must be contemplated. In planning the construction of a permanent link between two parts of a city separated by a firth, two alternatives were envisaged, a tunnel or a bridge; it was necessary to bear in mind many factors such as, for instance, possibility of bottlenecks and the entrance and at the exit of both undertakings because they had different layouts and traffic volumes according to their location. What about safety and reliability? And there could also be a potential future use for a railway. Was maintenance considered? A criterion “Maintenance” should have values for each project, most probably different. In scenarios related to logistics, such as the selection to purchase a fleet of cars for the police force, in addition to normal and known criteria addressing cost per unit, maintenance cost, speed, efficiency, etc., it must be considered reversal

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Complex and Complicated Scenarios

323

logistics, that is, use a criterion that takes into account values from manufacturers, that have established plans and policies to buy back their own vehicles and parts, at the end of their life cycle, to recover still useful components and for recycling materials. This is a practice that has been in use for decades now, especially in the automobile industry, textiles, electronics, etc. In conclusion, this section tries to emphasize the necessity of studying very thoroughly each project or alternative. In addition, with the mentioned brainstorming it is fundamental to analyze the most important factors that are related to each project or projects, and materialize their effects, consequences, or contributions by different criteria.

13.7.2.3

Weights

A weight is a metric aiming at defining the importance of criteria related to the objectives of the scenario. A value is assigned to each criterion. Naturally, not all criteria have the same weight (if they had, their effect will be zero or no significant). Obviously, a criterion that calls for maximizing the “Rate of Investment” (ROI), i.e., earnings on money invested, is more important than a criterion that calls for a similar percentage of female and male workers in a factory. But how much more important is it? How does the DM put weight on these two? What supports the decision to assign 25% to ROI and 15% to gender ratio? To make things worse, normally these weights are relative, and thus, if the problem has, say 14 criteria, how does he allocate each weight in order to get an aggregation of 1? There are several methods but none of them is 100% reliable or credible. Because of this, normally these weights are determined by experts, experienced people, assuming not bias, and working in good faith. Is that enough? Not really. Unfortunately, it is not uncommon to see cases where weights are influenced by vested interests or to personal and political reasons. For instance, it is known the case in mid last century, when a government choose a company to lay the railway tracks between two cities, separated by about 520 km as the crow flies. However, the company laid the tracks on 607 km, using a twisted route by adducing inexistent geological problems in the terrain. The reason was that the company offer was based on number of km of tracks, on of course, criterion investment was given a very high weight.

13.7.2.4

Criteria Units

Criteria normally have different units. A criterion for maximizing Return on Investment (ROI), uses percentages, probably the same for another criterion calling for people’s approval of each project. But what about a criterion calling for minimization of contaminants generated by each project, which are normally expressed in parts per million (ppm), or in mg/m3, or a criterion demanding minimization of labor

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(man-hours), or another calling for equalizing a certain amount of funds (Euros), or for distances (Km)? Obviously, to make comparisons between criteria it is necessary to normalize data, i.e., to generate dimensionless criteria. There are several methods for normalization and theoretically, none of them may influence the final result, that is, the results must be independent of the normalization system used, but in many cases, this does not happen. Usually, three or four methods are the most employed. (a) Using the sum of performance values in a row. aij

That is aij
=

8j

n

aij j=1

where aij = original performance value, and aij* = normalized performance value. That is, Find the sum of all performance values in a row and then divide each performance value by this total. (b) Using the largest value of the row aij
=

aij Max aij

8j

That is, Find the largest performance value in a row and divide each performance value by this largest value. (c) Using the Euclidean formula aij
=

aij √

n 1

ðaijÞ2

8j

Compute the denominator and then divide each performance value by this expression. d) Using the maximum/minimum ratio aij
=

aij - minðaijÞ maxðaijÞ - minðaijÞ

8j

Compute the difference between aij and aij(min) and divide this difference by the difference between aij(max) and aij(min).

13.7

Complex and Complicated Scenarios

13.7.2.5

325

Criteria Types

Criteria belong to two different species: “Quantitative or objective,” and “Qualitative or subjective.” The first refers to something that generally is easily measured and for which exist reliable values, for instance, m3 of water/day consumed by each project, cost of electric energy, total manpower needed, etc., as well as having defined limits, since our world is not unlimited. These limits may refer to man-hours, percentage for earnings, contamination allowed, maximum available funds, energy consumption, disposable income, etc. Qualitative criteria are those for which generally values are uncertain, do not exist, only words, expressions, intentions, levels of people satisfaction, etc. with several degrees between extremes as in approval or disapproval, happiness or sadness, good and bad, right and wrong, like and dislike, etc. Mathematical methods cannot normally work with these linguistic expressions; therefore, they must be converted into crisp numbers. This is not difficult because it can be done using appropriate numerical scales, albeit unfortunately, with high subjective content, and in addition, depending on the person doing the analysis, since one person may think and evaluate differently from another.

13.7.2.6

Cardinal Data

Data (performances) is imputed as cardinal numbers at the intersection of a column (alternatives or projects) and a row (criteria or restrictions). Projects, criteria, and performances constitute the decision matrix, which is the starting point for all MCDM methods. In this way, all data is condensed in a table or matrix. Performance values expressed as cardinal data represent the contribution of each project or alternative to the compliance of the criterion it intersects. Sometimes these values are obtained from formulas; consequently, a method must be able to handle mathematical formulas. For instance, there could be very large performance values, and they can be expressed in the usual units or in scientific notation. For example, 10,0000 Euros can be placed is this form or as 10E4. Same for small values; we can put 0.0000059 or 5.9E-5. Normally, when preparing the decision matrix there could be criteria with large values and others with very small. Examples of the first could be an investment in millions of Euros to acquire equipment. Examples of small values could be efficiency with values between “0” and “1”. Is it possible to mix large and small values? Yes, and this is one of the purposes of normalization.

13.7.2.7

Selecting a Method

It has been detailed the information required by most methods. Probably, one of the most difficult tasks, is to select the mathematical method to process data. There are

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many; therefore, choosing one of them is a MCDM problem by itself. One could say, well, it is a matter of running the same problem by different methods, and selecting the method that most approximates the objective. Unfortunately, it cannot be done, simply because we do not know what is that final result of the objective. See also Question 12.2.2 in Chap. 12.

13.7.2.8

Working with a Method

More often than not, given a problem, different methods provide different results for the same problem. If all of them start from the same data, use mathematical procedures and aim at the same objective, why results are different? Because subjectivity. That is, considering that distinctive methods use different weights (most of the time without any mathematical foundation), determining relationships between criteria and projects using diverse subjective assumptions (again based on personal preferences or experience), it would be a miracle if results coincide. In addition, most, if not all methods (except those based on Linear Programming) give results that are affected by the experience, or possible bias, or particular thinking of the person doing the analysis. For these reasons, two different independent DMs treating the same problem will probably get dissimilar results. Even if the same DM redoes the analysis in a time interval, results may be different because his/her thinking and appreciation may have changed. In the opinion of this author, this is the main problem that nowadays affects MCDM methods.

13.7.2.9

Difficulties than May Be Encountered in Interpreting a Solution

MCDM methods show results as a string of scores, from which the ranking of projects derives, as shown in Table 13.1 in boldphase, considering that the greater the score the higher a project’s importance in the ranking.

Table 13.1 Final results from a method

Criteria for evaluation C1 C2 C3 Scores Ranking

Projects or alternatives to be selected P1 P2 P3 P4 Cardinals a12 a13 a14 a11 a21 a22 a23 a24 a31 a32 a33 a34 0.23 0 0.19 0.23 P5 - P6 - P1/P4 - P3

P5

P6

a15 a25 a35 1.21

a16 a26 a36 0.89

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327

This is the typical outcome; however, different circumstances must be considered as follows: Projects Not Selected Project P2 received a “0” score meaning that it has not been selected out of the original list. The reason could be that it does not satisfy all criteria. Results with Ties Notice that projects P1 and P4 have the same score (0.23)—indicative of a tie— represented in the ranking by “/”, meaning that P1 = P4 or that P4 = P1 and placing the DM in a difficult position because he is not able to decide if adopting P1 or P4 for the third position in the ranking. Consequently, there must be a way for the DM to settle on one of them. See also Question 12.2.11 in Chap. 12. Aspects that Affect Ranking Response and Need for Sensitivity Analysis Most MCDM methods use a weight to quantify the importance of a criterion related to others (there are several definitions of weights and about what they mean, but here it is assumed that they are utilized to grade the relative importance of each criterion). Due to the fact that these weights are subjective (except when they are objective or endogenous, like those calculated using entropy or a statistical technique), the DM wonders how a change in the weight of a criterion, for instance, “Minimize custom duties,” affects other weights. That is, the DM can increase (or decrease) a weight figure, however, because these are relative values, its changing must affect other/s, and then one or more weights must be reduced (or increased). The DM may then wonder which, among the remaining weight/s he must decrease (or increase), for the summation to get unity. Consequently, sensitivity analysis, as good a technique as it is, does not help too much in this case, and in addition it is subject to subjective actions. What about if the DM considers the simultaneous variation of more than one criterion weight, for instance, for a criterion calling for “Maximizing demand” and another for a criterion calling for “Minimizing cost?” As per this author’s knowledge, there is no answer to these questions in methods using weights for criteria. In addition, the DM may ask himself how a variation of a cardinal data affects the ranking, for example, a change in “Unit price for equipment” criterion that rose 3.5%. Normally, rankings show a certain stability related to variations of criteria and cardinals, since most of the time these attributes may vary within certain limits, in minus or plus, and without altering the ranking, but how the DM know which these ranges are and their limits, provided that they exist? Some methods give a range of variation for which the ranking does not change, but not all methods do. See also Sect. 2.5.8. Close or Identical Scores Figures and DM Decision Assume that the result shows that project D = 1.01 and project A = 0.98, that is, both have close scores. Would it be acceptable for the DM to definitely accept project D as his first choice instead of project A because there is a little advantage of the first over the second?

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Probably not, as the difference is so small that a variation due to actual and external factors may change this ranking. This would be the case when projects are subject to criteria that are highly dependent on external factors and then uncontrollable, such as inflation, demand for the product to be manufactured by a project, weather, geological conditions, Return of Investments (ROI), risks, legal aspects, etc. Therefore, a method should give the DM more rational elements for him to judge than barely a little difference in score figures. See also Question 12.2.11 in Chap. 12. “Synergy” Between the DM and the Method All of the analyzed aspects demonstrate that MCDM is not an exact science even using scientific tools. It is a mix of DM knowledge, experience, and common sense, as well as correct modelling capacities and scope. However, even furnished with the best data, the method is not a mathematical equation, where values are imputed and results accepted without discussion, because are supported by theorems and axioms, as it would be if the DM were given the task to calculate the area of a right-angled triangle with a base of 5 cm and a height of 12 cm. In MCDM, the method only supports the DM in the treatment of thousands of possible combinations of projects, suggesting a solution that distillates a ranking of numbers, and is sometimes very confusing as in the case of ties. The method must be able to help the DM in giving numerical data, that he can translate into an educated conclusion for accepting or rejecting alternative D instead of alternative A, even when the mathematics of the method suggests it. It is necessary to remember that the task of the DM is to interpret, analyze, and consent or discard the results from the mathematical method, which is simply a tool; the reasoning is then left to the DM.

13.7.2.10

Role of Sensitivity Analysis

It can be roughly defined as the procedure that allows determining the impact on a solution when there are variations in some components. No MCDM study is complete without this analysis, and hence being short in getting all the answers to questions posed. Most MCDM methods provide sensitivity analysis just varying some parameters and seeing what happens with the ranking. Normally, this analysis is performed by selecting a criterion and applying a positive or negative variation to its weight or a threshold, and observing in a graphic how the ranking holds, and for long. In this way, the latter gives the range of variability of the change in the weight or threshold of the selected criterion. However, this is an activity that consists of giving successive values to a criterion weight until the graphic shows that the ranking changes. If we have to do the same for many criteria it can be very time-consuming and unreliable. In reality, this procedure appears to be an approximation, since increasing say 10% of the weight value of criterion “A” does not necessarily mean that the balance of criteria weights varies uniformly in the same amount. However, there is a more serious drawback to this system. When a weight is increased or decreased, in reality, the DM is not measuring the influence of this

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Using SIMUS for Decision-Making

329

change in the result, but how the ranking changes because errors in the appreciation of weights made by the DM regarding the true weights. There is no doubt that a system of projects, criteria, and cardinals have true weight values for the importance of criteria. For instance, the true value for criterion G could be 0.22; however, the DM assigned to this criterion a weight, say of 0.26. Therefore, when he plays around with this 0.26 value, he is examining a change in the ranking from an estimated weight, not the actual one. Sure, somebody will say “The problem is that we do not know the real value of this weight, and therefore we have to estimate one.” However, we really know it! The actual weight is an endogenous value that is generated by the original data and without any exogenous interference, and that can be obtained by different mathematical ways, as for instance using Shannon’s entropy concept (Shannon, 1948). A rare and valuable application of this can be seen in Zavadskas (2013). Consequently, if artificial weights are used, results may not be reliable. This system is applied in very well-known methods like ELECTRE, PROMETHEE, and TOPSIS, to name only the most used. The most popular MCDM method is AHP (Saaty, 1980), which is a generator of subjective weights that are further used in other methods. However, the problem with this last methodology is that it starts by making pairwise comparisons of criteria, using as a yardstick personal preferences regarding the main goal. Not only one analyst's preferences are debatable, but also another analyst may think differently, and consequently weights may be different. Also important, none of these methods can solve ties or quantitatively inform the DM about the convenience or not of accepting a ranking.

13.8

Using SIMUS for Decision-Making

There is no doubt that all mentioned problems could be avoided if in the MCDM scenario, the analysis stays clear of subjectivity, other than defining the number and quality of criteria. Most of the issues treated in preceding sections can be addressed by using Linear Programming (LP), the technique that in mid-twentieth century pioneered MCDM and for which his creator, Leonid Kantorovich (1939), was awarded the Noble Prize in Economics in 1956. LP does not need weights of any kind either for alternatives or for criteria, does not utilize personal preferences, thresholds, or functions to define the outweighing of one alternative over another, and no distances to the ideal, to define a compromise solution; in other words, its only requirement is the Initial Decision Matrix, and can work with hundreds of alternatives and criteria because it is based on a completely different approach from MCDM methods by using linear algebra and inequations, instead of equations, and thus, working with areas instead of lines. It determines a solution polygon (in two dimensions), a polyhedron (in three dimensions), or a polytope in “n” dimensions, where each dimension corresponds to a different alternative, therefore, there can be hundreds of mathematical dimensions.

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The set of criteria shapes this geometric figure, using straight lines (in two-dimension problems), planes (in three-dimensions problems), and hyper planes in more than three dimensions problems. Within this geometric figure are an infinite number of possible solutions, however, the DM is only interested in the best solution out of thousands, and that is achieved using a very efficient algorithm—the Simplex algorithm—developed by George Dantzig (1948), by executing the Solver program developed by Frontline Systems, which is in your computer, and that can be accessed from the “Data” tag. The advantage of LP is that it delivers optimal solutions, i.e., solutions that cannot be improved (if they exist), and then thoroughly complies with the Pareto principle. In addition, it simultaneously provides the solution for two problems. The first is called the “Primal” and the second is the “Dual,” and one is symmetric from the other, like the two sides of a coin. The primal gives the information we are looking for, that is a string of scores, and corresponding ranking—for selected projects or alternatives. The dual conveys very valuable information about the criteria significance, and it is the basis for performing sensitivity analysis, i.e., when conducting a postoptimality analysis of the solution reached by the primal. As Gass and Harris stated (1996) “The purpose (of dual) is to explore various scenarios about future conditions that may deviate from the initial method,” and add “A key part of postoptimality analysis is sensitivity analysis, which involves investigating the parameters, determining which ones are sensitive parameters and exploring the implications.” The underlined concepts are paramount in any sensitivity analysis because they point out three very important aspects: (a) To have a look at what can happen in the future (for whatever reasons). (b) To identify the most critical parameters, and most important, (c) To analyze how they can affect the performance of a project when it is built. Obviously, the reader is surely wondering why this technique is not being used, instead of dealing with heuristic programs as mentioned above. For his/her information, LP is actually being used by tens of thousands of companies around the world including the US Government. However, LP has two drawbacks: It works with only one objective, and it does not accept qualitative criteria. These are the reasons for the appearance of the heuristic methods. SIMUS (Sequential Interactive Method for Urban Systems), a method developed by the author of this paper (Munier, 2011b), is strongly based on LP, and can work with up to 100 criteria, of any type and mix, and up to 200 alternatives. It does not produce an optimal solution (if it exists), but, like the heuristic methods, delivers a compromise solution. SIMUS starts with the initial matrix with data, as in every other method. It works using the Simplex algorithm and the Solver program. It is a sequential method consisting of the following:

13.8

Using SIMUS for Decision-Making

331

1. Selecting a criterion to perform as an objective. To understand how a criterion can be used as an objective function, it is necessary to consider that criteria in LP are linear inequations and can be used indistinctly as criteria or as objective functions, since they have the same mathematical structure. In LP parlance, the left part of each inequation is known as Left Hand Side (LHS), and the right side, as Right-Hand Side (RHS). The only difference between a criterion and an objective is that the first has an RHS, which establishes the goal of that criterion, or limits to the inequation (for instance, maximizing or minimizing a resource), while the objective function expresses only a wish or aspiration (for instance, maximize benefits or minimize costs). The RHS, also called “independent term” is extremely useful to replicate actual situations related to funds, capacities, finance, the environment, social aspects, safety, risk, etc. Criteria are at the same time objectives, in the sense that they establish a goal, a target that needs to be satisfied. 2. Once the method is run its solution is automatically saved in an “Efficient Result Matrix” (ERM), and the utilized equation is returned as a criterion. 3. Another criterion is selected, used as objective function, processed by Solver, and the result is saved in the ERM. 4. Again, another criterion is selected and the same procedure is followed. When the ERM matrix is complete, it constitutes a matrix in which rows are Pareto efficient, i.e., a matrix containing a set of results that are optimal and cannot be improved. 5. In the ERM matrix, SIMUS adds up all scores in each column or project, and multiples this sum by a ratio between the number of criteria satisfied for each project to the total number of criteria. The result and its ranking are the solution to the problem, using the Weighted Sum Approach. 6. SIMUS analyzes each row and determines which project outranks another in value. The difference is placed in a square matrix called “Project Dominance Matrix” (PDM). 7. Summation of values in each row of the PDM gives the value for each dominant project. Summation of values in each column of the PDM gives the values for each subordinated project. The differences between dominant and subordinated values for each project produce scores for each one, and the corresponding ranking. The result and its ranking, is the solution of the problem, using the Outranking Approach. 8. Both, ERM and PDM solutions give different sets of scores because they use different procedures, however, their rankings are identical, and in fact each solution checks the other. Figure 13.2 portrays a simple problem solved by SIMUS, and it is a capture of its last screen. Observe in the ERM matrix the alternative scores in the blue solid line, and in the PDM matrix the alternative scores in the brown solid column. As can be seen, the scores have different values in the two matrices because they are the consequence of applying two different MCDM methods, however, notice that the two rankings are

1.46

1.54

Final Result (SC x NPF)

PDM Ranking

2.0

2.0

SE

Fig. 13.2 Capture on final SIMUS screen

Column sum of dominated projects

PV

SE

Dominant proj.

2.1

2.1

PV

Project Dominance Matrix (PDM) Dominated projects

ERM Ranking

2.43 3 0.60

SE - PV

SE - PV

C1 C2 C3 C4 C5

2.0

2.1

0.60

0.89

0.1 –0.1

Net dominance

1.11 0.72

1.12 0.67

Resume of shadow prices considering different objectives (in columns), against targets (in rows) Object. 1 Object. 2 Object. 3 Object. 4 Object. 5

Row sum of dominant projects

C1 C2 C3 C4 C5

2.57 3 0.60

Efficient Results Matrix (ERM) Normalized SE PV 0.57 0.43 1.00 1.00 1.00 1.00

C1 C2 C3 C4 C5

RHS 3

RHS 2

RHS 5

RHS 4

RHS 5 RHS 3

Increase Obj.1 0.08 0.18 Inc. Obj.2 1E+30 Inc. Obj.3 0.09 Inc. Obj.4 1E+30 Inc. Obj.5 0.18

RHS allowable limits Decrease Obj.1 0.12 0.08 Dec. Obj.2 0.03 Dec. Obj.3 0.34 Dec. Obj.4 0.03 Dec. Obj.5 0.32

13

Sum of Column (SC) Participation Factor (PF) Norm. Participation Factor (NPF)

Efficient Results Matrix (ERM) SE PV 0.56 0.41 0.00 1.45 0.81 0.00 1.18 0.00 0.00 0.86

C1 C2 C3 C4 C5 Objective function values per target 0.49 0.42 0.24 0.43 0.22

Initial Matrix (LHS (5x2)) SE PV 0.72 0.68 0.85 0.75 0.78 0.98 0.92 0.65 0.99 0.60

332 Best Practices: Modelling and Sensitivity Analysis in MCDM

13.9

Analyzing Variations in Criteria Limits (RHS)

333

identical, in both: SE > PV (both in red), and this happens even when there are many alternatives. The ERM matrix is shown in blue. When criterion C1 is removed from the Initial Decision Matrix (blue in Fig. 13.2), used as objective function Z1, and SIMUS run, two optimal values are found: 0.56, for alternative SE and 0.41 for alternative PV. Both are shown in the ERM matrix of Fig. 13.3. Therefore SE > PV. The dual can be seen in the green matrix, and it shows that objective function Z1 depends on binding criteria C3 (Manpower) and C5 (equipment). Both marginal values for their respective criteria indicate that for each unitary variation of the corresponding RHS, there is a change in the objective function, indicated by the shadow price. In this case, both criteria call for minimization and then, there will be a reduction in the objective function Z1 for unitary variation in RHS (3) and RHS (5).

13.9

Analyzing Variations in Criteria Limits (RHS)

It is not possible to increase or decrease indefinitely the significance of a criterion because normally it has limits in both senses, i.e., for increases and decreases. This is important because as long as the variation is kept within these limits, the marginal values hold, and the objective increase or decrease lineally, at a constant rate. Once these limits are surpassed, the criterion ceases to influence the objective. Therefore, the smaller the allowed variation, the larger the criticality of a criterion. Consequently, criterion (C5) “Minimization of equipment,” is the most critical, because the smallest variation over 5% will cause a change in the ranking. Thus, the corresponding shadow price indicates that a variation in this criterion may have a considerable influence on the unit cost (Z1). As can be appreciated, the DM has a wealth of information just by analyzing the shadow prices, consequently, this table makes this kind of analysis extremely powerful. Now, going to the table on the right an entering with objective Z1, there are two values. The left column indicates how much C3, whose RHS is 0.84, may increase without perturbing alternative SE, while on right the value 0.08 shows how much the same criterion may decrease. Consequently, criterion C3 may vary from a minimum of 0.84 - 0.08 = 0.76 to a maximum of 0.81 + 0.18 = 1.02. Considering that this criterion calls for minimization, it can safely decrease to 0.76 without altering the ranking. Similarly, for criterion C5 that also calls for minimization, it has RHS = 0.80, therefore, it can vary from a minimum of 0.80 - 0.12 = 0.68 to a maximum of 0.80 + 0.08 = 0.88, and decrease down to 0.68.

PV 0.41

Objective (Z1) C1 C2 Initial RHS C3 0.89 RHS (C3) = 0.84 C4 C5 0.59 RHS (C5) = 0.80 (0.18-0.08)/0.84=0.12 (0.12-0.08)/0.80=0.05

Percent variation 0.08

The smaller variation corresponds to criterion (C5),

Increase Decrease Z1 Z1 0.18 0.08 0.08 0.12

Resume of shadow prices considering objective 1

Fig. 13.3 Illustrates how to perform a sensitivity analysis, based on Fig. 13.2

Consequently, for 'Unit Cost' minimization objective (Z1), criterion (C5) (Generation Index) is more critical than criterion (C3) (Financial Index), meaning that especial care must be exerted on this criterion if the ranking is to be maintained

13

When criterion (C1) is used For objective (Z=1) there are two as objective function (Z1) shadow prices, one corresponding These are the original and thus it is the most critical optimal scores are 0.56 for to criterion (C3=0.89), and to RHS from the initial RHS (C3) may reach a maximum limit value of (0.84+0.18= 1.02) alternative SE and 0.41 for criterion (C5=0.59) decision materix and a minimum limit value of (0.84-0.08=0.76) alternative PV without ranking variation RHS (C5) may reach amaximum limit value of(0.80+0.08= 0.88) and a minimum limit value of (0.80-0.12=0.68) without ranking variation

SE Z1 0.56 Z2 Z3 Z4 Z5

EeeeeeRM ERM

334 Best Practices: Modelling and Sensitivity Analysis in MCDM

13.10

Analyzing Variations in Alternatives Scores

13.10

335

Analyzing Variations in Alternatives Scores

Given a set of values for alternatives scores, a very important question and the concern of many DMs, is determining how much they can change in order not to alter the optimal solution found; in other words, which is the variability they accept. Remember that for objective Z1, i.e., the minimization of unit costs, the program shows optimal scores of SE = 0.56 and PV = 0.41, which can be equated for instance to percentages of land used by each alternative. The objective function was: Z = 0:72 SE þ 0:41PV = 0:72 × 0:56 þ 0:68 × 0:41 = 0:68:

ð13:1Þ

Assume that the DM is concerned with unit cost performance variability. His question will probably be: Is it possible to vary the unit cost for a11 and for a12, in plus and in minus, but keeping the same score of 0.56, or optimal solution? To answer this question he checks Table 13.2, also produced by the method, which indicates how much each score (not each performance aij value) can vary without changing the optimal solution. Optimal SE score (0.56) can safely vary between 0.56 and 0.75. Optimal PV score (0.41) can safely very between 0.41 and 0.66. Since these scores correspond to the objective function, it is then possible to alter the aij performance values between certain limits. For instance, suppose that it is believed that the unit cost for SE will increase to 0.75 from 0.72 (see Table 13.3). Placing this new value in cell aij11 and running again the software it can be seen that the optimal scores of 0.56 and 0.41 have not changed and also the ranking has not changed. However, because of this change, now Z = 0.75, and since it represents the economic performance of the whole operation, it has deteriorated from 0.68, since we are minimizing. Can the DM put a new unit price lower than the original? No, it is not possible because the lower limit for SE score is equal to the original value. Unfortunately, there is not a calculation that tells us the limit values for alternative performance; therefore, this has to be done through trial and error. Keep in mind that the change that we did in SE unit price from 0.72 to 0.75 also changed the corresponding shadow prices and consequently the limits for RHS. As can be seen, SIMUS in its last screen supplies complete quantitative information about sensitivity on both, RHS and alternative scores variation, and this information is provided in tables where all criteria are analyzed. Consequently, all objectives, RHS, shadow prices and their variations, and alternative scores and their variations can be visualized in one screen, which facilitates comparisons, and most Table 13.2 Intervals of variation for alternative scores Scores Z function

SE Lower limit 0.56 0.68

Upper limit 0.75 0.80

PV Lower limit 0.41 0.68

Upper limit 0.66 0.85

336

13

Best Practices: Modelling and Sensitivity Analysis in MCDM

Table 13.3 Increasing unit cost for SE within its limits of variation

C1 C2 C3 C4 C5

Initial Matrix (LHS (5x2)) SE PV 0.75 0.68 0.85 0.75 0.78 0.98 0.92 0.65 0.99 0.60

C1 C2 C3 C4 C5

Efficient Results Matrix (ERM) PV SE 0.56 0.41 0.00 1.45 0.91 0.00 0.00 1.18 0.00 1.00

C1 C2 C3 C4 C5

Efficient Results Matrix (ERM) Normalized SE PV 0.57 0.43 1.00 1.00 1.00 1.00

Sum of Column (SC) Participation Factor (PF) Norm. Participation Factor (NPF)

2.57 3 0.60

2.43 3 0.60

Final Result (SC x NPF)

1.54

1.46

ERM Ranking

SE - PV

Project Dominance Matrix (PDM) Dominated projects Dominant proj.

SE

SE PV Column sum of dominated projects

Row sum of PV

dominant projects

2.1

2.1

Net dominance 0.1

2.0

–0.1

2.0 2.0

PDM Ranking

2.1

SE - PV

important, allows the DM to examine the consequences, advantages, and disadvantages of the selection done. In this example, it appears that the most critical criterion is (C5) which refers to “Equipment.” This is related to many aspects of the operation, such as consumption, maintenance, and unscheduled stops and most probably affects SE much more than PV.

References

13.11

337

Conclusion

This chapter attempted to shed some light on modelling actual scenarios, an issue that as per this author’s opinion, has not been seriously considered in the past, since most methods deal with very important themes like data uncertainty, new applications, new methods, solution stability, robustness, etc., but it appears that researchers do not pay attention to the inaccuracy subjectivity produces, other than some scholars who find it unacceptable, and with reason, for instance, when different methods deliver different results for a same problem. However, no solutions are proposed to deal with this problem. It is emphasized here that the largest culprit is the use of weights to quantify criteria importance in practically all methods. This author suggests coming back to the origins of MCDM which was born out of Linear Programming, a technique that does not need weights and consequently eliminates this subjectivity. In addition, and as a second objective, this chapter addresses the very important sensitivity issue and proposes using the dual property of LP to have a more effective control.

References Dantzig, G. (1948). Linear programming and extensions. United States Air Force. Gass, S., & Harris, C. (1996). Encyclopedia of operations research and management (p. 346). Kluwer Academic Publishers. Kantorovich, L. (1939). The best uses of economic resources. Lliso, P., & Munier, N. (2014). Multicriteria Decision-Making by Simus. http://decisionmaking. esy.es Munier, N. (2011a). A strategy for using multicriteria analysis in decision-making - A guide for simple and complex environmental schemes. Springer. Munier, N. (2011b). Procedimiento fundamentado en la Programación Lineal para la selección de alternativas en proyectos de naturaleza compleja y con objetivos múltiples. Ph.D. Thesis, Universidad Politécnica de Valencia. Saaty, T. (1980). Multicriteria decision making - The analytic hierarchy process. McGraw-Hill. Shannon, C. (1948). Mathematical theory of communication. The Bell System Technical Journal, 27, 379–423. Triantaphyllou, E., Shu, B., Nieto Sanchez, S., & Ray, T. (1998). Multi-criteria decision making: An operations research approach. In J. G. Webster (Ed.), Encyclopedia of electrical and electronics engineering (Vol. 15, pp. 175–186). Wiley. Zavadskas, E., Antucheviciene. J., Saparauskas, J., & Turkis, Z. (2013). MCDM methods WASPAS and MULTIMOORA: Verification of robustness of methods when assessing alternative solutions.

Chapter 14

Some Complex and Uncommon Cases Solved by SIMUS

Abstract This chapter examines some uncommon and complex scenarios that illustrate the ability of LP and SIMUS to solve them. Normally, they are related to the need of considering a scenario as a whole and where peripheral considerations intervene and linked to the main objective.

14.1 14.1.1

Case Study: Simultaneous Multiple Contractors’ Selection for a Large Construction Project Background Information

This section deals with selecting contractors for large projects. Usually in a sizable undertaking, there are a main contractor and many subcontractors. Normally, the main contractor is chosen, and then individually the promoter or the Project Manager (PM) selects subcontractors. However, this case addresses the more complex task of selecting simultaneously all contractors and subcontractors and at all levels. In addition, there are some circumstances that must be considered, probably the most common is that each bidder is normally allowed to associate with another bidder, forming what is called a “Joint venture.” An MCDM method should then be able to handle these simultaneous selections and, for this type of associations, provide a means for the PM to analyze results and introduce modifications based on his experience, considering the most vulnerable aspects of each selected bidder. Large construction projects include many areas, namely civil construction, excavation, foundations, concrete production, electrical installations, roads construction, industrial installations, etc., as well as necessitating a wide selection of engineers in different disciplines, social workers, safety people, etc., each one with their own schedule, personnel, and times. To understand this complex issue, one must consider the characteristics of this scenario as: © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 N. Munier, Strategic Approach in Multi-Criteria Decision Making, International Series in Operations Research & Management Science 351, https://doi.org/10.1007/978-3-031-44453-1_14

339

340

14

Some Complex and Uncommon Cases Solved by SIMUS

First tier Main contractor selection

Second tier Land clearing

Third tier Excavation & Foundations Fourth tier Concrete

Fifth tier Equipment selection

Equipment transport

Equipment purchasing Equipment construction

Installation

Equipment testing

Siding Steel truss roof Flooring Industrial Electric Buildings Electric Painting

Fig. 14.1 Shows links and dependency between different areas

• There are different trades. Any of them receives a project from another trade, and when its own job is complete, delivers it to another trade, i.e., there is a clear precedence of activities. • In this scenario trades can be considered as clusters, since there are normally more than one company in each, for the same trade, and competing for the job. For instance, three different companies in the “Plumbing” trade. Consequently, there is a need to perform a decision-making within each cluster to select only one company. • Then, there is another decision scenario which consists of choosing the clusters. However, different from the conventional methods, all clusters must be selected. Figure 14.1 is a scheme of the degree of dependency between areas. The whole undertaking is normally planned, scheduled, and condensed in a master schedule, made at a high level, that is, without too much detail. It relates the different schedules submitted by the main contractor and subcontractors and is the official legal document for the whole work. Therefore, a construction work is organized as a network with hundreds of nodes and connections. See Fig. 14.2, and observe links between subcontractors as dashed lines. When the promoter has all the documentation pertaining to all areas of the project, he calls for international tenders to select the main contractor and each one of the subcontractors and suppliers. It is then possible for the promoter to invite three or four of the most renowned construction companies at world level, and at the same time invite a few selected contractors for each one of the different specialties. This scenario is illustrated in the next section.

14.1

Case Study: Simultaneous Multiple Contractors’ Selection for a. . .

341

Contractor A Civil construction 11

Power building construction

Foundation

Contractor C Electrical instrumentation Switch board Control panel 32

71

Testing

Turbo set installation

89

FF link

FS links between contractors C and A

Control panel installation and testing

61

Contractor B (overseas) Equipment manufacturing FS Link between contractors B and A Turbo set testing in factory Turbo set shipping

55

Fig. 14.2 Example of a partial network

14.1.2

The Case: Construction of a Large Power Plant

This case refers to an actual project that took place in a North American country. Purpose of this study: Select contractors and subcontractors using MCDM. Conditions: Different areas or fields for contractors and subcontractors; there is no limit for the number of areas, however in this example, and for printing reasons, only twelve are considered: 1. 2. 3. 4. 5. 6. 7. 8. 9.

General management and coordination (Main contractor) Land clearing Excavation and foundations Mechanical and electrical equipment Concrete Steel truss and roofing Siding Flooring Industrial electric network

342

14

Some Complex and Uncommon Cases Solved by SIMUS

10. Electrical for building 11. Industrial plumbers 12. Industrial painters Base documentation: Prepared by the promoter’s engineering planning department and private consultants and approved by stakeholders. Input for selection: Technical, Economic, Finance, Environmental, and Manpower data from proposals submitted by different bidders worldwide. Evaluation criteria: Prepared by the promoter’s engineering, environmental, financial departments, and private consultants. There were 80 criteria; a different set for each area, but it was convenient to add criteria that evaluate most bidders simultaneously. Common criterion could be “Cost,” “Referrals from industry,” “Prestige,” etc. It does not present an additional difficulty; however, they are not included in this example to facilitate comprehension of the method and reduce its length, as well as computing time. The criteria, identified as “Ci” where “i” = 1 to 80, are as follows: Main Contractor C1 Total construction budget, C2 Total cost for construction management, C3 Referrals from the construction industry, C4 Years in business, C5 Number of engineers (Civil, Mechanical, Electrical, etc.), C6 Other technical specialties (Geologists, Architects, Hydrogeologists, etc.), C7 Expertise on this type of project expressed as number of projects, C8 Own equipment to be used in the project (%), C9 Number of projects finished in the last fifteen years, C10 Amount in project value in hundreds of millions of US$ in the last 15 years, C11 Amount of working capital for this project in millions of US$, C12 Number of own workers that will be in the project, C13 Square meters built in the last three years, C14 Average age of equipment (in years), C15 Number of projects delayed more than 10% of initial schedule in the last 15 years, C16 Number of times taken to court in the last 15 years, C17 Number of lawsuits won, C18 Liability insurance in millions of US$. Land Clearing C19 Proposal budget (US$), C20 Referrals from the construction industry (Very good – Good – Regular), C21 Years in business, C22 Average age of equipment to utilize (Years).

14.1

Case Study: Simultaneous Multiple Contractors’ Selection for a. . .

343

Excavation and Foundations C23 Proposal budget (US$), C24 Referrals from the construction industry (Very good – Good – Regular), C25 Years in business, C26 Own equipment to be used in this project (%), C27 Number of projects finished in the last 15 years, C28 Number of projects delayed more than 10% of initial schedule in the last 15 years. Equipment C29 Proposal budget (US$), C30 Number of generation units, C31 Output of each unit (MW), C32 Standalone efficiency (%), C33 Cogeneration efficiency (%), C34 Water consumption (L/day). Exhaust Gas Contamination C35 NOx (g/kWh), C36 CO2 (millions of tons,) C37 Particulate (μg/m3), C38 Type of cooling system for generators (air-hydrogen), C39 Noise at 50 meters (dB), C40 Noise at 40 meters (dB), C41 Travel time from country of origin to jobsite (days), C42 Transformers and installation (US$). Concrete C43 Proposal budget (US$), C44 Concrete plant production (m3/day), C45 Concrete cooling plant capacity (m3/day), C46 Number of concrete mixers, C47 Capacity of concrete mixers (m3), C48 Frequency of concrete supply (m), C49 Power source, C50 Maintenance cost (US$/month). Roofing C51 Proposal budget (US$/ m2), C52 Thousands of square meters installed in the last five years, C53 Number of cranes, C54 Crew (man-hours/day).

344

14

Some Complex and Uncommon Cases Solved by SIMUS

Siding C55 Proposal budget, C56 Chemical treatment (US$), C57 Thousands of square meters installed in the last five years. Flooring C58 Proposal budget (US$/m2), C59 Thousands of square meters installed in the last five years, C60 Guarantee (Years), C61 Flooring maintenance service (US$/m2). Industrial Electric C62 Number of projects finished in the last 15 years, C63 Number of engineers, C64 Certified manpower (man-hours), C65 Liability insurance in millions of US$, C66 Supply of elevators and budget (US$), C67 Budget for transformers and wiring (US$), C68 Supply of power station control panels (US$), C69 Duration of installation of control panels and test (US$). Electrical for Buildings C70 Proposal budget (US$), C71 Testing (US$), C72 Number of projects finished in the last 15 years. Plumbing C73 Proposal budget (US$), C74 Amount executed in the last 15 years (US$), C75 Number of projects finished in the last 15 years. Painting C76 Proposal budget (US$), C77 Equipment to use, C78 Duration (days), C79 Number of projects finished in the last 15 years, C80 Amount executed in the last 15 years (US$). Naturally, these criteria were subject to maximization and minimization actions and not all of them apply to all proposals. The promoter received 85 sealed proposals, and after their opening, the pre-selecting process starts, based on compliance of general characteristics specified in tender documents. Many proposals were unaccepted because of incompleteness or lack of compliance of some items, and finally 31 bidders were preselected. It was a blind examination since bidders were assigned a code number and therefore, the Project Manager (PM) received no

14.1

Case Study: Simultaneous Multiple Contractors’ Selection for a. . .

345

Table 14.1 Bidders selected after a first screening

Construction companies

Land clearing

Excavation & Foundations

Mechanical and Electrical equipment

CCA

CCB

CCC

LCA

LCB

E&F 1

E&F 2

ME 1

ME 2

ME 3

1

2

3

4

5

6

7

8

9

10

Concrete suppliers

Steel truss and roofing

Siding

CON SUP 1 CON SUP 2 CON SUP 3 STR 1 STR 2 STR 3 STR 4 Si 1 Si 2 11

12

13

Flooring

IP 1 28

IP 2 29

15

Industrial electric

FLOO 1 FLOO 2 FLOO 3 20 21 22

Industrial plumbers

14

IE 1 23

IE 2 24

16

17

18

19

Buildings electric EINS 1 EINS 2 EINS 3 25 26 27

Industrial painters IPAIN. 1 IPAIN. 2 30 31

information about the different codes. He and his staff needed to select the best one within each area or field. Table 14.1 shows the code numbers (IDs) of the selected bidders. All this information was loaded into SIMUS and the result in scores for each bidder is displayed in Table 14.2. The result can be seen in the blue solid line, where the values identify the largest scores for each area. The higher the quantity, the more significant the trade. Based on this information SIMUS displays in bold the resulting ranking. Bidder 3 - Bidder 13 - - Bidder 7 - Bidder 16 - Bidder 9 – Bidder 20 - Bidder 4 Bidder 18 - Bidder 23 – Bidder 28 - Bidder 30

Therefore, the winners are: AREAS Construction Company (3): Concrete supplier (13): Excavation and Foundation (7): Steel truss and roofing (16): Mechanical and electrical equipment (9):

SELECTED CONTRACTOR CCC CON SUP 3 E&F 2 STR 3 ME 2

346

14

Some Complex and Uncommon Cases Solved by SIMUS

Table 14.2 Scores for each bidder and for areas CCA 1

CCB 2

0.26

0.23

ME 2

ME 3

CCC 3 0.64

LCA 4

LCB 5

E&F 1 6

E&F 2 7

ME 1 8

0.01

0.01

0.04

0.16

0.04

CON SUP 2 12

CON SUP 3 13

STR 1

STR 2

STR 3

14

15

16

0

0

0.11

IE 1

IE 2

9

10

CON SUP 1 11

0.09

0.01

0.01

0.01

0.20

STR 4

Si 1

Si 2

FLOO 1

FLOO 2

17

18

19

20

21

FLOO 3 22

23

24

0

0.01

0.01

0.03

0.02

0.02

0.01

0.01

EINS 1

EINS 2

EINS 3

IP 1

IP 2

25

26

27

28

29

IPAIN 1 30

IPAIN 2 31

0

0

0

0.01

0

0.11

0.08

Flooring (20): Land clearing (4): Siding (18): Industrial electric (23): Industrial plumbers (28): Industrial painters (30): Buildings electric (27):

FLOO 1 LCA Si 12 IF 1 IP 1 IPAIN 1 EINS

Analysis It proceeds now to performing a sensitivity analysis, i.e., determining the strength of each selection; it is performed using IOSA. The scores in the blue row (Table 14.2) permit shaping a ranking of importance amongst areas. The most important selection corresponds, as expected, to “Construction Companies.” Within this area, bidders CCA, CCB, and CCC have the highest scores. However, the best of them corresponds to number 3, “CCC” (score 0.64), which is preferred for this area. In the same manner the selected bidders for other areas are identified, totaling 12 bidders, which is equal to the number of areas. However, notice that only eleven trades are selected when there should be twelve; trade EINS is not included, because all its scores are 0; this is probably because none of the EINS bidders satisfy the problem, considering all criteria. The second most important area is number 13, “Concrete suppliers” (score 0.20), the third is number 7, “Excavation and Foundations (score 0.16),” the fourth is number 16, “Steel truss and Roofing” (score 0.11), the fifth is number 9, “Mechanical

14.1

Case Study: Simultaneous Multiple Contractors’ Selection for a. . .

347

Table 14.3 Criteria that define selection for bidders 18 and 19, which have the same scores Efficient Results Matrix (ERM) Normalized Bidder 15 ----------C52 C53 C54 C55 C56 C57 C58 C59 -----

Bidder 16

Bidder 17

Bidder 18

Bidder 19

Bidder 20

1.00 1.00 1.00 1.00 1.00 0.44

and electrical equipment” (score 0.09), and so on. This ranking gives an idea of the importance of each area, and where the utmost attention and control must be exerted to avoid delays and over costs. Is it possible that more than one bidder appears with the same score? It is not only possible but also highly probable. As seen in the blue row, there are several bidders with scores of 0.01, meaning that they get the same importance. However, most of these coincidences are in different areas, and consequently they are not significant. In some cases, there can be the same scores for one or more bidders within the same area; in this situation, it is necessary to compare these bidders by analyzing the criteria that form each selection from the ERM matrix. Not all criteria have the same importance and then examining them and their values for each bidder it is possible to choose the winner based on criteria importance. As an example, consider bidders 18 and 19, both in the “Siding” area, and both with the same score of 0.01. Table 14.3 is an extract of the ERM matrix in SIMUS final screen for these two bidders. Observe that selection of bidder 19 involves a score of “1.00” for criterion C55, and for bidder 18 the same score appears for criterion C 57. Now, criterion C55 refers to the proposed budget, while criterion C57 refers to the number of square meters installed in the last five years. Maybe bidder 19 budget is a little lower than bidder 18 budget, and that is probably the reason by which bidder 19 is selected by the method when the objective is to minimize costs. However, when the objective is to maximize the experience expressed by the amount of work done in the last five years, it clearly favors bidder 18. For this reason, the PM may prefer bidder 18. The same analysis can be performed for any other bidders.

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Some Complex and Uncommon Cases Solved by SIMUS

Table 14.4 Criteria that define selection for bidder 3 Bidder 1 Bidder 2

C1 C2 C3 C4 C5 C6 C7 C8 C9 C 10 C 11 C 12 C 13 C 14 C 15

0.33

0.67 0.38

Bidder 3

Bidder 4

Bidder 5

0.62 1.00

1.00 1.00 0.81 1.00

0.19 1.00 1.00 0.08

0.92 0.87 1.00

0.13

Let us analyze now in more detail because bidder 3 (CCC) was selected. Table 14.4 is an extract of the ERM matrix for bidder 1 (CCA), 2 (CCB), and 3 (CCC). SIMUS identified eight criteria that are responsible for selection of bidder 3. They are: C4, C5, C7, C8, C9, C13, C17, and C18 and showed the intensity of this support given by the score value. The weakest criterion is C13 “Square meters built in the last three years,” with a value of 0.13. The highest scores correspond to the other criteria with a unit value each. These scores represent the contribution of bidder 3 to each criterion, and the higher, the better. They come from the solution of the primal problem and consequently are optimal values. As an example, when criterion C18 “Liability insurance”1 is used as objective function (and then named Z18), this result is supported in turn by criterion C4 “Years in business” = 0.83, and C8 “Own equipment to be used in the project”= 0.24. These values are the shadow prices, which are not shown here for clarity reasons. Their meaning is that one-unit increment in criteria C4 and C8 produces, respectively, an increase of 0.83 units and 0.24 units in functional Z18. Consequently, each additional year in business (or expertise) and each additional own equipment added to the project modify positively the contractor liability insurance. Table 14.5 depicts the criteria responsible for each selection.

1

Liability insurance is any insurance policy that protects an individual or business from the risk that they may be sued and held legally liable for something such as malpractice, injury, or negligence (Investopedia).

14.1

Case Study: Simultaneous Multiple Contractors’ Selection for a. . .

349

Table 14.5 Selected bidders for each area with indication of the criteria responsible for that selection Areas Main contractor

Concrete suppliers

Excavation & foundations

Steet truss & Roofing

Mech. & elect. equipment

Bidder selected 3

13

7

16

9

Bidder coding CCA

Objective C4 C5 C7

Objective score 0.62 1 1

C8

0.81

C9

1

C13

0.13

C17 C18

1 1

C45

1

C47

1

C48

1

C49 C23 C24

1 0.33 1

C25 C27

1 1

STR3

C52

1

ME 2

C53 C54 C30

1 1 0.56

C31 C34

1 1 0.44 0.67 1 1.72

CON SUP 3

E&F2

Flooring

20

FLOO 1

Land clearing Siding

4 18

LCA Si

C59 C62 C21 C57

Industrial electric

23

IE 1

C64

1.67

28

IP 1

C75

1.36

Objective affecting each selection Years in business Number of engineers Expertise on this type of project Own equipment to be used (%) # of projects finished in the last 15 years Square meters built in the last 3 years Number of lawsuits won Liability insurance in millions of Euros Concrete cooling plant capacity (m3/day) Capacity of concrete mixers (m3) Frequency of concrete supply (m) Power source Proposal budget (Euros) Referrals from the construction industry Years in business # of projects finished in last 15 years Thousands of m2 installed in last five years Number of cranes Crew (man-hours/day) Number of generation units Output of each unit (MW) Water consumption (L/day) Warranty (Years) Number of engineers Years in business Proposed budget (Euros/ m2) Liability insurance in millions of Euros Equipment to be used (continued)

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Some Complex and Uncommon Cases Solved by SIMUS

Table 14.5 (continued) Areas Industrial plumbers Industrial painters Elect. installers (Buildings)

Bidder selected

Bidder coding

30

IPAINS 1 EINS 1

25

Objective

Objective score

Objective affecting each selection

C79

1.02

–

–

# of projects finished in last 15 years Years in business

Notice that bidders 14, 15, and 17 (area “Steel truss and roofing”) have a score of “0.” For that reason, for that area bidder 16 (score 0.11) was selected. For area “Electric installers (Buildings),” the three bidders 25, 26 and 27 have scores of “0.” This means that none of the three bidders qualified for the job, therefore a new call for tenders must be done for this sector.

14.1.3

Conclusion of this Case

It is believed that this type of simultaneous selection for contractors in large construction projects has not been addressed before in the literature. This case illustrates a quantitative approach based on strict mathematics, complemented with the expertise and know-how of the PM to perform this complex task.

14.2

14.2.1

Case Study: Quantitative Evaluation of Government Policies Regarding Penetration of Advanced Technologies Background Information

In this section an example of advanced technologies is used as the “resource” to be shared by different strata in modern society. Its objective is determining the set of policies that contribute the most to technology penetration and applicable to different society sectors. Once identified, experts can use the information for designing developing programs focused at each area. Flowchart in Fig. 14.3 indicates the procedure.

14.2

Case Study: Quantitative Evaluation of Government Policies. . .

* Areas to consider * Policies to be applied * Values estimate

* Disparity between areas * Identification of significant polices per area

* Input-Output analysis * Policy evaluation * Actions to improve

351

* Periodic checking * Reassessment

Fig. 14.3 Flowchart indicating proposed procedure

14.2.2

Process Structure

This process has the same structure as other MCDM scenarios, in the sense that it relates areas (projects or alternatives) subject to policies (criteria in MCDM parlance) that must be honored in a larger or lesser degree; usually, MCDM is concerned with finding the best alternative and establishing a ranking of all of them. This is not the case here, since there is no interest in finding which are the most important or significant areas, but in determining which are the different policies affecting each of them and to what extent. If there are differences between scores, they denote existent related discrepancy regarding penetration of advanced technologies between areas that are indicating that their adoption is not even. This is important, because supposedly the central government is interested in having a penetration as balanced as possible between all levels of society, however, it needs to know the degree of penetration in each one. Primarily, SIMUS identifies which are significant policies for each area; there may be several, only one, or none. From then on, IOSA finds the degree in which the variation of each policy in funds, resources, and regulations influences each area. The analyst can easily prepare a table from information provided by IOSA where variations experimented by the objective function (or output) are portrayed against simulated changes in the policies (or inputs). These simulations represent different support intensities that the government can adopt for each policy. This makes it possible to ascertain which are the best policies, by examining how penetrations developed in each area using the scores. From here, a set of measures of diverse type may be implemented to improve it. This static procedure is valid for a certain period. Normally, society takes some time for accepting or adopting new technologies, usually not in the short time but over some period, measured in years. Consequently, sometimes after implementation, reality may have changed, then, it is necessary to repeat the process periodically, naturally with updated recent values, and the method applied again to consider the dynamic aspect of the process in society. In this way, it is also possible to check how central originated policies have been positive or negative.

352

14.2.3

14

Some Complex and Uncommon Cases Solved by SIMUS

The Case

This case assumes that a government contemplates six areas in society to investigate for penetration of advanced technologies, deemed paramount for the country; obviously, it is just an example, since undoubtedly many more additional areas can be considered, for instance e-commerce, environment, construction, rural exploitations, ecosystems protection, circular economy, etc. The areas are: 1. 2. 3. 4. 5. 6.

Government Military Industry University School Agriculture

Each area has its own demands, interests, needs, and funding. These areas require evaluation, performed by their analysis through a set of policies. The purpose of this exercise is for determining the policies that contribute the most to facilitate the penetration in the society spectrum, as well as to learn about how it influences each area. These polices are taken from Sanjaya Lall and Morris Teubal (1998) seminal work entitled “Market-Stimulating Technology Policies in Developing Countries: A Framework with Examples from East Asia.” However, other than policies, areas, data, actions, and process, comments are exclusive responsibility of the author of this book. C1. Cost for developing and importing advanced technologies, C2. Promoting local innovation, C3. Identifying, adapting, and operating imported technologies, C4. Modifying imported technologies and adapting to market needs, C5. Promotion of horizontal technology absorption and diffusion, C6. Government supporting the learning process by the market, C7. Government functional intervention, C8. Government vertical intervention, C9. Market-friendly approach. Table 14.6 shows the six areas, the nine policies, the performances values at their intersections, and the action required by each policy. A team of experts using a 1 to 10 scale determined cardinal performance values. Thus, for area “Industry,” for instance, experts consider that at present, cost (C1) for developing new technology is in general very high (score 8). That the sector is not very keen in government promoting local innovation (C2) (score 6), as well as adapting and operating imported technologies (C3) (score 6), nor in modifying imported technology (C4) (score 6).

Government policies Cost for developing new technology Promoting local innovation Identifying, adapting, and operating imported technologies Modifying imported technologies and adapting to market needs Promotion of horizontal technology absorption and diffusion Government supporting the learning process by the market Government functional intervention Government vertical intervention Market-friendly approach

Table 14.6 Initial data

8

6

6 8 8

C7 C8 C9

7 6

6

7

10

10

University (Academia) 8 10 8

C6

6

Industry 8 6 6

10

6

C4

Military 10 8 7

C5

Government 10 7 8

Areas Policy ID C1 C2 C3

8

7

School 6 8 6

4

4

Agriculture 4 5 4

MAX MAX MAX

MAX

MAX

MAX

Policy action MIN MAX MAX

14.2 Case Study: Quantitative Evaluation of Government Policies. . . 353

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Some Complex and Uncommon Cases Solved by SIMUS

Fig. 14.4 Screen capture of SIMUS last screen for ERM result Project Dominance Matrix (PDM)

Dominant sectors Government Government Military Industry University School Agriculture Column sum of subordinated sectors

Subordinated sectors Military Industry University School 0.9

1.9 2.7 1.7 0.2 0.3

2.4 1.7 0.0 0.2

6.8

5.2

PDM Ranking

1.0 1.7

1.0 2.0 2.4

1.0 1.8 2.6 1.7

1.5 0.1 0.1

0.2 0.4

0.4

4.5

6.0

7.5

Agriculture 0.9 1.7 2.3 1.7 0.1

Row sum of dominant sectors 4.9 9.1 12.5 8.4 0.7 1.4

Net dominance –1.9 3.9 8.0 2.4 –6.9 –5.5

6.9

Industry – Military – University – Government – Agriculture - School

Fig. 14.5 Screen capture of SIMUS last screen for PDM result

In these three criteria, performance values are relatively low, and the experts postulate that the inherent cost for acquisition and equipment modification largely depends on the type of industrial activities and degree of automation, and therefore, it is difficult to make an estimate. In general, they think that advanced technologies depend on many factors, and not only on government goodwill. It can be appreciated that there is not too much in government promoting horizontal technology (C5) (score 0), nor enthusiastic about government supporting the learning process in industry (C6) (score 7), nor government functional intervention (C7) (score 7). It is not very keen either in government vertical intervention (C8) (score 6) and not interested in a government-friendly approach in the market (C9) (score 0). They think that the area does not need government intervention, and they prefer to go on their own. These are the opinions of experts in the industrial area who involved 15 CEO of the largest industries in the country. Similar analysis was made for the other areas, with the advice of government top ministers, universities chancellors, as well as from high-level educators, and CEOs of large agribusiness companies. However, experts thought that in general, the government idea is a good one; they were satisfied for being consulted, and eventually assigned each policy the maximization action, except, naturally for cost.

14.2

Case Study: Quantitative Evaluation of Government Policies. . .

355

Starting from Table 14.6, SIMUS gave the results shown in Fig. 14.4 in a screen capture of the ERM which the first solution is given by SIMUS. See also Fig. 14.5 with the result from PDM. Both rankings coincide. Scores are indicated in the blue solid row. Notice the disparity in magnitude of scores, which indicates that by far “Industry” (score 1.18) is the area where advanced technologies have the maximum penetration (probably with imported and own technology). It is followed by “Military” (score 0.66), which in general agrees with the government as evidenced by the performance values of Table 14.6 and with a minimum of (0.04) for “Schools.” This is not surprising because the chancellors in general are not very keen on the introduction of new technologies, by the argument that in many cases, they inhibit the thinking capacity of students since different types of calculators and laptops easily solve problems, and the reasoning is correspondingly waning. To examine policies or objectives, any of them can be selected; however, Z2 is chosen because it is related to the most “important” area: “Industry.” The ERM shows in the “Industry” column that its final score of (1.18) is due to the scores of four policies (in red) Z2 (0.40), Z6 (1.00), Z7 (1.00), and Z8 (0.26). Let us concentrate on Z2; its scores correspond to the four policies C3, C4, C5, and C7 (See Sect. 8.1), whose respective values are (0.75), (0.15), (0.15), and (0.35). That is, these are the four policies, out of nine, that make up the Z2 solution; they are called the “Binding criteria.” Consequently, the remaining policies do not have any effect on Z2, and thus their variation does not affect Z2 evolution. The numbers in the box Zi / Ci, are “Marginal values” or “Shadow prices” (See Sect. 8.1) for each policy, indicating how much the objective Z2 increases when the corresponding policy value increments in one unit. In this way, each time that the DM increments or decrements the value of a policy, the objective function increases or decreases according to its marginal value. Therefore, if the government decides to promote a certain policy assigning more funding, then the objective function will increase according to its marginal value. Since objective functions are tied to areas, it means that any increase in the corresponding objective will vary the area score. Consequently, analyzing all objective functions and their corresponding policies the government can have a picture of how its policies affect each area. To avoid confusion, changes in policies are called “Increments” and “Decrements,” while evolution or variations in the policy function are called “Increases” and “Decreases.” Now that the procedure has been explained, let us continue with this example. First, the analyst chooses the policy he wants to examine, in this case, say it is C3. IOSA performs this effortlessly (See Fig. 14.6). The objective ID is inputted in the “Objective Z” box and selected from a dropdown menu. Then, it will be 2. The policy ID that is, C3, is inputted in the “Criteria” box from a drop-down menu, then it will be 3.

356

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Some Complex and Uncommon Cases Solved by SIMUS

Fig. 14.6 IOSA tablet loaded with data for Z2 and C3 with 0.01 intervals

In the “Automatic increment” box, the DM indicates the intervals he wants to work with. In this case, he/she selects 0.01. He may input only the first value since the software will complete the series. He does the same for decrements or input the intervals he wants, since it may be equal or different for both. In addition, increments/decrements do not need to be uniform. The DM presses the purple (+) button in “Proposed intervals.” The selection that he has made will appear in the box below. He now presses the “Accept” key. When the software finishes computation, which will be shown by an illuminated “Save” key, the DM presses it, and the work will be saved in an Excel sheet in a file under the “Projects” label. Also in this file are the graphics, drawn by IOSA, of the evolution of Z2 for different variations C3, as seen in Fig. 14.7. Since the DM requested the analysis of both, increments and decrements there will be two dissimilar curves, but both starting at the same value of Z2 = 0.85. The curve at the left shows the evolution of Z2 for increments in C3, while the graphic at right depicts the evolution of Z2 for decrements in C3. Below, there are interval values which can be used as control. In the left curve observe that the slope changes after the first increment (break point), and that indicates that C3 has a very little leeway for increments, therefore there is no reason for more increments. That is, the rate of increase of Z2 as a function of C3 is only valid for the first interval. After that, another criterion is considered and it will produce a straight line as long as its shadow price varies within the allowable interval. When SIMUS/IOSA is saved in “Projects” it gives information of Z2 values corresponding to each interval, and then allowing the user to have information as

14.2

Case Study: Quantitative Evaluation of Government Policies. . .

357

Fig. 14.7 Evaluation of objective Z2 for C3 positive and negative increments Table 14.7 SIMUS shows score values for the first increment of 0.01 in C3 ERM Ranking Final Result (SC × NPF)

Industry 0.33

Military 0.65

University 1.18

Government 039

Agriculture 0.04

School 0.1

how they change as seen in Fig. 14.7. Just by hitting one of the “SIMUS” keys at the bottom, the scores for the corresponding interval will be displayed. For instance, for interval 1 the DM presses “SIMUS 1,” and the result will appear as shown in Table 14.7. Therefore, it is possible to know how the alternative scores change. In this example, just by incrementing criterion C3 in 0.01, observe that Government increases from 0.23 to 0.33, Military decreased from 0.66 to 0.65, there is no change for Industry, University, and School, and a small increase from 0.09 to 0.10 for “Agriculture.” Consequently, criterion C3 “Identifying, adapting and operating imported technologies” is important for the Government, and almost irrelevant for the other areas.

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Table 14.8 Evolution of policy Z2 for constant increments of C3 Promote local innovation Government

Military

Industry

University

School

Agriculture

Increases

Decreases

No change

No change

Decreases

Increases

0.85 0.86

0.23 0.33

0.66 0.65

1.18 1.18

0.39 0.39

0.04 0.04

0.09 0.10

+ 0.02

0.86

0.35

0.64

1.18

0.39

0.02

0.10

+ 0.03

0.87

0.36

0.63

1.18

0.39

0.02

0.10

+ 0.04

0.87

0.37

0.63

1.18

0.39

0.02

0.10

+ 0.05

0.88

0.37

0.62

1.18

0.39

0.02

0.09

Z2 values

+ 0.01

Intervals Original

Remember that Z2 was chosen because it has a value (0.40) in the “Industry column,” which is turn has the highest score. When policy Z2 is analyzed for area “Industry” the scores of all other areas change, because they are trade-off values. Then it is possible to build a table that shows the scores of each policy and for each increment and decrement. See Table 14.8. Objective Z2: Promote local innovation. Initial value = 0.85. This table allows for learning how each policy responds to changes. Observe what happens to Z2 and scores for successive 0.01 interval increments of policy C3: 1. 2. 3. 4. 5. 6.

The objective value increases from 0.85 to 0.88 or 3.6%. There is an increase in “Government” from 0.23 to 0.37 or 6%. There is a decrease in “Military” from 0.66 to 0.62, or 6%. There is no change in “Industry.” There is no change in “University.” There is no change in “Agriculture,” except for the fifth interval.

Consequently, incrementing C3 in five intervals increases the “Promote local innovation” policy and “Penetration of advanced technologies” in “Government” and a decrease in “Military” with no changes in other areas. Similarly, if the DM works with decrements, the partial variation can be seen in Table 14.9. Decrementing C3 by 0.01 intervals when studying the “Industry” area: 1. 2. 3. 4.

The policy value decreases from 0.85 to 0.81, or 5% There is a decrease in “Government” from 0.23 to 0.21 or 8.7%. There is an increase in “Military” from 0.66 to 0.73, or 10%. There is an increase in “Industry” from 1.18 to 1.6 or 36%.

14.2

Case Study: Quantitative Evaluation of Government Policies. . .

359

Table 14.9 Evolution of policy Z2 for constant decrements of C3 Promote local innovation Government

Military

Industry

University

School

Agriculture

Intervals

Z2 values

Decreases

Increases

Increases

Decreases

Increases

Decreases

Original

0.85

0.23

0.66

1.18

0.39

0.04

0.09

– 0.01

0.84

0.22

0.67

1.50

0.38

0.07

0.08

– 0.02

0.83

0.22

0.69

1.53

0.37

0.08

0.07

– 0.03

0.83

0.22

0.70

1.55

0.36

0.09

0.05

– 0.04

0.82

0.21

0.71

1.58

0.35

0.10

0.04

– 0.05

0.81

0.21

0.73

1.60

0.35

0.11

0.02

5. There is a decrease in “University” from 0.39 to 0.35 or 10%. 6. There is an increase in “Schools” from 0.04 to 0.11, or 175%. 7. There is a decrease in “Agriculture” from 0.09 to 0.02, or 77%. Notice that there are increases in “Military,” “Industry,” and “Schools” when decrementing C3 and one wonders why. One explanation could be that “Military” for different reasons is reluctant or not very convinced in adapting technology from other countries and they perhaps prefer to develop their own. Regarding “Industry,” it may have several explanations; for instance, foreign technology is not well adapted to the main industrial pattern of the country. Fracking or hydraulic fracture is unsuitable because the country does not have shale reserves in large volumes. Or, car assembly plants cannot use a robotic system due to its cost which is not justified, due to the small size of production. Or perhaps the country’s main production lies in exporting grains or meat, which is still using old and proven technologies. Regarding “Schools,” the reason could be that country experts in education consider that it is not very convenient to introduce, for instance, electronic calculators and tablets. The reason could be that they believe that they diminish the student’s capacity for solving elemental mathematical problems. These results may constitute a reason for the government to investigate the causes for them, and then have corrected or supported. It appears from Fig. 14.7 that Z2 reaches its upper limit faster incrementing than when decrementing. The government of course wants to ensure Z2 increases, because it increases innovation, but now, it appears that innovation is more sensitive to a decreasing acceptance than to its increase. Therefore, the government realizes that a policy pursuing adaptation of import technologies is probably not worth considering.

360

14.2.4

14

Some Complex and Uncommon Cases Solved by SIMUS

Analysis of Different Policies

In Sect. 14.2.2 it was exemplified how a policy is investigated. This section provides results and comments on the other policies. Policy C2 (Promoting local innovation) (already analyzed) • • • •

Strong increase in Government, Strong decrease in Military, Strong decrease in Schools, Slight increase in Agriculture. It appears to be not a beneficial policy. Policy C3 (Identifying, adapting, and operating imported technologies)

• • • •

Slight increase in Military, Sharp increase in Industry, Decrease in University, Slight increase in Schools. It appears to be a policy with medium or regular benefits. Policy C4 (Modifying imported technologies and adapting to market needs)

• • • • •

Slight increase in Government, Slight decrease in Military, Increase in Industry, Strong increase in School, Increase in Agriculture. It appears to be a very good policy. Policy C5 (Promotion of horizontal technology absorption and diffusion)

• • • • •

Strong decrease in Government, Strong increase in Military, Strong decrease in Industry, Strong decrease in Schools, Slight increase in Agriculture. It does not appear to be a beneficial policy. Policy C6 (Government supporting the learning process by the market). Acts together with policy 2. Policy C7 (Government functional intervention). Acts together with policy 2. Policy C8 (Government vertical intervention). Acts together with policy C2. Policy C9 (Market-friendly approach)

• Strong decrease in Government, • Strong increase in Military, • Strong decrease in Industry,

14.3

Case Study: Selecting Hydroelectric Projects in Central Asia

361

• Strong decrease in Schools, • Slight increase in Agriculture. It does not appear to be a beneficial policy. Consequently, according to this analysis the best policies are C4 and C3. It is obvious that restricting import of technology has a larger effect that encouraging it, and that quantitatively its consequences are more significant than in the latter. Consequently, it is beneficial for the country to discourage importing technology.

14.2.5

Conclusion of This Case

This case addresses a concept that it is believed relevant and related to the influence of advanced technologies in a country. Most probably, other researchers have addressed this subject, but this section proposes a methodology, based on MCDM, where diverse areas of the country life are evaluated, founded on a set of policies. However, there is no interest in learning which is the most favored area, but in determining which is the policy or policies that will bring the maximum benefits to the country considering all areas. The proposed mathematical model allows for determining quantitatively the relationship between objectives and policies, and in so doing it decides which the best policy is based on the benefits that it produces in each area. This is illustrated by an example.

14.3 14.3.1

Case Study: Selecting Hydroelectric Projects in Central Asia Background Information

The government of a Central Asia country aims at taking advantage of its mountainous and rough terrain, considering the abundant opportunities for hydroelectric development due to mighty rivers coming from the Himalayan Range. However, what it is an advantage is also a drawback for construction work, because of the high altitude, the very difficult access, the cold weather, and the necessity of wild life preservation. The government’s hydraulic department has spent years in surveying an area of considerable extension and has concluded that there is potential for the construction of 16 dams, and made an inventory of characteristics for each one of them, which are briefly condensed as follows: Site 1: Small basin and with many villages nearby, Site 2: Attractive project, however it has difficult access,

362

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Some Complex and Uncommon Cases Solved by SIMUS

Site 3: Site where a hydro-project is in advanced stage of construction, Site 4: Very narrow and long, which may present difficulties. It also has an active wild life, Site 5: Attractive. Preliminary studies found it economically feasible, especially because the short transmission lines needed to connect to the national electric network, Site 6: Good potential for power generation, however, it is subject to avalanches, Site 7: Very difficult terrain for accessing. It is also a site with abundant aquatic life, Site 8: Area with migratory fish, Site 9: There is risk here due to GLOF (Glacial Lake Outburst Flood), Site 10: Geological fault in the lake behind the dam area, Site 11: The lake behind the dam would flood agricultural land, which is currently heavily cultivated. There could be people opposition, Site 12: More geological research is necessary due to geological diversity in the area. Fish and wild life habitat, Site 13: Steep slope with material that accumulated at the foot (colluviums). There is population in a nearby resort area, Site 14: Economically feasible and with good possibilities because of sound and stable terrain, Site 15: Protected aquatic and wild life. Difficult access, Site 16: Very good accessibility and power generation potential. These sites or alternatives are subject to the following main criteria, but there are also more criteria not specified here, as: C1: Construction cost in millions of dollars, C2: Cost per kW/h (in US$), C3: Installed capacity (MW), C4: Ranking from the technical point of view related to economic evaluation based on benefit/cost analysis, C5: Ranking from the social and environmental point of view. This fact considers benefits for people, as well as disadvantages, and also the inevitable contamination during construction and the threat to the wild life, C6: Number of people that must be relocated and compensated. That is, in the area people live in small villages, normally at the shores of a large river. There is no doubt that the lake foamed behind the dam will flood their villages and farms. This means not only a monetary compensation, but also the necessity to relocate this people and build new dwellings for them. However, there could be people who are not happy with this arrangement and may protest and act against a project, C7: Transmission distance to national network, C8: Installed electric capacity (GW), C9: Geological problems, C10: Risk. In this example it is considered only one value for all risk, which is not very realistic. In a real scenario it would be necessary to have several different

14.3

Case Study: Selecting Hydroelectric Projects in Central Asia

363

criteria for risk such as personal risk, avalanches, work accidents, impossibility to work because of bad or severe weather, etc., C11: Engineering effort. It refers to the technical work needed for designing, planning, scheduling, and controlling the work, C12: People approval for each project, from the point of view of how it may affect their lives, C13: Construction camps. This is a big item; some projects may require only one camp while other may need three or four. These are necessary for workforce housing accommodations as well as catering, sanitary services, electricity, etc. Normally, hydro-projects in high country are located in remote places, sometimes only accessible by helicopters, therefore, everything, with the probable exception of drinking water, has to be transported to the job site. Usually, the work is structured by rotation cycles, that is, a person works seven weeks in a row and gets one week of time off. Remember that this person has to be periodically transported to the place where he was hired; therefore, the logistics is complex and the cost is high. Existing restrictions are as follows: (a) Whatever the selection, the final ranking must consider Site 3, since it is already underway and must be completed, therefore, it is not possible to put it at the same level of the other projects. Consequently, this condition is a restriction that must be incorporated into the Initial Decision Matrix. (b) Projects in Sites 9 and Sites 14 are in the same river basin, one upstream and the other downstream. That is, the project at Site 9 is at a higher altitude than the project at Site 14, but on the same river; consequently, the lake level of the latter depends on the water discharge of the former. This puts a limit to the capacity of the lake at site 14, which must be introduced into the model. (c) Site 14 and Site 16 have a precedence condition since the water discharged from the project at Site 14 is the only input for the project in Site 16. This has also to be modelled. (d) There is a maximum limit for unit cost of generated energy. (e) The total risk has been gaged in a scale from 1 to 5, being the latter, the maximum tolerated. (f) There is minimum limit for the energy to be generated. This limit is established as a function of economies of scale. (g) Due to social reasons, there is a maximum limit for people to be relocated. The Initial Decision Matrix is not reproduced here due to its size, which would make its reading difficult. Figure 14.8 is a capture of the SIMUS screen with the result. The blue solid row depicts the scores. Observe that stipulated restrictions are honored: Site 3: It is in the first place (score 5.22), not because it is the best, but for the reason that it is under construction and thus, it must be selected to complete it,

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Some Complex and Uncommon Cases Solved by SIMUS Efficient Results Matrix (ERM) Normalized

Z1 Z2 Z3 Z4 Z5 Z6 Z7. Z8 Z9 Z10 Z11 Z12 Z13

Site 1 Site 2 Site 3 0.51 0.5 0.47 0.25 0.2 0.2 0.51 0.63 0.47 0.23 0.31 0.31 0.09 0.2 0.47 0.47

Sum of columns (SC) 0.2 Participation Factor (PF) 2 Norm. Participation Factor (NPF) 0.15 Final Result (SC x NPF 0.03 ERM Ranking

0.4 2 0.15 0.06

5.22 13 1 5.22

Site 4 Site 5 Site 6 Site 7 Site 8 Site 9 Site 10 Site 11 Site 12 Site 13 Site 14Site 15 Site 16 0.38 0.11 0.01 0.5 0.47 0.07 0.25 0.06 0.19 0.25 0.2 0.2 0.2 0.25 0.49 0.37 0.49 0.47 0.07 0.23 0.06 0.01 0.23 0.23 0.08 0.29 0.2 0.20. 0.2 0.04 0.04 0.17 0.02 0.17 0.17 0.17 0.02 0.17 0.17 1.62 0.2 0.2 5 1 1 0.38 0.008 0.08 0.62 0.02 0.02

0.92 8 0.62 0.57

0.78 1.11 0.23 4 5 3 0.31 0.380.23 0.08 0.24 0.43 0.05

0.19 1 0.08 0.01

0.2 0 0 0.02

0 0 0 0

0.38 0.97 0.38 3 3 3 0.23 0.023 0.23 0.09 0.22 0.09

Site 3 – Site 4 – Site 7 – Site 9 – Site 8 – Site 15 – Site 14 – Site 16 – Site 2 – Site 10 – Site 1 – Site 6 – Site 12 – Site 11 – Site 13

Fig. 14.8 Capture of SIMUS final screen with results

Site 4: It is the second best with a score of 0.62 (in fact it is the best, because Site 3 was not selected but imposed). Site 7: It is in the third place with a score of 0.57, Site 14: Has the same score (0.09) than Site 16, since both are complementary, as instructed.

14.3.2

Conclusion of This Case

This case illustrates in a very simplified form a complex problem, in selecting which hydro-project to develop, out of a portfolio of potential undertakings. Its only purpose is to show in an actual case conditions and restrictions that always exist and must be entered into the model.

14.4

Case Study: Community Infrastructure Upgrading for Villages in Ghana

Abstract Years ago, the World Bank developed a plan to improve infrastructure in poor areas of five cities in Ghana. The plan aimed at providing basic infrastructure services such as water, sewerage, and electricity through soft loans to their inhabitants, subject to their capacity to pay them back. SIMUS was applied to select the areas that will receive the largest help. It is worth mentioning that the present work was nominated in 2002 for the Equator Prize, sponsored by the United Nation Development Program Equator

14.4

Case Study: Community Infrastructure Upgrading for Villages in Ghana

365

Initiative Knowledge Base, as a “A proposal by a consultant to improve access to infrastructure; apparently by relocating local communities”.

14.4.1

Background Information

At the end of the 20th Century the World Bank developed a soft loan program to upgrade infrastructure in five Ghanaian cities, namely: Accra, Sekondi-Takoradi, Kumasi, Tema, and Tamale. These cities have differences in population, as well as commercial and industrial activities. In applying SIMUS, the method made a selection of urban areas with the objective of providing the benefits of water, sewerage, and electricity to the maximum possible number of people and at the lowest cost. The plan aimed at: • Provision of basic infrastructure, i.e., drinking water, sanitation, as well as electricity to African communities that did not have those services, • Conditions vary from one urban area to another. For instance, there was a large variation in the cost of providing these services per hectare, • Priorities were established based on certain activities in some areas; Kumasi, for instance, is an urban center performing as an important hub for automotive and agriculture machinery repairs, • Financing comes from the World Bank. Population willingness to pay is especially considered in this case, as well as limits for hectares for upgrading and limits for per loan payments, • The five urban centers have different sizes, densities, and activities as well as varied abilities to pay. Most data have been taken from the World Bank publication: “Towards a National Slum Upgrading Program for Ghana” https://documents1.worldbank.org/curated/pt/989511468771578095/pdf/multipage.pdf

14.4.2

Areas and Data

Table 14.10 details data for the five areas. Observe that there are different values for each one regarding cost for upgrading, maximum number of hectares to be developed, density, etc., and thus, reflecting the characteristics of each one. However, the following conditions apply to them all. • • • •

Annual interest rate: 4% Payback period in years: 5 Population density/ha: 470 Maximum total payment/capita 51

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Table 14.10 Characteristics and data for each area Accra Maximum cost for upgrading (US$/ha) Minimum number of hectares to be developed Maximum number of hectares to be developed Density (people/ha) Total population in the area Maximum total investment (US$) Ability to pay (US$/month-person) Annual interest rate Cost per capita Annual payment per hectare (US$/ha) Cash flow (investment annual payments/hectare)

Sekondi-Takoradi Maximum cost for upgrading (US$/ha) Minimum number of hectares to be developed Maximum number of hectares to be developed Density (people/ha) Total population in the area Maximum total investment (US$) Ability to pay (US$/month-person) Annual interest rate Cost per capita Annual payment per hectare (US$/ha) Cash flow (investment annual payments)/hectare

Kumasi Maximum cost for upgrading (US$/ha) Minimum number of hectares to be developed Maximum number of hectares to be developed Density (people/ha) Total population in the area Maximum total investment (US$) Ability to pay (US$/month-person) Annual interest rate Annual payment per hectare (US$/ha) Cash flow (investment annual payments/hectare)

Year 1 18,860 26 35 427 14,9450 574,000 0.72 0.04 43 3684 -16,400 NPV = 0 Year 1 54 72 450 32,400 1,471,238 0.85 0.04 51 4605 -20,500 NPV = 0 Year 1 20,470 71 94 396 37,224 1,673,200 0.84 0.04 3998 -17,800 NPV = 0

2

3

4

5

Notice criteria duality

3684

3684

3684

3684

2

3

4

5

Notice criteria duality

4605

4605

4605

4605

2

3

4

5

Notice criteria duality

3998

3998

3998

3998

14.4

Case Study: Community Infrastructure Upgrading for Villages in Ghana

367

Cash flow is also computed for each area, and the calculated Net Present Value (NPV) for all of them is zero. Guadalajara Tema Maximum cost for upgrading (US$/ha) Minimum number of hectares to be developed Maximum number of hectares to be developed Density (people/ha) Total population in the area Maximum total investment (US$) Ability to pay (US$/month-person) Annual interest rate Cost per capita Annual payment per hectare (US$/ha) Cash flow (investment annual payments/hectare)

Tamale Maximum cost for upgrading (US$/ha) Minimum number of hectares to be developed Maximum number of hectares to be developed Density (people/ha) Total population in the area Maximum total investment (US$) Ability to pay (US$/month-person) Annual interest rate Cost per capita Annual payment per hectare (US$/ha) Cash flow (investment annual payments/hectare)

Year 1 32 42 517 21,714 882,000 0.76 0.04 46 4717 -21,000 NPV = 0

Year 1 53 70 512 35,840 1,575,000 0.82 0.04 49 5054 -22,500 NPV = 0

2

3

4

5

Notice criteria duality

4717

4717

4717

4717

2

3

4

5

Notice criteria duality

5054

5054

5054

5054

Figure 14.9 shows final results when SIMUS optimizes only one objective: “Maximize ability to pay.” Number of hectares to be developed in each city is shown in the blue solid row. Notice how all restrictions are honored by comparing the RHS values with the LHS values, and in accordance with the mathematical symbol indicating the type of action.

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Criteria Max hectares in Accra Min hectares in Accra Max hectares in Sekondi-Takoradi Min hectares in Sekondi-Takoradi Max hectares in Kumasi Min hectares in Kumasi Max hectares in Tema Min hectares in Tema Max hectares in Tamale Min hectares in Tamale Total cost in Accra (US$) Total cost in Sekondi-Takoradi (US$) Total cost in Kumasi (US$) Total cost in Tema (US$) Total cost in Tamale (US$) Per capita cost in Accra (US$)/ha Per capita cost in Sekondi-Tak. (US$)/ha Per capita cost in Kumasi (US$)/ha Per capita cost in Tema (US$)/ha Per capita cost in Tamale (US$)/ha Pop. density in Accra (people/ha) Pop. density in Sekondi-Tak. (people/ha) Pop. density in Kumasi (people/ha) Pop. density in Tema (people/ha) Pop. density in Tamale (people/ha) Total funds available this period (US$)

Some Complex and Uncommon Cases Solved by SIMUS Urban areas Sekondi Accra Takoradi Kumasi Tema Tamale 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 427 450 396 517 512 1 1 1 1 1

LHS 35 35 72 72 94 94 38 38 64 64 6.E+05 1.E+06 2.E+06 8.E+05 1.E+06 43 51 50 41 45 14945 32295 37224 19740 32900 6.E+06

RHS Restrictions and Limits d 35 t 26.25 d 72 t 54 d 94 t 70.5 d 42 t 31.5 d 70 t 52.5 d 6.E+05 d 1.E+06 d 2.E+06 d 9.E+05 d 2.E+06 d 51 d 51 d 51 d 51 d 51 d 16450 d 33840 d 44180 d 19740 d 32900 d 6.E+06

Sekondi Accra Takoradi Kumasi Tema Tamale 72 94 38 64 Hectares to be developed 35 Total number of people 14945 32295 37224 19740 32900 137,104 Total cost 6.E+05 1.E+06 2.E+06 8.E+05 1.E+06 6.E+06 Number of hectares available 313 Number of hectares assigned 303 Land use efficiency 97%

Fig. 14.9 Reproduction of SIMUS last screen

14.4.2.1

Analysis

Regarding People The selected plan would benefit 137,104 people out of 147,110 people living in the five areas, that is, a 93.2%. Regarding Hectares Kumasi: 94 hectares, the largest, benefitting 37,224 people, Sekondi-Takoradi: 72 hectares, benefitting 32,295 people,

14.4

Case Study: Community Infrastructure Upgrading for Villages in Ghana

369

Table 14.11 Exploitation of available land Urban area Kumasi SekondiTakoradi Tamale Tema Accra

Size of areas Maximum 94 72 70 42 35 Total available hectares: 313

Minimum 71 54

Assigned 94 72

Exploitation of available land 100% 100%

53 32 26

64 38 35 Total assigned hectares: 303

85.7% 95.2% 100% Total available land exploited: 97%

Tamale: 64 hectares, benefitting 32,900 people, Tema: 38 hectares, benefitting 19,740 people, Accra: 35 hectares, benefitting 14,945 people. Table 14.11 indicates how the result makes the best possible use of land. Three areas Kumasi, Sekondi-Takoradi and Accra benefit with a 100% development of their available land. Tamale gets 85.7% and Tema 95.2%. On a total of 313 available hectares, 303 can be developed, i.e., the method makes possible to develop almost 97% of available land. Regarding Costs (Table 14.10) The World Bank estimated cost/ha for upgrading each area, and from it, assessed the necessary investment, considering the average between minimum and maximum number of hectares in each one, since the number of hectares is different for each site. Total estimated cost or investment in each site is compared with values yielded my SIMUS, as follows: Kumasi: US$ 1,673,200. It is exactly as the World Bank investment plan, Sekondi-Takoradi: US$1,471,238. It is 0.32% less than the World Bank investment plan, Tamale: US$ 1,445,801. It is 8.23% less than World Bank investment plan, Tema: US$ 801,818. It is 9.09% less than World Bank investment plan, Accra: US$ 574,000. It is exactly as the World Bank investment plan, Total cost available by the World Bank: US$ 6,180,200, Total calculated cost: US$ 5,966,057, Total loan used is 96.5%.

14.4.3

Conclusion for This Case

An actual and complex case was solved using SIMUS. It related to the provision of basic infrastructure to five low-income areas in Ghana. The whole program

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objective, financed by the World Bank, was to select areas to develop, based on their respective ability to pay back the loan. A financial analysis was included to compute the cash flow in each settlement. The result produced the number of hectares to develop and reached a remarkable 97% of land use and utilizes 96.5% of available funding.

14.5

Case Study: Urban Development Study for the Extended Urban Zone of Guadalajara, According to Sustainability Indicators, Mexico

Abstract This study was nominated by the Stockholm Partnerships for Sustainable Cities organization, for its presentation in an International Conference in said city, sponsored by the City of Stockholm (Stockholm Partnerships, http://dictionary. sensagent.com/stockholm%20partnerships%20for%20sustainable%20cities/enen/, 2002). It refers to selecting the most appropriate projects out of 27 different urban infrastructure undertakings in the city of Guadalajara (México) and 7 satellite cities. These projects were grouped in clusters and subject to 78 conditional criteria, some of them including projects shared by several cities. SIMUS was applied with the sole objective of minimizing costs and selected 17 projects.

14.5.1

Background Information

Guadalajara is México’s second largest city. The Great Guadalajara conurbation is formed by the city of Guadalajara (Central City) and seven municipalities in the outskirts of the central city, with a population of 4.6 million. In 2001 this conglomerate decided to join efforts to solve infrastructure problems that affected the Central City and its Satellites. After a thorough study by the Urban Development Secretariat (SEDEUR) (2001), five areas were identified as follows: 1. 2. 3. 4. 5.

Thoroughfares, Sanitation, Storm water manifolds, Transportation, Domestic garbage.

27 projects were preselected by the SEDEUR with the study aiming at selecting the best projects subject to 78 criteria. These were conditional criteria, since they established a series of conditions that selected projects should comply. The first condition (Linking projects and municipalities) established that the final solution should contemplate a certain number of projects for each municipality, in

14.5

Case Study: Urban Development Study for the Extended Urban Zone. . .

371

order to have a fair share of available resources. This number was established and agreed through meetings with the different City Halls and was based on the population of each city; each city was given a minimum number of projects. To comply with this condition, eight criteria were used calling for minimization. Minimization in this content means that at a minimum, there should be certain number of projects for each city. The second condition (Linking cities with projects sharing) was related to certain infrastructure projects that were shared by several cities; it was specified the minimum number of cities for each project. The purpose was to avoid developing shared projects that do not affect certain cities. To comply with this condition 27 criteria were used and calling for minimization. Similar to the last point, this meant that each project must consider as a minimum a certain quantity of cities. The third condition (Linking projects and costs) was that the budget for all projects should be equal to a certain amount of funds; however, from the beginning it was known that this amount was short in 7.5% of total funds needed. To express this condition a new criterion was needed. The fourth condition (Linking areas and projects). Eight criteria called for minimization establishing the minimum number of projects related with each area. The fifth condition (Linking budget of each city and limit). That is, it meant that the amount to be spent for each city should be exactly the amount agreed. To comply with this condition, 6 criteria were needed. The sixth condition (Linking compliance to number of sustainable indicators related to each project). It called for minimization; meaning that at a minimum, each selected project should be related to a specified number of sustainable indicators, and this was established by experts. To comply with this condition 28 criteria were needed. In addition, the Initial Decision Matrix was a binary matrix where a “1” indicated relationship or membership between a project and a criterion, and “0” indicated its absence. When SIMUS was used to analyze this complex scenario it produced binary results, or a binary list of projects, that is, a “1” represents a project chosen, and a “0” its rejection. In total 17 projects were selected; obviously, no ranking is needed here. The data inputted to the decision matrix is succinctly enumerated here.

14.5.1.1

Projects

With 27 projects and 5 areas, it was necessary to work with clusters. These clusters were as follows: First cluster: Thoroughfares 1. 2. 3. 4.

Santa Anita and highway to Morelia, Low-cost thoroughfares, Solectron overpass, Road junction in Revolución Avenue,

372

5. 6. 7. 8. 9. 10. 11. 12.

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Some Complex and Uncommon Cases Solved by SIMUS

Overpass in Mariano Otero and Periférico, Vial solution in Glorieta Colón, Vial in López Mateos Av., Improving vial accesses, Urbanización Malecón Av., El Salto-Juanacatlán, 11 San Agustín, Improving road safety in Ixtlahuacán de los Membrillos.

Second cluster: Sanitation 13. 14. 15. 16. 17. 18. 19. 20.

San Gaspar sanitation manifold, Guadalupe-Gallo sanitation manifold, Cleaning El Ahogado creek, Cleaning Las Pintas channel, Cleaning and maintenance of dams and gabions, Topographic survey for the El Ahogado creek, Studies and projects for sanitation of El Ahogado creek, Channelling of rain water in 8 de Octubre Av.

Third cluster: Storm water manifolds. 21. 22. 23. 24. 25.

Poniente manifold-interceptor, Metropolitan Plan for manifolds, Guadalupe-Gallo, Atemajac manifold, San Juan de Dios manifold.

Fourth cluster: Transportation. 26. Origin-Destination study Fifth cluster: Domestic garbage. 27. Acquisition of a tire shredder. 14.5.1.2

Criteria

The criteria were also grouped in the following clusters: • • • • • •

8 municipalities, 27 projects shared by several cities, 1 Maximum amount of funds, 8 Maximum total available funds, 5 Different areas to consider, 23 Sustainable indicators applicable to each project as: 1. Land use, 2. Informal employment,

14.5

Case Study: Urban Development Study for the Extended Urban Zone. . .

3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

373

Services in dwellings, Modal transportation, Average travel time, Infrastructure investment, Solid waste generation, Disposal of solid waste, Houses destruction by disasters, Capital spending by local government, Employees in local government, Spending in repetitive contracts, Multiplier because of site development, Spending in dwelling infrastructure, Bicycle lanes, Bicycle users, Walking areas, Public transportation passengers, Total distance by public transportation, Income from public transportation, Cost of public transportation, Investment in public transportation, Contamination by public transportation.

14.5.1.3

Project by Municipalities Considering

1. Guadalajara, 2. Tlaquepaque, 3. Tonalá, 4. Zapopan, 5. El Salto, 6. Ixtlahuacán de los Membrillos, 7. Juanacatlán, 8. Tlajomulco de Zúñiga.

14.5.1.4

Projects that Are Shared for more than One Municipality

1. Santa Anita and highway to Morelia, 2. Low-cost thoroughfares, 3. Solectrón overpass, 4. Road junction in Revolución Avenue, 5. Overpass in Mariano Otero and Periférico, (continued)

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6. Vial solution in Glorieta Colón, 7. Vial in Lopez Mateos Av., 8. Improving vial accesses, 9. Urbanization Malecón Av.

14.5.1.5

Maximum Amounts Available for Municipality Considering

1. Guadalajara, 2. Tlaquepaque, 3. Tonalá, 4. Zapopan, 5. El Salto, 6. Ixtlahuacán de los Membrillos, 7. Juanacatlán, 8. Tlajomulco de Zúñiga.

Thus, and remembering that the “≥” symbol means “Greater or Equal than”: For the first cluster it uses the “≥” symbol. For the second cluster it uses the “≥” symbol. For the third cluster it uses the “=” symbol. For the fourth cluster it uses the “=” symbol. For the fifth cluster it uses the “=” symbol. For the sixth cluster it uses the “≥” symbol. Due to the size of the matrix, it has been partitioned showing only the first five projects and the respective LHS and RHS values. By the same token, only 24 criteria are shown, but the result naturally considers the whole 27 x 78 matrix.

14.5.1.6

Result

Figure 14.10 (reduced) shows that SIMUS selects only 17 projects because those are the only ones that satisfy all criteria. The result is: Projects selected: 1-4-5-6-7-10-11-13-14-19-20-21-23-24-25-26-27 All scores are binary and all of them have the unit value. All these results are optimum. Just to make sure that this is the case here, the DM has to compare the LHS column versus the RHS column. The values for each criterion must comply with the symbol. For instance, look at the ninth criterion on the left, the “Entronque Santa Ana y carretera a Morelia.” The requirement is to have 6 cities involved (RHS), and the method found that effectively 6 cities are selected (LHS). It doesn’t necessarily need to be the same value as in this case, if

14.5

Case Study: Urban Development Study for the Extended Urban Zone. . .

Thoroughfares Projects by areas Santa Ana Low Over Road PROJECTS: & Morelia cost pass junction in road to thorough- Solec- Revolución Morelia fares trón Avenue Project #: 1 2 3 4 Cost per project (x 000) 1,700 10,400 75,000 170,000 Asignación a cada ciudad CRITERIA Projects by municipalities Guadalajara 1 Tlaquepaque 1 1 1 Tonalá 1 Zapopan 1 1 El Salto Ixtlahuacán de los Membrillos Juanacatlán Tlajomulco de Zúñiga 1 1 Projects shared by municipalities SantaAna & Morelia road to Morelia Low cost thoroughfares Overpass Solectrón Road junction in Revolución Av. Overpass M. Otero and Periférico Vial solutionin Colon roundabout Roadway in Lopez Mateos Av. Improvement of vial access

Overpass M. Otero and Periférico 5 100,000

1

Max.amount for all municipalities Guadalajara Tlaquepaque Tonalá Zapopán El Salto Ixtlahuacán de los Membrillos Juanacatlán Tlajomulco de Zúñiga

375

Total amount projects x (000) 674,000

Total available projects x (000) 624,000

Number of projects after computaion LHS Choose: 7 at least 7 at least 4 at least 8 at least 4 at least 2 at least 2 at least 6 at least

Number of projects requested RHS t t t t t t t t

6 at least 4 at least 6 at least 4 at least 3 at least 2 at least 2 at least 5 at least Investment after computation x (000) LHS 674,000 277,680 79,872 56,784 169,104 14,352 3,744 1,872 20,592

t 6 MIN. t 4 MIN. t 6 MIN. t 4 MIN. t 3 MIN. t 2 MIN. t 2 MIN. t 5 MIN. Investment estimated x (000) RHS t 624,000 t 277,680 t 79,872 t 56,784 t 169,104 t 14,352 t 3,744 t 1,872 t 20,592

3 2 3 3 1 1 1 1

MIN. MIN. MIN. MIN. MIN. MIN. MIN. MIN.

Santa Ana Low Over Road Overpass & Morelia cost pass junction in M. Otero road to thorough- Solec- Revolución and Morelia fares trón Avenue Periférico 1 2 3 4 5 Projects to execute

Fig. 14.10 Partial initial decision matrix and final result

LHS shows a value larger than 6 is correct, but should not be lower than 6. However, if even one comparison is not honored, the result is not optimum. The blue solid row shows in “1 s” the projects to execute. Remember that the matrix has been partitioned, and thus, this result is part of the total.

14.5.2

Conclusion of This Case

An actual complex project was described. Its main characteristics are not only its size, 27 projects and 78 criteria, but the fact that these criteria were conditional, established as requirements: • Select at least a certain number of projects that must comply with each criterion, as per the DM wishes,

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• Select at least a certain number of cities that must comply with each criterion, in order to get a fair share of projects, • Use only the amount of funds assigned to each city, as per agreement between City Halls. • Select at least a certain number of projects that must fulfill each sustainable indicator criterion, as per expert’s opinion. SIMUS gave a list of 17 projects each one complying with all criteria requirements and with very stiff conditions, as happens in real-world projects.

14.6

Case Study: Selection of the Best Route Between an Airport and the City Downtown

Abstract The selection of a route amongst several is a very well-known problem solved by MCDM methods. Normally, a finite number of preselected alternatives or routes are subject to a set of criteria; the solution identifies the route that best satisfies the DMs. That is, it starts with well investigated and analyzed potential routes and with performance values for each alternative. Also, the criteria are well defined, in the sense that distances, costs, speed, orography, etc., are well defined.

14.6.1

Background Information

The case presented here is different to the normal problem mentioned above. In this case, there are not preselected routes or dedicated corridors between two points, but several potential paths formed by a series of roads, avenues and highways, connected by road junctions and roundabouts; that is, a route can be formed combining these different thoroughfares, already built. The problem consists then in “fabricating” the best route using several of these thoroughfares. Naturally, there could exist some combinations that are preferred because they are more direct, or with little traffic or a higher speed limit. The objective of this case is, given a set of thoroughfares, junctions, etc., to select some of them to connect the airport and the city. Once this is done, there are different construction works to be done as adapting and improving road junctions by building supporting structures such as bridges, viaducts, tunnels, etc. or coordinating street traffic lights by a synchronization called a “green wave,” etc. It is necessary to also consider that certain roads and avenues are not only in a path but in several, and therefore, the objective is to cover the distance between the airport and downtown, as fast as possible. This same principle is routinely employed by firefighters when they run to fight a fire. They do not look for a direct way, because it probably does not exist, but for the

14.6

Case Study: Selection of the Best Route Between an Airport and the. . .

377

best combination of streets and avenues to reach the site of the fire as soon as possible. The problem is that constructions cannot be duplicated; that is, if a junction is built to join two avenues, it does not make sense to build another similar junction on an avenue that is not on the selected route. However, if an avenue is, for instance, widened to four lanes, it pays to execute the same job if the avenue that is its continuation has only two lanes, because if not, there will be bottlenecks in the second one. Therefore, if one improvement is performed in the first avenue, the same or larger must be executed in the second. The second aspect, i.e., to determine which works are not to be replicated if unneeded, is the most difficult element to address in solving an MCDM problem of this nature, since it involves a series of conditioning that must be inputted in the Initial Decision Matrix subject to the same criteria. It is necessary to specify that if junction A is selected, then junction B which is in another path should not be and vice versa. For the same token, if junction C is on the same path as junction A, then it has to be considered for construction.

14.6.2

The Case

Figure 14.11 shows the different thoroughfares available to connect the downtown area and the city airport. As mentioned, there are seven most frequently used paths between these two extreme points, as follows, and indicated in black dashed lines. 1. Airport - Junction 2 - Roundabout - Junction 3 - Junction 4 (Alternative 2) - Road to be upgraded - Bridge on the river - Downtown. 2. Airport - Junction 2 – Roundabout - Junction 3 - Junction 4 (Alternative 1) Junction 6 - Junction 7 - Underpass in railway tracks - Downtown. 3. Airport - Junction 2 - Roundabout - Junction 4 (Alternative 2) - Road to be upgraded - Bridge on the river - Downtown 4. Airport - Junction 2 – Roundabout - Junction 4 (Alternative 1) - Junction 6 Junction 7 - Underpass in railways tracks - Downtown 5. Airport - Junction 1 - Junction 4 (Alternative 2) - Road to be upgraded - Bridge on the river - Downtown 6. Airport - Junction 1- Junction 4 (Alternative 1) - Junction 6 - Junction 7 Underpass in railway tracks - Downtown 7. Airport - Junction 1 - Junction 5 - Junction 6 - Junction 7 - Underpass in railways tracks - Downtown. These different routes are subject to the normal set of criteria regarding construction; however, there are additional criteria that in reality are conditioners. Thus: (a) Junction 1 and Junction 4 (Alternative 1) are inclusive, that is, if junction 1 is built, then junction 4 (Alt. 1) has also to be built,

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Some Complex and Uncommon Cases Solved by SIMUS

Airport

Junction 2 and Junction 1 are mutually exclusive Junction 2

Junction 3 Roundabout

Junction 1 Junct. 1 and Junct.4 (Alt. 1) are inclusive Junct. 1 and Junct. 4 (Alt. 2) are exclusive

Alt. 1 and Alt. 2 are mutually exclusive Junct. 4 alt.1 Junct. 4 alt.2 Road to be upgraded River Downtown

Junction 5

d Bridge

Railw

ay

Underpass Junction 6 Junct. 4 (Alt. 1) and Junction 6 are inclusive Underpass and Junction 7 are inclusive Junction 7 Junct. 6 and Junct. 7 are inclusive Solid red= Selected route Dashed black:= Potential thoroughfares * bing.com/images

Fig. 14.11 Shows the complete initial decision matrix for this problem

(b) Junction 4 and Alternative 1 are also inclusive as well as the bridge over the river, (c) Junction 4 and Alternative 2 are also inclusive, (d) Alternatives 1 and 2 in Junction 4 are exclusive. That is construction of one of them rules out construction of the other, (e) Junction 6, Junction 7, and the underpass are inclusive, (f) Junction 1 and Junction 2 are exclusive, Figure 14.12 shows the initials decision matrix. It is divided into two matrices, upper and lower. The upper includes the 7 criteria that establish conditionings as detailed above. The lower is formed by 8 criteria as follows: • • • • • • • •

Vehicle traffic/day, Expropriations to be done, Signalling, Construction of storm sewerage, Lighting, Earthwork, Additional investment for complementary works, Total investment per junction.

14.6

Case Study: Selection of the Best Route Between an Airport and the. . .

379

Minimize investment 3,569,780 2,256,920 1,096,362 3,045,896 7,329,980 3,256,980 282,300 2,569,820 459,300 2,236,785 Road junctions ID Junction 1 Junction 2 Junction 3 Junction 4 Road upg. Junction 4 Junction 5 Junction 6 Junction 7 Underpass Alt. 1 and bridge Alt. 2 a Junct. 1 and Junct. 4 Alt. 1 INCLUSIVE b Junct.4 Alt. 1 and Junct. 6 INCLUSIVE c Alt. 1 and Alt 2 in Junct. 4 EXCLUSIVE d Junct. 6 and 7, Underpass. INCLUSIVE e Junct. 1 and Junct. 2 EXCLUSIVE f Junct.6 and Junct.7 INCLUSIVE g Alt. 1, Alt. 2, and Junct. 5 EXCLUSIVE

Traffic delay Expropriations Signalling Rain storm sewerage Lighting Earthwork Sum of additional investments Total investments

1

1 1 1 1

1

1

1

1

1

1

1

1

1 This submatrix is the conditioning or membership matrix 1

52,611 39,845 358,930 3,590,000 256,420 125,699 425,800 228,963 526,930 963,111 1,411,089 4,532,891 3,668,009

1

1

0

1

1

This submatrix is the normal initial decision matrix 42,989 198,996 27,280 88,500 275,698 289,000 450,000 256,000 100,000 102,000 75,000 245,896 896,321 796,354 1,236,542 89,745 242,130 250,000 259,862 459,863 450,236 345,660 70,000 1,218,088 1,443,364 953,944 1,764,260 2,868,425 2,314,450 4,489,260 8,193,068 5,021,240 3,150,725

0

1

SIMUS results 0 0

0

84,620 55,700 1,236,589 456,987 562,314 569,872

511,927 2,143,576 1,593,186 125,699 236,987 904,917 1,236,987 231,842 1,919,065 2,120,513 2,319,548 351,844 7,198,379 4,690,333 2,778,848 125,896 16,491,331

1

1

120,000

LHS 2 2 1 3 1 2 1

= = = = = = =

RHS 2 2 1 3 1 2 1

t d d d d d d d

MIN MIN MIN MIN MIN MIN MIN MIN

1

Fig. 14.12 Conditioning and initial decision matrix with SIMUS results

The top of the table details the only objective to be optimized, in this case minimizing investments. Using SIMUS, there is no problem in adding as many other objectives as wished, for instance “Minimize investment for additional work,” but here the City Hall is only interested in total investments. Notice that the performance factors for the top of the matrix are all ones. These unit values indicate that there is a relationship or membership between each alternative and each of these criteria. For instance, there is a “1” at each intersection of Junction 1 and Junction 4, (Alt. 1) with a criterion that calls for equality between them. Consequently, the RHS or requirement must have the value 2 and thus indicating that for said criterion, there must be two selections. In another case, for instance in the third criterion, there is “1” at the intersections of Junction 4 (Alt. 1) and Junction 4 (Alt. 2) with a criterion that also calls for equality, however the RHS is 1, meaning that for said criterion only one junction must be selected. As another example, criterion 4 has also “1 s” at the intersection of Junction 6, Junction 7 and the railway bridge; the number 3 is placed in the RHS, meaning that for this criterion the three options or alternatives are inclusive, and therefore, the three must be executed. The lower portion of this table is the normal in MCDM problems. For instance, criterion “Traffic of vehicles per day” calls for minimization, and thus, the symbol is “≥” is used, meaning “As a minimum the traffic must be 485,230 vehicles.” The other seven criteria call for maximization and then the symbol “≤” is used, meaning “Equal or less than a certain upper limit.” Then for a criterion such as “Earthwork” it means that there is limit of US$ 2,861,655 for earthwork. When the system is solved, the result is shown in the blue solid row of Table 14.11. The result indicates that the best route, indicated in binary format, is: Junction 1 – Junction 4 (Alt. 1) – Junction 6 – Junction 7 and underpass (option 8). This path is indicated in red solid line.

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Some Complex and Uncommon Cases Solved by SIMUS

In addition, SIMUS indicates that the cost of the objective function Z to Minimize Investment is 16,491,331 Euros, which is less than half the original budget of 32,838,824 Euros. This is the optimum result, meaning that it is the best combination of thoroughfares between the airport and city downtown.

14.6.3

Conclusion of This Case

A complex urban case is presented to determine the best route between a city airport and the city center; however, it is different from selecting a route between several existing alternatives. In this example, it is assumed, as reality shows, that there are many different ways to travel from the airport to the city core and vice versa, and the objective was to combine different options to select the best mix of thoroughfares, cost-wise. The particularity of this example lies in the fact that there are no initial alternatives, and then, the best alternative must be “constructed” based on criteria and conditions if inclusivity and exclusivity between existing routes can be combined.

References Lall, S., & Teubal, M. (1998). Market-stimulating technology policies in developing countries: A framework with examples from East Asia. World Development., 26(8), 1369–1385. SEDEUR. (2001). Secretaría de Infraestructura y Obra Pública. Gobierno del Estado de Jalisco, México. Accessed from https://siop.jalisco.gob.mx/

Appendix

The Simplex Algorithm: Its Analysis—Progressive Partial Solutions Linear Programming is an iterative mathematical procedure to optimize an objective function subject to a set of criteria and is solved using the Simplex algorithm. In each iteration, the algorithm tries to improve the result of the precedent objective function until no more improvement is possible. When this happens, the Simplex has reached the optimum solution, which is also Pareto efficient. As mentioned, the Simplex starts in the origin of coordinates and in each iteration, it determines and enters a new basic variable or project and identifies and removes another, and thus keeping the coordinate system dimensions constant. Let us see now how the Simplex works and how it selects an entering project. This is the heart of the problem and that precludes selecting a project worse than others, and then preventing RR. Table A.1 shows the initial data of the example that was proposed in Chap. 6 and solved graphically (Fig. 6.3), consisting in selecting the best project between two sources of renewable energy: Solar (x1) and Photovoltaic (x2) for an undertaking aimed at building a renewable energy power plant. Its elements are: Z: Is the objective function; in this case, it calls for minimizing the total cost. Its equation is Z = 0.72 x1 + 0.68 x2 and then it is necessary to determine the values for x1 and x2 that make that possible. Cj: Cost related to each project, then: (C1 = 0.72), unit cost for project x1. (C2 = 0.68) Unit cost for project x2 Criteria = Inequalities. As can be seen, there are four inequalities that need to be converted into equalities. To attain this purpose a slack variable “+S” is added to the “≤” type © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 N. Munier, Strategic Approach in Multi-Criteria Decision Making, International Series in Operations Research & Management Science 351, https://doi.org/10.1007/978-3-031-44453-1

381

382

Appendix

Table A.1 Initial data

Unit cost (Cj)

Projects or projects Solar energy Photovoltaic x2 x1 0.72 0.68

Criteria Efficiency index Financial index Land use index Generation index

Project or projects contributions (aij) or performance values 0.85 0.75 0.78 0.98 0.92 0.65 0.99 0.60

Objective function Z = 0.72 x1 + 0.68 x2 (MIN)

Action MAX MIN MAX MIN

Action symbol ≤ ≥ ≤ ≥

Constant value (B) 1 0.84 0.94 0.80

inequations (calling for maximization), and another slack variable “-S” is added to the “≥” inequations (calling for minimization) as well as an artificial variable “+A” with a very high value attached (M). Thus: S1 and S2: Positive/Negative slack variables to convert inequations into equations. A1 to A2: Positive artificial values to be added to inequations calling for minimization. Their purpose is to facilitate an initial non-feasible solution, that is, starting from the origin of the coordinates system. The first inequality “Efficiency” calls for maximization, and its expression is: 0:85 x1 þ 0:75 x2 ≤ 1 The unit value in the second member of RHS is a constant and puts a limit to efficiency since it never can surpass it. Accordingly, the efficiency equation will be: 0:85 x1 þ 0:75 x2 þ S1 = 1 That is: S1 is the value that the inequality needs to become an equality. The second inequality “Financial index” calls for minimization, and its expression is: 0:78x1 þ 0:98 x2 ≥ 0:84 The value RHS = 0.84 in the second member establishes that the financial index must be a minimum 0.84, that is, no less than it. Consequently, the financial index equation will be:

Appendix

383

Table A.2 First Simplex tableau

0:78 x1 þ 0:98 x2 - S2 þ MA: That is, S2 is the value that is in excess over the equality, and that must be deducted. A similar procedure for the other two criteria. The Simplex algorithm starts by working with the two slack variables (with zero cost each) and with the two artificial variables (with a high value M each). aij: Are the performance values, that is, the contribution of each project to satisfy a criterion. Zj: The objective function for each criterion. In Z1 it is the sum of the products of aij times costs (aij × Cj), thus for project x1 it is Eq. (A.1). = 0:85 × 0 þ 0:78 × 0 þ 0:92 × 0 þ 0:99 × 0 = 0

ðA:1Þ

Table A.2 shows the first Simplex tableau. It is composed by: Column 1: Costs Cj Column 2: Value of the basic variable, including slacks and artificial |Column 3: bi or RHS Columns 4 to 11: Variables Column 12: Ratio: RHS/aij. Thus, for the first row it is 1/0.85= 1.18, for the third row is 0.94/0.92 = 1.02, and so on. The upper row is for costs. The immediate lower row identifies the variables. Notice that it starts with the real variables x1 and x2 and continues with slacks and artificial variables. Observe the respective real costs (0.72 and 0.68) for the real variables. For the slack variables, the cost is “0” and for the artificial variables the cost is a high value M.

384

Appendix

Objective function row: It is used to display the Z value in each iteration. For each row, it is the product of the RHS value of the variable times its cost. In this initial case—remember that it starts at the origin of coordinates—there is no feasible solution. For instance, the score of all variables is defined for their respective RHS, consequently, the Z value will be the sum of the products between these values and the costs. In this case 0 × 1 + 0 × 0.84 + 0 × 0.94 + 0 × 0.8 + M × 0.84 + M × 0.80 = 1.64 M. That is an extremely high cost. The last row is for the difference between the objective function cost and each variable cost (Zj-Cj) is called the “Index.” In this case, it is: For the first project: Z1-C1= 0 - 0.72 = -0.72 For the second project: Z2-C2 = 0 - 0.68 = -0.68 Now the algorithm determines the project that best improves the former result. The Simplex algorithm selects the best basic project to enter based on the index. Since the objective function calls for minimization, the best project will be that with the most negative (Zj - Cj) value, because it will produce the highest decrease. In this case (Z1 - C1) is selected in lieu of (Z2 - C2), and then the algorithm identifies x1 as the project to enter. The corresponding x1 column is the “Pivot column” and is identified by a solid arrow. This is the reason why LP does not produce RR, since a new project x3 for instance, that is worse than x1 and x2 will not be selected according to the index, because it will not have a (Zj - Cj ) negative value higher than other projects, but lower. Therefore, it will not affect the ranking. However, the Simplex may enter the new project but with a score that will put it at the right end of the ranking, and then not altering the other project preferences arrangement. The algorithm can even place x3 in an intermediate position if (Z3 - C3) is better than some other existing projects, however, in this case, the order of the ranking is not altered, and consequently there is no RR. Continuing with this reasoning, the Simplex must now select which variable must leave in order to preserve the mathematical dimensions. To do this, the Simplex performs the “Ratio test,” that is, the ratio between the constant term (bi) or RHS and the corresponding (ai1) as seen in column “Ratio” and selects the lowest. Why the lowest? Because the test determines the criterion that most limits the decrease in the value of the entering variable. In this case, it is criterion (4) “Generation index.” The corresponding row is called “Pivot row.” The intersection of the pivot column and pivot row is called “Pivot” (0.99). Table A.2 shows the first Simplex tableau where the described procedure can be visualized. Observe that the (Z) value is the parameter used to measure how the scenario improves when the right projects are chosen. Now, the method looks at how to improve this solution. A new tableau is needed (Table A.3) because the new axes have changed the coordinates of all (aij) performance values, and then, they must be recalculated, using the pivot and the following rule, as indicated in Table A.2.

Appendix

385

Table A.3 Second Simplex tableau

New aij* = aij – (the product of values in the diagonal) divided by (the pivot value). Thus: a12* = a12 - a11 ● a42/Pivot = 0.75 - (0.60 × 0.85/0.99) = 0.23 (see Table A.3). This rule is valid for all cells except for the cells that are on the pivot row. For these, the rule is simply to divide each one by the pivot: a42  = a42 =Pivot = 0:60=0:99 = 0:61: And the same for the original values located on the same pivot row. Notice the “0” value for the pivot column for C1 - Z1 indicating that the corresponding variable is now part of the solution since the objective value is equal at its cost. The same procedure for all other values and the result can be seen in the second tableau in Table A.3. The entering variable (x1) is now a unit vector (ceros and one, because it is a coordinate axis) and notice that the objective function (Z) has decreased from (1.64M) to (0.58), since Z = 0.72*0.61= 0.58. Simplex looks again for the best project vector to enter and detects that it corresponds to x2. The ratio test indicates that the leaving row will be now that of the second criterion “Financial index” and then the pivot is (0.51). Transforming all values again leads to Table A.4. Then project x2 is shown now as a unit vector because it is now another axis. The objective function is now 0.72 × 0.56 + 0.68 × 0.41 = 0.68, which is a considerable reduction in costs from the first tableau. Simplex looks for more negative values of the index, but finds none, which indicates that the process is finished. Naturally in an actual scenario with perhaps hundreds of projects, the process is repeated until no negative values are found in the index row.

386

Appendix

Table A.4 Third Simplex tableau

When the Simplex stops it indicates (if it exists) which are the coordinates of the optimum point (that equates to those of the tangent point between the polygon and the objective function in the graphic example of Fig. 6.1, Chap. 6. As can be seen the result from the graphic and that of the Simplex coincide. In both cases x1 = 0.56 and x2 = 0.41. In reality, this result is exact, while in the graphic it obviously is an approximation. It has been demonstrated that if a new project “n” “worse” than any of the projects in an existing ranking is introduced into the scenario, which calls for a minimization of the objective function, it will be not selected if its (Zn - Cn) value is lower than any other. For the same token, it will not be selected if its (Zn - Cn) value is larger than any other in a maximization problem; thus, its introduction will not produce any alteration in the ordering of the ranking. However, if its (Zn - Cn) value is lower (in a minimization objective) than some other project it can be eventually considered and then it will appear in the ranking preceding some existing projects, however, the ordering of the ranking will be preserved, and consequently there will be not RR. A complete and thorough explanation of the Simplex Tableau is found in Kothari (2009) and in MIT.

Demonstration of Absence of Rank Reversal in SIMUS Starting from an initial problem several scenarios are considered. Notice that these involve much harder conditions that are found in the literature on RR where, in general, only one scenario is examined at a time, while this section shows that more than one at the same time and even mixing different scenarios. The examples and results show that no RR is expected from SIMUS, which is expected since RR is impossible in Linear Programming as demonstrated in Sect. A.1.

Appendix

387

Solving a Problem with SIMUS Software Assume the initial matrix depicted in bold in Table A.5 with five projects (with projects six and seven being added later). The result of this exercise will be taken as reference. The system utilizes Euclidean normalization but it can also work with others such as considering the largest value of each row, or the sum of all values in each row, or the minmax; in either case the result is not affected. This case is solved using SIMUS and the result is shown in its last screen which is reproduced in Fig. A.1. Observe that SIMUS provides two series of scores in its ERM and PDM matrices which are obtained by different procedures. The ERM

Table A.5 Initial Decision Matrix with five projects Initial Decision Matrix

Added projeAdded project

Added projects

Project 1

Project 2

Project 3

Project 4

Project 5

Project 6 ‘worse’ than any original

Project 6 ‘better’ than any original

Project 7 = Project 3

Action

6200 3 20 4

6050 4.2 20 3

4800 2.5 21 2.5

5100 6.1 30 3

3800 3.10 32 5

3600 2.4 35 2.4

6500 6.5 18 5.5

4800 2.5 21 2.5

MAX MAX MIN MAX

Fig. A.1 Original ranking found by SIMUS in its two processes

388

Appendix

scores are seen in the solid blue row while the PDM scores are found in the solid brown PDM column, however, notice that ERM ranking and the PDM are identical. The ranking is: 4 ≽ 5 ≽ 3 ≽ 2 ≽ 1

Adding an Exact Copy of an Existing Project According to some researches, the most likely scenario for RR is when two projects are close or even identical. This section examines this case and also demonstrates that SIMUS is immune to it. For instance, a new project x3, identical to x1, is introduced in this problem. This case is solved using SIMUS and the result is shown in Fig. A.2 that reproduces its last screen. The ranking is: 4 ≽ 5 ≽ 3 ≽ 2 ≽ 1 Output: Original ranking preserved.

Efficient Results Matrix (ERM) Normalized Project 1 Project 2 Project 3 Project 4 Project 5 Project 6

Target 1 Target 2 Target 3 Target 4 Sum of Column (SC) Participation Factor (PF) Norm. Participation Factor (NPF) Final Result (SC x NPF)

0.32

0.68 1.00 1.00

0.00 0 0.00

0.32 1 0.25

0.68 1 0.25

1.00 1 0.25

1.00 1 0.25

0.00 0 0.00

0.00

0.08

0.17

0.25

0.25

0.00

Number of targets 4

ERM Ranking Project 4 - Project 5 - Project 3 - Project 2 - Project 1 - Project 6 Project Dominance Matrix (PDM) Dominated projects Dominant proj. Project 1 Project 2 Project 3 Project 4 Project 5 Project 6

Project 1 Project 2

0.3

Project 3

0.7

0.4

Project 4

1.0

1.0

1.0

Project 5

1.0

1.0

1.0

0.0

–3.0

0.3

0.3

0.3

1.3

–1.1

0.7

0.7

0.7

3.1

1.1

1.0

1.0

5.0

3.0

1.0

5.0

3.0

0.0

–3.0

1.0

Project 6 Column sum of dominated 3.0 projects

2.4

2.0

Row sum of dominant projects Net dominance

2.0

2.0

3.0

PDM Ranking Project 4 - Project 5 - Project 3 - Project 2 - Project 1 - Project 6

Fig. A.2 Adding an exact copy of an existing project

Appendix

389

Adding Project 6 “Worse” than Others Now project x6 is added, which apparently is worse than any other since its performances are lower in maximization and higher in minimization. Figure A.3 shows the result, which as seen replicates the ranking when there were only five projects. Project x6 has a “0 score” in the ERM matrix, meaning that it is not considered. The ranking is: 4 ≽ 5 ≽ 3 ≽ 2 ≽ 1≽ 6 Output: Original ranking preserved. Efficient Results Matrix (ERM) Normalized Project 1 Target 1

Project 2

Project 3

0.32

0.68

Project 4

Target 2

Project 5

Project 6

1.00

Target 3 Target 4

1.00

Sum of Column (SC)

0.00

0.32

0.68

1.00

1.00

0.00

Number of targets

Participation Factor (PF)

0

1

1

1

1

0

4

Norm. Participation Factor (NPF)

0.00

0.25

0.25

0.25

0.25

0.00

Final Result (SC x NPF)

0.00

0.25

0.25

0.00

0.08

0.17

ERM Ranking Project 4 - Project 5 - Project 3 - Project 2 - Project 1 - Project 6 Project Dominance Matrix (PDM) Dominated projects Dominant proj.

Row sum of

Project 1 Project 2 Project 3 Project 4 Project 5 Project 6

Project 1 Project 2

0.3

Project 3

0.7

0.4

Project 4

1.0

1.0

1.0

Project 5

1.0

1.0

1.0

0.0

–3.0

0.3

0.3

0.3

1.3

–1.1

0.7

0.7

0.7

3.1

1.1

1.0

1.0

5.0

3.0

1.0

5.0

3.0

0.0

–3.0

1.0

Project 6 Column sum of dominated projects

3.0

2.4

2.0

dominant projects Net dominance

2.0

2.0

3.0

PDM Ranking Project 4 - Project 5 - Project 3 - Project 2 - Project 1 - Project 6

Fig. A.3 Adding project 6 “worse” than others

390

Appendix

Adding New Project x7 Keeping Project x6 and with x3 = x6 = x7 Project x6 is kept equal to project x3, and project x7 is added also equal to x3 and x6. See Fig. A.4. Notice that project x6 and x7 have ‘0’ scores The ranking is: 4 ≽ 5 ≽ 3 ≽ 2 ≽ 1≽ 6 Output: Original ranking preserved.

Efficient Results Matrix (ERM) Normalized Project 1 Target 1

Project 2

Project 3

0.04

0.96

Target 2

Project 4

Project 5

Project 6

Project 7

0.00

1.00

Target 3 Target 4

1.00

Sum of Column (SC)

0.00

0.04

0.96

1.00

1.00

0.00

Participation Factor (PF)

0

1

1

1

1

0

0

Norm. Participation Factor (NPF)

0.00

0.25

0.25

0.25

0.25

0.00

0.00

Final Result (SC x NPF)

0.00

0.01

0.24

0.25

0.25

0.00

0.00

ERM Ranking Project 4 - Project 5 - Project 3 - Project 2 - Project 1 - Project 6 - Project 7

Project Dominance Matrix (PDM) Dominated projects Dominant proj. Project 1

Project 2

Project 3

Project 4

Project 5

Row sum of Project 6

Project 7 dominant projects

Project 1 Project 2

0.0

Project 3

1.0

0.9

Project 4

1.0

1.0

1.0

Project 5

1.0

1.0

1.0

Net dominance

0.0

–3.0

0.0

0.0

0.0

0.0

0.2

–2.7

1.0

1.0

1.0

1.0

5.7

3.7

1.0

1.0

1.0

6.0

4.0

1.0

1.0

6.0

4.0

Project 6

0.0

–3.0

Project 7

0.0

–3.0

Column sum of dominated 3.0 2.9 projects PDM Ranking

2.0

1.0

2.0

2.0

3.0

3.0

Project 4 - Project 5 - Project 3 - Project 2 - Project 1 - Project 6 - Project 7

Fig. A.4 Adding project x7 keeping project 6 and with x3 = x6 = x7

Appendix

391

Adding a New Project Identical to Other and Simultaneously Adding Another Considered the Best The ranking is: 4 ≽ 5 ≽ 3 ≽ 2 ≽ 1≽ 6≽ 7 Output: Original ranking preserved.

Deleting Project from the Original Project 3 is deleted. See Fig. A.6. The ranking is: 4 ≽ 5≽ 2 ≽ 1 Output: Original ranking preserved. Observe that even when project 3 was deleted, the order of the ranking holds.

Efficient Results Matrix (ERM) Normalized Project 1 Project 2 Project 3 Project 4 Project 5 Project 6 Project 7 Target 1

0.32

0.68

Target 2

1.00

Target 3 Target 4

1.00

Sum of Column (SC)

0.00

0.32

0.68

1.00

1.00

0.00

Participation Factor (PF)

0

1

1

1

1

0

0.00 0

Norm. Participation Factor (NPF)

0.00

0.25

0.25

0.25

0.25

0.00

0.00

Final Result (SC x NPF)

0.00

0.08

0.17

0.25

0.25

0.00

0.00

ERM Ranking Project 4 - Project 5 - Project 3 - Project 2 - Project 1 - Project 6 - Project 7 Project Dominance Matrix (PDM)

Dominant proj.

Dominated projects

Row sum of

Project 1 Project 2 Project 3 Project 4 Project 5 Project 6 Project 7

dominant projects

Net dominance

0.0

–3.0

Project 1 Project 2

0.3

0.3

0.3

0.3

0.3

1.6

Project 3

0.7

0.4

–0.8

0.7

0.7

0.7

0.7

3.8

Project 4

1.0

1.0

1.0

1.8

1.0

1.0

1.0

6.0

Project 5

1.0

1.0

1.0

4.0

1.0

1.0

6.0

4.0

Project 6

0.0

–3.0

Project 7

0.0

–3.0

Column sum of dominated projects best

3.0

2.4

2.0

1.0

2.0

2.0

3.0

3.0

PDM Ranking Project 4 - Project 5 - Project 3 - Project 2 - Project 1 - Project 6 - Project 7

Fig. A.5 Adding a new project identical to other and simultaneously adding another is considered the best

392

Appendix Efficient Results Matrix (ERM) Project 1

Project 2

Project 3

Project 4

Project 5

Target 1

0.35

1.21

0.00

0.00

0.00

Target 2

0.00

0.00

0.00

1.22

0.00

Target 3

0.00

0.00

0.00

0.00

0.00

Target 4

0.00

0.00

0.00

0.00

1.63

Efficient Results Matrix (ERM) Normalized

Target 1

Project 1

Project 2

0.22

0.78

Project 3

Target 2

Project 4

Project 5

1.00

Target 3 Target 4

1.00

Sum of Column (SC)

0.22

0.78

0.00

1.00

Participation Factor (PF)

1

1

0

1

1.00 1

Norm. Participation Factor (NPF)

0.25

0.25

0.00

0.25

0.25

Final Result (SC x NPF)

0.06

0.19

0.00

0.25

0.25

ERM Ranking Project 4 - Project 5 - Project 2 - Project 1 - Project 3 Project Dominance Matrix (PDM) Dominated projects Dominant proj.

Project 1

Project 2

Project 1 Project 2

0.6

Row sum of Project 3

Project 4

Project 5

dominant projects

Net dominance

0.2

0.2

0.2

0.7

–1.9

0.8

0.8

0.8

2.9

0.9

0.0

–3.0

4.0

2.0

4.0

2.0

Project 3 Project 4

1.0

1.0

1.0

Project 5

1.0

1.0

1.0

1.0

2.6

2.0

3.0

2.0

Column sum of dominated projects

1.0

2.0

PDM Ranking Project 4 - Project 5 - Project 2 - Project 1 - Project 3

Fig. A.6 Deleting project from the original

Summary of Scenarios and Results Table A.6 summarizes the findings. As can be seen, SIMUS does not produce RR when different conditions are considered.

Appendix

393

Table A.6 Summary of results for different scenarios

Original project Section A.2.1 Table A.5 Adding an exact copy of an existing project Section A.2.2

Adding project 6 ‘worse’ than others Section A.2.3

Adding a new project equal to another two Section A.2.4

Adding a new project and adding another which is considered the best Section A.2.5

Deleting a project from the original Section A.2.6

Comments

Results referred to ranking

Action

Ranking

Original result considering only five projects

4≽5≽ 3≽ 2 ≽ 1 Figure A.1

Original ranking

Adding project x3 identical to project x1

4≽5≽ 3≽ 2 ≽ 1 Figure A.2

Ranking preserved

Adding a project x6 which apparently is worse than others

4≽5≽ 3≽2≽ 1≽ 6 Figure A.3

Ranking preserved

Observe that x6 which is worse than any other is placed at the end of the ranking

Adding project x7 equal to projects x3 and x6

4≽5≽ 3≽2≽ 1≽ 6≽7 Figure A.4

Ranking preserved

Observe that since project x7 is equal to project x6, it is placed at the end of the ranking

Adding a new project identical to another and simultaneously adding another is considered the best

4≽5≽ 3≽2≽ 1≽ 6≽7 Figure A.5

Ranking preserved

Project x3 is deleted

4 ≽ 5≽ 2≽1 Figure A.6

Ranking preserved

Observe that the original ranking is preserved although without project x3

Conclusion

The goal of this Appendix is to demonstrate that when LP is used for decisionmaking no RR is produced, and it was made evident by examining the original algebraic development proposed by Dantzig (cited) in his Simplex algorithm. It clearly reveals that the incorporation of a new project considered worse than existing projects, cannot alter the ranking order because the algorithm takes into account simultaneously the new project contribution (cost or benefit) as well as its performance. To say it in simpler terms, the algorithm works by analyzing and comparing opportunity costs and minimizes or maximizes them. It is a well-known fact that RR is also produced by deleting a project from the scenario, and when two projects are close or identical. These two scenarios were also examined in this section by modifying the original problem and solving each using SIMUS. Four different scenarios were considered even adding more than one project at a time and also mixing additions with identical projects. It is believed that the algebraic analysis performed and the examples proposed validate our claim.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 N. Munier, Strategic Approach in Multi-Criteria Decision Making, International Series in Operations Research & Management Science 351, https://doi.org/10.1007/978-3-031-44453-1

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Reference

Khotari, C. (2009). Quantitative techniques. Vikas Publishing House.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 N. Munier, Strategic Approach in Multi-Criteria Decision Making, International Series in Operations Research & Management Science 351, https://doi.org/10.1007/978-3-031-44453-1

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