Operations Research and Data Science in Public Services: 6th AIROYoung Workshop, Rome, Italy, February 23–25, 2022 (AIRO Springer Series, 11) 3031345452, 9783031345456

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AIRO Springer Series 11

Matteo Cosmi Lorenzo Peirano Alice Raffaele Marcella Samà   Editors

Operations Research and Data Science in Public Services 6th AIROYoung Workshop, Rome, Italy, February 23–25, 2022

AIRO Springer Series Volume 11

Editor-in-Chief Daniele Vigo, Dipartimento di Ingegneria dell’Energia Elettrica e dell’Informazione “Gugliemo Marconi”, Alma Mater Studiorum Università di Bologna, Bologna, Italy Series Editors Alessandro Agnetis, Dipartimento di Ingegneria dell’Informazione e Scienze Matematiche, Università degli Studi di Siena, Siena, Italy Edoardo Amaldi, Dipartimento di Elettronica, Informazione e Bioingegneria (DEIB), Politecnico di Milano, Milan, Italy Francesca Guerriero, Dipartimento di Ingegneria Meccanica, Energetica e Gestionale (DIMEG), Università della Calabria, Rende, Italy Stefano Lucidi, Dipartimento di Ingegneria Informatica Automatica e Gestionale “Antonio Ruberti” (DIAG), Università di Roma “La Sapienza”, Rome, Italy Enza Messina, Dipartimento di Informatica Sistemistica e Comunicazione, Università degli Studi di Milano-Bicocca, Milan, Italy Antonio Sforza, Dipartimento di Ingegneria Elettrica e Tecnologie dell’Informazione, Università degli Studi di Napoli Federico II, Naples, Italy

The AIRO Springer Series focuses on the relevance of operations research (OR) in the scientific world and in real life applications. The series publishes peer-reviewed only works, such as contributed volumes, lectures notes, and monographs in English language resulting from workshops, conferences, courses, schools, seminars, and research activities carried out by AIRO, Associazione Italiana di Ricerca Operativa – Optimization and Decision Sciences: http://www.airo.org/index.php/it/. The books in the series will discuss recent results and analyze new trends focusing on the following areas: Optimization and Operation Research, including Continuous, Discrete and Network Optimization, and related industrial and territorial applications. Interdisciplinary contributions, showing a fruitful collaboration of scientists with researchers from other fields to address complex applications, are welcome. The series is aimed at providing useful reference material to students, academic and industrial researchers at an international level. Should an author wish to submit a manuscript, please note that this can be done by directly contacting the series Editorial Board, which is in charge of the peer-review process. THE SERIES IS INDEXED IN SCOPUS

Matteo Cosmi · Lorenzo Peirano · Alice Raffaele · Marcella Samà Editors

Operations Research and Data Science in Public Services 6th AIROYoung Workshop, Rome, Italy, February 23–25, 2022

Editors Matteo Cosmi Department of Economics and Management University of Luxembourg Luxembourg, Luxembourg

Lorenzo Peirano Department of Economics and Management University of Brescia Brescia, Italy

Alice Raffaele Department of Mechanical, Energy and Management Engineering University of Calabria Rende, Italy

Marcella Samà Department of Civil, Computer Science and Aeronautical Engineering Roma Tre University Rome, Italy

ISSN 2523-7047 ISSN 2523-7055 (electronic) AIRO Springer Series ISBN 978-3-031-34545-6 ISBN 978-3-031-34546-3 (eBook) https://doi.org/10.1007/978-3-031-34546-3 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed 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

Organization

Program Committee Chairs Cosmi, Matteo

University of Luxembourg, Luxembourg Centre for Logistics and Supply Chain Management (LCL), Luxembourg, Luxembourg Peirano, Lorenzo University of Brescia, Department of Economics and Management, Brescia, Italy Raffaele, Alice University of Calabria, Department of Mechanical, Energy and Management Engineering, Rende, Italy Samà, Marcella Roma Tre University, Department of Civil, Computer Science and Aeronautical Engineering, Roma, Italy

Program Committee Members Bonadei, Francesca Cosmi, Matteo

University of Luxembourg, Luxembourg Centre for Logistics and Supply Chain Management (LCL), Luxembourg, Luxembourg Peirano, Lorenzo University of Brescia, Department of Economics and Management, Brescia, Italy Raffaele, Alice University of Calabria, Department of Mechanical, Energy and Management Engineering, Rende, Italy Samà, Marcella Roma Tre University, Department of Civil, Computer Science and Aeronautical Engineering, Roma, Italy

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Preface

This book contains the proceedings of the 6th AIROYoung Workshop, entitled “Operations Research and Data Science in Public Services”, which took place during February 23–25, 2022, in a hybrid format: in presence at Roma Tre University (Rome, Italy) and online by means of the Microsoft Teams platform. The AIROYoung Workshop is the main event organized by AIROYoung, the Young Chapter of AIRO (Italian Association of Operations Research), whose mission is to enhance collaboration among PhD students, postdoctoral researchers, and practitioners interested in Operations Research. Since its first edition in 2017, the AIROYoung Workshop has become an annual international occasion for young researchers to meet and present their work, in front of an increasing number of participants from both academia and industry. This volume contains five accepted contributions presenting both methodological and application-oriented results. In “Predicting the Empirical Hardness of Metric TSP Instances” by Gambardella et al., the authors propose how to classify the empirical hardness of instances of the Traveling Salesman Problem. To do so, they use four features depending only on the distributions of values in the cost matrix of the problem, and introduce the “tight triangle inequality index” feature, which indicates the percentage of triangle inequalities in the cost matrix satisfied with equality. In this work, a decision tree is designed for the prediction of the hardness of the instances, and an extensive dataset is built to evaluate its performance. In the latest decades, several variants of the Traveling Salesman Problem have been proposed and defined in the literature, such as the Vehicle Routing Problem, which finds many applications in the public sector and industry. For example, in “A Two-Phase Approach to Evaluate and Optimize an Interlibrary Loan Service: The Case Study of Provincia di Brescia” by Raffaele et al., the interlibrary loan service offered by the public body Provincia di Brescia is described as a Rich Vehicle Routing with pickup and delivery, backhauls, and time windows. Given the large number of libraries in the Province of Brescia, the authors apply the firstcluster-then-route philosophy by relying on both machine learning and mixed integer

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linear programming. Their approach is compared with the current implementation of the service on the real data of 2019. In “Optimising the Order Picking Process: The Case of a Retail Distribution Company in Italy” by Di Puglia Pugliese et al., instead, a multi-objective order picking problem is modeled as a variant of a Vehicle Routing Problem in which picking points are viewed as customers and trolleys handled by operators as vehicles. The authors propose two strategies based on weighting and -constrained methods and perform a computational study on reality-based instances to show that their approach outperforms the one currently applied by the company. Anyway, routing problems are not the only problems of great, practical interest for companies and researchers. To tackle situations inspired by real applications, other classical problems and modeling strategies have been employed in the following two works. In “Batch Delivery Considerations in Additive Manufacturing Machine Scheduling Problem” by Kucukkoc, the classical parallel unrelated machine scheduling problem is enriched with additional batch delivery constraints and dynamic processing times, related to the total volume and maximum height of the parts allocated to that batch. This problem, solved by using a customized genetic algorithm, finds here a practical application in the emerging 3D printing sector. Moreover, much attention is paid to current and contemporary issues, such as climate change, geopolitical events, and the use of energy resources. In “An LP Model for the Central Energy System” by Cordieri et al., the authors propose a mathematical model to optimize the expansion operations for the energy generation from the perspective of a Central Energy System. Extensions of the current energy network are considered, and constraints are implemented in order to limit the emissions of greenhouse gases. The performance of the model is tested on multiple instances and hardware architectures by making use of a commercial solver. As editors of the volume, we thank AIRO and AIROYoung, the invited lecturers, the industrial representatives and sponsors, the authors, and the researchers who spent their time in the review process to improve the quality of the selected contributions. A special thanks is due to the AIROYoung board and the Automation and Operations Research in Industry group of the Engineering Department of Roma Tre University, who organized the Workshop in Rome. Finally, we would like to express our gratitude to Springer for its strong support and cooperation during the event and the publishing process. The “Operations Research and Data Science in Public Services” Volume Editors. Luxembourg, Luxembourg Brescia, Italy Padova, Italy Roma, Italy

Matteo Cosmi Lorenzo Peirano Alice Raffaele Marcella Samà

Contents

Predicting the Empirical Hardness of Metric TSP Instances . . . . . . . . . . . . Luca Maria Gambardella, Stefano Gualandi, Monaldo Mastrolilli, and Eleonora Vercesi

1

A Two-Phase Approach to Evaluate and Optimize an Interlibrary Loan Service: The Case Study of Provincia di Brescia . . . . . . . . . . . . . . . . . . 17 Alice Raffaele, Matteo Zavatteri, Fabio Bazzoli, Marco Gussago, and Romeo Rizzi Optimising the Order Picking Process: The Case of a Retail Distribution Company in Italy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Luigi Di Puglia Pugliese, Francesca Guerriero, Giusy Macrina, Massimiliano Matteucci, and Veronica Mosca Batch Delivery Considerations in Additive Manufacturing Machine Scheduling Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Ibrahim Kucukkoc An LP Model for the Central Energy System . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Silvia Anna Cordieri, Michele Monaci, Paolo Paronuzzi, and Henrik Schwaeppe Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

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Editors and Contributors

About the Editors Matteo Cosmi is a postdoctoral researcher in Operations Research at Luxembourg Centre for Logistics and Supply Chain Management (LCL). Previously, he obtained his Ph.D. in Computer Science and Automation at Roma Tre University, working on operations research applications for food delivery problems, and was an Operations Research scientist at LINKS Foundation. His research focuses on models and algorithms for last-mile delivery, logistics, and supply chain. He is a reviewer for international scientific journals and conferences and was part of the organizing committee for the 6th AIROYoung Workshop. Lorenzo Peirano is a postdoctoral researcher and contract lecturer in Operations Research at the University of Brescia. He received his Ph.D. in Analytics for Economics and Business at the University of Bergamo in 2020, and his research is focused on the optimization of logistics, maritime and air transportation, network and combinatorial problems, and machine learning. Since February 2021, he has been a joint coordinator of AIROYoung, the young chapter of AIRO. Alice Raffaele is a postdoctoral researcher at the University of Calabria and visiting researcher at the University of Padova. She received her Ph.D. in Operations Research from the University of Trento and the University of Verona, where she is currently a contract lecturer. Her research interests are combinatorial optimization, routing problems, enumeration problems, temporal networks, and operations research in education. Since March 2019, she has been part of the board of AIROYoung, the young chapter of AIRO, of which she has been a joint coordinator since February 2021. She collaborates with MaddMaths! the main Italian website to disseminate mathematics. Marcella Samà is an Associate Professor in Operations Research at Roma Tre University, where she also received her Ph.D. in Operations Research. Her research focuses on models and algorithms for public service problems, with specific regard to xi

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transit and logistics. She is a reviewer for several international scientific journals, IET Intelligent Transportation Systems associate editor, guest editor for Transportation Research Part E, and organizing committee chair for the 6th AIROYoung Workshop.

Contributors Fabio Bazzoli Sistema Bibliotecario Sud Ovest Bresciano, Chiari, Italy Silvia Anna Cordieri Università di Bologna, Bologna, Italy Luigi Di Puglia Pugliese Istituto di Calcolo e Reti ad Alte Prestazioni, Consiglio Nazionale delle Ricerche, Rende, Italy Luca Maria Gambardella Faculty of informatics, USI, IDSIA - USI-SUPSI, Lugano, Switzerland Stefano Gualandi Dipartimento di Matematica “F. Casorati”, Università degli Studi di Pavia, Lugano, Switzerland Francesca Guerriero Department of Mechanical, Energy and Management Engineering, University of Calabria, Rende, Italy Marco Gussago Ufficio Biblioteche, Provincia di Brescia, Brescia, Italy Ibrahim Kucukkoc Department of Industrial Engineering, Balikesir University, Balikesir, Turkey Giusy Macrina Department of Mechanical, Energy and Management Engineering, University of Calabria, Rende, Italy Monaldo Mastrolilli IDSIA - USI-SUPSI, Lugano, Switzerland Massimiliano Matteucci Gruppo PAC2000A Conad, Umbria, Italy Michele Monaci Università di Bologna, Bologna, Italy Veronica Mosca Department of Mechanical, Energy and Management Engineering, University of Calabria, Rende, Italy Paolo Paronuzzi Università di Bologna, Bologna, Italy Alice Raffaele Department of Mechanical, Energy and Management Engineering, University of Calabria, Rende, Italy Romeo Rizzi Department of Computer Science, University of Verona, Verona, Italy Henrik Schwaeppe RWTH Aachen University, Aachen, Germany Eleonora Vercesi Dipartimento di Matematica “F. Casorati”, Università degli Studi di Pavia, Lugano, Switzerland Matteo Zavatteri Department of Mathematics, University of Padova, Padova, Italy

Predicting the Empirical Hardness of Metric TSP Instances Luca Maria Gambardella, Stefano Gualandi, Monaldo Mastrolilli, and Eleonora Vercesi

Abstract In this work, we propose to classify the empirical hardness of metric Travelling Salesman Problem (TSP) instances by using four features that depend only on the distribution of values in the cost matrix. The first three features are the standard deviation, the skewness, and the Gini index of the cost coefficients. The fourth feature is based on a new statistic, called the tight triangle inequality index, which gives the percentage over all possible triangle inequalities in the cost matrix that are satisfied with equality. As a dataset, we used over 45 000 instances, both collected from the literature and randomly generated, which we labeled as easy or hard by using the number of branch and bound nodes of Concorde. After computing the four features for all the instances in the training set, we designed a decision tree that predicts the empirical hardness of instances in the test set with high values of the Matthew Correlation Coefficient prediction score. Keywords Travelling salesman problem · Binary decision trees

1 Introduction The Travelling Salesman Problem (TSP) is probably the most popular problem in combinatorial optimization, and it is stated as follows: “Given a set of cities along with the cost of travel between each pair of them, the TSP is to find the cheapest way L. M. Gambardella Faculty of informatics, USI, IDSIA - USI-SUPSI, Lugano, Switzerland e-mail: [email protected] S. Gualandi · E. Vercesi (B) Dipartimento di Matematica “F. Casorati”, Università degli Studi di Pavia, Lugano, Switzerland e-mail: [email protected] S. Gualandi e-mail: [email protected] M. Mastrolilli IDSIA - USI-SUPSI, Lugano, Switzerland e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Cosmi et al. (eds.), Operations Research and Data Science in Public Services, AIRO Springer Series 11, https://doi.org/10.1007/978-3-031-34546-3_1

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of visiting all the cities and returning to the starting point” [2]. Any permutation of the cities induces a tour (or circuit), and the tour that has the minimum cost is called an optimal tour. An instance of the TSP with n cities is represented by a weighted graph G = (V, E), where the vertices are defined by the set V := {1, . . . , n}, and the edges by the set E := {{i, j} | i, j ∈ V } ⊂ V × V . The distances between pairs of cities are given by a cost function c : E → R+ . We denote by ci j the length of to denote all possible costs. In an edge {i, j} ∈ E, and we use matrix C ∈ Rn×n + this paper, we are interested in the Metric TSP, which is defined whenever the cost function gives a metric, that is when the following three axioms are satisfied: (i) ci j = 0 ⇔ i = j, for all {i, j} ∈ E; (ii) ci j = c ji , for all {i, j} ∈ E; (iii) the triangle inequality ci j ≤ cik + c jk holds for all triples {i, j}, {i, k}, { j, k} ∈ E. For metric instances, we refer to the cost vector obtained by flatting the upper-triangular matrix C, that is, c = triu(C). Although the TSP is NP-Hard even in the metric case [14], actual hard-to-solve small instances are poorly available in the literature. Concorde, which is the state-ofthe-art branch and cut solver for the TSP [1], can solve to optimality all the instances of the TSPLIB with n ≤ 1380 in less than a minute [18]. Chapter 16 in [2] shows that random Euclidean TSP sampled on the unit square can be solved to optimality in a few seconds for instances with up to 3 000 nodes. Only recently, small (with n ≤ 50) rectilinear TSP instances are introduced in [23], which are challenging for Concorde. A computational method to generate hard-to-solve metric TSP instances is introduced in [22], along with the Hard-TSPLIB, which is a collection of hard instances for exact methods. This computational method can be viewed as a “complexification procedure”: starting from a TSP instance given by a cost vector c1 , the procedure generates a second cost vector c2 corresponding to a TSP instance being more demanding for Concorde. However, a clear relationship between the structure of the cost matrix C and the empirical hardness of TSP instances is still missing. To tackle this research question, we design a classification algorithm based on the training of a binary decision tree [15] that can predict the empirical hardness of metric TSP instances. The decision tree is trained using over 45 000 metric instances generated with the code introduced in [22]. The instances are labeled as easy or hard according to their size n and the number of branch and bound nodes required by Concorde for actually solving the instance. The decision tree uses four features that are directly computed from the cost vector c. The first three features are the standard deviation, the skewness, and the Gini index of the cost coefficients. The fourth feature is based on a new statistic called the tight triangle inequality index, which gives the percentage of all possible triangle inequalities in the cost matrix that are satisfied with equality. The Gini index and the tight triangle inequality index are used in this context for the first time. Our computational results with different combinations of these four features show that our scores are capable to distinguish easy/hard instances with a fixed number of nodes. We also create a single dataset of instances with various numbers of nodes and train a decision tree on it. In this latter case the performances decrease and further studies should be conducted. Note that the performances are evaluated in terms of the Matthew Correlation Coefficient (MCC) [7], a prediction score which is appropriate

Predicting the Empirical Hardness of Metric TSP Instances

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for unbalanced datasets; other popular prediction scores, such as the accuracy and the AUC, are not suitable for unbalanced datasets. The MCC score is strongly related to chi-square statistic and ranges between –1 and 1, where 1 represents a perfect prediction, 0 is a random prediction, and –1 indicates total disagreement between prediction and observation. Related Works. Several works in the literature have studied the features that make TSP instances hard to solve in practice for a given class of algorithms. We describe next the main works that consider empirical hardness for exact algorithms, but there is also a scientific interest in hardness for heuristics (e.g., see [20, 21]). In [5], the authors observe a clear phase transition from easy to hard TSP instances related to the standard deviation of the values ci j in the cost matrix C. In [10], the phase transition is related to a tour value  and the area A occupied by n nodes, according to the ratio √  . Note that this ratio is only reliable when  is the optimal tour (or an almost n·A optimal tour) and when the cities are defined as points in a metric space where the notion of “area” is given (e.g., the Euclidean space in 2D). Unfortunately, the notion of area is undefined for general metric spaces. In [16], the authors report computational experiments with instances generated with a random uniform distribution and an exponential random distribution: the two types of distributions lead to TSP instances with two different levels of hardness. Interestingly, the level of empirical hardness is defined in terms of the dimension of the branch and bound tree. Along the same line of research, two families of TSP instances are introduced in [9], where the empirical hardness is related to the amount of perturbation added to the cost matrix: a moderate amount of perturbation makes an instance very hard, while no perturbation and a significant perturbation reduce the complexity. Instead of classifying instances according to features such as the number of branch and bound nodes or the runtime, a Random Forest is used in [4] to classify instances from the TSPLIB as either easy  and to distinguish the instances or hard. The main idea is to predict the quantity √n·A between easy and hard according to their distance to the transition phase conjectured in [10]. Unfortunately, with this approach, there are instances such as the gr120 from the TSPLIB which is classified as hard, even if it is solved by Concorde in less than a second. Outline. The outline of this paper is as follows. Section 2 formally introduces the cost features that we use in our study. Section 3 describes how we trained binary decision trees to predict hard instances. In Sect. 4, we report our extensive computational results that validate our approach to predict the empirical hardness of a given metric TSP instance by only using features of the cost vector c. We conclude with a perspective on future works.

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2 Cost Features of the Metric TSPs In this section, we introduce the features of the cost vector c we use to predict the empirical hardness of metric TSP instances. The first two features we consider were already used in the literature: the standard deviation is used in [4, 10], while the skewness is used in [4]. The third feature is the Gini index [11], a well-known score in statistics, which we use for the first time to predict the empirical hardness of TSP instances. The fourth feature we introduce is the tight triangle inequality index, which gives the percentage of all possible triangle inequalities in the cost matrix that are satisfied with equality. In the following paragraphs, we describe in detail the four features used in the context of the metric TSP instances. Standard Deviation. In our work, we consider the normalized standard deviation of the cost vector c, which is defined as follows   2   1  ci j ci j 1  − c¯ , where c¯ = . SD(c) :=  |E| {i, j}∈E max(c) |E| {i, j}∈E max(c)

(1)

Note that due to the normalization factor, we have 0 ≤ SD(c) ≤ 1. The standard deviation of the cost matrix C was already successfully explored for asymmetric TSP instances in [5], where the author reports a phase transition related to this measure: both small and large standard deviations lead to easy instances, while values in the middle lead to hard instances. Skeweness. The sample skewness is computed as the Fisher-Pearson coefficient of skewness of the cost vector: SK(c) :=

m3 3 2

m2

, where m k =

1  (ci j − c)k . |E| {i, j}∈E

(2)

The skewness measures the displacement of the weights in a distribution: for unimodal distributions, SK < 0 indicates that the tail is on the left side of the distribution, while SK that the tail is on the right. Our computational results show that hard instances are associated with right-skewed distributions. Figure 1 shows the empirical cumulative distribution of the values in the cost vector c for four instances. We can observe that the cumulative distributions associated with the Hard-TSPLIB instances have edge cost values concentrated toward low values. Note that having a lot of small costs close to a few higher ones is a commonly believed source of hardness for branch-and-bound algorithms. Gini Index. The Gini index is a measure of statistical dispersion that quantifies the inequality among values of a frequency distribution. This index was introduced by Gini to measure the income (or wealth) inequality within a population [11]. When the values of the distribution are non-negative, the Gini index ranges in the interval

Predicting the Empirical Hardness of Metric TSP Instances 1.0

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1.0

0.5

swiss42 TSPLIB Hard-TSPLIB

0.0 0

100

200

0.5

bayg29 TSPLIB Hard-TSPLIB

0.0

300

0

100

200

300

400

Fig. 1 Empirical cumulative distribution of edge cost values of two instances of the TSPLIB and the Hard-TSPLIB

0.5

GI index TSPLIB Hard-TSPLIB

0.4

0.3

0.2

0.1

0.0 gr24

bayg29 bays29 swiss42 dantzig42 gr48

hk48

eil51

brazil58

st70

pr76

Fig. 2 Gini index–GI(c) of 11 instances of the TSPLIB, along with the corresponding Hard-TSPLIB instances

[0, 1], where 0 means perfect equality and 1 represents perfect inequality. We use the Gini index to measure the dispersion (or inequality) of the cost vector values: GI(c) :=

    1 ci j − cvw  . 2 c¯ |E|2 {i, j}∈E {v,w}∈E

(3)

Figure 2 compares the GI of the instances of the TSPLIB with those of the HardTSPLIB. We note that hard instances always have a larger value of this index, and this motivates its introduction as a feature of our predictor. Tight Triangle Inequality Index. Given a Metric TSP instance with the associated cost vector c, the Tight triangle inequality Index (TI for short) is defined as the number of triangle inequalities satisfied with equality with respect to all possible triangle inequalities.

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TI index TSPLIB Hard-TSPLIB

25

20

15

10

5

0 gr24

bayg29 bays29 swiss42 dantzig42 gr48

hk48

eil51

brazil58

st70

pr76

Fig. 3 Tight triangle inequality index–TI(c) of 11 instances of the TSPLIB, along with the corresponding Hard-TSPLIB instances

n TI(c) :=

i=1

n j=i+1

n k= j+1

1ci j =cik +c jk + 1cik =ci j +c jk + 1c jk =ci j +cik

 . 3 · n3

(4)

Note that for any triple of vertices (i, j, k), at most one of the three corresponding triangle inequalities can be tight. To prove this, assume that we have both ci j = cik + c jk and c jk = cik + ci j . Then ci j = cik + c jk = cik + cik + ci j , that would imply ci j = 0. However, this is unfeasible, as we are in the metric case, that is, ci j > 0 for all i = j. Thus, we have 0 ≤ TI(c) ≤

1 . 3

Figure 3 compares TI(c) instances of the TSPLIB with the corresponding instances in the Hard-TSPLIB [22]. In all these instances, except for gr48, TI(c) is higher, and for instances swiss42 and brazil58, is close to the upper bound of 33.3%. Note that, according to [22], it is not possible to solve brazil58 using Concorde within a time limit of 24h. In [22], the authors also observed that hard-to-solve instances are highly structured, with multiple costs of the same weights and connected edges with the same sum with respect to the weight. Our idea is hence that TI(c) should encode this high regularity of the costs. Another reason to associate the empirical hardness of TSP instances with high values of TI is that any triple (i, j, k) satisfied with equality ci j = cik + c jk makes the associated vector c to lay on a face of the metric polytope

Predicting the Empirical Hardness of Metric TSP Instances

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PM E T := {c ∈ Rm | ci j ≤ cik + c jk , ∀i, j, k ∈ V, and ci j ≥ ε, ∀{i, j} ∈ E}, where ε  1. Being on a face means being close to the outside PM E T , that is, being close to non-metric instances. Non-metric instances are known to be computationally harder than metric ones. Note that in the general TSP constant approximation is not possible in polynomial time. Otherwise, the Hamiltonian cycle would be solvable in polynomial time, and therefore P = NP (e.g., see Theorem 9.10 page 226 in [8]).

3 Learning Decision Trees Our goal is to design a decision tree that can infer a relation among the features SD(c), SK(c), GI(c), and TI(c) versus the hardness label (0 = easy, 1 = hard). Indeed, we want to evaluate if these four features can correctly detect hard instances. We selected decision trees mainly because they are interpretable. However, instead of computing a single decision tree, we perform extensive numerical tests to assess the expected performance of a decision tree trained with a small subset of every possible instance, and we report the average performance. Each decision tree is trained using the scikit-learn library [17] with default parameter values. In the following paragraphs, we describe the datasets we have generated to run our experiments using the code provided by [22], and we detail the prediction scores used to evaluate each fitted decision tree. Dataset. To create the dataset we proceed as follows: First, we sample 5 000 metric TSP instances using the hit-and-run algorithm [19], as discussed in [22]. We repeat this procedure for n ∈ {20, 25, 30, 35, 40}. Second, we apply the complexification procedure presented in [22] to all the instances with a time limit of 7 days. However, when the time limit expires, we are unable to guarantee a complete conversion of a sampled instance to its optimal hard version. Third, we solve both easy and hard instances using Concorde, with 10 different seeds, and we collect the number of Branch & Bound nodes (B&B) and the total runtime. We decided to label as easy all the instances that take an average number of B&B nodes equal to or less than the 5-percentile of the complexified instances. In this way, we label as easy also the easiest instances of the complexified set, and if, by chance, we sample an instance that requires a lot of B&B nodes to be solved to the optimum, this is hence labeled as hard. To build a fair training set, we should know the true distribution of hard metric TSP instances within the metric TSP instances space. As this is unknown, we test five different percentage p of hard instances in each dataset, that is, p ∈ {0.01, 0.05, 0.1, 0.2}. For each n, for each p, and for each i ∈ {1, . . . , 10}, we i containing 2 000 instances. Each dataset contains 100 p % of hard build a dataset Dnp instances, and the remaining are easy. To train a decision tree, we divide each dataset i into training and test sets, with a test size of 33%. Finally, we train a binary Dnp i decision tree for each dataset Dnp . Although we know that the standard procedure

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for machine learning tasks is to use k-fold cross-validation, we decided to proceed as discussed to better control each dataset dimension and the percentage of contained hard instances. Prediction Score: Matthew Correlation Coefficient. To evaluate the performance of each decision tree, we use the Matthew Correlation Coefficient (MCC), which is defined as follows. Given the confusion matrix and two classes, positive P and negative N , MCC is computed as TP × TN − FP × FN , MCC := √ (TP + FP)(TP + FN)(TN + FP)(TN + FN) where T stands for true (correctly classified) and F stands for false (misclassified). Note that −1 ≤ MCC ≤ 1, where –1 means perfect discordance, 1 means perfect accordance, and 0 means that predictions are random. The motivation for using MCC for unbalanced datasets instead of accuracy or F1-score is that with the latter a naive predictor can achieve a high score by simply classifying all the instances as the more frequent class (i.e., “easy” in this case). For in-depth motivations supporting the use of MCC for unbalanced datasets, we refer the interested reader to [6]. Implementation Details. All the computations are executed on an HPC cluster running CentOS, having for each node an Intel CPU with 32 physical cores working at 2.1 GHz and 768 GB of RAM. To parallelize the computations as much as possible for our tests, we limit the execution of each job to 16 GB of RAM, and we reserve one core for the sampling procedure, ten cores for the complexification procedure, and one core for running Concorde on each instance. The code used for the complexification procedure is the same as in [22], while the remaining pipeline is implemented in Python.

4 Computational Results We have designed two types of experiments to validate the impact of the four features introduced in Sect. 2 in predicting the empirical hardness of metric TSP instances. The first class of tests targets instances of a fixed size n, while the second class of tests is independent of n. Note that all of our cost features, except SK, range between 0 and 1, and, thus, we expect them to be independent with respect to the number of nodes n of the TSP instance. i Decision Trees for Fixed n. For each sub-sampled dataset Dn, p , we perform the following steps: (i) we split the dataset into training and test sets, (ii) we fit it with a binary decision tree, and (iii) we collect the MCC prediction scores on the test set. Finally, we average over the 10 decision trees indexed by i for each pair of n and p. Table 1 shows the results of our trained decision tree on the sub-sampled datasets i with four different combinations of predictors. First of all, we study the impact Dnp of using only SD(c) and SK(c). These features were already used to predict hard-

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Table 1 MCC prediction scores: mean and standard deviations (in parenthesis). Each entry gives the average over 10 datasets for the corresponding n and p; we highlight in bold the best results for each group Features n p = 0.01 p = 0.05 p = 0.10 p = 0.20 SD+SK SD+SK+GI SD+SK+TI SD+SK+GI+TI SD+SK SD+SK+GI SD+SK+TI SD+SK+GI+TI SD+SK SD+SK+GI SD+SK+TI SD+SK+GI+TI SD+SK SD+SK+GI SD+SK+TI SD+SK+GI+TI SD+SK SD+SK+GI SD+SK+TI SD+SK+GI+TI

20 20 20 20 25 25 25 25 30 30 30 30 35 35 35 35 40 40 40 40

0.223 (0.17) 0.355 (0.19) 0.612 (0.12) 0.750 (0.13) 0.256 (0.15) 0.384 (0.10) 0.505 (0.23) 0.694 (0.22) 0.447 (0.25) 0.660 (0.19) 0.851 (0.12) 0.875 (0.09) 0.462 (0.19) 0.654 (0.20) 0.822 (0.13) 0.844 (0.14) 0.377 (0.15) 0.594 (0.13) 0.791 (0.15) 0.815 (0.12)

0.400 (0.08) 0.498 (0.05) 0.834 (0.05) 0.868 (0.05) 0.519 (0.07) 0.604 (0.07) 0.829 (0.04) 0.882 (0.02) 0.624 (0.07) 0.778 (0.08) 0.960 (0.03) 0.961 (0.03) 0.712 (0.08) 0.842 (0.07) 0.964 (0.03) 0.967 (0.02) 0.644 (0.04) 0.766 (0.06) 0.937 (0.03) 0.953 (0.03)

0.517 (0.05) 0.568 (0.05) 0.850 (0.03) 0.888 (0.02) 0.641 (0.07) 0.712 (0.05) 0.887 (0.02) 0.919 (0.03) 0.719 (0.03) 0.807 (0.03) 0.962 (0.02) 0.965 (0.01) 0.756 (0.03) 0.876 (0.03) 0.968 (0.01) 0.970 (0.02) 0.734 (0.04) 0.852 (0.05) 0.967 (0.02) 0.966 (0.02)

0.550 (0.04) 0.630 (0.03) 0.917 (0.02) 0.930 (0.01) 0.650 (0.04) 0.728 (0.05) 0.912 (0.02) 0.922 (0.03) 0.794 (0.03) 0.878 (0.02) 0.972 (0.01) 0.970 (0.02) 0.835 (0.03) 0.923 (0.02) 0.977 (0.01) 0.982 (0.01) 0.759 (0.02) 0.875 (0.02) 0.976 (0.01) 0.977 (0.01)

ness in [4], together with other predictors, such as the maximum and minimum distance between nodes, mean, and kurtosis. Once adding GI(c), we observe a small improvement. On the contrary, considering SD(c), SK(c), and TI(c) leads to better predictions, suggesting that indeed the tight index represents an important feature of hard instances. In any case, considering all four predictors together yields the best performances in terms of MCC. Moreover, the larger the percentage of hard instances in the test set, the better the decision tree predicts the empirical hardness. Figure 4 shows a decision tree that was trained for an instance with n = 40 and p = 10%. The blue leaf nodes are those corresponding to the “hard” class, while the yellow nodes correspond to “easy” instances. Note that a first level is defined by the TI: if this score is close to zero, instances are more likely to be classified as easy. Note also that hard instances have higher TI (e.g, see node 7). Global Decision Tree. For the second class of tests, which are independent of n, we i datasets, including instances of size n = 45 collected all the instances of the Dnp and n = 50, for which we report additional results in the Appendix A.1. Namely, we define a new dataset D as

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node 1 imp.:0.00 [1200,0]

node 4 imp.:0.00 [1,0]

node 2 imp.:0.07 SD≤ 0.271

node 3 imp.:0.04 GI≤ 0.263

node 15 imp.:0.00 [0,1]

node 5 imp.:0.03 GI≤ 0.439 node 6 imp.:0.02 SK≤ 0.063

node 10 imp.:0.00 [0,8]

node 16 imp.:0.00 [2,0]

node 7 imp.:0.32 TI≤ 0.155 node 9 imp.:0.00 [0,122]

node 8 imp.:0.20 SK≤ 0.062

node 14 imp.:0.44 GI≤ 0.355

node 12 imp.:0.00 [1,0]

node 13 imp.:0.00 [0,4]

node 11 imp.:0.00 [1,0]

Fig. 4 Decision tree with n = 40 and p = 10%

D=



i Dnp .

n,i, p

where the three indices ranges as follows: n ∈ {20, 25, 30, 35, 40, 45, 50}, i ∈ {1, 2, . . . , 10}, and p ∈ {0.01, 0.05, 0.1, 0.2}. Using this large dataset, we train a single global decision tree with scikit-learn using all the default parameters except the tree depth which is limited to 5, to make the trained decision tree easier to interpret. By running 10-fold cross-validation, we obtain an average MCC of 0.964. Thus, we retain our scores to be appropriate in distinguishing between hard and easy instances whenever the easy instances are sampled randomly for the metric cone. To further asses the capability of our scores in distinguishing easy and hard instances, we use our trained DT to predict a different test set. To this purpose, we used another collection of instances available in the literature and described in the next paragraph. Recently, new metric TSP instances were published with different purposes, such as, to introduce hard-to-solve instances in [13, 22, 23], and to study the integrality gap of the subtour elimination constraint polytope in [3, 12, 22]. We have collected all these published instances that have less than 100 nodes, for a total of 282 instances. Using the number fo B&B nodes of Concorde as discriminator criteria, we labeled as “easy” the instances proposed in [3, 12, 18], and as “hard” all the instances in [13, 22, 23]. Table 2 reports the results of our global classification tree for the 282 published instances. First of all, we note a decrease in the performances of the classifier with respect to the one tested with the original dataset. More specifically, the MCC is 0.076, meaning no correlation.

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Table 2 Results on published instances achieved by the global decision tree Family Hard? Correctly predicted TSPLIB Hougardy, 2014 Benoit & Boyd, 2008 Hougardy & Zhong 2020 Zhong, 2021 Vercesi, Gualandi, Mastrolilli, Gambardella, 2021

No No No Yes Yes Yes

4/26 1/17 0/91 17/17 90/91 39/40

Looking more deeply at the values obtained, we observe that our global decision tree is better at predicting hard instances than easy instances. We conjecture three reasons for explaining this observation. The first reason is that our training set only contains easy instances which come from a random sampling procedure, which are thus completely unstructured. On the contrary, the instances proposed in [3, 12] are highly structured, but they are classified as easy for our criteria. Indeed, our global decision tree is unable to capture some features of a regular easy instance. The second reason is that our training set contains instances with at most 50 nodes, while the easy instances of the test set have mostly more than 50 nodes. The third reason for explaining the results of Table 2 might be an implicit consequence of the original motivations of this work: we are focused on cost features that can predict the empirical hardness of an instance, disregarding the features of easy instances. Hence, we did not study easy instances carefully, and we did not try to understand why they are easy. However, the effort to understand hardness is more compelling: knowing the true sources of empirical hardness may lead to the design of ad-hoc algorithms for each specific feature. In addition, it might help to prove or disprove conjectures on the integrality gap of certain polytopes, such as, for instance, the 43 integrality gap conjecture for the subtour elimination polytope. Lastly, we are aware that supervised learning methods, as DTs are, are good at detecting procedural approaches. However, we believe that the difficulty in generalizing of our procedure is mainly due to the three reasons already discussed. As a matter of fact, our classifier correctly predicts also the instances in [13, 23], which are generated by other researchers and with a procedure different than those presented in [22]. For completeness, the global decision tree is reported in Fig. 7 in Appendix A.4.

5 Conclusions and Future Perspectives In this work, we have classified the empirical hardness of the metric TSP with a binary decision tree trained using features that depend exclusively on the distribution of values in the cost vector. The decision trees trained for a given fixed size n show

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statistically outstanding performance for the MCC prediction score. We also trained a global decision tree independent of n by putting all the instances (sampled and complexified) together. We test it in two different ways. First, we do standard 10Fold cross-validation, achieving good performances on average. Then, we test the global decision tree in trying to predict different families of instances yet published in the literature. In this task, our decision tree performs poorly, only predicting correctly the hard instances. In future work, we can improve our work in several directions. First, the procedure introduced in [22] requires a long runtime for large values of n. Hence, we were unable to go over n = 50 despite a very large time limit. Hence, we plan to explore other methods to generate hard-to-solve instances. Furthermore, it would be of interest to mix the dataset already provided with instances generated in another way. This may prevent a potential bias due to the capability of supervised learning of detecting procedural approaches. Second, the cost features we considered are good predictors for hard instances, but they are weaker in predicting easy instances. Thus, we plan to study new features able to detect easiness.

A. Additional Computational Results A.1 Partial Results with n = 45, 50 Note that the procedure introduced in [22] is time-consuming, and the instances we have generated are hard-to-solve. Thus, when n = 45, 50, we were unable to provide enough hard-to-solve instances to build datasets with specific percentages of the hard-to-solve instances. More specifically, for n = 45, we were able to generate and run Concorde 10 times for 266 instances, while with n = 50, only for 75. Thus, to perform our analysis, we reduce the size of the sub-sampled datasets, setting it equal to 1250 when n = 45 and 350 when n = 50. Then, we repeat the pipeline described in Sects. 3 and 4, obtaining Table 3, which is similar to Table 1. Although the datasets have smaller sizes, we draw the same conclusions as in Sect. 4. Note that when n = 50, the performances decrease, both globally and when compared to the usage of the four scores together with SD + SK + TI. We motivate this fact by the small dimension of the training set (only 230 instances).

A.2 Distribution of Numbers of B&B Nodes of Concorde Figure 5 shows the distribution of the number of B&B nodes in log10 scale of the instances we used for training and testing the decision tree. More specifically, we report the number of B&B nodes for n ∈ {20, 25, 30, 35, 40, 45, 50} of the sam-

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Table 3 MCC prediction scores: mean and standard deviations (in parenthesis) for n = 45, 50. Each entry gives the average over 10 datasets for the corresponding n and p; we highlight in bold the best results for each group Features n p = 0.01 p = 0.05 p = 0.10 p = 0.20 SD+SK SD+SK+GI SD+SK+TI SD+SK+GI+TI SD+SK SD+SK+GI SD+SK+TI SD+SK+GI+TI

45 45 45 45 50 50 50 50

0.532 (0.31) 0.589 (0.32) 0.854 (0.13) 0.823 (0.12) 0.370 (0.46) 0.269 (0.42) 0.600 (0.49) 0.670 (0.44)

0.745 (0.07) 0.830 (0.05) 0.970 (0.03) 0.970 (0.03) 0.715 (0.14) 0.765 (0.28) 0.949 (0.07) 0.949 (0.07)

0.776 (0.02) 0.889 (0.04) 0.978 (0.01) 0.977 (0.01) 0.789 (0.10) 0.894 (0.06) 0.973 (0.04) 0.965 (0.07)

0.800 (0.04) 0.926 (0.02) 0.986 (0.01) 0.988 (0.01) 0.834 (0.05) 0.941 (0.04) 0.995 (0.01) 0.992 (0.01)

log10 (B& B) nodes

4

3

2

1

. pl

ha rd

sa m 50

50

.

rd

m pl

ha

sa 45

45

.

rd ha 40

sa m pl 40

.

rd

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sa 35

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.

rd

pl

ha

m sa 30

30

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rd

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ha

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m sa 20

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Fig. 5 Boxplots of the distribution of numbers of B&B nodes required by Concorde to solve the TSP instances to optimality. Note that the time reduces the impact on the number of nodes for instances with n ≥ 30

pled instances, and the hard version of each sampled instance, using the procedure described in [22]. Note that sampled instances require always less than 10 nodes, while hard instances need thousands. The decrease in n = 50 is just apparent: the really hard instances are not solved to optimality within the time limit.

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10

Decision tree depth

8

6

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0 20

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35 40 Number of nodes

45

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Fig. 6 Boxplots of the decision tree depth for size n of the instances

A.3 Decision Tree Depth Versus Size of the TSP Instance Figure 6 shows the distribution of the depth of 40 decision trees we train for each number of nodes n and each percentage p. We observe that the depth decreases while increasing the number of nodes n.

A.4 The Global Decision Tree Figure 7 shows the global decision tree described in Sect. 4. We colored leaf nodes classified as easy with orange, while the ones classified as hard with blue. For each node, we reported a label (node number), the impurity of the node (imp.), and, for the non-leaf-nodes, the splitting attribute: The leftmost node means “less or equal”, while the rightmost one means “greater”. As an example, considering node 13: the splitting value on GI means that in node 14, GI ≤ 0.434, while in node 21, GI > 0.434. Blue nodes are the ones voting for “hard”, while easy nodes are the ones voting for “easy”. Note that a first level is defined by the TI: if this score is close to zero, instances are more likely to be classified as easy. Note also that most of our conjectures are confirmed: hard instances tend to have higher TI (see, e.g nodes 27, 39), non-negative SK (see node 17, where greater of –0.034 is required, and thus most of the instances have non-negative SK), high GI (see, e.g nodes 21 and children). In some nodes, also the phase transition of SD is confirmed [5] (see, e.g nodes 19, 20, 31, 38, and 39).

node 7 imp.:0.13 TI≤ 0.057

node 21 imp.:0.29 TI≤ 0.267

node 15 node 9 imp.:0.04 imp.:0.01 GI≤ 0.330 SK≤ −0.034

node 14 imp.:0.03 SD≤ 0.245

node 13 imp.:0.04 GI≤ 0.434

node 18 node 22 imp.:0.17 imp.:0.48 GI≤ 0.380 SD≤ 0.745

node 12 imp.:0.08 SD≤ 0.258

node 25 imp.:0.07 TI≤ 0.289

node 30 node 33 imp.:0.37 imp.:0.42 SD≤ 0.278 SD≤ 0.277

node 29 imp.:0.29 GI≤ 0.339

Fig. 7 Global decision tree trained on all the available instances

node 38 node 39 node 41 node 42 imp.:0.36 imp.:0.10 imp.:0.00 imp.:0.00 [46, 143] [11, 193] [6, 0] [0, 1]

node 40 imp.:0.24 SD≤ 0.353 node 35 imp.:0.2 [100, 20]

node 37 imp.:0.24 TI≤ 0.167

node 36 imp.:0.26 SD≤ 0.324

node 28 imp.:0.45 TI≤ 0.141

node 10 node 11 node 16 node 17 node 19 node 23 node 24 node 31 node 32 node 34 node 20 node 26 node 27 imp.:0.44 imp.:0.00 imp.:0.41 imp.:0.01 imp.:0.09 imp.:0.42 imp.:0.33 imp.:0.49 imp.:0.29 imp.:0.00 imp.:0.29 imp.:0.46 imp.:0.48 [2, 1] [38, 0] [7, 17] [56, 9435] [42, 821] [59, 135] [12, 54] [1, 255] [38, 10] [24, 111] [15, 9] [83, 60] [27, 36]

node 5 node 6 node 8 imp.:0.00 imp.:0.00 imp.:0.00 [1, 0] [0, 3] [0, 2]

node 4 imp.:0.37 TI≤ 0.058

node 2 node 3 imp.:0.00 imp.:0.22 [33517, 0] SD≤ 0.236

node 1 imp.:0.00 TI≤ 0.056

node 0 imp.:0.37 TI≤ 0.070

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References 1. Applegate, D., Bixby, R., Cook, W., Chvátal, V.: On the solution of traveling salesman problems (1998) 2. Applegate, D.L., Bixby, R.E., Chvatal, V., Cook, W.J.: The Traveling Salesman Problem: A Computational Study. Princeton University Press (2006) 3. Benoit, G., Boyd, S.: Finding the exact integrality gap for small traveling salesman problems. Math. Oper. Res. 33(4), 921–931 (2008) 4. Cárdenas-Montes, M.: Predicting hardness of travelling salesman problem instances. In: Conference of the Spanish Association for Artificial Intelligence, pp. 68–78. Springer (2016) 5. Cheeseman, P.C., Kanefsky, B., Taylor, W.M.: Where the really hard problems are. In: IJCAI, vol. 91, pp. 331–337 (1991) 6. Chicco, D., Jurman, G.: The advantages of the matthews correlation coefficient (MCC) over F1 score and accuracy in binary classification evaluation. BMC Genom. 21(1), 1–13 (2020) 7. Cramér, H.: Mathematical Methods of Statistics, 1946. Department of Mathematical SU (1946) 8. Demaine, E., Gasarch, W., Hajiaghayi, M.: Computational Intractability: A Guide to Algorithmic Lower Bounds. MIT Press (2022) 9. Fischer, T., Stützle, T., Hoos, H., Merz, P.: An analysis of the hardness of TSP instances for two high performance algorithms. In: Proceedings of the Sixth Metaheuristics International Conference, pp. 361–367 (2005) 10. Gent, I.P., Walsh, T.: The TSP phase transition. Artif. Intell. 88(1–2), 349–358 (1996) 11. Gini, C.: Variabilità e mutabilità: contributo allo studio delle distribuzioni e delle relazioni statistiche [Fasc. I.]. Tipogr. di P. Cuppini (1912) 12. Hougardy, S.: On the integrality ratio of the subtour LP for Euclidean TSP. Oper. Res. Lett. 42(8), 495–499 (2014) 13. Hougardy, S., Zhong, X.: Hard to solve instances of the Euclidean traveling salesman problem. Math. Program. Comput. 1–24 (2020) 14. Kannan, R., Monma, C.L.: On the computational complexity of integer programming problems. In: Optimization and Operations Research, pp. 161–172. Springer (1978) 15. Mitchell, T.M., Mitchell, T.M.: Machine Learning, vol. 1. McGraw-Hill, New York (1997) 16. Osorio, M.A., Pinto, D.: Hard and easy to solve TSP instances, pp. 26–30. XXX Aniversario del Programa Educativo de Computación, BUAP (2003) 17. Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., Vanderplas, J., Passos, A., Cournapeau, D., Brucher, M., Perrot, M., Duchesnay, E.: Scikit-learn: machine learning in Python. J. Mach. Learn. Res. 12, 2825–2830 (2011) 18. Reinelt, G.: TSPLIB - a traveling salesman problem library. INFORMS J. Comput. 3(4), 376– 384 (1991) 19. Smith, R.L.: Efficient Monte Carlo procedures for generating points uniformly distributed over bounded regions. Oper. Res. 32(6), 1296–1308 (1984) 20. Smith-Miles, K., van Hemert, J., Lim, X.Y.: Understanding TSP difficulty by learning from evolved instances. In: International Conference on Learning and Intelligent Optimization, pp. 266–280. Springer (2010) 21. van Hemert, J.I., Urquhart, N.B.: Phase transition properties of clustered travelling salesman problem instances generated with evolutionary computation. In: International Conference on Parallel Problem Solving from Nature, pp. 151–160. Springer (2004) 22. Vercesi, E., Gualandi, S., Mastrolilli, M., Gambardella, L.M.: On the generation of metric TSP instances with a large integrality gap by branch-and-cut. Math. Program. Comput. 15(2), 389–416 (2023) 23. Zhong, X.: Lower bounds on the integraliy ratio of the subtour LP for the traveling salesman problem (2021). arXiv preprint arXiv:2102.04765

A Two-Phase Approach to Evaluate and Optimize an Interlibrary Loan Service: The Case Study of Provincia di Brescia Alice Raffaele, Matteo Zavatteri, Fabio Bazzoli, Marco Gussago, and Romeo Rizzi Abstract We analyze the case study of Provincia di Brescia, an Italian public body managing libraries in the province of Brescia, in Northern Italy. The public body offers an interlibrary loan service organized as follows. Libraries are divided into established groups, each associated with a fixed route. According to a predetermined calendar, each library is visited by a courier a few times a week. The transportation of the items in the network is performed by an external logistics firm, selected through a reverse auction. To assess the current implementation of the service in terms of routing and transportation costs, we propose a two-phase approach based on machine learning and mixed-integer linear programming, and we evaluate it on the data of 2019. Then, we discuss its applicability in reality, and provide a few insights. Keywords Public sector · Interlibrary loan service · Machine learning · Clustering · Mixed-integer linear programming · Rich vehicle routing problem · Case study

A. Raffaele (B) Department of Mechanical, Energy and Management Engineering, University of Calabria, Rende, Italy e-mail: [email protected] M. Zavatteri Department of Mathematics, University of Padova, Via Trieste 63, 35131 Padova, Italy e-mail: [email protected] F. Bazzoli Sistema Bibliotecario Sud Ovest Bresciano, Via Ospedale Vecchio 8, 25032 Chiari, Italy e-mail: [email protected] M. Gussago Ufficio Biblioteche, Provincia di Brescia, Via Milano 13, 25100 Brescia, Italy e-mail: [email protected] R. Rizzi Department of Computer Science, University of Verona, Strada le Grazie 15, 37134 Verona, Italy e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Cosmi et al. (eds.), Operations Research and Data Science in Public Services, AIRO Springer Series 11, https://doi.org/10.1007/978-3-031-34546-3_2

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1 Introduction and Motivation The interlibrary loan is a service allowing users of a library to obtain copies of items from other libraries. It includes the phases of borrowing, lending, and document delivery of the materials [4]. Through such a service, libraries can share and exchange their documentary heritage, making more resources available to their users. This is precious when, for instance, libraries are very small, or located in little towns far from the cities. Nowadays, different libraries belong to the same network, sharing the same information system and accessing their own catalogue as well as the ones of other libraries. Given the several free services they offer, public libraries can definitely be defined as a form of public good (see, e.g., [9]). The direct recipient of the work of public libraries is the single user, who has the opportunity to read books, consult newspapers and magazines, view DVDs, and much more, generally without having to pay anything. This also allows individuals from less wealthy families to make use of these goods, in order to access common resources for free. Indirectly, another recipient is the whole community of people itself [2]. Indeed, public libraries help connect and empower citizens; engage in the creation, transmission, and preservation of cultural materials; offer opportunities for people of all ages to learn and develop literacies; provide equitable access to information and services; benefit society; and strive to manage their resources efficiently and effectively, to be able to respond to all users’ needs [8]. The problem of designing, developing, implementing, and monitoring an interlibrary loan service falls precisely into this latter aspect. In this work, which corrects and completes the one in Raffaele [18], we deal with the real case of Provincia di Brescia,1 a public body in Northern Italy managing libraries in the provinces of Brescia and Cremona. In Sect. 2, we describe the current implementation of the interlibrary loan service offered by Provincia di Brescia. In Sect. 3, we review the state of the art related to management and routing of interlibrary loan services. In Sect. 4, we propose a two-phase approach, based on machine learning and mixed-integer linear programming, to tackle the problem. In Sect. 5, we evaluate it on the available data related to loans that occurred in 2019. In particular, we compare the results with the actual costs sustained by Provincia di Brescia. In Sect. 6, we discuss the applicability of the approach, and we provide a few insights for concrete improvements.

2 The Interlibrary Loan Service of Provincia di Brescia Rete Bibliotecaria Bresciana, the library network of Provincia di Brescia, has been active since 1998. Through these years, the consolidation and development of shared services among the libraries allowed Provincia di Brescia to sign a plurality of agreements with a few neighboring entities, such as Rete Bibliotecaria Cremonese, the library network of the province of Cremona. By means of an interlibrary loan service, 1

https://www.provincia.brescia.it.

A Two-Phase Approach to Evaluate and Optimize …

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Fig. 1 Organization of the libraries into clusters, each one depicted with a different colour; the red-house icon is the depot (https://shorturl.at/axEFR)

any user registered in any library of the two networks can access the resources offered by all libraries. Ufficio Biblioteche (i.e., in English, Libraries Office) is the department of Provincia di Brescia which promotes and coordinates the libraries by managing the information systems and the data collected, and also the pickup/delivery operations related to interlibrary loans. Overall, Ufficio Biblioteche supervises 329 municipal, prison, and scholastic libraries in the provinces of Brescia and Cremona, spread geographically over two areas of about 4800 and 1800 km2 , respectively, in Northern Italy. Libraries are divided into nine established groups, or clusters, manually defined by Ufficio Biblioteche (see Fig. 1). A user can search for any items owned by any libraries in the network on the online OPAC platform2 Once a desired item is found, the user can place a request to obtain a copy of it, and also specify the target library where to pick up the item. After a few days, the user receives a notification indicating that the desired copy has been delivered to the specified library. What happens between the moments when the user sends the request and receives this loan ready notice? If the target library owns a copy of the item and this is not on loan, that copy is directly made ready to be borrowed, and thus the interlibrary loan service is not exploited. Otherwise, the request has to be satisfied by another library in the network owning a copy of the item. This is managed as follows. The request appears in the internal management software used by all the librarians in the network. In particular, all libraries owning an available copy of the desired item are ranked according to their closeness to the target library and their clusters. However, as for the service is implemented now, it is not granted that the source library will be neither the closest one nor one in the same cluster, but it will simply be the first one accepting to satisfy the request. Then, the item of the source library is prepared to be sent to the target library. All ready-for-transit copies are put into different boxes, according to the 2

https://opac.provincia.brescia.it.

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cluster each target library belongs to. These boxes also include copies of items that must be brought back to their original libraries, once users have returned them at the end of the loan period. Then, the source library is visited by a courier, which simultaneously picks up all the boxes and delivers the others which were addressed to that library. Ready-for-transit copies are manually set by librarians as in-transit in the management software. The courier temporarily brings all the collected boxes to a depot. In the following days, boxes are rearranged and delivered to their target libraries through other couriers. When target libraries receive the boxes, librarians empty them and prepare the ready-to-be-borrowed copies, notifying the respective users. Note that, at the moment, the management software does not record when copies are picked up by couriers or when they arrive at the depot. Indeed, couriers are neither librarians nor employees of Provincia di Brescia, and they do not have access to the internal management software. Ufficio Biblioteche has an agreement with an external logistics firm, selected through a reverse auction, which handles the transportation of the copies in the whole network through a fleet of vehicles, by travelling a series of weekly routes. In particular, each cluster of libraries is associated with a fixed route, which is scheduled in all its stops and depends on the day of the week. Each library is visited by a courier a few times a week, according to a calendar manually predetermined by considering historical data and libraries opening hours. Both routes and calendars are manually designed by Ufficio Biblioteche, which imposes them on the logistics firm. Couriers always visit all the libraries in the fixed route, even if there are no boxes to deliver or pick up. All routes start from and end in the depot. Every year, the outsourcing of transport required by the interlibrary loan service costs Ufficio Biblioteche a fixed amount, corresponding to the offer made by the logistics company during the reverse auction. To the users, the service is free. But how efficient and convenient is this implementation in terms of routing and transportation costs? In what follows, to provide a first evaluation on the current implementation of the interlibrary loan service of Provincia di Brescia, we will focus only on the eight clusters in the province of Brescia, as required by Ufficio Biblioteche. Indeed, including the libraries in the province of Cremona would imply to consider a secondary depot, located on the border between the two provinces. This secondary depot interacts as an interchange warehouse with the main one in Brescia, but data about these internal exchanges are not available.

3 Related Work In 1982, Merry [15] presented the practicability and financial benefits of several transport schemes for interlibrary lending in the United Kingdom. The main costs of the service were identified into the rental of the vehicles used to deliver the service and the drivers’ wages. In 1989, Min [16] dealt with the delivery operations in the city library systems of Columbus and Franklin (Ohio, US). He proposed a three-step solution algorithm to cluster the libraries, assign a courier to each cluster, and obtain a route for each cluster by solving an extended formulation of TSP. However, this

A Two-Phase Approach to Evaluate and Optimize …

21

did not include the temporal dimension and did not correspond to reality. In 2006, Apte and Mason [2] redesigned the implementation of the service offered by the San Francisco Public Library. They formulated two optimization models to account presorting of material to avoid double handling, cross docking to reduce cycle time of delivery, and workload balancing among delivery routes, but no temporal constraints. To tackle the considered problem, other variants of the Vehicle Routing Problem (VRP) have also been studied. When we deal with distributing and/or collecting items, we talk about VRP with Pickup and Delivery (PDVRP, see Parragh et al. [17] for more details). Subramanian et al. [19] studied the simultaneous PDVRP (i.e., pickup and delivery operations occur at the same visit). They applied a parallel algorithm, with a multi-start procedure and a Variable Neighborhood Descent method, on benchmarks available in the literature. Angelelli and Mansini [1] formulated the simultaneous PDVRP with time windows (PDVRPTW) by also including backhauls (i.e., products to be delivered are loaded at the single depot in the network and picked-up products are transported back to the depot). They developed a Branch-and-Price approach, which they tested on Solomon benchmark sets properly modified, with 20 customers per instance. The problem can also be modelled as a Periodic VRP (PVRP, [3]), where libraries are visited a few times a week, by accommodating service choice in the delivery operations (see, e.g., [7, 11]). In the Paired many-to-many PDVRP (see, e.g., [5]), demands between libraries are paired, and commodities stored at each node are presorted and bundled into groups according to their destinations. However, requests may not be fulfilled by visiting each library only once. Rich VRPs (RVRP) deal with complex constraints and objectives, stimulated by the complex characteristics of real-life VRPs (see Lahyani et al. [13] for more details). Indeed, real instances are typically larger than benchmark ones. In RVRP, distribution companies usually divide customers into clusters to deal with smaller instances and be more efficient, similar to what was done by Mint [16]. Žuni´c et al. [22] proposed a three-phase VRPTW hybrid approach to heuristically cluster and optimally serve the customers of some distribution companies in Bosnia and Herzegovina. Later, in [21], they developed a two-phase approach where first they used a Firefly algorithm to cluster the customers, and then they proposed an algorithm based on 2-opt to compute the routes. They did not include pickup/delivery operations.

4 A Two-Phase Approach In this section, we define the interlibrary loan problem of Provincia di Brescia as a RVRP with pickup and delivery, backhauls, and time windows. Appendix A summarizes the main characteristics of the problem by following the taxonomy introduced by Lahyani et al. [13] to classify RVRP. Given the large number of libraries in the province of Brescia, equal to 210, we apply the first-cluster-then-route philosophy, widely used to tackle VRPs (see, e.g., [10]). In this way, we reduce the size of the problem by obtaining several instances to solve in parallel. Note that this is exactly

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what is already done manually by Ufficio Biblioteche. In Raffaele [18], only the routing part was tackled.

4.1 Phase 1: Clustering Let L := {1, . . . , L} be the set of libraries distributed over the province of Brescia. Each library  ∈ L is associated with a latitude and a longitude value. Let z be the point corresponding to the only depot in the network. Let di j be the distance between any pair i, j of libraries. Similarly, let di z be the distance between the library i and z. The main goals are to compute the ideal number of clusters to group similar libraries together, and to assign each library to exactly one cluster. In what follows, we describe two alternative ways to solve this subproblem. The k-Means Algorithm As first approach, we consider a well-known unsupervised learning technique from Machine Learning (ML), known as k-means algorithm or generalized Lloyd’s algorithm [6, 14], used to solve the k-means clustering problem. Given a set of n points in the d-dimensional space Rd and an integer k, k-means clustering asks to compute a set of k points in Rd called centers, such that the mean squared distance from each point to its nearest center is minimized. The k-means clustering problem is NP- Hard. Lloyd’s algorithm allows finding a local minimum by implementing the following simple iterations: 1. first, it considers any set of k points as centers and, for each one of them, it computes the corresponding neighborhood (i.e., the set of points for which that center is the nearest neighbor); 2. then, for each neighborhood, it computes the centroid, moves the center in that position, and updates the neighborhood itself. The procedure is iterated until some convergence condition is met. The algorithm returns the set of k centroids and the assignment of the points to the computed clusters. The average complexity is O(kn N ), where N is the number of iterations. To apply the k-means algorithm, we need first to determine the optimal number of clusters k. We consider two different methods: the Elbow method [20] and the Average Silhouette (AS) method [12]. The Elbow method simply looks at the mean squared distance from each point to its nearest center as a function of the number of clusters. Given different values of k (e.g., from 2 to 10), for each k it computes the total mean squared distance (also called as distortion score). Then, it plots the curve obtained by putting the k values on the x-axis and the total mean squared distances on the y-axis. By looking at the plot, one should be able to identify a “bend”, called Elbow point, in correspondence of a particular k. This would be the optimal value to choose, since higher values should not improve the total mean squared distance. However, in case the plotted curve is fairly smooth, the Elbow point could be ambiguous. The AS method can be used as an alternative. Similarly to the Elbow method, it considers different values of k. For each k, first it solves the corresponding k-means

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23

clustering problem. Then, for each point in the set, it computes the so-called AS coefficient as the difference between the average distance with the other points in the same cluster and the mean nearest-cluster distance. This value is then normalized by the maximum value between the two. Finally, the method computes the average silhouette coefficient of all points to obtain a score, which evaluates the quality and the density of the clusters. This score ranges between −1 (i.e., completely incorrect clustering) and 1 (i.e., good clustering). The optimal number of k is thus the one obtaining the highest score. Integer Linear Programming The problem of clustering the libraries in the province of Brescia can also be tackled by means of Integer Linear Programming (ILP). Differently from k-means clustering, we also take into account the location of the depot z. The main idea is to let the model select the k centroids of the clusters, by taking into account their distances from the depot, besides the intra-distances within the clusters. We introduce a binary variable y for each library  ∈ L, whose value is 1 when  is a centroid, and a binary variable xi, for each pair of libraries i,  ∈ L, whose value is 1 when i is in the cluster with centroid . The mathematical formulation is the following. min

 ∈L

s.t.





dz y +

(1)

di xi

i,∈L

y = k

(2)

∈L

xi ≤ y ,  xi = 1,

∀i,  ∈ L

(3)

∀i ∈ L

(4)

∀i,  ∈ L

(5)

∈L

xi , y ∈ {0, 1}

The objective function in (1) minimizes the total intra-cluster distance plus the distances of the centroids from the depot. Exactly k libraries are chosen as centroids in Constraint (2). In Constraints (3), a library can be assigned to another library only if this is a centroid. Constraints (4) assign each library to exactly one centroid. Note that k is not a variable but is given as input (as result of the Elbow/AS method).

4.2 Phase 2: Routing Given any cluster among the ones obtained in Phase 1, we consider a time horizon of a day. Let K := {1, . . . , K } be the fleet of available vehicles. Vehicles are heterogeneous, with different capacities q and working times w. All vehicles start their routes from the same depot z, to which they also return at the end. Let L := {1, . . . , L} be the set of libraries to be served in a particular cluster. Each library  can have p items to be picked-up or d items to be delivered. According to the kind of operations

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performed, the service time s to perform the operations varies. Each vehicle is able to do at least one type of operation. Let K ⊆ K be the subset of vehicles in the fleet able to perform the operations required by library . All items to be delivered are already available at the depot z at the beginning of the day, whereas all items picked-up by vehicles’ couriers are brought to the depot z at the end of the day. Each library  has to be served by exactly one vehicle v during its opening window [a , b ], unless the courier of v owns a copy of the keys of . In that case, the service can be performed also after hours, by spending an additional time S to exploit the keys. Let L ⊆ L be the set of libraries whose keys are available, through some copies, to the whole fleet of vehicles. We can model the problem as a complete graph G := (V, A), where V := {z} ∪ L and A is the set of arcs (i, j) such that both i and j belong to V. Every arc is characterized by two values: its travel cost ci j and its travel time ti j , representing the distance from i to j and the time required to go through the arc, respectively. The goal is to build an optimal schedule, i.e., to assign a vehicle to each library and define the route of each vehicle, satisfying all constraints and minimizing the total travelled distance and the number of vehicles used. The problem is modelled as a single-depot PDVRPTW with backhauls, heterogeneous fleet, and drivers regulations, by means of Mixed Integer Linear Programming (MILP). We introduce: a binary variable u v for each vehicle v ∈ K, whose value is 1 when v is used; a binary variable xv for each library  ∈ L and for each vehicle v ∈ K, whose value is 1 when  is served by v; a binary variable h v for each library  ∈ L and for each vehicle v ∈ K, whose value is 1 when the courier of v uses a copy of the keys of  to provide the service requested; a binary variable yi jv for each arc (i, j) ∈ A and for each vehicle v ∈ K, whose value is 1 when v goes through the arc (i, j); a continuous variable li jv for each arc (i, j) ∈ A and for each vehicle v ∈ K, representing the occupied capacity of v when going through arc (i, j); a continuous variable τi for each node i ∈ V, representing the time when service starts in i; two continuous variables αv and Ωv for each vehicle v ∈ K, representing the start and end time, respectively, of the work of v. The MILP formulation is the following. 

min



ci j yi jv + M

v∈K (i, j)∈A

s.t.



(7)

yiv = xv

∀v ∈ K, ∀ ∈ L

(8)

yjv = xv

∀v ∈ K, ∀ ∈ L

(9)

yzv = u v

∀v ∈ K

(10)

yzv = u v

∀v ∈ K

(11)

∀ ∈ L \ L , ∀v ∈ K

(12)



(i,)∈A



(, j)∈A



∈L



(6)

uv

v∈K

∀ ∈ L

v∈K

xv = 1



∈L

h v = 0

A Two-Phase Approach to Evaluate and Optimize … h v ≤

 j∈L

yjv

⎛

τ ≥ a ⎝1 − τ ≤ b + M

v∈K



v∈K

h v ⎠

h v ⎛

τ j ≥ τ + s + tj + S



yzv ⎠

v∈K

 v∈K

⎛

h ,v − M ⎝1 −

 v∈K

αv ≤ τ − tz yzv + M (1 − xv ) Ωv ≥ τ + s + Sh v + tz yzv − M (1 − xv ) li jv ≤ qv yi jv   l zv = d xv

∈L

(13)

∀ ∈ L

(14)

∀ ∈ L

(15)

∀(z, ) ∈ A

(16)

∀(, j) ∈ A

(17)

∀v ∈ K

∀(i, j) ∈ A, ∀v ∈ K

(18) (19) (20) (21)

∀v ∈ K

(22)

∀v ∈ K

(23)

∀ ∈ L, ∀v ∈ K

(24)

∀v ∈ K

(25) (26) (27) (28)

⎞

Ωv − αv ≤ wv u v



∀v ∈ K, ∀ ∈ L ⎞



τ ≥ τz + tz − M ⎝1 −

∈L

25

⎞ yjv ⎠

∀v ∈ K, ∀ ∈ L ∀v ∈ K, ∀ ∈ L

∈L

lzv =



(i,)∈A



∈L

liv +

p xv



i∈V

( p − d )yiv =

u v ∈ {0, 1}, αv , Ωv ≥ 0 xv , h v ∈ {0, 1} yi jv ∈ {0, 1}, li jv ≥ 0 τi ≥ 0

 (, j)∈A

ljv

∀ ∈ L, ∀v ∈ K ∀(i, j) ∈ A, ∀v ∈ K ∀i ∈ V

The objective function in (6) minimizes the total travelled distance and the number of vehicles used, by exploiting a large positive constant M. Constraints (7) state that every library must be served by exactly one vehicle. Constraints (8)–(11) are needed to form a tour starting and ending in the depot z. A courier cannot exploit a copy of the keys of a library when this is not available (Constraints (12)) and, even when it is, the copy may not be used (Constraints (13)). Constraints (14)–(15) ask for a library to be served either during its opening hours or by exploiting a copy of its keys. Constraints (16)–(17) are needed to compute the start of service in the libraries. Constraints (18) guarantee that vehicles do not work more than their maximum working time, and Constraints (19)–(20) define their start and end time of work. Finally, Constraints (21)–(24) handle vehicles capacity. Differently than in Raffaele [18], here we do not introduce a binary variable g for each library  ∈ L, in order to allow a library not to be assigned to any vehicle, but we strictly require each library to be visited. Thus, also the objective function slightly differs from the one in [18], which takes into account the number of unserved customers.

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5 Experimental Evaluation We apply our two-phase approach to the real case of Provincia di Brescia, we report on the results obtained, and we compare them with the actual costs sustained by the public body. We exploited the data related to all interlibrary loans occurring in the province of Brescia from 2019–01–02 to 2019–12–30.3

5.1 Application of Phase 1: Clustering Figure 2a and b show the application of the Elbow method and of the AS Method, respectively, to clustering the libraries in the province of Brescia. As mentioned at the beginning of Sect. 4.1, each library is associated with a latitude and longitude, thus it is described by a point in R2 . By looking at the former, we can identify the Elbow point in correspondence of k = 4. Instead, by considering the results in the latter, the highest scores are obtained with k = 3 and k = 5. Since, in the second phase, we are going to exactly solve instances of a variant of VRP, we also need to consider the size of the clusters. The average number of libraries per cluster is 70, 52.5, and 42, respectively for k = 3, 4, and 5. By taking into account all of these aspects, we decided to set the number of clusters equal to 5. Then, we applied k-means clustering and ILP as described in Sect. 4.1 to get two sets of 5 clusters.

5.2 Instances By considering the two sets of clusters obtained, the weekly calendar of the visits of the libraries, their opening hours, and the data related to loans occurred on 252 d in 2019, we derived two sets of instances, ML_k5 (k-means clustering + MILP) and ILP_k5 (ILP + MILP). Also, we generated a third set, BS_k8, by taking into account the clusters currently used. An instance refers to a specific day and a specific cluster, indicating which libraries in that cluster require some pickup/delivery operations on that day. Table 1 summarizes the main features of the three sets, where the size of an instance is the number of libraries to be served.

5.3 Application of Phase 2: Routing We implemented the model described in Sect. 4.2 in Python and solved it over all instances by using Gurobi 9.5.1 on a machine with Ubuntu 20.04.2 LTS (GNU/Linux 3

We point out that users’ names were all anonymized to guarantee their privacy.

A Two-Phase Approach to Evaluate and Optimize …

(a) Elbow method (k = 2, . . . , 10).

27

(b) AS method (k = 3, 4, 5).

Fig. 2 k-means clustering of the libraries in the province of Brescia. a Shows the distortion scores normalized from 1 to 10 and the time needed to train the model. b Shows the AS coefficient for each k and the density of the clusters Table 1 Sets of instances derived from the data of 2019 Set name Number of Number of Min size clusters instances ML_k5 ILP_k5 BS_k8

5 5 8

1209 1208 1509

3 2 1

Max size

Avg size

30 29 25

16.03 16.04 12.84

5.4.0-90-generic x86_64), 16 core, 2 GHz, 32 GB RAM.4 We set a time limit of 60 min for each instance. We relied on Open Source Routing Machine5 to get shortest distances and times between any two points in the network. Table 2 shows the results obtained for each set of instances: the number of clusters, the total distance travelled, the number of instances optimally solved by the time limit, and minimum, maximum, the average gap of remaining instances, the total running times taken (in hours) for the whole set and for instances solved to optimality, and the average time (in minutes) to solve an instance to optimality. Overall, the computation took about ≈ 240.35 h ≈ 10 d. In every set, most instances are optimally solved. For each set, we aggregated the solutions obtained in the instances according to the date, in order to compute the number of vehicles used on a same day. Figure 3 shows, per each set, the percentages of days in 2019 when different numbers of vehicles are used. We can see that, in 57.94% of the days in 2019, the solutions of BS_k8 require 7 vehicles. Instead, the solutions of both ML_k5 and ILP_k5 use at most 6 vehicles (5.18 on average) on a same day. The overall best result, in terms of total distance travelled and average number of vehicles used on a same day, is obtained by the ML_k5 set, with a total of 218631 km travelled and at most 6 vehicles used on a same day.

4

The virtual machine is hosted by the Centro Piattaforme Tecnologiche (CPT) of the University of Verona (https://cpt.univr.it/en). 5 http://project-osrm.org/.

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Table 2 Results obtained by Gurobi over the three sets of instances Set name

Number of clusters

Total distance (km)

Number of optimal instances

Min gap (%)

Max gap (%)

Avg gap (%)

Total runtime (h)

Total runtime optimal (h)

Avg runtime optimal (min)

ML_k5

5

218631

1113 (92.06%)

0.02

0.18

0.11

128.05

32.05

1.73

ILP_k5

5

219761

1163 (96.27%)

0.09

0.43

0.21

64.78

19.78

1.02

BS_k8

8

220709

1463 (96.95%)

0.18

0.25

0.22

47.52

1.52

0.06

4 20.24%

6 38.49%

5 41.27%

(a) ML k5 (avg: 5.18).

4 20.63%

6 38.49%

5 40.87%

(b) ILP k5 (avg: 5.18).

5 20.63% 7 6 57.94% 21.43%

(c) BS k8 (avg: 6.37).

Fig. 3 Distribution of the number of vehicles used on different days in the solutions of the instance sets

Unfortunately, the information about the actual total distance travelled by vehicles in 2019 is not available. Anyway, we estimate it as follows. We consider the fixed routes established by Ufficio Biblioteche. Then, by taking into account the weekly calendar, every time a route is travelled on a particular day, we compute its theoretical distance by adding the shortest paths between the points in the route. In the end, we get a quite optimistic value, which nonetheless allows us to make a comparison. The theoretical total distance travelled by the vehicles in 2019 is 228945 km. In the BS_k8 set, by simply applying Gurobi on the current clusters adopted by Provincia di Brescia, we get an improvement of at least 8236 km by using 7 vehicles. Instead, the ML_k5 set improves the theoretical total distance of at least 10314 km by using 6 vehicles. About the costs, we know that the amount paid by Provincia di Brescia in 2019 was 296460 e.6 For each of the three sets of instances, we can provide a rough estimate of the cost as follows. We assume Provincia di Brescia buys the needed vehicles to carry out the service. The two types of vehicles typically used in this case are Fiat Talento and Citroen Jumpy, both having an average cost of about 32000 e per 6

https://www.provincia.brescia.it/sites/default/files/allegati/documenti/22843/dd_91_2016.pdf.

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29

Table 3 Comparison between an estimate of the costs of the three sets of instances and the actual cost sustained by Provincia of Brescia in 2019 Set name Cost estimate (e) Estimate versus actual ML_k5 ILP_k5 BS_k8

190711.29 190826.61 218990.02

–105748.71 e (–35.67%) –105633.39 e (–35.63%) –77469.98 e (–26.13%)

unit. Then, we consider vehicles’ fuel consumption. Since, in urban and extra urban roads, a Fiat Talento usually consumes 6.9 lt/100 km,7 whereas a Citroen Jumpy 6 lt/100 km,8 we take an average value of 6.5 lt/100 km. The fuel cost is set to 1.57 e/lt, which was the average price in Italy in 2019.9 We assume the useful life of a vehicle corresponds to about 200000 km. In all the three sets, the average distance travelled by a vehicle in one year is between 31000 and 36600 km. Thus, we assume the bought vehicles will last at least 3 years, and that every year the total distance travelled is the same. Finally, we set the salary of the couriers to 1450 e/month.10 We get: vehicles_cost := 32000 · number _vehiclesset consumption_cost_ per _3_year s := (total_distanceset /(100/6.5)) · 1.57 · 3 courier s_salar y_ per _3_year s := 1450 · 12 · nb_vehiclesset · 3. set_cost_ per _year := (vehicles_cost + consumption_cost + courier s_salar y)/3.

Table 3 shows the resulting estimates for the three sets, compared with the amounts paid by the public body in 2019.

6 Discussion and Conclusions Our two-phase approach shows that there is space for improvement in the current implementation of the interlibrary loan service by Provincia di Brescia. The number of clusters into which to divide the libraries could be decreased to 5. Also the number of vehicles could be reduced, to travel at least 8236 km less. Even if the public body keeps the current eight clusters, results in Sect. 5.3 indicate that the travelled distance and the cost could still be decreased. In our approach, vehicles only visit the libraries which actually need the service. Instead, in the current implementation, a courier “blindly” follows a fixed route by visiting every library, even if it does not require any operations. This could be simply improved as follows: 7

https://www.auto-data.net/en/fiat-talento-van-1.6-ecojet-125hp-l2h2-24706. https://www.auto-data.net/en/citroen-jumpy-iii-m-1.6-bluehdi-95hp-43908. 9 https://dgsaie.mise.gov.it/prezzi-annuali-carburanti. 10 https://www.glassdoor.co.in/Salaries/italy-driver-salary-SRCH_IL.0,5_IN120_KO6,12.htm. 8

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during the night, before the start of the working day, a query to the database could be performed to retrieve, for each cluster, only the in-transit or ready-for-transit items, together with their corresponding target and source libraries, respectively. Then, for each cluster, an instance of the routing subproblem could be obtained and solved to get a feasible (hopefully optimal) route. The predetermined weekly calendar could be theoretically discharged, and clusters could change as well day by day. However, even if this dynamic system may be better in terms of savings, relying on fixed clusters and a fixed calendar may still be a preferable method according to couriers and/or librarians. Indeed, if routes completely change every day, couriers would not be able to exploit their previous knowledge on roads or towns, and some delays could occur. Also, without a known-in-advance visiting time, librarians would not have a fixed schedule to follow, in order to prepare the items to be picked-up on time. Anyway, we believe our two-phase approach can be a good compromise. Since we include time-window constraints about libraries opening hours, we expect that, on the same day in different weeks, the schedule to follow would not drastically change. Moreover, we suggest Provincia di Brescia to start keeping track of the status of picked-up items when they arrive at the depot. This would require the external logistics company to be connected to the information system of Ufficio Biblioteche. Nonetheless, it would allow users to have more information about their reservations, and Ufficio Biblioteche to monitor any delays in the delivery of the service due to items that have been at the depot for too long. In order for they to understand whether and where there are slowdowns in the whole service, we also performed some statistical analysis on the real data of 2019, of which we report a sample in Appendix B. Given this analysis, a straightforward extension could be to redefine the weekly calendar by computing new frequencies of visits. Finally, another future direction may be to integrate the libraries in the province of Cremona to manage the exchanges at the border of the two provinces.

A. The Interlibrary Loan Problem of Provincia di Brescia as a RVRP Table 4 shows the main characteristics of the problem of optimizing the interlibrary loan service of Provincia di Brescia by following the taxonomy introduced by Lahyani et al. [13] to classify RVRP.

B. Statistical Analysis Performed on the Data of 2019 To evaluate the service not only in terms of routing costs, we performed a statistical analysis on the data of 2019. For each library in every cluster, we computed the average and median pickup/delivery copies and days of delay, the ratio between

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31

Table 4 Characteristics of the interlibrary loan problem of Provincia di Brescia, according to the taxonomy by Lahyani et al. [13] Feature Characteristics Input data Decision management components Depot Operation type Load splitting Planning period Multiple use of vehicles Vehicles

Time constraints

Time window structure Incompatibility constraints Specific constraints Objective function Formulation Real case study and instances Methods

Deterministic Location (clustering) and routing Single Pickup and delivery Backhauls Splitting not allowed Single Single trip/Multi-trip Heterogeneous Fixed number Not compartmentalized Capacity constraints No loading policy Drivers regulations Restriction on customer Restriction on depot Service time Single time window No Grouping libraries Drivers may own copies of libraries keys Single objective Yes Yes Two-phase approach (ML/ILP + MILP)

pickup/delivery copies and days of delay, and the ratio between the actual and the theoretical number of visits of a courier, as shown in Fig. 4.

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(a) Main statistics about pickup/delivery copies and days of delay.

(b) Comparison between pickup/delivery copies.

(c) Comparison between pickup/delivery days of delays. Fig. 4 Samples of the statistical analysis performed on a cluster of libraries currently adopted by Provincia of Brescia

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References 1. Angelelli, E., Mansini, R.: The vehicle routing problem with time windows and simultaneous pick-up and delivery. In: Quantitative Approaches to Distribution Logistics and Supply Chain Management, pp. 249–267. Springer (2002) 2. Apte, U.M., Mason, F.M.: Analysis and improvement of delivery operations at the San Francisco Public Library. J. Oper. Manag. 24(4) (2006) 3. Beltrami, E.J., Bodin, L.D.: Networks and vehicle routing for municipal waste collection. Networks 4(1), 65–94 (1974) 4. Boucher, V.: Interlibrary Loan Practices Handbook. ALA (1997) 5. Huey-Kuo Chen, Huey-Wen Chou, Che-Fu Hsueh, and Yen-Ju Yu. The paired many-to-many pickup and delivery problem: an application. TOP, 23(1), 2015 6. Faber, V.: Clustering and the continuous k-means algorithm. Los Alamos Sci. 22(138144.21), 67 (1994) 7. Francis, P., Smilowitz, K., Tzur, M.: The period vehicle routing problem with service choice. Transp. Sci. 40(4), 439–454 (2006) 8. Fraser-Arnott, M.: Exploring public library identity through mission statements. Public Library Quarterly 41(3), 236–256 (2022) 9. Galluzzi, A.: Biblioteche pubbliche tra crisi del welfare e beni comuni della conoscenza. rischi e opportunità. Bibliotime 14(3) (2011) 10. Gillett, B.E., Miller, L.R.: A heuristic algorithm for the vehicle-dispatch problem. Oper. Res. 22(2), 340–349 (1974) 11. Huerta-Muñoz, D.L., Archetti, C., Fernández, E., Perea, F.: The heterogeneous flexible periodic vehicle routing problem: mathematical formulations and solution algorithms. Comput. Oper. Res. 141 (2022) 12. Kaufman, L., Rousseeuw, P.J.: Finding Groups in Data: An Introduction to Cluster Analysis. Wiley (2009) 13. Lahyani, R., Khemakhem, M., Semet, F.: Rich vehicle routing problems: from a taxonomy to a definition. Eur. J. Oper. Res. 241(1) (2015) 14. Lloyd, S.: Least squares quantization in PCM. IEEE Trans. Inf. Theory 28(2), 129–137 (1982) 15. Merry, K.J.: Transport schemes for interlibrary lending: the UK experience. Interlend, Rev (1982) 16. Min, H.: The multiple vehicle routing problem with simultaneous delivery and pick-up points. Transp. Res. Part A: General 23(5), 377–386 (1989) 17. Parragh, S.N., Doerner, K.F., Hartl, R.F.: A survey on pickup and delivery problems. J. für Betriebswirtschaft 58(1), 21–51 (2008) 18. Raffaele, A.: From optimization to listing: theoretical advances in some enumeration problems. PhD thesis, University of Trento (2022) 19. Subramanian, A., Drummond, L.M.A., Bentes, C., Ochi, L.S., Farias, R.: A parallel heuristic for the vehicle routing problem with simultaneous pickup and delivery. Comput. Oper. Res. 37(11), 1899–1911 (2010). Metaheuristics for Logistics and Vehicle Routing 20. Thorndike, R.L.: Who belongs in the family. In: Psychometrika. Citeseer (1953) 21. Žuni´c, E., Delali´c, S., Donko, D., Šupi´c, H.: Two-phase approach for solving the rich vehicle routing problem based on firefly algorithm clustering. In: Proceedings of Sixth International Congress on Information and Communication Technology, pp. 253–262. Springer Singapore, , Singapore (2022) 22. Žuni´c, E., Donko, D., Šupi´c, H., Delali´c, S.: Cluster-based approach for successful solving real-world vehicle routing problems. In: 2020 15th Conference on Computer Science and Information Systems (FedCSIS) (2020)

Optimising the Order Picking Process: The Case of a Retail Distribution Company in Italy Luigi Di Puglia Pugliese, Francesca Guerriero, Giusy Macrina, Massimiliano Matteucci, and Veronica Mosca

Abstract A correct warehouse management is a key activity for an enterprise; among the involved processes, there is the order picking management, that includes the withdrawn of items ordered by customers, their packing, and shipment. In this paper the order picking problem is modelled as a variant of the vehicle routing problem: picking points can be viewed as customers to be visited, while trolleys handled by operators can be seen as vehicles. The proposed model has been defined in such a way to represent the specific scenario of an Italian retail distribution company. A preliminary computational study on real-based instances is carried out. Numerical results show that the implemented methods outperform the strategy currently applied by the company. Keywords Order picking · Warehouse optimisation · Vehicle routing problem · Picker routing problem

L. Di Puglia Pugliese Istituto di Calcolo e Reti ad Alte Prestazioni, Consiglio Nazionale delle Ricerche, Rende, Italy e-mail: [email protected] F. Guerriero · G. Macrina (B) · V. Mosca Department of Mechanical, Energy and Management Engineering, University of Calabria, Rende, Italy e-mail: [email protected] F. Guerriero e-mail: [email protected] V. Mosca e-mail: [email protected] M. Matteucci Gruppo PAC2000A Conad, San Giustino, Umbria, Italy e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Cosmi et al. (eds.), Operations Research and Data Science in Public Services, AIRO Springer Series 11, https://doi.org/10.1007/978-3-031-34546-3_3

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1 Introduction Warehouse management is one of the most difficult and crucial operations in logistics [1]: it includes the phase of goods handling, their storage, inventory management, goods collection, order consolidation, and their distribution and delivery. Among these activities, picking is one of the most important processes, as it involves high labour costs. Indeed, it is estimated that the cost of picking and distribution accounts for almost the 80% of the entire logistic cost. Aboelfotoh et al. [2] split picking in four sub-activities, i.e., (i) picker movement towards the location of the item, (ii) searching for the item on the shelf, (iii) item retrieval, and, if necessary, (iv) item sorting. The movement of the operator is the most expensive activity: it is estimated that it accounts for 60% of the picking costs and occupies 50% of the picking time. There are different methods to make the withdrawal; however, the most common picking methodologies are reported in what follows: – Single Order Picking: a single order is assigned to a single picker, who will retrieve in the warehouse the goods which compose the order. – Batch Picking: different orders are merged in a “batch”; each batch has a capacity, and will be assigned to a single picker; among the strategies implemented for the consolidation of batches [1], there is the First Come—First Serve (FCFS), in which orders are clustered according to their arrival time, or Time Window Batching, that is orders arrived in the same time window are batched together. – Item Picking: each picker is assigned to a specific item, and he will be able to collect only that kind of good; collected goods then will be carried in to the packing area, in which they will be consolidated in their shipments. – Zone Picking: warehouse is split into different zones, and each picker is assigned to a specific zone; every operator can pick goods only from the assigned zone. – Wave Picking: there is a picker who picks orders that are planned for a specific temporal slot; indeed, in this strategy, to better coordinate warehouse operations, picking is planned for specific periods of time during the day. – Zone-Batch Picking: it is the combination of the Zone Picking and the Batching strategies. – Zone-Batch-Wave Picking: it is the combination of the Zone-Batching and the Wave Picking strategies. Given its practical importance, the order picking problem has attracted the attention of many researchers. It is often approached as a special case of the Traveling Salesman Problem (TSP) [3, 4], and several heuristics (e.g., S-Shape, Largest Gap, Combined etc.) [2, 5] and meta-heuristic algorithms [6, 7] have been developed for its solution. The identification of the shortest route to visit all the required locations in the warehouse, known in the scientific literature as Picker Routing Problem—PRP [3], is a crucial issue in the order picking process. Briant et al. [3] studied the Joint Order Batching and Picker Routing Problem (JOBPRP) where the total walked distance to collect items is minimized. The key decision in this problem is how to group

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37

customers’ orders, i.e., the Order Batching Problem (OBP), since the item location determines the route in the warehouse. In the studied problem, the authors assumed that each picker uses a trolley to carry a maximum number of boxes (with a given capacity), where the withdrawn items are stored. The route of the operator is determined by solving the PRP. Aerts et al. [7] proposed a new formulation for the JOBPRP, dealing with the Clustered Vehicle Routing Problem (CluVRP): a variant of the capacitated VRP, in which customers are grouped into clusters, and each cluster needs to be visited by the same vehicle. Items that belong to the same order are forced to be processed together (the “order integrity rule”). To solve this problem, a two-level variable neighborhood search (2level-VNS) meta-heuristic is applied. Chen and Lin [8] addressed the JOBPRP defining a two-phase algorithm: in the first phase, the orders are merged to obtain batches; in the second one, the route planning for pick-ups is determined. The picking route is obtained using a greedy algorithm, which works as follows: once a list of items is assigned to a picker, the visiting tour starts from the warehouse entrance; hence, the algorithm chooses from the list the next location to be visited using a greedy criterion, i.e., the nearest one. Afterwards, the closest order-picking point from the current position of the picker from the list is selected as next destination. The process is repeated until each item of the list is visited, and the picker returns to the starting point. In the grocery retail industry, Chabot et al. [9] proposed a complex order picking problem under weight, fragility, and category constraints (OPPWFCC); each item has its weight: once the total weight of the products transported on the hand-lift truck exceeds a threshold value, heavy items can no longer be collected, and the picker is only allowed to pick light items; fragile products cannot be placed underneath heavy products, hence heavy products must be positioned on the bottom of the pallet, light products on the top instead. This problem aims at minimizing the total distance traveled under the assumption that the distance between the two sides of an aisle is large, such that the picker cannot pick items from both sides at the same time. Su et al. [10] considered the specific scenario of a multi-block warehouse. This layout is more difficult to manage than the ordinary one, thus a Steiner TSP model is used to mathematically represent the PRP. The authors proposed some enhanced versions of classical approaches proposed for the TSP to handle the problem under study. As in [7, 9], in this paper, we model the order picking problem as a variant of the VRP, in which the items to be withdrawn are customers to be visited, each one characterised by an item demand. As in [9] we take into account the fragility constraints. However, in our problem we consider a heterogeneous fleet of pickers combined with the presence of the fragility constraints, modeled as precedence constraints. In addition, we consider two objective function, i.e., the minimization of both the number of pickers and the makespan. The proposed formulation is used to solve the real-life instances derived from the case study. The aim is to determine a better way to split the order into the lists involving fewer resources, and to find routes that allow us to reduce the time required for the picking activities, compared with the current performance achieved

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by the company. The remainder of the paper is organised as follows. In Sect. 2 we introduce the mathematical model, in Sect. 3 we present the computational study with its results; Sect. 4 concludes the paper.

2 Mathematical Model In this Section we present the mathematical formulation of the picking problem. In our framework, a customer order, composed of several items, must be divided in several lists. Each list is assigned to a picker who must visit all the locations where the items, belonging to the assigned list, are located and collect them. This problem is formulated as a variant of the VRP, in which the items are all the locations that must be visited, and the lists represent the tours. The objective is to minimize both the makespan and the number of employed pickers. Firstly, we introduce the sets and parameters of the problem, then we explain the model with its variables and constraints. Let O be the set of items composing the order associated with a customer; a picking point i ∈ N is associated with each product of the order, where N represents the set of all picking points to be visited. The parameter si represents the required time to collect the product located at i, while di is the amount of product that has to be collected. Each product belongs to a product category: let K be the set of categories, k(i) represents the category associated with the product located at the picking point i. In order to respect the fragility constraints, hence, avoiding that fragile items are placed on the bottom of the support, product categories are ordered. Let k(i) and k( j) be two product categories that are associated to the picking point i and j, respectively, and such that k( j) > k(i). If i and j are visited by the same picker, then it is not possible to visit the picking point i after the picking point j. The picker can use two different types of support to collect products, i.e., pallets or rolls. We identify each support as a vehicle, and the index t ∈ { p, r } indicates the type of vehicle used, p identifies the pallet, whereas r identifies the roll. Every support has its maximum capacity expressed in volume, in particular Q p is the pallet capacity, whereas Q r is the roll capacity; sot represents the required time to pick up the support of type t. Each route starts at the location ot where the operator picks the support of type t, and it ends at the bay o where the products of the order are collected to be shipped. The parameter ti j represents the shortest time from location i to location j. The objective is to visit each picking point i ∈ N using the minimum number of vehicles f (x) and minimizing the makespan f (τ ), or the necessary time to complete the route with the longest completion time. For the sake of clarity, the sets and parameters are listed in what follows. Sets and Parameters: – O; set of items associated to an order; – N ; set of picking points associated to the items o ∈ O, e.g., |N | ≡ |O|;

Optimising the Order Picking Process …

– – – – – – – – – – –

39

t; type of vehicles: t = p pallets, t = r rolls; ot ; origin of the vehicles of type t; o; destination of vehicles; K ; set of product categories; k(i), i ∈ N ; product category of the item associated to the picking point i ∈ N ; di , i ∈ N ; demand of the picking point i ∈ N expressed in volume; si , i ∈ N ; picking time required to load item demand associated to the picking point i ∈ N ; sot ; retrieving time necessary for the vehicle of type t; Q p ; volume capacity of vehicle of type p; Q r ; volume capacity of vehicle of type r ; ti j , i ∈ N ∪ {o p , or }, j ∈ N ∪ {o}; the shortest travel time from picking point i to picking point j.

The definition of the decision variables is given in what follows – xit j , ∀i, j : k(i) ≡ k( j), t ∈ {r, p}, binary variable that assumes value of 1 if the vehicle of typology t crosses the arc (i, j) where i and j belong to the same product category, 0 otherwise; – yit j , ∀i, j : k( j) > k(i), t ∈ {r, p}, binary variable that assumes value of 1 if the vehicle of typology t visits the picking point j immediately after visiting the picking point i, 0 otherwise; – u i , ∀i ∈ N ∪ {o}, continuous non-negative variable that indicates the residual capacity of vehicles at the picking point i ∈ N ∪ {o}; – τi , ∀i ∈ N ∪ {o}, continuous non-negative variable that indicates the arrival time at picking point i ∈ N ∪ {o}. Minimize f (τ ) = τo  p  Minimize f (x) = xo p j + xor r j j∈N

s.t.



  t

z∈N ∪{o}

(1b)

j∈N

xitz −

z∈N ∪{o}

(1a)



x tf i +

f ∈N ∪{ot }

xitz +

 t



yit j −

j∈N



t yhi =0

∀i ∈ N , t

(1c)

∀i ∈ N

(1d)

∀j ∈ N

(1e)

∀j ∈ N

(1f)

∀i ∈ N , j ∈ N ∪ {o}

(1g)

∀j ∈ N

(1h)

∀j ∈ N

(1i)

h∈N

yit j = 1

j∈N

  p u j ≤ Q p + M 1 − xo p j   u j ≤ Q r + M 1 − xor r j      t t u j ≤ u i − di + M 1 − yi j + xi j t

  p τ j ≥ so p + to p j − M 1 − xo p j   τ j ≥ sor + tor j − M 1 − xor r j

t

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L. Di Puglia Pugliese et al.  τ j ≥ τi + si + ti j − M 1 −

  t

yit j +



 xit j

∀i ∈ N , j ∈ N ∪ {o}

(1j)

t

x, y ∈ {0, 1}, u, τ ≥ 0

(1k)

The Eq. (1b) define flow conservation constraints. The Eq. (1c) impose that each picking point i ∈ N has to be visited only one time. The Eqs. (1d)–(1f) define residual capacity u of vehicles. The Eqs. (1g)–(1i) define the arrival time τ to the picking point i ∈ N and to the arrival point o. The domain of decision variables is defined by (1j). In order to obtain a set of efficient solutions, we adopt two strategies which allow to deal with a single-objective model, i.e.: 1. Minimising the convex combination of the two functions; 2. Using the -constraint method. For the first method, we minimize the function: f (λ) = λ ∗ f (τ ) + (1 − λ) ∗ f (x),

(2)

where 0 ≤ λ ≤ 1 is used to obtain the weight associated with each function. Based on the values assigned to λ, different efficient solutions can be determined. The entire Pareto front can be obtained only if it is convex. In general, the Pareto front is not convex, hence only supported solutions can be obtained. To overcome this issue, we consider the second technique, i.e., the -constraint method, which allows to obtain single-objective sub-problems from the original problem, adding further constraints [11]. In particular, we obtain the new problem by transforming one of the objectives into a constraint [12]. The single-objective function maintained is the makespan f (τ ); thus, the number of departures f (x) is handled as a constraint. Minimize f (τ ) s.t. f (x) ≤  x∈X

(3a) (3b) (3c)

In order to define the -constraint method, we first determine both an upper bound (UB) and a lower bound (LB) on the value of f (x) by solving the model (1b)– (1j) twice: first the number of departures is maximized (i.e., [max f (x)|x ∈ X ]), second f (x) is minimized (i.e., [min f (x)|x ∈ X ]), where X is the solution space of x, considering the constraints (1c)–(1k). Then, we solve different instances, by selecting an integer value of  (see constraint (3b) from the interval [L B; U B]). Since the number of items in the lists is known and finite, fixing a bound on the number of trolleys to be used is reasonable. In addition, in view of the fact that the number of trolleys can assume only integer values, the number of sub-problems,

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aimed at minimizing the makespan imposing this bound, that are generated by the -constraint method, is limited. Solving these -constraint problems [13], we find the Pareto solution set, that constitutes the Pareto Front.

3 Computational Results In this Section, we present the numerical results collected on instances generated by considering a real case study. We implement the model with the convex combination (2), named CC and the -constraints formulation (3a)–(3c), named -C. Referring to the CC method, since the order of magnitude for f (x) and f (λ) are different, we multiply f (x) by 100. The aim is to compare the solution obtained by solving CC and -C with the performance currently achieved by the company. We implemented both CC and -C in Java language and solved using Cplex 12.10 on an Intel(R) Core(TM) I7-11850H CPU at 2.50 GHz 64 GB of RAM machine.

3.1 Case of Study The company resorts to the Single Order Picking strategy for warehouse operations: each customer sends some orders, the orders are divided into n lists, and each list is assigned to a single operator. The lists are generated by a software system, which considers both the volume of the items and the maximum trolley capacity. Once a list is assigned to a picker, the operator can switch to another list only when every item of the current list is withdrawn, hence, batch picking is not allowed. The adopted strategy to move in the warehouse is the Picker-to-Parts: there are mostly traditional shelves, so the picker moves towards the pick-up stations. The operators are guided by a Voice Picking System, that indicates, once one item is withdrawn, in which station to move. To move within the warehouse each picker must follow an S-Shape heuristic, while to move among the same aisle the operator follows a Z-Shape heuristic. Each aisle can be crossed just in one direction; this also prevents collisions between different operators within the same corridor. The customer sends one or more orders. If the order is small enough, it can be processed as a single list; otherwise it is split into different lists; each list can contain a maximum number of items (based on the maximum volume that can be carried in a single tour). There are more than 9,000 products in the warehouse: each product is located in a single picking point, and picking points are uniquely identified. To solve the problem, the warehouse has been modeled as a graph; each aisle has a direction of travel which must necessarily be respected (this is also done for safety reasons, to avoid accidents between pickers who work in the same aisle); for this reason, the nodes of the graph are connected by arcs. In the implemented model, customers

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represent the picking point to be visited, while the vehicles represent the two different types of support that the operators can pick up: in fact, they can pick 2 pallets or 3 rolls, hence, we have these two types of vehicles, each one with its own capacity. In our preliminary computational study, presented in Sect. 3.2, we tested the proposed model on orders containing different numbers of products derived from the real-case study. In particular, we considered 3 small real orders provided by the company: the first one contains 12 products, the second one 21, and the last one contains 23 products.

3.2 Numerical Results Comparison Between CC Method and -C Method: The first order tested is the one containing 12 products. We tested the model with the CC method, considering different values for λ; the obtained results are shown in Table 1. In particular, in the first column we report the value of λ, in the second and third columns the value of the objective functions f(x) and f(τ ), respectively, in the last one the resolution time [seconds]. Looking at Table 1, for λ → 0.1, i.e., we give more importance to the minimisation of departures, we obtain a makespan of about 654 s by activating just one route. For λ → 0.9, i.e., giving more importance to the makespan, 4 different routes are activated, reaching a makespan of about 279 s. When considering λ → 0.5, i.e., each function has the same importance, a mid-term result is obtained, activating 3 different routes with a makespan of about 315 s. We test the same instance by using the -C method, we depict the related results in Table 2. We made our tests by varying  in the range defined in Sect. 2, however, in Table 2 we depict the results up to the maximum value after that -C does not find any new non-dominated solution. In this case, the lower bound and upper bound values on the number of departures are equal to 1 and 12, respectively, however, we reported the results from 1 to 9. As shown in Table 2, when we set the value of  equal to the lower bound on the number of departures, i.e.,  = 1, we obtain a makespan of about 654 s. We

Table 1 Results of the CC Method on the order containing 12 products λ value f(x) f(τ ) Resolution time λ = 0.9 λ = 0.8 λ = 0.7 λ = 0.6 λ = 0.5 λ = 0.4 λ = 0.3 λ = 0.2 λ = 0.1

4 4 3 3 3 2 1 1 1

278.84 278.84 314.92 314.92 314.92 422.09 654.08 654.08 654.08

180.00 369.00 720.00 780.00 720.00 399.00 320.00 54.00 6.00

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Table 2 Results of the -C method on the order containing 12 products  f(x) f(τ ) Resolution time 1 2 3 4 5 6 7 8 9

1 2 3 4 5 6 7 8 9

654.08 422.09 314.92 278.84 274.72 267.52 261.04 258.12 252.64

6.00 120.00 123.00 180.00 120.00 60.00 30.00 60.00 6.00

Fig. 1 Pareto front obtained with CC and -C methods

observe that the same efficient solution is obtained by the CC method considering λ equal to 0.1, 0.2, and 0.3. By varying the value of , we determine a set of efficient solutions, for which the minimum value of the makespan (i.e., 252.64 sec) is obtained when 9 routes are considered. The same makespan is found for higher  values (i.e.,  = 10, 11, 12). Thus these last three solutions are dominated by the others (i.e., they do not belong to the Pareto front). We observe that with the CC method we do not obtain this Pareto optimal solution. Figure 1 depicts the Pareto front obtained with CC and -C methods. Looking at Fig. 1 it is evident that the two methods find a similar Pareto front, with the exception of more additional efficient solutions found by -C. We have tested our strategies also on two additional instances with 21 and 23 items. In particular, for the CC method we have imposed λ = 0.5, while for the -C method, as for the instances with 12 products, we depict the results obtained by varying  from the minimum value which allows us to find a feasible solution (i.e., 3 and 1 for the instances with 21 and 23 items, respectively), to the maximum value after that -C does not find any new non-dominated solution. Since Cplex does not find the optimal solution in short computing times, we imposed a time limit of 30 min. The collected results are depicted in Table 3 for the CC method, and Table 4 for the -C method. We highlight that all the solutions are not the optimal ones.

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Table 3 Results of the CC Method on the orders containing 21 and 23 products Number of products λ f(x) f(τ ) 21 23

0.5 0.5

4 3

552.27 512.54

Table 4 Results of the -C method on the orders containing 21 products and 23 products. Due to the capacity constraints, no feasible solutions are found for the instance with 21 products, when  is set equal to 1 and 2 21 Products  3 4 5 6 7 8 9

f(x) 3 4 5 6 7 8 9

23 Products f(τ )



f(x)

f(τ )

727.03 561.87 469.21 424.86 402.64 359.31 339.86

1 2 3 4 5 6 7 8

1 2 3 4 5 6 7 8

1286.66 701.83 512.53 436.02 391.45 364.73 346.04 329.79

Table 5 Comparison among the solutions adopted by the company and the results obtained by applying the proposed methods Products in the order Current solution CC -C f(x) f(τ ) f(x) f(τ ) f(x) f(τ ) 12

3

450.28

3

314.92

21

5

793.60

4

552.27

23

2

1201.66

3

512.54

1 2 3 3 4 5 1 2 3

654.08 422.09 314.92 727.03 561.87 469.21 1286.66 701.83 512.53

Comparison with the Real Solutions: In this section we make a comparison of the solutions obtained using our model, on the three real orders, with the strategy currently adopted by the considered company. Focusing on the -C method, for comparison we have taken into account the three feasible solutions obtained with the first three lowest values of . Table 5 summarizes the results. Looking at Table 5, it is clear that the solutions found by our models outperform in terms of efficiency the ones currently adopted by the company. Focusing on the order with 12 products, the CC method finds a solution that activates the same number of routes as the company solution, but the makespan is equal to about 315 sec, whereas the makespan of the company solution is about 30% higher. Looking at -C method solutions, when we impose  = 1, the solution has

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one route with a makespan of about 654 s; then, imposing  = 2, the model activates two routes, i.e., one route less than the company solution, with a lower makespan, while when we impose  = 3 we obtain the same solution as the one found with the CC method in terms of number of employed pickers. However, even if the same number of routes are activated, the makespan of the company solution is about 30% higher than the -C solution one. Overall, our solutions outperform the one currently adopted by the distribution company considered in this study. Looking at the results collected on the order containing 21 products, it is possible to highlight that overall the solutions found by the CC and the -C methods are more effective and efficient than the one adopted by the company, since they present a lower makespan and use less resources. In particular, the makespan of the solution found with the CC method is 1.4 times lower than the one of the current solution, using 4 routes; while with the -C method, when we impose  = 3, the number of activated routes is 3 with a makespan of 727.03 sec, i.e., about 1.1 times lower than the value of makespan of the current solution, while when we impose  = 4 and  = 5, the makespan is about 1.4 and 1.7 times lower than the current solution makespan, respectively. Focusing on the order containing 23 products, the solution currently adopted requires 2 routes, whereas the one found by the CC method considers 3 routes. In the current solution the makespan is equal to 1201.66 sec, whereas using the CC method it decreases of about the 57%. Using the -C method, when we impose  = 1, the solution found activates one route with a makespan that is about 85 s higher than the current one; when we impose  = 2 we obtain a reduction of the makespan of about the 41.6% with respect to the company solution, finally, when  = 3 we obtain the same solution as the one found with the CC method.

4 Conclusions and Future Works In this work, we modeled an order picking problem as a variant of the vehicle routing problem, to represent a real warehouse framework. We implemented and then tested our model on small real instances, based on the customers’ orders, provided by an Italian retail distribution company. We proposed two methods and compared our configurations to the ones currently adopted by the company. The results are promising, since in the majority of the cases our methods find more suitable solutions than those currently adopted. However, even if the considered instances has a smallsize, i.e., the considered orders contains at most 23 items, the model is not able to find an optimal solution within an imposed time limit of 30 min for all the instances. Hence, due to the complexity of the problem, as a future work we want to implement an effective and efficient meta-heuristic to solve larger instances; in fact, the orders managed by the considered company may contain more than 200 items. In addition, innovative solutions for the warehouse management can be considered for the future. Due to the technology advances, using autonomous ground vehicles (AGVs) in the

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order picking process is a promising and challenging solution, see, e.g., [14–16]. Hence, our model can be extended considering AGVs systems to compare traditional and innovative warehouse management solutions. Acknowledgements This research was partly supported by the grant under the project “SOluzioni efficienti di Logistica Industriale per la Distribuzione Organizzata (SOLIDO)”—CUP C22C21000990008.

References 1. De Koster, R., Le-Duc, T., Roodbergen, K.J.: Design and control of warehouse order picking: a literature review. Eur. J. Oper. Res. 182(2), 481–501 (2007) 2. Aboelfotoh, A., Singh, M., Suer, G.: Order batching optimization for warehouses with clusterpicking. Proc. Manufact. 39, 1464–1473 (2019) 3. Briant, O., Cambazard, H., Cattaruzza, D., Catusse, N., Ladier, A.L., Ogier, M.: An efficient and general approach for the joint order batching and picker routing problem. Eur. J. Oper. Res. 285(2), 497–512 (2020) 4. Scholz, A., Henn, S., Stuhlmann, M., Wäscher, G.: A new mathematical programming formulation for the single-picker routing problem. Eur. J. Oper. Res. 253(1), 68–84 (2016) 5. De Koster, M.B.M., Van der Poort, E.S., Wolters, M.: Efficient orderbatching methods in warehouses. Int. J. Prod. Res. 37(7), 1479–1504 (1999) 6. Henn, S., Koch, S., Doerner, K.F., Strauss, C., Wäscher, G.: Metaheuristics for the order batching problem in manual order picking systems. Bus. Res. 3(1), 82–105 (2010) 7. Aerts, B., Cornelissens, T., Sörensen, K.: The joint order batching and picker routing problem: modelled and solved as a clustered vehicle routing problem. Comput. Operat. Res. 129, 105168 (2021) 8. Chen, R.C., Lin, C.Y.: An efficient two-stage method for solving the order-picking problem. J. Supercomput. 76(8), 6258–6279 (2020) 9. Chabot, T., Lahyani, R., Coelho, L.C., Renaud, J.: Order picking problems under weight, fragility and category constraints. Int. J. Prod. Res. 55(21), 6361–6379 (2017) 10. Su, Y., Li, M., Zhu, X., Li, C.: Steiner TSP based on aisle as a unit for order picking. Comput. Ind. Eng. 168, 108026 (2022) 11. De Santis, M., Eichfelder, G., Patria, D.: On the exactness of the -constraint method for biobjective nonlinear integer programming. Oper. Res. Lett. 50(3), 356–361 (2022) 12. He, L., Zhang, L.: A bi-objective optimization of energy consumption and investment cost for public building envelope design based on the -constraint method. Energy Build. 266, 112133 (2022) 13. Grandinetti, L., Guerriero, F., Laganà, D., Pisacane, O.: An approximate -constraint method for the multi-objective undirected capacitated arc routing problem. In: International Symposium on Experimental Algorithms, pp. 214–225. Springer, Berlin (2010) 14. Aldarondo, F.J., Bozer, Y.A.: Expected distances and alternative design configurations for automated guided vehicle-based order picking systems. Int. J. Prod. Res. 60(4), 1298–1315 (2022). https://doi.org/10.1080/00207543.2020.1856438 15. Boccia, M., Masone, A., Sterle, C., Murino, T.: The parallel AGV scheduling problem with battery constraints: a new formulation and a matheuristic approach. Eur. J. Oper. Res. (2022). https://doi.org/10.1016/j.ejor.2022.10.023 16. Löffler, M., Boysen, N., Schneider, M.: Picker routing in AGV-assisted order picking systems. INFORMS J. Comput. 34(1), 440–462 (2022). https://doi.org/10.1287/ijoc.2021.1060

Batch Delivery Considerations in Additive Manufacturing Machine Scheduling Problem Ibrahim Kucukkoc

Abstract As a disruptive technology, additive manufacturing (AM) is attracting both researchers and practitioners thanks to its manufacturing ability. Several AM technologies have been developed based on the needs of the industry to fabricate parts through different materials, such as polymers, ceramics and metals. Having a powderbed fusion AM process, Selective Laser Melting (SLM) enables the direct production of highly customized complex geometries in high-quality lightweight metals. This study focuses on the scheduling of multiple SLM machines to produce part orders received from geographically dispersed customers. Each order may contain more than one part, which may be processed in any batch on any suitable machine. However, the delivery of an order requires that all individual part orders received from the same customer have been completed, to minimize delivery costs. Although there is limited research on multiple AM machine scheduling problems in the literature, none of them considered batch delivery. Following the detailed description of the problem, a genetic algorithm (GA) approach has been proposed to solve a numerical example and the results are presented. Keywords Additive manufacturing · Genetic algorithm · Scheduling · Selective laser melting · Batch delivery

1 Introduction The utilization of additive manufacturing (AM, shortly), or 3D printing, technologies has yielded a major advancement in the manufacturing industry. Rapid evolution has been acquired not only in the product design and fabrication processes but also in the supply chains thanks to the advantages provided by additive manufacturing techniques [1]. Among these advantages are design flexibility, high strength-to-weight ratio, improved reliability and reduced lead times [2]. I. Kucukkoc (B) Department of Industrial Engineering, Balikesir University, Balikesir, Turkey e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Cosmi et al. (eds.), Operations Research and Data Science in Public Services, AIRO Springer Series 11, https://doi.org/10.1007/978-3-031-34546-3_4

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AM is a common name for several technologies, which build parts by joining materials layer-by-layer through a 3D model. AM also provides material efficiency as the parts are not fabricated by removing material from a block, as opposed to subtractive manufacturing technologies [3, 4]. Various AM techniques have been developed to fabricate parts using different materials. These techniques, each of which offers different surface finishes and physical properties, can be grouped into six main categories [5, 6]: Extrusion-based systems such as fused deposition modelling, direct ink writing, inkjet printing, stereolithography, direct light projection and powder-bed fusion. The basic characteristics of these techniques have been explained briefly as follows. Fused deposition modelling is an extrusion-based system, mainly used for the fabrication of polymer materials via extruding filament through a heated nozzle [5]. Inkjet printing is adaptable to a wide range of materials (including biomaterials and functional ceramics) as the liquid droplets are deposited onto substrates selectively via a nozzle [7, 8]. Direct-ink writing is another extrusion-based additive manufacturing method which uses an ink composed of the constitutive material [9]. A computer-controlled laser beam is used in stereolithography to solidify the resin in the pattern. The first layer is photo-polymerized, the support platform is moved away, and the second layer is recoated with liquid resin. These steps are repeated until the object is fabricated [10]. Direct light projection is a type of stereolithography, but it uses a projection of visible ultraviolet light. The layer is flashed at once across the whole resin [11]. Selective laser melting (SLM) and selective laser sintering are powder-bed fusion techniques developed to fabricate metal components using metal powders. A highpower laser selectively melts and fuses a thin layer of the powder material based on the 2D section of a 3D model created. A new thin layer of material is deposited via a roller and the laser melts the material to form the new layer. That process is repeated until the final product is obtained eventually [5, 12]. As an emerging technology in the market, producing components through SLM machines requires enormous investments. That results in a requirement for efficient tools and methodologies to schedule their operations and manufacturing decisions on these high-cost resources. This paper addresses the parallel AM machine scheduling problem considering batch delivery constraints. Orders with one or more parts received from the customers (usually geographically dispersed) are scheduled on the parallel AM machines. All parts in the same order need to be completed to consider that the order has been completed and is ready to dispatch. This reflects real-world customer expectations as it helps reduce transportation costs. A genetic algorithm based meta-heuristic has been developed for the solution of the addressed problem and a numerical example is provided. The remainder of the paper is organized as follows. Section 2 provides a brief overview of the related studies in the literature while Sect. 3 presents the definition of the problem, together with a schematic representation. Section 4 gives the details of the proposed solution. The results for a numerical example retrieved from Kucukkoc [3] have been reported in Sect. 5. Finally, the paper has been concluded and some future research directions have been pointed out in Sect. 6.

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2 Related Work Batch scheduling problems have attracted many academics as well as practitioners mainly due to the wide application area of batch processes in the industry. SLM machines are capable of producing more than one part simultaneously, which represents a batch process involving several decisions to be handled properly. As shown by Li et al. [2], unit production costs can be reduced significantly through efficient planning and scheduling in SLM technology. The pioneering works on the scheduling AM machines have been conducted by Kucukkoc et al. [13] and Li et al. [2]. Kucukkoc et al. [13] introduced the AM machine scheduling problem for the first time in the literature and proposed a genetic algorithm (GA) approach to maximize the utilization of SLM machines. Li et al. [2] modelled the AM machine scheduling problem for the first time, aiming to minimize the average production cost. Two efficient heuristics, called best-fit and adapted bestfit, are proposed and their performances have been compared to those obtained from the execution of the proposed model by CPLEX. These have been followed and several heuristics/meta-heuristics have been developed to solve single and multiple AM machines to optimize certain objectives under various constraints. Chergui et al. [4] addressed the multiple AM machine scheduling problem and handled it in two stages, namely part grouping and batch scheduling. Fera et al. [14] proposed a genetic algorithm to schedule single AM machine to minimize the time as well as cost. Dvorak et al. [15] aimed to minimize the number of tardy parts and also considered 2D packing constraints on multiple AM machines. Kucukkoc et al. [16] proposed a GA-based approach to minimize maximum lateness in a multiple AM machine environment. Kucukkoc [3] addressed the single and multiple AM machine scheduling problem and proposed the first MILP models to minimize the makespan. Li et al. [6] introduced the dynamic order acceptance and scheduling problem in AM and produced “a strategy-based metaheuristic decision-making approach” to maximize the average profit-per-unit-time. Kucukkoc et al. [17] considered the 2D nesting constraints in the multiple AM machine scheduling problem and aimed to minimize total tardiness. Kucukkoc [18] investigated the relationship between nesting and scheduling and showed that optimal nesting does not guarantee optimal scheduling. Che et al. [19] proposed a MILP model and a simulated annealing approach to schedule AM machines considering 2D nesting constraints allowing orientation selection of parts. Aloui and Hadj-Hamou [20] proposed models aiming to minimize the total delay and maximize the utilization of the AM machines. A reinforcement learning-based meta-heuristic algorithm was proposed by Alicastro et al. [21] with no nesting considerations. Rohaninejad et al. [22] proposed a bi-objective model to minimize the makespan and total tardiness penalty. A hybrid meta-heuristic algorithm, i.e. a non-dominated sorting genetic algorithm (NSGA-II) with a learningbased local search, is also developed in the same work. Kapadia et al. [23] studied the order acceptance and scheduling problem in AM and proposed random-keys based genetic algorithms to fully or partially accept orders. Castillo-Riviera [24] attempted to maximize the utilization of AM machines. Arık [25] considered the

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post-processing assembly operations as well as the AM machine scheduling problem and proposed a MILP model and a heuristic with a simple local search mechanism. Altekin and Bukchin [26] explored the trade-off between cost and makespan in AM machine scheduling problems and proposed a MILP model and applied an efficient frontier approach to get non-dominated solutions. Hu et al. [27] addressed the parallel unrelated AM machine scheduling problem considering 2D nesting, part orientation and unequal release times of parts, what are called practical constraints in the corresponding study. Zehetner and Gansterer [28] proposed a central decision-making approach to multi-site batching problem in additive manufacturing, where customers and AM facilities are located dispersedly. Batching and sequencing problems have been solved in sequence with an integrated approach of MILP and GA. The works conducted by Kim and Kim [29] aimed to minimize makespan in parallel identical AM machine environment where parts have multiple processing alternatives. Kim and Kim [30] proposed three meta-heuristics to minimize makespan in a parallel non-identical AM machine environment considering setup times between material changes. However, the major limitation of both works is that the processing time of a batch has not been calculated via a function. Instead, it was assumed the maximum of processing times belonging to the parts assigned to the batch. Wu et al. [31] handled the online scheduling of AM machines, where parts arrive dynamically via a cloud-based platform, and proposed a heuristic algorithm to minimize the average cost per volume of material. One can refer to Oh et al. [32] for a review of the works on AM machine scheduling problems. As clearly seen from this survey, there is no study which considers the batch delivery schema in AM machine scheduling problems. This paper contributes to the literature by introducing the batch delivery considerations into parallel unrelated AM machine scheduling problem and proposing a GA approach to solve it. The aim is to minimize the total tardiness of orders, each may contain more than one part based on the requirements of the customers. The next section defines the problem and provides a schematic representation to illustrate its characteristics.

3 Problem Definition Parallel unrelated AM machine scheduling problem with batch delivery considerations is introduced in this paper. An order is a set of parts (i ∈ I ) to be fabricated on unrelated SLM machines (m ∈ M) located in parallel to each other. Machines have different specifications, i.e., speed, capacity, area (M Am ), cost parameters and building platform dimensions (width—M W m , length—M L m and height—M H m ). Orders (o ∈ O) have release times and due dates (Dd o ), which are the same for all parts in the same order. Parts have different area (ai ), volume (vi ) and dimensions (width—wi , length—li and height—h i ). Parts received from different customers in

Batch Delivery Considerations in Additive Manufacturing Machine …

51

Fig. 1 The schematic representation of the parallel unrelated AM machine scheduling problem with batch delivery

several orders are grouped in a mixed way and produced on AM machines in batches (or jobs), which are processed sequentially. Each machine can process one job at a time and the parts in the same job are fabricated simultaneously. Once the production has started it is not possible to add any new part to (or remove any completed part from) the batch until the whole job is completed. An order can be shipped upon all parts belonging to that order are completed. So that lateness of an order is calculated by subtracting its due date (Dd o ) from completion time (Co ) and order tardiness is the maximum of 0 and order lateness. The aim is to minimize total tardiness (T ), calculated as in Eq. (1). Figure 1 represents the basic schematic representation of the problem studied in this work. T =



max{0, |Co − Dd o |}

(1)

o∈O

One unique feature of the SLM process is that the processing time of a batch (P T m j ) is calculated via a function [2, 3] as given in Eq. (2). P T m j = S E T m · Zm j + V T m ·

 i∈I

  vi · X m ji + H T m · max h i · X m ji ; ∀m ∈ M, j ∈ J i∈I

(2) where X m ji and Z m j are binary decision variables. X m ji equals 1 if part i is assigned to job j on machine m, 0 otherwise. Z m j equals 1 if job j on machine m is utilized, 0 otherwise. P T m j is composed of three main components: (i) machine set-up time, (ii) time spent by laser scanning or forming material per unit of volume and (iii) time spent to spread per layer of material for each 2D section of the model. V T m represents the time required to form per unit volume of material and H T m represents the powder-layering time. These parameters are machine specific as the problem has an unrelated machine environment.

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One basic assumption of the problem is that there is only one type of material that all parts are fabricated from. Also, it is assumed that preemptions are not allowed, which means production cannot stop until the job finishes. Breakdowns and failures are ignored and the machines are available at the time of 0. The next section provides the details related to the proposed GA approach.

4 Proposed GA Approach As shown by Kucukkoc [3], the problem studied here has significant differences from the well-known batch scheduling problems. Specifically, in classical batch scheduling problems [33, 34], the processing time of a batch is usually the maximum of the processing times of parts included in the batch. However, in AM scheduling problem, the processing time of a batch is calculated dynamically based on the total volume and maximum height of the parts allocated to that batch. Grouping parts with similar heights into the same batch may provide an advantage to reduce the sum of job processing times and so makespan with efficient planning. However, as shown by Kucukkoc [3], CPLEX struggles to optimally solve the majority of the test instances even with the objective of minimizing the makespan. As contextualized above, the objective of minimizing order tardiness is a more complex objective than the makespan and requires a sophisticated (meta)heuristic solution approach to handle all these aspects properly. The GA approach adopted in this study is modified from Kucukkoc [16, 17] to minimize the total tardiness of all orders. However, nesting constraints have been relaxed in this work to keep the problem complexity at a minimum. The general outline of the proposed approach is presented in Fig. 2. The chromosomes in the initial population are generated randomly, to keep diversity. Each gene corresponds to a part number and the chromosomes are permutation coded. Each chromosome is decoded following the decoding procedure provided in Algorithm-1: The general outline 1 Generate random chromosomes to build initial population 2 Decode chromosomes (Algorithm-2) 3 Iter ← 1 4 while iter ≤ maxIter 5 Tournament selection 6 Permutation crossover 7 Mutation 8 Form new generation sustaining elitism 9 iter ← iter+1 10 endwhile 11 Report the best individual in the population

Fig. 2 The pseudocode of GA’s general outline

Batch Delivery Considerations in Additive Manufacturing Machine …

53

Algorithm-2: The decoding procedure 1 Add all parts into AP based on their sequence on chromosome 2 // start from the first job on each machine Set 3 WHILE ( 4 TO ) //start from the first machine FOR ( 5 , 6 FOREACH ( in ) //try each part 7 and and (( and IF ( and ) or ( ))) 8 Add part to 9 10 Remove part from 11 ENDIF 12 ENDFOREACH 13 14 ENDFOR 15 ENDWHILE 16 Constitute orders and calculate order tardiness values 17 Return total tardiness and assignment solutions

Fig. 3 The pseudocode of the decoding procedure (adapted from [16, 17])

Fig. 3. Genetic operators are applied until a pre-determined number of iterations is exceeded, and the best individual is eventually reported as the solution to the problem. Parents, determined via tournament selection, undergo crossover and mutation. Tournament size is determined based on population size. A random cutting point is picked, and the permutation-based crossover is applied keeping the feasibility (avoiding duplication of the genes). The winners of the tournament selection also continue with the mutation stage, where either swap or insert operation is applied with equal probability. Two randomly determined genes are swapped in the swap procedure, while a randomly selected gene is moved to a randomly selected location in the insert procedure. Any duplicated children are eliminated, and the fitness value of each offspring is calculated by applying the decoding procedure. The new population is formed by replacing the worst chromosome with the best child until there is no child which has a fitness value better than any individual in the population. The best chromosomes in the population are kept for future generations (elitism) by this way. One can refer to Kucukkoc [16, 17] for detailed explanations of crossover and mutation operations, which are not given here due to page limits. The decoding procedure is performed considering the sequence of the part numbers on the chromosome. Parts are picked one by one in the order that they appear on the chromosome and allocated to jobs on machines based on the decoding procedure provided. AP is the list of parts not assigned to a job yet, nm is the number of machines and Im j is the list of parts assigned to job j on machine m.

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5 Numerical Results A numerical example is solved using the proposed GA approach and the results are reported in this section. The data used for the numerical example is retrieved from medium-size test instances by Kucukkoc [3] and presented in Table 1, where height, width and length are in mm, area is in mm2 , volume is mm3 , release and due dates are in hr. There are 18 parts in total, received from 7 orders (or customers), to be processed on two AM machines in parallel. The algorithm was coded in JAVA and run on Intel® Core™ i7-6700HQ CPU @2.60 GHz equipped with 16 GB of RAM. The parameters of the algorithm have been determined through preliminary tests and the number of iterations, population size, crossover rate and mutation rate are set to 10,000, 30, 0.8 and 0.2, respectively. Figure 4 presents the change in the objective values through different sets of parameters during parameter setting. It is also noticeable that the average fitness values increase with higher mutation rates, which shows the exploration effect of the mutation operator. The algorithm was run three consecutive times and the best solution obtained by GA (within 10 s) has been presented in Table 2. As seen from the table, there are six batches (three on each machine) utilized to fabricate 18 parts in total and the batches Table 1 Data used for the test problems Order

Part

Height

Width

Length

Area

Volume

Release

Due Date

1

P1

16.7

18.8

16.0

300.8

1573.8

0.00

300.0

1

P2

8.8

6.7

22.8

152.8

421.3

0.00

300.0

1

P4

7.4

21.6

3.9

84.2

285.2

0.00

300.0

1

P5

27.3

14.9

4.1

61.1

583.3

0.00

300.0

2

P3

20.3

9.3

2.1

19.5

147.8

8.00

350.0

2

P9

20.4

20.5

1.0

20.5

278.5

8.00

350.0

2

P13

28.4

21.8

15.3

333.5

7347.6

8.00

350.0

2

P14

10.9

23.2

3.2

74.2

333.6

8.00

350.0

3

P6

25.8

23.2

12.9

299.3

3282.5

0.00

500.0

3

P7

14.5

22.2

6.7

148.7

1265.5

0.00

500.0

4

P8

3.5

24.6

15.3

376.4

723.3

8.00

300.0

4

P12

12.8

5.9

21.0

123.9

866.1

8.00

300.0

4

P15

14.1

19.6

13.7

268.5

2956.0

8.00

300.0

5

P10

23.3

6.8

13.4

91.1

1051.8

0.00

400.0

5

P11

26.3

4.0

5.0

20.0

201.9

0.00

400.0

6

P16

3.9

3.6

3.1

11.2

32.5

10.00

300.0

6

P17

3.2

10.6

13.1

138.9

265.6

10.00

300.0

7

P18

24.6

7.3

12.6

92.0

1387.4

10.00

500.0

Batch Delivery Considerations in Additive Manufacturing Machine …

55

Fig. 4 Minimum and average fitness values over 20 runs for parameter setting

have been formed in a mixed way. For example, job-2 on machine-1 processes parts 7, 10, 14 and 18 belonging to four orders, i.e. 3, 5, 2 and 7, respectively. Table 3 presents the calculation of the fitness value (total tardiness) by obtaining the lateness of each order. The lateness of an order is the maximum of the part lateness values belonging to that order. Part lateness is the difference between the completion time of the job in which the part is fabricated and the due date of the order that the part belongs. As seen in the table, there is only one tardy order, i.e. order-2 with 30.4744 h. Therefore, the fitness value of the solution is 30.4744 h. When the performance of the algorithm has been investigated, it has been observed that the fitness value was reduced from >120 h to 30.4744 h (from the first to the Table 2 Allocation of parts to batches and machines Machine M1

M2

Job-1

Job-2

Job-3

Parts assigned

3,4,5,9,11,12,15,16

7,10,14,18

6

Processing time (hr)

194.4625

151.2381

129.1111

Start time (hr)

10.00

204.4625

355.7006

Completion time (hr)

204.4625

355.7006

484.8117

Parts assigned

1,2,17

8

13

Processing time

87.4742

26.8239

256.1763

Start time

10.00

97.4742

124.2981

Completion time

97.4742

124.2981

380.4744

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I. Kucukkoc

Table 3 The calculation of fitness value based on the assignment solution Order

Part

Due date (hr)

1

P1

300.0

3

– 202.5257

P4

204.4625

– 95.5375

204.4625

– 95.5375

204.4625

– 145.5375

P9

204.4625

– 145.5375

P13

380.4744

30.4744

P14

355.7006

5.7006

500.0

484.8117

– 15.1883

355.7006

– 144.2994

300.0

124.2981

– 175.7019

204.4625

– 95.5375

P3

P6

350.0

P8 P15 P10

400.0

P11 6

P16

300.0

P17 7

– 95.5375

– 202.5257

97.4742

P12 5

Order lateness

97.4742

P7 4

Part lateness

P2 P5 2

Completion time (h)

P18

500.0

204.4625

– 95.5375

355.7006

– 44.2994

204.4625

– 195.5375

204.4625

– 95.5375

97.4742

– 202.5258

355.7006

– 144.2994

30.4744

– 15.1883 – 95.5375

– 44.2994 – 95.5375 – 144.2994

last iteration). Moreover, the average fitness value in the initial population was over 400 h, which was also reduced to 30.4744 when the algorithm has been terminated.

6 Conclusions This paper addressed parallel unrelated AM machine scheduling problem in metal additive manufacturing considering batch delivery of orders. The problem has been defined with a schematic representation and a GA-based approach has been proposed to minimize the total tardiness of orders. The tardiness of an order is characterized by the maximum tardiness of the parts in that order. The proposed approach can solve the problem within short CPU times. However, it tends to increase when the problem size and so the search space increases, due to the nature of the NP-hard problems. Although the proposed algorithm has a promising performance, a comprehensive computational study is required to verify this argument. The authors’ ongoing work aims to model the problem mathematically and compare the performance of the GA with it. New metaheuristics (e.g., ant colony, tabu search, artificial bee colony etc.) can also be developed and their results can be

Batch Delivery Considerations in Additive Manufacturing Machine …

57

compared to that of GA. The simultaneous availability of the machines at time 0 can be considered a limitation of the work, which can be eliminated via modifying the proposed approach in future studies. Acknowledgements The author acknowledges the financial support received from Balikesir University Scientific Research Projects department under grant number BAP-2022-086.

References 1. Li, Z., et al.: The influence of scan length on fabricating thin-walled components in selective laser melting. Int. J. Mach. Tools Manuf 126, 1–12 (2018) 2. Li, Q., Kucukkoc, I., Zhang, D.Z.: Production planning in additive manufacturing and 3D printing. Comput. Oper. Res. 83, 157–172 (2017) 3. Kucukkoc, I.: MILP models to minimise makespan in additive manufacturing machine scheduling problems. Comput. Oper. Res. 105, 58–67 (2019) 4. Chergui, A., Hadj-Hamou, K., Vignat, F.: Production scheduling and nesting in additive manufacturing. Comput. Ind. Eng. 126, 292–301 (2018) 5. Ryan, K.R., et al.: Additive manufacturing (3D printing) of electrically conductive polymers and polymer nanocomposites and their applications. eScience 2(4): 365–381 (2022) 6. Li, Q., et al.: A dynamic order acceptance and scheduling approach for additive manufacturing on-demand production. Int. J. Adv. Manuf. Technol. 105(9), 3711–3729 (2019) 7. Qian, Q., et al.: 3D reactive inkjet printing of bisphenol A-polycarbonate. Addit. Manuf. 54, 102745 (2022) 8. Derby, B.: Additive manufacture of ceramics components by inkjet printing. Engineering 1(1), 113–123 (2015) 9. Kachit, M., et al.: Direct-ink writing and compression behavior by in situ micro-tomography of architectured 316L scaffolds with a two-scale porosity. J. Market. Res. 20, 1341–1351 (2022) 10. Melchels, F.P.W., Feijen, J., Grijpma, D.W.: A review on stereolithography and its applications in biomedical engineering. Biomaterials 31(24), 6121–6130 (2010) 11. Huh, J.T., et al.: Chapter 74 - Three-dimensional bioprinting for tissue engineering. In: Lanza, et al., R., (eds.), Principles of Tissue Engineering, 5th edn., pp. 1391–1415 Academic 12. Liu, F., et al.: Optimising the process parameters of selective laser melting for the fabrication of Ti6Al4V alloy. Rapid Prototyping J. 24(1), 150–159 (2018) 13. Kucukkoc, I., Li, Q., Zhang, D.Z.: Increasing the utilisation of additive manufacturing and 3D printing machines considering order delivery times. In: 19th International Working Seminar on Production Economics. Innsbruck, Austria (2016) 14. Fera, M., et al.: A modified genetic algorithm for time and cost optimization of an additive manufacturing single-machine scheduling. Int. J. Ind. Eng. Comput. 9(4), 423–438 (2018) 15. Dvorak, F., Micali, M., Mathieu, M.: Planning and scheduling in additive manufacturing. Intel. Artif. 21, 40–52 (2018) 16. Kucukkoc, I., et al.: Scheduling of multiple additive manufacturing and 3D printing machines to minimise maximum lateness. In: 20th International Working Seminar on Production Economics. Innsbruck, Austria (2018) 17. Kucukkoc, I., Li, Z., Li, Q.: 2D nesting and scheduling in metal additive manufacturing. In: Communications in Computer and Information Science, pp. 97–112 (2021) 18. Kucukkoc, I.: Metal additive manufacturing: nesting versus scheduling. In: Optimization and Data Science: Trends and Applications. Springer International Publishing, Cham (2021) 19. Che, Y., et al.: Machine scheduling with orientation selection and two-dimensional packing for additive manufacturing. Comput. Oper. Res. 130, 105245 (2021)

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An LP Model for the Central Energy System Silvia Anna Cordieri, Michele Monaci, Paolo Paronuzzi, and Henrik Schwaeppe

Abstract In this paper we present a mathematical model for the generation expansion operations to optimize the performance of a Central Energy System (CES) system. To ensure that the proposed generation operations are compatible with the network, some extensions (or updates) of the existing network are considered as well. All these decisions are evaluated both from an economic viewpoint, using the objective function of the problem, and from an environmental perspective, as specific constraints related to GHG gases (measured in C O2 eq) emissions are imposed in the formulation. Keywords Central energy system · Linear programming · Expansion planning

1 Introduction The energy sector has been a widely discussed topic during the last years. Given the recent worsening of climate conditions, worldwide governments, especially in European countries, had implemented new directives in order to limit the consequences of this problem. Nevertheless, the most recent energy crisis highlighted the strong dependency that energy systems still have on conventional energy sources and the consequences implied by this strategy also from a political point of view. Disruptive structural developments are still necessary to deliver on the European Union’s COP21 commitments [6]. In this sense, a strategy adopted to achieve a S. A. Cordieri (B) · M. Monaci · P. Paronuzzi Università di Bologna, Bologna, Italy e-mail: [email protected] M. Monaci e-mail: [email protected] P. Paronuzzi e-mail: [email protected] H. Schwaeppe RWTH Aachen University, Aachen, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Cosmi et al. (eds.), Operations Research and Data Science in Public Services, AIRO Springer Series 11, https://doi.org/10.1007/978-3-031-34546-3_5

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decrease of Green House Gasses (GHG) emissions is the decarbonization of the energy sector, in spite of an increase of renewable energy sources. An efficient planning for future energy systems must also comprise the coupling of energy sectors as well as inter-dependencies of generation and transmission grid infrastructures [13]. The energy coupling concept consists in the integration of power and gas from a consumer perspective, thus leading to an electrification of the thermal and mobility sector. However, these strategies come with a significant change in the topology of energy generation that risks to compromise the stability of the transmission system. The system more and more often nears boundaries of safe operation, thus increasing the probability of undesired effects, e.g., loss of synchrony and voltage collapse [1]. Thus, adequate expansion and planning become essential. In the context of integrated energy systems, the Central Energy System (CES) represents all the entities that are relevant in the decision making process currently regulated by the Transmission System Operator. The CES focuses on large-scale generation, like central power plants, power to gas plants and wind power plants as well as the transmission grid. All these entities are necessary to fulfill the final consumer’s demand, on an hourly basis every day. The perfect planning of this system is a problem associated with significant challenges. Thus, providing CES planners with a decision support system able to compute efficient strategies to reach the aforementioned goals is fundamental. This paper presents a Linear Programming (LP) model to define the generation expansion operations needed to improve the performance of the CES. The model includes extensions (or updates) of the existing network, to avoid inconsistencies between the proposed generation operations and the transmission network. All these decisions are evaluated both from an economic viewpoint, using the objective function of the problem, and from an environmental perspective, as specific constraints related to GHG gases (measured in C O2 emissions) emissions are imposed in the formulation. The goal of the presented formulation is to model scenarios that represent an ideal future framework of the grid, towards which grid managers should strive. Given the size of the instances to be solved, the formulation introduces some modelling approximations, so that the resulting model can be solved with “reasonable” computing time and memory. The remainder of this paper is organized as follows. Section 2 presents a literature review of the topics discussed in this paper, while Sect. 3 formally presents an LP model for the CES problem. Section 4 shows the numerical tests. Finally, Sect. 5 gives some conclusions and outlines possible research perspectives.

2 Literature Review The generation expansion planning and the transmission expansion planning problems have gained interest within the scientific community recently. These problems have been investigated from different perspectives, modelling different constraints and objective, and have been attacked with different solution methods [8]. Franken et al. [7] concentrated on the transmission expansion planning problem, explicitly

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addressing the Alternating Current (AC) systems and Phase Shifting Transformers (PSTs) as well as operational congestion management interventions. The authors proposed a multi-stage approach, based on Mixed Integer Linear Programming (MILP), aimed at determining the operating point of power flow controlling devices while minimising expansion costs. Hörsch et al. [9] introduced the Linear Optimal Power Flow (LOPF) algorithm, which uses a linearization of the AC load flow equations to optimize generator dispatch in a network subject to loading constraints of the network branches. The authors showed that this formulation is computationally more efficient than the angle formulation when solved by means of a commercial linear programming solver. Müller et al. [10] implemented an approach for evaluating multi-modal energy systems, showing the advantages of coupled addressing electricity, heat, fuel and mobility sectors. The modeling framework enables system planners to optimally plan future investments in a detailed transition pathway of the energy system of a country, considering politically defined climate goals. Additional transmission system expansion measures are identified by applying a heuristic method, which reinforces the transmission grid in order to make use of existing infrastructure. Schwaeppe et al. [12] proposed a mathematical model for the single-stage generation and transmission expansion planning, that is intended to provide a general framework for multi-energy planning. To reflect constraints in the transmission network a power transfer distribution factor (PTDF) approach is used. Other papers are more focused on the optimization of the generation part. Elsido et al. [5] introduced a MILP model and a two-stage optimization algorithm for determining the most profitable synthesis and design of Combined Heat and Power units within a district heating network with heat storage, while taking into account the optimal scheduling of the units over the year. Bischi et al. [3] developed a MILP model for optimizing the daily schedule of cogeneration systems and networks of heat and power plants. Given the significant computational time needed to solve the model on a weekly time horizon, the authors introduce a heuristic rolling-horizon algorithm, in which a sequence of weekly MILP submodels is solved, while considering production and consumption estimates based on demand profiles from historical data. Finally, some papers in the literature try to detect the correct integration of storage systems in the grid framework. Aghaebrahimi et al. [2] demonstrated that, despite high cost of energy storage systems, their presence in the grid framework may increase the total income from wind energy sales. Bordin et al. [4] addressed the coupling of renewable energy sources and storage systems, focusing on battery degradation costs, and proposed LP models for the optimal management of off-grid systems. Almost all the references in the literature focus on specific parts of the CES problem, or use modelling strategies that are different from those we adopt. To the best of our knowledge, the first attempt to give a comprehensive view of CES by using the LOPF strategy, was made by Schwaeppe et al. [11]. Our formulation extends the model proposed in [11] by including additional details related to the technologies used in the generation part, such as renewable plants and storage systems.

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3 Methodology In the following, a generic, single-stage expansion planning model for multi-energy systems is described. The formulation is intended to be described as pure linear program. The problem definition involves several technologies and consumers that are described by a set of nodes. These nodes are located in different areas, and are interlinked by a set of arcs representing the grid lines. The objective of the problem is to minimize a cost function that takes into account both the operational cost (including production costs and transmission costs), and costs for installing new facilities. In case the consumers’ demands cannot be fully satisfied, a large penalty value is incurred in the objective function; similarly, costs for oversupply are taken into account. Finally, the formulation introduces a cap on the allowed C O2 emissions, and evaluates a preliminary analysis of the expansion of the transmission network. In our formulation, we consider a set of candidate lines among the existing ones that can be further expanded up to a certain value, in terms of capacity. More precisely, the problem consists of a set N = {1, ..., n} of electric nodes. Each node j ∈ N has a subset M( j) of available energy production technologies (both traditional ones and renewable ones). Specifically, M( j) ⊆ I , where I = {W, O, P, B, F} is the set of technologies (W stands for wind turbines, O for offshore wind turbines, P for photovoltaic systems, B for batteries, and F for thermal power plants). A set T = {1, ..., } of time slots is given. The transmission network topology is defined by a set E = {1, ..., m} of edges, where each edge e ∈ E is represented as e = (u, w) where (u, w) ∈ N and u < w. In addition to the electric power transmission grid, there is a thermal power system, that is characterized by a set R = {1, . . . , r } of thermal nodes. For each node k ∈ R, a set of available technologies MT (k) ⊆ J ∪ L ∪ Q is given, where J = {H, S} is a set of purely thermal technologies (H stands for solar heating   and S for thermal storage); L = {i = k(i), j (i) : k(i) ∈ R, j (i) ∈ N } is a set of technologies converting electric energy from node j ∈ N to heat for node k ∈ R (e.g.heat pumps, rods, boilers), and Q = {i = k(i), j (i) : k(i) ∈ R, j (i) ∈ N } is a set of technologies obtaining heat for node k ∈ R while producing electric energy for node j ∈ N . The behavior of each technology associated with a given node is described by a specific set of constraints and parameters. Installing a technology i at some node j is associated with an installation cost CiCj per unit of capacity and with the maximal that can be installed. For each battery i = B and node j ∈ N the unit capacity Yimax j p production cost C B j to charge or discharge is given. For each power plant i = F p and node j ∈ N , the unit production cost C F j and an efficiency parameter η F j are given. The thermal storage in each node k ∈ R is characterized by a unit production p cost C Sk to charge or discharge. For each thermal plant i ∈ MT (k) and node k ∈ R, p the unit production cost is Cik , whereas C −j and C +j denote the costs for power oversupply and deficit in node j ∈ N , respectively (note that C +j >> C −j ). Finally, for what concerns the network, we are given a set E cand ⊆ E of candidate lines for transmission corridor expansion. Each candidate line g ∈ E cand has an installation cost CgV to build the new line.

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To model the problem, we introduce variables xi jt representing the amount of power generated/withdrawn by technology i located at node j ∈ N during time slot t ∈ T . In addition, variables yi j represent the capacity that is installed for technology i at node j ∈ N . For each time slot t ∈ T and each node j ∈ N , the curtailed power due to oversupply or congestion is given by variable s −jt , and the power deficit is defined by variable s +jt . Variable u g represents the expanded fraction of candidate line along edge g ∈ E cand . For each t ∈ T and node j ∈ N in which a battery is installed, we introduce a variable x Bc jt to denote the amount of energy that is used to charge the battery, and a variable x Bd jt for the amount of energy taken from the battery. For each node k ∈ R, the amount of thermal energy generated/withdrawn by technology i during time slot t ∈ T is given by variable qikt , while variable xi j (i)t represents the amount of electric energy converted to heat from node j, through c denotes the amount of energy used to technology i in time slot t ∈ T . Variable q Skt d is charge the thermal storage located at node k during time slot t, and variable q Skt used for amount of energy that is taken from the thermal storage located at node k during time slot t. Finally, there are some additional variables to detect the costs in the objective function. Variable c P R O D computes the total production cost, variable c I N ST computes the total installation cost, variable cT R AN computes the total transportation cost, variable c SL AC K computes the total opportunity cost for undersupply/oversupply and variable cetT computes the transmission cost for edge e ∈ E during time slot t ∈ T . All these figures are set to their value by constraints (1)–(4). cP RO D =



⎛ ⎝

t∈T

+





p C B j (x Bd jt

− x Bc jt ) +

j∈N :B∈M( j)

⎛ ⎝

t∈T

c I N ST =



p

k∈R:S∈J

j∈N i∈M( j)

⎞ x F jt ⎠ ηF j ⎞

p CF j

j∈N :F∈M( j)

d c C Sk (q Skt − q Skt )+

 



CiCj yi j +

 





Cik qikt ⎠ p

(1)

k∈R i∈MT (k):i= S



C Cik yik +

CgV u g

(2)

g∈E cand

k∈R i∈MT (k)

cT R AN =



cetT

(3)

 C +j s +jt + C −j s −jt

(4)

t∈T e∈E

c SL AC K =

t∈T j∈N

For time step t ∈ T the electric demand D jt to be supplied at node j ∈ N is given. Constraints (5) compute the power injected in the grid (z jt ) in every node j ∈ N during time slot t ∈ T , while constraints (6) impose the power balance on the grid.

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z jt =



xi jt + D jt + s +jt + s −jt ∀ j ∈ N , ∀t ∈ T

i∈M( j)



z jt = 0 ∀t ∈ T

(5) (6)

j∈N

Constraints (7)–(8) limit the slack variables of the problem, while constraints (9) limit the maximum capacity of renewable energy that can be installed in a node. 0 ≤ s +jt ≤ −D jt ∀ j ∈ N , ∀t ∈ T s −jt yi j ≤

Yimax j

(7)

≤ 0 ∀ j ∈ N , ∀t ∈ T

(8)

∀ j ∈ N ∪ R, ∀i ∈ M( j) ∪ MT (k)

(9)

Parameter E g represents the total emission related to installation of candidate lines in set E cand . Moreover, any technology i installed at a node j ∈ N ∪ R is y characterized by the equivalent C O2 emission per unit of capacity installed (E i j ) x and per unit of power generated/withdrawn (E i j ). These values are related to variable ei j that represents the total emissions at node j due to technology i, while variable eik defines the total emissions at node k due to technology i. Constraints (10)–(11) define the emissions produced for every technology, while constraints (12) limit the total C O2 emissions to a maximum value E M I SS I O Ntotal . 

ei j =

y

(10)

y

(11)

E ixj xi jt + E i j yi j ∀ j ∈ N , ∀i ∈ M( j)

t∈T

eik =   j∈N i∈M( j)



E ikx qikt + E ik yik ∀k ∈ R, ∀i ∈ MT (k)

t∈T

ei j +

 

k∈R i∈MT (k)

eik +



E g u g ≤ E M I SS I O Ntotal

(12)

g∈E cand

Constraints (13) define the actual available energy for each technology i located at node j and producing renewable energy (i.e., for each i ∈ {W, O, P}), that exploits a renewable energy source according to the normalized feed-in F E E D I Ni jt at time t ∈ T. xi jt = F E E D I Ni jt yi j ∀t ∈ T, ∀ j ∈ N , ∀i ∈ M( j) ∩ {W, O, P}

(13)

Constraints (14)–(15) limit the charge and discharge values of batteries, respectively. Constraints (16) impose that the actual maximum storage capacity wmax B j of a battery located at a node j is a fraction of the maximum value C A PB j , while constraints (17) define the amount of energy injected/withdrawn in each battery at node j, according to efficiency parameters to charge or discharge ηcB j and ηdB j , respectively. Constraints (18) define the amount of energy h B jt to be stored in a battery located at a node j during time slot t, whereas constraints (19) impose this figure to be between a minimum value W Bmin j and a maximum value, depending on the installed capacity.

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In our model, we assume the initial and final values for variables h B jt must coincide. y Though we fixed these values to 2B j , different values can be used to model different scenarios. 0 ≤ x Bd jt ≤ y B j ∀t ∈ T, ∀ j ∈ N : B ∈ M( j) −y B j ≤ x Bc jt ≤ 0 ∀t ∈ T, ∀ j w max B j = C A PB j y B j ∀ j x B jt = x Bd jt + x Bc jt ∀t ∈ T, ∀ j h B jt = h B j,t−1 −

(14)

∈ N : B ∈ M( j)

(15)

∈ N : B ∈ M( j)

(16)

∈ N : B ∈ M( j)

(17)

− x Bc jt ηcB j ∀t ∈ T, ∀t ≥ 1, ∀ j ∈ N : B ∈ M( j)

(18)

max W Bmin j ≤ h B jt ≤ w B j ∀t ∈ T, ∀ j ∈ N : B ∈ M( j) yB j h B j0 = ∀ j ∈ N : B ∈ M( j) 2 yB j ∀ j ∈ N : B ∈ M( j) h B j = 2

(19)

x Bd jt ηdB j

(20) (21)

Each edge e ∈ E has a maximum capacity Fe in terms of flow, a unit transmission cost ceT and a constant line reactance (p.u.) X e . Moreover, an incidence matrix is given, whose generic element K je , for each edge e ∈ E and node j ∈ V , takes value 1 if edge e starts on node j, –1 if edge e ends on node j, and 0 otherwise. Cec represents a cycle matrix, that takes value 1 if edge e ∈ E is element of cycle c ∈ {1, .., |E| − |N | + 1}, –1 if edge e ∈ E is reversed element of cycle c ∈ {1, .., |E| − |N | + 1}, 0 otherwise. Variable f et describes the power flow along edge e ∈ E for each time slot t ∈ T . Constraints (22)–(23) represent the Kirchhoff laws that regulate the grid. Constraints (24)–(25) set a limit to the power flows, respect the limit of the grid lines. Constraints (26) define the transmission costs. z jt =



K je f et ∀t ∈ T, ∀ j ∈ N

(22)

Cec X e f et = 0 ∀t ∈ T, ∀c ∈ {1, .., E − N + 1}

(23)

−Fe ≤ f et ≤ Fe ∀e ∈ E − E cand , ∀t ∈ T

(24)

−Fg − u g Rg ≤ f gt ≤ Fg + u g Rg ∀g ∈ E cand , ∀t ∈ T

(25)



e∈E

e∈E

−cetT

≤

CeT f et

≤

cetT

∀e ∈ E, ∀t ∈ T

(26)

Constraints (27) ensure that the thermal demand Hkt of each node k ∈ R is satisfied for each time step t ∈ T . Constraints (28) limit the amount of electric energy converted to thermal energy for technologies of group L, while constraints (29)– (30) compute the amount of thermal energy generated/withdrawn respectively of technologies of group L and of solar heating, where C O Pit is the coefficient of performance for each conversion technology i and time slot t T , and F E E D I N H kt is

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the normalized feed-in for node k producing thermal energy passively from external sources.  qikt = Hkt ∀k ∈ R, ∀t ∈ T (27) i∈MT (k)

−yi j (i) ≤ xi j (i)t ≤ 0 ∀i ∈ L , ∀ j ∈ N , ∀t ∈ T = −xi j (i)t C O Pit ∀i ∈ L , ∀ j ∈ N , ∀k ∈ R, ∀t ∈ T

(28) (29)

q H kt = −y H k F E E D I N H kt ∀k ∈ R : H ∈ MT (k), ∀t ∈ T

(30)

qik(i)t

The thermal storage in each node k is characterized by a minimal and maximin and C A PSk , respectively. Efficiency mal storage energy capacity, denoted by W Sk c d s parameters η Sk , η Sk and η Sk are associated with charge, discharge, and for hourly standing, respectively, for storage at each node k. Constraints (31)–(33) limit the amount of energy used to charge/discharge the thermal storage, and the maximal energy capacity w max Sk for a thermal storage located at a node k. Constraints (34) define the amount of thermal energy generated/withdrawn for storage systems. Constraints (35) limit the energy stored in thermal storage systems h Skt located at a node k during time slot t, to maximum and minimum values, while constraints (36)–(37) state the energy stored in thermal storage systems for the first time period to the given value W Sk0 , and for all the other time periods. d ≤ ySk ∀t ∈ T, ∀k ∈ R : S ∈ MT (k) 0 ≤ q Skt c −ySk ≤ q Skt ≤ 0 ∀t ∈ T, ∀k max w Sk = C A PSk ySk ∀k d c q Skt = q Skt + q Skt ∀t ∈ T, ∀k min max W Sk ≤ h Skt ≤ w Sk ∀t ∈ T, ∀k

∈ R : S ∈ MT (k) ∈ R : S ∈ MT (k)

(32) (33)

∈ R : S ∈ MT (k)

(34)

∈ R : S ∈ MT (k) ∀k ∈ R : S ∈ MT (k)

(35) (36)

c − q Skt ηcSk ∀t ∈ T, ∀k ∈ R : S ∈ MT (k)

(37)

h Sk0 = W Sk0 h Skt = h Sk,t−1 ηsSk −

d q Skt ηdSk

(31)

Moreover, for each power plant i ∈ Q the backpressure coefficient B PCi is given. Constraints (38) limit the amount of electric energy converted to thermal energy for technologies of group Q, while constraints (39) define this value. 0 ≤ xi j (i)t ≤ yi j (i) ∀i ∈ Q, ∀ j ∈ N , ∀t ∈ T

(38)

qik(i)t B PCi = xi j (i)t ∀i ∈ Q, ∀ j ∈ N , ∀k ∈ R, ∀t ∈ T

(39)

The following constraints (40)–(45) define the variables of the problem.

An LP Model for the Central Energy System

67

Table 1 Sets, parameters and variables of the model Sets t ∈ T = {1, ..., }

Set of time slots

j ∈ N = {1, ..., n}

Set of nodes

I = {W, O, P, B, F}

Set of technologies

M( j) ⊆ I

Subset of technologies available at node j ∈ N

e ∈ E = {1, ..., m}

Set of edges

k ∈ R = {1, ..., r }

Set of thermal nodes

J = {H, S}

Set of purely thermal technologies (H stands for solar heating, S for thermal storage)

  L = {i = k(i), j (i) : k(i) ∈ R, j (i) ∈ N }   Q = {i = k(i), j (i) : k(i) ∈ R, j (i) ∈ N }

Set of technologies converting electric energy from node j ∈ N to heat for node k ∈ R Set of technologies obtaining heat for node k ∈ R while producing electric energy for node j ∈ N

MT (k) ⊆ J ∪ L ∪ Q

Subset of technologies available at node k ∈ R

g ∈ E cand ⊆ E

Set of candidate lines for transmission corridor expansion

Parameters p

CBj

Unit production cost to charge or discharge of battery at node j

CF j

Unit production cost of power plant at node j

p

ηF j

Efficiency of power plant at node j

CiCj

Installation cost per unit of capacity of technology i at node j

Yimax j

Maximum capacity of technology i that can be installed at node j

CgV

Installation cost to build new line g

p C Sk p Cik − C j and C + j

Unit production cost to charge or discharge at node k Unit production cost of thermal plant i at node k Costs for power oversupply and deficit at node j

D jt

Electric demand to be supplied at node j for time step t

Eg

Total emission related to installation of candidate line g

y Ei j

Equivalent C O2 emission per unit of capacity installed of technology i at node j

E ixj

Equivalent C O2 emission per unit of power generated/withdrawn by technology i at node j

F E E D I Ni jt

Normalized feed-in for each technology i at node j at time slot t

C A PB j

maximum storage capacity that can be installed at node j

W Bmin j

Minimum electric storage at node j

ηcB j and ηdB j

Efficiency to charge or discharge for battery at node j

Fe

Maximum capacity in terms of flow for edge e

ceT

Unit transmission cost of edge e

Xe

Line reactance p.u. of edge e

K je

Incidence matrix for edge e and node j

Cec

Cycle matrix for edge e and cycle c

Hkt

Thermal demand for node k and time step t

(continued)

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Table 1 (continued) C O Pit

conversion performance for technology i and time slot t

F E E D I N H kt

Normalized feed-in from external sources for node k

min W Sk

Minimum thermal storage at node k

C A PSk

Maximum thermal storage at node k

ηcSk , ηdSk and ηsSk

Efficiency for charging, discharging, and for hourly standing of thermal storage at node k

W Sk0

Initial value of thermal storage at node k

B PCi

Backpressure coefficient for each power plant i

Variables xi jt

Amount of power generated/withdrawn by technology i at node j during time slot t

yi j

Capacity installed for technology i at node j

+ s− jt and s jt ug

Slack variables of over/under supply for time slot t and node j Expanded fraction of candidate line along edge g

x cB jt and x dB jt

Amount of energy used to charge/discharge the battery at node j during time slot t

yik

Capacity installed for technology i at termal node k

qikt

Thermal energy generated/withdrawn by technology i at node k during time slot t

xi j (i)t

Electric energy converted to heat from node j, through technology i during time slot t

c q Skt

Energy used to charge the thermal storage at node k during time slot t

d q Skt

Energy taken from the thermal storage at node k during time slot t

cP RO D

Total production cost

c I N ST

Total installation cost

cT R AN

Total transportation cost

c S L AC K

Total opportunity cost for undersupply/oversupply

T cet

Transmission cost for edge e during time slot t

ei j

Total emissions at node j due to technology i

eik

Total emissions at node k due to technology i

E M I SS I O Ntotal

Total C O2 emissions

w max Bj

Maximum storage capacity of a battery installed at node j

h B jt

Energy to be stored in a battery located at node j during time slot t

f et

Power flow along edge e during time slot t

w max Sk

Maximum energy capacity for thermal storage at node k

h Skt

Thermal storage systems at node k during time slot t

An LP Model for the Central Energy System

69

0 ≤ xi jt ≤ yi j ∀ j ∈ N , ∀i ∈ M( j), ∀t ∈ T

(40)

z jt ≶ 0 ∀ j ∈ N , ∀t ∈ T f et ≶ 0 ∀e ∈ E, ∀t ∈ T ≶ 0 ∀ j ∈ N : B ∈ M( j), ∀t ∈ T

(41) (42) (43)

x Skt ≶ 0 ∀k ∈ R : S ∈ MT (k), ∀t ∈ T 0 ≤ u g ≤ 1 ∀g ∈ E cand

(44) (45)

x B jt

Finally, the objective function (46) minimizes the total costs calculated through constraints (1)–(4). min c P R O D + c I N ST + cT R AN + c SL AC K

(46)

Table 1 summaries all the elements of the model.

4 Numerical Tests We tested the model on two instances on three types of hardware architectures, using in all cases Gurobi as LP solver. Table 2 reports the main characteristics of the instances and of the hardware, as well as the results obtained in terms of computing time. Obviously, the time needed to compute an optimal solution depends on the instance and the machine used. While a solution for instance IEE-118 was found by all the three machines, when considering the larger instance (UC1-575), a solution could be found only using a large amount of memory, as the one available on machine IAEW. Instance IEE-118 was used for preliminary tests, which confirmed the relevant role of renewable sources and storage systems: in particular, 49.75% and 29.1% of the maximum capacity was installed, respectively. Instance UCI-575, represents a more realistic case, associated with Germany in year 2050. The optimal solution has a total cost of operation and expansion equal to 38.26 billion Euros p.a. The more significant capacity expansion is experienced in wind power (205 GW) and photovoltaic power (402 GW), while storage systems and conventional power plants are expanded by 61 GW and 86 GW, respectively. We notice that 99.5% of the demand is fulfilled, with a contribution of 84% given by renewable sources. A large percentage of C O2 emissions is produced by gas-fired power plants (66%), closely followed by grid expansion (23%).

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Table 2 Instances’ and hardware’s characteristics and obtained results Instance

Electric nodes

Electric branches

Heat nodes

Storages

Time steps

Variables

Constraints

IEE-118

118

165

296

106

8760

2.11e7

2.84e7

UCI-575

575

802

755

465

8760

6.87e7

9.35e7

IEE-118

UC1-575

cpu

ram

os

3:42 h

Does not compute

Intel Core i7-7700HQ 2.80GHz

32 GB

Windows10

CLAIX2018 2:10 h (16 cores)

Does not compute

2x Intel Skylake

192 GB

Linux

IAEW (16 cores)

11:25h

up to 2x56 threads

up to 1 TB

Linux

LAPTOP (4 cores)

1:26 h

5 Conclusions and Future Developments We proposed an LP model for the Central Energy System expansion planning, that minimizes both the operational cost and the installation cost. The model imposes limitations on the allowed C O2 emissions, while satisfying the customers’ demands. Moreover, the costs of oversupply or shortage are considered. To ensure that the proposed generation operations are compatible with the network, the problem takes into consideration also a preliminary analysis of the expansion of the transmission network. Moreover, we conducted some tests on two different instances, to detect the effectiveness of the methodology proposed, using Gurobi as a solver and three different types of machines. The results showed the high dependency of the memory required to find an optimal solution, on the number of time steps and number of cores used to solve the problem. In fact, while the smaller instance was successfully solved by all the machines, the larger one could be computed up to its optimum only by the machine with the larger memory available. Finally, given the complexity of the problem and the size of the instances of real scenarios, a heuristic may be necessary to create more competitiveness from a computational point of view. This is a preliminary work that requires additional research. From a modelling viewpoint, a further step in the direction of energy coupling consists in the integration of the gas network. From a computational perspective, the design and implementation of alternative solution methods, possibly based on the structure of the problem, could allow to solve larger instances and/or to reduce the computing time needed for obtaining an optimal (or near-optimal) solution. Acknowledgements This research is part of the project PlaMES (Integrated Planning of MultiEnergy Systems). PlaMES has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 863922. The content of this publication reflects only the authors’ view. The European Climate, Infrastructure and Environment Executive

An LP Model for the Central Energy System

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Agency (CINEA) is not responsible for any use that may be made of the information it contains. Thanks are also due to two anonymous reviewers for their constructive comments and remarks.

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Author Index

B Bazzoli, Fabio, 17

Mastrolilli, Monaldo, 1 Matteucci, Massimiliano, 35 Monaci, Michele, 59 Mosca, Veronica, 35

C Cordieri, Silvia Anna, 59 P Paronuzzi, Paolo, 59 D Di Puglia Pugliese, Luigi, 35

G Gambardella, Luca Maria, 1 Gualandi, Stefano, 1 Guerriero, Francesca, 35 Gussago, Marco, 17

R Raffaele, Alice, 17 Rizzi, Romeo, 17

S Schwaeppe, Henrik, 59

K Kucukkoc, Ibrahim, 47

V Vercesi, Eleonora, 1

M Macrina, Giusy, 35

Z Zavatteri, Matteo, 17

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Cosmi et al. (eds.), Operations Research and Data Science in Public Services, AIRO Springer Series 11, https://doi.org/10.1007/978-3-031-34546-3

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Subject Index

A Additive manufacturing, 47

B Batch delivery, 47 Binary decision trees, 2

C Case study, 17 Central energy system, 59 Clustering, 22 Concorde, 1

E Expansion planning, 60

M Machine learning, 17 Mixed-integer linear programming, 17

O Order picking, 35

P Picker routing problem, 36 Provincia di Brescia, 17 Public sector, 17

R Rich vehicle routing problem, 21

G Genetic algorithm, 47

S Scheduling, 47 Selective laser melting, 47

H Hardness, 1

T Travelling salesman problem, 1

I Interlibrary loan service, 17

V Vehicle routing problem, 35

L Linear programming, 60

W Warehouse optimisation, 35

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Cosmi et al. (eds.), Operations Research and Data Science in Public Services, AIRO Springer Series 11, https://doi.org/10.1007/978-3-031-34546-3

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