Postoptimal Analyses, Parametric Programming, and Related Topics: Degeneracy, Multicriteria Decision Making, Redundancy 9783110871203

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Postoptimal Analyses, Parametric Programming, and Related Topics

Tomas Gal

Postoptimal Analyses, Parametrie Programming, and Related Topics Degeneracy, Multicriteria Decision Making, Redundancy Second Edition

Walter de Gruyter . Berlin . New York 1995

Tomas Gal, Professor-Emeritus in Operations Research and Mathematics for Economists, FernUniversität Hagen, Germany With 185 tables and 46 figures The first edition was published by Walter de Gruyter in German ("Betriebliche Entscheidungsprobleme, Sensitivitätsanalysen und parametrische Programmierung"), 1973, and translated and published in English by McGraw-Hill International Book Company, 1979. @> Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability.

Library of Congress-in-Publication Data Gal, Tomas. Postoptimal analyses, parametric programming, and related topics : degeneracy, multicriteria decision making redundancy I Tomas Gal. - 2nd ed. [Betriebliche Entscheidungsprobleme, Sensitivitätsanalysen und parametrische Programmierung. English) p. cm. Includes bibliographical references. ISBN 3-11-014060-8 (acid-free paper) 1. Decision-making. 2. Linear programming. I. Title. T57.95.G3413 1995 94-12443 658.4'033 - dc20 CIP

Die Deutsche Bibliothek - Cata!oging-in-Publication Data GaI, Tomas: Postoptimal analyses, parametrie programming, and related topics : degeneracy, multicriteria decision making redundancy I Tomas Gal. 2. ed. - Berlin ; New York : de Gruyter, 1995 Einheitssacht. : Betriebliche Entscheidungsprobleme, Sensitivitätsanalyse und parametrische Programmierung < engl. > ISBN 3-11-014060-8

© Copyright

1994 by Walter de Gruyter & Co., D-10785 Berlin. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced in any form - by photoprint, microfilm, or any other means nor transmitted nor translated into a machine language without written permission from the publisher. Converted by: Knipp Satz und Bild digital, Dortmund - Printing: Gerike GmbH, Berlin. - Binding: D. Mikolai, Berlin. - Cover Design: Johannes Rother, Berlin . Printed in Germany.

Ta my grandchildren Thamas, Anna, Sandra, and Anique

Foreword

For some considerable time, linear programming has been one of the methods of operations research which has been widely known and much applied in Germany too, both in the literature ofthe subject and in practice. Its applications range from production planning through finance planning and from optimization of traffic networks through urban planning. The literature of linear programming includes textbooks of a strictly mathematical nature as weil as programmed textbooks for those with no previous knowledge of mathematics. An objection frequently heard to more extensive dissemination of the theories of linear programming in the practical field has been that data wh ich are available in practice are at once too inexact and too unreliable to provide the basis for the application of "exact" procedures like linear programming. This problem is the starting point of the present volume. The inexactitude and unreliability of existing data often cannot be disputed. Using conventional planning methods, the determination of the effects of these inaccuracies is frequently very difficult, if not impossible. In most cases, this is, however, possible using sensitivity analysis in the widest sense (i.e., including postoptimal analysis and parametric programming), and the amount of effort involved is reasonable. My colleague, Professor Gal has been involved in research in this field for many years. His decision to undertake the writing of an introduction to and interpretation of the area of linear programming which enables us to make statements on the possible effects of data changes, data inaccuracies, and decision changes on operational and other problems is, therefore, to be welcomed. Sensitivity analysis, as interpreted in the present volume, has been seen by experts - correctly, in my opinion - as the bridge between pure dissemination of information and decision making. There is, thus, ample justification for including a volume on such an important subject in the "Operations Research" series. As the present volume is likely to interest both those working in linear programming and research mathematicians, the chapters have been written primarily with practical application in mind, and an abridged mathematical version has been appended to each of them. It is hoped that this will increase the usefulness of the volume for a wide range of readers. Professor Dr H.-J. Zimmermann Aachen, July 1973

Preface to the (second) English edition

This edition aims at to bringing the book up-to-date and correcting some errors wh ich - in spite of almost endless efforts proofreadings again and again - still are found in the (first) English edition. Also, in the organization of References a change has been made, which should, hopefully, be of advantage to the reader: direct quotations ofliterature are put together as References to each of the chapters. At the end of the book a Bibliography is to be found, in which the quoted publications and such which deal directly or indirectly with the subject of the book, are listed. To offer the reader another help, an annotated bibliography is added. In order to keep the size of the book in some limits, the author decided to omit Section 10-8 (of the previous edition) "Parametric programming in the Transportation Problem" hinting, of course, to the relevant literature in the References to Chapter 10. The author notices with pleasure that his hope, as stated in the Preface to the (first) English edition, namely that " ... this will not remain the only work on parametric programming in English and ... inspiring others to write new and more comprehensive monographs on the subject", has been fulfilled. The reader will find several monographs on parametric programming written in English in the Bibliography at the end of the book (see also the References to Chapter 4). The organization and the structure of the book has been described and explained in the Preface to the German edition. From this point of view nothing has been changed. The author apologizes in this place for being wrong in the Preface to the (first) English edition (written in 1978). First, as is pointed out in a study on the history of parametric programming I, Gass and Saaty have not been the very first authors dealing with parametric programming. Second, claiming, in 1978, that " ... there has never been a conference on parametric programming anywhere in the world ... " was wrong: In July 1977, AY.Fiacco organized the first Conference on Data Perturbation in Washington, D.C. , which is ever since being held every year. I should take this opportunity of thanking my new/old publisher, W. de Gruyter, for taking care of publishing this new edition, while McGraw Hili forgot to notify me that the book had been sold out in 1983. I am indebted to Dr. F. Geue for his carefully reading the proofs and to Mr Th. Hanne for unifying the bibliography. My thanks go also to my second wife, Gisela, who helped me to formulate in some

I See[II,12jinChapter4.

x

Preface to the (second) English edition

places the English text and for whom I spoiled our stay in Spain by intensively working on this new edition instead of sitting on the beach. TomasGal Benidorm, Spain /Hagen, Germany, January 1994

Preface to the (first) English edition

Parametric programming was first developed over 20 years ago (the first papers by Saul I. Gass and Thomas L. Saaty date back to 1955). Since then, theories of linear and nonlinear parametric programming have been worked out, solution procedures for various cases have been developed, and parametric programming has also been used for solving problems of mathematical programming, such as decomposition, quadratic-, fractional-, and nonlinear programming. Parametric programming has also been applied in various fields of economics, chemistry, technology, agriculture, etc. It is, therefore, rather odd, that there are very few courses on parametric programming in existence and that, to the author's knowledge, there has never been a conference on parametric programming anywhere in the world; also, there has been no previous monograph on parametric programming in English. The manuscript of this book was originally written in Czech in the years 1968 through 1969. In 1973 it appeared in German. That is why the main theme of the work is linear parametric programming. The aims and objectives in writing this book are discussed in the Preface to the German Edition. The reader will also find there an explanation of why each chapter is divided into two parts. The original German edition of this book has here been revised with the aim of correcting errors and bringing the new English edition up-to-date. Since the first chapter of the German edition (on "Systems, Models and Systems Analysis") is now out of date, it has been omitted here. In its place appears a new chapter (Chap. 9) on "Multicriteria linear programming". Minor alterations have been in almost all chapters, but Chap. 2 (Chap. I in the present English edition), in particular, has been extensively revised to include some relations between convex polyhedrons and graphs as weil as a newly developed consideration of degeneracy. Chap. IO now includes a section on parametrization of transportation problems. The author would like to thank Dr W. Hummeltenberg for preparing this section. The writer regards it as a honor to find himselfthe author of the first monograph on parametric linear programming in English. It is to be hoped that this will not remain the only work on parametric programming in English and that the present book will make some contribution to the development, application, and dissemination of parametric programming, while at the same time inspiring others to write new and more comprehensive monographs on the subject. The bibliography has been something of a problem. The original German version contained a list of 392 works with an appendix containing a further 27. Since the appearance of the German book, something like 300 further titles have appeared. It was quite simply impossible to include these with the earlier titles in a single list. This would have meant altering all references throughout the book.

XII

Preface to the (first) English edition

The author, therefore, opted for presenting the new ti tl es in a separate list, in wh ich each number is indicated by a prime. Almost every day so me new title appears. In view of the sheer number of journals and books appearing, it is impossible ever to produce a complete list of references, let alone find all the relevant publications. We shall simply add new titles to the end of the list until such time as the book has actually gone to press. By now there exist about 500 direct or indirect references to parametric programming. The bibliography at the end of the book has been subdivided and is designed to provide the reader with a survey of relevant literat ure from various aspects. We make no claim to having provided a complete list. It has been put together in a way that seemed appropriate to the author, although this is possibly not the best conceivable arrangement. I should like to take this opportunity of thanking my publisher, McGrawHili, for their interest and their cooperation, especially Mr A. von Hagen, Ms B. Scholtz, his editorial asistent, and their colleagues. My thanks also go to Dr Geoffrey V. Davis for his careful translation and his patience in discussing the whole text with me. I am indebted to my wife Dana for reading the proofs. Tomas Gal Aachen, October 1977

Preface to the German edition

Economists and specialists in other fields often meet with failure when they first attempt to introduce linear programming (LP) into their operations. The reason for this is frequently one of the following factors: I. The difficulties wh ich have to be overcome in devising a suitable model. 2. The uncertainty and inaccuracy of the intial data, the Iinearizing of the de facta - nonlinear relationships, the neglect of time (dynamic) factors, the determining of originally stochastic data, etc. 3. The problem of evaluation and interpretation as weil as the application and exploitation of the results in practice. The aspects mentioned under 2 may possibly inspire specialists with a deep distrust of the result of a solved LP problem. It is for this reason that the optimal solution of a linear program mayaiso be considered the first step towards the solution of a given operational or similar problem. The question arises as to whether and to what extent the results of the solution of an LP problem may be of practical use, in spite of the "disadvantages" mentioned. A first reply to this would be that the optimal solution of the problem as such does, in fact, have more of an informative character. However, there do exist other possibilities of utilizing this solution and, in a certain sense, of eliminating the "disadvantages" mentioned in 2. Such is, for example, the aim of sensitivity analysis, by means of which (among other things) one may test in what region the values, say, ofthe right-hand side ofthe restrictions can be changed, so as to maintain the optimality of the optimal solution obtained. The extension of sensitivity analysis into parametric programming also enables us to compute all existing optimal basic solutions in relation to their dependence on the values of the components of the right-hand side. Approaches to the solution of these problems have been provided by the development of systems theory. On this theory are based systems analysis, systems synthesis, systems engineering, and other developments. Nevertheless, there is considerable confusion over the symbolism and vocabulary of these branches of systems theory. This makes it difficult to characterize them and distinguish them from one another in a few words. This is discussed more fully in the first chapter. I By way of a summary, let us consider a firm as a system: using the methods of systems analysis, it is then possible to devise a linear model which can serve as the basis for setting up of the LP model. The methods of sensitivity analysis or of parametric programming enable us to follow up the connection between the firm, I See the Preface to the (first) English edition.

XIV

Preface to the Gennan edition

the model, and the optimal solution. From the point of view of an expert (of an economist, for example), it is doubtless both important and necessary to be able to take into account as many of the changes occuring in the initial factors during the course of time as possible. The methods of parametric linear programming and sensitivity analysis, in association with the ideas of systems analysis, are particularly suited to this purpose. These methods are, thus, no longer the object of the investigation; they become it means. In the practical application of LP, three main complexes of problems may be distinguished. I. The setting-up of the linear model. 2. The computation of the optimal solution. 3. The evaluation, interpretation, and analysis of the optimal solution. The systems analysis approach is of assistance with the first and third complexes of problems. If we are dealing with a "normal" linear program, the computation of the optimal solution requires nothing but a computer. Sensitivity analysis and parametric programming are mainly, though not exclusively, of use in dealing with the third complex of problems. Moreover, parametric linear programming is an instrument which may possibly be applied to aB three complexes of problems. It is the aim of this book to elucidate the various methods of sensitivity analysis and parametric linear programming. The main emphasis, however, lies on the application of these methods to the most diverse purposes involved in the analysis of a model and/or of the corresponding optimal solution, as weB as on the combinations of different approaches to the solution of practical problems. We shaB also indicate other possible theoretical and practical applications of the methods described. It is assumed that the reader is familiar with the fundamentals of linear programming and, therefore, also with linear algebra (vector and matrix ca\culus, the theory of systems of linear equations, and inequalities). As an additional aid, the fundamental principles of LP have been briefty recapitulated in Chap. 2. 2 Each chapter is divided into two parts. In the first part, the problems are discussed with the aid of examples; the second part is then an abridged mathematical presentation which, however, makes no claim to being exhaustive. It is left to the reader to decide whether he 3 wishes to read both parts or only that wh ich corresponds more closely to his 4 own particular interests. The examples have been chosen with a view to helping the reader to arrive at a better understanding of the methods and problems described. They contain few unknowns and a few constraints. 2 See the Preface to the (first) English edition 3 Or she 4 Or her.

Preface to the Gennan edition

xv

The appendix offers an extensive list of references. Not all the authors and titles mentioned there have been quoted in the text. The aim was rather to provide a survey of the literature which is concemed directly or indirectly with the theory, solution procedures, and applications of sensitivity analysis and parametric programming. A selection of works has been incIuded, in addition, which deal with the theory and application of systems analysis as wel1 as a number of textbooks on linear programming which make reference to sensitivity analysis and parametric programming. The author of this book formerly worked in Prague and, after a short stay at the University of Louvain in Belgium, took up his present post at the RhineWestphalian Technical University of Aachen in the autumn of 1970. The preparation of the German text of this book from an earlier rough manuscript involved certain difficulties of a linguistic nature, which were overcome with the assistance of several members of the staff of the Department of Operations Research in Aachen. I should particularly like to thank Messrs. H. Gehring, W. Hummeltenberg and Dr U. Eckhardt, as wel1 as Miss I. Teutsch. I am also much indebted to the Head of the Department of Operations Research at the University of Aachen, Professor Dr H.-J. Zimmermann, for his friendly support during the preparation of this book and for enabling me to devote a considerable amount of effort to this time-consuming work. Final1y, I should like to express my thanks to my publisher, de Gruyter, for their cooperation and their understanding of my particular situation . Tomas Gal Aachen, March 1972

List of symbols

Symbol

Meaning

IR n

n-dimensional real space

O=(."" " .,)T= ( ]

column vector, a

E

IR n

row vector - vector a transposed

A = ('"

'" )

ami

' a mn

0=(0, .. " O)T

0=

an (m, n) matrix with elements aij, i = I, .. " m,j = I, .. " n, technological matrix, matrix of the coefficients of the variables null vector,

( ~: : :~) "," ..... 0 .. ,0

0 E

IR n

(m, n) null matrix

k

~ T ek =(0, .. ,,0,1,0, .. ,,0)

1=

( ~~:::~~) ""'" ....... 00 .. ,01

x

=(XI, .. "

C

= ( CI,

Z

= cT X =

xn)T

.. " C n

l

unit vector, e k

E

IR n

(m, m) identity matrix

vector of variables vector of objective function coefficients, or, briefty, cost vector

n

I: CjXj

objective function

j=1

b

=(bi, .. " bml

right-hand side of the constraints (Ax bE IR m

=b),

XVIII

List of symbols

p = {jl, ... ,jm} ... , ~mk)T. Let the linearly independent non-null vectors vj E IR m, j = I, ... , m, and the non-null vector a E IR m be given. Let aJ, ... , a m be the coordinates of a with respect to the basis

I = (eI, ... ,em),ei

E

IRm,ei

thejth unit vector;

then , obviously, a = ale l + .. . +ame m holds. Let the matrix 8- 1 be the inverse to matrix 8 = (vI, ... , vm). Then it follows that

where a=ulv I + ... +UmV m We then say that the vector a is transformed from the basis I into the basis 8 by the matrix 8- 1, or 8- l a is the vector a associated with basis 8. With the aid of an inverse 8- 1 each column äi of the matrix Ä or the column b of the problem (I-2a) can be transformed from the initial basis into basis 8; namely (1-9)

where .

T

yJ = (Ylj' ... , Ymj) .

(1-10)

Furthermore, 8- l b =

XB,

(I -11)

where

(1-12) and (1-13)

where Y = (y I , ... , y N ),

(1-14)

and see the "Notation note" at the end of Sec. 1-1. Definition 1-9 The vector XB E IR m is called a basic solution. If 8 is an optimal basis, XB is called an optimal basic solution.

30

Abridged mathematical presentation

Let the index-set J = {j Ij = 1, ... , N} of all variables be divided into subsets p and

0 for at least one jE {I, . .. , m+n}, j :f:; k. The proof follows immediately from Corollary II-5-1.

11-2-2 An algorithm for determining redundant constraints Consider the problem min Si, xeX

i

= I, ... , m,

(11-26)

and suppose that X :f:; 0 ( ~ X :f:; 0 ). Let x~O) E X be an optimal solution to (II-26) for some i E {I, ... , m}, i.e., xi is a nearby vertex to the ith inequality. Associated with B (or with p) denote 11 the set of subscripts j of the basic variables Xj; 12 the set of subscripts i of the basic slack variables Si; NI the set of subscripts j of the non basic variables Xj; N2 the set of subscripts i of the nonbasic slack variables Si . Without loss of generality, assurne that the subscripts j EIl and i E 12 are numbered equally with the rows of the corresponding simplex tableau. The algorithm consists of two parts: Part I: Determine a first solution x~O) EX. 14 Part 2: Starting with x~O) , solve (11-26) for every i = I, ... , m. Part I consists of the usual simplex method. Note According to the assumptions, X :f:; 0. Since, in X, the variables Xj are not sign-restricted, the initial solution as weil as each tableau is generated by a modified simplex procedure (cf. Sec. IV-4). If the variables are sign-restricted, use the usual simplex method. Suppose that in Part 2 a solution x~) is already generated. Then:

14 The subscript

"0"

indicates that the starting basis is denoted by 8 0 .

72

Abridged mathematical presentation

Step J. Regarding the tableau assigned to the solution x~) Si with i E N2 have already reached their minimum, i.e., min Si = 0

and

Si

E

X, the slack variables

NBY.

XEX

The corresponding inequalities (constraints) are binding or nonredundant. Step 2. Investigate in the tableau for x~) all rows with i E 12 with respect to the property (11-24). Denote by I' 2 ~ 12 the index-set of rows in which Redundancy Criterion 2 holds. Then

= s~ and

min Si iEI;

I

Si

BV

and the corresponding constraint is redundant. Step 3. In the "criterion rows" i E h - I~ for the remaining slack variables, there exists aij > 0 for at least one j (except the "1" of the corresponding unit column). Assume that this is the case in the rth row, i.e., r E 12 - I~ fixed . Determine

e Suppose e = determine

= max{ Urj I arj > 0}15 arp

> 0, P E N2 . In order to maintain the feasibility

r \ (p) ~"min -

Si ~

. {~I a' l p > O}, . mln

IEI,- I,

0 for all i, (11-27)

aip

3 1 Let Q~ln = ~r ; then it is possible to eliminate Sr. This implies that min Sr = rp o and Sr NBV; consequently, the corresponding constraint is binding (nonredundant) .

J!-,

v '# r, v E 12 - I~. Then Sv can be eliminated. This implies 32 Let Q~ln = vp min Sv = 0 and SV NBV, i.e., the corresponding constraint is binding. Step 4. Perform the procedure described in Step 3 for all i E 12 - I' 2. Step 5. List all those indices i for which the min Si is already known. If all indices i E 12 U N2 are listed, STOP. Otherwise go to Step 6. Note that until now no pivot step has been performed. Step 6. Assume that case 32 occurs, i.e., the minimum of Sr cannot be determined immediately. Perform a pivot step with the pivot a vp and generate by this a simplex tableau associated with B' and with the solution x~). Go back to Step 2, and consider B' and the indices i not yet listed. 15 This condition is not at all necessary. It suffices to choose any column j with arj > O. This note is based on a private communication from Dr Jan Teigen of the Erasmus University, Rotterdam (in 1978). The above selection should speed up the procedure.

Degenerate polytopes and degeneracy graphs

73

Note that it is possible to delete all the rows that correspond to redundant inequalities [13]. Therefore, before carrying out Step 6, delete all such rows.

11-3 Degenerate polytopes and degeneracy graphs In this section we shall deal with primal degenerate solutions and the associated degeneracy graphs. Let us stress that we shall present here only the main principles, which shall be needed in later chapters. More on this recently developed research area see, for example, [7,9, 12, 15,20]):6 Assume that all hyperplanes passing through avertex of X are Iinearly independent.

Definition 11-7 Avertex XO E Xis said to be a cr-degenerate vertex, cr being the degeneracy degree, 1 ::; cr < m, if n + cr hyperplanes pass through XO or cr basic variables in the complete basic feasible solution x~) associated with XO vanish, i.e., x(0) B

= ( YI"",Yo,Yo+I"

", Ym, 0 , ... , O)T

E

IR m+n ,

(11-28)

with Yi = 0 for i = I, ... , cr } Yi>O for i=cr+l , ... ,m,

(11-29)

holds; the indices of the variables are rearranged such that ji = i for all i = I, ... , m+n.

Definition 11-8 Let XO

E

X be a cr-degenerate vertex. Then the set

BO={B~lu=I, ... ,U},

U~I,

(11-30)

which is assigned to xO, is called the basis-set of XO and U the degeneracy power ofxo. It has been shown [15] that

2min {o.n}-1 (I n - cr I + 2) = Umin ::; U ::; U max = (n

~ cr )

(11-31)

(compare, for some values of cr and n, Table 2-14). Let us introduce three kinds of operators describing various types of basesexchanges. Given two neighboring bases B, B' and the associated tableaux T, T', respectively. Then the operators for a basis-exchange are:

16 The needed notation is in Sec. I

74 " f-

Abridged mathematical presentation

+ ---1"

using a positive pivot-element (positive pivot-step for short) using a negative pivot-element (negative pivot-step for short) using any nonzero pivot-element.

"f- - ---1" "f----1"

Definition /1-9 The representation graph G(X) of a polytope X is the (undirected) graph

(11-32)

G(X) := G = (V, E), where

= {B IBis a feasible basis of (1-7)} } E = {{B, B'} s;;; V I B f- + ---1 B'}

V

(11-33)

The (bases-) structure of a o-degenerate vertex XO can be studied using Definition /1-10 Let XO E Xc IRn be a o-degenerate vertex. Then the (undirected) graph

(11-34) where BO

E~

is given by (11-30)

= {{B~,B~, } s;;; BOIB~ f- + ---1 B~} u,u'

E

} (11-35)

{l, ... ,U}

is called the positive degeneracy graph (positive DG for short) of xO. If the operator is f- - ---1 , then the corresponding DG is called the negative DG of xO, notation G~ . If the operator is f----1, then the corresponding DG is called the general DG, notation GO. Let us now introduce some specific nodes of a DG 17 and their basicproperties. For this purpose partition the index-set 1= {i I i = I, ... , m } into:

= {ilYi = O},

(11-36)

IN = {ilYi >O}.

(11-37)

10

If, in tableau

T~,

there exists a nonbasic column "t" such that

V Yit ~ 0 ie!"

and

:3 Yit > 0

(11-38)

iel,

then column t is called transition column and the corresponding node is called a transition node. If, in T~, there is no transition column, then the corresponding node is called an internat node. Let B~ E BOassociated with the o-degenerate vertex XO E X be given. Denote by 17 The described properties hold for all three kinds ofthe DG's, unless otherwise specified.

Degenerate polytopes and degeneracy graphs

is feasible basis of (1-7) I

75

:1

B~ f-

+ ~ B}

(11-39)

B:eB" the set of all nodes associated with the neighboring vertices x' , t = I, ... , L, of xo. Then, the outer degree, da, of node B~ is given by (11-40)

The inner degree dj, dj or d i of B~, which depends on whether the corresponding DG is a positive, negative or generalOG, respectively, is given by dj = I{{B~,B~} ~ Ba I B~ f- + ~ B~}I { dj = I{{B~,B~} ~ BO I B~ f--~ B~}I di = I{{B~,B~} ~ BO I B~ f-~ B~}I

(11-41)

Note that if do~ I then B~ is transition node, if do = 0, then B~ is an internal node. If dj, dj, or di = 0, then B~ is a so-called isolated node in the corresponding positive, negative or generalOG, respectively. A frequently occuring problem in mathematical programming is the so-called neighborhood problem (N-problem for short). This is to determine all neighboring vertices of a cr-degenerate vertex XO E X . Various methods have been worked out to solving this problem (see the surveys, in [12,15]). These methods depend heavily on the applied pivot selection rules. Computer tests comparing various such rules showed [12J that such a pivot selection rule, which is quite efficient even in large scale problems, is the so called Transition-Node-Pivoting rule (TNP-rule for short), the principles of which are as folIows. Before starting this description, a few more notions from the theory ofthe OG's are needed [12,15,20]. Transition nodes of a OG, which are joined with node(s) assigned to the same neighboring vertex of xO , are called a set of transition nodes. Any subgraph Ö~of G~, that contains at least one node of every set of transition nodes, is called N-correspondence. The so-called N-condition is fullfilled (roughly said; for more details see [15]), if Ö~ is a connected graph and is connected with all nodes associated with all the neighbors of xo. If an N-correspondence is connected, then it satisfies the N-condition. Without loss of generality, suppose that all neighbors of XO are nondegenerate. Passing from B, of ,c to B( of xt' using j 4 Bt as pivot column, the known (primal) feasibility criterion is Q

~n = minie {~Yij I Yij > o} . I

It is also known that passing from (11-42) yields a set of indices

(Il-42)

,c to a cr-degenerate XO the feasibility criterion

76

Abridged mathematical presentation

{i

I

O ßik -00, if there does not exist

Yi _ max(--) Ak = { i.~;, 0

(3-11) ßik < 0

This makes clear that for the determination of the critical region (interval) of Ak in cases in which bk(Ak) = bk + Ak , the components Yi of the solution XB and the components ßik of the kth column of the matrix B- 1 are decisive. On the basis of (3-10) and (3-11), let us now determine the critical interval for A3 . In order to proceed systematically, we shall first go back to Table 1-2. Since it is a question of a change to the third element b3 = 121 of the vector b, the column ß3 , i.e. the column under the subsript 6 (Table 1-2) will be decisive for the critical interval A3 . First, find all positive elements ßik in this column and form the quotients from the elements Yi, which are directly opposite to them in the same row, and from the positive elements themselves. The quotients so formed are then provided with a minus sign, thus:

7.5 22.5 0.03' 0.3

--- ---

The larger of them determines ~k = -75. Now find the negative elements in the column ß3 and form the quotients in the same way as before:

5.5 21.5 -0.03 ' -0.1'

--- ---

The smaller of these quotients determines x'3 = 165. Hence, A3 = [- 75, 165 ]. Determine (briefly) the critical intervals for A2 and A4. We have

22.5} k = max { --1=-22.5, A2_ = +00; thus,

Sensitivity analysis with respeet to a single eomponent bi of the right hand side b

87

A2 = [-22.5,+00); and ~

_ = min { ---=-1 21.5} = 21.5 ; = -00, A4

thus A 4 = (-00,21.5]. Let Xj be a basic variable with the value Yi in the optimal solution (e.g., Xs = Y4 = 22.5). The dependence of the basic variable on the parameter Ak (derived in Sec. III) is given by (3-12) If we consider Table 1-2, the dependence of the basic variables on A3 will take on the form XI (A3) = 5.5 - O.03A3, X2(A3) = 7.5 + O.03A3, X3(A3) = 21 .5 - O.IA3, XS(A3) = 22.5 + O.3A3 with A3

E

A3 .

The dependence of the objective function value on parameter Ak in the case of bk(Ak) = bk + Ak (according to (III-39) is given by (p) ('

) -

(p)

,

zmax "'k - zmax + Uk"'k,

(3- 13)

where Uk is the value of the kth dual variable, i.e., it is the element in the criterion row wh ich appears in the kth column of the matrix B- I .4 Thus, for example, for AI (cf. Table 1-2), (p) Zmax(AI) -- 76.5 + 4.3AI

holds. Substituting an arbitrary value AI E AI into one of the relations in which we are interested, enables us to calculate the corresponding value of Yi(AI), i = I, ... , 4, or of Z~~x(AI) immediately. Now set A~ = XI = 225/26 and compute the corresponding value of the respective basic variables: • I I I I 225 154 XI(A) = - + - - = -::::::' 11.8 I 2 15 26 13 ' • 15 4 225 255 X?(A) = - + - = -26 : : : ' 9.9 ' - I 2 15 26 4 Thi s value is often ealled shadow priee. More on this, cf. Sees. 4-4-2 and IV-7-2.

88

Sensitivity analysis with respect to b

Table 3-1

Infeasible solution with AI: = 10 and passing to an optimal solution 5

11/5

0

-1/30

0

77/6

0

1130

0

61/6

16/5

0

-1/10

-I

107/2

-13/5*

I

3/10

0

-7/2

13/3

0

1/6

0

719/6

I

4/15

2 3 f-5

6

7

Y(A~ )

4

I

0

11/39

2/39

0

154/13

2

0

4/39

5/78

0

255/26

3

0

16/13

7/26

-I

1279/26

~4

1

-5/13

-3/26

0

35/26

0

5/3

2/3

0

*

X3(A I) =

*

X5(A I) =

43

16225

45

13225

114

1279

2 + 526 = 26 ~ 49.2, 2 - 526 = O.

By A~ = XI the value of x5(A~) has become zero. So, if we were to choose a value A~* = XI + e, e > 0 sufficiently smalI, we would obviously have x5(A~*) < 0, and this would make the solution primal infeasible. This is also the reason why the interval A k is called "critical". In substituting one of its boundary points, we reach "the limit of primal feasibility". Set, for example, AY = 10. Since AY ~ AI, the corresponding solution must necessarily be infeasible. Table 3-1 shows this primal infeasible solution as weil as the optimal solution regarding another optimal basis caIculated by means of a dual step. This all gives rise to the notion of sensitivity analysis with respect to the right-hand side in terms of (i): It is to find the corresponding critical interval Ak of a parameter Ak such that for all values Ak E Ak the optimal basis B does not change.

3-1-1 Geometrie meaning of a single parameter Let us consider the first condition from Exs. I-I and 3-1 with the parameter AI in the right hand side, i.e., x I + X2 ::; 13 + AI. The boundary line x I + X2 = 13 + AI of the half-plane (I) is in the original position for AI = 0, as shown in Fig. I-I . For

Sensitivity analysis with respect to a single component bj of the right hand side b

""-

""-

89

"

Ai" 8.6

Figure 3-1

1"1 > 0 , the line (I) moves into positions further away from the origin, for AI < 0 (I) moves nearer to the origin. This is shown in Fig. 3-1. From Section 2-4 we know that in point P of Fig. I-I, wh ich represents the optimal vertex (extreme point) of the original problem, the variables XI, X2, X3, and X5 are positive and X4 = X6 = O. If (I) is moved into a position such that it intersects with (3) within the segment P 5P6 (Fig. 3-1), then XI, X2, X3, and Xs will still be positive and we shall still have X4 = X6 = O. The values of the basic variables will, of course, vary according to the different positions of line (I). Thus, the values of the basic variables change, although the basis remains the same. If (I) is moved into the position shown in Fig. 3-1 by a dashed line passing through point P 6, then the intersection of (3) and (I) will be P6, which at the same time is also the intersection of (2), (3) and (I), (2). At point P6, obviously, XI > 0, X2 > 0, X3 > 0, X4 = Xs = X6 = O. This solution is degenerate, but it is sufficient to move line (I) slightly in the direction of the origin to obtain immediately X4 > O. In the basis, X5 is thus replaced by X4. This process was also illustrated numerically in the preceding section. We shall be confronted with a similar situation if (I) is moved to point Ps, since, here XI > 0, x2 > 0, x5 > 0, x3 = x4 = x6 =O. If (I) is moved however slightly c10ser to the origin, then we have XI > 0, X2 > 0, Xs > 0, x6 > O. In the basis, X3 is thus replaced by X6.

90

Sensitivity analysis with respect to b

Points Ps and P6 can, in a way, be seen as critical points. In each of these "critical points" we are at the separation point of two optimal bases associated with feasible values 00", . These critical points correspond exactly to the boundary points of the critical interval A I of the parameter" I. If,,~ = XI, i.e., "I becomes the critical value, the solution set of the corresponding problem will be formed by the polytope with the vertices PI, P2, Ps, P6. If ,,~* = .6: 1, the solution set will be formed by the polytope having the vertices PI, P2, Ps, P7 · Let us consider the following situation. If straight line (1) is moved so that it intersects with straight line (3) outside the segment Ps, P6, condition (I) becomes redundant. In order to get line (1) into such aposition, it is necessary that "I > XI, i.e. bl ("I) > 21.6. Setting "y = 10, i.e. b l = 23, we obtain the solution from Table 3-1. If, however, we omit the first condition for bl = 23, i.e., if the system of constraints were to have the form

(,,?)

("n

5xI -8xI 4xI XI

-

4X2:::;

20,

+ 22x2 :::; 121,

+

x2 2 8. 20, X2 2 0,

then we would obtain XI = 10, X2 = 9.81, X3 = 49.2, Xs = X6 = 0, and X4 would not be in the system. We shall now turn to the term "sensitivity". It should be pointed out that this term is being introduced here only in a restricted parametrie sense (cf. Dinkelbach [4]). In the case where (3-14) the critical interval A k defines the region (interval) of "stability" of the optimal solution Y("k) in respect of the change in bk according to (3-14). The "narrower" the interval Ak is, the more sensitive will be the optimal solution to the change in bk ("k) in the sense of (3-14). This means that, even with relatively small changes of bk> according to (3-14), primal feasibility is violated and, consequently, the corresponding solution is no longer optimal. Sensitivity analysis includes the determination of the A k for each k after the optimal solution ofthe original problem has been determined, whereas bk changes according to (3-14). As this takes place only after having determined the optimal solution, sensitivity analysis is a postoptimal investigation. In what folIows, we shall show that sensitivity analysis is not necessarily restricted to the case (3-14). However, in the most of commercial software to solving linear programming problems, there is a device, called RHS-ranging; this means sensitivity analysis in terms of (3-14).

Sensitivity analysis with respect to several components

91

3-2 Sensitivity analysis with respect to several components of the right-hand side depending on a scalar parameter Suppose that in Ex. I-I or Ex. 3-1 the eomponents of the right-hand side bare dependent on a seal ar parameter as folIows: b,(A) = 13-0.5A b2(A) = 20 (independent of A) b3(A) = 121 + 8A, b4 (A) = 8 + 0.6A. As we already know, the original problem given by (I-I) through (1-3) has a finite optimal solution. This was shown in Table 1-2. In other words, we would obtain this result if the value A = 0 were substituted into b(A) = (13 - 0.5A, 20, 121 + 8A, 8 + 0.6A)T. This allows us to determine the eritieal interval A for the parameter A starting from the optimal basis already eomputed, i.e., to earry out a (postoptimal) sensitivity analysis. We have 0.73 0 -0.03 Y(A) = B-1b(A) = (

~:;6 ~ _~:~3 _~

-2.6

I

~~ -0.5A)

0)

0.3

(

121 + 8A 8 + 0.6A

0

(-0.63 ) 5.5 - 0.63A) _ 7.5 + 0. 13A _ 0.13 ~ ( 21.5-3A -Xs+ -3 11.. 22.5 + 3.7A 3.7 If we denote the eoeffieient veetor of the parameter A by f = (-0.5,0,8, 0.6)T, then B- 1f = Pf = (-0.63,0.13, -3, 3.7)T. This yields Xs(A) = Xs + PfA. The dependenee of the objeetive funetion value on the parameter A beeomes

z~~x

=C~XS(A) = c~(xs + PfA) = z~~x + fm+IA,

f m+ 1

=c~Pf.

where

In our example, -

-

T

-

f m+1 = (3,8,0,0)(-0.63,0.13, -3,3.7) = 0.83, i.e.,

92

Sensitivity analysis with respect to b (p) Zmax(A) -- 76.5 - 0.83A.

In the preceding section, we demonstrated that the values L1Zj are independent of A. For this reason, the critical region (interval) A of the parameter A is defined by the primal feasibility condition only, i.e. XS(A) ;::: 0,

i.e. Xs

+ PfA;::: 0,

or - PfA ~ Xs

must hold. Solving this system of inequalities gives us the lower boundary point A and the upper boundary point X. of the critical region A of the parameter A. Here, y. max{---"':'}, { A= i.'f, >0 Pfi if there is no

Pfi > 0

. { - Yi} _ mm - < 0, A = { i.'f, Pfi +00 if there is no

Pfi < 0

-00

In our particular case, these critical values are 7.5

22.5

~ = max( - 0.13' - 3.7 ) '::: -6.1

_ 5.5 21.5 A = min(0.63' -3-) '::: 7.2

(precisely, -6.081)

(precisely, 7.16).

In this case, too, we speak of postoptimal sensitivity analysis, although we are not dealing directly with testing the sensitivity of the optimal solution to the changing of individual components of the right-hand side. In this extended sense, sensitivity analysis may be understood as the investigation of the influence of changes to the right-hand side on the optimal solution and on the value of the objective function within the framework of a found optimal basis. Sensitivity analysis is briefly touched on in many linear programming textbooks (cf., for example, [1,2,3,5,6,8]). A detailed discussion can be found in W. Dinkelbach [4], c. van de Panne [9, 10], F. Nozicka [8] and in other monographs to be found in the Bibliography at the end of this book.

93

Multiparametric sensitivity analysis

3-3 Multiparametric sensitivity analysis: Changing several components of the right-hand side depending on several parameters (on a vector parameter) The diverse ways in which a given firm, its divisions, and its environment can influence one another (if we consider only the right-hand side, as in this chapter) bring about changes not only to one component in the right-hand side but to several simultaneously. Insofar as a change may be expressed in terms of a single factor, this has been done in the case of the preceding section. Changing one component of the vector b can, however, cause subsequent changes in other components of b. If each of the components of the vector b is considered from this point of view one after the other, we shall see that we have to reckon with several influencing factors. If the dependences of the respective components of the right-hand side on the influencing factors we have mentioned are expressed in the form of linear functions, the "independent variables" being considered parameters, these functions will asume the following form:

In vector-matrix form

b(>") = b + FA, where

F=

(

fll,

fIS)

:

.

:

f ml ,

. •• ,

f ms

,>..

=(/"1, ... ,As)T.

Example 3-2 For the sake of simplicity and to facilitate geometrical representation, let us consider the case of a two-dimensional vector parameter>.. = (AI, A2)T. In the problem of Ex. 1-1, let

be given. Then,

94

Sensitivity analysis with respect to b

Figure 3-2

(p) zmax().) -- 76.5 - 0.83/1.) + 8.25A2.

From the condition XB(A) 2: 0 it follows that (1) (2) (3) (4)

+0.63AI -0.13AI +3 AI -3.7 AI

+

1.55A2 0.45A2 6.65A2 4.75A2

:S: :S: :S: :S:

5.5, 7.5, 21.5 22.5.

These four inequalities define the region Ac IR 2 . This region is shaded in Fig. 3-2. The boundary Iines correspond to the numbering of the inequalities. For any point P E A, i.e. , for any ordered pair (AI, A2) E A it is possible to determine the corresponding solution XB().) and the value of the objective function z~~x().) . Thus, for example, for A~ = 2, A; = 4" where ).* = (A~,~)T E

A

Multiparametrie sensitivity analysis

95

we have XI ():) = YI ():) = 5.5 - 0.63 X 2 + \.55 X 4 = 10.43, X2(X *) = Y2(X * ) = 7.5 + 0.13 X 2 + 0.45 X 4 = 9.56, X3(X*) = YJ(X*) = 2\.5 - 3 X 2 + 6.65 x4 = 42.1 , xs(X*) = Y4(X*) = 22.5 + 3.7 X 2 - 4.75 X 4 = 10.9, z~~x = 76.5 - 0.83 X 2 + 8.25 X 4 = 107.83. When applying such analyses in practice, it often proves difficult to determine the values of the components Ak of the vector parameter X such that the conditions XB(X) ~

0,

i.e.,

-

L fikAk ~ Yi , i = I, .. . , m, k=1

are satisfied. As with the suboptimal solution, it is of advantage to use an approximation region for this purpose. This region is a subset of the region A and all X E can be computed simply by means of a single equation. We have

e

e

e

o o

o

~I

o

k

o o

+ ... +as

o o

+ ... + a2s

o o

o

o o

~s

2s

Laj = I,aj ~ O. j=1

In our example,

s

= 2 ' ~1 = -6.08, XI = 7.17, k = -3.25, X2 = 4.57,

thus

(~~)

6

= al ( -6 08) + a2 (

-3~25 )

~

+ aJ (7 7 ) + a4 (

4.~7 )

,

al + a2 + aJ + a4 = I, aj ~ O,j = I, .. . , 4.

e

The subregion is indicated by double shading in Fig. 3-2. If X E A varies by passing through all points of A then z~lx(X) changes continuously along with X. Technical or economic analysis can make it important in particular problems to know the interval of the values of z~L(X) over A . This is equivalent to the problem: maximize (or minimize) the function feX)

= z~~x(X) -

z~~x

= fm+I ,IAI

+ .. . + fm+l,sA2

96

Sensitivity analysis with respect to b

subject to s

- LPfikAk ::;Yi,i = l, ... , m. k=1

In our particular case, this is the following problem: minimize5 f(A) = -0.83AI + 8.25A2 subject to constraints (I) through (4). It should be no ted that the "variables" AI, A2 are not sign-restricted, i.e., they can also take on negative values. In Table 3-2 the calculation of the minimum of the function f(A) is performed. In the second step, point P (cf. Fig. 3-2) is reached but the minimum is not yet reached, because one component in the last row is still positive. The optimum, which corresponds to point N in Fig. 3-2, is achieved in the third step. Point M in Fig. 3-2 has also been calculated (fourth step in Table 3-2). The variables SI , .. . , S4 are slacks, through which the system (1) through (4) was transformed into an equation system. Recall that Si ~ 0 for all i = 1, ... , 4. It is clear from Fig. 3-2 that the given function possesses no finite maximum; this could, of course, also be proven algebraically. In the given case, the minimum of the objective function is min z(p) (A) = 0, AEA max since

and min f(A) = -76.5, AEA so that min z(p) (A) = min f(A) + z(p) = -76.5 + 76.5 = O. AEA max AEA max As "profit", this value has only informative character. If we wish to calculate the values of the parameters AI, A2 for a certain value z~~x (A) = z*, it suffices to generate values AI, A2 which satisfy the equation max = f(A)

z * - z(p)

and the conditions s

-L PfikAk ::; Yi, i = 1, ... , m. k=1

5 The maximization can be carried out in a similar fashion .

Multiparametrie sensitivity analysis Table 3-2

97

Calculation of the minimum of the function f (A.) = -

Step 0

5

33

6AI + 4"" A2

A2

AI

SI

38

-93

330

s2

-8

-27

450

-\33

430

95

450

f- S3 S4

60* -74

-33/4

5/6 Step 1 f-SI s2 -7A I S4

AI -8.77* -44.73 - 2.217

S3 -0.63

57.7

0.\3

507.3 7. \7

0.017

-69.03

1.23

-6.4028 Step 2

0

980.3

-0.0139

sI

-5.9729 s3

PointP -6.577946

-7A2

-0.\\406844

0.07224334

f- S2

-5.10266\

3.365019*

AI

-0.2528517

0.17680608

s4

-7.874524

6.220532

526.2357

-0.7303548

0.45183776

-48.08935

Step 3 A2 -7 S3

sI -0.00452 -1.5\64

s2 -0.02\5 0.2972

AI f- S4

\.5582*

-\ .8486

-\86\ 13.23

-0.665 s2

Point N - Opt.

63.32

-0.05254

Step4

-7.414448

-11.15

0.0\525

-0.05

2\3.07934

-76.5 s4

PointM

A2

-0.02683

0.0029

-10.77

s3

-1.5018

0.99732

\92. \\

AI

-0.03444

-0.00979

-19.91

-1.1864

0.64\8

84.93

-0.1927

0.032\

-72.25

-7 S I

98

Sensitivity analysis with respect to b

For example, setting z* = 100, 100 - 76.5 = -0.83AI + 8.25A2 must hold and the conditions (I) through (4) must be satisfied. This is the case, for example, with A = (-2,2.64)T. In such a case, it is necessary to take into account all posible effects; this is done by caIculating the corresponding values of the basic variables and taking into account a large number of possible caIculations and considerations. With certain problems it is sensible to introduce parameters with the coefficient vectors f< = ek into the optimal solution. Each parameter thus occurs in one constraint with the coefficient equal to one. The matrix F then becomes an identity matrix, i.e., F = I. In our ex am pie,

Hence, Pf< = ßk or PF = 8 - 1 and fm+ l •k = Uk . For the analysis, we can then make direct use of the final tableau of the original problem, i.e.,

XB(A) = (

5.5) 2~:;

+

22.5

~

_~:~3

I

0.3

2.6

Z~~x(A) = 76.5 + 4 .3AI + 0.16Aj,f(A) = 4.3AI + 0.16Aj, and the feasible region A is defined by -0.73AI -0.26AI -3 .2 AI 2.6 AI

_~

( 0.73) (0) (-0.03) ( 0) }:;6 AI + A2 + A3 + A4,

- 0.03Aj ~ 5.5, - 0.03A3 ~ 7.5 + 0.1 AJ + A4 ~ 21.5, A2 - 0.3AJ ~ 22.5.

0

III Abridged mathematical presentation

111-1 General considerations Suppose that the problem: maximize z = cTx

(111-1)

Ax = b()')

(III-2)

x~

(III-3)

subject to

0,

where b()') = b + F).,

(III-4)

has an optimal solution XB associated with B for ). = o. The goal is then to determine a region A E IR', 0 E A, such that (111-1) through (111-3) has a finite optimal solution with the basis B for every ). E A and for ). 4. A the basis B is not primal feasible. Also F = (fik ), i = I, ... , m , k = I, ... , s, is a constant matrix, ). = (AI, ... , A,)T is a vector parameter. x, c E IR N , A = (aij), j = 1, ... , N, b()') E IR m. Assumptions (without 10ss of generalization) 1. All original conditions taking on the form (III-2) after the slacks have been introduced were inequalities of the type ~ . 2. The subscripts jE] be rearranged such that p = {I, ... , m},

..

(III-12)

The max z~~x().) or min z~~x().) overthe set (III-7) can be determined by means ).EA

).EA

of a modified simplex algorithm (cf. [7] and Sec. IV-4-1). At the same time, the "objective function" s

f()') = z~~x().) - z~~x =

L fm+l ,kAk

(1II-13)

k=1 is introduced and the optimum determined subject to (1II-7).

111-2 Special cases Case A Let

F = I, where I is an (m, m) identity matrix . Condition (III-2) then takes on the form

Ax = b + J).

(III-14)

or N

LaijXj=bi+Ai,i= I, ... ,m. j=1

(III-15)

The critical region is defined by

-B- 1).

::; XB

(III-16)

or m

-L k=1

ßikAk ::; Yi, i = I, ... , m,

(111-17)

Abridged mathematical presentation

102

where ßik is the ith eomponent of the kth eolumn of the matrix B- 1. Furthermore, m (p) ( ' ) -

(p)

zmax" - zmax

'"'

+L

, Uk/\,k,

(111-18)

um ) .

(III-19)

k=1

where UT

= eTB-I = (u I, B

.. . ,

Case B Let

F

= f,

i.e. , the matrix F "shrinks" to a veetor f.8 Constraints (III-2) then take on the form (III-20)

Ax = b+fA, or N

LaijXj=bi+fiA,i= I, ... ,m,

(III-21)

j=1

where A E IR is a seal ar parameter. The eritieal region A is defined by -B-1fA ~ XB -PfA ~ XB,

(III-22)

-PfiA ~ Yi, i = I, ... , m,

(III-23)

or

where Pf = (Pf l , ... , Pfm ) T.

Theorem 111-2 Suppose that eonstraints (IIl-2) of problem (111-1) through (111-4) take on the form (111-21) and that there is a finite optimal solution XB assoeiated with p for A = O. Partition the index set 1 = {i I i = I, ... , m } into the subsets 11, 12, 13 , so that for i EIl we have fi > 0, for i E 12 we have f i < 0 and for i E 13 we have fi = O. The eritieal region A is then defined by the following inequalities: (III-24) where ~

=

{ max { -Yj} JEI,

I1 = 0 _ A=

Pfj

~ ~

=-00

{ . { Yj}

(III-25)

mm-JEI,

Pfj _

12 = 0 ~ A = +00

8 This case is usually described in textbooks on linear programming (see the above cited literature).

Special cases

\03

holds.

Proof Consider PfiA.~-Yi, Vi EI l ,

(i)

i.e. ,

Yi. Pfi

I\.~--,IE

I I.

The solution of this system is, obviously, b A. ~ max{-~}. leI, Pfi

(ii)

Select all those inequalities from (111-1) for which i E 12 ; then we obtain A.

~ _l:!.., Vi Pfi

E

12.

The solution of this system is, obviously, A.

~ min{-l:!..} . ie I,

(iii)

Pfi

If we select all those inequalities from (III-I) for which i E 13, we obtain OA. ~ -Yi, which is satisfied for all A. E (-00, +00) (in the optimum, Yi ~ 0 for all i EI). Therefore, for

i E 13

we have

A. E (-00, +00).

(iv)

If I I :;:. 0, 12 :;:. 0, then the solution of (i) will be the intersection of the intervals (ii) and (iii), i.e., (III-24). If I I = 0 or 12 = 0, then the solution of (i) is the intersection of (iii) and (iv), i.e.,

or the intersection of (ii) and (iv), i.e., ~ ~

A. < +00.

QED.

From B- I b(A.) =B- I (b + fA.) = XB + PfA. it follows that the form of the dependence ofthe basic variables Xj"ji E p, i E I, on the scalar parameter A. is Xj, = Yi(A.) = Yi + PfiA., i = 1, . .. , m.

(III-26)

Suppose that, for example, ~ = - Yr fr and set A.*=-A.. Then,

xJ",(I\.,* )

Yr) = 0 . = Yr + Pfr(-Pf r

(III-27)

104

Abridged mathematical presentation

=

It obviously suffices to set A A* + E, E > 0 sufficiently smalI, in order to violate the constraints (III-23). Since A = A* + E makes the solution primal infeasible, another optimal basis could be computed by means of dual steps. The analysis of this possibility will be dealt with in Chap. 4. According to (III-16) and (III-17), for the scalar parameter A, (p) (1) (p) f ' Zmax 11. - Zmax + m+lll.,

(III-28)

where

(III-29) Theorem III-3 With regard to (III-28) and using the notation (III-29), the interval of the objective function values associated with the critical region A is given by: (i) for f m+1 > 0,

z~~x + fm+l~ ~ Z~~x(A) ~ z~~x + fm+1X;

(III-30)

(ii) for f m+1 < 0,

z~~x + fm+1X ~ Z~~x(A) ~ z~~x + fm+l~.

(III-31)

If f m+1 = 0, then Z~~x(A) is independent of A. If ~ = -00 or X = +00, then for f m+1 the value Z~~x(A) is correspondingly unbounded.

:f. 0

Proof According to (III-28), for f m+1 = 0, obviously, Z~~X 0, then (III-28) will be increasing and continuous over A. From the properties of a strictly monotonous increasing function it follows that inf Z~~X 0, it follows once again, on the basis ofthe properties of a strictly monotonous increasing function, that Z~~x(A) is unbounded from below. By analogy for f m+1 > 0 and X = +00, Z~~x(A) is unbounded from above. For f m+1 < 0, the situation is, obviously, converse. QED. Case C Consider the special case of f = ek, i.e.,

Ax = b + ekAk

(III-32)

or N

L j=1

aijXj

= bi , i = I, ... , m, i :f. k, 1 ~ k ~

m,

\05

Approximation region N

L akjXj = bk + Ak, k

fixed.

(III-33)

j=1 The critical region A k is then defined by the inequalities -B-1ekAk ::;; XB,

(III-34)

i.e., -ßikAk ::;; Yi, i = I, ... , m, k

fixed.

(III-35)

From B-1b(Ak) = B-1(b + ekAd = XB + ßk Ak it follows that the fonn of the dependence of the basic variables Xj,' ji E p, i E I, on the scalar parameter Ak is

(III-36) Theorem IIl-4 Assume that the constraints of the problem (III -I) through (III -4) take on the fonn (III-32) and that the problem has a finite optimal solution with respect to Ak O. Partition I {i I i I, ... , m } into 11, h, 13 so that for i E 11 we have ßik > 0, for i E 12 we have ßik < 0 and for i E 13 we have ßik = O. The critical region A k of the parameter Ak is then defined by the inequalities

=

=

=

(III-37) where ~k

Yi} - = mm . {- Yi} = maxI, {-ßik - ,Ak I, ßik IE

(III-38)

IE

holds. If 11 = 0 or 12 = 0, set ~k = -00 or X-k = +00. The proof is analogous to that of Theorem III-2. In this special case it is obvious that (p) ('

zmax

) -

II.k -

(p)

zmax

, + Ukll.k·

(III-39)

111-3 Approximation region In practical examples, the selection of the feasible values of the parameters from A involves considerable difficuIties. For the first approximation, it is, therefore, a good idea to introduce a subregion e of A, where e is defined by a single relation only in comparison to the m relations by which A is determined.

Theorem IIl-5 The region

e

which is defined by all convex combinations

106

Abridged mathematical presentation

0 = a,

A.s

0 2,2

2"

1"1 A.2

+ ... +as

+ a2 0 0

0 0

x"

0 0

+ ... + a2s

+as+, 0 2,s

0 0

0

0 0

0 x's (III-40)

2s :Laj = l,aj;;::O all J, j=' is a subset of the critical region A of the vector parameter>.. in the problem (III-I) through (III-4). The set e is called the approximation region. The proof is analogous to that of Theorem 11-4.

References

[I] Bank, B. , J. Guddat, D. Klaue, B. Kummer, K. Tammer: Nonlinear parametrie optimization, Akademie Verlag, Berlin 1982 [2] Charnes, A., W. W. Cooper: Management models and industrial applications of linear programming,2 Vols, J. Wiley, New York 1961 [3] Dantzig, G.B.: Linear programming and extensions, Princeton University Press, Princeton, N.J. 1963 f4] Dinkelbach, w.: Sensitivitätsanalysen und parametrische Programmierung, Springer, Heidelberg 1969 [5J Gass, S.1.: Linear Programming, 5th ed., McGraw Hili 1985 [6] Hadley, G. : Linear Programming, 2nd ed., Addison-Wesley, Reading, Mass., 1963 [7] Nedoma, J.: Nektere modifikace simplexove metody, WP Economic Institut of the Academy of Science, Prague 1969 [8] F. Nozicka, J. Guddat, H. Hollatz, B.Bank: Theorie der linearen parametrischen Optimierung, Akademie Verl. Berlin 1974 [9] Panne, van de, c.: Linear programming and related techniques, North Holland Publ. Co., Amsterdam 1971 [101 Panne, van de, c.: Methods for linear and quadratic programming, In : (H.Theil, ed.), Studies in Mathematical and Management Economics, Vol. 17 North-Holland Publ. Co., Amsterdam 1975

Chapter four 4 4-1

4-2

4-2-1 4-2-2 4-2-3 4-2-4 4-3 4-3-1 4-3-2 4-4 4-4-1 4-4-2 IV IV-l IV-2 IV-3 IV-3-1 IV-4 IV-4-1 IV-4-2 IV-4-3 IV-5 IV-5-1 IV-6 IV-6-1 IV-6-2 IV-7 IV-7-1 IV-7-2

Linear parametric programming with respect to b Changing the right-hand side with basis exchange Linear scalar parametric programming Changing a single component of the right-hand side with basis exchange Vector parametric linear programming with respect to the right-hand side Changing several components of the right-hand side with basis exchange Dependence on a scalar parameter Description of systematic parametrization for a sc al ar parameter Dependence on several parameters (on a vector parameter) Degeneracy Homogeneous multiparametric linear programming Problem (F) Problem (Fo) Sensitivity analysis and shadow prices under (primal) degeneracy Sensitivity analysis Shadow prices Abridged mathematical presentation Basic definitions and theorems The task . A solution procedure The application of the algorithm to the given multiparametric linear rogramming problem Some modifications of the simplex algorithm Linear programs with sign-unrestricted variables A special form of the additional restrictions Determination of the feasible bases in the auxiliary problems Special cases Linear parametric programming with a scalar parameter Homogeneous multiparametric linear programming Problem (F) Problem (Fo) Sensitivity analysis and shadow prices under degeneracy Sensitivity analysis Shadow prices References .

111

111

120 120 129 131 147 155 156 163 167 167 171 177 178 185 185 186 189 189 190 192 193 193 194 194 199 201 201 203 207

4 Linear parametric programming with respect to b Changing the right-hand side with basis exchange

In Sec. 3-1, we showed how, when we go beyond a boundary point of the critical region A the actual solution becomes primal infeasible. It is, however, sometimes possible to change the basis, so that the solution associated with the new basis becomes (formal) optimal again. Basis exchange does not necessarily imply that the result will be worse. This depends on the type of the parametric functions, on the structure of the model, etc. After a basis exchange necessitated by certain parameter values, the value of the objective function, for instance, can be higher or lower. Depending on the parameters, it is even possible for several optimal ba ses to exist (the solution is, thus, primal and dual feasible). In such cases, only a technical or economic analysis of the results with regard to the individual, optimal bases will enable us to select the particular optimal basis which proves to be the best in respect of the chosen objective function. I This procedure may be seen as one of the methods of an aposteriori system or model analysis, since it investigates the possible consequences of various changes in the given system (the model of which has been set up) caused by subsequently introducing parameters into the optimal solution. If the parameters should be introduced into the linear model immediately, we speak of an apriori systems analysis. The fundamentals of this range of problems will now be discussed using illustrative examples.

4-1 Linear scalar parametric programming Changing a single component of the right-hand side with basis exchange In Sec. 3-1, Ex. 3-1, we anal ysed the dependence of b I = 13 on AI according to the relation b l (AI) = 13 + AI within the framework ofthe optimal basis B =(a I, a 2 , _e4 , e 2 ), with Po = {I, 2, 3, 5 }. The critical interval A 1= [- 6.72, 8.65] was computed. In Fig. 3-1, we showed the total shift of the boundary line (I) associated with the interval AI. Finally, the ca se of A~ = 10 was also investigated; this value lies outside the interval A I (cf. Table 3-1 ).

I In this case too, it would be desirable to use so me sort of an interactive procedure.

Linear parametrie programming with respect to b

112

Since we are to deal with several bases, it is sensible to introduce the notation A~), P = 0, 1,2, ... , meaning a critical region associated with the basic-index p. Analogously with ~~),X~). Example 4- J Consider the following problem. In Ex. 3-1, determine all values of 11.( to which an optimal basis can be assigned ( i.e., a primal and dual feasible basis) where b l (AI) = b l + AI. In other words, on the real numbers axis on which the values of AI are ploued, determine a part such that an optimal basis exists to each AI from this part. Let K denote the set of all Al for which the above is valid. In Sec. 3-1, we have already shown that for AI = XI = X\O), with X~O) ~ 8.65, we obtain a "critical point" lying at the separation point of two optimal bases; for Al ~ 8.65, namely, XB(X~O» = (11.85,9.81,49.2,0) T. If we add this vector to Table 1-2 (Table 4-1), then the fourth row, which contains the null component of the vector XB(X~O» becomes the pivot row for a dual step.2 The new basis is denoted by BI and the old one by Bo . If we look at Table 4-1 with respect to BI, we could assume at first sight that XB is not feasible (the fourth component is negative). According to (III-36) or (3-12), i.e.,

xB(Ak) = XB + ßk Ak in our example, however, it holds that

XB(Al)=(:::~ )+(~)Al -8.65

onlyfor

AlEA~l).

1

Substituting an arbitrary value AI E (8.65, + 00 ) makes XB(Al) feasible. So, for example, for 1..; = 10 > 8.65 we have xB(A;) = (11.8,9.8,49.2, 1.35)T. According to (3-11) or (III-38) the critical region A ~I) was computed as folIows: (I)

~I

( -8.65) -(I) = max --1- = 8.65, AI = +00,

because the vector ßI =e4 contains no negative components. From this it follows that, for AI > 8.65 , no value AI exists for which at least one of the components of XB(AI) is reduced to zero. This means that for ascending AI there is no further

2 If we wish to proceed systematically, it is of advantage to mark the row in the given tableau with which the quotient (tower critical value) was determined by a lower bar circle or point. An upper bar circle will denote the row in which the quotient for the upper critical value was determined. Insofar as we are dealing with ascending or descending values of the parameter Ak exclusively, the corresponding row can simply be denoted by a circle (point). This has already been done in Table 4-1 .

Linear scalar parametrie programming Table 4-1

Initial tableau for a systematic parametrization and passing to BI 4

Po

113

6

XB(X\O)

xB

1

11/15

-1/30

5.5

11.8

2

4/15

1/30

7.5

9.8

Q3

A\O) -6.72 ::;; AI ::;; 8.65 6.28::;; bl(A I )::;; 21.65

3.2

-0.1

21.5

49.2

o~5

-2.6*

0.3

22.5

0

t.zj

13/3

2/3

114

114

zmax ( I)

PI

5

6

xB

XB(~\I)

A\I)

1

0.282

0.0513

11.8

11.8

2

0.103

0.0641

9.8

9.8

3

1.231

0.269

49.2

49.2

---74

-0.385

-0.115

-8.65

0

t.zj

5/3

2/3

114

(0)

A

= 76.5 +

13

3AI

8.65 ::;; AI < +00 21.65::;; bl(A I ) < +00

z~~x (AI)

114

= 114

(new) primal feasible basis. Therefore, we go back to the tableau for the basis Bo and set AI = !,,;o) "'" -6.72. This yields

XB(!"~O» = (0.573,5.71,0, 39.97)T. Since the third component is the null component, the third row becomes the pivot row for a dual step. Table 4-2 shows the initial tableau and passing to basis B 2 . According to Table 4-2,

'8(A,) =

-1.6) + (-0.3) (-21t _3~.3

A,.

-11

< A, < -6.72.

For AI = !,,\2) = -11, we have XB(!,,\2» = (2,0, 137, 10)T. The null component is in the second row, wh ich, however, contains no negative element. It is not, therefore, possible to pass to another optimal basis using the dual method. The process with inreasing AI is called the ascending process for short, that with falling values is called the descending process. Let us now analyze the computational results above. We first determined that to each AI from the interval [-11, +00) there exists a respective optimal basis. The

114

Linear parametrie programming with respect to b

Table 4-2

Passing to basis 8 2 4

Po

xB

6

xB(~iO»

,

A(Ol

I

11/15

-1/30

5.5

0.573

-6.72 ::; A, ::; 8.65

2

4/15

1/30

7.5

5.71

6.28::; b,(A,)::; 21.65

-0.1*

21.5

Qf-3

3.2

05

-2.6

0.3

22.5

39.97

~Zj

13/3

1/6

76.5

47.38

P2

3

4

xB

0

A\2l

I

-1/3

-1/3

-5/3

2

-11::;A,/..1 ~-II;

(8)

oj/"I ~ -1.6 ==> /..1 ~ -5;

-7/.. 1 ~ 87 ==> /..1 (2)

_

-

~

-(6)

(9)

-12.43; -

Zmax(>..)-112.3+9.6/.. 1,>"E A

(2)

.

According to Table 4-27, A (3) is defined by ~

-231;

(10)

9.8 ==> /..3

~

-153;

(11 )

-0.269/..3 ~ 49.2 ==> /..3

~

-182.5;

(12)

0.0769/..3

~

-0.0385/..3

11.8 ==> /..) ~

-/..1 + 0.115/..3

~

-8.65;

z~~x (>..) = 114 + 0.538/..3, >..

-(4) E

A (3) .

The inequalities, the numbering of which is identical ex ce pt for the opposite signs, point to the fact that two opposite half-planes (with common boundary lines) are involved. In Fig. 4-7, the regions A (P), P = 0, 1, 2,3, are shown. Point (I), (11), (III) are vertices, which were computed in Tables 4-21 through 4-23. In Fig. 4-7, the "faces" and the respective inequalities in the systems A (0), ... , A (3) are numbered equally. A further example, which will be dealt with in what folIows, is based on normal practical requirements, namely that the parameters should have specific content and are, thus, not meaningful over the whole range from -00 to + 00 13. For this reason, the condition G>.. ~ d, where G is a fixed (m, s) matrix and d E IR m a fixed vector, is added to the constraints Ax = b + F>... 13

If Ak refers, for example, to time (in days, months, ete.), then interest in the passage of time is naturally limited.

143

Vector parametric linear programming with respect to the right-hand side

Example 4-6 Maximize z = 3xI + 2X2 subject to

XI $ IO + AI + 2A2, X2 $ 2 - AI + A2, -XI + X2 ~ 4 - A3 , XI+ 2x2$12+AI -A3, XI ~ 0, X2 ~ 0, and

AI +A2 $ 10, AI + A2 + A3 ~ 20. Phase J Transform the linear inequality system into a linear equation system: XI

-AIX2 + AI + XI + X2 -XI + X2 XI + 2X2 - AI AI + AI + Xj

2A2 A2 A2

= 10, 2, + Xs = 20, - X6 + PI = 4, + X7 = 12, + Tli = 10, - Tl2 + P2 = 20, . .. ,7,Tli ~O,i = 1,2,pi ~O,i = 1,2.

+x3

=

+ X4

+ A3 + A3 A2 A2 + A3 ~O , j = 1,

The condition G~ $ d are simply added to the condition Ax = b + F~ and incIuded in the simplex tableau . In Tables 4-28 through 4-31 , the procedure for the computation of a feasible solution (X O, ~ 0) is shown. From Table 4-31, a feasible solution can obviously be computed in one step, I 0/3 , 4/3 , 46/3) T . Substituting ~ into the original problem (cf. , also, where ~ 0 Ex . 4-2) we obtain a " new" problem, the initial tableau of which is shown in Table 4-32. The constraints G~ $ d are not taken into account here, since these remain fixed and ~o satisfies them. If the "new" problem is solved, an optimal solution associated with Po = {I, 3, 4,5 , 6} results, wh ich can be found in Table 4-33. We have now computed Po E So· The critical region A (0) for Ro is given by

=(

- 2A2 - A3 AI - A2 AI + A2 - A3 AI - 2A3 -AI + A3

(l

$ - 2, $ 2, $ 8, $ -16, $ 12

144

Linear parametric programming with respect to b

Table 4-28

Initial tableau

XI

Xz

X6

AI

Az

1..3

Tlz

X3

I

0

0

-I

-2

0

0

10

X4

0

1

0

1

-I

0

0

2

x5

1

1

0

0

I

0

0

20

f-PI

-I

1

-I

0

0

1*

0

4

x7

1

2

0

-I

0

1

0

12

Tli

0

0

0

1

1

0

0

10

pz

0

0

0

1

1

1

-I

20

z -Ip;

1

-I

1

-I

-I

-2

1

-24

; =

b

I

Table 4-29 First iteration

XI

Xz

X6

AI

Az

Tlz

-I

-2

0

10

-I

0

2

X3

1

0

0

f- X4

0

1

0

1*

x5

1

1

0

0

1

0

20

-71.. 3

-I

1

-I

0

0

0

4

x7

2

1

1

-I

0

0

12

Tli

0

0

0

1

1

0

10

pz

1

-I

1

1

1

-1

20

-LPi

-1

1

-I

-I

1

- 16

-I

and AI AI

+ 1..2 + 1..2 + 1..3

$

10,

~

20.

Phase 2 According to Table 4-33 , neighbors can exist along the first, second, or fourth faces. The constraints GA $ d are, of course, also taken into account. The calculations can be found in Tables 4-34 through 4-39.

145

Vector parametrie linear programming with respect to the right-hand side Table 4-30

Second iteration

XI

x2

x6

X4

1.2

11z

X,

I

I

0

I

-3

0

12

~AI

0

I

0

I

-I

0

2

Xs

I

I

0

0

I

0

20

1.,

-I

I

-I

0

0

0

4

X7

2

2

I

-I

0

10

111

0

-I

0

-I

2

0

8

pz

I

-2

I

-I

2

-I

14

-LPi

-I

2

-I

-2

I

-14

x4

Az

112

f-

Table 4-31

1*

1

Third iteration

XI

Xz

x7

X,

I

I

0

I

-3

0

12

AI

0

I

0

I

-I

0

2

Xs

1

I

0

0

I

0

20

1.,

I

3

I

I

-I

0

14

~x6

2

2

I

I

-I

0

10

111

0

-I

0

-I

2

0

8

.. pz

-I

-4

-I

-2

3*

-I

4

-LPi

I

4

I

2

I

-4

~

-3

Before we proceed with the description of the procedure, let us state a fact that is described in detail in Sees. 2-2 and 11-2. When the min Si =s~ > 0 is reached for some i and Si in such a case is, of course, a basic variable (BV), then the following assertion holds: the ith constraint is strictly redundant with respect to A (p) and it can be deleted. In Table 4-37, we have a feasible solution with Ab k = 1,2,3, basic variables (i.e., a solution corresponding to a vertex of A (P». In this tableau, min SI = 16 > O. Hence, the first row can be deleted from the following computations. If it were

146 Table 4-32

Linear parametrie programming with respect to b Initial tableau for A, = A,0

I

-F

2

b(A,°)

b

3

I

0

-I

-2

0

10

16

4

0

I

I

-I

0

2

0

5

I

I

0

I

0

20

56/3

6

I

-I

0

0

-I

-4

34/3

7

I

2

-I

0

I

12

0

-3

-2

0

0

0

0

0

Table 4-33

Optimal solution associated with Bo - üF

Po

7

2

3

-I

-2

0

-2

-I

-2

16

4

0

I

I

-I

0

2

0

5

-I

-I

I

I

-I

8

56/3

6

-I

-3*

I

0

-2

-16

34/3

I

I

2

-I

0

I

12

0

dZj

3

4

-3

0

3

36

0

xB(A.o)

xB

necessary to continue the computations starting with Table 4-39, the second row could be deleted (min S3 = 8 > 0). From the computations given in Tables 4-34 through 4-39, it now follows that there is a neighbor only along the fourth face . This is PI = {I, 2, 3,4, 5}, so that Vo = {Po}, Wo = {pI }.lfwe take the pivot element marked by an asterisk in Table 4-33 into account, we obtain Table 4-40. According to Table 4-40, neighbors can exist along the first, second, third, and fourth faces. It follows from auxiliary calculations (which we need not go into here) that a neighbor exists only along the second and fourth faces. The latter leads to the node Po, which has already been Iisted in Vo , so that we only need consider the second face. From the second row of Table 4-40, it follows that P2 = {I, 2, 3, 5, 7}, so that V I = {P(h PI} and W I = {P2}. The tableau for P2 is shown in Table 4-41. As auxiliary calculations would show, V 2 = {Po, PI, P2} and W 2 = 0, so that V 2 = So. This finishes the process.

147

Veetor parametric linear programming with respeet to the right-hand side Table 4-34 AI

11. 2

11.3

OSI

0

-2

-I

-2

OS2

1*

-I

0

2

s3

I

I

-I

8

OS4

I

()

-2

-16

s5

-I

0

I

12

'11

I

I

()

10

'12

-I

-I

-I

-20

S2

11.2

11.)

Table 4-35

sI

0

-2

-I

-2

~AI

I

-I

()

2

s,

-I

2

-I

6

s4

-I

I

-2

-18

f-Ss

I

-1*

I

14

'11

-I

2

0

8

112

I

-2

-I

-18

4-2-4 Degeneracy Up till now we have tacitly assumed that, in performing the parametric procedure, no specific complications due to degeneracy occur. To c1arify such questions, let us investigate two special cases and their combinations: (a) when primal degeneracy occurs 14 , (b) when dual degeneracy occurs and (c) when both primal and dual degeneracy occur. Example 4-7 Maximize 14 Even A(p) can be degenerate in general. This case has no influenee on the proeedure at all. We shall. therefore, not eonsider it here. Cf. also Sees. 2-3 and 11-3.

148

Linear parametrie programming with respect to b

Table 4-36 S2

Ss

A3

sI

-2

-2

-3

-30

AI

0

-I

-I

-12

s3

1

2

I

34

s4

0

I

-I

-4

-I

-I

-I

-14

Tli

I

2

2

36

f- Tl2

-I

-2

-3*

S2

Ss

Tl2

-7 A2

-46

Table 4-37

0

-I

SI

-I

16

AI

1/3

-1/3

-1/3

1013

s3

213

4/3

1/3

56/3

f- S4

1/3

5/3*

-1/3

34/3

"-2

-2/3

-1/3

-1/3

413

Tli

1/3

2/3

213

16/3

-7 A3

1/3

2/3

-1/3

46/3

z = 3xI +x2, subject to

+ 3X2 2xI + X2 X,

XI

x,

S;

9+n

,+

A2,

S;

8 + AI - 2A2, 4 + AI + A2 ,

~

0, X2

S;

~

o.

According to the parametric procedure, we find that the given problem has an optimal solution with >,0 = (0, 0) T associated with Po = {I, 3, 5} (cf. Table 4-42). This solution is primal degenerate. As we know from Sec. 2-3 and II-3, it is possible to assign several different basic-indices to the vertex x" = (4, ol. However, in comparison with a "rigid" convex polyhedron X, the polytope in our

Vector parametrie linear programming with respect to the right-hand side

149

Table 4-38 S4

S2

112

1..)

2/5

1/5

-2/5

28/5

s)

2/5

-4/5

3/5

48/5

-tss

1/5

3/5

-1/5

34/5

1..2

-3/5

1/5

-2/5

18/5

115*

-2/5

4/5

4/5

115

-2/5

-1/5

54/5

S4

112

~Tl)

1..)

Table 4-39 Tl)

1..)

-2

I

-2

4

s3

-2

0

-I

8

Ss

-5

I

-I

6

3

-I

2

6

-t s 2

5

-2

4

4

A.J

-I

0

-I

10

1..2

case depends upon X, i.e., X(X). As follows from Table 4-42, vertex upon X as folIows : xO(X) = (4

XO

depends

+ 0.51..1 - 1..2, O)T .

The elements bj of the right-hand side depend upon X, as established in the given (initial) problem. The basic slack Xs for which it holds Xs =0, depends on X as folIows :

Thus, the degeneracy is removed for some admissible X ;f. o. From all this, it follows that the problem of primal degeneracy when parametrizing the right-hand side must be considered from another point of view. Recall that the aim of solving a parametric problem is to determine a region K. The procedure is then that we cover K by nonoverlapping regions, such that

150

Linear parametrie programming with respect to b

Table 4-40

PI 3 f--4

Optimal solution associated with BI - IF

7

6

-1/3

-2/3

-2/3

-2

1/3

26/3

1/3

4/3

-I

-2/3

- 10/3

-1/3*

xB

5

-2/3

-1/3

2/3

I

-1/3

40/3

~2

1/3

-1/3

-1/3

0

2/3

16/3

I

1/3

2/3

-1/3

0

-1/3

4/3

5/3

4/3

-5/3

0

1/3

44/3

~Zj

Table 4-41

Optimal solution associated with B2 4

P2

_ 2F

6

xB

3

-I

-I

-2

~7

-3

-I

-4

3

2

10

5

-2

-I

-2

3

I

20

2

I

0

I

-I

0

2

I

I

I

I

-I

-I

-2

~Zj

5

3

5

-5

-3

-2

-I

I

12

Table 4-42 -()F

4

2

Po

xB

3

5/2

-1/2

-3/2

-2

5/2

1

1/2

1/2

-1/2

1

1/2

5

-1/2

-1/2

-1/2

-2

-1/2

1/2

3/2

-3/2

3

1/2

~Zj

U

A(p)

=K .

pEG

Consider Table 4-42 and solve the auxiliary problem:

Vector parametrie linear programming with respect to the right-hand side

151

Table 4-43 AI

A2

OSI

-3/2

-2

f- S2

-1/2

Os,

-1/2

-2

AI

S2

f-SI

-5/2*

2

13

---t A2

-1/2

I

4

s3

-3/2

2

8

SI

S2

AI

-2/5

-4/5

-26/5

A2

-1/5

3/5

7/5

f- S 3

-3/5

4/5*

1/5

1*

5 4 0

Table 4-44

Table 4-45

From Table 4-45, it follows that min SI = 0 and SI NBV, s3 can be eliminated; thus, min S3 =0 and s3 NBY. With s3 = 0, the basic-index Po I = {I, 2, 3} results, which, from the point of view of the initial set X, is abasie-index assigned to the same vertex x O • However, from S3 = 0 , it follows that to A (0) the condition (4-8) is a binding constraint (cf. Sec. 2-2 or Sec. 11-2). Denoting A (01) the region adjacent to A (0), the condition (4-9) is a binding constraint. Hence, (4-8) and (4-9) define two opposite half-spaces (half-planes) and A(O), A(ol) are two nonoverlapping critical regions with the common supporting hyperplane (half-plane)

152

Linear parametric programming with respect to b

Table 4-46

_ "F

P2

2

3

4

-5

-2

3

3

-7

1

3

1

-2

-I

8

~Zj

5

2

-4

-2

16

Xs

1"1 + 4A2 = O.

If the given problem were now to be solved in full, the result would be the graph in Fig. 4-8. In conclusion, if primal degeneracy occurs, nothing changes in the procedure. It should be noted that in case of primal degeneracy, we obtain, in general, several critical regions (with the desired properties) assigned to the same vertex. An interesting question is to find an economic (technical) meaning of this fact. We leave the ans wer to the reader. Example 4-8 Maximize z= 2x l +x2

subject to XI +3X2 :::; 9+2AI +A2, 2xI + X2:::; 8 + AI - A2, XI ~ 0, X2 ~ O.

Assume that, in Phase I, the solution in Table 4-46 is generated 15. Only in the first row do there exist negative elements, and, solving the auxiliary problem, we would obtain min SI = 0 and SI NBY. The pivot element for a dual simplex step evidently cannot be determined uniquely; it is equally possible to choose either y 12 =-5 or Y13 =-2. Try, first, y 13 = -2. Then Po = {I, 2} results (Table 4-47). In Table 4-47, there is only one negative element in the first row and the pivot element Yl4 = -0.5 is defined uniquely. With this pivot element only, P2 again results. Thus, the procedure is finished and K = A (0) U A (2). The corresponding graph is shown in Fig. 4-9. Now, choose Yl2 = -5 as pivot element in Table 4-46. A dual pivot step leads to Table 4-48.

15 This is not. in fact, the case. For the discussion of this case it is, nevertheless. of advantage to start with P2 as initial node .

Vector parametric linear programming with respect to the right-hand side

153

Table 4-47

_ °F

Po

2

4

3

5/2

-1/2

-3/2

-3/2

7/2

I

1/2

1/2

-1/2

1/2

9/2

0

I

-I

I

9

t.zj

xB

Figure 4-9

Table 4-48

PI

- IF

4

3

xB

2

2/5

-1/5

-3/5

-3/5

7/5

I

-1/5

3/5

-1/5

1/5

19/5

0

1

-I

2

9

t.zj

Table 4-49 P3

4

1

- 'F

xB

2

2

1

-I

1

9

3

-5

-3

1

-4

-19

0

1

-I

I

9

t.zj

Regarding PI, in the corresponding tableau there are negative elements in both rows and the results of solving the auxiliary problem shows min Si = 0 and Si NBV, i = 1,2. The pivot elements Yl4 = -0.2, Y21 = -0.2 are determined uniquely. With y 14 , the basic-index P2 results; with Y21, a new basic-index P3 = {2, 3} results (Table 4-49). The only possibility from P3 is to go back to PI . Hence, K A (2)u A (I) U A (3) and the corresponding graph is in Fig. 4-10. Consider Table 4-47 associated with PO ' The dual solution is degenerate; consequently, there exists an alternative optimal solution (if all relevant conditions are

=

Linear parametrie programming with respect to b

154

Figure 4-10

Figure 4-11

Figure 4-12

fulfilled). Performing the corresponding primal simplex step, Table 4-48 results. However, Po and PI are not adjacent nodes, because the corressponding bases B o and BI are not neighboring optimal bases in the sense of the relevant definition (cf. Definition IV-3). The whole possihle graph is in Fig. 4-11 . Passing from Po to PI (which is not really feasible) is represented by a dashed line. This graph can be divided into two subgraphs as shown above. Let us conclude. If it is not possible to determine the pivot element for a dual simplex step uniquely, then: (a) dual degeneracy occurs in choosing any of the possible pivot elements and performing a corresponding dual step; (b) it is sufficient to choose any of the possible pivot elements in order to cover K by nonoverlapping critical regions; (c) by this, a subgraph of the whole possible graph is uniquely determined; (d) the whole possible graph need not be connected (see Ex. 4-2 and Fig. 4-7). The corresponding graph (Fig. 4-12) is not connected. Example 4-9 Maximize

subject to XI + 3X2 ~ 9 + 2AI + A2, 2xI + X2 ~ 8+ AI-2A2,

155

Homogeneous multiparametrie linear programming

G:

GOI:

~

GI":

Figure 4-13

XI

~ XI ~

4+ AI + A2, 0, X2 ~ o.

This example demonstrates a combination of primal and dual degeneracy. The computations are left to the reader. Two results are possible: K=

A(2) U A(4) U A(ol)

u

A(ü)

and K = A(5) UA(21) UA(4) U A(2) .

In both cases, K is covered by nonoverlapping critical regions. The whole possible graph G and its subgraphs GOI and G0 2 are drawn in Fig. 4_13 16 .

4-3 Homogeneous multiparametrie linear programming In decision making, in control and planning problems, in decision under uncertainty, in apriori and aposteriori system analysis, and on many other occasions, the case of b = 0 could be of considerable importance. Suppose, for example, that there is complete uncertainty about the elements of the vector b, i.e., about demand, supply, capacities, distribution, marketing, etc. (such a situation might occur in planning a new production center within an allocation problem). For decision problems in a firm, for possible adaptation maneuvers with the production program etc., familiarity with the spectrum of all possible values of bj for wh ich - ceteris paribus - there is always an optimal solution, would provide a useful source of information. If we are then in a position to represent the relevant endogenous and exogenous inftuences on bj in the form of linear functions of the parameters Aj (i = I, ... , s), the elements bj of the right-hand side b can be expressed as folIows: 16 The dashed line shows the connection between po and P2J which is achieved by a primal simplex step. Thus, po and P2J are not adjacent nodes.

156

Linear parametrie programming with respect to b

bi(~)

=L

fikA.k. i

= I, ... , m.

k=J

For these reasons, we shall show in this section how problems of the following type can be solved. Problem (F): Maximize z = cTx

subject to Ax=~,x;:::o .

Problem (F o ): Maximize z = cTx

subject to Ax = F o~, x ;:::

0, ~ ;::: 0,

where

b J , 0, ... ,0, 0, b2, ... , 0,

° °

Fo =

0, 0,

4-3-1

Problem (F)

Problem (F) is solved in basically the same way as the case with a vector parameter in the right-hand side described in Sec. 4-2. In this problem as weil, two phases are involved: in Phase I, a nontrivial solution (XO, ~ 0), XO ;::: 0, ~ ° =1= 0, is computed and in Phase 2 the nodes of G (or of Go) are found. Since Phase I and Phase 2 are treated differently, we shall explain the procedure using two examples (a formal discussion of this case can be found in Sec. IV-6). The first example illustrates Phase I, the second example Phase 2. Two examples are being used here because three parameters are involved in the course of Phase land only two in Phase 2. This allows a better overall view and, if necessary, we can represent the results geometrically. Example 4- 10 Maximize

subject to

157

Homogeneous multiparametrie linear programming Table 4-50

Initial simplex tableau

I

2

P

-I

f-4

2

-p

I

Table 4-51

-F

3

I

FA*

-I

-3

I

-I

3

0

-5

-6

I

10

I

3

-I

I

-3

-6/5

I

5*

-I

First iteration

I

3

4

f-p

-7/5

-I

-1/5

-2

11/5*

-+2

2/5

0

1/5

-I

-6/5

1/5

2

-p

7/5

I

1/5

2

-11/5

6/5

-I

-XI + x2 2 3/... 1 - /...2 +/...3, 2xI + 5X2 ~ 5/... 1 + 6/...2 - /"'3, XI 2 0,X2 2 O.

Phase I Set /...~ = /...~ = /...3 = I; briefty /... * = (l, I, I) T . In the initial tableau (Table 4-50), the matrix F remains and the right-hand side becomes F). * . The pivot element is determined in the usual way. After one pivot step with regard to the pivot element marked by an asterisk in Table 4-50, we obtain Table 4-51. According to Table 4-51 , the variable /...2 should enter the basis on account of the "p" row. But then the condition /...; = I is removed at the same time. The value X2 of the "basic variable" /...2 in the third iteration (Table 4-52) is determined as folIows: 3

X2 = LPfik/...~'/...~ = I,k = 1,3. k=1

In Table 4-52, a nontrivial feasible solution (XO, ).0): XO = (0, 28/11, 0, O)T,).o = (1, 16/11, I)T of the system Ax - F). = 0, x 2 0 was generated. The vector ). 0 is now substituted into the original problem (Table 4-53) and the ordinary LP solved. The optimal solution is given in Table 4-54. The critical region A (0) is defined by

-11/... 1 - 4/...2 - /...3 1O/... 1 -11/...2 +6/...3

~

~

0, 0

158

Linear parametrie programming with respect to b

Table 4-52

~A2

2

Table 4-53

Second iteration I

3

4

- 7/1 I

-5/1 I

-1/1 I

-10/1 I

I

-6/1 I

16/1 I

-4/1 I

- 6/1 I

1/1 I

-23/1 I

0

-5/1 I

28/1 I

Initial simplex tableau for A = AO

FAo

-F

I

2

3

P

-I

I

- I

-3

I

-I

28/1 I

4

2

5

0

-5

-6

I

140/1 I

-p

I

-I

I

3

- I

I

- 28/1 I

Table 4-54

Optimal solution associated with Bo

_ °F

FAo

Po

3

4

+-2

-2/7*

1/7

-11/7

-4/7

-1/7

5/7

1/7

10/7

-11/7

6/7

0

3/7

2/7

-1/7

- 15/7

5/7

28/1 I

I ~Zj

and (0) zmax

1 15 5 = 71. 1 + 71.2 -71.3 '

This ends Phase 1.

Example 4- J J Maximize

subject to XI -2xl 2xI xI

+ x2 2 + 3X2::; + 3X2::;

AI + 31.2, 81. 1 1.2, 51. 1 + 181.2, - 2X2 ::; -21. 1 + 31.2, xI 2 0, X2 2 O.

A

E

A

() 0 .

28/1 I

Homogeneous multiparametrie linear programming Table 4-55

159

Optimal solution for I.. = 1..*

_ °F

Po

4

5

I

-1/4

1/4

3/4

-19/4

4

0

2

1/6

1/6

-13/6

-17/6

5

0

3

-1/12

5/12

-5/12

-55/12

5

0

6

7112

1/12

-37/12

-47/12

7

0

6zj

1/12

7/12

-43/12

-125/12

14

0

xB(A. *)

xB

Phase I Since this example is concerned with describing Phase 2, it has been constructed in such a way that there is an optimal solution for 1..'; = 1..; = I ,i .e., A* = (1, I)T. This optimal solution is given in Table 4-55. The critical region A (0) is defined by

3 19 -AI - -1..2 44 13 17 --AI - -1..2 6 6 5 55 --AI - -1..2 12 12 37 47 --AI - -1..2 12 12

S;

0

S;

0

S;

0

S;

0

' '

' '

and (0)

43

125

Zmax(A) = 121.. 1 + \21..2,

AE A

(0)

.

Phase 2 In order to ascertain along which faces the region A(o) has neighboring regions, the fotlowing auxiliary problem is solved (for motivation see Sec. IV-6): minsj,i = 1,3, subject to 3Ar - 3Ai- 191..; + 191..2+ SI = -13Ar + 13Ai- 17A; + 17A2+ S2 = -Ar + Ai- In; + 111..2+ SJ = -37Ar + 37Ai- 47A; + 47A2+ S4 = Ar + Ai + 1..; + 1..2 = At~o,Ak~O,k= 1,2,sj~0,i = and

0, 0, 0, 0, I, 1, .. . ,4.

Linear parametrie programming with respect to b

160 Table 4-56 A~

At

A!

A2

OSI

3

-3

-19

19

0

s2

-13

13

-17

17

0

°S3

-I

1

-11

1I

0

s4

-37

37

-47

47

0

f-p

1

1

1

1

1*

Table 4-57

At

AI

Az

22*

16

38

19

s2

4

30

34

17

f- S3

10

12

22

11

f- S4

IO

84

94

47

-?A2+

1

1

I

1

f-S I

A; . Ak = 0, k = 1,2, where

Ak = A; - Ak· Note Here we are using the familiar technique of solving an LP with not signrestricted variables (cf., for example, [6, 7]). Tables 4-56 through 4-58 show the corresponding computations. Since there are negative elements in Table 4-55 in the first and third rows only, the question arises as to whether it is possible to eliminate Si for i = I or 3. The condition At . Ai;" = 0 means that At and Ai;" must not occur as basic variables at the same time. We already have a feasible solution in Table 4-57. Let us, therefore, try to minimize SI; the largest positive element is in the first column and Q~~n points to the first row. We have, thus, determined the pivot element. From Table 4-58, it then follows that since Ar, A~ are basic variables, the columns, to wh ich AI and A2 belong, will be ignored. Moreover, AI = 19/22 - 0 = 19/22, A2 = 3/22 - 0 = 3/22. Hence, min Si = si' > 0 and Si BV for i = 2, 3, 4, and SI =0 and SI NBY. On

Homogeneous multiparametrie linear programming

161

Table 4-58

A~

+ -?AI

A2

sI

8111

19111

1/22

19/22

s2

298/11

298/11

-2/11

149111

s3

52111

52111

-5111

26111

s4

844111

844/11

-5/11

422111

3111

-8111

-1/22

3/22

Ai Table 4-59

PI

Optimal solution associated with BI I

- IF

5

4

-4

-I

2

2/3

3

Xs

-3

-19

0

1/3

- 5/3

-3

0

-1/3

1/3

-2/3

-3

0

6

7/3

2/3

-4/3

-15

0

L'lZj

1/3

2/3

-10/3

- 12

0

the other hand, it follows that A (0) has a neighbor only along the first face . The adjacent node to Po is the node PI = {2, 3, 4, 6} and V o = {Po}, Wo = {PI}. As can be seen from this, the procedure of Phase 2 is basically the same as with "nonhomogeneous" parametric problems, except for a few technicalities. Choose PI E Wo ; after one pivot step, we obtain Table 4-59. The critical region A (I) is defined by -3AI 5 - - AI 3 2 - - AI 3 4 - - AI 3

and

+ 19A2

~

0,

-

3A2

~

0,

-

3A2

~

0,

- 15A2

~

0,

162

Linear parametric programming with respect to b

Table 4-60

1..+2

A~

At

Az

OSI

-3

3

19

-19

0

s2

-5/3

5/3

-3

3

0

Os 3

-2/3

2/3

-3

3

0

s4

-4/3

4/3

-15

15

0

f-p

1*

I

I

I

I

Table 4-61 A~

1..2

Ai 22**

-16

3

-4/3

14/3

5/3

4/3

-7/3

11/3

2/3

8/3

-41/3

49/3*

4/3

I

I

SI

6

s2

10/3

s3 f- S4

+

~AI

I

I

Table 4-62 A~

Ai

SI

422/49

422/49

s2

18/7

18/7

-2/7

9/7

s3

36/49

36/49

-11/49

18/49

Al

8/49

-41/49

3/49

4/49

Ai

41/49

90/49

-3/49

45/49

S4 48/49

211/49

The corresponding auxiliary problem is solved in Tables 4-60 through 4-62. From Table 4-61, it follows that it is possible to eliminate SI (the pivot element is marked with two asterisks); hence, there exists a solution with SI = 0 and SI NBY. Try to minimize S3 ; it cannot be the first column (At is BV; therefore, A.( cannot enter the basis at the same time). A positive element is in the 1..2 colunm.

Homogeneous multiparametrie linear programming

163

Determining Qmin in this column, the pivot element marked by an asterisk results. From Table 4-62, it follows that min Si = s;' > 0, Si BV for i = 2, 3 and 1"1 = 45/49, 1.2 = -4/49. Thus, the only neighboring region to A ()) exists along the first face; it is A (0) again. Hence, V) = {p(» p) }, W) = 0. This ends the process.

4-3-2 Problem (FD) Problem (Fo) differs from problem (F) in two respects. First,

FOA = (

bl:AI ) . bmA.m

If we assurne that the right-hand side b = (b), . .. , b m)T, b ::/- 0, is constant, then A in FOA can be regarded as a multiplicative parameter. Second, du ring the setting-up of a linear model, the elements bi of the right-hand side b become meaningful. If this interpretation has to be retained, then the constraint x~o

extended by A ~ 0 seems to be meaningful. Here too, the procedure will be explained using an example and we shall use two parameters in view of possible geometric representation.

Example 4-12 Maximize z = 4x) + 2X2 + 9X3 + 6X4, subject to x) +2X2+4x3+3x4 ~ 501.), 2x) + X2 + 4X3 + 3X4 ~ 801.2, Xj ~ O,j = I, .. . ,4, A.k ~ 0, k = 1,2. 0 ) ,b(A)= FOA.SetA*=(I,I)T ,l.e. . Phase I WehaveF o = ( 50 0 80 A.j*

* ="-i=

I

and set up the initial tableau. Since Phase 1 is analogous to that of problem (F), it need not be repeated here. In the given example, there exists an optimal solution for A = A* directly. This solution is indicated in Table 4-63. According to Table 4-63, Po = {3, 4}, A (0) is defined by -30A.)+16A.2~0,

(I)

(2) and

Linear parametrie programming with respect to b

164 Table 4-63

2

5

- °FD

6

Po

I

3

0.2

I

0.6

-0.2

-30

16

14

4

0.4

-I

-0.8

0.6

40

-48

8

t1zj

0.2

I

0.6

1.8

-30

144

174

FDA*

Table 4-64

Os, ~OS2

-30 40* I

P

s, -7A,

s,

A2

A,

16

I

-7 S2 A, A2

I

I

3/4

0

1/40

0

-1/40

I

23/44* 1/88

I

-1/88

I

44/23 -1/46

I I

0 I

-6/5

~s,

A, -7A2

I

1/5*

~p

0

I

-48

-20 I

s2

1/46

I

lOO1l1 6111 (i) 5111 400/23 8/23 (ii) 15/23

Z~~X(A) = 30A, - 144A2. Phase 2 Beeause of the eondition A ~ 0, the set of the admissible parameters in the parametrie spaee is restricted to the first orthant (in the example, quadrant) . A eritieal region A (p), whieh is defined by - PFA:S; 0 , forms a eone. From all possible eones A (p), we seleet only those Iying in the first quadrant. For this reason, the eondition (4-10)

Homogeneous multiparametrie linear programming

165

(5)

Figure 4-14

Table 4-65

Optimal solution associated with 8 1

2

- 'F

3

5

-5

-5

-3

150

-80

0

I

2

3

I

-50

0

0

2

10

9

6

-300

288

0

PI

I

6

-I

4 ßZj

xB

is added to the system - PFh ::; 0, ~ ~ o. For, if the intersection of A (p) and the first orthant is nonempty, the intersection of A (p) and (4-10) is also nonempty (formally, cf. Sec. IV-6). Thus, in our example, the neighboring regions are determined as folIows: (a) In Table 4-63 , there are negative elements in both rows; we, therefore, have to check whether min Si = 0 and Si NBV for i = 1,2. (b) The system (I), (2) is extended by the condition A.I + A.2 = I (cf. Table 4-64) and an artificial variable p is added to the latter for the calculations. In Fig. 4-14, the results are presented geometrically. From (i) (cf. Table 4-64) it follows that min S2 =0 and S2 NBV: from (ii) it follows that min SI =0 and SI NBY. This means that neighbors exist along the first and second faces. The corresponding basic-indices are PI = {4, 6}, P2 = {3, 5} and V" = {Po}, Wo = {PI, P2 }. Choose PI E Wo and, after having performed a pivot step, Table 4-65 results.

Linear parametrie programming with respect to b

166 Table 4-66

A,

A2

Os,

150

-80

0

s2

-50

0

0

f-p

I

1*

I

Table 4-67

A, 230*

80

s2

-50

0

~A2

I

I

f-S,

Table 4-68

s, ~A,

1/230

s2

5/23

A2

-1/230

8/23 400123

15/23

The critical region A (I) is defined by 150AI - 80A2

~

0,

-(I)

- 50AI

~

0,

(4)

AI

~

0,A2

~

0

and Z~~x(A)=300AI-288A2,

AE

A(I).

The auxiliary problem is solved in Tables 4-66 through 4-68. Since, in Table 4-65, there exist negative elements only in row I, we look for min SI. Hence, min SI = 0 and SI NBY. The only possible neighbor is PO. List VI = {Po, PI}, W I ={P2}. Choosing P2 E W I from Table 4-63, we obtain Table 4-69.

Sensitivity analysis and shadow prices under (prima!) degeneracy Table 4-69

P2

167

Optimal solution associated with 8 2 4

2

I

- 2F

6

xB

3

1/2

1/4

3/4

1/4

0

-116

0

5

-1 /2

5/4

-5/4

-3/4

-50

60

0

112

1/4

3/4

9/4

0

252

0

~Zj

The critical region A (2) is defined by -116A2

~

0,

-(2)

-50AI + 60A2 AI

~

0, A2

~

~

(6)

0,

0,

and

z~~x (>..) = -252A2,

>..

E

A (2).

Since, in Table 4-69, there are negative elements only in the second row, min S2 has to be determined. However, the uniquely defined pivot element for a dual step in this row leads back to Po. Hence, it is not necessary to solve the auxiliary problem. List: V 2 = {Po, Pb P2}, W 2 = 0 and this finishes the procedure.

4-4 Sensitivity analysis and shadow prices under (primal) degeneracy 17 4-4-1

Sensitivity analysis

As has been pointed out in Sec. 3-1, sensitivity analysis (with respect to the RHS b) in terms of (3-14) means to determine the critical interval A k such that for all Ak E A k fhe optimal basis B remains optimal. We learned in Secs. 2-3 or 11-3 that, with a (J-degenerate vertex xo, several bases are associated. Hence, if the optimal vertex Xo is degenerate, it is not possible to require that "the optimal basis remains optimal". We shall use a sm all illustrative example to show what the problem of sensitivity analysis underdegeneracy isoEven though the degeneracy ofthe polytope from Ex. 17 This section is situated in Chapter 4, because it is closely related to parametrie programming .

168

Linear parametric programming with respect to b

Table 4-70

< < < < < < < <
: 2>: 3>: 4>: 5>: 6>: 7>: 8>: 9>:

Numbering of the bases 3456 1356 2456 1245* 1246 1256* 1235 1236* 1234*

2-4 is caused excJusively by weakly redundant constraints, we use this polytope because of its simplicity. 18 Example 4-13 Consider maxz

= 3xI +4X2

subject to XI + 3X2 2xI + X2 2xI + 3X2 XI + X2 XI ~ 0, X2

15, 10, S; 18, S; 7, S; S;

~

o.

Before we start our analysis, let us introduce the following notation: Let B(,c)

= {B E

BO IBis dual feasible}

be the set of optimal bases associated with the cr-degenerate optimal vertex xo. It holds:

B(,c) ~ BO and in the most cases B(c)is a proper subset of BO • In Table 4-70 the numbering of bases associated with all vertices of our polytope Xis presented. To XO the bases Bi, i = 4, ... ,9, are assigned. The bases marked in Table 4-70 by an asterisk are the optimal bases.

18 Despite that there are so me special properties of sensitivity analysis when degeneracy is caused exclusively by weakly redundant constraints (see [131 and also Sec. 11-3, Theorems 11-6 through 11-8), they have no influence on the results presented in this section.

Sensitivity analysis and shadow prices under (primal) degeneracy

169

< 6> +-------+----------------------------------------------+--------------------+ B : 3 4 xB +-------+----------------------------------------------+--------------------+ I -0.2000 0.6000 3.0000

2 0.4000 -0.2000 4.0000 5 -0.8000 -0.6000 0.0000 6 -0.2000 -0.4000 0.0000 +-------+-------------------------------- -------------+--------------------+ : Dzj : 1.0000 1.0000 25.0000

Our task is now to determine Ai for each i E ,1, .. ., 4} . Let us start with 1.., and as starting optimal tableau choose Table 9. From tableau we have

t:::\6)

=

t:::, =-10

since this critical value is determined in the 2nd, nondegenerate row. We have (cf. Table

********** 0.0000 Row 3 < 8> PI+: 0.00 Way : < 6>:< Row 4 < 4>

8>

00

0.0000 Row 4 < 9> Way:
:
:
:-10.00 End 1 1[-10.00,

00

r2

0.0000 Row 3 < 8>

P2+:

0.0000 Row 4 < 9> 0.00

9>

Sensitivity analysis and shadow prices under (primal) degeneracy Table 4-71 Determination of Ai , i = I, . . . , 4, continued Way : < 6> :< Row 4


:


4>

Way:
:


L2

-5.0000 P2-: 1.00 Row I Row nondegenerate! Way:


End I 2( -5.00,

-5.00 00

1

n 00

P3+:

0.00

L3

0.0000 Row 3


1.25

P3-:

-8.0000 Row 2 Row nondegenerate ! Way :
:
: -8.00

End

I 31 -8.00,

00

J

1'4

P4+:

0.00

L4

0.0000 Row 4

< P4-:

4> 2.50 -2.0000 Row I Row nondegenerate!

173

174

Linear parametric programming with respect to b

Table 4-71 Determination of Ai, i = I, . .. ,4, continued Way: < 6>:< End 1 4[ -2.00, P+I P-I P+2 P-2 P+3 P-3 P+4 P-4

4> : -2.00

0.00 1.00 0.00 1.00 0.00 1.25 0.00 2.50

+

Pi

OZ

= ob:'" I

which is true , as a matter of fact, in nondegenerate cases. Consider Tables , , and (see above) and compare the "shadow prices" as given in these tableaux. In Table we have u I = I, U2 = I, u) = 0, U4 = O. In Table we have UI = 0, U2 = 0.25, U3 = 1.25, U4 = O. In Table we have UI =0,U2 =0,U3 = l,u4 = I. As is seen, to each basis another values of Ui are assigned. The question, hence, is: which are the "true" shadow prices? As has been shown in several publications (see, for ex am pIe [14] and the referenees quoted therein as weil as the bibliography at the end of this book), in degenerate cases there exist two-sided shadow prices, pi, pi. Here, pi means the shadow price of the ith resouree when bi increases by I, pi means the shadow price of the ith resouree when bj deereases by 1 . It has been shown ([I] and Sec. IV-7-2) that the two-sided shadow priees are closely related to sensitivity analysis with respeet to the RHS in terms of (3-14): In a tableau associated with an optimal basis: If Ai E [-a, 0] then pi is defined, if Ai E 10, b1then pi is defined, if Ai E [0, 0] then none of the shadow priees is defined; here a > 0 and a = 00 is possible, b > 0 and b = 00 is possible. Example 4-/4 Consider the LP from Ex . 4-13 . Choose Table ; here we have: AI

E

1-1O,0),A2

E

1-5,01,11.)

E

[0,+00),11.4

E

[0,+00),

Sensitivity analysis and shadow prices under (primal) degeneracy

175

< 4> +-------+----------------------------------------------+--------------------+ B: 3 6 xB +-------+----------------------------------------------+--------------------+

I 2

-0.5000 0.5000

1.5000 -0.5000

3.0000 4.0000

4

0.5000

-2.5000

0.0000

5

-0.5000

-1.5000

0.0000

+-------+-------------------------------- -------------+--------------------+

: Dzj:

0.5000

2.5000

25.0000

hence we have

P, = I, p; = I, p~ = 0, p~ = 0. Form the lists: LI = {PI' Pi", P3' p~ }of shadow prices wh ich are already known and L 2 = {pt, P2' P3", P4} of shadow prices which have to be determined. Pivot correspondingly to obtain Table . From we have:

pr

°

AI E 10, 00) ~ = 1.2 E [0, 0) ~ p; cannot be determined 1.3 E [-8, 0) ~ p:J = 1.25 A4 E [O , oo)~p;=O and update the lists LI and L2. Pivot to Table ; here we have 1.2 E [0,00) ~ 1.4

E

p; =0

P4 =? pt, PI' P3' P3" are 10,01 ~

Note that since in list LI, it is not necessary to determine the critical intervals for AI and AJ. The updated list L2 = { P4" } . Pivot therefore to Table from wh ich we have : 1.4 E 1-2, 01 ~

P4 = 2.5 .

Hence, list L2 = 0 and all shadow prices are determined:

pr = ° P, = I p;

=

p~ =

p~

=

° p; = °

°

I p:J = 1.25 P4 = 2.5

These are the "true" shadow prices for changing bj by + I or -I . Let us note that commercial LP software offering sensitivity analysis and shadow prices determination yield false results in case when primal degeneracy of the optimal solution occurs.

IV Abridged mathematical presentation

Consider the following problem: Determine a region K c;;;; IRs such that the problem of maximizing (IV-I) subject to Ax = b + F>.,

(lV-2)

x

(lV-3)

~

o.

has a finite, optimal solution for each >. E K, >. E IR s , and for >. E IR s - K the given problem has no optimal solution. Here, A = (aij), i = I, ... , m,j = I, ... , N, orA=(al, . . . ,aN),aiE IRm,C,XE IRN,b+F>'=b(>')E IRm,F=(fik),i=I, . . . , m, k = I, ... , s, >. = (AI , ... , As)T is a vector parameter. The quantities c, A, b, F are constant; let the rank of the matrix A be m. Denote (IV-4)

b(>') = b + F>.. The vector parameter should satisfy the additional constraints G>.

~

d

(lV-5)

where G = (gik), i = I , .. . , r, k = I, ... , sand d E IRr constant. In Chap. 2 we have already introduced the following notions and notations. Let I = {i I i = I , ... , m} , J = {j I j = I, ... , N } . Further, let ai', ... , ai", be a full system of linearly independent vectors of IR m , i.e. a basis from IR m . The set p = {j I, ... , jm} of the subscripts of the basis vectors is called the basic-index . The basis is then denoted by Band its inverse B- I . Let the elements of the inverse be ßik , the columns ßk . Further, let p c J or

.. . Let us further suppose that this degeneracy occurs for j = t, trip . If Yit > 0 exists for at least one i E I, it is possible to pass to another basis, say B", by one primal simplex step. But then basis B" will not be regarded as neighboring basis in the sense of Definition IV-3. Definition lV-4 Let BI and B2 be two optimal bases and A(I) and Am the corresponding critical regions, respectively, which are uniquely defined by (lV17). The critical regions A (I) and A (2) are called neighboring regions, iff Bland B2 are neighboring bases. Note For neighboring regions A (I) and A (2), obviously, A(I)nA(2)::I=0.

(lV-27)

Theorem lV-5 Two neighboring regions A (I) and A (2) c K* ~ IR s lie in opposite half spaces of IR s .

Proof Let Bland B 2 be neighboring bases and A (I) and A (2) the corresponding critical regions. The corresponding critical systems are then

Abridged mathematical presentation

182

z + IZ TX = 1f T >.. + z( I) m+1 ' IYX=IF>"+XBI, Z+2 ZTX = 2fT >..+z(2) m+1

(lV-28)

'

(IV-29) 25

2yx = 2F>.. + xB2.

Let Yrt < 0 be the pivot element for passing from BI to B2. The rth row of the system (lV-29) is obtained by dividing the rth row in (lV-28) by Yrt. Hence, 2 yr (>")

= I~r(>") Yrt

for aIl

>..

IR' .

E

(lV-30)

The rth constraint of the inequality system defining the critical region A (I) is Iyr (>") ~ 0 ~ IYr +

L

IfrkAk ~ 0,

k=1 i.e., with respect to BI , s

Iyr (>") ~ 0 ~ -

L

IfrkAk ~ IYr

(i)

k=1 and, for A (2) , I

2

S

Yr(>") I" I. Yr(>") = -I- ~ 0 ~ Yr + L frk Ak ~ 0, Yrt k=1

i.e., with respect to B2 , ~0~-

L

IfrkAk ~ Iyr . (ii) k=1 Obviously (i) and (ii) define two opposite half-spaces in IR' with the common hyperplane 2 yr (>")

s

-L

IfrkAk = Iyr.

(iii)

k=1 QED. Corollary IV-5-1 The critical region A (p) has a neighbor along its ith face if it is possible to pass to this neighbor by eliminating the ith basic variable from B by one dual simplex step. 25

Let the sequence of rows in (IV-28) and (lV-29) be the same, so that there is no need to renumber them when passing from one system to another.

183

Basic definitions and theorems

Proof Assurne that the numbering of the basic variables and that of the rows in (IV-28) is the same. The elements of the ith row of _I F can be regarded as the coefficients of the corresponding equation of the hyperplane which represents the "ith face" of A(I), i.e., - IfilAI - ... - IfisAs = Iyi . If the ith basic variable is to be eliminated by one dual simplex step (as Definition IV-3 requires), ).0 E A(I) must exist such that the ith element of xB I (). 0) = xB I + I F). 0 is zero and, at the same time, there must be at least one negative element in the ith row of Iy. QED. Corollary IV-5-2 The critical region A (p) has a neighboring region along its ith face (i E I) iff (I') )." E A(P) exists such that Xj,().O) = Yi().O) = 0 and (2') Yij < 0 for at least one je p.

The proof is straightforward. For the solution method, it is convenient to introduce an undirected graph G = (S, r) which is generated by problem (IV-I) through (lV-3).

Definition IV-6 An undirected graph G problem (IV-I) through (lV-3) iff

= (S,

r) is said to be generated by

(i) the set of nodes consists of subsets P = { j I, . . . , jrn} of the set J = {j I j = I, ... , N} such that PES iff B is an optimal basis to (IV-I) through (lV-3), and (ii) given {PI, P2} er; then P2 E r(PI) (and, vice versa, PI E iffB I and B2 are neighboring optimal bases. Let us call PI, P2 adjacent nodes.

np2»

Definition IV-7 Two nodes PI, P2 E S have the distance L'1 ,0< L'1 L'1 elements of P2 are different from the elements of PI.

~

m, ifprecisely

Theorem IV-6 Let).l,). 2 E K, ).1 i= ). 2, be two arbitrary admissible vector parameters, and let PIE S be given such that).l E A (I). Then, in the graph G = (S, r), there exists a path (p I, .. . , Pk), such that ). 2 E A (k).

Proof Let the segment between

).1, ). 2

E K be expressed in parametric form: (IV-31)

Obviously, ).(t) E K for all t E [0, 11, since K is convex. Substituting ). = ).(t) into (IV-I) through (lV-3) yields a parametric problem with the scalar parameter t. This parametrization is performed for 0 ~ t ~ I, starting with the basis BI and following the dual simplex method. The resulting sequence of optimal bases corresponds to the path (PI, ... , Pk) in the graph G = (S, V). QED. Graph G is evidently finite since, in (lV-2), only finitely many bases exist.

184

Abridged mathematical presentation

From what have been said up to now, it follows that the region K (or K*) can be determined by a set of regions A (p) that cover K (or K*) and that do not overl ap 26, i.e., U A (p) = K ( or ( U A (P» n M = K*). If there exists a dual degeneracy P

P

regarding any of the optimal bases B, then it is possible to pass from B to another optimal basis B' by one prima! simplex step (of course, if Yij > exists for ~Zj = O,j 4 p). However, Band B' are not then neighbors and p, p' are not adjacent nodes and A (p) and A (p) do overlap 27. If required, it is of course possible to find all existing optimal bases B (and, by this, all existing critical regions NP». Performing such a task would, however, almost cause an "explosion" of the corresponding computer time and the storage capacity would be strained. Thus, fordetermining K (or K*), it is, obviously, more efficient to deal only with nonoverlapping, neighboring, critical regions A(p). Note that, if dual degeneracy occurs, the generated graph need not be connected (see also Exs. 4-3 and 4-8). However, from Definition IV-6 and Theorem IV-6, it follows that generating nonoverlapping, neighboring regions, the corresponding subgraph Go (So, r o) is connected. Taking into account Theorems IV-2, IV-4, and IV-5 and Definitions IV-3 and IV-4 as weil, the following assertion obviously holds .

°

=

Theorem IV-7 Assume that in at least one optimal basis B dual degeneracy occurs. By dropping all nodes PES that are assigned to optimal bases by corresponding primat simplex steps, a subgraph Go = (So, r 0) of graph G, So O" .. ,XI=XI>O,AI =X I ;tO, ... ,At=Xt;tO,I+t=m; non basic variables, XI+I = ... = XN = 0, at+l, ... , A; =

A~~I =

aso

Denote v E IR m the basic solution and Vi the elements ofv, i.e., vI = XI, .. . , vI = XI, vt+1 = XI, ... , vm = Xt. The system (IV-73), transformed with respect to p, then has the form N

Xi -

L

s

YijXj =

j=I+1

L

j=I+1

PfikA~ = vi, i = I, ... , I,

(IV-75)

k=t+1

N

Ai -

L s

YijXj =

L

PfikA~ = vi , i = I + I, ... , m.

k=t+1

On account of XI+I = ... = XN = 0, ". = I, ... ,m, (lV-76) vi = "Pf ~ ikAk,1 k=t+1 from which it follows that vi ;t 0 for at least one i. This proves the following theorem. Theorem IV-8 Assurne that a solution of the system (IV-71) with the basic variables Ak " ... , Ak" Xj , ' ... , Xj,, .,, I ::::; t ::::; m, and with A~ .. , = ak,." ... , A~, = a s has been computed. Denote this solution (XO, XO) . Then the solution (XO, XO) is a nontrivial, feasible solution of (lV-71 ).

The second master problem is to determine all adjacent nodes to anode pES. The critical region A (p), defined by -B-IF'" < " < . - I , ... , m, _ 0 - "Pf· ~ ,kAk _ 0 , 1k=1

(IV-77)

forms a convex (polyhedral) cone in IR s . To determine the set Pp (cf. Step 3°°), the auxiliary problem (lV-43) has to be solved. In our case, this problem has the form min Si iE P"

(lV-78)

Homogeneous multiparametrie linear programming

197

subject to

-L PfikAk + si = 0, Si 20, i = I, ... , m.

(lV-79)

k=1

In order to ascertain for which of the Si 's the condition Urj ::; 0 \ij from Redundancy Criterion 2 (cf. Corollary II-5-1) is satisfied, we ought to include all Si 's as basic variables successively. In order to avoid this procedure, which increases the computations involved disproportionately, a modified auxiliary problem is solved on the basis of the following theorem.

°be an arbitrary but fixed real number and w = {A IR' I t I Ak I = w} .

Lemma IV-9 Let w > E

(IV-80)

k=1

Then, für every w > 0, A(P) 7: 0

Proo!

"~"

"~"

~

A(P)

(1

W 7: 0.

(IV-81)

is straightfürward. Let),," E A(P) . Define

WO

=

L

I Ak I > 0.

k=1

Set ljI = w/w" . Then, ),,"1jI E A(P) since Furthermore, ),, 0 ljI E W, since ljI

L

1jI>

°and A(P) is a (cünvex) cone.

I Ak I = IjIW" = W.

k=1

QED. Setting w = I, by Lemma IV-9 the following problem is established: minsi iEP"

(lV-82)

-L PfikAk + Si = 0, i = I, ... , m,

(IV-83)

subject to

k=1

(IV-84)

198

Abridged mathematical presentation

Theorem IV-IO

34

Consider the problem (lV-82) subject to s

-L PfikA~ + L PfikAi;" + Si = 0, i = I, ... , m, k=1

LA~+ LAi;" k=1

(lV-85)

k=1

(IV-86)

= I,

k=1

A~ . Ai;" = 0, k = I, .. . , s,

(IV-87)

= I, .. ., S, Si

A~ ~ 0, Ai;" ~ 0, k

~ 0, i

= I, ... , m.

(IV-88)

Let min s, = s;) ~ 0, s, BV, t E Pp fixed, be an optimal solution to (IV-82), (lV-85) through (lV-88). Then s7 is an optimal solution to (IV-82) through (lV-84).

Proof Define A~

=0 A~ -" Ai;",

k

= I, ... , s.

(IV-89)

Evidently, for each k E {I, ... , s },

°A~"Ai;"

=0 ~

s

s

k=1

k=1

L I Ak I = L °A~ + L "Ai;".

(lV-90)

k=1

With (IV-90), the relations (lV-86) and (lV-87) imply (lV-84). Setting (IV-89) into (IV-85), we obtain (lV-83). QED.

Note To determine the neighboring regions to A (p) 82), (IV-85) through (IV-88) has to be solved.

,

the auxiliary problem (lV-

We are now ready to present the procedure for solving (lV-68) through (IV-70). This procedure has two phases.

Phase I

Determine an arbitrary node p"

E

S" . In other words,

(a) determine a nontrivial feasible solution (X O, ~O) of the system

Ax -F~ =

(lV-91)

0,

x ~ 0;

(lV-92)

(b) determine the optimal solution of the problem of maximizing z = c T X,

(IV-93)

subject to

Ax

=F~(\ ~o #

x ~ o.

0

constant,

(IV-94) (IV-95)

34 The author is grateful to Dr H. Leberling and Dr W. Rödder, both of the University of Aachen (in 1978), für discussing the fünnulation and the proof of this theorem.

Homogeneous multiparametrie linear programming

199

In order to compute a nontrivial solution of the homogeneous, linear equation system (IV-91) with (lV-92), we proceed as folIows .

I. Setting Uk

= I for all k in Theorem IV-I 0, we have A; = .. . = A: = I ;briefty,

},,* = (1, ... , I)T.

2. Determine a feasible solution of the system Ax

= F}"*,x:2' o.

(IV-96)

3. If no feasible solution of (IV-96) exists, then enter the parameters Ak into the basis successively. This removes the constraint A~ = I for each of the parameters Ak in the basis (cf. (IV-76». 4. If we have found a solution with the basic variables Ak" ... , Ak" Xj" ... , Xj" and with A~,., = ... = A~, = I, Xj,_.. , = ... = Xj, = 0, then it is obviously a nontrivial solution of (IV-9I), (lV-92). Denote this solution by (XO, )" 0), XO :2' 0, }"° 7= O. 5. If there is no solution (xo,}"O) with the prescribed properties, then K = 0. As the case with K = {o}' 0 E IR s , is uninteresting, this case will be set equivalent to K =0 . Stop. 6. Substitute}" =}" ° into (IV-68) through (IV-70) and maximize (IV-97) subject to Ax

= F}"° , }"

x:2' o.

0

7= 0

constant,

(IV-98) (IV-99)

The optimal basic-index Po computed through the solution of (lV-97) through (IV-99) (if there is a nontrivial optimal solution) is the first (initial) node Po E So we are looking for. Go to Phase 2. 7. If no nontrivial, optimal solution of the problem (IV-97) through (lV-99) exists, then K = 0. Stop. Phase 2 Phase 2 is analogous to the Phase 2 described in the preceding sections. The only difference is that, as auxiliary problem, the problem (IV-82), (IV-85) through (IV-88) is solved.

IV-6-2 Problem (FD) Consider the problem (FD), i.e., determine a region K* c IR m such that for each K* c IR m the problem of maximizing

}" E

(lV-IOO) subject 10 Ax = F D }",

(lV-IOI)

200

Abridged mathematical presentation

x ~ O,A

~ 0,

(lV-I02)

has a finite nontrivial optimal solution and, for A E IR m - K*, the given problem has no nontrivial optimal solution. In addition,

° °

bl, 0, ... ,0,

0, b2, .. . , 0,

(lV-103)

Fo = 0, 0, .. . ,0, b m

*

is a constant matrix with bi 0, i = I, .. . , m, and A E IRrnis a vector parameter. This problem is treated as a special case for two reasons. First, bl AI b2 A2

(IV-I04)

FOA = bmAm

*

If we assume that the right-hand side b =(bi, . .. , bm)T, b 0, is constant, then A in FOA can be regarded as a multiplicative parameter. Second, there is a practical reason for the nonnegativity condition A ~ for the parameter A 35 . Denote K* = K n M, where M = { A E IR m I A ~ The assigned graph is then G~ = (S~, r~). This was introduced in Note 2 to Theorem IV-7. Phase I of the solution procedure for problem (Fo) is the same as that for problem (F), i.e., (lV-68) through (IV-70). In Phase 2, the following problems are solved as auxiliary problems:

°

° }.

(IV-82)

minsi ieP,

subject to m

-L PfikAk + Si = 0, i = I, .. ., m,

(IV-lOS)

k=1 m

LAk

= I,

(IV-I06)

k=1

Ak

~

0, k = I, ... , m, Si

~

0, i = I, ... , m.

(IV-I07)

The justification for extending (IV-lOS) by (IV-I06) follows immediately from Lemma IV-9 as a specialization of the statement to be found there. The corresponding formulation is given in Lemma IV-lI below. 35 This has to do with the technical (or economical) meaning of bi as weil as of A.i.

Sensitivity analysis and shadow prices under degeneracy

201

Lemma IV-I J Let w > 0 be an arbitrary but fixed real number and

W= {A E IR

m

I f)'k

= W, Ak ~ OVk} .

k=1

Then, for every w > 0,

A (p)

:f.

0 W n A (p)

:f.

0.

The proof is analogous to that of Lemma IV-9.

IV-7 Sensitivity analysis and shadow prices under degeneracy36 IV-7-1 Sensitivityanalysis Consider maxz

=cTx

(IV-I 08)

subjeet to Ax

= b,x ~ 0

(IV-I09)

and suppose that the optimal vertex XO is cr-degenerate. To Xo the set of bases BO (see (11-30» is assigned. Every B E BO is, by definition, a primal feasible basis. The set,

B(c) ~ BO , of primal and dual feasible (i.e. optimal) bases I) is, in general, a proper subset of BO , 2) induees a subgraph 0°(0+,0-) of GO(G~, G~)

ealled the general (positive, negative) optimum degeneraey graph of XO (o-DG for short), 3) some properties of o-DG's depend on c in (IV-l 08) 37. The proofs of these three assertions are to be found in [22] . If XO is nondegenerate, then, as has been pointed out in Sees. 3-1, III-I and III-2, sensitivity analysis with respeet to the RHS b in terms of 36 See foot note 17 37 This is the reason for using the subscript "(er' .

202

Abridged mathematical presentation

bi(Ad

= bi + Ai

is to determine the critical interval Ai 38 such that for all Ai E Ai the optimal basis remains optimal. Since, according to the above assumption, XO is the optimal, cr-degenerate vertex, we have more than one optimal basis associated with xo. Hence, the above "definition" of sensitivity analysis does not hold any more. As has been shown by several authors (see, for example, [9, 14, 15,21] and the references quoted therein) the overall critical region (interval) is given by K

Ai =

UAi k) = [~i' Xil,

(IV-I 10)

k=]

where k is the index of bases Bk E B(e) ( see also [23]). As has been pointed out at several places of Chapters 3 and 4, changing b implies, in general, a change of the shape of X. The only invariant information in nondegenerate cases is the optimal basis. In degenerate cases to~, the shape of X changes, in general , with Ai . Assume that we are looking for ~i and that ~fk) has been determined in the rth row of the tableau associated with Bk E Bie) . Assume further that Yr > 0, i.e., ~fk) is determined in a nondegenerate row. As follows from the preceding sections, there are three possibilities: I) There is no negative element in the rth row. This implies that there is no Ai < ~!k) for which there exists an optimal basis. Thus, ~i = ~fk) . 2) There exists Yrj < 0 for at least one j r1 p . Pivoting correspondingly, we would obtain a nondegenerate vertex. Thus, ~i =~!k). 3) -1 Alk) = - 0 0 . Thus '-1 A· = -1 Alk). Suppose that we have determined ~i and Xi. Then, for any Ai E (~i' Xi) there is at least one optimal basis Bk E B(c)' such that Ai E r~!kl, Xfk l ]. Summarizing, the above three points are equivalent with: Determine ~i' Xi such that for any Ai E [~i' Xil there is at least one optimal basis Bk E B(e) such that AI E [Alk) X(k)j -1' I . The following definition of sensitivity analysis with respect to the right-hand side becomes sensible. Definition IV-7 Suppose that XO is the optimal, cr-degenerate vertex to (IV-108), (IV-109). Then sensitivity analysis with respect to the right-hand side means to

38 This concerns also the more general case with a vector parameter >... Here we confine our analysis to the simplest case.

Sensitivity analysis and shadow prices under degeneracy

203

determine the overall critical region A 39 such that for all >. E A at least one basis BE B(C) remains optimal. Regarding sensitivity analysis with respect to the cost coefficients, the following definition seems to be sensible. Definition IV-8 Suppose that XO is the optimal, cr-degenerate vertex to (IV-l 08), (IV-l09). Then sensitivity analysis with respect to the cost coefficients means to determine the overall critical region K

T = UT(k)

(IV-lll)

k=1

such that for every t E T the set B(c) does not change. Note that Definitions IV-7 and IV-8 differ from each other. This is because, in Definition IV-7, changing the RHS the shape of the corresponding polytope (IV-I09) changes and as a consequence some of the optimal bases need not remain elements of B(~). In Definition IV-8, changing cost coefficients, the shape of the polytope does not change and there is no reason why so me of the optimal bases should become dual infeasible.

IV-7-2 Shadow prices Consider (IV-I 08), (IV-I 09) and assume that XO is the optimal vertex. I. Let XO be nondegenerate. Then

oz

Ui = Obi

(IV-I 12)

is the shadow price of the ith resource b i; it is called also marginal value or Lagrange multiplyer or dual value. It characterizes the change of the optimal value Zmax when bi changes by "one unit". From (3-13) it follows that, changing bi in terms of (3-14) by Ai = I, the value Zmax changes by Ui. Let us state that, if XO is nondegenerate, the shadow prices are defined uniquely in the optimal simplex tableau. 11. Let XO be cr-degenerate. Introduce +

oz

I

obi

p. = -

_ oz

and p. = I

obi

(IV-I 13)

as right- and left-partial derivatives [21 and let us note that, ifxo is nondegenerate, +

-

Ui = Pi = Pi'

(IV-I 14)

39 A is used in order to show that this definition is general and concerns either a vector or a scalar parameter (by specialisation).

204

Abridged mathematical presentation

Akgül [1] in his analysis of shadow priees under degeneraey introduees the funetion F(b)

= max { cT X I Ax ~ b, x ~ o},

(IV-I 15)

whieh is the known optimal value funetion. The funetion (IV-I 15) is a nondeereasing, eontinuous, pieeewise linear and eoneave funetion (see also Theorems IV-3 and IV-4). By (IV-I 15) the eonneetion between the theory of shadow priees and the RHS parametrie programming is established. Akgül shows that the direetional derivative of F at b in the direetion r is defined as F(b + Ar) - F(b) (lV-l 16) DrF(b) = lim . A.~O+ A Then p~

= DrF(b), Pr = -D(-nF(b).

(IV- I 17)

The positive shadow priee of the ith resouree is then defined by pt = De,F(b), r := e i, and the negative shadow priee by pi

=-D(-e,)F(b) ~ O.

Here pt is interpreted as the "maximal buying priee" for the ith resouree, and pi as the "least selling priee for the ith resouree". Denoting H(A)

=F(b + Ar),

Akgül shows that p~, Pr are the one-sided derivatives of H at A = 0, i.e. " p~ and Pr are nothing but slopes of H(A) at A = 0 " [I , p. 427]. He then shows that pt = mJn{ u(k)}

(lV-l 18) where k is the index of bases Bk E Bec )' From all this it follows that with respeet to a basis B Ai

E [~i'

Ai

E

0] then the slope is pi

[0, X;] then the slope is pr

E

B(c) it holds: if (IV-I 19)

Using the above relations we ean now roughly deseribe a proeedure of determining the two-sided shadow priees. Step 0: Start with an optimal basis B o degenerate vertex xO •

E

Bec ) assoeiated with the optimal,

0-

Sensitivity analysis and shadow prices under degeneracy

205

Step I: Determine A [0) for each i E {I, . .. , m } in the tableau associated with Bo . To each A!O) determine, according to (IV-I 19), the corresponding pt and pi, as far as possible40 . Introduce the lists: LI = {pt,pi} for those i for wh ich the shadow prices are already determined, and L2

= {pt, pi}

for those i for wh ich the shadow prices are not yet known. Step 2: According to the parametrie procedure, pivot on a negative pivot in a degenerate row to another tableau associated with basis Bk E B(e}' Step 3: In basis Bk repeat Step I with Bk instead of Bo and update the lists LI and L2. Note that, in Step I, and with respect to basis Bb it is not necessary to determine A fk) for those i for which both shadow prices are already in LI· Step 4: Repeat Steps I - 3 until L2 = 0 . Let us note that regarding the possible huge number of bases in BO , the determination of the two-sided shadow prices by (lV-1 18) may lead to superftuous calculations of bases. This is the main disadvantage of the method introduced by Knolmayer [20] (for other methods see, for example, [14] and the references quoted therein ).

40 If Ai

E

[0.0] the corresponding shadow prices cannot be determined.

References

[I) Akgül, M.: A note on shadow prices in linear programming. J. Oper. Res. 35 (1984) 425-431 [2) Aucamp, D.C., D. I. Steinberg: A note on shadow prices in linear programming, 1. Oper. Res. 33 (1982) 557-565 [3) Bank, B., J. Guddat, D. KlaUe, B. Kummer, K. Tammer: Nonlinear parametric optimization, Akademie Verlag, Berlin 1982 [4J Bereanu, D.: A property of convex piecewise linear functions with ap-plications to mathematical programming, Unternehmensforschung 9 (1965) 112-119 [5) Charnes, A., W.w. Cooper: Systems evaluation and repricing theorems, Managern . Sci.9 (1962) 209-228 [6) Collatz, W. Weuerling: Optimierungsaufgaben, 2nd ed., Springer, Berlin 1971 (7) Dantzig, G .B.: Linear programming and extensions, Princeton University Press, Princeton, NJ 1063 [8J Dinkelbach, W.: Sensitivitätsanalysen und parametrische Programmie rung, Springer, Berlin 1969 [9] Evans, J.R., N.R. Baker: Degeneracy and the (mis) interpretation of sensitivity analysis in linear programming, Decision. Sci. 13 (1982) 398-354 [101 Fucfk, 1., T. Gal : K otazce degenerace ve vychoz!m fesen! simplexovych uloh linearnfho programovan!, Ekon. Mal. Obzor 5 (1969) 295-303 [11] Gal, T. : A "historiogramme" of parametric programming, J. Opl. Res. Soc. 31 (1980) 449-451 [12) Gal, T. : A note on the history of parametric programming, J. Opl. Res. Soc. 34 (1983) 162-163 [13] Gal, T. : Weakly redundant constraints and their impact on postoptimal analyses in linear programming, EJOR 60 (1992) 315-326 [14) Gal, T. : Shadow prices and sensitivity analysis in linear programming under degeneracy : A state-of-the-art survey, EJOR 8 (1986) 59-71 [15] Gal, T.: Degeneracy graphs: Theory and application. An updated survey, In: (Gal, T., ed.) Degeneracy in mathematical programming, Annals of Operations Research, Vol. 46/47 Baltzer Publ. Co., Basel 1993, pp. 81-106 [16] Gal , T., J. Nedoma: Multiparametric linear programming, Managern. Sci. 18 (1972) 406-422 [17) Graves, R. L. : Parametric linear programming, In: Recent advances in mathematical programming (R.L.Graves and P. Wolfe, eds.) , McGraw Hili, New York 1963, pp. 201-210 [18] Guddat, J., F. Guerra Vasquez, H. Th. Jongen: Parametric optimiza tion: singularities, pathfollowing, and jumps, Woley & Sons, Chichester, and B. G. Teubner, Stuttgart, 1990 [19J Kern, w.: Die Empfindlichkeit linear geplanter Programme, In: (A. Angerman, ed.) Betriebsführung und Operations Research, Nowack, FrankfurtiM 1963, pp. 49-79 [20] Knolmayer, G.: How many-sided are shadow prices at degenerate primal optima? OMEGA 4 (1976) 493-494

208

References

(21) Knolmayer, G.: The effect of degeneracy on cost coefficient ranges and an algorithm to resolve interpretation problems, Oecision Sei. 15 (1984) 14-21 (22) Kruse, H.-J.: On some properties of o-degeneracy graphs, In : (Gal, T., ed.) Oegeneracy in mathematical programming, Annals of Operations Research, Vol. 46/47, Baltzer Pubt.. Co., Basel 1993, pp. 393-408 (23) Magnanti, TL, J.B. Orlin: Parametrie linear programming and anti-cycling pivoting rules, Math. Progr. 41 (1988) 317-325 (24) Manas,1. Nedoma: Finding all vertices of a convex polyhedron, Numer. Math. 12 (1968) 226-229 (25) Moeseke, P. van, G. Tintner: Base duality theorem for stochastic and parametrie programming, Unternehmensforschung 8 (1964) 75-79 (26) NoZicka, F., 1. Guddat, H. Hollatz, B. Bank: Theorie der linearen parametrischen Optimierung, Akademie Verlag, Berlin 1974 (27) Nykowski, I.: Zalesnosci pomiedzi r-parametrycznymi dual ny mi zadaniami programowania liniego, Przegl. Statist. 13 (1966) 311-323 [28] Ritter, K.: Über Probleme parameterabhängiger Planungsrechnung, OVL-Bericht No 238, J. Comp. Syst. Sei. I (1967) 44-54 (29) Sarkisjan, S.O.: Ob odnoj zadace parametriceskogo Iinejnogo programmirovanija, kogda zavisimost funkcionala od parametra nelinejnaja, Tr.vycisl.Centra AN Arm. SSR i Jerevanskoj lost. 2 (1964) 10-16 (30) R. E. Steuer, : Multiple criteria optimization: theory, computation, and application, Wiley & Sons, 1986 (31) Tlegenov, K.B., K.K.KaIcajev, P.P.Zapletin: Metody matematiceskogo programmirovanija, Nauka, Alma-Ata 1975 (32) Weinert, H.: Probleme der linearen Optimierung mit nichtIinear-ein-parametrischen Koeffizienten in der Ziel funktion, Math. Oper.-Forsch. Statist. I (1970) 21-43 (33) Weinert, H.:On uniqueness in parametrie linear programming problems with fixed matrix of constraints, Math. Oper. Forsch. Statist. 5 (1974) 177-189

Chapter fi ve

5

5-1 5-2 5-2-1 5-2-2 V

V-I V-2

Sensitivity analysis with respect to c Changing cost coefficients without basis exchange Changing single cost coefficients . Changing several cost coefficients Dependence on a scalar parameter Dependence on several parameters (on a vector parameter) Abridged mathematical presentation Special cases . . . Approximation region References . . . .

211 212 216 216 219 223 226 230 231

5 Sensitivity analysis with respect to c Changing cost coefficients without basis exchange

Let us suppose that we have a firm manufacturing n products, the market prices of wh ich are given as Cj and the variable unit costs of which are Pj. In a particular planning period, the firm has KF nonrecurring overhead expenses, which are independent of the output capacity. We can derive the objective function from these data, if the aim is the maximization of profit over the period: N

Z = 2)Cj - Pj)Xj - KF ·

(5-1)

j=1

As usual, the constraints contain the production technology, marketing conditions, etc. The production conditions contain quantitative data on production factors required and the number of production factors available, as weIl as da ta on the workload involved in each of the stages of production . Machine and labor capacity requirements involved in the separate stages of production are also included in the constraints. In this model, we assurne that the market prices and the variable unit costs are stable. If we suppose that the variable unit costs really do remain stable, then we can turn our attention to the possible changes in the market price. A firm is usually in a position to fix the prices for its products as action parameters within the general framework of its sales policy or the possibility of changes in the market price can lead it to take decisions at to the quantitative and qualitative organization of production. On the other hand, market adjustment can be taken into account in price policy, i.e., the reaction and counterreaction of the firms participating in the market. 1 Finally, a firm may have several goals, which are expressed in various objective functions (i.e., maximization of profit, profitability, etc.; see also Chap. 9). These and other cases can be tackled by the introduction of parameters into the objective function. By this the exogenous and endogenous inftuences on the system are described. If the market prices Cj are subject to change, they can then be expressed as a function of parameters t} in the general form Cj(t}). In the same way, the variable unit costs Pj can be expressed as a function of parameters tf in the general form pj(tf). Hence, the objective function from (5-1) becomes

I Questions of price policy are, of course, related to sales policy.

212 Table 5- 1

Sensitivity analysis with respect to c Initial tableau I

2

3

4

b

5

I

2

3

I

50

6

2

I

4

3

80

-4

-2

-9

-6

0

Table 5-2

Final tableau - optimal solution 2

5

6

eB

P

I

9

3

0.2

I

0.6

-0.2

14

6

4

0.4

-I

-0.8

0.6

8

0.2

1

0.6

1.8

174

~Zj

xB

N

z(t] , tJ) = L(cjCtj' )

-

Pj(tJ»Xj - K F.

(5-2)

j=1

Since only linear dependencies are being considered in this book, Cj(t}) and pj(t}) are linear functions of the parameters. For the sake of simplicity and in order to approach the general complex of problems as in the preceding chapters, we set Cj - Pj

=Cj,

and

z + KF

=Z

(5-3)

so that the objective function takes on the usual form N

z= LCjXj .

(5-4)

j=1

5-1 Changing single cost coefficients As usual, the range of problems indicated by the title of this section can best be illustrated by a small example. Example 5- J Let us take Ex. 2-1. The optimal solution is shown in Table 2-2. Here we had a (much simplified) four-product firm with two capacity restrictions. Initial and final tableaux are given in Tables 5-1 and 5-2.

213

Changing single cost coefficients

Denote by tj a parameter which expresses the change in the unit profit Cj in the form Cj(tj) = Cj + tj. Let us look first at a nonbasic cost coefficient (price) (j fi. p). In the optimal solution, ~Zj = PZj - Cj =

Let us take, for example,

eT yJ' - Cj,j fi. p fixed .

~ZI:

(5-5)

then,

~ZI = (9, 6)(0.2, O.4)T - 4 = 4.2 - 4 = 0.2.

If cjCtj) = Cj + tj ,j fi. p, then T .

T .

~Zj(tj) = eByJ - cjCtj) = eByJ - Cj - tj

= ~Zj -

(5-6)

tj , j fi. p fixed,

since the elements of the basic cost vector eB are independent of tj. For j = I, therefore,

In view of the relations both within the system itself and between the system and its environment, wh ich are inc1uded in the model, it is, therefore, disadvantageous to manufacture product PI with profit unit 4. Considerations on beginning the production of product PI (i.e., performing a suboptimal analysis) show that manufacturing a quantity unit of PI would entail a loss of 0.2 price units of the total profit. As we already know (cf. Chap. I), optimality is retained in maximization problems if ~Zj ;:: for all j. Therefore, we must also have ~zjCtj);:: for all j. For j = I we then have

°

°

0.2 - tl ;:: 0 => tl ::; 0.2. Thus, ifwe substitute any figure t~ ::; 0.2 for tl, the price CI(t~) = c~ will be such as to create with c~ the same quantitatively and qualitatively assembled optimal production program. Let tl = 0.2 be selected. Then, CI (tl) = 0. It then follows that product PI becomes "equally important" with products P3 , P4 , if an alternative optimal solution with P3, PI as "basic products" exists. In this new optimal basis, the total profit will stay at the level of 174 units and each convex linear combination of the two solutions yields a production program with a profit of 174 units. The alternative basic solutions are shown in the tableau in Table 5-3. If UI = U2 = 0.5 are the coefficients of the convex linear combination, the production program at a price of 4.2 per unit of PI will be T. 0.5(20,0, 10,0, 0,0)T + 0.5(0,0, 14, 8, 0,0)T = (10,O ,12,4,0,0)

Let us summarize. If Cj is a nonbasic cost coefficient (price) and if Cj(tj) then the critical region (interval) for the parameter tj is given by

=Cj + tj,

214

Sensitivity analysis with respect to c

Table 5-3

Optimal tableau for CI(~) 1

PI 3

2

0.2

3

=4.2

and the alternative solution

4

5

6

xB

1

1

0

0.6

-0.2

14

-1

0

1

-0.8

0.6

8

1.8

174

~4

0.4*

t.z/~)

0

1

0

0

0.6

3

0

1.5

I

-0.5

1

-0.5

10

-71

I

-2.5

0

2.5

-2

1.5

20

t.z/~)

0

1

0

0

1.8

174

0.6

tj ::; dZj'

(5-7)

Since the basic cost vector is not affected by the change in a nonbasic cost coefficient, the objective function value and other criterion elements will not change. Now assume that the unit price C4 changes in accordance with C4(t4) = C4 + t4. How and where does this change become apparent? First, we may note that C4 is an element of the basic vector Cs associated with p. From this, it follows that 4 will affect all expressions which contain cs. Thus

Z~~x(t4)

=C~(t4)XS

and dZj(t4)

=CS(t4)yj -

Cj

(for all j fi. p) will be influenced by t4. First, tet us analyze the criterion elements dZj ' For j = I, we have

= (9, 6+ (4)(0.2, O.4)T - 4 = 4.2 + 0.4t4 - 4 = 0.2 + 0.4t4 . For j = 2 (briefly), we have dZ2(t4) = I - t4 ; for j = 5, dZS(t4) = 0.6 - 0.8 t4; for j = 6, dZ6(t4) = 1.8 + 0.6t4 . In addition : dZ) = 0.2, Y2) = 0.4, dZ2 = I, Y22 = -1; dZs = 0.6, Y2S =-0.8; dZ6 = 1.8, Y26 = 0.6. Furthermore, dZ) (t4)

Z~~x(t4)

=(9,6 + t4)( 14,8)T = 174 + 8t4,

where z~~x = 174 and Y2 = X4 = 8. It obviously follows that a change in a basic cost coefficient also changes the reduced costs and the value of the objective function. The primat solution is independent of t4 . It further follows that only the dual feasibility can be violated by changing a basic cost coefficient. Thus, in order to maintain the optimality of a given optimal basis, dz/tr )

~

0, allj fi. p, rE p fixed,

obviously has to be valid.

215

Changing single cost coefficients

In the particular ca se under investigation, 0.2 + 0.4t4 ~ 0 ~ t4 ~ -0.5 } I t4~0~t4:=; I < < 0.6 - 0.8 t4 ~ 0 ~ t4:=; 0.75 ~ -0.5 - t4 - 0.75. 1.8 + 0.6t4

~

0

~

t4 :=; -3

The interval Tt) = [- 0.5, 0.75] is the critical region for t4 or the region of those values of t4 for which the optimal basis B is maintained, or it is the region of the stability of the basis B with respect to changing C4 . This basis is, thus, relatively sensitive to a change in the (basic) cost C4. Let us summarize. Let r E p be the subscript of a basic variable, Cr the corresponding cost coefficient and t r the parameter of which Cr is a linear function of the form cr(tr) = Cr + tr· For the sake of simplicity, let p = { I, ... , r, . .. , m} be the optimal basic-index and q> = {m+ I, ... , N} the index-set of the nonbasic variables. Then, (5-8) (5-9)

If we denote the lower endpoint of the interval T~P) by !r and the upper one by t r then ~z

~z

t =max{--J} y,,>o

-r

tr=min{--J}. y,, 0, Yi5 > 0 for at least one i = 1,2. We must now discover whether there actually are neighbors along these faces. This problem is equivalent to solving the auxiliary problem, as has been shown in Chap. 4. The solution procedure is described in Sec. 2-5 and for II-2.We, thus, solve minsj,j subject to

= 1,5,

249

Dependence on several parameters (on a vector parameter) Table 6-16

Optimal initial basis

3

5

I

Po

xB

3*

2

-1

24

I

I

0

10

~hl

-3

-I

0

-10

~h2

-I

1

1

10

~Zj

3

1

-I

10

~Z(I()

2

2

0

20

f-4 2 J J

J

Table 6-17a I1

12

OSI

3

I

3

OS5

I

-I

I

f-S 3

0

-1*

I

tl

s3

Table 6-17b

f-S I

3*

I

2

s5

I

-I

2

12

0

-1

1

3tl + t2 + SI = 3, tl - t2 + S5 = I, -t2+ S3=-I, Sj ~ O,j = 1,3,5. It should be noted that, in solving the auxiliary problem regarding parametrization of the cost coefficients, it is advantegeous to use the indices of the non basic columns for Sj . The auxiliary problem is solved in Tables 6-17a through 6-17c.

250

Linear parametrie programming with respect to c

Table 6-17c

5,

s3

t,

1/3

1/3

2/3

55

-1/3

-4/3

4/3

-I

I

0

t2

Table 6-18

Optimal solution associated with p,

4

p,

5

3 -1/3

xB

-+1

1/3

2/3

f-2

-1/3

1/3

~hl

1

I

-I

14

~h2

1/3

5/3

2/3

18

~Zj

-I

-I

0

-14

1/3*

8 2

J

J

From Table 6-17c, it follows that a neighbor in fact exists only along the first face. If we look at Table 6-16 in accordance with the corresponding pivot element (marked by an asterisk in Table 6-16), the new basic-index is PI = { I, 2}. Hence, we have the Iists V0 = {Po} , Wo = { PI}. In Table 6-18, the solution associated with PI is given. According to Table 6-18, the corresponding critical region T( I) is defined by 1

-tl - -t2 ::; -1,

3 2 -tl - -t2 ::; -1 3 2 tl - -t2 ::; 0, 3 and

z~~x(t) = -14 + 14tl + 18t2, tE T(I).

As follows from solving the auxiliary problem (not displayed here) neighbors only exist along the third and fourth faces. It folIows, further, that P2 = {I, 3}, which implies V I = { Po, pd, W I = {P2}. After carrying out the corresponding pivot step (from Table 6-18), we obtain Table 6-19 associated with basis B 2. The corresponding critical region T(2) is defined by

Dependence on several parameters (on a vector parameter) Table 6-19

Optimal solution associated with B 2

4

P2

251

2

5

xB

I

0

I

I

10

3

-I

I

3

6

~h ·1

0

2

3

20

~h2

1

I

-2

14

-I

0

-14

J

J

Mj

-I

o Figure 6-4

-t2 ::; -I, -2tl - t2 ::; -I, -3tl + 2t2 ::; 0, and

Z~~x(t)

=-14 + 20tl + 14t2, tE

T(2).

The solution of the auxiliary problem is left to the reader. As follows from these calculations, there is no new neighbor, so that V2 = {Po, PI, P2}, W2 = 0. This terminates the calculations, since K = T(o) u T(I) U T(2) has been generated. The region K and its partition into the critical regions is shown in Fig. 6-4. As can be seen from Fig. 6-4, there exists an absolute minimum of the function zmax(t) over K at the point t O = (2/3, I)T, zmax(tO) = 40/3. The coincidence of the

252

Linear parametrie programming with respect to c

minimum of the function z~~x(t) over the individual critical regions T(p) and of the absolute minimum does not always have to be the case of course. In general, over the critical regions T(p) , there exists a respective minimum of the function z~~x (t) (if the minimum exists at all), and, over K, there is then an absolute minimum (if zmax(t) over K is bounded from below). This absolute minimum is of economic importance and is known as the lower price limit. The lower price limit is usually that price (cost coefficient) at which " ... maintaining production brings a greater loss than would c\osing down the firm" ( cf. [Sf). Using the terminology of linear programming, we wouldsay that the lower price limit is the value of a cost coefficient (price) wh ich we must not fall below if - ceteris paribus - we are to have an optimal solution. If we have a problem with several parameters appearing in the objective function and we are looking for the lower price limit, then, in general, it is not necessary to compute the whole region K. In such cases, it is, as a rule sufficient to start from an optimal basis ad proceed in the direction of those neighbors which yield the largest negative increment in the value of the objective function. (W. Dinkelbach [3,4] has worked out a specifice procedure for this).

6-4 Homogeneous multiparametrie linear programming If, for instance, the maximization of profit is regarded as the goal, in certain cases the following questions may be put: What unit profits (or uni! prices) should be considered and how is the corresponding production program to be planned? In the terminology of linear programming, this means that the cost coefficents Cj are, as a matter of fact, unknown. The objective function can then be expressed in terms of its dependence on parameters, i.e., N

z(t) =

L L hjktkxj. j=1 k=1

The problem thus reads: compute all the parameter values tk for which (subject to the given constraints) there exists an optimal solution. If, for example, a critical region T(p) is found, then each s-vector (t~, ... ,t;)T E T(p) yields precisely one N-vector of the cost coefficients. We are, thus, dealing with a problem similar to that discussed in SecA-3 (Problem F or F D ). Since, in our case, the objective function written in vectormatrix form is

2 Cf. also [IOJ .

Homogeneous multiparametric linear programming

253

or z(t)

=(Hot)T x,

we speak here of problem (H) or (Ho), respectively. The matrix Ho is a diagonal matrix

Ho

=(

c.:I '

~'

0, 0,

... ,0, 0 )

.0,

'N

.

As in Sec. 4-3-1, we shall illustrate the problems involved from examples.

Example 6-5 Maximize z(t)

=(-tl + 2t2)XI + (tl -

5t2)X2, 3

subject to -4xI + X2 :s; 4, XI -5X2 :s; 5, XI ~ 0, X2 ~ O.

Phase 1 First we compute a dual feasible (nontrivial) solution (UO, t O ). Set t~ = t; = land introduce a dual artificial variable p. Table 6-20 shows the solution where the artificial variable has already been entered in the dual basis. The rows L'1 hY, k = 1,2 are formed by analogy with the row L'1 Zj . Here, h~ is the vector the components of which form the coefficients of the parameter tl, which are assigned to the basic variables. Similarly with h~ . The column L p plays the same role here as the row L Pi in problem (F). According to Table 6-20, X2 is first entered into the primal basis; this yields Table 6-21.ln this tableau, there is only one negative element in the column "LP", so that tl is introduced into the (dual) basis. The result of the corresponding pivot step is shown in Table 6-22. Note that if t I enters the (dual) basis, the condition t~ = I is removed. The value lr of this "basic variable" is, in general, determined as

3 By simple algebraic transformations, we obtain z(t) = tl(-XI + X2) + t2(2xl - 5X2). Denoting Zl = - XI + X2, Z2 = 2xI - 5X2, the function z(t) represents an overall function in a problem with two objective functions. This kind of problems is dealt with in Chap. 9.

254

Linear parametrie programming with respect to c

Table 6-20 Initial tableau I

2

3

I

f-4

b

:Ep

4

4

4

-I

5

-I

P

-5*

I

0

I

0

I

~h ·)

-I

-I

0

-1

~h2

5

2

0

2

~Zj

4

I

0

I

b

:Ep

J J

(t*)

Table 6-21

2

First pivot step

4

P

3

1/5

19/5

5

19/5

~2

-1/5

1/5

-1

1/5

1

0

1

0

1

-I

-4/5

)

f-~hj

~h2 J

~Zj

(t*)

-1/5

-4/5*

1

1

5

1

4/5

1/5

4

1/5

Thus, t) becomes t, = ~ and this yields tO =(5/4, I)T. Substituting tO into the original problem, after a few (primal) pivot steps we obtain an optimal solution, which is shown in Table 6-23. The critical region T(o) is defined by -4t, + 5t2 ::; 0, t, - 2t2 ::; 0 and

z~~x(t) = -5t, + IOt2 . This terminates Phase I.

Homogeneous multiparametrie linear programming Table 6-22

255

Second pivot step tl

4

3

19/4

-3/4

1/4

2

1/4

-1/4

-5/4

I

5/4

-1/4

-5/4

~h2

5/4

3/4

15/4

~Zj

5/4

3/4

15/4

4

xB

J

(t*)

Table 6-23

Optimal solution

Po

2

3

-19

4

24

I

-5

I

5

~hl

4

-I

-5

~h2

-5

2

IO

0

3/4

15/4

J

J

~Z· (to) J

Phase 2 As shown in Sec. IV-6 (cf. also Ex. 4-12), in this case we use as auxiliary problem the problem (lV-82), (IV-85) through (IV-88) (on account of the parameters, which are not sign-restricted), i.e.,

(6-1) subject to -4t1 + 4tl + 5t; - 5t2 + S2 = 0, t1 - tl - 2t; + 2t2 + S4 = 0, t1+tl +t;+t2 = I, t +>0->Ok-12 , tk , , , sJ·>0 - ,J' -24 , , k with

and t~ .

ti: = 0, k =

1,2.

256

Linear parametrie programming with respect to c

TabIe 6-24 I

-4

52

I;

f;

t+

OS4

1*

P

I

t;

4

5

-5

0

-I

-2

2

0

I

I

I

I

Table 6-25

t;

1-;

t;

54

S2

0

-3

3

4

0

-H~

-I

-2

2

I

0

f-p

2

-I

-I

I

3*

Table 6-26

I;

1-;

s4

s2

2

-2

-3

I

I;

1/3

-4/3

-1/3

2/3

-H~

2/3

-1/3

-1/3

1/3

In (6-1), j =4, for there are positiv elements only in the nonbasic column with the index 4 (cf. Table 6-23). The solution ofthe auxiliary problem is shown in Tables 6-24 through 6-26. From Table 6-26, it follows that min S4 =0 and S4 NBY. The node adja cent to the node Po = {I, 3} is, therefore, PI = {3, 4} . Hence, Vo = {Po}, Wo = {pI}. Table 6-27 shows the solution associated with PI. However, this solution, as such, is uninteresting, since both real variables here are x I = X2 =o. The critical region T( I) is defined by + 2t2

S;

0,

tl - 5t2 S;

0,

-tl

and the objective function value dependent on the parameter is zero for all tE T(I ).

The calculation of the auxiliary problem is left to the reader.

Homogeneous multiparametrie linear programming

257

Table 6-27 2

PI

I

3

-4

I

4

4

I

-5

5

~h ·1

I

-I

0

~hz

-2

5

0

J J

Table 6-28

xB

Solution associated with pz

3

I

pz

xB

2

-4

I

4

4

-19

5

25

~hl

-3

I

4

~hz

18

-5

-20

J

J

From these calculations, it follows that two adjacent nodes to PI exist: Po and P2 = {2, 4} . The corresponding solution for P2 is shown in Table 6-28. List: V I = {Po, PI} , W I = {P2} . According to Table 6-28, T(2) is defined by 3tl - 18t2 -tl + 5t2

:S; :S;

0, 0,

and

z~~x(t)

=4tl -

20t2·

As follows from the solution of the auxiliary problem, wh ich we have not set down here, the only node adjacent to P2 is PI . Then V 2 = {Po, PI , P2}, W 2 =0, i.e. So = S = V 2 and the problem is solved. Let us now have a brief look at Problem (UD). If the cost coefficients are only to be interpreted as nonnegative quantities and if, at the same time, there is complete uncertainty about their values, the objective function can be formulated as folIows : N

z(t) =

L j=1

CjtjXj;

258

Linear parametrie programming with respect to c

Table 6-29

Optimal solution for t = t*

4

Po

3

I

-1/2

1/2

1/2

2

1/2

1/2

3/2

5

3/2

-1/2

5/2

6hJl

-I

I

I

6hJ2

3/2

3/2

9/2

0

I

2

6z/t*)

xB

written in matrix-vector form , z(t) = (Hot)T X, where

C,. O• ... • O.

0 )

(

Ho =

and tj

O. O. H , O.

~

0 for all j.

CN

This case, too, will be illustrated by an example.

Example 6-6 Maximize z(t)

=2tlxI + 3t2x2,

subject to -XI xI 2xI XI ~

+ X2

~

+ X2

~

- X2 0, X2

~ ~

I, 2, 2, 0 , tl

~

0, t2

~

O.

Phase 1 We have

(~: ~) , z(t) = (Hot)T x. ) T, i.e., t~ = I, t; = I, and set up the initial tableau.

Ho =

=

Set t* (I, I Phase I is analogous to that of Problem (H). The example was chosen so that it has an optimal solution for t t*. This solution is given in Table 6-29. According to Table 6-29, Po = {I , 2, 5} and the critical region T(O) is defined by

=

259

Homogeneous multiparametrie linear programming Table 6-30

4

PI

5

xB

I

1/3

1/3

4/3

2

2/3

-1/3

2/3

3

-1/3

2/3

5/3

~h ·1

2/3

2/3

8/3

~h2

2

-I

2

J J

tl - 1.5t2 $; 0, i.e., 2tl - 3t2 -tl - 1.5t2 $; 0, i.e., 2tl + 3t2 tl ~ 0, t ~ 0,

$; $;

0, 0,

and (0) (t) zmax -

tl + 4 .5 t2, t

E T(O) .

Phase 2 The theory and the procedure for problem (H D ) are analogous to those of problem (F D) (cf. Sec. IV-6 and Ex. 4-12). The following auxiliary problem is then solved: min Sj,j = 3, 4, subject to 211 - 3t2 + S3 = 0, -2tl - 3t2 + S4 = 0, tl+t2=1, Ik ~ 0, k = 1,2, Sj ~ O,j

= 3,4.

As follows from the solution of this auxiliary problem (left to the reader), the min S4 = 12/5> 0 and S4 BV, min S3 = 0 and S3 NBY. This implies that only along the third face does there exist a neighor. The corresponding basic-index is PI = {I, 2, 3}. List: Vo = {Po}, Wo = {pI}. Choosing PI E Wo and performing the corresponding pivot step, we obtain Table 6-30. The critical region T( I) is defined by

-~tl - 2t2

$;

0, i.e., -

tl - 3t2

-~II + 12 $; 0, i.e., - 2tl + 312 tl · ~ 0, t2 ~ 0, and

$;

0,

$;

0,

260

Linear parametrie programming with respect to c (I)

zmax

(t) = :3tl 8

+ 2 t2, t E

T(I)

.

From the solution of the auxiliary problem it follows that min S4 = I > 0 and S4 BV; further, min S5 = 0 and S5 NBY. This implies that the only adjacent node to PI is again PO ' List: VI = {Po, pd, W I = 0. This terminates the procedure.

VI Abridged mathematical presentation

Consider the problem: Determine a region maximizing

K C IR'

such that the problem of

z(t) = cT (t)x,

(VI-I)

Ax= b,

(VI-2)

x

(VI-3)

subject to

~ 0,

where c(t) = c + Ht,

(VI-4)

has a finite optimal solution for each t E K ~ IR' ,and, for t E IR' - K., the given problem has no optimal solution . Here, c, x E IR N , bE IR m are constant vectors, A is a constant (m, N) matrix, H a constant (N, s) matrix, t is a vectorparameter, c(t) E IR N . In this case, too, auxiliary conditions Gt~d

(VI-5)

can be incIuded, where G is a constant (r, s) matrix and d E IRr is a constant vector. Formulae (V-6) through (V-27) also hold here, so that it is not necessary to repeat them here. An admissible vector parameter t is defined by analogy with Definition IV-I, and we obtain the corresponding definition by a simple exchange of t for A in Definition IV-I . Analogously to Sec. IV, K ~ IR" defines the region of all admissible vector parameters t E IR'. Here too, the region K is given as a union of the nonoverlapping critical regions T(P) ~ K., which are associated with the optimal bases B p , respectively. The following theorem is valid by analogy with Theorem IV-I.

Theorem VJ- J Suppose that there exists a finite optimal solution of the problem (VI-I) through (VI-4) for a constant tO E IR' . Then, either there exists a finite optimal solution of the problem (VI-I) through (VI-4) for any t E IR' or the given problem has an unbounded solution. Theorem VI-2 and VI-3 hold analogously to Theorems IV-3 and IV-4.

Theorem VJ-2 The optimal value function zmax(t), wh ich is defined over K ~ IR', is a convex function .

262

Abridged mathematical presentation

Theorem VI-3 The function z~~x(t) is linear over T(p) and, thus, continuous over

K. Definition VI-I Consider two arbitrary bases BI, B 2. These bases are said to be neighboring bases (neighbors for short) iff (i) there exists t* E K such that BI and B2 are optimal bases of problem (VI-I) through (VI-4) for t* E K, and (ii) it is possible to pass from BI to B2 (and conversely) by one primal simplex step.4 Definitions IV-4, IV-6 and Theorems IV-5, IV-6, IV-7 may be taken over without change, >. being correspondingly exchanged for t and A (p) for T(p).

Corollary Vl-I-I The critical region T(p) wh ich is defined by (V-25), has a neighbor along the jth face, j ri p, if it is possible to pass to this neighbor by entering the jth nonbasic variable from the system (V-I7) through (V-19) in the new basis, i.e., if the following conditions are met 1** there exists Yij > 0 for at least one i E I, and 2** there exists tE T(e) such that ~ Zj(t) = o.

VI-I A solution procedure In this case too, the algorithm described in Sec. VI is used to compute K or K* = K n M, M = {t E IR s I Gt ::; d},i.e., the procedure is carried out in two phases. In Phase 1, an arbitrary node Po E So is generated.5

Phase I I. Compute a feasible solution (UO, tO) of the system

ATu-Ht ~ c

(VI-6)

possibly with -Gt::; -d,

where u E IR m and t E IR s are variable vectors. The dual system (VI-6) can also be solved in the primal tableau; namely, determine a dual feasible solution of the following system: Ax = b, 4

(VI-7)

If it is possible to pass to another basis in one dual step, this new basis is not to be seen as a neighboring basis to the given basis. 5 If it proves necessary, it is quite possible to modify the theory and the algorithm for j(* ,as was the case in Sec. IV. For the sake of simplicity, only j( wh ich implies Go is taken into account.

Deseription of systematie parametrization with a seal ar parameter

=0, z - CT X + d T t =0, t ~ 0, v - HT X - G T t

263

(VI-8) (VI-9)

where t E IRr is a variable vector with nonnegative components. If (VI-6) has no solution, no dual feasible solution of the problem (VI-I) through (VIA) exists for any t E IRS, i.e., K = 0. 2. The vector tO, which was found to be the solution of the system (VI-6), is substituted into (VI-I) and the following problem is then solved: maximize w

=cT(to)x,

(VI-IO)

subject to Ax

= b, x ~ o.

(VI-lI)

If this problem has a primal feasible solution,6 an optimal basis B o of the problem (VI-IO), (VI-lI) exists, which at the same time is an optimal basis of the problem (VI-I) through (VI-4) regarding tO E T(o) . Since tO E T(o) , obviously, T(o) i= 0 , and, since T(o) ~ K, we also have K i= 0.

Phase 2 Although the set Qk is determined in the same way as in Sec. IV-3I (cf. Steps 1°° through 7°° of the algorithm in Section IV-3-1), the following system is investigated:

(VI-12) where S E IR N- rn is a slack variable vector and k H N or kCN is that part of matrix k H or vector k z, the elements of which are associated with the non basic real variables. In Qk we record the, as yet unlisted, adjacent nodes (neighboring regions) along the faces j 4 p, for which min Sj = 0 and Sj NBV regarding (VI-12). Analogously to the cases described in Sec. V, various special cases can be derived by specialization of Hand t here too.

VI-2 Description of systematic parametrization with a scalar parameter Systematic parametrization is a procedure for computing all admissible values of a scalar parameter t in the problem: maximize z(t)

=(c+ht)Tx,

subject to 6 Dual feasibility is al ready ensured by tO.

(VI-13)

Abridged mathematical presentation

264

Ax = b,x;::: 0,

(VI-14)

where h E IR N is a constant vector. Geometrically speaking, this is a systematic procedure for covering the interval K (of all admissible values of the parameter t) by critical regions (intervals) T(p) such that two mutually different critical regions do not overlap. Briefty, systematic parametrization is the computation of all neighbors starting from an optimal basis. Such a procedure has been described in several publications ( cf. the bibliography at the end of this book). If M = [1, t] denotes the total region for the parameter t,7 then K* = K nM is that part of K, which has points in common with M . The procedure is performed in two phases. Phase I deals with the computation of a first optimal basis Bo ; Phase 2 systematically computes all the neighbors starting from Bo '

Phase J I' . Convert the inequalities in the constraint set such that no artificial variables are needed,8 i.e., multiply all inequalities of type;::: by -I . 2'. Add the row ~hj = - hj to the initial tableau. 3'. Compute a dual feasible solution (XO, to ) in accordance with the dual algorithm. 3'-1. Add as many (dual) artificial variables to the initial tableau as the row ~Zj = -Cj has negative elements. Each auxiliary row is to be regarded as a row unit vector. The element I of this vector appears in the column, in which ~Zj =-Cj -

0, "'" vJ

,j

I:t:

p,

(VI-31)

with tk = t; - tk, k = I, "., s,

(VI-32)

(cf. Theorem IV-I 0).

VI-3-2 Problem (Ho) Consider problem (UD), i.e., determine the region K c IR N such that the problem of maximizing (VI-33) subject to Ax = b, x ~

0,

t ~

0,

t

E

IR N ,

(VI-34)

has a finite, nontrivial, optimal solution for each t E K* and this problem has no nontrivial optimal solution for any t E IRN - K*. Analogously to problem (F o ) (cf. Sec. 4-6-2), problem (UD) is also treated as aseparate case for two reasons . I. We have t TUTD=(Cltl, ... ,cNtN).

*

If the coefficients Cj 0 are constant, t in UDt can be regarded as a multiplicative parameter. 2. The nonnegativity condition t ~ 0 for the parameter depends on the particular technical meaning of the cost coefficients. Phase I of the solution procedure for problem (UD) is performed in the same way as in problem (U), i.e., (VI-15), (VI-16). Phase 2 is analogous to the Phase 2 described in the previous section, except that, as auxiliary problem, we solve the following task: (VI-35) subject to s

-L L1httk +

Sj

=0, Vj ri

p,

(VI-36)

k=1

s

Ltk = I , k=1

(VI-37)

270

Abridged mathematical presentation tk ~

0, k = I, " ' , s, Sj ~ 0, \ij 4 p,

(cf. Corollary IV-IO-I),

(VI-38)

References

[I J Arnoff, E.L., S.S.Sengupta: Sensitivity analysis: Parametric programming, In : Progr. of Operations Research I (1961) 175-180 [2) Candler, W.A .: A modified simplex solution for linear programming with variable prices, J.Farm.Econ . 38 (1956) 940-955 [3J Dinkelbach, w.: Sensitivitätsanalysen und parametrische Programmierung, Springer, Berlin 1969 [41 Dinkelbach, W., P. Hagelschuer: On multiparametric programming, Methods Oper. Res. VI (1968) 86-92 [51 Dragan, I, : Un algorithme pour la resolution de certain problemes parametriques, avec un seul parametre contenu dans la fonction economique, Rev. Roum. Mat. pur. Appl. 11 (1966) 447-451 (6) Gass, S.I., T.L.Saaty: The parametric objective function 2, Oper. Res. 3 (1955) 395401

[7) Gass, S.I., T.L. Saaty: The computational algorithm for the parametric objective function, Naval. Res. Log. Quart. 2 (\ 955) 39-45 [8) Hax, H.: Preisuntergrenzen im Ein- und Mehrproduktbereich, Zs. für Handelswirtsch. Forsch. 13 (1961) 424-449 [91 Saaty, T.L. , S.I.Gass: The parametric objective function I, Oper. Res. 2 (1954) 316319

[IOJ Urspruch, H.-D .: Parametrische lineare Programmierung und ihre Anwendungsmöglichkeiten, Master Thesis, Univ. Köln 1966 [11) Yu, P.L., M. Zeleny: Linear multiparametric programming by multicriteria simplex method, Managem. Sci . 23 (\976) 159-170

Chapter seven

7

7-1

7-2 VII VII-I VII-2

RIM parametric linear programming Simultaneous changing of the right-hand side and of the cost coefficients . . ........... . . . Dependence on a scalar parameter . . . . . . . . Dependence on several parameters (on a vector parameter) RIM-multiparametric linear programming with dependent parameters . . . . . . . . . Abridged mathematical presentation A solution procedure A special case References. . . .

275 275 285 299 302 303 307

7 RIM parametric linear programming Simultaneous changing of the right-hand side and of the cost coefficients

The interdependence of the sales price and the sales volume of goods being manufactured in a multi-product firm is familiar enough. Prices and/or costs and/or profit on the one hand and sales and/or demand and/or available output and/or quantities of raw material on the other may depend here on outside factors or on another. In such a system, we are taking into account fairly complicated relationships between parts of the firm (subsystems) and the firm (system) itself or between the system and its environment. The analysis of such relationships may be seen as the subsequent adaptation of production to certain changes wh ich have al ready occured or as an analysis, carried out in advance, which enables us to make predictions as to production or market conditions with regard to the given system and/or to make management decisions. If these influences are expressed by parameters in a linear model, the cost vector c and the vector b (the right-hand side) can, in general, be written as a function of two parameters t, ). which are dependent or independent of one another; i.e., c(t, ).), b(t, ).), where t is, say, an S-vector and ). an s-vector (we mayaiso have s = S). If a function F(t, ).) = 0 exists, I the vectors t, ). will be called dependent parameters. Otherwise they are called independent parameters. A generalized theory for the case of independent parameters is to be found in [4, 5], methods in [11. An application of this case to investment and lot-size problems is given in [3]. As usual, the problems indicated above will be discussed with the aid of illustrative examples.

7-1 Dependence on a scalar parameter Example 7-1

z(t)

Maximize

= Cl + t)x I + (I -

2t)X2,

subject to

We suppose, for simplicity, that this implicit function is solvable either for X = f(t) or t=

ljI (X) .

276

RIM parametrie linear programming

Table 7-1

Optimal solution for t = 'A =0 4

pr

hB

eB

P

3

I

I

I

-1/3

2/3

5/6

5/3

-2

I

2

2/3

-1/3

1/3

-1/3

i1zj

1/3

1/3

7/6

4/3

i1hj

-5/3

413

1/6

713

XB

XI + 2X2 ::; 1.5 + 'A, 2xI + X2::; 2 + 3'A, XI ~ 0, X2 ~ O. The tableau in Table 7-1 contains the optimal solution for t = 'A = O. We shall now consider four cases. I. Suppose that the parameter 'A is not inc1uded. This means, according to (V-40), -0.25 ::; t ::; 0.2, i.e., T(P) = [-0.25,0.2]. 2. Suppose that parameter t is not inc1uded. This means that, according to (III-25), -0.5 ::; 'A::; I, i.e., A (p) = [-0.5, 1]. 3. Assuming that t and 'A are dependent, each value t E T(p) influences c(t) = c + ht, but also b('A) = b + fA and vice versa. This means that a f(t, 'A) = 0 exists. The critical region R(p) is then given by all common ofthe regions T(p) and A (P) , i.e., R(p) = T(P) nA (p). Assume that the F(t, 'A) = 0 in our example takes on the simplest form t = 'A. Then, z(t) = (I + t)XI + (I - 2t)X2; XI +2X2 ::; 1.5 + t; 2xI +X2 ::; 2 + 3t; XI Select all those values of t for wh ich ~Zj(t) the same time. In our example, we have R(p) = [-0.25,0.2]

~

0 and XB(t)

~ 0

not only function "points" function

~O,

X2 ~O.

are fulfilled at

n [-0.5, 1] = T(p)

4. Assuming that t and 'A are independent of one another, t can take on arbitrary values of T(p) without thereby influencing XB('A); conversely, 'A E A (p) can be arbitrary without thereby influencing ~Zj(t). In other words, in this case there does not exist any function F(t, 'A) = 0 which expresses the mutual dependence of the parameters t and A. The critical region R(p) is then defined as all ordered pairs (t, 'A) such that t E T(p), 'A E A (p), i.e., as Cartesian product T(p) x A (p) (see also [4]). This situation is shown geometrically in Fig. 7-1 (b). Any ordered pair (l', ",*) from the rectangle in Fig. 7-I(b) yields ~Zj(t*) and Yi("'*) such

Dependence on a scalar parameter

277 A 1

(

)

!

0.5 - 0.25 .

o

..

0.2

t,A

~

(a)

- 0.5

0.2

t

-U.5 (b)

Figure 7-1

that the conditions ßzi.)

4.51.2 - AI + 1.2 + 21.11.2 - 41.1 1.2 - AT + 5A~ = 4 - AT + 5A~ - 2A. 11.2 + 0.51. 1 - 3.51.2, >. E R(o) .

= 4 + 1.5/1.( -

Since we are dealing with dependent parameters,

R(o) = A (0) n

T(O),

i.e., all conditions (I) through (5) must be met. Fig. 7-5 shows the critical region R(o). We must now find out which face of the R(o) neighbors exist along. This is done in the same way as in Chaps. 4 and 6. The auxiliary computations are carried out in Tables 7-8 through 7-11. Here, we proceed from te al ready familiar fact that, with respect to the right-hand side, at first only those slacks are used for

289

Dependence on several parameters (on a vector parameter) Table 7-8 AI

~I

°"'3

-I

0

8

11 *

0

-I

2

A2

PI

7

f-P2

-7

~2

0

-2

0

0

I

0~3

2

4

0

0

5

°"'5

3

-3

0

0

2

-~Pi

0

-19

I

I

-10

8

Table 7-9

~I

AI

"'3

f-PI

133/11 *

~A2

-7111

0

-1111

2/11

~2

-14/11

0

-2111

15/11

~3

50111

0

4/11

47/11

12/11

0

-3/11

28/11

-133/11

I

-8/11

-72/11

"'5 -~Pi

-I

8/11

72111

"minimization" to si = 0, which have at least one Yij < 0 in the corresponding rows. With respect to the cost coefficients, only those slacks are taken into account the corresponding columns of which contain Yij > O. To organize the computations beuer and to give us a better general view of the results, we introduce the following notations for the slack variables: _PF* I)..

+,p = Pz' ,

_PF*2)..

+ ~ = x~

(cf. (VII-IO), (VII-li» . With the ~i slacks, the subscript i is given by the rows of the tableau, with "'j the subscripts j 4. p are taken. The slacks of interest to us have been marked as usual in Table 7-8. It follows from Tables 7-8 through 7 -11 that neighbors can only exist along the "second" (dual) and the "fourth" (primal) faces . In contrast to the case where there is a vector parameter only in the righthand side or only in the cost coefficients exclusively, determining neighbors here

RIM parametrie linear programming

290 Table 7-10

~I

'l'3

-111133

--?AI

81133

-1119

-1119

f-~3

501133

121133*

'l's

12/133

A2

-451133

72/133 10119 2411133 2601133

Table 7-11

~I

~3

AI

-1/3

A2

1/6

7112

19/12

--? 'l'3

25/6

133112

241/12

3/2

45/12

667/76

'l'5

-2/3

-2/3

involves more computation. The present exaple will illustrate what this consists of. We have Po = {I, 2, 4}. Consider, first, face (2): along this face a neighor exists (cf. Table 7-10, where 'l'3 = 0). We must now discover the appropriate exchange of variables. For this we need the minimum of the quotients YiC~.) / Yi3, Yi3 > o. The right-hand side is dependent over)., however, so that the values Yi cannot be used directly. According to Table 7-10, for 'l'3 = 0 the vector parameter is ).' = (72/133, 10119)T. This).' must now be substituted into Yi().); namely, for all those i for which Yi3 > O. If we now look at Table 7-7, it follows that ßZ3().)

I

7

2 4 as expected; furthermore, , y,().)

72 11 10 +- x133 4 19

=-- - - x 1

10

39,

=0 72 10 +8x -

= O. 2 19 38 133 19 Hence, Y3().') / Y43 < y, ().') / Y13. The basic variable X4 is, therefore, replaced by the variable X3, so that p,

= - + - = -, Y3().) = -8 + 7 x

= {I, 2, 3}.

Dependence on several parameters (on a vector parameter) Table 7-12

Solution for P2 I

pz

291

_2f'

5

_2f2

xB

2

1/5

1/5

1/5

-3/5

I

3

-4/5

1/5

-4/5

-8/5

-2

4

24/5

-1/5*

-11/5

8/5

4

t.zj

-2/5

3/5

3/5

-9/5

3

t.h'

-7/5

-2/5

-2/5

6/5

-2

t.hJ2

11/5

1/5

1/5

-3/5

I

J

H.

Now look at face (4). According to Table 7-11, ~3 = 0 and ,,'; = -~,,,; = From Table 7-7, it follows that YZ(A'') = ~ + ~ - 2 x = 0 as expected, and, furthermore,

H

" I 7 2 11 19 241 ,1z3(A ) = -- + - x - + - x - = -

2 1 ,1ZS(A ) = - + 2 "

4 3 4 12 48' 3 2 3 19 35 - x - +- x - =16' 4 3 4 12

so that ,1Z3(A") 241 4 _ ,1ZS(A") 35 - - - = - x - =4.016, = -x4=8.75. YZ3 48 5 Yzs 16 Hence, the basic variable x I is replaced by the variable X3 and

pz = {2, 3,4} . The listing is thus: V o = {Po}, Wo = {PI,PZ}'

Select PZ E Wo . Starting from Table 7-7 and, after one dual step with the marked pivot element, we obtain Table 7-12. We now first have to ascertain whether there exists any solution at all of the systemS A(Z):

(6) 5 Regarding po, this question is superfluous, since X0 = (0, I) T exists here, so that, for X = XO , the given problem has a finite optimal solution. Hence, R(o) '# 0.

292

RIM parametrie linear programming

Table 7-13

Solution for Ps 4

I

Ps

_sfl

_'f2

xB

f-2

5

1*

-2

I

5

3

4

I

-3

0

2

5

-24

-5

11

-8

-20

14

3

-6

3

15

ßhJ·1

-11

-2

4

-2

-10

ßhJ2

7

I

-2

I

5

ßZj

4/"1 + 8A,2

10,

(4')

-I lAI + 8A,2 :s; 20;

(10)

~

T(2): -nI + IIA,2

~

2,

2A,1 - A,3 :s; 3.

(2) (8)

If there does, the possible neighbors are determined at the same time. If there does not, P2 must be deleted from the list Wo' If we perform these auxiliary computations (by the same procedure as in Tables 7-8 through 7-11) the result is: R(2) ~ 0 and, along face (4'), the neighbor R(o) with Po exists; along face (10), the neighbor R(S) with Ps = {2, 3, 5}, along face (2), the neighbor R(I) with PI, and along face (8bar) the neighbor R(6) with P6 ={3, 4, 5} exists (cf. Fig. 7-5). The lists are VI = {po,pÜ, W I = {PI,PS,P6}' From Table 7-12, it further follows that

z~~x().) = 3 + 0.4A,T + 0.6~ - 1.4A,1 A,2 - 2.6A,1 + 2.8A,2, boldA, E R(2). Choose PSEW 2. After one dual step from Table 7-12 with the pivot element marked, we obtain Table 7-13. From Table 7-\3, it fol\ows: A(S) : (13) :s; 2,

(7) (10')

Dependence on several parameters (on a vector parameter) Table 7-14

Solution for P6

f-4

_6fl

_6f2

I

-2

I

5

2

I

P6

293

5*

xB

3

-I

-I

-I

-I

-3

5

I

5

I

-3

5

öZj

-I

-3

0

0

0

öh·J1

-I

2

0

0

0

öhJ2

2

-I

0

0

0

(fI )

IIAI -7A2 $ 14, 2A I

-

A2 $

(8)

3.

The auxiliary ca1culations would show that first R(S) ::f:. 0 and that to R(S)there exists, along face (10'), the neighbor R(2) with P2; along face (8), the neighbor R(6) with P6 (cf. Fig. 7-5). This yields the Iists V2 = {Po,P2, PS}, W2 = {PI,P6}. It further follows from Table 7-13 that

z~~x(>.) = 15 - 4Af - A~ + 4A) A2 - 4AI + 2A2. >.

E

R(S).

Now choose P6 E W2. After one primal step from Table 7-13 with the pivot marked, we obtain Table 7-14 associated with P6. From Table 7-14 it folIows: A(6):

-2AI + A2 $ 5,

( 13)

3,

(15')

AI - 3A2 $ 5;

(6)

- AI + 2A 2 ~ I,

«(4')

AI + A2

~

T(6):

2A) -

A2

~

3.

(8')

For>. E R(6) , the objective function is completely independent of >.. It follows from the auxiliary computations that, at first, R(6) ::f:. 0; further, there exists, along

294

RIM parametrie linear programming

Table 7-15

Solution for P4 2

P4

_ 4r2

_4r l

4

xB

3

-4/5

1/5

-7/5

-4/5

-2

--71

1/5

1/5

-2/5

1/5

I

f-5

24/5 *

-1/5* *

7/5

-16/5

4

-14/5

1/5

-2/5

1/5

2

t1h·J1

11/5

1/5

-2/5

1/5

1

t1hJ2

-7/5

-2/5

4/5

-2/5

-2

t1zj

face (i4'), the neighbor R(4) with P4 = {I, 3, 5} and, along face R(2) with P2. This yields the lists

(8\ the neighbor

V 3 = {Po,P2,P5,P6}, W 3 = {PI,P4} . Choose P4 E W 3 . After one primal step, with the pivot element marked in Table 7-14, we obtain Table 7-15 . According to Table 7-15 : A(4) : ~

10,

(12 ' )

A2 ::; 5,

(13)

?AI - 16A2 ::; 20;

(9)

?A2 ~ 14,

- , (11 )

I.

(14)

7/q + 4A2 - 2AI +

T(4) : I IAI -

AI+ 2A2 ::;

The auxiliary computations would show that, along face (12'), there exists to R(4) the neighor R (3) with P3 = {I, 2, 5}; along (9), the neighbor R(7) with P7 = { I, 3, 4} ; along (fI'), the neighbor R(I) with PI; a nd, along ((4), the neighbor R (6) with P6 (cf. Fig. 7-5). Hence, V4

={Po,P2,P4,P5 , P6} , W4 ={PI,P3,P7} .

According to Table 7-15, we also have

z~~x ().)

= I + O.4(Ai + A~) -

AI A2 + I.4AI - 2.2A.2,).

E

R(4) .

Choose PI E W 4 . The corresponding tableau is given in Table 7-16. The auxiliary computations would show that R(I) 0 and :

*

Dependence on several parameters (on a vector parameter) Table 7-16

Solution for PI 4

PI

295

_Ifl

5

_lf2

xB

~2

-1/24

5/24

7/24

-2/3

5/6

I

5/24

-1/24

-11/24

1/3

5/6

-4/3

-4/3

+-3

1/6

1/6*

Lizj

1/12

7112

5/12

-5/3

10/3

Lih·J1

7/24

-11/24

-25/24

5/3

-5/6

-11/24

7/24

29/24

-4/3

-5/6

1

Lihj-

-7/6

A (I):

(9)

7/", - 16A2 ~ 20,

-IIAI + 8A2

~

20,

(10)

?AI + 8A2

~

8',

(I)

- ?AI + IIA2

~

2,

(2' )

~

14.

(CI)

T(I):

IIAI -

?A2

According to the auxiliary computations, there exists to R(I) the neighbors R(o) and R(2) along face (2') and the neighbors Rn) and R(4) along face (CI) (cf. Fig. 7-5). This yields the lists

Vs = {PO,PI,P2,P4,PS,P6}, Ws = {P3,P7} . Note As we have already shown, when finding adjacents to Po, the respective parameters determined must be substituted either into ßzi; the neighbor R(4) along face (12) (cf. Fig. 7-5). Hence, V6 = {pj,i = 0, ... , 6}, W6 = {p7}.

Further, z~~x(X) = 8 + 4.25AT - ~ - 1.25A, A2 - 9A, - 1.5A2, X

E

R(3).

Choosing P7 E W6 yields Table 7-18 after one primal step with the pivot element marked in Table 7-15 by two asterisks. According to Table 7-18, we have: A(7):

-

2A2 ::; I,

(3)

AI -

3A2 ::; 5,

(6)

~

20;

(9 ')

-TAl + llA2::; 2,

(2')

- A, + 2A.2 ::; l.

(14)

TA, -16A2 T(7):

297

Dependence on several parameters (on a vector parameter) Table 7-18

Solution for P7

2

P7

_7fl

5

_7f2

xB

3

4

I

0

-4

2

I

5

I

I

-3

5

4

-24

-5

-7

16

-20

dZj

2

I

I

-3

5

dhJl

7

I

I

-3

5

dhJ2

-11

-2

-2

6

-10

The auxiliary computations result in P7 having a single neighbor P4, so that no new neighbor has appeared (cf. Fig. 7-5). Hence, V7

= {pi, i = 0, ... , 7}, W 7 = 0.

The whole process is now terminated. Finally, in accordance with Table 7-18, we have

z~~x(>..)

=5 -

A.i - 6A.~ + 511.111.2 + 411. 1 -?A2, >..

E

R(7).

As in the case of b(>") (cf. Chap. 4) and the case of c(t) (cf. Chap. 6), the whole region 7

K=UR(i) i=O

forms a convex set. This is also apparent from Fig. 7-5 (cf. also Theorem VII-2).

VII Abridged mathematical presentation

Consider the problem: determine the domain of definition function

K ~ IR' of the linear (VII-I)

such that the problem of maximizing

=eT (t,).) . x

(VII-2)

=b(t, ).), x 2 0,

(VII-3)

z(t,).)

subject to

Ax

has a finite optimal solution for each ). E solution for any ). E IR' - K. We have f()') = d

K and the given problem has no optimal

+ 0)" e(t, ).) = e + H I t + F I )., b(t,).) = b + H 2t + F 2).,

(VII-4)

where H I is a constant (N, S) matrix, F I a constant (N, s) matrix, 0 a constant (S, s) matrix, F 2 a constant (m, s) matrix, d E IR' a constant vector, A a constant (m, N) matrix, e E IR N , b E IR m constant vectors, x E IR N a variable vector. Substituting (VII-4) into (VII-2) and (VII-3), we obain e(t,).) = e*

+ F' I)., b(t,).)

= b" + F*2).

(VII-5)

with e* = e + Hld, F*I = HIO + F I , b* = b + H 2d, F*2 = H 2 0 + F 2 .

(VII-6)

Thus, the problem (VII-2), (VII-3) reads : maximize z(t,).) = (e" + F' I).)T x

(VII-7)

Ax = b" + F *2)., X 2 o.

(VII-8)

subject to

Note Instead of t =f()'), the function ). = -0.25 . From this, the "basic condition" folIows, since all denominators have, thus, become positive expressions and, therefore, the numerators have to remain positive. This implies 8 + 27d 1 - 6d 1 + 14d 3 + 7d

~ ~ ~ ~

0 0 0 0

::::} ::::} ::::} ::::}

d d d d

~

s ~ ~

-8/27, 1/6, -1/14, -3/7,

314 Table 8-6

Sensitivity analysis with respect to the elements of the technological matrix A Solution associated with B2

P2

2

3

4

6

xB(d)

~5

4 +21d -3 + 2d

2(1 +4d) 3 +2d

2 (I -d) • -- 3 +2d

I-d --3 +2d

16 + 89d -3 + 2d

~l

11 -3 + 2d

4 -3 +2d

2 -3 +2d

1 -3+2d

47 -3+2d

~z/d)

20 (I - 3d) 3 +2d

S (I - 2d) 3 +2d

-3 +2d

10

470 -3 +2d

16 + 89d

~

2 (I + 14d) 3 + 2d

0 => d

~

-16/89,

from which d ~ 1/6 and d ~ -1/14 follow. Since -1/14 > -0.25 , we have

I

D(I)

1

= [-14 ' 6]·

With d l = -1/14, we obtain ßzs(dl) = 0, that provides the pivot column. We now have to find out whether at least one element of the vector yS(d) is positive. With d l -1/14,

=

Yls(dl) = 2 > 0, Y2s(d l ) = -14/5< 0, holds. The pivot element marked in Table 8-5 is, thereby, uniquely determined. After one primal step, we obain Table 8-6. In Table 8-6, the "basic condition" is determined by Y2(d) and ßZ6(d):

3 + 2d > 0 => d > -1.5. Further, we have (briefty) 16 1 1 1 d > -- d < - d < -- d < - 89' - 3' - 14' - 2' which implies 16

I

--~d~--.

89 14 Since, with d2 = - 16/89, we have YI (d2) = 0, we must check whether in the first row there exists at least one negative element setting d d2 . We have

=

YI2(d2) > 0, YI3(d 2 ) > 0, YI4(d2) < 0, YI6(d2) < O. Substituting d = d 2 into ßZ4(d) and into ßZ6(d), the pivot element marked in Table 8-6 is determined according to the dual rules. After one pivot step, we obtain Table 8-7.

Sensitivity analysis with respect to the elements of the technological matrix A Table 8-7

Solution associated with 8 3

P3

2

-74

----

3+ 2d 2 (I -d)

I +4d I-d

----

---

5

2

-

I-d

I-d

I -6d

5 (I - 2d)

I-d

2 (I -d)

--

2 (I -d)

I

-

I-d

5 (4-9d)

ßz/d)

5

3 4+21d 2 (I -d)

1

Table 8-8

315

6

xs(d)

1/2

----

0

16+89d 2 (I -d)

-

21

I-d

5 (68 - 173d)

5/2

2 (I -d)

Initial tableau with do =-5 1

4 -7p

b

3

1

-5

d

0

2

1*

-4

I+d

-I

1

4

-I-d

I

-I

-I

-p

2'

2

If we perfonn an analysis in Table 8-7 similar to that carried out for the previous tableaux, -00

- I

3

*

-3-2d

xB I

I

-I

I

2

-2

PI

I

3

xB(d)

f-4

-

I I+d

-

d * I+d

-

~2

-

I I+d

-

-I I+d

-

~z/d)

--

3 +2d I+d

-

-I I+d

-

2+d I+d I I+d I I+d

here which can influence the optimality and that is optimality, ~z2(d)

~z2(d).

In order to mantain

;;:: 0

must hold, wh ich implies d

~

-1.5.

Since -5 ~ d ~ 5 is, generally, valid for d, 0(0)

0(0)

becomes

= [-5,-1.5].

Now try to pass to a neighboring basis with d l = -1.5. Since ~z2(dl) = 0, this must be done by a primal step. For this, at least one positive element must be available in the second column. Since Yl2 = -I, it is questionable whether 1 + d remains positive. However, we have I + dl = 1 - 1.5 = -0.5, so that there is no positive element for the primal simplex step in column 2'. We, therefore, have to ask whether such a d exists at all, such that 1 + d > O. Obviously, all d > -1 are such. Of course, for all d E (-1.5, -I) the basis Bo is no longer dual feasible . But, with d > -1, we could pass to another basis and then examine it as to its optimality. This new basis is given in Table 8-10. First, examine the solution in Table 8-10 as to primal feasibility for d > -I. Set d =-I + €, € > 0 real number, and ask whether € > 0 exists such that y I (d(€» ;;:: 0 and Y2(d(€» ~ 0, i.e.,

Sensitivity analysis with respect to the elements of the technological matrix A Table 8-11

317

Solution for d > 0

4

I

PI ~3

I

I+d

d

d

I

2 Llz/d)

xB(d)

d

I d

1+ 2d d

d

2+d d 2

d 2 d

I

5

-s r-----

d

-2

Figure 8-1

2+(-1+10) I - - - - 2! 0, 2! 0, 1+(-1 +10) 1+(-1 +10) has to hold. By assumption, 10 > 0, so that both conditions are obviously met. From that, it follows that for 10 > 0 arbitrarily smalI, i.e., for d 2 > -I, the solution is actually primal feasible. Now, check the dual feasibility. Since ~z3(d) = -1/(1 + d) < 0 for all d > -I, it follows that the solution associated with plis not optimal. Now substitute d = -I + 10 into ~Zl (d), and we have (I + 210)/10 2! 0, and, since 10 > 0, we must have 10 2!0.5. This, again, implies 10 > 0 sufficiently smalI, and d > -I folIows. This analysis implies that the "critical" element is ~z3(d), since ~z3(d) < 0 for all d > -I. In order to pass to a new basis, a primal step must be applied; hence, we must have either Y13(d) = d/(I + d) ~ 0 or Y23(d) = - I /(1 + d) 2! 0 for d > -I. Since -I/(I+d) < 0 for all d> -I, substitute d = -1 + 10 into Y13(d). Then (-I + 10)/10 2! 0; since 10 > 0, we must have -I + 10 2! 0 =} 10 2! land d 2! O. With this pivot element, one primal step yields Table 8-1 I.

318

Sensitivity analysis with respect to the elements of the technological matrix A

Since d > 0, we must have, simultaneously, 1 + 2d ~ 0 and 2 + d ~ 0, i.e., d -0.5, d ~ -2, i.e., d > O. For all d > 0 and d :::; 5, the solution associated with P2 is, therefore, optimal. From Fig. 8-1, it is apparent that the objective function dependent on d (i.e. the optimal value function) is not continuous over the interval [-5,5]. ~

VIII Abridged mathematical presentation

The problem discussed here can be formulated as folIows: determine a region K of the parameter d k such that, for every d k E K, the problem of maximizing (VIII-I) subject to (A + A *D)x = b, x ~ 0,

(VIII-2)

has a finite optimal solution and, for any d k ri. K, the problem (VIII-I), (VIII-2) has no optimal solution. Here, A * is a constant (m, s) matrix, D is an (s, N) "parameter matrix", and d k are the "elements" of the matrix D. The simplest and most familiar case (cf., for example, [3]) is that of A * == a~r' D == d, i.e., only one element akr of the matrix A changes. Further, the matrix A * can contain only one nonnull column or row and D becomes a corresponding vector or even a scalar, i.e., it is a quest ion of changing a column or a row of the matrix A. If we view the analysis of the variations from the point of view of the columns ai of matrix A, then the simplest case is that in which the elements of a nonbasic vector change. If several of the elements aij to be changed belong to basic vectors, the question arises as to the efficient determination of a new inverse. A corresponding formula has been derived by Sherman and Morrison [13) . Gass [11) found the same formula in another way. Bodewig [2] and Egervary [5) report on a formula for a general case. Concerning more references, cf. the bibliography at the end of this book. Here, we use Bodewig's formula (2) for the determination of an inverse to a changed matrix B: (B + pqTrl = B- 1 -

where p, q E IR m are vectors.

B- 1 TB-I pq 1+ qTB-1p'

(VIII-3)

320

Abridged mathematical presentation

VIII-l Changing a column of matrix A Let

,qT

p=

= (0, ... ,0,1,0, ... ,0),

(VIII-4)

s

2: p~ld~

1=1

where d~ are elements of a vector parameter and p~ constant coefficients. Let 0,

°

0,

°

s

0, ... 0,

2: p~ld~,

1=1

V'= s

0, ... 0,

2: p~ld~,

1=1

The superscript k indicates that we are dealing with variations of the jk th column of matrix A. We can, then, write (VIII-5)

Suppose thatjkE pis fixed and the kth cohimn ofmatrix B corresponds to the jkth column of matrix A. In accordance with Sec. I, the elements of the inverse B- 1 are denoted by ßij' i, j, = I, ... , m. The expression 1 + q TB-I P from Bodewig's formula then becomes2

2:1 p~ld~ 1 + qTB-I P = I + (0, .. ., 0, 1,0, ... , 0)B- 1

(VIII-6)

s

2 For the sake of simplicity, we write only

2: instead of 2:. In the same way, 2: stands 1=1

.S

for

2:. ;=1

Changing a column of matrix A

321

(VIII-7)

so that, for an arbitrary element ßuv (d k) of matrix B- I(d k), ßkv I: I: ßuiP~d~ ~ k k' d k E IR" I+ ßkiPi(d(

i:

ßuv(d k) = ßuv -

i

holds. If d k E IR s becomes d

Cd =

(

~'

:::

(

IR and V becomes

E

~' PI.k. ~'

. ... .

. .. ..

(VIII-8)

. .

:::

~) d

. ....

.. .. .

0, .. . 0, Pmk 0, ... 0 i.e., p

=d (PIk. ... , Pmk)T, then (VIII-8) becomes dßkv I: ßuiPik ßuAd) = ßuv -

~ ,d 1+ d . ßkiPik

E

(VIII-9)

IR.

Using (VIII-8) and (VIII-9), all elements appearing in the simplex tableau can be represented as functions of d k. Since, as will be shown immediately, these derivations are, in fact, exercises in algebra, we leave them to the reader. 3 For example, from xB(d k) = B-I(dk)b

(VIII-IO)

it follows that (VIII-lI) or yu(d)

=Yu -

dYk I: ßuiPik ~ ,d I + d . ßkiPik

E

(VIII-I 2)

IR.

Similarly, from 3 In the previous editions, the author has not been lazy enough in algebra.

10

refuse these exercises

322

Abridged mathematical presentation

(VIII-l 3) from (VIII-14) from uT(d k) = C~B-I(dk),

(VIII-15)

and, finally, from z(p)

max

(d k)

=cTB-1(dk)b B '

(VIII-16)

the detailed, elementwise expressions can be easily derived.

VIII-2 Definition of a critical region Suppose that d = do exists, such that the problem: maximize Z

= cTx

subject to

=b, x ~ 0,

A( d)x

(VIII-I 7)

with A(d) = A + Cd

has a finite optimal solution associated with B. The conditions yu(d)

~

L1zj(d)

I,

(VIII-l 8)

0, for all j fi p

(VIII-19)

0, for all u ~

E

are necessary and sufficient to maintain the optimality of B. Note In the optimum, negative values can also appear among the dual variables We further suppose that the nonnegativity condition only applies to those elements Ui of the dual solution u T wh ich are to be non negative on the basis of the original constraints. These elements are then collected in the nonbasic variables with the subscripts j fi p. Let d E IR and denote Ui .

L ßkiPik = V, L ßuiPik = W, LYiPik = Z. From (VIlI-9) it is obvious that4 4 With I + dV = 0, the inverse B-1(d) would become a singular matrix.

(VIII-20)

323

Definition of a critical region

(VIII-21)

I +dV *0. For u =k, there folIows, from (VIII-I 2), Yk

( d)=~ 1+ dV'

(VIII-22)

The relation (VIII-22) is called the basic condition because the sign of Yk determines whether I + dV > 0 or I + dV < O. The basic condition will only be considered at the end of the following analysis. Case I Suppose Yu + d(yu V - Yk W) ~Zj

~

+ d(~zj V - ykjZ)

0, U ~

E

(VIII-23)

I,

(VIII-24)

O,j 4. p,

i.e., d(yu V - Yk W) d(~zj V

~

- ykjZ)

-Yu,

(VIII-25)

~ -~Zj.

(VlII-26)

Define the index-set 1+ c I, such that Yu V - Yk W > 0 holds for all u

E

1+.

Denote by u+ all u E 1+ . Define the index-set r c I, such that Yu V - Yk W < 0 holds for all u E r. Denote by u- all u Er. Analogously (and briefly), 0

([g(P!,o(P)I"0.

(Jg*(P'. o*(P)1 ,,0.

[g(P!, o(P'] =O) o(P)

V>O

VO

- ~ < g(P) v

-~ > o(P) v

V=O

[g(P). cfP)]

g(P) ::; _~ < o(p)

-

-

(_~. o(P>]

-

[d(P) _~)

v

g(P) < _~ ::; o(P)

-

v

V -~)

::; o*(P) -

-

• v

set d(€) = d* + €, € > 0 smalI, then we would not obtain a finite maximum. The question then arises as to whether € > 0 exists such that Yrld' + €) ~ 0 for at least one j ri. p, i.e., whether € > 0 exists such that (d*+€)ykjW * Yrj(d + €) = Yrj 1 + (d' + €)V

~

0

(VIII-31)

for at least one j ri. p. From (VIII -31) it follows that B r

*

*

'

Yrj + Yrjd V - d Ykj W +€(Yrj V - Ykj W) , 1 + d V +V€

~

0,

'--v---" A

i.e., B + €(YrjV - YkjW) ~ O. A+V€ From the assumption 1 + dV > 0, v > 0, it immediately follows that A + V€ > 0

~

€(ykjW - YrjV)

~

B.

Suppose that for j = t, tri. P constant, (VIII-32) is fulfilled and YktW - YrtV > O.

(VIII-32)

327

Changing a row of matrix A

Then, B

f~----­

(VIII-33)

Ykt W - Yrt V

With the pivot element Yrt(d* + E*), where f* satisfies the relation (VIII-33) as equation, it is possible to pass to a new simplex tableau in one dual step. This new tableau need not necessarily contain an optimal solution for d = d* + f*. It is, however, in accordance with the results from Sec. VIII-2, to determine d > d* + E*, so that the new tableau will contain an optimal solution, or to determine d > d* + E*, so that it will be possible to pass to a new optimal solution, or, finally, to find that for d > d* + E* there is no optimal solution at all. Let Q( I) > diP) be the lower endpoint of D(\) associated with the optimal basis BI . Then the problem is not defined for (Q(I), alP»~, and the functions x/d), zmax(d) become discontinuous functions. The analysis of the remaining possibilities for I + dV, V, YktW - YrtV, etc., would, in principle, lead to the same result, and a detailed account of these cases would be nothing more than an exercise in algebra on the part of the author. Moreover, for the case in wich the value ~zl(d*) = 0 ( for I 4. p fixed) results from substituting d* = dip) into ~zl(d), the analysis is, once again, analogous. In this case, it is a question of determining an E> 0 such that Yil(d* + f) > 0 for at least One i E I, since for d =d*, Yil(d*) ::; 0 for all i

E

I

(for the structure of the total region for d, cf. [1] ).

VIII-4 Changing a row of matrix A Let a change of matrix A be expressed by matrix 0,

0,

o

(VIII-34)

0,

0,

o

i.e., (VIIl-35)

328

Abridged mathematical presentation

where "r" means the subscript of the rth row (I ::; r::; m). Suppose that n is the number of real variables in the original system of constraints (before slacks and artificial variables were introduced). Then, in a , q~,n+ 1

= ... = q~N = 0,

for all t

= I, ... , s,

(VIII-36)

holds. The vectors p, q of Bodewig's formula (VIII-3) are as folIows : ~

T

P = (0, ... ,0, 1, 0, .. ., 0) , "

'-

m

(VIII-37) LetJ = {j I j = I , ... , n, n + I , ... , N } be the set of all subscripts, p ={j I, ... , jm} the basic-index, and, finally,