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English Pages 462 [464] Year 2010
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, ProfessorEmeritus 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 McGrawHill International Book Company, 1979. @> Printed on acidfree paper which falls within the guidelines of the ANSI to ensure permanence and durability.
Library of CongressinPublication 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 3110140608 (acidfree paper) 1. Decisionmaking. 2. Linear programming. I. Title. T57.95.G3413 1995 9412443 658.4'033  dc20 CIP
Die Deutsche Bibliothek  Cata!oginginPublication 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 3110140608
© Copyright
1994 by Walter de Gruyter & Co., D10785 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 uptodate 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 108 (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 uptodate. 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 righthand 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 righthand 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 settingup 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 timeconsuming 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
ndimensional 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
righthand side of the constraints (Ax bE IR m
=b),
XVIII
List of symbols
p = {jl, ... ,jm} ... , ~mk)T. Let the linearly independent nonnull vectors vj E IR m, j = I, ... , m, and the nonnull 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 (I2a) can be transformed from the initial basis into basis 8; namely (19)
where .
T
yJ = (Ylj' ... , Ymj) .
(110)
Furthermore, 8 l b =
XB,
(I 11)
where
(112) and (113)
where Y = (y I , ... , y N ),
(114)
and see the "Notation note" at the end of Sec. 11. Definition 19 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 indexset 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 II51.
1122 An algorithm for determining redundant constraints Consider the problem min Si, xeX
i
= I, ... , m,
(1126)
and suppose that X :f:; 0 ( ~ X :f:; 0 ). Let x~O) E X be an optimal solution to (II26) 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 (1126) 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 signrestricted, the initial solution as weil as each tableau is generated by a modified simplex procedure (cf. Sec. IV4). If the variables are signrestricted, 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 (1124). Denote by I' 2 ~ 12 the indexset 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, (1127)
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.
113 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 117 Avertex XO E Xis said to be a crdegenerate 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 ,
(1128)
with Yi = 0 for i = I, ... , cr } Yi>O for i=cr+l , ... ,m,
(1129)
holds; the indices of the variables are rearranged such that ji = i for all i = I, ... , m+n.
Definition 118 Let XO
E
X be a crdegenerate vertex. Then the set
BO={B~lu=I, ... ,U},
U~I,
(1130)
which is assigned to xO, is called the basisset 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 )
(1131)
(compare, for some values of cr and n, Table 214). 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 basisexchange are:
16 The needed notation is in Sec. I
74 " f
Abridged mathematical presentation
+ 1"
using a positive pivotelement (positive pivotstep for short) using a negative pivotelement (negative pivotstep for short) using any nonzero pivotelement.
"f  1" "f1"
Definition /19 The representation graph G(X) of a polytope X is the (undirected) graph
(1132)
G(X) := G = (V, E), where
= {B IBis a feasible basis of (17)} } E = {{B, B'} s;;; V I B f + 1 B'}
V
(1133)
The (bases) structure of a odegenerate vertex XO can be studied using Definition /110 Let XO E Xc IRn be a odegenerate vertex. Then the (undirected) graph
(1134) where BO
E~
is given by (1130)
= {{B~,B~, } s;;; BOIB~ f + 1 B~} u,u'
E
} (1135)
{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 f1, 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 indexset 1= {i I i = I, ... , m } into:
= {ilYi = O},
(1136)
IN = {ilYi >O}.
(1137)
10
If, in tableau
T~,
there exists a nonbasic column "t" such that
V Yit ~ 0 ie!"
and
:3 Yit > 0
(1138)
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 odegenerate 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 (17) I
75
:1
B~ f
+ ~ B}
(1139)
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 (1140)
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
(1141)
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 socalled isolated node in the corresponding positive, negative or generalOG, respectively. A frequently occuring problem in mathematical programming is the socalled neighborhood problem (Nproblem for short). This is to determine all neighboring vertices of a crdegenerate 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 TransitionNodePivoting rule (TNPrule 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 Ncorrespondence. The socalled Ncondition 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 Ncorrespondence is connected, then it satisfies the Ncondition. 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 (1142) yields a set of indices
(Il42)
,c to a crdegenerate XO the feasibility criterion
76
Abridged mathematical presentation
{i
I
O ßik 00, if there does not exist
Yi _ max() Ak = { i.~;, 0
(311) 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 (310) and (311), let us now determine the critical interval for A3 . In order to proceed systematically, we shall first go back to Table 12. 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 12) 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 (312) If we consider Table 12, 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 (III39) 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 12), (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. 442 and IV72.
88
Sensitivity analysis with respect to b
Table 31
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 f5
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 31 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 righthand 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.
311 Geometrie meaning of a single parameter Let us consider the first condition from Exs. II and 31 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 halfplane (I) is in the original position for AI = 0, as shown in Fig. II . For
Sensitivity analysis with respect to a single component bj of the right hand side b
""
""
89
"
Ai" 8.6
Figure 31
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. 31. From Section 24 we know that in point P of Fig. II, 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. 31), 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. 31 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 31. 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 (314) 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 (314). The "narrower" the interval Ak is, the more sensitive will be the optimal solution to the change in bk ("k) in the sense of (314). This means that, even with relatively small changes of bk> according to (314), 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 (314). 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 (314). However, in the most of commercial software to solving linear programming problems, there is a device, called RHSranging; this means sensitivity analysis in terms of (314).
Sensitivity analysis with respect to several components
91
32 Sensitivity analysis with respect to several components of the righthand side depending on a scalar parameter Suppose that in Ex. II or Ex. 31 the eomponents of the righthand side bare dependent on a seal ar parameter as folIows: b,(A) = 130.5A b2(A) = 20 (independent of A) b3(A) = 121 + 8A, b4 (A) = 8 + 0.6A. As we already know, the original problem given by (II) through (13) has a finite optimal solution. This was shown in Table 12. 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) = B1b(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 ~ ( 21.53A 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 righthand side. In this extended sense, sensitivity analysis may be understood as the investigation of the influence of changes to the righthand 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
33 Multiparametric sensitivity analysis: Changing several components of the righthand 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 righthand side, as in this chapter) bring about changes not only to one component in the righthand 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 righthand 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 vectormatrix form
b(>") = b + FA, where
F=
(
fll,
fIS)
:
.
:
f ml ,
. •• ,
f ms
,>..
=(/"1, ... ,As)T.
Example 32 For the sake of simplicity and to facilitate geometrical representation, let us consider the case of a twodimensional vector parameter>.. = (AI, A2)T. In the problem of Ex. 11, let
be given. Then,
94
Sensitivity analysis with respect to b
Figure 32
(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. 32. 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. 32. 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 signrestricted, i.e., they can also take on negative values. In Table 32 the calculation of the minimum of the function f(A) is performed. In the second step, point P (cf. Fig. 32) 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. 32, is achieved in the third step. Point M in Fig. 32 has also been calculated (fourth step in Table 32). 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. 32 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 32
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 fSI 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
1111 General considerations Suppose that the problem: maximize z = cTx
(1111)
Ax = b()')
(III2)
x~
(III3)
subject to
0,
where b()') = b + F).,
(III4)
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 (1111) through (1113) 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 (III2) after the slacks have been introduced were inequalities of the type ~ . 2. The subscripts jE] be rearranged such that p = {I, ... , m},
..
(III12)
The max z~~x().) or min z~~x().) overthe set (III7) can be determined by means ).EA
).EA
of a modified simplex algorithm (cf. [7] and Sec. IV41). At the same time, the "objective function" s
f()') = z~~x().)  z~~x =
L fm+l ,kAk
(1II13)
k=1 is introduced and the optimum determined subject to (1II7).
1112 Special cases Case A Let
F = I, where I is an (m, m) identity matrix . Condition (III2) then takes on the form
Ax = b + J).
(III14)
or N
LaijXj=bi+Ai,i= I, ... ,m. j=1
(III15)
The critical region is defined by
B 1).
::; XB
(III16)
or m
L k=1
ßikAk ::; Yi, i = I, ... , m,
(11117)
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,
(11118)
um ) .
(III19)
k=1
where UT
= eTBI = (u I, B
.. . ,
Case B Let
F
= f,
i.e. , the matrix F "shrinks" to a veetor f.8 Constraints (III2) then take on the form (III20)
Ax = b+fA, or N
LaijXj=bi+fiA,i= I, ... ,m,
(III21)
j=1
where A E IR is a seal ar parameter. The eritieal region A is defined by B1fA ~ XB PfA ~ XB,
(III22)
PfiA ~ Yi, i = I, ... , m,
(III23)
or
where Pf = (Pf l , ... , Pfm ) T.
Theorem 1112 Suppose that eonstraints (IIl2) of problem (1111) through (1114) take on the form (11121) 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: (III24) where ~
=
{ max { Yj} JEI,
I1 = 0 _ A=
Pfj
~ ~
=00
{ . { Yj}
(III25)
mmJEI,
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 (1111) 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 (IIII) 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., (III24). 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.
(III26)
Suppose that, for example, ~ =  Yr fr and set A.*=A.. Then,
xJ",(I\.,* )
Yr) = 0 . = Yr + Pfr(Pf r
(III27)
104
Abridged mathematical presentation
=
It obviously suffices to set A A* + E, E > 0 sufficiently smalI, in order to violate the constraints (III23). 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 (III16) and (III17), for the scalar parameter A, (p) (1) (p) f ' Zmax 11.  Zmax + m+lll.,
(III28)
where
(III29) Theorem III3 With regard to (III28) and using the notation (III29), 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;
(III30)
(ii) for f m+1 < 0,
z~~x + fm+1X ~ Z~~x(A) ~ z~~x + fm+l~.
(III31)
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 (III28), for f m+1 = 0, obviously, Z~~X 0, then (III28) 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
(III32)
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.
(III33)
j=1 The critical region A k is then defined by the inequalities B1ekAk ::;; XB,
(III34)
i.e., ßikAk ::;; Yi, i = I, ... , m, k
fixed.
(III35)
From B1b(Ak) = B1(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
(III36) Theorem IIl4 Assume that the constraints of the problem (III I) through (III 4) take on the fonn (III32) 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
=
=
=
(III37) where ~k
Yi}  = mm . { Yi} = maxI, {ßik  ,Ak I, ßik IE
(III38)
IE
holds. If 11 = 0 or 12 = 0, set ~k = 00 or Xk = +00. The proof is analogous to that of Theorem III2. In this special case it is obvious that (p) ('
zmax
) 
II.k 
(p)
zmax
, + Ukll.k·
(III39)
1113 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 IIl5 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 (III40)
2s :Laj = l,aj;;::O all J, j=' is a subset of the critical region A of the vector parameter>.. in the problem (IIII) through (III4). The set e is called the approximation region. The proof is analogous to that of Theorem 114.
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., AddisonWesley, 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 NorthHolland Publ. Co., Amsterdam 1975
Chapter four 4 41
42
421 422 423 424 43 431 432 44 441 442 IV IVl IV2 IV3 IV31 IV4 IV41 IV42 IV43 IV5 IV51 IV6 IV61 IV62 IV7 IV71 IV72
Linear parametric programming with respect to b Changing the righthand side with basis exchange Linear scalar parametric programming Changing a single component of the righthand side with basis exchange Vector parametric linear programming with respect to the righthand side Changing several components of the righthand 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 signunrestricted 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 righthand side with basis exchange
In Sec. 31, 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.
41 Linear scalar parametric programming Changing a single component of the righthand side with basis exchange In Sec. 31, Ex. 31, 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. 31, 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 31 ).
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 basicindex p. Analogously with ~~),X~). Example 4 J Consider the following problem. In Ex. 31, 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. 31, 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 12 (Table 41), 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 41 with respect to BI, we could assume at first sight that XB is not feasible (the fourth component is negative). According to (III36) or (312), 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 (311) or (III38) 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 41 .
Linear scalar parametrie programming Table 41
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 42 shows the initial tableau and passing to basis B 2 . According to Table 42,
'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 42
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
Qf3
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 427, 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 halfplanes (with common boundary lines) are involved. In Fig. 47, the regions A (P), P = 0, 1, 2,3, are shown. Point (I), (11), (III) are vertices, which were computed in Tables 421 through 423. In Fig. 47, 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 righthand side
Example 46 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 428 through 431 , the procedure for the computation of a feasible solution (X O, ~ 0) is shown. From Table 431, 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 . 42) we obtain a " new" problem, the initial tableau of which is shown in Table 432. 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 433. 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 428
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
fPI
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 429 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 433 , 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 434 through 439.
145
Vector parametrie linear programming with respect to the righthand side Table 430
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 431
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. 22 and 112. 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 437, 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 432
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 433
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 439, the second row could be deleted (min S3 = 8 > 0). From the computations given in Tables 434 through 439, 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 433 into account, we obtain Table 440. According to Table 440, 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 440, 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 441. 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 righthand side Table 434 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 435
sI
0
2
I
2
~AI
I
I
()
2
s,
I
2
I
6
s4
I
I
2
18
fSs
I
1*
I
14
'11
I
2
0
8
112
I
2
I
18
424 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 47 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. 23 and 113.
148
Linear parametrie programming with respect to b
Table 436 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 437
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 442). This solution is primal degenerate. As we know from Sec. 23 and II3, it is possible to assign several different basicindices 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 righthand side
149
Table 438 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 439 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 442, vertex upon X as folIows : xO(X) = (4
XO
depends
+ 0.51..1  1..2, O)T .
The elements bj of the righthand 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 righthand 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 440
PI 3 f4
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 441
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 442 ()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 442 and solve the auxiliary problem:
Vector parametrie linear programming with respect to the righthand side
151
Table 443 AI
A2
OSI
3/2
2
f S2
1/2
Os,
1/2
2
AI
S2
fSI
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 444
Table 445
From Table 445, it follows that min SI = 0 and SI NBV, s3 can be eliminated; thus, min S3 =0 and s3 NBY. With s3 = 0, the basicindex Po I = {I, 2, 3} results, which, from the point of view of the initial set X, is abasieindex assigned to the same vertex x O • However, from S3 = 0 , it follows that to A (0) the condition (48) is a binding constraint (cf. Sec. 22 or Sec. 112). Denoting A (01) the region adjacent to A (0), the condition (49) is a binding constraint. Hence, (48) and (49) define two opposite halfspaces (halfplanes) and A(O), A(ol) are two nonoverlapping critical regions with the common supporting hyperplane (halfplane)
152
Linear parametric programming with respect to b
Table 446
_ "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. 48. 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 48 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 446 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 447). In Table 447, 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. 49. Now, choose Yl2 = 5 as pivot element in Table 446. A dual pivot step leads to Table 448.
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 righthand side
153
Table 447
_ °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 49
Table 448
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 449 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 basicindex P2 results; with Y21, a new basicindex P3 = {2, 3} results (Table 449). 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. 410. Consider Table 447 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 410
Figure 411
Figure 412
fulfilled). Performing the corresponding primal simplex step, Table 448 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 IV3). The whole possihle graph is in Fig. 411 . 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. 42 and Fig. 47). The corresponding graph (Fig. 412) is not connected. Example 49 Maximize
subject to XI + 3X2 ~ 9 + 2AI + A2, 2xI + X2 ~ 8+ AI2A2,
155
Homogeneous multiparametrie linear programming
G:
GOI:
~
GI":
Figure 413
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 .
43 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 righthand 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,
431
Problem (F)
Problem (F) is solved in basically the same way as the case with a vector parameter in the righthand side described in Sec. 42. 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. IV6). 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 450
Initial simplex tableau
I
2
P
I
f4
2
p
I
Table 451
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
fp
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 450), the matrix F remains and the righthand 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 450, we obtain Table 451. According to Table 451 , 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 452) is determined as folIows: 3
X2 = LPfik/...~'/...~ = I,k = 1,3. k=1
In Table 452, 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 453) and the ordinary LP solved. The optimal solution is given in Table 454. 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 452
~A2
2
Table 453
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 454
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 455
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 455. 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. IV6): 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 456 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
fp
1
1
1
1
1*
Table 457
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
fS 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 456 through 458 show the corresponding computations. Since there are negative elements in Table 455 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 457. 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 458, 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 458
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 459
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 459. 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 460
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
fp
1*
I
I
I
I
Table 461 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 462 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 460 through 462. From Table 461, 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 462, 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.
432 Problem (FD) Problem (Fo) differs from problem (F) in two respects. First,
FOA = (
bl:AI ) . bmA.m
If we assurne that the righthand 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 settingup of a linear model, the elements bi of the righthand 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 412 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 463. According to Table 463, Po = {3, 4}, A (0) is defined by 30A.)+16A.2~0,
(I)
(2) and
Linear parametrie programming with respect to b
164 Table 463
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 464
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 (410)
Homogeneous multiparametrie linear programming
165
(5)
Figure 414
Table 465
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 (410) is also nonempty (formally, cf. Sec. IV6). Thus, in our example, the neighboring regions are determined as folIows: (a) In Table 463 , 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 464) and an artificial variable p is added to the latter for the calculations. In Fig. 414, the results are presented geometrically. From (i) (cf. Table 464) 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 basicindices 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 465 results.
Linear parametrie programming with respect to b
166 Table 466
A,
A2
Os,
150
80
0
s2
50
0
0
fp
I
1*
I
Table 467
A, 230*
80
s2
50
0
~A2
I
I
fS,
Table 468
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)=300AI288A2,
AE
A(I).
The auxiliary problem is solved in Tables 466 through 468. Since, in Table 465, 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 463, we obtain Table 469.
Sensitivity analysis and shadow prices under (prima!) degeneracy Table 469
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 469, 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.
44 Sensitivity analysis and shadow prices under (primal) degeneracy 17 441
Sensitivity analysis
As has been pointed out in Sec. 31, sensitivity analysis (with respect to the RHS b) in terms of (314) 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. 23 or 113 that, with a (Jdegenerate 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 470
< < < < < < < <
: 2>: 3>: 4>: 5>: 6>: 7>: 8>: 9>:
Numbering of the bases 3456 1356 2456 1245* 1246 1256* 1235 1236* 1234*
24 is caused excJusively by weakly redundant constraints, we use this polytope because of its simplicity. 18 Example 413 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 crdegenerate optimal vertex xo. It holds:
B(,c) ~ BO and in the most cases B(c)is a proper subset of BO • In Table 470 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 470 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. 113, Theorems 116 through 118), 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 471 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 471 Determination of Ai, i = I, . .. ,4, continued Way: < 6>:< End 1 4[ 2.00, P+I PI P+2 P2 P+3 P3 P+4 P4
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 twosided 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. IV72) that the twosided shadow priees are closely related to sensitivity analysis with respeet to the RHS in terms of (314): 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 . 413 . Choose Table ; here we have: AI
E
11O,0),A2
E
15,01,11.)
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 12, 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 (IVI) subject to Ax = b + F>.,
(lV2)
x
(lV3)
~
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 (IV4)
b(>') = b + F>.. The vector parameter should satisfy the additional constraints G>.
~
d
(lV5)
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 basicindex . 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 IV3. Definition lV4 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.
(lV27)
Theorem lV5 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
(lV28)
'
(IV29) 25
2yx = 2F>.. + xB2.
Let Yrt < 0 be the pivot element for passing from BI to B2. The rth row of the system (lV29) is obtained by dividing the rth row in (lV28) by Yrt. Hence, 2 yr (>")
= I~r(>") Yrt
for aIl
>..
IR' .
E
(lV30)
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 halfspaces in IR' with the common hyperplane 2 yr (>")
s
L
IfrkAk = Iyr.
(iii)
k=1 QED. Corollary IV51 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 (IV28) and (lV29) 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 (IV28) 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 IV3 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 IV52 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 (IVI) through (lV3).
Definition IV6 An undirected graph G problem (IVI) through (lV3) 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 (IVI) through (lV3), 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 IV7 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 IV6 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: (IV31)
Obviously, ).(t) E K for all t E [0, 11, since K is convex. Substituting ). = ).(t) into (IVI) through (lV3) 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 (lV2), 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. 43 and 48). However, from Definition IV6 and Theorem IV6, it follows that generating nonoverlapping, neighboring regions, the corresponding subgraph Go (So, r o) is connected. Taking into account Theorems IV2, IV4, and IV5 and Definitions IV3 and IV4 as weil, the following assertion obviously holds .
°
=
Theorem IV7 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 (IV73), 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,
(IV75)
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, (lV76) 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 IV8 Assurne that a solution of the system (IV71) 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 (lV71 ).
The second master problem is to determine all adjacent nodes to anode pES. The critical region A (p), defined by BIF'" < " < .  I , ... , m, _ 0  "Pf· ~ ,kAk _ 0 , 1k=1
(IV77)
forms a convex (polyhedral) cone in IR s . To determine the set Pp (cf. Step 3°°), the auxiliary problem (lV43) has to be solved. In our case, this problem has the form min Si iE P"
(lV78)
Homogeneous multiparametrie linear programming
197
subject to
L PfikAk + si = 0, Si 20, i = I, ... , m.
(lV79)
k=1
In order to ascertain for which of the Si 's the condition Urj ::; 0 \ij from Redundancy Criterion 2 (cf. Corollary II51) 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 IV9 Let w > E
(IV80)
k=1
Then, für every w > 0, A(P) 7: 0
Proo!
"~"
"~"
~
A(P)
(1
W 7: 0.
(IV81)
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 IV9 the following problem is established: minsi iEP"
(lV82)
L PfikAk + Si = 0, i = I, ... , m,
(IV83)
subject to
k=1
(IV84)
198
Abridged mathematical presentation
Theorem IVIO
34
Consider the problem (lV82) subject to s
L PfikA~ + L PfikAi;" + Si = 0, i = I, ... , m, k=1
LA~+ LAi;" k=1
(lV85)
k=1
(IV86)
= I,
k=1
A~ . Ai;" = 0, k = I, .. . , s,
(IV87)
= I, .. ., S, Si
A~ ~ 0, Ai;" ~ 0, k
~ 0, i
= I, ... , m.
(IV88)
Let min s, = s;) ~ 0, s, BV, t E Pp fixed, be an optimal solution to (IV82), (lV85) through (lV88). Then s7 is an optimal solution to (IV82) through (lV84).
Proof Define A~
=0 A~ " Ai;",
k
= I, ... , s.
(IV89)
Evidently, for each k E {I, ... , s },
°A~"Ai;"
=0 ~
s
s
k=1
k=1
L I Ak I = L °A~ + L "Ai;".
(lV90)
k=1
With (IV90), the relations (lV86) and (lV87) imply (lV84). Setting (IV89) into (IV85), we obtain (lV83). QED.
Note To determine the neighboring regions to A (p) 82), (IV85) through (IV88) has to be solved.
,
the auxiliary problem (lV
We are now ready to present the procedure for solving (lV68) through (IV70). 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~ =
(lV91)
0,
x ~ 0;
(lV92)
(b) determine the optimal solution of the problem of maximizing z = c T X,
(IV93)
subject to
Ax
=F~(\ ~o #
x ~ o.
0
constant,
(IV94) (IV95)
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 (IV91) with (lV92), we proceed as folIows .
I. Setting Uk
= I for all k in Theorem IVI 0, we have A; = .. . = A: = I ;briefty,
},,* = (1, ... , I)T.
2. Determine a feasible solution of the system Ax
= F}"*,x:2' o.
(IV96)
3. If no feasible solution of (IV96) 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. (IV76». 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 (IV9I), (lV92). 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 (IV68) through (IV70) and maximize (IV97) subject to Ax
= F}"° , }"
x:2' o.
0
7= 0
constant,
(IV98) (IV99)
The optimal basicindex Po computed through the solution of (lV97) through (IV99) (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 (IV97) through (lV99) 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 (IV82), (IV85) through (IV88) is solved.
IV62 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
(lVIOO) subject 10 Ax = F D }",
(lVIOI)
200
Abridged mathematical presentation
x ~ O,A
~ 0,
(lVI02)
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,
(lV103)
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
(IVI04)
FOA = bmAm
*
If we assume that the righthand 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 IV7. Phase I of the solution procedure for problem (Fo) is the same as that for problem (F), i.e., (lV68) through (IV70). In Phase 2, the following problems are solved as auxiliary problems:
°
° }.
(IV82)
minsi ieP,
subject to m
L PfikAk + Si = 0, i = I, .. ., m,
(IVlOS)
k=1 m
LAk
= I,
(IVI06)
k=1
Ak
~
0, k = I, ... , m, Si
~
0, i = I, ... , m.
(IVI07)
The justification for extending (IVlOS) by (IVI06) follows immediately from Lemma IV9 as a specialization of the statement to be found there. The corresponding formulation is given in Lemma IVlI 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 IVI 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 IV9.
IV7 Sensitivity analysis and shadow prices under degeneracy36 IV71 Sensitivityanalysis Consider maxz
=cTx
(IVI 08)
subjeet to Ax
= b,x ~ 0
(IVI09)
and suppose that the optimal vertex XO is crdegenerate. To Xo the set of bases BO (see (1130» 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 (oDG for short), 3) some properties of oDG's depend on c in (IVl 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. 31, IIII and III2, 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, crdegenerate 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,
(IVI 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 righthand side becomes sensible. Definition IV7 Suppose that XO is the optimal, crdegenerate vertex to (IV108), (IV109). Then sensitivity analysis with respect to the righthand 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 IV8 Suppose that XO is the optimal, crdegenerate vertex to (IVl 08), (IVl09). Then sensitivity analysis with respect to the cost coefficients means to determine the overall critical region K
T = UT(k)
(IVlll)
k=1
such that for every t E T the set B(c) does not change. Note that Definitions IV7 and IV8 differ from each other. This is because, in Definition IV7, changing the RHS the shape of the corresponding polytope (IVI09) changes and as a consequence some of the optimal bases need not remain elements of B(~). In Definition IV8, 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.
IV72 Shadow prices Consider (IVI 08), (IVI 09) and assume that XO is the optimal vertex. I. Let XO be nondegenerate. Then
oz
Ui = Obi
(IVI 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 (313) it follows that, changing bi in terms of (314) 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 crdegenerate. Introduce +
oz
I
obi
p. = 
_ oz
and p. = I
obi
(IVI 13)
as right and leftpartial derivatives [21 and let us note that, ifxo is nondegenerate, +

Ui = Pi = Pi'
(IVI 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},
(IVI 15)
whieh is the known optimal value funetion. The funetion (IVI 15) is a nondeereasing, eontinuous, pieeewise linear and eoneave funetion (see also Theorems IV3 and IV4). By (IVI 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) (lVl 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 onesided 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)}
(lVl 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 (IVI 19)
Using the above relations we ean now roughly deseribe a proeedure of determining the twosided 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 (IVI 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 twosided shadow prices by (lV1 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) 425431 [2) Aucamp, D.C., D. I. Steinberg: A note on shadow prices in linear programming, 1. Oper. Res. 33 (1982) 557565 [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 applications to mathematical programming, Unternehmensforschung 9 (1965) 112119 [5) Charnes, A., W.w. Cooper: Systems evaluation and repricing theorems, Managern . Sci.9 (1962) 209228 [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) 398354 [101 Fucfk, 1., T. Gal : K otazce degenerace ve vychoz!m fesen! simplexovych uloh linearnfho programovan!, Ekon. Mal. Obzor 5 (1969) 295303 [11] Gal, T. : A "historiogramme" of parametric programming, J. Opl. Res. Soc. 31 (1980) 449451 [12) Gal, T. : A note on the history of parametric programming, J. Opl. Res. Soc. 34 (1983) 162163 [13] Gal, T. : Weakly redundant constraints and their impact on postoptimal analyses in linear programming, EJOR 60 (1992) 315326 [14) Gal, T. : Shadow prices and sensitivity analysis in linear programming under degeneracy : A stateoftheart survey, EJOR 8 (1986) 5971 [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. 81106 [16] Gal , T., J. Nedoma: Multiparametric linear programming, Managern. Sci. 18 (1972) 406422 [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. 201210 [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. 4979 [20] Knolmayer, G.: How manysided are shadow prices at degenerate primal optima? OMEGA 4 (1976) 493494
208
References
(21) Knolmayer, G.: The effect of degeneracy on cost coefficient ranges and an algorithm to resolve interpretation problems, Oecision Sei. 15 (1984) 1421 (22) Kruse, H.J.: On some properties of odegeneracy graphs, In : (Gal, T., ed.) Oegeneracy in mathematical programming, Annals of Operations Research, Vol. 46/47, Baltzer Pubt.. Co., Basel 1993, pp. 393408 (23) Magnanti, TL, J.B. Orlin: Parametrie linear programming and anticycling pivoting rules, Math. Progr. 41 (1988) 317325 (24) Manas,1. Nedoma: Finding all vertices of a convex polyhedron, Numer. Math. 12 (1968) 226229 (25) Moeseke, P. van, G. Tintner: Base duality theorem for stochastic and parametrie programming, Unternehmensforschung 8 (1964) 7579 (26) NoZicka, F., 1. Guddat, H. Hollatz, B. Bank: Theorie der linearen parametrischen Optimierung, Akademie Verlag, Berlin 1974 (27) Nykowski, I.: Zalesnosci pomiedzi rparametrycznymi dual ny mi zadaniami programowania liniego, Przegl. Statist. 13 (1966) 311323 [28] Ritter, K.: Über Probleme parameterabhängiger Planungsrechnung, OVLBericht No 238, J. Comp. Syst. Sei. I (1967) 4454 (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) 1016 (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, AlmaAta 1975 (32) Weinert, H.: Probleme der linearen Optimierung mit nichtIineareinparametrischen Koeffizienten in der Ziel funktion, Math. Oper.Forsch. Statist. I (1970) 2143 (33) Weinert, H.:On uniqueness in parametrie linear programming problems with fixed matrix of constraints, Math. Oper. Forsch. Statist. 5 (1974) 177189
Chapter fi ve
5
51 52 521 522 V
VI V2
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 ·
(51)
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 (51) 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 52
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.
(52)
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
(53)
so that the objective function takes on the usual form N
z= LCjXj .
(54)
j=1
51 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. 21. The optimal solution is shown in Table 22. Here we had a (much simplified) fourproduct firm with two capacity restrictions. Initial and final tableaux are given in Tables 51 and 52.
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:
(55)
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 
(56)
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 53. 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 53
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'
(57)
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 basicindex and q> = {m+ I, ... , N} the indexset of the nonbasic variables. Then, (58) (59)
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. 25 and for II2.We, thus, solve minsj,j subject to
= 1,5,
249
Dependence on several parameters (on a vector parameter) Table 616
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
f4 2 J J
J
Table 617a I1
12
OSI
3
I
3
OS5
I
I
I
fS 3
0
1*
I
tl
s3
Table 617b
fS 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 617a through 617c.
250
Linear parametrie programming with respect to c
Table 617c
5,
s3
t,
1/3
1/3
2/3
55
1/3
4/3
4/3
I
I
0
t2
Table 618
Optimal solution associated with p,
4
p,
5
3 1/3
xB
+1
1/3
2/3
f2
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 617c, it follows that a neighbor in fact exists only along the first face. If we look at Table 616 in accordance with the corresponding pivot element (marked by an asterisk in Table 616), the new basicindex is PI = { I, 2}. Hence, we have the Iists V0 = {Po} , Wo = { PI}. In Table 618, the solution associated with PI is given. According to Table 618, 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 618), we obtain Table 619 associated with basis B 2. The corresponding critical region T(2) is defined by
Dependence on several parameters (on a vector parameter) Table 619
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 64
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. 64. As can be seen from Fig. 64, 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).
64 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 svector (t~, ... ,t;)T E T(p) yields precisely one Nvector of the cost coefficients. We are, thus, dealing with a problem similar to that discussed in SecA3 (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. 431, we shall illustrate the problems involved from examples.
Example 65 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 620 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 620, X2 is first entered into the primal basis; this yields Table 621.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 622. 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 620 Initial tableau I
2
3
I
f4
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 621
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 623. 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 622
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 623
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. IV6 (cf. also Ex. 412), in this case we use as auxiliary problem the problem (lV82), (IV85) through (IV88) (on account of the parameters, which are not signrestricted), i.e.,
(61) subject to 4t1 + 4tl + 5t;  5t2 + S2 = 0, t1  tl  2t; + 2t2 + S4 = 0, t1+tl +t;+t2 = I, t +>0>Ok12 , tk , , , sJ·>0  ,J' 24 , , k with
and t~ .
ti: = 0, k =
1,2.
256
Linear parametrie programming with respect to c
TabIe 624 I
4
52
I;
f;
t+
OS4
1*
P
I
t;
4
5
5
0
I
2
2
0
I
I
I
I
Table 625
t;
1;
t;
54
S2
0
3
3
4
0
H~
I
2
2
I
0
fp
2
I
I
I
3*
Table 626
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 (61), j =4, for there are positiv elements only in the nonbasic column with the index 4 (cf. Table 623). The solution ofthe auxiliary problem is shown in Tables 624 through 626. From Table 626, 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 627 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 627 2
PI
I
3
4
I
4
4
I
5
5
~h ·1
I
I
0
~hz
2
5
0
J J
Table 628
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 628. List: V I = {Po, PI} , W I = {P2} . According to Table 628, 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 629
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 matrixvector 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 66 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 629. According to Table 629, Po = {I , 2, 5} and the critical region T(O) is defined by
=
259
Homogeneous multiparametrie linear programming Table 630
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. IV6 and Ex. 412). 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 basicindex is PI = {I, 2, 3}. List: Vo = {Po}, Wo = {pI}. Choosing PI E Wo and performing the corresponding pivot step, we obtain Table 630. 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,
(VII)
Ax= b,
(VI2)
x
(VI3)
subject to
~ 0,
where c(t) = c + Ht,
(VI4)
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
(VI5)
can be incIuded, where G is a constant (r, s) matrix and d E IRr is a constant vector. Formulae (V6) through (V27) also hold here, so that it is not necessary to repeat them here. An admissible vector parameter t is defined by analogy with Definition IVI, and we obtain the corresponding definition by a simple exchange of t for A in Definition IVI . 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 IVI.
Theorem VJ J Suppose that there exists a finite optimal solution of the problem (VII) through (VI4) for a constant tO E IR' . Then, either there exists a finite optimal solution of the problem (VII) through (VI4) for any t E IR' or the given problem has an unbounded solution. Theorem VI2 and VI3 hold analogously to Theorems IV3 and IV4.
Theorem VJ2 The optimal value function zmax(t), wh ich is defined over K ~ IR', is a convex function .
262
Abridged mathematical presentation
Theorem VI3 The function z~~x(t) is linear over T(p) and, thus, continuous over
K. Definition VII 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 (VII) through (VI4) for t* E K, and (ii) it is possible to pass from BI to B2 (and conversely) by one primal simplex step.4 Definitions IV4, IV6 and Theorems IV5, IV6, IV7 may be taken over without change, >. being correspondingly exchanged for t and A (p) for T(p).
Corollary VlII The critical region T(p) wh ich is defined by (V25), 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 (VI7) through (V19) 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.
VII 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
ATuHt ~ c
(VI6)
possibly with Gt::; d,
where u E IR m and t E IR s are variable vectors. The dual system (VI6) can also be solved in the primal tableau; namely, determine a dual feasible solution of the following system: Ax = b, 4
(VI7)
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
(VI8) (VI9)
where t E IRr is a variable vector with nonnegative components. If (VI6) has no solution, no dual feasible solution of the problem (VII) 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 (VI6), is substituted into (VII) and the following problem is then solved: maximize w
=cT(to)x,
(VIIO)
subject to Ax
= b, x ~ o.
(VIlI)
If this problem has a primal feasible solution,6 an optimal basis B o of the problem (VIIO), (VIlI) exists, which at the same time is an optimal basis of the problem (VII) through (VI4) 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. IV3I (cf. Steps 1°° through 7°° of the algorithm in Section IV31), the following system is investigated:
(VI12) 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 (VI12). Analogously to the cases described in Sec. V, various special cases can be derived by specialization of Hand t here too.
VI2 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.
(VI13)
Abridged mathematical presentation
264
Ax = b,x;::: 0,
(VI14)
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,
(VI31)
with tk = t;  tk, k = I, "., s,
(VI32)
(cf. Theorem IVI 0).
VI32 Problem (Ho) Consider problem (UD), i.e., determine the region K c IR N such that the problem of maximizing (VI33) subject to Ax = b, x ~
0,
t ~
0,
t
E
IR N ,
(VI34)
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. 462), 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., (VI15), (VI16). Phase 2 is analogous to the Phase 2 described in the previous section, except that, as auxiliary problem, we solve the following task: (VI35) subject to s
L L1httk +
Sj
=0, Vj ri
p,
(VI36)
k=1
s
Ltk = I , k=1
(VI37)
270
Abridged mathematical presentation tk ~
0, k = I, " ' , s, Sj ~ 0, \ij 4 p,
(cf. Corollary IVIOI),
(VI38)
References
[I J Arnoff, E.L., S.S.Sengupta: Sensitivity analysis: Parametric programming, In : Progr. of Operations Research I (1961) 175180 [2) Candler, W.A .: A modified simplex solution for linear programming with variable prices, J.Farm.Econ . 38 (1956) 940955 [3J Dinkelbach, w.: Sensitivitätsanalysen und parametrische Programmierung, Springer, Berlin 1969 [41 Dinkelbach, W., P. Hagelschuer: On multiparametric programming, Methods Oper. Res. VI (1968) 8692 [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) 447451 (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) 3945 [8) Hax, H.: Preisuntergrenzen im Ein und Mehrproduktbereich, Zs. für Handelswirtsch. Forsch. 13 (1961) 424449 [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) 159170
Chapter seven
7
71
72 VII VIII VII2
RIM parametric linear programming Simultaneous changing of the righthand side and of the cost coefficients . . ........... . . . Dependence on a scalar parameter . . . . . . . . Dependence on several parameters (on a vector parameter) RIMmultiparametric 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 righthand side and of the cost coefficients
The interdependence of the sales price and the sales volume of goods being manufactured in a multiproduct 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 righthand 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 Svector and ). an svector (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 lotsize problems is given in [3]. As usual, the problems indicated above will be discussed with the aid of illustrative examples.
71 Dependence on a scalar parameter Example 71
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 71
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 71 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 (V40), 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 (III25), 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. 71 (b). Any ordered pair (l', ",*) from the rectangle in Fig. 7I(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 71
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. 75 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 78 through 711. Here, we proceed from te al ready familiar fact that, with respect to the righthand side, at first only those slacks are used for
289
Dependence on several parameters (on a vector parameter) Table 78 AI
~I
°"'3
I
0
8
11 *
0
I
2
A2
PI
7
fP2
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 79
~I
AI
"'3
fPI
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. (VIIIO), (VIIli» . 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 78. It follows from Tables 78 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 710
~I
'l'3
111133
?AI
81133
1119
1119
f~3
501133
121133*
'l's
12/133
A2
451133
72/133 10119 2411133 2601133
Table 711
~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 710, 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 righthand side is dependent over)., however, so that the values Yi cannot be used directly. According to Table 710, 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 77, 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 712
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 711, ~3 = 0 and ,,'; = ~,,,; = From Table 77, 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 77 and, after one dual step with the marked pivot element, we obtain Table 712. 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 713
Solution for Ps 4
I
Ps
_sfl
_'f2
xB
f2
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 78 through 711) 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. 75). The lists are VI = {po,pÜ, W I = {PI,PS,P6}' From Table 712, 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 712 with the pivot element marked, we obtain Table 713. From Table 7\3, it fol\ows: A(S) : (13) :s; 2,
(7) (10')
Dependence on several parameters (on a vector parameter) Table 714
Solution for P6
f4
_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. 75). This yields the Iists V2 = {Po,P2, PS}, W2 = {PI,P6}. It further follows from Table 713 that
z~~x(>.) = 15  4Af  A~ + 4A) A2  4AI + 2A2. >.
E
R(S).
Now choose P6 E W2. After one primal step from Table 713 with the pivot marked, we obtain Table 714 associated with P6. From Table 714 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 715
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
f5
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 714, we obtain Table 715 . According to Table 715 : 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. 75). Hence, V4
={Po,P2,P4,P5 , P6} , W4 ={PI,P3,P7} .
According to Table 715, 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 716. The auxiliary computations would show that R(I) 0 and :
*
Dependence on several parameters (on a vector parameter) Table 716
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. 75). 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. 75). 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 718 after one primal step with the pivot element marked in Table 715 by two asterisks. According to Table 718, 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 718
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. 75). Hence, V7
= {pi, i = 0, ... , 7}, W 7 = 0.
The whole process is now terminated. Finally, in accordance with Table 718, 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. 75 (cf. also Theorem VII2).
VII Abridged mathematical presentation
Consider the problem: determine the domain of definition function
K ~ IR' of the linear (VIII)
such that the problem of maximizing
=eT (t,).) . x
(VII2)
=b(t, ).), x 2 0,
(VII3)
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).,
(VII4)
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 (VII4) into (VII2) and (VII3), we obain e(t,).) = e*
+ F' I)., b(t,).)
= b" + F*2).
(VII5)
with e* = e + Hld, F*I = HIO + F I , b* = b + H 2d, F*2 = H 2 0 + F 2 .
(VII6)
Thus, the problem (VII2), (VII3) reads : maximize z(t,).) = (e" + F' I).)T x
(VII7)
Ax = b" + F *2)., X 2 o.
(VII8)
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 86
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
Id 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 85 is, thereby, uniquely determined. After one primal step, we obain Table 86. In Table 86, 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 86 is determined according to the dual rules. After one pivot step, we obtain Table 87.
Sensitivity analysis with respect to the elements of the technological matrix A Table 87
Solution associated with 8 3
P3
2
74

3+ 2d 2 (I d)
I +4d Id


5
2

Id
Id
I 6d
5 (I  2d)
Id
2 (I d)

2 (I d)
I

Id
5 (49d)
ßz/d)
5
3 4+21d 2 (I d)
1
Table 88
315
6
xs(d)
1/2

0
16+89d 2 (I d)

21
Id
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
Id
I
I
I
p
2'
2
If we perfonn an analysis in Table 87 similar to that carried out for the previous tableaux, 00
 I
3
*
32d
xB I
I
I
I
2
2
PI
I
3
xB(d)
f4

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 810. First, examine the solution in Table 810 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 811
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 81
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 81 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. 81, 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 (VIIII) subject to (A + A *D)x = b, x ~ 0,
(VIII2)
has a finite optimal solution and, for any d k ri. K, the problem (VIIII), (VIII2) 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 TBI pq 1+ qTB1p'
(VIII3)
320
Abridged mathematical presentation
VIIIl Changing a column of matrix A Let
,qT
p=
= (0, ... ,0,1,0, ... ,0),
(VIII4)
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 (VIII5)
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 TBI P from Bodewig's formula then becomes2
2:1 p~ld~ 1 + qTBI P = I + (0, .. ., 0, 1,0, ... , 0)B 1
(VIII6)
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
(VIII7)
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. ~'
. ... .
. .. ..
(VIII8)
. .
:::
~) d
. ....
.. .. .
0, .. . 0, Pmk 0, ... 0 i.e., p
=d (PIk. ... , Pmk)T, then (VIII8) becomes dßkv I: ßuiPik ßuAd) = ßuv 
~ ,d 1+ d . ßkiPik
E
(VIII9)
IR.
Using (VIII8) and (VIII9), 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) = BI(dk)b
(VIIIIO)
it follows that (VIIIlI) or yu(d)
=Yu 
dYk I: ßuiPik ~ ,d I + d . ßkiPik
E
(VIIII 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
(VIIIl 3) from (VIII14) from uT(d k) = C~BI(dk),
(VIII15)
and, finally, from z(p)
max
(d k)
=cTB1(dk)b B '
(VIII16)
the detailed, elementwise expressions can be easily derived.
VIII2 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
(VIIII 7)
with A(d) = A + Cd
has a finite optimal solution associated with B. The conditions yu(d)
~
L1zj(d)
I,
(VIIIl 8)
0, for all j fi p
(VIII19)
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 (VIlI9) it is obvious that4 4 With I + dV = 0, the inverse B1(d) would become a singular matrix.
(VIII20)
323
Definition of a critical region
(VIII21)
I +dV *0. For u =k, there folIows, from (VIIII 2), Yk
( d)=~ 1+ dV'
(VIII22)
The relation (VIII22) 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
(VIII23)
I,
(VIII24)
O,j 4. p,
i.e., d(yu V  Yk W) d(~zj V
~
 ykjZ)
Yu,
(VIII25)
~ ~Zj.
(VlII26)
Define the indexset 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 indexset 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
(VIII31)
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, (VIII32) is fulfilled and YktW  YrtV > O.
(VIII32)
327
Changing a row of matrix A
Then, B
f~
(VIII33)
Ykt W  Yrt V
With the pivot element Yrt(d* + E*), where f* satisfies the relation (VIII33) 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. VIII2, 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] ).
VIII4 Changing a row of matrix A Let a change of matrix A be expressed by matrix 0,
0,
o
(VIII34)
0,
0,
o
i.e., (VIIl35)
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,
(VIII36)
holds. The vectors p, q of Bodewig's formula (VIII3) are as folIows : ~
T
P = (0, ... ,0, 1, 0, .. ., 0) , "
'
m
(VIII37) LetJ = {j I j = I , ... , n, n + I , ... , N } be the set of all subscripts, p ={j I, ... , jm} the basicindex, and, finally,
........
Z(x 4
. . . _.f!' :Z(x~
Figure 94
Consider now the boundary point TEX that corresponds to Z(T) E Z in Fig. 94. The intersection of D and Z is nonempty and consists of more than one point. Hence, there is a feasible solution x such that it is possible to improve the values of both objective functions starting with the values z, (T) and z2(T). Consider the vertex Z(x 2) (or any of the boundary points Iying on the edges y(Z(x 2), Z(x 3» or y(Z(x'), Z(x 2 The intersection of D and Z consists of z(x 2) 2 itself. Starting with Z(x ) try to improve the value of z, (x). If this has to be done such that only feasible solutions x are to be taken into account, then one has to move along the edge y(Z(x'), Z(x 2 which implies that the value of Z2(X) becomes worse. And, vice versa, the same happens if we try to improve the value of Z2(X), i.e., this makes the value of z, (x) worse. The same is true for any points of y(Z(x 2 ), Z(x 3 and y(Z(x'), Z(x 2 and only for these edges. Hence, these edges consist of efficient points. Let us look for the corresponding efficient solutions. The edges mentioned are uniquely assigned to the edges y(x 2, x 3 ) and y(x', x 2) respectively. Hence, the set of all efficient solutions to the given LVMP is the set
»).
»,
»
»
E = y(x' , x2 ) U y(x 2, x 3 ). From this, we have learned what an efficient point in set Z and an efficient solution in set X are. What remains to be seen is how to determine set E in general.
A method for determining the set of all efficient solutions
343
H" Figure 95
Before offering a solution method, let us state that, between the homogeneous multiparametrie problem max(Ct)T x, t ~
(98)
0,
XEX
and the LVMP (94), there exists a onetoone correspondence, which is stated in the so called Efficiency Theorem (see Theorem IXI in Sec. IX2). This fact, which will be explained below by geometrie al means, is one of the reasons why multicriteria problems are briefty discussed in this book. We have already mentioned this relation in Chap. 6 (cf. Note below (VI18) in Sec. VI3). The efficiency theorem states: A solution XO E X is efficient if and only if there exists tO > 0 such that XO is an optimal solution to (98) with t = tO. Hence, (98) can be viewed as being a sort of a scalarization of the vector maximum problem (94). The set K* = M n K of all admissible parameters t > 0 to (98) yields the set of all x E E ~ X; here, M = {t E IR K I t> o} and K = UT(P), as defined in P
previous chapters. To illustrate the relation mentioned, let us first recall that a supporting hyperplane (in our case a supporting line) denoted by H in Fig. 93 is a straight line having at least one point in common with the boundary of X and there are no points of the relative interior of X in common with H. The particular supporting line H has the edge y(x 2 , x 3 ) as apart. A supporting hyperplane (line) H' in Fig. 94 corresponds to H. In Fig. 95 apart of Fig. 94 has been drawn . Consider Z(x 2 ); a supporting line H" through Z(x 2 ) has, as its normal, the vector t' . It should be noted that the values of tl ' and t2' are plotted on the axes of D, and that, for example, in 17], the proof of the efficiency theorem is based on the notion of separating hyperplanes. The normal t to any ofthe supporting lines crossing point Z(x 2 ) has the required property: t> o. We already know that Z(x 2 ) is an efficient point of Z. Consider
344
Multicriteria linear programming Solution associated with x I
Table 92
I
PI
x
3
xB
2
0.25
4
0.5*
0.5
2
5
2.5
0.5
14
I ~Zj
1.25
0.25
5 (max)
2 ~Zj
0.75
0.25
5
I
T
(0)
=(0,5),
xI
0.25
T
=(0,5,0, 2, 14),
zl(x
I
)=5,
z2(x
5
I
)=5
the edge y(Z(x 2), Z(x 3 », which we know to consist of efficient points. The only possible supporting line to this edge is denoted by H' in Fig. 95. The normal 10 this supporting line is tO > 0 at any point of this edge. Note that Z(x 2) and Z(x 3 ) have precisely one normal tO in common. This is used in the method we shall describe below ( see also Theorem IX2). Note that, setting t = tO > 0 into (98) and solving the corresponding linear program, we obtain two alternative basic optimal solutions x~O) and x~O) with the same value of the overall "combined" or "weighted" objective function: z(to) = t7zl(x) + t~Z2(X) = t~'(C:XI + C~X2) + t~(cix, + C~X2) ;
in oUf case, z(tO) = (t7 + t~)x I + (t7 + t~)X2 . Hence, if x* E X is an efficient solution, then t > 0 exists as anormal to a supporting hyperplane of Z crossing the efficient point Z(x*) and vice versa. Now, the only remaining question is how to compute set E. This we shall show with the aid of our example (see also [8]). Transform the system of linear inequalities (I) through (3) into a set of linear equatios by means of slack variables: XI + 4X2 + X3 XI + 2X2
+ X4
3xI + 2X2 Xj~O,j=
1, ... ,5.
= 20,
(I ')
= 12,
(2')
+ xs = 24
(3') (4')
A method for determining the set of all efticient solutions
345
Table 93
SI t2
0.5
0.625
s3
0
0.25
tl
0.5
0.375
In Table 92, the solution x;o) eorresponding to the vertex Xl (cf. Fig. 91) is given. means the same as l1hY in Chaps. 5 and 6. Note that Let us state that the eritieal region T( I) is defined by
l1zY
1.25t I + O.75t2 ~ 0, 0.25tl  0.25t2 ~ 0, tl 2: 0, t2 2: O. Note that, in T( I), there exists t > 0, henee, the eorresponding solution x(t) is effieient. In Sees. 64 and VI3, we learned how to determine neighboring eritieal regions in a homogeneous ease with the aid of solving the auxiliary problem minsj , j=1,3 subjeet to 1.25tl + 0.75t2 + SI 0.25tl  0.25t2 + S3 tl + t2 tk 2: 0, k = 1,2, Sj
= = = 2:
0, 0, I, O,j
= 1,3.
In the given eonneetion this subproblem is ealled the effieieney test. The solution of this subproblem is in Table 93. From Table 93, we obtain the following . I. x;O), i.e., vertex x I, is an effieient solution sinee there exists t l = (0.375, 0.625) T >0. 2. From what has been said on the geometrieal illustration of the effieieney theorem, it follows that the only existing neighbor to xl (min SI = 0 and SI NBV, min S3 = 0.25> 0 and S3 BV ) is an efficient neighbor x 2. 3. Finding the eorresponding pivot element in Table 92 (marked), the node P2 = {I, 2, 5} turns out to be adjaeent to PI. In Table 94, the solution assoeiated with P2 is presented. 4. The edge y(x I, x 2 ) is an effieient edge (see Lemma IX3 and Corollary IX22). 5. List: Vo = {pI}, Wo = {P2} ' Y" = { tl} , where the subseript eorresponds to PI, the superseript denotes the partieular t .
346
Multicriteria linear programming
Table 94
Solution associated with x2 3
P2 2
4
xB
0.5
4
2
4
2*
5
4
1.5
2.5
0
0.5
1.5
8
0.5
I
I
5 I
Mj
7
MI
2 T (0) T 2 2 x =(4,4), x2 =(4, 4,0,0,4), zl(x )=0, z2(x )=8
Table 95 S4
s3
0.5
0.25
II
0.25
0.375
t2
0.25
0.625
Let us summarize the results. To the vertex xl corresponds the efficient solution x\O) (i.e., xl is an efficient vertex of X), the only existing neighbor to Xl, that is also efficient, is x 2 and, moreover, the edge y(x l , x 2) k E. It should be noted that the algorithm (cf. [8]) described in Sec.lX is divided into two phases. Phase I finds a first efficient solution. To perform this part of the algorithm, various methods can be used (see, for example, [4,5, 13, 17, 19,20,21, 26, 28]) or Phase I in Sec. 64 can be applied. Phase 2 finds all efficient extreme points (or the corresponding solutions x~{») together with the assigned vectors t~. What we are now doing is carrying out Part I of Phase 2 of the algorithm. It should be noted that, for example, y(x l , x 2) k E can be determined by a simple inspection of some resuIts of Part I. This inspection forms Part 2 of Phase 2. The problem that now has to be solved is how to find all efficient neighbors of x 2 . This is the same task as in the homogeneous case of linear multiparametric programming, and involves solving the corresponding auxiliary problem (Tables 95 and 96). From Table 95 it folIows, as expected, that we have t l =(0.375, 0.675)T again. This, incidentally, confirms that y(x l , x 2) k E. Along the fourth face of T(2), the neighbor T(I) exists, since min S4 = 0 and S4 NBV; with the corresponding
347
A method for determining the set of all efticient solutions Table 96 S3 s3
2
0.5
tl
 0.5
0.25
t2
0.5
0.75
Table 97
Solution associated with x3
4
P3 2
0.75
xB
0.75
3
I
 0.5
0.5
6
3
2 .5
0.5
2
3
I
1.25
0.75
7
0.25
0.25
~Zj
~Zj
3
X
5
=(6,3)T,
(0)
x3
=(6,3,2,0,0)T,
3
zl(x )
= 3,
9 (max) 3
z2(x')
=9
pivot element this leads us back to Table 92. The corresponding adjacent node is already listed in V0 . From Table 96 we obtain the following.
I. There exists a new neighbor x 3 with P3 = {I, 2,3} (as follows from the marked pivot element in Table 94). The neighboring region T(3) exists along the third face of T(2) since min S3 = 0 and S3 NBY. 2. From t 2 = (0.25, 0.75)T > 0, it foilows that the edge y(x 2, x 3) is an efficient edge. 3. List: VI = {PI, P2} , W I = {P3}, Y I = {tl. t~, tn · After one pivot step with the element marked in Table 94, we obtain Table 97. The corresponding auxiliary problem has the solution given in Table 98. Note that using the element marked in this Table would eliminate tl . Hence, there is no other solution with tl > 0, t2 > 0 basic variables, so that no new tappears. From Table 98, we obtain (briefly) the following. 1. The only efficient neighbor to x 3 is x 2 ; y(x 2, x 3) ~ E as already known. 2. No new adjacent nodes exist. 3. List: V2 = {PI, P2, P3}, W 2 = 0, Y2 = {tl. t~, t~, tn ·
348
Multicrileria linear programming
Table 98 S5 S4
I
0.5
I,
1*
0.25
12
I
0.75
Since W 2 = 0, the procedure of Part I of Phase 2 is finished . In this very simple example, it is, as a matter of fact, no Ion ger necessary to perform Part 2 of Phase 2, since we already know the result. However, in order to show how to carry out Part 2, we shall act as though we did not know the result . Collect the vector parameters with identical superscripts from the last list Y2, i.e., t:, t~ and t~, t~. The corresponding subscripts show which of the efficient vertices form an efficient face (in oUf ca se an efficient edge). From this simple inspection, we have
as al ready known . Have a look at Fig. 93 and consider point Z(I) =(5, 9l, i.e., z,(I) =5, z2(1) = 9, where I = (2, 7)T. In this point, both ofthe given objective functions reach their best possible values simultaneously. However, the corresponding solution I is, unfortunately, not feasible; hence, this socalled ideal solution cannot be attained without violating the constraints. As we have al ready mentioned in the introduction to this chapter, the ideal solution cannot be attained in most cases (exceptions are the perfect solution, see Fig. 91, or no conflicting situation at all) . Having conflicting goals, any efficient solution represent a rational decision. However, in general, infinitely many efficient solutions (rational decisions) are available. The next question with regard to conflicting goals in practice is, therefore, how to find one (efficient) solution which all competing partners (e.g., members of the family, managers, etc.) can view as the best possible compromise decision. Since it would go beyond the confines of this book, we shall not deal with such specific questions, but merely hint again to socalled interactive approaches (see, for example, [6, 19,28,29]). Goal programming approaches are also sometimes used (see, for example, [2, 3]). For other approaches see also [26] and the bibliography at the end of this book .
Nonessential objective functions
349
Figure 96
92 Nonessential objective functions Consider the LYMP (94) and let us ask the question as to whether all the given objective functions are really needed for forming set E. In other words, does at least one objective function exist, the deletion of which does not change set E? Clearly there is some analogy between redundant constraints and such objective functions : in the former case, we may delete the redundant constraints without changing set X (see Sec. 112); in the latter, omitting some objective functions does not change set E (see [15]). In Fig. 96, five objective functions are considered. The set of all efficient solutons is the set E = y(x l , x 2) U y(x 2, x 3). Imagine that we omit the objective functions Z2(X), Z3(X) and Z4(X). As is apparent, set E remains the same. Set E will not change when omitting Z2, Z3 and Zs either. Thus, in any case, we may omit both Z2 and Z3 simultaneously without inftuencing E, but we cannot simultaneously omit Z4 and Zs. At least one of these objective functions must be retained if set E is to be the same. For, if we omit both Z4 and zs, set E is either determined by y(x I, x 2) if at least one of Z2, Z3 is considered, or set E reduces to the single vertex x I if both Z2 and Z3 are omitted at the same time. Hence, we can say that Zz and Z3 are certainly nonessential objective functions, because omiting them does not affect set E. Systems of objective functions of the type Z2, z3 are called strongly nonessential (or strongly Eredundant, by analogy with redundant constraints), since the whole system can be omitted without inftuencing set E. Systems like Z4 and Zs are weakly Eredundant or weakly nonessential, since any of the elements of such a system can be omitted, but not all of them. In Fig. 97 we have drawn the vectors e k for k = I, ... , 5. From the general theory of LYMP ( see, for example, [22, 23, 24, 25]), it follows that the cone qe l , e4) or C(e l ,es) determines E. The corresponding cone is the set of all nonnegative
Multicriteria linear programming
350
Figure 97
Figure 98
linear combinations of Cl, c4 or Cl ,c S, i.e. , C(c l ,c4) = {c E IR 2 I c = UIC I + U2C4, UI ~ 0, U2 ~ 0 } and similarly for C(c l , cs). In Fig. 98, we have drawn all c k , k = 1, . .. , 5. From this figure, it appears that using C(c l , c4) as "basic" cone, each vector c2, c3 can be represented as a non negative linear combination of c land c4 . This is also valid if C( Cl, cs ) is used. This again shows that c 2 and c 3 are "superfluous" for determining E and that Cl, c 4 or Cl, CS determine E uniquely. In determining set E or performing an interactive analysis, it may be of interest to know the nonessential objective functions and the systems of strongly or weakly nonessential objective functions . This knowledge could also be used in bargaining procedures for finding a compromise solution. We shall take an illustrative example in showing a convenient procedure. A comprehensive study ofthe corresponding theory is in [15,22,23] and the method is described in detail in [9, 10].
Example 92 The task is: in the LVMP ZI(X) Z2(X) "max" { Z3(X) Z4(X) zs(x)
X2 + X3 } X2 + X3 X2 + 4X3 XI + X3 = 3X2 + 4X3
= = = =
subject to
::; 0 X2 + 3X3::; 6 X2 + 2X3::; 9 2X2 + X3 ::; 12 Xj ~ O,j 1,2,3,
=
351
Nonessential objeclive funclions Table 99 1
2
3
4
5
0
0
0
1*
0
I
1
1
0
3
1
1
4
1
4
0
0
0
1
0
1*
1
0
3
1
1
4
0
4
0
0
0
1
0
I
1
1
0
3
2*
0
3
0
1
0
0
0
1
0
0
1
2.5
0
3.5
1
0
1.5
0
0.5
I
(i)
determine nonessential objective functions, systems of strongly and/or weakly nonessential objective functions, and, for determining E, use only those objective functions that are found to be essential in the first stage. The method of carrying out this task has three stages. The main idea of Stage I is based on the properties of the convex cones formed by the given e k , as briefly described above. To find a "basic" cone (ca lied spanning system) the matrix
0, 0, 0, I, 0) C=(e', .. . ,e5 )= ( 1,1,1,0,3 1, 1,4, 1,4
is rewritten in tabular form. Using the elimination method in combination with certain convenient criteria (see [15 J), this matrix (table) is transformed until a maximal system of unit vectors is obtained. This is carried out in Table 99. From (i) of Table 99, it follows that
e3 = 2.5e 2 + 1.5e' , e5 = 0.5e' + 3.5e2 . Hence, Z3(X) and Z5(X) are, surely, nonessential and can be omitted. This finds us a spanning system which consists of e' , e 2 , and e4 , and Stage I is finished .
352
Multicriteria linear programming
In Stage 2, set E is determined using zi (x), Z2(X), and Z4(X). Here, we need state only the results, since the method described in Sec. 91 is used again. 7 We obtain
where XO
= (0,0,2)T,x l = (0, 3, 3)T,x2 = (0,5,2)T .
Using a very simple modi/kation of the efficiency test as described in [810) we find a possible system of the minimum number of objective functions (called minimum cover) in the course of determining E. In our case, the objective functions ZI (x) and Z2(X) are found as possible minimum cover. This ends Stage 2. In Stage 3, a simple comparison is used which yields all possible minimal covers and the systems of strongly or weakly nonessential objective functions. Here, the individual optima of the elements of the minimal cover we have found are compared with the individual optima of the remaining objective functions . This means that, in our case, z)(x), Z4(X) form a strongly nonessential system, Z2(X) and Z5(X) form a weakly nonessential system and that another possible minimal cover consists of ZI (x) and Z5(X) (compare also Figs. 96 through 98).
93 Postefficient analysis In analogy to postoptimal analysis, postefficient analysis means introducing parameters into an LVMP after the set E has been found. Parameters can be introduced into the righthand side of the constraints, or into the matrix A, or into the objective functions coefficients of the single objective functions [9, 14, 24]. In connection with interactive approaches or  more generally with finding a compromise solution, the competing partners involved (e.g., managers) sometimes have the desire of changing some initial data. Such and similar questions prompted B. Roy [18] to propose his evolutive procedure, by which certain constraints are relaxed in order to "extend" the feasible set X. Since the corresponding analyses and methods are of a quite special nature, we shall not deal with these questions here.
7 Of course, any method which determines set E can be used.
IX Abridged mathematical presentation 8
IX 1 Introduction Solving a linear multicriteria problem, sometimes called a linear vector maximum problem (LVMP), generally means determining the set E of all efficient solutions. Several works have been devoted to finding an appropriate solution method (for references, see [19, 25, 28 J and the bibliography at the end of this book). All the methods of determining set E divide the corresponding procedures into two parts. In Part I, a first efficient solution is generated or it provides the result that no efficient solution exists at all. Part 2 of the methods consists of generating all efficient vertices and efficient faces starting with the first efficient vertex determined. In the most of these methods, special subprograms are needed to generate efficient faces . The only algorithms wh ich find the higherdimensional efficient faces by an appropriate simple inspection of the results found previously is Isermann's method [13] and the method described in Section IX3. In the organization of the procedure for generating all efficient vertices, we come up against a particular obstacle: degeneracy. Applying the results of Section 113, degenerate vertices can be handled easily. In practice, to solve an LVMP means to find a compromise (efficient) solution rather than to determine set E. We are not dealing with corresponding methods in this book; we refer the interested reader to [19, 25, 28] and the bibliography at the end of this book.
IX 2 Theoretical part In this and the following section, we shall use the notation already introduced in Chaps. 1,4, 5, and 6 and, especially, in Secs. I, IV6, and VI3. For the benefit of the reader, however, let us recall that
x = {x E
IR n lAx s; b, x ~ o}
X = {i E IR m+n I Äi = b, i
~ 0, i
(128) = (:)
,S E
IR m}
(129)
8 The author is obliged to Professor Joe G. Ecker of the Rensselaer Polytechnic Institute, Troy. N.Y. (in 1978), for his stimulating comments and advice on this section .
354
Abridged mathematical presentation
X~) =
(X: ) ,XB = BIb.
Let XU E X be a vertex of X associated with Bu and p, and recall that between XU and x~o) the correspondence is onetoone. Thus, in the following, we may use the term "vertex" for both XU and x~O). In order to simplify the notation and without causing any confusion, we set
IR n+m
:I
x ==
x.
If it should not be clear, from the given connection, wh ich x is meant, we shall always give an appropriate hint. Let (IXI) be K linear objective functions. According to the definition of X and of X, Cjk = 0 for all j = n+ I, ... , n+m, all k E {I , ... , K} evidently holds. The linear vector maximum problem (LVMP) is then to "max" Z(x) = C Tx, C = (Cl, .. . , c K ),
(IX2)
XE X
where
wh ich means finding the set of all efficient solutions, i.e., the set
E = {i E
X I there is no x
E
X such that C Tx
:;0:
CTi and C Tx;t: CTi}.
(IX3)
Consider the homogeneous multiparametric linear programming problem (MPLP) with respect to c (cf. also Secs. 64 and VI3) maxz=(Ct)Tx,t:;O:O,tE IR K ,
(IX4)
xd
0 such that XO E X is an optimal solution to (IX4) with respect to t = tO. Let x~) E X now be a vertex associated with the basis B with the basicindex p. Suppose X ;t: 0. Note that X ;t: 0 => X ;t: 0. Denote by H a supporting hyperplane to X . Let x~O) for B u, u = I, . .. , U be some vertices of X and denote by
u (0) , ( u0» =yx YX l
. . . ,x (0» u
=
{
u
'" (0) 'L...,.Au=I,Au:;O: 0 allu } xEXlx=L...,.AuX u '"
u=1
u=1
Theoretical part
355
(0) (0) the convex hul I 0 f xI , ... , X u . For our purposes define a face F of X as folIows .
Definition IXI Let F be a convex polyhedron with the dimensions I ~ dirn F ~ m + n  I with the following properties:
. (i) Iet x(0) I , . .. , X (0) 0 f F ; u be a ll thevertlces (ii) if F = "'u lu=1 (x(o» u ' then Fis bounded·, (iii) if x E F exists such that it is not possible to represent F in the form given in (ii), then F is unbounded. Denote the corresponding convex polyhedral set by PO. Then F (or FO) is said to be a face of X, iff H exists such that FeH (or FO c HO). Note By definition, an edge between two neighboring vertices of X is also a face. By higherdimensional faces we shall understand faces with the dimension dimF ~ 2. Theorem IX2 (Corollary to Theorem IXI) All x E F are efficient solutions to (IX2), iff there exists XO > 0 such that x~O) are, for B u , u = I, ... , U, optimal solutions to (IX4) with respect to t =t O • The proof is to be found in [8]. Corollary IX21 Let us denote by Int F the relative interior of F. Then x EInt F is an efficient solution to (lX2), iff all x E F are efficient solutions to (lX2).
This assertion follows immediately from the proof of Theorem IX2. Definition IX2 The basis B is said to be an efficient basis, iff the corresponding complete basic feasible solution x~) is an efficient solution to (lX4), or equivalently, iff there exists t" > 0 satisfying Theorem IXI such that X~) is an optimal solution to (IX4) wit respect to t = t*.
Suppose that B is an efficient basis associated with the basicindex p. Denote by (lX6) the elements of the criterion row for the kth objective function in the simplex tableau associated with the basis B. The region K
T(p) = {tE IRKI_ L.:1zftk ~O,j ri. P,tk >0 all k}
(lX7)
k=1
defines the set of t E IR K such that for all t E T(P), the solution is an efficient vertex of (lX2) or it is an optimal solution to (lX4). Definition IX3 Two vertices
X(I O ) ,
xi
O)
are said to be efficient neighbors, iff
356
Abridged mathematical presentation
(i) x\O) and x~O) are neighboring vertices in the usual sense; (ii) y(x\O), x~O») is an efficient edge, i.e. , all x E y(x;O), x~O») are efficient solutions to (IX2) .
In accordance with the following Definition IX4, assigne an undirected graph G = (S, r) to (IX2) as in Chaps. 1,4, and 6. Definition IX4 An undirected graph G = (S, r) is said to be generated by an LVMP (lX2) iffthe node set Sand the edgeset r satisfy the following conditions: PES, iff B is an efficient basis; between two nodes PI, P2 E S, there exists an edge iff the corresponding basic solutions x;O), x~O) are efficient neighbors in the sense of Definition IX3; let nodes PI , P2 be called adjacent nodes; (iii) to every node PES at least one t satisfying Theorem IX2 is assigned. (i) (ii)
Definition IX5 Consider two efficient bases Bland B2. The bases BI, B 2 are said to be efficient neighboring bases, iff the corresponding nodes PI , P2 E S are adjacents. Definition IX6 The sets T(I) and T(2) uniquely defined by (lX7) are said to be neighboring regions, iff BI, B 2 corresponding to T(I), T(2), respectively, are efficient neighboring bases. The solution procedure and the relevant theory for problem (lX4) have already been discussed in Sec. VI. For the sake of simplicity, and without loss of generality, suppose that P {n+ I, ... , n+m } ~ j = 1, . .. , n are the subscripts of the nonbasic variables associated with p. According to this assumption, the system of constraints of the auxiliary problem , i.e., of
=
mi!1 Sj,j fi p,
(IX8)
XE X
is the system
~Z:tl  . ..  ~zftK + SI = 0,
~Z~tl  . ..  ~z~tK + Sn = 0, tl + ... + tK = I.
(IX9)
Since, by assumption, x~) is an efficient vertex associated with P , according to Theorem IXI there must exist a solution t* > 0 to (lX9) . Suppose that the solution t * > 0 to (IX9) is associated with basis B, B being an (n+ I, n+ 1) regular matrix. Suppose, further, that the basic variables in system (IX9) transformed into basis B consist of the variables
Theoretical part
357
tl, "', tv, V
~
K, SI, ,'" sr, r ~ n, v + r = n + I,
and denote the corresponding vectors by T T t B = (tl, .. " tv) ,SB = (SI, .. ',Sr) ,
The nonbasic variables associated with the basis Bare then the variables
and denote the corresponding vectors by T
T
t N = (tv+l, "', tK) ,sN = (Sr+l, ""sn) ,
where N is an (n+ I, KI) matrix, Denote by ~zf the elements of the system of linear equations (lX9) transformed into basis B, Setting Sr+1 = '" = Sn = 0, we obtain A v+1 A K tl + uZI tv+1 + .. , + uZI tK = t*l > 0 ,
A v+1 A K tv + uZ v tK = t*v > 0 , v tv+1 + '" + uZ
(lXIO)
SI + ~z~!: tv+1 + .. , + ~Z~I tK = s~ ~ 0,
(IXlI) From (lX1 0) it follows that K
tk
= t~ 
~ ~ Z~th, k = I, .. "
h=v+1 Evidently, there exists t~ > 0 for all h
V
~ K.
= v+l,
(lX12)
.. " K such that tk > 0 for all k
=
I, .. " v,
Note This can be shown straightforwardly, since
(i) ~z~ ~ 0 for all h implies tk > 0 for all k and (ii) ~z~ > 0 for at least one h implies that, since, by assumption, t~ > 0 for all k, there must exist th = € > 0 sufficiently smalI, that satisfies (lX12) with tk > o for all k,
xi
xi
Let O ) and O ) be neighboring efficient vertices associated with the neighboring efficient bases BI and B2, respectively, to which the neighboring regions T(I) and T(2), respectively, are assigned, From the corresponding definitions of the respective types of neighbors and from Theorem IV5 it then folIows, with SI = 0, 1 = r+ I, " " n, that
Abridged mathematical presentation
358
(IX13) h=v+1
h=v+1
i.e., the representation of the basic variables tb k = I, .. . , v ~ K, is identical in the corresponding feasible solutions to the system of linear inequalities defining T( I) and T(2) associated with the basicindices PI and P2, respectively. Furthermore, the set T(I) (cf. (lV26» is a convex polyhedron in the parametric space and let T(2) be a neighboring convex polyhedron. From what has been said before it follows that the basic feasible solution t* =
(t~) I tN
~ 0
to (lX9) defines a vertex common to T(I) and T(2). Note that, from the theory of parametric programming as presented in the previous chapters, it follows that
UT(P)
=K, K ~ JRK,
P
is a convex polyhedral and connected set. Let the vertex t * satisfying (IX13) be a nondegenerate vertex; this avoids the difficulties mentioned in Chap. 2. Note If v = K, then, evidently, t~ > 0 for all k = I, . .. , K exist and it is the solution to (IX9) associated with both efficient bases BI, B2. Let t* be a nondegenerate vertex common to T(I) and T(2). Then, evidently, Lemma IX3 holds. Lemma IX3 The vertices x\O) and x~O) are efficient neighbors, iff there exists t* =
(t~) = (t~) > 0 tL t~N
N
with t~ = t~ > 0 and t~ = t~ = respect to t = t * ~ o.
0
such that both x\O) and x~O) solve (lX4) with
Note If the special case occurs in which vertex t" is degenerate, then : (i) Lemma IX3, evidently, holds in the sense of a sufficient condition, and (ii) in the procedure (see Sec. IX3), some special precautions have to be taken, as described especially in Chap. 2 of this book. From Theorem IX2 and Lemma IX3, Corollary IX22 immediately folIows. Corollary IX22 All x E F are efficient solutions to (IX2), iff there exists t* ~ o in the sense of Lemma IX3 such that x~) for all Bu with u = I, ... , U, solve (IX4) with respect to t = t* ~ o.
This is the theory we need. Now let us describe the procedure.
A solution procedure
359
IX3 A solution procedure9 In order to find the set E of all efficient solutions to (lX2), we proceed in two phases. 1o Phase J:
Determine an efficient complete feasible basic solution x~;) to (lX2) associated with the basicindex PO ' Phase 2 consists of two parts: Part J: Starting with Po determine all nodes of the graph G = (S, r). Part 2: Based on the results of Part I, determine the higherdimensional faces .
Let us now describe some details: Phase J
This is performed as described by J.G .Ecker and A.I.Kouada [4, 5], or by H. Isermann [13] or Phase I of the algorithm as described in Sec. VI32 can be used as weil. Phase 2 Part I
Suppose that in the hth step of Part I of Phase 2, the efficient solution x~() associated with B u has been found. The corresponding simplex tableau, called the master associated with the basicindex Pu E S, is thereby generated. From this master, it is easy to derive the conditions determining T(u) (cf. (IX7». The questions to be answered now are as folIows. I. Determine r(pu), i.e., find all nodes adjacent to node Pu. In other words, find all vertices (basic solutions) which are efficient neighbors to the vertex x~() . 2. Determine the corresponding vector parameters t satisfying Theorem IX2 and Lemma IX3 ( Corollary IX22).
Note
Due to Corollary IX22, it suffices to generate t s with t N = o.
At this stage, our problem is an applied parametrie problem. Therefore, according to the algorithm for solving multiparametrie linear programming problems (MPLP) as desribed in the previous chapter, we proceed by the following steps. 9 There are several and various algorithrns for generating the set E of all efticient solutions (see, especially, [19]). The presented algorithrn has also been written for a pe, the code is available with the author. 10 At this point it should be noted that  as follows frorn a private cornrnunication frorn A.GJeroslow 1978  the algorithrn to be described, as weil as those for rnultipararnetric linear prograrnrning, belongs 10 the dass of NPalgorithrns, as detined, for exarnple, in 111.
Abridged mathematical presentation
360
I. Find, in the master associated with Pu, all columns with j ri Pu in which there exists at least one positive element Yij. Denote by Pu the set of subscripts j for which this condition is satisfied. 2. Solve the auxiliary problem, called the Etest regarding Pu, i.e., (IX14) subject to K
L L1zf t
k
+ Sj = 0 for all j ri Pu,
(lX15)
k=1 K
Ltk = I,
(IXI 6)
k=1
tk ;::: 0 for all k, Sj
;:::
0 for all j ri Pu.
Denote by Pu ~ Pu the set of subscripts j ri Pu such that for all j E Pu the min 0 and Sj nonbasic variable (NBV). Due to Corollary IX22, the solutions
Sj
=
t =
(:!) ;: :
0,
corresponding to the "optimal" solutions with min Sj = 0 and Sj NBV, j E Pu, have the properties required by Theorem IX2 and Corollary IX22. 3. Introduce the lists V h = {Pu}, Wh = WhI U ['(Pu)  V h and Y h = {t~} ,V h consists of nodes such that, for each of them, the corresponding master is already generated and, consequently, all (efficient) adjacent neighbors to all these nodes are known. Wh consists of those nodes, the neighbors of which have not yet been explored; consequently, we do not know yet if there exist basicindices among these neighbors that have not yet been listed. List Yh consists of the vector parameters t~ ;::: 0 which have been found to be assigned to the nodes Pu E V h . 4 . Choose Pv E Wh and perform a primal simplex step in the master associated with Pu in order to obtain the master associated with Pv. 5 . If W h+ 1 = 0, go to Part 2 of Phase 2. Otherwise go back to Step I with Pu' instead of Pu .
Phase 2 Part 2
=
6. Suppose that Wh 0. Compare t~ E Y~ ( for all u) and collect those which are identical. Due to Theorem IX2, Corollary IX21, and Corollary IX22, the corresponding vertices associated with the respective nodes generate an efficient face. The union of all efficient faces defines the set E of all efficient solutions to (IX2) (compare P9, 25]).
Special cases
361
IX4 Special cases IX41
"Dual" degeneracy
Suppose that x~) E E, i.e., there exists an admissible parameter vector t ~ 0 to the corresponding auxiliary problem (lX14) through (IX16). Assume that in the column j' E P fixed in the master associated with p L1z} = 0 for all k = I, ... , K
(lXI7)
holds. Then, K
L Otk ~ 0, j = j',
(IXI 8)
k=1
is satisfied by arbitrary tk for all k; hence, also by tk > 0 for all k. Thus, column j' has to be considered in the master for generating set Pu,
IX42 Set E is not compact Let X ":t 0 be not compact, Two cases are considered:
f
~
X be an unbounded face (see Definition IXI).
Case A: P' 0 exists such that x Eint F" is an optial solution to (lX14) with respect to t = tO. Case B Let x~) E F(} be a vertex of X associated with p, i.e., x~) is an efficient solution, iff there exists t(} > 0, etc. according to Theorem IXI. The objective function Zk(X) is c1early unbounded over FO for at least one k E {I, .. . , K}, say kO, which implies that Yij ~ 0 for all i E {I, ... , m },j 4 p fixed, and L1Zjk ~ 0, as follows from the theory of linear programming. Set t = tO into (lX4) and solve. Evidently, the solution is x~) with L1z/t(}) = O. This implies xi=YiYijxj>Owithxj>O, alliE {I , ... ,m}.
(lX19)
By assumption, Yij ~ 0 and let x~) be (primal) nondegenerate. With Xj > 0, the solution x~)(Yj) Eint FO, and, at the same time, x~)(Yj) is an optimal solution to (lX4) with respect to t = t(}. This implies that for all x E FO there exists tO > 0 such that x solves (lX4) with respect to t = tO . Hence, FO is efficient.
362
Abridged mathematical presentation
Implications for the procedure: none.
Note In the subprogram (IX14) through (lX16), let t O 20 corresponding to the basicindex p be found, and Sj,. = O,jo fi p. Look at the master associated with p, column jo. Here, Yij" ~ 0 for all i. This signifies that FO is unbounded. From (IX19), it follows that FO is efficient. If min Sj .. > 0, then c1early FO is not efficient.
References
[I] Aho, A.Y., J.E. Hopcroft, lD.Ullman, The design and analysis of computer algorithms, AddisonWesley, Reading Mass 1974, pp. 372 f [2J Charnes, A.: Goal programming and multiple objective optimization, Part I, Work .Paper, Center of cybernetic studies, University of Texas, Austin, 1975 [3J Charnes, A.: Goal programming and multiple objective optimization, Part 2, Managern. Sci . Res. Rep. No 381 , CarnegieMellon Univ., 1975 [4J Ecker, J.G., l.A.Kouada: Finding efficient points for linear multiple objective programs, Math . Progr. 8 ( 1975) 375377 [5 JEcker, lG., l.A.Kouada: Generating all efficient faces for multiple objective linear programs, CORE DP 1975 [6J Fandei, G .: Optimale Entscheidungen bei mehrfacher Zielsetzung, Lecture Notes in Economics and mathematical systems No 76, Springer, Berlin 1972 (7) Focke, J.: VektormaximumProblem und parametrische Optimierung, Math. Oper.Forsch. Statist. 4 (1973) 365369 [8] Ga), T. : A general method for determining the set of all efficient solutions to a linear vectormaximum problem, Eur. J. Oper. Res. I (1977) 307322 [91 Gal, T., H. Leberling: Relaxation analysis in multicriteria linear programming: An introduction, In: Advances in Operations Research (M.Roubens, ed.), North Holland Publ. Co., Amsterdam 1977 [10) Gal, T., H. Leberling: Redundant objective functions in linear vectormaximum problems and a method for determining them, Eur. J. Oper. Res. I (1977) 176184 [111 Gal, T., H. Leberling: Über unwesentliche Zielfunktioen in linearen Vektormaximum Problemen, In : Proc. in Operations Research ( H.N.Dathe et al. , eds.) , Physica Verl. , Würzburg 1976, pp. 14141 1121 Geoffrion, A.M .: Proper efticiency and the theory of vectormaximization, J.Math.Anal. Appl. 22 ( 1968) 618630 (13) Isermann, H .: The enumeration of the set of all efticient solutions for a multiple objcctive program, Oper. Res. Quart. 28 ( 1977) 71 1725 [141 Kornbluth, J.H.: Duality, indifference and sensitivity analysis in multiobjective linear programming, Oper. Res. Quart. 25 (1975) 599614 [I5J Leberling, H. : Zur Theorie der linearen Vektormaximumprobleme, PhDThesis, RWTH Aachen, February 1977 (16) Pareto, Y. : Cours d ' economie politique, Lausanne, Switzerland, Rouge 2 Vols, 189697 [17] Philip, J.: Aigorithms for the veclormaximum problem , Math.Progr . 2 ( 1972) 207229 [181 Roy, B.: From optimization on a tixed set to multicriteria decision aid, In : Proc. on MCDM. Lecture Notes in Economics and mathematical Systems No 123 (M .Zeleny, ed .) , Springer, New York 1976, pp. 283286 [19J Steuer, R. E: Multiple Criteria optimization : Theory, Computation, and application, Wiley 1986
364
References
1201 Steuer, R.E. : AOBASE: An adjacent efficient bases algorithm for solving veetormaximum and interval weighted sums linear programming problems, J.Marketing Res. 12 (1975) 454455 [21] Steuer, R.E.: AOSENB : An algorithm for solving linear programming problems with interval objective function coefficients, College of Business and economics WP, University of Kentucky 1976 [22] Wets, R., C. Witzgall : Algorithms for frames and Iinearity spaces of cones, J.Res. Nat. Bur. Stand. B. Math.Math.Phys. 17,1967 123] Wets, R., C. Witzgall : Towards an algebraic characterization of convex polyhedral cones, Numer. Math . 12 (1968) 134138 [24] Wolf, K.: Sensitivity analysis in vectormaximum problems: A eonedominance representation, PhOthesis, FernUniversität Hagen, 1988 [25] Yu, P.L. : Multiple criteria decision making. Concepts, techniques, and extensions, Plenum New York, London 1985 [26] Yu, P.L. (ed.): Forming winning strategies, An integrated theory of habitual domains, Springer, Berlin, Heidelberg, New York 1990 [27] Yu, P.L., M. Zeleny: The set of all nondominated solutions in linear cases and a multicriteria simplex method, J.Math.AnaI.Appl. 49 (1975) 430468 [28] Zeleny, M.: Multiple criteria decision making, McGraw Hili, New York 1982 (29) Zionts, S., J. Walenius: An interactive programming method for solving the multiple criteria problem, Managern. Sei . 22 (1976) 652663
Chapter ten
10 101 1011 1012 1013 102 103
Possible applications of sensitivity analysis and linear parametric programming . . . . . . . . . Changing the values of basic variables . . . . Changing the values of the basic slack variables Changing the values of basic real variables Nonbasic variables and the righthand side Inconsistency of the constraint set Some remarks on redundancy and parametrization References. . . . . . . . . . . . . .
367 367 367 368 368 369 373 379
10 Possible applications of sensitivity analysis and linear parametrie programming
In this last chapter, we shall indicate, very briefly, different possibilities of applying sensitivity analysis, and linear parametric programming in various investigations, decision making, etc.
101 Changing the values of basic variables If, for whatever practical reasons, there is the need to change the value of some of the optimal values of the basic variables, this can be done easily by changing I. The righthand side, because xB = B 1b (see Chaps. 3 and 4); 2. The elements aij of the matrix A ( see Chap. 8).
1011
Changing the values of the basic slack variables
If there is the need to change the optimal values of some of the basic slack variables, this can easily be done on the basis of the following theorem. Let, here, Ps c P denote the set of subscripts of the basic slack variables. It is then ovious that P  Ps = PR is the set of subscripts of the basic real variables.
Theorem 101 Let Xj" jr E Ps, r E I fixed, be a slack variable assigned to the kth inequality ~ in the original constraints system and let Xj, be a basic variable with the value Yr in the optimal solution. If the value Yr then has to be changed, it suffices to change the element bk of bin Ax =b according to bk(J."k) =bk + Ab Ak E [ Yr , + 00); the optimal value of the objective function does not change. The proof is straightforward. If Xj,. is a surplus variable which has been assigned to the kth inequality C: in the original system of constraints, and if its value in the optimal solution is Yr, then obviously Ak E ( 00, Yr ]. Since changing the value of a single basic slack variable does not influence the values of the other basic variables, it is possible to change the values of several basic slack variables simultaneously without affecting the optimal value of the objective function or the values of the remaining basic variables .
368
Possible applications of sensitivity analysis and linear parametrie programming
1012 Changing the values of basic real variables Consider the basic real variable Xj, ,jr E PR, r E I fixed, and let its value Yr be changed. From what has been said in Chaps. 3 and 4, it follows immediately that such a change can be brought about by changing an arbitrary component b i of the vector b insofar, of course, as Pfi :1= 0 or ßik :1= o. Assurne we now have a change of bk by Ak; at the same time, Ak E [6:k' Xk1. Assurne that the value Yr of the variable Xj"jr E PR fixed, is to be changed to the value g by varying the component bk of b. From Yr(Ad = Yr + PfrkAk and xl: = Yr(Ak) = g, or Yr(Ak) = Yr + ßrkAb it follows that
or g = Yr + ßrkAb i.e., Ak = g  Yr , ßrk ßrk
:1=
O.
If, at the same time, Ak E [6:k' Xk] = A~), the proposed change can be carried out within the framework of the basis B. If, however, Ak 4. A~), we either have to select another value g' or pass to another optimal basis (if such exists). The case using several parameters is similar, though somewhat more complicated. We shall not go into this case here and leave the corresponding derivations to the reader.
1013 Nonbasic variables and the righthand side In Chap.2, we discussed the problems of suboptimal solutions or the influence of non basic variables on the solution. In Ex. 21, we ascertained that 20 units of quantity of product PI could be manufactured at most. Quantitatively speaking, this requirement, which might, for example, be caused by market demand, could be much higher, however. It would then be desirable to carry out not only the investigations envisaged in Chap. 2, but also an analysis of the possibilities of extending the feasible region of a nonbasic variable by suitable interventions in the model. For the sake of simplicity, changes to the righthand side were used for this. The different possibilities of extending the feasible region of a non basic variable should be derived from the quantity Q~ln' since, from the formula
369
Inconsistency of the constraint set
Q~ln = m.in {~IYik > Yik I
o},
it follows that when Yik is constant, the upper endpoint of the feasible region changes with Yi. If a change in bi can be brought about in practice, then the endpoint Q ~ln can, thus, easily be shifted.
102 Inconsistency of the constraint set In practical applications of linear programming, the number of columns and rows of matrix A can rise to something in order of 104 or much more. It can quite easily occur that, in the construction of such a largescale model, the system of constraints is contradictory. In other words, the solution set can be empty, i.e., the given problem has no solution whatsoever. We know that, as a rule, this fact becomes apparent in the result by the optimality criterion (P zT 2 0) and the feasibility criterion (XB 20) being satisfied, while there is at least one artificial variable with a positive value among the basic variables. Let us call such a solution quasioptimal for shorl. Without wishing to go into farreaching theoretical considerations, the above mentioned fact can be explained as folIows . In Chap.I, we gave abrief description of the "twophase simplex method". We showed there that, introducing artificial variables Pi, the objective function can be represented as the sum of two functions ZN and Zp (cf. (127) through (I28b»: ZN
n
I:>N
j=1
j=n+1
= LCjXj +0 L
u:>m
Xj,Zp
= LPi. i=1
If the problem max ZE = ZN + Zp, S.l. Ax = b, x 2 0, is to have a finite, optimal solution, then we must necessarily have Zp = 0, i.e., Pi =0 for all i = I, ... , u ~ m. If this is not the case, i.e., Pi > 0 for at least one i, the solution set of the original constraints set is empty. We then say that this system is inconsistenl. Suppose that, in the problem: maximize Z = cTx
subject to Ax = b,x 20, the solution set is empty. We shall then have to ask ourselves how the inconsistency of the constraints can be removed (if possible at all). A proposal how to solve this problem may be discussed using the following example.
Example 101 Maximize
370
Possible applications of sensitivity analysis and linear parametrie programming
Figure 101
subject to XI + x2 ~ 7 5xI + 3X2 :s; 15
8xl + 3X2 XI ~
0, X2
~
24
~
O.
Fig. 101 represents the solution set and, as this figure shows, the solution set is empty. The solution set could obviously be converted into a nonempty set either by changing the slope (changing aij) of one or more suitable boundary lines or by changing one or more elements bi of b. Since we showed in Chap. 8, that, although the changes of aij are basically possible, they do involve a large amount of computation, we can try to remove the inconsistency ofthe constraints by changing bi. For this we have, first, to calculate the quasioptimal solution of the given problem . The initial tableau is shown in Table 101 and the final tableau in Table 102. At first sight, the simplest way of removing the inconsistency consists in introducing b(},.) = b + E},., i.e., bi(},.) = bi + Ai for all i. If, namely, Pi = Ys > 0, i E p, then it obviously suffices to set< = y,. Since ß' = eS, we obtain Ys(A~') = 0, and, by means of one dual simplex step, the artificial variable Pi is eliminated from the basis B. If we apply this procedure to all artificial variables in the basis successively, the solution we obtain is, admittedly, degenerate, but it is feasible . This has removed the inconsistency.
Inconsistency of the constraint set Table 101
371
Initial tableau I
2
3
b
4
PI
I
I
I
0
7
5
5
3
0
0
15
8
3
0
I
24
7
4
I
I
31
p?  ~Pj
Table 102
Final tableau  quasioptimal solution xB(A. *)
Po
I
3
PI
2/3
I
0
1/3
2
I
1/3
0
0
2
5/3
0
0
1/3
5
0
1/3
0
5
P2
12
0
I
I
19
0
I
I
0
~Pj
38/3
I
I
413
I
0
Table 103
4
xB
5
°EA.
11
I
4/3
Optimal solution XB(A.*)
IEA.
PI
I
5
xB
3
2/3
1/3
2
0
1
1/3
0
2
5/3
1/3
5
5
0
1/3
0
4
12
1
9
0
0
I
1
l1zj
7/3
2/3
10
10
0
2/3
0
We can now have a look at this procedure using our example. Setting b(A) = b + EA, we obtain in the quasioptimal solution the term PEA; this we already included in Table 102. Set A~ = 2, A; = 0, Ai = 9; this yields XB(A" ) = (0, 5, O)T. In the first (dual) step, PI is eliminated, in the second P2. The corresponding tableau is shown in Table 103. With A* = (2, 0, _9)T, the vector b(A*) = (5,15, 15)T, and the new solution set is shown in Fig. 102. The values of the parameters Aj , i = 1,2,3, can, of course, be chosen ditferently as (2, 0, 9l within the framework of the critical region. This depends on the meaning ofbj and, especially, on whether the values Aj found first can also be factually substantiated.
372
Possible applications of sensitivity analysis and linear parametric programming
Figure 102
Table 104
xI as basic variable with CI(t~ ::: 10/3
P2
4
3
1/18
5/18
3/2
I
5/1 8
1/18
2
5/36
7/36
25/4
0
7/36
5/36
I
1/12
1/12
3/4
0
1/12
1/12
~Zj
7/36
17/36
47/4
0
17/36
7/36
~hj
1/12
1/12
3/4
0
1/12
1/12
5
2n.
xB
Incidentally, in this case there does not exist any neighbor, since, according to Table 103, all Yij are positive. If we would, nevertheless, like the variable x I, for instance, to become a basic variable, we have two methods open to us: (i) we can change the values of suitable aij ; (ii) we can change so me cost coefficients Cj' For the sake of simplicity we shall opt for changing Cj' For CI (t I) = CI + tl , we obtain, from table 103, the critical region ( 00, 7/3 J for tl. If we set tl = 7/3, we obtain ~Zl(t~) ::: 0 and XI can be inc\uded in the basis; on the other hand, X4 is eliminated. The result is shown in Table 104. In the new basis B2, of course, all sorts of other analyses can be carried out, such as were described in the preceding chapters. Finally, we should point out that it is not necessary to set b(X) ::: b + EX. Whether b(X) ::: b + FX, F t:. E, or b(A.) = b + fA. is chosen will depend on the specific situation.
Some remarks on redundancy and parametrization
373
Figure 103
103 Some remarks on redundancy and parametrization In Secs. 25 and 112, we dealt with redundancy of linear inequalities. With regard to an LP, there are so me constraints which can be weakly or strongly redundant. An algorithm for determining such constraints is to be found in Sec. 112, and Ex. 22 illustrates this. In order to show so me relationships between redundancy and parametric programming, we shall start with an illustrative example.
Example 102 Maximize the profit Z=
2X I+X2
subject to he capacity restrictions XI + 4X2
~
24,
(I)
5xI + 6X2
~
50,
(2)
4xI + X2 ~ 36,
(3)
2xI + 3X2 ~ 50,
(4)
XI~0,X2~0.
In Fig. 103, the feasible set X and the optimal solution are displayed. For convenience, denote the slack variables by si, i = I, ... , 4. The optimal solution is shown in Table 105. Note that SI = Xn+l = X4, S2 = Xn+2 = Xs, etc. Then, Po = {I, 2, 4, 6}.
374
Possible applications of sensitivity analysis and linear parametrie programming
Table 105
Optimal solution xB
of
0.74
11 .05
1.58 Ö
0.21
0.26
1.05
0.53
XI
0.05
0.32
8.74
0.26
s4
0.53
0.16
29.37
2.26
0.11
0.37
18.53
1.05
Po
s2
sI
0.79
x2
~Zj
s3
Assurne that constraint (4) represents a capital restriction, constraints (I), (2), (3) machinecapacity restrictions. Looking at the optimal solution in Table 105, we see that S4 = 29.37, i.e., there is some "released" financial source. From what has been said in the preceding sections, we know that it is no problem to adjust b4 = 50 formally by subtracting 29.37 from 50. Then S4 becomes zero in the optimal solution and we could say that under such conditions the "superftuous" capital has been fully used . The question of course arises (and only this question) as to how this "released" capital should be used and whether it is sensible simply to adjust b4 as indicated. Let us, therefore, put the question another way. We would like to use that "released" capital for relaxing at least one of the machinecapacity restrictions in order to manufacture more and thus to increase the profit. Note In this small illustrative example, we naturally do not deal with marketing. We simply ass urne that all we produce can be sold. Other questions can be left to the reader, such as whether it is worth investing the "released" capital to enlarge the machine capacities on the basis of a comparison of the "profit effect" gained and the money invested and many other specific and/or economic questions. In this connection the reader may be referred, for example, to riO). Denote by A the amount of money (in some convenient monetary unit), which we will use from the source b4 = 50 and suppose that bj(A) = bj + fjA, i
= I, .. . ,4,A ~ O.
Here, [f;) = ith capacity/money, i = 1,2,3, [A) = money in the same units as b4, and f4 represents the share of the "released" capital that has to be used for investments (relaxing the machine capacities). Let f = (0.79,4.47, 1.58,1.I6)T. The vector °f transformed into basis Bo is already attached to Table 105. Let us now ask whether there is some redundant constraint among (I), (2), (3), because, if such a case occured, any relaxation of such a capacity would be pointless.
375
So me remarks on redundancy and parametrization Table 106
0.21
0.36
27
Note that we al ready know, from Fig. 103, which of the constraints is redundant. However, in general cases, it is not possible to draw the corresponding picture and, therefore, we shall do this in order to "simulate" areal case, as though we did not know wh ich of the constraints is redundant. Using the results of Sees. 25 and 112, we may state: min S2 = min S3 = 0 and S2, S3 nonbasic variables (NBV); hence, constraints (2) and (3) are nonredundant or binding. Let us forget fand A for a while. Perform a "normal" sensitivity analysis of the optimal solution and call the corresponding parameters tj, i = I, . . . ,4. We obtain (see Table 105): tl E
[11.05, +00), t2
E
[5,14], t3
E
[14.93,4.04], t4
E
[29.37, +00).
The following assertion is trivially true and need not be proven formaIly. n
Theorem 102 Ifthe kth constraint in
L: ajjxj
~ bj, k E {I, ... , m}, is redundant,
j=1
then in bk(tk) = bk + tk the parameter tk becomes either!k = 00, or tk = +00. Note that this assertion cannot be converted. Hence, !k = 00 or tk = +00 is a necessary, but not sufficient condition if the kth inequality is to be redundant. From what we have found above, we may suspect that constraints (I) and (4) are redundant. To find out whether this is true, use the results of Sees. 25 and 112, particularly Redundancy Criterion 2. Try to minimize SI. As pivot element Yl3 results. Hence SI can be eliminated, which implies that there exists a solution with min SI = 0 and SI NBY. Thus, constraint (I) is nonredundant or binding. Try to minimize S4 . Then, the same pivot element Yl3 results. It is, therefore, necessary to perform at least one pivot step. In order to perform this as concisely as possible, transform only the fourth row of Table 105 with respect to the pivot element y 13 . We obtain Table 106. Hence, min S4 27 > 0 and S4 BY. This implies that constraint (4) is strongly redundant. This is valid, of course, with A = O. Now perform the parametrie procedure regarding fand A with the following auxiliary condition : proceed until at least one of the constraints (I), (2), (3) becomes redundant or constraint (4) becomes nonredundant. With respect to A ~ o we have, from Table 105,
=
AE [0,7]
=A(O) .
376
Possible applications of sensitivity analysis and linear parametrie programming
Table 107 Ir
PI
sI
s3
XB
S2
1.27
0.93
14
x2
0.27
0.07
4
0. 11
xI
0.07
0.27
8
0.37
S4
0.67
0.33
22
1 .27
0.13
0.47
20
0.84
t.zj
2
'A. E [7,18.171 = A(I)
Before proceeding with the parametrie procedure, let us find out how the redundancy of (4) depends upon 'A.. Setting 'A. = X,(o) = 7, we obtain, from Table 105, S4(x'(o» = 29.37  2.26 x 7 = 13.53 and, in Table 106, we obtain I 1.16 instead of 27. Since S4('A.) is a linear function, we have min S4(0) = 27 > 0, s4(27) = 11.16> 0, i.e., constraint (4) remains strongly redundant for all 'A. E A (0). Constraints (2) and (3) remain binding for all t E A (0) since S2 S3 and both are NBV, so that their values are independent of 'A. regarding Po. Furthermore, SI ('A.) = 11 .05  1.58'A., SI (x,(O» = 0, but SI ('A.) does not reach its minimum regarding
= =
°
PO' Since none of the constraints (I), (2), (3) has yet become redundant and constraint (4) nonredundant, the auxiliary conditions are not yet met, so that we now perform a (dual) step with the pivot element Yl2 = 0.79. This yields Table 107. From Table 107 it follows that min S2('A.) = 14 + 2/.... With 'A.' = 7, 'A." = 18.17, we obtain min S2('A.') = 0, min S2('A.") = 22.34. Hence, for 'A. = 'A.' , constraint (2) is weakly redundant and, with increasing 'A., i.e., for all 'A. E (7, 18.7], constraint (2) is strongly redundant. This finishes the parametrie procedure with respect to he auxiliary conditions. Nevertheless, let us have a look at the constraint (4) : min S4('A.) = 22 1.27'A., i.e., min S4('A.' ) = 13.11 > 0, min S4 ('A." ) =0. Since, if we proceed with the parametric procedure, the pivot row would be the fourth row, S4 would become NBV and min S4 = O. Hence, the auxiliary condition, that constraint (4) has to be nonredundant, is also met.
Some re marks on redundancy and parametrization
377
The particular aspect we have dealt with above was designed to show that, among other things, redundancy can be used as a stoprule for parametrie programming. At the very end, let us note that for transportation problems, in general for any kind, and even more general, for integer programming problems, there are worked out corresponding theories and procedures for parametrizing the corresponding models (see, for example, [1 , 2, 4  9], [3] and the references quoted therein, and the bibliography at the end of this book).
References
rI J [21
(3) (4)
15J [6J
[71
[81 [9J [101
Balachandran. v.. G.L.Thompson : An operator theory of parametrie programming for the generalized transportation problem, Part I : Basic theory, Naval. Res. Log. Quart. 22 (1975) 79100 Balas, E., P. Ivanescu (Hammer): On the transportation problem. VI. Stability of the optimal solution with respect to cost variations, Commun. Acad . RPR 13 (1963) 325331 Bank, B., J. Guddat. D. Klatte, B. Kummer, K. Tammer: Nonlinear parametrie optimization, Akademie Verlag, Berlin 1982 Frank, C.R. : Parametric programming in integers, In : Oper. Res. Verfahren (R. Henn , ed.), Hain, Meisenheim am Glan 1967, Vol.IlI, pp. 167180 Klein, D., S. Holm : Integer programming postoptimal analysis with cutting planes, Managern. Sei. 25 (1979) 6472 Marsten, R. E., T. L. Morin : Parametrie integer programming: The righthand side case, Annals of Discrete Math. I (1977) 375390 Noltemeier, H.: Sensitivitätsanalyse bei diskreten linearen Optimierungsproblemen, Lecture Notes in Operations Research and Mathem. Systems, Springer, Berlin 1970 Srinivasan, v., G.L. Thompson : An operator theory of parametric programming for the transportation problem I, Naval. Res. Log. Quart. 19 (1972) 205225 Srinivasan. v., G. L. Thompson : An operator theory of parametrie programming for the transportation problem 11, Naval. Res. Log. Quart. 19 (1972) 227 252 Zimmermann, H.J., T. Gal: Redundanz und ihre Bedeutung für betriebliche Optimierungsentscheidungen, Zs. für Betriebsw. 45 (1975) 221236
Annotated Bibliography
The numbers in [ ] correspond to the numbering in the Bibliography (p. 385f.) Degeneracy [16,22.42.48. 117, 169,201,288,326, 349. 350, 354, 356  360, 570  572, 586,618,619,651,685,800,900,933, 967,999, 1029] Monographs on Parametric Programming [69,226,416,650,694,903] MuItikriteria Decision Making [47,65,72,76,83,85,88,89,92,103, 121,122,130, 164. 167, 170,171, 174, 178,182,195,212,225,228,229,232, 234, 251, 272, 273, 290, 294  296, 298  300, 343. 348, 361  363, 384, 387,397,434,436,453,456,483.484, 490  494, 498, 506. 520, 528, 531, 542,543,549,576,577,597,641,676, 686.736.740,742,746,747, 755, 757, 758,778 780,789,790,801,802, 819, 858  866, 918, 930, 932, 935, 943, 957, 964  966, 975, 985  998, 1007  1010. 1016, 1018, 1021  1024) Parametrization and degeneraey [9, 619] and redundaney 1344] ofbottleneek LP [814] of eomplementarity problems [189, 529, 591,602,645,620,674,718,725,94] of dynamie systems [9311 of fraetional programming [657, 753. 803, 9141 of geometrie programming [209, 210. 743] of integer programming [115. 116. 142, 156.250.302,322,388,472.473,51 I. 512.515.539.605.631.639,680,690, 691,693,707.749,750,786.872,907, 908] of linear programming
Theory [91,99,137,138,173,226,
247.312,351,353,372.373,345, 405,411,412,440,469,470,503 505,530,565,589,614,615,633, 663,665,675,699.709,730.738, 739.751,773,781.793,794,799, 810,813,839,846,856,903.929. 948,958,961,982.1012,1031] Methods regarding the righthand side b sealar parameter [75. 151, 162, 226,
332.441,485,530,533,541, 616,626,677,809,813,835, 848,890,891,924,925,968, 972,920] veetor parameter [340, 364, 403, 596,845,1001,1003] regarding the eost eoeffieients c sealar parameter 1152, 153, 226,
263,375,445,544,697,733, 809,835,890,891,972] veetor parameter [340, 364, 370 372, 642, 643, 653, 654, 695 698,892,903,1005J Regarding the eoeffieient matrix A sealar parameter [31, 102, 155, 199,
208,214,226,313,314,323, 383,535,552,553,622,669, 722,833,844.9031 veetor parameter [258, 260, 333 336,338,339,412,563] regarding b + c sealar parameter [261,383,471,
700,708.796,854,874,945, 947J veetor parameter [337, 346, 723. 903.948, 1004, 1006]
382 regarding c + b + A [190 192,226,312,514,515,556, 637,903,944] with upper bounded variables [735] with nonlinear functions of parameter(s) [245, 246, 312, 798, 946,949,981] Usage lor solving 01 complementarity problems [189, 291] decomposition [I, 375, 420, 604, 90 I J fractional programming [486] fuzzy LP [154,163] integer programming [547, 607,193] linear equations system [242J multicriteria programming [89, 103, 167, 171, 174, 222, 225, 226, 228, 232,319,348,432,746,755,757, 758,995,998, 1007, 1009J quadratic problems [241, 732, 939) stochastic shipping problems [19 J Applications in binpacking [365J chemistry [459,627] decision theory [704] economy 1119,271,282,315,316,522, 551,621,700,825  827, 919, 10131 education [480J farm decision [394,462, 510, 564, 634, 635,638,728,841) fish industry [868] metallurgy [628J military decisions (845) inventory [971) portfolio selection [716] ofnetwork type problems [14,15,219, 488,489,583,656,715,737,770,792, 885) of nonlinear programming Theory [35,138  141, 177,270,276, 278,287,303,308,309,311,377, 385,386,398,415,416,469,523, 524, 526, 557  562, 569, 582, 594, 600,624,745,752,882,883,898, 899, 928, 960] Methods in convex [413,926) general [772]
Annotated Bibliography integer [70, 71) linear fractional [11, 36, 38, 39, 159, 161,421,502,629,913] nonconcave 1774) nonconvex [90,376) nonlinear fractional[224] quadratic [419,534,417,601,688, 726, 879J strictly concave [385, 386] usages lor .I'olving 01 complementarity problems [646J fractional [20, 221,487,518,973,974] fractional integer [495] nonconvex [466,499) nonlinear equations systems [451] quadratic [501,590,640,721,724, 726,731,734,837,880,906,976) reliability problems [176J of quadratic programming [93, 534, 921, 9271 of transportation type problems [57  61, 238,455,497,818,53  56,320,547, 849  851] Redundancy [125,274,342,347,352,532,608, 636,887, 888, 895, 10 191 Sensitivity Analysis general view (considerations, approach) [2,84,227,318,516,763,764,804, 828,916,917J and degeneracy [16, 22, 42, 288, 356, 570  572] and redundancy [356] of assignement problem [741] of computer simulation [7911 of complementarity problems [243,423, 509,717) of data envelopment analysis (175) of decision models [474] of dynamic programming [687J offuzzy problems [549,606,683,701, 702, 884J of game theory [4, 158, 374J of generalized equations [554] of geometrie programming [220, 593, 7711 of global optimization [437J
383
Annotated Bibliography ofinputoutput model 149, 2851 of integer programming [79, 134, 186,
404,513,567,581,692,760,812,829, 840, 940, 979J of linear fractional [10, 52] of linear least squares problems 161 1 of linear programming (LP) [24, 25, 28, 74,41,100,112.135.160, 188.203, 226.227,269,279,315,321,329331,333,334.391,410,450.548,617, 669,678,679, 710, 719, 765  769, 775,782  784, 797. 822, 834. 836,847,889,951  956, 980J of Markovprocesses [327,435,655] of multicriteria problems 165, 195, 361 363,434,438,576,577,975] of network type problems [30, 196. 197, 219, 283,284,326.395,396,433.464, 481,496,611,670,977.9841 of nonlinear programming [33, 34, 43 46.127,149,194.216,239,289,304, 305,306,324,328,378,407,418,442, 448,449,479,500,525  527,592, 603,678,689,762,777,805,830832, 878, 881, 882. 938, 1025  1028 J of quadratic programming 13. 128,404, 414,418,521,9051 of queueing problems [8701 of salesman problem [5781 of stochastic programming [32, 264 267.546,806,820,8231 of stochastic networks [51, 978] of transportation type problems [536 538] Applicatiolls in chemistry [111,475,6731
°
decision analysis 1828] economy (investment, budgeting, ete.)
121,118,119,120,213,218,235, 326,422,476.477,548, 573,666, 727,817,852,950] farm deeision 1132, 644, 649, 711, 728, 759,824,959] gravity model [2811 inventory 123] location problems 1166, 5951 pallet loading [244J radiology [869. 897J shipping problem [19) systems design 1785] waterresource [307,712,713] Systems Theory and Analysis (4  8,18,29,40,67,82,86,87, 104110,113,133,143,144, 183,276,277, 280, 317, 333  336, 341. 402,406, 408,424  427,429  431,439.454, 458,508.550,632,664,807,815,821, 855,857,873,876,923] Textbooks of linear programming or of OR in whieh sensitivity analysis and/or parametrie programming is diseussed at least briefly and mostly for the sealar eases
126,80,136,172,180, 181,184, 200, 240,30 1,355,367  369, 382, 392, 400,401,428,447,457,463,519,575, 584,585,609,671 , 721,838,877.886, 903,922,934,937,1014,1020] of multieriteria deeision making 166,483, 484,490.549,742,866,894, 918,932, 943,964,990,991,1007, 1009)
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Index
A
B
abbreviated form of the simplex tableau, 18 activity vector, 4 additional constraint, 177, 300 adjacent basicindex, 139 nodes, 60, 138, 183, 356 admissible parameter, 114, 178 region 114 region of parameters in the RHS, 114, 178 vector parameter, 100, 178 algorithm for the multiparametric RHSproblem, 186, to compute the set of all efficient solutions, 346 to determine redundant constraints, 71 alternative solutions, 26, analysis postefficient, 352 postoptimal, 311 sensitivity with respect to b 83, 88 with respect to c 209 suboptimal, 40 analysis relaxation, 352 anticycling rule, 76 approximation region, of non basic variables, 44, 68 of parameters, 95, 105,229 apriori systems analysis, 111 artificial variable, 5 ascending proccss, 113 augmented objective function, 15 auxiliary calculations, 146 conditions, 190, problem, 150,192, 248, 360
basic condition, 313, 323 cost vector, 31 solution, 6, 9,29, complete, 12 dual, 13 optimal, 29 variable, 6, 28 changing the value of, 367 dependence of parameters, 103 dual, 20 primal, 20 basicindex, 7,28, adjacent, 139 optimal, 7 basis, 6, 28, efficient, 355 basis exchange with respect to parameters in matrix A, 325 basisset of adegenerate vertex, 73 binding constraint, 71 boundary hyperplane of the admissible region of parameters, 133
c capacity restriction, 13 changes of matrix A, 319 of a column of matrix A, 320 of a row of matrix A, 327 the righthand side and the cost coefficients, 273 the value of the basic variables, 367 the value of the basic real variables, 368 the value of the basic slack variables, 367 coefficient vector of a parameter in the RHS, 91
Index
434 complete basic feasible solution, 9,12, 31 complete basic solution, 9, 31 compromise decision, 348 compromise programming, 340 compromise solution, 339 computation of set of all efficient solutions, 355 constraints, 1/4 additional, 177, 300 binding, 71 convex hull, 27,341 polyhedral set, 11, 27 polytope, 11, 27 set, 27 cost coefficients, 5 crash methods, 15 criterion elements, 32 criterion row, 32 critical interval, 85 determination of, 86 point, 89, 112 region, 179 of a parameter in the cost coefficients, 262 of a parameter in the RHS, 85, 102 of the parameters in the RHS, 100 of parameters in the matrix A, 312, 322 value, 86, 114 cycling, 54
D degeneracy and parametric programming, 147, degeneracy degree, 56, 73 graph, 54 power, 73 degenerate convex polyhedron, 54, 73 solution dual, 23,33 primal, 33 vertex , 573 dependent parameters, 275 descending process, 113
dominance region, 341 dual basic solution, 13 dual degeneracy, and parametric programming, 153 in multicriteria programming, 361 in vectormaximum problems, 261 dual degenerate solution, 33 feasibility, 32 problem, 13,19 simplex method, 22 solution, 20 value, 203 duality theorem, 32
E efficiency test, 352, 360, 356 efficiency theorem, 343, 354 efficient basis, 355 face, 355 neighboring vertices, 355 neighboring regions, 356 neighboring bases, 356 points, 342 solutions, 354 the set of all 339, 355 determination of 346 vertex, 355 elimination method GaussJordan 17 entering variable, 18 extending the feasible region of nonbasic variables, 368 extreme point, 11
F face of a convex polytope, 355 01' the admissible region of parameters, 133 feasibility criterion, 15 feasible solution, 6, 28
G general degeneracy graph, 74
435
Index general solution, 12, 30 geometrie meaning of an LP, 11 of parameters in the eosl eoeffieients, 243 of slaek variables, 49 of parameters in the RHS, 88 goal programming, 348 graph, 138, 183 generated by the parametrie problem, 138,183 generated by a multieriteria problem, 356
H homogeneous multiparametrie problem, 354 homogeneous multiparametric programming with respeet to the RHS, 155,194 with respeet to the cost eoeffieients, 252,266
linear veetormaximum problem, 353 lower priee limit, 252
M marginal value, 171, 203 master problem, 196 master tableau, 192 mathematical optimum, 117 matrix generator, 6 method for determining the set of all efficient solutions, 359 minimum cover, 352 modified pivot row, 17 multiatribut decision making, 337 multicriteria decision making, 337 linear programming, 337 programming, 267 multiparametric RHSproblem algorithm for, 186 sensitivity analysis of the righthand side, 93 multiplicative parameter, 163, 200, 269
I ideal solution, 348 identity matrix, 5 inconsistency of the solution set, 369 independent parameters, 275 individual maxima in multieriteria linear programming, 338 initial basic solution, 6 basis, 6, 236 simplex tableau, 5 inner degree of anode, 75 interaetive approaehes, 340 internal node, 59,74 internal point methods, 15 inverse of a ehanged matrix, 319 isolated internal node, 62 transition node, 63, 76
L Lagrange multiplyer, 203 leaving variable, 18 linear multicriteria problem, 337
N nearby vertex, 69 negative degeneracy graph, 74 pivotstep, 57, 73 shadow price, 204, neighborhood problem, 75 neighboring bases, 60,115,181,239,262 efficient bases, 356 regions, 133, 181, 356 vertices, 60, 355 node inner degree, 75 internal, 59,74 internal isolated, 62 of a graph, 2/21 transition isolated, 63, 76 nodes adjacent, 60, 138, 183, 356 nodeset, 17 non basic cost, )0 variables, 6 influence of, 40, 43
Index
436 nonessential objective functions, 349 strongly, 349 weakly, 349 nonnegativity conditions, 4 nontrivial feasible solution, 196 Ncondition, 75 Ncorrespondence, 75 Nproblem, 60, 75 Ntreemethod, 60
o objective function dependence on nonbasic variables, 66 optimal basis, 28, 33 optimal decision, 40 optimal solution, 21, 28 optimal value function with respect to the cost coefficients, 318 with respect to the RHS, 115, 180 optimality criterion, 15 optimum degeneracy graph negative, 20 I general, 201 positive, 20 I outer degree of anode, 74 overall critical interval, 171 , 202 determination of, 171
p parameter, admissible region in the RHS , 114, 178 matrix, 319 coefficient vector, 91 parametric analysis postefficient, 352 form, 84 programming with respect to the RHS, 111 , 120 with respect to the cost coefficients, 236 Paretooptimal solutions, 339 perfect solution, 338 phase two of the simplex algorithm, 15 pivot column, 17
pivot element, 17 pivot row, 17 positive degeneracy graph, 58, 74 optimum degeneracy graph, 20 I pivotstep, 57 , 73 shadow price, 204, possible graph in parametric programming, 154 postefficient analysis, 352 preference ordering, 339 primal degeneracy and parametric programming, 148 degenerate solution, 33 feasibility, 32 problem, 19 problem (FD), 163, 199 problem (F), 156,194 problem (HD), 253, 269 problem (H), 266, 253
R real variables, 4 reduced costs, 32 redundancy, 68 and parametrization, 373 redundancy criterion I, 69 redundancy criterion 2, 70 redundant constraint 50 determination of, 71 strongly, 52 weakly, 52 region of admissible parameters in the RHS, 114,178 removing the incosistency of the solution set, 369 representation graph of adegenerate polytope, 60, 73 restriction capacity, 13 reverse simplex method, 48 RIM multiparametrie linear program with dependent parameters, 373 RIM parametric linear program with independent parameters, 275, 300
437
Index
s scalarization of a multicriteria problem, 343,354 sensitivity analysis with respect to the RHS, 83, 88 under degeneracy 167, 20 I with respect to the cost coefficients, 203 with respect to matrix A, 311 with respect to c, 209 separating hyperplane, 134 set of all efticient solutions, 340 determination of, 340 all feasible solutions, 11, 27 optimal bases of adegenerate vertex, 168 shadow price, 87, 171, 203 shadow prices under degeneracy, 171, 203 true, 175 twosided, 174,204 signunrestricted variables, 189, simplex algorithm, 14 simplex tableau abbreviated form of, 18 initial, 5 slack variable, 5 solution basic, 6, 9, 29 basic optimal, 29 complete basic 9 compromise, 340 dual basic, 13 feasible, 6, 28 general, 12, 30 suboptimal, 39 solution procedure for the RIM parametric problem, 302 for parametrization of the cost coefficients, 248, 262 for the RHS parametric problem, 185, to find the set of all efficient solutions, 359 solution set, I1 solution vector. I I solutions alternative, 26
stalling. 54 strongly Eredundant objective functions, 349 nonessential objective functions, 349 redundant constraint, 52, 68 suboptimal analysis, 40 suboptimal solution. 39,41,66 suboptimality, 65 supporting hyperplane, 134, 343, 354, surplus variables, 5 systematic parametrization in the cost coefticients, 263 in the RHS, 129,
T TNPrule, 75, 76 transition column, 74, transition node, 59, 74 transition node pivoting rule, 75 transition point in parametric programming, 114 twosided shadow prices determination of, 205
u utility function, 339
v value function 115, 180 variable artificial, 5 basic. 6,28 entering, 18 leaving, 18 vector parameter admissible, 100, 178 vertex degenerate, 73 of a polytope, 11, 27
w weakly Eredundant objective functions, 349 nonessential objective functions, 349 redundant constraint, 52, 70, 76 and degeneracy, 76