Theory and Methods of Vector Optimization, Volume 1: The Theory and Methods of Vector Optimization

196 Pages·2020·2.176 MB·English

by  Yu. K. Mashunin

Theory and Methods
of Vector Optimization
(Volume One)

Theory and Methods
of Vector Optimization
(Volume One)
By
Yu. K. Mashunin
Theory and Methods of Vector Optimization (Volume One)

By Yu. K. Mashunin

This book first published 2020

Cambridge Scholars Publishing

Lady Stephenson Library, Newcastle upon Tyne, NE6 2PA, UK

British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library

Copyright © 2020 by Yu. K. Mashunin

All rights for this book reserved. No part of this book may be reproduced,
stored in a retrieval system, or transmitted, in any form or by any means,
electronic, mechanical, photocopying, recording or otherwise, without
the prior permission of the copyright owner.

ISBN (10): 1-5275-4831-7
ISBN (13): 978-1-5275-4831-2
TABLE OF CONTENTS

INTRODUCTION
VOLUME 1. THE THEORY AND METHODS OF VECTOR
OPTIMIZATION
CHAPTER 1
VECTOR PROBLEMS IN MATHEMATICAL PROGRAMMING
(VPMP)
1.1. Problems in defining vector optimization 6
1.2. A case study of vector optimization 8
CHAPTER 2
THE THEORETICAL BASES OF VECTOR OPTIMIZATION
2.1. The basic concepts and definitions of vector optimization 14
2.1.1. The normalization of criteria in vector problems 14
2.1.2. The relative evaluation of criterion 15
2.1.3. The relative deviation 16
2.1.4. The relative level 17
2.1.5. Prioritizing one criterion over others in VPMP 18
2.1.6. The vector criterion prioritized over other criteria in VPMP 18
2.1.7. The job of the vector in prioritizing one criterion over another
19
2.2. The axiomatics of vector optimization 20
2.3. Principles of optimality in a VPMP solution 22
2.4. Theoretical results that are bound to the axiomatics of vector
optimization 24
2.4.1. The properties of one-criteria and vector problems in linear
programming 25
2.4.2. Definitions according to the theory of continuous and convex
functions 26
2.4.3. Theoretical results of vector optimization 27
2.5. A geometrical interpretation of the axiomatics and principles of
optimality in a VPMP solution 34
2.6. Conclusions on the theoretical foundations of vector optimization.
38
vi Table of Contents

CHAPTER 3.
METHODS FOR SOLVING PROBLEMS OF VECTOR
OPTIMIZATION
3.1. Geometrical solutions to linear vector problems 39
3.2. The algorithm and the practice of solving vector problems in
mathematical programming (VPMP) with equivalent criteria 43
3.2.1. Algorithm 1. Solving a vector problem in mathematical
programming with equivalent criteria 43
3.2.2. The practice of solving a vector problem in linear
programming 46
3.2.3. The practice of solving a vector problem of fractional
programming 53
3.2.4. Practicing the solution to a vector problem in non-linear
programming 57
3.3. The methodology for solving a problem in vector optimization with
the given prioritized criterion 59
3.3.1. Solving a problem in vector optimization with the given priority
59
3.3.2. Algorithm 2. Solving a problem in vector optimization with the
given priority 61
3.3.3. Theoretical analysis of the maximin problem with criterion
priorities and weight coefficients 66
3.3.4. The practice of solving VPLP with the given priority 71
3.4. The methodology for solving the problem of vector optimization
with a given value of the objective function 74
3.4.1. Algorithm 3. Select a point from the Pareto set with the given
value of the objective function
3.4.2. An example of the choice of a point from the Pareto set with the
given value of the objective function 76
3.5. Test examples of vector problems in linear programming 79
CHAPTER 4
RESEARCH AND ANALYSIS OF APPROACHES TO PROBLEM-
SOLVING IN VECTOR OPTIMIZATION
4.1. Solutions to vector problems in mathematical programming
(VPMP) based on the folding of criteria 83
4.1.1. Introduction to the study of vector problems in mathematical
programming 83
4.1.2. Solutions to VPMP based on the folding of criteria (methods of
the 1-th type) 84
Theory and Methods of Vector Optimization (Volume One) vii
4.1.3. VPMP solution-methods using criteria restrictions (methods of
the 2-th type) 86
4.1.4. Methods of target programming (methods of the 3-th type) 87
4.1.5. Methods based on searching for a compromise solution
(methods of the 4-th type) 88
4.1.6. Methods based on human-machine procedures of decision-
making (methods of the 5-th type) 89
4.2. Analysis of the results of the solution in a test example and
solution-methods for VPMP 90
4.2.1. An analysis of the results of the solution in a test example 90
4.2.2. An analysis of the results of testing solution-methods for VPMP
91
4.2.3. Conclusions on the methods for solving vector problems 94
CHAPTER 5
THE THEORY OF VECTOR PROBLEMS IN MATHEMATICAL
PROGRAMMING WITH INDEPENDENT CRITERIA
5.1. A definition of vector problems of mathematical programming
with independent criteria 96
5.2. Vector problems in linear programming with independent
criteria 97
5.2.1. Vector linear-programming problems with independent
criteria 97
5.2.2. The practice of solving vector problems in linear-programming
with independent criteria 100
5.2.3. Vector problems in linear programming with independent
criteria, in modelling economic hierarchical systems 104
5.3. Two-level hierarchical systems developing in dynamics uniformly
and proportionally 107
5.3.1. The theory of two-level hierarchical systems developing the
dynamics uniformly and proportionally 107
5.3.2. The practical solution to a two-level hierarchical system which
develops in dynamics uniformly and proportionally 111
CHAPTER 6
THE DUALITY OF VECTOR PROBLEMS IN LINEAR
PROGRAMMING (VPLP)
6.1. The duality of a problem in linear-programming 115
6.2. VPLP with a maximum vector of the objective function
(VPLPmax) and the duality problem 117
6.2.1. Construction of dual VPLPmax 117
viii Table of Contents

6.2.2. The algorithm of decision PLPmin with a set of restrictions119
6.2.3. Algorithm 4. The solution to PLP with a set of restrictions 120
6.2.4. An algorithm for solving PLP in a set of restrictions with a
restriction priority 122
6.2.5. Algorithm 5. The solution to PLP in a set of restrictions with a
restriction priority 123
6.2.6. Duality theorems in VPLPmax 125
6.2.7. Duality VPLPmax in test examples 127
6.2.8. The solution to direct and dual VPLP with a prioritized
criterion (MATLAB) 131
6.2.9. An analysis of dual problems on the basis of a Lagrange
function 134
6.3. VPLP with a minimum vector of the objective function (VPLPmin)
and a problem dual to it 137
6.3.1. Construction of dual VPLPmin 137
6.3.2. The algorithm for decision PLPmax with a set of restrictions
139
6.3.3. Algorithm 6. Decision PLPmax in a set of restrictions 140
6.3.4. Algorithm 7. The solution to PLP in a set of restrictions with a
restriction priority 141
6.3.5. Duality theorems in VZLPmin 141
6.3.6. Duality VPLPmin in test examples 143
6.4. Duality in VPLP with a set of restrictions 145
6.4.1. An analysis of duality in VPLP with a set of restrictions 145
6.4.2. Algorithm 8. The solution to direct VPLP with equivalent
criteria 146
6.4.3. Algorithm 9. The solution to dual VPLP with equivalent
criteria 149
6.4.4. An analysis of duality in VPLP with a set of restrictions on the
basis of a Lagrange function 154
CHAPTER 7
THE THEORY OF MANAGEMENT DECISION-MAKING BASED ON
VECTOR OPTIMIZATION
7.1. Management decision-making: problems and methods 156
7.1.1. Problems with the theory of management decision-making 156
7.1.2. The definition of a management decision 158
7.1.3. The classification of management decisions 159
7.1.3. Decision-making 161
7.2. A model of decision-making under uncertainty 162
Theory and Methods of Vector Optimization (Volume One) ix
7.2.1. The conceptual formulation of the problem of decision-making
162
7.2.2. An analysis of the current ("simple") methods of decision-
making 164
7.2.3. Transforming a decision-making problem into a vector
problem of mathematical programming 165
7.3. The technology of decision-making under conditions of certainty
and uncertainty 167

Bibliography 179