Progress in Mathematics: Probability Theory, Mathematical Statistics, and Theoretical Cybernetics

130 Pages·1971·4.038 MB·English

by  V. V. Kolbin (auth.), R. V. Gamkrelidze (eds.)

PROGRESS IN
MATHEMATICS
Volume 11
Probability Theory, Mathematical Statistics,
and Theoretical Cybernetics
PROGRESS IN MATHEMATICS
Translations of ltogi Nauki - Seriya Maternatika
1968: Volume 1 - Mathematical Analysis
Volume 2 - Mathematical Analysis
1969: Volume 3 - Probability Theory, Mathematical
Statistics, and Theoretical
Cybernetics
Volume 4 - Mathematical Analysis
Volume 5 - Algebra
1970: Volume 6 - Topology and Geometry
Volume 7 - Probability Theory, Mathematical
Statistics, and Theoretical
Cybernetics
Volume 8 - Mathematical Analysis
1971 : Volume 9 - Algebra and Geometry
Volume 10 - Mathematical Analysis
Volume 11 - Probability Theory, Mathematical
Statistics, and Theoretical
Cybernetics
In preparation:
Volume 12 - Algebra and Geometry
Volume 13 - Probability Theory, Mathematical
Statistics, and Theoretical
Cybernetics
Volume 14 - Algebra, Geometry, and Topology
PROGRESS IN
MATHEMATICS
Volume 11
Probability Theory, Mathematical Statistics,
and Theoretical Cybernetics
Edited by
R. V. Gamkrelidze
V. A. Steklov Mathematics Institute
Academy of Sciences of the USSR, Moscow
Translated from Russian by J. S. Wood
c:E'PLENUM PRESS • NEW YORK-LONDON • 1971
The original Russian text was published for the All-Union Institute of
Scientific and Technical Information in Moscow in 1970 as a volume of
Itogi Nauki - Seriya Maternatika
EDITORIAL BOARD
R. V. Gamkrelidze, Editor-in-Chief
N. M. Ostianu, Secretary
P. S. Aleksandrov V. N. Latyshev
N. G. Chudakov Yu. V. Linnik
M. K. Kerimov M. A. Naimark
A. N. Kolmogorov S. M. Nikol'skii
L. D. Kudryavtsev N. Kh. Rozov
G. F. Laptev V. K. Saul'ev
Library of Congress Catalog Card Number 67-27902
ISBN-13: 978-1-4684-3311-1 e-ISBN-13: 978-1-4684-3309-8
DOl: 10.1007/978-1-4684-3309-8
The present translation is published under an agreement with
Mezhdunarodnaya Kniga, the Soviet book export agency
© 1971 Plenum Press, New York
Softcover reprint of the hardcover 1st edition 1971
A Division of Plenum Publishing Corporation
227 West 17th Street, New York, N. Y. 10011
United Kingdom edition published by Plenum Press, London
A Division of Plenum Publishing Company. Ltd.
Davis House (4th Floor), 8 Scrubs Lane, Harlesden, NW10 6SE, England
All rights reserved
No part of this publication may be reproduced in any form
without written permission from the publisher
Preface
This volume contains two review articles: "Stochastic Pro
gramming" by Vo V. Kolbin, and "Application of Queueing-Theoretic
Methods in Operations Research, " by N. Po Buslenko and A. P.
Cherenkovo
The first article covers almost all aspects of stochastic
programming. Many of the results presented in it have not pre
viously been surveyed in the Soviet literature and are of interest
to both mathematicians and economists. The second article com
prises an exhaustive treatise on the present state of the art of the
statistical methods of queueing theory and the statistical modeling
of queueing systems as applied to the analysis of complex systems.
Contents
STOCHASTIC PROGRAMMING
V. V. Kolbin
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
§ 1. The Geometry of Stochastic Linear Programming
Problems. . . . . . . . . . . . . . . . . . . . 5
§ 2. Chance-Constrained Problems . . . . . . . . . 8
§ 3. Rigorous Statement of stochastic Linear
Programming Problems . . . . . . . . . . 16
§ 4. Game-Theoretic Statement of Stochastic Linear
Programming Problems. . . . . . . . 18
§ 5. Nonrigorous Statement of SLP Problems . . . 19
§ 6. Existence of Domains of Stability of the
Solutions of SLP Problems . . . . . . . . . 29
§ 7. Stability of a Solution in the Mean. . . . . . . . . . . . 30
§ 8. Dual Stochastic Linear Programming Problems. . . 37
§ 9. Some Algorithms for the Solution of Stochastic
Linear Programming Problems . . . . . . . . . . 40
§ 10. Stochastic Nonlinear Programming: Some
First Results . . . . . . . . . . . . . . . . . . . . . . 42
§ 11. The Two-Stage SNLP Problem. . . . . . . . . . . . 47
§ 12. Optimality and Existence of a Plan in Stochastic
Nonlinear Programming Problems. 58
Literature Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
APPLICATION OF QUEUEING-THEORETIC METHODS
IN OP ERA TIONS RESEARC H
N. P. Buselenko and A. P. Cherenkov ... 77
1. Use of the Analytical Methods of Queueing Theory
in Operations Research. . . . . . . . . . ...... . 77
vii
viii CONTENTS
§ 1. Application of Queueing-Theoretic Methods
for the Analysis of Communication and
Control System . . . . . . . . . . . . . . . . . . 78
§2. Transportation Problems . . . . . . . . . . . . . . . • . • 83
§3. Application of Queueing Theory in Production
Scheduling . . . . . . • . . . . . . . . . • . . . . . . . 91
§4. Organization of Commerce and Food Services. . . . . 94
§5. Planning of Public Medical Services. . . . . . . . . • . 96
§6. other Applications: Monographs and Surveys. . . . . 97
II. Application of Statistic Modeling (Monte Carlo Method)
for the Solution of Problems in Queueing Theory. • . . 98
§ 1. General Considerations. . . . . . . . . . . . . . . . . . . . 98
§2. Production Technology and Scheduling. . . . . . . • .. 104
§3. Transportation . . . . . . . . . . . . . . . . . . . . . . . .. 106
§4. other Applications. . . . . . . . . . . . . . . . . . . . . .. 109
§5. Monographs . . . . . . . . . . . . . . . . . • . . . . . . . .. 111
Literature Cited . . . . . . . • . . . . . . . . . . . . . . . . . . . .. 111
Stochastic Programming
V. V. Kolbin
The present article is a general survey of the problems of
stochastic programming. It is based on lectures delivered by the
author to graduating students of the Cybernetics Section of the Econo
mics Department of Leningrad State University (LGU) in 1967 and
1968.
The author was aided substantially by LGU graduate students
G. Tsel'mer, V. N. Tanskaya, and L. Bonits, to whom he extends his
deepest appreciation.
We shall abide by the earlier surveys [9, 12, 13] with regard
to the description of terminology and classification.
Our primary concern will be those areas of stochastic pro
gramming which have not received warranted attention in the So
viet literature, viz., duality considerations in stochastic linear
programming (SLP) problems, special algorithms for the solution
of stochastic programming problems, and, of course, stochastic
nonlinear programming (SNLP).
Introduction
Almost any problem in applied mathematics may be assigned
to one of the following two classes. The first class comprises
"descriptive" problems, in which mathematical methods are used
to process information on some investigated effect and to deduce
certain laws of the effect from other laws.
The second class includes "optimization" problems, in which
the optimum, in some particular sense, is chosen from a set of fea
sible solutions.
1
2 V. V. KOLBIN
Besides the above division of applied mathematical problems,
they can be classified by other criteria as well. In particular, a
logical division recognizes deterministic and stochastic problems.
In the course of solution of the latter an extensive mathematical
discipline has emerged in the guise of probability theory.
However, until lately probabilistic methods have been re
stricted exclusively to the solution of problems of the descriptive
type. Stochastic optimization problems have only begun to receive
attention in the last decade. The same is true of the stochastic
variants of optimal programming problems.
Nevertheless, stochastic optimal programming is an exceed
ingly important and promiSing branch of applied mathematics, cer
tainly on no small account of the fact that solutions in practice are
always subject to some measure of uncertainty. It is also clear that
stochastic programming problems are bound to be far more com
plex than their deterministic counterparts.
In the development of optimal stochastic programming pro
blems, therefore, it is totally unrealistic to hope for the rapid at
tainment of sufficiently general and effective results.
In light of the foregoing it is essential in the future treatment
of stochastic programming to systematize the work published to
date and to consolidate it into some kind of more or less unified
mathematical theory. The present article represents an attempt
in this direction.
In the majority of practical mathematical programming prob
lems the coefficients involved are subject to variations. The
nature of these variations can be twofold:
1. The possible values of the coefficients are specified in the
form of a function of one or more parameters with known
domains of variation; the mathematical description of
such problems is embodied in parametric linear program
ming models.
2. The coefficients of the problems obey a certain probabilis
tic distribution.
The branch of applied mathematics that takes account of the
aforementioned character of the variation of the coefficients has
come to be known as stochastic programming.
STOCHASTIC PROGRAMMING 3
The literature devoted to stochastic programming encom
passes approximately two hundred journal articles. These may be
divided into five groups according to the problem areas covered in
the articles.
The first group of articles is concerned with the search for
various statements ofSLP problems. They include the papers of
Dantzig [64, 65], Dantzig and Madansky [67], Madansky [111,112],
Charnes and Cooper [48, 49, 50], Tintner [163, 164, 165], Kataoka
[98], Vajda [173, 174], Bereanu [28, 29, 30], Yosifescu and Theo
dorescu [199, 200], and others.
The second group of articles is devoted to extensions of the
concepts of the stability of solutions and duality to stochastic pro
gramming problems. These problems have been treated in papers
by Madansky [110, 114, 115], Tintner [166,167], Tintner, Sengupta,
and Rao [170], Vajda [173, 174], Sengupta [136], Sengupta and Ku
mar [137], Sengupta and Tintner [138], Hadley [89], Williams [194,
195], Bereanu [29, 30], Arbuzova [1,3], and Arbuzova and Danilov[4].
In the articles of the third group the authors have endeavored
to reduce stochastic programming problems to deterministic pro
blems devoid of random coefficients. These include papers by
Dantzig [65], Ferguson and Dantzig [78], Charnes [43, 44], Charnes
and Cooper [48], Elmaghraby [73-75], Vajda [173,174], Kataoka [98],
Hadley [89], and Reiter [132].
In the fourth group of articles special algorithms have been
developed for the solution of SLP problems; the authors include
Dantzig [65], Ferguson and Dantzig [78], Madansky [114, 115], El
maghraby [73-75], Williams [193-195], Soldatov [11-13], Karaoka
[98], and others.
Finally, the last group of papers is concerned with nonlinear
stochastic programming. Only a few papers have been published
in this area, chiefly those by Mangasarian [117], Mangasarian and
Rosen [118], and Hanson [91].
Consider the set S = {X:AX:::; b, X? O}, where A = (aij) is an
(m x n)-dimensional matrix, b = (~) is an m-dimensional vector,
and x = (Xj) is a point in n-dimensional Euclidean space.
The set S geometrically represents the intersection of m
half-spaces and the positive hyperorthant. We shall assume that S
is nonempty and bounded.