This book is devoted to the problems of stochastic (or probabilistic) programming. The author took as his basis the specialized lectures which he delivered to the graduates from the economic cybernetics department of Leningrad University beginning in 1967. Since 1971 the author has delivered a specialized course on Stochastic Programming to the gradu ates from the faculty of applied mathematics/management processes at Leningrad University. The present monograph consists of seven chapters. In Chapter I, which is of an introductory character, consideration is given to the problems of uncertainty and probability, used for modelling complicated systems. Fundamental indications for the classification of stochastic pro gramming problems are given. Chapter II is devoted to the analysis of various models of chance-constrained stochastic programming problems. Examples of technological and applied economic problems of management with chance-constraints are given. In Chapter III two-stage stochastic programming problems are investigated, various models are given, and these models are qualitatively analyzed. In the conclusion of the chapter consideration is given to: the transport problem with random data, the problem of the determination of production volume, and the problem of planning the flights of aircraft as two-stage stochastic programming problems. Multi-stage stochastic programming problems are investigated in Chapter IV. The dependencies between prior and posterior decision rules and decision distributions are given. Dual problems are investigated."

Rapid methodological progress is now taking place in the USSR in the solution of the problems of developing both society and economy. A considerable proportion of the total methodological problems of the USSR economy are dealt with in the present monograph. This work is intended for economists, managers and specialists in methodology, sociology and applied mathematics, and it may also be useful to researchers into operations as well as to politicians, philosophers and wide circles of readers interested in the present and future problems of the USSR economy. Readers will find here, I hope, answers to many questions. At the same time this work can be used as a manual for students and post-graduate students investigating countries with centrally planned economies. For his monograph the author has used the material originally developed for a special course of lectures called "Macromodels of Planning." Some sections of the book correspond to the subjects of courses on "Mathematical Programming" and "Operations Research" as well as to the subjects of special courses on "Methods of Vector Optimization," "Stochastic Programming," "Parametric Programming" and "Decomposition Methods of Programming," read by the author from 1971 to 1976 to the graduates and post graduates of the department of applied mathematics and management processes at Leningrad University."