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.

I. Risk and Uncertainty in Problems of Planning and Management.- 1. Uncertainty and Probability in the Problems of Planning and Management for Complicated Systems.- 2. Various Probabilistic Approaches Used for the Description of Complicated Systems.- 3. Basic Indications for the Classification of Stochastic Programming Problems.- II. Chance-Constrained Stochastic Programming Problems.- 4. Model and Qualitative Analysis of Chance-Constrained Stochastic Programming Problems.- 5. Charnes and Cooper Deterministic Equivalents.- 6. Deterministic Equivalents to Chance-Constrained Stochastic Programming Problems.- 7. Applications of Chance-Constrained Stochastic Programming Problems: Examples.- III. Two-Stage Stochastic Programming Problems.- 8. Model of a Two-Stage Stochastic Programming Problem.- 9. Two-Stage Stochastic Programming Problem Analysis.- 10. Some Partial Models of Two-Stage Stochastic Programming Problems.- 11. The Non-Linear Two-Stage Stochastic Programming Problem.- 12. Methods for the Solution of Two-Stage Stochastic Programming Problems: Examples.- 13. Applications of Two-Stage Stochastic Programming Problems: Examples.- IV. Multi-Stage Stochastic Programming Problems.- 14. Models of Multi-Stage Stochastic Programming Problems.- 15. Qualitative Analysis of Multi-Stage Stochastic Problems with Posterior Decisive Rules.- 16. Prior Decision Rules in Multi-Stage Stochastic Programming Problems.- 17. Duality in Multi-Stage Stochastic Programming.- 18. Applications of Multi-Stage Stochastic Programming Problems: Examples.- V. The Game Approach to Stochastic Programming Problems.- 19. The Game Model of Stochastic Programming Problems.- 20. Partial Cases of the Game G (En+, F,g).- VI. Problems of the Existence of a Solution and Its Optimality in Stochastic Programming Problems.- 21. Dual Linear Stochastic Programming Problems.- 22. Optimality and Existence of the Solution in Stochastic Programming Problems.- 23. Investigations into One Stochastic Programming Problem.- 24. The Definition of the Set of Feasible Plans in the Hanson Problem.- VII. Problems of the Stability of Solutions in Stochastic Programming Problems.- 25. Stability of the Solutions in Stochastic Programming Problems.- 26. The ?-Stability of the Solution at the Mean.- 27. Stability of the Solutions in Non-Linear Stochastic Programming Problems.- 28. The Stability of Planning and Functioning at the i-th Constraint.- 29. The Investigation of Absolute Planning Stability.- Conclusion.