# Optimal Decisions

### Principles of Programming

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Optimal Decisions: Principles of Programming deals with all important problems related to programming. This book provides a general interpretation of the theory of programming based on the application of the Lagrange multipliers, followed by a presentation of the marginal and linear programming as special cases of this general theory. The praxeological interpretation of the method of Lagrange multipliers is also discussed. This text covers the Koopmans' model of transportation, geometric interpretation of the programming problem, and nature of activity analysis. The solution of the problem by marginal analysis, Hurwitz and the Bayes-Laplace principles, and planning of production under uncertainty are likewise deliberated. This publication is a good source for researchers and specialists intending to acquire knowledge of the principles of programming.

## Table of Contents

Foreword

Introduction. Praxeology and the Theory of Programming

Chapter 1. Typical Models of Programming

1. The Routing Problem

2. The Transportation Problem

3. The Koopmans' Model of Transportation

4. The Allocation Problem

5. The Mix Problem

6. A Dynamic Problem: Production and Stocks

7. Another Dynamic Problem: Storing of Products

8. Investment Programming: The Choice of Investment Variants

9. Investment Programming: Allocation of Investment

10. Investment Programming: Distribution of Investment Over Time

11. Classification of Programming Models

Chapter 2. The General Principles of the Theory of Programming

1. Mathematical Formulation of the General Problem of Programming

2. Geometric Interpretation of the Programming Problem

3. The Method of Indeterminate Lagrange Multipliers. The Dual Program

4. Generalization: The Case when the Balance Relationships are in the Form of Inequalities

Chapter 3. Marginal Programming

1. The Method and the Geometric Interpretation of the Solution of a Marginal Programming Problem

2. Conditions for the Existence of a Solution to a Marginal Programming Problem

3. Examples of Marginal Programs

4. Programming Production when there are n Factors of Production

Chapter 4. Linear Programming

1. Mathematical Formulation of the Problem of Linear Programming

2. Geometrical Interpretation of Linear Programming. The Concept of the Simplex Method

3. The Basic Theorem of the Theory of Linear Programming. Duality in Linear Programming

4. The Simplex Method

5. Examples of Applications of the Simplex Methods

6. Solution of the Dual Problem

7. The Criterion of Optimality of the Solution

Chapter 5. Activity Analysis

1. The Nature of Activity Analysis

2. Maximization of Production and Minimization of Costs

3. The Problem of Joint Production

4. The Generalized Problem of Optimizing Production

5. Examples of Application of the Method of Activity Analysis

Chapter 6. Programming for Multiple Objectives

1. The Efficient Program

2. The Solution of the Problem by Marginal Analysis

3. Multiple Objectives and Linear Programming

Chapter 7. Programming Under Uncertainty

1. Optimal Allocation of Production Among Different Plants

2. The Case of Limited Productive Capacity of Plants

3. The Choice of Optimal Productive Capacity for a New Plant

4. Planning of Production Under Uncertainty

5. Planning Production when the Acceptable Risk is Limited

6. The Neo-Classical Theory of Risk

7. Planning of Production on the Basis of the Neo-Classical Theory of Risk. The Choice Preference Function

8. Criticism of the Neo-Classical Theory. The Method of Marginal Probability

Chapter 8. Dynamic Programming of Purchases and Stocks Under Certainty

1. Optimal Purchase Batch

2. First Generalized Variant of the Problem of Purchases and Stocks

3. The Case when Purchases are not Necessarily Equal

4. The Case of Restricted Warehouse Capacity

5. The Case when Withdrawal from Stock is not Evenly Distributed Over Time

Chapter 9. Dynamic Programming of Purchases and Stocks Under Uncertainty

1. The Case when the Probability of a Reserve Stock being Insufficient (Risk Coefficient) is Equal to a Given Value. Normal Probability Distribution

2. The Case when the Probability Distribution of Demand is a Poisson Distribution

3. The Case when Demand has a "Rectangular" Probability Distribution

4. Determining the Optimum Level of the Risk Coefficient and of the Reserve in Relation to the Stockholding Cost and the Cost of Shortage

Chapter 10. Dynamic Programming of Production Under Certainty

1. Determination of Optimal Production Over Time by Variation Calculation

2. An Example of Dynamic Programming of Production

Chapter 11. Dynamic Programming of Production Under Uncertainty

1. The Case when Aggregate Demand is a Random Variable with a Known Probability Distribution

2. Determination of the Probability Distribution of Aggregate Demand

3. The Solution of the Problem of Optimal Use of Sources of Electric Power

Chapter 12. Programming Under Complete Uncertainty

1. General Remarks on the Theory of Strategic Games

2. Programming Under Complete Uncertainty as a Game Played by Man Against Nature

3. The Hurwitz and the Bayes-Laplace Principles

4. Savage's Principle of the Minimax Effects of a False Decision

5. Determining the Optimum Stock of Raw Material on the Basis of the Theory of Strategic Games

6. The Equivalence of Linear Programming with the Two-Person Zero-Sum Game

7. The Minimax Principle and Collective Decisions

Bibliography

Index

## Product details

- No. of pages: 302
- Language: English
- Copyright: © Pergamon 1971
- Published: January 1, 1971
- Imprint: Pergamon
- eBook ISBN: 9781483148960