Mary W. Cooper

Non-linear integer programming

Abstract An exact method for solving all-integer non-linear programming problems with a separable non-decreasing objective function is presented. Dynamic programming methodology is used to efficiently search candidate hypersurfaces for the optimal feasible integer solution. An efficient computational and storage scheme exists and initial calculations give very promising results.

A dynamic programming algorithm for multiple-choice constraints

Abstract This report concerns a discrete mathematical programming problem in which the variables are binary or integer, the objective function separable or factorable, and the constraints are in either of two classes: linear or multiple-choice constraints. The problem is solved using a dynamic programming approach with fathoming by bounds and by infeasibility.

Some Simple Examples

This chapter focuses on an approach to problem solving that is typical of dynamic programming. It discusses the application of dynamic programming as a problem-solving technique. The chapter presents a problem in division for a given known and positive quantity b that is to be divided into n parts in such a way that the product of the n parts is to be a maximum. It describes the solution of this problem by a dynamic programming approach. It also discusses a simple equipment replacement problem.

Applications of Dynamic Programming

This chapter presents a number of actual applications of dynamic programming to practical problems, in addition to some new potential applications. The chapter highlights the variety of different kinds of problems and different areas of endeavor, to which dynamic programming methodology has been applied and can be applied. The different applications are of varying degrees of detail and complexity. The chapter discusses the use of dynamic programming in expansion of electric power systems, municipal bond coupon schedules, the design of a hospital ward, optimal scheduling of excess cash investment, animal feedlot optimization, optimal investment in human capital, and optimal crop supply.

One-dimensional Dynamic Programming: Analytic Solutions

This chapter focuses on the principles of dynamic programming to a large number of mathematical problems that have one thing in common, namely, in the expressions for the recursion relations, the maximization or minimization can be performed without recourse to numerical tabulation of the stage returns and the optimal return functions. Instead, the optimal return functions, g s (λ s ) can be represented by a mathematical formula and, indeed, classical optimization methods can be employed to obtain these representations. There are many activities where it makes sense to consider minimizing a maximum value of some measure rather than a pure minimization of some objective. The chapter discusses a problem in which a concave function was minimized over a closed convex set in each of the one-dimensional subproblems.

Stochastic Processes and Dynamic Programming

This chapter focuses on the difference and similarities between the application of dynamic programming to deterministic and stochastic processes. The essential difference between that kind of process and a stochastic process is that the state resulting from a decision is not predetermined. It can only be described by some known probability distribution function that depends upon the initial state and in some cases the decision that has been made. The chapter highlights a stochastic allocation problem—discrete case and a stochastic allocation problem—continuous case. Dynamic programming has proven to be of great utility in the solution of inventory problems. An important problem in the literature of management science is concerned with scheduling the production of some product over a planning horizon so that the combined expected cost of production and carrying inventory will be minimized. The stochastic nature of the problem resides in the fact that the demand in any period has to be considered to be a random variable.

Dynamic Programming and the Calculus of Variations

This chapter focuses an optimization problem, where a set X will consist of a set of functions. It presents the definition of a functional on Ζ as a function J (·) such that to each function in X , it associates a single real number. It is the optimization of functional that constitutes the class of problems that is analyzed and solved by the calculus of variations. The chapter discusses computational solution of variational problems by dynamic programming. It describes the use of the Euler–Lagrange equation. The chapter also discusses the way in which a variation differs from a differential.

One-dimensional Dynamic Programming: Computational Solutions

This chapter discusses the use of tabular function representation, both for the original problem statement and for the optimal return functions and optimal solution functions. It focuses on the use of tables to represent the optimal return functions and the optimal decision variable functions. A table is a perfectly acceptable representation of a function of a real variable, as it satisfies, in every way, the definition of a function. The chapter explains the computational effectiveness of dynamic programming. It discusses an equipment replacement model to highlight the nature of computational solutions when the original problem data are more conveniently expressed in terms of functions which are given as tables. The chapter discusses some integer where the variables are constrained to be integers in one or more unusual ways to highlight the versatility of the tabular approach to solving problems through dynamic programming.

Reduction of State Dimensionality and Approximations

This chapter discusses several techniques for dealing with the dimensionality problem, using both exact and approximate methods. The most fruitful area of research in dynamic programming is to find more effective means to deal with the vexing problem of high dimensionality. The use of Lagrange multipliers originated in efforts to solve problems in the calculus of variations, which were also constrained by an equality relationship involving the unknown function. The chapter highlights polynomial approximation in dynamic programming. It describes a new method for reduction of dimensionality that uses dynamic programming in the context of an exact search process that greatly extends the utility of dynamic programming for an important class of problems.

Functional Equations: Basic Theory

The dynamic programming is a way of structuring certain problems so that a certain methodology can be used. This being the case, the properties that an optimization problem must possess need to be known in advance so that its initial mathematical formulation can be converted into an equivalent formulation which is amenable to dynamic programming methodology. Problems to which dynamic programming has been applied are usually stated in the following terms. A physical, operational, or conceptual system is considered to progress through a series of consecutive stages. At each stage, the system can be described or characterized by a relatively small set of parameters called the state variables or state vector. At each stage, and no matter what state the system is in, one or more decisions must be made. These decisions may depend on either stage or state, or both. The key elements that one associates with a dynamic programming problem are stages, states, decisions, transformations, and returns.