The Application Potential of Integer Programming

Abstract

Linear programming has received much attention as a tool for managerial decision making in business and government. It provides the decision maker with a precise and simple framework for defining his problem and a quick and simple means of obtaining an optimal solution to that problem. In addition, linear programming has the advantages of being applicable to a wide range of managerial problems, being able to consider simultaneously each of the multiple goals and relationships which may be represented by a complex problem, and providing information about the relative significance of each constraint with respect to the objective (shadow prices). However, because of the simplicity inherent in the linear programming model, there are many managerial problems to which it cannot readily be applied. Such problems may be too complex for a modeling approach, or may contain relationships which cannot be represented by linear functions, or may be characterized by probabilistic relationships or perhaps relationships which change over time. Most managers could probably cite which the simple linear programming model would be unable to consider if applied to problems in their area of authority. To cope with complexities of this sort, several variations and extensions of linear programming have been developed. Quadratic programming and integer programming have been developed to attack problems of nonlinearity of functional relationships. Stochastic programming deals with problems containing probabilistic relationships. Dynamic programming can be adapted to problems in which time or the sequence of events is an important consideration. However, it can be stated that, in general, what these techniques contribute in terms of more accurate representation of actual problems is often sacrificed in the form of great difficulty in reaching an optimal solution. This paper concentrates upon a particular class of extensions of the traditional linear programming model-those problems containing integer variables which are restricted to a value of either zero or one. These variables are referred to as variables, and the formulation and solution of problems containing such variables is referred to as programming. Dichotomous-integer variables generally indicate discrete changes in objective or constraint functions, or the presence or absence of some condition or decision. Several well-known linear and integer programming problems are presented in this paper and then extended by means of adding dichotomousinteger variables representing additional considerations or complicating factors which are typical of many real world problems. The purpose of the paper is to demonstrate that integer programming, particularly dichotomous-integer programming, provides a powerful means of * Assistant professor of accounting at the College of Business Administration, University of Texas at Austin.