The Substitution of Analytic for Simulation Algorithms: A Response

Abstract

D EVELOPMENT of the Monte Carlo algorithm compared here with Mixed Integer programming (MIP) can be traced to Swedish work begun in 1964 by Lindgren and to Carlsson [2], who published the first mimeographed paper in 1965. Work on the first English version [3] began in 1966, and at that time, when the roles of these embryonic Monte Carlo programs (MCP) were being discussed, the decision to go ahead was to some extent based on a weariness of waiting for mathematicians to produce a workable MIP. It was anticipated, however, that a successful MIP would be developed, and MCP was not designed primarily to compete with MIP or LP. Although both types of can be handled by MCP, there remains a substantial number of problems which are well suited for handling by MCP. The matrix used by Candler, Cartwright, and Penn [1] as an example is a tiny designed to demonstrate a large number of examples in a small space. As these authors point out, the is very suitable for solution by MIP. Solution of the by maximizing a single objective function, however, fails to bring out one aspect of MCP which is possibly its most significant characteristic-its ability to handle multiple objectives. Departure from a simple profit maximizing model to one, for example, which handles different levels of investment immediately transforms the model into a multiple objective problem. The existence of multiple and often conflicting objectives has led to a number of attempts at ranking and aggregating objectives to produce a single utility that could be incorporated in LP or MIP. Financial objectives can with some degree of success be aggregated in this way, but the method is less effective when applied to heterogeneous objectives. Even the small matrix immortalized here by Candler, et al. as Thompson's problem has two obvious objectives: profit maximization and investment minimization. And suppose a third objective to be sought is the production for sale of the maximum number of product lines possible (a plausible objective in the manufacture of some consumer goods). It will no longer be sufficient in handling this to use an algorithm which calculates an optimum. Rather, a means of calculating the position of the discrete Efficiency surface relating the three objectives is required. The Monte Carlo algorithms developed in Sweden and England are capable of estimating the position of the Efficiency surface (or hyper-