A "Short-Cut" Method for the Complete Solution of Game Theory and Feed-Mix Problems

Abstract

Previous methods of solving linear programming problems have always had to revert to the simplex method, after the first two or three most promising activities have been located. The present paper shows that this is unnecessary, and presents a method of solving game theory and programming problems without using the simplex method. Despite its title the new method may, in large problems, involve the same amount of computing as the simplex method. A BRIEF DISCUSSION of the special terms used in the title provides a convenient summary of this paper. The adjective has been used to signify that the method presented here is an extension of the graphical short-cut methods previously presented by Waugh and Burrows [5] and Boles [1]. By complete is meant that with this method it is unnecessary to revert to the simplex method at the end of the short-cut; the short-cut leads to the solution of the problem, even when there is a large number of rows and columns. The game theory problems referred to are two-person zero-sum games, and the feed-mix problems refer to programming problems in which all resource supplies are positive, and prices are either all positive or all negative. The notation to be employed will now be described and then the short-cut method will be introduced in conjunction with a discussion of a small game theory problem. 1. NOTATION