The computer solution of problems in integer programming.

Abstract

The thesis is concerned largely with Gomory s Method of Integer Forms whereby an integer programming problem is solved by a combination of linear programming operations and the addition of new constraints. Chapter I describes the theory behind the method. It deals with the techniques of linear programming when the use of floating point and its associated rounding and truncation errors are avoided and describes the way in which new constraints can be generated and added during solution of the problem. Chapter 2 deals with the author'~ experimemts in integer programming. Parts I to 3 are concerned with the linear programming method which was developed partly to deal with numerical problems and partly to facilitate the choice of constraints. Part 4 deals with experiments with different criteria for choosing constraints. Chapter 3 is concerned with two algorithms. The first is essentially the lexicographic method advocated by Haldi and Isaacson. An independent approach has provided an insight into it which led to the development of the second algorithm. In this the objective function is replaced by approximations to it with smaller coefficients in order to obtain an approximate solution more rapidly. A restriction is then placed on the objective function and a search made for a better solution. Chapter 4 compares the two algorithms of Chapter 3 with those of certain other authors. It is concluded that the systematic method of choosing constraints used in the author s algorithms enables them to be regarded as special forms firstly of a branch and bound algorithm and secondly of a backtrack method. a As corollary it is suggested that some of the techniques used in the author·s algorithms to speed up solution could be applied to these other methods.