Bydraes tot die oplossing van die veralgemeende knapsakprobleem

Abstract

In this thesis contributions to the solution of the generalised knapsack problem are given and discussed. Attention is given to problems with functions that are calculable but not necessarily in a closed form. Algorithms and test problems can be used for problems with closed-form functions as well. The focus is on the development of good heuristics and not on exact algorithms. Heuristics must be investigated and good test problems must be designed. A measure of convexity for convex functions is developed and adapted for concave functions. A test problem generator makes use of this measure of convexity to create challenging test problems for the concave, convex and mixed knapsack problems. Four easy-to-interpret characteristics of an S-function are used to create test problems for the S-shaped as well as the generalised knapsack problem. The in uence of the size of the problem and the funding ratio on the speed and the accuracy of the algorithms are investigated. When applicable, the in uence of the interval length ratio and the ratio of concave functions to the total number of functions is also investigated. The Karush-Kuhn-Tucker conditions play an important role in the development of the algorithms. Suf- cient conditions for optimality for the convex knapsack problem with xed interval lengths is given and proved. For the general convex knapsack problem, the key theorem, which contains the stronger necessary conditions, is given and proved. This proof is so powerful that it can be used to proof the adapted key theorems for the mixed, S-shaped and the generalised knapsack problems as well. The exact search-lambda algorithm is developed for the concave knapsack problem with functions that are not in a closed form. This algorithm is used in the algorithms to solve the mixed and S-shaped knapsack problems. The exact one-step algorithm is developed for the convex knapsack problem with xed interval length. This algorithm is O(n). The general convex knapsack problem is solved by using the pivot algorithm which is O(n2). Optimality cannot be proven but in all cases the optimal solution was found and for all practical reasons this problem will be considered as being concluded. A good heuristic is developed for the mixed knapsack problem. Further research can be done on this heuristic as well as on the S-shaped and generalised knapsack problems.