Change-making problem

About: Change-making problem is a research topic. Over the lifetime, 1052 publications have been published within this topic receiving 35492 citations.

Knapsack Problems: Algorithms and Computer Implementations

Abstract: Introduction knapsack problem bounded knapsack problem subset-sum problem change-making problem multiple knapsack problem generalized assignment problem bin packing problem. Appendix: computer codes.

A Linear Programming Approach to the Cutting-Stock Problem

Abstract: The cutting-stock problem is the problem of filling an order at minimum cost for specified numbers of lengths of material to be cut from given stock lengths of given cost. When expressed as an integer programming problem the large number of variables involved generally makes computation infeasible. This same difficulty persists when only an approximate solution is being sought by linear programming. In this paper, a technique is described for overcoming the difficulty in the linear programming formulation of the problem. The technique enables one to compute always with a matrix which has no more columns than it has rows.

Integer Programming with a Fixed Number of Variables

Abstract: It is shown that the integer linear programming problem with a fixed number of variables is polynomially solvable. The proof depends on methods from geometry of numbers.

Fast Approximation Algorithms for the Knapsack and Sum of Subset Problems

Abstract: Given a positive integer M and n pairs of positive integers (p~, cD, , (p. , c.), maximize the s u m ~ ~p~ subject to the cons t ramts~ ~c, < M and ~, = 0 or 1 This is the well-known 0/1 knapsack problem An algorithm is presented which finds for any 0 < e < 1 an approximate solution P satisfying (P* P)/P* < ~, where P* is the desired optimal sum Moreover, for any fixed e, the algorithm has time complexity 0(n log n) and space complexity O(n) Modification of the algorithm for the unbounded knapsack problem where the ~,'s can be any nonnegative integer results in a O(n) computing time A hnear-time algorithm is also obtained for a special class of 0/1 knapsack problems having the property that p,/c, is the same for all 1 < z < n

Multistage Cutting Stock Problems of Two and More Dimensions

Abstract: In earlier papers [Opns. Res. 9, 849-859 1961, and 11, 863-888 1963] the one-dimensional cutting stock problem was discussed as a linear programming problem. There it was shown how the difficulty of the enormous number of columns occurring in the linear programming formulation could be overcome by solving a knapsack problem at every pivot step. In this paper higher dimensional cutting stock problems are discussed as linear programming problems. The corresponding difficulty of the number of columns cannot in general be overcome for there is no efficient method for solving the generalized knapsack problem of the higher dimensional problem. However a wide class of cutting stock problems of industry have restrictions that permit their generalized knapsack problem to be efficiently solved. All of the cutting stock problems that yield to this treatment are ones in which the cutting is done in stages. In treating these practical cutting problems, one often encounters additional conditions that affect the solution. An example of this occurs in the cutting of corrugated boxes, which involves an auxiliary sequencing problem. This problem is discussed in some detail, and a solution described for the sequencing problem under given simplifying assumptions.