In this paper we present a heuristic based upon genetic algorithms for the multidimensional knapsack problem. A heuristic operator which utilises problem-specific knowledge is incorporated into the standard genetic algorithm approach. Computational results show that the genetic algorithm heuristic is capable of obtaining high-quality solutions for problems of various characteristics, whilst requiring only a modest amount of computational effort. Computational results also show that the genetic algorithm heuristic gives superior quality solutions to a number of other heuristics.

A Genetic Algorithm for the Multidimensional Knapsack Problem

Optimization Techniques with FORTRAN

The multidimensional 0–1 knapsack problem: An overview

The resource allocation problem seeks to find an optimal allocation of a fixed amount of resources to activities so as to minimize the cost incurred by the allocation. A simplest form of the problem is to minimize a separable convex function under a single constraint concerning the total amount of resources to be allocated. The amount of resources to be allocated to each activity is treated as a continuous or integer variable, depending on the cases. This can be viewed as a special case of the nonlinear programming problem or the nonlinear integer programming problem.

Resource Allocation Problems

Many systems are required to perform a series of missions with finite breaks between any two consecutive missions. To improve the probability of system successfully completing the next mission, mai...

Selective maintenance scheduling under stochastic maintenance quality with multiple maintenance actions

This paper presents an efficient solution algorithm for the multiconstraint zero-one knapsack problem through a branch and bound search process. The algorithm has been coded in FORTRAN; and a group of thirty 5-constraint knapsack problems with 30-90 variables were run on IBM 360/75 using two other codes as well, in order to compare the computational efficiency of the proposed method with that of the original Balas and an improved Balas additive algorithms. The computational results show that the proposed method is markedly faster with regard to the total as well as the individual solution times for these test problems, and such superiority becomes more evident as the number of variables and the difficulty of the problems increase. The results also indicated that the original Balas method is extremely inefficient for the type of problems being considered here. The total solution time for the thirty problems is 13 min for the proposed method, 109 min for the improved Balas algorithm, and over 380 min for the original Balas algorithm. Extension of the solution algorithm to the generalized knapsack problem is also suggested.

A Branch and Bound Method for the Multiconstraint Zero-one Knapsack Problem

This paper presents an optimization procedure which would offer a much simpler and faster procedure than dynamic programming in reaching optimal solutions for a special class of resource allocation problems. The solution method is based upon an incremental analysis and does not require further computation beyond the conversion of a payoff table to a table of marginal payoffs by simple subtractions. The optimality of the incremental solution will be demonstrated by a heuristic proof with several examples; and a numerical problem to illustrate the use of incremental analysis as well as to compare it with the solution procedure of dynamic programming will also be given.

A New Application of Incremental Analysis in Resource Allocations

A branch and bound algorithm is developed in this paper for a class of discrete resource allocation problems with many constraints. The algorithm exploits certain properties of the objective function, and has been shown to be much faster and thus far more efficient than both dynamic programming and exhaustive search methods. The strength of this optimization procedure lies not only in the ease with which an appropriate solution can be identified for evaluation, but, more importantly, in the simplicity and rapidity with which upper bounds for the entire problem as well as for any subproblems can be calculated. A detailed description of the solution procedure is presented in the paper; and a numerical example to illustrate the application of the algorithm is also provided.

https://link.springer.com/content/pdf/10.1057%2Fjors.1977.77.pdf

Wei Shih

Papers

A Branch and Bound Method for the Multiconstraint Zero-one Knapsack Problem

A New Application of Incremental Analysis in Resource Allocations

A Branch and Bound Procedure for a Class of Discrete Resource Allocation Problems with Several Constraints