A multi-criteria approach to approximate solution of multiple-choice knapsack problem

Ewa M Bednarczuk,Przemysław Pyzel,Janusz Miroforidis

doi:10.1007/s10589-018-9988-z

Abstract

We propose a method for finding approximate solutions to multiple-choice knapsack problems. To this aim we transform the multiple-choice knapsack problem into a bi-objective optimization problem whose solution set contains solutions of the original multiple-choice knapsack problem. The method relies on solving a series of suitably defined linearly scalarized bi-objective problems. The novelty which makes the method attractive from the computational point of view is that we are able to solve explicitly those linearly scalarized bi-objective problems with the help of the closed-form formulae. The method is computationally analyzed on a set of large-scale problem instances (test problems) of two categories: uncorrelated and weakly correlated. Computational results show that after solving, in average 10 scalarized bi-objective problems, the optimal value of the original knapsack problem is approximated with the accuracy comparable to the accuracies obtained by the greedy algorithm and an exact algorithm. More importantly, the respective approximate solution to the original knapsack problem (for which the approximate optimal value is attained) can be found without resorting to the dynamic programming. In the test problems, the number of multiple-choice constraints ranges up to hundreds with hundreds variables in each constraint.

Highlights

The multi-dimensional multiple-choice knapsack problem (M MC K P) and the multiple-choice knapsack problem (MC K P) are classical generalizations of the knapsack problem (K P) and are applied to modeling many real-life problems, e.g., in project portfolio selection [21,29], capital budgeting [24], advertising [27], component selection in IT systems [16,25], computer networks management [17], adaptive multimedia systems [14], and other
A new approximate method of solving multiple-choice knapsack problems by replacing the budget constraint with the second objective function has been presented. Such a relaxation of the original problem allows to the smart scanning of the decision space by quick solving of the binary linear optimization problem
Let us note that our method can be used for finding an upper bound for the multi-dimensional multiple-choice knapsack problem (M MC K P) via the relaxation obtained by summing up all the linear inequality constraints [1]

Summary

B Przemysław Pyzel

Be found without resorting to the dynamic programming. In the test problems, the number of multiple-choice constraints ranges up to hundreds with hundreds variables in each constraint. Keywords Knapsack · Multi-objective optimization · Multiple-choice knapsack · Linear scalarization

Introduction

Multi-objective optimization problems

Decomposition

Computational experiments

Conclusions and future works