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

TL;DR

This paper introduces a novel, computationally efficient method for approximating solutions to the multiple-choice knapsack problem by transforming it into a bi-objective optimization problem and solving scalarized versions explicitly.

Contribution

The paper presents a new approach that transforms the multiple-choice knapsack problem into a bi-objective problem and solves scalarized forms explicitly using closed-form formulae, improving computational efficiency.

Findings

Approximate solutions are obtained with accuracy comparable to greedy and exact algorithms.

The method effectively handles large-scale problems with hundreds of constraints and variables.

Approximate solutions can be found without dynamic programming.

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.