An Integer Programming Model for Binary Knapsack Problem with Value-Related Dependencies among Elements

TL;DR

This paper introduces an integer programming model for a variant of the Binary Knapsack Problem where item values depend on other items, using fuzzy graphs to handle imprecise value-related dependencies.

Contribution

It presents a novel integer programming formulation for BKP with value dependencies modeled via fuzzy graphs, addressing imprecision in dependency strengths.

Findings

Model effectively captures imprecise value dependencies.

Fuzzy graph approach provides a flexible framework.

Potential for improved decision-making in knapsack problems.

Abstract

Binary Knapsack Problem (BKP) is to select a subset of an element (item) set with the highest value while keeping the total weight within the capacity of the knapsack. This paper presents an integer programming model for a variation of BKP where the value of each element may depend on selecting or ignoring other elements. Strengths of such Value-Related Dependencies are assumed to be imprecise and hard to specify. To capture this imprecision, we have proposed modeling value-related dependencies using fuzzy graphs and their algebraic structure.