A Dynamic Programming Algorithm for Solving Bi-Objective Fuzzy Knapsack Problem

This paper considers bi-objective knapsack problem with fuzzy weights, says bi-objective fuzzy knapsack problem (BOFKP). Here we introduce an index which gives the possibility of choosing the item (weights and knapsack availability are fuzzy in nature) for knapsack with crisp capacity such that both the objective value are optimized. A methodology using dynamic programming technique has been introduced in this paper with an algorithm which gives the optimal solution for single objective fuzzy knapsack problem (FKP) with some possibility. Using this methodology an algorithm is given to find the Pareto frontier in case of bi-objective fuzzy knapsack problem. Compromise ratio method for decision-making under fuzzy environment has been used to find the compromise solution. The possibility index gives an idea to choose the solution according to decision-maker's choice. An illustrative example is given to demonstrate the methodology.