Pareto Optimal Solutions of the Fuzzy Bicriteria Sheet Metal Problem

Abstract

Objectives: An algorithm has been developed to find Pareto Optimal solutions of the fuzzy bicriteria sheet metal problem with pairwise nesting of designs. Methods and Statistical Analysis: The sheet metal problem has been solved by many workers, all of whom have considered the entities of cost and time as crisp numbers. However, in practical situations since cost and time are imprecise, the present work considers them as interval fuzzy numbers. Ordering between overlapping interval numbers is obtained by applying a fuzzy membership approach and a modified Hungarian algorithm is developed to obtain fuzzy Pareto Optimal solutions of the bicriteria problem. The newly developed algorithm is explained by a numerical example. Findings and Results: The set of both fuzzy Pareto optimal and other solutions obtained by applying the proposed algorithm, provide the Decision maker a lot of flexibility in making decisions. He can select the solution according to his priority. From amongst the fuzzy Pareto Optimal solutions obtained, he can select the solution which minimizes the cost or the solution which minimizes the time or take the middle path and select the solution which minimizes both cost and time as much as possible. Apart from the three fuzzy Pareto Optimal solutions, other solutions obtained by the proposed method can also be selected by the decision maker as per requirement and conditions. The problem being NP hard, it is very difficult and expensive to find fuzzy Pareto Optimal solutions of the bicriteria problem by analytical methods. The newly developed algorithm is not only easy to understand and implement but also gives good fuzzy Pareto optimal solutions. Improvements: The method can also be applied to costs and times being triangular and trapezoidal fuzzy numbers and it can be extended to nesting of up to three designs on a sheet.