Comparative Analysis of Membership Functions in Multi Objective Fuzzy Linear Programming for the Apparel Industry

Abstract

Decision making in the apparel industry often involves trade-offs between multiple conflicting objectives under uncertain environments. While linear programming (LP) is widely used for profit maximization and cost minimization, it cannot effectively handle imprecision in objective and constraints. Fuzzy Multi Objective Linear Programming (FMOLP) addresses this limitation by incorporating uncertainty through membership functions. This study presents a comparative analysis of different membership functions specifically linear and trapezoidal in the development of an FMOLP model for an apparel manufacturing company producing polo, basic, mock-neck T shirts, singlets and short pants. Unlike the reference study, which used all available raw materials to maximize profit without accounting for excess production, the proposed FMOLP model aims to maximize profit while minimizing total production, overproduction, material wastage, and unnecessary costs. Overproduction is specifically defined as output that exceeds the allowable level after accounting for a 4% defect rate. To evaluate the impact of different membership functions, each LP models were solved separately to construct a pay-off matrix, establishing the lower and upper bounds of each objective functions. The FMOLP models were developed using linear and trapezoidal membership functions and solved using Julia. The results show that the choice of membership function significantly influences model performance. FMOLP model with linear membership function achieved a 25% reduction in total production, while with the trapezoidal membership function achieved a higher 35.2% reduction compared with the reference study. For overproduction, with the linear and trapezoidal membership function achieved reductions of 61% and 75.8%, respectively. These findings demonstrate the effectiveness of incorporating appropriate membership functions in FMOLP and highlight the advantages of the trapezoidal approach for improving production planning in the apparel industry.