Amir Fatehi Kivi - Young Researchers and Elite Clud, KhalkhalBranch Islamic Azad University, Khalkhal, Iran,
Amir Aydin Atashi Abkenar - Young Researchers and elite Clud, KhalkhalBranch Islamic Azad University, Khalkhal, Iran,
Hossin Alipuor - Assistant Professor, Industrial Eng. Dept., Khalkhal Branch, Islamic Azad University, Khalkhal, Iran

The aim of lot sizing problems is to determine the periods where production takes place and the quantities to be produced in order to satisfythe customer demand while minimizing the total cost. Due to its importance on the efficiency of the production and inventory systems, Lotsizing problems are one of the most challenging production planning problems and have been studied for many years with differentmodeling features. In this paper, we propose a fuzzy mathematical model for the single-item capacitated lot-sizing problem in closed-loopsupply chain. The possibility approach is chosen to convert the fuzzy mathematical model to crisp mathematical model. The obtained crispmodel is in the form of mixed integer linear programming (MILP), which can be solved by existing solver in crisp environment to findoptimal solution. Due to the complexity of the problems harmony search (HS) algorithm and genetic algorithm (GA) have been used tosolve the model for fifteen problem. To verify the performance of the algorithm, we computationally compared the results obtained by thealgorithms with the results of the branch-and-bound method. Additionally, Taguchi method was used to calibrate the parameters of themeta-heuristic algorithms. The computational results show that, the objective values obtained by HS are better from GA results for largedimensions test problems, also CPU time obtained by HS are better than GA for Large dimensions

Lot-sizing, Harmony search, Returned products