Production Lot Sizing Problem with the Lead Time

The issue in lot sizing problem is to plan production processes, so that mean the production quantities must be equal to customer demand quantities such that the inventory cost and setup production cost is minimized. In this work, we use the model Multi Level Capacitated Lot Sizing problem with consideration the Lead times, which means that the problem of finding a feasible solution is complex. For this, we propose a new formula in comparison with the classic model. The efficiency of the new formula is demonstrated and infeasible solutions are solved by a heuristic method that's based on Lagrangian relaxation. Computational tests conducted in 1000 instances with up to 40 components and 16 periods have shown that optimal solutions were obtained on average 96.43% of the large instances. For the improvement of the best solution, the heuristic is able to find the efficiency with 97.62% on average. The solution quality is evaluated through initial iterations, the average solution time provided by Lagrangian relaxation is less than 0.43s