Optimization of a Production Lot Sizing Problem with Quantity Discount

Dynamic lot sizing problem is one of the significant problem in industrial units and it has been considered by many researchers. Considering the quantity discount in purchasing cost is one of the important and practical assumptions in the field of inventory control models and it has been less focused in terms of stochastic version of dynamic lot sizing problem. In this paper, stochastic dynamic lot sizing problem with considering the quantity discount is defined and formulated. Since the considered model is mixed integer non-linear programming, a piecewise linear approximation is also presented. In order to solve the mixed integer non-linear programming, a branch and bound algorithm are presented. Each node in the branch and bound algorithm is also MINLP which is solved based on dynamic programming framework. In each stage in this dynamic programming algorithm, there is a sub-problem which can be solved with lagrangian relaxation method. The numeric results found in this study indicate that the proposed algorithm solve the problem faster than the mathematical solution using the commercial software GAMS. Moreover, the proposed algorithm for the two discount levels are also compared with the approximate solution in mentioned software. The results indicate that our algorithm up to 12 periods not only can reach to the exact solution, it consumes less time in contrast to the approximate model.