Lot sizing with nonlinear production cost functions

Abstract

In this study, we consider di erent variations of the lot sizing problem encountered in many real life production, procurement and transportation systems. First, we consider the deterministic lot sizing problem with piecewise concave production cost functions. A piecewise concave function can represent quantity discounts, subcontracting, overloading, minimum order quantities, and capacities. Computational complexity of this problem was an open question in the literature. We develop a dynamic programming (DP) algorithm to solve the problem and show that the problem is polynomially solvable when number of breakpoints of the production cost function is xed and the breakpoints are time-invariant. We observe that the time complexity of our algorithm is as good as the complexity of existing algorithms in the literature for the special cases with capacities, minimum order quantities, and subcontracting. Our algorithm performs quite well for small and medium sized instances. For larger instances, we propose a DP based heuristic to nd a good quality solution in reasonable time. Next, we consider the stochastic lot sizing problem with controllable processing times where processing times can be reduced in return for extra compression cost. We assume that the compression cost function is a convex function in order to re ect the increasing marginal cost of larger reductions in processing times. We formulate the problem as a second-order cone mixed integer program, strengthen the formulation and solve it by a commercial solver. Moreover, we obtain some convex hull and computational complexity results. We conduct an extensive computational study to see the e ect of controllable processing times in solution quality and observe that even with small reductions in processing times, it is possible to obtain a less costly production plan. As a nal problem, we study the multistage stochastic lot sizing problem with nervousness considerations and controllable processing times. System nervousness is one of the main problems of dynamic solution strategies developed for stochastic lot sizing problems. We formulate the problem so that the nervousness of the system is restricted by some additional constraints and parameters. Mixing and continuous mixing set structures are observed as relaxations of our formulation. We develop valid inequalities for the problem based on these structures and computationally test these inequalities.