A linearisation approach to the stochastic dynamic capacitated lotsizing problem with sequence-dependent changeovers

We present a mixed-integer linear programming formulation that simultaneously optimises lot sizes and production sequences on a capacity constrained machine with sequence-dependent changeovers subject to stochastic dynamic demand while at the same time satisfying a fill rate constraint. To tackle the non-linearity of the exact formulation, we introduce a piecewise linearisation technique both for the expected inventory on hand and for the backorder functions that uses the target service level and the parameters of the demand distribution to assign breakpoints to the most promising intervals of the linearisation domain. We show that our strategy leads to lower cost and to more conservative production plans, in comparison to techniques recommended by earlier research. In addition, we discuss why any breakpoint selection strategy that does not exclude the concave region for $t \geq 2 $t≥2, is prone to be outperformed by the approach we present. Finally, we propose a Relax-and-Fix with Fix-and-Optimize heuristic, and show based on the broad set of instances from Haase, Knut, and Alf Kimms [2000. “Lot sizing and scheduling with sequence-dependent setup costs and times and efficient rescheduling opportunities.” International Journal of Production Economics 66 (2): 159–169], that it is more effective than a state-of-the-art solver in terms of run time and solution quality.