Two-stage stochastic programming formulations for the multi-plant lot-sizing problem with uncertain demands

This paper addresses the multi-plant capacitated lot sizing problem (MPCLSP) under uncertain demands through two-stage stochastic programming. The MPCLSP consists of determining production and possible product transfers between plants to meet customers' demand within the planning horizon, ensuring that plants' capacities are not exceeded, while minimising setup, production, transfer, and inventory costs. Stochastic programming techniques are are well suited to modelling uncertainty in the optimisation of production processes during production planning. In particular, the two-stage stochastic programming considers that uncertain data are observed at one particular moment in time. Although stochastic programming is widely applied in lot sizing variants, it has not yet been explored for the MPCLSP. Uncertainty has only been considered in the MPCLSP via robust optimisation. Unlike robust optimisation, which protects decisions against worst-case data realisations, two-stage stochastic programming explicitly models the distribution of demand and quantifies the value of incorporating uncertainty, offering more flexible decisions for multi-plant coordination.

We propose three two-stage stochastic programming formulations in which customer demands are represented by a finite set of discrete scenarios with known probabilities. The models differ according to which decisions are taken before and after demand realisation. All variants consider the setup decisions to be first-stage, due to the nature of involving fixed costs and employee hiring. The first model considers that both production and transfer decisions are first-stage, and the second-stage consists of the inventory decisions only. The second model considers that only production decisions are first-stage, while decisions of transfer and inventory are second-stage. Finally, the third model considers that all production, transfer, and inventory decisions are second-stage.

Computational experiments were conducted on 126 generated instances with increasing numbers of products, plants, periods, and scenarios. With all formulations, optimal solutions were found within reasonable computation times. As expected, models incorporating a greater number of second-stage decisions required more time to be solved, particularly as the number of scenarios increases. Additionally, the quality of the stochastic solutions was assessed using the Value of the Stochastic Solution (VSS) and the Expected Value of Perfect Information (EVPI). These metrics demonstrate the benefit of explicitly accounting for demand uncertainty compared to the deterministic expected-value model.