The impact of initial inventory assumptions on cost performance in lot-sizing models

Most discrete-time, dynamic lot-sizing research assumes zero or given initial inventories and assigns zero value to final inventories. These assumptions influence the cost performance significantly. For instance, under capacity constraints, many products need to be produced initially to refill inventories, which causes capacity conflicts or desynchronizes the replenishments of products that should be replenished jointly. We use single-level lot-sizing, joint replenishment, warehouse scheduling, and multi-level capacitated lot-sizing on existing test sets to examine how the assumption of zero initial inventories compares to a different method that sets the initial inventories endogenously through a cycle condition. This condition enforces re-installing selected initial inventory levels as final inventories at the end of the planning horizon. It helps to determine initial and final inventories that minimize costs and create repeatable schedules that mitigate the end-of-horizon effect and improve the (de-)synchronization of lots through the lack of forced setups in the first period. The problems are modeled as mixed-integer linear programs and evaluated in a deterministic and a rolling horizon setting. Furthermore, we formulate the cyclic single-level capacitated lot-sizing problem as a stochastic dynamic program. We prove worst-case cost performance bounds for the lot-sizing problems compared to their cyclic counterparts. The results show increasing potential for savings the more the models rely on (de-)synchronizing replenishments because of capacity constraints or product interdependencies. Even with long planning horizons for which the impact of initial inventories on total costs diminishes, using the cycle condition leads to considerable savings through better (de-)synchronization and better smoothing.