The multiple-family ELSP with safety stocks

The multiple-family economic lot scheduling problem with safety stocks (MFELSP-SS) with normally distributed, time-stationary demand is considered in a manufacturing setting where the relevant costs include family setup costs, item setup costs, and inventory holding costs for both cycle and safety stocks. A family is a subset of the items that share a common family setup with its associated setup cost and setup time. Each item within the family may have its own setup time and setup cost. The families form a partition of the set of items manufactured on a single facility. The safety stock level for any item is a function of the time interval between production runs for the item, the service level specified, and the variance of its demand. We consider safety stocks explicitly in the formulation, as their holding costs vary nontrivially with the model's decision variables. The MFELSP-SS differs from multilevel inventory models with family setups in that the former assumes noninstantaneous inventory replenishment and considers the cost of holding safety stocks; the latter assumes instantaneous replenishment and does not directly assess the impact of safety stock levels on the total cost. An efficient solution procedure is developed for this model. Properties of the nonconvex feasible space are identified and used in the solution approach. The solution to the mathematical model comprises the basic period length, the family multipliers, and the item multipliers that give the lowest total cost of setups and carrying inventory. The family multipliers and item multipliers are restricted to integer powers of two.