# Supplier Replacement Model in a One-Level Assembly System under Lead-Time Uncertainty
## 2 Related Work
A few years later, new and more efficient models were developed for multi-period planning with known and constant demand. Generalized models were provided, but they were limited to OLAS. For example, in [10; 36], the authors have considered the MRP parameterization problem by optimizing planned lead times with a view to minimizing the expected total cost including the backlogging cost for finished products as well as the holding cost for components. In [37; 38], the authors optimized the same expected total cost and gave the optimal safety stock and optimal safety times. However, serious weaknesses in the approach were that it is only valid if all component procurement lead times follow the same probability distribution and if all components have the same unit inventory holding cost. The optimization approach was generalized in [3] to consider procurement lead times that are independent, but not necessarily identically distributed in order to minimize the expected total cost. In continuation of this work, various approaches [36; 39; 40; 41; 42] have attempted to extend this model to study other replenishment policies (L4L, POQ, EOQ, etc.) and to include setup cost. For the case of multi-period planning under uncertainty of both demand and lead times, [PERSON] [43] studied OLAS to develop a simulation model coupled with a hybrid algorithm. The algorithm is based on simulated annealing to optimize order quantities and planned lead times. In [44], the authors proposed a stochastic linear programming model to study a multi-product problem with several OLAS under both demand and lead-time uncertainty. Both these models are very interesting, but the quality of the optimal solutions depends on the number of scenarios that increases very quickly. As can be seen in Table 1, all the papers concentrate on ordering policies by optimizing order quantities, order release dates and planned lead times [33]. To the best of our knowledge, in the field of OLAS with multi-period planning no papers have attempted to co-optimize purchasing policies and pricing strategies even though both are crucial to successful supply chain management. Most of the literature on supplier diversification and responsive pricing has focused on the single-item purchasing case. Interested readers can refer to the-state-of-the-art surveys of [45; 46; 47; 48] to obtain more details on existing pricing models, and to the recent review paper [49] which provides an excellent overview of contributions to this issue. In the field of inventory control for assembly systems in an uncertain environment, the issue of supplier diversification and supplier selection/replacement strategies has not been sufficiently studied. As aforementioned, [PERSON] et al. in [13] studied an assembly system with one product stochastic demand. The finished product was assembled from two components. The authors extended the model introduced in [50] to consider delivery of the required components by a joint supplier or by two different suppliers. They demonstrated the optimality of base-stock when one supplier is selected. A few years later, the same authors [51] replaced lead-time uncertainty by yield uncertainty and came up with propositions under which it can be economically beneficial for the company to multi-source rather than use a single supplier. [PERSON] and [PERSON] in [52] considered an ATO environment to study random yields and price-dependent demand in single-period planning. They modeled the assembly of a finished product from \(n\) components and analyzed the effects of uncertainty on the optimal solution, defined by the production quantities-product pricing dyad. In this paper, we extend the model proposed in [3] in order to investigate an OLAS under lead-time uncertainty and to examine the benefit of paying suppliers an additional purchase cost (APC) in order to reduce the costs of component lead-time uncertainty. In [3], a one-level assembly system was considered with several suppliers and random supplier lead times.