# Supplier Replacement Model in a One-Level Assembly System under Lead-Time Uncertainty
## 2 Related Work
In [15; 16; 17], the authors use the same mathematical model with the same assumptions but in a multi-objective context. Later, in the article by [PERSON] and [PERSON] [18], the model was extended to authorize the assembly of the finished product before the due date if all the components at level 1 are available. A Branch-and-Bound procedure (B&B) was developed to optimize the planned lead times. Based on this approach, a joint model for replenishment and equipment maintenance planning that considered system deterioration was proposed in [19; 20]. For assembly systems with three levels in the BOM, [PERSON] proposed in [21] a continuous model based on an approximate decomposition technique to optimize inventory and backlogging costs. The main limitation of these studies is that they are limited to assembly systems with less than three levels in the BOM. The past few years have seen a huge research effort to propose more general models. To the best of our knowledge, only two studies have proposed mathematical formulations to model one-known-demand planning for multi-level assembly systems: [PERSON] and [PERSON] in [22] for the assemble-to-order and [PERSON] et al. [23] for the configure-to-order environments. Even though these two studies model the dependency between levels and offer the potential to analyze replenishment planning for assembly systems with more than two levels in the nomenclature, there are still a number of unanswered questions for the case of multi-period planning where there is dependency between inventory levels at different periods. Multi-period planning models in a stochastic environment with random lead times were investigated in many studies. Most models are limited to single-item replenishment with the assumption that there is no order crossover. The order crossover is a very well-known effect when the same type of order released at period \(t\) is delivered after the order released at the period \(t+1\). Despite it being a very real phenomenon in real-world replenishment planning and inventory control, in the literature, the models proposed often make the assumption that there is no crossover, because of the difficulty in calculating the probabilities of different states for a crossover effect (see [4; 24; 25; 26; 27] and their related work for well-known models, and [28; 29; 30; 31] for real examples). The multi-period models take into account the dependency between periods: at some point in time the backlog is covered by stocks of previous periods and vice versa [32]. For assembly systems with stochastic lead times and multi-period replenishment planning, the research was often done for the case of a known constant rather than a dynamic demand [33]. This can be explained by the fact that the order crossover is very hard to model for dynamic demand [28]. The choice of a known constant demand allows the crossover phenomenon to be neglected because the orders are of the same size, equal the demand, thus it does not matter which order is delivered on each period. A number of studies were built on this assumption to propose a one-period planning approach that is equivalent to multi-period planning with a known constant demand and stochastic lead times. [PERSON] in the work [34] was one of the first authors who proposed an economic order quantity (EOQ) approach for an OLAS under lead time uncertainty, but with only a two-period planning model. [PERSON] et al. [35] has formulated the inter-dependency between the inventories of different components in assembly systems and developed an optimal \((Q,r)\) policy for an OLAS. They assumed that: (i) the finished product is composed of several types of components; (ii) the assembly capacity is unlimited; (iii) the demand is constant and (iv) the probability distribution of procurement lead time for each component is given. Their proposed continuous model determines the optimal \((Q,r)\) policy and minimize the average total cost, which represents the sum of average inventory, backlogging and setup costs.