# Supplier Replacement Model in a One-Level Assembly System under Lead-Time Uncertainty
## 2 Related Work  
The literature includes several papers that study supply planning under uncertainty: most focus on demand uncertainty, and there has been little attempt to address the question of lead-time uncertainty. Here, we do not claim to provide an exhaustive review of the literature but a broad overview of important existing approaches in the field of supply planning under lead-time uncertainty. This first analysis of the literature confirms that most existing work focuses on one-customer demand planning and one-period planning with specific structures. [PERSON] [4] was one of the first authors to investigate one-demand replenishment planning for one-level production under lead-time uncertainty. He proved that the problem of planned lead-time calculation can be easily resolved by using the well-known Newsvendor model. [PERSON] in [5] generalized the approach to the case of multi-stage linear systems, and [PERSON] et al. in [6] proved that the problem for serial inventory systems is equivalent to the one that calculates the best base-stock levels. [PERSON] in [7] also dealt with multi-stage systems but with backlogging costs in intermediate stages. The author proposed a recursive procedure to calculate the expected total cost and then optimize planned lead times based on the convexity of the cost function. The literature covering assembly systems under lead-time uncertainty includes several studies that consider one-customer demand planning. [PERSON] [8] was among the first authors to study replenishment planning for one-level assembly systems (OLAS) under lead time uncertainty. In that paper, an assembly system for a finished product was considered. The finished product was assembled from two components. The lead time followed a stochastic Poisson distribution for the first component and a negative binomial distribution for the second component. Like [PERSON] in [8], [PERSON] in [9] developed a model to study OLAS under a known demand and uncertain lead times, but for several types of components. The objective was to minimize the expected total cost composed of holding cost and backlogging cost. An exact method was proposed to obtain optimal order release dates. The main limitation of this method is that it is only valid for certain types of lead-time distributions (exponential, uniform or normal). Several other works have set out to demonstrate the convexity of the objective function for the case of OLAS [10] or to optimize it using Newsboy formulae with specific assumptions on lead-time probability distributions [11; 12]. For the case of a single stochastic demand, [13] studied an assembly system with one finished product and two components, each purchased from its own supplier. The distributions of probabilities were of a specific type on two periods: an order is delivered at the time \(t\) with probability \(p\) otherwise it is delivered at the time \(t+1\). The authors optimized the ordered quantity purchased from each supplier. In [14], the authors developed a mathematical model to calculate the expected total cost, which is the sum of the inventory holding costs for components and sub-assemblies as well as the backlogging cost for the finished product, for two-level assembly systems. The components of the second level of the bill of materials (BOM) are necessary to assemble sub-assemblies. Then the sub-assemblies of the second level are used to assemble the finished product at the first level. The lead times for components and sub-assemblies are discrete random variables. The authors assumed that the finished product is assembled, at the earliest, on due date (Just-In-Time policy). The problem is more complex than for one-level assembly systems because of the necessity to calculate convolutions of probabilities for both level 1 and 2. A GA was proposed to minimize the expected total cost by searching for appropriate (if possible optimal) order release dates for the components at the second level.