# Supplier Replacement Model in a One-Level Assembly System under Lead-Time Uncertainty
## 3 Problem Description
It was developed to measure the effects of such a policy and to optimize the total cost composed of purchasing, holding and backlogging costs. Our model is based on existing models for replenishment planning under lead-time uncertainties. It assumes that assembly system capacity is infinite and that demand for the finished product is known. For each component, the lead time may take several values with given corresponding probabilities. Models of this type have already been formulated in [36; 40] for one-level multi-period problems and in [15; 22] for multi-level one-period assembly systems with random lead times. We will introduce additional decision variables and cost dealing with purchasing policies. Instead of optimizing only planned lead times, we will also optimize the choice of purchasing policy. Every purchasing policy \(j\) for a given supplier \(i\) has a purchasing cost (\(PC_{i}^{j}\)). This paper uses the following notations (see Table 2):  
As the basic model, i.e., without taking into account PC, we will use the model proposed in [3] which consists of minimizing the expected cost composed of the sum of the component holding cost and backlogging cost of the finished product:  
\[\min EC(X)=\sum_{i=1}^{n}h_{i}\left(x_{i}-E(L_{i})\right)+H\sum_{k\geq 0} \left(1-\prod_{i=1}^{n}F_{i}(x_{i}+k)\right) \tag{1}\]  
s.t. \[ \begin{array}{ll}1\leq x_{i}\leq u_{i}&\forall i=1,...,n\\ x_{i}\in\mathbb{N}&\forall i=1,...,n\end{array}  \tag{2}\]  
where:  
\[X=\left(x_{1},x_{2},...,x_{i},...,x_{n}\right) \tag{3}\]  
**Remark 1**. : _In this paper, we extend the model developed in [3] and treat the problem differently. Nevertheless, the techniques calculating component holding (\(HC(X,Y)\)) and backlogging (\(BC(X,Y)\)) costs are unchanged. These cost functions are explained in detail with mathematical proofs in [3, 53]._  
To solve this non-linear optimization problem in Equations (1)-(3), a B&B was developed in [3]. We will extend this model by including the APC explained above. In other words, we integrate prices (purchase costs) as additional decision variables into this model. Our model with the new objective function in Equation (4) is expressed as follows: \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameters** & \\ \hline \(i\) & supplier index, \\ \(j\) & purchase price index, \\ \(h_{i}\) & unit stock cost of the product purchased from supplier \(i\), \\ \(b\) & unit backlog cost for one finished product, \\ \(H=b+\sum_{i=1}^{n}h_{i}\) & global holding and backlog cost for one finished product, \\ \(L_{i}^{j}\) & random lead time for supplier \(i\) under purchasing price \(j\), \\ \(u_{i}^{j}\) & maximum lead time \(i\) under purchase price \(j\); i.e., \(1\leq L_{i}^{j}\leq u_{i}^{j}\). \\ \hline
**Variables** & \\ \hline \(x_{i}^{j}\) & planned lead time for supplier \(i\) with purchase price \(j\), \\  & (planned lead time is equal to order release date if due date is equal to zero), \\ \(y_{i}^{j}\) & binary variable taking the value \(1\) if using purchase price \(j\) for supplier \(i\). \\ \hline
**Functions** & \\ \hline \(PC_{i}^{j}\) & purchase cost under the purchase price \(j\) for supplier \(i\), \\ \(E(L_{i}^{j})\) & expected lead-time value for supplier \(i\) under purchase price \(j\), \\ \(F_{i}^{j}(x_{i}^{j})\) & cumulative distribution function of lead time for supplier \(i\) under purchase price \(j\), \\ \(p_{i}^{j}(k)\) & probability of having a lead time equal to \(k\) for supplier \(i\) under purchase price \(j\). \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Notation. \[ \begin{split}\min EC(X,Y)&=\sum_{i=1}^{n}\sum_{j=0}^{u_{i }^{0}-1}PC_{i}^{j}\cdot y_{i}^{j}+\sum_{i=1}^{n}\sum_{j=0}^{u_{i}^{0}-1}y_{i}^{j }\cdot h_{i}\left(x_{i}^{j}-E(L_{i}^{j})\right)\\ &\quad+H\cdot\sum_{k\geq 0}\left(1-\prod_{i=1}^{n}\left(\sum_{j=0} ^{u_{i}^{0}-1}y_{i}^{j}\cdot F_{i}^{j}(x_{i}^{j}+k)\right)\right)\end{split}  \tag{4}\]  
s.t.