## 3 Problem Formulation
### Mathematical Model  
A mathematical model is developed to determine an optimal set of suppliers, order quantities, and vehicle order collection routes. The total cost is defined as the sum of purchasing, ordering, inventory, and transportation costs. Transportation (routing) cost is determined by multiplying the transportation cost per unit distance and the distance matrix values. There is also a fixed cost added to the transportation cost when a vehicle starts a route. The purchasing cost consists of the discounted total amount of the products ordered from a supplier in a period. Ordering cost can be defined as the cost of placing a product from a supplier. Inventory costs may be preferable in order to benefit from the price discount offered by suppliers. \begin{table}
\begin{tabular}{l} \hline \hline Demand size for all components is deterministic. \\ Suppliers have limited capacities that may change in each time period. \\ A supplier can supply a complete or partial set of components. \\ A component can be purchased from multiple suppliers. \\ Supplier locations are fixed throughout the planning horizon. \\ Suppliers offer different prices, discount ratios, and discount intervals. \\ Discount ratios are determined based on the total amount of components ordered from a supplier in a given period. \\ An order placed at the end of each period is received at the beginning of the next period. \\ Transportation costs are independent of the quantity transported and depend only on the total distance of the route. \\ The number of vehicles in the fleet are constant. \\ Each vehicle may have a different fixed transportation capacity. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Assumptions. Let \(J:=\{1,\ldots,j\}\) be the set of components to be purchased, let \(I:=\{1,\ldots,i\}\) be the set of suppliers to choose from, and a depot (buyer) indexed by 0, let \(T:=\{1,\ldots,t\}\) be the set of supply periods, let \(F:=\{1,\ldots,f\}\) be the fleet of identical vehicles available for the service, and let \(K:=\{1,\ldots,k\}\) be the set of discount intervals offered from suppliers. Each component \(i\in I\) can be purchased from a group of suppliers \(j\in J\) depending on whether they have the component. Each component can be supplied from more than one supplier, and the order size of a component is determined based on the capacity of the supplier and total cost of the component. The \(s_{ijt}\) unit of the component \(j\in J\) can be found at supplier \(i\in I\). The distances of suppliers to the buyer are shown in the distance matrix \(D_{mn}\). There is a fixed cost per period for each vehicle as independent of distance traveled. The notation used to express equations of the mathematical model is as follows:  
Notation:  
**Indices:**  
\(i\): suppliers (\(i=1,2,\ldots,I\))  
\(j\): components (\(j=1,2,\ldots,J\))  
\(k\): discount intervals (\(k=1,2,\ldots,K\))  
\(t\): planning periods (\(t=1,2,\ldots,T\))  
\(f\): vehicles (\(f=1,2,\ldots,F\))  
\(m\): starting supplier (\(m=1,2,\ldots,I+1\))  
\(n\): ending supplier (\(n=1,2,\ldots,I+1\))  
**Decision Variables:**  
\[Q_{ijt}\ =\ \text{Order quantity for component $j$ from supplier $i$ in period $t$ (units)}\]  
\[R_{ftmn}=\left\{ \begin{array}{ll}1,&\text{if vehicle $f$ visits supplier $n$ after supplier $m$ in period $t$}\\ 0,&\text{otherwise}\end{array} \right.\]  
\[I_{ji}=\ \text{inventory level at the beginning of period $t$ for component $j$ (units)}\]  
\(T_{it}=\text{total amount ordered from supplier $i$ before discount in period $t$ (unit of money)}\)  
\(V_{it}=\text{total amount ordered from supplier $i$ after discount in period $t$ (unit of money)}\)  
\[U_{itk}=\left\{ \begin{array}{ll}1,&\text{if discount interval $k$ is applied by supplier $i$ in period $t$}\\ 0,&\text{otherwise}\end{array} \right.\]  
\[X_{it}=\left\{ \begin{array}{ll}1,&\text{if any component is ordered from supplier $i$ in period $t$}\\ 0,&\text{otherwise}\end{array} \right.\]  
\[Y_{ft}=\left\{