## 3 Problem Formulation
### Mathematical Model
\begin{array}{ll}1,&\text{if vehicle $f$ visits any supplier in period $t$}\\ 0,&\text{otherwise}\end{array} \right.\]  
**Parameters:**  
\(M\): a large number  
\(d_{fl}\): demand for component \(j\) in period \(t\) (units)  
\(s_{ijt}\): capacity of supplier \(i\) for component \(j\) in period \(t\) (units)  
\(P_{ij}\): undiscounted unit price of component \(j\) from supplier \(i\) (unit of currency/unit)  
\(p_{\bar{k}}\): discount ratio of supplier \(i\) for \(k\)th discount interval (%)  
\(K_{\bar{k}}\): lower limit of \(k\)th discount interval of supplier \(i\) (unit of currency)  
\(h_{j}\): unit holding cost of component \(j\) (unit of currency/unit/period)  
\(o_{i}\): fixed ordering cost for supplier \(i\) (unit of currency/order)  
\(D_{mn}\): distance matrix between the depot and suppliers (distance unit)  
\(c\): unit transportation cost per unit distance (unit of currency/distance unit)\(w\): fixed cost of a vehicle per time period (unit of currency/period)  
\(b_{f}\): maximum loading capacity in weight for vehicle \(f\) (weight unit)  
\(u_{j}\): unit weight of component \(j\) (weight unit/unit)  
**Objective Function:**  
\[ \begin{array}{rl}\mbox{Min TC}=&\left(\sum\limits_{i=1}^{I}\sum\limits_{t=1}^ {T}V_{it}+(o_{i}X_{it})\right)+\left(\frac{1}{2}\sum\limits_{j=1}^{J}\sum \limits_{t=1}^{T}h_{j}\left(I_{it}+I_{jt+1}\right)\right)+\left(w\sum\limits_ {f=1}^{F}\sum\limits_{t=1}^{T}Y_{ft}\right)\\ &+\left(c\sum\limits_{f=1}^{F}\sum\limits_{t=1}^{T}\sum\limits_{t=1}^{I}\sum \limits_{\begin{array}{c}I\\ m=0\end{array}}^{I}\sum\limits_{\begin{array}{c}I\\ n=0\end{array}}^{I}D_{mn}R_{ftmn}\right)\\ &m\
eq n\quad n\
eq m\end{array}  \tag{1}\]  
subject to  
\[I_{it+1}=I_{it}-d_{it}+\sum\limits_{i=1}^{I}Q_{it}\qquad\qquad j=1,2,\ldots,J; \ t=1,2,\ldots,T \tag{2}\]  
\[I_{j1}=0\qquad j=1,2,\ldots,\ J \tag{3}\]  
\[I_{jT+1}=0\quad j=1,2,\ldots,\ J \tag{4}\]  
\[Q_{ijt}\leq s_{ijt}\qquad i=1,2,\ldots,I;\ j=1,2,\ldots,J;\ t=1,2,\ldots,T \tag{5}\]  
\[\sum\limits_{j=1}^{J}Q_{ijt}\leq MX_{it}\qquad i=1,2,\ldots,I;\ t=1,2,\ldots,T \tag{6}\]  
\[T_{it}=\ \sum\limits_{j=1}^{J}Q_{ij}P_{ij}\qquad i=1,2,\ldots,I;\ t=1,2, \ldots,T \tag{7}\]  
\[V_{it}=\ \sum\limits_{k=1}^{K}T_{it}U_{it}p_{ik}\quad i=1,2,\ldots,I;\ t=1,2, \ldots,T \tag{8}\]  
\[K_{ik}+M(U_{lik}-1)\leq T_{it}<K_{ik+1}+M(1-U_{ilk})\ i=1,2,\ldots,I; \tag{9}\]  
\[ \begin{array}{rl}\sum\limits_{k=1}^{K}U_{ilk}=1\qquad\qquad i=1,2,\ldots,I; \ t=1,2,\ldots,T\end{array}  \tag{10}\]  
\[ \begin{array}{rl}\sum\limits_{n=0}^{I}\ R_{ftmn}\leq 1\qquad\ f=1,2,\ldots,F;\ t=1,2, \ldots,T;\ m=0,1,\ldots I\\ n\
eq m\end{array}  \tag{11}\]  
\[ \begin{array}{rl}\sum\limits_{m=1}^{I}\ \sum\limits_{\begin{array}{c}n=1\\ n\
eq m\end{array}}^{I}R_{ftmn}\leq M\sum\limits_{n=1}^{I}R_{ft(l+1)n}\qquad f=1,2,\ldots,F;\ t=1,2,\ldots,\ T\\ n\
eq m\end{array}  \tag{12}\]  
\[ \begin{array}{rl}\sum\limits_{m=1}^{I+1}\ R_{ftmn}=\sum\limits_{\begin{array}[] {c}m=1\\ m\
eq n\end{array}}^{I+1}R_{ftmn}&f=1,2,\ldots,F;\ t=1,2,\ldots,T;\ n=1,\ldots,\ I+1\\ m\
eq n\end{array}  \tag{13}\]\[R_{ftmn}+R_{ftmn}\leq 1\quad f=1,2,\ldots,F;\ t=1,2,\ldots,T;\ m,n=1,2, \ldots,I \tag{14}\] \[\sum_{f=1}^{F}\sum_{ \begin{array}{c}m=1\\ m\
eq n\end{array} }^{l+1}R_{ftmn}=X_{nt}\ t=1,2,\ldots,\ T;\ n=1,2,\ldots,I\] (15) \[\sum_{ \begin{array}{c}m=1\\ m=1\end{array} }^{l+1}\sum_{ \begin{array}{c}n=1\\ n\
eq n\end{array} }^{l}R_{ftmn}\left(\sum_{j=1}^{l}u_{j}Q_{mjt}\right)\leq b_{ f}\ f=1,2,\ldots,F;\ t=1,2,\ldots,T\] (16) \[\sum_{ \begin{array}{c}n=1\end{array} }^{l}R_{ft(l+1)n}\leq MY_{ft} \ f=1,2,\ldots,F;\ t=1,2,\ldots,T \tag{17}\]  
In the integer programming model above, Equation (1) states the objective function which is the sum of purchasing, ordering, inventory holding, fixed vehicle, and transportation costs. Equation (2) is the set of inventory balance equations for all components and time periods. Although Equation (3) states that there is zero inventory at the beginning of the first period, this restriction can easily be removed if necessary.