## 4 A Numerical Example  
A numerical example is introduced to illustrate how the developed model works. The numerical example includes 4 suppliers, 2 vehicles, 4 components, and 3 discount intervals for each supplier and 3 time periods. Unit transportation cost per unit distance is 10; fixed cost of a vehicle is 20; maximum loading capacity for vehicle f is 250; unit weight of component \(j\) is 3; unit holding costs of component \(j\) are h\({}_{1}\) = 10, h\({}_{2}\) = 20, h\({}_{3}\) = 5, h\({}_{4}\) = 10; and ordering costs for suppliers are o\({}_{1}\) = 10, o\({}_{2}\) = 20, o\({}_{3}\) = 15, o\({}_{4}\) = 25. Tables 2-7 show the values used in the numerical example for demand sizes of components, suppliers capacities, undiscounted unit prices, distance matrix, price discount ratios, and price discount interval lower limits respectively. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Component/Period** & **1** & **2** & **3** \\ \hline
**1** & 10 & 20 & 40 \\
**2** & 20 & 10 & 30 \\
**3** & 30 & 30 & 10 \\
**4** & 20 & 40 & 10 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Demand sizes for components (\(d_{\beta}\)). By using the numerical example data, the mathematical model is solved in the LINGO 9.0 program. The optimal solution of the model is given in the following tables. In Table 8, it is seen how much of the components are taken from suppliers in each period. \begin{table}
\begin{tabular}{c c c c c} \hline
**Supplier/Supplier** & **1** & **2** & **3** & **4** & **5** \\ \hline
**1** & 0 & 20 & 30 & 50 & 30 \\
**2** & & 0 & 30 & 40 & 20 \\
**3** & & 0 & 10 & 40 \\
**4** & & & & 0 & 30 \\
**5** = depot & & & & 0 \\ \hline \end{tabular}
\end{table} 
Table 6: Price discount ratios (\(p_{\bar{k}}\)). \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline  & & \multicolumn{3}{c}{**Period 1**} & \multicolumn{3}{c}{**Period 2**} & \multicolumn{3}{c}{**Period 3**} \\ \hline
**Suppliers/Components** & **1** & **2** & **3** & **4** & **1** & **2** & **3** & **4** & **1** & **2** & **3** & **4** \\ \hline
**1** & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 10 & 20 & 0 & 0 \\
**2** & 0 & 15 & 0 & 0 & 5 & 10 & 0 & 20 & 5 & 10 & 0 & 0 \\
**3** & 0 & 0 & 12 & 15 & 0 & 0 & 12 & 15 & 0 & 0 & 8 & 10 \\
**4** & 10 & 5 & 18 & 5 & 20 & 0 & 20 & 5 & 20 & 0 & 0 & 0 \\
**Total** & 10 & 20 & 30 & 20 & 25 & 10 & 32 & 40 & 35 & 30 & 8 & 10 \\ \hline \end{tabular}
\end{table} 
Table 7: Price discount interval lower limits (\(K_{\bar{k}}\)). \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline  & & \multicolumn{3}{c}{**Period 1**} & \multicolumn{3}{c}{**Period 2**} & \multicolumn{3}{c}{**Period 3**} \\ \hline
**Supplier/Component** & **1** & **2** & **3** & **4** & **1** & **2** & **3** & **4** & **1** & **2** & **3** & **4** \\ \hline
**1** & 10 & 20 & 5 & 10 & 10 & 20 & 5 & 10 & 10 & 20 & 5 & 10 \\
**2** & 5 & 15 & 10 & 20 & 5 & 15 & 10 & 20 & 5 & 15 & 10 & 20 \\
**3** & 4 & 30 & 12 & 15 & 4 & 30 & 12 & 15 & 4 & 30 & 12 & 15 \\
**4** & 20 & 10 & 22 & 30 & 20 & 10 & 22 & 30 & 20 & 10 & 22 & 30 \\ \hline \end{tabular}
\end{table} 
Table 3: Supplier capacities (\(s_{\mathit{i}\mathit{i}\mathit{i}}\)). \begin{table}
\begin{tabular}{c c c c c} \hline
**Supplier/Discount Interval** & **1** & **2** & **3** & **4** \\ \hline
**1** & 1 & 0.8 & 0.65 & 0.55 \\
**2** & 1 & 0.75 & 0.65 & 0.5 \\
**3** & 1 & 0.9 & 0.8 & 0.7 \\
**4** & 1 & 0.85 & 0.7 & 0.55 \\ \hline \end{tabular}
\end{table} 
Table 8: Order quantity for components (\(Q_{\mathit{i}\mathit{i}\mathit{i}}\)). \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline  & & \multicolumn{3}{c}{**Period 1**} & \multicolumn{3}{c}{**Period 2**} & \multicolumn{3}{c}{**Period 3**} \\ \hline
**Supplier/Component** & **1** & **2** & **3** & **4** & **1** & **2** & **3** & **4** & **1** & **2** & **3** & **4** \\ \hline
**1** & 10 & 20 & 5 & 10 & 10 & 20 & 5 & 10 & 10 & 20 & 5 & 10 \\
**2** & 5 & 15 & 10 & 20 & 5 & 15 & 10 & 20 & 5 & 15 &Table 9 shows the undiscounted total amounts of the components ordered from each supplier.