# Supplier Replacement Model in a One-Level Assembly System under Lead-Time Uncertainty
## 5 Experimental results
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Instance** & **Average Number** & **Mean Gap** & **Mean Gap \({}^{*}\)** & **CPU Time** \\
**Family** & **of Iterations** & **(\%)** & **from BKS (\%)** & **(s)** \\ \hline
10 & 148.06 & 162.23 & 11.09 & 0.31 \\
20 & 172.08 & 160.86 & 44.01 & 0.46 \\
30 & 306.38 & 178.86 & 44.44 & 0.59 \\
40 & 251.00 & 151.65 & 85.33 & 0.65 \\
50 & 385.06 & 132.32 & 102.62 & 0.73 \\
60 & 533.07 & 135.62 & 108.24 & 0.88 \\
70 & 593.01 & 148.96 & 108.99 & 0.99 \\
80 & 698.12 & 158.73 & 105.74 & 1.11 \\
90 & 728.58 & 161.89 & 105.09 & 1.27 \\
100 & 837.98 & 135.69 & 145.88 & 1.59 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Results obtained by \(GA_{P}\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Instance** & **Average Number** & **Mean Gap** & **Mean Gap \({}^{*}\)** & **CPU Time** \\
**Family** & **of Iterations** & **(\%)** & **from BKS (\%)** & **(s)** \\ \hline
10 & 38.23 & 147.26 & 19.04 & 0.28 \\
20 & 128.26 & 159.75 & 44.50 & 0.38 \\
30 & 275.96 & 178.03 & 44.87 & 0.48 \\
40 & 251.50 & 151.65 & 85.33 & 0.55 \\
50 & 368.22 & 131.95 & 102.88 & 0.63 \\
60 & 472.86 & 135.45 & 108.37 & 0.74 \\
70 & 570.92 & 148.98 & 108.97 & 0.86 \\
80 & 647.76 & 158.66 & 105.80 & 0.97 \\
90 & 697.04 & 161.99 & 105.00 & 1.08 \\
100 & 844.43 & 135.71 & 145.85 & 1.41 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Results obtained by genetic algorithm (GA). Figure 3 illustrates the evolution of the gap over iterations of \(GA_{H+p}\) on an instance with 100 components (\(gap_{gen}=\frac{best_{salgen}-best_{sal000}}{best_{sal000}}\cdot 100\)). In the first iteration, the gap is 86.81%, whereas at the 1000\({}^{th}\) iteration the gap is reduced to 1.31%. The gap finally reaches 0.00% at iteration 2132. Ultimately, 98.48% of the improvement is achieved during the first 1000 iterations. Another set of tests had already been done, we adapted B&B from [39] to policy 0 and applied it directly to the instances with a small number of suppliers (up to 50). For larger instances, the adapted B&B was used as a beam search algorithm (only the most promising nodes are explored) to reduce the calculation time. Thus, the results were not exact but approximate. The tests show that the gap between the beam search and the optimal solution is less than 2.5%. The quality of the beam algorithm is very good, as the beam search found optimal solutions for 491 of the 500 tests executed. Note that the optimal solutions found by B&B or beam search are also optimal for our problem for policy 0 and feasible for other policies, so they can be used as upper bounds in our study. In order to verify the pertinence of our approach and assess the efficiency of the genetic algorithm, we compare the results of our \(GA_{H+P}\) model with the upper and lower bounds (see Table 7). The table lists the names of the instance families (corresponding to the number of suppliers) along with the mean gaps from the lower and upper bounds. As each family has 100 instances, the mean gaps in percentage format are calculated as follows:  
\[\text{mean GAP}(GA_{H+P},\text{LB})=\left(\frac{\sum_{k=1}^{K}\frac{GA_{H+P}^{ k}-LB^{k}}{LB^{k}}}{K}\right)\times 100 \tag{15}\]  
\[\text{mean GAP}(GA_{H+P},\text{UB})=\left(\frac{\sum_{k=1}^{K}\frac{UB^{k}-GA_ {H+P}^{k}}{GA_{H+P}^{k}}}{K}\right)\times 100 \tag{16}\]  
where \(GA_{H+P}^{k}\), \(LB^{k}\) and \(UB^{k}\) are the result of GA, lower bound and upper bound to the problem on instance \(k\), respectively and \(K\)=100 (instances by family). Figure 3: Evolution of the gap to the best known solution (BKS) over iterations. The mean gap between the lower bound and the \(GA_{H+P}\) is less than 15%, and smaller gaps are observed for instances with 20 to 50 suppliers. Note: the mean gap between the upper bound and the \(GA_{H+P}\) is very large, ranging between 145.89% and 194.95%. Thus, the solutions without the pricing policies give very poor upper bounds. This shows the huge impact of the applied pricing policies for lead-time uncertainty reduction on the total cost reduction.