## 4 Matheuristics Development
### Step 2. Order Picking Makespan Minimization Problem (Using \(y_{ijpb}\) as Input from Step 1)
The number of solutions varied from one to seven for the 1 st layout (instances 1-10), from three to seven for the 2 nd layout (instances 41-50), and from one to three for the third layout (instances 81-90). The average time spent on solving was 284.79 s and varied from 133.52 s to 302.81 s for the first layout (instances 1-10). The average time spent on solving was 301.65 s and varied from 300.77 s to 303.28 s for the 2 nd layout (instances 41-50). The average time spent solving was 302.79 s and varied from 300.86 s to 304.85 s for the 3 rd layout (instances 81-90). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Instance**} & \multirow{2}{*}{**Orders**} & \multicolumn{2}{c}{**Distribution**} & \multirow{2}{*}{**Pickers**} & \multirow{2}{*}{**Locations**} & \multirow{2}{*}{**Repeated**} & \multirow{2}{*}{**Buffers**} & \multirow{2}{*}{**Products in the Order**} \\  & & **Centre** & & & & & \\  & & **Layout** & & & & & \\ \hline
1–10 & 5 & & & & & & \\
11–20 & 10 & & & & & & \\
21–30 & 15 & & & & & & \\
31–40 & 20 & & & & & & \\ \hline
41–50 & 5 & & & & & & \\
51–60 & 10 & & & & & & \\
61–70 & 15 & & 2 & 4 & 120 & 48 & 6 & \(1,\ldots,5\) \\
71–80 & 20 & & & & & & \\ \hline
81–90 & 5 & & & & & & \\
91–100 & 10 & & 3 & 8 & & & \\
101–110 & 15 & & & & & & \\
111–120 & 20 & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Test data. Table 3 reports the results of the matheuristics' 1 st step solution--the buffer free time minimization problem--for five orders. The search was completed, and all instances were solved optimally. The number of solutions varied from 1 to 3 for the first layout (instances 1-10), from 1 to 5 for the second layout (instances 41-50), and from 1 to 19 for the third layout (instances 81-90). The average time spent in solving was 3.56 s and varied from 2.49 s to 11.82 s for the 1 st layout (instances 1-10). The average time spent solving was 2.60 s and varied from 1.75 s to 8.69 s for the second layout (instances 41-50). The average time spent solving was 15.93 s and varied from 3.41 s to 44.19 s for the third layout (instances 81-90). Table 3 also reports the results of the matheuristics' second step solution--order picking makespan minimization problem after solving buffers free time minimization problem--for five orders. The search was completed only for the seventh instance was so only for this instance was an optimal solution found. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multicolumn{8}{c}{**Step 1: Pickers’ Free Time Minimization Problem**} & \multicolumn{4}{c}{**Step 2: Order Picking Makespan Minimization Problem after Solving Pickers Free Time Minimization Problem**} \\ \hline \multirow{3}{*}{**Instance**} & \multirow{3}{*}{**Number of Solutions**} & **Time Spent in Solving Solving [s]** & **Engine Extraction [s]** & **Number of Solutions** & **Comparison with the Lower Bound** & **Time Spent in Solving [s]** & **Engine Extraction [s]** \\ \cline{1-1}  & & 2 & 2.54 & 0.96 & 1.58 & 7 & 42.19\% & 301.13 & 257.25 & 43.88 \\  
\hline \hline \end{tabular}
\end{table} 
Table 2: Results for 2 steps matheuristics starting with solving pickers’ free time minimization problem for 5 orders. For 1-10 instances, the average gap to the lower bound was 44.84% and varied from 0.00% to 100.00%. For 41-50 instances, the average gap to the lower bound was 54.08% and varied from 25.00% to 66.93%. For 81-90 instances, the average gap to the lower bound was 63.48% and varied from 39.51% to 100.00%. The number of solutions varied from one to seven for the first layout (instances 1-10), from three to seven for the second layout (instances 41-50), and from one to three for the third layout (instances 81-90). The average time spent on solving was 285.22 s and varied from 139.13 s to 302.61 s for the 1 st layout (instances 1-10). The average time spent on solving was 301.31 s and varied from 300.78 s to 302.36 s for the second layout (instances 41-50).