## 4 Case Illustration
 } & \multicolumn{8}{c}{**Processing Times on Resources (min)**} \\ \cline{5-12}  & & & & **S1** & **S2** & **S3** & **S4** & **S5** & **S6** & **S7** & **S8** & **S9** \\ \hline A & 81 & 2800 & 63.64 & 34 & 33 & 31 & 24 & 32 & 37 & 35 & 44 & 36 \\ B  
Step B05: By dividing each product's processing time on a resource by the column total of the respective resource, the processing times (\(p_{ij}\)) under resources are normalized (see Table 4). Steps B06 and B07: In the same manner, the maximum available capacities of the bottlenecks and the required capacities of slack resources are determined and normalized (\(q_{ij}\)) (see Table 4). Step B08: By multiplying the normalized processing times (\(p_{ij}\)) obtained in Step B05 and the normalized feasible capacities (\(q_{ij}\)) obtained in Step B07, a reciprocal decision matrix (\(h_{i}\)) is derived (see Table 4). Step B09: A priority sequence for the products is derived by multiplying the normalized contribution margin for each product by the weighted vector of profit (0.667) and then adding the priority of the resources, which is established by the multiplication of the resulting vector of resources (0.333) and the resulting normalized matrix of the capacity and processing times in the preceding step. The new priority sequence, seq2, is lined up as G > A > B > F > E > C > D. This sequence relies on the fact that the relative weight of the contribution margins dominates the priority for products \(G\), \(F\), and \(E\), in contrast to the relative weight of the resources. Steps B10-a and B10-b: Products are allocated to the initial MPS according to the priority sequence obtained in Step A02 because the PMP has been identified as a single-bottleneck environment (see Table 5). A temporary product mix is formed as 81A, 52B, 57C, 69E, 290D, 51G, and 0F. This actually implies a branching point, from which the solution procedure can proceed in two directions. The first direction is taken in the situation where the PMP is characterized as a multi-bottleneck environment, followed by Step B11. The second direction relates to the single-bottleneck environment, which subsequently branches out depending on a comparison of the priority sequences from A02 and B09. The second direction is picked up in this case because it relates to a single-bottleneck environment. Due to the discordant priority sequences from A02 and B09, the process continues to step B12. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**SBU**} & \multicolumn{2}{c}{**B04**} & \multicolumn{5}{c}{**B05—Normalized Processing Times (\(p_{ij}\))**} & \multicolumn{5}{c}{**B08**} & \multicolumn{1}{c}{**B09**} \\ \cline{2-13}  & **(\(p_{i}\))** & **S1** & **S2** & **S3** & **S4** & **S5** & **S6** & **S7** & **S8** & **S9** & **(\(h_{i}\))** & **(\(W_{i}\))** & **seq2** \\ \hline A & 0.114 & 0.081 & 0.083 & 0.081 & 0.084 & 0.083 & 0.082 & 0.083 & 0.081 & 0.082 & 0.219 & 0.149 & 2 \\ B & 0.116 & 0.095 & 0.095 & 0.094 & 0.094 & 0.094 & 0.096 & 0.095 & 0.094 & 0.096 & 0.189 & 0.140 & 3 \\ C & 0.118 & 0.112 & 0.111 & 0.113 & 0.111 & 0.112 & 0.111 & 0.111 & 0.113 & 0.112 & 0.161 & 0.132 & 6 \\ D & 0.118 & 0.124 & 0.123 & 0.123 & 0.122 & 0.122 & 0.122 & 0.123 & 0.124 & 0.123 & 0.146 & 0.127 & 7 \\ E & 0.150 & 0.153 & 0.153 & 0.154 & 0.153 & 0.153 & 0.153 & 0.154 & 0.153 & 0.153 & 0.117 & 0.139 & 5 \\ F & 0.162 & 0.189 & 0.188 & 0.188 & 0.190 & 0.189 & 0.189 & 0.190 & 0.189 & 0.095 & 0.140 & 4 \\ G & 0.223 & 0.246 & 0.246 & 0.247 & 0.247 & 0.247 & 0.246 & 0.245 & 0.246 & 0.072 & 0.173 & 1 \\ \hline
**B07 (\(q_{j}\)):** & 0.116 & 0.110 & 0.106 & 0.079 & 0.106 & 0.121 & 0.117 & 0.121 & 0.121 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Normalized processing time (\(p_{ij}\)).