# Linking Scheduling Criteria to Shop Floor Performance in Permutation Flowships
## 4 Computational Results
### Ranking of Scheduling Criteria  
In this section, we further try to explore the trade-off among the different criteria by answering the following question: Once we choose certain scheduling criterion according to the aforementioned ranking, how are the gains (or losses) that we can expect in the different shop floor performance measures when we switch from one scheduling criterion to another. More formally, we intend to quantify the difference between picking one scheduling criterion or another for a given shop floor performance measure. To address this issue, we define the \(RD_{PM}\) or Relative Deviation with respect to a given \(PM\) (performance measure) in the following manner. \[RD(A)_{PM}=\frac{PM(S_{A})-PM(S_{A^{+}})}{PM(S_{A^{+}})}\cdot 100 \tag{5}\]  
where \(PM(S_{A})\) is the value of \(PM\) obtained for the sequence \(S_{A}\) which minimises scheduling criterion \(A\). Analogously, \(S_{A^{+}}\) is the sequence obtained by minimising scheduling criterion \(A^{+}\), being \(A^{+}\) the scheduling criterion ranking immediately behind \(A\) for the performance measurement \(PM\). When \(A\) is the scheduling criterion ranking last for \(PM\), then \(RD\) is set to zero. Note that this definition of \(RD\) allows us to obtain more information than the mere rank of scheduling criteria. For instance, let us consider the scheduling criteria \(A\), \(B\), and \(C\), which rank (ascending order) with respect to the performance measure \(PM\) in the following manner: \(B\), \(C\), \(A\). This information (already obtained in Section 4.1) simply states that \(B\) (\(C\)) is more aligned that \(C\) (\(A\)) with respect to performance measure \(PM\), but does not convey information on whether there are substantial differences between the three criteria for \(PM\), or not. This information can be obtained by measuring the corresponding \(RD\): If \(RD(B)\) is zero or close to zero, it implies that \(B\) and \(C\) yield similar values for \(PM\), and therefore there is not so much difference (with respect to \(PM\)) between minimizing \(B\), or \(C\). In contrast, a high value of \(RD(C)\) indicates a great benefit (with respect to \(PM\)) when switching from minimizing \(A\) to minimizing \(C\). Since \(RD\) is defined for a specific instance, we use the Average Relative Deviation (\(ARD\)) for comparison across the testbed, consisting in averaging the RDs. The results of the experiments for the different testbeds with respect to ARD are shown in Tables 12-15, together with the rank of each criterion for each problem size. In addition, the cumulative ARD of the scheduling criteria for each shop floor performance measures are shown in Figure 2 for the different testbed. In view of the results, we give the following comments. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline  & & & \multicolumn{2}{c}{LC} & \multicolumn{2}{c}{MC} & \multicolumn{2}{c}{HC} & \multicolumn{2}{c}{NC} \\ \(n\) & \(m\) & **TH** & **ACT** & **WIP** & **TH** & **ACT** & **WIP** & **TH** & **ACT** & **WIP** & **TH** & **ACT** & **WIP** \\ \hline
5 & 5 & **100.0** & **100.0** & **100.0** & **100.0** & **100.0** & **67.97** & **7.29** & **0.03** & **0.00** & **100.0** & **100.0** & **100.0** \\
5 & 10 & **100.0**  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c c c} \hline  & & & \multicolumn{4}{c}{\(TH\)} & \multicolumn{4}{c}{\(ACT\)} & \multicolumn{4}{c}{\(WIP\)} \\ \(n\) & \(m\) & \(C_{max}\) & \(\sum C_{f}\) & \(\sum ITH_{j}\) &  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c c c} \hline  & & & \multicolumn{4}{c}{\(TH\)} & \multicolumn{4}{c}{\(ACT\)} & \multicolumn{4}{c}{\(WIP\)} \\ \(n\) & \(m\) & \(C_{max}\) & \(\sum C_{f}\) & \(\sum ITH_{j}\) & \(\sum IT_{j}\) &  
* \(\sum{ITH_{j}}\) emerges as an interesting criterion as its performance is only marginally worse than \(C_{max}\) with respect to \(TH\)--particularly in the NC testbed, see Figure 2a, but it obtains better values regarding \(ACT\) and \(WIP\).