## 3 The Optimality Box
### The Largest Relative Perimeter of the Optimality Box
_Algorithms_**2018**, _11, 66_

\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \(n\) & \(|B|\) & **N-Fix Jobs** & **Laws** & \(\Delta\) **(\%)** & \(\Delta_{mid-p}\) **(\%)** & \(\mbox{Per}\ \mathcal{OB}(\pi_{k_{b}},T)\) & \(\Delta_{mid-p}/\Delta\) & \(\Delta_{max}/\Delta\) & **CPU-Time (s)** \\ \hline
**1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** \\ \hline \multicolumn{10}{c}{**Class 2**} \\ \hline
50 & 1 & 0 & 1.2.3 & 1.023598 & 2.401925 & 1.027 & 2.346551 & 1,395708 & 0,020781 \\
100 & 1 & 0 & 1.2.3 & 0.608379 & 0.995588 & 0.9948 & 1.636461 & 1.618133 & 0.047795 \\
500 & 1 & 0 & 1.2.3 & 0.265169 & 0.482631 & 0.9947 & 1.820092 & 1.630094 & 0.215172 \\
1000 & 1 & 0 & 1.2.3 & 0.176092 & 0.252525 & 0.9952 & 1.434053 & 1.427069 & 0.535256 \\
5000 & 1 & 0 & 1.2.3 & 0.111418 & 0.14907 & 0.9952 & 1.33793 & 1.089663 & 7.096339 \\
10,000 & 1 & 0 & 1.2.3 & 0.117165 & 0.13794 & 0.9948 & 1.177313 & 1.004612 & 2.58328 \\ \hline \multicolumn{10}{c}{**Class 3**} \\ \hline
50 & \multicolumn{1}{c}{**Minimum**} & \multicolumn{1}{c}{0.11418} & 0.13794 & 0.9947 & 1.177131 & 1.004612 & 0,020781 \\
**Average** & \multicolumn{1}{c}{0.383637} & 0.736613 & 0.99494 & 1.6254 & 1.399944 & 5.533104 \\
**Maximum** & \multicolumn{1}{c}{1.023598} & 2.401925 & 0.9952 & 2.346551 & 1.630094 & 25.28328 \\ \hline \multicolumn{10}{c}{**Class 3**} \\ \hline
50 & \multicolumn{1}{c}{3} & 1 & 1.2.3 & 0.636163 & 0.657619 & 1.171429 & 1.033727 & 1.004246 & 0.047428 \\
100 & \multicolumn{1}{c}{3} & 1 & 1.2.3 & 1.705078 & 1.789222 & 1.240238 & 1.1049349 & 1.009568 & 0.066329 \\
500 & \multicolumn{1}{c}{3} & 1 & 1.2.3 & 0.332547 & 0.382898 & 1.205952 & 1.154112 & 1.138869 & 0.249044 \\
1000 & \multicolumn{1}{c}{3} & 1 & 1.2.3 & 0.286863 & 0.373247 & 1.400833 & 1.301132 & 1.101748 & 0.421837 \\
5000 & \multicolumn{1}{c}{3} & 1 & 1.2.3 & 0.246609 & 0.323508 & 1.380833 & 1.311825 & 1.140728 & 2.51218 \\
10,000 & \multicolumn{1}{c}{3} & 1 & 1.2.3 & 0.26048 & 0.338709 & 1.098572 & 1.300324 & 1.095812 & 5.46782 \\ \hline \multicolumn{10}{c}{**Minimum**} & \multicolumn{1}{c}{0.246609} & 0.323508 & 1.098572 & 1.03727 & 1.004264 & 0.047428 \\ \multicolumn{10}{c}{**Average**} & \multicolumn{1}{c}{0.577957} & 0.644201 & 1.249643 & 1.191295 & 1.0818286 & 1.460773 \\
**Maximum** & \multicolumn{1}{c}{1.705078} & 1.789222 & 1.400833 & 1.311825 & 1.140728 & 5.46782 \\ \hline \multicolumn{10}{c}{**Class 4**} \\ \hline
50 & \multicolumn{1}{c}{3} & 1 & 1 & 0.467885 & 0.497391 & 1.17369 & 1.063064 & 1.035412 & 0.043454 \\
100 & \multicolumn{1}{c}{3} & 1 & 1 & 0.215869 & 0.226697 & 1.317222 & 1.05016 & 1.031564 & 0.067427 \\
500 & \multicolumn{1}{c}{3} & 1 & 1 & 0.128445 & 0.15453 & 1.424444 & 1.203083 & 1.17912 & 0.256617 \\
1000 & \multicolumn{1}{c}{3} & 1 & 1 & 0.11304 & 0.118882 & 1.307738 & 1.068077 & 1.048258 & 0.50344 \\
5000 & \multicolumn{1}{c}{3} & 1 & 1 & 0.076917 & 0.085504 & 1.399048 & 1.111631 & 1.046061 & 2.612428 \\
10,000 & \multicolumn{1}{c}{3} & 1 & 1 & 0.067836 & 0.076221 & 1.591905 & 1.123606 & 1.114005 & 4.407236 \\ \hline \multicolumn{10}{c}{**Minimum**} & \multicolumn{1}{c}{0.067836} & 0.076221 & 1.17369 & 1.05016 & 1.031564 & 0.043454 \\
**Average** & \multicolumn{1}{c}{0.178043} & 0.193204 & 1.369008 & 1.10327 & 1.074836 & 1.3151 \\
**Maximum** & \multicolumn{1}{c}{0.467885} & 0.497391 & 1.591905 & 1.203083 & 1.17912 & 4.407236 \\ \hline \multicolumn{10}{c}{**Class 5**} \\ \hline
50 & \multicolumn{1}{c}{3} & 2 & 1.2.3 & 1.341619 & 1.508828 & 1.296195 & 1.124632 & 1.035182 & 0.049344 \\
100 & \multicolumn{1}{c}{3} & 2 & 1.2.3 & 0.700955 & 0.867886 & 1.271976 & 1.238149 & 1.037472 & 0.070402 \\
500 & \multicolumn{1}{c}{3} & 2 & 1.2.3 & 0.182378 & 0.241735 & 1.029 & 1.32546 & 1.296414 & 0.255643 \\
1000 & \multicolumn{1}{c}{3} & 2 & 1.2.3 & 0.098077 & 0.11073 & 1.473451 & 1.129006 & 1.104537 & 0.509969 \\
5000 & \multicolumn{1}{c}{3} & 2 & 1.2.3 & 0.074599 & 0.084418 & 1.204435 & 1.13624 & 1.056254 & 2.577595 \\
10,000 & \multicolumn{1}{c}{3} & 2 & 1.2.3 & 0.064226 & 0.074749 & 1.359181 & 1.163846 & 1.042676 & 5.684847 \\ \hline \multicolumn{10}{c}{**Minimum**} & \multicolumn{1}{c}{0.064226}

## 7 Concluding Remarks

The uncertain problem \(1|p_{i}^{L}\leq p_{i}\leq p_{i}^{Ul}|\sum C_{i}\) continues to attract the attention of the OR researchers since the problem is widely applicable in real-life scheduling and is commonly used in many multiple-resource scheduling systems, where only one of the machines is the bottleneck and uncertain.