# Blocking Cyclic Job-Shop Scheduling Problems
## 5 Experimental Settings and Results
\begin{table}
\begin{tabular}{r r r r r r r r r r r r r r r} \hline \hline \multirow{2}{*}{**K**} & \multirow{2}{*}{**J**} & \multirow{2}{*}{**\#Cons**} & \multirow{2}{*}{**\#Var**} & \multicolumn{3}{c}{**Cyclic JSS: Zero Setups**} & \multicolumn{3}{c}{**Anticipatory Setups**} & \multicolumn{3}{c}{**Nonanticipatory Setups**} \\ \cline{3-16}  & & & & **BF** & **LB** & **Gap** & **Time** & **BF** & **LB** & **Gap** & **Time** & **BF** & **LB** & **Gap** & **Time** \\ \hline \multirow{4}{*}{**9**} & **3** & 733 & 190 & 257 & 257 & 0 & 7.35 & 390 & 390 & 0 & 18.6 & 337 & 337 & 0 & 17.3 \\  & **4** & 1313 & 289 & 270 & 270 & 0 & 5.09 & 441 & 441 & 0 & 6.92 & 335 & 335 & 0 & 7.77 \\  & **5** & 2157 & 406 & 535 & 164 & 69.27 & 3600 & 701 & 207 & 70.42 & 3600 & 660 & 166 & 74.80 & 3600 \\  & **6** & 3319 & 541 & 566 & 391 & 30.92 & 3600 & - & 444 & - & 3600 & - & 437 & - & 3600 \\ \hline \multirow{4}{*}{**12**} & **3** & 988 & 253 & 258 & 258 & 0 & 14.94 & 373 & 373 & 0 & 14.53 & 354 & 354 & 0 & 22.83 \\  & **4** & 1769 & 385 & 335 & 335 & 0 & 308.59 & 468 & 468 & 0 & 684.56 & 431 & 431 & 0 & 386.49 \\  & **5** & 2904 & 541 & 529 & 189 & 64.21 & 3600 & 695 & 258 & 62.88 & 3600 & 741 & 214 & 71.08 & 3600 \\  & **6** & 4465 & 721 & - & 190 & - & 3600 & - & 223 & - & 3600 & - & 206 & - & 3600 \\ \hline \multirow{4}{*}{**15**} & **3** & 1243 & 316 & 313 & 313 & 0 & 11.07 & 449 & 449 & 0 & 9.15 & 411 & 411 & 0 & 18.75 \\  & **4** & 2225 & 481 & 344 & 213 & 38.08 & 3600 & 457 & 251 & 44.99 & 3600 & 462 & 242 & 47.62 & 3600 \\  & **5** & 3651 & 676 & 568 & 330 & 41.90 & 3600 & 761 & 385 & 49.41 & 3600 & - & 285 & - & 3600 \\  & **6** & 5611 & 901 & - & 171 & - & 3600 & - & 200 & - & 3600 & - & 195 & - & 3600 \\ \hline \multirow{4}{*}{**18**} & **3** & 1498 & 379 & 275 & 275 & 0 & 17.42 & 365 & 365 & 0 & 26.56 & 360 & 360 & 0 & 26.95 \\  & **4** & 2681 & 577 & 364 & 201 & 44.78 & 3600 & 519 & 315 & 39.36 & 3600 & 484 & 180 & 62.76 & 3600 \\  & **5** & 4398 & 811 & 600 & 171 & 71.55 & 3600 & - & 210 & - & 3600 & 721 & 175 & 75.79 & 3600 \\  & **6** & 6757 & 1081 & - & 184 & - & 3600 & - & 198 & - & 3600 & - & 193 & - & 3600 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The performance of proposed MIP approaches on the test problem instances (#_Cons: number of constraints; #Var: number of variables; BF: best found; LB: lower bound_). In order to further understand the impact of sequence-dependent setup times, we considered a constant setup time with the following two options: (a) **MIN**: use the minimum set up time in a problem instance as the standard setup time between all jobs and (b) **MAX**: use the maximum set up time in a problem instance as the standard setup time between all jobs. Furthermore, to account for random effects in the data, four different datasets were generated with the same problem sizes. Table 4 shows the results of the comparisons with these changes for five different datasets for each problem size. The first three columns show replication (**Rep**), machines (**K**), and jobs (**J**), and the following columns are split by the best feasible solution, lower bounds, and gaps. Within each, the results are split by the type of setup, i.e., no setups (**ZS**), anticipatory, and nonanticipatory. For the case with setups, the results are further split by sequence-dependent setups (**SD**), standard maximum setups (**MAX**), and standard minimum setups (**MIN**). The results showed, as expected, that cycle times increased with increasing amounts of setup time. Furthermore, the MIP models for which setups were used led to more instances where solutions could not be found. For example, **MAX** compared to **ZS** clearly showed two instances where solutions were not found. Interestingly, **MIN** was able to find two solutions where no solution was found in the case of **ZS**. The differences in cycle times in the systems with sequence-dependent and standard setups demonstrated the importance of considering the sequence-dependent setups in blocking cyclic job-shop systems.