4.2 Relax & Repair Heuristics
4.3.1 Influence of the Percentage of Fixed Connections on the Relative
First, we analyze the influence of the parameterk(i.e. the percentage of fixed “important”
connections). To this end, we compare the relative error of different variants of Fsfs-Fix andPriority-Repair, varying only in the parameter k. For our analysis, we use five different valuesk∈ {0,25,50,75,100} wherek= 0 represents a no-wait policy and k= 100an all-wait policy.
As it turned out during the case study, values of kwhich yield good results for small source delays lead to large relative errors for large source delays and vice versa. Hence we combine different variants of Fsfs-Fixin an approach calledBest-Fsfs-Fix: For each input instance, runFsfs-Fixwith different values forkand for each single scenario,
take the best solution. Analogously,Best-Repair denotes a similar approach based onPriority-Repair. Formally:
Algorithm 4.8: Best-Fsfs-Fix Input: An instanceiof (DM).
Step 1: Solveiby running Fsfs-Fixwithk∈ {0,25,50,75,100} and obtain solutions(xk, zk, g)with objective values Fk.
Step 2: Letk∗= argmin
k∈{0,25,50,75,100}
Fk.
Output: A solution(xk∗, zk∗, g) for iwith objective valueFk∗.
Algorithm 4.9: Best-Repair Input: An instanceiof (DM).
Step 1: Solveiby running Priority-Repairwith k∈ {0,25,50,75,100}
and obtain solutions(xk, zk, gk) with objective valuesFk. Step 2: Letk∗= argmin
k∈{0,25,50,75,100}
Fk.
Output: A solution(xk∗, zk∗, gk∗) for iwith objective valueFk∗.
We analyze three classes of delay scenarios: For the class “small source delays”, in each scenario, we randomly generated ten source delays between 60 and 180 seconds. The class “large source delays” contains scenarios in which we generated ten source delays between 1 500 and 1 800 seconds. All scenarios in the class “mixed source delays” feature ten randomly generated source delays between 180 and 900 seconds (as in Section 3.5).
For each class of delay scenarios and for each observation period, we generated about 400 different delay scenarios and solved each resulting instance of the delay management problem both exactly and by invoking each of the heuristics presented in this chapter.
We start with analyzing Fsfs-FixandPriority-Repair on the class of small source delays. The relative errors for different variants ofFsfs-Fixare summarized in Table 4.6, the average relative error is depicted in Figure 4.7. For all observation periods, both the maximal and the average relative error decrease when k grows. For Priority-Repair, the relative errors are given in Table 4.8 while the average relative error also
4.3 Numerical Results
is illustrated in Figure 4.9. For all observation periods, the average relative error of Priority-Repairdecreases with growingk, too, and apart from some exceptions, the same is true for its maximal relative error. Hence, for the data set of our case study, the best fixed rule when small source delays occur is an all-wait policy.
If we consider large source delays, the situation changes radically. For this case, the relative errors of Fsfs-Fix and Priority-Repair are summarized in Tables 4.10 and 4.12, while the average relative errors are depicted in Figures 4.11 and 4.13. For both heuristics and for each observation period, the average relative errors grow withk, and with only two exceptions, the same holds for the maximal relative errors. Hence, for the data set of our case study, the best fixed rule when large source delays occur is a no-wait policy.
Finally, we compare different variants of Fsfs-FixandPriority-Repairon the third class of delay scenarios. We present an overview of the relative errors in Table 4.14 and Table 4.16 and depict the average relative errors in Figure 4.15 and Figure 4.17. As in the case of large source delays, both the average and the maximal relative errors increase when kgrows. Hence, again a no-wait strategy yields the lowest average relative error and minimizes the maximal relative error, too.
Due to the fact that the parameter k has such a large effect on the relative error (and since this effect in the case of small source delays is in diametrical opposition to the effect in the case of large source delays), it is reasonable not to use one fixed variant of Fsfs-FixorPriority-Repair, but to use theBest-Fsfs-Fixor Best-Repair approach. Hence, when comparing Fsfs-Fix and Priority-Repair with other heuristics in Section 4.3.2 and in Section 4.3.3, we always use Best-Fsfs-Fix and Best-Repairinstead of Fsfs-Fix andPriority-Repair.
k= 0 k= 25 k= 50
hours max avg max avg max avg
2 3.7100 0.2169 0.4337 0.0319 0.4151 0.0204 4 2.1139 0.1870 0.5945 0.0301 0.3085 0.0170 6 1.8266 0.1964 0.6300 0.0415 0.3791 0.0255 8 2.8212 0.1731 0.4453 0.0321 0.3340 0.0185 10 1.6615 0.1953 0.6218 0.0459 0.3257 0.0193 12 2.0049 0.1854 0.7937 0.0341 0.7937 0.0220
k= 75 k= 100 Best-Fsfs-Fix
hours max avg max avg max avg
2 0.4151 0.0103 0.4151 0.0046 0.4151 0.0042 4 0.2293 0.0050 0.2293 0.0040 0.2293 0.0035 6 0.3791 0.0098 0.3791 0.0079 0.3791 0.0077 8 0.2597 0.0075 0.2060 0.0042 0.2060 0.0039 10 0.3257 0.0087 0.3257 0.0063 0.3257 0.0053 12 0.5814 0.0118 0.5814 0.0094 0.5814 0.0083
Table 4.6: Average and maximal relative error of Fsfs-Fixfor small source delays.
Figure 4.7: Average relative error ofFsfs-Fix, depending on the size of the observation period, for small source delays.
4.3 Numerical Results
k= 0 k= 25 k= 50
hours max avg max avg max avg
2 4.7921 0.3357 4.7921 0.1309 2.8715 0.0830
4 6.1489 0.3829 6.1383 0.2055 6.1383 0.1794
6 6.6541 0.3679 6.0740 0.2020 6.0740 0.1822
8 9.0051 0.5140 8.7283 0.3533 6.5679 0.1841
10 12.9513 0.3043 12.9513 0.1485 12.9513 0.1211
12 8.8329 0.3238 7.3069 0.1562 7.3069 0.1360
k= 75 k= 100 Best-Repair
hours max avg max avg max avg
2 2.8715 0.0729 2.8715 0.0672 2.8715 0.0663
4 6.1383 0.1674 6.1383 0.1663 6.1383 0.1655
6 8.5783 0.1731 8.9225 0.1721 6.0740 0.1637
8 6.5679 0.1731 6.5679 0.1698 6.5679 0.1695
10 12.9513 0.1106 12.9513 0.1082 12.9513 0.1058 12 10.3362 0.1324 10.7562 0.1314 7.3069 0.1223
Table 4.8: Average and maximal relative error of Priority-Repairfor small source delays.
Figure 4.9: Average relative error of Priority-Repair, depending on the size of the observation period, for small source delays.
k= 0 k= 25 k= 50
hours max avg max avg max avg
2 2.5852 0.3921 2.5688 0.4270 4.2746 0.4748
4 4.8944 0.9151 5.2519 1.0235 5.7679 1.1799
6 6.6746 1.2786 7.1859 1.4408 8.3953 1.7256
8 8.9313 1.4631 9.7857 1.7034 10.2950 2.0201
10 12.9596 1.4490 13.3854 1.6730 14.4160 2.0317 12 11.2851 1.4867 11.9408 1.7509 14.5262 2.0686
k= 75 k= 100 Best-Fsfs-Fix
hours max avg max avg max avg
2 4.2746 0.4996 4.5889 0.5272 2.5677 0.3843
4 5.9564 1.2830 6.0409 1.3292 4.8944 0.9104
6 8.6063 1.9001 8.7733 1.9886 6.6746 1.2728
8 13.5434 2.2881 14.0488 2.3845 8.9313 1.4579 10 16.5247 2.3475 17.4129 2.4824 12.9596 1.4444 12 20.2093 2.4272 20.3932 2.5746 11.2851 1.4812
Table 4.10: Average and maximal relative error of Fsfs-Fixfor large source delays.
Figure 4.11: Average relative error of Fsfs-Fix, depending on the size of the observa-tion period, for large source delays.
4.3 Numerical Results
k= 0 k= 25 k= 50
hours max avg max avg max avg
2 1.2704 0.0675 1.2704 0.1095 1.8971 0.1654
4 6.9360 0.2546 7.6836 0.3988 8.6981 0.5761
6 13.3661 0.8108 14.4953 1.0696 22.4031 1.4974 8 30.0758 1.5590 33.6386 1.9273 39.2582 2.6344 10 40.8221 1.9558 42.3855 2.3405 45.6342 3.1980 12 36.0509 2.0466 39.1093 2.4290 42.9223 3.3764
k= 75 k= 100 Best-Repair
hours max avg max avg max avg
2 2.3358 0.2300 2.7501 0.2718 1.2499 0.0638
4 10.0314 0.7724 10.0764 0.8908 6.9360 0.2518 6 24.9630 1.8892 26.6913 2.1468 13.3661 0.8106 8 39.5246 3.2449 40.5419 3.6115 30.0758 1.5327 10 42.2398 4.0302 46.5789 4.4183 33.6390 1.8997 12 88.5609 4.5261 89.7079 5.0700 36.0509 1.9356
Table 4.12: Average and maximal relative error of Priority-Repairfor large source delays.
Figure 4.13: Average relative error of Priority-Repair, depending on the size of the observation period, for large source delays.
k= 0 k= 25 k= 50
hours max avg max avg max avg
2 1.6042 0.1758 1.6336 0.1494 1.8881 0.1680 4 1.8154 0.2265 1.7595 0.2282 1.7510 0.2500 6 1.9983 0.2391 1.9786 0.2518 2.3513 0.2864 8 2.2311 0.2369 2.6057 0.2411 2.6869 0.2838 10 1.3684 0.2542 1.6049 0.2676 1.8055 0.2929 12 4.0549 0.2584 4.7230 0.2771 5.1889 0.2960
k= 75 k= 100 Best-Fsfs-Fix
hours max avg max avg max avg
2 1.8881 0.1807 1.9070 0.1840 1.6042 0.1410 4 1.9851 0.2684 1.9851 0.2738 1.7510 0.2050 6 2.3513 0.3146 2.3513 0.3169 1.9786 0.2184 8 2.7407 0.3058 2.7642 0.3099 2.2311 0.2131 10 2.2144 0.3199 2.2314 0.3238 1.2877 0.2298 12 5.2352 0.3219 5.2352 0.3254 4.0549 0.2373
Table 4.14: Average and maximal relative error of Fsfs-Fix for mixed source delays.
Figure 4.15: Average relative error of Fsfs-Fix, depending on the size of the observa-tion period, for mixed source delays.
4.3 Numerical Results
k= 0 k= 25 k= 50
hours max avg max avg max avg
2 1.0800 0.1237 1.0187 0.0987 1.5671 0.0913
4 7.7600 0.3161 7.7646 0.3297 7.7622 0.3429
6 47.7743 0.8774 49.4055 0.9417 52.8792 1.0278
8 150.5909 2.0991 155.7212 2.2107 168.3575 2.3881 10 113.1986 1.0459 116.7790 1.1275 125.8791 1.2727 12 271.3629 1.3865 280.3381 1.4734 299.4943 1.5723
k= 75 k= 100 Best-Repair
hours max avg max avg max avg
2 1.7714 0.1180 1.7714 0.1324 0.7567 0.0548
4 7.7622 0.4198 7.8109 0.4350 7.7600 0.2418
6 53.8926 1.2053 54.0041 1.2221 47.7743 0.7844
8 173.1206 2.6307 174.1113 2.6767 150.5909 2.0010 10 131.4135 1.5572 131.9608 1.5972 113.1986 0.9568 12 315.5096 1.9773 316.6856 2.0038 271.3629 1.2905
Table 4.16: Average and maximal relative error of Priority-Repairfor mixed source delays.
Figure 4.17: Average relative error of Priority-Repair, depending on the size of the observation period, for mixed source delays.