Small-sized instances

To show the influence of the valid inequalities presented in this paper on computation times and bounds derived from the linear relaxation (LP bound), we tested all 2⁷ = 128 possible combinations of sets of valid inequalities defined in Table 7.4. For each combi-nation we use 10 different random number seeds for optimization to get distinct solution paths with different computation times, because the seed is used for tie breaking in the preprocessing and solution process of the solver. Table 7.6 summarizes a selection of settings, properties and results. Depending on the size of the instances we set an up-per bound for the transition time (Trans., in minutes) between two consecutive trips to reduce the number of arcs (more precisely, reduce the number of waiting arcs) of the underlying graph. The maximum number of conductors |C| is set to 4 for all small-sized instances. The number of nodes |V |, number of arcs |A| of the graph, maximum number of parallel arcs |P | and the optimal solution value (Obj.) found by the model are also presented in Table 7.6.

Figure 7.4 displays the results of all tested runs as box plots. For instances CSPAR03a, CSPAR04a, CSPAR03b and CSPAR04b we skip set S7, because there are no trips with

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Figure 7.4: Mean computation times depending on selected sets of inequalities

Table 7.6: Small-sized instances

Instance Trans. |C| |V | |A| |P | Obj.

CSPAR01a ∞ 4 94 212 1 46,050

CSPAR02a ∞ 4 108 422 2 46,500

CSPAR03a ∞ 4 130 435 - 45,800

CSPAR04a ∞ 4 138 458 - 45,800

CSPAR01b 120 4 188 350 2 68,700

CSPAR03b 120 4 216 514 - 52,700

CSPAR02b 120 4 212 532 3 69,200

CSPAR04b 120 4 230 542 - 68,700

an attendance rate of 100%. The dashed line shows the computation time for solving the original problem without valid inequalities. The box plot consists of boxes and whiskers, whereby the bottom and top of the boxes represent the first and third quartiles, the line inside the box the second quartile (median). The upper whisker extends from the top of the box to the largest value no further than 1.5 times the distance between the first and third quartiles. The lower whisker is defined analogously. Outlying points represent data beyond the end of the whiskers.

We will describe the plot in the upper left corner (CSPAR01a) a little bit more in detail. For instance CSPAR01a we perform 2⁷ · 10 = 1280 runs, because there are 128 combinations of sets of valid inequalities and we run each combination 10 times with different random number seeds to reduce the influence of coincidence. The measured computation times of all 10 runs are averaged so that each combination delivers one mean computation time value. Computation times of all 64 combinations where set S1 is added to the basic arc flow model (regardless of the state of all other sets) form the light gray filled box above “S1” (right box plot). The dark gray filled box is represented by computation times of all 64 combination omitting set S1 (left box plot).

Figure 7.4 clearly demonstrates that valid inequalities rather increase computation times for small problem sizes (e.g. CSPAR02a, CSPAR03a and CSPAR04a) in contrast to larger instances, for which computations times rather are improved. We can derive that some sets of valid inequalities have a positive influence on the computation time, others not. Sets S4 and S7 seem to affect the computation time in a positive way, whereas set S1 can be omitted for fast solutions. In the case of S7 all quartiles for all instances are better if S7 is added to the basic model. Set S4 performs in a similar way.

Almost all quartiles are better if S4 is activated. In contrast, omitting set S1 delivers better quartiles except for instance CSPAR01a. The activation of S6 appears to have a slight positive impact on computation times for instances with attendance rate less than

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100% (CSPAR04a, CSPAR04b). For instances with trips that have to be attended by a conductor (100%), set S7 has the biggest influence on computation times (CSPAR01a, CSPAR02a, CSPAR01b, CSPAR02b).

An extract from the test results delivering good computation times is presented in Table 7.7. We use various criteria for filtering the best combinations. First, we delete all combinations where the computation times for two or more instances exceed the compu-tation time for the basic arc flow model. Second, we sort all remaining combinations by increasing total computation time as well as decreasing accumulated percentage devia-tion from the computadevia-tion time of the basic arc flow model. The intersecdevia-tion of the first 10 positions of both rankings are the 5 combinations presented in Table 7.7. Combina-tions of S1, ..., S7 are binary coded, e.g. 0000001 means only S7 is added to the arc flow model. All values represent mean computation times for 10 runs with different random number seeds. Best mean values of all combinations are highlighted in bold. There are only three bold numbers, because these combinations provide near-best values over all instances. All other best mean values are presented in Table 7.8 (columns “Best mean”) and are highlighted in bold in Table 7.11 in the supplementary material.

Table 7.7: Computation times for small-sized instances (extract) Instances 0000000 0001001 0001011 0011001 0011011 0111011

CSPAR01a 35.27 12.18 11.98 15.85 17.57 16.14

CSPAR02a 30.28 19.51 13.71 28.90 19.36 13.84

CSPAR03a 55.68 43.47 62.46 56.35 53.36 38.42

CSPAR04a 67.90 76.41 49.46 59.24 81.50 62.68

CSPAR01b 308.53 50.67 53.12 65.27 59.05 84.58

CSPAR03b 271.99 166.06 159.03 203.73 181.50 141.35

CSPAR02b 146.09 27.72 33.74 25.92 29.43 33.53

CSPAR04b 3,059.05 778.39 428.08 713.74 634.57 671.68

Table 7.8 summarizes best computation times for all small-sized instances. Column

“0000000” shows the mean of computation times for the basic arc flow model without any valid inequality. We compare the best of all computation times (Best overall) as well as the best of all means (Best mean) with the basic arc flow model. All best cases are represented by the binary coding of valid inequality sets S1, ..., S7 (VI), the computation time (t) and the improvement of the computation time compared to the basic arc flow model (∆t). As you can see, computation time can be improved by 31 − 86.01% on average. In some cases the computation time can be reduced by up to 94.79%. Here again, combinations with set S1 omitted and set S7 activated are the best for most instances.

Table 7.8: Best computation times for small-sized instances

Best overall Best mean

Instances 0000000 VI t ∆t VI t ∆t

CSPAR01a 35.27 1111101 6.20 -82.42% 1011101 8.22 -76.69%

CSPAR02a 30.28 0100101 7.70 -74.57% 0001011 13.71 -54.72%

CSPAR03a 55.68 0000001 25.89 -53.50% 0111011 38.42 -31.00%

CSPAR04a 67.90 0000011 16.17 -76,19% 0110011 35.78 -47.30%

CSPAR01b 308.53 0100111 19.41 -93.71% 0101001 50.22 -83.72%

CSPAR03b 271.99 0110111 53.47 -80.34% 0001111 123.49 -54.60%

CSPAR02b 146.09 0010111 16.06 -89.01% 0100001 23.14 -84.16%

CSPAR04b 3,059.05 0011011 159.32 -94.79% 0001011 428.08 -86.01%