Computational results

We now present computational results for the instances just described. We start by studying the performance of the two models proposed for each variant of the problem. To this end, we have run a subset of the instances with a limit on the running time of 1800 seconds. Table 2 shows the percentage of instances for which the implicit compartment assignment model is superior to the explicit compartment assignment model. Table 3 shows that after the 1800 seconds the implicit models usually yield smaller percentage gaps between the upper and lower bounds.

Table 2: Percentage of best (or equal) results favorable to the Implicit Compartment Assign-ment models

Problem Criteria

Upper Bound Lower Bound Time (s)

Split-Split 85 97 50

Split-Unsplit 83 99 55

Unsplit-Split 69 92 54

Unsplit-Unsplit 75 92 66

Table 3: Average gaps for the Explicit vs Implicit Compartment Assignment models after 1800 seconds of running time

Problem Average gap (%) Explicit Implicit Split-Split 11.1 2.8 Split-Unsplit 0.7 0.4 Unsplit-Split 3.0 0.1 Unsplit-Unsplit 4.3 3.9

With these results we can now put the relative performance of each of the models into perspective, and we can proceed to obtain solutions for larger and more challenging

in-stances. To this end, we have evaluated the implicit compartment assignment models with a two-hour time limit. We have also applied the findings of Coelho and Laporte [6] to order the input data with respect to the demand of the customers, in such a way that customers are relabeled consecutively in non-increasing order of their demand. Input ordering was first proposed by Jans and Desrosiers [21, 22] and was shown to improve the lower bound of the problem. Labeling the customers according to their demand has proved to be the most successful choice among those implemented and tested by Coelho and Laporte [6].

We start this detailed analysis with instances containing a single product, in order to evaluate the effect of having several compartments on the performance of the algorithm.

As in Cornillier et al. [8], single period instances were used. The results for all four categories of the problem are summarized in Table 4. Several instances with up to 50 customers were solved to optimality. Even some of the largest instances considered, which contain up to 50 customers, four compartments, and 14 vehicles were solved to optimality.

Likewise, instances with up to 50 customers, three compartments and 18 vehicles were also solved for some variants of the problem to optimality in relatively small running times.

One can observe that the problem becomes more difficult when it is more constrained, to the point where our algorithm could not find any feasible solution within two hours of computing time for some instances of the unsplit-unsplit case. This is also reflected by the size of the gaps and by the increased running time towards the right of the table.

We have then considered instances defined over several periods. Average results over all four variants of the problem are presented in Table 5. These instances are considerably more difficult to solve than their single-period counterpart, as reflected by the gaps and running times. The algorithm often could not identify any feasible solution for instances with more than 20 customers. For the 20 instances contained in Table 5, we observe that, as is the case for the results of Table 4, the solutions of the first two variants of the problem are quite similar, and so are those of the last two. Moreover, for each pair, the more restrictive case seems to be easier to solve, which is reflected by smaller average

Table 4: Summary of the results on single-period single-product instances

Instance

Split-Split Split-Unsplit Unsplit-Split Unsplit-Unsplit

Solution Gap (%) Time (s) Solution Gap (%) Time (s) Solution Gap (%) Time (s) Solution Gap (%) Time (s)

MCD-10-1-3-5-1 1643.65 0.00 0 1643.65 0.00 0 2081.07 0.00 0 2081.07 0.00 0

MCD-10-1-4-4-1 1550.36 0.00 0 1550.36 0.00 0 2249.41 0.00 0 2249.41 0.00 0

MCD-20-1-3-8-1 2630.80 0.00 2 2630.80 0.00 2 4107.46 0.00 18 4107.46 0.00 10

MCD-20-1-4-7-1 2225.80 0.00 1 2225.80 0.00 0 3296.98 0.00 5 3296.98 0.00 5

MCD-30-1-3-12-1 3551.00 0.00 37 3551.00 0.00 23 5974.96 0.00 2385 5978.76 1.17 2696

MCD-30-1-4-9-1 3192.10 0.00 32 3192.10 0.00 19 4552.04 0.00 204 4552.04 0.00 143

MCD-40-1-3-15-1 4375.41 0.00 477 4375.41 0.00 216 9928.79 23.10 5897 8367.21 13.98 5816

MCD-40-1-4-12-1 4179.92 0.00 258 4179.92 0.00 165 6198.55 4.04 4636 6129.95 4.08 4662

MCD-50-1-3-18-1 5265.05 0.00 2340 5265.05 0.00 910 22812.70 58.19 7203 − − 7205

MCD-50-1-4-14-1 5077.01 0.00 1386 5077.01 0.00 681 10784.88 27.07 7202 7371.66 6.54 6783

gaps and running times. These similar results are, however, an effect of the structure of the problem obtained by imposing unsplit compartments rather than a consequence of the model.

Table 5: Summary of the results on multi-period single-product instances

Instance Split-Split Split-Unsplit Unsplit-Split Unsplit-Unsplit

Solution Gap (%) Time (s) Solution Gap (%) Time (s) Solution Gap (%) Time (s) Solution Gap (%) Time (s)

MCD-10-1-3-5-3 3976.58 0.00 7 3976.58 0.00 8 7548.75 0.00 12 7548.75 0.00 8

MCD-10-1-4-4-3 3972.38 0.00 8 3972.38 0.00 7 7069.04 0.00 8 7069.04 0.00 8

MCD-20-1-3-8-3 6676.11 8.06 6924 6602.30 6.26 5699 13750.96 11.12 7200 13714.56 6.77 6169

MCD-20-1-4-7-3 6829.03 7.52 7105 6842.83 6.93 5653 13653.58 8.82 7200 13627.58 7.36 7200

Finally, we have solved the most general instances containing several products, periods, vehicles and compartments. These instances contain up to 20 customers, three products, five compartments, eight vehicles, and five periods. As expected, the size of the instances for which optimal solution can be obtained decreases, as is shown in Table 6.

A transversal analysis over the last three tables allows us to derive some comments on the relative difficulty of each variant of the problem. We observe that the average increase on the solution cost (or on the upper bound when optimality is not achieved) of each

Table 6: Summary of the results on multi-period multi-product instances

Instance

Split-Split Split-Unsplit Unsplit-Split Unsplit-Unsplit

Solution Gap (%) Time (s) Solution Gap (%) Time (s) Solution Gap (%) Time (s) Solution Gap (%) Time (s)

MCD-10-2-3-9-3 5622.00 0.00 370 5622.00 0.00 196 12109.86 0.00 81 12109.86 0.00 59

MCD-10-2-4-5-1 9010.03 4.69 7200 9023.85 3.41 7200 18588.52 0.50 3245 18588.52 0.63 2349

MCD-10-2-5-4-1 5443.75 0.00 15 5443.75 0.00 14 11749.56 0.00 32. 11749.56 0.00 18

MCD-10-2-5-4-3 8852.51 0.69 2848 8851.71 0.74 2301 18407.28 0.00 512 18407.28 0.00 312

MCD-10-3-3-6-1 5537.40 0.00 26 5537.60 0.00 27 11358.74 0.00 48. 11358.74 0.00 30

MCD-10-3-3-6-3 8497.38 0.17 1864 8497.38 0.00 1512 17956.90 0.00 177 17956.90 0.00 107

MCD-10-3-4-5-3 8189.95 2.33 4953 8225.93 2.09 5017 18504.90 0.00 971 18504.90 0.00 1474

MCD-10-3-4-5-5 10963.30 11.42 7200 11036.36 12.36 7200 25478.76 0.00 1812 25478.76 0.00 1335

MCD-10-3-4-8-5 7061.88 0.00 967 7064.25 0.00 770 16643.96 0.00 686 16643.96 0.00 365

MCD-10-3-5-4-1 10796.9 7.57 7200 10828.36 6.71 7200 25036.18 0.35 5026 25036.18 0.44 3633

MCD-20-2-4-8-5 11441.85 24.80 7201 − − 7200 23088.38 13.14 7201 23314.18 12.86 7201

MCD-20-2-5-7-5 9740.60 11.73 7200 15525.20 17.90 7201 22376.88 10.12 7200 22503.28 10.36 7201 MCD-20-3-3-6-1 14342.60 15.70 7201 8913.44 4.71 7200 35011.42 9.29 7201 34794.42 8.41 7201 MCD-20-3-3-6-3 9012.13 4.69 6947 15126.16 14.67 7201 21280.90 7.48 7200 21245.30 5.88 7201

MCD-20-3-5-4-3 − − 7202 − − 7200 35833.52 13.89 7202 35773.32 12.67 7203

variant of the problem with respect to the most general scenario, i.e., the split-split case, is quite stable. The split-unsplit case often yields solutions that are marginally more expensive than the split-split variant, while solving either the split or the unsplit-unsplit case approximately doubles the cost. Moreover, moving from the unsplit-unsplit-split to the unsplit-unsplit variant yields no significant difference in costs over the instances we have tested.

6 Conclusion

We have introduced, classified, modeled and solved a wide range of routing problems with several compartments used for the delivery of several products spanning several periods, with the aim of minimizing routing and inventory costs over the network. This problem typically arises in the distribution of petroleum products to gas stations. We have developed two models for each variant of the problem, and we have shown how to

adapt these models to handle specific versions of the problem described in the operational research literature. Extensive computational results on a set of benchmark instances show that optimal solutions can be proved for instances containing up to 50 customers, four compartments and 14 vehicles. However, multi-period instances are clearly more challenging. When several periods are considered, instances with up to 20 customers, and several products, compartments, and vehicles can be solved optimally.

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