The use of the smaller vehicles should increase cost. However, this may be offset by the advantages of using lighter vehicles, e.g., reducing emissions. To examine the potential cost increase due to a decrease in vehicle capacity we experimented with three different settings, consisting of 10%, 20% and 30% reduction in vehicle capacity. The details of the experiments are in the Appendix.
Table 11 summarizes the results for the three experimental settings. More instances are likely to become infeasible as the capacity decreases. In columns two, four and six we report the number of solved instances and the number of feasible instances for each of the three settings. Columns three, five, and seven contain the average increase in cost, in value and as percentage, when compared to the base case. The cost increase is computed only on instances that were solved in the base case as well as in the reduced capacity setting.
The number of feasible instances decreases from 90 to 88 if vehicle capacity is reduced by 10%, and to 73 if vehicle capacity is reduced by 30%. We observe that the infeasible instances tend to appear in instances with a low number of customers. A 10% reduction in vehicle capacity causes an average cost increase of 3.6%, whereas a 30% reduction in vehicle capacity causes an average cost increase of 15.8%.
Finally, we experimented with capacity reductions for the base case when extending due dates withα= 2. We compare this case with the base case in order to validate if the flexibility in due dates can balance the decrease in vehicle capacity. The results of these experiments are summarized in Table 12 and the details of the experiments are in the Appendix. Despite the reduced vehicle capacity, average savings are achieved. This is explained by the fact
that extending the deadlines offsets the decrease in vehicle capacity. The number of feasible instances decreases from 90 to 84 only if vehicle capacity is reduced by 30%. A 10% reduction in vehicle capacity, coupled with α = 2, causes an average saving of 12.4%, whereas a 30%
reduction in vehicle capacity, coupled with α= 2, causes a saving of 1.1%.
10% capacity reduction 20% capacity reduction 30% capacity reduction Num. of
10 8(8) 124.9(3.7) 2(2) 287.3(7.7) 0(0) 0(0)
15 10(10) 296.5(8.5) 10(10) 572.3(15.7) 2(2) 970.7(31.6)
20 10(10) 198.3(4.7) 10(10) 485.3(11.3) 10(10) 1017.6(23.6)
25 10(10) 146.5(3.1) 10(10) 347.7(7.5) 10(10) 899(18.6)
30 10(10) 123.7(2.8) 10(10) 276.9(6.1) 10(10) 533(11.7)
35 10(10) 169.9(3.4) 10(10) 431.3(8.7) 5(10) 886.1(16.7)
40 8(10) 110.3(2.2) 5(10) 276.7(5.4) 3(10) 688.9(12.3)
45 7(10) 67.8(1.2) 7(10) 122.2(2.2) 3(10) 433.1(7.1)
50 5(10) 28.7(0.4) 5(10) 107.4(1.5) 4(10) 220.2(3.2)
Total 78(88) 69(82) 48(73)
Average 151.9(3.6) 354.9(8.1) 731.6(15.8)
Table 11: Base case with 10%, 20% and 30% capacity reduction
10% capacity reduction 20% capacity reduction 30% capacity reduction Num. of
10 10(10) -371.7(-10.6) 10(10) -62.6(-1.5) 4(4) 267(8.7)
15 10(10) -371.6(-10.2) 10(10) -122.1(-3.2) 10(10) 227.7(6.6)
20 10(10) -622.4(-15.1) 10(10) -477.7(-11.7) 10(10) -153(-4.1)
25 10(10) -665.9(-14.3) 10(10) -506.4(-10.9) 10(10) -148.6(-3.6)
30 10(10) -759.4(-16.1) 10(10) -607.6(-12.8) 8(10) -445.3(-9.1)
35 7(10) -612.1(-12.1) 9(10) -569.5(-11.5) 4(10) 36.6(0.6)
40 6(10) -496.4(-9.1) 6(10) -354.2(-6.6) 2(10) 139(2.8)
45 4(10) -573(-9.6) 5(10) -500.2(-8.2) 3(10) -87.2(-1.6)
50 4(10) -735.9(-10.7) 4(10) -708.1(-10.3) 2(10) -379.2(-5.8)
Total 71(90) 74(90) 53(84)
Average -569.1(-12.4) -410.1(-8.5) -72.2(-1.1)
Table 12: Base case withα = 2, 10%, 20% and 30% capacity reduction
4. Conclusions
The MVRPD captures the operations of many distributors that deliver products from a CDC to customers, within predetermined due dates. The distribution activities are planned over several days, the main decisions relate to which customers to visit on each day and in what order. Products kept at the warehouse entail inventory holding costs. The MVRPD aims to balance transportation cost, inventory cost, as well as penalty costs, incurred as a result of unserved demand within the planing horizon. As such, the MVRPD balances the flexibility of choosing to serve a customer within an interval of consecutive days with the cost of keeping the its demand at the CDC.
We proposed three formulations for the MVRPD: a flow based formulation, a flow based formulation with assignment variables and a load based formulation. For each of the three formulations we developed a series of valid inequalities. We generated a test bed for the MVRPD, which we used to examine the performance of each of the formulations. The load based formulation substantially outperformed the two other formulations, in terms of the number of solved instances and of the average runtimes.
We performed a number of analyses with the aim of understanding how the MVRPD solutions would be altered to accommodate changes in the input parameters. Through a series of experiments we demonstrated how the model may provide managerial insights with respect to such changes. Based on the results of the considered instances, we conclude that substantial cost savings can be achieved by extending customer due dates. Such additional flexibility allows for improved routing decisions, that yield a considerable reduction in the travelled distance. Despite the fact that the due dates are exogenous parameters, in reality distribution companies periodically negotiate contracts with their customers. Therefore, the MVRPD is paramount in quantifying the added value of extending due dates. Our experiments also showed that using additional vehicles does not yield a substantial reduction in the considered operational costs. Finally, our results indicated that reducing vehicle capacity causes a cost increase but this can be balanced by due dates flexibility. Future research could incorporate environmental factors in the model, while accounting for more complex distribution networks.
Acknowledgments
One of the authors gratefully acknowledges funding provided by the Canadian Natural Sci-ences and Engineering Research Council under grant 436014-2013.
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Appendix
Num. of customers
Num. of solved (feasible)
Average gap
Average runtime (solved)(sec)
110 8(8) 0% 0
15 10(10) 0% 0.2
20 10(10) 0% 6.1
25 10(10) 0% 22.3
30 10(10) 0% 96.2
35 10(10) 0% 621.5
40 8(10) 0.6% 4236.1
45 7(10) 0.8% 4460.6
50 5(10) 1.5% 6135
Total 78(88)
Average 1323.7
Table 13: Computational results for base case with 10% capacity reduction
Num. of customers
Num. of solved (feasible)
Average gap
Average runtime (solved)(sec)
10 2(2) 0% 0
15 10(10) 0% 0
20 10(10) 0% 7.3
25 10(10) 0% 23.9
30 10(10) 0% 211.2
35 10(10) 0% 1126.6
40 5(10) 1.7% 6462.9
45 7(10) 2.4% 4088.6
50 5(10) 2.4% 6552.2
Total 69(82)
Average 1556.3
Table 14: Computational results for base case with 20% capacity reduction
Num. of customers
Num. of solved (feasible)
Average gap
Average runtime (solved)(sec)
10 0(0) 0
15 2(2) 0% 0
20 10(10) 0% 7.1
25 10(10) 0% 130.9
30 10(10) 0% 423
35 5(10) 1.4% 6992
40 3(10) 4% 9389
45 3(10) 3.2% 7965.7
50 4(10) 4.1% 8091.2
Total 48(73)
Average 2604.1
Table 15: Computational results for base case with 30% capacity reduction
Num. of customers
Num. of solved (feasible)
Average gap
Average runtime (solved)(sec)
10 10(10) 0% 0
15 10(10) 0% 6
20 10(10) 0% 54.3
25 10(10) 0% 384.1
30 10(10) 0% 455.6
35 7(10) 1% 5060.1
40 6(10) 2.4% 5365.2
45 4(10) 6.1% 7049.6
50 4(10) 4.9% 7088.5
Total 71(90)
Average 1875.6
Table 16: Computational results for base case with 10% capacity reduction and α= 2
Num. of customers
Num. of solved (feasible)
Average gap
Average runtime (solved)(sec)
10 10(10) 0% 0
15 10(10) 0% 5.8
20 10(10) 0% 29.5
25 10(10) 0% 202.5
30 10(10) 0% 738
35 9(10) 0.3% 4517.3
40 6(10) 3.2% 6115.5
45 5(10) 4.9% 6400.2
50 4(10) 6.8% 6781.2
Total 74(90)
Average 1976.1
Table 17: Computational results for base case with 20% capacity reduction α= 2
Num. of customers
Num. of solved (feasible)
Average gap
Average runtime (solved)(sec)
10 4(4) 0% 0
15 10(10) 0% 5.6
20 10(10) 0% 78.6
25 10(10) 0% 458.4
30 8(10) 0.5% 2915.2
35 4(10) 2.9% 7948.8
40 2(10) 6.7% 9114.2
45 3(10) 6.6% 9099.8
50 2(10) 8.2% 8889.1
Total 53(84)
Average 2336.8
Table 18: Computational results for base case with 30% capacity reduction α= 2