We have performed several experiments to test the behavior of our VMNS matheuristic for different values of parameter α, which defines how much time the algorithm invests performing local searches. In the limit whereα →0 the algorithm becomes a pure B&C.
Table 5 shows the results obtained by the algorithm in medium-sized scenarios (n = 40)
Table 3: Results of the Variable MIP Neighborhood Search (VMNS) matheuristic
Scenario Upper bound Improvement over B&C (%) Lower bound Gap (%) Time (s)
20 / 6 / 1 3788.3 0.0 3788.3 0.0 4
20 / 6 / 2 4237.0 0.0 4237.0 0.0 4
20 / 6 / 3 4237.0 0.0 4237.0 0.0 7
20 / 12 / 1 8004.8 0.0 7261.7 9.3 10800
20 / 12 / 2 8848.7 −0.2 8161.2* 8.5 10800
20 / 12 / 3 7771.7 0.1 7771.7* 2.6 10800
20 / 18 / 1 10829.2 0.1 9137.5* 17.4 10800
20 / 18 / 2 11995.3 0.0 10430.1* 13.4 10800
20 / 18 / 3 11377.7 0.3 9670.6 15.0 10800
40 / 6 / 1 6910.6 0.0 6910.6 0.0 63
40 / 6 / 2 8820.4 0.0 8820.4 0.0 22
40 / 6 / 3 7547.1 0.0 7547.1* 2.3 10800
40 / 12 / 1 13919.2 16.7 10701.6 23.1 10800
40 / 12 / 2 15437.2 1.2 13407.7 13.1 10800
40 / 12 / 3 13570.6 5.7 10941.9* 20.6 10800
40 / 18 / 1 19551.6 38.5 13170.6 32.6 10800
40 / 18 / 2 21007.1 38.5 16177.0 23.0 10800
40 / 18 / 3 20020.3 51.8 13053.2 34.8 10800
60 / 6 / 1 9982.3 0.0 9982.3 0.0 9439
60 / 6 / 2 11946.3 0.0 11946.3 0.0 361
60 / 6 / 3 10651.0 0.2 9646.0 9.4 10800
60 / 12 / 1 20318.3 22.7 14599.0* 28.3 10800
60 / 12 / 2 20507.7 6.9 17407.6 15.1 10800
60 / 12 / 3 18801.1 27.0 14578.4 22.5 10800
60 / 18 / 1 28296.8 55.0 17784.2 37.2 10800
60 / 18 / 2 30604.8 54.1 21566.9 29.5 10800
60 / 18 / 3 28840.0 51.5 17478.4 39.4 10800
Average 13993.4 13.70 10981.5 14.7 8366.7
∗ indicates an improvement with respect to the B&C lower bound.
Table 4: Time required by B&C to find its best solution, and time required by VMNS to find a solution at least as good
Scenario B&C time (s) VMNS time (s) Fraction of time required (%)
20 / 6 / 1 2 2 100.0
20 / 6 / 2 1 1 100.0
20 / 6 / 3 1 2 200.0
20 / 12 / 1 8187 56 0.7
20 / 12 / 2 1700 0 0.0
20 / 12 / 3 689 120 17.4
20 / 18 / 1 7405 257 3.5
20 / 18 / 2 10373 0 0.0
20 / 18 / 3 10319 1279 12.4
40 / 6 / 1 1 3 300.0
40 / 6 / 2 8 10 125.0
40 / 6 / 3 4380 1 0.0
40 / 12 / 1 9512 83 0.9
40 / 12 / 2 8267 884 10.7
40 / 12 / 3 5959 193 3.2
40 / 18 / 1 9964 163 1.6
40 / 18 / 2 9901 109 1.1
40 / 18 / 3 6659 89 1.3
60 / 6 / 1 634 68 10.7
60 / 6 / 2 115 168 146.1
60 / 6 / 3 10086 2430 24.1
60 / 12 / 1 9399 353 3.8
60 / 12 / 2 9440 280 3.0
60 / 12 / 3 6343 235 3.7
60 / 18 / 1 9475 272 2.9
60 / 18 / 2 10007 88 0.9
60 / 18 / 3 9055 271 3.0
Average 5847.5 274.7 39.8
when α takes the value of 0.5, 1.0, and 2.0. We can observe that there is no apparent trend in the table regarding α, and that results are very similar among them, with an average deviation of only 1%. This means that the VMNS is very robust with respect to this parameter.
Table 5: Sensitivity analysis for parameter α
Scenario UB α= 0.5 UBα = 1.0 UBα= 2.0 Maximum deviation (%)
40 / 6 / 1 6910.6 6910.6 6910.6 0.0
40 / 6 / 2 8820.4 8820.4 8820.4 0.0
40 / 6 / 3 7547.1 7547.1 7547.1 0.0
40 / 12 / 1 13911.6 13919.2 13911.6 0.1
40 / 12 / 2 15358.1 15437.2 15488.6 0.8
40 / 12 / 3 13570.6 13570.6 13569.3 0.0
40 / 18 / 1 18914.5 19551.6 19197.9 3.4
40 / 18 / 2 21063.2 21007.1 21821.7 3.9
40 / 18 / 3 19889.8 20020.3 19969.9 0.7
Average 13998.4 14087.1 14137.5 1.0
To further show this point, in Figure 6 we present the evolution of the solution over time for different versions of the algorithm, for scenarios with H = 12. This figure shows two interesting things. First, looking at how similar the evolution of the solution is for the different values ofα, it confirms that the algorithm is not very sensitive to this parameter.
This means that it is not necessary to spend too much effort calibrating this value on future implementations of this algorithm. Second, irrespective of the value of α, the figure also shows that the VMNS algorithm helps reaching very good solutions in significantly less time than B&C, and that it works specially well during the first minutes of execution.
Extending the maximum running time yields only very marginal improvements.
6 Conclusions
In this paper we have solved a practical problem arising in the ATM management and replenishments in Santiago, Chile. This rich and challenging problem shares much of its structure with the IRP, but some features makes it unique. We have modeled the problem and solved it by branch-and-cut. In order to efficiently tackle large instances, we have derived and formalized a Variable MIP Neighborhood Search heuristic, which takes advantage of the MIP implementation. Our VMNS is shown to significantly improve the solutions using only a fraction of the time. As a positive side effect, our heuristic has also improved several lower bounds, helping closing the gap.
The VMNS has several features that make it appropriate to solve other classes of problems.
First, it relies on the mathematical formulation of the problem, and helps solving it when the instance size is excessive for a branch-and-bound (or branch-and-cut) method to tackle it. Second, deriving neighborhoods is as simple as imposing a few new constraints on the model, which are to be designed as problem-specific, even though our neighborhoods can be used as general frameworks for many distribution problems. Third, based on a single parameter, one can drive the VMNS to act more like a pure local search heuristic or like a pure exact method. Finally, given enough time, the VMNS is guaranteed to terminate and is capable of proving optimality (relying on the underlying branch-and-bound or -cut method), thus, being an exact algorithm.
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Execution time (h)
Upperbound(M$)
15 50
20 25 30 35 40 45
0 1 2 3 0 1 2 3 0 1 2 3
Scenario (40/12/1) Scenario (40/12/2) Scenario (40/12/3)
Alpha = 0.5 Alpha = 1.0 alpha = 2.0 B&C
! = 0.5 ! = 1.0 ! = 2.0
Figure 6: Convergence of the algorithms