3.13 Computational Results
3.13.2 Lagrangian Decomposition Approaches
We use the volume algorithm as described by Haouari and Siala [86] for approx-imately solving the Lagrangian dual problem, which is configured as follows. La-grangian multipliers are initialized by λk,e = ce/|C|, ∀k ∈ C, ∀e ∈ E, ensuring a positive lower bound already in the first iteration. The target value T is set to
3.13 Computational Results
Table 3.10: Number of instances per set where the solution to the linear relaxation is integral. Only those instances are considered where (dCol)LP could be solved withing 7200 CPU-seconds usingD(k,d0).
Variant Set # (MCF)LP (Col)LP (dCol)LP
T = 1.1zUB with zUB being the actual upper bound unless the actual lower bound zLB > 0.9T in which case T is multiplied by 1.1. We initially set ρ = 0.1 and α= 0.01. After 20 consecutive non-improving iterations,ρ is multiplied by 0.67 in case it is greater than 10−4 and by 1.1 in an improving iteration if ρ < 1. If zLB did not improve by more than 1% within the last 100 iterations and ifα >10−5, we multiply α by 0.85. The volume algorithm is terminated if dzLBe =zUB, after 250 consecutive non improving iterations, or if the maximum time limit is reached.
For the sequential hybrid Lagrangian method (SEQ) from Section 3.11, we setN = 30 and βmin = 0.01, βmax = 0.4, γ = 0.05, and δ = 100 for the interleaved hybrid (INT) whereβ is initially set toβ = 0.1.
Furthermore, we memorize hash-values of candidate solutions which have already
Chapter 3 The bmax-Survivable Network Design Problem
been used as starting solutions to avoid unnecessary runs of VND. These hash-values are also used to ensure that the N solutions stored in the sequential approach are pairwise different.
Table 3.11 details average relative gaps and corresponding standard deviations for LD, SEQ, and INT, while Table 3.12 shows median values of their individual CPU-times.
Table 3.11: Average gaps and corresponding standard deviations in % for LD, SEQ, and INT for the SST variant ofbmax-SNDP. Best values are marked bold.
Variant Set LD SEQ INT
SST,bmax= 0
ClgSE-I1 2.00 (2.62) 1.63 (2.38) 1.63 (2.38) ClgSE-I2 15.37 (4.65) 9.13 (4.65) 9.13 (4.65) ClgSE-I3 9.11 (4.84) 7.09 (4.84) 7.09 (4.84) ClgN1B-I1 5.37 (5.76) 2.61 (2.21) 2.56 (2.18) ClgN1B-I2 2.35 (2.34) 1.29 (1.46) 1.29 (1.46) SST,bmax= 30
ClgSE-I1 2.53 (3.39) 1.92 (2.60) 1.75 (2.36) ClgSE-I2 21.59 (4.78) 13.35 (4.78) 13.18 (4.82) ClgSE-I3 10.92 (4.13) 7.19 (4.13) 7.08 (4.02) ClgN1B-I1 6.76 (4.52) 3.36 (2.84) 2.91 (1.72) ClgN1B-I2 3.74 (4.71) 1.54 (1.85) 1.54 (1.85) SST,bmax= 50
ClgSE-I1 3.27 (4.80) 2.17 (3.05) 2.17 (3.05) ClgSE-I2 23.58 (5.13) 16.96 (5.13) 16.59 (5.00) ClgSE-I3 9.58 (4.01) 6.19 (4.01) 6.09 (3.97) ClgN1B-I1 6.03 (3.63) 3.84 (2.57) 2.96 (1.95) ClgN1B-I2 3.01 (3.75) 1.59 (1.83) 1.59 (1.83) SST,bmax= 100
ClgSE-I1 2.93 (3.54) 2.23 (2.64) 2.22 (2.63) ClgSE-I2 29.12 (7.13) 20.36 (7.13) 19.13 (4.96) ClgSE-I3 14.51 (6.38) 10.40 (6.38) 10.33 (6.33) ClgN1B-I1 7.48 (4.34) 4.84 (3.43) 3.65 (2.26) ClgN1B-I2 2.76 (3.23) 1.79 (2.00) 1.79 (2.00)
We conclude that both SEQ as well as INT significantly reduce the resulting opti-mality gap for all used settings and instance sets. Furthermore, the gaps obtained by INT are always smaller than or equal to those of SEQ.
The average relative difference between the lower bounds of SEQ and INT, respec-tively, compared to LD is strictly smaller than 0.07% except for the set ClgSE-I1 in the SST variant withbmax(k) = 100, ∀k∈C2, where it is 0.18%. Since the relative difference between the lower bounds obtained by SEQ and INT is always smaller than 0.01% we conclude that a smaller gap implies the generation of better feasible solutions.
From Table 3.12 we observe that both SEQ and INT significantly increase the
neces-3.13 Computational Results
Table 3.12: Median CPU-times for LD, SEQ, and INT for the SST variant ofbmax -SNDP. Best values are marked bold.
Variant Set LD SEQ INT
sary CPU-time with SEQ usually being the fastest among these two. However, since its overhead compared to SEQ is not too high, INT can be recommended among the three Lagrangian variants to compute high quality solutions with relatively tight bounds in reasonable time.
3.13.3 Metaheuristics
In the following, we compare the metaheuristic methods introduced in Section 3.10.
The VNS approach is terminated after 25 iterations of the outermost, largest shaking move. For GRASP we chose α = 0.25 and generated 30 initial solutions. Further-more, an absolute time limit of 7200 CPU-seconds has been used for all experi-ments.
Table 3.13 depicts average relative improvements of the obtained solution values as well as corresponding standard deviations for VNS and GRASP compared to the simpler VND. Each experiment has been repeated 30 times. Table 3.14 reports on average CPU-times and corresponding standard deviations.
Chapter 3 The bmax-Survivable Network Design Problem
Table 3.13: Average relative improvements over VND and corresponding standard deviations in % for VNS and GRASP considering the SST variant of bmax-SNDP for 30 runs per instance. Best values are marked bold.
Variant Set VND−GRASPGRASP [%] VND−VNSVNS [%]
SST,bmax= 0
We conclude that both metaheuristics outperform the simpler VND with respect to the obtained solutions in the majority of cases. While the solutions obtained by the GRASP approach are sometimes worse than those of VND, this is obviously impossible for the VNS approach. With respect to the average improvement over VND, the VNS clearly outperforms GRASP in 24 out of 28 cases while its runtime is approximately equal.
3.13 Computational Results
Table 3.14: Average CPU-times in seconds and corresponding standard deviations for GRASP and VNS for the SST variant of bmax-SNDP. 30 runs per instance. Best values are marked bold.
Variant Set GRASP VNS
In the following, we compare the objective values of the solutions derived by INT and VNS to those of (dCol) for a representative subset of the so far considered configurations. Table 3.15 reports average relative objective values solutions as well as corresponding standard deviations in percent.
Overall, we conclude that each method is able to compute high quality solutions.
The interleaved Lagrangian hybrid approach yields slightly better primal solutions than VNS and has the advantage of additionally providing a lower bound and thus a gap on the maximum distance to an optimal solution. VNS, however, can be used