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Analysis of the Computational Results

Dans le document Multicommodity Network Flow Problems (Page 27-36)

5 Computational Experiments

5.3 Analysis of the Computational Results

We first analyze the results obtained with different strategies to approximate v(EP+). We compare the following approaches:

• LAG, the Lagrangian strategy performed in Steps 1a to 1d of the Lagrangian dual optimization procedure presented in Section 3.2.

• LPS, the LP-based strategy performed in Steps 2a to 2c of the Lagrangian dual opti-mization procedure.

• LAG+LPS, the whole Lagrangian dual optimization procedure, combining the two previous strategies.

• LAG+SUB, the same as strategy LAG, except that the solution to the single La-grangian subproblem in Step 1d is replaced by a call to an adaptation of the subgradient method described in Section 3.3, where both π and β multipliers are adjusted.

Table 1 summarizes the computational results obtained for all instances. Lower and upper bound GAPs and CPU times in seconds are reported on average for the four network dimen-sions (|N|,|A|) = (20,75),(20,100),(25,100),(25,150) (each class contains 24 instances), as well as for the 96 instances in each class, Small (|K| = 25), Medium (|K| = 50) and Large (|K|= 100).

These results show that all the strategies generate similar lower bounds, with a slight edge for the combined approach LAG+LPS. We observed that, for some instances, LAG

(|N|,|A|) LAG LPS LAG+LPS LAG+SUB (20,75) 0.93, 1.03, 16 0.93, 1.03, 14 0.94, 1.02, 30 0.93, 1.03, 13829 (20,100) 0.94, 1.02, 22 0.93, 1.04, 40 0.94, 1.02, 62 0.94, 1.02, 24391 (25,100) 0.92, 1.01, 17 0.93, 1.04, 26 0.93, 1.01, 44 0.92, 1.01, 31380 (25,150) 0.92, 1.05, 36 0.92, 1.07, 33 0.92, 1.04, 69 0.92, 1.05, 75612 Small 0.93, 1.03, 23 0.93, 1.05, 28 0.93, 1.02, 51 0.93, 1.03, 36303 (20,75) 0.93, 1.01, 33 0.94, 1.03, 37 0.94, 1.01, 70 0.93, 1.01, 17769 (20,100) 0.94, 1.01, 13 0.94, 1.03, 32 0.94, 1.01, 45 0.94, 1.01, 31774 (25,100) 0.93, 1.02, 34 0.93, 1.04, 28 0.93, 1.02, 62 0.93, 1.02, 34363 (25,150) 0.91, 1.02, 27 0.91, 1.05, 65 0.91, 1.02, 92 0.91, 1.02, 80222 Medium 0.93, 1.02, 27 0.93, 1.04, 41 0.93, 1.02, 68 0.93, 1.01, 41302 (20,75) 0.95, 1.01, 19 0.96, 1.01, 50 0.96, 1.00, 69 0.95, 1.01, 5374 (20,100) 0.95, 1.00, 19 0.95, 1.01, 66 0.95, 1.00, 85 0.95, 1.00, 14613 (25,100) 0.94, 1.01, 40 0.95, 1.01, 127 0.95, 1.00, 166 0.94, 1.01, 25644 (25,150) 0.94, 1.01, 188 0.95, 1.03, 165 0.95, 1.00, 353 0.94, 1.01, 43103 Large 0.95, 1.01, 67 0.95, 1.02, 102 0.95, 1.00, 168 0.95, 1.01, 22184

Table 1: Strategies to approximate v(EP+): lower bound GAP, upper bound GAP, CPU produces better lower bounds than LPS, while for other instances, the opposite is true.

Thus, by combining the two strategies, we obtain better overall lower bounds. Although strategy LAG generates better upper bounds than LPS on average, the same observation holds for the upper bounds: no dominance exists across all instances, which implies that the combination of the two strategies produces better overall upper bounds. This can be seen on the average values for some of the instance classes, for example Small instances with size (20,75) for both the lower bound and the upper bound GAPs and Small instances with size (25,150) for the upper bound GAP. As shown in column LAG+SUB, the subgradient method does not help improving the bounds and its computing times are prohibitive. These results justify solving a single Lagrangian subproblem instead of a call to the subgradient method in Step 1d of the Lagrangian dual optimization procedure.

Table 2 displays the lower bound GAPs and the CPU times in seconds to compute the different lower bounds (excluding the times for upper bound computations), averaged for each problem class as in Table 1, for each combination of model and lower bounding method described in Section 5.2 (with the exception of BB whose results are shown below in Table 4).

From these results, we draw the following conclusions:

• As expected, formulation BS is weak, producing lower bound gaps around 25% on average. The B&B method of CPLEX at the root node reduces these gaps by 5-10%, thanks to its preprocessing and cutting-plane procedures. The extended models, on the other hand, improve these gaps by 15-20%, which shows the strength of the extended forcing constraints (40).

• The Lagrangian dual optimization procedure provides effective lower bound

approx-(|N|,|A|) Algo BS ES ES+ EP+ LD 0.75, 0 0.92, 0 0.93, 7 0.94, 28 (20,75) LP 0.75, 0 0.93, 1 0.93, 1 0.93, 65 BB0 0.83, 0 0.94, 2 0.95, 2 0.95, 13 LD 0.74, 0 0.91, 0 0.94, 9 0.94, 49 (20,100) LP 0.74, 0 0.92, 1 0.92, 1 0.92, 56 BB0 0.84, 0 0.94, 2 0.95, 3 0.95, 18 LD 0.74, 0 0.92, 0 0.92, 4 0.93, 40 (25,100) LP 0.74, 0 0.92, 1 0.92, 2 0.92, 273

BB0 0.84, 1 0.94, 3 0.95, 3 0.95, 25 LD 0.70, 0 0.91, 1 0.92, 10 0.92, 68 (25,150) LP 0.70, 0 0.91, 3 0.92, 4 0.92, 204

BB0 0.79, 1 0.92, 5 0.94, 8 0.94, 43 LD 0.73, 0 0.92, 0 0.93, 8 0.93, 46 Small LP 0.73, 0 0.92, 2 0.92, 2 0.92, 150

BB0 0.83, 1 0.94, 3 0.95, 4 0.95, 25 LD 0.76, 0 0.92, 1 0.93, 9 0.94, 68 (20,75) LP 0.76, 0 0.93, 3 0.93, 4 0.93, 124

BB0 0.83, 1 0.94, 5 0.95, 7 0.95, 28 LD 0.73, 0 0.93, 0 0.94, 4 0.94, 42 (20,100) LP 0.73, 0 0.94, 6 0.94, 7 0.94, 228

BB0 0.83, 1 0.94, 9 0.95, 11 0.96, 39 LD 0.70, 1 0.92, 1 0.93, 9 0.93, 60 (25,100) LP 0.70, 1 0.93, 7 0.93, 8 0.93, 433

BB0 0.79, 2 0.93, 12 0.95, 16 0.95, 54 LD 0.66, 1 0.90, 2 0.91, 4 0.91, 90 (25,150) LP 0.66, 1 0.91, 25 0.92, 35 0.92, 1294

BB0 0.76, 3 0.91, 28 0.92, 49 0.93, 140 LD 0.71, 1 0.92, 1 0.93, 7 0.93, 65 Medium LP 0.71, 1 0.93, 10 0.93, 14 0.93, 520

BB0 0.80, 2 0.93, 14 0.94, 21 0.95, 65 LD 0.80, 1 0.95, 2 0.95, 3 0.96, 64 (20,75) LP 0.80, 1 0.96, 12 0.96, 18 0.96, 310

BB0 0.87, 1 0.96, 18 0.96, 17 0.96, 54 LD 0.78, 1 0.95, 1 0.95, 2 0.95, 72 (20,100) LP 0.78, 1 0.95, 24 0.95, 35 0.95, 1169

BB0 0.82, 3 0.96, 34 0.96, 40 0.96, 89 LD 0.74, 1 0.94, 1 0.94, 2 0.95, 123 (25,100) LP 0.74, 1 0.95, 39 0.95, 63 0.95, 1095

BB0 0.80, 3 0.95, 54 0.95, 59 0.95, 130 LD 0.76, 1 0.94, 3 0.94, 3 0.95, 189 (25,150) LP 0.76, 1 0.95, 67 0.95, 106 0.95, 1103

BB0 0.81, 7 0.95, 81 0.95, 107 0.95, 227 LD 0.77, 1 0.95, 2 0.95, 3 0.95, 112 Large LP 0.77, 1 0.95, 36 0.95, 56 0.95, 919 BB0 0.83, 4 0.96, 47 0.96, 56 0.96, 125

imations, independently of the formulation. As can be seen from column ES, the subgradient method provides a tight approximation (within 1% on average) of the the-oretical bound v(ES) computed by the LP solver of CPLEX. It is also noteworthy that CPLEX provides only slight improvements (on the order of 1-2%) by adding its sophisticated preprocessing and cutting-plane features at the root node (method BB0).

• For the same model, the Lagrangian dual optimization procedure is in general signif-icantly faster than the LP solver of CPLEX and the difference in computing times increases with the number of commodities. In particular, method LD solves the ex-tended models for Medium and Large instances much faster than LP. For the same instances, the computing times for LD are also generally better than those for BB0, except for model EP+ where the CPU times are similar.

• Formulation ES produces gaps around 5-10% on average, with better results on Large instances. Small improvements (on the order of 1-2%) are obtained by adding segment-based cutset inequalities through formulation ES+. When adding to this last model the point-based cutset inequalities, a similar behavior is observed.

We now compare the upper bounds obtained by the Lagrangian heuristic method of Section 4, LH, with those computed by the B&B method of CPLEX, BB, with a limit of 1 hour. Table 3 displays three measures for each class of instances: Nfeas, the number of instances in the corresponding class for which each method-model combination found a feasible solution; Nopti, the number of instances in the corresponding class for which each method-model combination provided a certificate of optimality; CPU, the computing time in seconds for each method-model combination. We note that, independently of the model used, method LH always generates a feasible solution. Moreover, the best solution it generates has never been shown optimal for any of the instances. Hence, the values of Nfeas and Nopti are easy to interpret for methodLH: for 100% of the instances, LHfound feasible, but non-optimal solutions.

These results show that, in spite of being the weakest model in terms of the quality of its LP relaxation bound, BS gives the best performance for computing upper bounds with the B&B method of CPLEX. In particular, it is the only model for which BB finds feasible solutions to all instances. In contrast, the strongest model EP+ identifies a feasible solution within 1 hour for only 117 of the 288 instances. Formulations ES and ES+ show intermediate results, with 252 and 277 instances, respectively, for which they could find feasible solutions. The performance in terms of CPU times and number of optimal solutions found are similar: BSis generally faster than the other models and is able to prove optimality for a larger number of instances. The only exception is for Small instances for which model ES+ is slightly better than both BS and ES, which indicates that the addition of cutset inequalities can help in solving the problem, at least for instances with few commodities.

Overall, these results show that, although the extended models generate much stronger lower bounds than the basic model, their size is an issue for a stand-alone MIP solver like CPLEX and that decomposition methods must be used to exploit their strength. Table 3 also shows that the Lagrangian heuristic is fast, its CPU times being one to two orders of magnitude smaller than those of the B&B method of CPLEX, .

(|N|,|A|) Algo BS ES ES+ EP+ (20,75) BB 24, 20, 1066 24, 20, 1260 24, 21, 935 24, 8, 3175

LH 24, 0, 0 24, 0, 1 24, 0, 8 24, 0, 30 (20,100) BB 24, 18, 1311 24, 18, 1489 24, 19, 1307 24, 8, 3222

LH 24, 0, 0 24, 0, 0 24, 0, 9 24, 0, 62 (25,100) BB 24, 16, 1924 24, 13, 2361 24, 15, 2151 22, 0, 3613

LH 24, 0, 0 24, 0, 3 24, 0, 7 24, 0, 44 (25,150) BB 24, 5, 3103 24, 9, 2774 24, 8, 2828 12, 0, 3632

LH 24, 0, 0 24, 0, 1 24, 0, 10 24, 0, 69 Small BB 96, 59, 1851 96, 60, 1971 96, 63, 1805 82, 16, 3411

LH 96, 0, 0 96, 0, 1 96, 0, 9 96, 0, 51 (20,75) BB 24, 12, 2099 24, 10, 2545 24, 12, 2295 17, 2, 3584

LH 24, 0, 0 24, 0, 2 24, 0, 10 24, 0, 70 (20,100) BB 24, 9, 2492 24, 10, 2554 24, 9, 2509 17, 2, 3578

LH 24, 0, 0 24, 0, 1 24, 0, 5 24, 0, 45 (25,100) BB 24, 8, 2914 23, 6, 2937 24, 8, 3008 1, 0, 3644

LH 24, 0, 1 24, 0, 3 24, 0, 11 24, 0, 62 (25,150) BB 24, 3, 3274 18, 2, 3419 20, 3, 3364 0, 0, 3730

LH 24, 0, 1 24, 0, 3 24, 0, 5 24, 0, 92 Medium BB 96, 32, 2695 89, 28, 2864 92, 32, 2794 35, 4, 3634

LH 96, 0, 1 96, 0, 2 96, 0, 8 96, 0, 67 (20,75) BB 24, 20, 968 21, 12, 2685 24, 15, 2193 0, 0, 3642

LH 24, 0, 1 24, 0, 3 24, 0, 5 24, 0, 69 (20,100) BB 24, 11, 2337 21,5, 3382 24, 6, 3152 0, 0, 3680

LH 24, 0, 1 24, 0, 7 24, 0, 8 24, 0, 85 (25,100) BB 24, 8, 2742 14, 0, 3642 22, 1, 3597 0, 0, 3721

LH 24, 0, 1 24, 0, 18 24, 0, 19 24, 0, 166 (25,150) BB 24, 3, 3385 11, 0, 3648 19, 1, 3662 0, 0, 3818 LH 24, 0, 1 24, 0, 155 24, 0, 161 24, 0, 353 Large BB 96, 42, 2358 67, 17, 3339 89, 22, 3151 0, 0, 3715 LH 96, 0, 1 96, 0, 46 96, 0, 48 96, 0, 168

Table 3: Upper bounds: Nfeas, Nopti, CPU

As a further comparison between BB and LH, Table 4 shows the results obtained with the best method-model combination for each of the two methods. For BB, as just seen in Table 3, the best model isBS, while forLH, we selectedEP+ as the best model. Indeed, the incremental strategy used when the Lagrangian heuristic is combined with the Lagrangian dual optimization procedure (see Section 4.3) guarantees that the upper bound obtained after solving EP+ (Steps 1d, 2c and 3) dominates any other upper bound found during the course of the Lagrangian heuristic. In practice, we observed that the upper bound found when solvingES is already very good, as it is about 1% away from the best feasible solution found by the Lagrangian heuristic when solvingEP+. Nonetheless, EP+ is to be preferred, as it produces the best lower and upper bounds, with a modest additional computational effort, as shown in Table 3. Table 4 summarizes the results obtained with the two method-model combinations, BB-BS and LH-EP+. Three performance measures are provided: the lower bound GAP, the upper bound GAP and the CPU times in seconds.

These results show that, on Small and Medium instances, the upper bounds obtained by LHare on average within 2% of the best known solutions. On Large instances, the Lagrangian heuristic generally computes the best known upper bounds, with BB being 1% away from them on average, and even 3% away on average on the largest instances with (25,150). The computational effort to obtain such effective upper bounds is reasonable, as the CPU time is typically on the order of 1 minute on all instances. On Large instances with size (25,150), the computing time is around 5 minutes on average. The lower bounds computed by the B&B of CPLEX are, as expected, better than the Lagrangian lower bounds, but only slightly so. In particular, for Small, Medium and Large instances with size (25,150), the two lower bounds are close, and gets closer as the number of commodities increases. Indeed, the final gaps produced by the Lagrangian heuristic are on average better for Large instances with size (25,150).

6 Conclusions

We have considered the piecewise linear integer multicommodity network flow problem (PMFI). We have introduced formulations that exploit the integrality of the flows by us-ing discretization. We have shown that the basic model obtained by discretization can be viewed as a particular case of the basic segment-based formulation introduced in [12]. We have strengthened the discretized models either by adding valid inequalities derived from cutset inequalities or by using flow disaggregation techniques, obtaining a model similar to the so-called extended (segment-based) formulation introduced in [12].

When comparing the relative strength of the different formulations, our main results state that:

• Discretization provides stronger cutset inequalities than those obtained from segment-based models.

• Discretization, when combined with flow disaggregation, does not improve upon the LP relaxation of the extended segment-based model.

We have exploited these results by deriving a reformulation of the problem that combines the strength of both techniques: cutset inequalities based on discretization and flow

disag-(|N|,|A|) Algo-Model

(20,75) BB-BS 1.00, 1.00, 1066 LH-EP+ 0.94, 1.02, 30 (20,100) BB-BS 0.99, 1.00, 1311

LH-EP+ 0.94, 1.02, 62 (25,100) BB-BS 1.00, 1.00, 1924

LH-EP+ 0.93, 1.01, 44 (25,150) BB-BS 0.97, 1.00, 3103

LH-EP+ 0.92, 1.04, 69 Small BB-BS 0.99, 1.00, 1851

LH-EP+ 0.93, 1.02, 51 (20,75) BB-BS 0.99, 1.00, 2099

LH-EP+ 0.94, 1.01, 70 (20,100) BB-BS 0.98, 1.00, 2492

LH-EP+ 0.94, 1.01, 45 (25,100) BB-BS 0.97, 1.00, 2914

LH-EP+ 0.93, 1.02, 62 (25,150) BB-BS 0.92, 1.02, 3274

LH-EP+ 0.91, 1.02, 92 Medium BB-BS 0.97, 1.01, 2695

LH-EP+ 0.93, 1.02, 68 (20,75) BB-BS 1.00, 1.00, 968

LH-EP+ 0.96, 1.00, 69 (20,100) BB-BS 0.98, 1.00, 2337

LH-EP+ 0.95, 1.00, 85 (25,100) BB-BS 0.97, 1.01, 2742

LH-EP+ 0.95, 1.00, 166 (25,150) BB-BS 0.95, 1.03, 3385

LH-EP+ 0.95, 1.00, 353 Large BB-BS 0.98, 1.01, 2358

LH-EP+ 0.95, 1.00, 168

Table 4: Upper bounds: lower bound GAP, upper bound GAP, CPU

gregation with segment-based variables. In order to overcome the large size of the resulting model, we developed an efficient and effective Lagrangian relaxation method to compute lower and upper bounds.

Computational experiments on a large set of randomly generated instances allowed us to compare the relative efficiency of the different modeling alternatives (flow disaggregation plus addition of cutset inequalities with or without discretization), when used within the Lagrangian relaxation approach. The results derived from these experiments show the La-grangian relaxation method is both efficient and effective. For all instances, it produces lower and upper bounds in relatively small computing times and with gaps on the order of 5-10%.

On all instances, Lagrangian lower bounds are computed in less time than that required by CPLEX to solve the LP relaxation, with similar gaps; moreover, high-quality upper bounds are obtained in reasonable time, while the B&B method of CPLEX on any of the tested models often does not converge to optimality within the one-hour time limit.

This work opens the way for many research avenues. The models that we study are generic and include as special cases a large number of problems with applications in trans-portation and logistics, but also in other areas such as telecommunications and production planning. To the best of our knowledge, apart from the references already cited, no other work on reformulations by discretization has been performed on such problems. It would be interesting to investigate the impact of discretization on such problems, as well as on other problems with a similar structure. The formulations we have introduced involve a large number of variables and constraints. We handled the large size of the models by developing Lagrangian relaxation methods. It would be interesting to investigate other approaches, such as column-and-cut generation (recent examples of such methods on problems similar to the PMFI include [14, 15, 18]).

Acknowledgments

We are grateful to Serge Bisaillon for his help with implementing and testing the algorithms.

We also gratefully acknowledge financial support for this project. The work of the first author has been supported by NSERC (Canada) under grant 184122-09. The work of the second author has been supported by the Funda¸c˜ao para a Ciˆencia e Tecnologia under project PEst-OE/MAT/UI0152.

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