objective 1 Benchmark 2tsp-150, |PF*|=4701
6. Concluding remarks
The aim of this paper is to apply the recently proposed improved version of the augmented ε-constraint method (AUGMECON2), which is suitable for general MOIP problems, to two popular MOCO problems, namely, the Multi-Objective Traveling Salesman Problem (MOTSP) and the Multi-Objective Set Covering Problem (MOSCP). Although AUGMECON2 is designed for the general case, here it is applied to bi-objective problems confined by benchmark-data availability.
For the MOTSP case the proposed method was a combination of a general purpose MOIP model (AUGMECON2), with a Branch-and-Cut-and-Heuristic model (BCHTSP) available in GAMS model library. It was found that the ε-constraint sub-problem is solved almost as many times as the cardinality of the Exact Pareto Set, which is a very favourable characteristic for a generation approach (no redundant iterations). Obviously, the BCHTSP model is appropriately modified in order to solve the ε-constraint sub-problem. Relying on the efficiency of the modified BCHTSP which is used as a subroutine, the AUGMECON2 method is able to effectively calculate the Exact Pareto Set in 24-60h wall clock time for every instance of our test bed. A novel feature of our implementation is the parallelization of the AUGMECON2 loop into indicatively three threads. In general, the AUGMECON2 method is appropriate for parallelization as the main loop can be divided into independent segments.
In our work it was reaffirmed that MOTSP is among the hardest MOCO problems. Even bi-objective instances with 100 cities have not been solved exactly in the literature. To the best of our knowledge our work is the first one that generates the exact Pareto set for 16 popular MOTSP instances with 2 objectives and 100 cities, studied intensively in the literature. In general, our approach is among the few implementations able to solve the MOTSP exactly (i.e. produce the exact Pareto set). We also created and solve exactly a bi-objective problem with 150 cities but probably this is the upper limit for the exact solution of bi-objective problems with our method and the current hardware. Moving to three objective functions, the difficulty of generation of the exact Pareto front escalates dramatically and the upper limit seems to be 15-20 cities, which make the use of exact algorithms prohibitive even for small size multi-objective TSP problems. We think that a great contribution of our work is that the data sets and the results are available in https://sites.google.com/site/kflorios/motsp for the interested readers.
Having the exact Pareto set for the BOTSP we were able to assess the effectiveness of state-of-the-art Multi-Objective Metaheuristics (MOMHs) previously utilized to approximately solve the same 16 datasets. The MOMHs are evaluated, using the two set coverage and convergence metrics exploiting the information of the Exact Pareto Set. In our case the coverage metric is actually the percentage of POS
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found by the MOMH. The coverage metric in the MOTSP problems varied from 25% to 90% depending on the type of instances. Euclidean instances were better approximated by MOMH techniques. Random matrix instances showed poor performance for MOMHs. The mixed type instances yielded approximations better than random matrix but worse than Euclidean instances, as expected. With respect to the convergence metric, we found that, in general, state-of-the-art MOMHs approximate very well the Exact Pareto Set. The magnitude of the convergence metric with respect to the true Pareto Front found by our work, was either of the order of 10-4 or 10-5, depending on the MOMH type and the instance type.
Another important finding which is in accordance with similar results from other researchers in MOCO, is that the number of supported POS is only a small proportion among the generated POS. Consequently, the POS produced using the weighting method (that produces only supported POS) is a remarkable underestimation of the true Pareto set for the MOTSP.
Regarding the BOSCP, AUGMECON2 succeeded in solving the previously unsolved benchmarks (instances 201a and 201b) of the MOCOlib for the MOSCP problem. In total, 42 out of 44 benchmarks were exactly solved, leaving only 2 datasets unsolved (in a 24 hours time limit).
In general, for both kinds of problems, namely MOTSP and MOSCP, the effectiveness of the AUGMECON2 method is reflected on the fact that for each benchmark the number of model solved is very close to the cardinality of the Pareto set, indicating good performance and computational economy.
In order to contribute to the testing of relevant algorithms (MOMH or exact algorithms) for the MOTSP and the MOSCP a web site was created that gathers all the datasets and the results, as well as source code in Fortran implementing AUGMECON2 and GAMS implementing modified BCHTSP for the ε-constraint sub-problem.
Extension of our approach, AUGMECON2-BCHTSP to three objective TSPs and massive parallelization (using more than 3 threads for computations) is studied. The optimal allocation of computational load for many processors in the bi-objective case is an interesting problem. Also, parallelization of AUGMECON2 for three objective problems is more delicate, since only the outer loop can be parallelized safely. Perhaps, the research stream with the most potential for the exact solution of MOTSP is to substitute the BCHTSP part of the AUGMECON2-BCHTSP algorithm with a fast dedicated exact solver like CONCORDE [64] or TSP1 [33] but this needs nontrivial programming. The ε-constraint sub-problem has to be programmed inside CONCORDE or TSP1 which requires effort but would be worthwhile. Also, comparison of the AUGMECON2 method with other exact schemes for general MOIP problems as well as the specific MOTSP seems promising (e.g. methods of Lemesre et al. [65] and Dächert et al. [66]).
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