This chapter is to examine the efficiency and effectiveness of the proposed algorithms. We designed a series of computational experiments. The algorithms were implemented in Java and tested on a personal computer with a Pentium D 2.8GHz CPU and 512 MB memory running Microsoft Windows XP. The test data were obtained by randomly generating points with real coordinates on a 100×100 square plane. The costs (setup time between jobs) are thereby defined in the Euclidean distance (rounded to integer).
To analyze the efficiency of our binary integer program, we use CPLEX to implement the two formulations IP_Latency_TSP and BIP_Latency_TSP to delivery man problem which mentioned in Chapter 2. For each problem size, five instances were executed. The results are shown on the Table1. The “optimal” row indicates the number of instances out of each five that were optimally solved, and the “Avg Time” row gives the average execution time for each five instances. It is clear that the binary formulation permits shorter execution times than the integer program. Although our binary program is faster than the integer program, we do not claim absolute superiority. As mentioned in Chapter 2, the integer program provides good structures for other research purposes.
The second part of our experiments is to investigate the efficiency and effectiveness of the dynamic programming algorithm and the proposed approximation approaches. The number of the orders is a given constant and the weight of each order
w
i is generated from the uniform interval [1, 10]. Each job is also randomly distributed to the orders.In the computational study, there are different numbers of nodes represented by n and different numbers of orders represented by K. One hundred instances were generated for each case, and the average of these one hundred instances was presented as the computational result. The experiment analysis basically consists of three parts of comparisons. The first part is the performance comparison between the dynamic
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programming algorithm and all other meta-heuristic methods. Two performance indices are used: the elapsed run time and the relative errors (|opt-apprx|/max{opt,apprx}) (Ausiello et al., 2005). Due to the difficulty of obtaining optimal solutions, within a reasonable time frame, by the dynamic programming algorithm as the problem size increase, the second and third parts only conduct the performance comparisons among the specified meta-heuristic methods. The numerical statistics are displayed through Tables 2 to 9.
Small-scaled problems, n = 10, 14, 18, 20, 22, or 24 and K = 3 or 5, are shown in Table 2 and Table 3. The tables illustrate the average run time of each case by different algorithms adopted in the research. Because the exact dynamic programming algorithm could provide the optimal solutions within a reasonable time frame for small problems, error ratio and hit frequency of approximate algorithms are also depicted in the tables.
The local search methods, both H1_LS and H2_LS, tend to be easily trapped in the local optima in the experiments. Consequently, the overall hit frequencies and error ratios of the local search approaches appear to be inferior to other methods. In addition, if we examine the computational results closely, we may observe the superiority of the 2-Opt strategy for constructing neighborhood structures in the local search approaches. Similar verification could also be applied to the tabu search algorithm. Therefore, for simplification, we only adopt TS_2Opt to represent tabu search for the rest of all the computational analyses.
Tabu search method demonstrates the ascendancy, in hit frequency, for small-scaled problems. Optimal solutions could be obtained for most of the test cases by the tabu search method. On the other hand, for GA and ILS, they both exhibit high hit frequencies with low error ratios, but they work less efficiently than the tabu search in terms of runtime. As the number of jobs rises (n exceeds 18), the exact dynamic programming algorithm requires a longer run time than all approximation methods. The low error ratios are below 0.3 percent for all heuristic methods. This implies that the number of orders does not affect the performance of these algorithms when the problem size is small.
Tables 4 to 6 exhibit some medium-scaled problems, as the number of nodes is greater than twenty (n = 30, 40, 50, or 60, K = 3, 6, or 15). Because the dynamic programming algorithm fails to produce optimal solutions within a desirable time frame, the performances of the heuristic methods are compared with each other. The “win”
columns in the tables indicate how may times a heuristic method is able to provide the best approximate solution. The “deviation” columns display the deviation of each individual approximation solution away from the “best” one. Some observations, for various n and various K, are listed as follows.
1. Different numbers of nodes, n: The numbers recorded in the “win” columns drop as
n increases for GA and tabu search methods; on the contrary, ILS has the
ascendancy with n. When n = 30, tabu search was able to obtain the best approximation most often, while GA and ILS could also provide the solution more than 50 times with the deviation bounded within 1%. When n ≥ 40, ILS ascended as the leader among all tested heuristic methods. Such phenomenon is aggrandized as the number of nodes increases.
2. Different numbers of orders, K: The run times and deviations for all three heuristic methods increase with the number of orders. Moreover, GA provided less “best”
solutions as the number of orders increases. This implies that GA performs worse when more orders are received. On the contrary, such performance deterioration along with ascending K is not evident for either tabu search or ILS.
Tables 7 to 9 summarize the statistics for the large-scaled cases. We examined four different numbers of nodes (n = 70, 80, 90, or 100), along with various numbers of orders (K = 6, 15, or 25). It is easy to see that ILS outperforms all other meta-heuristic methods when coping with large-scaled cases. While genetic algorithm failed to improve the solution qualities for large scaled problems, it also required a much longer execution time in comparison with other meta-heuristic approaches adopted in the research. Tabu search seems to work efficiently by accomplishing one hundred jobs in
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ten seconds. However, tabu search also failed to improve the solution even if the number of iteration is raised to 100,000 times. This fact implies that the tabu search might not find a better neighbor solution when the problem size increases. Therefore, compared with the tabu search approach, ILS effectively improves the solution quality with a moderate compromise of elapsed run time.