### Table 1: Evidence about some of the advantages in solving SCOPs via meta- heuristics instead of using exact classical methods.

2006

"... In PAGE 2: ... In contrast, approaches based on metaheuristics are capable of finding good and sometimes optimal solutions to problem instances of realistic size, in a generally smaller computation time. Table1 lists some papers in the literature providing evidence about the advantages in solving SCOPs1 via metaheuristics instead of using exact classical methods. This survey paper is the first attempt to put under a unifying view the several applications of metaheuristics to SCOPs, and it should contribute in balancing the literature, where a number of surveys and books about solving SCOPs via classical techniques exist, but none about using metaheuristics, despite the re- search literature is already quite rich.... In PAGE 2: ... Finally, section 6 highlights the conclusions. 1Legend for the SCOPs of Table1 : VRPSD = vehicle routing problem with stochastic demands, SCP = set covering problem, TSPTW = traveling salesman problem with stochastic time windows, PTSP = probabilistic traveling salesman problem, SSP = shop scheduling problem, SDTCP = stochastic discrete time-cost problem, SOPTC = sequential ordering problem with time constraints, VRPSDC = vehicle routing problem with stochastic demands and customers.... ..."

### Table 1. Meta-heuristic classification.

"... In PAGE 5: ... Similarly, scatter search was accompanied by adaptive memory elements as a result of being associated with early tabu search ideas, but this connection is likewise only beginning to be pursued. A few proponents of simulated annealing and genetic algorithms have recently gone farther in modifying the original conceptions than indicated in Table1 , to propose the inclusion of elements of adaptive memory as embodied in tabu search. Such proposals are often described by their originators as hybrid methods, due to their marriage of aspects from different frameworks.... ..."

### TABLE II. A COMPARISON AMONG THE META-HEURISTICS RECENTLY PUBLISHED WITH RESPECT TO THE QUALITY OF THE SOLUTION IN RPE INDEX.

### Table 1: p-median heuristic references (the types are: Constructive heuristics (CH), Local Search (LS), Mathematical Programming (PM) and MetaHeuristics (MH)).

"... In PAGE 16: ... An interesting idea of flnding the partition of L, and thus the number of subproblems, by using dynamic programming is developed in the DEC procedure. 7 Conclusions Table1 presents an overview on the development of heuristics for solving the p-median problem (PMP). We should ask a basic question given the nature of this survey: Has the advent of metaheuristics advanced the state-of-the-art signiflcantly? Based on a large body of empirical... ..."

Cited by 1

### Table 1. Comparisons of results over ten seeds

1998

"... In PAGE 6: ... The 600,000 value was appropriate for the larger Armour and Buffa problem while the smaller Bazaraa problem converged in much fewer number of solutions searched. Objective function values from the perimeter metric are in Table1 , where the best, median, worst and standard deviation over ten random seeds are shown. The twenty department Armour and Buffa (A amp;B) problem was studied with maximum aspect ratios of 10, 7, 5, 4, 3 and 2, which represent problems ranging from lightly constrained to extremely constrained.... ..."

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### Table 1 Internal comparisons

"... In PAGE 11: ... First, a comparison is made between the sequential methods involved in the cooperative search. Results are displayed in Table1 . The FFve meta-heuristics equally solve the problems of type C that have clustered customers.... ..."

### Table 2 lists the average runtimes of various approaches over the sam e 52 instances: the first six (i.e. IP, GA-1, GA-2, GA-3, GA-4 and GA-5) were run on a different Pentium III PC, while the follow ing two (i.e. EDA and LCS) on a similar Pentium 4 2.0G H z PC. Obviously, the IP is much slow er than any of the above meta-heuristics. A m ong these meta-heuristic methods, our algorithm takes no m ore time although an accurate com parison in terms of runtime is difficult due to the different environm ents (i.e. m achines, com pilers and programming languages) in use. For exam ple, the genetic algorithm s are coded in C and the EDA is coded in C++. The com parison in terms of the num ber of evaluations is also difficult because the other algorithm s evaluate each candidate solution as a w hole, while our algorithm evaluates partial solutions as well.

"... In PAGE 13: ... Table2 : Comparison of the average runtime of various approaches. 4.... ..."

### Table 5. Comparison of algorithms for formulation (PR) of example 4. Method* Standard

2000

"... In PAGE 20: ...ractional l1k. Since this lower bound is greater than the current upper bound of 68.01, the search stops. Table5 shows the comparison with other algorithms when the problem (9) -(11) is reformulated as the MINLP problem (PR) with the convex hull representation for the disjunctions. Note that the proposed BB algorithm and the standard BB yield the same lower bound (62.... In PAGE 32: ... Comparison of branch and bound methods for example 4. Table5 . Comparison of algorithms for formulation (PR) of example 4.... ..."

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### Table 5: Comparison of solution quality over time for Solomon apos;s problems (long run, with a discrepancy limit of a2 =5)

1998

"... In PAGE 12: ... However, it does appear that the problems with time windows require a higher discrepancy limit than those without(around 5 as opposed to around 2). Table5 shows how well a simple constraint programming technique such as LNS performs in comparison with the best Operations Research meta-heuristic techniques: the number of vehicles and distance is reduced to approximately the same level as TAI using a roughly equivalent amount of CPU time. RT TAI LNS Class CPU Quality CPU Quality CPU Quality 315 12.... ..."

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