A new set of hybrid heuristics for the quadratic assignment problem

Transcription

1 A new set of hybrid heuristics for the quadratic assignment problem André de São Thiago Moreira Valdir Agustinho de Melo Paulo Oswaldo Boaventura Netto Programa de Engenharia de Produção, COPPE/UFRJ CP , Rio de Janeiro, RJ, Brazil A new set of hybrid heuristics is proposed for the quadratic assignment problem, through the use of a matrix defined by Picard and Queyranne in the 70 s. This matrix has the advantage of guaranteeing the feasibility of a solution through the structural arrangement of the corresponding entries. Its use integrates the concept of pseudo-feasible solutions defined by Rangel. A modified GRASP using a taboo search, a reactive GRASP and an adaptive memory GRASP take profit of this structure. Among the result evaluation techniques utilized, a semi-qualitative comparison by pairs of results is applied. Keywords: Quadratic Assignment Problem, metaheuristics, matrix structures. 1. Introduction 1.1 A vision of the quadratic assignment problem (QAP) The classic (symmetric) QAP is the problem of allocating facilities to positions in a pairwise dependent situation, where the cost is obtained through a quadratic function depending on materials flows and distances. A graph-theoretical approach allows us to consider the QAP as the problem of allocating the vertices of a complete graph K n (f) (facilities) to those of another K n (d) (positions), the flows and distances values being associated to the edges of the corresponding graphs. Example: The permutation corresponds to (Fig. 1.1): 1 2 f 14 d 21 f 12 d 23 K n (d) 4 1 f 13 d 24 f 24 d K n (f) f 34 d f 23 d 34 Fig. 1.1: Facility allocation in the QAP A graph K n has n vertices and n(n 1)/2 edges. So the entries of each QAP data matrix can be associated to their respective edges, while the vertex positions can be permuted to obtain different solutions. The QAP objective function can be expressed in terms of permutations, (BS78) as min Z = c i N j N iπ(i)jπ(j) A QAP solution can be associated either to a vertex- (as above) or to an edge-permutation. But, as the edges cannot be independently allocated, not every edge-permutation corresponds to a feasible solution. [1747]

2 As an example, let us consider the symmetric instance (Land05), where n is the instance order; then N = n(n 1)/2 is the length of the vectors corresponding to the upper data matrix triangles: n = 5 (N = 10); D = [28, 33, 22, 20, 27, 40, 25, 30, 15, 18]; F = [10, 3, 1, 15, 6, 8, 4, 0, 10, 3], where D and F contain these entries in lexicographic order We can build a matrix Q = FD T with these vectors, for this instance, where the entry indices correspond to the lexicographic orderings of the two vectors (Table 1.1): F D Table 1.1: Cost coefficient matrix for a QAP instance This matrix contains every possible term of the QAP objective function. 1.2 The Picard-Queyranne matrix We propose a new hybrid heuristic for the quadratic assignment problem, through the use of a matrix defined by Picard and Queyranne (here referenced as PQ-matrix) in the 70 s. It represents a QAP instance through a graph H made by n copies of the complete graph K n. We can build on that structure n! different maximal cliques, each one being associated to a solution (Fig. 1.2): ( 2143) Fig. 1.2: Solution generation through the Picard-Queyranne matrix The PQ-matrix for a PQA instance corresponds to the Kronecker product of the instance data matrices: f11d V(H) = F D =. fn1d f1nd. f D nn (1.1) [1748]

3 It can also be obtained from Q by rebuilding n(n 1) n-matrices, each Q row corresponding to the content of the upper triangle of the corresponding matrix when token in lexicographic order. As we are dealing with symmetric QAPs, these matrices are then symmetrized. We can observe that two different PQ-matrices can be built (they could be called the FD- and the DF-PQmatrices), the second being obtained by interchanging F and D positions in (1.1). An important property of the PQ-matrix is that all cost coefficients of a feasible solution can be found along the same rows and columns. Example: Table 1.2 shows the lower submatrix triangle of the Land05 FD-PQ-matrix. Within it, we represent the solution (4,1,3,2,5), which corresponds to the pair set (14,34,24,54,31,21,51,23,53,52) whose elements are bold-underlined both on the matrix Q above (where they correspond to the edge-permutation (3,8,6,10,2,1,4,5,9,7)) and also in the PQ-matrix below. We call these edge-permutations relaxed solutions. Within the PQ-matrix structure, these elements appear on the same rows and columns for each row of submatrices and for each column of submatrices, characterizing this solution as being feasible Table 1.2: The PQ-matrix for Land05 So to obtain a feasible solution we have to take n 1 entries within a given submatrix column (all on the same internal column, which defines the first element) or within a given submatrix row (all on the same internal row, which defines the last element). In the example we can choose positions 1, 3, 2 and 5, all in column 4 of the first submatrix column, or we can choose the last submatrix row (positions 4, 1, 3 and 2) and within it the fifth row. Anyway, the vertexpermutation is (4,1,3,2,5). We have to choose the remaining 6 elements of our edge-permutation in the indicated positions in order to the solution to be feasible (we then obtain a feasible relaxed solution). We call the first n 1 elements the head of a permutation, the remaining ones forming its tail. We can make different tail allocations and obtain other (infeasible) solutions with the same elements in the first n - 1 positions. These solutions will be called pseudo-feasible solutions (Ra00a). [1749]

4 The pseudo-feasible solutions given by chosen line and column are the neighborhood where we will make the GRASP local search, where we look for a new feasible solution. In this process we obtain an infeasible solution by doing a 2-exchange and then we look for a new feasible solution having this 2-exchange. 2. The algorithms 2.1 GRASP (Greedy Randomized Adaptive Search Procedure) is a multistart metaheuristic proposed by Feo and Resende (FR95), composed by two steps: start solution generation, using a randomized greedy approach; local search, using convenient neighborhood and search techniques. The classical GRASP is a memoryless multistart greedy-randomized algorithm for whose details we refer to (FR95). An initial solution is obtained by the general procedure below (construction phase), where a probabilistic parameter α is applied in order to randomize a greedy element selection: Procedure GRASP( ); 1 Data Reading( ); 2 While stop criterion not satisfied do 3 BuildRandomizedGreedySolution (solution); 4 LocalSearch (solution,neighborhood(solution)); 5 AtualizeSolution (solution,bestsolutionfound); 6 EndWhile; 7 Return (bestsolutionfound); EndProcedureGRASP; The local search applied to the solution considers a neighborhood structure, which has a number of possible definitions, according to the problem structure and, even for a given problem, to the theoretical resources that are utilized. A general vision of this search is LocalSearch (solution,neighborhood(solution)); 1 While solution not locally optimal do 2 Find best solution t in the set Neighborhood (solution); 3 s = t; 4 EndWhile; EndLocalSearch; The GRASP variants studied here are: normal, reactive and adaptive memory GRASP versions (PR00, FG99); a classic taboo search as the local search heuristic (Gl89a-b); an upper value limit or filter for the initial solutions (Ra00b). A summary of these variants is presented below, where we also describe the normal GRASP here used, which is based on PQ-matrix structure (GRASP-PQ). The stopping rules were based chiefly on iteration number (here we used either 2000 or iterations) but also, for some tests, on processing time and on a target to be attained. Comparisons were made with two GRASP variants of the literature: the GRASP by Resende (RR02) the GRASP with robust tabu search by Taillard (Ta91); [1750]

5 At this stage of the research we were chiefly interested in the efficiency of the proposed search in what concerns the finding of new solutions. So the comparison with Taillard s algorithms was done by taking the same number of iterations (2000 or 20000), disregarding the time processing advantage which allows that algorithm to work with much more iterations to obtain better results. 2.2 The basic GRASP and the PQ-matrix A mixed random and minimax strategy was used to find the initial solutions for GRASP-PQ. In the algorithm, the positions indicated by letters correspond to submatrices and those indicated by numbers, to matrix elements. Procedure GRASP-PQ ( ) 1 Data reading ( ) 2 While solution not completely built do 3 Choose one among the α% greater flows f ab, not yet allocated; 4 Choose the least distance d xy, which allows for a feasible solution; 5 Allocate f ab to d xy doing P a y; P b x; < P a and P b are y and x image positions respectively > End While. The local search is done on the set of pseudo-feasible solutions corresponding to each submatrix, as follows: While the initial solution is improved do 1 For every solution tail submatrix do 2 For every row/column where the cost coefficient is lesser than that of the initial solution do 3 - generate neighbor solution; 4 - calculate cost of neighbor solution; 5 - actualize best neighbor solution; 6 End (for) 7 End (for) 8 If better neighbor solution < initial solution then Initial solution better neighbor solution; else Initial solution not improved End While 2.3 The modified GRASP variants The general idea of modifying GRASP was the classical one of giving it some type of memory (result recall) in order to allow the algorithm to use some of the experience acquired in previous trials. The reactive GRASP (PR00) examines the results obtained through a block of iterations in order to modify the assigned probabilities of a set of specified values for the random-greedy proportion parameter α of GRASP. We worked with ten values and with 100-iteration blocks. The adaptive memory GRASP (FG99) builds an elite solution set where the algorithm seeks comparison with each new starting solution in order to know if it is sufficiently different (in the sense of the existance of equal image elements) from these solutions, which allows the algorithm to enlarge its exploring of the solution set. A strategy is defined to actualize the elite solutions. We found it useful to work with 20% difference values. [1751]

6 These variants use strategies designed to give some memory to the basic memoryless GRASP, which is done through the actualization of parameter α (for reactive search) and through building of an elite solution set (for adaptive search). In the first scheme a previously defined set A of α values is used (usually we have A = m = 10) and an α value is chosen with probability p i for each iteration. The scheme begins with p i = 1/m for all I and the p i are actualized by using information given by iteration blocks usually of 100 iterations each. If M i is the solution value average corresponding to α i value and F(s*) is the best solution value until the moment, we have from where F(s*) q i = (i = 1,..., m) M i (2.1) m p = i q i / q j= 1 j. (2.2) Higher M i values improve, then, the probability of the corresponding α i be selected. The elite set S for adaptive (memory) tabu search is built by defining cost bounds best(s) and worst(s), which allow a solution to be admitted in S if its cost is better than worst(s) and it is also sufficiently different (in terms of pair composition) form every solution in S, by replacing this worst solution. Then the elite solutions are the basis for a more complete exploration of the solutions universe. For some problems the new solutions are built by using elements from elite solutions, but here we only used them to guarantee the pair difference. For reactive GRASP the commands to be included in the basic GRASP / tabu are sufficiently obvious to authorize us to avoid the repetition of the algorithm; on the other hand, the procedure for verifying differences between solutions is While (i < R) and (counter nsimil) do 1 j = 1; 2 count = 0; 3 While (j n) and (count nsimil) do 4 If (initialsolution[j] = S[i][j]) then 5 count = count + 1; 6 j = j+1; 7 EndWhile; 8 i = i+1; 9 EndWhile; 10 If (contador > nsemelhanca) then RejectSolution; 11 EndIf. 2.4 The tabu search Every variant but the basic GRASP utilized a simple tabu search strategy. Tabu search is a local search procedure which accepts the worsening of solution values in order to escape from local optima. The basic local search does not admit such strategy and as a consequence the solution path can be allowed to return to previously found solutions. This can be avoided through the use of a tabu list where a solution attribute stays during a given number of iterations (tabu time). A number of improvements have been implemented over this basic scheme, in order to deal with different situations (Glover (Gl77), (Gl86), (Gl89a), (Gl89b), Hansen (Ha86), Werra and Hertz (Wh89)). We formulated a GRASP / tabu algorithm by modifying the algorithm presented in 2.3 through the introduction of a tabu condition in Step 3 and the inclusion of present solution [1752]

7 attributes after Step 7. These attributes are the last pair whose cost coefficients were interchanged and the corresponding cost coefficients. 3. Computational results 3.1 The performance tests were run on three platforms: C compiler for Windows in a PC with AMD 2.6 XP processor (basis for reported times); C compiler for Windows in a PC with Pentium IV 3.2 processor; C compiler for Linux in a workstation with 2 Xeon 2.8 processors. The Pentium IV times were about 0.76 of equivalent AMD times and 0.85 of equivalent Xeon times. The parameter α took the values {0.1, 0.3, 0.5}, the results shown in Tables 3.1 (a) and (b) being the best of these options. The percent errors are calculated on the basis of the optimal or best known value for the instance. The tests used either 2000 or iterations, the second option being indicated by a (*) added to the instance name. We used instances from QAPLIB collection (QAPLIB): nug12 to 30, chr12a to 25a, tai12a to 50a, rou12 to 20, scr12 to 20 and els19 and also instances dre15 to 24 from Drezner (DHT04) collection (total 37 instances). For each instance we used ten seeds randomly chosen from the prime number list between 1 and 2,000,000 (Pi06). The seeds utilized were , 11399, , 2963, , , 7331, 797, and The results presented in the tables are the corresponding averages, with the exception of the Minimal Error column which gives the best result for all seeds. The column Average Improvement % gives the percent value improvement over the initial value, on the basis of this value. The column Average Gap % gives the percent gap value between the best value obtained and the best value known, on the basis of this one. When using GRASP with adaptive memory, solutions were considered for admission to the elite list through a value criterion, but the decision of using or not a solution used a pair similarity criterion with an upper limit of 20%, established by previous testing. 3.2 Discussion Tables 3.1 (a) and (b) below present the results obtained with the basic GRASP-PQ, the simple tabu version (GRASP-TB), the reactive tabu version (GRASP-RT), the adaptive memory version (GRASP-AM) and the Resende version (GRASP-RR). We can observe that the pure versions (PQ and RR) are less precise as their percent errors are greater than those of TB, RT and AM. RT seems to be more precise than AM. The least errors, both average and minimum, correspond to TB (23 and 19 cases respectively), against RT (12 and 12) and AM (13 and 10). There was a significant occurrence of zero errors for all techniques. On the other hand, RR and TB find more best solutions (13 cases each), while RT won in 7 cases. In what concerns speed PQ is much faster than RR and, among the other variants, AM is the faster as it discards initial solutions not sufficiently different from those in the elite list. The strategy seems then fairly efficient, even if we used a single selection criterion instead of a double one involving also quality of the initial solution. [1753]

8 Instance Error (%) Minimum error (%) Processing time, sec (AMD 2.6) Nsol Variant PQ TB RT AM RR PQ TB RT AM RR PQ TB RT AM RR PQ TB RT AM RR Nug ,21 6,50 6,38 5,25 2, Nug14 0 0,02 0, ,57 16,82 15,78 15,73 5, Nug ,53 24,77 22,45 21,43 7, Nug16a 0, ,59 35,08 34,32 30,81 11, Nug17 0,09 0,03 0,09 0,02 0, ,32 49,76 50,02 47,55 16, Nug18 0,28 0,12 0,1 0, ,49 65,37 58,45 62,18 20, Nug20 0, ,25 112,20 102,40 101,95 35, Nug21 0, , ,82 154,97 142,46 150,48 47, Nug25* 0,01 0 0,01 0, , , ,3 409, , Nug30* 0,29 0,13 0,17 0,40 0,08 0, , ,9 7984,0 1002,6 2994, Scr12 0, ,53 2,76 2,68 2,18 2, Scr15 6, , ,97 8,64 9,19 8,32 8, Scr20 7,54 0,003 0,03 0,01 0,01 4, ,90 33,55 28,46 30,50 36, Rou ,61 8,90 7,86 6,71 2, Rou , ,96 27,76 24,67 25,26 7, Rou20* 0,006 0,02 0,01 0,16 0, ,02 128, ,23 909,33 120,12 311, Els19 0, , ,67 70,96 62,41 65,28 33, Chr12a 1, , ,23 0,56 0,68 0,56 2, Chr15a 11,57 0 0,04 0,52 0 7, ,01 23,64 17,49 2,60 66, Chr18a* 40,68 0,19 0,52 7,05 2,83 32,67 0 0,18 0 0,18 7,22 44,16 37,29 3,98 165, Chr20a* 78,91 1,07 1,86 6,45 0,56 54, ,46 0 9,54 45,21 44,84 4,86 266, Chr20b* 47,45 3,35 4,83 8,40 4,35 36,38 0 4,18 6,44 3,83 6,69 41,11 40,66 4,75 242, Chr22a* 10,78 0,72 0,79 1,76 1,81 8,97 0 0,32 0 1,59 13,69 91,35 67,80 9,22 495, Chr22b* 17,33 1,51 1,66 2,68 1,58 15,95 0,84 0,94 1,78 1,36 13,46 78,34 83,64 8,58 450, Chr25a* 112,76 7,02 7,24 12,21 10,48 100,32 0 1,84 2,05 10,22 15,48 163,56 138,40 14,81 954, Tai12a ,72 8,97 8,60 6,40 2, Tai15a 0,13 0 0,02 0,02 0, ,55 25,32 26,37 23,75 6, Tai17a 0,44 0,35 0,28 0,49 0, ,49 50,20 47,19 49,75 13, Tai20a* 0,58 0,22 0,50 0,80 0,37 0, ,47 0,30 101, ,67 112,88 106,56 297, Tai25a* 1,08 0,64 0,79 1,43 1,67 0,55 0,15 0,37 1,02 1,55 264, , ,8 340,31 897, Tai30a* 1,32 1,29 1,21 1,74 1,60 0,93 1,02 0,56 0,85 1,53 611, ,3 7392,5 902, , Tai40a 2,44 2,28 2,42 2,31 2,35 1,80 1,76 2,08 1,99 2,09 181, , ,2 3719,5 969, Tai50a 2,80 2,72 2,73 2,71 2,81 2,27 2,56 2,41 2,42 2,52 436, ,1 8416, , Dre15 57, ,42 49, ,44 2,72 2,61 2,64 5, Dre18 83,32 3,43 1,45 1,45 4,34 68, ,65 6,69 6,06 6,36 15, Dre21* 87,19 7,92 15,79 30,17 26,81 71,91 0 7, ,79 8,57 140,19 100,57 10,93 319, Dre24* 70,81 25,10 25,76 41,92 33,43 54, ,85 33,33 20,29 260,71 186,56 20,98 602, Table 3.1 (a): Comparison among GRASP variants [1754]

9 Instance Average improvement % Average gap % Variant PQ TB RT AM RR PQ TB RT AM RR Nug12 19,81 20,56 22,67 18,13 17,19 5,53 4,66 4,89 3,95 5 Nug14 19,23 20,20 19,78 18,46 16,52 6,00 4,84 4,87 4,52 5 Nug15 21,73 22,70 22,33 20,92 19,22 5,66 4,24 4,24 3,92 5 Nug16a 18,08 20,19 20,19 18,85 17,61 5,36 4,69 4,69 4,35 5 Nug17 21,26 21,65 21,92 20,50 17,64 4,99 4,15 4,07 3,85 4 Nug18 20,31 21,07 21,18 20,55 17,77 4,91 4,25 4,22 4,09 4 Nug20 20,37 20,42 21,06 20,55 17,57 4,77 3,95 4,00 3,90 4 Nug21 24,53 25,17 25,29 24,83 22,99 5,07 4,38 4,30 4,22 4 Nug25* 21,44 22,13 21,92 21,93 19,17 4,58 3,75 3,76 3,76 4 Nug30* 19,74 21,03 20,95 20,95 19,14 4,48 3,94 3,92 3,68 4 Scr12 30,45 40,50 39,11 32,28 33,32 19,58 6,73 6,76 5,55 6 Scr15 30,51 40,28 42,15 39,67 40,67 19,38 9,65 9,41 8,84 9 Scr20 31,38 45,50 44,34 43,83 38,84 21,85 8,86 9,05 8,90 8 Rou12 17,41 17,67 17,05 14,08 13,86 5,40 5,10 5,08 4,18 5 Rou15 16,77 17,35 17,47 16,53 12,04 7,05 6,39 6,42 6,13 6 Rou20* 14,12 15,97 15,69 15,59 13,22 4,82 4,58 4,60 4,46 5 Els19 56,82 59,91 59,45 57,37 56,99 24,14 19,33 19,93 19,33 25 Chr12a 29,51 52,86 59,89 49,43 52,48 53,18 31,26 33,73 27,93 25 Chr15a 34,64 72,62 69,37 69,30 55,58 60,13 34,85 34,07 33,89 28 Chr18a* 33,90 73,49 72,10 71,88 59,23 60,03 41,34 40,52 37,58 39 Chr20a* 16,45 63,33 66,66 66,70 53,16 53,79 31,60 30,97 27,89 32 Chr20b* 13,10 62,23 66,25 66,19 48,88 54,32 26,70 25,85 23,36 27 Chr22a* 28,72 56,06 52,79 52,84 42,16 32,47 12,00 11,86 10,99 11 Chr22b* 15,76 49,25 51,94 51,86 38,67 37,12 10,73 10,79 9,95 11 Chr25a* 15,24 68,85 67,32 67,24 58,26 45,91 32,82 32,93 29,80 31 Tai12a 20,16 20,53 19,66 16,54 20,15 8,45 8,02 8,10 6,75 7 Tai15a 14,66 15,02 16,71 15,64 11,07 4,79 4,50 4,56 4,25 4 Tai17a 15,84 16,09 16,25 15,40 11,24 5,16 4,98 5,08 4,58 5 Tai20a* 13,97 14,52 15,97 15,57 12,58 5,80 5,53 5,08 4,80 5 Tai25a* 13,79 14,39 14,13 14,11 11,36 4,04 4,64 4,50 3,90 4 Tai30a* 12,49 12,60 12,65 12,62 9,62 3,65 3,57 3,69 3,14 3 Tai40a 12,55 12,40 12,42 12,42 8,95 2,56 2,62 2,49 2,59 3 Tai50a 11,70 11,95 11,79 11,79 8,05 2,13 2,12 2,13 2,15 3 Dre15 36,15 53,36 58,26 54,84 39,63 44,41 43,27 43,88 41,33 38 Dre18 27,60 57,87 61,47 60,12 47,06 46,91 42,50 43,73 42,70 42 Dre21* 25,69 62,27 60,66 60,42 44,57 43,96 44,37 40,28 32,85 34 Dre24* 35,01 62,73 61,72 61,59 45,05 43,49 35,35 34,62 26,69 31 Table 3.1 (b): Comparison among GRASP variants The average improvement was better for TB (for 19 instances) and RT (for 17 ones). On the other hand, the average gap results were much better for AM with 30 better results, while RR had 7. All these results point to the interest of further research on algorithm implementation, chiefly on the development of AM. 3.3 Comparison with GRASP by Taillard The table below shows a performance comparison between the GRASP-TB here presented and the robust tabu search by Taillard (Ta91). Whereas the speed of this last algorithm is outstanding for any such comparison, its precision was inferior with relation to that of GRASP-TB when tested in similar conditions. The tests were done on 37 instances with a defined target as stopping criterion, which was the optimal best value known for the instance. GRASP-TB presented lesser error for 17 instances, while GRASP-Taillard was better for 11 instances, a tie being observed for 9 others (NBR row). [1755]

10 Instance 4. Pair evaluation Error (%) (Taillard) Error (%) (GRASP/Tabu) Instance Error (%) (Taillard) Error (%) (GRASP/Tabu) Nug Chr18a* 0,10 0,19 Nug14 0,02 0,02 Chr20a* 3,89 1,07 Nug15 0,02 0 Chr20b* 4,42 3,35 Nug16a 0 0 Chr22a* 0,85 0,72 Nug17 0,03 0,03 Chr22b* 1,47 1,51 Nug18 0,11 0,12 Chr25a* 8,48 7,02 Nug20 0,07 0 Tai12a 0 0 Nug21 0,35 0 Tai15a 0,09 0 Nug25* 0 0 Tai17a 0,42 0,35 Nug30* 0,04 0,13 Tai20a* 0,33 0,22 Scr Tai25a* 0,54 0,64 Scr15 0,39 0 Tai30a* 0,44 1,29 Scr20 0,08 0,003 Tai40a 1,49 2,28 Rou Tai50a 1,80 2,72 Rou15 0,07 0 Dre15 14,84 0 Rou20* 0,01 0,02 Dre18 6,81 3,43 Els19 0,36 0 Dre21* 1,40 7,92 Chr12a 0 0 Dre24* 22,98 25,10 Chr15a 0,12 0 NBR A set of criteria can be used to compare the performance of a set of k algorithms by examining the different orders observed on the same parameter values. If a non-decreasing order for a parameter p is used we can say that, according to the values obtained by two algorithms, that if p(i) < p(j), then the algorithm i obtained a better result than j according to p. A sequence of C k,2 comparisons can then be made. If we have more than a parameter a given sequence can present discordances with respect to another sequence: then the distance between two sequences is the number of such discordances. The relative error between two sequences is the percent ratio distance / C k,2. We have four evaluation parameters: the average gap between final values and best value known, the complement of the average improvement from initial and final values, the execution time and the number of better solutions obtained. This last one will be presented on a suitably modified form, I(nsol, error) = 10 (error%) + nsol -1, in order to allow for cases where the optimal or best known solution is not found by the algorithm. The five algorithms under comparison were (1) GRASP-PQ, (2) GRASP-TB, (3) GRASP-RT, (4) GRASP-AM and (5) Rangel/Resende GRASP. As an example, the instance Nug15 gives the following results: A B C D E (a) avg gap 5,66 4,24 4,24 3,92 5,00 4 3,92 2 4,24 3 4,24 5 5,00 1 5,66 (b) (1-avg improv) 78,27 77,30 77,67 79,08 83, , , , , ,48 (c) I(n,sol) 0,100 0,040 0,040 0,042 0, , , , , ,100 (d) (t,sec) 3,53 24,77 22,45 21,43 7,86 1 3,53 5 7, , , ,77 The right side of the table gives the non-increasing ordering which can be found in the left side. A numerical code was used, based on the relations {(+1, >),( -1, <),(0, =)}, in order to obtain a numerical table which we can use also to compare the parameters a b c d [1756]

11 By summing pairs of rows we can evaluate discordances (corresponding to zeroes). The last column entries count their number: if they correspond, for example, to more than 75% of C 5,2 = 10, the parameter utility would be questionable. On the other hand, by considering columns we are testing the ability of pairs of algorithms to distinguish instances by difficulty, based on the current criteria Discord ab ac ad bc bd cd Discord The results for the instances below consider only pairs with 3 or less discordancies, where the indicated ordering is the same for the different parameter pairs (e.g. 23 cannot be compared through their results with Nug15 owing to different ordering scores, two +1 and two -1): Instance nug chr15a nug chr20a chr25a nug tai20a tai30a dre dre dre dre tai40a tai50a Σ We can see that GRASP-PQ is less efficient than GRASP-TB and, almost everywhere, than GRASP- RT. GRASP-AM shows itself better than GRASP-PQ but in two cases. Every algorithm but GRASP- PQ is better than algorithm (5). GRASP-RT was somewhat better than GRASP-TB and GRASP-AM was much better than (5). 5. Conclusions This paper presents some research results on a new neighborhood for local search, that of pseudofeasible QAP solutions, accessible through a structure known as the Picard-Queyranne matrix. A local search was applied to the solutions generated by a classical GRASP. A tabu search strategy was also used within the local search and two memory variants, reactive and adaptive memory GRASPs, were also tested. Comparisons were made with Taillard s robust taboo search (in what concerns the precision) and with the Resende GRASP. The results obtained show interest in the exploration of the pseudo-feasible solutions, avalable as a new neighborhood through the use of the Picard-Queyranne matrix. While some implementations need to be improved, we think this approach to develop new heuristics presents an interest for further research, both in theory and new algorithms. [1757]

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