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Rodríguez Rueda, David; Cotta, Carlos; Fernández Leiva, Antonio J. A Memetic Algorithm for Designing Balanced Incomplete Blocks International Journal of Combinatorial Optimization Problems and Informatics, vol. 2, núm. 1, enero-abril, 2011, pp. 14-22 International Journal of Combinatorial Optimization Problems and Informatics Morelos, México Available in: http://www.redalyc.org/src/inicio/ArtPdfRed.jsp?iCve=265219618003

International Journal of Combinatorial Optimization Problems and Informatics ISSN (Printed Version): 2007-1558 [email protected] International Journal of Combinatorial Optimization Problems and Informatics México

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© International Journal of Combinatorial Optimization Problems and Informatics, Vol. 2, No. 1, JanApril 2011, pp. 14-22. ISSN: 2007-1558.

A Memetic Algorithm for Designing Balanced Incomplete Blocks David Rodríguez Rueda1, Carlos Cotta2, Antonio J. Fernández Leiva2 1 Universidad Nacional Experimental del Táchira (UNET) Laboratorio de Computación de Alto Rendimiento (LCAR) San Cristóbal, Venezuela [email protected] 2 Universidad de Málaga ETSI Informática, Campus de Teatinos 29071 Málaga, Spain. {ccottap, afdez}@lcc.uma.es Abstract. The generation of balanced incomplete block designs (BIBD) is a hard constrained combinatorial problem whose applications are manifold. Although the BIBD problem can be easily formulated as a combinatorial optimization problem, its resolution still constitutes a formidable challenge for solving techniques. In this work we devise a memetic algorithm (MA) for tackling the BIBD problem. This MA features a heuristic recombination operator based on greedy procedures and a local search method embedded in the evolutionary cycle. An extensive empirical evaluation is done using 86 different instances of the problem. The results indicate that the two mentioned components of the MA contribute synergistically to the performance of the algorithm, whose results are better than those of other metaheuristics such as genetic algorithms, hill climbing, and tabu search (which was the previous incumbent for the problem). Keywords: Balanced incomplete block designs; combinatorial optimization problem; genetic algorithms; metaheuristics; memetic algorithm.

1 Introduction Block design generation is a conspicuous combinatorial problem of great complexity [26]. Among the different variants of the problem, the so-called Balanced Incomplete Block Designs (BIBDs) –defined as an arrangement of v distinct objects into b blocks such that each block contains exactly k distinct objects, each object occurs in exactly r different blocks, and every two distinct objects occur together in exactly λ blocks (for k,r, λ > 0)– is a popular one. The construction of BIBDs was initially attacked in the area of experiment design [12, 13]; however, nowadays BIBD can be applied to a variety of fields such as cryptography [3] and coding theory [11], among others. BIBD generation is a NP-hard problem [5] that provides an excellent benchmark since it is scalable and has a wide variety of problem instances, ranging from easy instances to very difficult ones. The scalability of the problem as well as its difficulty make it an adequate setting to test the behavior of different techniques/algorithms. As it will be discussed in Sect. 2.2, complete methods (including exhaustive search) have been applied to the problem although this remains intractable even for designs of relatively small size [9]. The application of metaheuristics thus seems to be more appropriate to attack larger problem instances. Indeed, some work in this line [16, 19] has provided evidence of the potential of heuristic approaches –in particular local search, which seems to outperform some genetic-algorithm approaches– for this problem. In this work we approach the BIBD problem from a memetic perspective, and show how a populationbased approach can be synergistically combined with a local search method, thus providing a powerful solver capable of outperforming the remaining approaches, finding solutions to instances that the latter could not solve.

Received Sep 10, 2010 / Accepted Nov 5, 2010

Rodríguez Rueda, Cotta, and Fernández Leiva / A Memetic Algorithm for Designing Balanced Incomplete Blocks. IJCOPI Vol. 2, No. 1, Jan-April 2011, pp. 14-22

2 Background This section provides a formulation of the problem and an overview of how it has been tackled in the literature.

2.1 Formulation A BIBD can be specified by five parameters ‹v,b,r,k,λ›. Using this notation, a ‹v,b,r,k,λ›-BIBD problem consists of dividing a set of v objects into b subsets of k < v objects each, such that each object in v belongs to r different subsets and any pair of objects in v share membership to exactly b < λ subsets. A standard way of representing the solution to such a problem is in terms of its incidence matrix Μ≡{mij}v×b, which is a v×b binary matrix where mij ∈ {0,1} is equal to 1 if the i th object is contained in the j th block, and 0 otherwise; thus, it is easy to see that a matrix representing a feasible solution has exactly r ones per row, k ones per column, and the scalar product of any pair of different rows is λ. Figure 1 shows configurations of the incidence matrix Μ representing possible solutions to a ‹8,14,7,4,3›-BIBD and a symmetric (i.e., b = v) ‹7,7,3,3,1›-BIBD, respectively. 0 1 0 0 1 0 1 1

0 1 1 0 0 1 1 0

0 0 1 0 1 0 1 1

1 1 1 0 0 0 0 1

0 1 1 1 1 0 0 0

1 0 1 1 1 0 0 0

1 1 0 1 0 0 1 0

1 0 0 1 0 0 1 1

0 0 1 1 0 1 1 0

0 1 0 1 0 1 0 1

0 0 0 1 1 1 0 1

1 0 0 0 1 1 1 0

1 0 1 0 0 1 0 1

1 1 0 0 1 1 0 0

0 1 1 0 0 1 0

1 0 1 0 1 0 0

0 0 1 1 0 0 1

1 1 0 0 0 0 1

0 0 0 0 1 1 1

1 0 0 1 0 1 0

0 1 0 1 1 0 0

Fig. 1. (Left) a ‹8,14,7,4,3›-BIBD; (Right) a ‹7,7,3,3,1›-symmetric BIBD Notice that the five parameters defining a ‹v,b,r,k,λ›-BIBD are interrelated and satisfy the following two relations: bk = vr and λ(v-1) = r(k-1). This means that we could just define an instance by using three parameters ‹v,k,λ›. and compute b and r in terms of the former three. However, although these relations restrict the set of admissible parameters for a BIBD, such an admissibility is a necessary yet not sufficient condition to guarantee the existence of a BIBD [4, 7].

2.2 Related work The BIBD problem has been tackled by a number of different techniques in the literature with different success. A survey of known results can be found in [26]. Early approaches were based on complete methods based on mathematical programming [24, 10] or constraint programming [8, 18, 14] among others [25, 22]. Regarding constraint programming approaches, it is worth mentioning that one of the key points of interest of the problem is it symmetrical nature (i.e., rows and columns can be permuted and objects can be relabeled). In this sense, Puget [18] proposed to combine methods for symmetry breaking in order to solve the problem, and Meseguer and Torras [14] explored two strategies (namely, a heuristic for variable selection and a domain pruning procedure) for exploiting the symmetry of the problem. We will return to the symmetry issue later on. More recently, heuristic methods have been also applied to the problem. Thus, the generation of BIBDs is formulated by Bofill et al. [1] as a combinatorial optimization problem tackled with a neural network. A simulated annealing algorithm endowed with this neural network (NN-SA) was shown to offer better performance than an analogous hybridization with mean field annealing. Later, these results were improved by Prestwich [16, 17], using a constrained local search (CLS) approach that considered different schemes for adding symmetry breaking constraints. In turn, these results have been further improved by Rodríguez et al. in a recent work [19]. Rodríguez et al. have considered the use of both local search methods (hill climbing and tabu search) and population-based techniques (genetic algorithms) for the BIBD. Two different neighborhood structures were defined, based on bit-fipping and position-swapping respectively. It was shown that the swap-based neighborhood was overly superior to the flip-based neighborhood, and that tabu search (TS) offered the best performance, being not only capable of beating hill climbing and genetic algorithms in more than 78% of the instances, but also solving more instances than NN-SA and CLS. Recently Yokoya and Yamada [6] have proposed a

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Rodríguez Rueda, Cotta, and Fernández Leiva / A Memetic Algorithm for Designing Balanced Incomplete Blocks. IJCOPI Vol. 2, No. 1, Jan-April 2011, pp. 14-22 mathematical model based on the constraints that a valid design for a BIBD has to satisfy; they consider a tabu search-based constructive algorithm and use the commercial linear programming solver CPLEX for the evaluation of candidate solutions. The basic process consists of building the block by adding, in each iteration, a new row that satisfies all the constraints with respect to the rows previously added.

3 Memetic Approaches to the ‹v,b,r,k,λ›-BIBD problem Memetic algorithms (MAs) constitute a metaheuristic optimization paradigm based on the systematic exploitation of knowledge about the problem being solved, and the synergistic combination of ideas taken from other population-based and trajectory-based metaheuristics. This section describes the particular incarnation of MAs considered in this work, whose more prominent features are the use of an embedded local-searcher based on TS, as well as the utilization of a heuristic recombination mechanism. Before entering into these components of the MA, let us firstly discuss its general structure in next subsection.

3.1 MA Structure Our MA uses a genetic algorithm (GA) as an underlying population-based search mechanism. This GA is not a mere diversification device for multi-start local search though. On the contrary, it incorporates intensifying components (namely the heuristic recombination operator) which play an important role in the discovery of new improved solutions. The general scheme of the MA is depicted in Algorithm 1. The input of this algorithm consists of the problem parameters (i.e., v, b, r, k, λ), as well as the algorithm parameters, i.e, the genetic operators and their associated parameters (such as, e.g., application rates); the output of the algorithm will be the best individual found during the search process. As it can be seen, at a high-level the MA resembles a steady state GA. Selection is done using binary tournament for breeding and replacement of the worst individual in the population. To keep diversity in the population, duplicated solutions are not accepted in the population, and a re-starting mechanism is introduced to re-activate the search whenever stagnation takes place. Algorithm 1: Pseudo code of the memetic algorithm. 1 begin 2 for i ← to popsize do 3 pop[i] ← GenerateMatrix(v,b,r); 4 Evaluate(pop[i]); 5 endfor while numEvals < maxEvals do 6 if rand < pX then 7 parent1 ← TournamentSelect(pop); 8 parent2 ← TournamentSelect(pop); 9 10 offspring ← Recombine(parent1 , parent2 ); 11 else 12 offspring ← TournamentSelect(pop); 13 endif 14 offspring ← Mutate(offspring, pM); 15 if rand < pLS then offspring ← LocalSearch(offspring); 16 Evaluate(offspring); 17 pop ← Replace(pop, offspring); 18 19 if stagnation (ηi ) then pop ← Restart(pop, ƒ %); 20 endw 21 end

This is done by keeping a fraction ƒ % of the top individuals in the current population, and refreshing the rest of the population with random individuals. This procedure is triggered after a number of ηi evaluations with no improvement of the current best solution. Mutation is performed by swapping two positions (with values 0 and 1) within the same row. Notice that this does not change the unitation of the row and –as it will be shown later– neither do the recombination operator nor the local search method. Hence row constraints must be enforced during the initialization stage.

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Rodríguez Rueda, Cotta, and Fernández Leiva / A Memetic Algorithm for Designing Balanced Incomplete Blocks. IJCOPI Vol. 2, No. 1, Jan-April 2011, pp. 14-22

3.2 Objective Function As it is often done when dealing with constraint satisfaction problems, BIBD constraints are mapped to scalar functions which indicate whether the corresponding constraint is violated or not, and in the former case which the degree of violation (i.e., the distance to feasibility) is. More precisely, for the general case of the instance ‹v,b,r,k,λ›, the following objective function (to be minimized) is defined: v

b

i =1

j =1

v −1

v

f 〈 v, b , r , k , λ 〉 (M ) = ∑ϕ ir ( M ) + ∑ϕ ′jk ( M ) + ∑ ∑ ϕij′′λ (M ) i =1 j = i +1

(1)

where b

v

b

j =1

i =1

k =1

ϕir ( M ) = r − ∑mij ;ϕ jk '(M ) = k − ∑mij ;ϕijλ '′( M ) = λ − ∑mik m jk

(2)

If the instance is satisfiable, a global optimum is a configuration M * such that f 〈 v ,b , r , k ,λ 〉 ( M * ) = 0 .

3.3 Recombination Operator Given the representation of solutions chosen, a simple recombination operator based on uniform crossover would work, provided it uses rows rather than bits as the basic unit of information for building the offspring (otherwise, given the mutation operator used by the algorithm, restoring the unitation of a row to the target value would be much more difficult, requiring some lucky mutations and crossovers; a simple bit-flip mutation could be used, but its combination with plain bitwise uniform crossover was shown in [19] to perform worse than the swap-based counterpart). We will refer to this recombination operator as UX in the following. There are two issues with this operator. The first one is its blindness: it randomly shuffles the rows of the parents, which can be good as a diversifying strategy. However, harder problem instances may require a more intensive approach, e.g., based on a smart choice of the rows to build the offspring. The second issue is the fact that UX is not symmetry-aware: two different matrices can represent the same solution via a permutation of the rows, yet UX would treat them as different solutions and blindly mix the rows to produce an infeasible solution (i.e., a solution with repeated rows). Of course, the solutions need not be identical for this problem to take place. To account for these two issues, a greedy recombination operator (GrX) has been devised. Its functioning is shown in Algorithm 2; in this case the parents being recombined are supplied as input, and a child solution is obtained as output. Algorithm 2: Pseudo code of the recombination algorithm. 1 begin 2 rowSet ← 0; 3 for i ← to numParents do InsertRows(parenti, rowSet); 4 child.row[1] ←ExtractRandom(rowSet); 5 for i ← 2 to v do 6 for each row in rowSet do Test(child, row, i); 7 child.row[i] ← Best(rowSet); 8 endfor 9 endfor 10 end

As it can be seen, GrX starts by creating a set with all available rows in the parents and then (if there are enough different rows; otherwise standard UX is invoked), selecting at random an initial row, and subsequently testing all available rows, picking the one which violates less scalar-product constraints. Of course this incurs in a computational cost that must be accounted for. To do this, notice that scalar products can be stored in a cache once computed, and hence each time a new row is to be added only scalar products with respect to the row selected in the previous step must be computed. If there are ρ rows in the set, the number of computations is (ρ−1)+…+(ρ −v+1)=(ρ −v/2)(v−1). Since the total number of scalar products in a full evaluation is v(v−1)/2, the cost of GrX is in each case equivalent to 2ρ/v−1 evaluations. This is included in the total computational toll of the algorithm for comparison purposes.

3.4 Local Search Method The local search method is based on the TS approach in [19]. This algorithm works on the swap neighborhood of the current solution, which is fully explored to select the best non-tabu move. A move is tabu if it attempts to reverse a previous swap mij ↔ mik stored in the tabu list. To prevent cycling, the tabu

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Rodríguez Rueda, Cotta, and Fernández Leiva / A Memetic Algorithm for Designing Balanced Incomplete Blocks. IJCOPI Vol. 2, No. 1, Jan-April 2011, pp. 14-22 tenure –i.e., the number of iterations tabu move stays in the list– is chosen randomly in the range [β/2,3β/2], where β = vbr. The tabu status of a move can be overridden if the aspiration criteria is fulfilled, namely, finding a solution better than the current best solution found so far. When used as a stand-alone technique, TS intensifies the search returning to the best-so-far solution if no improvement takes place after ηi evaluations. No intensification mechanism is used when TS is invoked within the MA, since the depth of search is kept low (fixed to (vb)2 neighbors computed in this case) to prevent a high computational cost. This cost is computed by considering that the size of the neighborhood is vr(b-r), and that evaluating a neighbor requires computing v-1 scalar products and 2 column constraints.

4 Experimental results The experiments has been done on 86 instances taken from [16, 19, 1] where vb ≤ 1000 and k ≠ 3. This corresponds to the hardest instances reported therein, since the cases where k = 3 were easily solvable. We refer to these works for a listing of the particular ‹v,b,r,k,λ› parameters defining each instance. All algorithms have been run 30 times per problem instance. When used as an stand-alone algorithm TS is limited to explore ηv = 2.107 neighbors. The GAs and MAs consider the equivalent number of full evaluations in each case (see Sect. 3.4). The number of evaluations without improvement to trigger intensification in TS or re-starting in GA is ηi = ηv/10. Other parameters of the GA are population size=100, crossover and mutation probabilities pX=.9 and pM = 1/ℓ (where ℓ = vb is the size of individuals) respectively, and ƒ % = 10% . In the case of the MA, pLS is set to 1/2000 (preliminary experiments indicate this setting provides a good tradeoff between the fraction of the computational cost devoted to local search and the quality of solutions attained). Table 1. Results of TS (30 runs per instance). x , σ, B and S denote, respectively, the fitness average value, the standard deviation, the best obtained result, and the number of times that a problem instance solution is obtained ID 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

x ±σ 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 2.40 ± 1.96 0.00 ± 0.00 0.37 ± 1.97 0.00 ± 0.00 5.20 ± 0.98 2.57 ± 3.48

B

S

ID

0

30

23

0

30

24

0

30

25

0

30

26

0

30

27

0

30

28

0

30

29

0

30

30

0

30

31

0

30

32

0

30

33

0

30

34

0

30

35

0

30

36

0

30

37

0

30

38

0

12

39

0

30

40

0

29

41

0

30

42

4

0

43

0

18

44

x ±σ 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 7.00 ± 1.57 10.27 ± 1.36 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 7.33 ± 1.70 4.83 ± 0.97 0.00 ± 0.00 39.50 ± 22.56 0.00 ± 0.00 0.00 ± 0.00 13.17 ± 1.85 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 17.10 ± 1.27 46.77 ± 28.39

B

S ID

0 30 45 0 30 46 0 30 47 0 30 48 4

0 49

8

0 50

0 30 51 0 30 52 0 30 53 0 30 54 4

0 55

4

0 56

0 30 57 0

6 58

0 30 59 0 30 60 8

0 61

0 30 62 0 30 63 0 30 64 14 0 65 0

8 66

x ±σ 2.13 ± 2.00 29.80 ± 1.72 0.27 ± 1.00 6.43 ± 1.02 0.00 ± 0.00 3.90 ± 1.16 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 21.10 ± 1.33 0.00 ± 0.00 36.13 ± 1.67 40.00 ± 0.00 92.77 ± 2.95 32.10 ± 1.54 0.00 ± 0.00 0.00 ± 0.00 26.60 ± 1.20 3.07 ± 1.69 5.17 ± 1.04 0.00 ± 0.00 0.00 ± 0.00

B

S

ID

0 14

67

25 0

68

0 28

69

4

0

70

0 30

71

0

2

72

0 30

73

0 30

74

0 30

75

18 0

76

0 30

77

33 0

78

40 0

79

86 0

80

28 0

81

0 30

82

0 30

83

24 0

84

0

7

85

4

0

86

x ±σ 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 14.93 ± 1.46 20.70 ± 1.27 30.67 ± 1.81 4.40 ± 0.80 0.00 ± 0.00 2.00 ± 2.00 10.23 ± 1.43 37.13 ± 1.67 0.00 ± 0.00 0.00 ± 0.00 5.33 ± 1.16 3.80 ± 11.40 119.70 ± 2.99 142.23 ± 3.96 0.00 ± 0.00 28.07 ± 1.55 58.17 ± 2.44

B

S

0 30 0 30 0 30 10 0 17 0 27 0 4

0

0 30 0 15 8

0

33 0 0 30 0 30 4

0

0 27 115 0 132 0 0 30 26 0 51 0

0 30 0 30

Tables 1-3 show detailed results for each instance. As a summary of performance, TS is capable of solving 59 instances (68.60%), more than both GAUX and GAGrX which solve 46 (53.49%) and 48 (55.81%) respectively. Notice nevertheless that the GA featuring the heuristic recombination operator performs better, not just in number of instances solved but also in the average quality of solutions: GAUX performs better (with statistical significance at the standard 0.05 level using a Wilcoxon ranksum test) than GAGrX in 1.16% of instances, whereas the latter beats the former in 59.30% of instances. The MAs are however capable of performing comparably or even better than TS (and hence better than the GAs). While MAUX is capable of solving as many instances as TS, MAGrX solves 4 more for a total of 73.26% of

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Rodríguez Rueda, Cotta, and Fernández Leiva / A Memetic Algorithm for Designing Balanced Incomplete Blocks. IJCOPI Vol. 2, No. 1, Jan-April 2011, pp. 14-22 instances. Regarding the median quality of solutions, MAGrX is beaten by TS in 2.33% of instances. As to MAUX vs TS, they beat each other in 8.14% of instances. Quite interestingly, MAUX beats MAGrX in 4.65% of instances. These percentages are quite small given the number of instances and do not appear to indicate any MA exhibits a clear trend of superiority in median quality over other memetic techniques. It is however interesting to note again the higher success rate of MAGrX, which can be interpreted in light of GrX being a heavy operator whose usage is computationally costly, expending resources that in not so-hard instances can be better devoted to LS, but also giving the algorithm an edge when it comes to solve these very hard instances. Table 2. Results of GAs (30 runs per instance) using UX (GA UX ) and GrX (GA GrX ) GA UX

ID 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43

x ±σ 0.00 ± 0.00 0.33 ± 1.80 0.00 ± 0.00 0.27 ± 1.00 0.00 ± 0.00 0.80 ± 1.60 0.00 ± 0.00 7.47 ± 7.06 3.70 ± 1.77 11.00 ± 6.44 4.20 ± 2.69 0.40 ± 1.20 5.53 ± 5.60 0.67 ± 1.49 0.00 ± 0.00 3.80 ± 1.85 7.93 ± 1.71 1.70 ± 1.95 43.77 ± 3.89 3.37 ± 1.94 10.83 ± 2.30 15.93 ± 3.05 3.33 ± 1.94 9.37 ± 9.66 0.00 ± 0.00 0.53 ± 1.36 15.83 ± 2.68 20.63 ± 2.98 2.23 ± 2.11 0.27 ± 1.00 3.27 ± 2.06 5.10 ± 2.07 16.00 ± 2.25 12.73 ± 2.84 5.97 ± 1.68 83.57 ± 4.86 1.10 ± 1.83 0.10 ± 0.54 25.47 ± 3.20 0.67 ± 1.49 1.13 ± 1.89 2.67 ± 2.07 33.50 ± 4.05

GA GrX

B

S

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 36 0 7 11 0 0 0 0 10 17 0 0 0 0 12 6 0 73 0 0 16 0 0 0 26

30 29 30 28 30 24 30 14 5 7 8 27 15 25 30 5 0 17 0 7 0 0 7 15 30 26 0 0 14 28 8 3 0 0 1 0 22 29 0 25 22 11 0

x ±σ 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 1.20 ± 3.60 0.13 ± 0.72 3.73 ± 5.72 0.97 ± 1.76 0.00 ± 0.00 1.20 ± 3.60 0.00 ± 0.00 0.00 ± 0.00 2.00 ± 2.00 4.93 ± 1.31 0.00 ± 0.00 29.53 ± 8.75 0.27 ± 1.00 7.33 ± 1.72 12.10 ± 3.17 0.27 ± 1.00 2.63 ± 6.80 0.00 ± 0.00 0.00 ± 0.00 10.53 ± 1.84 14.00 ± 2.68 0.40 ± 1.23 0.00 ± 0.00 1.07 ± 1.77 3.50 ± 1.86 12.57 ± 1.69 10.07 ± 1.41 1.63 ± 2.01 67.77 ± 3.74 0.13 ± 0.72 0.00 ± 0.00 20.23 ± 1.89 0.00 ± 0.00 0.13 ± 0.72 0.27 ± 1.00 24.83 ± 1.97

GA UX

B 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 6 6 0 0 0 0 8 7 0 60 0 0 15 0 0 0 20

S ID 30 30 30 30 30 30 30 27 29 21 23 30 27 30 30 15 1 30 1 28 0 1 28 26 30 30 0 0 27 30 22 6 0 0 18 0 29 30 0 30 29 28 0

44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86

x ±σ 92.00 ± 5.92 9.73 ± 2.22 41.50 ± 4.36 8.07 ± 1.91 15.23 ± 3.29 0.13 ± 0.72 14.43 ± 2.91 3.87 ± 2.14 7.67 ± 2.36 1.50 ± 2.00 39.27 ± 4.08 2.40 ± 2.30 52.93 ± 4.37 37.77 ± 2.16 142.77 ± 7.54 52.70 ± 4.57 0.53 ± 1.36 36.17 ± 7.53 48.23 ± 6.65 11.17 ± 2.15 15.03 ± 2.36 2.43 ± 2.17 6.43 ± 2.45 1.10 ± 2.17 3.07 ± 2.49 1.87 ± 2.22 27.00 ± 3.85 35.50 ± 4.72 58.17 ± 6.17 14.30 ± 2.70 0.27 ± 1.00 16.20 ± 3.54 21.83 ± 3.42 70.43 ± 6.51 10.00 ± 2.82 7.87 ± 2.43 22.63 ± 4.23 72.73 ± 23.97 169.80 ± 8.77 216.73 ± 10.87 2.97 ± 2.63 46.50 ± 5.71 92.20 ± 9.47

19

GA GrX

B

S

79 4 33 4 7 0 8 0 4 0 32 0 45 31 124 42 0 0 29 6 11 0 0 0 0 0 18 27 48 9 0 10 17 60 6 4 14 0 147 198 0 32 80

0 0 0 0 0 29 0 5 0 19 0 14 0 0 0 0 26 1 0 0 0 13 2 23 11 17 0 0 0 0 28 0 0 0 0 0 0 1 0 0 12 0 0

x ±σ 81.33 ± 4.53 7.57 ± 1.82 38.47 ± 2.38 6.60 ± 1.43 12.63 ± 2.02 0.00 ± 0.00 10.10 ± 1.80 2.07 ± 1.95 4.27 ± 2.00 0.00 ± 0.00 30.93 ± 2.16 0.67 ± 1.49 48.43 ± 2.36 40.03 ± 0.18 119.20 ± 6.34 44.47 ± 3.15 0.00 ± 0.00 36.23 ± 3.00 37.87 ± 3.36 10.37 ± 1.60 13.33 ± 1.94 0.50 ± 1.28 1.80 ± 2.09 0.00 ± 0.00 2.63 ± 2.06 0.47 ± 1.20 26.27 ± 2.90 33.23 ± 3.27 46.40 ± 3.30 13.40 ± 2.56 0.00 ± 0.00 11.20 ± 2.24 20.40 ± 2.72 57.07 ± 3.57 8.33 ± 2.23 7.47 ± 1.87 15.60 ± 2.01 74.33 ± 18.29 159.10 ± 5.22 187.20 ± 9.49 0.87 ± 1.59 45.80 ± 2.99 82.80 ± 4.92

B

S

70 4 33 4 9 0 6 0 0 0 27 0 45 40 101 36 0 31 30 6 10 0 0 0 0 0 20 28 39 6 0 6 13 50 4 4 11 15 143 166 0 40 74

0 0 0 0 0 30 0 14 4 30 0 25 0 0 0 0 30 0 0 0 0 26 17 30 11 26 0 0 0 0 30 0 0 0 0 0 0 0 0 0 23 0 0

Rodríguez Rueda, Cotta, and Fernández Leiva / A Memetic Algorithm for Designing Balanced Incomplete Blocks. IJCOPI Vol. 2, No. 1, Jan-April 2011, pp. 14-22 Table 3. Results of MAs (30 runs per instance) using UX (MA UX ) and GrX (MA GrX ) MA UX

ID 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43

x ±σ 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 1.60 ± 1.96 0.00 ± 0.00 0.53 ± 2.87 0.00 ± 0.00 3.73 ± 1.61 0.20 ± 1.08 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 6.03 ± 1.22 8.87 ± 1.82 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.13 ± 0.72 7.13 ± 1.52 5.20 ± 1.11 0.00 ± 0.00 31.67 ± 24.20 0.00 ± 0.00 0.00 ± 0.00 12.30 ± 1.29 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 18.23 ± 1.80

MA GrX

B

S

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 4 0 0 0 0 4 4 0 0 0 0 10 0 0 0 15

30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 18 30 29 30 4 29 30 30 30 30 0 0 30 30 30 29 0 0 30 10 30 30 0 30 30 30 0

x ±σ 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 1.33 ± 1.89 0.00 ± 0.00 1.07 ± 3.99 0.00 ± 0.00 4.43 ± 1.50 0.40 ± 1.58 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 6.70 ± 1.35 9.90 ± 1.68 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 7.63 ± 1.70 5.03 ± 1.38 0.00 ± 0.00 41.00 ± 21.08 0.00 ± 0.00 0.00 ± 0.00 12.83 ± 1.86 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 17.77 ± 1.63

MA UX

B

S

ID

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 6 0 0 0 0 4 0 0 0 0 0 10 0 0 0 14

30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 20 30 28 30 2 28 30 30 30 30 0 0 30 30 30 30 0 1 30 5 30 30 0 30 30 30 0

44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86

x ±σ 42.70 ± 26.34 2.30 ± 2.02 28.07 ± 2.66 0.80 ± 1.60 6.63 ± 1.60 0.00 ± 0.00 4.87 ± 1.12 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 19.73 ± 1.81 0.00 ± 0.00 35.97 ± 2.07 40.00 ± 0.00 93.80 ± 2.97 31.60 ± 2.40 0.00 ± 0.00 5.27 ± 9.04 26.80 ± 2.02 3.00 ± 1.84 5.83 ± 1.39 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 15.63 ± 1.78 21.53 ± 1.96 30.83 ± 2.46 5.20 ± 1.25 0.00 ± 0.00 3.97 ± 1.56 10.57 ± 2.22 38.17 ± 2.03 0.40 ± 1.20 0.00 ± 0.00 6.37 ± 1.60 11.33 ± 17.32 120.23 ± 6.13 144.10 ± 4.59 0.00 ± 0.00 29.30 ± 2.79 59.03 ± 2.96

MA GrX

B

S

0 0 20 0 4 0 4 0 0 0 16 0 32 40 88 27 0 0 23 0 4 0 0 0 0 0 13 18 26 4 0 0 6 34 0 0 4 0 109 135 0 21 54

7 13 0 24 0 30 0 30 30 30 0 30 0 0 0 0 30 22 0 8 0 30 30 30 30 30 0 0 0 0 30 3 0 0 27 30 0 21 0 0 30 0 0

x ±σ 48.03 ± 23.97 2.53 ± 1.93 29.00 ± 2.32 1.47 ± 1.93 6.67 ± 1.27 0.00 ± 0.00 4.50 ± 1.61 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 21.33 ± 2.26 0.00 ± 0.00 37.20 ± 2.23 40.00 ± 0.00 92.07 ± 3.89 32.93 ± 2.10 0.00 ± 0.00 7.40 ± 10.47 26.97 ± 2.12 3.63 ± 1.74 5.80 ± 1.85 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 15.13 ± 2.11 21.33 ± 1.64 31.73 ± 2.53 5.10 ± 1.01 0.00 ± 0.00 2.90 ± 2.20 10.13 ± 1.73 39.20 ± 1.94 0.27 ± 1.00 0.00 ± 0.00 6.60 ± 1.85 13.93 ± 18.31 120.23 ± 5.67 143.13 ± 5.30 0.00 ± 0.00 29.10 ± 2.10 59.17 ± 3.94

B

S

0 0 24 0 4 0 0 0 0 0 16 0 32 40 83 28 0 0 22 0 0 0 0 0 0 0 11 17 28 4 0 0 6 36 0 0 0 0 109 133 0 23 51

5 11 0 19 0 30 2 30 30 30 0 30 0 0 0 0 30 20 0 5 1 30 30 30 30 30 0 0 0 0 30 10 0 0 28 30 1 19 0 0 30 0 0

5 Conclusions and Future Work We have explored the application of memetic algorithms to the BIBD problem with very positive results. The synergy in the combined use of a heuristic recombination operator and a TS-based local searcher is supported by the empirical results: algorithms with GrX solve more instances than their counterparts with UX, and MAs also outperform their GA counterparts. It is nevertheless interesting to note that MAs are complex techniques whose performance rests on a number of tradeoffs such as the depth and intensity of local search (check, e.g., [21]) and the cost of other heuristic add-ons (e.g., the recombination operator in this case). A full sensitivity analysis of the relevant parameters is a worthy line of future research. Results

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Rodríguez Rueda, Cotta, and Fernández Leiva / A Memetic Algorithm for Designing Balanced Incomplete Blocks. IJCOPI Vol. 2, No. 1, Jan-April 2011, pp. 14-22 regarding the local search intensity are currently available, and these will be complemented with a thorough analysis of other factors such as its interaction with multi-parent versions of the recombination operator. Also, as already mentioned, Yokoda and Yamada [6] have recently proposed a very effective constructive approach to tackle the BIBD problem. As future work we will consider the attainment of hybrid cooperative strategies involving the aforementioned approach with and thechniques described in this work. Let us finally mention that there exist other variants of the BIBD problem, e.g., partially BIBDs, randomized block designs, pairwise balanced designs, regular graph designs, and maximally balanced maximally uniform designs, among others [20, 15, 23, 2]. We plan to attack some of these variants in the future as well.

Acknowledgements The second and third authors were partially supported by Spanish MICINN project under contract TIN2008-05941 (NEMESIS project).

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