An Adaptive Local Search Based Genetic Algorithm for Solving Multi ...

An Adaptive Local Search Based Genetic Algorithm for Solving Multi-objective Facility Layout Problem Kazi Shah Nawaz Ripon, Kyrre Glette, Mats Høvin, and Jim Torresen Department of Informatics, University of Oslo, Norway {ksripon,kyrrehg,matsh,jimtoer}@ifi.uio.no

Abstract. Due to the combinatorial nature of the facility layout problem (FLP), several heuristic and meta-heuristic approaches have been developed to obtain good rather than optimal solutions. Unfortunately, most of these approaches are predominantly on a single objective. However, the real-world FLPs are multiobjective by nature and only very recently have meta-heuristics been designed and used in multi-objective FLP. These most often use the weighted sum method to combine the different objectives and thus, inherit the well-known problems of this method. This paper presents an adaptive local search based genetic algorithm (GA) for solving the multi-objective FLP that presents the layouts as a set of Pareto-optimal solutions optimizing both quantitative and qualitative objectives simultaneously. Unlike the conventional local search, the proposed adaptive local search scheme automatically determines whether local search is used in a GA loop or not. The results obtained show that the proposed algorithm outperforms the other competing algorithms and can find near-optimal and nondominated solutions by optimizing multiple criteria simultaneously. Keywords: Adaptive local search, Multi-objective facility layout problem, Pareto-optimal solution, Multi-objective evolutionary optimization.

1 Introduction The objective of the FLP is the optimum arrangement of facilities/departments on a factory floor, such that the costs associated with projected interactions between these facilities can be optimized. These costs may reflect minimizing the total cost of transporting materials (material handling cost) or maximizing the adjacency requirement between the facilities. In essence, FLP can be considered as a searching or optimization problem, where the goal is to find the best possible layout. In fact, the problem falls into the category of multi-objective optimization problems (MOOPs). On one hand, many researchers handle the problem as one of optimizing product flow by minimizing the total material handling costs. On the other hand, it can also be considered as a design problem by considering qualitative information on how different activities, like safety, management convenience, and assistant power supply are related from the viewpoint of adjacency. Facility layout planning plays an important role in the manufacturing process and has a serious impact on a company’s profitability. A good layout will help any company to improve its business performance and can reduce up to 50% of the total operating expenses [1]. Although the FLP is an inherently MOOP, it has traditionally K.W. Wong et al. (Eds.): ICONIP 2010, Part I, LNCS 6443, pp. 540–550, 2010. © Springer-Verlag Berlin Heidelberg 2010

An Adaptive Local Search Based GA for Solving Multi-objective FLP

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been solved considering only one objective, either qualitative or quantitative feature of the layout. Quantitative approaches involve primarily the minimization of material handling (MH) costs between various departments. Qualitative goals aiming at placing departments that utilize common materials, personnel, or utilities adjacent to one another, while separating departments for the reasons of safety, noise, or cleanliness. In general, minimization of the total MH costs is often used as the optimization criterion in FLP. However, the closeness rating, hazardous movement or safety, and the like are also important criteria in FLP. Based on the principle of MOOP, obtaining an optimal solution that satisfies all the objectives is almost impossible. It is mainly for the conflicting nature of objectives, where improving one objective may only be achieved when worsening another objective. Surprisingly, however, there is a little attention to the study of multi-objective FLP. Although dealing with multiple objectives has received attention over the last few years [2], [3], these approaches are still considered limited, and mostly dominated by the unrealistic weighted sum method. In this method, multiple objectives are added up into a single scalar objective using weighted coefficients. However, there are several disadvantages of this technique [4]. Firstly, as the relative weights of the objectives are not exactly known in advance and cannot be pre-determined by the users, the objective function that has the largest variance value may dominate the multi-objective evaluation. As a result, inferior non-dominated solutions with poor diversity will be produced. Secondly, the user always has to specify the weight values for objectives and sometimes these will not have any relationship with the importance of the objectives. Thirdly, a single solution is obtained at one time. Also, the layout designer based on his/her past experience randomly selects the layout having multiple objectives. This restricts the designing process completely designer dependent and thus, the layout varies from designer to designer. To overcome such difficulties, Pareto-optimality [5] has become an alternative to the classical weighted sum method. Previously, we proposed a Pareto-optimal based genetic approach for solving the multi-objective FLP [6]. Later we extend this approach by incorporating local search [7]. To our best knowledge, these are the first and only available approaches to find Pareto-optimal layouts for the multi-objective FLP. Due to the combinatorial nature of the FLP, optimal algorithms are often found not to be suited for large FLP instances. Thus, the interest lies in the application of heuristic and meta-heuristic methods to solve large problems. Among these approaches, GA seems to become quite popular in solving FLP [3]. GA is an efficient, adaptive and robust search process for optimizing general combinatorial problems and therefore is a serious candidate for solving the FLP. However, there is also a limitation in applying GA to multi-objective FLP - GA can do global search in the entire space, but there is no way for exploring the search space within the convergence area generated by the GA loop. Therefore, it is sometimes insufficient in finding optimal solution for complex and large search space, which is very usual in real-world FLP. In certain cases, GA performs too slowly to be practical. A very successful way to improve the performance of GA is to hybridize it with local search techniques. Local search techniques are used to refine the solutions explored by GA by searching its vicinity for the fittest individuals and replacing it if a better one is found. However, hybridization with local search may degrade the global search ability of a multi-objective GA and require much computational time than conventional GA [8]. This is because most of the computation time is spent by local search.

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This paper proposes an adaptive local search based GA for solving the multiobjective FLP to demonstrate the importance of finding a good balance between GA and local search technique. The proposed adaptive local search scheme automatically determines whether local search is used in a GA loop or not. The experimental results are compared to the results found by several existing and meta-heuristic approaches, and our previously proposed two Pareto-optimal based multi-objective evolutionary approaches [6], [7]. Computational experience shows that the proposed approach performs very well to improve the best fitness value, average number of generations, and convergence behaviors of average fitness values. Concurrently, it can find a set of Pareto-optimal layouts optimizing both quantitative and qualitative objectives simultaneously throughout the entire evolutionary process. The rest of the paper is organized as follows. In Section 2, the importance of Pareto-optimality in the FLP is introduced. Section 3 presents the related works. The justification of adaptive local search and the implementation details has been discussed in Section 4. Section 5 is to design the proposed approach. The proposed approach has been experimentally verified in Section 6. Finally, chapter 7 provides a conclusion of the paper.

2 Importance of Pareto-Optimality in FLP Real-life scientific and engineering problems typically require the search for satisfactory solution for several objectives simultaneously. It is also common that conflicts exist among the objectives. In such problems, the absolute optimal solution is absent, and the designer must select a solution that offers the most profitable trade-off between the objectives instead. The objective of such MOOPs is to find a set of Pareto-optimal solutions, which are the solutions to which no other feasible solutions are superior in all objective functions. Historically, FLPs have been solved for either quantitative or qualitative goodness of the layout. In general, real-life FLPs are multi-objective by nature and they require the decision makers to consider a number of criteria involving quantitative and qualitative objectives before arriving at any conclusion. Thus, instead of offering a single solution, giving options and letting decision makers choose between them based on the current requirement is more realistic and appropriate.

3 Related Works Several researches have been done in the FLP for the past few decades. These approaches can be divided into exact algorithms and heuristic algorithms. The exact methods such as the branch-and-bound and cutting plane algorithm have been successfully applied to FLP when the number of facilities is less than 15. However, problem of size larger than 15 cannot be solved optimally in reasonable time. Recently, meta-heuristic approaches such as simulated annealing (SA), GA, tabu search (TS) and ant colony optimization (ACO) have been successfully applied to solve large FLP. Among these approaches, GA has been shown to be effective in solving FLPs, especially in solving the large-scale problems [6]. However, research in this area typically focused on single objective. To date, there are only a few attempts to tackle the multi-objective FLP using GA. However, they used weighted sum method. Thus, they ignored the prospects of Pareto-optimal solutions in solving the multi-objective FLP. Interested readers should consult [3], [9] for a detailed review. In recent time,


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various methods for hybridizing GA using conventional local search techniques have been suggested [10]. The combination of local search heuristics and GA is also a promising approach for finding near-optimal solutions to the FLPs [7], [11]. However, most of the local search techniques used in these approaches are applied to all individuals of the population within each generation of a GA without any generalization or analysis with respect to their convergence characteristics and reliabilities. For improving these limitations in the application of local search, a local search technique with adaptive scheme can be a better alternative.

4 Adaptive Local Search in Multi-objective FLP Although GA has proven to be successful in producing quite good results when applied individually, near-optimal and top quality solutions seem to require the combined efforts of local search techniques and GA [10]. Local search algorithms are improvement heuristics that search in the neighborhood of the current solution for a better one until no further improvement can be made. If the search space is too large, GA has inherent difficulties to converge in the global optimum with an adequate precision. Local search techniques can remove this limitation by finding any local optimum with great precision, using information from the neighboring candidate solutions [12]. The synergy between both methods can therefore give rise to a family of hybrid algorithms, simultaneously global and precise. The GA globally explores the domain and finds a good set of initial estimates, while the local search further refines these solutions in order to locate the nearest, best solution. In order to apply local search, we have to specify an objective function to be optimized by the search. This specification is straightforward in a single objective optimization problem. Unfortunately, the situation in the MOOP is unclear since the number of such studies in the field has been small. A weighed sum of multiple objectives is often used for local search in hybrid multi-objective GA [8], which is not realistic. In this work, we extend the 1opt local search [12] for multi-objective FLP by incorporating domination strategy [5]. The general outline of this algorithm is given in Algorithm 1. Algorithm 1. 1-opt Local Search for MOFLP set fit1 = MH cost of the current chromosome 1

set fit 2 = CR score of the current chromosome for i = 1 to N do repeat select another gene j randomly such that i ≠ j construct a new layout by swapping i and j 1

find fit1 = MH cost of the new layout 2

find fit 2 = CR cost of the new layout 2

if the current layout is dominated by the new layout replace the current one with the new layout else continue with the current one end if until any improvement in the current layout end for

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In most studies, local search is applied to the solutions of each generation of GA. As a result, the local search technique has to examine a large number of solutions for finding a locally optimum solution from each initial solution generated by genetic operations. It is nothing but mere waste of CPU time. For decreasing the computation time spent by local search, we implemented an adaptive local search technique that will only search around the convergence area produced by GA loop instead of applying to all individuals. This technique helps generating new individuals having certain high fitness values like the superior individuals generated by GA. The proposed adaptive local search scheme is based on similarity coefficient method (SCM) [10] to consider the similarity of individuals of a GA population. We modify the basic idea of SCM for multi-objective FLP environment. We can calculate the similarity coefficient SCpq between two chromosomes p and q as follows:

SC pq =

(

∑ nk =1 ∂ f pk , f qk

)

(1) n Where, k is the index of location in the layout, fpk is the facility at location k in chromosome p, and n is the number of facilities.

⎧1, if f pk = f qk ∂ f pk , f qk = ⎨ ⎩0, otherwise

(

)

(2)

The average similarity coefficient ( SC ) for all individuals can be expressed as follows: SC =

∑ np−=q1 ∑ nq = p +1 SC pq

(3)

n Assuming that a pre-defined threshold value ( β ), the modified 1-opt local search method described in Algorithm 1 will be automatically invoked in a GA loop by the following condition: ⎪⎧apply local search to GA loop, if SC > β (4) ⎨ ⎪⎩apply GA alone, otherwise

5 The Proposed Approach Figure 1 presents the flowchart for a complete evolutionary cycle of the proposed multi-objective FLP using adapting local search. In this approach, we used the nondominated sorting genetic algorithm 2 (NSGA 2) [13] as the multi-objective GA. 5.1 Chromosome Representation and Genetic Operators

In this study, the FLP has been presented as a quadratic assignment problem (QAP), which assigns n equal area facilities to the same number of locations. An integer string representation is used to represent the layouts. The solution is represented as a string of length n, where n is the number of facilities. The integers denote the facilities and their positions in the string denote the positions of the facilities in the layout. For crossover and mutation, we follow the concept described by Suresh et. al. [14] and swap mutation, respectively. Their implementation details can be found in [6].


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Fig. 1. Flowchart for the multi-objective FLP using adaptive local search

5.2 Fitness Function

We separately apply material handling (MH) costs and closeness relationship (CR) among various departments as quantitative and qualitative objective respectively. They can be expressed in the following mathematical model: n n

n n

F1 = ∑ ∑ ∑ ∑ fik d jl X ij X kl i =1 j =1 k =1l =1

(5)

n n n n

F2 = ∑ ∑ ∑ ∑ Wijkl X ij X kl i =1 j =1k =1l =1

(6)

Subject to n

∑ X ij = 1, j=1,2,….,n

(7)

∑ X ij = 1, i =1 X tij = 0 or 1

(8)

i =1 n

i=1,2,….,n

(9)

⎧r , if locations j and l neighbours Wijkl = ⎨ ik (10) otherwise ⎩0, Where, i, k are facilities; j, l are locations in the layout; Xij : 1 or 0 variable for locating i at j; fik is the flow cost for unit distance from facility i to k; djl is the distance from location j to l; rik : CR value when departments i and k are neighbours with common boundary and n is the number of facilities in the layout. Here, the aim of the first fitness function is to minimize the MH costs (Eq.(5)). The second fitness function tries to maximize the adjacency requirement based on CR value (Eq.(6)).

6 Experimental Results and Analysis The proposed algorithm is tested for benchmark instances taken from published literature [6]. The problems are composed of 6, 8, 12, 15, 20, 30, 42, 72 and 100 facilities.

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We used the last digits to indicate the number of facilities in each problem. Very few benchmark problems are available for multi-objective FLP, particularly, in the case of CR score. Thus, we have ourselves created the test data sets for CR score where no such data exists. The experiments are conducted using 50 chromosomes and 40 generations for problems with up to 15 facilities; and 100 chromosomes and 80 generations for problems with more than 15 facilities. However, for justifying the convergence behaviour of the proposed adaptive local search, later we run the three multi-objective FLP approaches less than the mentioned generations. The probabilities of crossover, mutation and the pre-defined coefficient (β) are 0.9, 0.3, and 0.9 (90%), respectively. We use traditional tournament selection with tournament size of 2. Each benchmark problem is tested 30 times with different seeds. Then each of the final generations is combined and non-dominated sorting is performed to constitute the final nondominated solutions.

HU & Wang

Without LS

With LS

43 (47.5) 107 (118.8)

50.6 (50.6) 126.7 (126.7)

43 (43) 116 (116)

43 (43) 107 (107)

43 (43) 107 (107.8)

43 (43) 107 (107.75)

43 43 (43) 107 107 (107.34)

Proposed Adaptive LS

GESA

43 (43) 107 (107)

Best Known

TAA

43 (44.2) 107 (107)

FATE

43 (44.2) 107 (113.4)

DISCON

Biased Sampling

43 (44.2) 107 (110.2)

FLAC

CRAFT

43 (44.2) 109 8 (114.4) 6

H63-66

Problem (naug n) H63

Table 1. Comparison with existing algorithms for MH cost

43 (43) 107 (107)

(287) 289 289 287 287 287 (289.97) (289.36) (290.6) (292.57) (290.87) 573 617 578 583 575 585 597 660.8 596 575 575 575 575 15 575 (588.7) (632.6) (600.2) (600) (580.2) (585) (630.8) (660.8) (596) (575.18) (576.4) (677.01) (601.16) 1285 1285 1286 1286 1384 1319 1324 1304 1303 1376 1436.3 1414 1285 1285 (1288.13) 20 (1400.4) (1345) (1339) (1313) (1303) (1416.4) (1436.3) (1414) (1287.38) (1290.5) (1290.6) (1289.08) 3059 3244 3161 3148 3093 3079 3330 3390.6 3326 3062 3064 3062 3062 30 3062 (3079.05) (3267.2) (3206.8) (3189.6) (3189.6) (3079) (3436.4) (3390.6) (33269) (3079.32) (3075.1) (3081.02) (3081.07) 12

301 304 289 289 (317.4) (310.2) (296.2) (293)

289 295 326.2 314 (289) (322.2) (326.2) (314)

Since almost all FLP algorithms try to optimize single criteria only (mainly minimizing the MH cost), first we compare the MH costs obtained by our approach with the existing single objective approaches. The values provided in Table 1 show the MH cost for the best layouts by some existing algorithms. From this table, it can be easily found that performance of the proposed approach is superior or equivalent to other approaches. Most importantly, the incorporation of adaptive local search helps the algorithm to achieve the new best solutions for naug15 and naug30, and also to reduce the gaps between the best and the average MH costs. Table 2 shows the performance statistics of the proposed adaptive local search based multi-objective FLP approach with the Pareto-optimal based multi-objective approach with conventional local search [7] and without local search [6] in the context of MH cost and CR score. Note that, for both single and multiple objectives, we used the same results obtained by our approach. The results shown in the table indicate that the adaptive local search based approach clearly outperforms the others.


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Indeed, it achieves the new best MH cost for naug20, naug30, sko42, sko72, and will100. Also, in case of CR score, it finds better result for ds6, ct9, naug20, naug30, sko72, and will100. Furthermore, the average values for both objectives considerably improve. This can be more justified by Fig. 2, where the convergence behavior of the proposed and previous methods over generations for both objectives is depicted. From the figures, it can be found that the incorporation of adaptive local search reduces the gaps between the best and average values than that of the competing approaches. Table 2. Comparison with exiting MOFLP approaches Pr. Adaptive LS With LS Without LS Adaptive LS ds8 With LS Without LS Adaptive LS singh6 With LS Without LS Adaptive LS singh8 With LS Without LS Adaptive LS ct9 With LS Without LS Adaptive LS naug20 With LS Without LS Adaptive LS naug30 With LS Without LS Adaptive LS sko42 With LS Without LS Adaptive LS sko72 With LS Without LS Adaptive LS Will100 With LS Without LS ds6

MH Cost

CR Score

Best

Avg

Best

Avg

Time (sec)

92 92 96 179 179 179 94 94 94 179 179 179 4818 4818 4818 1285 1286 1286 3059 3062 3062 15484 15642 15796 65002 65544 66034 271862 273862 273988

92 94.96 96.8 200.04 199.649 209.84 95.25 96.66 98.28 187.6 191.013 199.84 4819.6 4820.03 4822.9 1288.13 1289.08 1290.6 3079.05 3081.07 3081.02 15863.42 16095.3 16876.56 65946.2 66086.3 67658.33 276954.52 279073.12 280126.62

56 48 48 82 82 82 56 56 48 82 82 82 92 90 90 202 198 186 332 312 292 372 380 370 604 592 602 1128 1109 1084

52.16 44.24 43.40 76.34 75.26 70.3 52.96 51.305 40.48 73.1 72.99 73.1 80.14 77.09 74.79 187.25 176.4 172.56 298.75 272.36 254.05 346.12 328.24 325.13 576.5 518.15 536.8 1017.72 979.25 977.42

0.128 0.139 0.128 0.208 0.291 0.2 0.132 0.138 0.132 0.192 0.22 0.19 0.273 0.315 0.27 15.814 24.714 13.997 63.933 96.42 56.025 235.287 308.918 207.13 1750.993 1942.56 1605.96 3960.21 4864.65 3804.84

Optimal solutions after 50% of the generations 74% 22% 16% 68% 0% 0% 82% 18% 15% 72% 0% 0% 72% 0% 0% 64% 0% 0% 68% 0% 0% 69% 0% 0% 61% 0% 0% 53% 0% 0%

The required time for a complete evolutionary cycle mentioned in Table 2 also shows that the proposed method is able to optimize both MH cost (minimize) and CR score (maximize) from the first to the last generation faster than the others. From the table, we can find that the adaptive local search based approach takes less time than the conventional local search based approach, and the difference is very significant. For obvious reason, the proposed approach takes slightly more time than the approach

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without local search. Despite that it is very important to note that the performance of our proposed approach with adaptive local search is much better than the other two approaches. In fact, after performing half of the scheduled generations, the proposed approach finds the best values for both objectives in case of more than 50% of the populations for all test problems. Where as, at this point the performances of the other two approaches are not satisfactory enough. They can find the best values only for the problems with 6 facilities and the number is also small. Table 2 also summarizes the percentages of the optimal solutions obtained by each algorithm.

(a) material handling cost (for singh8)

(b) closeness rating scale (for ds8)

Fig. 2. Two objectives over generations

As mentioned earlier, we run the proposed algorithm for 60% of the scheduled generations for all test problems to test their convergence behaviors. The experimental results suggest that after 50% of the scheduled generations, it starts finding the known best values and it almost convergences for around 60% of the total population. Fig. 2 also shows this tendency. For this reason, the required time for the proposed approach will be less than the values mentioned in Table 2. However, for fair comparison, we mentioned the time for the same number of generations for all approaches. Thus, the proposed adaptive local search appears to be highly effective, and the additional coding effort and time required in comparison to the approach without local search is definitely justified.

(a) ct9

(b) naug30

Fig. 3. Final Pareto-optimal front


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MOEAs do not try to find one optimal solution but all the trade-off solutions, and deal with two goals: convergence and diversity. Non-dominated solutions of the final generation obtained by each alternative are shown in Fig. 3 to illustrate the convergence and diversity of the layouts. It is worthwhile to mention that in all cases, most of the solutions of the final population are Pareto-optimal. In the figures, the occurrences of the same non-dominated solutions are plotted only once. From these figures, it can be observed that the final solutions produced by the proposed method are more spread and well converged than previous approaches. And for this reason, it is capable of finding extreme solutions.

7 Conclusion In this paper, we propose an adaptive local search based evolutionary approach for solving the multi-objective FLP, which is very rare in the literature. The proposed approach is composed of the multi-objective GA, the improved 1-opt local search method and the modified adaptive local search scheme using SCM. Unlike conventional local search method, the adaptive local search scheme automatically determines whether local search is used in a GA loop or not. A comparative analysis clearly show that the introduction of adaptive local search helps well to improve the best fitness value, average number of generations, and convergence behaviors of fitness values. In addition, it can find the near-optimal and non-dominated layout solutions, which are also the best-known results especially when applied to larger instances (n >15). Furthermore, it is capable of finding a set of Pareto-optimal layout solutions that optimizes both MH cost and CR score simultaneously throughout the entire evolutionary process and provides a wide range of alternative layout choices for the designers.

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