A Hybrid Genetic Search Based Approach to Solve

Asia Pacific Management Review 15(2) (2010) 301-312

www.apmr.management.ncku.edu.tw

A Hybrid Genetic Search Based Approach to Solve Single Period Facility Layout Problem Surya Prakash Singha,∗, Renduchintala Raghavendra Kumar Sharmab a

Department of Management Studies, Indian Institute of Technology Delhi, New Delhi, India Department of Industrial and Management Engineering, Indian Institute of Technology Kanpur, India

b

Accepted 7 December 2009

Abstract Single period facility layout problem (SFLP) is a well known combinatorial optimization problem and is traditionally formulated as a quadratic assignment problem (QAP) which is a NP-Hard problem. Therefore, heuristic and meta-heuristic are widely used to solve it. This paper proposes a new hybrid genetic search based approach (referred as GA_SC_PEM) to solve SFLP. We propose a new crossover and mutation scheme. Empirical investigation was conducted and it was found that the proposed approach gives promising solution in a reasonable computational time. Keywords: Facility layout problem, quadratic assignment problem, combinatorial optimization, genetic algorithm 1. Introduction Combinatorial optimization problems arising from both practice and theory pose a real challenge and continuously draw attention of the researchers, practitioners and academicians around the world over last few decades. Many optimization problems belong to the class NPhard and cannot be solved optimally in polynomial time. One such combinatorial problem is SFLP. SFLP is a static layout fixed for one time period (does not change over the time period) and is referred as single period facility layout problem. On the other hand, layout which changes from one period to the other is termed as dynamic facility layout problem (DFLP). Paper by Rosenblatt (1985) can be referred for DFLP. The SFLP is a well researched problem and formulated by Koopmans and Beckman (1957). They called it quadratic assignment problem (QAP). The name is quadratic because the objective function is a second degree function of the variables and the constraints are linear. The nature of the QAP is NP-hard (Sahni and Gonzalez, 1976). SFLP formulated as QAP is a difficult problem to solve optimally in a reasonable computational time (Burkard and Stratmann, 1978; Burkard and Rendl, 1984). The solution of the large sized SFLP is significant as a number of important problems such as backboard wiring; scheduling and control panel layout can be formulated as QAP. QAPs of size 20 and more are presently not solvable by the use of exact implicit enumeration techniques. Thus, the interest lies in the development of new heuristic and metaheuristic approaches to solve large sized SFLP. This paper proposes a new hybrid genetic search based approach for solving the SFLP. Similarly, other approaches that are similar to GA are simulated annealing (SA), tabu search (TS), and ant colony algorithm (ACO). The organization of the paper is as follows. The literature relevant to the application of GA in solving SFLP is reviewed in Section 2. SFLP is formulated in Section 3. An overview ∗

Corresponding author. E-mail: [email protected]

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S. P. Singh, R. R. K. Sharma / Asia Pacific Management Review 15(2) (2010) 301-312

of GA is given in Section 4. Proposed GA based heuristic is presented in Section 5, while Section 6 deals with the computational experiments and its results. Finally, in Section 7 conclusion and future research direction are provided. 2. Literature review Papers that deal with the research work related to the SFLP, applying GA, are considered for brief discussion. Cohoon et al. (1991) were the first who proposed a distributed GA for the floor plan design problem. A year after, Tam (1992) proposed a genetic search based method for solving facility layout problem (FLP) but for the mixed integer linear program (MILP) based formulation of SFLP. Tate and Smith (1995) were the first to propose GA particularly for the QAP modeled SFLP. In this implementation, solutions are encoded as permutations and the initial population is therefore composed of in random permutations. To promote the selection of better individuals for reproduction they used a rank based selection mechanism. The crossover operator adopted by them is shown in Figure 1 and is summarized below. (a) If a facility is assigned to the same location for both parents, it remains at the same location for the offspring. (b) Unassigned locations are scanned from left to right. For an unassigned location, pick up facility at random among those that occupied that location in the parents. Once an object is assigned it is no longer considered in future random choices. (c) The remaining facilities are assigned to the unassigned locations. Parent 1

3

1

2

7

4

6

5

Parent 2

2

1

4

3

6

7

5

After crossover Step 1

1

5

Step 2

3

1

4

7

6

Step 3

3

1

4

7

6

5 2

5

Figure 1. Example of a crossover operator used in Tate and Smith (1995). The mutation operator used by Tate and Smith (1995) consists of selecting two sites at random and reversing the order of the assignments between those two. Suresh et al. (1995) tested three standard crossover operators and proposed a new crossover operator to avoid infeasible solutions in the population. Tavakkoli-Moghaddam and Shayan (1998) analyzed the suitability of genetic operator for solving FLP. Ahuja et al. (2000) pointed out the drawbacks of crossover scheme proposed by Tate and Smith (1995) and further suggested the application of GA for the QAP and reported its computational behaviour. Their GA incorporates several greedy principles in its design, and named it the greedy GA. They used construction heuristic for generating initial population and formed a new crossover scheme as well as new special purpose immigration scheme. Balakrishnan et al. (2003) describes that FLP is a well researched one and a few effective and user friendly approaches have been proposed but for most of the part the effective algorithms are not user friendly and also the user friendly methods are not effective in handling the intricacies such as unequal area 302


departments. FACOPT, a heuristic approach has been proposed by them, which uses GA to solve FLP. Application and theory of GA show that it can be useful in solving combinatorial optimization problem (see Goldberg, 1989). GA provides good solutions for small instances of the QAP. However, a pure genetic approach has its shortcomings for combinatorial problems. While the pure GA based approach still yields good solutions for small problems, it cannot find the best known solution even for the problems of size 20 and more. For larger problems of size up to 100, simple GA cannot really compete with hybrid GA. Despite this fact, the results reported by Tate and Smith (1995), Azadivar and Wang (2000), and Balakrishnan et al. (2003) are encouraging. In this research, a new hybrid GA is proposed for solving the SFLP. For a good coverage of theoretical and practical issues on GA based SFLP one can refer to the papers listed in Table 1. Also, for more comprehensive survey of QAP formulated SFLP, papers by Burkard et al. (1999), Singh and Sharma (2006), Duman and Or (2007), Loiola et al. (2007), Singh and Sharma (2008a, 2008b), and Sharma and Singh (2008) can be referred. Table 1. List of GA based SFLP papers. Reference

QAP MIP

Tam (1992)

√

Banerjee and Zhou (1995)

√

Tate and Smith (1995)

√

Kochhar and Heragu (1998) Islier (1998)

√ √

Rajashekaran et al. (1998)

√

Mak et al. (1998)

√

Mckendall et al. (1999)

√

Kochhar and Heragu (1999) Gau and Meller (1999) Azadivar and Wang (2000)

√ √

Al-Hakim (2000) Ahuja et al. (2000)

√

Wu and Appleton (2000)

√

Lee et al. (2003)

√

Balakrishnan et al. (2003)

√

3. Problem formulation This section describes the formulation of QAP. Let us assign “n” facilities to “n” locations with the cost being proportional to the flow between the facilities multiplied with their distances. The objective is to allocate each facility at a location such that total cost is minimized as given in (1). Thus we are given two matrices, the flow matrix F ik and distance matrix D jl . An instance of SFLP with input flow and distance matrix is denoted by QAP. 303


Equation (2) ensures that only one facility can be located at one location and (3) ensures that each location can only be assigned to one facility. Equation (4) shows that the variables are binary. The QAP is written as: n

n

n

n

Min TF = ∑ ∑ ∑ ∑ F ik ∗ D jl ∗ X ij ∗ X kl i =1 j =1 k =1l =1 i ≠ k j ≠l

n ∑ X ij = 1 ∀ j = 1,..., n i =1 n ∑ X ij = 1 ∀ i = 1,..., n j =1 X ij ∈ {0,1}

(1)

(2) (3) (4)

∀ i, j = 1,..., n

X ij = 1

if facility “i” is located to location “j”.

X ij = 0

if facility “i” is not located to location “j”.

This above formulated problem is called QAP and was first given by Koopmans and Beckman (1957). The QAP problem up to 20 facilities is claimed to be solved optimally (Diponegoro and Sarkar 2003). Beyond that the computational time becomes intractable. Thus, heuristic and meta-heuristic approaches are used to obtained solutions for large size problems. 4. Genetic Algorithm: An overview The GA imitates the process of evolution. Each feasible solution is treated as an individual, and the fitness of an individual is measured by the objective function value. Population is equivalent to a set of solutions. Two of the solutions (also called parents) are selected and subjected to breed/crossover. A crossover operator makes this possible. The healthy (having good fitness) offspring replaces the weaker parent in the population pool. The mutation operator is used to improve the solution space by diversifying it. The algorithm continues until a predetermined number of generations are reached. Generally, GA has not gained the acceptance of the operations research community as did other meta-heuristic approaches viz. TS, SA, and ACO. The reason for this is often times GA may generate infeasible solutions (refer Bean, 1994). This paper proposes a new GA based heuristic approach for solving SFLP. The crossover operator is designed in such a way that it guarantees to produce feasible solution during crossover. Similarly, a new mutation scheme is proposed here which, instead of randomly mutating complete solution string, only mutates some facilities. The new crossover and mutation scheme are named as swapped crossover and K-pairwise exchange mutation respectively and described in the following section. 5. Proposed GA based heuristic This section describes the proposed genetic search based approach to solve SFLP. The proposed genetic search based method along with the swapped crossover and K-pairwise exchange mutation operator are described here. The proposed GA based heuristic approach is referred as genetic search with swapped crossover and K-pairwise exchange mutation and denoted as GA_SC_PEM through the paper.

5.1 Swapped crossover

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The proposed GA crossover viz. swapped crossover operator which is designed for SFLP always generates a feasible child after crossover. In this paper, we applied single point crossover between two parents. The working methodology of the swapped crossover scheme can best be viewed from Figure 2 which shows the crossover scheme for problem size n = 5. The following are the steps of the proposed crossover scheme used in the paper. Step 1. Select parent say “P” from the parent pool. Step 2. Generate random number say “r” between 1 and “n”, such that 1 < r ≤ n. Step 3. If “r” is any number except first and last facility then consider “r” as facility located at first location in the child 1. Step 4. For child 1, facility from “r” to “n” of parent “P” will be the facility located at locations from 1 to n-(r-1). Step 5. Similarly, facility from 1 to (r-1) of parent “P” will be the facility for child 1 from location (n-r) to n Step 6. Similarly, repeat Step 1 to 5 for all parents to complete the crossover. Crossover point

Parent 1

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Parent 2

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After crossover Child1

4

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Child 2

1

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3

Figure 2. Single point swapped crossover. After crossover is done, old parent pool is updated with the new one. The new thing associated with the swapped crossover is that it works with one parent instead of two. This new way of crossover method generates feasible solution string and checking the feasibility of the solution is no more required. Crossover scheme proposed by Tam (1992), and Tate and Smith (1995) generates infeasible child after crossover which need a separate routine to make infeasible layout solution into feasible layout. The proposed crossover scheme avoids the generation of infeasible solution.

5.2 K-pairwise exchange mutation In the proposed K-pairwise exchange mutation scheme, K% of nC2 (total number of pairs of facilities that can be exchanged together) pair of facilities of the SFLP solution are exchanged with their locations. If K is taken 100%, then the proposed mutation scheme exchanges all possible nC2 pairs of facilities. In this research work GA_SC_PEM is applied for five different values of K i.e. 20%, 40%, 60%, 80% and 100%. For example if n = 5 and K = 40% then 5C2 is equal to 10. Hence, four pairs of facilities (40% of 5C2) are interchanged. To check an effect of proposed K-pairwise exchange mutation scheme on the performance of GA, five different values of K are considered here. Figure 3 shows a schematic diagram of proposed K-pairwise exchange mutation scheme and its working methodology. In Figure 3, the proposed K-pairwise exchange mutation scheme is shown for two parents for K = 20% where two (since, 20% of 5C2 is 2) randomly selected pair of facilities interchange their locations. In Figure 3, facilities (4, 3) and (2, 1) interchange there locations resulting in the 305


new solution string. So, the proposed mutation scheme updates the population pool of GA as per the procedure described in Figure 3. Parent 1

4

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Parent 2

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After mutation New_Parent 1

3

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New_Parent 2

3

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2

1

Figure 3. K-pairwise exchange mutation scheme. 6. Computational experiments and results Proposed heuristic GA_SC_PEM is coded in C and executed on a Pentium IV HCL machine of 2.4 GHz having 512 MB RAM. The computational experiments of GA_SC_PEM have been conducted on the data set provided by various authors given in QAPLIB which is now maintained by the University of Pennsylvania. QAPLIB is an online version of Burkard et al. (1997) and is available at http://www.seas.upenn.edu/qaplib/inst.html. These data set includes SFLP instances involving facilities from 8 to 256 (i.e. n = 8 to 256). These test problems are labeled as what is given in QAPLIB along with the problem size. Forty instances of size from n = 16 to n = 256 are taken randomly to test the performance of GA based heuristic GA_SC_PEM. Since the performance of GA_SC_PEM is parameter dependent therefore the parameters that are used in GA_SC_PEM are summarized in Table 2. For each parameter setting, proposed GA_SC_PEM based heuristic is executed five times for different values of SEED (it is an integer value used as starting point with pseudo-random number to generate population pool). The crossover probability (pc) is set at 0.9 while mutation probability (pm) is set at 0.1. The same value pc and pm is set for two different population pool of sizes of 500 and 1000, which is shown in Table 2. The generation limit is set at 1000 for all executions with mutation limit of 50 and 100 for the population pool of 500 and 1000 respectively (see Table 2). Table 3 and 4 present results of GA_SC_PEM for different K values along with the average CPU time for all forty instances for population size 500 and 1000 respectively. Best results for the population sizes of 500 and 1000 are reported in Table 5 along with the percentage optimal gap (%OG) from the best known value. Table 2. Parameters used in the GA_SC_PEM. Population size

Values for K-pariwise exchange mutation (PEM)

pc

pm

Generation limit

Mutation limit

1

2

3

4

5

500

20%

40%

60%

80%

100% 0.9

0.1

1000

50

1000

20%

40%

60%

80%

100% 0.9

0.1

1000

100

306


S. No. Instance 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

Esc16a Esc16b Esc16c Esc16h Had16 Had20 Scr20 Tai20a Tai20b Chr22a Chr22b Bur26a Bur26b Bur26c Bur26d Bur26e Bur26f Bur26g Bur26h Nug30 Lipa30a Esc32c Esc32d Esc32e Esc32f Esc32g Tai35b Lipa40a Lipa50a Lipa60a Tai64c Lipa70a Lipa80a Lipa90a Sko100c Sko100d Sko100e Wil50 Wil100 Tai256c

Table 3. Result of GA_SC_PEM for population size 500. GA Based Heuristic Solution 20% 40% 60% 80% 38 36 34 34 146 146 146 146 85 85 85 84 502 498 502 498 1932 1936 1928 1890 3637 3621 3606 3515 76327 74294 78797 70419 401101 388924 393564 376548 136333609 136634872 136456498 138600534 4412 4425 4446 4220 4504 4386 4312 3847 6238104 6224326 5812486 6224124 4123178 4135628 4100544 4092764 5716278 5716278 5809342 5602875 4148764 4162780 4075493 4018324 5973216 5762986 5724538 5712092 3916254 3981624 3981762 3803076 10410278 10416234 10428765 10394434 7256810 7258726 7235728 7211862 3544 3480 3457 3457 15608 15846 14926 14642 812 786 786 684 310 274 238 226 36 34 34 34 2 2 2 2 6 6 6 6 304684312 304522106 304522106 302342312 43102 43218 40218 37892 72916 72908 68264 65120 147862 149268 147246 136298 2314836 2321564 2139936 2139936 180214 179834 174298 173612 272896 275124 274186 272618 368914 368904 368374 367954 160214 160132 161138 159836 160198 161196 161864 160142 159232 161234 159896 161784 52682 54678 51486 52468 302634 315628 318142 309818 50987628 50986724 50894838 50439816 307

100% 34 146 85 506 1946 3638 70647 401010 136100412 4426 4371 5570301 4098276 5612897 4028646 5539658 3925146 10452389 7328762 3664 15026 724 238 34 2 6 284614368 40122 67248 145728 2139936 175286 273724 367182 161728 161286 161718 53872 315782 51012098

Avg. CPU 19 17 18 16.25 16.75 17.25 17 18.25 43.8 16.25 14.25 18.14 17.20 19.10 17.50 18.0 19.30 18.50 19.10 23.50 21.50 21.70 23.50 25.60 20.70 21.30 29.70 35.60 43.10 51.30 60.20 72.10 80.0 93.0 135.0 141.0 140.0 150.0 239.1 478.2


S. No.

Instance

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

Esc16a Esc16b Esc16c Esc16h Had16 Had20 Scr20 Tai20a Tai20b Chr22a Chr22b Bur26a Bur26b Bur26c Bur26d Bur26e Bur26f Bur26g Bur26h Nug30 Lipa30a Esc32c Esc32d Esc32e Esc32f Esc32g Tai35b Lipa40a Lipa50a Lipa60a Tai64c Lipa70a Lipa80a Lipa90a Sko100c Sko100d Sko100e Wil50 Wil100 Tai256c

Table 4. Result of GA_SC_PEM for population size 1000. GA Based Heuristic Solution 20% 40% 60% 80% 100% 34 36 34 34 34 146 146 146 146 146 86 86 84 84 84 498 501 498 498 498 1916 1908 1908 1924 1872 3635 3618 3606 3636 3576 59326 58126 59164 59214 56325 395672 395886 401221 399798 364602 127975157 125286585 127419606 128267768 125470555 3888 3609 3568 3511 3123 3884 4379 4283 4441 3105 6211424 6211424 5687246 5640214 5570301 3912314 3910324 3932474 3931246 3910324 5683242 5686326 5642118 5686326 5583146 4065826 4008314 3946128 3964214 3929213 5626124 5626124 5546324 5582416 5539658 3824167 3819042 3914023 3806286 3803076 10489312 10489312 10478342 10476124 10394434 7213876 7246327 7348127 7326820 7211862 3457 3519 3489 3521 3223 14068 13826 13642 13642 13610 794 684 684 724 652 310 286 218 224 200 32 32 34 34 34 2 2 2 2 2 6 6 6 6 6 291563486 294618242 284456286 284614368 284456286 33816 32809 32672 33248 32422 64523 64902 63728 63678 63623 110625 109802 109876 110234 109562 1948646 1987236 1987236 2001672 1876836 180234 174032 174346 176826 171534 258927 258674 260124 262348 257643 367014 366978 367198 367914 366397 158464 152436 160186 159862 160910 151874 158236 160786 160436 160786 158268 161396 152326 160524 160872 50786 54896 54162 51602 539018 296847 300214 309786 309764 290410 48592216 47635764 49967856 49979964 49826438 308

Avg. CPU 53.0 57.50 68.25 73.0 67.25 74.50 68.25 67.25 73.20 85.25 79.40 41.35 44.20 49.15 57.30 64.10 68.50 57.30 81.40 91.50 87.35 91.0 140.3 98.70 119.0 147.2 238.1 131.4 134.3 319.3 206.4 241.0 306.2 303.1 400.5 376.8 409.3 421.2 351.3 1064


S. No. 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

Table 5. SFLP solution obtained by GA_SC_PEM. GA_SC_PEM solution Instance OS / BKS Population size Population 500 size 1000 Esc16a 34 34 34 Esc16b 146 146 146 Esc16c 84 84 84 Esc16h 498 498 498 Had16 1860 1890 1872 Had20 3461 3576 3515 Scr20 55015 70419 56325 Tai20a 351741 376548 364602 Tai20b 122455319 136100412 125286585 Chr22a 3078 4220 3123 Chr22b 3097 3847 3105 Bur26a 5426670 5570301 5570301 Bur26b 3817852 4092764 3910324 Bur26c 5426795 5602875 5583146 Bur26d 3821225 4018324 3929213 Bur26e 5386879 5712092 5539658 Bur26f 3782044 3803076 3803076 Bur26g 10117172 10394434 10394434 Bur26h 7098658 7211862 7211862 Nug30 3062 3457 3223 Lipa30a 13178 14642 13610 Esc32c 642 684 652 Esc32d 200 226 200 Esc32e 32 34 32 Esc32f 2 2 2 Esc32g 6 6 6 Tai35b 283315445 284614368 284456286 Lipa40a 31538 37892 32422 Lipa50a 62093 65120 63623 Lipa60a 107218 136298 109562 Tai64c 1855928 2139936 1876836 Lipa70a 169755 173612 171534 Lipa80a 253195 272618 257643 Lipa90a 360630 367182 366397 Sko100c 147862 159836 152436 Sko100d 149576 160142 151874 Sko100e 149150 159232 152326 Wil50 48816 51486 50786 Wil100 273038 302634 290410 Tai256c 44759294 50439816 47635764 309

% OG 0 0 0 0 0.64 1.5 2.3 3.6 2.3 1.4 0.25 2.6 2.8 2.8 2.8 0.5 2.7 1.5 1.5 5.2 3.2 1.5 0 0 0 0 0.4 2.8 2.4 2.1 1.12 1.04 1.7 1.6 3.09 1.5 2.1 4.0 6.36 6.4


From Table 3, we can see that GA_SC_PEM gives better result for K value set at 40%, 60%, and 80%. On the other hand when the K value is set at 100%, the GA_SC_PEM gives better result only in three instances viz. Tai20b, Tai35b, and Lipa90a. Similarly, when the K value is set at 20%, GA_SC_PEM gives better result only in two instances viz. Esc16c, and Chr22a. Instances such as Esc16a, Esc16b, Esc16h, Esc32e, Esc32f, and Esc32g are unaffected by the K value and we get same solution for other values of K. Similarly, Table 4 also show that in most of the cases GA_SC_PEM gives better result when the K value is set at 40%, 60%, and 80%. When the K value is 100%, we get better result only in three instances viz. Bur26e, Tai35b, and Lipa50a. On the other hand when K value is set at 20%, GA_SC_PEM gives better result only in two instances i.e. Had16, and Chr22b. GA_SC_PEM give same solution for instances Esc16a, Esc16b, Esc16h, Esc32e, Esc32f, and Esc32g for other values of K. Table 3 and 4 clearly show that performance of GA_SC_PEM largely depends on the parameters such as population pool, and K value. The affect of population pool size can be seen from Table 5. GA_SC_PEM gives best results in almost all cases except one i.e. Had20 when the population pool size is kept at 1000. In this paper, we have not tested the performance of GA_SC_PEM on other parameters viz. Pc, Pm, generation limit, and mutation limit. Overall the proposed GA_SC_PEM approach for SFLP provides promising solution in reasonable computational time and we are hopeful that the proper selection of GA parameters, good crossover and mutation strategy will improve the performance of GA_SC_PEM. 7. Conclusions and future research directions In this paper, we proposed GA based heuristic GA_SC_PEM to solve SFLP with swapped crossover embedded with K-pairwise exchange mutation scheme. Performance of the proposed approach is tested for forty instances taken from QAPLIB. On testing it was found that the proposed approach gives promising solution in reasonable CPU time. It was also found that the quality of the solution largely depends on the parameter setting of GA such as pc, pm, population size, K value, and SEED number. Optimizing the GA parameters could be another area of research work. We found that even the changing the SEED value changes the solution quality to a large extent. Optimal selection of GA parameters for GA_SC_PEM will be the future research work where we would like to focus.

Acknowledgement We would like to put on record their appreciation to anonymous learned referees for their valuable suggestions, which have substantially enhanced the quality of the paper over earlier versions.

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