運用延伸人工染色體基因演算法求解 單列機台佈置問題
Amalia Utamima1, 2 歐陽超1, *
1國立臺灣科技大學工業管理系
2Department of Information System, Institut Teknologi Speuluh Nopember
摘 要
單列機台佈置問題 (single row facility layout problem, SRFLP) 是一個 NP-Complete的問題，該問題之目標值是希望將兩兩機台間距離之和最小化。
延伸人工染色體基因演算法 (extended artificial chromosome genetic algorithm, eACGA) 是結合基因演算法 (genetic algorithm, GA) 及分佈估計演算法 (es- timation of distribution algorithm, EDA)。該方法在解決生產排程問題上獲得了 不錯的成果。本研究修改 eACGA 之方法並用來解 10 個 SRFLP 標竿問題，計 算結果顯示 eACGA 較 GA 或 EDA 更可獲得較好之目標值及較低誤差值。
關鍵詞：單列機台佈置問題，基因演算法，分佈估計演算法。
SOLVING SINGLE ROW FACILITY LAYOUT PROBLEM USING EXTENDED ARTIFICIAL CHROMOSOME GENETIC ALGORITHM
Amalia Utamima1, 2 Chao Ou-Yang1, *
1Department of Industrial Management National Taiwan University of Science and Technology
Taipei City, Taiwan 106, R.O.C.
2Department of Information System Institut Teknologi Speuluh Nopember
Surabaya 60111, Indonesia
Key Words: single row facility layout, estimation distribution algorithm, genetic algorithm.
ABSTRACT
The layout positioning problem of facilities on a straight line is known as Single Row Facility Layout Problem (SRFLP). The objective of SRFLP, categorized as NP-Complete problem, is to arrange the layout such that the sum of distances between all facilities’ pairs can be minimized.
Extended Artificial Chromosome Genetic Algorithm (eACGA) is a prom- ising algorithm that has been proposed recently. eACGA extends the probabilistic model in Estimation of Distribution Algorithms (EDAs) and then hybridize it with Genetic Algorithms (GAs). eACGA is proven to produce an excellent solution for scheduling problem. In this paper, we
*通訊作者：歐陽超，e-mail: [email protected]
Corresponding author: Chao Ou-Yang, e-mail: [email protected]
modify the eACGA to solve SRFLP. Computational results on benchmark problems show the effectiveness of eACGA for solving SRFLP.
I. INTRODUCTION
The Facility Layout Problem (FLP) is concerned with the arrangement of a number of facilities in a given space to satisfy an objective function. Facility layout designed ef- fectively can decrease manufacturing lead time and increase throughput. Thus, it increases the overall productivity and efficiency of the plant. The Single Row Facility Layout Problem (SRFLP) is considered when multiple-products with different production volumes and different process routings need to be manufactured. The objective is to set up the facilities so that it can minimize the sum of the dis- tances between all facility pairs [1]. Because of SRFLP is proven to be a NP-Complete problem, the exact methods applied to large instances of the problem are time consum- ing; hence heuristic methods have been built to acquire a near optimal solution to the problem [2].
Recent researches have already tried to solve SRFLP with meta-heuristic approaches. Solimanpur et al. [3] use Ant algorithm to overcome SRFLP. Samarghandi et al.
[4, 2] use Particle Swarm Optimization and Tabu algo- rithms, respectively, to figure out an efficient solution to SRFLP. Datta et al. [5] offer permutation-based genetic algorithm for better solution to SRFLP.
Estimation of distribution algorithms (EDAs) are stochastic optimization techniques that explore the space of potential solutions by exploiting the inter-variable de- pendency and sampling probabilistic models of promising candidate solutions. Because of their effectiveness, many optimization problems have been resolved with good competitive results [6].
EDAs improve the solution quality efficiently in the first few runs, but the diversity lost grows rapidly as more iterations are run. To maintain the diversity, hybridiza- tion with meta-heuristic algorithm is needed. EDAs will be used to characterize the parental solutions and then search around the current solution space. After that, meta-heuristics might introduce new solutions into the population to maintain diversity, which can avoid the pre- mature convergence of EDAs [7].
The Extended Artificial Chromosome Genetic Algo- rithm (eACGA) is derived from artificial chromosomes with a genetic algorithm (ACGA), an algorithm that joins EDA and GA in effective manner [7]. ACGA was used
for chip resistor scheduling [8] in order to accelerate the convergence rate in GA. Research has proposed eACGA as a solution for scheduling problems. In [7], ACGA is improved upon as eACGA and employed to solve permu- tation flowshop scheduling problems. eACGA collects not only the univariate probabilistic model, like the ACGA just considered, but also the bivariate probabilistic model.
eACGA seems very powerful since it considers both uni- variate and bivariate statistical information.
In this paper, we propose eACGA to solve SRFLP.
We first define SRFLP and describe every step to build eACGA. Experimental results for 10 benchmark prob- lems are conducted to verify the effectiveness of the pro- posed method.
II. PROBLEM DEFINITION
This research considers SRFLP with different sizes of facilities. The length li of each facility i and a n × n matrix C = [cij], where cij is the traffic loads between facilities i and j, are given. The distance between two facilities is supposed to be taken between their mid-points.
ABSMODEL, proposed by Heragu & Kusiak [9], is a well-known model for solving SRFLP. ABSMODEL is illustrated in equation (1).
1
1 1
minz= Σ Σni=− nj i= +c dij ij (1)
s.t.: 1( ) ;
2
1, 2, , 1; 1, ,
0; 1, 2, , 1; 1, ,
ij i j ij
ij
d l l s
i n j i n
d i n j i n
≥ + +
= − = +
≥ = − = +
… …
… …
In the ABSMODEL, dij stands for the distance be- tween the centers of the facilities i and j, while sij is the necessary clearance or gap between the two facilities.
Heragu & Kusiak [9] defined dij as equation (2).
2
i j
ij ij
l l
d + D
= + (2)
Dij is the space between facilities i and j. Note that Dij
is not necessarily equal to sij. If facility k is placed be- tween facilities i and j with sij = 0 then Dij = lk. Based on
Stop yes g = max generation?
Elitism Performed Fitness Calculation
Replacement Sampling
Univariate and Bivartate Probabilistic Modeling
Selection EDA
yes g mod 2 = 0
Initialization Start
no
no GA Crossover
Mutation
Fig. 1 eACGA framework
[1, 2, 4, 5], the authors assume sij between facilities is 0.
We adopt the similar assumption for this research that sij is also assumed 0.
III. METHODOLOGY
1. General Procedure of eACGA
The general procedure of eACGA is shown in Fig. 1.
We modify the eACGA framework taken from [7] to be more suitable with our code to solve SRFLP. The eACGA procedure starts with the initialization of all vari- ables. A population consisting of a number of chromo- somes is also initialized randomly. The details of the chromosome structure will be explained in section 2.
The main iteration starts with fitness calculation of every chromosome. Next, a decision to choose between extended EDA or GA procedure is made. If g, which represents the current generation, can be divided by 2, then extended EDA procedure is executed, otherwise GA pro- cedure is performed.
Extended EDA procedure starts with a selection proc- ess that attempts to choose chromosomes with better fit-
Table 1 Chromosome Illustration
i 1 2 3 4 5 6 7 8 X[i] 5 3 4 1 8 2 7 6
ness values. Chromosomes fitness are sorted first and a group of chromosomes with better fitness are selected.
The probabilistic modeling forms the ordinal (univariate model) and dependency (bivariate model) matrices from selected chromosomes. These probabilistic modeling will be explained in the next section.
After two probabilistic models are established, a group of chromosomes for the next generation are gener- ated with the sampling process. On the other side, GA procedure consists of the crossover and mutation processes.
The crossover rate and mutation rate decide whether chromosomes will mate and mutate, respectively. We use 0.7 for the crossover rate and 0.3 for the mutation rate.
Roulette wheel selection chooses the parental chromo- somes. Two-point crossover operator mates two chro- mosomes with better fitness values as the parent. Muta- tion then probably occurs and chooses between three mu- tations operators, which flip, swap, or slide, and that are to be undertaken.
The replacement step replaces the parental chromo- somes with their offspring. This step is only done once in each generation. The elitism strategy is also done in this step. The elitism rate is 10% of the population size. A group of new generation chromosomes is sent to be proc- essed into the next iteration.
2. Extended Artificial Chromosome
In a problem containing n facilities labeled as F1, F2, ..., Fn, a sequence X contains permutation numbers of all the labels. X can be represented by assigning facility X[i] to location i, where i ∈ [1,n]. Different sequences in X represent different layout solutions.
The representation of chromosome is shown in Table 1. The number of a chromosome’s location is adjusted with the number of facilities in the benchmark problems.
Table 1 shows the example of chromosome representation with 8 facilities. X[2] = 3 means assigning facility 3 to location 2.
In every generation g, a group of C chromosomes which have better fitness are selected. The chromosomes are labeled as X1, X2, …, and XC, where C is half of popu- lation size. Distribution of parental chromosomes is in- terpreted by sampling new solutions from the univariate
and bivariate probabilistic model [7]. The solutions gen- erated by the two probabilistic models are known as ex- tended artificial chromosomes. We adopt a similar prob- abilistic model like described in [7]. The univariate or ordinal probabilistic model, φi[i](g) in equation (3) shows the importance of facilities in the sequence. It represents how many times facility i is placed before or at position [i]
at current generation g. Aki[i] will set to 1 if facility i is placed before or at position [i], otherwise it will set to 0.
C k
[ ]( ) k 1A , i 1,i[i] , n; 1, ,
i i g k C
φ = Σ = = … = … (3)
The bivariate or dependency probabilistic model ψij in (4) represents how many times facility j is placed im- mediately after facility i. B will set to 1 if facility j is kij placed next to facility i, otherwise it will set to 0. We replace the 0 values in φi[i](g) and ψij(g) with 1/C for main- taining the diversity of eACGA.
C k
k 1 ij
( ) B , i, j 1, , n; 1, ,
ij g k C
ψ = Σ= = … = … (4)
Let Pi[i](g) be the probability value of facility i at position [i]. Selecting facility i has better probability value than other facilities when both probabilistic models’
statistical information is used, according to [7], φi[i](g) is multiplied by ψij(g). This will summarize the probability values of all unassigned facilities that could be set at posi- tion [i] [7].
For every offspring O1, O2, …, and O2C, some meth- ods are used to assign facilities to a specified location.
Selecting a facility at the first location, our approach is randomly picking the first facility that appears in the C chromosomes, that is X[1]. For example to assign the first facility to O1, if X3 is picked randomly, then the first facil- ity is from X3[1]. We think that this approach is more promising than the one proposed in [7], which selects fa- cility randomly. For assigning the remaining facilities, we adopt the formula in [7] that is equation (5).
[ ] [ ]
f [ ]
( ) ( )
( ) ( ( ) ( ))
i i ij
i i
i i ij
g g
P g
g g
φ ψ
φ ψ
∈Ω
= ×
Σ × (5)
where,
[i] = 2, 3, …, n; i = 1,2, …, n;
Ω: set of unassigned facilities
A selection method to select a facility from Ω and to set it on position [i] is modified from [7]. This selection
method is done in producing every offspring. The fol- lowing pseudo code demonstrates the assignment proce- dure.
Ω: The set of unassigned facilities.
F: The set of assigned facilities. F is empty in the begin- ning.
θ: A random probability is drawn from U (0,1) i: A selected facility by proportional selection k: The element index of the facility’s position n: The number of facilities
1:
2:
3:
4:
5:
6:
7:
F Φ
for k=2 until n do θ U(0,1)
Select a facility i satisfy θ ≤ Pi[i], where i Є Ω F(k) i
Ω Ω\i end for
IV. COMPUTATIONAL RESULTS
Three scenarios are held in the experiments to test our proposed method. First, we compare the proposed method with basic EDA and GA to check its robustness when solving SRFLP. We use the ten benchmark prob- lems, P4, LW5, S8, S8H, S9, S9H, S10, S11, LW11, and P15 as fully described in [1]. The number of facilities in each problem is labeled as n. In the first scenario, we perform 10 runs for each algorithm and consider the minimum result and error rate for each algorithm. The error rate percentages are obtained by comparing the mean values in 10 runs and the minimum values achieved in all three algorithms. Subsequently, in the second scenario, we also compare minimum values achieved or that could be called Objective Function Value (OFV) with previous works [1, 4, 5]. In the last scenario, the runtime com- parison of eACGA with EDA and GA is also conducted.
The results from the first, second, and third scenarios are provided in Table 2, 3, and 4 respectively.
We utilize similar parameter settings for eACGA, EDA, and GA. The size of the population and generation vary depending on the number of facilities; the higher the number of facilities, the more populations and generations are needed to get an optimal solution. We set population size as 4n in each problem. We adjust the maximum number of generations to 10n. The elitism rate is 10 per- cent for each generation of eACGA and GA.
Compared to recent researches, the minimum record
Table 2 Comparison of eACGA with EDA and GA
EDA GA eACGA
Problem n
min error(%) min error(%) min error(%) P4 4 638.0 0.00% 638.0 0.00% 638.0 0.00%
LW5 5 151.0 0.73% 151.0 0.00% 151.0 0.00%
S8 8 801.0 1.95% 801.0 0.82% 801.0 0.17%
S8H 8 2324.5 0.86% 2324.5 0.59% 2324.5 0.21%
S9 9 2469.5 2.23% 2469.5 1.94% 2469.5 0.28%
S9H 9 4695.5 0.47% 4695.5 0.30% 4695.5 0.10%
S10 10 2781.5 2.88% 2793.50 4.72% 2781.5 0.34%
S11 11 6933.5 4.93% 6953.50 4.82% 6933.5 0.56%
LW11 11 6933.5 4.71% 7024.50 4.25% 6933.5 1.16%
P15 15 6534.0 6.42% 6556.00 9.71% 6305.0 2.12%
Table 3 Comparative Result with previous papers
OFV OFV OFV OFV Problem n
Amaral (2006) Samarghandi et al.
(2010) Datta et al. (2011) eACGA Gap P4 4 638.0 638.0 - 638.0 0.00%
LW5 5 151.0 151.0 151.0 151.0 0.00%
S8 8 801.0 801.0 - 801.0 0.00%
S8H 8 2324.5 2324.5 2324.5 2324.5 0.00%
S9 9 2469.5 2469.5 - 2469.5 0.00%
S9H 9 4695.5 4695.5 - 4695.5 0.00%
S10 10 2781.5 2781.5 2781.5 2781.5 0.00%
S11 11 6933.5 6933.5 6933.5 6933.5 0.00%
LW11 11 6933.5 6933.5 6933.5 6933.5 0.00%
P15 15 6305.0 6305.0 - 6305.0 0.00%
or OFV achieved by eACGA shows equal performance with OFV of [1, 4, 5] in all benchmark problems. This achievement results in a 0.00% gap if compared to OFV of previous papers.
Table 2 demonstrates the comparison of statistical properties between the proposed method, EDA, and GA.
The bold records imply the minimum values found among all algorithms. We can observe in Table 2 that eACGA minimum values are superior for problem S10 until P15 compared to eACGA, while for problem P4 until S9H it has equal minimum values. Compared to EDA, eACGA has lower minimum values in P15. Moreover, we can see that error rates of eACGA always have the lowest values in all problems compared to EDA and GA.
Table 3 presents the minimization values labeled as objective function value (OFV) in previous papers, com- pared to eACGA. These values are very important since SRFLP is aimed to minimize its objective function. OFV from previous papers are obtained from Amaral [1], Samarghandi et al. [4], and Datta et al. [5]. The dash (‘-‘)
Table 4 Runtime comparison of eACGA
EDA GA eACGA Problem n
runtime(s) runtime(s) runtime(s) P4 4 0.042 0.066 0.062 LW5 5 0.090 0.124 0.121 S8 8 0.388 0.487 0.516 S8H 8 0.385 0.491 0.519 S9 9 0.560 0.697 0.762 S9H 9 0.618 0.693 0.760 S10 10 0.786 0.938 1.063 S11 11 1.062 1.251 1.449 LW11 11 1.069 1.257 1.448 P15 15 2.846 3.228 4.030
in Table 3 means the results are not available in the com- pared references.
Compared to EDA and GA, eACGA mostly provides the lowest error rate and the best minimum values.
Therefore, the consequence is that the runtime value is worse than the two others as drawn in Table 4. The pro- gram code was run with Matlab, using Intel Core 2 Duo
CPU and 2GB of RAM. However, the runtime values are still reasonable to accept since they only spend approxi- mately 0.062 seconds for the lowest problem number and 4 seconds for the highest one.
V. DISCUSSION AND CONCLUSION
This paper has proposed the eACGA to find the opti- mal solution for solving the NP-Complete SRFLP. We modify eACGA and compare the results with GA and EDA. Computational results show that eACGA performs better in getting the minimum values and minimum error for ten benchmark problems listed.
Instead of using one probabilistic model like in EDA, eACGA uses two. EDA uses a univariate probabilistic model, while eACGA applies univariate and bivariate probabilistic models. The use of variable interactions in bivariate probabilistic models could represent better indi- vidual information for the EDA part in eACGA [7]. It allows eACGA to perform better than EDA.
Artificial chromosomes generated from global statis- tical information characterize the distribution of promising solutions in the search space. Therefore, the combination of artificial chromosomes with genetic operators could improve the solution quality. This also makes eACGA perform better than standard GA, which just uses ordinary genetic operators.
Moreover, compared to previous algorithms published in the literature, eACGA achieves equal performance in all 10 benchmark problems. The 0% gap indicates that eACGA always succeeds in reaching the minimum value in given benchmark data. Thus, we can conclude that eACGA successfully solves SRFLP in order to achieve minimum cost with good performance.
REFERENCES
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Manuscript Received: Apr. 26, 2012 First Revision Received: May. 31, 2012 Second Revision Received: Jun. 07, 2012 and Accepted: Jun. 29, 2012
