USING EDGE HISTOGRAM MODELS TO SOLVE FLOW SHOP ...

CHAPTER 13 USING EDGE HISTOGRAM MODELS TO SOLVE FLOW SHOP SCHEDULING PROBLEMS WITH PROBABILISTIC MODELBUILDING GENETIC ALGORITHMS

Shigeyoshi Tsutsui† and Mitsunori Miki‡ †Department of Management and Information Science, Hannan University 5-4-33 Amamihigashi, Matsubara, Osaka 580-5802, Japan E-mail: [email protected] ‡Department of Knowledge Engineering and Computer Sciences, Doshisha University, l-3 Tatara, Miyakodani, Kyo-tanabe, Kyoto, 610-0321, Japan [email protected] In evolutionary algorithms based on probabilistic modeling, the offspring population is generated according to the estimated probability density model of the parent instead of using recombination and mutation operators. In this chapter, we have proposed a probabilistic model-building genetic algorithms (PMBGAs) for solving flow shop scheduling problems using edge histogram based sampling algorithms (EHBSAs). The effectiveness of introducing the tag node (TN) in a string representation is also discussed.

1. Introduction Genetic Algorithms (GAs)7 are widely used as robust searching schemes in various real world applications, including function optimization, optimum scheduling, and many combinatorial optimization problems. Traditional GAs start with a randomly generated population of candidate solutions (individuals). From a current population, better individuals are selected by the selection operators. The selected solutions produce new candidate solutions by applying recombination and mutation operators.

S. Tsutsui & M. Miki

Recently, there has been a growing interest in developing evolutionary algorithms based on probabilistic models.13,19 In this scheme, the offspring population is generated according to the estimated probabilistic model of the parent population instead of using traditional recombination and mutation operators. The model is expected to reflect the problem structure, and as a result it is expected that this approach provides a more effective mixing capability than recombination operators in traditional GAs. These algorithms are called probabilistic modelbuilding genetic algorithms (PMBGAs) or estimation of distribution algorithms (EDAs). In a PMBGA, better individuals are selected from an initially randomly generated population like in standard GAs. Then, the probability distribution of the selected set of individuals is estimated and new individuals are generated according to this estimate, forming candidate solutions for the next generation. The process is repeated until the termination conditions are satisfied. Many studies on PMBGAs have been performed in discrete (mainly binary) domain and there are several attempts to apply PMBGAs in continuous domain. However, a few studies on PMBGAs in permutation representation domain are found. In previous studies,21,22 we have proposed an approach to PMBGAs in permutation representation domains, focusing on solving the Traveling Salesman Problem (TSP). In this approach, we developed a symmetrical edge histogram matrix (EHM) from the current population, where an edge is a link between two nodes in a string. We then sample nodes of a new string according to the edge histogram matrix. We called this method the edge histogram based sampling algorithm (EHBSA). We proposed two types of EHBSAs, an edge histogram based sampling algorithm without template (EHBSA/WO) and an edge histogram based sampling algorithm with template (EHBSA/WT). EHBSA/WO uses only the learned edge histogram matrix to generate new candidate solutions, whereas EHBSA/WT uses a template selected from the population of promising candidate solutions as a starting point and modifies the template to generate a new solution. EHBSA/WO and EHBSA/WT were applied to several benchmark instances of the TSP with and without the use of local search. The results showed that the EHBSA/WT has performed significantly better than

Using Edge Histogram Models to Solve Flow Shop Scheduling Problems

EHBSA/WO and other PMBGAs proposed in the past for solving this type of problems. EHBSA had also proven to provide significantly better results than some other popular two-parent recombination techniques for permutations. Experimental results indicated that another advantage of EHBSA is the use of significantly smaller population sizes than those that are necessary with most other evolutionary algorithms for permutation problems. In this chapter, we extend the EHBSAs to solving a flow shop scheduling problem, a typical, well known problem in the area of scheduling. In a flow shop scheduling problem, each string represents a sequence of jobs to be processed. For example, string s = {1, 2 3, 0} means that job 1 is first processed, then jobs 2, 3, and 0 follow in this sequence. In this case, there are four edges, i.e., 1–>2, 2–>3, 3–>0, and 0–>1. Thus, in the flow shop scheduling problem, each edge is directional and the edge histogram matrix becomes asymmetrical. This is a big difference from previous study.21,22 In Section 2 of this chapter, a brief review of PMBGAs is given. In Section 3, the EHBSAs for flow shop scheduling problems are described. The empirical analysis is given in Section 4. In Section 5, introducing the tag node in a string representation aiming to improve the performance of EHBSAs is discussed. Section 6 concludes the chapter. 2. A Brief Review of PMBGAs According to Pelikan, et al.,13 these PMBGAs in binary string representation can be classified into three classes depending on the complexity of models they use; (1) no interactions, (2) pairwise interactions, and (3) multivariate interactions. In models with no interactions, interactions among variables are treated independently. Algorithms in this class work well on problems which have no interactions among variables. These algorithms include the population based incremental learning (PBIL) by Baluja,1 compact GA (cGA) by Harik, et al.8 and univariate marginal distribution algorithm (UMDA) by Muehlenbein & Paass11. In pairwise interactions, some pairwise interactions among variables are considered. These algorithms include the mutual-information-maximization input clustering (MIMIC)

S. Tsutsui & M. Miki

algorithm by De Bonet, et al.,6 the algorithm using dependency trees by Baluja & Davies.2 In models with multivariate interactions, algorithms use models that can cover multivariate interactions. Although the algorithms require increased computational time, they work well on problems which have complex interactions among variables. These algorithms include extended cGA (ECGA) by Harik9 and Bayesian optimization algorithm (BOA) by Pelikan, et al.12,14 Studies to apply PMBGAs in continuous domains have also been done. These include continuous PBIL with Gaussian distribution by Sebag & Ducoulombier 15 and a real-coded variant of PBIL with iterative interval updating by Servet, et al.16 The UMDA and MIMIC were introduced in continuous domain. None of the above algorithms cover interactions among the variables. In EGNA by Larranaga, et al.,10 a Gaussian network learns to estimate a multivariate Gaussian distribution of the parent population. Two density estimation models, i.e., the normal distribution, and the histogram distribution, are discussed by Bosman & Thierens.3 These models are intended to cover multivariate interaction among variables. It is reported that the normal distribution models have shown good performance by Bosman & Thierens.4 A normal mixture model combined with a clustering technique is introduced to deal with non-linear interactions by Bosman & Thierens.5 An evolutionary algorithm using marginal histogram models in continuous domain was proposed by Tsutsui, et al.20 A study on PMBGAs in permutation domains is found in Ref. 18. In it, PMBGAs are applied to solving TSP using two approaches. PMBGAs are also applied to solve job shop scheduling problems and graph matching problems.19 Tsutsui, et al. proposed EHBSAs, several variants of PMBGAs based on learning and sampling an edge histogram matrix for solving permutation problems.21,22 3. Edge Histogram Based Sampling Algorithm for Flow Shop Scheduling This section describes how the edge histogram based sampling algorithm (EHBSA) can be used to (1) model promising solutions and (2) generate new solutions by simulating the learned model.

Using Edge Histogram Models to Solve Flow Shop Scheduling Problems

3.1. The Basic Description of the Algorithm An edge is a link or connection between two nodes and has important information about the permutation string. Some crossover operators, such as Edge Recombination (ER)26 and enhanced ER (eER)17 which are used in traditional two-parent recombination, use the edge distribution only in the two parents strings. The basic idea of the edge histogram based sampling algorithm (EHBSA) is to use the edge histogram of the whole population in generating new strings. The algorithm starts by generating a random permutation string for each individual population of candidate solutions. Promising solutions are then selected using any popular selection scheme. An edge histogram matrix (EHM) for the selected solutions (population) is constructed and new solutions are generated by sampling based on the edge histogram model. New solutions replace some of the old ones and the process is repeated until the termination criteria are met. 3.2. Developing Edge Histogram Matrix for Flow Shop Scheduling Previous study21 proposed a symmetrical edge histogram matrix. In this chapter, we represent it as EHM(S). In a scheduling problem, an edge in a string is directional. Thus, we must consider an asymmetrical edge histogram matrix EHM(A). Let string of kth individual in population P(t) at generation t be represented as stk=(πtk(0), πtk(1), ..., πtk(L–1)). (πtk(0), πtk(1), ..., and πtk(L– 1)) are the permutation of (0, 1, ..., L–1) representing a possible job sequence, where L is the length of the permutation. An asymmetric edge histogram matrix EHM(A)t (etij) (i, j =0,1, .., L–1) of population P(t) consists of L2 elements as follows: ⎧⎪ e t i, j = ⎨ ⎪⎩

∑k =1 δ i, j ( s kt ) + ε N

0

if i ≠ j if i = j

(1)

where N is the population size, δij(stk) is a delta function defined as ⎧ 1 if ∃h [h ∈ {0,1,L L − 1} ⎪ ∧ π kt (h) = i ∧ π kt ((h + 1)mod L) = j ] ⎪ 0 othersise ⎩

δ i , j ( skt ) = ⎨

(2)

S. Tsutsui & M. Miki

and ε (ε > 0) is a bias to control pressure in sampling nodes just like those used for adjusting the selection pressure in the proportional selection in GAs. The average number of edges of element etij (ij) in EHM(A)t is LN/(L2–L) = N/(L–1). So, ε is determined by a bias ratio Bratio (Bratio>0) of this average number of edges as ε=

N B ratio L −1

(3)

A smaller value of Bratio reflects the real distribution of edges in sampling of nodes and a bigger value of Bratio will give a kind of perturbation in the sampling. An example of EHM(A)t is shown in Fig. 1. st 1 = ( 0, st 2 = ( 1, st 3 = ( 3, st 4 = ( 4, st 5 = ( 2,

1, 3, 4, 0, 1,

2, 4, 2, 3, 3,

P(t)

3, 2, 1, 1, 4,

4) 0) 0) 2) 0)

0 2.05 1.05 2.05 0.05 1.05 0 2.05 2.05 0.05 1.05 2.05 0 1.05 1.05 0.05 1.05 0.05 0 4.05 3.05 0.05 2.05 0.05 0 EHM(A)t

Fig. 1. Examples of symmetrical and asymmetric edge histogram matrices for N = 5

3.3. Sampling Methods In this subsection, we describe how to sample a new string from the edge histogram matrix EHM(A)t. As for EHM(A)t in Ref. 21, there are two types of sampling methods; one is an edge-histogram based sampling algorithm without template (EHBSA/WO), and the other an edgehistogram based sampling algorithm with template (EHBSA/WT). 3.3.1. EHBSA/WO In a symmetrical EHM such as used in the symmetrical TSP, the absolute positions (loci) of a string have no meaning. For example, string s1 = (0, 1, 2, 3, 4) and string s2 = (4, 0, 1, 2, 3) represent the same tour. However, in a asymmetrical EHM such as for scheduling problems, these two strings represent two completely different solutions. Thus, we must consider how to determine the initial position and what node we assign to

Using Edge Histogram Models to Solve Flow Shop Scheduling Problems

the position. Here, we propose EHBSA/WO/T and EHBSA/WO/R.23

two

types

of

EHBSA/WO,

EHBSA/WO/T: Let us represent a new individual permutation by c[]. In EHBSA/WO/T, the initial sampling position is always the first position, i.e. c[0]. The value for c[0] is taken from a pseudo template individual PT[] which is taken from the current population P(t) randomly, and a new individual permutation c[] is generated straightforwardly as follows: 1. Set the position counter p ← 0 2. Choose a pseudo template PT[] from P(t) 3. Obtain the first node as c[0] ← PT[0] 4. Construct a roulette wheel vector rw[] from EHM(A)t as rw[j] = etc[p],j (j=0, 1, .., L–1) 5. Set to 0 previously sampled nodes in rw (rw[ c[i] ] = 0 for i =0, 1, .., p) L −1 6. Sample the next node c[p+1] with probability rw[ x] ∑ j =0 rw[ j ] using roulette wheel rw[] 7. Update the position counter p ← p+1 8. If p1) cut points are applied to the template randomly. When n cut points are obtained for the template, the template should be divided into n segments. Then, we choose one segment randomly and sample nodes for the segment. Nodes in other n–1 segments remain unchanged. We denote this sampling method by EHBSA/WT/n. Since average length of one segment is L/n, EHBSA/WT/n generates new strings that are different L/n nodes on average from their templates. Fig. 2 shows an example of EHBSA/WT/3. In this example, nodes of new string from after cut[2] and before cut[1] are the same as the nodes of the template. New nodes are sampled from cut[1] up to, but not including, cut[2] based on the EHM(A)t. The sampling method for EHBSA/WT/n is as follows: 1. 2. 3. 4. 5. 6.

Choose a template T[] from P(t). Obtain sorted cut point array cut[0], cut[1], .., cut[n–1] randomly. Choose a cut point cut[l] by generating random number l ∈ [0, n–1]. Copy nodes in T[] to c[] from after cut[(l+1) mod n] and before cut[l]. Set the position counter p ← cut[l]–1. Construct a roulette wheel vector rw[] from EHM(A)t as rw[j] = etc[p],j (j=0, 1, .., L–1).

Using Edge Histogram Models to Solve Flow Shop Scheduling Problems

7. Set to 0 copied and previously sampled nodes in rw[] (rw[ c[i] ] = 0 for i =cut[(l+1) mod n], .., p) . 8. Sample next node c[(p+1) mod L] with L −1 probability rw[ x] ∑ j =0 rw[ j ] using roulette wheel rw[]. 9. Update the position counter p ← (p+1) mod L. 10. If (p+1) mod L = cut[(l+1) mod n], go to Step 6. 11. Obtain a new individual string c[]. cut[0] cut[1] segment0

cut[2] segment1

segment2

template T[]

sampling EHM(A)t

new string c[] Fig. 2. An example of EHBSA/WT/3

Let f(x) be the probability density function of the length of a segment to be sampled in EHBSA/WT/n. Then, f(x) is obtained as22 f ( x) =

x⎞ (n − 1) ⎛ ⎜1 − ⎟ L ⎝ L⎠

n −2

.

(6)

Fig. 3 shows the probability density function f(x). For n = 2, f(x) is uniformly distributed on [0, L]. Thus, the length of a segment to be sampled in EHBSA/WT/2 is uniformly distributed on [0, L]. When the length of the segment is small, the EHBSA/WT/2 samples a small number of nodes, performing a kind of local search improvement over the template individual. On the other hand, when the length is large, the EHBSA/WT/2 samples a large number of nodes, performing a kind of global search improvement over the template individual or produces a new string. Thus, we can expect the EHBSA/WT/2 to work by balancing global and local improvements. For n > 2, short segments are more likely to occur.

S. Tsutsui & M. Miki

1 4.0 L 3.5 n=5

3.0 2.5

n=4

f(xx ) 2.0 1.5

n=2

1.0 n=3

0.5 0.0 0.0

0.2

0.4

0.6

0.8

x

1.0 L

Fig. 3. Probability density function f(x)

4. Empirical Study This section applies EHBSA/WO and EHBSA/WT to the flow shop scheduling problem, which is one of the most popular scheduling problems. 4.1. Experimental Methodology

4.1.1. Evolutionary Models The evolutionary model is the same as the model used for symmetrical EHM(S)t in Ref. 21 as follows: The Evolutionary Model for EHBSA/WT: Let the population size be N, and let it, at time t, be represented by P(t). The population P(t+1) is produced as follows (Fig. 4):

Using Edge Histogram Models to Solve Flow Shop Scheduling Problems

1. Edge distribution matrix EHM(A)t described in Subsection 3.2 is developed from P(t). 2. A template individual T[] is selected from P(t) randomly. 3. EHBSA/WT described in Subsection 2.3.2 is performed using EHM(A)t and T[], and generate a new individual c[]. 4. The new c[] individual is evaluated. 5. If c[] is better than T[], then T[] is replaced with c[], otherwise T[] remains, forming P(t+1). P(t) t = t+1

T N

EHM(A)t c

Select better one T

Fig. 4. Evolutionary model for EHBSA/WT

The Evolutionary Model for EHBSA/WO/T and EHBSA/WO/R: The evolutionary model for EHBSA/WO is basically the same as the model for EHBSA/WT except EHBSA/WO uses a pseudo template PT[]. The Evolutionary Model for Two-parent Recombination Operators: To compare the performance of proposed methods with the performance of traditional two-parent recombination operators, we designed an evolutionary model for two-parent recombination operators. For fair comparison, we design it as similar as possible to that of the EHBSA. We generate only one child from two parents. Using one child from two parents is already proposed for designing the GENITOR algorithm by Whitley et al.25 In our generational model, two parents are selected from P(t) randomly. No bias is used in this selection. Then we apply a recombination operator to produce one child. This child is compared

S. Tsutsui & M. Miki

with its parents. If the child is better than the worst parent, then the parent is replaced with the child. 4.1.2. Flow Shop Scheduling Problems and Performance Measures General assumptions of flow shop scheduling problems can be described as follows: Jobs are to be processed on multiple machines sequentially. There is one machine at each stage. Machines are available continuously. A job is processed on one machine at a time without preemption, and a machine processes no more than one job at a time. In this chapter, we assume that L jobs are processed in the same order on m machines. This means that our flow shop scheduling problem is the L-job and mmachine sequence problem. The purpose of this problem is to determine the sequence of L jobs. This sequence is denoted by a permutation string of {0, 1, ..., L–1}. The problem is to find a permutation which minimizes the makespan (i.e., the completion time of all jobs). As test problems, we generated two flow shop scheduling problems, 20-job and 10-machine, and 30-job and 10-machine problems. In designing each problem, we specified the processing time of each job at each machine as a random integer in the interval [1, 99]. We compared EHBSA with popular order-based two-parent recombination operators, namely, the original order crossover OX,24 the enhanced edge recombination operator eER,17 and the partially mapped crossover.7 20 runs were performed. Each run continued until the population has converged, or evaluations reached Emax. Values of Emax were 300,000. The performance is measured by the minimum makespan (best), mean of the minimum makespan (mean) in 20 runs and standard deviation of the minimum makespan (std). Population sizes of 50, 100, 200 were used for EHBSA, and 50, 100, 200, 400, 800, and 1600 for other operators, respectively. As to the bias ratio Bratio in Eq. 3, a Bratio value of 0.02 was used. 4.1.3. Blind Search In solving scheduling problems using GAs, mutation operators play an important role. Several types of mutation operators were proposed. Also,

Using Edge Histogram Models to Solve Flow Shop Scheduling Problems

it is well known that combining GAs with local optimization methods or heuristics greatly improve the performance of the algorithms. However, in this experiment, we use no mutation and no heuristic in order to see the pure effect of applying the proposed algorithms. Thus, the algorithm is a blind search. 4.2. Empirical Analysis of Results Results on the 20-job and 10-machine problem are shown in Table 1. EHBSA/WO/T and EHBSA/WO/R showed obviously poorer performance compared with EHBSA/WT and other two-parent recombination operators. Comparing EHBSA/WO/T and EHBSA/WO/R, we can see that the EHBSA/WO/R is better than EHBSA/WO/T. As was found in Refs 21 and 22 with TSP, EHBSA/WT shows much better performance than EHBSA/WO. In EHBSA/WT/2 with N = 50, the mean scored 1521.4, the best value in the experiment. In the other operators, PMX showed good performance. The mean of PMX with N = 1600 was 1522.5. OX also showed relatively good performance showing mean = 1523.4 with N = 1600. eER which showed good performance in TSP21 showed poor performance in this problem. Comparing the performance of EHBSA/WT with other operators, EHBSA/WT is almost the same as PMX and OX. One big difference between EHBSA/WT and PMX/OX is that EHBSA/WT requires a smaller population size to work than PMX/OX. Results on the 30-job and 10-machine problem are shown in Table 2. Again, the performance of EHBSA/WT is much better than EHBSA/WO. Comparing the performance of EHBSA/WT with other operators, EHBSA/WT is almost the same as PMX and OX, again showing that EHBSA/WT requires a smaller population size to work than PMX/OX. From the results described above, we can see that EHBSA/WT works fairly well in flow shop scheduling problems used in this chapter. It has almost the same performance with popular traditional two-parent recombination operators, OX and PMX, and has much better performance than eER. One interesting feature of EHBSA/WT is that it requires smaller population size than traditional two parent recombination operators. This may be an important property of

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EHBSA/WT. In our experiments, we used a blind search. When we combine EHBSA/WT/n with some heuristics, it would work well with a smaller population size. Table 1. Results on 20-job and 10-machine problem EHBSA EHBSA/WO/T

EHBSA/WO/R

EHBSA/WT/2

EHBSA/WT/3

EHBSA/WT/4

OX

eER

PMX

Performance Mesure best mean std best mean std best mean std best mean std best mean std best mean std best mean std best mean std

50 1540 1563.4 11.8 1523 1538.8 6.6 1519 1521.4 2.5 1519 1524.4 2.8 1520 1525.7 3.7 1533 1566.1 16.2 1567 1593.9 11.6 1542 1562.8 12.9

100 1538 1563.3 14.2 1519 1536.1 8.1 1519 1523.7 5.3 1520 1524.3 4.4 1920 1526.0 5.1 1523 1549.5 12.9 1550 1584.2 15.6 1523 1546.6 8.6

Population Size 200 400 1550 1571.0 11.2 1539 1552.1 8.3 1522 1530.6 3.8 1524 1528.6 2.6 1520 1529.8 5.6 1520 1520 1537.1 1529.1 9.6 7.0 1576 1553 1588.8 1598.0 7.9 13.1 1523 1521 1534.9 1529.0 7.3 5.1

800

1600

1518 1524.6 3.8 1584 1606.3 11.6 1519 1525.7 4.3

1517 1523.4 3.6 1595 1613.1 9.0 1518 1522.5 2.0

5. The Effect of Applying the Tag Node In this section, we explore the effect of introducing the tag node (TN) in a string representation aiming to improve the performance of EHBSAs. In a scheduling problem where a solution is represented by a permutation string, the performance of each string is tightly linked not only to the relative sequence of nodes (jobs) but also to the absolute position of nodes in the string. Since EHM, described in Section 2, has no explicit information on the absolute positions of nodes in each string, it may be useful to introduce additional information on the absolute position of each node in a string. The tag node (TN) proposed in this section is an approach to introduce information on the absolute position of each node in a string. In addition to normal nodes, we add a TN to each permutation string. We

Using Edge Histogram Models to Solve Flow Shop Scheduling Problems

Table 2. Results on 30-job and 10-machine problem EHBSA EHBSA/WO/T

EHBSA/WO/R

EHBSA/WT/2

EHBSA/WT/3

EHBSA/WT/4

OX

eER

PMX

Performance Mesure best mean std best mean std best mean std best mean std best mean std best mean std best mean std best mean std

50 2161 2193.8 20.2 2116 2157.1 21.8 2086 2105.4 8.4 2087 2106.8 8.9 2097 2107.8 6.8 2133 2154.5 13.2 2191 2220.1 19.2 2117 2137.3 14.3

100 2173 2204.1 10.7 2158 2182.3 13.1 2100 2112.2 5.7 2098 2111.0 5.4 2099 2110.5 7.2 2117 2137.0 11.1 2172 2211.1 19.4 2100 2122.8 10.2

Population Size 200 400 2177 2211.7 17.5 2193 2206.0 9.2 2106 2120.3 7.8 2107 2117.9 3.9 2092 2116.3 7.5 2107 2100 2124.3 2115.4 10.0 6.4 2208 2219 2226.0 2231.9 13.2 9.2 2087 2087 2111.1 2105.3 9.1 10.9

800

1600

2087 2107.2 10.0 2214 2244.0 12.3 2087 2094.7 10.3

2132 2141.5 6.1 2229 2253.0 13.9 2096 2114.3 6.5

call a string with a TN a virtual string (VS). String length of a VS is LVS = L+1, where L is the length of real string (RS; string without TN). The TN in VS works as a tag in a permutation string to indicate the first node (job), i.e., the next node following the TN is assumed to be the first node (job) in the solution. The TN is a virtual node because it does not correspond to any real nodes, or jobs. Fig. 5 shows how to obtain RS from VS. In this case, L = 6. We can use any symbol to represent the TN in a VS. However, for implementation convenience, we use an integer number to represent it in a VS. Let us consider the string in Fig. 5, for example. In this example, string length L = 6 and the length of VS LVS = 7. Nodes 0, 1, …, 5 represent real nodes. Then, we assign number 6 to the TN (see Fig. 6). Thus, a VS of length L+1 is represented as a permutation {0, 1, …, L–1, L} corresponding the numbers 0, 1, …, L–1 to real nodes and number L to the TN. With this representation, we do not need any special modification in basic EHBSAs.

S. Tsutsui & M. Miki

VS

0

2

4

3

TN

1

RS (real string)

1

5

0

2

4

3

5

LVS = 7

L=6

Fig. 5. An example of a VS

VS

0

2

4

3

6

1

RS (real string)

1

5

0

2

4

3

5

LVS = 7

L=6

Fig. 6. Representation of the TN in a VS

The effect of applying the TN was tested using the 20-job and 10machine problem in Section 3. The experimental settings are the same as those in Section 3. Table 3 shows results without and with the TN at evaluations = 200,000. The important observation is that in all experiments with EHBSA/WT, except for EHBSA/WT/2 with population size = 50, values of the mean with TN were better than those without TN, although the difference is not so remarkable. On average, the value of the mean with TN is smaller than those without TN by 0.21%. To see the statistical difference of mean values between the models with and without the TN, t values are presented in the table. Since we did 20 runs for each experiment, the value of df = 39. When we use the t-test with the value of 0.05 for the level of significance, t values over 1.6849 satisfy the test. Tables 3 shows only results at evaluations = 200,000. Fig. 7 shows the convergence process of EHBSA/WT/3 with population size = 50 between evaluations = [0, 300,000]. From this figure, we can see that values of mean with TN converge faster than those without TN, taking significantly smaller values.

Using Edge Histogram Models to Solve Flow Shop Scheduling Problems

Since EHM has no explicit information on the absolute position of each node, the TN in EHBSA works as a tag that indicates the initial position in a string. Thus, as we intended, it improves the performance of a problem where in addition to the relative position of each node, the absolute position of each node effects the performance. Here, we must note that without using the TN, EHBSA can somehow maintain information about absolute position of each node in a string of the population. That is because, for example in EHBSA/WT, nodes in a new string that are not sampled inherit the same absolute positions from its template string. Introducing the TN helps the managing of information on absolute position more, and thus increases the performance. Finally, since the TN is treated as just a normal node in algorithms, there is no special processing necessary in introducing the TN. The difference of computational complexity with and without TN is influenced only by the string length; without TN the length is L, and with TN the length is L+1. Table 3. The effect of applying the TN EHBSA EHBSA/WT/2 EHBSA/WT/3 EHBSA/WT/4

TN without with without with without with

mean 1523.2 1523.6 1526.5 1521.8 1528.5 1523.8

50 std 3.3 3.6 3.6 2.6 3.7 3.9

t -0.3591

4.6501

3.7890

Population Size 100 mean std t 1526.8 3.7 4.4776 1522.0 2.8 1526.9 3.9 2.9068 1523.9 2.4 1528.4 4.9 0.3764 1527.9 4.0

mean 1534.0 1526.5 1531.6 1527.1 1532.7 1529.9

200 std 2.8 3.4 3.5 3.3 5.6 4.2

t 7.3937

4.1187

1.7704

6. Conclusions In this chapter, we extended the EHBSAs to solving a flow shop scheduling problem, a typical, well known problem in the area of scheduling. In the flow shop scheduling problem, the edge histogram matrix becomes asymmetrical. This is a big difference from previous studies on EHBSAs. The results showed that EHBSA/WT also worked well on the flow shop scheduling problems and performed significantly better than EHBSA/WO. Comparing the performance of EHBSA/WT

S. Tsutsui & M. Miki

with some other popular two-parent recombination operators for permutations, EHBSA/WT is almost the same as PMX and OX. One big advantage of EHBSA is the use of significantly smaller population sizes than those that are necessary with PMX and OX as was observed in the previous studies. Although we used a blind search in this study, if we combine EHBSA/WT with some heuristics, it would work well with a smaller population size. Thus, we can confirm that the EHBSA also works well on flow shop scheduling problems. We also confirmed that applying the tag node (TN) in a string representation enhances the performance of the EHBSAs. Despite the promising results, the chapter should be understood as one of the first steps toward scalable solution of scheduling problems with PMBGAs, because there are many opportunities for further research related to the proposed algorithms. The effect of parameter values of Bratio, number of cut points of the template n, and size of population N, on the performance of the algorithm must be further investigated. We experimented with EHBSAs using a blind search to test the pure mixing capability of the proposed algorithms. But we must still test the algorithms with appropriate heuristics in problems with large numbers of nodes.

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Fig. 7. Convergence process of EHBSA/WT/3 with and without TN

Using Edge Histogram Models to Solve Flow Shop Scheduling Problems

Acknowledgments The authors gratefully acknowledge Prof. Martin Pelikan for his valuable comments on this work. This research is partially supported by the Ministry of Education, Culture, Sports, Science and Technology of Japan under Grant-in-Aid for Scientific Research number 13680469, and a grant to RCAST at Doshisha University from the Ministry of Education, Culture, Sports, Science and Technology of Japan. Reference 1.

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