Chung Yuan Christian University

Intelligent Manufacturing System Midterm Report

10274014陳柏廷

10274015廖文璿

10274054彭惠華

10274059薛來德

4-30-2014

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1.introduction ................................................................................................................. 3

1.1. Single Row Facility Layout Problem (SRFLP) Definition ........................ 3

1.2. Estimation of Distribution Algorithm (EDAs) .......................................... 3

1.3. Study Objectives and Problem Definition ................................................. 3

2. mathematical formulation ...................................................................................... 5

2.1. Single Row Facility Layout Problem (SRFLP) ......................................... 5

2.1.1. Notations ........................................................................................ 5

2.1.2. Assumptions ................................................................................... 5

2.2. Enhanced Single Row Facility Layout Problem ........................................ 6

2.2.1. Notations ........................................................................................ 6

2.2.2. Assumptions ................................................................................... 7

3. Problem solving methods ....................................................................................... 8

3.1. Assumptions & Notations .......................................................................... 9

3.2. extended Chromosomes Genetic Algorithm (eACGA) ........................... 10

3.3. EDAhybrid ............................................................................................... 12

3.4. EDA Part .................................................................................................. 12

3.5. PSO Part ................................................................................................... 16

3.6. Tabu Search Part ...................................................................................... 16

4. results analysis ..................................................................................................... 19

4.1. Parameter setting ...................................................................................... 19

4.2. EDAHybrid for SRFLP............................................................................ 20

4.3. EDAHybrid for enhanced SRFLP ........................................................... 21

5. conclusion and improvement ............................................................................... 22

TABLE OF CONTENTS

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5.1. Improvement ............................................................................................ 22

6. References ............................................................................................................ 24

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1. INTRODUCTION

1.1. Single Row Facility Layout Problem (SRFLP) Definition

SRFLP is a method where multiple products with different production volumes and

different process routings need to be manufactured. The objective of SRFLP is to set

up facilities so that sum of distances between all facility pairs can be minimized.

1.2. Estimation of Distribution Algorithm (EDAs)

EDAs are stochastic optimization techniques that explore that space of potential

solutions by exploiting inter variable dependency and sampling probabilistic models of

promising candidate solutions problem definition (Hauschild & Pelikan, 2011). EDA

are predicted to be potentially effective to solve SRFLP (Ou-Yang & Utamima, 2013).

As more and more iteration is to be run, EDAs will suffer diversity loss that grows

rapidly; however, EDAs improve the solution quality in the first few runs efficiently.

1.3. Study Objectives and Problem Definition

The proposed study by (Ou-Yang & Utamima, 2013) creates a new Hybrid Estimation

of Distribution Algorithm (EDA Hybrid), an algorithm which consist of hybridization

of Estimation of Distribution Algorithm (EDAs), Particle Swarm Optimization (PSO),

and Tabu Search Algorithm to surmount the basic and enhanced SRFLP. PSO is used

as metaheuristic algorithm for maintaining the diversity of EDA. Tabu Search explores

the global best value achieved in every iteration. Other hybridization algorithm,

extended Artificial Chromosomes Genetic Algorithm (eACGA) is built as a benchmark.

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Objectives of this research are as follows (Ou-Yang & Utamima, 2013):

1. Develop EDA hybrid, a new meta-heuristic algorithm which is the hybridization

of Estimation of Distribution Algorithm Particle Swarm Optimization and Tabu Search

Algorithm.

2. To design an enhanced Single Row Facility Layout Problem that considers more

constraints which include not only flow, length, and clearance space, but also the

installation cost and safety reason.

3. To apply EDAhybrid algorithm to solve the basic and enhanced SRFLP.

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2. MATHEMATICAL FORMULATION

2.1. Single Row Facility Layout Problem (SRFLP)

The research proposed (Ou-Yang & Utamima, 2013) considers SRFLP with different

sizes of facilities. The objective function is to minimize 𝑍 which stands for sum of the

distances between all facility pairs. ABSMODEL, proposed by Heragu and Kusiak

(1991), is well known model for solving SRFLP. ABSMODEL is illustrated in Eq. (1).

𝑚𝑖𝑛 𝑧 = ∑ ∑ 𝑇𝑖𝑗𝑑𝑖𝑗

𝑛

𝑗=𝑖+1

𝑛−1

𝑖=1

𝑠. 𝑡: 𝑑𝑖𝑗 ≥1

2(𝐼𝑖 + 𝐼𝑗) + 𝑠𝑖𝑗 (1)

𝑑𝑖𝑗 ≤ 0; 𝑖 = 1,2, … , 𝑛 − 1; 𝑗 = 𝑖 + 1, … , 𝑛

𝑑𝑖𝑗 =1

2(𝐼𝑖 + 𝐼𝑗) + 𝐷𝑖𝑗 (2)

2.1.1. Notations

𝐼𝑖 The length of each facility i.

𝐼𝑗 The length of each facility j.

𝑇𝑖𝑗 Traffic loads between facilities i and j.

𝑑𝑖𝑗 Distance between the centers of the facilities i and j.

𝑠𝑖𝑗 The necessary clearance or gap between the two facilities.

𝐷𝑖𝑗 The space between facilities i and j.

2.1.2. Assumptions

𝑛 × 𝑛 𝑚𝑎𝑡𝑟𝑖𝑥 𝑇 = [𝑇𝑖𝑗] are given.

𝑑𝑖𝑗 are defined as equation (2).

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The distance between two facilities is supposed to be taken between their mid points.

2.2. Enhanced Single Row Facility Layout Problem

Combining the safety constraint and the fixed cost concept, we could enhance the

SRFLP objective function to be more extensive and more applicable in real case by

considering the safety constraint and installation cost (Ou-Yang & Utamima, 2013).

Enhanced SRFLP is an enhanced model of single row facility layout which considers

length of facilities, traffic loads between facilities, installation cost of each facility and

also safety constraints. The objective function of Enhanced SRFLP is shown in the

following equation:

min 𝑧 = ∑ ∑ 𝑇𝑖𝑗𝑑𝑖𝑗𝐵𝑖𝑗 + ∑ ∑ 𝛿𝑖𝑥𝐶𝑖𝑥

𝑛

𝑥=1

𝑛

𝑖=1

𝑛

𝑗=𝑖+1

𝑛−1

𝑖=1

s. t. : 𝑑𝑖𝑗 ≥1

2(𝐼𝑖 + 𝐼𝑗) + 𝑠𝑖𝑗 (3)

𝑑𝑖𝑗 ≤ 0; 𝑖 = 1,2, … , 𝑛 − 1; 𝑗 = 𝑖 + 1, … . , 𝑛

2.2.1. Notations

𝐼𝑖 The length of each facility i.

𝐼𝑗 The length of each facility j.

𝑇𝑖𝑗 Traffic loads between facilities i and j.

𝑑𝑖𝑗 Distance between the centers of the facilities i and j.

𝑠𝑖𝑗 The necessary clearance or gap between the two facilities.

𝐷𝑖𝑗 The space between facilities i and j.

𝛿𝑖𝑥 The permutation matrix variable (equals to 1 if facility i is assigned to location x,

otherwise its equals to 0).

𝐶𝑖𝑥 The construction cost of assigning facility i on location x.

𝐵𝑖𝑗 The interactive cost of assigning facility i on the location neighboring facility j.

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2.2.2. Assumptions

To calculate z, the penalty cost will be given to a solution violating the safety constraint.

𝐵𝑖𝑗 is related to safety constraint handling.

𝐵𝑖𝑗 equals to penalty cost if facility i cannot be adjacent to facility j due to safety

constraint, otherwise it equals to 1.

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3. PROBLEM SOLVING METHODS

Start

Develop modified eACGA ,EDA,GA to

deal with SRFLP problem

Set the developed model as a benchmark

Performance decreased on larger problem ?

Test the benchmarked

method

Develop a PSO and EDA algorithm

Performance cannot reach near optimal solution?

Add a local search using Tabu search

Named the proposed method

EDAHybrid

End

Yes

Yes

Figure 1. The flow of the research

The figure above shows the big picture of the research done by the researchers

(Ou-Yang & Utamima, 2013). Before they develop the EDAHybrid, they first develop

some models to be the benchmark of the proposed method. The research first, develop

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eACGA, EDA, GA method to deal with SRFLP problem. After the test, it is found that

the proposed benchmark cannot produce a good near optimal solutions. Afterwards, the

researchers choose EDA to be combined with PSO as it is found that PSO can perform

better than GA. On the second test using EDA PSO Hybrid Algorithm it is found that

the results of the solution is still not optimal or as the researchers expected. Then, finally

the researchers develop a local search using Tabu search to solve the performance

problem; and named it EDAHybrid.

3.1. Assumptions & Notations

In this research by (Ou-Yang & Utamima, 2013) the assumptions and notations can be

defined as below:

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 𝜖 [1, 𝑛] . Different sequences in X represent different layout

solutions. The representation of chromosome is shown in Table 1.

i 1 2 3 4 5

X[i] 5 3 2 1 4

Table 1 Chromosome Illustration

Source:(Ou-Yang & Utamima, 2013)

Table 1 shows the example of chromosome representation with 5 facilities.

X[2] = 3 means assigning facility 3 to location 2.

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3.2. extended Chromosomes Genetic Algorithm (eACGA)

Start

Initialize all variables

Initialize population of chromosomes

randomly

g mod 2=0

Selection Decide cross over rate and mutation rate to be 0.7 and 0.3 respectively.

Yes No

Univariate and Bivariate

Probabilistic Modelling

Sampling

EDA GA

Crossover and Mutation process

Population Replacement

Fitness Calculation

Set elitism strategy to be 10%

g = max generation?

Stop

Yes

No

Figure 2. eACGA Flowchart

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The eACGA method proposed by (Ou-Yang & Utamima, 2013) procedures starts

with initialization of all variables. Afterwards we will initialize the population of

chromosomes randomly. If g, the current generation of iteration is able to be divided by

2, then EDA procedure is executed, otherwise GA procedure is executed.

The EDA procedure is started with the selection of chromosomes with better

fitness values. Afterwards the selected chromosomes that are grouped is being applied

probabilistic models form by the ordinal (univariate model) and dependency (bivariate

model) matrices.

GA procedure start with assigning the crossover and mutation rate to be 0.7 and

0.3 respectively. The crossover rate and mutation rate decide whether chromosomes

mate and mutate, respectively. Roulette wheel selection is used to choose the parental

chromosomes. Two point crossover operator mates two chromosomes with better

fitness values to be the parents for the next iteration. The mutation then probably occurs

and chooses between three mutations operators that flip, swap, or slide, and that is to

be undertaken.

The chromosomes solutions afterwards will replace the previous iteration

populations. The fitness value of each chromosomes is calculated. Develop an elitism

strategy with the rate of 10%. If the generation have not exceeds the maximum iteration,

then the group of the new generation chromosomes are to be processed into the next

iteration.

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3.3. EDAhybrid

Figure 3. EDAHybrid Pseudocode

Source: (Ou-Yang & Utamima, 2013)

3.4. EDA Part

In every generation g, a group of C particles which have better fitness are selected. The

particles are labeled as X1, 𝑋2, … . , 𝑋𝐶 , where C is half of population size. The

distribution of parental particles are interpreted by sampling new solutions from the

univariate and bivariate probabilistic model (Ou-Yang & Utamima, 2013).

1. Set up and initialize EDAhybrid parameters;

2. Initialize swarm P randomly;

3. Compute pBest and gBest;

4. for g = 1 until maxGen do

5. if g is divisible by 2

6. Select S;

7. Calculate Univariate probabilistic model(S);

8. Calculate Bivariate probabilistic model(S);

9. Sampling ();

10. else

11. Calculate velocity of P();

12. Update P();

13. end if

14. Calculate fitness value;

15. Update pBest and gBest;

16. TabuSearch(gBest);

17. if tabuSol < gBest

18. gBest ← tabuSol;

19. end if

20. if g is not divisible by 2

21. Elitism();

22. end if

23. end for

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(Ou-Yang & Utamima, 2013) develop their own univariate probabilistic while the

bivariate probabilistic is obtained from the previous research. The univariate model,

∅𝑖[𝑖] in equation (4) shows the importance of facilities in the sequence. It represents

how many time facility i is placed at position [i] at current generation. Ai[i]k is set to

facility 1 if facility i is placed at position [i], otherwise it is set to 0.

∅𝑖[𝑖] = ∑ Ai[i]k𝐶

𝑘=1 (4)

where i=1,….,n; k=1,….,C.

Source:(Ou-Yang & Utamima, 2013)

The bivariate or dependency probabilistic model 𝜓𝑖𝑗 in Equation (5), represents

how many times facility j is placed immediately after facility i. 𝐵𝑖𝑗𝑘 is set to 1 if facility

j is placed next to facility I, otherwise it is set to 0..

𝜓𝑖𝑗 = ∑ 𝐵𝑖𝑗𝑘𝐶

𝑘=1 (5)

where i,,j=1,….,n; k=1,….,C.

Source:(Ou-Yang & Utamima, 2013)

Let 𝑃𝑖[𝑖] be the probability value of assigning facility i at position [i]. Selecting

facility i has better probability value than other facilities when both probabilistic models’

statistical information is used. ∅𝑖[𝑖] is added by 𝜓𝑖𝑗 . They will summarize tehe

probability values of all unassigned facilities that could be set at position [i].

For every particles’ offspring O1, 𝑂2, … , 𝑂2𝐶, some methods are used to assign

facilities to a specified location. Selecting a facility at the first location, our approach

picks randomly the first facility that appears in the group of selected C particles, that is

X[1]. We think that this approach is more promising than the one proposed in previous

research, which generates facility randomly.

For assigning the remaining facilities, the authors modified formula from previous

research which multiplying ∅𝑖[𝑖] with 𝜓𝑖𝑗. They change the formula to become adding

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∅𝑖[𝑖] with 𝜓𝑖𝑗, that is Eq.(6). The reason is because add operation, not multiplication

one, gives better result when tested in EDAhybrid.

P𝑖[𝑖] =∅𝑖[𝑖]+𝜓𝑖𝑗

∑ (∅𝑖[𝑖]+𝜓𝑖𝑗)𝑖𝜖Ω (6)

where Pi[i]: the probability value of assigning facility i at position [i], [i] = 2, 3, . . ., n;

i,j = 1, 2, . . ., n, Ω = set of unassigned facilities. The flowchart shown in Fig. 4

demonstrates the assignment flowchart for placing facilities into location 2 until n.

Where:

𝛺 = 𝑠𝑒𝑡 𝑜𝑓 𝑢𝑛𝑎𝑠𝑠𝑖𝑔𝑛𝑒𝑑 𝑓𝑎𝑐𝑖𝑙𝑖𝑡𝑖𝑒𝑠.

𝐹 = 𝑠𝑒𝑡 𝑜𝑓 𝑎𝑠𝑠𝑖𝑔𝑛𝑒𝑑 𝑓𝑎𝑐𝑖𝑙𝑖𝑡𝑖𝑒𝑠. 𝐹 𝑖𝑠 𝑒𝑚𝑝𝑡𝑦 𝑎𝑡 𝑏𝑒𝑔𝑖𝑛𝑛𝑖𝑛𝑔

𝜃 = 𝑟𝑎𝑛𝑑𝑜𝑚 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑑𝑟𝑎𝑤𝑛 𝑓𝑟𝑜𝑚 𝑈(0,1)

𝑖 = 𝑎 𝑠𝑒𝑙𝑒𝑐𝑡𝑒𝑑 𝑓𝑎𝑐𝑖𝑙𝑖𝑡𝑦 𝑏𝑦 𝑝𝑟𝑜𝑝𝑜𝑝𝑟𝑡𝑖𝑜𝑛𝑎𝑙 𝑠𝑒𝑙𝑒𝑐𝑡𝑖𝑜𝑛

𝑘 = 𝑡ℎ𝑒 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝑖𝑛𝑑𝑒𝑥 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑎𝑐𝑖𝑙𝑖𝑡𝑦′𝑠 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛

𝑛 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑐𝑖𝑙𝑖𝑡𝑖𝑒𝑠

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Start

Sampling the population

Set assigned facilities = φ

k≤ n

Set φ using univariate random (0,1) distribution

Set assigned facilities to location

to proportional location i

Set element index of

the facility’s position=k. k=2

Yes

Select a facility

which satisfy φ <Pi[i] ,where i Є Ω

EndNo

Figure 4. Flowchart of assignment of facilities

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3.5. PSO Part

PSO procedure starts when current generation is odd and begins with initialization of

all related parameters. Next, the velocities calculation of every particle is operated. If

the velocity is larger, the particle is more likely to change to a new permutation

sequence. The velocity update formula remains the same like provided in previous

research, while the particle update process is changed. EDAhybrid uses permutation

based particle updating based on concept from previous research.

3.6. Tabu Search Part

Start

TabuSolution=BestSolution

i<=n

Construct Swaplist

Calculate CostFor every move

Move is tabu &

Move’s cost > tabusolution

Give move penalty cost

Update tabulist

Calculate minimum cost of currentSolution

currentSolution < tabuSolution

tabuSolution = currentSolution

tabuSolution < BestSolution

BestSolution = tabuSolution

End

Y

Y

Y

N

N

N

N

Y

Figure 5 Tabu Search Algorithm

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Figure 6. Tabu Search Pseudocode

Source:(Ou-Yang & Utamima, 2013)

The best solution from individual generated by EDA or PSO part in every

generation is the input for Tabu Search. Fig. 6 provides the pseudocode of Tabu Search

in EDAhybrid. The Tabu Search procedure is performed for n (equal to the number of

facilities) generations. Swaplist is constructed in every tabu generation; this process is

shown in line 3 of Fig. 4. Swaplist consists of three columns, the first and second

columns are the specified facility that will be swapped and the last column provides the

cost of that move.When calculating a move’s cost, the algorithm checks whether the

move is taboo or not by looking in tabulist. If the move is listed in tabulist and the

move’s cost is greater than tabuSolution, a penalty cost is added; otherwise the move is

accepted without a penalty cost. The algorithm records the move with minimum cost in

line 8. Tabulist is updated in line 9. If the currentSolution is better than tabuSolution,

then tabuSolution is updated. The iterations continue to run until n generations.

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At the end of this local search, the algorithm checks whether tabuSolution, as the

output of Tabu Search, can find better solution than the current best solution. This part

is shown in line 14 until 16. If tabuSolution succeeds to get lower cost, then the current

best solution is replaced with tabuSolution.

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4. RESULTS ANALYSIS

4.1. Parameter setting

The problem set some parameter which is general, such as population size and

generation size. Population and generation size here is based on the size of the problem.

The larger the number of facilities, the more population and generation are needed to

get an optimal solution. The number of facilities here is represented by n. The number

of generation and population size here can be described in the table below:

Number of

facilities n

Problem Size Generation Size

4-15 facilities 4n 10n

20 facilities 5n 350 generations

30 facilities 6n 800 generations

Table 2 Generation and Problem Size in Regards of number of Facilities

The elitism used in the proposed problem by (Ou-Yang & Utamima, 2013) are

10% which cited from the previous research. To determine the PSO parts parameter the

researchers use such equation to determine:

w = (maxgen − currentgen)/maxgen (7)

Source: (Ou-Yang & Utamima, 2013)

Regarding the accelerators c1, c2 the researchers also site from the previous

research which is [2.05,2.05] respectively. Tabu search in here is used as the local

search, so that the algorithm is set to become faster with n generations, swaplist size

equal to 2(n − 1) and tabu tenure is [n/5].

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4.2. EDAHybrid for SRFLP

In the research by (Ou-Yang & Utamima, 2013), the researchers uses 15 problem and

compare EDAHybrid with another algorithm. The researchers in here measure the error

rates percentages using such function:

error(%) =𝑚𝑒𝑎𝑛− min𝑎𝑙𝑙

min𝑎𝑙𝑙× 100% (8)

The researchers also using Matlab function for standard deviation to compute the

standard deviation. The obtained value will be returned onto standard deviation using

function in Eq.(9).

s = (1

𝑛−1∑ (𝑋𝑖 − )2𝑛

𝑖=1 )

1

2 (9)

Several table below are the results of the simulation using Matlab code obtained from

the researcher’s results (Ou-Yang & Utamima, 2013):

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4.3. EDAHybrid for enhanced SRFLP

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5. CONCLUSION AND IMPROVEMENT

The paper proposed by (Ou-Yang & Utamima, 2013) delivers a new metaheuristic

algorithm named Hybrid Estimation of Distribution Algorithm (EDAHybrid), which

consist of hybridization of EDA, PSO, and Tabu Search. The proposed algorithm runs

with EDA and PSO alternatively every two generations and then Tabu Search as a local

search is added at the end of every iteration.

Computational results of 15 benchmark problems shows that the performance of

the proposed algorithm always achieved near optimum solution in basic SRFLP. The

error rates of the proposed algorithm is proven to be better than eACGA, EDA, PSO,

and GA. The proposed metaheuristic develop two probabilistic model in the

EDAHybrid Algorithm which is univariate and bivariate model. EDAHybrid proves to

perform better than eACGA because it adds Tabu Search as a local search. Enhanced

SRFLP problem is designed to make basic SRFLP closer to real case by adding more

constraints. Compared to other algorithms, EDAHybrid can get the lowest minimum

and error rates in enhanced SRFLP.

5.1. Improvement

In the proposed paper by (Ou-Yang & Utamima, 2013) they always stated that

their algorithm achieved an optimum value, however it is known that in NP-Complete

Problem there is no possibility that metaheuristic can reach an optimum solution.

The proposed research can be improved in the way the metaheuristic worked. We

can use another algorithm to combine with EDAHybrid instead of PSO, we can use

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another algorithm such as the recently proposed bat algorithm. Another method is to

tweak the parameter of the algorithm to find another possible better solutions.

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6. REFERENCES

Hauschild, M., & Pelikan, M. (2011). An introduction and survey of estimation of

distribution algorithms Swarm and Evolutionary Computation (Vol. 1, pp.

111-128).

Ou-Yang, C., & Utamima, A. (2013). Hybrid Estimation of Distribution Algorithm

for solving Single Row Facility Layout Problem. Computers & Industrial

Engineering, 66(1), 95-103. doi: 10.1016/j.cie.2013.05.018