Simulation and Genetic to FLP

8/12/2019 Simulation and Genetic to FLP

1/15

INT. J. PROD.RES.,2000, VOL. 38, NO. 17, 43694383

Facility layout optimization using simulation and genetic algorithms

FARHAD AZADIVAR{* and JOHN (JIAN) WANG{

Traditionally, t he objective of a facility layout problem has been to minimize thematerial handling cost of the manufacturing system. While it is important toreduce the amount of material handling, the traditional methods do not addressthe actual time at which the material is transported. In todays short cycle timeproduction environments, the timing of material movement may have a biggerimpact on the productivity of the system than its cost. In this paper, a facilitylayout optimization technique is presented that takes into consideration the

dynamic characteristics and operational constraints of the system as a whole,and is able to solve the facility layout design problem based on a systems per-formance measures, such as the cycle time and productivity. Each layout solutionis presented in the form of a string that is suitable for analysis by a geneticalgorithm technique. These solutions are then translated into simulation modelsby a specially designed automated simulation model generator. Geneticalgorithms are used to optimize the layout for manufacturing eectivenesswhile simulation serves as a system performance evaluation tool. Combinedwith a statistical comparison technique to reduce the simulation burden, thetest results demonstrate that the proposed approach overcomes the limitations

of traditional layout optimization methods and is capable of nding optimal ornear optimal solutions.

1. Introduction

The facility layout problem in a manufacturing setting is dened as the

determination of the relative locations for, and allocation of, the available space

among a given number of workstations. Although most facility layout solutions

have, in the past, focused on minimizing the amount of transportation, the eect

of a given layout design on the production function of a manufacturing system is

much more than just the cost of material handling. While material handling cost

remains critical, shorter cycle times have become much more important in todays

manufacturing systems. In other words, when a certain material is moved is as

important, if not more important, as how much it costs to move it. Rapid develop-

ment of new products, coupled with short delivery times demanded by customers,

are the bases of the time-based competitive strategies rapidly being adopted by

leading rms in many industries. Responsive delivery without inecient excess

inventory and short manufacturing cycle times are the practical considerationsthat have strong impacts on the layout design and should be incorporated into the

layout design process as genuine concerns.

International Journal of Production Research ISSN 00207543 print/ISSN 1366588X online # 2000 Taylor & Francis Ltd

http://www.tand f.co.uk/journals

{Department of Industrial and Manufacturing Systems Engineering, Kansas StateUniversity, Manhattan, KA 66506, USA.

{Talus Solutions Inc., Waterstone, Suite 300, 4751 Best Road, Atlanta, GA 30337-5609,USA.

* To whom correspondence should be addressed. Present address: College of Engineering,University of Massachusetts, Dartmouth, MA 02747-2300, USA. e-mail: [email protected]


2/15

Because of complexity of the manufacturing systems, usually closed-form

analytical expressions for objective functions do not exist. During the past three

decades, a variety of approaches have been proposed to deal with facility layout

optimization problems. In order to come up with analytical objective functions, most

of these approaches limited themselves within the assumption that the volume ofmaterial ow between workstation pairs is xed and resources are always available.

In some cases, there have been a few attempts to take the dynamic characteristics of

systems into considerations as well. Techniques such as dynamic programming

(Rosenblatt 1986), and fuzzy logic theory (Cheng and Gen 1996) have been used

to model such uncertainties.

However, we believe that, in order to account for all the impacts a layout design

has on the performance of a system, a more detailed model of the system needs to be

considered for evaluation of the performance measures. To accomplish this, we use

computer simulation. The problem with computer simulation models is that they do

not yield themselves easily to optimization processes. In this paper, it is proposed to

use an integrated solution procedure that optimizes facility layout designs using

simulation as the means of evaluating the objective function. This provides an addi-

tional exibility in optimization because, in addition to the usual quantitative vari-

ables, evaluation by simulation allows consideration of qualitative decision variables

that analytical objective functions are not equipped to incorporate.

One of the promising methods of optimizing problems whose performances are

evaluated by a simulation model, especially when qualitative variables are involved,is the use of Genetic Algorithms (GA). Azadivar and Tompkins (1999) proposed a

simulation model generator with a GA-based optimum seeking algorithm capable of

optimizing simulation models whose performances are functions of qualitative and

structural decision variables of the system. Zhang (1997) extended the technique to

more general exible manufacturing systems. The work presented here is a method-

ology that is based on this approach for facility layout design where the objective

function is a measure of an actual system performance rather than just the volume of

materials handled.

2. Problem statement

Consider a manufacturing system consisting ofm workstations in which n types

of parts, each requiring a set of tasks (operations), are to be processed. A work-

station may consist of a single machine, a cell of several machines, an inspection

centre, a paint booth, etc. The parts require processing on dierent subsets of the m

workstations and have dierent processing times in each workstation. Each work-

station has its own queuing discipline and breakdown distribution. The system iseither a pull or a push type. In addition, let the area of the shop oor, the area

required by each workstation, the time delay in each workstation, capacity and speed

of the material handling devices, and the precedence constraints of tasks be given. A

desired design for the system requires an arrangement of these workstations into the

shop oor such that a certain measure of performance is optimized.

The main assumptions for this problem are as follows.

. The work areas of workstations are rectangular in shape and their orientations

are known.. Every workstation works only one part at a time.. Every transporter carries only one type of part at a time.

4370 F. Azadivar and J. Wang


3/15

. The operations are not pre-emptable.

. The operating sequences of tasks are the same for the same part types.

. The objective of the facility layout design is to minimize some measure of the

system performance (e.g. production completion time of the parts produced in

the system, while preserving the stated constraints).Although the procedure being described in this paper is suitable for all types of

layouts, here we describe the procedure for free layout problems, which are the

most general (and the most dicult) facility layout systems. A free layout is dened

as follows.

There is a set ofm workstations, denoted byfMig, i 1; 2;. . . ; m. The area thateach workstation occupies is restricted to be rectangular and is characterized by its

length li, width wi and length and width clearances of cli and cwi, respectively. A

facility layout solution for a given m-workstation plant consists of a boundedrectangle, R, partitioned by horizontal and vertical line segments into m non-over-

lapping rectangular regions, denoted by fri, i1; 2;. . . ; m. Each region ri, withwidthxiand length yi, must be large enough to accommodate one workstation Mi

plus its clearances.

3. Use of genetic algorithm in facility layout design

Genetic Algorithms (GA), proposed by Holland (1975), are heuristic search and

optimization techniques that imitate the natural selection and biological evolution-ary process. In a GA approach to optimization, feasible solutions to the problem are

encoded in data structures in the form of a string of decision choices that resemble

chromosomes. The algorithm maintains a population of individuals or chromosomes

(solutions) that evolve as chromosomes are created and discarded. Each chromo-

some comprises a number of genes (decision choices), that describe various aspects of

a particular solution. The layout design corresponding to each chromosome is char-

acterized by its tness, which is measured by its objective function value. A genera-

tion consisting of surviving individuals of the previous population and new

individuals or ospring is generated through reproduction by means of crossover,

mutation, and selection of their parents chromosomes.

An eective layout of workstations can signicantly cut down manufacturing

lead times. Unfortunately, the complexity of this task increases exponentially as

the number of workstations increases. There are n! dierent ways of arranging n

workstations into n locations. If all workstations are of equal area, or can be physi-

cally interchanged without altering the overall adjacency or distance relationships

among the remaining workstations, it is easy to specify, in advance, a nite number

of potential sites for these workstations to occupy. Given this, the layout problemcan be modelled as a quadratic assignment problem (QAP). If we allow workstations

unequal areas, their respective dimensions and the clearance requirements between

them will determine the distance between two workstations. In such a case, since

distances between locations are not equal and cannot be predetermined, it becomes

extremely dicult even to describe feasible solutions.

During the past three decades, numerous heuristic methods have been developed

to obtain some good, rather than optimal, solutions for layout problems. The pri-

mary diculties associated with these problems are the vast number of possible

physical layouts, and the existence of many relatively poor local optima. For such

a problem, one might expect parallel search methods to perform better than strictly

4371Facility layout optimization


4/15

serial searches, and randomized search methods to perform better than greedy or

enumerative searches. Genetic algorithms combine both of these attributes in a

parallel, stochastic heuristic.

As a powerful and broadly applicable stochastic search and optimization tech-

nique, genetic algorithms have successfully been applied in various areas of industrialengineering, such as production scheduling and sequencing, reliability design, vehicle

routing and scheduling, group technology, transportation, and many others. The

technique has also been applied to the facility layout problem (Tate and Smith 1995,

Cheng and Gen 1996, Meller and Gau 1996, Tam 1992). However, these published

works are mostly material handling cost driven and do not put enough emphasis on

the performance measures that are time driven and are complex functions of the

layout design (e.g. production rate, cycle time). To evaluate these complex measures,

simulation modelling is often the only feasible method. The approach proposed here,

which combines simulation modelling and a genetic algorithm, provides a unique

opportunity to address this issue.

3.1. String representation of a layout design

Most of the concepts in modelling layout problems for application of genetic

algorithms in this work have been adopted from Cheng and Gen (1996). A brief

description of their approach in representing these problems and using genetic

algorithm operators is presented here.

A free layout type facility design can be represented as a slicing structure. Slicingis the process of cutting a rectangular region into two smaller rectangular regions by

either a horizontal or a vertical line segment (gure 1(a)). The line segment is called

the cut-line. The slicing operations are repeated for each newly formed rectangle,

with the slice-line direction chosen to be perpendicular to the previous slice line. A

slicing structure is constructed by recursively partitioning a rectangle R(i.e. the oor

plan) in such a way that each rectangular partition in the slicing structure cor-

responds to the space allocated to a workstation.

An equivalent representation of a slicing structure is a slicing tree. A slicing tree is

a binary tree, which shows the recursive partitioning process that generates a slicing


*

1

* + +

111 2

1222

11

1

1

2

22 2

12* 21* 12+ 21+

(a) Slicing Structure

(b) Slicing Tree

(c) Reverse Polish Expression

Figure 1. Slicing structure, slicing tree and reverse Polish expression.


5/15

structure. Let the operation of a horizontal cut and a vertical cut be denoted by the

position symbols * and, respectively. Each symbol is explained pictorially in gure1(b). Each internal node represents the way a rectangular partition is cut. Partitions

reserved for workstations reside in the leaves of the tree. Each leaf is assigned a

unique integer corresponding to the identier (id) of a workstation. If we recursivelyprint out the left subtree and the right subtree, then the position symbol of a slicing

tree, a postx expression called Reverse Polish Expression, is obtained (gure1(c)).

This representation method yields itself very well to the coding scheme of chro-

mosomes in genetic algorithms. A chromosome will then have m dierent work-

station numbers and m1 position symbols where m is the number ofworkstations. Slicing structures comprising mgiven workstations can be represented

by slicing trees or Reverse Polish Expressions over the symbol setP f1; 2;. . . ; m; *; g (Gen and Chen 1997). Figure 2 demonstrates the process

of constructing a layout from its Reverse Polish Expression for a 6-station facility

layout problem.

3.2. Crossover

We employed a special form of crossover operation as suggested by Cheng and

Gen (1996) to preserve the feasibility of solutions. In this operation, an ospring

chromosome is generated by adopting workstation numbers from one parent and

position symbols from the other. An example of crossover operation is shown in

gures 3 and 4. Suppose we have two parents, p1 and p2. The crossover operatorcopies the workstation numbers from parents p1 into the corresponding positions in

an ospring o. Then it copies position symbols from p2, by scanning from left to

right, to complete the ospring o.

3.3. Mutation

Random altering, inverting and swapping are used as a mutation operation

(that is, altering a position symbol to the opposite one (gure 5), inverting a

sequence of adjacent position symbols or a sequence of adjacent facility numbers

(gure 6), swapping two adjacent position symbols or two adjacent facility numbers

(gure 7). Mutation performed in this way can also guarantee to generate legal

ospring.

3.4. Selection

The task of selection in the genetic algorithms is to allocate the reproductive

opportunities to each chromosome such that the chromosomes with higher tness

value are more likely to survive to the next generation. Selection directs a genetic

algorithm search toward promising regions in the search space. The degree to whichthe better chromosomes are favoured is dened as the selection pressure. Typically,

higher selection pressure indicates that more of the high tness chromosomes are

selected.

Tournament selection (Brindle 1981, Goldberg 1989) is a selection approach with

both random and deterministic sampling features. This method randomly chooses a

set of chromosomes and picks out the one with the highest tness value (the winner)

for reproduction. The number of chromosomes in the set is called tournament size.

Usually, tournaments are held between pairs of chromosomes (tournament size 2),and the selection process is repeated until a desired size of reproduction set has been

formed. The tournament selection is ecient, simple to code, has no scaling problem,



6/15

and is capable of adjusting its selection pressure. This selection method has been

employed in this work.

In a facility layout design process based on simulation and genetic algorithm, the

tness functions are almost always stochastic. The tness value of the chromosome,

which is the output of a simulation experiment, could be viewed as one realization of

a random variable whose mean corresponds to the presumed true response. Since the

selection process is based on tness values, random tness functions cause the selec-

tion process itself to be random as well. However, the precision of the tness values

can be improved by replicating the simulation experiment and obtaining a narrower


+

+

+ *

*1 2 3

54

6

cut-point

12+345+6**+

+

+ *

*1 2 3

54

6

(12+) 345+6**

+

*

54

6

+

54

1 2

3

45+6*

1 2

3

45+

6

1

6

54

3

2

Figure 2. Constructing a layout from its reverse Polish expression.


7/15

condence interval. The more replications are performed the better will be the accu-

racy of the solution obtained. Welshs (1938) statistical comparison was employed

for determining the winners of the tournaments.

Since simulation experiments are usually very time consuming, a variable number

of replications per solution are used. This allows us to make fewer replications when

the dierence in the tness values is relatively large, and to save a larger number of

replications for points where the signal-to-noise ratio is small. The process for using

a variable sample size is as follows.


1 2 * 3 4 5 * 6 + + *

1 2 + 3 4 5 * 6 * * +

3 6 1 + 2 5 4 * * * +

P1

o

P2

Figure 3. Crossover operation.

m2

m3m4

m5

m6

*

*

*

+

+

1

2

3

5

4

6

m1

1 (b) Parent 2 p2

m1 m2

m3

m4

m5

m6

*

*

*

+

+

1 2 3

54

6

*

(c) Offspring

m1

m2

m3m4

m5

m6

*

*

* +

+1 2 3

54

6

(a) Parent 1 p1

Figure 4. Layout after crossover.


8/15


1

6

54

3

2

+

+

+ *

*1 2 3

54

6

Parent p1 (12+345+6**+)

1 2 + 3 4 5 + 6 * * +

1 2 * 3 4 5 + 6 * * +(a) Altering operation

4 5

3

6

1

2

+

+

* *

*1 2 3

54

6

(b) Offspring o1 after altering

Figure 5. Altering operation.

1 2 + 3 4 5 + 6 * * +

1 2 + 5 4 3 + 6 * * +(a) Inverting operation

1 2

5

4 3

6

(b) Offspring o2 after inverting

+

+

+ *

*1 2

3

5

4

6

Figure 6. Inverting operation.


9/15

. The tness values of two chromosomes are rst compared based on ve repli-

cations of their corresponding simulation models. A test of signicance is

performed to make sure the dierence between tness values is signicant. Ifso, the chromosome with the inferior tness value is considered the loser and

does not move on to the next generation.

. If the dierence is not signicant, one more replication is made for each

chromosome and the comparison is repeated. This process continues until

either the dierence becomes signicant (as a result of a reduction in the con-

dence interval due to the larger number of replications) or a limit of 20

replications per point is reached.

3.5. System owchart

The general owchart of the algorithm is given in gure 8. In the algorithm,

simulation is considered as a function evaluator, and its output is regarded as the

tness of the chromosome. The algorithm starts with an initial set of random

solutions generated by the optimization module. A new generation is formed by

selecting parent chromosomes from the current generation and modifying them

with crossover and mutation operators. Then, chromosomes in the new genera-tion are evaluated by simulation, their representation strings and tness are stored

in a standard data structure called a hashing table, which is very ecient in

searching for identical elements. In each evaluation process, the hashing table is

rst probed to nd out if the same chromosome has been tested in previous

generations; if not, a simulation experiment is run to get the tness value.

Otherwise, the tness value found in the tness hashing table is simply assigned

to the chromosome. A similar approach has been employed by Zhang (1997) and

has shown a signicant reduction in simulation runs resulting in savings of up to

45% 70% of expensive CPU time. Such benets have been also observed in thisimplementation.


1 2 + 3 4 5 + 6 * * +

1 2 + 3 4 5 + 6 * + *(a) Swapping operation

1 2

3

4 5

6

*

+

+ +

*1 2 3

54

6

(b) Offspring o3 after swapping

Figure 7. Swapping operation.


10/15

4. System architecture

The system consists of a GA package, a simulation package, an automatic

simulation model generator, and a graphical user interface. The graphical user inter-

face is used to input the information on workstations and parts, dimensional

constraints of the shop oor and GA parameters. The simulation is considered a

function evaluator (objective function). The genetic algorithm systematicallysearches and generates alternative layout designs according to the decision criterion

specied by the user. The simulation model generator then creates and executes

simulation models recommended by the GA and returns the results to the GA.

This iteration between the generator and GA continues until all the chromosomes

in the generation converge to one structure, or the limit on the number of genera-

tions to consider (set according to the available time and resources) is reached.

5. Numerical exampleThe test problem is described as follows: A manufacturing system consists of

eight workstations, and two lift trucks. Four dierent types of parts come into the

system randomly with an inter-arrival time following a certain distribution. The

parts require processing on dierent subsets of eight workstations and have dierent

distributions of processing times on each operation. Lift trucks are used to move

parts from one workstation to another according to the pre-dened processing

sequences for each part type. The area requirements of each workstation are given

in table 1. Tables 2, 3 and 4 provide more information about the manufacturing

system. The shop oor is a 90 90 (m) square area. The objective is to nd a freelayout design for the system to result in an overall shorter average cycle time for all


start

initial

population

converged ?

end

yes

no

selectionadd one more

replication

comparing

chromosomes

crossover

mutation

update fitness

table

evaluation

significant?

yes

no

search in

fitness table

find?

run simulation

no

get fitness

yes

Figure 8. Computation owchart of the algorithm.


11/15


Workstation Length Width Clearance Clearance Total areaidentication (X-axis) (Y-axis) in X-direction in Y-direction required

1 9 17 1 3 10202 18 18 2 2 2020

3 26 16 4 4 30204 32 9 8 1 40105 45 28 5 2 50306 16 8 4 2 20107 36 17 4 3 40208 16 9 4 1 2010

Table 1. Geometric constraints on workstations.

SpeedNumber (m/min) Policy Capacity

2 10.0 FCFS 1

Table 2. Transporter information.

Part Inter-arrival Batch Maximum Start Total numberidentication distribution (min) size batch time of processes

1 EXPONENTIAL(12) 1 100 0 42 EXPONENTIAL(14) 1 100 0 43 CONSTANT(8) 1 100 0 34 EXPONENTIAL(14) 1 100 0 4

Table 3. Parts information.

Part Part Routing Processing timeidentication name sequences distribution

1 Part 1 Machine 1 NORM(1,0.5)Machine 4 NORM(1,0.2)Machine 6 NORM(1,0.2)Machine 7 NORM(6,1)

2 Part 2 Machine 4 NORM(1,0.5)

Machine 1 CONSTANT(1)Machine 3 NORM(0.5,0.1)Machine 5 CONSTANT(1)

3 Part 3 Machine 7 NORM(1,0.2)Machine 4 NORM(1.5,0.5)Machine 8 NORM(4,1)

4 Part 4 Machine 5 CONSTANT(1)Machine 6 NORM(2,0.5)Machine 2 NORM(2,0.5)

Machine 3 CONSTANT(4)

Table 4. Routing information.


12/15

parts. The evolutionary environment for this GA experiment is given as follows. The

population size is 30, the crossover rate is 0.60, the mutation rate is 0.008, and the

maximum number of generations allowed is 50. A typical solution (the best chromo-

some obtained from one pass of the genetic algorithm) is shown as follows:

2; 1; *; 3; *; 7; 6; *; 4; *; 8; ; 5; *;

which has an average cycle time of 940.88 minutes.

This best layout is depicted in gure 9. The evolutionary process is shown in

gure 10.


Width90

Length 90

80

80

m5

m8

m4

m6

m7

m3

m1

m2

Figure 9. The llayout of the Example

Evolutionary Process

0

200

400

600

800

1000

1200

1400

1600

1800

1 611

16

21

26

31

36

41

46

generation

cycletim

e

averagecycle time

minimum

cycle time

Figure 10. Evolutionary process of GA in obtaining the optimum layout.


13/15

From the evolutionary process, we can see that before the 11th generation, the

solution with the minimum cycle time still has a cycle time greater than the nal

minimum and the whole generation has an average cycle time of above 1200 minutes.

The worst cycle time found during this period has an average cycle time of 2101.53

minutes. Under pressure of selection, chromosomes evolve gradually while under theguidance of genetic algorithm the solutions in the search process move slowly to

some promising regions. The average cycle time improves from the initial value of

1665.81 to 948.67 in the 20th generation. The solutions in the set converge gradually

and, after the 25th generation, the variation among the chromosomes in the set

diminishes and all the solutions converge to only one alternative. As shown in the

history of the best tness values, the best chromosome is found in the 21st genera-

tion, with an average cycle time of 940.88, which is less than half of the average cycle

time that resulted from the worst system.

6. Comparison of proposed and traditional methods

As mentioned earlier, in traditional facility layouts, material handling cost is the

major concern of the layout design. Eort is spent to reduce unnecessary part move-

ments between workstations. In todays rapidly changing global market, while

material handling cost remains critical, the development of new products and

quick customer delivery are playing a key role when competing with other factors.

Responsive delivery without inecient excess inventory, short manufacturing cycletime, and other practical considerations have strong impacts on the layout design

and should be incorporated into the layout design process. Since traditional methods

only consider the volume of materials handled, changing other factors such as the

number of transporters or processing time on various machines does not aect the

solution. If the optimum layout does indeed change with the change in these par-

ameters, it will be an indication of the need for considering the actual performance of

interest rather than just the volume of material handling. To assess this, experiments

were conducted to investigate the eects of varying factors, such as the number of

material handling units and machines capacities, and the changes in the results were

observed.

A stable manufacturing system with eight workstations is chosen as a base model,

in which each workstation has a medium level work loading (utilization between

30% 70% ). In the rst experiment, the eect of machine performance on layout

design was studied. The processing speed of a machine was changed while keeping all

other factors the same. In the second set of experiments, the eect of the number of

material handling resources on the layout design was investigated by varying the

number of available lift trucks with all other settings of the manufacturing systemkept unchanged.

To examine the eect of machines capacities, two slightly dierent manufactur-

ing systems were chosen and the objective of the optimization process was set as a

minimization of the average cycle time. Basically, the two models are the same except

that the processing speed of one of the machines in system 1 is lower than a similar

machine in system 2. All the other parameter settings (number of simulation runs,

population size, maximum generation, etc) were kept the same for both systems. The

experiment showed that the results are dierent for these two systems. Figure 11( a)

shows this evolutionary process. Obviously, with traditional methods the answer

provided for both systems would have been the same.



14/15

Similar tests were conducted with material handling resources. This time the

manufacturing system was tested with dierent numbers of material handling

resources. In the two test cases, the number of transporters was varied while all

other parameters were kept the same. The test results showed that, again the

optimum layout changed with this change. The evolutionary process for this test

is shown in gure 11(b).

7. Conclusions

This paper presented an approach for solving facility layout optimizationproblems for manufacturing systems with dynamic characteristics and qualitative

and structural decision variables. The proposed approach integrates genetic algor-

ithms, computer simulation and an automated simulation model generator with a

user-friendly interface. Since GA is capable of solving the combinatorial optimiza-

tion problems, and the simulation is capable of modelling and evaluating the per-

formance of complex systems, this combination enables us to optimize eciently the

facility layout design of such systems. The proposed method considers the opera-

tional policies, resources and time requirements of all aspects of the process toovercome the limitations of traditional layout optimization methods.

Although this method cannot guarantee an optimum solution, empirical tests indi-

cate it is able to make considerable improvements in the value of the objective

function.

Additional work in this area can improve the performance of the process. In

particular, to preserve the feasibility of a solution, the search proposed in this work

uses a particular crossover strategy that allows only the position symbols to trade

places. Other techniques can be investigated to take full advantage of all crossover

methods that can still preserve feasibility. Furthermore, other methods of mutation

can also be investigated.


0

200

400

600

800

1000

12001400

1600

1800

2000

1 11 21 31 41Number of Generations

(b)

AverageCycleTimeofthePopulation

Two Transporters

Three Transporters

0

500

1000

1500

2000

1 11 21 31 41

Number of Generations

(a)

AverageCycleTime

System 1

Syste 2

Figure 11. The eect of change in system parameters on the optimum layout.


15/15

References

AZADIVAR,F.and TOMPKINS,G ., 1999, Simulation optimization with quantitative variablesand structural model changes: a genetic algorithm approach. European Journal ofOperational Research, 113, 169182.

BRINDLE, A

.

, 1981, Genetic algorithms for function optimization. PhD dissertation,University of Alberta.CHENG, R . and GEN, M., 1996, Genetic search for facility layout design under interows

uncertainty. Japanese Journal of Fuzzy Theory and Systems, 8, 335346.GEN,M.and CHENG,R ., 1997, Genetic Algorithms and Engineering Design (Wiley).GOLDBERG,D. E., 1989, Genetic Algorithms in Search, Optimization, and Machine Learning

(MA: Addison-Wesley).HOLLAND, J. H., 1975, Adaptation in Natural and Articial Systems (Ann Arbor: The

University of Michigan Press).MELLER, R . D. and GAU, K. Y., 1996, The facility layout problem: recent and emerging

trends and perspectives.Journal of Manufacturing Systems, 15, 351366.ROSENBLATT,M., 1986, The dynamic of plant layout. Management Science, 32, 7685.TAM, K. Y., 1992, Genetic algorithms, function optimization, and facility layout design.

European Journal of Operational Research, 63, 322346.TATE, D. and SMITH, A., 1995, A genetic approach to the quadratic assignment problem.

Computers and Operations Research, 22, 7383.WELCH,B.L., 1938, The signicance of the dierence between two means when the popula-

tion variances are unequal. Biometrika, 25, 350362.ZHANG, J., 1997, Genetic algorithm based simulation optimization of exible manufacturing

systems. PhD dissertation, Kansas State University.


Simulation and Genetic to FLP

Documents