2nd International Conference on Recent Advances in Railway Engineering (ICRARE-2009) Iran university of science and Technology Tehran I.R. Iran -Sep 27-28, 2009
Genetic Application in a Facility Location Problem with Random Demand within Queuing Framework Seyed Hamid Reza Pasandideh, Seyed Taghi Akhavan Niaki Assistant Professor ,Department of Mechanic and Industrial Engineering, Islamic Azad University,Qazvin Branch , Iran ,e-mail:
[email protected], Phone: +98 912 1721820 Professor, Department of Industrial Engineering, Sharif University of Technology, Tehran, Iran ,e-mail:
[email protected]
locations of the facilities in a system such that the sum of system operating costs is minimized. For example, Li et al. (1999) developed a dynamic programming model to find the location of web proxies with minimum cost. The stochastic queue median (SQM) of Berman et al. (1985) considers a mobile server such as an emergency response unit, in which in response to each demand call (e.g., patients), the available sever (e.g., ambulance) travels to the demand location to provide services. Another objective of the facility layout and location problem is to determine the minimum number of storage facilities among a discrete set of location sites such that the probability of each customer being covered is not less than a critical value. The literature within the subject of emergency services also includes many works that extend the probabilistic location set covering problem (PLSCP) (Revelle and Hogan 1989). For example, Marianov and Revelle (1994) developed the PLSCP, which models each geographic region as multi-server queuing system. The flow-capturing model introduced by Hodgson (1990) is another closely related subject. Locating gas stations, convenience stores, and billboards are some applications of the flow-capturing model (Berman et al. 1995 and Hodgson & Berman 1997), in which sometimes the server may be congested (Berman 1995). As an example, Shavandi and Mahlooji (2006) presented a fuzzy location allocation model for congested systems.
Abstract In many service and industrial applications of the facility location problem, the number of required facilities along with allocation of the customers to the facilities are the two major questions that need to be answered. In this paper, a facility location problem with stochastic customer demand and immobile servers is studied. Two objectives considered in this problem are: (1) minimizing the average customer waiting time and (2) minimizing the average facility idle-time percentage. We formulate this problem using queuing theory and solve the model by a genetic algorithm within the desirability function framework. Several examples are presented to demonstrate the applications of the proposed methodology. Keywords: Facility Location; Queuing Theory; Genetic Algorithm; Desirability Function 1. Introduction Facility layout and location problems have been the subject of analysis since the seventeenth century (Francis et al. 1992). Even though these problems have received considerable attention over the years, it was not until the emergence of the interest in operations research and management science that the subject received renewed attention in a number of disciplines. Currently, there exists a strong interdisciplinary interest in facility layout and location problems. Mathematicians, operation researchers, architects, computer scientists, economists, engineers from several disciplines, management scientists, technical geographer, transportation system designers, regional scientists, and urban planners have discovered a commonality of interest in a concern for the layout and location of the facilities. Each brings different interpretations and different solutions to the problem. One of the objectives of the facility layout and location problem is to find the
They utilized fuzzy theory to develop a queuing maximal covering location allocation model which they called the fuzzy queuing maximal covering location allocation model. Wang et al. (2004), motivated by applications to locating servers in communication networks and automated teller machines, presented several models for the facilities
177
location subject to congestion. These models were developed for situations in which immobile service facilities were congested by stochastic demand originating from nearby customer locations. They suggested using heuristic procedures to find good solutions for those models that were more challenging. Server allocation models have also been studied in the manufacturing area (Shanthikumar and Yao 1987). One of the problems in manufacturing environments is to optimally allocate a number of servers to the work centers such that the throughput of the queuing network is maximized. A variation of this problem is introduced by Green and Guha (1995). The model presented in this paper is related to a facility location problem with allocation of immobile servers. We model the servers as a classic M/M/1 queuing system. In locating the facilities, we take both the customer waiting time and facility idle-time percentage into account. The investigated problem of this paper is different from the SQM problem in a sense that there are many immobile server locations. The rest of the paper is organized as follows. Section 2 describes the problem. In section 3 the problem is modeled. The concept of desirability function is described in section 4. The genetic algorithm along with its steps is proposed in section 5. In order to demonstrate the proposed model and examine the applicability of the solution method some numerical examples are solved in section 6. Finally, the conclusion and some recommendations for future research come in section 7.
facilities idle times. In fact, two goals are involved to satisfy both the customers and the facilities owner. 3. Problem Modeling The parameters and the variables of the model are: m : Number of customer nodes. n : Number of potential facility nodes. k: Number of servers that are on-duty ( k n) .
t ij : The travelling time from customer i to facility node j, i=1, ,m and j=1, ,n. T [t ij ] : The travelling time matrix. : Demand rate of service requests from customer node i ,i=1, ,m. : The common service rate of each server. i
j
: Demand rate at open facility j, j=1, ,n.
w j : Expected
waiting
time
of
customers
assigned to facility node j, j=1, ,n. 0 j : The probability of the server being idle at open facility (idle probability) j, j=1, , n. z1 : Sum of traveling and waiting time.
z2 : Average idle probabilities for all facilities. 1 if a facility is opened at node j , yj 0 otherwise
x ij
1 if customer i is assigned to facility j , 0 otherwise
Then, the aggregate travelling time of the customers per unit time ( T1 ) is obtained by: m
2. Problem Definition and Assumptions Consider a facility location and server allocation problem in which the demand is stochastic and the servers are immobile with limited service capacity. In other words, there exists a service system in which the customers with uncertain demands travel to a facility with a permanent location to receive service. The automated teller machines (ATMs) and internet mirror sites are examples of the system under consideration. In order to model the problem, we make the following assumptions: 1. The service request of each customer follows an independent Poisson stream. 2. Each open facility has only one server with exponential service times. The objective is to find a set of facility locations that minimizes the expected total time of customers traveling to and waiting time at the facilities along with the minimization of the
n
T1
t xij
(1)
i ij i 1 j 1
Since an open facility behaves like a M/M/1 queue (Gross and Harris 1998), the expected waiting time at an open facility site j is
wj
1
where
j
m j
i
xij .Thus, the aggregate waiting time
i 1
of customers per unit time ( T2 ) is: m
n
n
T2
i i 1 j 1
w j xij = j 1
j
(2) j
Thus, the first objective is the sum of traveling and waiting time ( z1 T1 T2 ) that must be minimized. According to the characteristics of a M/M/1 queue, the idle probability at open facility 178
j is
1
0j
y ij
. Hence, we need to
j
minimize z2 as the average of
0j
n
n
Min z 1
j 1
(j
Min z 2
j
j
VAR (
yj
ij
)
2 i
variance
such that we can relate the
average response to the p independent variables by Eq (5). (5) f i (x 1 ,..., x p ) i 1,..., q i
j 1
Subject to: n
yj
E ( ij ) = 0 and
mean
)
n
k A desirability function, d i ( yi ) , assigns
j 1 n
xij
1 , i 1,..., m
numbers between 0 and 1 to the possible value of each response yi . The value of
(3)
j 1
xij
y j , i 1,..., m , j 1,..., n
d i ( yi ) increases as the desirability of the
m i
xij
,j
corresponding response increases. We define the overall desirability, D , by the geometric mean of the individual desirability values shown in Eq . (6).
1,..., n
i 1
m j
i
xij , j
1,..., n
i 1
yj xij
1,..., n i
response, yi , and x1 , . . . , xp . The parameter ni is the maximum number of observations for each of the q responses and ij is an error term with
))
1
1,..., q , j
and f i denotes the relationship between the ith
n
(1 (
i
where
j
t x ij
,
yij is the jth observation on the ith response
i ij i 1 j 1
ij
s for all j.
This will be the second objective of the model. In short, the mathematical programming model of the problem at hand becomes: m
f i (x 1 ,..., x p )
{0,1}
{o,1} , i 1,..., m , j
D
,
(d 1 ( y 1 ) d 2 ( y 2 ) ... d q ( y q ))
1 q
(6)
1,..., n Note that if a response
The first constraint of model (3) sets an upper limit for the maximum number of opened facilities. The second and the third constraints ensure that each customer demand is satisfied by only one open facility. The fourth constraint guarantees that the input to each server to be less than its capacity. Finally, the fifth constraint is the input of each server.
yi is completely
undesirable, i.e., d i ( yi ) = 0, then the overall desirability value is 0. Depending on whether a particular response yi is to be maximized, minimized, or assigned a target value, different desirability functions can be used. Derringer and Suich (1980) introduced a useful class of desirability functions. There are two types of transformation from yi to d i ( yi ) , namely one-sided and twosided transformation. We employ the one-sided transformation when yi is to be maximized or minimized, and two-sided transformation when yi is to be assigned a target value. Since we want to minimize the objectives in model (3), the one-sided transformation is employed in this paper. In this case, the individual desirability of the responses are defined as
4. Desirability Function Since there are two objectives in model (3), one needs to take advantages of a multiobjectives optimization method. The desirability function approach is one of the most widely used methods in industry for dealing with the optimization of multiple-response problems (Derringer and Suich 1980). This method assigns a score to a set of responses and chooses factor settings that maximize that score. In order to describe the desirability function approach mathematically, suppose each of the q response variables are related to p independent variables by Eq. (4) (4)
179
1, di (y i )
ui ui 0,
yi yi li
1. Population size: It is the number of the chromosomes or scenarios that will be kept in each generation and we denote it by N. 2. Crossover rate: This is the probability of performing a crossover in the GA methods, denoted by Pc . 3. Mutation rate: This is the probability of performing mutation in the GA method denoted by Pm .
li ,
s
, li yi
yi
ui
(7)
ui
In equation (7), li represents a small enough value for the response and u i is the upper bound for the response y i . The exponent s determines
5.2. Chromosome In the GA method, we present a chromosome by a matrix that has m rows and n columns. Each column shows the number of potential facility nodes and each row of the matrix show the number of the customer nodes.
how strictly the target value is desired. For s 1 , the desirability function increases linearly towards li , for s 1 the function is convex, and for s
1 the function is concave.
5. Genetic Algorithm The formulation given in (3) is a nonlinear-multiobjective integer-programming model. These characteristics cause the model to be hardly solved by exact methods (Gen 1997). Accordingly, we need a heuristic search algorithm to solve the problem. Historically, among the search algorithms, genetic algorithm (GA) has been successful in solving models similar to the model in Eq. (3) (Gen 1997). The usual form of GA was described by Goldberg (1989). Genetic algorithms are stochastic search techniques based on the mechanism of natural selection and natural genetics. GA differs from conventional search techniques in a sense that it starts with an initial set of random solutions called population. Each individual in the population is called a chromosome, representing a solution to the problem at hand. The chromosomes evolve through successive iterations, called generations. During each generation, the chromosomes are evaluated using some measures of fitness. To create the next generation, new chromosomes, called offspring, are formed by either crossover operator or mutation operator. A new generation is formed according to the fitness values of the chromosomes. After several generations, the algorithm converges to the best chromosome. In the next subsections, we demonstrate the steps required to solve the model given in Eq. (3) by a genetic algorithm.
5.3. Initial Population In this stage, a collection of chromosomes is randomly generated. For each row of the chromosome matrix, only one nonzero value of 1 is generated. The mechanism of generating 1 and 0 is random. For example, if m=3 and n=4, then Figure 1 shows a chromosome matrix. Since model (3) has an assignment structure, we generate N chromosome in this manner. Therefore, we can expect the constraints to be more satisfied and the algorithm starts with good solutions.
5.4. Evaluation In a GA method, as soon as a chromosome is generated, we need to assign a fitness value to it. In optimization problems, it is the value of the objective function. Since there are some constraints in the model of the problem given in Eq. (3), some generated chromosomes may not be feasible. In the last few years, several methods were proposed for handling constraints by GAs, most of which have serious drawbacks. While some may give infeasible solution or require many additional parameters, others are problemdependent. The most popular approach in GA to handle constraints is to use penalty functions that penalize infeasible solutions by reducing their fitness values in proportion to their degree of constraint violation (Gen 1997). In this research, we use the penalty function approach. If a constraint is satisfied then its penalty is set to zero. Otherwise, the penalty assumes a non-zero value. The penalty functions
5.1. Initial Conditions The required initial information to start the GA is:
180
used for each constraint of model (3) have been selected by trial-and-error approach and are listed in Table 1.
the selected row, we replace 1 with 0 and then another element on this row will be selected, randomly. At the end, the value of the selected element will change to 1. As an example, Figure 3 shows a graphical representation of the mutation operation on a randomly selected chromosome. In this example, we first generate an integer 3 that is between 1 and 3. Then, the "1"-element in the third row and fourth column is replaced with zero. If the next random digit is 3, then the element in the third row and third column is replaced with 1.
Since the desirability of the two objectives T1 and T2 are d1 and d 2 respectively, then the total desirability for a chromosome x will be D ( x ) d 1d 2 and for the ideal chromosome the total desirability becomes 1. The penalties p1 , p 2 , p 3 , and p4 are defined so that their values will be between 0 and 1. The more satisfied a constraint, the closer the value of its penalty to zero becomes. The total penalty (P) for an unfeasible chromosome x is defined as:
5.7. Emigration In addition to the crossover and mutation operations, we consider an extra operation named emigration to avoid fast convergence (Gen 1997). This operation classifies the similar chromosomes to some groups and randomly selects the group that has the maximum number of similar chromosomes. The emigration operation is performed if the number of similar chromosomes is greater than a percentage of N (say 5%). In the emigration operation some of chromosomes are deleted and replaced by new chromosomes. The percentage of the deleted chromosomes will be a random number between 0 and 1.
4
P (x )
(f h
ph )
, h 1, 2,3, 4
(8)
h 1
In equation (8), f h determines the importance of each constraint. The value of f h would be between 0 and 1 and is determined when the problem is solved. Finally, the fitness value of a chromosome x is defined as: f ( x) D( x) P( x) (9)
5.5. Crossover In a crossover process, it is necessary to mate pairs of chromosomes to create offspring. This is done by a random selection of a pair of chromosomes from the generation with probability Pc . For crossover operations, we first generate an integer random number between 1 and m. Let this number be a. Then, we replace the first row in the chromosome matrix of the first parent with its ath row; resulting in the first offspring. The second offspring is generated by replacing the first row of the second parent chromosome by the first row of the first parent chromosome. Fig. 2 shows a graphical representation of the crossover operation in which a=2.
5.8. Chromosome Selection In the next phase of the genetic algorithm, the chromosomes for the next generation are selected. This process is based on the fitness value of each chromosome. We select N chromosomes among the parents and offspring with the best fitness values. 5.9. Stopping Criteria The last step in the GA methodology is to check if the method has found a solution that is good enough to meet the user s expectations. Stopping criteria is a set of conditions such that when satisfied, a good solution is obtained. In this research stopping criteria is defined as the number of generations. When the algorithm reaches a predefined number of generations, the algorithm will be stopped.
5.6. Mutation Mutation is the second operation in the GA methods for exploring new solutions. In mutation, we replace a gene with a randomly selected number within the boundaries of the parameter. In this research, we generate a random number between 0 and 1 and whenever this number becomes less than p m , the chromosome is selected for mutation. Then, one of the rows of the selected chromosome is chosen randomly. In
6. Numerical Examples Several examples with different sizes are considered to demonstrate the applicability of the proposed method. The first five columns of Table 2 show these examples along with their
181
properties. For each example, to fine tune the parameters of the GA algorithm ( p m , p c and N), the concept of experimental design is used. In this research, we use the central composite face centered (CCF) 33 design for parameter adjustment (Montgomery 2004). As an example, the raw (not coded) data of the parameters for problem No. 4 is presented in Table 3. In order to find the best possible combination of the GA
f ( pc , p m , N ) 0.268339 0.645843 p m
parameters, the regression analysis is then employed to fit the fitness function f ( p c , p m , N ) on the coded data. This function is maximized for the values of pc and p m between 0 and 1 and N between 6 and 114. The estimated fitness function is obtained as:
0.102775 pc
0.228212 pc2 6.10244E
05N
2
0.549188 p m2
0.00786579 N
0.0395000 p m
pc
0.00316759 p m N
0.000661111pc N n
xij
Subject to:
0
pc
0 6
pm 1 N 114
1
xij
(10)
y j , i 1,..., m , j
1,..., n
i
xij
,j
1,..., n
i 1 m j
i
xij , j
1,..., n
i 1
d1
u1 z1 , u1 l1
d2
u2 z 2 , u 2 l2 m
0
d1
0
1
d2
u1 ,
100000 , l1 u1
1 , l1
j
i t ij xij + i 1 j 1
,
1
n
n
z1
j 1
j
n
(1 ( z2
(
j
))
j 1
)
n
yj j 1
yj xij
{0,1} {o,1} , i 1,..., m , j
,
1,..., n
The results of Table 2 show that The proposed GA is more efficient than the Lingo software. In other words, while the proposed GA obtains solutions with almost the same quality as Lingo, it requires much less CPU time. While the LINGO software cannot solve large-size problems within reasonable amount of CPU time (e.g. problem 12), the proposed GA is able to solve.
d1d 2
Subject to: n
yj
(11)
m
N :integer Model (10) can be easily solved by Lingo resulting in p m 1 , pc 0.78 , N 76, and f 0.8176 . Similarly, the tuned input parameters along with the optimal fitness function values of the other problems are determined. It should be mentioned that for all problems the values of matrix T and parameter i are generated randomly and the decision variables are obtained using the proposed method. As an example, the value of the decision variables in problem 2 are x11 1, x12 1, x13 1 , y 1 1, and y 3 1 . The other variables are all equal to zero. To save space, we withdraw the values of the decision variables in other examples. To compare the efficiency of the proposed algorithm in terms of both fitness function value and CPU time of execution, we suggest models that can be solved by Lingo software. For model (3), which is replaced by (11) using the desirability approach, Table 2 summarizes the results. Moreover, to show the convergence of the proposed GA algorithm, the diagram of the fitness function in terms of generation number for problem No. 4 is presented in Figure 4.
Max
1 , i 1,..., m
j 1
k
j 1
182
0
0
Variables." Journal of Quality Technology 12: 214-219. 5. Francis, R.L., Megginis, L.F., and White, J.A., (1992). "Facility Layout and Location: An Analytical Approach." 2nd Edition, Prentice-Hall, Englewood Cliffs, NJ. 6. Gen, M., (1997). "Genetic Algorithm and Engineering Design." John Wiley & Sons, New York, NY. 7. Goldberg, D., (1989). "Genetic Algorithms in Search: Optimization and Machine Learning." Addison-Wesley, Reading, MA, USA. 8. Green, L. and Guha, D., (1995). "Note on the Efficiency of Imbalance in Multi-Facility Multi-Server Service Systems." Management Science 41: 179-187. 9. Gross, D. and Harris, C.M., (1998). "Fundamental of Queuing Theory." 3rd Edition, Wiley-Interscience, New York, NY. 10. Hodgson, M.J., (1990). "A FlowCapturing Location-Allocation Model." Geographical Analysis 22: 270-279. 11. Hodgson, M.J. and Berman, O., (1997). "A Billboard Location Model." Geographical and Environmental Modeling 1: 25-46. 12. Li, B., Golin, M.J., Italiano, G.F., Deng, X., and Sohraby, K., (1999). "On the Optimal Placement of Web Proxies in the Internet." In Proceeding of the INFOCOM 99 Conference, 1282-1290. 13. Marianov, V. and Revelle, C., (1994). "The Queering Probabilistic Location Set Covering Problem and Some Extensions." SocioEconomic Planning Sciences 28: 167-178. 14. ,MontgomeryD.C., (2004). "Response Surface Methodology." Wiley, New York.
7. Conclusion and Future Research In this paper a facility location problem with stochastic customer demand and immobile servers with two objective functions using queuing theory and desirability function approaches were first modeled and shown to be a nonlinear-multiobjective integer-programming model. Then, a GA was proposed to solve the model. In order to demonstrate the applicability of the proposed methodology and to compare its performance with that of LINGO software, the tuned GA was then employed to several numerical problems. The results obtained from the numerical problems showed that while both the proposed method and LINGO share almost the same quality of solution, the CPU execution time of the proposed procedure is much less than the one of the LINGO software. For future research, we recommend the followings: The immobile servers may have different service rates. A budget constraint can be added to the problem. In addition to the exponential distribution for the customer demands, other probability distributions can be investigated. In these cases, instead of the M/M/1 queuing system, new models should be employed. For the bi-objective problem formulation, instead of the desirability function approach, other methods such as goal programming may be used. In addition to GA other meta-heuristic algorithms like simulated annealing may be utilized. Some of the parameters of the problem may be considered random or fuzzy.
15. Revelle, C. and Hogan, K., (1989). "The Maximum Reliability Location Problem and Reliable p-Center Problem: Derivatives of the Probabilistic Location Set Covering Problem." Annals of Operations Research 18: 155-173. 16. Shanthikumar, J.G. and Yao, D.D., (1987). "Optimal Server Allocation in a System of Multi-Server Stations." Management Science 33: 1173-1191. 17. Shavandi, H. and Mahlooji, H., (2006). "A Fuzzy Queuing Location Model with a Genetic Algorithm for Congested Systems." Applied Mathematics and Computation 181: 440456. 18. Wang, Q., Batta, R., and Rump, C.M., (2004). "Facility Location Models for Immobile Servers with Stochastic Demand." Naval Research Logistic, 51: 137-152.
8. References 1. Berman, O., (1995). "The Maximizing Market-Size Discretionary Facility Location Problem with Congestion." Socio-Economic Planning Sciences 29: 39-46. 2. Berman, O., Hodgson, M.J., and Krass, D., (1995). "Flow Interception Problems in Facility Location: A Survey of Applications and Methods." Ed. Z. Drezner, Springer Series in Operations Research, Springer, New York, NY. 3. Berman, O., Larson, R.C., and Chiu, S.S., (1985). "Optimal Server Location on a Network Operating as an M/G/1 Queue." Operations Research 33: 746-771. 4. Derringer, G. and Suich, R., (1980). "Simultaneous Optimization of Several Response 183
1 0 0 0 0 0 1 0 0 1 0 0 Figure 1. A Sample Chromosome
Table 1. Penalty Functions for the Constraints
Constraint
Penalty function
Description
n
( st
1
yj) k j 1
p1 1 (
-
)
n k
m
A1 m
m
xij , i 1,..., n
Bj p2
2nd
j 1
1 (
3
p3
1 (
Cij ) i 1 j 1
)
m n
parent 1
parent 2
p4
1 (
j 1
1 0 0 0 0 1 0 0
1
xij
Bj
y then C ij
otherwise
C ij
0 1 0
m
Dj 4
1 then B j
otherwise then if
n
th
Aj
if
n
( rd
)
n
m
j 1
n
)
if
x
i ij
,j
1,..., n
then D j
i 1
otherwise then D j
offspring 1
0
0 1 0 0 0 1 0 0
0 0 0 1
0 0 0 1
0 1 0 0 0 1 0 0 0 0 1 0
1 0 0 0 0 1 0 0 0 0 1 0
offspring 2
Figure 2. An Example of the Crossover Operation
0 0 0 1
0 0 0 1
0 0 0 1 0 0 0 1
0 0 0 1 0 0 1 0
Figure 3. An Example of the Mutation Operation
184
1
Table 2. The Numerical Problems with Performance Comparison Problem No.
m
n
1 2 3 4 5 6 7 8 9 10 11 12
3 14 30 30 40 45 50 55 60 65 100 150
4 20 20 20 30 35 40 45 50 55 65 65
5 20 17 35 35 35 52 60 70 60 120 150
k
GA
Lingo
GA
2 8 5 12 20 15 7 6 6 8 8 15
0.7562 0.9071 0.9544 0.8176 0.8796 0.9415 0.8171 0.9696 0.9285 0.8488 0.8126 0.7098
0.7562 0.9069 0.9546 0.8177 0.8797 0.9416 0.8617 0.9696 0.9285 0.8488 0.8125 ------
0
0
0.40 0.53 0.74 1.21 1.75 2.52 2.75 3.02 3.41 5.12 40.6
1.80 3.13 10.3 31.3 41.2 54.5 63.5 72.1 95.6 515 -----
Table 3. Row Data for Problem No.4 Row
pc
pm
N
Fitness Function
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1 0.5 1 0 0 0.5 0 0.5 0.5 0.5 1 1 0.5 0.5 0.5 0 0.5 0.5 0.5 1
1 0.5 0 1 0 0.5 0 0.5 0.5 1 0 1 0.5 0 1 0.5 0.5 0.5 0.5 0.5
6 60 114 114 6 60 114 60 60 6 6 114 60 60 60 60 6 114 6 114
0.5295 0.8175 0.8175 0.4568 0.3459 0.6333 0.3632 0.8174 0.8173 0.4568 0.404 0.8175 0.8175 0.8173 0.8174 0.4285 0.4568 0.7079 0.8174 0.8176
185
Lingo
0.9
Fitness Value
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
1
3
5
7
9
11
13
15
17
19
21
23
25
27
29
31
33
35
Generation Number Figure 4. Convergence Diagram of Problem No. 4
186
37
39
41
43
45
47
49
