Download Optimization of fuzzy nonlinear programming with Gaussian

Technical Journal of Engineering and Applied Sciences
Available online at www.tjeas.com
©2014 TJEAS Journal-2014-4-04/328-338
ISSN 2051-0853 ©2014 TJEAS
Optimization of fuzzy nonlinear programming with
Gaussian membership parameters via Genetic
Algorithm
A.A. Noura1, S.A. Khaleghi1, A.R. Hajihosseini1
1
Department of Mathematics,
Kerman Branch, Islamic Azad University,
Kerman, Iran
ABSTRACT: The aim of this paper is to solve the fuzzy nonlinear optimization problem (FNOP) with
fuzzy programming parameters through genetic algorithm. Main approach is based on Gaussian
membership function(GMF) for FNOP with linear fuzzy parameters. From theoretical point of view, we
justify the fact that linear combination of GMFs remain Gaussian, so together with using fuzzy ranking
of Liu and Wang can be attained a genetic algorithm and then applied it to exert a new solution
technique for linear parametric FNOP. Finally numerical examples are provided to show the
effectiveness of the proposed method, the proposed method is more effective for fuzzy quadratic
programming and fuzzy linear programming with fuzzy parametric.
Keywords: Fuzzy, nonlinear programming, Optimization, Ranking, Genetic algorithms.
INTRODUCTION
Over the past decades, solving fuzzy (linear and nonlinear) optimization problems was one of the fundamental
search subjects in the field of fuzzy sets and systems. The primary works of fuzzy mathematical programming have
been presented by Zimmermann (1978, 1983). Since then, many papers have appeared on fuzzy linear
programming. H.J. Zimmermann solves fuzzy linear programming with several objective functions [10]. L. Campos,
J.L. Verdegay proposed the method for solving fuzzy linear programming by ranking of fuzzy number [12]. S
Chanas used of parametric programming in fuzzy linear programming [16]. Afterwards, many authors considered
various types of fuzzy linear programming problems and proposed several approaches for solving these problems.
When nonlinear programming is convex, we can obtain a global optimal solution by some convex programming
techniques, e.g. the sequential quadratic programming. Otherwise, i.e. when it is non-convex, because it is difficult
to find a global optimal solution, we search an approximate optimal solution by some approximate solution methods
such as genetic algorithms or simulated annealing. So in the fuzzy nonlinear case the situation is quite different, as
there is a wide variety of specific and both practically and theoretically relevant nonlinear problems, each having a
different solution method. The many scientists solved the fuzzy nonlinear programming problems such as Jinquan
Li, Shuang Feng [2] ,Yahia zare mehrjerdi[3],Pandivan Vast [4]. Here a genetic algorithm procedure is used to find
the solution of fuzzy nonlinear programming with defining the Gaussian membership function in parametric form for
fuzzy numbers.
The paper is organized in 7 Sections. In Section 2, we give some necessary notations and definitions of fuzzy set
theory and fuzzy arithmetic. Section 3 provides a discussion of Gaussian fuzzy numbers and a ranking function for
ordering them. We define a fuzzy nonlinear optimization problem (FNOP) in Section 4, and focus on solving theses
problems. In section 5, we propose a genetic algorithm model for solving FNOP. We present numerical examples
and compared results with crisp solutions in section 6. Finally, we conclude in Section 7.
.
PRELIMINARIES OF FUZZY NUMBER
Definition: A fuzzy number ũ in parametric form is a pair ( (x),
the following requirements:
1) (x) is a bounded monotonic increasing left continuous function,
(x)) of function
(x),
(x), 0≤x≤1 which satisfies
Tech J Engin & App Sci., 4 (4): 328-338, 2014
2)
(x) is a bounded monotonic decreasing left continuous function,
3)
(x) ≤
(x), 0≤x≤1
A crisp number α is simply represent by
(x) = (x) = α, 0≤x≤1.By using the extension principle [18], the addition
and scalar multiplication of fuzzy number are defined by
(u+v)(x)=sup min x=r+s {u(r), v(s)},
(1)
(cu)(x)=u(x/c), c≠0,
(2)
Where u&v are be membership functions of , and cєR. Equivalently, for arbitrary fuzzy parametrics form
=
( (x), (x)), =( (x), (x)) and constant real number cєR, the addition and scalar multiplication can be achieved as
follows:
=
(
(x)+ (x),
(3a)
) (x)= (x)+ (x),
(
(x) =c (x), (
(x) =c
(3b)
)(x)= c (x), c 0
(x), (
(3c)
)(x)= c (x),, c
(3d)
FUZZY NUMBER WITH GAUSSIAN MEMBERSHIP FUNCTION
2
Now let
is a Gaussian fuzzy number as µ(x)=exp{-k(x-α) }, fuzzy number in parametric form can be transform
as
( (x),
, α+
(x))= (α-
For example, consider
=
)
(4)
2
as a fuzzy number with Gaussian membership function as µ(x)=exp{-(x-1) },which in
parametric form is (u1(x), u2(x))=( 1-
) , 0≤x≤1. Gaussian membership function and
1+
parametric form of fuzzy number for = are shown in Fig.1 and Fig.2.
5
1
4
0.9
0.8
3
0.7
2
0.6
0.5
1
0.4
0
0.3
-1
0.2
-2
0.1
0
-2
-1
0
1
2
3
Fig.1 Gaussian membership function
-3
0
4
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Fig.2 parametric form of fuzzy number
1
Now we show that the edition and scalar multiplication of Gaussian numbers will be Gaussian numbers:
2
2
let u(r)= exp{-k1(r-m) } ,v(s)= exp{-k2(s-n) } are two Gaussian membership functions of
,
fuzzy numbers,
sequentially. The parametric membership functions of
( (x),
(x))= (m-
, m+
,
are
) and ( (x),
(x))= (n-
, n+
).
By
using extension principle, we have
2
2
(u+v)(x)=sup min x= r+ s {u(r) ,v(s)}= sup min x= r+ s { exp{-k1(r-m) }, exp{-k2(s-n) }}=
2
2
Sup min x= r+ s {exp {-k1(r-m) }, exp{-k2(x-r-n) }}
329
Tech J Engin & App Sci., 4 (4): 328-338, 2014
1
m=1
n=2
0.9
0.8
0.7
sup min[u(r),v(s)]
0.6
0.5
0.4
0.3
0.2
0.1
0
-4
-3
-2
-1
0
1
Fig.3
2
3
4
5
6
Pay attention to Fig.3, the supermom is occurred in the interaction point u(x) and v(x). so
2
2
2
2
exp{-k1(r-m) }=exp{-k2(x-r-n) }→ k1(r-m) =k2(x-r-n) → r-m=
Then substituted r in the one of membership functions
2
or
2
u(r)= exp{-k1(r-m) }= exp{-k1(
(x-r-n) → r=
, we get
2
-m) }=exp{
(x-(m+n)) }
Therefore
2
(u+v)(x)=exp {
(x-(m+n)) }.
(5)
Which is proved that addition of two GMFs is a GMF.
2
Also we can show that scalar multiplication of GMF is a GMF: suppose u(x) = exp{-k1(x-m) } is be membership
functions of and c єR-{0}, by using extension principle, we have
2
(cu)(x)=u(x/c) = exp{-k1(x/c-m) }= exp{-
2
(x-cm) }.
(6)
By using the Equations (3), we attain parametric form of (5) and (6) as follow
(u+v)(x)= (m+ n-
, m+ n+
),
(8)
(cu)(x)= (cm-
), c
(9)
(cu)(x)= (cm+
), c≤0,
(10)
We calculate fuzzy objective function and fuzzy constraints using formulations (8) & (9) &(10).
3.1 Fuzzy ranking
A simple method for ordering fuzzy numbers consists in the definition of a ranking function F, mapping each fuzzy
number to the real number R, where a natural order exists. suppose S={
,
,
,
}is a set of n fuzzy
numbers,and the ranking function F is a mapping from S to the real numbers R, i.e. F:S→R.then for any distinct
pair of fuzzy numbers
,
єS,the ranking function can be defined as
If
F(
)<F(
);then
<
If
F(
)=F(
);then
=
330
Tech J Engin & App Sci., 4 (4): 328-338, 2014
If
F(
)>F(
);then
>
This implies for example, that if F(
.the higher
is,the larger F(
)>F(
),the fuzzy number
is numerically greater than fuzzy number
) is.A useful technique for ranking fuzzy numbers is liou and wang ranking
function[7], defined by
(ũ)=
(y) dy+(1-
)
(y) dy
(11)
Where L(.),R(.) are left and right shape functions of ũ, respectively and
represents the degree of optimism of a decision maker.
Theorem 1. if ũ is a GMF in parametric form then
Proof. let ũ= ( (x), (x))=(m-
(ũ)= m+
, m+
є[0,1] is the index of optimism which
(m+
).
) is a GMF, In this method, we take
(x) .we have
(ũ)=
=m+ m
(x) dx+(1-
+
)
(x) dx=
dx= m+ m
(ũ)= m+
(m+
)dx+(1-
+
)(
)=m+
),
(m+
)
) dx
). Thus
(12)
MODELING FORMULATION
A constrained optimization problem can be mathematically formulated as
Minimize f (x)
Subject to gj (x) ≤ 0, j = 1,…, m
(13)
n
x∈ R
n
n
where f (x) is the objective function and gj(x) , j = 1,……,m are the constraints which defined on R and X={xєR │
gj(x) ≤ 0, j = 1,…, m} is feasible space. This problem will be solved for the values x belonging to the feasible space
that satisfy the constraints and minimize the function f. If exist a feasible point x* that satisfying all the constraints,
such that f (x*) ≤ f (x) for each feasible point x, then x* is an optimal solution. if f (x) and gj (x) j=1,2,…,m are linear
functions, the model describes a linear optimization problem otherwise the model becomes a nonlinear
optimization problem. Now we consider a class of nonlinear programming problem with fuzzy coefficients as
follows:
Minimize
Subject to
≤ ci
, i=1…m,
(14)
x є [l,u], 0≤l≤u
n
Where x = (x1 ,..., xn )∈ R , [lj ,uj ] ⊂ R(j= 1,...,n) and gij(x), ft (x), ( t=1,2,…,k,j=1,2,…,n, i=1…m) are linear or
nonlinear functions. The significance of this model is that the fuzzy parametric numbers are retractable, gij(x), ft (x),(
t=1,2,…,k ,j=1,2,…,n, i=1…m) functions are arbitrary and the problem is bounded. This method easily is solved
fuzzy quadratic problems (FQM) and fuzzy linear programming and some FNOPs compared with other methods.
For example, a special class of (14) can be address as following
Min Z=
S. to
+1/2
i=1,2,…,m
(15)
Theorem 2.by the properties of GMFs, the fuzzy quadratic problem(15) can be transformed as following
331
Tech J Engin & App Sci., 4 (4): 328-338, 2014
Min
=
+1/2
S. to
, i=1,2,..,m
0
Where
(16)
,
=(
),
=(
),
=(
),
=(
),i=1,2,..,m are
GMFs.
Proof .We define GMFs in parametric form
,
,
, i=1,2,…,m,
,
,
(17)
Because
and using (9), we have
, Then using of (8), we have
=(
)=(
(18)
Sequentially, we can obtain
=(
)= (
)
(
(19)
),
i=1,2,..,m,
(20)
By using formulations (17),(18),(19),(20), the problem(14) can be written as follows.
Min
(x) =
+1/2
S.to
, i=1,2,..,m
0
Where
,
=(
),
=(
),
=(
),
=(
),(i=1,2,..,m) are
defined in (17),(18),(19),(20).
Theorem 2. Assume the fuzzy linear programming as following
Min Z=
S. to
i=1,2,…,m
(21)
by the properties of GMFs, the fuzzy linear problem(21) can be transformed as following
332
Tech J Engin & App Sci., 4 (4): 328-338, 2014
Min
(x) =
S.to
, i=1,2,..,m
0
Where
(22)
,
=(
),
=(
),
=(
),(i=1,2,..,m) are GMFs.
.
Proof . This is similar to the proof of theorem 1.
Now we have a fuzzy programming problem in parametric form that the objective function is a GMF and right hand
and left hand of each constrain is a GMF in parametric form. We utilize the ranking equation (12) to compare fuzzy
number in (16). Here a genetic algorithm is used to find the solution of fuzzy optimization problem (14) after that is
transformed.
1. Algorithm for solving FNOP through a Genetic Algorithm
Now, we illustrate the genetic algorithm model used for solving FNOP. The applied algorithm which is needed to
solve the FNOP in the programmed environment (MATLAB 2010) is written and executed.
Step0. Convert the coefficients of objective function and the constraints into fuzzy number using GMFs, and
calculate the objective function and left and right side of the constraints using equations(8)&(9),(10).Set the
parameters, population size (popsize), mutation rate (
),selection rate( ),crossover rate( ),the maximum
generation (max gen) and initialize number of generations gen=0.
Step1. Select initial population at random for genetic algorithm and then one should code the decision variable ’x’
into binary string with finite length. We randomly take a string
of leng
for evrery variable
whose
characteristics are either 0 or 1.
Step2. Obtain the decoded values for the current population generated. if the current population are satisfy of all
constraints, go to step 3.otherwise choose the rest of the population randomly so that all the constraints are
satisfied. We use to satisfy of constraints the fuzzy ranking function (12).
Step3. Calculate objective value for each chromosome and compute the fitness value using the equation (12).
Step4. Selection: Parents selection is made on the basis of fitness function, individuals with the best fitness values
are chosen more often. The best fitness value of an individual had the more likely that the individual will be selected
for recombination. The selection of mating parents is done by roulette wheel selection.
Step5. Implement crossover and mutation operator on selected members from the old generation. Crossover
combines information from two parents that two children have a resemblance to each parent. Standard crossovers
such as one-point and two-point, are used in GA model and bit-wise mutation operator performed here.
Step6. If the termination condition is fulfilled, stop, otherwise. Set t=t+1 and go to step 2.
6. Numerical Examples
Example 5.1 Consider the following nonlinear problem [1]:
Min f(x)= x1x2x3
Subject to 2ox1x2+6x1x3+6x2x3≤108
(19)
x2+x3≤3
0≤xi≤10 ,i=1,2,3
The optimal solution is = (2.9952, 1.0031, 1.9968),f( )=5.9999. Now, the fuzzy version of the problem is
Min
x1x2x3
x1 x2+ x1x3+ x2x3≤
(20)
x2+ x3≤
0≤xi≤10 ,i=1,2,3
Fuzzy coefficients of objective function and constraints are presented in Table. 3.
333
Tech J Engin & App Sci., 4 (4): 328-338, 2014
Table.3 Fuzzy number representations
The later convert the coefficients of objective function and constraints into fuzzy numbers in parametric form, we
calculate the objective function and constraints using equations (8)&(9)&(10).The fuzzy problem (20) is transformed
as following
Max
=(x1x2x3- x1x2x3
, x1x2x3+ x1x2x3
)
S.t. (20x1x2+6x1x3+6x2x3-(x1x2+x1x3+ x2x3
,108
(x2+x3-( x2+x3)
, 20x1x2+6x1x3+6x2x3) +(x1x2+x1x3+ x2x3
)
≤(108-
( 21)
)≤(3
, x2+x3+( x2+x3)
,3
)
0≤xi≤10 ,i=1,2,3,0≤x≤1
Now we apply genetic algorithm on new problem(21).
The parameter values used in genetic algorithm for solving fuzzy nonlinear problem (21) were set as follows:
ps=0.40,pc=0.40,pm=0.03, Maxgen=100,Popsize=70. The method had 5 runs. The results obtained in 5 runs are
shown in table.2, and the best results of
obtainded in five runs are shown in Fig.4.
Table.2 The results obtained in 5 runs for the fuzzy problem (21).
Genoptim
al
Time(
Sec.)
at µ=1
for
left side of
constraints
(93.4874,2.7
585)
for
hand side of
constraints
(162.88,5.38)
(2.5211,1.1519,1.
6066)
µ(x)=
40
56
5.6656
(3.2128,0.9059,1.
8826)
µ(x)=
22
49
5.6495
(104.7327,2.
7885)
(162.88,5.38)
(3.0906,0.9852,1.
8496)
µ(x)=
10
70
5.6319
(106.1288)
(162.88,5.38)
(3.2139,0.8094,2.
1530)
µ(x)=
48
68
5.6008
(103.9996,2.
9624)
(162.88,5.38)
(3.3293,0.8607,1.
9387)
µ(x)=
85
76
5.5554
(106.0495,2.
7994)
(162.88,5.38)
This genetic algorithm method is executed on this problem (21) with same above parameters. The best results of
fitness functions (equation (12) for objective function) in a run at grate of membership value µ=1 are shown in Fig.5
334
Tech J Engin & App Sci., 4 (4): 328-338, 2014
Membership of objective functions
1
u(f(x))
0.9
e-0.0549(x-5.6656)
2
e-0.0333(x-5.6495)
2
0.8
e-0.0315(x-5.6319)
2
0.7
e-0.0319(x-5.6008)
2
e-0.0324(x-5.5554)
2
0.6
0.5
0.4
0.3
0.2
0.1
0
-10
-5
0
5
f(x)
10
15
20
Best fit of each generation using Liou&Wang ranking function
Fig.4 Optimal values obtained from solving problem (21).
6
5
4
3
2
1
0
0
10
20
30
40
50
60
Generation
70
80
90
100
Fig.5 The curves for the best value obtained in 100 generations in a run for the fuzzy problem (21).
Example 5.2 Consider the following nonlinear programming problem:
2
2
Min f(x)=-4x1+x1 -2x1x2+2x2
2x1+x2≤6
(22)
x1-4x2≤0
0≤x1≤5, 0≤x2≤6
The optimal solution is =(2.4615,1.0769),and f( )=-6.7692.Now,the fuzzy version of the problem is
Min
2
=- x1+ x1 - x1x2+ x2
x1+ x2≤
2
(23)
x1- x2≤
0≤x1≤5, 0≤x2≤6
335
Tech J Engin & App Sci., 4 (4): 328-338, 2014
Table.3 Fuzzy number representation
Fuzzy coefficients of objective function and constraints are presented in Table 3.
The later
convert the coefficients of objective function and constraints into fuzzy numbers in parametric form, we calculate
the objective function and constraints using equations (8) & (9) & (10). We have
2
2
2
2
2
2
2
2
Min
=(-4x1+x1 -2x1x2+2x2 - ( x1+x1 +x1x2+x2 )
, -4x1+x1 -2x1x2+2x2 - ( x1+x1 +x1x2+x2 )
)
S.t.
(2x1+x2-
(x1+x2
(x1-x2)
, 2x1+x2-
(x1+x2
, x1-4x2+
(x1-x2)
)≤(6-
,6
)≤(
(24)
(x1-4x2-
)
0≤x1≤5 ,0≤x2≤6 .0≤x≤1.
Now we apply genetic algorithm on new problem (24).
The parameter values used in genetic algorithm for solving fuzzy nonlinear problem (24) were set as follows:
ps=0.40,pc=0.30, pm=0.04, Maxgen=100,Popsize=50.The method had 5 runs. The results obtained in 5 runs are
shown in table.4 and the best results of
obtainded in five runs are shown Fig.6.
Table.4 The results obtained in 5 runs for fuzzy problem (24).
Genoptim
al
Time(
Sec.)
at µ=1
for left
side of
constraints
(-0.1957,2.1878)
for
hand side of
constraints
(10.25,1.25)
(2.3366,1.1311)
µ(x)=
72
49
-6.6137
(2.3988,1.1243)
µ(x)=
43
80
-6.7069
(-0.0781,2.0984)
(10.25,1.25)
(2.5093,0.9333)
µ(x)=
44
62
-6.6824
(-0.0481,1.2239)
(10.25,1.25)
(2.4794,0.9736)
µ(x)=
48
59
-6.7022
(-0.0676,1.415)
(10.25,1.25)
(2.3811,1.1469)
µ(x)=
64
90
-6.6858
(-.0964,2.1845)
(10.25,1.25)
336
Tech J Engin & App Sci., 4 (4): 328-338, 2014
Membership of objective functions
1
u(f(x))
0.9
e-0.0649(x+6.6137)
2
e-0.0600(x+6.6858)
2
0.8
e-0.0548(x+6.7069)
2
0.7
e-0.0322(x+6.6824)
2
e-0.0676(x+6.7022)
2
0.6
0.5
0.4
0.3
0.2
0.1
0
-25
-20
-15
-10
-5
0
5
10
f(x)
Fig.6 Optimal values obtained from solving problem (24).
Best fit of each generation using Liou&Wang ranking function
This genetic algorithm method is executed on this problem (24) with same above parameters. The best results of
fitness function in a run at grate of membership value µ=1 are shown in Fig.7
0
-1
-2
-3
-4
-5
-6
-7
0
10
20
30
40
50
60
Generation
70
80
90
100
Fig.7 The curves for the best value obtained in 100 generations in a run for the fuzzy problem (24).
CONCLUSION
In this paper, we proposed a approach to find a fuzzy optimal solution the class of fuzzy nonlinear programming
problems. In this method, the fuzzy coefficients of problem suppose Gaussian fuzzy numbers in parametric form.
The fuzzy problem transformed in simply other fuzzy problem in parametric form by attributes of GMFs. Then a
genetic algorithm applied on fuzzy problem in parametric form using a fuzzy ranking function. The proposed
337
Tech J Engin & App Sci., 4 (4): 328-338, 2014
approach easily solved fuzzy quadratic programming and fuzzy linear programming and many FNOPs. Illustrative
numerical examples were provided to demonstrate the feasibility and efficiency of the proposed method.
REFERENCES
[1] Ali F. Jameel and Amir Sadeghi,2012, Solving Nonlinear Programming Problem in Fuzzy Environment,Int.J.Contemp.Math.Sciences,Vol.7,
no.4,159-170
[2] Jinquan Li,Shuang Feng, Honghai M,2012, A kind of nonlinear programming problem based on mixed fuzzy relation equations constraints.
Physics Procedia,volume 33. pages 1717-1724.
[3] Yahia Zare Mehrjerdi,2011, Solving fractional programming problem through fuzzy goal setting and approximation. Applied soft computing.
Volume 11,issue 2, pages 1735-1742
[4] Pandian Vasant,2013.Hibrid LS-SA-PS methods for solving fuzzy nonlinear programming problems. Mathematical and computer
modeling,volume 57,issue 1-2, pages 180-188.
[5] H.J.Zimmermann(1979). Describtion and optimization of fuzzy systems. international journal of General Systems.2:209-215.
[6] Z. Michalewicz (1992).Genetic Algorithm +Data Structures=Evolution programs. Springer Verlag.
[7] Y. J. Lai and C.L. Hwang,1992.Fuzzy Mathematical Programming Methods and applications. Springer, Berlin, 1,2
[8] C.A. Coello, D.V. Veldbuizen, G.B. Lamont(2002).Evolutionary Algorithm for Solving Multi-objective problem Kluwer Academic/Plenum
publishers, Newyork.
[9] D.E.Goldberg(1989).Genetic Algorithms in Search,optimization,and Machine Learning.Addisson-wesley.
[10] H.J.Zimmermann,1978.Fuzzy Programming and Linear Programming with Several Objective Functions,Fuzzy set and systems,1.45-55.
[11] Y.Deng,Z.Zhenfu and Liu Qi,2006.Ranking Fuzzy Numbers with an Area Method using radius of Gyration,Computers and mathematics with
Applications 51.1127-1136.
[12] L.Campos,J.L.Verdegay,1989.Linear programming problems and ranking of fuzzy numbers,Fuzzy Sets and Systems 32.1-11.
[13] H.Tanaka,T.Okuda,K.asai(1974).on fuzzy mathematical programming .journal of cybernetics,3(4):37-46.
[14] R.E.Bellman and L.A.Zadeh,1970.Decision-making in a fuzzy environment. Management Sci,Vol.17,No.4.pp.141-164.
[15]Iyengar.P, 2010.Non-linear Programming;Introduction,IEOR,Handout 19,16 October.
[16] S Chanas,1983.The use of parametric programming in fuzzy linear programming, Fuzzy sets and Systems11. 243-251.
[17] S.H.Nasseri,2008.A new method for solving fuzzy linear programming problems by solving linear programming, Applied Mathematical
Sciences Journal,Vol.50.pp.2473-2480.
[18] D.Dubois and H.prade,1980. Fuzzy sets and Systems:theory and Applications. Academic press,Newyork.
338