SAUSSUREA (ISSN: 0373-2525), 2017
Volume 6(1):PP. 118-127
RESEARCH ARTICLE
Application of Genetic Algorithm For Solving of Nonlinear Fuzzy
Programming Problems With Linear Constraints
*Abbas Akramia, Mohammad Mehdi Hosseinia,b and Syed Mehdi Karbassia
a
Department of Applied Mathematics, Faculty of Mathematics, Yazd University of Iran, Yazd,
Iran.
b
Department of Applied Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar
Kerman University of Iran, Kerman, Iran.
* Contact author
Abstract
In this paper, we consider nonlinear fuzzy optimization problems with linear constraints. In general,
conventional methods for solving nonlinear programming problems are not practical. Because in most
cases objective function and constraints are not continuous and differentiable or the problem may have
large size. Recently, genetic algorithms (GAs) are used to solve many real-world problems and
have received a great deal of attention for their ability as optimization techniques in optimization
problems. In this work, we convert nonlinear fuzzy optimization problems with linear constraints into a
crisp model by ranking function and then the obtained optimization problem is solved with GA.
Keywords: Fuzzy numbers, Nonlinear fuzzy optimization, Ranking function, Genetic algorithm.
1. Introduction
Fuzzy set theory has been applied to many disciplines such as control theory and operational research,
mathematical modeling and industrial applications. The concept of fuzzy optimization in general was first
proposed by (Tanaka et al, 1974). (Zimmerman, 1978) proposed the first formatting of fuzzy linear
programming. An optimal solution of nonlinear fuzzy programming problems introduced by (Kumar and
Kaur, 2010; Kheirfam, 2011). In their works they have taken all coefficients and decision variables to be
fuzzy numbers and all the constraints to be linear. (Behara and Nayak, 2012; Pathak and Pirzada, 2011)
have developed KKT conditions for solving nonlinear fuzzy programming problems with continuous and
differentiable objective function and constraints but in their works, the problems at most had three
decision variables.
In many actual problems especially in the complex industrial systems, there exist many kinds of fuzzy
nonlinear problems and they cannot be described and solved by traditional methods, so, the research on
novel methods for nonlinear fuzzy programming is not only important in the fuzzy optimization theory
but also great and wide value in application to the production planning and scheduling problems. In real
space, many authors investigated GA in optimization (Bunnang, 2005) applied the GA for constrained
global optimization in continuous variable, but in fuzzy environment it is not completely considered.
(Hassanzadeh et al., 2011) proposed a GA method for optimization problems with fuzzy relation
www.saussurea.org
SAUSSUREA (ISSN: 0373-2525), 2017
Volume 6(1):PP. 118-127
RESEARCH ARTICLE
constraints using max-product composition. Recently, in fuzzy linear optimization (Thakur et al., 2014;
Lin, 2008, Deep et al., 2011) are solved the problems with GA.
This paper is organized as follows: in section 2, some basic definitions and arithmetic operations of
triangular fuzzy numbers are reviewed. In section 3 formulation of nonlinear fuzzy programming
problems and application of ranking function for solving nonlinear fuzzy optimization problems with
linear constraints are discussed. In section 4, GA method is applied for solving crisp problems. In section
5, to demonstrate the effectiveness of the proposed method, some examples are solved. The conclusion
appears in section 6.
2. Preliminaries
In this section, some definitions and notations of fuzzy set theory are reviewed.
Definition 2.1 A fuzzy subset of (universal set) is specified by
that is called
membership function. Where the value
at displays the grade of membership of in .
Definition 2.2 The membership function of a triangular fuzzy number is defined as:
x a

A (x )   b  a
c  x
 c  b
a  x b
b  x c
Where it can define as triplet
.
Definition 2.3 A triangular fuzzy number
is siad to be nonnegative if and only if (iff)
Definition 2.4 Let
be a set of fuzzy numbers that defined on set of real numbers. A ranking function
is a function
which maps each fuzzy number into the real line. Let
be a triangular fuzzy number then
.
Definition 2.5 Let
and
be two triangular fuzzy numbers then
i.
iff
.
ii.
iff
.
iii.
iff
.
Definition 2.6 Let
i.
ii.
iii.
iv.
and
be two triangular fuzzy numbers then
.
.
.
.
Lemma 2.1 (Kaur and Bhardwaj, 2012) Let R is any linear function. Then
implies
. Further if
and
then
.
www.saussurea.org
iff
which
SAUSSUREA (ISSN: 0373-2525), 2017
Volume 6(1):PP. 118-127
RESEARCH ARTICLE
3. Fuzzy nonlinear programming
We consider the fuzzy nonlinear programming problem as following:
Max (or Min)
(3.1)
Where:
and
are triangular fuzzy numbers and
are crisp variables.
We use the ranking function on the problem (3.1) to get
Max (or Mini)
(3.2)
This is equal to:
Max (or Min)
(3.3)
Suppose that
then we have:
Max (or Min)
(3.4)
Lemma 3.1 According to the above relations we conclude that the optimal solutions of (3.1) and (3.4) are
equivalent.
Definition 3.1 Let:
.
www.saussurea.org
SAUSSUREA (ISSN: 0373-2525), 2017
Volume 6(1):PP. 118-127
RESEARCH ARTICLE
We say that
is an optimal solution of the problem (3.1), for a minimization problem, if there exist no
such that
.
Example 3.1(Behara and Nayak, 2012) we consider the following nonlinear fuzzy programming problem:
The authors have used the Lagrange's function for this problem as follows:
.
Then by using the KKT conditions, the optimal solution was obtained:
.
In section 5 we show that the optimal solution of this problem with GA the same as the solution that
obtained with KKT conditions.
4. Genetic Algorithm (GA)
GAs are a part of evolutionary computing, which is a rapidly growing area of artificial intelligence.
They are a robust and flexible approach that can be applied to a wide range of learning and optimization
problems. GAs generate solutions to optimization problems using techniques inspired by natural
evolution, such as inheritance, mutation, selection, and crossover. GAs replicate the survival of the fittest
among individuals over consecutive generation for solving a problem. Each generation consists of a
population of character strings that are similar to the chromosome that we see in our DNA. Each
individual represents a point in a search space and a possible solution. The individuals in the population
are then made to go through a process of evolution.
4.1. Implementation
In a GA, a population of individuals to an optimization problem is evolved toward better solutions.
Each candidate solution has a set of properties which can be mutated and altered; solutions are
represented in binary as strings of 0s and 1s, but other encodings are also possible. The evolution usually
starts from a population of randomly generated individuals, and is an iterative process, with the
population in each iteration called a generation. In each generation, the fitness of every individual in the
population is evaluated; the fitness is usually the value of the objective function in the optimization
problem being solved. The more fit individuals are randomly selected from the current population, and
each individual’s genome is modified to form a new generation. The new generation of candidate
solutions is then used in the next iteration of the algorithm. Commonly, the algorithm terminates when
either a maximum number of generations has been produced, or a satisfactory fitness level has been
reached for the population (Kaur and Bhardwaj 2012).
The Genetic algorithm evolves through these three operators:
1. Selection which equates to survival of the fittest.
www.saussurea.org
SAUSSUREA (ISSN: 0373-2525), 2017
Volume 6(1):PP. 118-127
RESEARCH ARTICLE
2. Crossover which represents mating among individuals.
3. Mutation which introduces random modifications.
A generic selection procedure may be implemented as follows:
1. The fitness function is evaluated for each individual, providing fitness values.
2. The population is sorted by descending fitness values.
3. Accumulated normalized fitness values are computed. The accumulated fitness of the last individual
should be 1.
4. A random number R between 0 and 1 is chosen.
5. The selected individual is the first one whose accumulated normalized value is greater than R.
6. Fitness may be determined by an objective function or by a subjective judgment.
Crossover is a genetic operator used to vary the programming of a chromosome or chromosomes from
one generation to the next. Crossover is a process of taking more than one parent solutions and producing
a child solution from them.
Mutation is used to maintain genetic diversity from one generation of a population of genetic algorithm
chromosomes to the next. Mutation alters one or more gene values in a chromosome from its initial state.
In mutation, the solution may change entirely from the previous solution. Hence Genetic Algorithm can
come to better solution by using mutation (Tseng and Huang 2006).
4.2. Basic Genetic Algorithm:
1. [Start] Generate random population of N chromosomes (suitable solutions for the problem).
2. [Fitness] Evaluate the fitness f(x) of each chromosome x in the population.
3. [New population] Create a new population by repeating following steps until the new population is
complete.
a) [Selection] Select two parent chromosomes from a population according to their fitness (the better
fitness, the bigger chance to be selected)
b) [Crossover] with a crossover probability crossover the parents to form a new offspring. If no crossover
was performed, offspring is an exact copy of parents.
c) [Mutation] with a mutation probability mutate new offspring at each locus (position in chromosome).
d) [Accepting] Place new offspring in a new population.
4. [Replace] Use new generated population for a further run of algorithm.
5. [Test] if the end condition is satisfied, stops, and returns the best solution in current population
6. [Loop] Go to step 2.
5. Numerical examples
By using the ranking function the equivalent crisp model of the example 3.1 is obtained as follows:
www.saussurea.org
SAUSSUREA (ISSN: 0373-2525), 2017
Volume 6(1):PP. 118-127
RESEARCH ARTICLE
The optimal solution of example 3.1 by using the GA which is run in the Matlab software is:
.
So, we see that optimal solution of example 3.1 with KKT conditions and GA are same. But conventional
methods such as KKT method are not efficient for problems with high dimensions. Therefore using
heuristic methods such as GA is preferred.
Now in the next section, we solve two examples of nonlinear fuzzy programming that have ten decision
variables and it is not possible to solve them with traditional methods.
Example5.1: Consider the following nonlinear fuzzy optimization problem with high dimension:
.
(5.1)
Where
is a vector of triangular fuzzy numbers and
fuzzy numbers as below:
is a fuzzy matrix and
The equivalent crisp problem by using ranking function is:
www.saussurea.org
is a fuzzy vector with triangular
SAUSSUREA (ISSN: 0373-2525), 2017
Volume 6(1):PP. 118-127
RESEARCH ARTICLE
Where
matrix with crisp elements as below:
are real vectors and
The solution of the above problem with GA is:
.
By substituting this solution into the problem (5.1) gives:
.
Example5.2: Consider the following fuzzy nonlinear optimization problem:
Where,
.
is a fuzzy matrix and
is a fuzzy vector with triangular fuzzy numbers as follows:
By using ranking function we have:
www.saussurea.org
is a
SAUSSUREA (ISSN: 0373-2525), 2017
Volume 6(1):PP. 118-127
RESEARCH ARTICLE
Where:
and
Also
.
is a matrix with crisp elements as below:
The optimal solution of this problem with GA is:
,
By substituting this solution into fuzzy form of the problem; gives:
.
6. Conclusion
In this paper, we obtained the optimal solution of the nonlinear fuzzy programming problems with linear
constraints. First, we convert the problem into a crisp model and then the crisp form was solved by the
method of genetic algorithm. Conventional methods for such problems with large dimensions are not
usually effective. Thus, heuristic methods such as genetic algorithms can be tailored for solving problems
with large dimensions. Illustrated numerical examples approve the effectiveness of the proposed method.
7. Acknowledgement
This research was financially supported by the center of Excellence for Robust Intelligent systems of
Yazd University. We are grateful to Yazd University, for their useful collaboration.
www.saussurea.org
SAUSSUREA (ISSN: 0373-2525), 2017
Volume 6(1):PP. 118-127
RESEARCH ARTICLE
References
[1] Behera S.K, Nayak, J.R. (2012).” Optimal Solution of Fuzzy Nonlinear Programming Problems with
Linear Constraints”. International Journal of Advances in Science and Technology, Vol.4 (2), pp. 43-91.
[2] Bunnang D, Sun M. (2005). “Genetic algorithm for constrained global optimization in continuous
variables”. Mathematics and Computation, Vol.171 (4), pp.604-636.
[3] Deep K, Singh KP, Kansal ML, Mohan C. (2011). “An interactive method using genetic algorithm for
multi-objective optimization problems modeled in fuzzy environment”. Expert Systems with
Applications, Vol.38 (10), pp.1659-1667.
[4] Hassanzadeh R, Khorram E, Mahdavi I, Mahdavi Amiri N. (2011). “A genetic algorithm for
optimization problems with fuzzy relation constraints using max-product composition”. Applied Soft
Computing, Vol.11 (9), pp.551-560.
[5] Kaur S, Bhardwaj V. (2012). “Study of Genetic Algorithms”. International Journal of Science and
Research, Vol.12 (20), pp.1260-1263.
[6] Kheirfam B. (2011). “A Method for solving fully fuzzy quadratic programming problems”. Acta
universitatis Apulensis, Vol.27 (7), pp.69-76.
[7] Kumar A, Kaur J, Singh P. (2010). “Fuzzy Optimal Solution of Fully Fuzzy Linear Programming
Problems with Inequality Constraints”. International Journal of Applied Mathematics and Computer
Sciences, Vol.6 (3), pp.37-40.
[8] Lin FT. (2008). “A Genetic Algorithm for Linear Programming with Fuzzy Constraints”. Journal of
information science and engineering, Vol.24 (), pp.801-817.
[9] Liou TS, Wang M J. (1992). “Ranking fuzzy numbers with integral value”. Fuzzy sets and systems,
Vol.50 (5), pp.247-255.
[10] Nasseri SH. (2008). “Fuzzy Nonlinear optimization”. The Journal of Nonlinear Science and
Applications, Vol.1 (1), pp.230-235.
[11] Pathak VD, Pirzada UM. (2011). “Necessary and Sufficient Optimality Conditions for Nonlinear
Fuzzy Optimization Problem”. International Journal of Mathematical Science Education, Vol.4 (4), pp.1 –
16.
[12] Tanaka H, Okuda T, Asai K. (1974). “On Fuzzy Mathematical Programming”. The Journal of
Cybernetic, Vol.3 (2), pp.37-46.
[13] Thakur M, Meghwani SS, Jalota H. (2014). “A modified real coded genetic algorithm for constrained
optimization”. Applied Mathematics and Computation, Vol.235 (1), pp.292-317.
[14] Tseng SU, Huang YM, Lin CC. (2006). “Genetic algorithm for delay and degree constrained”.
Multimedia broadcasting on overlay networks, Vol.29, pp.
[15] Zimmerman H J. (1978). “Fuzzy programming and linear programming with several objective
functions”. Fuzzy sets and systems, Vol.1 (1), pp.45-55.
www.saussurea.org