Application of Genetic Algorithm For Solving of Nonlinear Fuzzy Programming Problems With Linear Constraints

Transcription

1 Abstract Application of Genetic Algorithm For Solving of Nonlinear Fuzzy Programming Problems With Linear Constraints *Abbas Akrami a, Mohammad Mehdi Hosseini a,b and Syed Mehdi Karbassi a 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 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

2 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 21 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 22 The membership function of a triangular fuzzy number is defined as: x a a x b b a A ( x ) c x b x c c b Where it can define as triplet Definition 23 A triangular fuzzy number is siad to be nonnegative if and only if (iff) Definition 24 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 25 Let and be two triangular fuzzy numbers then i iff ii iff iii iff Definition 26 Let and be two triangular fuzzy numbers then i ii iii iv Lemma 21 (Kaur and Bhardwaj, 2012) Let R is any linear function Then iff which implies Further if and then

3 3 Fuzzy nonlinear programming We consider the fuzzy nonlinear programming problem as following: Max (or Min) (31) Where: and are triangular fuzzy numbers and are crisp variables We use the ranking function on the problem (31) to get Max (or Mini) (32) This is equal to: Max (or Min) Suppose that (33) then we have: Max (or Min) (34) Lemma 31 According to the above relations we conclude that the optimal solutions of (31) and (34) are equivalent Definition 31 Let:

4 We say that is an optimal solution of the problem (31), for a minimization problem, if there exist no such that Example 31(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 41 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

5 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) 42 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 31 is obtained as follows:

6 The optimal solution of example 31 by using the GA which is run in the Matlab software is: So, we see that optimal solution of example 31 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 Example51: Consider the following nonlinear fuzzy optimization problem with high dimension: (51) Where is a vector of triangular fuzzy numbers and is a fuzzy matrix and is a fuzzy vector with triangular fuzzy numbers as below: The equivalent crisp problem by using ranking function is:

7 Where are real vectors and is a matrix with crisp elements as below: The solution of the above problem with GA is: By substituting this solution into the problem (51) gives: Example52: 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:

8 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,

9 References [1] Behera SK, Nayak, JR (2012) Optimal Solution of Fuzzy Nonlinear Programming Problems with Linear Constraints International Journal of Advances in Science and Technology, Vol4 (2), pp [2] Bunnang D, Sun M (2005) Genetic algorithm for constrained global optimization in continuous variables Mathematics and Computation, Vol171 (4), pp [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, Vol38 (10), pp [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, Vol11 (9), pp [5] Kaur S, Bhardwaj V (2012) Study of Genetic Algorithms International Journal of Science and Research, Vol12 (20), pp [6] Kheirfam B (2011) A Method for solving fully fuzzy quadratic programming problems Acta universitatis Apulensis, Vol27 (7), pp69-76 [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, Vol6 (3), pp37-40 [8] Lin FT (2008) A Genetic Algorithm for Linear Programming with Fuzzy Constraints Journal of information science and engineering, Vol24 (), pp [9] Liou TS, Wang M J (1992) Ranking fuzzy numbers with integral value Fuzzy sets and systems, Vol50 (5), pp [10] Nasseri SH (2008) Fuzzy Nonlinear optimization The Journal of Nonlinear Science and Applications, Vol1 (1), pp [11] Pathak VD, Pirzada UM (2011) Necessary and Sufficient Optimality Conditions for Nonlinear Fuzzy Optimization Problem International Journal of Mathematical Science Education, Vol4 (4), pp1 16 [12] Tanaka H, Okuda T, Asai K (1974) On Fuzzy Mathematical Programming The Journal of Cybernetic, Vol3 (2), pp37-46 [13] Thakur M, Meghwani SS, Jalota H (2014) A modified real coded genetic algorithm for constrained optimization Applied Mathematics and Computation, Vol235 (1), pp [14] Tseng SU, Huang YM, Lin CC (2006) Genetic algorithm for delay and degree constrained Multimedia broadcasting on overlay networks, Vol29, pp [15] Zimmerman H J (1978) Fuzzy programming and linear programming with several objective functions Fuzzy sets and systems, Vol1 (1), pp45-55