Fuzzy Stochastic Linear Fractional Programming based on Fuzzy Mathematical Programming

In this paper, we consider a Fuzzy Stochastic Linear Fractional Programming problem (FSLFP). In this problem, the coefficients and scalars in the objective function are the triangular fuzzy number and technological coefficients and the quantities on the right side of the constraints are fuzzy random variables with the specific distribution. Here we change an FSLFP problem to an equivalent deterministic Multi-objective Linear Fractional Programming (MOLFP) problem. Then by using Fuzzy Mathematical programming approach transformed MOLFP problem is reduced single objective Linear programming (LP) problem. A numerical example is presented to demonstrate the effectiveness of the proposed method.


Introduction
In real-world problems, often face problems to take decisions that optimise the department/equity ratio, profit/cost, inventory/sales, actual cost/standard cost, output/employee, student/cost, nurse/patient ratio, as financial and corporate planning, production planning, marketing and media selection, university planning and student admissions, health care and hospital planning, etc. such problems can be solved through Linear Fractional Programming (LFP) problems. In practice, due to the errors of measurement or vary with market conditions, etc., some or all of the coefficients are not exact. These situations can be modelled through Fuzzy Linear Fractional Programming (FLFP). Most of the FLFP problems can be modelled and solved by fuzzy goal programming approach [1][2][3][4][5][6][7][8], but very few authors considered FLFP problem where fuzzy coefficients are fuzzy numbers. Pop and Stancu Minasian [9], analysed a method to solve the fully fuzzified LFP problem, where all the variables and parameters are represented by triangular fuzzy numbers and Veeramani and Sumathi [10], proposed a method to solve FLFP problem where the cost of the objective function, the resources and the technological coefficients are triangular fuzzy numbers.
In some significant real-world problems, one has to base decisions on information which is both fuzzily imprecise and probabilistically uncertain [11][12][13][14][15][16][17][18]. As an example, consider a production problem that is cast into a linear program format. Assume that the second member components of constraints are demands which are random variables while coefficients of the technological matrix are given by experts who prefer to express them as fuzzy numbers, in a way to couple their vague perceptions with hard statistical data. Fuzzy Stochastic Linear Programming aims at grappling with such hybrid situations.
On the other hand, the case of fuzzy stochastic multiobjective linear programming Problems has been considered by utilising the possibility programming approach as well as the chance-constrained approach [1,6]. In this paper, we consider the FSLFP problem with fuzzy objective coefficient and technological coefficient and resources as fuzzy random variables, where the fuzzy coefficient is triangular fuzzy numbers. First the given FSLFP problem is transformed into a deterministic MOLFP problem. This transformation is obtained by using Zadeh extension principle. By using Fuzzy Mathematical programming approach transformed MOLFP problem is reduced single objective Linear Programming (LP). Different sections of the paper are organised as follows: In Section 2, we review some concepts of fuzzy numbers. In Section 3, the method of converting LFP problem into an LP problem is discussed. The procedure for transforming MOLFP into MOLP problem and Fuzzy Mathematical programming technique is presented in Section 4. In Section 5, the method for solving FSLFP problem using Fuzzy Mathematical Programming approach is developed. The proposed procedure illustrated through a numerical example in Section 6.

Preliminaries
In this section, we recall some basic definitions involving fuzzy sets, fuzzy numbers and operations on fuzzy numbers are outlined. Definition 2.1: Let X denote a universal set. Then a fuzzy subsetÃ of X is defined by its membership function which assigns a real number μÃ(x) in the interval [0, 1] to each element x ∈ X, where μÃ(x) represents the grade of membership of x inÃ. Thus, the nearer the value of μÃ(x) is unity, the higher the grade of membership of x inÃ. A fuzzyÃ subset can be characterised as a set of ordered pairs of element x and its grade μÃ(x) and is often written asÃ (1)Ã is normal and convex fuzzy set.
The membership function μÃ ofÃ can be expressed as

Trapezoidal Fuzzy Number
In this section, the membership function of Triangular Fuzzy Number (TFN) is presented. Definition 2.3:ã is a TFN ifã = {a, a 0 ,ā}, where a is the least possible value, a 0 is the main value, andā is the highest possible value. The triangular shaped membership function is μÃ(a; θ), θ ∈ (0, 1], where θ is the maximum value of the membership function, i.e. when a = a 0 . Then Therefore,

Random Variables
This section covers univariate random variables.

Definition 2.4:
Let ( , F, P) be a probability space. If X : → R is a real-valued function have as its domain elements of , then X is called a random variable.

Definition 2.5:
A random variable is called discrete if its range consists of a countable (possibly infinite) number of elements.
Discrete random variables are characterised by a Probability Mass Function (PMF) which gives the probability of observing a particular value of the random variable.

Definition 2.6:
The probability mass function for a discrete random variable X is defined as f (x) = P(x), for all x ∈ R(X) and f (x) = 0, for all x / ∈ R(X), where R(X) is the range of X (i.e. the values for which X is defined).

Definition 2.7:
A random variable is called continuous if its range is uncountably infinite and there exists a non-negative-valued function f (x) defined or all x ∈ (−∞, ∞) such that for any event B ⊂ R(X), P(X) = ∫ x∈B f (x)dx and f (x) = 0, for all x ∈ R(X), where R(X) is the range of X (i.e. the values for which X is defined).
The PMF of a discrete random variable is replaced with the probability density function (pdf) for continuous random variables.

Definition 2.9: The Cumulative Distribution Function (CDF) for a random variable X is defined as
The cumulative distribution function is used for both discrete and continuous random variables. When X is a discrete random variable, the CDF is for c ∈ (−∞, ∞) and when X is a continuous random variable, the CDF is for x ∈ (−∞, ∞).

Linear Fractional Programming Problem
The linear fractional programming problem can be written as where j = 1, . . . , n, A ∈ R m×n , b ∈ R m , c j , d j ∈ R n and r, s ∈ R−. For some of x, Q(x) may be equal to zero. To avoid such cases, one requires that either For convenience, assume that LFP satisfies the condition that Theorem 3.1: Assume that no point (z; 0) with z ≥ 0 is feasible for the following linear programming problem Now assume that the condition (9), then the LFP (8) us equivalent to linear programming problem (10).

Consider the following two related problems
and where (11) is obtained from (8) by the transformation t = 1 Q(x) , y = tx and (12) differs from (11) by replacing the equality constraint Q y t = 1 by an inequality constraint tQ y t ≤ 1.

The problem (8) is said to be standard concave-convex programming problem, if P(x) is concave on D with P(γ ) ≥ 0 for some γ ∈ D and Q(x) is convex and positive on D.
Theorem 3.2: Let for some γ ∈ D, P(γ ) ≥ 0, if (8) achieve to a (global) maximum at x = x * , then (12) achieve to a (global) maximum at (t, y) = (t * , y * ), where y * t * = x * and The objective functions are equal at this point. (8) is a standard concave-convex programming problem which reaches a (global) maximum at a point x * , then the corresponding transformed problem (12) attains the same maximum value at a point (t * , y * ) Where y * t * = x * . Moreover (12) has a concave objective function and a convex feasible set.

P(x) is concave, Q(x) is concave and positive on D and P(x) is negative for each
where −P(x) is convex and positive. Now, with the applications of the Theorem (3.1) and under the present hypotheses the fractional programming (8) transformed to the following linear programming problem

Multi-Objective Linear Fractional Programming problem
The MOLFP problem can be written as follows: Let I be the index set such that and By using the transformation y = tx(t > 0), Theorem 3.2 and 3.3, MOLFP problem (15) may be written as follows In classical linear programming with objective functions represented by fuzzy sets, the complete solution set (y,t) from theoretically well-defined membership function expression μ Q (y, t) = ∩ k i=1 μ k (y, t), Zimmermann [3] proved that, if μ Q (y, t) had a unique maximum value μ Q (y * , t * ) = max μ Q (y, t), then (y * , t * ) which is an element of a complete solution set (y, t) can be derived by solving a classical linear programming with one variable λ. The complete solution set is composed of all those solution vectors which results μ Q (y, t) > 0. If no solution vector (y, t) can result μ Q (y, t) > 0, we say that the complete solution set does not exist. If a complete solution set contains all solutions vectors with μ Q (y, t) > 0 and if an (y * , t * ) with the unique μ Q (y * , t * ) = max μ Q (y, t) exists, it must be included in the complete solution set. If i ∈ I, then membership function of each objective function can be written as If i ∈ I c , then membership function of each objective function can be written as Using Zimmermann's min operator the model (16) transformed to the crisp model as

Fuzzy Stochastic Linear Fractional Programming problem
where x j , j = 1, . . . , n denotes the vector of decision variables,c j andd j are fuzzy coefficients whiler ands are two fuzzy scalars.ã s ij andb s i are normally distributed fuzzy random variables. Thus, by incorporating predetermined tolerance measures β i , i = 1, . . . , m, and by use the chance-constrained approach, the set of fuzzy stochastic constraint problem (20) can be transformed to their deterministic fuzzy equivalents as follows [7,19]: Now let E(.) and Var(.) is represents the mean and the variance of random variables respectively, then the relationship (21) can be rewritten as follows: for i = 1, . . . , m. So problem (20) becomes as follows: In a fuzzy decision making situation, it is to be assumed that the mean and variance associated with the fuzzy random variablesh i are triangular fuzzy numbers, which are considered as follows: Also the expected values E(b s i ), i = 1, . . . , m, is shown as follows: For fuzzy coefficientsc j andd j , we havẽ Therefore, the problem (23) can be written as ).

Conclusion
In the paper, a method of solving the FSLFP problems, where the cost of the objective function is triangular fuzzy numbers and the resources and technological coefficients are fuzzy random variables, is proposed. In the proposed method, FSLFP problem is transformed into a Multi Objective Linear Fractional programming (MOLFP) problem and the resultant problem is converted to a LP problem, using Fuzzy Mathematical programming method. The proposed approach can be extended for solving linear fractional programming problems, where the cost of the objective function, the resources and the technological coefficients are trapezoidal fuzzy numbers or non-linear membership functions and solving Fuzzy Multi-objective linear fractional programming problems.

Disclosure statement
No potential conflict of interest was reported by the authors.