Goal Programming for Solving Fractional Programming Problem in Fuzzy Environment

Anil Kumar Nishad, Shiva Raj Singh^*

Department of Mathematics, Banaras Hindu University, Varanasi, India.

DOI: 10.4236/am.2015.614208                

Abstract

This paper is comprised of the modeling and optimization of a multi objective linear programming problem in fuzzy environment in which some goals are fractional and some are linear. Here, we present a new approach for its solution by using α-cut of fuzzy numbers. In this proposed method, we first define membership function for goals by introducing non-deviational variables for each of objective functions with effective use of α-cut intervals to deal with uncertain parameters being represented by fuzzy numbers. In the optimization process the under deviational variables are minimized for finding a most satisfactory solution. The developed method has also been implemented on a problem for illustration and comparison.

Keywords

Fuzzy Sets, Trapezoidal Fuzzy Number (TFN), Multi-Objective Linear Programming Problem (MOLPP), Multi-Objective Linear Fractional Programming Problem (MOLFPP)

Received 16 October 2015; accepted 28 December 2015; published 31 December 2015

1. Introduction

The modeling of a real life optimization problem in general needs to address several objective functions and hence become a multiobjective programming problem in a natural way. The goal programming developed by Charnes and Cooper [1] emerged a powerful tool to solve such multiobjective programming problems. Since commencement of the goal programming technique, it has been enriched by many research workers such as Lee [2] , Ignizio [3] [4] and many more. It undoubtedly established that goal programming has been one of the major breakthroughs in dealing with multi objective linear programming problems but still it fails to deal with situations when parameters are imprecise or vague. On the other hand, the development of fuzzy set by Zadeh [5] motivated Zimmermann [6] to give another approach of solving multi objective programming as fuzzy program- ming. Thus a new dimension of goal programming was introduced as fuzzy goal programming by Narsimhan [7] [8] and Ignigio [4] . However, one of the major problems which are faced by decision makers is the modeling of ill conditioned optimization problems or the problems where the coefficients are imprecise and vague. Thus the classical mathematical programming methods of optimization failed to model such problems. Bellman and Zadeh [9] gave a concept that the constraints and goals in such situations may be viewed as fuzzy sets.

Further, in many practical optimization problems the decision making becomes further complicated in situations when multiple objectives are conflicting and non commensurate or imprecise in nature. Thus such method based on goal programming needs the additional information from decision makers for priority structure of various goals and their respective aspiration levels. In view of resolving this difficulty of setting appropriate priority and aspiration levels to various objective functions, Mohanty and Vijayraghavan [10] gave a fuzzy approach to multiobjective linear programming problem to get an equivalent goal programming problem by developing a method to compute appropriate priority levels. Kuwano [11] gave a α-optimal solution to fuzzy multiobjective linear programming problem using goal programming approach. At the same time, Chanas and Kuchta [12] also considered the imprecision problem in multi objective optimization by considering interval valued objective functions. The theory of fuzzy goal programming was further enriched by Chen and Tsai [13] , Stancinlesu et al. [14] in view of providing more satisfying solutions. Thus a popular min-max approach in goal programming was studied by Lin [15] , Yaghoobi and Tamiz [16] and Cheng et al. [17] . However, on application side Soliman et al. [18] , Mishra and Singh [19] and Bharati et al. [20] used fuzzy goal programming model in agricultural sector. Recently fuzzy goal programming problems with interval coefficients and interval weights have been studied by Sen and Pal [21] and Hossein Hajiagha [22] . The readers may get a review of linear programming in works of Lotfi et al. [23] and Marbini and Tavana [24] . Recently Cheng [25] used a deviation degree measure of fuzzy numbers and applied a weighted max-min method to solve a fuzzy multi objective linear programming problem.

In problem of production planning, financial engineering and in several other areas, there are situations where one has to optimize the efficiency of the system, and thus the objective functions become ratio of two objective functions and give rise to fractional programming problem. Further there may be more such fractional objectives and thus may become a multi objective fractional programming problem. Luhandjula [26] gave a fuzzy approach to multi objective linear fractional programming problem in fuzzy environment which has been further developed by Chakraborty and Gupta [27] , Pal et al. [28] , Pop and Minasian [29] and Cui et al. [30] . The subject has been vastly envisaged by several workers and thus various approaches have been developed to solve fractional programming problems by fuzzy goal programming method given by Mehrjerdi [31] , Singh and Kumar [32] , Biswas and Dewan [33] and Ohta and Yamaguchi [34] . The solution of a fractional programming problem with interval valued coefficients has been studied by Pal and Sen [35] and Effati and Pakdaman [36] . Singh et al. [32] considered the solution of a combination of fuzzy multi objective linear programming problem and linear fractional programming problem using a goal programming approach. Thus motivated with above studies, we have extended the work of Ohta and Yamaguchi [34] to solve the fractional goal programming problem with imprecise parameters by computing the appropriate priority and weight to each goals to find optimal solution. The work done has been organized in various sections as follows. Section 2 provides the preliminaries of the subject to make the study self sufficient. Section 3 deals with the a cut presentation of multi objective linear programming problem followed by a cut presentation of linear fractional programming problem with its solution procedure. The methods developed in Sections 3 and 4 have been implemented on a numerical problem given in Section 5 followed by result and discussion placed in Section 6.

2. Preliminaries

Definition 1. Fuzzy Set

Let X is a collection of objects denoted by x, then a fuzzy set in X is a set of ordered pairs: , where is called the membership function or grade of membership of x in that maps X to the membership space.

Definition 2. Fuzzy Number

A Fuzzy set of real line ℝ with membership function is called a fuzzy number, if it holds following axioms.

1) is normal set.

2) is convex fuzzy set.

3) is upper semicontinuous.

4) is bounded.

Definition 3. Trapezoidal Fuzzy Number (TFN)

A trapezoidal fuzzy number with parameters denoted by as given in Figure 1 is a FS on real line ℝ whose membership function is defined as follows:

If in a trapezoidal FN we take then it becomes a triangular fuzzy number (TFN) with the parameters.Definition 4. α-cut of a fuzzy numberLet be a fuzzy number defined on X and number be any numbers, then α-cut of a fuzzy num-ber is a crisp set and denoted by, is defined as which is a crisp interval.Therefore, a a-cut of a triangular fuzzy number denoted by can be represented by the following interval,It is important to note that if we put then, above fuzzy number turns out to be a crisp real number.Definition 5. α-cut of a LR-fuzzy numberLet be a LR-fuzzy number denoted by, then its α-cut is defined as which is a crisp interval.3. Multi-Objective Goal Programming Formulation with α Cut of the Fuzzy NumbersLet us consider a multi-objective optimization problem with n decision variables, m constraints and k objective functions,

(1)

Figure 1. Membership function of TFN.

where, and are n dimensional and m dimensional vectors respectively, is a m ´ n matrix with fuzzy parameter and and are fuzzy numbers. Since the above problem (1) have fuzzy coefficients which have possibilistic distribution in an uncertain intervals and hence may be approximated in terms of its α-cut intervals.

Let be α-cut interval of fuzzy number defined by the definition (4).

Where and are the lower and upper bound of the α-cut interval of fuzzy number. Since the coefficients of the objective function are fuzzy numbers, α-cut interval of can be defined as

,

where and is given as in definition of α-cut interval. Thus can be represented as a closed interval, such that.

Now the lower and upper bound for the respective α-cut intervals of the objective function are defined as

(2)

. (3)

In the next step, we to construct a membership function for the maximization type objective function, and can be replaced by the upper bound of its α-cut interval i.e.

. (4)

Similarly to construct a membership function for minimization type objective function, and can be replaced by the lower bound of its α-cut interval that is

. (5)

And the constraint inequalities

(6)

(7)

can be written in terms of α-cut values as

and the fuzzy equality constraint

(8)

can be transformed into two inequalities as

Thus the undertaken maximization problem is transformed in to the following multi objective linear programming problem (MOLPP) as

(9)

Now consider the transformation of objectives to fuzzy goals by means of assigning an aspiration level to each of them. Thus applying the goal programming approach, the problem (9) can be transformed in to fuzzy goals by taking certain aspiration levels and introducing under deviational variables to each of the objective functions. In proposed method the above maximization type objective function, is transformed as

(10)

where, is under deviational variables and is aspiration level for the k^th goal and the highest acceptable level for the k^th goal and the lowest acceptable level are ideal and anti-ideal solutions and are computed as for appropriate values of

(11)

(12)

Now using min-sum goal programming method, the above fuzzy goal programming problem is converted in to single objective linear programming problem as follows.

Find so as to

(13)

Here, Z represents the achievement function and the weights w_k attached to the under deviational variables, an

(14)

4. Fractional Goal Programming Formulation with α-Cut of the Fuzzy Parameters

Let us consider a fractional optimization problem with n decision variables, m constraints and

(15)

where, and are n dimensional and m dimensional vectors respectively, is a m ´ n matrix with fuzzy parameter, and, , , and are fuzzy numbers.

It is also to assume that.

Since, above problem (15) have fuzzy coefficients which have possibilistic distribution in an uncertain intervals and hence the problem can be written in terms of its α-cut intervals.

Now the lower and upper bound for the respective α-cut intervals of the objective function are defined as

(16)

. (17)

In the next step, we to construct a membership function for the maximization type objective function, and can be replaced by the upper bound of its α-cut interval i.e.

(18)

(19)

Similarly we construct a membership function for minimization type objective function, and can be obtained by replacing the upper bound by lower bound of its α-cut interval as

(20)

(21)

And the constraint inequalities and equalities are transformed as defined in the Equation (6), (7) and (8).

Now the undertaken maximization problem is transformed in to the following linear programming problem (LPP) as

(22)

Further consider the conversion of objectives to fuzzy goals by means of assigning an aspiration level to the objective function. Thus applying the goal programming method, the problem (22) can be transformed in to fuzzy goal by taking certain aspiration levels and introducing under deviational variables to the objective function. In proposed method the above maximization type objective function, is transformed as

(23)

where, is under deviational variables and is aspiration level for the k^th objective goal and the highest acceptable level for the objective goal and the lowest acceptable level are ideal and anti-ideal solutions and are computed as for appropriate values of

(24)

(25)

Now using min-sum goal programming method, the above fuzzy goal programming problem is converted in to single objective linear programming problem as follows.

Find so as to

(26)

Here Z represents the achievement function and the weights attached to the under deviational variable, and are defined as in Equation (14).

Linearization of Membership Goal

For simplicity to solve the problem (27) we linearize the membership goal which is non-linear in nature and can be write in the following form

(27)

where.

Introducing the expression of from Equation (18), the above goal can be written as

(28)

where and.

Now by using the method of variable change given by Kornbluth and Steuer [36] the goal in the expression (28) can be linearized as follows.

Let, then the linear form of the (28) can be written as

(29)

with and.

Now in decision making, to minimize the negative deviational variable in the expression (27) means we are going to maximize the membership goal function, and also minimize which is a non linear term.

It may be noted that when membership goal is fully achieved, the value of negative deviational variable becomes zero (), and when achievement of membership goal is zero at this time value of negative deviational variable is one () for k^th objective. The involvement of in the solution leads to impose the following constraints to model the problem

. (30)

Now for a given value of α, under the framework of Goal Programming, (min-sum Goal programming) [10] , the problem under consideration can be presented as.

Find X so as to

(31)

Here Z represents the achievement function and the weights w_i attached to the under deviational variable, and are defined as in Equation (14).

5. Numerical Illustration

In view of illustrating the developed method in previous section, we consider the modelling and optimization of a problem of electronic component maker dealing in domestic and overseas markets as undertaken by Ohta and Yamaguchi [18] . The company wishes to make a mid-term production plan for three months. The company has two types of products, A and B, estimated and anticipated prices and expected gross margins products are shown in Table 1. The time required for individual products in the process and amount used of the principle materials are shown in Table 2 with amount of the expected demand shown in Table 3.

The company cherishes the idea of determine the Amount of production which satisfies the following goals and other fuzzy number with good balance as shown in Table 4.

Supposing that the amount of domestic production is x₁ and the amount of overseas production is x₂ for product A and the amount of domestic production is x₃ and the amount of overseas production is x₄ for product B. Now using data from Table 1 to Table 4, the mathematical formulation for all the fuzzy goals of the under taken problem of production system are as follows.

Table 1. Price and grass margin of each product ($).

Table 2. The time required in the process (hours) and principle materials (units).

Table 3. Expected demand.

Table 4. Aspiration levels of goals and other fuzzy numbers.

(32)

Solving the above problem by proposed method as described in section 2, first we replace the fuzzy numbers in coefficients by their α-cuts and thus above multi-objective linear fractional programming problem (32) is transformed into the following problem

(33)

Now to consider the solution of above problem (33), we apply the developed fuzzy fractional goal programming method developed in section 4 and 5 and consider its solution for, and compute various required parameters. Aspiration level of fraction goal (Z₁ and Z₂) is given as, and, and weight of fraction goal is calculated as defined in (14) i.e., , We also compute (i = 3, 4, 5) for other linear goals as defined in (14), and aspiration level for other goals (i = 3, 4, 5) is given in Table 3. Now on implementation levels and weight to the goals, the above fractional programming problem can be equivalently written in linear programming problem given as below.

(34)

where

and.

The above linear programming problem (34) has been solved by the MATLAB^®, and optimal solution for decision variables are obtained as

.

And the values of objective functions are Z₁ = 0.149 or 14.9%, Z₂ = 0.713 or 71.3%, Z₃ = 15885.52, Z₄ = 604.04, Z₅ = 18447.96.

6. Results and Discussion

The developed method uses the a-cut representation of fuzzy numbers which deals with imprecision in optimization problem. We compare the results obtained by the proposed method with the results of Ohta and Yamaguchi [33] in terms of satisfaction of various goals. The achievements of various goals by method of Ohta and Yamaguchi are Z₁ = 13.53%, Z₂ = 64.99%, Z₃ = (14504.9, 15003.7), Z₄ = (580.8, 586.8), Z₅ = (18082.21, 18383.8), whereas by the proposed method the achievements of goal are Z₁ = 14.9%, Z₂ = 71.3%, Z₃ = 15885.52, Z₄ = 604.04, Z₅ = 18447.96. Clearly the level of satisfaction of each goal by the proposed method is higher than the previous results. The proposed method has a further advantage that in general it is a difficult task to set priority weight for various goals in multi-objective programming problem. The situation becomes more tedious when the goals are conflicting in nature. It is hard to set a definite weight for a fractional goal obtained in modeling by taking the ratio of two objective functions. The proposed method also computes the appropriate weight to each goal and hence provides a better solution.

Thus for modeling the optimization problems having vagueness and imprecision in information with fuzzy optimization approach various methods are available in literature for various situations. The fuzzy optimization problems are classified in various categories, such as problems with fuzzy coefficients in constraints, fuzzy coefficients in objective functions and problems with fuzzy inequalities. The proposed method is more suitable to find the optimal solutions of the problems having L-R number fuzzy coefficients to various field of production planning problem, transportation problem, and other real world multi-objective programming problems.

NOTES

^*Corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

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