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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.

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 [

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 [

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 [

Definition 1. Fuzzy Set

Let X is a collection of objects denoted by x, then a fuzzy set

Definition 2. Fuzzy Number

A Fuzzy set

1)

2)

3)

4)

Definition 3. Trapezoidal Fuzzy Number (TFN)

A trapezoidal fuzzy number with parameters

If in a trapezoidal FN we take

where

Let

Where

where

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

In the next step, we to construct a membership function for the maximization type objective function

Similarly to construct a membership function for minimization type objective function

And the constraint inequalities

can be written in terms of α-cut values as

and the fuzzy equality constraint

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

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

where^{th} goal and the highest acceptable level for the k^{th} goal and the lowest acceptable level

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

Here, Z represents the achievement function and the weights w_{k} attached to the under deviational variables

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

where

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

In the next step, we to construct a membership function for the maximization type objective function

Similarly we construct a membership function for minimization type objective function

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

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

where^{th} objective goal and the highest acceptable level for the objective goal and the lowest acceptable level

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

Here Z represents the achievement function and the weights

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

where

Introducing the expression of

where

Now by using the method of variable change given by Kornbluth and Steuer [

Let

with

Now in decision making, to minimize the negative deviational variable

It may be noted that when membership goal is fully achieved, the value of negative deviational variable becomes zero (^{th} objective. The involvement of

Now for a given value of α, under the framework of Goal Programming, (min-sum Goal programming) [

Find X so as to

Here Z represents the achievement function and the weights w_{i} attached to the under deviational variable

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 [

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

Supposing that the amount of domestic production is x_{1} and the amount of overseas production is x_{2} for product A and the amount of domestic production is x_{3} and the amount of overseas production is x_{4} for product B. Now using data from

Product A | Product B | |
---|---|---|

Price | ||

Domestic | ||

Overseas | ||

Grassmargin | ||

Domestic | ||

Overseas |

Product A | Product B | |
---|---|---|

Process | ||

Materials |

Product A | Product B | |
---|---|---|

Domestic | 720 - 780 | 420 - 480 |

Overseas | 900 - 1200 | 600 - 800 |

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

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_{1} and Z_{2}) is given as

where

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 Z_{1} = 0.149 or 14.9%, Z_{2} = 0.713 or 71.3%, Z_{3} = 15885.52, Z_{4} = 604.04, Z_{5} = 18447.96.

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 [_{1} = 13.53%, Z_{2} = 64.99%, Z_{3} = (14504.9, 15003.7), Z_{4} = (580.8, 586.8), Z_{5} = (18082.21, 18383.8), whereas by the proposed method the achievements of goal are Z_{1} = 14.9%, Z_{2} = 71.3%, Z_{3} = 15885.52, Z_{4} = 604.04, Z_{5} = 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.

Anil KumarNishad,Shiva RajSingh, (2015) Goal Programming for Solving Fractional Programming Problem in Fuzzy Environment. Applied Mathematics,06,2360-2374. doi: 10.4236/am.2015.614208