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Multi-objective fuzzy linear programming and its application in transportation model.

Abstract

In this study, the solution procedure of Multi-objective Fuzzy Linear Programming Problem (MOFLPP) with mixed constraints and its application in solid transportation problem, is going to be presented. There are two parts in this paper. In the first part, a Multi-objective Linear Programming Problem MOLPP) with fuzzy coefficients occurring in constraints and objective functions and fuzzy constraint goals, has been considered. Here fuzzy constraint goals and coefficients of objective and constraint functions are characterised by Triangular Fuzzy Numbers (TFNs). Using Bellman and Zadeh's (1970) multicriteria fuzzy decision-making process, the very problem has been converted to a crisp non-linear programming problem. Then it has been solved using fuzzy decisive set method. In other part, a linear multi-objective solid transportation problem with mixed constraint as well as additional restriction in fuzzy environment is considered. In this transportation problem, cost coefficients of objective functions and additional restriction function, the supply, demand and conveyance capacity have been expressed as TFNs. This MOFLPP is solved by fuzzy decisive set method as before. Numerical examples have been provided for two parts to illustrate the solution procedure.

Keywords and Phrases: Fuzzy multi-objective linear programming, Solid transportation problem, Triangular fuzzy number.

1. Introduction

Generally, in a Multi-Objective Linear Programming Problem (MOLPP), coefficients (of objective and constraint functions) as well as constraint goals are assumed to be fixed in value. But there are many practical situations where this assumtions are not valid. These coefficients as well as constraint goals may not be well defined due to lack of information of data and/or uncertain market situations. For this reasons, the different coefficients and constraint goals may be chacterised by fuzzy numbers.

The idea of fuzzy set was first proposed by Zadeh [11], as a mean of handling uncertainty that is due to imprecision rather than to randomness. After that Bellman and Zadeh [11] proposed that a fuzzy decision might be defined as the fuzzy set, defined by the intersection of fuzzy objective and constraint goals. From this view point , Tanakka and Asai [6], Zimmermann [7] introduced fuzzy linear programming problem in fuzzy environment. Tong [14], Gasimov and Yenilmez [12] among others, considered single objective mathematical programming with all fuzzy parameters. Tong considered the fuzzy linear programming problem with fuzzy constraints. After defuzzification he solved the so-obtained crisp problem by fuzzy decisive set method proposed by Sakawa and Yano [10]. Gasimov and Yenilmez considered fuzzy linear programming (FLP) problem with less than type constraints. In their paper Coefficients of constraints were taken as fuzzy numbers. They solved it by fuzzy decisive set method and modified sub-gradient method. Lai-Hawng [15] considered MOLPP with all parameters, having a triangular possibility distribution. They used an auxiliary model and it was solved by multi-objective linear programming methods. Chanas, [3] proposed a fuzzy programming in multi-objective linear programming and it was solved by parametric approach. Zimmermann [7] proposed a fuzzy multicriteria decision making set, defined as the intersection of all fuzzy goals and constraints.

There are so many Transportation models where FLP have been applied. Bit et al [1] considered fuzzy programming approach to multicriteria decision making transportation problem in which the constraints are of equality types. Latter Bit et al [2] also considered a fuzzy programming approach to multi-objective solid transportation problem in which the supply, demand and capacity constraints are of equality and inequality types. They solved it by fuzzy programming technique. Das et al [13] considered the multi-objective transportation problem with interval cost, source and destination parameters. They converted the interval cost, source and destination parameters into deterministic one and finally it was solved by fuzzy programming technique.

In this paper, we have proposed a MOFLPP with mixed constraints in which right hand side of constraints are fuzzy numbers. Using Bellman and Zadeh's fuzzy decision-making process, the MOFLPP is converted into an equivalent crisp LPP. Then it is solved by simplex method. Next we have also considered MOFLPP with coefficients of objective as well as constraint functions and right hand sides of constraints are TFNs. Converting it into an equivalent crisp non-linear programming problem, it is also solved by fuzzy decisive set method.

We have also considered an application of MOFLPP on a transportation model .We have considered a multi-objective solid transportation model with an additional restriction and mixed constraints in which coefficients of objective functions, additional restriction function, demand, supply and conveyance capacities are expressed as TFNs. It is then solved by fuzzy decisive set method as shown before.

2. Triangular Fuzzy Number (TFN)

Let F([??]) be a set of all triangular fuzzy number in a real line [??]. A triangular fuzzy number [??] ([member of] F([??])) is a normal and convex fuzzy set with the following membership function [[mu].sub.[??]] : [??] [right arrow] [1,0] (which satisfies both normality i.e [[mu].sub.[??]] ([??]) = 1 for at least one [??] [member of] R and convexity i.e [[mu].sub.[??]]([[??].sup./]) [greater than or equal to] min.([[mu].sub.[??]]([[??].sub.1]), [[mu].sub.[??]]([[??].sub.2])) where [[mu].sub.[??]]([??]) [member of] [1,0] and [for all] [[??].sup./] [member of] [[[??].sub.1],[[??].sub.2]]).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

[FIGURE 1 OMITTED]

It is parameterized by a triplet ([a.sub.1],[a.sub.2],[a.sub.3]) where [a.sub.1],[a.sub.3] are the lower and upper limits of support of [??] and [a.sub.2] is the pick value of [??] (fig.-1). Triangular fuzzy numbers are very often used in different application (e.g. fuzzy controls, managerial decision making, business and finance, social sciences etc.).

More generally, the left [??] and right [??] are branches of the TFN. They can be denoted by left [??] = ([a.sub.1], [a.sub.2], [a.sub.2]) and right [??] = ([a.sub.2], [a.sub.2], [a.sub.3]).

i) The left TFN [??] =([a.sub.1], [a.sub.2], [a.sub.2]) (fig.-2) is suitable to represent positive large or words with similar meaning (e.g. old age, big profit , high risk, etc.) provided that [a.sub.2] > [a.sub.1] . It is represented by the following membership functions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

= [[x - [a.sub.1]]/[[a.sub.2] - [a.sub.1]]] for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

= 1 for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

[FIGURE 2 OMITTED]

ii) The right TFN [??] = ([a.sub.2] ,[a.sub.2] ,[a.sub.3]) (fig.-3) [??] is suitable to represent positive small or words with similar meaning (e.g. young age, small cost, small risk etc.) provided that [a.sub.3] > [a.sub.2]. It is represented by the following membership functions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

= [[[a.sub.3] - x]/[[a.sub.3] - [a.sub.2]]] for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

= 0 for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

[FIGURE 3 OMITTED]

Note: A TFN = ([a.sub.1],[a.sub.2],[a.sub.3]) is positive (negative) if [a.sub.1] [greater than or equal to] 0 ([a.sub.3] < 0).

3. Multi-Objective Linear Programming Problem (MOLPP) with Fuzzy Resources

The General Multi-Objective Linear Programming Problem (GMOLPP) with mixed constraints may be written as follows: Minimize Z = [[Z.sup.1],[Z.sup.2],[Z.sup.3], ......, [Z.sup.K]] (3.1)

subject to [n.summation over (j=1)][a.sub.ij][x.sub.j] [greater than or equal to] [b.sub.i] for i = 1,2,3, ......, [m.sub.1]

[n.summation over (j=1)][a.sub.ij][x.sub.j] [less than or equal to] [b.sub.i] for i = [m.sub.1]+1, [m.sub.1]+2, ......, [m.sub.2]

[n.summation over (j=1)][a.sub.ij][x.sub.j] = [b.sub.i] for i = [m.sub.2]+1, [m.sub.2]+2, ......, m

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where

[Z.sup.k] = [n.summation over (j=1)][c.sup.k.sub.j][x.sub.j], k = 1, 2, 3, ......, K

3.1 MOLPP with fuzzy resources

When constraint goals are TFNs, (3.1) becomes Minimize Z =[[Z.sup.1],[Z.sup.2],[Z.sup.3], ......, [Z.sup.K]] (3.2)

subject to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for i = 1,2,3, ......, [m.sub.1]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for i = [m.sub.1]+1, [m.sub.1]+2, ......, [m.sub.2]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for i = [m.sub.2]+1 , [m.sub.2]+2, ......, m

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where

[Z.sup.k] = [n.summation over (j=1)][c.sup.k.sub.j][x.sub.j], k = 1, 2, 3, ......, K

We will accept some assumptions.

Assumption1: [[??].sub.i] are considered as the following positive TFNs:

Left TFN [[??].sub.i] = ([b.sub.i] - [b.sup.0.sub.i], [b.sub.i], [b.sub.i]) with tolerance [b.sup.0.sub.i] (<[b.sub.i]) for [n.summation over (j=1)] [a.sub.ij][x.sub.j] [greater than or equal to] [[??].sub.i]; 1,2,3, ......, [m.sub.1];

RightTFN [[??].sub.i] = ([b.sub.i], i, [b.sub.i] + [b.sup.0.sub.i]) with tolerance [b.sup.0.sub.i] (>0) for [n.summation over (j=1)] [a.sub.ij][x.sub.j] [less than or equal to] [[??].sub.i]; = [m.sub.1]+1,[m.sub.1]+2, ......, [m.sub.2];

and TFN [[??].sub.i] = ([b.sub.i]- [b.sup.l.sub.i],[b.sub.i],[b.sub.i]+ [b.sup.r.sub.i]) with tolerances [b.sup.l.sub.i](< [b.sub.i]), [b.sup.r.sub.i] (> 0) for [n.summation over (j=1)] [a.sub.ij][x.sub.j] = [[??].sub.i],

i = [m.sub.2]+1 , [m.sub.2]+2, ......, m.

The problem (3.2) becomes with extreme tolerances as Minimize Z =[[Z.sup.1],[Z.sup.2],[Z.sup.3], ......, [Z.sup.K]] (3.3)

subject to [n.summation over (j=1)][a.sub.ij][x.sub.j] [greater than or equal to] [b.sub.i] - [b.sup.0.sub.i] i = 1,2,3, ......, [m.sub.1]

[n.summation over (j=1)][a.sub.ij][x.sub.j] [less than or equal to] [b.sub.i] + [b.sup.0.sub.i] i = [m.sub.1]+1, [m.sub.1]+2, ......, [m.sub.2]

[n.summation over (j=1)][a.sub.ij][x.sub.j] [greater than or equal to] [b.sub.i] - [b.sup.l.sub.i] i = [m.sub.2]+1, [m.sub.2]+2, ......, m

[n.summation over (j=1)][a.sub.ij][x.sub.j] [less than or equal to] [b.sub.i] + [b.sup.r.sub.i] i = [m.sub.2]+1, [m.sub.2]+2, ......, m

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where

[Z.sup.k] = [n.summation over (j=1)][c.sup.k.sub.j][x.sub.j], k = 1, 2, 3, ......, K

3.2 Fuzzy Programming Technique for the Solution of MOLPP with Fuzzy Resources

The MOFLPP can be considered as a Vector Minimum Problem (VMP). Let [L.sub.k] and [U.sub.k] be the lower and upper bound for the k-th objective, where

[L.sub.k] = aspired level of achivement for the k-th objective function, and

[U.sub.k] = highest acceptable level of achivement for the k-th objective function.

When the aspiration levels for each objective have been specified, we formed a fuzzy model. Our next step is to transform the fuzzy model into a crisp model (i.e a coventional LPP). The details of the foregoing steps may be presented as follows:

Algorithm

Step-1. Solve the MOLPPs (3.1) and (3.3) as a single objective LPP using each time only one objective and ignore all others.

Step-2. From the results of step-1, determine the corresponding value for every objective functions at each solutions.

Step-3. Find upper and lower bounds (i.e [U.sub.k] and [L.sub.k]) for kth objective from the 2k objective values derived in step-2.

Step-4. The initial fuzzy model is equivalent to following:

Find [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.4)

so as to satisfy

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for k = 1, 2, 3, ......, K.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] i = 1,2,3, ......, [m.sub.1]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] i = [m.sub.1]+1, [m.sub.1]+2, ......, [m.sub.2]

[n.summation over (j=1)][a.sub.ij][x.sub.j] [congruent to] [b.sub.i] i = [m.sub.2]+1, [m.sub.2]+2, ......, m

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Here membership functions for fuzzy constraints of (3.4) are defined as: (for kth constraints [[??].sub.k] (k = 1, 2, 3, ......, K)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for [n.summation over (j=1)][c.sup.k.sub.j][x.sub.j] [less than or equal to] [L.sub.k], k = 1, 2, 3, ......, K.

= [[[U.sub.k] - [n.summation over (j=1)][c.sup.k.sub.j][x.sub.j]]/[[U.sub.k] - [L.sub.k]]] for [L.sub.k] [less than or equal to] [n.summation over (j=1)][c.sup.k.sub.j][x.sub.j] [less than or equal to] [U.sub.k],

= 0 for [n.summation over (j=1)][c.sup.k.sub.j][x.sub.j] > [U.sub.k],

(for the ith constraints [[??].sub.i] (i = 1,2,3, ......, [m.sub.1]))

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for [b.sub.i] [less than or equal to] [n.summation over (j=1)][a.sub.ij][x.sub.j] - [b.sup.0.sub.i]

= [[[n.summation over (j=1)][a.sub.ij][x.sub.j] - [b.sub.i]]/[b.sup.0.sub.i]] for [n.summation over (j=1)][a.sub.ij][x.sub.j] - [b.sup.0.sub.i] [less than or equal to] [b.subi] [less than or equal to] [n.summation over (j=1)][a.sub.ij][x.sub.j]

= 0 for [b.sub.i] > [n.summation over (j=1)][a.sub.ij][x.sub.j]

(for the ith constraints [[??].sub.i] (i = [m.sub.1]+1, [m.sub.1]+2, ......, [m.sub.2]))

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for [b.sub.i] < [n.summation over (j=1)][a.sub.ij][x.sub.j]

= [[[b.sub.i] - [n.summation over (j=1)][a.sub.ij][x.sub.j]]/[b.sup.0.sub.i]] for [n.summation over (j=1)][a.sub.ij][x.sub.j] [less than or equal to] [b.sub.i] [less than or equal to] [n.summation over (j=1)][a.sub.ij][x.sub.j] + [b.sup.0.sub.i]

= 1 for [b.sub.i] [greater than or equal to] [n.summation over (j=1)][a.sub.ij][x.sub.j] + [b.sup.0.sub.i]

(for the ith constraints [[??].sub.i] (i = [m.sub.2]+1, [m.sub.2]+2, ......, m))

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for [b.sub.i] > [n.summation over (j=1)][a.sub.ij][x.sub.j] + [b.sup.l.sub.i]

= [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for [n.summation over (j=1)] [a.sub.ij][x.sub.j] < [b.sub.i] [less than or equal to] [n.summation over (j=1)] [a.sub.ij][x.sub.j] + [b.sup.l.sub.i]

= [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for [n.summation over (j=1)][a.sub.ij][x.sub.j] - [b.sup.r.sub.i] [less than or equal to] [b.sub.i] < [n.summation over (j=1)][a.sub.ij][x.sub.j]

= 0 for [b.sub.i] [less than or equal to] [n.summation over (j=1)][a.sub.ij][x.sub.j] - [b.sup.r.sub.i]

Step-5. Using the max-min operator (as Zimmermann [7]) crisp LPP for (3.2) is formulated as follows:

Max [lambda] (3.5)

subject to, [n.summation over (j=1)][c.sup.k.sub.j][x.sub.j] + [lambda]([U.sub.k] - [L.sub.k]) [less than or equal to] [U.sub.k] for k = 1, 2, 3, ......, K.

[n.summation over (j=1)][a.sub.ij][x.sub.j] - [lambda] [b.sup.0.sub.i] [greater than or equal to] [b.sub.i], for i = 1,2,3, ......, [m.sub.1]

[n.summation over (j=1)][a.sub.ij][x.sub.j] + [lambda] [b.sup.0.sub.i] [less than or equal to] [b.sub.i], for i = [m.sub.1]+1, [m.sub.1]+2, ......, [m.sub.2]

[n.summation over (j=1)][a.sub.ij][x.sub.j] - [lambda] [b.sup.l.sub.i] [greater than or equal to] [b.sub.i] - [b.sup.l.sub.i], for i = [m.sub.2]+1, [m.sub.2]+2, ......, m

[n.summation over (j=1)][a.sub.ij][x.sub.j] + [lambda] [b.sup.r.sub.i] [less than or equal to] [b.sub.i] + [b.sup.r.sub.i], for i = [m.sub.2]+1, [m.sub.2]+2, ......, m

0 [less than or equal to] [lambda] [less than or equal to] 1, [x.sub.j] [greater than or equal to] 0 . j=1,2,3, ......, n

It can be solved by any simplex method.

Numerical Example1:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1)

subject to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], i, j, k = 1, 2 and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

[[??].sub.2] = [??] = (9, 10, 10) respectively constraint goals.

To solve this problem, we first solve the following four Sub-Problems (SPs):

Minimize [Z.sup.11] = 5[[??].sub.1]+3[[??].sub.2] ... (1.1) subject to 2[[??].sub.1]+4[[??].sub.2] [greater than or equal to] 20, [[??].sub.1]+[[??].sub.2] [greater than or equal to] 10, [[??].sub.1], [[??].sub.2] [greater than or equal to] 0;

Minimize [Z.sup.12] = 5[[??].sub.1]+3[[??].sub.2] ..........(1.2) subject to 2[[??].sub.1]+4[[??].sub.2] [greater than or equal to] 18, [[??].sub.1]+[[??].sub.2] [greater than or equal to] 9, [[??].sub.1], [[??].sub.2] [greater than or equal to] 0;

Minimize [Z.sup.21] = 2[[??].sub.1]+7[[??].sub.2] ......(1.3) subject to 2[[??].sub.1]+4[[??].sub.2] [greater than or equal to] 20, [[??].sub.1]+[[??].sub.2] [greater than or equal to] 10, [[??].sub.1], [[??].sub.2] [greater than or equal to] 0;

Minimize [Z.sup.22] = 2[[??].sub.1]+7[[??].sub.2] ..........(1.4) subject to 2[[??].sub.1]+4[[??].sub.2] [greater than or equal to] 18, [[??].sub.1]+[[??].sub.2] [greater than or equal to] 9, [[??].sub.1], [[??].sub.2] [greater than or equal to] 0;

So the optimal solutions of (1.1), (1.2), (1.3) and (1.4) are [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], respectively.

So [L.sub.1] = min [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]= min{30, 27, 50, 45} = 27

and [U.sub.1] = max [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = max{30, 27, 50, 45} = 50 [L.sub.2] = min [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = min{70, 63, 20, 18} = 18

and [U.sub.2] = max [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = max{70, 63, 20, 18} = 70

Following the step-4, the problem (1) is equivalent to

Find {[x.sub.j] , j = 1, 2.} (1.5) so as to satisfy [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Here membership functions for fuzzy constraints of (1.5) are defined as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]) for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

= [50 - [(5[x.sub.1] + 3[x.sub.2])]/23] for 27 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

= 0 for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

= [70 - [(2[x.sub.1] + 7[x.sub.2])]/52] for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

= 0 for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

= [2[x.sub.2] + 4[x.sub.2] - 20]/2] for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

= 0 for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

= [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

= 0 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

Using the max-min operator (as Zimmermann [7]) crisp LPP for (1) is formulated as follows:

Max [lambda] (1.6) 5[[??].sub.1]+3[[??].sub.2]+23[lambda] [less than or equal to] 50, 2[[??].sub.1]+7[[??].sub.2]+52[lambda] [less than or equal to] 70, 2[[??].sub.1]+4[[??].sub.2] -2[lambda] [greater than or equal to] = 20, [[??].sub.1]+[[??].sub.2] -[lambda] [greater than or equal to] 10, 0 [less than or equal to] [lambda] [less than or equal to] 1, [[??].sub.1],[[??].sub.2] [greater than or equal to] 0.

So, optimal solution of MOFLPP (1) are [[??].sup.*.sub.1] =4.758065; [[??].sup.*.sub.2] = 5.645161; [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with aspiration level [[lambda].sup.*] = 0.403226.

4. MOLPP with Fuzzy Coefficients and Fuzzy Resources

When the objective function's coefficients, technological coefficients and also right hand side of constraints are fuzzy numbers then (3.1) becomes Minimize Z =[[Z.sup.1],[Z.sup.2],[Z.sup.3], ......, [Z.sup.K]] (4.1)

subject to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for i = 1,2,3, ......, [m.sub.1]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for i = [m.sub.1]+1, [m.sub.1]+2, ......, [m.sub.2]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for i = [m.sub.2]+1, [m.sub.2]+2, ......, m

[x.sub.j] [greater than or equal to] 0 . j=1, 2, 3, ......, n

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], k = 1, 2, 3, ......, K

Assumption1: Fuzzy objective and constraints coefficients are considered as the following positive TFN's:

Right TFN [c.sup.k.sub.j] = ([c.sup.k.sub.j], [c.sup.k.sub.j], [c.sup.k.sub.j] + [p.sup.k.sub.j]) with tolerance [p.sup.k.sub.j] (> 0) for the objective function [n.summation over (j=1)][[??].sup.k.sub.j][x.sub.j] for k = 1, 2, 3, ......, K.

Left TFNs [[??].sub.ij] = ([a.sub.ij] - [d.sup.0.sub.ij], [a.sub.ij], [a.sub.ij]) with tolerance [d.sup.0.sub.ij] (< [a.sub.ij]) and [[??].sub.i] = ([b.sub.i]-[b.sup.0.sub.i], [b.sub.i], [b.sub.i]) with tolerance [b.sup.0.sub.i](< [b.sub.i]) for [n.summation over (j=1)][[??].sub.ij] [x.sub.j] [greater than or equal to] [[??].sub.i]; i = 1,2,3, ......, [m.sub.1].

Right TFNs [[??].sub.ij] = ([a.sub.ij] , [a.sub.ij] , [a.sub.ij] + [d.sup.0.sub.ij]) with tolerance [d.sup.0.sub.ij] (> 0) and [[??].sub.i] = ([b.sub.i], [b.sub.i], [b.sub.i] + [b.sup.0.sub.i] with tolerance [b.sup.0.sub.i] (> 0) for [n.summation over (j=1)][[??].sub.ij] [x.sub.j] [less than or equal to] [[??].sub.i]; i =[m.sub.1]+ 1, [m.sub.1+2], ......, [m.sub.2]. and TFNs [[??].sub.ij] = ([a.sub.ij] - [d.sup.l.sub.ij], [a.sub.ij], [a.sub.ij] + [d.sup.r.sub.ij]) with tolerances [d.sup.l.sub.ij](< [a.sub.ij]), [d.sup.r.sub.ij] (> 0) and [[??].sub.i] = ([b.sub.i] - [b.sup.l.sub.i], [b.sub.i], [b.sub.i] + [b.sup.r.sub.i]) with tolerances [b.sup.l.sub.ij] (< [b.sub.i]), [b.sup.r.sub.ij] (> 0) for [n.summation over (j=1)][[??].sub.ij][x.sub.j] = [[??].sub.i]; i = [m.sub.2+1], [m.sub.2]+2, ......, m.

For the calculation of upper ([U.sub.k]) and lower ([L.sub.k]) bounds of the k-th(k = 1, 2, 3, ......, K) objective function, we first construct the following eight sub-problems(4.2 - 4.9):

Minimize [Z.sup.k1] = [n.summation over (j=1)][c.sup.k.sub.j][x.sub.j] (4.2)

subject to [n.summation over (j=1)][a.sub.ij][x.sub.j] [greater than or equal to] [b.sub.i], for i = 1,2,3, ......, [m.sub.1]

[n.summation over (j=1)][a.sub.ij][x.sub.j] [less than or equal to] [b.sub.i], for i = [m.sub.1]+1, [m.sub.1]+2, ......, [m.sub.2]

[n.summation over (j=1)][a.sub.ij][x.sub.j] = [b.sub.i], for i = [m.sub.2]+1, [m.sub.2]+2, ......, m

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Minimize [Z.sup.k2] = [n.summation over (j=1)]([c.sup.k.sub.j] + [p.sup.k.sub.j])[x.sub.j] (4.3)

subject to same constraints of (4.2)

Minimize[Z.sup.k3] = [n.summation over (j=1)][c.sup.k.sub.j][x.sub.j] (4.4)

subject to [n.summation over (j=1)][a.sub.ij][x.sub.j] [greater than or equal to] [b.sub.i]-[b.sup.0.sub.i], for i = 1,2,3, ......, [m.sub.1]

[n.summation over (j=1)][a.sub.ij][x.sub.j] [less than or equal to] [b.sub.i] + [b.sup.0.sub.i], for i = [m.sub.1]+1, [m.sub.1]+2, ......, [m.sub.2]

[n.summation over (j=1)][a.sub.ij][x.sub.j] [greater than or equal to] [b.sub.i] - [b.sup.l.sub.i], for i = [m.sub.2]+1, [m.sub.2]+2, ......, m

[n.summation over (j=1)][a.sub.ij][x.sub.j] [less than or equal to] [b.sub.i] + [b.sup.r.sub.i], for i = [m.sub.2]+1, [m.sub.2]+2, ......, m

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Minimize [Z.sup.k4] = [n.summation over (j=1)][c.sup.k.sub.j] + [p.sup.k.sub.j])[x.sub.j] (4.5)

subject to same constraints of (4.4)

Minimize [Z.sup.k5]= [n.summation over (j=1)][c.sup.k.sub.j][x.sub.j] (4.6)

subject to [n.summation over (j=1)]([a.sub.ij] - [d.sup.0.sub.ij])[x.sub.j] [greater than or equal to] [b.sub.i], for i = 1,2,3, ......, [m.sub.1]

[n.summation over (j=1)]([a.sub.ij] - [d.sup.0.sub.ij])[x.sub.j] [less than or equal to] [b.sub.i], for i = [m.sub.1]+1, [m.sub.1]+2, ......, [m.sub.2]

[n.summation over (j=1)]([a.sub.ij] - [d.sup.l.sub.ij])[x.sub.j] [greater than or equal to] [b.sub.i], for i = [m.sub.2]+1, [m.sub.2]+2, ......, m

[n.summation over (j=1)]([a.sub.ij] + [d.sup.r.sub.ij])[x.sub.j] [less than or equal to] [b.sub.i], for i = [m.sub.2]+1, [m.sub.2]+2, ......, m

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Minimize [Z.sup.k6] = [n.summation over (j=1)][c.sup.k.sub.j] + [p.sup.k.sub.j])[x.sub.j] (4.7)

subject to same constraints of (4.6).

Minimize [Z.sup.k7] = [n.summation over (j=1)][c.sup.k.sub.j][x.sub.j] (4.8)

subject to [n.summation over (j=1)]([a.sub.ij] - [d.sup.0.sub.ij])[x.sub.j] [greater than or equal to] [b.sub.i] - [b.sup.0.sub.i], for i = 1,2,3, ......, [m.sub.1]

[n.summation over (j=1)]([a.sub.ij] + [d.sup.0.sub.ij])[x.sub.j] [less than or equal to] [b.sub.i] + [b.sup.0.sub.i], for i = [m.sub.1]+1, [m.sub.1]+2, ......, [m.sub.2]

[n.summation over (j=1)]([a.sub.ij] - [d.sup.l.sub.ij])[x.sub.j] [greater than or equal to] [b.sub.i] - [b.sup.l.sub.i], for i = [m.sub.2]+1, [m.sub.2]+2, ......, m

[n.summation over (j=1)]([a.sub.ij] + [d.sup.r.sub.ij])[x.sub.j] [less than or equal to] [b.sub.i] + [b.sup.r.sub.i], for i = [m.sub.2]+1, [m.sub.2]+2, ......, m

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Minimize [Z.sup.k8] = [n.summation over (j=1)]([c.sup.k.sub.j] + [p.sup.k.sub.j])[x.sub.j] (4.9)

subject to same constraints of (4.8).

4.1 Fuzzy Programming Technique for the Solution of MOLPP with Fuzzy Coefficients and Fuzzy Resources

The MOFLPP can be considered as a vector minimum problem. Let [L.sub.k] and [U.sub.k] be the lower and upper bound for the k-th objective, where

[L.sub.k] = aspired level of achivement for the k-th objective function, and [U.sub.k] = highest acceptable level of achivement for the k-th objective function.

When the aspiration levels for each objectives have been specified, we formed a fuzzy model. Our next step is to transform the fuzzy model into a crisp model The foregoing steps may be presented as follows:

Step-1. Solve the MOLPPs (4.2), (4.3), (4.4), (4.5), (4.6), (4.7), (4.8) and (4.9) for each kth objectives (k = 1, 2, 3, ...., K).

Step-2. From the results of step-1, determine the corresponding value for every objective function at each solution.

Step-3. Find upper and lower bounds (i.e [U.sub.k] and [L.sub.k]) for kth objective from the 8k objective values derived in step-2, as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] k= 1, 2, 3, ......, K 1 [less than or equal to] r [less than or equal to] K 1 [less than or equal to] s [less than or equal to] 8

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] k= 1, 2, 3, ......, K 1 [less than or equal to] r [less than or equal to] K 1 [less than or equal to] s [less than or equal to] 8

Step-4. The initial fuzzy model becomes (in terms of aspiration levels with each objectives)

Find [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.10) so as to satisfy

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for k = 1, 2, 3, ......, K.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] i = 1,2,3, ......, [m.sub.1]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] i = [m.sub.1]+1, [m.sub.1]+2, ......, [m.sub.2]

[n.summation over (j=1)][a.sub.ij][x.sub.j] [congruent to] [b.sub.i] i = [m.sub.2]+1, [m.sub.2]+2, ......, m

Here the membership functions for the fuzzy constraints of (4.10) are defined as:

(for kth constraints [[??].sub.k] (k = 1, 2, 3, ......, K)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]) for [U.sub.k] < [n.summation over (j=1)][c.sup.k.sub.j][x.sub.j]

= [[[U.sub.k] - [n.summation over (j=1)][c.sup.k.sub.j][x.sub.j]]/[[n.summation over (j=1)][p.sup.k.sub.j] [x.sub.j] + M]] for [n.summation over (j=1)][c.sup.k.sub.j][x.sub.j] [less than or equal to] [U.sub.k] [less than or equal to] [n.summation over (j=1)](c.sup.k.sub.j] + [p.sup.k.sub.j])[x.sub.j] + M

= 1 for [U.sub.k] [greater than or equal to] [n.summation over (j=1)]([c.sup.k.sub.j] + [p.sup.k.sub.j]) [x.sub.j] + M

where M = [U.sub.k] - [L.sub.k], k = 1, 2, 3, ......, K.

(for the ith constraints [[??].sub.i] (i = 1,2,3, ......, [m.sub.1]))

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for [b.sub.i] [less than or equal to] [n.summation over (j=1)]([a.sub.ij] - [d.sup.0.sub.ij])[x.sub.j] - [b.sup.0.sub.i]

= [n.summation over (j=1)][a.sub.ij][x.sub.j] - [b.sub.i]/[n.summation over (j=1)] [d.sup.0.sub.ij][x.sub.j] + [b.sup.0.sub.i] for [n.summation over (j=1)]([a.sub.ij] - [d.sup.0.sub.ij]) [x.sub.j] - [b.sup.0.sub.i] [less than or equal to] [n.summation over (j=1)][a.sub.ij][x.sub.j]

= 0, for [b.sub.i] > [n.summation over (j=1)][a.sub.ij][x.sub.j]

(for the ith constraints [[??].sub.i] (i = [m.sub.1]+1, [m.sub.1]+2, ......, [m.sub.2]))

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for [b.sub.i] < [n.summation over (j=1)][a.sub.ij][x.sub.j]

= [[[b.sub.i] - [n.summation over (j=1)][a.sub.ij][x.sub.j]]/[[[n.summation over (j=1)][d.sup.0.sub.ij][x.sub.j]]+[b.sup.0.sub.i]]],

for [n.summation over (j=1)][a.sub.ij][x.sub.j]] [less than or equal to] [b.sub.i] [less than or equal to] [n.summation over (j=1)] ([a.sub.ij] + [d.sup.0.sub.ij])[x.sub.j] + [b.sup.0.sub.i]

= 1, for [b.sub.i] [greater than or equal to] [n.summation over (j=1)]([a.sub.ij] + [d.sup.0.sub.ij])[x.sub.j] + [b.sup.0.sub.i]

(for the ith constraints [[??].sub.i] (i = [m.sub.2]+1, [m.sub.2]+2, ......, m))

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for [b.sub.i] < [n.summation over (j=1)] ([a.sub.ij] - [d.sup.l.sub.ij]) [x.sub.j] - [b.sup.r.sub.i]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for [n.summation over (j=1)] ([a.sub.ij] - [d.sup.l.sub.ij]) [x.sub.j] - [b.sup.r.sub.i] [less than or equal to] [b.sub.i] [less than or equal to] [n.summation over (j=1)] [a.sub.ij][x.sub.j]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for [n.summation over (j=1)] [a.sub.ij] [x.sub.j] [less than or equal to] [b.sub.i] [less than or equal to] [n.summation over (j=1)] ([a.sub.ij] + [d.sup.r.sub.ij]) [x.sub.j] + [b.sup.l.sub.i]

= 0, for [b.sub.i] > [n.summation over (j=1)] ([a.sub.ij] + [d.sup.0.sub.ij]) [x.sub.j] + [b.sup.l.sub.i]

Step-5. Using the max-min operator (as Zimmermann [15]) crisp LPP for (4.1) is formulated as follows:

Max [lambda] (4.11)

subject to, [n.summation over (j=1)] ([c.sup.k.sub.j] + [lambda][p.sup.k0.sub.j])[x.sub.j] + [lambda] ([Z.sup.k.sub.U] - [Z.sup.k.sub.L]) [less than or equal to] [Z.sup.k.sub.L] for k = 1, 2, 3, ..., K.

[n.summation over (j=1)] ([a.sub.ij] - [lambda][d.sup.0.sub.ij])[x.sub.j] - [lambda] [b.sup.0.sub.i] [greater than or equal to] [b.sub.i], for i = 1,2,3, ......, [m.sub.1]

[n.summation over (j=1)] ([a.sub.ij] + [lambda][d.sup.0.sub.ij])[x.sub.j] + [lambda][b.sup.0.sub.i] [less than or equal to] [b.sub.i], for i = [m.sub.1]+1, [m.sub.1]+2, ......, [m.sub.2]

[n.summation over (j=1)] ([a.sub.ij] - (1 - [lambda]) [d.sup.l.sub.ij]) [x.sub.j] + [lambda][b.sup.r.sub.i] [less than or equal to] [b.sub.i] + [b.sup.r.sub.i], for i = [m.sub.2]+1, [m.sub.2]+2, ......, m

[n.summation over (j=1)] ([a.sub.ij] + (1 - [lambda])[d.sup.r.sub.ij])[x.sub.j] - [lambda][b.sup.l.sub.i] [greater than or equal to] [b.sub.i] - [b.sup.l.sub.i], for i = [m.sub.2]+1, [m.sub.2]+2, ......, m

0 [less than or equal to] [lambda] [less than or equal to] 1, [x.sub.j] [greater than or equal to] 0 j=1,2,3, ......, n

Note: The constraints in problem (4.11) containing cross product terms [lambda][x.sub.j](j = 1, 2, 3, ..., n) which are not convex. Therefore the solution of this problem requires the special approach adopted for solving general non-convex application problems. It may be solved by fuzzy decisive set method [12].

4.2 The algorithm of the fuzzy decisive set method (Sakawa and Yano [12]):

This method is based on the idea that for a fixed value of [lambda], the problem (4.11) are linear programming problem. Obtains the optimal solution [[lambda].sup.*] to the problem (4.11) is equivalent to determining the maximum value of [lambda] so that the feasible set is nonempty. The algorithm to this method for the problem (4.11) is presented below.

Algorithm:

Step1- Set [lambda] = 1 and test whether a feasible set satisfying the constraints of the problem (31) exist or not, using phase one of the Simplex method. If a feasible set exist, set [lambda] = 1. Otherwise, set [[lambda].sup.L] = 0 and [[lambda].sup.R] = 1 and go to the next step.

Step2- For the value of [lambda] = [[[lambda].sup.L] + [[lambda].sup.R]]/2 up date the value of [[lambda].sup.L] and [[lambda].sup.R] using the bisection method as follows:

[[lambda].sup.L] = [lambda] if feasible set is non-empty for [lambda],

[[lambda].sup.R] = [lambda] if feasible set is empty for [lambda].

Consequently for each [lambda], test whether a feasible set of the problem (4.11) exists or not exists , using phase one of the simplex method and determine the maximum value of [[lambda].sup.*] satisfying the constraints of the problem (4.11).

Numerical Example 2:

Minimize [Z.sup.1] = [[??].sup.1.sub.1] [[??].sub.1] + [[??].sup.1.sub.2] [[??].sub.2] (2)

Minimize [Z.sup.2] = [[??].sup.2.sub.1] [[??].sub.1] + [[??].sup.2.sub.2] [[??].sub.2]

subject to [[??].sub.11] [[??].sub.1] + [[??].sub.12] [[??].sub.2] [greater than or equal to] [[??].sub.1] [[??].sub.21] [[??].sub.1] + [[??].sub.22] [[??}.sub.2] [greater than or equal to] [[??].sub.2] [x.sub.1], [x.sub.2] [greater than or equal to] 0

where [[??].sup.1.sub.1] = [??] = (5, 5, 6); [[??].sup.1.sub.2] = [??] = (3, 3, 4.5) and [[??].sup.2.sub.1] = [??] = (2, 2, 4); [[??].sup.2.sub.2] = [??] = (7, 7, 7.5) respectively for objective coefficients.

[[??].sub.11] = [??] = (1.5,2,2); [[??].sub.12] = [??] = (3,4,4) [[??].sub.12] = [??] = (.5,1,1); [[??].sub.22] = [??] = (1,1,1) respectively for technological coefficients.

and [[??].sub.1] = [??] = (18, 20, 20); [[??].sub.2] = [??] = (9, 10, 10) respectively for constraint goals.

To solve the problem (2), we first solve the following sixteen sub-problems:

Minimize [Z.sub.11] = 5[x.sub.1]+3[x.sub.2], -- (2.1) subject to

2[[??].sub.1]+4[[??].sub.2] [greater than or equal to] 20, [[??].sub.1]+[[??].sub.2] [greater than or equal to] 10, [[??].sub.1], [[??].sub.2] [greater than or equal to] 0;

Minimize [Z.sup.12] = 6[[??].sub.1]+4.5[[??].sub.2], -- (2.2) subject to same constraints of (2.1).

Minimize [Z.sub.13] = 5[[??].sub.1]+3[[??].sub.2], -- (2.3) subject to

2[[??].sub.1]+4[[??].sub.2] [greater than or equal to] 18, [[??].sub.1]+[[??].sub.2] [greater than or equal to] 9, [[??].sub.1], [[??].sub.2] [greater than or equal to] 0;

Minimize [Z.sub.14] = 6[[??].sub.1]+4.5[[??].sub.2] -- (2.4) subject to same constraints of (2.3).

Minimize [Z.sup.15] = 5[[??].sub.1]+3[[??].sub.2], -- (2.5) subject to

1.5[[??].sub.1]+3[[??].sub.2] [greater than or equal to] 20, 0.5[[??].sub.1]+1[[??].sub.2] [greater than or equal to] 10, [[??].sub.1], [[??].sub.2] [greater than or equal to] 0;

Minimize [Z.sub.16] = 6[[??].sub.1]+4.5[[??].sub.2], -- (2.6) subject to same constraints of (2.5).

Minimize [Z.sup.17] = 5[[??].sub.1]+3[[??].sub.2], -- (2.7) subject to

1.5[[??].sub.1]+3[[??].sub.2] [greater than or equal to] 18, 0.5[[??].sub.1]+1[[??].sub.2] [greater than or equal to] 9, [[??].sub.1], [[??].sub.2] [greater than or equal to] 0;

Minimize [Z.sup.18] = 6[[??].sub.1]+4.5[[??].sub.2], -- (2.8) subject to same constraints of (2.7).

Minimize [Z.sup.21] = 2[[??].sub.1] + 7[[??].sub.2], -- (2.9) subject to

2[[??].sub.1]+4[[??].sub.2] [greater than or equal to] = 20, [[??].sub.1]+[[??].sub.2] [greater than or equal to] 10, [[??].sub.1], [[??].sub.2] [greater than or equal to] 0 ;

Minimize [Z.sup.22] = 4[[??].sub.1]+7.5[[??].sub.2], -- (2.10) to same constraints of (2.9).

Minimize [Z.sup.23] = 2[[??].sub.1]+7[[??].sub.2], -- (2.11) subject to

2[[??].sub.1]+4[[??].sub.2] [greater than or equal to] 18, [[??].sub.1]+[[??].sub.2] [greater than or equal to] 9, [[??].sub.1], [[??].sub.2] [greater than or equal to] 0 ;

Minimize [Z.sup.24] = 4[[??].sub.1]+7.5[[??].sub.2] -- (2.12) subject to same constraints of (2.11).

Minimize [Z.sup.25] = 2[[??].sub.1]+7[[??].sub.2], -- (2.13) subject to

1.5[[??].sub.1]+3[[??].sub.2] [greater than or equal to] 20, 0.5[[??].sub.1]+1[[??].sub.2] [greater than or equal to] 10, [[??].sub.1], [[??].sub.2] [greater than or equal to] 0 ;

Minimize [Z.sup.26] = 4[[??].sub.1]+7.5[[??].sub.2], -- (2.14) subject to same constraints of (2.13).

and

Minimize [Z.sup.27] = 2[[??].sub.1]+7[[??].sub.2], -- (2.15) subject to

1.5[[??].sub.1]+3[[??].sub.2] [greater than or equal to] 18 0.5[[??].sub.1]+1[[??].sub.2] [greater than or equal to] 9, [[??].sub.1], [[??].sub.2] [greater than or equal to] 0;

Minimize [Z.sup.28] = 4[[??].sub.1]+7.5[[??].sub.2], -- (2.16) subject to same constraints of (2.15).

so the optimal solutions of the sub-problems ((2.1) - (2.16)) are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

So,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

According to step-4 formulating membership functions and following step-5, crisp LPP of (2) is formulated as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.17)

Using gradient based non- linear programming package the optimal solutions be [[??].sup.*.sub.1] = 10.88221, [[??].sup.*.sub.2] = 2.041447, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and aspiration level [[lambda].sup.*] = 0.4539063.

The problem (2.17) may also be solved by the fuzzy decisive set method. For [lambda] = 1 , the problem can be written as

6[[??].sub.1]+4.5[[??].sub.2] [less than or equal to] 27

4[[??].sub.1]+7.5[[??].sub.2] [less than or equal to] 18

1.5[[??].sub.1]+3[[??].sub.2] [greater than or equal to] 22

.5[[??].sub.1]+[[??.sub.2] [greater than or equal to] 11

[[??].sub.1], [[??].sub.2] [greater than or equal to] 0.

Since the feasible set is empty, by taking [[lambda].sup.L] =0 & [[lambda].sup.R] = 1, the new value of [lambda] = [[0 + 1]/2] = 1/2 is tired.

For [lambda] = 1/2 = .5, the problem (2.17) can be written as

5.5[[??].sub.1]+3.75[[??].sub.2] [less than or equal to] 63.5 3[[??].sub.1]+3.25[[??].sub.2] [less than or equal to] 44 1.75[[??].sub.1]+3.5[[??].sub.2] [greater than or equal to] 21 0.75[[??].sub.1]+[[??].sub.2] [greater than or equal to] 10.5 [[??].sub.1], [[??].sub.2] [greater than or equal to] 0.

Since the feasible set is empty, by taking [[lambda].sup.L] =1 & [[lambda].sup.R] = 1/2, the new value of [lambda] = [0 + [1/2]/2] = 1/4 is tired. And so on.

The following values of [lambda] are obtained in the next 25th iterations:

[lambda] = 0.25; [lambda] = 0.375; [lambda] = 0.4375; [lambda] = 0.46875; [lambda] = 0.453125; [lambda] = 0.4609375; [lambda] = 0.4570312; [lambda] = 0.4550781; [lambda] = 0.4541016; [lambda] = 0.4536133; [lambda] = 0.4538574; [lambda] = 0.4539795; [lambda] = 0.4539185; [lambda] = 0.4538879; [lambda] = 0.4539032; [lambda] = 0.4539109; [lambda] = 0.4539070; [lambda] = 0.4539051; [lambda] = 0.4539060; [lambda] = 0.4539065; [lambda] = 0.4539063; [lambda] = 0.4539064; [lambda] = 0.4539063.

Consequently, we obtain the optimal value [[lambda].sup.*] = 0.4539063 at the 25th iteration by using the fuzzy decisive set method and solutions of the problem (2) are

[[??].sup.*.sub.1] = 10.88221, [[??].sup.*.sub.2] = 2.041448, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] =60.53539, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = 36.05455 and aspiration Level [[lambda].sup.*] = 0.4539063.

5. Application in Transportation Models

The following model adopted from Bit, at al [3] is used to show that the above MOFLPP can be employed to solve the multi-objective transportation problems.

Consider m origins (or sources) [O.sub.i] (i = 1, 2, ......, m) and n destinations [D.sub.j] (j = 1, 2, ..., n). At each origin [O.sub.i], let [a.sub.i] be the amount of homogeneous product which we want to transport to n destinations [D.sub.j] to satify the demands for [b.sub.j] units of the product there. Let [e.sub.k] (k =1, 2, ...., K) be the units of product which can be carried by K different modes of transport called conveyance, such as trucks, air freight, freight train, ship, etc. A penalty [p.sup.p.sub.ijk] associated with transportation of a unit of the product from source i to destination j by means of k-th conveyance for the p-th criterion. The penalty could represent transportation cost, deterioration amount, quantity goods delivered, under used capacity, etc. A variable [x.sub.ijk] represents the unknown quantity to be transported from origin [O.sub.i] to destination [D.sub.j] by means of the k-th conveyance.

A general Multi-Objective solid transportation model with mixed constraints, written as follows:

Minimize Z =[ [Z.sup.1], [Z.sup.2], [Z.sup.3], ......, [Z.sup.P] (5.1)

Subject to:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [Z.sup.p] = [summation over (i[member of]I)] [summation over (j[member of]J)] [summation over (k[member of]K)] [c.sup.p.sub.ijk] [x.sub.ijk], p = 1, 2, 3, ......, P.

It is noted that the restriction on total delivery time is necessary for transportation of perishable goods, delivery of emergency supplies, etc. We are now adding an additional restriction to the above model that the total delivery time ([summation over (i)] [summation over (j)] [summation over (k)] [t.sub.ijk] [x.sub.ijk]) is not more than T units. Here [t.sub.ijk] represents delivery time of unit item of transportation from i-th zone to j-th zone by means of k-th conveyance for the p-th criterion. In the above model, penalties, supply and demand amount, etc. are assumed to be fixed in value. In general, transportation penalties, delivery time, demand and supply amount are somewhat uncertain (non-stochastic) imprecise and vague in nature. So in real life situation, to depict this nature, all the parameters in the above model may be taken as fuzzy numbers.

Then the above model (5.1) in fuzzy environment may be rewritten as

Minimize Z =[[Z.sup.1], [Z.sup.2], [Z.sup.3], ......, [Z.sup.P] (5.2)

Subject to:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It is a MOFLPP. It can be solved as before.

Numerical Example3:

5.1 Fuzzy Multi-Objective Solid Transportation Model with Restriction on Total delivery Time

Minimize [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3)

Subject to:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

[[??].sub.ijk] = 0, for i [member of] 1, 2, 3; j [member of] 1, 2, 3; k [member of] 1, 2, 3

where three penalties and delivery time are given as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

[Here [c.sup.1.sub.111] = (9,9,10), [c.sup.1.sub.112] = (12,12,12.5), [c.sup.1.sub.113] = (9,9,10) and similar representation for other elements.]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], respectively for constraint goals.

The optimal solution of the above fuzzy transportation problem is

[x.sup.*.sub.121] = 4.805142, [x.sup.*.sub.122] = 0.6301453, [x.sup.*.sub.132] = 1.872076, [x.sup.*.sub.211] = 5.556556, [x.sup.*.sub.212] = 2.136081, [x.sup.*.sub.222] = 0.08009255, [x.sup.*.sub.223] = 1.099346, [x.sup.*.sub.232] = 0.5889684, [Z.sup.1*] ([x.sup.*]) = 97.09676, [Z.sup.2*] ([x.sup.*]) = 58.50913, [Z.sup.3*] ([x.sup.*]) = 58.08856 with aspiration level [[lambda].sup.*]=0.3073629.

6. Conclusion

In this paper, we have proposed two types of FLPP. One is FLPP with fuzzy number resources and another is a FLPP with fuzzy number coefficients and resources. A procedure is developed for solving said FLPPs. It is also applied to a fuzzy solid transportation problem. This procedure may be very helpful for any fuzzy multi criteria decision making problem.

Received July 27, 2004, Accepted December 31, 2004

References

[1] A. K. Bit, M. P. Biswal and S. S. Alam, Fuzzy programming approach to multiobjective solid transportation problem, Fuzzy Sets and Systems 57 (1993) 183-194.

[2] A. K. Bit, M. P. Biswal and S. S. Alam, Fuzzy programming approach to multicriteria decision making transportation problem, Fuzzy sets and systems 50 (1992), 135-141.

[3] D. Chanas, Fuzzy programming in multiobjective linear programming- a parametric approach, Fuzzy Set and system 29 (1989) 303-313

[4] D. Klingman, and R. Russell, The transportation problems with mixed constraints. , Operational Research Quarterly 25 (1974) 447-455.

[5] G. M. Appa, The transportation problems with its variants, Operational Research Quarterly 24 (1973), 79-99.

[6] H. Tanaka, K. Asai, Fuzzy linear programming problems with fuzzy numbers. , Fuzzy Sets and Systems 13 (1984), 1-10.

[7] H. J. Zimmermann, Fuzzy programming and linear programming with several objective functions., Fuzzy sets and System 1 (1978), 45-55.

[8] Ishibuchi;Tanaka, Multiobjective programming in optimization of the interval objective function., European Journal of Operational Research 48 (1990), 219-225.

[9] M. Oheigeartaigh, A fuzzy transportation Algorithm, Fuzzy Sets and Systems 8 (1982), 235-243.

[10] M. Sakawa, and H. Yano, Interactive decision making for multi-objective linear fractional programming problems with fuzzy parameters. , Cybernetics Systems 16 (1985) 377-394.

[11] R. E. Bellman and L. A. Zadeh, Decision making in a fuzzy environment. Management Science 17 (1970), B141-B164.

[12] R. N. Gasimov and K. Yenilmez, Soving fuzzy linear programming with linear membership funtions., Turk J Math 26 (2002), 375-396.

[13] S. K. Das, A. Goswami, and S. S. Alam, Multiobjective transportation problem with interval; cost, source and destination parameters., European Journal of Operational Research 117 (1999), 100-112.

[14] S. Tong, Interval number and fuzzy number linear programming. , Fuzzy Sets and Systems 66 (1994), 301-306.

[15] Y. J. Lai and C. L. Hawng, Fuzzy Mathematical Programming, Lecture notes in Economics and Mathematical systems., Springer-Verlag, (1992).

Department of Mathematics, Bengal Engineering college (Deemed University), Howrah, West-Bengal, Pin 711103, India

* E-mail: roy_t_k@yahoo.co.in
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