fuzzy linear programs with trapezoidal fuzzy numbers

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Ann Oper Res (2006) 143: 305–315

DOI 10.1007/s10479-006-7390-1

Fuzzy linear programs with trapezoidal fuzzy numbers

K. Ganesan · P. Veeramani

C Springer Science + Business Media, Inc. 2006

Abstract The objective of this paper is to deal with a kind of fuzzy linear programming

problem involving symmetric trapezoidal fuzzy numbers. Some important and interesting

results are obtained which in turn lead to a solution of fuzzy linear programming problems

without converting them to crisp linear programming problems.

Keywords Fuzzy numbers . Ranking . Fuzzy linear programming

AMS subject classification: 90C05 . 90C70

Bellman and Zadeh (1970) proposed the concept of decision making in fuzzy environments.

After the pioneering work on fuzzy linear programming by Tanaka et al. (1974, 1984) and

Zimmermann (1974), several kinds of fuzzy linear programming problems have appeared in

the literature and different methods have been proposed to solve such problems. Numerous

methods for comparison of fuzzy numbers have been suggested in the literature. Maleki,

Tata and Mashinchi (1996, 2000) used the Rouben’s method of comparison of fuzzy numbers

and obtained an optimal solution. In this paper, we introduce a new type of fuzzy arithmeticfor symmetric trapezoidal fuzzy numbers and propose a method for solving fuzzy linear

programming problems without converting them to crisp linear programming problems.

This paper is organized as follows: In section 1, we give the definitions of fuzzy linear

programming, symmetric trapezoidal fuzzy numbers and some related results of fuzzy arith-

metic on symmetric trapezoidal fuzzy numbers. In section 2, we prove fuzzy analogues of

some important theorems of linear programming. A numerical example involving symmetric

trapezoidal fuzzy numbers is also given to illustrate the theory developed in this paper.

K. Ganesan ()Department of Mathematics, S. R. M Institute of Science and Technology, Deemed University,

Chennai–603 203, India

e-mail: gansan [email protected]

P. Veeramani

Department of Mathematics, Indian Institute of Technology, Madras, Chennai–600 036, India

e-mail: [email protected]

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306 Ann Oper Res (2006) 143: 305–315

In this paper we restrict ourself to symmetric trapezoidal fuzzy numbers due to the com-

plication involved in the multiplication of general fuzzy numbers. It is to be noted that if

a, b, c are intervals, then c(a + b) need not be even equivalent to (ca + cb), under the existing

interval arithmetic (and hence also for symmetric trapezoidal fuzzy numbers). But by using

the modified fuzzy arithmetic for symmetric trapezoidal fuzzy numbers we can establish thedistributive law (up to equivalence) and hence we are able to prove the main results of this

paper.

1. Preliminaries

The aim of this section is to present some notations, notions and results which are of useful

in our further consideration.

Definition 1.1. A fuzzy set a on R is said to be a symmetric trapezoidal fuzzy number if there exist real numbers a1, a2, a1 ≤ a2 and h > 0 such that

a( x) =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

x

h+

h − a1

h, for x ∈ [a1 − h, a1]

1, for x ∈ [a1, a2]

− x

h+

a2 + h

h, for x ∈ [a2, a2 + h]

0, otherwise

We denote it by a = [a1, a2, h, h]. When h = 0; a = [a1, a2]. We use F (S) to denote the set

of all symmetric trapezoidal fuzzy numbers.

Let a = [a1, a2, h, h] and b = [b1, b2, k , k ] be two symmetric trapezoidal fuzzy numbers.

Then the arithmetic operations on a and b are given by:

(i) Addition: a + b = [a1, a2, h, h] + [b1, b2, k , k ] = [a1 + b1, a2 + b2, h + k , h + k ].

(ii) Subtraction: a − b = [a1, a2, h, h] − [b1, b2, k , k ] = [a1 − b2, a2 − b1, h + k , h + k ].

(iii) Multiplication: ab = [a1, a2, h, h][b1, b2, k , k ]

=

a1 + a2

2

b1 + b2

2

− w,

a1 + a2

2

b1 + b2

2

+ w, | a2k + b2h |, | a2k + b2h |

,

where w =

β − α

2

, α = min(a1b1, a1b2, a2b1, a2b2) and β = max(a1b1, a1b2, a2b1,

a2b2).

From (iii), it is clear that

λa =

[λa1, λa2, λh, λh], for λ ≥ 0

[λa2, λa1, −λh, −λh], for λ < 0.

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Ann Oper Res (2006) 143: 305–315 307

Remark 1.1. Depending upon the need, one can also use a smaller w in the definition of

multiplication involving symmetric trapezoidal fuzzy numbers.

Definition 1.2. Let a = [a1, a2, h, h] and b = [b1, b2, k , k ] be two symmetric trapezoidal

fuzzy numbers.Define the relation as

a b (or b a) if and only if either(a1 − h) + (a2 + h)

2<

(b1 − k ) + (b2 + k )

2

that is

a1 + a2

2<

b1 + b2

2

(in this case, we also write a ≺ b)

or

a1 + a2

2=

b1 + b2

2, b1 < a1 and a2 < b2.

or

a1 + a2

2=

b1 + b2

2, b1 = a1, a2 = b2 and h ≤ k .

(in the above two cases, we also write a ≈ b and say that a and b are equivalent)

Remark 1.2. Two symmetric trapezoidal fuzzy numbers [a1, a2, h, h], [b1, b2, k , k ] are

equivalent if and only if

a1 + a2

2=

b1 + b2

2.

In this case, we just write [a1, a2, h, h] ≈ [b1, b2, k , k ] and it is to be noted that a1 need not

be equal to b1 or a2 need not be equal to b2, but [a1, a2, h, h] − [b1, b2, k , k ]≈ [−α,α, h + k , h + k ], where α = (b2 − a1) ≥ 0.

The proofs of the following results are straightforward.

Proposition 1.1. For any symmetric trapezoidal fuzzy numbers a, b and c, we have (i)

c(a + b) ≈ (ca + cb) and (ii) c(a − b) ≈ (ca − cb).

Theorem 1.2. (i) The relation is a partial order relation on the set of symmetric trape-

zoidal fuzzy numbers.

(ii) The relation is a linear order relation on the set of symmetric trapezoidal fuzzynumbers.

(iii) For any two symmetric trapezoidal fuzzy numbers a and b; if a b, then

a (1 − λ)a + λb b, for all λ, 0 ≤ λ ≤ 1.

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308 Ann Oper Res (2006) 143: 305–315

Definition 1.3. For any symmetric trapezoidal fuzzy number ˜ x , let us define ˜ x 0 if there

exist a ≥ 0 and h ≥ 0 such that ˜ x [−a, a, h, h]. We also denote [−a, a, h, h] by 0. Note

that 0 is equivalent to [0, 0, 0, 0] = 0. It is easy to see that if ˜ x ˜ y, then ( ˜ x − ˜ y) 0.

Remark 1.3. If ˜ x ≈ 0, then ˜ x is said to be a zero symmetric trapezoidal fuzzy number. Itis to be noted that if ˜ x = 0, then ˜ x ≈ 0, but the converse need not be true. If ˜ x ≈ 0 (that

is ˜ x is not equivalent to 0), then ˜ x is said to be a non-zero symmetric trapezoidal fuzzy

number. It is to be noted that if ˜ x ≈ 0, then ˜ x = 0, but the converse need not be true. If ˜ x 0

and ˜ x ≈ 0, then ˜ x is said to be a positive symmetric trapezoidal fuzzy number and is denoted

by ˜ x 0.

Definition 1.4. Let F (S) be the set of all symmetric trapezoidal fuzzy numbers.

The model

max ˜ z ≈n

j=1

c j ˜ x j

subject to

n j=1

aij

˜ x j bi , i = 1, 2, 3, · · · , m0,

n j =1

aij

˜ x j bi , i = m0 + 1, m0 + 2, m0 + 3, · · · , m (1)

and ˜ x j 0 for all j = 1, 2, 3, · · · , n,

where aij

∈ R, c j , ˜ x j , bi ∈ F (S), i = 1, 2, 3, · · · , m, j = 1, 2, 3, · · · , n

is called a fuzzy linear programming problem.

Definition 1.5. Any x = ( ˜ x1, ˜ x2, ˜ x3, · · · , ˜ xn ) ∈ F n (S), where each ˜ x i ∈ F (S), which satisfies

the constraints and non-negativity restrictions of (1) is said to be a fuzzy feasible solution to (1).

Definition 1.6. Let Q be the set of all fuzzy feasible solutions of (1). A fuzzy feasible

solution x0 ∈ Q is said to be a fuzzy optimum solution to (1) if cx0 cx for all x ∈ Q,

where c = (c1, c2, · · · , cn ) and cx = c1 ˜ x1 + c2 ˜ x2 + · · · + cn ˜ xn .

Definition 1.7. Let x = ( ˜ x1, ˜ x2, ˜ x3, · · · , ˜ xn ). Suppose x solves Ax ≈ b. If all

˜ x j ≈ [−α j , α j , h j , h j ] for some α j ≥ 0and h j ≥ 0, then x issaidtobea fuzzy basic solution.

If ˜ x j ≈ [−α j , α j , h j , h j ] for α j ≥ 0 and h j ≥ 0, then x has some non-zero components, say

˜ x1, ˜ x2, ˜ x3, · · · , ˜ xk , 1 ≤ k ≤ n. Then Ax ≈ b can be written as:

a1 ˜ x1 + a2 ˜ x2 + a3 ˜ x 3 + · · · + ak ˜ xk + ak +1[−βk +1, βk +1, hk +1, hk +1]

+ ak +2[−βk +2, βk +2, hk +2, hk +2] + · · · + an [−βn , βn , hn , hn ] ≈ b.

If the columns a1, a2, a3, · · · , ak corresponding to these non-zero components ˜ x1, ˜ x2, · · · , ˜ xk

are linearly independent, then x is said to be a fuzzy basic solution.

Remark 1.4. Given a system of m simultaneous fuzzy linear equations involving symmetric

trapezoidal fuzzy numbers in n unknowns (m ≤ n) Ax ≈ b ; b ∈ F m (S), where A i s a (m × n)

real matrix and rank of A is m.Let B beany(m × m) matrix formed by m linearly independent

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Ann Oper Res (2006) 143: 305–315 309

columns of A. Let x B = B−1b = ( ˜ x1 ˜ x2 . . . ˜ xk )T . Then x = ( ˜ x1, ˜ x2, ˜ x3, · · · , ˜ xk , 0, 0, · · · , 0)

is a fuzzy basic solution. In this case, we also say that x B is a fuzzy basic solution.

2. Main results

Now we are in a position to prove fuzzy analogues of some important theorems of linear

programming. Any fuzzy linear programming problem can be converted to its standard

form as:

max ˜ z ≈ cx subject to Ax ≈ b and x 0, (2)

where A is an (m × n) real matrix, b, c, x are (m × 1), (1 × n), (n × 1) fuzzy matrices

consisting of symmetric trapezoidal fuzzy numbers.

2.1. Improving a fuzzy basic feasible solution

Let B = (b1, b2, . . . , bm ) form a basis for the columns of A. Let x B = B−1b be a fuzzy

basic feasible solution and the fuzzy value of the objective function ˜ z is given by ˜ z0 ≈ c B x B ,

where c B = (c B1, c B2, c B3, . . . , c Bm ) be the cost vector corresponding to x B . Assume that

a j =m

i=1 yij

bi = y j B and the symmetric trapezoidal fuzzy number ˜ z j =m

i =1 c Bi yij=

c B y j are known for every column vector a j in A, which is not in B. Now we shall examine

the possibility of finding another fuzzy basic feasible solution with an improved fuzzy value

of ˜ z by replacing one of the columns of B by a j .

Theorem 2.1. Let x B = B−1b is a fuzzy basic feasible solution of (2). If for any column

a j in A which is not in B, the condition (˜ z j − c j ) ≺ 0 hold and yij

> 0 for some

i , i ∈ {1, 2, 3, · · · , m} then it is possible to obtain a new fuzzy basic feasible solution by

replacing one of the columns in B by a j .

Proof: Suppose that x B = ( ˜ x B1, ˜ x B2, ˜ x B3, · · · , ˜ x Bm ) be a fuzzy basic feasible solution with

k positive components such that

Bx B ≈ b or x B = B−1b.

where ˜ x Bi = [αi , βi , hi , hi ], αi ≤ βi , hi ≥ 0, for i = 1, 2, 3, . . . , m

andαi + βi

2> 0, for i = 1, 2, 3, . . . , k

αi

+ βi

2= 0, for i = k + 1, k + 2, . . . , m.

That is

˜ x Bi 0, for i = 1, 2, 3, . . . , k

˜ x Bi = [−βi , βi , hi , hi ], for i = k + 1, k + 2, . . . , m.

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310 Ann Oper Res (2006) 143: 305–315

Now equation Bx B ≈ b becomes

k

i =1

˜ x Bi bi + [−βk +1, βk +1, hk +1, hk +1]bk +1 + [−βk +2, βk +2, hk +2, hk +2]bk +2

+ · · · + [−βm , βm , hm , hm ]bm ≈ b.

That is

k i =1

˜ x Bi bi +

mi =k +1

[−βi , βi , hi , hi ]bi ≈ b. (3)

Then for any column a j of A which is not in B, we write

a j =

mi=1

yij

bi = y1 j b1 + y2 j b2 + · · · + yr j br + · · · + ym j bm = y j B.

We know that if the basis vector br for which yr j = 0 is replaced by a j of A, then the new

set of vectors (b1, b2, · · · , br −1, a j , br +1, · · · , bm ) still form a basis.

Now for yr j = 0 and r ≤ k , we can write

br =a j

yrj−

mi =1i=r

yij

yrjbi =

a j

yrj−

k i=1i=r

yij

yrjbi −

mi =k +1

yij

yrjbi .

Equation (3) becomes

k i=1i =r

˜ x Bi bi + ˜ x Br br +

mi=k +1

[−βi , βi , hi , hi ]bi ≈ b.

⇒

k i=1i=r

˜ x Bi bi +˜ x Br

yrja j −

˜ x Br

yrj

k i=1i=r

yij bi −˜ x Br

yrj

mi=k +1

yij bi +

mi =k +1

[−βi , βi , hi , hi ]bi ≈ b.

⇒

k i=1i=r

˜ x Bi −

˜ x Br

yrj

yij

bi +

˜ x Br

yrj

a j +

mi=k +1

[−βi , βi , hi , hi ] −

˜ x Br

yrj

yij

bi ≈ b.

Since ˜ x Bi = [−βi , βi , hi , hi ], for i = k + 1, k + 2, · · · , m, we have

k i=1i =r

˜ x Bi − ˜ x Br yrj

yij bi + ˜ x Br yrj

a j +

mi=k +1

˜ x Bi − ˜ x Br yrj

yij bi ≈ b.

⇒

mi=1i=r

˜ x Bi −

˜ x Br

yrj

yij

bi +

˜ x Br

yrj

a j ≈ b ⇒

mi=1i=r

x Bi bi + x Br a j ≈ b,

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Ann Oper Res (2006) 143: 305–315 311

where

˜ x Bi =

˜ x Bi −

˜ x Br

yrj

yij

, i = r and

˜ x Br =

˜ x Br

yrj

(4)

which gives a new fuzzy basic solution to Ax ≈ b.

We shall show that this new fuzzy basic solution is also feasible. This requires that˜ x Bi −

˜ x Br

yrj

yij

0, i = r (5)

and

˜ x Br

yrj

0. (6)

Select yrj > 0 such that˜ x Br

yrj

≈ mini

˜ x Bi

yij

: yij

> 0

. Then

˜ x Br

yrj

˜ x Bi

yij

⇒

αr

yrj

,βr

yrj

,hr

yrj

,hr

yrj

αi

yij

,βi

yij

,hi

yij

,hi

yij

⇒

αi

yij

−βr

yrj

,βi

yij

−αr

yrj

,hr

yrj

+hi

yij

,

hr

yrj +

hi

yij ˜0.

⇒

⎧⎨⎩

αi

yij

−βr

yrj

+

βi

yij

− αr

yrj

2

⎫⎬⎭ ≥ 0 ⇒

αi + βi

yij

−

αr + βr

yrj

≥ 0

⇒

˜ x Bi

yij

−˜ x Br

yrj

0.

Hence the new fuzzy basic solution is a fuzzy basic feasible solution.

After the replacement of basis vectors, the new basis matrix is ˆ B = (b1, b2, . . . , bm ),

where bi = bi for i = r and br = a j . The new fuzzy basic feasible solution is x B , where x Bi =

˜ x Bi − ˜ x Br

yrj y

ij

, i = r and x Br = ˜ x Br

yrjare the basic variables.

Theorem 2.2. If x B = B−1b is a fuzzy basic feasible solution of (2) with ˜ z0 ≈ c B x B as

the fuzzy value of the objective function and if

x B is another fuzzy basic feasible solution

with

˜ z ≈

c B

x B obtained by admitting a non-basic column vector a j in the basis for which

(˜ z j

− c j

) ≺ 0 and yij

> 0 for some i, i ∈ {1, 2, 3, . . . , m} , then˜ z ˜ z0

.

Proof: Let x B be a fuzzy basic feasible solution and ˜ z0 ≈ c B x B . Let a j be the column

vector introduced in the basis for which ( ˜ z j − c j ) ≺ 0. Let br be the column vector removed

from the basis and x B be a new fuzzy basic feasible solution, then x Bi = ( ˜ x Bi −˜ x Br yrj

yij

),

i = r and x Br =˜ x Br yrj

.

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312 Ann Oper Res (2006) 143: 305–315

Since c Bi = c Bi , i = r andc Br = c j , the new fuzzy value of the objective function is

˜ z ≈c Bx B ≈

m

i=1 c Bi˜ x Bi ≈

m

i=1i=r c Bi˜ x Bi +c Br ˜ x Br

≈

mi=1i=r

c Bi

˜ x Bi −

˜ x Br

yrj

yij

+ c j

˜ x Br

yrj

≈

mi=1i=r

c Bi

˜ x Bi −

˜ x Br

yrj

yij

+ c Br

˜ x Br −

˜ x Br

yrj

yrj

+ c j

˜ x Br

yrj

(7)

≈

mi=1

c Bi ˜ x Bi − ˜ x Br

yrj

yij + c j

˜ x Br

yrj

≈

mi=1

c Bi ˜ x Bi −˜ x Br

yrj

mi=1

c Bi yij+ c j

˜ x Br

yrj

, by proposition (1.1)

≈ ˜ z0 −˜ x Br

yrj

˜ z j + c j

˜ x Br

yrj

≈ ˜ z0 −˜ x Br

yrj

(˜ z j − c j )

Since yrj > 0, (˜ z j − c j ) ≺ 0 and˜ x Br yrj

0, let˜ x Br yrj

=

αr

yrj,

βr

yrj, hr

yrj, hr

yrj

0, αr ≤ βr

and (˜ z j − c j ) = [α j , β j , h j , h j ] ≺ 0, α j ≤ β j .

Now˜ x Br

yrj

(˜ z j − c j ) ≈

αr

yrj

,βr

yrj

,hr

yrj

,hr

yrj

[α j , β j , h j , h j ]

≈ αr + βr

2 yrj

α j + β j

2 − w, αr + βr

2 yrj

α j + β j

2 + w,βr h j + β j hr

yrj

,

βr h j + β j hr

yrj

Also

⎧⎪⎪⎨⎪⎪⎩

αr + βr

2 yrj

α j + β j

2

− w +

αr + βr

2 yrj

α j + β j

2

+ w

2

⎫⎪⎪⎬⎪⎪⎭

=

αr + βr

2 yrj

α j + β j

2

≤ 0, since

αr + βr

2 yrj

≥ 0 and

α j + β j

2

< 0.

So equation (7) becomes z ˜ z0. Hence the new fuzzy basic feasible solution gives the

improved fuzzy value of the objective function.

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Ann Oper Res (2006) 143: 305–315 313

2.2. Unbounded solution

We have seen that for a column vector a j of A which is not in B, for which (˜ z j − c j ) ≺ 0 and

yij

> 0, for some i , is alone considered for inserting into the basis. Let us now discuss the sit-

uation when there exists an a j such that (˜ z j − c j ) ≺ 0 and yij ≤ 0, for all i = 1, 2, 3, · · · , m.If ˜ x = [ x1, x2, h, h] 0 and λ > 0, then λ ˜ x = [λ x1, λ x2, λh, λh] 0. Now λ can be

made sufficiently large so that λ ˜ x ˜ y for any symmetric trapezoidal fuzzy number ˜ y. If

λ > 0, (˜ z j − c j ) ≺ 0, then λ(˜ z j − c j ) ≺ 0. Now the proof of the following theorem follows

easily.

Theorem 2.3. Let x B = B−1b be a fuzzy basic feasible solution of (2). If there exist an a j of

A which is not in B such that (˜ z j − c j ) ≺ 0 and yij

≤ 0 , for all i, i ∈ {1, 2, 3, · · · , m} then

the fuzzy linear programming problem (2) has an unbounded solution.

2.3. Conditions of optimality

As in the classical linear programming problems, we can prove that the process of inserting

and removing vectors from the basis matrix will lead to any one of the following situations:

(i). there exist j such that (˜ z j − c j ) ≺ 0, yij

≤ 0, i = 1, 2, 3, · · · , m or

(ii). for all j , (˜ z j − c j ) 0.

In the first case, we get an unbounded solution and if the second case occurs , it is easy to

show that the fuzzy linear programming problem has a fuzzy optimal solution.

Theorem 2.4. If x B = B−1b is a fuzzy basic feasible solution of (2) and if (˜ z j − c j ) 0 for

every column a j of A, then x B is a fuzzy optimal solution to (2).

2.4. A numerical example

A company produces three products P1, P2 and P3. These products are processed on three

different machines M 1, M 2 and M 3. The time required to manufacture one unit of each

product and the daily capacity of the machines are given below:

Time per unit (minutes) Machine capacity

Machines P1 P2 P3 (min/day)

M 1 12 13 12 490

M 2 14 − 13 470

M 3 12 15 − 480

Note that the time availability can vary from day to day due to break down of machines,

overtime work etc. Finally the profit for each product can also vary due to variations in price.

At the same time the company wants to keep the profit somewhat close to Rs .14 for P1,

Rs .13 for P2 and Rs .16 for P3. The company wants to determine the range of each product

to be produced per day to maximize its profit. It is assumed that all the amounts produced

are consumed in the market.

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314 Ann Oper Res (2006) 143: 305–315

Since the profit from each product and the time availability on each machine are uncertain,

the number of units to be produced on each product will also be uncertain. So we will model

the problem as a fuzzy linear programming problem. We use symmetric trapezoidal fuzzy

numbers for each uncertain value.

Profit for P1 which is close to 14 is modelled as [13, 15, 2, 2]. Similarly the other parame-ters are also modelled as symmetric trapezoidal fuzzy numbers taking into account the nature

of the problem and other requirements. So we formulate the given fuzzy linear programming

problem as

max ˜ z ≈ [13, 15, 2, 2] ˜ x1 + [12, 14, 3, 3] ˜ x2 + [15, 17, 2, 2] ˜ x3

subject to 12 ˜ x1 + 13 ˜ x2 + 12 ˜ x3 [475, 505, 6, 6], 14 ˜ x1 + 13 ˜ x3 [460, 480, 8, 8],

12 ˜ x1 + 15 ˜ x2 [465, 495, 5, 5] and ˜ x1 0, ˜ x2 0, ˜ x3 0.

Now the standard form of the fuzzy linear programming problem becomes

max ˜ z ≈ [13, 15, 2, 2] ˜ x1 + [12, 14, 3, 3] ˜ x2 + [15, 17, 2, 2] ˜ x3

subject to 12 ˜ x1 + 13 ˜ x2 + 12 ˜ x3 + s1 ≈ [475, 505, 6, 6], 14 ˜ x1 + 13 ˜ x3 + s2

≈ [460, 480, 8, 8],

12 ˜ x1 + 15 ˜ x2 + s3 ≈ [465, 495, 5, 5] and ˜ x1 0, ˜ x2 0, ˜ x3 0, s1 0,

s2 0, s3 0, where s1, s2 and s3 are the slack fuzzy variables. That is

max ˜ z ≈ cx subject to Ax ≈ b and x 0, where

A =

⎛⎜⎜⎝a1 a2 a3 a4 a5 a6

12 13 12 1 0 0

14 0 13 0 1 0

12 15 0 0 0 1

⎞⎟⎟⎠ , b =

⎛⎝ [475, 505, 6, 6]

[460, 480, 8, 8]

[465, 495, 5, 5]

⎞⎠, x =

˜ x1 ˜ x2 ˜ x3 s1 s2 s3

and c = ([13, 15, 2, 2], [12, 14, 3, 3], [15, 17, 2, 2], 0, 0, 0).

Initial Iteration: The initial fuzzy basic feasible solution is given by x B = B−1b, where

B =

⎛⎝1 0 0

0 1 0

0 0 1

⎞⎠ , x =

⎛⎝ s1

s2

s3

⎞⎠ , b =

⎛⎝ [475, 505, 6, 6]

[460, 480, 8, 8]

[465, 495, 5, 5]

⎞⎠ and ˜ x1 = [0, 0, 0, 0],

˜ x2 = [0, 0, 0, 0], ˜ x3 = [0, 0, 0, 0], s1 = [475, 505, 6, 6] , s2 = [460, 480, 8, 8] ,

s3 = [465, 495, 5, 5] and the fuzzy value of the objective function is ˜ z ≈ [0, 0, 0, 0]. Now

(˜ z3 − c3) ≈ [−17, −15, 2, 2] ≺ 0.

First Iteration: By theorems (2.1) and (2.2), we get a new fuzzy basic feasible solution

˜ x1 = [0, 0, 0, 0], ˜ x2 = [0, 0, 0, 0], ˜ x3 = 46013

, 48013

, 813

, 813 , s1 =

41513

, 104513

, 17413

, 17413 ,

s2 = [0, 0, 0, 0] , s3 = [465, 495, 5, 5] with the improved fuzzy value of the objective func-tion, ˜ z ≈

6890

13, 8150

13, 1096

13, 1096

13

. Here also there are some (˜ z j − c j ) ≺ 0.

Second Iteration: Proceeding in a similar way,we get a new fuzzy basic feasible solution

˜ x1 = [0, 0, 0, 0], ˜ x2 =

415169

, 1045169

, 174169

, 174169

, ˜ x3 =

46013

, 48013

, 813

, 813

,

s1 = [0, 0, 0, 0] , s2 = [0, 0, 0, 0] , s3 =

62910169

, 77430169

, 3455169

, 3455169

with the improved fuzzy

value of the objective function, ˜ z ≈

94235169

, 120265169

, 19819169

, 19819169

.

Springer



Ann Oper Res (2006) 143: 305–315 315

Here (˜ z j − c j ) 0, for all j . Hence by theorem (2.4), the current fuzzy basic feasible

solution is a fuzzy optimal solution.

Acknowledgement The authors would like to thank the referees for their critical comments and valuable

suggestions which helped the authors to improve the presentation of this paper. Also the authors would like tothank Professor Dan Butnariu, Department of Mathematics, University of Haifa, Israel, for the critical reading

of the preliminary version of this paper and for many valuable suggestions for the improvement of this paper.

The first author would like to thank his supervisor Professor S.Elumalai for his valuable suggestions.

References

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141–164.

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intelligent and Cognitive Systems FSS Sponsored by IEE and ISRF, Tehran, Iran 145–148.Maleki, H.R., M. Tata and M. Mashinchi. (2000). Linear Programming with Fuzzy Variables. Fuzzy Sets and

Systems 109, 21–33.

Tanaka, H., T. Okuda, and K. Asai. (1974). On Fuzzy Mathematical Programming. Journal of Cybernetics 13,

37–46.

Tanaka, H. and K. Asai. (1984). Fuzzy Linear Programming Problems with Fuzzy numbers. Fuzzy Sets and

Systems 13, 1–10.

Zimmermann, H.J. (1974). Optimization in fuzzy environment. presented at XXI Int. TIMES and 46th ORSA

Conference, San Juan, Puerto Rioco.

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