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Under non-random uncertainty, a new idea of finding a possibly optimal solution for linear programming problem is examined in this paper. It is an application of the intuitionistic fuzzy set concept within scope of the existing fuzzy optimization. Here, we solve a linear programming problem (LPP) in an intuitionistic fuzzy environment and compare the result with the solution obtained from other existing techniques. In the process, the result of associated fuzzy LPP is also considered for a better understanding.

Linear programming is part of a very important area of applied mathematics called “optimization techniques”. They may deal with hundreds of variables simultaneously, but they fail to handle imprecise data. Fuzzy linear programming was introduced to capture this imprecision in linear programming problem (LPP). Later, various modification methods have appeared from different interpretations. Intuitionistic fuzzy set (IFS) developed by Atanassov [

The concept of maximizing decision under uncertainty was proposed by Bellman and Zadeh [

Recent years have witnessed growing interest in the study of decision-making problems with intuitionistic fuzzy sets/numbers [

Angelov [

In this paper, our aim is to propose a method to solve IFLPP when both the co-efficient matrix of the constraints and the cost co-efficients are represented as triangular intuitionistic fuzzy numbers and compare it with the case when both of them are triangular fuzzy numbers. Each problem is first converted into an equivalent crisp linear programming problem with the help of (α-β)-cut [

This work seeks to study extensively the existing fuzzy [

The paper is organized into seven sections. After a brief introductory section we present some basic concepts necessary for the development of a mechanism for solving intuitionistic fuzzy linear programming problems in Section 2. In Section 3, we discuss the mathematical formulation of our proposed technique to solve IFLPP when both the coefficient matrix of the constraints and cost coefficients are represented by triangular intuitionistic fuzzy numbers. In Section 4, we develop an algorithm and illustrate the same with some numerical examples. In Section 5, a comparative study between triangular intuitionistic fuzzy and triangular fuzzy environment is presented. Proposed result is also compared with other fuzzy [

Definition 1 [

A ˜ = { 〈 x j , μ A ˜ ( x j ) , ν A ˜ ( x j ) 〉 : x j ∈ U }

where the functions μ A ˜ : U → [ 0,1 ] and ν A ˜ : U → [ 0,1 ] respectively define the degree of membership and the degree of non-membership of an element x j ∈ U , such that they satisfy the following condition:

0 ≤ μ A ˜ ( x j ) + ν A ˜ ( x j ) ≤ 1, ∀ x j ∈ U ;

known as intuitionistic condition. The degree of acceptance μ A ˜ ( x ) and of non-acceptance ν A ˜ ( x ) can be arbitrary.

Definition 2 [

0 ≤ π A ˜ ( x ) ≤ 1 ; ∀ x j ∈ U .

When π A ˜ ( x ) = 0 , ∀ x ∈ U , i.e., μ A ˜ ( x ) + ν A ˜ ( x ) = 1 , A ˜ becomes a fuzzy set. Therefore, a fuzzy set is a special intuitionistic fuzzy set.

Definition 3 [

Definition 4 [

Definition 5 [

Definition 6 [

A ˜ α , β = { x j ∈ U : μ A ˜ ( x j ) ≥ α , ν A ˜ ( x j ) ≤ β } .

Thus, the ( α , β ) -cut of an intuitionistic fuzzy set to be denoted by A ˜ ( α , β ) , is defined as the crisp set of elements x which belong to A ˜ at least to the degree α and which does not belong to A ˜ at most to the degree β .

Definition 7 [

1) An intuitionistic fuzzy subset of the real line ℜ ;

2) Normal, i.e., ∃ x 0 ∈ ℜ such that μ A ˜ j ( x 0 ) = 1 (so ν A ˜ j ( x 0 ) = 0 );

3) Convex for the membership function, i.e.,

μ A ˜ j ( λ x 1 + ( 1 − λ ) x 2 ) ≥ min { μ A ˜ j ( x 1 ) , μ A ˜ j ( x 2 ) } ; ∀ x 1 , x 2 ∈ ℜ , λ ∈ [ 0 , 1 ] ;

4) Concave for the non-membership function, i.e.,

ν A ˜ j ( λ x 1 + ( 1 − λ ) x 2 ) ≤ max { ν A ˜ j ( x 1 ) , ν A ˜ j ( x 2 ) } ; ∀ x 1 , x 2 ∈ ℜ , λ ∈ [ 0 , 1 ] .

Definition 8 [

μ A ˜ t ( x ) = { x − a + l l w a ; a − l ≤ x < a a + r − x r w a ; a ≤ x ≤ a + r 0 ; otherwise , (1)

and

ν A ˜ t ( x ) = { ( a − x ) + u a ( x − a + l ) l ; a − l ≤ x < a ( x − a ) + u a ( a + r − x ) r ; a ≤ x ≤ a + r 1 ; otherwise ; (2)

where l, r are called spreads and a is called mean value. w a and u a represent the maximum degree of membership and minimum degree of non-membership respectively such that they satisfy the condition

0 ≤ w a ≤ 1, 0 ≤ u a ≤ 1 and 0 ≤ w a + u a ≤ 1.

Definition 9 [

Let S be the set of all intuitionistic fuzzy feasible solutions. Any vector x 0 ∈ S is said to be an intuitionistic fuzzy optimum solution if C x 0 ≥ C x ∀ x ∈ S where C = ( c 1 , c 2 , ⋯ , c n ) and C x = c 1 x 1 + c 2 x 2 + ⋯ + c n x n .

Definition 10 [

AI ( A ˜ α , β I ≤ B ˜ α , β I ) = m 2 − m 1 2 ( w 1 + w 2 ) .

Intuitionistic fuzzy optimization (IFO), a method of optimization under uncertainty, is put forward on the basis of intuitionistic fuzzy sets due to Atanassov [

There is no additional assumption about the nature of cost of decision variables and constraints. According to different considerations, distinct IFLPP could be obtained. We consider the case in which cost of decision variables and co-efficient matrix of constraints are represented as triangular intuitionistic fuzzy numbers and it is checked with a numerical example.

max Z ˜ = c ˜ I x = ∑ k = 1 n c ˜ k I x k

subject to

∑ k = 1 n A ˜ j k I x k ≤ B ˜ j I ; 1 ≤ j ≤ m , x k ≥ 0 ; 1 ≤ k ≤ n where , x = ( x 1 , x 2 , ⋯ , x n ) ′ .

Which is equivalent to,

max Z ˜ = ∑ k = 1 n ( c a , c l , c r ) k I x k

subject to

∑ k = 1 n ( a a , a l , a r ) j k I x k ≤ ( b a , b l , b r ) j I ; j = 1 , 2 , ⋯ , m .

Now, to solve the above IFLPP, first we find ( α , β ) -cut [

A ˜ α , β I = { A β I ; if α < ( 1 − β ) w a 1 − u a A α I ; if α > ( 1 − β ) w a 1 − u a A β I or A α I ; if α = ( 1 − β ) w a 1 − u a (3)

where 0 ≤ α ≤ w a , u a ≤ β ≤ 1 such that 0 ≤ α + β < 1 and 0 ≤ w a + u a < 1 .

Case 1: When

α < ( 1 − β ) w a 1 − u a , A ˜ α , β I = A β I ; (4)

Now, according to the definition of TIFN, A ˜ β is a closed interval [

A L ( β ) = ( a − l a ) + l a ( 1 − β ) 1 − u a , and A R ( β ) = ( a + r a ) − r a ( 1 − β ) 1 − u a .

Then, the above IFLPP reduces to the following,

max Z ˜ = ∑ k = 1 n [ ( c a − c l ) + c l ( 1 − β ) 1 − u a , ( c a + c r ) − c r ( 1 − β ) 1 − u a ] k x k

subject to

∑ k = 1 n [ ( a a − a l ) + a l ( 1 − β ) 1 − u a , ( a a + a r ) − a r ( 1 − β ) 1 − u a ] j k x k ≤ [ ( b a − b l ) + b l ( 1 − β ) 1 − u a , ( b a + b r ) − b r ( 1 − β ) 1 − u a ] j ;

i.e.,

max Z ˜ = [ ( c a − c l ) + c l ( 1 − β ) 1 − u a , ( c a + c r ) − c r ( 1 − β ) 1 − u a ] 1 x 1 + [ ( c a − c l ) + c l ( 1 − β ) 1 − u a , ( c a + c r ) − c r ( 1 − β ) 1 − u a ] 2 x 2 + ⋯ + [ ( c a − c l ) + c l ( 1 − β ) 1 − u a , ( c a + c r ) − c r ( 1 − β ) 1 − u a ] n x n

subject to

[ ( a a − a l ) + a l ( 1 − β ) 1 − u a , ( a a + a r ) − a r ( 1 − β ) 1 − u a ] 11 x 1 + [ ( a a − a l ) + a l ( 1 − β ) 1 − u a , ( a a + a r ) − a r ( 1 − β ) 1 − u a ] 12 x 2 + ⋯ + [ ( a a − a l ) + a l ( 1 − β ) 1 − u a , ( a a + a r ) − a r ( 1 − β ) 1 − u a ] 1 n x n ≤ [ ( b a − b l ) + b l ( 1 − β ) 1 − u a , ( b a + b r ) − b r ( 1 − β ) 1 − u a ] 1 ;

[ ( a a − a l ) + a l ( 1 − β ) 1 − u a , ( a a + a r ) − a r ( 1 − β ) 1 − u a ] 21 x 1 + [ ( a a − a l ) + a l ( 1 − β ) 1 − u a , ( a a + a r ) − a r ( 1 − β ) 1 − u a ] 22 x 2 + ⋯ + [ ( a a − a l ) + a l ( 1 − β ) 1 − u a , ( a a + a r ) − a r ( 1 − β ) 1 − u a ] 2 n x n ≤ [ ( b a − b l ) + b l ( 1 − β ) 1 − u a , ( b a + b r ) − b r ( 1 − β ) 1 − u a ] 2 ;

⋮

[ ( a a − a l ) + a l ( 1 − β ) 1 − u a , ( a a + a r ) − a r ( 1 − β ) 1 − u a ] m 1 x 1 + [ ( a a − a l ) + a l ( 1 − β ) 1 − u a , ( a a + a r ) − a r ( 1 − β ) 1 − u a ] m 2 x 2 + ⋯ + [ ( a a − a l ) + a l ( 1 − β ) 1 − u a , ( a a + a r ) − a r ( 1 − β ) 1 − u a ] m n x n ≤ [ ( b a − b l ) + b l ( 1 − β ) 1 − u a , ( b a + b r ) − b r ( 1 − β ) 1 − u a ] m .

Now, using the concept of comparison between interval numbers [

AI ( A ˜ α , β I ≤ B ˜ α , β I ) = m 2 − m 1 2 ( w 1 + w 2 ) ≥ 0 and max Z ˜ = ∑ k = 1 n c ˜ α , β I x k

is constructed as,

max Z ˜ = ∑ k = 1 n c ˜ α , β I x k = ∑ k = 1 n 1 2 [ c L ( β ) + c R ( β ) ] x k .

Hence, the above IFLPP can be reformulated as:

max Z ˜ = 1 2 [ ( c a − c l ) + c l ( 1 − β ) 1 − u a + ( c a + c r ) − c r ( 1 − β ) 1 − u a ] 1 x 1 + 1 2 [ ( c a − c l ) + c l ( 1 − β ) 1 − u a + ( c a + c r ) − c r ( 1 − β ) 1 − u a ] 2 x 2 + ⋯ + 1 2 [ ( c a − c l ) + c l ( 1 − β ) 1 − u a + ( c a + c r ) − c r ( 1 − β ) 1 − u a ] n x n

i.e.,

max Z ˜ = [ c a + c r − c l 2 β − u a 1 − u a ] 1 x 1 + [ c a + c r − c l 2 β − u a 1 − u a ] 2 x 2 + ⋯ + [ c a + c r − c l 2 β − u a 1 − u a ] n x n

subject to

[ b a + b r − b l 2 β − u a 1 − u a ] 1 − [ a a + a r − a l 2 β − u a 1 − u a ] 11 x 1 − [ a a + a r − a l 2 β − u a 1 − u a ] 12 x 2 − ⋯ − [ a a + a r − a l 2 β − u a 1 − u a ] 1 n x n ≥ 0

i.e.,

[ a a + a r − a l 2 β − u a 1 − u a ] 11 x 1 + [ a a + a r − a l 2 β − u a 1 − u a ] 12 x 2 + ⋯ + [ a a + a r − a l 2 β − u a 1 − u a ] 1 n x n ≤ [ b a + b r − b l 2 β − u a 1 − u a ] 1 ;

[ a a + a r − a l 2 β − u a 1 − u a ] 21 x 1 + [ a a + a r − a l 2 β − u a 1 − u a ] 22 x 2 + ⋯ + [ a a + a r − a l 2 β − u a 1 − u a ] 2 n x n ≤ [ b a + b r − b l 2 β − u a 1 − u a ] 2 ;

⋮

[ a a + a r − a l 2 β − u a 1 − u a ] m 1 x 1 + [ a a + a r − a l 2 β − u a 1 − u a ] m 2 x 2 + ⋯ + [ a a + a r − a l 2 β − u a 1 − u a ] m n x n ≤ [ b a + b r − b l 2 β − u a 1 − u a ] m .

Hence, solve the required equivalent crisp LPP using standard optimization methods.

Case 2: When

α > ( 1 − β ) w a 1 − u a , A ˜ α , β I = A α I ; (5)

Now, according to the definition of TIFN, A ˜ α is a closed interval [

A ˜ α = [ A L ( α ) , A R ( α ) ] ,

where A L ( α ) = ( a − l a ) + l a α w a , and A R ( α ) = ( a + r a ) − r a α w a .

The given problem reduces to the following:

max Z ˜ = ∑ k = 1 n [ ( c a − c l ) + c l α w a , ( c a + c r ) − c r α w a ] k x k

subject to

∑ k = 1 n [ ( a a − a l ) + a l α w a , ( a a + a r ) − a r α w a ] j k x k ≤ [ ( b a − b l ) + b l α w a , ( b a + b r ) − b r α w a ] j ;

i.e.,

max Z ˜ = [ ( c a − c l ) + c l α w a , ( c a + c r ) − c r α w a ] 1 x 1 + [ ( c a − c l ) + c l α w a , ( c a + c r ) − c r α w a ] 2 x 2 + ⋯ + [ ( c a − c l ) + c l α w a , ( c a + c r ) − c r α w a ] n x n

subject to

[ ( a a − a l ) + a l α w a , ( a a + a r ) − a r α w a ] 11 x 1 + [ ( a a − a l ) + a l α w a , ( a a + a r ) − a r α w a ] 12 x 2 + ⋯ + [ ( a a − a l ) + a l α w a , ( a a + a r ) − a r α w a ] 1 n x n ≤ [ ( b a − b l ) + b l α w a , ( b a + b r ) − b r α w a ] 1 ;

[ ( a a − a l ) + a l α w a , ( a a + a r ) − a r α w a ] 21 x 1 + [ ( a a − a l ) + a l α w a , ( a a + a r ) − a r α w a ] 22 x 2 + ⋯ + [ ( a a − a l ) + a l α w a , ( a a + a r ) − a r α w a ] 2 n x n ≤ [ ( b a − b l ) + b l α w a , ( b a + b r ) − b r α w a ] 2 ;

⋮

[ ( a a − a l ) + a l α w a , ( a a + a r ) − a r α w a ] m 1 x 1 + [ ( a a − a l ) + a l α w a , ( a a + a r ) − a r α w a ] m 2 x 2 + ⋯ + [ ( a a − a l ) + a l α w a , ( a a + a r ) − a r α w a ] m n x n ≤ [ ( b a − b l ) + b l α w a , ( b a + b r ) − b r α w a ] m .

Now, utilizing the concept of comparison between interval numbers [

index (AI) of A ˜ α , β I ≤ B ˜ α , β I is defined by AI ( A ˜ α , β I ≤ B ˜ α , β I ) = m 2 − m 1 2 ( w 1 + w 2 ) ≥ 0 .

The objective can now be restated as,

max Z ˜ = ∑ k = 1 n c ˜ α , β I x k = ∑ k = 1 n 1 2 [ c L ( α ) + c R ( α ) ] x k .

Hence, the above FLPP is reformulated to:

max Z ˜ = [ c a + c r − c l 2 w a − α w a ] 1 x 1 + [ c a + c r − c l 2 w a − α w a ] 2 x 2 + ⋯ + [ c a + c r − c l 2 w a − α w a ] n x n

subject to

[ b a + b r − b l 2 w a − α w a ] 1 − [ a a + a r − a l 2 w a − α w a ] 11 x 1 − [ a a + a r − a l 2 w a − α w a ] 12 x 2 − ⋯ − [ a a + a r − a l 2 w a − α w a ] 1 n x n ≥ 0

i.e.,

[ a a + a r − a l 2 w a − α w a ] 11 x 1 + [ a a + a r − a l 2 w a − α w a ] 12 x 2 + ⋯ + [ a a + a r − a l 2 w a − α w a ] 1 n x n ≤ [ b a + b r − b l 2 w a − α w a ] 1

[ a a + a r − a l 2 w a − α w a ] 21 x 1 + [ a a + a r − a l 2 w a − α w a ] 22 x 2 + ⋯ + [ a a + a r − a l 2 w a − α w a ] 2 n x n ≤ [ b a + b r − b l 2 w a − α w a ] 2

⋮

[ a a + a r − a l 2 w a − α w a ] m 1 x 1 + [ a a + a r − a l 2 w a − α w a ] m 2 x 2 + ⋯ + [ a a + a r − a l 2 w a − α w a ] m n x n ≤ [ b a + b r − b l 2 w a − α w a ] m .

Hence, we solve the required equivalent crisp LPP in standard optimization methods.

It appears as a simple fuzzy LPP which depends on α and w a only. β is not explicitly used in its formulation.

Case 3: When

α = ( 1 − β ) w a 1 − u a , A ˜ α , β I = A α I or A β I ; (6)

We can choose anyone of the above two formulations.

Input: An Intuitionistic fuzzy LPP in mathematical form.

Output: Converging solution and corresponding decision.

Step 1: Calculate separately the α-cut and β-cut of each TIFN as follows:

Let A ˜ I = 〈 a , l a , r a ; w a , u a 〉 then

A ˜ α = [ A L ( α ) , A R ( α ) ] ,

where

A L ( α ) = ( a − l a ) + l a α w a and A R ( α ) = ( a + r a ) − r a α w a .

similarly, A ˜ β = [ A L ( β ) , A R ( β ) ] , which can be calculated as A L ( β ) = ( a − l a ) + l a ( 1 − β ) 1 − u a , and A R ( β ) = ( a + r a ) − r a ( 1 − β ) 1 − u a .

Step 2: Now depending on the above calculations, we find (α-β)-cut of each TIFN i.e., A ˜ α , β I as follows:

A ˜ α , β I = { A β I ; if α < ( 1 − β ) w a 1 − u a A α I ; if α > ( 1 − β ) w a 1 − u a A β I or A α I ; if α = ( 1 − β ) w a 1 − u a (7)

Step 3: Accordingly, we have the formulation:

max Z ˜ = ∑ k = 1 n c ˜ k α , β I x k

subject to ∑ k = 1 n A ˜ α , β I x k ≤ B ˜ α , β I ; 1 ≤ j ≤ m .

Step 4: For the constraints, utilizing the concept of comparison between interval numbers [

AI ( A ˜ α , β I ≤ B ˜ α , β I ) = m 2 − m 1 2 ( w 1 + w 2 ) ≥ 0 .

Step 5: For the objective function max Z ˜ = ∑ k = 1 n c ˜ α , β I x k is constructed as max Z ˜ = ∑ k = 1 n c ˜ α , β I x k = ∑ k = 1 n 1 2 [ c L ( α or β ) + c R ( α or β ) ] x k .

Step 6: Solve the ordinary Linear programming problem using simplex technique.

To illustrate the same let us consider the problem as in the following.

Example 1: Let us consider an IFLPP as in the following:

max f ( x 1 , x 2 ) = ( 2,1,2 ) I x 1 + ( 3,1,1 ) I x 2 s .t . ( 1,1,2 ) I x 1 + ( 2,1,3 ) I x 2 ≤ ( 4,2,1 ) I ( 3,2,3 ) I x 1 + ( 1,1,2 ) I x 2 ≤ ( 6,1,1 ) I (8)

Here, c ¯ 1 = ( 2 , 1 , 2 ) . let α = 0.9 , β = 0.09 , α ∈ [ 0,0.925 ) , β ∈ [ 0.07,1 ] , We first calculate ( α , β ) -cut of c ¯ 1 .

c ^ α = [ c L ( α ) , c R ( α ) ] ,

c ^ L ( 0.9 ) = 1 + 1 × 0.9 0.925 = 1.9729 ,

c ^ R ( 0.9 ) = 4 − 2 × 0.9 0.925 = 2.0541 ,

c ˜ α = c ^ 0.9 = [ 1.9729 , 2.0541 ] . (9)

c ^ β = [ c L ( β ) , c R ( β ) ] ,

c ^ L ( 0.09 ) = 1 + ( 1 − 0.09 ) × 1 1 − 0.07 = 1.9784 ,

c ^ R ( 0.09 ) = 4 − ( 1 − 0.09 ) × 2 1 − 0.07 = 2.0431 , (10)

Since, α < 1 − β 1 − u a ω a , c ˜ 0.9 , 0.09 I = c ^ 0.09 = [ 1.9784 , 2.0431 ] .

For c ¯ 2 = ( 3,1,1 ) , ( α , β ) -cut is [ 2.978,3.022 ] .

Similarly,

For ( 1,1,2 ) , ( α , β ) -cut is [ 0.978,1.044 ] ;

For ( 2,1,3 ) , ( α , β ) -cut is [ 1.978,2.066 ] ;

For ( 4,2,1 ) , ( α , β ) -cut is [ 3.956,4.022 ] ;

For ( 3,2,3 ) , ( α , β ) -cut is [ 2.956,3.066 ] ; and finally

For ( 6,1,1 ) , ( α , β ) -cut is [ 5.978,6.022 ] .

Hence, the associated FLPP becomes the following:

max f 1 = [ 1.9784,2.0431 ] x 1 + [ 2.978,3.022 ] x 2

or,

max f 1 ∗ = 1 2 ( 1.9784 + 2.0431 ) x 1 + 1 2 ( 2.978 + 3.022 ) x 2 = 2.01075 x 1 + 3.0 x 2

subject to the constraints

[ 0.978,1.044 ] x 1 + [ 1.978,2.066 ] x 2 ≤ [ 3.956,4.022 ] (11)

AI ( A ˜ I < B ˜ I ) = 3.956 + 4.022 2 − 0.978 x 1 + 1.978 x 2 + 1.044 x 1 + 2.066 x 2 2 2 [ 1.044 x 1 + 2.066 x 2 − 0.978 x 1 − 1.978 x 2 2 + 4.022 − 3.956 2 ] ≥ 0

i.e.,

3.989 − 1.011 x 1 − 2.022 x 2 ≥ 0

i.e.,

1.011 x 1 + 2.022 x 2 ≤ 3.989

&

[ 2.956,3.066 ] x 1 + [ 0.978,1.044 ] x 2 ≤ [ 5.978,6.022 ]

i.e.,

3.011 x 1 + 1.011 x 2 ≤ 6 (12)

Hence, the LPP assumes the form:

max f 1 ∗ = 2.01075 x 1 + 3.0 x 2 s .t . 1.011 x 1 + 2.022 x 2 ≤ 3.989 3.011 x 1 + 1.011 x 2 ≤ 6.0 x 1 , x 2 ≥ 0

The solution of the IFLPP for different values of α , β is presented in

Example 2: Let us consider another intuitionistic fuzzy LPP as in the following:

max f ( x 1 , x 2 ) = ( 1,1,1 ) I x 1 + ( 1,1,2 ) I x 2 s .t . ( 1,1,2 ) I x 1 + ( 2,2,1 ) I x 2 ≤ ( 3,2,1 ) I ( 2,1,2 ) I x 1 + ( 3,1,2 ) I x 2 ≤ ( 4,1,2 ) I (13)

The solution of this IFLPP for different values of α , β is presented in

In Section 4, instead of TIFN if we take both the co-efficient matrix and cost co-efficient as TFN, then according to the proposed method with the help of α-cut

Sr. No. | 1 − β 1 − u a | α | β | w a | u a | x 1 | x 2 | z |
---|---|---|---|---|---|---|---|---|

1 | 0.5667 | 0.50 | 0.49 | 0.890 | 0.10 | 1.575082 | 0.7672818 | 5.793250 |

2 | 0.6220 | 0.55 | 0.44 | 0.890 | 0.10 | 1.578146 | 0.8135342 | 5.895164 |

3 | 0.6670 | 0.55 | 0.40 | 0.890 | 0.10 | 1.580664 | 0.8528314 | 5.983002 |

4 | 0.6800 | 0.57 | 0.32 | 0.860 | 0.00 | 1.581395 | 0.8644747 | 6.009238 |

5 | 0.7000 | 0.60 | 0.30 | 0.860 | 0.00 | 1.582524 | 0.8826509 | 6.050380 |

6 | 0.7202 | 0.51 | 0.35 | 0.890 | 0.10 | 1.583771 | 0.9030222 | 6.096753 |

7 | 0.7550 | 0.63 | 0.32 | 0.860 | 0.10 | 1.585652 | 0.9343457 | 6.168583 |

8 | 0.7780 | 0.60 | 0.30 | 0.880 | 0.10 | 1.586969 | 0.9567404 | 6.220313 |

9 | 0.8670 | 0.60 | 0.35 | 0.720 | 0.25 | 1.592100 | 1.0474000 | 6.433050 |

10 | 0.9420 | 0.80 | 0.19 | 0.850 | 0.14 | 1.596500 | 1.1312700 | 6.633199 |

11 | 0.9647 | 0.80 | 0.18 | 0.840 | 0.15 | 1.597889 | 1.1576960 | 6.697068 |

12 | 0.9733 | 0.70 | 0.27 | 0.720 | 0.25 | 1.598000 | 1.1681000 | 6.722000 |

13 | 0.9750 | 0.70 | 0.22 | 0.720 | 0.20 | 1.598504 | 1.1698840 | 6.726640 |

14 | 0.9780 | 0.90 | 0.09 | 0.925 | 0.07 | 1.598683 | 1.1734580 | 6.735325 |

15 | 1.0000 | 1.00 | 0.00 | 1.000 | 0.00 | 1.600000 | 1.2000000 | 6.800000 |

Sr. No. | 1 − β 1 − u a | α | β | w a | u a | x 1 | x 2 | z |
---|---|---|---|---|---|---|---|---|

1 | 0.6444 | 0.57 | 0.42 | 0.850 | 0.100 | 1.580866 | 0.8560424 | 5.990228 |

2 | 0.7071 | 0.60 | 0.30 | 0.800 | 0.010 | 1.585366 | 0.9295393 | 6.157520 |

3 | 0.9420 | 0.40 | 0.35 | 0.420 | 0.310 | 1.597158 | 1.1433040 | 6.662241 |

4 | 0.9444 | 0.70 | 0.15 | 0.720 | 0.100 | 1.598337 | 1.1665580 | 6.718564 |

5 | 0.9800 | 0.80 | 0.02 | 0.810 | 0.001 | 1.599243 | 1.1850090 | 6.763429 |

6 | 0.9800 | 0.90 | 0.02 | 0.910 | 0.000 | 1.599341 | 1.1866550 | 6.767442 |

Sr. No. | 1 − β 1 − u a | α | β | w a | u a | x 1 | x 2 | z |
---|---|---|---|---|---|---|---|---|

1 | 0.5667 | 0.50 | 0.49 | 0.890 | 0.10 | 1.902323 | 0.0 | 1.902323 |

2 | 0.6220 | 0.55 | 0.44 | 0.890 | 0.10 | 1.913659 | 0.0 | 1.913659 |

3 | 0.6670 | 0.55 | 0.40 | 0.890 | 0.10 | 1.923148 | 0.0 | 1.923148 |

4 | 0.6800 | 0.57 | 0.32 | 0.860 | 0.00 | 1.925926 | 0.0 | 1.925926 |

5 | 0.7000 | 0.60 | 0.30 | 0.860 | 0.00 | 1.930233 | 0.0 | 1.930233 |

6 | 0.7220 | 0.51 | 0.35 | 0.890 | 0.10 | 1.935016 | 0.0 | 1.935016 |

7 | 0.7550 | 0.63 | 0.32 | 0.860 | 0.10 | 1.942285 | 0.0 | 1.942285 |

8 | 0.7780 | 0.60 | 0.30 | 0.880 | 0.10 | 1.947418 | 0.0 | 1.947418 |

9 | 0.8441 | 0.60 | 0.35 | 0.740 | 0.23 | 1.962487 | 0.0 | 1.962487 |

10 | 0.9420 | 0.80 | 0.19 | 0.850 | 0.14 | 1.985707 | 0.0 | 1.985707 |

11 | 0.9647 | 0.80 | 0.18 | 0.840 | 0.15 | 1.991252 | 0.0 | 1.991252 |

12 | 0.9733 | 0.70 | 0.27 | 0.720 | 0.25 | 1.993369 | 0.0 | 1.993369 |

13 | 0.9750 | 0.70 | 0.22 | 0.720 | 0.20 | 1.993789 | 0.0 | 1.993789 |

14 | 0.9780 | 0.90 | 0.09 | 0.925 | 0.07 | 1.994530 | 0.0 | 1.994530 |

15 | 1.000 | 1.00 | 0.00 | 1.000 | 0.00 | 2.000000 | 0.0 | 2.000000 |

we can reformulate it as a crisp LPP and find a solution of the problem. For that let us consider the same Example 1. The solution of the said problem is given in

For α ≥ 0.5 and β ≤ 0.5 example 1 and example 2 approaches towards a limiting solution as 1 − β 1 − u a tend to 1 as shown in first and third tables.

Moreover, solution of our approach is convergent. The solution obtained from our proposed approach for solving a LPP in an intuitionistic fuzzy environment is better for same values of α than the fuzzy environment which is shown from the first, second and fourth tables. If we defuzzify this IFLPP then we obtain optimal solution of the corresponding crisp LPP. The solution of our IFLPP is quite close to the optimal solution of the associated crisp LPP.

Actually, when available information is not sufficient, the evaluation of the membership and non-membership functions together gives satisfactory result than considering any one of the membership value or the non-membership value. In which case, there remains a part indeterministic on which hesitation survives. Certainly, fuzzy optimization is unable to deal with such hesitation since in this case membership and non-membership functions are complement to each other. Here, in our proposed ( α , β ) -cut technique, sum of membership degree and non-membership degree is always taken as strictly less than one and hence hesitation is considered. Consequently, in our proposed method for solving IFLPP converge rapidly than fuzzy environment as seen in

Sr. No. | α | x 1 | x 2 | z |
---|---|---|---|---|

1 | 0.10 | 1.553863 | 0.4879819 | 5.218856 |

2 | 0.20 | 1.562298 | 0.5942085 | 5.427466 |

3 | 0.30 | 1.571429 | 0.7142857 | 5.678571 |

4 | 0.40 | 1.397908 | 0.9442094 | 5.861195 |

5 | 0.57 | 1.579151 | 0.8290688 | 5.929755 |

6 | 0.60 | 1.397908 | 0.9442094 | 5.861195 |

7 | 0.70 | 1.586969 | 0.9567404 | 6.220313 |

8 | 0.80 | 1.593472 | 1.0729320 | 6.493382 |

9 | 0.90 | 1.594648 | 1.0950200 | 6.546117 |

10 | 0.99 | 1.599460 | 1.1890700 | 6.773329 |

Decisive set method [ | Modified subgradient method [ | Zimmermann’s extended approach [ | Our proposed approach An IF approach |
---|---|---|---|

x 1 = 1.1474 | x 1 = 1.1475 | x 1 = 1.470526 | x 1 = 1.598683 |

x 2 = 0.7508 | x 2 = 0.7514 | x 2 = 0.9410526 | x 2 = 1.173458 |

α = 0.5491 | α = 0.90 | ||

β = 0.08421 | β = 0.09 | ||

Z * = 4.5474 | Z * = 4.5492 | Z * = 5.7642098 | Z * = 6.735325 |

Decisive set method [ | Modified subgradient method [ | Zimmermann’s extended approach [ | Our proposed approach An IF approach |
---|---|---|---|

x 1 = 1.690331 | x 1 = 1.45804 | x 1 = 1.333333 | x 1 = 1.99453 |

x 2 = 0.0 | x 2 = 7.8 × 10 − 8 | x 2 = 0.0 | x 2 = 0.0 |

α = 0.5556 | α = 0.90 | ||

β = 0.142 | β = 0.09 | ||

Z * = 1.690331 | Z * = 1.45804 | Z * = 1.333333 | Z * = 1.99453 |

Concept of an intuitionistic fuzzy set can be viewed as an alternative approach to define a fuzzy set in cases where available information is not sufficient for the definition of an imprecise concept. In general, the theory of intuitionistic fuzzy set is a generalization of the theory of fuzzy set. Therefore, it is expected that intuitionistic fuzzy sets would perform effectively the task of simulation of human decision-making processes and any activities requiring human expertise and knowledge, which are inevitably imprecise or not totally reliable. As proposed we have tried to obtain a solution of an intuitionistic fuzzy LPP using ( α , β ) -cut method. With different simple problems it is tested and significant improvements over existing techniques have been noticed in each case. However, an analytical proof of the same could not be possible to be constructed because of the subjective nature of membership or non-membership functions of TIFN’s used in the representation of the original problem.

There is considerable scope for research in this domain. This includes, in particular, an attempt to find solution for a class of IFLPP without converting them to crisp LPP and to compare other existing fuzzy and intuitionistic fuzzy optimization techniques.

This research is supported by UGC SAP-DRS Phase-III Programme at the Department of Mathematics. UGC’s financial support is highly appreciated.

The authors declare no conflicts of interest regarding the publication of this paper.

Kabiraj, A., Nayak, P.K. and Raha, S. (2019) Solving Intuitionistic Fuzzy Linear Programming Problem—II. International Journal of Intelligence Science, 9, 93-110. https://doi.org/10.4236/ijis.2019.94006