Tackling the fuzzy multi-objective linear fractional problem using a parametric approach

2.1 Fuzzy numbers and intervals

Definition 1 (boldialpha- cut). Suppose A ˜ is a fuzzy set in X and α ∈ [0, 1]. The α-cut of the fuzzy set A ˜ is the crisp set A ˜ α given by: A ˜ α = { x ∈ X : μ A ( x ) ⩾ α } .

Definition 2 (Triangular fuzzy numbers). Assume A ˜ is a fuzzy set. A triangular fuzzy number A ˜ is defined as: μ A ˜ ( x ) = μ A ( x ; a , b , c ) = { ( x - a ) / ( b - a ) , x ∈ [ a , b ) ( c - x ) / ( c - b ) , x ∈ [ b , c ] 0 , x > c or x < a .

For the triangular fuzzy number A ˜ , A ˜ α = [ A α L , A α U ] = [ a + α ( b - a ) , c - α ( c - b ) ] .

In the literature, there exists several methods to rank fuzzy numbers. Chen and Hwang [8] reviewed 20 ranking methods. See Rao [23] for a more recent method to rank fuzzy numbers.

A fuzzy number can be represented uniquely by its α - cut. Therefore, in this paper, arithmetic operations in addition to ranking of fuzzy numbers are defined in terms of the α - cuts of the fuzzy numbers.

According to Wu [32], the following proposition is true for any two fuzzy numbers A ˜ = [ A α L , A α U ] and B ˜ = [ B α L , B α U ] .

Proposition 1. A ˜ ⩾ α B ˜ if and only if A α L ⩾ B α L and A α U ⩾ B α U , ∀ α ∈ [ 0 , 0.5 ] .

In the above, α ∈ [0, 0.5] means the α - cuts include the elements with small membership degrees. To compensate for this shortcoming, referential fuzzy numbers are used.

Definition 3 (Referential fuzzy number). Let A ˜ be a triangular fuzzy number with the membership function μ A ˜ ( x ) = μ A ( x ; a , b , c ) , then its referential fuzzy number is a triangular fuzzy number denoted by A ˜ * and defined as follow. μ A ˜ * ( x ) = μ A ˜ * ( x ; a + 2 b 3 , b , 2 b + c 3 ) = { . 05 ( x - a ) / ( b - a ) , x ∈ [ a + 2 b 3 , b ) 0.5 ( c - x ) / ( c - b ) , x ∈ [ b , 2 b + c 3 , ] 0 , otherwise

In addition, A ˜ α * = [ a + 2 α ( b - a ) , c - 2 α ( c - b ) ] , ∀ α ∈ [ 1 3 , 1 2 ] .

Proposition 2. A ˜ α * ⊂ A ˜ α , ∀ α > 0 .

According to the above proposition, ∃ intervals V₁ and V₂ such that:

A ˜ α * ∪ V 1 ∪ V 2 = A ˜ α and μ A ˜ ( x ) < μ A ˜ ( y ) , ∀x ∈ V₁ ∪ V₂ and ∀ y ∈ A ˜ α * ; this means the number of elements with low degree of membership functions are decreased in A ˜ α * compared to A ˜ α .

Definition 4 (Ranking of fuzzy numbers). Let A ˜ * and B ˜ * be referential fuzzy numbers related to triangular fuzzy numbers A ˜ and B ˜ , respectively. Consider A ˜ α * = [ A α * L , A α * U ] and B ˜ α * = [ B α * L , B α * U ] , ∀ α ∈ [ 1 3 , 1 2 ] . Fuzzy number B ˜ is said to be smaller than A ˜ and denoted by A ˜ ⩾ α B ˜ if only if A α * L ⩾ B α * L and A α * U ⩾ B α * U , ∀ α ∈ [ 1 3 , 1 2 ] .

Definition 5 (Arithmetic of intervals). Assume A = [A ^L , A ^U ], B = [B ^L , B ^U ] are intervals, and k ⩾ 0 is a scalar. Therefore, addition, multiplication, and division on the intervals are defined as following. A + B = [ A L + B L , A U + B U ] , - A = [ - A U , - A L ] , kA = [ kA L , kA U ] , AB = [ min { A L B L , A L B U , A U B L , A U B U } , max { A L B L , A L B U , A U B L , A U B U } ] , A B = [ min { A L B L , A L B U , A U B L , A U B U } , max { A L B L , A L B U , A U B L , A U B U } ] .

2.2 Linear fractional programming

Consider the general form of the linear fractional programming problem (LFPP) as follows:

Minimize C T X + p D T X + q subject to Ω = { AX ⩽ b , X ⩾ 0 } ,(1) where A ∈ ℝ m × n , b ∈ ℝ m × 1 , C , D ∈ ℝ n × 1 , p , q ∈ ℝ . In addition, D ^T X + q > 0, ∀ X = (X₁, …, X _n ) ∈ Ω. It is additionally assumed that feasible region Ω is a regular set i.e. a non-empty and compact set. Notify X ⩾ 0 means X _i  ⩾ 0, for i = 1, …, n.

Using variable transformations t = 1 D T X + β , Y = tX changes the (1) into:

Minimize C T Y + pt subject to φ = { AY - bt ⩽ 0 , D T Y + qt = 1 , Y , t ⩾ 0 } .(2)

Lemma 1 ([7]). In (2), variable t cannot be zero.

Lemma 2 ([7]). If ( Y ¯ , t ¯ ) ∈ φ , then Y ¯ t ¯ ∈ Ω .

Theorem 1 ([7]). If (Y^*, t^*) is optimum for problem (2), then X * = Y * t * is optimum for (1).

Consider the general form of MOPP as follows:

Minimize X ∈ S { F 1 ( X ) , … , F k ( X ) } ,(3) where S is a non-empty and bounded feasible region.

Definition 6. In (3), X^* ∈ S is an efficient solution if ∀X ∈ S, then ∃j∈ { 1, …, k } such that F _j  (X^*) < F _j  (X).

Definition 7. In (3), X^* ∈ S is an ɛ- weakly efficient solution if ∀X ∈ S, then ∃j∈ { 1, …, k } such that F _j  (X^*) - ɛ ⩽ F _j  (X), where ɛ = inf {δ > 0 : F _j  (X^*) - δ ⩽ F _j  (X)}.

Weighted sum approach is a classical method which transforms the (3) into:

Minimize X ∈ S ∑ i = 1 k w i F i ( X ) ,(4) where w _i  ⩾ 0, i = 1, …, k.

Theorem 2. The optimal solution X^* of (4) is an efficient solution for (3).

Proof. Let us assume X^* not be an efficient solution for (3). Therefore, ∃ X ¯ ∈ S such that:

F i ( X * ) ⩾ F i ( X ¯ ) , i = 1, …, k, ∃j∈ { 1, …, k } so that F j ( X * ) > F j ( X ¯ ) . Therefore, ∑ i = 1 k w i F i ( X * ) > ∑ i = 1 k w i F i ( X ¯ ) . This contradicts the optimality of X^* for (4). ■

Max-min technique transforms the (3) into:

Minimize β s . t φ = S ∪ { F i ( X ) ⩽ β , i = 1 , … , k } .(5)

Theorem 3. Let (X^*, β^*) be the unique optimal solution of (5), then X^* is an efficient solution for (3).

Proof. Let X^* not be an efficient solution for (3). Therefore, ∃ X ¯ ∈ S such that:

F i ( X ¯ ) ⩽ F i ( X * ) , i = 1 , … , k , ∃ j ∈ { 1 , … , k } so that F j ( X ¯ ) < F j ( X * ) .(6)

Since (X^*, β^*) is feasible point of φ, then F _i  (X^*) ⩽ β^*. Thus, from the (6):

F i ( X ¯ ) ⩽ β * , i = 1 , … , k , ∃ j ∈ { 1 , … , k } so that F j ( X ¯ ) < β * .(7)

Let us define:

θ ¯ = max { F i ( X ¯ ) , i = 1 , … , k } .(8)

Thus, it directly comes from (7) and (8) that: F i ( X ¯ ) ⩽ θ ¯ , θ ¯ ⩽ β * .(9)

Thus, X ¯ ∈ S and F i ( X ¯ ) ⩽ θ ¯ implies that ( X ¯ , θ ¯ ) ∈ φ . Hence, feasibility of ( X ¯ , θ ¯ ) and θ ¯ ⩽ β * contradicts the unique optimality of (X^*, β^*). The proof is then complete. ■

Dinkelbach [11] showed that a factional programming can be changed into a non-fractional program. In other word, he proved the following two problems are equivalent.

Minimize X ∈ S f ( X ) g ( X ) .(10)

Minimize X ∈ S f ( X ) - θ g ( X ) .(11) where f (X) and g (X) are continuous functions, and S is a non-empty compact feasible region, and θ is a parameter.

Theorem 3. If X^* is optimum for (10), then (10) and (11) are equivalent if θ = f ( X * ) g ( X * ) .

Proof. To proof, we need to demonstrate that the optimal solution of (10) is also optimum for (11), and vice versa. Since X^* is optimum for (10), then we have: (i) θ = f ( X * ) g ( X * ) ⩽ f ( X ) g ( X ) , ∀ X ∈ S . In addition, (ii) f (X^*) - θg (X^*) = 0. Now, let us assume X^* not be optimum for (11). Then, the (ii) results: ∃ X ¯ ∈ S such that: f ( X ¯ ) - θ g ( X ¯ ) < 0 . Thus, positivity of g ( X ¯ ) results: θ = f ( X * ) g ( X * ) > f ( X ¯ ) g ( X ¯ ) . This contradicts the (i) ; this contradiction proves X^* is also optimum for (11).

Now, let us assume X ¯ is optimum for (11), but X ¯ is not optimum for (10). Since X ¯ is optimum for (11), then we have: (iii) 0 = f ( X * ) - f ( X * ) g ( X * ) g ( X * ) ⩾ f ( X ¯ ) - f ( X * ) g ( X * ) g ( X ¯ ) . Since X ¯ is not optimum for (10), then we conclude: (iv) f ( X * ) g ( X * ) 〈 f ( X ¯ ) g ( X ¯ ) ⇒ f ( X ¯ ) - f ( X * ) g ( X * ) g ( X ¯ ) 〉 0 . This contradicts the (iii); this means X ¯ is also optimal for (10).

■

This algorithm summarizes the procedure of finding efficient solutions for the multi-objective linear fractional programming problem with fuzzy coefficients or with interval coefficients.

Initial Step. Define the referential fuzzy numbers, and then specify their α - cuts .

Step 1. Specify α ∈ [ 1 3 , 1 2 ] , w _i  ∈ [0, 1], i = 1, …, k, and v₁, v₂ ∈ [0, 1].

Step 2. Formulate (14).

Step 3 . Determine θ i * = Minimize X ∈ S F i L ( X ) and γ i * = Minimize X ∈ S F i U ( X ) , i = 1, …, k.

Step 4. Formulate (15).

Step 5 . Formulate (18) and find X^* as the optimal solution.

Step 6. Calculate ɛ = min { max { F i L ( X * ) - θ i * , F i U ( X * ) - γ i * } , i = 1 , … , k } as the criteria to evaluate the accuracy of the solution obtained.

Step 7. Set X^* as a weakly ɛ - efficient solution for (12). And, consider [ F i L ( X * ) , F i U ( X * ) ] as an α - acceptable interval for the value of i ^th objective function of (12).

Hint 2. In Step 5 of the algorithm, if we formulate (19), then the solution resulted from the max-min technique is obtained.

4.1 Example 1 (An illustrative example)

Minimize { F ˜ 1 = - 1 ˜ X 1 + 3 ˜ X 2 + 2 ˜ 1 ˜ X 1 + 2 ˜ X 2 + 1 ˜ , F ˜ 2 = 5 ˜ X 1 + 2 ˜ X 2 + 2 ˜ 2 ˜ X 1 + 3 ˜ X 2 + 1 ˜ } subject to S ˜ = { 2 ˜ X 1 + 1 ˜ X 2 ⩽ 4 ˜ , 3 ˜ X 1 - 2 ˜ X 2 ⩽ 5 , ˜ 1 ˜ X 1 + 2 ˜ X 2 ⩽ 3 ˜ , - 1 ˜ X 1 - 3 ˜ X 2 ⩽ - 2 ˜ , X 1 , X 2 ⩾ 0 } ,(20) where

μ 1 ˜ ( X ) = μ 1 ˜ ( X ; 0.5 , 1 , 1.25 ) , μ 2 ˜ ( X ) = μ 2 ˜ ( X ; 1 , 2 , 3 ) , μ 3 ˜ ( X ) = μ 3 ˜ ( X ; 2.5 , 3 , 4 ) , μ 4 ˜ ( X ) = μ 4 ˜ ( X ; 3.5 , 4 , 4.5 ) , μ 5 ˜ ( X ) = μ 5 ˜ ( X ; 4 , 5 , 6 ) .

Referential fuzzy numbers related to the above fuzzy numbers are defined as following:

μ 1 ˜ * ( X ) = μ 1 ˜ * ( X ; 2.5 3 , 1 , 3.25 3 ) , μ 2 ˜ * ( X ) = μ 2 ˜ * ( X ; 5 3 , 2 , 7 3 ) , μ 3 ˜ * ( X ) = μ 3 ˜ * ( X ; 8.5 3 , 3 , 10 3 ) , μ 4 ˜ * ( X ) = μ 4 ˜ * ( X ; 11.5 3 , 4 , 12.5 3 ) , μ 5 ˜ * ( X ) = μ 5 ˜ * ( X ; 14 3 , 5 , 16 3 ) .

The α–cuts of the corresponding referential fuzzy coefficients are given as below, ∀ α ∈ [ 1 3 , 1 2 ] .

[ 1 ˜ * ] α = [ α + 0.5 , 1.25 - 0.5 α ] , [ 2 ˜ * ] α = [ 2 α + 1 , 3 - 2 α ] , [ 3 ˜ * ] α = [ α + 2.5 , 4 - 2 α ] , [ 4 ˜ * ] α = [ α + 3.5 , 4.5 - α ] , [ 5 ˜ * ] α = [ 2 α + 4 , 6 - 2 α ] .

Step 1. Set α = 0.4, w₁ = w₂ = 0.5.

Step 2. We formulate (14) for this example as follows:

Minimize { [ F 1 L ( X ) , F 1 U ( X ) ] = [ - 1.05 X 1 + 2.9 X 2 + 1.8 1.05 X 1 + 2.2 X 2 + 1.05 , - 0.9 X 1 + 3.2 X 2 + 2.2 0.9 X 1 + 1.8 X 2 + 0.9 ] , [ F 2 L ( X ) , F 2 U ( X ) ] = [ 4.8 X 1 + 1.8 X 2 + 1.8 2.2 X 1 + 3.2 X 2 + 1.05 , 5.2 X 1 + 2.2 X 2 + 2.2 1.8 X 1 + 2.9 X 2 + 0.9 ] } subject to S = { 1.8 X 1 + 0.9 X 2 ⩽ 3.9 , 2.2 X 1 + 1.05 X 2 ⩽ 4.1 , 2.9 X 1 - 2.2 X 2 ⩽ 4.8 , 3.2 X 1 - 1.8 X 2 ⩽ 5.2 , 0.9 X 1 + 1.8 X 2 ⩽ 2.9 , 1.05 X 1 + 2.2 X 2 ⩽ 3.2 , - 1.05 X 1 - 3.2 X 2 ⩽ - 2.2 , - 0.9 X 1 - 2.9 X 2 ⩽ - 1.8 , X 1 , X 2 ⩾ 0 } .(21)

Step 3.

θ 1 * = Minimize X ∈ S F 1 L ( X ) = 0.1265 ,

γ 1 * = Minimize X ∈ S F 1 U ( X ) = 0.4087 , and

θ 2 * = Minimize X ∈ S F 2 L ( X ) = 0.7745 ,

γ 2 * = Minimize X ∈ S F 2 U ( X ) = 1.0551 .

Step 4. The (15) is formed for this example as follows:

Minimize X ∈ S { [ - 1.1828 X 1 + 2.6217 X 2 + 1.6672 , - 1.2678 X 1 + 2.4643 X 2 + 1.8322 ] , [ 3.0961 X 1 - 0.6784 X 2 + 0.9868 , 3.3008 X 1 - 0.8598 X 2 + 1.2504 ] } .(22)

Step 5. For this example, (18) is formulated as follows:

Minimize X ∈ S v 1 ( w 1 ( - 1.1828 X 1 + 2.6217 X 2 + 1.6672 ) + w 2 ( 3.0961 X 1 - 0.6784 X 2 + 0.9868 ) ) + v 2 ( w 1 ( - 1.2678 X 1 + 2.4643 X 2 + 1.8322 ) + w 2 ( 3.3008 X 1 - 0.8598 X 2 + 1.2504 ) )(23)

We solve the (23) for different values of v _i and w _i , i = 1, 2 and the results are summarized in Tables 1, 2 and 3.

Max-min technique i.e. the (19) is formed for this example as follows:

Minimize β s . t S ∪ { w 1 ( 1.9133 X 1 + 1.9433 X 2 + 2.654 ) ⩽ β , w 2 ( 2.033 X 1 + 1.6045 X 2 + 3.0831 ) ⩽ β , β ⩾ 0 . }(24)

For (w₁, w₂) = (0.5, 0.5) , (1, 0) , (0, 1), the above problem is solved and the solution resulted is: X * = ( 0 , 0.6875 ) .

4.1.1 Numerical analysis

In Table 4, the extreme points of feasible region S and values of F ˜ 1 ( X ) and F ˜ 2 ( X ) at extreme points are listed.

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Table 4

Extreme points and their objective function values for the fuzzy multi-objective problem for α = 0.4

Extreme point X ˆ	F ˜ 1 ( X ˆ ) = [ F 1 L ( X ˆ ) , F 1 U ( X ˆ ) ]	F ˜ 2 ( X ˆ ) = [ F 2 L ( X ˆ ) , F 2 U ( X ˆ ) ]
(0, 1.4545)	[1.4160, 1.9483]	[0.7745, 1.0551]
(1.5144, 0.7318)	[0.5487, 0.8879]	[1.5448, 2.0328]
(1.7541, 0.2295)	[0.1836, 0.4688]	[1.8841, 2.5040]
(1.6983, 0.1303)	[0.1265, 0.4087]	[1.9577, 2.6109]
(0, 0.6875)	[1.4805, 2.0585]	[0.9348, 1.2829]

Let X^* = (0, 0.6875), and ɛ = 0.2278. In this case, [ F 1 L ( X * ) - ɛ , F 1 U ( X * ) - ɛ ] = [ 1.2527 , 1.8307 ] , and [ F 2 L ( X * ) - ɛ , F 2 U ( X * ) - ɛ ] = [ 0.7068 , 1.0551 ] .

As we observe numerically, ∀ extreme point X ˆ : F 2 L ( X * ) - ɛ < F 2 L ( X ˆ ) and F 2 U ( X * ) - ɛ ⩽ F 2 U ( X ˆ ) . Therefore, convexity of S and pseudoconvexity of F 2 L ( X ) , F 2 U ( X ) result that: F 2 L ( X * ) - ɛ < F 2 L ( X ) and F 2 U ( X * ) - ɛ ⩽ F 2 U ( X ) , ∀ X ∈ S . As the consequence, F ˜ 2 ( X * ) - ɛ = [ F 2 L ( X * ) - ɛ , F 2 U ( X * ) - ɛ ] ⩽ [ F 2 L ( X ) , F 2 U ( X ) ] = F ˜ 2 ( X ) , ∀ X ∈ S . This indicates X^* is a weakly ɛ - efficient solution for the fuzzy multi-objective problem.

Let X^* = (1.6983, 0.1303), and ɛ = 0 . In this case, [ F 1 L ( X * ) , F 1 U ( X * ) ] = [ 0.1265 , 0.4087 ] , and [ F 2 L ( X * ) , F 2 U ( X * ) ] = [ 1.9577 , 2.6109 ] . As we observe numerically, ∀ extreme point X ˆ : F 1 L ( X * ) < F 1 L ( X ˆ ) and F 1 U ( X * ) < F 1 U ( X ˆ ) . Therefore, convexity of S in addition to pseudoconvexity of F 1 L ( X ) , F 1 U ( X ) results that: F 1 L ( X * ) < F 2 L ( X ) and F 1 U ( X * ) ⩽ F 1 U ( X ) , ∀ X ∈ S . As the consequence,

F ˜ 1 ( X * ) = [ F 1 L ( X * ) , F 1 U ( X * ) ] ⩽ [ F 1 L ( X ) , F 1 U ( X ) ] = F ˜ 1 ( X ) , ∀ X ∈ S . This indicates X^* is an efficient solution for the fuzzy multi-objective problem. In a same way, it is showed numerically X^* = (0, 1.4545) is an efficient solution.

Apart from that we applied the multiobj documentation of MATLAB R2016b and the results are depicted in figures below Figs. 1–4. According to the average of the Pareto fronts, average of the values of the objective functions at the efficient solutions, it can be concluded that:

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Fig. 1

Pareto front resulted from multiobj where objective 1 = F 1 L , objective 2 = F 2 L .

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Fig. 2

Pareto front resulted from multiobj where objective 1 = F 1 L , objective 2 = F 2 U .

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Fig. 3

Pareto front resulted from multiobj where objective 1 = F 1 U , objective 2 = F 2 L .

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Fig. 4

Pareto front resulted from multiobj where objective 1 = F 1 U , objective 2 = F 2 U .

F 1 ( X ) ∈ [ 0.2857 , 1.6841 ] or F 2 ( X ) ∈ [ 0.9256 , 2.2653 ] , ∀ Pareto solution X .(25)

According to the Tables 1, 2 and 3, the proposed solutions of this study are X¹ = (1.6983, 0.1303) and X² = (0, 1.4545) i.e. the solution with the lowest ɛ. Obviously, these two solutions satisfy the (25) ; this means the solutions proposed are Pareto, efficient, for this example.

In this section, an example with triangular fuzzy numbers taken from Nayak and Ojha [20] is considered. In their approach, the AHP technique is utilized to specify the values of weights. In order to make a fair comparison, we use the same weights.

Maximize { F ˜ 1 ( X ) = ( 4 , 7 , 11 ) X 1 + ( 6 , 9 , 15 ) X 2 + ( 3 , 5 , 9 ) X 3 + ( 2 , 5 , 7 ) ( 2 , 5 , 6 ) X 1 + ( 4 , 5 , 7 ) X 2 + ( 3 , 6 , 8 ) X 3 + ( 2 , 5 , 8 ) , F ˜ 2 ( X ) = ( 6 , 9 , 11 ) X 1 + ( 5 , 7 , 9 ) X 2 + ( 8 , 10 , 12 ) X 3 + ( 3 , 4 , 5 ) ( 5 , 8 , 12 ) X 1 + ( 3 , 5 , 7 ) X 2 + ( 8 , 12 , 14 ) X 3 + ( 4 , 5 , 8 ) , F ˜ 3 ( X ) = ( 3 , 5 , 12 ) X 1 + ( 5 , 7 , 8 ) X 2 + ( 8 , 11 , 15 ) X 3 + ( 2 , 5 , 6 ) ( 1 , 3 , 4 ) X 1 + ( 2 , 8 , 14 ) X 2 + ( 3 , 9 , 12 ) X 3 + ( 1 , 3 , 5 ) } subject to ( 1 , 3 , 5 ) X 1 + ( 2 , 5 , 7 ) X 2 + ( 3 , 5 , 6 ) X 3 ⩽ ( 6 , 8 , 12 ) .(26)