Solving fully fuzzy linear programming problems with flexible constraints based on a new order relation

1 Introduction

Based on the idea of fuzzy decision proposed by Bellman and Zadeh [1], Tanaka et al. and Zimmermann introduced linear programming problems in a fuzzy environment [8, 18]. After then, many researchers turned their attention to the fuzzy linear programming (FLP) problems. All kinds of method for solving FLP problems had been proposed [10, 15-30]. There were two important solution approaches to the FLP problems. One solution approach was to convert the fuzzy model, with fuzzy parameters or fuzzy variables, into some normal linear programming(s) which could be solved by the classical simplex method. The other one was solving the FLP directly by the modified simplex method based on some special ranking functions [24-28, 32]. In both of these two solution approaches, ranking method of fuzzy numbers [35-38] always play an important role when dealing with the FLP problem. With respect to the practical application, the readers can refer to [39-43]. In fact, FLP was widely applied in operational research, optimization management and fuzzy control.

The FLP is said to be fully fuzzy linear programming (FFLP), if all the parameters and variables are considered as fuzzy numbers. The FFLP problems have drew many researchers’ attention, after proposed by J.J. Buckley et al. [11] in 2000. To our knowledge, there are several existing methods for solving the FFLP problems [3-7, 31-33]. We summarize them as follows.

(i) Multi-objective linear programming (MOLP) method [11, 33-34]

J.J. Buckley and T. Feuring [11] studied the FFLP problems for the first time. The authors proposed three methods to verify the feasibility of the constraints and the fuzzy-valued objective function was equivalently transformed into three sub-objective functions. So the FFLP problem might be converted into its equivalent crisp MOLP problem and solved.

In [33], based on a new lexicographic ordering on triangular fuzzy numbers, a novel algorithm was proposed to deal with the FFLP problem by converting it to its equivalent MOLP problem and then it was solved by the lexicographic method. In [34], similarly, according to the defined order relation, a FFLP problem with L-R fuzzy numbers was converted into a classical MOLP problem and then solved.

(ii) Particle swarm optimization (POS) method [14]

A. Baykasoglu and T. Gocken [14] proposed the POS algorithm to solve the FFLP problems which the parameters and variables were represented by symmetric triangular fuzzy numbers. The ranking method based on left and right dominance (Chen et al. [2]) was used to rank the objective function values and determine the feasibility of the constraints in the PSO algorithm [14].

(iii) Lexicographic order method [4,12, 4,12]

S.M. Hashemi et al. [12] introduced the concepts of possibilistic mean value and standard deviation. The lexicographic order was introduced for comparison of fuzzy numbers. Then, a two-phase method was used to solve the FFLP problem by converting it into the equivalent linear programming problems.

In [4], all general fuzzy numbers were transformed into its nearest symmetric triangular fuzzy numbers and then the lexicographic order between symmetric triangular fuzzy numbers was constructed. Based on this lexicographic order, every FFLP model was converted into two crisp complex linear programming problems. Applying this method, the authors were able to find the fuzzy approximate solution of the FFLP problems.

(iv) Ranking function method [3, 31]

A. Kumar et al. and T. Allahviranloo et al. [3, 7, 31] used ranking function to solve FFLP problems with all parameters represented by triangular, trapezoidal or L-R fuzzy numbers, respectively. Usually, an order relation for comparing two arbitrary fuzzy numbers was introduced, according to the given ranking function. After then, a fully fuzzy linear programming could be equivalently changed into a crisp one, whose optimal solution could be easily found. Consequently the exact fuzzy optimal solution was obtained.

(v) Fuzzy simplex method [32]

I.U. Khan et al. [32] developed the fuzzy simplex method to FFLP problem. The basic arithmetic operations, including addition, substraction, multiplication and division, between triangular fuzzy numbers, were defined. The authors employed a ranking function to compare two arbitrary triangular fuzzy numbers. Since arithmetic operations and ranking method were given, the fuzzy simplex method could be carried out for obtaining the fuzzy optimal solutions, if they existed. In [32], the division was not the inverse operation of the multiplication.

In fact, order relation, which is introduced for comparing fuzzy numbers, paly an important role in the solution method to FFLP problem. In this paper, we define the concepts of expected value, expected interval and the degree in which a ˜ is bigger than b ˜, where a ˜ , b ˜ are two arbitrary fuzzy numbers. According to these concepts, we may rank two fuzzy numbers reasonable. Furthermore, a new order relation on the set of all fuzzy numbers is given. Based on the new order relation, new algorithms for solving FFLP problems are proposed.

In this paper, we consider two types of fully fuzzy linear programming problems, named FFLP1 and FFLP2. The constraints of FFLP1 are ordinary inequalities while those of FFLP2 are flexible inequalities. Both of the solution methods to these two types of FFLP problems are developed.

The remaining of the paper unfolds as follows. In Section 2, some basic definitions and the new order relation, together with its properties, are given. Two types of FFLP problems are introduced in this section too. In Sections 3 and 4, the algorithms for solving FFLP1 and FFLP2 are proposed respectively. In Section 5, a numerical example is given to illustrate the feasibility and efficiency of the proposed algorithms. Simple discussion on the proposed method is made in Section 6 while conclusion drawn in Section 7.

2 Fully fuzzy linear programming (FFLP)

The concepts of feasible region, feasible solution, optimal solution and optimal value of the FFLP are similar to that of the crisp linear programming. We introduce these concepts by the following definitions.

Definition 3.1. In the above FFLP1 problem, D = { x ˜ ∈ ( TF ( R ) + ) n | A ˜ ⊗ x ˜ ⪯ b ˜ , } is said to be the feasible region, and any x ˜ ∈ D is said to be the feasible solution.

Definition 3.2. x ˜ * ∈ D is said to be the optimal solution of FFLP1, if z ( x ˜ ) ⪯ z ( x ˜ * ) holds for all x ˜ ∈ D. The corresponding objective function value z ( x ˜ * ) is said to be the optimal value.

In order to solve the FFLP1, we try to change it into a crisp linear programming equivalently.

Lemma 3.1. a ˜ ⪯ b ˜ ⇔ EV ( a ˜ ) ≤ EV ( b ˜ ).

Proof. a ˜ ⪯ b ˜ if and only if E b ˜ + - E a ˜ - E b ˜ + - E a ˜ - - ( E b ˜ - - E a ˜ + ) ≥ 1 2 , if and only if E a ˜ - + E a ˜ + ≤ E b ˜ - + E b ˜ + , if and only if EV ( a ˜ ) ≤ EV ( a ˜ ) . □

Theorem 3.1. The model FFLP1 is equivalent to the following model LP1. ( LP 1 ) max EV ( z ( x ˜ ) ) = EV ( c ˜ ⊗ x ˜ ) s . t . { EV ( a ˜ i 1 ⊗ x ˜ 1 ⊕ a ˜ i 2 ⊗ x ˜ 2 ⊕ ⋯ ⊕ a ˜ in ⊗ x ˜ n ) ≤ EV ( b ˜ i ) , x ˜ j ∈ TF ( ℝ ) + , i = 1 , 2 , ⋯ , m , j = 1 , 2 , ⋯ , n .(3)

Proof. Suppose the feasible regions of FFLP1 and LP1 are D _FFLP1 and D _LP1 respectively. By the Lemma 3.1, we get a ˜ i 1 ⊗ x ˜ 1 ⊕ a ˜ i 2 ⊗ x ˜ 2 ⊕ ⋯ ⊕ a ˜ in ⊗ x ˜ n ⪯ b ˜ i if and only if EV ( a ˜ i 1 ⊗ x ˜ 1 ⊕ a ˜ i 2 ⊗ x ˜ 2 ⊕ ⋯ ⊕ a ˜ in ⊗ x ˜ n ) ≤ EV ( b ˜ i ) , i = 1, 2, ⋯ , m. So FFLP1 and LP1 have the same feasible region, i.e. D _FFLP1 = D _LP1.

If x ˜ * is the optimal solution of FFLP1, then z ( x ˜ ) ≤ z ( x ˜ * ) for all x ˜ in D _FFLP1. According to the Lemma 3.1, we may obtain that EV ( z ( x ˜ ) ) ≤ EV ( z ( x ˜ * ) ) for all x ˜ in D _LP1 = D _FFLP1. This shows that x ˜ * is the optimal solution of LP1.

In the similar way we may prove that any optimal solution of LP1 is also that of FFLP1. Hence the model FFLP1 is equivalent to LP1.□

Lemma 3.2. Let a ˜ = ( a 1 , a 2 , a 3 ) ∈ TF ( ℝ ). Then E a ˜ - = 1 3 ( a 1 + 2 a 2 ), E a ˜ + = 1 3 ( 2 a 2 + a 3 ), and EV ( a ˜ ) = 1 6 ( a 1 + 4 a 2 + a 3 ).

Proof. Suppose the parametric form of a ˜ is ( a _ ( r ) , a ¯ ( r ) ), r ∈ [0, 1]. Then a _ ( r ) = a 1 + r ( a 2 - a 1 ) , a ¯ ( r ) = a 3 - r ( a 3 - a 2 ) .

So we obtain that E a ˜ - = ∫ 0 1 r a _ ( r ) dr ∫ 0 1 rdr = 2 ∫ 0 1 r ( a 1 + r ( a 2 - a 1 ) ) dr = 1 3 ( a 1 + 2 a 2 ) , E a ˜ + = ∫ 0 1 r a ¯ ( r ) dr ∫ 0 1 rdr = 2 ∫ 0 1 r ( a 3 - r ( a 3 - a 2 ) ) dr = 1 3 ( 2 a 2 + a 3 ) and then EV ( a ˜ ) = 1 2 ( E a ˜ - + E a ˜ + ) = 1 6 ( a 1 + 4 a 2 + a 3 ) . □

Theorem 3.2. LP1 is a crisp linear programming.

Proof. Let c ˜ j = ( c j l , c j c , c j r ), x ˜ j = ( x j l , x j c , x j r ), a ˜ ij = ( a ij l , a ij c , a ij r ), b ˜ i = ( b i l , b i c , b i r ), i = 1, 2, ⋯ , m, j = 1, 2, ⋯ , n. Since x ˜ j ∈ TF ( R ) + , j = 1 , 2 , ⋯ , n, the results of z ( x ˜ ) = c ˜ ⊗ x ˜ and ∑ j = 1 n a ˜ ij x ˜ j = a ˜ i 1 ⊗ x ˜ 1 ⊕ a ˜ i 2 ⊗ x ˜ 2 ⊕ ⋯ ⊕ a ˜ in ⊗ x ˜ n, i = 1, 2, ⋯ , m may be calculated by the arithmetic operations on TF (R). Let X = ( x 1 l , x 1 c , x 1 r , x 2 l , x 2 c , x 2 r , ⋯ , x n l , x n c , x n r ) .

The results of c ˜ ⊗ x ˜ and ∑ j = 1 n a ˜ ij x ˜ j are still triangular fuzzy numbers, and written as c ˜ ⊗ x ˜ = ( f l ( X ) , f c ( X ) , f r ( X ) ) , and ∑ j = 1 n a ˜ ij x ˜ j = ( g i l ( X ) , g i c ( X ) , g i r ( X ) ) , where all the functions f l ( X ) , f c ( X ) , f r ( X ) , g i l ( X ), g i c ( X ) and g i r ( X ) are linear functions with variable vector X, i = 1, 2, ⋯ , m .

By the Lemma 3.2, we get EV ( c ˜ ⊗ x ˜ ) = 1 6 ( f l ( X ) + 4 f c ( X ) + f r ( X ) ) . EV ( ∑ j = 1 n a ˜ ij x ˜ j ) = 1 6 ( g i l ( X ) + 4 g i c ( X ) + g i r ( X ) ) , EV ( b ˜ i ) = 1 6 ( b i l + 4 b i c + b i r ) .

So the model LP1 may be written as ( LP 1 ′ ) max 1 6 ( f l ( X ) + 4 f c ( X ) + f r ( X ) ) s . t . { 1 6 ( g i l ( X ) + 4 g i c ( X ) + g i r ( X ) ) ≤ 1 6 ( b i l + 4 b i c + b i r ) , x j l ≥ 0 , x j l ≤ x j c ≤ x j r , i = 1 , 2 , ⋯ , m , j = 1 , 2 , ⋯ , n .(4) which is a crisp linear programming with decision variable vector X = ( x 1 l , x 1 c , x 1 r , ⋯ , x n l , x n c , x n r ) . □

From Theorem 3.1 and 3.2, it is easy to get the following remark.

Remark 3.1. FFLP1 has optimal solution if and only if the linear programming LP1’ has optimal solution. Moreover, x ˜ * = ( ( x 1 l * , x 1 c * , x 1 r * ) , ( x 2 l * , x 2 c * , x 2 r * ) , ⋯ , ( x n l * , x n c * , x n r * ) ) is the optimal solution of FFLP1 if and only if X * = ( x 1 l * , x 1 c * , x 1 r * , ⋯ , x n l * , x n c * , x n r * ) is the optimal solution of LP1’.

Based on the above-mentioned theorems, we develop Algorithm 1 for solving FFLP1 below.

Algorithm 1.

Step 1. Suppose c ˜ j = ( c j l , c j c , c j r ), x ˜ j = ( x j l , x j c , x j r ), a ˜ ij = ( a ij l , a ij c , a ij r ), b ˜ i = ( b i l , b i c , b i r ), i = 1, 2, ⋯ , m, j = 1, 2, ⋯ , n. Then, by the Theorem 3.1 and 3.2, we change the FFLP1 into its equivalent crisp linear programming LP1’.

Step 2. Solving the linear programming LP1’ obtained in step 1.

Step 3. If LP1’ has no optimal solution, then FFLP1 has no optimal solution and stop. Otherwise, suppose X * = ( x 1 l * , x 1 c * , x 1 r * , ⋯ , x n l * , x n c * , x n r * ) is one of the optimal solutions of LP1’ and go to step 4.

Step 4. Generate the optimal solutions x ˜ * of FFLP1 by X ^* as follows and stop. x ˜ * = ( x ˜ 1 * , x ˜ 2 * , ⋯ , x ˜ n * ) = ( ( x 1 l * , x 1 c * , x 1 r * ) , ( x 2 l * , x 2 c * , x 2 r * ) , ⋯ , ( x n l * , x n c * , x n r * ) ) .

Let α ∈ [0, 1], we denote the following programming as FFLP _α . ( FFLP α ) max z ( x ˜ ) = c ˜ 1 ⊗ x ˜ 1 ⊕ c ˜ 2 ⊗ x ˜ 2 ⊕ ⋯ ⊕ c ˜ n ⊗ x ˜ n s . t . { a ˜ i 1 ⊗ x ˜ 1 ⊕ a ˜ i 2 ⊗ x ˜ 2 ⊕ ⋯ ⊕ a ˜ in ⊗ x ˜ n ⪯ α b ˜ i , x ˜ j ∈ TF ( ℝ ) + , i = 1 , 2 , ⋯ , m , j = 1 , 2 , ⋯ , n .(5)

Next, we will introduce a theorem which helps us to develop the method for solving FFLP _α .

Theorem 4.1. Let a ˜ = ( a 1 , a 2 , a 3 ) and b ˜ = ( b 1 , b 2 , b 3 ) are two triangular fuzzy numbers, α ∈ [0, 1]. Then a ˜ ⪯ α b ˜ if and only if 2 b 2 + b 3 - a 1 - 2 a 2 b 3 - b 1 + a 3 - a 1 ≥ α .

Proof. a ˜ ⪯ α b ˜ if and only if P ( b ˜ , a ˜ ) ≥ α , if and only if E b ˜ + - E a ˜ - ≥ 0 and E b ˜ + - E a ˜ - E b ˜ + - E a ˜ - - ( E b ˜ - - E a ˜ + ) ≥ α .

Since E b ˜ + - E a ˜ - = 1 3 ( 2 b 2 + b 3 ) - 1 3 ( a 1 + 2 a 2 ) , E b ˜ + - E a ˜ - E b ˜ + - E a ˜ - - ( E b ˜ - - E a ˜ + ) = 2 b 2 + b 3 - a 1 - 2 a 2 b 3 - b 1 + a 3 - a 1 , and b 3 - b 1 + a 3 - a 1 ≥ 0 , 0 ≤ α ≤ 1 , so we get a ˜ ⪯ α b ˜ if and only if 2 b 2 + b 3 - a 1 - 2 a 2 b 3 - b 1 + a 3 - a 1 ≥ α . □

We can solve FFLP _α by the following Algorithm 2, when the parameter α is fixed in the interval [0, 1].

Algorithm 2.

Step 1. Suppose c ˜ j = ( c j l , c j c , c j r ), x ˜ j = ( x j l , x j c , x j r ), a ˜ ij = ( a ij l , a ij c , a ij r ), b ˜ i = ( b i l , b i c , b i r ), i = 1, 2, ⋯ , m, j = 1, 2, ⋯ , n. Then the FFLP _α may be written as: max ( f l ( X ) , f c ( X ) , f r ( X ) ) s . t . { ( g i l ( X ) , g i c ( X ) , g i r ( X ) ) ⪯ α ( b i l , b i c , b i r ) , x ˜ j ∈ TF ( ℝ ) + , i = 1 , 2 , ⋯ , m , j = 1 , 2 , ⋯ , n .(6) where f l ( X ) , f c ( X ) , f r ( X ) , g i l ( X ) , g i c ( X ) , g i r ( X ) are linear functions with variable vector X = ( x 1 l , x 1 c , x 1 r , x 2 l , x 2 c , x 2 r , ⋯ , x n l , x n c , x n r ) , i = 1, 2, ⋯ , m .

Step 2. By Theorem 3.1, 3.2 and 4.1, the model obtained in step 1 may be transformed into its equivalent crisp linear programming as follows. ( LP α ) max 1 6 ( f l ( X ) + 4 f c ( X ) + f r ( X ) ) s . t . { 2 b i c + b i r - g i l ( X ) - 2 g i c ( X ) b i r - b i l + g i r ( X ) - g i l ( X ) ≥ α , x j l ≥ 0 , x j l ≤ x j c ≤ x j r , i = 1 , 2 , ⋯ , m , j = 1 , 2 , ⋯ , n .(7)

Step 3. Solving the linear programming LP _α obtained in step 2.

Step 4. If LP _α has no optimal solution, then FFLP _α has no optimal solution and stop. Otherwise, suppose X α * = ( x 1 l * , x 1 c * , x 1 r * , ⋯ , x n l * , x n c * , x n r * ) is one of the optimal solutions of LP _α and go to step 5.

Step 5. Generate the optimal solution x ˜ α * of FFLP _α by X α * as follows and stop. x ˜ α * = ( x ˜ 1 * , x ˜ 2 * , ⋯ , x ˜ n * ) = ( ( x 1 l * , x 1 c * , x 1 r * ) , ( x 2 l * , x 2 c * , x 2 r * ) , ⋯ , ( x n l * , x n c * , x n r * ) ) .

Theorem 4.2. Let α₁, α₂ ∈ [0, 1] and α₁ ≤ α₂. x ˜ α 1 * and x ˜ α 2 * are the optimal solutions of FFLP _{α 1} and FFLP _{α 2} , respectively. Then z ( x ˜ α 1 * ) ⪰ z ( x ˜ α 2 * ) .

Proof. Suppose LP _{α 1} and LP _{α 2} are the equivalent linear programmings of FFLP _{α 1} and FFLP _{α 2} , respectively. The feasible regions of LP _{α 1} and LP _{α 2} are D _{α 1} and D _{α 2} , respectively.

In fact, z ( x ˜ α 1 * ) ⪰ z ( x ˜ α 2 * ) if and only if EV ( z ( x ˜ α 1 * ) ) ≥ EV ( z ( x ˜ α 2 * ) ) .

Here, EV ( z ( x ˜ α 1 * ) ) and EV ( z ( x ˜ α 2 * ) ) are the optimal objective values of LP _{α 1} and LP _{α 2} , respectively. In addition, LP _{α 1} and LP _{α 2} have the same objective function. Hence, we only need to prove that D _{α 2}  ⊆ D _{α 1} below.

If X α = ( x 1 l , x 1 c , x 1 r , ⋯ , x n l , x n c , x n r ) ∈ D α 2, then 2 b i c + b i r - g i l ( X ) - 2 g i c ( X ) b i r - b i l + g i r ( X ) - g i l ( X ) ≥ α 2 , and x j l ≥ 0 , x j l ≤ x j c ≤ x j r , i = 1, 2, ⋯ , m, j = 1, 2, ⋯ , n . Since α ₂ ≥ α ₁, we get 2 b i c + b i r - g i l ( X ) - 2 g i c ( X ) b i r - b i l + g i r ( X ) - g i l ( X ) ≥ α 1 , and x j l ≥ 0 , x j l ≤ x j c ≤ x j r , i = 1, 2, ⋯ , m, j = 1, 2, ⋯ , n . This indicates X _α  ∈ D _{α 1} . Hence, D _{α 2}  ⊆ D _{α 1} . □

We divide [ 1 2 , 1 ] into n subintervals equally, where n is a positive integer. Take α k = 1 2 + k 2 n and suppose x ˜ α k * is the optimal solution of FFLP _{α k} , k = 0, 1, ⋯ , n. It is clear that 1 2 = α 0 ≤ α 1 ≤ ⋯ ≤ α n = 1 . From Theorem 4.2, we get z ( x ˜ α 0 * ) ⪰ z ( x ˜ α 1 * ) ⪰ ⋯ ⪰ z ( x ˜ α n * ) , and EV ( z ( x ˜ α 0 * ) ) ≥ EV ( z ( x ˜ α 1 * ) ) ≥ ⋯ ≥ EV ( z ( x ˜ α n * ) ) .

Define the membership function of z ( x ˜ ) as follow: μ ( z ( x ˜ ) ) = { ( i ) 0 , if EV ( z ( x ˜ ) ) < EV ( z ( x ˜ α n * ) ) , ( ii ) EV ( z ( x ˜ ) ) - EV ( z ( x ˜ α n * ) ) EV ( z ( x ˜ α 0 * ) ) - EV ( z ( x ˜ α n * ) ) , if EV ( z ( x ˜ α n * ) ) ≤ EV ( z ( x ˜ ) ) ≤ EV ( z ( x ˜ α 0 * ) ) , ( iii ) 1 , if EV ( z ( x ˜ ) ) > EV ( z ( x ˜ α 0 * ) ) .

Then we may get 1 = μ ( z ( x ˜ α 0 * ) ) ≥ μ ( z ( x ˜ α 1 * ) ) ≥ ⋯ ≥ μ ( z ( x ˜ α n * ) ) = 0 .

Let β _k  = 2α _k  - 1, k = 0, 1, ⋯ , n. then 0 = β 0 ≤ β 1 ≤ ⋯ ≤ β n = 1 .

If min k { | β k - μ ( z ( x ˜ α k * ) ) | } = | β k 0 - μ ( z ( x ˜ α k 0 * ) ) |, we take x ˜ α k 0 * as the fuzzy optimal solution of fully fuzzy linear programming FFLP2.

Below, we propose Algorithm 3 for solving FFLP2.

Algorithm 3.

Step 1. Fixed a positive integer n and calculate the value of α _k by α k = 1 2 + k 2 n, k = 0, 1, ⋯ , n.

Step 2. Solving FFLP _{α 0} , FFLP _{α 2} , ⋯, FFLP _{α n} by Algorithm 2 respectively.

Step 3. Suppose x ˜ α k * is one of the optimal solutions of FFLP _{α k} , k = 0, 1, ⋯ , n. Calculate the optimal objective values z ( x ˜ α 0 * ), z ( x ˜ α 1 * ) , ⋯, z ( x ˜ α n * ), and then give the membership functions μ ( z ( x ˜ ) ) .

Step 4. Calculate β _k  = 2α _k  - 1 and μ ( z ( x ˜ α k * ) ), k = 0, 1, ⋯ , n.

Step 5. Calculate min k { | β k - μ ( z ( x ˜ α k * ) ) | }.

Step 6. Suppose min k { | β k - μ ( z ( x ˜ α k * ) ) | } = | β k 0 - μ ( z ( x ˜ α k 0 * ) ) | .

Then the fuzzy optimal solution of FFLP2 is x ˜ * = x ˜ α k 0 *.

Remark 4.1. Here, we only consider the situation that every fully fuzzy linear programming FFLP _{α k} (k ∈ {0, 1, ⋯ , n}) has one optimal solution at least, in Step 3.

In this section we illustrate the proposed method by a numerical example.

Example 5.1. Consider the following fully fuzzy linear programming with flexible constraints: ( FFLP 21 ) max z ( x ˜ ) = ( 45 , 50 , 52 ) ⊗ x ˜ 1 ⊕ ( 56 , 60 , 63 ) ⊗ x ˜ 2 s . t . { ( 9 , 10 , 12 ) ⊗ x ˜ 1 ⊕ ( 3 , 5 , 6 ) ⊗ x ˜ 2 ≾ ( 180 , 200 , 210 ) , ( 0 , 1 , 1 ) ⊗ x ˜ 1 ⊕ ( 2 , 3 , 4 ) ⊗ x ˜ 2 ≾ ( 45 , 50 , 55 ) , ( 3 , 4 , 6 ) ⊗ x ˜ 1 ⊕ ( 3 , 4 , 5 ) ⊗ x ˜ 2 ≾ ( 90 , 100 , 120 ) , x ˜ 1 , x ˜ 2 ∈ TF ( ℝ ) + .(8)

Solution:

Step 1. Take n = 10, and then α k = 1 2 + k 20, k = 0, 1, ⋯ , n.

Step 2. Suppose x ˜ j = ( x j l , x j c , x j r ) , j = 1 , 2. The fully fuzzy linear programming FFLP _{α k} may be written as: ( FFLP α k ) max z ( x ˜ ) = ( 45 , 50 , 52 ) ⊗ ( x 1 l , x 1 c , x 1 r ) ⊕ ( 56 , 60 , 63 ) ⊗ ( x 2 l , x 2 c , x 2 r ) s . t . { ( 9 , 10 , 12 ) ⊗ ( x 1 l , x 1 c , x 1 r ) ⊕ ( 3 , 5 , 6 ) ⊗ ( x 2 l , x 2 c , x 2 r ) ⪯ α k ( 180 , 200 , 210 ) , ( 0 , 1 , 1 ) ⊗ ( x 1 l , x 1 c , x 1 r ) ⊕ ( 2 , 3 , 4 ) ⊗ ( x 2 l , x 2 c , x 2 r ) ⪯ α k ( 45 , 50 , 55 ) , ( 3 , 4 , 6 ) ⊗ ( x 1 l , x 1 c , x 1 r ) ⊕ ( 3 , 4 , 5 ) ⊗ ( x 2 l , x 2 c , x 2 r ) ⪯ α k ( 90 , 100 , 120 ) , ( x 1 l , x 1 c , x 1 r ) , ( x 2 l , x 2 c , x 2 r ) ∈ TF ( ℝ ) + .(9) k = 0, 1, ⋯ , n . We solve these fully fuzzy linear programming models by Algorithm 2.