A fuzzy multi-objective linear programming with interval-typed triangular fuzzy numbers

1 Introduction

Optimization problems include objectives and constraints. Deterministic optimization problems have been well studied, but they are limited and inadequate to exactly express the real problem [1]. In our daily life, many complicated problems involve uncertain data in economics, social sciences, medical diagnosis, natural sciences and many other fields. Accordingly, fuzzy optimal control and multi-objective linear programming with fuzzy parameters have been playing an increasing role in uncertain systems [2, 3, 4, 5, 6].

Recently, several researchers have considered the issues of expressing the coefficients of objectives and constraints in multi-objective linear programming. There are several well-known theories to describe uncertainty such as fuzzy set theory, possibility theory, probability theory and other mathematical tools. For instance, fuzzy linear programming with fuzzy coefficients are considered by many authors [7, 8, 9]. Wang and Wang [10] have proposed a fuzzy multi-objective linear programming with fuzzy-numbered cost coefficients and transformed the problem into a multi-objective problem with parametrically interval-valued cost coefficients by utilizing membership functions. Different algorithms [11, 12] are developed to solve multi-objective linear programming problems based on interval-valued cost coefficients. Combining fuzziness and randomness in an optimization problem, many models are considered such as fuzzy random chance-constrained programming model [13, 14] and multi-objective linear programming model with fuzzy random coefficients [15]. Liu and Liu [14] have established an expected value model and developed a hybrid intelligent algorithm of the fuzzy random multi-objective programming problem. Li et al. [15] have presented a genetic algorithm using the compromise approach for solving a fuzzy random multi-objective programming problem. However, the probability distributions of parameters may be unknown [16]. Considering uncertainties as interval analysis, multi-objective robust optimization approaches are developed which wouldn’t involve any probability distribution [17]. In recent years, regarding coefficients as interval numbers [18, 19], many multi-objective programming with interval parameters [20, 21, 22, 23] have been discussed in detail. The approach proposed by Hajiagha et al. [21] proves to be flexible especially in ill-defined information circumstance. Moreover, the approach has less computational complexity than the literature [23] when the number of objective functions increases.

As pointed out by Chiang [24], Transportation problems prove to be better to express the parameters as interval-valued fuzzy numbers instead of normal fuzzy numbers. To find solution of a linear multi-objective transportation problem with parameters represented as interval-valued fuzzy numbers, Gupta and Kumar [25] propose a linear ranking function method via signed distance among interval-valued fuzzy numbers and obtain the non-dominated solution of the transformed crisp linear programming model. As a general form of fuzzy number linear programming, Farhadinia [26] introduces a formulation of interval-valued trapezoidal fuzzy number linear programming problems and presents its primal simplex algorithm in fuzzy sense via signed distance ranking function.

The description of incomplete and vague information has received more and more attentions recently. Interval-valued trapezoidal fuzzy numbers [27, 28, 29] are introduced to express the attributes and applied to solve the actual multi-attribute group decision making problems. As the complexity of information in the real world is increasing, interval-valued fuzzy numbers, which have many advantages in decision making and multi-objective programming fields, still have its limits to process the vague information. In this paper, we offer the notion of interval-typed triangular fuzzy numbers, which could be regarded as an extension of interval-valued triangular fuzzy numbers and interval numbers respectively. Moreover, interval-typed triangular fuzzy numbers may be not interval-valued triangular fuzzy numbers. Like traditional fuzzy sets, Interval-typed triangular fuzzy numbers could also be exploited to extend many practical applications in reality [30]. Considering the fact that both interval-valued fuzzy numbers and interval numbers are more general and better to express incomplete and vague information [27, 28, 29], we try to propose an effective algorithm of multi-objective linear programming in which the coefficients of objective functions and constraints are stated as interval-typed triangular fuzzy numbers.

The rest of this paper is organized as follows. In section 2, we review some basic definitions related to interval-typed triangular fuzzy numbers and several useful operators. In section 3, we give multi-objective linear programming (ITF-MOLP) on the basis of interval-typed triangular fuzzy numbers and discuss some interesting properties. Particularly, the decomposition theorem about an interval-typed triangular fuzzy number is presented. In section 4, we establish an algorithm of the ITF-MOLP. By introducing the cut set of interval-typed triangular fuzzy numbers, the objective function is transformed into the maximization of the sum of membership degrees of each objective, where the membership degree of each objective is given based on the deviation from optimal solutions of individual objective. Utilizing the dominance possibility criterion, the comparison between two interval-typed triangular fuzzy numbers in constraints is transformed to normal inequalities. Accordingly, the ITF-MOLP is finally converted to a linear programming that could be solved by existing methods. Section 5 gives some illustrated examples. Moreover, the comparisons with existing approaches are made and the sensitive analysis concerning the optimal solution of linear programming is also investigated. Section 6 concludes the results and points out further research.

2 Preliminaries

Suppose that 𝓡 is the set of all real numbers and 𝓡⁺ is the set of all positive real numbers. Some basic concepts used in this paper are given in this section.

The level (h^L, h^U)-interval-valued trapezoidal fuzzy number is given as follows.

As for the definitions of interval-valued fuzzy numbers, one could also consult the references [27, 28, 29]. In this paper, we introduce the definition of interval-typed triangular fuzzy numbers as follows.

3 ITF-MOLP problem

As is well known, in classical transportation problem we always consider minimizing the costs of transporting several products. However, the complexity of the social environment in most real world problems requires the explicit consideration of objective functions other than cost [25]. Moreover, these objectives are frequently in conflict, measured in different scales and difficult to combine in one overall utility function. For instance, in real transportation problem the total transportation and implementation cost, the environmental impact and the distribution time need to be minimized respectively while the average delivery rate requires to be maximized.

Solving fuzzy linear programming problems, whose parameters in objects and constraints are considered as interval-valued fuzzy numbers meanwhile the decision variables are assumed to nonnegative crisp values, has received increasingly attention in recent years [25, 26, 33]. In this section, we consider a multi-objective linear programming problem (ITF-MOLP) whose all parameters except crisp decision variables are taken as interval-typed triangular fuzzy numbers. The ITF-MOLP problem could be considered as an extension of the multi-objective linear programming [25] whose parameters are assumed to be a special normal interval-valued triangular fuzzy numbers.

Next, we investigate an ITF-MOLP problem. An ITF-MOLP problem could be stated as follows:

s.t.

Where x_j ∈ 𝓡 denotes decision variable (j = 1, 2, ⋯, n) and parameters r̃_lj, ã_ij and b̃_i are all interval-typed triangular fuzzy numbers. Denote r̃_lj = [r̃_lj1, r̃_lj2], ã_ij = [ã_ij1, ã_ij2], b̃_i = [b̃_i1, b̃_i2], φ̃_tj = [φ̃_tj1, φ̃_tj2], d̃_t = [d̃_t1, d̃_t2], in which r̃_lj1, r̃_lj2, ã_ij1, ã_ij2, b̃_i1, b̃_i2, φ̃_tj1, φ̃_tj2, d̃_t1 and d̃_t2 are all triangular fuzzy numbers, l = 1, ⋯, k; i = 1, 2, ⋯, m; j = 1, 2, ⋯, n; t = 1, 2, ⋯, q.

From the viewpoint of linguistic model, maximizing or minimizing a certain objective f̃_l means the maximization or minimization of the objective in fuzzy environment. Because of the existence of interval-typed triangular fuzzy numbers, the ITF-MOLP problem is not well-defined. That is, the meaning of maximizing or minimizing f̃_l, (l = 1, 2, ⋯, k) is not clear and the constraints do not define a deterministic feasible set. Particularly, if the coefficients of multi-objective functions and constraints are interval numbers, then the ITF-MOLP degenerates to a multi-objective linear programming with interval coefficients [21].

To deal with the maximization or minimization of the multi-objectives and compare with interval-valued fuzzy numbers, many authors [25, 26] introduce a ranking method to compute the signed distance from interval-valued fuzzy number to y-axis as follows. Thus the multi-objective problem would be convenient for computation.

Noting that I(𝓡) = {[a^L, a^R]|a^L, a^R ∈ 𝓡, a^L ≤ a^R} denote the set of all closed interval numbers on 𝓡.

Indeed, an interval number [a^L, a^R] ∈ I(𝓡) is a special fuzzy number, whose membership degree could be stated as follows:

The center, a^C of interval number [a^L, a^R] is defined as follows:

Obviously, each one of a^C, a^L and a^R can be determined by two other scalars in Equation (19).

By Definition 3.2, the following properties are easily obtained.

The λ-cut cet of an interval-typed triangular fuzzy number [A^L, A^U] is defined as [(A^L)_λ, (A^U)_λ].

The decomposition theorem [34] of a fuzzy set, which characterizes the relationship among all the cut sets of a fuzzy set and the given fuzzy set, has been developed as follows:

4 An algorithm of ITF-MOLP

The algorithm of ITF-MOLP given in formulas (9 – 12) is a multi-stage procedure.

1. The ITF-MOLP problem is decomposed to a set of k linear programming problems based on interval-typed triangular fuzzy numbers. Each problem optimizes one of the k objective functions associated with constraints set. For a given objective l(l = 1, ⋯, k), this problem is given as follows:

   max(min)f~l=∑j=1nr~ljxj (26)

   s.t.

   ∑j=1na~ijxj⪯b~i,i=1,2,⋯,m∑j=1nφ~tjxj⪰d~t,t=1,2,⋯,qxj≥0,j=1,2,⋯,n. (27)

   Where r̃_lj, ã_ij, b̃_i, φ̃_tj and d̃_t are all interval-typed triangular fuzzy numbers. Denote r̃_lj = [r̃_lj1, r̃_lj2], ã_ij = [ã_ij1, ã_ij2], b̃_i = [b̃_i1, b̃_i2], φ̃_tj = [φ̃_tj1, φ̃_tj2], d̃_t = [d̃_t1, d̃_t2], in which r̃_lj1, r̃_lj2, ã_ij1, ã_ij2, b̃_i1, b̃_i2, φ̃_tj1, φ̃_tj2, d̃_t1 and d̃_t2 are all triangular fuzzy numbers, i = 1, 2, ⋯, m; j = 1, 2, ⋯, n; t = 1, 2, ⋯, q.

   Then the objective function f̃_l could be written as follows:

   f~l=[∑j=1nr~lj1xj,∑j=1nr~lj2xj] (28)

   For a given λ-level, λ ∈ (0, 1], the λ-cut set of the objective function f̃_l is obtained by Propositions 3.3 and 3.4 as follows:

   (f~l)λ=[∑j=1n(r~lj1)λxj,∑j=1n(r~lj2)λxj] (29)

   Note that (r~lj1)λ=[(r~lj1)λL,(r~lj1)λR] and (r~lj2)λ=[(r~lj2)λL,(r~lj2)λR] in Equation (29). It follows from Definition 3.1 and Proposition 3.1 that

   (f~l)λ=[∑j=1n(r~lj1)λLxj,∑j=1n(r~lj1)λRxj],[∑j=1n(r~lj2)λLxj,∑j=1n(r~lj2)λRxj] (30)

   If the original objective is to maximize f̃_l, l ∈ {1, 2, ⋯, k}, then a decision maker with pessimistic and conservative attitude would require the maximization of the lower and center of “interval set” (f̃_l)_λ. From Definition 3.2 and Proposition 3.4, the solution of model (26) could be presented as the optimal solution of the following bi-objective problem based on a given λ–level:

   max((f~l)λL,(f~l)λC) (31)

   where the lower bound of (f̃_l)_λ, denoted by (f~l)λL, is given as,

   (f~l)λL=ω1∑j=1n(r~lj1)λLxj+ω2∑j=1n(r~lj2)λLxj, (32)

   the center of (f̃_l)_λ, denoted by (f~l)λC, is given as,

   (f~l)λC=ω1∑j=1n(r~lj1)λCxj+ω2∑j=1n(r~lj2)λCxj. (33)

   And the weight ω_i satisfies the conditions:

   ω1+ω2=1,ωi≥0,i=1,2. (34)

   Remark

   Note that (f̃_l)_λ is not a classical interval set. Thus the lower bound of (f̃_l)_λ considers the lower bounds of interval sets both [∑j=1n(r~lj1)λLxj,∑j=1n(r~lj1)λRxj]  and  [∑j=1n(r~lj2)λLxj,∑j=1n(r~lj2)λRxj] , and the center of (f̃_l)_λ considers the center points of interval sets [∑j=1n(r~lj1)λLxj,∑j=1n(r~lj1)λRxj]  and  [∑j=1n(r~lj2)λLxj,∑j=1n(r~lj2)λRxj] . Moreover, parameters ω₁ and ω₂ represent the relative importance weight of [∑j=1n(r~lj1)λLxj,∑j=1n(r~lj1)λRxj] and [∑j=1n(r~lj2)λLxj,∑j=1n(r~lj2)λRxj] , respectively. Particularly, an extremely conservative decision maker would select parameters ω₁ = 1 and ω₂ = 0.

   If the original objective is to minimize f̃_l, l ∈ {1, 2, ⋯, k}, then the solution of model (26) could be given as the optimal solution of the following bi-objective problem:

   min((f~l)λR,(f~l)λC) (35)

   Where

   (f~l)λC=ω1∑j=1n(r~lj1)λCxj+ω2∑j=1n(r~lj2)λCxj, (36)

   (f~l)λR=ω1∑j=1n(r~lj1)λRxj+ω2∑j=1n(r~lj2)λRxj, (37)

   and both ω₁ and ω₂ are satisfied to Equation (34).

   The comparison between fuzzy numbers can be carried out using dominance possibility criterion (DPC) or strong dominance possibility criterion (SDPC) [35, 36]. In this paper, we adopt dominance possibility criterion (DPC) to compare the relationship between two triangular fuzzy numbers. If both fuzzy numbers ã = (a, a₀, a) and b̃ = (b, b₀, b) are triangular fuzzy numbers, then

   Poss(a~⪰Db~)=Poss(b~⪯Da~)=1,ifa0≥b0a¯−b_(a¯−a0)+(b0−b_),ifb0≥a0,a¯≥b_0,ifb_≥a¯ (38)

   Poss(ã ⪰ b̃) means the possibility that the maximum value of ã is greater than or equal to the minimum value of b̃ [36].

   For μ ∈ (0, 1],

   Poss(a~⪰Db~)≥μ

   is equivalent to the inequalities below:

   a¯≥b_(1−μ)a¯+μa0≥(1−μ)b_+μb0. (39)

   Note that ∑j=1na~ijxj⪯Db~i in (27) is well-defined. It follows that

   ∑j=1na~ijxj=[∑j=1na~ij1xj,∑j=1na~ij2xj].

   Let

   b~i=[b~i1,b~i2].

   Since ∑j=1na~ij1xj,∑j=1na~ij2xj , b̃_i1, and b̃_i2 are all triangular fuzzy numbers, for a given tolerance measure μ, μ ∈ (0, 1], the constraint

   ∑j=1na~ijxj⪯Db~i (40)

   could be transformed into the following conditions:

   Poss(∑j=1na~ij2xj⪯Db~i2)≥μPoss(∑j=1na~ij1+a~ij22xj⪯Db~i1+b~i22)≥μ (41)

   Here, Poss(∑j=1na~ij2xj⪯Db~i2) represents the possibility of the proposition that the upper bound ∑j=1na~ij2xj of ∑j=1na~ijxj is no larger than the upper bound b̃_i2 of b̃_i under the dominance possibility criterion. Moreover, Poss(∑j=1na~ij1+a~ij22xj⪯Db~i1+b~i22) means the possibility of the proposition that the center ∑j=1na~ij1+a~ij22xj  of  ∑j=1na~ijxj is less than or equal to the center b~i1+b~i22 of b̃_i under the DPC. In this way, we can appropriately characterize the linguistic term of the inequality (27).

   And in the same way the constraint

   ∑j=1nφ~tjxj⪰d~t (42)

   could be transformed into the following conditions:

   Poss(∑j=1nφ~tj1xj⪰Dd~t1)≥μPoss(∑j=1nφ~tj1+φ~tj22xj⪰Dd~t1+d~t22)≥μ (43)

   Finally, by solving the problem (26) based on Equations (26-43) for each objective function, the range of optimal objective functions are determined as

   (f~l)λ∗∈[(f~l)λ∗L,(f~l)λ∗R],

   where (f~l)λ∗R=(f~l)λ∗L+2((f~l)λ∗C−(f~l)λ∗L) .

2. Consider the lth objective again. If the objective is a maximization type, its membership function could be defined as follows:

   μ~l(x)=1,if(f~l(x))λL∨(f~l(x))λC≥(f~l)λ∗R(f~l(x))λL∨(f~l(x))λC−(f~l)λ∗L(f~l)λ∗R−(f~l)λ∗L,if(f~l)λL∨(f~l(x))λC≤(f~l)λ∗R (44)

   In the same way, for a minimization type objective, the membership function can be given as follows:

   μ~l(x)=1,if(f~l(x))λR∧(f~l(x))λC≤(f~l)λ∗L(f~l)λ∗R−(f~l(x))λR∧(f~l(x))λC(f~l)λ∗R−(f~l)λ∗L,if(f~l)λ∗L≤(f~l(x))λR∧(f~l(x))λC (45)

   Lemma 4.1

   From Equations (44) and (45) it always holds that μ̃_l(x) ≤ 1.

   Proof

   Suppose that μ̃_l(x) > 1. Then based on Equation (44), it follows that

   μ~l(x)>1⇒(f~l(x))λL∨(f~l(x))λC−(f~l)λ∗L(f~l)λ∗R−(f~l)λ∗L>1

   1. (f~l(x))λL∨(f~l(x))λC−(f~l)λ∗L>(f~l)λ∗R−(f~l)λ∗L

   2. (f~l(x))λL∨(f~l(x))λC>(f~l)λ∗R

   3. (f~l(x))λL>(f~l)λ∗R  or(f~l(x))λC>(f~l)λ∗R

   4. (f~l(x))λR>(f~l)λ∗R .

   This contradicts with the optimality of (f~l)λ∗R .

   Based on Equation (45), it follows that

   μ~l(x)>1⇒(f~l)λ∗R−(f~l(x))λR∧(f~l(x))λC(f~l)λ∗R−(f~l)λ∗L>1

   1. (f~l)λ∗R−(f~l(x))λR∧(f~l(x))λC>(f~l)λ∗R−(f~l)λ∗L

   2. (f~l(x))λR∧(f~l(x))λC<(f~l)λ∗L

   3. (f~l(x))λR<(f~l)λ∗Lor(f~l(x))λC<(f~l)λ∗L

   4. (f~l(x))λL<(f~l)λ∗L .

   This contradicts with the optimality of (f~l)λ∗L .

   It completes the proof.□

3. For a given λ-level, λ ∈ (0, 1], the problem (9-12) is transformed into the linear programming based on interval-typed fuzzy numbers as follows:

   max{μ~1(x),μ~2(x),⋯,μ~k(x)} (46)

   s.t.

   μ~l(x)≤1,l=1,2,⋯,k∑j=1na~ijxj⪯b~i,i=1,2,⋯,m∑j=1nφ~tjxj⪰d~t,t=1,2,⋯,qxj≥0,j=1,2,⋯,n. (47)

4. For a given λ-level, λ ∈ [0, 1], the problem (46) is transformed into the form below:

   max∑l=1kμ~l(x) (48)

   s.t.

   μ~l(x)≤1,l=1,2,⋯,k∑j=1na~ijxj⪯b~i,i=1,2,⋯,m∑j=1nφ~tjxj⪰d~t,t=1,2,⋯,qxj≥0,j=1,2,⋯,n. (49)

5. For a given λ-level and μ-tolerance measure, λ, μ ∈ (0, 1], the problem (48) is transformed into the single objective programming as follows:

   max∑l=1kμ~l(x) (50)

   s.t.

   μ~l(x)≤1,l=1,2,⋯,kPoss(∑j=1na~ij2xj⪯Db~i2)≥μPoss(∑j=1na~ij1+a~ij22xj⪯Db~i1+b~i22)≥μPoss(∑j=1nφ~tj1xj⪰Dd~t1)≥μPoss(∑j=1nφ~tj1+φ~tj22xj⪰Dd~t1+d~t22)≥μxj≥0,j=1,2,⋯,n. (51)