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

Abstract A multi-objective linear programming problem (ITF-MOLP) is presented in this paper, in which coefficients of both the objective functions and constraints are interval-typed triangular fuzzy numbers. An algorithm of the ITF-MOLP is provided by introducing the cut set of interval-typed triangular fuzzy numbers and the dominance possibility criterion. In particular, for a given level, the ITF-MOLP is converted to the maximization of the sum of membership degrees of each objective in ITF-MOLP, whose membership degrees are established based on the deviation from optimal solutions of individual objectives, and the constraints are transformed to normal inequalities by utilizing the dominance possibility criterion when compared with two interval-typed triangular fuzzy numbers. Then the equivalent linear programming model is obtained which could be solved by Matlab toolbox. Finally several examples are provided to illuminate the proposed method by comparing with the existing methods and sensitive analysis demonstrates the stability of the optimal solution.


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 elds. 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 coe cients 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 coe cients are considered by many authors [7][8][9]. Wang and Wang [10] have proposed a fuzzy multi-objective linear programming with fuzzy-numbered cost coecients and transformed the problem into a multi-objective problem with parametrically interval-valued cost coe cients by utilizing membership functions. Di erent algorithms [11,12] are developed to solve multiobjective linear programming problems based on interval-valued cost coe cients. 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 coe cients [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 coe cients 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 exible especially in ill-de ned 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 nd 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 multiobjective programming elds, still have its limits to process the vague information. In this paper, we o er the notion of interval-typed triangular fuzzy numbers, which could be regarded as an extension of intervalvalued 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 e ective algorithm of multiobjective linear programming in which the coe cients 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 de nitions 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 nally 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.

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. a triplet (a , a , a ). The membership function µ A (x) is de ned as

De nition 2.1. [31] A triangular fuzzy number A is de ned as
where a , a , a ∈ R and a ≤ a ≤ a ; its membership function µ A (x) is fuzzy convex, showing that the membership degree of element x belonging to A; a represents the value for which µ A (a ) = , and a and a are the most extreme values on the left and on the right of the fuzzy number A respectively with membership µ A (a ) = µ A (a ) = . If a = a = a , then A is reduced to a real number.
De nition 2.2. [30] Let A = (a , a , a ) and B = (b , b , b ) be two triangular fuzzy numbers. Then the operations with these fuzzy numbers are de ned as follows: (v) λA = (λa , λa , λa ) for any positive scalar λ ∈ R.

De nition 2.3. [30]
Let I denote a nite index set, {A i = (a i , a i , a i )|i ∈ I} be a family of triangular fuzzy numbers, and and represent the supremum and in mum operator on the real set R respectively. Then The level (h L , h U )-interval-valued trapezoidal fuzzy number is given as follows.

De nition 2.4. [32] A level
is an interval-valued fuzzy set on R with the lower trapezoidal fuzzy numberÃ L expressing bỹ and the upper trapezoidal fuzzy numberÃ U expressing bỹ where a L ≤ a L ≤ a L ≤ a L , a U ≤ a U ≤ a U ≤ a U , < h L ≤ h U ≤ , a U ≤ a L and a L ≤ a U . Moreover,Ã L ⊆Ã U . Moreover, if a L = a L , a U = a U and h L = h U = , thenÃ is a normal interval-valued triangular fuzzy number.
As for the de nitions of interval-valued fuzzy numbers, one could also consult the references [27][28][29].In this paper, we introduce the de nition of interval-typed triangular fuzzy numbers as follows. Remark. An interval-valued triangular fuzzy number is a special interval-typed triangular fuzzy number, however, an interval-typed triangular fuzzy number may be not an interval-valued triangular fuzzy number.

De nition 2.5. [30] An interval-typed triangular fuzzy number is a fuzzy interval
De nition 2.6. [30] Let [A L , A U ] and [B L , B U ] be two interval-typed triangular fuzzy numbers. Then the operations with them are de ned as follows:

be a collection of interval-typed triangular fuzzy numbers, where I denotes a nite index set. Then
i∈I

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 con ict, measured in di erent scales and di cult 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 intervalvalued triangular fuzzy numbers.
Next, we investigate an ITF-MOLP problem. An ITF-MOLP problem could be stated as follows: . . .
are all triangular fuzzy numbers, l = , · · · , k; i = , , · · · , m; j = , , · · · , n; t = , , · · · , q. From the viewpoint of linguistic model, maximizing or minimizing a certain objectivef 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-de ned. That is, the meaning of maximizing or minimizingf l , (l = , , · · · , k) is not clear and the constraints do not de ne a deterministic feasible set. Particularly, if the coe cients of multi-objective functions and constraints are interval numbers, then the ITF-MOLP degenerates to a multi-objective linear programming with interval coe cients [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. Lemma 3.1. [26] LetÃ be a normal interval-valued fuzzy number. The signed distance of from O(y−axis) is given as: De nition 3.
Noting that 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 de ned as follows: Obviously, each one of a C , a L and a R can be determined by two other scalars in Equation ( ).

De nition 3.2. Let
Then the operations between them are de ned as follows: By De nition . , the following properties are easily obtained.
and p be a scalar. Then the following statements hold.
Then the above equations hold by De nition 3.2.
The λ-level (λ-cut cet) of a triangular fuzzy number A = (a , a , a ) is the interval number de ned by where λ ∈ ( , ]. Proposition 3.2. [34] Let A = (a , a , a ) and B = (b , b , b ) be two triangular fuzzy numbers. Then the following statements are satis ed.
be a triangular fuzzy number, u i ∈ R + , i = , , · · · , n. Then the following statement holds: Proof. It is easily concluded by Proposition . .
The λ-cut cet of an interval-typed triangular fuzzy number

De nition 3.3. Let T denote an index set,
where and represent the supremum and in mum operator on the real set R respectively.
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: where G λ is the λ-cut set of a fuzzy set G. Similarly, the decomposition theorem about an interval-typed triangular fuzzy number could be obtained as follows.

Proposition 3.4. Let [A L , A U ] be an interval-typed triangular fuzzy number. Then
where Proof. Let A L = (a L , a L , a L ) and A U = (a U , a U , a U ) be two triangular fuzzy numbers , a L ≤ a U . By De nition . and Lemma . , we have where (A L ) λ and (A U ) λ are interval numbers. Therefore

An algorithm of ITF-MOLP
The algorithm of ITF-MOLP given in formulas ( − ) is a multi-stage procedure. STEP1. 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 = , · · · , k), this problem is given as follows: Wherer lj ,ã ij ,b i ,φ tj andd t are all interval-typed triangular fuzzy numbers.
andd t are all triangular fuzzy numbers, i = , , · · · , m; j = , , · · · , n; t = , , · · · , q. Then the objective functionf l could be written as follows: For a given λ-level, λ ∈ ( , ], the λ-cut set of the objective functionf l is obtained by Propositions . and . as follows: Note that ( ). It follows from De nition . and Proposition . that If the original objective is to maximizef l , l ∈ { , , · · · , k}, then a decision maker with pessimistic and conservative attitude would require the maximization of the lower and center of "interval set" (f l ) λ . From De nition . and Proposition . , the solution of model ( ) could be presented as the optimal solution of the following bi-objective problem based on a given λ−level: where the lower bound of (f l ) λ , denoted by (f l ) L λ , is given as, the center of (f l ) λ , denoted by (f l ) C λ , is given as, And the weight ω i satis es the conditions: Remark. Note that (f l ) λ is not a classical interval set. Thus the lower bound of (f l ) λ considers the lower bounds of interval sets both parameters ω = and ω = .
If the original objective is to minimizef l , l ∈ { , , · · · , k}, then the solution of model ( ) could be given as the optimal solution of the following bi-objective problem: Where and both ω and ω are satis ed to Equation ( ). 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 a = (a, a , a) andb = (b, b , b) are triangular fuzzy numbers, then Poss(ã b ) means the possibility that the maximum value ofã is greater than or equal to the minimum value ofb [36].
For µ ∈ ( , ], Poss(ã Db ) ≥ µ is equivalent to the inequalities below: Note that could be transformed into the following conditions: Finally, by solving the problem ( ) based on Equations ( − ) for each objective function, the range of optimal objective functions are determined as . STEP2. Consider the lth objective again. If the objective is a maximization type, its membership function could be de ned as follows: In the same way, for a minimization type objective, the membership function can be given as follows: This contradicts with the optimality of (f l ) *R λ . Based on Equation ( ), it follows that This contradicts with the optimality of (f l ) *L λ . It completes the proof.
(49) STEP5. For a given λ-level and µ-tolerance measure, λ, µ ∈ ( , ], the problem ( ) is transformed into the single objective programming as follows: Consider the problem 1. This problem is further transformed into the following model: By ( ), the model above can be equivalently transformed into the following form: , ω = ω = . , then the above model is stated as follows: x ≥ Problem 2 is further transformed into the following model: . Similarly, Problem can be solved and the optimal solution is x * = ( . , ). Therefore, (f ) *L λ = . , (f ) *C λ = . and (f ) *R λ = . .

Ch. Li
Using model ( ), the problem above is transformed into the following problem: The optimal solution of the above linear programming is obtained as follows: x * = ( . , ). And µ (x * ) + µ (x * ) = .
If we adopt di erent λ-levels of ITF-MOLP and µ-tolerance measure, the optimal solutions of the given problem are easily obtained by the algorithm proposed in this paper.
Example . [37] In a competitive business environment, a company produces two products I and II. Product I is manufactured approximately 1 hour by both machines A and B. Product II is manufactured approximately 2 hours and 1 hours by machine A and B, respectively. Subject to many factors such as machine breakdown, waiting for material, bottleneck, the available time of machine A is approximately 4 hours and 2 hours for machine B. In addition, product I is needed to be mixed after processing on both machines A and B. The estimated mixing time for product I is 2 hours. The available time for mixing is approximately 3 hours. The prices for product I and II are 2 and 1 dollar(s) per kilogram, respectively. The management of the company wants to determine how much to produce for each product to maximize the total revenue. Let x be the amount of product I to be produced and x the amount of product II to be produced. Therefore, we have the following linear programming problem: Van Hop [37] assumes that all parameters are in form of normal-symmetric-triangular fuzzy numbers with the left and right spreads equal to 0.5. In this regards, the above parameters can be represented as the following Van Hop [37] has solved the above problem and obtained the optimal solution X H* = (x H* , x H* ) = ( . , . ). Applying the signed distance ranking function to the above problem together with introducing the slack variables, Farhadinia [26] has given the optimal solution X F* = (x * , x * ) = ( . , . ).
Let parameter ω = ω = . . The levels λ and µ are very important parameters in the proposed models. It is necessary to know variation in the range of the optimal solution with the change of the parameters. Utilizing our proposed method, we obtain di erent optimal solutions of Example 2 with di erent values of λ and µ, which is shown in Table 1. The candidate values of levels λ and µ are respectively selected in the ranges [ . , . ] and [ . , ] based on the decision makers' opinion. Moreover, we investigate the sensitive analysis with di erent levels λ and µ as shown in Table 2.  Tables 1 and 2 could be explained as follows. Firstly, it can be seen from Table 1 that the optimal solution X * = (x * , x * ) = ( . , . ) when the levels λ and µ are set as . and respectively. The optimal solution accords with the optimal solution given by Farhadinia [26]. Secondly, the optimal solution values would vary under di erent levels of λ and µ, which allows decision makers' objective attitude in the process of solving the linear programming problem. These results could help decision makers identify desired schemes. Finally, Table 2 tells us that for a small perturbation of the levels λ and µ, the variation of the optimal solution is very small. For example, when λ = . and µ = . , the optimal solution in Example 2 is X * = ( . , .

The results shown in
). Meanwhile, the optimal solution is X * = ( . , . ) as λ = . + − and µ = . + − , and the corresponding optimal solution is X * = ( . , . ) while λ = . − − and µ = . − − . Therefore, the perturbation of the levels of λ and µ would not lead to unstable tendency of the optimal solution, which shows the robustnes of the results.
Note that the level of constraint violation (or satisfaction) provides one tool for comparing optimal solutions. If we investigate the tightly of constraints of the above linear programming in optimal solutions. For example, we select X * = ( . , . ), X * = ( . , .
) and X * = ( . , . ) with di erent levels in our proposed algorithm, take X F* = ( . , . ) given by Farhadinia [26] and discuss X H* = ( . , . ) obtained by Van Hop [37]. The results are summarized in Table 3. As pointed out by Farhadinia [26], it is shown in Table 3 that the feasibility level of constraint violation in X * and X F* are intuitionally more controllable than that in Van Hop's solution X H* . In addition, the feasibility level of constraint violation in X * and X * are also better than Van Hop's optimal solution. From the obtained results, it is clear that our proposed algorithm could gives better solution than Farhadinia and Van Hop's methods.
Example . Consider the following linear programming: Where the values of parameters are given as follows, Utilizing our proposed method, the linear programming problem could be converted into the following programming: (57) Applying the signed distance ranking function together with the slack variables x and x , Farhadinia [26] transforms the above problem into the following problem: MinZ =c x +c x +˜ x +˜ x x + x = .
Then by Matlab software, we could obtain the optimal solutions with di erent parameters ω , ω , λ and µ, which is shown in Table 4. With the increase of λ and µ, there is a decreasing trend for the optimal solution. Particularly, when levels λ = . and µ = , the optimal solution is X * = ( . , . ). This approximately agrees with the optimal solution X F* = ( . , . ) obtained by Farhadinia's signed distance method [26]. Table 5 presents the sensitive analysis with di erent levels of the parameters. It is shown that the optimal solution is stable, whose variety may be ignored with the perturbation of the levels of λ and µ.

Conclusions
In this paper, we consider multi-objective linear programming problems on the basis of interval-typed triangular fuzzy numbers. An algorithm of the proposed problem is established. By introducing the λ−cut set of interval-typed triangular fuzzy numbers and the dominance possibility criterion to compare with two interval-typed fuzzy numbers, the problem is equivalently transformed into maximization of the sum of membership degrees of each original objective when λ level is given in advance. These membership degrees are obtained based on the deviation from optimal solutions of individual objectives, and the constraints are transformed to classical inequalities by utilizing the dominance possibility criterion. The linear programming model could be solved by Matlab toolbox. It could be noted that the proposed algorithm for a problem constituting k objectives turns out solving ( k + ) linear programming problems. Three illustrated examples are given to validate the feasibility of the proposed algorithm.
Further studies will explore generalized multi-objective linear programming based on interval-typed fuzzy numbers, in which the coe cients of multi-objective functions and constraints may be stated as di erent typed fuzzy numbers such as interval-typed trapezoidal fuzzy numbers and interval-valued fuzzy numbers. It would be interesting to consider di erent tolerance measures for each constraint. Also, in order to more e ciently cope with optimal decision making and optimal control in uncertain systems, we will develop the application of the proposed multi-objective linear programming models.