ON GENERATING FUZZY PARETO SOLUTIONS IN FULLY FUZZY MULTIOBJECTIVE LINEAR PROGRAMMING VIA A COMPROMISE METHOD

. In the present paper, it is unified and extended recent contributions on fully fuzzy multi-objective linear programming, and it is proposed a new method for obtaining fuzzy Pareto solutions of a fully fuzzy multiobjective linear programming problem. For its formulation, triangular fuzzy numbers and variables are combined with fuzzy partial orders and fuzzy arithmetic, and no ranking functions are required. By means of solving related crisp multiobjective linear problems, it is provided algorithms to generate fuzzy Pareto solutions; in particular, to generate compromise fuzzy Pareto solutions, what is a novelty in this field

(FFLP) problems with inequality constraints, with triangular fuzzy numbers and not necessarily symmetric, via solving a crisp multiobjective linear programming problem.This method does not require ranking functions, and has been extended to linear programming problem with parameterized fuzzy numbers by Arana-Jiménez and Sánchez-Gil [8].Reader can find very recent applications and extensions to solve fully fuzzy minimax mixed integer linear programming and maximal covering location problems in Arana-Jiménez et al. [6,10].
As an extension of that commented above, some models require decision maker to address not only one objective, but several objectives at the same time.That is, a model with two or more objectives, which have to be optimized, with conflicts among the objectives, what derives a multiobjectve programming problem.The Pareto optimality in multiobjective programming is a well known concept of a solution to this type of problem, with important applications in optimal control, economics, engineering, decision theory, among others (see [3,7]).
Recently, in a conference paper, Arana-Jiménez [5] has advanced a natural extension of such model to fully fuzzy multiobjective linear programming, with the introduction of the fuzzy Pareto solutions.To this matter, Author has proposed a method to generate fuzzy Pareto solutions by means of related crisp multiobjective programming problems.The proposal does not require ranking functions, and then is different from that given by Bharati et al. [13], who proposed the concept of Pareto-optimal solution suggested by Jimenez and Bilbao [27].Some applications can be found in Data Envelopment Analysis by Mehlawat et al. [38].In that conference paper [5], Author comments that proofs and examples are omitted and will be presented in a paper (extended version).In this work, and as an extension of the proposals advanced by Arana-Jiménez [5], we address the challenge of studying a linear optimization model where all variables and data can be fuzzy numbers, that is, a fully fuzzy multiobjective linear programming problem ((FFMLP), for short).To this matter, and with no ranking functions, we proof results that derive a method to to get a set of fuzzy Pareto solutions.Furthermore, since the decision maker can require a very reduced set of fuzzy Pareto solutions, even only one in some cases, we propose a new method based on a compromise method, as well as corresponding algorithm to get such fuzzy Pareto solution.
The structure is as follows.In next section, we present notations, arithmetic and partial orders on fuzzy numbers.Later, in Section 3, we formulate the fully fuzzy multiobjective linear programming problem, and relate its fuzzy Pareto solutions to Pareto solutions of auxiliary crisp multiobjective programming problems, as advanced in Arana-Jiménez [5].Then, and based on the previous relations, in Section 4, we provide algorithms to generate fuzzy Pareto solutions; in particular, an algorithm to attain a compromise fuzzy Pareto solution for (FFMLP).To illustrate this latter, in Section 5 we present a numerical application.Finally, we conclude the paper and present future works.

Preliminaries on arithmetic and partial order on fuzzy numbers
As usual in the literature, we consider a fuzzy set on R  as a mapping  : R  → [0, 1].Each fuzzy set  has associated a family of -level sets, which are described as []  = { ∈ R  | () ≥ } for any  ∈ (0, 1], and its support as () = { ∈ R  | () > 0}.The 0-level of  is defined as the closure of (), that is, [] 0 = (()).A very useful type of fuzzy set to model parameters and variables are the fuzzy numbers.Following Dubois and Prade [18,19], a fuzzy set  on R is said to be a fuzzy number if  is normal, this is there exists  0 ∈ R such that ( 0 ) = 1, upper semi-continuous function, convex, and (iv) [] 0 is compact.F  denotes the family of all fuzzy numbers.The -levels of a fuzzy number can be represented by means of real interval, that is, []  = [  ,   ] ∈ K  ,   ,   ∈ R, with K  is the set of real compact intervals.There exist many families of fuzzy numbers that have been applied to model uncertainty in different situations.some of the most popular are the L-R, triangular, trapezoidal, polygonal, gaussian, quasi-quadric, exponential, and singleton fuzzy numbers.The reader is referred to Báez-Sánchez et al. [11], Hanss [26] and Stefanini et al. [42] for a complete description of these families and their representation properties.Among them, we point out triangular fuzzy numbers, because of their easy modeling and interpretation (see, for instance, [18,28,29,34,42]), and whose definition is as follows.
Definition 1.Given a fuzzy number ã = ( − , â,  + ) whose membership function is then, it is said to be a triangular fuzzy number (TFN for short).
In terms of -levels, if we consider a triangular fuzzy number ã = ( − , â,  + ), then its -levels are as follows: for all  ∈ [0, 1].This means that triangular fuzzy number are well determined by three real numbers  − ≤ â ≤  + .A unique triangular fuzzy number is characterized by means of the previous formulation of -levels, such as Goestschel and Voxman [24] established.The set of all TFNs is denoted as T F .Many optimization problems requires conditions about the nonpositivity or nonnegativity on some parameters and variables involeved.To this matter, a fuzzy number ã is said to be nonnegative or nonpositive if ã0 ≥ 0 or ã0 ≤ 0, repectively.So, in the case that ã is a TFN, then ã nonnegative (nonpositive, respectively) if and only if  − ≥ 0 ( + ≤ 0, respectively).
Classical arithmetic operations on intervals are well known, and can be referred to Moore [39,40] and Alefeld and Herzberger [1].A natural extension of these arithmetic operations to fuzzy numbers ,  ∈ F  can be found described in Liu [32] and Ghaznavi et al. [23], where the membership function of the operation  * , with * ∈ {+, •}, is defined by Furthermore, the previous arithmetic operations can be provided by means of their -levels as follows (see, [23], Thm.2.6).For any  ∈ [0, 1]: []  = [min{  ,   }, max{  ,   }], T F is closed under addition and multiplication by scalar.The above operations (2) and (3) are straightforward particularized to triangular fuzzy number as follows.Given ã = ( − , â,  + ), b = ( − , b,  + ) ∈ T F and  ∈ R, then However, T F is not closed under the multiplication operation (4) (see, for instance, the examples in [2]).To avoid this situation, it is usual to apply a different multiplication operation between TFNs, such as those referenced in Kaufmann and Gupta [28], Kumar et al. [31], Khan et al. [29] and Arana-Jiménez [4], which can be considered as an approximation to the multiplication given in (1).To this regard, in Arana-Jiménez [4] readers can find a discussion.Then, we provide the following multiplication operation, which is used throughout the text: In the case that ã or b is a nonnegative TFN, then the previous multiplication is reduced (see, for instance, [28,31]).For instance, if b is nonnegative, then And if ã and b are nonnegative, then ãb = (︁ To compare two fuzzy numbers, there exist several definitions based on interval binary relations (see e.g., [25]) which provides partial orders in fuzzy sets (see, e.g., [43,45]).
In a minimization process, and through the paper, we refer ( ≺ ) ⪯ −  as a fuzzy number  is (strictly) dominated by a fuzzy number , or equivalently,  (strictly) dominates .In a similar way, we define ≻, ⪰ − .In case of TFNs, the previous definition can be really reduced, as recently Arana-Jiménez and Blanco [6] have proved: The relations ≻, ⪰ − are obtained in a similar manner.Note that to say that ã is nonnegative (previously defined) is equivalent to write ã ⪰ − 0 = (0, 0, 0).

Fully fuzzy multiobjective linear problem
Consider a fuzzy vector z = (z 1 , . . ., z ) For the sake of simplicity, we write z = (z  )  =1 .In a same manner, , and so on.Following Arana-Jiménez [5], let us present a formulation of a Fully Fuzzy Multiobjective Linear Problem, as well as a concept for its solutions.
where z is the fuzzy-valued vector objective function, each c = (c 1 , . . ., c ) ∈ (T F )  is the fuzzy vector with the coefficients of the th component of the fuzzy-valued vector function, x = (x 1 , . . ., x ) is the fuzzy vector with the fuzzy decision variables, and ã and b are the fuzzy technical coefficients.Since we deal with (FFMLP) without any kind of ranking function, it is necessary to define a nondominated fuzzy solution concept, as follows.

A proposal to generate fuzzy Pareto solutions for (FFMLP)
In the literature, we can find several methods to generate Pareto solutions of a multiobjective linear problem (see [3] and the bibliography therein).Most popular methods are based on scalarization, such as the weighted problems.These usually produce a set of solutions, whose extension can be partially controlled by the election of a set of weights.The compromise methods allow us to reduce and orient the election of Pareto solutions.

Fuzzy Pareto solutions via weighted problems
The formulation of a weighted problem can be as follows.Given (CMLP) and  = ( 11 ,   14)- (19).
In Arana-Jiménez [5], it was advanced that we can generate a set of fuzzy Pareto solutions of (FFMLP) by means of optimal solutions of the previous weighted problems.Such result was presented, but no proof was provided.Thus, following, we write the result with a proof.

Proof. Let us consider the minimization case in both problems
is an optimal solution of the weighted optimization problem (CMLP)  , then, it follows that  is a Pareto solution of (CMLP) (see [3], for instance).Then, by Theorem 2, we have that x = ( −  , x ,  +  ) ∈ T F ,  = 1, . ., , is a fuzzy Pareto solution of (FFMLP).For the maximization case we proceed similarly, and the proof is complete.
The previous result allows us to outline a method to get fuzzy Pareto solutions for (FFMLP) problem in the minimization (maximization) case of (FFMLP) by means of the solutions of the minimization (maximization) case of the weighed problems (CMLP)  ℎ .Therefore, given  ∈ N and a set of weights   , it is obtained a set  of fuzzy Pareto solutions for (FFMLP) problem.

Compromise fuzzy Pareto solution
By now, and thanks to the previous method, we provide a set  of fuzzy Pareto solutions for (FFMLP) to the decision maker.Now, the decision maker can apply additional criterion to the set  to choose some elements if necessary.In the literature on multiobjective optimization, we find some criterion to select 'the best' objective function value among the nondominated set (usually, the images of the Pareto or weakly Pareto solutions).Thanks to Theorem 2, we can perform a similar method for (FFMLP) by means of (MCLP).In this way, and among the different methods to do this, we can use compromise programming to get a compromise solution [46].In this method, the procedure for obtaining a compromise solution is to minimize a distance between the potential optimal solution and the utopia or ideal score in the criterion space (see also [17,37,46]), where distances are defined on   .Let us recall that the utopia score is obtained by the optimization (minimization or maximization in (MCLP)) of each component of the objective function, which usually is not attained by any feasible point (see [3]).For further information on compromise solution methods, we refer Marler and Arora [37], who offer a survey of methods to compute Pareto solutions in multiobjective optimization.
On the other hand, let us consider the lexicographic weighted Tchebycheff method.It is considered a min-max method, and not a compromise method, such as reader can verify in [37].This method depends on a collection of weights.However, the lexicographic weighted Tchebycheff method provides a modification by Tind and Wiecek [44], in which all weights are equal, eliminates the possibility of non-unique solutions and guarantees a Pareto solution (see [44]).Such as Marler and Arora [37] describe, and in summary, in this particular case the method is as follows.First, calculate the utopia objective function value, and then minimize the  ∞ distance between the non-dominated scores and the utopia score for the multiobjective optimization program.Then, include this result as a new constraint, and minimize  1 distance between the non-dominated scores and the utopia score.In order to not confuse this particular case with the general lexicographic weighted Tchebycheff method, we will refer as Tind-Wiecek lexicographic Tchebycheff method from now on.Observe that this method, in essence, provides a solution that is as close as possible (by means of distances) to the utopia point, what links with the definition of compromise solution.Then, in our opinion, we can refer the solution given by the Tind-Wiecek lexicographic Tchebycheff method as a compromise solution.
The previous steps of the Tind-Wiecek lexicographic Tchebycheff method are determined by means of the following algorithm, depending of the optimization (minimization or maximization) case in (FFMLP), as follows in Algorithms (1 and 2, respectively).
Step 5 End Thus, we refer the computed output by the previous algorithm as a compromise fuzzy Pareto solution for (FFMLP).To prove that such output is really a fuzzy Pareto solution for (FFMLP), we provide the following result.

Numerical application
To illustrate the previous algorithm to compute a compromise fuzzy Pareto solution, let us consider the following fully fuzzy multiobjective programming problem, from another used by Khan et al. [29,30], also by Arana-Jiménez [4], where two fuzzy objective functions have been included.
Steps Outputs from Algorithm 2 (maximization case) Step 1 (  .Then, it is applied Algorithm 2, in the maximization case, and then the results are obtained, step by step, given in Table 3. Note that in Step 1 in Table 3, the obtained vector with utopia scores can be interpreted as two fuzzy utopia scores (2.457143, 19.354839, 36.562500)Note that the two fuzzy utopia scores are less than or equal to (in the fuzzy sense ⪯ − ) the corresponding z1 and z2 .Furthermore, each lower extreme of the two fuzzy utopia scores coincides with the corresponding lower extreme of z1 and z2 , respectively.Then, the compromise fuzzy Pareto solution for the fully fuzzy multiobjective programming problem is given by ( x1 , x2 , x3 ), and the fuzzy value of the fuzzy-valued objective function is ( z1 , z2 ).
The computations have been made in R (see https://www.r-project.org), and using the lpSolve package for solving Linear Programs.

Conclusions
An equivalence between a (FFMLP) problem and a crisp multiobjective lineal programming problem is established, without loss of information and without ranking functions.As results, methods to obtain fuzzy Pareto solutions for (FFMLP) has been provided; in particular, a compromise fuzzy Pareto solution is obtained by an algorithm which considers an adaptation of lexicographic weighted Tchebycheff method.
As future works, the techniques presented will be extended to generate fuzzy Pareto solutions in interval and fuzzy fractional programming with applications to economy, among others, as well as to inverse Data Envelopment Analysis with fuzzy data in inputs and outputs.