Lexicographic Methods for Fuzzy Linear Programming

: Fuzzy Linear Programming (FLP) has addressed the increasing complexity of real-world decision-making problems that arise in uncertain and ever-changing environments since its introduction in the 1970s. Built upon the Fuzzy Sets theory and classical Linear Programming (LP) theory, FLP encompasses an extensive area of theoretical research and algorithmic development. Unlike classical LP, there is not a unique model for the FLP problem, since fuzziness can appear in the model components in different ways. Hence, despite ﬁfty years of research, new formulations of FLP problems and solution methods are still being proposed. Among the existing formulations, those using fuzzy numbers (FNs) as parameters and/or decision variables for handling inexactness and vagueness in data have experienced a remarkable development in recent years. Here, a long-standing issue has been how to deal with FN-valued objective functions and with constraints whose left- and right-hand sides are FNs. The main objective of this paper is to present an updated review of advances in this particular area. Consequently, the paper brieﬂy examines well-known models and methods for FLP, and expands on methods for fuzzy single- and multi-objective LP that use lexicographic criteria for ranking FNs. A lexicographic approach to the fuzzy linear assignment (FLA) problem is discussed in detail due to the theoretical and practical relevance. For this case, computer codes are provided that can be used to reproduce results presented in the paper and for practical applications. The paper demonstrates that FLP that is focused on lexicographic methods is an active area with promising research lines and practical implications.


Introduction
Bellman and Zadeh's [1] celebrated work on fuzzy decision-making gave birth to Fuzzy Mathematical Programming. Particularly, Fuzzy Linear Programming (FLP) became one of the most studied and developed fields in the Fuzzy Sets and Systems area. FLP combines classical Linear Programming (LP) theory (a fundamental tool for optimization) and Fuzzy Sets theory (a rigorous mathematical theory for modeling the imprecision of ill-defined concepts, such as those that arise from natural language). FLP makes it possible to take into account the inaccuracies of human thinking within optimization processes and, thus, expands the application of classical LP. The influence of the context in the decision-making process has been recently studied in [2] with special emphasis on FLP.
The concept of a fuzzy set plays a pivotal role in Bellman and Zadeh's [1] theory. Zadeh [3] defined fuzzy sets as classes of objects with a continuum of grades of membership, and extended the classical notions of union, intersection, inclusion, etc. to these sets. Formally, a fuzzy setÃ in a space of points X is characterized by a membership function µÃ, which associates with each point x ∈ X a real number in the interval [0, 1]; the value of µÃ x at x is termed grade of membership of x inÃ, and the nearer this value to unity, the higher the grade of membership of x inÃ.
Bellman and Zadeh [1] viewed decisions as the confluence of goals and constraints. How this confluence is modeled in a fuzzy sense is still a topic of research. However, the interpretation of confluence as an intersection of fuzzy sets representing goals and constraints by means of the min operator has become traditional (The intersection of fuzzy sets can be more generally defined via t-norms [4].). Thus, according to this interpretation, given a fuzzy goalG and a fuzzy constraintC in a space of alternatives X,G andC are combined to form a fuzzy decision setD =G ∩C with membership function µD x = min µG x , µC x for x ∈ X. The optimal decision is taken as any alternative that maximizes µD; hence, µD x * = max x∈X µD x . This idea was adopted by Zimmermann [5] and Tanaka et al. [6], who formally initiated and developed FLP.
Taking as a starting point the classical formulation of the LP problem (1), we are faced with a situation in which the goal of the decision-maker and the problem constraints are better expressed as fuzzy sets.
Zimmermann [5] proposed problem (2) as a fuzzified version of (1), in which denotes fuzzy inequality and has the linguistic interpretation "essentially smaller than or equal to" and z 0 is an aspiration level that is chosen by the decision-maker.
Zimmermann [5] introduced the auxiliary variable α and obtained the equivalent problem max α : µz(x) ≥ α, µã 1 (x) ≥ α, . . . , µã m (x) ≥ α, x j ≥ 0 for j = 1, . . . , n . Tanaka et al. [6] proposed an algorithm to solve a fuzzified version of (1) given by max g(x) : a 1 (x) b 1 , . . . , a m (x) b m , x j ≥ 0 for j = 1, . . . , n , where has the same interpretation as in Zimmermann's [5] method, and g(x) is a membership function obtained by normalizing the objective function z(x); i.e., g( Verdegay [7] generalized the approaches of Tanaka et al. [6] and Zimmermann [5] by demonstrating that their solutions are particular values in the fuzzy solution set obtained through the parametric LP problem (3). Note that, for all values of α ∈ [0, 1], conventional LP problems are obtained, whose corresponding solutions can be integrated into a fuzzy solution via the representation theorem for fuzzy sets. A general view on the theoretical developments obtained from [7] has been presented in [8].
In order to better address the inexactness and vagueness of data when modeling real-world problems, it has been suggested to use fuzzy numbers (FNs) as parameters in FLP problems [9]. This modeling approach raises the issue of comparing FNs that appear as objective function values and constraint right-hand/left-hand side values. In [9], the authors obtained an equivalent non-linear programming formulation and proposed a specific solution algorithm. The use of linear ranking functions to compare FNs and transform the FLP problems into crisp ones has become widespread due to the simplicity of the resulting crisp transformations. For example, in [10], the authors considered FLP problems with FNs in both sides of the constraints, and proposed several auxiliary crisp LP problems that were obtained as a result of applying known linear ranking functions. A fuzzy multi-objective LP problem was addressed in [11], in which each objective function may come from a different decision-maker. By using an appropriate linear ranking function for each decision-maker, it is possible to capture each decision-maker's individual ranking attitude and, thus, transform the problem into a crisp multi-objective one. Kaur and Kumar [12] defined the product of unrestricted LR-type FNs, and proposed a method for FLP with fuzzy parameters and fuzzy decision variables that applies Yager's ranking function to the objective function values and to both sides of the fuzzy constraints. On a similar line, Yang et al. [13] proposed two methods for solving FLP problems: one method solves FLP problems with fuzzy inequality constraints and the other FLP problems with flexible fuzzy inequality constraints. Both of the methods use the expected value of an FN as the linear ranking function for making the crisp transformations.
Despite being widely used, linear ranking functions have little discriminative capabilities, as they may map different FNs into the same real number, which, in turn, may yield unreasonable results [14]. Lexicographic ranking criteria can overcome this issue by defining multiple comparison indices arranged in a hierarchical fashion [15]. This fact has motivated the development of lexicographic methods for FLP. In this paper, we contribute, with an updated review, to this particular area of FLP. We do it by first establishing a general framework for lexicographic FLP that organizes existing models and solution methods in a way that they can be well studied, applied in real-world problems, and eventually extended from the perspectives of other theories in the field of Soft Computing. Consequently, in Section 2, we give an overview of FNs, linear ranking functions, lexicographic ranking criteria, and their use in FLP. Thereafter, we review some lexicographic methods for FLP. A lexicographic method for the Fuzzy Linear Assignment (FLA) problem has also been included due to the theoretical and practical relevance of Linear Assignment (LA) problems. Lexicographic methods for fuzzy multi-objective linear programming are reviewed in Section 3. Concluding remarks and ideas for possible extensions of the presented methods are provided in Section 4.

Fuzzy Linear Programming: Lexicographic Approaches
This section surveys several methods for solving FLP problems based on the use of lexicographic criteria for ranking FNs. We first introduce the concepts of FN, trapezoidal FN (TrFN), and LR-type FN. To avoid an unnecessary lengthening of the paper, we shall not present the arithmetic operations for FNs; the reader is referred to Dubois and Prade's [16] work on LR-type FNs and the textbooks [17,18]. Note that addition and multiplication operations for FNs shall be denoted generically by ⊕ and ⊗, respectively; however, their definitions may be different according to specific FLP problems, solution methods, and applications.
As ranking FNs is an important step in the solution of FLP problems, we shall introduce, only in a general sense, two ranking methodologies that are relevant to the subsequent discussion. A detailed analysis of several ranking methods and their rationality, as well as an introduction to this topic, can be found in [19].

Fuzzy Numbers
A real FNã is any fuzzy subset of the real line R, whose membership function µã is: In addition, we may have e = −∞, or f = g, or e = f , or g = h, or h = +∞ [16]. We denote the set of all FNs by F R .
A special type of FN is the trapezoidal FN (TrFN), whose membership function is given by and it is denoted byã = e, f , g, h . Clearly, when f = g a TrFN reduces to a triangular FN (TFN). Dubois and Prade [16] introduced the LR-type representation of FNs by considering the so-called left and right reference functions L and R, respectively. L (or R) is a reference function if (1) L(x) = L(−x), (2) L(0) = 1, and (3) L is nonincreasing on [0, +∞). Symbolically, an LR-type FN is written as a = f , g, α, β LR and its membership function takes the form where α and β are the left and right spreads ofã, respectively. If L(x) = R(x) = max{0, 1 − |x|}, then the LR-type FNã reduces to a TrFNã = e, f , g, h , where e = f − α and h = g + β. We may also writeã = e, f , h when referring to a TFN, andã = f , α, β LR in the case of LR-type FNs without a flat. Figure 1 depicts the graphical representation of an LR-type FN and a TrFN.

Ranking Functions and Lexicographic Ranking Criteria
At some point in solving fuzzy decision-making problems, two or more FNs representing evaluations of available alternatives need to be compared. As a result of such comparisons, a ranking of the alternatives is obtained, which allows for the selection of the best one. Several methodologies for ranking FNs have been proposed [19]. However, because different FNs may overlap each other, ranking them properly remains a challenging problem in Fuzzy Sets theory.
One way of ranking FNs consists in the use of a linear ranking function R : F R → R to map FNs into real numbers and derive the ranking of the FNs from the comparison of the corresponding real numbers. However, different FNs may be mapped into the same real number and, therefore, R may fail to distinguish between different FNs, which in turn could lead to deciding for worse alternatives. Lexicographic ranking criteria use multiple comparison indices and thus improve the discriminative capabilities of linear ranking functions. A rather general lexicographic ranking criterion, which needs to be particularized for the specific decision-making situation at hand, is given in Definition 1. Examples of such a criterion can be found in [20,21].

Definition 1.
For an arbitrary FNã with p defining parameters, let f k (for k = 1, . . . , p) be p linear functions of the parameters ofã. Assume that the coefficient matrix corresponding to the p linear functions is non-singular. Furthermore, let ≤ lex denote the lexicographic order relation on R p . Given any two FNsã andb (with the same type of membership functions) the strict inequalityã ≺b holds if and only if f k (ã) The lexicographic order relation satisfies the total order properties; that is, is reflexive, transitive, anti-symmetric and complete [22].

Lexicographic Methods for Solving Fuzzy Linear Programming Problems
A model for a FLP problem in which the parameters and/or the decision variables take on FNs is formulated in Equation (4). Such a type of FLP problem is termed Fully Fuzzy Linear Programming (FFLP) problem, and was initially introduced in [23].
We shall consider that all of the parametersc j ,ã ij andb i , and decision variablesx j are FNs with the same type of membership functions. Variations of this model appear when, e.g., the decision variables are assumed crisp, in which case one deals with a fuzzy number linear programming problem, or when the decision variables and the right-hand side of the constraints are FNs and the remaining parameters are crisp, in which case one deals with a fuzzy variable linear programming problem. Other variations might appear as well; e.g., when only some parameters and/or decision variables are assumed as FNs.
Having presented the necessary concepts, we survey in this subsection several lexicographic methods for solving FLP problem (4). However we do it within a general framework, as the existing methods share many of their characteristics (see Table A1 in Appendix A).
To solve FLP problem (4), we must first clarify what we mean by minimizing FN-valued functions and how fuzzy inequality constraints are handled. Thus, in accordance with the ranking methodologies already discussed, we may say that either min ã,b =ã iff R ã ≤ R b , or, by . The latter approach overcomes the indistinguishability issues of the linear ranking functions, and it leads to lexicographic . A frequent approach for handling fuzzy inequality constraints has been however to defineã b by means of a partial order, such as the use of the crisp element-wise inequality between the corresponding components of the parametric representation of the FNs in both sides of the fuzzy inequality constraints, or replacing ≤ lex with the usual element-wise inequality ≤ between two vectors (see, e.g., [24][25][26][27]). Thus, leaving aside the cases that simply use linear ranking functions, the following crisp models can be obtained as possible transformations of FLP problem (4); we have assumed that ij ⊗x j , I eq contains the indices of the fuzzy equality constraints and I ineq the indices of the fuzzy inequality constraints.
lex min or lex max f k z k=1,...,p where ≤ p is a partial order for FNs and each g k is a linear function of the parameters of the FNs that may or may not be the same as f k . Lastly, the fuzzy equality constraints are defined element-wise from the corresponding parametric representations of the FNs in both sides of the constraints. In order to solve either of the models, Algorithm 1 for classical lexicographic optimization is used [28].

Algorithm 1: Lexicographic optimization (minimization case).
Data: Feasible set X of (Model 1) or (Model 2) and objective functions f k , k = 1, . . . , p. Initialization: Define X 1 := X and let k := 1; Solve the single-objective optimization problem has a unique optimal solution then stop and return the solution; else if (P k ) is unbounded then stop, the lexicographic optimization problem is unbounded; else if k=p then stop, the set of optimal solutions of the lexicographic optimization problem is  [29] approach corresponds with (Model 1) for p = 2. They proposed a two-phase method to solve FFLP problems with inequality constraints in which the parameters and decision variables are symmetric LR-type FNs; i.e., all of the fuzzy parameters and decision variables have the formã = f , α L . They used a specific lexicographic ranking criterion based on the possibilistic mean Assuming a maximization problem, in the first phase of their method, the set of all feasible solutions that maximize the mean value of the fuzzy objective function is obtained, and in the second phase, the standard deviation of the fuzzy objective function is minimized subject to the set obtained in the first phase. Hashemi et al.'s [29] method is however limited for real applications, as it relies too much on the form of the FNs and the proposed lexicographic criterion (see [25,29] for detailed illustrations of each phase).

Hosseinzadeh Lotfi et al.'s Method
Hosseinzadeh Lotfi et al.'s [30] approach corresponds with (Model 2) for p = 2. The authors proposed a method for solving FFLP problems having only equality constraints. They used the nearest symmetric TFN approximation of an arbitrary FN in parametric formã = a(r), a(r) (r ∈ [0, 1]), and a lexicographic criterion for ranking symmetric TFNs to find an approximate solution of the original FFLP problem. The nearest symmetric TFN ofã is given by a − α, a, a + α , where a = 1 We see that this criterion is similar to that of Hashemi et al.'s [29] for ranking symmetric LR-type FNs.

Kaur and Kumar's Methods
Kaur and Kumar [31] presented a lexicographic method for solving a variation of FLP problem (4) in which only the coefficients of the objective function are considered FNs. The authors used Farhadinia's [21] lexicographic criterion for ranking LR-type FNs. Their method was extended to a fully fuzzy environment in [32] but considering only TrFNs, fuzzy equality constraints and a different lexicographic ranking criterion. A version of this latter method for FFLP with unrestricted LR-type FNs in the problem parameters and decision variables is given in [33]. Kaur and Kumar's approaches [31][32][33] correspond with (Model 2) for p = 4.

Ezzati et al.'s Method
Ezzati et al. [34] noticed that Hosseinzadeh Lotfi et al.'s [30] method can only yield approximate solutions of FFLP problems. They proposed a new lexicographic criterion for ranking TFNs, and a method for solving FFLP problems with equality constraints in which the problem parameters take on unrestricted TFNs and the decision variables taken on non-negative TFNs. According to their ranking criterion, given any two TFNsã = a 1 , a 2 , Their approach corresponds with (Model 2) for p = 3. A similar method, but with LR-type FNs, is given in [35]. A more efficient version of Ezzati et al.'s [34] method is given in [36] for solving FFLP problems having only equality constraints. To handle fuzzy inequality constraints, Ezzati et al. [34] introduced non-negative triangular fuzzy slack and surplus variables as needed. However, given any two TFNsã andc withã c, there does not always exist a TFNb, such that a ⊕b =c. Furthermore, Bhardwaj and Kumar [37] noticed that, in the case of Ezzati et al.'s [34] method, the introduction of non-negative fuzzy slack and surplus variables leads to a contradiction with Ezzati et al.'s [34] ranking criterion and therefore yields unfeasible solutions. A modified version of Ezzati et al.'s [34] method that handles fuzzy inequality constraints properly is given in [38]. Ezzati et al.'s [34] method has been applied to transportation, investment [34] and supply chain [39] problems in fuzzy environments.

Das et al.'s Method
Das et al. [40] proposed a method for solving FFLP problems with equality and inequality constraints having unrestricted trapezoidal fuzzy parameters and non-negative trapezoidal fuzzy decision variables. Assuming TrFNs in the parametric formã = e, f , g, h , Das et al.'s [40] lexicographic ranking criterion for the objective function values defines f 1 ã = e − f , f 2 ã = f , f 3 ã = ( f + g)/2 and f 4 ã = h − g. However, the handling of fuzzy inequality constraints was not consistent with this lexicographic ranking criterion. Ebrahimnejad and Verdegay [25] noticed this fact and modified Das et al.'s [40] method to handle fuzzy inequality constraints correctly. The modified approach [25] exactly corresponds with (Model 2) for p = 4 whenã i ≤ pbi if and only if f k ã i ≤ f k b i for k ∈ {1, . . . , 4}. Thus, a lexicographic criterion is used with the objective function values and a partial order with the fuzzy inequality constraints. Das et al. [40] illustrated their method by solving production planning and diet problems in fully fuzzy environments.

Lexicographic Method
Pérez-Cañedo and Concepción-Morales [14] proposed a method to solve FFLP problems with equality and inequality constraints in which the problem parameters and decision variables can take on unrestricted LR-type FNs. The approach followed in [14] corresponds exactly with (Model 1). Originally, in this method, the same lexicographic ranking criterion is used with the objective function values and the fuzzy inequality constraints. However, different lexicographic ranking criteria for the objective function and the inequality constraints can be easily incorporated by a proper redefinition of the g k functions. The authors proposed the following transformation of those constraints to handle the lexicographic constraints in (Model 1): for positive real values of and M sufficiently small and large, respectively. The authors illustrated their method by means of simple fuzzy production planning problem. An application in fuzzy linear regression can be found in [41]. Recently, this method was extended to a fully intuitionistic fuzzy environment in [42].

Illustrative Example and Comparative Analysis
Consider the following FFLP problem taken from [14].
From its formulation, we notice that the methods in Table A1 (with the exception of [14]) cannot be used to solve this problem. On solving the above FFLP problem with method [14] and the lexicographic ranking criterion [21], we obtainx 1 = 12.  Table 1 shows two solutions obtained with alternative methods for handling fuzzy inequality constraints. Figure 2 depicts the fuzzy values of the solutions that are given in Table 1 and the one obtained with the lexicographic method [14]; considering that this is a maximization problem, it can be seen that the lexicographic method yielded an intuitively better solution.
(  Table 1 and the one obtained with the lexicographic method [14].

Fuzzy Linear Assignment Problem
The LA problem is a combinatorial optimization problem, in which a decision-maker wishes to assign n resources (persons or machines) to n tasks, on a one-to-one basis, at minimum cost. The LA problem has n! feasible assignments, which makes complete enumeration procedures extremely inefficient. Kuhn [43] proposed a primal-dual algorithm, known as the Hungarian Method, which solves LA problems with worst-case time complexity of O(n 4 ). The Hungarian Method made possible for the first time to tackle high-dimensional LA problems in a more efficient way.
In real-world decision-making situations, the coefficients of the LA problem are often obtained from decision-makers and experts as subjective evaluations of potential assignments whose costs are only approximately known. In the framework of Fuzzy Sets theory, these cost coefficients can be taken as FNs. Hence, it makes sense to formulate and solve a fuzzy version of the classical LA problem. Consequently, let S n denote the set of all the permutations of I = {1, 2, . . . , n } and C = c ij n×n , c ij ∈ F (R), a matrix comprising the FNs (costs) of the potential assignments. The objective is to find a permutationφ ∈ S n that minimizes n ∑ i=1c iϕ(i) =c 1ϕ(1) ⊕c 2ϕ(2) ⊕ · · · ⊕c nϕ(n) . Thus, the FLA problem can be formulated as in Equation (5).
The FLP formulation of the FLA problem is given by Equation (6).
For simplicity, it shall be assumed that all of the cost coefficients are TrFNs; although, a more general approach considering different types of FNs simultaneously has been recently investigated in [44]. There are several methods for solving FLA problems; the reader is referred to [44] for a description of some of those methods. Here, we present three methods that can be regarded as representative of the existing ones.

Kumar and Gupta's Method
Kumar and Gupta [45] transformed the FLA problem into a crisp one by using a linear ranking function, which, for an arbitrary TrFNã = e, f , g, h , has the form R(ã) = (e + f + g + h)/4. Thus, according to their method, LA problem (7) is obtained.
s.t. the same constraints of problem (6).
The solution of LA problem (7) is obtained by any of the existing classical assignment methods.

Baykasoglu et al.'s Method
Although Baykasoglu et al.'s [46] method was originally proposed to solve FLA problems with triangular fuzzy coefficients in the context of multiple-attribute group decision-making, an extension to the case of trapezoidal fuzzy coefficients is straightforward.
Let C = [c ij ] n×n denote the fuzzy cost matrix of the FLA problem, where eachc ij = e ij , f ij , g ij , h ij is a TrFN for i, j ∈ I. By using Baykasoglu et al.'s [46] method, the FLA problem is transformed into a four-objective crisp LA problem, as in Equation (8), with C e = e ij n×n , C f = f ij n×n , C g = g ij n×n and C h = h ij n×n being the coefficient matrices of each of the objective functions. Compromise programming [47] with a preferred metric is then used to solve the four-objective crisp LA problem.

Lexicographic Method
As noticed previously, the use of a linear ranking function for making crisp transformations does not guarantee optimal solutions for the FLP problem under consideration. Knowing that lexicographic ranking criteria have better discriminative capabilities than linear ranking functions, it seems natural to use Definition 1 in order to transform FLA problem (5) into lexicographic linear assignment (LLA) problem (9). lex min Theorem 1. Ifφ is an optimal assignment for LLA problem (9), thenφ is also an optimal assignment for FLA problem (5).
Proof. Similar to that of [44].
To solve LLA problem (9), the authors in [44] adhered to Burkard et al.'s [48] general algebraic framework. Burkard et al. [48] assumed that the cost coefficients of the LA problem are elements of a totally ordered commutative semigroup H, * , ≤ H with internal composition * and total order relation ≤ H fulfilling the following properties: 1. ∀x, y, z ∈ H, x ≤ H y =⇒ x * z ≤ H y * z (≤ H is compatible with * ); 2. ∀x, y ∈ H with x ≤ H y ∃z ∈ H such that x * z = y.
Burkard et al. [48] also stated and proved an optimality criterion, and proposed a general solution algorithm that can be seen as a generalization of the Hungarian Method.
It can be shown that R p , +, ≤ lex is a totally ordered commutative semigroup that fulfills the above-mentioned properties. Hence, LLA problem (9) can be optimally solved within Burkard et al.'s [48] algebraic framework; Theorem 1 guarantees that an optimal solution of LLA problem (9) is optimal for FLA problem (5) as well.

Lexicographic Methods for Fuzzy Multi-Objective Linear Programming
The Fully Fuzzy Multi-Objective Linear Programming (FFMOLP) problem is given by Equation (10). The first reference to this topic is the work of Buckley et al. [53], where the authors used an evolutionary algorithm to find approximate solutions. Recent works on this topic mainly use linear ranking functions for making crisp transformations [54,55]. In what follows, we review two lexicographic methods.    [34] lexicographic ranking criterion f 1 ã = a 2 , f 2 ã = a 1 − a 3 , and f 3 ã = a 1 + a 3 . output : Values of the decision variablesx j . Initialization: Define X 1 := X and let k : Solve the single-objective optimization problems (F k ) and denote the corresponding optimal solutions asx (t) for t = 1, . . . , r.

Fuzzy Epsilon-Constraint Method
Pérez-Cañedo et al. [15] showed that the transformation of FFMOLP problems into crisp multi-objective linear programming problems by means of linear ranking functions does not guarantee Pareto optimal fuzzy solutions. Thus, taking as a starting point the lexicographic method for fully fuzzy single-objective linear programming presented in Section 2.3.6, in [15], the authors extended the classical definitions of dominance and Pareto optimality to the fuzzy case using lexicographic ranking criteria, and proposed a fuzzy epsilon-constraint method that guarantees Pareto optimal fuzzy solutions. In what follows, we briefly present their approach only considering the minimization case. Definition 2 ([15]). Let be given as in Definition 1 and X denote the set of all feasible fuzzy solutions of FFMOLP problem (10).

Definition 3.
A decision vectorx ∈ X is said to be Pareto optimal fuzzy solution of FFMOLP problem (10) if there does not exist anotherỹ ∈ X, such thatỹ dominatesx.
By introducing auxiliary variablesz,s + t ,s − t , real-and FN-valued parameters λ t and˜ t for t ∈ {1, . . . , r}\{q}, respectively, FFMOLP problem (10) is transformed into fully fuzzy single-objective linear programming problem (11), which is solved using the lexicographic method from Section 2.3.6. Again, it is assumed that all of the fuzzy parameters and decision variables have the same type of membership functions. In this formulation,M is a constant FN, e.g.,M = 0, m, m, m LR with m sufficiently large, whose sole purpose is to guarantee the existence ofz so that the first fuzzy equality constraint holds for optimal values ofx,s + ands − [15]. Theorem 2 ([15]). Let (x,s − ,s + ,z) be an optimal fuzzy solution of FFLP problem (11) with λ t > 0 ∀t ∈ {1, . . . , r}\{q}. Subsequently,x is a Pareto optimal fuzzy solution of FFMOLP problem (10).
In [15], the authors exemplified their method by solving a three-objective fully fuzzy project crashing problem. This method has been recently used in [57] to solve a fully fuzzy multi-objective berth allocation problem.

Illustrative Example and Comparative Analysis
Yang et al. [56] illustrated their lexicographic method by solving the following FFMOLP problem, in which all of the parameters and decision variables are TFNs. minz (1) = (7, 10, 11) ⊗x 1 ⊕ (8, 10,13)  According to Yang et al. [56], their method yieldsx 1 = (0, 3, 3.3743),x 2 = (1.5, 2, 2.2503) with objective functions valuesz (1) = (12, 50, 66.3712) andz (2) = (6, 23, 40.5008). However, the authors must have done some rounding or miscalculation, as this solution is not satisfying the problem constraints exactly. Yang et al.'s [56] method only yields one solution to the FFMOLP problem; hence, it lacks the necessary flexibility in real-world applications. On the other hand, the fuzzy epsilon-constraint method [15], equipped with Ezzati et al.'s [34] lexicographic ranking criterion, yields multiple Pareto optimal fuzzy solutions, as shown in Table 3. Figure 4 depicts a portion of the Pareto front of the FFMOLP problem corresponding to the solutions in Table 3 plus Yang et al.'s [56] solution. From the objective functions values in Table 3, the reader may notice that all of the solutions are tied with respect to Ezzati et al.'s [34] first ranking index; hence, to distinguish one solution from another, we must resort to the other indices and, in this case, the second index is enough to conclude that no solution dominates another.

Concluding Remarks
In this paper, we have presented a review of existing lexicographic methods for FLP. These methods can be grouped into two main models. Enclosed by (Model 1) are those methods that handle fuzzy inequality constraints by using lexicographic ranking criteria (we have only found two methods of this kind in the existing literature) and enclosed by (Model 2) are those that use partial orders or introduce fuzzy slack and surplus variables to handle the fuzzy inequality constraints. In the future, it can be expected that these methods will be extended to deal with FLP problems in which the parameters and/or decision variables take on generalizations of FNs, such as interval-valued FNs, intuitionistic FNs, interval-valued intuitionistic FNs, and so on; an interesting topic in this regard is the hybridization with Rough Sets theory (see, e.g., [58] and references therein). Such generalizations allow for incorporating other levels of imprecision, e.g., by considering that membership grades may not be fixed real numbers but belong to some closed subintervals of [0, 1], there may be hesitation with regard to the values of some parameters, and data may be inconsistent due to the existence conflicting values.
Currently, in the case of FFMOLP, there are only two lexicographic methods available. Hence, we believe that new lexicographic methods for FFMOLP will emerge as extensions of classical ones given the many applications of classical multi-objective optimization. Particularly, a fully fuzzy extension of classical compromise programming is an interesting research line to explore in connection with fuzzy-valued distance functions. Here too, other methods using generalizations of FNs as parameters and/or decision variables can be expected.
Lexicographic methods for FLP are quite recent. Although they have been used in some real-world decision-making situations, their usefulness and advantages in this regard over other established approaches have yet to be explored in a deeper way. Therefore, researchers that are interested in this topic may consider not only the generalizations commented here, but real-world applications as well. In this regard, a fundamental issue to address is the non-existence of suitable software for rapid development and implementation of real-world applications. A step in this direction has been recently taken in [59], where the authors presented an open-source R package for FLP. A similar package, or perhaps an extension, could incorporate the lexicographic methods presented in this review, thus making these methods available for use as well.
In this review, we have only considered lexicographic methods for FLP. The non-linear case has not been arbitrarily left out. In fact, to the best of our knowledge, no research has been conducted on the use of lexicographic methods for Fuzzy Non-linear Programming (FNLP). FNLP is a research area with many real-world applications (see, e.g., [60], where Fuzzy Sets theory and genetic algorithms are combined to solve a congestion management problem). We believe that lexicographic methods may also prove useful in this area. Acknowledgments: The authors are grateful to the editors and the anonymous reviewers for their constructive comments and suggestions.

Conflicts of Interest:
The authors declare no conflict of interest.