A Revised Version of a Lexicographical-based Method for Solving Fully Fuzzy Linear Programming Problems with Inequality Constraints

Ezzati et al. (A new algorithm to solve fully fuzzy linear programming problems using the MOLP problem. Appl Math Model. 2015;39(12):3183–3193) introduced a lexicographic criterion for ranking triangular fuzzy numbers (TFNs), and proposed a method to solve fully fuzzy linear programming (FFLP) problems based on the lexicographic method of multi-objective optimisation; the authors assumed that fuzzy inequality constraints can be transformed into fuzzy equality constraints by introducing non-negative fuzzy slack and surplus variables. They illustrated the proposed method by means of a fully fuzzy investment problem. Bhardwaj and Kumar (A note on “A new algorithm to solve fully fuzzy linear programming problems using the MOLP problem”. Appl Math Model. 2015;39(19):5982–5985) demonstrated that introducing fuzzy slack and surplus variables is mathematically incorrect, and showed that the solution of the fuzzy investment problem is unfeasible. Towards the end of their paper, they claimed that there is no feasible solution to the fuzzy investment problem when considering Ezzati et al.’s ranking criterion. In this paper, we propose a revised version of Ezzati et al.’s method whereby the optimal solution of FFLP problems with equality and inequality constraints can be obtained. Furthermore, by using the revised method, we show that feasible solutions of the fuzzy investment problem actually exist, and therefore Bhardwaj and Kumar’s claim is false. To show the applicability of the revised method, we also consider a fully fuzzy project scheduling problem with budget constraint.


Introduction and Preliminaries
Since the foundational works on fuzzy decision-making and optimisation by Bellman and Zadeh [1], Tanaka et al. [2] and Zimmermann [3] fuzzy linear programming (FLP) has experimented a considerable development and has spread into almost every area of decision-making. For example, methodologies of FLP and applications in fuzzy assignment, scheduling, transportation, matrix games, recommender systems and logistics network design problems with a single objective function or multiple ones are reported in [4][5][6][7][8][9][10][11][12][13][14]. Recent surveys by Ebrahimnejad and Verdegay [15] and Nasseri et al. [16] offer a detailed exposition of several FLP methodologies, from the very beginning of FLP to its modern approaches.
One such approach was developed by Ezzati et al. [17] for solving fully fuzzy linear programming (FFLP) problems; i.e. FLP problems in which all parameters and decision variables take on fuzzy numbers. Specifically, Ezzati et al. [17] considered FFLP problems with unrestricted triangular fuzzy parameters and non-negative triangular fuzzy decision variables. The authors found the motivation for their proposal in the shortcomings and limitations of the methods of Hosseinzadeh Lotfi et al. [18] and Kumar et al. [19]. Ezzati et al. [17] noticed that in the case of Hosseinzadeh Lotfi et al.'s [18] method, it is necessary to approximate all the problem parameters to their corresponding nearest symmetric triangular fuzzy numbers (TFNs); therefore, the obtained solutions are approximate and do not satisfy the problem constraints exactly. In [19], on the other hand, a ranking function was used to transform the FFLP problem into a crisp linear programming problem. Since by using a ranking function different fuzzy numbers may be mapped into the same real number, the information encompassed by the fuzzy numbers is lost and therefore cannot be used to guide the search for an optimal solution. To overcome these shortcomings and limitations, Ezzati et al. [17] introduced a new lexicographic criterion for ranking TFNs and transformed the FFLP problem into a three-objective crisp linear programming problem; the lexicographic method of classical multi-objective optimisation [20, chap. 5, pp. 128-130] was subsequently used to solve the problem. Recently, Ebrahimnejad [21] proposed a more efficient formulation of Ezzativ et al.'s [17] method. However, Ebrahimnejad [21] approached FFLP problems having only equality constraints. We shall show shortly that Ezzati et al.'s [17] method cannot be used to solve FFLP problems with inequality constraints; considering the vast number of FFLP applications, this is a serious limitation from a practical viewpoint.
Before discussing on the shortcomings and limitations of Ezzati et al.'s [17] method, it is convenient to present some basic definitions concerning TFNs; the reader may refer to Ezzati et al.'s [17] paper for the introductory concepts on fuzzy numbers and other related definitions.

Definition 1.1 ([17]):
A fuzzy numberṽ = v l , v c , v u is said to be a TFN if its membership function is given by: is an arbitrary real number.

Definition 1.3 ([17]):
, v u 2 be two TFNs, addition ⊕, subtraction and multiplication ⊗ ofṽ 1 andṽ 2 are defined as follows: Ezzati et al. [17] illustrated their method for solving FFLP problems by means of three examples. In particular, the last example is a fully fuzzy investment problem in which a corporation seeks to allocate ($25, $30, $40) million to its four subsidiaries. A minimal level of funding has been established for each subsidiary; these levels are ($2, $3, $5), ($4, $5, $6), ($5, $8, $9) and ($7, $8, $14) million, respectively. Each subsidiary can conduct three projects. A rate of return, as a percent of investment, has been estimated for each project, and a limited level of investment has been fixed for each project.
The order relation is defined as follows [see 17, Definition 2.6].
Towards the end of their paper, Bhardwaj and Kumar [22] claimed that it is not possible to find any feasible fuzzy solution of FFLP problem (1) according to the ranking criterion of Ezzati et al. [17]. However, the reader can easily verify thatx 11 = (0, 0, 0),x 12 = (0, 0, 5), Bhardwaj and Kumar [22] focused their discussion on two points: (1) to prove that transforming the fuzzy inequality constraints into fuzzy equality constraints by means of non-negative triangular fuzzy slack and surplus variables leads to a contradiction with Ezzati et al's [17] ranking criterion, and (2) to show, as we did above, the existence of this contradiction in the solution of FFLP problem (1) given in [17]. Although Bhardwaj and Kumar [22] successfully addressed both points, the authors did not provide a method for solving FFLP problems with inequality constraints considering the ranking criterion of Ezzati et al. [17]; so, until now, this issue has remained unsolved. Based on the above discussion, the present paper makes the following contributions: • A revised version of Ezzati et al.'s [17] method for solving FFLP problems with inequality constraints that overcomes the shortcomings and limitations pointed out in [22]. • Bhardwaj and Kumar's [22] claim on the unfeasibility of Ezzati et al.'s [17] fully fuzzy investment problem is shown to be false. • Three numerical examples that discuss practical problems show the applicability of the revised method in real-world problems.
The rest of this paper is organised as follows. In Section 2, based on a recent methodology for handling fuzzy inequality constraints [23], we make some corrections to Ezzati et al.'s [17] method that overcome the shortcomings and limitations pointed out by Bhardwaj and Kumar [22]. This new revised version of Ezzati et al.'s [17] method is illustrated in Section 3 by solving the fully fuzzy investment problem and a fully fuzzy project scheduling problem with budget constraint. In the case of the fully fuzzy investment problem, it is shown that, contrary to Bhardwaj and Kumar's [22] claim, this problem has infinitely many optimal fuzzy solutions. A comparison with existing methods is carried out in Section 4. Concluding remarks and directions for future research are provided in Section 5.

Revised Version of Ezzati et al.'s Method
The general form of the FFLP problem with arbitrary triangular fuzzy parameters and nonnegative triangular fuzzy decision variables can be formulated as follows: To facilitate the following discussion, we rewrite the definition of the order relation , given by Ezzati et al. [17], as in Definition 2.1.

Definition 2.1:
Let ≤ lex be the lexicographic order relation in 3 two arbitrary TFNs. We sayṽ 1 is relatively less thanṽ 2 , which is We sayṽ 1 is relatively less than or equal toṽ 2 , which is denoted byṽ 1 In what follows, we will go through the steps of a method for solving FFLP problem (3) and point out the differences with the method of Ezzati et al. [17] and other approaches from the literature.
Step (3) can be written as: (4) is rewritten as follows: is a non-negative TFN, for j = 1, 2, . . . , n Remark 2.2: At this point, Ezzati et al. [17] introduce non-negative triangular fuzzy slack and surplus variables to transform each fuzzy inequality constraint of FFLP problem (5) into a fuzzy equality constraint. This transformation has also been used, for example, in [7,24]. However, as shown by Bhardwaj and Kumar [22] and Gupta et al. [25], such a transformation is not mathematically correct; therefore, we do not use it here and proceed to Step 3.
Step 3. Denote by I e , I le and I ge the index sets of the fuzzy equality, less-than-or-equal-to and greater-than-or-equal-to constraints of FFLP problem (5). By using Definitions 1.2 and 2.1, transform FFLP problem (5) into the following lexicographic optimisation problem: . . , n Step 4. By introducing binary variables y ik , for i ∈ I le ∪ I ge and k ∈ {1, 2, 3}, transform lexicographic optimisation problem (6) into crisp mixed 0-1 lexicographic linear programming (MLLP) problem (7), y ik ∈ {0, 1}, for i ∈ I le ∪ I ge and k ∈ {1, 2, 3} (7h) for positive real values of and Lsufficiently small and large, respectively.
Step 5. Solve MLLP problem (7) by using the lexicographic method of classical multiobjective optimisation [20, chap. 5, pp. 128-130] to obtain an optimal solution x l * j , x c * j and x u * j , and put their values intox * j = (x l * j , x c * j , x u * j ) to obtain an optimal fuzzy solution of FFLP problem (3).
Step 6. Evaluate n j=1c j ⊗x * j to obtain the optimal fuzzy objective value of FFLP problem (3).

Theorem 2.1 ([23]): MLLP problem (7) is equivalent to FFLP problem (3).
Proof: The proof can be divided into two parts. Firstly, it is necessary to show that FFLP problem (3) is equivalent to lexicographic optimisation problem (6); secondly, that problem (6) is equivalent to MLLP problem (7). We omit the first part as it is easily shown by contradiction using Definition 2.1. For the second part, it is sufficient to show that the lexicographic constraints of problem (6) are equivalent to constraints (7b)-(7h) of MLLP problem (7). Thus, for the ≤ lex −type constraints, we analyse the following cases: (1) if y ik = (1, * , * ) for k = 1, 2, 3, where * means 0 or 1, then by substituting into constraint set (7b)-(7d) it follows that ≤ b c i − n j=1 m c ij ≤ L which implies n j=1 m c ij < b c i hence, from Definition 2.1, the ≤ lex -type constraints of problem (6) are satisfied.
, then it follows that y i1 = 0, y i2 = 1 (similar to the previous case) and −L + y i3 ≤ s i3 ≤ Ly i3 ; these inequalities are satisfied for y i3 = 1, with a sufficiently large positive value of L so that s i3 ∈ [−L + ε, L].
The remaining cases and the proof for the ≥ lex -type constraints can be shown to hold in a similar manner.

Remark 2.4:
Ezzati et al. [17] assumed that fuzzy inequality constraints can be transformed into equality constraints by means of non-negative triangular fuzzy slack and surplus variables. They based their results on this false assumption; therefore, their Theorem 3.1 is valid only for FFLP problems having only equality constraints. Steps 3 and 4 of the revised method and Theorem 2.1 give an appropriate way in which Ezzati et al.'s [17] ranking criterion can be used for handling fuzzy inequality constraints.

Numerical Examples
In this section, we illustrate the revised method by means of three examples. In the first example, the fully fuzzy investment problem of Ezzati et al. [17] is solved and shown to have infinitely many optimal fuzzy solutions, the second example is a fully fuzzy project scheduling problem and third one is a fully fuzzy project crashing problem. We remark that before the introduction of the revised method these problems could not be solved considering Ezzati et al.'s [17] ranking criterion; thus, they are illustrative of the advantages of the revised method. The examples in this section were solved using the linear programming modeller PuLP version 1.6.0 [27] and CBC 1 version 2.9.9 [28] on a computer with an Intel ® Core TM i3-4005U @ 1.70 GHz x4 and 4GB RAM running Ubuntu 18.04.4.    (138.9998, 248.9999, 374.0003). The reader is encouraged to verify that the obtained optimal fuzzy solution does satisfy all the constraints of FFLP problem (1) with respect to Ezzati et al.'s [17] lexicographic ranking criterion.
An alternative optimal fuzzy solution is given byx 31 = (3, 5, 6),x 43 = (4, 4, 7.0001) and the other variables take on the same values as in the previous solution. As happens with classical linear programming, it can be shown that the convex combination of the two optimal fuzzy solutions is also optimal for FFLP problem (1). So, FFLP problem (1) is not an unfeasible problem as Bhardwaj and Kumar [22] claimed; it actually has infinitely many optimal fuzzy solutions.
The feasible solution of FFLP problem (1) given in the introduction of the present paper is obtained by imposing the integer condition on all variables of the above MLLP problem. Lastly, it is important to mention that if we use the element-wise inequality in place of ≤ lex , then the resulting optimisation problem becomes unfeasible; this may have motivated Bhardwaj and Kumar's [22] claim. Example 3.2: (Fully fuzzy project scheduling problem): A project is composed of interrelated activities, whose structure can be represented by a directed acyclic graph (see, e.g. Figure 1). The set of vertices V represent events and the set of edges A represent activities. Formally, a project can be represented by the triplet (V, A, D), where A ⊂ V × V andd ij ∈ D is the time required for the completion of activity (i, j) ∈ A. In what follows, it is assumed that the duration of each activity is estimated by a non-negative TFN. Ift j denotes the fuzzy time of event j ∈ A, then for each activity (i, j) ∈ A we must have thatt i ⊕d ij t j to ensure that all precedence relationships hold.
In the case of the project depicted in Figure 1, we have V = {1, 2, . . . , 9}; A and D are shown in Table 1.
To schedule a project, we must find the shortest time interval in which all precedence relationships are satisfied; therefore, the fully fuzzy mathematical programming model for  Table 1. Data of the project network depicted in Figure 1. Table 2. Solution of the fully fuzzy project scheduling problem (8).  Table 3. Data of the fully fuzzy project crashing problem (9).  Table 4. Solution of the fully fuzzy project crashing problem (9), up to two decimal places, obtained by using the revised method.
problem with inequality constraints into a crisp one. This transformation is carried out by applying R(·), EV(·) or any other ranking function to the objective function and to both sides of the inequality constraints of the FFLP problem. An optimal solution for the resulting crisp problem is then considered optimal for the fuzzy one. However, the ranking functions may map different fuzzy numbers into the same real number, and therefore have little discriminative capabilities. In this sense, using ranking functions to solve FFLP problems may not yield satisfying solutions.
Nasseri et al. [38], on the other hand, have used the partial orderṽ 1 to handle the fuzzy inequality constraints, and R(·) with the objective function; therefore, they have used two different approaches for the inequality of TFNs. We note that, with respect to Nasseri et al.'s [38] partial order, ifṽ 1 ṽ 2 , then R(ṽ 1 ) ≤ R(ṽ 2 ), but the converse is not true. Ebrahimnejad and Verdegay [15, chap. 4, p. 298] claimed that it is not correct to use two different approaches for the inequality of two fuzzy numbers in the same problem. Although we share their view in general, we also recognise that it can be open to discussion, since decision-makers might be interested in checking solution feasibility and comparing objective values by different criteria; in any case, the choice of these criteria must be carefully justified on an application basis. In this paper, we choose to use the same lexicographic criterion for the problem objective function and its inequality constraints because our main objective is to present a revised version of Ezzati et al.'s [17] method that overcomes its shortcomings and limitations. It should be noted that the revised method presented in Section 2 can be straightforwardly modified to account for cases in which decision-makers check feasibility and compare objective values by different lexicographic criteria.
Stanojević et al. [45], based on the work of Dong and Wan [10], used the interval expectation of trapezoidal fuzzy numbers and a partial order for intervals to transform the FFLP problem into a two-objective crisp linear programming problem. The weighted sum approach was subsequently used to transform the multi-objective problem into a single-objective one.
In the following section, we compare the revised method with the existing methods [38,40,41,45].

Discussion on the Results Obtained by Using Existing Methods
The results obtained by using the existing methods [41, sec. 5.4.2] and [40, Algorithm 1] to solve FFLP problem (8) are shown in Table 5. We see from those results that most event times are crisp instead of fuzzy; hence, they do not properly represent the inherent fuzziness of project scheduling problem (8). Furthermore, it seems unreasonable for this particular problem that the most optimistic and possible values of the few fuzzy event times are zero. We therefore conclude that the existing methods [ [38] method does yield a truly fuzzy solution of FFLP problem (8); actually, the project completion timet 9 = (63, 77, 88) is exactly the same obtained by using the revised method; these results are however unfeasible according to Ezzati et al.'s [17] ranking criterion.  Stanojević et al.'s [45] method yields the results shown in the last column of Table 5. Those results were obtained by setting parameters α = 0.2 and β = 0.5 in Stanojević et al.'s [45] method. Parameter β weights the objective functions and α is the acceptance degree for the violation of the fuzzy inequality constraints. By fixing β = 0.5 and varying α, we observed that values of α over 0.25 lead to unbounded problems in this particular situation. As in the case of Nasseri et al. [38], Stanojević et al.'s [45] solution is unfeasible according to Ezzati et al.'s [17] ranking criterion.
As for the fully fuzzy project crashing problem (9), a similar conclusion, as given above, can be drawn with respect to the use of the existing methods [41, sec. 5

Conclusions
In this paper, we confirmed the results of Bhardwaj and Kumar [22] that demonstrate that the method proposed by Ezzati et al. [17] for solving FFLP problems cannot be used to find optimal solutions of FFLP problems with inequality constraints. We proposed a revised version of Ezzati et al.'s [17] method that overcomes its shortcomings and limitations. By using the revised method, we showed that Bhardwaj and Kumar's [22] claim on the unfeasibility of the fully fuzzy investment problem of Ezzati et al. [17] is false, and this problem actually has infinitely many optimal fuzzy solutions. To further illustrate the applicability of the revised method, a fully fuzzy project scheduling problem with budget constraint was solved. The example problems could not be solved by using Ezzati et al.'s [17] original method.
Future research will focus on using the revised method to solve real-world FFLP problems, assessing its efficiency and proposing improvements to its implementation. Taking into account that most FLP problems of practical interest have inequality constraints, and the interpretation of fuzzy inequality constraints is a major concern for decision-makers, we believe that the suitability of the revised method to address diverse application problems with fuzzy inequality constraints such as recommender systems for improving health conditions [12], multi-item solid transportation problems [7] and reverse logistics network design problems [11] is worthy of investigation.
Linear Programming, Computer Vision and Pattern Recognition. He has published academic papers in peer-reviewed journals and conferences.