2.2.2. Fuzzified Models

Model (7) generally defines the class of fuzzy mathematical programming problems that use the fuzzy coefficients *b* and *c* to describe the uncertainty of the real system on which the feasible set and objective function depend, respectively.

$$\begin{array}{ll}\max & f(\mathbf{x}, \tilde{c}),\\\text{s.t.} & \mathbf{x} \in X(\tilde{b}), \end{array} \tag{7}$$

Problems belonging to this class are known as mathematical programming problems with fuzzy coefficients. Depending on the type of the constraints that define the feasible set *X*( *b*) a solution approach to (7) provides either crisp optimal values (real numbers) or "optimal" fuzzy sets values for the decision variables. In the second case, the fuzziness of the decision variables cannot be concluded from the model; sometimes, it is ignored, especially when the focus is exclusively put on the objective function's fuzzy set value, and sometimes, it is derived with respect to the endpoints of the support of the fuzzy set of optimal objective values.

Whenever the values of the decision variables are fuzzy sets the model should be formalized as

$$\begin{array}{ll}\max & f(\check{\mathbf{x}}, \hat{\mathbf{c}}),\\ \text{s.t.} & \check{\mathbf{x}} \in X(\check{\mathbf{b}}), \end{array} \tag{8}$$

in order to explicitly show the fuzziness of *x*-es. In fact, Model (8) describes the so-called full fuzzy (FF) mathematical programming problems. Again, particular forms of the objective function and/or constraints in Model (8) contribute to classifying it as full fuzzy linear programming model, full fuzzy multiple objective model, full fuzzy fractional model, etc. Similar concepts and models exist in an intuitionistic fuzzy (IF) environment.

We will use the following abbreviations related to mathematical programming: LP for linear programming; LFP for linear fractional programming; MO for multiple objective; TP for transportation problem. More elaborate abbreviations can be obtained by concatenating some of the basic abbreviations. For instance FIF-LP stands for full intuitionistic fuzzy linear programming.

#### **3. The Necessity of the Extension Principle**

The extension principle was created to generalize the crisp mathematical concepts to concepts compatible with the fuzzy set theory. Neglecting the extension principle when dealing with optimization problems whose models use fuzzy numbers either move the derived fuzzy solutions out of their proper bounds or damage their shapes.

The solution approaches to Model (8) a priori impose certain shapes for the decision variables (see FIS in Figure 1), mainly the same shape as the shapes used for the coefficients, generally fuzzy numbers. Further on, they formally use the fuzzy arithmetic theory to evaluate the expressions of the objective function and constraints with respect to the fuzzy coefficients and variables, and then propose different methods relying on fuzzy numbers optimization and ranking.

**Figure 1.** Decision variables shapes with respect to the two types of fuzzy mathematical programming and how they comply with the extension principle.

Figure 2 visually presents the current position on which the extension principle is mainly involved in the solution approaches to mathematical programming with fuzzy numbers; and the desired position on which the extension principle should be generally reinstated.

**Figure 2.** Current versus desired roles of the extension principle in mathematical programming with fuzzy numbers.

Comparing Models (7) and (8) and the solution approaches found in the literature, it is noticeable that problems with fuzzy coefficients and their corresponding full fuzzy problems share some hidden characteristics visible through certain solution approaches. To derive solution values to Model (7), one must use crisp decision variables (see C in Figure 1) through the computation, and can easily ignore the ranges of their values whenever the main focus is put on the objective function, i.e., on finding the fuzzy set that in the best way describes the possible values of the objective function. In some cases, paying more attention to the values of the decision variables, one can see that they describe fuzzy sets that easily fit in Model (8). Consequently, the border between the class of problems with fuzzy coefficients and the class of full fuzzy problems cannot be clearly stated, and solution approaches to problems with fuzzy coefficients might be extremely useful in solving full fuzzy problems, especially when they strictly comply with the extension principle. Figure 1

visually presents how the shapes of the decision variables in Models (7) and (8) comply with the extension principle.

By the literature review given in the next section, we aim to present the existing approaches to both classes of fuzzy mathematical programming problems, grouping them by the models they solve and their methodological particularities, and also conclude about the negative effects of eluding the extension principle in the crucial stages of the optimization process.

#### **4. Literature Review from the Extension-Principle-Based Perspective**

LP models both crisp and fuzzy are widely studied due to their simplicity. Ghanbari et al. [23] surveyed the literature on fuzzy LP problems discussing both the mathematical models and the solutions. Fractional programming models are the next most used models since they offer a relatively simple generalization of linear programming toward hard nonlinearity. Stanojevi´c et al. [24] reviewed the literature on LFP problems with fuzzy numbers. Transportation problems in both forms linear and fractional are also widely studied in the literature due to their particularities that bring consistent simplifications to the general solution approaches.

The main facts that make a difference among solution approaches to full fuzzy optimization problems are related to the way of ranking the fuzzy numbers when optimizing the objective function and/or evaluating the fuzzy constraints.

We organize our literature review in three subsections that cover the three classes of problems mentioned above and restrict our attention to those papers that are relevant to our discussion, which is focused on how the presence of the extension principle in formulating the solution concept contributes to the relevance of the derived fuzzy set solutions.

#### *4.1. Fuzzy Transportation Problems*

A full fuzzy linear TP consists of minimizing the fuzzy cost of transferring fuzzy amounts of goods from sources to destinations with fuzzy demands. Model (9)

$$\begin{array}{ll}\min & \sum\_{i=1}^{m} \sum\_{j=1}^{n} \widetilde{\varsigma}\_{ij} \widetilde{\chi}\_{ij} \\ \text{s.t.} & \\ & \sum\_{j=1}^{n} \widetilde{\chi}\_{ij} = \widetilde{a}\_{i\prime} & i = \overline{1, m\_{\prime}} \\ & \sum\_{i=1}^{m} \widetilde{\chi}\_{ij} = \widetilde{b}\_{j\prime} & j = \overline{1, n}\_{\prime} \\ & \widetilde{\chi}\_{ij} \ge \widetilde{0} & i = \overline{1, m}\_{\prime}, j = \overline{1, n}\_{\prime} \end{array} \tag{9}$$

summarizes the essential details, assuming that there are *m* sources with the amount of available goods expressed by the fuzzy quantities *ai*, *i* = 1, *m* and *n* destinations with the fuzzy demands *bj*, *j* = 1, *n*. The decision variables *<sup>x</sup>ij*, *i* = 1, *m*, *j* = 1, *n*, represent the amount of goods transported from the source *i* to the destination *j*, with a transport cost per unit equal to *cij*.

Liu and Kao [25] and Liu [26] proposed solution approaches to linear and linear fractional TPs with fuzzy coefficients, respectively, that fit to Model (7). Their solution concept is based on defining the fuzzy set of the optimal solution values of the objective function through a membership function that complies to the extension principle. Equation (10) shows their membership function adapted to the notation used in Model (7).

$$\mu\_{\tilde{f}}(z) = \begin{cases} \max\_{\substack{(b,c)|z=\max\_{\mathbf{x}\in X(b)} f(\mathbf{x},c)}} \left( \mu\_{\left(\overline{b},\overline{c}\right)}(b,c) \right), & \exists b,c|z = \max\_{\mathbf{x}\in X(b)} f(\mathbf{x},c), \\ 0, & \text{otherwise}, \end{cases} \tag{10}$$

where *<sup>μ</sup>*(*b*,*c*)(*b*, *c*) = min*<sup>μ</sup>b*(*b*), *<sup>μ</sup>c*(*c*), and *<sup>μ</sup>b*(*b*) and *μc*(*c*) represent the membership functions of the coefficients *b* and *c*, respectively.

Neither Liu and Kao [25] nor Liu [26] reported the shapes of the fuzzy sets of the optimal values of the decision variables. Later on, based on Liu and Kao's solution concept [25], the optimal values of the decision variables were empirically disclosed by a Monte Carlo simulation in [27] to be trapezoidal-like FNs whenever the problem's coefficients were TrFNs. In that way, it was proved that the problems solved by Liu and Kao [25] belong to the class of problems that can be modeled using fuzzy decision variables with a priori unknown shapes (see FUS in Figure 1). Stanojevi´c and Stanojevi´c [28] introduced an extension-principle-based formulation for the fuzzy set value of each decision variable and proposed mathematical models able to derive numerical approximations to the membership functions of the fuzzy sets representing the best values for the decision variables. Formulation (11) represents their definition for each decision variable *<sup>x</sup>i*, *i* = 1, *n* adapted to the notation used in Model (7).

$$\mu\_{\overline{x}\_i}(t) = \begin{cases} \max\_{(b,c)|t} \max\_{\substack{(b,c) \in X(b) \\ y \in X(b)}} \left( \mu\_{\left(\overline{b,c}\right)}(b,c) \right), & \exists b, c | t = \arg\_i \max\_{y \in X(b)} f(y,b), \\ 0, & \text{otherwise}, \end{cases} \tag{11}$$

where arg*i* max *y*∈*<sup>X</sup>*(*b*) *f*(*y*, *b*) represents the optimal value of the crisp scalar decision variable *yi* obtained when maximizing *f*(*y*, *b*) over the feasible set *<sup>X</sup>*(*b*), and *<sup>μ</sup>*(*b*,*c*)(*b*, *c*) has the same meaning as in the definition of the membership function of optimal solution values (10). The membership function of the fuzzy set *x* that collects the crisp values of the vector *x* of the decision variables, i.e.,

$$\mu\_{\vec{x}}(t) = \begin{cases} \max\limits\_{(b,c)\mid t} \max\limits\_{y \in X(b)} f(y,b) \left(\mu\_{\left(\vec{b},\vec{c}\right)}(b,c)\right), & \exists b,c \left| t = \arg\max\_{y \in X(b)} f(y,b), \\ 0, & \text{otherwise}, \end{cases} \right. \tag{12}$$

was first given in [29] for the general linear case. The graph of the membership function given in (11) represents in fact the projection of the graph of the membership function (12) on the plane (*Oxi*,*Oα*), *i* = 1, *n*, where the axes *Oxi* and *Oα* contain the values of the variable *xi* and the membership degrees *α*, respectively.

The first step in solving a general crisp TP is reserved to a simple balancing procedure. In the case of solving fuzzy TPs, the situation is quite different: some approaches can be applied only to balanced problems; several approaches focus only on balancing a general fuzzy TP; and also some approaches can be applied to unbalanced problems. Unfortunately, the solution provided by an approach applied to an unbalanced problem is far from being similar to the solution derived after a balancing procedure is applied. Mishra and Kumar [30] proposed a balancing procedure for FIF-TP that is an adaptation of Kumar and Kaur's method [31] to balance FF-TPs. Any solution approach to fuzzy TP based on a solution concept that is in accordance to the extension principle does not need a fuzzy balancing procedure, since crisp TPs are solved and their optimal solutions build the final fuzzy set solution to the original fuzzy problem. The methodology proposed in Stanojevi´c and Stanojevi´c [28] to solve FIF-TPs discussed this issue in detail.

Kumar and Kaur [31], after balancing the fuzzy transportation problem, proposed two approaches to solved it. In both approaches, they first applied algebraic operators on the fuzzy numbers to derive the fuzzy number values of the objective function and left-hand sides of the constraints; and then focused on the optimization of the objective function. In the first method, Kumar and Kaur applied a ranking function on the TrFN representing the objective function, obtained a crisp expression of the components of the involved TrFNs, and optimized it subject to a conjunctive constraint system. In the second method, they solved four crisp TPs, one for each component of the TrFN representing the objective function. None of these methods respects the extension principle.

Singh and Yadav [32] proposed several algorithms to determine a basic feasible solution to a balanced FIF-TP. All their algorithms are analogous to the well known methods developed for crisp TPs. This simple analogy also discards any possibility to follow the extension principle.

Stanojevi´c and Stanojevi´c [28] improved Liu and Kao's approach [25] by proposing a mathematical model with disjunctive constraint system and then extended it to solving TPs with TrIFN as parameters. They compare their results with the results reported by Kumar and Hussain [33], Ebrahimnejad and Verdegay [34] (who formulated their solution approach based on an accuracy function used for ordering the TrIFNs), and Mahmoodirad et al. [35]. All of these papers addressed FIF-TPs. The experiments showed that, due to its compliance with the extension principle, the method proposed in [28] found a fuzzy set solution with a wider support and smaller optimal values for the objective function that had to be minimized than the methods introduced in [33–35].

Further investigations are necessary to conclude about Mahajan and Gupta's approach [36] for solving a FIF-MO-TP problem. Using an accuracy function on each objective, they first reduced the problem to a crisp MO-TP, and developed an algorithm to solve it. To handle the intuitionistic fuzzy constraints, they used linear, exponential, and hyperbolic membership functions. Unifying the arithmetic and optimization, and formulating an extension-principle-based approach to solve the same problem, possibly better results can be derived.

Table 1 reports results found in the literature for three numerical examples of full fuzzy TPs. The first example uses trapezoidal fuzzy numbers, whereas the next two of them consider intuitionistic fuzzy numbers for all coefficients and decision variables. The results derived with a full respect to the extension principle correspond to the references written in bold. All full fuzzy TPs were minimization problems, and all optimal fuzzy numbers that comply to the extension principle are clearly smaller than those that elude it. For the first example, the values with maximal amplitude are the same in both references, but the second reference reports smaller minimal values with nonzero membership function. For the second and third examples, one can notice that smaller minimal values with nonzero membership function were obtained by the solution approach that strictly followed the extension principle. Figure 3 shows graphically the results reported in Table 1 for the second example.


**Table 1.** Comparative numerical results for full fuzzy and full intuitionistic fuzzy TPs.

**Figure 3.** Graphic representation of the results reported in Table 1, example 2. Symbol ∗ marks the reference that introduced the approach that comply to the extension principle.

#### *4.2. Fuzzy Linear Programming Problems*

A full fuzzy LP problem is generally described by Model (13)

$$\begin{array}{ll}\max & \tilde{\varepsilon}^{T}\tilde{\mathbf{x}}\\ \text{s.t.} & \tilde{A}\tilde{\mathbf{x}} \le \tilde{b},\\ & \tilde{\mathbf{x}} \ge 0. \end{array} \tag{13}$$

Several variants of Model (13) are sometimes addressed in the literature—for instance, those that minimize the objective function and/or use equality constraints to define the feasible set, but they are not different in nature, and their approaches can be easily adapted to solve Model (13) as well.

Perez-Canedo et al. [37] surveyed the literature on fuzzy linear programming, paying attention to those papers that used a lexicographic method to rank the fuzzy numbers. Their up-to-date review shows that fuzzy LP relying on lexicographic methods is an active research area with a wide range of applications in practice.

We briefly survey several papers on fuzzy linear programming, including notes from the extension-principle-based perspective.

Hosseinzadeh Lotfi et al. [38] used lexicographic method and fuzzy approximate solutions to FF-LP problems with TFNs. They transformed all TFNs coefficients in symmetric TFNs and assumed that all decision variables were symmetric TFNs. They also used a special ranking function on fuzzy numbers, and obtained a crisp MO-LP to solve. They did not involve the extension principle in the optimization step, and derived only one efficient solution to the crisp MO-LP.

Khan et al. [39] introduced a simplex-like technique for solving FF-LP problems. They involved a ranking function in the Gaussian elimination process and developed a flexible easy and reasonable solution algorithm. The use of any ranking function supposes defuzzification before optimization that is not in the desired accordance to the extension principle.

To transform FF-LP problems into crisp LPs, Kumar et al. [40] used a ranking function in order to compare the FN values of the objective function and a component-wise comparison of the left and right hand sides of the constraints. They finally solved one single objective linear programming problem deriving optimal values for all components of all decision variables. The use of two distinct methods to compare fuzzy quantities embedded in the same model is against the essence of the extension principle.

Ezzati et al. [41] addressed FF-LP problems by applying first fuzzy arithmetic to the TFN values of the decision variables and coefficients in order to derive the TFN value of the objective function; then, they constructed a three-objective crisp problem and solved it by a lexicographic method. Stanojevi´c and Stanojevi´c [29] proposed empirical solutions to FF-LP problems based on a Monte Carlo simulation that fully comply with the extension principle. They analyzed one of Ezzati et al.'s examples and conclude that Ezzati et al.'s methodology fails to follow the extension principle.

Noticing a shortcoming in the solution approach introduced in [41], Bhardwaj and Kumar [42] corrected and improved that approach by proposing a proper transformation able to replace the fuzzy inequality constraints by equivalent equality constraints. The effort put to rewrite the inequalities in a convenient form is unnecessary whenever the extension principle is applied.

The subsequent papers have come to our attention and require further examination. Das et al. [43] introduced a new method to solve FF-LP problems with TrFNs. Their method was based on solving a mathematical model derived from the MO-LP problem and lexicographic ordering method. They solved real-life problems as production planning and diet problem to illustrate the applicability of their approach.

Pérez-Cañedo and Concepción-Morales [44] derived a unique optimal fuzzy value to a FF-LP problem with inequality constraints containing unrestricted LR flat fuzzy coefficients and decision variables. In [45], they ranked LR-type IFNs using a lexicographic criterion, and introduced a method to derive solutions to FIF-LP problems with unique optimal values.

Khalifa [46] addressed FF-LP problems with coefficients and variables expressed by LR fuzzy numbers. He transformed the original fuzzy problem into a three-objective crisp LP problem, and solved it using a classic weighted sum method. The derived crisp solution was next used to construct the fuzzy solution to the original problem.

Khalili Goudarzi et al. [47] proposed a solution approach to FF-MILP problems. First, they formulate a crisp three-objective problem, and find the positive and negative ideal solutions to each objective. Then, they determined a linear membership function for each objective, and constructed a new achievement function defined as a convex combination of the lower bound for satisfaction degree of all objectives, and the weighted sum of the satisfaction degrees of all objectives. In this way, their approach aimed to ensure a balanced compromise solution.

Hamadameen and Hassan [48] introduced a compromise solution approach to FF-MO-LP problems based on a revised simplex method and a Gaussian elimination method.

Table 2 presents the numerical results derived in the literature to a maximization full fuzzy LP problem with triangular fuzzy coefficients and decision variables. Analyzing the reported results one can notice the improvement brought by the approach that complies to the extension principle: the optimal objective function value with maximal amplitude is significantly greater for the third reference than for the other two references. A significant difference can be seen observing the optimal values of the decision variables: the supports of the fuzzy numbers derived by the approach described in the third reference are much wider than those derived by the approaches introduced in the first two references. All values belonging to the supports of the reported fuzzy numbers are feasible values with certain nonzero membership degrees. The narrow supports seen for the first two references show that many relevant feasible values of the decision variables were ignored by the approaches that eluded the extension principle in the optimization step. Figure 4 gives a graphic representation of the results reported in Table 2 for the optimal objective values.

**Table 2.** Comparative numerical results for a full fuzzy LP problem.


**Figure 4.** Graphic representation of optimal objective values reported in Table 2. Symbol ∗ marks the reference that introduced the approach that comply to the extension principle.

#### *4.3. Fuzzy Linear Fractional Programming Problems*

A full fuzzy LFP problem differs from a full fuzzy LP problem only through its objective function. However, the nonlinearity of the objective function brings certain complications to the solution approaches. Model (14)

$$\begin{array}{ll}\max & \frac{\widetilde{c}^T \widetilde{\chi} + \widetilde{c}\_0}{\widetilde{d}^T \widetilde{\chi} + \widetilde{d}\_0} \\ \text{s.t.} & \widetilde{A} \widetilde{\chi} \le \widetilde{b}\_{\prime} \\ & \widetilde{\chi} \ge 0, \end{array} \tag{14}$$

has a linear fractional objective function that has to be maximized subject to linear constraints. Due to the presence of the denominator, an additional condition assuming its strictly positiveness over the feasible set has to be imposed. This condition does not reduce the generality of the problem, since the case of an objective function with strictly negative denominator can be equivalently transformed to a case complying to the initially imposed condition.

Except the methodology for solving a linear fractional transportation problem proposed by Liu [26], no other attempts were made in the literature to approach the fractional programming problems with an exclusive focus on the extension principle. All of the results reported in the papers that are briefly described below might be amended by an approach that fully comply to the extension principle. The analogical treatment of the FF-LFP problems inspired from the approaches to FF-LP problems is the main fact that underpins this conclusion.

Pop and Stancu-Minasian [49] transformed a FF-LFP problem into a crisp MO-LFP problem, and applied Buckley and Feuring's approach [50] to derive the final solution. Stanojevi´c and Stancu-Minasian [51] evaluated the fuzzy inequalities of a FF-LFP problem, and using a generalized form of Charnes-Cooper transformation linearized the original problem. Zadeh's principle was used exclusively for the fuzzy arithmetic between coefficients and variables. Das et al. [52] developed an algorithm for solving the FF-LFP problem. They used a generalized form of Charnes–Cooper transformation and obtained a crisp MO-LP model solved by a lexicographic method. For their numerical experiments, they used real-life problems, and for comparison, they used the approaches introduced in [49,51]. Any linearization of a fractional expression in fuzzy environment is not conformed with the extension principle if the same fuzzy quantity appears in both numerator and denominator of the ratio.

Stanojevi´c et al. [53] evaluated the approximation error that arises when the exact membership function of a ratio of TFNs is replaced by the membership function of a TFN, and its applications to decision-making. They mentioned for the first time the importance of using crisp decision variables in the optimization models even when fuzzy solutions were desired, which was a step toward unification of full fuzzy models with no a priori shapes for the decision variables and models with fuzzy coefficients and constraints that dictate the fuzziness of the decision variables.

Chinnadurai and Muthukumar [54] proposed a numerical approach to solving LFP problems in a fuzzy environment. They obtained (*<sup>α</sup>*,*<sup>r</sup>*)-acceptable optimal values by solving crisp bi-objective LFP problems. Their procedure was improved by Ebrahimnejad et al. [55], who modified the optimization model, and ensured the non-negativity of the fuzzy valued decision variables. A new extension principle-based solution concept able to embed the (*<sup>α</sup>*,*<sup>r</sup>*)-acceptable optimality is desired in order to assure further improvements.

Kaur and Kumar [56] discussed the shortcoming of papers [49,51] that resides in an improper interpretation of the fuzzy constraints of FF-LFP models. To avoid the constraint interpretation, they used Yager's ranking approach [57]. An effective way to remove the constraints interpretation from a solution approach is to transfer them in crisp environment, as proved in [28] for unbalanced FF-TP problems.

Arya et al. [58] are the first authors that addressed FF-MO-LFP problems. Their approach analogically follows Chakraborty and Gupta's method [59] designed to solve MO-LFP problems via fuzzy goals, and it is essentially based on a generalized Charnes– Cooper transformation whose arithmetic is in discordance with the extension principle.

Loganathan and Ganesan [60] transformed the original FF-LFP problem into a FF-LP problem and then replaced all coefficients and variables by their parametric forms derived from the *α*-cuts of triangular fuzzy numbers. They solved the parametric problem using a simplex-like algorithm, and derived parametric solutions. The fuzzy solutions were further derived numerically for different values of the parameter *α* ∈ [0, 1]. Because of the linearization step, the paper by Loganathan and Ganesan [60] might be affected by not complying to the extension principle.
