2.1.1. Fuzzy Numbers

Fuzzy numbers (FNs) are special cases of fuzzy sets. A fuzzy set *A* of the universe of real numbers *R* is called a fuzzy number if and only if: (i) it is fuzzy normal and fuzzy convex; (ii) the membership function *μA* is upper semicontinuous; and (iii) its support, i.e., *x* ∈ *<sup>R</sup>*|*μA*(*x*) > 0 is bounded. Similarly, an intuitionistic fuzzy set of *R* is an intuitionistic fuzzy number (IFN) if and only if its membership function fulfills all conditions in the definition of a fuzzy number; the nonmembership function is fuzzy concave and lower semicontinuous; and its support *x* ∈ *<sup>R</sup>*|*<sup>ν</sup>A*(*x*) < 1 is bounded.

An LR flat fuzzy number is defined using two reference functions for the left and right sides of the fuzzy number, respectively. The reference functions *L* and *R* are both defined on the interval [0, <sup>∞</sup>), take values from the interval [0, 1], and have two essential characteristics: (i) *L*(0) = *R*(0) = 1; and (ii) both *L* and *R* are nonincreasing on [0, <sup>∞</sup>). We refer the reader to the book of Dubois and Prade [14] for more details.

In what follows, we are interested in triangular and trapezoidal fuzzy and intuitionistic fuzzy numbers. The graph of the nonzero piece of the membership function of a triangular fuzzy number (TFN) forms a triangle with the abscissa, and is generally expressed by its components, as a triple (*<sup>a</sup>*1, *a*2, *<sup>a</sup>*3), *a*1 ≤ *a*2 ≤ *a*3. The interval (*<sup>a</sup>*1, *<sup>a</sup>*3) is the support of the fuzzy set and the component *a*2 is the value with the maximal amplitude. Similarly, the graph of the nonzero piece of the membership function of a trapezoidal fuzzy number (TrFN) forms a trapezoid with the abscissa, and is generally expressed by its components, as a quadruple (*<sup>a</sup>*1, *a*2, *a*3, *<sup>a</sup>*4), *a*1 ≤ *a*2 ≤ *a*3 ≤ *a*4. The interval (*<sup>a</sup>*1, *<sup>a</sup>*4) is the support of the fuzzy set and all values in the interval [*<sup>a</sup>*1, *<sup>a</sup>*4] have the maximal amplitude.

A triangular intuitionistic fuzzy number (TIFN) is generally denoted by

$$
\bar{A}^I = (a\_1, a\_2, a\_3; a\_1', a\_2', a\_3'). \tag{3}
$$

Its first three components are related to the membership function (that is identical to a membership function of a triangular fuzzy number), and last three related to the nonmembership function. Similarly, a trapezoidal intuitionistic fuzzy number (TrIFN) is generally described by

$$
\dot{A}^{I} = (a\_1, a\_2, a\_3, a\_4; a\_1', a\_2', a\_3', a\_4'), \tag{4}
$$

first four components being related to the membership function that is in fact a membership function of a trapezoidal fuzzy number.

#### 2.1.2. The Extension Principle

Bellman and Zadeh [15] introduced the concepts of fuzzy decisions and fuzzy constraints, and proposed a principle to aggregate them. The fuzzy arithmetic was developed with the help of the extension principle mentioned by Zadeh from the beginning in [1].

According to this principle, the fuzzy set *B* of the universe *Y* that is the result of evaluating the function *f* at the fuzzy sets *<sup>A</sup>*1, *<sup>A</sup>*2, ... , *Ar* over their universes *X*1, *X*2, ... , *Xr* is defined through its membership function as

$$\mu\_{\overline{\mathcal{B}}}(y) = \begin{cases} \sup\_{(\mathbf{x}\_1, \dots, \mathbf{x}\_r) \in f^{-1}(y)} \left( \min \left\{ \mu\_{\overline{A}\_1}(\mathbf{x}\_1), \dots, \mu\_{\overline{A}\_r}(\mathbf{x}\_r) \right\} \right), & f^{-1}(y) \neq \mathcal{O} \\\ 0, & \text{otherwise.} \end{cases} \tag{5}$$

See also Zimmerman [16] for more details.

Fuzzy addition and subtraction of triangular and trapezoidal fuzzy numbers both yield triangular and trapezoidal numbers, respectively. Strictly following the extension principle, neither fuzzy multiplication nor division of triangular/trapezoidal fuzzy numbers yield triangular/trapezoidal numbers. This issue is generally overcome using a relative innocent approximation that replaces the exact results by those triangular/trapezoidal numbers that keep the extreme values (i.e., the endpoints of their support, and the values with maximal amplitude) the same.

Diniz et al. [17] discussed the optimization of a fuzzy-valued function using Zadeh's extension principle. The objective function was a Zadeh's extension of a function with respect to a parameter and an independent variable. Kupka [18] introduced some results on the approximation of Zadeh's extension of a given function, and studied the quality of the approximation with respect to the choice of the metric on the space of the fuzzy sets.

#### *2.2. Mathematical Programming Models*

#### 2.2.1. The General Crisp Model

Model (6) generally defines a crisp mathematical programming problem that consists in maximizing the objective function *f* that depends on the coefficients *c*, and the decision variables *x* over the feasible set *X*(*b*) that depends on the coefficients *b*.

$$\begin{array}{ll}\max & f(\mathbf{x}, \mathbf{c}),\\\text{s.t.} & \mathbf{x} \in X(b), \end{array} \tag{6}$$

Under additional assumptions (imposed on the objective function and/or the constraints), Model (6) becomes convex (if the objective function is convex over the feasible set that is also convex); linear (if both the objective function and constraint system are defined by linear expressions, i.e., *f*(*<sup>x</sup>*, *c*) = *<sup>c</sup>Tx*, and *<sup>X</sup>*(*<sup>A</sup>*, *b*) = {*x*|*Ax* ≤ *b*, *x* ≥ <sup>0</sup>}, with *A*, *b*, and *c* being matrices of certain dimensions); linear fractional (when the objective function is a ratio of linear functions); mixed integer (if some decision variables take integer and/or binary values); multiple objective (if the values of the objective function are vectors), etc. Specific models have specific solution methods. The basic references here are [19] for linear programming, [20] for multiple objective programming, [21] for linear fractional programming, and [22] for integer programming.
