*2.1. Fuzzy-Interval Linear Programming (FILP)*

On account of parameter uncertainties and objective inconsistency in multi-objective programming [38–40], the FILP model can well handle the uncertainty parameters denoted by interval numbers, and also coordinate the conflicts among different objective functions by introducing membership function λ, which makes the resulting solutions more scientific and reliable. The model is summarized as follows [26]:

$$\text{Max}\lambda^{\pm} \tag{1a}$$

Subject to:

$$\mathcal{C}\_{\mathcal{S}}^{\pm} X^{\pm} \ge f\_{\mathcal{S}}^{-} + \lambda^{\pm} (f\_{\mathcal{S}}^{+} - f\_{\mathcal{S}}^{-}) \quad \text{g} = 1, 2, \dots, m \tag{1b}$$

$$\mathcal{C}\_{\hbar}^{\pm}X^{\pm} \le f\_{\hbar}^{+} - \lambda^{\pm}(f\_{\hbar}^{+} - f\_{\hbar}^{-}) \quad \hbar = m+1, \ldots, n \tag{1c}$$

$$A\_i^{\pm} X^{\pm} \le B\_i^{\pm} \quad i = 1, 2, \dots, k \tag{1d}$$

$$X^{\pm} \ge 0 \tag{1e}$$

$$0 \le \lambda^{\pm} \le 1 \tag{1f}$$

It is worth noting that:

$$\mathbb{C}\_{\mathcal{S}}^{\pm} X^{\pm} = \text{Max } f\_{\mathcal{S}}^{\pm} \quad \text{g} = 1, 2, \dots, m \tag{2a}$$

$$\mathbb{C}\_{\hbar}^{\pm}X^{\pm} = \text{Min }f\_{\hbar}^{\pm} \quad \hbar = m+1, \ \cdots, \ n \tag{2b}$$

where *C* ± *g* ∈ *R* ± 1 <sup>1</sup>×*<sup>t</sup>* , *C* ± *h* ∈ *R* ± 2 <sup>1</sup>×*<sup>t</sup>* , *A* ± *i* ∈ *R* ± 3 <sup>1</sup>×*<sup>t</sup>* , *X* <sup>±</sup> ∈ *R* ± 4 *<sup>t</sup>*×<sup>1</sup> , and *R* ± *<sup>e</sup>* means a set of interval numbers (*e* ∈ [1, 2, 3, 4]), *g* and *h* are core markers for maximizing and minimizing the objective functions individually, and *i* is the index of the constraints. *f* −, *f* <sup>+</sup> are the lower and upper bounds of *f* ±, and *λ* ± is the membership function in fuzzy decision-making. The larger the *λ* ± is, the more credible the calculation result would be; on the contrary, the smaller *λ* ± would lead to less credible results.
