**3.2 The lexicographic goal programming approach (LGP)**

The LGP is an extension of linear programming (LP), was originally introduced by [21] and further presented by [22], and others. This technique was developed to handle multi-criteria situations within the general framework of LP. In the variant of the lexicographic Goal Programming, the objectives are ranked in order of priority, as the relative importance given to them by the decision maker. The mathematical formulation corresponding to this variant consists of a vector the deviations ordered on different objectives, which implies a minimization in the order of the different priority levels q [23]. The mathematical program is written as follows:

$$l \text{ } l \text{ex.} \, \text{Min.} \, \underline{\text{L}} = [l \mathbf{1}(\boldsymbol{\delta}^-, \boldsymbol{\delta}^+), \dots, lq(\boldsymbol{\delta}^-, \boldsymbol{\delta}^+)] \tag{5}$$

Subject to:

$$\begin{aligned} \mathbf{a\_{i\downarrow}x\_{\mathfrak{j}}} + \boldsymbol{\delta}^{-} \mathbf{1}\_{\mathfrak{j}} + \boldsymbol{\delta}^{+} \boldsymbol{\chi} &\leq \mathbf{g\_{i}};\\ \mathbf{a\_{2\circ}x\_{\mathfrak{j}}} + \boldsymbol{\delta}^{-} \mathbf{2}\_{\mathfrak{j}} + \boldsymbol{\delta}^{+} \boldsymbol{\varrho} &\leq \mathbf{g\_{i}};\\ &\dots\\ \mathbf{a\_{n\circ}x\_{\mathfrak{j}}} + \boldsymbol{\delta}^{-} \mathbf{n} + \boldsymbol{\delta}^{+} \boldsymbol{\chi} &\leq \mathbf{g\_{n}};\\ \boldsymbol{\delta}^{-} \mathbf{i}\_{\mathfrak{i}}, \boldsymbol{\delta}^{+} \boldsymbol{\varrho} &\geq \mathbf{0} \text{ pour } (\mathfrak{i} = \mathbf{1}, 2, \dots, \mathbf{n}) \end{aligned} \tag{7}$$

(7)

$$\mathbf{x}\_{\mathbf{j}} \ge \mathbf{0} \text{ pour}(\mathbf{j} = \mathbf{1}, \mathbf{2}, \dots, \mathbf{n})$$
