A group decision making model based on goal programming with fuzzy hierarchy: an application to regional forest planning

Abstract

In this paper a multi-criteria group decision making model is presented in which there is a heterogeneity among the decision makers due to their different expertise and/or their different level of political control. The relative importance of the decision makers in the group is handled in a soft manner using fuzzy relations. We suppose that each decision maker has his/her preferred solution, obtained by applying any of the techniques of distance-based multi-objective programming [compromise, goal programming (GP), goal programming with fuzzy hierarchy, etc.]. These solutions are used as aspiration levels in a group GP model in which the differences between the unwanted deviations are interpreted in terms of the degree of achievement of the relative importance amongst the group members. In this way, a group GP model with fuzzy hierarchy (Group-GPFH) is constructed. The solution for this model is proposed as a collective decision. To show the applicability of our proposal, a regional forest planning problem is addressed. The objective is to determine tree species composition in order to improve the values achieved by Pan-European indicators for sustainable forest management. This problem involves stakeholders with competing interests and different preference schemes for the aforementioned indicators. The application of our proposal to this problem allows us to be able to comfortably address all these issues. The results obtained are consistent with the preferences of each stakeholder and their hierarchy within the group.

Appendices

Appendix 1: An extended goal programming model with fuzzy hierarchy

The corresponding specifications introduced in Table 1 lead to the following models:

• Conventional mathematical programming (CMP):

  $$\begin{aligned} \begin{array}{l} \min \;\; \alpha _1 n_1 + \beta _1 p_1 \\ \mathrm {s.t.} \\ \quad f_1 (x) + n_1 - p_1 = f_1^* \\ \quad n_1 ,p_1 \ge 0,\quad n_1 \times p_1 = 0, \\ \quad x \in X. \\ \end{array} \end{aligned}$$

  the objective is assumed to be minimized, then we have $n_1=0$ and $p_1=f_1(x)-f_1^*$. By substituting this expression in the objective function of (CMP), the conventional mathematical programming model is obtained.

• Weighting model (WM):

  $$\begin{aligned} \begin{array}{l} \min \;\; \sum \limits _{i = 1}^m {\left( -\alpha _i n_i + \alpha _i p_i \right) } \\ \mathrm {s.t.} \\ \quad f_i (x) + n_i - p_i = 0 ,\quad i = 1,\ldots ,m, \\ \quad n_i, \quad p_i \ge 0,\quad n_i \times p_i = 0,\quad i = 1, \ldots ,m, \\ \quad x \in X. \\ \end{array} \end{aligned}$$

  In (WM) we have $f_i(x)=p_i-n_i$ and substituing in the objective function we obtain the usual weighted model.

• Compromise Programming ($L_1$ metric):

  $$\begin{aligned}&\min \;\; \sum \limits _{i = 1}^m {\left( \gamma _i W_i n_i +\eta _i W_i p_i \right) } \end{aligned}$$
  $$\begin{aligned}&\mathrm {s.t.} \\&\quad f_i (x) + n_i - p_i = f_i^* ,\quad i = 1,\ldots ,m, \\&\quad n_i ,p_i \ge 0,\quad n_i \times p_i = 0,\quad i = 1, \ldots ,m, \\&\quad x \in X. \\ \end{aligned}$$

  In this case we have $\gamma _i=0$ for minimizing objectives and $\eta _i=0$ for maximizing objectives; $W_i$ is a preferential as well as normalizing factor.

• Compromise Programming ($L_\infty$ metric):

  $$\begin{aligned} \begin{array}{l} \min \;\; D \\ \mathrm {s.t.} \\ \quad \gamma _i W_i n_i +\eta _i W_i p_i \le D \\ \quad f_i (x) + n_i -p_i = f_i^* ,\quad i = 1,\ldots ,m, \\ \quad n_i ,p_i \ge 0,\quad n_i \times p_i = 0,\quad i = 1, \ldots ,m, \\ \quad x \in X. \\ \end{array} \end{aligned}$$

• linear GP with Archimedian weights:

  $$\begin{aligned} \begin{array}{l} \min \;\; \sum \limits _{i = 1}^m {\left( \alpha _i n_i + \beta _i p_i \right) } \\ \mathrm {s.t.} \\ \quad f_i (x) + n_i - p_i = t_i ,\quad i = 1,\ldots ,m, \\ \quad n_i ,p_i \ge 0,\quad n_i \times p_i = 0,\quad i = 1, \ldots ,m, \\ \quad x \in X. \\ \end{array} \end{aligned}$$

• MINMAX GP:

  $$\begin{aligned} \begin{array}{l} \min \;\; D \\ \mathrm {s.t.} \\ \quad \alpha _i n_i +\beta _i p_i \le D \\ \quad f_i (x) + n_i -p_i = t_i ,\quad i = 1,\ldots ,m, \\ \quad n_i ,p_i \ge 0,\quad n_i \times p_i = 0,\quad i = 1, \ldots ,m, \\ \quad x \in X. \\ \end{array} \end{aligned}$$

• Extended GP:

  $$\begin{aligned} \begin{array}{l} \min \;\; \left( 1-\lambda \right) D + \lambda \sum \limits _{i = 1}^m {\left( \alpha _i n_i + \beta _i p_i \right) } \\ \mathrm {s.t.} \\ \quad \alpha _i n_i +\beta _i p_i \le D \\ \quad f_i (x) + n_i -p_i = t_i ,\quad i = 1,\ldots ,m, \\ \quad n_i ,p_i \ge 0,\quad n_i \times p_i = 0,\quad i = 1, \ldots ,m, \\ \quad x \in X. \\ \end{array} \end{aligned}$$

Parameter $\lambda$ weights the importance attached to the minimizing of the weighted sum of unwanted desviation variables. For $\lambda \in \left( 0,1\right)$ we have intermediate solutions between the solutions provided by de MINMAX GP model $\left( \lambda =0 \right)$ and the Archimedian GP model $\left( \lambda =1 \right)$.

• Archimedean GP with Fuzzy Hierarchy:

  $$\begin{aligned} \begin{array}{l} \min \;\; \lambda \sum \limits _{i = 1}^m \displaystyle \gamma _{i}\frac{n_i}{u_i} + \displaystyle \eta _{i}\frac{p_i }{u_i} - \left( 1- \lambda \right) \mathop {\mathop {\mathop {\sum }\limits _{i,j = 1}}_{i \ne j}}\limits ^m \; \sum \limits _{l=1}^{3} \, b_{\tilde{R}_l \left( {i,j} \right) } \mu _{\tilde{R}_l} \left( {h_i,h_j} \right) \\ \mathrm {s.t.} \\ \\ f_i (x) + n_i^{} - p_i^{} = t_i ,\quad i = 1,\ldots ,m, \\ \\ \displaystyle 1 - \left( \frac{h_i }{u_i } - \frac{h_j}{u_j } \right) \ge \mu _{\tilde{R}_1} (h_i,h_j), \quad \mathrm{{if}} \quad {b_{\tilde{R}_1 \left( i,j \right) }} = 1, \\ \\ \displaystyle \frac{1 - \displaystyle \left( \frac{h_i }{u_i } - \frac{h_j}{u_j } \right) }{2} \ge \mu _{\tilde{R}_2} (h_i,h_j),\quad \mathrm{{if }} \quad b_{\tilde{R}_2} \left( i,j \right) = 1, \\ \\ \displaystyle \frac{{h_j }}{{u_j }} - \frac{{h_i} }{{u_i }} \ge \mu _{\tilde{R}_3} (h_i,h_j),\quad \mathrm{{if }} \quad {b_{\tilde{R}_3 \left( {i,j} \right) }} = 1, \\ \quad i,j = 1, \ldots , m, \\ \\ \quad 0 \le \mu _{\tilde{R}_l} (h_i,h_j) \le 1,\quad l = 1,2,3, \\ \\ \quad n_i ,p_i \ge 0,\quad n_i \times p_i = 0,\quad i = 1, \ldots ,m, \\ \\ \quad x \in X. \\ \end{array} \end{aligned}$$

• MINMAX GP with Fuzzy Hierarchy:

  $$\begin{aligned}&\min \;\;\lambda D - \; \left( 1- \lambda \right) \mathop {\mathop {\mathop {\sum }\limits _{i,j = 1}}_{i \ne j}}\limits ^m \; \sum \limits _{l=1}^{3} \, b_{\tilde{R}_l \left( {i,j} \right) } \mu _{\tilde{R}_l} \left( {h_i,h_j} \right) \\&\mathrm {s.t.} \\&\displaystyle \gamma _{i}\frac{n_i}{u_i} + \displaystyle \eta _{i}\frac{p_i}{u_i} \le D \\&f_i (x) + n_i^{} - p_i^{} = t_i ,\quad i = 1,\ldots ,m, \\&\displaystyle 1 - \left( \frac{h_i }{u_i } - \frac{h_j}{u_j }\right) \ge \mu _{\tilde{R}_1} (h_i,h_j),\quad \mathrm{{if}} \quad {b_{\tilde{R}_1 \left( i,j \right) }} = 1, \\&\displaystyle \frac{1 - \displaystyle \left( \frac{h_i }{u_i } - \frac{h_j}{u_j } \right) }{2} \ge \mu _{\tilde{R}_2} (h_i,h_j),\quad \mathrm{{if }} \quad b_{\tilde{R}_2} \left( i,j \right) = 1, \\&\displaystyle \frac{{h_j }}{{u_j }} - \frac{{h_i} }{{u_i }} \ge \mu _{\tilde{R}_3} (h_i,h_j) ,\quad \mathrm{{if }} \quad {b_{\tilde{R}_3 \left( {i,j} \right) }} = 1, \\&\quad i,j = 1, \ldots , m, \\&\quad 0 \le \mu _{\tilde{R}_l} (h_i,h_j) \le 1,\quad l = 1,2,3, \\&\quad n_i ,p_i \ge 0,\quad n_i \times p_i = 0,\quad i = 1, \ldots ,m, \\&\quad x \in X. \\ \end{aligned}$$

Appendix 2

See Tables 6, 7 and 8.

Table 6 Value of indicators for different tree species

Table 7 Anti-ideal and ideal indicators

Table 8 Bounds for different tree species