Preference elicitation for a ranking method based on multiple reference profiles

Abstract

Multiple criteria decision aid methodologies support decision makers (DM) facing decisions involving conflicting objectives. DM’s preferences should be captured to provide meaningful recommendations. Preference elicitation aims at incorporating DM’s preferences in decision models. We propose a new preference elicitation tool for a ranking model based on reference points (RMP—Ranking with Multiple Profiles). Our methodology infers an RMP model from a list of pairwise comparisons provided by the DM. The inference algorithm makes use of a Mixed Integer mathematical programming formulation. We prove the applicability by performing extensive numerical experiments on datasets whose size corresponds to real-world problem.

Appendix: Invariance with respect to Irrelevant alternative, the case of pairwise comparison methods

Consider a multiple criteria ranking method in which a weak preference relation $\succsim$ is constructed on the set of alternatives, based on a weighted voting of criteria, and where the ranking is defined computing on $\succsim$ the net flow score of alternatives [5]. More precisely, we consider the relation $\succsim$ defined on $\mathcal{A}$ as follows. for any pair $x,y \in \mathcal{A}$:

$$\begin{aligned} x \succsim y \Leftrightarrow \sum _{j:x_j\ge y_j}w_j \ge \sum _{j:y_j\ge x_j}w_j \end{aligned}$$

where $x=(x_1, ..., x_m)$ and $y=(y_1, ..., y_m)$. The relation $\succsim$ is exploited to rank alternatives using the net flow score $NF(x), \; x \in \mathcal{A}$: $NF(x)=|f_l(x)|-|f_e(x)|$ where $f_l(x)$ ($f_e(x)$, respectively) represents the leaving flow of x (the entering flow of x, respectively), and is defined by $f_l(x) = \{ y\in \mathcal{A} : x \succsim y \}$ ($f_e(x) = \{ y\in \mathcal{A} : y \succsim x \}$, respectively).

Consider a small example involving 3 criteria (to be maximized) and 6 alternatives ($\mathcal{A} = \{ a,b,c,d,e,f \}$) with the alternatives evaluations described in Table 6.

Table 6 Small example

Suppose that the DM is able to provide preference information concerning $\mathcal{A}^* \subset \mathcal{A}$ a reference set of alternatives $\mathcal{A}^* = \{ a, b, c, d \}$ through the form of a ranking on $\mathcal{A}^*$ : $a \succ b \succ c \succ d$. Note that the informational content of this ranking boils down to the fact that none of the three criteria is a majority alone (i.e, $w_1+w_2> w_3$, $w_1+w_3> w_2$ and $w_2+w_3> w_1$). Hence, any inference program which would compute the criteria weights from this ranking will find weights compatible with these three inequalities. The computation of the ranking on $\mathcal{A}^*$ using such weights using the net flow score is provided in Table 7, and leads to the ranking: $a \succ b \succ c \succ d$.

Table 7 Outranking relation on ${\mathcal {A}}^*$

Suppose now that we want to compute the ranking on the whole set $\mathcal{A}$ (including e and f), based on the weights inferred from the ranking on $\mathcal{A}^*$($a \succ b \succ c \succ d$), i.e., using weights such that $w_1+w_2> w_3$, $w_1+w_3> w_2$ and $w_2+w_3> w_1$. This is provided in Table 8 bellow, and leads to a ranking in which b is ranked first, a is ranked second, then e and f at equal rank, then c then d. It appear that b is ranked better than a, in contradiction with the initially provided preference ranking.

Hence, it appears that, when using such ranking method, the ranking on $\mathcal{A}$ using the weights inferred from the ranking on $\mathcal{A}^*$ (provided by the DM) does not necessarily extend the ranking on $\mathcal{A}^*$. Such observation, makes it difficult to use such ranking method in a disagregation perspective.

Table 8 Outranking relation on ${\mathcal {A}}$