On the Influence of Altering the Action Set on PROMETHEE II’s Relative Ranks

Abstract

For some multi-criteria decision aid methods, the relative ranks of two actions may be inverted when the original set is altered. This phenomenon is known as rank reversal. In this contribution, we formalise rank reversal for the Promethee II method. The aim is not to debate about the legitimacy of such effect but rather to derive the exact conditions for its occurrence when one or more actions are added or removed from/to the original set. These conditions eventually lead us to: (1) assess whether rank reversal between a given pair of actions is, at all, possible, and (2) characterise the evaluations of the actions that have to be added or removed to induce rank reversal. We also propose two metrics that express the “strength” of and the “sensitivity” towards rank reversal. Finally, we show on a toy example how they could be used in a decision making process.

A Domains of extreme values for the \({{z_h}\left( {x}\right) }\) Function

The possible contribution of one additional action x to the rank reversal of two actions $a_i$ and $a_j$ is mainly represented by the corresponding value of the ${{z_h}\left( {x}\right) }$ function (defined in Sect. 3). Assuming that $a_i$ is initially better ranked than $a_j$, i.e., ${{\varDelta \phi }\left( {a_i,a_j}\right) } = {{\phi }\left( {a_i}\right) } - {{\phi }\left( {a_j}\right) } > 0$, then the lower the value of ${{z_h}\left( {x}\right) }$, the more the additional action x (with its evaluation $g_h(x)$) contributes to $a_i$ and $a_j$’s rank reversal. Depending on the actions’ evaluation difference on criterion h, $\varDelta g_h (a_i, a_j) = g_h(a_i) - g_h(a_j)$, we observe different scenarios, yielding different characteristic values for the minimum and maximum of ${{z_h}\left( {x}\right) }$. The following table presents, for a given criterion h, the main possible scenarios. However, as the ${{z_h}\left( {x}\right) }$ function also depends on the preference parameters, $q_h$ and $p_h$, other “degenerated” cases may also occur. These are not represented here for the sake of readability. Each row of the table corresponds to one scenario, depending on the evaluation difference ${{z_h}\left( {x}\right) }$ given in the first column. The second column schematically depicts the shape of the corresponding $z_h$ function as a function of the evaluation ${{g_h}\left( {x}\right) }$ of the added action x on criterion h. For the first row, for instance, the plot shows that the addition action evaluation (that contributes the most to rank reversal of $a_i$ and $a_j$) are located in between $g_h(a_i)$ and $g_h(a_j)$. The third column provides extreme values associated to the scenario:

• $\smash {\underline{z}}_h = \min _{{{g_h}\left( {x}\right) } \in [0,1]} {{z_h}\left( {a_i,a_j;x}\right) }$

• $\smash {\overline{z}}_h = \max _{{{g_h}\left( {x}\right) } \in [0,1]} {{z_h}\left( {a_i,a_j;x}\right) }$

• $\smash {\underline{\mathcal {G}}}_h = {{\mathrm{arg\,min}}}_{{{g_h}\left( {x}\right) } \in [0,1]} {{z_h}\left( {a_i,a_j;x}\right) }$

• $\smash {\overline{\mathcal {G}}}_h = {{\mathrm{arg\,max}}}_{{{g_h}\left( {x}\right) } \in [0,1]} {{z_h}\left( {a_i,a_j;x}\right) }$

$\smash {\underline{z}}_h$ and $\smash {\overline{z}}_h$, respectively, indicate the minimum and maximum value of the function $z_h(a_i,a_j,;x)$. $\smash {\underline{\mathcal {G}}}_h$ and $\smash {\overline{\mathcal {G}}}_h$ represent the corresponding interval(s) of the additional action’s evaluation $g_h(x)$ that lead to the minimum and maximum values. While the two first inform the user to what extent the pair of actions $(a_i,a_j)$ may suffer from rank reversal, the two last ones indicate for what values of $g_h(x)$ rank reversal is provoked ($\smash {\underline{\mathcal {G}}}_h$) or, on the contrary, what values of $g_h(x)$ do “stabilise” most strongly the relative ranking of $a_i$ with respect to $a_j$.