Robust ordinal regression for decision under risk and uncertainty

Abstract

We apply the Robust Ordinal Regression (ROR) approach to decision under risk and uncertainty. ROR is a methodology proposed within multiple criteria decision aiding (MCDA) with the aim of taking into account the whole set of instances of a given preference model, for example instances of a value function, which are compatible with preference information supplied by the Decision Maker (DM) in terms of some holistic preference comparisons of alternatives. ROR results in two preference relations, necessary and possible; the necessary weak preference relation holds if an alternative is at least as good as another one for all instances compatible with the DM’s preference information, while the possible weak preference relation holds if an alternative is at least as good as another one for at least one compatible instance. To apply ROR to decision under risk and uncertainty we have to reformulate such a problem in terms of MCDA. This is obtained by considering as criteria a set of quantiles of the outcome distribution, which are meaningful for the DM. We illustrate our approach in a didactic example based on the celebrated newsvendor problem.

Appendices

Appendix 1: Constraints of the ELECTRE$^{GKMS}$ method

In this Appendix, we shall give more details on the technical constraints of the ELECTRE$^{GKMS}$ method as well as on the constraints translating the preference information provided by the DM.

• Preference information on comparisons between alternatives $a^{*},b^{*}\in A^{R}\subseteq A$:

  • For all $(a^{*},b^{*}) \in A^R$ such that $a^{*} S b^{*}$:

    $$\begin{aligned} C(a^{*},b^{*})= \sum _{j=1}^m \psi _j(a^{*},b^{*}) \ge \lambda \quad \text{ and } \quad g_j(b^{*}) - g_j(a^{*}) < v_j(a^{*}), \ j=1,\ldots ,m, \end{aligned}$$

    where $\psi _{j}(a,b)=\varphi _{j}(a,b)\times w_j$ for all j and for all $(a,b)\in A\times A$.

  • For all $(a^{*},b^{*}) \in A^R$ such that $a^{*} S^C b^{*}$:

    $$\begin{aligned} C(a^{*},b^{*})= \sum _{j=1}^m \psi _j(a^{*},b^{*}) < \lambda \quad \text{ or } \quad \exists j \in G : g_j(b^{*}) - g_j(a^{*}) \ge v_j(a^{*}), \end{aligned}$$

    which can be modeled as:

    $$\begin{aligned} C(a^{*},b^{*})= \sum _{j=1}^m \psi _j(a^{*},b^{*}) < \lambda + M_0(a^{*},b^{*}) \quad \text{ and } \quad g_j(b^{*}) - g_j(a^{*}) \ge v_j(a^{*}) - \delta M_j(a^{*},b^{*}), \end{aligned}$$

    where $M_j(a^{*},b^{*}) \in \{0,1\}, \ j=0,\ldots ,m$, $\displaystyle \sum _{j=0}^m M_j(a^{*},b^{*}) \le m$ and $\delta$ is an auxiliary variable equal to a big positive value (i.e. $\delta \ge max_j \{ x_j^{m_j} - x_j^0\}$).

    A vector M of binary variables is used to express that either the concordance test or non-discordance test has to be negative. Notice that if $M_j(a^{*},b^{*}) = 0$, then a corresponding j-th condition causes discordance with statement $a^{*} S b^{*}$. On the contrary, if $M_j(a^{*},b^{*}) = 1$, then the respective condition is always satisfied, not making veto. However, there has to be at least one non-equality for which $M_j(a^{*},b^{*}) = 0$, as $\displaystyle \sum _{j=0}^m M_j(a^{*},b^{*}) \le m.$

• Limitations on the values of inter-criteria parameters: $\lambda , v_j(a), \forall a \in A, \,$ and $w_j \ j = 1, \ldots , m$:

  • The range of allowed values of a concordance threshold, $0.5 \le \lambda \le 1.0$,

  • Normalization of the marginal concordance indices for all criteria so that the indices corresponding to the largest difference in evaluations of two alternatives on each criterion $(g_j(a^*_j) - g_j(a_{j,*}) = x_j^{m_j} - x_j^0)$ sum up to 1:

    $$\begin{aligned} \sum _{j=1}^m \psi _j(a^*_j,a_{j,*})=1 \quad \text{ with } a^*_j,a_{j,*} \in A,\quad \ j=1,\ldots ,m. \end{aligned}$$

    As we previously normalized weights of the criteria so that they sum up to 1, each weight is now understood as a maximal share of each criterion in the global concordance index. Consequently, $w_j = \psi _j(a^*_j,a_{j,*}), \ j=1,\ldots ,m.$

  • Lower bounds on the values of veto thresholds $v_j(a), \ j = 1, \ldots , m, \ \forall a \in A$:

    • $v_j(a)$ needs to be larger than corresponding preference threshold (the greatest value of a preference threshold $p_{j}^{R}(a)$ allowed by the DM), $v_j(a) > p_j^{R}(a), \ j \in G_1$,²

    • $v_j(a)$ is required to be larger than $g_j(b) - g_j(a)$ for all pairs of alternatives (a, b) for which the DM stated that the difference between $g_j(a)$ and $g_j(b)$ is non-significant ($g_j(a) \le g_j(b)$), $a \sim _j b, \ j \in G_2$ ³ (we do not refer here to pairs $(a^{*},b^{*})$ such that $a^{*} \succ _j b^{*}$, because even if the DM states that the difference between $g_j(a^{*})$ and $g_j(b^{*})$ is relevant, the veto threshold $v_j$ can still be less, equal, or larger than $g_j(a^{*})-g_j(b^{*})$).

  • The function determining the values of veto thresholds $v_j(a)$ is required to be monotone non-decreasing with respect to $g_j(a)$:

    $$\begin{aligned} v_j(a) \ge v_j(b) \text{ if } g_j(a) > g_j(b) \quad \text{ and } \quad v_j(a) = v_j(b) \text{ if } g_j(a) = g_j(b), \ j=1,\ldots ,m. \end{aligned}$$

    If the DM wishes to model the veto threshold with a constant value (not dependent on $g_j(a)$), we would skip the monotonicity constraints and replace $v_j(a)$ with $v_j$ in all aforementioned formulas.

• Restrictions concerning the value of marginal concordance indices $\psi _j(a,b), \ j = 1, \ldots , m$:

  • $\psi _j(a,b) = 0$ if $g_j(b) - g_j(a) \ge p_j^{R}(a)$, for all $(a,b) \in A \times A$,

  • $\psi _j(a,b) > 0$ if $g_j(a) - g_j(b) > -p_{j}^{L}(a)$ for all $(a,b) \in A \times A$,

  • $\psi _j(a,b) = \psi _j(a^*_j,a_{j,*})$ if $g_j(a) - g_j(b) \ge -q_{j}^{L}(a)$, for all $(a,b) \in A \times A$;

  • $\psi _j(a,b) < \psi _j(a^*_j,a_{j,*})$ if $g_j(b) - g_j(a) > q_{j}^{R}(a)$, for all $(a,b) \in A \times A$,

  • $\psi _j(a,b) = 0$ if $b \succ _j a$,

  • $\psi _j(a,b) = \psi _j(a^*_j,a_{j,*}), \ \psi _j(b,a) = \psi _j(a^*_j,a_{j,*})$ if $a \sim _j b$.

• Monotonicity of the functions of marginal concordance indices $\psi _j(a,b), \ j = 1, \ldots , m$:

  • If, according to the preferences of the DM, indifference $q_j$ and preference $p_j$ thresholds for criterion $g_j$ are not dependent on $g_j(a)$, then $\forall a,b,c,d \in A$, and for $j=1,\ldots ,m$:

    • $\psi _j(a,b) \ge \psi _j(c,d) \text{ if } g_j(a) - g_j(b) > g_j(c) - g_j(d),$

    • $\psi _j(a,b) = \psi _j(c,d) \text{ if } g_j(a) - g_j(b) = g_j(c) - g_j(d).$

  • If according to the preferences of the DM indifference $q_j(a)$ and preference $p_j(a)$ thresholds depend on the value of $g_j(a)$, then, for $j=1,\ldots ,m$:

    • $\psi _j(a,c) \ge \psi _j(b,c) \text{ if } g_j(a) > g_j(b) \text{ for } \text{ all } a, b, c \in A,$

    • $\psi _j(a,c) = \psi _j(b,c) \text{ if } g_j(a) = g_j(b) \text{ for } \text{ all } a, b, c \in A,$

    • $\psi _j(a,b) \ge \psi _j(a,c) \text{ if } g_j(b) < g_j(c) \text{ for } \text{ all } a, b, c \in A,$

    • $\psi _j(a,b) = \psi _j(a,c) \text{ if } g_j(b) = g_j(c) \text{ for } \text{ all } a, b, c \in A.$

The whole set of monotonicity and normalization constraints together with the constraints translating the preference information provided by the DM is denoted by $E^{A^{R'}}$:

$$\begin{aligned} \left. \begin{array}{l} \text{ Pairwise } \text{ comparison } \text{ stating } a^{*} S b^{*} \text{ or } a^{*} S^C b^{*}: \\ { \quad } C(a^{*},b^{*})= \sum _{j=1}^m \psi _j(a^{*},b^{*}) \ge \lambda \text{ and } g_j(b^{*}) - g_j(a^{*}) + \varepsilon \le v_j(a^{*}), \ j=1,\ldots ,m,\\ \quad \quad \quad \text{ if } a^{*}S b^{*}, \text{ for } (a^{*},b^{*})\in A^R, \\ { \quad } C(a^{*},b^{*})= \sum _{j=1}^m \psi _j(a^{*},b^{*}) + \varepsilon \le \lambda + M_0(a^{*},b^{*}) \text{ and } g_j(b^{*}) - g_j(a^{*}) \ge v_j(a^{*}) - \delta M_j(a^{*},b^{*}),\\ \quad \quad \quad \text{ if } a^{*}S^C b^{*}, \text{ for } (a^{*},b^{*})\in A^R, \\ { \quad \quad \quad } M_j(a^{*},b^{*}) \in \{0,1\}, \ j=0,\ldots ,m, \ \sum _{j=0}^m M_j(a^{*},b^{*}) \le m, \\ \text{ Values } \text{ of } \text{ inter-criteria } \text{ parameters: } \\ { \quad } 0 \le \lambda \le 1,\\ { \quad } \sum _{j=1}^m \psi _j(a^*_j,a_{j,*})=1, \text{ for } \text{ all } j=1,\ldots ,m : \ (g_j(a^*_j)=x_j^{m_j}) \text{ and } (g_j(a_{j,*}) = x_j^0) \\ \quad \quad \quad \text{ with } a^*_j,a_{j,*} \in A, \ j=1,\ldots ,m, \\ { \quad } v_j(a) \ge p_{j}^{R}(a) + \varepsilon , \ j=1,\ldots ,m, \ \text{ for } \text{ all } a \in A,\\ { \quad } v_j(a) \ge g_j(b) - g_j(a) + \varepsilon \text{ if } a \sim _j b, \text{ and } g_j(a) \le g_j(b), \ j \in G_2,\\ { \quad } v_j(a) \ge v_j(b) \text{ if } g_j(a) > g_j(b), \ j=1,\ldots ,m, \ \text{ for } \text{ all } (a,b) \in A \times A,\\ { \quad } v_j(a) = v_j(b) \text{ if } g_j(a) = g_j(b), \ j=1,\ldots ,m, \ \text{ for } \text{ all } (a,b) \in A \times A,\\ \text{ Values } \text{ of } \text{ marginal } \text{ concordance } \text{ indices } \text{ conditioned } \text{ by } \text{ intra-criterion } \text{ preference } \text{ information: } \\ { \quad } \psi _j(a,b) = 0 \text{ if } g_j(a) - g_j(b) \le -p_{j}^{R}(a), \text{ for } \text{ all } (a,b) \in A \times A, \ j \in G_1,\\ { \quad } \psi _j(a,b) \ge \varepsilon \text{ if } g_j(a) - g_j(b) > -p_{j}^{L}(a), \text{ for } \text{ all } (a,b) \in A \times A, \ j \in G_1,\\ { \quad } \psi _j(a,b) = \psi _j(a^*_j,a_{j,*}) \text{ if } g_j(a) - g_j(b) \ge -q_{j}^{L}(a) , \text{ for } \text{ all } (a,b) \in A \times A, \ j \in G_1,\\ { \quad } \psi _j(a,b) + \varepsilon \le \psi _j(a^*_j,a_{j,*}) \text{ if } g_j(a) - g_j(b) < -q_{j}^{R}(a) , \text{ for } \text{ all } (a,b) \in A \times A, \ j \in G_1,\\ { \quad } \psi _j(a,b) = \psi _j(a^*_j,a_{j,*}), \ \psi _j(b,a) = \psi _j(a^*_j,a_{j,*}) \text{ if } a \sim _j b, \ j \in G_2,\\ { \quad } \psi _j(a,b) = 0 \text{ if } b \succ _j a, \ j \in G_2,\\ \text{ Monotonicity } \text{ of } \text{ the } \text{ functions } \text{ of } \text{ marginal } \text{ concordance } \text{ indices: } \\ \quad \text{ If } \text{ the } \text{ thresholds } \text{ for } g_j \hbox { are not dependent on} g_j(a): \\ { \quad } \psi _j(a,b) \ge \psi _j(c,d) \text{ if } g_j(a) - g_j(b) > g_j(c) - g_j(d), \text{ for } \text{ all } a,b,c,d \in A, \ j=1,\ldots ,m,\\ { \quad } \psi _j(a,b) = \psi _j(c,d) \text{ if } g_j(a) - g_j(b) = g_j(c) - g_j(d), \text{ for } \text{ all } a,b,c,d \in A, \ j=1,\ldots ,m,\\ \quad \text{ If } \text{ the } \text{ thresholds } \text{ for } g_j \hbox { are dependent on} g_j(a): \\ { \quad } \psi _j(a,c) \ge \psi _j(b,c) \text{ if } g_j(a) > g_j(b), \text{ for } \text{ all } a, b, c \in A, \ j=1,\ldots ,m,\\ { \quad } \psi _j(a,c) \ge \psi _j(b,c) \text{ if } g_j(a) = g_j(b), \text{ for } \text{ all } a, b, c \in A, \ j=1,\ldots ,m,\\ { \quad } \psi _j(a,b) \ge \psi _j(a,c) \text{ if } g_j(b) < g_j(c), \text{ for } \text{ all } a, b, c \in A, \ j=1,\ldots ,m,\\ { \quad } \psi _j(a,b) \ge \psi _j(a,c) \text{ if } g_j(b) = g_i(c), \text{ for } \text{ all } a, b, c \in A, \ j=1,\ldots ,m.\\ \end{array} \right\} E^{A^{R'}} \end{aligned}$$

Analogously to what has been done in Sect. 4.1, to verify the feasibility of this set of constraints $E^{A^{R'}}$ by linear programming, one has to transform the strict inequality constraints into weak inequalities involving a variable $\varepsilon$. $E^{A^{R'}}$ has the form of 0-1 mixed integer program (MIP). If $E^{A^{R'}}$ is feasible and $\varepsilon ^{*}>0$, where $\varepsilon ^* = max \ \varepsilon$, subject to $E^{A^{R'}}$, then there exists at least one instance of the preference model compatible with the preference information provided by the DM.

Appendix 2: Computation of the relations $S^N$ and $S^P$

Given a pair of alternatives $(a,b) \in A \times A$, and the following sets of constraints

$$\begin{aligned} \left. \begin{array}{l} E^{A^{R'}}\\ C(a,b)= \displaystyle \sum _{i=1}^n \psi _i(a,b) + \varepsilon \le \lambda + M_0(a,b) \text{ and } g_i(b) - g_i(a) \ge v_i(a) - \delta M_i(a,b),\\ \displaystyle \sum _{i=0}^n M_i(a,b) \le n, \ M_i(a,b) \in \{0,1\}, \ i=0,\ldots ,n \end{array} \right\} E^{S^N}(a,b) \end{aligned}$$

and

$$\begin{aligned} \left. \begin{array}{l} E^{A^{R'}}\\ C(a,b)= \displaystyle \sum _{i=1}^n \psi _i(a,b) \ge \lambda \text{ and } g_i(b) - g_i(a) + \varepsilon \le v_i(a), \ i=1,\ldots ,n \end{array} \right\} E^{S^P}(a,b) \end{aligned}$$

we have the following:

• $aS^{N}b$ iff $E^{S^{N}}(a,b)$ is infeasible or $\varepsilon ^{N}(a,b)\le 0$, where $\varepsilon ^{N}(a,b)=\max \varepsilon$ subject to $E^{S^{N}}(a,b)$,

• $aS^{P}b$ iff $E^{S^{P}}(a,b)$ is feasible and $\varepsilon ^{P}(a,b)>0$, where $\varepsilon ^{P}(a,b)=\max \varepsilon$ subject to $E^{S^{P}}(a,b)$.