# Robust ordinal regression for decision under risk and uncertainty

• Original Paper
• Published:

## 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.

This is a preview of subscription content, log in via an institution to check access.

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

## Notes

1. Supposing that all criteria are gain criteria, a dominates b if $$g_{j}(a)\ge g_{j}(b)$$ for all $$j=1,\ldots ,m$$, and there exists at least one j such that $$g_{j}(a)>g_{j}(b)$$.

2. $$G_{1}$$ is the subset of criteria G on which the DM has expressed direct preferences of the type 3 and 4 shown in Sect. 4.2.

3. $$G_{2}$$ is the subset of criteria G on which the DM has expressed indirect preferences of the type 5 and 6 shown in Sect. 4.2.

## References

• Angilella S, Greco S, Matarazzo B (2010) Non-additive robust ordinal regression: a multiple criteria decision model based on the Choquet integral. Eur J Oper Res 201(1):277–288

• Anscombe FJ, Aumann RJ (1963) A definition of subjective probability. Ann Math Stat 34(1):199–205

• Bell DE, Raiffa H, Tversky A (1988) Decision making: descriptive, normative, and prescriptive interactions. Cambridge University Press, Cambridge

• Bewley T (2002) Knightian decision theory: part I. Decis Econ Finance 25:79–110

• Branke J, Deb K, Miettinen K, Słowiński R (2008) Multiobjective optimization, LNCS, vol 5252. Springer, Heidelberg

• Branke J, Greco S, Słowiński R, Zielniewicz P (2015) Learning value functions in interactive evolutionary multiobjective optimization. Evol Comput IEEE Trans 19(1):88–102

• Branke J, Corrente S, Greco S, Słowiński R, Zielniewicz P (2016) Using Choquet integral as preference model in interactive evolutionary multiobjective optimization. Eur J Oper Res. 250(3):884–901 doi:10.1016/j.ejor.2015.10.027

• Brans JP, Vincke Ph (1985) A preference ranking organisation method: the PROMETHEE method for MCDM. Manag Sci 31(6):647–656

• Chambers CP (2007) Ordinal aggregation and quantiles. J Econ Theory 137(1):416–431

• Choquet G (1953) Theory of capacities. Annales de l’Institut Fourier 5:131–295

• Corrente S, Greco S, Słowiński R (2012) Multiple criteria hierarchy process in robust ordinal regression. Decis Support Syst 53(3):660–674

• Corrente S, Greco S, Słowiński R (2013a) Multiple criteria hierarchy process with ELECTRE and PROMETHEE. Omega 41:820–846

• Corrente S, Greco S, Kadziński M, Słowiński R (2013b) Robust ordinal regression in preference learning and ranking. Mach Learn 93:381–422

• Corrente S, Greco S, Kadziński M, Słowiński R (2014) Robust ordinal regression. Wiley Encyclopedia of Operations Research and Management Science, pp 1–10

• Corrente S, Greco S, Ishizaka A (2015) Combining analytical hierarchy process and choquet integral within non-additive robust ordinal regression. Omega. doi:10.1016/j.omega.2015.07.003

• Deb K (2001) Multi-objective optimization using evolutionary algorithms, vol 16. Wiley

• Dias L, Mousseau V (2006) Inferring ELECTREs veto-related parameters from outranking examples. Eur J Oper Res 170(1):172–191

• Edgeworth FY (1888) The mathematical theory of banking. J R Stat Soc 51(1):113–127

• Ellsberg D (1961) Risk, ambiguity, and the Savage axioms. Q J Econ, pp 643–669

• Figueira J, Greco S, Ehrgott M (2005a) Multiple criteria decision analysis: state of the art surveys. Springer, Berlin

• Figueira J, Mousseau V, Roy B (2005b) ELECTRE methods. Multiple criteria decision analysis: state of the art surveys, pp 133–153

• Figueira J, Greco S, Słowiński R (2008) Identifying the “most representative” value function among all compatible value functions in the GRIP. In: Proceedings of the 68th EURO Working Group on MCDA, Chania

• Figueira J, Greco S, Słowiński R (2009) Building a set of additive value functions representing a reference preorder and intensities of preference: GRIP method. Eur J Oper Res 195(2):460–486

• Figueira JR, Greco S, Słowiński R, Roy B (2013) An overview of ELECTRE methods and their recent extensions. J Multi Criteria Decis Anal 20:61–85

• Giarlotta A, Greco S (2013) Necessary and possible preference structures. J Math Econ 49(2):163–172

• Gilboa I, Maccheroni F, Marinacci M, Schmeidler D (2010) Objective and subjective rationality in a multiple prior model. Econometrica 78(2):755–770

• Gilboa I, Schmeidler D (1989) Maxmin expected utility with non-unique prior. J Math Econ 18(2):141–153

• Greco S, Matarazzo B, Słowiński R (2001) Rough sets theory for multicriteria decision analysis. Eur J Oper Res 129(1):1–47

• Greco S, Mousseau V, Słowiński R (2008) Ordinal regression revisited: multiple criteria ranking using a set of additive value functions. Eur J Oper Res 191(2):415–435

• Greco S, Matarazzo B, Słowiński R (2010) Dominance-based rough set approach to decision under uncertainty and time preference. Ann Oper Res 176(1):41–75

• Greco S, Kadziński M, Mousseau V, Słowiński R (2011) ELECTRE$$^{GKMS}$$: robust ordinal regression for outranking methods. Eur J Oper Res 214(1):118–135

• Greco S, Matarazzo B, Slowinski R (2013) Beyond Markowitz with multiple criteria decision aiding. J Bus Econ 83(1):29–60

• Greco S, Mousseau V, Słowiński R (2014) Robust ordinal regression for value functions handling interacting criteria. Eur J Oper Res 239(3):711–730

• Hammond JS, Keeney RL, Raiffa H (1999) Smart choices: a practical guide to making better decisions, vol 226. Harvard Business Press

• Jacquet-Lagrèze E, Siskos Y (1982) Assessing a set of additive utility functions for multicriteria decision-making, the UTA method. Eur J Oper Res 10(2):151–164

• Jacquet-Lagrèze E, Siskos Y (2001) Preference disaggregation: 20 years of MCDA experience. Eur J Oper Res 130(2):233–245

• Jorion P (2007) Value at risk: the new benchmark for managing financial risk, 3rd edn. McGraw-Hill

• Kadziński M, Greco S, Słowiński R (2011) Selection of a representative value function in robust multiple criteria ranking and choice. Eur J Oper Res 217(3):541–553

• Kadziński M, Greco S, Słowiński R (2012) Extreme ranking analysis in robust ordinal regression. Omega 40(4):488–501

• Kadziński M, Tervonen T (2013a) Robust multi-criteria ranking with additive value models and holistic pair-wise preference statements. Eur J Oper Res 228(1):169–180

• Kadziński M, Tervonen T (2013b) Stochastic ordinal regression for multiple criteria sorting problems. Decis Support Syst 55(11):55–66

• Keeney RL, Raiffa H (1993) Decisions with multiple objectives: preferences and value tradeoffs. Wiley, New York

• Khouja B (1999) The single-period (news-vendor) problem: literature review and suggestions for future research. Omega 27(5):537–553

• Knight FH (1921) Risk, uncertainty and profit. Harper & Row, New York

• Lahdelma R, Hokkanen J, Salminen P (1998) SMAA—stochastic multiobjective acceptability analysis. Eur J Oper Res 106(1):137–143

• Langlois RN, Cosgel MM (1993) Frank Knight on risk, uncertainty, and the firm: a new interpretation. Econ Inq 31(3):456–465

• Manski CF (1988) Ordinal utility models of decision making under uncertainty. Theory Decis 25(1):79–104

• Markowitz H (1952) Portfolio selection. J Finance 7(1):77–91

• Matos MA (2007) Decision under risk as a multicriteria problem. Eur J Oper Res 181(3):1516–1529

• Miller GA (1956) The magical number seven, plus or minus two: some limits on our capacity for processing information. Psychol Rev 63(2):81

• Mousseau V, Figueira JR, Dias L, Gomes da Silva C, Climaco J (2003) Resolving inconsistencies among constraints on the parameters of an MCDA model. Eur J Oper Res 147(1):72–93

• Mousseau V, Dias L (2004) Valued outranking relations in ELECTRE providing manageable disaggregation procedures. Eur J Oper Res 156(2):467–482

• Petruzzi N, Dada M (1999) Pricing and the newsvendor problem: a review with extensions. Oper Res 47(2):183–194

• Qin Y, Wang R, Vakharia AJ, Chen Y, Seref MMH (2011) The newsvendor problem: review and directions for future research. Eur J Oper Res 213(2):361–374

• Rostek M (2010) Quantile maximization in decision theory. Rev Econ Stud 77(1):339–371

• Roy B (1993) Decision science or decision-aid science? Eur J Oper Res 66(2):184–203

• Roy B (1996) Multicriteria methodology for decision aiding. Kluwer, Dordrecht

• Russo JE, Schoemaker PJH (1989) Decision traps: ten barriers to brilliant decision-making and how to overcome them. Doubleday Publishing Co., New York

• Savage LJ (1954) The foundations of statistics. Wiley, New York

• Schmeidler D (1989) Subjective probability and expected utility without additivity. Econ J Econ Soc 57:571–587

• Słowiński R, Greco S, Matarazzo B (2009) Rough sets in decision making. In: Meyers RA (ed) Encyclopedia of complexity and systems science. Springer, New York, pp 7753–7786

• Słowiński R, Greco S, Matarazzo B (2015) Rough set methodology for decision aiding. Chapter 22. In: Kacprzyk J, Pedrycz W (eds) Handbook of computational intelligence, Springer, Berlin, pp 349–370

• Von Neumann J, Morgenstern O (1944) Theory of games and economic behavior. Princeton University Press, Princeton

## Acknowledgments

The first two authors wish to acknowledge funding by the “Programma Operativo Nazionale” Ricerca & Competitivitá “2007–2013” within the project “PON04a2 E SINERGREEN-RES-NOVAE” and by the FIR of the University of Catania BCAEA3 “New developments in Multiple Criteria Decision Aiding (MCDA) and their application to territorial competitiveness”.

## Author information

Authors

### Corresponding author

Correspondence to Salvatore Corrente.

## 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), $$v_j(a) > p_j^{R}(a), \ j \in G_1$$,Footnote 2

• $$v_j(a)$$ is required to be larger than $$g_j(b) - g_j(a)$$ for all pairs of alternatives (ab) for which the DM stated that the difference between $$g_j(a)$$ and $$g_j(b)$$ is non-significant ($$g_j(a) \le g_j(b)$$), $$a \sim _j b, \ j \in G_2$$ Footnote 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 $$E^{S^{P}}(a,b)$$.

## Rights and permissions

Reprints and permissions

Corrente, S., Greco, S., Matarazzo, B. et al. Robust ordinal regression for decision under risk and uncertainty. J Bus Econ 86, 55–83 (2016). https://doi.org/10.1007/s11573-015-0801-5