Multiple criteria sorting with a set of additive value functions

European Journal of Operational Research 207 (2010) 1455–1470

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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor

Decision Support

Multiple criteria sorting with a set of additive value functions Salvatore Greco a,*, Vincent Mousseau b, Roman Słowin´ski c a

Faculty of Economics, University of Catania, Corso Italia, 55, 95129 Catania, Italy LGI, Ecole Centrale Paris, Grande Voie des Vignes, 92 295 Châtenay Malabry, France c ´ University of Technology, 60-965 Poznan ´ , and Systems Research Institute, Polish Academy of Sciences, 01-447 Warsaw, Poland Institute of Computing Science, Poznan b

a r t i c l e

i n f o

Article history: Received 27 January 2009 Accepted 13 May 2010 Available online 12 June 2010 Keywords: Multiple criteria sorting Additive value function Disaggregation-aggregation procedure

a b s t r a c t We present a new multiple criteria sorting method that aims at assigning actions evaluated on multiple criteria to p pre-defined and ordered classes. The preference information supplied by the decision maker (DM) is a set of assignment examples on a subset of actions relatively well known to the DM. These actions are called reference actions. Each assignment example specifies a desired assignment of a corresponding reference action to one or several contiguous classes. The set of assignment examples is used to build a preference model of the DM represented by a set of general additive value functions compatible with the assignment examples. For each action a, the method computes two kinds of assignments to classes, concordant with the DM’s preference model: the necessary assignment and the possible assignment. The necessary assignment specifies the range of classes to which the action can be assigned considering all compatible value functions simultaneously. The possible assignment specifies, in turn, the range of classes to which the action can be assigned considering any compatible value function individually. The compatible value functions and the necessary and possible assignments are computed through the resolution of linear programs.  2010 Elsevier B.V. All rights reserved.

1. Introduction Real-world decision problems usually involve multiple points of view on evaluation of actions they concern. These points of view are relevant for a decision maker (DM) who is either a single person or a collegial body. They are formally represented by real-valued functions called criteria. Multiple criteria decision aiding aims at recommending a decision which is consistent with a value system of the DM. This recommendation is considered within one of the following main problematics [10]: choice, ranking or sorting. Given set A of actions, the choice problematic consists in selection of a subset of actions (as small as possible) being judged as the most satisfactory of the whole set. The ranking problematic consists in establishing a preference pre-order (either partial or complete) on the set of actions. The sorting problematic concerns ordinal classification of actions and consists in assignment of each action to one of pre-defined and ordered classes. In this paper, we are considering the multiple criteria sorting problematic. Our multiple criteria sorting model used to work out a recommendation is a set of value functions. Multiple attribute value theory (see [6]) is a well-established theory considering compensatory preference models to represent how DMs account * Corresponding author. E-mail addresses: [email protected] (S. Greco), [email protected] (V. Mousseau), [email protected] (R. Słowin´ski). 0377-2217/$ - see front matter  2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2010.05.021

for tradeoffs among criteria. Several value based sorting methods have been proposed in the literatures: namely [1,5,8,7,12,13] (see also [14] for a review). However, a major difficulty when implementing value based sorting models comes from the fact that the value function must be elicited almost directly from the DM. Therefore, several authors (see, e.g., [1,13]) proposed methodologies that avoid direct elicitation of the value function, and instead elicit the value function indirectly from assignment examples (typical actions that should be assigned to particular classes) supplied by the DM. The value function is then inferred through a certain form of regression on assignment examples. Such indirect elicitation is usually called disaggregation. In particular, the method proposed in [7] involves indirect specification of preferences through assignment of reference actions, and provides possible assignments for non-reference actions based on a set of piecewise-linear additive value functions. The new multiple criteria value based sorting method proposed in this paper is called UTADISGMS. This method requires that the DM expresses his/her preferences providing a set of assignment examples on a subset of actions (s)he knows relatively well. Each assignment example specifies a desired assignment of a corresponding reference action to one or several contiguous classes. The set of assignment examples is a preference information used to build a preference model of the DM. Here, the underlying preference model is a set of general additive value functions compatible with the assignment examples. For each action a, the

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method computes two kinds of assignments to classes, concordant with the DM’s preference model: the necessary assignment and the possible assignment. The necessary assignment specifies the range of classes to which the action can be assigned considering all compatible value functions simultaneously. The possible assignment specifies, in turn, the range of classes to which the action can be assigned considering any compatible value function individually. The compatible value functions and the necessary and possible assignments are computed through the resolution of linear programs. Comparing to the existing methods and, in particular, to the method proposed in [7], we consider possible and necessary assignments which no previous method did consider. Moreover, instead of an additive model with piecewise linear marginal value functions, our method considers all non-decreasing marginal value functions. Finally, our method makes it possible for the DM to provide imprecise assignment examples (interval assignments). This method is intended to be used interactively, that is, the DM can provide progressively new assignment examples. It also supports the DM when the set of assignment examples is inconsistent in the sense that there is no additive value function compatible with the provided assignment examples. The DM may also specify qualitative confidence levels for assignment examples, e.g., ‘‘absolutely sure”, ‘‘sure”, ‘‘mildly sure”. In such a case, the resulting possible and necessary assignments are expressed as ranges of classes that correspond to the different confidence levels. The paper is organized as follows. Notation and definitions are introduced in the next section. Section 3 considers sorting procedures grounded on a single value function, and studies the relation between two particular sorting procedures: ‘‘threshold-based” and ‘‘example-based”. Simultaneous handling of all value functions compatible with assignment examples requires to study sorting procedures taking into account a set of value functions. Such a study is provided in Section 4. Section 5 presents the new sorting method called UTADISGMS. Section 6 is devoted to the case where the DM provides assignment example that cannot be represented by an additive value function. Section 7 proposes an extension of the method that takes into account assignment examples with different confidence judgments. Section 8 presents application of the UTADISGMS method on a real-world multiple criteria sorting problem, emphasizing the interaction between the method and the DM. The last Section groups conclusions and outlines some proposals for future research. Proofs of propositions can be found in Appendices A, B, C. 2. Notation and definitions

 A = {a1, a2, . . ., ai, . . ., am} – a finite set of m actions to be assigned to classes,  g1, g2, . . ., gn – n evaluation criteria, g j : A ! R for all j 2 G = {1, 2, . . ., n},  C1, C2, . . ., Cp – p pre-defined and ordered classes, where Ch+1  Ch( is a complete order on the set of classes), h = 1, . . ., p  1, moreover, H = {1, . . ., p},  X j ¼ fxj 2 R : g j ðai Þ ¼ xj ; ai 2 Ag – a set of all different evaluations on gj, j 2 G, m  x0j ; x1j ; . . . ; xj j -ordered values of X j ; xkj < xkþ1 ; k ¼ 0; 1; . . . ; mj  1; j mj 6 m. In order to represent DM’s preferences, we shall use an additive value function U: n X j¼1

uj ðxkj Þ 6 uj ðxkþ1 Þ; j

k ¼ 0; 1; . . . ; mj  1; j 2 G:

To normalize the value function so that U(a) 2 [0, 1], "a 2 A, we set:

9 uj ðx0j Þ ¼ 0; 8j 2 G > = n P mj > uj ðxj Þ ¼ 1 ;

ð3Þ

j¼1

In this paper, we are interested in sorting procedures that aim at assigning each action to one class or to a set of contiguous classes. The procedures we are considering are value driven sorting procedures, that is, they use a single value function U, or a set of value functions, to decide the assignments in such a way that if U(a) > U(b), then a is assigned to a class not worse than b. 3. Sorting with a single additive value function Given a value function U defined as (1), two different sorting procedures can be considered:  threshold-based value driven sorting procedure, in which the frontiers between consecutive classes are defined by thresholds on the utility scale,  example-based value driven sorting procedure, in which classes are implicitly delimited by some assignment examples. Definition 3.1. A threshold-based value driven sorting procedure U is driven by a value function U and its associated thresholds bh , h 2 H, in the following way: U

U

a 2 A is assigned to class C h ; denoted as a ! C h ; iff UðaÞ 2 ½bh1 ; bh ½; U

where bh1 is the minimum value for an action to be assigned to U class Ch, and bh is the supremum value for an action to be assigned U to class Ch. Obviously, we set b0 ¼ 0 and, moreover, we impose U U bh1 < bh ; 8h 2 H. Such a threshold-based value-driven sorting procedure has been used in the UTADIS method [1,13]. In order to define an example-based value driven sorting procedure, let us consider preference information provided by the DM in terms of m* assignment examples on a subset of actions A* # A, where m* = jA*j.

uous classes C LDM ða Þ ; C LDM ða Þþ1 ; . . . ; C RDM ða Þ ; LDM ða Þ 6 RDM ða Þ. Each such action is called a reference action. A* # A is called the set of reference actions. An assignment example is said to be precise if LDM(a*) = RDM(a*), and imprecise, otherwise. Definition 3.3. Given a value function U, a set of assignment examples is said to be consistent with U iff

8a ; b 2 A ;





Uða Þ P Uðb Þ ) RDM ða Þ P LDM ðb Þ:

ð1Þ

ð4Þ

Observe that (4) can also be expressed in a equivalent way as

8a ; b 2 A ; LDM ða Þ > RDM ðb Þ ) Uða Þ > Uðb Þ: In fact, the contrapositive of (4) is

uj ðg j ðaÞÞ;

ð2Þ

Definition 3.2. An assignment example consists of an action a* 2 A* for which the DM defined a desired assignment a ! h i h i C LDM ða Þ ; C RDM ða Þ , where C LDM ða Þ ; C RDM ða Þ is an interval of contig-

We shall use the following notation:

UðaÞ ¼

where the marginal value functions uj are defined by uj ðxkj Þ, such that:





LDM ðb Þ > RDM ða Þ ) Uða Þ < Uðb Þ;

ð5Þ

S. Greco et al. / European Journal of Operational Research 207 (2010) 1455–1470

such that if we switch the role of a* and b*, we get 

Now, let us study the relationship between the threshold-based (Definition 3.1) and the example-based (Definition 3.4) sorting procedures.



LDM ða Þ > RDM ðb Þ ) Uða Þ > Uðb Þ: In this section, we suppose that the DM provides a set of assignment examples consistent with U. Definition 3.4. An example-based value driven sorting procedure is driven by a value function U and its associated assignment examples a* 2 A*. It assigns an action a 2 A to an interval of classes h i C LU ðaÞ ; C RU ðaÞ , in the following way:

 n o LU ðaÞ ¼ Max f1g [ LDM ða Þ : Uða Þ 6 UðaÞ; a 2 A ;

ð6Þ

 n o RU ðaÞ ¼ Min fpg [ RDM ða Þ : Uða Þ P UðaÞ; a 2 A :

ð7Þ

Such an example-based sorting procedure has been used in [8] along with ahlinear valuei function. The following lemma states that the interval C LU ðaÞ ; C RU ðaÞ is not empty. Proposition 3.1. For any action a 2 A n A*, the example-based sorting h i procedure provides an assignment a ! C LU ðaÞ ; C RU ðaÞ , such that U

U

L (a) 6 R (a). Proof. See Appendix A.1. h Proposition 3.2. The example-based sorting procedure assigns each h i reference action a* 2 A* to an interval of classes C LU ða Þ ; C RU ða Þ , such that:

LU ða Þ P LDM ða Þ;

ð8Þ

RU ða Þ 6 RDM ða Þ:

ð9Þ

Proposition 3.3. Consider the case where the DM provides precise assignment examples only, i.e., LDM(a*) = RDM(a*), for each a* 2 A*. Assuming the use of a single value function U in the example-based sorting procedure, if we choose, for each h = 1, . . ., p  1, the threshold U

bh in the interval Maxa !C h fUða Þg; Mina !C hþ1 fUða Þg, then we obtain a threshold-based sorting procedure that restores the assignment examples and assigns each non-reference action a 2 A n A* to a h i single class in the interval C LU ðaÞ ; C RU ðaÞ stemming from the examplebased sorting procedure. Proof. See Appendix A.3. h Fig. 1 illustrates Proposition 3.3 in a sorting example involving 3 classes and 18 reference actions. In this case, assignment examples are such that a1 ; a2 ; a3 ; a4 ; a5 and a6 are assigned to class C1; a7 ; a8 ; a9 ; a10 ; a11 ; a12 are assigned to C2, and the remaining actions are assigned to C3. Consider a non-reference action a0 2 A n A*. According to the value of U(a0 ), the example-based sorting procedure assigns a0 as follows:     

If If If If If

Uða0 Þ 6 Uða6 Þ, then a0 ? C1. Uða0 Þ 2Uða6 Þ; Uða7 Þ½, then a0 ? [C1,C2]. Uða0 Þ 2 ½Uða7 Þ; Uða12 Þ, then a0 ? C2. Uða0 Þ 2Uða12 Þ; Uða13 Þ½, then a0 ? [C2, C3]. Uða0 Þ P Uða13 Þ, then a0 ? C3.

Consistently with Proposition 3.3, it appears that if we fix the thresholds such that b1 2Uða6 Þ; Uða7 Þ and b2 2Uða12 Þ; Uða13 Þ, the threshold-based sorting procedure will assign a new action a0 consistently with the example-based sorting procedure, i.e.:  If Uða0 Þ 6 Uða6 Þ then a0 ? C1.  If Uða0 Þ 2Uða6 Þ; Uða7 Þ½, then a0 ? C1 or C2, depending whether U

Proof. See Appendix A.2. h

1457

Uða0 Þ < b1 or not.  If Uða0 Þ 2 ½Uða7 Þ; Uða12 Þ then a0 ? C2.

Fig. 1. Intervals for the thresholds separating classes (precise assignments).

Fig. 2. Intervals for the thresholds separating classes (imprecise assignments).

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 If Uða0 Þ 2Uða12 Þ; Uða13 Þ[then a0 ? C2 or C3, depending whether U Uða0 Þ < b2 or not.  If Uða0 Þ P Uða13 Þ, then a0 ? C3.

Definition 4.3. Given a set A* of assignment examples and a corresponding set U A of compatible value functions, for each a 2 A, we define the following indices of classes:

Proposition 3.4. Consider the case where the DM provides possibly imprecise assignment examples, i.e., LDM(a*) 6 RDM(a*) for each a* 2 A*. Assuming the use of a single value function U in the example-based sorting procedure, if we choose, for each h = 1, . . ., p  1,

 minimum possible class:  n o LUP ðaÞ ¼ Max f1g [ LDM ða Þ : 8U 2 U A ; Uða Þ 6 UðaÞ; a 2 A ;

U

the threshold bh

U

in the interval Iðbh Þ ¼Maxa :RDM ða Þ6h fUða Þg;



Mina :LDM ða Þ>h fUða Þg½, with

U bh

<

U bhþ1 ,

then we obtain a threshold-

 minimum necessary class:

 n LUN ðaÞ ¼ Max f1g [ LDM ða Þ : 9U 2 U A

based sorting procedure that assigns each reference action a* 2 A* to h i a single class in the interval C LDM ða Þ ; C RDM ða Þ , and assigns each nonreference action a 2 AnA* to a single class in the interval   C LU ðaÞ; C RU ðaÞ stemming from the example-based sorting procedure. Proof. See Appendix A.4. h Fig. 2 illustrates Proposition 3.4 in an example involving 5 classes and 11 reference actions. Hence, Propositions 3.3 and 3.4 show that the example-based and the threshold-based sorting procedures provide consistent results, the former being more general than the latter, as it implicitly considers intervals of possible thresholds instead of single values for the thresholds. 4. Sorting with a set of value functions In this section, we consider a value driven sorting procedure based on a set U of value functions, and we focus on the example-based sorting procedure. Definition 4.1. Considering a set A* of assignment examples, the set U A of compatible value functions is defined by all value functions U, satisfying U(a*) > U(b*) for all a*, b* 2 A* such that RDM(b*) < LDM(a*). Definition 4.2. Given a set A* of assignment examples and a corresponding set U A of compatible value functions, for each a 2 A, we define the possible assignment CP(a) as a set of indices of classes Ch for which there exists at least one value function U 2 U A assigning a to Ch, and the necessary assignment CN(a) as a set of indices of classes Ch for which all value functions U 2 U A assign a to Ch, that is:

n h io C p ðaÞ ¼ h 2 H : 9U 2 U A for which h 2 LU ðaÞ; RU ðaÞ ; n h io C N ðaÞ ¼ h 2 H : 8U 2 U A it holds h 2 LU ðaÞ; RU ðaÞ :

ð10Þ ð11Þ

It is obvious to observe that for each a 2 A it holds:

C P ðaÞ ¼

[ h U2U A

C N ðaÞ ¼

i LU ðaÞ; RU ðaÞ ;

\ h

i LU ðaÞ; RU ðaÞ ;

for which Uða Þ 6 UðaÞ; a 2 A

o

;

 maximum necessary class:

 n RUN ðaÞ ¼ Min fpg [ RDM ða Þ : 9U 2 U A for which UðaÞ 6 Uða Þ; a 2 A

o ;

 maximum possible class:  n o RUP ðaÞ ¼ Min fpg [ RDM ða Þ : 8U 2 U A ; UðaÞ 6 Uða Þ; a 2 A : The following proposition states how the above indices are related to CP(a) and CN(a). Proposition 4.2. Given a set A* of assignment examples and a corresponding set U A of compatible value functions, for each a 2 A, it holds:  LUp ðaÞ 6 LUN ðaÞ,  RUN ðaÞ 6 RUP ðaÞ,  LUP ðaÞ 6 RUP ðaÞ. Moreover, for any a 2 A, we have:    C p ðaÞ # LUP ðaÞ; RUP ðaÞ ,  if LUN ðaÞ 6 RUN ðaÞ, then C N ðaÞ ¼ ½LUN ðaÞ; RUN ðaÞ,  if LUN ðaÞ > RUN ðaÞ, then CN(a) = ;. Proof. See Appendix B.1. h In this context, two important questions arise:  Under which condition on value functions U 2 U A we have CN(a) – ;?  Under which condition on value functions U 2 U A the following ‘‘no jump property” holds? 0

If h; h 2 C P ðaÞ;

0

with h 6 h ;

00

then h 2 C p ðaÞ;

8h00 2 ½h; h0 : ð13Þ

ð12Þ

U2U A

C N ðaÞ # C P ðaÞ: Proposition 4.1. For any a 2 A there exists U 2 U A and values of the thresholds bh, h 2 H, such that for any h 2 CP(a) the threshold-based sorting procedure assigns a to class Ch, for each h 2 [LU(a), RU(a)]. n Proof. Since C P ðaÞ ¼ h 2 H : 9U 2 U A for which h 2 ½LU ðaÞ; RU ag, remembering that according to Proposition 3.4, for all U 2 U A there exist values for the thresholds bh, h 2 H, such that for any h 2 CP(a) the threshold-based sorting procedure assigns a to class Ch, for each h 2 [LU(a), RU(a)], we get the thesis. h

Let us observe that the first question is related to the robustness of the assignment, in the sense that it concerns classes to which a is assigned by all compatible value functions U 2 U A . The second question regards instead the presence of a ‘‘jump” in the classes of the possible assignment. In fact, the ‘‘no jump property” (13)  is equivalent to C P ðaÞ ¼ LUP ðaÞ; RUP ðaÞ , which means that no class between the worst, C LU ðaÞ, and the best, C RU ðaÞ, is absent in the posP P sible assignment. These two questions are studied in Propositions 4.3, 4.4–4.6. Proposition 4.3. CN(a) R ; if and only if the following strict continuity condition is satisfied: for all a*, b* 2 A*, such that LDM(a*) > RDM(b*), there are no two compatible value functions U; U 0 2 U A for which U(a) P U(a*) and U0 (b*) P U0 (a).

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Proof. See Appendix B.2. h The following condition is important for the absence of jumps in the classes of the possible assignment. Definition 4.4 (Weak continuity). For all a*, b* 2 A*, such that LDM(a*) > RDM(b*) + 1, if there exist U 0 ; U 00 2 U A for which U0 (a) P U0 (a*), and U00 (b*) P U00 (a) then there exists U 000 2 U A for which U000 (a*) P U000 (a) P U000 (b*). Proposition 4.4. The ‘‘no jump property” (13) holds and, therefore, C P ðaÞ ¼ ½LUP ðaÞ; RUP ðaÞ, if and only if the weak continuity (4.4) holds. Proof. See Appendix B.3. h Corollary 4.1. If the set U A of compatible value functions satisfies the weak continuity (4.4), the example-based sorting procedure assigns  each reference action a* 2 A* to an interval of classes LUP ða Þ; h i RUP ða Þ # LDM ða Þ; RDM ða Þ . Proof. See Appendix B.4. h Proposition 4.4 is a very general result. As we are interested in additive value functions, in order to specialize the result of Proposition 4.4 to this case, we introduce the concept of a convex set of value functions. Definition 4.5. A set U of value functions is convex if for all U; U 0 2 U and for all a 2 ½0; 1; aU þ ð1  aÞU 0 2 U. Since a convex set of value functions satisfies weak continuity, we get the following result. Proposition 4.5. If the set U A of compatible value functions is  convex, then C P ðaÞ ¼ LUP ðaÞ; RUP ðaÞ , i.e., the ‘‘no jump property” (13) holds. Proof. See Appendix B.5. h Since the set of all additive compatible value functions is convex, we get the following result. Proposition 4.6. If U A is the set of all additive compatible value   functions, then C P ðaÞ ¼ LUP ðaÞ; RUP ðaÞ , i.e., the ‘‘no jump property” (13) holds. Proof. See Appendix B.6. h

5. A new sorting method using a set of additive value functions The new UTADISGMS method is an ordinal regression method for multiple criteria sorting problems using a set of additive value functions (1) as a preference model. One of its characteristic features is that it takes into account the set of all value functions compatible with the preference information provided by the DM. Moreover, it considers general non-decreasing marginal value functions instead of piecewise linear only. The DM is supposed to provide preference information in form of possibly imprecise assignment examples on a reference set A*, i.e., for all a* 2 A* the DM defines a desired assignment a ! h i C LDM ða Þ ; C RDM ða Þ . According to (5), a value function is compatible if "a*, b* 2 A*, LDM(a*) > RDM(b*) ) U(a*) > U(b*). Thus, we can state that, formally, a general additive compatible value function is an

additive value function UðaÞ ¼ ing linear constraints:

Pn

j¼1 uj ðg i ðaÞÞ

satisfying the follow-

9 > > > ðk1Þ  > >  > uj ðxk Þ  u ðx Þ P 0; j ¼ 1; . . . ; n; k ¼ 2; . . . ; m ðA Þ; > j j j > > >  mj =   m ðA Þ 1 uj ðxj Þ P 0; uj ðxj Þ P uj ðxj Þ; j ¼ 1; . . . ; n; EA ; > 0 > uj ðxj Þ ¼ 0; j ¼ 1; . . . ; n; > > > > > n P > mj > > uj ðxj Þ ¼ 1; ; 



Uða Þ > Uðb Þ if LDM ða Þ > RDM ðb Þ;

8a ; b 2 A ;

j¼1

where, as stated in Section 2, x0j ¼ mina2A g j ðaÞ and m   xj j ¼ maxa2A g j ðaÞ; xk 2 X ðA Þ; k ¼ 1; . . . ; m ðA Þ with Xj(A*) # Xj j j j being the set of all different evaluations of reference actions from A* on criterion gj, j = 1, . . ., n, and mj(A*) = jXj(A*)j. The values  xk j ; k ¼ 1; . . . ; mj ðA Þ are increasingly ordered, i.e., ðmj ðA Þ1Þ

2 x1 j < x j <    < xj

mj ðA Þ

< xj

:

Let us observe that constraints (EA*) are equivalent to:

9    Uða Þ P Uðb Þ þ e if LDM ða Þ > RDM ðb Þ; 8a ;b 2 A ; > > >     > ðk1Þ  > k  > uj xj  uj xj P 0; j ¼ 1;. .. ;n; k ¼ 2;. .. ;m ðA Þ; > > > > >       m > > m ðA Þ =  0 j 1 uj xj P 0; uj xj 6 uj xj ; j ¼ 1;. .. ; n; EA ;   > > > > uj x0j ¼ 0; j ¼ 1; .. .; n; > > > > >   > n P > mJ > > uj xj ¼ 1; ;

j¼1

with e > 0. Thus, to check if the set of all compatible value functions U A is not empty it is sufficient to verify that e* > 0, where   0 e* = max e, s.t. constraints EA . Remark that, due to the above formulation of the ordinal regression problem, no linear interpolation is required to express the marginal value of any reference action. Thus, one cannot expect that increasing the number of characteristic points will bring some ‘‘new” compatible additive value functions. In consequence, UTADISGMS considers all compatible additive value functions, while classical UTADIS deals with a subset of the whole set of compatible additive value functions, more precisely, the subset of piecewise linear additive value functions relative to the considered characteristic points. A similar approach taking into account a set of characteristic points has been considered in [7]. The advantage of considering all criteria values as characteristic points involves, however, an increased computation cost, because all the values correspond to variables uj(gj(a*)), a* 2 A*, j = 1, . . ., n,   0 of constraints EA . As all above optimization problems are linear programs, this increase is easily manageable, so it does not represent a major problem. Moreover, in general, the reference set is composed of a relatively small number of actions which are well known to the DM. The relatively small cardinality of the reference set implies that the number of characteristic points, and, conse  0 quently, the number of variables in constraints EA , requires a computational effort which is much lower than the capacity of popular linear programming solvers. Notice also that taking all criteria values as characteristic points increases the degree of freedom in assessing the compatible value functions. This feature is not, however, a disadvantage of our methodology. Indeed, we are looking for robust conclusions and, therefore, we want to explore the whole space of compatible value functions. This is not the case when a limited set of characteristic points and a linear interpolation between them is considered. In fact, the selection of these

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characteristic points is rather arbitrary, which prevents obtaining robust solutions being independent of this selection. In other words, some possible assignments could be ignored if their corresponding compatible value functions would not be considered for the reason of such selection of characteristic points. On the basis of the set of all compatible value functions U A , we can define two binary relations in the set of all actions A:  necessary weak preference relation %N , in case U(a) P U(b) for all compatible value functions,  possible weak preference relation %P , in case U(a) P U(b) for at least one compatible value function. Using necessary weak preference relation %N and possible weak preference relation %P , we can redefine indices LUP ðaÞ; LUN ðaÞ; RUN ðaÞ and RUP ðaÞ from Definition 4.3 as follows:  minimum possible class:

 n o LUP ðaÞ ¼ Max f1g [ LDM ða Þ : a %N a ; a A ;

ð14Þ

 minimum necessary class:

 n o LUN ðaÞ ¼ Max f1g [ LDM ða Þ : a %P a ; a A ;

ð15Þ

 maximum necessary class:

 n o RUN ðaÞ ¼ Min fpg [ RDM ða Þ : a %P a; a A ;

ð16Þ

ðmj ðA [fa;bgÞ1Þ

mj ðA [fa;bgÞ

< xj

ð17Þ

:

Then, the characteristic points of uj(), j = 1, . . ., n, are in x0j ; xk j ; m k ¼ 1; . . . ; mj ðA [ fa; bgÞ; xj j . Let us consider the following  ordinal regression constraints (E(a, b)) and (E(b, a)), with uj xk ; j ¼ 1; . . . ; n; k ¼ 1; . . . ; mj ðA [ j mj fa; bgÞ; xj and e as variables:

9 > > > > >    Uða Þ P Uðb Þ þ e if LDM ða Þ P RDM ðb Þ; 8a ; b 2 A ; > > > > >     > > ðk1Þ > k > uj xj  uj xj P 0; j ¼ 1; . . . ; n; > > > > > >  > = k ¼ 2; . . . ; mj ðA [ fa; bgÞ; ðEða; bÞÞ;         m m ðA [fa;bgÞ > P 0; uj xj 6 uj xj j ; j ¼ 1; . . . ; n; > uj x1 > j > > > >   > > > 0 > uj xj ¼ 0; j ¼ 1; . . . ; n; > > > > > >   > n P > mj > > uj xj ¼ 1; ;

UðaÞ P UðbÞ;

j¼1

j¼1

The above constraints depend on the pair of actions a, b 2 A because their evaluations gj(a) and gj(b) give coordinates for two of m*(A* [ {a, b}) characteristic points of marginal value function uj(),for each j = 1, . . ., n. Let us suppose that the polyhedron defined by constraints (E(a, b)), as well as the one defined by constraints (E(b, a)), are not empty. In this case we have that:

ð18Þ

where e(b, a) = max e, s.t. constraints (E(b, a)), and

a %P b () eða; bÞ > 0;

The above definitions can be used directly to calculate the possible and the necessary assignments of any action on the base of the necessary weak preference relation %N and possible weak preference relation %P representing the preferences of the DM. Necessary weak preference relation %N and possible weak preference relation %P can be calculated as follows [3,2]. For all actions a, b 2 A, let Xj(A* [ {a, b}) # Xj be the set of all different evaluations of actions from A* [ {a, b} on criterion gj, j = 1, . . ., n, and  mj(A* [ {a, b}) = jXj(A* [ {a, b})j. The values xk j 2 X j ðA [ fa; bgÞ; k ¼  1; . . . ; mj ðA [ fa; bgÞ, are increasingly ordered, i.e., 2 x1 j < x j <    < xj

9 > > > > >     > DM DM    Uða Þ P Uðb Þ þ e if L ða Þ > R ðb Þ; 8a ; b 2 A ; > > > > > >     > > ðk1Þ > > uj xk  u P 0; j ¼ 1; . . . ; n; x j > j j > > > > > >  > = k ¼ 2; . . . ; mj ðA [ fa; bgÞ; ðEðb; aÞÞ:        m > m ðA [fa;bgÞ > j > x x uj x1 ; u 6 u ; j ¼ 1; . . . ; n; > j j j j j > > > > >   > > > > uj x0j ¼ 0; j ¼ 1; . . . ; n; > > > > > > >  m > n P > j > > uj xj ¼ 1; ; UðbÞ P UðaÞ þ e;

a %N b () eðb; aÞ 6 0;

 maximum possible class:

 n o RUP ðaÞ ¼ Min fpg [ RDM ða Þ : a %N a; a A :

and

ð19Þ

where e(a, b) = max e, s.t. constraints (E(a, b)). Therefore, on the basis of the above results, the following example-based sorting procedure can be proposed: 1. Ask the DM for sorting examples. 2. Verify that the set of compatible value functions U A is not empty, by verifying that e* > 0, where e* = max e s.t. constraints (EA*)0 . 3. Calculate the necessary and the possible weak preference relations a %N a; a %P a ; a %N a and a %P a, according to (18) and (19), by solving (4  m*  m) LP problems, i.e., max e, s.t. constraints (E(a, a*)), and max e, s.t. constraints (E(a*, a)), for each couple of pairs of actions (a, a*), (a*, a), a 2 A, a* 2 A*. 4. Calculate for each action a 2 A the indices LUP ðaÞ; LUN ðaÞ; RUN ðaÞ and RUP ðaÞ using (14)–(17). 5. Specify for each a 2 A its possible assignment C P ðaÞ ¼  U  LP ðaÞ; RUP ðaÞ . 6. Specify for each a 2 A its necessary assignment, which is   C N ðaÞ ¼ LUN ðaÞ; RUN ðaÞ in case LUN ðaÞ 6 RUN ðaÞ, and CN(a) = ;, otherwise. Observe that the above procedure is an example-based sorting procedure rather than a threshold-based procedure. It is also possible to define a threshold-based sorting procedure as follows. In a threshold-based sorting procedure, a pair (U, b), where U is a value function and b = [b1, . . ., bp1], is a vector of thresholds bh, h = 1, . . ., p  1, such that 0 < b1 < b2 < . . . < bp1 < 1, is consistent, if for all a* 2 A*

bRDM ða Þ > Uða Þ P bLDM ða Þ1 :

ð20Þ

On the basis of (20), we can state that, formally, a pair (U, b), where P U is an additive value function UðaÞ ¼ nj¼1 uj ðaÞ and b = [b1, . . ., bp1] is a vector of thresholds, is compatible, if it satisfies the following constraints:

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Uða Þ P bLDM ða Þ1

) ; 8a 2 A

Uða Þ < bRDM ða Þ     ðk1Þ uj xk  uj xj P 0; j

j ¼ 1; . . . ; n;

9 > > > > > > > > > > > > > > > > > > > > > > > > > > =

k ¼ 2; . . . ; mj ðA Þ;       m  m ðA Þ  uj x1 P 0; uj xj 6 uj xj j ; j ¼ 1; . . . ; n; j EAT : >   > > > > uj x0j ¼ 0; j ¼ 1; . . . ; n; > > > > >  m > n P > j > > uj xj ¼ 1; > > > j¼1 > > > > > > b1 > 0; bp1 < 1; > > > ; bh > bh1 ; h ¼ 2; . . . ; p  1: Let us also consider the following constraints:

9 > > > > > > > > > R ða Þ > > >     > > ðk1Þ > > uj xk x  u P 0; j ¼ 1; . . . ; n; > j j j > > > >  > > k ¼ 2; . . . ; mj ðA Þ; > > >       m > =  0 m ðA Þ j 1 uj xj P 0; uj xj 6 uj xj ; j ¼ 1; . . . ; n; EAT : > >   > > > uj x0j ¼ 0; j ¼ 1; . . . ; n; > > > > > >   n > P > mj > > uj xj ¼ 1; > > > j¼1 > > > > > b1 P e; bp1 6 1  e; > > > > ; bh  bh1 P e; h ¼ 2; . . . ; p  1: 9 Uða Þ P bLDM = ða Þ1 ; 8a 2 A b DM   Uða Þ P e ;

    0 Constraints EAT and EAT are equivalent if e > 0. Thus, to check if the set of all compatible pairs (U, b) is not empty it is suf  0 ficient to verify that e* > 0, where e* = max e s.t. constraints EAT . A compatible pair (U, b) assigns action a 2 A to some class Ck    with k P h, h 2 H, if it satisfies constraints EAT ða ! C Ph Þ obtained   by adding to constraints EAT the constraint U(a) P bh1. Analogously, a compatible pair (U, b) assigns action a 2 A to some class    Ck with k 6 h, h 2 H, if it satisfies constraints EAT ða ! C 6h Þ ob  the constraint U(a) < bh. tained by adding to constraints EA T From a computational point of view, it is more convenient to    express constraints EAT ða ! C Ph Þ as constraints obtained by add  0 the two further constraints: e > 0 and U(a) P bh1. ing to EAT Consequently, to verify that there is a compatible pair (U, b) assigning action a 2 A to some class Ck with k P h, h 2 H, it is enough to verify that e*(a ? CPh) > 0, where e*(a ? CPh) = max e s.t. con 0 straints EA plus the constraint U(a) P bh1 Instead, all compatT ible pairs (U, b) assign action a 2 A to some class Ck with k P h, h 2 H, if e(a ? CPh) 6 0, where e(a ? CPh) = max e s.t. constraints   0 EAT plus the constraint U(a) 6 bh1. Analogously, from a computational point of view, it is more   as constraints convenient to express constraints EA T ða ! C 6h Þ  0 obtained by adding to EA the two further constraints: e > 0 T and bh  U(a) P e. Consequently, to verify that there is a compatible pair (U, b) assigning action a 2 A to some class Ck with k 6 h, h 2 H, it is enough to verify that e*(a ? C6h) > 0, where



0

e*(a ? C6h) = max e s.t. constraints EA plus the constraint T bh  U(a) P e. Instead, all compatible pairs (U, b) assign action a 2 A to some class Ck with k 6 h, h 2 H, if e(a ? C6h) 6 0, where e(a ? C6h) = max e s.t. constraints

 0 EA plus the constraint T

bh  U(a) 6 0. On the basis of the above considerations, indices LUP ðaÞ; LUN ðaÞ; RUN ðaÞ and RUP ðaÞ can be redefined as follows:  minimum possible class:

LUP ðaÞ ¼ Maxðf1g [ fh 2 H : e ða ! C Ph Þ 6 0gÞ;

ð21Þ

 minimum necessary class:

LUN ðaÞ ¼ Maxðf1g [ fh 2 H : e ða ! C Ph Þ > 0gÞ;

ð22Þ

 maximum necessary class:

RUN ðaÞ ¼ Minðfpg [ fh 2 H : e ða ! C 6h Þ > 0gÞ;

ð23Þ

 maximum possible class:

RUP ðaÞ ¼ Minðfpg [ fh 2 H : e ða ! C 6h Þ 6 0gÞ:

ð24Þ

Therefore, the following threshold-based sorting procedure can be proposed: 1. Ask the DM for sorting examples. 2. Verify that the set of compatible pairs (U, b) is not empty. 3. Calculate for each action a 2 A the indices LUP ðaÞ; LUN ðaÞ; RUN ðaÞ and RUP ðaÞ using (21)–(24). 4. Specify for each a 2 A its possible assignment C P ðaÞ ¼  U LP ðaÞ; RUP ðaÞ . 5. Specify for each a 2 A its necessary assignment, which is C N ðaÞ ¼ LUN ðaÞ; RUN ðaÞ in case LUN ðaÞ 6 RUN ðaÞ and CN(a) = ;, otherwise. 6. Management of inconsistencies 6.1. Analysis of incompatibility Let us consider now the case where there is no value function U in an example-based sorting procedure, or equivalently, no pair (U, b) in a threshold-based sorting procedure, compatible with the assignment examples. We say, this is the case of incompatibil  ity. In such a case, the polyhedron generated by constraints EA is empty or, equivalently, the maximal value of e satisfying con  0 is not positive. Such a case may occur in one of EA

straints

the following situations:  the exemplary assignments of the DM do not match the additive model,  the DM may have made an error in his/her statements; for example stating that LDM(a) > RDM(b) while b dominates a (that is, gj(b) > gj(a), "j 2 G),  the statements provided by the DM are contradictory because his/her exemplary assignments are unstable, some hidden criteria are taken into account, etc. In such a case, the DM may want either to pursue the analysis with the incompatibility or to identify its reasons in order to remove it and, therefore, to define a new set of exemplary assign 0 ments whose corresponding constraints EA generate a nonext

empty polyhedron. Let us consider below the two possible solutions.

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6.1.1. Situation where incompatibility is accepted If the DM wants to pursue the analysis with the incompatibility, (s)he has to accept that some of his/her exemplary assignments of reference actions will not be reproduced by any value function. Note that, from a formal point of view, if the polyhedron generated by (EA*) is empty, then possible and necessary assignments CP(a) and CN(a) are meaningless for all a 2 A. Thus, the acceptance of the inconsistency means that the DM does not change the exemplary assignments and computes CP(a) and CN(a) on new con 0 straints EA differing from original constraints (EA*)0 by an ext

additional constraint on the acceptable total error:

9    Uða Þ P Uðb Þ þ e if LDM ða Þ > RDM ðb Þ; 8a ; b A ; > > >     > > ðk1Þ >  uj xj P 0; j ¼ 1; . . . ; n; uj xk > j > > > >  > k ¼ 2; . . . ; mj ðA Þ; > >       m > > m ðA Þ j 1 6 uj xj ; j ¼ 1; . . . ; n; = A 0 uj xj P 0; uj xj ; E   > ext > > uj x0j ¼ 0; j ¼ 1; . . . ; n; > > > >  m > n P > > j > uj xj ¼ 1; > > > j¼1 > > ; ext ePe ;

Min ! f ¼

X

v a;b

a;b2A :LDM ðaÞ>RDM ðbÞ

s:t: 8 UðaÞ P UðbÞ þ M v a;b P e if LDM ðaÞ > RDM ðbÞ; 8a; b 2 A ; > > >     > > ðk1Þ > >  uj xj P 0; j ¼ 1; . . . ; n; k ¼ 2; . . . ; m ðA Þ; uj xk > j > >      m > > m ðA Þ > > uj x1 P 0; uj xj 6 uj xj j ; j ¼ 1; . . . ; n; > j > >   < uj x0j ¼ 0; j ¼ 1; . . . ; n; > > >   > n P > j > > uj xm ¼ 1; > j > > j¼1 > > > > > v a;b 2 f0; 1g; > > : e P e0 ;

  0 where eext 6 e*, such that e* = max e s.t. constraints EA , so that ext   0 A the resulting new constraints E are feasible. Notice that eext ext

is a negative value because e* is negative due to the incompatibility of the exemplary assignments given by the DM. For any pair (a, b) 2 A, constraints (E(a, b)ext) and (E(b, a)ext) can be built as constraints (E(a, b)) and (E(b, a)), taking into account   0   0 EA rather than EA . Then, preference relations %0P and %0N ext

can be computed by maximizing e s.t. constraints (E(a, b)ext), and (E(b, a)ext), respectively. Obviously, the necessary and possible rankings resulting from these computations will not fully respect the exemplary assignments provided by the DM, i.e., there is at least one couple a*, b* 2 A* such that LDM(a*) > RDM(b*) and nevertheless U(a) 6 U(b). Of course, also in this context indices LUP ða Þ; LUN ða Þ; U RN ða Þ and RUP ða Þ maintain all their main properties stated in Proposition 4.2. 6.1.2. Situation where incompatibility is not accepted If the DM does not want to pursue the analysis with the incompatibility, it is necessary to identify the troublesome exemplary assignments responsible for this incompatibility, so as to remove some of them. Remark that there may exist several sets of pairwise   comparisons which, once removed, make constraints EA feasible. Hereafter, we outline the main steps of a procedure which identifies these sets. Recall that the exemplary assignments of reference actions are   represented in the ordinal regression constraints EA by linear constraints. Hence, identifying the troublesome pairwise comparisons of reference actions amounts at finding a minimal subset of   constraints that, once removed from EA , leads to constraints generating a non-empty polyhedron of compatible value functions. The identification procedure is to be performed iteratively since there may exist several minimal subsets of this kind. Let associate with each couple of actions a, b 2 A* such that LDM(a) > RDM(b) a new binary variable va,b, i.e., va,b = 0 or va,b = 1. Using these binary variables, we rewrite the first constraint of (EA*) as follows:

UðaÞ  UðbÞ þ Mv a;b > 0 if LDM ðaÞ > RDM ðbÞ;

where M > 1. Remark that if va,b = 1, then the corresponding constraint is satisfied whatever the value function is, which is equivalent to elimination of this constraint. Therefore, identifying a minimal subset of troublesome pairwise comparisons can be performed by solving the following mixed 0–1 linear program:

ð25Þ

ð26Þ

where e0 is a small positive value. The optimal solution of (26) indicates one of the subsets of smallest cardinality being the cause of incompatibility. Alternative subsets of this kind can be found by solving (26) with additional constraint that forbids finding again the same solution. Let f* be the optimal value of the objective function of (26) and v a;b the values of the binary variables at the optimum. Let also S1 ¼ fða; bÞ 2 A  A : LDM ðaÞ > RDM ðbÞ and v a;b ¼ 1g. The additional constraint has then the form

X

v a;b 6 f   1:

ð27Þ

ða;bÞ2S1

Continuing in this way, we can identify other subsets, possibly all of them. These subsets of pairwise comparisons are to be presented to the DM as alternative solutions for removing incompatibility. Such a procedure has been described in [9]. Notice that (26) is a mixed integer linear program, so it is clearly not as easy to solve as a simple linear program. Mousseau et al. [9] propose an LP-based algorithm to solve this problem, which proved empirically to be rather efficient. In any case, when the number of reference actions increases, solving (26) can become computationally difficult. Observe, however, that very often the actions in the reference set are not numerous, because they should be rather well known to the DM for (s)he is enough confident with respect to their assignment. Nevertheless, if problem (26) appears to be computationally intractable, one can ask the DM to indicate which constraints (s)he would accept to remove and which constraints (s)he would not accept to remove. In this way, the number of 0-1 variables can be reduced to the number of constraints that the DM is ready to remove.

7. Specifying assignments with gradual confidence levels The UTADISGMS method presented in Section 5 is intended to support the DM in an interactive process. Indeed, defining the set of exemplary assignments of reference actions can be difficult for the DM, even if intervals of classes (imprecise assignments) are accepted. Therefore, one way to reduce the difficulty of this task is to permit the DM an incremental specification of exemplary assignments. This way of proceeding allows the DM to control the evolution of the necessary and possible assignments. Another way of reducing the difficulty of the DM’s task is to extend the UTADISGMS method so as to account for different h  confidence levels given to exemplary assignments. Let LDM 1 ða Þ;

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h i h i DM DM DM DM      RDM be embed1 ða Þ L2 ða Þ; R2 ða Þ    Ls ða Þ; Rs ða Þ ded sets of DM’s exemplary assignments of reference actions h i DM   a* 2 A*. To sets of exemplary assignments LDM t ða Þ; Rt ða Þ ; t ¼   generating sets of 1; . . . s; a 2 A , correspond constraints EAt compatible value functions U tA . Sets U tA ; t ¼ 1; . . . ; s, are embedded in the same order as the sets of the exemplary assignments h i  DM 1 2 s    LDM t ða Þ; Rt ða Þ ; t ¼ 1; . . . ; s; a 2 A , i.e., U A U A    U A . We suppose that U sA – ; and, therefore, as U 1A U 2A    U sA we have U tA – ;, for all t = 1, . . ., s. If U sA – ;, we consider exemplary h i  DM    where assignments only until LDM p ða Þ; Rp ða Þ ; a 2 A , p ¼ maxft : U tA – ;g. Definition 7.1. Given sets of exemplary assignments h i  DM    LDM ða Þ; R ða Þ ; t ¼ 1; . . . ; s; a 2 A , and corresponding sets t t U tA

of compatible value functions of level t, for each a 2 A, the following indices have to be considered:  minimum possible class:  n o  DM t    ; Lt;U P ðaÞ ¼ Max f1g [ Lt ða Þ : 8U 2 U A ; Uða Þ 6 UðaÞ; a 2 A

ð28Þ

relations %Pt and %Nt , we should proceed as follows. For all pairs (a, b) 2 A, constraints (Et(a, b)) and (Et(b, a)) can be obtained from   constraints EAt by adjoining the constraints relative to the characteristic points introduced by actions a, b that possibly do not belong to A*. For each t = 1, . . ., s, and binary preference relations %Nt and %Pt , we have

a%Nt b () et ðb; aÞ 6 0; where

 n DM t  Lt;U P ðaÞ ¼ Max f1g [ Lt ða Þ : 9U 2 U A 

for which Uða Þ 6 UðaÞ; a 2 A



o

;

ð29Þ

et ðb; aÞ ¼ max e; s:t: constraints ðEt ðb; aÞÞ;

ð34Þ

and

a%Pt b () et ða; bÞ > 0; where

 minimum necessary class:



In order to compute possible and necessary assignments C tP ðaÞ and C tN ðaÞ; t ¼ 1; . . . ; s, we can use an example-based sorting procedure, as well as a threshold-based sorting procedure. Using example-based sorting procedure, we should proceed as follows. For all a, b 2 A, we say that there is a necessary weak preference relation of level t, denoted by a%Nt b ðt ¼ 1; . . . ; sÞ, if for all value functions U 2 U tA , we have U(a) P U(b). Analogously, for all a, b 2 A, we say that there is a possible weak preference relation of level t, denoted by a%Pt b ðt ¼ 1; . . . ; sÞ, if for at least one value function U 2 U tA , we have U(a) P U(b). In order to compute possible and necessary weak preference

et ða; bÞ ¼ max e; s:t: constraints ðEt ða; bÞÞ:

ð35Þ

Each time we pass from %t1 to %t ; t ¼ 1; . . . ; s  1, we add to    A Et1 and, consequently, to (Et1(a, b)), new constraints concern

DM  ing pairs (a*, b*) 2 A*  A*, such that LDM t ða Þ > Rt ðb Þ but not

 maximum necessary class:

Rt;U P ðaÞ

 n t  ¼ Min fpg [ RDM t ða Þ : 9U 2 U A for which UðaÞ 6 Uða Þ; a 2 A



o ;

ð30Þ

 maximum possible class:  n o  DM t    Rt;U : P ðaÞ ¼ Min fpg [ Rt ða Þ : 8U 2 U A ; UðaÞ 6 Uða Þ; a 2 A

ð31Þ

Definition 7.2. Given sets of exemplary assignments h i  DM DM    Lt ða Þ; Rt ða Þ ; t ¼ 1; . . . ; s; a 2 A , and corresponding sets U tA of compatible value functions of level t, for each a 2 A, we define the possible assignment C tP ðaÞ of level t, as the set of indices of class Ch for which there exists at least one value function U 2 U tA assigning a to Ch, and the necessary assignment C tN ðaÞ of level t, as the set of indices of class Ch for which all value functions U 2 U tA assign a to Ch, i.e., for t = 1, . . ., s:

n h io ¼ h 2 H : 9U 2 U tA for which h 2 LU ðaÞ; RU ðaÞ ; n h io C tN ðaÞ ¼ h 2 H : 8U 2 U tA it holds h 2 LU ðaÞ; RU ðaÞ :

C tP ðaÞ

Proposition 7.1. For all t = 2, . . ., s, and for all a 2 A, t1;U t1;U Lt;U ðaÞ and Rt;U ; ðaÞ; P ðaÞ P LP P ðaÞ 6 Rp

and thus

C tP ðaÞ # C pt1 ðaÞ: Proof. See Appendix C.1. h

ð32Þ ð33Þ

DM N P  LDM t1 ða Þ > Rt1 ðb Þ, thus the computations of %t and %t , proceed iteratively. Binary preference relations %Nt and %Pt ; t ¼ 1; . . . ; s, satisfy the following properties:

Proposition 7.2. It holds:    

%Nt # %Pt , %Nt is a complete preorder (i.e., transitive and strongly complete), %Pt is strongly complete and negatively transitive, %Nt1 # %Nt and %Pt # %Pt1 , t = 2, . . . , s.

For the proof of Proposition 7.2, see [3]. On the basis of binary preference relations %Nt and %Pt ; t ¼ 1; . . . ; s, it is possible to calculate iteratively the indices (28)–(30) in the same way as in Section 5. 8. Illustrative example In this section, a real-world multiple criteria sorting problem will be presented followed by the application of the UTADISGMS software. A transport company is about to classify 76 buses into 4 pre-defined and preference ordered classes C1  C4, such that C1 will group the buses being in the worst technical state, needing a vary major revision, C2 will group the buses being in the lower-intermediate technical state, needing a major revision, C3 will group the buses being in the upper-intermediate technical state, needing a minor revision, and C4 will group the buses being in the best technical state, needing no revision. The buses were evaluated according to a total of 8 quantitative criteria reflecting their performance and technical parameters. The names and types of the criteria are given in Table 1, and the performance matrix for a subset of buses is presented in Table 2 (see [11]).

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Table 1 Table of criteria.

Table 4 Possible assignments in the first iteration.

Code

Name

Type

LUP

g1 g2 g3 g4 g5 g6 g7 g8

Maximum speed Compression pressure Blacking Torque Summer fuel consumption Winter fuel consumption Oil consumption Horse power

Gain Gain Cost Gain Cost Cost Cost Gain

C1 C1 C1

C1 C2 C3

C1

C4

a6, a23, a40, a60, a62, a69 a8, a14, a17, a19 a2, a10, a12, a21, a24, a26, a27, a30, a34, a36, a38, a39, a45, a46, a47, a48, a50, a53, a63, a66, a67 a3, a9, a11, a15, a16, a20, a31, a52, a58, a68, a70

C2 C2 C2

C2 C3 C4

a1 a25, a28, a42 a4, a13, a22, a33, a37, a41, a43, a44, a55, a56, a59, a64, a65, a71, a73, a74, a75

C3 C3

C3 C4

a5 a32, a35, a51, a54, a57, a61, a76

C4

C4

a7, a18, a29, a49, a72

Table 2 Performance matrix. Bus

g1

g2

g3

g4

g5

g6

g7

g8

a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 .. . a74 a75 a76

90 76 63 90 85 72 88 87 90 60 .. . 87 86 88

2.52 2.11 1.98 2.48 2.45 2.20 2.50 2.48 2.56 1.95 .. . 2.48 2.50 2.52

38 70 82 49 52 73 50 56 16 95 .. . 52 55 46

481 420 400 477 460 425 480 465 486 400 .. . 465 456 472

21.8 22.0 22.0 21.9 21.8 23.1 21.6 22.8 26.5 23.3 .. . 21.9 22.0 21.8

26.4 25.5 24.8 25.1 25.2 27.4 24.7 27.6 27.3 24.8 .. . 24.6 25.1 23.8

0.7 2.7 3.7 1.0 1.4 2.8 1.1 1.4 0.2 4.4 .. . 1.4 1.5 1.1

145 110 101 138 130 112 140 135 150 96 .. . 135 130 141

We will apply the example-based sorting procedure to this multiple criteria sorting problem. In Step 1, the diagnostic expert, assisted by a decision analyst, is able to make a more or less precise assignment of few buses to the classes of technical state. The state of these buses, called reference buses, is relatively well known to the expert, so they will serve to provide assignment examples in successive iterations. In the first iteration, the expert provides a typical example for each one of the four classes, and makes, moreover, an imprecise assignment of two other reference buses to the two intermediate classes. This preference information is given in Table 3. In Step 2, we verify that for this initial preference information the set of compatible value functions U A is not empty, by verifying that e* > 0, where e* = max e s.t. constraints (EA*)0 . In Step 3, using (18) and (19), we calculate the necessary and the possible weak preference relations a%N a ; a%P a ; a %N a and a %P a, by solving (4  6  76) LP problems, i.e., (6  76) LP problems max e s.t (E(a, a*)) and (6  76) LP problems max e s.t. (E(a*, a)), for each couple of pairs (a, a*) (a*, a), a 2 A, a* 2 A*. In Step 4, for each action a 2 A, we calculate LUP ðaÞ; LUN ðaÞ; RUN ðaÞ and RUP ðaÞ using (14)–(17). In Steps 5 and 6, on the basis of indices LUP ðaÞ; LUN ðaÞ; RUN ðaÞ and RUP ðaÞ for each action a 2 A possible assignment CP(a) and its necessary assignment CN(a) are specified. The results of Steps 4, 5 and 6 are presented in Tables 4 and 5. To obtain them, we used the UTADISGMS method implemented on the Deci-

1

RUP

1

Assigned buses

Table 5 Necessary assignments in the first iteration. LUN

RUN

Assigned buses

C1 C2 C3 C4

C1 C2 C3 C4

a6, a23, a40, a60, a62, a69 a1 a5 a7, a18, a29, a49, a72

1

1

Table 6 Additional exemplary assignments in the second iteration. Reference bus

Lower class (LDM)

Upper class (RDM)

a30 a35

C1 C3

C1 C3

sionDeck Platform (http://decision-deck.sourceforge.net/) as an open source software. Next iterations were performed in the same way. The resulting possible assignments review all possible consequences of the preference information on sorting of the whole set of buses (see Table 4). One can see that the inferred model has assigned all reference buses to their respective desired classes: a1 is assigned to C2, a5 to C3, etc. For most of non-reference buses the possible assignment is non-univocal. Precisely, we have 9 non-reference buses assigned to single class C1 or C4, 12 and 38 buses assigned, respectively, to the range of two and three contiguous classes, and 11 busses which can be possibly assigned to all four classes. In the first iteration, when the preference information is yet rather poor, the possible weak preference relation used to assess the necessary assignments is very rich, i.e., for most of pairs of buses (ai, aj) it is true ai %P aj as well as aj %P ai . Therefore, in most 1

C 1N ðaÞ

Table 3 Exemplary assignments of some reference buses in the first iteration. Reference bus

Lower class (LDM)

Upper class (RDM)

a1 a5 a7 a28 a40 a42

C2 C3 C4 C2 C1 C2

C2 C3 C4 C3 C1 C3

1

cases, one can observe that RUN ðaÞ < LUN ðaÞ and, consequently, ¼ ;. Table 5 summarizes the non-empty necessary assignments in the first iteration. The necessary assignment is nonempty for the 4 reference buses and for 9 non-reference ones assigned to the extreme classes. Remember that the method does not guarantee that all imprecise assignments of reference actions are repeated in the necessary assignments. For example, the range of necessary assignment for bus a28 is empty because for some compatible value functions it is assigned to C2 and for some others to C3, but there is no compatible value function for which the assignment of a28 to both classes would be feasible. Obviously, for every bus ai it holds C 1N ðai Þ # C 1P ðai Þ.

S. Greco et al. / European Journal of Operational Research 207 (2010) 1455–1470

The UTADISGMS method is intended to be used interactively, so that the DM and the analyst could provide assignment examples incrementally, looking at consequences of the introduced preference information on the possible assignments of all actions. Let us suppose that considering the results of the first iteration, our diagnostic expert feels confident that a30, a19, and a35 should be assigned to class C1, C2 and C3, respectively. It appears, however, that the introduced preference information is inconsistent, and no additive value function is compatible with the reference assignments. The method states that the newly introduced assignments (a30 ? C1) and (a19 ? C2) cannot be represented together by any additive value function. Then, the DM may want either to pursue the analysis with this incompatibility, or to remove it. Suppose that our DM will remove one of the two inconsistent exemplary assignments, precisely the one concerning a19, so that the additional Table 7 Possible and necessary assignments in the second iteration. LUP

2

RUP

2

Assigned buses

C1 C1 C1

C1 C2 C3

C1

C4

a6, a19, a23, a30, a40, a47, a60, a62, a63, a69 a8, a14, a17 a2, a10, a11, a12,a15, a20, a21, a24, a26, a27, a31, a34, a36, a38, a39, a45, a46, a48, a50, a53, a66, a67, a70 a3, a9, a16, a52, a58, a68

C2 C2 C2

C2 C3 C4

a1 a25, a28, a41, a42, a43, a64 a4, a13, a22, a33, a37, a44, a55, a56, a59, a65, a71, a73, a74, a75

C3 C3

C3 C4

a5, a35 a32, a51, a4, a57, a61, a76

C4

C4

a7, a18,a29, a49, a72

2 LUN

2 RUN

Assigned buses

C1 C2 C3 C4

C1 C2 C3 C4

a6, a19, a23, a30, a40, a47, a60, a62, a63, a69 a1 a5, a35 a7, a18, a29, a49, a72

Table 8 Additional exemplary assignments in the third iteration. Reference bus

Lower class (LDM)

Upper class (RDM)

a26 a74

C2 C4

C2 C4

Table 9 Possible and necessary assignments in third iteration. LUP

3

RUP

3

Assigned buses

C1 C1 C1

C1 C2 C3

C1

C4

a6, a19, a23, a30, a40, a47, a60, a62, a63, a69 a8, a14, a17, a27 a2, a10, a11, a12, a15, a20, a21, a24, a34, a36, a38, a39, a45, a46, a48, a50, a53, a66, a67, a70 a3, a9, a16, a52, a58, a68

C2 C2 C2

C2 C3 C4

a1, a26 a25, a28, a31, a41, a42, a43, a64 a4, a13, a22, a33, a37, a44, a55, a56, a59, a65, a71, a73, a75

C3 C3

C3 C4

a5, a35 a32, a51, a54, a61

C4

C4

a7, a18, a29, a49, a57, a72, a74, a76

3 LUN

3 RUN

Assigned buses

C1 C2 C3 C4

C1 C2 C3 C4

a6, a19,a23, a30, a40, a47, a60, a62, a63 a1, a26 a5, a35 a7, a18, a29, a49, a57, a72, a74, a76

1465

exemplary assignments are those presented in Table 6. In consequence, the system becomes consistent, and the corresponding results are presented in Table 7. One can observe that the assignments are more tight with the growth of the preference information. There are 10 non-reference buses, for which the possible assignments have changed with respect to the first iteration (those buses are marked in bold in the table). For example, a11 and a41 are possibly assigned to classes C1, C2 and C3 but not to C4, as previously. As far as necessary assignments are concerned, there are 5 additional buses assigned to the same class by all compatible value functions: reference bus a35 and 4 non-reference buses have been assigned to class C1. In the third iteration, the diagnostic expert provides two additional exemplary assignments (see Table 8). As a consequence (see Table 9), 4 non-reference buses got a tighter range of possible assignments (those buses are marked in bold in the table). Moreover, two of them are necessarily assigned to a non-empty range of classes which they were not in the previous iteration. The illustration of the use of the UTADISGMS method will be stopped at this stage. The DM may be satisfied with the results, or may want to pursue the iterative process, either by adding some new assignments of reference buses or by revising the previous judgments, which would result in new possible and necessary assignments.

9. Conclusions In this paper, we presented a new multiple criteria sorting method called UTADISGMS, which assigns actions into ordered classes. The underlying preference model is a set of monotone non-decreasing additive value functions. The method applies to the case in which the classes are explicitly defined by thresholds on the value that separate consecutive classes, and to the case where classes are implicitly defined through the assignment examples. In order to specify the sorting model, the DM is asked to provide assignment examples (desired assignment for specific actions) which yield a set of compatible additive value functions. Taking into account the set of assignment examples as a preference information, the method provides for each action a two intervals of classes: the possible assignment corresponds to the interval of classes to which action a can be assigned considering any compatible value functions individually, while the necessary assignment corresponds to the interval of classes to which action a can be assigned considering all compatible value functions simultaneously. The necessary and possible assignments are computed through the resolution of linear programs. The UTADISGMS method has some new desirable characteristics and provides some new capabilities for the analyst and for the DM, which did not exist in any previous method. First, it considers possible and necessary assignments which no previous method did consider. Second, UTADISGMS makes use of a very general and flexible preference model as it considers all non-decreasing marginal value functions (rather than piecewise linear marginal value functions). Third, when computing assignments, it takes into account all value functions compatible with the assignment examples provided by the DM (and not only a single ‘‘representative” value function). Finally, the method makes it possible to account for both the threshold-based and example-based sorting procedures. We envisage the following possible developments of this method in the future:  consideration of other types of preference information, such as intensity of preferences and tradeoffs,  extension of the method to group decision (a preliminary idea has been given in [4]),

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 application of the method to interactive optimization problems,  application of the method to some real-world sorting problems, such as bankruptcy risk or global risk evaluation, and comparison with classical ordinal regression methods (UTADIS, MHDIS, etc.), as well as with other competitive methods, like discriminant analysis, dominance-based rough set approach, neural nets or support vector machines,  improve the interactive process in order to better support the DM in the elaboration of robust recommendations. Acknowledgements

A.3. Proof of Proposition 3.3 Proposition 3.3. Consider the case where the DM provides precise assignment examples only, i.e., LDM(a*) = RDM(a*), for each a* 2 A*. Assuming the use of a single value function U in the example-based sorting procedure, if we choose, for each h = 1, . . ., p  1, the threshold U

bh in the interval Maxa !C h fUða Þg; Mina !C hþ1 fUða Þg, we obtain a threshold-based sorting procedure that restores the assignment examples and assigns each non-reference action a 2 AnA* to a single h i class in the interval C LU ðaÞ ; C RU ðaÞ stemming from the example-based sorting procedure.

The authors acknowledge the support of the COST Action IC0602 ‘‘Algorithmic Decision Theory”, www.algodec.org. Moreover, the third author wishes to acknowledge financial support from the Polish Ministry of Science and Higher Education, Grant N519 314435. The authors also thank Milosz Kadzinski who implemented the UTADISGMS method on the DecisionDeck Platform and run the calculations of the illustrative example. The comments of two anonymous referees permitted to improve previous versions of the paper.

Proof. As we suppose the assignment examples to be consistent with U, we have "a*, b* 2 A*, U(a*) P U(b*) ) RDM(a*) P LDM(b*). Moreover, since all assignment examples are precise (LDM(a*) = RDM(a*)), we will use LDM(a*) to refer to the category to which the DM wishes to assign a*. Therefore, the fact that assignment examples are consistent with U can be reformulated as "a*, b* 2 A*, U(a*) P U(b*) ) LDM(a*) P LDM(b*). If we choose the U

threshold values bh in the interval Maxa !C h fUða Þg;  Mina !C hþ1 fUða Þg, for each h=1, . . ., p-1, then any a 2 A for which h h U U UðaÞ 2 bh1 ; bh will be assigned by the threshold-based sorting

Appendix A. Proofs of propositions from Section 3 A.1. Proof of Proposition 3.1 Proposition 3.1. For any action a 2 AnA*, the sorting h example-based i procedure provides an assignment a ! C LU ðaÞ ; C RU ðaÞ , such that LU(a) 6 RU(a).

procedure to category Cit h (see Definition 3.1). As a consequence, the assignment examples will be restored by the threshold-based sorting procedure. Considering a non-reference action a 2 AnA*, we distinguish two cases:

Proof. Consider the sets A*+ A* and A* A* defined as A* = {a* 2 A*: U(a*) 6 U(a)} and A*+ = {a* 2 A*: U(a*) P U(a)}. If A* = ;, then LU(a) = 1 and, consequently, LU(a) 6 RU(a). Analogously, if A*+ = ;, then RU(a) = p and, consequently, LU(a) 6 RU(a). Now let us suppose that A*+ – ; and A* – ; and

(i) $h 2 {1, . . ., p} such that Maxa !C h fUða Þg, (ii) $h 2 {1, . . ., p  1} such that Mina !C hþ1 fUða Þg½.

LU ðaÞ > RU ðaÞ:

Case (i):

ð36Þ *

*

*+

A.2. Proof of Proposition 3.2 Proposition 3.2. The example-based sorting procedure assigns h i each reference action a* 2 A* to an interval of classes C LU ða Þ ; C RU ða Þ , such that: 

L ða Þ P L

DM



ða Þ;

ð37Þ

RU ða Þ 6 RDM ða Þ:

ð38Þ

Proof. Observe that taking into account sets A* and A*+ introduced in the proof of 3.1, we have that a* 2 A* and a* 2 A*+, such that A*+ – ; and A* – ;. Therefore, according to (6) and (7):

n o LU ða Þ ¼ Max LDM ða0 Þ : Uða0 Þ 6 Uða Þ; a0 2 A ; n o RU ða Þ ¼ Min RDM ða0 Þ : Uða0 Þ P Uða Þ; a0 2 A : 0

0

 UðaÞ 2 Maxa !C h fUða Þg;

*+

(36) means that there exists a 2 A and a 2 A such that LDM(a*) = LU(a), RDM(a*+) = RU(a) and LDM(a*) > RDM(a*+). As the set of assignment examples is supposed to be consistent with U, LDM(a*) > RDM(a*+) implies that U(a*) > U(a*+). However, as a*+ 2 A*+ and a* 2 A*, it holds that U(a*) 6 U(a*+), which contradicts the hypothesis and concludes the proof. h

U

 UðaÞ 2 Mina !C h fUða Þg;

ð39Þ

Considering Definition 3.4, we have LU(a) = RU(a) = h, i.e., a is assigned to class Ch by the example-based sorting procedure. In h i this case, the interval C LU ðaÞ ; C RU ðaÞ boils down to Ch.   Moreover, since UðaÞ 2 Mina !C h fUða Þg; Maxa !C h fUða Þg , a will be assigned to class Ch by the threshold-based sorting procedure, whatever the value of the thresholds bh1 2 Maxa !C h1 fUða Þg; Mina !C h fUða Þg and bh 2Maxa !C h fUða Þg; Mina !C hþ1 fUða Þg. Case (ii): Considering Definition 3.4, we have LU(a) = h and RU(a) = h + 1, i.e., a is assigned imprecisely to interval of classes [Ch, Ch+1] by the example-based sorting procedure.   Moreover, since UðaÞ 2 Maxa !C h fUða Þg; Mina !C hþ1 fUða Þg , a will be assigned by the threshold-based sorting procedure to class Ch+1 if bh 2Maxa!C h fUða Þg; UðaÞ, or to class Ch if   bh 2 UðaÞ; Mina !C hþ1 fUða Þg . h

A.4. Proof of Proposition 3.4

ð40Þ 0

Since U(a*) 6 U(a*), then LDM(a*) 2 {LDM(a* ): U(a* ) 6 U(a*), a* 2 A*} 0 0 0 and, therefore, Max {LDM(a* ): U(a* ) 6 U(a*), a* 2 A*} P LDM(a*), i.e., U DM U L (a*) P L (a*). Analogous proof holds for R (a*) 6 RDM(a*). h

Proposition 3.4. Consider the case where the DM provides possibly imprecise assignment examples, i.e., LDM(a*) 6 RDM(a*) for each a* 2 A*. Assuming the use of a single value function U in the example-based sorting procedure, if we choose, for each

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  U U h = 1, . . ., p  1, the threshold bh in the interval I bh ¼ i i U U Maxa :RDM ða Þ6h fUða Þg; Mina :LDM ða Þ>h fUða Þg , with bh < bhþ1 , we obtain a threshold-based sorting procedure that assigns each reference h i action a* 2 A* to a single class in the interval C LDM ða Þ ; C RDM ða Þ , and assigns each non-reference action a 2 AnA* to a single class in the h i interval C LU ðaÞ ; C RU ðaÞ stemming from the example-based sorting

and

i i U bk 2 Maxa :RDM ða Þ6k fUða Þg; Mina: LDM ða Þ>k fUða Þg ; we have U

u

Maxa :RDM ða Þ6k1 fUða Þg < bk1 6 Uða0 Þ < bk

6 Mina :LDM ða Þ>k fUða Þg:

procedure. From Proof. The proof will consist of the five following points:   U 1. The set of possible values for bh , I bh , is a non-empty interval.

Maxa :RDM a 6k1 fðUða Þg < Uða0 Þ; 0

U

2. If bh 6 Maxa :RDM ða Þ6h fUða Þg, then the threshold-based sorting 0 U procedure assigns each a* 2 A* such that bh 6 Uða0 Þ 6  Maxa :RDM ða Þ6h fUða Þg inconsistently with the statement of the 0 DM, in the sense that a* is assigned to a class Ck with DM *0 k > R (a ). U 3. If bh > Mina :LDM ða Þ>h fUða Þg, then the threshold-based sorting 0 procedure assigns each a* 2 A* such that Mina :LDM ða Þ>h U  0 fUða Þg 6 Uða Þ < bh inconsistently with the statement of the 0 DM, in the sense that a* is assigned to a class Ck with DM *0 k < L (a ).   U U 4. If bh 2 I bh ; h ¼ 1; . . . ; p  1, then the threshold-based sorting procedure assigns each a* 2 A* to a single class Ck in the interval h i C LDM ða Þ ; C RDM ða Þ , and each a 2 AnA* in a single class Ck in the h i interval C LU ðaÞ ; C RU ðaÞ . h i 5. For each a* 2 A*, and for each C k 2 C LDM ða Þ ; C RDM ða Þ , there exist   U U thresholds bh 2 I bh ; h ¼ 1; . . . ; p  1, such that the threshold-based sorting procedure assigns a* to Ck; for each a 2 AnA*, h i and for each C k 2 C LU ðaÞ; C RU ðaÞ , there exist thresholds   U U bh 2 I bh ; h ¼ 1; . . . ; p  1, such that the threshold-based sorting procedure assigns a to Ck. Proof of 1.  Consider b*, c* 2 A* such that Uðb Þ ¼ Maxa :RDM ða Þ6h fUða Þg and   Uðc Þ ¼ Mina :LDM ða Þ>h fUða Þg. Therefore, we have LDM(c*) > h P RDM(b*). As we suppose the assignment examples provided by the DM are consistent with U (see (4)), we have U(b*) < U(c*), i.e., Maxa :RDM ða Þ6h fUða Þg < Mina :LDM ða Þ>h fUða Þg. Proof of 2. 0 U Consider a* 2 A* such that bh 6 Uða0 Þ 6 Maxa :RDM ða Þ6h fUða Þg. 0 U 0 Since bh 6 Uða Þ, according to Definition 3.1, a* should be assigned to class Ck with k>h. 0 As Uða0 Þ 6 Maxa :RDM ða Þ6h fUða Þg, we have RDM(a* ) 6 h. There0 fore, the threshold-based sorting procedure assigns a* inconsistently with the DM’s statement. Proof of 3. 0 U Consider a* 2 A* such that Mina :LDM ða Þ>h fUða Þg 6 Uða0 Þ < bh . U *0 0 Since Uða Þ < bh , according to Definition 3.1, a should be assigned to class Ck with k6h. 0 As Mina :LDM ða Þ>h fUða Þg 6 Uða0 Þ, we have LDM(a* ) > h. There0 fore, the threshold-based sorting procedure assigns a* inconsistently with the DM’s statement. Proof of 4. h h 0 U U Consider a* 2 A* such that Uða0 Þ 2 bk1 ; bk and, therefore, for 0

Definition 3.1, a* ? Ck. Since, by hypothesis,

i i U bk1 2 Maxa :RDM ða Þ6k1 fUða Þg; Mina: LDM ða Þ>k1 fUða Þg ;

we get RDM(a* ) P k, while from

Uða0 Þ < Mina :LDM ða Þ>k fUða Þg; h i 0 we get LDM(a* ) 6 k. Thus, C k 2 C LDM ða0 Þ ; C RDM ða0 Þ .

h h U U Analogously, for a 2 AnA* for which UðaÞ 2 bk1 ; bk and, h i consequently, a ? Ck, we have C k 2 C LU ðaÞ ; C RU ðaÞ . Proof of 5. h i 0 Consider a* 2 A* such that C k 2 C LDM ða0 Þ ; C RDM ða0 Þ . U Let us fix the thresholds bh ; h ¼ 1; . . . ; p  1, such that: U

 bh 6 Uða0 Þ for each h < k, U

 bh > Uða0 Þ for each h P k, U

U

with bh < bhþ1 . 0 For the above thresholds, a* is assigned by the threshold-based procedure to class Ck. Hence, we need to prove that there always exists a set of thresholds defined as above which verify

i   i U U bh 2 I bh ¼ Maxa :RDM ða Þ6h fUða Þg; Mina :LDM ða Þ>h fUða Þg ; h ¼ 1; . . . ; p  1:

ðiÞ

Consider h < k. For (i) we have U

Uða0 Þ P bh > Maxa :RDM ða Þ6h fUða Þg; 0

which holds because RDM(a* ) P k > h. Consider h P k. For (i) we have u

Uða0 Þ < bh 6 Mina :LDM ða Þ>h fUða Þg; 0

which holds because LDM(a* ) 6 k 6 h.

h

Appendix B. Proofs of propositions from Section 4 B.1. Proof of Proposition 4.2 Proposition 4.2. Given a set A* of assignment examples and a corresponding set U A of compatible value functions, it holds, for each a 2 A,  LUP ðaÞ 6 LUN ðaÞ,  RUN ðaÞ 6 RUP ðaÞ,  LUP ðaÞ 6 RUP ðaÞ. Moreover, for any a 2 A, we have:    C P ðaÞ # LUP ðaÞ; RUP ðaÞ ,  if LUN ðaÞ 6 RUN ðaÞ, then C N ðaÞ ¼ ½LUN ðaÞ; RUN ðaÞ,  if LUN ðaÞ > RUN ðaÞ, then CN(a) = ;.

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Proof. Consider

8  Aall ðaÞ ¼ fa 2 A : 8U 2 U A ; Uða Þ 6 UðaÞg; > > >  < Aone ðaÞ ¼ fa 2 A : 9U 2 U A for which Uða Þ 6 UðaÞg; 8a 2 A: þ    > > Aall ðaÞ ¼ fa 2 A : 8U 2 U A ; Uða Þ P UðaÞg; > : þ Aone ðaÞ ¼ fa 2 A : 9U 2 U A for which Uða Þ P UðaÞg:  þ The proof in case one or more among A all ðaÞ; Aone ðaÞ; Aall ðaÞ, and Aþ ðaÞ are empty is simple and left to the reader. Therefore, in one  þ þ the following, we suppose that A all ðaÞ; Aone ðaÞ; Aall ðaÞ, and Aone ðaÞ are all not empty.  Observe that A all ðaÞ # Aone ðaÞ from which we get:

n

o

LUP ðaÞ ¼ Maxa 2A LDM ða Þ : 8U 2 U A ; Uða Þ 6 UðaÞ n o 6 Maxa 2A LDM ða Þ : 9U 2 U A for which Uða Þ 6 UðaÞ

Proof. Let us observe that if for a 2 A, CN(a) = ;, then there exist U; U 0 2 U A such that

h i 0 0 LU ðaÞ; RU ðaÞ \ ½LU ðaÞ; RU ðaÞ ¼ ;: 0

We have (43) if LU ðaÞ > RU ðaÞ or LU ðaÞ > RU ðaÞ. Let us suppose, without loss of the generality, that we have (43) because 0 0 LU ðaÞ > RU ðaÞ. Observe that LU ðaÞ > RU ðaÞ implies LU(a) > 1 and U0 R ðaÞ < p, and therefore it is

n o LDM ðc Þ : Uðc Þ 6 UðaÞ; c 2 A – ;; and

n o RDM ðc Þ : U 0 ðc Þ P U 0 ðaÞ; c 2 A – ;: 0

LU ðaÞ > RU ðaÞ is equivalent to the fact that there exist a*, b* 2 A* for which LDM(a*) > RDM(b*), U(a) P U(a*) and U0 (b*) P U0 (a), such that

¼ LUN ðaÞ:

n o LU ðaÞ ¼ max LDM ðc Þ : Uðc Þ 6 UðaÞ; c 2 A P LDM ða Þ

þ Analogously, from Aþ all ðaÞ # Aone ðaÞ, we get:

n o RUN ðaÞ ¼ Mina 2A RDM ða Þ : 9U 2 U A ; for which UðaÞ 6 Uða Þ n o 6 Mina 2A RDM ða Þ : 8U 2 U A ; UðaÞ 6 Uða Þ ¼ RUP ðaÞ:

> n o 0  RDM ðb Þ ¼ min RDM ðc Þ : U 0 ðc Þ P U 0 ðaÞ; c 2 A ¼ RU ða Þ:

þ For all U 2 U A and for all a 2 A 2 Aþ all ðaÞ and a all ðaÞ we have * *+ DM * DM *+ U(a ) 6 U(a) 6 U(a ) which implies L (a ) 6 R (a ). Therefore,

n o n o max LDM ða Þ : ða Þ 2 A 6 min RDM ða Þ : a 2 Aþ all all ðaÞ :

ð43Þ 0

Thus, there do not exist U; U 0 2 U A for which U(a) P U(a*) and U0 (b*) P U0 (a), i.e., strict continuity holds, if and only if for all 0 0 U; U 0 2 U A ; ½LU ðaÞ; RU ðaÞ \ ½LU ðaÞ; RU ðaÞ – ;, and consequently CN(a) – ;. h

ð41Þ B.3. Proof of Proposition 4.4

From the definition of LUP ðaÞ and RUP ðaÞ, we get

n o LUP ¼ max LDM ða Þ : a 2 A all ðaÞ ;

Proposition 4.4. The   no jump property (13) holds, and therefore C P ðaÞ ¼ LUP ðaÞ; RUP ðaÞ , if and only if the weak continuity (4.4) holds.

and

n o RUP ðaÞ ¼ min RDM ða Þ : a 2 Aþ all ðaÞ ;

Proof. Let us start by proving that weak continuity (4.4) is necessary for the no jump property (13). For contradiction, let us suppose that weak continuity does not hold and thus:

such that (41) gives (a) there exist a*, b* 2 A* such that LDM(a*) > RDM(b*) + 1, and U 0 ; U 00 2 U A for which U0 (a) P U0 (a*) and U00 (b*) > U00 (a), but (b) there does not exist U 000 2 U A for which U000 2 (a*) P U000 (a) P U000 (b*).

LUP ðaÞ 6 RUP ðaÞ: Observe that for any a 2 A and for any U 2 U A ðaÞ; LU ðaÞ P LUP ðaÞ and RU ðaÞ 6 RUP ðaÞ, i.e.,

h

i   LU ðaÞ; RU ðaÞ # LUP ðaÞ; RUP ðaÞ :

ð42Þ

For (12), (42) gives

  C P ðaÞ # LUP ðaÞ; RUP ðaÞ : Observe that for any a 2 A,

n o LUN ðaÞ ¼ max LU ðaÞ : U 2 U A ; and

n o RUN ðaÞ ¼ min RU ðaÞ : U 2 U A : For (12), if LUN ðaÞ 6 RUN ðaÞ, then C N ðaÞ ¼ ½LUN ðaÞ; RUN ðaÞ. Instead, again for (12), if LUN ðaÞ > RUN ðaÞ, then CN(a) = ;. h

Consider



U þA ða; a Þ ¼ fU 2 U A : UðaÞ P Uða Þg; 

U A ða; a Þ ¼ fU 2 U A : Uðb Þ P UðaÞg: Since the set of assignment examples is consistent with all value functions U 2 U A , according to (4), LDM(a*) > RDM(b*) implies that for all U 2 U A we have U(a*) > U(b*). Therefore, (b) implies that  for all U 2 U A ; UðaÞ P Uða Þ > Uðb Þ or U(a*) > U(b*) P U(a). This   þ     means that U A ða; a Þ [ U A ða;b Þ ¼ U A and U þ A ða;a Þ \ U A ða; b Þ ¼ ;. 0 þ  0 Since from (a) U 0 ðaÞ P U 0 ða Þ; U þ  ða; a Þ – ; because U 2 U  ða; a Þ. A A    ða; b Þ – ; because Analogously, since from (a) U 00 ðb Þ P U 00 ðaÞ; U  A   þ  þ   U 00 2 U  A ; ða; b Þ. For all U 2 U A ða; a Þ and U 2 U A ða; b Þ we have

n o þ LU ðaÞ ¼ Max LDM ðc Þ : U þ ðc Þ 6 U þ ðaÞ; c 2 A P LDM ða Þ; and

B.2. Proof of Proposition 4.3

n o   RDM ðb Þ P Min RDM ðc Þ : U  ðc Þ P U  ðaÞ; c 2 A ¼ RU ðaÞ;

Proposition 4.3. CN(a) – ; if and only if the following strict continuity condition is satisfied: for all a*, b* 2 A* such that LDM(a*) > RDM(b*) there are no two compatible value functions U; U 0 2 U A for which U(a) P U(a*) and U0 (b*) P U0 (a).

such that, remembering that LDM(a*) > RDM(b*) + 1, there is no U 2 U A for which [LU(a), RU(a)] \ [RDM(b*) + 1,LDM(a*)  1] – ;, i.e., for all U 2 U A and h 2 [RDM(b*) + 1,LDM(a*)  1], h R [LU(a), RU(a)].

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This means that no jump property (13) does not hold. Thus we proved that (4.4) is necessary for the no jump property (13). Now we prove that (4.4) is sufficient for the no jump property (13) to hold. Two cases are possible: (1) for all a*, b* 2 A* such that LDM(a*) > RDM(b*) + 1, there do not exist U 0 ; U 00 2 U A for which U0 (a) P U0 (a*) and U00 (b0 ) P U00 (a), (2) there exist a*, b* 2 A* and U 0 ; U 00 2 U A such that LDM(a*) > RDM(b*) and U0 (a) P U0 (a*) and U00 (b*) P U00 (a).

such that

h

 in case (2), we would have

n o 000 RU ðaÞ ¼ min RDM ðc Þ : U 000 ðc Þ P U 000 ðaÞ; c 2 A 

6 RDM ðd Þ < LDM ða Þ;

Case (1). A necessary condition for the presence of a class jump in the possible assignment of a 2 A is that there must exist 0 00 U 0 ; U 00 2 U A such that LUP ðaÞ > RUP ðaÞ þ 1. This is possible only if condition (1) is not verified, i.e., if there exists a*, b* 2 A*, such that LDM(a*) > RDM(b*) + 1, and U 0 ; U 00 2 U A for which U0 (a) P U0 (a*) and U00 (b*) P U00 (a). In fact, in this case,

n o 0 LU ðaÞ ¼ Max LDM ðc Þ : U 0 ðc Þ 6 U 0 ðaÞ; c 2 A P LDM ða Þ; and

n o 00  RDM ðb Þ P Min RDM ðc Þ : U 00 ðc Þ P U 00 ðaÞ; c 2 A ¼ RU ðaÞ; such that, remembering that LDM(a*) > RDM(b*) + 1, 0



0

00

00

½RDM ðb Þ þ 1; LDM ða Þ  1 \ ð½LU ðaÞ; RU ðaÞ [ ½LU ðaÞ; RU ðaÞÞ ¼ ;: In conclusion, if condition (1) is verified, there is no class jump in the possible assignment. Case (2). For what is said in the previous point, in this case, there could be a jump over some classes Ch belonging to [LDM(b*) + 1,RDM(a*)  1]. However, for weak continuity (4.4), we have that there exists U 000 2 U A for which U000 (a*) P U000 (a) P U000 (b*), and thus

n o 000  LU ðaÞ ¼ max LDM ðc Þ : U 000 ðc Þ 6 U 000 ðaÞ; c 2 A P LDM ðb Þ; and

n o 000  RU ðaÞ ¼ min RDM ðc Þ : U 000 ðc Þ P U 000 ðaÞ; c 2 A 6 RDM ðb Þ;

DM

L

U 000



U 000

ðb Þ 6 L ðaÞ 6 R ðaÞ 6 R

DM

ða Þ:

If there does not exist any other d* 2 A* different from a* and b* 000 such that U000 (a*) P U000 (d*) P U000 (b*), we have LU ðaÞ ¼  DM U 000 DM  L ðb Þ and R ðaÞ ¼ R ða Þ, so that 000

000



½LU ðaÞ; RU ðaÞ ¼ ½LDM ðb Þ; RDM ða Þ; and, therefore, there is no jump over classes Ch belonging to [LDM(b*),RDM(a*)] and, a fortiori, over classes Ch belonging to [LDM(b*) + 1,RDM(a*)  1]. If instead there exists some other d* 2 A* different from a* and b*, such that U000 (a*) P U000 (d*) P U000 (b*), there could continue to be some jump over classes Ch belonging to [LDM(b*) + 1, RDM(a*)  1], if (1) U000 (a) P U000 (d*) and LDM(d*) > RDM(b*) + 1, or (2) U000 (a) 6 U000 (d*) and RDM(d*) < LDM(a*)  1. In fact:

n o 000 LU ðaÞ ¼ max LDM ðc Þ : U 000 ðc Þ 6 U 000 ðaÞ; c 2 A PL



DM

ðd Þ > R

h

i  RDM ðd Þ þ 1; LDM ða Þ  1 h 0 i h 000 i 0 000 \ LU ðaÞ; RU ðaÞ [ LU ðaÞ; RU ðaÞ ¼ ;:

However, observe that for weak continuity (4.4), we have that there exist U iv ; U v 2 U A for which Uiv(a*) P Uiv(a) P Uiv(d*) and Uiv(d*) P Uiv(a) P Uiv(b*), such that we can repeat the same reasoning until there is no more space for some jump. h B.4. Proof of Corollary 4.1 Corollary 4.1. If set U A of compatible value functions satisfies the weak continuity (4.4), the example-based sorting procedure assigns each reference action a* 2 A* to an interval of classes ½LUP ða Þ; RUP ða Þ # ½LDM ða Þ; RDM ða Þ. Proof. Since we are supposing that weak continuity (4.4) holds, for Proposition 4.4, C p ðaÞ ¼ ½LUP ða Þ; RUP ða Þ. Moreover, due to Proposition 3.2, for each compatible value function U 2 U A , the corresponding example-based sorting procedure assigns each reference action a* 2 A* to an interval of classes such that

LU ða Þ P LDM ða Þ; U



R ða Þ 6 R

DM

ð44Þ



ða Þ:

ð45Þ

Therefore, also

LUP ða Þ P LDM ða Þ; 6R

DM

which gives

ð46Þ



ða Þ;

½LUP ða Þ; RUP ða Þ # ½LDM ða Þ; RDM ða Þ.

ð47Þ h

B.5. Proof of Proposition 4.5 Proposition 4.5. If the set U A  of compatible value functions is convex, the C p ðaÞ ¼ LUP ðaÞ; RUP ðaÞ , i.e., the no jump property (13) holds. Proof. We prove that if the set U A of compatible value functions is convex, then weak continuity (4.4) is satisfied, because, on the basis of Proposition 4.4, this is sufficient for no jump property (13). Let us suppose that there exist a*, b* 2 A* such that LDM(a*) > RDM(b*) + 1 and U; U 0 2 U A for which U(a) P U(a*) and U0 (b*) P U0 (a). Since U A is convex, then for all a 2 [0, 1], also aU þ ð1  aÞU 0 2 U A . Thus we look for U00 = aU + (1  a)U0 such that U00 (a*) > U00 (a) > U00 (b*), that is

aUða Þ þ ð1  aÞU 0 ða Þ > aUðaÞ þ ð1  aÞU 0 ðaÞ   > aUðb Þ þ ð1  aÞU 0 ðb Þ ðiÞ:

 in case (1), we would have

DM

such that

RUP ða Þ

such that, for Proposition 3.1,

i   RDM ðb Þ þ 1; LDM ðd Þ  1 h 00 i h 000 i 00 000 \ LU ðaÞ; RU ðaÞ [ LU ðaÞ; RU ðaÞ ¼ ;;



ðb Þ þ 1;

Let us remember that since the set of assignment examples is consistent, from LDM(a*) > RDM(b*) + 1 we get U(a*) > U(b*) and U0 (a*) > U0 (b*). Thus, from (i) we get

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S. Greco et al. / European Journal of Operational Research 207 (2010) 1455–1470

aUða Þ þ ð1  aÞU 0 ða Þ > aUðaÞ þ ð1  aÞU 0 ðaÞ

Proof. Let us prove that for all t = 2, . . ., s, and for all a 2 A,

()

t1U Lt;U ðaÞ: P ðaÞ P LP

a½Uða Þ  U 0 ða Þ  UðaÞ þ U 0 ðaÞ > U 0 ðaÞ  U 0 ða Þ () 0

0

0

Remembering that U tA # U t1 A , we have

0

a½U ða Þ  U ðaÞ þ UðaÞ  Uða Þ < U ða Þ  U aÞ 





   a 2 A : 8U 2 U tA ; Uða Þ 6 UðaÞ    a 2 A : 8U 2 U t1 A ; Uða Þ 6 UðaÞ ;

() 0



0

U ða ÞU aÞ a < ½U0 ða ÞU ; 0 ðaÞþUðaÞUða Þ

DM   and, consequently, taking into account that LDM t ða Þ P Lt1 ða Þ for all a* 2 A*,  n o  DM t    Lt;U P ðaÞ ¼ Max f1g [ Lt ða Þ : 8U 2 U A ; Uða Þ 6 UðaÞ; a 2 A

and

aUðaÞ þ ð1  aÞU 0 ðaÞ > aUðb Þ þ ð1  aÞU 0 ðb Þ ()

P

a½UðaÞ  U 0 ðaÞ  Uðb Þ þ U 0 ðb Þ > U 0 ðb Þ  U 0 ðaÞ

 n o  t1    Max f1g [ LDM ¼ LPt1;U ðaÞ: t1 ða Þ : 8U 2 U A ; Uða Þ 6 UðaÞ; a 2 A

() 0



0

U ðb ÞU ðaÞ a > ½UðaÞUðb :  ÞþU 0 ðb ÞU 0 ðaÞ

Analogous proof holds for

00

0

This means that if U = aU + (1  a)U and we fix 0



0

0



0

t1;U Rt;U ðaÞ: P ðaÞ 6 RP

U ða Þ  U ðaÞ U ðb Þ  U ðaÞ >a> ;   ½U 0 ða Þ  U 0 ðaÞ þ UðaÞ  Uða Þ ½UðaÞ  Uðb Þ þ U 0 ðb Þ  U 0 ðaÞ

For Proposition 4.6, we have that

we get U00 (a*) > U00 (a) > U00 (b*), which concludes the proof. h

h i   t1 t;U t1;U C tP ðaÞ ¼ Lt;U ðaÞ; Rt1;U ðaÞ ; P ðaÞ; RP ðaÞ and C P ðaÞ ¼ LP P such that, for

B.6. Proof of Proposition 4.6

t1U t1;U Lt;U ðaÞ and Rt;U ðaÞ; P ðaÞ P LP P ðaÞ P RP

Proposition 4.6. If U A is the set of all additive compatible value   functions, then C P ðaÞ ¼ LUP ðaÞ; RUP ðaÞ , i.e., the no jump property (13) holds. Proof. On the basis of Proposition 4.5, we have simply to prove that if U A is the set of all additive compatible value functions, then it is convex. Observe that, for any U; U 0 2 U A and all a, b 2 A, if

UðaÞ ¼

n X

uj ðg j ðaÞÞ P

j¼1

n X

uj ðg j ðbÞÞ ¼ UðbÞ;

j¼1

and

U 0 ðaÞ ¼

n X

u0j ðg j ðaÞÞ P

j¼1

X

u0j ðg j ðbÞÞ ¼ U 0 ðbÞ;

j¼1

then also n n X X ðauj ðg j ðaÞÞ þ ð1  aÞu0j ðg j ðaÞÞ P auj ðg j ðbÞÞ þ ð1  aÞu0j ðg j ðbÞÞ: j¼1

j¼1

Therefore, if U; U 0 2 U A are additive compatible value functions, P also U 00 ðxÞ ¼ nj¼1 u00j ðg j ðaÞÞ such that for all j 2 G u00j ðxÞ ¼ ðauj ðg j ðxÞÞ þ ð1  aÞu0j ðg j ðxÞÞ is an additive compatible value function. Since U00 = aU + (1  a)U0 , then if U A is the set of all additive compatible value functions, it is convex. h

Appendix C. Proofs of propositions from Section 7 C.1. Proof of Proposition 7.1 Proposition 7.1. For all t = 2, . . ., s, and for all a 2 A, t1;U t1;U Lt;U ðaÞ and Rt;U ðaÞ; P ðaÞ P LP P ðaÞ 6 RP

and thus

C tP ðaÞ # C Pt1 ðaÞ:

we conclude that

C tP ðaÞ # C t1 P ðaÞ:



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