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A multicriteria assignment procedure for a nominal sorting problematic
European Journal of Operational Research 138 (2002) 349–364 www.elsevier.com/locate/dsw
A multicriteria assignment procedure for a nominal sorting problematic Julien Leger *, Jean-Marc Martel Facult e des Sciences de l’Administration, Universit e Laval, Sainte-Foy, Que., Canada G1K 7P4
Abstract This paper addresses the problem of the assignment of an action in the particular case where the categories defined a priori are not ordered. In multicriteria literature, it is the nominal sorting problematic. We propose a simple and easy method to understand the procedure of assignment to tackle this problematic. Afterwards, we shall illustrate the proposed method by using an example that is inspired by a research carried out the construction sector of the work accident prevention field. 2002 Elsevier Science B.V. All rights reserved. Keywords: Multicriteria method; Nominal sorting problematic; Classification; Assignment; Similarity
1. Introduction The human brain constantly tries to classify individuals and objects. In order to be able to treat all the available information, the human being must set categories that allow putting together individuals or objects having common characteristics. In general, a name is given to the different categories. In zoology, for example, we can find the mammal category and the vertebrate category. Thus, an individual or object is identified according to its belonging to such or such a category (Chandon and Pinson, 1981). The classification methods are divided into two groups: the automatic classification method and the assignment method. The automatic classification methods (‘‘clustering’’) consist in regrouping individuals (or objects) into a restricted number of categories. These categories must be, on the one hand, as few as possible and on the other hand, as homogeneous as possible. These methods are based on the notion of non-supervised learning. In non-supervised learning, we have individuals but we have no information on their belonging to categories. We regroup the individuals into categories in order to make the distances between the individuals within a same category the shortest and the distances between the centres of the different categories the longest.
*
Corresponding author. E-mail address: [email protected] (J.-M. Martel).
0377-2217/02/$ - see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 0 1 ) 0 0 2 5 1 - X
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The assignment method allows affecting an individual to a defined category a priori. For example, the categories may have been defined by using an automatic classification method based on historical data. Thus, we know the individuals whose assignment categories are known. Based on these individuals, we can define assignment standards. These assignment methods are based on the notion of supervised learning. Under supervised learning, we dispose a set of individuals whose belonging is known and serve to calibrate the parameters of the method. Once this adjustment is done, the method is able to assign an unknown individual to the most plausible category. In the literature on multicriteria decision aids, we sometimes find it handy to refer to the retained decisional problematic since it determines the nature of the recommendation coming from a multicriteria analysis. The assignment methods or procedures are associated to the sorting problematic (or of the segmentation). This type of problematic is found quite frequently in practice as shown in some of the following examples (Yu, 1992). ‘‘Trichotomic admission on file’’ (Moscarola, 1977): The actions are the files of the candidates that apply for a job. The problem is to develop an examination procedure to facilitate the work of the decision maker. The procedure should allow to sort the files into the three following ordered categories: admission, research for additional information and refusal. ‘‘Granting of credits’’ (Fabre, 1985; Bergeron et al., 1996; Zopounidis and Doumpos, 1998): A financial organisation is constantly faced with requests for credit. Each request is analysed taking into account data such as: the financial report, the qualitative aspects related to the company’s policies and culture. The problem consists in creating a system capable of treating the information in real-time and of giving an opinion for each of the requests. The system will affect the actions in one of the following ordinate categories: a credit approved immediately, a rather favorable credit and a rejected credit. ‘‘Student evaluation’’ (Massaglia and Ostanello, 1991): In this case, the actions are the students who must take a series of tests (oral and written). Each student is evaluated taking into account the results obtained in each test. The problem consists in separating the students into partially ordered categories: very good, good, quite good, rather good, rather bad, bad, very bad and uncertain. ‘‘Allocation of resources’’ (Dekempener and Gervasi, 1985; Vansnick, 1989): Each year, a public organism must treat several requests concerning the maintenance and/or the repairing of the apartments that it owns. The total sum of the requests is often much higher that the allocated budget for the year. The problem consists in creating an automatic tool allowing to allocate the funds to the various requests. The categories are the different priority levels for the maintenance or the repairing of the apartments that must be defined beforehand. All these examples concern the ordinal sorting problematic. However, it is also frequent to face decisionmaking situations found in the nominal sorting problematic. Here are two examples of problems that belong to this type of problematic. ‘‘Help in the orientation of student’’ (Bloch, 1984): The problem consists in helping in the orientation of the students. We know that these are different types of training in various sectors that it is very difficult for a student to choose his future orientation. The objective is to conceive a system that would propose the training best required for each student. ‘‘Help in medical diagnosis’’ (Belacel, 1998): The problem consists in identifying the disease(s) of the patients using their symptoms. One must build a system that will allow the identification of the possible disease(s) by considering the symptoms. In this case, the categories are the various possible diseases (example: acute leukaemia). These categories are not ordered. In this paper, we are interested in the assignment methods of each action (individual, object, . . .) in the case where the categories defined a priori are not ordered, i.e., the nominal sorting problematic. This document is divided into four sections. In Section 2, we present the sorting problematic by putting the accent on the one concerned by the nominal sorting. Section 3 is essentially devoted to two existing methods capable of treating problems related to the nominal sorting problematic. In Section 4, we propose a method
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capable of treating this type of problem and a numerical example illustrating the proposed method is presented in Section 5.
2. The sorting problematic We can conceive the problematic as the way the decision aid is contemplated, i.e., the way of formulating a problem in order to reach results allowing to shed light on a decision. According to Roy (1985), it is possible to distinguish four reference problematics: description, choice, ranking and sorting problematics. This paper deals with the sorting problematic and, more specifically, the nominal sorting. The sorting problematic consists in assigning each one of the potential actions to one or more categories while examining the intrinsic value of the action. The categories are defined according to pre-established norms of assignment independently from all the actions to be taken. A specific norm of assignment must be associated to each category. Therefore, the assignment of any action to one of the categories is solely founded on the analysis of the action with regards to the assignment norms and not on its relative value with regards to other actions. According to Bana e Costa (1992), one must distinguish between two significantly different situations in the sorting problematic. Effectively, there are cases where all the categories have a purely nominal structure, i.e., there is no order between the categories, and the cases where the decisional context imposes an ordered structure on the categories. In the ordinal sorting problematic, the categories are completely ordered. They are characterized by a set of reference objects or prototypes (typical actions). Often for this type of sorting, the procedure of assignment is formulated in the following manner: any action judged to be between the two limits (higher and lower) of a category, must be assigned to the category in question. Fig. 1 represents schematically the ordinal sorting problematic. In the nominal sorting problematic, the categories are not ordered. The assignment procedure associated to a category must be formulated in the following manner: any action judged to be sufficiently similar to at least one typical action must be assigned to the category to which this typical action belongs. Fig. 2 represents schematically the nominal sorting problematic. When either only one superior and one inferior reference action or only one typical action defines a category, it is then a question of monoprofile categories. However, a category may also be defined by a set of reference objects or prototypes. In this case, we refer to multiprofile categories. This second case is harder to treat, but it offers more richness from the modelisation of the categories point of view. This implies that it is possible for an action to be assigned in different manners to a category. In practice, it may be useful to use thresholds (of indifference, of similarity, of resemblance, of dissimilarity). In multicriteria decision aid, it may seem realistic to wish to set a value under which there is indifference between two actions and over which there is strict preference. Often, there is an intermediate zone in which there is hesitation between preference and indifference. It is the pseudo-criterion concept introduced by Roy and Vincke (1984). When considering the nominal sorting problematic, this definition of thresholds (of indifference and of preference) must be slightly modified (Henriet, 1995). For example, in the
Fig. 1. The ordinal sorting problematic.
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Fig. 2. The nominal sorting problematic.
nominal sorting problematic framework, the notion of preference does not really appropriate and it should be replaced by one of similarity. A threshold may be perceived as a tool allowing to consider the imprecision and uncertainty related to the evaluations. In the TRINOMFC method (which will be presented later in this paper), we are in the presence of a similarity threshold (ST) and of a dissimilarity threshold (DT). The ST represents the maximum value between two actions so that the two actions may be judged to be sufficiently similar. On the other hand, the DT is the minimal value between two actions from which we may assert that an action is significantly dissimilar from another. The sorting problematic is used very little as compared with the choice and ranking problematics (Yu, 1992). However, a certain number of research and application papers have treated this problematic. In multicriteria analysis, certain methods were conceived to treat the sorting problematic. Among these methods, we find TRICHOM (Moscarola, 1977) which is a trichotomic method that is limited to problems with three ordered categories. The generalisation with more than three categories may be treated by the NTOMIC methods (Massaglia and Ostanello, 1991) or ELECTRE TRI (Yu, 1992). As compared to the ordinal sorting, the nominal sorting is a little harder to deal with and it has been so till quite recently. We propose the following formulation of the nominal sorting problematic: Let A a finite and not empty set of objects to assign to the different categories, F ¼ fg1 ; . . . ; gn g a set of n criteria, n P 1, C ¼ fC 1 ; . . . ; C K g the group of predefined categories h; K > 1, Bh ¼ fbhp j p ¼ 1; . . . ; Lh and h ¼ 1; . . . ; Kg the set of typical objects of the hth category where bhp repreSK h sents the pth typical object of the hth category and let B ¼ h¼1 B . Each object of A and B is defined by its performances evaluated on all the criteria. Hence, 8a 2 A; we have gðaÞ ¼ ðg1 ðaÞ; . . . ; gn ðaÞÞ, where g1 ðaÞ is the evaluation of an object a according to the g1 ðj ¼ 1Þ criterion and for 8bhp 2 Bh B, we have gðbhp Þ ¼ ðg1 ðbhp Þ; . . . ; gn ðbhp ÞÞ with p ¼ 1; . . . ; Lh and h ¼ 1; . . . ; K, where Lh is the number of typical object of the hth category. Also, a 2 C h means we assign action a to category h. The question raised is the following: since we have object a 2 A, to which category should we assign it where comparisons are made between its evaluation profile gðaÞ and those of the typical objects gðbhp Þ. This assignment must be made on the basis of a similarity index between the profiles.
3. Two methods for nominal sorting problematic We shall now briefly present two methods of assignment in the nominal sorting problematic. These methods are very recent since they only go back to 1998. To our knowledge, no other methods based on the concepts of outranking relations have been developed to deal with the nominal sorting problematic. There
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are some methods based on rough data, but we do not present them in this paper. The first one by Perny (1998), is a multicriteria filtering method based on the concordance and non-discordance principles. The second one by Belacel (1998) called PROcedure d’Affectation Floue dans le cadre de la problematique du Tri Nominal problem (PROAFTN), is a multicriteria fuzzy classification procedure for nominal sorting problematic. For the two methods, the authors assume that the categories are definite a priori and that each category is characterized by one or several typical actions (reference objects or prototypes). Also, they assume that the different criteria are determined and constitute a coherent criterion family.
3.1. Perny’s method (1998) With this method, a comparison by pairs is made in order to verify if an action a is close to or similar to a typical action representing a typical element of a category. In fact, the concordance and discordance are computed for each pair of actions. First of all, we must build an indifference relation or else an outranking relation for which the symmetric part plays the role of the indifference relation. In order to be able to present different judgements of preference, the preference relations S and I are used. Let a and bhp be two actions, the information on the preferences is expressed by H ða; bhp Þ 2 ½0; 1 representing the confidence in proposition aHbhp , where H 2 fS; Ig. Let us consider that the thresholds of preference pj and of indifference qj , where qj 6 pj . The function Sj (that is the outranking relation for criterion j) is defined by n o pj min gj ðbhp Þ gj ðaÞ; pj n o: Sj ða; bhp Þ ¼ pj min gj ðbhp Þ gj ðaÞ; qj The outranking relation Sj ða; bhp Þ is very useful since it represents the indifference at the same time. Consequently, n o Ij ða; bhp Þ ¼ min Sj ða; bhp Þ; Sj ðbhp ; aÞ : The principle of similarity simply uses the rule of the majority to decide between each pair of actions a and bhp . For a pair of actions ða; bhp Þ, we obtain Ij ða; bhp Þ 2 ½0; 1 ; j ¼ 1; . . . ; n. These numbers n are aggregated in the following manner: CI ða; bhp Þ ¼
n X
wj Ij ða; bhp Þ;
j¼1
to obtain aPglobal index CI ða; bhp Þ measuring the global agreement against the proposition aIbhp , where wj ( P 0 and nj¼1 wj ¼ 1) is the relative importance of the criterion j. The principle of discordance introduces a complementary notion in the aggregation process. At this level, we introduce a veto threshold vj which is the maximum value of a difference gj ðbhp Þ gj ðaÞ which is compatible with the proposition aHbhp , where H 2 fS; Ig. The veto is characterised by the following condition: 8ða; bhp Þ 2 A B; ð9j 2 f1; . . . ; ng; gj ðbhp Þ gj ðaÞ > vj Þ ) :ðaHbhp Þ; where : represents the logic operator of the negation.
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The introduction of this notion of discordance allows measuring the degree to which criterion j is strongly opposed to aIbhp . The discordance relation associated to the outranking relation for criterion j is defined as follows: !! gj ðbhp Þ gj ðaÞ pj S h Dj ða; bp Þ ¼ min 1; max 0; : vj p j The discordance relation corresponding to the indifference is defined by DIj ða; bhp Þ ¼ DIj ðbhp ; aÞ ¼ maxðDSj ða; bhp Þ; DSj ðbhp ; aÞÞ: Also, DI ða; bhp Þ ¼ 1
n Y
1 DIj ða; bhp Þ
a=n
;
j¼1
where a 2 ½1; n is a technical parameter introduced to modify the degree of synergy between the criteria. When using the principle of concordance and of non-discordance, a proposition of the type aIbhp is true if and only if the coalition of the criteria in agreement with this proposition is strong enough and that there be no discordant coalition with this proposition. This principle may be expressed as follows: n o Iða; bhp Þ ¼ min CI ða; bhp Þ; 1 DI ða; bhp Þ : Then, we calculate the membership degree of the action to each category, dða; C h Þ, as follows: n o dða; C h Þ ¼ max Iða; bh1 Þ; Iða; bh2 Þ; . . . ; Iða; bhLh Þ ; h ¼ 1; . . . ; K: Action a is assigned to the category according to the following rule: a 2 C t () dða; C t Þ ¼ max dða; C 1 Þ; dða; C 2 Þ; . . . ; dða; C K Þ :
3.2. The Belacel (1998) method The method of Belacel (1998), named PROAFTN, is a multicriteria decision aid method to deal with the problems of fuzzy multicriteria assignment. This method allows building a fuzzy indifferent relation by generalising the concordance and discordance ratings used in the ELECTRE III method. Following this, we determine the category of assignment of a given object. Belacel et al. (1999) and Belacel (2000) are another good references for this method. The indifference relation of synthesis is based on the concordance/discordance rule as it is done in (Perny and Roy, 1992). In adopting this rule in the nominal sorting context, the result is: ‘‘when action a is judged to be indifferent with a typical action bhp based on sufficiently important majority of criteria majority principle and there is no criterion that puts its veto against the affirmation ‘‘a is indifferent to bhp ’’ principle of respect of the minorities, then the action a is indifferent to bhp ’’. In practice, the performances of the typical actions are often more easily given under the form of intervals. For each criterion gj , the interval ½gj1 ðbhp Þ; gj2 ðbhp Þ is associated to each reference action bhp with gj1 ðbhp Þ 6 gj2 ðbhp Þ.
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Let Iða; bhp Þ be the indifference rating between the action a and the typical action bhp ; h ¼ 1; . . . ; K and p ¼ 1; . . . ; Lh . Yet, Iða; bhp Þ is based on concordance and non-discordance principles. More precisely, ! n n
Y
whj X h h h h Iða; bp Þ ¼ wj Cj ða; bp Þ 1 Dj ða; bp Þ ; j¼1
j¼1
where • Cj ða; bhp Þ; j ¼ 1; . . . ; n, is the degree to which the criterion gj is in favour of the indifference relation between a and bhp . Cj ða; bhp Þ is calculated using two discrimination thresholds djþ ðbhp Þ and dj ðbhp Þ used to take into account of fuzzy data. • Dj ða; bhp Þ; j ¼ 1; . . . ; n, is the degree to which the criterion gj is against the indifference relation between a h h and bhp . Dj ða; bhp Þ is calculated using two veto thresholds vþ j ðbp Þ and vj ðbp Þ used to consider the values for h which a is very different from bp . Pn • whj is the relative importance of the criterion j of the category C h , whj P 0 and j¼1 whj ¼ 1. Following this, we calculate the membership degrees of the action a to hth category, dða; C h Þ, as follows: n o dða; C h Þ ¼ max Iða; bh1 Þ; Iða; bh2 Þ; . . . ; Iða; bhLh Þ ; h ¼ 1; . . . ; K: The action a is assigned to the category according to the following rules: a 2 C t () dða; C t Þ ¼ max dða; C 1 Þ; dða; C 2 Þ; . . . ; dða; C K Þ :
3.3. Remarks concerning the two methods These two methods have certain similarities. In Belacel’s method, the typical actions are given using the interval of the form ½gj1 ðbhp Þ; gj2 ðbhp Þ with gj1 ðbhp Þ 6 gj2 ðbhp Þ. This last interval has a link with Perny’s method if we suppose that gj ðbhp Þ ¼ ðgj1 ðbhp Þ þ gj2 ðbhp ÞÞ=2 with qj ðbhp Þ ¼ gj2 ðbhp Þ gj ðbhp Þ ¼ gj ðbhp Þ gj1 ðbhp Þ. The preference thresholds and the discrimination thresholds are very similar since they represent the marker between the weak indifference and non-indifference. The Perny method uses an indifference threshold while that of Belacel does not use any. Belacel’s idea of using an interval to determine the typical actions seems very interesting. In practice, it may prove to be difficult to give a precise value for the evaluation of a typical action according to a criterion. However, the indifference threshold in Perny’s method plays the same role as the interval used by Belacel. The Belacel method seems to have had some success in the field of medical diagnosis aid. However, these methods are quite complex to use in the sense that they necessitate several thresholds (preference, indifference and veto thresholds). These thresholds, in particular the veto thresholds, may be quite difficult to obtain. Also, are these thresholds introduced to model the preferences also appropriate to compute similarity index in the nominal sorting problematic? In the nominal context there is no preference notion and the criteria are not being maximized or minimized.
4. The proposed method Our objective is to develop a method of assignment within the framework of nominal sorting that may be simple and easy to understand for a potential user; in particular, a method that will not necessitate the
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determination of veto thresholds. It implies that less information is needed than the two other methods presented above. We propose a procedure for which the assignment is based on the determination of a similarity rating between an object and each one of the categories as characterised by its typical objects. For each criterion, a function representing the similarity index will be introduced in order to model the concept of similarity between two objects. Such a function is defined for each criterion and its value varies between 0 and 1; this value will be so much greater than that the similarity between the two actions of this criterion is greater. Different types of criteria functions (see Appendix A) are available to model with more flexibility the notion of similarity between two objects. However, we are free to choose another function. We have named this method TRI NOMinal base sur des Fonctions Criteres (TRINOMFC). The proposed method necessitates the use of certain thresholds. Consequently, STjh ðbhp Þ and DTjh ðbhp Þ represent, respectively, the similarity threshold and the non-similarity threshold associated to the typical object bhp according to criterion j. Let SIjh ða; bhp Þ be the similarity index between the object a and the typical object bhp of the hth category according to criterion j. Let SI h ða; bhp Þ be the similarity index between the object a and the typical object bhp of the hth category, all the criteria taken simultaneously. Finally, let MI h ða; C h Þ be the membership index between the object a and the hth category. To measure the similarity index between the two objects a and b leads us to answer the following question: Is object a sufficiently similar to object b? In the affirmative, the similarity index is a value close to 1. If not, the similarity index is closer to 0. Since the objects are evaluated according to n criteria, we must start by calculating a similarity index SIjh ða; bhp Þ according to each criterion j. Starting from these n similarity indices (between an object and a typical object according to the n criteria), we shall be able to calculate a membership index aggregating these n similarity indices. We must make the same calculations for each one of the typical objects and with this information we shall be able to calculate a membership index between an object and each one of the categories. It is natural to assign an object to a category for which the similarity is greater, but this index must reach a minimal level of similarity. The similarity indices must possess certain characteristics (Smadja, 1998). In fact, they must respect three axioms: (i) The similarity between two objects is a positive measure, at most equal to 1. (ii) Only two identical objects have a similarity equal to 1. (iii) The similarity between two objects does not depend on the order in which they are presented. It is important to mention that the axiom (iii) is debatable because some authors consider that the similarity relation as not necessarily symmetric. In this work, we assume the similarity relation to be symmetric. Let us consider the true-criterion (the criterion function no. 1 in Appendix A) that is expressed analytically as follows: ( 1 if gj ðaÞ gj ðbhp Þ ¼ 0; SIjh ða; bhp Þ ¼ 0 if gj ðaÞ gj ðbhp Þ 6¼ 0: For this type of criterion, it is evident that SIjh ða; bhp Þ is a similarity index since the three axioms are verified. In fact, axiom (i) is satisfied since SIjh ða; bhp Þ 2 f0; 1g. The axiom (ii) is also satisfied since SIjh ða; bhp Þ ¼ 1 if and only if gj ðaÞ ¼ gj ðbhp Þ. Since jgj ðaÞ gj ðbhp Þj ¼ jgj ðbhp Þ gj ðaÞj, it is clear that SIjh ða; bhp Þ ¼ SIjh ðbhp ; aÞ and the axiom (iii) is verified. Proceeding in a similar manner, it easy to verify that the criterion function no. 3 (in Appendix A) satisfies all three axioms.
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The other criteria functions (criteria functions no. 2, no. 4 and no. 5 in Appendix A) satisfy axioms (i) and (iii). However, due to the use of similarity and dissimilarity thresholds, axiom (ii) is not verified as expressed. This is easy to verify with a counter-example. In fact, let gj ðbhp Þ ¼ gj ðaÞ þ
STjh ðbhp Þ : 2
Thus, gj ðaÞ gj ðbhp Þ ¼ gj ðaÞ
STjh ðbhp Þ gj ðaÞ þ 2
! ST h ðbh Þ j p < STjh ðbhp Þ: ¼ 2
Then, SIjh ða; bhp Þ ¼ 1 because jgj ðaÞ gj ðbhp Þj < STjh ðbhp Þ. Nevertheless, objects a and bhp are not identical. The use of thresholds makes the axiom (ii) not to be strictly satisfied. When considering indifference thresholds, it is evident (and it is the sense of these thresholds) that two actions whose evaluations differ from less than these thresholds are considered as identical in the sense of preference. It is the same thing for the similarity thresholds. We shall therefore be able to extend the meaning of axiom (ii) to include the similarity thresholds. Then, axiom (ii) may be defined as follows: ðii0 ) Only two similar objects have a similarity equal to 1. Two objects a and bhp are considered similar on criterion j if jgj ðaÞ gj ðbhp Þj < STjh ðbhp Þ. This modification is justified by the fact that the evaluations of the objects are not of absolute precision. While considering this new definition of axiom (ii), we can conclude that the criterion function no. 2 satisfies axiom ðii0 ) and consequently verify three axioms (modified) characterising a similarity index. We may proceed in the same way for the criteria functions no. 4 and no. 5 in order to verify if they satisfy the axioms. Even if for certain criteria functions, we must have recourse to axiom ðii0 ), we shall nevertheless qualify the indices obtained on this basis of similarity indices. We have judged that the use of thresholds (similarity and/or dissimilarity thresholds) is appropriate in the calculation of the similarity index. An important property that the typical objects of the categories must have is that: the similarity index between two typical objects of a same category must be superior to the similarity index between two typical objects from different categories. Let a 2 A an object to be assigned. The first step of the TRINOMFC method consists in determining one of the criteria functions that is found in Appendix A (that are the adaptations of the criteria functions of THE E) for each criterion. In fact, the ‘‘user’’ is free to define himself a function if that seems to PROME him more appropriate to characterise the differences of evaluation. For each category C h and for each criterion j, we determine a criterion function SIjh : A B ! ½0; 1 ; j ¼ 1; . . . ; n and h ¼ 1; . . . ; K that represents this similarity index. This function must represent the manner for which the similarity index between a and bhp decreases with the difference jgj ðaÞ gj ðbhp Þj. Because of the chosen functions, we will have to eventually determine parameters such as the similarity threshold STjh ðbhp Þ or dissimilarity threshold DTjh ðbhp Þ. A similarity threshold STjh ðbhp Þ represents the maximal value of the difference between two objects so that the two objects may be judged to be similar. On the other hand, the dissimilarity threshold DTjh ðbhp Þ is the minimal value of the difference between two objects from which we may conclude that an object is totally dissimilar to an other one. The similarity as well as the dissimilarity is never absolute. These thresholds play an equivalent role to indifference and preference thresholds but does not have the same significance. It is reasonable to take into account the imperfections in the evaluations of the objects according to each criterion to have recourse to similarity thresholds (STjh ðbhp Þ as if jgj ðaÞ gj ðbhp Þj 6 STjh ðbhp Þ then SIjh ¼ 1) and of dissimilarity ðDTjh ðbhp Þ as if jgj ðaÞ gj ðbhp Þj > DTjh ðbhp Þ then SIjh ¼ 0Þ. Fig. 3 illustrates a similarity index.
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Fig. 3. Similarity index.
After having determined the criteria functions and, if necessary, the thresholds, the second step consists in calculating the local similarity index, that is the calculation of the SIjh ða; bhp Þ; j ¼ 1; . . . ; n; p ¼ 1; . . . ; Lh and h ¼ 1; . . . ; K. These calculations allow us to obtain the similarity index between the object a and the typical objects according to each of the criteria taken individually. For a given category (let us suppose category h), we obtain a chart as shown in Table 1. With this information, we can go on to step three which consists in calculating, for each category, the similarity index between the object a and each one of the typical objects bhp of this category taken individually. This global similarity index is noted as SI h ða; bhp Þ; p ¼ 1; . . . ; Lh and h ¼ 1; . . . ; K. To calculate SI h ða; bhp Þ, we make the aggregation of n local similarity indices, SIjh ða; bhp Þ; j ¼ 1; . . . ; n, corresponding to the different similarity index between a and bhp according to each criterion. A weighted sum is used for this aggregation: SI h ða; bhp Þ ¼
n X
whj SIjh ða; bhp Þ;
ð1Þ
j¼1
where whj is the relative importance of criterion j of the category C h (like in Belacel, 1998). We obtain a chart like the one illustrated in Table 2 for each one of the categories C h ; h ¼ 1; . . . ; K. Table 1 Local similarity index between object a and the typical objects of category h SIjh ða; bhp Þ bh1 h b .2
ðp ¼ 1Þ ðp ¼ 2Þ
.. bhLh ðp ¼ Lh Þ
g1 ðj ¼ 1Þ
g2 ðj ¼ 2Þ
...
gn ðj ¼ nÞ
SI1h ða; bh1 Þ h h .SI1 ða; b2 Þ
SI2h ða; bh1 Þ h h .SI2 ða; b2 Þ
... ... . . . ...
SInh ða; bh1 Þ h h ..SIn ða; b2 Þ . SInh ða; bhLh Þ
.. SI1h ða; bhLh Þ
.. SI2h ða; bhLh Þ
Table 2 Global similarity index between the object a and the typical objects for category h SI h ða; bhp Þ bh1 ðp ¼ 1Þ h b .. 2 ðp ¼ 2Þ . bhLh ðp ¼ Lh Þ
SI h ða; bh1 Þ h h ..SI ða; b2 Þ . SI h ða; bhLh Þ
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Thus, we have the global similarity index between the object a and each one of the typical objects for each one of the categories. The fourth step consists in determining the membership index between the object a and each one of the categories, which means that we find values of MI h ða; C h Þ; h ¼ 1; . . . ; K. For this purpose, a min rule (pessimistic rule) is used since in this manner we can say that in the worst case the similarity rating between a and C h is therefore n o MI h ða; C h Þ ¼ min SI h ða; bhp Þ ; h ¼ 1; . . . ; K: ð2Þ p2f1;...;Lhg
In the fifth step, we must assign object a to the category for which the membership index between the object and the category is maximal. Then, we simply need to find the maximum of the values of MI h ða; C h Þ; h ¼ 1; . . . ; K. This idea is represented by the following expression: a 2 C t () MI t ða; C t Þ ¼ max MI h ða; C h Þ 8h ¼ 1; . . . ; K : ð3Þ We can suppose that the values of MI h ða; C h Þ; h ¼ 1; . . . ; K, are distinct, which implies that each object is affected to exactly one category. The fourth and the fifth steps may be represented by a global maximin rule: min fSI h ða; bhp Þg : ð4Þ a 2 C t () MI t ða; C t Þ ¼ max h2f1;...;Kg
p2f1;...;Lhg
In the TRINOMFC method, an object is assigned to exactly one category. We do not define any cutting threshold k. However, nothing prevents from defining such a threshold when an object cannot be assigned to one category. In general, such a threshold of cutting is superior to 0.5. While considering cutting threshold k, we obtain the following result: t t t h h a 2 C () MI ða; C Þ ¼ max min fSI ða; bp Þg and MI t ða; C t Þ P k: ð5Þ h2f1;...;Kg
p2f1;...;Lhg
We can be conduct to case of several categories with a rule of the type a 2 C t () MI t ða; C t Þ P k and we also have maxh2f1;...;Kg fminp2f1;...;Lhg gfSI h ða; bhp Þgg then we will have one, several or no category. The sixth step is very simple since it indicates that we must repeat the same process with all the objects of the A group so that each object may be assigned to a category.
4.1. Commentary on the method The TRINOMFC method offers great flexibility while allowing to have several profiles to represent different categories. We thus talk about the multiprofile categories. In practice, it is realistic to believe that an object may belong to a category in several ways. In the ordinal sorting problematic, it is very difficult to include the multiprofile notion (Yu, 1992). In ELECTRE TRI, for example, each category may be represented by only one superior and only one inferior action of reference. We use a maximin rule to identify the category for which the object should be assigned. When using this rule, we can say that we have an attitude that is conservative since the min forces us to take the minimal similarity index between the object and the typical objects for each one of the categories. However, other rules could also have been used like a maximax rule. This rule finds, for a given object, the maximal index with each one of the categories. The maximum of these indices is taken to know the category to which the object will be affected. The maximax rule would express itself as follows:
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a2C
t
t
t
() MI ða; C Þ ¼ max
h2f1;...;Kg
max fSI
p2f1;...;Lhg
h
ða; bhp Þg
:
ð6Þ
The Hurwicz rule could also have been used. It follows an attitude that is a little more nuanced as shown in the following expression: a 2 C t () MI t ða; C t Þ n o n o a max SI h ða; bhp Þ þ ð1 aÞ min SI h ða; bhp Þ ; ¼ max h2f1;...;K g
p2f1;...;Lhg
p2f1;...;Lhg
ð7Þ
where a 2 ½0; 1 is a coefficient of relative optimism. In this case, we give a certain weight a to the typical object having the maximal similarity index and the complementary weight 1 a to the typical object having the minimal similarity index. To sum up, the Hurwicz rule is a mixture of maximin and maximax rules. Also, we may use a weighted sum. More precisely, it suffices to give a degree of likelihood to each one of the typical objects Let wh ðbhp Þ be the degree of likelihood of typical object bhp . For each one of PLhof ahcategory. h the categories, p¼1 w ðbp Þ ¼ 1. We would then be led to the following rule: ( ) X t t t h h h h w ðbp Þ SI ða; bp Þ : ð8Þ a 2 C () MI ða; C Þ ¼ max h2f1;...;K g
p
In the particular case, where a category is represented by only one typical object ðbh Þ, we obtain SI ða; C h Þ ¼ SI h ða; bh1 Þ. For each one of the rules of assignment, we suppose that the values of MI h ða; C h Þ are unique. However, it is very possible that two values of MI h ða; C h Þ are identical. In such a situation, it is suggested to apply a second rule (Hurwicz, maximax, . . .) in order to overcome the impasse if we wish that an action be assigned to exactly one category. In the case where there are several typical objects in each category, it is very important to verify the property that the categories must have. The property is the following: the similarity index between the typical objects of a same category must be superior to the similarity index between the typical objects of different categories, that is h
SI h ðbhp ; bhp0 Þ > SI h ðbh0q ; bhp Þ 8p; p0 2 f1; . . . ; Lh g; 8h0 6¼ h 2 f1; . . . ; k g and 8q 2 f1; . . . ; Lh0 g: This one must be verified before using the TRINOMFC method if we wish to obtain an effective procedure of assignment.
5. Numerical illustration In order to illustrate the proposed procedure, we were inspired by the case of an application done in collaboration with the ‘‘Institut de recherche en sante et securite au travail‘‘ (IRSST). In the construction sector, different types of accidents may occur. For a given worker, we shall try to identify the type of accidents he is most likely to be a victim of. Let us consider three types of accidents: falls ðC 1 Þ, eye injuries ðC 2 Þ and deep cuts ðC 3 Þ. Each type of accident is characterised by several typical casualties. In fact, C 1 is characterised by two typical casualties and C 2 is characterised by a typical casualty while C 3 is characterised by three typical casualties. Also, three criteria are retained: the task done by the worker ðg1 Þ, the worker’s trade ðg2 Þ and the type of building site ðg3 Þ. Each criterion has several modalities.
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The typical casualties are determined as follows: gðb11 Þ ¼ (Surface preparation; Electrician, Electricity renovation); gðb12 Þ ¼ (Surface preparation; Roofer; Roof renovation); gðb21 Þ ¼ (Product application; Painter; Interior renovation); gðb31 Þ ¼ (Material installation; Day labourer; Exterior house finishing); gðb32 Þ ¼ (Material installation; Plumber; Plumbing service); gðb33 Þ ¼ (Setting up a temporary installation; Heavy equipment operator; Civil engineering excavation). In order to be able to go from the qualitative to the quantitative data, one has to use crossed tables. The interpretation of these tables is very simple. For each column, the relative frequencies represent the distribution of the casualties for the modality of a criterion according to the type of accidents. This distribution is called the conditional distribution according to the type of accidents taking into account the modality of a given criterion. The frequencies relative to a column are conditional frequencies. When using crossed tables determined by a study on the health and security at work, we obtain the following typical casualties: • gðb11 Þ ¼ ð11; 17; 15Þ, • gðb12 Þ ¼ ð11; 18; 11Þ, • gðb21 Þ ¼ ð10; 13; 18Þ, • gðb31 Þ ¼ ð15; 14; 12Þ, • gðb32 Þ ¼ ð15; 16; 11Þ, • gðb33 Þ ¼ ð16; 15; 14Þ. For each criterion, we agree to use criterion function no. 5 (Appendix A). In Table 3, we find the weights of the criteria ðwhj Þ as well as the similarity thresholds ðSTjh ðbhp ÞÞ and those of dissimilarity ðDTjh ðbhp ÞÞ. The typical casualties satisfy the property the similarity index between two typical objects of the same category must be superior to the similarity index between two typical objects of different categories. The results of these calculations are found in Table 4. In order to use the method, let us consider three workers a, b and c who are represented by the following profiles: • gðaÞ ¼ (Assembling a temporary instalation, Day labourer, Civil engineering excavation). • gðbÞ ¼ (Application of product, Roofer, Roof renovation). • gðcÞ ¼ (Material installation, Electrician, Interior renovation). • • • • • •
Table 3 Weights, similarity and dissimilarity thresholds Criterion 1 b1p b2p b3p
Criterion 2
Criterion 3
wh1
ST1h
DT1h
wh2
ST2h
DT2h
wh3
ST3h
DT3h
0.4 0.1 0.5
1 1 1
4 6 2
0.3 0.2 0.3
1 1 1
4 2 5
0.3 0.7 0.2
1 1 1
3 6 4
Table 4 The similarity index between the typical objects of each category b11 b12 b21 b31 b32 b33
b11
b12
b21
b31
b32
b33
1 0.70 0.52 0.22 0.30 0.43
0.70 1 0.10 0.27 0.42 0.22
0.40 0.40 1 0.30 0.15 0.23
0.10 0.30 0.22 1 0.93 0.93
0.30 0.50 0.02 0.93 1 0.87
0.50 0.10 0.28 0.93 0.87 1
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Table 5 The similarity index between the workers and the types of accidents C1 C2 C3
a
b
c
0.00 0.48 0.79
0.70 0.10 0.21
0.30 0.72 0.65
While using crossed tables, we obtain the following vectors: • gðaÞ ¼ ð16; 14; 14Þ. • gðbÞ ¼ ð10; 18; 11Þ. • gðcÞ ¼ ð15; 17; 18Þ. After having applied the method and using the maximin rule, we obtain the Table 5: We consider a maximin method with a cutting threshold of k ¼ 0; 5. Then, a 2 C t () MI t ða; C t Þ ¼ max min fSI h ða; bhp Þg and MI t ða; C t Þ P 0; 5: h2f1;...;Kg
p2f1;...;Lhg
Consequently, the workers a, b and c are affected to the categories C 3 ; C 1 , and C 2 , respectively. So these workers can be oriented towards an appropriate preventive program.
6. Conclusion The TRINOMFC method is, according to us, very simple because it is based only on the similarity index and does not refer to discordance or to the veto threshold. We believe that, with the nominal sorting, the notion of preference and of concepts which were introduced to model it are not very appropriate; the criteria are not being maximized or minimized. It is more a question of similarity than of preference. Since our method is very simple, it is also easy to understand for a potential user. Finally, we hope that the nominal sorting problematic will get more interest from researchers and practitioners. There are many situations that concern this problematic, so that it would be justified to give more attention to classification procedures.
Appendix A The criterion function no. 1, the true-criterion 1 if gj ðaÞ gj ðbhp Þ ¼ 0; h h SIj ða; bp Þ ¼ 0 if gj ðaÞ gj ðbhp Þ 6¼ 0:
The criterion function no. 2, the quasi-criterion 1 if jgj ðaÞ gj ðbhp Þj 6 STjh ðbhp Þ; SIjh ða; bhp Þ ¼ 0 if jgj ðaÞ gj ðbhp Þj > STjh ðbhp Þ:
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The criterion function no. 3, the pre-criterion 1 jgj ðaÞ gj ðbhp Þj=DTjh ðbhp ÞÞ if jgj ðaÞ gj ðbhp Þj 6 DTjh ðbhp Þ; h h SIj ða; bp Þ ¼ 0 if jgj ðaÞ gj ðbhp Þj > DTjh ðbhp Þ:
The criterion function 8 <1 SIjh ða; bhp Þ ¼ 1=2 : 0
no. 4, the pseudo-criterion if jgj ðaÞ gj ðbhp Þj 6 STjh ðbhp Þ; if STjh ðbhp Þ < jgj ðaÞ gj ðbhp Þj 6 DTjh ðbhp Þ; if jgj ðaÞ gj ðbhp Þj > DTjh ðbhp Þ:
The criterion function no. 5, an other pseudo-criterion 8 1 if jgj ðaÞ gj ðbhp Þj 6 STjh ðbhp Þ; > > < DT h ðbh Þ jg ðaÞ g ðbh Þj j j p p j SIjh ða; bhp Þ ¼ if STjh ðbhp Þ < jgj ðaÞ gj ðbhp Þj 6 DTjh ðbhp Þ; DTjh ðbhp Þ STjh ðbhp Þ > > :0 if jg ðaÞ g ðbh Þj > DT h ðbh Þ: j
j
p
j
p
References Bana e Costa, C., 1992. Structuration, Construction et Exploitation d’un Modele d’Aide a la Decision, These de doctorat, Universidade Tecnica de Lisboa. Belacel, N., 1998. La methode PROAFTN d’affectation multicritere: Fondement et application dans le domaine d’aide au diagnostic medical Serie: Mathematique de la gestion, Universite Libre de Bruxelles. Belacel, N., 2000. Multicriteria assignment method PROAFTH: Methodology and medical application. European Journal of Operational Research 125 (1), 176–184. Belacel, N., Vincke, Ph., Boulassel, M.R., 1999. Application of PROAFTN method to assist astrocytic tumors diagnosis using the parameters generated by computer-assisted microscope analysis of cell image. Innovative and Technology in Biology and Medicine 20 (4), 239–244.
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Bergeron, M., Martel, J.-M., Twarabimenye, P., 1996. The evaluation of corporate loan applications based on the MCDA. Journal of Euro-Asian Management 2 (2), 16–46. Bloch, M., 1984. Une aide a l’orientation des eleves: Les test du miroir, Memoire de DEA Methodes Scientifiques de Gestion, Universite de Paris Dauphine. Chandon, J.L., Pinson, S., 1981. In: Analyse typologique: Theorie et applications. Masson, Paris, p. 254. Dekempener, D., Gervasi, F., 1985. Deux cas pratiques d’aide a la decision Memoire de fin d’etudes d’ingeniorat commercial Universite de Mons. Fabre, G., 1985. Application d’une methode multicritere pour la classification de dossiers de credit, Memoire de DEA Methodes Scientifiques de Gestion, Universite de Paris Dauphine. Henriet, L., 1995. Probleme d’affectation et methodes de classification, Memoire de DEA, Universite de Paris Dauphine. Massaglia, M., Ostanello, 1991. N-TOMIC: A support system for multicriteria segmentation problems. In: Korhonen, P. (Ed.), International Workshop on Multicriteria Criteria Decision Support, Helsinki 7-11, 1989. Lectures Notes in Economics and Mathematical Systems, Vol. 356. Spinger, Berlin, pp. 167–174. Moscarola, J., 1977. Aide a la decision en presence de criteres multiples fondee sur une procedure trichotomique: Methodologie et application, These de 3eme cycle Sciences des Organisations, Universite de Paris Dauphine. Perny, P., 1998. Multicriteria filtering methods based on concordance and non-discordance principles. Annals of Operations Research 80, 137–165. Perny, P., Roy, B., 1992. The use of fuzzy outranking relations in preference modelling. Fuzzy Sets and Systems 49, 33–53. Roy, B., 1985. Methodologie Multicritere d’Aide a la decision. Economica, Paris. Roy, B., Vincke, Ph., 1984. Relational systems of preference with one or more pseudo-criteria: Some new concepts and results. Management Sciences 30 (11), 1323–1335. Smadja, A., 1998. Segmenter ses Marches. Presses Polytechniques et Universitaires Romandes, Lausanne. Vansnick, J.-C., 1989. Application of multicriteria decision-aid to allocating budget for building repairs and maintenance. In: Tabucanon, M.T., Chankong, V. (Eds.), Proceedings of the International Conference on Multiple Criteria Decision Making: Applications in Industry and Service. Asian Institute of Technology, Bangkok, 6–8 December, pp. 629–642. Yu, W., 1992. Aide multicritere a la decision dans le cadre de la problematique du tri, These de doctorat, Universite Paris Dauphine. Zopounidis, C., Doumpos, M., 1998. Developing a multicriteria decision support system for financial classification problems: The finclos system. Optimization Methods and Software 8, 277–304.
A multicriteria assignment procedure for a nominal sorting problematic Julien Leger *, Jean-Marc Martel Facult e des Sciences de l’Administration, Universit e Laval, Sainte-Foy, Que., Canada G1K 7P4
Abstract This paper addresses the problem of the assignment of an action in the particular case where the categories defined a priori are not ordered. In multicriteria literature, it is the nominal sorting problematic. We propose a simple and easy method to understand the procedure of assignment to tackle this problematic. Afterwards, we shall illustrate the proposed method by using an example that is inspired by a research carried out the construction sector of the work accident prevention field. 2002 Elsevier Science B.V. All rights reserved. Keywords: Multicriteria method; Nominal sorting problematic; Classification; Assignment; Similarity
1. Introduction The human brain constantly tries to classify individuals and objects. In order to be able to treat all the available information, the human being must set categories that allow putting together individuals or objects having common characteristics. In general, a name is given to the different categories. In zoology, for example, we can find the mammal category and the vertebrate category. Thus, an individual or object is identified according to its belonging to such or such a category (Chandon and Pinson, 1981). The classification methods are divided into two groups: the automatic classification method and the assignment method. The automatic classification methods (‘‘clustering’’) consist in regrouping individuals (or objects) into a restricted number of categories. These categories must be, on the one hand, as few as possible and on the other hand, as homogeneous as possible. These methods are based on the notion of non-supervised learning. In non-supervised learning, we have individuals but we have no information on their belonging to categories. We regroup the individuals into categories in order to make the distances between the individuals within a same category the shortest and the distances between the centres of the different categories the longest.
*
Corresponding author. E-mail address: [email protected] (J.-M. Martel).
0377-2217/02/$ - see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 0 1 ) 0 0 2 5 1 - X
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The assignment method allows affecting an individual to a defined category a priori. For example, the categories may have been defined by using an automatic classification method based on historical data. Thus, we know the individuals whose assignment categories are known. Based on these individuals, we can define assignment standards. These assignment methods are based on the notion of supervised learning. Under supervised learning, we dispose a set of individuals whose belonging is known and serve to calibrate the parameters of the method. Once this adjustment is done, the method is able to assign an unknown individual to the most plausible category. In the literature on multicriteria decision aids, we sometimes find it handy to refer to the retained decisional problematic since it determines the nature of the recommendation coming from a multicriteria analysis. The assignment methods or procedures are associated to the sorting problematic (or of the segmentation). This type of problematic is found quite frequently in practice as shown in some of the following examples (Yu, 1992). ‘‘Trichotomic admission on file’’ (Moscarola, 1977): The actions are the files of the candidates that apply for a job. The problem is to develop an examination procedure to facilitate the work of the decision maker. The procedure should allow to sort the files into the three following ordered categories: admission, research for additional information and refusal. ‘‘Granting of credits’’ (Fabre, 1985; Bergeron et al., 1996; Zopounidis and Doumpos, 1998): A financial organisation is constantly faced with requests for credit. Each request is analysed taking into account data such as: the financial report, the qualitative aspects related to the company’s policies and culture. The problem consists in creating a system capable of treating the information in real-time and of giving an opinion for each of the requests. The system will affect the actions in one of the following ordinate categories: a credit approved immediately, a rather favorable credit and a rejected credit. ‘‘Student evaluation’’ (Massaglia and Ostanello, 1991): In this case, the actions are the students who must take a series of tests (oral and written). Each student is evaluated taking into account the results obtained in each test. The problem consists in separating the students into partially ordered categories: very good, good, quite good, rather good, rather bad, bad, very bad and uncertain. ‘‘Allocation of resources’’ (Dekempener and Gervasi, 1985; Vansnick, 1989): Each year, a public organism must treat several requests concerning the maintenance and/or the repairing of the apartments that it owns. The total sum of the requests is often much higher that the allocated budget for the year. The problem consists in creating an automatic tool allowing to allocate the funds to the various requests. The categories are the different priority levels for the maintenance or the repairing of the apartments that must be defined beforehand. All these examples concern the ordinal sorting problematic. However, it is also frequent to face decisionmaking situations found in the nominal sorting problematic. Here are two examples of problems that belong to this type of problematic. ‘‘Help in the orientation of student’’ (Bloch, 1984): The problem consists in helping in the orientation of the students. We know that these are different types of training in various sectors that it is very difficult for a student to choose his future orientation. The objective is to conceive a system that would propose the training best required for each student. ‘‘Help in medical diagnosis’’ (Belacel, 1998): The problem consists in identifying the disease(s) of the patients using their symptoms. One must build a system that will allow the identification of the possible disease(s) by considering the symptoms. In this case, the categories are the various possible diseases (example: acute leukaemia). These categories are not ordered. In this paper, we are interested in the assignment methods of each action (individual, object, . . .) in the case where the categories defined a priori are not ordered, i.e., the nominal sorting problematic. This document is divided into four sections. In Section 2, we present the sorting problematic by putting the accent on the one concerned by the nominal sorting. Section 3 is essentially devoted to two existing methods capable of treating problems related to the nominal sorting problematic. In Section 4, we propose a method
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capable of treating this type of problem and a numerical example illustrating the proposed method is presented in Section 5.
2. The sorting problematic We can conceive the problematic as the way the decision aid is contemplated, i.e., the way of formulating a problem in order to reach results allowing to shed light on a decision. According to Roy (1985), it is possible to distinguish four reference problematics: description, choice, ranking and sorting problematics. This paper deals with the sorting problematic and, more specifically, the nominal sorting. The sorting problematic consists in assigning each one of the potential actions to one or more categories while examining the intrinsic value of the action. The categories are defined according to pre-established norms of assignment independently from all the actions to be taken. A specific norm of assignment must be associated to each category. Therefore, the assignment of any action to one of the categories is solely founded on the analysis of the action with regards to the assignment norms and not on its relative value with regards to other actions. According to Bana e Costa (1992), one must distinguish between two significantly different situations in the sorting problematic. Effectively, there are cases where all the categories have a purely nominal structure, i.e., there is no order between the categories, and the cases where the decisional context imposes an ordered structure on the categories. In the ordinal sorting problematic, the categories are completely ordered. They are characterized by a set of reference objects or prototypes (typical actions). Often for this type of sorting, the procedure of assignment is formulated in the following manner: any action judged to be between the two limits (higher and lower) of a category, must be assigned to the category in question. Fig. 1 represents schematically the ordinal sorting problematic. In the nominal sorting problematic, the categories are not ordered. The assignment procedure associated to a category must be formulated in the following manner: any action judged to be sufficiently similar to at least one typical action must be assigned to the category to which this typical action belongs. Fig. 2 represents schematically the nominal sorting problematic. When either only one superior and one inferior reference action or only one typical action defines a category, it is then a question of monoprofile categories. However, a category may also be defined by a set of reference objects or prototypes. In this case, we refer to multiprofile categories. This second case is harder to treat, but it offers more richness from the modelisation of the categories point of view. This implies that it is possible for an action to be assigned in different manners to a category. In practice, it may be useful to use thresholds (of indifference, of similarity, of resemblance, of dissimilarity). In multicriteria decision aid, it may seem realistic to wish to set a value under which there is indifference between two actions and over which there is strict preference. Often, there is an intermediate zone in which there is hesitation between preference and indifference. It is the pseudo-criterion concept introduced by Roy and Vincke (1984). When considering the nominal sorting problematic, this definition of thresholds (of indifference and of preference) must be slightly modified (Henriet, 1995). For example, in the
Fig. 1. The ordinal sorting problematic.
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Fig. 2. The nominal sorting problematic.
nominal sorting problematic framework, the notion of preference does not really appropriate and it should be replaced by one of similarity. A threshold may be perceived as a tool allowing to consider the imprecision and uncertainty related to the evaluations. In the TRINOMFC method (which will be presented later in this paper), we are in the presence of a similarity threshold (ST) and of a dissimilarity threshold (DT). The ST represents the maximum value between two actions so that the two actions may be judged to be sufficiently similar. On the other hand, the DT is the minimal value between two actions from which we may assert that an action is significantly dissimilar from another. The sorting problematic is used very little as compared with the choice and ranking problematics (Yu, 1992). However, a certain number of research and application papers have treated this problematic. In multicriteria analysis, certain methods were conceived to treat the sorting problematic. Among these methods, we find TRICHOM (Moscarola, 1977) which is a trichotomic method that is limited to problems with three ordered categories. The generalisation with more than three categories may be treated by the NTOMIC methods (Massaglia and Ostanello, 1991) or ELECTRE TRI (Yu, 1992). As compared to the ordinal sorting, the nominal sorting is a little harder to deal with and it has been so till quite recently. We propose the following formulation of the nominal sorting problematic: Let A a finite and not empty set of objects to assign to the different categories, F ¼ fg1 ; . . . ; gn g a set of n criteria, n P 1, C ¼ fC 1 ; . . . ; C K g the group of predefined categories h; K > 1, Bh ¼ fbhp j p ¼ 1; . . . ; Lh and h ¼ 1; . . . ; Kg the set of typical objects of the hth category where bhp repreSK h sents the pth typical object of the hth category and let B ¼ h¼1 B . Each object of A and B is defined by its performances evaluated on all the criteria. Hence, 8a 2 A; we have gðaÞ ¼ ðg1 ðaÞ; . . . ; gn ðaÞÞ, where g1 ðaÞ is the evaluation of an object a according to the g1 ðj ¼ 1Þ criterion and for 8bhp 2 Bh B, we have gðbhp Þ ¼ ðg1 ðbhp Þ; . . . ; gn ðbhp ÞÞ with p ¼ 1; . . . ; Lh and h ¼ 1; . . . ; K, where Lh is the number of typical object of the hth category. Also, a 2 C h means we assign action a to category h. The question raised is the following: since we have object a 2 A, to which category should we assign it where comparisons are made between its evaluation profile gðaÞ and those of the typical objects gðbhp Þ. This assignment must be made on the basis of a similarity index between the profiles.
3. Two methods for nominal sorting problematic We shall now briefly present two methods of assignment in the nominal sorting problematic. These methods are very recent since they only go back to 1998. To our knowledge, no other methods based on the concepts of outranking relations have been developed to deal with the nominal sorting problematic. There
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are some methods based on rough data, but we do not present them in this paper. The first one by Perny (1998), is a multicriteria filtering method based on the concordance and non-discordance principles. The second one by Belacel (1998) called PROcedure d’Affectation Floue dans le cadre de la problematique du Tri Nominal problem (PROAFTN), is a multicriteria fuzzy classification procedure for nominal sorting problematic. For the two methods, the authors assume that the categories are definite a priori and that each category is characterized by one or several typical actions (reference objects or prototypes). Also, they assume that the different criteria are determined and constitute a coherent criterion family.
3.1. Perny’s method (1998) With this method, a comparison by pairs is made in order to verify if an action a is close to or similar to a typical action representing a typical element of a category. In fact, the concordance and discordance are computed for each pair of actions. First of all, we must build an indifference relation or else an outranking relation for which the symmetric part plays the role of the indifference relation. In order to be able to present different judgements of preference, the preference relations S and I are used. Let a and bhp be two actions, the information on the preferences is expressed by H ða; bhp Þ 2 ½0; 1 representing the confidence in proposition aHbhp , where H 2 fS; Ig. Let us consider that the thresholds of preference pj and of indifference qj , where qj 6 pj . The function Sj (that is the outranking relation for criterion j) is defined by n o pj min gj ðbhp Þ gj ðaÞ; pj n o: Sj ða; bhp Þ ¼ pj min gj ðbhp Þ gj ðaÞ; qj The outranking relation Sj ða; bhp Þ is very useful since it represents the indifference at the same time. Consequently, n o Ij ða; bhp Þ ¼ min Sj ða; bhp Þ; Sj ðbhp ; aÞ : The principle of similarity simply uses the rule of the majority to decide between each pair of actions a and bhp . For a pair of actions ða; bhp Þ, we obtain Ij ða; bhp Þ 2 ½0; 1 ; j ¼ 1; . . . ; n. These numbers n are aggregated in the following manner: CI ða; bhp Þ ¼
n X
wj Ij ða; bhp Þ;
j¼1
to obtain aPglobal index CI ða; bhp Þ measuring the global agreement against the proposition aIbhp , where wj ( P 0 and nj¼1 wj ¼ 1) is the relative importance of the criterion j. The principle of discordance introduces a complementary notion in the aggregation process. At this level, we introduce a veto threshold vj which is the maximum value of a difference gj ðbhp Þ gj ðaÞ which is compatible with the proposition aHbhp , where H 2 fS; Ig. The veto is characterised by the following condition: 8ða; bhp Þ 2 A B; ð9j 2 f1; . . . ; ng; gj ðbhp Þ gj ðaÞ > vj Þ ) :ðaHbhp Þ; where : represents the logic operator of the negation.
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The introduction of this notion of discordance allows measuring the degree to which criterion j is strongly opposed to aIbhp . The discordance relation associated to the outranking relation for criterion j is defined as follows: !! gj ðbhp Þ gj ðaÞ pj S h Dj ða; bp Þ ¼ min 1; max 0; : vj p j The discordance relation corresponding to the indifference is defined by DIj ða; bhp Þ ¼ DIj ðbhp ; aÞ ¼ maxðDSj ða; bhp Þ; DSj ðbhp ; aÞÞ: Also, DI ða; bhp Þ ¼ 1
n Y
1 DIj ða; bhp Þ
a=n
;
j¼1
where a 2 ½1; n is a technical parameter introduced to modify the degree of synergy between the criteria. When using the principle of concordance and of non-discordance, a proposition of the type aIbhp is true if and only if the coalition of the criteria in agreement with this proposition is strong enough and that there be no discordant coalition with this proposition. This principle may be expressed as follows: n o Iða; bhp Þ ¼ min CI ða; bhp Þ; 1 DI ða; bhp Þ : Then, we calculate the membership degree of the action to each category, dða; C h Þ, as follows: n o dða; C h Þ ¼ max Iða; bh1 Þ; Iða; bh2 Þ; . . . ; Iða; bhLh Þ ; h ¼ 1; . . . ; K: Action a is assigned to the category according to the following rule: a 2 C t () dða; C t Þ ¼ max dða; C 1 Þ; dða; C 2 Þ; . . . ; dða; C K Þ :
3.2. The Belacel (1998) method The method of Belacel (1998), named PROAFTN, is a multicriteria decision aid method to deal with the problems of fuzzy multicriteria assignment. This method allows building a fuzzy indifferent relation by generalising the concordance and discordance ratings used in the ELECTRE III method. Following this, we determine the category of assignment of a given object. Belacel et al. (1999) and Belacel (2000) are another good references for this method. The indifference relation of synthesis is based on the concordance/discordance rule as it is done in (Perny and Roy, 1992). In adopting this rule in the nominal sorting context, the result is: ‘‘when action a is judged to be indifferent with a typical action bhp based on sufficiently important majority of criteria majority principle and there is no criterion that puts its veto against the affirmation ‘‘a is indifferent to bhp ’’ principle of respect of the minorities, then the action a is indifferent to bhp ’’. In practice, the performances of the typical actions are often more easily given under the form of intervals. For each criterion gj , the interval ½gj1 ðbhp Þ; gj2 ðbhp Þ is associated to each reference action bhp with gj1 ðbhp Þ 6 gj2 ðbhp Þ.
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Let Iða; bhp Þ be the indifference rating between the action a and the typical action bhp ; h ¼ 1; . . . ; K and p ¼ 1; . . . ; Lh . Yet, Iða; bhp Þ is based on concordance and non-discordance principles. More precisely, ! n n
Y
whj X h h h h Iða; bp Þ ¼ wj Cj ða; bp Þ 1 Dj ða; bp Þ ; j¼1
j¼1
where • Cj ða; bhp Þ; j ¼ 1; . . . ; n, is the degree to which the criterion gj is in favour of the indifference relation between a and bhp . Cj ða; bhp Þ is calculated using two discrimination thresholds djþ ðbhp Þ and dj ðbhp Þ used to take into account of fuzzy data. • Dj ða; bhp Þ; j ¼ 1; . . . ; n, is the degree to which the criterion gj is against the indifference relation between a h h and bhp . Dj ða; bhp Þ is calculated using two veto thresholds vþ j ðbp Þ and vj ðbp Þ used to consider the values for h which a is very different from bp . Pn • whj is the relative importance of the criterion j of the category C h , whj P 0 and j¼1 whj ¼ 1. Following this, we calculate the membership degrees of the action a to hth category, dða; C h Þ, as follows: n o dða; C h Þ ¼ max Iða; bh1 Þ; Iða; bh2 Þ; . . . ; Iða; bhLh Þ ; h ¼ 1; . . . ; K: The action a is assigned to the category according to the following rules: a 2 C t () dða; C t Þ ¼ max dða; C 1 Þ; dða; C 2 Þ; . . . ; dða; C K Þ :
3.3. Remarks concerning the two methods These two methods have certain similarities. In Belacel’s method, the typical actions are given using the interval of the form ½gj1 ðbhp Þ; gj2 ðbhp Þ with gj1 ðbhp Þ 6 gj2 ðbhp Þ. This last interval has a link with Perny’s method if we suppose that gj ðbhp Þ ¼ ðgj1 ðbhp Þ þ gj2 ðbhp ÞÞ=2 with qj ðbhp Þ ¼ gj2 ðbhp Þ gj ðbhp Þ ¼ gj ðbhp Þ gj1 ðbhp Þ. The preference thresholds and the discrimination thresholds are very similar since they represent the marker between the weak indifference and non-indifference. The Perny method uses an indifference threshold while that of Belacel does not use any. Belacel’s idea of using an interval to determine the typical actions seems very interesting. In practice, it may prove to be difficult to give a precise value for the evaluation of a typical action according to a criterion. However, the indifference threshold in Perny’s method plays the same role as the interval used by Belacel. The Belacel method seems to have had some success in the field of medical diagnosis aid. However, these methods are quite complex to use in the sense that they necessitate several thresholds (preference, indifference and veto thresholds). These thresholds, in particular the veto thresholds, may be quite difficult to obtain. Also, are these thresholds introduced to model the preferences also appropriate to compute similarity index in the nominal sorting problematic? In the nominal context there is no preference notion and the criteria are not being maximized or minimized.
4. The proposed method Our objective is to develop a method of assignment within the framework of nominal sorting that may be simple and easy to understand for a potential user; in particular, a method that will not necessitate the
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determination of veto thresholds. It implies that less information is needed than the two other methods presented above. We propose a procedure for which the assignment is based on the determination of a similarity rating between an object and each one of the categories as characterised by its typical objects. For each criterion, a function representing the similarity index will be introduced in order to model the concept of similarity between two objects. Such a function is defined for each criterion and its value varies between 0 and 1; this value will be so much greater than that the similarity between the two actions of this criterion is greater. Different types of criteria functions (see Appendix A) are available to model with more flexibility the notion of similarity between two objects. However, we are free to choose another function. We have named this method TRI NOMinal base sur des Fonctions Criteres (TRINOMFC). The proposed method necessitates the use of certain thresholds. Consequently, STjh ðbhp Þ and DTjh ðbhp Þ represent, respectively, the similarity threshold and the non-similarity threshold associated to the typical object bhp according to criterion j. Let SIjh ða; bhp Þ be the similarity index between the object a and the typical object bhp of the hth category according to criterion j. Let SI h ða; bhp Þ be the similarity index between the object a and the typical object bhp of the hth category, all the criteria taken simultaneously. Finally, let MI h ða; C h Þ be the membership index between the object a and the hth category. To measure the similarity index between the two objects a and b leads us to answer the following question: Is object a sufficiently similar to object b? In the affirmative, the similarity index is a value close to 1. If not, the similarity index is closer to 0. Since the objects are evaluated according to n criteria, we must start by calculating a similarity index SIjh ða; bhp Þ according to each criterion j. Starting from these n similarity indices (between an object and a typical object according to the n criteria), we shall be able to calculate a membership index aggregating these n similarity indices. We must make the same calculations for each one of the typical objects and with this information we shall be able to calculate a membership index between an object and each one of the categories. It is natural to assign an object to a category for which the similarity is greater, but this index must reach a minimal level of similarity. The similarity indices must possess certain characteristics (Smadja, 1998). In fact, they must respect three axioms: (i) The similarity between two objects is a positive measure, at most equal to 1. (ii) Only two identical objects have a similarity equal to 1. (iii) The similarity between two objects does not depend on the order in which they are presented. It is important to mention that the axiom (iii) is debatable because some authors consider that the similarity relation as not necessarily symmetric. In this work, we assume the similarity relation to be symmetric. Let us consider the true-criterion (the criterion function no. 1 in Appendix A) that is expressed analytically as follows: ( 1 if gj ðaÞ gj ðbhp Þ ¼ 0; SIjh ða; bhp Þ ¼ 0 if gj ðaÞ gj ðbhp Þ 6¼ 0: For this type of criterion, it is evident that SIjh ða; bhp Þ is a similarity index since the three axioms are verified. In fact, axiom (i) is satisfied since SIjh ða; bhp Þ 2 f0; 1g. The axiom (ii) is also satisfied since SIjh ða; bhp Þ ¼ 1 if and only if gj ðaÞ ¼ gj ðbhp Þ. Since jgj ðaÞ gj ðbhp Þj ¼ jgj ðbhp Þ gj ðaÞj, it is clear that SIjh ða; bhp Þ ¼ SIjh ðbhp ; aÞ and the axiom (iii) is verified. Proceeding in a similar manner, it easy to verify that the criterion function no. 3 (in Appendix A) satisfies all three axioms.
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The other criteria functions (criteria functions no. 2, no. 4 and no. 5 in Appendix A) satisfy axioms (i) and (iii). However, due to the use of similarity and dissimilarity thresholds, axiom (ii) is not verified as expressed. This is easy to verify with a counter-example. In fact, let gj ðbhp Þ ¼ gj ðaÞ þ
STjh ðbhp Þ : 2
Thus, gj ðaÞ gj ðbhp Þ ¼ gj ðaÞ
STjh ðbhp Þ gj ðaÞ þ 2
! ST h ðbh Þ j p < STjh ðbhp Þ: ¼ 2
Then, SIjh ða; bhp Þ ¼ 1 because jgj ðaÞ gj ðbhp Þj < STjh ðbhp Þ. Nevertheless, objects a and bhp are not identical. The use of thresholds makes the axiom (ii) not to be strictly satisfied. When considering indifference thresholds, it is evident (and it is the sense of these thresholds) that two actions whose evaluations differ from less than these thresholds are considered as identical in the sense of preference. It is the same thing for the similarity thresholds. We shall therefore be able to extend the meaning of axiom (ii) to include the similarity thresholds. Then, axiom (ii) may be defined as follows: ðii0 ) Only two similar objects have a similarity equal to 1. Two objects a and bhp are considered similar on criterion j if jgj ðaÞ gj ðbhp Þj < STjh ðbhp Þ. This modification is justified by the fact that the evaluations of the objects are not of absolute precision. While considering this new definition of axiom (ii), we can conclude that the criterion function no. 2 satisfies axiom ðii0 ) and consequently verify three axioms (modified) characterising a similarity index. We may proceed in the same way for the criteria functions no. 4 and no. 5 in order to verify if they satisfy the axioms. Even if for certain criteria functions, we must have recourse to axiom ðii0 ), we shall nevertheless qualify the indices obtained on this basis of similarity indices. We have judged that the use of thresholds (similarity and/or dissimilarity thresholds) is appropriate in the calculation of the similarity index. An important property that the typical objects of the categories must have is that: the similarity index between two typical objects of a same category must be superior to the similarity index between two typical objects from different categories. Let a 2 A an object to be assigned. The first step of the TRINOMFC method consists in determining one of the criteria functions that is found in Appendix A (that are the adaptations of the criteria functions of THE E) for each criterion. In fact, the ‘‘user’’ is free to define himself a function if that seems to PROME him more appropriate to characterise the differences of evaluation. For each category C h and for each criterion j, we determine a criterion function SIjh : A B ! ½0; 1 ; j ¼ 1; . . . ; n and h ¼ 1; . . . ; K that represents this similarity index. This function must represent the manner for which the similarity index between a and bhp decreases with the difference jgj ðaÞ gj ðbhp Þj. Because of the chosen functions, we will have to eventually determine parameters such as the similarity threshold STjh ðbhp Þ or dissimilarity threshold DTjh ðbhp Þ. A similarity threshold STjh ðbhp Þ represents the maximal value of the difference between two objects so that the two objects may be judged to be similar. On the other hand, the dissimilarity threshold DTjh ðbhp Þ is the minimal value of the difference between two objects from which we may conclude that an object is totally dissimilar to an other one. The similarity as well as the dissimilarity is never absolute. These thresholds play an equivalent role to indifference and preference thresholds but does not have the same significance. It is reasonable to take into account the imperfections in the evaluations of the objects according to each criterion to have recourse to similarity thresholds (STjh ðbhp Þ as if jgj ðaÞ gj ðbhp Þj 6 STjh ðbhp Þ then SIjh ¼ 1) and of dissimilarity ðDTjh ðbhp Þ as if jgj ðaÞ gj ðbhp Þj > DTjh ðbhp Þ then SIjh ¼ 0Þ. Fig. 3 illustrates a similarity index.
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Fig. 3. Similarity index.
After having determined the criteria functions and, if necessary, the thresholds, the second step consists in calculating the local similarity index, that is the calculation of the SIjh ða; bhp Þ; j ¼ 1; . . . ; n; p ¼ 1; . . . ; Lh and h ¼ 1; . . . ; K. These calculations allow us to obtain the similarity index between the object a and the typical objects according to each of the criteria taken individually. For a given category (let us suppose category h), we obtain a chart as shown in Table 1. With this information, we can go on to step three which consists in calculating, for each category, the similarity index between the object a and each one of the typical objects bhp of this category taken individually. This global similarity index is noted as SI h ða; bhp Þ; p ¼ 1; . . . ; Lh and h ¼ 1; . . . ; K. To calculate SI h ða; bhp Þ, we make the aggregation of n local similarity indices, SIjh ða; bhp Þ; j ¼ 1; . . . ; n, corresponding to the different similarity index between a and bhp according to each criterion. A weighted sum is used for this aggregation: SI h ða; bhp Þ ¼
n X
whj SIjh ða; bhp Þ;
ð1Þ
j¼1
where whj is the relative importance of criterion j of the category C h (like in Belacel, 1998). We obtain a chart like the one illustrated in Table 2 for each one of the categories C h ; h ¼ 1; . . . ; K. Table 1 Local similarity index between object a and the typical objects of category h SIjh ða; bhp Þ bh1 h b .2
ðp ¼ 1Þ ðp ¼ 2Þ
.. bhLh ðp ¼ Lh Þ
g1 ðj ¼ 1Þ
g2 ðj ¼ 2Þ
...
gn ðj ¼ nÞ
SI1h ða; bh1 Þ h h .SI1 ða; b2 Þ
SI2h ða; bh1 Þ h h .SI2 ða; b2 Þ
... ... . . . ...
SInh ða; bh1 Þ h h ..SIn ða; b2 Þ . SInh ða; bhLh Þ
.. SI1h ða; bhLh Þ
.. SI2h ða; bhLh Þ
Table 2 Global similarity index between the object a and the typical objects for category h SI h ða; bhp Þ bh1 ðp ¼ 1Þ h b .. 2 ðp ¼ 2Þ . bhLh ðp ¼ Lh Þ
SI h ða; bh1 Þ h h ..SI ða; b2 Þ . SI h ða; bhLh Þ
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Thus, we have the global similarity index between the object a and each one of the typical objects for each one of the categories. The fourth step consists in determining the membership index between the object a and each one of the categories, which means that we find values of MI h ða; C h Þ; h ¼ 1; . . . ; K. For this purpose, a min rule (pessimistic rule) is used since in this manner we can say that in the worst case the similarity rating between a and C h is therefore n o MI h ða; C h Þ ¼ min SI h ða; bhp Þ ; h ¼ 1; . . . ; K: ð2Þ p2f1;...;Lhg
In the fifth step, we must assign object a to the category for which the membership index between the object and the category is maximal. Then, we simply need to find the maximum of the values of MI h ða; C h Þ; h ¼ 1; . . . ; K. This idea is represented by the following expression: a 2 C t () MI t ða; C t Þ ¼ max MI h ða; C h Þ 8h ¼ 1; . . . ; K : ð3Þ We can suppose that the values of MI h ða; C h Þ; h ¼ 1; . . . ; K, are distinct, which implies that each object is affected to exactly one category. The fourth and the fifth steps may be represented by a global maximin rule: min fSI h ða; bhp Þg : ð4Þ a 2 C t () MI t ða; C t Þ ¼ max h2f1;...;Kg
p2f1;...;Lhg
In the TRINOMFC method, an object is assigned to exactly one category. We do not define any cutting threshold k. However, nothing prevents from defining such a threshold when an object cannot be assigned to one category. In general, such a threshold of cutting is superior to 0.5. While considering cutting threshold k, we obtain the following result: t t t h h a 2 C () MI ða; C Þ ¼ max min fSI ða; bp Þg and MI t ða; C t Þ P k: ð5Þ h2f1;...;Kg
p2f1;...;Lhg
We can be conduct to case of several categories with a rule of the type a 2 C t () MI t ða; C t Þ P k and we also have maxh2f1;...;Kg fminp2f1;...;Lhg gfSI h ða; bhp Þgg then we will have one, several or no category. The sixth step is very simple since it indicates that we must repeat the same process with all the objects of the A group so that each object may be assigned to a category.
4.1. Commentary on the method The TRINOMFC method offers great flexibility while allowing to have several profiles to represent different categories. We thus talk about the multiprofile categories. In practice, it is realistic to believe that an object may belong to a category in several ways. In the ordinal sorting problematic, it is very difficult to include the multiprofile notion (Yu, 1992). In ELECTRE TRI, for example, each category may be represented by only one superior and only one inferior action of reference. We use a maximin rule to identify the category for which the object should be assigned. When using this rule, we can say that we have an attitude that is conservative since the min forces us to take the minimal similarity index between the object and the typical objects for each one of the categories. However, other rules could also have been used like a maximax rule. This rule finds, for a given object, the maximal index with each one of the categories. The maximum of these indices is taken to know the category to which the object will be affected. The maximax rule would express itself as follows:
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a2C
t
t
t
() MI ða; C Þ ¼ max
h2f1;...;Kg
max fSI
p2f1;...;Lhg
h
ða; bhp Þg
:
ð6Þ
The Hurwicz rule could also have been used. It follows an attitude that is a little more nuanced as shown in the following expression: a 2 C t () MI t ða; C t Þ n o n o a max SI h ða; bhp Þ þ ð1 aÞ min SI h ða; bhp Þ ; ¼ max h2f1;...;K g
p2f1;...;Lhg
p2f1;...;Lhg
ð7Þ
where a 2 ½0; 1 is a coefficient of relative optimism. In this case, we give a certain weight a to the typical object having the maximal similarity index and the complementary weight 1 a to the typical object having the minimal similarity index. To sum up, the Hurwicz rule is a mixture of maximin and maximax rules. Also, we may use a weighted sum. More precisely, it suffices to give a degree of likelihood to each one of the typical objects Let wh ðbhp Þ be the degree of likelihood of typical object bhp . For each one of PLhof ahcategory. h the categories, p¼1 w ðbp Þ ¼ 1. We would then be led to the following rule: ( ) X t t t h h h h w ðbp Þ SI ða; bp Þ : ð8Þ a 2 C () MI ða; C Þ ¼ max h2f1;...;K g
p
In the particular case, where a category is represented by only one typical object ðbh Þ, we obtain SI ða; C h Þ ¼ SI h ða; bh1 Þ. For each one of the rules of assignment, we suppose that the values of MI h ða; C h Þ are unique. However, it is very possible that two values of MI h ða; C h Þ are identical. In such a situation, it is suggested to apply a second rule (Hurwicz, maximax, . . .) in order to overcome the impasse if we wish that an action be assigned to exactly one category. In the case where there are several typical objects in each category, it is very important to verify the property that the categories must have. The property is the following: the similarity index between the typical objects of a same category must be superior to the similarity index between the typical objects of different categories, that is h
SI h ðbhp ; bhp0 Þ > SI h ðbh0q ; bhp Þ 8p; p0 2 f1; . . . ; Lh g; 8h0 6¼ h 2 f1; . . . ; k g and 8q 2 f1; . . . ; Lh0 g: This one must be verified before using the TRINOMFC method if we wish to obtain an effective procedure of assignment.
5. Numerical illustration In order to illustrate the proposed procedure, we were inspired by the case of an application done in collaboration with the ‘‘Institut de recherche en sante et securite au travail‘‘ (IRSST). In the construction sector, different types of accidents may occur. For a given worker, we shall try to identify the type of accidents he is most likely to be a victim of. Let us consider three types of accidents: falls ðC 1 Þ, eye injuries ðC 2 Þ and deep cuts ðC 3 Þ. Each type of accident is characterised by several typical casualties. In fact, C 1 is characterised by two typical casualties and C 2 is characterised by a typical casualty while C 3 is characterised by three typical casualties. Also, three criteria are retained: the task done by the worker ðg1 Þ, the worker’s trade ðg2 Þ and the type of building site ðg3 Þ. Each criterion has several modalities.
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The typical casualties are determined as follows: gðb11 Þ ¼ (Surface preparation; Electrician, Electricity renovation); gðb12 Þ ¼ (Surface preparation; Roofer; Roof renovation); gðb21 Þ ¼ (Product application; Painter; Interior renovation); gðb31 Þ ¼ (Material installation; Day labourer; Exterior house finishing); gðb32 Þ ¼ (Material installation; Plumber; Plumbing service); gðb33 Þ ¼ (Setting up a temporary installation; Heavy equipment operator; Civil engineering excavation). In order to be able to go from the qualitative to the quantitative data, one has to use crossed tables. The interpretation of these tables is very simple. For each column, the relative frequencies represent the distribution of the casualties for the modality of a criterion according to the type of accidents. This distribution is called the conditional distribution according to the type of accidents taking into account the modality of a given criterion. The frequencies relative to a column are conditional frequencies. When using crossed tables determined by a study on the health and security at work, we obtain the following typical casualties: • gðb11 Þ ¼ ð11; 17; 15Þ, • gðb12 Þ ¼ ð11; 18; 11Þ, • gðb21 Þ ¼ ð10; 13; 18Þ, • gðb31 Þ ¼ ð15; 14; 12Þ, • gðb32 Þ ¼ ð15; 16; 11Þ, • gðb33 Þ ¼ ð16; 15; 14Þ. For each criterion, we agree to use criterion function no. 5 (Appendix A). In Table 3, we find the weights of the criteria ðwhj Þ as well as the similarity thresholds ðSTjh ðbhp ÞÞ and those of dissimilarity ðDTjh ðbhp ÞÞ. The typical casualties satisfy the property the similarity index between two typical objects of the same category must be superior to the similarity index between two typical objects of different categories. The results of these calculations are found in Table 4. In order to use the method, let us consider three workers a, b and c who are represented by the following profiles: • gðaÞ ¼ (Assembling a temporary instalation, Day labourer, Civil engineering excavation). • gðbÞ ¼ (Application of product, Roofer, Roof renovation). • gðcÞ ¼ (Material installation, Electrician, Interior renovation). • • • • • •
Table 3 Weights, similarity and dissimilarity thresholds Criterion 1 b1p b2p b3p
Criterion 2
Criterion 3
wh1
ST1h
DT1h
wh2
ST2h
DT2h
wh3
ST3h
DT3h
0.4 0.1 0.5
1 1 1
4 6 2
0.3 0.2 0.3
1 1 1
4 2 5
0.3 0.7 0.2
1 1 1
3 6 4
Table 4 The similarity index between the typical objects of each category b11 b12 b21 b31 b32 b33
b11
b12
b21
b31
b32
b33
1 0.70 0.52 0.22 0.30 0.43
0.70 1 0.10 0.27 0.42 0.22
0.40 0.40 1 0.30 0.15 0.23
0.10 0.30 0.22 1 0.93 0.93
0.30 0.50 0.02 0.93 1 0.87
0.50 0.10 0.28 0.93 0.87 1
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Table 5 The similarity index between the workers and the types of accidents C1 C2 C3
a
b
c
0.00 0.48 0.79
0.70 0.10 0.21
0.30 0.72 0.65
While using crossed tables, we obtain the following vectors: • gðaÞ ¼ ð16; 14; 14Þ. • gðbÞ ¼ ð10; 18; 11Þ. • gðcÞ ¼ ð15; 17; 18Þ. After having applied the method and using the maximin rule, we obtain the Table 5: We consider a maximin method with a cutting threshold of k ¼ 0; 5. Then, a 2 C t () MI t ða; C t Þ ¼ max min fSI h ða; bhp Þg and MI t ða; C t Þ P 0; 5: h2f1;...;Kg
p2f1;...;Lhg
Consequently, the workers a, b and c are affected to the categories C 3 ; C 1 , and C 2 , respectively. So these workers can be oriented towards an appropriate preventive program.
6. Conclusion The TRINOMFC method is, according to us, very simple because it is based only on the similarity index and does not refer to discordance or to the veto threshold. We believe that, with the nominal sorting, the notion of preference and of concepts which were introduced to model it are not very appropriate; the criteria are not being maximized or minimized. It is more a question of similarity than of preference. Since our method is very simple, it is also easy to understand for a potential user. Finally, we hope that the nominal sorting problematic will get more interest from researchers and practitioners. There are many situations that concern this problematic, so that it would be justified to give more attention to classification procedures.
Appendix A The criterion function no. 1, the true-criterion 1 if gj ðaÞ gj ðbhp Þ ¼ 0; h h SIj ða; bp Þ ¼ 0 if gj ðaÞ gj ðbhp Þ 6¼ 0:
The criterion function no. 2, the quasi-criterion 1 if jgj ðaÞ gj ðbhp Þj 6 STjh ðbhp Þ; SIjh ða; bhp Þ ¼ 0 if jgj ðaÞ gj ðbhp Þj > STjh ðbhp Þ:
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The criterion function no. 3, the pre-criterion 1 jgj ðaÞ gj ðbhp Þj=DTjh ðbhp ÞÞ if jgj ðaÞ gj ðbhp Þj 6 DTjh ðbhp Þ; h h SIj ða; bp Þ ¼ 0 if jgj ðaÞ gj ðbhp Þj > DTjh ðbhp Þ:
The criterion function 8 <1 SIjh ða; bhp Þ ¼ 1=2 : 0
no. 4, the pseudo-criterion if jgj ðaÞ gj ðbhp Þj 6 STjh ðbhp Þ; if STjh ðbhp Þ < jgj ðaÞ gj ðbhp Þj 6 DTjh ðbhp Þ; if jgj ðaÞ gj ðbhp Þj > DTjh ðbhp Þ:
The criterion function no. 5, an other pseudo-criterion 8 1 if jgj ðaÞ gj ðbhp Þj 6 STjh ðbhp Þ; > > < DT h ðbh Þ jg ðaÞ g ðbh Þj j j p p j SIjh ða; bhp Þ ¼ if STjh ðbhp Þ < jgj ðaÞ gj ðbhp Þj 6 DTjh ðbhp Þ; DTjh ðbhp Þ STjh ðbhp Þ > > :0 if jg ðaÞ g ðbh Þj > DT h ðbh Þ: j
j
p
j
p
References Bana e Costa, C., 1992. Structuration, Construction et Exploitation d’un Modele d’Aide a la Decision, These de doctorat, Universidade Tecnica de Lisboa. Belacel, N., 1998. La methode PROAFTN d’affectation multicritere: Fondement et application dans le domaine d’aide au diagnostic medical Serie: Mathematique de la gestion, Universite Libre de Bruxelles. Belacel, N., 2000. Multicriteria assignment method PROAFTH: Methodology and medical application. European Journal of Operational Research 125 (1), 176–184. Belacel, N., Vincke, Ph., Boulassel, M.R., 1999. Application of PROAFTN method to assist astrocytic tumors diagnosis using the parameters generated by computer-assisted microscope analysis of cell image. Innovative and Technology in Biology and Medicine 20 (4), 239–244.
364
J. Leger, J.-M. Martel / European Journal of Operational Research 138 (2002) 349–364
Bergeron, M., Martel, J.-M., Twarabimenye, P., 1996. The evaluation of corporate loan applications based on the MCDA. Journal of Euro-Asian Management 2 (2), 16–46. Bloch, M., 1984. Une aide a l’orientation des eleves: Les test du miroir, Memoire de DEA Methodes Scientifiques de Gestion, Universite de Paris Dauphine. Chandon, J.L., Pinson, S., 1981. In: Analyse typologique: Theorie et applications. Masson, Paris, p. 254. Dekempener, D., Gervasi, F., 1985. Deux cas pratiques d’aide a la decision Memoire de fin d’etudes d’ingeniorat commercial Universite de Mons. Fabre, G., 1985. Application d’une methode multicritere pour la classification de dossiers de credit, Memoire de DEA Methodes Scientifiques de Gestion, Universite de Paris Dauphine. Henriet, L., 1995. Probleme d’affectation et methodes de classification, Memoire de DEA, Universite de Paris Dauphine. Massaglia, M., Ostanello, 1991. N-TOMIC: A support system for multicriteria segmentation problems. In: Korhonen, P. (Ed.), International Workshop on Multicriteria Criteria Decision Support, Helsinki 7-11, 1989. Lectures Notes in Economics and Mathematical Systems, Vol. 356. Spinger, Berlin, pp. 167–174. Moscarola, J., 1977. Aide a la decision en presence de criteres multiples fondee sur une procedure trichotomique: Methodologie et application, These de 3eme cycle Sciences des Organisations, Universite de Paris Dauphine. Perny, P., 1998. Multicriteria filtering methods based on concordance and non-discordance principles. Annals of Operations Research 80, 137–165. Perny, P., Roy, B., 1992. The use of fuzzy outranking relations in preference modelling. Fuzzy Sets and Systems 49, 33–53. Roy, B., 1985. Methodologie Multicritere d’Aide a la decision. Economica, Paris. Roy, B., Vincke, Ph., 1984. Relational systems of preference with one or more pseudo-criteria: Some new concepts and results. Management Sciences 30 (11), 1323–1335. Smadja, A., 1998. Segmenter ses Marches. Presses Polytechniques et Universitaires Romandes, Lausanne. Vansnick, J.-C., 1989. Application of multicriteria decision-aid to allocating budget for building repairs and maintenance. In: Tabucanon, M.T., Chankong, V. (Eds.), Proceedings of the International Conference on Multiple Criteria Decision Making: Applications in Industry and Service. Asian Institute of Technology, Bangkok, 6–8 December, pp. 629–642. Yu, W., 1992. Aide multicritere a la decision dans le cadre de la problematique du tri, These de doctorat, Universite Paris Dauphine. Zopounidis, C., Doumpos, M., 1998. Developing a multicriteria decision support system for financial classification problems: The finclos system. Optimization Methods and Software 8, 277–304.