Inconsistent preferences in verbal decision analysis

Transcription

1 Chapter 43 Inconsistent preferences in verbal decision analysis Alexey B. Petrovsky Institute for System Analysis Russian Academy of Sciences,Russia Abstract There are many practical MCDA problems where objects are described with inconsistent and contradictory attributes, which convolution is either impossible or mathematically incorrect. Another source of inconsistencies and contradictions in MCDA area is different preferences of several decision makers. For instance, such problems arise when objects are estimated differently by several experts upon multiple qualitative criteria. In this paper, new methods for ordering and classifying multi-attribute objects by contradictory preferences of several decision makers are suggested. These methods are based on the theory of multiset metric spaces. The proposed techniques are applied to case studies: ranking companies and a competitive selection of projects, which have inconsistent multi-attribute descriptions. Keywords group decision analysis, inconsistent individual preferences, multisets, multiset metric spaces 1. Introduction There are many applied problems in various areas where the objects under analysis are to be sorted or ordered while taking into account their properties. Usually objects are characterized with many diverse features (attributes) that may be quantitative and qualitative. Verbal 773

2 Creativity and innovation in decision making and decision support 2 decision analysis (VDA) deals with verbal (nominative, ordinal) data on all stages of the problem analysis and solution without transformation into a numerical form. The most important features of VDA are as follows: the problem description with a professional language, which is natural and habitual for a decision maker (DM); an examination of information consistency; a logical and psychological basis of decision rules; an explanation of intermediate and final results [Larichev and Moshkovich, 1997]. In the case of individual rational choice, when decision rules are based on judgements of only a single decision maker, the consistency of DM subjective preferences is postulated as preferences transitivity. So, special facilities for discovering and removing possible inconsistencies and contradictions within single DM judgements are included in VDA methods. A situation, where decision rules are based on judgements of several independent decision makers, is principally different due to variety and inconsistency of DM subjective preferences. As a result, individual decision rules may be similar, diverse, or contradictory. For example, this case arises when the considered objects are estimated differently by several experts upon multiple criteria. Such kinds of peculiarities would not be agreed or excluded but have to be included into VDA procedures. Various classifying and ranking techniques have been developed in MCDA area (see, for example, [Roy, 1996], [Larichev and Moshkovich, 1997], [Pawlak and Slowinsky, 1994], [Zadeh, 1965]). But usually they don t take into account contradictions and inconsistencies of object descriptions and DM preferences. Multiplicity and redundancy of attributes, which describe the decision problem, lead to an additional hardness of solving such problem because a lot of data is to be processed simultaneously without nonnumerical transformations (such as data averaging, mixing, weighting, and so on). A convenient mathematical model for representing multi-attribute objects is a multiset or set with repeating elements [Knuth, 1969], [Yager, 1986], [Petrovsky, 1994, 2001, 2003]. In this paper, methods for ordering and classifying objects that are described with inconsistent and contradictory verbal attributes are considered. These methods are based on the theory of multiset metric spaces. Objects are ordered by their closeness with regard to any ideal object in a 774

3 Inconsistent preferences in verbal decision analysis multiset metric space. An objects classification is built with the general selection rule that approximates diverse (and may be contradictory) individual sorting rules of several experts. The suggested techniques are applied to practical problems that are ranking companies and a competitive selection of projects, which are estimated by several experts upon many qualitative criteria. 2. Multiset model for a multi-attribute object representation Plurality and redundancy of data, which are characterized objects, alternatives, situations, and their properties, are peculiar for many applied tasks of classification and ordering. Let us consider two practical examples (ranking companies, a project selection) where objects (companies, projects) are described by many qualitative attributes, and different attributes may be repeated within the object description. Let A ={A 1,...,A n } be a collection of n objects evaluated by k experts upon m qualitative criteria Q 1,Q 2,...,Q m. A criteria list depends on the aim of analysis. Each criterion has a nominative or ordinal scale of verbal estimates Q s ={q s es }, e s =1,,h s, s=1,,m. Ordinal estimates are ordered from the best to the worst as q s 1 q s 2 q s hs. For example, in the case of ranking companies, a company activity in any market sector may be evaluated as follows: Q 1. Business activity ; Q 2. Total profit ; Q 3. Gross sales ; Q 4. Net sales ; Q 5. Number of projects ; Q 6. Number of employees ; Q 7. Professional skills of the staff, and so on [Who, 2000]. Different criteria may have a different relative importance (weight) for various cases. Criterion estimates q s es may be quantitative or qualitative. For a convenience and simplicity of company s evaluation and comparison, sometimes it is useful to transform a quantitative scale into a qualitative one with a small number of grades. For instance, scales of criteria Q 1 Q 7 may be the following: q s 1 very large; q s 2 large; q s 3 medium; q s 4 small. While several experts estimate each company A i, some of attributes q s es can be repeated several times. Obviously, estimations of different experts may be diverse and even contradictory. Thus, a company is 775

4 Creativity and innovation in decision making and decision support 2 presented as a multi-attribute object with repeated qualitative attributes. And there exist two ideal objects (may be virtual) the best one and the worst one that has correspondingly the highest and the lowest estimates by all criteria. Need to order objects from the best to the worst. Consider another practical case of competitive project selection, for instance, for a scientific program. The questionnaire for the project estimation by an expert includes the following criteria: Q 1. The project contribution to the program goals ; Q 2. Long-range value of the project ; Q 3. Novelty of the approach to solve the task ; Q 4. Qualification level of the team ; Q 5. Resources available for the project realization ; Q 6. Peculiarity of the project results [Larichev et al, 1989]. Each criterion has a nominative or ordered scale of verbal estimates. The scale of the criterion Q 4. Qualification level of the team looks like this: 1 q 4 the team is one of the best by an experience and qualification level; q 2 4 the team has an experience and qualification level sufficient for the project realization; 3 q 4 the team has an experience and qualification level insufficient for the project realization; q 4 4 an experience and qualification level of the team are unknown. Several experts evaluate each project with all criteria Q 1,...,Q m and make one of the following recommendations for sorting the project: r 1 to approve the project; r 2 to reject the project; r 3 to consider the project later after improving. Obviously, different experts can evaluate one and the same project differently. The individual expert recommendations for the project approval may also coincide or not. The inconsistencies of individual sorting rules may be caused, for instance, by errors in the expert classification of objects, the incoherence between experts estimates of objects and decision classes, the intransitivity of expert judgements, and by other reasons. Need to find a simple general classification rule 776

5 Inconsistent preferences in verbal decision analysis that approximates a large family of expert sorting rules in the best way and assign objects the given classes with the admitted accuracy. There are various ways of how to find decision rules for ordering or classification of multi-attribute objects [Roy, 1996], [Larichev and Moshkovich, 1997], [Pawlak and Slowinsky, 1994]. But in our cases we have the principal bottleneck that is caused by a necessity to take into account not coincided assessments of all individual experts. There exist several copies of one and the same object (company, project) with qualitative multiple criteria estimations given by several experts. So, it is necessary to aggregate all of these individual expert preferences (may be contradictory) into a single (and simple) group preference. In the cases above, a description of object A i, i=1,,n consists of several groups of attributes (criteria) Q 1,...,Q m, and some of attribute values may occur more than once because several experts evaluate every object. Usually a multi-attribute object A i is represented as a vector or cortege in the Cartesian space Q=Q 1 Q m, where Q s ={q s es } is a continuos or discreet scale of s-th attribute. Our cases are more complicated because one and the same object A i is represented now as a group of k vectors {q i (1),,q i (k) } with q i (j) =(q i1 e1(j),,q im em(j) ), j=1,,k. And this vector group is to be considered as a whole in spite of a possible incomparability of separate vectors q i (j). A collection of such complex multi-attribute objects has an overcomplicated structure that is very hard for an analysis. So, instead of space Q=Q 1 Q m, let us now define a consecutive attribute scale a set G={Q 1,...,Q m } that consists of m attribute groups Q s ={q s es }, and represent an object A i as the following set of repeating attributes: A i = {k Ai (q 1 1 ) q 1 1,,k Ai (q 1 h1 ) q 1 h1,, k Ai (q m 1 ) q m 1,,k Ai (q m hm ) q m hm }. (1) Here k A (q s es ) is a number of attribute q s es, which is equal to a number of experts evaluated the object A i with the criteria estimate 777

6 Creativity and innovation in decision making and decision support 2 q es s Q s ; the sign denotes that there are k Ai (q es es s ) copies of attribute q s in the description of object A i. An object A i may be rewritten also in the other form as A i ={k Ai (x 1 ) x 1,,k Ai (x h ) x h } (2) if elements of the set G={x 1,...,x h } are defined as follows: x 1 =q 1 1,x 2 =q 2 1,, x h =q h1 1 1, x h 1 1 =q 1 2,, xh 1 h =q h2 2 2,, x h1... h m 11 =q 1 m,, xh... h =q hm 1 m m, where h=h h m. In the cases, an object (company, project) A i is presented as a set of repeating elements (criteria estimates) x j or as a multiset over the domain G={Q 1,,Q m }. Multi-attribute objects are considered now as points of multiset metric space (A, d). The theoretical model of multisets is very appropriated to represent and analyze a collection of objects that is described with many qualitative attributes and may exist in several copies with various values of attributes. 3. Ordering multi-attribute objects by closeness to ideal object Represent a company A i described by many repeated qualitative attributes as a multiset (1) over the domain G={Q 1,,Q m }. So, the problem of ordering multi-attribute objects is reduced to ordering multisets. The simplest way of comparing and ordering multisets is a multiset arrangement by an inclusion. In this case, an i-th object A i is better than a j-th object A j (A i A j ), if the inclusion A i A j is fulfilled for multisets. The last statement is equal to the demand k Ai (q es s )k Aj (q es s ) for all q es s G. But this situation is very rare practically. Let us consider now multi-attribute objects as points of multiset metric space (A, d), for instance, with the Hamming-type distance between points d 11 (A,B)=m(AB). In our case, this distance may be presented as m hs es es d 11 (A,B) = m(ab) = s 1ws e 1 k A( qs ) kb ( qs ), (3) s 778

7 Inconsistent preferences in verbal decision analysis where w s 0 is a coefficient of a relative importance of criterion Q s. If all criteria Q s are equally important, then all w s =1. The evaluation of criteria importance is not discussed in this paper. The multisets A max = {k q 1 1,0,,0, k q 2 1,0,,0,, k q m 1,0,,0}, (4) A min = {0,,0,k q 1 h1, 0,,0,k q 2 h2,, 0,,0,k q m hm } (5) correspond to the best A max and the worst A min objects (possibly, not-existent), k is a number of experts. These objects are called also the ideal and anti-ideal solutions or referent points. We shall arrange objects with respect to their closeness to the best object A max. An object A i is said to be better then an object A j (A i A j ), if a multiset A i is more close to the multiset A max, that is if d 11 (A max, A i ) d 11 (A max, A j ). (6) Let us order all objects by values of their distances from the best object A max. If d 11 (A max,a i )=d 11 (A max,a j ) for certain objects A i and A j, then these objects will be equivalent or incomparable. So the obtained ranking will be nonstrict. While each object A i is estimated by all of k experts by all of m criteria, it s easy to show that the distance (3) between multisets A max and A i may be rewritten in the following form: m hs es es d 11 (A max, A i ) = s 1ws k A ( qs ) k Ai ( qs ) = 2 m s 1 ws [k k Ai (q s 1 )]. e s 1 max Now, the condition (6) for objects comparisons looks as follows: an object A i is better then an object A j (A i A j ), if 779

8 Creativity and innovation in decision making and decision support 2 m s 1ws k Ai (q 1 s ) s 1ws k Aj (q 1 s ). m Thus, the rule of ordering multi-attribute objects by the values of their distances is replaced by the rule of ordering with respect to the values of weighted sum S 1 Ai = s w s k Ai (q 1 s ) of the first (best) objects estimates by all criteria Q s. The greater a sum S 1 Ai, the better an object A i. For some objects A ir, r=1, t, the equalities d 11 (A max,a i1 )= =d 11 (A max,a it ) may be fulfilled instead of inequality (6). In this case, there is a partial ordering where objects A i1,,a it occupy one and the same place. In order to arrange these equivalent or incomparable objects from the best to the worst inside the group, let us find the values of weighted sum S 2 Air = s w s k Air (q 2 s ) of the second objects estimates by all criteria Q s. So, we say that an object A iu is better then an object A iv, if m s 1ws k Aiu (q 2 m s ) s 1ws k Aiv (q 2 s ). If sums S Air 2 will be equal for some objects A irp, then we shall order these objects from the best to the worst inside the subgroup by the values of weighted sum S Airp 3 = s w s k Airp (q s 3 ) of the third objects estimates by all criteria Q s. This procedure is repeated until all objects of a collection A={A 1,...,A n } will be ordered wholly. A procedure of ordering multi-attribute objects with respect to their closeness to the worst object A min may be constructed in the same way. An object A i is said to be better then an object A j (A i A j ), if a a multiset A i is more far from the multiset A min, that is if d 11 (A min,a i )d 11 (A min,a j ). As above, the objects A i and A j will be equivalent or incomparable, if d 11 (A min,a i )=d 11 (A min,a j ). Note that ordering objects with respect to the best and to the worst ones may be diverse. 780

9 Inconsistent preferences in verbal decision analysis 4. Classification of multi-attribute objects based on contradictory rules Let us present project A i with a multiple criteria estimates as multiset of type (1): A i = {k Ai (q 1 1 ) q 1 1,,k Ai (q h1 1 ) q h1 1,, k Ai (q 1 m ) q 1 m,,k Ai (q hm m ) q hm m, k Ai (r 1 ) r 1,,k Ai (r f ) r f }. (7) drawn from the domain G={Q 1,...,Q m,r} where k Ai (q es s ) and k Ai (r t ) are equal to numbers of independent experts who has estimated the object A i with the attribute q es s and has given the recommendation r t. Every group of criteria estimates Q s ={q es s }, s=1,,m; e s =1,,h s is the family of object properties. The group of attributes R={r t }, t=1,,f is the set of expert recommendations for assigning each object the specific class X t. We may consider also the representation (7) of object A i as an individual expert sorting rule: IF conditions, THEN decision. (8) The arguments in the formula (7) are associated with the sorting rule (8) as follows. The antecedent term conditions includes the various combinations of criteria estimates q es s. The consequent term decision denotes the object A i assignment the class X t, if any condition is fulfilled. For instance, the object A i is said to belong to the class X t if k Ai (r t )k Ai (r p ) for all pt, or if k Ai (r t ) pt k Ai (r p ). In order to simplify the problem, assume that beforehand the collection of objects A ={A 1,...,A n } is sorted out only by two classes X a and X b, and these classes are formed as sums of multisets. In this case, the classes X a and X b are represented as multisets X t = {k Xt (q 1 1 ) q 1 1,,k Xt (q h1 1 ) q h1 1,,k Xt (q 1 m ) q 1 m,, k Xt (q hm m ) q hm m, k Xt (r a ) r a, k Xt (r b ) r b }, and may be aggregated as a sum of the following multisets (t=a,b) m X t = s 1 X ts + X tr, X ts = i I A t is, X tr = i I A t ir. Here the index subsets I a I b ={1,,n}, I a I b =. The demand to sort objects out of two classes is not the principle restriction. Whenever objects are to be classified by more than two classes, it is possible to divide the collection A into two groups, then into subgroups, and so on. For instance, the competitive projects can be classified as the 781

10 Creativity and innovation in decision making and decision support 2 projects approved and not approved, then the not approved projects can be divided into the projects rejected and considered later, and so on. Let us consider now the problem of object selection as the problem of multiset classification in a metric space. The main idea of approximating a large family of contradictory sorting rules with a compact decision algorithm or a simple decision rule can be formulated as follows. For every group of attributes Q 1,...,Q m,r, the pairs of new multisets would be generated such that the multisets within each pair are the mostly coincident with the initial sorting objects out of the classes X a and X b. Combinations of the attributes, that define the pairs of the new generated multisets, produce the generalized decision rule for objects classification. Let us form new multisets over the same domain G, which consists of categorical multisets R a = {k Xa (r a ) r a, k Xb (r a ) r a }, R b = {k Xa (r b ) r b, k Xb (r b ) r b }, and substantial multisets Q as = j J Q as j, Q bs = j J Q bs j, Q j ={k Xa (q j s ) q j s, k Xb (q j s ) q j s }. Here the index subsets J as J bs ={1,,h s }, J as J bs =. Obviously, the binary decomposition of the categorical multisets R a, R b is identical to the possibly best object partition X ar, X br. So, in a metric space of multisets (A,d), the distance d(r a, R b )=d* is the biggest of all possible distances with regard to all attribute groups for the given family of expert judgements. In the case of the ideal classification without inconsistencies of individual sorting rules, the maximal distance in different metric spaces is equal correspondingly to d * 11 =kn, d * 21 =1/h, d * 31 =1. Thus, in every s-th attribute group Q s, need to find the best binary decomposition of new substantial multisets such that these submultisets are placed the mostly far in a multiset metric space (A,d). Now the problem of multi-attribute objects classification is transformed into the following m optimization problems: d(q as, Q bs ) maxd(q as, Q bs ) = d(q as *, Q bs *). (9) The solution of every problem (9) is a set of attributes within s-th group, which combination forms the generalized decision rule for the objects categorization: 782

11 Inconsistent preferences in verbal decision analysis IF (q iu Q tu *) AND (q iv Q tv *) AND AND (q iw Q tw *), THEN Object A i X t. Attribute q is Q ts *, that provides the demanded level of approximation rate L s L 0, L s =d(q as *, Q bs *)d*, is to be included in the final classification rule. Remark that the decision rules for classifying objects into diverse categories X a and X b, generally speaking, are quite different. 5. Conclusion In this paper, we have suggested the tools for ordering and classifying a collection of objects described by many qualitative attributes, when several copies of object may exist. These tools are based on the theory of multiset metric spaces. The multiset approach allows us to discover, present and utilize the available information contained in the object descriptions, to interpret the obtained results and their peculiarities, especially for inconsistent objects properties and contradictory DM preferences. The proposed techniques were applied to analyze the real-life cases. About 50 Russian companies in the area of information and telecommunication technologies were ranked upon multiple qualitative criteria by using the ordering method [Who, 2000]. 30 companies were selected as leading high-tech companies, 10 companies as leading developers of software, and 10 as the mostly dynamic developing companies. The method of contradictory classification was tested on the base of expert conclusions related to the State Scientific and Technological Program on High-temperature-Superconductivity. More than 250 projects had been considered by experts and about 170 projects had been approved. The following generalized decision rules for selecting projects had been found [Petrovsky, 2001]: The team must be one of the best or have the experience and qualification level sufficient for the project realization (the estimates q 4 1 or q 4 2 ; the approximation rate L s 0,65); The project is to be very important or important for achievement of the major program goals, the team must be one of the best or have the experience, qualification level, and resources sufficient for the 783

12 Creativity and innovation in decision making and decision support 2 project realization (the estimates q 1 1 or q 1 2, and q 4 1 or q 4 2, and q 5 1 or q 5 2 ; the approximation rate L s 0,55). The last decision rule completely coincides with the heuristic rule mentioned in [Larichev et al, 1989]. 6. Appendix. Basis concepts of multiset theory A multiset (also called a bag) is a known notion that is used in combinatorial mathematics and other fields. Let us review briefly a theory of multisets and metric spaces of multisets [Petrovsky, 2003]. A multiset A drawn from a crisp (ordinary) set G={x 1,x 2,...,x j,...} with different elements is defined as the following collection of elements groups A={k A (x 1 ) x 1,,k A (x j ) x j, }={k A (x) x xg, k A Z + }. Here k A : GZ + ={0,1,2, } is called a counting function of multiset, which defines the number of times that the element x i G occurs in the multiset A, and this is indicated with the symbol. A multiset becomes a crisp set when k A ()= A (), where A ()=1, if x, and A ()=0, if x. If all multisets of family A={A 1,A 2, } are composed from the elements of set G then G is said to be a generic domain for a family A. A crisp set SuppA={x xg, SuppA ()= ()} is named a support set of the multiset A. The multiset cardinality = x k A () is defined as a total number of all copies of its elements, and the multiset dimensionality //= x A ()= SuppA is defined as a total number of different elements. The maximal value of the counting function hgta=max xg k A () is called a height of the multiset A, and an element x A* =argmax xg k A () is called a peak of multiset A. The multiset is named the empty multiset, if k (x)=0, and the maximum multiset Z, if k Z ()=max AA k A (), xg. We shall call the multiset [h] constant if k [h] (x)=h=const, x [h]. So, the empty multiset is the constant multiset [0] of the height 0. And any crisp set, including the domain G, and support set, is the constant multiset [1] of the height 1. Consider the rules for comparing multisets. Multisets and are said to be equal (=) if k A ()=k B (), xg. Multisets and are called equicardinal if = ; are called equidimensional if /A/=/B/; 784

13 Inconsistent preferences in verbal decision analysis and are called equivalued if multisets are equicardinal and equidimensional. The equal multisets are equivalued. The converse is invalid in general. We say that multiset is contained or included in a multiset () if k ()k A (), xg. Then the multiset is called a submultiset of the multiset, and the multiset is called an overmultiset of the. Multisets and are said to be homogeneously equivalent or S- equivalent ( ), if their support sets coincide (SuppA=SuppB), and there exists a one-to-one correspondence f between the multiset components of the same name: k ()=f(k A ()), xg. Multisets and are said to be heterogeneously equivalent or D-equivalent (), if their support sets are equivalent (SuppA~SuppB) or equal, and there exists a one-to-one correspondence f between the multiset components of different names: k ( i )=f(k A ( j )), x i,x j G. Here f is an integervalued function with the range Z +. The S- and D-equivalent multisets are equidimensional //=//. One of two S-equivalent multisets is always a submultiset of the other, but this statement is invalid for D- equivalent multisets. D-equivalent multisets can be transformed into S- equivalent multisets by renaming the elements i j in one of the multisets. Special cases of S-equivalency are equal multisets; shifted multisets, for which k ()=k A ()+s; and stretched or proportional multisets, for which k ()=qk A (), s0, q1 are integers. Any constant multiset [h] is a multiset that is shifted by h1 units, or stretched by h times with respect to its support set Supp [h]. An important special case of D-equivalency is equicomposed multisets with equal heteronymous components k A ( i )=k B ( j ), x i,x j G. Equal multisets are equicomposed, whereas the converse is invalid. Let us define operations with multisets: the union AB = {k AB (x) x k AB (x)=max(k A (x), k B (x))}; the intersection AB = {k AB (x) x k AB (x)=min(k A (x), k B (x))}; the arithmetic addition A+B = {k A+B (x) x k A+B (x)=k A (x)+k B (x)}; the arithmetic subtraction AB = {k AB (x) x k AB (x)=k A (x)k AB (x)}; 785

14 Creativity and innovation in decision making and decision support 2 the symmetric difference AB = {k AB (x) x k AB (x)= k A (x)k B (x) }; the complement A = Z A = { k A ( x ) x ( x ) =k Z (x) k A (x)}; k A the multiplication by a scalar (reproduction) h A = {k h A (x) x k h A (x)=hk A (x), hn}; the arithmetic multiplication = {k A (x) x k A (x) = k A (x)k B (x)}; the arithmetic power = {k A n k (x) x k (x) A A = (k A (x)) }; the direct product AB = {k AB x i, x j k AB =k A (x i )k B (x j ), x i A, x j B}; the direct power (A) n = { k n x ( A) 1,,x k n = k ( A) A (x 1 ) k A (x n ), x i A}, where x 1,,x n is a cortege of n elements. Note that the multiplication of multiset by a scalar may also be represented as the sum of h multisets A: h A=A+ +A, or as the product of a constant multiset [h] and a multiset A: h A= [h] A. The support sets of operations with multisets are defined as follows: Supp(AB) = Supp(A+B) = (SuppA)(SuppB); Supp(AB) = Supp ( ) = (SuppA)(SuppB); Supp(AB) = (Supp(AB))(Supp(B)); Supp(h A) = Supp( ) = SuppA; Supp(AB) = (SuppA)(SuppB); Supp() = (SuppA). The left equalities in the first line generalize analogue equalities for the support sets of two multisets [Knuth, 1969], [Yager, 1986], and two fuzzy sets [Zadeh, 1965]. A kind of duality of operations is characteristic for multisets. Multiset union and arithmetic addition, intersection and arithmetic multiplication, multiplication by a scalar and raising to an arithmetic power form pairs of similar operations with respect to mapping Supp:Z + {0,1}, which defines passage from multisets to sets. And there is another kind of duality of multiset union and intersection, 786

15 Inconsistent preferences in verbal decision analysis arithmetic addition and arithmetic subtraction under the operation of multiset complement: A B = A B, A B = A B ; A B = A B = B, A B = A +B, A B = B. In the case of multisets, these equalities generalize the de Morgan laws for sets. Note that some properties of operations, which exist for sets, are absent for multisets. And at the same time new properties of operations appear that have no analogues for sets arise, for instance, A+ A =Z, Z A =A; A A Z; A A ; A A A A. In general, the operations of arithmetic addition, multiplication by a scalar, arithmetic multiplication, and raising to arithmetic powers are not defined in the theory of sets. Analogues of these operations may be operations with vectors a+b=(a 1 +b 1,,a n +b n ), ha=(ha 1,,ha n ), and matrixes A+= a ij +b ij mn, ha= ha ij mn, A= a ij b ij mn. The last operation is different from a traditional matrix multiplication. The operation of multiset selection suggested in [Yager, 1986] is a special case of multiset arithmetic multiplication, where one of the factors is an ordinary set. When multisets are reduced to sets, the operations of arithmetic multiplication and raising to arithmetic powers degenerate into a set intersection, but the operations of set arithmetic addition and set multiplication by a scalar will be impracticable. A family of multisets closed under operations of union, intersection, addition and complement is said to be an algebra L(Z) of multisets with the maximal multiset Z as the unit, and the empty multiset as the zero of the algebra. A non-negative real-valued function m(a) is called a measure of multiset A if it is defined on the algebra L(Z) and has a properties of strong additivity m( i A i )= i m(a i ); weak additivity m( i A i )= i m(a i ) for A i A j =; weak monotonicity m(a)m(b)ab; continuity lim m(a i)=m( lim A i); i i symmetry m(a)+m( A )=m(z); elasticity m(h A)=hm(A). Different metric spaces (A,d) may be determined for the same collection of objects by introducing the following types of distances between multisets: d 1p (A,B) = [m(ab)] 1/p ; d 2p (A,B) = [m(ab)m(z)] 1/p ; d 3p (A,B) = [m(ab)m(ab)] 1/p, 787

16 Creativity and innovation in decision making and decision support 2 where p0 is an integer. The distances d 2p (A,B) and d 3p (A,B) satisfy the normalization condition 0d(A,B)1. Note, that due to the continuity of the multiset measure, the distance d 3p (A,B) is undefined for A=B=. So d 3p (,)=0 by the definition. The measure of multiset A may be determined in the various ways, for instance, as a linear combination of counting functions: m(a)= i w i k A (x i ), w i >0. In this case, the distances are as follows: d 1 (A,B) = wi k xig d 2p (A,B) = w' i k xig A A ( xi ) kb( xi ) ( xi ) kb( xi ) 1/ p 1/ p ;, w' i =1/ 1/ p h w j j1 wi k A( xi ) kb ( xi ) xig d 3p (A,B) =. wimax[ k A( xi ),kb ( xi )] xig The distance d 1 (A,B) is a Hamming-type distance between objects, which is traditional for many applications. The distance d 2 (A,B) characterizes a difference between two objects related to common properties of all objects as a whole. And the distance d 3 (A,B) reflects a difference related to properties of only both objects. For any fixed p, the metrics d 1 and d 2 are the continuous and uniformly continuous functions, the metric d 3 is the piecewise continuous function almost everywhere on the corresponding metric space. In the case of sets, d 11 (A,B)=m(AB) is called the Frechet distance, and d 31 (A,B)=m(AB)/m(AB) is called the Steinhaus distance. The following indexes of multiset similarities are connected with the distances: s 1 (A,B) = 1m(AB)m(Z); s 2 (A,B) = m(ab)m(z); s 3 (A,B) = m(ab)m(ab). The functions s 1, s 2, s 3 generalize for multisets the known nonmetric indexes of object similarity such as the simple matching coefficient, Russel and Rao measure of similarity, Jaccard coefficient or Tanimoto measure. Various properties of multisets and multiset metric spaces are considered and discussed in (Petrovsky, 2003). ; 788

17 Inconsistent preferences in verbal decision analysis 7. References Knuth D.E. The art of computer programming. Vol.2. Seminumerical algorithms. Addison-Wesley, Reading, MA, Larichev O.I., Moshkovich E.M. Verbal decision analysis for unstructured problems. Kluwer Academic Publishers, Boston, Larichev O.I., Prokhorov A.S., Petrovsky A.B., Sternin M.Yu., Shepelev G.I. The experience of planning of the basic research on the competitive base. Vestnik of the USSR Academy of Sciences, no.7, 1989, p (in Russian). Pawlak Z., Slowinsky R. Rough set approach to multi-attribute decision analysis. European Journal of Operational Research, no.72, 1994, p Petrovsky A.B. An axiomatic approach to metrization of multiset space. In: Multiple Criteria Decision Making, G.H.Tzeng, H.F.Wang, U.P.Wen, P.L.Yu (eds). Springer-Verlag, New York, 1994, p Petrovsky A. Method for approximation of diverse individual sorting rules. Informatica, vol.2, no.1, 2001, p Petrovsky A.B. Spaces of sets and multisets. Editorial URSS, Moscow, 2003 (in Russian). Roy B. Multicriteria methodology for decision aiding. Kluwer Academic Publishers, Dordrecht, Who is the most intellectual in Russia? Rating leading Russian developers of hi-tech. Company, no.47(143), 2000, p (in Russian). Yager R.R. On the theory of bags. International Journal of General Systems, vol.13, 1986, p Zadeh L.A. Fuzzy sets. Information and Control, vol.8, no.3, 1965, p