IN GROUP-decision-making (GDM) analysis, the preference

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1 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART B: CYBERNETICS, VOL 0, NO 5, OCTOBER 00 An Approach to Solve Group-Decision-Making Problems With Ordinal Interval Numbers Zhi-Ping Fan and Yang Liu Abstract The ordinal interval number is a form of uncertain preference information in group decision making (GDM), while it is seldom discussed in the existing research This paper investigates how the ranking order of alternatives is determined based on preference information of ordinal interval numbers in GDM problems When ranking a large quantity of ordinal interval numbers, the efficiency and accuracy of the ranking process are critical A new approach is proposed to rank alternatives using ordinal interval numbers when every ranking ordinal in an ordinal interval number is thought to be uniformly and independently distributed in its interval First, we give the definition of possibility degree on comparing two ordinal interval numbers and the related theory analysis Then, to rank alternatives, by comparing multiple ordinal interval numbers, a collective expectation possibility degree matrix on pairwise comparisons of alternatives is built, andanoptimizationmodelbasedonthismatrixisconstructed Furthermore, an algorithm is also presented to rank alternatives by solving the model Finally, two examples are used to illustrate the use of the proposed approach Index Terms Alternative ranking, decision analysis, group decision making (GDM), ordinal interval numbers, possibility degree I INTRODUCTION IN GROUP-decision-making (GDM) analysis, the preference information on alternatives provided by each expert is often aggregated to form a collective opinion Then, ranking of the alternatives or selection of the best alternative(s) is based on the derived collective opinions [] In practical GDM problems, the preference information provided by experts can be expressed in multiple forms Commonly used preference formats include utility values [] [5], multiplicative preference relations [6], [7], fuzzy preference relations [], [7], [8], linguistic variables [], [9] [], interval numbers [], [], preference rankings [5] [8], and ordinal interval numbers [9] [] A lot of approaches have been found to solve GDM problems [], [5], [], [], [], whereas research on the GDM problem with ordinal interval numbers is seldom conducted Due to the estimation inaccuracies, lack of knowledge, and people s lim- Manuscript received January 9, 009; revised June 0, 009 and October, 009; accepted December, 009 Date of publication February 7, 00; date of current version September 5, 00 This work was supported in part by the National Science Fund for Distinguished Young Scholars of China under Proect , by the National Science Foundation for Excellent Innovation Research Group of China under Proect 70700, and by the National Natural Science Foundation of China under Proects and This paper was recommended by Associate Editor Q Shen The authors are with the Department of Management Science and Engineering, School of Business Administration, Northeastern University, Shenyang 000, China ( zpfan@mailneueducn; liuy@mailneueducn) Digital Obect Identifier 009/TSMCB ited expertise related with the problem domain, the preference rankings or ranking ordinals of alternatives provided by experts are often uncertain [9] [], [] For example, a consumer wants to buy a car among the four color cars (black, white, blue, and yellow) His/her preferences are the following: The black one is ranked top, the white one is top, the blue one is second or third, and the yellow one is bottom Such preferences are in the form of ordinal interval number Therefore, the GDM problem with ordinal interval numbers is an important research topic González-Pachón and Romero [9] earlier proposed an approach to solve the GDM problem with ordinal interval numbers In their approach, an interval goal programming model is constructed to aggregate the ordinal interval numbers provided by experts into a collective opinion The ranking ordinals of alternatives can be obtained by solving the model Furthermore, González-Pachón et al [0] proposed an approach to solve the multiple-criterion decision-making problem with both ordinal interval numbers and pairwise comparison matrix The approach is also based on an interval goal programming model Wang et al [] developed a preference aggregation approach to solve the GDM problem with preference rankings of alternatives In their study, the example provided by González-Pachón and Romero [9] is used Prior studies have significantly advanced GDM analysis with ordinal interval numbers However, maybe it is difficult to solve the GDM problem with ordinal interval numbers when the approach of González-Pachón and Romero [9] is used This is because their approach is based on an integer programming model If the number of alternatives or experts is great, then the number of constraints of the model is also great Thus, it is necessary to develop an effective algorithm if the scale of the model is great Furthermore, in the study of Wang et al [], how to process ordinal interval numbers is not involved On the other hand, we have found several approaches to compare interval numbers [], [] [7], fuzzy numbers [8] [0], and uncertain linguistic terms [] Since the ordinal interval number involved in this paper is a special and discrete interval and is different from a traditional interval number, a fuzzy number, or an uncertain linguistic term in nature, the existing approaches [], [], [] [0] could not be directly used to compare ordinal interval numbers Therefore, to solve the GDM problem with ordinal interval numbers, it is necessary to investigate the comparison of ordinal interval numbers The obective of this paper is to develop a new approach to solve the GDM problem with ordinal interval numbers Since the ordinal interval number is a special and discrete interval and is different from a traditional interval number, ordinal 08-9/$ IEEE 转载

2 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART B: CYBERNETICS, VOL 0, NO 5, OCTOBER 00 interval numbers cannot be simply added That is, it is difficult to aggregate multiple ordinal interval numbers directly Thus, to process ordinal interval numbers, we first give a definition of possibility degree on comparing two ordinal interval numbers The definition is based on the consideration that every ranking ordinal in an ordinal interval number is thought to be uniformly and independently distributed in its interval Then, to ranking alternatives, a collective expectation possibility degree matrix on pairwise comparisons of alternatives is built, and the optimization model based on this matrix is constructed Furthermore, a theorem-based analysis for solving the model is given, and the algorithm for ranking alternatives using ordinal interval numbers is developed The rest of this paper is arranged as follows Section II gives the definition of the ordinal interval number Section III presents the definition of possibility degree on comparing ordinal interval numbers and the related theory analysis In Section IV, by constructing the collective expectation possibility degree matrix and the optimization model, the algorithm for ranking alternatives using ordinal interval numbers is presented In Section V, two examples are used to illustrate the use of the proposed approach Lastly, Section VI summarizes and highlights the main features of this paper II ORDINAL INTERVAL NUMBER The following definition of the ordinal interval number is used throughout this paper Definition : Let Z + be the positive integer set An ordinal interval number is expressed in r [r L,r L,,r U, r U ], where r L,r U Z +, r L r U r L and r U are the lower and upper bounds of ordinal interval number r, respectively Particularly, if r L r U, then ordinal interval number r reduced to a ranking ordinal For simplicity of representation, ordinal interval number r [r L,r L,,r U,r U ] is expressed as r [r L,r U ] Without loss of generality, in this paper, r represents a set of possible ranking ordinals of an alternative For example, r [, ] that represents the possible ranking ordinals of an alternative may be first, second, or third Therefore, the smaller r L or r U is, the better the corresponding alternative will be In this paper, let positive integer r (r Z + ) denote the verity ranking ordinal of an alternative Usually, r exists exactly and obectively Because of the estimation inaccuracies, lack of knowledge, and experts limited expertise related with the problem domain, what is obtained about the ranking ordinal of an alternative can be an estimated ordinal interval r [r L,r U ], which covers the verity ranking ordinal r It can be regarded that any possible ranking ordinal in the interval r has the equal possibility to cover the verity ranking ordinal r Thus, it can be assumed that the estimated interval r is static, and the verity ranking ordinal of an alternative r is a random variable in the corresponding interval r Since there are (r U r L ) possible ranking ordinals in r [r L,r U ], r is thought to be uniformly distributed on the (r U r L ) ranking ordinals Therefore, the possibility of r r is equal to /(r U r L ), where r is a certain ranking ordinal, r L r r U, r Z + Based on which, the following section will Fig Ordinal interval numbers [, ] and [, ] give the definition of possibility degree on comparing ordinal interval numbers III COMPARING ORDINAL INTERVAL NUMBERS For simplicity, consider two ordinal interval numbers ã [a L,a U ] and b [b L,b U ] Let them represent the possible ranking ordinals of two alternatives S a and S b, respectively Let a and b be the verity ranking ordinals of alternatives S a and S b, respectively, where a L a a U, b L b b U, and a, b Z + Suppose that a and b are uniformly and independently distributed in ã and b, respectively Note p a<b and p a>b as the possibility degrees of a<band a>b, respectively Note p ab as the possibility degree of a b, ie, p ab is the possibility degree that alternatives S a and S b are ranked at the same position Let and pã b be the possibility degree of ã superior to b (noted as ã b) and that of ã inferior to b (noted as ã b), respectively Particularly, if a b, then we regard that the cases of a<band a>bmay occur simultaneously with the same possibility degree 05p ab Consequently, there are p a<b +05p ab and pã b p a>b +05p ab Weuse an example to illustrate this circumstance in the following Consider two ordinal interval numbers ã [, ] and b [, ] (see Fig ) Suppose that a and b are uniformly and independently distributed in ã and b, respectively Thus, instances of a, a, b, and b occur with the possibility degrees of 05 Furthermore, there are four possible cases on comparing a and b: )a and b ;)a and b ; ) a and b ; and ) a and b, which are shown in Table I By integrating the four cases in Table I, we can know that 0875 and p b ã 05 Without loss of generality, in the following analysis, we assume that a L b L holds Thus, there are three possible position relationships between ã [a L,a U ] and b [b L,b U ], which are shown in Figs The first case is when ã and b are disoint, ie, a L a U <b L b U The second one is when ã overlaps with b, ie, a L b L a U <b U The third one is when b is inside of ã, ie, a L b L b U a U For the first case (see Fig ), we always have a<b a, b because of a L a U <b L b U Thus, the possibility degree of ã b is For the second case (see Fig ), we consider three possible situations As for the first situation, if a [a L,b L ), then the possibility degree of a<bis one As for the second situation, if a [b L,a U ] and b [b L,a U ], then the possibility degree of a<b is 05 For the third situation, if a [b L,a U ] and b (a U,b U ], then the possibility degree of a<bis one Consequently, the possibility degree of a<bin the case as shown in Fig can be expressed by ( ) p a<b p a [a L,b L ) + p a [b L,a U ] 05pb [b L,a U ] + p b (a U,b U ] ()

3 FAN AND LIU: APPROACH TO SOLVE GDM PROBLEMS WITH ORDINAL INTERVAL NUMBERS 5 TABLE I FOUR POSSIBLE CASES ON COMPARING ã AND b Fig Disoint case of ã and b Fig Inclusion case of ã and b Fig Overlapping case of ã and b where p a [a L,b L ) is the possibility degree of a [a L,b L ), p a [b L,a U ] is the one of a [b L,a U ], p b [b L,a U ] is the one of b [b L,a U ], and p b (a U,b U ] is the one of b (a U,b U ] In (), since random variables a and b are thought to be uniformly and independently distributed in intervals [a L,a U ] and [b L,b U ],wehavep a [a L,b L ) (b L a L )/(a U a L ), p a [b L,a U ] (a U b L )/(a U a L ), p b [b L,a U ] (a U b L )/(b U b L ), and p b (a U,b U ] (b U a U )/(b U b L ) Therefore, by (), the possibility degree of ã b can be expressed by bl a L a U a L + ( a U b L ) a U a L ( 05 au b L b U b L + bu a U ) b U b L Similarly, for the third case (see Fig ), the possibility degree of a<bcan be expressed by () p a<b p a [a L,b L ) +05p a [b L,b U ] () where p a [a L,b L ) and p a [b L,b U ] are the possibility degrees of a [a L,b L ) and a [b L,b U ], respectively Furthermore, by (), the possibility degree of ã b can be expressed by bl a L a U a L +05 bu b L a U a L () Based on the aforementioned analysis, we give the definition of possibility degree that one ordinal interval number is superior to the other Definition : Let ã [a L,a U ] and b [b L,b U ] be two arbitrary ordinal interval numbers The possibility degree of ã superior to b (ã b) is given by (5), shown at the bottom of the next page Accordingly, the possibility degree of b superior to ã ( b ã) is given by 0, ( )( ) if a L a U <b L b U a p b ã 05 U b L a U b L, if a L b L a U <b U a U a L b U b L 05 bu b L a U a L + au b U a U a L, if al b L b U a U (6) Here, the possibility degree can also be regarded as the probability that one ordinal interval number is superior to the other Based on Definition, we give the following theory analysis Theorem : Let ã [a L,a U ] and b [b L,b U ] be two arbitrary ordinal interval numbers; for any case among those in Figs, we have + p b ã Proof: For Figs, the following three cases are considered ) If a L a U <b L b U (see Fig ), according to Definition, we have and p b ã 0; then, + p b ã ) If a L b L a U <b U (see Fig ), according to Definition, we have + p b ã bl a L ( a U a U a L + b L ) a U a L ( 05 au b L b U b L + bu a U ) b U b L ( a U b L )( a U b L ) +05 a U a L b U b L bl a L ( a U a U a L + b L ) a U a L ( a U b L b U b L + bu a U ) b U b L bl a L a U a L + au b L a U a L

4 6 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART B: CYBERNETICS, VOL 0, NO 5, OCTOBER 00 we know Fig 5 b L a L a U b U ) If a L b L b U a U (see Fig ), according to Definition, we have + p b ã bl a L a U a L +05 bu b L a U a L +05 bu b L a U a L + au b U a U a L bl a L a U a L + bu b L a U a L + au b U a U a L au a L a U a L Lemma : Let ã [a L,a U ] and b [b L,b U ] be two arbitrary ordinal interval numbers; if and only if a L + a U b L + b U, then 05 Particularly, if and only if a L + a U b L + b U, then 05 Proof: We prove the sufficient condition and the necessary condition First, we prove the sufficient condition, ie, if a L + a U b L + b U, then 05; particularly, if a L + a U b L + b U, then 05 There are four possible position relationships between ordinal interval numbers ã [a L,a U ] and b [b L,b U ] under the condition of a L + a U b L + b U : ) a L a U <b L b U (see Fig ); ) a L b L a U <b U (see Fig ); ) a L b L b U a U and a U b U b L a L (see Fig ); ) b L a L a U b U and a L b L b U a U (see Fig 5) According to Definition, we will prove that the conclusion follows for each aforementioned case ) If a L a U <b L b U (see Fig ), then we know by (5) Therefore, we also have > 05 ) If a L b L a U <b U (see Fig ), we have (b U b L a U )/(b U b L )< Furthermore, by (5), bl a L ( a U a U a L + b L ) a U a L ( 05 au b L b U b L + bu a U ) b U b L bl a L ( a U a U a L + b L ) a U a L ( b U b L a U ) (b U b L ) ( b L a L a U a L + au b L ) a U a L ( b U b L a U ) (b U b L ) bu b L a U (b U b L ) > bu b L (b U b L ) 05 ie, > 05 follows ) If a L b L b U a U (see Fig ), by (5), we know bl a L a U a L +05 bu b L a U a L bl a L + b U a U a U a L If a L + a U <b L + b U, then we have ((b L a L + b U a U )/(a U a L + )) > 05Ifa L + a U b L + b U, then we have 05+05((b L a L + b U a U )/(a U a L + )) 05 Therefore, we have 05 for a L + a U b L + b U ; particularly, 05 for a L + a U b L + b U ) If b L a L a U b U (see Fig 5), by (6), we know 05 au a L b U b L + bu a U b U b L bl + b U a L a U b U b L If a L + a U <b L + b U, then we have 05+05((b L + b U a L a U )/(b U b L + )) > 05 If a L + a U b L + b U, then we have 05+05((b L + b U a L a U )/(b U b L + )) 05 Therefore, we have 05 for a L + a U b L + b U ; particularly, 05 for a L + a U b L + b U, ( )( ) if a L a U <b L b U b L a L a U a L + a U b L 05 au b L a U a L b U b L + bu a U, if a L b L a U <b U b U b L (5) b L a L a U a L +05bU b L a U a L, if al b L b U a U

5 FAN AND LIU: APPROACH TO SOLVE GDM PROBLEMS WITH ORDINAL INTERVAL NUMBERS 7 Second, we prove the necessary condition, ie, if 05, then a L + a U b L + b U ; particularly, if 05, then a L + a U b L + b U According to Definition, there are six formulas for in different cases Therein, four cases may satisfy the condition of 05; they are the following: ) ; ) bl a L a U a L + ( 05 au b L b U b L + ( a U b L ) a U a L bu a U b U b L ) bl a L a U a L +05 bu b L a U a L ; ) 05 au a L b U b L + bu a U b U b L ) ; We will prove that the conclusion follows for each aforementioned case ) If, by Definition (see Fig ), we know a L a U <b L b U, and then, we have a L + a U <b L + b U ) If ((b L a L )/(a U a L + ))+((a U b L )/ (a U a L ))(05((a U b L )/(b U b L + ))+(b U a U )/(b U b L + )) > 05, by Definition (see Fig ), we know a L b L a U <b U or a L b L and a U <b U ; then, we have a L + a U <b L + b U ) By Definition (see Fig ), we know bl a L a U a L +05 bu b L a U a L bl a L + b U a U a U a L for a L b L b U a U If 05, wehaveb L a L + b U a U 0, ie, a L + a U b L + b U Particularly, if 05, then b L a L + b U a U 0, ie, a L + a U b L + b U ) By Definition (see Fig 5), we know 05 au a L b U b L + bu a U b U b L bl + b U a L a U b U b L for b L a L a U b U If 05, wehaveb L + b U a L a U 0, ie, a L + a U b L + b U Particularly, if 05, then b L + b U a L a U 0, ie, a L + a U b L + b U Theorem : Let ã [a L,a U ], b [b L,b U ], and c [c L,c U ] be three arbitrary ordinal interval numbers; we have the following conclusions ) If 05 and p b c 05, then pã c 05 ) If 05and p b c 05, then pã c 05 ) If 05, p b c 05, and pã c 05, then p b c 05 Proof: We will prove conclusions ), ), and ) First, we prove conclusion ) If 05 and p b c 05, according to Lemma, we know a L + a U b L + b U and b L + b U c L + c U Obviously, we have a L + a U c L + c U Furthermore, by a L + a U c L + c U, we have pã c 05 according to Lemma Second, we prove conclusion ) If 05and p b c 05, according to Lemma, we know a L + a U b L + b U and b L + b U c L + c U Obviously, we have a L + a U c L + c U Furthermore, by a L + a U c L + c U, we have pã c 05 according to Lemma Finally, we prove conclusion ) If 05, p b c 05, and pã c 05, according to Lemma, we know a L + a U b L + b U, b L + b U c L + c U, and a L + a U c L + c U Obviously, we have a L + a U b L + b U and b L + b U c L + c U Furthermore, by a L + a U b L + b U and b L + b U c L + c U, we have p b c 05 according to Lemma Theorem shows the transitivity of the comparison of ordinal interval numbers This theorem is important for rankings of ordinal interval numbers According to Theorem, we have the following corollary Corollary : If p b ã and p b c p c b, then pã c p c ã Proof: According to Theorem, we know + p b ã and p b c + p c b If p b ã and p b c p c b, we have 05 and p b c 05 According to Theorem, we know pã c 05 Furthermore, by Theorem, we know pã c + p c ã Thus,wehavepã c p c ã Let two ordinal interval numbers ã [a L,a U ] and b [b L,b U ] denote the possible ranking ordinals of two alternatives S a and S b, respectively We may give the following definition of alternative ranking based on Definition Definition : If the possibility degree of ã b is (or p b ã 0), then alternative S a is preferable to S b, denoted by S a S b (or S b S a ); if the possibility degree of ã b is 0 05 (or p b ã 05), then alternative S a is equivalent to S b, denoted by S a S b (or S a S b ); and if the possibility 05 degree of ã b is 05 < <, then the possibility degree that alternative S a is preferable to S b is, denoted by S b S a By Definition and Theorem, we have the following corollary on the transitivity of ranking orders of alternatives Corollary : Let the ordinal interval numbers ã [a L,a U ], b [b L,b U ], and c [c L,c U ] denote the possible ranking ordinals of three alternatives S a, S b, and S c, respectively If S a S b and S b S c, then S b S c, where 05 p b c pã c, 05 p b c, and 05 pã c IV PROPOSED APPROACH Let S {S,S,,S m } be a finite set of alternatives and E {E,E,,E n } be a finite set of experts Let

6 8 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART B: CYBERNETICS, VOL 0, NO 5, OCTOBER 00 w (w,w,,w n ) T be the weight vector of the experts w i denotes the important degree of expert E i, where n i w i, w i 0, i,,,n Usually, weights of experts are determined by the decision maker or manager The determination of the weights is based on the consideration of the authority and status of each expert If important degrees of experts are indifferent, then all the weights are equal, ie, w w w n /n [9] In GDM analysis, the alternatives (S,S,,S m ) need to be ranked from the best to the worst, using the preference information given by the experts In this paper, we assume that the experts express their preference information on the alternatives in the form of ordinal interval numbers Suppose that ordinal interval numbers r (i) [r (i)l,r (i)u ], r (i) [r (i)l,r (i)u ],, r m (i) [r m (i)l,r m (i)u ] are the intervals of the possible ranking ordinals of m alternatives, which are provided by expert E i, i,,,n To rank the alternatives using ordinal interval numbers, according to Definition, an m m possibility degree matrix P (i) on pairwise comparisons of all the ordinal interval numbers provided by E i can be constructed as follows: P (i) r (i) r (i) r (i) m where p (i) k r (i) r (i) r m (i) p (i) p (i) p (i) m p (i) p (i) p (i) m p (i) m p (i) m p (i) mm, denotes the possibility degree of r(i),,,m By Theorem, we know p (i) k + p(i) k i,,,n r (i) k,, k for, k {,,,m} Thus, matrix P (i) is also in the form of fuzzy preference relation (refer to [7] and [8]) There are several methods for solving GDM problems with fuzzy preference relations, such as the methods based on the ordered weighted averaging (OWA) operator [8], [], [] and the methods based on calculations of net flows [] [5] However, when the OWA operator is used, it is necessary to define the membership functions of linguistic quantifiers representing fuzzy maorities and to select carefully their parameters in advance Moreover, due to the fact that the parameters are determined subectively, the aggregation and selection processes may be not optimal [], [] On the other hand, although the methods based on calculations of net flows [] [5] have simple computation processes, they neglect the consideration of consistency In the following, we will give a new approach to determining the ranking of alternatives based on fuzzy preference relations To aggregate the preference information provided by each expert into a collective preference, we give the following definition of the collective expectation possibility degree that one alternative is superior to the other Definition : Let p (i) k be the possibility degree of r(i) r (i) k or S S k with regard to expert E i and w i be the weight of expert E i, where n i w i, w i 0, i,,,n Then, the collective expectation possibility degree of S S k, ie, p k, is given by n p k w i p (i) k,,k,,,m (7) i By Definition, the collective expectation possibility degree matrix P on pairwise comparisons of alternatives is obtained, ie, S S S m P S S S m p p p m p p p m p m p m p mm where p k is the collective expectation possibility degree of alternative S superior to alternative S k,, k,,,m Matrix P is also in the form of fuzzy preference relation, where p k + p k for any and k Definition 5: Let d be the ranking value of alternative S,,,,m, where 0 d and m d The greater d is, the better alternative S will be Then, an m m matrix H [h k ] m m on pairwise comparisons of alternatives is given by S S S m S h h h m H S h h h m S m h m h m h mm where the elements in matrix H are h k d /(d + d k ),, k,,,m Obviously, there is h k + h k for any and k Therefore, matrix H [h k ] m m is also in the form of fuzzy preference relation [8], and h k represents a preference relationship between alternatives S and S k Concretely, we have S S k for 05 <h k, S k S for 0 h k < 05, ors S k for h k 05 It can be seen that matrix H has multiplicative consistency that satisfies h k h ki h i h k h ik h i for any i,, and k [8] On the other hand, for matrix H, wehavethe following conclusion on the transitivity of the ranking order of alternatives Theorem : Let S {S a,s a,,s am } be a subset of alternative set S, S S, which is composed of m (m m) alternatives There is no recycling chain in the ranking order of alternatives derived from matrix H [h k ] m m, ie, there is no such a ranking order of the alternatives, S a S a S am S a, with at least one strict preferable relation Here, 05 h a a,h a a,,h a(m ) a m,h am a, and symbol denotes that one alternative is preferable to or equivalent to the other Proof: We use reduction to absurdity to prove Theorem Let S {S a,s a,,s am } be a subset of alternative set S, S S, which is composed of m (m m) alternatives Suppose that there exists a recycling chain S a S a S am S a with at least one strict preferable

7 FAN AND LIU: APPROACH TO SOLVE GDM PROBLEMS WITH ORDINAL INTERVAL NUMBERS 9 relation Then, by Definition 5, we know 05 h a a, 05 h a a,,05 h a(m ) a m, 05 h am a, and there is at least one strict preferable relation, where 05< h a a () ( {,,,m }) or 05<h am a By h k d /(d + d k ), we know d d k if 05 h k Since 05 h,h,,h (m )m,h m, there is d d d m d Furthermore, we have d d d m d Therefore, we have h a a h a a h a(m ) a m h am a 05 This contradicts the hypothesis of 05 <h a a () ( {,,,m }) or 05 < h am a Therefore, Theorem follows Since matrix H has multiplicative consistency and transitivity of ranking orders of alternatives, with regard to the element p k of matrix P, it is desirable to determine the ranking value d such that p k h k,,k,,,m (8) Substitute d /(d + d k ) for h k ; then, (8) can be changed into d (d + d k )p k,,k,,,m (9) It can be seen from (9) that the greater p k is, the greater d will be and the better the corresponding alternative S will be Based on (9), the deviation degree or the distance between d and (d + d k )p k is given by z k [d (d + d k )p k ],,k,,,m (0) Based on (0), the total deviation degree is given by m m z [d (d + d k )p k ] () k Apparently, z is the explicit function of d,,,,m To obtain the ranking values of alternatives, according to (), a constrained optimization model is set up as follows: m m min z [d (d + d k )p k ] () st k m d () d 0,,,,m () For the convenience of analysis, () () can be expressed in the form of matrix as follows: min z d T Qd (5) st e T d (6) d 0 (7) where d (d,d,,d m ) T, e (,,,) T, and Q [q i ] m m The elements in matrix Q are m q ii p ki, i,,,m (8) k k i q i ( p i p i ), i,,,,m; i (9) Theorem : If there exists at least one inequality d p k (d + d k ) for any and k, then matrix Q is positive definite and is also nonsingular or invertible Proof: According to (8) and (9), we know that matrix Q is symmetric, and all of its principal elements are not smaller than zero, and all nonprincipal elements are not greater than zero Since there exists at least one inequality d p k (d + d k ) for any and k, we know z>0 by () Therefore, the symmetry of matrix Q and the definition of positive definite quadratic form determine that Q is positive definite By the property of the positive definite matrix, Q is nonsingular or invertible, ie, Q exists To solve the optimization model which is composed of (5) (7), we first ignore the nonnegative constraint (7) and then set up the following Lagrangian function: L d T Qd +λ(e T d ) (0) where λ is the Lagrangian multiplier Let L/ d 0 and L/ λ 0; then, we have Qd + λe () e T d () If matrix Q is invertible, then solutions to () and () can be obtained as follows: d Q e e T Q e λ e T Q e () () where d is the derived ranking vector on the alternatives The greater d is, the better the corresponding alternative S will be In general, the ranking vector determined by () has practical meanings only if it satisfies the nonnegative constraint (7), ie, d 0 The problem is to prove that d must satisfy the nonnegative constraint (7) In the following, we give the detailed discussion Lemma : Let F [f i ] m m be an m m matrix satisfying f i < 0 for i Then, F [0] m m (ie, F is a nonnegative matrix) if and only if all principal minor determinants F are greater than zero Proof: The proof of Lemma can refer to the study in [5] Theorem 5: If there exists at least one inequality d p k (d + d k ) for any and k, then Q [0] m m, ie, Q is a nonnegative matrix Proof: If there exists at least one inequality d p k (d + d k ) for any and k, then Q is a positive definite matrix satisfying q i 0, i, and q ii > 0 Therefore, by Lemma, we know that Q [0] m m, ie, Q is a nonnegative matrix Theorem 6: Let d be the ranking vector obtained by () If there is at least one inequality d p k (d + d k ) for any and k, then we have d > 0 Proof: Theorem 5 shows that Q [0] if d p k (d + d k ) for any and k Letv i denote the ith row vector of

8 0 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART B: CYBERNETICS, VOL 0, NO 5, OCTOBER 00 Q Then, v i 0; moreover, v i e>0, ie, there is at least one nonzero element in v i Suppose that v i 0, ie, Q contains the row vector in which all elements are zero; then, Q must be a singular matrix (Q (Q ) does not exist) This contradicts the fact that Q exists with all principal minor determinants greater than zero Let u m i v ie e T Q e Since v i e>0, then u>0 Therefore, by (), we know that d Q e/e T Q e>0holds Based on the aforementioned analysis, the following algorithm is proposed to rank alternatives using ordinal interval numbers Step ) Set up the possibility degree matrices by (5) and (6), ie, P (i) [p (i) k ] m m, i,,,n Step ) Determine the collective expectation possibility degree matrix on pairwise comparisons of alternatives by (7), ie, P [p k ] m m Step ) Determine the matrix Q [q i ] m m by (8) and (9) Step ) Determine the ranking values of alternatives by (), ie, d,,,,m Step 5) Determine the ranking order of the alternatives according to the derived ranking values According to matrix P, the possibility degree that one alternative is preferable to the other is marked in the ranking order For the convenience of further analysis, several indexes of deviation between the final ranking positions of alternatives and the original ordinal interval numbers provided by experts are introduced as follows Note R as the ranking position of alternative S, R {,,,m} LetD T, DU, and DM denote the disagreement time between R and r (i) on alternative S, the total disagreement unit between R and r (i) maximum disagreement unit between R and r (i) They are respectively calculated by D T D U D M where m i m i max, and the, respectively σ (i) (5) ( n (i) { n () σ (i) n (i) v (i) ) + v (i),n (),,n (m),v () {, if R r (i) 0, if R r (i),v () { R r (i)u, if R >r (i)u 0, if R r (i)u { 0, if R r (i)l r (i)l R, if R <r (i)l },,v (m) (6) (7) TABLE II PREFERENCE INFORMATION ON ALTERNATIVES PROVIDED BY THE FIVE EXPERTS V I LLUSTRATIVE EXAMPLES In this section, two examples are used to illustrate the use of the proposed approach Example : Eastsoft is one of the top five software companies in China It offers a rich portfolio of businesses, mainly including industry solutions, product engineering solutions, and related software products and platform and services It is dedicated to becoming a globally leading IT solutions and services provider through continuous improvement of organization and process, competence development of leadership and employees, and alliance and open innovation To improve the operation and competitiveness capability in the global market, Eastsoft plans to establish a strategic alliance with a transnational corporation After numerous consultations, four transnational corporations would like to establish a strategic alliance with Eastsoft; they are HP (S ),PHILIPS(S ),EMC (S ), and SAP (S ) To select the desirable strategic alliance partner, five experts (E, E, E, E, and E 5 ) are invited to participate in the decision analysis, who come from the operation management department, the engineering management department, the finance department, the human resources department, and the business process outsourcing department of Eastsoft, respectively The preference information on the potential alliance partners provided by the five experts is in the form of ordinal interval number, which is presented in Table II The weight vector of experts provided by the decision maker is w (0, 0, 0, 0, 0) T To select the most desirable strategic alliance partner, the computation process and results using the proposed approach are summarized as follows By (5) and (6), the five possibility degree matrices, namely, P (), P (), P (), P (), and P (5), are set up as follows: P () P () r () r () r () r () r () r () r () r ()

9 FAN AND LIU: APPROACH TO SOLVE GDM PROBLEMS WITH ORDINAL INTERVAL NUMBERS P () P () P (5) r () r () r () r () r (5) r (5) r (5) r (5) r () r () r () r () r (5) r (5) r (5) r (5) By (7), the collective expectation possibility degree matrix P can be obtained, ie, P S S S S S S S S By (8) and (9), the matrices Q and Q can be obtained, respectively, ie, Q Q By (), the ranking values of the four alternatives can be obtained, ie, d 068, d 0850, d 006, and d 05 According to the derived ranking values and matrix P, the ranking order of the alternatives with possibility degrees is S S S S Example : Consider a GDM problem with ordinal interval numbers, which was investigated by González-Pachón and Romero [9] In the GDM problem, there are four experts (E, E, E, and E ) and four alternatives (S, S, S, and S ) The preference information on alternatives provided by the experts is in the form of ordinal interval number, which is presented in Table III Suppose that the weight vector of experts is w (05, 05, 05, 05) T To rank the alternatives, the proposed approach is used, and the procedure is summarized in the following TABLE III PREFERENCE INFORMATION ON ALTERNATIVES PROVIDED BY THE EXPERTS TABLE IV RANKING RESULTS OBTAINED BY THE TWO APPROACHES By (5) and (6), the four possibility degree matrices, namely, P (), P (), P (), and P (), are set up as follows: P () P () P () P () r () r () r () r () r () r () r () r () r () r () r () r () r () r () r () r () By (7), the collective expectation possibility degree matrix P can be obtained, ie, P S S S S S S S S Based on matrix P, using (8), (9), and (), the ranking values of the four alternatives can be obtained, ie,

10 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART B: CYBERNETICS, VOL 0, NO 5, OCTOBER 00 TABLE V COMPARISON OF SEVERAL INDEX VALUES OBTAINED BY THE TWO APPROACHES TABLE VI COMPARISON OF TOTAL INDEX VALUES OBTAINED BY THE TWO APPROACHES d 086, d 0780, d 00, and d 05 Thus, we can obtain the ranking order of the alternatives, ie, S S S S We can compare the result obtained by the proposed approach and the result obtained by González-Pachón and Romero [9], as shown in Table IV The approach proposed by González-Pachón and Romero is based on an interval goal programming model The obective of the model is to minimize the consensus measure of the collective ranking result and the individual preference of each expert It can be seen from Table IV that the ranking results between the two approaches only have minor difference By (5) (7), a comparison of index values obtained by the aforementioned two approaches is shown in Table V Based on the indexes in Table V, we can obtain the calculation results of DT, DU, and max{d M,,, }, as shown in Table VI From Tables IV VI, we can make a conclusion that the ranking result obtained by the proposed approach is also another optimal solution obtained by the approach of González-Pachón and Romero In fact, using the approach of González-Pachón and Romero, we can find that there are multiple optimal solutions, where one optimal solution is R, R, R, and R, ie, the ranking of alternatives is S S S S Itis necessary to point out that the approach proposed in this paper is simple Using the proposed approach, the possible degree matrices can be directly obtained by (5) and (6) Although a constrained optimization model is used to obtain the ranking values of alternatives, the solution to the model can be directly obtained by () However, in González-Pachón and Romero s approach, the interval goal programming model is actually an integer programming problem If the number of alternatives or experts is great, then the number of constraints will be great For example, in the example investigated by González- Pachón and Romero, the interval goal programming model has 6 constraints (see, Table and model (7)[9]) If the scale of the problem is great, it is difficult to solve the problem; usually, an effective algorithm is needed On the other hand, using the proposed approach, we cannot only give the ranking order of all alternatives but also give the collective expectation possibility degree that one alternative is preferable to the other in the ranking order VI CONCLUSION AND DISCUSSION This paper has presented a new approach to solve GDM problems with ordinal interval numbers In the approach, the collective expectation possibility degree matrix is constructed through the comparison of ordinal interval numbers Based on the matrix, an optimization model is set up to compute the ranking vector of alternatives According to the derived ranking vector, the ranking order of the alternatives can be obtained The proposed approach has distinct characteristics as discussed in the following First, in the proposed approach, the definition of possibility degree on comparing two ordinal interval numbers is given It is a new idea and lays a good foundation for solving GDM problems with ordinal interval numbers Second, the proposed approach has a clear logic and a simple computation process By constructing the collective expectation possibility degree matrix, the ranking vector of alternatives can be directly obtained by a simple and straightforward formula Third, using the proposed approach, not only the ranking of alternatives can be obtained but also the collective expectation possibility degree that one alternative is preferable to the other in the ranking order can be given Thus, alternative ranking outcomes with the interpretation of possibility degree are important for solving GDM problems with ordinal interval numbers; it is conductive to support the decision making of decision makers It is important to highlight that, since the proposed approach is new and different from the existing approaches, it can give experts or decision analysts one more choice for identifying the appropriate group decision model(s) to solve GDM problems with ordinal interval numbers In addition to supplementing the existing approaches, the proposed approach is also important for developing and enriching theories and methods of GDM problems with ordinal interval numbers In terms of future research, the proposed approach can be extended to multiple-attribute GDM problems with ordinal interval numbers It can also be extended to support the situation where the preference information in GDM is in multiple forms such as crisp numbers, interval numbers, ordinal interval numbers, and so on

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note on the reciprocity in the aggregation of fuzzy preference relations using OWA operators, Fuzzy Sets Syst, vol 7, no, pp 7 8, Jul 00 [] M Behzadian, R B Kazemzadeh, A Albadvi, and M Aghdasi, PROMETHEE: A comprehensive literature review on methodologies and applications, Eur J Oper Res, vol 00, no, pp 98 5, 00 [] B Roy, The outranking approach and the foundations of ELECTRE methods, Theory Decis, vol, no, pp 9 7, Jul 99 [5] Y M Wang and Z P Fan, Fuzzy preference relations: Aggregation and weight determination, Comput Ind Eng, vol 5, no, pp 6 7, Aug 007 Zhi-Ping Fan received the BE and MS degrees in industrial automation and the PhD degree in control theory and applications from Northeastern University (NEU), Shenyang, China, in 98, 986, and 996, respectively He was a Research Fellow with the City University of Hong Kong, Kowloon City, Hong Kong, in 00, 00, and 00 He is currently a Professor with the Department of Management Science and Engineering, School of Business Administration, NEU He is the author or coauthor of more than 0 papers published in international ournals His current research interests include decision analysis, operations research, and knowledge management Yang Liu received the BE degree and the MS degree in management science and engineering from Dalian University of Technology, Dalian, China, in 00 and 00, respectively He is currently working toward the PhD degree at Northeastern University (NEU), Shenyang, China He is also currently a Lecturer with the Department of Management Science and Engineering, School of Business Administration, NEU He is the author or coauthor of more than ten papers published in international and local ournals His current interests include decision analysis and operations research