A preference aggregation method through the estimation of utility intervals

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1 Available online at Computers & Operations Research 32 (2005) A preference aggregation method through the estimation of utility intervals Ying-Ming Wang a;b;, Jian-Bo Yang a, Dong-Ling Xu a a Manchester School of Management, UMIST, P.O. Box 88, Manchester M60 1QD, UK b School of Public Administration, Fuzhou University, Fuzhou, Fujian, , PR China Abstract In this paper, a preference aggregation method is developed for ranking alternative courses of actions by combining preference rankings of alternatives given on individual criteria or by individual decision makers. In the method, preference rankings are viewed as constraints on alternative utilities, which are normalized, and linear programming models are constructed to estimate utility intervals, which are weighted and averaged to generate an aggregated utility interval. A simple yet pragmatic interval ranking method is used to compare and/or rank alternatives. The nal ranking is generated as the most likely ranking with certain degrees of belief. Three numerical examples are examined to illustrate the potential applications of the proposed method. Scope and purpose The aggregation of preference rankings has wide applications in group decision making, social choice, committee election and voting systems. The purpose of this paper is to develop a preference aggregation method through the estimation of utility intervals, in which preference rankings are associated with utility intervals that are estimated using linear programming models and aggregated using the simple additive weighting method.? 2004 Elsevier Ltd. All rights reserved. Keywords: Group decision making; Preference ranking; Preference aggregation; Utility estimation; Linear programming; Extreme point; Interval numbers This research was supported by the National Natural Science Foundation of China (NSFC) under the Grant No and also in part by Fok Ying Tung Education Foundation under the Grant No and by the UK Engineering and Physical Sciences Research Council under the Grant No. GR/N65615/01. Corresponding author. Manchester School of Management, UMIST, P.O. Box 88, Manchester M60 1QD, UK. Tel.: ; fax: address: msymwang@hotmail.com (Y.-M. Wang) /$ - see front matter? 2004 Elsevier Ltd. All rights reserved. doi: /j.cor

2 2028 Y.-M. Wang et al. / Computers & Operations Research 32 (2005) Introduction How to aggregate individual preferences or a set of ordinal rankings into a group preference or consensus ranking is a typical group decision making problem. It has wide applications in social choice, committee election and voting systems. A large amount of research has already been conducted in this area. Borda [1] was the rst to examine the ordinal ranking problem for choosing candidates in an election and proposed a method of marks to rank candidates according to the sum of ranks assigned by voters to each candidate (see also Black [2]). Kendall [3] was the rst to study the problem in a statistical framework. He approached it as a problem of estimation. In other words, if there is an agreement among observers and their judgments are accurate, then the question is how to estimate the true ranking. The solution is to rank candidates according to the sum of ranks, which is precisely equivalent to Borda s method of marks. So, the method is frequently referred to as Borda Kendall (BK) method, which is probably the most widely used technique in determining a consensus ranking due to its computational simplicity. An alternative approach to solve the ordinal ranking problem is the majority rule, which is also the simplest form of group consensus. Many researchers advocated that any method used to draw a consensus from preferences supplied by individuals should satisfy some social welfare axioms, which were originally put forward by Arrow [4]. Inada [5,6] worked out conditions under which the simple majority decision rule is a social welfare function satisfying Arrow s social welfare axioms. Further work by Bowman and Colantoni [7,8] and by Blin and Whinston [9] showed that the majority rule problem under transitivity could be solved via integer programming models. Kemeny and Snell [10] studied the ordinal ranking problem using distance measures. They proposed a set of axioms which any distance measure should satisfy and proved its existence and uniqueness. This set of axioms is very similar to those given by Arrow. Using their distance functions, Kemeny and Snell proposed the median and mean rankings (l 1 and l 2 metrics) as acceptable forms of consensus. Bogart [11,12] generalized Kemeny and Snell s theory of distance to partial orders. Both transitive and intransitive orderings were considered, the existence of a unique distance function was proved, and its form was determined. Cook and Seiford [13] also developed a theory of distance between complete ordinal rankings (no ties) and came up with six axioms which are similar to those proposed by Kemeny and Snell and by Bogart. They further proved the existence of a unique distance function satisfying the stated axioms and derived its form, which was a l 1 metric. The corresponding consensus ranking was dened to minimize the total absolute distance (disagreement), which leaded to a median ranking. Such a median ranking problem could be expressed as a linear programming or linear assignment model. In the presence of ties, Cook and Seiford [14] showed that the Borda Kendall method could not perform well as claimed. They thus extended it using a l 2 distance approach and put forward a minimum variance method, which turned out to be equivalent to the BK method if ties are not allowed. Armstrong et al. [15] also studied the case of ties and showed that the l 1 distance function was also unique for a general quasi-linear case. Cook and Kress [16,17] and Ali et al. [18] extended the l 1 distance measure to ordinal rankings with intensity of preferences and worked out an integer linear programming method to acquire a consensus ranking. Barzilai et al. [19] demonstrated that the Kemeny and Snell model (l 1 metric) could be equivalently expressed as a generalized network model. Cook et al. [20] presented a general framework for associating value or worth with ordinal ranks and developed a general model

3 Y.-M. Wang et al. / Computers & Operations Research 32 (2005) for distance-based consensus. Recently, Gonzalez-Pachon and Romero [21,22] extended l 1 and l 2 distance measure to any l p metric (1 6 p 6 ) and presented a goal programming (GP) and an interval goal programming approach to aggregate complete and incomplete ordinal rankings, respectively. Linares and Romero [23] utilized the above GP approach to aggregate individual preference weights. In other framework, Beck and Lin [24] developed a procedure called the maximize agreement heuristic (MAH) to arrive at a consensus ranking that maximizes agreement among decision makers (DMs) or voters. Cook and Kress [25,26] proposed a data envelopment analysis (DEA) model for aggregating preference rankings, which was shown to be equivalent to the Borda Kendall method in certain special circumstances, and an extreme-point approach for obtaining weighted ratings in ordinal multicriteria decision making, respectively. Green et al. [27] used the cross-eciency evaluation method in DEA to aggregate preference rankings. Puerto et al. [28] also discussed the problem of aggregating ordinal rankings using an extreme point approach. The literature review shows that existing methods for preference aggregation either view ordinal rank positions directly as utilities or associate a value/weight with an ordinal rank position. In fact, a range of utilities instead of a single utility could be assigned to an ordinal rank position. However, no one has ever attempted to associate utility intervals with ordinal rankings to generate an aggregated ranking in group decision situations. It is argued that an ordinal ranking may be interpreted or understood as constraints on utility. Dierent DMs may produce dierent utility ranking estimates. Such estimates only provide preference information rather than concrete numerical values. Thus, they may be used to estimate relative utilities and their ranges. A consensus ranking may be generated on the basis of relative utility intervals. This paper is devoted to developing a linear programming method to generate an aggregated ranking from a set of ordinal rankings by estimating a utility interval for each alternative and every ordinal ranking. The rest of the paper is organized as follows. In Section 2, a linear programming model is proposed to estimate utility intervals. Section 3 introduces a simple yet practical preference ranking method for interval utilities. This is followed by three numerical examples, which are used to show the validity and potential applications of the proposed method. The paper is concluded in Section Estimation of utility intervals Suppose m DMs or electoral committees rank or vote on n alternative courses of action. Each ranking is denoted by a vector R i =(r i1 ;:::;r in )(i =1;:::;m) which ranks n alternatives/candidates from the best to the worst as (A i1 ;A i2 ;:::;A in ), where r ij is the rank given to the jth alternative/candidate by the ith DM or electoral committee, and A ij is the alternative or candidate being ranked at the jth place (j =1;:::;n). The relative weight of each DM or electoral committee is denoted by w i (i =1;:::;m), which satises the normalization condition m i=1 w i = 1. The nal aggregated ranking is denoted by R =(r 1 ;r 2 ;:::;r n ), where r j is the rank given to the jth alternative/candidate. Two questions arise naturally as to how to rank alternatives and how to aggregate the rankings of all DMs. It is not uncommon to use utilities for ranking alternatives. Therefore, a ranking may be interpreted or understood as the utility-based ranking of alternatives. The latter question is what is going to be investigated in this paper. Utility is a relative measure and as such normalized utility is used in this paper.

4 2030 Y.-M. Wang et al. / Computers & Operations Research 32 (2005) Let U i =(u i1 ;:::;u in )(i =1;:::;m) be a normalized utility vector, namely, n u ij =1; i=1;:::;m; j=1 (1) where u ij is the utility of the jth alternative perceived by the ith DM. Preference rankings may include the following three kinds of order relations in terms of utility: (1) strict order u ij u ik if the alternative A ij is strictly preferred to the alternative A ik, or expressed as u ij u ik jk where jk is a small positive number; (2) weak order u ip u iq if A ip is not inferior to A iq ; (3) indierence relation (tie) u is = u it if A is is indierent to A it. The following pairs of linear programming models are thus constructed to estimate the utility interval for each alternative: min=max u ij (2) s:t: u iij u iij+1 j;j+1 ; i j ;i j+1 S ; (3) u iip u iip+1 0; i p ;i p+1 W ; (4) u iir u iir+1 =0; i r ;i r+1 I ; (5) n u iij =1; j=1 (6) u iij 0; j =1;:::;n; (7) where S ; W and I are, respectively, the index sets of those alternatives with strict and weak orders and indierence relations in the ranking (A i1 ;A i2 ;:::;A in ), and j;j+1 is a small positive number. The optimal objective values of the above pair of LP models consist of the permissible utility interval for u ij, which we denote by [uij;u L ij]. U Repeating the above solution process for each utility u ij (j =1;:::;n), all the utility intervals that are perceived by the ith DM can be obtained, which we denote by U i =([ui1 L ;uu i1 ];:::;[ul in;uin]). U It represents the utility estimation of the ith DM. From other ordinal rankings, similar LP models can be built in the same way. Accordingly, the overall utility interval for each alternative or candidate can be aggregated as the weighted average of the utility intervals perceived by all DMs as follows: m uj L = w i uij; L j =1;:::;n; (8) u U j = i=1 m w i uij; U i=1 j =1;:::;n; where w i (i=1;:::;m) is the relative weight of the ith DM, u L j and u U j form the overall utility interval [u L j ;u U j ](j =1;:::;n), based on which the nal aggregated ranking can be generated. (9)

5 Y.-M. Wang et al. / Computers & Operations Research 32 (2005) In certain special cases, the analytical solutions of LP models (2) (7) can be obtained. The main results are summarized as follows: Case 1: The solution space D 1 = {U =(u 1 ;:::;u n ) T u 1 u 2 u n 0; n i=1 u i =1}, which corresponds to a complete weak order, has the following matrix of extreme points (Carrizosa et al. [29]; Marmol et al. [30]; and Puerto et al. [28]): u 1 1 1=2 1=3 1=n u 2 0 1=2 1=3 1=n E = u =3 1=n ; (10) =n =n u n where each column vector constitutes a vertex of the hyper-polygon D 1. The corresponding utility intervals for u i are [1=n; 1]; i=1; [0; 1=i]; i=2;:::; n: (11) It must be pointed out that the above matrix of extreme points was ever used by Carrizosa et al. [29], Marmol et al. [30] and Puerto et al. [28] to estimate the weight intervals of criteria in MADA, but has never been utilized to generate the utility intervals of alternatives or candidates. Case 2: The solution space D 2 ={U =(u 1 ;:::;u n ) T u i u i+1 i ;u i 0;i=1;:::;n 1; n i=1 u i=1}, which corresponds to a complete strict order, has the following matrix of extreme points: u = = =n u = = =n E = u = =n u n n n n n + 1=n =2 1=3 1=n =2 1=3 1=n = =3 1=n ; (12) n n n n =n where n i n j ; i=1;:::;n 1; i = j=1 0; i= n; (13)

6 2032 Y.-M. Wang et al. / Computers & Operations Research 32 (2005) n 1 =1 i i : i=1 As a matter of fact, if we let n i u u i n j = i i ; i=1;:::;n 1; i = j=1 u n ; i= n: (14) (15) Then solution space D 2 becomes ˆD 2 = {U =(u 1 ;:::;u n) T u 1 u 2 u n 0; n i=1 u i = }, whose matrix of extreme points is times (10). So (12) is true. When 1 = = n 1 =, formulas (13) and (14) can be simplied by i =(n i); i =1;:::;n; (16) =1 n(n 1)=2: The corresponding utility intervals for u i are given by (17) [ 1 + 1=n; 1 + ]; i=1; [ i ; i + 1=i]; i=2;:::;n: (18) Case 3: Hyper-polygon D 3 ={U =(u 1 ;:::;u n ) T u 1 u k = =u l u n 0; n i=1 u i = 1}, which corresponds to an incomplete weak order that includes some ties, has the following matrix of extreme points: u 1 1 1=2 1=(k 1) 1=l 1=l 1=(l +1) 1=n u 2 0 1=2 1=(k 1) 1=l 1=l 1=(l +1) 1=n u =(k 1) 1=l 1=l 1=(l +1) 1=n u k =(k 1) 1=l 1=l 1=(l +1) 1=n u k =l 1=l 1=(l +1) 1=n E = : (19) u l =l 1=l 1=(l +1) 1=n u l =(l +1) 1=n u l =n =n u n

7 Y.-M. Wang et al. / Computers & Operations Research 32 (2005) The corresponding utility intervals for u i are given by [1=n; 1]; i=1; [0; 1=i]; i=2;:::;k 1; [0; 1=l]; i= k;:::;l; [0; 1=i]; i= l +1;:::;n: (20) Case 4: Hyper-polygon D 4 = {U =(u 1 ;:::;u n ) T u i u i+1 i ;u k = = u l ;u n 0;i=1;:::;k 1; 1;:::;n 1; n i=1 u i =1}, which corresponds to an incomplete strict order that includes some ties, has the following matrix of extreme points: =2 1 +=(k 1) 1 +=l 1 +=l 1 +=(l+1) 1 +=n 2 2 +=2 2 +=(k 1) 2 +=l 2 +=l 2 +=(l+1) 2 +=n =(k 1) 3 +=l 3 +=l 3 +=(l+1) 3 +=n k 1 k 1 k 1 +=(k 1) k 1 +=l k 1 +=l k 1 +=(l+1) k 1 +=n k k k k +=l k +=l k +=(l+1) k +=n E = ; l l l l +=l l +=l l +=(l+1) l +=n l+1 l+1 l+1 l+1 l+1 l+1 +=(l+1) l+1 +=n l+2 l+2 l+2 l+2 l+2 l+2 l+2 +=n n n n n n n n +=n (21) where n l k i n j + k j ; i=1;:::;k 1; j=1 j=1 n l n j ; i= k;:::;l 1; i = j=1 n i n j ; i= l;:::;n 1; j=1 0; i= n; (22)

8 2034 Y.-M. Wang et al. / Computers & Operations Research 32 (2005) k 1 n 1 =1 i i i i : i=1 i=l Especially when 1 = = k 1 = l = = n 1 =, the above formulas (22) and (23) can be simplied as follows: (n l + k i); i =1;:::;k 1; (n l); i = k;:::;l 1; i = (24) (n i); i = l;:::;n 1; 0; i= n; k(k 1)+(n l)(n + l 1) =1 : (25) 2 The corresponding utility intervals for u i are [ 1 + 1=n; 1 + ]; i=1; [ i ; i + 1=i]; i=2;:::;k 1; [ i ; i + 1=l]; i= k;:::;l; [ i ; i + 1=i]; i= l +1;:::;n: (26) Case 5: Hyper-polygon D 5 = {U =(u 1 ;:::;u n ) T u k = = u l ; u i 0;i=1;:::;n; n i=1 u i =1}, which corresponds to a partial indierence relation, has the following matrix of extreme points: u u u u k u k u k =(l k +1) 1=(l k +1) 0 0 E = : (27) u l =(l k +1) 1=(l k +1) 0 0 u l u l u n u n (23)

9 Y.-M. Wang et al. / Computers & Operations Research 32 (2005) The corresponding utility intervals for u i are given by [0; 1]; i=1;:::;k 1; [0; 1=(l k + 1)]; i= k;:::;l; [0; 1]; i= l +1;:::;n: (28) 3. Comparison and ranking of interval utilities In preference aggregation, the nal aggregated ranking is based on the comparison and ranking of overall utilities, which are all interval numbers. A simple yet practical ranking method is thus needed for ranking interval numbers. The comparison rule for interval numbers proposed by Ishibuchi and Tanaka [31] is probably one of the most prominent and simplest ways in the analysis and comparison of interval numbers. But it sometimes fails when one interval number is included in another one. Moreover, due to the existence of uncertainty, the preference ranking among interval numbers is not likely to be 100% certain. So, it would be more desirable to give the degrees of preference in the ranking of interval utilities. Kundu [32] dened a fuzzy leftness relationship between two interval numbers, which was to some extent able to reect the degree of one interval number to be superior or inferior to another one. But it required the assumption that all the interval numbers are independent and uniformly distributed. Sengupta and Pal [33] dened an acceptability index, which was also designed to reect the grade of acceptability of one interval number to be inferior to another one. But the index was totally based on the midpoints of interval numbers. Our experience told us that the use of midpoints to compare or rank interval numbers was sometimes inconvincible and not easy to be accepted. In view of this, we developed a simple yet practical and more rational preference ranking method of interval numbers, which makes no assumption and makes no use of the midpoints of interval numbers. The approach is summarized as follows. Let a =[a 1 ;a 2 ] and b =[b 1 ;b 2 ] be two interval utilities, whose possible relationships are as shown in Fig. 1. We refer to the degree of one interval utility being greater than another one as the degree of preference. Accordingly, we have the following denitions. Denition 1. The degree of preference of a over b (or a b) is dened as P(a b)= max(0;a 2 b 1 ) max(0;a 1 b 2 ) : (29) (a 2 a 1 )+(b 2 b 1 ) a < b a > b a < b a > b a 1 -b 2 a 2 -b 1 0 a-b 0 a 1 -b 2 a 2 -b 1 a-b a 1 -b 2 0 a 2 -b 1 a-b (a) (b) (c) Fig. 1. Relationships between two interval utilities a and b.

10 2036 Y.-M. Wang et al. / Computers & Operations Research 32 (2005) The degree of preference of b over a (or b a) can be dened in the same way. That is P(b a)= max(0;b 2 a 1 ) max(0;b 1 a 2 ) : (30) (a 2 a 1 )+(b 2 b 1 ) It is obvious that P(a b)+p(b a)=1 and P(a b)=p(b a) 0:5 when a = b, i.e. a 1 = b 1 and a 2 = b 2. Denition 2. If P(a b) P(b a), then a is said to be superior to b to the degree of P(a b), denoted by a P(a b) b; ifp(a b)=p(b a)=0:5, then a is said to be indierent to b, denoted by a b; IfP(b a) P(a b), then a is said to be inferior to b to the degree of P(b a), denoted by a P(b a) b. A more general denition can be stated as follows: Denition 3. If P(a b) P(b a), then a is said to be superior to b to the degree of P(a b); if P(a b) P(b a) 6, then a is said to be indierent to b; IfP(a b) P(b a), then a is said to be inferior to b to the degree of P(b a), where is a very small positive number predetermined by DM. Since P(a b) + P(b a) = 1, the above Denition 3 may still be equivalently stated as follows: Denition 4. If P(a b) 0:5 +, then a is said to be superior to b to the degree of P(a b); if P(a b) [0:5 ; 0:5 + ], then a is said to be indierent to b; IfP(a b) 0:5, then a is said to be inferior to b to the degree of P(b a), where is a very small positive number predetermined by DM. Denition 2 can be viewed as a special case of Denitions 3 and 4 with = = 0. Moreover, it must be pointed out that Denitions 2 4 are dened for benet interval numbers because larger utility is always preferred. About the degree of preference, we have the following properties. Property 1. P(a b) = 1 if and only if a 1 b 2. Property 2. If a 1 b 1 and a 2 b 2, then P(a b) 0:5 and P(b a) 6 0:5. Property 3. If b is nested in a, i.e. a 1 6 b 1 and a 2 b 2, then P(a b) 0:5 if and only if (a 1 + a 2 )=2 (b 1 + b 2 )=2. Property 4. If P(a b) 0:5+ and P(b c) 0:5+, then P(a c) 0:5+. Property 1 shows that if two interval utilities do not overlap, then the one on the upper end will 100% dominate the one on the lower end. Property 2 is quite similar to the comparison rule for

11 Y.-M. Wang et al. / Computers & Operations Research 32 (2005) ranking interval numbers, but the former provides information on the degree of preference of one interval utility over the other, while the latter does not provide such information. Property 3 shows how to compare two interval utilities when one interval utility is included in the other. Property 4 shows that if a b and b c, then a c. That is, the preference relations dened by Denitions 2 4 are transitive. With the help of transitivity, a complete ranking order for interval utilities could be achieved. The ranking process is outlined below: Step 1: Calculate the matrix of the degrees of preference u 1 u 2 u n u 1 p 12 p 1n u 2 p 21 p 2n (31) P D = ;..... p n1 p n2 where u n p ij = P(u i u j )= max(0;uu i u L j ) max(0;u L i u U j ) (u U i u L i )+(uu j u L j ) ; i;j =1;:::;n; i j: (32) u i =[ui L ;ui U ] is the overall utility of ith alternative (i =1;:::;n). Symbol means that the value does not have to be calculated. Step 2: Calculate the matrix of preference relation u 1 u 2 u n u 1 m 12 m 1n u 2 m 21 m 2n (33) M PL = ;..... u n p n1 p n2 where { 1 if pij 0:5+ ; m ij = 0 if p ij 6 0:5+ ; i; j =1;:::;n; i j: (34) is a threshold determined by DM. If there is no particular requirement, it may be set to be zero. Step 3: Calculate the sum of the elements of each row in the above matrix of preference relation and generate the nal aggregated ranking R =(r 1 ;:::;r n ). The ith alternative is ranked higher than the jth alternative if the sum for the ith row is larger than that for the jth row. According to transitivity (Property 4), the alternative/candidate being ranked at the rst is preferred to all the other alternatives, so its sum of row is maximal. The alternative or candidate being ranked at the last is undoubtedly inferior to all the others, so its sum of row is zero. The other alternatives or candidates can be easily ranked according to their sums of row.

12 2038 Y.-M. Wang et al. / Computers & Operations Research 32 (2005) Table 1 Voting among three political parties by 60 voters Political Number of voters Totals Rank party (Borda Kendall) A B C Note: 1 = the best, 3 = the worst. 4. Numerical examples In this section, three numerical examples are examined to illustrate the potential applications of the proposed method in ordinal preference aggregations. Example 1. Consider a voting problem of 60 voters voting among three political parties. The voting results are shown in Table 1, which was investigated by Hwang and Lin [34] and Gonzalez-Pachon and Romero [21]. According to Borda Kendall method, since political party B receives the lowest sum of 111, it is ranked to be the best, while political party C with the highest sum of 127 is ranked to be the worst. Therefore, the nal aggregated ranking is given by B A C. Gonzalez-Pachon and Romero developed a goal programming (GP) method to generate three dierent rankings. Using a linear weighted GP model, they got A B C if ties were permitted; otherwise A B C. For a MINMAX or Chebyshev GP model, they got B A C. The utility interval estimation method is now used to aggregate preference rankings. For the ranking A B C provided by 23 voters, the following pair of linear programming models can be used to estimate the relative utilities of three political parties: min=max u 1A s:t: u 1A u 1B 0; u 1B u 1C 0; u 1A + u 1B + u 1C =1; u 1A ;u 1B ;u 1C 0: The solutions to the above LP models are u L 1A =1=3 and uu 1A = 1, which constitute the utility interval for u 1A or u 1A [1=3; 1]. Substituting u 1B or u 1C for u 1A in the objective function, we get u 1B [0; 1=2] and u 1C [0; 1=3]. Note that the utility intervals can be generated directly from formula (11). Table 2 presents all the utility estimates that are generated from Table 1 using formula (11). The aggregated

13 Y.-M. Wang et al. / Computers & Operations Research 32 (2005) Table 2 Utility interval estimates corresponding to the preference rankings of Table 1 Political Number of voters Weighted average Rank party utility A [1/3,1] [0,1/3] [0,1/2] [0,1/2] [0,1/3] [0.1278, ] 1 B [0,1/2] [1/3,1] [1/3,1] [0,1/3] [0,1/2] [0.1056, ] 2 C [0,1/3] [0,1/2] [0,1/3] [1/3,1] [1/3,1] [0.1000, ] 3 Table 3 The matrix of degrees of preference among interval utilities u A, u B and u C p ij u A u B u C u A u B u C utility, i.e. weighted average utility, of each political party can be calculated as follows: u A = [ 1 17 ; 1] [0; 1 3 ] [0; 1 10 ] [0; 1 2 ] [0; 1 ]=[0:1278; 0:6222]; 3 u B = [0; 1 17 ] [ 1 3 ; 1] [ 1 10 ; 1] [0; 1 3 ] [0; 1 ]=[0:1056; 0:6306]; 2 u C = [0; 1 17 ] [0; 1 2 ] [0; 1 10 ] [ 1 3 ; 1] [ 1 ; 1]=[0:1000; 0:5806]: 3 The ranking method described in Section 3 is used to compare and rank the three political parties. Table 3 records their degrees of preference, from which it is clear that u A is superior to both u B and u C to a degree of 50.68% and 53.56%, respectively, and u B over u C to a degree of 52.76%. 50:68% 52:76% So, a complete ranking for interval utilities u A, u B and u C is u A u B u C. Accordingly, the nal aggregated ranking for the three parties A, B and C can be understood as A 50:68% B 52:76% C or A B C for short, which is exactly the same as the rank obtained using Gonzalez-Pachon and Romero s linear weighted GP model, but our ranking provides information on the degrees of preference and is thus easier to be understood and accepted. In the above discussion, it was assumed that all the preference rankings provided by the voters are weak orders. If all the preference rankings are assumed to be strict orders, then the following pairs of LP models can be used to estimate the utility intervals of the three political parties from the preference ranking A B C given by 23 voters min=max u 1i s:t: u 1A u 1B AB ;

14 2040 Y.-M. Wang et al. / Computers & Operations Research 32 (2005) Table 4 Utility interval estimates under the assumption of strict orders for Example 1 Political party Number of voters A [ 1 + ; 1 ] 3 [0; 1 ] 3 [; 1 1 ] 2 2 [; 1 1 ] 2 2 [0; 1 ] 3 B [; 1 1 ] 2 2 [1 + ; 1 ] 3 [1 + ; 1 ] 3 [0; 1 ] 3 [; 1 1 ] 2 2 C [0; 1 ] 3 [; 1 1 ] 2 2 [0; 1 ] 3 [1 + ; 1 ] 3 [1 + ; 1 ] 3 u 1B u 1C BC ; u 1A + u 1B + u 1C =1; u 1A ;u 1B ;u 1C 0; where i = A; B; C and AB and BC are two given small positive numbers (also called discriminant factors). If AB and BC are assumed to be equivalent, say, AB = BC =, then formulas (16) (18) can be used to derive the utility intervals directly. Table 4 shows all the utility estimates for the three political parties under the assumption of strict orders. Accordingly, the overall utility for each political party can be calculated as follows: u A = [ ; 1 ]+ [0; ]+ [; ]=[23 =[0: :5833; 0:6222 0:9]; ; ] u B = 23+8 [; ]+ [ ; 1 ] [0; 1 3 ]=[ ; ] =[0: :8333; 0:6306 0:7417]; u C = 23+2 [0; 1 17 ] [; ]+ [ 1 + ; 1 ]=[ =[0:1+0:5833; 0:5806 0:8583]: ; ] The corresponding aggregated rankings under dierent values of are summarized in Table 5, from which it is clear that under the assumption of weak orders the political party A is slightly superior to the other two parties, but under the strict order assumption, the political party B becomes more and more superior to the others as the discriminant factor increases. When takes its maximum, i.e. max = 1, the overall relative utilities degenerate to point estimates (crisp numbers) and we can 3 then be 100% sure that the political party B is superior to the political party A and A to the political party C. In this example, max = 1 is due to the constraint of =1 n(n 1)=2=1 3 0 (see 3 formula (17) for detail). Since the discriminant factor may take values within a closed interval [0; max ], we recommend taking its maximum so that the dominant relations between alternatives/candidates can be determined to the best degrees of preference.

15 Y.-M. Wang et al. / Computers & Operations Research 32 (2005) Table 5 The aggregated rankings under dierent values of for Example 1 Utility Ranking u A u B u C 0 [0.1278, ] [0.1056, ] [0.1000, ] 50:68% 52:76% u A u B 0.1 [0.1861, ] [0.1889, ] [0.1583, ] 51:89% 54:78% u B u A 0.2 [0.2444, ] [0.2722, ] [0.2167, ] 58:31% 57:82% u B u A 0.3 [0.3028, ] [0.3556, ] [0.2750, ] 100% 79:18% u B u A 1/3 [0.3222, ] [0.3833, ] [0.2944, ] 100% 100% u B u A u C u C u C u C u C Table 6 Rankings of the nine proposals by a group of six experts Proposal Group member Totals Rank (Borda Kendall) A B C D E F G H I Note: 1= highest, 9= lowest. Example 2. Consider a preference aggregation problem where a group of six experts is required to rank nine proposals individually. The rankings given by each expert are presented in Table 6, which was investigated by Islei and Lockett [35]. This problem is dicult to deal with since several ties are included in the preference ranking provided by the sixth group member. Dierent approaches may produce dierent aggregated rankings. Traditional Borda Kendall and distance methods all use the average value instead of ties. For example, both proposals D and F are ranked at the seventh place. But in theory they should occupy two ranking positions 7 and 8, so their average value, 7.5, would be used to replace the original ranking number 7 given by the sixth group member. As such, the ties among proposals E, G and H can be dealt with in the same way. The corresponding Borda Kendall count and aggregated ranking are presented in the last two columns in Table 6.

16 2042 Y.-M. Wang et al. / Computers & Operations Research 32 (2005) Table 7 Utility interval estimates corresponding to the preference rankings of Table 6 Proposal Group member Weighted average utility A [0,1/4] [1/9,1] [0,1/7] [1/9,1] [0,1/2] [1/9,1] [0.0556, ] B [0,1/9] [0,1/7] [0,1/9] [0,1/4] [0,1/7] [0,1/9] [0, ] C [0,1/2] [0,1/2] [0,1/5] [0,1/2] [1/9,1] [0,1/2] [0.0185, ] D [1/9,1] [0,1/8] [1/9,1] [0,1/5] [0,1/4] [0,1/8] [0.0370, 0.45] E [0,1/5] [0,1/9] [0,1/2] [0,1/3] [0,1/3] [0,1/6] [0, ] F [0,1/6] [0,1/6] [0,1/8] [0,1/8] [0,1/9] [0,1/8] [0, ] G [0,1/3] [0,1/5] [0,1/3] [0,1/9] [0,1/5] [0,1/6] [0, ] H [0,1/8] [0,1/4] [0,1/4] [0,1/6] [0,1/6] [0,1/6] [0, ] I [0,1/7] [0,1/3] [0,1/6] [0,1/7] [0,1/8] [0,1/3] [0, ] However, in the utility estimation method, ties are easy to deal with. For the preference ranking A C I E G H D F B provided by the sixth group member, it is sucient to solve the following pairs of LP models: min=max u 6i s:t: u 6A u 6C 0; u 6C u 6I 0; u 6I u 6E 0; u 6E u 6G =0; u 6G u 6H =0; u 6H u 6D 0; u 6D u 6F =0; u 6F u 6B 0; u 6A + u 6B + u 6C + u 6D + u 6E + u 6F + u 6G + u 6H + u 6I =1; u 6A ;u 6B ;u 6C ;u 6D ;u 6E ;u 6F ;u 6G ;u 6H ;u 6I 0; where i=a;:::;i, respectively. The optimal solutions to the above LP models can be obtained directly from formula (19) or(20). Table 7 records all the utility interval estimates. The weighted average utilities are presented in the last column, from which the aggregated ranking can be generated as 56:88% 53:49% 65:49% 55:02% 51:95% 52:51% 56:42% 51:46% u A u C u D u E u G u I u H u B u F, which means A C D E G I H B F. If the preference rankings provided by a group of the six experts are assumed to be all strict orders, then for the rst to fth experts formulas (16) (18) can be used to estimate the utilities of the nine

17 Y.-M. Wang et al. / Computers & Operations Research 32 (2005) Table 8 Utility interval estimates under the assumption of strict orders for Example 2 Proposal Group member A [5; 1 4] 4 [1 +4; 1 28] 9 [2; 1 22 ] 7 7 [1 +4; 1 28] 9 [7; 1 11] 2 [ ; 1 15] 9 9 B [0; 1 4] 9 [2; 1 22 ] 7 7 [0; 1 4] 9 [5; 1 4] 4 [2; 1 22 ] 7 7 [0; 1 20 ] 9 9 C [7; 1 11] 2 [7; 1 11] 2 [4; 1 16 ] 5 5 [7; 1 11] 2 [1 +4; 1 28] 9 [4; 1 6] 2 D [ 1 +4; 1 28] 9 [; 1 7 ] 8 2 [1 +4; 1 28] 9 [4; 1 16 ] 5 5 [5; 1 4] 4 [; 1 3 ] 8 2 E [4; [0; 1 4] 9 [7; 1 11] 2 [6; 1 6] 3 [6; 1 6] 3 [2; 1 4 ] 6 3 F [3; 1 3] 6 [3; 1 3] 6 [; 1 7 ] 8 2 [; 1 7 ] 8 2 [0; 1 4] 9 [; 1 3 ] 8 2 G [6; 1 6] 3 [4; 1 16 ] 5 5 [6; 1 6] 3 [0; 1 4] 9 [4; 1 16 ] 5 5 [2; 1 4 ] 6 3 H [; 1 7 ] 8 2 [5; 1 4] 4 [5; 1 4] 4 [3; 1 3] 6 [3; 1 3] 6 [2; 1 4 ] 6 3 I [2; ] [6; 1 3 6] [3; 1 6 3] [2; ] [; ] [3; ] proposals, which are recorded in the second to sixth columns of Table 8, where max = But for the sixth expert, his preference ranking needs to be transformed into the following pairs of LP models: min=max u 6i s:t: u 6A u 6C ; u 6C u 6I ; u 6I u 6E ; u 6E u 6G =0; u 6G u 6H =0; u 6H u 6D ; u 6D u 6F =0; u 6F u 6B ; u 6A + u 6B + u 6C + u 6D + u 6E + u 6F + u 6G + u 6H + u 6I =1; u 6A ;u 6B ;u 6C ;u 6D ;u 6E ;u 6F ;u 6G ;u 6H ;u 6I 0; where i = A;:::;I and max = 1. The optimal solutions to the above LP models are shown 20 in the last column of Table 8. Accordingly, the weighted average utilities of the nine proposals can

18 2044 Y.-M. Wang et al. / Computers & Operations Research 32 (2005) Table 9 The weighted average utility interval estimates for dierent values of Proposal 0 1/180 1/72 1/45 1/36 A [0.0556, ] [0.0785, ] [0.1129, ] [0.1473, ] [0.1703, ] B [0, ] [0.0083, ] [0.0208, ] [0.0333, ] [0.0417, ] C [0.0185, ] [0.0491, ] [0.0949, ] [0.1407, ] [0.1713, ] D [0.0370, 0.45] [0.0546, ] [0.0810, ] [0.1074, ] [0.1250, ] E [0, ] [0.0231, ] [0.0579, ] [0.0926, ] [0.1157, ] F [0, ] [0.0083, ] [0.0208, ] [0.0333, ] [0.0417, ] G [0, ] [0.0204, ] [0.0509, ] [0.0815, ] [0.1019, ] H [0, ] [0.0176, ] [0.0440, ] [0.0704, ] [0.0880, ] I [0, ] [0.0157, ] [0.0394, ] [0.0630, ] [0.0787, ] be expressed as follows: u A =[ ; ]=[0: :1296; 0: :8571]; u B =[ 3 2 ; ]=[1:5; 0:1448 3:4180]; 189 u C =[ ; 8 u D =[ ; 9 u E =[ 25 6 ; 37 u F =[ 3 2 ; ]=[0: :5; 0: :7]; ]=[0:037+3:16667; 0:45 11:3667]; ]=[4:16667; 0:2741 5:2556]; ]=[1:5; 0:1366 3:0833]; u G =[ ; ]=[3:6667; 0:2241 3:9556]; u H =[ 19 6 ; ]=[3:1667; 0:1875 3:1389]; u I =[ 17 6 ; ]=[2:8333; 0:2073 3:7421]: 252 The aggregated utility intervals and rankings for dierent values of are summarized in Tables 9 and 10, respectively. It can be seen from Table 10 that as the discriminant factor increases the degrees of preference of proposals A over C, I over H, and B over F, become increasingly smaller, while the degrees of preference of A and C over D, D over E, E over G, G over H and I, and H and I over F and B become larger. When approaches its maximum 1, the preference 36 relations between A and C, H and I, and B and F are all reversed. The aggregated ranking at this time C A D E G H I F B is exactly the same as the ranking generated using Borda Kendall method, but the above rankings provide the valuable degrees of preference over the ranking of the proposals.

19 Y.-M. Wang et al. / Computers & Operations Research 32 (2005) Table 10 The aggregated rankings corresponding to Table 9 Ranking =0 56:88% 53:49% 65:49% 55:02% 51:95% 52:51% 56:42% 51:46% u A u C u D u E u G u I u H u B =1=180 56:76% 55:05% 66:66% 55:64% 52:86% 52:26% 59:90% 51:42% u A u C u D u E u G u I u H u B =1=72 56:41% 59:51% 66:10% 57:46% 55:38% 51:60% 69:75% 51:20% u A u C u D u E u G u I u H u B =1=45 55:27% 74:53% 67:74% 63:48% 64:51% 50:41% 94:00% 50:50% u A u C u D u E u G u H u I u B =1=36 50:67% 100% 85:71% 100% 100% 58:38% 100% 52:87% u C u A u D u E u G u H u I u F u B u F u F u F u F Example 3. Consider an incomplete preference ranking problem, which was investigated by Gonzalez-Pachon and Romero [22]. In the problem the set of alternatives is A = {x 1 ;x 2 ;x 3 ;x 4 } and the set of experts is Q = {Q 1 ;Q 2 ;Q 3 ;Q 4 }. The partial rankings given by each expert are as follows: Q 1 : x 1 x 4 x 3 ; Q 2 : x 3 x 2 x 1 and x 4 x 1 ; Q 3 : x 2 x 1 and x 4 x 3 ; Q 4 : x 1 x 3 : From Q 1 : x 1 x 4 x 3, we have the following pairs of LP models: min=max u 1i s:t: u 11 u 14 ; u 14 u 13 ; u 11 + u 12 + u 13 + u 14 =1; u 11 ;u 12 ;u 13 ;u 14 0; where i =1;:::;4 and is a given small positive number ( 6 max =1=3). The optimal solutions are presented in the second column of Table 11. From Q 2 : x 3 x 2 x 1 and x 4 x 1, we solve the following pairs of LP models: min=max u 2i s:t: u 23 u 22 ; u 22 u 21 ; u 24 u 21 ; u 21 + u 22 + u 23 + u 24 =1; u 21 ;u 22 ;u 23 ;u 24 0;

20 2046 Y.-M. Wang et al. / Computers & Operations Research 32 (2005) Table 11 Utility interval estimates for alternatives x 1 x 4 Alternative Expert Weighted average utility Q 1 Q 2 Q 3 Q 4 x 1 [2; 1 ] [0; 1 ] 4 [0; 1 ] 2 [0; 1 ] 2 [1 ; x 2 [0; 1 3] [; 1 ] 2 [; 1 ] [0; 1] [ 1 ; 7 5 ] x 3 [0; 1 3 ] [2; 1 2] [0; 1 2 ] [0; 1 2 ] [1 2 ; 7 8 ] x 4 [; ] [; 1 3] [; 1 ] [0; 1] [ 3 4 ; ] where i =1;:::;4 and 6 max =1=4. The corresponding optimal solutions are shown in the third column of Table 11. From Q 3 : x 2 x 1 and x 4 x 3, we solve the following pairs of LP models: min=max u 3i s:t: u 32 u 31 ; u 34 u 33 ; u 31 + u 32 + u 33 + u 34 =1; u 31 ;u 32 ;u 33 ;u 34 0; where i =1;:::;4 and 6 max = 1. The corresponding utility intervals are presented in the fourth 2 column of Table 11. The incomplete preference information x 1 x 3 given by expert Q 4 is used to solve the following pairs of LP models: min=max u 4i s:t: u 41 u 43 =0; u 41 + u 42 + u 43 + u 44 =1; u 41 ;u 42 ;u 43 ;u 44 0; where i =1;:::;4. The corresponding utility intervals are presented in the fth column of Table 11. The weighted average utility of each alternative is presented in the last column. The nal aggregated rankings are shown in Table 12, from which it is clear that under the assumption of weak orders (ties are allowed) x 4 x 3 x 2 x 1, but under the assumption of strict/strong orders, x 4 x 3 x 2 x 1. This is dierent from the ranking x 2 x 3 x 4 x 1 obtained by Gonzalez-Pachon and Romero [22]. However, a close look at the original information may reveal that our ranking could be more convincing. In fact, two out of the four experts rank x 4 ahead of x 3 whilst the other two experts do not have any direct preference between x 4 and x 3. So it is indeed questionable to generate a consensus ranking to put x 3 ahead of x 4. It is equally questionable to put x 2 ahead of x 3 because one

21 Y.-M. Wang et al. / Computers & Operations Research 32 (2005) Table 12 The aggregated rankings under dierent values of for Example 3 Utility Ranking u x1 u x2 u x3 u x4 0 [0, ] [0, 0.875] [0, 0.875] [0, 0.875] 60:87% u x4 u x3 u x2 u x1 0.1 [0.05, ] [0.05, 0.75] [0.05, 0.775] [0.075, ] 50:44% 50:88% 61:54% u x4 u x3 u x2 0.2 [0.1, ] [0.1, 0.625] [0.1, 0.675] [0.15, 0.65] 51:16% 52:27% 62:69% u x4 u x3 u x [0.125, 0.375] [0.125, ] [0.125, 0.625] [0.1875, ] 51:72% 53:33% 63:64% u x4 u x3 u x2 u x1 u x1 u x1 expert actually says that x 3 should be ahead of x 2 and the rest do not provide explicit preferences between x 2 and x 3 at all. 5. Concluding remarks The aggregation of preference rankings has wide applications in social choice, committee election and voting systems. Although a large amount of research has been conducted, there has been no attempt to associate utility intervals with ordinal rankings to generate an aggregated ranking in group decision situations. In this paper, a utility interval estimation method is proposed, where ordinal rankings are interpreted as constraints on utilities and each ordinal ranking corresponds to a set of utility interval estimates. Linear programming models are developed to estimate the ranges of utility intervals. A simple additive weighting method is used to aggregate utility intervals. A simple yet practical preference ranking method for interval numbers is used to generate an aggregated ranking. The proposed method is clear in concept, simple in computation and able to provide the degrees of preference as additional information to the rankings generated. The aggregated rankings thus represent more reliable yet convincing preference relations among alternatives. This feature has been shown using the three dierent types of numerical examples, which demonstrate the validity and potential applications of the proposed method. Acknowledgements The authors would like to thank the Editor, Gilbert Laporte, and two anonymous referees for their constructive comments and suggestions that helped to improve the quality of the paper. References [1] Borda JC. Memoire sur les elections au scrutin. Histoire de l Academie Royale de Science; Paris: 1784 (Translated in the political theory of condorcet. Sommerlad F, Mclean I. Social studies. Working paper 1/89, Oxford, 1989). [2] Black D. The theory of committees and elections. Cambridge: Cambridge University Press; [3] Kendall M. Rank correction methods, 3rd ed. New York: Hafner; 1962.

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