- Email: [email protected]
A selection method for a preferential election
Applied Mathematics and Computation 163 (2005) 107–116 www.elsevier.com/locate/amc
A selection method for a preferential election A.A. Foroughi
a,*
, D.F. Jones b, M. Tamiz
b
a
b
Department of Mathematics, University of Qom, Qom 37165, Iran Department of Mathematics, University of Portsmouth, Buchingham Building, Lion Terrace, Portsmouth PO1 3HE, UK Received 17 August 2003; accepted 30 October 2003
Abstract This paper deals with a recently proposed method for a ranked voting system. It is based on a data envelopment analysis model. It is shown that the number of problems to be solved and the number of constraints in the model can be decreased. In addition, the winner can be selected with a fewer efforts and in some cases without solving any problems. The obtained results also illustrate the importance of selecting a proper norm in the model. Ó 2004 Elsevier Inc. All rights reserved. Keywords: Data envelopment analysis; Voting; Efficiency
1. Introduction In a preferential election a voter selects a subset of the candidates and places them in a ranked order. The problem to be considered is how the candidates are ranked and the winner is selected using the number of first, second, third, etc. place votes each candidate received. A well known method for this purpose is to impose a predetermined set of weights on each candidates’s standing and use a total score:
*
Corresponding author. E-mail address: [email protected] (A.A. Foroughi).
0096-3003/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2003.10.055
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Zi ¼
t X
yij uj ;
j¼1
for each candidate i ði ¼ 1; 2; . . . ; nÞ, where yij is the number of jth place votes candidate i receives, and uj ðj ¼ 1; . . . ; tÞ is the sequence of weights given to the jth place vote. Cook and Kress [3] consider that schemes involving an imposed set of weights do not provide a fair overall assessment. They propose using data envelopment analysis (DEA) [2,4] and present the following DEA/assurance region (AR) [4,8] model Zp ¼ max
t X
ð1Þ
ypj uj ;
j¼1
s:t:
t X
yij uj 6 1;
i ¼ 1; 2; . . . ; n;
j¼1
uj ujþ1 P dðj; Þ;
j ¼ 1; . . . ; t 1;
ut P dðt; Þ where dð:; Þ is called the discrimination intensity function that is non-negative and non-decreasing in a non-negative and satisfies dð:; 0Þ ¼ 0. This problem is solved for each candidate p ðp ¼ 1; 2; . . . ; nÞ to find their scores using the weights obtained by themselves. Noting the existing problems of selecting discrimination functions dð:; Þ, Green et al. [5] propose using the idea of cross-evaluation in DEA, and selecting all dð:; Þ equal to zero. Hashimoto [6] also proposes using the DEA exclusion model, introduced by Andersen and Petersen [1], with dð:; Þ ¼ , which is a positive non-Archimedean infinitesimal. In a recent paper by Obata and Ishii [7], another model is proposed to discriminate efficient points obtained by (1). They use the model: 1 ¼ min kuk; b Z P t X ypj uj ¼ 1; s:t:
ð2Þ
j¼1 t X
yij uj 6 1;
i ¼ 1; 2; . . . ; n; i 6¼ p;
j¼1
uj ujþ1 P dðj; Þ; ut P dðt; Þ;
j ¼ 1; . . . ; t 1;
A.A. Foroughi et al. / Appl. Math. Comput. 163 (2005) 107–116
109
where k:k is a certain norm. This problem is solved for each candidate p b are used as the discrimination scores, and the candidate ðp ¼ 1; 2; . . . ; nÞ, Z P with the highest score is selected as the winner. Note that n problems each with n þ t 1 constraints must be solved to find the winner by this model. It is shown in this paper that solving (2) is not necessary for many candidates, and also many of the constraints in the model are redundant, so the number of problems to be solved as well as the number of constraints are decreased. In addition, selection of the norm is important in this method. For the l1 -norm and the l1 -norm the models are linear programming problems and it is shown in this paper that the winner can be selected with less effort and in some cases without solving any problems. For the l1 -norm, it is shown that the winner is the one with the biggest aggregate number of votes (with the weights all equal to one), which illustrates the disadvantage of using this norm for the model.
2. Preliminary discussions and basic definitions Let Y be an n t matrix with component yij as defined before. Each candidate i corresponds to the ith row of Y , say yi . Also let U ¼ fujuj ujþ1 P dðj; Þ; j ¼ 1; . . . ; t 1; ut P dðt; Þg be the assurance region of the weights. The following definitions are used in this paper: Definition 1. A candidate p is efficient, corresponding to the given weights-set U , if and only if there does not exist another yi such that yi u > yp u for all u 2 U . Definition 2. A candidate p is convex-efficient (c-efficient), corresponding to U , if and only if there exists u 2 U such that yp u P yi u, for all i ði ¼ 1; 2; . . . ; nÞ. For each case, the vector yp corresponding to the candidate p is called ARefficient or AR-c-efficient, respectively. In this paper, it is assumed that dðj; Þ ¼ , in which is a positive nonArchimedean infinitesimal and is used only for discriminating between strong and weak efficiency. Now set u0 ¼ Au, in which A is a t t matrix with the component ajj ¼ 1, ajjþ1 ¼ 1, for all j, and zeroes elsewhere. With this transformation U is transformed to U 0 ¼ fu0 ju0 P eg in which e is a vector of ones. Let A1 be the 1 inverse of A, with the components a1 rj , which can be obtained as: arj ¼ 0 when 1 1 r > j, and arj ¼ 1 when j P r. By noting that yi u ¼ yi A Au ¼ yi0 u0 , with Y 0 ¼ YA1 , the previous definitions are seen to be equivalent with the following: Definition 10 . A candidate p (or equivalently yp0 ) is (strongly) efficient if and only if there does not exist another yi0 such that yi0 P yp0 and yi0 6¼ yp0 .
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Definition 20 . A candidate p (or yp0 ) is (strongly) convex-efficient (c-efficient) if and only if there exists u0 > 0 such that yp0 u0 P yi0 u0 , for all i ði ¼ 1; 2; . . . ; nÞ. Note that there exists u0 > 0 such that yp0 u0 P yi0 u0 for all i ði ¼ 1; 2; . . . ; nÞ, if and only if there dose not exist a convex combination of yi0 which dominates yp0 . Let E and Ec be the sets of all efficient (corresponding to U ) and c-efficient candidates, respectively. Obviously each c-efficient point is efficient so Ec E. Similarly Nc and N are used for the index-sets of inefficient and c-inefficient candidates that is: N ¼ I E and Nc ¼ I Ec , in which I ¼ f1; 2; . . . ; ng, and N Nc .
3. Reducing the model By applying the conclusions of the above discussions, (2) is transformed to the following equivalent model 1 ¼ min kA1 u0 k; b Z P s:t:
t X
ð3Þ
0 0 ypj uj ¼ 1;
j¼1 t X
yij0 u0j 6 1;
i ¼ 1; 2; . . . ; n; i 6¼ p;
j¼1
u0j P ;
j ¼ 1; . . . ; t:
This problem gives a score to each c-efficient candidate and is infeasible for each c-inefficient point. On the other hand, all constraints corresponding to inefficient candidates are redundant in the model. In addition, since the original model is a DEA/AR model, a large number of inefficient points are expected. Hence determining inefficient points before solving the model would reduce the number of constraints as well as the number of problems to be solved. For this purpose, using Definition 20 , a test is used to determine all inefficient candidates (N) without solving any problem. Indeed, for each candidate p, the search for the existence of another candidate i such that yi0 P yp0 is undertaken using the following test Test 1. Let J ¼ f1; . . . ; tg, and I ¼ f1; 2; . . . ; ng. For every p 2 I set 0 Ip ¼ I fpg and if maxi2Ip fminj2J fyij0 ypj gg P 0 then remove p from I and repeat this for all the other candidates.
A.A. Foroughi et al. / Appl. Math. Comput. 163 (2005) 107–116
111
After this we need only the constraints corresponding to the elements in the last I obtained from this process. Indeed, inefficient candidates and also the repeated constraints are deleted from further considerations. Assuming that there are no ties between efficient candidates, this set is equal to E, and without this assumption it is a subset of E. Since the ties between the candidates can be determined easily, we may assume that there is not any tie between the candidates.
4. Solving the model It is assumed first that ¼ 0, and the non-zero value for is used only when necessary. Using this and the above discussions (3) can be written as follows 1 ¼ min kA1 u0 k; b Z P t X 0 0 ypj uj ¼ 1; s:t:
ð4Þ
j¼1 t X
yij0 u0j 6 1;
i 2 Ep ;
j¼1
u0j P 0;
j ¼ 1; . . . ; t;
in which Ep is E fpg, and p 2 E. Noting the selection of , some of (weakly) c-inefficient points may give a greater score than some c-efficient points. But it is shown that when the greatest score obtained by (4) is unique (between existing candidates after Test 1), it is strongly c-efficient and can be selected as the winner. To prove this, let p be such candidate. By contradiction assume that there exists a convex combinaP 0 0 tion of other candidates that dominates p, that is y ' k y i p i , which i2Ip P 0 i2Ip ki ¼ 1, and all ki P 0. Also let u be the optimal solution of (4) corresponding to this p. Then X 1 ¼ yp0 u0 6 ki yi0 u0 ) yi0 u0 ¼ 1 for all ki 6¼ 0; i2Ip
and this means that u0 is a feasible solution for the problems corresponding to b 6 Z b which is a all these i (which at least one exists by the assumption), so Z P i contradiction. Now consider (4) P and let us first use the l1 -norm. By the definition of A we have: kA1 u0 k1 ¼ tj¼1 ju0j . By u00j ¼ ju0j and yj00 ¼ 1j yj0 , (4) is transformed to the following model
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A.A. Foroughi et al. / Appl. Math. Comput. 163 (2005) 107–116 t X 1 ¼ min u00j ; b Z j¼1
ð5Þ
P
s:t:
t X
00 00 ypj uj ¼ 1;
j¼1 t X
yij00 u00j 6 1;
i 2 Ep ;
j¼1
u0j P 0;
j ¼ 1; . . . ; t;
To select the winner, (5) should be solved for all efficient points. The following test can be used to find some optimal solutions of this model without solving it 00 00 Test 2. For each j 2 f1; . . . ; tg, let ypj ¼ maxi2E fyij g. Then, the optimal 00 00 00 b ¼ y , if y P y for all j 2 f1; . . . ; tg. solution of (5) is Z pj pj pj P
Proof. By the assumptions, u00j ¼ y 001 and u00j ¼ 0, for all j 6¼ j , is a feasible pj
solution to (5). Now for every feasible solution to (5), for the candidate p, we have 1¼
t X
00 00 ypj uj 6
j¼1
t X
00 00 00 ypj uj ¼ ypj
j¼1
which gives the result.
t X j¼1
u00j )
t X j¼1
u00j P
1 00 ypj
h
In addition, the following test can help to decrease the number of candidates to be considered: Test 20 . For each candidate i, b 6 maxfy 00 g: Z i ij j2J
Using this test, there is no need to solve the problem when the maximum possible score for a candidate is less than the biggest score obtained before. Test 3. For the l1 -norm, it is shown that the optimal value of (4) corresponding to the winner can be found without solving any problem. Indeed, using the relationship between the ui ðu1 P u2 P P ut Þ, kuk1 ¼ u1 , so (4) is transformed to the following model
A.A. Foroughi et al. / Appl. Math. Comput. 163 (2005) 107–116 t X 1 ¼ min u0j ; b Z j¼1
113
ð6Þ
P
t X
s:t:
0 0 ypj uj ¼ 1;
j¼1 t X
yij0 u0j 6 1;
i 2 Ep ;
j¼1
u0j P 0;
j ¼ 1; . . . ; t:
We show that the optimal solution of (6) for the winner is b ¼ y 0 ¼ maxi2I fy 0 g. Indeed for this p by (6) Z P pt it 1 ¼ yp0 u0 6
X
! u0j ypt0 )
j
X
u0j P
j
1 : ypt0
On the other hand u0t ¼ y10 , u0j ¼ 0; j 6¼ t, is a feasible solution to (6), which pt gives Z^P ¼ ypt0 . It is sufficient to show that this is the best score. For this pur0 pose, let p be an arbitrary candidate. Using the assumptions and by replacing p with p0 in (6) we have
1¼
yp0 0 u0
6
X j
! u0j
yp0 0 t
6
X j
! u0j
ypt0 )
X j
u0j P
1 b 0 6 Z b: )Z P P ypt0
In brief, using the above discussions, to select the winner by (3) we can first remove all inefficient points with an easy test. This decreases the number of constraints as well as the number of problems that are necessary to be solved. Then, with this assumption that dðj; Þ ¼ 0 for all j, (3) is solved. If the biggest score is unique, its corresponding candidate is strongly c-efficient and is selected as the winner, else another test is necessary to distinguish between the candidates that share the biggest score. For example (3) may be solved by selecting some values for dðj; Þ, for example dðj; Þ ¼ j, and then solve it only for these candidates. On the other hand, selection of the norm is important in this model. For the l1 -norm and the l1 -norm the model is a linear programming problem and other tests are devised to simplify the solution process. For the l1 -norm it is shown that the winner can be selected without solving any problem. This indeed shows the disadvantage of using this norm for the model, since it is equivalent to a fixed-weight model with equal weights.
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5. Example Consider the following data from [7]: Candidate
First rank
Second rank
The data for Y (n ¼ 6, t ¼ 2) A 32 B 28 C 13 D 20 E 27 F 30 G 0
10 20 36 27 19 8 30
Candidate
First rank
First + second rank
32 28 13 20 27 30 0
42 48 49 47 46 38 30
The data for Y A B C D E F G
0
Test 1. Using this data, the candidates D and E (both dominated by B), F (dominated by A), and G (dominated by all), are determined as inefficient. Therefore, we need only to solve the problems 1 t ¼ min k½u01 þ u02 ; u02 k; b ZP
ð7Þ
s:t: 32u01 þ 42u02 6 ð¼; if p ¼ AÞ1; 28u01 þ 48u02 6 ð¼; if p ¼ BÞ1; 13u01 þ 49u02 6 ð¼; if p ¼ CÞ1; u0j P 0;
j ¼ 1; 2;
for the candidates p 2 fA; B; Cg, to obtain the same results which is given by the original model (2) with solving 8 problems each with 8 constraints. Now we consider the following simpler norms:
A.A. Foroughi et al. / Appl. Math. Comput. 163 (2005) 107–116
115
l1 -norm: To use Test 2, Y 00 should be found first: Candidate
First rank
1 (First + second 2
rank)
00
The data for Y (for c-efficient candidates) A 32 B 28 C 13
21 24 24.5
b 0 ¼ 32 and Z^ 0 ¼ 24:5, can be found. On the other By Test 2 the scores Z A C 0 b 6 28 < Z b 0 , so there is no need to solve its hand by Test 2 we know that Z B A corresponding problem and the winner is selected to be A. Note that the number of problems to be solved may change by altering the order of selection of candidates to be considered. For example if we first find the score of A then there is no need to find the scores of B and C, by using Test 20 . If we find the score of B in the first instance we still need to find the score of A and even C (if its score is found before A and depending on the obtained score of B). In the worst case by finding the score of C first and then B and finally A, all the scores should be found (but only for B the problem is needed to be solved). Hence the order of selection may compromise the usefulness of Test 20 . To overcome this, it is proposed that first all candidates are sorted in the order of their highest possible scores, that is the maximum values in the rows of Y 00 as was shown in Test 20 . By this ordering, it can be seen that there is no need to solve problems corresponding to inefficient points even if we do not use Test 2. l1 -norm: For this norm candidate C, corresponding to the biggest value in the last column of Y 0 , is selected as the winner.
6. Conclusions This paper considers selection of the winner in a preferential election, by simplifying a recently proposed model. The model is based on DEA/AR so a large number of inefficient candidates is expected. Noting that there is no need to solve the model for inefficient candidates and that the constraints corresponding to these candidates are redundant, a test is proposed to determine them before using the model. This test is based on a transformation on the given data which makes it possible to use the definition of efficiency instead of AR-efficiency. Some methods are proposed to simplify solving the remaining problems. For the l1 -norm and the l1 -norm, the model is a linear programming problem and other tests are devised that simplify solving the model. For the l1 -norm, it
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A.A. Foroughi et al. / Appl. Math. Comput. 163 (2005) 107–116
is shown that the winner can be selected without solving any problems. This indeed shows the disadvantage of this norm for the model, since it is equivalent to a fixed-weight model with equal weights.
References [1] P. Andersen, N.C. Petersen, A procedure for ranking efficient units in data envelopment analysis, Management Science 39 (10) (1993) 1261–1264. [2] A. Charnes, W.W. Cooper, E.L. Rhodes, Measuring the efficiencies of decision making units, European Journal of Operational Research 2 (6) (1978) 429–444. [3] W.D. Cook, M. Kress, A data envelopment model for aggregating preference ranking, Management Science 36 (11) (1990) 1302–1310. [4] W.W. Cooper, L.M. Seiford, K. Tone, Data Envelopment Analysis, A comprehensive text with models, Applications, references and DEA-solver software, Kluwer Academic Publishers, Boston, 2000. [5] R.H. Green, J.R. Doyle, W.D. Cook, Preference voting and project ranking using DEA and cross-evaluation, European Journal of Operational Research 90 (1996) 461–472. [6] A. Hashimoto, A ranked voting system using a DEA/AR exclusion model: A note, European Journal of Operational Research 97 (1997) 600–604. [7] T. Obata, H. Ishii, A method for discriminating efficient candidates with ranked voting data, European Journal of Operational Research 151 (2003) 233–237. [8] R.G. Thompson, F.D. Singleton jr., R.M. Thrall, B.A. Smith, Comparative site evaluations for locating a high-energy physics lab in Texas, Interfaces 16 (6) (1986) 35–49.
A selection method for a preferential election A.A. Foroughi
a,*
, D.F. Jones b, M. Tamiz
b
a
b
Department of Mathematics, University of Qom, Qom 37165, Iran Department of Mathematics, University of Portsmouth, Buchingham Building, Lion Terrace, Portsmouth PO1 3HE, UK Received 17 August 2003; accepted 30 October 2003
Abstract This paper deals with a recently proposed method for a ranked voting system. It is based on a data envelopment analysis model. It is shown that the number of problems to be solved and the number of constraints in the model can be decreased. In addition, the winner can be selected with a fewer efforts and in some cases without solving any problems. The obtained results also illustrate the importance of selecting a proper norm in the model. Ó 2004 Elsevier Inc. All rights reserved. Keywords: Data envelopment analysis; Voting; Efficiency
1. Introduction In a preferential election a voter selects a subset of the candidates and places them in a ranked order. The problem to be considered is how the candidates are ranked and the winner is selected using the number of first, second, third, etc. place votes each candidate received. A well known method for this purpose is to impose a predetermined set of weights on each candidates’s standing and use a total score:
*
Corresponding author. E-mail address: [email protected] (A.A. Foroughi).
0096-3003/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2003.10.055
108
A.A. Foroughi et al. / Appl. Math. Comput. 163 (2005) 107–116
Zi ¼
t X
yij uj ;
j¼1
for each candidate i ði ¼ 1; 2; . . . ; nÞ, where yij is the number of jth place votes candidate i receives, and uj ðj ¼ 1; . . . ; tÞ is the sequence of weights given to the jth place vote. Cook and Kress [3] consider that schemes involving an imposed set of weights do not provide a fair overall assessment. They propose using data envelopment analysis (DEA) [2,4] and present the following DEA/assurance region (AR) [4,8] model Zp ¼ max
t X
ð1Þ
ypj uj ;
j¼1
s:t:
t X
yij uj 6 1;
i ¼ 1; 2; . . . ; n;
j¼1
uj ujþ1 P dðj; Þ;
j ¼ 1; . . . ; t 1;
ut P dðt; Þ where dð:; Þ is called the discrimination intensity function that is non-negative and non-decreasing in a non-negative and satisfies dð:; 0Þ ¼ 0. This problem is solved for each candidate p ðp ¼ 1; 2; . . . ; nÞ to find their scores using the weights obtained by themselves. Noting the existing problems of selecting discrimination functions dð:; Þ, Green et al. [5] propose using the idea of cross-evaluation in DEA, and selecting all dð:; Þ equal to zero. Hashimoto [6] also proposes using the DEA exclusion model, introduced by Andersen and Petersen [1], with dð:; Þ ¼ , which is a positive non-Archimedean infinitesimal. In a recent paper by Obata and Ishii [7], another model is proposed to discriminate efficient points obtained by (1). They use the model: 1 ¼ min kuk; b Z P t X ypj uj ¼ 1; s:t:
ð2Þ
j¼1 t X
yij uj 6 1;
i ¼ 1; 2; . . . ; n; i 6¼ p;
j¼1
uj ujþ1 P dðj; Þ; ut P dðt; Þ;
j ¼ 1; . . . ; t 1;
A.A. Foroughi et al. / Appl. Math. Comput. 163 (2005) 107–116
109
where k:k is a certain norm. This problem is solved for each candidate p b are used as the discrimination scores, and the candidate ðp ¼ 1; 2; . . . ; nÞ, Z P with the highest score is selected as the winner. Note that n problems each with n þ t 1 constraints must be solved to find the winner by this model. It is shown in this paper that solving (2) is not necessary for many candidates, and also many of the constraints in the model are redundant, so the number of problems to be solved as well as the number of constraints are decreased. In addition, selection of the norm is important in this method. For the l1 -norm and the l1 -norm the models are linear programming problems and it is shown in this paper that the winner can be selected with less effort and in some cases without solving any problems. For the l1 -norm, it is shown that the winner is the one with the biggest aggregate number of votes (with the weights all equal to one), which illustrates the disadvantage of using this norm for the model.
2. Preliminary discussions and basic definitions Let Y be an n t matrix with component yij as defined before. Each candidate i corresponds to the ith row of Y , say yi . Also let U ¼ fujuj ujþ1 P dðj; Þ; j ¼ 1; . . . ; t 1; ut P dðt; Þg be the assurance region of the weights. The following definitions are used in this paper: Definition 1. A candidate p is efficient, corresponding to the given weights-set U , if and only if there does not exist another yi such that yi u > yp u for all u 2 U . Definition 2. A candidate p is convex-efficient (c-efficient), corresponding to U , if and only if there exists u 2 U such that yp u P yi u, for all i ði ¼ 1; 2; . . . ; nÞ. For each case, the vector yp corresponding to the candidate p is called ARefficient or AR-c-efficient, respectively. In this paper, it is assumed that dðj; Þ ¼ , in which is a positive nonArchimedean infinitesimal and is used only for discriminating between strong and weak efficiency. Now set u0 ¼ Au, in which A is a t t matrix with the component ajj ¼ 1, ajjþ1 ¼ 1, for all j, and zeroes elsewhere. With this transformation U is transformed to U 0 ¼ fu0 ju0 P eg in which e is a vector of ones. Let A1 be the 1 inverse of A, with the components a1 rj , which can be obtained as: arj ¼ 0 when 1 1 r > j, and arj ¼ 1 when j P r. By noting that yi u ¼ yi A Au ¼ yi0 u0 , with Y 0 ¼ YA1 , the previous definitions are seen to be equivalent with the following: Definition 10 . A candidate p (or equivalently yp0 ) is (strongly) efficient if and only if there does not exist another yi0 such that yi0 P yp0 and yi0 6¼ yp0 .
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Definition 20 . A candidate p (or yp0 ) is (strongly) convex-efficient (c-efficient) if and only if there exists u0 > 0 such that yp0 u0 P yi0 u0 , for all i ði ¼ 1; 2; . . . ; nÞ. Note that there exists u0 > 0 such that yp0 u0 P yi0 u0 for all i ði ¼ 1; 2; . . . ; nÞ, if and only if there dose not exist a convex combination of yi0 which dominates yp0 . Let E and Ec be the sets of all efficient (corresponding to U ) and c-efficient candidates, respectively. Obviously each c-efficient point is efficient so Ec E. Similarly Nc and N are used for the index-sets of inefficient and c-inefficient candidates that is: N ¼ I E and Nc ¼ I Ec , in which I ¼ f1; 2; . . . ; ng, and N Nc .
3. Reducing the model By applying the conclusions of the above discussions, (2) is transformed to the following equivalent model 1 ¼ min kA1 u0 k; b Z P s:t:
t X
ð3Þ
0 0 ypj uj ¼ 1;
j¼1 t X
yij0 u0j 6 1;
i ¼ 1; 2; . . . ; n; i 6¼ p;
j¼1
u0j P ;
j ¼ 1; . . . ; t:
This problem gives a score to each c-efficient candidate and is infeasible for each c-inefficient point. On the other hand, all constraints corresponding to inefficient candidates are redundant in the model. In addition, since the original model is a DEA/AR model, a large number of inefficient points are expected. Hence determining inefficient points before solving the model would reduce the number of constraints as well as the number of problems to be solved. For this purpose, using Definition 20 , a test is used to determine all inefficient candidates (N) without solving any problem. Indeed, for each candidate p, the search for the existence of another candidate i such that yi0 P yp0 is undertaken using the following test Test 1. Let J ¼ f1; . . . ; tg, and I ¼ f1; 2; . . . ; ng. For every p 2 I set 0 Ip ¼ I fpg and if maxi2Ip fminj2J fyij0 ypj gg P 0 then remove p from I and repeat this for all the other candidates.
A.A. Foroughi et al. / Appl. Math. Comput. 163 (2005) 107–116
111
After this we need only the constraints corresponding to the elements in the last I obtained from this process. Indeed, inefficient candidates and also the repeated constraints are deleted from further considerations. Assuming that there are no ties between efficient candidates, this set is equal to E, and without this assumption it is a subset of E. Since the ties between the candidates can be determined easily, we may assume that there is not any tie between the candidates.
4. Solving the model It is assumed first that ¼ 0, and the non-zero value for is used only when necessary. Using this and the above discussions (3) can be written as follows 1 ¼ min kA1 u0 k; b Z P t X 0 0 ypj uj ¼ 1; s:t:
ð4Þ
j¼1 t X
yij0 u0j 6 1;
i 2 Ep ;
j¼1
u0j P 0;
j ¼ 1; . . . ; t;
in which Ep is E fpg, and p 2 E. Noting the selection of , some of (weakly) c-inefficient points may give a greater score than some c-efficient points. But it is shown that when the greatest score obtained by (4) is unique (between existing candidates after Test 1), it is strongly c-efficient and can be selected as the winner. To prove this, let p be such candidate. By contradiction assume that there exists a convex combinaP 0 0 tion of other candidates that dominates p, that is y ' k y i p i , which i2Ip P 0 i2Ip ki ¼ 1, and all ki P 0. Also let u be the optimal solution of (4) corresponding to this p. Then X 1 ¼ yp0 u0 6 ki yi0 u0 ) yi0 u0 ¼ 1 for all ki 6¼ 0; i2Ip
and this means that u0 is a feasible solution for the problems corresponding to b 6 Z b which is a all these i (which at least one exists by the assumption), so Z P i contradiction. Now consider (4) P and let us first use the l1 -norm. By the definition of A we have: kA1 u0 k1 ¼ tj¼1 ju0j . By u00j ¼ ju0j and yj00 ¼ 1j yj0 , (4) is transformed to the following model
112
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ð5Þ
P
s:t:
t X
00 00 ypj uj ¼ 1;
j¼1 t X
yij00 u00j 6 1;
i 2 Ep ;
j¼1
u0j P 0;
j ¼ 1; . . . ; t;
To select the winner, (5) should be solved for all efficient points. The following test can be used to find some optimal solutions of this model without solving it 00 00 Test 2. For each j 2 f1; . . . ; tg, let ypj ¼ maxi2E fyij g. Then, the optimal 00 00 00 b ¼ y , if y P y for all j 2 f1; . . . ; tg. solution of (5) is Z pj pj pj P
Proof. By the assumptions, u00j ¼ y 001 and u00j ¼ 0, for all j 6¼ j , is a feasible pj
solution to (5). Now for every feasible solution to (5), for the candidate p, we have 1¼
t X
00 00 ypj uj 6
j¼1
t X
00 00 00 ypj uj ¼ ypj
j¼1
which gives the result.
t X j¼1
u00j )
t X j¼1
u00j P
1 00 ypj
h
In addition, the following test can help to decrease the number of candidates to be considered: Test 20 . For each candidate i, b 6 maxfy 00 g: Z i ij j2J
Using this test, there is no need to solve the problem when the maximum possible score for a candidate is less than the biggest score obtained before. Test 3. For the l1 -norm, it is shown that the optimal value of (4) corresponding to the winner can be found without solving any problem. Indeed, using the relationship between the ui ðu1 P u2 P P ut Þ, kuk1 ¼ u1 , so (4) is transformed to the following model
A.A. Foroughi et al. / Appl. Math. Comput. 163 (2005) 107–116 t X 1 ¼ min u0j ; b Z j¼1
113
ð6Þ
P
t X
s:t:
0 0 ypj uj ¼ 1;
j¼1 t X
yij0 u0j 6 1;
i 2 Ep ;
j¼1
u0j P 0;
j ¼ 1; . . . ; t:
We show that the optimal solution of (6) for the winner is b ¼ y 0 ¼ maxi2I fy 0 g. Indeed for this p by (6) Z P pt it 1 ¼ yp0 u0 6
X
! u0j ypt0 )
j
X
u0j P
j
1 : ypt0
On the other hand u0t ¼ y10 , u0j ¼ 0; j 6¼ t, is a feasible solution to (6), which pt gives Z^P ¼ ypt0 . It is sufficient to show that this is the best score. For this pur0 pose, let p be an arbitrary candidate. Using the assumptions and by replacing p with p0 in (6) we have
1¼
yp0 0 u0
6
X j
! u0j
yp0 0 t
6
X j
! u0j
ypt0 )
X j
u0j P
1 b 0 6 Z b: )Z P P ypt0
In brief, using the above discussions, to select the winner by (3) we can first remove all inefficient points with an easy test. This decreases the number of constraints as well as the number of problems that are necessary to be solved. Then, with this assumption that dðj; Þ ¼ 0 for all j, (3) is solved. If the biggest score is unique, its corresponding candidate is strongly c-efficient and is selected as the winner, else another test is necessary to distinguish between the candidates that share the biggest score. For example (3) may be solved by selecting some values for dðj; Þ, for example dðj; Þ ¼ j, and then solve it only for these candidates. On the other hand, selection of the norm is important in this model. For the l1 -norm and the l1 -norm the model is a linear programming problem and other tests are devised to simplify the solution process. For the l1 -norm it is shown that the winner can be selected without solving any problem. This indeed shows the disadvantage of using this norm for the model, since it is equivalent to a fixed-weight model with equal weights.
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5. Example Consider the following data from [7]: Candidate
First rank
Second rank
The data for Y (n ¼ 6, t ¼ 2) A 32 B 28 C 13 D 20 E 27 F 30 G 0
10 20 36 27 19 8 30
Candidate
First rank
First + second rank
32 28 13 20 27 30 0
42 48 49 47 46 38 30
The data for Y A B C D E F G
0
Test 1. Using this data, the candidates D and E (both dominated by B), F (dominated by A), and G (dominated by all), are determined as inefficient. Therefore, we need only to solve the problems 1 t ¼ min k½u01 þ u02 ; u02 k; b ZP
ð7Þ
s:t: 32u01 þ 42u02 6 ð¼; if p ¼ AÞ1; 28u01 þ 48u02 6 ð¼; if p ¼ BÞ1; 13u01 þ 49u02 6 ð¼; if p ¼ CÞ1; u0j P 0;
j ¼ 1; 2;
for the candidates p 2 fA; B; Cg, to obtain the same results which is given by the original model (2) with solving 8 problems each with 8 constraints. Now we consider the following simpler norms:
A.A. Foroughi et al. / Appl. Math. Comput. 163 (2005) 107–116
115
l1 -norm: To use Test 2, Y 00 should be found first: Candidate
First rank
1 (First + second 2
rank)
00
The data for Y (for c-efficient candidates) A 32 B 28 C 13
21 24 24.5
b 0 ¼ 32 and Z^ 0 ¼ 24:5, can be found. On the other By Test 2 the scores Z A C 0 b 6 28 < Z b 0 , so there is no need to solve its hand by Test 2 we know that Z B A corresponding problem and the winner is selected to be A. Note that the number of problems to be solved may change by altering the order of selection of candidates to be considered. For example if we first find the score of A then there is no need to find the scores of B and C, by using Test 20 . If we find the score of B in the first instance we still need to find the score of A and even C (if its score is found before A and depending on the obtained score of B). In the worst case by finding the score of C first and then B and finally A, all the scores should be found (but only for B the problem is needed to be solved). Hence the order of selection may compromise the usefulness of Test 20 . To overcome this, it is proposed that first all candidates are sorted in the order of their highest possible scores, that is the maximum values in the rows of Y 00 as was shown in Test 20 . By this ordering, it can be seen that there is no need to solve problems corresponding to inefficient points even if we do not use Test 2. l1 -norm: For this norm candidate C, corresponding to the biggest value in the last column of Y 0 , is selected as the winner.
6. Conclusions This paper considers selection of the winner in a preferential election, by simplifying a recently proposed model. The model is based on DEA/AR so a large number of inefficient candidates is expected. Noting that there is no need to solve the model for inefficient candidates and that the constraints corresponding to these candidates are redundant, a test is proposed to determine them before using the model. This test is based on a transformation on the given data which makes it possible to use the definition of efficiency instead of AR-efficiency. Some methods are proposed to simplify solving the remaining problems. For the l1 -norm and the l1 -norm, the model is a linear programming problem and other tests are devised that simplify solving the model. For the l1 -norm, it
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is shown that the winner can be selected without solving any problems. This indeed shows the disadvantage of this norm for the model, since it is equivalent to a fixed-weight model with equal weights.
References [1] P. Andersen, N.C. Petersen, A procedure for ranking efficient units in data envelopment analysis, Management Science 39 (10) (1993) 1261–1264. [2] A. Charnes, W.W. Cooper, E.L. Rhodes, Measuring the efficiencies of decision making units, European Journal of Operational Research 2 (6) (1978) 429–444. [3] W.D. Cook, M. Kress, A data envelopment model for aggregating preference ranking, Management Science 36 (11) (1990) 1302–1310. [4] W.W. Cooper, L.M. Seiford, K. Tone, Data Envelopment Analysis, A comprehensive text with models, Applications, references and DEA-solver software, Kluwer Academic Publishers, Boston, 2000. [5] R.H. Green, J.R. Doyle, W.D. Cook, Preference voting and project ranking using DEA and cross-evaluation, European Journal of Operational Research 90 (1996) 461–472. [6] A. Hashimoto, A ranked voting system using a DEA/AR exclusion model: A note, European Journal of Operational Research 97 (1997) 600–604. [7] T. Obata, H. Ishii, A method for discriminating efficient candidates with ranked voting data, European Journal of Operational Research 151 (2003) 233–237. [8] R.G. Thompson, F.D. Singleton jr., R.M. Thrall, B.A. Smith, Comparative site evaluations for locating a high-energy physics lab in Texas, Interfaces 16 (6) (1986) 35–49.