A Note on A Solution Method to the Problem. Proposed by Wang in Voting Systems

Transcription

1 Applied Matheatical Sciences, Vol. 5, 0, no. 6, A Note on A Solution Method to the Proble Proposed by Wang in Voting Systes F. Hosseinzadeh Lotfi Departent of Matheatics, Science and Research Branch Islaic Azad Uniersity, Tehran, Iran R. Fallahnead Departents of Matheatics, Khorraabad Branch Islaic Azad Uniersity, Khorraabad, Iran Abstract Recently Wang et al. [N. S. Wang, R. H. Yi, D. Liu, A solution ethod to the proble proposed by Wang in oting systes, Journal of Coputational and Applied Matheatics, (008), 063] proposed a ethod for soe odifications on the ethod proposed by Wang and Chin [Y.M. Wang, K.S. Chin, Discriinating DEA efficient candidates by considering their least relatie total scores, Journal of Coputational and Applied Matheatics, 06 (007), 09 5] in oting systes. In this article, ith to siple exaples, e sho that theirs ethod cannot be capable in ranking oting systes in all cases. Keyords: Data Enelopent Analysis; Rank Voting Systes. Wang et al. s Method In oting systes, one candidate ay receie different otes in different ranking places. The total score of each candidate is the eighted su of the otes that the Corresponding author: Eail: r.fallahnead@gail.co (Reza Fallahnead) Address: Departent of Matheatics, Khorraabad Branch, Islaic Azad Uniersity, Kiloeter 5, Tehran road, Khorraabad, Lorestan, Iran

2 305 F. Hosseinzadeh Lotfi and R. Fallahnead candidate receies in different places. Eeryone that has the bigger total score has the greater rank. Therefore the key point is the deterination of the eights associated ith different places. In a oting syste, each oter selects candidates aong n candidates ( n ) and ranks the fro top to the th place. Let ( =,..., ) be the relatie iportance eight associated ith the place and i be the otes of the candidate i that it being ranked in the th place. Wang and Chin [] discriinate efficient candidates by considering their least relatie total scores. Wang et al. [] acclaied that in the proposed ethod by Wang and chin [], the least relatie total scores and the best relatie total scores are not easured ithin the sae range. So for soling the entioned proble, they firstly introduced a concept of irtual orst in candidate (VWC) ith the otes = in i{i }, =,...,. Then, they proposed odels () and () for obtaining the beast and the least relatie total scores: Max Min Y α ax i Y α in i = = ε = = ε = i = i,... i i, i =,...,n... i =,..., n Where, α is the best relatie total score for VWC hich, can be obtained fro the folloing odel: Max α = = = i in i, i =,..., n... ε They also entioned that ε ust be in range of [ 0, δ ], here i = δ = in {/ /( ( / ))}. i () () (3)

3 Data enelopent analysis 3053 In this paper ith siple counter exaples, e ill sho that the Wang ethod in soe cases is inalid.. To Counter Exaples Counter Exaple : Suppose that 0 oters are asked to rank 4 out of 7 candidates A~G. The otes each candidate receies are shon in Table. Table : Votes receied by 0 candidates Candidates Place Place Place 3 Place 4 A 0 6 B 4 0 C 0 D 0 E 0 4 F 0 0 G VWC Table : Ealuating Candidates by Models () and () Candidates Best relatie total score, Least relatie total score, ε = 0.00 ε = 0.00 A B C D E F G We can see that VMC = (0,0,0,0), and therefore by using the odel (3), α = 0. No, let use odel () and () for candidates. In table, e can see the results. We can see that both candidates A and B hae the best relatie total score equals and the least relatie total score equal But e see that they hae different otes. Therefore, e see that by using Wang et al. s ethod [], e cannot rank candidates A and B.

4 3054 F. Hosseinzadeh Lotfi and R. Fallahnead It is ery possible that in a oting syste it happens that VMC=(0,,0). In this case, e can see that α = 0. Therefore, in odels and 3, the constraints α i ill be abundant, and ithout soling any proble, optial solutions can be obtained. In this case, in optiality, e hae = =... =, = ε. So, it is possible that soe efficient candidates hae the sae least relatie total scores. Counter Exaple : Suppose again that 0 oters are asked to rank 4 out of 7 candidates A~G. The otes each candidate receies are shon in Table 3. Table 3: Votes receied by 0 candidates Candidates Place Place Place 3 Place 4 A 3 5 B 4 C 0 D 0 0 E F 0 0 G 0 0 VWC Table 4: Ealuating Candidates by Models () and () Candidates Best relatie total Least relatie total score ε = score ε = A B C D E F G Least relatie total score ε = In table 4, e see that although in this exaple, VMC = (0,0,0,0). But if e get ε = 0.00, then candidates can be ranked. But if ε = 0, then their proposed ethod cannot discriinate efficient candidates. Therefore, e ust odify the interal for the acceptable interal for ε fro the [ 0, δ ] to the ( 0, δ ]. Since in [], e see [ 0, δ ] in seeral cases, it is not see that it is a type error.

5 Data enelopent analysis 3055 References [] N. S. Wang, R. H. Yi and D. Liu, A solution ethod to the proble proposed by Wang in oting systes, Journal of Coputational and Applied Matheatics, (008), 063. [] Y.M. Wang, K.S. Chin, Discriinating DEA efficient candidates by considering their least relatie total scores, Journal of Coputational and Applied Matheatics, 06 (007), Receied: May, 0