The analysis of scoring matrix in voting system with multiple criteria and multiple places

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1 0 rd International Conference on Advanced Management Science IPEDR vol.9 (0) (0) IACSIT Press, Singaore The analysis of scoring matrix in voting system with mltile criteria and mltile laces Fh-Hwa Franlin LIU +, and Ling-Chan TSAI Deartment of Indstrial Engineering and Management, National Chiao Tng University Box 7, 00 Daxe Road, Hsinch City, Taiwan 000, Reblic of China. Abstract. Decision gros sally emloy voting systems that reflect the democratic content of that gro. The isse of a gro of decision-maers raning candidates with mltile criteria is interesting. A gro rans the effects of those criteria by sing a secific method. For each criterion, a gro votes for candidates with mltile laces. This stdy develoed a rocedre to determine the common set of weights of the laces in those criteria. Candidates were sbseqently raned according to their sm of weighted votes that they earned in all the laces of those criteria. Keywords: Decision analysis; gro decisions; voting system; data enveloment analysis. Introdction Gro decision is always a significant isse in daily life. In decision sciences, two distinct methodologies are basically resent in gro decision-maing: mlti-criteria decision-maing (MCDM) and social choice (SC) theory []. If the MCDM methodology is sed in gro decision-maing, the analytic hierarchy rocess (AHP) [] is ossibly one of the otimal choices. The method rovides a comrehensive and rational framewor to strctre a decision roblem, to reresent and qantify its elements (criteria), to relate the criteria to overall goals, and to evalate alternative soltions. Once the strctre is constrcted, decision maers se a system of airwise comarisons to measre the comonent weights of the strctre, and finally to ran the alternatives in the decision. The analytic networ rocess (ANP) is a more general form of the AHP sed in mlti-criteria decision analysis [,]. In a gro decision-maing environment, aggregating the reference of individals into a consenss rating is reqired. Combining the nmerical airwise comarison jdgments of individals is necessary to form a jdgment for the gro, of which two methods exist: consenss vote and geometrics mean [5]. Consenss vote reqires the gro to research an agreement on the vale of each entry in a matrix of airwise comarisons, thogh this is sally difficlt to achieve. A serior aroach is to regard the geometric mean. [5,6,7]. Frthermore, Leng and Cao [8] arged that the airwise jdgments in AHP are ambigos, and that the determined weights may be meaningless. However, Saaty [9] attribted the ran reversal of AHP to the assmtion of hierarchic decomosition, which is jdged to rodce arbitrary reslts [0,,,]. This henomenon has not yet been flly resolved and may never be becase the aggregation of references transosed from nits of different scales cannot be easily interreted and even qestioned, according to the French school []. In addition, the scale of airwise comarisons is sosedly fndamental in the mind, yet no rles exist for how a transformation to sch a scale occrs. The transformation of an individal of a set of weights + Corresonding athor. Tel.: ; fax: address: fli@mail.nct.ed.tw (F-H F LIU); nell.iem9g@nct.ed.tw (L-C TSAI). 0

2 to the fndamental scale cold change over time. This indicates to the decision maer their consistency, recommending revisions of comarisons that may be de to a manal error for setting the comarisons, occasionally forced de to the er limitation of the comarison scale. However, Ishizaa and Labib [5] mentioned that the decision maer mst answer a significantly larger nmber of qestions, which may be relatively comlex. For large roblems, the ability to calclate the riority vales from all the accetable vales, based on consistency level, renders combinations of airwise comarisons nractical de to the large nmber of ossible different comarison matrices. Decision maers may not have formed a strong oinion on a articlar jdgment; ths the airwise matrix might be incomlete. The aroach is chiefly qalitative and difficlt to jdge and is argably more brdensome to imlement than the voting raning aroach [6] according to both the data reqirement and validation viewoint. The Borda cont [7] assigns oints to candidates according to their osition in the referred order of the voters. Each voter selects and rans candidates from the set of n candidates ( is no more than n). The candidates are laced as the first, the second, and so on to the th lace. In the simlest Borda cont form, a vote of the first, the second, and so on, and the th laces are resectively conted as, ( ), and so on [8]. The Borda cont can be restated as an arithmetic rogress that starts with oints and with one oint decreasing ntil the last lace is assigned with one oint. Finally, candidates are sbseqently raned according to their aggregated scores, which is the sm of the weighted votes. Coo and Kress [9] roosed a rocedre to ran candidates in a referential election by emloying data enveloment analysis (DEA) [0,] to maximize the ga between consective weights of the scoring. For an imortant class of discrimination intensity fnctions, the rocedre is eqivalent to a scoring rle. In articlar if discrimination intensity fnctions in their model are eqal ε, a non-archimedean infinitesimal ositive constant, the rocedre is eqivalent to Borda s rle. Li and Tsai [] considered that the seqences of weights are arithmetic rogression (AP) and geometric rogression (GP). Sch as in Olymic Games, the jdges grade the first three athletes from a large nmber of nations (candidates) in each cometition event, and award medals of gold, silver, and bronze. The weights of gold, silver, and bronze are (,, ) in AP model. The AP model is also eqivalent to Borda s rle. This stdy develoed a rocedre to determine the common set of weights of the laces in those criteria. According the Borda s rle and fixed order of weights roerty, we can obtain the weights of the lace by simle formlations. Candidates are sbseqently raned according to the sm of the weighted votes they earned at all the laces in those criteria. The remainder of this aer is organized as follows: Section introdces fixed order of weights roerty which is indicated the relationshis between criteria and laces; Section introdces the alied rocedre; Section illstrates a nmerical examle; and lastly, Section 5 rovides a conclsion.. Fixed order of weights roerty This stdy introdced an interesting isse that a gro of decision-maers ran candidates with the c criteria. For each criterion, the gro votes for candidates at laces. The criteria can be classified into different classes according to their imortance: j =,, c. Let the weights of laces are V = ( v i i =,..., ) c and the weights of criteria are U = ( j j =,..., c ). Hence, the scoring matrix W = V T U = [ w c ij i =,...,, j =,..., c], the weights of the laces in those c criteria. Note, the serscrits and c are the total nmbers of laces and criteria in the voting system. Candidates are sbseqently raned according to the sm of the weighted votes they earned at all the laces in those criteria. Let b ij denote the total nmber of votes that Candidate received at the ith lace on the criteria in Class j. The score obtained by the candidate is Z : Z = c j= i= w c ij bij Li and Tsai [] consider the fixed order of weights (FOW) roerty that vote weights on the criteria at each lace wold have the orders as the criteria rans and frthermore, the vote weights of a lace shold be higher than that vote weights on the lower laces. In other words, the elements of the scoring matrix are in the orders sch as c w c w c c w w c w c w w c w w i i 05 c ic c c c (), the scoring weights

3 c on each criterion j ossess the relationshi w j w c j w j c, and the scoring weights on each lace i ossess the relationshi w c i w c i w ic c. The relationshi between every two adjacent laces and two c c adjacent classes is assmed simly as w ic wi +,, i =,,. The weight of the ith lace on the last c c class wic is no less than the weight of the (i + ) lace on Class wi +,. The relationshis between weights are shown in Table. Table The general scoring matrix v c c w Criteria c c j c c w j c w c Place v i c wi c ij w w c ic v w c c w j c c c Base on AP model and Borda cont, the gas of consective are. Let w c = x. Then w c = x, w = x + (c ), and w c c = ( )[x + (c )]. Becase w c w c we can obtain x ( )( c ), x = ( )( c ) +, and w c c c = ( )( c ) + c. Hence, v =, v i = i +, = ( ) (c ) + c, j c = c +. In general, the weights of votes vi = i +, i =,...,, the weights of criteria c j = c j +, j =,..., c, and w c ij = ( i + )( c j + ), i =,...,, j =,..., c. The reslts of weights are eqal Li and Tsai [] when normalized the reslts of weights in their model. Frthermore, the scoring matrix can be obtained. Finally, the aggregate score Z cold be calclated by eqation(). The decision maer can se the maximm aggregate score to divide into each aggregate score of candidates for obtain the vales of scores between zero to one.. The alied rocedre Firstly, the gro votes for the c criteria at laces for their imact to evalate the candidates. The criterion with a higher imact is classified into Class, Class for a lower one, and so on. For examle, five criteria are resent: A, A,, A 5 raned in different classes according to their imortances by the to decision maers/managers or other MCDM methodologies. Thereafter, for each criterion, the gro votes for the n candidates with laces. The vales b ij, i =,,, j =,, c, =,, n cold be collected. Finally, the roosed model cold be sed to obtain the weights of laces and calclate the aggregate score of each candidate. In or model, we rovide a roerty to find the normalized scoring matrix in general voting systems. Decision maer can se the eqation v i = i +, i =,..., and c j = c j +, j =,..., c to rodce the scoring matrix wij c = ( i + )( c j + ), i =,...,, j =,..., c easily. The eqation j c = c j + is the minimm mltile vale of scoring matrix to satisfy the constraints of fixed order of weights. Decision maer also can adjst the mltile sch as c j = c j + + a to find the scoring matrix for satisfying the roerty of fixed order of weights. Emloy the fixed order of weights roerty, the vote weights of freqent voting cases are arranged in Table. Table Vote weights c w ij of several voting cases c w c No. of lace Nmber of criteria c= c= c= c=

4 = = = v v v v v v v v v Nmerical examle One comany wants to select three sly chain members from ten candidates. Six decision maers are resent in the selection committee. The director managers choose the imortance of criteria before the review meeting. Their orders of criteria are cost (Class ), delivery (Class ), qality (Class ), and flexibility (Class ). In the review meeting, the decision maers vote the three laces for candidates according to their references for each criterion. The collected data are shown in Table. Table Votes table Criterion Cost (j = ) Delivery (j = ) Qality (j = ) Flexibility (j = ) Place i st nd rd st nd rd st nd rd st nd rd A B C D E F G H I J Using the scoring matrix in Table, the aggregate scores of candidates are shown in Table. The normalized scores of candidates are as same as the reslts of aggregates score in Li and Tsai []. Finally, the decision maer can choose Candidate A, B, and C to be their sly chain members. Table Aggregate scores and rans Candidate b ij Candidate Score Z Normalized score Ran A 95 B C D E F G

5 5. Conclsion and discssion H I J This stdy obtains the weight relationshis between the nmbers of criteria and the nmbers of laces when the evalated criteria are ordered according to their level of imortance by sing a MCDM method or the reference of a riority committee/manager. The rocedre avoids the comlex comarison roblems in traditional AHP and maes the evalated rocess easily in decision maing roblems. 6. References [] B. Srdjevic. Lining analytic hierarchy rocess and social choice methods to sort gro decision-maing in water management. Decis. Sort Syst. 007, : 6-7. [] T. L. Saaty. The Analytic Hierarchy Process: Planning, Priority Setting, Resorce Allocation. McGraw-Hill, New Yor, 980. [] T. L. Saaty. The Analytic Networ Process: Decision Maing With Deendence and Feedbac, nd ed. RWS Pblications, Pennsylvania, 00. [] T. L. Saaty. Theory and Alications of the Analytic Networ Process: Decision Maing with Benefits, Oortnities, Costs and Riss, rd ed. RWS Pblications, Pennsylvania, 005. [5] T. L. Saaty, and J. S. Shang. Gro decision-maing: head-cont verss intensity of reference. Socio. Econ. Plan. Sci. 007, : -7. [6] J. Aczél, and T. L. Saaty. Procedres for synthesizing ratio jdgements. J. Math. Psychol. 98, 7: 9-0. [7] A. Yosefic, and A. Hadi-Vencheh. An integrated gro decision maing model and its evalation by DEA for atomobile indstry. Exert Syst. Al. 00, 7: [8] L. C. Leng, and D. Cao. On the efficacy of modeling mlti-attribte decision roblems sing AHP and Sinarchy. Er. J. Oer. Res. 00, : 9-9. [9] T. L. Saaty. Axiomatic fondation of the analytic hierarchy rocess. Manage. Sci. 986, : [0] V. Belton, and T. Gear. On a short-coming of Saaty's method of analytic hiearachies. Omega-Int. J. Manage. S. 98, : 8-0. [] V. Belton, and T. Gear. The legitimacy of ran reversal - a comment. Omega-Int. J. Manage. S. 985, : -. [] J. S. Dyer. Remars on the analytic hierarchy rocess. Manage. Sci. 990, 6: [] S. R. Watson, and A. Freeling. Assessing attribte weights. Omega-Int. J. Manage. S. 98, 0: [] B. Roy. Mlticriteria Methodology for Decision Analysis. Klwer Academic Pblishers, Dordrecht, 996. [5] A. Ishizaa, and A. Labib. Review of the main develoments of the analytic hierarchy rocess. Exert Syst. Al. 0, 8: 6-5. [6] F.-H. F. Li, and H.-L. Hai. The voting analytic hierarchy rocess method for selecting slier. Int. J. Prod. Econ. 005, 97: [7] J.-C. Borda. Mémoire sr les élections a scrtin. Paris, 78. [8] D. Blac. The Theory of Committees and Elections. Cambridge University Press, Cambridge, 958. [9] W. D. Coo, and M. Kress. A data enveloment model for aggregating reference ranings. Manage. Sci. 990, 6: 0-0. [0] A. Charnes, W. W. Cooer, and E. Rhodes. Measring the efficiency of decision maing nits. Er. J. Oer. Res. 978, : 9-. [] W. W. Cooer, L. M. Seiford, and K. Tone. Data Enveloment Analysis: A Comrehensive Text with Models, Alications, References and DEA-Solver Software, nd ed. Sringer, Boston, 006. [] F.-H. F. Li, and L.-C. Tsai. The voting theories with arithmetic and geometric rogression, 0 (woring aer). 08