Issues in Information Systems Volume 14, Issue 1, pp , 2013

Transcription

1 Issues n Informton Systems Volume 4, Issue, pp.58-65, 0 RITI REVIEW OF POPUR MUTI-RITERI DEISION MKING METHODOGIES Yong. Shn, Frns Mron Unversty, yshn@fmron.edu Seungho. ee, Ulsn ollege, shlee@u. Sun G hun, lbm Stte Unversty, sunghun@lsu.edu Dlsng hung, Governors Stte Unversty, dhung@govst.edu STRT nlyt Herrhy Proess (HP) s powerful quntttve mesurement method n the mult-rter deson mkng (MDM) re. It hs been used n bnks, mnufturng systems, orgnztonl performne evluton, nd proet seleton. Despte ths wdespred usge, mny studes rgue tht rnk reversl phenomenon s unpreventble when ny HP method s ppled. Ths pper presents tht the rnk reversl ours n other populr MDM pprohes s well, suh s ord-kendll (K), Tehnque for Order Preferene by Smlrty to Idel Soluton (TOPSIS), nd the smple ddtve weghtng (SW) pprohes. For vodng rnk reversl phenomenon n deson mtres where suh reversls should not tke ple, ths pper ttempts to llustrte tht the proposed lest ommon multple (M) pproh suessfully ddresses these rnk reversl problems n most populr MDM pprohes n deson support systems re. Keywords: Multple rter Deson Mkng (MDM), nlyt Herrhy Proess (HP), Rnk reversl, nd est ommon Multple (M) method INTRODUTION Over the pst three dedes, the nlyt herrhy proess (HP) developed by Sty [0] hs beome very populr multple rter deson mkng (MDM) tehnque. HP hs been ppled to dverse felds of study suh s softwre evluton, mnufturng systems, orgnztonl performne evluton, ustomer requrement rtng, nd fnnl ndustres. However, for nerly the sme durton, HP hs lso been rtzed for rnk reversls when deson lterntve s dded or dropped, frst noted by elton et l. []. In order to vod rnk reversl phenomenon n the HP where suh reversls should not tke ple, mny other dfferent mthemtl pprohes [, 4, 8, ] hve been proposed. It s noteble tht none of these methods hve resolved ths problemt phenomenon nd there re stll on-gong debtes on how to vod rnk reversls. Rnk reversls re lso found n mny other populr MDM pprohes s well, suh s ord-kendll [7] for ordnl preferene mesurements, Tehnque for Order Preferene by Smlrty to Idel Soluton (TOPSIS) [6], nd the smple ddtve weghtng (SW) pproh [9]. Ths pper proposes n lterntve method tht yelds relble deson nd the preservton of rnkng of lterntves when deson opton s dded or dropped. The purpose of ths pper s to llustrte tht the rnk reversls our n other MDM pprohes nd present tht the proposed pproh suessfully ddresses these rnk reversl problems of the most populr MDM pprohes n deson support systems re. RNK REVERS PHENOMENON IN HP ND PROPOSED M PPROH Rnk reversl n the HP 58

2 Issues n Informton Systems Volume 4, Issue, pp.58-65, 0 Wthn frmework of HP [0], deson problem s deomposed nto herrhy of the gol, rter, subrter, nd fnlly lterntves t the bottom of the herrhy. The rtng method usng elements n Tble s rred out s follows: ) the rter re ompred prwse to derve ther normlzed weghts whh sum to one, ) the lterntves lso re ompred prwse wth respet to eh rteron to derve ther reltve lol prortes, fnlly, ) the lol lterntve prortes re ggregted wth the rter weghts to produe the overll prortes for the lterntves n deson mtr. Tble. The fundmentl sle Intensty of preferene on n bsolute sle 5 7 9, 4, 6, 8 Defnton Equl preferene Moderte preferene of one over others Essentl or strong preferene Very strong preferene Etreme preferene Intermedte vlues when ompromse s needed Sty et l. [] presents the rnk reversl phenomenon usng three lterntves,, nd wth respet to three rter,, nd, respetvely (Tble ). The numbers to lterntves re ssgned one t tme by usng the rtngs mode. However, t s lso ssumed tht the dt hve been derved from onsstent udgment mtres wth prwse omprsons n order to vod ny other possble effets from nonsstent mesurements n the deson problem. Tble. Deson problem set lterntve (/) 9 Deson rter (/) 9 (/) 8 9 Tble. ddng n lterntve n the dstrbutve HP ddton lterntve rter weghts omposte Weghts D opy of D (/) / 9/ / /0 9/0 /0 9/0 (/) 9/ / / 9/ / / / (/) 8/8 9/8 /8 8/7 9/7 /7 9/ Rnk 59

3 Issues n Informton Systems Volume 4, Issue, pp.58-65, 0 D ner opy of D /9 9/9 /9 8/9 9/ / / / 8/6 9/6 /6 8/ To obtn the lol prortes of lterntves n the deson mtr, the orgnl HP [0], lso known s the dstrbutve HP uses prnpl Egenvlue Method (EM), whh requres tht one dds the mesurement vlues under eh rteron, nd dvdes eh mesurement by the sum of the mesurements wth respet to ll other rter. They ssume tht the three rter hve n equl weght of mportne nd derved rnkng for, nd. ordng to Tble, before D s dded, the rnkng between nd n the orgnl deson mtr s >. However, ddng D whh s opy of, hnges the preferene of over. The rnk s reversed, wth (.56) preferred to (.95). lso when we ntrodue n lterntve D whh s smll hnge of wth respet to, dentl to, nd smll hnge of wth respet to, we hve , nd s preferred to. The followng emple nvolves three lterntves, nd wth respet to two rter nd respetvely. The rteron weghts re /0 nd 7/0. Tble 4 shows tht the rnkng between nd n the orgnl deson mtr s (.8) > (.79) before s removed. However, when we remove n lterntve, we hve , nd now s preferred to. Tble 4. Removng n lterntve lterntve rter weghts (/0) (7/0) omposte weghts Rnk s s known for most multple omprson deson mkng problems, n order to get rd of the dmensons of dfferent deson ttrbutes, normlzton s neessry. Two emples of the rnk reversl seem to dept tht the rnk reversl s presumbly used by proedurl flws of the normlzton method. The lterntve pproh to yeld the most relble ntl rnkng nd to preserve the rnkng s proposed n net seton. proposed est ommon Multple (M) pproh Shn et l. [, 4] propose n lterntve pproh tht onverts ll mesurement vlues of lterntves to the ommensurte vlues by multplyng lest ommon multple (M) of ll olumn sums of rter n the deson mtr. efore the omposte weghts of ll lterntves re omputed, mtr, s multpled by, lest ommon multple of ll olumn sums of rter, where " () Now the weght vetor of rter ( ) s gven by [ ] T. Then, multplyng the rter weght vetor by the revsed vlue mtr yelds the followng dt mtr, X. 60

4 Issues n Informton Systems Volume 4, Issue, pp.58-65, X () Fnlly, the normlzed omposte weghts of lterntves re obtned from the followng equton, T X... ' () euse of the onverted mtr of the unfed ommensurte unt, rnk reversl problems n the HP n be prevented wthout dustng the weghts of rter or wonderng bout struturl or funtonl dependeny nd ndependeny. To verfy the vldty of our proposed pproh, the net prgrphs present the results of M mode by re-emnng the deson mtres used n two emples n the prevous seton. Tble 5 shows tht the rnkngs n the orgnl deson mtr re > > before D s dded. Now, the rnkngs re preserved s D > > nd > > D >, regrdless of whether new lterntve, D ( opy or ner opy of lterntve ), s dded. In Tble 6, the rnkng between nd n the orgnl deson mtr s > before s removed. When we remove n lterntve, we hve , nd now s stll preferred to. Tble 5. ddng n lterntve n M Mode pproh ddton lterntve rter weghts omposte Weghts Rnk () () () *98/ 9*98/ 8*98/ *98/ *98/ 9*98/ *98/ *98/ *98/ D opy of *98/ 9*98/ 8*98/ *98/ *98/ 9*98/8 0.0 *98/ *98/ *98/ D 9*98/ *98/ 9*98/8 0.0 D ner opy of *98/ 9*98/ 8*98/ *98/ *98/ 9*98/8 0. *98/ *98/ *98/ D 8*98/ *98/ 8*98/ Tble 6. Deletng n lterntve n M Mode pproh

5 Issues n Informton Systems Volume 4, Issue, pp.58-65, 0 rter weghts lterntve M M omposte weghts Rnk (/0) (7/0) 57*00/00 *00/0 0.8 *00/00 4*00/ *00/00 *00/ *00/00 *00/ *00/00 4*00/ RNK REVERSS IN OTHER MDM TEHNIQUES There re wde rnge of MDM problem soluton tehnques, vryng n omplety nd possble solutons. Eh method hs ts own strength, weknesses nd possbltes to be ppled. mong them, t s obvous tht most populr MDM pprohes, suh s ord-kendll, TOPSIS nd the SW method, lso suffer from rnk reversl. In ths seton we llustrte tht the rnk reversls our n these populr MDM pprohes. ord-kendll method Multple rter desons re lso ommonly used to formulte onsensus rnkng problems. onsensus rnkng hs strong nterdsplnry nture nd s onsdered n orgnztonl senes, mrketng reserh, nd mngement sene. The ord-kendll (K) method [5] s the most wdely used tool n determnng onsensus rnkng beuse of ts omputtonl smplty. It uses weghted ordnl rnkng model n whh eh of set of n lterntves ws gven n ordnl rnk on set of rter []. Suppose there re m voters who vote on n nddtes. Eh nddte wll reeve some votes t dfferent rnkng ples. The K method ssgns the frst rnkng ple mrk of one, the seond rnkng ple mrk of two, nd so on. The totl sore (Z ) eh nddte reeves n be omputed by ggregtng the results from the smple equton, Z v ", the votes tht eh nddte reeves th rnkng ple. The best nddte wll be the one wth the lest totl sore. Tble 6 shows votng problem n whh 60 voters vote on three poltl nddtes. fter the votng s ompleted, s the best nddte, followed by nddtes nd. lerly, nddte s the most undesrble nddte nd n be dropped out of further votes system. Suppose nddte s dropped nd the 60 voters preferenes on the nddtes nd remn unhnged. Then, the result shows tht nddte s no longer better nddte thn. The rnkng between nd s reversed. Smlrly, f s dropped for whtever reson, the rnkng between remnng nddtes nd s reversed. Tble 7 lso shows the proposed M method preserves the orgnl rnkngs n ll ses. Tble 7. Votng mong three poltl nddtes by 60 voters lterntves Number of votes Totl K K M M Sore verge Rnk Weghts Rnk.0 >> 0.9 >>

6 Issues n Informton Systems Volume 4, Issue, pp.58-65, *> 0.54 > *> > Smple ddtve Weghtng (SW) method Smple ddtve Weghtng (SW), whh s lso known s weghted lner sorng method s smple nd the most frequently used multple ttrbute deson mkng (MDM) tool [6]. The SW s used for MDM problem wth n lterntves nd m deson ttrbutes (rter). Eh lterntve s evluted wth respet to the m ttrbutes. The evluton sore s normlzed by the followng equton (4) to elmnte the dmensonl unts of the ttrbutes. " " "# "# "#,,, n;,, m (4) The overll ssessment vlue of eh lterntve s omputed by multplyng the evluton vlue gven to the lterntve of tht ttrbute wth the weghts of reltve mportne dretly ssgned by deson mker followed by summng of the produts for ll rter. The greter the overll prorty vlue, the better the lterntve. It s esly noted tht the SW method s lso subet to the rnk reversl problem when n lterntve s dded or dropped. In Tble 8, wth SW method, when new lterntve E s dded, the rnkng between lterntves nd D s reversed. When n lterntve D s dropped out from the orgnl set of lterntves, the rnkng between nd s reversed wth the lterntve beomng the best lterntve. However, t n be observed tht the proposed M method preserves the orgnl rnkngs n both ses. Tble 8. Deson mtr for SW method lterntves rter eghts 4 SW SW M M (/6) (/) (/) (/6) Weghts Rnk Weghts Rnk >>D> >>D> D *>D>>>E >>D>>E D E (new) *>> 0.60 >> Tehnque for Order Preferene by Smlrty to Idel Soluton (TOPSIS) method 6

7 Issues n Informton Systems Volume 4, Issue, pp.58-65, 0 The TOPSIS method s gol bsed pproh for fndng the lterntve tht s losest to the del soluton [6]. The generl TOPSIS proess wth 5 steps s lsted s follows: From bs struture of deson problem wth n lterntves nd m deson ttrbutes, normlzed ttrbute vlues (r " ) re omputed by r " " ". The weghted normlzed ttrbute vlues (v " ) re omputed by v " w r ". Then the postve del soluton (m v " є J) nd the negtve del soluton (mn v " є J) re determned by the deson mker. Net, lulte the seprton mesure (S, Eulden dstne) of eh lterntve from the postve del one wth S v " v seprton mesure (S )of eh lterntve from the negtve del one wth S v " v TOPSIS weghts (reltve loseness) of lterntve to the del soluton re omputed wth S lrger the S vlue, the better the performne of the lterntves., nd the. Fnlly, the S S. The onsder the deson mtr used wth SW method n Tble 8 gn. In Tble 9, the result of TOPSIS method ndtes tht ths method s lso not free from the rnk reversl when n lterntve s ntrodued or removed. When new lterntve E s dded to the orgnl set of the four lterntves, the rnkngs between nd s well s nd D re both reversed. When D s removed from the orgnl deson mtr, the rnkng between nd s reversed. ddtonlly, regrdng the rnkng of the orgnl set of the four lterntves, the TOPSIS method yelds n lterntve s the best, whh s dfferent from tht omputed by the SW method (n lterntve s the best). However, the proposed M method provdes onsstent rnkng of n orgnl set of lterntves nd preserves the orgnl rnkngs where n lterntve s dded or dropped out. Tble 9. Deson mtr of four lterntves wth respet to four ttrbutes rter weghts 4 TOPSIS TOPSIS M M lterntves (/6) (/) (/) (/6) Weghts Rnk Weghts Rnk >>>D >>D> D *>>D>>E >>D>>E D E (new) *>> 0.60 >> ONUSIONS There s no doubt tht the HP s powerful method n mult-rter deson mkng re. It s ssumed tht the HP wll ontnue to be useful for mny future ses s t hs been n the pst. Despte ths wdespred usge, the 64

8 Issues n Informton Systems Volume 4, Issue, pp.58-65, 0 HP stll suffers from some theoretl dsputes. Rnk reversls re lso found n mny other well-known MDM methods. Mny studes rgue tht the rnk reversl phenomenon s unpreventble when ny MDM method s ppled. Some show tht ths phenomenon seems to be n nherted problem when deson mker dels wth rter whh re mesured on dfferent unts. The vlble wde rnge of MDM problem soluton tehnques, vryng n omplety nd possble solutons, onfuses potentl deson mkers. Eh method hs ts own strengths, weknesses nd possbltes to be ppled. Ths n use nonsstent rnkng problem used by dfferent MDM methods. s seen n the seton of TOPSIS method, prmry rtsm of MDM methods s tht due to the dfferenes mong dfferent tehnques, nonsstent results re obtned when ppled to the sme deson problem. It s mportnt tht good MDM method must not yeld the rnkng reversls when n lterntve s dded nd removed. Even though the proposed method does not suffer from those problems, t s more mportnt tht ddtonl reserh n deson nlyss s neessry to produe the relble rnkngs one my trust. REFERENES. rzl, J. & Golny,. (994). HP rnk reversl, normlzton nd ggregton rules, INFOR, (), elton, V. & Ger, T. (98). On shortomng of Sty s method of nlyt herrhes. Omeg, (), ook, W.D., Kress, M., & Seford,.M. (997) generl frmework for dstne-bsed onsensus n ordnl rnkng models, Europen Journl of Opertonl Reserh 96, Dyer, J. & Wendell, R. (985). rtque of the nlyt Herrhy Proess. Tehnl Report, 84/85, 4-4. Deprtment of Mngement. The Unversty of Tes t ustn. 5. Hwng,.., & n, M.J. (987). Group deson mkng under multple rter. eture Notes n Eonoms nd Mthemtl Systems, 8, Sprnger, erln. 6. Hwng,.., & Yoon, M. M (98). Multple ttrbute Deson Mkng: Methods nd ppltons, Sprnger- Verlg, erln. 7. Kendll, M. (96). Rnk orreton Methods, rd Ed. Hfner, New York. 8. ootsm, F. (999). Mult-rter deson nlyss v rto nd dfferene udgment. Kluwer dem Publshers: Dordreht. 9. Mrmon, K.R. (968). Deson mkng mong multple ttrbute lterntves: survey nd onsoldted pproh. RND Memorndum, RM-48-RP. The Rnd orporton, Snt Mon,. 0. Sty, T.. (980). The nlyt herrhy proess. MGrw-Hll Interntonl: New York.. Sty, T.. & Sgr, M. (009). n essy on rnk preservton nd reversl. Mthemtl nd omputer Modellng, 49, Shoner,., Wedley, W.., & hoo, E.U. (99). unfed pproh to HP wth lnkng pns. Europen Journl of Opertonl Reserh, 64, Shn, Y.. & ee, S.H. (0). Note on n pproh to preventng rnk reversls wth ddton or deleton of n nlyt Herrhy Proess. Journl of Eduton Revew (), Shn, Y.. & ee, S.H. (0). Remrks on Rnkngs by Dfferent Methods n HP. Interntonl Journl of usness Dsplnes. (forthomng). 65