Fuzzy TOPSIS Based on α Level Set for Academic Staff Selection

Transcription

1 Gadg Busess ad Maagemet Joural Vol. No., 57-70, 007 Fuzzy TOPSIS Based o evel Set for Academc Staff Selecto Nazrah Raml Nor Azzah M. Yacob Faculty of Iformato Techology ad Quattatve Scece Uverst Tekolog MARA (UTM, Malaysa Emal: azrahr@pahag.utm.edu.my ABSTRACT Ths paper apples the fuzzy TOPSIS based o -level solvg a mult-crtera decso-makg problem. The basc prcple of fuzzy TOPSIS s that the chose alteratves should have the shortest dstace from the fuzzy postve deal soluto ad the farthest dstace from the fuzzy egatve deal soluto whch ca be determed by calculatg the closeess coeffcet. I ths paper, the -level set of the fuzzy closeess coeffcet s calculated at eleve levels. The closeess coeffcet ca be preseted as a fuzzy umber whch geerates a more accurate fuzzy estmato of the relatve closeess. A emprcal study of academc staff selecto s coducted to llustrate ts applcato. Keywords: academc staff selecto, fuzzy TOPSIS, mult-crtera decso-makg Itroducto Mult-crtera decso-makg (MCDM refers to screeg, prortsg, rakg or selectg a set of alteratves uder depedet, commesurate ad coflctg crtera (Hwag & Yoo, 98. A MCDM problem s charactersed by the ratgs of each alteratve wth respect to each crtero ad the weghts gve to each crtero. Classcal MCDM methods assume that the ratgs of alteratves ad the weghts ISSN Uverst Tekolog MARA (UTM, Malaysa 57

2 Gadg Busess ad Maagemet Joural of crtera are crsp values. However, the real world, formato sources maybe uquatfable, complete, uobtaable ad partal gorace (Che & Hwag, 99. Hece, the classcal MCDM caot hadle problems whch the values of the ratgs are lgustc terms represeted by fuzzy sets. I order to cope wth such a problem, fuzzy MCDM was developed ad appled. The geeral use of fuzzy set theory MCDM s dscussed Che ad Hwag (99, Rbero (996 ad Robert ad Fuller (996, whle specfc fuzzy MCDM methods ca be foud Hsu ad Che (997, Che (000, Cheg, Cha ad Huag (00 ad Wag ad Poh (00. Rbero (996 proposed fuzzy decso makg wth partal preferece whle Hsu ad Che (997 appled fuzzy credblty relato (FCR method rakg alteratves uder multple crtera. I aother study, Che (000 exteded the cocept of techque for order performace by smlarty to deal soluto (TOPSIS for solvg MCDM problems fuzzy evromet. The classcal TOPSIS method was frst proposed by Hwag ad Yoo (98 based o the cocept that the chose alteratve should have the shortest dstace from the postve deal soluto ad the farthest dstace from the egatve deal soluto. Ispred by Che s approach, Wag ad Elhag (006 proposed a fuzzy TOPSIS based o level set brdge rsk assessmet. Fdgs by Wag ad Elhag (006 led to a fuzzy relatve closeess for each alteratve compared to a crsp relatve closeess for each alteratve usg TOPSIS method as Che (000. Crsp relatve closeess provdes oly oe possble soluto to a fuzzy MCDM problem whch caot reflect the whole pcture of ts all soluto ad, therefore, the advatage of collectg fuzzy data becomes uapparet. Ths paper appled the fuzzy TOPSIS based o level set as Wag ad Elhag (006 selectg the academc staff at the Departmet of Mathematcs ad Statstcs Uverst Tekolog MARA (UTM Pahag. Ths study shows a alteratve way of selectg academc staff UTM Pahag by usg aalytcal method compared to dvdual percepto ad huma tuto the tradtoal process of selecto. The result ca help the maagemet choosg the best caddate based o the vague ad mprecse performace ratgs by the decso maker. 58

3 Fuzzy TOPSIS Based o evel Set for Academc Staff Selecto Prelmares Defto A tragular fuzzy umber N ~ ca be defed by a trplet (,,. The membershp fucto μ ( x s defed as follows. N ~ μ ~ N ( x Defto x x = 0, x, x, otherwse The -level sets or -cuts of fuzzy umber P ~ s defed as { x X, μ ( x } ad [ 0,] P = P ~. Fuzzy Topss TOPSIS, oe of the kow classcal MCDM methods, was frst developed by Hwag ad Yoo (98 for solvg a MCDM prcple. The basc prcple of TOPSIS s that the chose alteratves should have the shortest dstace from the postve deal soluto ad the farthest dstace from the egatve deal soluto. The performace ratgs ad the weghts of the crtera TOPSIS method are gve as crsp values. Sce huma udgemet s ofte vague ad uable to estmate wth a exact umercal value, crsp data are adequate to model real lfe stuatos. Hece, Che (000 exteded the cocept of TOPSIS ad proposed a methodology for solvg MCDM fuzzy evromet. The procedure of TOPSIS method as Che (000 s as follows: ~ X = ~ x where x~ s a lgustc varable ad ca be descrbed by tragular fuzzy umber,. Buld a fuzzy decso crtera matrx ( m ~ x = ( a, b, c. 59

4 Gadg Busess ad Maagemet Joural. Normalse the fuzzy decso matrx X ( x m ( r m ~ = ~ ~ as R = ~ where r a b c =,, * * c c c * c * = max c. ~ V = v, =,,, m, =,,, where v ~ = r~ ( w~, w ~ s the relatve. Costruct the weghted ormalsed decso matrx as [ ] m weght of the -th crtera ad w =. 4. Determe the postve (A* ad egatve (A - deal solutos as * * * * A = ( v~, v ~,, v~ A = ( v~, ~ v,, v~ ~* =,, ~ v = 0,0,0. = where v ( ad ( 5. Calculate the separato measure by usg the -dmesoal Eucldea dstace as defed Che (000. The separato of each alteratve from A* ad A - s gve as: ~ * ~* d = d v~ ~, v d = d v~, v~ where d = ( ad ( = [ ] ~ =, are tragular umbers. ( m ~, ~ = ( m + ( m + ( m, m ~ = ( m, m m ad (,, 6. Calculate the relatve closeess of each alteratve to the deal soluto. The relatve closeess of the alteratve A wth respect to A* s defed as: ~ d RC = ~ d + d ~ * 7. Rak the alteratves accordg to the relatve closeess to the deal soluto. The bgger the RC, the better the alteratve A. The best alteratve s the oe wth the greatest relatve closeess to the deal soluto. 60

5 Fuzzy TOPSIS Based o evel Set for Academc Staff Selecto Fuzzy Topss Based o evel Set Che (000 ~ defes the Eucldea dstace of two tragular fuzzy m, m m ~ =, as umbers m = ( ad (,, [ ] d( m ~, ~ = ( m + ( m + ( m. Ths Eucldea dstace s a crsp value. Based o the TOPSIS procedure dscussed before, the Eucldea dstace of each alteratve to the postve ad egatve deal soluto s both crsp whch leads to a crsp pot estmate for the relatve closeess of each alteratve. Sce the fuzzy MCDM problem s defuzzfed to a crsp value at the early stage, the the advatage of collectg fuzzy data becomes uapparet. To overcome the shortcomgs, Wag ad Elhag (006 proposed a fuzzy TOPSIS based o level set. The fuzzy TOPSIS based o level set s preseted as below. ~ X = ~ x et ( m = w, w be fuzzy weghts. ~ x = ( a, b, c ad ~ ( w, w w ~ W ~ ~ ~ be a fuzzy decso matrx ad (,, w =, for =,,,m ad =,,, are tragular fuzzy umbers. The ormalsed decso matrx ca be wrtte as ~ R = ~ where [ r ] m ~ = a b c r,, * * c * c c ad c * = max c for B ; w a a a r ~ =,, c b c ad a = m a for C ; B ad C are the set of beeft crtera ad cost crtera respectvely. Normalsed are stll tragular fuzzy umbers. et [ ] r~ U ( r = ( r ( r ad ( w ( w,( w, U [ ] = be the level sets of r~ ad w ~. Based o the defto of relatve closeess Che (000, the relatve closeess of the alteratve ca be equvaletly wrtte as 6

6 Gadg Busess ad Maagemet Joural RC = r r + ( r = = =, for =,,K,m, ( w w ( w U ad ( r r ( r U for =,,K,. RC s a terval whose lower ad upper bouds ca be captured usg the followg par of fractoal programmg models: ( RC = m r r + ( r = = = s.t. ( w w ( w U, ( r r ( r U ( RC U = max for =,, K,. r r + ( r = = = s.t ( w w ( w U, ( r r ( r U for =,, K,. RC s mootocally creasg fuctos of r ( =,,,, whch meas RC reaches ts maxmum ad mmum at r ( U = r ad r ( = r respectvely. Therefore, the par of fractoal programmg models ca be smplfed as ( RC = m ( r ( r + w ( r = = ( = 6

7 Fuzzy TOPSIS Based o evel Set for Academc Staff Selecto s.t ( w w ( w U for =,, K,. ( RC U = m U ( r U ( r + w ( r = = U ( = s.t ( w w ( w U for =,, K,. I order to select the best alteratve, the fuzzy relatve closeess RC has to be defuzzfed. The smplest defuzzfcato method based o level set s bult by applyg the averagg level cut (AC (Oussalah, 00. By usg AC, the defuzzfed value of RC for N level sets ca be calculated by ( N ( RC + ( RC * RC = AC U N =. Academc Staff Selecto UTM Pahag: A Emprcal Study The tradtoal process of the academc staff selecto UTM Pahag s based o the tervewers or decso makers dvdual percepto o the caddates. Although, a gudele cosstg of four crtera (voce toe, appearace, presetato ad audo vsual has bee prepared by the maagemet, there has bee o aalyss ever doe o t. The selecto has almost bee based o huma tuto whch s vague, ucerta ad mmeasurable. No proper measuremet has bee doe to support the decso. Therefore, ths study s coducted as a alteratve way of selectg academc staff UTM Pahag by usg a aalytcal method. The preset study attempted to apply the fuzzy TOPSIS based o level set method o the selecto of a academc staff the Departmet of Mathematcs ad Statstcs UTM Pahag. The method was appled o the recet selecto exercse. Ths process s demostrated below. The Departmet of Mathematcs ad Statstcs UTM Pahag wated to select a academc staff amog three caddates, A, A ad A. A commttee of three decso makers, D, D ad D evaluated the 6

8 Gadg Busess ad Maagemet Joural caddates agast fve beeft crtera whch are academc qualfcato (C, oral commucato sklls (C, Eglsh profcecy (C, selfcofdece (C 4 ad teachg sklls (C 5. The herarchcal structure of ths decso-makg problems s show Fgure. The relatve mportace weghts of the fve crtera are descrbed usg lgustc varables wth hedges such as very low, low, medum low, medum, medum hgh, hgh ad very hgh as show Table. The ratgs are also charactersed by lgustc varable wth hedges such as very poor, poor, medum poor, far, medum good, good ad very good (Table. Commttee C Academc Qualfcato C Commucato Sklls C Eglsh Profcecy C 4 Self Cofdece C 5 Teachg Sklls A A A Fgure : The Herarchcal Structure Table : gustc Varables for the Relatve Importace Weghts of Fve Crtera gustc Varable Fuzzy Number Very low (V ( 0, 0, 0. ow ( ( 0, 0., 0. Medum low (M ( 0., 0., 0.5 Medum (M ( 0., 0.5, 0.7 Medum hgh (MH ( 0.5, 0.7, 0.9 Hgh (H ( 0.7, 0.9,.0 Very hgh (VH ( 0.9,.0,.0 64

9 Fuzzy TOPSIS Based o evel Set for Academc Staff Selecto Table : gustc Varables for the Ratgs gustc Varable Fuzzy Number Very poor (VP (0, 0, Poor (P (0,, Medum poor (MP (,, 5 Far (F (, 5, 7 Medum good (MG (5, 7, 9 Good (G (7, 9, 0 Very good (G (9, 0, 0 Results The three decso makers expressed ther opo o the mportace of weghts of fve crtera ad the ratgs of each caddate wth respect to the fve crtera depedetly. Tables ad 4 show the assessmet provded by the three decso makers ad the aggregated fuzzy umber. The calculato of the aggregated fuzzy umber s based o Che (000, p. 5 whch states that for K decso makers, the aggregated fuzzy umber of each crtero ad the relatve weght of each crtero are defed as ~ K x [ ~ x ( ~ x ( ( ~ K = x ] ad w ~ [ w~ ( w~ = + ( + ( + w~ ]. K K Table : The Relatve Importace of Weghts of the Crtera by Decso Maker D D D Aggregated fuzzy umber C VH VH VH (0.9,.0,.0 C VH H H (0.77, 0.9,.0 C H H H (0.7, 0.9,.0 C 4 H VH MH (0.7, 0.87, 0.97 C 5 MH VH H (0.7, 0.87,

10 Gadg Busess ad Maagemet Joural Table 4: Ratgs of Caddates wth Respect to the Crtera by the Decso Maker Crtera Caddates Decso maker Aggregated fuzzy umber D D D C A MG MG G (5.7, 7.7, 9. A G MG G (6., 8., 9.7 A G G G (7, 9, 0 C A MG F F (.7, 5.7, 7.7 A G G G (7, 9, 0 A G MG MG (5.7, 7.7, 9. C A F F MG (.7, 5.7, 7.7 A G G VG (7.6, 9., 0 A G MG MG (5.7, 7.7, 9. C 4 A F F G (4., 6., 8 A G MG G (6., 8., 9.7 A G MG G (6., 8., 9.7 C 5 A F F G (.7, 5.7, 7.7 A G MG G (7, 9, 0 A G MG G (6., 8., 9.7 Table 5 shows the ormalsed fuzzy decso matrx ad fuzzy weghts. Table 5: The Normalsed Fuzzy Decso Matrx ad Fuzzy Weghts C C C C 4 C 5 A (0.57, 0.77, (0.7, 0.57, (0.7, 0.57, (0.44, 0.65, (0.7, 0.57, A (0.6, 0.8, (0.7, 0.9, (0.76, 0.9, (0.65, 0.86, (0.7, 0.9, A (0.7, 0.9, (0.57, 0.77, (0.57, 0.77, (0.65, 0.86, (0.6, 0.8, Weght (0.9,.0, (0.77, 0.9, (0.7, 0.9, (0.7, 0.87, (0.7, 0.87,

11 Fuzzy TOPSIS Based o evel Set for Academc Staff Selecto The fuzzy relatve closeess at dfferet -level s show Table 6. Table 6: -level Set of the Fuzzy Relatve Closeess of the Three Caddates Caddates A A A 0 [0.46, 0.84] [0.68, 0.987] [0.6, 0.956] 0. [0.447, 0.795] [0.7, 0.978] [0.64, 0.944] 0. [0.467, 0.777] [0.7, 0.968] [0.66, 0.9] 0. [0.487, 0.759] [0.74, 0.957] [0.68, 0.98] 0.4 [0.507, 0.740] [0.76, 0.946] [0.70, 0.905] 0.5 [0.57, 0.7] [0.78, 0.94] [0.7, 0.89] 0.6 [0.548, 0.70] [0.799, 0.9] [0.74, 0.878] 0.7 [0.568, 0.684] [0.89, 0.9] [0.76, 0.864] 0.8 [0.588, 0.666] [0.88, 0.9] [0.78, 0.85] 0.9 [0.608, 0.647] [0.858, 0.889] [0.80, 0.86] [0.68, 0.68] [0.877, 0.877] [0.8, 0.8] Therefore, the fuzzy relatve closeess for caddates A, A ad A are RC = [0.46, 0.68, 0.84], RC = [0.68, 0.877, 0.987] ad RC = [0.6, 0.8, 0.956] respectvely ad t s show Fgure. The defuzzfed values ad the rakg of each caddate are show Table 7. A A A Relatve closeess Fgure : The Fuzzy Relatve Closeess 67

12 Gadg Busess ad Maagemet Joural Table 7: Defuzzfed Value ad Rakg of the Three Caddates Caddates Defuzzfed value Rak A 0.64 A A Table 8: Comparsos of Results Caddates Che s approach Fuzzy TOPSIS based o -level Relatve closeess Defuzzfed value A A A Dscusso I ths study, the fuzzy TOPSIS based o -level set s appled solvg problem of academc staff selecto. The relatve closeess of each alteratve s preseted as a fuzzy umber ad s calculated at -level usg Mathcad software. The fuzzy relatve closeess s accurate eough by usg the -level whch are at = 0, 0., 0., 0., 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, ad. The more -level values used, the more accurate fuzzy relatve closeess wll be. The defuzzfed values computed usg AC as Ousallah (00 are RC A = 0.64, RC A = ad RC A = 0.807, whch gve the rakg of A > A > A as Table 7. Table 8 shows a comparso of results usg fuzzy TOPSIS by Che (000 ad the fuzzy TOPSIS based o -level set. Che s approach obtaed the relatve closeess as 0.55 for A, 0.7 for A ad 0.69 for A whch are sgfcatly lower tha the defuzzfed values. Although Che s approach leads to the same rakg, t produces oly a crsp value for the relatve closeess whle the fuzzy TOPSIS based o -level set geerates a more accurate fuzzy estmato of the relatve closeess. The fuzzy TOPSIS based o -level set method also defuzzfed the mprecse values at the ed of the process ad ot at the very begg of the process whch s the ratoale of usg fuzzy method. The fuzzy TOPSIS based o -level set method also geerates a more accurate fuzzy estmato of the relatve closeess compared to Che s approach. 68

13 Fuzzy TOPSIS Based o evel Set for Academc Staff Selecto Cocluso Sce mult-crtera decso problems geerally volve ucertaty, fuzzy MCDM has bee wdely appled solvg real world decso-makg problem. Ths paper has preseted a fuzzy TOPSIS based o -level solvg problem of a academc staff selecto the Departmet of Mathematcs ad Statstcs UTM Pahag. Based o the vague ad mprecse performace ratgs by the decso makers, the maagemet ca select the best caddate wth fve evaluated crtera whch are academc qualfcato, commucato sklls, Eglsh profcecy, self cofdece ad teachg sklls. Ths method s based o the holstc crtera of all the caddates ad ot oly by lookg at oe or two crtera that may mpress the decso makers. It combes all the crtera of the caddates wth the decso makers descrpto. Up tll ow, there has bee o exact mechasm staff selecto that looks to all the crtera as suggested by ths method. Therefore, ths aalytcal method s hoped to be a alteratve way of selectg academc staff UTM Pahag compared to dvdual percepto ad huma tuto the tradtoal process. For future studes, t s suggested that a system ca be developed selectg staff UTM based o the TOPSIS algorthm. The comg system ca be appled ot oly for selecto of academc staff but also to the o-academc staff. Refereces Che, C.T. (000. Exteso of the TOPSIS for group decso makg uder fuzzy evromet. Fuzzy Sets ad Systems, 4, -9. Che, S.J. & Hwag, C.. (99. Fuzzy multple attrbute decso makg: Methods ad applcatos. Berl: Sprger-Verlag. Cheg, S., Cha, C.W. & Huag, G.H. (00. A tegrated mult crtera decso aalyss ad exact mxed teger lear programmg approach for sold waste maagemet. Egeerg Applcatos of Artfcal Itellgece, 6, Hsu, H.M. & Che, C.T. (997. Fuzzy credblty relato method for multple crtera decso makg problems. Itellget System, 96,

14 Gadg Busess ad Maagemet Joural Hwag, C.. & Yoo, K.P. (98. Multple attrbute decso makg: Methods ad applcatos. Berl: Sprger. Oussalah, M. (00. O the compatblty betwee defuzzfcato ad fuzzy arthmetc operatos. Fuzzy Sets ad Systems, 8, Rbero, R.A. (996. Fuzzy multple attrbute decso makg: A revew ad ew preferece elctato techque. Fuzzy Sets ad Systems, 78, Robert, C.C. & Fuller, R. (996. Fuzzy multple crtera makg: Recet developmets. Fuzzy Sets ad Systems, 78, 9-5. Wag, W. & Poh, K.. (00, October 9-. Fuzzy MCDM based o cofdece aalyss. Paper preseted at the Teth Iteratoal Assocato for Fuzzy Set Maagemet ad Ecoomy Cogress, eo, Spa. Wag, Y.M. & Elhag, T.M.S. (006. Fuzzy TOPSIS method based o alpha level sets wth a applcato to brdge rsk assessmet. Expert Systems wth Applcatos,,