DECISION MAKING WITH THE OWA OPERATOR IN SPORT MANAGEMENT

Transcription

1 ESTYLF08, Cuecas Meras (Meres - Lagreo), 7-9 de Septembre de 2008 DECISION MAKING WITH THE OWA OPERATOR IN SPORT MANAGEMENT José M. Mergó Aa M. Gl-Lafuete Departmet of Busess Admstrato, Uversty of Barceloa Av. Dagoal 690, Barceloa, Spa, {jmergo, amgl}@ub.edu Abstract We aalyze the use of the OWA operator the selecto of huma resources sport maagemet. We use dfferet busess decso makg techques for dog so. We wll cosder the use of the Hammg dstace, the adequacy coeffcet ad the dex of maxmum ad mmum level. By usg the OWA operator, we ca parameterze these decso makg techques from the maxmum to the mmum result depedg o the terests of the decso maker. We wll develop a llustratve example about the decso makg process to follow the selecto of a football player for a team. Keywords: Aggregato operators, OWA operator, Decso makg, Sport maagemet. INTRODUCTION The selecto of the most approprate huma resources sports lke football, basketball, etc. represets a fudametal problem for ts good developmet. The eterprse eeds to aalyze how to select the best player accordg wth ther terests. I order to solve ths problem, the compay has to develop a selecto process whch they have to compare the dfferet characterstcs of each avalable caddate foud the market wth ther deals. Amog the great varety of studes exstg selecto, ths work wll focus o the models developed [5,9] about selecto of huma resources, the models developed [6,-3] about selecto of facal products ad the models developed [7-8] about selecto of players sport maagemet. Note that a very useful survey about dfferet decso makg methods ca be foud [4]. Oe problem about these selecto dexes s that they are eutral agast the atttudal character of the decso maker. The, whe developg the selecto process, we caot mapulate the results accordg to the terests of the decso maker. Ths problem becomes mportat stuatos where we wat to uder estmate or over estmate the decsos order to be more or less prudet agast the ucerta factors affectg the future. Oe commo method for aggregatg the formato cosderg the decso atttude of the decso maker s the ordered weghted averagg (OWA) operator troduced [4]. Sce ts appearace, the OWA operator has bee studed by dfferet authors such as [- 3,0-3,5-6]. Recetly, ths problem has bee cosdered other decso makg problems such as facal maagemet ad huma resource maagemet [2-3]. The am of the paper cossts developg ew selecto dexes that clude the atttudal character of the decso maker for the selecto of huma resources sport maagemet. These ew dexes wll cosst combg the Hammg dstace, the adequacy coeffcet ad the dex of maxmum ad mmum level, wth the OWA operator because the, the eutralty of the old methods wll be chaged by the OWA operator. We wll troduce the selecto of huma resources sport maagemet, the ordered weghted averagg dstace (OWAD) operator [2], the ordered weghted averagg adequacy coeffcet (OWAAC) [2] ad the ordered weghted averagg dex of maxmum ad mmum level (OWAIMAM) [2]. We wll also develop a llustratve example order to see the usefuless of the ew approach. Depedg o the aggregato operator used, we wll see that the results may be dfferet leadg to dfferet decsos. Ths paper s orgazed as follows. I Secto 2 we brefly revew some basc cocepts. I Secto 3, we aalyze the process to follow the selecto of huma resources sport maagemet. I Secto 4 we develop a llustratve example of the problem. We focus o the selecto of huma resources a football team. Fally, Secto 5 we summarze the ma coclusos of the paper. XIV Cogreso Español sobre Tecologías y Lógca fuzzy

2 ESTYLF08, Cuecas Meras (Meres - Lagreo), 7-9 de Septembre de PRELIMINARIES 2.. OWA OPERATOR The OWA operator troduced [4] provdes a parameterzed famly of aggregato operators whch have bee used may applcatos [-3,0-6]. I the followg, we provde a defto of the OWA operator as troduced by [4]. Defto. A OWA operator of dmeso s a mappg f: R R that has a assocated weghtg vector W of dmeso such that the sum of the weghts s ad w [0,]. The: f (a, a 2,, a ) = w j b j where b j s the jth largest of the a. From a geeralzed perspectve of the reorderg step we ca dstgush betwee the descedg OWA (DOWA) ad the ascedg OWA (AOWA) operator. Note that the weghts of ths two operators are related by w j = w* j+, where w j s the jth weght of the DOWA ad w* j+ the jth weght of the AOWA operator. By choosg a dfferet mafestato of the weghtg vector, we are able to obta dfferet types of aggregato operators such as the maxmum, the mmum, the average ad the weghted average [4]. Other famles of OWA operators ca be foud [,5] BUSINESS DECISION MAKING TECHNIQUES I the lterature, we fd a wde rage of methods for busess decso makg [-9,-6]. I ths paper, we wll focus o the Hammg dstace, the adequacy coeffcet ad the dex of maxmum ad mmum level. Frst, we wll cosder the formulato whe all the weghts w j, are equal. We wll use two sets A ad B. Defto 2. A ormalzed Hammg dstace (NHD) of dmeso s a mappg d H : R R R such that: d H (A,B) = a b = where a ad b are the th argumets of the sets A ad B respectvely. Defto 3. A ormalzed adequacy coeffcet (NAC) of dmeso s a mappg K: R R R such that: () (2) ( k K(P k P) = [ ( µ + µ ] (3) ) = where µ ad µ k are the th argumets of the sets P ad P k respectvely. Defto 4. A ormalzed dex of maxmum ad mmum level (NIMAM) of dmeso s a mappg K: R R R such that: η(p,p j ) = ( j) ( j µ ( u) µ ( u) + ( 0 ( µ ( v) µ ( v)) ) ) u + v (4) u v where µ ad µ k are the th argumets of the sets P ad P j, j s the jth alteratve cosdered, ad u ad v are the characterstcs to be cosdered wth the Hammg dstace ad the adequacy coeffcet, respectvely. Sometmes, whe ormalzg these measures, t s better to gve dfferet weghts to each dvdual argumet. The, these measures are kow as the weghted Hammg dstace (WHD), the weghted adequacy coeffcet (WAC) ad the weghted dex of maxmum ad mmum level (WIMAM). Defto 5. A WHD of dmeso s a mappg d WH : R R R that has a assocated weghtg vector W of dmeso such that the sum of the weghts s ad w [0,]. The: d WH (A,B) = w a b (5) = where a ad b are the th argumets of the sets A ad B respectvely. Defto 6. A WAC of dmeso s a mappg K: R R R that has a assocated weghtg vector W of dmeso such that the sum of the weghts s ad w [0,]. The: w = K(P k P) = [ ( µ + µ ] (6) where µ ad µ k are the th argumets of the sets P ad P k respectvely. Defto 7. A WIMAM of dmeso s a mappg K: R R R that has a assocated weghtg vector W of dmeso such that the sum of the weghts s ad w [0,]. The: 2 XIV Cogreso Español sobre Tecologías y Lógca fuzzy

3 ESTYLF08, Cuecas Meras (Meres - Lagreo), 7-9 de Septembre de 2008 ( j) Z ( u) µ ( u) µ ( u) + u η(p,p j ) = ( j) Z ( v) 0 ( µ ( v) µ ( v)) v [ ] where µ ad µ k are the th argumets of the sets P ad P j, j s the jth alteratve cosdered, ad u ad v are the characterstcs to be cosdered wth the Hammg dstace ad the adequacy coeffcet, respectvely. 3 SELECTION OF HUMAN RESOURCES WITH THE OWA OPERATOR IN SPORT MANAGEMENT The reaso for usg the OWA operators the selecto of huma resources sport maagemet appears because the decso maker wats to take the decso wth a certa degree of optmsm or pessmsm rather tha wth a eutral posto. Due to the fact that the tradtoal methods used sport maagemet for ths purpose [7-8] are eutral agast the atttude of the decso maker, the troducto of the OWA operators these models may chage the eutralty ad reflect decsos wth dfferet degrees of optmsm ad pessmsm. These techques ca be used a lot of stuatos but the geeral deas about t are the possblty of uder estmate or over estmate the problems. The process to follow the selecto of huma resources sport maagemet wth the OWA operator, s smlar to the process developed [5-9] wth the dfferece that the strumets used wll clude the OWA operator the selecto process. Note that smlar models that use the OWA operator have bee developed for other selecto problems [2]. The 5 steps to follow are: Step : Aalyss ad determato of the sgfcat characterstcs of the avalable caddates for the team. Theoretcally, t wll be represeted as: C = {C, C 2,, C,, C }, where C s the th characterstc to cosder of the caddate. (7) Step 3: Fxato of the real level of each characterstc for all the dfferet caddates cosdered. That s: P k = Table 2: Avalable players C C 2 C C µ µ 2 µ µ wth k =, 2,, m; where P k s the kth caddate expressed by a fuzzy subset, C s the th characterstc to cosder ad µ [0,]; =,,, s the valuato betwee 0 ad for the th characterstc of the kth caddate. Step 4: Comparso betwee the deal player ad the dfferet caddates cosdered, ad determato of the level of removal usg the OWA operator. That s, chagg the eutralty of the results to over estmate or uder estmate them. I ths step, the objectve s to express umercally the removal betwee the deal player ad the dfferet caddates cosdered. For dog ths, t ca be used the dfferet avalable selecto dexes such as the Hammg dstace, the adequacy coeffcet, the dex of maxmum ad mmum level, etc. [5-9,2-3]. Step 5: Adopto of decsos accordg to the results foud the prevous steps. Fally, we should take the decso about whch perso select. Obvously, our decso wll cosst choosg the caddate wth the best results accordg to the dex used. 3.. USING THE OWA OPERATOR AND THE HAMMING DISTANCE Whe makg the comparso betwee the deal player ad the avalable caddates Step 4, t s possble to use dfferet techques. Frst, we wll cosder the combato betwee the OWA operator ad the Hammg dstace that t s kow as the OWAD operator. It ca be defed as follows. Defto 8. A OWAD operator of dmeso, s a mappg OWAD: R R R that has a assocated weghtg vector W, wth the sum of the weghts equal to ad w j [0,] such that: Step 2: Fxato of the deal levels of each sgfcat characterstc order to form the deal player. That s: OWAD(P,P k ) = w j D j (8) Table : Ideal player C C 2 C C P = µ µ 2 µ µ where P s the deal player expressed by a fuzzy subset, C s the th characterstc to cosder ad µ [0,]; =, 2,,, s the valuato betwee 0 ad for the th characterstc. where D j represets the jth largest of the µ µ, ad k =,2,,m. Note that from a geeralzed perspectve of the reorderg step t s possble to dstgush betwee ascedg ad descedg orders. The weghts of these operators are related by w j = w* j+, where w j s the jth weght of the XIV Cogreso Español sobre Tecologías y Lógca fuzzy 3

4 ESTYLF08, Cuecas Meras (Meres - Lagreo), 7-9 de Septembre de 2008 ascedg OWAD (AOWAD) ad w* j+ the jth weght of the descedg OWAD (DOWAD) operator. As we ca see, the OWAD operator s commutatve, mootoc, bouded ad dempotet. By choosg a dfferet mafestato of the weghtg vector, we are able to obta dfferet types of aggregato operators. For example, the maxmum dstace s foud whe w = ad w j = 0 for all j. The mmum s foud whe w = ad w j = 0 for all j. The NHD s obtaed whe w j = / for all j. The WHD s foud whe the ordered posto of s the same tha the ordered posto of j. Note that the case of te the fal result, t could be used the decso the secod best or worst result, ad so o USING THE OWA OPERATOR AND THE ADEQUACY COEFFICIENT I ths Secto, we troduce the use of the OWA operator the selecto of huma resources sport maagemet wth the adequacy coeffcet. We wll call t the OWAAC. It ca be defed as follows. Defto 9. A OWAAC operator of dmeso, s a mappg OWAAC: R R R that has a assocated weghtg vector W, wth w j [0,] ad the sum of the weghts s equal to, such that: OWAAC(P k P) = w j K j where K j represets the jth largest of the [ ( - µ + µ )], ad k =,2,,m. Note that the fal result wll be a umber betwee [0,], beg the maxmum possble result. Note also that from a geeralzed perspectve of the reorderg step we ca dstgush betwee descedg ad ascedg orders. The weghts of these operators are related by w j = w* j+, where w j s the jth weght of the descedg OWAAC (DOWAAC) ad w* j+ the jth weght of the ascedg OWAAC (AOWAAC) operator. Note also that the OWAAC operator s commutatve, mootoc, bouded ad dempotet. Note that t s possble to use dfferet types of aggregato operators such as the maxmum, the mmum, the NAC, the WAC, the step-owaac, the S- OWAAC, the cetered-owaac, etc. Aalogously to the OWAAC operator, we ca suggest a equvalet removal dex that t s a dual of the OWAAC because Q(P k P) = - K(P k P). We wll call t the ordered weghted averagg dual adequacy coeffcet (OWADAC). (9) Defto 0. A OWADAC operator of dmeso, s a mappg OWADAC: R R R that has a assocated weghtg vector W, wth w j [0,] ad the sum of the weghts s equal to, the: OWADAC(P k P) = w j Q j (0) where Q j represets the jth largest of the [0 (µ - µ )], ad k =,2,,m. The fal result wll be a umber betwee [0,]. Note that ths case we usually select the lowest value as the best result. I ths case, we ca also dstgush betwee the descedg OWADAC (DOWADAC) ad the ascedg OWADAC (AOWADAC) operator ad t s also possble to obta dfferet famles of aggregato operators. Aother terestg ssue to cosder s the ufcato pot the selecto of huma resources sport maagemet. As t has bee explaed [3], the ufcato pot appears whe the results obtaed the Hammg dstace are the same tha the results obtaed the adequacy coeffcet. I the ew methods suggested ths paper, we also fd the ufcato pot whe the OWAD ad the OWAAC accomplsh the theorems explaed [3]. Note that t s possble to fd a total ufcato pot or a partal ufcato pot [3]. I the followg, we brefly show the ma theorem whe usg the OWA operator. Theorem. Assume OWAD(P,P k ) s the selecto of huma resources sport maagemet wth the OWAD operator ad OWADAC(P k P) the selecto of huma resources sport maagemet wth the OWADAC operator. If µ µ for all, the: OWAD(P,P k ) = OWADAC(P k P) () Proof. Let OWAD(P,P k ) = w j µ µ ad OWADAC(P k P) = w [0 ( µ µ )] j Sce µ µ for all, [0 (µ - µ )] = (µ - µ ) for all, the OWADAC(P k P) = w ( µ µ ) = OWAD(P,P k ) j Aalysg ths theorem, we could geeralze t for all the players - alteratves cosdered the decso problem. 4 XIV Cogreso Español sobre Tecologías y Lógca fuzzy

5 ESTYLF08, Cuecas Meras (Meres - Lagreo), 7-9 de Septembre de 2008 The theorem that explas ths geeralzato s very smlar to theorem () wth the dfferece that ow we cosder all the characterstcs ad all the players k USING THE OWA OPERATOR AND THE INDEX OF MAXIMUM AND MINIMUM LEVEL I ths Secto, we develop a dex for the selecto of huma resources sport maagemet that uses the OWA operator the dex of maxmum ad mmum level. We wll call t the OWAIMAM. It s defed as follows. Defto. A OWAIMAM operator of dmeso, s a mappg OWAIMAM: R R R that has a assocated weghtg vector W, wth w j [0,] ad the sum of the weghts s equal to, such that: S(P k P) = w j S j (2) where S j represets the jth largest of all the µ µ ad the [0 (µ - µ )]; wth k =,2,,m. Note that from a geeralzed perspectve of the reorderg step we could dstgush betwee descedg (DOWAIMAM) ad ascedg (AOWAIMAM) orders. I ths case, t s also possble to use dfferet types of aggregato operators such as the maxmum, the mmum, the NIMAM ad the WIMAM. Aalogously to the OWAIMAM operator, we ca suggest a equvalet removal dex that t s a dual of the OWAIMAM because R(P k P) = - S(P k P). We wll call t the ordered weghted averagg dual dex of maxmum ad mmum level (OWADIMAM). Aother terestg ssue to cosder s the ufcato pot the selecto of huma resources sport maagemet for the dex of maxmum ad mmum level. As t has bee explaed [3], these stuatos, the dex of maxmum ad mmum level becomes the Hammg dstace. Note that t s possble to fd a total ufcato pot ad a partal ufcato pot [3]. 4 ILLUSTRATIVE EXAMPLE We wll develop a decso makg problem about selecto of players a football team. We wll assume that a football team s lookg for a player for a specfc posto that they eed to cover. Obvously, the market there are a lot of choces but ot all of them ca be cosdered as a real choce because hs team does ot wat to sell t. The, the experts of the team eed to cosder all the alteratves ad develop a frst geeral selecto order to aalyze detal the real alteratves that ca be acqured by the team. Step : Aalyss ad determato of the sgfcat characterstcs for the team. Assume that a football team wats to select a player for a forward posto ad t has 4 caddates P, P 2, P 3, P 4, wth dfferet characterstcs. It s cosdered for each characterstc a property. Step 2: Fxato of the deal level for each sgfcat characterstc. It s defed the deal player as: Table 3: Characterstcs of the deal player C C 2 C 3 C 4 C 5 P * = Step 3: Fxato of the real level of each characterstc for all the dfferet caddates cosdered. For each of these characterstcs, t s foud the followg formato: Table 4: Avalable caddates C C 2 C 3 C 4 C 5 P P P P Step 4: Comparso betwee the deal worker ad the dfferet caddates cosdered usg the OWA operator. We wll cosder the NHD, the WHD, the OWAD ad the AOWAD operator. I ths example, we assume that the compay decdes to use the followg weghtg vector: W = (0, 0, 0 2, 0 3, 0 3). If we elaborate the selecto process wth the Hammg dstace, we wll get the results show Table 5. Table 5: Aggregated results wth the Hammg dstace NHD WHD OWAD AOWAD P P P P I ths case, our decso wll cosst selectg the caddate wth the smallest dstace. The, we wll select P 4 as t gves us the lowest dstace the four cases. If we develop the selecto process wth the adequacy coeffcet, we wll get the followg results show Table 6. Note that ths case we select the alteratve wth the hghest result beg P 4 the optmal player for the team. XIV Cogreso Español sobre Tecologías y Lógca fuzzy 5

6 ESTYLF08, Cuecas Meras (Meres - Lagreo), 7-9 de Septembre de 2008 Table 6: Aggregated results wth the adequacy coeffcet NAC WAC OWAAC AOWAAC P P P P Aalogously to ths dex, we ca obta ts equvalet removal dex. I a abbrevated form, ths dex ca be obtaed by usg Q(P k P) = - K(P k P). The results are show Table 7. Table 7: Results wth the dual adequacy coeffcet NDAC WDAC OWADAC AOWADAC P P P P Fally, f we use the IMAM the selecto process as a combato of the Hammg dstace ad the adequacy coeffcet, we wll get the followg. Note that ths example, we wll assume that the characterstcs C ad C 2 have to be treated wth the adequacy coeffcet ad the other three characterstcs wth the Hammg dstace. Its resoluto s show Table 8. Table 8: Aggregated results wth the IMAM NIMAM WIMAM OWAIM AOWAIM P P P P The, our decso wll cosst select P 4 because t s the caddate wth the smallest removal to the deal. Note that ths example, the results obtaed by usg the Hammg dstace ad the IMAM are equal. The reaso s that ths example fulfls the codtos to eter a stuato of ufcato pot as explaed [3]. The, these stuatos, the Hammg dstace ad the IMAM have the same formulato. Aalogously to ths dex, we ca obta ts equvalet approxmato dex. I a abbrevated form, ths dex ca be obtaed by usg R(P k P) = - S(P k P). The results are show Table 9. Table 9: Aggregated results wth the dual IMAM NDIM WDIM OWADIM AOWADIM P P P P As we ca see, depedg o the partcular type of aggregato operator used, the results may lead to dfferet decsos. 5 CONCLUSIONS We have aalysed the selecto of huma resources sport maagemet. We have used a ew methodology that uses the OWA operator wth dfferet measures frequetly used busess decso makg such as the Hammg dstace, the adequacy coeffcet ad the dex of maxmum ad mmum level. We have see that the ma advatage of usg the OWA operator ths type of problems s the possblty of uder or over estmate the results. The, the decso maker s able to take decsos cosderg ts atttudal character. I future research, we expect to develop further extesos of these problems by addg ew characterstcs ad applyg t to other decso makg problems. Ackowledgemets We would lke to thak the aoymous referees for ther valuable commets that have mproved the qualty of the paper. Refereces [] G. Belakov, A. Pradera, T. Calvo, Aggregato Fuctos: A Gude for Practtoers, Sprger- Verlag, New York, [2] H. Bustce, F. Herrera, J. Motero, Fuzzy Sets ad ther Extesos: Itellget Systems from Decso Makg to Data Mg, Web Itellgece ad Computer Vso, Sprger-Verlag, [3] T. Calvo, G. Mayor, R. Mesar. Aggregato Operators: New Treds ad Applcatos, Physca- Verlag, New York, [4] J.R. Fguera, S. Greco, M. Ehrgott, Multple crtera decso aalyss: State of the art surveys, Sprger, Bosto, [5] J. Gl-Aluja. The teractve maagemet of huma resources ucertaty, Kluwer Academc Publshers, Dordrecht, XIV Cogreso Español sobre Tecologías y Lógca fuzzy

7 ESTYLF08, Cuecas Meras (Meres - Lagreo), 7-9 de Septembre de 2008 [6] A.M. Gl-Lafuete. Fuzzy logc facal aalyss, Sprger, Berl, [7] J. Gl-Lafuete. El ídce del máxmo y mímo vel e la optmzacó del fchaje de u deportsta, I X Cogreso Iteracoal de la Asocacó Europea de Dreccó y Ecoomía de la Empresa (AEDEM), Reggo Calabra, Italy, pp , 200. [8] J. Gl-Lafuete, Algortmos para la exceleca: Claves para el éxto e la gestó deportva, Ed. Mlladoro, Vgo, [9] A. Kaufma, J. Gl-Aluja, Itroduccó de la teoría de los subcojutos borrosos a la gestó de las empresas, Ed. Mlladoro, Satago de Compostela, 986. [0] N. Karayas. Soft Learg Vector Quatzato ad Clusterg Algorthms Based o Ordered Weghted Aggregato Operators, IEEE Trasactos o Neural Networks, Pág , [] J.M. Mergó. New extesos to the OWA operators ad ther applcato busess decso makg, Upublshed thess ( Spash), Departmet of Busess Admstrato, Uversty of Barceloa, [2] J.M. Mergó, A.M. Gl-Lafuete, Usg the OWA operators the selecto of facal products, I Proceedgs of the 4st CLADEA Cogress, Motpeller, Frace, CD-ROM, [3] J.M. Mergó, A.M. Gl-Lafuete, Ufcato pot methods for the selecto of facal products, Fuzzy Ecoomc Revew 3, Pág , [4] R.R. Yager. O Ordered Weghted Averagg Aggregato Operators Mult-Crtera Decso Makg, IEEE Trasactos o Systems, Ma ad Cyberetcs B 8, Pág , 988. [5] R.R. Yager. Famles of OWA operators, Fuzzy Sets ad Systems 59, Pág , 993. [6] R.R. Yager, J. Kacprzyk. The Ordered Weghted Averagg Operators: Theory ad Applcatos, Kluwer Academc Publshers, Norwell, 997. XIV Cogreso Español sobre Tecologías y Lógca fuzzy 7