, July 6-8, 2011, London, U.K. Fuzzy TOPSIS Method n the Selecton of Investment Boards by Incorporatng Operatonal Rsks Elssa Nada Mad, and Abu Osman Md Tap Abstract Mult Crtera Decson Makng (MCDM) nvolves not only attrbutes that are precse or crsp, but also values that are not determnstc. Currently, Fuzzy TOPSIS presents a soluton for decson makers when dealng wth real world data that are usually mult attrbutes and nvolves a complex decson makng process. In ths work, an applcaton of ths method s demonstrated n the selecton of Investment Boards by takng nto account the operatonal rsks nvolved. Index Terms Fuzzy TOPSIS, Mult Crtera Decson Makng (MCDM), Operatonal Rsks I. INTRODUCTION Nowadays, dynamcs and rsky global fnancal envronment had caused the stock nvestors to become more beware durng nvestng process. Hence, nvestment assessment s mportant to mmune the nvested stocks from exposed by that rsk. In most of real world stuatons, usually decson makers are confronted wth multple crtera to be consdered before any decson can be made. Ths s the case of Mult Crtera Decson Makng (MCDM); a case wth the am to fnd the overall preferences among the avalable alternatves. In addton, when the attrbutes are not determnstc; the fuzzy logc approach s usually adopted. One of the most popular methods n MCDM s the Technque for Order Preference by Smlarty or TOPSIS. Hence, n the case of attrbutes that are not determnstc, fuzzy TOPSIS method wll be used. The theores were appled for choosng the best board n Bursa Malaysa stock tradng nvestment. It wll choose based on three crtera whch are Volume Tradng Stock, Market Valuaton and Value of the stock. Manuscrpt receved January 7, 2011; revsed February,9, 2011. Frst author, Elssa Nada Mad currently work as a lecturer at Faculty of Informatcs, Unversty Sultan Zanal Abdn, Terengganu, Malaysa. Her major feld s n Fuzzy Set Theory,(correspondng author phone: +6012-6654563, Fax: 609-6673412,e-mal: elssa@unsza.edu.my). Abu Osman Md. Tap currently work as a Professor at Dept. of Informaton Systems, Internatonal Islamc Unversty Malaysa. Hs major feld s n Topology and Algebra. (correspondng author phone: +603-61965646, Fax: +603-61965179, emal: abuosman@kct.u.edu.my). Many busness frms only focus on common fnancal rsks (nterest rate rsk, market rsk and etc.) to mantan ther busness to keep runnng. However, the queston of how the frms manage ther busness operatons s a major factor to mantan ther performance [1]. In real world, dynamc and rsky global fnancal envronment had caused stock nvestors to be more aware of the nvestment process. Operatonal rsks are one crucal factor that determnes the fnal outcomes of an nvestment hence makng decson makng process crtcal n order to avod expected and unexpected losses. As a result, managng operatonal rsks s usually done by a frm for the purpose of addng value by reducng the rsks assocated wth the frm s earnngs [1]. That s the reason why we propose the ncorporaton of operatonal rsks n our study. The rest of ths paper s organzed as follows. Secton 2 outlnes related works on Fuzzy TOPSIS whereas Secton 3 presents the prelmnares on Fuzzy TOPSIS. An example applcaton of the model s descrbed n Secton 4, followed by the concluson n Secton 5. II. RELATED WORKS TOPSIS was proposed by Hwang and Yoon n 1981 [2]. In ths method, the man concept s that the most preferred alternatve should have the shortest dstance from the Postve Ideal Soluton (PIS) and the longest dstance from the Negatve Ideal Soluton (NIS) [3]. Based on Wang and Elhag [4], PIS s the one that maxmzes the beneft crtera and mnmzes the cost crtera, whle the NIS functons n the opposte way. As opposed to the orgnal applcaton of TOPSIS where the weght of the crtera and the ratngs of alternatves are known precsely, many real-lfe decson problems are confronted wth unquantfable, ncomplete and non-obtanable nformaton [5] that make precse judgment mpossble. Ths s when fuzzy TOPSIS comes nto play where the crtera weghts and alternatve ratngs are gven by lngustc varables, expressed by fuzzy numbers. In the year 2000, Chen [6] had used an algorthm of a group mult-crtera decson makng that s composed of the followng steps n Table I [6] :
, July 6-8, 2011, London, U.K. TABLE 1 STEPS OUTLINING THE ALGORITHM OF A GROUP MULTI-CRITERIA DECISION MAKING Step Remarks 1. Identfy the evaluaton crtera (Usually done by a commttee of decson-makers) 2. Choose approprate lngustc varables (based on the mportance weght of the crtera) and the lngustc ratngs for alternatves wth respect to the crtera 3. Aggregate the weght of the crtera to get the aggregated fuzzy weght ŵ j of crteron C j and pool the decson makers opnons to get the aggregated fuzzy ratng x j of alternatve A under crteron C j. 4. Construct the fuzzy decson matrx and the normalzed fuzzy decson matrx 5. Construct the weghted normalzed fuzzy decson matrx 6. Determne the FPIS and NPIS 7. Calculate the dstance of each alternatve from FPIS and NPIS, respectvely 8. Calculate the closeness coeffcent of each alternatve 9. Determne the rankng order of all alternatves accordng to the closeness coeffcents. III. PRELIMINARIES Ths secton brefly outlnes some basc defntons of fuzzy sets from [7 10]-: Defnton 3.1. A fuzzy set à n a unverse of dscourse X s characterzed by a membershp functon µ à (x)whch assocates wth each element x n X a real number n the nterval [0,1].The functon value µ à (x) s termed the grade of membershp of x n à [7]. Defnton 3.2. A trangular fuzzy number can be defned by a trplet (n 1, n 2, n 3 ) shown n Fg. 1. The membershp functon µ ñ (x) s defned as [8] : Fg. 1. A trangular fuzzy number n Defnton 3.3. Let m =(m 1,m 2,m 3 ) and =(n 1,n 2,n 3 ) be two trangular fuzzy numbers. If m = n, then m 1 =n 1, m 2 =n 2 and m 3 =n 3. Defnton 3.4. D s called a fuzzy matrx, f at least an entry n D s a fuzzy number [9]. Defnton 3.5. A lngustc varable s a varable whose values are lngustc terms [10]. The concept of lngustc varable s very useful n dealng wth stuatons whch are too complex or too ll-defned to be reasonably descrbed n conventonal quanttatve expressons [9]. For example, weght s a lngustc varable and ts values are very low, low, medum, hgh, very hgh, etc. These lngustc values can also be represented by fuzzy numbers. IV. THE PROPOSED METHOD In ths study, TOPSIS method s used n the determnaton of fnal rankng from a group of nvestment boards. The method s calculated as follows: Let MCDM problem has n alternatves A 1, A 2,...,A n, and m crtera, C 1, C 2,...,C m. Each alternatve wll take a consderaton wth respect to crteron m. The ratngs of crtera can be concsely expressed n matrx format as and, where ( = 1,...,; j = 1,...,n) and (j = 1,...,n) are the fuzzy ratng of alternatve A ( = 1,...,m) wth respect to crteron C j (j = 1,...,m) and the weght of crteron C j (j = 1,...,m), respectvely. The method s calculated usng the followng steps : (a) Decson matrx, s normalzed va Eq. (2): Weghted normalzed decson matrx s formed: (2) : (1) (3)
, July 6-8, 2011, London, U.K. (b) Postve deal soluton (PIS) and negatve deal soluton (NIS) are determned: TABLE II LINGUISTIC VARIABLES FOR THE IMPORTANCE WEIGHT OF EACH CRITERION (c) The dstance of each alternatve from PIS and NIS are calculated usng Eucldean dstance formula: : (4) (5) (6) Very Not Important (VNI) (0, 0, 0.1) Not Important (NI) (0, 0.1, 0.3) Somewhat Not Important (SNI) (0.1, 0.3, 0.5) Medum (M) (0.3, 0.5, 0.7) Somewhat Important (SI) (0.5, 0.7, 0.9) Important (I) (0.7. 0.9, 1.0) Very Important (VI) (0.9, 1.0, 1.0) (7) TABLE III LINGUISTIC VARIABLES FOR THE RATINGS (d) The closeness coeffcent of each alternatve s calculated: d CC =, = 1, 2,..m. (8) * d + d (e) By comparng determned. CC values, the rankng of alternatves are V. APPLICATION OF FUZZY TOPSIS IN THE SELECTION OF INVESTMENT BOARDS ON BURSA MALAYSIA Investors may want to evaluate the performance of the stocks to nclude n ther portfolo. In ths case, the frst step s they have to choose whch boards they want to nvest wth respect to operatonal rsk whch mght be exst n each of the stocks. Kng [1] states that there are three crtera that wll provde good nsghts n the evaluaton of stock performance wth respect to operatonal rsks. The three crtera are Market Valuaton, Stock Tradng Volume and Stock Tradng Value. The data used here were gathered from the Bursa Malaysa webste startng from January 2005 untl December 2006. In ths study, the scale of the mportance of varous crtera and scale of the prortes were expressed n the form of lngustc varables. Lngustc varables are presented as trangular fuzzy number as n Tables II and III. Level of mportance of each crteron can be obtaned drectly or ndrectly usng pared comparsons. In ths study, t s proposed that the decson-makers use the lngustc varables (see Table II and III) to assess the mportance of each crtera and the alternatve prortes for the crtera. Very Not Poor (VNP) (0, 0, 1) Poor (P) (0, 1.3) Medum Poor (MP) (1,3,5) Far (F) (3, 5, 7) Medum Good (MG)) (5, 7, 9) Good (G) (7. 9, 10) Very Good (VG) (9, 10,0) The three alternatves of Investment Boards on Bursa Malaysa are as follows : 1) The Man Board, (A 1). 2) The Second Board, (A 2). 3) The MESDAQ Market, (A 3) In addton, the three crtera to be consdered n ths study are : 1) Market valuaton (RM bllon), (C 1 ) 2) Stock Tradng Volume (mllon unts), (C 2 ) 3) Stock Tradng Value (RM mllon), (C 3 ) The next stage nvolves sx steps as outlned n Secton V. Step 1: The decson-makers use the lngustc weghtng varables (See Table II) for determnng the level of mportance of crtera and the results are summarzed n Table IV.
, July 6-8, 2011, London, U.K. TABLE IV THE IMPORTANCE WEIGHT OF THE CRITERIA the range system, 1 represents the hghest value n upward movement where 0 represents the lowest value. D 1 D 2 D 3 C 1 I VI M C 2 VI VI VI C 3 VI I M Step 2: Decson makers use the lngustc weghtng varables (See Table III) to determne the prorty of each crteron and the alternatve s summarzed n Table V. TABLE V THE RATINGS OF THE THREE CANDIDATES UNDER ALL CRITERIA Crtera Alternatve The decson maker D 1 D 2 D 3 C 1 A 1 VG VG MG A 2 G MG F A 3 G G G C 2 A 1 G VG MG A 2 VG G VG A 3 MG MG F C 3 A 1 G G MG A 2 G MG MG A 3 F G G Step 3: Changng the lngustc evaluaton (shown n Table IV and V) to the trangular fuzzy numbers (Table VI) and then buld a fuzzy decson matrx and determne the weght for each crteron as presented n Table VII. TABLE VI FUZZY DECISION MATRIX A 1 (7.7, 9.0, 7.9) (7, 8.7, 7.9) (6.3, 8.3, 7.9) A 2 (5, 7, 8.7) (8.3, 9.7, 10) (5.7, 7.7, 3.9) A 3 (7, 9, 10) (4.3, 6.3, 8.3) (5.7, 7.7, 9.0) TABLE VII FUZZY WEIGHT FOR ALL CRITERIA Weght (0.63, 0.80, 0.90) (0.9, 1.0, 1.0) (0.63, 0.80, 0.90) Step 4: Construct a normalzed fuzzy decson matrx as shown n Table VIII. The step of data normalzaton s necessary to overcome dfferences between the unts. Normalzaton also enables valuaton measure n the same range of values whch s usually between zero and one. In TABLE VIII FUZZY NORMALIZED DECISION MATRIX FOR THE SELECTION OF STOCK LISTINGS OF THE BOARDS A 1 (0.8, 0.9, 1.0) (0.7, 0.9, 1.0) (0.7, 0.9, 1.0) A 2 (0.5, 0.7, 0.9) (0.8, 1.0, 1.0) (0.6, 0.8, 1.0) A 3 (0.7, 0.9, 1.0) (0.4, 0.6, 0.8) (0.6, 0.8, 0.9) Step 5: Construct a Weghted Normalzed Fuzzy Decson Matrx as shown n Table IX. To get mult crtera ndex, data from each of the crtera need to be aggregated. Varous methods can be done to mplement them. An example of ths s to use the weghted mean. There are two methods for calculatng weghted mean, frst s an arthmetc mean and second s by usng geometrc mean. Index based on arthmetc mean s generally more popular because of easly understood and mplemented. TABLE IX WEIGHTED NORMALIZED FUZZY DECISION MATRIX FOR SELECTION OF BOARD STOCK LISTING A 1 (0.5, 0.7, 0.9) (0.6, 0.9, 1.0) (0.4, 0.7, 0.9) A 2 (0.3, 0.6, 0.8) (0.8, 1.0, 1.0) (0.4, 0.6, 0.9) A 3 (0.4, 0.7, 0.9) (0.4, 0.6, 0.8) (0.4, 0.6, 0.8) Table IX s a weghted normalzed fuzzy decson matrx, takng nto account the weghts as determned by decson makers. The next step s to get the Fuzzy Postve Ideal Solutons (FPIS), (A * ) and Fuzzy Negatve Ideal Solutons (FNIS), (A - ). Step 6: To assgn both the deal solutons, the method used by Chen [4] s adopted as t can easly be understood. Consequently, the FPIS (A * ) and FNIS (A - ) are defned as the followng : A * = (1, 1, 1) and A - = (0, 0, 0). After gettng the deal solutons, the next step s to calculate the dstance of the alternatves from (A * ) and (A - ) usng. Equaton (6) and (7), respectvely. Step 7: After calculatng the range of alternatves to (A * ) and (A - ), the next step s to obtan the correlaton coeffcents between the three alternatves. The calculaton s done usng Equaton (8). The results are shown n Table X. TABLE X THE CORRELATION COEFFICIENT OF EACH ALTERNATIVE Alternatve Correlaton Coeffcent Man Board 0.67 Second Board 0.62 MESDAQ Market 0.63 Based on Table X, t can be seen that the correlaton coeffcents of the frst alternatve, namely the Man Board s of the hghest value followed by MESDAQ Market and the Second Board. Correlaton coeffcents for the Man
, July 6-8, 2011, London, U.K. Board s of 0.67, whle the Second Board and MESDAQ Market, each has value 0.62 and 0.63. Based on the correlaton coeffcents, an alternatve to selectng the frms lsted on the Man Board should be the frst choce, followed by selectng the frms lsted on the MESDAQ Market and the last one s to select a frm on the Second Board. In essence, the greater the value of the correlaton coeffcent ndcates the prortes of the decson to be made. Ths method not only allows the decson maker to provde the rank of each alternatve, but also shows the degree of lkelhood of alternatve selecton as llustrated n Table X. It should be noted that our results are based out of the three crtera set out earler ths analyss (market valuaton, stock tradng volume and stock tradng value). From Table X, t s also apparent that the correlaton coeffcents for the Second Board and MESDAQ only dffer by 0.01. However, although the dfference s only one percent, the result s sgnfcant for the decson makers n determnng the order of the rankng. Therefore, the mplementaton of fuzzy TOPSIS n ths scenaro s really effectve n real world applcatons. The proposed method s very approprate when dealng wth subjectve assessment of the real envronment that s full of uncertantes. VI. CONCLUSION In ths paper, fuzzy TOPSIS was appled n the selecton of the best nvestment boards accordng to three crtera by ncorporatng operatonal rsks n nvestment. Frst crtera s market valuaton, second crtera s stock tradng volume and thrd crtera s stock tradng value. Results obtaned from the relatve closeness to the deal solutons were used to rank the preference order n the selecton of nvestment boards for stock exchanges. Clearly, the applcaton of fuzzy set theory n conjuncton wth TOPSIS s effectve n order to provde a more realstc soluton to the process of decson makng n stock nvestment. REFERENCES [1] Kng, J.L. : Operatonal Rsks : Measurement and Modellng, John Wley and Sons Ltd., New York, 2001 [2] Hwang, C.L, Yoon, K. : Multple Attrbutes Decson Makng Methods and Applcatons, Sprnger, Berln Hedelberg (1981) [3] Oprcovck, S., Tzeng, G.H. : Compromse Soluton by MCDM Methods : A Comparatve Analyss of VIKOR and TOPSIS. European Journal of Operatonal Research, 156, 445 455 (2004) [4] Wang, Y.M, Elhag, T.M.S. : Fuzzy TOPSIS Method Based on Alpha Level Sets wth an Applcaton to Brdge Rsk Assessment. Expert Systems wth Applcatons, 31, 309 319 (2006) [5] Olcer, A.I, Odabas, A.Y. : A New Fuzzy Multple Attrbutve Group Decson Makng Methodology and ts Applcaton to Populaton/Maneuverng System Selecton Problem. European Journal of Operatonal Research, 166, 93 114 (2005) [6] Chen, C-T. : Extenson of the TOPSIS for Group Decson-Makng Under Fuzzy Envronment. Fuzzy Sets and Systems, 114, 1 9 (2000) [7] Zadeh, L.A. : Fuzzy Sets, Informaton and Control, 8, 338 353 (1965) [8] Kauffman, A., Gupta, M.M. : Introducton to Fuzzy Arthmetc Theory and Applcatons. Van Nostrand Renhold, New York, 1985 [9] Buckley, J.J. : Fuzzy Herarchcal Analyss. Fuzzy Sets and Systems, 17 233 247 (1985) [10] Zadeh, L.A. : The Concept of A Lngustc Varable and Its Applcaton to Approxmate Reasonng. Informaton Scence, 8, 199 249 (I), 301 357(II) (1975)