Australia Joural of Basic ad Applied Scieces, 8(11) Special 2014, Pages: 16-22 AENSI Jourals Australia Joural of Basic ad Applied Scieces ISSN:1991-8178 Joural home page: www.abasweb.com Explorig the Hammig Distace Method with Shao Etropy Cocept: Measurig i Performace Appraisal 1 MD Saad, R., 1 Ahmad, M.Z., Phd ad 2 Jusoh, M.S., Phd 1 Uiversity Malaysia Perlis (UiMAP), Istitute of Egieerig Mathematics, Kampus Tetap Pauh Putra, 02600 Arau, Perlis, Malaysia 2 Uiversity Malaysia Perlis (UiMAP), School of Techopreeurship ad Busiess Iovatio, 01000 Kagar, Perlis, Malaysia A R T I C L E I N F O Article history: Received 25 Jue 2014 Received i revised form 8 July 2014 Accepted 10 August May 2014 Available olie 30 August 2014 Keywords: Multi criteria decisio makig, fuzzy method, Hammig distace, Shao s etropy cocept A B S T R A C T Performace appraisal ofte bee regards as oe of essetial process i determiig the curret performace of the employee. However, to idetify the best performace of a employee is ot a easy process. There are several aspects that eed to be cosider to esure that the selectio process are doe i fair udgmets with o existece of bias or error. Thus, the mai purpose of this paper is to determie the best employee performace by usig Hammig distace method with Shao etropy cocept. I this case, Shao etropy cocept is used to determie the criteria importat plus fuzzy set theory is also icorporated to hadle the ucertaity that exist. A real world data from a istitute of local uiversity i Malaysia is applied. Based from fial results, a rakig of the alteratives is made ad future research is suggested to improve the proposed method. 2014 AENSI Publisher All rights reserved. To Cite This Article: MD Saad, R., Ahmad, M.Z., Phd ad Jusoh, M.S., Phd., Explorig the Hammig Distace Method with Shao Etropy Cocept: Measurig i Performace Appraisal. Aust. J. Basic & Appl. Sci., 8(11): 16-22, 2014 INTRODUCTION Performace appraisal ca be used as a idicator i measurig the performace of the employee effectiveess ad efficiecy i a istitutio or orgaizatio (Aggarwal & Thakur, 2013). Evetually, it ca be employed i performig some sigificace process for istace measurig the past performace of a employee, rewardig process ad also predictig ad creatig the goal for future performace ad developmet of a employee (Sapra, 2012). Sice this process icluded multiple criteria that eed to be evaluated, oe solutio is by applyig multi-criteria decisio makig (MCDM) method. MCDM methods usually ca be used i evaluatig, selectig or rakig a fiite set of available alteratives agaist multiple ad coflictig criteria (Chag et al., 2013). Distace measure method is oe of MCDM method that recogized for its proficiecy ad capability i measurig related distace measure exemplified by similarity ad proximity. Oe of the wellkow distace measure methods is Hammig distace method. Hammig distace method (Hammig, 1950), ca be used to calculate the differece betwee two sets or elemets, for istace the distace betwee the extremes of the itervals i fuzzy set theory (Caós et al., 2011). Cosequetly, this method ca be used i solvig the decisio makig problem by determiig the distace values betwee two elemets which are the alterative ad ideal alterative. I this case, the ideal alterative is a virtual alterative created by the decisio makers that will be used as a idicator of the alterative performace. Usually, the ideal alterative will possessed the highest marks for each criterio. I this paper, the ideal alterative marks will be assessed based o the decisio makers preferece. Through the distace values, the order ad rakig of the alteratives is made. This rakig is useful for the selectio of the best alterative i which the alterative with the less distace value will be the oe to be selected. Apart from Hammig distace method, the applicatio of fuzzy set theory (Zadeh, 1965) also embedded with problem solvig capabilities. The use of this theory are relevat with this cases, as the decisio makers ofte ecouter with icosistecies of huma udgmet durig the evaluatio process especially whe ivolvig the qualitative criteria such as leadership, persoality ad creativity that usually defie i subective ad vagueess data. Plus, i certai coditio, the iformatio that exists is uobtaiable, uquatifiable, icompleted ad i partial igorace coditio which hider the evaluatio process (Yeh & Deg, 1997). Therefore, the use of fuzzy set theory is clearly matched to solve this problem. Correspodig Author: MD Saad, R., Uiversity Malaysia Perlis (UiMAP), Istitute of Egieerig Mathematics, Kampus Tetap Pauh Putra, 02600 Arau, Perlis, Malaysia
17 MD Saad, R. et al, 2014 Australia Joural of Basic ad Applied Scieces, 8(11) Special 2014, Pages: 16-22 I certai circumstaces, the rakig of the alteratives caot properly be made due to the same distace values obtai by two or more alteratives which resulted, them to share the same rakig place. Alteratively, to solve this problem, the decisio maker ca prioritize the criteria by assigig relative weight for each oe of them. The criteria with the highest weight show that this criterio plays a importat role i decidig the best alterative. Shao etropy cocept is oe of umerous methods that ca be used to calculate the criteria weight. Shao s etropy cocept (Shao & Weaver, 1947) is kow as a geeral measure of ucertaity i iformatio expressed i terms of probability theory (Wag et al., 2007). This cocept ca be used i measurig the relative differeces itesities of criteria i represetig the average itrisic iformatio trasmitted to the decisio makers (Zeley, 1996). Meawhile, etropy weight is as a parameter that ca be used to idetify the differeces betwee the alteratives with respect to a certai criteria (Wag & Lee, 2009). The mai obective of this paper is to solve the performace appraisal process usig the Hammig distace method with Shao etropy cocept. Ispired by the previous algorithm proposed by Caós et al. (2011), the researchers exteded ad improved the existig algorithm with the additio of etropy based weight. Liguistic terms captured i a form of triagular fuzzy umbers are used to express the criteria ad criteria weight evaluatio. The remaiig of this paper is orgaized as follows. I sectio 2, the algorithm for the proposed approach amely Hammig distace method with Shao etropy cocept is explaied. Sectio 3; validate the proposed approach by usig real data i oe of local uiversity i Malaysia. The discussio o the obtaied results also preseted. Fially, the last sectios coclude the paper. Hammig Distace Method With Shao Etropy Cocept: The proposed method which is Hammig distace method with Shao etropy cocept ca be used i solvig ay decisio makig problem. However, for certai circumstaces, some adustmet might be made to harmoize with the proposed algorithm. As for the essetial requiremet, the evaluatio process is doe by m decisio makers, E E1, E2,, Em i the form of liguistic variables by usig triagular fuzzy umbers. The basic steps for the proposed algorithm are preseted as follows: Step 1: Costruct a decisio matrix for ideal alterative. The decisio matrix for ideal alterative that represets the optimum values of selectio criteria C C1, C2,, C is give as follow: I v1, v2,, v. (1) Step 2: Costruct a decisio matrix for alteratives. The decisio matrix for a set of m possible alteratives, A A1, A2,, Am where x i represet the liguistic assessmet o the utility ratigs of alterative A i 1,2,, m with respect to selectio criteria C C1, C2,, C, is give as follow: C C C 1 2 A x x x 1 A x x x D A x x x 11 12 1 2 21 22 2 m m1 m2 m, Step 3: Costruct a decisio matrix for weight (criteria importace). The weightig matrix for criteria weight, w which represets the relative importace of selectio criteria C 1,2,, 1 2 give by the decisio maker, is ufolded as follow: C C C E w w w 1 E w w w W E w w w 11 12 1 2 21 22 2 m m1 m2 m. i Step 4: Costruct a iterval-valued fuzzy umber. The iterval performace matrix for alteratives, ideal alteratives ad criteria weight, are derived by usig α-cut of triagular fuzzy umber ad shows as follows respectively: i) The iterval decisio matrix for the ideal alterative: i (2) (3)
18 MD Saad, R. et al, 2014 Australia Joural of Basic ad Applied Scieces, 8(11) Special 2014, Pages: 16-22 l u l u l u I v1, v 1, v2, v 2,, v, v. (4) ii) The iterval decisio matrix for alteratives ratig: i u l u l u x11, x 11 x12, x 12 x1, x1 l u l u l u x21, x21 x22, x22 x2, x2 (5) D. l u l u l u xm 1, xm 1 xm2, xm2 x m, x m iii) The iterval decisio matrix for criteria weight: L U L U L U w11, w 11 w12, w 12 w1, w1 L U L U L U w21, w21 w22, w22 w2, w2 W (6), L U L U L U wm 1, wm1 wm 2, wm 2 w m, w m where 0 1. The use of differet α value shows the degree of cofideces i the decisio makers evaluatios. The higher α values shows a higher cofidece level i decisio makers which meas, the decisio makers evaluatio are earer to the possible value of a 2 for the respective triagular fuzzy umbers [7]. Step 5: Calculatig of criteria weight The criteria weight is calculated by usig Shao s etropy cocept. The iterval valued fuzzy umber is trasformed ito crisp umber before usig Shao s etropy cocept. Step 5.1: Calculate the crisp value of iterval weight [12]: l u wi wi wi, (7) 2 Cosequetly, a crisp value matrix represetig a relative weight of each criterio from the decisio makers evaluatio is expressed as: w11 w12 w1 w21 w22 w 2 W, (8) wm1 wm 2 wm Step 5.2: Calculate etropy values e as: m e k w l w, 1,, (9) i i i1 where k is costat ad let 1 k l m. If 0 Step 5.3: Calculate the degree of diversificatio, w, the w l w is equal to 0. i d : d 1 e, 1,, (10) Step 5. 4: Calculate the criteria weight, k 1 k w : d w. (11) d Step 6: Calculatig the distace values. The distace values are calculated by usig weighted Hammig distace method: l l u u d I, D w v x w v x. (12) WHD i i 1 Step 7: Rakig the alteratives. Step 8: Repeat step 4, 5, 6 ad 7 for differet values of α. The use of differet α values may yield differet results of the rakig of the alteratives. i i
19 MD Saad, R. et al, 2014 Australia Joural of Basic ad Applied Scieces, 8(11) Special 2014, Pages: 16-22 Step 9: Select the best alterative. Noted that this algorithm represeted for oe expert. If there is more tha oe expert, the average distace value betwee the m decisio makers is idetify ad use as fial result. Performace Appraisal: I this case, the score from the performace appraisal reports from oe of istitute i Uiversity Malaysia Perlis, UiMAP for the year 2010 is applied to validate the proposed approach. 21 officer assessed which is the academic officer, will be assessed based o 14 criteria. Sice the evaluatio marks are i the forms of crisp umbers, those marks are fuzzified ito liguistic variables. These liguistic variables are represets by a set of liguistic terms captured i the forms of triagular fuzzy umbers which is build based o the discussio ad prefereces from two decisio makers. The liguistic terms that represet the rage of the performace are ragig from terrible to excellet. As for the liguistic variables for criteria weight, it is ispired from Wag ad Lee (2009), sice the decisio makers fid it s suitable ad appropriate to be use. Discussio: The iformatio ad summarizatio of these steps are preseted as follows: Step 1: The ideal alterative vector (1) is build based o the liguistic variables preseted i Table ad Fig. 1. Evetually, both the decisio makers had give the same ad highest evaluatio score for the ideal alterative which is Excellet performace for all 14 criteria. Step 2: Similar with the previous step, by usig the proposed liguistic variables, the decisio matrix of the alteratives agaist 14 criteria is build. Appedix 1 shows the evaluatio score o each alterative by two decisio makers. Fig. 1: The fuzzy liguistic variables for each criterio. Table 1: Fuzzy liguistic terms ad respective fuzzy umbers for each criterio. Liguistic Terms Fuzzy Numbers Terrible (T) (0, 0, 1) Almost Terrible (AT) (0, 1, 2) Very poor (VP) (1, 2, 3) Poor (P) (2, 3, 4) Almost fair (AF) (3, 4, 5) Fair (F) (4, 5, 6) Almost Good (AG) (5, 6, 7) Good (G) (6, 7, 8) Very Good (VG) (7, 8, 9) Almost Excellet (AE) (8, 9, 10) Excellet (E) (9, 10, 10) Step 3: By usig the liguistic variable as show i Table 2 ad Fig. 2, the weightig matrix (3) that represeted the 14 criteria weight is build. Table 3 marked the criteria weight evaluatio score by two decisio makers. Table 2: Fuzzy liguistic terms ad respective fuzzy umbers for each criteria weight. Liguistic Terms Fuzzy Numbers Very Low (VL) (0, 0, 0.2) Low (L) (0.05, 0.2, 0.35) Medium Low (ML) (0.2, 0.35, 0.5) Medium (M) (0.35, 0.5, 0.65) Medium High (MH) (0.5, 0.65, 0.8) High (H) (0.65, 0.8, 0.95) Very High (VH) (0.8, 1, 1)
20 MD Saad, R. et al, 2014 Australia Joural of Basic ad Applied Scieces, 8(11) Special 2014, Pages: 16-22 1 VL L ML M MH H VH 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 Fig. 2: The fuzzy liguistic variables for each criteria weight. Table 3: Criteria weight evaluatio for the year 2010. C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 DM1 MH H H H H VH H MH MH H VH VH H H DM2 H MH VH H H H MH MH MH MH VH H MH H Step 4: The iterval valued fuzzy umbers of performace matrixes for the ideal alterative (4), the alteratives (5) ad criteria weights (6) are derived by usig α-cut of triagular fuzzy umbers. Step 5: To determie the criteria weight, Shao etropy cocept (7-11) will be used. The etropy based weights for each criterio at each α level are determied. Table 4 marked the criteria weight at α=0. Accordig to the Shao etropy cocept, C11 is cosidered as the most importat criteria followed by C3, C6 ad C12. Table 4: Result of criteria weight at α=0. α C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 0 0.058 0.058 0.104 0.083 0.083 0.104 0.058 0.033 0.033 0.058 0.125 0.104 0.058 0.042 Step 6: The distace values for the ideal alterative ad the alteratives are obtaied by usig weighted Hammig distace method (12). Step 7: Accordig to the distace values, the alteratives are rak i ascedig order. The alterative with the least distace values is cosidered as the best alterative. Step 8: The steps 4 to 7 are repeated by usig differet α values, 0,1. Table 5 illustrated the rakig of the alteratives at α=0. Usually, for some alteratives, their rakigs are chagig accordig to α value. Table 5: Rakig of the alteratives at α=0. α 1 2 3 4 5 6 7 8 9 10 11 0 A3 A8 A13, A4 A9 A22 A12, A10, A19 A17 A20 α 12 13 14 15 16 17 18 19 20 21 0 A1 A11 A5 A6 A21 A2 A14 A7 A18 A15 Step 9: Based o the rakig of the alteratives, the best performace of academic officers ca be idetify at the differet α values. Apparetly, based o Table 5, alterative A3 is likely to be cosidered as the best alterative. Coclusio: I this paper, Hammig distace method with Shao etropy cocept is proposed to select the best performace of academic officer i UiMAP, Malaysia. The use of criteria weight will help to idetify which criteria are the most importat criteria i solvig this problem. I the proposed method, the applicatio of fuzzy set theory is also embedded sice most of the criteria assessmets are doe i qualitative ad subective measuremet. As a result from the applicatio of the proposed method, the rakig of the academic officers based o their overall evaluatio o 14 criteria is made. The fial results show that the most importat criterio is belogig to C11, while the performace from academic officer, A3 is cosidered as the best performace. Aside from solvig performace appraisal problem, the proposed method also ca be used i solvig ay decisio makig problem. I upcomig research, we would like to exted this research by usig the appropriate method i evaluatig ideal alterative hece improvig the proposed algorithm.
21 MD Saad, R. et al, 2014 Australia Joural of Basic ad Applied Scieces, 8(11) Special 2014, Pages: 16-22 REFERENCES Aggarwal, A., G.S.M. Thakur, 2013. Techiques of performace appraisal-a review. Iteratioal Joural of Egieerig ad Advaced Techology (IJEAT), 2(3): 617-621. Sapra, N., 2012. Curret treds i performace appraisal. Iteratioal Joural of Research i IT & Maagemet, 2(2): 1203-1211. Chag, Y.H., C.H. Yeh, Y.W. Chag, 2013. A ew method selectio approach for fuzzy group multicriteria decisio makig. Applied Soft Computig, 13: 2179-2187. Hammig, R.W., 1950. Error detectig ad error correctig codes. Bell System Techical Joural, 29(2): 147-160. Caós, L., T. Casasús, E. Crespo, T. Lara, J.C. Pérez, 2011. Persoel selectio based o fuzzy methods. Revista De Matemática: Teoría Y Aplicacioes, 18(1): 177-192. Zadeh, L.A., 1965. Fuzzy sets. Iformatio ad Cotrol, 8(3): 338-353. Yeh, C.H., H. Deg, 1997. A algorithm for fuzzy multi-criteria decisio makig. Proceedigs of the 1997 IEEE Iteratioal Coferece o Itelliget Processig Systems, 2, 1564-1568. doi: 10.1109/ICIPS.1997.669295. Shao, C.E., W. Weaver, 1947. The mathematical theory of commuicatio. Urbaa: The Uiversity of Illiois Press. Wag, P., H.D. Lee, M.C.S. Chag, 2007. A fuzzy TOPSIS Approach with etropy measure for decisiomakig problem. Proceedigs of the IEEE Iteratioal Coferece o Idustrial Egieerig ad Egieerig Maagemet, 2007, 124-128. doi: 10.1109/IEEM.2007.4419164. Zeley, M., 1996. Multiple criteria decisio makig. New York: Spriger. Wag, T.C., H.D. Lee, 2009. Developig a fuzzy TOPSIS approach based o subective weights ad obective weights. Expert Systems with Applicatios, 36: 8980-8985. Chaghooshi, A.J., M.R. Fathi, M. Kashef, 2012. Itegratio of fuzzy Shao s etropy with fuzzy TOPSIS for idustrial robotic system selectio. Joural of Idustrial Egieerig ad Maagemet, JIEM, 5(1): 102-114. Appedix 1: Decisio makers 1 ad 2 evaluatio o alteratives performace agaist the criteria for the year 2010. DM C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 A1 D1 ME ME ME VG VG ME ME ME ME ME ME ME ME ME D2 ME ME ME ME VG ME ME VG ME ME ME ME ME ME A2 D1 VG VG VG VG VG ME ME ME VG VG ME VG ME VG D2 ME VG ME VG ME ME ME ME ME VG ME ME ME VG A3 D1 ME ME ME ME ME ME ME ME ME ME ME ME ME ME D2 ME ME ME ME ME ME E ME ME ME ME E ME ME A4 D1 ME ME ME ME ME ME ME ME ME ME ME ME ME ME D2 E ME ME ME ME E ME VG VG ME ME ME ME ME A5 D1 VG ME ME VG VG ME ME ME ME ME ME VG VG ME D2 ME ME ME ME ME ME ME ME ME ME ME ME ME VG A6 D1 VG VG VG VG VG ME ME ME VG ME VG ME ME ME A7 D1 VG VG VG VG VG ME VG VG VG VG VG ME ME ME D2 ME ME ME ME VG ME VG ME ME ME VG ME ME ME A8 D1 ME ME ME ME ME ME ME ME ME ME ME VG ME ME D2 E ME E ME ME ME ME ME E E ME ME ME ME A9 D1 VG ME ME ME VG ME ME ME ME ME ME ME ME ME D2 E ME ME ME ME ME ME ME ME ME ME ME ME ME A10 D1 ME ME VG VG VG ME ME ME ME ME ME ME ME ME A11 D1 ME ME ME VG VG ME ME ME ME ME ME VG VG ME D2 ME ME ME ME ME ME ME ME ME ME ME ME VG ME A12 D1 ME ME ME ME VG ME ME ME ME ME ME ME ME ME D2 ME ME ME ME VG ME ME VG ME ME ME ME ME ME A13 D1 ME ME ME ME ME ME ME ME ME ME ME ME ME ME A14 D1 VG VG VG VG VG ME ME VG ME VG VG ME ME ME D2 ME VG ME ME VG ME ME VG ME ME VG ME ME VG A15 D1 VG VG VG VG VG VG VG VG ME ME VG VG ME ME D2 VG ME ME VG ME ME ME VG ME ME VG VG ME ME A16 D1 ME ME ME VG VG ME ME ME ME ME ME ME ME ME D2 ME ME ME ME ME ME ME VG ME ME ME ME ME ME A17 D1 VG VG VG VG VG ME ME VG ME VG VG ME ME ME D2 VG ME ME VG ME ME ME ME VG ME VG VG ME VG
22 MD Saad, R. et al, 2014 Australia Joural of Basic ad Applied Scieces, 8(11) Special 2014, Pages: 16-22 Cotiue Appedix 1: Decisio makers 1 ad 2 evaluatio o alteratives performace agaist the criteria for the year 2010. DM C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 A18 D1 ME ME ME ME ME ME ME ME ME ME ME ME ME ME A19 D1 ME ME ME VG VG ME ME ME ME ME ME VG ME ME A20 D1 VG VG VG VG VG ME ME ME ME VG ME VG ME VG D2 ME ME ME ME VG E ME ME ME ME ME ME ME VG A21 D1 ME ME ME ME VG ME ME ME ME ME ME ME ME ME D2 ME ME ME ME ME ME VG ME ME ME ME ME ME ME