Research Article A Modified TOPSIS Method Based on D Numbers and Its Applications in Human Resources Selection

Transcription

1 Mathematcal Problems Egeerg Volume 016, Artcle ID , 14 pages Research Artcle A Modfed TOPSIS Method Based o D Numbers ad Its Applcatos Huma Resources Selecto Lguo Fe, 1 Yog Hu, Fuyua Xao, 1 Luyua Che, 1 adyogdeg 1,,3,4 1 School of Computer ad Iformato Scece, Southwest Uversty, Chogqg , Cha Bg Data Decso Isttute, Ja Uversty, Tahe, Guagzhou 51063, Cha 3 Isttute of Itegrated Automato, School of Electroc ad Iformato Egeerg, X a Jaotog Uversty, X a, Shaax , Cha 4 School of Egeerg, Vaderblt Uversty, Nashvlle, TN 3735, USA Correspodece should be addressed to Yog Deg; prof.deg@hotmal.com Receved 9 February 016; Accepted 8 Aprl 016 Academc Edtor: Rta Gamber Copyrght 016 Lguo Fe et al. Ths s a ope access artcle dstrbuted uder the Creatve Commos Attrbuto Lcese, whch permts urestrcted use, dstrbuto, ad reproducto ay medum, provded the orgal work s properly cted. Multcrtera decso-makg (MCDM) s a mportat brach of operatos research whch composes multple-crtera to make decso. TOPSIS s a effectve method hadlg MCDM problem, whle there stll exst some shortcomgs about t. Upo facg the MCDM problem, varous types of ucertaty are evtable such as completeess, fuzzess, ad mprecso result from the powerlessess of huma begs subjectve judgmet. However, the TOPSIS method caot adequately deal wth these types of ucertates. I ths paper, a D-TOPSISmethodsproposedforMCDMproblembasedoaeweffectveadfeasble represetato of ucerta formato, called D umbers. The D-TOPSIS method s a exteso of the classcal TOPSIS method. Wth the proposed method, D umbers theory deotes the decso matrx gve by experts cosderg the terrelato of multcrtera. A applcato about huma resources selecto, whch essetally s a multcrtera decso-makg problem, s coducted to demostrate the effectveess of the proposed D-TOPSIS method. 1. Itroducto Multcrtera decso-makg (MCDM) or multple-crtera decso aalyss s a mportat brach of operatos research that deftely uses multple-crtera decsomakg evromets [1, ]. I daly lfe ad professoal learg, there exst geerally multple coflctg crtera whch eed to be cosdered makg decsos ad optmzato [3, 4]. Prce ad sped are typcally oe of the ma crtera wth regard to a large amout of practcal problems. However, the factor of qualty s geerally aother crtero whch s coflct wth the prce. For example, the cost, safety, fuel ecoomy, ad comfort should be cosdered as the ma crtera upo purchasg a car. It s the most beeft for us to select the safest ad most comfortable oe whch has the bedrock prce smultaeously. The best stuato s obtag the hghest returs whle reducg the rsks to the most extet wth regard to portfolo maagemet. I addto, the stocks that have the potetal of brgg hgh returs typcally also carry hgh rsks of losg moey. I servce dustry, there s a couple of coflcts betwee customer satsfacto ad the cost to provde servce. Upo makg decso, t wll be compellg f multple-crtera are cosdered eve though theycamefromadarebasedosubjectvejudgmetof huma. What s more, t s sgfcat to reasoably descrbe the problem ad precsely evaluate the results based o multple-crtera whe the stakes are hgh. Wth regard to the problem of whether to buld a chemcal plat or ot ad where the best ste for t s, there exst multple-crtera that eed to be cosdered; also, there are multple partes that wll be affected deeply by the cosequeces. Costructg complex problems properly as well as multple-crtera take to accout explctly results more reasoable ad better decsos. Sgfcat achevemets ths feld have bee made sce the begg of the moder multcrtera decso-makg (MCDM) dscple

2 Mathematcal Problems Egeerg the early 1960s. A varety of approaches ad methods have bee proposed for MCDM. I [5], a ovel MCDM method amed FlowSort-GDSS s proposed to sort the falure modes to prorty classes by volvg multple decso-makers, whch has the robust advatages sortg falures. I the feld of multple objectve mathematcal programmg, Evas ad Yu [6, 7] proposed the vector maxmzato method amed at approxmatg the odomated set whch s orgally developed for multple objectve lear programmg problems. Torrace [8] used elaborate tervew techques to deal wth the problem MCDM, whch exst for elctg lear addtve utlty fuctos ad multplcatve olear utlty fuctos. Ad there are may other methods, such as best worst method [9], characterstc objects method [10], fuzzy sets method [11 13], rough sets [14], ad aalytc herarchy process [15 17]. I [18], the authors am to systematcally revew the applcatos ad methodologes of the MCDM techques ad approaches, whch s a good gudace for us to fully uderstad the MCDM. Techque for order preferece by smlarty to deal soluto (TOPSIS), whch s proposed [19 3], s a rakg method cocepto adapplcato.thestadardtopsismethodologyamsto select the alteratves whch have the shortest dstace from thepostvedealsolutoadthelogestdstacefromthe egatve deal soluto at the same tme. The postve deal soluto maxmzes the beeft attrbutes ad mmzes the cost attrbutes, whereas the egatve deal soluto maxmzes the cost attrbutes ad mmzes the beeft attrbutes. The TOPSIS methodology s appled wdely MCDM feld [4 7], especally the fuzzy exteso of lgustc varables [8 31]. It s obvous that the metoed approaches play a role uder some specfc crcumstaces, but, the practcal applcatos, they show more ucertates due to the subjectve judgmet of experts assessmet. I order to effectvely hadle varous ucertates volved the MCDM problem, a ew represetato of ucerta formato, called D umbers [3], s preseted ths paper. It s a exteso of Dempster-Shafer evdece theory. It gves the framework of oexclusve hypotheses, appled to may decso-makg problems uder ucerta evromet [33 38]. Comparg wth exstg methods, D umbers theory ca effcetly deote ucerta formato ad more cocde wth the actual codtos. Therefore, ths paper, to well address these ssues TOPSIS method, a exteded verso s preseted based o D umbers amed D-TOPSIS, whch cosders the terrelato of multcrtera ad hadles the fuzzy ad ucerta crtera effectvely. The D-TOPSIS method ca represet ucerta formato more effectvely tha other group decso support systems based o classcal TOPSIS method, whch caot adequately hadle these types of ucertates. A applcato has bee coducted usg the D-TOPSIS method huma resources selecto, ad the result ca be more reasoable because of ts cosderato about the terrelato of multple-crtera. The remader of ths paper s costtuted as follows. Secto troduces the Dempster-Shafer theory ad ts basc rules ad some ecessary related cocepts about D umbers theory ad ts dstace fucto ad TOPSIS. The proposed D-TOPSIS method s preseted Secto 3. Secto 4 coducts a applcato huma resources selecto based o D-TOPSIS. Cocluso s gve Secto 5.. Prelmares.1. Dempster-Shafer Evdece Theory. Dempster-Shafer evdece theory [39, 40], whch s frst developed by Dempster ad later exteded by Shafer, s used to maage varous types of ucerta formato [41 44], belogg to the category of artfcal tellgece. As a theory wdely appled uder the ucerta evromet, t eeds weaker codtos ad has a wder rage of use tha the Bayesa probablty theory. Whe the gorace s cofrmed, Dempster-Shafer theory couldcoverttobayesatheory,sotsofteregardedas a exteso of the Bayesa theory. Dempster-Shafer theory has the advatage to drectly express the ucertaty by assgg the probablty to the subsets of the uo set composed of multple elemets, rather tha to each of the sgle elemets. Besdes, t has the ablty to combe pars of bodes of evdece or belef fuctos to geerate a ew evdece or belef fucto [45, 46]. The decso-makg or optmzato real system s very complex wth complete formato [47 49]. Wth the superorty dealg wth ucerta formato ad the practcablty egeerg, a umber of applcatos of D-S evdece theory have bee publshed the lterature dcatg ts wdespread for fault dagoss [50, 51], patter recogto [5 54], suppler selecto [55, 56], ad rsk assessmet [57, 58]. Also, t exerts a great effect o combg wth other theores ad methods such as fuzzy umbers [59], decso-makg [60], ad AHP [61 63]. Moreover, based o the Dempster-Shafer theory, the geeralzed evdece theory has bee proposed by Deg to develop the classcal evdece theory [64] to hadle coflctg evdece combato [65]. It should be oted that the combato of depedet evdece s stll a ope ssue [66, 67]. For a more detaled explaato of evdece theory, some basc cocepts are troduced as follows. Defto 1 (frame of dscermet). A frame of dscermet s a set of alteratves perceved as dstct aswers to a questo. Suppose U s the frame of dscermet of researchg problem, a fte oempty set of N elemets that are mutually exclusve ad exhaustve, dcated by U={E 1,E,...,E,...,E } (1) ad deote U as the power set composed of N elemets of U, ad each elemet of U s regarded as a proposto. Based o the two coceptos, mass fucto s defed as below. Defto (mass fucto). For a frame of dscermet U, a mass fucto s a mappg m from U to [0, 1], formally defed by m: U [0, 1] ()

3 Mathematcal Problems Egeerg 3 y y 1 1 ZD LD MD HD CD ZD LD MD HD CD O (a) Dempster-Shafer evdece theory x O (b) D umber theory x Fgure 1: The framework of DSET ad DNT. satsfyg m (0) =0, m (A) =1, A U where 0 s a empty set ad A represets the propostos. I Dempster-Shafer theory, m s also amed as basc probablty assgmet (BPA), ad m(a) s amed as assged probablty, presetg how strog the evdece supports A. A s regarded as a focal elemet whe m(a) > 0,adtheuo of all focal elemets are called the core of the mass fucto. Cosderg two peces of evdece from dfferet ad depedet formato sources, deoted by two BPAs m 1 ad m, Dempster s rule of combato s used to derve a ew BPA from two BPAs. Defto 3 (Dempster s rule of combato). Dempster s rule of combato, also kow as orthogoal sum, s expressed by m=m 1 m, defed as follows: wth 1 { m (A) = 1 K m 1 (A 1 )m (A ), A=0; A { 1 A =A { 0, A = 0 K= m 1 (A 1 )m (A ), A 1 A =0 where K s a ormalzato costat, called coflct coeffcet of two BPAs. Note that the Dempster-Shafer evdece theory s oly applcable to such two BPAs whch satsfy the codto K<1. (3) (4) (5).. D Number Theory. D umber theory, proposed by Deg [3], s a geeralzato of Dempster-Shafer evdece theory. A wde rage of applcatos have bee publshed based o t, especally the ucerta evromet ad MCDM [33]. I the classcal Dempster-Shafer theory, there are several strog hypotheses o the frame of dscermet ad basc probablty assgmet. However, some shortcomgs stll exst whch lmt the represetato of some types of formato as well as the restrcto of the applcato practce. D umber theory, cosdered as a exteso ad developed method, makes the followg progress. Frst, Dempster-Shafer evdece theory deals wth the problem about the strog hypotheses, whch meas that elemets the frame of dscermet are requred to be mutually exclusve. I geeral, the frame of dscermet s determed by experts, always volvg huma beg s subjectve judgmets ad ucertaty. Hece, the hypothess s hard to meet. For example, there are fve achor pots zero depedece [ZD], low depedece [LD], moderate depedece [MD], hgh depedece [HD], ad complete depedece [CD] correspodg to depedece levels avalable to aalysts to make judgmets. It s evtable that there exst some overlaps of huma beg s subjectve judgmets. D umber theory s more sutable to the actual stuato based o the framework of oexclusve hypotheses. The dfferece betwee Dempster-Shafer theory ad D umber theory about ths s show Fgure 1. Secod, the problem solved by D umber theory wthout aother hypothess of Dempster-Shafer theory s related to basc probablty assgmet. I Dempster-Shafer theory, the sum of BPAs must be equal to 1, whch meas that the experts have to make all the judgmets ad the gve the assessmet results. Nevertheless, o the oe had, t would be dffcult to satsfy some complex evromet. O the other had, from tme to tme, t would be uecessary ad redudat to

4 4 Mathematcal Problems Egeerg meet the hypothess, whe the framework does ot cota overall stuatos. From ths pot of vew, D umber theory allows the completeess of formato, havg the ablty to adapt to more cases. Thrd, compared wth Dempster-Shafer theory, D umber theory s more sutable to the framework. I Dempster- Shafer theory, the BPA s calculated through the power setoftheframeofdscermet.itshardtoworkwhe there are too may elemets t, ad t eve ca ot be acceptedwhetheumberoftheframeworksstoohgh to use Dempster s rule of combato to some degree. D umber theory emphaszes the set of problem domas tself. Ucerta formato s represeted by D umbers so that the fuso would have less calculato ad allows arbtrary framework. Eve so, D umbers theory s stll preferable may cases, for the advatages of all the three pots above, ot just mprovg oe certa aspect. It s defed as follows. Defto 4 (D umber). Let a fte oempty set Ω deote the problem doma. D umber fucto s a mappg formulated by wth D: Ω [0, 1] (6) D (0) =0, D (B) 1 B Ω ad, compared wth the mass fucto, the structure of the expresso seems to be smlar. However, D umber theory, the elemets of Ω do ot requre to be mutually exclusve. I addto, beg cotrary to the frame of dscermet U cotag overall evets, Ω s sutable to complete formato by B Ω D(B) < 1. (7) Furthermore, for a dscrete set Ω={b 1,b,...,b,...,b }, where b R,adwhe =j, b =b j.aspecalformofd umbers ca be expressed by D({b 1 }) = V 1 D({b }) = V. D({b }) = V. D({b }) = V or smply deoted as D = {(b 1, V 1 ), (b, V ),...,(b, V ),..., (b, V )},wherev >0ad =1 V 1. Below s the combato rule, a kd of addto operato to combe two D umbers. Defto 5 (two D umbers rule of combato). Suppose D 1 ad D are two D umbers, dcated by (8) D 1 ={(b 1 1, V1 1 ),...,(b1, V1 ),...,(b1, V1 )}, (9) D ={(b 1, V 1 ),...,(b j, V j ),...,(b m, V m )}, ad the combato of D 1 ad D, whch s expressed as D= D 1 D, s defed as follows: wth D (b) = V (10) b= b1 +b j V = (V1 + V j )/, C m { =1 C= m, =1 m =1 m { { =1 ( V1 + V j ), ( V1 + V j ( V1 + V j ( V1 + V j )+ )+ )+ m =1 m ( V1 c + V j ), ( V1 + V c ), ( V1 c + V j )+ =1 ( V1 + V c )+ V1 c + V c, =1 =1 =1 =1 V 1 =1, V 1 <1, V 1 =1, V 1 <1, m m m m V j =1; V j =1; V j <1; V j <1, (11)

5 Mathematcal Problems Egeerg 5 where V 1 c = 1 =1 V1 ad V c = 1 m V j.notethat superscrpt the above equatos s ot expoet whe D umbers are more tha two. It must be poted out that the combato operato defed Defto 5 does ot preserve the assocatve property. It s clear that (D 1 D ) D 3 = D 1 (D D 3 ) = (D 1 D 3 ) D. I order that multple D umbers ca be combed correctly ad effcetly, a combato operato for multple D umbers s developed as follows. Defto 6 (multple D umbers rule of combato). Let D 1,D,...,D be Dumbers, μ j s a order varable for each D j, dcated by tuple μ j,d μj, ad the the combato operato of multple D umbers s a mappg f D,suchthat f D (D 1,D,...,D ) = [ [D λ1 D λ ] D λ ], (1) where D λ s D μj of the tuple μ j,d μj havg the th lowest μ j. I the meawhle, a aggregate operato s proposed o thsspecal D umbers, as such. Defto 7 (D umbers tegrato). For D={(b 1, V 1 ), (b, V ),...,(b, V ),...,(b, V )}, the tegratg represetato of D s defed as I (D) = =1 b V. (13).3. Dstace Fucto of D Numbers. Aewdstacefucto to measure the dstace betwee two D umbers has bee proposed [68]. I D umbers theory, there s o compulsve requremet that the frame of dscermet s a mutually exclusve ad collectvely exhaustve set. So a relatve matrx s used to represet the relatoshp of D umbers. The defto of relato matrx s show as follows Relatve Matrx ad Itersecto Matrx Defto 8. Let L ad L j deote the umber ad umber j of lgustc costats, S j represet the tersecto area betwee L ad L j,adu 1 s the uo area betwee L ad L j. The defto of oexclusve degree E j ca be show as follows: E j = S j U j. (14) U 1 L1 L S 1 L L +1 S +1 L 1 S 1 Fgure : Example for lgustc costats. Next, the relatve matrx ca be costructed for these elemets based o E j : 1 E 1 E 1 E 1 E 1 1 E E.... R= E 1 E 1 E. (15) [.... ] [ E 1 E E 1 ] For example, suppose there are lgustc costats whch are show Fgure. The oexclusve degree betwee two D umbers ca be obtaed by E j based o the tersecto area S j ad the uo area U j of two lgustc costats L ad L j. Defto 9. After obtag the relatve matrx R betwee two subsets whch belog to Ω, the defto of the tersecto degree of two subsets ca be show as follows: I(S 1,S )= E j S 1 S, (16) where = j ad S 1,S Ω. deotes the frst elemet s row umber of set S 1 the relatve matrx R ad j s the frst elemet s colum umber of set S. S 1 expresses the cardalty of S 1 ad S represets the cardalty of S.I partcular, whe =j, I= Dstace betwee Two D Numbers. It s kow that D umbers theory s a geeralzato of the Dempster-Shafer theory. The body of D umbers ca be cosdered as a dscrete radom varable whose values are Ω by a probablty dstrbuto d. Therefore, D umbercabeseeasavectord the vector space. Thus, the dstace fucto betwee two D umbers ca be defed as follows. Defto 10. Let d 1 ad d be two D umbers o the same frame of dscermet Ω, cotag N elemets whch are ot requred to be exclusve to each other. The dstace betwee d 1 ad d s d D-umber(d1,d ) = 1 ( d 1 d ) T D I( d 1 d ), (17) where D ad I are two ( N N )-dmesoal matrxes. L

6 6 Mathematcal Problems Egeerg The elemets of D caberepresetedas D (A, B) = A B A B, (A,B Ω ). (18) The elemets of I ca be deoted as I (A, B) = E j A B, ( =j), (A,B Ω ), (whe =j, I=1), (19) where deotes the frst elemet s row umber of set S 1 the relatve matrx R ad j s the frst elemet s colum umber of set S..4. TOPSIS Method. Techque for order preferece by smlarty to deal soluto (TOPSIS), whch s proposed [19], s a rakg method whch s appled to MCDM problem. The stadard TOPSIS method s desged to fd alteratves whch have the shortest dstace from the postve deal solutoadthelogestdstacefromtheegatvedeal soluto. The postve deal soluto attempts to seek the maxmzato of beeft crtera ad the mmum of the cost crtera, whereas the egatve deal soluto s just the opposte. Defto 11. Costruct a decso matrx D=(x m ),whch cludes alteratves ad crtera. Normalze the decso matrx r j = x j m x j, =1,...,m;,...,. (0) To obta the weghted decso matrx usg the assocated weghtstomultplythecolumsoftheormalzeddecso matrx A=V(m), V j =w j r j, =1,...,m;,...,, (1) where w j s the weght of jth crtero. Determe the postve deal ad egatve deal solutos. The deftos of the postve deal soluto, represeted as A +, ad the egatve deal soluto, represeted as A,are show as follows: A + ={V + 1, V+,...,V+ } ={(max V j j K b )(m V j j K c )} A ={V 1, V,...,V } ={(m V j j K b )(max V j j K c )}, () where K b deotes the set of beeft crtera ad K c represets the set of cost crtera. Calculate the separato measures betwee the exstg alteratves ad the postve deal ad egatve deal solutos. The separato measures that are determed by Eucldea dstace, S + ad S, of each alteratve from the postve deal ad egatve deal solutos, respectvely, are show as S + S = = (V + j V j), =1,...,m;,...,, (V j V j), =1,...,m;,...,. Obta the relatve closeess to the deal soluto: S (3) C = S +S +, =1,...,m. (4) Sort the alteratves based o the relatve closeess to the deal soluto. If alteratves have hgher C,twllbemore sgfcat ad should be assged hgher prorty. 3. The Modfed TOPSIS Method Based o D Numbers TOPSIS s a effectve methodology to hadle the problem multcrtera decso-makg. D umberstheorysaew represetato of ucerta formato, whch ca deote the more fuzzy codtos. So the combato of TOPSIS ad D umbers s a ew expermet to make decsos a ucerta evromet. Next, we wll propose the modfed TOPSIS method amed D-TOPSIS to deal wth some Gorda kots MCDM Costruct the Decso Matrx Defto 1. Suppose there s a matrx D=(x m ),whchs costructed by alteratves ad crtera. Obta the weght for each crtero of the matrx, ad assg the weght to correspodg crtero to determe the weghted matrx A=V(m): V j =w j x j, =1,...,m;,...,, (5) where w j s the weght for j crtero. Normalze the matrx to get the decso matrx: d j = V j m V j, =1,...,m;,...,. (6) 3.. Determe D Numbers ad Defe Iterrelato betwee Ther Elemets. I Secto 3.1, the decso matrx has bee costructed; the t wll be trasformed to D umbers as d 1 ({1}) =d 11 d 1 ({}) =d 1 d 1 ({}) =d 1 d ({1}) =d 1 d ({}) =d d ({}) =d. (7) [ d. ] [ d m ({1}) =d m1 d m ({}) =d m d m ({}) =d m ]

7 Mathematcal Problems Egeerg 7 The terrelato betwee crtera s cosdered the D-TOPSIS method for more reasoable ad more effectve decso-makg, whch s defed as follows. Defto 13. Let I j deote the fluece relato from crtero to crtero j. LetS j represet the terrelato betwee crtero ad crtero j,whchcaalsobeseeas the tersecto of crtero ad crtero j.the,oegves the defto of S based o I show as follows: S j =S j = 1 (I j +I j ). (8) Defto 14. Let U j deote the uo set betwee crtero ad crtero j. LetW represet the weght of crtero from the comprehesve vews of four experts. The, oe determes the defto of U based o I ad weghts of crtero ad crtero j show as follows: U j =U j =W +W j I j. (9) 3.3. The Methodology for Proposed D-TOPSIS. Frstly, determethepostvedealadegatvedealsolutos.the postve deal soluto, deoted as D A +,adtheegatve deal soluto, deoted as D A, are defed as follows: Determe postve deal solutos D-TOPSIS Decso matrx Trasform decso matrx to D umbers Dstace fucto of D umbers Determe egatve deal solutos Calculate the dstace betwee each soluto ad postve deal ad egatve deal solutos Calculate the relatve closeess ad rak D A + ={d + 1,d+,...,d+ } ={(max d j j K b )(m d j j K c )}, D A ={d 1,d,...,d } (30) Fgure 3: The flow chart of D-TOPSIS. ={(m d j j K b )(max d j j K c )}, where K b s the set of beeft crtera ad K c s the set of cost crtera. Secodly, obta the separato measures of the exstg alteratves from the postve deal ad egatve deal solutos. The separato measures based o the dstace fucto of D umbers, D S + ad D S,ofeachalteratvefrom the postve deal ad egatve deal solutos, respectvely, are derved from D S + D S 1 = ( d + j d j ) 1 = ( d j d j ) T T D I( d + j d j ), =1,...,m;,...,, D I( d j d j ), =1,...,m;,...,. (31) Fally, calculate the relatve closeess to the deal soluto: D S D C = D S +D S +, =1,...,m. (3) Rak the alteratves accordg to the relatve closeess to the deal soluto: the alteratves wth hgher D C are assumed to be more mportat ad should be gve hgher prorty. The flow chart of D-TOPSIS s show Fgure A Applcato for Huma Resources Selecto Based o D-TOPSIS A mport ad export tradg compay plas to recrut a departmet maager who must satsfy ther varous requremets [69]. There are some relevat test tems provded by the huma resources departmet of the compay for selectg the best caddate. The test tems clude two great aspects: the objectve ad the subjectve aspects. I addto, the objectve aspect s dvded to two sdes. The frst oe s kowledge test whch cludes laguage test, professoal test, ad safety rule test. The other oe s skll test whch has the tems of professoal sklls ad computer sklls. The subjectve aspect s determed by the correspodg tervews cludg pael tervew ad 1-o-1 tervew. Now, 17 caddates are qualfed for the test, ad four experts rate all the caddates tervews. The test results for objectve ad subjectve attrbutes are show Tables 1 ad. What s more, the weghts of all the tems from four experts arealsoshowtable3. The flow chart of the process to select the best caddate sshowfgure4.next,wewllllustratethespecfcsteps

8 8 Mathematcal Problems Egeerg Table 1: The scores of the objectve aspects. Number Caddates Objectve attrbutes Kowledge test Skll test Laguage test Professoal test Safety rule test Professoal sklls Computer sklls 1 JamesB.Wag Carol L. Lee KeeyC.Wu RobertM.Lag SophaM.Cheg LlyM.Pa AboC.Hseh Frak K. Yag Ted C. Yag Sue B. Ho Vcet C. Che Rosemary I. L Ruby J. Huag George K. Wu Phlp C. Tsa Mchael S. Lao Mchelle C. L Table : The scores of the subjectve aspects from dfferet experts for tervew. Subjectve attrbutes Number Expert 1 Expert Expert 3 Expert 4 Pael 1-o-1 Pael 1-o-1 Pael 1-o-1 Pael 1-o

9 Mathematcal Problems Egeerg 9 Table 3: Weght for dfferet test tems from dfferet experts. Number Attrbutes Weght Expert 1 Expert Expert 3 Expert 4 1 Laguage test Professoal test Safety rule test Professoal sklls Computer sklls Pael tervew o-1 tervew Italzato Obta the objectve ad subjectve tests score of each caddate Determe the weght of each attrbute derved from experts Step 1 Costruct the decso matrx Step Trasform the attrbute matrx to D umbers Step 3 Determe the postve deal ad egatve deal solutos Step 4 Calculate the dstace betwee each soluto ad postve deal ad egatve deal solutos based o D umbers dstace fucto Step 5 Calculate the relatve closeess to the deal soluto ad rak Fgure 4: The flow chart of huma resources selecto. about how to select the best oe from the 17 caddates for thecompayusgtheewproposed D-TOPSIS method. Step 1. Costruct the attrbute matrx. Frstly, we calculate the comprehesve scores of each caddate combg the four experts advce the tervews. Ad the results are show Table 4. The, we ca obta the weghted overall results of ths testfromtheobjectveadsubjectveaspectsbasedotables 1,3,ad4,whchsshowTable5adcabeseeasthe decso matrx. Step. TrasformdecsomatrxtoD umbers ad obta the terrelato betwee these crtera. From Step 1, the decso matrx has be determed. Now, we eed to trasform the matrx to D umbers. Frstly, ormalze the decso matrx for each tem of each caddate show Table 6. We wll represet each test tem usg a, b,

10 10 Mathematcal Problems Egeerg Table 4: The comprehesve scores from dfferet experts for the tervew. Number Subjectve attrbutes Pael tervew 1-o-1 tervew b a g c Fgure 5: The etwork chart of terrelato betwee dfferet crtera. d f e c, d, e, f,adg for coveece. The terrelato betwee 7 dfferet crtera s show Table 7. The, the uo set U ad tersecto I ca be obtaed from the experts scorg ad experece ad s show Table 8 based o Deftos 13 ad 14. Ad, Fgure 5, the terrelato betwee dfferet crtera ca be represeted by the etwork. The dfferet sze of each ode deotes the weght of dfferet crtera from multple experts, whle the wdth of the edge reflects the terrelato of the dfferet crtera some ways. Step 3. Obta the postve deal solutos D A + ad egatve deal solutos D A based o (30). We select the postve deal ad egatve deal solutos from Table 6. The postve deal soluto s determed by the hghest score of each attrbute; smlarly, the egatve deal soluto s defed by the lowest score of each attrbute. Ad theresultsareshowtable9. Step 4. Calculate the dstace betwee each soluto ad postve deal ad egatve deal solutos based o (31). From the above steps, the postve deal solutos D A + ad egatve deal solutos D A have bee obtaed. Next, we wll calculate the dstace from each alteratve scheme to D A + ad D A by (17), respectvely. The results are show Table 10. Step 5. Calculate the relatve closeess ad rak. I ths step, we calculate the relatve closeess to the deal soluto of each attrbute by (3). Fally, sort each caddate bythecloseessvalues.thedstacesadrakgresultsare show Table 10. The best caddate ca be selected easly based o the rakg results. It s worth otg that the rakg results wll be dfferet depedg o two factors: (1) the scores objectve ad subjectve tests of each crtero ad () the terrelato ad weghts amog dfferet crtera. Ad the major advatages of D-TOPSIS are reflected two aspects. Frstly, t ca keep the valdty of the tradtoal TOPSIS method. I addto, the relatoshp betwee multattrbutes s cosdered for the more reasoable results. The effectveess of D-TOPSIS ca be demostrated by the applcato. 5. Cocluso I ths paper, a ew TOPSIS method called D-TOPSIS s proposed to hadle MCDM problem usg D umbers to exted the classcal TOPSIS method. I the proposed method, the decso matrx determato from MCDM problem ca be trasformed to D umbers, whch ca effectvely represet the evtable ucertaty, such as completeess ad mprecso due to the subjectve assessmetofhumabegs.adtherelatoshpbetwee multattrbutes s cosdered the process of decsomakg, whch s more grouded realty. A example of huma resources selecto s coducted ad llustrates the effectveess of the proposed D-TOPSIS method. I future research, the theoretcal framework of the D-TOPSIS eeds to be creasgly perfected. For example, how to scetfcally produce the relatoshp betwee multcrtera should be further vestgated. Also, the proposed method should be utlzed other applcatos to further verfy ts effectveess.

11 Mathematcal Problems Egeerg 11 Table 5: The weghted overall scores of the test. Objectve attrbutes Number Kowledge test Skll test Subjectve attrbutes Laguage test Professoal test Safety rule test Professoal sklls Computer sklls Pael tervew 1-o-1 tervew Table 6: Costructg D umbers of each caddate. Number Laguage test Professoal test Safety rule test Professoal sklls Computer sklls Pael tervew 1-o-1 tervew Table 7: The terrelato betwee 7 dfferet crtera. Relato a b c d e f g a b c d e f g Table 8: The uo set ad tersecto betwee dfferet crtera. S U j j a b c d e f g a b c d e f g

12 1 Mathematcal Problems Egeerg Table 9: The postve deal solutos D A + ad egatve deal solutos D A. Ideal soluto Laguage test Professoal test Safety rule test Professoal sklls Computer sklls Pael tervew 1-o-1tervew D A D A Table 10: The relatve closeess ad rakg results by D-TOPSIS method. Number D S + D S D C Rak Competg Iterests There s o coflct of terests ths paper. Authors Cotrbutos Yog Deg desged ad performed research. Lguo Fe ad YogHuwrotethepaper.LguoFeadYogHuperformed the computato. Yog Deg, Lguo Fe, Fuyua Xao, ad Luyua Che aalyzed the data. All authors dscussed the resultsadcommetedothepaper.lguofeadyoghu cotrbuted equally to ths work. Ackowledgmets The work s partally supported by Natoal Natural Scece Foudato of Cha (Grat os , , ad ) ad Cha State Key Laboratory of Vrtual Realty Techology ad Systems, Behag Uversty (Grat o. BUAA-VR-14KF-0). Refereces [1] A. Marda, A. Jusoh, ad E. K. Zavadskas, Fuzzy multple crtera decso-makg techques ad applcatos two decades revew from 1994 to 014, Expert Systems wth Applcatos,vol.4,o.8,pp ,015. []W.Pedrycz,R.Al-Hmouz,A.Morfeq,adA.S.Balamash, Buldg graular fuzzy decso support systems, Kowledge- Based Systems,vol.58,pp.3 10,014. [3] S. Badyopadhyay ad R. Bhattacharya, Fdg optmum eghbor for routg based o mult-crtera, mult-aget ad fuzzy approach, Joural of Itellget Maufacturg, vol. 6, o. 1, pp. 5 4, 013. [4] S. Yao ad W.-Q. Huag, Iduced ordered weghted evdetal reasog approach for multple attrbute decso aalyss wth ucertaty, Iteratoal Joural of Itellget Systems,vol.9, o.10,pp ,014. [5] F. Loll, A. Ishzaka, R. Gamber, B. Rm, ad M. Messor, FlowSort-GDSS a ovel group mult-crtera decso support system for sortg problems wth applcato to FMEA, Expert Systems wth Applcatos, vol.4,o.17-18,pp , 015. [6] J. P. Evas ad R. E. Steuer, A revsed smplex method for lear multple objectve programs, Mathematcal Programmg, vol. 5, o. 1, pp. 54 7, [7] P. L. Yu ad M. Zeley, The set of all odomated solutos lear cases ad a multcrtera smplex method, Joural of Mathematcal Aalyss ad Applcatos,vol.49,o.,pp , [8] G. W. Torrace, Decsos wth multple objectves: prefereces ad value tradeoffs, Health Servces Research, vol. 13, o. 3, p. 38, [9] J. Rezae, Best-worst mult-crtera decso-makg method, Omega,vol.53,pp.49 57,015. [10] W. Sałabu, The characterstc objects method: a ew dstacebased approach to multcrtera decso-makg problems, Joural of Mult-Crtera Decso Aalyss, vol.,o.1-,pp , 015. [11] E. K. Zavadskas, J. Atuchevcee, S. H. R. Hajagha, ad S. S. Hashem, The terval-valued tutostc fuzzy MULTI- MOORA method for group decso makg egeerg, Mathematcal Problems Egeerg, vol.015,artcleid , 13 pages, 015. [1] W. Jag, Y. Luo, X.-Y. Q, ad J. Zha, A mproved method to rak geeralzed fuzzy umbers wth dfferet left heghts ad rght heghts, Joural of Itellget ad Fuzzy Systems,vol. 8, o. 5, pp , 015. [13] G. Lee, K. S. Ju, ad E.-S. Chug, Robust spatal flood vulerablty assessmet for Ha Rver usg fuzzy TOPSIS wth cut level set, Expert Systems wth Applcatos,vol.41,o.,pp , 014. [14] D. Lag, W. Pedrycz, D. Lu, ad P. Hu, Three-way decsos based o decso-theoretc rough sets uder lgustc assessmet wth the ad of group decso makg, Appled Soft Computg Joural,vol.9,pp.56 69,015. [15] B.L.Golde,E.A.Wasl,adP.T.Harker,Aalytc Herarchy Process, vol. 113, Sprger, New York, NY, USA, 003. [16] H. Nguye, S. Z. M. Dawal, Y. Nukma, H. Aoyama, K. Case, ad Y. Deg, A tegrated approach of fuzzy lgustc

13 Mathematcal Problems Egeerg 13 preferece based AHP ad fuzzy COPRAS for mache tool evaluato, PLOS ONE,vol.10,o.9,artclee ,015. [17] Y. Deg, Fuzzy aalytcal herarchy process based o caocal represetato o fuzzy umbers, Joural of Computatoal Aalyss ad Applcatos,vol.,o.,pp.01 8,017. [18] E. K. Zavadskas, J. Atuchevcee, Z. Tursks, ad H. Adel, Hybrd multple crtera decso makg methods: a revew of applcatos egeerg, Sceta Iraca, vol.3,o.1,pp. 1 0, 016. [19] C. L. Hwag ad K. Yoo, Multple Attrbute Decso Makg, Sprger, Berl, Germay, [0] S. M. Mousav, S. A. Torab, ad R. Tavakkol-Moghaddam, A herarchcal group decso-makg approach for ew product selecto a fuzzy evromet, Araba Joural for Scece ad Egeerg,vol.38,o.11,pp ,013. [1] Z. Yue, TOPSIS-based group decso-makg methodology tutostc fuzzy settg, Iformato Sceces, vol. 77, pp , 014. [] A. Suder ad C. Kahrama, Mmzg evrometal rsks usg fuzzy topss: locato selecto for the tu faculty of maagemet, Huma ad Ecologcal Rsk Assessmet, vol. 1, o. 5, pp , 015. [3] M.H.Ahmad,M.A.Ahmad,adS.A.Sadatsakkak, Thermodyamc aalyss ad performace optmzato of rreversble Carot refrgerator by usg mult-objectve evolutoary algorthms (MOEAs), Reewable ad Sustaable Eergy Revews, vol. 51, artcle 4609, pp , 015. [4] B. Vahda, S. M. Mousav, R. Tavakkol-Moghaddam, ad H. Hashem, A ew desg of the elmato ad choce traslatg realty method for mult-crtera group decso-makg a tutostc fuzzy evromet, Appled Mathematcal Modellg,vol.37,o.4,pp ,013. [5] M. Xa ad Z. Xu, A ovel method for fuzzy mult-crtera decso makg, Iteratoal Joural of Iformato Techology ad Decso Makg,vol.13,o.3,pp ,014. [6] K. Khall-Damgha, S. Sad-Nezhad, ad M. Tavaa, Solvg mult-perod project selecto problems wth fuzzy goal programmg based o TOPSIS ad a fuzzy preferece relato, Iformato Sceces,vol.5,pp.4 61,013. [7] Y. Km, E.-S. Chug, S.-M. Ju, ad S. U. Km, Prortzg the best stes for treated wastewater stream use a urba watershed usg fuzzy TOPSIS, Resources, Coservato ad Recyclg, vol. 73, pp. 3 3, 013. [8] E. K. Zavadskas, Z. Tursks, ad V. Bagočus, Mult-crtera selecto of a deep-water port the Easter Baltc Sea, Appled Soft Computg,vol.6,pp ,015. [9] G. Lee, K. S. Ju, ad E.-S. Chug, Group decso-makg approach for flood vulerablty detfcato usg the fuzzy VIKOR method, Natural Hazards ad Earth System Sceces, vol. 15, o. 4, pp , 015. [30] S.Dadelo,Z.Tursks,E.K.Zavadskas,adR.Dadelee, Multcrtera assessmet ad rakg system of sport team formato based o objectve-measured values of crtera set, Expert Systems wth Applcatos,vol.41,o.14,pp ,014. [31] S.-M. Yu, H. Zhou, X.-H. Che, ad J.-Q. Wag, A multcrtera decso-makg method based o Heroa mea operators uder a lgustc hestat fuzzy evromet, Asa- Pacfc Joural of Operatoal Research, vol.3,o.5,artcle , Artcle ID , 015. [3] Y. Deg, D umbers: theory ad applcatos, Joural of Iformato ad Computatoal Scece, vol.9,o.9,pp.41 48, 01. [33] X. Deg, Y. Hu, Y. Deg, ad S. Mahadeva, Suppler selecto usg AHP methodology exteded by D umbers, Expert Systems wth Applcatos,vol.41,o.1,pp ,014. [34] X. Deg, Y. Hu, Y. Deg, ad S. Mahadeva, Evrometal mpact assessmet based o D umbers, Expert Systems wth Applcatos,vol.41,o.,pp ,014. [35] G. Fa, D. Zhog, F. Ya, ad P. Yue, A hybrd fuzzy evaluato method for curta groutg effcecy assessmet based o a AHP method exteded by D umbers, Expert Systems wth Applcatos,vol.44,pp ,016. [36] N. Rkhtegar, N. Masour, A. A. Oroumeh, A. Yazda- Chamz, E. K. Zavadskas, ad S. Kldeė, Evrometal mpact assessmet based o group decso-makg methods mg projects, Ecoomc Research-Ekoomska Istrazvaja, vol. 7, o. 1, pp , 014. [37]X.Deg,X.Lu,F.T.S.Cha,R.Sadq,S.Mahadeva,ad Y. Deg, D-CFPR: D umbers exteded cosstet fuzzy preferece relatos, Kowledge-Based Systems,vol.73,pp.61 68, 015. [38] H.-C. Lu, J.-X. You, X.-J. Fa, ad Q.-L. L, Falure mode ad effects aalyss usg D umbers ad grey relatoal projecto method, Expert Systems wth Applcatos, vol. 41, o. 10, pp , 014. [39] A. P. Dempster, Upper ad lower probabltes duced by a multvalued mappg, Aals of Mathematcal Statstcs, vol. 38, pp , [40] G. Shafer, A Mathematcal Theory of Evdece,vol.1,Prceto Uversty Press, Prceto, NJ, USA, [41] B. She, Y. Lu, ad J.-S. Fu, A tegrated model for robust multsesor data fuso, Sesors, vol. 14, o. 10, pp , 014. [4] Y. Deg, S. Mahadeva, ad D. Zhou, Vulerablty assessmet of physcal protecto systems: a bo-spred approach, Iteratoal Joural of Ucovetoal Computg, vol.11,o.3-4, pp. 7 43, 015. [43] G. Kabr, S. Tesfamaram, A. Fracsque, ad R. Sadq, Evaluatg rsk of water mas falure usg a Bayesa belef etwork model, Europea Joural of Operatoal Research,vol.40,o. 1, pp. 0 34, 015. [44] Y. L, J. Che, F. Ye, ad D. Lu, The mprovemet of DS evdece theory ad ts applcato IR/MMW target recogto, Joural of Sesors,vol.016,ArtcleID190379,15pages, 016. [45] W. Jag, Y. Yag, Y. Luo, ad X. Q, Determg basc probablty assgmet based o the mproved smlarty measures of geeralzed fuzzy umbers, Iteratoal Joural of Computers, Commucatos ad Cotrol,vol.10,o.3,pp ,015. [46] W. Jag, J. Zha, D. Zhou, ad X. L, A method to determe geeralzed basc probablty assgmet the ope world, Mathematcal Problems Egeerg,Ipress. [47] W.-B. Du, Y. Gao, C. Lu, Z. Zheg, ad Z. Wag, Adequate s better: partcle swarm optmzato wth lmted-formato, Appled Mathematcs ad Computato, vol.68,pp , 015. [48] D. Cheg, R.-X. Hao, ad Y.-Q. Feg, Embeddg eve cycles o folded hypercubes wth codtoal faulty edges, Iformato Processg Letters,vol.115,o.1,pp ,015. [49] Y. Deg, A threat assessmet model uder ucerta evromet, Mathematcal Problems Egeerg,vol.015,Artcle ID87804,1pages,015.

14 14 Mathematcal Problems Egeerg [50] O. Basr ad X. Yua, Ege fault dagoss based o multsesor formato fuso usg Dempster-Shafer evdece theory, Iformato Fuso,vol.8,o.4,pp ,007. [51] W. Jag, B. We, C. Xe, ad D. Zhou, A evdetal sesor fuso method fault dagoss, Advaces Mechacal Egeerg,vol.8,o.3,pp.1 7,016. [5] I. Bloch, Some aspects of Dempster-Shafer evdece theory for classfcato of mult-modalty medcal mages takg partal volume effect to accout, Patter Recogto Letters,vol.17, o. 8, pp , [53] Y.Deg,Y.Lu,adD.Zhou, Amprovedgeetcalgorthm wth tal populato strategy for symmetrc TSP, MathematcalProblemsEgeerg,vol.015,ArtcleID1794,6pages, 015. [54] Z.-G.Lu,Q.Pa,adJ.Dezert, Abelefclassfcatorulefor mprecse data, Appled Itellgece,vol.40,o.,pp.14 8, 014. [55] X.Zhag,Y.Deg,F.T.S.Cha,adS.Mahadeva, Afuzzy exteded aalytc etwork process-based approach for global suppler selecto, Appled Itellgece, vol. 43, o. 4, pp , 015. [56]J.Ma,W.Lu,P.Mller,adH.Zhou, Aevdetalfuso approach for geder proflg, Iformato Sceces, vol. 333, pp. 10 0, 016. [57] L. Su, R. P. Srvastava, ad T. J. Mock, A formato systems securty rsk assessmet model uder the Dempster-Shafer theory of belef fuctos, Joural of Maagemet Iformato Systems,vol.,o.4,pp ,006. [58] W. Jag, C. Xe, B. We, ad D. Zhou, A modfed method for rsk evaluato falure modes ad effects aalyss of arcraft turbe rotor blades, Advaces Mechacal Egeerg, vol. 8,o.4,pp.1 16,016. [59] H.-C. Lu, L. Lu, ad Q.-L. L, Fuzzy falure mode ad effects aalyss usg fuzzy evdetal reasog ad belef rule-based methodology, IEEE Trasactos o Relablty, vol. 6, o. 1, pp.3 36,013. [60] Y. Yag ad D. Ha, A ew dstace-based total ucertaty measure the theory of belef fuctos, Kowledge-Based Systems,vol.94,pp ,016. [61] A. Awasth ad S. S. Chauha, Usg AHP ad Dempster- Shafer theory for evaluatg sustaable trasport solutos, Evrometal Modellg & Software,vol.6,o.6,pp , 011. [6] A. Frkha ad H. Moalla, Aalytc herarchy process for multsesor data fuso based o belef fucto theory, Europea Joural of Operatoal Research,vol.41,o.1,pp ,015. [63] X. Su, S. Mahadeva, P. Xu, ad Y. Deg, Depedece Assessmet Huma Relablty Aalyss Usg Evdece Theory ad AHP, Rsk Aalyss, vol. 35, o. 7, pp , 015. [64] Y. Deg, Geeralzed evdece theory, Appled Itellgece, vol. 43,o.3,pp ,015. [65] W. Jag, B. We, X. Q, J. Zha, ad Y. Tag, Sesor data fuso based o a ew coflct measure, Mathematcal Problems Egeerg,Ipress. [66] X. Su, S. Mahadeva, W. Ha, ad Y. Deg, Combg depedet bodes of evdece, Appled Itellgece,vol.44,o. 3, pp , 016. [67] X. Su, S. Mahadeva, P. Xu, ad Y. Deg, Hadlg of depedece Dempster-Shafer theory, Iteratoal Joural of Itellget Systems,vol.30,o.4,pp ,015. [68] M. L, Y. Hu, Q. Zhag, ad Y. Deg, A ovel dstace fucto of D umbers ad ts applcato product egeerg, Egeerg Applcatos of Artfcal Itellgece,vol.47,pp.61 67, 016. [69] H.-S. Shh, H.-J. Shyur, ad E. S. Lee, A exteso of TOPSIS for group decso makg, Mathematcal ad Computer Modellg,vol.45,o.7-8,pp ,007.

15 Advaces Operatos Research Advaces Decso Sceces Joural of Appled Mathematcs Algebra Joural of Probablty ad Statstcs The Scetfc World Joural Iteratoal Joural of Dfferetal Equatos Submt your mauscrpts at Iteratoal Joural of Advaces Combatorcs Mathematcal Physcs Joural of Complex Aalyss Iteratoal Joural of Mathematcs ad Mathematcal Sceces Mathematcal Problems Egeerg Joural of Mathematcs Dscrete Mathematcs Joural of Dscrete Dyamcs Nature ad Socety Joural of Fucto Spaces Abstract ad Appled Aalyss Iteratoal Joural of Joural of Stochastc Aalyss Optmzato