Research Article A Selection Method Based on MAGDM with Interval-Valued Intuitionistic Fuzzy Sets

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1 Mathematical Problems i Egieerig Volume 2015, rticle ID , 13 pages Research rticle Selectio Method Based o MGDM with Iterval-Valued Ituitioistic Fuzzy Sets Gai-Li Xu, 1,2 Shu-Pig Wa, 2 ad Xiao-La Xie 3 1 College of Sciece, Guili Uiversity of Techology, Guili , Chia 2 College of Iformatio Techology, Jiagxi Uiversity of Fiace ad Ecoomics, Nachag , Chia 3 College of Iformatio Sciece ad Egieerig, Guili Uiversity of Techology, Guili , Chia Correspodece should be addressed to Gai-Li Xu; jiali0706@126.com Received 3 December 2014; ccepted 11 May 2015 cademic Editor: Julie Brucho Copyright 2015 Gai-Li Xu et al. This is a ope access article distributed uder the Creative Commos ttributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. s the cloud computig develops rapidly, more ad more cloud services appear. May eterprises ted to utilize cloud service to achieve better flexibility ad react faster to market demads. I the cloud service selectio, several experts may be ivited ad may attributes (idicators or goals) should be cosidered. Therefore, the cloud service selectio ca be regarded as a kid of Multiattribute Group Decisio Makig (MGDM) problems. This paper develops a ew method for solvig such MGDM problems. I this method, the ratigs of the alteratives o attributes i idividual decisio matrices give by each expert are i the form of iterval-valued ituitioistic fuzzy sets (IVIFSs) which ca flexibly describe the prefereces of experts o qualitative attributes. First, the weights of experts o each attribute are determied by extedig the classical gray relatioal aalysis (GR) ito IVIF eviromet. The, based o the collective decisio matrix obtaied by aggregatig the idividual matrices, the score (profit) matrix, accuracy matrix, ad ucertaity (risk) matrix are derived. multiobjective programmig model is costructed to determie the attribute weights. Subsequetly, the alteratives are raked by employig the overall scores ad ucertaities of alteratives. Fially, a cloud service selectio problem is provided to illustrate the feasibility ad effectiveess of the proposed methods. 1. Itroductio Cloud computig [1 4] is the latest computig paradigm that delivers hardware ad software resources as virtualizatio services i which users are free from the burde of worryig about the low-level system admiistratio details. I recet years, cloud computig is developig rapidly ad has provided eterprises with may advatages such as flexibility, busiess agility, ad pay-as-you-go cost structure. s a result, may eterprises with limited fiacial ad huma resources are icreasigly adoptig cloud computig to deliver their busiess services ad products olie to exted their busiess markets. I may domais, multiple cloud services ofte supply similar fuctioal properties. For example, i Customer Relatioship Maagemet (CRM), CRM veders offer fuctioally equivalet cloud services, such as Microsoft Dyamic CRM, Salesforce Sales Cloud, SP Sales o Demad, ad Oracle Cloud CRM. However, for eterprises, which lack cloud computig kowledge, it isdifficulttoselectaappropriatecadidatefromaset of fuctioally equivalet cloud services. Therefore, it is ecessary for eterprises to ivite several related experts to evaluate the potetial cadidates from several idicators (attributes), such as paymet, performace, reputatio, scalability, ad security. The selectio of cloud services has attracted attetio ad may methods have bee preseted to guide eterprises i selectig the cloud services. Roughly, these methods may be divided ito two categories ad briefly reviewed as follows, respectively. The first category is the Multiattribute Decisio Makig (MDM) methods. ccordig to the key performace idicators defied by Siegel ad Perdue [5], Garg et al. [6] proposed the cloud service rakig framework usig the alytic Hierarchy Process (HP) techique. Mezel et al. [7] utilized the alytic Network Process (NP) to develop amulticriteriacomparisomethodwhichisusedtoselect

2 2 Mathematical Problems i Egieerig Ifrastructure-as-a-Service (IaaS). Limam ad Boutaba [8] preseted a trustworthiess-based service selectio method based o the Multiple ttribute Utility Theory (MUT). By employig the Elimiatio ad Choice Expressig Reality (ELECTRE) method, Silas et al. [9] developed a cloud service selectio middleware to help cloud users select desired cloud service. Saripalli ad Pigali [10] discussed Simple dditive Weightig (SW) methods to rak alteratives i a decisio problem of cloud service adoptio. Zhao et al. [11] suggested a SW-based service searchig ad schedulig algorithm to obtai a set of raked services. The secod category is the optimizatio approaches. Chag et al. [12] desiged a dyamic programmig algorithm by maximizig the overall survival probability to select cloud storage providers. Sudareswara et al. [13] selected cloud service with a greedy algorithm method which ca make experts retrieve iformatio fast. I order to help service providers to select Software-as-a-Service (SaaS) services with multiteats, He et al. [14] explored three types of optimizatio algorithm, icludig iteger programmig, skylie, ad greedy algorithm, ad proposed a quality of service- (QoS-) drive optimizatio framework. By miimizig costs ad risks, Martes et al. [15] costructed a scalable mathematical decisio model to select cloud service. Yag et al. [16] built amarkovdecisioprocessmodeltoguarateetheearoptimal performace i a chagig eviromet by dyamically adjustig the compoets of a service compositio. The aforemetioed methods seem to be effective ad applicable for selectig cloud services. However, they have the followig shortcomigs. (1) The decisio makig i methods [13 15] is sigle Multiattribute Decisio Makig (MDM); that is, oly oe expert participates i the decisio makig ad gives assessmet iformatio of alteratives with respect to several attributes. Sice every expert is good at oly some fields rather tha all fields, the reliability of some iformatio give by the expert is alittledoubtful. (2) Curret methods [9, 12] are more focused o quatitative attributes measured via precise umerical values, such as respose time, storage space, ad latecy time. Nevertheless, i cloud service, some qualitative attributes (such as reputatio ad security) usually play importat roles, but they do ot gai eough attetio. (3) I existig methods [12, 13, 16], the assessmet values (attribute values) are crisp umbers, which is somewhat urealistic. Due to the iheret vagueess of huma prefereces as well as the fuzziess ad ucertaityofobjects,itismoresuitabletoexpress the assessmet values as fuzzy umbers [17 20]. (4) The attribute weights provided by experts are give a priori i methods[8, 10, 11], which always caot avoid subjective radomess of the expert s preferece. Furthermore, with icreasig complexity i may real decisio situatios, it is difficult for expert to provide precise ad complete preferece iformatioduetotimepressureadlackofdata. Oe of the reasos leadig to the above shortcomigs is that the fuzziess ad ucertaity are ot fully cosidered durig the decisio makig process. The fuzzy set (FS) theory itroduced by Zadeh [21] is a very useful tool to describe fuzzy ad ucertai iformatio. Based o the FS theory, taassov [22] preseted the ituitioistic fuzzy set (IFS), which cosiders the membership (satisfactio) degree, omembership (dissatisfactio) degree, ad hesitat degree simultaeously. Subsequetly, taassov ad Gargov [23] geeralizedifsadpresetediterval-valuedituitioistic fuzzy set (IVIFS) that describes the membership ad omembership degrees as itervals. Compared with the FS ad IFS, IVIFS is more suitable to express the fuzziess ad ucertaity ad has bee widely used i may fields [24 28]. ccordig to IVIFS theory, to overcome the aforemetioed shortcomigs, we ivestigate the cloud service selectio problems with IVIFSs ad develop a ovel method. The proposed method has the followig key characteristics. (1) The selectio of cloud service is regarded as a Multiattribute Group Decisio Makig (MGDM) problem that several experts are ivited to evaluate the potetial cloud services, whereas it is cosidered as a sigle MDM problem i methods [6 11]. With icreasig complexity ad the limit kowledge owed by sigle expert, i order to icrease the quality of cloud service, it is more reasoable ad reliable for eterprises to ivite multiple experts to participate i makig decisio. (2) The assessmet values give by experts are expressed as IVIFSs. Compared with the crisp umber, IVIFS is more flexible to measure the qualitative attributes sice IVIFS cosiders membership, omembership, ad hesitat degrees which are expressed as itervals. dditioally, it is easier for experts to supply assessmet values with IVIFSs i the icreasig ucertai ad complex eviromet. (3) By extedig the classical gray relatioal aalysis (GR) [29] ito IVIF eviromet, a ew approach is proposed to determie the weights of experts. otable characteristic of the proposed approach is that the obtaied weights of each expert are differet with respect to differet attributes. (4) For MGDM problems with icomplete iformatio o attributes, a multiobjective programmig model is costructed to objectively determie the attribute weights, which ca avoid the subjective radomess appearig i the methods [8, 10, 11]. Moreover, it is easier for experts to give partial iformatio o attribute weights tha to assig a crisp umber to the attribute weights. The rest of this paper ufolds as follows. Some prelimiaries about IVIFSs ad the classical GR method are

3 Mathematical Problems i Egieerig 3 itroduced i Sectio 2. ISectio 3, aewmethodisproposed to solve MGDM problems with IVIFSs ad icomplete attribute weight iformatio. I additio, a framework of decisio supportig system (DSS) is costructed. I Sectio 4, a cloud service selectio example is provided to illustrate the applicability of the proposed method ad compariso aalysis is coducted. Fially, the coclusios are discussed i Sectio Prelimiaries I this sectio, we itroduce some basic cocepts related to iterval-valued fuzzy set (IVIFS) ad gray relatioal aalysis (GR) Iterval-Valued Ituitioistic Fuzzy Set Defiitio 1 (see [23]). Let X = {x 1,x 2,...,x } be a oempty set of the uiverse. IVIFS i X is defied as ={(x i,[μ L (x i),μ R (x i)], [V L (x i),v R (x i)]) x i X}, where [μ L (x i), μ R (x i)] ad [V L (x i), V R (x i)] deote the itervals of membership degree ad omembership degree of elemet x i, respectively,satisfyigμ R (x i)+v R (x i) 1, 0 μ L (x i) μ R (x i) 1, ad 0 V L (x i) V R (x i) 1forall x i X. π (x i)=[1 μ R (x i) V R (x i), 1 μ L (x i) V L (x i)] is called the iterval-valued ituitioistic hesitat degree of IVIFS. Forayx i X,ifμ L (x i)=μ R (x i) ad V L (x i)= V R (x i),the isreducedtoaifs. Xu [30] calledthepair α =(μ α (x i ), V α (x i )) a itervalvalued ituitioistic fuzzy umber (IVIFN) ad deoted a IVIFN by α = ([a, b], [c, d]), where[a, b] [0, 1], [c, d] [0, 1], b+d 1. Defiitio 2 (see [30]). Let α 1 = ([a 1,b 1 ], [c 1,d 1 ]), α 2 = ([a 2,b 2 ], [c 2,d 2 ]), ad α = ([a, b], [c, d]) be three IVIFNs; the (1) ( α) c = ([c, d], [a, b]); (2) α 1 + α 2 =([a 1 +a 2 a 1 a 2,b 1 +b 2 b 1 b 2 ], [c 1 c 2,d 1 d 2 ]); (3) λ α =([1 (1 a) λ, 1 (1 b) λ ], [c λ,d λ ]), λ > 0. Defiitio 3 (see [30]). Let α j = ([a j,b j ], [c j,d j ]) (j = 1, 2,...,)be a collectio of IVIFNs. If IVIFW ω ( α 1, α 2,..., α )= (1) ω j α j, (2) the the IVIFW is called a iterval-valued ituitioistic fuzzy weighted averagig (IVIFW) operator of dimesio, whereω =(ω 1,ω 2,...,ω ) T is a weight vector of α j with ω j [0, 1] ad ω j = 1. The aggregated value determied by the IVFW operator is also a IVIFN; that is, IVIFW ω ( α 1, α 2,..., α ) =([ 1 [ [ [ c ω j j, d ω j (1 a j ) ω j, 1 j ] ]). (1 b j ) ω j ], ] Defiitio 4 (see [30]). Let α = ([a, b], [c, d]) be a IVIFN. The (3) s ( α) = 1 (a+b c d), (4) 2 h ( α) = 1 (a+c+b+d) (5) 2 are, respectively, called the score fuctio ad accuracy fuctio of the IVIFN α, wheres( α) [ 1, 1] ad h( α) [0, 1] ca be cosidered as et membership ad accuracy degree, respectively. Sice s( α) [ 1, 1], whe may score fuctios are aggregated with liear weighted summatio method, it maybe appears that positive score fuctios are offset by egative score fuctios. Therefore, we ormalize the score fuctioadmakeitbelogto[0, 1]. Give a variable y [ 1, 1], if we defie f (y) = y+1 2, (6) the f(y) caot oly retai the mootoicity of the variable y but also map y for [0, 1]. Hece,wemodifythescore fuctio i Defiitio 4 ad defie a ew score fuctio of IVIFN α. Defiitio 5. Let α = ([a, b], [c, d]) be a IVIFN. The s ( α) = 1 (s ( α) + 1) (7) 2 is called a ormalized score fuctio, where s( α) = (1/2)(a c+b d).obviously,s ( α) [0, 1]. Defiitio 6. Let α = ([a, b], [c, d]) be a IVIFN. The γ ( α) = 1 h( α) (8) is called a ucertaity fuctio, where h( α) = (1/2)(a + c + b+d). Let α = ([a, b], [c, d]) be a assessmet value of the cloud service x withrespecttotheattribute(idicator) α.thethe ormalized score fuctio s ( α) ad the ucertaity fuctio γ( α) ca be, respectively, iterpreted as the et profit ad risk provided by cloud service x o attribute α.hece,the bigger the s ( α) ad the smaller the γ( α),thebetterthecloud service x. I the followig, a order relatioship betwee IVIFNs is give.

4 4 Mathematical Problems i Egieerig Defiitio 7. Let α 1 = ([a 1,b 1 ], [c 1,d 1 ]), α 2 = ([a 2,b 2 ], [c 2, d 2 ]) be two IVIFNs; the (1) If s ( α 1 )<s ( α 2 ),the α 1 < α 2. (2) If s ( α 1 )=s ( α 2 ),the (i) If γ( α 1 )=γ( α 2 ),the α 1 = α 2. (ii) If γ( α 1 )>γ( α 2 ),the α 1 < α 2. Defiitio 8. Let α 1 = ([a 1,b 1 ], [c 1,d 1 ]), α 2 = ([a 2,b 2 ], [c 2, d 2 ]) be two IVIFNs; the Euclidea distace betwee α 1 ad α 2 is defied as follows: d( α 1, α 2 ) = 1 2 (a 1 a 2 ) 2 +(b 1 b 2 ) 2 +(c 1 c 2 ) 2 +(d 1 d 2 ) Gray Relatio alysis. GR Theorem is a importat part of Gray Theorem developed by Deg [29]. GR ivestigates ucertai relatioship betwee oe mai factor ad all other factors i a system ad has bee used i a wide variety of decisio makig eviromets, such as supplier selectio [31], material selectio [32], ad water protectio strategy evaluatio [33]. The details of the classical GR method are preseted as follows. (i) Calculate the ormalized decisio matrix. Let F = (f ij ) m be a decisio matrix. The ormalized matrix R =(r ij ) m is calculated as (9) max i f ij f ij { max i f ij mi i f ij r ij = f { ij mi i f ij { max i f ij mi i f ij i=1, 2,...,m;, 2,...,; j cost attributes i=1, 2,...,m;, 2,...,; j beefit attributes. (10) (ii) Geerate comparability sequeces r i = (r i1,r i2,..., r i ) (i = 1, 2,...,m) ad a referece sequece is r 0 = (r 01,r 02,...,r 0 ). For example, we ca take r 0j = max i {r ij }(, 2,...,). (iii) Compute the gray relatioal coefficiet betwee the comparability sequece r i ad the referece sequece r 0 by the followig formula: d +τd + ξ(r ij,r 0j )= d(r ij,r 0j )+τd + (11) (i = 1, 2,...,m;, 2,...,), where d(r ij,r 0j )= r ij r 0j, d = mi 1 i m mi 1 j d(r ij,r 0j ), d + = max 1 i m max 1 j d(r ij,r 0j ),adτ [0,1] is a distiguishig coefficiet. Usually, τ=0.5. (iv) Calculate the gray relatioal grade betwee r i ad r 0 ; that is, ξ i = ω j ξ ij, (12) where ω = (ω 1,ω 2,...,ω ) T is a weight vector satisfyig ω j [0, 1] ad ω j = 1. The bigger the ξ i,thecloserthe sequece r i to the sequece r Novel Method for MGDM with IVIFSs ad Icomplete ttribute Weight Iformatio I this sectio, a ew method is proposed to hadle MGDM with IVIFSs. The proposed method icludes determiatio of the weights of experts ad idetificatio of attribute weights. Let = { 1, 2,..., m } be the set of m feasible alteratives, let U={u 1,u 2,...,u } be the set of attributes, ad let E = {e 1,e 2,...,e t } be the set of decisio makers (DMs). ssume that ω = (ω 1,ω 2,...,ω ) T is a attribute weight vector, where ω j [0, 1] ad ω j = 1. Let the idividual decisio matrix give by expert e k be F k k = ( f ij ) k m, where f ij = ([a k ij,bk ij ], [ck ij,dk ij ]) is a IVIFN for the alterative i with respect to attribute u j.ithis paper, [a k ij,bk ij ] ad [ck ij,dk ij ] provided by the expert e k are, respectively, the satisfactio (agreeig) degree iterval ad dissatisfactio (disagreeig) degree iterval of the ith cloud service i with respect to the jth attribute (idicator) u j Determie the Weights of Experts by the Exteded GR Method. Due to the fact that each expert is skilled i some fields rather tha all fields, it is more reasoable that the weights of each expert with respect to differet attributes should be assiged differet values. However, the weights of each expert obtaied with the existig methods [34 37] are the same. Let λ k j be the weight of expert e k with respect to attribute u j. Geerally, for the attribute u j,theclosertheattribute values of all alteratives give by expert e k are to those give by all other t 1 experts, the more similar the iformatio provided by the expert e k is to that implied by the group. Cosequetly, the weight of expert e k should be assiged a greater value. Bearig this idea i mid, we preset a ovel method to determie the weights of experts by extedig classical GR method. Give the decisio matrices F k =( f k ij ) m (k = 1, 2,..., k t), the elemets f ij cabeormalizedas

5 Mathematical Problems i Egieerig 5 r k ij f k ij = { { (f { k ij )c i=1, 2,...,m;, 2,...,; j beefit attributes i=1, 2,...,m;, 2,...,; j cost attributes. (13) 3.2. Itegrate Idividual Decisio Matrices ito a Collective Matrix. fter the weights of experts are obtaied, idividual decisio matrix R k = ( r kij ) m cabeitegrateditoa collective matrix R =( r ij ) m with IVIFW operator, where The ormalized decisio matrices ca be deoted by R k = ( r k ij ) m (k = 1, 2,...,t). Let r k j = ( r k 1j, rk 2j,..., rk mj ) be the referece sequece ad let all other sequeces r l j = ( r l 1j, rl 2j,..., rl mj )(l = 1, 2,...,t, l =k)be comparability sequeces. The, for the attribute u j, the gray relatioal coefficiet betwee r k j ad rl j with respect to alterative i is defied as ξ lk ij =ξ( rl ij, rk ij )= d k j +τd k+ j d( r l ij, rk ij )+τdk+ j, (14) where d( r l ij, rk ij ) is the distace betwee rl ij ad r k ij (see (9)), d k j = mi 1 l s,l=k mi 1 i m d( r l ij, rk ij ), dk+ j = max 1 l s,l=k max 1 i m d( r l ij, rk ij ),adτ=0.5. Thus, the matrix of gray relatioal coefficiet betwee r l ij ad r k ij is costructed as ξ k j =(ξlk ij ) (t 1) m, (15) where l=1, 2,...,t, l=k. The gray relatioal grade betwee r k j ad rl j is calculated as η( r l j, rk j )= 1 m m ξ lk i=1 ij. (16) The gray relatioal grade η( r k j, rl j ) describes the degree of closeess betwee sequece r k j ad sequece rl j.iother words, η( r k j, rl j ) idicates the similarity degree betwee the iformatio give by DM e k ad that give by DM e l o attribute u j. For the attribute u j, the average gray relatioal grade betwee DM e k ad all other DMs e l (l D, l = k) is computed as η k j = 1 t 1 t l=1,l=k γ( r l j, rk j ). (17) Thus, the larger the η k j is, the more similar the iformatio give by the expert e k is to that implied by the group. Therefore, the bigger the λ k j. ccordigly, the weight of expert e k with respect to attribute u j, deoted by λ k j,cabe defied as λ k j = η k j t l=1 ηl j. (18) r ij = t r k ij λk j k=1 =([1 [ t k=1 t (c k j ij )λk k=1, (1 a k j ij )λk, 1 t k=1 (d k ij )λk j]). t k=1 For coveiece, we deote r ij by (1 b k ij )λk j], (19) r ij =([a ij,b ij ],[c ij,d ij ]). (20) By employig (5), (7),ad(8),thescorematrix,accuracy matrix, ad ucertaity matrix of matrix R are, respectively, obtaied as follows: S =(s ij ) m, H =(h ij ) m, γ =(γ ij ) m, (21) where s ij =s ( r ij ), h ij =h( r ij ),adγ ij =γ( r ij ). Utilizig the weighted summatio method, we ca derive the overall score fuctio, accuracy fuctio, ad ucertaity fuctio of alterative i as s i = h i = γ i = ω j s ij, (22) ω j h ij, (23) ω j γ ij. (24) Iftheattributeweightsarekowiadvace,thealteratives ca be raked ad selected accordig to Defiitio 7. I what follows, a ew multiobjective liear programmig model is costructed to determie the attribute weights Idetify the ttribute Weights by a New Multiobjective Liear Programmig Model. Due to the ucertaity of decisio makig eviromet ad the limited kowledge possessed by experts, experts oly may supply partial iformatio about attribute weights. Namely, the iformatio of the attribute weights is icomplete. Let D be the set of icomplete iformatio o attribute weights. ccordig to Defiitio 7, thebiggertheoverallscore fuctio (i.e., profit fuctio) s i ad the smaller the overall

6 6 Mathematical Problems i Egieerig ucertaity fuctio (i.e., risk fuctio) γ i of the alterative i, the better the alterative i. Therefore, by maximizig the overall score fuctios ad miimizig the overall ucertaity fuctios, a multiobjective programmig is built to objectively determie the weights of attributes: max {s 1,s 2,...,s m } mi {γ 1,γ 2,...,γ m } s.t. ω D. (25) By the max-mi method for solvig multiobjective programmig [38], (25) cabecovertedas max mi s.t. {mi i s i } {max i γ i } ω D. (26) From the relatioship betwee γ i ad h i (see (8)), whe γ i reaches maximum, h i reaches miimum. ccordigly, miimizig the maximum amog γ i is equivalet to maximizig the miimum amog h i. Therefore, (26) cabetrasformed as max max s.t. {mi i s i } {mi i h i } ω D. (27) ssume that y=mi i s i, x=mi i h i ;wehaves i yad h i x.thus,byemployig(22)-(23), (27) caberewritteas max y max s.t. x s ij w j y h ij w j x ω D. i=1, 2,...,m i=1, 2,...,m (28) By the liear weighted summatio method, (28) ca be coverted ito the followig sigle objective programmig model: max {py + (1 p)x} s.t. s ij w j y h ij w j x ω D, i=1, 2,...,m i=1, 2,...,m (29) where p [0,1]represets the relative importace of the two objects. If 0 p < 0.5, the experts are pessimistic ad are more cocered about ucertaity fuctio (i.e., risk) tha score fuctio (i.e., profit); if 0.5 <p 1, the experts are optimistic ad are more cocered about profit tha risk; if p=0.5, the experts cosidered that profit is as importat as risk. By solvig (29), the vector of attribute weights ω = (ω 1,ω 2,...,ω m ) T ca be obtaied Decisio Process ad lgorithm for MGDM Problems with IVIFSs. Basedotheaboveaalysis,thealgorithmad decisio process for MGDM problems are summarized as follows. Step 1. The experts establish the idividual decisio matrix R k =( r k ij ) m with IVIFSs ad supply the set of iformatio o the attribute weights D. Step 2. Calculate the weight of expert e k by (13) (18), where k=1, 2,...,t;, 2,...,. Step 3. Itegrate all idividual decisio matrix R k ito a collective matrix R =( r ij ) m by (19). Step 4. Derive the score matrix S, accuracy matrix H, ad ucertaity matrix γ of the matrix R by (20)-(21). Step 5. Determie the weight vector of attributes ω accordig to (29). Step 6. Compute the overall score s i ad ucertaity γ i of alteratives i by (22) ad (24). Step 7. Rak the alteratives ad select the best oe accordig to Defiitio The Framework Decisio Support System Based o MGDM with IVIFSs. s the scale of decisio makig icreases, the procedure solvig a MGDM may be complicated. I this case, a decisio supportig system (DSS), which is a class of computer-based iformatio system icludig kowledge-based systems [39, 40], ca be formulated to help experts improve their decisio-makig level ad quality through problem aalysis, establishmet of models, ad simulatio of decisio-makig process i a humacomputer iteractio way. Figure 1 depicts a framework of DSS desiged i this paper for MGDM with IVIFSs. s show i Figure 1,theDSScosistsofthreemodules: User iterface, Kowledge base ad Model base. Geerally, the user iterface establishes a iteractio betwee experts ad iputs the basic decisio iformatio, such as attributes, alteratives ad assessmet values of alteratives oattributes.themaifuctioofkowledgebaseisto help experts perform iformatio trasformatio ad store the correspodig iformatio. For example, the ratigs of alteratives o attributes give by experts are trasformed

7 Mathematical Problems i Egieerig 7 Descriptio of the problems Kowledge base Costruct IVIF idividual decisio matrices Normalize the idividual decisio matrices Select the referece ad comparability sequeces Compute the gray relatioal coefficiet Determiatio module of experts weights Calculate the gray relatioal grade Calculate the average gray relatioal grade Decisio group User iterface Derive the experts weights by ormalizig the average gray relatioal grade Itegrate the idividual decisio matrices ito a collective matrix Compute the score, accuracy, ad ucertaity matrices Costruct a biobjective programmig model Determiatio module of attributes weights Obtai the weights of attributes by solvig the model Compute the overall score values ad ucertaity values Rak alteratives ad select the best oe Rakig order module of alteratives Model base Decisio supportig Figure 1: Framework of iterval-valued ituitioistic fuzzy MGDM decisio supportig system. ito IVIF forms from which idividual decisio matrices with IVIFSs are costructed ad used for the ext calculatio procedure. Model base ivolves the methods, such as exteded GR ad objective programmig as metioed above. Thus, the rakig of alteratives ca be deduced ad the optimal decisio ca be derived by DSS. 4. Cloud Service Selectio Problem ad Compariso alysis I this sectio, a real cloud service selectio problem is give to illustrate the applicatio of the proposed method. Meawhile,thecomparisoaalysisisalsocoductedto show the superiority of the proposed method Cloud Service Provider Selectio Problem ad the Solutio Process. Duetothelimitedtechologyadcapital, a eterprise itself may be uable to build the cloud platform ad tries to seek a cloud service to realize its CRM. fter the market research ad prelimiary screeig, there are four potetial cloud services for further evaluatio, icludig SP Sales o Demad ( 1 ), Salesforce Sales Cloud ( 2 ), Microsoft Dyamic CRM ( 3 ) ad Oracle Cloud CRM ( 4 ). Four experts (e 1,e 2,e 3,e 4 ) are ivited to evaluate these cloud services o five idicators (attributes), icludig performace (u 1 ), paymet (u 2 ), reputatio (u 3 ), scalability (u 4 ), ad security (u 5 ). I terms of each attribute, each expert has preseted his (her) ormalized evaluatio iformatio forfourcloudservicesitables1 4. The preferece relatio set of attributes iformatio supplied by experts is as follows: D= { { ω 1 2ω 2 ; 0.05 ω 2 ω 4 0.1; ω 5 2ω 3 ; ω 1 { 0.4; ω 1 +ω 2 +ω 3 0.3; ω j = 1; ω j 0 } }. } (30)

8 8 Mathematical Problems i Egieerig Table1:IVIFdecisiomatrixR 1. ([0.55, 0.65], 1 [0.15, 0.25]) 2 [0.25, 0.35]) ([0.55, 0.65], 3 [0.15, 0.25]) ([0.35, 0.55], 4 [0.35, 0.45]) u 1 u 2 u 3 u 4 u 5 ([0.35, 0.55], ([0.65, 0.75], ([0.55, 0.75], [0.35, 0.45]) [0.15, 0.25]) ([0.15, 0.35], [0.15, 0.35]) ([0.75, 0.85], ([0.15, 0.25], [0.65, 0.75]) [0.45, 0.55]) ([0.55, 0.85], [0.15, 0.15]) ([0.15, 0.25], [0.55, 0.75]) Table 2: IVIF decisio matrix R 2. ([0.25, 0.45], [0.45, 0.55]) ([0.45, 0.65], [0.25, 0.35]) [0.35, 0.55]) ([0.10, 0.40], [0.30, 0.50]) ([0.70, 0.80], [0.10, 0.20]) ([0.50, 0.60], [0.20, 0.30]) ([0.20, 0.30], [0.50, 0.60]) ([0.45, 0.55], 1 [0.25, 0.45]) ([0.35, 0.55], 2 [0.30, 0.40]) ([0.45, 0.65], 3 [0.25, 0.35]) 4 [0.35, 0.55]) u 1 u 2 u 3 u 4 u 5 ([0.30, 0.40], ([0.55, 0.65], ([0.55, 0.65], [0.40, 0.60]) [0.10, 0.15]) [0.05, 0.25]) ([0.10, 0.30], [0.30, 0.70]) ([0.60, 0.80], [0.10, 0.20]) ([0.10, 0.20], [0.60, 0.80]) ([0.25, 0.35], [0.35, 0.45]) ([0.65, 0.75], ([0.05, 0.15], [0.65, 0.75]) ([0.25, 0.35], [0.55, 0.65]) ([0.45, 0.65], [0.25, 0.35]) [0.35, 0.55]) ([0.15, 0.35], [0.25, 0.45]) ([0.65, 0.85], ([0.55, 0.65], [0.15, 0.35]) ([0.25, 0.45], [0.45, 0.55]) Step 1. See Tables 1 4. Step 2. Calculate the weights of experts. We take the weights of experts o u 1 as a example, that is, λ k 1 (k = 1, 2, 3, 4), to illustrate the calculatig process of the experts weights. The calculatig processes for λ k 1 are as follows: (i) Select the referece sequece ad comparability sequeces. Selectig r 1 1 as a referece sequece ad r2 1, r3 1, r4 1 as comparability sequeces, where r 1 1 =( r1 11, r1 21, r1 31, r1 41 )=(([0.55, 0.65], [0.15, 0.25]), ([0.35, 0.45], [0.25, 0.35]), ([0.55, 0.65], [0.15, 0.25]), ([0.35, 0.55], [0.35, 0.45])) ; r 2 1 =( r2 11, r2 21, r2 31, r2 41 )=(([0.45, 0.55], [0.25, 0.45]), r 4 1 =( r4 11, r4 21, r4 31, r4 41 )=(([0.65, 0.75], [0.15, 0.25]), ([0.40, 0.50], [0.40, 0.50]), ([0.40, 0.50], [0.30, 0.40]), ([0.30, 0.40], [0.40, 0.50])). (31) (ii) Compute the gray relatioal coefficiet matrix. By (14)-(15), the gray relatioal coefficiet matrix is derived as ξ 1 1 =( ). (32) (iii) Calculate the gray relatioal grades. ccordig to (16),we have ([0.35, 0.55], [0.30, 0.40]), ([0.45, 0.65], [0.25, 0.35]), ([0.35, 0.45], [0.35, 0.55])) ; r 3 1 =( r3 11, r3 21, r3 31, r3 41 )=(([0.45, 0.75], [0.15, 0.25]), ([0.45, 0.55], [0.25, 0.45]), ([0.25, 0.45], [0.35, 0.45]), ([0.35, 0.45], [0.25, 0.45])) ; η( r 2 1, r1 1 )=1 4 η( r 3 1, r1 1 )=1 4 η( r 4 1, r1 1 )=1 4 4 ξ 21 i=1 4 ξ 31 i=1 4 ξ 41 i=1 i1 = , i1 = , i1 = (iv) Determie the average relatioal grade. (33)

9 Mathematical Problems i Egieerig 9 Table3:IVIFdecisiomatrixR 3. ([0.45, 0.75], 1 [0.15, 0.25]) ([0.45, 0.55], 2 [0.25, 0.45]) ([0.25, 0.45], 3 [0.35, 0.45]) 4 [0.25, 0.45]) u 1 u 2 u 3 u 4 u 5 ([0.35, 0.55], ([0.60, 0.70], ([0.55, 0.65], [0.25, 0.35]) [0.10, 0.20]) [0.05, 0.25]) ([0.25, 0.45], [0.35, 0.45]) ([0.65, 0.85], ([0.15, 0.25], [0.55, 0.75]) ([0.40, 0.50], [0.30, 0.40]) ([0.50, 0.70], [0.10, 0.30]) ([0.10, 0.30], [0.50, 0.70]) Table 4: IVIF decisio matrix R 4. ([0.15, 0.25], [0.65, 0.75]) ([0.55, 0.75], [0.15, 0.25]) ([0.25, 0.35], [0.45, 0.65]) ([0.35, 0.55], [0.25, 0.45]) ([0.65, 0.75], [0.15, 0.25]) ([0.65, 0.85], ([0.15, 0.25], [0.55, 0.75]) ([0.65, 0.75], 1 [0.15, 0.25]) ([0.40, 0.50], 2 [0.40, 0.50]) ([0.40, 0.50], 3 [0.30, 0.40]) ([0.30, 0.40], 4 [0.40, 0.50]) u 1 u 2 u 3 u 4 u 5 ([0.30, 0.40], ([0.75, 0.85], ([0.50, 0.60], [0.30, 0.40]) [0.10, 0.30]) ([0.10, 0.20], [0.20, 0.30]) ([0.60, 0.70], [0.10, 0.30]) ([0.10, 0.30], [0.60, 0.70]) [0.45, 0.55]) ([0.55, 0.85], [0.15, 0.15]) ([0.15, 0.25], [0.55, 0.75]) ([0.20, 0.30], [0.40, 0.60]) ([0.40, 0.50], [0.20, 0.30]) ([0.20, 0.30], [0.40, 0.50]) ([0.15, 0.25], [0.45, 0.65]) ([0.65, 0.75], ([0.55, 0.65], [0.25, 0.35]) [0.45, 0.55]) By usig (17), the average relatioal grade betwee expert e 1 ad all other three experts is obtaied as η 1 1 = 1 3 Similarly, we ca get t l=2 η( r l 1, r1 1 )= (34) λ 1 1 = , λ 2 1 = , λ 3 1 = , λ 4 1 = (36) η 2 1 = , η 3 1 = , (35) η 4 1 = By employig (18), the weights of four experts with respect to u 1 are derived as: The calculatig processes for the weights of experts with respect to other attributes are omitted, ad the results are show i Table 5. Step 3. Itegrate idividual decisio matrices R k (k = 1, 2, 3, 4) ito a collective decisio matrix R =( r ij ) m by (19), that is, R ([0.531, 0.685], [0.170, 0.288]) ([0.326, 0.482], [0.320, 0.440]) ([0.639, 0.743], [0.093, 0.182]) ([0.535, 0.663], [0.062, 0.234]) ([0.176, 0.381], [0.308, 0.511]) ([0.389, 0.514], [0.292, 0.420]) ([0.153, 0.332], [0.236, 0.425]) ([0.339, 0.439], [0.375, 0.476]) ([0.214, 0.341], [0.482, 0.628]) ([0.665, 0.796], [0.074, 0.178]) =( ). ([0.423, 0.572], [0.249, 0.353]) ([0.658, 0.809], [0.070, 0.191]) ([0.568, 0.790], [0.099, 0.182]) ([0.459, 0.636], [0.209, 0.310]) ([0.556, 0.687], [0.154, 0.289]) ([0.338, 0.467], [0.331, 0.484]) ([0.126, 0.251], [0.599, 0.749]) ([0.109, 0.239], [0.561, 0.736]) ([0.284, 0.385], [0.385, 0.553]) ([0.249, 0.378], [0.480, 0.595]) (37) Step 4. Derive the score matrix, accuracy matrix ad ucertaity matrix. By (5), (7) ad (8), the score matrix, accuracy matrix ad ucertaity matrix of the collective matrix R are computed as: S =( ),

10 10 Mathematical Problems i Egieerig Table 5: The weights of each expert with respect to differet attributes. u 1 u 2 u 3 u 4 u 5 e e e e H = ( ), γ = ( ) ( ) (38) Step 5. Determie the attribute weights. By (28), the followig liear programmig model is costructed: ( ) w w w w w 5 y; w w w w w 5 y; w w w w w 5 y; w w w w w 5 y; w w w w w 5 x; { w w w w w 5 x; max {py + (1 p)x} w w w w w 5 x; w w w w w 5 x; w 1 2w 2 ;0.05 w 2 w 4 0.1; w 5 2w 3 ; w 1 0.4; w 1 +w 2 +w 3 0.3; w ; w 1 +w 2 +w 3 +w 4 +w 5 = 1; { { w 1,w 2,w 3,w 4,w 5 0. (39) Set p=0.5adsolve(39) with Simplex Method.The mai compoets for (39) are as follows: y=0.405, x=0.779, w 1 = , w 2 = , w 3 = 0.05, w 4 = , w 5 = (40) Step 6. Compute the overall score ad overall ucertaities of each alterative. Utilizig (22) ad (24),we ca calculate the overall scores ad ucertaities of all alteratives which are show i Table 6. Step 7. Rak alteratives i term of Defiitio 7.Theresultof rakig is also listed i Table 6. From Table 6, it ca be see that alterative 3 is the best oe, that is, Microsoft Dyamic CRM is the best cloud service Sesitivity alysis for Parameter p. I above example, we get the computatio results by a give a priori (p = 0.5).However,theattributeweightsmayvaryasthevalueof weightig coefficiet p chages, which may result i differet decisio results. Hece, it is ecessary to do the sesitivity aalysis for parameter p. The results of sesitivity aalysis are depicted i Figure2. s show i Figure 2, whe the value of parameter p chages from 0 ad 1, although the overall scores of four providers chage slightly, the rakigs amog the four cloud services remai uchaged. 3 is first, followed by 1 ad followed by 2 ad the 4 is raked i the last all alog. Therefore, we ca use (29) freely.

11 Mathematical Problems i Egieerig 11 Table 6: The overall scores, accuracies, ad rakig of alteratives. lterative Score Ucertaity Rakig Compariso alysis with the Method Usig the Score Fuctio Give by Xu [30]. I the above cloud service selectio example, if the scores of alteratives are computed with the score fuctio give by Xu [30] (see(4)), the the score matrix is give as The scores of alteratives p S =( ) (41) Figure 2: The overall scores of four cadidate providers with respect to p. The accuracy matrix retais uchaged. Puttig the score matrix S ad accuracy matrix H ito (29), we get the followig programmig model: w w w w w 5 y; w w w w w 5 y; w w w w w 5 y; 0.005w w w w w 5 y; w w w w w 5 x; { w w w w w 5 x; max {py + (1 p)x} w w w w w 5 x; w w w w w 5 x; w 1 2w 2 ;0.05 w 2 w 4 0.1; w 5 2w 3 ; w 1 0.4; w 1 +w 2 +w 3 0.3; w ; w 1 +w 2 +w 3 +w 4 +w 5 = 1; { { w 1,w 2,w 3,w 4,w 5 0. (42) Still let p = 0.5,byemployigtheLigoSoft,wefid that (42) has o feasible solutio. Thus, the rakig order of alteratives caot be obtaied. This shows that itroducig the ormalized score fuctio proposed i this paper is very importat. 5. Coclusios I order to stad out i the fierce competitio, more ad more eterprises begi to select cloud service as oe of their developmet strategy. Cloud service selectio ca be regarded as a kid of MGDM. I this paper, we have studied the cloud service selectio problems with IVIFSs ad icomplete iformatio o attribute weights. ovel MGDM method was proposed to solve this kid of GDM problems. There are followig three dramatic features i the proposed method. (1) The assessmet values of alteratives o attributes are i the form of IVIFSs which ca help experts express their prefereces more flexibly.

12 12 Mathematical Problems i Egieerig (2) By extedig the classical GR method ito IVIF eviromet, a ew approach is preseted to determie the weights of experts. Furthermore, the weights of each expert obtaied are differet o differet attributes, which is much closer to the real-world decisio situatio. (3) multiobjective programmig model is costructed to derive the weights of attributes. The future work of this study is to apply the proposed method to other maagemet areas, such as risk ivestmet, material selectio ad so o. Coflict of Iterests The authors declare that there is o coflict of iterests regardig the publicatio of this paper. ckowledgmets The authors would like to thak ssociate Professor. Jiu-Yig Dog for improvig the liguistic quality of this paper. This research was supported by the Natioal Natural Sciece Foudatio of Chia (os , , , ad ), the Natioal 863 Pla Project ( ), the Humaities Social Sciece Programmig Project of Miistry of Educatio of Chia (o. 09YGC630107), the Natural Sciece Foudatio of Jiagxi Provice of Chia (os BB ad 20142BB201011), Twelve five Programmig Project of Jiagxi provice Social Sciece (2013) (o. 13GL17), the Natural Sciece Foudatio of Guagxi (GXNSF) ( ad ), Sciece Foudatio of Guagxi Educatioal Committee (YB ) ad the Excellet Youg cademic Talet Support Program of Jiagxi Uiversity of Fiace ad Ecoomics. Refereces [1] L. Su, H. Dog, F. K. Hussai, O. K. Hussai, ad E. 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