Multi attribute decision making method research based on interval vague set and TOPSIS method

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1 Techological ad Ecoomic Developmet of Ecoomy ISSN: (Prit) (Olie) Joural homepage: Multi attribute decisio makig method research based o iterval vague set ad TOPSIS method Peide Liu To cite this article: Peide Liu (2009) Multi attribute decisio makig method research based o iterval vague set ad TOPSIS method, Techological ad Ecoomic Developmet of Ecoomy, 5:3, To lik to this article: Published olie: 2 Oct 200. Submit your article to this oural Article views: 565 Citig articles: 73 View citig articles Full Terms & Coditios of access ad use ca be foud at

2 Techological ad ecoomic developmet OF ECONOMY Baltic Joural o Sustaiability (3): MULTI-ATTRIBUTE DECISION-MAKING METHOD RESEARCH BASED ON INTERVAL VAGUE SET AND TOPSIS METHOD Peide Liu Shadog Ecoomic Uiversity, Iformatio Maagemet School, Ji a 25004, Chia Peide.liu@gmail.com Received 3 March 2009; accepted 20 August 2009 Abstract. This paper proposed a method to resolve the multi-attribute decisio-makig problem usig TOPSIS method based o attribute weights ad attribute values are all iterval vague value. Firstly, based o the operatio rules of the iterval Vague value, the iterval Vague attribute value is made by weighted operatio, ad the ideal ad egative ideal solutios are calculated based o the score fuctio. The the distace of iterval Vague value is defied, as well as the distace betwee each proect ad the ideal, ad egative ideal solutios. The relative adacet degree is calculated by TOPSIS method, the the order of the proects is cofirmed accordig to the relative adacet degree. Fially, a case is used to show the process of the method this paper proposed ad the validity of this method is proved. Keywords: iterval Vague set, score fuctio, TOPSIS, multi-attribute decisio-makig. Referece to this paper should be made as follows: Liu, P Multi-attribute decisio-makig method research based o iterval vague set ad TOPSIS method, Techological ad Ecoomic Developmet of Ecoomy 5(3): Itroductio Decisio-makig is the process of fidig the best optio from all of the feasible alteratives. Sometimes, decisio-makig problems cosiderig several criteria are called multi-criteria decisio-makig (MCDM) problems. The MCDM problems may be divided ito two kids. Oe is the classical MCDM problems (Hwag ad Yoo 98; Kaklauskas et al. 2006; Zavadskas et al. 2008a, b; Li et al. 2008; Gievičius et al. 2008), amog which the ratigs ad the weights of criteria are measured i crisp umbers. Aother is the fuzzy multiple criteria decisio-makig (FMCDM) problems (Bellma ad Zadeh 970; Wag et al. 2003; Liu ad Wag 2007; Liu ad Du 2008; Wei ad Liu 2009; Ji et al. 2007; Liu ad Gua 2008, 2009), amog which the ratigs ad the weights of criteria evaluated o imprecisio, subective ISSN prit/issn olie doi: /

3 454 P. Liu. Multi-attribute decisio-makig method research based... ad vagueess are usually expressed by liguistic terms, fuzzy umbers or ituitio fuzzy umbers. Ataasov (986, 989) proposed Ituitio fuzzy set theory i 986. Gau ad Buehrer (993) proposed the cocept of Vague set at 993. Bustice ad Burillo (996) proposed that Vague set was ituitio fuzzy set ad uified the ituitio fuzzy set ad the Vague set. As the Vague set (Bustice ad Burillo 996) took the membership degree, o-membership degree ad hesitacy degree ito accout, ad has more ability to deal with ucertai iformatio tha traditioal fuzzy set, lots of scholars pay attetios to the research of Vague set. Ataassov ad Gargov (989) exteded the ituitio vague set ad proposed the cocept of iterval ituitio vague set, also amed iterval Vague set. Iterval Vague set has more ability to express vagueess ad ucertaity. At preset, the cocered researches of Vague set are focused o the operatio rules of the iterval Vague set (Ataasov 994), correlatio degree (Bustice ad Burillo 995; Hog 998), ad topological structure (Modal ad Samata 200). There are less researches i the area of multi-attribute decisio-makig. Xu ad Che (2007) proposed iterval ituitio prefer iformatio operator ad hybrid itegrated operator, ad proposed a ugemet method which the prefered iformatio of the decisio maker is a iterval-valued ituitioistic udgmet matrices. But the attributes weights of this method are geeral real umbers ad have some limitatio. Zhou ad Wu (2006) proposed a multiple criteria decisio method based o iterval value Vague sets of distace. But this method s attributes weights are also geeral real umbers. Wag (2006) proposed a multi-criteria decisio-makig method based o iterval-value vague sets with ucertai iformatio. It firstly calculates the commo real umber attribute weight through evidece theory, the defies the ideal solutio ad the egative ideal solutio, ad costructs oliear model base o distace. This method is essetial for iterval ituitio decisio makig with weights of real umber type. Multi-attribute iterval Vague set decisio makig methods that the weights are also iterval vague value are rare. I this paper, the attribute value ad weight are all vague value, a method usig TOPSIS method to solve the multiattribute decisio makig problem is proposed. Firstly, accordig to the operatio rules of the iterval value, the iterval vague attribute value has doe weighted calculatio. Ad the ideal ad egative solutios are cofirmed based o the score fuctio. The the distace of the iterval vague value is defied, ad the distaces betwee each proect ad the ideal ad egative ideal solutios are calculated. Accordig to the TOPSIS method, the relative adacet degrees are calculated. The the order of the proects is cofirmed accordig to the relative adacet degrees. 2. Evaluatio method 2.. The defiitio ad the essetial operatios of the iterval Vague set 2... The defiitio of the iterval Vague set Defiitio (Ataasov 986): Suppose discourse domai X = {x, x 2,..., x }, a Vague set A is described by true membership fuctio t A ad false membership fuctio i discourse domai, X, t A : X [0,], f A : X [0,]. Where, t A (x i ) is the lower boud that affirms the mem-

4 Techological ad Ecoomic Developmet of Ecoomy, 2009, 5(3): bership exported by the evidece that support x i, f A (x i ) is the lower boud that egates the membership exported by the evidece that support x i ad t A (x i ) f A (x i ). The membership of the elemet x i i Vague set A is defied by a subiterval [t A (x i ), f A (x i )] i iterval [0,], ad this iterval is called the Vague value of x i i set A. To Vague set A, the represetatio forms are as follows: [ t ( x ), f ( x ) x x X X A i A i ]/, whe is discrete, i i A = i= [ t ( x), f ( x) / x, x X X X A A ] whe is cotiuous. () x X, π A ( x) = t A ( x) f A ( x) is called Vague degree of x compared with Vague set A. π A ( x) shows the hesitatio degree or ucertai degree. Obviously, 0 π A ( x), x X. Because of the ucertaity ad complexity of the decisio, the values of t A (x) ad f A (x) are difficult to express by accurate real umbers value. The iterval values are more flexible tha the real umber values, extedig t A (x) ad f A (x) from real umber value to iterval value ituitio set ca get the iterval Vague set. Obviously, this set is much stroger to show the ucertai data ad ituitio data. The iterval Vague value is deoted as x = < t, f > x x, where t t t f = [, ] [ 0, ], = [ f, f ] [ 0, ] x x x x x x, ad the followig equatio is satisfied: t f. x x π A ( x) = [, ] t A ( x) f A ( x) = [ t A ( x) f A ( x), t A ( x) f A ( x)] is called the hesitacy degree of the iterval Vague value The essetial operatios of the iterval Vague set (Gau ad Buehrer 993, Xu ad Che 2007, Li ad Rao 200) Suppose there is a iterval Vague value x t f = <, > = < [ t, t ],[ f, f ] > x x x x x x, y t f = <, > = < [ t, t ],[ f, f ] >, where t, f, t, f [ 0, ] ad y y y y y y x x y y x x y y t f, t f. The followig operatio rules ad relatios ca be received: x = < f, t > = < [ f, f ],[ t, t ] >, (2) x x x x x x x y = < t t t t, f f > = (3) x y x y x y < [ t t t t, t t t t ],[ f f, f f ] >, x y x y x y x y x y x y x y = < t t, f f f f > = < (4) < [ t t, t t ],[ f f f x y x y x y f, f f f f ] >, x y x y x y x y x y x y λ λ λ λ λ x = < [ ( t ), ( t ) ],[( f ), ( f ) ] > λ 0. (5) x x x x

5 456 P. Liu. Multi-attribute decisio-makig method research based... The results of the operatio are all iterval vague value. Accordig to the operatio rules, the followig relatios ca be received: () x y = y x, (2) x y = y x, (3) λ( x y ) = λx λ y λ 0, (4) λ x λ x = ( λ λ ) x λ, λ The vague distaces the iterval Vague values Suppose: X is a discourse domai of elemets, A ad B are Vague sets, ad { A A A A } { B B B B } A = < [ t ( x), t ( x)],[ f ( x), f ( x)] > x X. B = < [ t ( x), t ( x)],[ f ( x), f ( x)] > x X. the the distace betwee Vague sets A ad B has differet defiitio, oe of the defiitio i literature (Zhou ad Wu 2006) is show as follows: d( A, B) = max( t ( x ) t ( x ), t ( x ) t ( x ), A i B i A i B i i=, f ( x ) f A f( x( ) x) ) f ( x ), f ( x ) f ( x ) ) i B i A i B i. (6) The defiitio i literature (Wag, 2006) is show as follows: d( A, B) = ( t ( x ) t ( x ) t ( x ) t ( x ) f ( x A i B i A i B i A 4 i= f ( x ) f ( x ) π ( x ) π ( x ) π ( x ) π A i B i A i B i A i B ( x i ) ), ) f ( x ) i B i (7) where π π ( x) = t ( x) f ( x), π ( x) = t ( x) f ( x), A A A A A A ( x) = t ( x) f ( x), π ( x) = t ( x) f ( x). B B B B B B Obviously, the defiitio i equatio (6) is lock of the sigificatio of distace. Otherwise, it oly adopts the iterval vague value to get a part of iformatio. So this paper adopts the equatio (7) to defie the distace of Vague set A ad B The descriptio of decisio problems based o iterval Vague set Defiitio 3: there is a multi-attribute decisio makig problem, suppose A = {A, A 2,..., A m } is a decisio proect set, C = {C, C 2,..., C } is the attribute set of the proects. Suppose the character of decisio proect A i to attribute set C is deoted by iterval Vague set: = < t f, >, where, t = [ t, t ] [ 0, ] deotes the degree that decisio proect A i φ

6 Techological ad Ecoomic Developmet of Ecoomy, 2009, 5(3): satisfies attribute C. f f = f [, ] [ 0, ] deotes the degree that the decisio proect A i does ot satisfy attribute C ad the follows is satisfied: t f. Suppose, the attribute weight is W = ( w, w,, w ) 2, where, w is deoted by the iterval Vague value as < t, f >, ad t = [ t, t ] [ 0, ], f = [ f, f ] [ 0, ], t f w w w w w w w w w w. Usig the above attribute weights ad iterval vague value of differet attributes to determie the order of the proects The score fuctio I order to compare the iterval vague value, the score fuctio of the iterval Vague value is show as follows. The iterval value vague set is the extesio of real umber vague. So the real umber Vague set score fuctio i existece is exteded appropriately i order to costruct the iterval Vague set score fuctio. Suppose there is a Vague value < [ t, t ],[ f, f ] >, accordig to the defiitio of x x x x the score fuctio S of Che ad Ta (994) ad the characteristic of the iterval umber, the score fuctio of the iterval Vague value is defied as follows: S t t f f = 2 2. (8) Compare the score fuctio S value, determie the iterval Vague value. The bigger the value of the score fuctio S, the bigger the correspodig iterval Vague value. But whe the value of the score fuctio is equivalet ad the umber is more tha two, this method caot do the udgmet. Hog ad Choi (2000) aalysed the deficiecy of the score fuctio, ad added the precise fuctio H. Accordig to the characteristics of iterval vague value, the precise fuctio is defied as follows: H t t f f = 2 2. (9) It shows the precisio of the membership situatio that is reflected by the iterval Vague set. Whe the values of the score fuctio S are the same, the precise fuctio H is compared. If H is bigger, the correspodig iterval vague value is also bigger. Usig a example Liu (2004) prove that the decisio makig ca be doe more reasoably through aalyzig the crowd that drop their rights. The thought of this method is show as follows: through aalyzig the hesitacy degree π of the decisio makig, divide it ito three parts accordig to the result of the votig model: t π, f π, π 2, they deote the ratio of the amout of the supporter to that of the obector ad that of the waiver. So, the support t degree is t t π, ad the obect degree is f f t f f π * *. t =, f =, π π 2 2 * π =. The the defiitio of the modified score fuctio S is as follows: 2

7 458 P. Liu. Multi-attribute decisio-makig method research based... * * * * * * * * * S = t t π ( f f π ) = ( t f )( π ). (0) Whe the values of the score fuctio S are the same ad the precise fuctio H are also the same, we use Sto cofirm the iterval vague value. The bigger S is, the bigger is the correspodig iterval vague value The processes of the multi-attribute decisio makig based o the iterval Vague set ad the TOPSIS TOPSIS is used to cofirm the order of the evaluatio obects i virtue of the ideal solutio ad the egative ideal solutio of the multi-attribute problems (Yue 2003). The ideal solutio is a best solutio that is assumed (marked as V ). Each of its idicator value is the best value of the optioal schemes. The egative solutio is aother worst solutio that is assumed (marked as V ). Each of its idicator value is the worst value of the optioal proect. V ad V are compared with each proect i the origial proect set. The distace iformatio of them is used to be the stadard to cofirm the order of the proects i X. () Weight the attribute value of each proect Accordig to Eq.(4), calculate the followig value: where, b w t f - - = φ = <, > = < [ t, t ],[ f, f ] >, () b b b b b b t = [ t t, t t ], (2) b w w f = [ f f f f, f f f f ]. (3) b w w w w (2) Calculate the score fuctio Accordig to Eq.(8), Eq.(9) ad Eq.(0), calculate the score fuctio, precise fuctio ad corrected fuctio of b separately. (3) Cofirm the ideal solutio ad the egative solutio of the evaluatio obect The iterval ituitioistic fuzzy set ideal solutio V ad egative ideal solutio V is show as follows: max_ i = max( S ), (4) V = < [ t, t ],[ f, f ] > = < [ t, t ],[ f V V V V b b max_ max_ i, f ] >, (5) bmax_ bmax_ mi_ i = mi( S ), (6) i V = < [ t, t ],[ f, f ]] > = < [ t, t ],[ f, f ] >. (7) V V V V b b mi_ mi_ (4) Calculate the distace betwee each proect ad the ideal solutio Accordig to Eq.(7), the distace betwee each proect ad the ideal solutio is calculated as follows: bmi_ bmi_

8 Techological ad Ecoomic Developmet of Ecoomy, 2009, 5(3): d = ( t t t t f f f i b V 4 = π π π π ). b V b V f b V b V b V 459 (8) (5) Calculate the distace betwee each proect ad the egative ideal solutio Accordig to Eq.(7), the distace betwee each proect ad the egative ideal solutio is calculated as follows: d = ( t t t t f f f i b V 4 = π π π π ). b V b V f b V b V b V (6) Cofirm the relative adacet degree The relative adacet degree of the evaluatio obect ad the ideal solutio is: C i (9) di = i = m d d.(, 2,, ). (20) i i Accordig to the relative adacet degree, the order of the evaluatio obects ca be cofirmed ad the best appropriate obect ca be selected. 3. Applicatio example A compay iteds to select a perso to take the departmet maager positio. Four aspects of the cadidate are evaluated by the experts. The four aspects are ideological ad moral quality (C ), professioal ability (C 2 ), creative ability (C 3 ) ad kowledge rage (C 4 ). The experts give evaluatio data ad weights to each aspects. Ad they are all deoted by the iterval Vague value, amely, the iterval umber of the support degree is give, ad the iterval umber of the obect degree is also give. The evaluatio data ad attribute weight is show by Tables ad 2. The order of the 3 cadidates must be cofirmed. Table. The evaluatio data of a differet cadidate give by the experts C C 2 C 3 C 4 S ([0.6,0.7],[0.2,0.3]) ([0.5,0.6],[0.,0.3]) ([0.6,0.7],[0.2,0.3]) ([0.3,0.4],[0.5,0.6]) S2 ([0.4,0.5],[0.3,0.4]) ([0.7,0.8],[0.,0.2]) ([0.5,0.6],[0.3,0.4]) ([0.4,0.5],[0.3,0.4]) S3 ([0.5,0.6],[0.3,0.5]) ([0.3,0.4],[0.3,0.5]) ([0.6,0.7],[0.,0.3]) ([0.5,0.6],[0.,0.3]) Table 2. The attribute weight give by the experts C C 2 C 3 C 4 W ([0.3,0.4],[0.5,0.6]) [0.6,0.7],[0.,0.2]) ([0.6,0.7],[0.2,0.3]) ([0.3,0.4],[0.5,0.6])

9 460 P. Liu. Multi-attribute decisio-makig method research based... The process to cofirm the order is show as follows: () Calculate the weighted b ([0.8,0.28],[0.60,0.72]) ([0.30,0.42],[0.9,0.44]) ([0.36,0.49],[0.36,0.5]) ([0.09,0.6],[0.75,0.84]) b = ([0.2,0.20],[0.65,0.76]) ([0.42,0.56],[0.9,0.36]) ([0.30,0.42],[0.44,0.58]) ([0.2,0.20],[0.65,0.76]) ([0.5,0.24],[0.65,0.80]) ([0.8,0.28],[0.37,0.60]) ([0.36,0.49],[0.28,0.5]) ([0.5,0.24],[0.55,0.72]) (2) The score fuctio is calculated as follows S = H = (3) Cofirm the ideal solutio ad the egative solutio All the idicators are beefit type. The ideal solutio ad the egative solutio are cofirmed as follows: V = {([0.8,0.28],[0.6,0.72]) ([0.42,0.56],[0.9,0.36]) ([0.36,0.49],[0.28,0.5]) ([0.5,0.24],[0.55,0.72])}. V = {([0.2,0.2],[0.65,0.76]) ([0.8,0.28],[0.37,0.6]) ([0.3,0.42],[0.44,0.58]) ([0.09,0.6],[0.75,0.84])}. (4) Calculate the distace betwee each proect ad the egative solutio d = (0.0825, , ), d = ( , , ). (5) Calculate the relative adacet degree: C = (0.4884, , ). (6) Cofirm the order of the cadidates Accordig to the relative adacet degree, the order of 3 cadidates ca be cofirmed: S S S. 2 3 So the secod cadidate S 2 is the best oe.

10 Techological ad Ecoomic Developmet of Ecoomy, 2009, 5(3): Coclusio As the decisio makig is fuzzy ad ucertai, the iterval Vague set s ability to express the fuzziess ad ucertaity is stroger. It is easier to express the decisio iformatio usig the iterval vague value. This paper explored the multi-attribute decisio makig problem based o the iterval vague value. Firstly, accordig to the operatio rules of the iterval vague value, do weighted operatio to the iterval Vague attribute value. Ad the ideal ad egative ideal solutios are cofirmed o the basis of the score fuctio. The the distace of the iterval vague value is defied ad the distaces betwee each proect ad the ideal ad egative ideal solutios are calculated. The relative adacet degree is calculated by the TOPSIS method. Accordig to the relative adacet degree, the order of the proect is cofirmed. This paper offers a ew efficiet approach to deal with the multi-attribute decisio makig problems based o the iterval vague value. The theory of the iterval Vague set is eriched ad developed, ad the traditioal TOPSIS method is exteded ad exteds its applicatio boud. The other applicatio of the iterval vague set is waitig to be researched ad explored. Ackowledgmet The authors gratefully ackowledge the fiacial support from Nature Sciece Foudatio of Shadog Provice (No.Y2007H23). The authors also would like to express appreciatio to the aoymous reviewers for their very helpful commets o improvig the paper. Refereces Ataasov, K. T Ituitioistic fuzzy sets, Fuzzy Sets ad Systems 20(): doi:0.06/s065-04(86) Ataasov, K. T More o ituitioistic fuzzy sets, Fuzzy Sets ad Systems 33(): doi:0.06/065-04(89) Ataasov, K. T Operators over iterval-valued ituitioistic fuzzy sets, Fuzzy Sets ad Systems 64(2): doi:0.06/065-04(94)9033-x. Ataasov, K. T.; Gargov, G Iterval-valued ituitioistic fuzzy sets, Fuzzy Sets ad Systems 3(3): doi:0.06/065-04(89) Bellma, R. E.; Zadeh, L. A Decisio-makig i a fuzzy eviromet, Maagemet Scieces 7: doi:0.287/msc.7.4.b4. Bustice, H.; Burillo, P Correlatio of iterval-valued ituitioistic fuzzy sets, Fuzzy Sets ad Systems 74(2): doi:0.06/065-04(94) Bustice, H.; Burillo, P Vague sets are Ituitioistic fuzzy Sets, Fuzzy Sets ad Systems 79: doi:0.06/065-04(95) Che, S. M.; Ta, J. M Hadlig multi-criteria fuzzy decisio-makig problems based o vague set theory, Fuzzy Sets ad Systems 67(2): doi:0.06/065-04(94) Gau, W. L.; Buehrer, D. J Vague sets, IEEE TrasSystMaCyber 23(2): doi:0.09/

11 462 P. Liu. Multi-attribute decisio-makig method research based... Gievičius, R.; Podvezko, V.; Raslaas, S Evaluatig the alterative solutios of wall isulatio by multicriteria methods, Joural of Civil Egieerig ad Maagemet 4(4): doi:0.3846/ Hog, D. H A ote o correlatio of iterval-valued ituitioistic fuzzy sets, Fuzzy Sets ad Systems 95(): 3 7. doi:0.06/s065-04(96) Hog, D. H.; Choi, C. H Multi-criteria fuzzy decisio-makig problems based o vague set theory, Fuzzy Sets ad Systems 44():03 3. doi:0.06/s065-04(98) Hwag, C. L.; Yoo, K. 98. Multiple Attribute Decisio Makig: Methods ad Applicatio. Spriger, New York. Ji, F.; Liu, P. D.; Zhag, X The evaluatio study of huma resources based o ituitioistic fuzzy set ad TOPSIS method, Joural of Iformatio ad Computatioal Sciece 4(3): Kaklauskas, A.; Zavadskas, E. K.; Raslaas, S.; Gievičius, R.; Komka, A.; Maliauskas, P Selectio of low-e widows i retrofit of public buildigs by applyig multiple criteria method COPRAS: A Lithuaia case, Eergy ad Buildigs 38(5): doi:0.06/.ebuild Li, F.; Rao, Y Weighted multi-criteria decisio makig based o vague sets, Computer Sciece 28(7): Li, Y. H.; Lee, P. C.; Chag, T. P.; Tig, H. I Multi-attribute group decisio-makig model uder the coditio of ucertai iformatio, Automatio i Costructio 7(6): doi:0.06/.autco Liu, H.W Vague set methods of multi-criteria fuzzy decisio makig, Systems Egieerig-theory & Practice 5(5): Liu, P. D.; Du, Z. W Applicatio of E-commerce risk assessmet research with weight ukow TOP- SIS method, i Iteratioal Symposium o Iformatio Processig, MAY 22 23, 2008, Liu, P. D.; Gua, Z. L A approach for multiple attribute decisio-makig based o Vague sets, Joural of Harbi Egieerig Uiversity 30(): Liu, P. D.; Gua, Z. L Research o group decisio makig based o the vague set ad hybrid aggregatio operators, Joural of Wuha Uiversity of Techology 30(0): Liu, P. D.; Wag, T. J A method for multiple attribute decisio makig with triagular fuzzy Number ad partial attribute weight iformatio, Joural of Iformatio ad Computatioal Sciece 4(3): Modal, T. K.; Samata, S. K Topology of iterval-valued ituitioistic fuzzy sets, Fuzzy Sets ad Systems 9(3): doi:0.06/s065-04(98) Wag, J. Q Multi-criteria iterval ituitioistic fuzzy decisio-makig approach with icomplete certai iformatio, Cotrol ad Decisio : Wag, Y. J.; Lee, H. S.; Li, K Fuzzy TOPSIS for multi-criteria decisio-makig, Iteratioal Mathematical Joural 3: Wei, Y. Q.; Liu, P. D Risk evaluatio method of high-techology based o ucertai liguistic variable ad TOPSIS method, Joural of Computers 4(3): Xu, Z. S.; Che, J A approach to group decisio makig based o iterval-valued ituitioistic udgmet matrices, Systems Egieerig-Theory & Practice 4: Yue, C. Y Decisio Makig Theory ad Method. Beig: Sciece press. Zavadskas, E. K.; Turskis, Z.; Tamošaitieė, J.; Maria, V. 2008a. Multicriteria selectio of proect maagers by applyig grey criteria, Techological ad Ecoomic Developmet of Ecoomy 4(4): doi:0.3846/

12 Techological ad Ecoomic Developmet of Ecoomy, 2009, 5(3): Zavadskas, E. K.; Kaklauskas, A.; Turskis, Z.; Tamošaitieė, J. 2008b. Selectio of the effective dwellig house walls by applyig attributes values determied at itervals, Joural of Civil Egieerig ad Maagemet 4(2): doi:0.3846/ Zhou, Z.; Wu, Q. Z Multi-criteria Decisio-makig based o iterval-value vague sets, Trasactios of Beig Istitute of Techology 8: DAUGIAKRITERINĖS SPRENDIMO PRIĖMIMO PROBLEMOS SPRENDIMAS taikant NEAPIBRĖŽTĄ INTERVALĄ IR TOPSIS METODĄ P. Liu Satrauka Straipsye siūlomas daugiakriteriės spredimo priėmimo problemos spredimas TOPSIS metodu, kai kriterų reikšmigumai ir reikšmės yra itervaliiai dydžiai. Iš pradžių, audoatis procedūromis, ustatomos svertiės itervaliių dydžių reikšmės, paskui apskaičiuoami idealiai teigiamas ir idealiai egiamas sprediiai. Toliau ustatomi itervalų dydžiai, apskaičiuoami atstumai tarp kiekvieos alteratyvos ir idealiai teigiamo ir idealiai eigiamo sprediių. TOPSIS metodu apskaičiuoami satykiiai atstumai iki miėtų idealių sprediių ir alteratyvos išrikuoamos į eilę. Galiausiai kokrečiu pavyzdžiu demostruoamas skaičiavimo procesas ir patvirtiamas siūlomo metodo pagrįstumas. Reikšmiiai žodžiai: itervaliės eapibrėžtos reikšmės, fukca, TOPSIS, daugiakriteriis spredimų priėmimas. Peide LIU (Chia, 966) graduated from the Southeast Uiversity ad obtaied the bachelor degree i electroic techology. The he obtaied his master degree i iformatio processig i the Southeast Uiversity. At preset, he is studyig his i-service doctor of iformatio maagemet i Beig Jiaotog Uiversity. His mai research fields are techology ad iformatio maagemet, decisio support ad electroic-commerce. He was egaged i the techology developmet ad the techical maagemet i the Ispur compay a few years ago. Now he is a full-time professor i Shadog Ecoomic Uiversity ad assistat director of the Eterprise s Electroic-commerce Egieerig Research Ceter of Shadog.