Dynamic multi-attribute decision making model based on triangular intuitionistic fuzzy numbers

Transcription

1 Scientia Iranica B (20) 8 (2), Sharif University of Technology Scientia Iranica Transactions B: Mechanical Engineering wwwsciencedirectco Research note Dynaic ulti-attribute decision aing odel based on triangular intuitionistic fuzzy nubers Y Chen a,b, B Li a,c, a State Key Laboratory of Robotics and Syste (HIT), Harbin, 5000, China b School of Mechanical & Electrical Engineering, Shandong University at Weihai, Weihai, , China c Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen, 58055, China Received 28 February 200; revised 4 Noveber 200; accepted 22 January 20 KEYWORDS Dynaic ulti-attribute decision aing ethod; Triangular intuitionistic fuzzy nubers; Entropy; Intuitionistic fuzzy sets Abstract Triangular Intuitionistic Fuzzy Nubers (TIFNs) express ore abundant and flexible inforation than Triangular Fuzzy Nubers (TFNs) The ain purpose of this paper is to propose a Dynaic Multi- Attribute Decision Maing (DMADM) odel on the basis of TIFNs, to solve the DMADM proble, where all the decision inforation taes the for of TIFNs A new distance easure between two TIFNs is developed to aid in deterining attribute weights, using the entropy ethod An aggregation operator, the weighted arithetic averaging operator on TIFNs (TIFN-WAA), is presented to aggregate the decision inforation with TIFNs Finally, the effectiveness and applicability of the proposed DMADM odel, as well as analysis of coparison with another odel, are illustrated with an investent exaple 20 Sharif University of Technology Production and hosting by Elsevier BV Open access under CC BY-NC-ND license Introduction Multiple-Attribute Decision Maing (MADM) ethods have been extensively applied to various areas, such as society, anageent science, econoics, ilitary research and public adinistration [ 5] However, ost MADM ethods focus on decision aing probles at the sae period, such as those proposed by Ye [6] who developed a MADM odel with interval-valued, intuitionistic, fuzzy nubers, and Jasowsi et al [7] who presented an extended fuzzy AHP odel for group decision aing, at the sae period Greco et al [8,9] and Blaszczynsi et al [0] extended the rough set theory into a ulti-attribute decision aing ethod, and Hu et al [] also extended a rough set MADM odel to solve a ulti-attribute decision aing proble at the sae period Corresponding author E-ail address: libingsgs@hiteducn (B Li) Sharif University of Technology Production and hosting by Elsevier BV Open access under CC BY-NC-ND license Peer review under responsibility of Sharif University of Technology doi:006/jscient In any decision areas, such as ulti-period investent and personnel dynaic exaination, the decision inforation is usually collected at different periods Thus, it is necessary to develop soe dynaic decision aing odels to deal with these ulti-period and ulti-attribute decision aing probles (also nown as Dynaic Multi-Attribute, Decision Maing (DMADM) probles [2]) Recently, research on DMADM probles has received soe attention [2 5] Xu [2] developed a ulti-period and ulti-attribute decision aing odel based on a siple additive weighting ethod Lin et al [3] proposed a dynaic ulti-attribute decision aing odel, where the attribute values are firstly aggregated into an overall evaluation value at each period, then all evaluation values are aggregated into an overall score of all alternatives Xu and Yager [4] investigated a dynaic ulti-attribute decision aing proble where the decision inforation taes the for of the interval uncertain inforation Wei [5] developed two aggregation operators to solve a dynaic ulti-attribute decision aing proble where the decision inforation also taes the for of the interval uncertain inforation All existing research focuses on DMADM probles where the decision inforation taes the for of a real nuber or interval uncertain inforation Nevertheless, in any practical cases, the available decision inforation is usually difficult to judge precisely; instead, they can be easily characterized by soe fuzzy linguistic ters, such as good, poor and so on In addition, triangular intuitionistic fuzzy nubers in the Intuitionistic Fuzzy sets (IFs) can not only deal with vagueness inforation, but also express ore

2 Y Chen, B Li / Scientia Iranica, Transactions B: Mechanical Engineering 8 (20) abundant and flexible inforation than triangular fuzzy nubers [6,7] Therefore, the ain purpose of this paper is to propose a Dynaic Multi-Attribute Decision Maing (DMADM) odel on the basis of TIFNs to solve DMADM probles where the decision inforation taes the for of TIFNs In addition, for the purpose of dealing with decision inforation with TIFNs, this paper also extends the entropy ethod to calculate attribute weights in the DMADM odel 2 Preliinaries Soe basic definitions of Intuitionistic Fuzzy Sets (IFSs) and Triangular Intuitionistic Fuzzy Nubers (TIFNs) are reviewed Definition An intuitionistic fuzzy set, A, in the universe of discourse X is defined with the for [8]: A = { x, µ A (x), ν A (x) x X}, where µ A : X [0, ], ν A : X [0, ], and with the condition: 0 µ A (x) + ν A (x), x X The nubers µ A (x) and ν A (x) denote ebership and nonebership degrees of x, with respect to A, respectively In addition, we call: π A (x) = µ A (x) ν A (x), the degree of hesitancy of x, with respect to A [9] Especially, if π A (x) = 0, µ A (x) =, ν A (x) = 0, for all x X, then the IFSs A is reduced to a fuzzy set Definition 2 A triangular intuitionistic fuzzy nuber α = (a, b, c); µ α, ν α is a special intuitionistic fuzzy set on a real nuber set R, whose ebership function and nonebership function are defined as follows [8]: (x a)µ α /(b a) if a x < b, µ µ α (x) = α if x = b, () (c x)µ α /(c b) if b < x c, 0 Otherwise, and: [b x + ν α (x a)]/(b a) if a x < b, ν ν α (x) = α if x = b, (2) [x b + ν α (c x)]/(c b) if b < x c, Otherwise The values µ α and ν α represent the axiu ebership degree and the iniu non-ebership degree, respectively Definition 3 Let α = (a, b, c ); µ α, ν α and α 2 = (a 2, b 2, c 2 ); µ α2, ν α2 be two triangular intuitionistic fuzzy nubers and λ is a real nuber Soe arithetical operations are defined as follows [20]: (a + a α α 2 = 2, b + b 2, c + c 2 );, (3) µ α + µ α2 µ α µ α2, ν α ν α2 λα = (λa, λb, λc ); ( µ α ) λ, (ν (λc, λb, λa ); ( µ α ) λ, (ν α2 ) λ if λ 0 α2 ) λ (4) if λ < 0 Definition 4 Let α i = (a i, b i, c i ) (i =,, n) be a set of triangular fuzzy nubers, and w = (w, w 2,, w n ) T be the weight vector of α i (i =,, n), then we call: TFN-WAA w ( α, α 2,, α n ) = n = w i a i, n w i b i, n w i α i, n w i c i, (5) the weighted averaging operator on TFNs (TFN-WAA) Definition 5 Let α i = (a i, b i, c i ); µ αi, ν αi be a set of triangular intuitionistic fuzzy nuber and w = (w, w 2,, w n ) T be the weight vector of α i (i =,, n), then we call: TFN-WAA w ( α, α 2,, α n ) n n n = w j a j, w j b j, w j c j ; j= j= j= n n ( µ εj ) w j, j= j= (ν εj ) w j, (6) the weighted arithetic averaging operator on TIFNs (TIFN- WAA) 3 Weight easures using entropy ethod Weight easures have a direct relationship with the distance easure between two fuzzy nubers [2] In order to especially deal with decision inforation with triangular intuitionistic fuzzy nubers, this paper proposes a new distance easure Definition 6 Let à = ã; µã, νã and B = b; µ B, ν B be two arbitrary triangular intuitionistic fuzzy nubers where ã and b are two triangular fuzzy nubers with λ-cut representations, ã λ = [a L (λ), a R (λ)] and bλ = [b L (λ), b R (λ)] The distance between à and B is defined as follows: d(ã, B) = + 0 al (λ) b L (λ) 2 + (a R (λ) b R (λ)) 2 dλ 2 ((µ à µ B) 2 + (νã ν B) 2 + (µã + νã µ B ν B) 2 ) An MADM decision aing proble usually includes a set of possible alternatives A i (i =, 2,, ), which is based on a set of n evaluation attributes C j (j =, 2,, n) The final noralized decision atrix is expressed as D = [ rij ] where r ij = (a ij, b ij, c ij ); µ ij, ν ij is the noralized rating value of the alternative A i with respect to the attribute C j TIFNs can enable decision aers to assess the alternatives in different diensions For exaple, a triangular intuitionistic fuzzy nuber r ij = G; 07, 02 = (065, 08, 095); 07, 02 can be interpreted as the expert can not only use the linguistic variable ( Good ) to rate the ith alternative, with respect to the jth attribute, but can also provide the confidence level of 07 at which the expert believes the ith alternative belongs to Good, as well as the non-confidence level of 02, at which the expert does not believe the ith alternative belongs to Good Vector M j, as shown in Eq (8), can be used to express deviations of the rating values, with respect to their average rating values: M j = d( r j, r j ),, d( r ij, r j ),, d( r j, r j ), (8) (7)

3 270 Y Chen, B Li / Scientia Iranica, Transactions B: Mechanical Engineering 8 (20) where d( r ij, r j ) is the distance between two TIFNs defined in Definition 6, and the average rating value r j = (α j, β j, η j ); µ j ν j can be obtained as follows: r j = (α j, β j, η j ); µ j ν j = ( r j r ij r j ) (j =, 2,, n), where: α j = a ij, β j = b ij, η j = c ij, µ j = ( µ ij ) /, ν j = (ν ij ) / The noralized vector for vector M j is calculated as follows: M j = [ρ i ] = d( r ij, r j ), ax d( r ij, r j ) i =, 2,, (0) i The entropy easure with the jth attribute, C j can be described as follows: e j = ρ i ln() ln ρ i () ρ i ρ i Then the entropy weight of the jth attribute can be obtained as follows: w j = e j (j =, 2,, n) (2) n n e = 4 Dynaic ulti-attribute decision aing odel A dynaic, ulti-attribute decision aing proble can be described as follows: the ost desirable alternative needed to be found fro a set of feasible alternatives, A i (i =, 2,, ), evaluated, with respect to a set of n attributes, C j (j =, 2,, n), by a group of p decision aers for the periods, Let w j (t ) represents the attribute weight of the jth attribute, C j, at the th period And let λ(t ) be the period weight of the th period The average decision atrix at different periods can be denoted as follows: D(t ) = { x ij (t )} C A (t ) C 2 (t ) C n (t ) x A (t ) x 2 (t ) x n (t ) 2 = x 2 (t ) x 22 (t ) x 2n (t ), (3) A x (t ) x 2 (t ) x n (t ) where x ij (t ) indicates the average rating values of the ith alternative, A i, with respect to the jth attribute, C j (t ), at the th period, and x ij (t ) = (a ij (t ), b ij (t ), c ij (t )); µ ij (t ), ν ij (t ) 4 DMADM odel with TIFNs This paper attepts to propose a DMADM odel with TIFNs based on priority attributes The DMADM odel firstly (9) aggregates the rating values of each alternative, with respect to each attribute at the th period, into an overall rating vector of each alternative, with respect to the th period, then builds all overall rating vectors for each period into a new decision atrix of each alternative, with respect to period Finally, the final evaluation value is calculated based on the new decision atrix (the flowchart is shown in Figure ) Based on a priority attribute, the DMADM odel can obtain not only the final evaluation value of each alternative, but also the iddle evaluation value of each alternative at each period The iddle evaluation value will provide the assistant function of deterining the ost desirable alternative A procedure for the DMADM odel with TIFNs, based on the priority attribute, is discussed in the following section 42 Procedure for DMADM odel based on priority attribute Step : Calculate the average rating values evaluated by p decision aers for each period, and obtain the noralized decision atrix at each period Let ζ ijs (t ) = (a ijs (t ), b ijs (t ), c ijs (t ), µ ijs (t ), ν ijs (t )) be the orig of the ith alternative, with respect to the jth attribute, evaluated by sth decision aer at the th period The average rating value of the ith alternative, A i, with respect to the jth attribute, C j, for p decision aers can be obtained as follows: x ij (t ) = (a ij (t ), b ij (t ), c ij (t )); µ ij (t ), ν ij (t ) = ζ ij (t ) ζ ij2 (t ) ζ ijp (t ) (4) p To obtain the noralized decision atrix Ñ(t ) = { r ij (t )} n, one can eploy the following noralized transforation: a ij (t ) r ij (t ) = c +, b ij(t ) j c +, c ij(t ) j c + ; j r ij (t ) = µ ij (t ), ν ij (t ) c j c ij (t ), µ ij (t ), ν ij (t ) cj b ij (t ),, j Ω B, cj a ij (t ), ;, j Ω C, (5) where c + j = ax i cij (t ), c j = in i aij (t ), and Ω B, Ω C are the benefit and cost attribute sets, respectively Step 2: Calculate the attribute weights at each period as w j (t ) (j =, 2,, n) based on the noralized decision atrices, using the entropy ethod with TIFNs Step 3: Aggregate the noralized decision atrices, Ñ(t ) = { r ij (t )} n, for all periods into a final decision atrix, R = { ξ i } Utilizing the TIFN-WAA operator in Definition 5; ξ i = TIFN-WAA wj (t )( r i (t ), r i2 (t ),, r in (t )) = w (t ) r i (t ) w 2 (t ) r i2 (t ) w n (t ) r in (t ), to aggregate all noralized decision atrices Ñ(t ) = { r ij (t )} n, into a final decision atrix, R = { ξ i }

4 Y Chen, B Li / Scientia Iranica, Transactions B: Mechanical Engineering 8 (20) Figure : Flowchart of the DMADM odel based on priority attribute Table : Original rating values evaluated by three decision aers at the t period DM DM 2 DM 3 A G; 07, 02 G; 08, 0 G; 09, 0 A 2 VG; 08, 0 VG; 09, 0 VG; 08, 02 A 3 G; 07, 02 VG; 06, 04 G; 08, 0 A 4 VG; 06, 03 VG; 07, 02 VG :, 06, 04 A 5 G; 07, 02 VG; 07, 03 G; 06, 03 A G; 06, 03 VG; 07, 02 G; 06, 04 A 2 G; 05, 04 MG; 07, 02 MG; 07, 02 A 3 VG; 08, 0 VG; 09, 0 VG; 08, 02 A 4 G; 08, 0 G; 07, 02 G; 08, 0 A 5 G : 07, 03 VG; 07, 02 G; 08, 0 A F; 08, 02 F; 08, 0 F; 07, 02 A 2 F; 07, 02 MP; 05, 04 F; 08, 0 A 3 MP; 08, 0 MP; 09, 0 MP; 08, 02 A 4 MP; 07, 02 F; 06, 03 MP; 05, 04 A 5 F; 06, 03 MG; 07, 02 F; 08, 0 C C 2 C 3 Table 2: Original rating values evaluated by three decision aers at the t 2 period C DM DM 2 DM 3 A G; 08, 0 G; 09, 0 G; 08, 02 A 2 VG; 08, 0 VG; 07, 02 VG; 09, 0 A 3 G; 09, 0 VG; 05, 04 G; 08, 0 A 4 G; 08, 02 G; 07, 02 VG; 05, 04 A 5 VG; 08, 02 VG; 08, 0 VG; 07, 02 A VG; 08, 0 G; 05, 04 VG; 06, 03 A 2 G; 07, 02 G; 08, 0 VG; 05, 04 A 3 G; 09, 0 G; 08, 02 G; 07, 02 A 4 G; 06, 03 MG; 05, 03 G; 07, 02 A 5 VG; 08, 0 VG; 08, 02 G; 05, 04 A F; 08, 0 MP; 05, 04 F; 07, 02 A 2 MP; 08, 02 F; 05, 02 MP; 09, 0 A 3 MP; 08, 02 F; 05, 04 MP; 07, 02 A 4 F; 09, 0 F; 08, 0 F; 08, 02 A 5 MG; 07, 02 F; 08, 0 MG; 07, 02 C 2 C 3 Step 4: Define γ + = (γ +,, γ +,, γ + K )T and γ = (γ,, γ,, γ K )T as the triangular intuitionistic fuzzy positive ideal solution (TIFPIS) and the triangular intuitionistic fuzzy negative ideal solution (TIFNIS), respectively According to the noralized decision atrix, we now that γ + = (,, );, 0 are the largest TIFNs, and γ + = (0, 0, 0); 0, are the sallest TIFNs In addition, for convenience of depiction, we denote the alternative, A i (i =, 2,, ), by A i = ( ξ i, ξ i2,, ξ ik ) Step 5: Calculate the period weights in the decision atrix, R = { ξ i }, as λ(t ), using the entropy ethod with TIFNs Step 6: Calculate the distance between the alternative and the TIFPIS (γ + ), and the distance between the alternative and the TIFNIS (γ ) according to the following distance easure defined in Definition 6: K D(A i, γ + ) = λ(t )d ξ i, γ +, = (i =, 2,, ), K D(A i, γ ) = = λ(t )d ξ i, γ, (i =, 2,, ) (6) Step 7: Calculate the closeness coefficient, CC i (i =, 2,, ), of all alternatives and ran all alternatives, A i (i =, 2,, ), according to the closeness coefficient: D(A i, γ ) CC i = D(A i, γ + ) + D(A i, γ ), (i =, 2,, ) (7) 5 Illustrative exaple One exaple [4] is provided to deonstrate the effectiveness and applicability of the DMADM odel based on a priority attribute An investent copany wants to invest an aount of oney There are five possible copanies, A i (i =, 2,, 5), in which to invest the oney: () A is a car copany; (2) A 2 is a food copany; (3) A 3 is a coputer copany; (4) A 4 is an ars copany; and (5) A 5 is a TV copany A group of decision aers is fored with three decision aers, DM s (s =, 2, 3) Each possible copany will be evaluated across three attributes with regard to: () econoic benefit (C ); (2) social benefit (C 2 ); and (3) environental pollution (C 3 ), where C and C 2 are benefit attributes and C 3 is a cost attribute The rating values of five possible copanies, with respect to three attributes, are represented by TIFNs, and the three decision aers construct the original rating values at three periods, as listed in Tables 3 5 Raning alternatives using the DMADM odel with TIFNs The calculation procedure for the DMADM odel with TIFNs, based on a priority attribute, is described step by step, as below: Step : Calculate the average rating values evaluated by three decision aers, according to Eq (4), and obtain the noralized decision atrices using Eq (5) The noralized decision atrices, Ñ(t ) = { r ij (t )} 5 3 ( =, 2, 3) for three periods are shown in Tables 4 6 Step 2: Calculate the attribute weights at each period Using Eq (2), the attribute weights, w j (t ) (j =, 2, 3), for three periods, are shown in Tables 4 6

5 272 Y Chen, B Li / Scientia Iranica, Transactions B: Mechanical Engineering 8 (20) Table 3: Original rating values evaluated by three decision aers at the t 3 period C DM DM 2 DM 3 A VG; 08, 0 G; 06, 03 VG; 08, 0 A 2 G; 07, 02 VG; 05, 04 G; 08, 02 A 3 VG; 09, 0 VG; 08, 02 VG; 08, 0 A 4 VG; 08, 02 VG; 05, 04 G; 08, 0 A 5 VG; 07, 02 VG; 09, 0 VG; 08, 02 A G; 07, 0 VG; 05, 03 G; 08, 0 A 2 VG; 08, 0 G; 06, 03 VG; 08, 02 A 3 G; 08, 02 G; 08, 0 G; 07, 02 A 4 G; 08, 0 G; 09, 0 G; 08, 02 A 5 G; 07, 02 VG; 08, 0 G; 08, 0 A MP; 07, 02 P; 05, 04 MP; 07, 02 A 2 MP; 08, 0 MP; 08, 0 MP; 08, 0 A 3 F; 07, 02 MP; 05, 04 F; 08, 02 A 4 MP; 05, 04 F; 08, 0 F; 08, 0 A 5 F; 08, 0 F; 08, 0 F; 08, 0 Step 3: Utilize the TIFN-WAA operator to aggregate the noralized decision atrices Ñ(t ) = { r ij (t )} 5 3 into a final decision atrix, R = { ξ i } 5 3, as shown in Table 7 Step 4: Define the TIFPIS, γ +, and TIFNIS, γ, by: γ + = (((,, );, 0), ((,, );, 0), ((,, );, 0)) T ; γ = (((0, 0, 0); 0, ), ((0, 0, 0); 0, ), ((0, 0, 0); 0, )) T Step 5: Calculate the period weights, λ(t ) ( =, 2, 3), in the decision atrix R = { ξ i } 5 3, as shown in Table 7 Step 6: Calculate the distance between the alternative and the TIFPIS (γ + ), and the distance between the alternative and the TIFNIS (γ ) using Eq (6), respectively Step 7: Using Eq (7), the closeness coefficient, CC i (i =, 2, 3), can be obtained The distances, closeness coefficient and raning order of five alternatives are tabulated in Table 8 We can see that the raning order is A 3 A 5 A 2 A A 4, where indicates the relation preferred to C 2 C 3 52 Coparison analysis If we do not consider the ebership and non-ebership degrees in TIFNs, ie µ ij = and ν ij = 0, in the original rating values, then the rating values with TIFNs in Tables 3 will be reduced to values with TFNs, and the DMADM proble with TIFNs will also be reduced to the DMADM proble with TFNs To verify the effectiveness and applicability of the proposed DMADM odel with TIFNs, we also ran the alternatives using the DMADM odel with TFNs, which is based on the noralized Euclidean distance and the weighted averaging operator on TFNs (TFN-WAA) in Definition 4 Table 9 shows the final raning results obtained by the two DMADM odels As shown in Table 9, it is not very difficult to observe that the ost desirable alternative, A 3, presented by the DMADM odel with TFNs, is the sae as the result obtained through the DMADM odel with TIFNs, but the raning order obtained by the DMADM odel with TFNs is soewhat different fro the order presented through the DMADM odel with TIFNs These results show that the ebership and non-ebership degrees in TIFNs play an iportant role in deterining the raning order in the DMADM odel with TIFNs, and they also provide ore exact and abundant decision inforation Therefore, we can draw a conclusion that the raning order obtained by the DMADM odel with TIFNs is ore reliable than the order presented through the DMADM odel with TFNs 53 Sensitivity analysis Sensitivity analysis (SA) is the investigation of soe potential changes and errors of rating values and their ipact on the final raning order [22] In this paper, soe sensitivity analyses are conducted to investigate the ipact of changing the ebership and non-ebership degrees in the rating values on the alternatives raning order A slight variation in the original rating values evaluated by decision aers goes as follows: (aijs (t x ijs (t ) = ), b ijs (t ), c ijs (t ));, (8) µ ijs (t ) + qh, ν ijs (t ) qh where q = σ j /h,,, 0,,, σ j /h, h is the step size, and σ j, σ j (j =, 2, 3) are the variation intervals of the ebership and non-ebership degrees, with respect to three attributes Table 4: Noralized decision atrix, Ñ(t ), at the t period C C 2 C 3 A (065, 08, 095); 0883, 026 (07, 08667, 09667); 06366, (03077, 04, 0574); 077, 0587 A 2 (08,, ); 0843, 026 (055, 07, 085); 06443, 0252 (03333, 04444, 06667); 06893, 02 A 3 (07, 08667, 09667); 076, 02 (08,, ); 0843, 026 (04, 0574, ); 0843, 026 A 4 (08,, ); 06366, (065, 08, 095); 077, 026 (03636, 05, 08); 06085, A 5 (07, 08667, 09667); 06698, 0262 (07, 08667, 09667); 07379, 087 (02857, 03636, 05); 076, 087 w(t ) Table 5: Noralized decision atrix, Ñ(t 2 ), at the t 2 period C C 2 C 3 A (065, 08, 095); 0843, 026 (07627, 09492, ); 0658, (0467, 05556, 08333); 06893, 02 A 2 (08,, ); 0883, 026 (079, 0884, 0983); 06893, 02 (04545, 0625, ); 07846, 0587 A 3 (07, 08667, 09667); 07846, 0587 (066, 0836, 0966); 0883, 0587 (04545, 0625, ); 06893, 0252 A 4 (07, 08667, 09667); 06893, 0252 (0602, 07627, 0953); 06085, 0262 (03846, 05, 0743); 0843, 026 A 5 (08,, ); 077, 0587 (07627, 09492, ); 07286, 02 (03333, 0467, 05556); 07379, 0587 w(t 2 )

6 Y Chen, B Li / Scientia Iranica, Transactions B: Mechanical Engineering 8 (20) Table 6: Noralized decision atrix, Ñ(t 3 ), at the t 3 period C C 2 C 3 A (075, 09333, 09833); 0748, 0442 (079, 0884, 0983); 06893, 0442 (03333, 05, ); 06443, 0252 A 2 (07, 08667, 09667); 06893, 0252 (07627, 09492, ); 0748, 087 (03, 04286, 075); 08, 0 A 3 (08,, ); 0843, 026 (066, 0836, 0966); 077, 0587 (025, 03333, 05); 06893, 0252 A 4 (075, 09333, 09833); 07286, 02 (066, 0836, 0966); 0843, 026 (025, 03333, 05); 07286, 0587 A 5 (08,, ); 0883, 0587 (079, 0884, 0983); 077, 026 (02308, 03, 04286); 08, 0 w(t 3 ) Table 7: Final decision atrix, R{ ξ i } t t 2 t 3 A (05628, 0702, 0835); 0727, 0208 (06382, 08023, 094); 07249, 090 (0656, 07896, 0988); 0709, 0699 A 2 (05289, 0676, 082); 07096, (0662, 08394, 0992); 07545, 0672 (06028, 07646, 0938); 07446, 073 A 3 (06504, 08346, 09934); 0824, 038 (063, 07742, 09757); 07798, 0806 (05943, 07449, 08425); 07827, 0662 A 4 (05866, 0742, 09); 0705, 094 (05696, 0753, 0872); 0726, 026 (05749, 079, 0836); 0774, 062 A 5 (05655, 07033, 085); 077, 0953 (06523, 0833, 0876); 07423, 077 (06049, 07568, 0827); 07988, 0287 λ(t ) Table 8: Distances, closeness coefficient and raning order of five alternatives Alternatives D(A i, γ + ) D(A i, γ ) CC i Ran A A A A A Table 9: Final raning results obtained by the two DMADM odels The DMADM Model with TFNs The DMADM Model with TIFNs CC i Raning CC i Raning A A A A A Figure 2: Raning order sensitivity to the ebership and non-ebership degrees with respect to the first attribute, C As shown in Figures 2 4, the raning order of all alternatives will reain constant when the variation values of the ebership and non-ebership degrees, with respect to the three attributes, vary in the range fro 006 to 004 But over the range, the raning order of the two alternatives A and A 2 will change with the ebership and non-ebership degrees, with respect to the three attributes It deonstrates that the alternatives A and A 2 are ore sensitive to ebership and non-ebership degrees than the other three alternatives 6 Conclusions In this paper, a DMADM odel with triangular intuitionistic fuzzy nubers is presented to deal with vagueness and uncertain inforation Using an aggregation operator (TIFN- WAA), a procedure for the DMADM odel with TIFNs, based on the priority attribute, is developed to solve the DMADM proble, where all attribute values are expressed in triangular intuitionistic fuzzy nubers In addition, a new distance easure between two TIFNs is proposed to deterine the entropy weights in the DMADM odel Another DMADM odel with TFNs is established to copare with the DMADM odel with TIFNs The coparison results Figure 3: Raning order sensitivity to the ebership and non-ebership degrees with respect to the second attribute, C 2 deonstrate that the DMADM odel with TIFNs can provide a ore reliable raning order than the DMADM odel with TFNs

7 274 Y Chen, B Li / Scientia Iranica, Transactions B: Mechanical Engineering 8 (20) Figure 4: Raning order sensitivity to the ebership and non-ebership degrees with respect to the third attribute, C 3 Although the exaple provided here is for selecting a desirable investent copany, the proposed odel can be applied to any different fields Acnowledgeents The wor was supported by the National Natural Science Foundation of China (Grant No , ) and State Key Laboratory of Robotics and Systes (HIT) (Project No SKLRS-200-ZD-02) References [] Cavallaro, F Fuzzy TOPSIS approach for assessing theral-energy storage in concentrated solar power (CSP) systes, Applied Energy, 87(2), pp (200) [2] Sun, CC and Lin, GR Using fuzzy TOPSIS ethod for evaluating the copetitive advantages of shopping websites, Expert Systes with Applications, 36(9), pp (2009) [3] Wang, YM and Elhag, TM Fuzzy TOPSIS ethod based on alpha level sets with an application to bridge ris assessent, Expert Systes with Applications, 3(2), pp (2006) [4] Yang, ZL, Bonsall, S and Wang, J Use of hybrid ultiple uncertain attribute decision aing techniques in safety anageent, Expert Systes with Applications, 36(2), pp (2009) [5] Thaer, A, Jarvis, J, Buggy, M and Sahed, A A novel approach to aterials selection; strategy case study; wave energy extraction ipulse turbine blade, Materials and Design, 29, pp (2008) [6] Ye, F An extended TOPSIS ethod with interval-valued intuitionistic fuzzy nubers for virtual enterprise partner selection, Expert Systes with Applications, 37(0), pp (200) [7] Jasowsi, P, Biru, S and Bucon, R Assessing contractor selection criteria weights with fuzzy AHP ethod application in a group decision environent, Autoation in Construction, 9(2), pp (200) [8] Greco, S, Matarazzo, B and Slowinsi, R Rough sets ethodology for sorting probles in the presence of ultiple attributes and criteria, European Journal of Operational Research, 38(2), pp (2002) [9] Greco, S, Matarazzo, B and Slowinsi, R Rough approxiation by doinance relations, International Journal of Intelligent Systes, 7, pp 53 7 (2002) [0] Blaszczynsi, J, Greco, S and Slowinsi, R Multi-criteria classification-a new schee for application of doinance-based decision rules, European Journal of Operational Research, 8, pp (2007) [] Hu, Q, Yu, D and Guo, M Fuzzy preference based rough sets, Inforation Sciences, 80, pp (200) [2] Xu, ZS On ulti-period, ulti-attribute decision aing, Knowledge- Based Systes, 2(2), pp 64 7 (2008) [3] Lin, YH, Lee, PC and Ting, HI Dynaic ulti-attribute decision aing odel with grey nuber evaluations, Expert Systes with Applications, 35(4), pp (2008) [4] Xu, ZS and Yager, RR Dynaic intuitionistic fuzzy ulti-attribute decision aing, International Journal of Approxiate Reasoning, 48, pp (2008) [5] Wei, GW UDWGA operator and its application to dynaic ultiple attribute decision aing, In IITA International Conference on Services Science, Manageent and Engineering,, pp 8 84 (2009) [6] Li, DF TOPSIS-based nonlinear-prograing ethodology for ultiattribute decision aing with interval-valued intuitionistic fuzzy sets, IEEE Transactions on Fuzzy Systes, 8(2), pp (200) [7] Li, DF, Chen, GH and Huang, ZG Linear prograing ethod for ulti-attribute group decision aing using IF sets, Inforation Sciences, 80(9), pp (200) [8] Atanassov, K, Intuitionistic Fuzzy Sets: Theory and Applications, Springer Physica-Verlag, Berlin (999) [9] Szidt, E and Kacprzy, J Distances between intuitionistic fuzzy sets, Fuzzy Sets and Systes, 4, pp (2000) [20] Wang, JQ and Zhang, Z Aggregation operators on intuitionistic trapezoidal fuzzy nuber and its application to ulti-criteria decision aing probles, Journal of Systes Engineering and Electronics, 20(2), pp (2009) [2] Guha, D and Charaborty, D A theoretical developent of distance easure for intuitionistic fuzzy nubers, International Journal of Matheatics and Matheatical Sciences, pp 25 (200) Article ID [22] Baird, BF, Managerial Decisions Under Uncertainty, an Introduction to the Analysis of Decision Maing, Wiley, New Yor (989) Yuan Chen received the MS degree in Mechanical Design and Theoretic fro Hubei University of Technology, China, in 2004 Since 2007, he has been a PhD candidate at the Harbin Institute of Technology Shenzhen Graduate School, China Since 2004, he has been a lecturer in the School of Mechanical and Electrical Engineering of Shandong University at Weihai His priary research interest focused on Cooing robots, CAD/CAM etc Bing Li received his Bachelor and Master degrees at echanical engineering fro Liaoning Technical University, China in 993 and 996 In 200 he got his PhD in Mechanical Engineering fro the Hong Kong Polytechnic University Currently Bing Li is Dean and Professor of School of Mechanical Engineering and Autoation, Shenzhen Graduate School, Harbin Institute of Technology His research interests focus on robotic echaniss, parallel robot, PKM, etc