Research Article Interval-Valued Intuitionistic Fuzzy Ordered Weighted Cosine Similarity Measure and Its Application in Investment Decision-Making

Transcription

1 Hdaw Complexty Volume 2017 Artcle ID pages Research Artcle Iterval-Valued Itutostc Fuzzy Ordered Weghted Cose Smlarty Measure ad Its Applcato Ivestmet Decso-Makg Dogha Lu 12 Xaohog Che 1 ad Da Peg 2 1 School of Busess Cetral South Uversty Chagsha Cha 2 Departmet of Mathematcs Hua Uversty of Scece ad Techology Xagta Cha Correspodece should be addressed to Dogha Lu; doghalu@126.com Receved 13 October 2016; Revsed 23 December 2016; Accepted 11 Jauary 2017; Publshed 6 February 2017 Academc Edtor: Ja Hao Copyrght 2017 Dogha Lu 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. We preset the terval-valued tutostc fuzzy ordered weghted cose smlarty (IVIFOWCS) measure ths paper whch combes the terval-valued tutostc fuzzy cose smlarty measure wth the geeralzed ordered weghted averagg operator. The ma advatage of the IVIFOWCS measure provdes a parameterzed famly of smlarty measures ad the decso maker ca use the IVIFOWCS measure to cosder a lot of possbltes ad select the aggregato operator accordace wth hs terests. We have studed some of ts ma propertes ad partcular cases such as the terval-valued tutostc fuzzy ordered weghted arthmetc cose smlarty (IVIFOWACS) measure ad the terval-valued tutostc fuzzy maxmum cose smlarty (IVIFMAXCS) measure. The IVIFOWCS measure ot oly s a geeralzato of some smlarty measure but also t ca deal wth the correlato of dfferet decso matrces for terval-valued tutostc fuzzy values. Furthermore we preset a applcato of IVIFOWCS measure to the group decso-makg problem. Fally the exstg smlarty measures are compared wth the IVIFOWCS measure by a llustratve example. 1. Itroducto The smlarty measure s a mportat tool for measurg the degree of smlarty betwee two objects whch s very useful some areas such as decso-makg mache learg patter recogto ad medcal dagoss [1 6]. Over the past several decades a varety of smlarty measures have bee troduced ad vestgated [7 14] based o tutostc fuzzy sets (IFSs) [15]. For example L ad Cheg [9] vestgated smlarty measures o IFSs ad showed how these measures may be used patter recogto problems. Later Lag ad Sh [10] troduced several ew smlarty measures o IFSs ad dscussed the relatoshps betwee these measures. Hug ad Yag [11] preseted a method to calculate the dstace betwee IFSs based o the Hausdorff dstace ad used ths dstace to geerate several smlarty measures betwee IFSs. Furthermore Hug ad Yag [12] preseted two ew smlarty measures betwee IFSs whch have bee foud to satsfy some smlarty measure axoms. Oe of may smlarty measures s the cose smlarty measure based o Bhattacharyya s dstace [13] whch s defed as the er product of two vectors dvded by the product of ther legths. Ye [14] proposed a cose smlarty measure betwee IFSs ad appled t to medcal dagoss ad patter recogto. However some cases the degrees of membershp or omembershp are sometmes assumed ot exactly as a umber but as a whole terval; Ataassov ad Gargov [16] troduced the cocept of terval-valued tutostc fuzzy sets (IVIFSs). Furthermore Xu [17] developed some smlarty measures of tutostc fuzzy sets ad appled them to patter recogto. Ye [18] proposed a cose smlarty measure for IVIFSs ad appled t to multple attrbute decso-makg problems. Whe smlarty measures are wdely used decsomakg problems the mportace of ordered posto of each degreeofsmlartyshouldbeemphaszed.iotherwords

2 2 Complexty the hgher the degree of smlarty the hgher the weght whch should be assged to t; a very useful techque s the ordered weghted averagg (OWA) operator. The OWA operator s troduced by Yager [19] whch s a very wellkow aggregato operator that provdes a parameterzed famly of aggregato operators cludg the maxmum the mmum ad the average as specal cases. The promet characterstc of the OWA operator s the reorderg step. Sce t has appeared the OWA operator has bee wdely exteded to other aggregato evromets cludg lgustc evromet (Mergó ad Casaovas [20] We ad Zhao [21] ad Zhou ad Che [22 23]) fuzzy evromet (Mergó ad Gl-Lafuete [24] Xu [25]) tutostc fuzzy evromet (L [26] Zeg ad Su [27] ad Zhou et al. [28 29]) ad terval-valued tutostc fuzzy evromet (Letal.[30]Yuetal.[31]adZhouetal.[32])ad used areas such as decso-makg ad eural etworks (Yager [33] Mergó ad Gl-Lafuete [34] ad Zhou et al. [35 38]). The am of ths paper s to troduce the tervalvalued tutostc fuzzy ordered weghted cose smlarty (IVIFOWCS) measure. It combes the tervalvalued tutostc fuzzy cose smlarty measure wth the geeralzed OWA operator. A more complete formulato of the cose smlarty measure s obtaed because t ca cosder parameterzed famles of operators that clude the maxmum the mmum ad the average as specal cases. Usg the advatage of IVIFOWCS measure carelevetheflueceofudulylargeorudulysmall devatos o the aggregato results. Ths measure provdes a robust formulato that cludes a wde rage of partcular cases such as the terval-valued tutostc fuzzy ordered weghted arthmetc cose smlarty (IVI- FOWACS) measure the terval-valued tutostc fuzzy ordered weghted quadratc cose smlarty (IVIFOWQCS) measure the terval-valued tutostc fuzzy ordered weghted geometrc cose smlarty (IVIFOWGCS) measure the terval-valued tutostc fuzzy maxmum cose smlarty (IVIFMAXCS) measure the terval-valued tutostc fuzzy mmum cose smlarty (IVIFMINCS) measure the terval-valued tutostc fuzzy ormalzed cose smlarty (IVIFNCS) measure the terval-valued tutostc fuzzy ormalzed arthmetc cose smlarty (IVIFNACS) measure ad the terval-valued tutostc fuzzy ormalzed geometrc cose smlarty (IVIFNGCS) measure. The decso maker s able to cosder a wde rage of scearos ad select the oe that s accordace wth hs terests. The paper s orgazed as follows. I Secto 2 we brefly revew the cocepts of IFSs IVIFSs the cose smlarty measure for IVIFSs ad the OWA operator. I Secto 3 we troduce the IVIFOWCS measure; some propertes ad dfferet famles of the IVIFOWCS measures are aalyzed. Secto 4 develops a applcato the group decso-makg problem. Secto 5 gves a umercal example. Secto 6 summarzes the ma coclusos of the paper. 2. Prelmares 2.1. Basc Cocepts of IFSs ad IVIFSs Defto 1. Let X={x 1 x 2...x } be a fte uversal set; IFs A X s defed as A={(x μ A (x ) ] A (x ) x X)} (1) where μ A (x ) ] A (x ):X [01]arethe membershp fucto ad omembershp fucto respectvely such that 0 μ A (x )+] A (x ) 1 x X. Assume π A (x ) = 1 μ A (x ) ] A (x ) x X;the π A (x ) s called the hestato degree of whether x belogs to A or ot. It s obvous that 0 π A (x ) 1 x X. For coveece we call α = (μ α ] α π α ) a tutostc fuzzy umber (IFN) ad deote the module of α as α = μα 2 + ]2 α +π2 α. Defto 2. Let X={x 1 x 2...x };IVIFs A X s defed as A ={(x μ A (x ) V A (x ) x X}wheretervalsμ A (x )= [μ AL (x ) μ AU (x )] [0 1] ad V A (x )=[V AL (x ) V AU (x )] [0 1] deote the membershp degree ad omembershp degree of the elemet x to the set A respectvely. For each x X the hestacy degree of a terval tutostc fuzzy set A s defed as follows: π A (x )=[π AL (x )π AU (x )] =[1 μ AU (x ) V AU (x )1 μ AL (x ) V AL (x )]. A terval-valued tutostc fuzzy umber (IVIFN) α [μ αl μ αu ] [V αl V αu ] [π αl π αu ]);wedeotethemoduleof α as α = (μ αl ) 2 +(μ αu ) 2 +(V αl ) 2 +(V αu ) 2 +(π αl ) 2 +(π αu ) 2. Let A = ([μ A L μ A U ] [] A L ] A U ]) ad B = ([μ BL μ BU ] [] BL ] BU ]) be two IVIFNs; the operatos are defed as follows (Ye [18]): (1) A+ B = ([μ AL +μ BL μ ALμ BL μ AU +μ BU μ AUμ BU ] [] AL ] BL ] AU ] BU ]). (2) λ A = ([1 (1 μ AL )λ 1 (1 μ AU )λ ] [] λ AL ] λ AU ]) λ>0. (3) A = B f μ AL =μ BL μ AU =μ BU ] AL = ] BL ad ] AU = ] BU The OWA Operator. The OWA operator s a aggregato operator that provdes a parameterzed famly of aggregato operators that cludes the maxmum the mmum ad the average as specal cases. It ca be defed as follows. Defto 3. AOWAoperatorofdmeso s a mappg OWA: R R thathasaassocatedweghtgvectorw wth w j [0 1] ad w j = 1suchthatOWA(a 1 a 2...a )= w jb j whereb j s the largest jth of the argumets a 1 a 2...a. Note that the OWA operator s commutatve mootoc bouded ad dempotet. (2)

3 Complexty 3 Yager [39] developed the geeralzed OWA (GOWA) operator whch s defed as follows. Defto 4. AGOWAoperatorsamappgGOWA:R R that has a assocated weghtg W wth w j [0 1] ad w j =1 ad a parameter λ ( + ) ad λ =0such that GOWA (a 1 a 2...a ) w j b j λ ) (3) where b j s the largest jth of the argumets a 1 a 2...a. We kow that the GOWA operator s also commutatve mootoc bouded ad dempotet (Yager [39]). We ca obta a group of partcular cases. For example f λ = 1 the the GOWA operator s reduced to the OWA operator. If λ 0 the ordered weghted geometrc averagg (OWGA) operator s obtaed. If λ = 1 the ordered weghted harmoc averagg (OWHA) operator s formed Cose Smlarty Measures for IVIFSs Defto 5. Let X={x 1 x 2...x } assume that there are two IVIFSs A = {x [μ AL (x ) μ AU (x )] [] AL (x ) ] AU (x )] x X}ad B={x [μ BL (x ) μ BU (x )] [] BL (x ) ] BU (x )] x X} ad a cose smlarty measure betwee two IVIFSs A ad B s defed as follows: C IVIFS (A B) = 1 =1 μ AL (x )μ BL (x )+μ AU (x )μ BU (x )+] AL (x ) ] BL (x )+] AU (x ) ] BU (x )+π AL (x )π BL (x )+π AU (x )π BU (x ) μal 2 (x )+μau 2 (x )+] 2 AL (x )+] 2 AU (x )+πal 2 (x )+πau 2 (x ) H (4) where H = μ 2 BL (x )+μ 2 BU (x )+] 2 BL (x )+] 2 BU (x )+π 2 BL (x )+π 2 BU (x ). The cose smlarty measure betwee A ad B satsfes the followg propertes: (1) 0 C IVIFS (A B) 1. (2) C IVIFS (A B) = C IVIFS (B A). (3) C IVIFS (A B) = 1 f A = B.e. μ AL (x ) = μ BL (x ) μ AU (x )=μ BU (x ) ] AL (x )=] BL (x ) ad ] AU (x )= ] BU (x ). (=12...). 3. Iterval-Valued Itutostc Fuzzy Ordered Weghted Cose Smlarty Measure I ths secto we wll troduce the IVIFOWCS measure whch s a smlarty measure that uses the cose smlarty measure for IVIFS the GOWA operator The IVIFOWCS Measure. Let A = (α j ) m Ω B = (β j ) m Ω be two terval-valued tutostc fuzzy matrces α j = ([μ αj Lμ αj U] [] αj L ] αj U]) β j = ([μ βj L μ βj U] [] βj L ] βj U]) are IVIFNs for all j ad assume that α j = (α 1j α 2j...α mj ) T ad β j = (β 1j β 2j...β mj ) T for j = We ca defe the IVIFOWCS measure as follows. Defto 6. A IVIFOWCS measure of dmeso s a mappg IVIFOWCS: Ω Ω Rthat has a assocated weghtg vector W wth w j [0 1] ad w j =1such that IVIFOWCS (A B) w j (C IVIFS (α σ(j) β σ(j) )) λ ) where C IVIFS (α j β j ) s the cose smlarty measure betwee IVIFS α j ad β j ad (σ(1)σ(2)...σ())s ay permutato of (12...)suchthat C IVIFS (α σ(j 1) β σ(j 1) ) C IVIFS (α σ(j) β σ(j) ) j=23... Remark 7. If = 1 A ad B the IVIFOWCS measure reduces to the cose smlarty measure for IVIFS (Ye [18]). Example 8. Let (5) (6) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) Aα 1 α 2 α 3 ) ) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ])

4 4 Complexty ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) Bβ 1 β 2 β 3 ) ). ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) (7) By (4) we ca get C IVIFS (α 1 β 1 ) = C IVIFS (α 2 β 2 ) = adC IVIFS (α 3 β 3 ) = The C IVIFS (α σ(1) β σ(1) ) = C IVIFS (α σ(2) β σ(2) ) = adC IVIFS (α σ(3) β σ(3) ) = If W = ( ) by usg the IVIFOWCS measure we ca obta the cose smlarty measures correspodg to some specal cases of the parameter λwhchareshow Table 1. Proof. Let IVIFOWCS ((α 1 β 1 )(α 2 β 2 )...(α β )) w j (C IVIFS (α σ(j) β σ(j) )) λ ) IVIFOWCS ((β 1 α 1 )(β 2 α 2 )...(β α )) (10) 3.2. Propertes of the IVIFOWCS Measure. The IVIFOWCS measure s commutatve mootoc bouded dempotet oegatve ad reflexve. These propertes are show wth the followg theorems. Theorem 9 (commutatvty-gowa aggregato). Let A = (α 1 α 2...α ) Ωad Bβ 1 β 2...β ) Ω. If ((α 1 β 1 )...(α β )) s ay permutato of the argumets (( α 1 β 1 )...( α β )) theivifowcs((α 1 β 1 )... (α β )) = IVIFOWCS(( α 1 β 1 )...( α β )). Proof. Let IVIFOWCS ((α 1 β 1 )...(α β )) w j (C IVIFS (α σ(j) β σ(j) )) λ ) IVIFOWCS (( α 1 β 1 )...( α β )) w j (C IVIFS ( α σ(j) β σ(j) )) λ ) Because ((α 1 β 1 )...(α β )) s a permutato of the argumets (( α 1 β 1 )...( α β ))adwekowc IVIFS (α σ(j) β σ(j) ) = C IVIFS ( α σ(j) β σ(j) ) for all j wehaveivifowcs((α 1 β 1 )...(α β )) = IVIFOWCS(( α 1 β 1 )...( α β )). Thewe complete the proof of Theorem 9. Theorem 10 (commutatvty-smlarty measure). Let A = (α 1 α 2...α ) Ω B = (β 1 β 2...β ) Ω;the. (8) w j (C IVIFS (β σ(j) α σ(j) )) λ ) Because C IVIFS (α j β j )=C IVIFS (β j α j )thec IVIFS (α σ(j) β σ(j) )=C IVIFS (β σ(j) α σ(j) ) for all j. Ths completes the proof of Theorem 10. Theorem 11 (mootocty). Let Aα 1 α 2...α ) Ω B = (β 1 β 2...β ) Ωadγ = (γ 1 γ 2...γ ) Ω;f C IVIFS (α j β j ) C IVIFS (α j γ j ) for all jthe IVIFOWCS ((α 1 β 1 )(α 2 β 2 )...(α β )) IVIFOWCS((α 1 γ 1 )(α 2 γ 2 )...(α γ )). Proof. Let IVIFOWCS ((α 1 β 1 )(α 2 β 2 )...(α β )) w j (C IVIFS (α σ(j) β σ(j) )) λ ) IVIFOWCS ((α 1 γ 1 )(α 2 γ 2 )...(α γ )) w j (C IVIFS (α σ(j) γ σ(j) )) λ ).. (11) (12) Because C IVIFS (α j β j ) C IVIFS (α j γ j )thec IVIFS (α σ(j) β σ(j) ) C IVIFS (α σ(j) γ σ(j) ) for all j. Ths completes the proof of Theorem 11. Theorem 12 (boudary). Let Aα 1 α 2...α ) Ωad Bβ 1 β 2...β ) Ωfor all j;the m j C IVIFS (α j β j ) IVIFOWCS ((α 1 β 1 )(α 2 β 2 )...(α β )) = IVIFOWCS ((β 1 α 1 )(β 2 α 2 )...(β α )). (9) IVIFOWCS ((α 1 β 1 )(α 2 β 2 )...(α β )) max C IVIFS (α j β j ). j (13)

5 Complexty 5 Table1:TheIVIFOWCSmeasuresresult. λ IVIFOWCS(A B) λ IVIFOWCS(A B) Proof. If max j C IVIFS (α j β j )=pad m j C IVIFS (α j β j )=q otcg w j =1the IVIFOWCS ((α 1 β 1 )(α 2 β 2 )...(α β )) ( w j (C IVIFs (α σ(j) β σ(j) )) λ ) w j (p) λ ) p λ w j ) =p IVIFOWCS ((α 1 β 1 )(α 2 β 2 )...(α β )) ( w j (C IVIFs (α σ(j) β σ(j) )) λ ) w j (q) λ ) q λ w j ) ThewecompletetheproofofTheorem12. =q. (14) Theorem 13 (dempotecy). Let Aα 1 α 2...α ) Ωad Bβ 1 β 2...β ) Ω;fC IVIFs (α j β j )=c(c s a costat) for all jtheivifowcs((α 1 β 1 ) (α 2 β 2 )...(α β )) = c. Proof. Let IVIFOWCS ((α 1 β 1 )(α 2 β 2 )...(α β )) w j (C IVIFS (α σ(j) β σ(j) )) λ ). (15) Because C IVIFS (α j β j )=cfor all j wehavec IVIFS (α σ(j) β σ(j) )=c 2... The IVIFOWCS((α 1 β 1 ) (α 2 β 2 )...(α β )) = c. Theorem 14 (oegatvty). Let Aα 1 α 2...α ) Ω ad Bβ 1 β 2...β ) Ω;the IVIFOWCS ((α 1 β 1 )(α 2 β 2 )...(α β )) 0. (16) Proof. It s straghtforward ad thus omtted. Theorem 15 (reflexvty). Let Aα 1 α 2...α ) Ω;the IVIFOWCS((α 1 α 1 ) (α 2 α 2 )...(α α )) = 0. Proof. Let IVIFOWCS ((α 1 α 1 )(α 2 α 2 )...(α α )) w j (C IVIFS (α σ(j) α σ(j) )) λ ). (17) Because C IVIFS (α σ(j) α σ(j) ) = 0 for all j the IVIFOWCS(α σ(j) α σ(j) )= Ths completes the proof of Theorem Famles of the IVIFOWCS Measures. By usg dfferet cases of the weghtg vector W ad parameter λweareable to obta a wde rage of partcular types of the IVIFOWCS measure Aalyzg the Parameter λ. By choosg dfferet cases of the parameter λ the IVIFOWCS measure we ca obta dfferet types of cose smlarty measure such as the terval-valued tutostc fuzzy ordered weghted arthmetc cose smlarty (IVIFOWACS) measure the tervalvalued tutostc fuzzy ordered weghted quadratc cose smlarty (IVIFOWQCS) measure ad the terval-valued tutostc fuzzy ordered weghted geometrc cose smlarty (IVIFOWGCS) measure. Remark 16. If λ=1 the the IVIFOWCS measure s reduced to IVIFOWACS measure: IVIFOWACS (A B) = w j C IVIFS (α σ(j) β σ(j) ) (18) where (σ(1) σ(2)... σ()) s ay permutato of (12... )suchthat C IVIFS (α σ(j 1) β σ(j 1) ) C IVIFS (α σ(j) β σ(j) ) j=23... (19) Remark 17. If λ=2 the the IVIFOWCS measure becomes the IVIFOWQCS measure: IVIFOWQCS (A B) w j (C IVIFS (α σ(j) β σ(j) )) 2 ) 1/2 (20)

6 6 Complexty Table 2: Ideal alteratve. u 1 u 2 u 3 u e k y k 1 y k 2 y k 3 y k where (σ(1)σ(2)...σ()) s ay permutato of (12... ) suchthatc IVIFS (α σ(j 1) β σ(j 1) ) C IVIFS (α σ(j) β σ(j) ) j= Remark 18. If λ 0 the the IVIFOWCS measure s reduced to the IVIFOWGCS measure: IVIFOWGCS (A B) = (C IVIFS (α σ(j) β σ(j) )) w j (21) where (σ(1)σ(2)...σ()) s ay permutato of (12... ) suchthatc IVIFS (α σ(j 1) β σ(j 1) ) C IVIFS (α σ(j) β σ(j) ) j= Aalyzg the Weghtg Vector W. By choosg a dfferet mafestato of the weghtg vector the IVI- FOWCS measure we are able to obta dfferet types of cose smlarty measures such as IVIFMAXCS measure IVIFMINCS measure IVIFNCS measure IVIFNACS measure ad IVIFNGCS measure. Remark 19. If w 1 =1ad w j =0for all j =1 the IVIFOWCS measure s reduced to IVIFMAXCS measure. Remark 20. If w =1ad w j =0for all j = the IVIFOWCS measure s reduced to IVIFMINCS measure. Remark 21. If w j =1/for all j the IVIFOWCS measure s reduced to IVIFNCS measure. Specally f λ = 1 we ca get the IVIFNACS measure; f λ=2 we ca get the IVIFNQCS measure; f λ 0the IVIFNCS measure s reduced to IVIFNGCS measure. 4. Multple Attrbute Group Decso-Makg wth the IVIFOWCS Measure I ths paper we cosder a decso-makg applcato of the IVIFOWCS measure the selecto of vestmets uder ucertaty. Let A = {A 1 A 2...A m } be a set of alteratves ad U={u 1 u 2...u } be the set of attrbutes. Let E = {e 1 e 2...e t } be the set of decso makers. Each decso maker provdes hs ow payoff matrx A (k) = (a (k) j ) m (k = 12...t)wherea (k) j s gve by the decso maker e k Eforthealteratvea Awthrespecttothe attrbute u j U. The based o the IVIFOWCS measure we propose a method wth the IVIFOWCS measure group decsomakg whch volves the followg steps. Step 1. Form the deal alteratve by gvg the deal levels of each characterstc whch s show Table 2 where Y (k) = ) s the deal characterstc of e k. (y (k) j Step 2. Calculate the IVIFCS measure betwee each preferece vector α (k) provded by the decso maker e k ad Y (k) y (k) 1 y(k) 2...y(k) ); the formula s gve as follows: C IVIFS (α (k) Y (k) )= 1 μ (k) α jl μ (k) y +μ (k) L α ju μ (k) y + ] (k) U α jl ] (k) y + ] (k) L α ju ] (k) y +π (k) U α jl π (k) y +π (k) L α ju π (k) y U (μ (k) α jl ) 2 +(μ (k) α ju ) 2 +(] (k) α jl ) 2 +(] (k) α ju ) 2 +(π (k) α ju ) 2 +(π (k) α ju ) 2 H (22) where H = (μ (k) y L ) 2 +(μ (k) y U ) 2 +(] (k) y L ) 2 +(] (k) y U ) 2 +(π (k) y L ) 2 +(π (k) y U ) 2. Step 3. Utlze the IVIFOWCS measure t k=1 [ w k (C IVIFS (α σ(k) Y σ(k) )) λ ] (23) to aggregate the IVIFCS measure to the collectve value S A of the alteratve A where(σ(1) σ(2)... σ(t)) s ay permutato of (12...k)suchthat C IVIFS (α σ(k 1) Y σ(k 1) ) C IVIFS (α σ(k) Y σ(k) ) =12...m. (24) Step 4. Rak all the alteratves A ( = 12...m) accordace wth the collectve values S A descedg order ad select the best oe of them. 5. Illustratve Example 5.1. A Illustrato of the Proposed IVIFOWCS Measure. I the followg we are gog to develop a bref example of the ew approach a group decso-makg problem about vestmet selecto. Assume a decso maker wats to vest moey a compay; after aalyzg the market he cosders sx possble alteratves: (1) Ivest a chemcal compay called A 1 (2) Ivest a food compay called A 2 (3) Ivest a computer compay called A 3 (4) Ivest a car compay called A 4 (5) Ivest a furture compay called A 5 (6) Ivest a pharmaceutcal compay called A 6

7 Complexty 7 Table 3: Characterstcs of the vestmets-expert 1. 1 ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) 2 ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) 3 ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) 4 ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) 5 ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) 6 ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) Table 4: Characterstcs of the vestmets-expert 2. 1 ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) 2 ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) 3 ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) 4 ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) 5 ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) 6 ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) Table 5: Characterstcs of the vestmets-expert 3. 1 ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) 2 ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) 3 ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) 4 ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) 5 ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) 6 ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) Table 6: Ideal strategy. Y (1) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) Y (2) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) Y (3) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) Table 7: C IVIFS measure. Y (1) Y (2) Y (3) A A A A A A The decso maker has brought together a group of experts. Each expert revews the formato of the vestmets from sx characterstcs U={u 1 u 2...u 6 }: u 1 : beefts the short term u 2 : beefts the md term u 3 : beefts the log term u 4 :rskofthevestmet u 5 : dffculty of the vestmet u 6 :otherfactors Three experts provde ther opos about the vestmets;theresultsareshowtables3 5. Accordg to ther objectves the compay experts establsh the deal vestmets show Table 6. Utlze (4) to calculate the cose smlarty measure betwee each decso maker s preferece vector α (k) ad hs deal preferece vector Y (k) ;theweobtathec IVIFS (α (k) Y (k) ) Table 7. Usg (23) to aggregate cose smlarty measure to thecollectvevalues A of all the alteratve A.Forcoveece we assume that the experts weghtg vector W =

8 8 Complexty Table 8: Aggregated results. IVIFMAXCS IVIFMINCS IVIFNACS IVIFNQCS IVIFNGCS IVIFOWCS A A A A A A Table 9: Orderg of the strateges. IVIFMAXCS IVIFMINCS IVIFNACS IVIFNQCS IVIFNGCS IVIFOWCS Orderg Table 10: Characterstcs of the vestmets-expert 1. 1 ( ) ( ) ( ) ( ) ( ) ( ) 2 ( ) ( ) ( ) ( ) ( ) ( ) 3 ( ) ( ) ( ) ( ) ( ) ( ) 4 ( ) ( ) ( ) ( ) ( ) ( ) 5 ( ) ( ) ( ) ( ) ( ) ( ) 6 ( ) ( ) ( ) ( ) ( ) ( ) Table 11: Characterstcs of the vestmets-expert 2. 1 ( ) ( ) ( ) ( ) ( ) ( ) 2 ( ) ( ) ( ) ( ) ( ) ( ) 3 ( ) ( ) ( ) ( ) ( ) ( ) 4 ( ) ( ) ( ) ( ) ( ) ( ) 5 ( ) ( ) ( ) ( ) ( ) ( ) 6 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ad λ=3;wedevelopdfferetmethodsbased o the IVIFOWCS measure for selecto of a vestmet. For example we cosder IVIFMAXCS IVIFMINCS IVIF- NACS IVIFNQCS ad IVIFNGCS measures. The results are show Table 8. As we ca see the best alteratve s A 1.Thatstosaythe optmal alteratve for the vestor s the chemcal compay. Depedg o the partcular cases of the IVIFOWCS measure used the orderg of the compaes s dfferet. Therefore whch vestmet wll be selected for the decso maker may also dffer. We establsh a orderg of the vestmets for some partcular case Table A Comparso Aalyss wth the Exstg Method Usg IFOWCS Measure. Furthermore order to demostrate the reasoablty of the IVIFOWCS measure we use the tutostc fuzzy ordered weghted cose smlarty (IFOWCS) measuretosolvethesamellustratveexample.forcomparso the trasformato from IVIFNs to IFNs s carred out by substtutg each terval value wth the mea value of ts upper ad lower lmts. The values of the llustrated example that have bee coverted are show Tables Zhou et al. [28] troduced the IFOWCS measure. Now weusetheifowcsmeasuretoaggregatethecollectvevalue S A of all the alteratve A. For comparso we stll assume

9 Complexty 9 Table 12: Characterstcs of the vestmets-expert 3. 1 ( ) ( ) ( ) ( ) ( ) ( ) 2 ( ) ( ) ( ) ( ) ( ) ( ) 3 ( ) ( ) ( ) ( ) ( ) ( ) 4 ( ) ( ) ( ) ( ) ( ) ( ) 5 ( ) ( ) ( ) ( ) ( ) ( ) 6 ( ) ( ) ( ) ( ) ( ) ( ) Table 13: Ideal strategy. Y (1) ( ) ( ) ( ) ( ) ( ) ( ) Y (2) ( ) ( ) ( ) ( ) ( ) ( ) Y (3) ( ) ( ) ( ) ( ) ( ) ( ) Table 14: Aggregated results. IFMAXCS IFMINCS IFNACS IFNQCS IFNGCS IFOWCS A A A A A A IFMAXCS IFMINCS IFNACS IFNQCS IFNGCS IFOWCS Table 15: Orderg of the strateges. Orderg Frst Secod Thrd Fourth Ffth Sxth Fgure 1: Varato of the collectve results of sx alteratves wth parameter λ. that the experts weghtg vector W = ( ) ad λ= 3 ad the dfferet partcular cases of the IFOWCS measure for the aggregato results are show Table 14. We establsh a orderg of the alteratves for each specal IFOWCS measure; the results are show Table 15. Obvously the frst alteratve each orderg s stll the optmal choce; ths reveals the valdty of the proposed method ths paper. Ad we see that each aggregato measure may also lead to dfferet results the IFOWCS measure. Itheedweaalyzehowthedfferetparametervalueλ plays a role the IVIFOWCS measure. Cosderg dfferet value of λ: thecollectveoverall values of sx alteratves are show Fgure 1.

10 10 Complexty 6. Coclusos I ths paper we preseted the IVIFOWCS measure by combg the terval-valued tutostc fuzzy cose smlarty measure wth the geeralzed OWA operator whch s very useful to deal wth the decso formato uder ucerta stuatos. Moreover we have studed some of ts ma propertes ad preseted a umercal example of the ew approach to see the applcato of the IVIFOWCS measure a vestmet decso-makg problem. The ma advatage of the IVIFOWCS measure provdes a parameterzed famly of aggregato operators ad smlarty measure. I addto the IVIFNs used ths paper are sutable for expressg evaluato; the decso maker ca use the IVIFOWCS measure to cosder a lot of possbltes ad select the aggregato operator that s accordace wth hs terests. I future research we expect to develop further extesos by addg ew characterstcs the problem such as probablstc aggregatos. Competg Iterests The authors declare o coflct of terests regardg the publcato for the paper. Ackowledgmets ThsresearchsfullysupportedbytheKeyIteratoal Collaborato Project of the Natoal Nature Scece Foudato of Cha (o ) a grat from Natoal Natural Scece foudato of Hua (2015JJ6041) Natoal Natural Scece Foudato of Cha ( ) ad Natoal Socal Scece Fud of Cha (15BTJ028). Refereces [1] Shy-Mg Che Measures of smlarty betwee vague sets Fuzzy Sets & Systemsvol.74o.2pp [2] S.-M. Che Smlarty measures betwee vague sets ad betwee elemets IEEE Trasactos o Systems Ma ad Cyberetcs Part B: Cyberetcsvol.27o.1pp [3] D. H. Hog ad C. Km A ote o smlarty measures betwee vague sets ad betwee elemets Iformato Scecesvol.115 o. 1 4 pp [4] S. Sat ad R. Ja Smlarty measures IEEE Trasactos o Patter Aalyss ad Mache Itellgece vol.21o.9pp [5] E. Szmdt ad J. Kacprzyk A ew cocept of a smlarty measure for tutostc fuzzy sets ad ts use group decso makg Proceedgs of the Iteratoal Coferece o Modelg Decsos for Artfcal Itellgece pp SprgerTsukubaJapaJuly2005. [6] E. Szmdt ad J. Kacprzyk A applcato of tutostc fuzzy set smlarty measures to a mult-crtera decso makg problem Proceedgs of the Iteratoal Coferece o Artfcal Itellgece ad Soft Computg pp August [7] H.-W. Lu New smlarty measures betwee tutostc fuzzy sets ad betwee elemets Mathematcal & Computer Modellgvol.42o.1-2pp [8] G. A. Papakostas A. G. Hatzmchalds ad V. G. Kaburlasos Dstace ad smlarty measures betwee tutostc fuzzy sets: a comparatve aalyss from a patter recogto pot of vew Patter Recogto Lettersvol.34o.14pp [9] D. L ad C. Cheg New smlarty measures of tutostc fuzzy sets ad applcato to patter recogtos Patter Recogto Lettersvol.23o.1 3pp [10] Z. Lag ad P. Sh Smlarty measures o tutostc fuzzy sets Patter Recogto Lettersvol.24o.15pp [11] W.-L. Hug ad M.-S. Yag Smlarty measures of tutostc fuzzy sets based o Hausdorff dstace Patter Recogto Lettersvol.25o.14pp [12] W.-L. Hug ad M.-S. Yag O smlarty measures betwee tutostc fuzzy sets Iteratoal Joural of Itellget Systemsvol.23o.3pp [13] A. Bhattacharyya O a measure of dvergece betwee two multomal populatos Sakhya vol. 7 o. 4 pp [14] J. Ye Cose smlarty measures for tutostc fuzzy sets ad ther applcatos Mathematcal & Computer Modellg vol. 53 o. 1-2 pp [15] K.T.Ataassov Itutostcfuzzysets Fuzzy Sets & Systems vol.20o.1pp [16] K. T. Ataassov ad G. Gargov Iterval valued tutostc fuzzy sets Fuzzy Sets & Systemsvol.31o.3pp [17] Z. Xu Some smlarty measures of tutostc fuzzy sets ad ther applcatos to multple attrbute decso makg Fuzzy Optmzato ad Decso Makg vol.6o.2pp [18] J. Ye Iterval-valued tutostc fuzzy cose smlarty measures for multple attrbute decso-makg Iteratoal Joural of Geeral Systemsvol.42o.8pp [19] R. R. Yager O ordered weghted averagg aggregato operators multcrtera decsomakg IEEE Trasactos o Systems Ma ad Cyberetcs vol.18o.1pp [20] J. M. Mergó ad M. Casaovas Decso makg wth dstace measures ad lgustc aggregato operators Iteratoal Joural of Fuzzy Systems vol. 12 o. 3 pp [21] G. We ad X. Zhao Some depedet aggregato operators wth 2-tuple lgustc formato ad ther applcato to multple attrbute group decso makg Expert Systems wth Applcatosvol.39o.5pp [22] L. Zhou ad H. Che A geeralzato of the power aggregato operators for lgustc evromet ad ts applcato group decso makg Kowledge-Based Systemsvol.26pp [23] L. Zhou ad H. Che The duced lgustc cotuous ordered weghted geometrc operator ad ts applcato to group decso makg Computers & Idustral Egeerg vol. 66 o. 2 pp [24] J. M. Mergó ad A. M. Gl-Lafuete Fuzzy duced geeralzed aggregato operators ad ts applcato mult-perso decso makg Expert Systems wth Applcatosvol.38o. 8 pp [25] Z. Xu Fuzzy ordered dstace measures Fuzzy Optmzato ad Decso Makgvol.11o.1pp [26] D.-F. L The GOWA operator based approach to multattrbute decso makg usg tutostc fuzzy sets Mathematcal ad Computer Modellg vol. 53 o. 5-6 pp

11 Complexty 11 [27] S. Zeg ad W. Su Itutostc fuzzy ordered weghted dstace operator Kowledge-Based Systemsvol.24o.8pp [28] L. Zhou Z. Tao H. Che ad J. Lu Itutostc fuzzy ordered weghted cose smlarty measure Group Decso & Negotatovol.23o.4pp [29] L. Zhou F. J H. Che ad J. Lu Cotuous tutostc fuzzy ordered weghted dstace measure ad ts applcato to group decso makg Techologcal ad Ecoomc Developmet of Ecoomyvol.22o.1pp [30] Y.LY.DegF.T.S.ChaJ.LuadX.Deg Amproved method o group decso makg based o terval-valued tutostc fuzzy prortzed operators Appled Mathematcal Modellg vol. 38 o pp [31] D. Yu Y. Wu ad T. Lu Iterval-valued tutostc fuzzy prortzed operators ad ther applcato group decso makg Kowledge-Based Systems vol. 30 o. 6 pp [32] L. Zhou Z. Tao H. Che ad J. Lu Cotuous tervalvalued tutostc fuzzy aggregato operators ad ther applcatos to group decso makg Appled Mathematcal Modellg. Smulato ad Computato for Egeerg ad Evrometal Systemsvol.38o.7-8pp [33] R. R. Yager O ordered weghted averagg aggregato operators multcrtera decso makg-readgs fuzzy sets for tellget systems Readgs Fuzzy Sets for Itellget Systemsvol.18o.1pp [34] J. M. Mergó ad A. M. Gl-Lafuete New decso-makg techques ad ther applcato the selecto of facal products Iformato Scecesvol.180o.11pp [35] L.-G. Zhou ad H.-Y. Che Geeralzed ordered weghted logarthm aggregato operators ad ther applcatos to group decso makg Iteratoal Joural of Itellget Systems vol.25o.7pp [36] L.ZhouH.CheadJ.Lu Geeralzedpoweraggregato operators ad ther applcatos group decso makg Computers & Idustral Egeergvol.62o.4pp [37] L.ZhouH.CheadJ.Lu Geeralzedmultpleaveragg operators ad ther applcatos to group decso makg Group Decso ad Negotatovol.22o.2pp [38] L. Zhou ad H. Che Geeralzed ordered weghted proportoal averagg operator ad ts applcato to group decso makg Iformatcavol.25 o.2pp [39] R. R. Yager Geeralzed OWA aggregato operators Fuzzy Optmzato ad Decso Makg vol.3o.1pp

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