Some Aggregation Operators For Bipolar-Valued Hesitant Fuzzy Information

Transcription

1 Some Aggregatio Operators For Bipolar-Valued Hesitat Fuzzy Iformatio Tahir Mahmood Departmet of Mathematics, Iteratioal Islamic Uiversity Islamabad, Pakista Florezti Smaradache Departmet of Mathematics, Uiversity of New Mexico, 705 Gurley Aveue, Gallup, NM 87301, USA Kifayat Ullah Departmet of Mathematics, Iteratioal Islamic Uiversity Islamabad, Pakista Qaisar Kha Departmet of Mathematics, Iteratioal Islamic Uiversity Islamabad, Pakista

2 Abstract I this article we defie some aggregatio operators for bipolar-valued hesitat fuzzy sets. These operatios icludes bipolar-valued hesitat fuzzy ordered weighted averagig (BPVHFOWA) operator alog with bipolarvalued hesitat fuzzy ordered weighted geometric (BPVHFOWG) operator ad their geeralized forms. We also defie hybrid aggregatio operators ad their geeralized forms ad solved a decisio makig problem o these operatio.

3 Keywords Bipolar-valued hesitat fuzzy sets (BPVHFSs) Bipolar-valued hesitat fuzzy elemets (BPVHFEs) BPVHFOWA operator BPVHFOWG operator BPVHFHA operator BPVHFHG operator Score fuctio ad decisio makig (DM).

4 Itroductio At ay level of our life decisio makig plays a essetial role. It is a very famous research field ow days. Everyoe eeds to take decisio about the selectio of best choice at ay stage of his life. Usig ordiary mathematical techiques we are uable to solve DM problems. To deal with problems related to differet kid of ucertaities, L. A. Zadeh [37] i 1965 iitiated the cocept of fuzzy sets (FSs). After his idea of FSs, researchers started to thik about differet extesios of FSs ad some advaced forms of FSs have bee established. Some of these extesios are itervalvalued fuzzy set (IVFS) [5], ituitioistic fuzzy set (IFS) [1], hesitat fuzzy set (HFS) [22] ad bipolar-valued fuzzy set (BVFS) [15] are some well kow sets. Later o these ew extesios of FSs have bee extesively used i decisio makig [5], [16].

5 As differet advaced forms of FSs came oe after aother, scietist started to merge two kids of fuzzy iformatio i a sigle set. The idea was quite useful ad some very iterestig extesios of FSs have bee defied. These extesios iclude ituitioistic hesitat fuzzy sets (IHFSs), iter-valued hesitat fuzzy sets (IVHFSs) ad bipolar-valued hesitat fuzzy sets (BVHFSs).The idea of mergig differet kid of fuzzy sets was quite useful ad very shortly some ew advaced forms of FSs have bee established which are itervalued ituitioistic hesitat fuzzy sets (IVIHFSs), cubic hesitat fuzzy sets (CHFSs) ad bipolar-valued hesitat fuzzy sets (BPVHFSs). Tahir M [25] itroduced BPVHFSs, a ew extesio of FSs ad a combiatio of HFSs ad BVFSs. BPVHFSs have affiliatio fuctios (membership fuctio) i terms of set of some values. The positive affiliatio fuctio is a set havig values i the iterval [0, 1] which coveys the satisfactio extet of a elemet belog to the give set. While the egative affiliatio fuctio is a set of some values i [ 1, 0] which coveys the egative or couter satisfactio degree of a elemet belog to give set.

6 Tahir M [25] defies some basic operatios for BPVHFSs ad proved some iterestig results. He also defies aggregatio operators for BPVHFSs ad the used these operators i DM. I our article we apply some order o previously defied bipolar-valued hesitat fuzzy weighted averagig ad weighted geometric operators by defiig bipolar-valued hesitat fuzzy ordered weighted averagig ad bipolar-valued hesitat fuzzy ordered weighted geometric operators alog with their geeralized operators. We also defied some hybrid aggregatio operators o BPVHFSs alog with their geeralized forms. Fially we did solve a DM problem usig these ewly defied aggregatio operators ad get very useful results. This article cosists of 4 sectios with sectio oe as itroductio. I sectio two we recall the defiitio of BPVHFSs, their properties ad some aggregatio operators of BPVHFSs. Sectio three cotai BPVHFOWA operators, BPVHFOWG operators, BPVHFHA operators ad BPVHFHG operators.

7 We also solve some examples o these defied operatios. I the last sectio we solve a DM problem usig the defied operatios i sectio three. Fially we fiish our article by addig a coclusio to it.

8 Prelimiaries This sectio cosists of the defiitio of BPVHFS ad some aggregatio operators o BPVHFSs. We also add some properties of BPVHFSs to this sectio ad we recall the cocept of score fuctio for BPVHFSs. Defiitio 1: [25] For ay set Ϫ, the BPVHFS B o some domai of Ϫ is deoted ad defied by: B = ϗ, (Њ + ϗ, Њ (ϗ)) : ϗ Ϫ Where Њ + : Ϫ [0,1] is a fiite set of few distict values i the iterval [0, 1]. It coveys the satisfactio extet of ϗ correspodig to BPVHFS B ad Њ : Ϫ [ 1,0] is a fiite set of few distict values i the iterval [-1, 0]. It coveys the implicit couter or egative property of ϗ correspodig to BPVHFS B. Here Њ = {Њ + ϗ, Њ (ϗ)} is a BPVHFE. The set of all BPVHFEs is deoted by Ф. Cosider two BPVHFSs: A = 8 4-Mar-17 B = ϗ, (Њ + A ϗ, (Њ A ϗ ) : ϗ Ϫ ϗ, (Њ + B ϗ, (Њ B ϗ ) : ϗ Ϫ

9 The set operatios for BPVHFSs are defied as: A B = Ϟ Њ A ϗ Њ B ϗ : (Њ + A Њ+ B) ϗ, (Њ A Њ B) ϗ. A B = Ϟ Њ A ϗ Њ B ϗ : (Њ + A Њ+ B) ϗ, (Њ A Њ B) ϗ. A c = ϗ, Њ + A ϗ c, Њ A ϗ c : ϗ Ϫ. Њ + A ϗ c = 1 Ϟ: Ϟ Њ + A ϗ, Њ A ϗ c = 1 Ϟ: Ϟ Њ A ϗ. A B (ϗ) = {Ϟ 1 + Ϟ 2 Ϟ 1 Ϟ 2 : Ϟ 1 Њ + A ϗ, Ϟ 2 Њ + B ϗ, Ϟ 1 Ϟ 2 : Ϟ 1 Њ A ϗ, Ϟ 2 Њ B ϗ }. A B (ϗ) = {Ϟ 1 Ϟ 2 Ϟ 1 Њ + A ϗ, Ϟ 2 Њ + B ϗ, ( Ϟ 1 Ϟ 2 Ϟ 1 Ϟ 2 ): Ϟ 1 Њ A ϗ, Ϟ 2 Њ B ϗ }. For ay > 0 A(ϗ) = {1 1 Ϟ : Ϟ Њ + A ϗ, ( Ϟ): Ϟ Њ A ϗ }. A ϗ = {Ϟ : Ϟ Њ + A ϗ, 1 1 Ϟ : Ϟ Њ A ϗ }. 9 4-Mar-17

10 Defiitio 2: [25] Let Њ į (į = 1,2,3,4 ) be a set of BPVHFEs ad let ω = ω 1, ω 2, ω 3, ω 4, ω T be the WV of Њ į (į = 1,2,3,4 ) with [0,1] ad σ = 1, the 1. BPVHFWA Operator is a fuctio Ψ Ψ such that BPVHFWA Њ 1, Њ 2, Њ = (Њ į ) : = 1 1 Ϟ į Ϟ į Њ + i, Ϟ į : Ϟ į Њ i 2. BPVHFWG Operator is a fuctio Ψ Ψ such that BPVHFWG Њ 1, Њ 2, Њ = Њ į = Ϟ į : Ϟ į Њ + i, 1 1 Ϟ į : Ϟ į Њ i 10 4-Mar-17

11 3. A GBPVHFWA Operator is a fuctio Ψ Ψ such that GBPVHFWA Њ 1, Њ 2, Њ = Њ į 1 = 1 1 Ϟ į 1 : Ϟ į Њ + i, 1 1 Ϟ į 1 : Ϟ į Њ i wįth > 0 4. A GBPVHFWG Operator is a fuctio Ψ Ψ such that GBPVHFWG Њ 1, Њ 2, Њ = 1 Њ į = 1 1 ς 1 1 Ϟ į 1 : Ϟ į Њ + į, ቐ 1 ൭ς ቆ ൬ Mar-17

12 Defiitio 3: [25] Let Њ =< Њ +, Њ > be a BPVHFE, the the Score fuctio (Accuracy Fuctio) of Њ is deoted ad defied by: S Њ = 1 l Њ (ξ Њ + + ξ Њ ) Where ξ Њ + is the sum of elemets of Њ + ad ξ Њ is the sum of elemets of Њ, l Њ is the legth of Њ ad S Њ [ 1, 1]. Remark 1: [25] Legth of Њ + ad Њ are ot ecessarily equal. For two BPVHFEs Њ 1 ad Њ 2 if S Њ 1 < S(Њ 2 ), the Њ 1 is said to be mior tha Њ 2 i.e. Њ 1 < Њ 2. S Њ 1 > S(Њ 2 ), the Њ 1 is said to bigger tha Њ 2 i.e. Њ 1 > Њ 2. S Њ 1 = S(Њ 2 ), the Њ 1 is idifferet (similar) to Њ 2 deoted by Њ 1 ~Њ Mar-17

13 Ordered Weighted Ad Hybrid Operators For BPVHFSS I this sectio we defie BPVHFOWA operators, BPVHFOWG operators, BPVHFHA operators ad BPVHFHG operators. We also explai these operatios with the help of examples. Defiitio 4: Let Њ į (į = 1,2,3,4 ) be a set of BPVHFEs ad Њ σ(į) the į th largest amog them. Let ω = ω 1, ω 2, ω 3, ω 4, ω T be the aggregatio associated weight vector of Њ į (į = 1,2,3,4 ) with [0,1] ad σ = 1. The 1. A BPVHFOWA operator is a fuctio BPVHFOWA: Ψ Ψ, such that BPVHFOWA Њ 1, Њ 2, Њ = (Њ σ(į) ) 2. A BPVHFOWG operator is a fuctio BPVHFOWG: Ψ Ψ, such that BPVHFWG Њ 1, Њ 2, Њ = Њ σ(į) 13 4-Mar-17

14 Theorem 1: Let Њ į (į = 1,2,3,4 ) be a set of BPVHFEs. The their aggregated value determied by usig BPVHFOWA operator or BPVHFOWG operator is a BPVHFE ad BPVHFOWA Њ 1, Њ 2, Њ = 1 : + 1 Ϟ σ(į) Ϟ σ(į) Њ σ į, Ϟ σ(į) : Ϟ σ(į) Њ σ(i) BPVHFWG Њ 1, Њ 2, Њ = + Ϟ σ į : Ϟ σ į Њ σ į, 1 1 Ϟ σ į Ϟ σ(į) Њ σ(į) 14 4-Mar-17

15 Defiitio 2: Let Њ į (į = 1,2,3,4 ) be a set of BPVHFEs ad Њ σ(į) the į th largest amog them. Let ω = ω 1, ω 2, ω 3, ω 4, ω T be the aggregatio associated weight vector of Њ į (į = 1,2,3,4 ) with [0,1] ad σ = 1. The A GBPVHFOWA operator is a fuctio GBPVHFOWA: Ψ Ψ, such that GBPVHFOWA Њ 1, Њ 2, Њ = = 1 1 Ϟ σ(į) Њ σ(į) : Ϟ σ(į) Њ σ(į) wįth > 0, 15 4-Mar-17

16 2. A GBPVHFOWG operator is a fuctio GBPVHFOWG: Ψ Ψ, such that GBPVHFOWG Њ 1, Њ 2, Њ = 1 1 = 1 Њ σ(į) 1 1 Ϟ σ(į) 1 wįth > 0 : Ϟ σ(į) Њ i +, 16 4-Mar-17

17 S Њ 3 = 1 l Њ 3 Clearly as So ξ + Њ3 + ξ Њ3 = = 0.7 S Њ 1 < S(Њ 3 ) < S(Њ 2 ) Њ σ 1 = Њ 2 = 0.5, 0.6, 0.2, 0.1, Њ σ 2 = Њ 3 = 0.9, 0.8, 0.2, 0.1 ad Њ σ 3 = Њ 1 = 0.1, 0.2, 0.3, 0.2 Now GBPVHFOWA 1 Њ 1, Њ 2, Њ 3 = 3 Њ σ į = , , , , , , , { , 0.2, 17 4-Mar-17

18 GBPVHFOWG 1 Њ 1, Њ 2, Њ 3 = 3 Њ σ(į) = , , , , , , , , { , 18 4-Mar-17

19 Defiitio 5: Let Њ į (į = 1,2,3,4 ) be a collectio of BPVHFEs, ω = ω 1, ω 2,, ω T is their weight vector with ω į 0,1 ad σ ω į = 1, is the balacig coefficiet ad ω = ω 1, ω 2, ω 3, ω 4, ω T be the aggregatio associated weight vector of Њ į (į = 1,2,3,4 ) with [0,1] ad σ = 1. The 1. The BPVHFHA operator is a mappig BPVHFHA: Ψ Ψ such that BPVHFHA Њ 1, where Њ 2, Њ = 1 = 1 Ϟ σ(į) : ω į Њ σ į Ϟ σ(į) Њ + σ į, Ϟ σ(į) : Њ σ į is the r th largest of Њ = w k Њ k (k = 1, 2, 3,, ) Ϟ σ(į) Њ σ(i) 2. The BPVHFHG operator is a mappig BPVHFHA: Ψ Ψ such that BPVHFHG here ሷ Њ 1, ሷ Њ 2, = ሷ Њ = Ϟሷ σ į : Ϟሷ σ į Њሷ + σ į ሷ Њ σ(į), 1 Њሷ σ į is the r th largest of Њ ሷ w = Њ k k (k = 1, 2, 3,, ) 19 1 ሷ Ϟ σ į : Ϟሷ σ(į) Њሷ σ(į)

20 3. A GBPVHFA operator is a fuctio GBPVHFHA: Ψ Ψ, such that GBPVHFHA here = 1 Њ 1, Њ 2, Њ = ω į Њ σ(į) 1 Ϟ σ(į) 1 : Ϟ σ(į) Њ + σ(į) 1, 1 wįth > 0 Њ σ į is the m th largest of Њ = w k Њ k (k = 1, 2, 3,, ) 1 Ϟ σ(į) 4. A GBPVHFHG operator is a fuctio GBPVHFHG: Ψ Ψ, such that GBPVHFHG here ሷ Њ 1, = 1 1 ሷ Њ σ į ሷ Њ 2, ሷ Њ = ሷ Ϟ σ(į) ሷ Њ σ(į) is the m th largest of ሷ Њ = Њ k w k 1 with > 0 : Ϟሷ σ(į) Њሷ + i, 1 k = 1, 2, 3,, 1 ሷ Ϟ σ į ρ 1 1 ω į : Ϟ σ(į) Њ σ(į) : Ϟሷ σ(į) Њሷ σ(į) Note: Examples o these operatios ca be solved i aalogous way. 20

21 Multi-Attribute Decisio Makig Based O Bipolar Valued Hesitat Fuzzy Sets: I this sectio we describe a brief techique to deal with MADM problems usig BPVHF aggregatio operator i which a DM provide iformatio i bipolar valued hesitat fuzzy decisio matrix ad every elemet is characterized by BPVHFE. Cosider that we have substitutes Ѧ į (į = 1,2,3,4, ) with m attributes ϗ j j = 1,2,3, m ad assume that w = (w 1, w 2,, w m ) be the weight vector such that w j 1, 1, j = 1,2,3 m ad σ j=1 ω j = 1. The decisio makers assiged values i the form of BPVHFEs (Њ įj ) for the substitutes Ѧ į uder the attributes ϗ j i the state of beig aoymous. The DM method is based o the followig steps: Step 1: This is the step of decisio matrix formatio as each alterative Ѧ į has assiged some values i the form of BPVHFEs (Њ įj ) uder some attributes ϗ j. Step 2: I this step, BPVHFE α į (į = 1,2,3,, ) ca be obtaied for the substitutes Ѧ į (į = 1,2,3,, ) by usig bipolar-valued hesitat fuzzy hybrid operators. Step 3: By applyig score fuctio (accuracy fuctio) we get the accuracy value S α į, į = 1,2,3,, of α į (į = 1,2,3,, 4).. 21

22 Step 4: To get the most suitable alterative Ѧ į (į = 1,2,3,, ), we established a order i the score values S α į, į = 1,2,3,,. Example 3: A cricket board eeds a head coach for their cricket team. The cricket board advertised the post i a ewspaper ad a umber of cadidates applied for the post. Based o their history i the cricket field iitially 4 cadidates are called for a iterview. The cricket board is goig to appoit a coach who possesses the qualities like hard workig, Creative, Committed ad skillful. For the better future of cricket i the coutry, the cricket board eeds to appoit the most suitable head coach. Let Ѧ = {Ѧ 1, Ѧ 2, Ѧ 3, Ѧ 4 } be the set of substitutes ad Ϫ = {ϗ 1, ϗ 2, ϗ 3, ϗ 4 } be the set of attributes ad let w = 0.25, 0.23, 0.35,0.17 T be the weight vector of the attributes Ϫ į (į = 1,2,3,4) ad ω = 0.3, 0.4, 0.2, T be the aggregatio associated weight vector. 22

23 Coclusio I our treatise we successfully apply aggregatio operators of BPVHFSs i a DM problem. The results clearly idicate that either we use BPVHF hybrid averagig operators or BPVHF hybrid geometric operators, we get the same results. Hece these two aggregatio operatio ca be very useful i DM especially i two-sided DM. Future research will ivolve the geeralizatio to bipolar-valued hesitat eutrosophic iformatio ad its aggregatio operators. 23

24 Refereces [1] Ataassov K, ituitioistic fuzzy sets, Fuzzy Sets ad Systems, 20 (1986) [2] Ataassov K. ad Gargov G, iterval valued ituitioistic fuzzy sets, fuzzy sets ad systems, 31 (1989) [3] Chauha A. ad Vaish R, a comparative study o decisio makig methods with Iterval Data, Hidawi Publishig Corporatio Joural of Computatioal Egieerig (2014). [4] Che T, A comparative aalysis of score fuctios for multiple criteria decisio-makig i ituitioistic fuzzy settigs, Iformatio Scieces, 181 (2011) [5] Che N, Xu Z. S, ad Xia M, Iterval-valued hesitat preferece relatios ad their applicatios to group decisio makig, Kowledge-Based Systems, 37 (2013) [6] Che N, Xu Z. S, properties of iterval-valued hesitat fuzzy sets, joural of itelliget ad fuzzy systems (2014). [7] De S. K, Biswas R ad Roy A.R, Some operatios o ituitioistic fuzzy sets, Fuzzy Sets ad Systems (2000) [8] Dubois D, Prade H, Fuzzy sets ad systems: theory ad applicatios, Academic Press NewYork [9] Dubois D, Prade H, Iterval-valued Fuzzy Sets, Possibility Theory ad Imprecise Probability, Istitute of research i computer sciece from Toulouse Frace. [10] Farhadiia B, iformatio measures for hesitat fuzzy sets ad iterval-valued hesitat fuzzy sets, Iformatio Scieces [11] Herrera-Viedma, Aloso E, Chiclaa S, Herrera F, A cosesus model for group decisio makig with icomplete fuzzy preferece relatios, IEEE Trasactios o Fuzzy Systems

25 [12] Hog, D. H. ad Choi, Multi-criteria fuzzy decisio makig problems based o vague set theory, Fuzzy Sets ad Systems, 21, [13] Hwag C.L, ad Yoo K, Multiple Attribute Decisio Makig Methods ad Applicatio Spriger New York [14] Klutho S, Mathematical Decisio Makig a Overview of the Aalytic Hierarchy Process [15] Lee K. M, Bipolar-Valued Fuzzy sets ad their operatios, Proc. It. Cof. o itelliget Techologies, Bagkok, Thailad, 2000, [16] Li D, Multiattribute decisio makig models ad methods usig ituitioistic fuzzy sets, Joural of Computer ad System Scieces, 70, [17] Liu, H.W ad Wag G.J, multi-criteria decisio makig methods based o ituitioistic fuzzy sets, Europea joural of operatioal research [18] Muhammad A, Saleem A ad Kifayat U. Bipolar Fuzzy Soft Sets ad its applicatio i decisio makig problem, Joural of itelliget ad fuzzy system, 27 (2014) [19] Pohekar S. D ad Ramachadra M, Applicatio of multi- criteria decisio makig to sustaiable eergy plaig a review, Reewable ad Sustaiable Eergy Reviews [20] Rui D. S. N. ad Pierre L, Bipolarity i huma reasoig ad effective decisio makig, iteratioal joural of Itelliget Systems, 23 (2008) [21] Shabir M, Kha I. A, Iterval-valued fuzzy ideals geerated by a iterval valued fuzzy subset i ordered semigroups, Mathware ad soft computig [22] Torra V, Narukawa Y, O hesitat fuzzy sets ad decisio, I The 18th IEEE Iteratioal Coferece o Fuzzy Systems, Jeju Islad, Korea [23] Torra V, Hesitat fuzzy sets, Iteratioal Joural of Itelliget Systems [24] Tahir M, Muhammad M, O bipolar-valued fuzzy subgroups, World Applied Scieces Joural 27(12) (2013)

26 [25] Tahir M, Kifayat U, Qaiser K, bipolar-valued hesitat fuzzy sets ad their applicatios i multiattribute decisio makig, iteratioal joural of algebra ad statistics, Submitted. [26] Wag W ad Xi X, Distace measure betwee ituitioistic fuzzy sets, Patter Recogitio Letters, 26 (13) [27] Wu D, Medel J. M, A comparative study of rakig methods, similarity measures ad ucertaity measures for iterval type-2 fuzzy sets, Iformatio Scieces [28] Xia M, Xu Z.S, Hesitat fuzzy iformatio aggregatio i decisio makig, iteratioal joural of approximate reasoig [29] Xia M. M, Xu Z. S, Che, N, Some hesitat fuzzy aggregatio operators with their applicatio i group decisio makig, Group Decisio ad Negotiatio [30] Xia M. M, Xu Z. S, Maagig hesitat iformatio i GDM problems uder fuzzy ad multiplicative preferece relatios, Iteratioal Joural of Ucertaity, Fuzziess ad Kowledgebased Systems [31] Xu Z. S, A method based o liguistic aggregatio operators for group decisio makig with liguistic preferece relatios, Iformatio Scieces [32] Xu Z. S, Yager R. R, Some geometric aggregatio operators based o ituitioistic fuzzy sets, Iteratioal Joural of Geeral Systems 35, [33] Xu Z. S, Ituitioistic fuzzy aggregatio operators, IEEE Trasactios o Fuzzy Systems [34] Xu Z ad Che J, a approach to group decisio makig based o iterval-valued ituitioistic fuzzy judgmet matrices, System Egieer-Theory ad Practice [35] Ye J, Improved method of multicriteria fuzzy decisio-makig based o vague sets, Computer-Aided Desig 39 (2) [36] Ye J, Multicriteria fuzzy decisio-makig method based o a ovel accuracy fuctio uder iterval-valued ituitioistic fuzzy eviromet, Expert Systems with Applicatios vol. 36 o [37] Zadeh L, Fuzzy sets, Iformatio ad Cotrol 8 (3)

27 [38] Zadeh L, The cocept of a liguistic variable ad its applicatio to approximate reasoig-i, Iformatio Scieces vol. 8 (3) [39] Zhag X, Liu P. D, Method for multiple attribute decisio-makig uder risk with iterval umbers, Iteratioal Joural of Fuzzy Systems [40] Zhag Z, iterval-valued ituitioistic hesitat fuzzy aggregatio operators ad their applicatio i group decisio-makig, joural of applied mathematics [41] Zheg P, A ote o operatios of hesitat fuzzy sets, Iteratioal Joural of Computatioal Itelligece Systems, 8(2) [42] Zhu B, Xu Z. S ad Xia M. M, Hesitat fuzzy geometric Boferroi meas, Iformatio Scieces