https://doi.org/10.31449/inf.v45i3.3025 Informatica 45 (2021) 447–461 447 Some Picture Fuzzy Aggregation Operators Based on Frank t-norm and t-conorm: Application to MADM Process Mijanur Rahaman Seikh and Utpal Mandal Department of Mathematics, Kazi Nazrul University, Asansol-713 340, India E-mail: mrseikh@ymail.com, utpalmandal2204@gmail.com Keywords: picture fuzzy sets, frank operations, picture fuzzy frank aggregation operator, decision support system Received: December 11, 2019 In this paper, we develop some new operational laws and their corresponding aggregation operators for picture fuzzy sets (PFSs). The PFS is a powerful tool to deal with vagueness, which is a generalization of a fuzzy set and an intuitionistic fuzzy set (IFS). PFSs can model uncertainty in situations that consist of more than two answers like yes, refusal, neutral, and no. The operations of t-norm and t-conorm, developed by Frank, are usually a better application with its flexibility. From that point of view, the concepts of Frank t-norm and t-conorm are introduced to aggregate picture fuzzy information. We propose some new operational laws of picture fuzzy numbers (PFNs) based on Frank t-norm and t-conorm. Further, with the assistance of these operational laws, we have introduced picture fuzzy Frank weighted averaging (PFFWA) operator, picture fuzzy Frank order weighted averaging (PFFOWA) operator, picture fuzzy Frank hybrid averaging (PFFHA) operator, picture fuzzy Frank weighted geometric (PFFWG) operator, picture fuzzy Frank order weighted geometric (PFFOWG) operator, picture fuzzy Frank hybrid geometric (PFFHG) operator and discussed with their suitable properties. Then, with the help of PFFWA and PFFWG operators, we have presented an algorithm to solve multiple-attribute decision making (MADM) problems under the picture fuzzy environment. Finally, we have used a numerical example to illustrate the flexibility and validity of the proposed method and compared the results with other existing methods. Povzetek: Prispevek se ukvarja z operatorji mehkih množic na osnovi Frankove t-norme in t-konorme. 1 Introduction In real-life situations, the fuzzy set theory [48] plays a vital role in handling the vagueness of human choices. Then continuous efforts are paid for further generalization of fuzzy set theory. The IFS theory is one of such general- izations, introduced by Atanssov [2]. IFS is characterized by a degree of membership and degree of non-membership such that their sum does not exceed one. However, IFSs are insufficient to handle the possibility with more than two an- swers as just yes-no type. Consider the case of usual voting where one has the choices like a vote for, vote against, ab- stain from voting, and refusal. To deal with such situations with high accuracy, Cuong and Krienovich [8] conveyed a novel concept of picture fuzzy set (PFS). PFS is character- ized by a membership degree, a non-membership degree, and a neutrality degree such that their sum is less than or equal to one. Cuong [9] examined few properties of PFSs and introduced distance measures between PFSs. Recently, some research models have been developed on the picture fuzzy (PF) environment. Dinh and Thao [10] introduced some distance measures and dissimilarity mea- sures between PFSs and applied them to MADM problems. Wang and Li [33] proposed a hesitant fuzzy set in the PF environment and developed picture hesitant fuzzy aggre- gation operators. Wei [39] extended the TODIM method to MADM problems under the PF environment. Wei [40] developed some similarity measures between PFSs such as cosine measure, set-theoretic cosine similarity measure, grey similarity measure and applied these to building ma- terial recognition and mineral field recognition. Dutta and Ganju [11] introduced decomposition theorems of PFSs and defined the extension principle for PFSs. Wei [35] introduced PF cross-entropy as an extension of the cross- entropy of fuzzy sets. Xu et al. [45] developed some aggre- gation operators for fusing PF information. Dutta [12] ap- plied distance measure between PFSs in medical diagnosis. Singh [27] proposed correlation coefficients for PFSs and gave the geometrical interpretation for PFSs. Son [28] and Thong [29] introduced several clustering algorithms with PFSs. Le et al. [21] proposed some dissimilarity measures under PF information and applied them to decision-making problems. Wei et al. [36] introduced the projection mod- els for the MADM problem with PF information. Wei and Gao [42] developed the generalized dice similarity mea- sure under PF environment and applied them to building material recognition. Zeng et al. [50] proposed the ex- ponential Jensen PF divergence measure and applied it in multi-criteria group decision making. Several researchers proposed information aggregation operators under the PF environment [3, 16, 17, 22, 30, 32, 43, 51]. Garg [14] pre- sented some PF aggregation operators and applied them to 448 Informatica 45 (2021) 447–461 M.R. Seikh et al. multi-criteria decision making. Wei [38] presented cosine similarity measures for PFS and applied them to strategic decision making. Wei [41] proposed PF Hamacher aggre- gation operators and applied them to the MADM process. Khan et al. [18] investigated the information aggregation operators method under the PF environment with the help of Einstein norm operations. Khan et al. [19] introduced a series of logarithmic aggregation operators under the PF environment. Wang et al. [34] proposed Muirhead mean operators under PF environment and applied them to eval- uate the financial investment risk. A fascinating generalization of probabilistic and Lukasiewicz t-norm and t-conorm [23] are Frank t-norm and t-conorm [13], which form an ordinary and adequately flexible family of the continuous triangular norm. The employment of a specific parameter makes the Frank t- norm and t-conorm more resilient along with the proce- dure of fusion of information. Several works [1, 20] can be found in the literature related to Frank t-norm and t- conorm. The functional equations of Frank and Alsina are thoroughly studied by Calvo et al. [4] for two classes con- taining commutative, associative, and increasing binary op- erators. Exploring the additive generating function (AGF) of Frank t-norms, Yager [46] launched a framework in ap- proximate reasoning with Frank t-norms. Casasnovas and Torrens [5] introduced a novel axiomatic approach to the scalar cardinality of Frank t-norms, and they further estab- lished the properties of other standard t-norms. Compar- ing between the Frank t-norms and the Hamacher t-norms up to an extent, Sarkoci [26] concluded that two different t-norms belong to the same family. Xing et al. [44] in- troduced aggregation operators for Pythagorean fuzzy set based on Frank t-norm and t-conorm and then applied them to solve MADM problems. Zhou et al. [52] investigated some Frank aggregation operators of interval-valued neu- trosophic numbers and analyzed a case study of select- ing agriculture socialization. Qin and Liu [24] introduced Frank aggregation operators for a triangular interval type- 2 fuzzy set and applied it to solve multiple attribute group decision making (MAGDM) problems. Qin et al. [25] de- veloped some hesitant fuzzy aggregation operators based on Frank t-norm operations. Evidently, a general t-norm and t-conorm can be used for shaping both the intersection and union of PFS. The PFS is compatible to reveal uncertain information. Since the Frank aggregation operators involve a parameter so the operators make the information process more flexible and strong. The investigation on the applications of Frank op- erators is rare, specifically in the area of information ag- gregation and decision making. Keeping this in mind, it is worthy to prolong Frank t-norm and t-conorm to handle the PF environment. With such motivation of aforemen- tioned analysis, we have introduced new operational rules of PFNs based on Frank operators and exhibited their char- acteristics. In this paper, we have introduced some new opera- tional laws for PFNs based on Frank t-norm and Frank t- conorm. Then using these operational laws, we have devel- oped Frank t-norm and t-conorm based PFFWA, PFFOWA, PFHWA, PFFWG, PFFOWG, and PFFHG aggregation op- erators. We have also investigated some of their desirable properties. Utilizing PFFWA and PFFWG operators, we have developed an algorithm to solve an MADM problem under the PF environment. To illustrate the validity and su- periority of the proposed method, a numerical example is considered, solved, and the obtained results are compared with other existing well-known methods. The rest of the paper is organized as follows. In Section 2, some basic definitions and preliminaries are recalled, which help us to make the concept about the present article. In Section 3, some new operational laws for PFNs based on Frank t-norm and t-conorm have been proposed, and using those operational laws, some new ag- gregation operators are defined in the PF environment. An algorithm to solve the decision-making problems based on Frank aggregation operators is presented in Section 4. In Section 5, we have checked the validity of the proposed method through a real-life example. Section 6 analyze the effect of the parameters on the decision-making result. Sec- tion 7 presents a useful comparison between the results of our proposed method and other significant models. Finally, the conclusion is made in Section 8. 2 Preliminaries In this section, we recall some basic definitions and prelim- inaries. Definition 2.1. [6, 7] Let us considerX as a universal set. The PFS ~ P over the universal setX is interpreted as ~ P =fhx; ~ P (x); ~ P (x); ~ P (x)ijx2Xg where ~ P : X ! [0; 1]; ~ P : X ! [0; 1] and ~ P : X! [0; 1] are called the positive degree of membership, neutral degree of membership and the negative degree of membership to the set ~ P respectively, with the condition 0 ~ P (x) + ~ P (x) + ~ P (x) 1 for everyx2X. Also the degree of hesitancy forx2 X is defined as ~ P (x) = 1 ~ P (x) ~ P (x) ~ P (x): For our convenience, we denote p = ( p ; p ; p ) as a picture fuzzy number (PFN). Definition 2.2. [6, 37] Let p = ( p ; p ; p ) and q = ( q ; q ; q ) be two PFNs over the universal set X and > 0 be any real number, then the corresponding oper- ations are defined as follows: 1.p q; if p q ; p q and p q 2.p W q=(maxf p ; q g;minf p ; q g;minf p ; q g): 3.p V q=(minf p ; q g;maxf p ; q g;maxf p ; q g): 4.p c = ( p ; p ; p ): 5.p^q=(minf p ; q g;maxf p ; q g;maxf p ; q g): 6.p_q=(maxf p ; q g;minf p ; q g;minf p ; q g): 7.p q = ( p + q p q ; p q ; p q ): 8.p q = ( p q ; p + q p q ; p + q p q ): 9.p = (1 (1 p ) ; p ; p ): 10.p = ( p ; 1 (1 p ) ; 1 (1 p ) ): Some Picture Fuzzy Aggregation Operators Based on. . . Informatica 45 (2021) 447–461 449 Definition 2.3. [16] The score function of the PFN p = ( p ; p ; p ) is defined by ( p) = 1 + p p 2 where ( p)2 [0; 1]: Definition 2.4. [16] The accuracy function of the PFNp = ( p ; p ; p ) is defined by r(p) = p + p where ( p)2 [ 1; 1]: According to Definitions 2.3 and 2.4, ifp = ( p ; p ; p ) andq = ( q ; q ; q ) be any two PFNs then 1. If ( p)> ( q) thenp>q; 2. If ( p)< ( q) thenpr(q); thenp>q; – Ifr(p) =r(q); thenp =q: Wei [37] introduced the PF aggregation operators depicted in the upcoming definitions. Definition 2.5. Letp i = ( pi ; pi ; pi ) (i = 1; 2;:::;n) be a number of PFNs. Then the aggregated value of them using PF weighted averaging (PFWA) operator is also a PFN and PFWA(p 1 ;p 2 ;:::;p n ) = n L i=1 (w i p i ) = 1 n Q i=1 (1 pi ) wi ; n Q i=1 pi wi ; n Q i=1 pi wi ; where w = (w 1 ;w 2 ;:::;w n ) t be the weight vector of p i (i = 1; 2;:::;n);w i 2 [0; 1] and n P i=1 w i = 1. Definition 2.6. Letp i = ( pi ; pi ; pi ) (i = 1; 2;:::;n) be a number of PFNs. The PF order weighted averaging (PFOWA) operator of dimension n is a function p n ! p such that, PFOWA(p 1 ;p 2 ;:::;p n ) = n L i=1 (w i p (i) ) = 1 n Q i=1 (1 p (i) ) wi ; n Q i=1 p (i) wi ; n Q i=1 p (i) wi ; where w = (w 1 ;w 2 ;:::;w n ) t be the weight vector of p i (i = 1; 2;:::;n); w i 2 [0; 1] and n P i=1 w i = 1, ( (1); (2);:::; (n)) is the permutation of (i = 1; 2;:::;n); for which p (i 1) p (i) for all i = 1; 2;:::;n: In the following, we recall the definition of Frank t-norm and t-conorm. Definition 2.7. [13] Let us assume that a and b be two real numbers. Then, Frank t-norm and Frank t-conorm are defined by, Fra(a;b)=log r 1 + (r a 1)(r b 1) r 1 Fra 0 (a;b)=1 log r 1 + (r 1 a 1)(r 1 b 1) r 1 respectively, where (a;b)2 [0; 1] [0; 1] andr6= 1: Based on limit theory, we observe some interesting re- sults [31]: 1. If r ! 1; then Fra 0 (a;b) ! a + b ab and Fra(a;b)! ab. Therefore, if r tends to 1 the the Frank sum and Frank product reduced to the proba- bilistic sum and probabilistic product. 2. If r ! 1; then Fra 0 (a;b) ! minfa +b; 1g and Fra(a;b)! maxf0;a +b 1g. So, forr tends to infinity the Frank sum and the Frank product reduced to the Lukasiewicz sum and Lukasiewicz product. EXAMPLE 1. Leta = 0:29,b = 0:56 andr = 4, then, Fra(0:29; 0:56) = log 4 1 + (4 0:29 1)(4 0:56 1) 4 1 =0.1276. Fra 0 (0:29; 0:56)= 1 log 4 1 + (4 1 0:29 1)(4 1 0:56 1) 4 1 =0.8723. 3 Picture fuzzy Frank aggregation operators In this section, we develop some operational rules under the PF environment with the assistance of Frank t-norm and t-conorm. Further, we propose the PFFWA, PFFOWA, PFFHWA, PFFWG, PFFOWG and PFFHWG aggregation operators using our developed operational rules. Definition 3.1. Letp = ( p ; p ; p ),p 1 = ( p1 ; p1 ; p1 ) and p 2 = ( p2 ; p2 ; p2 ) be any three PFNs, r > 1 and > 0 be any real number. Then Frank t-norm and t-conorm operations of PFNs are defined as: 1. p 1 p 2 = 1 log r 1 + (r 1 p 1 1)(r 1 p 2 1) r 1 ; log r 1 + (r p 1 1)(r p 2 1) r 1 ; log r 1 + (r p 1 1)(r p 2 1) r 1 ! : 2. p 1 p 2 = log r 1 + (r p 1 1)(r p 2 1) r 1 ; 1 log r 1 + (r 1 p 1 1)(r 1 p 2 1) r 1 ; 1 log r 1 + (r 1 p 1 1)(r 1 p 2 1) r 1 ! : 3. p = 1 log r 1 + (r 1 p 1) (r 1) 1 ; log r 1 + (r p 1) (r 1) 1 ; log r 1 + (r p 1) (r 1) 1 ! : 450 Informatica 45 (2021) 447–461 M.R. Seikh et al. 4. p = log r 1 + (r p 1) (r 1) 1 ; 1 log r 1 + (r 1 p 1) (r 1) 1 ; 1 log r 1 + (r 1 p 1) (r 1) 1 ! : EXAMPLE 2. Let p 1 = (0:60; 0:20; 0:08) and p 2 = (0:50; 0:20; 0:15) be two PFNs, then by using Frank op- erations on PFNs as defined in Definition 3.1, for r = 3 and = 4 we have 1. p 1 p 2 =(0:7325; 0:0270; 0:0074). 2. p 1 p 2 =(0:2674; 0:9729; 0:9925): 3. 4p 1 =(0:9947; 0:0002; 0): 4. p 1 4 =(0:0421; 0:7999; 0:5819): THEOREM 3.1. Letp = ( p ; p ; p ),p 1 = ( p1 ; p1 ; p1 ) and p 2 = ( p2 ; p2 ; p2 ) be any three PFNs, r > 1 and ; 1 ; 2 be any three positive real numbers, then we have 1. p 1 p 2 =p 2 p 1 ; 2. p 1 p 2 =p 2 p 1 ; 3. (p 1 p 2 ) =p 1 p 2 ; 4. 1 p 2 p = ( 1 + 2 )p; 5. (p 1 p 2 ) =p 1 p 2 ; 6. p 1 p 2 =p 1+ 2 : Proof: For three PFNs p = ( p ; p ; p ), p 1 = ( p1 ; p1 ; p1 ) andp 2 = ( p2 ; p2 ; p2 ) and; 1 ; 2 > 0, according to Definition 3.1, we can obtain 1. p 1 p 2 = 1 log r 1 + (r 1 p 1 1)(r 1 p 2 1) r 1 ; log r 1 + (r p 1 1)(r p 2 1) r 1 ; log r 1 + (r p 1 1)(r p 2 1) r 1 ! = 1 log r 1 + (r 1 p 2 1)(r 1 p 1 1) r 1 ; log r 1 + (r p 2 1)(r p 1 1) r 1 ; log r 1 + (r p 2 1)(r p 1 1) r 1 ! =p 2 p 1 : 2. p 1 p 2 = log r 1 + (r p 1 1)(r p 2 1) r 1 ; 1 log r 1 + (r 1 p 1 1)(r 1 p 2 1) r 1 ; 1 log r 1 + (r 1 p 1 1)(r 1 p 2 1) r 1 ! = log r 1 + (r p 2 1)(r p 1 1) r 1 ; 1 log r 1 + (r 1 p 2 1)(r 1 p 1 1) r 1 ; 1 log r 1 + (r 1 p 2 1)(r 1 p 1 1) r 1 ! =p 2 p 1 : 3. (p 1 p 2 )= 1 log r 1 + (r 1 p 1 1)(r 1 p 2 1) r 1 ; log r 1 + (r p 1 1)(r p 2 1) r 1 ; log r 1 + (r p 1 1)(r p 2 1) r 1 ! = 1 log r 1 + (r 1 p 1 1) (r 1 p 2 1) (r 1) 2 1 ; log r 1 + (r p 1 1) (r p 2 1) (r 1) 2 1 ; log r 1 + (r p 1 1) (r p 2 1) (r 1) 2 1 ! : Now, p 1 p 2 = 1 log r 1 + (r 1 p 1 1) (r 1) ; log r 1 + (r p 1 1) (r 1) ; log r 1 + (r p 1 1) (r 1) ! 1 log r 1 + (r 1 p 2 1) (r 1) ; log r 1 + (r p 2 1) (r 1) ; log r 1 + (r p 2 1) (r 1) ! = 1 log r 1 + (r 1 p 1 1) (r 1 p 2 1) (r 1) 2 1 ; log r 1 + (r p 1 1) (r p 2 1) (r 1) 2 1 ; log r 1 + (r p 1 1) (r p 2 1) (r 1) 2 1 ! : Therefore, (p 1 p 2 ) =p 1 p 2 : 4. 1 p 2 p = 1 log r 1 + (r 1 p 1) 1 (r 1) 1 ; log r 1 + (r p 1) 1 (r 1) ; log r 1 + (r p 1) 1 (r 1) ! 1 log r 1 + (r 1 p 1) 2 (r 1) 2 ; Some Picture Fuzzy Aggregation Operators Based on. . . Informatica 45 (2021) 447–461 451 log r 1 + (r p 1) 2 (r 1) 2 ; log r 1 + (r p 1) 2 (r 1) 2 ! = 1 log r 1 + (r 1 p 1) 1+ 2 (r 1) 1+ 2 ; log r 1 + (r p 1) 1+ 2 (r 1) 1+ 2 ; log r 1 + (r p 1) 1+ 2 (r 1) 1+ 2 ! = ( 1 + 2 )p: 5. (p 1 p 2 ) = log r 1 + (r p 1 1)(r p 2 1) r 1 ; 1 log r 1 + (r 1 p 1 1)(r 1 p 2 1) r 1 ; 1 log r 1 + (r 1 p 1 1)(r 1 p 2 1) r 1 ! = log r 1 + ((r p 1 1)(r p 2 1)) (r 1) 2 1 ; 1 log r 1 + ((r 1 p 1 1)(r 1 p 2 1)) (r 1) 2 1 ; 1 log r 1 + ((r 1 p 1 1)(r 1 p 2 1)) (r 1) 2 1 ! = log r 1 + (r p 1 1) (r 1) ; 1 log r 1 + (r 1 p 1 1) (r 1) ; 1 log r 1 + (r 1 p 1 1) (r 1) ! log r 1 + (r p 2 1) (r 1) ; 1 log r 1 + (r 1 p 2 1) (r 1) ; 1 log r 1 + (r 1 p 2 1) (r 1) ! =p 1 p 2 : 6. p 1 p 2 = log r 1 + (r p 1) 1 (r 1) 1 1 ; 1 log r 1 + (r 1 p 1) 1 (r 1) 1 1 ; 1 log r 1 + (r 1 p 1) 1 (r 1) 1 1 ! log r 1 + (r p 1) 2 (r 1) 2 1 ; 1 log r 1 + (r 1 p 1) 2 (r 1) 2 1 ; 1 log r 1 + (r 1 p 1) 2 (r 1) 2 1 ! = log r 1 + (r p 1) 1+ 2 (r 1) 1+ 2 1 ; 1 log r 1 + (r 1 p 1) 1+ 2 (r 1) 1+ 2 1 ; 1 log r 1 + (r 1 p 1) 1+ 2 (r 1) 1+ 2 1 ! =p 1+ 2 : 3.1 Picture fuzzy Frank arithmetic aggregation operators Definition 3.2. Let p i = ( pi ; pi ; pi )(i = 1; 2;:::;n) be a number of PFNs. Then PFFWA operator is a function p n !p such that, PFFWA(p 1 ;p 2 ;:::;p n ) = n M i=1 w i p i where w = (w 1 ;w 2 ;:::;w n ) t be the weight vector of p i (i = 1; 2;:::;n);w i 2 [0; 1] and n P i=1 w i = 1. Hence, we get consequential theorem that follows the Frank operations on PFNs. THEOREM 3.2. Let p i = ( pi ; pi ; pi )(i = 1; 2;:::;n) be a number of PFNs, then aggregated value of them using PFFWA operator is also a PFN, and PFFWA(p 1 ;p 2 ;:::;p n ) = n L i=1 w i p i = 1 log r 1 + n Q i=1 (r 1 p i 1) wi ; log r 1 + n Q i=1 (r p i 1) wi ; log r 1 + n Q i=1 (r p i 1) wi ! . Proof: We prove this theorem by the method of mathe- matical induction. Forn = 2, based on Frank operations of PFNs we get the corresponding result PFFWA(p 1 ;p 2 ) = 2 L i=1 w i =w 1 p 1 w 2 p 2 = 1 log r 1 + (r 1 p 1 1) w1 (r 1) w1 1 ; log r 1 + (r p 1 1) w1 (r 1) w1 1 ; log r 1 + (r p 1 1) w1 (r 1) w1 1 ! L 1 log r 1 + (r 1 p 2 1) w2 (r 1) w2 1 ; log r 1 + (r p 2 1) w2 (r 1) w2 1 ; log r 1 + (r p 2 1) w2 (r 1) w2 1 ! 452 Informatica 45 (2021) 447–461 M.R. Seikh et al. = 1 log r 1 + 2 Q i=1 (r 1 p i 1) wi ; log r 1 + 2 Q i=1 (r p i 1) wi ; log r 1 + 2 Q i=1 (r p i 1) wi ! [* 2 P i=1 w i = 1] Hence, the result is valid forn = 2. Let us assume that, the given result is true for n = s: Therefore, we have, PFFWA(p 1 ;p 2 ;:::;p s ) = s L i=1 w i p i = 1 log r 1 + s Q i=1 (r 1 p i 1) wi ; log r 1 + s Q i=1 (r p i 1) wi ; log r 1 + s Q i=1 (r p i 1) wi ! Now, forn =s + 1 PFFWA(p 1 ;p 2 ;:::;p s ;p s+1 ) = s+1 L i=1 w i p i = s L i=1 w i p i L w s+1 p s+1 = 1 log r 1 + s Q i=1 (r 1 p i 1) wi (r 1) s P i=1 wi 1 ! ; log r 1 + s Q i=1 (r p i 1) wi (r 1) s P i=1 wi 1 ! ; log r 1 + s Q i=1 (r p i 1) wi (r 1) s P i=1 wi 1 !! L 1 log r 1 + (r 1 p s+1 1) ws+1 (r 1) ws+1 1 ; log r 1 + (r p s+1 1) ws+1 (r 1) ws+1 1 ; log r 1 + (r p s+1 1) ws+1 (r 1) ws+1 1 ! = 1 log r 1 + s+1 Q i=1 (r 1 p i 1) wi ; log r 1 + s+1 Q i=1 (r p i 1) wi ; log r 1 + s+1 Q i=1 (r p i 1) wi ! [as s+1 P i=1 w i = 1] Therefore, the result is true forn =s + 1 if it is true for n = s. Also it is true forn = 2. Hence, by the method of induction the given result is true for any natural numbern. THEOREM 3.3. (Idempotency Property). If p i = ( pi ; pi ; pi )(i = 1; 2;:::;n) be a number of identical PFNs, i.e., p i = p for all i, wherep = ( p ; p ; p ), then PFFWA(p 1 ;p 2 ;:::;p n ) =p: Proof: Sincep i =p for alli then, we have PFFWA(p 1 ;p 2 ;:::;p n ) = 1 log r 1 + n Q i=1 (r 1 p i 1) wi ; log r 1 + n Q i=1 (r p i 1) wi ; log r 1 + n Q i=1 (r p i 1) wi ! = 1 log r 1 + n Q i=1 (r 1 p 1) wi ; log r 1 + n Q i=1 (r p 1) wi ; log r 1 + n Q i=1 (r p 1) wi ! = 1 log r 1 + (r 1 p 1) n P i=1 wi ; log r 1 + (r p 1) n P i=1 wi ; log r 1 + (r p 1) n P i=1 wi ! = 1 log r 1 + (r 1 p 1) ; log r 1 + (r p 1) ; log r 1 + (r p 1) ! = p ; p ; p =p: Hence the result follows. THEOREM 3.4. (Boundedness property). Let p i = ( pi ; pi ; pi )(i = 1; 2;:::;n) be a num- ber of PFNs. Let p = minfp 1 ;p 2 ;:::;p n g and p + = maxfp 1 ;p 2 ;:::;p n g: Then, p PFFWA(p 1 ;p 2 ;:::;p n ) p + : Proof: Let p i = ( pi ; pi ; pi )(i = 1; 2;:::;n) be a number of PFNs. Let p = minfp 1 ;p 2 ;:::;p n g = ( ; ; ) and p + = maxfp 1 ;p 2 ;:::;p n g = ( + ; + ; + ): We have = min k f p k g; = max k f p k g; = max k f p k g; + = max k f p k g; + = min k f p k g and + = min k f p k g: Now, 1 log r 1 + n Q i=1 (r 1 ( ) 1) wi 1 log r 1 + n Q i=1 (r 1 p i 1) wi 1 log r 1 + n Q i=1 (r 1 ( + ) 1) wi ; log r 1 + n Q i=1 (r ( + ) 1) wi log r 1 + n Q i=1 (r p i 1) wi Some Picture Fuzzy Aggregation Operators Based on. . . Informatica 45 (2021) 447–461 453 log r 1 + n Q i=1 (r ( ) 1) wi ; log r 1 + n Q i=1 (r ( + ) 1) wi log r 1 + n Q i=1 (r p i 1) wi log r 1 + n Q i=1 (r ( ) 1) wi : Therefore,p PFFWA(p 1 ;p 2 ;:::;p n ) p + : THEOREM 3.5. (Monotonicity property) Letp i andp 0 i (i=1, 2,. . . , n) be two sets of PFNs, if p i p 0 i for all i, then PFFWA(p 1 ;p 2 ;:::;p n ) PFFWA(p 0 1 ;p 0 2 ;:::;p 0 n ): Proof: Sincep i p 0 i for alli = 1; 2;:::;n, then, we have pi 0 pi ; pi 0 pi and pi 0 pi for alli = 1; 2;:::;n. Now, (r 1 p i 1) wi (r 1 0 p i 1) wi ) log r 1 + n Q i=1 (r 1 p i 1) wi log r 1 + n Q i=1 (r 1 0 p i 1) wi ) 1 log r 1 + n Q i=1 (r 1 p i 1) wi 1 log r 1 + n Q i=1 (r 1 0 p i 1) wi : Similarly, it can be shown that log r 1 + n Q i=1 (r p i 1) wi log r 1 + n Q i=1 (r 0 p i 1) wi and log r 1 + n Q i=1 (r p i 1) wi log r 1 + n Q i=1 (r 0 p i 1) wi : Therefore,PFFWA(p 1 ;p 2 ;:::;p n ) PFFWA(p 0 1 ;p 0 2 ;:::;p 0 n ): Now, we would like to introduce PFFOWA operator. Definition 3.3. Letp i = ( pi ; pi ; pi )(i = 1; 2;:::;n) be a number of PFNs. The PFFOWA operator of dimensionn is a functionp n !p such that, PFFOWA(p 1 ;p 2 ;:::p n ) = n M i=1 w i p (i) where w = (w 1 ;w 2 ;:::;w n ) t be the weight vec- tor of p i (i = 1; 2;:::;n); w i 2 [0; 1] and n P i=1 w i = 1, ( (1); (2);:::; (n)) is the permutation of (i = 1; 2;:::;n); for which p (i 1) p (i) for all i = 1; 2;:::;n: Based on Frank product of PFNs the following theorem is developed. THEOREM 3.6. Let p i = ( pi ; pi ; pi )(i = 1; 2;:::;n) be a number of PFNs. The PFFOWA operator of dimen- sionn is a functionp n !p with the corresponding weight vector w = (w 1 ;w 2 ;:::;w n ) t such that w i 2 [0; 1] and n P i=1 w i = 1: Then, PFFOWA(p 1 ;p 2 ;:::;p n ) = n L i=1 w i p (i) = 1 log r 1 + n Q i=1 (r 1 p (i) 1) wi ; log r 1 + n Q i=1 (r p (i) 1) wi ; log r 1 + n Q i=1 (r p (i) 1) wi ! where ( (1); (2);:::; (n)) are the permutation of (i = 1; 2;:::;n) for which p (i 1) p (i) for all i = 1; 2;:::;n: With the help of PFFOWA operator we can easily prove the following properties. THEOREM 3.7. (Idempotency property). If p i = ( pi ; pi ; pi )(i = 1; 2;:::;n) be a number of PFNs all are identical, i.e., p i = p for all i. Then, PFFOWA(p 1 ;p 2 ;:::;p n ) =p: THEOREM 3.8. (Boundedness Property). Let p i = ( pi ; pi ; pi )(i = 1; 2;:::;n) be a num- ber of PFNs. Let p = minfp 1 ;p 2 ;:::;p n g and p + = maxfp 1 ;p 2 ;:::;p n g: Then, p PFFOWA(p 1 ;p 2 ;:::;p n ) p + : THEOREM 3.9. (Monotonicity Property). Let p i and p 0 i (i=1, 2,. . . , n) be two sets of PFNs, if p i p 0 i for all i, then PFFOWA(p 1 ;p 2 ;:::;p n ) PFFOWA(p 0 1 ;p 0 2 ;:::;p 0 n ): THEOREM 3.10. (Commutative Property). Let p i and p 0 i (i=1, 2,. . . , n) be two sets of PFNs, then PFFOWA(p 1 ;p 2 ;:::;p n ) = PFFOWA(p 0 1 ;p 0 2 ;:::;p 0 n ) where p 0 i is any permu- tation ofp i (i = 1; 2;:::;n): In Definition 3.2, we find that the weights associated with the PFFWA operator are the simplest form of PF value and in Definition 3.3 the weights associated with the PF- FOWA operator is the original form of the ordered posi- tions of the PF values. In this way, the weights disclosed in the PFFWA and PFFOWA operators, present various per- spectives which are conflicting with one another. But, these perspectives are deliberated to be the same in a general ap- proach. Only to be rescued of such drawback, we now in- troduce PFFHA operator. Definition 3.4. Let p i = ( pi ; pi ; pi )(i = 1; 2;:::;n) be a number of PFNs. The PFFHA operator of dimension 454 Informatica 45 (2021) 447–461 M.R. Seikh et al. n is a functionp n !p such that, PFFHA(p 1 ;p 2 ;:::;p n ) = n M i=1 w i _ p (i) = 1 log r 1 + n Y i=1 (r 1 _ p (i) 1) wi ; log r 1 + n Y i=1 (r _ p (i) 1) wi ; log r 1 + n Y i=1 (r _ p (i) 1) wi ! where w = ( w 1 ; w 2 ;:::; w n ) t is the aggregation associ- ated weight vector, n P i=1 w i = 1;w = (w 1 ;w 2 ;:::;w n ) t be the weight vector of p i (i = 1; 2;:::;n); w i 2 [0; 1] and n P i=1 w i = 1: _ p (i) is thei th biggest weighted PF values of _ p i ( _ p i = nw i p i ;i = 1; 2;:::;n);n is the balancing coeffi- cient. Deduction 3.1. When w = ( 1 n ; 1 n ;:::; 1 n ) t ; then _ p i = n 1 n p i = p i for i = 1; 2;:::;n: Then the PFFHA operator degenerates into PFFOWA operator. If w = ( 1 n ; 1 n ;:::; 1 n ) t ; then PFFHA operator reduces to PFFWA operator. Hence, PFFWA and PFFOWA operators are a specific type of PFFHA operator. Thus, PFFHA operator is a generalization of both the PFFWA and PFFOWA oper- ators, which reflects the degrees of the stated disagreements and their organized situations. 3.2 Picture fuzzy Frank geometric aggregation operators Definition 3.5. Let p i = ( pi ; pi ; pi )(i = 1; 2;:::;n) be a number of PFNs. Then PFFWG operator is a function p n !p such that, PFFWG(p 1 ;p 2 ;:::;p n ) = n O i=1 (p i ) wi where w = (w 1 ;w 2 ;:::;w n ) t be the weight vector of p i (i = 1; 2;:::;n);w i 2 [0; 1] and n P i=1 w i = 1. Hence, we get consequential theorem that follows the Frank operations on PFNs. THEOREM 3.11. Letp i = ( pi ; pi ; pi )(i = 1; 2;:::;n) be a number of PFNs, then aggregated value of them using PFFWG operator is also a PFN, and PFFWG(p 1 ;p 2 ;:::;p n ) = n N i=1 (p i ) wi = log r 1 + n Q i=1 (r p i 1) wi ; 1 log r 1 + n Q i=1 (r 1 p i 1) wi ; 1 log r 1 + n Q i=1 (r 1 p i 1) wi ! . Proof: The proof of this theorem emulates from Theorem 3.2. The following properties may be easily proved by PF- FWG operator. THEOREM 3.12. (Idempotency Property). If p i = ( pi ; pi ; pi )(i = 1; 2;:::;n) be a number of iden- tical PFNs, i.e., p i = p for all i. Then, PFFWG(p 1 ;p 2 ;:::;p n ) =p: THEOREM 3.13. (Boundedness Property). Let p i = ( pi ; pi ; pi )(i = 1; 2;:::;n) be a num- ber of PFNs. Let p = minfp 1 ;p 2 ;:::;p n g and p + = maxfp 1 ;p 2 ;:::;p n g: Then, p PFFWG(p 1 ;p 2 ;:::;p n ) p + : THEOREM 3.14. (Monotonicity Property). Let p i and p 0 i (i=1, 2,. . . , n) be two sets of PFNs, if p i p 0 i for all i, then PFFWG(p 1 ;p 2 ;:::;p n ) PFFWG(p 0 1 ;p 0 2 ;:::;p 0 n ): Now, we would like to introduce PFFOWG operator. Definition 3.6. Letp i = ( pi ; pi ; pi )(i = 1; 2;:::;n) be a number of PFNs. The PFFOWG operator of dimensionn is a functionp n !p such that, PFFOWG(p 1 ;p 2 ;:::p n ) = n O i=1 (p (i) ) wi where w = (w 1 ;w 2 ;:::;w n ) t be the weight vec- tor of p i (i = 1; 2;:::;n); w i 2 [0; 1] and n P i=1 w i = 1, ( (1); (2);:::; (n)) are the permutation of (i = 1; 2;:::;n); for which p (i 1) p (i) for all i = 1; 2;:::;n: The following theorem is developed based on Frank prod- uct operation on PFNs using PFFOWG operator. THEOREM 3.15. Letp i = ( pi ; pi ; pi )(i = 1; 2;:::;n) be a number of PFNs. The PFFOWG operator of dimen- sionn is a functionp n !p with the corresponding weight vector w = (w 1 ;w 2 ;:::;w n ) t such that w i 2 [0; 1] and n P i=1 w i = 1: Then, PFFOWG(p 1 ;p 2 ;:::p n ) = n N i=1 (p (i) ) wi = log r 1 + n Q i=1 (r p (i) 1) wi ; 1 log r 1 + n Q i=1 (r 1 p (i) 1) wi ; 1 log r 1 + n Q i=1 (r 1 p (i) 1) wi ! where ( (1); (2);:::; (n)) are the permutation of (i = 1; 2;:::;n) for which p (i 1) p (i) for all i = 1; 2;:::;n: Some Picture Fuzzy Aggregation Operators Based on. . . Informatica 45 (2021) 447–461 455 The following properties can be investigated by using PFFOWG operator. THEOREM 3.16. (Idempotency property). If p i = ( pi ; pi ; pi )(i = 1; 2;:::;n) be a number of PFNs all are identical, i.e., p i = p for all i. Then, PFFOWG(p 1 ;p 2 ;:::;p n ) =p: THEOREM 3.17. (Boundedness Property). Let p i = ( pi ; pi ; pi )(i = 1; 2;:::;n) be a num- ber of PFNs. Let p = minfp 1 ;p 2 ;:::;p n g and p + = maxfp 1 ;p 2 ;:::;p n g: Then, p PFFOWG(p 1 ;p 2 ;:::;p n ) p + : THEOREM 3.18. (Monotonicity Property). Let p i and p 0 i (i=1, 2,. . . , n) be two sets of PFNs, if p i p 0 i for all i, then PFFOWG(p 1 ;p 2 ;:::;p n ) PFFOWG(p 0 1 ;p 0 2 ;:::;p 0 n ): THEOREM 3.19. (Commutative Property). Let p i and p 0 i (i=1, 2,. . . , n) be two sets of PFNs, then PFFOWG(p 1 ;p 2 ;:::;p n ) = PFFOWG(p 0 1 ;p 0 2 ;:::;p 0 n ) where p 0 i is any permu- tation ofp i (i = 1; 2;:::;n): In Definition 3.5, we find that the weights associated with the PFFWG operator are in the simplest form of PF value and in Definition 3.6 the weights associated with the PFFOWG operator are in the actual form of the ordered po- sitions of the PF values. In this way, the weights disclosed in the PFFWG and PFFOWG operators, present various perspectives which are conflicting with one another. But, these perspectives are deliberated to be the same in a gen- eral approach. Only to be rescued of such drawback, we at this moment introduce PFFHG operator. Definition 3.7. Let p i = ( pi ; pi ; pi )(i = 1; 2;:::;n) be a number of PFNs. The PFFHG operator of dimension n is a functionp n !p such that, PFFHG(p 1 ;p 2 ;:::;p n ) = n N i=1 ( _ p (i) ) wi = log r 1 + n Q i=1 (r _ p (i) 1) wi ; 1 log r 1 + n Q i=1 (r 1 _ p (i) 1) wi ; 1 log r 1 + n Q i=1 (r 1 _ p (i) 1) wi ! where w = ( w 1 ; w 2 ;:::; w n ) t is the aggregation associ- ated weight vector, n P i=1 w i = 1;w = (w 1 ;w 2 ;:::;w n ) t be the weight vector of p i (i = 1; 2;:::;n); w i 2 [0; 1] and n P i=1 w i = 1: _ p (i) is thei th biggest weighted PF values of _ p i ( _ p i = nw i p i ;i = 1; 2;:::;n);n is the balancing coeffi- cient. Deduction 3.2. When w = ( 1 n ; 1 n ;:::; 1 n ) t ; then _ p i = n 1 n p i = p i for i = 1; 2;:::;n: Then the PFFHG operator degenerates into PFFOWG operator. If w = ( 1 n ; 1 n ;:::; 1 n ) t ; then PFFHG operator reduces to PFFWG operator. Hence, PFFWG and PFFOWG operators are specific types of PFFHG operator. Thus, PFFHG opera- tor is a generalization of both the PFFWG and PFFOWG operators, which reflects the degrees of the stated disagree- ments and their organized situations. 4 Model for MADM using picture fuzzy data In this section, we introduce a novel approach for decision- making problems using PF information manipulating PF- FWA and PFFWG operators, where attribute values are PFNs and attribute weights are real numbers. For an MADM problem, let F = fF 1 ;F 2 ;:::;F m g be a dis- crete set of m alternatives to be selected and H = fH 1 ;H 2 ;:::;H n g be the arrangement of attributes to be assessed. Let w =fw 1 ;w 2 ;:::;w n g be the weight vec- tor of the attributes H j (j = 1; 2;:::;n) where w k (k = 1; 2; 3;:::;n) are all real numbers such that w k > 0 and n P k=1 w k = 1. Assume that P = ( ij ) m n = (( ij ; ij ; ij )) m n is the PF decision matrix, where ij is the possible value for which the alternative F i satis- fies the attribute H j where ij + ij + ij 1 and ij ; ij ; ij 2 [0; 1]. To achieve the final ranking of the alternatives, we pro- pose an algorithm which is shown in the following. 4.1 Algorithm The proposed MADM problem with PF data based on the proposed PFFWA and PFFWG operators is now presented as follows: Step I: Construct the PF decision matrix P = ( ij ) m n = (( ij ; ij ; ij )) m n . Step II: Transform the matrix P = ( ij ) m n = (( ij ; ij ; ij )) m n into a normalize PF matrixP 0 = ( 0 ij ) m n = (( 0 ij ; 0 ij ; 0 ij )) m n by Equation (1). 0 ij = ( ij ; ij ; ij ); ifH j is benefit attribute; ( ij ; ij ; ij ); ifH j is cost attribute. (1) Step III: Calculate the collective information k of the al- ternativeF k by Equations (2) and (3). k = PFFWA( 0 k1 ; 0 k2 ;:::; 0 kn ) = n M j=1 (w j kj ) = 1 log r 1 + n Y i=1 (r 1 0 p kj 1) w j ; log r 1 + n Y i=1 (r 0 p kj 1) w j ; log r 1 + n Y i=1 (r 0 p kj 1) w j ! : (2) 456 Informatica 45 (2021) 447–461 M.R. Seikh et al. and k = PFFWG( 0 k1 ; 0 k2 ;:::; 0 kn ) = n O j=1 ( kj ) w j = log r 1 + n Y i=1 (r 0 p kj 1) w j ; 1 log r 1 + n Y i=1 (r 1 0 p kj 1) w j ; 1 log r 1 + n Y i=1 (r 1 0 p kj 1) w j ! : (3) Step IV: Compute the score function ( i ) for each alter- native using Definition 2:3. Step V: The optimal decision is to selectF k if ( k ) = max l f( l )g. 5 Numerical illustration In this section, we are willing to sketch a numerical prob- lem to illustrate the possible assessment of commercializa- tion with the help of PF data. Suppose a renowned multi-tasking company has decided to utilize a part of its total annual profit in some improve- ment of the company’s good-will. The managing board has selected some alternative choices to invest the fund, such as 1. F 1 : Air conditioning and furnishing the whole floor. 2. F 2 : Purchasing of some advanced gadgets. 3. F 3 : Constructing a parking zone. 4. F 4 : Advertising. 5. F 5 : Security facility. Now, since each alternative satisfies different requirements so, confusion arises to make a decision. Thereby, the man- aging board has determined the following considerable at- tributes, – H 1 : Enhancement of profit. – H 2 : Customer’s benefit. – H 3 : Maintenance cost. – H 4 : Ecofriendliness. Now the decision making in this case is difficult be- cause each alternative promises to maximize a different at- tribute. The managing board defines the weight vector of the attributeH j (j = 1; 2; 3; 4) as (0:30; 0:25; 0:20; 0:25). Meanwhile, H 1 ;H 2 ;H 4 are benefit attributes and H 3 is a cost attribute. Assume that the alternative F i with re- spect to the attribute H j is expressed as PF matrix P = ( ij ) m n = (( ij ; ij )) m n . The assessment for the al- ternatives are shown in the Table 1. In order to select the most preferable alternativeF i (i = 1; 2; 3; 4; 5) we exploit the PFFWA and PFFWG operators to develop an MADM theory with PF data, which can be evaluated as follows: Step 1: We input the PF decision matrix given in Table 1. Step 2: By normalizing of PF decision matrix with the help of Equation (1) we get the matrixN. Step 3: We taker = 2 and use PFFWA operator to com- pute overall performance values i (i = 1; 2; 3; 4; 5) of alternativesF i ’s using Equation (2) – 1 = (0:6188; 0:1800; 0:0879) – 2 = (0:6517; 0:1827; 0:1214) – 3 = (0:5441; 0:2861; 0:0559) – 4 = (0:6006; 0:2100; 0:0713) – 5 = (0:5823; 0:1456; 0:1478): Step 4: We compute the values of the score functions us- ing Definition 2.3, ( i )(k = 1; 2; 3; 4; 5) of the overall PFNs i (i = 1; 2; 3; 4; 5) as – ( 1 ) = 0:7654 – ( 2 ) = 0:7651 – ( 3 ) = 0:7441 – ( 4 ) = 0:7646 – ( 5 ) = 0:7172: Therefore, with respect to the score values, we rank all the alternatives asF 1 >F 2 >F 4 >F 3 >F 5 : Step 5: Therefore, F 1 should be selected as the most preferable alternative by the company. Again, if PFFWG operator is used instead of PFFWA operator, then the problem can be solved similarly as above. Step 1: We input the PF decision matrix given in Table 1. Step 2: The normalized matrix is same as the matrixN: Step 3: We taker = 2 and use PFFWG operator to com- pute overall performance values i (i = 1; 2; 3; 4; 5) by Equation 3 of the alternativesF i ’s. – 1 = (0:4210; 0:2959; 0:1067) – 2 = (0:5968; 0:2118; 0:1277) – 3 = (0:2922; 0:4242; 0:0580) – 4 = (0:3057; 0:3727; 0:0778) – 5 = (0:4545; 0:2031; 0:1600): Step 4: We compute the values of the score function using Definition 2.3, ( i )(i = 1; 2; 3; 4; 5) of the overall PFNs i (i = 1; 2; 3; 4; 5) as – ( 1 ) = 0:6571 – ( 2 ) = 0:7345 – ( 3 ) = 0:6170 – ( 4 ) = 0:6139 – ( 5 ) = 0:6472: Some Picture Fuzzy Aggregation Operators Based on. . . Informatica 45 (2021) 447–461 457 H 1 H 2 H 3 H 4 F 1 (0.60,0.25,0.12) (0.91,0.03,0.05) (0.22,0.20,0.38) (0.12,0.59,0.05) F 2 (0.72,0.15,0.10) (0.32,0.40,0.20) (0.11,0.15,0.70) (0.75,0.12,0.10) F 3 (0.80,0.10,0.04) (0.09,0.70,0.05) (0.08,0.60,0.07) (0.70,0.20,0.07) F 4 (0.85,0.05,0.04) (0.76,0.15,0.07) (0.09,0.70,0.07) (0.09,0.53,0.12) F 5 (0.71,0.10,0.11) (0.56,0.20,0.19) (0.09,0.50,0.09) (0.69,0.03,0.24) Table 1: Picture fuzzy decision matrix Therefore, with respect to the score values, we rank all the alternatives asF 2 >F 1 >F 5 >F 3 >F 4 : Step 5: Therefore, F 2 should be selected as the most preferable alternative by the company. As we have demonstrated above, the score values of the alternatives are different from each other. But the ranking orders corresponding to various alternatives are the same, and the preferable alternative is alwaysF 2 : N = 0 B B B B @ (0:60; 0:25; 0:12) (0:91; 0:03; 0:05) (0:38; 0:20; 0:22) (0:12; 0:59; 0:05) (0:72; 0:15; 0:10) (0:32; 0:40; 0:20) (0:70; 0:15; 0:11) (0:75; 0:12; 0:10) (0:80; 0:10; 0:04) (0:09; 0:70; 0:05) (0:07; 0:60; 0:08) (0:70; 0:20; 0:07) (0:85; 0:05; 0:04) (0:76; 0:15; 0:07) (0:07; 0:70; 0:09) (0:09; 0:53; 0:12) (0:71; 0:10; 0:11) (0:56; 0:20; 0:19) (0:09; 0:50; 0:09) (0:69; 0:03; 0:24) 1 C C C C A Next, we will show how the parameterr affects the rank- ing results obtained by utilizing PFFWA and PFFWG op- erators. 6 Analysis of the effect of the parameterr on decision making We can utilize different values of the operational param- eter r, for ranking the given alternatives in our proposed method. For exploring the flexibility and sensitivity of the param- eterr, we fix different values ofr to categorize the novel numerical MADM example. Depending on PFFWA opera- tor and PFFWG operator, the consequences of ranking or- ders of the alternativesF 1 ;F 2 ;F 3 ;F 4 ;F 5 for different val- ues of the parameterr are shown in the Table 2 and Table 3. To provide a better view of the aggregation results, we show the results of the rankings of the alternatives by the proposed PFFWA and PFFWG operators in Figure 1(a) and Figure 1(b) respectively. From Table 2 and Figure 1(a) we can easily see that when 3 r 10;r = 15; 20; 25; 50 the aggregation score values using PFFWA operator with different param- eter r are different, but the ranking orders of the alterna- tivesF i (i = 1; 2; 3; 4; 5) are same and the ranking order is F 2 > F 1 > F 4 > F 3 > F 5 . However, whenr = 2; we obtain F 1 > F 2 > F 4 > F 3 > F 5 and in that case the optimal alternative isF 1 : From Table 3 and Figure 1(b), we can see that the ag- gregation score values using PFFWG operator with dif- ferent parameter r are different, but the optimal alter- native is always F 2 : When 2 r 9; we obtained F 2 > F 1 > F 5 > F 3 > F 4 ; when r = 10; we get F 2 > F 1 > F 5 > F 3 F 4 and for r = 15; 20; 25; 50; we obtainedF 2 >F 1 >F 5 >F 4 >F 3 : Hence, the over- all best alternative isF 2 : In general, different decision-makers can set different values of the parameterr based on their preferences. In this MADM problem based on PFFWA and PFFWG operators, we can notice that for PFFWG operator the rank- ing orders of the alternatives can be changed by the varia- tion of values of the parameter r: Therefore, the PFFWG operator has responded more tor in this MADM method. At the same time, in correspondence with PFFWA opera- tor according to different values of working parameter r; ranking forms can be changed. So PFFWA operator is less responsive tor in this case of the MADM procedure. 7 Comparison analysis In order to verify the utility of the proposed method and to pursue its advantages, we compare our proposed Frank ag- gregation operators with other existing well-known aggre- gation operators under the PF environment. The compara- tive results are shown in Table 4. We compare our proposed method with PFWA operator [37] and PFWG operator [37]. Making a comparison with PFWA or PFWG operators introduced by Wei [37], we can find that PFWA or PFWG operator is only a particular case of our proposed opera- tors when the parameter r ! 1: Therefore, indeed, our introduced procedures are more generalized. Moreover, our proposed operators, based on Frank t-norm and Frank t-conorm are more nourished and can adopt the relation- ship between various arguments. Also, our proposed op- erators present the Lukasiewicz product and Lukasiewicz sum when the parameterr!1: Therefore, we have ar- 458 Informatica 45 (2021) 447–461 M.R. Seikh et al. r ( 1 ) ( 2 ) ( 3 ) ( 4 ) ( 5 ) Ranking order Optimal alternative 2 0.7654 0.7651 0.7441 0.7646 0.7172 F 1 >F 2 >F 4 >F 3 >F 5 F 1 3 0.7610 0.7640 0.7396 0.7594 0.7153 F 2 >F 1 >F 4 >F 3 >F 5 F 2 4 0.7581 0.7632 0.7365 0.7559 0.7140 F 2 >F 1 >F 4 >F 3 >F 5 F 2 5 0.7558 0.7626 0.7342 0.7532 0.7130 F 2 >F 1 >F 4 >F 3 >F 5 F 2 6 0.7541 0.7622 0.7324 0.7511 0.7122 F 2 >F 1 >F 4 >F 3 >F 5 F 2 7 0.7526 0.7618 0.7309 0.7493 0.7115 F 2 >F 1 >F 4 >F 3 >F 5 F 2 8 0.7514 0.7614 0.7297 0.7478 0.7110 F 2 >F 1 >F 4 >F 3 >F 5 F 2 9 0.7503 0.7611 0.7286 0.7465 0.7105 F 2 >F 1 >F 4 >F 3 >F 5 F 2 10 0.7494 0.7609 0.7277 0.7454 0.7101 F 2 >F 1 >F 4 >F 3 >F 5 F 2 15 0.7459 0.7599 0.7243 0.7412 0.7085 F 2 >F 1 >F 4 >F 3 >F 5 F 2 20 0.7436 0.7593 0.7221 0.7385 0.7075 F 2 >F 1 >F 4 >F 3 >F 5 F 2 25 0.7419 0.7588 0.7204 0.7364 0.7068 F 2 >F 1 >F 4 >F 3 >F 5 F 2 50 0.7370 0.7575 0.7160 0.7308 0.7046 F 2 >F 1 >F 4 >F 3 >F 5 F 2 Table 2: Effect of the parameterr on decision making result using PFFWA operator r ( 1 ) ( 2 ) ( 3 ) ( 4 ) ( 5 ) Ranking order Optimal alternative 2 0.6571 0.7345 0.6170 0.6139 0.6472 F 2 >F 1 >F 5 >F 3 >F 4 F 2 3 0.6613 0.7361 0.6220 0.6197 0.6514 F 2 >F 1 >F 5 >F 3 >F 4 F 2 4 0.6641 0.7372 0.6255 0.6238 0.6541 F 2 >F 1 >F 5 >F 3 >F 4 F 2 5 0.6663 0.7380 0.6282 0.6269 0.6562 F 2 >F 1 >F 5 >F 3 >F 4 F 2 6 0.6679 0.7386 0.6304 0.6294 0.6577 F 2 >F 1 >F 5 >F 3 >F 4 F 2 7 0.6693 0.7391 0.6322 0.6315 0.6590 F 2 >F 1 >F 5 >F 3 >F 4 F 2 8 0.6704 0.7395 0.6337 0.6333 0.6601 F 2 >F 1 >F 5 >F 3 >F 4 F 2 9 0.6714 0.7398 0.6351 0.6349 0.6611 F 2 >F 1 >F 5 >F 3 >F 4 F 2 10 0.6732 0.7401 0.6363 0.6363 0.6619 F 2 >F 1 >F 5 >F 3 F 4 F 2 15 0.6755 0.7412 0.6408 0.6415 0.6648 F 2 >F 1 >F 5 >F 4 >F 3 F 2 20 0.6776 0.7419 0.6438 0.6450 0.6668 F 2 >F 1 >F 5 >F 4 >F 3 F 2 25 0.6791 0.7424 0.6461 0.6476 0.6682 F 2 >F 1 >F 5 >F 4 >F 3 F 2 50 0.6833 0.7437 0.6528 0.6551 0.6720 F 2 >F 1 >F 5 >F 4 >F 3 F 2 Table 3: Effect of the parameterr on decision making result using PFFWG operator rived at the decision that all of the arithmetic and geometric aggregation operators for PFNs are contained in PF Frank aggregation operators, concerning the different values ofr: If we modify the value of the parameterr in the problem, we get different ranking results for the alternatives. For ex- ample, if we modify the value of the parameter r from 2 to 50, then using PFFWG operator we get the score values of the alternatives as ( 1 ) = 0:6833; ( 2 ) = 0:7437; ( 3 ) = 0:6528; ( 4 ) = 0:6551 and ( 5 ) = 0:6720: Obviously, it can be obtained that the ranking position of the alternative F 4 changed from a bad position to a good position. But the PFWA and the PFWG operators are inde- pendent of the parameterr. So, the ranking order obtained with the help of those operators remains the same. Based on the above comparison analysis, the approach in the present study is proved to be more flexible, compatible, and reliable than other existing procedures to control PF environment based MADM problems. 8 Conclusions In this paper, we have studied MADM problems using PF information. We have developed Frank operations for PFSs and proposed a series of new aggregation operators, like, PFFWA operator, PFFOWA operator, PFFHA opera- tor, PFFWG operator, PFFOWG operator, and PFFHG op- erator. Then, we have proposed an algorithm to deal with the MADM problem under the PF environment by using the PFFWA operator and the PFFWG operator. Finally, we have compared our proposed method with the existing ap- proaches to exhibit its benefits and applicability. In further research, we can study some new extensions of PFS, such as complex PFS, rough PFS. We can also extend them to other decision-making methods, such as COPRAS method [49], TOPSIS method [15], VIKOR [47] method, and so on, and apply them to deal with some real- life decision-making problems. We shall continue to in- vestigate PF aggregation operators with the help of various t-norms and t-conorms. Acknowledgement The author, Utpal Mandal, would like to thank the Coun- cil of Scientific and Industrial Research (CSIR), India, for granting the financial support to continue this research work under the Junior Research Fellowship (JRF) scheme with sanctioned Grant No. 09/1269(0001)/2019-EMRI, Dated 02/07/2019. Some Picture Fuzzy Aggregation Operators Based on. . . Informatica 45 (2021) 447–461 459 (a) Score of alternatives whenr 2 [2;50] based on PF- FWA operator (b) Score of alternatives whenr 2 [2;50] based on PF- FWG operator Figure 1: Pictorial representation of the ranking of the alternatives with different values ofr Aggregation Operators ( 1) ( 2) ( 3) ( 4) ( 5) Ranking order Optimal alternative PFWA [37] 0.7731 0.7671 0.7522 0.7737 0.7206 F4 >F1 >F2 >F3 >F5 F4 PFWG [37] 0.6494 0.7312 0.6085 0.6040 0.6394 F2 >F1 >F5 >F3 >F4 F2 Proposed method 0.6571 0.7345 0.6170 0.6139 0.6472 F2 >F1 >F5 >F3 >F4 F2 Table 4: Comparison table References [1] C. Alsina, M. Frank, B. 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