JAISCR, 2014, Vol. 4, No. 1, pp. 57 – 77 DOI 10.2478/jaiscr-2014-0025 GROUP DECISION MAKING USING INTERVAL-VALUED INTUITIONISTIC FUZZY SOFT MATRIX AND CONFIDENT WEIGHT OF EXPERTS Sujit Das1 , Samarjit Kar2 , Tandra Pal3 1 Department of Computer Science and Engineering, Dr. B. C. Roy Engineering College, Durgapur-6, India sujit [email protected] 2 Department of Mathematics, National Institute of Technology, Durgapur-9, India kar s [email protected] 3 Department of Computer Science and Engineering, National Institute of Technology, Durgapur-9, India [email protected] Abstract This article proposes an algorithmic approach for multiple attribute group decision making (MAGDM) problems using interval-valued intuitionistic fuzzy soft matrix (IVIFSM) and confident weight of experts. We propose a novel concept for assigning confident weights to the experts based on cardinals of interval-valued intuitionistic fuzzy soft sets (IVIFSSs). The confident weight is assigned to each of the experts based on their preferred attributes and opinions, which reduces the chances of biasness. Instead of using medical knowledgebase, the proposed algorithm mainly relies on the set of attributes preferred by the group of experts. To make the set of preferred attributes more important, we use combined choice matrix, which is combined with the individual IVIFSM to produce the corresponding product IVIFSM. This article uses IVIFSMs for representing the experts’ opinions. IVIFSM is the matrix representation of IVIFSS and IVIFSS is a natural combination of interval-valued intuitionistic fuzzy set (IVIFS) and soft set. Finally, the performance of the proposed algorithm is validated using a case study from real life. 1 Introduction Molodtsov [1] introduced soft set theory as a generic mathematical tool for dealing with uncertain problems, which cannot be handled using traditional mathematical tools such as theory of probability [2], theory of fuzzy sets [3], rough set theory [4], and the interval mathematics. All these theories have their own inherent difficulties due to the inadequacy of the parameterization. Molodtsov’s soft set [1] is free from such kind of difficulties, which can be used for approximate description of objects without any restriction. Due to this absence of restric- tion on the approximate description, soft set theory has been emerging as a convenient and easily applicable tool in practice. Since its introduction, soft set theory has been successfully applied in many different fields such as decision making [5-12], data analysis [13], forecasting [14], simulation [15], optimization [16], texture classification [17], etc. Maji et al. [18] presented some operations for soft sets and also discussed their properties. They presented the concept of fuzzy soft set (FSS) [19] which is based on a combination of the fuzzy set and soft set mod- 58 els. Yang et al. [20] introduced the concept of the interval-valued fuzzy soft set (IVFSS) by combining the interval-valued fuzzy set and soft set and then explained a decision making algorithm based on IVFSS. Decision making problems were solved first by Maji and Roy [11] using soft sets. They [12] studied a soft set theoretic approach to deal with decision making and introduced the concept of choice value. Combining soft set [1] with intuitionistic fuzzy set [25, 33], Maji et al. [21–23] introduced the concept of intuitionistic fuzzy soft sets (IFSS). The suitability of IFSS for the decision making applications is found in [24]. Das and Kar [31] proposed an algorithmic approach for group decision making based on IFSS. The authors [31] have used cardinals of IFSS as a novel concept for assigning confident weight to the set of experts. Cagman and Enginoglu [6-7] pioneered the concept of soft matrix to represent a soft set. Mao et al. [30] presented the concept of intuitionistic fuzzy soft matrix (IFSM) and applied it in group decision making problems. Jiang et al. [26] introduced IVIFSS by combining interval-valued intuitionistic fuzzy set and soft set and discussed their properties. Jiang et al. [9] presented an adjustable approach to IFSS based decision making by using level soft sets of IFSSs. Feng et al. [27] extended the level soft sets method to interval-valued fuzzy soft sets. Qin et al. [28] generalized the approaches introduced by Feng et al. [27] and Jiang et al. [9]. They [28] defined the notion of reduct IFSSs and presented an adjustable approach for decision making based on IVIFSS. Zhang et al. [29] investigated the decision making problems based on IVIFSS. They [29] developed an adjustable approach to IVIFSSs based decision making using level soft sets. Recently, Das et al. in [32] have introduced IVIFSM and applied it to multiple attribute decision making problems. Das and Kar [34-35] have also introduced the concept of intuitionistic multi fuzzy soft set (IMFSS) and hesitant fuzzy soft set (HFSS) and applied them in decision making problem. Preceding discussion narrates that a few decision making approaches [26, 28, 29, 32] have been developed by the researchers in the context of IVIFSS. In [26], Jiang et al. introduced IVIFSSs and discussed their various operations and properties. In [28], authors presented an adjustable approach to IVIFSS based decision making using reduct IFSS and level soft sets of IFSS. Zhang et al. Tambouratzis T., Souliou D., Chalikias M., Gregoriades A. in [29] developed an adjustable approach to IVIFSS based decision making. They [29] also defined the concept of the weighted IVIFSS, where they emphasized the importance of the parameters in IVIFSS. Then they presented a decision making approach using weighted IVIFSS. Authors introduced IVIFSM in [32] and studied a novel algorithmic approach focussing on the choice parameters of different experts. Since decision makers/experts express their opinions based on the available information and domain of expertise of different experts are different, so a prioritising mechanism of various experts are necessary to introduce a quality decision making paradigm. Due to lack of information or limited domain knowledge, experts often prefer to express their opinions only for a subset of attributes instead of the entire set of attributes. As a result, some decision makers may express their opinions to more numbers of attributes, while others show their interest only for a few attributes. Normally, more importance can be assigned to those experts who encompass more attributes. In another case, an expert who is more confident about some set of attributes can be assigned more importance. For having better decision making outcome in MAGDM problems, importance or weight of various decision makers imparts a huge significance in the entire process. But as per our knowledge, this mechanism can be found only in [31], where authors have used a novel concept in terms of confident weight assigning mechanism of experts in the framework of IFSS. None have implemented the idea presented in [31] for group decision making using IVIFSS. Motivated by this idea, we have introduced a MAGDM algorithm using confident weight assigning of experts in the context of IVIFSS. The proposed algorithm mainly focuses on the choice parameters/attributes of various experts and computes the confident weight of an expert based on her prescribed opinions. We have used cardinals of IVIFSS for measuring the weight. The proposed confident weight also reduces the chance of biasing. Once the weights are assigned to individual experts’ opinions, this article present a consensus reaching approach based on IVIFSM, combined choice matrix, product IVIFSM, score, and accuracy values. We have used a case study related to heart disease diagnosis, where two cases are shown. Case I shows the final outcome without assigning any weight, whereas case II shows the result by assign- 59 MAXIMISING ACCURACY AND EFFICIENCY OF. . . Table 1. Tabular representation of (F{A} , E) / U E s1 s2 s3 s4 s5 d1 1 1 0 0 0 d2 1 0 1 0 0 d3 0 1 0 0 0 d4 0 0 1 0 0 d5 0 0 0 0 0 ing weight. As per the experimental result, these two cases show different ordering of diseases. Rest of this article is organized as follows. In section 2, we briefly review some basic notions and background of soft sets, FSSs, IVFSSs, IVIFSSs, and IVIFSMs. Section 3 presents IVIFSM and a few operations on it. The cardinal of IVIFSS is introduced in section 4 followed by the proposed algorithmic approach in section 5. A case study has been illustrated in section 6 to verify the practicability and effectiveness of the proposed method. Then a brief discussion on the results is given in section 7. Finally, key conclusions are drawn in section 8. 2 Preliminaries This section briefly reviews some basic concepts related with this article. 2.1 Soft Set Let U be an initial universal set and E be a set of parameters. Let P(U) denotes the power set of U and A ⊆ E. A pair (F{A} , E) is called a soft set [1] over U, where F{A} is a mapping given by F{A} : E → P(U)such that F{A} (e) = 0/ if e ∈ / A. In other words, a soft set over U is a mapping from parameters to P(U), and it is not a set, but a parameterized family of subsets of U. For any parameter e ∈ A, F{A} (e) may be considered as the set of eapproximate elements of the soft set (F{A} , E). For illustration, let us consider the following example. Example 1. Let U be the set of five diseases (Viral fever, Malaria, Typhoid, Gastric ulcer, Pneumonia) given by U = {d1 , d2 , d3 , d4 , d5 } and E be the set of five symptoms given by E= {Temperature, Headache, Stomach pain, Cough, Chest pain} ={s1 , s2 , s3 , s4 , s5 }. Let A = {s1 , s2 , s3 } ⊂ E. Now consider that F{A} is a mapping given by, F{A} (s1 ) = {d1 , d2 } , F{A} (s2 ) = {d1 , d3 } , F{A} (s3 ) = {d2 , d4 } . Then the soft set (F{A} , E) = {(s1 , d1 , d2 ) , / , (s5 , {0}) / . (s2 , d1 , d3 ),(s3 , {d2 , d4 }) , (s4 , {0}) A fuzzy soft set can also be represented in the form of a two dimensional table. Table 1, given below, represents the soft set (F{A} , E). 2.2 Fuzzy soft set Let U be an initial universe and E be a set of parameters (which are fuzzy variables). Let FS(U) denotes (the set of ) all fuzzy sets of U and A ⊂ E. � A pair F{A} , E is called a fuzzy soft set (FSS) [19] over U, where F�{A} is a mapping given by / e∈ 0if /A F�{A} : E → FS (U) such that F�{A} (e) = � � where 0/ is null fuzzy set. A fuzzy soft set is a parameterized family of fuzzy subsets of U. Its universe is the set of all fuzzy sets of U, i.e., FS(U). A fuzzy soft set can be considered a special case of a soft set because it is still a mapping from parameters to a universe. The difference between fuzzy soft set and soft set is that in a fuzzy soft set, the universe to be considered is the set of fuzzy subsets of U. Example 2. Let U and E remain same as in Example 1. Here A = {s1 , s2 , s3 , s5 } ⊂ E. Let F�{A} (s1 ) = {d1 /0.2, d2 /0.4, d3 /0.9, d4 /0.7} , F�{A} (s2 ) = {d2 /0.8, d3 /0.1, d4 /0.7} , and F�{A} (s3 ) = {d1 /0.6, d2 /0.2, d3 /0.8} , F�{A} (s5 ) = {d1 /0.6, d2 /0.7, d3 /0.5, d4 /0.8} . Then the fuzzy soft set is given by (s1 , {d1 /0.2, d2 /0.4, d3 /0.9, d4 /0.7}) , (s 2 , {d2 /0.8, d3 /0.1, d4 /0.7}) , (s (F�{A} , E) = ) d2 /0.2, d3 /0.8}) , ( 3 , {d1 /0.6, � / , s4 , {0} (s5 , {d1 /0.6, d2 /0.7, d3 /0.5, d4 /0.8}) Tabular representation of the fuzzy soft set (F�{A} , E) is shown in Table 2. . d1 0.2 0 0.6 0 0.6U2 E s1 T., s2 sTambouratzis s4 s5 Souliou D., Chalikias M., Grego 3 0.9 0.1 0.8 0 0.5 d3 0.4 0.8 0.2 0 0.7 d 2 Table F�{A} , E) d1 2. Tabular 0.2 representation 0 0.6 0 of (0.6 s2 s3 s4 s5 U E s1 0.7 0.7 0 0 0.8 d 0.1 00.8 0.6 0 0.5d24/ d0.2 3 0.4 0.8 0.2 0 0.7 d1 00.9 0.6 0s 0s 0 0s dF�5{A}E, E)0s Table 2.0.8 Tabular of 0.7 0.7representation 0 0.8(dU d0.4 4 1 2 3 5 0.9 0.1 0.8 0s4 0.5 0.2 00 0.7 d2 3 0 0 0 0 0 d d 0.2 0 0.6 0 0.6 / 5 0.7Souliou 0.7D., Chalikias 0 0 M., Gregoriades 0.8 d41 0.9 0.1 0.8 0 0.5 d3 T., A. 60 s2 s3 2.3sTambouratzis s5 U E s1 fuzzy0.2 soft set (IFSS) 4 Intuitionistic 0.4 0.8 0 d 2 0 0 0 0 00.7 d5 0.7 0.7 0 0 0.8 d4 d1 0.2 0 0.6 0 0.6 0.9initial 0.1 universe 0.8 0and 0.5 2.3 Let dU3 be an E be a set 0 0 0fuzzy0 soft0set (IFSS) d5 Intuitionistic 0.4 0.8 0.2 0 0.7 d2 0.7 0.7 0 0 0.8 d 4 � of IFS (U) the set of softdenotes set (IFSS) Table 2. Tabular representation of (initial F0.9 {A} , E) Let U be universe and0parameters. EIntuitionistic be a set Let fuzzy 0.1 0.82.3 d3 an 0 fuzzy0 sets0of U0and A 0 ⊂ E. A d0.5 5 all intuitionistic 2.3 Intuitionistic fuzzy soft set (IFSS) / of parameters. (U) 0denotes set 0.7IFS0.7 0Letthe d4 Let U0.8 be of an initial universe and E be a set U E s1 all s2 intuitionistic s3 s4 fuzzy s5 sets of U pair (F�{A} , E) A is called an intuitionistic fuzzy soft and A ⊂ E. 0 0 and0 Eofbe0parameters. 0 d5 Let IFS (U) denotes the set of Let U be an initial universe a set 2.3 Intuitionistic soft set (IFSS) d1 0.2 pair 0 (F�0.6, E)0 is called 0.6 an intuitionistic is a mapping set (IFSS) [22] over fuzzy U, where F�{A} fuzzy soft {A} all intuitionistic fuzzy sets of U and A ⊂ E. A of parameters. Let IFS (U) denotes the set of 0.4 0.8 0.2 0 0.7 d2 � �be � : called E initial → IFS(U) so that (e)asoft =set 0/ given FE) Let U an and F�E{A} be {A} is a�by, mapping set (IFSS) [22] over where pair (F , is anuniverse intuitionistic fuzzy all intuitionistic fuzzy sets U, of U and F A{A} ⊂ set E. A 2.3 Intuitionistic fuzzy soft (IFSS) {A} 0.9 � 0.1 0.8 � 0 0.5 d3 � �/{A} / isU, of eparameters. IFS (U) denotes setset, of if ∈ A,(e) where 0Let null intuitionistic fuzzy IFS(U) so fuzzy that F =� 0/ over pair (Fgiven , E)by, is F called: Ean→intuitionistic soft mapping set (IFSS) [22] where F�{A} is athe 0.7 {A} 0.7 0Let {A} 0.8initial universe d4 U0 be an and E be a set all intuitionistic fuzzy sets of U and A ⊂ E. A i.e., the membership value of x, µ(x) = 0; the non� is nullF�{A} is mapping [22] if0 e of ∈ / A, where intuitionistic fuzzy set, : Eof→ IFS(U) so that F�{A} (e) = � 0/ by, F�{A} 0 over 0 U,0/ where 0 Let IFS d5 set0 (IFSS) parameters. (U)agiven denotes the set � pair= (0; F{A} , E) is �called intuitionistic fuzzy soft membership value of x,an ν(x) = 1 and the indeter� � �the i.e.,F membership value of x, µ(x) the non/ : E → IFS(U) so that F (e) = 0 given by, ifUe and ∈ / A,Awhere is null intuitionistic fuzzy set, {A}intuitionistic fuzzy sets of {A} all ⊂ of E. 0/ A � / is a mapping ministic part π(x)U,=where 0, ∀x ∈F�{A} 0. set (IFSS) [22] x,over membership value of x, ν(x) = 1 and the indeter� � i.e., the membership value of x, µ(x) = 0; the non/ if e ∈ / A, where 0 is null intuitionistic fuzzy set, pair soft (F{A}set , E)(IFSS) is called an intuitionistic fuzzy soft 2.3 Intuitionistic fuzzy �in �{A} � : E → IFS(U) so that F (e) = � 0/ given by, F / ministic part of x, π(x) = 0, ∀x ∈ 0. Example 3. Let U, A and E are same as Example {A} value of x, ν(x) = 1 and the indeteri.e., the membership of x, U, µ(x)where = 0;membership the is a mapping set (IFSS) value [22] over F�{A}nonLet U bemembership an initial universe E and be= aE1 are set 2. take if indetereLet ∈ /inus A, where 0/π(x) is null fuzzy set, / ministic part of x,� = 0, intuitionistic ∀x ∈ � 0. value ofFand x, andsame the Example 3.by,Let U, ν(x) A Example �{A} : the E→ IFS(U) soasthat F�{A} (e) = � 0/ given of parameters.ministic Let2.IFS (U) denotes set of i.e., valuedof/(0.4, x, µ(x) = 0; /(0.9, the non/ part π(x) = 0, �∀x ∈ � 0. Let of us x, take {d F�{A} the (s1 )membership =fuzzy 0.5), 0.3), 0.1), d4 /(0.7, 0.2)} Example 3. Let1 /(0.2, U, A and E are as ind3Example 2 same / isE.null if e of ∈ / A, set, all intuitionistic fuzzy sets U where and A 0⊂ A intuitionistic membership value of x, ν(x) = 1 and the indeter2. Letdus take 3.an Let U,{d A1and E fuzzy are same as i.e., membership value ofinx, µ(x) 0; {d the nonF�{A} (s )the = /(0.2, 0.5), dsoft 0.3), /(0.9, 0.1), d4 /(0.7, pair (F�{A} , E)Example is called 1intuitionistic 2 /(0.4, 3= �Example / 0.8), d4 /(0.7, 0.2)} , ministic of x, π(x) = 0, ∀x ∈,� 0. F = 0.2), d0.2)} 2 /(0.8, 3 /(0.1, {A} (s2 ) part 2. Let us take membership value of x, ν(x) = 1 and the indeter� � is a mapping set (IFSS) [22] overF�U, (s where F {d (s ) = /(0.2, 0.5), d /(0.4, 0.3), /(0.9, 0.1), d4 /(0.7, 0.2)} F 1 1 2 {A} {A} �{A} 0.8), d43/(0.7, 0.2)} , 0.3), Example 3. {d Let U, A and Ed2are same as din 2 ) = {d 2 /(0.8, 3 /(0.1, )= /(0.2, 0.6), d33 Example /(0.8, 0.1)} , and F {A}ministic � 1 /(0.6, / (s part of�dx,0.2), π(x)d= 0, ∀x ∈ 0. � � � {d (s ) = /(0.2, 0.5), /(0.4, 0.3), d /(0.9, 0.1), d /(0.7, 0.2)} , F / :E→ so that F{A} given by, F{A} {A} 1 IFS(U) 1 2 (e) = 0 3� 4 2. Let us take {d F (s ) = /(0.8, 0.2), d /(0.1, 0.8), d /(0.7, � 23 /(0.8, �{A} F{A}Example (s3 ) = {d3.1 /(0.6, 0.3), d2 /(0.2, 0.6), das , and {d20.1)} = Example 0.1), d32 /(0.7, 0.2), d43 /(0.5,0.2)} 0.2), d, 4 /(0.8, 0.2)} 5 ) in 1 /(0.6, Let U, A and E are F same � {A} (s / if e ∈ / A, where 0 is null intuitionistic fuzzy set, � F{A} (s2 ) = {d2 /(0.8, 0.2), d3 /(0.1, 0.8), d4� {d,11/(0.6, (s310.2)} ={d /(0.2,0.3), 0.5),dd22/(0.2, /(0.4, 0.3), d /(0.9, 0.1), d4 /(0.7, 0.2)} F�/(0.7, {A}(s 2. Let take ))= {A} {d )= d2 /(0.7,FThen 0.2), dthe 0.2), d4 /(0.8, 0.2)} . 0.6), d33/(0.8, 0.1)} , and i.e., the membershipF�value x,us µ(x) = 0;0.1), the non1 /(0.6, 3 /(0.5, {A} (s5of IFSS is given below. � =�{d d2 /(0.2, 0.6), d3� /(0.8, , and F{A} (sof3 ) x, 1 /(0.6, membership value = 1 0.3), and the indeter{d (s520.1)} = /(0.8, 0.2), /(0.1, 0.8), /(0.7,0.2), 0.2)}d,4 /(0.8, 0.2)} (s ))= dd23/(0.7, 0.2), FF�{A} {dgiven (sIFSS 0.5), d2 /(0.4, d3{d /(0.9, d4 /(0.7, 0.2)} , dd34/(0.5, Fν(x) {A}0.3), 12/(0.6, 1 ) = is 1 /(0.2, {A} , {d0.1), (s10.1), Then the below. 1 /(0.2, 0.5), d2 /(0.4, 0.3), d3 /(0.9, 0.1), d4 /(0. � / ministic part of x, π(x) = 0, ∀x ∈ 0. � 0.2), d4 /(0.8, . 0.2), d0.6), F{A} (s5 ) = {d1 /(0.6, 0.1), d2 /(0.7, 0.2), d3F�/(0.5, d0.1)} (s , {d0.2)} 0.8), 0.2)}) , {d (sthe = /(0.6, 0.3), d3,/(0.8, , and 20.2)} 2 /(0.8, 3 /(0.1, 4 /(0.7, 3 )0.3), {d12,/(0.8, F�{A} (s2 ) = (s 0.2),0.5), d3 /(0.1, 0.8), dIFSS , dd24/(0.2, given below. d13is {d1 /(0.2, dThen /(0.4, /(0.9, 0.1), /(0.7, 0.2)}) 4 /(0.7, 2{A} Example 3. Let U, A and E are same as in Example {d (s , /(0.6, 0.3), d /(0.2, 0.6), d /(0.8, 0.1)}) , � 3 1 2 3 , E) ={dd14(s (3�F/(0.1, , {d2 /(0.8, 0.2), dF /(0.7, 0.2)}) Then the IFSS is given below. ( {A}(s = (s d30.2), = /(0.6, d2,0.5), /(0.7,d20.2), d30.3), /(0.5, d40.1), /(0.8,d40.2)} , {d)10.1), /(0.2, /(0.4, /(0.9, /(0.7 5 )0.8), {A} 1 {d21 /(0.6, 0.3), d2 /(0.2, 0.6), d3 /(0.8, 0.1)} , and F� 2. Let us take {A} (s3 ) � / , 0 , s {d (s , /(0.6, 0.3), d /(0.2, 0.6), d /(0.8, 0.1)}) , 4 � 3 1 2 3 , E)(s=1 , {d1(/(0.2,)0.5), d2 /(0.4, 0.3), d3 /(0.9, (F{A} (s /(0.8, 0.2)}) 0.2), d3, /(0.1, 0.8),. d4 /(0.7, 0.2)}) , 0.1), d42 /(0.7, 2 , {d the is given {ds1,/(0.6, (s5 ) =0.3), 0.1), d2 /(0.7, 0.2), dIFSS 0.2), dbelow. 0.2)} . F� d2 /(0.4, d0/ 3 /(0.9, 0.1), d4Then /(0.7, 0.2)} , (s F�{A} (s1 ) = {d1 /(0.2, 0.5), 3 /(0.5, 4 /(0.8, � {A} {d , /(0.6, 0.1), d /(0.7, 0.2), /(0.5,0.1)}) 0.2), d,4 /(0. (s , 5 1 2 {d , /(0.6, 0.3), d /(0.2, 0.6), dd33/(0.8, {d (s , /(0.8, 0.2), d /(0.1, 0.8), d /(0.7, 0.2)}) , � 4 3 1 2 2 3 4 = ( , E) ( F ) 2 {A} {d , /(0.2, 0.5), d /(0.4, 0.3), d /(0.9, 0.1), d4 /(0. (s 1 1 2 3 (sthe /(0.6, 0.3), d0.2)} /(0.2, d30.2), /(0.8, �{A} , E) 3 , {d 1(s 20.1), {d /(0.6, /(0.7, d30.1)}) /(0.5, d4 /(0.8, 0.2)}) 0/ ,0.2), , s4 , � .(F� , E) is =Then IFSS below. F�{A} (s2 ) = {d(2F /(0.8, 0.2), d3 /(0.1, d4given ,d20.6), 5 , is 1/(0.7, )0.8), ( Tabular representation of the IFSS {A} {d (s , /(0.8, 0.2), d /(0.1, 0.8), d /(0.7, 0.2)}) , 2 2 4 3 0/ , (s , {d /(0.2, 0.5), s4 , � {d , /(0.6, 0.1), d /(0.7, 0.2), d /(0.5, 0.2), d /(0.8 (s 5 1 2 3 4 d /(0.4, 0.3), d /(0.9, 0.1), d /(0.7, 0.2)}) , shown in Table 3. 1 1 2 3 4 d /(0.2, 0.6), d /(0.8, 0.1)} , and F�{A} (s3 ) = {d1 /(0.6, 0.3), 3 (s 0.6), d3 /(0.8, 0.1)}) , �{A}(,F� , E) is Tabular representation of the IFSS 3 , {d)1 /(0.6, 0.3), d2 /(0.2, 2 E) = ( F {A} ( (s5 , {d1 /(0.6, d2 /(0.7, 0.2), d3 /(0.5, 0.2), d4 /(0.8, 0.2)}) {d2 /(0.8, (s ,0.1), 0.2), d3 /(0.1, 0.8), d4 /(0.7, 0.2)}) , Table0.2), 3. d32/(0.5, 0/ , of the IFSS (F�{A} , E)is s4 , � 0.1), d2in/(0.7, 0.2), d /(0.8, 0.2)} . F�{A} (s5 ) = {d1 /(0.6,shown Tabular representation 4 (s3 , {d)1 /(0.6, 0.3), dTable d3 /(0.8, 0.1)}) , of (F� , E) . Tabular 2 /(0.2, (F�{A} , E) = 3.0.6), ( {A} 3.5 , {drepresentation Tabular representation of � the IFSS (F�shown d3 /(0.5, 0.2), d4 /(0. 1 /(0.6, 0.1), d2 /(0.7, 0.2), {A} , E)inisTable (s / , 0 , s Then the IFSS is given below. 4 � , E) Table 3. Tabular representation of ( F shown in Table 3. {A} {d1 /(0.6, 0.1), d/Tabular /(0.7, 0.2), d3 /(0.5, 0.2), d4 /(0.8, 0.2)}) (s5 ,0.3), representation of the IFSS (F�{A} , E) iss 2E d3 /(0.9, 0.1), d4 /(0.7, 0.2)}) , (s1 , {d1 /(0.2, 0.5), d2 /(0.4, � s s s U s , E) Table 3. Tabular representation of ( F 1 2 3 5 {A} 4 / shown in Table 3. , {d2 /(0.8, 0.2), drepresentation /(0.7, 0.2)}) , (0.2,0.5) 3 /(0.1, 0.8), d4of �{A} (s2Table d 0 (0.6,0.3) 0 (0.6,0.1) , E) 3. Tabular ( F 1 � s2 d /(0.8, s0.1)}) s5 is U E Tabular representation of the 3 IFSS,s(4F{A} , E) (s 0.3),s1d2 /(0.2, 0.6), 3 (F�{A} , E) = . ( 3 , {d)1 /(0.6, (0.8,0.2) (0.2,0.6) 0 (0.7,0.2) d/2 0(0.4,0.3) d (0.2,0.5) 0 (0.6,0.3) (0.6,0.1) shown in Table 3. 1 s2 s3of (F� s4, E) s5 U Table E sTabular 1 � 3. representation / , 0 , s / {A}0 4 (0.1,0.8) (0.8,0.1) (0.5,0.2) d3 0(0.9,0.1) (0.4,0.3) (0.8,0.2) (0.2,0.6) (0.7,0.2) d 2 d (0.2,0.5) 0 (0.6,0.3) 0 (0.6,0.1) s s s s U E s 1 2 3 41 5 (0.7,0.2) (0.7,0.2) 0 0 (0.8,0.2) d 0.1), d /(0.7, 0.2), d /(0.5, 0.2), d /(0.8, 0.2)}) (s5 , {d1 /(0.6, 4 40 2 3 (0.8,0.1) (0.9,0.1) (0.5,0.2)(0.8,0.2) (0.2,0.6) 0 (0.7,0.2) d3 Table 3. Tabular representation (F� (0.4,0.3) d1 (0.2,0.5) 0 (0.1,0.8) (0.6,0.3) 0d/2 of(0.6,0.1) {A} , E) 0 0 d5 0 s(0.8,0.2) (0.7,0.2) (0.7,0.2) 0U d4 s02 s03 s05 1 (0.9,0.1) (0.1,0.8) (0.8,0.1) 0s4 (0.5,0.2) (0.4,0.3) (0.8,0.2) 0d3 E(0.7,0.2) d2 E) is Tabular representation of the IFSS (F�{A} ,(0.2,0.6) 0(0.1,0.8) 0(0.8,0.1) 00dd41 (0.5,0.2) 0(0.7,0.2) 0 (0.7,0.2) d5 / (0.2,0.5) 0 (0.6,0.3) (0.6,0.1) 0 00 (0.8,0.2) (0.9,0.1) shown in Table 3.d3 U E s1 s2 s s s 3 4 5 (0.4,0.3) (0.8,0.2) (0.2,0.6) 0 0 0 00 (0.7,0.2) 0 (0.7,0.2) (0.7,0.2) 0 0dd52 (0.8,0.2) d4 d1 (0.2,0.5) 0 (0.6,0.3) 0 (0.6,0.1) (0.9,0.1) (0.1,0.8) (0.8,0.1) 0 (0.5,0.2) d 0 0 0 0 3 0 d5 Table 3. Tabular representation of (F�{A} , E) d2 (0.4,0.3) (0.8,0.2) (0.2,0.6) 0 (0.7,0.2) (0.7,0.2) (0.7,0.2) 0 0 (0.8,0.2) d4 d3 (0.9,0.1) (0.1,0.8) (0.8,0.1) 00 (0.5,0.2)0 0 0 0 d5 / d4 (0.7,0.2) (0.7,0.2) 0 0 (0.8,0.2) s2 s3 s4 s5 U E s1 d 0 0 0 0 0 d1 (0.2,0.5) 0 5 (0.6,0.3) 0 (0.6,0.1) (0.4,0.3) (0.8,0.2) (0.2,0.6) 0 (0.7,0.2) d2 (0.9,0.1) (0.1,0.8) (0.8,0.1) 0 (0.5,0.2) d3 (0.7,0.2) (0.7,0.2) 0 0 (0.8,0.2) d4 0 0 0 0 0 d5 / MAXIMISING ACCURACY AND EFFICIENCY OF. . . parameters. IV IF(U) denotes the set ofset all(IVIFSS) intervalMAXIMISINGvalued ACCURACY AND EFFICIENCY OF. . . intuitionistic fuzzy sets of U. Let Let Abe⊆anE.initial A andOF. E .be MAXIMISINGU ACCURACY AND universe EFFICIENCY . a set of � 2.4 Interval-valued intuitionistic fuzzy soft pair (F�{A} , E) is an IVIFSS [26] over U, where F parameters. IV IF(U) {A} denotes the set of all intervalMAXIMISING ACCURACY AND EFFICIENCY OF. . . MAXIMISING ACCURACY AND EFFICIENCY OF. . . set (IVIFSS) is a mapping, given by fuzzy softvalued intuitionistic fuzzy sets of U. Let A ⊆ E. A 2.4 Interval-valued intuitionistic 61 (F�{A} IVIFSS [26] over U, F�{A} 2.4 pair Interval-valued intuitionistic Let U, E) be is ananinitial universe andfuzzy E where be soft a set of set (IVIFSS) is aset mapping, given bydenotes the set of all interval(IVIFSS) parameters. IV IF(U) 2.4 Interval-valued intuitionistic fuzzy soft Let U be an initial universe beIV a set of2.4 Interval-valued intuitionistic fuzzy soft F�{A}and : EE→ IF(U). valued be intuitionistic fuzzy setsand of E U.be Leta A E. A set (IVIFSS) 2.4 parameters. Interval-valued intuitionistic fuzzy soft an initial universe set⊆of (IVIFSS) IV IF(U) denotes the set of all interval-Let U�set � pair (F{A}IV, E) isF{A} andenotes IVIFSS [26] where F�{A} : E → IV IF(U). set (IVIFSS) IF(U) the setover of allU,intervalvalued intuitionistic sets ofisU. aLetparameterized A ⊆ E.parameters. A An fuzzy IVIFSS family of MAXIMISING ACCURACY AND EFFICIENCY OF. . . Let U be an initial universe and E be a set of Let U be an initial universe and E be a set of is aintuitionistic mapping, given by sets � valued fuzzy of U. Let A ⊆family E. A of pairU(F�be , IF(U) E) is andenotes IVIFSS [26] over parameters. IVan the set of {A} {A}parameters. Let initial universe and E all beU,intervalawhere set ofF An IVIFSS is adenotes parameterized interval-valued intuitionistic fuzzy subsets of U. IV IF(U) the set of all interval� pairinterval-valued (F{A} , E) is an IVIFSS [26] over U, where F� of U. is aintuitionistic mapping, given bysetsthe valued fuzzy of set U. of Let ⊆ E. Aof parameters. IV IF(U) denotes allA intervalintuitionistic valued intuitionistic fuzzy setsfuzzy of U. subsets Let A ⊆{A} E. A Thus, its universe is the set all interval-valued � mapping, givenFby : Ethe →set IV IF(U). {A}is pair (F�{A} , E) is an IVIFSS [26]ofover F�{A} � valued intuitionistic fuzzy sets U. U, Letwhere A ⊆ E. A is a pair Thus, its universe of all interval-valued (F{A} , E) is an IVIFSS [26] over U, where F�{A} intuitionistic fuzzy sets of U, i.e., IV IF(U). � � is a mapping, given by pair (F{A} , E) is an IVIFSS U, where F{A} intuitionistic fuzzy U, i.e., IV IF(U). F�{A} : E [26] → IVover IF(U). is a mapping, givenissets bya ofparameterized An IVIFSS family of �{A} : E → IV IF(U). F is a mapping, given by fuzzy subsets of U. ∀ε ∈ A, F(ε) interval-valued intuitionistic ∀ε ∈ A, F(ε) An IVIFSS is a parameterized family of F�{A} : E → IV IF(U). Thus, its universe is the set of all interval-valued F�{A}a : parameterized E → IV IF(U). family of An IVIFSS is interval-valued intuitionistic fuzzy subsets of U.is an interval-valued intuitionistic set of U. intuitionistic fuzzy setfuzzy of U. �{A}an intuitionistic sets U, i.e.,subsets IVfuzzy IF(U). Fis : Einterval-valued → IV IF(U). interval-valued intuitionisticof fuzzy of U. Thus,IVIFSS its universe isparameterized the set of all interval-valued An is a family of F(ε) is expressed as: An IVIFSS is a parameterized family of F(ε) is expressed as: its universe is the all interval-valued intuitionistic fuzzy offuzzy U, i.e.,subsets IV family IF(U). ∀ε set ∈ A,ofF(ε) interval-valued of U. An IVIFSSintuitionistic is a sets parameterized of Thus, interval-valued intuitionistic fuzzy subsets of U. intuitionistic fuzzy sets of U, i.e., IV IF(U). Thus, its universe is the set of all interval-valued interval-valued intuitionistic fuzzy subsets of U. Thus, its universe is the set of all fuzzy interval-valued ∀ε ∈ A, F(ε) is an interval-valued intuitionistic set of U. l r l F(ε) = {< x, [µ (x), µF(ε) (x)], [νIV (x), νrF(ε) (x)] > |x ∈ X}. intuitionistic fuzzy is setstheof set U, i.e., IVinterval-valued IF(U). Thus, its universe of all ) ) F(ε F(ε F(ε) ∀ε ∈ A, intuitionistic fuzzy sets of U, i.e., IF(U). is expressed as: l r F(ε) =of {< [µlF(ε µrF(εset) (x)], [νF(ε) is an interval-valued intuitionistic fuzzy of U. ) (x), intuitionistic fuzzy sets U,x,i.e., IV IF(U). F(ε) (x), νF(ε) (x)] > |x ∈ X}. ∀ε ∈ A, F(ε) is anExample interval-valued fuzzyassetin of U. ∀ε ∈ A,EF(ε) 4. Let U,intuitionistic A and are same Example F(ε) is expressed as: ∀ε ∈ A, F(ε) F(ε) is expressed as: 3. Let usintake l l is an interval-valued intuitionistic fuzzy set of Example 4. Let U, A and E U. are same F(ε) = {<Example x, µrF(ε ) (x)], νrF(ε) is anas interval-valued fuzzy of(x)] U.)> |x ∈(X}. ) [νF(ε) ((x),set { [µF(ε() (x),intuitionistic F(ε) expressed as: is an isinterval-valued intuitionistic fuzzy set of U. (0.4, 0.5) (0.7 (0.2, 0.4) F(ε) is )expressed 3. Let us take l r l r �{A} , d3 / , d2 / (s = d1 / as: = {< x, [µ > |x( ∈1X}. F(ε ) (x), µF(ε ) FExample ) ) { ) (x)],([νF(ε) (x), νF(ε) (x)] ( ( F(ε)F(ε) is expressed as: 4. l Let U, A E arel same as rin Example (0.3, 0.4) (0.1 (0.5, r and0.6) {< x,(0.4, [µF(ε0.5) (x), ν0.8) ) (x), µF(ε ) (x)], [νF(ε) (0.2, 0.4) F(ε)3.=Let (0.7, (0.6, 0.7 F(ε) (x)] > |x)∈ X}. ) { ( ( ( us take � , d , d d (s d / / / / 1 1 2 3 4 lF{A} r) = l same asr in Example Example 4. Let U, A and E are (0.6, 0.8) (0.1, 0.2), (0.5 l r l r ) ) ( ( { ( F(ε) = {< x, [µF(ε ) (x), µF(ε ) (x)], [νF(ε) (x), νF(ε) (x)] > |xF(ε) X}. (0.5, 0.6) (0.1, 0.2) 0.3 = (x), νF(ε) (x)] > ,|xd(0.2, F�∈{A} (s )(0.3, = x,U, d[µ0.4) /A )and / X}.(0.7 ) (x)], 2F(ε 3 /in (x), 4∈ (0.4, 0.5) (0.2, 0.4) Example 4.2{< Let EµF(ε are same,[νdasF(ε) Example l r l r take (0.1, 0.2) (0.7, 0.8) (0.2 � , d , d F / / (s ) = d / F(ε)3.=Let {< us x, [µ (x), µ (x)], [ν (x), ν (x)] > |x ∈ X}. 2 1 1 ) ) 0.6) )} ) 3 ( (0.1 F(ε )( (F(ε) ){A} ( ( ( { ) ( F(ε ) { F(ε) (0.3, 0.4) (0.5, 3. Let us(take ) { ( ) ( )} Example 4. Let U, A and(0.2, E are same as(0.6, in Example 0.8) (0.1, 0.2), (0.5, 0.7) (0.4, 0.5) (0.7, 0.8) (0.6, 0.7) 0.4) Example 4. Let U, A and E are same as in Example (0.6, 0.7) (0.2, 0.3) ) { ( ( �{A} ( 0.4) ( 0.5) ( (0.6 F�{A} d2in / Example , d / 3 ) = {d1 /(0.2, d4,/d4 / ) , d2 /(0.4, , ) , ), d(3 /(0.7, =F d1A /(s =are dsame /d3(s 1 )Let F�{A} 2) 2 / ,as 3, Let 0.8 3. Let us (s take Example 4. U, and E (0.3, 0.4) (0.1, 0.2) (0.2, 0.3) (0.5, 0.6) 3. us take (0.6, 0.8) (0.1, 0.2), (0.5 � (0.2, 0.3), d2 /,(0.2, (0.6, 0.7), d3 / , d / (0.1 (0.1, 0.2) )F{A}F�(s(1 )(s= )(0.7, 0.3) d1{ / d0.8) ) ) )} { ( ( ( d = / / 2 2 3 4 ) ) ( ( ( {A} 0.4))} (0.1,(0.2 0.2 3. Let us take { (0.2, ) ( )} ( 0.4){ )( (0.4, ( 0.5) ) and ( 0.8) (0.5,(0.2, (0.1, 0.2)( (0.7, 0.8) )0.6)(0.6, (0.7, 0.7) (0.3,(0.4, 0.5) (0.7 ) , d2 / ( (0.6,(0.1, ) , d3F�/ ( (s ){ )(d4 / ( 0.4) )} (0.6, 0.8) 0.2), (0.5, 0.7) , F�{A} (s 1 ) = {d1 / ( ) ) { ( ( ) ) ( ( ( , d , d = d / / / (0.2, 0.3) (0.6, 0.8) 0.7) � 3 {A} , d4(0.1, ,0.6) F{A} (s2 ) = � d2 /(0.5, /1 0.2) 0.6) (0.3, 0.4) 0.3) (0.2, 0.4) 0.5) (0.7, 0.8)1 (0.6,(0.5, (0.6, 0.7) 2 (0.1 0.7) (0.5.0.7) (0.4 0.2),0.4) (0.5, 0. 0.8) (0.2, 0.3), (0.6 , d(0.2, = 0.2) d,1d/2 /, d3 /(0.4, (0.7, 0.8) (0.2, 0.3) 2 /)(s 3 / , d /, d /(0.1, d,3d d4 /(0.6, , (0.3, F�{A} (s1 ) = d1F /{A} (s3 )(0.1, �/(s �,{A} , d F ) = d / F , d / = d / 5 1 2 3/ ( 2 2 3 4 {A} (0.6, 0.7) (0.1, 0.2) (0.2, 0.3) ) ) )} { {( (0.5, ( ( 3 0.6) (0.3, 0.4) (0.1, 0.2) (0.2, 0.3) ) ) { (0.1, ( )} ( ) ) ( ( ( (0.1, 0.2) (0.2, 0.3) (0.2 (0.7,(0.6, 0.8)0.7) (0.2, 0. 0.2)0.3) (0.2, (0.1 (0.6,(0.6, 0.8) 0.7) (0.1,(0.2, 0.2),0.3) (0.5,(0.6, 0.7) 0.8) ) , d3 /,( ) , dF )}(0.6, ((s ) ) (3 / (0.1, 0.2), (d4 / (0.5 {= d(20.8) F�{A} F , (s /( / � �{A} 2) = 4 and , d , ) / d , d , (s3{) d=2and d1 /(0.6, / / 2 {A}3 (0.2, 2 (0.1, (0.1,(0.2, 0.2) (0.7,(0.6, 0.8) 0.3) 0.8){ 0.2),0.7) 0.7) (0.2, 0.3)) 0.8 (0.6, 0.7)0.2)(by 0.8), d)/( (0.6, ) (=(0.5,d(0.1, 0.3), d ( 0.2) ) { ( (0.1, ( (0.7, ( (0.2 the F�{A} (s2 ) = {d2 / ( , d4(s ,)is given /(3 )Then F�{A} , d) / IVIFSS 3/ ( 2/ 3 1{ ) ) )} (0.6, 0.7) (0.5.0.7) (0.4 ) ( ( ( (0.6, 0.7) (0.5.0.7) (0.4, 0.6) (0.7, 0 (0.1, 0.7) 0.2) (0.7, 0.8) (0.2,0.8) 0.3) (0.2, 0.3) 0.7) 0.2 and (0.2, 0.3) (0.6, �( (0.6, , d2 /(0.6,)} , d3 /(0.1,(0.6 / ) 5 ) = d1)} (0.6, 0.7) (0.2, 0.3) dF��/{A} , d , d (s / (s / / 2 3 4 ( (5 ) = )d,1d/2)/ ( (0.1, ( 0.2) )and ( ( F�{A} (s3 ) = {d1F , d,3) , /{{A} (0.1, 0.2) (0.2, (0.2, 0.3) , d3 /(0.1, (0.2 F{A} (s , d20.8) =0.8) d10.6) / { / 0.3) 0 3 )(0.2, (0.6,(0.5.0.7) 0.7) (0.1, (0.2,(0.6, 0.3) 0.2) (0.6, 0.7) 0.7) (0.2, 0.3) (0.6, () (0.7, () ( (0.3) (0.2, 0.3) {(0.4, ( )(0.6,. 0.7) ( )(0.1 �{A} F �{A} F , d , d , (s ) = d / / / (s ) = d / , d / , d / , d / 3 1 2 3 (0.2, 0.4) (0.4, 0.5) 5 1 2 3 4 0.3) (0.6, (0.5.0.7) and (0.2,(0.1, 0.3) 0.2) (0.6,(0.2, 0.7) 0.3) (0.1, 0.2) 0.2) , d2 / , d3 / (0.4,,0.6 d3 d(1 given / , d(0.1, s1 , 0.7) the IVIFSS is by F�{A}and (s(5 )Then = d(0.2, / 1{ 2 / 0.6) ) ) ) )} { ( ( (0.5, (0.3, ) ) 0.4) ( ( ( ) 0.3 (0.1, 0.2) (0.2, 0.3) (0.2, Then the IVIFSS is given by and ( ) { ( ( (0.6, 0.7) (0.5.0.7) (0.4, 0.6) (0.7, 0.8) (0.6, 0.7) (0.4 ) , dby ) , d3F�/ ( (s ) = )} .(0.5.0.7)(0.1,, d0.2), (0.6,, d0.8) F�{A} (s5 )Then = {dthe 1/ ( 2/ ( IVIFSS is given d1 /)s,2d, 4 /d(2 / (0.1, 5 2/ 3/ {A} (0.2, , d , d4 / (0.1, 0.2) (0.2, 0.3) 0.3) 0.2) (0.6, 0.7) (0.5.0.7) (0.4, 0.6) (0.7, 0.8) 3 )(0.2, 0.3) { ( ( (0.7, 0.8) )(0.2 ( ,isd (0.1, the IVIFSS given by(0.1, 0.2) , d2 / , d3Then . F�{A} (s5 ) = d1 / / / 0.2) 4{ (0.2, 0.4) (0.4, 0.5) ( ) ) ( ( (0.1, 0.2) ( {(0.2, 0.3) (0.2, 0.2) ) 0.3) ) 0.7) (, d2 / (0.2, 0.3) ) ,( ( (s1 , d1 /(0.1, d (0.6, ) ) )}) { ( ( (0.4) (s (0.4, ( F�{A} , E))= , d33 ( is given (0.5, 0.6) (0.3, 0.4) Then the IVIFSS by (0.2, 0.5) (0.7, 0.8) , d , d / / Then the IVIFSS is given by 3 1 2 ( ) ) { ( ( (0.2, 0.4) (0.4, 0.5) 0.8) (0.2,,0.3) ) d4 / ( d)3 /(0.6,(0.7) (0.6, / {(0.7, 2 (0.6, (0.1, 0.2), , d( d4 /0.8) ,0.7) d1 / sis1 ,given )( 0.4) Then the IVIFSS by s1 , d1 /, d2 /(0.5, 0.6) , d 3 / ( (0.3, (0.1, 0.2) (0.2, 0.4) (0.4, 0.5) s , d , d / / (0.5, 0.6) (0.3, 0.4) (0.1, 0.2) (0.2, 0.3) 3 ( )}) { )( ( ( , d)3 / , d ,(s4d2,1� / ,2)( (0.1, / )}) s1( )( )}) ( ( ) 0/{ (0.7, ) ) { {( ()0.2), d)2( 0.8) ( ( ( (0.5, 0.6) (0.3, 0.4) (0.1, 0.2), ( (0.5, ) (0.5, {( 0.7) ( (0.2,(0.6, ( (0.4, ({ (0.1, (0.6, 0.8) 0.2), 0.7) (0.6, 0.4) 0.8) , d /(0.4, 0.5) (0.7, 0.8) ) ) ( 0.7) (0.2, 0.3) , d/ d2 /(0.2, ) ,dd22/ (3 ),,E) )(0.6, ( 0.4) s2 ,0.2) d( ,,) d4 0.7) / , d(2)}) 4/ s , d / 0.5) ( (0.5.0.7) d d4 /(0.6, , 0.2), s1 , {sd21,/ ( (F�{A} 3 3,/= 0.8) (0.1, , d, / 1 1 (0.1, (0.7, 0.8) (0.2, 0.3) 3 1 2 , d , d / / s (0.5, 0.6) (0.3, 0.4) (0.1, 0.2) (0.2, 0.3) (0.2, 0.4) ) ) ( (0.4, 0.5) 0.8) (0.6, (0.1, 0.2)),) 0.3) , d4,/d333 s(0.7, ,0.7) d) , 5d(0.7, / 1 0.8) (0.5, 0.6) (0.3, 0.4) ( ) )}) { ( ( ( 2( 2{ 3 / 2(0.2, (0.2, 0.3) (0.6, 0.7) ( )}) { ( ( (0.1, 0.2) (0.2, 0.3) ( ) )}) { ( ( , d d d , , d / / / / s ( ) ( ( 1 1 2 3 4 0.2) )(0.8) (0.1, (0.7, 0.8) (0.2, 0.3) (0.6, 0.8) (0.1, 0.2), (0.5, 0.7) (0.5,(0.6, 0.6) 0.7) 0.4) (0.1, 0.2) (0.2, ). {(0.6, ( =(� {s3 , ( (F�{A} , E) (0.6, 0.8) (0.1, 0.2), (0.6, 0.7) 0.3) 0.8) d2 /(0.3, , ) 0.3), d((0.6, d= � ) , d3 /,( ) , d4 /, d( )}) ( 1/ 3 / s4(0.2, s , , d / ( F , E) / , 0 , 2 2 (0.6, 0.7) (0.2, 0.3) , d4 d / / � {A} 2 2 3 (0.2, 0.3) (0.6, 0.7) (0.1, 0.2) s , d , d , , d / / / ( F , E) = � (0.1, 0.2) (0.7, 0.8) (0.2, 0.3) 3 1 2 3 (0.6, 0.8) (0.1, 0.2), (0.5, 0.7) {A} ,(E)is Tabular representation of( the IVIFSS( ) ) { s3( , d) ,( d(0.6, / )}) ({d /)( (0.1, 0.2), d) (0.7, 0.8) {A}( 1{ 2 /F(0.1, 3/ ) ) ( ( ( (0.2, 0.3) 0.7) 0.2) s , d , d , , / / ( 2 2 3 4 ( (0.2,(0.6, 0.3)0.7) (0.6,(0.5.0.7) 0.7) s4 , � (0.2) (0.6, (0.6, 0.7) ) (0.2, 0.8) 0.3) Table 4.) (0.6, 0.7) , d2 / (0.2, (F�{A} , E) = ,d ,0.3) d1)}) / ) s50.8) .0.3) ( ) , d2)/ ( (0.7, ),,E) {d10//{(, (0.1, ( (0.2, (shown F�{A} = s3 ,( din , / 3 ) )}) ( ( ( ( � � s , d , d33 , d / / (0.1, 0.2) (0.2, 0.3) 3 1 2 / s , 0 , / s , 0 , (0.2, 0.3) (0.6, 0.7) (0.1, 0.2) 4 (0.6, 0.7) (0.2, 0.3) (0.6, 0.8) 4 � (0.2, 0.3) (0.6, 0.7) (F{A} , E) = . (0.6, 0.7) (0.5.0.7) (0.4, 0.6) (0.7, 0.8) ( ) /( ) ) (( ( ({ )( s3 , sd51,/ d1 / d ,,) {, d2 /,( ( 3 /, d d2 /(0.6, 0.7) ,) d4 / ) ( ( (0.5.0.7) 3 (0.1, 0.2) (0.6, 0.7) (0.2, 0.3) 0.2) s4 , � 0/ ) , � � (0.1, (0.2, 0.3) (0.2, 0.3) (0.1, 0.2) ( (0.6, 0.7) (0.5.0.7) (0.4, 0.6) / s , 0 , , E)is Tabular representation of the IVIFSS( F 4 {A} , d2 / )}) ,,dd)3 // , d1{ / )( ( , d / s5 ( / { ( ) , d( (s , � s , ) d1 / ( ( 0.3) (0.1, 0.2)(0.7, (0.2, 2 (0.4, 3 ) 4 / , 4 0 shown in Table 4. (0.2, (0.6, 0.7)5 (0.5.0.7) 0.6) 0.3) 0.8)(0.2, (0.1, 0.2) 0.3) (0.6, 0.7) (0.5.0.7) � ) ) ) )}) { ( ( ( ( ( s , d / , d / , d / , d / , E)is Tabular representation of the IVIFSS( F 5 1 2 3 4 (0.2, {A} , d2 / , d3 , d1 / s50.3) (0.1, 0.7) 0.2) (0.2, 0.3) representation (0.1, 0.8) 0.2) (0.6, (0.5.0.7) (0.4, 0.6) (0.7, 0.3) Table Tabular of the IVIFSS( shown in , d2 / , d3 /� , d4(0.1, / 0.2)F�{A} , E)is (0.2, s5 , 4.d1 / (0.1, 0.2) 0.3) (0.1, 0.2) , E)is Tabular representation of (0.2, the 0.3) IVIFSS( F{A} in Table 4.(0.2, Tabular representation of the IVIFSS(F�{A} , E)is shown Tabular representation of the IVIFSS(F�{A} , E)is inofTable 4. shown Table shown 4. Tabularinrepresentation the IVIFSS( F�{A} , E)is shown in Table 4. shown in Table 4. / U E ( ) ( ) ( ) ( ) s s s s s 1 2 3 4 5 (0.4, 0.5) (0.6, 0.8) (0.2, 0.3) (0.5.0.7) ( ) (0 ) Tambourat d2 ( (0.2, 0.4) ) (0.6, 0.7) (0.6, 0.7) � (0.3, 0.4) (0.1, 0.2) (0.6, 0.7) (0.2, 0.3) ) representation0of (F{A} , E) d1 ) ( ( ((0.5, 0.6) ) ( 0 Table 4.)Tabular 0.3) 0.8) / Tambourat (0.4,)0.6) ( (0.7, 0.8)) ( (0.1, 0.2), ) ( (0.2,(0.6, ) ( (0.1, 0.2) s s U E s 0T., d 1 2 �{A} 3 (0.4, 0.5) (0.6, 0.8) 0.3) (0.5.0.7) Tambouratzis Tambouratzis T.,(0.2, Souliou T., D., Souliou Chalikias Tambouratzis D., Chalikias M., Gregoriades M., Souliou Gregoriades A. D., Chalikias A.3 M., Table 4. Tabular representation of ( F , E) ( ) ( ) G 62 (0.2, (0.1, 0.2) ) ( (0.7, 0.8) ) (0.2, (0.1, d2 0.4) 0.2) 0 (0.6,0.3) 0.7) ) ((0.3, ( 0.4) ) ( (0.1, 0.2) d) (0.6, 0.7) ) 0 (0.2, 0.3) /1 ) ( ( (0.6, (0.5,s1representation 0.6) ) ( of s(2F (0.2, 0.3) ) 0.7) (0.5, Table 0.7) (0.7, �{A} , ) s0.8) U E 4. ( Tabular E) ( 3 ( ( (0.6, 0.8)0 ) ( ) d4 0 (0.4, 0.6) (0.1, 0.2), (0.7, 0.8) � (0.4, 0.5) (0.6, 0.8) (0.2, 0.3) � � Table 4.Table Tabular representation representation Table of 4.(FTabular , E) (F{A} representation , E) of (F{A} ,(0.2, E) 0.4) (0.6,0.2) 0.7) 0 d4.3 Tabular (0.2, 0.3) (0.2, 0.3) (0.1, {A}of / d 2 d1E 0.3) (0.1, 0.2) (0.3,s10.4) (0.1,s020.2) (0.6, s 0.7) ) U) ( (0.1, 0.2) ) ( (0.7, 0.8) (0 (0.2, ) ( ) ( / d5 / / ( (0.5, 0.6) 0 ) ( ( ) ( (0.2, ) 0 0 0)30.3) s (0.6,U0.7) Es s s (0.5, 0.7) s s s s (0.7, s 0.8) s s s (0.1, 0.2), s (0.7, 0.8) s (0.6, 0.8) U E U E s (0.2, 0.4) 0.7) 1 3 5 3)5 (0.6, 4 5 (0.2,) 0.8) 0.3) ( d4 ( )1 ) 2 ( 2 1( ) ( ( 400.5) (( ( d)33 2 4 )(0.4, 0) )00.8) (0.1, 0.2) (0.7, 0.2) (0.5, 0.6) (0.2, 0.3) (0.2, 0.4) (0.2,(0.2, 0.4) 0.3) (0.2, 0.4) (0.6,12 0.7)( (0.1, (0.6, 0.7) (0.6, 0.7) (0.6, (0.6, 0.7) 0.7) (0.6, 0.7) (0.3, 0.4) (0.1, 0.2) (0.6, 0.7) ) (0.2, 0.3) (0.1, 0.2) ) ( ) d1 d1 d1 0 0 0 0 0( 0 ) ( ) ( (0.6, 0.7) (0.5, 0.7) (0.5, 0.6) (0.5, 0.6) (0.5, 0.6) (0.2, 0.3) (0.2, 0.3) (0.1, (0.2, 0.2) (0.1, 0.3) 0.2) (0.1, 0.2) (0.4, 0.5) (0.6, 0.8) (0.2, 0.8) ( d5 ( ) (0) ( ( ) ( ) )0 ( ( dd)42 )(0.7, ) ( 00.8) (( ) (0.1, 0) ) 0.2),( 0 (0.6,0)0.3) 3 (0.2, 0.3) (0.2, 0.3) (0.3, 0.4) (0.1, 0.2) (0.6, 0.7) (0.4, 0.5) (0.4, 0.5) (0.6, 0.8) (0.6, (0.4, 0.8) 0.5)(0.2, 0.3) (0.2, 0.3) (0.5.0.7) (0.6, 0.8) (0.5.0.7) (0.2, 0.3) (0.5.0.7) (0.1, 0.2) ) (0.7, 0.8) (0.1, 0.2) ) ( ) ( d2 0 0( 0 d2 d2 ( ) d5 0.7) 0 0 0 (0.3, 0.4) (0.3, 0.4) (0.1, 0.2) (0.1, (0.3, 0.2) 0.4) (0.6, 0.7) (0.6, (0.1, 0.2) (0.2, (0.6, 0.3) (0.2, 0.7) 0.3) (0.2, 0.3) (0.6, 0.8) (0.1, 0.2), (0.7, 0.8) ) (0.5, ( ) ) 0.7) ( ) 0.7) (( ) ( ) ( ( ) ( )) (( d)3 )(0.6, ( ( 0) 4 0.8) (0.1, 0.2) (0.7, 0.8) (0.1, 0.2) (0.4, 0.6) (0.4, 0.6) (0.6, 0.8) (0.1, 0.2), (0.7, 0.8) (0.7, 0.8) (0.1, 0.2), (0.1, (0.7, 0.2), 0.8)(0.6, 0.8) (0.6, (0.4, 0.6) ) 0( (0.2, 0 d3 0 0.3) ) ( (0.2, 0.3) d3 d3 0.3) 0.2) (0.7,(0.8) (0.7, 0.8) (0.6, (0.1,(0.2) (0.1, (0.7, (0.1, 0.2))(0.1, 0.2) (0.2, (0.5, 3 (Interval-valued Intuitionistic 00.7) )0.2)( ) ( )0.8) ) 0.2) ((0.1, ) ( (0.2, ((0.1, )0.3) )00.7) ( (0.2, 0.3) 00) dd54 0.3)0.8) (0.2, 0.3) (0.7, 0.8) (0.6,� 0.7) (0.6, 0.7) (0.5, 0.7) (0.5, (0.6, 0.7) 0.7) 3 Interval-valued (0.7, 0.8) (0.5, 0.7) (0.2, (0.7, Intuitionistic d4 dFuzzy 0 0 uni-0 0 initial 0 0 Let (F{A} , E) bed(0.2, an IVIFSS over the 4 4 Soft (0.2, 0.3) (0.2, 0.3) Matrix 0.3) (0.2, (0.2, 0.3) 0.3) (0.2, 0.2) (0.1, 0.2) 0 (0.1, 0.2) 0 d5 0.3)Soft(0.1, 0Matrix Fuzzy andA ⊆ E. d5verse d5 set d5 U. 0Let E 0 be a 0 of 0parameters 0 0 0 0 0 0 0 0 0 0 Interval-valued Intuitionistic0 �{A} , E) be an IVIFSS over the3initial Let ( F uni� Then a subset of U × E is uniquely Let defined (F{A} , E)by be an IVIFSS over the initial uniFuzzy Soft Matrix verse U. Let E be a set of parameters andA ⊆ E. verse U. Let E be andA ⊆ E. 3 Interval-valued Intuitionistic � RA = {(u, e) : e ∈ A, u ∈ F{A} (e)},which is called a a set of parameters a (subset ×IVIFSS E is uniquely by 3 Interval-valued 3Then Interval-valued Interval-valued Intuitionistic Intuitionistic a subset 3of U × E Intuitionistic is uniquely Then defined by, Soft Let F�{A} E)ofbeUan over the defined initial uniFuzzy Matrix membership function of relation of (F�{A} , E). The �{A} (e)},which is called a R = {(u, e) : e ∈ A, u ∈ F � A verse U. LetaE be a set of parameters andA ⊆ E. RA Fuzzy =Soft {(u,Matrix e) : e Matrix ∈ A,Fuzzy u ∈ F{A} (e)},which is called Fuzzy Soft Soft Matrix �{A} RA is written as µ : U × E →Int([0, 1]) deLet Fand E) IVIFSS over thefunction initial unirelation (F�,{A} ,be E). The of R Then a (of subset Uan× E membership is uniquely defined by �{A} , E). AThe membership function ofE of relation of ( F � � � verse U. Let be a set of parameters andA ⊆ E. Let (fined F�{A} Let, E) (F be , an E) IVIFSS be an IVIFSS Let over ( F the over initial , E) the be uniinitial an IVIFSS uniover the initial uniR is written as µ : U × E →Int([0, 1]) and deRA u ∈ F{A} (e)},which is called a {A} {A} by A = {(u, e) : e ∈ A, RLet as µRAofU. : parameters ULet×andA →Int([0, 1]) and deA is Then a subset of UE.× E membership is uniquely function defined by verse U. Leta Esetbeof averse set EEbe set of ⊆ fined E. parameters andA ⊆ verse U. E written be parameters ⊆aandA E. by � The of relation of (F{A} , E). { Then �{A} (e)},which is called a fineda by R = {(u, e) : e ∈ A, u ∈ F subset ×uniquely E ais subset uniquely of rU defined × E isby uniquely defined by Then aThen subset of U ×ofE Uis defined by { l l r A RA is{ν written as µl RAν: U × Er(u)}] →Int([0, 1]) and de[{µ (u), µ (u)}, (u), i f e ∈ A r [{µ µF� (e) (u)}, function {νlF� (e) (u), ��{(u, (e) F�{A} F�(u), F�of �{A} RAµe) RA = {(u, = :{(u, e ∈e)A,:ue ∈∈ F�A,Ru(e)},which ∈=FF (e)},which e) :ise called ∈ A, is u a∈ called F�{A}fined (e)},which a (u,by is called aThe {A} (e) {A}((e) F�{A} (e) , E). membership of νF�{A} (e) (u)}] i f relation F {A}(e) {A} {A} µ e) = RA (u, e) ={ {A}Al {A} R A l [{µ µ̸∈rF�,A, (u)}, {ν (u), νrFµ�0],RA of i f e ∈ A1]) and de[0, i:fU(u)}] e× ̸∈ E A, →Int([0, Rof written as The , E). membership The E). function of The membership function relationrelation of (F�{A}of, E). (F�{A} relation F�{A} [0, 0],membership i(u), f (efunction A is F�{A} (e)of F�{A} (e) { {A} (e) l{A} (e)(u), µr µ (u, e) = R [{µ (u)}, {νlF� (e) (u), νrF� (e) (u)}] i f A by 1]) andF�{A} RA is written RA is written as µRA as : Uµ× E:AU →Int([0, is×written 1])asand :U and × deEfined →Int([0, deRAR AdeF�{A} (e) (e) A, µR1]) [0, 0], iEf →Int([0, e ̸∈ µ (u, e) = RA where Int([0, for the set of all{A}closed {A} { 1])stands fined byfined by Int([0, 1])stands fined by [0, 0], i f e ̸∈ lA, where for the set subintervals of all closed l l of 1].(u), [µµF�r�(e) (u), µrF� (e) and r [{µF�[0, (e) (u)}, {ν(u)] �{A} (e) (u), νF�{A} (e) (u)}] i f { { { (e) F F A A l r {A} {A} µr Rall (u, e) l l 1])stands l the l {ν l (u), r (u)}, r where set closed subintervals of [0,(u)}, [µ µ[ν (u)] and [{µInt([0, [{µ [{µ (u), µrF� (u), µr1].for {ν(u)}, (u), (u), νrFof µ(u), ν(e) i=f(u)}] {ν eνr∈lF1])stands Ai f(e) e(u), ∈i fAare νefor (u)}] e∈ A closed where Int([0, the set i fofthe allinterval�l�AA(u)}] �F�(e) �{A} (u), (u)] respectively � F �[0, 0], F (e) (e) (e)e)F�= (e) F�(e) F�(e) F�̸∈ F�{A} (e) F�{A}µ(e) F�{A} (e) (e) A{A} {A} {A} (e) lF{A} {A} {A} (e) {A}A, {A} � F (e) F r µRA (u, e) µRsubintervals = (u, e) = (u, {A} {A} RA[0, 1]. [µ A l r of (u), µ (u)] and r̸∈ A, subintervals of [0, 1]. [µF�membership (u), µF� (e) (u)] and 0], i[0, fν e0], i f e ̸∈ A, areF [0, 0], i f e ̸∈ �Arespectively F�AA, (e) (e) valued intuitionistic fuzzy and non(u), (u)] theInt([0, interval[νlF� [0, (e) A where 1])stands for Athe set of all closed F�{A} (e) l r {A} (e) l r (u), νdegrees (u)] the interval[ν membership ofare therespectively object u associated with interval[νF� (e) (u), νF� (e) (u)] are respectively the �{A} Fnon(e)and F�{A}set of (e) [0, 1]. [µlF� (e) (u), µrF� (e) (u)] and where where Int([0, Int([0, 1])stands for thefor setInt([0, the of set all 1])stands of closed all closed for subintervals the of all closed valued intuitionistic fuzzy membership {A}where {A}1])stands parameter e. A A valued intuitionistic fuzzy membership and nonl1]. (u), l µr (u), lthe r subintervals subintervals of intuitionistic [0, 1]. of [0, [µsubintervals [µfuzzy of 1]. (u)] µrF�[0, and (u)] [µand (u),(u), µnonand l and r(u)] (u)] valued membership �A (e) ν are respectively the intervalF�A (e) F (e) (e) F�A (e) F�Aof F�[ν membership degrees the object u associated with A (e) A �{A}, ..., membership degrees of the object u associated F (e) (e) F�{A} , x x } and E = {e , e , ..., en }.with For Let U = {x 1 2 m 1 2 l νr (u),(u)] membership degrees of(u), the associated with (u), νrF� are (u)] respectively the νrF�object interval(u)] the uintervalare respectively the e. interval[νlF� (e)[ν [νlF�are(e)respectively th valued intuitionistic fuzzy membership and non� � the parameter the parameter e. F (e) (e) (e) (e) F simplicity, if we take the [i j] entry of the relation {A} {A} {A} {A} {A} {A} the parameter e.valued land nonr of the u associated with valued valued intuitionistic intuitionistic fuzzy membership fuzzy intuitionistic membership and nonand fuzzynonmembership � ai j = [{µdegrees F µrF� object {νl..., (x i )}, i ), νF� (e ) (xi )}], , (e ...,j ) (x xmi}),and E =(x{e Letas� U = {x1 ,Fx�{A} �{A}e 2For 1 , e2 , F (enj}. ) For {A} (e j ) {A} j , x , ..., x } and E = {e , e , ..., e }. Let U = {x the2 parameter membership membership degrees of1the 2object membership of the uobject degrees u associated with of thewith u associated massociated 1object n e. with ..., ethe For Let degrees U = {x if weistake the [iasj]th entry of the relation then defined 1 , x2 , ..., xm } and E =th{e1 , e2 ,simplicity, n }.matrix the parameter the parameter e. e. if we the take parameter simplicity, the[i j]e.[ithj]entry entry ofas� relation �the xm }),and e }. For r Let U 1 , lx 2 , ..., (x 1 , e{ν 2 , l..., athe [{µ µr E(e = (x{e i j=={x i )}, simplicity, if we take the of F relation �{A} (enj ) (xi ), νF�{A} (e j ) (xi )}], F�{A} (e j ) ri a11F�{A} j )· · · a1n F a th l r l 12 � simplicity, we take of the relation , x=2a,i{x ...,1= x, xm[{µ and E}=and =21r,{e x21,e..., en 2}.,x..., For eand For Ethen = {e ..., eriis }. Forthe [iasj](xentry Let U =Let {x Let U{e(x =1E,{x 1as� 2}, ..., m} ni}. 1, e 2 ,if n), F µ,..., (x )}, (x ν )}], l �xm l{ν ie), �U the matrix defined � F as� a j= [{µ (x )}, {ν (x ), F�{A} (e(x F�{A} (e a(x aj )22 ·i(x · · )}, a2n{νl F (e)entry F (e[{µ ) l νrelation ijthe ri )}], r th thµif th j )ithe 21 j ) i ), {A} � � � (e ) (e ) F�the F F as� a = (x ), µ (e (e ) F�{A} F simplicity, simplicity, if wei j take if we thetake [i{A} j] simplicity, the we take [i j] relation of the relation entry of j j j [i j] ofentry j i j i i {A} {A} {A} . F�.{A} (e. j ) . �{A} . (e j ) (xi ), νF�{A} (e j ) (xi )}], =j ) [� ai jl]m×n F�{A} (e F l l r r l l l r r r r .. 11 .. a12 .(x then the matrix isas� as{ν(x(x�i i),),νµ��(xithen �then �i ), ·.. a1n F� as� ai j F as� aiF�j = adefined = [{µ [{µmatrix (x (xF µ(x = [{µ {ν (xas ), ν(x (xithe )}, {ν (x )}], (xi ),aν the is defined ij)}, i�)}, i)}], as(e j ) .· ·i )}], �i ), µ �){A} (e j ) F�i{A} (e j ) F�F (ej )j ) F�{A} (e jmatrix ) Fi�{A} (eisj ) defined F�{A} (ej )j ) F{A} (e jF)F{A} {A}(e {A} (e j ) F{A} (e jF {A} {A}(e aa21 a · · a2n m1 a22 m2 · ··· a mn then thethen matrix the matrix is defined is defined asthen the as matrix is defined as ] = [� a . a a · · · a . . . i j m×n a a · · · a . a a · · · a 11 12 1n 11 12 1n 11 12 1n . . . . . . . . of order The a21 an a22IVIFSM · · · a2n[32] a11 a12a11 · · ·a12 aa1na· 21 · ·a aa1n22· · ··a·a aa12 ·a1n matrix iscalled · ·above 11 · 2n 21 22 2n a · · · a a m1 m2 mn [� a ] = ., E) over � .. the .. IVIFSS i j m×n . . .. (F{A} to a22= a21 · · ·a22 a.2n.· ·. · .a.2n . a21. a22m× ·.· n·acorresponding [� a[� 2n . ai jai]j21m×n ]m×n = . . . . . . . . . . .. .. .. .. U. [� ai j ]m×n[� a= a.i j ] . .. [� .. ... =. .. i j ] m×n. =. . m×n . . The above matrix is called an IVIFSM [32] of order .. .. . .. . .... . . .. . . . am1 am2 · · · amn . . . � · · ··a·m1 a·mnaam2 m to the IVIFSS (F{A} , E) over Example corresponding to Example m1 a·m1 · · ·am2 aamn · ·am2 aamnm2 · ·n· corresponding amn5. The IVIFSM am1 am2am1 mn× abovebelow. matrix is called an IVIFSM [32] of order U. 4The is given mIVIFSM ×of n corresponding The above The matrix above is matrix called isan called The IVIFSM above an IVIFSM [32] matrix is [32] order called of order an [32] of orderto the IVIFSS (F�{A} , E) over The above matrix isis called anof IVIFSM [32] order The above matrix called an IVIFSM [32] of order Example 5. The IVIFSM corresponding to Example �{A} , E) m × n corresponding mm ×× n corresponding to×the n corresponding IVIFSS (F�{A} to IVIFSS (over F�{A} , E) over to themIVIFSS (Fthe over , E)the over �U. n corresponding to IVIFSS ( F , E) 4 is given below. {A} � to the IVIFSS (F{A} , E) over U. U.m × n corresponding U. Example 5. The IVIFSM corresponding to Example 3 Interval-valued Intuitionistic Fuzzy Soft Matrix U. U. 4 is given below. Example Example 5. The IVIFSM 5. The IVIFSM corresponding Example corresponding 5. The to Example IVIFSM to Example corresponding to Example Example 5. The IVIFSM corresponding to Example 4 is given 4 isbelow. given below. 4 is given below. Example 5. The IVIFSM corresponding to Example 4 is given below. 4 is given below. (0.2, 0.4) (0,) 0) ( (0.6, 0.7) (0,) 0) ( (0.6, 0.7) ) ( ) ( ) ( (0.2, 0.4) (0, 0) (0.6, 0.7) (0, 0) (0.6, 0.7)0.2) MAXIMISING ACCURACY AND EFFICIENCY OF. . . (0.5, 0.6) (0, 0) (0.2, 0.3) (0, 0) (0.1, MAXIMISING ACCURACY AND EFFICIENCY ( 0.6) ) ( )(0.2, ( 0.3) )(0,(0)OF. . . )(0.1, ( 0.2) (0.5, EFFICIENCY (0, 0) MAXIMISING ACCURACY AND OF. . . (0.4, 0.5) (0.6, 0.8) (0.2, 0.3) (0, 0) (0.5.0.7 ( ) ( ) ( ) ( ) ( ) MAXIMISING AND OF. (0.4, 0.5) ( (0.6, ) ( ACCURACY ) 0.3) ( EFFICIENCY ) 0) ( .. ) ( 0.8) (0.2, (0, (0.5.0.7) (0.1, 0.2) 0) 0)) ( )(0.2, ( (0, 0) (0.6, ) 0.7) ( 0.7) ) (0, ( (0, ( 0.3 ( (0.3, 0.4) )(0.2, 0.4) (0.6, (0.6, 0.7 ) (0) )(0.6, ( (MAXIMISING (0.2, 0.4) (0, 0) (0.6, 0.7) (0, (0.3, 0.4) (0.1, 0.2) 0.7) (0, (0.2, 0.3) MAXIMISING MAXIMISING ACCURACY ACCURACY AND EFFICIENCY AND EFFICIENCY OF. . . OF. . . ACCURACY AND EFFICIENCY OF. . . ( ) ( ) ( ) ( ) ( ) 63 (0.5, 0.6) (0, 0) (0.2, 0.3) (0, 0) (0.1, 0.2 (0.4, 0.6 (0, 0) (0.6, 0.8) (0.1, 0.2), (0.7, 0.8) ) ) ( ) ( ) ( ) ( ( ) (( (0, 0) ) ) (((0.2, 0.3) ) (( (0.5, 0.6) ) )( ( (0, (0.2, 0.4)[� (0, 0) (0.6, (0, (0.6,)0.7) (0.7) 0) ai j ]= (0.4, 0.6) (0, 0) (0.6, 0.8) (0.1, 0.2), (0.7, 0.8) ( ) ( ) ( ) ( (0.2, 0.4) (0, 0) (0.6, 0.7) (0, (0.4, 0.5) (0.6, 0.8) (0.2, 0.3) (0, 0) (0.5.0 AND EFFICIENCY OF. (0.2, 0.3 (0,. .0)(0.2, (0.1, (0.7,(0, 0.8) 0.2) 0.3) (MAXIMISING 0.6) (0, 0) (0.2, 0) ACCURACY (0.1, 0.2)0.2) [� a ]= (0.4, 0.5) (0.6, 0.8) (0 ) 0) (0.3) ) ( )((0.1, ( ( )(0.1, ( (0.5, (( ) )( ( )( () ) ( ) 0) ( ij ) ( ) ( )(0.3, (( ) ) (0.2) )(0, )0.4) ) () ) 0) ( (0.2, ) 0.3) ( (0.5, 0.6) (0, 0.3) (0,0 (0.1, 0.2) (0.6, 0.7) (0, (0.2, (0.2, (0.7, 0.8) (0.1, 0.2) ( ) ( ) ( ) ( ) ( ) (0.7, 0.8) ) ( (0, 0) ) ( ( (0, 0) (0.5, 0.7) (0.6, 0.7) (0.4, 0.5) (0.6, 0.8) (0.2, 0.3) (0, 0) (0.5.0.7) ) ) ( ) ( ) ( ) ( ( (0.2, 0.4) (0.6, 0.7) (0, (0, 0) 0.4) (0.6, 0.7) (0.2, 0.4) (0, 0) (0, 0) (0.6, (0.2, (0.6, 0.7) (0, 0)0.2) ) ((0.6, 0.7) (0.3, (0.1, (0.6, 0.7) ) ((0, (0 0.7) 0)( (0.1, (((0.6, (0.4,0.2), 0.5)0)) (0.6, 0.8) (0.2, (0.4,(00 (0, (0.7, 0)0.3))0.8) 0.8) (0.7, 0.8) (0, (0, (0.5, 0.7) (0.6, 0.7) ( )(0.1, )0) (0)(0.2, ( (0.6, (0.3, 0.4) (0.1, 0.2) 0.7) (0, 0) (0.2, 0.3) (0.5, 0.6) (0, 0) (0.2, 0.3) (0, 0) 0.2) (0.5, 0.6) (0, 0) (0.2, 0.3) (0, 0) (0.5, 0.6) (0.1, 0.2) (0, 0) 0.3) (0, [� a ] = ( (0.6, 0.8) (0.1, 0.2), (0.7, 0.8) (0.1, 0.2) (0, (0, 0) (0.2, 0.3) (0.2, 0.3) i j ) (0.4) ) ( ( ( ( ( )) (() ))(((0.7, ) ) () ) ) ) ( ) ) )) ( (0.1, ) ((0.6, ) (0.2, ) (0.2, (0, 0.7) (0.3, (0.1, 0.2) (0.6, ( 0) (( )(((0.1, ([� )(((0,(0 ((0.1, 0 (0, 0))0.7) 0.2) 0.8) a0.2) ]( = i j 0.3) ) ) ( ) ( ( 0.2) (0, 0) (0, 0) (0.2, (0.2, 0.3) ( ) ( ) ( ) ( (0.4, 0.6) (0, 0) (0.6, 0.8) (0.1, 0.2), (0.7, 0.8) (0.4, 0.5) (0.6, 0.8) (0.2, 0.3) (0, 0) (0.5.0.7) (0.4, 0.5) (0.6, 0.8) (0.2, 0.3) (0, 0) (0.4, 0.5) (0.5.0.7) (0.6, 0.8) (0.2, 0.3) (0 ( (0.1, 0.2) (0.7, (0.1, 0.2) ( (0, 0)) (0, 0)) ( (0,) 0) ( (0,) 0) ( (0.6, 0.7) (0, 0) ) ( ( ) ) ( ) ( ( . [� ai j ] = (0.5, 0.6) (0, 0) (0.2, 0.3) (0, ( (0.6, 0.8) (0.1, 0.2), (0.7, 0.8) (0.7, 0.8 (0, 0) (0, 0) (0.5, 0.7) (0.6, )0.3) ( ) ( (0 ( (0.6, 0.4) (0.2, 0.3) (0,( 0))(0.3, (0.1,0.7) 0.2) (0.7, 0.2) 0.8) (0.1, 0.2) a0) ]= (0.1, (0.6, (0, (0.2, (0, 0) 0.4) (0.2, (0.1, 0.2) 0.7)0)) (0, 0) (0, 0) (0, (0.5, 0.7) 0) (0.6, 0.7) (0,0) 0) i0.7) j( ( (0.3, ) )[� ) ( (0.1, ( (0.3,)0.4) )0) )0.3) )0)) (( )(0) ( )( ((0.2, ) ( ) ((0, ) ((0.2, )0.2) (( ) ( ) ((0, ( (0, (0, (0, 0) (0, 0) (0, (0.4, 0.5) (0.6, 0.8) (0.2, 0.3) (0 ( (0.1, 0.2) (0.7, (0.1, 0.2) (0.1, 0.2 (0, 0) (0, 0) 0.3) 0.3) ) ( ) ( ) ( ( )0) ( )(0.4, ( ) (0, (0)(0.7, ) ) 0) ( (0.7, 0.8) 0) (0,(0)0.8) (0.5,0.2), 0.7) (0.6, 0.7) (0.4, 0.6) 0.6) (0, 0) 0) (0, ( (0.6, 0.8) (0.6, 0.8) (0.1, 0.2),(0.6, (0.1, 0.2), (0.7, 0.8) (0.1, (0.7, 0.8) 0.8) (0, (0, (0.2, 0.3) (0.2, 0.3) (0, (0, (0, 0) (0, 0) (0, 0) ) ( ) ( ) ( ( (0.3, 0.4) (0.1, 0.2) (0.6, 0.7) (0 (0.6, (0,. 0) (0, 0) (0, 0) (0, (0, (0.5, 0) 0.7) 0) 0.7) (0,[� 0) . [� ai j ] = [� ai j ] = ai j ] = (0, (0, ) (0)( ) ( ) ( ( (0.1, 0.2) (0.1, 0.2) (0, 0) (0, 0)0.2) (0.2,0.8) 0.3) (0.2, (0.2, (0.1, (0.7,)( 0.8)( (0.1, (0.7, 0.8) (0.1, (0.1, 0.2) (0, 0) 0.2) (0,0.3) 0) ) )0.2) ( (0.3) ) ((0, ) ) ) )(0.2, () )(0.7, ( 0) ((0)(0, )(0) 0.2) ( ) )(0, (0, ( (0,( ) ) ((0, ( (0.1, ) () ( (0.7, ) 0.3) ( (0.6, 0.8) (0.1, 0.2), 0.8) (0, 0) (0.2, 0.3) (0.2, 0.3) (0, (0, 0) 0) 0) 0) (((0.6, (0, 0) ) ( ) ( ) ( ( 3.1 Choice Matrix and Combined Choice (0, 0) 0)0.7) 0) 0.8) [� a0) (0.7,(0, 0.8) (0.7, 0)(0, (0, (0, (0, 0) (0, (0, 0) (0.5, 0.7) (0.5, (0.5, 0.7) (0, 0) (0.6, 0.7) (0, (0.6, 0.7) (0, 0) 0) (0, 0) i (0, j ] =0) 0) (0, 0.7) ( (0.1, 0.2) (0.7, 0.8) (0.1, 0.2) (0, (0, 0) (0, 0) (0, 0) 3.1 Choice Matrix and Combined Choice 0)( (0, 0.3) ) ( (0, 0) ) ( 0) (0, 0) ) ( (0, (0, 0) (0, 0) (0, 0) 0) (0.1, 0.2) (0.1, 0.2) 0) (0, (0, (0, 0) (0, 0) (0.2, 0.3) (0.2, 0.3) (0.2, (0.2, 0.3) (0.2, 0.3) (0.2, 0.3) Matrix ) ((0.6, ( ( ) () ( ) 3.1()Choice ) ) ( ) ( ) ( ) ( ( ) ( ) ( ) ( (0, (0, 0) (0.5, 0.7) Choice (0, 0) (0, 0)0.7) (0, 0) (0, 0) and Combined (0, 0) (0, 0) Matrix 0) (0, (0, 0) 0)3.1(0, 0) (0, 0) (0, 0) (0, 0) (0, 0) (0,Matrix 0) (0, 0)0) (0, (0, Choice Matrix and Combined Choice (0, 0) (0, (0, 0) (0.2, 0.3) (0.2, 0.3) (0,(0) Matrix ( (0, 0) ) ( (0, 0) ) ( (0, 0) Choice Matrix is0)a (0, square matrix whose rows (0, 0) 0) (0, (0, 0) Choice (0, 0) (0, 0) (0, 0) (0,(0, 0)0)) 3.1 Choice (0, Matrix and Combined 0) Matrix (0, 0) (0, 0) (0, 0) (0, 0) 3.1 Choice Combined Choice Choice Matrixboth is aindicate square matrix whoseMatrix and columns Ifrows β whose is and a rows Matrix Choice Matrixparameters. is a square matrix (0, 0) (0,rows 0) (0, 0) (0, 0) Matrix Choice Matrix matrix whose andMatrix columns both indicate parameters. If β isisaa square 3.1 Choice Choice 3.1 Matrix Choice Matrix and Combined and Combined Choice Choice 3.1 Choice Matrix and Combined Choice and columns both indicate parameters. If β is a choice matrix, then its element β(i, j) is defined as is a square matrix whose rows and columns both indicate parameters. If β is a choice matrix, then its element β(i, j)Matrix is defined asisa defined Choice Matrix square matrix whose rows choice β(i, j)is as Matrix Matrix follows: 3.1its element Choice Matrix and Combined Choice and columns both indicate parameters. If βmatrix, is a then choice matrix, then its element β(i, j) is defined as and columns both indicate parameters. If β is a follows: follows: { choice matrix, then its element β(i, j) is defined as Matrix ChoiceChoice MatrixMatrix is a square is a square matrix matrix whose whose rows rows Choice Matrix is a square matrix whose rows follows: th matrix, choice then its element is defined as {1) when (1, 1), (1, icolumns and jthboth parameters areβ(i, thej)choice of bo th P{ th follows: and columns and columns both indicate both indicate parameters. parameters. If β is If a β is and a indicate parameters. Ifchoice βparameters is aparameters th { th (1, 1), (1, 1) when i and j parameters are the oft = β(i, j) th P th Choice Matrix is a square matrix whose rows follows: (1, (0, 1), (1, 1) when i and j parameters are the choice parameters of both th th P = β(i, j) (1, 1), (1, 1) when i and j parameters are the Pmatrix, 0), (0,defined 0)(0,otherwise, i.e.,then when at leastatβ(i, one i theand th j paramet th choi { matrix, = element β(i, j) choice choice matrix, then itsthen element its β(i, j) is choice as (0, its element j)ofone isthe defined as isthdefined as β(i, j) = th j) β(i, 0), 0) otherwise, i.e., when least of i and j param th th both parameters. Ifand β isjat aleast (1, 1), (1, 1) when i (0, and0), j (0, parameters areand the columns choice parameters of both the makers 0) otherwise, i.e.,{when atindicate least one ofdecision the i parameters parameters (0, 0), (0, 0) otherwise, i.e., when th follows: follows: β(i, j)Pfollows: = 1), 1) when ith and jthdefined areone theof choi Pmatrix, th (1,then th (1, choice its element β(i, j) is = i and j parameters is not the choice as j) the (0, 0), (0, 0) otherwise, i.e., when at least β(i, one of { { { (0, 0), (0, 0) otherwise, i.e., when at least one of th In 1)combined matrix, indicate of choice the decision makers. th follows: (1, 1) 1),when (1, iwhen (1, 1), (1, 1) ith and ijthth and parameters jth parameters are theIn are choice therows choice parameters parameters of when both ofthe both decision and the jth decision parameters makersmakers are the choi combined choice matrix, rows indicate P P(1, 1),(1, P In combined choice matrix, rows indicate = j) = = β(i, j) β(i, j) β(i, th th th th { choice parameters of single decision maker, where combined choice matrix, rows indicate In combined matrix, rows indicate (0, 0),choice (0, (0, 0) 0),otherwise, (0, 0) otherwise, i.e., In when i.e., atwhen leastatchoice one least of one the iof(1, (0, the 0), ij of (0, 0) otherwise, i.e., when at least one of th and and parameters j when parameters is the isjthnot choice the choice parameters single decision maker, where 1), (1, 1) ithnot and parameters In combined choice matrix, rows indicate are the choi Pmaker, choice parameters of combined single decision where = β(i, j) columns indicate choice parameters (obchoice parameters of single decision maker, where choice parameters of single decision maker, where columns indicate combined choice parameters (ob(0, 0), 0) otherwise, when at least one of th choice parameters of(0,single decision i.e., maker, where columns indicate combined choice parameters (obcolumns indicate combined choice parameters (ob- parameters columns indicate combined choice parameters In combined In combined choice choice matrix, rows indicate rows(obindicate In combined choice rows indicate tained by matrix, the intersection oftained their parameters sets) by the intersection ofmatrix, their sets) columns indicate combined choice parameters (obtained by the intersection of where their parameters tainedmaker, by the intersection of makers. their sets) parameters sets) maker, where tained choice by the parameters intersection ofsingle their parameters sets) choice parameters of single of decision decision maker, where choice parameters of single decision of decision of decision makers. In by combined choice ofmatrix, rows indicate tained the intersection their parameters sets) of decision makers. of decision makers. of decision makers. columns columns indicate indicate combined combined choice choice parameters parameters (ob- (obcolumns indicate combined choice parameters (obchoice parameters of decision makers.of single decision maker, where tained tained by the by intersection the intersection of theirofparameters their parameters sets) sets) tained by the intersection of theirand parameters sets) 3.2 Product of IVIFSM Combined columns indicate combined choice parameters (ob3.2 Product of IVIFSM and Combined 3.2 Product of IVIFSM and Combined of of decision makers. makers. decision makers. 3.2decision Product of IVIFSM and of Combined Choice Matrix tained byCombined the intersection of theirand parameters sets) 3.2 Product IVIFSM and 3.2 Product of IVIFSM Combined Choice Matrix Choice Matrix Choice Matrix of decision makers. Choice Product of Matrix IVIFSM and combined choice maChoice Matrix 3.2 Product Product 3.2 of Product of IVIFSM of IVIFSM and Combined and Combined 3.2 Product of IVIFSM and Combined Product IVIFSM and combined choiceofmaIVIFSM and combined ma-of Product of choice IVIFSM and combined choice matrix is possible if the number columns of IVIFSM Product of IVIFSM and combined choice maChoice Choice Matrix Matrix Choice Matrix Product of IVIFSM and combined choice ma� be trix isIVIFSM possible if the number of columns of IVIFSM 3.2 Product of IVIFSM Combined trix is possible if the number columns A equal to of theIVIFSM number of rowsand of the combined trix isofpossible ifofthe number of columns trix is possible if the number of columns of IVIFSM � choice � beProduct � A be equal to the number of rows of the combined A equalProduct toofthe number of rows of the combined Choice Matrix trix is possible if the number of columns of IVIFSM IVIFSM of IVIFSM and combined and combined choice mamaProduct of IVIFSM and combined choice machoice matrixβ. Then A and β are said to be con� be equal to the number ofA�rows A of the combined be equal to the number of�rows of the combined � � choice matrixβ. Then A and β are said to be con� choice matrixβ. Then A and β are said to be contrix is possible trix is possible if the if the number of columns of columns of IVIFSM ofof IVIFSM trix is possible if the number IVIFSM formable for product (Aof⊗columns β) and of their prodA number be equal to the number rows the combined �and � and Product ofthe IVIFSM machoice matrixβ. ThenconA and combined β are saidchoice to be conchoice matrixβ. Then A βof are said to �to �rows � be equal � befor �product �equal formable for the uct (A ⊗the β) be andproduct theirof�prodformable the choice product (matrixβ. A ⊗ β) and their prodA A to equal number to the number of ofof rows the combined of the combined A be number rows of the combined ( A ⊗ β) is called IVIFSM. The product � Then A and β are said to be contrix is possible if the number of columns of IVIFSM �formable for their the product (A ⊗ β)�and their prodformable forsaid the (A β) and prod�product �⊗ �Then � �⊗⊗product A β) called IVIFSM. The product uct (A β) is matrixβ. called product The product choice choice matrixβ. Then A and IVIFSM. A β� are and β uct are to(said be⊗conto is be conmatrixβ. Then A and βby areAβ. said to be con(choice is also denoted simply �A � �β) A be equal to the number of rows of the combined uct ( A ⊗ β) is called product IVIFSM. The product formable for the product ( A ⊗ β) and their prod� � � � � � � � uct ( A ⊗ β) is called product IVIFSM. The product ( A ⊗ β) is also denoted simply by Aβ. (formable A ⊗ β)formable is for alsothe denoted for product thesimply product (A ⊗by β)(Aβ. Aand ⊗ β)their andprodtheir prodformable for the product (A ⊗ β)�and their prod� � � � � � choice matrixβ. Then A and β are said to be con-, ( A ⊗ β) is also denoted simply by Aβ. If A = [� a ] and β = [ β ] , then Aβ = [� c i j m×nproduct jk n×p ik ]m×p A ⊗product β) is also called product IVIFSM. � �The �⊗ �is⊗called ��⊗ (([A ⊗ β) is denoted simply by Aβ. uctA A uct (A β) uct is called product IVIFSM. IVIFSM. The product The product uct (A β) is ,called product IVIFSM. The product � � � �(= � � If A = [� a ] and β = [ β ] then Aβ = [� c ] , If [� aiβ) ] and β = β ] , then Aβ = [� c ] , formable (ikA m×p ⊗ β)�and prodim×p j m×n n×p the product jk for where m×n n×p �⊗ � �� � their (A β)jk is also by Aβ. � ⊗ β)(A �isj⊗also �denoted � iksimply �A [� ais β = [simply β jk ]n×pby , then = [� cik ]m×p , (where A β) is denoted also denoted simply simply by Aβ. by Aβ. (IfA ⊗=�β) denoted Aβ.Aβ i j ]also m×n and where � uct ( A ⊗ β) is called product IVIFSM. The product � � If A = [� ai j ]m×n and β = [β jk ]n×p , then Aβ n= [� cik ]m×p l ,,µl }, maxn min{µr , µr }], where [max min{µ � � � � � � � � � � � � � � ( A ⊗ β) is also denoted simply by Aβ. j=1 j=1 � If A = [� IfaiAj ]m×n = [� aand [� ai j= =and [lβ jkβ β, jkthen ]n×p Aβ , then [� cAβ ]m×p = [� cn×p ,ikrn]m×p ,A =Aβ ]m×n βn =, [a�βi jjk ]n×p Aβ = [� c ] a�i j ,�β jk Ifβ A = [� a]= and β== , Ifthen [� cikand ]m×p i j ]m×n n×p ik[β β jkr , then il j ][m×n r jk,]µ . where ik rm×p [maxnj=1 min{µ [max , µ�r n}], c�ik= la�i j , µ�l }, }],min{µ }, maxnj=1 min{µ nn max j=1 min{µ ll , νll a�}, j=1 n min{ν r ,,νµrr }] a�i j � a�i j , µ� i j min β where where where β β β [min min{ν jk jk [max }], min{µ , µ }, max min{µ jk jk � where � � j=1 j=1 � � a � � a j=1 j=1 � � � a . = c � � a If A = [� a ] and β = [ β ] , then Aβ = [� c ] , i j i j . c�ik = i j ββjkjk i jjk n×p ββjk j lm×n ik m×p n l ,i rjk , νr }] l }, minn ikmin{ν r[min r n }]min{ν nj=1 min{νla� . ν }, min min{ν , ν , ν � [min = c r l �β j=1 l ik}, max n min{µ j=1r l n � a�i j n� r r � ij j ij β β jk nnmin{µ βl rjk, µ r}],where r }], n lmin{µ l j=1 n},j=1 n a�rimin{µ l, µ jk n min{ν r , ν r }] [min min{ν ,,νµ��ll a�}], }, min jk l},,[max [max nj=1 j=1 j=1 j=1 [max [max min{µ , µ , µ min{µ , µ }], min{µ , µ max µ max min{µ }, max � a � � a � � a � a i j i j i j i j r r n l l n β β β β j=1 a� j=1 a�i,j µ � j=1 j=1 �i j max a�i j � � � a� a�i j}],� jk � ij β jk a}, β jk min{µ βjkjk β[max β jk j=1 βjkjkjk , µ min{µ jk i j � . = c j=1 j=1 � � a � � a �ikr =. c�ik = c�ik = n n n6.. nExample Example product i j r βfor n lik l n ri j l rβ ,jkrνl c nn of IVIFSM l,ljkν,rνthe ll }] rr , νrr }] . n min{ν [min min{ν }, min min{ν [min , ν }] , ν }] min{ν min{ν , ν� la�}, ,min νn�l nj=1 },j=1 min min{ν min{ν }, min min{ν [min j=1[min [max min{µ , µ }], min{µ , µ }, max � . c = j=1 j=1 j=1 j=1 j=1 � � a � � a � � � � � � a a a � � � a a ik j=1 j=1 � � a i j � � a i j n i j i j i j i j i j i j Example for min{ν the product ofi}] l ,6.νlβExample r , νrmatrix jIVIFSM Example 6. Example for of IVIFSM β jkββjkjkis β[min β jk min{ν βand jk }, choice shown in Step 5i j ofββjkjk jk product jkβ jk combined min the � . = c Example for the product j=1 j=1 6. nExample a�i j � ik a�i j � n of IVIFSM r , νr }] β combined shown injkla�i Step min{ν , ν�l 5},of min min{ν [min and combined choice matrix is shownand in Step 5 ofβ jk choice both matrix the caseisj=1 studies, explained in Section 6. j=1 � � a i j j and combined choice matrixβ jkis shown in Step 5 ofβ jk both theof case studies, explained in Section both the case studies, explained inthe 6. Example Example 6. Example 6. Example for thefor product product of IVIFSM IVIFSM Example 6. Example for6.the product of IVIFSM Example 6.Section Example for the IVIFSM bothproduct the case of studies, explained in Section 6. and combined and combined choice choice matrix matrix is shown is shown in Stepfor in5 Step of and of combined choice matrix is shown in Step 5 of Example 6. Example the 5product of Example 6. IVIFSM Example and combined choice matrix is shown in Step 5for of the product of IVIFSM both the both case thestudies, case studies, explained explained in Section in Section 6.matrix 6. is shown both the in case studies, explained in Section 6. and combined choice Step 5 of and combined choice matrix is shown in Step 5 of both the case studies, explained in Section 6. both the case studies, explained in Section 6. both the case studies, explained in Section 6. 3.3 � and B� are said to be conTwo IVIFSMs A formable for addition, if they have the same order � and Addition of IVIFSMs Two IVIFSMs A B� are said to be order conformable for addition, if they same and after addition, the sum is have also the an IV IFSM of 3.3 IVIFSMs Addition of IVIFSMs formable addition, theyto the same �for and afterA addition, theifsaid sum also an IV IFSM � of �ishave � order Two and B� are be con- Tambourat Tambourat the same order. Now if both A = (� a3.3 B = (bof i j ) and i j )IVIFSMs Tambourat Addition � � � Tambourat and after addition, the sum is also an IV IFSM of the same order. Now if both A = (� a ) and B = ( b � � i j i j) formableTwo forbeaddition, iforder they have the same order � IVIFSMs A and B are said to be conthe same m × n, then the addition of A and Tambouratzis T., Souliou D., Chalikias M., Gregoriades A. 64 � = (� � and B� are said to be conthe same order. Now both A aaddition B�IVIFSMs band �(� Two 3.3 IVIFSMs i j ) and iof j) A be same order mthey × n, then the of=A and after addition, the sum also an IFSM ofAddition �ifis⊕ �if and formable for addition, have the same order B� isthe denoted by A B isIV defined by formable for A if they have D., theChalikias same order Tambouratzis T., Souliou M., Gregoriades A. �addition, 3.3 Addition of IVIFSMs �A be the same order m the addition of � �sum �= � and B� is denoted by ⊕=× B�(� and is defined the same Now if both an, )also and B (IFSM bby IVIFSMs A B� are said to}be con{ 3.3 of the IVIFSMs and order. after addition, theA isi j then an IV of and iTwo j )Addition and after addition, sum is also an IV IFSM of l , µl }, �max {µ r }], } � ⊕ B� and ��raare { B�order is denoted bythen A is defined by 3.3 Addition oforder. IVIFSMs [ max {µ , µ formable for addition, if they have the same order � � Two IVIFSMs A and B said to be con� � a � b b � be the same m × n, the addition of A and the same Now if both A = (� a ) and B = ( b ) i j i j i j i j � ij the same if and both{µ and Bbe =∀i, (conbij.j ) li j , µl Now ri j )}], Two IVIFSMs Amax B�Arr= are to} ai j ]⊕[� bi j ], where c�i j = [� ci j ] = [� max {µorder. },the ,(� µasaid { [[formable lafor lbi j }, and after addition, isaihave also an IFSM of addition, ifsum they same order ij, ν j, νrbithe j}] IV � � 3.3 Addition of IVIFSMs � � � min {ν min {ν � � Two IVIFSMs A and B are said to be conB� is denoted by A ⊕ B and is defined by l l r r be the same order m × n, then the addition of A and � ] = [� a ]⊕[ b ], where c = [� c a a ∀i, j.) addition, if they have order be the same order ofsame A{µand bl i j}, max i j, µ j , µb ij ij m× i j n, then thei jaddition j }],same � lfor r i= riijthe � � [[formable max {µ the order. Now if both A (� a ) and B = ( b and after addition, the sum is also an IV IFSM of a a i j b b , ν }, min {ν , ν }] min {ν ij j� aii jjalsobii jan formable fordenoted addition, have � the i j bybi iA {issame B� iscondenoted B� and defined j by IV IFSM j ⊕the � ⊕� and afterlaaddition, sumis} is of = [� aifIVIFSMs ]⊕[ bB�i j ], where c�order = by [� ci j ]Two ∀i, j. A and B�defined are to be i jthey i j said B� is by A and � l r r � � � the same order. both = aaddition ) and B�of=A(� band) l ,[µ l the rif n, be same × i j }] ,order νbiraNow },,mµmin {νA , ν(� min {ν Example the{µ addition � and after addition, the sum isExample also an IV[for IFSM of max },operation max }],then aorder. ai the i j {µ j{(� same Now A = ab} (l bii jj ) ii jj ) and B j i j is formable for7.addition, if they have the aisame bifi jboth bi jorder j the { l = � � � � � B is denoted by A ⊕ B and is defined by [ max {µ , µ }, max {µrai j , µrbi j }], be the same order m × n, then the addition of A � ] = [� a ]⊕[ b ], where c = [� c ∀i, j. � Example 7. Example for the addition operation is � j i j shown iafter j if both ja ai j �band l }, order r }], theisame order. Now A = i(� and = (b[{ν l IV,{µ l l the r exr rn, ij imax j )the � in Step 6i j )of of[Bis�min both case studies, and addition, the sum also an IFSM of be same m × then the addition of A and , µ max {µ , µ � ν }, min {ν , ν }] [� c ] = [� a ]⊕[ b ], where c = � �biis aijji jdenoted l l ai j B bj i defined biijj byiajA biajand i j iis i j is j { [ min ⊕ B� by Example 7. Example for the addition min {νrari j , νrbri j }] � � �of= shown in Step both case ex� ci j ] = [� aisame binithen where c�i jof = ∀i,{ν j.alai j ,, νµblbi j }, [� be the same order m × n,� the 6 addition A the order. Now if of both A (� aand and B� boperation j ]⊕[ j ], B�=lis(denoted by A ⊕ B�rand is defined by i j )the istudies, j )l [ max {µ }, max {µai j , µbi j }], r plained Section 6. i j ij ,studies, ν[� }]{ [ min � �the ai j , ν ]⊕[min bi jex], {ν where c�ibji= [� ciaA baiijj}, shown ininorder Step 6×of of both the {ν case j�i]j= j { B�Example is denoted7. by ⊕ B�same and is defined by l ,ν l }, min beA m n, then the addition of and plained Section 6. [ max {µ µ {µrarraiijj,,νµrbrrbi j}] }], Example for the addition operation is {ν [ min {ν Example 7. Example forc�the=addition operation is max al i j , µbl i j }, � [ max {µ max {µrai j , µrbiijj }], { } � ] = [� a ]⊕[ b ], where [� c � � ai j lbi j ij ij ij ij B is denoted by A ⊕ B and is ldefined by l plained in Section 6. � r] = }, min {νrai j , νrbi j }] [ min studies, {νlai j , νlbiex[� arin }], ]⊕[ bi j6], of where c�i j =the case [� ci jexshown Step of both shown in Step of of both the case [for max {µ , µlbistudies, }, max {µ Example Example the operation is Example aiaddition ai j , µbiij j7. 3.47.6 Complement of IVIFSM }the j{ [ minoperation {νai j , νbi jj }, j for addition is min {νai j , νbi j }] Example ai j ]⊕[� bi j ], where c�i j = [� ci j ] = [� ∀i, j. l l r r r in plained Section max {µstudies, , µra bi,j }, max {µai j , µ6.bi j }], 3.4 Complement oflathe plained in Section 6. 6 �of of[ min ,IVIFSM νlb[icase }, min {ν ν }] {ν a shown in Step both exi j b i j i j shown in Step 6 of offorboth studies, exi j 7. j Example Example the addition is ai j ]⊕[bi j ], where c�i j = [� ci j ] = [� ∀i,the j. case operation r , νr }]for the 7.denoted Example addition operation is Complement of anofIVIFSM (� alaiExample is ,)νlbi j }, in min {ν [ min {ν 3.4 Complement IVIFSM ai6j of i jj m×n plained in Section 6. plained Section 6.bi j of both the case studies, exshown in Step Example 7. Example the,where addition operation is (� Complement of(� IVIFSM aishown is Step denoted j )m×n 3.4 Complement of both IVIFSM in 6 of of the case studies, exby (� ai jfor )cm×n aan representai j )m×n is the matrix plained in Section 6. 3.4 Complement of IVIFSM c Example 7. Example for the addition operation is shown in Stepby 6 of of both the case studies, exComplement of an IVIFSM (� a ) is denoted plained in Section 6. i j m×n (� a ) ,where (� a ) is the matrix representai j m×n i j m×n intuitionistic tion6.of fuzzy soft of anofIVIFSM (� ai j )m×n is denoted 3.4 Complement Complement IVIFSM shown incthe Stepinterval-valued 6 ofofofIVIFSM both the case studies, explained3.4 in Section Complement by (� a ) ,where (� a isintuitionistic the matrix representacfuzzy i j i ja)m×n tion of the interval-valued soft c m×n � Complement of an IVIFSM (� ) is denoted by (� a ) ,where (� a ) is the matrix representaj m×n set (F{A} , E). (� ai j )m×n i is the matrix i jComplement i jofm×n 3.4representation IVIFSM plained in Section 6. m×n Complement of an IVIFSM (� a c 3.4 Complement of IVIFSM i j )m×n is denoted � tion of the interval-valued intuitionistic fuzzy soft c set (F{A} ,iE). (� aisiIVIFSM )m×nmatrix is (� the matrix representation tion of cthe soft interval-valued intuitionistic fuzzy soft j the by (� aComplement ,where (� aIVIFSM representaComplement an a ) is denoted i j )m×n j )of m×n of the interval-valued intuitionistic fuzzy set i j m×n 3.4 of by (� a�i j )m×n ,whereof(� representaComplement an amatrix is denoted c i j )IVIFSM m×n is the(� ca i j )m×n � set ( F , E). (� a ) is the matrix representation set ( F , E). (� a ) is the matrix representation i j c cthe Complement ofm×n an IVIFSM (� ai j )m×n is denoted i)j ofm×n of interval-valued intuitionistic fuzzy set {A} {A} Complement IVIFSM cthe soft tion of interval-valued intuitionistic fuzzy soft bythe (� ai j3.4 )F� ,where (� a is the matrix representa( , E)and is defined as tion of interval-valued intuitionistic fuzzy soft by (� a ) ,where (� a ) is the matrix representai j m×n {A} m×n i j cm×n i j m×n ¬ Complement of IVIFSM (� ai defined denoted of fuzzy the fuzzy soft set c an ,interval-valued j )m×n isintuitionistic by (� a�i j )interval-valued ,where (� representa� of the soft set cE)and m×n is the matrix cai j )intuitionistic m×n ( F is as �tion {A} set ( F , E). (� a ) is the matrix representation set ( F , E). (� a ) is the matrix representation ¬ of,( the fuzzy soft Complement of an)IVIFSM (� of the intuitionistic fuzzy softinterval-valued j m×n j ) m×nis[( )aci j])m×n [(is tion )isc ]idefined {A} ,where c ( cof{A} by (� ai j )cm×n (� aiiinterval-valued representa(denoted F�c){A} E)and as intuitionistic m×n the matrix tion ther interval-valued fuzzy soft �¬c(� c (by F is cj )jc, E)and l adefined ras)matrix l ¬) rintuitionistic lrepresentation � c of the interval-valued intuitionistic fuzzy soft rset, 1−νl ]. ) ] [( ) ] [( ( ( {A} a ,where (� ) is the representaset ( F , E). (� a ) is the matrix � i i j m×n i j of the interval-valued intuitionistic fuzzy soft set {A} m×n c c c c , ν = [1−µ )m×n = µ , µ , ν , 1−µ ], [1−ν (� ai,jE). m×n c F{A} (� a ) is the matrix representation � tion of set the (interval-valued intuitionistic fuzzy soft ) ) ] [( ) )c ]ar i j [( ( ( i j a a a a a a a m×n set ai j )cm×n is the ij ij i�j c(F{A} , E). i j (� icj cr i j matrix lrepresentation c l r l r l ij tion ofm×n the = interval-valued intuitionistic fuzzy soft ( F E)and is c ,,( l]defined r as l fuzzy rsoft set, 1−νr ]. of the interval-valued intuitionistic c ) ) ] [( ) ) [( ( , ν = [1−µ ) µ , µ ν , 1−µ ], [1−ν (� a c {A} � ¬ i j � c c c c a a a a a a a ai ji,j 1−µlai j ] , ν = [1−µ ) = µ , µ , ν (� a (F(¬F{A} E)and defined as set ainterval-valued is the matrix representation i j intuitionistic ij i ji jthe i j ai j ij ij ij of, E). the(� fuzzy soft set i j )is m×n a a a a {A} of interval-valued intuitionistic fuzzy soft set m×n i j i j i j c c , E). (� l r l r r l r l c � � set ( F a ) is the matrix representation ) )c],([1−ν )c ]a , 1−νa ]. ( as)c ] , [( is defined iµ j m×n , µ {A} = , νa((iF =c [1−µ , , E)and νa[( 1−µ (� ai j )m×n �j ¬]c {A} of the (interval-valued intuitionistic fuzzy j ) set i j is ij ij F c , E)and l definedr asai j l ai j r ) )aicj ] as [( aisoft F�¬c {A}[( , E)and {A} c is(defined c (fuzzy) c i¬ of the interval-valued intuitionistic soft set , ν = [1−µrai j , 1−µlai j ] (� a ) = µ , µ , ν j ) ) ] [( ) [( ( ( m×n a a a a c ij c ij c ij c ilj )c ] c l r l r r l r ((� F�a , E)and is defined as Cardinal IVIFSS Car] , [(νal ), c1−ν )].] ¬ i{A} cand r ) r l , µisa defined ,Set , ν 1−µ ],c[1−ν c , (µaof j )m×n = 4 (F�cµ{A} a [( aµillaj i,j ) )ascν]aof )aci j (4 )[1−µ = [(Car, (νrraiajCar(� ai j )= j, E)and) j ( i[( j i j c = [1−µrai j , 1−µlai j ] Cardinal Set cm×n raii jj IVIFSS laiijj and c] Set and ) )cIVIFSS ]l [(c 4 ¬)c i Cardinal ( l)c ] ci[( ( of , ν = [1−µ ) = µ , µ , ν , 1−µ (� a i j r r r l r l c ai j ai j ai j ai j ai j ] ] l = [1−µ [( ,)c µl( )c ],r [( ν )c (, rν)c m×n r a , 1−µ , 1−νai j ]. ai j a=i j )m×n dinal ν aSet µ4l =c,Cardinal µrµai j l,Score [1−ν ,ilj1−νl ].ari j ], [1−ν (� ai j )c (� i jr , νof IVIFSS i j [1−µ jdinal],Carl a= r a,i1−µ r l ai j Score and a Cardinal a a aIVIFSS Set of[1−ν ,a νa = [1−µ , a µa , ν4 ]. Cara a , 1−µ a ], a , 1−νaand dinalµaScore 4 Cardinal Set of IVIFSS and Cardinal Score 4 Cardinal Set of IVIFSS and(F�Cardinal Score cardinal set of IVIFSS (F�{A} , E) is denoted 4 , The Cardinal Set of IVIFSS and CarThe cardinal set of IVIFSS E) is denoted {A} � dinal Score 4 Cardinal Set of IVIFSS and CarThe cardinal set of IVIFSS ( F , E) is denoted � {A} 4 Cardinal and Carand defined by byCar(cdinal F{A} , E)Score 4byScore Cardinal Setof ofIVIFSS IVIFSS dinal and defined by and (c F�{A} , E)Set The cardinal set of IVIFSS (F�{A} , E) is denoted �{A} , E) The cardinal set of IVIFSS ( F is denoted � dinal Score , E) and defined by by (c F dinal Score {A} The of IVIFSS dinal Score , E) andset defined by (F� , E) is denoted by (c F�{A}cardinal The cardinal set of IVIFSS (F�{A} {A} , E) is denoted � by by (cF� , E) and defined m×n aai j ) a (� i j m×n =i j ij ij ij ij ij ij ij ij ij ij ij ij ij ij {A}of IVIFSS The cardinalofset (F{A}denoted , E) is denoted l defined rby F� E){[µ and The cardinal set IVIFSS set (F�{A} E) is (c F�(c E), = (x), µcby (x)], [νlcF� (x), νrcF� (x)]}x : x ∈ E}. The cardinal of ,IVIFSS (F�{A} , E) is by denoted {A} defined by (c F�,{A} F�{A} cF�{A} {A} , E) and {A} l r l r � � � The cardinal set of IVIFSS ( F , E) is denoted and by � (x), by(c(c F{A} (c ,defined E) = {[µ µcr F�{A}(x)], [νcl F� (x), νcr F�l (x)]}xr : x ∈ E}.l {A} �{A} , E), E) and defined by by F�{A} {A} , E) and defined by by F (c F r c F l {A}, E) (x), µcF�: x (x)], [νcF� (x), νcF� (x)]}x : x ∈ E}. F�,{A} {[µcF�{A}by (x), µcF�{A} (x)], (c [νFc�{A} (x),=ν{[µF�{A} (x)]}x ∈ E}. cF�{A} {A} {A} E), E) and=defined by (cF�(c interval-valued intuitionistic F�It{A} is rcan {A} l r {A} {A} l {A} r {A} l � � (cF�{A} ,(x), E) =ν{[µlcF� (x)]}x (x), µrc:F�x ∈ (x)], [νllcF� (x), νrrcF� (x)]}x : x ∈ E}. (cF{A} , E) = {[µcF� (x), µcF� (x)], [ν E}. � � {A} {A} (x)]}x : x ∈ E}. {A} {A} (c F , E) = {[µ (x), µ (x)], [ν (x), ν fuzzy set over E. The membership function cF{A} cF{A} {A} {A} {A} is cF�{A} cF�{A} cF�{A} interval-valued cF�{A} l an l interval-valued r l r Itintuitionistic l � It r r is an intuitionistic r l l r l r (c F , E) = {[µ (x), µ (x)], [ν (x), ν (x)]}x : x ∈ E}. (c(c F�{A} , E) = {[µ (x), µ (x)], [ν (x), ν (x)]}x : x ∈ E}. {A} (x), µcF�: x(x)] and non-membership function ItcF� set is interval-valued intuitionistic cF�{A}[ν cF�{A}(x), c�F�{A} cF�an cF�{A} ν[µcF cF�{A} µ F�{A} , E) =fuzzy (x), (x)]}x ∈ E}. c{[µ F�{A} c(x)], F�{A} over The {A} {A} fuzzy over The membership function c{A} F�{A} E. cF�{A} It set isfunction an E.interval-valued intuitionistic cF�{A}membership {A} rinterval-valued l r isfunction interval-valued intuitionistic It= {[µ isl r over an r an l Itintuitionistic �{A} �{A}fuzzy set E. The membership l [ν of (c F , E) are respectively ν (c F , E) (x), µ (x)], [ν (x), ν (x)]}x : x ∈ E}. [µ (x), µ (x)] and non-membership fuzzy set over E. The membership function [µcF�It interval-valued (x)]cFintuitionistic non-membership (x), �and �{A}function �{A} cF�{A} �{A} is cµF�an intuitionistic cF cF {A} It is an {A} cF {A} fuzzy set over E. The membership function cr{A} F�{A} interval-valued l {A} (x), r l fuzzy set over E. The membership function [µ (x)] and non-membership function µ [µ (x)] and non-membership function defined byµ an interval-valued intuitionistic fuzzy over function lcF�{A} (x), �{A} set fuzzyIt set is over The function F�{A}E. The membership of (cnon-membership F�{A} , E) are respectively (x), µνrccFF��{A} (x)] and function [µ rcrmembership llcl FE. �{A} , E) [ν cF�{A} [ν (x)] of (c F are (x), ν r {A} {A}respectively [µ (x)] and non-membership function (x), µ {A} l r � � l r [µ (x), µ (x)] and non-membership function � � � c F c F r l set over E. The membership function l r (d) [µfuzzy (x)] and non-membership function (x), µ �{A} cF{A} cF {A} ancF�ν{A} interval-valued intuitionistic [ν (x)]µ of (c F�{A} , E) are respectively (x), {A} (x), �{A} , E) {A} (d) µ defined byν (x)] of (c F are lcF�{A}respectively rcF�{A} ∑d∈U ∑ cIt F�{A}[νcFis cF�{A} d∈U � � (x)] of (c F , E) are respectively (x), ν [ν � � F{A} F{A} {A}r rdefined l � l�{A} {A} r rcF by lcF{A} F ∀x ∈ E �{A} (x), (x)] non-membership function , µcF� (x) = (x) µcfunction [ν(x)] (x)] of (cof F�{A} , E) are, E) respectively (x), νand � fuzzy over E. The membership defined by =cF{A} (x), ν (x)] (c F are respectively �{A} [ν[µlcF�cF�{A} (c F , E) are respectively (x), νrcF�µ[ν �{A} ��{A} of �set l r (d) c F c F cc F c F � {A} {A} |U| |U| {A} defined by c F F (d) µ µ defined by ∑ ∑ {A} {A} {A} {A} d∈U d∈U � � r by (x)] and non-membership l F{A} F{A} defined l l r (d) (x), µ(x)] r µfunction defined ∀x ∈ E , µrcF� (x) = ∑d∈U µrr� (d) (x) = ∑µd∈U (d) µE) byof (c∑F�d∈U µlFl� (d) ∑ [νlcF� [µby , are respectively (x), cF�{A} cF�{A}νdefined � d∈U {A} c F � � � {A} {A} F F F l {A} (d) r {A}|U| cF{A} {A} (d) µ(d) µF�{A} (d) ∑µd∈U ∑d∈U|U| l r ∑ r l r {A} l l µ and � ∑ F , µ ∈E (x) = (x) = µ d∈U d∈U µ µν�lr (x) (d) {A} ∀x r∈ {A} ∑d∈UofµF� (c(d) = (x) ∑respectively �{A} � � d∈U �{A}(x) ∀x µlcµF��= (d) [νcby (x)] ,r E)µrF,� µare F ∑(x), {A} ∑d∈U , µccFF��{A}E(x) = ∀x ∈ E = lF{A} rF{A}|U| cr(d) F�{A} |U| {A} F l F{A} d∈U defined lc � � � |U| |U| F µ (d) c F (d) µ c F F l r {A} {A} (x) = ∈ Eν (d)∀x ∈{A}E (x) = = r∑d∈U F� , µcF� (x) {A}∑d∈U ∀x� {A} µl � (x) ={A} µµccF�F νF� (d) �{A} ∑d∈U|U| F{A}|U| , µcF� |U| ∀x|U| ∈ E ∑ (x)|U| = {A} {A} , µrc=F�{A}(x) d∈U and {A} F�{A} l |U| |U| cF{A} defined {A} µby l= r E {A} |U| ∀x ∈ E. , µ ∀x ∈ (x) = (x) l r , νcF� (x) = (x) = νcF� (d) cF�d∈U cF�d∈U (d) µ l r ∑ ∑ {A} and {A} µ � {A} {A} |U| |U| |U| |U| � ν (d) (d) ν ∑ ∑ F F d∈U d∈U and and {A} {A} F�{A} F�{A} r and l l l , µ(d) r l ∀x ∈E νd∈U ∀x ∈ E. , νrcF� (x) = ∑d∈U νrr� (d) (x) =(d) µ µ andµcF�{A} (x) = ν (d) ∑ �{A} (x) = ∑ and ∑ � d∈U d∈U c F l c F � � � F{A} (d) |U| ∑ F{A} |U| F{A} ∑d∈U |U| ν(d) νFF�{A} (d) l l r(d) r (d) r |U| ∑ l ν{A} r {A} d∈U � ν (d) µl � and ∑ ν ν F (x)ν= , ν (x) = (x) = ν ∈ E. ∑d∈U ∑d∈U rF� ∀x{A}∈ E rcF�{A} d∈U {A} l µd∈U = d∈U F�{A} lF F�{A}(x) �,{A} νlcF��{A} ∀x �{A}νr(x) (d) l l � (d) rF ∑ ∑ , νcF� (x) = = ∀xν∈{A} ∀x ∈ E. c(d) |U| |U| cF{A} d∈U r � , ν ν (x) = (x) = E. ν |U| |U| (d) F F ∑ ∑ c F {A}(x) = r d∈U {A} |U| νcr F� (x) E. � cF�{A} , {A} � |U| ∀x ∈{A} νl � (x) = νcclF�{A} , νcF� |U| (x)|U| = ∀x|U| ∈= E. d∈U|U| lF{A}(d) rF{A}(d) F�{A} {A} ν |U| cF{A} and {A} ν ν ∑ |U| ∑ , ν (x) = (x) = ∀x ∈ E. d∈U d∈U � � F� F� lcF{A} rcF{A} |U|{A} |U|{A} ∀x ∈ E. , ν (x) = l r = (d) and ν∑ �{A} (x) � c F cFd∈U (d) ν ∑ {A} ν � d∈U |U| |U| � F{A} F{A} r νl � (x) = l , ν r (d) (x) = ∀x ∈ E. ν (d) ν ∑ ∑ � d∈U d∈U c F cF{A} {A} |U| F�{A} F�{A} r νl � (x) = |U| ∀x ∈ E. , ν (x) = � cF{A} cF{A} |U| |U| F ∈ IV IFS(U), cF ∈ cIV IFS(U),E = {A}the cardinality of the {A}the |U|is theU.cardinality of the universe |U|is Here |U|is Here universe U. cardinality ofHere the universe MAXIMISING ACCURACY AND EFFICIENCY OF. . . U. |U|is Here the cardinality of the universe U. � {x , x , ..., x },and A ⊆ E, then cF is presented |U|is Here the cardinality of the universe U. 1 2 n {A} � � � The sets of all cardinal sets of IVIFSS ( F , E) The sets of all cardinal setsHere The of IVIFSS sets (all F{A} cardinal sets |U|isofthe cardinality of of theIVIFSS universe(FU. {A} , E) {A} , E) � {A} The sets of all cardinal sets of IVIFSS ( F Here |U|is the cardinality of the universe U. �{A} , E) HereU|U|is cardinality theIFS(E). universe in Table 5. by ofover over U ofis all�denoted IFS(E). (F{A} ,Let E) cardinal sets cIV of IVIFSS over is the denoted cIV U is U.denoted Let by The cIVsets IFS(E). Let by over U of isall(Fthe denoted by cIV IFS(E). , E) � The �sets of all�cardinal sets of IVIFSS �{A} , E)U. Let |U|is Here of the universe {A}cardinality The sets cardinal sets of IVIFSS ( F � � � F ∈ IV IFS(U), cF ∈ cIV IFS(U),E = , E) The cardinal cF sets of IVIFSS ( F over U is denoted by cIV IFS(E). Let F�{A}sets∈ ofIVall F IFS(U), ∈ cIV ∈ IFS(U),E IV IFS(U), = cF ∈ cIV IFS(U),E = {A} {A} {A} {A}�{A} � {A} {A} {A} F ∈ IV IFS(U), cF ∈ cIV IFS(U),E = over UCardinal is denoted by cIV Let over U is denoted by cIV IFS(E). Let {A}IFS(E). {A} Table 5. set of IVIFSS � � � � � � E) The of all cardinal of cF IVIFSS (F{A} , 65 over isxn },and denoted cIV IFS(E). F{A} IV ∈ cIV IFS(U),E = ACCURACY EFFICIENCY OF. .Let . A ⊆ E, then {x , ..., xisnIFS(U), },and A� ⊆cF E,sets then is presented {A} {xMAXIMISING {x A ⊆by E,AND then , x2{A} , ...,isxnpresented },and cF presented 1 , xsets 2∈ {A} 1 , xU 2 , ..., 1cF {A} � � � � F{A} IVxIFS(U), cF ∈ cIVcF IFS(U),E = F{A}∈ ∈ IV IFS(U), = cF {A} over ∈ IFS(U),E 1 n {A} {x x22∈,U..., A= ⊆{A} E, then is presented 1 ,cIV n },and � {A} � is denoted by cIV IFS(E). Let F�{A} ∈ IV IFS(U), cF cIV IFS(U),E in 1Table 5.xn },and A ⊆ E, then {A} {x , x , ..., cF {A} is presented in Table 5. in Table 5. � {A} in {x , x22, ...,5.is xn },and Ax ⊆2 E, cF is presented xn E x1 A ⊆ E, then � 1Table �{A} �then , ..., x },and cF presented � F ∈ IV IFS(U), cF ∈ cIV IFS(U),E(= n {A} {A} {x1 , x2 , ..., xn },and A ⊆ E, {x then is presented in Table 5. (1 , x2cF ) ( ) {A} in Table 5. l r l r l Table 5. Cardinal set of IVIFSS Table 5. Cardinal set of IVIFSS Table 5. Cardinal set of IVIFSS µcofF�5. µTable (x5.2 ), (x in Table � {A} is presentedµ � (xn ), µ the(xuniverse 1 ), µcF� U.(x1 ) 2 ) cF {x1 , x2 , ..., },and A µ⊆cF�E, then in TableHere 5. |U|is the cardinality Cardinal cxF�n{A} cF{A} {A} {A} set of IVIFSS {A} �{A} Table 5. Cardinal set of IVIFSS c F . . . Table 5. Cardinal set of IVIFSS in Table 5. � r l r , E) The E sets of all cardinalx1setsνlof IVIFSS ( F Eν E x21 of IVIFSS x2 2 ),xn (x)1 ),5.( νcCardinal (x1 ),set νcF� xx11(x νlcF� xx22(xn ),xν E c) Table �{A} �{A} �{A} Table(5. Cardinal setr of cIVIFSS ((x )) ( )l ( ( 2l ),l ( F F F 1 r {A} {A} l l l r r l r r lx2µr (x (x over U is µdenoted cIV (x(x ), x), ( µcllµF� � (xx(x µ11), (x1 ), µby (x )IFS(E). µccFF�� Let (x21), µccFF�� (x21EE ) Table (x ), µ1 )c2F�) )(xn )( µcll F� x2(x2µ{A} 1µcrrµ 2 crr F� n ), F� cF�{A} 1 � cF�nof 5. Cardinal set IVIFSS cF�{A} cF{A} cr2F�{A} cF�µ ( ) ( ( ) ( {A} {A}) {A} {A} (x 1 1 2 ( ) {A} {A} {A} {A} {A} {A} {A} µ µ (x ), µ (x ) ), (x � � � � 1 1 2 l l r � l r l rccrF � � c F F�{A} c F c F c F � � � c F c F . . . . . . {A} F ∈ IV IFS(U), cF ∈ cIV IFS(U),E = {A} {A} c F µ µ (x ), µ (x ) (x ), µ (x c F c F {A} E {A} x), l �{A} µF�c{A} µcl F�2{A} (x µν{A} (x ), 2 )� {A} {A} {A} {A} (x l r EFFICIENCY l 1(x lcF�), r 1x), r ), (x ), l µ xcrr2νF(x E1 ),){A} xl2x11), MAXIMISING ACCURACY OF. .ν. ll 1lAND 2 νrr n (x 11 ), 11)), r ( r) � c F � � cF�2 cν cF� (x ), (x (x ), ν (x ν (x ν (x (x (x ), (x ν (x ), ν ν ν ν {A} {A} {A} c F F 2 l l r ( ( ) 1 2 1 2 n 2 n n � {A} {A} {A} {A} {A} � � ) (x ( )} ) ( I f a1lj = {µ � cF�then (x ),� µcF�(is(xpresented {νcF�µrr cc(x ��{A} ��{A} �{A} cFν cl F cl F� (x νcrrcFF�r{A} (x (x l 11 ), j ), c), F�{A} cF�{A} FF F�{A} cF�{A} ccFF cF�{A} crx cxF�ν {A} (x 1 l jl )}, l� l crrcF {A} {A} r 2 {A} {A} �{A} �{A} � 1�), 2 ), {A} cF�{A} νjlν µ),(x (xA1 ),⊆µcFrcE, (x1j) cF (x12),), (x µrc(x (xn )νl � ν lccF lccF {x1 , x2 , ..., xnµ},and 1ν(x 2ν F (x ν (x ), (x ), ν (x ), 21c))FE n{A} � � � � {A} µ µcF {A} {A} µ µ (x µ (x ), µ ) {A} {A} {A}(x {A} {A} 1 1 2 2 c F c F F F � � � � � 2 2 n (x ), (x ), (x ), (x ν ν � � � {A} {A} {A} F c F F c F c F �{A} 1 {A} �F� cF{A} cF2{A}) cF ( .(. . cFlc{A}Fc�F{A} F{A} lcF {A} {A} )rcF�cF�{A} 1{A} ) c( � �{A}( 5. l {A} cF {A} (x �{A}l ( l cl F�lr{A}{A} r )crrF�{A} ( l )µl rc{A} µ (x (x ), r l 1 ), µ �r{A} (xr1 ) l r.µ. .�{A} r in Table c F l ν r l r 2 ( ) � � l (xµ1 ), ν2, ν F�I jf)}, (x1 2j{ν ), (x ),lν{ν (x(x νthe a(x = {µl(x µν(xcF�F�j )} (x ), ν �cF{A} (x j )}) cν F I(x f a1n, I f a1 j = {µ (xj j ),= (x� j..., )}, {ν {µ (x jc), (x ),(xµcardinal ννccFjF�), )} 2), n ), jn)}, {A} {A} ( 1),j c= j(x j ),(x j ), ), ), ν�{A} ν cF{A}(x then set �ν (x �c{A} r µrrc(x rj ), νrrcF F� F� c(x F�llc{A} 2 ), F�U. cF�1, cF�{A}cF� F�{A} F F�{µ I{A} = {µllclcis (x {ν (x F {A} {A} cardinality {A} {A} �{A} �{A} {A} cF{A} of the {A} 1 cF{A}ccF {A} |U|is Herefor the universe I ff aa111j1jjc= (x(x µF{A} )}, {ν (x )} (x jj )}), νcFr�{A} (xn �(x �jl{A} �{A} �2{A} cF�{A} cF�c{A} l cjj ), r jj )}, rccF lcc), rccνF j ), jν j ), νcjjF� (x F F � � � � �(x � �{A} {A} {A} (x(x {A} (x {A} F F c F F F c F F c c ), (x ν I f a = {µ ), µ (x )}, {ν (x ), ν )} {A} {A} {A} {A} {A} {A} {A} 1 1 2 ( uniquely ) j� jF� � cF�{A} cF F�{A} j Table Cardinal set of F�{A} )cthe cF�c{A} cF cF�{A} j set characterized by2,rset a n,is matrix [a1i jjj]cardinal {A} (j IVIFSS = 1,= 2,{A}..., n,{A}iscthen cardinal is r then j = ...,),5. n, the for = 1, then for the set 1×n , jE) of2,(x all cardinal sets oflcardinal IVIFSS (�..., F�{A} I ffor a1The {µlc1, µ (x )}, {ν (x ), ν (x )} l r l r ( ) j = 1, 2, ..., n, then the cardinal set is j = sets j j j for j = 1, 2, ..., n, then the cardinal set is � � � cF F{A} a cF{A} f a],which {µis (xcFj{A} ), µthe (xbyj )}, (x ),i ..., νcF�n, = (x j )} j= j2, [a called characterized by(x a)},matrix [ai),j ]= r = j matrix =characterized 1,l[a is(x )} uniquely matrix characterized [a a {νmatrix = 11 12 ...{A}aIby 1n cLet F�{A}cardinal cF�{A} F�{A} i j ]1×n j ]1×n xfor xl n�[ai j(x Echaracterized x11auniquely over U is denoted by cIV IFS(E). Iuniquely {µ ), µrcthen {νcardinal ν1×n uniquely by ]1×n 2f a1 j c= uniquely characterized bya the a matrix [ajii jj ]set = j{A} jmatrix j �{A} (x 1×n F�{A} cF�{A} c F cF{A} ( ) ( ) ( ) 1×n � l r l r l r [a a ... a ],which is called the cardinal matrix uniquely characterized by a matrix [a for[a11 j = 1, 2, ..., n, then the cardinal set is ] = [a a ... a ],which is called the a cardinal ... a ],which matrix is called the cardinal matrix � � 11 12 1n cardinal set c F over E. i j 11 12 1n [a a ... a ],which is called the cardinal matrix F{A}12 ∈ of IV1nthe IFS(U), ∈ cIV IFS(U),E = µcF� (xcF µ µ ), µ (x ) (x ), µ (x ) (x ), µ (x ) 1×n {A} 11 12 1n 1 1 2 2 n n {A}cjF� = 1, 2, ..., n, then 11 12 ... a1n 1n ],which �{A} [a11 a12 is called matrix the cardinal set�cn,F�isthen cF�{A} cF�{A} cF�{A} the cardinal {A} {A} for �{A} set cF�l {A} of the cardinal set =... 1, ..., thethecardinal set is [a aj12 a1n2, ],which isover called cardinal matrix uniquely characterized by aE.of matrix [aiisj ]1×n of {x the,cardinal thecF cardinal set=cFcl�F{A} over overfor E.11 cF .F�{A} of the cardinal set c.F. {A} over E.l E. � r r r xn },and A ⊆ E, then presented 1 x2 , ..., {A} of the cardinal set c over E. {A} (xuniquely (x2 ), ν�matrix (x νcF�score νcF� by characterized acof [ai j ]set 1 ), νcF� (x 1 ), a cardinal 2 ),de{A} νcF� (xn ), νcF� (xn ), �= is of set F 1×ncF c{A} F�{A}cardinal uniquely characterized by a matrix [a ] = the over E. [a11 ina12Table ... aCardinal ],which is the matrix {A}cardinal {A} {A}called [36] {A} {A} i j 1n {A} �is{A}de1×n Cardinal score [36] ofa cardinal a cardinal cF� is deCardinal [36] set set cF�{A} deisofdeCardinal score a[acardinal Cardinal set cscore F�{A} is [36] a cardinal setscore cF�{A} isofcalled thea12 cardinal matrix (5. set[36] ) �{A}ofover 11 a12 ... a1n ],which [a11 ...score a�1n ],which is called the cardinal matrix of the cardinal cFby E. {A} is deCardinal [36] of a cardinal set c F � {A} ) and defined as noted S(c F � l r l r � {A} � � ) and defined as noted by S(c F Cardinal score [36] of a cardinal set c F F ) and defined as noted by S(c {A} � ) and defined as ) and defined as noted F{A} noted by S(c F {A} is deI f a1by (x ), µ (x )}, {ν (x ), ν (x )} {A} j =S(c{µ j j j j {A} of the cardinal set c F over E. � � � � � � {A} cF{A} set of IVIFSS cF{A} of the by cardinal set) cand F{A}defined over E.as cF{A}5. Cardinal cF Table noted S(cF{A} Cardinal score [36] of a cardinal set cF�{A} noted by S(cF�{A} {A} is de{A} ) and defined as forby S(c j E=F� 1,)2,and ..., n, then the score cardinal is deCardinal [36]set of aiscardinal set cscore F�{A} of a cardinal set cF�{A} is de(( [36] noted xCardinal xn ) xas 1 2 {A} ) ) ) ( ( defined (( ( ) ) ( 1 ( uniquely 1characterized byµr by a (x matrix [a = (x ),noted ldefined r � by l �{A} )µland(x) r ), µr l r �{A} iand j ]1×n defined as S(c F ) as noted S(c F 1 µlcl F� (x µ µ ), ) µ (x ) (x (x 1 S(c F + µ (x) − ) = 1 1 2 2 n l l r r cF� ∑ νnc)F� (x) l − ∑ νcF�{A} (x) (∑ {A}rr �� rl lclF�{A} 1is called cF�{A} cF�{A} cF cµ F�{A} c∑ F�{A} 1 � {A}νl {A} {A} {A} (x) r l r (x) + µ (x) − (x) − ν S(c F ) = 2 S(c[aF�11 S(c F µ + µ (x) − ν µ (x) (x) − + ν µ (x) (x) − . ν (x) − ν (x) . )12 = ) = l r � ... a ],which the cardinal matrix {A} ∑ ∑ ∑ � {A}ac {A} � ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ � � 1n x∈E .(x) x∈E x∈E F{A} . µ. cl F�− cr(x) F�c{A} µ�{A} (x) ν12) νcx∈E .(x S(c �− cF��{A} (x) ccF�)F�{A} F�r{A} clcF�F c{A} F�{A} S(c (x) µ), (x) − x∈E νcllcFF��{A} x∈E (x) − ∑ ν = r µ(x rF l+ {A} {A} {A}� {A} {A}ν)lcF�= 2( F 2 + �x∈E �{A} lccF rccF {A} ∑ ∑ ∑ ∑ (x ), ν ), (x ), ν (x ), (x ), ν ν �ν �{A} x∈E c F F F F c c x∈E x∈E x∈E x∈E x∈E x∈E x∈E � {A} � {A} (x) {A} (x) − x∈E 1 1 2 2 n n F F c F � {A} {A} 2 {A} S(c F ) = µ + µ (x) − ν ν {A} � � {A} � � � 2 of the cardinal set over cE. F{A} cF{A} cF{A} ccFF cF{A} cF{A} {A} ∑ ∑ ∑ ∑ x∈E x∈E x∈E x∈E � � � {A} {A} x∈E x∈E x∈E x∈E x∈E x∈E x∈E x∈E c F c F c F 1 {A} {A} 2 (x∈E {A} l r l r x∈E x∈E x∈E ) ( � ( ) S(cF{A} ) = µcF� (x) + ∑ µcF� (x) −�∑ νcF� (x) − Example . 8. Suppose the IVIFSM [P1 (i, j)]5×5 for ∑ ∑ νcF�{A} (x) 1 Suppose {A} set {A} ofr a cardinal l l1 cF{A} is r {A} 2 score de-(x Cardinal [36] lfor �= Example 8. the IVIFSM (i, − j)]5×5 x∈E x∈E x∈E x∈E � I f a1 j =8. {µSuppose (x ), µ (x )}, {ν (x ), ν )} Example the IVIFSM Example [P 8. Suppose the IVIFSM [P (i, j)] for (i, j)] IVIFSS ( F j j j j , E) is as below. l r lgiven r 1(x) 1 5×5 1 5×5 S(c F ) µ (x) +−∑ µrcνF�[P νlcF� (x) − ∑ ν � � � � {P } � 1 −∑ ν ∑). for cS(c F{A}F cF{A} ) = cF�{A} ∑ µcFc�F{A}(x) + Example µ{A} (x) (x) (x) 1 5×5 cF�{A} 8. Suppose the IVIFSM [P (i, j)] for {A} {A} 1 5×5 ∑ ∑ ∑ � 2 � � � � ( {A} ) ( ( � c F c F ) and defined as noted by S(c F c F Example Suppose IVIFSM [P1x∈E (i, j)]5×5 for IVIFSS (F�8.{P , E) given below. x∈Eis as {A} x∈E x∈E) {A} Example the IVIFSM [P1{A} (i,0.4), j)]x∈E IVIFSS (F{P IVIFSS 2(the F{Px∈E , E) given below. , E) is{A}as given below. {A}is as8. 5×5 for (0.5, 0.6), 1 }Suppose 1} 1} x∈Eis as x∈E (0.3, IVIFSS (F{P , E) given below. (0.3.0.7), {P } 1 for j = 1, 2, ..., n, then the cardinal set is } ) ( ) ( ( 1 ) ( ) () ) () ( ) (( ( () ( IVIFSS F�)1 } ,(E) is as given Example 8. IVIFSS Suppose j)]5×5below. for (the F�(0.3.0.7), , E) is)[Pas1((i,given (0.2, (0.4, 0.5) 0.4), (0.1, 0.2) below. {PIVIFSM ( 0.4) )0.6), ( (0.3, (0.5,) 0. (0.3.0.7), 1 } by a matrix (0.0, 0.0), (0.5, 0.6), 0.6), (0.5, 0.6), {P(0.3, (0.5, (0.3, 0.4), 0.4), (0.3.0.7), (( ) ( )(0.5, () uniquely characterized [a ] = ) ( ) ( ( i j � 1×n (0.5, (0.3, 0.4), (0.3.0.7), Example 8. Suppose the IVIFSM [P j)]5×5 for IVIFSS (F{P1 } , E)(is as given below. ) (0. ) ( ) (for )(0.1, ( (0.2, (8. Suppose ) 1 (i, (0.4, 0.5), (0.3, 0.7), (0.3, 0.6), (0.2, (0.4, 0.5) (0.1, 0.2) (0.5, 0. (0.3, 0.4), (0.3.0.7), (0.0, 0.0) (0.2, 0.4) (0.2, 0.4) 0.4) (0.2, 0.4) (0.4, 0.5) (0.4, 0.5) (0.1, 0.2) 0.2) Example the IVIFSM [P (i, j)] 5×5 [a11 a12 ... a11n ],which the matrix ( (0.4, ( 0.3) is(as () ( ) 0.5) (0.2) ) () ) ( (( l is called )) (r(cardinal ) ( ( IVIFSS ) (() (0.2, (0.2, 0. 0.5) )( ( )below. )0.6), (0.1, �{P (0.5, )(0.1, (0.5, 0.6), (0.3,)0.4), (0.3.0.7), (r1) F given (0.3, 0.3) } , E) l (0.4, 1(0.3, � ( ) ( ) ( (0.4, 0.5), (0.3, 0.7), (0.3, 0 ( ) ( ) ( ) (0.2, 0. (0.4, 0.5) (0.1, 0.2) � (0.4, 0.5), (0.3, 0.7), (0.3, 0.5), 0.6), 0.7), (0.3, 0.6), (0.3, 0.7), (0.3, 0.4), (0.5, 0.6), (0.5, 0.6), (0.0, 0.0), (0.3.0.7), (0.3, 0.7), (0.0, 0.0), S(c F{A}cardinal µF�c{A} (x) + µ (x) − ν (x) − ν (x) . )= IVIFSS ( F , E) is as given below. ∑ ∑ ∑ ∑ of the set c over E. � � � � ( ) ( } ) (0.4, ( (0.3, )(0.4, ( 0.6), ) cF{A} (0.4, 0.5) cF(0.1, cF{A} ( F{A} 1{A} x∈E {P (0.4, 0.5), 0.7), (0.3, 0 (0.3, 0.6), 0.6), (0.2, 0.4) (0.2, 0.4) 0.2) 2x∈E (0.1, x∈E x∈E )0.2) (0.3, 0.5) (0.2, 0.3) (0.1, (0.5, 0. (0.3, 0.4), (0.3.0.7), (0.4, 0.5), (0.3, 0.7), (0.3, 0 (0.3, 0.5) (0.2, 0.3) 0.5)0.4) (0.2, 0.3) (0.1, 0.3) (0.1, 0.5) (0.2, (0.2, (0.0, 0.0) 0.2) (0.1, (0.1, 0.2) (0.0, 0.0) {)0.3) P) j)} =( )(0.3, ( ) ((0.4) )(0.2, ( ) )(( ) () ( ) ( (( () ))((((0.4, 1 (i, ) ( ( ( ) ( ) ) (0.3, 0.5) (0.2, 0.3) (0.1, 0 ( ( ) ( ) ( ( ) ( ) ) ( ) (0.3, 0.4) 0.3) (0.2, 0.4) set (0.5, 0.6), (0.5, 0.6), (0.3, 0.4), de-(0.3, Cardinal score of a0.5), cardinal cF�{A} ( ) ( ) ( ) ) ( ) ( ( (0.4, 0.5), (0.3, 0.7), (0.3, 0.6), (0.3, 0.7), is (0.3.0.7), [36](0.4, (0.3, 0.6), (0.4, 0.6), (0.4, 0 (0.2, 0. (0.4, 0.5) (0.1, 0.2) 0.5) (0.2, 0.3) (0.1, 0.6), (0.4, 0.6), (0.4, 0.6), (0.4, 0.6), (0.3, 0.7), (0.3, 0.6), (0.3, 0.7), (0.0, 0.0), (0.3, 0.6), (0.4, 0.6), (0.4, 0.6), (0.4, 0.6), (0.0, 0.0), (0.2, 0.5), (0.2, 0.5), (0.4, 0.8), ( ) ( ) ( (0.3, 0.6), (0.4, 0.6), (0.4, 0 { P1 (i, (0.4, j)} =0.5) { P1 (i, j)} = �Suppose ,0.4) ) and (0.2, 0.4)(0.2, (0.2, 0.4) 0.2) as{ P1 (i, defined noted by j)} S(c F {)0.4) P110.3) (i, j)} = Example 8. = the IVIFSM [P(0.2, j)] for(0.3, (0.3, 0.4) (0.2, 0.3) 0 (0.4, 0.5), (0.3, 0.7), (0.3, 0.6), (0.4, 0.6), (0.4, (0.3, 0.5) (0.2, (0.1, 0.3) (0.1, 0.2) 0.4) (0.2, 0.3) (0.2, 0.4) (0.2, 0.5) (0.2, 0.3) (0.1, 0.3) (0.1, 0.2) (0.0, 0.0) (0.3, 0.4) 0.3) (0.2, (0.2, 0.4) (0.0, 0.0) {A} 1 (i, 5×5(0.1, ( ( ) ( ) ( 0.4) (0.3, 0.4) (0.1, 0.2) (((0.3, )(0.3, () ) ( ( ) ( ) ( ) ( ( ) ) ( ) ( ) ( ) ( ) ( ) ) ( ) ( ) ( )) ( ( ) ( ) ( ) ( ( { P (i, j)} = (0.3, 0.4) (0.2, 0.3) (0.2, 0 ) ) ( ) ( ( 1 � ) ( ) ( ( (0.4, 0.5), (0.3, 0.7), (0.3, 0.6), (0.3, 0.7), IVIFSS (F{P1 E) is(0.3, as given below. (0.3, 0.5), (0.4, 0.8), 0.5) (0.1, 00 (0.3, 0.4) (0.2, 0.3) (0.2, (0.2, 0.5), (0.2,(0.0, 0.5), (0.2, 0.5), (0.4, 0.8), (0.0, 0.0), 0.6), (0.4, 0.6), (0.4, 0.6), (0.4, 0.6), 0.0), (0.2, 0.5), (0.2, 0.5), (0.2, 0.5), (0.4, 0.8), } , (0.2, 0.3), (0.2, 0.3), (0.3, 0.5), 0.6), (0.4, 0.6), (0.4, 0.6), (0.4, 0.6), ) ( ) ( ( (0.2, (0.2, 0.5), (0.4, 0.8), , { P1 (i, j)} = ) (0.2, ) ) (0.0, ( ) (0.2, ( ) 0.5) ((0.3, ((0.3, ((0.1, (0.3, {( P j)}0.4) = 1 (i, (0.3, 0 (0.3, 0.4) (0.1, 0.2) (0.2, 0.3) (0.1, 0.3) (0.1, 0.2) (0.3, 0.6), (0.4, 0.6), (0.4, (0.2, (0.2, 0.5), (0.4, 0.8), (0.3, (0.3, 0.4) (0.3, 0.4) (0.1, 0.2) (0.0, 0.0) (0.3, 0.4) (0.3, 0.3) (0.2, 0.4) 0.4) 0.0) 0.4) 0.4) 0.2) 0.4) (0.5, (0.4, 0.7) (0.2, 0.4) ) ))(((0.3, ) ( 0.6) ) ( ( (0.0, 0.0), (0.5, 0.6), (0.5, 0.6), (0.3, 0.4), (0.3.0.7), ( ) ( ) ( ) ( ) ) ( ) ) ( ) ( ) ( ) ( ) )( ) ( ( ) ( ) ( ( ) ( (0.3, ( (0.3, 0.4) (0.1, 0.2) 0.4) (0.2, 0.3) (0.2, 0.4) (0.2, 0.4) { P (i, j)} = 1 ) ( ) ( ( ) ( ) ( ( ) (00 ) ( (0.2, (0.2, 0.3), (0.3, 0.5), 1 0.4) 0.3) (0.3, (0.3, 0.4) (0.1, 0.2) (0.0, 0.0), (0.3, 0.4), (0.2, 0.3), (0.2, 0.3), 0.0), 0.5), (0.2, 0.5), (0.3, 0.4), (0.2, 0.5), (0.2,(0.0, 0.3), 0.5), (0.2, 0.3), 0.8), 0.5), (0.3, 0.6), 0.6), (0.4, 0.6), (0.4, 0.6), l(0.3, r (0.2, l (0.3, r (0.4, then µ(0.4, (0.0, 0.0) (0.2, 0.4) (0.2, 0.4) (0.4, 0.5) (0.1, 0.2) ) ( ) ( ( (0.2, 0 (0.2, 0.3), (0.3, 0.5), S(cF�{A} ) = (x) + µ (x) − ν (x) − ν (x) . { P∑ j)} = (0.2, (0.2, 0.5), 0.8), (0.2, 0.5), 0.5), ∑ ∑ (0.3, ∑0.6) ( ((0.4, ) (0.3, ((0.5, ) (0.3, ( ) (0.0, ( ) 1 (i,) cF�(0.4, cF�{A} c(0.2, F�{A}0.2) cF� (0.5, (0.4, 0.7) (0.2, 0.4) {A} {A}0.4) 0.5), (0.4, 0.8), (0.2, 00 (0.2, 0.3), (0.3, 0.5), (0.0, 0.0) (0.4, 0.5) 0.0) 0.7) 0.4) 0.4) 0.4) (0.4, 0.5) (0.3, 0.4) (0.5, 0.6) (0.4, 0.7) (0.2, 0.4) 2 (0.2, 0.3) (0.2, 0.4) (0.2, 0.4) x∈E (0.1, x∈E x∈E x∈E (0.4, 0.5), (0.3, 0.7), (0.3, 0.6), (0.3, 0.7), (0.0, 0.0), (0.5, (0.4, 0.7) (0.2, 0.4) ( ) ( ) ( ) ( ) ( ) l 0.2) ) )(0.3, ( 0.4) )0.3(+0.4) ) (x0.4) (1 ) = (0.3 ( µ(0.3, + 0.4 (0.3, + 0.4 + 0.3)/5(0.3, = 0.34 (0.3, 0.5), (0.3, cF�0.3), 0.4)(0.2, (0.1, 0.2) (0.5, (0.4, 0.7) (0.2, 0.4) (0.2,(0.2, 0.3),0.3) (0.2, (0.3, 0.4), (0.0, 0.0), ( (0.1, {P1 } 0.5), ( (0.2, 0.5), (0.2, 0.5), (0.4, 0.8), then ) (0 ) ( ) ) ( then then (0.3, 0.5) (0.1, 0.3) (0.1, 0.2) (0.0, 0.0) ) ( ) ( ( r (0.2, ( 0.4) ) (0.3, ( ) (0.5, (then ) (0.4, ( ) +(0.0, ( ) (x ) = (0.7 + 0.5 + 0.6 + 0.8 0.5)/5 = 0.62 µ(0.2, 1 (0.2, 0 (0.2, 0.3), (0.3, 0.5), (0.3, 0.4), (0.2, 0.3), 0.3), 0.5), (0.4, 0.7) 0.6) 0.5) 0.0) � cF{P1 } 0.6), then (0.3, (0.3, 0.4) (0.3, 0.4)(0.4, (0.1, (0.3,IVIFSM 0.6), (0.4, 0.6), (0.4, 0.6), (0.0,)0.0), Example 8. Suppose the [P1 (i, j)]( for 0.2) 5×5 ll ) (x l0.4 + 0.4) 0.4 + ) ( ) ( ( µ ) = (0.3 + 0.4 + 0.3 + 0.4 + 0.3)/5 = 0.34 , P1 (i, j)} = µl � {(x µ ) = (0.3 + 0.3 + (x 0.3)/5 ) = (0.3 = 0.34 + 0.4 + 0.3 + 0.4 + 0.3)/5 = 0.34 1 l (0.5, (0.4, 0.7) 1 (0.2, 0.4) 0.4) (x ++ 0.3 + 0.3 +0.6) 0.1 + 0.2)/5 =0.0) 0.2= 0.34 0.4), 0 11))= (0.4, 0.5) (0.5, 0.7) clcF F�� µν(0.4, =(0.1 (0.3 0.4 + 0.3 + 0.4 +(0.0, 0.3)/5 cF{P1 } cF�{P1 } (0.2, thenIVIFSS (0.2, 0.3) (0.2, 0.4) (0.2, 0.4) {P1}} (x (F�{P1 } , E) given0.4) below. (0.3, (0.2, 0.3), 0.3), (0.3, 0.5), � 1 (0.2, lccF is(as (0.3, �{P ) ) ( ( ) ( ) ( {P11 }} (x ) =) F µ (0.3 + 0.4 + 0.3 + 0.4 + 0.3)/5 = 0.34 r {P r r 1 )0.5)/5 1 (x ) ) 0.8 ( )(0.7 ( )+0.4) (then ) () =(0.2, (+ = + 0.5 + 0.6 + + 0.5)/5 = 0.62 µ (x ) = (0.7 + 0.5 0.6 + 0.8 + (x 0.5)/5 (0.7 = 0.62 + 0.5 0.6 0.8 + = 0.62 µ µ 0.5) crrr+ F�� 1 (x ) = (0.2 + 0.5 + 0.4 + 0.2 + 0.4)/5 = 0.34 ν (0.0, 0.0), (0.2, 0.5), (0.2, 0.5), 0.5), (0.4, 0.8), 1 1 {P } 1 � (0.4, (0.5, 0.6) (0.4, 0.7) (0.2, � � 1 ccF F�{P cF{P {P11}} 0.6), then (x11 ) = (0.7 +(0.5, 0.5 + 0.6 + 0.8 +(0.0, 0.5)/5 = 0.62 µ(0.5, 0.0), 0.6), 0.4), (0.3.0.7), µlcF�cF{P(x (0.3 + 0.4 + 0.3 + 0.4 +1 }0.3)/5(0.3, = 0.34 1} 1) = rccF F�{P ) = (0.7 + 0.5 + 0.6 + 0.8 + 0.5)/5 = 0.62 µ l 1 }} (x {P l l (0.0, 0.0) (0.3, 0.4) (0.3, 0.4) (0.3, 0.4) (0.1, 0.2) 1 {P } 1 (x+1))0.2)/5 =)(0.3 (0.1 +(0.2, 0.3 + + 0.3 + + 0.1(+(0.0, 0.2)/5 = 0.34 0.2 ν 1 (x1 ) = (0.1 + νc0.1 0.3 0.3 0.2) + + ) = (0.1 =0.5) 0.2 + 0.3)+ 0.3 0.1 0.2 F� ) ) 0.4 (=+ (νlccl+ )(x0.2)/5 (+ 1( } 0.4) 0.0) 0.4) (0.1, �{P then 0.4 0.3 0.3)/5 = F� 0.3 + 0.2)/5 11 ) = F�{P {P11 } (x (x =)(0.1 0.3 + +(0.0, = 0.2 ( ( (0.4, ) (µν(0.2, ( +(0.3, ) 0.1(+ ) + 0.5 + 0.6 + 0.8 + 0.5)/5 = 0.62 µrcF�cF�{P(x 1 } 1 ) = (0.7 1 }) clcF l 0.0), 0.4), (0.2, 0.3), (0.2, 0.3), (0.3, 0.5), � {P }} (x {P F 1 ν ) = (0.1 + 0.3 + 0.3 + 0.1 + 0.2)/5 = 0.2 r �{P11 } (x µc(0.2 (x0.5 = (0.3 + 0.4 +=(0.3, 0.3 + 0.4 + 0.3)/5 = 0.34 r 1 } (x ) = r0.2 1 )0.4)/5 {P (0.4, 0.5), 0.7), (0.3, 0.6), (0.3, 0.7), (0.0, 0.0), 1) + = (0.2 + 0.5 + 0.4 + 0.2 + 0.4)/5 = 0.34 ν r + 0.4 + + (x 0.4)/5 ) (0.2 = 0.34 + 0.5 + 0.4 + 0.2 + = 0.34 ν ν � c F 1 1 1 {P } F � r (x = (0.7 + 0.5 + 0.6 + 0.8 + 0.5)/5 = 0.62 µν(0.5, l 0.4) {P1 } c0.4 F�{P11 }+ (0.0, 0.0) (0.4, 0.5) 0.6) (0.4, 0.7) (0.2, c(x F�{P 11 ) (x ) = (0.2 + 0.5 + 0.4 + 0.2 + 0.4)/5 = 0.34 µ = (0.3 + 0.4 + 0.3 + 0.3)/5 = 0.34 νlcF�cF�{P(x 0.3 + 0.3 + 0.1 + = 0.2 1 } 1 ) = (0.1 + 1 })0.2)/5 c F r 1 � {P } (0.3, 0.5) (0.2, 0.3) (0.1, 0.3) (0.1, 0.2) (0.0, 0.0) {P11 }} (x ) = (0.2 + 0.5 + 0.4 + 0.2 + 0.4)/5 = 0.34 � c F r ν c F {P 1 {P1 } + 0.5 {P1 } ) (+ 0.6 + 0.8) ( lcF�{P1 } = 0.62 ) ( ) ( ) (x1() = r(0.7 + 0.5)/5 µcF� ν (x ) = (0.1 + 0.3 + 0.3 + 0.1 + 0.2)/5 = 0.2 1 then r 1 (0.3, 0.6), (0.4, 0.6), 0.6), (0.4, 0.6), (0.0, 0.0), {P+ 0.5 + 0.4 0.2(x+ =+0.34 νcF� (x c0.8 F�{P1 }+ = (0.7 0.5 + 0.6 +(0.4, 0.5)/5 = 0.62 µcF+ 1 } , 1 ) = (0.2 1 )0.4)/5 = � {P1 } { P1 (i, j)} l {P r 1 } + 0.3 +(0.2, (0.3, 0.4) 0.3) (0.2, 0.4) (0.2, 0.4) (0.0, 0.0) ν (x ) = (0.1 0.3 + 0.1 + 0.2)/5 = 0.2 l (x ) = (0.2 + 0.5 + 0.4 + 0.2 + 0.4)/5 = 0.34 ν 1 1 ) ( ) ( ) ( ( ) ) ( l µ � (x1 ) c= � + 0.4 + 0.3 + 0.4 + 0.3)/5 = 0.34 �{P(0.3 c F F {P1 }+ 0.2)/5 = 0.2 νcF� 0.8), (x1 ) = (0.1 +0.5), 0.3 + 0.3 +(0.2, 0.1 cF{P1 } 1 } (0.2, 0.5), 0.5), (0.2, (0.4, (0.0, 0.0), {P1 } (x = (0.2 + 0.5 + 0.4 + 0.2 + 0.4)/5 = 0.34 +1 )0.5 + 0.6 + 0.8 + 0.5)/5 = 0.62 µrcF� (x1ν) r=� (0.7 r (0.3, (0.3, 0.4) (0.3, 0.4) (0.1, 0.2) (0.0, 0.0) 0.4) ) =((0.2 + 0.5 +)0.4(+ 0.2 + 0.4)/5 = 0.34 cF{P1 } ( ν � (x1 ) {P1 } ) ) ( ) ( F c {P } 1 (0.0, 0.0), (0.3, 0.4), (0.2, 0.3), (0.2, 0.3), (0.3, 0.5), νlcF� (x1 ) = (0.1 + 0.3 + 0.3 + 0.1 + 0.2)/5 = 0.2 {P1 } (0.0, 0.0) (0.4, 0.5) (0.5, 0.6) (0.4, 0.7) (0.2, 0.4) νrcF� (x1 ) = (0.2 + 0.5 + 0.4 + 0.2 + 0.4)/5 = 0.34 {P1 } then ∑ µlcF� {P1 } µrcF� {P1 } νlcF� {P1 } ∑ (x1 ) = (0.3 + 0.4 + 0.3 + 0.4 + 0.3)/5 = 0.34 (x1 ) = (0.7 + 0.5 + 0.6 + 0.8 + 0.5)/5 = 0.62 (x1 ) = (0.1 + 0.3 + 0.3 + 0.1 + 0.2)/5 = 0.2 νrcF� (x1 ) {P1 } = (0.2 + 0.5 + 0.4 + 0.2 + 0.4)/5 = 0.34 ∑ ∑ 5 66 Similarly, Algorithmic Approach Tambouratzis T., Souliou D., Chalikias M.,Tamb Greg Tambouratzis T., Souliou D., Chalikias Tambouratzis Gregoriades A.T., SouliouA. D., Ch Tambouratzis T., Souliou D.,M., Chalikias M., Gregoriades The steps for the proposed approach areM.,given Tambouratzis Tambouratzis T., Souliou T.,D., Souliou Chalikias D., M., Chalikias Gregoriades Gregoriades A. A. below. Tambouratzis Souliou D., Chalikias M., Gregoriades A. Step 1: Similarly, Opinions of aT., set of experts / decision l makers , p20.28, , ...,νprµkr} for a )given setνl of alter(xr 2 ) = 0.28, µrcF� l(x2 ) = νµlclPF�� =r(x{p (x(x =0.44 (x2 ) = 0.3, νrcF� (x2 ) 22)1= 0.3, 2 )2= l r µ(x ll r0.5, r (x l r0.5, (x l 0.28, �){A}= c(x F�c{A} cF�2{A} F c F c2F�){A}= 0.44 {A} {A} (x ) = µ ) = 0.5, ν (x (x ) ) = = 0.3, 0.28, ν µ (x ) = ) = 0.44 0.5, ν 0.3, ν µ µ {A} {A} (x ) = 0.28, µ (x ) = 0.5, ν (x ) = 0.3, ν (x ) = 0.44 µ 2 2 2 2 2 2 natives D = {d , d , ..., d } and a set of attributes 2 2 2 2 l l r r l l r r � � � � � � � � 1 2 m � � � � c F c F c F c F c F c c F F c F c(x F{A} c(x F{A} c(x F{A} c(x F{A} {A} {A} {A} l 0.44 {A} {A} {A} r r (x2µ)c= 0.28, =cF�0.28, (xµ2µl)c= 0.5, )=c= 0.5, (x )µcr= 0.3, ))c= 0.3, (x2{A} ν)µcνl= = 0.44 µcF� {A} l r 2) µ 2ν 2ν 23ν 2 )) � � � � � � (x ) 0.32, (x = 0.52, (x ) = 0.26, ν (x ) = 0.42 Similarly, (x = 0.32, µ (x ) = 0.52, ν (x ) = 0.26, ν (x F F F F F F 3 3 3 3 3 {A} {A}r l r ll rr {s1c(x l represented r interval{A}l {A} �{A} �{A} l 0.32, r30.52, S{A} = ,Fc�{A} s{A} , ..., sn }(xare using cF c�F cF�3{A} F3�32{A} cF cF�{A} cF�{A} {A} (x ) = 0.32, µ (x ) = 0.52, ν (x (x ) ) = = 0.26, µ ν (x ) ) = = 0.42 ν (x ) = 0.26, ν (x ) = 0.42 µ{A} µ {A} (x ) = 0.32, µ (x ) = 0.52, ν (x ) = 0.26, ν ) = 0.42 µ 3 3 3 3 3 3 3 3 3 l l r r l l r r � � � � � � � � � � �{A} � c F c c F F c F c c F F c F c F c F c F c F {A} {A} {A} {A} l r l r {A} {A} {A} (x3µ)c= 0.32, (x ) µ = 0.32, (x µ ) = 0.52, (x ) ν = 0.52, (x ν ) = 0.26, (x ) ν = 0.26, (x ν ) = 0.42 (x ) = 0.42 µcF� cF{A} 3 3 cF�(x4 ){A} 3=valued 3l cF�(x4 ){A} 3= 0.24, �{A} µ3 �cr F�{A} � (x4{A} )3 = c0.3, 0.56, νµrintuitionistic (xmatrices. =0.56, 0.38 fuzzy 0.3, µνrcFc�F�soft (x νlcF� (x4 ) = 0.24, νrcF� (x4 ) F F�{A} 4))= l� {A} 4) = {A} l l µ(x r�r{A} ν �)) = l(x0.24, r 20.56, F ccFF µllc4FF�){A} =µc4Fr)0.28, ) l= νc(x =r4µνc)0.3, (x 0.44 (x =l(x 0.3, (xrµc4Fc){A} =r �(x 0.56, ν)µlc0.5, == 0.3, (x (x = 0.38 ν)lcF= (x{A} ) =40.24, νrcF� (x µ{A} 2 )(x 2(x 2 )(x {A} {A}ν {A} = 0.3, µ = 0.56, = 0.24, (x 0.38 µ 4{A} 4))ν 4�4{A} 4{A} 4 ) = 0.38 � � � 4 4 l r l r � � � � � � � F c F c F � � � c F c c F F c F c F F {A} {A} {A} c(x F{A} c(x F{A} c(x F{A} c(x F{A} {A}4µ {A} {A} {A} {A} {A} {A} (x4µ)c= 0.3, 0.3, (x )c= 0.56, =cF�0.56, (x )c= 0.24, =Step 0.24, (x2: )c{A} = 0.38 0.38 µcF� {A} 4 )µc= 4) ν 4ν 4) ν 4ν 4 ) = matrix � � � � � � Cardinal of each IVIFSM is exF F c F F F F l r l r {A} {A} 0.32, µc{A} (x ) ={A}0.52, ν {A} νc{A} (x3 ) =is0.42 µcF�{A} (x3 ) ={A} 3 ) = 0.26, �{A} (xIVIFSM Cardinal the [p F�{A} 3matrix of cF Cardinal of theis IVIFSM [p (i, j)]5×5 is 1F�(i, 5×5 {A} j)]matrix {A} plored and then cardinal Cardinall Cardinal matrix ofmatrix ther IVIFSM Cardinal [p matrix of the IVIFSM [p1 (i,score (i, j)] is j)]5×5 iscomputed. 1 of the IVIFSM [p (i,= j)] 1 l 5×5 r 1) 5×5 is [p (i, j)] = [a a ... a ], where (x ) = 0.3, µ (x ) = 0.56, ν (x 0.24, ν (x ) = 0.38 µmatrix CardinalCardinal matrix of the of IVIFSM the IVIFSM [p (i, j)] [p (i, is j)] is [p (i, j)] = [a a ... a ], where 1 11 12 15 4 4 4 4 1 5×5 1 5×5 11 12 15 �{A} = [a 1×5 cF� F F�[p [p (i, j)]c1×5 a12=...[a a{A} where j)]cF1×5 a123: ...1 cCardinal a�{A} {A}1= (i,[aj)] a], 1×5 ... [p a 1 (i, ], where Step score is multiplied with the cor111×5 11 15 ], where [p1 (i,1 j)][p =j)] [a1×5 ...11a15 a11 ],15where ...12a15 ], 15 where 1 (i, 11 a= 12 [a 12 )[(( )( [( 1×5 )) (( )) (( the ) ( )] (0.32, ) )( [( [( )IVIFSM ( ) (responding ( to produce )(normalized ( )] ) (IV) ( 0.62) )(0.28, ( IVIFSM ) (1 (i,0.62) [( (0, 0) (0.3, 0.56) (0.32, 0.52) 0.5) (0.34, 0. (0.28, 0.5) (0.34, ] [ Cardinal matrix of the [p j)] is )] )] ) (0.56) ) ( )a� ( = ) ( 0.56) ) ( 5×5 )( ) 1×5 ( 0.62) ) ( 0.5) [( (0.34, [([p10.62) (i, (0.34, j)] = (0.28, (0.3, 0.56) (0.32, 0.52) (0.28, 0.5) (0.34, 0.62) (0.32, 0.52) (0.3, (0, 0) (i, j)] [p (0, 0) (0.3, (0.32, 0.52) (0.28, 0.5) 1 IFSM. Let be an IVIFSM and cardi1×5 (i j) (0, 0) (0.24, 0.38) (0.26, 0.42) (0.3, 0.44) (0.2, 0.34) j)]11×5 = (i, j)] = [p1 (i, [p [p (0, 0) (0, 0) (0.3, 0.56) (0.3, 0.56) (0.32, 0.52) (0.32, 0.52) (0.28, 0.5) (0.28, 0.5) (0.34, 0.62) (0.34, 0.62) (0.26, 0. (0.3, 0.44) (0.2, 0.34) m×n (i, j)] = [p 1 1 1×5 (i, j)]1×5 = = [a110.34) a12 (0.2, ... a15 ], where 1×5 (0.26, (0.3, 0.44) 0.34) (0.2, (0.3, 0.44)(0.3, 0.44) (0.26, 0.42) (0.24, 0.38) (0, 0)0.42) = j)] [p1 (i, j)][p (0, 0) (0.24, 0.38) (0.24, 0.38) 0.42) 0.34) 1 (i, nal(0.2, score is(0.26, h,0.42) then the 0.38) normalized IVIFSM, 1×5 1×5 0) (0,de0) (0.24, (0.24, 0.38)(0, (0.26, 0.42) (0.26, (0.3, 0.44) (0.2, 0.34) (0.2, 0.34)(0.3, 0.44) ) ( ) defined ( ) (by NIV IFSM , is )( [( noted byNIV IFSM [� a(i(0, = )] j) ] 0) �{A} )0.5) ]Cardinal [ (0.3, (0.32, 0.52) (0.28, (0.34, 0.62)score S(c �{A} 0.56) Cardinal F is score S(c F ) is � � [p1 (i, j)] ∗ a�F(i{A} m and j = 1, 2, ..., Cardinal score S(c Fscore ) isS(c scorehS(c ) is , for i = 1, 2, ..., 1×5 = Cardinal F�{A} ) isCardinal j)(0.26, m×n 0.42) (0,n.0)) (0.24, 0.38) (0.3, 0.44) (0.2, 0.34) CardinalCardinal score S(cscore F�{A}{A} )S(c is F�{A} ) is ( ( P and ) ) ( ( ( ) combined 4: µr� Choice j) l l β(i, r r (x) 1 ∑matrix l ) ) (1 ( 1l S(cF�{A} ) = 12 ∑ µlcF� 1r(x)Step +∑ (x) − ν (x) − ν ∑ S(c F ) = µ (x) + µ (x) − ν (x) ∑ ∑ ∑ x∈E x∈E x∈E x∈E r l l r r l r {A} � � � x∈E x∈E x∈E P l l r � � 2 − cF− cdecision F{A} cF{A} cF�{A} cνeach F�{A} cF�{A} {A} {A} S(cF )Cardinal = +µ)∑�isx∈ES(c )r=− νc− µc∑ +�x∈E µcC∑ (x) −� ∑(x) (x) − ∑x∈E νcF� (x) ∑ ∑ )score ∑choice �l{A} matrix β(i, of of νthe S(c2 F�{A} (x) + µ2c∑F�x∈E (x) νc∑ (x) {A} ∑(x) ∑(x) x∈E x∈E 1= l µ2cF�S(c r µcF lx∈E l (x) r νcj) r (x) F �(x) x∈E x∈E x∈E x∈E F�(x) F�{A} F�{A} F�(x) cF�{A} c F c F F {A} µ {A} {A}1µ {A}F{A} S(cF�{A}{A} µ (x) + (x) µ + − (x) ν − ν − (x) ν − (x) ν (x) )S(c = F12�{A} =x∈E ∑ ∑ ∑ ∑ ∑ ∑ ∑) x∈E ∑ {A} {A} {A} {A} x∈E x∈E x∈E x∈E x∈E x∈E x∈E �{A} �0.43 2 cF�{A}= (3.44 cF�{A} cF�{A} cF�{A} − 2.58) cF�{A} = 1= cF�(3.44 {A} computed in the con{A} makersP {p1 , − pc2F2.58) , ..., pk= }cFare 1 1 = 0.43 2 1 = − 2.58) ) (3.44=( −0.43 2.58) = 0.43 = 2 (3.44 − 2.58) = 0.432 1 2 (3.44 1= 2 = 2 (3.44 =−2 2.58) (3.44 = −12.58) 0.43 = 0.43 text of interval-valued intuitionistic fuzzy set based l r l r � S(cF{A} ) = 2 ∑x∈E µcF� (x) + ∑x∈E µcF� (x) − ∑x∈E νcF� (x) − ∑x∈E νcF� (x) on their choice parameters /{A} attributes. {A} {A} {A} = 12 (3.44 − 2.58) = 0.43 Step 5: Product IV IFSM (P ) for each deci- Similarly,Similarly, Similarly, Similarly, Similarly, IV IFSM 5 Algorithmic Approach5Approach Algorithmic Approach 5 Algorithmic Approach 5 Algorithmic sion maker is calculated by multiplying the normalAlgorithmic Approach 5 Algorithmic 5 5Algorithmic Approach Approach IV IFSMwith its choice matrix. are given The steps for the proposedized approach given Thearesteps forcombined the proposed approach The stepsThe for the proposed approach The aresteps givenfor the proposed approach are given steps for the proposed approach are given The steps The forsteps the proposed forbelow. the proposed approachapproach are givenare given below. MAXIMISING ACCURACY EFFICIENCY .. Step 6: Summation ofAND these product IVOF. IFSMs is Algorithmic Approach below.5 below. below. below. below. theexperts resultant IFSM (RIVofIFSM ). of experts / decision Step 1: Opinions of a set of / IV decision Step 1: Opinions a set Step 1: The Opinions of athe setproposed of experts Step /1:experts decision Opinions of a set of experts / decision Step 1: Opinions of a set of / decision steps for approach are given Pset = of {p1experts , ..., p/ kdecision } for a given set P of=alterStep 1: Step Opinions 1: Opinions of a makers setofofaexperts /, pdecision makers {p1 , p2 , ..., pk } for a given set of altermakers P makers = {p1 , pP2 ,=...,{pp1k,}pfor P= {p a given seta2 of alterp2 ,alter...,7:pk }Weight for a given alter-alternative d {i = pkmakers }{dfor given set 1 , of Step W (diset ) ofofeach 2 , ..., below. natives D = , d , ..., d } and a set of attributes makers Pmakers = {p1P {p , p= , ..., p , p } , for ..., a p given } for set a given of alterset of alternatives D = {d1 , d2 , ..., dm } and a set ofi attributes 1 2 1k 2 k natives Dnatives = {d12, dD2 ,= natives Dset =mof {d1attributes ..., dm1 ,}d2and admset of attributes ,1, d22, , ..., d } and a set of attributes {d , ..., } and a m ...,using m} is estimated by adding the membership {sad1am ,set s}set ..., }set areofrepresented natives D natives = {d =, ..., {d1d,mSd}2=,of , dOpinions and ..., ofsexperts aattributes attributes S = {s1interval, s2 , ..., sn } are represented using interval2 ,and Step 1: of decision 1D S = {s Susing =n {s1interval..., sn21}, sare represented , s2using ,/ ..., sninterval}and arenon-membership represented using interval{s s } are represented 1 , sS 2 ,= values of the entries of the re2 , ..., n valued intuitionistic fuzzy soft matrices. S = {s1 ,makers Ss2=, ..., {ss1nP ,}s= , are ..., s represented } are represented using intervalusing intervalvalued intuitionistic fuzzy soft matrices. 2 {p1n, p2 , ..., pk } for a given set of alterth valued intuitionistic fuzzy softfuzzy matrices. valued intuitionistic fuzzyrow soft(imatrices. valued intuitionistic soft matrices. spective row). valued intuitionistic valued matrices. natives intuitionistic D =fuzzy {d1 ,Step dsoft ...,matrices. and a set of attributes 2 ,fuzzy 2:dm }soft Cardinal matrix of each Step IVIFSM is ex- matrix of each IVIFSM is ex2: Cardinal Step 2: Cardinal matrix of each Step IVIFSM 2: Cardinal is exmatrix IVIFSM ex- S(di ) of di , such Step 2: Cardinal matrix of each IVIFSM is exD, compute theisscore 8:∀dof S = {s , s , ..., s } are represented using intervali ∈each 2matrix nplored andof then cardinal is Step 2: Step Cardinal 2:1 Cardinal of matrix each IVIFSM each IVIFSM is ex-score is Step ex-computed. plored and then cardinal score is computed. ploredvalued andplored then cardinal is soft computed. plored then cardinal and thenscore cardinal score is and computed. that, score is computed. intuitionistic fuzzy matrices. plored and plored thenand cardinal then cardinal score score is computed. Step is 3:computed. Cardinal score is multiplied with cor- score is multiplied with the corStep 3: the Cardinal Step 3: Cardinal score ismatrix multiplied Step with 3: the Cardinal corscore is multiplied Step 3: Cardinal scoreofisIVIFSM multiplied withis the cor2: Cardinal Cardinal each IVIFSM exl IVl with rthe corto produce the normalized Step 3: Step Step Cardinal 3: score isresponding score multiplied is multiplied with the with corthe corresponding IVIFSM IV) = {(µ − ν νriIV)}/2, the di ∈normalized D ∀i. S(d i i i ) +] (µito−produce ] [ [ responding IVIFSM to produce the responding normalized IVIFSM IVto produce the normalized responding IVIFSM to produce the normalized IVplored cardinal score is(ithe computed. ]IVIFSM ] [and then [ ] produce [IFSM. � Let be an IVIFSM and cardia responding responding IVIFSM to produce to the normalized normalized IVIVIFSM. Let a�(i j) m×n be an IVIFSM and cardij) m×n ]be [ a�(i]j) Let [ IFSM. Let IFSM. an IVIFSM andLet cardibe an IVIFSM and cardia�and IFSM. be anthen IVIFSM cardia� (i j)Step (ian j)score m×n m×n ) > the S(dnormalized then dem×n nal is h, the normalized IVIFSM, de� IFSM. Let IFSM. be IVIFSM be an IVIFSM and cardiand cardia�3: a j ) ∀d j ∈ D, nal9: scoreIf is S(d h, ithen IVIFSM, Step score is multiplied with the cor(i Let j)Cardinal (i j) m×n m×n nal score nal is h,score thenisthe normalized nal IVIFSM, score is deh, then the IVIFSM, deh, thenbytheN normalized IVIFSM, de- normalized alternatived is selected. If ∃ j, such that,S(d noted , is defined byN [� a ] = nal scoreresponding nalis score h, then isIVIFSM h, thethen normalized the normalized IVIFSM, IVIFSM, dedei i noted by N , is defined byN a= IV IFSM IFSM (i j)IV IFSM to produce the normalized IV-is[ IV IV IFSM)[� (i j) ] = ] byNIV IFSM [� noted by] noted NIV IFSM noted by N isIV[]defined [� a](iIV = defined a(i j) ] = by][, N , is]byN defined [� a,and ]= IV IFSMbyN IFSM j) ]IV IFSM IFSM (i j) [ [ S(d ), where i = ̸ j for highest score value, then de[ � h ∗ a , for i = 1, 2, ..., m j = 1, 2, ..., n. noted byIFSM. noted by ,NIV isa�IFSM , is (ibe defined byN byN [� aIV ] and =[� a(icardi� j) m×n , for i = 1, 2, ..., m and j = 1, 2, ..., n. Let an IV IFSM IFSM (i j) j) ] =j h ∗ a (idefined j) 2, m×n ]for [ h ∗ a� []j)NIVhIFSM m×n �(iand h=∗ma1, i= 1, ...,i = mj) 1, and jIVIFSM 2, n.,1, for2, i..., = n. 1, 2, ...,(i m and j = 1, 2, ..., n. ∗,a� , for 2, ..., j= j) ..., (i j) m×n cision is made according to their accuracy values as m×n h ∗ a�(i j)(inal hm×n ∗m×n a�,(ifor i = 1, , for 2, ..., i = m 1, and 2, ..., j m = and 1, 2, ..., j = n. 1, 2, ..., n. score is h, then the normalized IVIFSM, deP j) m×n and 4: combined Step 4: matrix β(i, j)Step Choice matrix β(i, j)P and combined PChoice P and combined P described in step 10. and combined Step noted 4: Step Choice matrix β(i, j) Step 4: Choice matrix β(i, j) Choice matrix β(i, j)IFSM by] 4: NIVChoice , is defined [� a(icombined PCand IFSM j) ] = of matrix β(i, j) of each the decision [ Choice combined and combined Step 4: Step 4: matrix β(i, matrix j)PCP and β(i,byN j)PIV matrix β(i, j)PC of each of the decision PCchoice Pchoice C of each choice hmatrix β(i, j) choice matrix β(i, j) of each of the decision of value the decision choice matrix β(i, j) each of the decision � ∗ a , for i = 1, 2, ..., m and j = 1, 2, ..., n. P P 10: Accuracy H(d ), i = 1, 2, ..., m, is C of m×n makersP = , ...,the pk }decision are Step computed in the conchoice matrix choice(i j)β(i, matrix j) C β(i, of j) each of{p each the makersP = {p 1 , pdecision 2of 1 , p2 , ..., pk }iare computed in the conmakersP = {p1 , p2 ,=...,{pp1k,}pare makersP = {p computed in the con, p , ..., p } are computed in the conmakersP , ..., p } are computed in the con1 2 k 2 k defined as P text of intuitionistic fuzzy based makersPStep makersP = {p14:, p= {p , ..., computed pinterval-valued computed inj)the conin the con- text of set interval-valued intuitionistic fuzzy set based and combined Choice matrix 2 , ..., 1p,kp} 2are k } areβ(i, text of interval-valued intuitionistic text fuzzy ofset interval-valued based intuitionistic fuzzy set based text of interval-valued intuitionistic fuzzy set based P C on their choice parameters / attributes. text of interval-valued text of interval-valued intuitionistic intuitionistic fuzzy set fuzzy based set based on their choice parameters / attributes. choice matrix β(i, j)/ attributes. of each of choice the decision on their choice parameters on their parameters / attributes. {(µli + νli ) + (µri + νri )}/2, di ∈ D ∀i. H(di ) = on their choice parameters / attributes. on their makersP on choice theirparameters choice attributes. / attributes. = {p1 ,parameters pStep pk }Product are computed in the 2 ,/ ..., 5: IV IFSM (PIVcon) for 5: each deci- IV IFSM (PIV IFSM ) for each deciProduct IFSM Step Step 5: Product IFSM (P 5: Product IV IFSM for each deciStep 5:IVProduct IV (P ) for each deci- (P IVIFSM IFSM )Step IV IFSM ) for each deciIV IFSM text of interval-valued intuitionistic fuzzy set based maker is calculated by multiplying the normalStep 5: Step Product 5: Product IV IFSMsion IV (PIFSM (P ) for each ) for decieach deci) > H(d ) ∀calculated j for which S(d ) = S(d the If H(d sion maker by multiplying normalIFSM IV IFSM i jis j ), desion maker is calculated byIVmultiplying sion the maker normalis calculated by multiplying the normal- i sion maker is calculated by multiplying the normalon their choice parameters / attributes. ized IV IFSMwith its combined choice matrix. sion maker sionismaker calculated is calculated by multiplying by multiplying the normalthe normalfinedized in Step 9, alternative is selected. If H(d IV IFSMwith its dcombined choice matrix. i) = ized IV IFSMwith its combined choice izedmatrix. IVchoice IFSMwith its combined choice matrix.i ized IV IFSMwith its combined matrix. ized IV IFSMwith ized IV its combined its combined choice matrix. choice matrix. H(d )for any j, then d and d both are selected. Step 5:IFSMwith Product IV IFSM (P ) for each decij i j IV IFSM Step 6: Summation of these product IFSMs is StepIV6: Summation of these product IV IFSMs is Step 6: Summation of these by product Step IVproduct 6: IFSMs Summation isIFSMs of isthese product IV IFSMs is Step 6:isSummation of multiplying these IV sion maker calculated the normalthe resultant IV (R ). 6: Summation of these product IV IFSMs is Step 6: Step Summation of these product IVIFSM IFSMs is the resultant IV IFSM (RIV IFSM ). IV IFSM the resultant IFSM (R resultant IV IFSM (RIV IFSM ). ).(RIVthe the resultant IVits IFSM ). IV IFSM IFSM izedresultant IVIV IFSMwith combined the IV(RIFSM the resultant IV IFSM ). IV IFSM ).choice matrix. IV IFSM(R Step 6: Summation of these product IV IFSMs is the resultant IV IFSM (RIV IFSM ). the set of five symptoms (Chest pain, Palpitations, Dizziness, Fainting, Fatigue) given by E = {s1 , s2 , s3 , s4 , s5 } . Suppose that ACCURACY a group of three experts MAXIMISING AND EFFICIENCY OF. . . P = {p1 , p2 , p3 } 6are StepCase 7: Weight Study W (di ) of each alternative di {i = monitoring the symptoms of a patient as per 1, 2, ..., m} is estimated by adding the membership theirLet knowledgebase reach }a consensus D = {d1 , d2 ,to dvalues be entries the setofabout of 3 , d4 , dof 5 the and non-membership thefive rewhich stage is more likely to appear for the patient, th stages of heart disease (Stage ‘I’, Stage ‘II’, Stage spective row (i row). where expert p1 is aware of symptoms (s1 , s2 , s3 , ‘III’, Stage ‘IV’, and Stage ‘V’). Patients belongsStep is aware symptoms , s )),ofand compute the (s score di , psuch 4 ), p28:∀d 1 s2 , sS(d 3 i ∈ D,of ing to Stage ‘I’ are assumed not to 3be5i affected by is aware of symptoms (s1 , s2 , s4 , s5 ). According to that, heart disease. Patients belonging to Stage ‘II’ are the symptoms or parameters observed by the three in initial stage, belonging to Stage ‘III’ lpatients l experts, assume have in IV-are = {(µ (µri the − νriinformation )}/2, di ∈ D ∀i. S(di )we i − νto i) + in more unsafe stage than in stage ‘II’ and so on. IFSSs Patients to, E), Stage are ) (>F�{p‘V’ S(d ) ∀dinj the ∈ last D, stage then Step 9:belonging If(F�{pS(d , E), 1} i 2} j of heart disease which is unrecoverable. Let E be alternatived i is selected. If ∃ j, such that,S(di ) = and set of five the symptoms (Chest pain, Palpitations, S(d j ), where i ̸= j for� highest score value, then de(F{p3 } , E) Dizziness, Fainting, Fatigue) given by cision is made according to their accuracy values as for experts in p1step , p2 , and described 10. p3 respectively. E = {s1 , s2 , s3 , s4 , s5 } . Let IVIFSMs of thevalue IVIFSSs Stepthe10: Accuracy H(di ), i = 1, 2, ..., m, is Dizziness, Fainting, Fatigue) given by E = {s1 , s2 , s3 , s4 , s5 } . Suppose that a group of three experts Tambouratzis T., Souliou D., Chalikias M., Gregoriades67 A. P = {p1 , p2 , p3 } are monitoring the symptoms of a patient as per their knowledgebase to reach a consensus about which stage is more likely to appear for the patient, where expert p1 is aware of symptoms (s1 , s2 , s3 , s4 ), p2 is aware of symptoms (s1 s2 , s3 , s5 ), and p3 is aware of symptoms (s1 , s2 , s4 , s5 ). According to the symptoms or parameters observed by the three experts, we assume to have the information in IVIFSSs (F�{p1 } , E), (F�{p2 } , E), and (F�{p3 } , E) for experts p1 , p2 , and p3 respectively. Let the IVIFSMs of the IVIFSSs defined asthat Suppose a group of three experts (F�{p1 } , E), (F�{p2 } , E), (F�{p3 } , E) (F�{p1 } , E), (F�{p2 } , E), (F�{p3 } , E) H(d ) = {(µl + νl ) + (µr + νr )}/2, d ∈ D ∀i. i i i P= i {p1 , ip2 , pi3 } are respectively, are respectively, ( ) ( ) ( ) ( ) ( ) ∀ symptoms j for whichofS(d )= S(das de-(0.5, 0.6), ) ( (0.5, If H(d ( are monitoring a i patient i ) > H(dthe j )(0.3.0.7), j ), per (0.3, 0.4), 0.6), (0.3.0.7), (0.0, 0.0), (0.3, 0.4), fined knowledgebase in Step9, alternative di is selected. If H(d their to reach a consensus about i) = 0.0) 0.4) (0.1, (0.4, ) 0.2) ( (0.0, )0.5) ( (0.1, 0.2) ) ( (0.4, 0.5) ) ( (0.2, 0.4) ) ( (0.2, ) ( ( H(d j )for any then di 0.5), and to d j appear both(0.3, are which stage isj,more likely forselected. the patient,(0.3, 0.6), (0.4, 0.7), (0.3, 0.7), (0.0, 0.0), (0.4, 0.5), (0.3, 0.7), p1 is(0.3, where expert aware of symptoms (s , s , s , 0.5) (0.2, 0.3) (0.1, 0.3) (0.1, 0.2) (0.0, 0.0) 1 2 3 ( ) ( ) ( ) ( ) 0.5) ( ) 0.3) (0.3, (0.2, ( ) ( 0.6), (0.4, (0.0, 0.0), s4 ), p2 is aware of(0.3, symptoms (s1 s(0.4, s5 ), and p3(0.4, 0.6), 2 , s3 ,0.6), , 0.6), (0.3, 0.6), (0.4, 0.6), {p1 (i, j)} = (0.3,AND 0.4) 0.3) (0.0, 0.0) ) {p1 (i,)j)}(=(0.2, MAXIMISING ACCURACY OF. . . ) (to(0.2, 0.4) is aware of symptoms (s1 ,EFFICIENCY s2), s4(, s5(0.2, ). According 0.4) ) ( ( (0.3, 0.4) ) ( (0.2, 0.3) ( (0.2, 0.5), (0.2, 0.5), (0.2,OF. 0.5), (0.0, 0.0), (0.4,AND 0.8),EFFICIENCY the symptoms or parameters observed by MAXIMISING ACCURACY . .the three (0.2, 0.5), (0.4, 0.8), 0.4) (0.3, 0.4) (0.3, 0.4) (0.0, 0.0) (0.1, 0.2) (0.3, experts, we assume to have the information in IV ) ) ( ) ( ) ( ) ( ( (0.3, 0.4) (0.1, 0.2) ( (0.0, 0.0), (0.3, 0.4), (0.2, 0.3), (0.2, 0.3), (0.3, 0.5), ) ( IFSSs (0.2, 0.3), (0.3, 0.5), ) ) ( ) ( ) ( ) ( ( (0.0, 0.0) (0.4, 0.5) (0.5, 0.6) (0.4, 0.7) 0.4)(F� , E), (F�(0.2, , E), {p (0.0, 0.0), (0.2, 0.5), (0.5, 0.6), (0.3, 1 } 0.5), {p2 } (0.4, 0.7) (0.2, 0.4)(0.5, 0.7), ( (0.3, 0.4) ) ( (0.3, 0.4) ) ( (0.3, 0.4) ) ( (0.0, 0.0) ) ( (0.2, 0.3) ) and ( (0.3, 0.5), ) ( (0.5, 0.6), ) ( (0.2, 0.5), ) ( (0.0, 0.0), ) ( (0.5, 0.7), ) (0.4, (0.4, (0.3, 0.4) (0.3, 0.7), 0.4) ) ( (0.3, (0.3, 0.5), 0.4) ) ( (0.0, (0.0,0.0), 0.0) ) ( (0.4, (0.2, 0.5), 0.3) ) (F�0.5), {p3 } , E) ) ( (0.2, ( (0.3, (0.3, 0.4) (0.0, 0.0) (0.2, 0.3) 0.3) 0.5) ( (0.4, 0.5), ) ( (0.4, 0.7), ) ( (0.3, 0.5), ) ( (0.0, 0.0), ) ( (0.4, 0.5), ) for experts p and 0.5) p3 respectively. 0.6), (0.7, 0.3) 0.8), (0.2, 0.3) 0.4), (0.0, 1 , p2 , (0.3, (0.3,0.6), 0.4) ) (0.0,0.0), 0.0) ) ( (0.4, (0.2, (0.2, (0.3, {p2 (i, k)} = ( (0.3, , ) ( ) ( ) ( 0.4) (0.1, (0.3, 0.4), 0.5) ) ( (0.0, ( of (0.3, 0.6), (0.7, 0.2) 0.8), ) ( (0.2, (0.0, 0.0) 0.0), ) ( (0.2, (0.4, 0.4) 0.6), ) ) ( Let the IVIFSMs the IVIFSSs , {p2 (i, k)} = (0.2, 0.4) 0.6), (0.5, 0.2) 0.7), (0.3, 0.5) 0.6), (0.0, 0.0) 0.0), (0.5, (0.3, (0.1, (0.3, (0.0, (0.2, 0.6), 0.4) ) ( ) ( ) ( ) ( ) ( (0.1, 0.2), E), 0.2) (0.2, 0.6), 0.3) ) ( (0.0, (F�0.6), F�{p(0.1, , E) (F� (0.2, (0.5, 0.7), ) ( (0.3, (0.0, 0.0) 0.0), ) ( (0.3, (0.5, 0.4) 0.6), ) {p1 ( } , E), {p2 } ) (( 3} (0.4, 0.5), (0.2, 0.3), (0.4, 0.7), (0.0, 0.0), (0.4, 0.5), ( (0.1, 0.2) ) ( (0.1, 0.2) ) ( (0.2, 0.3) ) ( (0.0, 0.0) ) ( (0.3, 0.4) ) (0.2, 0.5), 0.4) (0.4, 0.3), 0.5) (0.1, 0.7), 0.3) (0.0, 0.0), 0.0) (0.3, 0.5), 0.4) are respectively, (0.4, (0.2, (0.4, (0.0, (0.4, ( (0.2, 0.4) ) ((0.4, 0.5) ) ((0.1, 0.3) (0.3.0.7), (0.3, 0.4), (0.5, 0.6), ( ) ( ) ( ) (0.3, (0.1, 0.2) (0.4, 0.5) (0.2, 0.4) 0.5), (0.2, 0.7), (0.0, 0.0), )( ( (( ) ) )( ( (0.3, 0.4) 0.3) (0.0, 0.0)0.6), (0.4,0.5), 0.5),) (0.3, 0.7), (0.3, 0.0), (0.2, 0.7), ) ( (0.1, ) ( (0.0, ) ( (0.3, (0.3, (0.3, 0.5), (0.0, 0.0), (0.3,0.7), 0.5) ) ((0.1, (0.2, 0.3) ) ( (0.1, (0.0, 0.0)0.3) 0.4) (((0.3, ) ( (0.3, 0.3) ) ( (0.0, ) (0.2, 0.3) 0.5) 0.0) (0.3,0.7), 0.6),) ( (0.3, (0.4, 0.6), (0.4, 0.6), 0.5), 0.0), ) ( (0.0, ) ( (0.3, {p1 (i, j)} = (0.5, 0.6), (0.0, 0.0), (0.3,0.7), 0.4) ) ((0.5, (0.2, 0.3) ) ( (0.2, 0.3) (0.3, 0.5) (0.0, 0.0)0.4) (((0.2, {p3 (i, l)} = ) ( ) ( ) (0.1, 0.3) (0.3, 0.4) (0.0, 0.0) (0.2, 0.5), (0.2, 0.5), (0.4, 0.8), ( (0.5, 0.7), ) ( (0.5, 0.6), ) ( (0.0, 0.0), ) {p3 (i, l)} = (0.4, 0.5), (0.0, 0.0), (0.3, (0.3, 0.4)) )( ( (0.1,0.6), 0.2) ))(((0.4, 0.3) (0.3, 0.4) (0.0, 0.0)0.4) (((0.1, ) 0.3) (0.3, 0.4) (0.0, 0.0) ( (0.2, (0.4, 0.6), (0.4, 0.5), (0.0, 0.0), (0.2, 0.3), (0.2, 0.3), (0.3, 0.5), ) ) ( ) ( 0.0), 0.6), (0.1, 0.3) (0.3, 0.4) (0.0, 0.0)0.6) (0.5, (0.4, 0.7)) ( (0.0, (0.2,0.2), 0.4) ) ( (0.4, ( (0.2, (0.0, 0.0) ) (0.3, 0.4) (0.5, 0.7) (0.0, 0.0), (0.4, 0.6), (0.1, 0.2), (0.0, 0.0) (0.3, 0.4) (0.5, 0.7) Here i (1,2,...,5) is used to represent an alternative, i.e., a stage ofisheart and theanindices Here i (1,2,...,5) useddisease to represent alterj, k, l (1,2,...,5) are used for attributes, i.e., sympnative, i.e., a stage of heart disease and the indices toms. are are the input information, by the j, k, l These (1,2,...,5) used for attributes,given i.e., sympexperts. toms. These are the input information, given by the ) (0.0, ( 0.0) ) (0.3, ( 0.4) ) (0.5, 0.6), (0.0, 0.0), ) ( ) (0.0, 0.0) ) 0.6),0.4) ) (0.2, ) (0.3, ( (0.2, ( 0.4), ) ) ( ( 0.4) (0.0, 0.0), (0.2, 0.6) 0.4), (0.3,(0.3, 0.6),0.7), ( (0.3, ) ( (0.5, ) (0.4,(0.1, 0.8), (0.7, 0.8), (0.0, 0.0) (0.5, ) (0.3, ( 0.4)0.2) ( 0.6) ( ) ()(0.1, ) ) (0.1, 0.2) 0.2) (0.0, 0.0), 0.8),0.6), ( (0.4,(0.4, ) ( (0.7, 0.8), ) (0.5,(0.2, 0.6),0.4) (0.4, 0.6), . (0.0, 0.0) (0.1, 0.2) (0.1, 0.2) ) ( ) ( ) ( ) ( ) (0.3, 0.4) (0.3, 0.4) (0.0, 0.0), (0.2, 0.5), ( (0.5, 0.6), ) ( (0.4, 0.6), ) . (0.3, 0.7), (0.3, 0.4), (0.0, 0.0) (0.3, 0.4) (0.3, 0.4) (0.3, 0.4) ( 0.5) ) ( ) (0.2, ( 0.3) ) ()(0.4, ) 0.7),0.4), (0.0, ) 0.0), ) ( (0.3, 0.4), ( (0.3,(0.3, (0.1, 0.5) 0.4), (0.2,(0.4, 0.3),0.5) (0.4, 0.3) (0.0, 0.0) ) ) ( ( (0.2, (0.4, 0.4), 0.5) (0.5, 0.3), 0.6) (0.1, (0.2, (0.4, 0.5) (0.5, 0.6) ( ) ( ) ( ) ( ) ( ) ( , (0.5, 0.6), (0.2, 0.4) (0.3, 0.6), (0.1, 0.3) (0.4, 0.6), (0.2, 0.4) (0.2, 0.5), (0.3, 0.4) (0.2, 0.3), (0.5, 0.6) . j, k, l (1,2,...,5) are used for attributes, i.e., symp{p3 (i, l)} = (0.1, 0.3) (0.3, 0.4) (0.0, 0.0) (0.3, 0.4) (0.3, 0.4) ( ) ( ) ( ) (are the input ) ( ) by toms. These information, given the (0.4, 0.6), (0.4, 0.5), (0.0, 0.0), (0.3, 0.7), (0.3, 0.4), experts. ( (0.2, 0.3) ) ( (0.3, 0.4) ) ( (0.0, 0.0) ) ( (0.2, 0.3) ) ( (0.4, 0.5) ) In this case study, we consider two cases. In the first (0.1, 0.4), (0.2, 0.3), (0.0, 0.0), (0.4, 0.6), (0.1, 0.2), case, we use non-normalized IVIFSM and normalTambouratzis T., Souliou M.,0.5) Gregoriades A. (0.4, (0.5, 0.6)D., Chalikias (0.0, 0.0) (0.3, 0.4) (0.5, 0.7) 68 ized IVIFSM in the second case. Here i (1,2,...,5) is used to represent an alter- Case 1: It uses non-normalized IVIFSMs as input. 6 Case native, i.e., aStudy stage of heart disease and the indices [Step 2 & Step 3]: These steps are not applicable in Case1. j, k, l (1,2,...,5) are used for attributes, i.e., sympLet D = {d , d , d3 , d4 , d5 } be the set of five toms. These are 1the2input information, given by the stages of heart disease (Stage ‘I’, Stage ‘II’, Stage experts. ‘III’, Stage ‘IV’, and Stage ‘V’). Patients belongIn weassumed consider not twoto cases. In the first ingthis to case Stagestudy, ‘I’ are be affected by case, use non-normalized IVIFSM and normalheart we disease. Patients belonging to Stage ‘II’ are ized IVIFSM the second case. to Stage ‘III’ are in initial stage,inpatients belonging in more stage than in stage ‘II’ and so on. Case 1: Itunsafe uses non-normalized IVIFSMs as input. s{ p p } Patients belonging to Stage ‘V’ are in the last stage [Step 2 & Step 3]: are not applicable These ( 1 ,is 1 )unrecoverable. , steps ( 1 , 1 ) , Let ( 0E, 0be) , of heart disease which in Case1. ( 1 , 1 ) ( 1 , 1 ) the set of five symptoms (Chest pain, Palpitations, ( 0 ,0 ) Fatigue) [Step 4] The combined Dizziness, Fainting, ( 1 , 1choice ) , given ( 1 by , 1 ) , forp(10, ,p02), , matrices and p3 are respectively ( 1 ,1 ) ( 1 ,1 ) ( 0 ,0 ) E = {s1 , s2 , s3 , s4 , s5 } . 2 [Step 4] The combined choice matrices for p1 , p2 , and p3 are respectively 3 ( 1 ,1 ) , ( 1 ,1 ), ( 0 ,0 ), s{ p1 } ( 1 , 1 ) ( 1 , 1 ) ( 0 , 0 ) Suppose that a group of three experts ( 1 ,1 ) , ( 1 ,1 ), ( 0 ,0 ), =( 1{p P , 11 ), p2 , p3 }( 1 , 1 ) ( 0 , 0 ) ( 0 ,0 ) , ( 0 ,0 ), ( 0 ,0 ), symptoms are monitoring the of a patient as per ( 0 ,0 ) ( 0 ,0 ) ( 0 ,0 ) their knowledgebase to reach a consensus about which stage is more likely to appear for the patient, s where expert p1 is aware of symptoms (s1 , { sp2, ps3} , ( 1 , 1 ) , (s1(s12 ,,s13), ,s5 ), and ( 0 ,p03 ) , s4 ), p2 is aware of symptoms , 11), s2 , s4, s(51)., 1According ) ( 0 ,to 0) is aware of symptoms ( 1 (s ( 1 , 1 ) , observed the symptoms or parameters , the (three 0 ,0 ), ( 1 , 1 )by experts, we assume ( 1have , 1 ) the information ( 1 , 1 ) in( 0IV,0 ) to IFSSs ( 1 ,1 ) , ( 1 ,1 ) , ( 0 ,0 ) , s { p(F �} {p � 1} ,( E), 1 , 1 ()F{p2 } , E), (1 ,1 ) (0 ,0 ) 1 2 and ( 0 ,0 ) , ( 0 ,0 ) , ((F � , 0 }), E) (0 ,0 ) 3 0{p ( 1 ,1 ) , ( 1 ,1 ) , for experts p1 , p2 , and p3 respectively. ( 1 ,1 ) ( 1 ,1 ) Let the IVIFSMs of the IVIFSSs (0 ,0 ), (0 ,0 ) (0 ,0 ), (0 ,0 ) 1 (0 ,0 ), (0 ,0 ) (0 ,0 ), (0 ,0 ) (1 ,1 ) , (1 ,1 ) (1 ,1 ) , (1 ,1 ) (0 ,0 ), (0 ,0 ) (1 ,1 ) , (1 ,1 ) (0 ,0 ), ( 0 , 0 ) (0 ,0 ), (0 ,0 ) (1 ,1 ) , ( 1 , 1 ) (0 ,0 ) , (0 ,0 ) 3 s (F�{p1 } , E), (F�{p2 } , E), (F�{p3 } , E) { p (1 ,1 ), (1 ,1 ) (1 ,1 ), (1 ,1 ) ( 0 ,0 ) , ( 0 ,0 ) ( 0 ,0 ) , ( 0 ,0 ) (1 ,1 ), (1 ,1 ) (0 ,0 ), (0 ,0 ) ( 0 ,0 ) , ( 0 ,0 ) ( 0 ,0 ) , ( 0 ,0 ) (1 ,1 ), (1 ,1 ) (0 ,0 ), (0 ,0 ) p2 } ( 1 ,1 ) , ( 1 ,1 ) , ( 1 ,1 ) , ( 0 ,0 ) , ( 0 ,0 ) , are respectively, ( 1 , 1 ) ( 1 , 1 ) ( 1 , 1 ) ( 0 , 0 ) ( 0 , 0 ) ( ( 1 , 1 ) , ) (( 1 , 1 ) , ()1 , 1() , ( 0 , 0 )), ( ( 0 , 0 ) , ) (0.3.0.7), (0.5, (0.3, 0.4), 0.6), (0.5, 0.6), ( 1 ,1 ) ( 1 ,1 ) ( 1 ,1 ) ( 0 ,0 ) ( 0 ,0 ) (0.1, 0.2) (0.4, 0.5) (0.2, 0.4) (0.2, 0.4) ( ( 0 , 0 ) , ) (( 0 , 0 ) , () 0 , 0() , ( 0 , 0 )) , ( ( 0 ,0 ), ) 0.5), (0.3, 0.7), (0.3, 0.6), (0.3, 0.7), s{ p 3 } (0.4, 0 ) ( 0 ,0 ) ( 0 ,0 ) ( 0 ,0 ) ( 0 ,0 ) ( 0 ,0.5) (0.3, ( ) ( (0.2, 0.3) ) ( (0.1, 0.3) ) ( (0.1, 0.2) ) 1 ) , ( 1(0.4, , 1 ) , 0.6), ( 0 , 0 ) , (0.4, ( 0 , 00.6), ), (0.3, ( 1 ,0.6), ( 1 , 1 ) , (0.4, 0.6), ( 1 ,1 ) ( 1 ,1 ) ( 1 ,1 ) ( 0 ,0 ) ( 0 ,0 ) {p1 (i, j)} = (0.3, 0.4) ( 0.4) ) ( (0.2, 0.3) ) ( (0.2, 0.4) ) ) ( (0.2, (0.4, ( 1 , 1 ) , ( 1 , 1 ) , ( 1 , 1 ) , ( 0 , 0 ) , (0.2, ( 0 , 00.5), ), (0.2, 0.5), (0.2, 0.5), 0.8), (0.1, 1 ) ( 1(0.3, 1 , 1 ) (0.3, ( 0 , 0 ) (0.3, ) () ( ( 1 ,0.2) ) ) ( ( 0 , 00.4) ( 0.4) ) ( , 1 ) 0.4) ) (0.0, 0.0), ( (0.0, 0.0) ) (0.0, 0.0), ( (0.0, 0.0) ) (0.0, 0.0), , ( (0.0, 0.0) ) (0.0, 0.0), (0.0, 0.0) ) ( (0.0, 0.0), (0.3, 0.4), (0.2, 0.3), (0.2, 0.3), (0.3, 0.5), [Step 5](0.2, The0.4) product (as per the0.7) rule of multiplication of IVIFSMs (non-normalized) and combined choic (0.0, 0.0) (0.4, 0.5) (0.5, 0.6) of IVIFSMs) (0.4, matrices are given below. ( Tambouratzis T., Souliou D., Chalikias M., Gregoriades69 A. MAXIMISING ACCURACY AND EFFICIENCY OF. . . [Step 5] The product (as per the rule of multiplication of IVIFSMs) (non-normalized) ) ( ) ( ( of IVIFSMs (0.2, 0.5), (0.5,below. 0.6), (0.3, 0.5), are given and combined choice matrices (0.3, 0.4) ) ( (0.3, 0.4) ) ( (0.3, 0.4) ( (0.3, 0.5), (0.4, 0.7), (0.4, 0.5), (0.3, (0.3.0.7), (0.3,0.4), (0.5,0.6), (0.5,0.6), (0.0,0.0), (0.2,0.3) (0.2, 0.3) 0.5) ) ((0.0,0.0) ( ) ( (0.1,0.2) (0.4,0.5) (0.2,0.4) (0.2,0.4) (0.3, 0.6), (0.7, 0.8), (0.2, 0.4), {p2 (i, k)}= (0.4,0.5), (0.3,0.7), (0.3,0.6), (0.3,0.7), (0.0,0.0), (0.3, 0.4) (0.1, 0.2) (0.3, 0.5) ( ( ( (0.2,0.3) ) (0.3,0.5) (0.1,0.3) (0.1,0.2) ) (0.0,0.0) (0.2, 0.6), (0.5, 0.7), (0.3, 0.6), (0.3,0.6), (0.4,0.6), (0.4,0.6), (0.4,0.6), (0.0,0.0), s { p (i , j )} (0.1, 0.2) (0.1, 0.2) (0.2, 0.3) (0.3,0.4) ( (0.2,0.3) ) (0.0,0 .0) (0.2,0.4) ( ( (0.2,0.4) ) 0.5), (0.2,0.5), (0.2, 0.3), (0.0,0.0), (0.4,0.7), (0.4,0.8), (0.4, (0.2,0.5), (0.2,0.5), (0.1,0.2) (0.2, (0.3,0.4) (0.3,0.4) (0.3,0.4) (0.0,0.0) 0.4) (0.4, 0.5) (0.1, 0.3) { p1 } 1 {p3 (i, l)} = ( ( ( ( (0.3, 0.5), (0.3, 0.4) (0.3, 0.7), (0.2, 0.3) (0.5, 0.7), (0.1, 0.3) (0.4, 0.6), (0.2, 0.3) (0.1, 0.2), (0.5, 0.7) ) ( ) ( ) ( ) ( ) ( (0.0, 0.0), (0.0, 0.0) (0.4, 0.6), (0.3, 0.4) (0.3,0.5), (0.5,0.6), (0.2,0.5), (0.0,0.0), (0.5,0.7), 2 (0.1,0.2) (0.1,0.2) (0.2,0.3) (0.0,0.0) (0.3,0.4) In this case study, we consider two cases. In the first (0.4,0.5), (0.2,0.3), (0.4,0.7), (0.0,0.0), (0.4,0.5), non-normalized case, we use and normal IVIFSM (0.2,0.4) (0.4,0.5) (0.1,0.3) (0.0,0.0) (0.3,0.4) ized IVIFSM in the second case. (0.5,0.7), (0.5,0.7), (0.0,0.0), (0.0,0.0) (0.4,0.7), (0.4,0.7), (0.0,0.0), [Step 2 & Step 3]: These steps are not applicable (0.2,0.3) (0.2,0.3) (0.0,0.0) in Case1. (0.7,0.8), (0.7,0.8), (0.0,0.0), (0.1,0.2) (0.1,0.2) (0.0,0.0) [Step 4] The combined choice matrices for p1 , p2 , (0.5,0.7), (0.5,0.7), (0.0,0.0), (0.1,0.2) (0.1,0.2) (0.0,0.0) and p3 are respectively (0.4,0.7), (0.4,0.7), (0.0,0.0), (0.1,0.3) (0.1,0.3) (0.0,0.0) (0.2,0.3) Case 1: It uses non-normalized IVIFSMs as input. (0.2,0.3) (0.2, 0.3), (0.5, 0.6) (1,1), (1,1) (1,1), (1,1) (1,1), { p2 } (1,1) (0,0), (0,0) (1,1), (1,1) (0.3,0.4) Here i (1,2,...,5) used to (0.3,0.4) represent an alter (0.3,0.4) is (0.0,0.0) (0.2,0.3) (0 .3,0.5), (0.0,0.0), (0.4,0.5), native, i.e., a(0.4,0.5), stage of (0.4,0.7), heart disease and the indices (0.3,0.5) (0.2,0.3) (0.2,0.3) (0.0,0.0) (0.3,0.4) j, k, l (1,2,...,5) are used for attributes, i.e., symp (0.3,0.6), (0.7,0.8), (0.2,0.4), (0.0,0.0), (0.4,0.6), i , k )} are the input s { p (These toms. information, given by the (0.3,0.4) (0.1,0.2) (0.3,0.5) (0.0,0.0) (0.2,0.4) experts. (0.2,0.6), (0.5,0.7), (0.3,0.6), (0.0,0.0), (0.5,0.6), { p2 p3 } (0,0), (0,0), (0,0), (0,0), (0,0), (0.0,0.0), (0.0,0.0) (0,0) (0,0) (0,0) (0,0) (0,0) ) ) ( ) ( ) ( (0.5.0.7), (0.5.0.7), (0.0,0.0), (0.0,0.0), (0.5.0.7), (0.0, (0.2, 0.7), 0.0), (0.3, 0.6), (0.2, 0.4), (0.1,0.2) (0.1,0.2) (0.0,0.0) (0.0,0.0) (0.1,0.2) (0.3, 0.4) ) ( (0.5, 0.6) ) (0.0, 0.0) (0.1, 0.3) ) (0.4,0.7), (0.4,0.7), 0), ( (0.0,0.0), (0.4,0.7), ( (0.0,0.) 0.5), 0.0), (0.4, 0.8), (0.7, 0.8), (0.3, (0.0, (0.1,0.2) (0.1,0.2) (0.0,0.0) (0.0,0.0) (0.1,0.2) 0.5) (0.3, (0.1, 0.2) (0.4,0.6),) (0.4,0.6), 0.0) (0.0,0.0), (0.4,0.6), ( (0.0, ) ( (0.0,0.0), ) ( (0.1, 0.2) ) (0.5, 0.6), (0.2,0.3) (0.0, (0.5, 0.6), (0.2,0.3) 0.0), (0.0,0.0) (0.0,0.0) (0.2,0.3) (0.4, 0.6), (0.4,0.8), 0.0) (0.0,0.0), (0.0,0.0), (0.4,0.8), (0.3, 0.4) (0.4,0.8), (0.3, 0.4) ) (0.0, (0.3, 0.4) ( ) ( ) ( ) (0.1,0.2) (0.1,0.2) (0.0,0.0) (0.0,0.0) (0.1,0.2) 0.0), (0.3, 0.7), (0.3, (0.4, (0.0, 0.4), 0.5), (0.3,0.5), (0.3,0.5), (0.0,0.0), (0.0,0.0), (0.3,0.5), (0.3, 0.5) ) 0.4) 0.0) ) ( (0.4, ) ( (0.2, 0.3) ) ( (0.0, (0.2,0.4) (0.2,0.4) (0.0,0. 0) (0.0,0.0) (0.2,0.4) (0.3,0.5), (0.2,0.3), (0.2,0.3) , (0.3,0.4), (0.2,0.4) (0.4,0.7) (0.5,0.6) (0.4,0.5) ( ) ) ( (0.5, 0.7), (0.0, 0.0), ) ( (0.0, 0.0) ) ( (0.2, 0.3) ) (0.4, 0.5), (0.0, 0.0), s (1,1), (0.0, (0,0), 0.4) (1,1), (0,0), (0.3, (1,1), 0.0) ) ) ( ) ( (1,1) (1,1) (0,0) (0,0) (1,1) (0.0, 0.0), (0.4, 0.6), (1,1), (1,1), (0,0), (0,0), (1,1), 0.0) ) ( (0.2,0.4) ) ( (0.0, (1,1) (1,1) (0,0) (0,0) (1,1)) (0.0, 0.0), (0.5, 0.6), (1,1), (1,1), (0,0), (0,0), (1,1), 0.0) (0.3,0.4) (1,1) (1,1) (0,0) (0,0) (1,1) ) ) ( (0.0, ) ( 0.0), (0,0), 0.5), (1,1), (0,0), (0.4, (1,1), (1,1),(0.0, (1,1) (0.0, (0,0) 0.4) 0.0) (1,1) (0,0) (0.3, (1,1) ) ( (1,1), (1,1) (1,1), (1,1) (1,1), (1,1) (0,0), (0 ,0) (1,1), (1,1) (0.5,0.7), (0.2,0.3) (0 .4,0.7), (0.2,0.3) (0.7,0.8), (0.1,0.2) (0.5,0.7), (0.1,0.2) s{ p1p3 } (0,0), (0,0) (0,0), (0,0) (0,0), (0,0) (0,0), (0,0) (0.1, 0.4), (0.4, 0.5) (1,1), (1,1) (1,1), (1,1) (1,1), (1,1) (0,0), (0,0) (0,0), (0,0) (0,0), (0,0) (0,0), (0,0) (0,0), (0,0) (0,0), (1,1), (0,0), (0,0) (1,1) (0,0) (0.0,0.0), (0.0,0.0) (0.0,0.0), (0.0,0.0) (0.0,0.0), (0.0,0.0) (0.0,0.0), (0.0,0.0) (0.4,0.7), (0.0,0.0), (0.1,0.3) (0.0,0.0) , . 70 6 Tambouratzis T., Souliou D., Chalikias M., Gregoriades A. s{ p1 p 2 } Case Study (0.3,0.5), (0.2,0.7), (1,1), (1,1), (0.0,0.0), (0.3,0.6), (0.2,0.4), (0.3,0.4) (0.1,0.3) (0.0,0.0) (0.3,0.4) (0.5,0.6) (1,1) (1,1) Let D (0.3,0.7), = {d1 , d2,(0.3,0.5), d3 , d4 , d 5 } (0be the set of five (1,1), .0,0.0), (0.3,0.6), (0.4,0.8), (0.2,0.4), (0.7,0.8), (1,1), (1,1), (1,1), (0.3,0.5), (0.2,0.7), (0.0,0.0), (1,1) (1,1) stages of heart disease (Stage Stage ‘II’, Stage (0.2,0.3) (0.3,0.5) ‘I’, (0.1,0.2) (0.1,0.2) (0.0,0.0) (1,1) (1,1) (0.3,0.4) (0.1,0.3) (0.0,0.0) (0.3,0.4) (0.5,0.6) ‘III’, Stage ‘IV’, and Stage ‘V’). Patients belong (0.5,0.7), (0,0), (0,0), (0.5,0.6), (0.0,0.0), (0.5,0.6), (0.4,0.6), { p 3 (i , l )} (0.3,0.7), (0.3,0.5), (0 .0,0.0), (0.4,0.8), (0.7,0.8), s{ p 3 } (1,1), (1,1), (0.0,0.0) (0.3,0.4)by (0.3,0.4) (0.1,0.3) (1,1) (0.3,0.4) assumed ing to Stage ‘I’ are to be affected (1,1) (0.3,0.5) not (0.2,0.3) (0,0) (0,0) (0.0,0.0) (0.1,0.2) (0.1,0.2) (0.4,0.6), (0.4,0.5), (0.0,0.0), (0.3,0.7), (0.3,0.4), (1,1), (1,1), (0,0), (0,0), (0.5,0.7), (0.5,0.6), (0.0,0.0), (0.5,0.6), (0.4,0.6), heart disease. Patients belonging to Stage ‘II’ are { p 3 (i , l )} (0.2,0.3) (0.3,0.4) (0.0,0.0) (0.2,0.3) (0.4,0.5) s{ p 3 } (1,1) (1 ,1) (0.1,0.3) (0.3,0.4) (0.0,0.0) (0.3,0.4) (0.3,0.4) (0,0) (0,0) in initial stage, patients belonging to Stage ‘III’ are (0.1,0.2), (0.4,0.6), (0.0,0.0), (0.2,0.3), (0.1,0.4), (1,1), (1,1), (1,1), (0.4,0.6), (1,1), (0.4,0.5), (0.0,0.0), (0.3,0.7), (0.3,0.4), in more unsafe stage than in stage ‘II’ and so on. (0.0,0.0) (0.5,0.6) (0.4,0.5) (1,1) (0.2,0.3) (1,1) (0.5,0.7) (0.3,0.4) (0.0,0.0) (0.2,0.3) (0.4,0.5) (1,1) (1 ,1) (0.3,0.4) Patients belonging to Stage ‘V’ are in the last stage (0.1,0.2), (0.4,0.6), (0.0,0.0), (0.3,0 (1,1), .7), (0.3,0.7), (0.0,0.0), (0.2,0.3), (0.1,0.4), (0.3,0.7), (1,1), (0.3,0.4) (0.0,0.0) (0.1,0.3) of heart disease is unrecoverable. Let E be (0.5,0.7)which (1,1) (0.4,0.5) (0.1,0.3) (0.5,0.6) (0.1,0.3) (1,1) (0.0,0.0) (0.7,0.8), (0.7,0.8), (0.7,0.8), (0.0,0.0), the set of five symptoms (Chest pain, Palpitations, (0.3,0 .7), (0.3,0.7), (0.3,0.7), (0.0,0.0), (0.0,0.0) (0.1,0.2) (0.1,0.3) (0.1,0.2) (0.1,0.3) (0.1,0.2) (0.0,0.0) Dizziness, Fainting, Fatigue) given by (0.1,0.3) (0.5,0.7), (0.5,0.7), (0.5,0.7), (0.0,0.0), (0.7,0.8), (0.7,0.8), (0.7,0.8), (0.0,0.0), (0.0,0.0) E = {s1 , s2 , s3 , s4 , s5 } . (0.1,0.3) (0.1,0.2) (0.1,0.3) (0.1,0.2) (0.1,0.3) (0.1,0.2) (0.0,0.0) (0.4,0.7), (0.4,0.7), (0.4,0.7), (0.0,0.0), (0.5,0.7), (0.5,0.7), (0.5,0.7), (0.0,0.0), (0.2,0.3) (0.1,0.3) (0.2,0.3) (0.0,0.0) (0.1,0.3) (0.1,0.3) Suppose that a group of three experts (0.2,0.3) (0.0,0.0) (0.4,0.6), (0.4,0.6), (0.4,0.6), (0.0,0.0), (0.4,0.7), (0.4,0.7), (0.4,0.7), (0.0,0.0), (0.3,0.4) (0.2,0.3) (0.3,0.4) OF. (0.2,0.3) (0.3,0.4) (0.2,0.3) (0.0,0.0) (0.0,0.0) MAXIMISING ACCURACY AND EFFICIENCY .. , p , p } P = {p 1 2 3 (0.4,0.6), (0.4,0.6), (0.4,0.6), (0.0,0.0), (0.3,0.4) (0.3,0.4) (0.0,0.0) [Step The sum of the IVIFSMs are6]monitoring theproduct symptoms of aispatient as per (0.3,0.4) (1,1), (0,0), s{ p1 p 2 } (0,0) (1,1) (1,1), (0,0), (0,0), (1,1), (1,1) (0,0) (1,1) (0,0) (0,0), (1,1), (0,0), (0,0), (0,0) (0,0) (1,1) (0,0) (1,1), (0,0), (0,0), (0,0), (1,1) (0,0) (0,0) (0,0) (1,1), (1,1), (0,0), (0,0), (1,1) (0,0) (1,1) (0,0) (0.0,0.0), (1,1), (0,0), (1,1) (0,0) (0.0,0.0) (0.0,0.0), (0.0,0.0), (0.0,0.0) (0.0,0.0) (0.0,0.0), (0.0,0.0), (0.0,0.0) (0.0,0.0) (0.0,0.0), (0.0,0.0), (0.0,0.0) (0.0,0.0) (0.0,0.0), (0.0,0.0), (0.0,0.0) (0.0,0.0) (0.0,0.0), (0.0,0.0) (0,0), (0,0) (0,0), (0,0), (0,0) (0,0) (0,0), (0,0), (0,0) (0,0) (0,0), (0,0), (0,0) (0,0) (0,0), (0,0), (0,0) (0,0) (0,0), (0,0) their knowledgebase to product reach aIVIFSMs consensus [Step 6] The sum of the is about (0 .5 .0 .7 ), (0 .5 ,0 .7 ), (0 .5 ,0 .7 ), (0patient, which stage is more likely to appear for the (0 .1 ,0 .2 ) (0 .1 ,0 .2 ) (0 .0 ,0 .0 ) (0 .0 ,0 .0 ) .1 ,0 .2 ) (0 .2 ,0 .3 ) (0 .2 ,0 .3 ) (0 .4 ,0 .7 ), (0 .4 ,0 .7 ), (0 .0 ,0 .0 ), (0 .0 ,0 .0 ), (0 .4 ,0 .7 ), (0 .4 ,0 .7 ), (0 .4 ,0 .7 ), expert p is aware of symptoms (s , s .0,.7s),3, (0 .5 ,0 .7 ), (0 .5 ,0 .7 ), (0 .5 .0 .7 ), where (0 .5 .01.7 ), (0 .0 ,0 .0 ), (0 .0 ,0 .0 ), 1(0 .5 2 (0 .1 ,0 .2 ) (0 .1 ,0 .2 ) (0 .0 ,0 .0 ) (0 .0 ,0 .0 ) (0 .1 ,0 .2 ) (0 .2 ,0 .3 ) (0 .2 ,0 .3 ) .1 ,0p.2 ) symptoms .1 ,0 .2 p ) 3 (0 .2 ,0 .3 ) (0 .2 ,0 .3 ) s2,0,.0s3) ,s5 ), aware (0 .1 ,0 .2of (0 .0 ,0 .0 ) (s (01.0 (0and 4 ), 2 ) is s(0 (0 .4 ,0 .6 ), (0 .4 ,0 .6 ), (0 .0 ,0 .0 ), (0 .0 ,0 .0 ), (0 .4 ,0 .6 ), (0 .7 ,0 .8 ), (0 .7 ,0 .8 ), (0 .4 ,0 .7 ), of .4 ,0 .7 ), (0 .0 ,0 .0 ), , s (0 .0 ,0 .0 ), (0symptoms (0 .4 ,0 .7 ),to (0 .4 ,0 .7 ), (0 .4 ,0 .7 ), is aware (s , s , s ). According (0 .2 ,0 .3 ) (0 .2 ,0 .3 ) (0 .0 ,01.0 ) 2 4(0 .05,0 .0 ) (0 .2 ,0 .3 ) (0 .1 ,0 .2 ) (0 .1 ,0 .2 ) (0 .1 ,0 .2 ) (0 .1 ,0 .2 ) (0 .0 ,0 .0 ) (0 .0 ,0 .0 ) (0 .1 ,0 .2 ) (0 .2 ,0 .3 ) (0 .2 ,0 .3 ) three (0 .4symptoms ,0 .8 ), (0 .4 ,0or .8 ),parameters (0 .4 ,0 .8 ), (0 .5 ,0 .7 ), (0 .5 ,0 .7 ), (0 .0 ,0 .0 ), observed (0 .0 ,0 .0 ), by the the .4 ,0 .6 ), (0 .4 ,0 .6 ), (0 .0 ,0 .0 ), (0 .0 ,0 .0 ), (0 .4 ,0 .6 ), (0 .7 ,0 .8 ), (0 .7 ,0 .8 ), (0 (0 .1 ,0 .2 ) we (0 .1 ,0 .2 ) to (0 .0 ,0 .0 ) the (0 .0 ,0 .0 ) (0 .1in ,0 .2 ) (0 .1 ,0 .2 ) (0 .1 ,0 .2 ) have experts, (0 .2 ,0 .3 ) (0assume .2 ,0 .3 ) (0 .0 ,0 .0 ) (0information .0 ,0 .0 ) (0 .2 ,0 .3IV) (0 .1 ,0 .2 ) (0 .1 ,0 .2 ) (0 .3 ,0 .5 ), (0 .3 ,0 .5 ), (0 .0 ,0 .0 ), ( 0 .0 ,0 .0 ), (0 .3 ,0 .5 ), (0 . 4 ,0 .7 ), (0 .4 ,0 .7 ), (0 .4 ,0 .8 ), (0 .4 ,0 .8 ), (0 .0 ,0 .0 ), (0 .0 ,0 .0 ), (0 .4 ,0 .8 ), (0 .5 ,0 .7 ), (0 .5 ,0 .7 ), IFSSs .2 ,0 .4 ) (0 .2 ,0 .4 ) (0 .0 ,0 .0 ) (0 .0 ,0 .0 ) (0 .2 ,0 .4 ) (0 .1 ,0 .3 ) (0 .1 ,0 .3 ) (0 .0 ,0 .0� ) (0 .0 ,0 .0 ) (0 .1 ,0 .2 ) (0 .1 ,0 .2 ) (0 .1 ,0 .2 ) (0 .1 ,0 .2 ) (0 .1 ,0(.2 F�) 1 },(0 E), , E), (0 .3 ,0(.7F),{p 2 } (0 .0 ,0 .0 ), (0 .0 ,0 .0 ), (0 .5 ,0 .7 ), (0 .5 ,0 .7 ), (0 .3 ,0 .7 ), (0 .3 ,0 .7 ),{p (0 .5 .0 .7 ), (0 .5 .0 .7 ), (0 .0 ,0 .0 ), (0 .0 ,0 .0 ), [Step 6] The sum of the product IVIFSMs is (0 .3 ,0 .5 ), .1 ,0 .3 ) (0 (0 .2 ,0 .4 ) (0 .7 ,0 .8 ), (0 .3 ,0 .7 ), (0 .1 ,0 .2 ) (0 .1 ,0 .3 ) (0 .5 ,0 .7 ), (0 .7 ,0 .8 ), (0 .1 ,0 .3 ) (0 .1 ,0 .2 ) (0 .4 ,0 .7 ), (0 .5 ,0 .7 ), (0 ,0 .3 ) (0 .2 .1 ,0 .3 ) (0 .4 ,0 .6 ), (0 .4 ,0 .7 ), .3 ,0 .4 ) (0 (0 .2 ,0 .3 ) (0 .4 ,0 .6 ), (0 .3 ,0 .4 ) and (0 .3 ,0 .5 ), (0 .1 ,0 .3 ) (0 .2 ,0 .4 ) (0 .7 ,0 .8 ), .3 ,0 .7 ), (0 (0 .1 ,0 .2 ) (0 .1 ,0 .3 ) (0 .5 ,0 .7 ), (0 .7 ,0 .8 ), (0 1 .1 ,0 .32) (0 .1 ,0 .2 ) (0 .4 ,0 .7 ), (0 .5 ,0 .7 ), .2 ,0 .3 ) (0 (0 .1 ,0 .3 ) (0 .4 ,0 .6 ), (0 .4 ,0 .7 ), .3 ,0 {p 1 }.4 ) (0 (0 .2 ,0 .3 ) (0 .0 ,0 .0 ), (0 .1 ,0 .3 ) (0 .0 ,0 .0 ) (0 .7 ,0 .8 ), (0 .3 ,0 .7 ), (0 .1 ,0 (0{p 3 }.2 ) .1 ,0 .3 ) (0 .5 ,0 .7 ), (0 .7 ,0 .8 ), (0 .13,0 .3 ) (0 .1 ,0 .2 ) (0 .4 ,0 .7 ), (0 .5 ,0 .7 ), (0 .2 ,0 .3 ) (0 .1 ,0 .3 ) (0 .4 ,0 .6 ), (0 .4 ,0 .7 ), .3 ,0 {p 2 }.4 ) (0 (0 .2 ,0 .3 ) (0 .4 ,0 .6 ), (0 .4 ,0 .6 ), (0 .0 ,0 .0 ), (0 .3 ,0 .4 ) (0 .0 ,0 .0 ) (F� , E) ( 0 .0 ,0 .0 ), (0 .0 ,0 .0 ) (0 .0 ,0 .0 ) (0 .0 ,0 .0 ), (0 .0 ,0 .0 ), (0 ,0 .0 ) (0 .0 .0 ,0 .0 ) (0 .0 ,0 .0 ), (0 .0 ,0 .0 ), (0 .0 ,0 .0 ) (0 .0 ,0 .0 ) (0 .0 ,0 .0 ), (0 .0 ,0 .0 ), (0 ,0 .0 ) (0.0 .0 ,0 .0 ) (0 .0 ,0 .0 ), (0 .0 ,0 .0 ), .0 {p,03 .0 } ) (0 (0 .0 ,0 .0 ) for experts p , p , and p respectively. Let the IVIFSMs of the IVIFSSs (F� , E), (F� are respectively, (0 .3 ,0 .4 ) ( , E), (F� , E) (0 .0 ,0 .0 ), (0 .5 ,0 .7 ), (0 .0 ,0 .0 ) , (0 .0 ,0 .0 ) (0 .2 ,0 .3 ) (0 .0 ,0 .0 ) (0 .0 ,0 .0 ), (0 .4 ,0 .7 ), (0 .0 ,0 .0 ), (0 .0 ,0 .0 ), (0 .5 ,0 .7 ), (0 .0 ,0 .0 ) , (0 .0 ,0 .0 ) (0 .0 ,0 .0 ) (0 .2 ,0 .3 ) (0 (0 .0 ,0 .0 ) (0 .2 ,0 .3 ) .0 ,0 .0 ) (0 .0 , 0 .0 ), (0 .7 ,0 .8 ), (0 .0 ,0 .0 ), (0 .0 ,0 .0 ), (0 .4 ,0 .7 ), (0 .0 ,0 .0 ), (0 .0 ,0 .0 ) (0 .1 ,0 .2 ) (0 .0 ,0 .0 ) (0 .0 ,0 .0 ) (0 .2 ,0 .3 ) (0 .0 ,0 .0 ) (0 .0 ,0 .0 ), (0 .5 ,0 .7 ), (0 .0 ,0 .0 ), .0 , 0 .0 ), (0 .7 ,0 .8 ), (0 .0 ,0 .0 ), (0 .0 ,0 .0 ) (0 .1 ,0 .2 ) (0 .0 ,0 .0 ) (0 (0 .0 ,0 .0 ) (0 .1 ,0 .2 ) (0 .0 ,0 .0 ) (0 .0 ,0 .0 ), (0 .4 ,0 .7 ), (0 .0 ,0 .0 ), (0 .0 ,0 .0 ), (0 .5 ,0 .7 ), (0 .0 ,0 .0 ), (0 .0 ,0 .0 ) (0 .1 ,0 .3 ) (0 .0 ,0 .0 ) (0 .0 ,0 .0 ) (0 .1 ,0 .2 ) (0 .0 ,0 .0 ) (0 .3 ,0 .7 ), (0 .5 ,0 .7 ), (0 .5 ,0 .7 ), (0 .3 ,0 .5 ), (0 . 4 ,0 .7 ), (0 .4 ,0 .7 ), (0 .0 ,0 .0 ), (0 .4 ,0 .7 ), (0 .0 ,0 .0 ), (0 .2 ,0 .3 ) (0 .1 ,0 .2 ) (0.0.2,0,0.0.4)) (0(0.1.1,0,0.2.3) ) (0(0.1.1,0,0.2.3) ) (0(0.1.0,0,0.3.0) ) (0 (0 .1 ,0 .3 ) (0 .0 ,0 .0 ) (0 .0 ,0 .0 ), (0 .7 ,0 .8 ), (0 .7 ,0 .8 ), (0 .7 ,0 .8 ), (0 .4 ,0 .7 ), (0 .4 ,0 .7 ), (0 .0 ,0 .0 ), (0 .5 ,0 .7 ), (0 .5 ,0 .7 ), (0 .3 ,0 .7 ), (0 .5 ,0 .7 ), (0 .5 ,0 .7 ), (0 .0 ,0 .0 ) (0 .1 ,0 .2 ) (0 .1 , 0 .2 ) (0 .1 ,0 .2 ) (0 .2 ,0 .3 ) (0 .1 ,0 .2 ) (0 .0 ,0 .0 ) (0 .1 ,0 .2 ) (0 .1 ,0 .2 ) (0 .1 ,0 .3 ) (0 .2 ,0 .3 ) (0 .1 ,0 .2 ) (0 .0 ,0 .0 ), (0 .7 ,0 .8 ), (0 .7 ,0 .8 ), (0 .5 ,0 .7 ), (0 .7 ,0 .8 ), (0 .4 ,0 .6 ), (0 .0 ,0 .0 ), (0 .7 ,0 .8 ), (0 .7 ,0 .8 ), (0 .7 ,0 .8 ), (0 .4 ,0 .7 ), (0 .4 ,0 .7 ), . (0 ,0 .0 ) (0 .1 ,0 .2 ) (0 .1 ,0 .2 ) (0 .1 ,0 .3 ) (0 .1 ,0 .2 ) ( 0 .2 ,0 .3 ) (0 .0 .0 ,0 .0 ) (0 .1 ,0 .2 ) (0 .1 , 0 .2 ) (0 .1 ,0 .2 ) (0 .2 ,0 .3 ) (0 .1 ,0 .2 ) (0 .0 ,0 .0 ), (0 .5 ,0 .8 ), (0 .5 ,0 .8 ), (0 .4 ,0 .7 ), (0 .5 ,0 .7 ), (0 .4 ,0 .8 ), (0 .0 ,0 .0 ), (0 .7 ,0 .8 ), (0 .7 ,0 .8 ), (0 .5 ,0 .7 ), (0 .7 ,0 .8 ), (0 .4 ,0 .6 ), (0 ,0 .0 ) (0 .1 ,0 .2 ) (0 .1 ,0 .2 ) (0 .2 ,0 .3 ) (0 .1 ,0 .2 ) (0 .1 ,0 .2 ) . (0.0 .0 ,0 .0 ) (0 .1 ,0 .2 ) (0 .1 ,0 .2 ) (0 .1 ,0 .3 ) (0 .1 ,0 .2 ) ( 0 .2 ,0 .3 ) (0 .0 ,0 .0 ), (0 .4 ,0 .7 ), (0 .4 ,0 .7 ), (0 .4 ,0 . 6 ), (0 .4 ,0 .7 ), (0 .3 ,0 .5 ), (0 .0 ,0 .0 ), (0 .5 ,0 .8 ), (0 .5 ,0 .8 ), (0 .4 ,0 .7 ), (0 .5 ,0 .7 ), (0 .4 ,0 .8 ), .0 ,0 .0 ) (0 .1 ,0 .3 ) (0 .1 ,0 .3 ) (0 .3 ,0 .4 ) (0 .1 ,0 .3 ) (0 .2 ,0 .4 ) (0 (0 .0 ,0 .0 ) (0 .1 ,0 .2 ) (0 .1 ,0 .2 ) (0 .2 ,0 .3 ) (0 .1 ,0 .2 ) (0 .1 ,0 .2 ) (0 .0 ,0 .0 ), (0 .4 ,0 .7 ), (0 .4 ,0 .7 ), (0 .4 ,0 . 6 ), (0 .4 ,0 .7 ), (0 .3 ,0 .5 ), .3 ) (0 .1 ,0 .3 ) Souliou .4 ) Chalikias ) Gregoriades (0 .3 ,0D., (0 .1 ,0 .3M., (0 .2 ,0 .4 ) A. Tambouratzis T., (0 .0 ,0 .0 ) (0 .1 ,0 ) ( ) ( ) ( ) ( ) D., Chalikias M., Gregori Souliou (0.3.0.7), (0.3, 0.4), (0.5, 0.6), (0.5, 0.6), Tambouratzis (0.0,T.,0.0), 0.2) ) ( (0.4, 0.5) ) ( (0.2, 0.4) ) ( (0.2, 0.4) ) ( (0.0, 0.0) ) ( (0.1, [Step 7] The of various stages of heart dis weights (0.4, 0.5), (0.3, 0.7), (0.3, 0.6), (0.3, 0.7), (0.0, 0.0), ease, W (di ), i= 1, 2, ...,[Step 5 are 7] calculated as follows: The weights of various stages of heart dis0.5) ) ( (0.2, 0.3) ) ( (0.1, 0.3) ) ( (0.1, 0.2) ) ( (0.0, 0.0) ) ( (0.3,ease, W0.5 (di ),+i 0.5, = 1, 2, ..., 5 are calculated as follows: + 0.5 [0.5 +0.6), 0.3 + (0.3, (0.4, 0.6), (0.4, 0.6), (0.4, 0.6), (0.0, 0.0), [ ] , {p1 (i, j)}=0.7 [2.3, 3.5], + 0.7 + 0.7 + 0.7 + 0.7], [0.5 + 0.5 + 0.3 + 0.5 + 0.5, (0.3, 0.4) ) ( (0.2, 0.3) (0.2, 0.4) [ (0.2, 0.4) ]) ( (0.0, 0.0) ) W (d1 ) = = . ) ( ) ( ( [0.1 + 0.1 [0.6, 1.2] +0.8), 0.1 + 0.2 0.1, + 0.7 + 0.7 + 0.7 + 0.7], (0.0, 0.0), (0.2,3.5], 0.5), . (0.2, 0.5), = [2.3, (0.2, 0.5), (0.4, +0.7 W (d ) = 1 +[0.1 + 0.2 0.2 + 0.3 0.2] [0.6, 1.2] + 0.4) 0.1 + 0.1 + 0.2 + 0.1, (0.0, 0.0) (0.3, 0.4) (0.3, 0.4) (0.3, 0.2)+ 0.3 ( (0.1, ) ) ( ) ( ) ( ) ( 0.2 + 0.2 + 0.3 + 0.3 + 0.2] (0.0, 0.0), (0.3, 0.4), (0.2, 0.3), (0.2, 0.3), (0.3, 0.5), Similarly, (0.0, 0.0) (0.4, 0.5) (0.5, 0.6) (0.4, 0.7) (0.2, 0.4) Similarly, [ [ ] ] [2.9, 4.5], [3.0, 3.7], [ ] , ] , W (d3 )[= W (d2 ) = [0.6, 1.1] [0.6, 1.2] [2.9, 4.5], [3.0, 3.7], [ ] (d2 ) = [ ] (d3 ) = W ,W , [2.3, 3.8], [1.9, [0.6, 1.1]3.2], ] . [ [0.6, 1.2] ] W (d4 ) = , W (d5 )[= [0.6, 1.1] [0.8, 1.7] [2.3, 3.8], [1.9, 3.2], W (d4 ) = , W (d5 ) = . [0.6, 1.1] [0.8, 1.7] [Step 8] Scores of various stages of heart disease [Step 7] The[[weights of various stages[[of heart dis-]] ]] [ ] [ ] [2.9, [2.9, 4.5], 4.5], [3.0, [3.0, 3.7], 3.7], [2.9, 4.5], [3.0, 3.7], ease, W (d ), i = 1, 2, ..., 5 are calculated as follows: i = = ,, W W(d (d33))= = ,, W (d2 ) = , W (d3 ) = , W W(d (d22)) [0.6,1.1] 1.1] ]] [0.6,1.2] 1.2] ]] [[ [0.6, [[ [0.6, [ [0.6, 1.1] ] [ [0.6, 1.2] ] [0.5 + 0.5 + 0.3 + 0.5 + 0.5, [3.2], ] [2.3, [2.3,3.8], 3.8], [1.9, [1.9,3.2], [2.3, 3.8], [1.9, 3.2], [2.3, .3.5], 0.7 W W(d (d44)) = =0.7 + 0.7 + 0.7,+ ,W W (d (d5+ = = . W (d4 ) = , W (d5 ) = . 5))0.7], W (d1 ) = [0.6, = . [0.6,1.1] 1.1] [0.8, [0.8, 1.7] 1.7] [0.6, 1.1] [0.8, 1.7] [0.6, 1.2] [0.1 + 0.1 + 0.1 + 0.2 + 0.1, MAXIMISING ACCURACY AND EFFICIENCY OF. . . 71 0.2 + 0.2 + 0.3 + 0.3 + 0.2] [Step [Step 8] 8] Scores Scores of of various various stages stages of of heart heart disease disease [Step 8] Scores of various stages of heart disease are arecomputed computedand andgiven givenbelow. below. are computed and given below. Similarly, [Step 6] The sum of the product IVIFSMs is [ ] [ ] S(d S(d11))= S(d1 ) = {(2.3 − 0.6) + (3.5 − 1.2)}/2 = 2 ={(2.3 {(2.34.5], − −0.6) 0.6)+ +(3.5 (3.5− −1.2)}/2 1.2)}/2 = 22 [2.9, [3.0,= 3.7], W (d ) = , W (d ) = , 2 3 S(d S(d22))= S(d2 ) = {(2.9 − 0.6) + (4.5 − 1.1)}/2 = 2.85 ={(2.9 {(2.9− −0.6) 0.6)+ +(4.5 (4.5− −1.1)}/2 1.1)}/2 = =2.85 2.85 [ [0.6, 1.1] ] [ [0.6, 1.2] ] S(d S(d33))= S(d3 ) = {(3.0 − 0.6) + (3.7 − 1.2)}/2 = 2.45 ={(3.0 {(3.0 − − 0.6) 0.6) + + (3.7 (3.7 − − 1.2)}/2 1.2)}/2 = = 2.45 2.45 [2.3, 3.8], [1.9, 3.2], W (d ) = , W (d ) = . 4 5 S(d S(d44))= S(d4 ) = {(2.3 − 0.6) + (3.8 − 1.1)}/2 = 2.2 ={(2.3 {(2.31.1] − −0.6) 0.6)+ +(3.8 (3.8− −1.1)}/2 1.1)}/2 = 2.2 2.2 [0.6, [0.8,= 1.7] S(d S(d55))= S(d5 ) = {(1.9 − 0.8) + (3.2 − 1.7)}/2 = 1.3 ={(1.9 {(1.9− −0.8) 0.8)+ +(3.2 (3.2− −1.7)}/2 1.7)}/2= =1.3 1.3 [Step 8] Scores of various stages of heart disease maximum,the thepatient patient [Step 9] Since score of d2 is maximum, the patient [Step [Step9] 9]Since Sincescore scoreof ofdd22 isismaximum, are computed and given below. under underconsideration considerationbelongs belongsto toStage Stage‘II’, ‘II’,i.e., i.e.,initial initial under consideration belongs to Stage ‘II’, i.e., initial stage stage heart disease as(3.5 per perthe the1.2)}/2 collective collective opinions stage of heart disease as per the collective opinions S(d1of )of=heart {(2.3disease − 0.6)as + − = 2opinions of of the the group group of of experts. experts. of the group of experts. S(d2 ) = {(2.9 − 0.6) + (4.5 − 1.1)}/2 = 2.85 S(d32: )2:=ItIt{(3.0 0.6) + (3.7 − 1.2)}/2 = 2.45 Case Case uses uses− normalized normalized IVIFSMs. IVIFSMs. Case 2: It uses normalized IVIFSMs. S(d4 ) = {(2.3 − 0.6) + (3.8 − 1.1)}/2 = 2.2 S(d5 ) = {(1.9 − 0.8) + (3.2 − 1.7)}/2 = 1.3 [Step [Step2]: 2]: Cardinal Cardinalmatrix matrixof ofeach eachIVIFSM IVIFSMand andthe the [Step 2]: Cardinal matrix of each IVIFSM and the is maximum, the patient [Step 9] Since score of dscore corresponding corresponding cardinal cardinal are aregiven givenbelow. below. corresponding cardinal score are given below. 2score under consideration belongs to Stage ‘II’, i.e., initial Cardinal Cardinal matrix matrix [p [p11(i, (i,j)] j)]1×5 for for IFSM IFSM [p [p11(i, (i,j)] j)] isis Cardinal matrix [p1 (i, j)]1×5 for IFSM [p1 (i, j)] is stage of heart disease as per1×5 the collective opinions (as (asshown shownin inExample Example8) 8) (as shown in Example 8) of the group of experts. [( [( )) (( )) (( )) [( (( ) ))((( )] )] ) ( (0.34, (0.34,0.62) 0.62) (0.28, (0.28,0.5) 0.5) (0.32, (0.32,0.52) 0.52) (0.34, (0.3, (0.3,0.56) 0.56) 0.62) (0.28, (0, (0,0) 0)0.5) (0.32, 0.5 Case 2: It uses normalized IVIFSMs. [p11(i, [p1 (i, j)]1×5 = [p (i,j)] j)]1×5 = 1×5= (0.2, (0.2,0.34) 0.34) (0.3, (0.3,0.44) 0.44) (0.26, (0.26,0.42) 0.42) (0.2, (0.24, (0.24, 0.34) 0.38) 0.38) (0.3, (0, (0,0) 0) 0.44) (0.26, 0.4 [Step 2]: Cardinal matrix of eachscore IVIFSM andthe thecorresponding corresponding cardinal score isS(cand and cardinal isS(c F�F�{p = and the corresponding cardinal score isS(cF�{p1 } ) = {p11}the }))= corresponding cardinal score are given below. 0.43. 0.43. 0.43. Cardinal matrix [p1 (i, j)] [p (i,j)] j)] and is Similarly, Similarly, cardinal cardinal matrices matrices for for [p [p j)] and Similarly, cardinal matrices for [p2 (i, j)] and 1(i, 22(i, 1×5 for IFSM (as in Example 8) [p [pshown (i, (i, j)]and j)]and the the corresponding corresponding cardinal cardinal scores scores are are [p3 (i, j)]and the corresponding cardinal scores are 33 respectively, respectively, [( )( ) ( respectively,) ( ) ( )] (0.34, 0.62) (0.28, 0.5) (0.32, 0.52) (0.3, 0.56) (0, 0) )] )) (() ( )) (( )) (( )) (( [( )] ) ( [( [p1 (i, j)]1×5 = [( (0.2, 0.34) (0.3, 0.44) 0.42) (0, 0) 0.54) (0, 0) (0.28, 0.54) (0.28, 0. (0.46, 0.62) (0.46, 0.62) (0.32, 0.54) (0.32, 0.54) (0.44, (0.44, 0.54) (0.28, 0.54) (0.24, (0,0.38) 0) (0.32, 0.54) (0.46, 0.62) (0.26, [p [p [p22(i, (i,j)] j)]1×5 = = (i, j)] = 2 1×5 0.38) (0, (0.22, 0.36) (0.22, 0. (0.22, (0.22, 0.32) (0.24, (0.24, 0.38) (0.26, (0.26, 0.38) (0.22,1×5 0.36) (0,0) 0) (0.24,0.38) 0.38) (0.22,0.32) 0.32) �{p } ) = and the corresponding cardinal score isS(c F 1 [( [( )) (( )) (( )) (( [( )) (() ( )] )] ) ( ) 0.43. (0.32, (0.32,0.54) 0.54) (0.36, (0.36,0.58) 0.58) (0, (0,0) 0) (0.34, (0.34, (0.32, 0.6) 0.6)0.54) (0.34, (0.34, (0.36, 0.52) 0.52) 0.58) (0, 0) [p [p3 (i, j)]1×5 = [p33(i, (i,j)] j)]1×5 = 1×5= (0.26, 0.4) 0.4) for [p(0.26, (0.26, 0.4) (0, (0,0) 0) (0.28, (0.28, (0.26, 0.38) 0.38) 0.4) (0.34, (0.34, (0.26, 0.44) 0.44) 0.4) (0, 0) Similarly, cardinal (0.26, matrices (i, j)]0.4) and 2 [pS(c j)]and corresponding cardinal �F�{p 3 (i,F =the 0.68 0.68 and andS(c S(cF�F�{p =0.42. 0.42.scores are S(c S(cF�{p2 } ) = 0.68 and S(cF�{p3 } ) = 0.42. {p22}}))= {p33}}))= respectively, )( ) ( ) ( ) ( )] [( (0, 0) (0.28, 0.54) (0.46, 0.62) (0.32, 0.54) (0.44, 0.54) [p2 (i, j)]1×5 = (0, 0) (0.22, 0.36) (0.22, 0.32) (0.24, 0.38) (0.26, 0.38) [p3 (i, j)]1×5 = [( (0.32, 0.54) (0.26, 0.4) ) ( (0.36, 0.58) (0.26, 0.4) S(cF�{p2 } ) = 0.68 and S(cF�{p3 } ) = 0.42. ) ( (0, 0) (0, 0) ) ( (0.34, 0.6) (0.28, 0.38) )( (0.34, 0.52) (0.34, 0.44) )] MAXIMISING ACCURACY AND EFFICIENCY OF. . . 72 Tambouratzis T., Souliou D., Chalikias M., Gregoriades A. [Step 7] 3]:The Theweights normalized of various IVIFMSs stagesare of given heart disbelow. ease, W (di ), i = 1, 2, ..., 5 are calculated as follows: N p1 ( i+ , j0.3 )] + [0.43 * p0.5, IFSM 1 ( i , j )]5 5 [0.5 +[0.5 0.5 + [ ] 0.7 + (0.22,0.26), 3.5], 0.7 + 0.7 + 0.7 + 0.7], (0.13.0.30), (0.13,0.17), [2.3, W (d1 ) = . 0.2 + 0.1, = [0.6, 1.2] [0.1 0.1 + 0.1 + (0.04,0.09) (0.17,0.22) (0.09,0.17) + 0.2 + 0.2 + 0.3 + 0.3 + 0.2] (0.17,0.22), (0.13,0.30), (0.13,0.26), (0.22,0.26), (0.09,0.17) (0.13,0.30), (0.09,0.13) (0.04,0.13) (0.04,0.09) .22) (0.13,0 Similarly, (0.17,0.26), (0.17,0.26), [ (0.13,0.29), ] (0.17,0.26), [ ] (0.09,0.13) ( 0.09,0.17) (0.09,0.17) [2.9, 4.5], [3.0, 3.7], (0.13,0.17) , , W(d3 ) = W (d2 ) = 1.1] [0.6, 1.2] (0.09,0.22), [ [0.6, ] [ ] (0.17,0.34), (0.09,0.22), (0.09,0.22), [2.3, 3.8], [1.9, 3.2], (0.13,0.17) (0.13,0.17) (0.13,0.17) (0.04,0.09) W (d4 ) = , W (d ) = . 5 1.1] [0.6, [0.8, 1.7] (0.13,0.22), (0.09,0.13), (0.09,0.13), (0.13,0.17), (0.17,0.30) (0.22,0.26) (0.17,0.22) of heart disease (0.09,0.17) [Step 8] Scores of various stages (0.0,0.0), (0.0,0.0) (0.0,0.0), (0.0,0.0) (0.0,0.0), (0.0,0.0) , .0), (0.0,0 (0.0,0.0) (0.0,0.0), (0.0,0.0) are computed and given below. N IVIFSM [ p 2 (i , j )] [0.68 * p 2 (i , j )]55 S(d1 ) = {(2.3 − 0.6) + (3.5 − 1.2)}/2 = 2 (0.20,0.34), (0.34,0.41), (0.14,0.34), − 0.6) + (4.5 S(d2 ) = {(2.9 − 1.1)}/2 = 2.85 (0.20,0.27) (0.20,0.27) (0.20,0.27) S(d3 ) = {(3.0 − 0.6) + (3.7 − 1.2)}/2 = 2.45 (0.27,0.34), (0.27,0.48), (0.20,0.34), S(d4 ) = {(2.3 − 0.6) + (3.8 − 1.1)}/2 = 2.2 34) (0.20,0. − (0.14,0.20) S(d5 ) = {(1.9 − 0.8) + (3.2 1.7)}/2 = 1.3 (0.14,0.20) (0.20,0.41), (0.48,0.54), (0.14,0.27), score the patient [Step 9] Since of d2 ismaximum, (0.20,0.27) (0.07,0.14) (0.20 ,0.34) under consideration belongs to Stage ‘II’, i.e., initial (0.14,0.41), (0.34,0.48), (0.20,0.41), stage of heart disease as per the collective opinions (0.07,0.14) (0.07,0.14) (0.14,0.20) of the group of experts. (0.27,0.34), (0.14,0.20), (0.27,0.48), Case 2: It uses normalized IVIFSMs. (0.14,0.27) (0.27,0.34) (0.07,0.20) (0.0,0.0), (0.0,0.0) (0.0,0.0), (0.0,0.0) (0.0,0.0), (0.0,0.0) (0.0,0.0), (0.0,0.0) (0.0,0.0), (0.0,0.0) (0.34,0.48), (0.14,0.20) (0.27,0.34), (0.20,0.27) (0.27,0.41), , (0.14,0.27) (0.34,0.41), (0.20,0.27) (0.27,0.34), (0.20,0.27) N IVIFSM [matrix p3 ( i , j )] [0.42IVIFSM * p3 ( i , j )]and [Step 2]: Cardinal ofeach 5 5 the corresponding cardinal score are given below. (0.13,0.21), (0.08,0.29), (0.0,0.0), (0.13,0.25), (0.08,0.17), (0.0,0.0) (0.13,0.17) (0.21,0.25) Cardinal matrix [p1 (i, j)]1×5 for IFSM [p1 (i, (0.04,0.13) j)] is (0.13,0.17) (0.13,0.29), (as shown in Example 8) (0.13,0.21), (0.0,0.0), (0.17,0.34), (0.29,0.34), ( )( ) ( (0.04,0.08)) ( )] (0.13,0.21) (0.0,0.0) [( (0.08,0. 13) ) (0.04,0.08) (0.34, 0.62) (0.28, 0.5) (0.32, 0.52) (0.3, 0.56) (0, 0) [p1 (i, j)]1×5 = (0.21,0.29), (0.21,0.25), (0.0,0.0), (0.21,0.25), (0.17,0.25), (0.3, 0.44) (0.26, 0.38) . (0, 0) (0.2, 0.34) 0.42) (0.24, (0.04,0.13) (0.13,0.17) (0.0,0. 0) (0.17,0.21), (0.17,0.25), score and the corresponding cardinal isS(cF�{p1 } )(0.0,0.0), = (0.08,0.13) (0.13,0.17) (0.0,0.0) 0.43. (0.04,0.08), (0.17,0.25), (0.0,0.0), Similarly, cardinal matrices for [p2 (i, j)] and (0.13,0.17) [p3 (i, j)]and the corresponding scores are cardinal (0.0,0.0) (0.21,0.29) respectively, (0.13,0.17) (0.13,0.17), (0.17,0.21) (0.08,0.13), (0.04,0.17), (0.21,0.25) (0.17,0.21) (0.13,0.17) (0.13,0.29), (0.08,0.13) )( ( ) ( p1 , p2 , and p) ) ( matrices for experts [(4]: Combined choice [Step 3 are computed as in Case 1. )] (0.44, 0.54) (0, 0) (0.28, 0.54) (0.46, 0.62) (0.32, 0.54) [p2 (i, j)]1×5 [Step = 5]: Now the corresponding product of normalized IVIFSM and combined choice matrices are as follows. (0.26, 0.38) (0, 0) (0.22, 0.36) (0.22, 0.32) (0.24, 0.38) [p3 (i, j)]1×5 = [( (0.32, 0.54) (0.26, 0.4) ) ( (0.36, 0.58) (0.26, 0.4) S(cF�{p2 } ) = 0.68 and S(cF�{p3 } ) = 0.42. ) ( (0, 0) (0, 0) ) ( (0.34, 0.6) (0.28, 0.38) )( (0.34, 0.52) (0.34, 0.44) )] Tambouratzis T., Souliou D., Chalikias M., Gregoriades73 A. MAXIMISING ACCURACY AND EFFICIENCY OF. . . [Step 4]: The Combined choice matrices [Step 3]: normalized IVIFMSs are given for beexpertsp ,p , andp are computed as in Case 1. low. 1 2 3 [Step 5]: Now the corresponding product of normalized IVIFSM and combined choice matrices are as follows. (0.13.0.30), (0.04,0.09) (0.17,0.22), (0.13,0.22) (0.13,0.29), N IVIFSM { p1 (i , j )} (0.13,0.17) (0.17,0.34), (0.04,0.09) (0.13,0.22), (0.09,0.17) (0.13,0.17), (0.17,0.22) (0.13,0.30), (0.09,0.13) (0.22,0.26), (0.09,0.17) (0.13,0.26), (0.04,0.13) (0.17,0.26), (0.09,0.13) (0.09,0.22), (0.13,0.17) (0.17,0.26), (0.09,0.17) (0.09,0.22), (0.13,0.17) (0.09,0.13), (0.09,0.13), (0.17,0.30) (0.22,0.26) (0.22.0.30) (0.04,0.09) (0.17,0.30) (0.04,0.09) (0.17,0.29) (0.09,0.13) (0.17,0.34) (0.04,0.09) (0.13,0.22) (0.09,0.17) s{ p2p3 } (1,1), (0.22,0.26), (0.0,0.0), (0.09,0.17) (0.0,0.0) (1,1) (1,1), (0.13,0.30), (0.0,0.0), (0.04,0.09) (0.0,0.0) (1,1) (0.17,0.26), (0.0,0.0), (1,1), s{ p1 } (0.09,0.17) (0.0,0.0) (1,1) (0.09,0.22), (0.0,0.0), (1,1), (1,1) (0.13,0.17) (0.0,0.0) (0,0), (0.13,0.17), (0.0,0.0), (0.17,0.22) (0.0,0.0) (0,0) (0.22.0.30) (0.04,0.09) (0.17,0.30) (0.04,0.09) (0.17,0.29) (0.09,0.13) (0.17,0.34) (0.04,0.09) (0.13,0.22) (0.09,0.17) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (1,1), (1,1) (1,1), (1,1) (0,0), (0,0) (0,0), (0,0) (0,0), (0,0) (0,0), (0,0) (1,1), (1,1) (1,1), (1,1) (0,0), (0,0) (0,0), (0,0) (0,0), (0,0) (0,0), (0,0) (0,0), (0,0), (0,0), (0,0) (0,0) (0,0) (1,1), (1,1) (1,1), (1,1) (1,1), (1,1) (1,1), (1,1) (0,0), (0,0) (0.22.0.30) (0.04,0.09) (0.17,0.30) (0.04,0.09) (0.17,0.29) (0.09,0.13) (0.17,0.34) (0.04,0.09) (0.13,0.22) (0.09,0.17) s{ p1p3 } (0.20,0.34), (0.20,0.27) (0.27,0.34), (0.20,0.34) (0.20,0.41), N IVIFSM { p2 (i , k )} (0.20,0.27) (0.14,0.41), (0.07,0.14) (0.27,0.34), (0.14,0.27) (0.34,0.41), (0.20,0.27) (0.27,0.48), (0.14,0.20) (0.14,0.34), (0.20,0.27) (0.20,0.34), (0.14,0.20) (0.0,0.0), (0.0,0.0) (0.0,0.0), (0.0,0.0) (0.48,0.54), (0.07,0.14) (0.34,0.48), (0.07,0.14) (0.14,0.27), (0.20,0.34) (0.20,0.41), (0.14,0.20) (0.0,0.0), (0.0,0.0) (0.0,0.0), (0.0,0.0) (0.14,0.20), (0.27,0.48), (0.0,0.0), (0.27,0.34) (0.07,0.20) (0.0,0.0) (0.34,0.48) (0.14,0.20) (0.27,0.48) (0.14,0.20) (0.48,0.54) (0.07,0.14) (0.34,0.48) (0.07,0.14) (0.27,0.48) (0.07,0.20) (1,1), (0.34,0.48), (0.14,0.20) (1,1) (1,1), (0.27,0.34), (1,1) (0.20,0.27) (0.27,0.41), (1,1), (0.14,0.27) s{ p2 } (1,1) (0,0), (0.34,0.41), (0.20,0.27) (0,0) (1,1), (0.27,0.34), (0.20,0.27) (1,1) (1,1), (1,1) (1,1), (1,1) (0,0), (0,0) (0,0), (0,0) (1,1), (1,1) (1,1), (1,1) (1,1), (1,1) (0,0), (0,0) (0,0), (0,0) (0,0), (0,0) (1,1), (1,1) (0,0), (0,0) (1,1), (0,0), (1,1), (1,1) (0,0) (1,1) (0.34,0.48) (0.14,0.20) (0.27,0.48) (0.14,0.20) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.34,0.48) (0.14,0.20) (0.27,0.48) (0.14,0.20) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.48,0.54) (0.07,0.14) (0.34,0.48) (0.07,0.14) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.48,0.54) (0.07,0.14) (0.34,0.48) (0.07,0.14) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.27,0.48) (0.0,0.0) (0.27,0.48) (0.0,0.0) (0.07,0.20) (0.0,0.0) (0.07,0.20) (0.0,0.0) (0,0), (0,0) (0,0), (0,0) (0,0), (0,0) (0,0), (0,0) (0,0), (0,0) 74 Tambouratzis T., Souliou D., Chalikias M., Gregoriades A. [Step 4]: (0.13,0.21), Combined choice matrices for (0.08,0.29), (0.0,0.0), (0.13,0.25), computed as in Case 1. expertsp1 ,p2 , (0.13,0.17) andp3 are (0.04,0.13) (0.0,0.0) (0.13,0.17) s{ p1p2 } (1,1), (1,1), (1,1), (0,0), (0,0), (0.08,0.17), s p2 } (0.21,0.25) (1,1) (0,0) (0,0) (1,1) { p1(1,1) (0.13,0.29), (1,1), .13,0.21), (0.0,0.0), (0.08,0.17), (0.17,0.34), (0.13,0.21), (0 (0.08,0.29), (0.0,0.0), of (0.13,0.25), (1,1), (1,1), (1,1), (0,0), (0,0), (0,0), (0,0), (1,1), (1,1), [Step 5]: Now nor- (0.29,0.34), the corresponding (0.13,0.21) product (0.21,0.25) (0.08,0.13) (1,1) (0.13,0.17) (0.04,0.13) (0.0,0.0) (0.0,0.0) (0.04,0.08) (0.13,0.17) (0.04,0.08) (1,1) (1,1) (1,1) (1,1) (1,1) (0,0) (0,0) (0,0) (0,0) malized IVIFSM and combined choice matrices are 21,0.25), (0.17,0.25), (0.17,0.34), (0.21,0.29), (0,0), (0.13,0.21), (0.0,0.0), (0.0,0.0), (0. (0.29,0.34), (0.13,0.29), (0.21,0.25), (1,1), (0,0), (1,1), (0,0), (1,1), (0,0), (0,0), (0,0), (0,0), s{ p3 } N IVIFSM { p3 (i , l )} as follows. (0.13,0.17) (0.13,0.17) (0.13,0.21) (0.0,0.0) (0.0,0.0) (0.13,0.17) (0.04,0.13) (1,1) (0,0) (0.08,0.13) (0.04,0.08) (0.04,0.08) (0,0) (1,1) (0,0) (1,1) (0,0) (0,0) (0,0) (0,0) (0.13,0.17), (0.17,0.25), (1,1), (0.21,0.29), (0.17,0.21), (0.21,0.25), (0.0,0.0), (0.0,0.0), (0.13,0.29), (0.21,0.25), (0.17,0.25), (0,0), (1,1), (0,0), (1,1), (0,0), (0,0), (0,0), (0,0), (0,0), N IVIFSM { p3 (i , l )} (0.04,0.13) (0.13,0.17) (0.13,0.17) (0.0,0.0) (0.0,0.0) (0.08,0.13) (0.13,0.17) (0.17,0.21) (0.13,0.17) s{ p3} (1,1) (0,0) (1,1) (0,0) (1,1) (0,0) (0,0) (0,0) (0,0) (0,0) (0.08,0.13) (0 (1,1), .04,0.08), (0.17,0.25), (0.17,0.25), (0.17,0.21), (0.0,0.0), (0.0,0.0), (0.08,0.13), (0.13,0.29), (0.04,0.17), (0.13,0.17), (1,1), (1,1), (1,1), (1,1), (1,1), (0,0), (0,0), (0,0), (0,0), 1) (0,0) (1,1) (0.08,0.13) (0.13,0.17) (0.13,0.17) (0.0,0.0) (0.0,0.0) (0.21,0.25) (0.08,0.13) (0.17,0.21) (0.17,0.21) (1,1) (1, (1,1) (0,0) (0,0) (0,0) (0.21,0.29) (1,1) (1,1) (0.04,0.08), (0.17,0.25), (0.13,0.29) (1,1), (0.0,0.0), (0.08,0.13), (1,1), 0) (1,1), (0,0), (0,0), (0.13,0.29) (0.0,0.0) (0.13,0.29) (0.04,0.17), (0.0,0. (1,1) (1,1) (0,0) (0,0) (0.0,0.0) (0.21,0.29) (0.13,0.17) (0.04,0.13) (1,1) (0.0,0.0) (0.21,0.25) (0.17,0.21) (0.04,0.13) (0.04,0.13) (0.0,0.0) (0.29,0.34) (0.0,0.0) (0.13,0.29) (0.29,0.34) (0.13,0.29) (0.29,0.34) (0.13,0.29) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.04,0.08) (0.04,0.13) (0.04,0.08) (0.04,0.13) (0.04,0.08) (0.04,0.13) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.21,0.29) (0.29,0.34) (0.21,0.29) (0.29,0.34) (0.21,0.29) (0.29,0.34) (0.0,0.0) (0.0,0.0) (0.04,0.08) (0.04,0.08) (0.0,0.0) (0.0,0.0) (0.04,0.13) ) (0.04,0.13) (0.0,0.0) (0.0,0.0) (0.04,0.08) (0.04,0.13 0.0) (0.17,0.29) (0.0, (0.21,0.29) (0.17,0.29) (0.21,0.29) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.21,0.29) (0.17,0.29) (0.08,0.13) (0.0,0.0) (0.0,0.0) (0.08,0.13) (0.08,0.13) (0.0,0.0) (0.04,0.13) (0.04,0.13) (0.04,0.13) (0.0,0.0) (0.17,0.25) (0.17,0.29) (0.17,0.25) (0.17,0.29) (0.17,0.25) (0.17,0.29) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.0,0.0) MAXIMISING ACCURACY AND EFFICIENCY OF. . . (0.0,0.0) (0.08,0.13) (0.13,0.17) (0.08,0.13) (0.13,0.17) (0.08,0.13) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.13,0.17) (0.17,0.25) (0.17,0.25) (0.17,0.25) (0.0,0.0) (0.0,0.0) [Step 6] The sum of these product IVIFSMs is (0.13,0.17) (0.13,0.17) (0.0,0.0) (0.0,0.0) (0.13,0.17) [Step 6] The sum of these product IVIFSMs is [Step (0.22.0.30) (0.22.0.30) (0.0,0.0) (0.0,0.0) (0.22.0.30) (0.34,0.48) (0.34,0.48) (0.0,0.0) ( 0.34,0.48) (0.0,0.0) 6] The sum of these product IVIFSMs is (0.04,0.09) (0.04,0.09) (0.0,0.0) (0.0,0.0) (0.04,0.09) (0.14,0.20) (0.14,0.20) (0.0,0.0) (0.14,0.20) (0.0,0.0) (0.17,0.30) (0.17,0.30) (0.0, 0.0) (0.0,0.0) (0.17,0.30) (0.27,0.48) (0.27,0.48) (0.0,0.0) (0.27,0.4 8) (0.0,0.0) (0.22.0.30) (0.22.0.30) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.22.0.30) (0.34,0.48) (0.34,0.48) (0.0,0.0) ( 0.34,0.48) 0) (0.0,0.0) (0.04,0.09) (0.14,0.20) (0.0,0.0) (0.14,0.2 (0.0,0.0) (0.04,0.09) (0.14,0.20) (0.04,0.09) (0.0,0.0) (0.04,0.09) (0.04,0.09) (0.0,0.0) (0.0,0.0) (0.04,0.09) (0.14,0.20) (0.14,0.20) (0.0,0.0) (0.14,0.20) (0.0,0.0) (0.17,0.29) (0.0,0.0) (0.0,0.0) (0.17,0.29) (0.48,0.54) (0.48,0.54) (0.0,0.0) (0.48,0.54) (0.0,0.0) (0.17,0.29) (0.17,0.30) (0.17,0.30) (0.0, 0.0) (0.0,0.0) (0.17,0.30) (0.0,0.0) (0.27,0.4 8) (0.0,0.0) (0.27,0.48) (0.27,0.48) (0.0,0.0) (0.09,0.13) 7,0.14) (0.07,0.14) .0,0.0) (0.07,0.14) (0 (0.09,0.13) (0.0,0.0) (0.0,0.0) (0.09,0.1 3) (0.0 (0.04,0.09) (0.04,0.09) (0.0,0.0) (0.0,0.0) (0.04,0.09) (0.14,0.20) (0.14,0.20) (0.0,0.0) (0.14,0.2 0) (0.0,0.0) (0.17,0.34) (0.17,0.34) (0.0,0.0) (0.0,0.0) (0.17,0.34) (0.34,0.48) (0.34,0.48) (0.0,0.0) (0.34,0.48) (0.0,0.0) (0.17,0.29) (0.17,0.29) (0.0,0.0) (0.0,0.0) (0.17,0.29) (0.48,0.54) (0.48,0.54) (0.0,0.0) (0.48,0.54) (0.0,0.0) (0.04,0.09) (0.09,0.13) (0.0,0.0) (0.0,0.0) (0.07,0.14) (0.0,0.0) (0.07,0.14) (0.0,0.0) (0.04,0.09) (0.07,0.14) (0.09,0.13) (0.04,0.09) (0.0,0.0) (0.0,0.0) (0.09,0.1 3) (0.0 7,0.14) (0.07,0.14) (0.0,0.0) (0.07,0.14) (0 .0,0.0) (0.13,0.22) (0.13,0.22) (0.0,0.0) (0.0,0.0) (0.13,0.22) (0.27,0.48) (0.27,0.48) (0.0,0.0) (0.27,0.4 8) (0.0,0.0) (0.17,0.34) (0.17,0.34) (0.0,0.0) (0.0,0.0) (0.17,0.34) (0.34,0.48) (0.34,0.48) (0.0,0.0) (0.34,0.48) (0.0,0.0) 0) (0.0,0.0) (0.07,0.20) (0.0,0.0) (0.07,0.2 (0.0,0.0) (0.09,0.17) (0.07,0.20) (0.09,0.17) (0.0,0.0) (0.09,0.17) (0.04,0.09) (0.04,0.09) (0.0,0.0) (0.0,0.0) (0.04,0.09) (0.07,0.14) (0.07,0.14) (0.0,0.0) (0.07,0.14) (0.0,0.0) (0.0,0.0) (0.0,0.0) (0.34,0.48) (0.34,0.48), (0.13,0.29) (0.34,0.48) (0.22,0.30) (0.13,0.29) (0.13,0.29) (0.13,0.29) (0.0,0.0) (0.0,0.0) (0.13,0.22) (0.13,0.22) (0.13,0.22) (0.27,0.48) (0.27,0.48) (0.0,0.0) (0.27,0.4 8) (0.0,0.0) (0.04,0.13) (0.0,0.0) (0.0,0.0) (0.04,0.09) (0.09,0.17) (0.04,0.13) (0.04,0.09) (0. 04,0.13) (0.14,0.20) (0.04,0.09) (0.04,0.13) (0.09,0.17) (0.0,0.0) (0.0,0.0) (0.09,0.17) (0.07,0.20) (0.07,0.20) (0.0,0.0) (0.07,0.2 0) (0.0,0.0) (0.29,0.34) (0.29,0.34) (0.29,0.34) (0.0,0.0) (0.0,0.0) (0.29,0.48) (0.29,0.48) (0.29,0.34) (0.27,0 .48) (0.17,0.30) (0.13,0.29) (0.13,0.29) (0.13,0.29) (0.0,0.0) (0.0,0.0) (0.34,0.48) (0.34,0.48), (0.13,0.29) (0.34,0.48) (0.22,0.30) (0.04,0.08) .20) (0.04,0 .09) (0.04,0.08) 04,0.08) (0.04,0.08) (0.0,0.0) (0.0,0.0) (0.04,0.08) (0. (0.04,0.08) (0.14,0 (0.04,0.13) (0.04,0.13) (0.04,0.13) (0.0,0.0) (0.0,0.0) (0.04,0.09) (0.04,0.09) (0. 04,0.13) (0.14,0.20) (0.04,0.09) (0.21,0.29) (0.21,0.29) (0.0,0.0) (0.0,0.0) (0.48,0.54) (0.48,0.54) (0.21,0.29) (0.48,0 .54), (0.17,0.29) (0.21,0.29) (0.29,0.34) (0.29,0.34) (0.29,0.34) (0.0,0.0) (0.0,0.0) (0.29,0.48) (0.29,0.48) (0.29,0.34) (0.27,0 .48) (0.17,0.30) . (0.04,0.13) .14) (0.09,0.13) (0.04,0.13) (0.04,0.13) (0.04,0.13) (0.0,0.0) (0.0,0.0) (0.04,0.13) (0.04,0.13) (0.07,0 (0.04,0.08) (0. 04,0.08) (0.04,0.08) (0.0,0.0) (0.0,0.0) (0.04,0.08) (0.04,0.08) (0.04,0.08) (0.14,0 .20) (0.04,0 .09) (0.17,0.29) (0.17,0.29) (0.17,0.29) (0.0,0.0) (0.0,0.0) (0.34,0.48) (0.34,0.48) (0.17,0.29) (0.34,0.48) (0.17,0.34) (0.21,0.29) (0.21,0.29) (0.21,0.29) (0.0,0.0) (0.0,0.0) (0.48,0.54) (0.48,0.54) (0.21,0.29) (0.48,0 .54), (0.17,0.29) . (0.08,0.13) (0.08,0.13) (0.0,0.0) (0.0,0.0) (0.04,0.09) (0.04,0.0 9) (0.08,0.13) (0.07,0.14) (0.04,0.09) (0.04,0.13) (0.08,0.13) (0.04,0.13) (0.04,0.13) (0.0,0.0) (0.0,0.0) (0.04,0.13) (0.04,0.13) (0.04,0.13) (0.07,0 .14) (0.09,0.13) (0.17,0.25) (0.17,0.25) (0.17,0.25) (0.0,0.0) (0.0,0.0) (0.27,0.48) (0.27,0.48) (0.17,0.25) (0.27,0 .48) (0.13,0.22) (0.17,0.29) (0.17,0.29) (0.17,0.29) (0.0,0.0) (0.0,0.0) (0.34,0.48) (0.34,0.48) (0.17,0.29) (0.34,0.48) (0.17,0.34) .20) (0.09,0.17) (0.0,0.0) (0.0,0.0) (0.07,0.17) (0.13,0.17) (0.13,0.17) (0.07,0.17) (0.13,0.17) (0.07,0 (0.13,0.17) (0.08,0.13) (0.08,0.13) (0.08,0.13) (0.0,0.0) (0.0,0.0) (0.04,0.09) (0.04,0.0 9) (0.08,0.13) (0.07,0.14) (0.04,0.09) (0.17,0.25) (0.17,0.25) (0.17,0.25) (0.0,0.0) (0.0,0.0) (0.27,0.48) (0.27,0.48) (0.17,0.25) (0.27,0 .48) (0.13,0.22) (0.13,0.17) ( 0.13,0.17) (0.13,0.17) (0.0,0.0) (0.0,0.0) (0.07,0.17) (0.07,0.17) (0.13,0.17) (0.07,0 .20) (0.09,0.17) [Step 7] Now the weights of various stages of heart disease W ( d i ), i 1, 2, ..., 5 are calculated as follows: [Step 7] Now of 3various ofheart disease W ( d i ), i 1, 2, ..., 5 are calculated as follows: weights 0.34stages 0.2 2, 0.34 0.1 [0.34the 0.48 0.48 0.29 0.48 0.30], 0.340.0 0.14300.34 0.2 2, [0.340.04 [0.04 .14 0.04, 0.48 0.48 0.29 0.48 0.30], 0.09 0.1 3 0.20 0.0 9] W ( d 1 ) 0.09 [0.04 0.04 0.0 4 0 .14 0.04, Similarly, 0.09 0.09 0.1 3 0.20 0.0 9] W ( d1 ) [1.31, 2.08], Similarly, W ( d 2 ) , W (d3 ) [0.3, 0.53] 2.08], [1.31,2.07], [1.36, , W (d ) WW( d( d4 )2 ) , W ( d 5 3) [0.3, 0.53] [0.27, 0.54] [1.82, 2.2], [0.28, 0.66] , [1.82, 2.2], [1.11,1.91], ., [0.43, 0.66] [0.28,0.88] [1.36, 2.07], [1.11,1.91], W ( d 4 ) , W ( d 5 ) [0.43, 0.88] . [0.27, 0.54] [1.37, 2.0 3], [0 .3, 0.6] . [1.37, 2.0 3], [0 .3, 0.6] . Tambouratzis T., Souliou D., Chalikias M., Gregoriades75 A. MAXIMISING ACCURACY AND EFFICIENCY OF. . . [Step [Step 7] 6] Now The sum the weights of theseof product variousIVIFSMs stages ofisheart disease W (di ), i = 1, 2, ..., 5 are calculated as follows: [0.34 + 0.34 + 0.13 + 0.34 + 0.22, 0.48 + 0.48 + 0.29 + 0.48 + 0.30], W (d1 ) = [0.04 + 0.04 + 0.04 + 0.14 + 0.04, = 0.09 + 0.09 + 0.13 + 0.20 + 0.09] = Similarly, [ [1.37, 2.03], [0.3, 0.6] ] . ] [1.31, 2.08], , W (d2 ) = [ [0.3, 0.53] ] [1.82, 2.2], W (d3 ) = , [ [0.28, 0.66] ] [1.36, 2.07], W (d4 ) = , [ [0.27, 0.54] ] [1.11, 1.91], W (d5 ) = . [0.43, 0.88] [ [Step 8] Scores of various stages of heart disease are computed as follows: S(d1 ) = {(1.37 − 0.3) + (2.03 − 0.6)}/2 = 1.25 S(d2 ) = {(1.31 − 0.3) + (2.08 − 0.53)}/2 = 1.28 S(d3 ) = {(1.82 − 0.28) + (2.2 − 0.66)}/2 = 1.54 S(d4 ) = {(1.36 − 0.27) + (2.07 − 0.54)}/2 = 1.31 S(d5 ) = {(1.11 − 0.43) + (1.91 − 0.88)}/2 = 0.86 lack of information or limited domain knowledge, experts often prefer to express their opinions only for a subset of attributes instead of the entire attribute set. In that case, cardinal score will be more when an expert provides opinion about more number of attributes and less when she provides opinion about less number of attributes. This is very much similar to our real life situations. This also removes the biasness and adds more credibility to the final decision. Our algorithm does not use any medical knowledgebase, so it does not depend on the correctness of the knowledgebase. We give more importance on the parameter selection of experts by finding initial choice matrices and then combined choice matrix. We rely on experts’ opinions and then investigate the collective opinion by a systematic procedure based on cardinal matrix. As our approach is guided by a confident weight assigning mechanism, so chances of biasing is less in our model. Among two cases, Case 1 does not use the confident weight, while Case 2 uses it. As per collective opinion of a group of experts, the ordering of the diseases, i.e., stages of heart disease is given below in Table 6. The result shows different ordering in second case as we have considered the experts’ confident weights. Case I produces the final outcome as d2 , i.e., stage ‘II’ of heart disease and Case II produces d3 , i.e., stage ‘III’ of heart disease as per the collective opinion. Table 6. Ordering of diseases in different cases Case I Case II [Step 9] Since score of d3 is maximum, the patient under consideration belongs to Stage ‘III’ as per the collective opinions of the group of experts. 7 Discussion of Results This study proposes a decision making methodology under interval-valued intuitionistic fuzzy environment, where a confident weight is assigned to each of the experts based on prescribed opinion. The confident weight for each expert is computed by deriving the cardinal score of their corresponding IVIFSSs. The opinion of an expert with more cardinal score is more important than others and consequently the opinion of that expert contributes a vital role in decision making process. When an expert is more confident about her opinion, more cardinal score is assigned to that expert. Due to 8 d 2 > d 3 > d4 > d1 > d5 d 3 > d 4 > d2 > d1 > d5 Conclusion This article has proposed an algorithmic approach for multiple attribute group decision making using IVIFSM based on confident weight assigning mechanism of experts. Firstly, we have presented IVIFSM and some of its relevant operations. Next we have proposed the confident weight assigning mechanism of experts using cardinal score in the context of interval-valued intuitionistic fuzzy environment. Our proposed algorithm is based on combined choice matrix, product IVIFSM, cardinal matrix, score function, and accuracy function, MAXIMISING ACCURACY AND EFFICIENCY OF. . . 76 [23] P.K. Maji,the A.R. Roy, R. Biswas, which yields collective opinionOn of intuitionistic a group of fuzzymakers. soft sets,Another Journal ofimportant Fuzzy Mathematics, decision aspect of vol. this 12, no. 3, pp. 669-683, 2004 study is that we have not used the concept of medi[24] P.K. Maji, An application of intuitionistic fuzzy cal knowledgebase in our decision making process. soft sets in a decision making problem, in IEEE The case study is related with medical diagnosis, International Conference on Progress in Informatwhere we have used opinions of a group of experts ics and Computing (PIC), vol. 1, December 10–12, about a common set of symptoms. IVIFSM has 2010, pp. 349–351 been used to represent the opinions. We have per- [25] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets formed a comparative analysis in the case study to and Systems, vol. 20, pp. 87- 96, 1986 demonstrate the effectiveness of using weight as- [26] Y. Jiang, Y. Tang, Q. Chen, J. Tang, signing procedure. Future scope H. of Liu, this research Interval-valued intuitionistic fuzzy soft sets and work might be to investigate the application of rotheir properties, Comput. Math. Appl., vol. 60, pp. bustness in MAGDM in the framework of IVIFSS. 906–918, 2010 Also researchers might focus on various properties [27] F. Feng, Y. Li, V. Leoreanu-Fotea, Application of of interval-valued fuzzy matrix level soft sets in intuitionistic decision making basedsoft on intervaland then apply to suitable uncertain decision valued fuzzythem soft sets, Comput. Math. Appl., vol. making problems. 60, pp. 1756–1767, 2010 [28] H. Qin, X. Ma, T. Herawan, J. M. Zain, An adjustable approach to interval-valued intuitionistic References fuzzy soft sets based decision making, in N. T. C. G. Kim, andtheory A. Janiak (Eds.), Proceed[1] Nguyen, D. Molodtsov, Soft set – first results, Comings of ACIIDS, 2011, LNAI, vol. 6592, pp. put. Math. Appl. vol. 37, no. (4–5), pp. 80–89, 19–31, Springer-Verlag Berlin Heidelberg 1999 [29] Z. Zhang, C. Wang, D. Tian, K. Li, A novel ap[2] proach S.R.S. to Varadhan, Probability Theory, fuzzy American interval-valued intuitionistic soft Mathematical Society, 2001 set based decision making, Applied Mathematical Modelling, vol. 38, no. 4, pp. 1255-1270, 2014 [3] L. A. Zadeh, Fuzzy sets, Information and Control, [30] J. Mao, D. 338-353, Yao, C. Wang, vol. 8, pp. 1965 Group decision making methods based on intuitionistic fuzzy soft matri[4] ces, Z. Pawlak, sets, International Journal of Applied Rough Mathematical Modelling, vol. 37, pp. Computer Science, vol. 11, pp. 341–356, 1982 6425–6436, 2013 [5] S. Kar, S. Das, P. K. Ghosh, Applications of neuro fuzzy systems: A brief review and future outline, Applied Soft Computing Journal, vol. 15, pp. 243259, 2014 [6] N. Cagman, S. Enginoglu, Soft matrix theory and its decision making, Computers and Mathematics with Applications, vol. 59, no. 10, pp. 3308–3314, 2010 [7] N. Cagman, S. Enginoglu, Soft set theory and uni– int decision making, European Journal of Operational Research, vol. 207, no. 2, pp. 848–855, 2010 [8] F. Feng, Y. B. Jun, X. Liu, L. Li, An adjustable approach to fuzzy soft set based decision making, Journal of Computational and Applied Mathematics, vol. 234, no. 1, pp. 10–20, 2010 [9] Jiang, Y. Tang, Q. Chen, An adjustable approach to intuitionistic fuzzy soft sets based decision making, Applied Mathematical Modelling, vol. 35, no. 2, pp. 824–836, 2011 Tambouratzis T., Souliou D., Chalikias M., Gregoriades A. [31] Das, L. S. Group decision mak[10] S. Z. Kong, Gao,Kar, L. Wang, Comment on A fuzzy ing in theoretic medical approach system: to decision An intuitionissoft set making tic fuzzy Journal soft set approach, Applied Soft problems, of Computational and Applied Computing, vol. 223, 24.no.pp. 2014, Mathematics, vol. 2, pp.196–211, 540–542, 2009 http://dx.doi.org/10.1016/j.asoc.2014.06.050 [11] P. K. Maji, A. R. Roy, R. Biswas, An application of Das, soft M. setsB.inKar, a decision problem, Com[32] S. T. Pal, S.making Kar, Multiple Attribute puters and Mathematics with Applications, vol. 44, Group Decision Making using Interval-Valued Inno. (8-9), pp. 1077–1083, 2002 tuitionistic Fuzzy Soft Matrix, Proc. of IEEE In- (FUZZ[12] ternational A. R. Roy,Conference P. K. Maji,on A Fuzzy fuzzy Systems soft set theoretic IEEE), Beijing, July 6-11, 2014, pp. 2222 – 2229, approach to decision making problems, Journal of doi: 10.1109/FUZZ-IEEE.2014.6891687 Computational and Applied Mathematics, vol. 203, no. 2, pp. 412–418, 2007 [33] S. Das, M. B. Kar, S. Kar, Group Multi Criteria [13] Decision Y. Zou, Z.Making Xiao, Data approaches soft usinganalysis Intuitionistic Multi of Fuzzy sets under incomplete information, KnowledgeSets, Journal of Uncertainty Analysis and ApplicaBased1:10, Systems, 21, no. 8, pp. 941–945, 2008 tions, 2013,vol. doi:10.1186/2195-5468-1-10 [14] Z. Xiao, K. Gong, Y. Zou, A combined forecasting [34] S. Das, S. Kar, Intuitionistic Multi Fuzzy Soft approach based on fuzzy soft sets, Journal of ComSet and its Application in Decision Making, in P. putational and Applied Mathematics, vol. 228, no. Maji et al. (Eds.), Proceedings of Fifth Interna1, pp. 326–333, 2009 tional Conference on Pattern Recognition and Ma[15] chine J. Kalayathankal G. S. 2013, Singh,Lecture A fuzzy soft Intelligence and (PReMI), Notes flood alarm model, Mathematics and587–592 Computers in in Computer Science, vol. 8251, pp. Simulation, vol. 80, no. 5, pp. 887–893, 2010 [35] S. Das, S. Kar, The Hesitant Fuzzy Soft Set and [16] its D. Application V. Kovkov, in V.Decision M. Kolbanov, D. in A.Proceedings Molodtsov, Making, Soft sets theory-based optimization, of of International Conference on Facets Journal of UncerComputer and Systems Sciences International, vol. tainties and Applications (ICFUA), 2013, Lecture 46, no.in6,Computer pp. 872–880, 2007Springer, unpublished Notes Science, [17] M. M. Mushrif, S. Sengupta, A. K. Ray, Texture [36] S. Chen, J. Tan, multisetcriteria classification usingHandling a novel, soft theory fuzzy based decision-making problemsinbased on vague setS.theclassification algorithm, P. J. Narayanan, K. ory, Fuzzy and Systems, vol. 67, no.2,ofpp. Nayar, H. Sets Y. Shum (Eds.), Proceedings the1637th 172, Asian1994 Conference on Computer Vision, 2006, Lecture Notes in Computer Science, vol. 3851, pp. 246–254 [18] P. K. Maji, R. Biswas, A. R. Roy, Soft sets theory, Comput. Math. Appl., vol. 45, pp. 555–562, 2003 [19] P. K. Maji, R. Biswas, A. R. Roy, Fuzzy soft sets, J. Fuzzy Math., vol. 9, no. 3, pp. 589–602, 2001 [20] X. B. Yang, T. Y. Lin, J. Y. Yang, Y. Li, D. Y. Yu, Combination of interval-valued fuzzy set and soft set, Comput. Math. Appl., vol. 58, pp. 521–527, 2009 [21] P. K. Maji, More on intuitionistic fuzzy soft sets, in H. Sakai, M. K. Chakraborty, A. E. Hassanien, D. Slezak, W. Zhu (Eds.), Proceedings of the 12th International Conference on Rough Sets, Fuzzy Sets, Data Mining and Granular Computing (RSFDGrC 2009), Lecture Notes in Computer Science, vol. 5908, pp. 231-240 [22] P.K. Maji, R. Biswas, A.R. Roy, Intuitionistic fuzzy soft sets, Journal of Fuzzy Mathematics vol. 9, no. 3, pp. 677–692, 2001 MAXIMISING ACCURACY AND EFFICIENCY OF. . . which yields opinionOn ofintuitionistic a group of [23] P.K. Maji,the A.R.collective Roy, R. Biswas, fuzzymakers. soft sets,Another Journal ofimportant Fuzzy Mathematics, decision aspect of vol. this 12, no. 3, pp. 669-683, 2004 study is that we have not used the concept of medical knowledgebase in our decision making process. [24] P.K. Maji, An application of intuitionistic fuzzy in aisdecision problem,diagnosis, in IEEE The soft casesets study related making with medical International Conference onof Progress Informatwhere we have used opinions a groupinof experts ics and Computing (PIC), vol. 1, December 10–12, about a common set of symptoms. IVIFSM has 2010, pp. 349–351 been used to represent the opinions. We have per- [25] K. Atanassov, Intuitionistic Fuzzy Sets formed a comparative analysisfuzzy in thesets, case study to and Systems, vol. 20, pp. 87- 96, 1986 demonstrate the effectiveness of using weight as- [26] Y. Jiang, Y. Tang, Q. Chen, J. Tang, signing procedure. Future scope H. of Liu, this research Interval-valued intuitionistic fuzzy soft sets work might be to investigate the application ofand rotheir properties, Comput. Math. Appl., vol. 60, pp. bustness in MAGDM in the framework of IVIFSS. 906–918, 2010 Also researchers might focus on various properties [27] F. Feng, Y. Li, V. Leoreanu-Fotea, Application of of interval-valued intuitionistic fuzzy soft matrix level soft sets in decision making based on intervaland then apply to suitable uncertain decision valued fuzzythem soft sets, Comput. Math. Appl., vol. making problems. 60, pp. 1756–1767, 2010 [28] H. Qin, X. Ma, T. Herawan, J. M. Zain, An adjustable approach to interval-valued intuitionistic References fuzzy soft sets based decision making, in N. T. C. G. Kim, andtheory A. Janiak (Eds.), Proceed[1] Nguyen, D. Molodtsov, Soft set – first results, Comings of ACIIDS, 2011, LNAI, vol. 6592, pp. put. Math. Appl. vol. 37, no. (4–5), pp. 80–89, 19–31, Springer-Verlag Berlin Heidelberg 1999 [29] Z. Zhang, C. Wang, D. Tian, K. Li, A novel ap[2] proach S.R.S. to Varadhan, Probability Theory, fuzzy American interval-valued intuitionistic soft Mathematical Society, 2001 set based decision making, Applied Mathematical vol. 38, no. 4, Information pp. 1255-1270, [3] Modelling, L. A. Zadeh, Fuzzy sets, and2014 Control, [30] J.vol. Mao, D. 338-353, Yao, C. Wang, 8, pp. 1965 Group decision making methods based on intuitionistic fuzzy soft matri[4] ces, Z. Pawlak, sets, International Journal of Applied Rough Mathematical Modelling, vol. 37, pp. Computer Science, vol. 11, pp. 341–356, 1982 6425–6436, 2013 [5] S. Kar, S. Das, P. K. Ghosh, Applications of neuro fuzzy systems: A brief review and future outline, Applied Soft Computing Journal, vol. 15, pp. 243259, 2014 [6] N. Cagman, S. Enginoglu, Soft matrix theory and its decision making, Computers and Mathematics with Applications, vol. 59, no. 10, pp. 3308–3314, 2010 [7] N. Cagman, S. Enginoglu, Soft set theory and uni– int decision making, European Journal of Operational Research, vol. 207, no. 2, pp. 848–855, 2010 [8] F. Feng, Y. B. Jun, X. Liu, L. Li, An adjustable approach to fuzzy soft set based decision making, Journal of Computational and Applied Mathematics, vol. 234, no. 1, pp. 10–20, 2010 [9] Jiang, Y. Tang, Q. Chen, An adjustable approach to intuitionistic fuzzy soft sets based decision making, Applied Mathematical Modelling, vol. 35, no. 2, pp. 824–836, 2011 Tambouratzis T., Souliou D., Chalikias M., Gregoriades77 A. [10] S. Z. Kong, Gao,Kar, L. Wang, Comment on A makfuzzy [31] Das, L. S. Group decision soft set making ing in theoretic medical approach system: to decision An intuitionisproblems, of Computational and Applied tic fuzzy Journal soft set approach, Applied Soft Mathematics, vol. 2, pp.196–211, 540–542, 2009 Computing, vol. 223, 24.no.pp. 2014, http://dx.doi.org/10.1016/j.asoc.2014.06.050 [11] P. K. Maji, A. R. Roy, R. Biswas, An application of soft sets in a decision making problem, Com[32] S. Das, M. B. Kar, T. Pal, S. Kar, Multiple Attribute puters and Mathematics with Applications, vol. 44, Group Decision Making using Interval-Valued Inno. (8-9), pp. 1077–1083, 2002 tuitionistic Fuzzy Soft Matrix, Proc. of IEEE In(FUZZ[12] ternational A. R. Roy,Conference P. K. Maji,on A Fuzzy fuzzy Systems soft set theoretic IEEE), Beijing, July 6-11, 2014, pp. 2222 – 2229, approach to decision making problems, Journal of doi: 10.1109/FUZZ-IEEE.2014.6891687 Computational and Applied Mathematics, vol. 203, no. 2, pp. 412–418, 2007 [33] S. Das, M. B. Kar, S. Kar, Group Multi Criteria [13] Decision Y. Zou, Z. Xiao, Data approaches soft Making usinganalysis Intuitionistic Multi of Fuzzy sets under incomplete information, KnowledgeSets, Journal of Uncertainty Analysis and ApplicaBased1:10, Systems, 21, no. 8, pp. 941–945, 2008 tions, 2013,vol. doi:10.1186/2195-5468-1-10 [14] Z. Xiao, K. Gong, Y. Zou, A combined forecasting [34] S. Das, S.based Kar,onIntuitionistic Multi Fuzzy Soft approach fuzzy soft sets, Journal of ComSet and its Application in Decision Making, in P. putational and Applied Mathematics, vol. 228, no. Maji et al. (Eds.), Proceedings of Fifth Interna1, pp. 326–333, 2009 tional Conference on Pattern Recognition and Ma[15] chine J. Kalayathankal G. S. 2013, Singh,Lecture A fuzzy soft Intelligence and (PReMI), Notes flood alarm model, Mathematics and Computers in in Computer Science, vol. 8251, pp. 587–592 Simulation, vol. 80, no. 5, pp. 887–893, 2010 [35] S. Das, S. Kar, The Hesitant Fuzzy Soft Set and [16] D. V. Kovkov, V. M. Kolbanov, D. A. Molodtsov, its Application in Decision Making, in Proceedings Soft sets theory-based optimization, Journal of of International Conference on Facets of UncerComputer and Systems Sciences International, vol. tainties and Applications (ICFUA), 2013, Lecture 46, no. 6, pp. 872–880, 2007 Notes in Computer Science, Springer, unpublished [17] M. M. Mushrif, S. Sengupta, A. K. Ray, Texture [36] S. Chen, J. Tan, multisetcriteria classification usingHandling a novel, soft theory fuzzy based decision-making problemsinbased on vague setS.theclassification algorithm, P. J. Narayanan, K. ory, Fuzzy and Systems, vol. 67, no.2,ofpp. Nayar, H. Sets Y. Shum (Eds.), Proceedings the1637th 172, 1994 Asian Conference on Computer Vision, 2006, Lecture Notes in Computer Science, vol. 3851, pp. 246–254 [18] P. K. Maji, R. Biswas, A. R. Roy, Soft sets theory, Comput. Math. Appl., vol. 45, pp. 555–562, 2003 [19] P. K. Maji, R. Biswas, A. R. Roy, Fuzzy soft sets, J. Fuzzy Math., vol. 9, no. 3, pp. 589–602, 2001 [20] X. B. Yang, T. Y. Lin, J. Y. Yang, Y. Li, D. Y. Yu, Combination of interval-valued fuzzy set and soft set, Comput. Math. Appl., vol. 58, pp. 521–527, 2009 [21] P. K. Maji, More on intuitionistic fuzzy soft sets, in H. Sakai, M. K. Chakraborty, A. E. Hassanien, D. Slezak, W. Zhu (Eds.), Proceedings of the 12th International Conference on Rough Sets, Fuzzy Sets, Data Mining and Granular Computing (RSFDGrC 2009), Lecture Notes in Computer Science, vol. 5908, pp. 231-240 [22] P.K. Maji, R. Biswas, A.R. Roy, Intuitionistic fuzzy soft sets, Journal of Fuzzy Mathematics vol. 9, no. 3, pp. 677–692, 2001
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