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Exponential and non-Exponential Based Generalized Similarity Measures for Complex Hesitant Fuzzy Sets with Applications

Exponential and non-Exponential Based Generalized Similarity Measures for Complex Hesitant Fuzzy... FUZZY INFORMATION AND ENGINEERING 2020, VOL. 12, NO. 1, 38–70 https://doi.org/10.1080/16168658.2020.1779013 ORIGINAL ARTICLE Exponential and non-Exponential Based Generalized Similarity Measures for Complex Hesitant Fuzzy Sets with Applications Tahir Mahmood, Ubaid ur Rehman and Zeeshan Ali Department of Mathematics and Statistics, International Islamic University Islamabad, Islamabad, Pakistan ABSTRACT ARTICLE HISTORY Received 26 March 2020 The purpose of this manuscript is to explore the notion of a com- Revised 30 May 2020 plex hesitant fuzzy set (CHFS), as a generalization of the hesitant Accepted 1 June 2020 fuzzy set (HFS) and complex fuzzy set (CFS) to cope with the uncer- tain and complicated information in the real-world decision. CHFS KEYWORDS contains truth grades in the form of a subset of the unit disc in Complex fuzzy set; complex the complex plane. The operational laws of the explored notion are hesitant fuzzy sets; similarity also described. Further, the exponential based generalized similarity measures measures, without exponential based generalized similarity mea- sures, and their important characteristics are also explored. These similarity measures are applied in the environment of pattern recog- nition and medical diagnosis to evaluate the proficiency and feasi- bility of the established measures. We also solved some numerical examples using the established measures. To examine the reliability and validity of the proposed measures by comparing it with existing measures. The advantages, comparative analysis, and graphical rep- resentation of the explored measures and existing measures are also discussed in detail. 1. Introduction The theory of fuzzy set (FS) was firstly explored by Zadeh [1] in 1965 which successfully applied in different fields. FS contains one function, called truth grade, belonging to the unit interval. FS has gained extensive achievement and various researchers have utilized it in the environment of medical diagnosis [2–4], pattern recognition [5], decision making [6], and clustering algorithm. Moreover, the concept of interval-valued FS (IVFS) was estab- lished by Zadeh [7], which contains the grade of truth in the form of some closed subinterval of the unit interval. Couso et al. [8] defined a formal relational study of similarity and dissim- ilarity measures between FSs. The SM between FSs and between elements is described by Lee-Kwang et al. [9]. Pramanik and Mondal [10] presented weighted fuzzy SM based on tan- gent function and its discussed application to medical diagnosis. Some new SMs on FSs are defined by Wang [11]. Kwon [12] also defined SM based on FSs. A new approach to fuzzy distance measure and SM between generalized fuzzy numbers was described by Guha and CONTACT Tahir Mahmood tahirbakhat@iiu.edu.pk © 2020 The Author(s). Published by Taylor & Francis Group on behalf of the Fuzzy Information and Engineering Branch of the Operations Research Society of China & Operations Research Society of Guangdong Province. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/ licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. FUZZY INFORMATION AND ENGINEERING 39 Chakraborty [13]. Kakati [14] explored a note on the new similarity measure for FSs. Some SMs based on FSs are presented by Hesamian [15] to find about the closeness between two objects. Various researchers arise a question, what will happen when the range of FS changes to complex numbers form a unit disc in a complex plan instead of a real number. Ramot et al. [16] introduced the idea of complex FS (CFS), which contains the truth grade in the form of a complex number by a member of a unit disc in the complex plane. CFS deals with two dimensions in a single set. CFS is a powerful procedure to illustrate the belief of a human being in the formation of grades. Bi et al. [17] described complex fuzzy arithmetic aggregation operators. Adaptive image restoration by a novel neuro-fuzzy approach using CFSs is presented by Li [18]. A systematic review of CFSs and logic is described by Yazdan- bakhsh and dick [19]. Dai [20] wrote some comments on complex fuzzy logic. Jun and Xin [21] applied CFSs to BCK/BCI-algebra. The orthogonality between CFSs and its application to signal detection is described by Hu et al. [22]. Hu et al. [23] also defined distances of CFSs and continuity of CF operations. In the real decision making procedure, it is hard to set up the membership degree of FS due to the insufficiency of knowledge or data, hesitation, and many other reasons. To over- come such kind of issues Torra [24] investigated the notion of the hesitant fuzzy set (HFS) which contains the grade of truth in the form of a subset of the unit interval. HFS is the generalization of FS to deal with uncertain and more complicated information in real deci- sion theory. Xu and Xia [25] explored distance and SMs for HFSs. Liao and Xu [26] described subtraction and division operation over HFSs. Decomposition theorems and extension prin- ciples for HFSs are explored by Alcantud and Torra [27]. Bishti and Kumar [28] defined fuzzy time series forecasting method based on HFSs. Novel distance and SMs on HFSs with appli- cation to clustering analysis presented by Zhang and Xu [29]. Alcantud and Giarlotta [30] proposed an extension of Torra’s concept of HFSs. Farhadinia and Herrera-Viedma [31] defined multiple criteria group decision-making method based on extended HFSs with unknown weight information. Distance and SMs between HFSs and their application in pattern recognition were stated by Zeng et al. [32]. In real-life problems, we come across many situations where we need to quantify the uncertainty existing in the data to make optimal decisions. Exponential based similarity measures and without exponential based similarity measures are important tools for han- dling uncertain information present in our day-to-day life problems. Different measures, such as similarity, exponential, distance, entropy, and inclusion, process the uncertain infor- mation, and enable us to reach some conclusion. Recently, these measures have gained much attention from many authors due to their wide applications in various fields, such as pattern recognition, medical diagnosis, clustering analysis, and image segment. All the existing approaches of decision-makers, based on exponential based similarity measures and without exponential based similarity measures, in FS, CFS, and HFS theories, deal with membership functions belonging to a unit interval in the form of a subset in the concept of HFS. In CHFS theory, membership degrees are complex-valued and are represented in polar coordinates. These all notions worked effectively, but when a decision-maker faced such kinds of information which contains two-dimensional information in a single-set. For i2π(0.3) i2π(0.6) i2π(0.2) i2π(0.2) instance, 0.9e , 0.7e , 0.3e , 0.1e , then the existing all notions are failed. For coping with such kind of problems, the CHFS is a proficient technique to resolve 40 T. MAHMOOD ET AL. realistic decision problems in the environment of fuzzy set theory. CHFS is more power- ful and more general than existing notions like HFS, CFS, and FS to cope with awkward and complicated information in real-life decisions. Because these all notions are the special cases of the explored CHFS. The advantages of the presented CHFS are discussed below: (1) When we choose the imaginary parts of the CHFS as zero, then the CHFS is reduced into HFS which is in the form of {0.9, 0.7, 0.3, 0.1}. (2) When we choose the CHFS as a singleton set, then the CHFS is reduced into CFS which i2π(0.3) is in the form of 0.9e . (3) When we choose the CHFS as a singleton set and the imaginary parts as zero, then the CHFS is reduced into FS which is in the form of {0.9}. Motivated by the above challenges and keeping the advantages of the CHFS, in this manuscript, some key contributions are made: (1) To explore the novel approach of the complex hesitant fuzzy set (CHFS), which is the generalization of the hesitant fuzzy set (HFS) and complex fuzzy set (CFS) to cope with the uncertain and complicated information in the real-world decision. CHFS contains truth grades in the form of subset of the unit disc in the complex plane. Operational laws of the explored notion are also described and verified with the help of some numerical examples. (2) To present some similarity measures is called exponential based similarity measures, without exponential based similarity measures, generalized similarity measures and their important characteristics are also explored. (3) These similarity measures are utilized in the environment of pattern recognition and medical diagnosis to evaluate the proficiency and feasibility of the established mea- sures. We solve some numerical examples using the established measures. (4) To examine the reliability and validity of the proposed measures by comparing with existing measures. The advantages, comparative analysis, and graphical representa- tion of the explored measures and existing measures are also discussed in detail. The graphical interpretation of the explored works is discussed with the help of Figure 1. The remainder of this manuscript is organized as follows: In Section 2, the notion of FSs, CFSs, HFSs are review. In Section 3, the purpose of this manuscript is to explore the notion of the complex hesitant fuzzy set (CHFS), as a mixture of the hesitant fuzzy set (HFS) and complex fuzzy set (CFS) to cope with the uncertain and complicated information in the real-world decision. CHFS contains truth grades in the form of a subset of the unit disc in a complex plane. The operational laws of the explored notion are also described. In Section 4, the exponential based similarity measures, without exponential based similarity measures, generalized similarity measures, and their important characteristics are also explored. In Section 5, these similarity measures are applied in the environment of pattern recognition and medical diagnosis to evaluate the proficiency and feasibility of the established mea- sures. We solve some numerical examples using the established measures. To examine the reliability and validity of the proposed measures by comparing it with existing measures. FUZZY INFORMATION AND ENGINEERING 41 Figure 1. The geometrical representation of the explored approach. The advantages, comparative analysis, and graphical representation of the explored mea- sures and existing measures are also discussed in detail. The conclusion of this manuscript is discussed in Section 6. 2. Preliminaries In this part of the article, we review basic definitions like FS, CFS, and HFS. Throughout this article X represents a fix set. Definition 1: [1]:AFS E is of the form: E = {(x, μ (x))|x ∈ X} with a condition 0 ≤ μ (x) ≤ 1, where μ (x) represents the grade of truth. Throughout this E E article, the collection of all FSs on X are denoted by FS(X).The pair E = (x, μ (x)) is called fuzzy number (FN). Definition 2: [16]:ACFS E is of the form: E = {(x, μ (x)|x ∈ X} i2π(ω (x)) where μ (x) = γ (x).e represents the complex-valued truth grade in the form of E E i2π(ω (x)) polar coordinate, where γ (x), ω (x) ∈ [0, 1]. Further, the pair E = (x, γ (x).e ) is E γ E called complex fuzzy number (CFN). Definition 3: [24]:AHFS E is of the form: E = {(x, μ (x))|x ∈ X} where μ (x)is the set of different finite values in [0, 1] representing the grade of truth for each element x ∈ X. Further, the pair E = (x, μ (x)) is called hesitant fuzzy number (HFN). E 42 T. MAHMOOD ET AL. Definition 4: [25]: For any two HFSs E and F, the similarity measure S(E, F) satisfies the following conditions: (1) 0 ≤ S(E, F) ≤ 1; (2) S(E, F) = 1 ⇔ E = F; (3) S(E, F) = S(F, E). Definition 5: [25]: For any two HFSs E and F, the distance measure d(E, F) satisfies the following conditions: (1) 0 ≤ d(E, F) ≤ 1; (2) d(E, F) = 1 ⇔ E = F; (3) d(E, F) = d(F, E). From the above analysis, we obtain that the S(E, F) = 1 − d(E, F). 3. Complex Hesitant Fuzzy Sets In this portion, we presented the idea of complex hesitant fuzzy sets (CHFSs) and its some properties. Definition 6: A CHFS E is of the form: E = { (x, μ (x))|x ∈ X} where i2π(ω (x)) E j μ (x) = γ (x).e , j = 1, 2, 3, ... , n E E i2π(ω (x)) i2π(ω (x)) i2π(ω (x)) γ γ γ E 1 E2 En = γ (x).e , γ (x).e , ... , γ (x).e E E E 1 2 n represented the complex-valued truth grade which is subset of unit disc in complex plane i2π(ω (x)) Ej with a condition γ (x), ω (x) ∈ [0, 1]. Further, E = (x, γ (x).e ) is called complex E γ E j E j hesitant fuzzy number (CHFN). i2π(ω (x)) i2π(ω (x)) γ γ E F j j Definition 7: Let E = (x, γ (x).e ) and F = (x, γ (x).e ) be two CHFNs. E F j j Then i2π( 1−ω (x) ) Ej (1) c(γ (x)) = (x, 1 − γ (x) .e ) ; E E i2π(max(ω (x),ω (x))) γ γ E F j j (2) E ∪ F = (x,max(γ (x), γ (x)).e ) ; E F j j i2π(min(ω (x),ω (x))) γ γ E F j j (3) E ∩ F = (x,min(γ (x), γ (x)).e ) . E F j j The notion of CHFS is an extensive powerful technique to cope with uncertain and awk- ward information in realistic decision theory. The CHFS contains the grade of supporting in the form of a subset of the unit disc in the complex plane, whose entities in the form of polar FUZZY INFORMATION AND ENGINEERING 43 coordinates. Basically, the CHFS contains two-dimension information in a single set. The presented CHFS is more general than existing drawbacks, whose detailed and justifications are discussed are below: In Definitions (6) and (7), if we choose the imaginary parts as zero, then the explored notion is converted for HFS, which is presented by Torra [24]. Similarly, if we choose the CHFS as a singleton set, then the CHFS is converted for CFS, which is presented by Ramot et al. [16]. Further, if we choose the CHFs as a singleton set and the imaginary part is zero, then the CHFS is converted for FS, which is explored by Zadeh [1]. Due to its structure, it makes powerful and proficient to cope with uncertain and unreliable information in real decision theory. Example 1: Let i2π(0.3) i2π(0.6) i2π(0.4) i2π(0.5) i2π(0.6) (x , 0.9e , 0.7e ), (x , 0.3e , 0.8e , 0.5e ), 1 2 E = i2π(0.8) i2π(0.5) i2π(0.1) i2π(0.6) (x , 0.6e ), (x , 0.7e , 0.9e , 0.3e ) 3 4 and i2π(0.6) i2π(1) i2π(0.3) (x , 0.8e , 0.1e ), (x , 0.2e ), 1 2 F = , i2π(0.5) i2π(0.8) i2π(0.5) i2π(0.6) i2π(0.2) (x , 0.6e , 0.7e ), (x , 1e , 0.7e , 0.9e ) 3 4 be two CHFSs. Then the operational laws are defined as i2π(0.7) i2π(0.4) i2π(0.6) i2π(0.5) i2π(0.4) 0.1e , 0.3e , 0.7e , 0.2e , 0.5e , (1) E = ; i2π(0.2) i2π(0.5) i2π(0.9) i2π(0.4) 0.4e , 0.3e , 0.1e , 0.7e i2π(0.6) i2π(1) i2π(0.4) i2π(0.5) i2π(0.6) 0.9e , 0.7e , 0.3e , 0.8e , 0.5e , (2) E ∪ F = ; i2π(0.8) i2π(0.8) i2π(0.5) i2π(0.6) i2π(0.6) 0.6e , 0.7e , 1e , 0.9e , 0.9e i2π(0.3) i2π(0.6) i2π(0.3) 0.8e , 0.1e , 0.2e , E ∩ F = . i2π(0.5) i2π(0.5) i2π(0.1) i2π(0.2) 0.6e , 0.7e , 0.7e , 0.3e Theorem 1: Let E,Fand G ∈ CHFS(X) then the following holds (1) c(c(E)) = E (2) i. E ∪ F = F ∪ E ii. E ∩ F = F ∩ E (3) i. (E ∪ F) ∪ G = E ∪ (F ∪ G) ii. (E ∩ F) ∩ G = E ∩ (F ∩ G) (4) E ∪ (F ∩ G) = (E ∪ F) ∩ (E ∪ G) (5) E ∩ (F ∪ G) = (E ∩ F) ∪ (E ∩ G) i2π(ω (x)) i2π(ω (x)) γ E j j Proof: In this theorem we have E = (x, γ (x).e ), F = (x, γ (x).e ) and E F j j i2π(ω (x)) G = (x, γ (x).e ) j 44 T. MAHMOOD ET AL. (1) By Definition 7 we have i2π( 1−ω (x) ) i2π(ω (x)) γ E Ej c(E) = c(x, γ (x).e ) = (x, 1 − γ (x) .e ) , then E E j j i2π( 1−(1−ω (x)) ) Ej c(c(E)) = (x, 1 − (1 − γ (x)) .e ) i2π(ω (x)) Ej = (x, γ (x).e ) = E. (2) By Definition 7 we have i2π(max(ω (x),ω (x))) γ γ E F j j E ∪ F = (x,max(γ (x), γ (x)).e ) E F j j i2π(max(ω (x),ω (x))) γ γ F E j j = (x,max(γ (x), γ (x)).e ) F E j j = F ∪ E. i2π(min(ω (x),ω (x))) γ γ E F j j E ∩ F = (x,min(γ (x), γ (x)).e ) E F j j i2π(min(ω (x),ω (x))) γ γ F E j j = (x,min(γ (x), γ (x)).e ) F E j j = F ∩ E (3) i. By Definition 7 we have i2π(max(ω (x),ω (x))) γ γ E F j j E ∪ F = (x,max(γ (x), γ (x)).e ) E F j j To prove that (E ∪ F) ∪ G = E ∪ (F ∪ G).As i2π(max(ω (x),ω (x))) γ γ E F j j E ∪ F = (x,max(γ (x), γ (x)).e ) then E F j j i2π(max(max(ω (x),ω (x)),ω (x))) γ γ γ E F G j j (E ∪ F) ∪ G = (x,max(max(γ (x), γ (x)), γ (x)).e ) E F G j j j i2π(max(ω (x),max(ω (x),ω (x)))) γ γ γ E F j j j = (x,max(γ (x),max(γ (x), γ (x))).e ) E F G j j j = E ∪ (F ∪ G). ii. By Definition 7 we have i2π(min(ω (x),ω (x))) γ γ E F j j E ∩ F = (x,min(γ (x), γ (x)).e ) E F j j To prove that (E ∩ F) ∩ G = E ∩ (F ∩ G).As i2π(min(ω (x),ω (x))) γ γ E F j j E ∩ F = (x,min(γ (x), γ (x)).e ) then E F j j i2π(min(min(ω (x),ω (x)),ω (x))) γ γ γ E F G j j (E ∩ F) ∩ G = (x,min(min(γ (x), γ (x)), γ (x)).e ) E F G j j j i2π(min(ω (x),min(ω (x),ω (x)))) γ γ γ E F G j j j = (x,min(γ (x),min(γ (x), γ (x))).e ) E F G j j j = E ∩ (F ∩ G). FUZZY INFORMATION AND ENGINEERING 45 (4) By definition 7 we have i2π(min(ω (x),ω (x))) γ γ F G j j F ∩ G = (x,min(γ (x), γ (x)).e ) F G j j Then i2π(max(ω (x),min(ω (x),ω (x)))) γ γ γ E F G j j j E ∪ (F ∩ G) = (x,max(γ (x),min(γ (x), γ (x))).e ) E F G j j j Next we have i2π(max(ω (x),ω (x))) γ γ E F j j E ∪ F = (x,max(γ (x), γ (x)).e ) E F j j and i2π(max(ω (x),ω (x))) γ γ E G j j E ∪ G = (x,max(γ (x), γ (x)).e ) E G j j then (E ∪ F) ∩ (E ∪ G) ⎧ ⎛ ⎛ ⎛ ⎞ ⎞ ⎞ ⎫ max(ω (x), ω (x)), ⎪ γ γ ⎪ E F ⎪ j j ⎪ ⎝ ⎝ ⎠ ⎠ ⎪   i2π min ⎪ ⎨ ⎜ ⎟ ⎬ max(ω (x), ω (x)) max(γ (x), γ (x)), ⎜ E F γ γ ⎟ E G j j j j = x,min .e ⎜ ⎟ ⎪ ⎝ max(γ (x), γ (x)) ⎠ ⎪ E G ⎪ j j ⎪ ⎪ ⎪ ⎩ ⎭ i2π(max(ω (x),min(ω (x),ω (x)))) γ γ γ E F G j j j = (x,max(γ (x),min(γ (x), γ (x))).e ) E F G j j j Finally we obtain E ∪ (F ∩ G) = (E ∪ F) ∩ (E ∪ G). (5) By Definition 7 we have i2π(max(ω (x),ω (x))) γ γ j j F ∪ G = (x,max(γ (x), γ (x)).e ) F G j j Then i2π(min(ω (x),max(ω (x),ω (x)))) γ γ γ E F G j j j E ∩ (F ∪ G) = (x,min(γ (x),max(γ (x), γ (x))).e ) E F G j j j Next we have i2π(min(ω (x),ω (x))) γ γ E F j j E ∩ F = (x,min(γ (x), γ (x)).e ) . E F j j and i2π(min(ω (x),ω (x))) γ γ E ∩ G = (x,min(γ (x), γ (x)).e ) E G j j 46 T. MAHMOOD ET AL. then (E ∩ F) ∪ (E ∩ G) ⎧ ⎛ ⎛ ⎛ ⎞ ⎞ ⎞ ⎫ min(ω (x), ω (x)), ⎪ γ γ ⎪ E F ⎪ j j ⎪ ⎝ ⎝ ⎠ ⎠ ⎪   i2π max ⎪ ⎨ ⎜ ⎟ ⎬ min(ω (x), ω (x)) min(γ (x), γ (x)), ⎜ E F γ γ ⎟ E G j j j j = x,max .e ⎜ ⎟ ⎪ ⎝ min(γ (x), γ (x)) ⎠ ⎪ E G ⎪ j j ⎪ ⎪ ⎪ ⎩ ⎭ i2π(min(ω (x),max(ω (x),ω (x)))) γ γ γ E F G j j j = (x,min(γ (x),max(γ (x), γ (x))).e ) E F G j j j Finally we obtain E ∪ (F ∩ G) = (E ∪ F) ∩ (E ∪ G). Example 2: Let i2π(0.3) i2π(0.6) i2π(0.4) i2π(0.5) i2π(0.6) (x , 0.9e , 0.7e ), (x , 0.3e , 0.8e , 0.5e ), 1 2 E = i2π(0.8) i2π(0.5) i2π(0.1) i2π(0.6) (x , 0.6e ), (x , 0.7e , 0.9e , 0.3e ) 3 4 i2π(0.6) i2π(1) i2π(0.3) (x , 0.8e , 0.1e ), (x , 0.2e ), 1 2 F = , i2π(0.5) i2π(0.8) i2π(0.5) i2π(0.6) i2π(0.2) (x , 0.6e , 0.7e ), (x , 1e , 0.7e , 0.9e ) 3 4 and i2π(0.1) i2π(0.1) i2π(0.9) i2π(0.4) (x , 0.2e , 0.7e ), (x , 0.3e ,1e ), 1 2 G = i2π(0.8) i2π(0.2) i2π(0.6) i2π(0.3) i2π(1) (x , 0.9e , 0.4e ), (x , 0.7e , 0.8e , 0.9e ) 3 4 be CHFSs. Then i2π(0.7) i2π(0.4) i2π(0.4) i2π(0.5) i2π(0.6) 0.1e , 0.3e , 0.3e , 0.8e , 0.5e , (1) c(E) = ; i2π(0.2) i2π(0.5) i2π(0.9) i2π(0.4) 0.4e , 0.3e , 0.1e , 0.7e i2π(0.3) i2π(0.6) i2π(0.6) i2π(0.5) i2π(0.4) 0.9e , 0.7e , 0.7e , 0.2e , 0.5e , (2) c(c(E)) = ; i2π(0.8) i2π(0.5) i2π(0.1) i2π(0.6) 0.6e , 0.7e , 0.9e , 0.3e i2π(0.6) i2π(1) i2π(0.4) i2π(0.5) i2π(0.6) 0.9e , 0.7e , 0.3e , 0.8e , 0.5e , (3) i. E ∪ F = i2π(0.8) i2π(0.8) i2π(0.5) i2π(0.6) i2π(0.6) 0.6e , 0.7e , 1e , 0.9e , 0.9e and i2π(0.6) i2π(1) i2π(0.4) i2π(0.5) i2π(0.6) 0.9e , 0.7e , 0.3e , 0.8e , 0.5e , E ∪ F = i2π(0.8) i2π(0.8) i2π(0.5) i2π(0.6) i2π(0.6) 0.6e , 0.7e , 1e , 0.9e , 0.9e this implies that E ∪ F = F ∪ E. i2π(0.3) i2π(0.6) i2π(0.3) 0.8e , 0.1e , 0.2e , ii. E ∩ F = i2π(0.5) i2π(0.5) i2π(0.1) i2π(0.2) 0.6e , 0.7e , 0.7e , 0.3e and i2π(0.3) i2π(0.6) i2π(0.3) 0.8e , 0.1e , 0.2e , F ∩ E = i2π(0.5) i2π(0.5) i2π(0.1) i2π(0.2) 0.6e , 0.7e , 0.7e , 0.3e this implies that E ∩ F = F ∩ E. FUZZY INFORMATION AND ENGINEERING 47 (4) i. We have i2π(0.6) i2π(1) i2π(0.9) i2π(0.5) i2π(0.6) 0.9e , 0.7e , 0.3e ,1e , 0.5e , (E ∪ F) ∪ G = i2π(0.8) i2π(0.8) i2π(0.6) i2π(0.6) i2π(1) 0.9e , 0.7e , 1e , 0.9e , 0.9e And i2π(0.6) i2π(1) i2π(0.9) i2π(0.4) 0.8e , 0.7e , 0.3e ,1e , F ∪ G = i2π(0.8) i2π(0.8) i2π(0.6) i2π(0.6) i2π(1) 0.9e , 0.7e , 1e , 0.8e , 0.9e This implies that i2π(0.6) i2π(1) i2π(0.9) i2π(0.5) i2π(0.6) 0.9e , 0.7e , 0.3e ,1e , 0.5e , E ∪ (F ∪ G) = i2π(0.8) i2π(0.8) i2π(0.6) i2π(0.6) i2π(1) 0.9e , 0.7e , 1e , 0.9e , 0.9e Finally we obtain (E ∪ F) ∪ G = E ∪ (F ∪ G). ii. Next we have i2π(0.1) i2π(0.1) i2π(0.3) 0.2e , 0.1e , 0.2e , (E ∩ F) ∩ G = i2π(0.5) i2π(0.5) i2π(0.1) i2π(0.2) 0.6e , 0.7e , 0.7e , 0.3e And i2π(0.1) i2π(0.1) i2π(0.3) 0.2e , 0.1e , 0.2e , F ∩ G = i2π(0.5) i2π(0.2) i2π(0.5) i2π(0.3) i2π(0.2) 0.6e , 0.4e , 0.7e , 0.7e , 0.9e This implies that i2π(0.1) i2π(0.1) i2π(0.3) 0.2e , 0.1e , 0.2e , E ∩ (F ∩ G) = i2π(0.5) i2π(0.5) i2π(0.1) i2π(0.2) 0.6e , 0.7e , 0.7e , 0.3e Finally we obtain (E ∩ F) ∩ G = E ∩ (F ∩ G). (5) We have i2π(0.1) i2π(0.1) i2π(0.3) 0.2e , 0.1e , 0.2e , F ∩ G = i2π(0.5) i2π(0.2) i2π(0.5) i2π(0.3) i2π(0.2) 0.6e , 0.4e , 0.7e , 0.7e , 0.9e 48 T. MAHMOOD ET AL. Then i2π(0.3) i2π(0.6) i2π(0.4) i2π(0.5) i2π(0.6) (x , 0.9e , 0.7e ), (x , 0.3e , 0.8e , 0.5e ), 1 2 E ∪ (F ∩ G) = i2π(0.8) i2π(0.2) i2π(0.5) i2π(0.3) i2π(0.6) (x , 0.6e , 0.4e ), (x , 0.7e , 0.9e , 0.9e ) 3 4 Next we have i2π(0.6) i2π(1) i2π(0.4) i2π(0.5) i2π(0.6) 0.9e , 0.7e , 0.3e , 0.8e , 0.5e , E ∪ F = i2π(0.8) i2π(0.8) i2π(0.5) i2π(0.6) i2π(0.6) 0.6e , 0.7e , 1e , 0.9e , 0.9e and i2π(0.3) i2π(0.6) i2π(0.9) i2π(0.5) i2π(0.6) (x , 0.9e , 0.7e ), (x , 0.3e ,1e , 0.5e ), 1 2 E ∪ G = i2π(0.8) i2π(0.2) i2π(0.6) i2π(0.3) i2π(1) (x , 0.9e , 0.4e ), (x , 0.7e , 0.9e , 0.9e ) 3 4 then i2π(0.3) i2π(0.6) i2π(0.4) i2π(0.5) i2π(0.6) 0.9e , 0.7e , 0.3e , 0.8e , 0.5e , (E ∪ F) ∩ (E ∪ G) = i2π(0.8) i2π(0.2) i2π(0.5) i2π(0.3) i2π(0.6) 0.6e , 0.4e , 0.7e , 0.9e , 0.9e Finally we obtain E ∪ (F ∩ G) = (E ∪ F) ∩ (E ∪ G). (6) We have i2π(0.6) i2π(1) i2π(0.9) i2π(0.4) 0.8e , 0.7e , 0.3e ,1e , F ∪ G = i2π(0.8) i2π(0.8) i2π(0.6) i2π(0.6) i2π(1) 0.9e , 0.7e , 1e , 0.8e , 0.9e Then i2π(0.6) i2π(1) i2π(0.4) i2π(0.4) 0.8e , 0.7e , 0.3e , 0.8e , E ∩ (F ∪ G) = i2π(0.8) i2π(0.5) i2π(0.1) i2π(0.6) 0.6e , 0.7e , 0.8e , 0.3e Next we have i2π(0.3) i2π(0.6) i2π(0.3) 0.8e , 0.1e , 0.2e , E ∩ F = i2π(0.5) i2π(0.5) i2π(0.1) i2π(0.2) 0.6e , 0.7e , 0.7e , 0.3e and E then i2π(0.6) i2π(1) i2π(0.4) i2π(0.4) 0.8e , 0.7e , 0.3e , 0.8e , (E ∩ F) ∪ (E ∩ G) = i2π(0.8) i2π(0.5) i2π(0.1) i2π(0.6) 0.6e , 0.7e , 0.8e , 0.3e Finally we obtain E ∩ (F ∪ G) = (E ∩ F) ∪ (E ∩ G). FUZZY INFORMATION AND ENGINEERING 49 4. The Generalized Similarity Measures Based on CHFSs In the part of the paper, we proposed SMs established on the exponential function. We also proposed SMs without exponential function. Definition 8: Let E and F be two CHFSs on X. Then similarity measure (SM) between E and Fis identified by S (E, F), which satisfies the following properties (1) 0 ≤ S (E, F) ≤ 1; (2) S (E, F) = 1if and only if E = F; (3) S (E, F) = S (F, E). c c Definition 9: Let E and F be two CHFS on X. Then the exponential based generalized SM is calculated as ⎡ ⎡ ⎤ ⎤ 1 λ 1 λ 1− |γ (x )−γ (x )| ∨ |ω (x )−ω (x )| E k F k γ k γ k j j  E F ⎢ ⎢ j j ⎥ ⎥ j=1 j=1 S (E, F) = 2 − 1 ⎣ ⎣ ⎦ ⎦ k=1 where λ> 0, and ∨ described the maximum operation. In Definitions (8) and (9), if we choose the imaginary parts will be zero, then the explored notion is converted for HFS. Similarly, if we choose the CHFS is a singleton set, then the CHFS is converted for CFS. Further, if we choose the CHFs is a singleton set and the imaginary part will be zero, then the CHFS is converted for FS. Due to its structure, it make powerful and proficient to cope with uncertain and unreliable information in real decision theory. Theorem 2: The SM S (E, F) satisfy the following properties (1) 0 ≤ S (E, F) ≤ 1; (2) S (E, F) = 1 if and only if E = F; 1 1 (3) S (E, F) = S (F, E). c c 1 λ 1 λ Proof: 1. Since |γ (x ) − γ (x )| ∈ [0, 1] and |ω (x ) − ω (x )| ∈ [0, 1] then E k F k γ k γ k j j E F j j j=1 j=1 1 λ 1 λ |γ (x ) − γ (x )| ∨ |ω (x ) − ω (x )| ∈ [0, 1] this implies that for k = 1we E F γ γ j k j k E k F k j j j=1 j=1 have 1 λ 1 λ 1− |γ (x )−γ (x )| ∨ |ω (x )−ω (x )| E 1 F 1 γ 1 γ 1 j j E F j j j=1 j=1 2 − 1 ∈ [0, 1] For k = 2 1 1 λ λ 1− |γ (x )−γ (x )| ∨ |ω (x )−ω (x )| E 2 F 2 γ 2 γ 2 j j  E F j j j=1 j=1 2 − 1 ∈ [0, 1] 50 T. MAHMOOD ET AL. By doing this process we obtain ⎡   ⎤ 1 λ 1 λ 1− |γ (x )−γ (x )| ∨ |ω (x )−ω (x )| E k F k γ k γ k j j  E F ⎢ j j ⎥ j=1 j=1 ⎣ 2 − 1⎦ ∈ n[0, 1] k=1 ⎡ ⎤ 1 λ 1 λ 1− |γ (x )−γ (x )| ∨ |ω (x )−ω (x )| E F γ γ k k k k j j  E F ⎢ j j ⎥ j=1 j=1 ⇒ 0 ≤ 2 − 1 ≤ n ⎣ ⎦ k=1 ⎡   ⎤ 1 λ 1 λ 1− |γ (x )−γ (x )| ∨ |ω (x )−ω (x )| E k F k γ k γ k 1 j j E F j j ⎢ ⎥ j=1 j=1 ⇒ 0 ≤ 2 − 1 ≤ 1 ⎣ ⎦ k=1 ⎡ ⎡ ⎤ ⎤ 1 λ 1 λ 1− |γ (x )−γ (x )| ∨ |ω (x )−ω (x )| E F γ γ k k k k j j  E F ⎢ ⎢ j j ⎥ ⎥ j=1 j=1 ⇒ 0 ≤ 2 − 1 ≤ 1 ⎣ ⎣ ⎦ ⎦ k=1 ⇒ 0 ≤ S (E, F) ≤ 1. 2. By Definition 7 we have ⎡ ⎡   ⎤ ⎤ 1 λ 1 λ 1− |γ (x )−γ (x )| ∨ |ω (x )−ω (x )| γ γ E k F k E k F k 1  j j ⎢ ⎢ j j ⎥ ⎥ j=1 j=1 S (E, F) = 2 − 1 ⎣ ⎣ ⎦ ⎦ k=1 ⎛ ⎞ ⎡ ⎡ ⎤ ⎤ 1 λ λ λ λ ((|γ (x ) − γ (x )| +|γ (x ) − γ (x )| + ... +|γ (x ) − γ (x )| )∨) E k F k E k F k E k F k 1 1 2 2 ⎝ ⎠ n 1− λ λ λ ⎢ ⎢ ⎥ ⎥ 1 (|ω (x ) − ω (x )| +|ω (x ) − ω (x )| + ... +|ω (x ) − ω (x )| ) γ k γ k γ k γ k γ k γ k ⎢ ⎢  E F E F E F ⎥ ⎥ 1 1 2 2 ⇔ S (E, F) = 2 − 1 ⎣ ⎣ ⎦ ⎦ k=1 ⎛ ⎞ ⎡ ⎡ 1 λ λ λ ((|γ (x ) − γ (x )| +|γ (x ) − γ (x )| + ... +|γ (x ) − γ (x )| )∨) E 1 F 1 E 1 F 1 E 1 F 1 1 1 2 2 ⎝ ⎠ 1− 1 λ λ λ ⎢ ⎢ 1 (|ω (x ) − ω (x )| +|ω (x ) − ω (x )| + ... +|ω (x ) − ω (x )| ) γ 1 γ 1 γ 1 γ 1 γ 1 γ 1 E F E F E F ⎢ ⎢ 1 1 2 2 ⇔ S (E, F) = 2 − 1 ⎣ ⎣ ⎛ ⎞ 1 λ λ λ ((|γ (x ) − γ (x )| +|γ (x ) − γ (x )| + ... +|γ (x ) − γ (x )| )∨) E 2 F 2 E 2 F 2 E 2 F 2 1 1 2 2 ⎝ ⎠ 1− 1 λ λ λ (|ω (x ) − ω (x )| +|ω (x ) − ω (x )| + ... +|ω (x ) − ω (x )| ) γ 2 γ 2 γ 2 γ 2 γ 2 γ 2 E F E F E F 1 1 2 2 + 2   − 1 + ... ⎛ ⎞ ⎤ ⎤ 1 λ λ λ ((|γ (x ) − γ (x )| +|γ (x ) − γ (x )| + ... +|γ (x ) − γ (x )| )∨) E n F n E n F n E n F n 1 1 2 2 ⎝ ⎠ 1− 1 λ λ λ ⎥ ⎥ (|ω (x ) − ω (x )| +|ω (x ) − ω (x )| + ... +|ω (x ) − ω (x )| ) γ n γ n γ n γ n γ n γ n E F E F E F ⎥ ⎥ 1 1 2 2 +2 − 1 ⎦ ⎦ i2π(ω (x )) i2π(ω (x )) E F k k j j Now as E = F ⇔ μ (x ) = μ (x ) for k = 1, 2, ... , n ⇔ γ (x )e = γ (x )e E k F k E k F k j j i2π(ω (x )) i2π(ω (x )) E k F k j j for k = 1, 2, ... , n ⇔ γ (x ) = γ (x ) and e = e for k = 1, 2, ... , n. Then E k F k j j ! " 1 1 1−0 1−0 1−0 ⇔ S (E, F) = [2 − 1 + 2 − 1 + ... + 2 − 1] ⇔ S (E, F) = 1. c FUZZY INFORMATION AND ENGINEERING 51 3. We have ⎡ ⎡   ⎤ ⎤ 1 λ 1 λ 1− |γ (x )−γ (x )| ∨ |ω (x )−ω (x )| E k F k γ k γ k E F 1 j j j j ⎢ ⎢ ⎥ ⎥ j=1 j=1 S (E, F) = 2 − 1 ⎣ ⎣ ⎦ ⎦ k=1 ⎡ ⎡ ⎤ ⎤ 1 λ 1 λ 1− |−γ (x )+γ (x )| ∨ |−ω (x )+ω (x )| F E γ γ k k k k j  F E ⎢ ⎢ j j ⎥ ⎥ j=1 j=1 = 2 − 1 ⎣ ⎣ ⎦ ⎦ k=1 ⎡ ⎡   ⎤ ⎤ 1 λ 1 λ 1− |−(γ (x )−γ (x ))| ∨ |−(ω (x )−ω (x ))| F k E k γ k γ k F E 1 j j j ⎢ ⎢ ⎥ ⎥ j=1 j=1 = 2 − 1 ⎣ ⎣ ⎦ ⎦ k=1 ⎡ ⎡ ⎤ ⎤ 1 λ 1 λ 1− |γ (x )+γ (x )| ∨ |ω (x )+ω (x )| F E γ γ k k k k j  F E ⎢ ⎢ j j ⎥ ⎥ j=1 j=1 = 2 − 1 ⎣ ⎣ ⎦ ⎦ k=1 = S (F, E). Remark 1: If λ = 1 then the exponential based generalized SM becomes ⎡ ⎤ 1 1 1− |γ (x )−γ (x )|∨ |ω (x )−ω (x )| E F γ γ k k k k j j  E F ⎢ j j ⎥ 1 j=1 j=1 S (E, F) = 2 − 1 ⎣ ⎦ k=1 Definition 10: Let E and F be two CHFS on X. Then we can also calculate exponential based generalized SM as follows ⎡ ⎡   ⎤ ⎤ 1 λ 1 λ 1− |γ (x )−γ (x )| + |ω (x )−ω (x )| E k F k γ k γ k 2 2 1 j j E F j j ⎢ ⎢ ⎥ ⎥ j=1 j=1 S (E, F) = 2 − 1 ⎣ ⎣ ⎦ ⎦ k=1 where λ> 0. Theorem 3: The SM S (E, F) satisfy the following properties (1) 0 ≤ S (E, F) ≤ 1; (2) S (E, F) = 1 if and only if E = F; 2 2 (3) S (E, F) = S (F, E). c c 1 λ 1 λ Proof: 1. Since |γ (x ) − γ (x )| ∈ [0, 1] and |ω (x ) − ω (x )| ∈ [0, 1] E k F k γ k γ k j j E F 2 2 j j j=1 j=1 1 λ 1 λ then |γ (x ) − γ (x )| + |ω (x ) − ω (x )| ∈ [0, 1] this implies that for E F γ γ k k k k 2 j j 2 E F j j j=1 j=1 52 T. MAHMOOD ET AL. k = 1wehave 1 1 λ λ 1− |γ (x )−γ (x )| + |ω (x )−ω (x )| E 1 F 1 γ 1 γ 1 2 j j 2 E F j j j=1 j=1 2 − 1 ∈ [0, 1] For k = 2 1 λ 1 λ 1− |γ (x )−γ (x )| + |ω (x )−ω (x )| E 2 F 2 γ 2 γ 2 2 j j 2 E F j j j=1 j=1 2 − 1 ∈ [0, 1] By doing this process we obtain ⎡ ⎤ 1 λ 1 λ 1− |γ (x )−γ (x )| + |ω (x )−ω (x )| E k F k γ k γ k 2 j j 2 E F ⎢ j j ⎥ j=1 j=1 2 − 1 ∈ n[0, 1] ⎣ ⎦ k=1 ⎡ ⎤ 1 λ 1 λ 1− |γ (x )−γ (x )| + |ω (x )−ω (x )| E k F k γ k γ k 2 j j 2 E F j j ⎢ ⎥ j=1 j=1 ⇒ 0 ≤ 2 − 1 ≤ n ⎣ ⎦ k=1 ⎡   ⎤ 1 λ 1 λ 1− |γ (x )−γ (x )| + |ω (x )−ω (x )| E k F k γ k γ k 1 2 j j 2 E F j j ⎢ ⎥ j=1 j=1 ⇒ 0 ≤ 2 − 1 ≤ 1 ⎣ ⎦ k=1 ⎡ ⎡ ⎤ ⎤ 1 λ 1 λ 1− |γ (x )−γ (x )| + |ω (x )−ω (x )| E F γ γ k k k k 2 j j 2 E F ⎢ ⎢ j j ⎥ ⎥ j=1 j=1 ⇒ 0 ≤ 2 − 1 ≤ 1 ⎣ ⎣ ⎦ ⎦ k=1 ⎡ ⎡   ⎤ ⎤ 1 λ 1 λ 1− |γ (x )−γ (x )| + |ω (x )−ω (x )| E k F k γ k γ k 1 2 j j 2 E F j j ⎢ ⎢ ⎥ ⎥ j=1 j=1 ⇒ 0 ≤ 2 − 1 ≤ 1 ⎣ ⎣ ⎦ ⎦ k=1 ⎡ ⎡ ⎤ ⎤ 1 λ 1 λ 1− |γ (x )−γ (x )| + |ω (x )−ω (x )| E F γ γ k k k k 2 j j 2 E F ⎢ ⎢ j j ⎥ ⎥ j=1 j=1 ⇒ 0 ≤ 2 − 1 ≤ 1 ⎣ ⎣ ⎦ ⎦ k=1 ⇒ 0 ≤ S (E, F) ≤ 1. 2. By definition 7 we have ⎡ ⎡   ⎤ ⎤ 1 1 λ 1 λ 1− |γ (x )−γ (x )| + |ωγ (x )−ωγ (x )| E k F k k k 1 2 j j 2 E F ⎢ ⎢ j j ⎥ ⎥ j=1 j=1 S (E, F) = 2 − 1 ⎣ ⎣ ⎦ ⎦ k=1 ⎛ ⎞ ⎡ ⎡ ⎤ ⎤ 1 λ λ λ ((|γ (x ) − γ (x )| +|γ (x ) − γ (x )| + ... +|γ (x ) − γ (x )| )+) E k F k E k F k E k F k 2 1 1 2 2 ⎝ ⎠ 1− 1 λ λ λ ⎢ ⎢ ⎥ ⎥ 1 (|ω (x ) − ω (x )| +|ω (x ) − ω (x )| + ... +|ω (x ) − ω (x )| ) γ k γ k γ k γ k γ k γ k 2 2 E F E F E F ⎢ ⎢ ⎥ ⎥ 1 1 2 2 ⇔ S (E, F) = 2 − 1 c ⎣ ⎣ ⎦ ⎦ k=1 ⎛ ⎞ ⎡ ⎡ 1 λ λ λ ((|γ (x ) − γ (x )| +|γ (x ) − γ (x )| + ... +|γ (x ) − γ (x )| )+) E 1 F 1 E 1 F 1 E 1 F 1 2 1 1 2 2 2 2 ⎝ ⎠ 1− 1 λ λ λ ⎢ ⎢ 1 (|ω (x ) − ω (x )| +|ω (x ) − ω (x )| + ... +|ω (x ) − ω (x )| ) γ 1 γ 1 γ 1 γ 1 γ 1 γ 1 2 2 E F E F E F ⎢ ⎢ 1 1 2 2 ⇔ S (E, F) = 2 − 1 ⎣ ⎣ n FUZZY INFORMATION AND ENGINEERING 53 ⎛ ⎞ 1 λ λ λ ((|γ (x ) − γ (x )| +|γ (x ) − γ (x )| + ... +|γ (x ) − γ (x )| )+) E 2 F 2 E 2 F 2 E 2 F 2 2 1 1 2 2 ⎝ ⎠ 1− λ λ λ (|ω (x ) − ω (x )| +|ω (x ) − ω (x )| + ... +|ω (x ) − ω (x )| ) γ 2 γ 2 γ 2 γ 2 γ 2 γ 2 2 E F E F E F 1 1 2 2 + 2 − 1 + ... ⎛ ⎞ ⎤ ⎤ 1 λ λ λ ((|γ (x ) − γ (x )| +|γ (x ) − γ (x )| + ... +|γ (x ) − γ (x )| )+) E n F n E n F n E n F n 2 1 1 2 2 ⎝ ⎠ 1− 1 λ λ λ ⎥ ⎥ (|ω (x ) − ω (x )| +|ω (x ) − ω (x )| + ... +|ω (x ) − ω (x )| ) γ n γ n γ n γ n γ n γ n 2 E F E F E F 1 1 2 2 ⎥ ⎥ + 2   − 1 ⎦ ⎦ i2π(ω (x )) i2π(ω (x )) E F k k j j Now as E = F ⇔ μ (x ) = μ (x ) for k = 1, 2, ... , n ⇔ γ (x )e = γ (x )e E k F k E k F k j j i2π(ω (x )) i2π(ω (x )) E k F k j j for k = 1, 2, ... , n ⇔ γ (x ) = γ (x ) and e = e for k = 1, 2, ... , n. Then E k F k j j ! " 2 1−0 1−0 1−0 ⇔ S (E, F) = [2 − 1 + 2 − 1 + ... + 2 − 1] ⇔ S (E, F) = 1. 3. We have ⎡ ⎡   ⎤ ⎤ 1 1 λ λ 1− |γ (x )−γ (x )| + |ω (x )−ω (x )| E k F k γ k γ k 2 j j 2 E F j j ⎢ ⎢ ⎥ ⎥ j=1 j=1 S (E, F) = 2 − 1 ⎣ ⎣ ⎦ ⎦ k=1 ⎡ ⎡ ⎤ ⎤ 1 λ 1 λ 1− |−γ (x )+γ (x )| + |−ω (x )+ω (x )| F E γ γ k k k k 2 j 2 F E ⎢ ⎢ j j ⎥ ⎥ j=1 j=1 = 2 − 1 ⎣ ⎣ ⎦ ⎦ k=1 ⎡ ⎡   ⎤ ⎤ 1 1 λ λ 1− |−(γ (x )−γ (x ))| + |−(ω (x )−ω (x ))| F k E k γ k γ k 2 j 2 F E j j ⎢ ⎢ ⎥ ⎥ j=1 j=1 = 2 − 1 ⎣ ⎣ ⎦ ⎦ k=1 ⎡ ⎡ ⎤ ⎤ 1 λ 1 λ 1− |γ (x )+γ (x )| + |ω (x )+ω (x )| F E γ γ k k k k 2 j 2 F E ⎢ ⎢ j j ⎥ ⎥ j=1 j=1 = 2 − 1 ⎣ ⎣ ⎦ ⎦ k=1 = S (F, E). Remark 2: If λ = 1 then the exponential based generalized SM becomes ⎡ ⎤ 1 1 1− |γ (x )−γ (x )|+ |ω (x )−ω (x )| E F γ γ k k k k 2 j j 2 E F ⎢ j j ⎥ 2 j=1 j=1 S (E, F) = 2 − 1 ⎣ ⎦ k=1 Definition 11: Let E and F be two CHFSs on X. Then without exponential based generalized SMs are calculated as follows # $ ⎡ ⎡ ⎤ ⎤ 1 λ 1 λ 1 − |γ (x ) − γ (x )| ∨ |ω (x ) − ω (x )| E k F k γ k γ k j=1 j j j=1 E F 1   j j ⎣ ⎣ ⎦ ⎦ # $ S (E, F) = 1  λ 1  λ 1 + |γ (x ) − γ (x )| ∨ |ω (x ) − ω (x )| k=1 E k F k γ k γ k j=1 j j j=1 E F j j 54 T. MAHMOOD ET AL. ⎡ ⎤ 1 λ 1 λ (γ (x ) ∧ γ (x )) + (ω (x ) ∧ ω (x )) E F γ γ k k k k 1 j=1 j j j=1 E F j j ⎣ ⎦ S (E, F) = 1  1 λ λ (γ (x ) ∨ γ (x )) + (ω (x ) ∨ ω (x )) E k F k γ k γ k j=1 j j j=1 E F k=1 j j ⎡ ⎛ ⎞ ⎤ 1/λ % & (ω (x ) ∧ ω (x )) (γ (x ) ∧ γ (x )) γ k γ k 1 1 E F E F j k j k j j ⎣ ⎝ ⎠ ⎦ S (E, F) = α + β cc cc λ λ (γ (x ) ∨ γ (x )) (ω (x ) ∨ ω (x )) E F γ γ j k j k E k F k j=1 j j k=1 where λ> 0and α , β ∈ [0, 1] such that α + β = 1. cc cc cc cc Theorem 4: The SM S (E, F) satisfies the following properties (1) 0 ≤ S (E, F) ≤ 1; (2) S (E, F) = 1if and only if E = F; 3 3 (3) S (E, F) = S (F, E). c c 1 1 λ λ Proof: 1. Since |γ (x ) − γ (x )| ∈ [0, 1] and |ω (x ) − ω (x )| ∈ [0, 1] then E k F k γ k γ k j j E F j j j=1 j=1 1 λ 1 λ |γ (x ) − γ (x )| ∨ |ω (x ) − ω (x )| ∈ [0, 1] and denominator will always E F γ γ k k k k j j  E F j j j=1 j=1 greater than numerator, then for k = 1wehave # $ ⎡ ⎤ 1 1 1 − |γ (x ) − γ (x )|∨ |ω (x ) − ω (x )| E 1 F 1 γ 1 γ 1 j=1 j j j=1 E F j j ⎣ ⎦ # $ ∈ [0, 1] 1 1 1 + |γ (x ) − γ (x )|∨ |ω (x ) − ω (x )| E 1 F 1 γ 1 γ 1 j j E F j=1  j=1 j j For k = 2 # $ ⎡ ⎤ 1  1 1 − |γ (x ) − γ (x )|∨ |ω (x ) − ω (x )| E 2 F 2 γ 2 γ 2 j=1 j j j=1 E F j j ⎣ ⎦ # $ ∈ [0, 1] 1  1 1 + |γ (x ) − γ (x )|∨ |ω (x ) − ω (x )| E 2 F 2 γ 2 γ 2 j=1 j j j=1 E F j j By doing this process we obtain # $ ⎡ ⎤ 1 1 1 − |γ (x ) − γ (x )|∨ |ω (x ) − ω (x )| E k F k γ k γ k j j E F j=1  j=1 j j ⎣ ⎦ # $ ∈ n[0, 1] 1 1 1 + |γ (x ) − γ (x )|∨ |ω (x ) − ω (x )| E F γ γ k=1 k k k k j=1 j j  j=1 E F j j # $ ⎡ ⎤ 1 1 1 − |γ (x ) − γ (x )|∨ |ω (x ) − ω (x )| E F γ γ k k k k j=1 j j  j=1 E F j j ⎣ ⎦ ⇒ 0 ≤ # $ ≤ n 1  1 1 + |γ (x ) − γ (x )|∨ |ω (x ) − ω (x )| k=1 E k F k γ k γ k j=1 j j j=1 E F j j # $ ⎡ ⎤ 1 1 1 − |γ (x ) − γ (x )|∨ |ω (x ) − ω (x )| E k F k γ k γ k j=1 j j j=1 E F 1   j j ⎣ ⎦ # $ ⇒ 0 ≤ ≤ 1 1  1 1 + |γ (x ) − γ (x )|∨ |ω (x ) − ω (x )| k=1 E k F k γ k γ k j=1 j j j=1 E F j j FUZZY INFORMATION AND ENGINEERING 55 # $ ⎡ ⎡ ⎤ ⎤ 1 λ 1 λ 1 − |γ (x ) − γ (x )| ∨ |ω (x ) − ω (x )| E k F k γ k γ k j j E F 1  j=1  j=1 j j ⎣ ⎣ ⎦ ⎦ ⇒ 0 ≤ # $ ≤ 1 n 1 λ 1 λ 1 + |γ (x ) − γ (x )| ∨ |ω (x ) − ω (x )| E F γ γ k=1 k k k k j=1 j j  j=1 E F j j ⇒ S (E, F). 2. By definition 7 we have # $ ⎡ ⎡ ⎤ ⎤ 1 λ 1 λ 1 − |γ (x ) − γ (x )| ∨ |ω (x ) − ω (x )| E k F k γ k γ k j=1 j j j=1 E F 1   j j ⎣ ⎣ ⎦ ⎦ # $ S (E, F) = n 1 λ 1 λ 1 + |γ (x ) − γ (x )| ∨ |ω (x ) − ω (x )| k=1 E k F k γ k γ k j=1 j j  j=1 E F j j ⎡ ⎡  ⎤ ⎤ 1 λ λ λ ((|γ (x ) − γ (x )| +|γ (x ) − γ (x )| + ... +|γ (x ) − γ (x )| )∨) E F E F E F 1 k 1 k 2 k 2 k  k  k 1 − n 1 λ λ λ ⎢ ⎢ ⎥ ⎥ (|ω (x ) − ω (x )| +|ω (x ) − ω (x )| + ... +|ω (x ) − ω (x )| ) 1 γ k γ k γ k γ k γ k γ k ⎢ ⎢  E F E F E F ⎥ ⎥ 3 1 1 2 2 ⇔ S (E, F) = ⎢ ⎢  ⎥ ⎥ 1 λ λ λ ⎣ n ⎣ ((|γ (x ) − γ (x )| +|γ (x ) − γ (x )| + ... +|γ (x ) − γ (x )| )∨) ⎦ ⎦ E1 k F1 k E2 k F2 k E k F k k=1 1 + 1 λ λ λ (|ω (x ) − ω (x )| +|ω (x ) − ω (x )| + ... +|ω (x ) − ω (x )| ) γ k γ k γ k γ k γ k γ k E F E F E F 1 1 2 2 ⎡ ⎡ 1 λ λ λ ((|γ (x ) − γ (x )| +|γ (x ) − γ (x )| + ... +|γ (x ) − γ (x )| )∨) E 1 F 1 E 1 F 1 E 1 F 1 1 1 2 2 1 − ⎢ ⎢ λ λ λ (|ω (x ) − ω (x )| +|ω (x ) − ω (x )| + ... +|ω (x ) − ω (x )| ) 1 γE 1 γF 1 γE 1 γF 1 γE 1 γF 1 ⎢ ⎢ 3 1 1 2 2 ⇔ S (E, F) = ⎢ ⎢ 1 λ λ λ ⎣ n ⎣ ((|γ (x ) − γ (x )| +|γ (x ) − γ (x )| + ... +|γ (x ) − γ (x )| )∨) E 1 F 1 E 1 F 1 E 1 F 1 1 1 2 2 1 + λ λ λ (|ω (x ) − ω (x )| +|ω (x ) − ω (x )| + ... +|ω (x ) − ω (x )| ) γE 1 γF 1 γE 1 γF 1 γE 1 γF 1 1 1 2 2 1 λ λ λ ((|γ (x ) − γ (x )| +|γ (x ) − γ (x )| + ... +|γ (x ) − γ (x )| )∨) E 2 F 2 E 2 F 2 E 2 F 2 1 1 2 2 1 − 1 λ λ λ (|ω (x ) − ω (x )| +|ω (x ) − ω (x )| + ... +|ω (x ) − ω (x )| ) γ 2 γ 2 γ 2 γ 2 γ 2 γ 2 E F E F E F 1 1 2 2 + + ... 1 λ λ λ ((|γ (x ) − γ (x )| +|γ (x ) − γ (x )| + ... +|γ (x ) − γ (x )| )∨) E 2 F 2 E 2 F 2 E 2 F 2 1 1 2 2 1 + 1 λ λ λ (|ω (x ) − ω (x )| +|ω (x ) − ω (x )| + ... +|ω (x ) − ω (x )| ) γ 2 γ 2 γ 2 γ 2 γ 2 γ 2 E F E F E F 1 1 2 2 ⎤ ⎤ 1 λ λ λ ((|γ (x ) − γ (x )| +|γ (x ) − γ (x )| + ... +|γ (x ) − γ (x )| )∨) E n F n E n F n E n F n 1 1 2 2 1 − 1 λ λ λ ⎥ ⎥ (|ω (x ) − ω (x )| +|ω (x ) − ω (x )| + ... +|ω (x ) − ω (x )| ) γ n γ n γ n γ n γ n γ n E F E F E F ⎥ ⎥ 1 1 2 2 ⎥ ⎥ 1 λ λ λ ⎦ ⎦ ((|γ (x ) − γ (x )| +|γ (x ) − γ (x )| + ... +|γ (x ) − γ (x )| )∨) E n F n E n F n E n F n 1 1 2 2 1 + 1 λ λ λ (|ω (x ) − ω (x )| +|ω (x ) − ω (x )| + ... +|ω (x ) − ω (x )| ) γ n γ n γ n γ n γ n γ n E F E F E F 1 1 2 2 i2π(ω (x )) i2π(ω (x )) E F k k j j Now as E = F ⇔ μ (x ) = μ (x ) for k = 1, 2, ... , n ⇔ γ (x )e = γ (x )e E k F k E k F k j j i2π(ω (x )) i2π(ω (x )) E k F k j j for k = 1, 2, ... , n ⇔ γ (x ) = γ (x ) and e = e for k = 1, 2, ... , n. E F k k j j Then ! " 1 ⇔ S (E, F) = [1 + 1 + ... + 1] ⇔ S (E, F) = 1. 3. We have # $ ⎡ ⎡ ⎤ ⎤ 1 λ 1 λ 1 − |γ (x ) − γ (x )| ∨ |ω (x ) − ω (x )| E k F k γ k γ k j j E F 1  j=1  j=1 j j ⎣ ⎣ ⎦ ⎦ S (E, F) = # $ n 1 λ 1 λ 1 + |γ (x ) − γ (x )| ∨ |ω (x ) − ω (x )| E F γ γ k=1 k k k k j=1 j j  j=1 E F j j # $ ⎡ ⎡ ⎤ ⎤ 1 λ 1 λ 1 − |−(γ (x ) − γ (x ))| ∨ |−(ω (x ) − ω (x ))| F k E k γ k γ k j=1 j j j=1 F E j j ⎣ ⎣ ⎦ ⎦ = # $ n 1 λ 1 λ 1 + |−(γ (x ) − γ (x ))| ∨ |−(ω (x ) − ω (x ))| F E γ γ k=1 j k j k F k E k j=1  j=1 j j 56 T. MAHMOOD ET AL. # $ ⎡ ⎡ ⎤ ⎤ 1 λ 1 λ 1 − |(γ (x ) − γ (x ))| ∨ |(ω (x ) − ω (x ))| F k E k γ k γ k j j F E 1  j=1  j=1 j j ⎣ ⎣ ⎦ ⎦ = # $ n 1 λ 1 λ 1 + |(γ (x ) − γ (x ))| ∨ |(ω (x ) − ω (x ))| F E γ γ k=1 k k k k j=1 j j  j=1 F E j j = S (F, E). Theorem 5: The SM S (E, F) satisfy the following properties (1) 0 ≤ S (E, F) ≤ 1; (2) S (E, F) = 1 if and only if E = F; 4 4 (3) S (E, F) = S (F, E). c c 1 λ 1 λ Proof: 1. Since (γ (x ) ∧ γ (x )) ∈ [0, 1], (ω (x ) ∧ ω (x )) ∈ [0, 1], E k F k γ k γ k j j E F j j j=1 j=1 1 1 λ λ (γ (x ) ∨ γ (x )) ∈ [0, 1], (ω (x ) ∨ ω (x )) ∈ [0, 1] and denominator is E k F k γ k γ k j j E F j j j=1 j=1 always greater then nominator. Thus for k = 1wehave 1  λ 1  λ (γ (x ) ∧ γ (x )) + (ω (x ) ∧ ω (x )) E 1 F 1 γ 1 γ 1 j=1 j j j=1 E F j j ∈ [0, 1] 1 λ 1 λ (γ (x ) ∨ γ (x )) + (ω (x ) ∨ ω (x )) E 1 F 1 γ 1 γ 1 j=1 j j j=1 E F j j For k = 2 1 λ 1 λ (γ (x ) ∧ γ (x )) + (ω (x ) ∧ ω (x )) E 2 F 2 γ 2 γ 2 j=1 j j  j=1 E F j j ∈ [0, 1] 1  1 λ λ (γ (x ) ∨ γ (x )) + (ω (x ) ∨ ω (x )) E 2 F 2 γ 2 γ 2 j=1 j j j=1 E F j j By doing this process we obtain 1 λ 1 λ (γ (x ) ∧ γ (x )) + (ω (x ) ∧ ω (x )) E F γ γ j k j k E k F k j=1  j=1 j j ∈ n[0, 1] 1  λ 1  λ (γ (x ) ∨ γ (x )) + (ω (x ) ∨ ω (x )) E k F k γ k γ k j=1 j j j=1 E F k=1 j j 1  1 λ λ (γ (x ) ∧ γ (x )) + (ω (x ) ∧ ω (x )) E k F k γ k γ k j=1 j j j=1 E F j j ⇒ 0 ≤ ≤ n 1 λ 1 λ (γ (x ) ∨ γ (x )) + (ω (x ) ∨ ω (x )) E k F k γ k γ k j=1 j j j=1 E F k=1   j j ⎡ ⎤ 1  λ 1  λ (γ (x ) ∧ γ (x )) + (ω (x ) ∧ ω (x )) E k F k γ k γ k 1 j=1 j j j=1 E F j j ⎣ ⎦ ⇒ 0 ≤ ≤ 1 1 λ 1 λ (γ (x ) ∨ γ (x )) + (ω (x ) ∨ ω (x )) E F γ γ k k k k j=1 j j  j=1 E F k=1 j j ⎡ ⎡ ⎤ ⎤ 1 1  1 λ λ (γ (x ) ∧ γ (x )) + (ω (x ) ∧ ω (x )) E k F k γ k γ k 1 j=1 j j j=1 E F j j ⎣ ⎣ ⎦ ⎦ ⇒ 0 ≤ ≤ 1 1 λ 1 λ (γ (x ) ∨ γ (x )) + (ω (x ) ∨ ω (x )) E F γ γ k k k k j=1 j j  j=1 E F k=1 j j ⇒ 0 ≤ S (E, F) ≤ 1. 2. By definition 7 we have ⎡ ⎤ 1 1  λ 1  λ λ (γ (x ) ∧ γ (x )) + (ω (x ) ∧ ω (x )) E k F k γ k γ k 1  j=1 j j  j=1 E F j j ⎣ ⎦ S (E, F) = 1  λ 1  λ (γ (x ) ∨ γ (x )) + (ω (x ) ∨ ω (x )) E k F k γ k γ k j=1 j j  j=1 E F k=1 j j FUZZY INFORMATION AND ENGINEERING 57 ⎡ ⎤ 1 λ λ λ ((γ (x ) ∧ γ (x )) + (γ (x ) ∧ γ (x )) + ... + (γ (x ) ∧ γ (x )) )+ E k F k E k F k E k F k 1 1 2 2 n 1 λ λ λ ⎢ ⎥ ((ω (x ) ∧ ω (x )) + (ω (x ) ∧ ω (x )) + ... + (ω (x ) ∧ ω (x )) ) 1 γ k γ k γ k γ k γ k γ k E F E F E F ⎢  1 1 2 2 ⎥ ⇔ S (E, F) = ⎢ ⎥ 1 λ λ λ ⎣ ⎦ n ((γ (x ) ∨ γ (x )) + (γ (x ) ∨ γ (x )) + ... + (γ (x ) ∨ γ (x )) )+ E k F k E k F k E k F k 1 1 2 2 k=1 1 λ λ λ ((ω (x ) ∨ ω (x )) + (ω (x ) ∨ ω (x )) + ... + (ω (x ) ∨ ω (x )) ) γ k γ k γ k γ k γ k γ k E F E F E F 1 1 2 2 ⎡ ⎡ 1 λ λ λ ((γ (x ) ∧ γ (x )) + (γ (x ) ∧ γ (x )) + ... + (γ (x ) ∧ γ (x )) )+ E 1 F 1 E 1 F 1 E 1 F 1 1 1 2 2 1 λ λ λ ⎢ ⎢ ((ω (x ) ∧ ω (x )) + (ω (x ) ∧ ω (x )) + ... + (ω (x ) ∧ ω (x )) ) 1 γ 1 γ 1 γ 1 γ 1 γ 1 γ 1 ⎢ ⎢  E F E F E F 4 1 1 2 2 ⇔ S (E, F) = ⎢ ⎢ 1 λ λ λ ⎣ n ⎣ ((γ (x ) ∨ γ (x )) + (γ (x ) ∨ γ (x )) + ... + (γ (x ) ∨ γ (x )) )+ E 1 F 1 E 1 F 1 E 1 F 1 1 1 2 2 1 λ λ λ ((ω (x ) ∨ ω (x )) + (ω (x ) ∨ ω (x )) + ... + (ω (x ) ∨ ω (x )) ) γ 1 γ 1 γ 1 γ 1 γ 1 γ 1 E F E F E F 1 1 2 2 1 λ λ λ ((γ (x ) ∧ γ (x )) + (γ (x ) ∧ γ (x )) + ... + (γ (x ) ∧ γ (x )) )+ E 2 F 2 E 2 F 2 E 2 F 2 1 1 2 2 1 λ λ λ ((ω (x ) ∧ ω (x )) + (ω (x ) ∧ ω (x )) + ... + (ω (x ) ∧ ω (x )) ) γ 2 γ 2 γ 2 γ 2 γ 2 γ 2 E F E F E F 1 1 2 2 + + ... 1 λ λ λ ((γ (x ) ∨ γ (x )) + (γ (x ) ∨ γ (x )) + ... + (γ (x ) ∨ γ (x )) )+ E 2 F 2 E 2 F 2 E 2 F 2 1 1 2 2 1 λ λ λ ((ω (x ) ∨ ω (x )) + (ω (x ) ∨ ω (x )) + ... + (ω (x ) ∨ ω (x )) ) γ 2 γ 2 γ 2 γ 2 γ 2 γ 2 E F E F E F 1 1 2 2 ⎤ ⎤ 1 λ λ λ ((γ (x ) ∧ γ (x )) + (γ (x ) ∧ γ (x )) + ... + (γ (x ) ∧ γ (x )) )+ E n F n E n F n E n F n 1 1 2 2 1 λ λ λ ⎥ ⎥ ((ω (x ) ∧ ω (x )) + (ω (x ) ∧ ω (x )) + ... + (ω (x ) ∧ ω (x )) ) γ n γ n γ n γ n γ n γ n E F E F E F ⎥ ⎥ 1 1 2 2 + ⎥ ⎥ 1 λ λ λ ⎦ ⎦ ((γ (x ) ∨ γ (x )) + (γ (x ) ∨ γ (x )) + ... + (γ (x ) ∨ γ (x )) )+ E n F n E n F n E n F n 1 1 2 2 1 λ λ λ ((ω (x ) ∨ ω (x )) + (ω (x ) ∨ ω (x )) + ... + (ω (x ) ∨ ω (x )) ) γ n γ n γ n γ n γ n γ n E F E F E F 1 1 2 2 i2π(ω (x )) i2π(ω (x )) E k F k j j Now as E = F ⇔ μ (x ) = μ (x ) for k = 1, 2, ... , n ⇔ γ (x )e = γ (x )e E k F k E k F k j j i2π(ω (x )) i2π(ω (x )) E F k k j j for k = 1, 2, ... , n ⇔ γ (x ) = γ (x ) and e = e for k = 1, 2, ... , n. Then E k F k j j ! " 1 ⇔ S (E, F) = [1 + 1 + ... + 1] ⇔ S (E, F) = 1. 3. We have ⎡ ⎤ 1 1 λ 1 λ (γ (x ) ∧ γ (x )) + (ω (x ) ∧ ω (x )) E F γ γ 1 j k j k E k F k j=1  j=1 j j ⎣ ⎦ S (E, F) = 1  λ 1  λ (γ (x ) ∨ γ (x )) + (ω (x ) ∨ ω (x )) E k F k γ k γ k j=1 j j j=1 E F k=1 j j ⎡ ⎤ 1 1 λ 1 λ (γ (x ) ∧ γ (x )) + (ω (x ) ∧ ω (x )) F E γ γ 1 j k j k F k E k j=1  j=1 j j ⎣ ⎦ 1  λ 1  λ (γ (x ) ∨ γ (x )) + (ω (x ) ∨ ω (x )) F k E k γ k γ k j=1 j j j=1 F E k=1 j j = S (F, E). Theorem 6: The SM S (E, F) satisfy the following properties (1) 0 ≤ S (E, F) ≤ 1; (2) S (E, F) = 1if and only if E = F; 5 5 (3) S (E, F) = S (F, E). c c λ λ λ Proof: 1. Since (γ (x ) ∧ γ (x )) ∈ [0, 1], (ω (x ) ∧ ω (x )) ∈ [0, 1], (γ (x ) ∨ γ (x )) E F γ γ E F k k k k k k j j E F j j j j (γ (x )∧γ (x )) E F k k j j ∈ [0, 1], (ω (x ) ∨ ω (x )) ∈ [0, 1] this implies that α ∈ [0, 1] and γ k γ k cc E F λ j j (γ (x )∨γ (x )) E k F k j j 58 T. MAHMOOD ET AL. (ω (x )∧ω (x )) γ γ k k E F j j β ∈ [0, 1]. Thus for k = 1wehave cc (ω (x )∨ω (x )) γ γ k k E F j j % & (ω (x ) ∧ ω (x )) (γ (x ) ∧ γ (x )) γ 1 γ 1 1 E F E 1 F 1 j j j j α + β ∈ [0, 1] cc cc λ λ (γ (x ) ∨ γ (x )) (ω (x ) ∨ ω (x )) E 1 F 1 γ 1 γ 1 j j E F j=1 j j For k = 2 % & (ω (x ) ∧ ω (x )) (γ (x ) ∧ γ (x )) γ 2 γ 2 1 E F E 2 F 2 j j j j α + β ∈ [0, 1] cc cc λ λ (γ (x ) ∨ γ (x )) (ω (x ) ∨ ω (x )) E 2 F 2 γ 2 γ 2 j j E F j=1 j j By doing this process we obtain ⎛ ⎞ % & (ω (x ) ∧ ω (x )) (γ (x ) ∧ γ (x )) γ k γ k 1 E F E k F k j j j j ⎝ ⎠ α + β ∈ n[0, 1] cc cc λ λ (γ (x ) ∨ γ (x )) (ω (x ) ∨ ω (x )) E k F k γ k γ k j j E F j=1 j j k=1 ⎛ ⎞ % & (ω (x ) ∧ ω (x )) (γ (x ) ∧ γ (x )) γ k γ k 1 E F E F k k j j j j ⎝ ⎠ ⇒ 0 ≤ α + β ≤ n cc cc λ λ (γ (x ) ∨ γ (x )) (ω (x ) ∨ ω (x )) E k F k γ k γ k j j E F j=1 j j k=1 ⎡ ⎛ ⎞ ⎤ % & (ω (x ) ∧ ω (x )) (γ (x ) ∧ γ (x )) γ k γ k 1 1 E F E F k k j j j j ⎣ ⎝ ⎠ ⎦ ⇒ 0 ≤ α + β ≤ 1 cc cc λ λ (γ (x ) ∨ γ (x )) (ω (x ) ∨ ω (x )) E F γ γ k k k k j j E F j=1 j j k=1 ⎡ ⎛ ⎞ ⎤ % & (ω (x ) ∧ ω (x )) (γ (x ) ∧ γ (x )) γ k γ k 1 1 E F E k F k j j j j ⎣ ⎝ ⎠ ⎦ ⇒ 0 ≤ α + β ≤ 1 cc cc λ λ (γ (x ) ∨ γ (x )) (ω (x ) ∨ ω (x )) E k F k γ k γ k j j E F j=1 j j k=1 ⇒ 0 ≤ S (E, F) ≤ 1. 2. By definition 7 we have ⎡ ⎛ ⎞ ⎤ 1 % & λ λ (ω (x ) ∧ ω (x )) (γ (x ) ∧ γ (x )) γ k γ k 1 1 E k F k E F j j j j ⎣ ⎝ ⎠ ⎦ S (E, F) = α + β cc cc λ λ (γ (x ) ∨ γ (x )) (ω (x ) ∨ ω (x )) E k F k γ k γ k j j E F j=1 k=1 j j ⎡ ⎛ λ λ λ (γ (x ) ∧ γ (x )) + (γ (x ) ∧ γ (x )) + ... + (γ (x ) ∧ γ (x )) 1 1 E k F k E k F k E k F k 5 1 1 2 2 ⎣ ⎝ ⇔ S (E, F) = α cc λ λ λ (γ (x ) ∨ γ (x )) + (γ (x ) ∨ γ (x )) + ... + (γ (x ) ∨ γ (x )) E k F k E k F k E k F k 1 1 2 2 k=1 j=1 && λ λ λ λ (ω (x ) ∧ ω (x )) + (ω (x ) ∧ ω (x )) + ... + (ω (x ) ∧ ω (x )) γ k γ k γ k γ k γ k γ k E F E F E F 1 1 2 2 + β cc λ λ λ (ω (x ) ∨ ω (x )) + (ω (x ) ∨ ω (x )) + ... + (ω (x ) ∨ ω (x )) γ k γ k γ k γ k γ k γ k E F E F E F 1 1 2 2 ⎡ ⎡ ⎡ λ λ λ 1 1 (γ (x ) ∧ γ (x )) + (γ (x ) ∧ γ (x )) + ... + (γ (x ) ∧ γ (x )) E 1 F 1 E 1 F 1 E 1 F 1 5 1 1 2 2 ⎣ ⎣ ⎣ ⇔ S (E, F) = α cc λ λ λ (γ (x ) ∨ γ (x )) + (γ (x ) ∨ γ (x )) + ... + (γ (x ) ∨ γ (x )) E 1 F 1 E 1 F 1 E 1 F 1 1 1 2 2 j=1 λ λ λ (ω (x ) ∧ ω (x )) + (ω (x ) ∧ ω (x )) + ... + (ω (x ) ∧ ω (x )) γ 1 γ 1 γ 1 γ 1 γ 1 γ 1 E F E F E F 1 1 2 2 +β cc λ λ λ (ω (x ) ∨ ω (x )) + (ω (x ) ∨ ω (x )) + ... + (ω (x ) ∨ ω (x )) γ 1 γ 1 γ 1 γ 1 γ 1 γ 1 E F E F E F 1 1 2 2 λ λ λ 1 (γ (x ) ∧ γ (x )) + (γ (x ) ∧ γ (x )) + ... + (γ (x ) ∧ γ (x )) E 2 F 2 E 2 F 2 E 2 F 2 1 1 2 2 + α cc λ λ λ (γ (x ) ∨ γ (x )) + (γ (x ) ∨ γ (x )) + ... + (γ (x ) ∨ γ (x )) E 2 F 2 E 2 F 2 E 2 F 2 1 1 2 2 j=1 λ λ λ (ω (x ) ∧ ω (x )) + (ω (x ) ∧ ω (x )) + ... + (ω (x ) ∧ ω (x )) γ 2 γ 2 γ 2 γ 2 γ 2 γ 2 E F E F E F 1 1 2 2 +β + ... cc λ λ λ (ω (x ) ∨ ω (x )) + (ω (x ) ∨ ω (x )) + ... + (ω (x ) ∨ ω (x )) γ 2 γ 2 γ 2 γ 2 γ 2 γ 2 E F E F E F 1 1 2 2   FUZZY INFORMATION AND ENGINEERING 59 λ λ λ 1 (γ (x ) ∧ γ (x )) + (γ (x ) ∧ γ (x )) + ... + (γ (x ) ∧ γ (x )) E n F n E n F n E n F n 1 1 2 2 + α cc λ λ λ (γ (x ) ∨ γ (x )) + (γ (x ) ∨ γ (x )) + ... + (γ (x ) ∨ γ (x )) E n F n E n F n E n F n 1 1 2 2 j=1 j=1 &&& λ λ λ (ω (x ) ∧ ω (x )) + (ω (x ) ∧ ω (x )) + ... + (ω (x ) ∧ ω (x )) γ n γ n γ n γ n γ n γ n E F E F E F 1 1 2 2 +β cc λ λ λ (ω (x ) ∨ ω (x )) + (ω (x ) ∨ ω (x )) + ... + (ω (x ) ∨ ω (x )) γ n γ n γ n γ n γ n γ n E F E F E F 1 1 2 2 i2π(ω (x )) i2π(ω (x )) E k F k j j Now as E = F ⇔ μ (x ) = μ (x ) for k = 1, 2, ... , n ⇔ γ (x )e = γ (x )e E k F k E k F k j j i2π(ω (x )) i2π(ω (x )) E F k k j j for k = 1, 2, ... , n ⇔ γ (x ) = γ (x ) and e = e for k = 1, 2, ... , n.and E k F k j j α , β ∈ [0, 1] such that α + β = 1. Then cc cc cc cc ⇔ S (E, F) = 1. 3. We have ⎡ ⎛ ⎞ ⎤ % & (ω (x ) ∧ ω (x )) (γ (x ) ∧ γ (x )) γ k γ k 1 1 E F E F k k j j j j ⎣ ⎝ ⎠ ⎦ S (E, F) = α + β cc cc λ λ (γ (x ) ∨ γ (x )) (ω (x ) ∨ ω (x )) E F γ γ k k k k j j E F j=1 j j k=1 ⎡ ⎛ ⎞ ⎤ % & (ω (x ) ∧ ω (x )) (γ (x ) ∧ γ (x )) γ k γ k 1 1 F E F E k k j j j j ⎣ ⎝ ⎠ ⎦ = α + β cc cc λ λ (γ (x ) ∨ γ (x )) (ω (x ) ∨ ω (x )) F E γ γ k k k k j j F E j=1 j j k=1 = S (F, E). Remark 3: If λ = 1 then without exponential based generalized SMs become # $ ⎡ ⎤ 1 1 1 − |γ (x ) − γ (x )|∨ |ω (x ) − ω (x )| E F γ γ j k j k E k F k 1  j=1  j=1 j j ⎣ ⎦ S (E, F) = # $ 1 1 1 + |γ (x ) − γ (x )|∨ |ω (x ) − ω (x )| k=1 E k F k γ k γ k j=1 j j j=1 E F j j 1  1 (γ (x ) ∧ γ (x )) + (ω (x ) ∧ ω (x )) E k F k γ k γ k 1 j=1 j j j=1 E F j j S (E, F) = 1 1 (γ (x ) ∨ γ (x )) + (ω (x ) ∨ ω (x )) E k F k γ k γ k j j E F j=1  j=1 k=1 j j ⎛ ⎞ % & (ω (x ) ∧ ω (x )) (γ (x ) ∧ γ (x )) γ k γ k 1 1 E F E F j k j k j j ⎝ ⎠ S (E, F) = α + β cc cc n  (γ (x ) ∨ γ (x )) (ω (x ) ∨ ω (x )) E F γ γ k k k k j j E F j j j=1 k=1 Now we defined exponential based weighted generalized SMs and without exponential based weighted generalized SMs. Definition 12: Let E and F be two CHFS on X. Then the exponential based weighted generalized SM is calculated as ⎡ ⎡   ⎤ ⎤ 1 λ 1 λ 1− |γ (x )−γ (x )| ∨ |ω (x )−ω (x )| E k F k γ k γ k j j E F j j ⎢ ⎢ ⎥ ⎥ j=1 j=1 S (E, F) = w 2 − 1 ⎣ ⎣ ⎦ ⎦ k=1 60 T. MAHMOOD ET AL. where λ> 0, and ∨ described the maximum operation and w ∈ [0, 1] be the weight of each element x for k = 1, 2, 3, .., n such that w = 1. k k k=1 Remark 4: If λ = 1 then the exponential based weighted generalized SM becomes ⎡   ⎤ 1 1 1− |γ (x )−γ (x )|∨ |ω (x )−ω (x )| E k F k γ k γ k E F j j j j ⎢ ⎥ j=1 j=1 S (E, F) = w 2 − 1 ⎣ ⎦ k=1 Definition 13: Let E and F be two CHFS on X. Then we can also calculate the exponential based weighted generalized SM as follows ⎡ ⎡ ⎤ ⎤ 1 λ 1 λ 1− |γ (x )−γ (x )| + |ω (x )−ω (x )| E k F k γ k γ k j j  E F ⎢ ⎢ j j ⎥ ⎥ j=1 j=1 S (E, F) = w 2 − 1 ⎣ k ⎣ ⎦ ⎦ k=1 where λ> 0. Remark 5: If λ = 1 then the exponential based weighted generalized SM becomes ⎡ ⎤ 1 1 1− |γ (x )−γ (x )|+ |ω (x )−ω (x )| γ γ E k F k k k j j  E F ⎢ j j ⎥ j=1 j=1 S (E, F) = w 2 − 1 ⎣ ⎦ k=1 Definition 14: Let E and F be two CHFSs on X. Then without exponential based weighted generalized SMs are calculated as follows # $ 1 ⎡ ⎡ ⎤ ⎤ 1  1 λ λ 1 − |γ (x ) − γ (x )| ∨ |ω (x ) − ω (x )| E F γ γ j k j k E k F k j=1  j=1 j j ⎣ ⎣ ⎦ ⎦ # $ S (E, F) = w 1  1 λ λ 1 + |γ (x ) − γ (x )| ∨ |ω (x ) − ω (x )| E F γ γ k=1 j k j k E k F k j=1  j=1 j j ⎡ ⎤ 1  1 n λ λ (γ (x ) ∧ γ (x )) + (ω (x ) ∧ ω (x )) E F γ γ k k k k j=1 j j  j=1 E F j j ⎣ ⎦ S (E, F) = w 1  λ 1  λ (γ (x ) ∨ γ (x )) + (ω (x ) ∨ ω (x )) E k F k γ k γ k j=1 j j j=1 E F k=1   j j ⎡ ⎛ ⎞ ⎤ 1 % & (ω (x ) ∧ ω (x )) (γ (x ) ∧ γ (x )) γ γ 1 E k F k E k F k j j j j ⎣ ⎝ ⎠ ⎦ S (E, F) = w α + β k cc cc w λ λ (γ (x ) ∨ γ (x )) (ω (x ) ∨ ω (x )) E k F k γ k γ k j j E F j=1 j j k=1 where λ> 0and α , β ∈ [0, 1] such that α + β = 1. cc cc cc cc Remark 6: If λ = 1 then without exponential based weighted generalized SMs become # $ ⎡ ⎤ 1 1 1 − |γ (x ) − γ (x )|∨ |ω (x ) − ω (x )| E k F k γ k γ k j j E F j=1  j=1 j j ⎣ ⎦ S (E, F) = w # $ 1 1 1 + |γ (x ) − γ (x )|∨ |ω (x ) − ω (x )| E F γ γ k=1 k k k k j=1 j j  j=1 E F j j ⎡ ⎤ 1 1 (γ (x ) ∧ γ (x )) + (ω (x ) ∧ ω (x )) E k F k γ k γ k j=1 j j j=1 E F j j ⎣ ⎦ S (E, F) = w 1 1 (γ (x ) ∨ γ (x )) + (ω (x ) ∨ ω (x )) E k F k γ k γ k j=1 j j j=1 E F k=1   j j FUZZY INFORMATION AND ENGINEERING 61 ⎛ ⎞ % & (ω (x ) ∧ ω (x )) γ γ (γ (x ) ∧ γ (x )) k k 1 E k F k E F j j j j ⎝ ⎠ S (E, F) = w α + β k cc cc (γ (x ) ∨ γ (x )) (ω (x ) ∨ ω (x )) E F γ γ j k j k E k F k j j k=1 j=1 5. Application In this portion, the SMs and WSMs are applied to two cases which are pattern recognition and medical diagnosis. We evaluate the performance of the SMs in dealing with different practical world problems. 5.1. Pattern Recognition Example 3: Let X = {x , x , x , x } be a set and four know patterns E (j = 1, 2, 3, 4) which 1 2 3 4 j are given in the form of CHFSs as follows i2π(1) i2π(0.1) i2π(0.6) i2π(0.3) i2π(0.2) (x , 0.9e , 0.1e ), (x , 0.4e , 0.7e , 0.5e ), 1 2 E = i2π(0.6) i2π(0.8) i2π(0.7) (x , 0.2e ), (x , 0.3e , 0.2e ) 3 4 i2π(0.5) i2π(0.4) i2π(0.5) (x , 0.3e ), (x , 0.6e ,1e ), 1 2 E = i2π(0.8) i2π(0.7) i2π(0.6) i2π(0.1) (x , 0.2e , 0.2e ), (x , 0.9e , 0.7e ) 3 4 i2π(0.3) i2π(0.1) i2π(0.2) i2π(0.6) (x , 0.6e ), (x , 0.2e , 0.4e , 0.3.e ), 1 2 E = i2π(0.1) i2π(0.2) i2π(0.8) i2π(0.4) (x , 0.8e ,1e ), (x , 0.5e , 0.8e , ) 3 4 i2π(0.9) i2π(1) i2π(0.5) i2π(0.2) i2π(0.4) (x , 0.3e ,1e , 0.9e ), (x , 0.4.e , 0.7e ), 1 2 E = i2π(1) i2π(0.2) i2π(0.6) (x , 0.1e ), (x , 0.2e , 0.9e ) 3 4 Next let an unknown pattern which need to be identify i2π(0.1) i2π(0.5) i2π(0.6) i2π(0.5) (x , 0.3e , 0.9e ), (x , 0.5e , 0.6e ), 1 2 E = i2π(0.9) i2π(0.7) i2π(0.4) i2π(1) (x , 0.4e ), (x , 0.8e , 0.2e ,1e ), 3 4 In Table 1 we calculated the proposed SMs from E to E (j = 1, 2, 3, 4). The motive of this issue is to find that the unknown pattern E belong to which of the Pattern E (j = 1, 2, 3, 4). From the calculation stated in Table 1, we obtained the following consequences 1 2 (1) The similarity degree between E and E is a massive one as got by SMs, SM ,SM ,and c c 3 1 2 SM . So by the principle of the maximum degree of similarity the SMs SM ,SM and c c c SM allot the unknown pattern E to the pattern E . 4 5 (2) The similarity degree between E and E is a massive one as got by SMs, SM ,and SM . c c 4 5 So by the principle of the maximum degree of similarity the SMs SM and SM allot the c c unknown pattern E to the pattern E . If we let the weight of each element x (k = 1, 2, 3, 4) are 0.1, 0.2, 0.3 and 0.4 respectively. Then the calculation of proposed SMs are stated in Table 2. From the calculation stated in Table 2, we obtained the following consequences 62 T. MAHMOOD ET AL. Table 1. Calculation of proposed SMs for λ = 1and (α , β ) = (0.5, 0.5). cc cc Similarity measures E E E E Ranking 1 2 3 4 S (E, E ) 0.4928 0.5109 0.3218 0.4503 E ≥ E ≥ E ≥ E j 2 1 4 3 S (E, E ) 0.5316 0.5686 0.4309 0.555 E ≥ E ≥ E ≥ E j 2 4 1 3 S (E, E ) 0.4175 0.4281 0.2536 0.3801 E ≥ E ≥ E ≥ E j 2 1 4 3 S (E, E ) 0.4465 0.4448 0.2529 0.5003 E ≥ E ≥ E ≥ E j 4 2 1 3 S (E, E ) 0.4366 0.4259 0.2516 0.4667 E ≥ E ≥ E ≥ E j 4 1 2 3 Table 2. Calculation of proposed WSMs for Example (3) based on λ = 1and (α , β ) = (0.5, 0.5). cc cc Similarity Measures E E E E Ranking 1 2 3 4 S (E, E ) 0.5164 0.4902 0.3065 0.4259 E ≥ E ≥ E ≥ E j 1 2 4 3 S (E, E ) 0.5543 0.5537 0.4157 0.5253 E ≥ E ≥ E ≥ E j 1 2 4 3 S (E, E ) 0.4368 0.4082 0.2405 0.3594 E ≥ E ≥ E ≥ E j 1 2 4 3 S (E, E ) 0.4717 0.4514 0.2772 0.4725 E ≥ E ≥ E ≥ E j 4 1 2 3 S (E, E ) 0.4588 0.4367 0.2776 0.432 E ≥ E ≥ E ≥ E j 1 2 4 3 Figure 2. Graphical representation of established SMs for Example 3. 1 2 (1) The similarity degree between E and E is a massive one as got by SMs, SM ,SM , c c w w 3 5 1 SM and SM . So by the principle of the maximum degree of similarity the SMs SM , c c c w w w 2 3 5 SM ,SM ,andSM allot the unknown pattern E to the pattern E . c c c w w w (2) The similarity degree between E and E is a massive one as got by SM, SM .Sobythe principle of the maximum degree of similarity the SM SM allot the unknown pattern E to the pattern E . The ranking of the proposed SMs and WSMs are also stated in Tables 1 and 2 respectively. The graphical representation of the proposed SMs is shown in Figure 1 andproposedWSMs are shown in Figure 2. FUZZY INFORMATION AND ENGINEERING 63 Table 3. Calculation of proposed SMs for λ = 1and (α , β ) = (0.5, 0.5). cc cc Similarity measures D D D D Ranking 1 2 3 4 S (P,D ) 0.5396 0.5083 0.4023 0.2781 D ≥ D ≥ D ≥ D j 1 2 3 4 S (P,D ) 0.6447 0.5759 0.4881 0.3554 D ≥ D ≥ D ≥ D j 1 2 3 4 S (P,D ) 0.4588 0.4335 0.336 0.2173 D ≥ D ≥ D ≥ D j 1 2 3 4 S (P,D ) 0.5249 0.3981 0.2614 0.1474 D ≥ D ≥ D ≥ D j 1 2 3 4 S (P,D ) 0.5713 0.3956 0.2675 0.1488 D ≥ D ≥ D ≥ D j 1 2 3 4 5.2. Medical Diagnosis The symptoms of different diseases are different. The medical diagnosis depends on the victim’s symptoms which show what type of disease a victim has. The multiple symptoms of a victim represent a symptom set and a set of diseases can represent by different diseases. Example 4: Let a set of diagnoses D = {D (Coronovirous),D (Pneumonia),D (Flu),D 1 2 3 4 (Chestproblem)} and a set of symptoms X ={x (shortofbreath), x (Fever), x (cough), 1 2 3 x (chestpain)}. The victim’s symptoms can be showed in the form of CHFSs as below i2π(1) i2π(1) i2π(0.5) i2π(0.8) i2π(0.3) i2π(0.4) (x , 0.9e , 0.5e , 0.5e ), (x , 0.5e , 0.7e , 0.4e ), 1 2 P = i2π(0.6) i2π(0.9) i2π(0.3) (x , 0.2e , 0.8e ), (x , 0.1e ) 3 4 The symptoms of each disease D (j = 1, 2, 3, 4) canbeshowedinCHFSs asbelow i2π(0.5) i2π(0.8) i2π(0.7) i2π(0.9) (x , 1e , 0.8e ), (x , 0.5e , 0.6e ), 1 2 D (coronovirous) = i2π(0.7) i2π(1) i2π(0.7) i2π(0.1) (x , 0.8e , 0.6e , 0.9e ), (x , 0.1e ), 3 4 i2π(0.2) i2π(0.7) i2π(0.9) (x , 0.1e ), (x , 0.6e , 0.4e ), 1 2 D (Pneumonia) = i2π(0.6) i2π(0.6) i2π(0.4) i2π(0.2) (x , 0.4e , 0.5e ), (x , 0.3e , 0.4e ) 3 4 i2π(0.0) i2π(0.2) i2π(0.5) (x , 0.1e ), (x , 0.3e , 0.2e ), 1 2 D (Flu) = i2π(0.8) i2π(0.7) i2π(0.6) i2π(0.2) i2π(0.4) (x , 1e , 0.6e , 0.9e ), (x , 0.1e , 0.2e ) 3 4 i2π(0.1) i2π(0.2) i2π(0.2) i2π(0.2) (x , 0.2e , 0.3e ), (x , 0.1.e , 0.0e ), 1 2 D (Chestpain) = i2π(0.3) i2π(0.9) i2π(0.7) i2π(0.6) (x , 0.1e ), (x , 1e , 0.9e , 0.5e ) 3 4 In Table 3 we calculated the proposed SMs from P to D (j = 1, 2, 3, 4). The motive of this issue is to know about the disease of the victim that what disease a victim has in the above four diseases D (j = 1, 2, 3, 4). From the calculation stated in Table 3, we obtained that the similarity degree between P and D is a massive one as got by all SMs. So by the principle of the maximum degree of similarity, we can say that a victim has coronavirus. If we let the weight of each element x (k = 1, 2, 3, 4) are 0.1, 0.2, 0.3 and 0.4 respectively. Then the calculation of proposed SMs are stated in Table 4. From the calculation stated in Table 4, we obtained the following consequences 2 4 (1) The similarity degree between P and D is a massive one as got by SMs, SM ,SM , c c w w 5 2 4 and SM . So by the principle of the maximum degree of similarity the SMs SM ,SM c c c w w w and SM give us that a victim has coronavirus. w 64 T. MAHMOOD ET AL. Table 4. Calculation of proposed WSMs for Example (4) based on λ = 1and (α , β ) = (0.5, 0.5). cc cc Similarity measures D D D D Ranking 1 2 3 4 S (P,D ) 0.5633 0.5821 0.4749 0.2692 D ≥ D ≥ D ≥ D j 2 1 3 4 S (P,D ) 0.6802 0.6525 0.5662 0.3376 D ≥ D ≥ D ≥ D j 1 2 3 4 S (P,D ) 0.4852 0.5013 0.4022 0.2099 D ≥ D ≥ D ≥ D j 2 1 3 4 cw S (P,D ) 0.5106 0.4423 0.3089 0.1332 D ≥ D ≥ D ≥ D j 1 2 3 4 S (P,D ) 0.5812 0.4433 0.3163 0.1327 D ≥ D ≥ D ≥ D j 1 2 3 4 Figure 3. Graphical representation of established WSMs for Example 3. (2) The similarity degree between P and D is a massive one as got by SMs, SM ,and 3 1 3 SM . So by the principle of the maximum degree of similarity the SMs SM ,andSM c c c w w w provide that a victim has pneumonia. The ranking of the proposed SMs and WSMs are also stated in Tables 3 and 4 respectively. The graphical representation of the proposed SMs is shown in Figure 2 andproposedWSMs are shown in Figure 3. 6. Comparison Our goal to expand the new SMs is to deal with new kinds of data such as CHFSs and exist- ing data CFSs, HFSs, and FSs. In this portion, we expressed the benefits of proposed SMs by comparing with existing SMs. The geometrical representations of the information’s of example 4 and example 5, are discussed in Figures 4–7. Example 5: Let X = {x , x , x , x } be a set and four know patterns E (j = 1, 2, 3, 4) which 1 2 3 4 j are given in the form of HFSs as follows (x , {0.1, 0.4, 0.6}), (x , {0.4, 1}), 1 2 E = (x , {0.7, 0.8}), (x , {0.2}), 3 4 FUZZY INFORMATION AND ENGINEERING 65 Figure 4. Graphical representation of established WSMs for Example 4. Figure 5. Graphical representation of established WSMs for Example 4. (x , {0.3, 0.5, 0.8}), (x , {0.4., 0.6}), 1 2 E = (x , {0.1, 0.3}), (x , {0.2, 0.4, 0.7}) 3 4 (x , {0.6, 0.8}), (x , 0.1, 0.2, 0.6), 1 2 E = (x , {0.7}), (x , {0.4, 0.5}) 3 4 (x , {0.1, 0.3}), (x , {0.4, 0.8}), 1 2 E = (x , {0.4, 0.7, 0.8}), (x , {0.4}), 3 4 66 T. MAHMOOD ET AL. Figure 6. Graphical representation of the comparison of the establish SMs with existing SMs for Example 5. Figure 7. Graphical representation of the comparison of the establish SMs with existing SMs for Example 3. Next let an unknown pattern which need to be identify (x , {0.2, 0.3, 0.6}), (x , {0.8, 1}), 1 2 E = (x , {0.5, 0.8}), (x , {1}) 3 4 Nowwehavethat e = 1 then the data given in the HFSs converted into the CHFSs as follows i2π(0.0) i2π(0.0) i2π(0.0) i2π(0.0) i2π(0.0) (x , 0.1e , 0.4e , 0.6e ), (x , 0.4e , 0.1e ), 1 2 E = i2π(0.0) i2π(0.0) i2π(0.0) (x , 0.7e , 0.8e ), (x , 0.2.e ) 3 4 FUZZY INFORMATION AND ENGINEERING 67 Table 5. Comparison between Proposed SMs and Existing SMs for Example (5) and λ = 1, (α , β ) = cc cc (1, 0). Method Score value Ranking Xu and Xia [25] s (E, E ) =0.7083, s (E, E ) = 0.6042s (E, E ) = 0.6083, s (E, E ) = 0.7 E ≥ E ≥ E ≥ E g 1 g 2 g 3 g 4 1 4 3 2 Zeng et al. [32] s (E, E ) = 0.8375, s (E, E ) = 0.6792s (E, E ) = 0.6167, s (E, E ) = 0.7979 E ≥ E ≥ E ≥ E h 1 h 2 h 3 h 4 1 4 2 3 Tang et al. [33] s (E, E ) = 0.6319, s (E, E ) = 0.3652s (E, E ) = 0.2333,s (E, E ) = 0.5541 E ≥ E ≥ E ≥ E 1 2 3 4 1 4 2 3 h h h h 1 1 1 1 Proposed SM S (E, E ) =0.6664, S (E, E ) = 0.5127S (E, E ) =0.3563, S (E, E ) = 0.4819 E ≥ E ≥ E ≥ E 1 2 3 4 1 4 2 3 c c c c 2 2 2 2 Proposed SM S (E, E ) = 0.6664, S (E, E ) =0.5127 S (E, E ) = 0.3563, S (E, E ) = 0.5582 E ≥ E ≥ E ≥ E 1 2 4 5 1 4 2 3 c c c c 3 3 3 3 Proposed SM S (E, E ) = 0.6177, S (E, E ) = 0.4366S (E, E ) = 0.2833, S (E, E ) = 0.4775 E ≥ E ≥ E ≥ E 1 2 3 4 1 4 2 3 c c c c 4 4 4 4 Proposed SM S (E, E ) = 0.6694, S (E, E ) = 0.4115S (E, E ) = 0.2438, S (E, E ) = 0.4885 E ≥ E ≥ E ≥ E 1 2 3 4 1 4 2 3 c c c c 5 5 5 5 Proposed SM S (E, E ) = 0.6694, S (E, E ) = 0.4115S (E, E ) = 0.2438, S (E, E ) = 0.4885 E ≥ E ≥ E ≥ E c 1 c 2 c 3 c 4 1 4 2 3 Table 6. Comparison between proposed SMs with existing SMs for Example (3) and λ = 1. Method Score value Ranking Xu and Xia [25] Failed Failed Zeng et al. [32] Failed Failed Tang et al. [33] Failed Failed 1 1 1 1 Proposed SM S (E, E ) = 0.4928,S (E, E ) =0.5109 S (E, E ) = 0.3218, S (E, E ) = 0.4503 E ≥ E ≥ E ≥ E 1 2 3 4 2 1 4 3 c c c c 2 2 2 2 Proposed SM S (E, E ) = 0.5316,S (E, E ) = 0.5686S (E, E ) = 0.4309, S (E, E ) = 0.555 E ≥ E ≥ E ≥ E 1 2 3 4 2 4 1 3 c c c c 3 3 3 3 Proposed SM S (E, E ) = 0.4175,S (E, E ) = 0.4281S (E, E ) = 0.2536, S (E, E ) = 0.3801 E ≥ E ≥ E ≥ E 1 2 3 4 2 4 1 3 c c c c 4 4 4 4 Proposed SM S (E, E ) = 0.4465,S (E, E ) = 0.4448S (E, E ) = 0.2529, S (E, E ) = 0.5003 E ≥ E ≥ E ≥ E 1 2 3 4 4 2 1 3 c c c c 5 5 5 5 Proposed SM S (E, E ) = 0.4366,S (E, E ) = 0.4259S (E, E ) = 0.2516, S (E, E ) = 0.4667 E ≥ E ≥ E ≥ E 1 2 3 4 4 1 2 3 c c c c i2π(0.0) i2π(0.0) i2π(0.0) i2π(0.0) i2π(0.0) (x , 0.3e , 0.5e , 0.8e ), (x , 0.4e , 0.6.e ), 1 2 E = i2π(0.0) i2π(0.0) i2π(0.0) i2π(0.0) i2π(0.0) (x , 0.1e , 0.3e ), (x , 0.2e , 0.4e , 0.7e ) 3 4 i2π(0.0) i2π(0.0) i2π(0.0) i2π(0.0) i2π(0.0) (x , 0.6e , 0.8e ), (x , 0.1e , 0.2e , 0.6e ), 1 2 E = i2π(0.0) i2π(0.0) i2π(0.0) (x , 0.7e ), (x , 0.4e , 0.5e ) 3 4 i2π(0.0) i2π(0.0) i2π(0.0) i2π(0.0) i2π(0.0) (x , 0.9.e ,1.e ,1.e ), (x , 0.1.e , 0.3.e ), 1 2 E = iπ(0.0) i2π(0.0) i2π(0.0) i2π(0.0) (x , 0.4e , 0.7e , 0.8e ), (x , 0.4e ) 3 4 And i2π(0.0) i2π(0.0) i2π(0.0) i2π(0.0) i2π(0.0) (x , 0.2e , 0.3e , 0.6e ), (x , 0.8e ,1e ), 1 2 E = i2π(0.0) i2π(0.0) i2π(0.0) (x , 0.5e , 0.8e ), (x , 1e ) 3 4 From Table 5 we can observe that the data in the form of FSs and HFSs are solvable through existing SMs in the literature. The data in Example 5 is in the form HFSs which we can convert to CHFSs by taking 1 = e and through proposed SMs we get the similarity between E and E (j = 1, 2, 3, 4) as shown in Table 5. But what about the data in the form of CFSs and CHFSs, these type of data are unsolvable through the existing SMs as shown in Table 6. The data in Example 3 are in the form of CHFSs so we can find the similarity between E and E (j = 1, 2, 3, 4) through proposed SMs. This means that our proposed SMs are the extension of the existing SMs. Through the proposed SMs we can find the similarity between FS, HFS, CFS, and CHFSs. The ranking of Example 5 is given in Table 5. The similar- ity degree between E and E is a massive one as got by all SMs in Example 5. The ranking of Example 3 given in Table 6 is slightly different than ranking given in Table 5. The simi- 1 2 3 larity degree between E and E is a massive one as got by SMs, SM ,SM ,andSM and the c c c 4 5 similarity degree between E and E is a massive one as got by SMs, SM and SM . c c 68 T. MAHMOOD ET AL. 7. Conclusion SMs are utilized to inspect the variation between the two objects. The purpose of this arti- cle is to define the fundamental notion of CHFSs which is the combination of HFS and CFS and also defined their fundamental properties. We explored SMs for CHFSs. We presented exponential based generalized SMs and without exponential based generalized SMs for CHFSs. We obtained some valuable remarks. Further, the established SMs are used in Pat- tern recognition and medical diagnosis to inspect the practicability and credibility of the established SMs. Furthermore, we solved numerical examples for established SMs to show the supremacy and integrity of the proposed SMs. At last, to assess the trustworthiness of the established SMs we represented the comparison of the established SMs with some existing SMs. Our future work is to explore the application of CHFNs in many other type of researches [34–48]. Disclosure statement No potential conflict of interest was reported by the authors. Notes on contributors Tahir Mahmood is Assistant Professor of Mathematics at Department of Mathematics and Statistics, International Islamic University Islamabad, Pakistan. He received his Ph.D. degree in Mathematics from Quaid-i-Azam University Islamabad, Pakistan in 2012 under the supervision of Professor Dr. Muham- mad Shabir. His areas of interest are Algebraic structures, Fuzzy Algebraic structures and Soft sets. He has more than 65 international publications to his credit and he has also produced 38 MS students. Ubaid ur Rehman, received the M.Sc. degrees in Mathematics from International Islamic University Islamabad, in 2018. Currently, He is a Student of MS in mathematics from International Islamic Univer- sity Islamabad, Pakistan. His research interests include similarity measures, distance measures, fuzzy logic, fuzzy decision making, and their applications. Zeeshan Ali, received the B.S. degrees in Mathematics from Abdul Wali Khan University Mardan, Pak- istan, in 2016. He received has M.S. degrees in Mathematics from International Islamic University Islamabad, Pakistan, in 2018. Currently, He is a Student of Ph.D. in mathematics from International Islamic University Islamabad, Pakistan. His research interests include aggregation operators, fuzzy logic, fuzzy decision making, and their applications. 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Hesitant fuzzy sets. Int J Intell Syst. 2010;25(6):529–539. [25] Xu Z, Xia M. Distance and similarity measures for hesitant fuzzy sets. Inf Sci (Ny). 2011;181(11): 2128–2138. [26] Liao H, Xu Z. Subtraction and division operations over hesitant fuzzy sets. J Intell Fuzzy Syst. 2014;27(1):65–72. [27] Alcantud JCR, Torra V. Decomposition theorems and extension principles for hesitant fuzzy sets. Inf Fusion. 2018;41:48–56. [28] Bisht K, Kumar S. Fuzzy time series forecasting method based on hesitant fuzzy sets. Expert Syst Appl. 2016;64:557–568. [29] Zhang X, Xu Z. Novel distance and similarity measures on hesitant fuzzy sets with applications to clustering analysis. J Intell Fuzzy Syst. 2015;28(5):2279–2296. [30] Alcantud JCR, Giarlotta A. Necessary and possible hesitant fuzzy sets: a novel model for group decision making. Inf Fusion. 2019;46:63–76. [31] Farhadinia B, Herrera-Viedma E. Multiple criteria group decision making method based on extended hesitant fuzzy sets with unknown weight information. Appl Soft Comput. 2019;78:310–323. [32] Zeng W, Li D, Yin Q. Distance and similarity measures between hesitant fuzzy sets and their application in pattern recognition. Pattern Recognit Lett. 2016;84:267–271. [33] Tang X, Peng Z, Ding H, et al. Novel distance and similarity measures for hesitant fuzzy sets and their applications to multiple attribute decision making. J Intell Fuzzy Syst. 2018;34(6):3903–3916. [34] Liu P, Mahmood T, Ali Z. Complex q-Rung Orthopair fuzzy aggregation operators and their applications in multi-attribute group decision making. Information. 2020;11(1):5. [35] Liu P, Ali Z, Mahmood T. A method to multi-attribute group decision-making problem with com- plex q-Rung Orthopair linguistic information based on Heronian mean operators. Int J Comp Intell Sys. 2019;12(2):1465–1496. 70 T. MAHMOOD ET AL. [36] Ullah K, Mahmood T, Ali Z, et al. 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Int J Fuzzy Syst. 2018;20(1):93–103. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Fuzzy Information and Engineering Taylor & Francis

Exponential and non-Exponential Based Generalized Similarity Measures for Complex Hesitant Fuzzy Sets with Applications

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FUZZY INFORMATION AND ENGINEERING 2020, VOL. 12, NO. 1, 38–70 https://doi.org/10.1080/16168658.2020.1779013 ORIGINAL ARTICLE Exponential and non-Exponential Based Generalized Similarity Measures for Complex Hesitant Fuzzy Sets with Applications Tahir Mahmood, Ubaid ur Rehman and Zeeshan Ali Department of Mathematics and Statistics, International Islamic University Islamabad, Islamabad, Pakistan ABSTRACT ARTICLE HISTORY Received 26 March 2020 The purpose of this manuscript is to explore the notion of a com- Revised 30 May 2020 plex hesitant fuzzy set (CHFS), as a generalization of the hesitant Accepted 1 June 2020 fuzzy set (HFS) and complex fuzzy set (CFS) to cope with the uncer- tain and complicated information in the real-world decision. CHFS KEYWORDS contains truth grades in the form of a subset of the unit disc in Complex fuzzy set; complex the complex plane. The operational laws of the explored notion are hesitant fuzzy sets; similarity also described. Further, the exponential based generalized similarity measures measures, without exponential based generalized similarity mea- sures, and their important characteristics are also explored. These similarity measures are applied in the environment of pattern recog- nition and medical diagnosis to evaluate the proficiency and feasi- bility of the established measures. We also solved some numerical examples using the established measures. To examine the reliability and validity of the proposed measures by comparing it with existing measures. The advantages, comparative analysis, and graphical rep- resentation of the explored measures and existing measures are also discussed in detail. 1. Introduction The theory of fuzzy set (FS) was firstly explored by Zadeh [1] in 1965 which successfully applied in different fields. FS contains one function, called truth grade, belonging to the unit interval. FS has gained extensive achievement and various researchers have utilized it in the environment of medical diagnosis [2–4], pattern recognition [5], decision making [6], and clustering algorithm. Moreover, the concept of interval-valued FS (IVFS) was estab- lished by Zadeh [7], which contains the grade of truth in the form of some closed subinterval of the unit interval. Couso et al. [8] defined a formal relational study of similarity and dissim- ilarity measures between FSs. The SM between FSs and between elements is described by Lee-Kwang et al. [9]. Pramanik and Mondal [10] presented weighted fuzzy SM based on tan- gent function and its discussed application to medical diagnosis. Some new SMs on FSs are defined by Wang [11]. Kwon [12] also defined SM based on FSs. A new approach to fuzzy distance measure and SM between generalized fuzzy numbers was described by Guha and CONTACT Tahir Mahmood tahirbakhat@iiu.edu.pk © 2020 The Author(s). Published by Taylor & Francis Group on behalf of the Fuzzy Information and Engineering Branch of the Operations Research Society of China & Operations Research Society of Guangdong Province. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/ licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. FUZZY INFORMATION AND ENGINEERING 39 Chakraborty [13]. Kakati [14] explored a note on the new similarity measure for FSs. Some SMs based on FSs are presented by Hesamian [15] to find about the closeness between two objects. Various researchers arise a question, what will happen when the range of FS changes to complex numbers form a unit disc in a complex plan instead of a real number. Ramot et al. [16] introduced the idea of complex FS (CFS), which contains the truth grade in the form of a complex number by a member of a unit disc in the complex plane. CFS deals with two dimensions in a single set. CFS is a powerful procedure to illustrate the belief of a human being in the formation of grades. Bi et al. [17] described complex fuzzy arithmetic aggregation operators. Adaptive image restoration by a novel neuro-fuzzy approach using CFSs is presented by Li [18]. A systematic review of CFSs and logic is described by Yazdan- bakhsh and dick [19]. Dai [20] wrote some comments on complex fuzzy logic. Jun and Xin [21] applied CFSs to BCK/BCI-algebra. The orthogonality between CFSs and its application to signal detection is described by Hu et al. [22]. Hu et al. [23] also defined distances of CFSs and continuity of CF operations. In the real decision making procedure, it is hard to set up the membership degree of FS due to the insufficiency of knowledge or data, hesitation, and many other reasons. To over- come such kind of issues Torra [24] investigated the notion of the hesitant fuzzy set (HFS) which contains the grade of truth in the form of a subset of the unit interval. HFS is the generalization of FS to deal with uncertain and more complicated information in real deci- sion theory. Xu and Xia [25] explored distance and SMs for HFSs. Liao and Xu [26] described subtraction and division operation over HFSs. Decomposition theorems and extension prin- ciples for HFSs are explored by Alcantud and Torra [27]. Bishti and Kumar [28] defined fuzzy time series forecasting method based on HFSs. Novel distance and SMs on HFSs with appli- cation to clustering analysis presented by Zhang and Xu [29]. Alcantud and Giarlotta [30] proposed an extension of Torra’s concept of HFSs. Farhadinia and Herrera-Viedma [31] defined multiple criteria group decision-making method based on extended HFSs with unknown weight information. Distance and SMs between HFSs and their application in pattern recognition were stated by Zeng et al. [32]. In real-life problems, we come across many situations where we need to quantify the uncertainty existing in the data to make optimal decisions. Exponential based similarity measures and without exponential based similarity measures are important tools for han- dling uncertain information present in our day-to-day life problems. Different measures, such as similarity, exponential, distance, entropy, and inclusion, process the uncertain infor- mation, and enable us to reach some conclusion. Recently, these measures have gained much attention from many authors due to their wide applications in various fields, such as pattern recognition, medical diagnosis, clustering analysis, and image segment. All the existing approaches of decision-makers, based on exponential based similarity measures and without exponential based similarity measures, in FS, CFS, and HFS theories, deal with membership functions belonging to a unit interval in the form of a subset in the concept of HFS. In CHFS theory, membership degrees are complex-valued and are represented in polar coordinates. These all notions worked effectively, but when a decision-maker faced such kinds of information which contains two-dimensional information in a single-set. For i2π(0.3) i2π(0.6) i2π(0.2) i2π(0.2) instance, 0.9e , 0.7e , 0.3e , 0.1e , then the existing all notions are failed. For coping with such kind of problems, the CHFS is a proficient technique to resolve 40 T. MAHMOOD ET AL. realistic decision problems in the environment of fuzzy set theory. CHFS is more power- ful and more general than existing notions like HFS, CFS, and FS to cope with awkward and complicated information in real-life decisions. Because these all notions are the special cases of the explored CHFS. The advantages of the presented CHFS are discussed below: (1) When we choose the imaginary parts of the CHFS as zero, then the CHFS is reduced into HFS which is in the form of {0.9, 0.7, 0.3, 0.1}. (2) When we choose the CHFS as a singleton set, then the CHFS is reduced into CFS which i2π(0.3) is in the form of 0.9e . (3) When we choose the CHFS as a singleton set and the imaginary parts as zero, then the CHFS is reduced into FS which is in the form of {0.9}. Motivated by the above challenges and keeping the advantages of the CHFS, in this manuscript, some key contributions are made: (1) To explore the novel approach of the complex hesitant fuzzy set (CHFS), which is the generalization of the hesitant fuzzy set (HFS) and complex fuzzy set (CFS) to cope with the uncertain and complicated information in the real-world decision. CHFS contains truth grades in the form of subset of the unit disc in the complex plane. Operational laws of the explored notion are also described and verified with the help of some numerical examples. (2) To present some similarity measures is called exponential based similarity measures, without exponential based similarity measures, generalized similarity measures and their important characteristics are also explored. (3) These similarity measures are utilized in the environment of pattern recognition and medical diagnosis to evaluate the proficiency and feasibility of the established mea- sures. We solve some numerical examples using the established measures. (4) To examine the reliability and validity of the proposed measures by comparing with existing measures. The advantages, comparative analysis, and graphical representa- tion of the explored measures and existing measures are also discussed in detail. The graphical interpretation of the explored works is discussed with the help of Figure 1. The remainder of this manuscript is organized as follows: In Section 2, the notion of FSs, CFSs, HFSs are review. In Section 3, the purpose of this manuscript is to explore the notion of the complex hesitant fuzzy set (CHFS), as a mixture of the hesitant fuzzy set (HFS) and complex fuzzy set (CFS) to cope with the uncertain and complicated information in the real-world decision. CHFS contains truth grades in the form of a subset of the unit disc in a complex plane. The operational laws of the explored notion are also described. In Section 4, the exponential based similarity measures, without exponential based similarity measures, generalized similarity measures, and their important characteristics are also explored. In Section 5, these similarity measures are applied in the environment of pattern recognition and medical diagnosis to evaluate the proficiency and feasibility of the established mea- sures. We solve some numerical examples using the established measures. To examine the reliability and validity of the proposed measures by comparing it with existing measures. FUZZY INFORMATION AND ENGINEERING 41 Figure 1. The geometrical representation of the explored approach. The advantages, comparative analysis, and graphical representation of the explored mea- sures and existing measures are also discussed in detail. The conclusion of this manuscript is discussed in Section 6. 2. Preliminaries In this part of the article, we review basic definitions like FS, CFS, and HFS. Throughout this article X represents a fix set. Definition 1: [1]:AFS E is of the form: E = {(x, μ (x))|x ∈ X} with a condition 0 ≤ μ (x) ≤ 1, where μ (x) represents the grade of truth. Throughout this E E article, the collection of all FSs on X are denoted by FS(X).The pair E = (x, μ (x)) is called fuzzy number (FN). Definition 2: [16]:ACFS E is of the form: E = {(x, μ (x)|x ∈ X} i2π(ω (x)) where μ (x) = γ (x).e represents the complex-valued truth grade in the form of E E i2π(ω (x)) polar coordinate, where γ (x), ω (x) ∈ [0, 1]. Further, the pair E = (x, γ (x).e ) is E γ E called complex fuzzy number (CFN). Definition 3: [24]:AHFS E is of the form: E = {(x, μ (x))|x ∈ X} where μ (x)is the set of different finite values in [0, 1] representing the grade of truth for each element x ∈ X. Further, the pair E = (x, μ (x)) is called hesitant fuzzy number (HFN). E 42 T. MAHMOOD ET AL. Definition 4: [25]: For any two HFSs E and F, the similarity measure S(E, F) satisfies the following conditions: (1) 0 ≤ S(E, F) ≤ 1; (2) S(E, F) = 1 ⇔ E = F; (3) S(E, F) = S(F, E). Definition 5: [25]: For any two HFSs E and F, the distance measure d(E, F) satisfies the following conditions: (1) 0 ≤ d(E, F) ≤ 1; (2) d(E, F) = 1 ⇔ E = F; (3) d(E, F) = d(F, E). From the above analysis, we obtain that the S(E, F) = 1 − d(E, F). 3. Complex Hesitant Fuzzy Sets In this portion, we presented the idea of complex hesitant fuzzy sets (CHFSs) and its some properties. Definition 6: A CHFS E is of the form: E = { (x, μ (x))|x ∈ X} where i2π(ω (x)) E j μ (x) = γ (x).e , j = 1, 2, 3, ... , n E E i2π(ω (x)) i2π(ω (x)) i2π(ω (x)) γ γ γ E 1 E2 En = γ (x).e , γ (x).e , ... , γ (x).e E E E 1 2 n represented the complex-valued truth grade which is subset of unit disc in complex plane i2π(ω (x)) Ej with a condition γ (x), ω (x) ∈ [0, 1]. Further, E = (x, γ (x).e ) is called complex E γ E j E j hesitant fuzzy number (CHFN). i2π(ω (x)) i2π(ω (x)) γ γ E F j j Definition 7: Let E = (x, γ (x).e ) and F = (x, γ (x).e ) be two CHFNs. E F j j Then i2π( 1−ω (x) ) Ej (1) c(γ (x)) = (x, 1 − γ (x) .e ) ; E E i2π(max(ω (x),ω (x))) γ γ E F j j (2) E ∪ F = (x,max(γ (x), γ (x)).e ) ; E F j j i2π(min(ω (x),ω (x))) γ γ E F j j (3) E ∩ F = (x,min(γ (x), γ (x)).e ) . E F j j The notion of CHFS is an extensive powerful technique to cope with uncertain and awk- ward information in realistic decision theory. The CHFS contains the grade of supporting in the form of a subset of the unit disc in the complex plane, whose entities in the form of polar FUZZY INFORMATION AND ENGINEERING 43 coordinates. Basically, the CHFS contains two-dimension information in a single set. The presented CHFS is more general than existing drawbacks, whose detailed and justifications are discussed are below: In Definitions (6) and (7), if we choose the imaginary parts as zero, then the explored notion is converted for HFS, which is presented by Torra [24]. Similarly, if we choose the CHFS as a singleton set, then the CHFS is converted for CFS, which is presented by Ramot et al. [16]. Further, if we choose the CHFs as a singleton set and the imaginary part is zero, then the CHFS is converted for FS, which is explored by Zadeh [1]. Due to its structure, it makes powerful and proficient to cope with uncertain and unreliable information in real decision theory. Example 1: Let i2π(0.3) i2π(0.6) i2π(0.4) i2π(0.5) i2π(0.6) (x , 0.9e , 0.7e ), (x , 0.3e , 0.8e , 0.5e ), 1 2 E = i2π(0.8) i2π(0.5) i2π(0.1) i2π(0.6) (x , 0.6e ), (x , 0.7e , 0.9e , 0.3e ) 3 4 and i2π(0.6) i2π(1) i2π(0.3) (x , 0.8e , 0.1e ), (x , 0.2e ), 1 2 F = , i2π(0.5) i2π(0.8) i2π(0.5) i2π(0.6) i2π(0.2) (x , 0.6e , 0.7e ), (x , 1e , 0.7e , 0.9e ) 3 4 be two CHFSs. Then the operational laws are defined as i2π(0.7) i2π(0.4) i2π(0.6) i2π(0.5) i2π(0.4) 0.1e , 0.3e , 0.7e , 0.2e , 0.5e , (1) E = ; i2π(0.2) i2π(0.5) i2π(0.9) i2π(0.4) 0.4e , 0.3e , 0.1e , 0.7e i2π(0.6) i2π(1) i2π(0.4) i2π(0.5) i2π(0.6) 0.9e , 0.7e , 0.3e , 0.8e , 0.5e , (2) E ∪ F = ; i2π(0.8) i2π(0.8) i2π(0.5) i2π(0.6) i2π(0.6) 0.6e , 0.7e , 1e , 0.9e , 0.9e i2π(0.3) i2π(0.6) i2π(0.3) 0.8e , 0.1e , 0.2e , E ∩ F = . i2π(0.5) i2π(0.5) i2π(0.1) i2π(0.2) 0.6e , 0.7e , 0.7e , 0.3e Theorem 1: Let E,Fand G ∈ CHFS(X) then the following holds (1) c(c(E)) = E (2) i. E ∪ F = F ∪ E ii. E ∩ F = F ∩ E (3) i. (E ∪ F) ∪ G = E ∪ (F ∪ G) ii. (E ∩ F) ∩ G = E ∩ (F ∩ G) (4) E ∪ (F ∩ G) = (E ∪ F) ∩ (E ∪ G) (5) E ∩ (F ∪ G) = (E ∩ F) ∪ (E ∩ G) i2π(ω (x)) i2π(ω (x)) γ E j j Proof: In this theorem we have E = (x, γ (x).e ), F = (x, γ (x).e ) and E F j j i2π(ω (x)) G = (x, γ (x).e ) j 44 T. MAHMOOD ET AL. (1) By Definition 7 we have i2π( 1−ω (x) ) i2π(ω (x)) γ E Ej c(E) = c(x, γ (x).e ) = (x, 1 − γ (x) .e ) , then E E j j i2π( 1−(1−ω (x)) ) Ej c(c(E)) = (x, 1 − (1 − γ (x)) .e ) i2π(ω (x)) Ej = (x, γ (x).e ) = E. (2) By Definition 7 we have i2π(max(ω (x),ω (x))) γ γ E F j j E ∪ F = (x,max(γ (x), γ (x)).e ) E F j j i2π(max(ω (x),ω (x))) γ γ F E j j = (x,max(γ (x), γ (x)).e ) F E j j = F ∪ E. i2π(min(ω (x),ω (x))) γ γ E F j j E ∩ F = (x,min(γ (x), γ (x)).e ) E F j j i2π(min(ω (x),ω (x))) γ γ F E j j = (x,min(γ (x), γ (x)).e ) F E j j = F ∩ E (3) i. By Definition 7 we have i2π(max(ω (x),ω (x))) γ γ E F j j E ∪ F = (x,max(γ (x), γ (x)).e ) E F j j To prove that (E ∪ F) ∪ G = E ∪ (F ∪ G).As i2π(max(ω (x),ω (x))) γ γ E F j j E ∪ F = (x,max(γ (x), γ (x)).e ) then E F j j i2π(max(max(ω (x),ω (x)),ω (x))) γ γ γ E F G j j (E ∪ F) ∪ G = (x,max(max(γ (x), γ (x)), γ (x)).e ) E F G j j j i2π(max(ω (x),max(ω (x),ω (x)))) γ γ γ E F j j j = (x,max(γ (x),max(γ (x), γ (x))).e ) E F G j j j = E ∪ (F ∪ G). ii. By Definition 7 we have i2π(min(ω (x),ω (x))) γ γ E F j j E ∩ F = (x,min(γ (x), γ (x)).e ) E F j j To prove that (E ∩ F) ∩ G = E ∩ (F ∩ G).As i2π(min(ω (x),ω (x))) γ γ E F j j E ∩ F = (x,min(γ (x), γ (x)).e ) then E F j j i2π(min(min(ω (x),ω (x)),ω (x))) γ γ γ E F G j j (E ∩ F) ∩ G = (x,min(min(γ (x), γ (x)), γ (x)).e ) E F G j j j i2π(min(ω (x),min(ω (x),ω (x)))) γ γ γ E F G j j j = (x,min(γ (x),min(γ (x), γ (x))).e ) E F G j j j = E ∩ (F ∩ G). FUZZY INFORMATION AND ENGINEERING 45 (4) By definition 7 we have i2π(min(ω (x),ω (x))) γ γ F G j j F ∩ G = (x,min(γ (x), γ (x)).e ) F G j j Then i2π(max(ω (x),min(ω (x),ω (x)))) γ γ γ E F G j j j E ∪ (F ∩ G) = (x,max(γ (x),min(γ (x), γ (x))).e ) E F G j j j Next we have i2π(max(ω (x),ω (x))) γ γ E F j j E ∪ F = (x,max(γ (x), γ (x)).e ) E F j j and i2π(max(ω (x),ω (x))) γ γ E G j j E ∪ G = (x,max(γ (x), γ (x)).e ) E G j j then (E ∪ F) ∩ (E ∪ G) ⎧ ⎛ ⎛ ⎛ ⎞ ⎞ ⎞ ⎫ max(ω (x), ω (x)), ⎪ γ γ ⎪ E F ⎪ j j ⎪ ⎝ ⎝ ⎠ ⎠ ⎪   i2π min ⎪ ⎨ ⎜ ⎟ ⎬ max(ω (x), ω (x)) max(γ (x), γ (x)), ⎜ E F γ γ ⎟ E G j j j j = x,min .e ⎜ ⎟ ⎪ ⎝ max(γ (x), γ (x)) ⎠ ⎪ E G ⎪ j j ⎪ ⎪ ⎪ ⎩ ⎭ i2π(max(ω (x),min(ω (x),ω (x)))) γ γ γ E F G j j j = (x,max(γ (x),min(γ (x), γ (x))).e ) E F G j j j Finally we obtain E ∪ (F ∩ G) = (E ∪ F) ∩ (E ∪ G). (5) By Definition 7 we have i2π(max(ω (x),ω (x))) γ γ j j F ∪ G = (x,max(γ (x), γ (x)).e ) F G j j Then i2π(min(ω (x),max(ω (x),ω (x)))) γ γ γ E F G j j j E ∩ (F ∪ G) = (x,min(γ (x),max(γ (x), γ (x))).e ) E F G j j j Next we have i2π(min(ω (x),ω (x))) γ γ E F j j E ∩ F = (x,min(γ (x), γ (x)).e ) . E F j j and i2π(min(ω (x),ω (x))) γ γ E ∩ G = (x,min(γ (x), γ (x)).e ) E G j j 46 T. MAHMOOD ET AL. then (E ∩ F) ∪ (E ∩ G) ⎧ ⎛ ⎛ ⎛ ⎞ ⎞ ⎞ ⎫ min(ω (x), ω (x)), ⎪ γ γ ⎪ E F ⎪ j j ⎪ ⎝ ⎝ ⎠ ⎠ ⎪   i2π max ⎪ ⎨ ⎜ ⎟ ⎬ min(ω (x), ω (x)) min(γ (x), γ (x)), ⎜ E F γ γ ⎟ E G j j j j = x,max .e ⎜ ⎟ ⎪ ⎝ min(γ (x), γ (x)) ⎠ ⎪ E G ⎪ j j ⎪ ⎪ ⎪ ⎩ ⎭ i2π(min(ω (x),max(ω (x),ω (x)))) γ γ γ E F G j j j = (x,min(γ (x),max(γ (x), γ (x))).e ) E F G j j j Finally we obtain E ∪ (F ∩ G) = (E ∪ F) ∩ (E ∪ G). Example 2: Let i2π(0.3) i2π(0.6) i2π(0.4) i2π(0.5) i2π(0.6) (x , 0.9e , 0.7e ), (x , 0.3e , 0.8e , 0.5e ), 1 2 E = i2π(0.8) i2π(0.5) i2π(0.1) i2π(0.6) (x , 0.6e ), (x , 0.7e , 0.9e , 0.3e ) 3 4 i2π(0.6) i2π(1) i2π(0.3) (x , 0.8e , 0.1e ), (x , 0.2e ), 1 2 F = , i2π(0.5) i2π(0.8) i2π(0.5) i2π(0.6) i2π(0.2) (x , 0.6e , 0.7e ), (x , 1e , 0.7e , 0.9e ) 3 4 and i2π(0.1) i2π(0.1) i2π(0.9) i2π(0.4) (x , 0.2e , 0.7e ), (x , 0.3e ,1e ), 1 2 G = i2π(0.8) i2π(0.2) i2π(0.6) i2π(0.3) i2π(1) (x , 0.9e , 0.4e ), (x , 0.7e , 0.8e , 0.9e ) 3 4 be CHFSs. Then i2π(0.7) i2π(0.4) i2π(0.4) i2π(0.5) i2π(0.6) 0.1e , 0.3e , 0.3e , 0.8e , 0.5e , (1) c(E) = ; i2π(0.2) i2π(0.5) i2π(0.9) i2π(0.4) 0.4e , 0.3e , 0.1e , 0.7e i2π(0.3) i2π(0.6) i2π(0.6) i2π(0.5) i2π(0.4) 0.9e , 0.7e , 0.7e , 0.2e , 0.5e , (2) c(c(E)) = ; i2π(0.8) i2π(0.5) i2π(0.1) i2π(0.6) 0.6e , 0.7e , 0.9e , 0.3e i2π(0.6) i2π(1) i2π(0.4) i2π(0.5) i2π(0.6) 0.9e , 0.7e , 0.3e , 0.8e , 0.5e , (3) i. E ∪ F = i2π(0.8) i2π(0.8) i2π(0.5) i2π(0.6) i2π(0.6) 0.6e , 0.7e , 1e , 0.9e , 0.9e and i2π(0.6) i2π(1) i2π(0.4) i2π(0.5) i2π(0.6) 0.9e , 0.7e , 0.3e , 0.8e , 0.5e , E ∪ F = i2π(0.8) i2π(0.8) i2π(0.5) i2π(0.6) i2π(0.6) 0.6e , 0.7e , 1e , 0.9e , 0.9e this implies that E ∪ F = F ∪ E. i2π(0.3) i2π(0.6) i2π(0.3) 0.8e , 0.1e , 0.2e , ii. E ∩ F = i2π(0.5) i2π(0.5) i2π(0.1) i2π(0.2) 0.6e , 0.7e , 0.7e , 0.3e and i2π(0.3) i2π(0.6) i2π(0.3) 0.8e , 0.1e , 0.2e , F ∩ E = i2π(0.5) i2π(0.5) i2π(0.1) i2π(0.2) 0.6e , 0.7e , 0.7e , 0.3e this implies that E ∩ F = F ∩ E. FUZZY INFORMATION AND ENGINEERING 47 (4) i. We have i2π(0.6) i2π(1) i2π(0.9) i2π(0.5) i2π(0.6) 0.9e , 0.7e , 0.3e ,1e , 0.5e , (E ∪ F) ∪ G = i2π(0.8) i2π(0.8) i2π(0.6) i2π(0.6) i2π(1) 0.9e , 0.7e , 1e , 0.9e , 0.9e And i2π(0.6) i2π(1) i2π(0.9) i2π(0.4) 0.8e , 0.7e , 0.3e ,1e , F ∪ G = i2π(0.8) i2π(0.8) i2π(0.6) i2π(0.6) i2π(1) 0.9e , 0.7e , 1e , 0.8e , 0.9e This implies that i2π(0.6) i2π(1) i2π(0.9) i2π(0.5) i2π(0.6) 0.9e , 0.7e , 0.3e ,1e , 0.5e , E ∪ (F ∪ G) = i2π(0.8) i2π(0.8) i2π(0.6) i2π(0.6) i2π(1) 0.9e , 0.7e , 1e , 0.9e , 0.9e Finally we obtain (E ∪ F) ∪ G = E ∪ (F ∪ G). ii. Next we have i2π(0.1) i2π(0.1) i2π(0.3) 0.2e , 0.1e , 0.2e , (E ∩ F) ∩ G = i2π(0.5) i2π(0.5) i2π(0.1) i2π(0.2) 0.6e , 0.7e , 0.7e , 0.3e And i2π(0.1) i2π(0.1) i2π(0.3) 0.2e , 0.1e , 0.2e , F ∩ G = i2π(0.5) i2π(0.2) i2π(0.5) i2π(0.3) i2π(0.2) 0.6e , 0.4e , 0.7e , 0.7e , 0.9e This implies that i2π(0.1) i2π(0.1) i2π(0.3) 0.2e , 0.1e , 0.2e , E ∩ (F ∩ G) = i2π(0.5) i2π(0.5) i2π(0.1) i2π(0.2) 0.6e , 0.7e , 0.7e , 0.3e Finally we obtain (E ∩ F) ∩ G = E ∩ (F ∩ G). (5) We have i2π(0.1) i2π(0.1) i2π(0.3) 0.2e , 0.1e , 0.2e , F ∩ G = i2π(0.5) i2π(0.2) i2π(0.5) i2π(0.3) i2π(0.2) 0.6e , 0.4e , 0.7e , 0.7e , 0.9e 48 T. MAHMOOD ET AL. Then i2π(0.3) i2π(0.6) i2π(0.4) i2π(0.5) i2π(0.6) (x , 0.9e , 0.7e ), (x , 0.3e , 0.8e , 0.5e ), 1 2 E ∪ (F ∩ G) = i2π(0.8) i2π(0.2) i2π(0.5) i2π(0.3) i2π(0.6) (x , 0.6e , 0.4e ), (x , 0.7e , 0.9e , 0.9e ) 3 4 Next we have i2π(0.6) i2π(1) i2π(0.4) i2π(0.5) i2π(0.6) 0.9e , 0.7e , 0.3e , 0.8e , 0.5e , E ∪ F = i2π(0.8) i2π(0.8) i2π(0.5) i2π(0.6) i2π(0.6) 0.6e , 0.7e , 1e , 0.9e , 0.9e and i2π(0.3) i2π(0.6) i2π(0.9) i2π(0.5) i2π(0.6) (x , 0.9e , 0.7e ), (x , 0.3e ,1e , 0.5e ), 1 2 E ∪ G = i2π(0.8) i2π(0.2) i2π(0.6) i2π(0.3) i2π(1) (x , 0.9e , 0.4e ), (x , 0.7e , 0.9e , 0.9e ) 3 4 then i2π(0.3) i2π(0.6) i2π(0.4) i2π(0.5) i2π(0.6) 0.9e , 0.7e , 0.3e , 0.8e , 0.5e , (E ∪ F) ∩ (E ∪ G) = i2π(0.8) i2π(0.2) i2π(0.5) i2π(0.3) i2π(0.6) 0.6e , 0.4e , 0.7e , 0.9e , 0.9e Finally we obtain E ∪ (F ∩ G) = (E ∪ F) ∩ (E ∪ G). (6) We have i2π(0.6) i2π(1) i2π(0.9) i2π(0.4) 0.8e , 0.7e , 0.3e ,1e , F ∪ G = i2π(0.8) i2π(0.8) i2π(0.6) i2π(0.6) i2π(1) 0.9e , 0.7e , 1e , 0.8e , 0.9e Then i2π(0.6) i2π(1) i2π(0.4) i2π(0.4) 0.8e , 0.7e , 0.3e , 0.8e , E ∩ (F ∪ G) = i2π(0.8) i2π(0.5) i2π(0.1) i2π(0.6) 0.6e , 0.7e , 0.8e , 0.3e Next we have i2π(0.3) i2π(0.6) i2π(0.3) 0.8e , 0.1e , 0.2e , E ∩ F = i2π(0.5) i2π(0.5) i2π(0.1) i2π(0.2) 0.6e , 0.7e , 0.7e , 0.3e and E then i2π(0.6) i2π(1) i2π(0.4) i2π(0.4) 0.8e , 0.7e , 0.3e , 0.8e , (E ∩ F) ∪ (E ∩ G) = i2π(0.8) i2π(0.5) i2π(0.1) i2π(0.6) 0.6e , 0.7e , 0.8e , 0.3e Finally we obtain E ∩ (F ∪ G) = (E ∩ F) ∪ (E ∩ G). FUZZY INFORMATION AND ENGINEERING 49 4. The Generalized Similarity Measures Based on CHFSs In the part of the paper, we proposed SMs established on the exponential function. We also proposed SMs without exponential function. Definition 8: Let E and F be two CHFSs on X. Then similarity measure (SM) between E and Fis identified by S (E, F), which satisfies the following properties (1) 0 ≤ S (E, F) ≤ 1; (2) S (E, F) = 1if and only if E = F; (3) S (E, F) = S (F, E). c c Definition 9: Let E and F be two CHFS on X. Then the exponential based generalized SM is calculated as ⎡ ⎡ ⎤ ⎤ 1 λ 1 λ 1− |γ (x )−γ (x )| ∨ |ω (x )−ω (x )| E k F k γ k γ k j j  E F ⎢ ⎢ j j ⎥ ⎥ j=1 j=1 S (E, F) = 2 − 1 ⎣ ⎣ ⎦ ⎦ k=1 where λ> 0, and ∨ described the maximum operation. In Definitions (8) and (9), if we choose the imaginary parts will be zero, then the explored notion is converted for HFS. Similarly, if we choose the CHFS is a singleton set, then the CHFS is converted for CFS. Further, if we choose the CHFs is a singleton set and the imaginary part will be zero, then the CHFS is converted for FS. Due to its structure, it make powerful and proficient to cope with uncertain and unreliable information in real decision theory. Theorem 2: The SM S (E, F) satisfy the following properties (1) 0 ≤ S (E, F) ≤ 1; (2) S (E, F) = 1 if and only if E = F; 1 1 (3) S (E, F) = S (F, E). c c 1 λ 1 λ Proof: 1. Since |γ (x ) − γ (x )| ∈ [0, 1] and |ω (x ) − ω (x )| ∈ [0, 1] then E k F k γ k γ k j j E F j j j=1 j=1 1 λ 1 λ |γ (x ) − γ (x )| ∨ |ω (x ) − ω (x )| ∈ [0, 1] this implies that for k = 1we E F γ γ j k j k E k F k j j j=1 j=1 have 1 λ 1 λ 1− |γ (x )−γ (x )| ∨ |ω (x )−ω (x )| E 1 F 1 γ 1 γ 1 j j E F j j j=1 j=1 2 − 1 ∈ [0, 1] For k = 2 1 1 λ λ 1− |γ (x )−γ (x )| ∨ |ω (x )−ω (x )| E 2 F 2 γ 2 γ 2 j j  E F j j j=1 j=1 2 − 1 ∈ [0, 1] 50 T. MAHMOOD ET AL. By doing this process we obtain ⎡   ⎤ 1 λ 1 λ 1− |γ (x )−γ (x )| ∨ |ω (x )−ω (x )| E k F k γ k γ k j j  E F ⎢ j j ⎥ j=1 j=1 ⎣ 2 − 1⎦ ∈ n[0, 1] k=1 ⎡ ⎤ 1 λ 1 λ 1− |γ (x )−γ (x )| ∨ |ω (x )−ω (x )| E F γ γ k k k k j j  E F ⎢ j j ⎥ j=1 j=1 ⇒ 0 ≤ 2 − 1 ≤ n ⎣ ⎦ k=1 ⎡   ⎤ 1 λ 1 λ 1− |γ (x )−γ (x )| ∨ |ω (x )−ω (x )| E k F k γ k γ k 1 j j E F j j ⎢ ⎥ j=1 j=1 ⇒ 0 ≤ 2 − 1 ≤ 1 ⎣ ⎦ k=1 ⎡ ⎡ ⎤ ⎤ 1 λ 1 λ 1− |γ (x )−γ (x )| ∨ |ω (x )−ω (x )| E F γ γ k k k k j j  E F ⎢ ⎢ j j ⎥ ⎥ j=1 j=1 ⇒ 0 ≤ 2 − 1 ≤ 1 ⎣ ⎣ ⎦ ⎦ k=1 ⇒ 0 ≤ S (E, F) ≤ 1. 2. By Definition 7 we have ⎡ ⎡   ⎤ ⎤ 1 λ 1 λ 1− |γ (x )−γ (x )| ∨ |ω (x )−ω (x )| γ γ E k F k E k F k 1  j j ⎢ ⎢ j j ⎥ ⎥ j=1 j=1 S (E, F) = 2 − 1 ⎣ ⎣ ⎦ ⎦ k=1 ⎛ ⎞ ⎡ ⎡ ⎤ ⎤ 1 λ λ λ λ ((|γ (x ) − γ (x )| +|γ (x ) − γ (x )| + ... +|γ (x ) − γ (x )| )∨) E k F k E k F k E k F k 1 1 2 2 ⎝ ⎠ n 1− λ λ λ ⎢ ⎢ ⎥ ⎥ 1 (|ω (x ) − ω (x )| +|ω (x ) − ω (x )| + ... +|ω (x ) − ω (x )| ) γ k γ k γ k γ k γ k γ k ⎢ ⎢  E F E F E F ⎥ ⎥ 1 1 2 2 ⇔ S (E, F) = 2 − 1 ⎣ ⎣ ⎦ ⎦ k=1 ⎛ ⎞ ⎡ ⎡ 1 λ λ λ ((|γ (x ) − γ (x )| +|γ (x ) − γ (x )| + ... +|γ (x ) − γ (x )| )∨) E 1 F 1 E 1 F 1 E 1 F 1 1 1 2 2 ⎝ ⎠ 1− 1 λ λ λ ⎢ ⎢ 1 (|ω (x ) − ω (x )| +|ω (x ) − ω (x )| + ... +|ω (x ) − ω (x )| ) γ 1 γ 1 γ 1 γ 1 γ 1 γ 1 E F E F E F ⎢ ⎢ 1 1 2 2 ⇔ S (E, F) = 2 − 1 ⎣ ⎣ ⎛ ⎞ 1 λ λ λ ((|γ (x ) − γ (x )| +|γ (x ) − γ (x )| + ... +|γ (x ) − γ (x )| )∨) E 2 F 2 E 2 F 2 E 2 F 2 1 1 2 2 ⎝ ⎠ 1− 1 λ λ λ (|ω (x ) − ω (x )| +|ω (x ) − ω (x )| + ... +|ω (x ) − ω (x )| ) γ 2 γ 2 γ 2 γ 2 γ 2 γ 2 E F E F E F 1 1 2 2 + 2   − 1 + ... ⎛ ⎞ ⎤ ⎤ 1 λ λ λ ((|γ (x ) − γ (x )| +|γ (x ) − γ (x )| + ... +|γ (x ) − γ (x )| )∨) E n F n E n F n E n F n 1 1 2 2 ⎝ ⎠ 1− 1 λ λ λ ⎥ ⎥ (|ω (x ) − ω (x )| +|ω (x ) − ω (x )| + ... +|ω (x ) − ω (x )| ) γ n γ n γ n γ n γ n γ n E F E F E F ⎥ ⎥ 1 1 2 2 +2 − 1 ⎦ ⎦ i2π(ω (x )) i2π(ω (x )) E F k k j j Now as E = F ⇔ μ (x ) = μ (x ) for k = 1, 2, ... , n ⇔ γ (x )e = γ (x )e E k F k E k F k j j i2π(ω (x )) i2π(ω (x )) E k F k j j for k = 1, 2, ... , n ⇔ γ (x ) = γ (x ) and e = e for k = 1, 2, ... , n. Then E k F k j j ! " 1 1 1−0 1−0 1−0 ⇔ S (E, F) = [2 − 1 + 2 − 1 + ... + 2 − 1] ⇔ S (E, F) = 1. c FUZZY INFORMATION AND ENGINEERING 51 3. We have ⎡ ⎡   ⎤ ⎤ 1 λ 1 λ 1− |γ (x )−γ (x )| ∨ |ω (x )−ω (x )| E k F k γ k γ k E F 1 j j j j ⎢ ⎢ ⎥ ⎥ j=1 j=1 S (E, F) = 2 − 1 ⎣ ⎣ ⎦ ⎦ k=1 ⎡ ⎡ ⎤ ⎤ 1 λ 1 λ 1− |−γ (x )+γ (x )| ∨ |−ω (x )+ω (x )| F E γ γ k k k k j  F E ⎢ ⎢ j j ⎥ ⎥ j=1 j=1 = 2 − 1 ⎣ ⎣ ⎦ ⎦ k=1 ⎡ ⎡   ⎤ ⎤ 1 λ 1 λ 1− |−(γ (x )−γ (x ))| ∨ |−(ω (x )−ω (x ))| F k E k γ k γ k F E 1 j j j ⎢ ⎢ ⎥ ⎥ j=1 j=1 = 2 − 1 ⎣ ⎣ ⎦ ⎦ k=1 ⎡ ⎡ ⎤ ⎤ 1 λ 1 λ 1− |γ (x )+γ (x )| ∨ |ω (x )+ω (x )| F E γ γ k k k k j  F E ⎢ ⎢ j j ⎥ ⎥ j=1 j=1 = 2 − 1 ⎣ ⎣ ⎦ ⎦ k=1 = S (F, E). Remark 1: If λ = 1 then the exponential based generalized SM becomes ⎡ ⎤ 1 1 1− |γ (x )−γ (x )|∨ |ω (x )−ω (x )| E F γ γ k k k k j j  E F ⎢ j j ⎥ 1 j=1 j=1 S (E, F) = 2 − 1 ⎣ ⎦ k=1 Definition 10: Let E and F be two CHFS on X. Then we can also calculate exponential based generalized SM as follows ⎡ ⎡   ⎤ ⎤ 1 λ 1 λ 1− |γ (x )−γ (x )| + |ω (x )−ω (x )| E k F k γ k γ k 2 2 1 j j E F j j ⎢ ⎢ ⎥ ⎥ j=1 j=1 S (E, F) = 2 − 1 ⎣ ⎣ ⎦ ⎦ k=1 where λ> 0. Theorem 3: The SM S (E, F) satisfy the following properties (1) 0 ≤ S (E, F) ≤ 1; (2) S (E, F) = 1 if and only if E = F; 2 2 (3) S (E, F) = S (F, E). c c 1 λ 1 λ Proof: 1. Since |γ (x ) − γ (x )| ∈ [0, 1] and |ω (x ) − ω (x )| ∈ [0, 1] E k F k γ k γ k j j E F 2 2 j j j=1 j=1 1 λ 1 λ then |γ (x ) − γ (x )| + |ω (x ) − ω (x )| ∈ [0, 1] this implies that for E F γ γ k k k k 2 j j 2 E F j j j=1 j=1 52 T. MAHMOOD ET AL. k = 1wehave 1 1 λ λ 1− |γ (x )−γ (x )| + |ω (x )−ω (x )| E 1 F 1 γ 1 γ 1 2 j j 2 E F j j j=1 j=1 2 − 1 ∈ [0, 1] For k = 2 1 λ 1 λ 1− |γ (x )−γ (x )| + |ω (x )−ω (x )| E 2 F 2 γ 2 γ 2 2 j j 2 E F j j j=1 j=1 2 − 1 ∈ [0, 1] By doing this process we obtain ⎡ ⎤ 1 λ 1 λ 1− |γ (x )−γ (x )| + |ω (x )−ω (x )| E k F k γ k γ k 2 j j 2 E F ⎢ j j ⎥ j=1 j=1 2 − 1 ∈ n[0, 1] ⎣ ⎦ k=1 ⎡ ⎤ 1 λ 1 λ 1− |γ (x )−γ (x )| + |ω (x )−ω (x )| E k F k γ k γ k 2 j j 2 E F j j ⎢ ⎥ j=1 j=1 ⇒ 0 ≤ 2 − 1 ≤ n ⎣ ⎦ k=1 ⎡   ⎤ 1 λ 1 λ 1− |γ (x )−γ (x )| + |ω (x )−ω (x )| E k F k γ k γ k 1 2 j j 2 E F j j ⎢ ⎥ j=1 j=1 ⇒ 0 ≤ 2 − 1 ≤ 1 ⎣ ⎦ k=1 ⎡ ⎡ ⎤ ⎤ 1 λ 1 λ 1− |γ (x )−γ (x )| + |ω (x )−ω (x )| E F γ γ k k k k 2 j j 2 E F ⎢ ⎢ j j ⎥ ⎥ j=1 j=1 ⇒ 0 ≤ 2 − 1 ≤ 1 ⎣ ⎣ ⎦ ⎦ k=1 ⎡ ⎡   ⎤ ⎤ 1 λ 1 λ 1− |γ (x )−γ (x )| + |ω (x )−ω (x )| E k F k γ k γ k 1 2 j j 2 E F j j ⎢ ⎢ ⎥ ⎥ j=1 j=1 ⇒ 0 ≤ 2 − 1 ≤ 1 ⎣ ⎣ ⎦ ⎦ k=1 ⎡ ⎡ ⎤ ⎤ 1 λ 1 λ 1− |γ (x )−γ (x )| + |ω (x )−ω (x )| E F γ γ k k k k 2 j j 2 E F ⎢ ⎢ j j ⎥ ⎥ j=1 j=1 ⇒ 0 ≤ 2 − 1 ≤ 1 ⎣ ⎣ ⎦ ⎦ k=1 ⇒ 0 ≤ S (E, F) ≤ 1. 2. By definition 7 we have ⎡ ⎡   ⎤ ⎤ 1 1 λ 1 λ 1− |γ (x )−γ (x )| + |ωγ (x )−ωγ (x )| E k F k k k 1 2 j j 2 E F ⎢ ⎢ j j ⎥ ⎥ j=1 j=1 S (E, F) = 2 − 1 ⎣ ⎣ ⎦ ⎦ k=1 ⎛ ⎞ ⎡ ⎡ ⎤ ⎤ 1 λ λ λ ((|γ (x ) − γ (x )| +|γ (x ) − γ (x )| + ... +|γ (x ) − γ (x )| )+) E k F k E k F k E k F k 2 1 1 2 2 ⎝ ⎠ 1− 1 λ λ λ ⎢ ⎢ ⎥ ⎥ 1 (|ω (x ) − ω (x )| +|ω (x ) − ω (x )| + ... +|ω (x ) − ω (x )| ) γ k γ k γ k γ k γ k γ k 2 2 E F E F E F ⎢ ⎢ ⎥ ⎥ 1 1 2 2 ⇔ S (E, F) = 2 − 1 c ⎣ ⎣ ⎦ ⎦ k=1 ⎛ ⎞ ⎡ ⎡ 1 λ λ λ ((|γ (x ) − γ (x )| +|γ (x ) − γ (x )| + ... +|γ (x ) − γ (x )| )+) E 1 F 1 E 1 F 1 E 1 F 1 2 1 1 2 2 2 2 ⎝ ⎠ 1− 1 λ λ λ ⎢ ⎢ 1 (|ω (x ) − ω (x )| +|ω (x ) − ω (x )| + ... +|ω (x ) − ω (x )| ) γ 1 γ 1 γ 1 γ 1 γ 1 γ 1 2 2 E F E F E F ⎢ ⎢ 1 1 2 2 ⇔ S (E, F) = 2 − 1 ⎣ ⎣ n FUZZY INFORMATION AND ENGINEERING 53 ⎛ ⎞ 1 λ λ λ ((|γ (x ) − γ (x )| +|γ (x ) − γ (x )| + ... +|γ (x ) − γ (x )| )+) E 2 F 2 E 2 F 2 E 2 F 2 2 1 1 2 2 ⎝ ⎠ 1− λ λ λ (|ω (x ) − ω (x )| +|ω (x ) − ω (x )| + ... +|ω (x ) − ω (x )| ) γ 2 γ 2 γ 2 γ 2 γ 2 γ 2 2 E F E F E F 1 1 2 2 + 2 − 1 + ... ⎛ ⎞ ⎤ ⎤ 1 λ λ λ ((|γ (x ) − γ (x )| +|γ (x ) − γ (x )| + ... +|γ (x ) − γ (x )| )+) E n F n E n F n E n F n 2 1 1 2 2 ⎝ ⎠ 1− 1 λ λ λ ⎥ ⎥ (|ω (x ) − ω (x )| +|ω (x ) − ω (x )| + ... +|ω (x ) − ω (x )| ) γ n γ n γ n γ n γ n γ n 2 E F E F E F 1 1 2 2 ⎥ ⎥ + 2   − 1 ⎦ ⎦ i2π(ω (x )) i2π(ω (x )) E F k k j j Now as E = F ⇔ μ (x ) = μ (x ) for k = 1, 2, ... , n ⇔ γ (x )e = γ (x )e E k F k E k F k j j i2π(ω (x )) i2π(ω (x )) E k F k j j for k = 1, 2, ... , n ⇔ γ (x ) = γ (x ) and e = e for k = 1, 2, ... , n. Then E k F k j j ! " 2 1−0 1−0 1−0 ⇔ S (E, F) = [2 − 1 + 2 − 1 + ... + 2 − 1] ⇔ S (E, F) = 1. 3. We have ⎡ ⎡   ⎤ ⎤ 1 1 λ λ 1− |γ (x )−γ (x )| + |ω (x )−ω (x )| E k F k γ k γ k 2 j j 2 E F j j ⎢ ⎢ ⎥ ⎥ j=1 j=1 S (E, F) = 2 − 1 ⎣ ⎣ ⎦ ⎦ k=1 ⎡ ⎡ ⎤ ⎤ 1 λ 1 λ 1− |−γ (x )+γ (x )| + |−ω (x )+ω (x )| F E γ γ k k k k 2 j 2 F E ⎢ ⎢ j j ⎥ ⎥ j=1 j=1 = 2 − 1 ⎣ ⎣ ⎦ ⎦ k=1 ⎡ ⎡   ⎤ ⎤ 1 1 λ λ 1− |−(γ (x )−γ (x ))| + |−(ω (x )−ω (x ))| F k E k γ k γ k 2 j 2 F E j j ⎢ ⎢ ⎥ ⎥ j=1 j=1 = 2 − 1 ⎣ ⎣ ⎦ ⎦ k=1 ⎡ ⎡ ⎤ ⎤ 1 λ 1 λ 1− |γ (x )+γ (x )| + |ω (x )+ω (x )| F E γ γ k k k k 2 j 2 F E ⎢ ⎢ j j ⎥ ⎥ j=1 j=1 = 2 − 1 ⎣ ⎣ ⎦ ⎦ k=1 = S (F, E). Remark 2: If λ = 1 then the exponential based generalized SM becomes ⎡ ⎤ 1 1 1− |γ (x )−γ (x )|+ |ω (x )−ω (x )| E F γ γ k k k k 2 j j 2 E F ⎢ j j ⎥ 2 j=1 j=1 S (E, F) = 2 − 1 ⎣ ⎦ k=1 Definition 11: Let E and F be two CHFSs on X. Then without exponential based generalized SMs are calculated as follows # $ ⎡ ⎡ ⎤ ⎤ 1 λ 1 λ 1 − |γ (x ) − γ (x )| ∨ |ω (x ) − ω (x )| E k F k γ k γ k j=1 j j j=1 E F 1   j j ⎣ ⎣ ⎦ ⎦ # $ S (E, F) = 1  λ 1  λ 1 + |γ (x ) − γ (x )| ∨ |ω (x ) − ω (x )| k=1 E k F k γ k γ k j=1 j j j=1 E F j j 54 T. MAHMOOD ET AL. ⎡ ⎤ 1 λ 1 λ (γ (x ) ∧ γ (x )) + (ω (x ) ∧ ω (x )) E F γ γ k k k k 1 j=1 j j j=1 E F j j ⎣ ⎦ S (E, F) = 1  1 λ λ (γ (x ) ∨ γ (x )) + (ω (x ) ∨ ω (x )) E k F k γ k γ k j=1 j j j=1 E F k=1 j j ⎡ ⎛ ⎞ ⎤ 1/λ % & (ω (x ) ∧ ω (x )) (γ (x ) ∧ γ (x )) γ k γ k 1 1 E F E F j k j k j j ⎣ ⎝ ⎠ ⎦ S (E, F) = α + β cc cc λ λ (γ (x ) ∨ γ (x )) (ω (x ) ∨ ω (x )) E F γ γ j k j k E k F k j=1 j j k=1 where λ> 0and α , β ∈ [0, 1] such that α + β = 1. cc cc cc cc Theorem 4: The SM S (E, F) satisfies the following properties (1) 0 ≤ S (E, F) ≤ 1; (2) S (E, F) = 1if and only if E = F; 3 3 (3) S (E, F) = S (F, E). c c 1 1 λ λ Proof: 1. Since |γ (x ) − γ (x )| ∈ [0, 1] and |ω (x ) − ω (x )| ∈ [0, 1] then E k F k γ k γ k j j E F j j j=1 j=1 1 λ 1 λ |γ (x ) − γ (x )| ∨ |ω (x ) − ω (x )| ∈ [0, 1] and denominator will always E F γ γ k k k k j j  E F j j j=1 j=1 greater than numerator, then for k = 1wehave # $ ⎡ ⎤ 1 1 1 − |γ (x ) − γ (x )|∨ |ω (x ) − ω (x )| E 1 F 1 γ 1 γ 1 j=1 j j j=1 E F j j ⎣ ⎦ # $ ∈ [0, 1] 1 1 1 + |γ (x ) − γ (x )|∨ |ω (x ) − ω (x )| E 1 F 1 γ 1 γ 1 j j E F j=1  j=1 j j For k = 2 # $ ⎡ ⎤ 1  1 1 − |γ (x ) − γ (x )|∨ |ω (x ) − ω (x )| E 2 F 2 γ 2 γ 2 j=1 j j j=1 E F j j ⎣ ⎦ # $ ∈ [0, 1] 1  1 1 + |γ (x ) − γ (x )|∨ |ω (x ) − ω (x )| E 2 F 2 γ 2 γ 2 j=1 j j j=1 E F j j By doing this process we obtain # $ ⎡ ⎤ 1 1 1 − |γ (x ) − γ (x )|∨ |ω (x ) − ω (x )| E k F k γ k γ k j j E F j=1  j=1 j j ⎣ ⎦ # $ ∈ n[0, 1] 1 1 1 + |γ (x ) − γ (x )|∨ |ω (x ) − ω (x )| E F γ γ k=1 k k k k j=1 j j  j=1 E F j j # $ ⎡ ⎤ 1 1 1 − |γ (x ) − γ (x )|∨ |ω (x ) − ω (x )| E F γ γ k k k k j=1 j j  j=1 E F j j ⎣ ⎦ ⇒ 0 ≤ # $ ≤ n 1  1 1 + |γ (x ) − γ (x )|∨ |ω (x ) − ω (x )| k=1 E k F k γ k γ k j=1 j j j=1 E F j j # $ ⎡ ⎤ 1 1 1 − |γ (x ) − γ (x )|∨ |ω (x ) − ω (x )| E k F k γ k γ k j=1 j j j=1 E F 1   j j ⎣ ⎦ # $ ⇒ 0 ≤ ≤ 1 1  1 1 + |γ (x ) − γ (x )|∨ |ω (x ) − ω (x )| k=1 E k F k γ k γ k j=1 j j j=1 E F j j FUZZY INFORMATION AND ENGINEERING 55 # $ ⎡ ⎡ ⎤ ⎤ 1 λ 1 λ 1 − |γ (x ) − γ (x )| ∨ |ω (x ) − ω (x )| E k F k γ k γ k j j E F 1  j=1  j=1 j j ⎣ ⎣ ⎦ ⎦ ⇒ 0 ≤ # $ ≤ 1 n 1 λ 1 λ 1 + |γ (x ) − γ (x )| ∨ |ω (x ) − ω (x )| E F γ γ k=1 k k k k j=1 j j  j=1 E F j j ⇒ S (E, F). 2. By definition 7 we have # $ ⎡ ⎡ ⎤ ⎤ 1 λ 1 λ 1 − |γ (x ) − γ (x )| ∨ |ω (x ) − ω (x )| E k F k γ k γ k j=1 j j j=1 E F 1   j j ⎣ ⎣ ⎦ ⎦ # $ S (E, F) = n 1 λ 1 λ 1 + |γ (x ) − γ (x )| ∨ |ω (x ) − ω (x )| k=1 E k F k γ k γ k j=1 j j  j=1 E F j j ⎡ ⎡  ⎤ ⎤ 1 λ λ λ ((|γ (x ) − γ (x )| +|γ (x ) − γ (x )| + ... +|γ (x ) − γ (x )| )∨) E F E F E F 1 k 1 k 2 k 2 k  k  k 1 − n 1 λ λ λ ⎢ ⎢ ⎥ ⎥ (|ω (x ) − ω (x )| +|ω (x ) − ω (x )| + ... +|ω (x ) − ω (x )| ) 1 γ k γ k γ k γ k γ k γ k ⎢ ⎢  E F E F E F ⎥ ⎥ 3 1 1 2 2 ⇔ S (E, F) = ⎢ ⎢  ⎥ ⎥ 1 λ λ λ ⎣ n ⎣ ((|γ (x ) − γ (x )| +|γ (x ) − γ (x )| + ... +|γ (x ) − γ (x )| )∨) ⎦ ⎦ E1 k F1 k E2 k F2 k E k F k k=1 1 + 1 λ λ λ (|ω (x ) − ω (x )| +|ω (x ) − ω (x )| + ... +|ω (x ) − ω (x )| ) γ k γ k γ k γ k γ k γ k E F E F E F 1 1 2 2 ⎡ ⎡ 1 λ λ λ ((|γ (x ) − γ (x )| +|γ (x ) − γ (x )| + ... +|γ (x ) − γ (x )| )∨) E 1 F 1 E 1 F 1 E 1 F 1 1 1 2 2 1 − ⎢ ⎢ λ λ λ (|ω (x ) − ω (x )| +|ω (x ) − ω (x )| + ... +|ω (x ) − ω (x )| ) 1 γE 1 γF 1 γE 1 γF 1 γE 1 γF 1 ⎢ ⎢ 3 1 1 2 2 ⇔ S (E, F) = ⎢ ⎢ 1 λ λ λ ⎣ n ⎣ ((|γ (x ) − γ (x )| +|γ (x ) − γ (x )| + ... +|γ (x ) − γ (x )| )∨) E 1 F 1 E 1 F 1 E 1 F 1 1 1 2 2 1 + λ λ λ (|ω (x ) − ω (x )| +|ω (x ) − ω (x )| + ... +|ω (x ) − ω (x )| ) γE 1 γF 1 γE 1 γF 1 γE 1 γF 1 1 1 2 2 1 λ λ λ ((|γ (x ) − γ (x )| +|γ (x ) − γ (x )| + ... +|γ (x ) − γ (x )| )∨) E 2 F 2 E 2 F 2 E 2 F 2 1 1 2 2 1 − 1 λ λ λ (|ω (x ) − ω (x )| +|ω (x ) − ω (x )| + ... +|ω (x ) − ω (x )| ) γ 2 γ 2 γ 2 γ 2 γ 2 γ 2 E F E F E F 1 1 2 2 + + ... 1 λ λ λ ((|γ (x ) − γ (x )| +|γ (x ) − γ (x )| + ... +|γ (x ) − γ (x )| )∨) E 2 F 2 E 2 F 2 E 2 F 2 1 1 2 2 1 + 1 λ λ λ (|ω (x ) − ω (x )| +|ω (x ) − ω (x )| + ... +|ω (x ) − ω (x )| ) γ 2 γ 2 γ 2 γ 2 γ 2 γ 2 E F E F E F 1 1 2 2 ⎤ ⎤ 1 λ λ λ ((|γ (x ) − γ (x )| +|γ (x ) − γ (x )| + ... +|γ (x ) − γ (x )| )∨) E n F n E n F n E n F n 1 1 2 2 1 − 1 λ λ λ ⎥ ⎥ (|ω (x ) − ω (x )| +|ω (x ) − ω (x )| + ... +|ω (x ) − ω (x )| ) γ n γ n γ n γ n γ n γ n E F E F E F ⎥ ⎥ 1 1 2 2 ⎥ ⎥ 1 λ λ λ ⎦ ⎦ ((|γ (x ) − γ (x )| +|γ (x ) − γ (x )| + ... +|γ (x ) − γ (x )| )∨) E n F n E n F n E n F n 1 1 2 2 1 + 1 λ λ λ (|ω (x ) − ω (x )| +|ω (x ) − ω (x )| + ... +|ω (x ) − ω (x )| ) γ n γ n γ n γ n γ n γ n E F E F E F 1 1 2 2 i2π(ω (x )) i2π(ω (x )) E F k k j j Now as E = F ⇔ μ (x ) = μ (x ) for k = 1, 2, ... , n ⇔ γ (x )e = γ (x )e E k F k E k F k j j i2π(ω (x )) i2π(ω (x )) E k F k j j for k = 1, 2, ... , n ⇔ γ (x ) = γ (x ) and e = e for k = 1, 2, ... , n. E F k k j j Then ! " 1 ⇔ S (E, F) = [1 + 1 + ... + 1] ⇔ S (E, F) = 1. 3. We have # $ ⎡ ⎡ ⎤ ⎤ 1 λ 1 λ 1 − |γ (x ) − γ (x )| ∨ |ω (x ) − ω (x )| E k F k γ k γ k j j E F 1  j=1  j=1 j j ⎣ ⎣ ⎦ ⎦ S (E, F) = # $ n 1 λ 1 λ 1 + |γ (x ) − γ (x )| ∨ |ω (x ) − ω (x )| E F γ γ k=1 k k k k j=1 j j  j=1 E F j j # $ ⎡ ⎡ ⎤ ⎤ 1 λ 1 λ 1 − |−(γ (x ) − γ (x ))| ∨ |−(ω (x ) − ω (x ))| F k E k γ k γ k j=1 j j j=1 F E j j ⎣ ⎣ ⎦ ⎦ = # $ n 1 λ 1 λ 1 + |−(γ (x ) − γ (x ))| ∨ |−(ω (x ) − ω (x ))| F E γ γ k=1 j k j k F k E k j=1  j=1 j j 56 T. MAHMOOD ET AL. # $ ⎡ ⎡ ⎤ ⎤ 1 λ 1 λ 1 − |(γ (x ) − γ (x ))| ∨ |(ω (x ) − ω (x ))| F k E k γ k γ k j j F E 1  j=1  j=1 j j ⎣ ⎣ ⎦ ⎦ = # $ n 1 λ 1 λ 1 + |(γ (x ) − γ (x ))| ∨ |(ω (x ) − ω (x ))| F E γ γ k=1 k k k k j=1 j j  j=1 F E j j = S (F, E). Theorem 5: The SM S (E, F) satisfy the following properties (1) 0 ≤ S (E, F) ≤ 1; (2) S (E, F) = 1 if and only if E = F; 4 4 (3) S (E, F) = S (F, E). c c 1 λ 1 λ Proof: 1. Since (γ (x ) ∧ γ (x )) ∈ [0, 1], (ω (x ) ∧ ω (x )) ∈ [0, 1], E k F k γ k γ k j j E F j j j=1 j=1 1 1 λ λ (γ (x ) ∨ γ (x )) ∈ [0, 1], (ω (x ) ∨ ω (x )) ∈ [0, 1] and denominator is E k F k γ k γ k j j E F j j j=1 j=1 always greater then nominator. Thus for k = 1wehave 1  λ 1  λ (γ (x ) ∧ γ (x )) + (ω (x ) ∧ ω (x )) E 1 F 1 γ 1 γ 1 j=1 j j j=1 E F j j ∈ [0, 1] 1 λ 1 λ (γ (x ) ∨ γ (x )) + (ω (x ) ∨ ω (x )) E 1 F 1 γ 1 γ 1 j=1 j j j=1 E F j j For k = 2 1 λ 1 λ (γ (x ) ∧ γ (x )) + (ω (x ) ∧ ω (x )) E 2 F 2 γ 2 γ 2 j=1 j j  j=1 E F j j ∈ [0, 1] 1  1 λ λ (γ (x ) ∨ γ (x )) + (ω (x ) ∨ ω (x )) E 2 F 2 γ 2 γ 2 j=1 j j j=1 E F j j By doing this process we obtain 1 λ 1 λ (γ (x ) ∧ γ (x )) + (ω (x ) ∧ ω (x )) E F γ γ j k j k E k F k j=1  j=1 j j ∈ n[0, 1] 1  λ 1  λ (γ (x ) ∨ γ (x )) + (ω (x ) ∨ ω (x )) E k F k γ k γ k j=1 j j j=1 E F k=1 j j 1  1 λ λ (γ (x ) ∧ γ (x )) + (ω (x ) ∧ ω (x )) E k F k γ k γ k j=1 j j j=1 E F j j ⇒ 0 ≤ ≤ n 1 λ 1 λ (γ (x ) ∨ γ (x )) + (ω (x ) ∨ ω (x )) E k F k γ k γ k j=1 j j j=1 E F k=1   j j ⎡ ⎤ 1  λ 1  λ (γ (x ) ∧ γ (x )) + (ω (x ) ∧ ω (x )) E k F k γ k γ k 1 j=1 j j j=1 E F j j ⎣ ⎦ ⇒ 0 ≤ ≤ 1 1 λ 1 λ (γ (x ) ∨ γ (x )) + (ω (x ) ∨ ω (x )) E F γ γ k k k k j=1 j j  j=1 E F k=1 j j ⎡ ⎡ ⎤ ⎤ 1 1  1 λ λ (γ (x ) ∧ γ (x )) + (ω (x ) ∧ ω (x )) E k F k γ k γ k 1 j=1 j j j=1 E F j j ⎣ ⎣ ⎦ ⎦ ⇒ 0 ≤ ≤ 1 1 λ 1 λ (γ (x ) ∨ γ (x )) + (ω (x ) ∨ ω (x )) E F γ γ k k k k j=1 j j  j=1 E F k=1 j j ⇒ 0 ≤ S (E, F) ≤ 1. 2. By definition 7 we have ⎡ ⎤ 1 1  λ 1  λ λ (γ (x ) ∧ γ (x )) + (ω (x ) ∧ ω (x )) E k F k γ k γ k 1  j=1 j j  j=1 E F j j ⎣ ⎦ S (E, F) = 1  λ 1  λ (γ (x ) ∨ γ (x )) + (ω (x ) ∨ ω (x )) E k F k γ k γ k j=1 j j  j=1 E F k=1 j j FUZZY INFORMATION AND ENGINEERING 57 ⎡ ⎤ 1 λ λ λ ((γ (x ) ∧ γ (x )) + (γ (x ) ∧ γ (x )) + ... + (γ (x ) ∧ γ (x )) )+ E k F k E k F k E k F k 1 1 2 2 n 1 λ λ λ ⎢ ⎥ ((ω (x ) ∧ ω (x )) + (ω (x ) ∧ ω (x )) + ... + (ω (x ) ∧ ω (x )) ) 1 γ k γ k γ k γ k γ k γ k E F E F E F ⎢  1 1 2 2 ⎥ ⇔ S (E, F) = ⎢ ⎥ 1 λ λ λ ⎣ ⎦ n ((γ (x ) ∨ γ (x )) + (γ (x ) ∨ γ (x )) + ... + (γ (x ) ∨ γ (x )) )+ E k F k E k F k E k F k 1 1 2 2 k=1 1 λ λ λ ((ω (x ) ∨ ω (x )) + (ω (x ) ∨ ω (x )) + ... + (ω (x ) ∨ ω (x )) ) γ k γ k γ k γ k γ k γ k E F E F E F 1 1 2 2 ⎡ ⎡ 1 λ λ λ ((γ (x ) ∧ γ (x )) + (γ (x ) ∧ γ (x )) + ... + (γ (x ) ∧ γ (x )) )+ E 1 F 1 E 1 F 1 E 1 F 1 1 1 2 2 1 λ λ λ ⎢ ⎢ ((ω (x ) ∧ ω (x )) + (ω (x ) ∧ ω (x )) + ... + (ω (x ) ∧ ω (x )) ) 1 γ 1 γ 1 γ 1 γ 1 γ 1 γ 1 ⎢ ⎢  E F E F E F 4 1 1 2 2 ⇔ S (E, F) = ⎢ ⎢ 1 λ λ λ ⎣ n ⎣ ((γ (x ) ∨ γ (x )) + (γ (x ) ∨ γ (x )) + ... + (γ (x ) ∨ γ (x )) )+ E 1 F 1 E 1 F 1 E 1 F 1 1 1 2 2 1 λ λ λ ((ω (x ) ∨ ω (x )) + (ω (x ) ∨ ω (x )) + ... + (ω (x ) ∨ ω (x )) ) γ 1 γ 1 γ 1 γ 1 γ 1 γ 1 E F E F E F 1 1 2 2 1 λ λ λ ((γ (x ) ∧ γ (x )) + (γ (x ) ∧ γ (x )) + ... + (γ (x ) ∧ γ (x )) )+ E 2 F 2 E 2 F 2 E 2 F 2 1 1 2 2 1 λ λ λ ((ω (x ) ∧ ω (x )) + (ω (x ) ∧ ω (x )) + ... + (ω (x ) ∧ ω (x )) ) γ 2 γ 2 γ 2 γ 2 γ 2 γ 2 E F E F E F 1 1 2 2 + + ... 1 λ λ λ ((γ (x ) ∨ γ (x )) + (γ (x ) ∨ γ (x )) + ... + (γ (x ) ∨ γ (x )) )+ E 2 F 2 E 2 F 2 E 2 F 2 1 1 2 2 1 λ λ λ ((ω (x ) ∨ ω (x )) + (ω (x ) ∨ ω (x )) + ... + (ω (x ) ∨ ω (x )) ) γ 2 γ 2 γ 2 γ 2 γ 2 γ 2 E F E F E F 1 1 2 2 ⎤ ⎤ 1 λ λ λ ((γ (x ) ∧ γ (x )) + (γ (x ) ∧ γ (x )) + ... + (γ (x ) ∧ γ (x )) )+ E n F n E n F n E n F n 1 1 2 2 1 λ λ λ ⎥ ⎥ ((ω (x ) ∧ ω (x )) + (ω (x ) ∧ ω (x )) + ... + (ω (x ) ∧ ω (x )) ) γ n γ n γ n γ n γ n γ n E F E F E F ⎥ ⎥ 1 1 2 2 + ⎥ ⎥ 1 λ λ λ ⎦ ⎦ ((γ (x ) ∨ γ (x )) + (γ (x ) ∨ γ (x )) + ... + (γ (x ) ∨ γ (x )) )+ E n F n E n F n E n F n 1 1 2 2 1 λ λ λ ((ω (x ) ∨ ω (x )) + (ω (x ) ∨ ω (x )) + ... + (ω (x ) ∨ ω (x )) ) γ n γ n γ n γ n γ n γ n E F E F E F 1 1 2 2 i2π(ω (x )) i2π(ω (x )) E k F k j j Now as E = F ⇔ μ (x ) = μ (x ) for k = 1, 2, ... , n ⇔ γ (x )e = γ (x )e E k F k E k F k j j i2π(ω (x )) i2π(ω (x )) E F k k j j for k = 1, 2, ... , n ⇔ γ (x ) = γ (x ) and e = e for k = 1, 2, ... , n. Then E k F k j j ! " 1 ⇔ S (E, F) = [1 + 1 + ... + 1] ⇔ S (E, F) = 1. 3. We have ⎡ ⎤ 1 1 λ 1 λ (γ (x ) ∧ γ (x )) + (ω (x ) ∧ ω (x )) E F γ γ 1 j k j k E k F k j=1  j=1 j j ⎣ ⎦ S (E, F) = 1  λ 1  λ (γ (x ) ∨ γ (x )) + (ω (x ) ∨ ω (x )) E k F k γ k γ k j=1 j j j=1 E F k=1 j j ⎡ ⎤ 1 1 λ 1 λ (γ (x ) ∧ γ (x )) + (ω (x ) ∧ ω (x )) F E γ γ 1 j k j k F k E k j=1  j=1 j j ⎣ ⎦ 1  λ 1  λ (γ (x ) ∨ γ (x )) + (ω (x ) ∨ ω (x )) F k E k γ k γ k j=1 j j j=1 F E k=1 j j = S (F, E). Theorem 6: The SM S (E, F) satisfy the following properties (1) 0 ≤ S (E, F) ≤ 1; (2) S (E, F) = 1if and only if E = F; 5 5 (3) S (E, F) = S (F, E). c c λ λ λ Proof: 1. Since (γ (x ) ∧ γ (x )) ∈ [0, 1], (ω (x ) ∧ ω (x )) ∈ [0, 1], (γ (x ) ∨ γ (x )) E F γ γ E F k k k k k k j j E F j j j j (γ (x )∧γ (x )) E F k k j j ∈ [0, 1], (ω (x ) ∨ ω (x )) ∈ [0, 1] this implies that α ∈ [0, 1] and γ k γ k cc E F λ j j (γ (x )∨γ (x )) E k F k j j 58 T. MAHMOOD ET AL. (ω (x )∧ω (x )) γ γ k k E F j j β ∈ [0, 1]. Thus for k = 1wehave cc (ω (x )∨ω (x )) γ γ k k E F j j % & (ω (x ) ∧ ω (x )) (γ (x ) ∧ γ (x )) γ 1 γ 1 1 E F E 1 F 1 j j j j α + β ∈ [0, 1] cc cc λ λ (γ (x ) ∨ γ (x )) (ω (x ) ∨ ω (x )) E 1 F 1 γ 1 γ 1 j j E F j=1 j j For k = 2 % & (ω (x ) ∧ ω (x )) (γ (x ) ∧ γ (x )) γ 2 γ 2 1 E F E 2 F 2 j j j j α + β ∈ [0, 1] cc cc λ λ (γ (x ) ∨ γ (x )) (ω (x ) ∨ ω (x )) E 2 F 2 γ 2 γ 2 j j E F j=1 j j By doing this process we obtain ⎛ ⎞ % & (ω (x ) ∧ ω (x )) (γ (x ) ∧ γ (x )) γ k γ k 1 E F E k F k j j j j ⎝ ⎠ α + β ∈ n[0, 1] cc cc λ λ (γ (x ) ∨ γ (x )) (ω (x ) ∨ ω (x )) E k F k γ k γ k j j E F j=1 j j k=1 ⎛ ⎞ % & (ω (x ) ∧ ω (x )) (γ (x ) ∧ γ (x )) γ k γ k 1 E F E F k k j j j j ⎝ ⎠ ⇒ 0 ≤ α + β ≤ n cc cc λ λ (γ (x ) ∨ γ (x )) (ω (x ) ∨ ω (x )) E k F k γ k γ k j j E F j=1 j j k=1 ⎡ ⎛ ⎞ ⎤ % & (ω (x ) ∧ ω (x )) (γ (x ) ∧ γ (x )) γ k γ k 1 1 E F E F k k j j j j ⎣ ⎝ ⎠ ⎦ ⇒ 0 ≤ α + β ≤ 1 cc cc λ λ (γ (x ) ∨ γ (x )) (ω (x ) ∨ ω (x )) E F γ γ k k k k j j E F j=1 j j k=1 ⎡ ⎛ ⎞ ⎤ % & (ω (x ) ∧ ω (x )) (γ (x ) ∧ γ (x )) γ k γ k 1 1 E F E k F k j j j j ⎣ ⎝ ⎠ ⎦ ⇒ 0 ≤ α + β ≤ 1 cc cc λ λ (γ (x ) ∨ γ (x )) (ω (x ) ∨ ω (x )) E k F k γ k γ k j j E F j=1 j j k=1 ⇒ 0 ≤ S (E, F) ≤ 1. 2. By definition 7 we have ⎡ ⎛ ⎞ ⎤ 1 % & λ λ (ω (x ) ∧ ω (x )) (γ (x ) ∧ γ (x )) γ k γ k 1 1 E k F k E F j j j j ⎣ ⎝ ⎠ ⎦ S (E, F) = α + β cc cc λ λ (γ (x ) ∨ γ (x )) (ω (x ) ∨ ω (x )) E k F k γ k γ k j j E F j=1 k=1 j j ⎡ ⎛ λ λ λ (γ (x ) ∧ γ (x )) + (γ (x ) ∧ γ (x )) + ... + (γ (x ) ∧ γ (x )) 1 1 E k F k E k F k E k F k 5 1 1 2 2 ⎣ ⎝ ⇔ S (E, F) = α cc λ λ λ (γ (x ) ∨ γ (x )) + (γ (x ) ∨ γ (x )) + ... + (γ (x ) ∨ γ (x )) E k F k E k F k E k F k 1 1 2 2 k=1 j=1 && λ λ λ λ (ω (x ) ∧ ω (x )) + (ω (x ) ∧ ω (x )) + ... + (ω (x ) ∧ ω (x )) γ k γ k γ k γ k γ k γ k E F E F E F 1 1 2 2 + β cc λ λ λ (ω (x ) ∨ ω (x )) + (ω (x ) ∨ ω (x )) + ... + (ω (x ) ∨ ω (x )) γ k γ k γ k γ k γ k γ k E F E F E F 1 1 2 2 ⎡ ⎡ ⎡ λ λ λ 1 1 (γ (x ) ∧ γ (x )) + (γ (x ) ∧ γ (x )) + ... + (γ (x ) ∧ γ (x )) E 1 F 1 E 1 F 1 E 1 F 1 5 1 1 2 2 ⎣ ⎣ ⎣ ⇔ S (E, F) = α cc λ λ λ (γ (x ) ∨ γ (x )) + (γ (x ) ∨ γ (x )) + ... + (γ (x ) ∨ γ (x )) E 1 F 1 E 1 F 1 E 1 F 1 1 1 2 2 j=1 λ λ λ (ω (x ) ∧ ω (x )) + (ω (x ) ∧ ω (x )) + ... + (ω (x ) ∧ ω (x )) γ 1 γ 1 γ 1 γ 1 γ 1 γ 1 E F E F E F 1 1 2 2 +β cc λ λ λ (ω (x ) ∨ ω (x )) + (ω (x ) ∨ ω (x )) + ... + (ω (x ) ∨ ω (x )) γ 1 γ 1 γ 1 γ 1 γ 1 γ 1 E F E F E F 1 1 2 2 λ λ λ 1 (γ (x ) ∧ γ (x )) + (γ (x ) ∧ γ (x )) + ... + (γ (x ) ∧ γ (x )) E 2 F 2 E 2 F 2 E 2 F 2 1 1 2 2 + α cc λ λ λ (γ (x ) ∨ γ (x )) + (γ (x ) ∨ γ (x )) + ... + (γ (x ) ∨ γ (x )) E 2 F 2 E 2 F 2 E 2 F 2 1 1 2 2 j=1 λ λ λ (ω (x ) ∧ ω (x )) + (ω (x ) ∧ ω (x )) + ... + (ω (x ) ∧ ω (x )) γ 2 γ 2 γ 2 γ 2 γ 2 γ 2 E F E F E F 1 1 2 2 +β + ... cc λ λ λ (ω (x ) ∨ ω (x )) + (ω (x ) ∨ ω (x )) + ... + (ω (x ) ∨ ω (x )) γ 2 γ 2 γ 2 γ 2 γ 2 γ 2 E F E F E F 1 1 2 2   FUZZY INFORMATION AND ENGINEERING 59 λ λ λ 1 (γ (x ) ∧ γ (x )) + (γ (x ) ∧ γ (x )) + ... + (γ (x ) ∧ γ (x )) E n F n E n F n E n F n 1 1 2 2 + α cc λ λ λ (γ (x ) ∨ γ (x )) + (γ (x ) ∨ γ (x )) + ... + (γ (x ) ∨ γ (x )) E n F n E n F n E n F n 1 1 2 2 j=1 j=1 &&& λ λ λ (ω (x ) ∧ ω (x )) + (ω (x ) ∧ ω (x )) + ... + (ω (x ) ∧ ω (x )) γ n γ n γ n γ n γ n γ n E F E F E F 1 1 2 2 +β cc λ λ λ (ω (x ) ∨ ω (x )) + (ω (x ) ∨ ω (x )) + ... + (ω (x ) ∨ ω (x )) γ n γ n γ n γ n γ n γ n E F E F E F 1 1 2 2 i2π(ω (x )) i2π(ω (x )) E k F k j j Now as E = F ⇔ μ (x ) = μ (x ) for k = 1, 2, ... , n ⇔ γ (x )e = γ (x )e E k F k E k F k j j i2π(ω (x )) i2π(ω (x )) E F k k j j for k = 1, 2, ... , n ⇔ γ (x ) = γ (x ) and e = e for k = 1, 2, ... , n.and E k F k j j α , β ∈ [0, 1] such that α + β = 1. Then cc cc cc cc ⇔ S (E, F) = 1. 3. We have ⎡ ⎛ ⎞ ⎤ % & (ω (x ) ∧ ω (x )) (γ (x ) ∧ γ (x )) γ k γ k 1 1 E F E F k k j j j j ⎣ ⎝ ⎠ ⎦ S (E, F) = α + β cc cc λ λ (γ (x ) ∨ γ (x )) (ω (x ) ∨ ω (x )) E F γ γ k k k k j j E F j=1 j j k=1 ⎡ ⎛ ⎞ ⎤ % & (ω (x ) ∧ ω (x )) (γ (x ) ∧ γ (x )) γ k γ k 1 1 F E F E k k j j j j ⎣ ⎝ ⎠ ⎦ = α + β cc cc λ λ (γ (x ) ∨ γ (x )) (ω (x ) ∨ ω (x )) F E γ γ k k k k j j F E j=1 j j k=1 = S (F, E). Remark 3: If λ = 1 then without exponential based generalized SMs become # $ ⎡ ⎤ 1 1 1 − |γ (x ) − γ (x )|∨ |ω (x ) − ω (x )| E F γ γ j k j k E k F k 1  j=1  j=1 j j ⎣ ⎦ S (E, F) = # $ 1 1 1 + |γ (x ) − γ (x )|∨ |ω (x ) − ω (x )| k=1 E k F k γ k γ k j=1 j j j=1 E F j j 1  1 (γ (x ) ∧ γ (x )) + (ω (x ) ∧ ω (x )) E k F k γ k γ k 1 j=1 j j j=1 E F j j S (E, F) = 1 1 (γ (x ) ∨ γ (x )) + (ω (x ) ∨ ω (x )) E k F k γ k γ k j j E F j=1  j=1 k=1 j j ⎛ ⎞ % & (ω (x ) ∧ ω (x )) (γ (x ) ∧ γ (x )) γ k γ k 1 1 E F E F j k j k j j ⎝ ⎠ S (E, F) = α + β cc cc n  (γ (x ) ∨ γ (x )) (ω (x ) ∨ ω (x )) E F γ γ k k k k j j E F j j j=1 k=1 Now we defined exponential based weighted generalized SMs and without exponential based weighted generalized SMs. Definition 12: Let E and F be two CHFS on X. Then the exponential based weighted generalized SM is calculated as ⎡ ⎡   ⎤ ⎤ 1 λ 1 λ 1− |γ (x )−γ (x )| ∨ |ω (x )−ω (x )| E k F k γ k γ k j j E F j j ⎢ ⎢ ⎥ ⎥ j=1 j=1 S (E, F) = w 2 − 1 ⎣ ⎣ ⎦ ⎦ k=1 60 T. MAHMOOD ET AL. where λ> 0, and ∨ described the maximum operation and w ∈ [0, 1] be the weight of each element x for k = 1, 2, 3, .., n such that w = 1. k k k=1 Remark 4: If λ = 1 then the exponential based weighted generalized SM becomes ⎡   ⎤ 1 1 1− |γ (x )−γ (x )|∨ |ω (x )−ω (x )| E k F k γ k γ k E F j j j j ⎢ ⎥ j=1 j=1 S (E, F) = w 2 − 1 ⎣ ⎦ k=1 Definition 13: Let E and F be two CHFS on X. Then we can also calculate the exponential based weighted generalized SM as follows ⎡ ⎡ ⎤ ⎤ 1 λ 1 λ 1− |γ (x )−γ (x )| + |ω (x )−ω (x )| E k F k γ k γ k j j  E F ⎢ ⎢ j j ⎥ ⎥ j=1 j=1 S (E, F) = w 2 − 1 ⎣ k ⎣ ⎦ ⎦ k=1 where λ> 0. Remark 5: If λ = 1 then the exponential based weighted generalized SM becomes ⎡ ⎤ 1 1 1− |γ (x )−γ (x )|+ |ω (x )−ω (x )| γ γ E k F k k k j j  E F ⎢ j j ⎥ j=1 j=1 S (E, F) = w 2 − 1 ⎣ ⎦ k=1 Definition 14: Let E and F be two CHFSs on X. Then without exponential based weighted generalized SMs are calculated as follows # $ 1 ⎡ ⎡ ⎤ ⎤ 1  1 λ λ 1 − |γ (x ) − γ (x )| ∨ |ω (x ) − ω (x )| E F γ γ j k j k E k F k j=1  j=1 j j ⎣ ⎣ ⎦ ⎦ # $ S (E, F) = w 1  1 λ λ 1 + |γ (x ) − γ (x )| ∨ |ω (x ) − ω (x )| E F γ γ k=1 j k j k E k F k j=1  j=1 j j ⎡ ⎤ 1  1 n λ λ (γ (x ) ∧ γ (x )) + (ω (x ) ∧ ω (x )) E F γ γ k k k k j=1 j j  j=1 E F j j ⎣ ⎦ S (E, F) = w 1  λ 1  λ (γ (x ) ∨ γ (x )) + (ω (x ) ∨ ω (x )) E k F k γ k γ k j=1 j j j=1 E F k=1   j j ⎡ ⎛ ⎞ ⎤ 1 % & (ω (x ) ∧ ω (x )) (γ (x ) ∧ γ (x )) γ γ 1 E k F k E k F k j j j j ⎣ ⎝ ⎠ ⎦ S (E, F) = w α + β k cc cc w λ λ (γ (x ) ∨ γ (x )) (ω (x ) ∨ ω (x )) E k F k γ k γ k j j E F j=1 j j k=1 where λ> 0and α , β ∈ [0, 1] such that α + β = 1. cc cc cc cc Remark 6: If λ = 1 then without exponential based weighted generalized SMs become # $ ⎡ ⎤ 1 1 1 − |γ (x ) − γ (x )|∨ |ω (x ) − ω (x )| E k F k γ k γ k j j E F j=1  j=1 j j ⎣ ⎦ S (E, F) = w # $ 1 1 1 + |γ (x ) − γ (x )|∨ |ω (x ) − ω (x )| E F γ γ k=1 k k k k j=1 j j  j=1 E F j j ⎡ ⎤ 1 1 (γ (x ) ∧ γ (x )) + (ω (x ) ∧ ω (x )) E k F k γ k γ k j=1 j j j=1 E F j j ⎣ ⎦ S (E, F) = w 1 1 (γ (x ) ∨ γ (x )) + (ω (x ) ∨ ω (x )) E k F k γ k γ k j=1 j j j=1 E F k=1   j j FUZZY INFORMATION AND ENGINEERING 61 ⎛ ⎞ % & (ω (x ) ∧ ω (x )) γ γ (γ (x ) ∧ γ (x )) k k 1 E k F k E F j j j j ⎝ ⎠ S (E, F) = w α + β k cc cc (γ (x ) ∨ γ (x )) (ω (x ) ∨ ω (x )) E F γ γ j k j k E k F k j j k=1 j=1 5. Application In this portion, the SMs and WSMs are applied to two cases which are pattern recognition and medical diagnosis. We evaluate the performance of the SMs in dealing with different practical world problems. 5.1. Pattern Recognition Example 3: Let X = {x , x , x , x } be a set and four know patterns E (j = 1, 2, 3, 4) which 1 2 3 4 j are given in the form of CHFSs as follows i2π(1) i2π(0.1) i2π(0.6) i2π(0.3) i2π(0.2) (x , 0.9e , 0.1e ), (x , 0.4e , 0.7e , 0.5e ), 1 2 E = i2π(0.6) i2π(0.8) i2π(0.7) (x , 0.2e ), (x , 0.3e , 0.2e ) 3 4 i2π(0.5) i2π(0.4) i2π(0.5) (x , 0.3e ), (x , 0.6e ,1e ), 1 2 E = i2π(0.8) i2π(0.7) i2π(0.6) i2π(0.1) (x , 0.2e , 0.2e ), (x , 0.9e , 0.7e ) 3 4 i2π(0.3) i2π(0.1) i2π(0.2) i2π(0.6) (x , 0.6e ), (x , 0.2e , 0.4e , 0.3.e ), 1 2 E = i2π(0.1) i2π(0.2) i2π(0.8) i2π(0.4) (x , 0.8e ,1e ), (x , 0.5e , 0.8e , ) 3 4 i2π(0.9) i2π(1) i2π(0.5) i2π(0.2) i2π(0.4) (x , 0.3e ,1e , 0.9e ), (x , 0.4.e , 0.7e ), 1 2 E = i2π(1) i2π(0.2) i2π(0.6) (x , 0.1e ), (x , 0.2e , 0.9e ) 3 4 Next let an unknown pattern which need to be identify i2π(0.1) i2π(0.5) i2π(0.6) i2π(0.5) (x , 0.3e , 0.9e ), (x , 0.5e , 0.6e ), 1 2 E = i2π(0.9) i2π(0.7) i2π(0.4) i2π(1) (x , 0.4e ), (x , 0.8e , 0.2e ,1e ), 3 4 In Table 1 we calculated the proposed SMs from E to E (j = 1, 2, 3, 4). The motive of this issue is to find that the unknown pattern E belong to which of the Pattern E (j = 1, 2, 3, 4). From the calculation stated in Table 1, we obtained the following consequences 1 2 (1) The similarity degree between E and E is a massive one as got by SMs, SM ,SM ,and c c 3 1 2 SM . So by the principle of the maximum degree of similarity the SMs SM ,SM and c c c SM allot the unknown pattern E to the pattern E . 4 5 (2) The similarity degree between E and E is a massive one as got by SMs, SM ,and SM . c c 4 5 So by the principle of the maximum degree of similarity the SMs SM and SM allot the c c unknown pattern E to the pattern E . If we let the weight of each element x (k = 1, 2, 3, 4) are 0.1, 0.2, 0.3 and 0.4 respectively. Then the calculation of proposed SMs are stated in Table 2. From the calculation stated in Table 2, we obtained the following consequences 62 T. MAHMOOD ET AL. Table 1. Calculation of proposed SMs for λ = 1and (α , β ) = (0.5, 0.5). cc cc Similarity measures E E E E Ranking 1 2 3 4 S (E, E ) 0.4928 0.5109 0.3218 0.4503 E ≥ E ≥ E ≥ E j 2 1 4 3 S (E, E ) 0.5316 0.5686 0.4309 0.555 E ≥ E ≥ E ≥ E j 2 4 1 3 S (E, E ) 0.4175 0.4281 0.2536 0.3801 E ≥ E ≥ E ≥ E j 2 1 4 3 S (E, E ) 0.4465 0.4448 0.2529 0.5003 E ≥ E ≥ E ≥ E j 4 2 1 3 S (E, E ) 0.4366 0.4259 0.2516 0.4667 E ≥ E ≥ E ≥ E j 4 1 2 3 Table 2. Calculation of proposed WSMs for Example (3) based on λ = 1and (α , β ) = (0.5, 0.5). cc cc Similarity Measures E E E E Ranking 1 2 3 4 S (E, E ) 0.5164 0.4902 0.3065 0.4259 E ≥ E ≥ E ≥ E j 1 2 4 3 S (E, E ) 0.5543 0.5537 0.4157 0.5253 E ≥ E ≥ E ≥ E j 1 2 4 3 S (E, E ) 0.4368 0.4082 0.2405 0.3594 E ≥ E ≥ E ≥ E j 1 2 4 3 S (E, E ) 0.4717 0.4514 0.2772 0.4725 E ≥ E ≥ E ≥ E j 4 1 2 3 S (E, E ) 0.4588 0.4367 0.2776 0.432 E ≥ E ≥ E ≥ E j 1 2 4 3 Figure 2. Graphical representation of established SMs for Example 3. 1 2 (1) The similarity degree between E and E is a massive one as got by SMs, SM ,SM , c c w w 3 5 1 SM and SM . So by the principle of the maximum degree of similarity the SMs SM , c c c w w w 2 3 5 SM ,SM ,andSM allot the unknown pattern E to the pattern E . c c c w w w (2) The similarity degree between E and E is a massive one as got by SM, SM .Sobythe principle of the maximum degree of similarity the SM SM allot the unknown pattern E to the pattern E . The ranking of the proposed SMs and WSMs are also stated in Tables 1 and 2 respectively. The graphical representation of the proposed SMs is shown in Figure 1 andproposedWSMs are shown in Figure 2. FUZZY INFORMATION AND ENGINEERING 63 Table 3. Calculation of proposed SMs for λ = 1and (α , β ) = (0.5, 0.5). cc cc Similarity measures D D D D Ranking 1 2 3 4 S (P,D ) 0.5396 0.5083 0.4023 0.2781 D ≥ D ≥ D ≥ D j 1 2 3 4 S (P,D ) 0.6447 0.5759 0.4881 0.3554 D ≥ D ≥ D ≥ D j 1 2 3 4 S (P,D ) 0.4588 0.4335 0.336 0.2173 D ≥ D ≥ D ≥ D j 1 2 3 4 S (P,D ) 0.5249 0.3981 0.2614 0.1474 D ≥ D ≥ D ≥ D j 1 2 3 4 S (P,D ) 0.5713 0.3956 0.2675 0.1488 D ≥ D ≥ D ≥ D j 1 2 3 4 5.2. Medical Diagnosis The symptoms of different diseases are different. The medical diagnosis depends on the victim’s symptoms which show what type of disease a victim has. The multiple symptoms of a victim represent a symptom set and a set of diseases can represent by different diseases. Example 4: Let a set of diagnoses D = {D (Coronovirous),D (Pneumonia),D (Flu),D 1 2 3 4 (Chestproblem)} and a set of symptoms X ={x (shortofbreath), x (Fever), x (cough), 1 2 3 x (chestpain)}. The victim’s symptoms can be showed in the form of CHFSs as below i2π(1) i2π(1) i2π(0.5) i2π(0.8) i2π(0.3) i2π(0.4) (x , 0.9e , 0.5e , 0.5e ), (x , 0.5e , 0.7e , 0.4e ), 1 2 P = i2π(0.6) i2π(0.9) i2π(0.3) (x , 0.2e , 0.8e ), (x , 0.1e ) 3 4 The symptoms of each disease D (j = 1, 2, 3, 4) canbeshowedinCHFSs asbelow i2π(0.5) i2π(0.8) i2π(0.7) i2π(0.9) (x , 1e , 0.8e ), (x , 0.5e , 0.6e ), 1 2 D (coronovirous) = i2π(0.7) i2π(1) i2π(0.7) i2π(0.1) (x , 0.8e , 0.6e , 0.9e ), (x , 0.1e ), 3 4 i2π(0.2) i2π(0.7) i2π(0.9) (x , 0.1e ), (x , 0.6e , 0.4e ), 1 2 D (Pneumonia) = i2π(0.6) i2π(0.6) i2π(0.4) i2π(0.2) (x , 0.4e , 0.5e ), (x , 0.3e , 0.4e ) 3 4 i2π(0.0) i2π(0.2) i2π(0.5) (x , 0.1e ), (x , 0.3e , 0.2e ), 1 2 D (Flu) = i2π(0.8) i2π(0.7) i2π(0.6) i2π(0.2) i2π(0.4) (x , 1e , 0.6e , 0.9e ), (x , 0.1e , 0.2e ) 3 4 i2π(0.1) i2π(0.2) i2π(0.2) i2π(0.2) (x , 0.2e , 0.3e ), (x , 0.1.e , 0.0e ), 1 2 D (Chestpain) = i2π(0.3) i2π(0.9) i2π(0.7) i2π(0.6) (x , 0.1e ), (x , 1e , 0.9e , 0.5e ) 3 4 In Table 3 we calculated the proposed SMs from P to D (j = 1, 2, 3, 4). The motive of this issue is to know about the disease of the victim that what disease a victim has in the above four diseases D (j = 1, 2, 3, 4). From the calculation stated in Table 3, we obtained that the similarity degree between P and D is a massive one as got by all SMs. So by the principle of the maximum degree of similarity, we can say that a victim has coronavirus. If we let the weight of each element x (k = 1, 2, 3, 4) are 0.1, 0.2, 0.3 and 0.4 respectively. Then the calculation of proposed SMs are stated in Table 4. From the calculation stated in Table 4, we obtained the following consequences 2 4 (1) The similarity degree between P and D is a massive one as got by SMs, SM ,SM , c c w w 5 2 4 and SM . So by the principle of the maximum degree of similarity the SMs SM ,SM c c c w w w and SM give us that a victim has coronavirus. w 64 T. MAHMOOD ET AL. Table 4. Calculation of proposed WSMs for Example (4) based on λ = 1and (α , β ) = (0.5, 0.5). cc cc Similarity measures D D D D Ranking 1 2 3 4 S (P,D ) 0.5633 0.5821 0.4749 0.2692 D ≥ D ≥ D ≥ D j 2 1 3 4 S (P,D ) 0.6802 0.6525 0.5662 0.3376 D ≥ D ≥ D ≥ D j 1 2 3 4 S (P,D ) 0.4852 0.5013 0.4022 0.2099 D ≥ D ≥ D ≥ D j 2 1 3 4 cw S (P,D ) 0.5106 0.4423 0.3089 0.1332 D ≥ D ≥ D ≥ D j 1 2 3 4 S (P,D ) 0.5812 0.4433 0.3163 0.1327 D ≥ D ≥ D ≥ D j 1 2 3 4 Figure 3. Graphical representation of established WSMs for Example 3. (2) The similarity degree between P and D is a massive one as got by SMs, SM ,and 3 1 3 SM . So by the principle of the maximum degree of similarity the SMs SM ,andSM c c c w w w provide that a victim has pneumonia. The ranking of the proposed SMs and WSMs are also stated in Tables 3 and 4 respectively. The graphical representation of the proposed SMs is shown in Figure 2 andproposedWSMs are shown in Figure 3. 6. Comparison Our goal to expand the new SMs is to deal with new kinds of data such as CHFSs and exist- ing data CFSs, HFSs, and FSs. In this portion, we expressed the benefits of proposed SMs by comparing with existing SMs. The geometrical representations of the information’s of example 4 and example 5, are discussed in Figures 4–7. Example 5: Let X = {x , x , x , x } be a set and four know patterns E (j = 1, 2, 3, 4) which 1 2 3 4 j are given in the form of HFSs as follows (x , {0.1, 0.4, 0.6}), (x , {0.4, 1}), 1 2 E = (x , {0.7, 0.8}), (x , {0.2}), 3 4 FUZZY INFORMATION AND ENGINEERING 65 Figure 4. Graphical representation of established WSMs for Example 4. Figure 5. Graphical representation of established WSMs for Example 4. (x , {0.3, 0.5, 0.8}), (x , {0.4., 0.6}), 1 2 E = (x , {0.1, 0.3}), (x , {0.2, 0.4, 0.7}) 3 4 (x , {0.6, 0.8}), (x , 0.1, 0.2, 0.6), 1 2 E = (x , {0.7}), (x , {0.4, 0.5}) 3 4 (x , {0.1, 0.3}), (x , {0.4, 0.8}), 1 2 E = (x , {0.4, 0.7, 0.8}), (x , {0.4}), 3 4 66 T. MAHMOOD ET AL. Figure 6. Graphical representation of the comparison of the establish SMs with existing SMs for Example 5. Figure 7. Graphical representation of the comparison of the establish SMs with existing SMs for Example 3. Next let an unknown pattern which need to be identify (x , {0.2, 0.3, 0.6}), (x , {0.8, 1}), 1 2 E = (x , {0.5, 0.8}), (x , {1}) 3 4 Nowwehavethat e = 1 then the data given in the HFSs converted into the CHFSs as follows i2π(0.0) i2π(0.0) i2π(0.0) i2π(0.0) i2π(0.0) (x , 0.1e , 0.4e , 0.6e ), (x , 0.4e , 0.1e ), 1 2 E = i2π(0.0) i2π(0.0) i2π(0.0) (x , 0.7e , 0.8e ), (x , 0.2.e ) 3 4 FUZZY INFORMATION AND ENGINEERING 67 Table 5. Comparison between Proposed SMs and Existing SMs for Example (5) and λ = 1, (α , β ) = cc cc (1, 0). Method Score value Ranking Xu and Xia [25] s (E, E ) =0.7083, s (E, E ) = 0.6042s (E, E ) = 0.6083, s (E, E ) = 0.7 E ≥ E ≥ E ≥ E g 1 g 2 g 3 g 4 1 4 3 2 Zeng et al. [32] s (E, E ) = 0.8375, s (E, E ) = 0.6792s (E, E ) = 0.6167, s (E, E ) = 0.7979 E ≥ E ≥ E ≥ E h 1 h 2 h 3 h 4 1 4 2 3 Tang et al. [33] s (E, E ) = 0.6319, s (E, E ) = 0.3652s (E, E ) = 0.2333,s (E, E ) = 0.5541 E ≥ E ≥ E ≥ E 1 2 3 4 1 4 2 3 h h h h 1 1 1 1 Proposed SM S (E, E ) =0.6664, S (E, E ) = 0.5127S (E, E ) =0.3563, S (E, E ) = 0.4819 E ≥ E ≥ E ≥ E 1 2 3 4 1 4 2 3 c c c c 2 2 2 2 Proposed SM S (E, E ) = 0.6664, S (E, E ) =0.5127 S (E, E ) = 0.3563, S (E, E ) = 0.5582 E ≥ E ≥ E ≥ E 1 2 4 5 1 4 2 3 c c c c 3 3 3 3 Proposed SM S (E, E ) = 0.6177, S (E, E ) = 0.4366S (E, E ) = 0.2833, S (E, E ) = 0.4775 E ≥ E ≥ E ≥ E 1 2 3 4 1 4 2 3 c c c c 4 4 4 4 Proposed SM S (E, E ) = 0.6694, S (E, E ) = 0.4115S (E, E ) = 0.2438, S (E, E ) = 0.4885 E ≥ E ≥ E ≥ E 1 2 3 4 1 4 2 3 c c c c 5 5 5 5 Proposed SM S (E, E ) = 0.6694, S (E, E ) = 0.4115S (E, E ) = 0.2438, S (E, E ) = 0.4885 E ≥ E ≥ E ≥ E c 1 c 2 c 3 c 4 1 4 2 3 Table 6. Comparison between proposed SMs with existing SMs for Example (3) and λ = 1. Method Score value Ranking Xu and Xia [25] Failed Failed Zeng et al. [32] Failed Failed Tang et al. [33] Failed Failed 1 1 1 1 Proposed SM S (E, E ) = 0.4928,S (E, E ) =0.5109 S (E, E ) = 0.3218, S (E, E ) = 0.4503 E ≥ E ≥ E ≥ E 1 2 3 4 2 1 4 3 c c c c 2 2 2 2 Proposed SM S (E, E ) = 0.5316,S (E, E ) = 0.5686S (E, E ) = 0.4309, S (E, E ) = 0.555 E ≥ E ≥ E ≥ E 1 2 3 4 2 4 1 3 c c c c 3 3 3 3 Proposed SM S (E, E ) = 0.4175,S (E, E ) = 0.4281S (E, E ) = 0.2536, S (E, E ) = 0.3801 E ≥ E ≥ E ≥ E 1 2 3 4 2 4 1 3 c c c c 4 4 4 4 Proposed SM S (E, E ) = 0.4465,S (E, E ) = 0.4448S (E, E ) = 0.2529, S (E, E ) = 0.5003 E ≥ E ≥ E ≥ E 1 2 3 4 4 2 1 3 c c c c 5 5 5 5 Proposed SM S (E, E ) = 0.4366,S (E, E ) = 0.4259S (E, E ) = 0.2516, S (E, E ) = 0.4667 E ≥ E ≥ E ≥ E 1 2 3 4 4 1 2 3 c c c c i2π(0.0) i2π(0.0) i2π(0.0) i2π(0.0) i2π(0.0) (x , 0.3e , 0.5e , 0.8e ), (x , 0.4e , 0.6.e ), 1 2 E = i2π(0.0) i2π(0.0) i2π(0.0) i2π(0.0) i2π(0.0) (x , 0.1e , 0.3e ), (x , 0.2e , 0.4e , 0.7e ) 3 4 i2π(0.0) i2π(0.0) i2π(0.0) i2π(0.0) i2π(0.0) (x , 0.6e , 0.8e ), (x , 0.1e , 0.2e , 0.6e ), 1 2 E = i2π(0.0) i2π(0.0) i2π(0.0) (x , 0.7e ), (x , 0.4e , 0.5e ) 3 4 i2π(0.0) i2π(0.0) i2π(0.0) i2π(0.0) i2π(0.0) (x , 0.9.e ,1.e ,1.e ), (x , 0.1.e , 0.3.e ), 1 2 E = iπ(0.0) i2π(0.0) i2π(0.0) i2π(0.0) (x , 0.4e , 0.7e , 0.8e ), (x , 0.4e ) 3 4 And i2π(0.0) i2π(0.0) i2π(0.0) i2π(0.0) i2π(0.0) (x , 0.2e , 0.3e , 0.6e ), (x , 0.8e ,1e ), 1 2 E = i2π(0.0) i2π(0.0) i2π(0.0) (x , 0.5e , 0.8e ), (x , 1e ) 3 4 From Table 5 we can observe that the data in the form of FSs and HFSs are solvable through existing SMs in the literature. The data in Example 5 is in the form HFSs which we can convert to CHFSs by taking 1 = e and through proposed SMs we get the similarity between E and E (j = 1, 2, 3, 4) as shown in Table 5. But what about the data in the form of CFSs and CHFSs, these type of data are unsolvable through the existing SMs as shown in Table 6. The data in Example 3 are in the form of CHFSs so we can find the similarity between E and E (j = 1, 2, 3, 4) through proposed SMs. This means that our proposed SMs are the extension of the existing SMs. Through the proposed SMs we can find the similarity between FS, HFS, CFS, and CHFSs. The ranking of Example 5 is given in Table 5. The similar- ity degree between E and E is a massive one as got by all SMs in Example 5. The ranking of Example 3 given in Table 6 is slightly different than ranking given in Table 5. The simi- 1 2 3 larity degree between E and E is a massive one as got by SMs, SM ,SM ,andSM and the c c c 4 5 similarity degree between E and E is a massive one as got by SMs, SM and SM . c c 68 T. MAHMOOD ET AL. 7. Conclusion SMs are utilized to inspect the variation between the two objects. The purpose of this arti- cle is to define the fundamental notion of CHFSs which is the combination of HFS and CFS and also defined their fundamental properties. We explored SMs for CHFSs. We presented exponential based generalized SMs and without exponential based generalized SMs for CHFSs. We obtained some valuable remarks. Further, the established SMs are used in Pat- tern recognition and medical diagnosis to inspect the practicability and credibility of the established SMs. Furthermore, we solved numerical examples for established SMs to show the supremacy and integrity of the proposed SMs. At last, to assess the trustworthiness of the established SMs we represented the comparison of the established SMs with some existing SMs. Our future work is to explore the application of CHFNs in many other type of researches [34–48]. Disclosure statement No potential conflict of interest was reported by the authors. Notes on contributors Tahir Mahmood is Assistant Professor of Mathematics at Department of Mathematics and Statistics, International Islamic University Islamabad, Pakistan. He received his Ph.D. degree in Mathematics from Quaid-i-Azam University Islamabad, Pakistan in 2012 under the supervision of Professor Dr. Muham- mad Shabir. His areas of interest are Algebraic structures, Fuzzy Algebraic structures and Soft sets. He has more than 65 international publications to his credit and he has also produced 38 MS students. Ubaid ur Rehman, received the M.Sc. degrees in Mathematics from International Islamic University Islamabad, in 2018. Currently, He is a Student of MS in mathematics from International Islamic Univer- sity Islamabad, Pakistan. His research interests include similarity measures, distance measures, fuzzy logic, fuzzy decision making, and their applications. Zeeshan Ali, received the B.S. degrees in Mathematics from Abdul Wali Khan University Mardan, Pak- istan, in 2016. He received has M.S. degrees in Mathematics from International Islamic University Islamabad, Pakistan, in 2018. Currently, He is a Student of Ph.D. in mathematics from International Islamic University Islamabad, Pakistan. His research interests include aggregation operators, fuzzy logic, fuzzy decision making, and their applications. 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Journal

Fuzzy Information and Engineering – Taylor & Francis

Published: Jan 2, 2020

Keywords: Complex fuzzy set; complex hesitant fuzzy sets; similarity measures

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