Article
Different Forms of Triangular Neutrosophic Numbers, De-Neutrosophication Techniques, and their Applications Avishek Chakraborty 1,4, Sankar Prasad Mondal 2, Ali Ahmadian 3,*, Norazak Senu 3, Shariful Alam 4 and Soheil Salahshour 5 1
Department of Basic Science, Narula Institute of Technology, Agarpara, Kolkata 700109, India;
tirtha.avishek93@gmail.com Department of Mathematics, Midnapore College (Autonomous), Midnapore, West Midnapore 721101, India; sankar.res07@gmail.com 3 Laboratory of Computational Sciences and Mathematical Physics , Institute for Mathematical Research, University Putra Malaysia, Serdang 43400 UPM, Malaysia; norazak@upm.edu.my 4 Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711103, India; salam50in@yahoo.co.in 5 Young Researchers and Elite Club, Mobarakeh Branch, Islamic Azad University, Mobarakeh, Iran; soheilsalahshour@yahoo.com * Correspondence: ahmadian.hosseini@gmail.com; Tel.: +60-12-668-7968 2
Received: 10 June 2018; Accepted: 03 July 2018; Published: 7 August 2018
Abstract: In this paper, we introduce the concept of neutrosophic number from different viewpoints. We define different types of linear and non-linear generalized triangular neutrosophic numbers which are very important for uncertainty theory. We introduced the de-neutrosophication concept for neutrosophic number for triangular neutrosophic numbers. This concept helps us to convert a neutrosophic number into a crisp number. The concepts are followed by two application, namely in imprecise project evaluation review technique and route selection problem. Keywords: linear and non-linear neutrosophic number; de-neutrosophication methods
1. Introduction 1.1. Theory of Uncertainty and Uncertainty Quantification Uncertainty theory playsanimportant role in modeling sciences and engineering problems. However, there is a basic question regarding how we can define or use the uncertainty concept in our mathematical modeling. Researchers around the globe defined many approaches to defining them, and give their various recommendations to using uncertainty theory. There are several literaturestudiesthatclassify some basic uncertain parameters. It should be noted that there is no unique reorientation of the uncertain parameter. For the problemΓ’€™s purpose or decision makersΓ’€™ choice, it can be varied and presented as a different application. We now, here, give some info about uncertain parameters, and show how they differ from eachother using the concept of uncertainty using some definition, flowcharts, and diagrams. In this paper, we recommend the researcher to take the uncertain parameter as a parametric interval valued neutrosophic number. Some basic differences between some uncertain parameters: If we take Interval number [1] then we can see, 1. 2.
The information belongs to a certain interval There is no concept of membership function
Symmetry2018, 10, 327; doi:10.3390/sym10080327
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If we take Fuzzy number [2,3], then we can see, 1. 2.
The concept of belongingness of the elements comes The use of membership function is present If we take Intuitionistic fuzzy number [4], then we can see,
1. 2.
The concept of belongingness and non-belongingness of the elementscomes The use of membership and non-membership function is present If we take Neutrosophic fuzzy number [5], then we can see,
1. 2.
The concept of truthiness, falsity, and indeterminacy of the elements comes The use of membership function for truthiness, falsity, and indeterminacy is present
Please follow the idea given in the flowchart below, as shown in Figure 1:
Figure 1. Flowchart for different uncertain parameter.
1.2. Neutrosophic Number Fuzzy systems (FSs) and Intuitionistic fuzzy systems (IFSs) cannot successfully deal with a situation where the conclusion is adequate, unacceptable, and decision-maker declaration is uncertain. Therefore, some novel theories are mandatory for solving the problem with uncertainty. The neutrosophic sets (NSs) [5] reflect on the truth membership, indeterminacy membership, and falsity membership concurrently, which is more practical and adequate than FSs and IFSs in commerce,which areuncertain, incomplete, and inconsistent in sequence. Single-valued neutrosophic sets are an extension of NSs which were introduced by Wang et al. [6]. Ye [7] introduced simplify neutrosophic sets, and Peng et al. [8,9] definite their novel operations and aggregation operators. Finally, there are different extensions of NSs, such as interval neutrosophic set [10], bipolar neutrosophic sets [11], and multi-valued neutrosophic sets [12,13]. The decision-making problem [14Γ’€“38]is very important in study, when it is with uncertainty. Although many researchers and scientists have worked in the recently developed neutrosophic method, and applied it in the field of decision making, there is, however, still some viewpoints regarding defining neutrosophic numbers in different forms, and their corresponding de-impreciseness is very important.
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1.3. Ranking and De-Impreciseness The ranking and de-impreciseness of the imprecise numbers are not a new concept.However, what is the basic concept of the above-said important results and what is the relation. Figure 2 shows the flowchart for de-impreciseness and ranking. Ranking is a concept where we can compare two imprecise numbers, and de-impreciseness is a technique where the imprecise number is converted to a crisp number. Somewhere, the decision maker takes the two concepts as the same. In this case, they convert the imprecise number into crisp number, and compares them on the basis of crisp value.
Figure 2. Flowchart for de-impreciseness and ranking.
1.4. Structure of the Paper The paper is organized as follows. In Section 1, the basic concept on imprecise set theory and neutrosophic set theory are discussed. Section 2 contains the preliminaries section. Section 3 goes for the known definition of neutrosophic sets and numbers. Single valued linear neutrosophic number and its variation are showing in Section 3. In Section 4, we address the basic concept of neutrosophic non-linear number and generalized neutrosophic number. In Section 5, the de-neutrosophication of linear neutrosophic triangular fuzzy number is performed. The PERT problem is considered in Section 6. The application in assignment problem, considering aproblem,is taken in Section 7. The conclusions are written in Section 8. 2. Neutrosophic Number Δƒ Definition 1. (Neutrosophic Set) A set Δ?‘† Δ?‘›Δ?‘’Δ?‘Λ in the universal discourse Δ?‘‹, which is denoted generically by Δƒ (Δ?‘Δ½), Δ?œ‡Δ?‘†Δƒ (Δ?‘Δ½), Δ?œ—Δ?‘†Δƒ (Δ?‘Δ½)]Γ’ŒΕ Γ’‹Ε½ Δ?‘Δ½ Γ’ˆˆ Δ?‘‹} , where Δ?‘Δ½ , is said to be a neutrosophic set if Δ?‘† Δ?‘›Δ?‘’Δ?‘Λ = {Γ’ŒΕ Δ?‘Δ½; [Δ?œ‹Δ?‘†Δƒ Δ?‘›Δ?‘’Δ?‘Λ Δ?‘›Δ?‘’Δ?‘Λ Δ?‘›Δ?‘’Δ?‘Λ (Δ?‘Δ½): Δ?‘‹ Γ’†’ [0,1] is called the truth membership function which represents the degree of confidence, Δ?œ‹Δ?‘†Δƒ Δ?‘›Δ?‘’Δ?‘Λ (Δ?‘Δ½): Δ?œ‡Δ?‘†Δƒ Δ?‘‹ Γ’†’ [0,1] is called the indeterminacy membership function which represents the degree of Δ?‘›Δ?‘’Δ?‘Λ (Δ?‘Δ½): Δ?‘‹ Γ’†’ [0,1] is called the falsity membership function which represents the degree of uncertainty, andΔ?œ—Δ?‘†Δƒ Δ?‘›Δ?‘’Δ?‘Λ scepticism on the decision given the decision maker. (Δ?‘Δ½), Δ?œ‡Δ?‘†Δƒ (Δ?‘Δ½)&Δ?œ—Δ?‘†Δƒ (Δ?‘Δ½) exhibits the following relation: Δ?œ‹Δ?‘†Δƒ Δ?‘›Δ?‘’Δ?‘Λ Δ?‘›Δ?‘’Δ?‘Λ Δ?‘›Δ?‘’Δ?‘Λ (Δ?‘Δ½) + Δ?œ‡Δ?‘†Δƒ (Δ?‘Δ½) + Δ?œ—Δ?‘†Δƒ (Δ?‘Δ½) Γ’‰Β€ 3 0 Γ’‰Β€ Δ?œ‹Δ?‘†Δƒ Δ?‘›Δ?‘’Δ?‘Λ Δ?‘›Δ?‘’Δ?‘Λ Δ?‘›Δ?‘’Δ?‘Λ Δƒ Definition 2. (Single Valued Neutrosophic Set) Neutrosophic set Δ?‘† Δ?‘›Δ?‘’Δ?‘Λ in the definition 2.3, is called a Single Δƒ Δƒ Valued Neutrosophic Set (Δ?‘†Δ?‘›Δ?‘’Δ?‘Λ ) if Δ?‘Δ½ is a single valued independent variable. Thus Δ?‘† Δ?‘›Δ?‘’Δ?‘Λ = (Δ?‘Δ½), (Δ?‘Δ½), (Δ?‘Δ½)]Γ’ŒΕ (Δ?‘Δ½), (Δ?‘Δ½)&Δ?œ— (Δ?‘Δ½) {Γ’ŒΕ Δ?‘Δ½; [Δ?œ‹Δ?‘†Δƒ Δ?œ‡ Δ?œ— Γ’‹Ε½ Δ?‘Δ½ Γ’ˆˆ Δ?‘‹} , where Δ?œ‹ Δ?œ‡ represents the truth, Δ?‘†Δƒ Δ?‘†Δƒ Δ?‘†Δƒ Δ?‘†Δƒ Δ?‘†Δƒ Δ?‘›Δ?‘’Δ?‘Λ Δ?‘›Δ?‘’Δ?‘Λ Δ?‘›Δ?‘’Δ?‘Λ Δ?‘›Δ?‘’Δ?‘Λ Δ?‘›Δ?‘’Δ?‘Λ Δ?‘›Δ?‘’Δ?‘Λ
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indeterminacy, and falsity membership function, respectively, as stated in definition 2.3, and also exhibits the same relationship as stated earlier. (Δ?‘Ž0 ) = 1, Δ?œ‡Δ?‘†Δƒ (Δ?‘?0 ) = 1 &Δ‘Δ?‘†Δƒ (Δ?‘?0 ) = 1, then If there exists three points,Δ?‘Ž0 , Δ?‘?0 &Δ?‘?0 , for which Δ?œ‹Δ?‘†Δƒ Δ?‘›Δ?‘’Δ?‘Λ Δ?‘›Δ?‘’Δ?‘Λ Δ?‘›Δ?‘’Δ?‘Λ Δƒ the Δ?‘†Δ?‘›Δ?‘’Δ?‘Λ is called neut-normal. Δƒ A Δ?‘† Δ?‘›Δ?‘’Δ?‘Λ is said to be neut-convex, which implies that it is a subset of a real line, by satisfying the following conditions: 1. 2. 3.
Γ’ŒΕ Δ?œŒΔ?‘Ž1 + (1 Γ’ˆ’ Δ?œŒ)Δ?‘Ž2 Γ’ŒΕ Γ’‰Δ½ Δ?‘šΔ?‘–Δ?‘›Γ’ŒΕ Δ?œ‹Δ?‘†Δƒ (Δ?‘Ž1 ), Δ?œ‹Δ?‘†Δƒ (Δ?‘Ž2 )Γ’ŒΕ Δ?œ‹Δ?‘†Δƒ Δ?‘›Δ?‘’Δ?‘Λ Δ?‘›Δ?‘’Δ?‘Λ Δ?‘›Δ?‘’Δ?‘Λ Γ’ŒΕ Δ?œŒΔ?‘Ž (1 Γ’ŒΕ (Δ?‘Ž ), (Δ?‘Ž2 )Γ’ŒΕ Δ?œ‡Δ?‘†Δƒ + Γ’ˆ’ Δ?œŒ)Δ?‘Ž Γ’‰Β€ Δ?‘šΔ?‘ŽΔ?‘Δ½Γ’ŒΕ Δ?œ‡ Δ?œ‡ 1 2 1 Δ?‘†Δƒ Δ?‘†Δƒ Δ?‘›Δ?‘’Δ?‘Λ Δ?‘›Δ?‘’Δ?‘Λ Δ?‘›Δ?‘’Δ?‘Λ Γ’ŒΕ Δ?œŒΔ?‘Ž1 + (1 Γ’ˆ’ Δ?œŒ)Δ?‘Ž2 Γ’ŒΕ Γ’‰Β€ Δ?‘šΔ?‘ŽΔ?‘Δ½Γ’ŒΕ Δ?œ—Δ?‘†Δƒ (Δ?‘Ž1 ), Δ?œ—Δ?‘†Δƒ (Δ?‘Ž2 )Γ’ŒΕ Δ?œ—Δ?‘†Δƒ Δ?‘›Δ?‘’Δ?‘Λ Δ?‘›Δ?‘’Δ?‘Λ Δ?‘›Δ?‘’Δ?‘Λ
where,Δ?‘Ž1 &Δ?‘Ž2 Γ’ˆˆ Γ’„? Δ?‘ŽΔ?‘›d Δ?œŒ Γ’ˆˆ [0,1]. Definition 3. (Single Valued Neutrosophic Number)Single Valued Neutrosophic Number (Δ?‘Β§Δƒ ) is defined as Δ?‘Β§Δƒ = Γ’ŒΕ [(Δ?‘?1 , Δ?‘ž1 , Δ?‘&#x;1 , Δ?‘ 1 ); Δ?›ΕΊ], [(Δ?‘?2 , Δ?‘ž 2 , Δ?‘&#x; 2 , Δ?‘ 2 ); Δ?›Λ], [(Δ?‘?3 , Δ?‘ž 3 , Δ?‘&#x; 3 , Δ?‘ 3 ); Δ?›ΕΎ]Γ’ŒΕ whereΔ?›ΕΊ, Δ?›Λ, Δ?›ΕΎ Γ’ˆˆ [0,1], the truth membership function (Δ?œ‹Δ?‘Β§Δƒ ): Γ’„? Γ’†’ [0, Δ?›ΕΊ] , the indeterminacy membership function (Δ?œ‡Δ?‘Β§Δƒ ): Γ’„? Γ’†’ [Δ?›Λ, 1] , and the falsity membership function (Δ?œ—Δ?‘Β§Δƒ ): Γ’„? Γ’†’ [Δ?›ΕΎ, 1] is given as: 1 1 Δ?œ‹Δ?‘Β§Δ?‘™ Δƒ (Δ?‘Δ½) Δ?‘? Γ’‰Β€ Δ?‘Δ½ Γ’‰Β€ Δ?‘ž 1 1 Δ?œ‹Δ?‘Β§Δƒ (Δ?‘Δ½) = { Δ?›ΕΊ Δ?‘ž1 Γ’‰Β€ Δ?‘Δ½ Γ’‰Β€ Δ?‘&#x;1 , Δ?œ‹Δ?‘Β§Δ?‘Λ Δƒ (Δ?‘Δ½) Δ?‘&#x; Γ’‰Β€ Δ?‘Δ½ Γ’‰Β€ Δ?‘ 0 otherwise 2 2 Δ?œ‡Δ?‘Β§Δ?‘™ Δƒ (Δ?‘Δ½) Δ?‘? Γ’‰Β€ Δ?‘Δ½ Γ’‰Β€ Δ?‘ž Δ?›Λ Δ?‘ž2 Γ’‰Β€ Δ?‘Δ½ Γ’‰Β€ Δ?‘&#x;2 Δ?œ‡Δ?‘Β§Δƒ (Δ?‘Δ½) = { 2 2 Δ?œ‡Δ?‘Β§Δ?‘Λ Δƒ (Δ?‘Δ½) Δ?‘&#x; Γ’‰Β€ Δ?‘Δ½ Γ’‰Β€ Δ?‘ 1 otherwise 3 3 Δ?œ—Δ?‘Β§Δ?‘™ Δƒ (Δ?‘Δ½) Δ?‘? Γ’‰Β€ Δ?‘Δ½ Γ’‰Β€ Δ?‘ž 3 3 Δ?›ΕΎ Δ?‘ž Γ’‰Β€Δ?‘Δ½Γ’‰Β€Δ?‘&#x; Δ?œ—Δ?‘Β§Δƒ (Δ?‘Δ½) = { 3 3 Δ?œ—Δ?‘Β§Δ?‘Λ Δƒ (Δ?‘Δ½) Δ?‘&#x; Γ’‰Β€ Δ?‘Δ½ Γ’‰Β€ Δ?‘ 1 otherwise
3. Single Valued Linear Neutrosophic Number 1.
Triangular Single Valued Neutrosophic number of Type 1: The quantity of the truth, indeterminacy and falsity are not dependent: A Triangular Single Valued Neutrosophic Δƒ Neu = (p1 , p2 , p3 ; q1 , q2 , q3 ; r1 , r2 , r3 ) whose truth membership, number of Type 1 is defined as A indeterminacy and falsity membership is defined as follows: Δ?‘Δ½ Γ’ˆ’ Δ?‘?1 whenΔ?‘?1 Γ’‰Β€ Δ?‘Δ½ < Δ?‘?2 Δ?‘?2 Γ’ˆ’ Δ?‘?1 1 when Δ?‘Δ½ = Δ?‘?2 Δ?‘‡Δ??Β΄ΔƒΔ?‘ Δ?‘’Δ?‘Λ (Δ?‘Δ½) = Δ?‘? Γ’ˆ’ Δ?‘Δ½ 3 whenΔ?‘?2 < Δ?‘Δ½ Γ’‰Β€ Δ?‘?3 Δ?‘?3 Γ’ˆ’ Δ?‘?2 { 0 otherwise
and Δ?‘Δ½ Γ’ˆ’ Δ?‘?1 whenΔ?‘?1 Γ’‰Β€ Δ?‘Δ½ < Δ?‘?2 Δ?‘?2 Γ’ˆ’ Δ?‘?1 1 when Δ?‘Δ½ = Δ?‘?2 Δ?‘‡Δ??Β΄ΔƒΔ?‘ Δ?‘’Δ?‘Λ (Δ?‘Δ½) = Δ?‘? Γ’ˆ’ Δ?‘Δ½ 3 whenΔ?‘?2 < Δ?‘Δ½ Γ’‰Β€ Δ?‘?3 Δ?‘?3 Γ’ˆ’ Δ?‘?2 { 0 otherwise
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π2 β π₯ when π1 β€ π₯ < π2 π2 β π1 0 when π₯ = π2 πΌπ΄Μπππ’ (π₯) = π₯ β π 2 whenπ2 < π₯ β€ π3 π3 β π2 { 1 otherwise and π₯ β π1 whenπ1 β€ π₯ < π2 π2 β π1 1 when π₯ = π2 ππ΄Μπππ’ (π₯) = π β π₯ 3 whenπ2 < π₯ β€ π3 π3 β π2 { 0 otherwise where, 0 β€ ππ΄Μπππ’ (π₯) + πΌπ΄Μπππ’ (π₯) + πΉπ΄Μπππ’ (π₯) β€ 3, π₯ β π΄Μπππ’ . The parametric form of the above type number is (π΄Μπππ’ )πΌ,π½,πΎ = [ππππ’1 (πΌ), ππππ’2 (πΌ); πΌπππ’1 (π½), πΌπππ’2 (π½); πΉπππ’1 (πΎ), πΉπππ’2 (πΎ)], where, ππππ’1 (πΌ) = π1 + πΌ(π2 β π1 ) ππππ’2 (πΌ) = π3 β πΌ(π3 β π2 ) πΌπππ’1 (π½) = π2 β π½(π2 β π1 ) πΌπππ’2 (π½) = π2 + π½(π3 β π2 ) πΉπππ’1 (πΎ) = π2 β πΎ(π2 β π1 ) πΉπππ’2 (πΎ) = π2 + πΎ(π3 β π2 ) here, 0 < πΌ β€ 1, 0 < π½ β€ 1, 0 < πΎ β€ 1 and 0 < πΌ + π½ + πΎ β€ 3 Example 1. Take π΄Μππ = (10,15,20; 14,16,22; 12,15,19). The parametric representation is πππ1 (πΌ) = 10 + 5πΌ πππ2 (πΌ) = 20 β 5πΌ πΌππ1 (π½) = 16 β 2π½ πΌππ1 (π½) = 16 + 6π½ πΉππ1 (πΎ) = 15 β 3πΎ πΉππ2 (πΎ) = 15 + 4πΎ Table 1 and Figure 3 show the value of πππ1 (πΌ), πππ2 (πΌ), πΌππ1 (π½), πΌππ1 (π½), πΉππ1 (πΎ), and (πΎ) πΉππ2 and graphical representation of triangular single valued neutrosophic numbers (TrSVNNs) respectively. Table 1. Value of πππ1 (πΌ) ,πππ2 (πΌ), πΌππ1 (π½), πΌππ1 (π½), πΉππ1 (πΎ), and πΉππ2 (πΎ).
πΆ, π·, πΈ π»π΅ππ (πΆ) π»π΅ππ (πΆ) π°π΅ππ (π·) π°π΅ππ (π·) ππ΅ππ (πΈ) ππ΅ππ (πΈ) 0 0.1 0.2 0.3
10 10 11 11.5
20 19.5 19 18.5
16 15.8 15.6 15.4
16 16.6 17.2 17.8
15 14.7 14.4 14.1
15 15.4 15.8 16.2
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0.4 0.5 0.6 0.7 0.8 0.9 1
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12 12.5 13 13.5 14 14.5 15
18 17.5 17 16.5 16 15.5 15
15.2 15 14.8 14.6 14.4 14.2 14
18.4 19 19.6 20.2 20.8 21.4 22
13.8 13.5 13.2 12.9 12.6 12.3 12
16.6 17 17.4 17.8 18.2 18.6 19
Figure 3. Graphical representation of TrSVNNs.
2.
Triangular Single Valued Neutrosophic Number of Type 2: The quantity of indeterminacy and falsity are dependent: A triangular single valued neutrosophic number (TrSVNN) of Type 2 is defined as Δ??Β΄ΔƒΔ?‘ Δ?‘’Δ?‘Λ = (Δ?‘?1 , Δ?‘?2 , Δ?‘?3 ; Δ?‘ž1 , Δ?‘ž2 , Δ?‘ž3 ; Δ?‘ΛΔ?‘ Δ?‘’Δ?‘Λ , Δ?‘ΕΔ?‘ Δ?‘’Δ?‘Λ ) whose truth membership, indeterminacy, and falsity membership are defined as follows: Δ?‘Δ½ Γ’ˆ’ Δ?‘?1 Δ?‘?2 Γ’ˆ’ Δ?‘?1 1 Δ?‘‡Δ??Β΄ΔƒΔ?‘ Δ?‘’Δ?‘Λ (Δ?‘Δ½) = Δ?‘? Γ’ˆ’ Δ?‘Δ½ 3 Δ?‘?3 Γ’ˆ’ Δ?‘?2 { 0
whenΔ?‘?1 Γ’‰Β€ Δ?‘Δ½ < Δ?‘?2 when Δ?‘Δ½ = Δ?‘?2 when Δ?‘?2 < Δ?‘Δ½ Γ’‰Β€ Δ?‘?3 otherwise
and Δ?‘ž2 Γ’ˆ’ Δ?‘Δ½ + Δ?‘ΛΔ?‘ Δ?‘’Δ?‘Λ (Δ?‘Δ½ Γ’ˆ’ Δ?‘ž1 ) whenΔ?‘ž1 Γ’‰Β€ Δ?‘Δ½ < Δ?‘ž2 Δ?‘ž2 Γ’ˆ’ Δ?‘ž1 Δ?‘ΛΔ?‘ Δ?‘’Δ?‘Λ when Δ?‘Δ½ = Δ?‘ž2 Δ??ΕΊΔ??Β΄ΔƒΔ?‘ Δ?‘’Δ?‘Λ (Δ?‘Δ½) = Δ?‘Δ½ Γ’ˆ’ Δ?‘ž2 + Δ?‘ΛΔ?‘ Δ?‘’Δ?‘Λ (Δ?‘ž3 Γ’ˆ’ Δ?‘Δ½) whenΔ?‘ž2 < Δ?‘Δ½ Γ’‰Β€ Δ?‘ž3 Δ?‘ž3 Γ’ˆ’ Δ?‘ž2 { 1 otherwise and
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π2 β π₯ + π¦πππ’ (π₯ β π1 ) whenπ1 β€ π₯ < π2 π2 β π1 π¦πππ’ when π₯ = π2 πΉπ΄Μπππ’ (π₯) = π₯ β π2 + π¦πππ’ (π3 β π₯) whenπ2 < π₯ β€ π3 π3 β π2 { 1 otherwise where, 0 β€ ππ΄Μπππ’ (π₯) + πΌπ΄Μπππ’ (π₯) + πΉπ΄Μπππ’ (π₯) β€ 2, π₯ β π΄Μπππ’ . The parametric form of the above type number is (π΄Μπππ’ )πΌ,π½,πΎ = [ππππ’1 (πΌ), ππππ’2 (πΌ); πΌπππ’1 (π½), πΌπππ’2 (π½); πΉπππ’1 (πΎ), πΉπππ’2 (πΎ)], where ππππ’1 (πΌ) = π1 + πΌ(π2 β π1 ) ππππ’2 (πΌ) = π3 β πΌ(π3 β π2 ) πΌπππ’1 (π½) =
π2 β π’πππ’ π1 β π½(π2 β π1 ) 1 β π’πππ’
πΌπππ’2 (π½) =
π2 β π’πππ’ π3 + π½(π3 β π2 ) 1 β π’πππ’
πΉπππ’1 (πΎ) =
π2 β π¦πππ’ π1 β πΎ(π2 β π1 ) 1 β π¦πππ’
πΉπππ’2 (πΎ) =
π2 β π¦πππ’ π3 + πΎ(π3 β π2 ) . 1 β π¦πππ’
Here, 0 < πΌ β€ 1, π’πππ’ < π½ β€ 1, π¦πππ’ < πΎ β€ 1 and 0 < π½ + πΎ β€ 1 and 0 < πΌ + π½ + πΎ β€ 2. Example 2. Take π΄Μππ = (10,15,20; 14,16,22; 0.4,0.5) The parametric representation is, πππ1 (πΌ) = 10 + 5πΌ πππ2 (πΌ) = 20 β 5πΌ πΌππ1 (π½) =
1 (52 β 10π½) 3
πΌππ1 (π½) = 12 + 10π½ πΉππ1 (πΎ) = 18 β 4πΎ πΉππ2 (πΎ) = 10 + 12πΎ. Table 2 and Figure 4 show the value of πππ1 (πΌ) , πππ2 (πΌ), πΌππ1 (π½), πΌππ1 (π½), πΉππ1 (πΎ), and πΉππ2 (πΎ) and graphical representation of type-2 TrSVNNs. Table 2. Value of πππ1 (πΌ) , πππ2 (πΌ), πΌππ1 (π½), πΌππ1 (π½), πΉππ1 (πΎ), and πΉππ2 (πΎ).
πΆ, π·, πΈ π»π΅ππ (πΆ) π»π΅ππ (πΆ) π°π΅ππ (π·) π°π΅ππ (π·) ππ΅ππ (πΈ) ππ΅ππ (πΈ) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
10 10.5 11 11.5 12 12.5 13 13.5 14 14.5
20 19.5 19 18.5 18 17.5 17 16.5 16 15.5
16 15.6667 15.3333 15. 14.6667 14.3333
16 17 18 19 20 21
16 15.6 15.2 14.8 14.4
16 17.2 18.4 19.6 20.8
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1
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15
15
14
22
14
22
Figure 4. Graphical representation of type-2 TrSVNNs. 3.
Triangular Single Valued Neutrosophic number of Type 3: The quantity of the truth, indeterminacy, and falsity are dependent: A TrSVNN of Type 3 is defined as Δ??Β΄ΔƒΔ?‘ Δ?‘’Δ?‘Λ = (Δ?‘?1 , Δ?‘?2 , Δ?‘?3 ; Δ?‘Β€Δ?‘ Δ?‘’ , Δ?‘ΛΔ?‘ Δ?‘’Δ?‘Λ , Δ?‘ΕΔ?‘ Δ?‘’Δ?‘Λ ), whose truth membership, indeterminacy, and falsity membership are defined as follows: Δ?‘Δ½ Γ’ˆ’ Δ?‘?1 whenΔ?‘?1 Γ’‰Β€ Δ?‘Δ½ < Δ?‘?2 Δ?‘?2 Γ’ˆ’ Δ?‘?1 Δ?‘Β€Δ?‘ Δ?‘’Δ?‘Λ when Δ?‘Δ½ = Δ?‘?2 Δ?‘‡Δ??Β΄ΔƒΔ?‘ Δ?‘’Δ?‘Λ (Δ?‘Δ½) = Δ?‘?3 Γ’ˆ’ Δ?‘Δ½ Δ?‘Β€Δ?‘ Δ?‘’Δ?‘Λ Δ?‘?3 Γ’ˆ’ Δ?‘?2 whenΔ?‘?2 < Δ?‘Δ½ Γ’‰Β€ Δ?‘?3 { 0 otherwise Δ?‘Β€Δ?‘ Δ?‘’Δ?‘Λ
and Δ?‘?2 Γ’ˆ’ Δ?‘Δ½ + Δ?‘ΛΔ?‘ Δ?‘’Δ?‘Λ (Δ?‘Δ½ Γ’ˆ’ Δ?‘?1 ) whenΔ?‘?1 Γ’‰Β€ Δ?‘Δ½ < Δ?‘?2 Δ?‘?2 Γ’ˆ’ Δ?‘?1 Δ?‘ΛΔ?‘ Δ?‘’Δ?‘Λ when Δ?‘Δ½ = Δ?‘?2 Δ??ΕΊΔ??Β΄ΔƒΔ?‘ Δ?‘’Δ?‘Λ (Δ?‘Δ½) = Δ?‘Δ½ Γ’ˆ’ Δ?‘?2 + Δ?‘ΛΔ?‘ Δ?‘’Δ?‘Λ (Δ?‘?3 Γ’ˆ’ Δ?‘Δ½) whenΔ?‘?2 < Δ?‘Δ½ Γ’‰Β€ Δ?‘?3 Δ?‘?3 Γ’ˆ’ Δ?‘?2 { 1 otherwise and Δ?‘?2 Γ’ˆ’ Δ?‘Δ½ + Δ?‘ΛΔ?‘ Δ?‘’Δ?‘Λ (Δ?‘Δ½ Γ’ˆ’ Δ?‘?1 ) whenΔ?‘?1 Γ’‰Β€ Δ?‘Δ½ < Δ?‘?2 Δ?‘?2 Γ’ˆ’ Δ?‘?1 Δ?‘ΛΔ?‘ Δ?‘’Δ?‘Λ when Δ?‘Δ½ = Δ?‘?2 Δ??ΕΊΔ??Β΄ΔƒΔ?‘ Δ?‘’Δ?‘Λ (Δ?‘Δ½) = Δ?‘Δ½ Γ’ˆ’ Δ?‘?2 + Δ?‘ΛΔ?‘ Δ?‘’Δ?‘Λ (Δ?‘?3 Γ’ˆ’ Δ?‘Δ½) whenΔ?‘?2 < Δ?‘Δ½ Γ’‰Β€ Δ?‘?3 Δ?‘?3 Γ’ˆ’ Δ?‘?2 { 1 otherwise where, 0 Γ’‰Β€ Δ?‘‡Δ??Β΄ΔƒΔ?‘ Δ?‘’Δ?‘Λ (Δ?‘Δ½) + Δ??ΕΊΔ??Β΄ΔƒΔ?‘ Δ?‘’Δ?‘Λ (Δ?‘Δ½) + Δ??Ε‘Δ??Β΄ΔƒΔ?‘ Δ?‘’Δ?‘Λ (Δ?‘Δ½) Γ’‰Β€ 1, Δ?‘Δ½ Γ’ˆˆ Δ??Β΄ΔƒΔ?‘ Δ?‘’Δ?‘Λ . The parametric form of the above
type
[Δ?‘‡Δ?‘ Δ?‘’Δ?‘Λ1 (Δ?›ΕΊ), Δ?‘‡Δ?‘ Δ?‘’Δ?‘Λ2 (Δ?›ΕΊ); Δ??ΕΊΔ?‘ Δ?‘’Δ?‘Λ1 (Δ?›Λ), Δ??ΕΊΔ?‘ Δ?‘’Δ?‘Λ2 (Δ?›Λ); Δ??Ε‘Δ?‘ Δ?‘’Δ?‘Λ1 (Δ?›ΕΎ), Δ??Ε‘Δ?‘ Δ?‘’Δ?‘Λ2 (Δ?›ΕΎ)], where Δ?›ΕΊ (Δ?‘? Γ’ˆ’ Δ?‘?1 ) Δ?‘Β€Δ?‘ Δ?‘’Δ?‘Λ 2 Δ?›ΕΊ Δ?‘‡Δ?‘ Δ?‘’Δ?‘Λ2 (Δ?›ΕΊ) = Δ?‘?3 Γ’ˆ’ (Δ?‘? Γ’ˆ’ Δ?‘?2 ) Δ?‘Β€Δ?‘ Δ?‘’Δ?‘Λ 3 Δ?‘‡Δ?‘ Δ?‘’Δ?‘Λ1 (Δ?›ΕΊ) = Δ?‘?1 +
number
is
(Δ??Β΄ΔƒΔ?‘ Δ?‘’Δ?‘Λ )Δ?›ΕΊ,Δ?›Λ,Δ?›ΕΎ =
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πΌπππ’1 (π½) =
π2 β π’πππ’ π1 β π½(π2 β π1 ) 1 β π’πππ’
πΌπππ’2 (π½) =
π2 β π’πππ’ π3 + π½(π3 β π2 ) 1 β π’πππ’
πΉπππ’1 (πΎ) =
π2 β π¦πππ’ π1 β πΎ(π2 β π1 ) 1 β π¦πππ’
πΉπππ’2 (πΎ) =
π2 β π¦πππ’ π3 + πΎ(π3 β π2 ) . 1 β π¦πππ’
Here, 0 < πΌ β€ π€πππ’ , π’πππ’ < π½ β€ 1, π¦πππ’ < πΎ β€ 1, and 0 < πΌ + π½ + πΎ β€ 1. Example 3. Take π΄Μππ = (14,16,22; 0.5,0.8,0.7) The parametric representation is, πππ1 (πΌ) = 14 + 4πΌ πππ2 (πΌ) = 22 β 12πΌ 5 πΌππ1 (π½) = 16 β π½ 2 15 πΌππ1 (π½) = 16 + π½ 2 20 πΉππ1 (πΎ) = 16 β πΎ 7 60 πΉππ2 (πΎ) = 16 + πΎ. 7 Table 3 and Figure 5 show the value of πππ1 (πΌ) , πππ2 (πΌ), πΌππ1 (π½), πΌππ1 (π½), πΉππ1 (πΎ) and πΉππ2 (πΎ). and Graphical representation of type-3 TrSVNNs Table 3. Value of πππ1 (πΌ), πππ2 (πΌ), πΌππ1 (π½), πΌππ1 (π½), πΉππ1 (πΎ) and πΉππ2 (πΎ).
πΆ, π·, πΈ π»π΅ππ (πΆ) π»π΅ππ (πΆ) π°π΅ππ (π·) π°π΅ππ (π·) ππ΅ππ (πΈ) ππ΅ππ (πΈ) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
14 14.4 14.8 15.2 15.6 16
22 20.8 19.6 18.4 17.2 16
16 15.75 15.5
16 16.75 17.5
16 16.2857 16.5714 16.8571
16 15.1429 14.2857 13.4286
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Figure 5.Graphical representation of type-3 TrSVNNs. Different Operational Laws of Two Triangular Neutrosophic Numbers: If Δ??Β΄ΔƒΔ?‘ Δ?‘’Δ?‘Λ and Δ??ΔΎΔƒΔ?‘ Δ?‘’Δ?‘Λ are two single valued neutrosophic numbers with nine components having truthmembership Δ?‘‡Δ??Β΄ΔƒΔ?‘ Δ?‘’Δ?‘Λ &Δ?‘‡Δ??ΔΎΔƒΔ?‘ Δ?‘’Δ?‘Λ , indeterminacymembership Δ??ΕΊΔ??Β΄ΔƒΔ?‘ Δ?‘’Δ?‘Λ &Δ??ΕΊΔ??ΔΎΔƒΔ?‘ Δ?‘’Δ?‘Λ , and falsitymembership Δ??Ε‘Δ??Β΄ΔƒΔ?‘ Δ?‘’Δ?‘Λ &Δ??Ε‘Δ??ΔΎΔƒΔ?‘ Δ?‘’Δ?‘Λ , respectively, such as: Δ??Β΄ΔƒΔ?‘ Δ?‘’Δ?‘Λ = Γ’ŒΕ Δ?‘Ž1 , Δ?‘Ž2 , Δ?‘Ž3 ; Δ?‘?1 , Δ?‘?2 , Δ?‘?3 ; Δ?‘?1 , Δ?‘?2 , Δ?‘?3 Γ’ŒΕ and Δ??ΔΎΔƒΔ?‘ Δ?‘’Δ?‘Λ = Γ’ŒΕ Δ?‘Ž4 , Δ?‘Ž5 , Δ?‘Ž6 ; Δ?‘?4 , Δ?‘?5 , Δ?‘?6 ; Δ?‘?4 , Δ?‘?5 , Δ?‘?6 Γ’ŒΕ, where a, band c are the scores given by the decision maker in the scale, ranging from lower limit Llto upper limit Ul. Γ’€Λ
Addition
Δ??ΕΔƒΔ?‘ Δ?‘’Δ?‘Λ = Δ??Β΄ΔƒΔ?‘ Δ?‘’Δ?‘Λ + Δ??ΔΎΔƒΔ?‘ Δ?‘’Δ?‘Λ = Γ’ŒΕ
Γ’€Λ
{Δ?‘šΔ?‘–Δ?‘›(Δ?‘Ž1 + Δ?‘Ž4 , Δ?‘ˆΔ?‘™ ) , Δ?‘šΔ?‘–Δ?‘›(Δ?‘Ž2 + Δ?‘Ž5 , Δ?‘ˆΔ?‘™ ), Δ?‘šΔ?‘–Δ?‘›(Δ?‘Ž3 + Δ?‘Ž6 , Δ?‘ˆΔ?‘™ )}; Γ’ŒΕ {Δ?‘šΔ?‘–Δ?‘›(Δ?‘?1 + Δ?‘?4 , Δ?‘ˆΔ?‘™ ) , Δ?‘šΔ?‘–Δ?‘›(Δ?‘?2 + Δ?‘?5 , Δ?‘ˆΔ?‘™ ), Δ?‘šΔ?‘–Δ?‘›(Δ?‘?3 + Δ?‘?6 , Δ?‘ˆΔ?‘™ )}; {Δ?‘šΔ?‘–Δ?‘›(Δ?‘?1 + Δ?‘?4 , Δ?‘ˆΔ?‘™ ) , Δ?‘šΔ?‘–Δ?‘›(Δ?‘?2 + Δ?‘?5 , Δ?‘ˆΔ?‘™ ), Δ?‘šΔ?‘–Δ?‘›(Δ?‘?3 + Δ?‘?6 , Δ?‘ˆΔ?‘™ )}
Negative of SVNNs
Δ?‘†ΔƒΔ?‘ Δ?‘’Δ?‘Λ = Γ’ˆ’Δ??Β΄ΔƒΔ?‘ Δ?‘’Δ?‘Λ = Γ’ŒΕ Γ’ˆ’Δ?‘Ž3 , Γ’ˆ’Δ?‘Ž2 , Γ’ˆ’Δ?‘Ž1 ; Γ’ˆ’Δ?‘?3 , Γ’ˆ’Δ?‘?2 , Γ’ˆ’Δ?‘?1 ; Γ’ˆ’Δ?‘?3 , Γ’ˆ’Δ?‘?2 , Γ’ˆ’Δ?‘?1 Γ’ŒΕ Γ’€Λ
Subtraction
ΔƒΔ?‘ Δ?‘’Δ?‘Λ = Δ??Β΄ΔƒΔ?‘ Δ?‘’Δ?‘Λ Γ’ˆ’ Δ??ΔΎΔƒΔ?‘ Δ?‘’Δ?‘Λ Δ??Λ = Δ??Β΄ΔƒΔ?‘ Δ?‘’Δ?‘Λ + (Γ’ˆ’Δ??ΔΎΔƒΔ?‘ Δ?‘’Δ?‘Λ ) {Δ?‘šΔ?‘ŽΔ?‘Δ½(Δ?‘Ž1 Γ’ˆ’ Δ?‘Ž6 , Δ??ΕΌΔ?‘™ ) , Δ?‘šΔ?‘ŽΔ?‘Δ½(Δ?‘Ž2 Γ’ˆ’ Δ?‘Ž5 , Δ??ΕΌΔ?‘™ ), Δ?‘šΔ?‘ŽΔ?‘Δ½(Δ?‘Ž3 Γ’ˆ’ Δ?‘Ž4 , Δ??ΕΌΔ?‘™ )}; Γ’ŒΕ = {Δ?‘šΔ?‘ŽΔ?‘Δ½(Δ?‘?1 Γ’ˆ’ Δ?‘?6 , Δ??ΕΌΔ?‘™ ) , Δ?‘šΔ?‘ŽΔ?‘Δ½(Δ?‘?2 Γ’ˆ’ Δ?‘?5 , Δ??ΕΌΔ?‘™ ), Δ?‘šΔ?‘ŽΔ?‘Δ½(Δ?‘?3 Γ’ˆ’ Δ?‘?4 , Δ??ΕΌΔ?‘™ )}; Γ’ŒΕ {Δ?‘šΔ?‘ŽΔ?‘Δ½(Δ?‘?1 Γ’ˆ’ Δ?‘?6 , Δ??ΕΌΔ?‘™ ) , Δ?‘šΔ?‘ŽΔ?‘Δ½(Δ?‘?2 Γ’ˆ’ Δ?‘?5 , Δ??ΕΌΔ?‘™ ), Δ?‘šΔ?‘ŽΔ?‘Δ½(Δ?‘?3 Γ’ˆ’ Δ?‘?4 , Δ??ΕΌΔ?‘™ )} Γ’€Λ
Multiplications
ΔƒΔ?‘ Δ?‘’Δ?‘Λ = Δ??Β΄ΔƒΔ?‘ Δ?‘’Δ?‘Λ Γ’ˆ’ Δ??ΔΎΔƒΔ?‘ Δ?‘’Δ?‘Λ Δ??Λ = Δ??Β΄ΔƒΔ?‘ Δ?‘’Δ?‘Λ + (Γ’ˆ’Δ??ΔΎΔƒΔ?‘ Δ?‘’Δ?‘Λ )
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{πππ₯(π1 β π6 , πΏπ ) , πππ₯(π2 β π5 , πΏπ ), πππ₯(π3 β π4 , πΏπ )}; = β© {πππ₯(π1 β π6 , πΏπ ) , πππ₯(π2 β π5 , πΏπ ), πππ₯(π3 β π4 , πΏπ )}; βͺ {πππ₯(π1 β π6 , πΏπ ) , πππ₯(π2 β π5 , πΏπ ), πππ₯(π3 β π4 , πΏπ )} β’
Multiplication by a constant
πΈΜπππ’ = π[π΄Μπππ’ ] = π Γ β©π1 , π2 , π3 ; π1 , π2 , π3 ; π1 , π2 , π3 βͺ = β©ππ1 , ππ2 , ππ3 ; ππ1 , ππ2 , ππ3 ; ππ1 , ππ2 , ππ3 βͺ β’
Inverse of SVNNs
1 β©π1 , π2 , π3 ; π1 , π2 , π3 ; π1 , π2 , π3 βͺ 1 1 1 1 1 1 1 1 1 =β© , , ; , , ; , , βͺ πππ (π, π, π) > 0 π3 π2 π1 π3 π2 π1 π3 π2 π1 1 1 1 1 1 1 1 1 1 = β© , , ; , , ; , , βͺ πππ (π, π, π) < 0 π1 π2 π3 π1 π2 π3 π1 π2 π3 πΉΜπππ’ = π΄Μβ1 πππ’ =
β’
Divisions
πΊΜπππ’ = Μπ΄πππ’ Γ· π΅Μπππ’ = π΄Μπππ’ Γ· π΅Μπππ’ 1 1 1 1 1 1 1 1 1 , , ; , , ; , , βͺ π6 π5 π4 π6 π5 π4 π6 π5 π4 π1 π1 π1 π2 π2 π2 π3 π3 π3 πππ ( , , , , , , , , ) , π4 π5 π6 π4 π5 π6 π4 π5 π6 π1 π1 π1 π2 π2 π2 π3 π3 π3 ππππ ( , , , , , , , , ) , ; π4 π5 π6 π4 π5 π6 π4 π5 π6 π1 π1 π1 π2 π2 π2 π3 π3 π3 max ( , , , , , , , , ) { π4 π5 π6 π4 π5 π6 π4 π5 π6 } π1 π1 π1 π2 π2 π2 π3 π3 π3 πππ ( , , , , , , , , ) , π4 π5 π6 π4 π5 π6 π4 π5 π6 π1 π1 π1 π2 π2 π2 π3 π3 π3 =β© ππππ ( , , , , , , , , ) , ; βͺ π4 π5 π6 π4 π5 π6 π4 π5 π6 π1 π1 π1 π2 π2 π2 π3 π3 π3 max ( , , , , , , , , ) { π4 π5 π6 π4 π5 π6 π4 π5 π6 } π1 π1 π1 π2 π2 π2 π3 π3 π3 πππ ( , , , , , , , , ) , π4 π5 π6 π4 π5 π6 π4 π5 π6 π1 π1 π1 π2 π2 π2 π3 π3 π3 ππππ ( , , , , , , , , ) , π4 π5 π6 π4 π5 π6 π4 π5 π6 π1 π1 π1 π2 π2 π2 π3 π3 π3 πππ₯ ( , , , , , , , , ) { π4 π5 π6 π4 π5 π6 π4 π5 π6 } = β©π1 , π2 , π3 ; π1 , π2 , π3 ; π1 , π2 , π3 βͺ Γ β©
Example 4. If π΄Μπππ’ = β©5, 10, 15; 2.5, 5, 7.5; 10, 17.5, 25βͺ and π΅Μπππ’ = β©4, 6, 8; 3, 6, 9; 1, 1.75, 2.5βͺ are two single valued neutrosophic numbers with independent truth, indeterminate, and false values in the scale of 0 π΄Μ to 25, then find the π΄Μπππ’ + π΅Μπππ’ , π΄Μπππ’ β π΅Μπππ’ , π΄Μπππ’ Γ π΅Μπππ’ , πππ’ πππ ππ΅Μπππ’ where k = 3. π΅Μπππ’
β’
Addition π΄Μπππ’ + π΅Μπππ’ = β©9, 16, 23; 5.5, 11, 16.5; 11, 19.25, 25βͺ,
β’
Subtraction π΄Μπππ’ β π΅Μπππ’ = β©0, 4, 11; 0, 0, 4.5; 7.5, 15.75, 24βͺ
β’
Multiplication
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Δ??Β΄ΔƒΔ?‘ Δ?‘’Δ?‘Λ Δ— Δ??ΔΎΔƒΔ?‘ Δ?‘’Δ?‘Λ = Γ’ŒΕ 20, 60, 120; 7.5, 30, 67.5; 10, 30.625, 62.5Γ’ŒΕ Γ’€Λ
Division Δ??Β΄ΔƒΔ?‘ Δ?‘’Δ?‘Λ Δ??ΔΎΔƒΔ?‘ Δ?‘’Δ?‘Λ
Γ’€Λ
= Γ’ŒΕ 0.625, 1.806, 3.75; 0.278, 1.0185, 2.5; 4, 11.5, 25Γ’ŒΕ,
Multiplication by a constant Δ?‘˜Δ??ΔΎΔƒΔ?‘ Δ?‘’Δ?‘Λ = Γ’ŒΕ 12, 18, 24; 9, 18, 27; 3, 5.25, 7.5Γ’ŒΕ
4. Neutrosophic Non-Linear Number and Generalized Neutrosophic Number 4.1. Single Valued Non-Linear Triangular Neutrosophic Number with Nine Components A single valued non-linear triangular neutrosophic number with nine components is defined as Δƒ Δ??Β΄Δ?‘ Δ?‘’Δ?‘Λ = (Δ?‘?1 , Δ?‘?2 , Δ?‘?3 ; Δ?‘ž1 , Δ?‘ž2 , Δ?‘ž3 ; Δ?‘&#x;1 , Δ?‘&#x;2 , Δ?‘&#x;3 ) , whose truth membership, indeterminacy, and falsity membership is defined as: Δ?‘Δ½ Γ’ˆ’ Δ?‘?1 Δ?‘Ž1 ( ) when Δ?‘?1 Γ’‰Β€ Δ?‘Δ½ < Δ?‘?2 Δ?‘?2 Γ’ˆ’ Δ?‘?1 1 when Δ?‘Δ½ = Δ?‘?2 Δ?‘‡Δ??Β΄ΔƒΔ?‘ Δ?‘’Δ?‘Λ (Δ?‘Δ½) = Δ?‘?3 Γ’ˆ’ Δ?‘Δ½ Δ?‘Ž2 ( ) whenΔ?‘?2 < Δ?‘Δ½ Γ’‰Β€ Δ?‘?3 Δ?‘?3 Γ’ˆ’ Δ?‘?2 { 0 Δ?‘œΔ?‘ΔΓ’„ŽΔ?‘’Δ?‘&#x;Δ?‘Β€Δ?‘–Δ?‘ Δ?‘’ and Δ?‘Δ½ Γ’ˆ’ Δ?‘ž1 Δ?‘?1 ( ) Δ?‘€Ò„ŽΔ?‘’Δ?‘›Δ?‘ž1 Γ’‰Β€ Δ?‘Δ½ < Δ?‘ž2 Δ?‘ž2 Γ’ˆ’ Δ?‘ž1 0 Δ?‘€Ò„ŽΔ?‘’Δ?‘› Δ?‘Δ½ = Δ?‘ž2 Δ??ΕΊΔ??Β΄ΔƒΔ?‘ Δ?‘’Δ?‘Λ (Δ?‘Δ½) = Δ?‘Δ½ Γ’ˆ’ Δ?‘ž3 Δ?‘?2 ( ) Δ?‘€Ò„ŽΔ?‘’Δ?‘› Δ?‘ž2 < Δ?‘Δ½ Γ’‰Β€ Δ?‘ž3 Δ?‘ž3 Γ’ˆ’ Δ?‘ž2 { 1 Δ?‘œΔ?‘ΔΓ’„ŽΔ?‘’Δ?‘&#x;Δ?‘Β€Δ?‘–Δ?‘ Δ?‘’ and Δ?‘Δ½ Γ’ˆ’ Δ?‘&#x;1 Δ?‘?1 ) Δ?‘€Ò„ŽΔ?‘’Δ?‘›Δ?‘&#x;1 Γ’‰Β€ Δ?‘Δ½ < Δ?‘&#x;2 Δ?‘&#x;2 Γ’ˆ’ Δ?‘&#x;1 0 Δ?‘€Ò„ŽΔ?‘’Δ?‘› Δ?‘Δ½ = Δ?‘&#x;2 Δ??Ε‘Δ??Β΄ΔƒΔ?‘ Δ?‘’Δ?‘Λ (Δ?‘Δ½) = Δ?‘Δ½ Γ’ˆ’ Δ?‘&#x;3 Δ?‘?2 ( ) Δ?‘€Ò„ŽΔ?‘’Δ?‘› Δ?‘&#x;2 < Δ?‘Δ½ Γ’‰Β€ Δ?‘&#x;3 Δ?‘&#x;3 Γ’ˆ’ Δ?‘&#x;2 {1 Δ?‘œΔ?‘ΔΓ’„ŽΔ?‘’Δ?‘&#x;Δ?‘Β€Δ?‘–Δ?‘ Δ?‘’ (
where, 0 Γ’‰Β€ Δ?‘‡Δ??Β΄ΔƒΔ?‘ Δ?‘’Δ?‘Λ (Δ?‘Δ½) + Δ??ΕΊΔ??Β΄ΔƒΔ?‘ Δ?‘’Δ?‘Λ (Δ?‘Δ½) + Δ??Ε‘Δ??Β΄ΔƒΔ?‘ Δ?‘’Δ?‘Λ (Δ?‘Δ½) Γ’‰Β€ 3, Δ?‘Δ½ Γ’ˆˆ Δ??Β΄ΔƒΔ?‘ Δ?‘’Δ?‘Λ . Note. If Δ?‘Ž1 , Δ?‘Ž2 , Δ?‘?1 , Δ?‘?2 , Δ?‘?1, Δ?‘?2 = 1 , then single valued non-linear triangular neutrosophic number with nine components will be converted into single valued linear triangular neutrosophic number with nine components. 4.2. Single Valued Generalized Triangular Neutrosophic Number with Nine Components A single valued triangular neutrosophic number with nine components is defined as Δ??Β΄ΔƒΔ?‘ Δ?‘’ = (Δ?‘?1 , Δ?‘?2 , Δ?‘?3 ; Δ?‘ž1 , Δ?‘ž2 , Δ?‘ž3 ; Δ?‘&#x;1 , Δ?‘&#x;2 , Δ?‘&#x;3 ), whose truth membership, indeterminacy, and falsity membership is defined as:
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π₯ β π1 whenπ1 β€ π₯ < π2 π2 β π1 π when π₯ = π2 ππ΄Μπππ’ (π₯) = π3 β π₯ π π3 β π2 whenπ2 < π₯ β€ π3 { 0 ππ‘βπππ€ππ π π
and π₯ β π1 whenπ1 β€ π₯ < π2 π2 β π1 0 when π₯ = π2 πΌπ΄Μπππ’ (π₯) = π₯ β π3 π π3 β π2 whenπ2 < π₯ β€ π3 { π ππ‘βπππ€ππ π π
and π₯ β π1 whenπ1 β€ π₯ < π2 π2 β π1 0 when π₯ = π2 πΉπ΄Μπππ’ (π₯) = π₯ β π3 π whenπ2 < π₯ β€ π3 π3 β π2 { π ππ‘βπππ€ππ π π
where, 0 β€ ππ΄Μπππ’ (π₯) + πΌπ΄Μπππ’ (π₯) + πΉπ΄Μπππ’ (π₯) β€ 3, π₯ β π΄Μπππ’ . 4.3. Single Valued Generalized Non-Linear Triangular Neutrosophic Number with Nine Components. A single valued non-linear triangular neutrosophic number with nine components is defined as π΄Μπππ’ = (π1 , π2 , π3 ; π1 , π2 , π3 ; π1 , π2 , π3 ) , whose truth membership, indeterminacy, and falsity membership is defined as: π₯ β π1 π1 π( ) whenπ1 β€ π₯ < π2 π2 β π1 π when π₯ = π2 ππ΄Μπππ’ (π₯) = π3 β π₯ π2 π( ) whenπ2 < π₯ β€ π3 π3 β π2 {0 ππ‘βπππ€ππ π and π₯ β π1 π1 π( ) whenπ1 β€ π₯ < π2 π2 β π1 0 when π₯ = π2 πΌπ΄Μπππ’ (π₯) = π₯ β π3 π2 π( ) when π2 < π₯ β€ π3 π3 β π2 {π ππ‘βπππ€ππ π and π₯ β π1 π1 π( ) π€βπππ1 β€ π₯ < π2 π2 β π1 0 π€βππ π₯ = π2 πΉπ΄Μπππ’ (π₯) = π₯ β π3 π2 π( ) π€βππ π2 < π₯ β€ π3 π3 β π2 {π ππ‘βπππ€ππ π where, 0 β€ ππ΄Μπππ’ (π₯) + πΌπ΄Μπππ’ (π₯) + πΉπ΄Μπππ’ (π₯) β€ 3, π₯ β π΄Μπππ’ .
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Note. if Δ?‘Ž1 , Δ?‘Ž2 , Δ?‘?1 , Δ?‘?2 , Δ?‘?1, Δ?‘?2 = 1 , then single valued generalized non-linear triangular neutrosophic number with nine components will be converted into single valued generalized linear triangular neutrosophic number with nine components. 5. De-Neutrosophication of Linear Neutrosophic Triangular Fuzzy Number De-Neutrosophication Using Removal Area Method Let us consider a linear neutrosophic triangular fuzzy number as follows: Δ??Β΄ΔƒΔ?‘ Δ?‘’ = (Δ?‘Ž, Δ?‘?, Δ?‘?; Δ?‘‘, Δ?‘’, Δ?‘“; Δ?‘”, Γ’„Ž, Δ?‘˜) whose pictorial representation is as follows. Firstly, we consider the graphical representation of linear neutrosophic triangular fuzzy number in Figure 6.
Figure 6. Linear neutrosophic number.
We consider an ordinary number Δ?‘˜ Γ’ˆˆ Δ?‘… and a fuzzy number Δ??Β΄ΔŒ for the lower triangle, then left side removal of Δ??Β΄ΔŒ with respect to k is Δ?‘…Δ?‘™ (Δ??Β΄ΔŒ, Δ?‘˜), defined as the area bounded by k and the left side of the fuzzy number Δ??Β΄ΔŒ. Similarly, the right side removal of Δ??Β΄ΔŒ with respect to k is Δ?‘…Δ?‘&#x; (Δ??Β΄ΔŒ, Δ?‘˜). Also consider an ordinary number Δ?‘˜ Γ’ˆˆ Δ?‘… and a fuzzy number Δ??ΔΎΔŒ for the left most upper triangle(Γ’ˆ†def), then the left side removal of Δ??ΔΎΔŒ with respect to k is Δ?‘…Δ?‘™ (Δ??ΔΎΔŒ , Δ?‘˜), defined as the area bounded by k and the left side of the fuzzy number Δ??ΔΎΔŒ .Similarly, the right side removal of Δ??ΔΎΔŒ with respect to k is Δ?‘…Δ?‘&#x; (Δ??ΔΎΔŒ , Δ?‘˜). A fuzzy number Δ??ΕΔŒ for the right most upper triangle(Γ’ˆ†ghk), then left side removal of Δ??ΕΔŒ with respect to k is Δ?‘…Δ?‘™ (Δ??ΕΔŒ , Δ?‘˜), defined as the area bounded by k and the left side of the fuzzy number Δ??ΕΔŒ .Similarly, the right side removal of Δ??ΕΔŒ with respect to k is Δ?‘…Δ?‘&#x; (Δ??ΕΔŒ , Δ?‘˜). Δ?‘… (Δ??Β΄ΔŒ,Δ?‘˜)+Δ?‘…Δ?‘&#x; (Δ??Β΄ΔŒ,Δ?‘˜) Δ?‘… (Δ??ΔΎΔŒ ,Δ?‘˜)+Δ?‘…Δ?‘&#x; (Δ??ΔΎΔŒ,Δ?‘˜) Δ?‘… (Δ??ΕΔŒ ,Δ?‘˜)+Δ?‘…Δ?‘&#x; (Δ??ΕΔŒ ,Δ?‘˜) Mean is defined as (Δ??Β΄ΔŒ, Δ?‘˜) = Δ?‘™ , Δ?‘…(Δ??ΔΎΔŒ , Δ?‘˜) = Δ?‘™ , Δ?‘…(Δ??ΕΔŒ , Δ?‘˜) = Δ?‘™ . 2
2
Then, we defined the defuzzification of a linear neutrosophic triangular fuzzy as ΔŒ
ΔŒ
ΔŒ
ΔŒ , Δ?‘˜) = Δ?‘…(Δ??Β΄,Δ?‘˜)+Δ?‘…(Δ??ΔΎ,Δ?‘˜)+Δ?‘…(Δ??Ε ,Δ?‘˜). Δ?‘…(Δ??Λ 3
For Δ?‘˜ = 0ΔΕΊŒ Δ?‘…(Δ??Β΄ΔŒ, 0) = Δ?‘…(Δ??ΔΎΔŒ , 0) = Δ?‘…(Δ??ΕΔŒ , 0) =
Δ?‘…Δ?‘™ (Δ??Β΄ΔŒ, 0) + Δ?‘…Δ?‘&#x; (Δ??Β΄ΔŒ, 0) 2 ΔŒ Δ?‘…Δ?‘™ (Δ??ΔΎ , 0) + Δ?‘…Δ?‘&#x; (Δ??ΔΎΔŒ , 0) 2 Δ?‘…Δ?‘™ (Δ??ΕΔŒ , 0) + Δ?‘…Δ?‘&#x; (Δ??ΕΔŒ , 0) 2
2
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Then, ΔŒ , 0) = Δ?‘…(Δ??Λ
Δ?‘…(Δ??Β΄ΔŒ, 0) + Δ?‘…(Δ??ΔΎΔŒ, 0) + Δ?‘…(Δ??ΕΔŒ , 0) 3
We take Δ??Β΄ΔŒ = (Δ?‘Ž, Δ?‘?, Δ?‘?), Δ??ΔΎΔŒ = (Δ?‘‘, Δ?‘’, Δ?‘“), Δ??ΕΔŒ = (Δ?‘”, Γ’„Ž, Δ?‘˜). Figure 7 shows the pictorial representation of de-neutrosophication.
(a)
(c)
(b)
(d)
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(e)
(f)
Figure 7. Pictorial representation of de-neutrosophication. (a) Area of trapezium OABR; (b) Area of trapezium OABR; (c) Area of trapezium OEDR;(d) Area of trapezium OEFR; (e) Area of trapezium OHGR: (f) Area of trapezium OHKR.
Then, (Δ?‘Ž+Δ?‘?) Δ?‘…Δ?‘™ (Δ??Β΄ΔŒ, 0) = Area of trapezium OABR= .1 2
(Δ?‘Ž+Δ?‘?) Δ?‘…Δ?‘™ (Δ??Β΄ΔŒ, 0) = Area of trapezium OABR= .1
Δ?‘…Δ?‘™ (Δ??ΔΎΔŒ , 0) = Area of trapezium OEDR=
2 (Δ?‘‘+Δ?‘’)
Δ?‘…Δ?‘&#x; (Δ??ΔΎΔŒ , 0) = Area of trapezium OEFR= Δ?‘…Δ?‘™ (Δ??ΕΔŒ , 0) = Area of trapezium OHGR= Hence, (Δ??Β΄ΔŒ, 0) =
Δ?‘…Δ?‘&#x; (Δ??Ε, 0) (Δ?‘Ž+2Δ?‘?+Δ?‘?) 4
= Area of trapezium OHKR=
2 (Δ?‘’+Δ?‘“) 2 (Δ?‘”+Γ’„Ž) 2 (Δ?‘˜+Γ’„Ž) 2
.1 .1 .1 . 1.
(Δ?‘‘+2Δ?‘’+Δ?‘“) (Δ?‘”+2Γ’„Ž+Δ?‘˜) , Δ?‘…(Δ??ΔΎΔŒ , 0) = , Δ?‘…(Δ??ΕΔŒ , 0) = .
ΔŒ , 0) = (Δ?‘Ž+2Δ?‘?+Δ?‘?+Δ?‘‘+2Δ?‘’+Δ?‘“+Δ?‘”+2Γ’„Ž+Δ?‘˜). So, Δ?‘…(Δ??Λ
4
4
12
Examples 5. Finding De-neutrosophication value of Neutrosophic number. Table 4 shows the de-neutrosophication value of Neutrosophic number.
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Table 4. De-neutrosophication value of Neutrosophic number.
Experiment No. Set 1 Set 2 Set 3 Set 4
Neutrosophic Number ΔŒ = (Δ?&#x;?, Δ?&#x;?, Δ?&#x;‘; Δ?&#x;Ž. Δ?&#x;“, Δ?&#x;?. Δ?&#x;“, Δ?&#x;?. Δ?&#x;“; Δ?&#x;?. Δ?&#x;?, Δ?&#x;?. Δ?&#x;•, Δ?&#x;‘. Δ?&#x;“) Δ?‘Β¨ ΔŒ = (Δ?&#x;Ž. Δ?&#x;“, Δ?&#x;?. Δ?&#x;“, Δ?&#x;?. Δ?&#x;“; Δ?&#x;Ž. Δ?&#x;‘, Δ?&#x;?. Δ?&#x;‘, Δ?&#x;?. Δ?&#x;?; Δ?&#x;Ž. Δ?&#x;•, Δ?&#x;?. Δ?&#x;•, Δ?&#x;?. Δ?&#x;?) Δ?‘Ε ΔŒ = (Δ?&#x;Ž. Δ?&#x;‘, Δ?&#x;?. Δ?&#x;?, Δ?&#x;?. Δ?&#x;–; Δ?&#x;Ž. Δ?&#x;“, Δ?&#x;?. Δ?&#x;“, Δ?&#x;?. Δ?&#x;“; Δ?&#x;Ž. Δ?&#x;–, Δ?&#x;?. Δ?&#x;•, Δ?&#x;?. Δ?&#x;•) Δ?‘Ε ΔŒ = (Δ?&#x;?, Δ?&#x;‘, Δ?&#x;“; Δ?&#x;Ž. Δ?&#x;“, Δ?&#x;?. Δ?&#x;“, Δ?&#x;?. Δ?&#x;“; Δ?&#x;?. Δ?&#x;?, Δ?&#x;?. Δ?&#x;•, Δ?&#x;’. Δ?&#x;“) Δ?‘Ε€
De-Neutrosophication Value 2.0083 1.45 1.533 2.425
6. PERT in Triangular Neutrosophic Environment and the Proposed Model The full form of PERT method is project evaluation and review technique, which is a project management tool used to schedule, organize, and coordinate tasks within a project. It is basically a method to analyze the tasks involved in completing a given project, especially the time needed to complete each task, and to identify the minimum time needed to complete the total project. PERT planning involves the following steps: 1. 2. 3. 4. 5. 6.
Identify the specific activities and milestones. Determine the proper sequence of the activities. Construct a network diagram. Estimate the time required for each activity. Determine the critical path. Update the PERT chart as the project progresses.
The main objective of PERT is to facilitate decision making and to reduce both the time and cost required to complete a project. PERT is intended for very large-scale, one-time, non-routine, complex projects with a high degree of dependency, projects which require a series of activities, some of which must be performed sequentially, and others that can be performed in parallel with other activities. PERT has been mainly used in new projects which have large uncertainty with respect to design of a structure, technology, and networking system. To take care of associated uncertainties, we introduced triangular neutrosophic environment for PERT activity duration. The three time estimates for activity duration are as follows: Optimistic time (Δ?‘œΔŒ): Generally, the shortest time in which the activity can be completed. It is common practice to specify optimistic time to be three standards deviations from the mean so that there is approximately a 1% chance that the activity will be completed within the optimistic time. Pessimistic time (Δ?‘?ΔŒ ): Generally, the longest time that an activity might require. Three standard deviations from the mean are commonly used for the pessimistic time. Most likely time (Δ?‘š ΔŒ): Generally, it is the completion time, in normal circumstances, having the highest probability. Note that this time is different from the expected time. Note 2. In Ref. [22], the authors introduced the concept of score and accuracy function to compute the crisp value of a trapezoidal neutrosophic number. In our proposed model, we choose all the three different times (optimistic, pessimistic, most likely) as triangular neutrosophic number. ΔŒ , 0) = To obtain the crisp value, we introduced the de-neutrosophication value Δ?‘…(Δ??Λ (Δ?‘Ž+2Δ?‘?+Δ?‘?+Δ?‘‘+2Δ?‘’+Δ?‘“+Δ?‘”+2Γ’„Ž+Δ?‘˜) 12
of triangular neutrosophic number(Δ?‘Ž, Δ?‘?, Δ?‘?; Δ?‘‘, Δ?‘’, Δ?‘“; Δ?‘”, Γ’„Ž, Δ?‘˜).
Now, the expected time and standard deviation can be calculated by the formula Δ??ΒΈΔ?‘—Δ?‘˜ = and Δ?œŽΔ?‘—Δ?‘˜ =
Δ?‘?Γ’ˆ’Δ?‘œ 6
Δ?‘œ+4Δ?‘š+Δ?‘? 6
, where o, p, and m are all crisp value of optimistic, pessimistic, and most likely time
estimations, respectively. Now, we use CPM method for further calculation of earliest/latest time, critical path, and float. In forward pass, starting with a time of zero for the first event, the computation proceeds from left to right, up to the final event. For any activity (Δ?‘–, Δ?‘—), let Δ??ΒΈΔ?‘†Δ?‘– denote the earliest time of event Δ?‘–, then Δ??ΒΈΔ?‘†Δ?‘— = Δ??ΒΈΔ?‘†Δ?‘– + Δ?‘ΔΔ?‘–Δ?‘— . If more than one activity enters an event, the earliest start time for that event is computed as Δ??ΒΈΔ?‘†Δ?‘— = max{Δ??ΒΈΔ?‘†Δ?‘– + Δ?‘ΔΔ?‘–Δ?‘— } for all activities emanating from node i entering into j.
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In case of backward pass, starting with the final node, the computation proceeds from right to left, up to the initial event. For any activity (Δ?‘–, Δ?‘—), let Δ??ΕΌΔ??Ε‘Δ?‘– denote the latest finished time of event i, then Δ??ΕΌΔ??Ε‘Δ?‘– = Δ??ΕΌΔ??Ε‘Δ?‘— Γ’ˆ’ Δ?‘ΔΔ?‘–Δ?‘— . If more than one activity enters an event, the latest finish time for that event is computed as Δ??ΕΌΔ??Ε‘Δ?‘– = min{Δ??ΕΌΔ??Ε‘Δ?‘— Γ’ˆ’ Δ?‘ΔΔ?‘–Δ?‘— } for all activities emanating from node j entering into i. After calculating the critical path, compute project length variance, which is the sum of the Δ?‘‡ Γ’ˆ’Δ?‘‡ variances of all the critical activities. Next, calculate the standard normal variable Δ?‘? = Δ?‘ Δ?‘’, where Δ?œŽ
Δ?‘‡Δ?‘ is the scheduled time to complete the project, andΔ?‘‡Δ?‘’ is the normal expected project length duration. Using a normal curve, we can estimate the probability of completing the project within a specified time. The steps of the said method are shown in Figure 8. We also set the numerical value for the said problem to show the importance of our method in Table 5.
Figure 8. Flowchart for the solution procedure.
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Table 5. Set of Neutrosophic value of the above problem.
A B C D E F G H I J K
Description Selection of Officer and Force Member Selection of Site and do Site Survey Selection of Arms Final Plan and Blueprint Bring Utilities to the Site Interview Acquisition and take delivery of arms Construct the battlefield Developed networking system Run the system Training for all
Predecessors Γ’€“ Γ’€“ A B B A C D A E,G,H F,I,J
Optimistic Time (1,2,3;0.5,1.5,2.5;1.2,2.7,3.5) (Δ?&#x;?, Δ?&#x;“, Δ?&#x;–; Δ?&#x;?, Δ?&#x;‘, Δ?&#x;”; Δ?&#x;’, Δ?&#x;•, Δ?&#x;—) (1,4,7;1,3,5;3.5,6,7.5) (1,3,5;0.5,2.5,3.5;2.5,4,6) (0.5,2.5,4.5;0.5,1.5,3.5;2,4,6) (2,4,6;1.5,2.5,3.5;3,5,7) (0.5,2.5,4.5;1,2,3;2,4,6) (1.5,3.5,5.5;1,2,3;3,4.5,6) (1,5,9;1.5,4.5,6.5;4,7,10) (0.5,3.5,6.5;0.5,2.5,4.5;3,5,7) (1,5,8;1.5,3.5,6.5;4,6,8.5)
Pessimistic Time (1,5,9;1.5,4.5,6.5;4,7,10) (1,2,3;0.5,1.5,2.5;1.5,2.5,3.5) (1,1.5,4;0.5,1,2.5;1.25,3,4.25) (1.5,3.5,5.5;1,2,3;3,4.5,6) (1,5,9;1.5,4.5,6.5;4,7.5,10.5) (1,2,3;0.5,1.5,2.5;1.2,2.7,3.5) (1,5,8;1.5,3,6.5;4,7,9) (0.5,3.5,6.5;0.5,2.5,4.5;3,5,7) (0.5,2.5,4.5;1,2,3;1.5,3.5,5.5) (1.5,2.5,3.5;1,1.5,3;2,3,4) (1,4,7;1,3,5;3.5,6,7.5)
Most Likely Time (1.5,3.5,5.5;1,2,3;3,4.5,6) (1,5,8;1.5,3,6.5;4,7,9) (1,5,9;1.5,4.5,6.5;4,7,10) (0.5,2.5,4.5;1,2,3;1.5,3.5,5.5) (1.5,2.5,3.5;1,1.5,3;2,3,4) (1,4,7;1,3,5;3.5,6,7.5) (1,5,9;1.5,4.5,6.5;4,7,10) (1,2,3;0.5,1.5,2.5;1.5,2.5,3.5) (Δ?&#x;?, Δ?&#x;“, Δ?&#x;–; Δ?&#x;?, Δ?&#x;‘, Δ?&#x;”; Δ?&#x;’, Δ?&#x;•, Δ?&#x;—) (1,2,3;0.5,1.5,2.5;1.2,2.7,3.5) (1,1.5,4;0.5,1,2.5;1.25,3,4.25)
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Draw the project network and find the probability that the project is completed in 16 days. Solution. Now, we solve the problem by the following steps, as shown in Table 6, Figures 9 and 10. Step-1. Table 6. The value of Δ??ΒΈΔ?‘—Δ?‘˜ and Δ?œŽΔ?‘—Δ?‘˜ 2 for the above problem.
Optimistic Time (o)
Pessimistic Time (p)
Most Likely Time (m)
2.26 4.92 4.67 2.96 2.75 3.83 2.83 3.33 5.42 3.50 4.88
5.42 2.00 1.71 3.33 5.54 2.26 4.92 3.50 2.67 2.42 4.67
3.33 4.92 5.42 2.67 2.42 4.67 2.00 2.26 4.92 2.00 1.71
Δ?‘ΕΉΔ?’‹Δ?’Œ Δ?’? + Δ?&#x;’Δ?’Ž + Δ?’‘ = Δ?&#x;” 3.50 4.43 4.68 2.83 3.00 4.13 2.63 2.65 4.63 2.32 2.73
Figure 9. The network diagram for the problem.
Δ??ˆΔ?’‹Δ?’Œ Δ?&#x;? Δ?’‘Γ’ˆ’Δ?’? Δ?&#x;? =( ) Δ?&#x;” 0.277 0.244 0.243 0.004 0.216 0.068 0.121 0.001 0.210 0.032 0.001
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Figure 10. Critical path analysis for the problem.
Therefore, the expected project duration is 15.9 days. Critical path AΓ’†’CΓ’†’GΓ’†’JΓ’†’K. Project length variance Δ?œŽ 2 = 0.962, standard deviation0.98. Probability that the project will be finished within 16 days is Δ?‘ƒ (Δ?‘Β§ Γ’‰Β€
16Γ’ˆ’15.9 0.98
) = Δ?‘ƒ(Δ?‘Β§ Γ’‰Β€ 0.1)
Area under the normal curve Δ?‘ƒ(Δ?‘Β§ Γ’‰Β€ 0.1) = 0.5 + Γ’ˆ…(0.1) = 0.5398 The related normal curve is drawn in Figure 11.
Figure 11. The normal curve for the above problem.
7. Application of Triangular Neutrosophic Fuzzy Number in Assignment Problem Using De-Neutrosophic Value The assignment problem is very important for transferring goods from one place to another place. In the assignment problem, if uncertainty occurs, then it is more complicated to solve. By the concept of impreciseness and its corresponding crispified value, we can easily handle the assignment problem. In this section, we take a route selection problem with neutrosophic cost data and solve the problem. We consider a problem of assigning three different trucks to three different destinations. The assigning costs that are the travelling costs in rupees are given here. How should the trucks be dispatched so as to minimize the total travelling cost? Note, that all the costs are triangular neutrosophic numbers.
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Let us consider that the transportation cost for the three trucks are neutrosophic in nature. For that viewpoint, we take that the cost of the three trucks are as follows in Table 1, in units of dollar. Each component represents the moneys in units of dollars. Here, red car denotes Truck 1, yellow car denotes Truck 2, and green car denotes Truck 3 as shown in the Figure 12.
Figure 12. Pictorial representation of the problem.
We apply the defuzzification result of triangular neutrosophic number from Table 7. Table 7. Neutrosophic value for the transportation costs.
Truck 1 Truck 2 Truck 3
Destination-1
Destination-2
Destination-3
(1,4,7;1,3,5;3.5,6,7.5) (1,2,3;0.5,1.5,2.5;1.5,2.5,3.5) (2,4,6;1.5,2.5,4.5;3,5,7)
(0.5,2.5,4.5;1,2,3;1.5,3.5,5.5) (1,1.5,4;0.5,1,2.5;1.25,3,4.25) (1,5,8;1.5,4.5,7.5;4,6.5,9)
(1,3,5;0.5,1.5,3.5;2,4,6) (1.5,2.5,3.5;1,1.5,3;2,3,4) (1,5,8;1.5,3,6.5;4,7,9)
ΔŒ , 0) = (Δ?‘Ž+2Δ?‘?+Δ?‘?+Δ?‘‘+2Δ?‘’+Δ?‘“+Δ?‘”+2Γ’„Ž+Δ?‘˜) to convert the numbers into a crisp number. Δ?‘…(Δ??Λ 12
Then, we have the following Table 8. Table 8. De-neutrosophication value for the transportation costs.
Truck 1 Truck 2 Truck 3
Destination-1
Destination-2
Destination-3
4.25 2.00 3.92
2.67 1.71 5.25
2.92 2.75 5.08
Now, we consider row minimum from each row, and subtract it from the other element (row-wise). Thus, we get Table 9. Table 9. Row minimum from each row, and subtract it from the other element (row-wise).
Truck 1 Truck 2 Truck 3
Destination-1 1.58 0.29 0
Destination-2 0 0 1.33
Destination-3 0.25 1.04 1.16
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Now, we consider column minimum from each column and subtract it from the other element (column-wise). Thus, we get Table 10. Table 10. Column minimum from each column and subtract it from the other element (column-wise).
Destination-1 1.58 0.29 0
Truck 1 Truck 2 Truck 3
Destination-2 0 0 1.33
Destination-3 0 0.79 0.91
Here, the minimum number of straight lines to cover all the zeros is 3 (which is also equal to the order of the matrix), as shown in Table 11. Table 11. Minimum number of straight lines to cover all the zeros.
Truck 1 Truck 2 Truck 3
Destination-1 1.58 0.29 0
Destination-2 0 0 1.33
Destination-3 0 0.79 0.91
From the tables 12, we see that if the Truck1 goes to Destination-3, Truck2 goes to Destination-2, and Truck3 goes to Destination-1, then the carrying is minimum. Table 12. Transformed table.
Destination-1 1.58 0.29 [Δ?&#x;Ž]
Truck 1 Truck 2 Truck 3
Destination-2 0 [Δ?&#x;Ž] 1.33
Destination-3 [Δ?&#x;Ž] 0.79 0.91
That means from the Figure 13 Truck-1 Γ’†’ Destination-3, Truck-2 Γ’†’ Destination-2, Truck-3 Γ’†’Destination-1. The corresponding Min cost = (3.92 + 1.71 + 2.92) = 8.55 units of dollar. Then, we get Table 13. Table 13. Neutrosophic value of destinations.
Truck 1 Truck 2 Truck 3
Destination-1 (1,4,7;1,3,5;3.5,6,7.5) (1,2,3;0.5,1.5,2.5;1.5,2.5,3.5) (2,4,6;1.5,2.5,4.5;3,5,7)
Destination-2 (0.5,2.5,4.5;1,2,3;1.5,3.5,5.5) (1,1.5,4;0.5,1,2.5;1.25,3,4.25) (1,5,8;1.5,4.5,7.5;4,6.5,9)
Destination-3 (1,3,5;0.5,1.5,3.5;2,4,6) (1.5,2.5,3.5;1,1.5,3;2,3,4) (1,5,8;1.5,3,6.5;4,7,9)
Ye [21] built up the concept of score function and accuracy function. The score function S and the accuracy function H are applied to compare the grades of triangular fuzzy numbers (TFNS). These functions show that greater is the value, the greater is the TFNS, and by using these, concept paths can be ranked. We apply the result of triangular neutrosophic number. Let, Δ??Β΄ΔŒ = (Δ?‘Ž, Δ?‘?, Δ?‘?; Δ?‘‘, Δ?‘’, Δ?‘“; Δ?‘”, Γ’„Ž, Δ?‘˜) be a triangular neutrosophic fuzzy number, then the score {8+(Δ?‘Ž+2Δ?‘?+Δ?‘?)Γ’ˆ’(Δ?‘‘+2Δ?‘’+Δ?‘“)Γ’ˆ’(Δ?‘”+2Γ’„Ž+Δ?‘˜)} function is defined as Δ?‘†(Δ??Β΄ΔŒ) = , and accuracy function is defined as {(Δ?‘Ž+2Δ?‘?+Δ?‘?)Γ’ˆ’(Δ?‘”+2Γ’„Ž+Δ?‘˜)} Δ??Ε₯(Δ??Β΄ΔŒ) = .
12
4
In order to make comparisons between two triangular neutrosophic values, Ye [21] presented the order relations between two triangular neutrosophic values. ΔŒ1 = (Δ?‘Ž1 , Δ?‘?1 , Δ?‘?1 ; Δ?‘‘1 , Δ?‘’1 , Δ?‘“1 ; Δ?‘”1 , Γ’„Ž1 , Δ?‘˜1 ) and Δ??Β΄ ΔŒ2 = (Δ?‘Ž2 , Δ?‘?2 , Δ?‘?2 ; Δ?‘‘2 , Δ?‘’2 , Δ?‘“2 ; Δ?‘”2 , Γ’„Ž2 , Δ?‘˜2 ) be two Let Δ??Β΄ triangular neutrosophic values, then the ranking method is defined as follows. ΔŒ1 ) > Δ?‘†(Δ??Β΄ ΔŒ2 ), then Δ??Β΄ ΔŒ1 > Δ??Β΄ ΔŒ2 (i) if Δ?‘†(Δ??Β΄
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ΔŒ1 ) = Δ?‘†(Δ??Β΄ ΔŒ2 ) and Δ??Ε₯(Δ??Β΄ ΔŒ1 ) > Δ??Ε₯(Δ??Β΄ ΔŒ2 ), then Δ??Β΄ ΔŒ1 > Δ??Β΄ ΔŒ2 (ii) if Δ?‘†(Δ??Β΄ We
apply
the
score
function
result
of
triangular
neutrosophic
number
Δ?‘†(Δ??Β΄ΔŒ) =
{8+(Δ?‘Ž+2Δ?‘?+Δ?‘?)Γ’ˆ’(Δ?‘‘+2Δ?‘’+Δ?‘“)Γ’ˆ’(Δ?‘”+2Γ’„Ž+Δ?‘˜)} 12
to convert the numbers into a crisp number.
Figure 13. Pictorial representation of the solution.
Then we have the following table, as shown in Table 14. Table14. Converted the numbers into a crisp number.
Truck 1 Truck 2 Truck 3
Destination-1
Destination-2
Destination-3
Γ’ˆ’0.92 0.00 Γ’ˆ’0.58
Γ’ˆ’0.33 Γ’ˆ’0.04 Γ’ˆ’1.42
Γ’ˆ’0.25 Γ’ˆ’0.08 Γ’ˆ’1.17
Take the most negative cost (Γ’ˆ’1.42), add it with all the elements of the matrix we get Table 15. Table 15. Corrosponing positive value table.
Truck 1 Truck 2 Truck 3
Destination-1
Destination-2
Destination-3
0.50 1.42 0.84
1.09 1.38 0.00
1.17 1.34 0.25
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Now, we consider row minimum from each row and subtract it from the other elements (row-wise). Thus, we get Table 16. Table 16. Row minimum from each row and subtract it from the other elements (row-wise).
Destination-1
Destination-2
Destination-3
0 0.08 0.84
0.59 0.04 0
0.67 0 0.25
Truck 1 Truck 2 Truck 3
Now, we consider column minimum from each column, and subtract it from the other elements (column-wise). Thus, we get Table 17. Table 17. Column minimum from each column, and subtract it from the other elements (column-wise).
Destination-1 Destination-2 Destination-3 Truck 1 Truck 2 Truck 3
0 0.08 0.84
0.59 0.04 0
0.67 0 0.25
Here, the minimum number of straight lines to cover all the zeros is 3(which is also equal to the order of the matrix), as shown in Table 18. Table 18. Minimum number of straight lines to cover all the zeros is 3.
Truck 1 Truck 2 Truck 3
Destination-1
Destination-2
Destination-3
0 0.08 0.84
0.59 0.04 0
0.67 0 0.25
From the Table 19, we see that if the Truck1 goes to Destination-1, Truck2 goes to Destination-3, and Truck3 goes to Destination-2, then the carrying is minimum. Table19. Decision table.
Truck 1 Truck 2 Truck 3
Destination-1
Destination-2
Destination-3
[0] 0.08 0.84
0.59 0.04 [0]
0.67 [0] 0.25
That means from the Figure 14 the destination is as follows Truck1 β Destination-1, Truck2 βDestination-3, Truck3 βDestination-2. The corresponding Min cost = (β0.92 β 1.42 β 0.08) = β2.42 units of dollar. Note: Since, using de-neutrosophic value, we observe that min cost is 8.55 units of dollar, whereas using score function, we get min cost in negative quantity that is loss, hence de-neutrosophication gives us a better result than the score function.
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Figure 14. Pictorial representation of the solution.
8. Conclusions The theory of uncertainty plays a key role in applied mathematical modeling. The concept of neutrosophic number is very popular nowadays. The formation and de-neutrosophication of the corresponding number can be very important for the researcher who deals with uncertainty and decision-making problems. In this paper, we construct the concept triangular neutrosophic number from different viewpoints, which is not defined earlier. We use the concept of linear and non-linear form with generalization of the pick value of truth, falsity, and indeterminacy functions by considering triangular neutrosophic numbers, which are very important for uncertainty theory. We introduced the de-neutrosophication concept for triangular neutrosophic numbers. This concept helps us to convert a neutrosophic number into a crisp number, which is surely helpful for decision-making problems. In future, we can extend the concept into different types of neutrosophic numbers, which can be more applicable in modeling with uncertainty. Author Contributions: Conceptualization, A.C., S.P.M. and A.A.; Methodology, N.S. and S.S.; Vdation, S.A. and S.S.; Formal Analysis, A.C., S.P.M. and A.A.; Investigation, N.S. and S.A. Acknowledgments: The authors acknowledge the financial support from Universiti Putra Malaysia under Putra-IPB grant: GP-IPB/2017/9542402. Conflicts of Interest: The authors declare no conflict of interest.
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