A soft set based VIKOR approach for some decision-making problems under complex neutrosophic environment

A soft set based VIKOR approach for some decision-making problems under complex neutrosophic environment

Engineering Applications of Artificial Intelligence 89 (2020) 103432 Contents lists available at ScienceDirect Engineering Applications of Artificia...
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Engineering Applications of Artificial Intelligence 89 (2020) 103432

Contents lists available at ScienceDirect

Engineering Applications of Artificial Intelligence journal homepage: www.elsevier.com/locate/engappai

A soft set based VIKOR approach for some decision-making problems under complex neutrosophic environment✩ Soumi Manna βˆ—, Tanushree Mitra Basu, Shyamal Kumar Mondal Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore 721102, WB, India

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Keywords: Complex neutrosophic soft set Complex neutrosophic aggregation Complex neutrosophic soft union Complex neutrosophic soft intersection Score function VIKOR method

ABSTRACT Applications in various fields of neutrosophic soft set theory enhance its appreciation to the researchers. Although, this uncertain solving tool can accomplish several types of real-life problems, but not able to deal with the decision-making problems in which considering an additional information of a corresponding parameter is needed to get a proper decision from a problem. But, complex neutrosophic soft sets can handle such type of decision-making problems. Consequently, in this study, we have given concentration on solving soft set based decision-making problems in the field of complex neutrosophic environment. Firstly, we have introduced some basic set-theoretic operations of complex neutrosophic sets including different types of unions, intersections and aggregations. Then, a new definition of score function of a complex neutrosophic number has been proposed. Additionally, the above-defined unions and intersections of complex neutrosophic sets have been stated for complex neutrosophic soft sets. Finally, by utilizing these proposed notions, we have offered a complex neutrosophic soft VIKOR approach to get a compromise optimal solution for single as well as multiple decision-maker based problems. Our proposed approach has been clarified by several real liferelated problems including medical diagnosis problem, sustainable manufacturing material selection problem, company’s manager selection problem, etc. The feasibility and effectiveness of our proposed approach have also been included in the article.

1. Introduction One of the important and challenging issues in our day-to-day life is the presence of uncertainty to be addressed. Neutrosophic set theory (Smarandache, 2003, 2005) is one of the innovative generalization of a bunch of theories, including, fuzzy set theory (Zadeh, 1965), intuitionistic fuzzy set theory (Atanassov, 1986), vague set theory (Gau and Buehrer, 1993) etc., through three independent membership grades (truth membership (𝑇 (π‘₯)), indeterminate membership (𝐼(π‘₯)) and false membership (𝐹 (π‘₯)) where, 𝑇 (π‘₯), 𝐼(π‘₯), 𝐹 (π‘₯) ∈ [0, 1] and 0 ≀ 𝑇 (π‘₯) + 𝐼(π‘₯) + 𝐹 (π‘₯) ≀ 3). In the last few decades, neutrosophic set has been improved very quickly and has been successfully used to solve various types of real-life problems. For instance, Ye (2015) used the cosine similarity measure of neutrosophic sets in medical diagnosis problem, Zhang et al. (2015) utilized the advantages of correlation coefficient of neutrosophic sets in solving multi-criteria decision-making problems, Wu et al. (2018) developed multi attribute group decision-making by using single-valued neutrosophic 2-tuple linguistic sets etc. All the above-aforementioned discussions are based on real-valued membership magnitudes like membership, non membership, indeterminate membership, which are limited in the closed interval [0, 1].

Then, Romot et al. (2002) extended a real-valued membership degree of a fuzzy set to a complex-valued membership degree (πœ‡πΆ (π‘₯)) by adding a second dimension 𝑒(π‘₯) as, πœ‡πΆ (π‘₯) = π‘Ÿ(π‘₯)𝑒𝑖𝑒(π‘₯) and introduced the concept of complex fuzzy set, where, π‘Ÿ(π‘₯) ∈ [0, 1] is the amplitude term and 𝑒(π‘₯), a real number, is the phase term. So, complex fuzzy set can be applied to the problems where adding a second decision information of an object over an attribute is needed to get a better and accurate result. For instance, in medical science, during disease diagnosis of a patient, the decision information about a symptom in a patient is perfect if the doctor will consider the two information β€˜belongingness of a symptom’ and β€˜time duration of the symptom’ together. In the earlier section, Romot et al. (2002) developed some basic set theoretic operations and properties of complex fuzzy sets along with some of its application fields. Then, Zhang et al. (2009) worked on complex fuzzy set theory, when the range of the phase term is limited to [0, 2πœ‹]. They also introduced 𝛿-equalities of complex fuzzy sets. After that, a systematic review of complex fuzzy sets was presented by the researchers Yazdanbakhsh and Disk (2018). Latter, Alkouri and Salleh (2012) proposed complex intuitionistic fuzzy set by using complex sense in the both of membership degree and non membership degree.

✩ No author associated with this paper has disclosed any potential or pertinent conflicts which may be perceived to have impending conflict with this work. For full disclosure statements refer to https://doi.org/10.1016/j.engappai.2019.103432. βˆ— Corresponding author. E-mail addresses: [email protected] (S. Manna), [email protected] (T.M. Basu), [email protected] (S.K. Mondal).

https://doi.org/10.1016/j.engappai.2019.103432 Received 19 July 2019; Received in revised form 8 November 2019; Accepted 18 December 2019 Available online xxxx 0952-1976/Β© 2019 Elsevier Ltd. All rights reserved.

S. Manna, T.M. Basu and S.K. Mondal

Engineering Applications of Artificial Intelligence 89 (2020) 103432

β€’ Besides, the main flavor of complex neutrosophic soft set is in its additional phase term. Because, through this idea, we can handle more complicated real-life related problems having a second dimension. But, in their illustrated decision-making problem, they do not give any idea about the phase part.

Then, Rani and Garg (2017) developed some distance measures to complex intuitionistic fuzzy sets such as Hamming distance, Hausdorff distance etc. Furthermore, Ali and Smarandache (2017) introduced complex neutrosophic set by using complex-valued truth membership, complex-valued indeterminate membership and complex-valued false membership. Then, Dat et al. (2019) solved some decision-making problems by using interval complex neutrosophic sets. Even though, probability theory, fuzzy set theory, vague set theory etc. can be effectively employed to cover up uncertain situations, Molodsov (1999) said that, due to the lack of parameterization, the above theories are not sufficient to solve all types of problems. Then, as a solution, he (Molodsov, 1999) proposed the idea soft set theory by utilizing the notion of parameterization. Then, several types of theoretical developments of soft set theory have been done by the researchers. For instance, Ali et al. (2009) developed some set-theoretic operations on soft sets, Babitha and Sunil (2010) introduced soft relations and soft functions, Majumdar and Samanta (2010) offered the notion of soft mappings, Cagman and Enginoglu (2010) introduced soft matrices associated with soft sets, etc. Besides, researchers have extended soft set theory to different environments. Fuzzy soft set theory (Basu et al., 2012; Paik and Mondal, 2019; Manna et al., 2017), intuitionistic fuzzy soft set theory (Manna et al., 2019), neutrosophic soft set theory (Basu and Mondal, 2015), trapezoidal interval type-2 fuzzy soft set theory (Manna et al., 2018) etc. are some important extensions of soft set theory. Then, several types of decision-making have been disposed by using soft set theory. For example, Basu et al. (2012) developed a balanced solution by using fuzzy soft set and used it in medical science, Manna et al. (2019) proposed an algorithm through generalized trapezoidal intuitionistic fuzzy soft sets and applied it in diabetes disease diagnosis problem, Karaaslan (2017) used single-valued neutrosophic redefined soft sets in clustering analysis system, etc. On the other hand, complex-valued generalization of soft set theory have also been developed by the researchers. For instance, Thirunavukarasu et al. (2017) introduced the concept of complex fuzzy soft set theory by taking all the parameters in a soft set in complex fuzzy sense, Selvachandran et al. (2016a) initiated complex vague soft set theory by taking all the parameters in a vague soft set in complex fuzzy sense, Broumi et al. (2017) introduced complex neutrosophic soft set theory by taking all the parameters in a soft set in complex neutrosophic sense, etc. Then, Kumar and Bajaj (2014) defined some distance measures and entropies on complex intuitionistic fuzzy soft sets and applied it in solving multi-criteria decision-making, Selvachandran et al. (2016a) solved pattern recognition problems by using some different types of distance measures of complex vague soft sets. Furthermore, the relations between complex vague soft sets have been thrived by Selvachandran et al. (2016b). After surveying the above literature, complex neutrosophic set theory, introduced by Ali and Smarandache (2017), first motivates us to use this concept in soft set theory, developed by Molodsov (1999), to solve some real-life based decision making problems. Then, we have studied complex neutrosophic soft set in decision-making, proposed by Broumi et al. (2017), from which some shortcomings have been raised in our mind such as,

Therefore, to fulfill these research gaps, we are motivated to develop some decision-making methods in soft set theory under complex neutrosophic environment to take a quality of decision from several types of real-life related problems. Moreover, aggregation operator is a very significant tool to take a decision from a group of evaluations. But till now, no aggregation operator exists on the basis of complex neutrosophic sets. Therefore, the introduction of aggregation operator on complex neutrosophic sets is required. However, to avoid complexity in solving real-life data based problems, transformation from an uncertain value into a real value by using a score function is very useful to handle complicated problems. But there is no definition of score function of a complex neutrosophic number. Besides, VIKOR method (Park et al., 2013) is a popular multi-criteria decision-making methodology which provides a compromise optimal solution of a decision-making problem. So, if this familiar optimization method is used in solving complex neutrosophic soft set based decision-making problems, then compromise optimal solution would be obtained as a solution instead of a general optimal solution. From the above aforementioned analysis, our main contributions can be highlighted as following: β€’ A new definition of score function of a complex neutrosophic number has been proposed. β€’ Two algorithmic approaches have been developed to solve complex neutrosophic soft set based decision-making problems by using VIKOR method. β€’ Some real-life related problems have been discussed and solved to illustrate the working progress of our proposed approach. The outline of this article is as follows. In Section 2, some basic terminologies have been recalled. In Sections 3 and 5, we have introduced some set-theoretic operations of complex neutrosophic sets and complex neutrosophic soft sets. In Section 4, a new definition of score function of a complex neutrosophic number has been proposed. After that, we have provided two algorithms for complex neutrosophic soft set based decision-making problems as given in Section 6. Then, some applications of our proposed approaches including company’s manager selection problem, sustainable manufacturing material selection problem and medical diagnosis problem etc. have been discussed and solved in Section 7. Finally in Section 8, the sensitivity analysis, validity and effectiveness of our proposed approach have been discussed. Section 9 contains the conclusion of this paper. 2. Preliminaries In this section, we have given some basic definitions and properties which have been exploited in our forthcoming discussions. In the whole paper, π‘ˆ and 𝐸 have been taken as the universal set and the parameter set respectively.

β€’ In a complex neutrosophic soft set based real-life decision-making problem, all the parameters may not be uniform in nature i.e., conflicting parameters may exist in the parameter set. But, in their approach, they do not give any process to handle this type of the parameters. β€’ Moreover, all the considered parameters in a decision-problem may not have equal weight. But, their approach can only be applied to the decision-making problems where all weights of the parameters are equal. So, when parameters will have unequal weights then, we cannot embed their algorithm to solve a complex neutrosophic soft set based problem.

2.1. Complex neutrosophic set Definition 1 (Ali and Smarandache, 2017). Complex neutrosophic set is a generalization of neutrosophic set by taking complex-valued neutrosophic membership grade instead of neutrosophic membership grade in a neutrosophic set. Mathematically, a complex neutrosophic set over the universe π‘ˆ can be denoted by 𝐢̃𝑁 and is defined as follows: 𝐢̃𝑁 = {(π‘₯, (𝑇 (π‘₯), 𝐼(π‘₯), 𝐹 (π‘₯)))|π‘₯ ∈ π‘ˆ } 2

S. Manna, T.M. Basu and S.K. Mondal

Engineering Applications of Artificial Intelligence 89 (2020) 103432 Table 1 Tabular representation of the CNS-Set (𝑓𝐢̃𝑁 , 𝐸).

where 𝑇 (π‘₯), 𝐼(π‘₯) and 𝐹 (π‘₯) are the complex-valued truth membership degree, complex-valued indeterminate membership degree and complex-valued false membership degree of π‘₯ ∈ π‘ˆ such that, 𝑇 (π‘₯) = 𝑝(π‘₯)𝑒

𝑖𝑒(π‘₯)

𝑖𝑣(π‘₯)

, 𝐼(π‘₯) = π‘ž(π‘₯)𝑒

, 𝐹 (π‘₯) = π‘Ÿ(π‘₯)𝑒

𝑖𝑀(π‘₯)

𝑠1

.

𝑝(π‘₯), π‘ž(π‘₯), π‘Ÿ(π‘₯) are the amplitude terms and 𝑒(π‘₯), 𝑣(π‘₯), 𝑀(π‘₯) are the phase terms of 𝑇 (π‘₯), 𝐼(π‘₯) and 𝐹 (π‘₯) respectively with 𝑝(π‘₯), π‘ž(π‘₯), π‘Ÿ(π‘₯) ∈ [0, 1] such that 0 ≀ 𝑝(π‘₯) + π‘ž(π‘₯) + π‘Ÿ(π‘₯) ≀ 3 and 𝑒(π‘₯), 𝑣(π‘₯), 𝑀(π‘₯) are real-valued.

𝑠3

𝑠4

𝑠5

𝑠6

𝑑1 (1𝑒𝑖2πœ‹ , 0𝑒𝑖0 , 0𝑒𝑖0 ) (0.2π‘’π‘–πœ‹βˆ•8 , 0.8π‘’π‘–πœ‹βˆ•5 , 0.9𝑒𝑖2πœ‹βˆ•5 ) (0.2𝑒𝑖6πœ‹βˆ•7 , 0.6π‘’π‘–πœ‹βˆ•5 , 0.7𝑒𝑖7πœ‹βˆ•9 ) 𝑑2 (0.9𝑒𝑖2πœ‹βˆ•3 , 0.5π‘’π‘–πœ‹βˆ•3 , 0.6π‘’π‘–πœ‹βˆ•3 ) (0.7π‘’π‘–πœ‹βˆ•4 , 0.3π‘’π‘–πœ‹βˆ•10 , 0.4𝑒𝑖3πœ‹βˆ•7 ) (0.5π‘’π‘–πœ‹ , 0.3π‘’π‘–πœ‹βˆ•5 , 0.4π‘’π‘–πœ‹βˆ•3 ) 𝑑3 (0.1π‘’π‘–πœ‹ , 0.8𝑒𝑖0 , 0.7π‘’π‘–πœ‹βˆ•7 ) (0.3π‘’π‘–πœ‹βˆ•6 , 0.6𝑒𝑖4πœ‹βˆ•5 , 0.8𝑒𝑖5πœ‹βˆ•6 ) (0.8𝑒𝑖3πœ‹βˆ•7 , 0.4π‘’π‘–πœ‹βˆ•7 , 0.2π‘’π‘–πœ‹ )

Now, to reduce the complexity in calculation, in this article, we have taken the range of the phase term as [0, 2πœ‹], followed by the article Zhang et al. (2009).

In that case, to deal with this medical problem, we can use complex neutrosophic soft set, since, in this soft set, two information of a symptom over a disease can be considered together through amplitude term and phase term.

Definition 2 (Ali and Smarandache, 2017; Zhang et al., 2009). The complement of a complex neutrosophic set 𝐢̃𝑁 = {(π‘₯, (𝑝(π‘₯)𝑒𝑖𝑒(π‘₯) , π‘ž(π‘₯)𝑒𝑖𝑣(π‘₯) , π‘Ÿ(π‘₯)𝑒𝑖𝑀(π‘₯) ))|π‘₯ ∈ π‘ˆ } over the universal set π‘ˆ , as defined in the above 𝑐 and is defined as follows: definition, is denoted by 𝐢̃𝑁

Now consider, π‘ˆ = {π‘π‘›π‘’π‘’π‘šπ‘œπ‘›π‘–π‘Ž (𝑑1 ), π‘£π‘–π‘Ÿπ‘Žπ‘™ 𝑓 π‘’π‘£π‘’π‘Ÿ (𝑑2 ), 𝑑𝑒𝑛𝑔𝑒𝑒 (𝑑3 )} be the set of three diseases (universal set) and 𝐸 = {𝑓 π‘’π‘£π‘’π‘Ÿ (𝑠1 ), β„Žπ‘’π‘Žπ‘‘π‘Žπ‘β„Žπ‘’ (𝑠2 ), π‘π‘œπ‘‘π‘¦ π‘π‘Žπ‘–π‘› (𝑠3 ), π‘π‘œπ‘’π‘”β„Ž (𝑠4 ), π‘€π‘’π‘–π‘”β„Žπ‘‘ π‘™π‘œπ‘ π‘  (𝑠5 ), π‘β„Žπ‘–π‘™π‘™ (𝑠6 )} be the corresponding symptoms over the above diseases (parameter set). Now, the β€˜effects of the symptoms over these diseases’ has been expressed by a complex neutrosophic soft set (𝑓𝐢̃𝑁 , 𝐸) as given in Table 1. Here, β€˜the availability of a symptom’ in a patient has been expressed through the amplitude term and β€˜time duration of the said symptom’ has been expressed through the phase term. All the data has been collected based on the 10 consecutive days. To express the data in terms of complex neutrosophic numbers, the value 2πœ‹ has been taken in the phase term instead of 10 days.

𝑐 𝐢̃𝑁 = (π‘Ÿ(π‘₯)𝑒𝑖(2πœ‹βˆ’π‘’(π‘₯)) , (1 βˆ’ π‘ž(π‘₯))𝑒𝑖(2πœ‹βˆ’π‘£(π‘₯)) , 𝑝(π‘₯)𝑒𝑖(2πœ‹βˆ’π‘€(π‘₯)) ).

Definition 3 (Ali and Smarandache, 2017). Let 𝐢̃𝐴 and 𝐢̃𝐡 be two complex neutrosophic sets over the universal set π‘ˆ where, 𝐢̃𝐴 = ( ) ( 𝑝𝐴 (π‘₯)𝑒𝑖𝑒𝐴 (π‘₯) , π‘žπ΄ (π‘₯)𝑒𝑖𝑣𝐴 (π‘₯) , π‘Ÿπ΄ (π‘₯)𝑒𝑖𝑀𝐴 (π‘₯) and 𝐢̃𝐡 = 𝑝𝐡 (π‘₯)𝑒𝑖𝑒𝐡 (π‘₯) , π‘žπ΅ (π‘₯) ) 𝑖𝑣 (π‘₯) 𝑖𝑀 (π‘₯) 𝑒 𝐡 , π‘Ÿπ΅ (π‘₯)𝑒 𝐡 , βˆ€π‘₯ ∈ π‘ˆ . Then, the distance measure between 𝐢̃𝐴 and 𝐢̃𝐡 is denoted by, 𝑑̃CN (𝐢̃𝐴 , 𝐢̃𝐡 ) and is defined as follows: 𝑑̃CN (𝐢̃𝐴 , 𝐢̃𝐡 ) = π‘šπ‘Žπ‘₯(π‘šπ‘Žπ‘₯(sup |𝑝𝐴 (π‘₯) βˆ’ 𝑝𝐡 (π‘₯)|, sup |π‘žπ΄ (π‘₯) βˆ’ π‘žπ΅ (π‘₯)|, π‘₯βˆˆπ‘ˆ

𝑠2

𝑑1 (1𝑒𝑖2πœ‹ , 0𝑒𝑖0 , 0𝑒𝑖0 ) (0.2π‘’π‘–πœ‹βˆ•8 , 0.4π‘’π‘–πœ‹βˆ•5 , 0.4𝑒𝑖2πœ‹βˆ•5 ) (0.2𝑒𝑖6πœ‹βˆ•7 , 0.5π‘’π‘–πœ‹βˆ•5 , 0.2𝑒𝑖7πœ‹βˆ•9 ) 𝑑2 (0.6𝑒𝑖2πœ‹βˆ•3 , 0.2π‘’π‘–πœ‹βˆ•3 , 0.1π‘’π‘–πœ‹βˆ•3 ) (0.1π‘’π‘–πœ‹βˆ•4 , 0.6π‘’π‘–πœ‹βˆ•10 , 0.3𝑒𝑖3πœ‹βˆ•7 ) (0.5π‘’π‘–πœ‹ , 0.1π‘’π‘–πœ‹βˆ•5 , 0.4π‘’π‘–πœ‹βˆ•3 ) 𝑑3 (0.8π‘’π‘–πœ‹ , 0𝑒𝑖0 , 0.2π‘’π‘–πœ‹βˆ•7 ) (0.3π‘’π‘–πœ‹βˆ•6 , 0.4𝑒𝑖4πœ‹βˆ•5 , 0.3𝑒𝑖5πœ‹βˆ•6 ) (0.1𝑒𝑖3πœ‹βˆ•7 , 0.8π‘’π‘–πœ‹βˆ•7 , 0𝑒𝑖0 )

π‘₯βˆˆπ‘ˆ

sup |π‘Ÿπ΄ (π‘₯) βˆ’ π‘Ÿπ΅ (π‘₯)|),

π‘₯βˆˆπ‘ˆ

π‘šπ‘Žπ‘₯(

2.3. Normalization process. (Ploskas and Papathanasiou, 2019)

1 1 sup |𝑒 (π‘₯) βˆ’ 𝑒𝐡 (π‘₯)|, sup |𝑣 (π‘₯) βˆ’ 𝑣𝐡 (π‘₯)|, 2πœ‹ π‘₯βˆˆπ‘ˆ 𝐴 2πœ‹ π‘₯βˆˆπ‘ˆ 𝐴

Definition 4 (Broumi et al., 2017). A pair (𝑓𝐢̃𝑁 , 𝐸) is said to be a complex neutrosophic soft set (CNS-set) over the universal set π‘ˆ if and only if 𝑓𝐢̃𝑁 a function from the parameter set 𝐸 to the set of all complex neutrosophic subsets of the set π‘ˆ . i.e., 𝑓𝐢̃𝑁 ∢ 𝐸 β†’ 𝑃𝐢̃𝑁 (π‘ˆ ), where 𝑃𝐢̃𝑁 (π‘ˆ ) is the set of all complex neutrosophic subsets of the set π‘ˆ .

A multi-criteria decision-making (MCDM) problem contains a set of alternatives ({𝐴1 , 𝐴2 , … , 𝐴𝑀 }) and a set of corresponding parameters ({𝐸1 , 𝐸2 , … , 𝐸𝑁 }). The rating of the alternatives are evaluated based on the associated parameters. Let, 𝐹𝑠𝑗 be the rating of an alternative 𝐴𝑠 ; 𝑠 = 1, 2, … , 𝑀 over a parameter 𝐸𝑗 ; 𝑗 = 1, 2, … , 𝑁. Now, in a decision-making problem, not all the parameters may be based on the same environment, therefore, they may not be commensurable to one another. Then, to handle these parameters, normalization process can be used by which, all the parameters become dimension less. Some popular normalization techniques are as follows: 𝐹 (1) Linear sum normalization. 𝑅𝑠𝑗 = βˆ‘π‘€ 𝑠𝑗 ; 𝑗 = 1, 2, … , 𝑁; 𝑠 =

Mathematical framework of the complex neutrosophic soft set (𝑓𝐢̃𝑁 , 𝐸) is as follows,

1, 2, … , 𝑀. (2) Linear max normalization. 𝑅𝑠𝑗 =

(𝑓𝐢̃𝑁 , 𝐸) = {(𝑒, 𝑓𝐢̃𝑁 (𝑒))|βˆ€π‘’ ∈ 𝐸},

evaluation over 𝑒𝑗 .

where, 𝑓𝐢̃𝑁 (𝑒) is the set of all complex neutrosophic 𝑒-approximate elements of the set π‘ˆ such that,

2.4. VIKOR method. (Park et al., 2013)

1 sup |𝑀 (π‘₯) βˆ’ 𝑀𝐡 (π‘₯)|)) 2πœ‹ π‘₯βˆˆπ‘ˆ 𝐴 2.2. Complex neutrosophic soft set

𝑠=1 𝐹𝑠𝑗

𝐹𝑠𝑗 𝐹𝑗+

; π‘€β„Žπ‘’π‘Ÿπ‘’, 𝐹𝑗+ is the ideal

VIKOR method is a well-known optimization technique which provides a compromise optimal solution by testing β€˜acceptable advantage’ and β€˜acceptable stability’ in decision-making. This method contains the following steps:

𝑓𝐢̃𝑁 (𝑒) = {(π‘₯𝑠 , (𝑇𝑓 (𝑒)(π‘₯𝑠 ), 𝐼𝑓 (𝑒)(π‘₯𝑠 ), 𝐹𝑓 (𝑒)(π‘₯𝑠 ))) |βˆ€ π‘₯𝑠 ∈ π‘ˆ } = {(π‘₯𝑠 , (𝑝𝑓 (𝑒)(π‘₯𝑠 )𝑒𝑖𝑒𝑓 (𝑒)(π‘₯𝑠 ) , π‘žπ‘“ (𝑒)(π‘₯𝑠 )𝑒𝑖𝑣𝑓 (𝑒)(π‘₯𝑠 ) , π‘Ÿπ‘“ (𝑒)(π‘₯𝑠 )𝑒𝑖𝑀𝑓 (𝑒)(π‘₯𝑠 ) ))}. Here, all the amplitude terms 𝑝𝑓 (𝑒)(π‘₯𝑠 ), π‘žπ‘“ (𝑒)(π‘₯𝑠 ), π‘Ÿπ‘“ (𝑒)(π‘₯𝑠 ) are limited in the closed interval [0, 1] with 0 ≀ 𝑝𝑓 (𝑒)(π‘₯𝑠 )+π‘žπ‘“ (𝑒)(π‘₯𝑠 )+π‘Ÿπ‘“ (𝑒)(π‘₯𝑠 ) ≀ 3 and all the phase terms 𝑒𝑓 (𝑒)(π‘₯𝑠 ), 𝑣𝑓 (𝑒)(π‘₯𝑠 ), 𝑀𝑓 (𝑒)(π‘₯𝑠 ) are limited in the closed interval [0, 2πœ‹].

Step 1. Define the ideal solution 𝐴+ = {𝐹1+ , 𝐹2+ , … , 𝐹𝑁+ }, where, 𝐹𝑗+ = π‘šπ‘Žπ‘₯𝑠 {𝐹𝑠𝑗 }. Step 2. Obtain the group of utility (utility measure) 𝑆𝑠 and individual regret (regret measure) 𝑅𝑠 of each of the alternatives as,

Example 1. Now, we have considered a problem from medical science. In medical science, various diseases may have a common symptom. For example, the symptoms fever, headache, body pain, cough, weight loss, chill etc. may be the caused of the diseases pneumonia, viral fever, dengue etc. Now, to describe the β€˜effect of a symptom’ over a disease in a patient, if a doctor gives his/her concentration on the information about the β€˜time duration of a symptom’ together with the information about the β€˜availability of a symptom’ over a disease, then this combined information (β€˜availability of a symptom’ and ’time duration of a symptom’) is more appropriate to detect the exact disease of the patient.

𝑆𝑠 =

𝑁 βˆ‘

𝑀𝑗 𝑑(𝐹𝑠𝑗 , 𝐹𝑗+ ); 𝑅𝑠 = π‘šπ‘Žπ‘₯𝑁 𝑀 𝑑(𝐹𝑠𝑗 , 𝐹𝑗+ ). 𝑗=1 𝑗

𝑗=1

𝑀 = {𝑀1 , 𝑀2 , … , 𝑀𝑛 } be the weights of the parameters such that, 𝑀𝑗 β‰₯ 0 βˆ‘ and 𝑛𝑗=1 𝑀𝑗 = 1. Step 3. Then, determine the VIKOR index 𝑄𝑠 of an alternative 𝐴𝑠 is defined as follows, 𝑄𝑠 = πœƒ ( 3

𝑆𝑠 βˆ’ π‘šπ‘–π‘›π‘  𝑆𝑠 𝑅𝑠 βˆ’ π‘šπ‘–π‘›π‘  𝑅𝑠 ) + (1 βˆ’ πœƒ) ( ) π‘šπ‘Žπ‘₯𝑠 𝑆𝑠 βˆ’ π‘šπ‘–π‘›π‘  𝑆𝑠 π‘šπ‘Žπ‘₯𝑠 𝑅𝑠 βˆ’ π‘šπ‘–π‘›π‘  𝑅𝑠

S. Manna, T.M. Basu and S.K. Mondal

Engineering Applications of Artificial Intelligence 89 (2020) 103432

β€’ Complex neutrosophic standard union.

where, πœƒ ∈ [0, 1] is the weight of strategy of the maximum group of utility.

If, βˆͺΜƒ 𝐹 is complex fuzzy standard union (βˆͺΜƒ π‘šπ‘Žπ‘₯ 𝐹 ), then complex

Step 4. Rank the alternatives based on the decreasing order of 𝑆𝑠 , 𝑅𝑠 and 𝑄𝑠 ; 𝑠 = 1, 2, … , π‘š.

neutrosophic union of 𝐢̃𝐴 and 𝐢̃𝐡 is called complex neutrosophic standard union, which is defined as, ( π‘–π‘šπ‘Žπ‘₯(𝑒𝐴 ,𝑒𝐡 ) , π‘šπ‘Žπ‘₯(π‘ž , π‘ž )π‘’π‘–π‘šπ‘Žπ‘₯(𝑣𝐴 ,𝑣𝐡 ) , Μƒ 𝐢̃𝐴 βˆͺΜƒ π‘šπ‘Žπ‘₯ 𝐴 𝐡 𝑁 𝐢𝐡 = π‘šπ‘Žπ‘₯(𝑝𝐴 , 𝑝𝐡 )𝑒 ) π‘–π‘šπ‘Žπ‘₯(𝑀 ,𝑀 ) 𝐴 𝐡 π‘šπ‘Žπ‘₯(π‘Ÿπ΄ , π‘Ÿπ΅ )𝑒 .

Step 5. Now, suppose The alternative 𝐴1 is the best alternative based on the VIKOR index 𝑄. Then, it will be a compromise optimal solution if the following two conditions are satisfied: (𝐢1) Acceptable advantage: 𝑄(𝐴2 ) βˆ’ 𝑄(𝐴1 ) β‰₯ 𝐷𝑄 (π‘Ž π‘‘β„Žπ‘Ÿπ‘’π‘ β„Žπ‘œπ‘™π‘‘ π‘£π‘Žπ‘™π‘’π‘’) = 1 , where, 𝐴2 is the second best alternative based on 𝑄. π‘€βˆ’1 (𝐢2) Acceptable stability: The alternative 𝐴1 is also the best for the both 𝑆 and 𝑅. If any one of 𝐢1 and 𝐢2 is fail, then a set of compromise optimal solutions will be obtained as a optimal solution as follows: β‡’ If the condition 𝐢1 is dissatisfied, then a set of 𝑀 β€² number of alβ€² ternatives {𝐴1 , 𝐴2 , … , 𝐴𝑀 } will be the compromise optimal solutions, β€² β€² 𝑀 where, 𝐴 is obtained from 𝑄(𝐴𝑀 ) βˆ’ 𝑄(𝐴1 ) < 𝐷𝑄 for the maximum value of 𝑀 β€² . β‡’ If the condition 𝐢2 is not satisfied then, 𝐴1 and 𝐴2 both will be the optimal solutions.

β€’ Complex neutrosophic algebraic sum. Μƒ ), then complex neuIf, βˆͺΜƒ 𝐹 is complex fuzzy algebraic sum (βˆ” 𝐹 trosophic union of 𝐢̃𝐴 and 𝐢̃𝐡 is called complex neutrosophic algebraic sum, which is defined as, ( ) 𝑒 𝑒 𝑒 𝑒 ( Μƒ 𝐢̃ = (𝑝 + 𝑝 βˆ’ 𝑝 .𝑝 )𝑒𝑖2πœ‹ 2πœ‹π΄ + 2πœ‹π΅ βˆ’ 2πœ‹π΄ . 2πœ‹π΅ , (π‘ž + π‘ž βˆ’ 𝐢̃𝐴 βˆ” 𝑁 𝐡 ( 𝐴 𝐡 𝐴 𝐡 𝐴 ) ( )𝐡 𝑀𝐴 𝑀𝐡 𝑀𝐴 𝑀𝐡 ) 𝑣𝐴 𝑣𝐡 𝑣𝐴 𝑣𝐡 π‘žπ΄ .π‘žπ΅ )𝑒𝑖2πœ‹ 2πœ‹ + 2πœ‹ βˆ’ 2πœ‹ . 2πœ‹ , (π‘Ÿπ΄ + π‘Ÿπ΅ βˆ’ π‘Ÿπ΄ .π‘Ÿπ΅ )𝑒𝑖2πœ‹ 2πœ‹ + 2πœ‹ βˆ’ 2πœ‹ . 2πœ‹ . β€’ Complex neutrosophic bold sum. If, βˆͺΜƒ 𝐹 is complex fuzzy bold sum (β‹“Μƒ 𝐹 ), then complex neutrosophic union of 𝐢̃𝐴 and 𝐢̃𝐡 is called complex neutrosophic bold sum, which is defined as, ( 𝐢̃𝐴 β‹“Μƒ 𝑁 𝐢̃𝐡 = π‘šπ‘–π‘›(1, 𝑝𝐴 + 𝑝𝐡 )π‘’π‘–π‘šπ‘–π‘›(2πœ‹,𝑒𝐴 +𝑒𝐡 ) , π‘šπ‘–π‘›(1, π‘žπ΄ + π‘žπ΅ ) ) π‘’π‘–π‘šπ‘–π‘›(2πœ‹,𝑣𝐴 +𝑣𝐡 ) , π‘šπ‘–π‘›(1, π‘Ÿπ΄ + π‘Ÿπ΅ )π‘’π‘–π‘šπ‘–π‘›(2πœ‹,𝑀𝐴 +𝑀𝐡 ) .

3. Some set-theoretic operations of complex neutrosophic sets Basic set-theoretic operations are very useful to deal with reallife problems. For example, suppose, three experts have given their opinions individually about an alternative over a parameter. Then, to get a single information about the alternative over that parameter, aggregation operator is needed. In 2009, Zhang et al. (Appendix B) defined some set-theoretic operations for complex fuzzy sets. Now, we have extended these ideas to complex neutrosophic sets. Here, we have used Smarandache’s 2003, 2005 neutrosophic operations (given in Appendix A) as follows: Let, 𝐢̃𝐴 and 𝐢̃𝐡 be two complex neutrosophic sets over the universal set π‘ˆ , where, 𝐢̃𝐴 = (𝑇𝐢̃𝐴 , 𝐼𝐢̃𝐴 , 𝐹𝐢̃𝐴 ) = (𝑝𝐴 𝑒𝑖𝑒𝐴 , π‘žπ΄ 𝑒𝑖𝑣𝐴 , π‘Ÿπ΄ 𝑒𝑖𝑀𝐴 ) and 𝐢̃𝐡 = (𝑇𝐢̃𝐡 , 𝐼𝐢̃𝐡 , 𝐹𝐢̃𝐡 ) = (𝑝𝐡 𝑒𝑖𝑒𝐡 , π‘žπ΅ 𝑒𝑖𝑣𝐡 , π‘Ÿπ΅ 𝑒𝑖𝑀𝐡 ).

Definition 7 (Complex Neutrosophic Bounded Difference). Complex neuΜƒ 𝑁 𝐢̃𝐡 trosophic bounded difference of 𝐢̃𝐴 and 𝐢̃𝐡 is denoted by, 𝐢̃𝐴 βŠ– and is defined as, ( Μƒ 𝑁 𝐢̃𝐡 = π‘šπ‘Žπ‘₯(0, 𝑝𝐴 βˆ’π‘π΅ )π‘’π‘–π‘šπ‘Žπ‘₯(0,𝑒𝐴 βˆ’π‘’π΅ ) , π‘šπ‘Žπ‘₯(0, π‘žπ΄ βˆ’π‘žπ΅ )π‘’π‘–π‘šπ‘Žπ‘₯(0,𝑣𝐴 βˆ’π‘£π΅ ) , 𝐢̃𝐴 βŠ– ) π‘šπ‘Žπ‘₯(0, π‘Ÿπ΄ βˆ’ π‘Ÿπ΅ )π‘’π‘–π‘šπ‘Žπ‘₯(0,𝑀𝐴 βˆ’π‘€π΅ ) . Definition 8 (Aggregation of π‘š Complex Neutrosophic Sets). Let, 𝐢̃𝑠 = (𝑝𝑠 𝑒𝑖𝑒𝑠 , π‘žπ‘  𝑒𝑖𝑣𝑠 , π‘Ÿπ‘  𝑒𝑖𝑀𝑠 ); 𝑠 = 1, 2, … , π‘š be π‘š complex neutrosophic sets over the universe π‘ˆ . Then, different types of aggregations of π‘š complex neutrosophic sets are defined as follows:

Definition 5 (Complex Neutrosophic Intersection). Then, complex neutrosophic intersection of 𝐢̃𝐴 and 𝐢̃𝐡 is denoted by, 𝐢̃𝐴 βˆ©Μƒ 𝑁 𝐢̃𝐡 and is defined as, 𝐢̃𝐴 βˆ©Μƒ 𝑁 𝐢̃𝐡 = (𝑇𝐢̃𝐴 βˆ©Μƒ 𝐹 𝐢̃𝐡 , 𝐼𝐢̃𝐴 βˆ©Μƒ 𝐹 𝐢̃𝐡 , 𝐹𝐢̃𝐴 βˆ©Μƒ 𝐹 𝐢̃𝐡 ), where, βˆ©Μƒ 𝐹 is the complex fuzzy intersection. Now,

β€’ Complex neutrosophic arithmetic mean aggregation of π‘š complex Μƒ 𝐴𝑀 𝐢̃2 βŠ• Μƒ 𝐴𝑀 β‹― neutrosophic sets 𝐢̃1 , 𝐢̃2 , … , πΆΜƒπ‘š is denoted by, 𝐢̃1 βŠ• 𝑁 𝑁 Μƒ 𝐴𝑀 πΆΜƒπ‘š and is defined as follows, βŠ• 𝑁 1 ( Μƒ 𝐴𝑀 𝐢̃2 βŠ• Μƒ 𝐴𝑀 β‹― βŠ• Μƒ 𝐴𝑀 πΆΜƒπ‘š = 1 (𝑝1 + 𝑝2 + β‹― + π‘π‘š )𝑒𝑖 π‘š (𝑒1 +𝑒2 +β‹―+π‘’π‘š ) , 𝐢̃1 βŠ•

β€’ Complex neutrosophic standard intersection. If, complex fuzzy intersection (βˆ©Μƒ 𝐹 ) is complex fuzzy standard Μƒ intersection (βˆ©Μƒ π‘šπ‘–π‘› 𝐹 ) then, complex neutrosophic intersection of 𝐢𝐴 and 𝐢̃𝐡 called complex neutrosophic standard intersection, which ( ) Μƒ can be denoted by, 𝐢̃𝐴 βˆ©Μƒ π‘šπ‘–π‘› 𝐢̃𝐡 , 𝐼𝐢̃𝐴 βˆ©Μƒ π‘šπ‘–π‘› 𝐢̃𝐡 , 𝐹𝐢̃𝐴 βˆ©Μƒ π‘šπ‘–π‘› 𝐢̃𝐡 𝑁 𝐢𝐡 = 𝑇𝐢̃𝐴 βˆ©Μƒ π‘šπ‘–π‘› 𝐹 𝐹 𝐹 and is defined as, ( π‘–π‘šπ‘–π‘›(𝑒𝐴 ,𝑒𝐡 ) , π‘šπ‘–π‘›(π‘ž , π‘ž )π‘’π‘–π‘šπ‘–π‘›(𝑣𝐴 ,𝑣𝐡 ) , Μƒ 𝐢̃𝐴 βˆ©Μƒ π‘šπ‘–π‘› 𝐴 𝐡 𝑁 𝐢𝐡 = π‘šπ‘–π‘›(𝑝𝐴 , 𝑝)𝐡 )𝑒 π‘–π‘šπ‘–π‘›(𝑀 𝐴 ,𝑀𝐡 ) . π‘šπ‘–π‘›(π‘Ÿπ΄ , π‘Ÿπ΅ )𝑒

𝑁 𝑁 𝑁 π‘š 1 1 π‘–π‘š (𝑣1 +𝑣2 +β‹―+π‘£π‘š ) 1 (π‘ž + π‘ž + β‹― + π‘ž )𝑒 , π‘š (π‘Ÿ1 2 π‘š π‘š 1 1 π‘–π‘š (𝑀1 +𝑀2 +β‹―+π‘€π‘š ) )

+ π‘Ÿ2 + β‹― + π‘Ÿπ‘š )

.

𝑒

β€’ Complex neutrosophic geometric mean aggregation of π‘š complex Μƒ 𝐺𝑀 𝐢̃2 βŠ• Μƒ 𝐺𝑀 β‹― neutrosophic sets 𝐢̃1 , 𝐢̃2 , … , πΆΜƒπ‘š is denoted by, 𝐢̃1 βŠ• 𝑁 𝑁 𝐺𝑀 Μƒ Μƒ βŠ•π‘ πΆπ‘š and is defined as follows, 𝑒1 𝑒2 π‘’π‘š 1βˆ•2 ( Μƒ 𝐺𝑀 𝐢̃2 βŠ• Μƒ 𝐺𝑀 ..βŠ• Μƒ 𝐺𝑀 πΆΜƒπ‘š = 𝐢̃1 βŠ• (𝑝1 .𝑝2 , … , π‘π‘š )1βˆ•2 𝑒𝑖2πœ‹( 2πœ‹ . 2πœ‹ ,…, 2πœ‹ ) ,

β€’ Complex neutrosophic product. If, βˆ©Μƒ 𝐹 is complex fuzzy product (̃◦𝐹 ), then complex neutrosophic intersection of 𝐢̃𝐴 and 𝐢̃𝐡 is called complex neutrosophic product which is defined as, ( ) (𝑣 𝑣 ) 𝑒𝐴 𝑒𝐡 ( 𝐴 𝐡 𝐢̃𝐴 β—¦Μƒ 𝑁 𝐢̃𝐡 (= 𝑝𝐴 .𝑝)𝐡 𝑒𝑖2πœ‹ 2πœ‹ . 2πœ‹ , π‘žπ΄ .π‘žπ΅ 𝑒𝑖2πœ‹ 2πœ‹ . 2πœ‹ , 𝑀𝐴 𝑀𝐡 ) π‘Ÿπ΄ .π‘Ÿπ΅ 𝑒𝑖2πœ‹ 2πœ‹ . 2πœ‹ .

𝑁

𝑁

𝑁𝑣

1 𝑣2

π‘£π‘š 1βˆ•2

(π‘ž1 .π‘ž2 , … , π‘žπ‘š )1βˆ•2 𝑒𝑖2πœ‹( 2πœ‹ . 2πœ‹ ,…, 2πœ‹ ) 𝑀1 𝑀2 π‘€π‘š 1βˆ•2 ) 𝑒𝑖2πœ‹( 2πœ‹ . 2πœ‹ ,…, 2πœ‹ )

, (π‘Ÿ1 .π‘Ÿ2 , … , π‘Ÿπ‘š )1βˆ•2

β€’ Complex neutrosophic harmonic mean aggregation of π‘š complex Μƒ 𝐻𝑀 𝐢̃2 βŠ• Μƒ 𝐻𝑀 β‹… neutrosophic sets 𝐢̃1 , 𝐢̃2 , … , πΆΜƒπ‘š is denoted by, 𝐢̃1 βŠ• 𝑁 𝑁 𝐻𝑀 Μƒ Μƒ βŠ• 𝑁 πΆπ‘š and is defined as follows, ( ) 𝑖2πœ‹ 2πœ‹ 2πœ‹π‘š ( + 𝑒 +β‹―+ 𝑒2πœ‹ 𝐻𝑀 𝐻𝑀 𝐻𝑀 π‘š 𝑒 Μƒ Μƒ Μƒ Μƒ Μƒ π‘š , 1 2 𝐢̃1 βŠ• = ( 1 1 )𝑒 𝑁 𝐢2 βŠ•π‘ ..βŠ•π‘ πΆπ‘š + +β‹―+ 𝑝1 𝑝1 𝑝2 π‘š ( ) π‘š

β€’ Complex neutrosophic bold intersection. If, βˆ©Μƒ 𝐹 is complex fuzzy bold intersection (β‹’Μƒ 𝐹 ), then complex neutrosophic intersection of 𝐢̃𝐴 and 𝐢̃𝐡 is called complex neutrosophic bold intersection which is defined as, ( 𝐢̃𝐴 β‹’Μƒ 𝑁 𝐢̃𝐡 = π‘šπ‘Žπ‘₯(0, 𝑝𝐴 + 𝑝𝐡 βˆ’ 1)π‘’π‘–π‘šπ‘Žπ‘₯(0,𝑒𝐴 +𝑒𝐡 βˆ’2πœ‹) , π‘šπ‘Žπ‘₯(0, π‘žπ΄ + π‘žπ΅ βˆ’ ) 1)π‘’π‘–π‘šπ‘Žπ‘₯(0,𝑣𝐴 +𝑣𝐡 βˆ’2πœ‹) , π‘šπ‘Žπ‘₯(0, π‘Ÿπ΄ + π‘Ÿπ΅ βˆ’ 1)π‘’π‘–π‘šπ‘Žπ‘₯(0,𝑀𝐴 +𝑀𝐡 βˆ’2πœ‹) .

𝑖2πœ‹

(

1 π‘ž1

+ π‘ž1 2

𝑖2πœ‹

𝑒

Definition 6 (Complex Neutrosophic Union). Complex neutrosophic union of 𝐢̃𝐴 and 𝐢̃𝐡 is denoted by, 𝐢̃𝐴 βˆͺΜƒ 𝑁 𝐢̃𝐡 and is defined as, 𝐢̃𝐴 βˆͺΜƒ 𝑁 𝐢̃𝐡 = (𝑇𝐢̃𝐴 βˆͺΜƒ 𝐹 𝐢̃𝐡 , 𝐼𝐢̃𝐴 βˆͺΜƒ 𝐹 𝐢̃𝐡 , 𝐹𝐢̃𝐴 βˆͺΜƒ 𝐹 𝐢̃𝐡 ), where, βˆͺΜƒ 𝐹 is the complex fuzzy union.

(

π‘š +β‹―+ π‘ž1

2πœ‹ + 2πœ‹ +β‹―+ 2πœ‹ 𝑣1 𝑣2 π‘£π‘š

)𝑒

π‘š π‘š

2πœ‹ 2πœ‹ 2πœ‹ 𝑀1 + 𝑀2 +β‹―+ π‘€π‘š

)

,(

1 π‘Ÿ1

π‘š + π‘Ÿ1 +β‹―+ π‘Ÿ1 π‘š 2

)

) .

Example 2. Let, π‘ˆ = {π‘₯1 , π‘₯2 , π‘₯3 } and 𝐢̃𝐴 = {π‘₯1 βˆ•(0.6𝑒𝑖2πœ‹ , 0.4π‘’π‘–πœ‹ , 0.5𝑒𝑖2πœ‹ ), π‘₯2 βˆ•(0.8π‘’π‘–πœ‹ , 0.3π‘’π‘–πœ‹ , 0.2𝑒𝑖2πœ‹ ), π‘₯3 βˆ•(0.3π‘’π‘–πœ‹βˆ•2 , 0.5𝑒𝑖2πœ‹ , 0.1π‘’π‘–πœ‹ )}, 𝐢̃𝐡 = {π‘₯1 βˆ•(0.4π‘’π‘–πœ‹ , 4

S. Manna, T.M. Basu and S.K. Mondal

Engineering Applications of Artificial Intelligence 89 (2020) 103432

Then, the score function of 𝐢̃𝑁 is denoted by, π‘†Μ‚π‘π‘Ÿ (𝐢̃𝑁 ) and is defined as follows: } 1 1{ [𝑝(π‘₯)+(2βˆ’(π‘ž(π‘₯)+π‘Ÿ(π‘₯)))]+ [𝑒(π‘₯)+(4πœ‹ βˆ’(𝑣(π‘₯)+𝑀(π‘₯)))] (1) π‘†Μ‚π‘π‘Ÿ (𝐢̃𝑁 ) = 6 2πœ‹ The score function of a complex neutrosophic number measures the accuracy of the number 𝐢̃𝑁 in favor of truth degree. ( ) ( Proposition 1. Let 𝐢̃𝐴 = 𝑝𝐴 𝑒𝑖𝑒𝐴 , π‘žπ΄ 𝑒𝑖𝑣𝐴 , π‘Ÿπ΄ 𝑒𝑖𝑀𝐴 and 𝐢̃𝐡 = 𝑝𝐡 𝑒𝑖𝑒𝐡 , ) π‘žπ΅ 𝑒𝑖𝑣𝐡 , π‘Ÿπ΅ 𝑒𝑖𝑀𝐡 be two complex neutrosophic numbers over the universal set π‘ˆ where, 𝑝𝐴 , π‘žπ΄ , π‘Ÿπ΄ , 𝑝𝐡 , π‘žπ΅ , π‘Ÿπ΅ ∈ [0, 1] with 0 ≀ 𝑝𝐴 + π‘žπ΄ + π‘Ÿπ΄ ≀ 3, 0 ≀ 𝑝𝐡 + π‘žπ΅ + π‘Ÿπ΅ ≀ 3 and 𝑒𝐴 , 𝑣𝐴 , 𝑀𝐴 , 𝑒𝐡 , 𝑣𝐡 , 𝑀𝐡 ∈ [0, 2πœ‹]. Then, the proposed score function (π‘†Μ‚π‘π‘Ÿ ) satisfies the following properties. (𝑖) 𝐢̃𝐴 ≀ 𝐢̃𝐡 ⇔ π‘†Μ‚π‘π‘Ÿ (𝐢̃𝐴 ) ≀ π‘†Μ‚π‘π‘Ÿ (𝐢̃𝐡 ). (𝑖𝑖) If 𝐢̃𝐴 = (1𝑒𝑖2πœ‹ , 0, 0), then π‘†Μ‚π‘π‘Ÿ (𝐢̃𝐴 ) = 1. (𝑖𝑖𝑖) If 𝐢̃𝐴 = (0, 1𝑒𝑖2πœ‹ , 1𝑒𝑖2πœ‹ ), then π‘†Μ‚π‘π‘Ÿ (𝐢̃𝐴 ) = 0. (𝑖𝑣) π‘†Μ‚π‘π‘Ÿ (𝐢̃𝐴 ) ∈ [0, 1].

Fig. 1. Venn diagram of a complex neutrosophic number.

Proof. (𝑖) From the Ali and Smarandache (2017) we have,

0.6π‘’π‘–πœ‹ , 0.8𝑒𝑖2πœ‹ ), π‘₯2 βˆ•(0.7π‘’π‘–πœ‹ , 0.3π‘’π‘–πœ‹ , 0.4𝑒𝑖2πœ‹ ), π‘₯3 βˆ•(0.1π‘’π‘–πœ‹ , 0.5𝑒𝑖2πœ‹ , 0.7π‘’π‘–πœ‹βˆ•2 )} be two complex neutrosophic sets over π‘ˆ . Then,

𝐢̃𝐴 ≀ 𝐢̃𝐡 ⇔ 𝑝𝐴 ≀ 𝑝𝐡 , π‘žπ΄ β‰₯ π‘žπ΅ , π‘Ÿπ΄ β‰₯ π‘Ÿπ΅ ; 𝑒𝐴 ≀ 𝑒𝐡 , 𝑣𝐴 β‰₯ 𝑣𝐡 , 𝑀𝐴 β‰₯ 𝑀𝐡

Μƒ (𝑖) 𝐢̃𝐴 βˆ©Μƒ π‘šπ‘–π‘› = {π‘₯1 βˆ•(0.4π‘’π‘–πœ‹ , 0.4π‘’π‘–πœ‹ , 0.5𝑒𝑖2πœ‹ ), π‘₯2 βˆ•(0.7π‘’π‘–πœ‹ , 0.3π‘’π‘–πœ‹ , 0.2𝑒𝑖2πœ‹ ), 𝑁 𝐢𝐡 π‘₯3 βˆ•(0.1π‘’π‘–πœ‹βˆ•2 , 0.4π‘’π‘–πœ‹ , 0.1π‘’π‘–πœ‹βˆ•2 )}.

⇔ 𝑝𝐴 ≀ 𝑝𝐡 , (1 βˆ’ π‘žπ΄ ) ≀ (1 βˆ’ π‘žπ΅ ), (1 βˆ’ π‘Ÿπ΄ ) ≀ (1 βˆ’ π‘Ÿπ΅ ); 𝑒𝐴 ≀ 𝑒𝐡 , (2πœ‹ βˆ’ 𝑣𝐴 ) ≀ (2πœ‹ βˆ’ 𝑣𝐡 ), (2πœ‹ βˆ’ 𝑀𝐴 ) ≀ (2πœ‹ βˆ’ 𝑀𝐡 ).

(𝑖𝑖) 𝐢̃𝐴 β—¦Μƒ 𝑁 𝐢̃𝐡 = {π‘₯1 βˆ•(0.24π‘’π‘–πœ‹ , 0.24π‘’π‘–πœ‹βˆ•2 , 0.4𝑒𝑖2πœ‹ ), π‘₯2 βˆ•(0.5π‘’π‘–πœ‹βˆ•2 , 0.09π‘’π‘–πœ‹βˆ•2 , 0.08𝑒𝑖2πœ‹ ), π‘₯3 βˆ•(0.03π‘’π‘–πœ‹βˆ•4 , 0.2π‘’π‘–πœ‹ , 0.07π‘’π‘–πœ‹βˆ•4 )}.

1 ⇔ π‘†Μ‚π‘π‘Ÿ (𝐢̃𝐴 ) = [(𝑝𝐴 + (1 βˆ’ π‘žπ΄ ) + (1 βˆ’ π‘Ÿπ΄ )) 6 1 + (𝑒 + (2πœ‹ βˆ’ 𝑣𝐴 ) + (2πœ‹ βˆ’ 𝑀𝐴 ))] 2πœ‹ 𝐴 1 ≀ [(𝑝𝐡 + (1 βˆ’ π‘žπ΅ ) + (1 βˆ’ π‘Ÿπ΅ )) 6 1 + (𝑒 + (2πœ‹ βˆ’ 𝑣𝐡 ) + (2πœ‹ βˆ’ 𝑀𝐡 ))] 2πœ‹ 𝐡 Μ‚ Μƒ = π‘†π‘π‘Ÿ (𝐢𝐡 )

(𝑖𝑖𝑖) 𝐢̃𝐴 β‹’Μƒ 𝑁 𝐢̃𝐡 = {π‘₯1 βˆ•(0π‘’π‘–πœ‹ , 0𝑒𝑖0 , 0.3𝑒𝑖2πœ‹ ), π‘₯2 βˆ•(0.5𝑒𝑖0 , 0𝑒𝑖0 , 0𝑒𝑖2πœ‹ ), π‘₯3 βˆ•(0𝑒𝑖0 , 0π‘’π‘–πœ‹ , 0𝑒𝑖0 )}. 𝑖2πœ‹ , 0.6π‘’π‘–πœ‹ , 0.8𝑒𝑖2πœ‹ ), π‘₯ βˆ•(0.8π‘’π‘–πœ‹ , 0.3π‘’π‘–πœ‹ , 0.4𝑒𝑖2πœ‹ ), Μƒ (𝑖𝑣) 𝐢̃𝐴 βˆͺΜƒ π‘šπ‘Žπ‘₯ 2 𝑁 𝐢𝐡 = {π‘₯1 βˆ•(0.6𝑒 π‘₯3 βˆ•(0.3π‘’π‘–πœ‹ , 0.5𝑒𝑖2πœ‹ , 0.7π‘’π‘–πœ‹ )}.

Μƒ 𝐢̃ (𝑣) 𝐢̃𝐴 βˆ” = {π‘₯1 βˆ•(0.76𝑒𝑖2πœ‹ , 0.76𝑒𝑖3πœ‹βˆ•2 , 0.9𝑒𝑖2πœ‹ ), π‘₯2 βˆ•(0.94𝑒𝑖3πœ‹βˆ•2 , 0.51 𝑁 𝐡 𝑒𝑖3πœ‹βˆ•2 , 0.52𝑒𝑖2πœ‹ ), π‘₯3 βˆ•(0.37𝑒𝑖5πœ‹βˆ•4 , 0.7𝑒𝑖2πœ‹ , 0.73𝑒𝑖5πœ‹βˆ•4 )}.

(𝑖𝑖) If 𝐢̃𝐴 = (1𝑒𝑖2πœ‹ , 0, 0) then, from Eq. (1) it is obvious that, π‘†Μ‚π‘π‘Ÿ (𝐢̃𝐴 ) = 1.

(𝑣𝑖) 𝐢̃𝐴 β‹“Μƒ 𝑁 𝐢̃𝐡 = {π‘₯1 βˆ•(1𝑒𝑖2πœ‹ , 1𝑒𝑖2πœ‹ , 1𝑒𝑖2πœ‹ ), π‘₯2 βˆ•(1𝑒𝑖2πœ‹ , 0.6𝑒𝑖2πœ‹ , 0.6𝑒𝑖2πœ‹ ), π‘₯3 βˆ•(0.4𝑒𝑖3πœ‹βˆ•2 , 0.9𝑒𝑖2πœ‹ , 0.8𝑒𝑖3πœ‹βˆ•2 )}.

(𝑖𝑖𝑖) If 𝐢̃𝐴 = (0, 1𝑒𝑖2πœ‹ , 1𝑒𝑖2πœ‹ ) then, from Eq. (1) it is obvious that, π‘†Μ‚π‘π‘Ÿ (𝐢̃𝐴 ) = 0.

Μƒ 𝐴𝑀 𝐢̃𝐡 = {π‘₯1 βˆ•(0.5𝑒𝑖3πœ‹βˆ•2 , 0.5π‘’π‘–πœ‹ , 0.65𝑒𝑖2πœ‹ ), π‘₯2 βˆ•(0.75π‘’π‘–πœ‹ , 0.3π‘’π‘–πœ‹ , (𝑣𝑖𝑖) 𝐢̃𝐴 βŠ• 𝑁 𝑖2πœ‹ 0.3𝑒 ), π‘₯3 βˆ•(0.2𝑒𝑖3πœ‹βˆ•4 , 0.45𝑒𝑖3πœ‹βˆ•2 , 0.4𝑒𝑖3πœ‹βˆ•4 )}.

(𝑖𝑣) Since, each of 𝑝𝐴 , π‘žπ΄ , π‘Ÿπ΄ ∈ [0, 1] and each of 𝑒𝐴 , 𝑣𝐴 , 𝑀𝐴 ∈ [0, 2πœ‹], then from Eq. (1) it is obvious that, π‘†Μ‚π‘π‘Ÿ (𝐢̃𝐴 ) ∈ [0, 1].

Μƒ 𝐺𝑀 𝐢̃𝐡 = {π‘₯1 βˆ•(0.49𝑒𝑖1.42πœ‹ , 0.49𝑒𝑖1.42πœ‹ , 0.63𝑒𝑖2πœ‹ ), π‘₯2 βˆ•(0.75π‘’π‘–πœ‹ , (𝑣𝑖𝑖𝑖) 𝐢̃𝐴 βŠ• 𝑁 0.3π‘’π‘–πœ‹ , 0.28𝑒𝑖2πœ‹ ), π‘₯3 βˆ•(0.17𝑒𝑖0.7πœ‹ , 0.45𝑒𝑖1.42πœ‹ , 0.26𝑒𝑖0.7πœ‹ )}.

Example 3. Let, 𝐢̃𝐴 = (0.7𝑒𝑖2πœ‹ , 0.4π‘’π‘–πœ‹ , 0.3𝑒𝑖2πœ‹ ) be a complex neutrosophic number over π‘ˆ . Then, by using Eq. (1), its score value is, { } 1 π‘†Μ‚π‘π‘Ÿ (𝐢̃𝐴 ) = 61 [0.7 + (2 βˆ’ (0.4 + 0.3))] + 2πœ‹ [2πœ‹ + (4πœ‹ βˆ’ (πœ‹ + 2πœ‹))] = 0.58. Let, 𝐢̃𝐡 = (0.2π‘’π‘–πœ‹ , 0.5𝑒𝑖2πœ‹ , 0.7𝑒𝑖2πœ‹ ) be another complex neutrosophic number over π‘ˆ . Then, by using Eq. (1), its score value is, { } 1 π‘†Μ‚π‘π‘Ÿ (𝐢̃𝐡 ) = 61 [0.2 + (2 βˆ’ (0.5 + 0.7))] + 2πœ‹ [πœ‹ + (4πœ‹ βˆ’ (2πœ‹ + 2πœ‹))] = 0.25.

Μƒ 𝐻𝑀 𝐢̃𝐡 = {π‘₯1 βˆ•(0.48𝑒𝑖4πœ‹βˆ•3 , 0.48π‘’π‘–πœ‹ , 0.5𝑒𝑖2πœ‹ ), π‘₯2 βˆ•(0.75π‘’π‘–πœ‹ , 0.3π‘’π‘–πœ‹ , (𝑖π‘₯) 𝐢̃𝐴 βŠ• 𝑁 𝑖2πœ‹ 0.27𝑒 ), π‘₯3 βˆ•(0.15𝑒𝑖2πœ‹βˆ•3 , 0.44𝑒𝑖4πœ‹βˆ•3 , 0.175𝑒𝑖2πœ‹βˆ•3 )}. Μƒ 𝑁 𝐢̃𝐡 = {π‘₯1 βˆ•(0.2π‘’π‘–πœ‹ , 0𝑒𝑖0 , 0𝑒𝑖0 ), π‘₯2 βˆ•(0.1𝑒𝑖0 , 0𝑒𝑖0 , 0𝑒𝑖0 ), π‘₯3 βˆ•(0.2𝑒𝑖0 , (π‘₯) 𝐢̃𝐴 βŠ– 0.1π‘’π‘–πœ‹ , 0π‘’π‘–πœ‹βˆ•2 )}. 4. A new definition of score function of a complex neutrosophic number (CNN) Some times in real-life decision-making problems, comparing uncertain evaluations such as fuzzy evaluations, intuitionistic fuzzy evaluations, neutrosophic evaluations, etc. is very necessary. In such cases, if we transform these uncertain values into some real values, then it will be easy to compare. Therefore, in this section, we have introduced a new definition of score function of a complex neutrosophic number to convert the uncertain number into some real number in the closed interval [0, 1].

5. Some necessary notions on complex neutrosophic soft sets Let, π‘ˆ = {π‘₯1 , π‘₯2 , … , π‘₯π‘š } be an universal set and 𝐸 = {𝑒1 , 𝑒2 , … , 𝑒𝑛 } be a set of parameters. Now, consider two complex neutrosophic soft sets (𝑓𝐢̃𝑁 , 𝐴) and (𝑔𝐢̃𝑁 , 𝐡) over π‘ˆ where, 𝐴, 𝐡 βŠ† 𝐸. Then, union operation and intersection operation of (𝑓𝐢̃𝑁 , 𝐴) and (𝑔𝐢̃𝑁 , 𝐡) can be defined as follows: Definition 10. Complex neutrosophic soft union (CNS-union) of two complex neutrosophic soft sets (𝑓𝐢̃𝑁 , 𝐴) and (𝑔𝐢̃𝑁 , 𝐡) is denoted by (𝑓𝐢̃𝑁 , 𝐴)βˆͺΜƒ 𝑁 (𝑔𝐢̃𝑁 , 𝐡) = (β„ŽπΆΜƒπ‘ , 𝐷) where, 𝐴 βˆͺ 𝐡 = 𝐷 and βˆ€π‘Ž ∈ 𝐷,

Definition 9. Let, 𝐢̃𝑁 = (𝑇 , 𝐼, 𝐹 ) = (𝑝(π‘₯)𝑒𝑖𝑒(π‘₯) , π‘ž(π‘₯)𝑒𝑖𝑣(π‘₯) , π‘Ÿ(π‘₯)𝑒𝑖𝑀(π‘₯) ) be a complex neutrosophic number where, 𝑝(π‘₯), π‘ž(π‘₯), π‘Ÿ(π‘₯) are the amplitude parts and 𝑒(π‘₯), 𝑣(π‘₯), 𝑀(π‘₯) are the phase parts of truth membership (𝑇 ), indeterminate membership (𝐼) and false membership (𝐹 ) of 𝐢̃𝑁 , 𝑝(π‘₯), π‘ž(π‘₯), π‘Ÿ(π‘₯) ∈ [0, 1] with 0 ≀ 𝑝(π‘₯) + π‘ž(π‘₯) + π‘Ÿ(π‘₯) ≀ 3 and 𝑒(π‘₯), 𝑣(π‘₯), 𝑀(π‘₯) ∈ [0, 2πœ‹].

βŽ§π‘“πΆΜƒ (π‘Ž) if π‘Ž ∈ 𝐴 βˆ’ 𝐡 βŽͺ 𝑁 β„ŽπΆΜƒπ‘ (π‘Ž) = βŽ¨π‘”πΆΜƒ (π‘Ž) if π‘Ž ∈ 𝐡 βˆ’ 𝐴 𝑁 βŽͺ βŽ©π‘“πΆΜƒπ‘ (π‘Ž)βˆͺΜƒ 𝑁 𝑔𝐢̃𝑁 (π‘Ž) if π‘Ž ∈ 𝐴 ∩ 𝐡

The Venn diagram of a complex neutrosophic number 𝐢̃𝑁 has been given in Fig. 1.

βˆͺΜƒ 𝑁 represents complex neutrosophic union of 𝑓𝐢̃𝑁 (π‘Ž) and 𝑔𝐢̃𝑁 (π‘Ž) as defined in Definition 6. 5

S. Manna, T.M. Basu and S.K. Mondal

Engineering Applications of Artificial Intelligence 89 (2020) 103432

β€’ If, βˆͺΜƒ 𝑁 is considered as complex neutrosophic standard union (βˆͺΜƒ π‘šπ‘Žπ‘₯ 𝑁 ), then the corresponding complex neutrosophic soft union is called complex neutrosophic soft standard union, which can be denoted by, (𝑓𝐢̃𝑁 , 𝐴)βˆͺΜƒ π‘šπ‘Žπ‘₯ 𝑁 (𝑔𝐢̃𝑁 , 𝐡) = (β„ŽπΆΜƒπ‘ , 𝐷). Μƒ ), β€’ If, βˆͺΜƒ 𝑁 is considered as complex neutrosophic algebraic sum (βˆ” 𝑁 then the corresponding complex neutrosophic soft union is called complex neutrosophic soft algebraic sum, which can be written Μƒ (𝑔 Μƒ , 𝐡) = (β„Ž Μƒ , 𝐷). as, (𝑓𝐢̃𝑁 , 𝐴)βˆ” 𝑁 𝐢𝑁 𝐢𝑁 β€’ If, βˆͺΜƒ 𝑁 is considered as complex neutrosophic bold sum (β‹“Μƒ 𝑁 ), then the corresponding complex neutrosophic soft union is called complex neutrosophic soft bold sum which can be written as, (𝑓𝐢̃𝑁 , 𝐴)β‹“Μƒ 𝑁 (𝑔𝐢̃𝑁 , 𝐡) = (β„ŽπΆΜƒπ‘ , 𝐷). Definition 11. Complex neutrosophic soft intersection (CNSintersection) of (𝑓𝐢̃𝑁 , 𝐴) and (𝑔𝐢̃𝑁 , 𝐡) is denoted by, (𝑓𝐢̃𝑁 , 𝐴)βˆ©Μƒ 𝑁 (𝑔𝐢̃𝑁 , 𝐡) = (𝐻𝐢̃𝑁 , 𝐷) where, 𝐴 ∩ 𝐡 = 𝐷 and βˆ€π‘Ž ∈ 𝐷, 𝐻𝐢̃𝑁 (π‘Ž) = 𝑓𝐢̃𝑁 (π‘Ž)βˆ©Μƒ 𝑁 𝑔𝐢̃𝑁 (π‘Ž). βˆ©Μƒ 𝑁 represents complex neutrosophic intersection of 𝑓𝐢̃𝑁 (π‘Ž) and 𝑔𝐢̃𝑁 (π‘Ž) as defined in Definition 5. Fig. 2. CNS ideal solution.

β€’ If, βˆ©Μƒ 𝑁 is considered as complex neutrosophic standard intersection (βˆ©Μƒ π‘šπ‘–π‘› 𝑁 ), then the corresponding complex neutrosophic soft intersection is called complex neutrosophic soft standard intersection, which can be denoted by, (𝑓𝐢̃𝑁 , 𝐴)βˆ©Μƒ π‘šπ‘–π‘› 𝑁 (𝑔𝐢̃𝑁 , 𝐡) = (𝐻𝐢̃𝑁 , 𝐷). β€’ If, βˆ©Μƒ 𝑁 is considered as complex neutrosophic product (̃◦𝑁 ), then the corresponding complex neutrosophic soft intersection is called complex neutrosophic soft product which can be denoted by, (𝑓𝐢̃𝑁 , 𝐴)̃◦𝑁 (𝑔𝐢̃𝑁 , 𝐡) = (𝐻𝐢̃𝑁 , 𝐷). β€’ If, βˆ©Μƒ 𝑁 is considered as complex neutrosophic bold intersection (β‹’Μƒ 𝑁 ), then the corresponding complex neutrosophic soft intersection is called complex neutrosophic soft bold intersection which can be denoted by, (𝑓𝐢̃𝑁 , 𝐴)β‹’Μƒ 𝑁 (𝑔𝐢̃𝑁 , 𝐡) = (𝐻𝐢̃𝑁 , 𝐷).

Complex neutrosophic soft ideal solution. Let, π‘ˆ = {π‘₯1 , π‘₯2 , … , π‘₯π‘š } be an universal set and 𝐸 = {𝑒1 , 𝑒2 , … , 𝑒𝑛 } be a corresponding parameter set. Now, consider (𝑓𝐢̃𝑁 , 𝐸) be a complex neutrosophic soft set over π‘ˆ such that, { } (𝑓𝐢̃𝑁 , 𝐸) = (𝑒1 , 𝑓𝐢̃𝑁 (𝑒1 )), (𝑒2 , 𝑓𝐢̃𝑁 (𝑒2 )), … , (𝑒𝑛 , 𝑓𝐢̃𝑁 (𝑒𝑛 )) { = (𝑒1 , {(π‘₯1 , π‘₯11 ), (π‘₯2 , π‘₯21 ), … , (π‘₯π‘š , π‘₯π‘š1 )}), (𝑒2 , {(π‘₯1 , π‘₯12 ), (π‘₯2 , π‘₯22 ), … , (π‘₯π‘š , π‘₯π‘š2 )}), … , } (𝑒𝑛 , {(π‘₯1 , π‘₯1𝑛 ), (π‘₯2 , π‘₯2𝑛 ), … , (π‘₯π‘š , π‘₯π‘šπ‘› )}) where, π‘₯𝑠𝑗 is the complex neutrosophic valued evaluation of an alternative π‘₯𝑠 over a parameter 𝑒𝑗 .

Example 4. Let π‘ˆ = {π‘₯1 , π‘₯2 , π‘₯3 } and 𝐸 = {𝑒1 , 𝑒2 , 𝑒3 }. Now assume that, (𝑓𝐢̃𝑁 , 𝐴) and (𝑔𝐢̃𝑁 , 𝐡) be two complex neutrosophic soft sets over π‘ˆ where, 𝐴 = {𝑒1 , 𝑒2 } and 𝐡 = {𝑒1 , 𝑒3 } such that, (𝑓𝐢̃𝑁 , 𝐴) = {(𝑒1 , (π‘₯1 βˆ•(0.3π‘’π‘–πœ‹βˆ•2 , 0.1π‘’π‘–πœ‹ , 0.6𝑒𝑖3πœ‹βˆ•2 ), π‘₯2 βˆ•(0.6𝑒𝑖3πœ‹βˆ•2 , 0.5π‘’π‘–πœ‹ , 0.4π‘’π‘–πœ‹βˆ•2 ), π‘₯3 βˆ•(0.8𝑒𝑖2πœ‹ , 0.3π‘’π‘–πœ‹ , 0.2π‘’π‘–πœ‹βˆ•2 ))), (𝑒2 , (π‘₯1 βˆ•(0.7π‘’π‘–πœ‹βˆ•4 , 0.3π‘’π‘–πœ‹ , 0.2π‘’π‘–πœ‹ ), π‘₯2 βˆ•(0.6𝑒𝑖2πœ‹ , 0.4π‘’π‘–πœ‹βˆ•2 , 0.1π‘’π‘–πœ‹ ), π‘₯3 βˆ•(0.4π‘’π‘–πœ‹ , 0.7𝑒𝑖2πœ‹ , 0.9𝑒𝑖2πœ‹ )))} and (𝑔𝐢̃𝑁 , 𝐴) = {(𝑒1 , (π‘₯1 βˆ•(0.4π‘’π‘–πœ‹ , 0.3π‘’π‘–πœ‹βˆ•2 , 0.5𝑒𝑖2πœ‹ ), π‘₯2 βˆ•(0.7𝑒𝑖3πœ‹βˆ•2 , 0.4π‘’π‘–πœ‹βˆ•2 , 0.2π‘’π‘–πœ‹ ), π‘₯3 βˆ•(0.7𝑒𝑖3πœ‹βˆ•2 , 0.4π‘’π‘–πœ‹ , 0.5π‘’π‘–πœ‹βˆ•4 ))), (𝑒3 , (π‘₯1 βˆ•(0.6π‘’π‘–πœ‹βˆ•2 , 0.4π‘’π‘–πœ‹ , 0.3π‘’π‘–πœ‹βˆ•2 ), π‘₯2 βˆ•(0.5π‘’π‘–πœ‹ , 0.4π‘’π‘–πœ‹βˆ•2 , 0.3π‘’π‘–πœ‹ ), π‘₯3 βˆ•(0.3π‘’π‘–πœ‹βˆ•2 , 0.8𝑒𝑖3πœ‹βˆ•2 , 0.8𝑒𝑖2πœ‹ )))}. Then,

Definition 12. Then, complex neutrosophic soft ideal solution (CNSideal solution) of a CNS-set (𝑓𝐢̃𝑁 , 𝐸) is the best reference solution (as given in Fig. 2) which can be defined by taking complex neutrosophic standard union among the evaluations of all the alternatives with respect to each of the parameters. Mathematically, it can be denoted by π‘₯+ and is defined as, π‘₯+ = {𝑓 +Μƒ (𝑒1 ), 𝑓 +Μƒ (𝑒2 ), … , 𝑓 +Μƒ (𝑒𝑛 )}

π‘–πœ‹ π‘–πœ‹ 2πœ‹ β€’ (𝑓𝐢̃𝑁 , 𝐴)βˆͺΜƒ π‘šπ‘Žπ‘₯ 𝑁 (𝑔𝐢̃𝑁 , 𝐡) = {(𝑒1 , (π‘₯1 βˆ•(0.4𝑒 , 0.3𝑒 , 0.6𝑒 ), π‘₯2 βˆ•(0.7𝑒𝑖3πœ‹βˆ•2 , 0.5π‘’π‘–πœ‹ , 0.4𝑒2πœ‹ ), π‘₯3 βˆ•(0.8𝑒𝑖2πœ‹ , 0.4π‘’π‘–πœ‹ , 0.5π‘’π‘–πœ‹βˆ•2 ))), (𝑒2 , (π‘₯1 βˆ•(0.7π‘’π‘–πœ‹βˆ•4 , 0.3π‘’π‘–πœ‹ , 0.2π‘’π‘–πœ‹ ), π‘₯2 βˆ•(0.6𝑒𝑖2πœ‹ , 0.4π‘’π‘–πœ‹βˆ•2 , 0.1π‘’π‘–πœ‹ ), π‘₯3 βˆ•(0.4π‘’π‘–πœ‹ , 0.7𝑒𝑖2πœ‹ , 0.9𝑒𝑖2πœ‹ ))), (𝑒3 , (π‘₯1 βˆ•(0.6π‘’π‘–πœ‹βˆ•2 , 0.4π‘’π‘–πœ‹ , 0.3π‘’π‘–πœ‹βˆ•2 ), π‘₯2 βˆ•(0.5π‘’π‘–πœ‹ , 0.4π‘’π‘–πœ‹βˆ•2 , 0.3π‘’π‘–πœ‹ ), π‘₯3 βˆ•(0.3π‘’π‘–πœ‹βˆ•2 , 0.8𝑒𝑖3πœ‹βˆ•2 , 0.8𝑒𝑖2πœ‹ )))}. Μƒ (𝑔 Μƒ , 𝐡) = {(𝑒 , (π‘₯ βˆ•(0.5𝑒𝑖1.25πœ‹ , 0.37𝑒𝑖1.25πœ‹ , 0.8𝑒2πœ‹ ), β€’ (𝑓𝐢̃𝑁 , 𝐴)βˆ” 𝑁 𝐢𝑁 1 1 π‘₯2 βˆ•(0.88𝑒𝑖1.88πœ‹ , 0.7𝑒𝑖1.25πœ‹ , 0.52𝑒𝑖1.25πœ‹ ), π‘₯3 βˆ•(0.9𝑒𝑖2πœ‹ , 0.58𝑒𝑖1.5πœ‹ , 0.6𝑒𝑖1.5πœ‹βˆ•2 ))), (𝑒2 , (π‘₯1 βˆ•(0.7π‘’π‘–πœ‹βˆ•4 , 0.3π‘’π‘–πœ‹ , 0.2π‘’π‘–πœ‹ ), π‘₯2 βˆ•(0.6𝑒𝑖2πœ‹ , 0.4π‘’π‘–πœ‹βˆ•2 , 0.1π‘’π‘–πœ‹ ), π‘₯3 βˆ•(0.4π‘’π‘–πœ‹ , 0.7𝑒𝑖2πœ‹ , 0.9𝑒𝑖2πœ‹ ))), (𝑒3 , (π‘₯1 βˆ•(0.6π‘’π‘–πœ‹βˆ•2 , 0.4π‘’π‘–πœ‹ , 0.3π‘’π‘–πœ‹βˆ•2 ), π‘₯2 βˆ•(0.5π‘’π‘–πœ‹ , 0.4π‘’π‘–πœ‹βˆ•2 , 0.3π‘’π‘–πœ‹ ), π‘₯3 βˆ•(0.3π‘’π‘–πœ‹βˆ•2 , 0.8𝑒𝑖3πœ‹βˆ•2 , 0.8𝑒𝑖2πœ‹ )))}. β€’ (𝑓𝐢̃𝑁 , 𝐴)β‹“Μƒ 𝑁 (𝑔𝐢̃𝑁 , 𝐡) = {(𝑒1 , (π‘₯1 βˆ•(0.7𝑒𝑖3πœ‹βˆ•2 , 0.4𝑒𝑖3πœ‹βˆ•2 , 1𝑒2πœ‹ ), π‘₯2 βˆ•(1𝑒𝑖2πœ‹ , 0.9𝑒𝑖3πœ‹βˆ•2 , 0.6𝑒𝑖3πœ‹βˆ•2 ), π‘₯3 βˆ•(1𝑒𝑖2πœ‹ , 0.7𝑒𝑖2πœ‹ , 0.7𝑒𝑖3πœ‹βˆ•4 ))), (𝑒2 , (π‘₯1 βˆ•(0.7π‘’π‘–πœ‹βˆ•4 , 0.3π‘’π‘–πœ‹ , 0.2π‘’π‘–πœ‹ ), π‘₯2 βˆ•(0.6𝑒𝑖2πœ‹ , 0.4π‘’π‘–πœ‹βˆ•2 , 0.1π‘’π‘–πœ‹ ), π‘₯3 βˆ•(0.4π‘’π‘–πœ‹ , 0.7𝑒𝑖2πœ‹ , 0.9𝑒𝑖2πœ‹ ))), (𝑒3 , (π‘₯1 βˆ•(0.6π‘’π‘–πœ‹βˆ•2 , 0.4π‘’π‘–πœ‹ , 0.3π‘’π‘–πœ‹βˆ•2 ), π‘₯2 βˆ•(0.5π‘’π‘–πœ‹ , 0.4π‘’π‘–πœ‹βˆ•2 , 0.3π‘’π‘–πœ‹ ), π‘₯3 βˆ•(0.3π‘’π‘–πœ‹βˆ•2 , 0.8𝑒𝑖3πœ‹βˆ•2 , 0.8𝑒𝑖2πœ‹ )))}. β€’ (𝑓𝐢̃𝑁 , 𝐴)βˆ©Μƒ π‘šπ‘–π‘› = {(𝑒1 , (π‘₯1 βˆ•(0.3π‘’π‘–πœ‹βˆ•2 , 0.1π‘’π‘–πœ‹βˆ•2 , 0.5𝑒𝑖3πœ‹βˆ•2 ), 𝑁 (𝑔𝐢̃𝑁 , 𝐡) π‘₯2 βˆ•(0.6𝑒𝑖3πœ‹βˆ•2 , 0.4π‘’π‘–πœ‹βˆ•2 , 0.2π‘’πœ‹βˆ•2 ), π‘₯3 βˆ•(0.7𝑒𝑖3πœ‹βˆ•2 , 0.3π‘’π‘–πœ‹ , 0.2π‘’π‘–πœ‹βˆ•2 )))}. β€’ (𝑓𝐢̃𝑁 , 𝐴)̃◦𝑁 (𝑔𝐢̃𝑁 , 𝐡) = {(𝑒1 , (π‘₯1 βˆ•(0.12π‘’π‘–πœ‹βˆ•4 , 0.03π‘’π‘–πœ‹βˆ•4 , 0.3𝑒3πœ‹βˆ•2 ), 𝑖9πœ‹βˆ•8 π‘–πœ‹βˆ•4 π‘–πœ‹βˆ•4 π‘₯2 βˆ•(0.42𝑒 , 0.2𝑒 , 0.08𝑒 ), π‘₯3 βˆ•(0.56𝑒𝑖3πœ‹βˆ•4 , 0.12π‘’π‘–πœ‹βˆ•2 , 0.1π‘’π‘–πœ‹βˆ•16 )))}. β€’ (𝑓𝐢̃𝑁 , 𝐴)β‹’Μƒ 𝑁 (𝑔𝐢̃𝑁 , 𝐡) = {(𝑒1 , (π‘₯1 βˆ•(0𝑒𝑖0 , 0𝑒𝑖0 , 0.1𝑒3πœ‹βˆ•2 ), π‘₯2 βˆ•(0.3π‘’π‘–πœ‹ , 0𝑒𝑖0 , 0𝑒𝑖0 ), π‘₯3 βˆ•(0.5𝑒𝑖3πœ‹βˆ•2 , 0𝑒𝑖0 , 0𝑒𝑖0 )))}.

𝐢𝑁

𝐢𝑁

𝐢𝑁

where, Μƒ π‘šπ‘Žπ‘₯ Μƒ π‘šπ‘Žπ‘₯ 𝑓 +Μƒ (𝑒𝑗 ) = π‘₯1𝑗 βˆͺΜƒ π‘šπ‘Žπ‘₯ 𝑁 π‘₯2𝑗 βˆͺ𝑁 ..βˆͺ𝑁 π‘₯π‘šπ‘— ; 𝑗 = 1, 2, … , 𝑛. 𝐢𝑁

𝑓 +Μƒ (𝑒𝑗 ) is called the complex neutrosophic soft ideal evaluation of a 𝐢𝑁 parameter 𝑒𝑗 and βˆͺΜƒ π‘šπ‘Žπ‘₯ 𝑁 is the complex neutrosophic standard union as defined in Definition 6. Example 5. Let us consider the complex neutrosophic soft set (𝑓𝐢̃𝑁 , 𝐸) of Example 1 (Table 1). Then, complex neutrosophic soft ideal solution (π‘₯+ ) of (𝑓𝐢̃𝑁 , 𝐸) is defined as follows, π‘₯+ = {(𝑠1 , (1𝑒𝑖2πœ‹ , 0.2π‘’π‘–πœ‹βˆ•3 , 0.2π‘’π‘–πœ‹βˆ•3 )), (𝑠2 , (0.3π‘’π‘–πœ‹βˆ•4 , 0.6π‘’π‘–πœ‹βˆ•5 , 0.4𝑒𝑖5πœ‹βˆ•6 )), (𝑠3 , (0.5π‘’π‘–πœ‹ , 0.8π‘’π‘–πœ‹βˆ•5 , 0.4𝑒𝑖7πœ‹βˆ•9 )), (𝑠4 , (0.1𝑒𝑖2πœ‹ , 0.8π‘’π‘–πœ‹βˆ•3 , 0.7π‘’π‘–πœ‹βˆ•3 )), (𝑠5 , (0.7π‘’π‘–πœ‹βˆ•4 , 0.8π‘’π‘–πœ‹βˆ•5 , 0.9π‘’π‘–πœ‹βˆ•6 )), (𝑠6 , (0.8π‘’π‘–πœ‹ , 0.6π‘’π‘–πœ‹βˆ•5 , 0.7π‘’π‘–πœ‹ ))}.

6. Decision-making problems based on soft set theory under complex neutrosophic environment Decision-making problems based on soft set theory have been studied by the researchers under different environments like, crisp environment (Cagman and Enginoglu, 2010), fuzzy environment (Basu et al., 2012) intuitionistic fuzzy environment (Manna et al., 2019), 6

S. Manna, T.M. Basu and S.K. Mondal

Engineering Applications of Artificial Intelligence 89 (2020) 103432

β€’ Nature of the parameters in the considered decision-making problem.

Table 2 Tabular form of the CNS-set (𝑓𝐢̃𝑁 , 𝐸).

π‘₯1 π‘₯2 . . π‘₯π‘š

𝑒1

𝑒2

...

𝑒𝑛

π‘₯11 π‘₯21 . . π‘₯π‘š1

π‘₯12 π‘₯22 . . π‘₯π‘š2

... ... ... ... ...

π‘₯1𝑛 π‘₯2𝑛 . . π‘₯π‘šπ‘›

Molodtsov’s soft set is a parameterized family of subsets of the universal set π‘ˆ . In this set, any parameterization can be considered to handle real-life problems. For instance, we can take the parameters with the help of words, sentences, functions, real numbers, etc. But, since, in a soft set based decision-making problem, not all the parameters come from same environment, therefore, conflicting criteria may exist in the parameter set. i.e., some of the parameters may be benefit criteria and some of the parameters may be cost criteria. For example, to select the best manufacturing company, the parameter β€˜investment cost’ is a cost parameter (negative sense) because, with respect to this parameter, smallest evaluation of a company is best whereas, the parameter β€˜availability of skilled workers’ is a benefit parameter (positive sense) because, with respect to this parameter, highest evaluation of a company is best. So, in a complex neutrosophic soft set based decision-making problem, there may exist conflicting criteria in the considered parameter set. Then, to deal with the conflicting criteria, firstly, we have equalized the sense of all the parameters i.e., we have converted the sense of all the parameters into either cost or benefit, by using complement operator. In this paper, we have converted the evaluations of all the alternatives with respect to each of the cost parameters into benefit sense by taking their complex neutrosophic complement.

Fig. 3. Graphical representation of complex neutrosophic numbers.

β€’ Weights of the considered parameters.

neutrosophic environment (Basu and Mondal, 2015), etc. However, in many real-life related decision-making problems, one dimensional membership magnitudes (fuzzy membership degree, intuitionistic fuzzy membership degree, neutrosophic membership degree, etc.), which represent only one information of an alternative over a parameter, are not sufficient. For example, to know whether a patient is suffered by a disease β€˜viral fever’ or not, it is necessary to know both the information β€˜temperature level’ and β€˜time duration’ of a related symptom β€˜fever’ in the patient. So in that case, these two information should be considered together to express the belongingness of the symptom β€˜fever’ over the disease β€˜viral fever’ in the patient. Though, neutrosophic environment is the most powerful uncertain controlling tool, but cannot handle our above discussed medical problem. In this situation, complex neutrosophic environment can be used to handle this problem through its amplitude term and phase term. Therefore, in this section, we have proposed two methodologies to solve decision-making problems by using complex neutrosophic soft set theory.

Weights of the parameters play an important role in a decision-making problem. But, in a decision-making problem, weights of the parameters may or may not be given at the initial stage. Moreover, an incomplete information about the weights of the parameters may be given in the decision-making problem. Then, to solve the problems, where, no information is given about the weights of the parameters or where, an incomplete information is given about the weights of the parameters, we will derive the exact weights of the parameters. If, π‘Š = {π‘Š1 , π‘Š2 , … , π‘Šπ‘› }; π‘Šπ‘— ∈ [0, 1] be the associated weights of βˆ‘ parameters, then they should satisfy the condition, 𝑛𝑗=1 π‘Šπ‘— = 1. So, based on these above discussions, our targets are as follows: (𝑖) Equalize the sense of all the considered parameters associated with the CNS-set (𝑓𝐢̃𝑁 , 𝐸) to deal with the conflicting criteria. (𝑖𝑖) Use a normalization process for handling non commensurable parameters. (𝑖𝑖𝑖) Derive the exact weights of the parameters, if the weights of the parameters are not given or an incomplete information about the weights of the parameters is given in the decision-making problem. (𝑖𝑣) Rank the alternatives based on the complex neutrosophic soft set (𝑓𝐢̃𝑁 , 𝐸).

6.1. Complex neutrosophic soft decision-making for single decision maker based problems 6.1.1. Problem description Consider, a set of π‘š alternatives as, π‘ˆ = {π‘₯1 , π‘₯2 , … , π‘₯π‘š } and a set of 𝑛 corresponding parameters as, 𝐸 = {𝑒1 , 𝑒2 , … , 𝑒𝑛 } (which are in complex neutrosophic sense). Now assume that, (𝑓𝐢̃𝑁 , 𝐸) be a complex neutrosophic soft set (CNS-set ) over π‘ˆ such that,

6.1.2. Complex neutrosophic soft VIKOR approach for single decision maker (CNSVISDM-approach) based problems To fulfill the above targets, now we have proposed an algorithm in complex neutrosophic soft set framework. In this algorithm we have used VIKOR approach (Park et al., 2013) to obtain a compromise optimal solution by testing β€˜acceptable advantage’ and β€˜acceptable stability’ in this decision-making problem. This compromise optimal solution would be closest to the ideal solution. Our proposed approach contains the following steps.

(𝑓𝐢̃𝑁 , 𝐸) = {(𝑒1 , 𝑓𝐢̃𝑁 (𝑒1 )), (𝑒2 , 𝑓𝐢̃𝑁 (𝑒2 )), … , (𝑒𝑛 , 𝑓𝐢̃𝑁 (𝑒𝑛 ))} where, 𝑓𝐢̃𝑁 (𝑒𝑗 ) = {π‘₯1 βˆ•π‘₯1𝑗 , π‘₯2 βˆ•π‘₯2𝑗 , … , π‘₯π‘š βˆ•π‘₯π‘šπ‘— }. Here, π‘₯𝑠𝑗 = (𝑇𝑠𝑗 , 𝐼𝑠𝑗 , 𝐹𝑠𝑗 ) ; 𝑠 = 1, 2, … , π‘š, 𝑗 = 1, 2, … , 𝑛 be a complex neutrosophic valued evaluation of an alternative π‘₯𝑠 over a parameter 𝑒𝑗 where, 𝑇𝑠𝑗 , 𝐼𝑠𝑗 and 𝐹𝑠𝑗 are the complex-valued truth membership degree, complexvalued indeterminate membership degree and complex-valued false membership degree such that, 𝑇𝑠𝑗 = 𝑝𝑠𝑗 𝑒𝑖𝑒𝑠𝑗 , 𝐼𝑠𝑗 = π‘žπ‘ π‘— 𝑒𝑖𝑣𝑠𝑗 , 𝐹𝑠𝑗 = π‘Ÿπ‘ π‘— 𝑒𝑖𝑀𝑠𝑗 ; 𝑝𝑠𝑗 , π‘žπ‘ π‘— , π‘Ÿπ‘ π‘— ∈ [0, 1] with 0 ≀ 𝑝𝑠𝑗 + π‘žπ‘ π‘— + π‘Ÿπ‘ π‘— ≀ 3 and 𝑒𝑠𝑗 , 𝑣𝑠𝑗 , 𝑀𝑠𝑗 ∈ [0, 2πœ‹]. The tabular representation of the complex neutrosophic soft set (𝑓𝐢̃𝑁 , 𝐸) has been given in Table 2. In Fig. 3, graphical representation of a complex neutrosophic number (π‘₯𝑠𝑗 ) has been given.

Algorithm I (CNSVISDM-approach). Step 1. Input the necessary substance. Input the selected π‘š alternatives (π‘ˆ = {π‘₯1 , π‘₯2 , … , π‘₯π‘š }) and 𝑛 corresponding parameters (𝐸 = {𝑒1 , 𝑒2 , … , 𝑒𝑛 }). Input the evaluations of all the alternatives with respect to 𝑛 parameters through a complex neutrosophic soft set (𝑓𝐢̃𝑁 , 𝐸) as given in Table 2. If the weights 7

S. Manna, T.M. Basu and S.K. Mondal

Engineering Applications of Artificial Intelligence 89 (2020) 103432 Table 3 Tabular form of the complex neutrosophic bounded difference soft set (π‘“πΆβŠ–Μƒ , 𝐸).

of the parameters are given in a problem then, input them (π‘Š = {π‘Š1 , π‘Š2 , … , π‘Šπ‘› }) also.

𝑁

Step 2. Equalization of the sense of all the parameters for dealing with conflicting criteria. To deal with the conflicting parameters in a complex neutrosophic soft set based decision-making problem, we have transformed the evaluations of all the alternatives over each of the cost parameters into benefit sense to uniform the sense of all the parameters. In this regard, (𝑖) First, identify the benefit parameters (with respect to these parameters, large evaluation of an alternative is better) and cost parameters (with respect to these parameters, small evaluation of an alternative is better). Let 𝐴 be the set of benefit parameters and 𝐡 be the set of cost parameters such that, 𝐴 βˆͺ 𝐡 = 𝐸 and 𝐴 ∩ 𝐡 = πœ™. (𝑖𝑖) Transform the evaluations of all the alternatives with respect to each of the parameters of the set 𝐡 into benefit sense by taking their complement. Mathematically, it can be defined as follows, βˆ€ 𝑒𝑗 ∈ 𝐡,

𝑐-denotes the complement of a complex neutrosophic evaluation π‘₯𝑠𝑗 as given in Definition 2.

𝐢𝑁

Then, from Definition 6 it is concluded that, Μƒ π‘šπ‘Žπ‘₯ Μƒ π‘šπ‘Žπ‘₯ 𝑓 +Μƒ (𝑒𝑗 ) = π‘₯1𝑗 βˆͺΜƒ π‘šπ‘Žπ‘₯ 𝑁 π‘₯2𝑗 βˆͺ𝑁 ..βˆͺ𝑁 π‘₯π‘šπ‘— 𝐢𝑁 ( = π‘šπ‘Žπ‘₯{𝑝1𝑗 , 𝑝2𝑗 , … , π‘π‘šπ‘— }π‘’π‘–π‘šπ‘Žπ‘₯{𝑒1𝑗 ,𝑒2𝑗 ,…,π‘’π‘šπ‘— } ,

πœ™π‘ π‘— =

Step 4. Determination of the bounded difference between 𝑓 +Μƒ (𝑒𝑗 ) and π‘₯𝑠𝑗 .

βŠ– 𝑑21 . . βŠ– π‘‘π‘š1

βŠ– 𝑑22 . . βŠ– π‘‘π‘š2

... ... ... ...

βŠ– 𝑑2𝑛 . . βŠ– π‘‘π‘šπ‘›

𝑒1

𝑒2

...

𝑒𝑛

π‘₯1

βŠ– π‘†Μ‚π‘π‘Ÿ (𝑑11 )

βŠ– π‘†Μ‚π‘π‘Ÿ (𝑑12 )

...

βŠ– π‘†Μ‚π‘π‘Ÿ (𝑑1𝑛 )

π‘₯2 . . π‘₯π‘š

βŠ– π‘†Μ‚π‘π‘Ÿ (𝑑21 ) . . βŠ– π‘†Μ‚π‘π‘Ÿ (π‘‘π‘š1 )

βŠ– π‘†Μ‚π‘π‘Ÿ (𝑑22 ) . . βŠ– π‘†Μ‚π‘π‘Ÿ (π‘‘π‘š2 )

... ... ... ...

βŠ– π‘†Μ‚π‘π‘Ÿ (𝑑2𝑛 ) . . βŠ–) π‘†Μ‚π‘π‘Ÿ (π‘‘π‘šπ‘›

By using Definition 7, we have obtained the complex neutrosophic bounded difference of the evaluation π‘₯𝑠𝑗 of an alternative π‘₯𝑠 with respect to a parameter 𝑒𝑗 from the corresponding complex neutrosophic soft ideal evaluation 𝑓 +Μƒ (𝑒𝑗 ).

+ βŠ– π‘†Μ‚π‘π‘Ÿ (𝑑𝑗 )

(2)

Step 7. Determination of the group utility value of an alternative π‘₯𝑠 (Utility measure). π‘Š (π‘₯ ) The group utility value of an alternative π‘₯𝑠 is denoted by, π‘†Μ‚π‘π‘Ÿ 𝑠 and is derived by adding the weighted values πœ™π‘ 1 , πœ™π‘ 2 , … , πœ™π‘ π‘› . Mathematically, it is defined as follows,

𝐢𝑁

βŠ– Mathematically, the resultant value is denoted by, 𝑑𝑠𝑗 and is defined βŠ– + Μƒ as, 𝑑𝑠𝑗 = (𝑓 Μƒ (𝑒𝑗 )βŠ–π‘ π‘₯𝑠𝑗 ) such that, 𝐢𝑁

βŠ– Μƒ 𝑁 π‘₯𝑠𝑗 ) 𝑑𝑠𝑗 = (𝑓 +Μƒ (𝑒𝑗 )βŠ– 𝐢𝑁

π‘–π‘šπ‘Žπ‘₯(0,𝑣+ 𝑗 βˆ’π‘£π‘ π‘— )

βŠ– π‘†Μ‚π‘π‘Ÿ (𝑑𝑠𝑗 )

So, by using the above equation, each of the evaluations πœ™π‘ π‘— ; 𝑠 = 1, 2, … , π‘š; 𝑗 = 1, 2, … , 𝑛 over each of the parameters now become unit less.

𝐢𝑁

π‘Š π‘†Μ‚π‘π‘Ÿ (π‘₯𝑠 ) =

𝑛 βˆ‘

π‘Šπ‘— πœ™π‘ π‘— ; 𝑠 = 1, 2, … , π‘š; 𝑗 = 1, 2, … , 𝑛

(3)

𝑗=1

,

where, π‘Šπ‘— is the weight of the parameter 𝑒𝑗 and πœ™π‘ π‘— the normalized bounded difference of an evaluation π‘₯𝑠𝑗 from the corresponding ideal evaluation 𝑓 +Μƒ (𝑒𝑗 ) in terms of score value. If the weights of the param𝐢𝑁 eters are not given or an incomplete information about the weights of the parameters is given then, we have to determine the exact weights of the parameters by using Section 6.2. So, from Eq. (3) it is concluded that, the alternative π‘₯𝑠 which has π‘Š is best. smallest value of π‘†Μ‚π‘π‘Ÿ

)

= (π‘ΜŒπ‘ π‘— π‘’π‘–π‘’ΜŒ 𝑠𝑗 , π‘žΜŒπ‘ π‘— π‘’π‘–π‘£ΜŒπ‘ π‘— , π‘ŸΜŒπ‘ π‘— π‘’π‘–π‘€ΜŒ 𝑠𝑗 ) The resultant soft set is called complex neutrosophic bounded difference soft set. It is denoted by, (𝑓 βŠ– , 𝐸) and its tabular form has been 𝐢̃𝑁 given in Table 3. Step 5. Derivation of score value of each of the entries of the soft set (𝑓 βŠ– , 𝐸) (given in Table 3). 𝐢̃𝑁 Now, by using our proposed score function of a complex neutrosophic number (Definition 9), we have deduced the score value (π‘†Μ‚π‘π‘Ÿ ) βŠ– of each of the entries 𝑑𝑠𝑗 = (π‘ΜŒπ‘ π‘— π‘’π‘–π‘’ΜŒ 𝑠𝑗 , π‘žΜŒπ‘ π‘— π‘’π‘–π‘£ΜŒπ‘ π‘— , π‘ŸΜŒπ‘ π‘— π‘’π‘–π‘€ΜŒ 𝑠𝑗 ) in the complex neutrosophic bounded difference soft set (𝑓 βŠ– Μƒ , 𝐸) as follows:

Step 8. Determination of the individual regret value of an alternative π‘₯𝑠 (Regret measure). π‘šπ‘Žπ‘₯ and is The individual regret of an alternative π‘₯𝑠 is denoted by, π‘†Μ‚π‘π‘Ÿ derived by taking the maximum weighted value over all the weighted values of πœ™π‘ 1 , πœ™π‘ 2 , … , πœ™π‘ π‘› . Mathematically, it is defined as follows,

𝐢𝑁

βŠ– π‘†Μ‚π‘π‘Ÿ (𝑑𝑠𝑗 )

βŠ– 𝑑1𝑛

π‘₯2 . . π‘₯π‘š

6.2 Then, we have applied the linear max normalization technique βŠ– to normalize an entry π‘†Μ‚π‘π‘Ÿ (𝑑𝑠𝑗 ) associated with Table 4 as follows:

π‘šπ‘Žπ‘₯{π‘Ÿ1𝑗 , π‘Ÿ2𝑗 , … , π‘Ÿπ‘šπ‘— }π‘’π‘–π‘šπ‘Žπ‘₯{𝑀1𝑗 ,𝑀2𝑗 ,…,π‘€π‘šπ‘— } ) +) + 𝑖𝑒+ 𝑗 , π‘ž + 𝑒𝑖𝑣𝑗 , π‘Ÿ+ 𝑒𝑖𝑀𝑗 ; 𝑗 = 1, 2, … , 𝑛 = (𝑝+ 𝑗 𝑗 𝑗𝑒

π‘–π‘šπ‘Žπ‘₯(0,𝑀+ 𝑗 βˆ’π‘€π‘ π‘— )

...

+ βŠ– π‘†Μ‚π‘π‘Ÿ (𝑑𝑗 ) = π‘šπ‘Žπ‘₯π‘š 𝑆̂ (𝑑 βŠ– ) 𝑠=1 π‘π‘Ÿ 𝑠𝑗

π‘šπ‘Žπ‘₯{π‘ž1𝑗 , π‘ž2𝑗 , … , π‘žπ‘šπ‘— }π‘’π‘–π‘šπ‘Žπ‘₯{𝑣1𝑗 ,𝑣2𝑗 ,…,π‘£π‘šπ‘— } ,

π‘šπ‘Žπ‘₯(0, π‘Ÿ+ 𝑗 βˆ’ π‘Ÿπ‘ π‘— )𝑒

𝑒𝑛

βŠ– 𝑑12

Step 6. Normalization process to deal with non commensurable parameters. In a complex neutrosophic soft set based decision-making problem, there may exist non commensurable criterion in the parameter set. Then, to deal with the non commensurable data, we have used the linear max normalization technique (given in the preliminary Section 2.3) to make all the parameters unit less. 6.1 Firstly, we have derived the maximum score value over all the alternatives with respect to a parameter 𝑒𝑗 . Mathematically it is denoted + (𝑑 βŠ– ) and is defined as follows: by, π‘†Μ‚π‘π‘Ÿ 𝑗

Μƒ π‘šπ‘Žπ‘₯ Μƒ π‘šπ‘Žπ‘₯ 𝑓 +Μƒ (𝑒𝑗 ) = π‘₯1𝑗 βˆͺΜƒ π‘šπ‘Žπ‘₯ 𝑁 π‘₯2𝑗 βˆͺ𝑁 ..βˆͺ𝑁 π‘₯π‘šπ‘—

, π‘šπ‘Žπ‘₯(0, π‘žπ‘—+ βˆ’ π‘žπ‘ π‘— )𝑒

...

βŠ– 𝑑11

These score values have been presented in Table 4 which is named as score valued matrix.

Step 3. Derivation of complex neutrosophic soft ideal evaluation of each of the parameters. Since our goal is, select the alternative which satisfies all the parameters with maximum evaluation level therefore, by using Definition 12, we have derived the complex neutrosophic soft ideal evaluation of each of the parameters, denoted by 𝑓 +Μƒ (𝑒𝑗 ); 𝑗 = 1, 2, … , 𝑛, over the complex 𝐢𝑁 neutrosophic soft set (𝑓𝐢̃𝑁 , 𝐸) as follows:

= (π‘šπ‘Žπ‘₯(0, 𝑝+ 𝑗 βˆ’ 𝑝𝑠𝑗 )𝑒

𝑒2

π‘₯1

Table 4 βŠ– Tabular form of the score valued matrix (π‘†Μ‚π‘π‘Ÿ (𝑑𝑠𝑗 ))π‘šΓ—π‘› .

(𝑓𝐢̃𝑁 (𝑒𝑗 ))𝑐 = {π‘₯1 βˆ•π‘₯𝑐1𝑗 , π‘₯2 βˆ•π‘₯𝑐2𝑗 , … , π‘₯π‘š βˆ•π‘₯π‘π‘šπ‘— }

π‘–π‘šπ‘Žπ‘₯(0,𝑒+ 𝑗 βˆ’π‘’π‘ π‘— )

𝑒1

} 1 1{ [π‘ΜŒπ‘ π‘— + (2 βˆ’ (π‘žΜŒπ‘ π‘— + π‘ŸΜŒπ‘ π‘— ))] + [π‘’ΜŒ + (4πœ‹ βˆ’ (π‘£ΜŒ 𝑠𝑗 + π‘€ΜŒ 𝑠𝑗 ))] = 6 2πœ‹ 𝑠𝑗

π‘šπ‘Žπ‘₯ π‘†Μ‚π‘π‘Ÿ (π‘₯𝑠 ) = max{π‘Šπ‘— πœ™π‘ π‘— }; 𝑠 = 1, 2, … , π‘š; 𝑗 = 1, 2, … , 𝑛 𝑗

8

(4)

S. Manna, T.M. Basu and S.K. Mondal

Engineering Applications of Artificial Intelligence 89 (2020) 103432 1 ) 1 (π‘Ÿ + π‘Ÿ2𝑗 + β‹― + π‘Ÿπ‘šπ‘— )𝑒𝑖 π‘š (𝑀1𝑗 +𝑀2𝑗 +β‹―+π‘€π‘šπ‘— ) π‘š 1𝑗 Now to evaluate the exact weights of the parameters, we have followed the following steps. 6.2.1 After equalization of all the parameters (by using Step 2 of Algorithm I) associated with the CNS-set (𝑓𝐢̃𝑁 , 𝐸), determine the mean potentiality of each of the parameters by using Eq. (6). 6.2.2 Evaluate the complex neutrosophic distance measure (𝑑𝑠𝑗 ) between the evaluation π‘₯𝑠𝑗 of an alternative π‘₯𝑠 ; 𝑠 = 1, 2, … , π‘š with respect to the parameter 𝑒𝑗 and the mean potentiality (π‘šπ‘’π‘— ) of the parameter 𝑒𝑗 as follows,

Μ‚ 𝑠 ) of an alternative π‘₯𝑠 . Step 9. Derivation of the compromising index 𝐢(π‘₯ The compromising index of an alternative π‘₯𝑠 over all the parameters can be defined by the following equation: Μ‚ 𝑠) = πœƒ 𝐢(π‘₯

π‘Š (π‘₯ ) βˆ’ 𝑆̂ π‘Š (π‘₯ ) π‘†Μ‚π‘π‘Ÿ 𝑠 𝑠 π‘π‘Ÿ

+ (1 βˆ’ πœƒ)

π‘Š (π‘₯ ) βˆ’ 𝑆̂ π‘Š (π‘₯ ) π‘†Μ‚π‘π‘Ÿ 𝑠 𝑠 π‘π‘Ÿ

π‘šπ‘Žπ‘₯ (π‘₯ ) βˆ’ 𝑆̂ π‘šπ‘Žπ‘₯ (π‘₯ ) π‘†Μ‚π‘π‘Ÿ 𝑠 𝑠 π‘π‘Ÿ

(5)

π‘šπ‘Žπ‘₯ (π‘₯ ) βˆ’ 𝑆̂ π‘šπ‘Žπ‘₯ (π‘₯ ) π‘†Μ‚π‘π‘Ÿ 𝑠 𝑠 π‘π‘Ÿ

π‘Š (π‘₯ ) = min 𝑆̂ π‘Š (π‘₯ ), 𝑆̂ π‘Š (π‘₯ ) = max 𝑆̂ π‘Š (π‘₯ ), 𝑆̂ π‘šπ‘Žπ‘₯ (π‘₯ ) = where, π‘†Μ‚π‘π‘Ÿ 𝑠 𝑠 π‘π‘Ÿ 𝑠 𝑠 𝑠 π‘π‘Ÿ 𝑠 𝑠 π‘π‘Ÿ π‘π‘Ÿ π‘šπ‘Žπ‘₯ min𝑠 𝑆̂ (π‘₯𝑠 ), 𝑆̂ π‘šπ‘Žπ‘₯ (π‘₯𝑠 ) = max𝑠 𝑆̂ π‘šπ‘Žπ‘₯ (π‘₯𝑠 ). π‘π‘Ÿ

π‘π‘Ÿ

π‘π‘Ÿ

Here, the parameter πœƒ ∈ [0, 1] indicates the weight of the strategy of maximum group of utility and (1 βˆ’ πœƒ) indicates the weight of the strategy of minimum individual regret.

𝑑𝑠𝑗 = 𝑑̃CN (π‘šπ‘’π‘— , π‘₯𝑠𝑗 ) where, 𝑑̃CN is the complex neutrosophic distance measure as given in Definition 3. 6.2.3 After that, derive the total complex neutrosophic distance of all the alternatives with respect to the parameter 𝑒𝑗 . Mathematically, it is denoted by, 𝑑̃𝑒𝑗 and is defined as,

Step 10. Ranking of the alternatives based on the compromising index Construct the ranking order of π‘š alternatives based on the descendπ‘Š ), regret measure (𝑆̂ π‘šπ‘Žπ‘₯ ) and ing order of their utility measure (π‘†Μ‚π‘π‘Ÿ π‘π‘Ÿ Μ‚ compromising index (𝐢). Therefore, we will get three ranking results. Assume that, the alternative π‘₯𝐿 has minimum compromising index value among π‘š alternatives. Then, the alternative π‘₯𝐿 will be a compromise optimal solution if the following two conditions are satisfied:

𝑑̃𝑒𝑗 =

π‘š βˆ‘

𝑑̃CN (π‘šπ‘’π‘— , π‘₯𝑠𝑗 )

(7)

𝑠=1

6.2.4 Then, the weighted complex neutrosophic distance function of this complex neutrosophic soft set based decision-making problem is Μƒ denoted by, 𝐷(π‘Š ) and is constructed as follows. Μƒ 𝐷(π‘Š )=

𝑛 βˆ‘

𝑑̃𝑒𝑗 π‘Šπ‘— =

π‘š 𝑛 βˆ‘ βˆ‘

𝑑𝑠𝑗 π‘Šπ‘— =

π‘š 𝑛 βˆ‘ βˆ‘

𝑑̃CN (π‘šπ‘’π‘— , π‘₯𝑠𝑗 )π‘Šπ‘—

𝑗=1 𝑠=1

𝑗=1 𝑠=1

𝑗=1

Case-I. Weights of the parameters are incompletely known: Based on the Xue et al. (2016), the possible types of incomplete information about the weights of the parameters are as follows. For 𝑗 β‰  𝑗′, (𝑖) Weak ranking: 𝑃 1 = {π‘Šπ‘— β‰₯ π‘Šπ‘— β€² }; (𝑖𝑖) Strict ranking: 𝑃 2 = {π‘Šπ‘— βˆ’ π‘Šπ‘— β€² β‰₯ 𝛽𝑗 β€² , 𝛽𝑗 β€² β‰₯ 0}; (𝑖𝑖𝑖) Ranking of differences: 𝑃 3 = {π‘Šπ‘— βˆ’ π‘Šπ‘— β€² β‰₯ π‘Šπ‘˜ βˆ’ π‘Šπ‘˜β€² , 𝑗 β€² β‰  π‘˜ β‰  π‘˜β€² }; (𝑖𝑣) Ranking with multiples: 𝑃 4 = {π‘Šπ‘— β‰₯ 𝛽𝑗 β€² π‘Šπ‘— β€² , (0 ≀ 𝛽𝑗 β€² ≀ 1)}; (𝑣) Interval forms: 𝑃 5 = {𝛽𝑗 ≀ π‘Šπ‘— ≀ 𝛽𝑗 + πœ€π‘— , (0 ≀ 𝛽𝑗 ≀ 𝛽𝑗 + πœ€π‘— ≀ 1)}. Let 𝑃 be the set of these five possible categories of information about the weights of the parameters such that, 𝑃 = 𝑃 1βˆͺ𝑃 2βˆͺ𝑃 3βˆͺ𝑃 4βˆͺ𝑃 5. Now, we have established an optimization model to derive the exact weights of the parameters as following: βˆ‘ βˆ‘ ⎫ Μƒ Μƒ π‘šπ‘Žπ‘₯ 𝐷(π‘Š ) = 𝑛𝑗=1 π‘š 𝑠=1 𝑑CN (π‘šπ‘’π‘— , π‘₯𝑠𝑗 )π‘Šπ‘— βŽͺ (8) 𝑠.𝑑., π‘Š ∈ 𝑃 ; ⎬ βˆ‘ βŽͺ 0 ≀ π‘Šπ‘— ≀ 1 π‘Žπ‘›π‘‘ 𝑛𝑗=1 π‘Šπ‘— = 1 ⎭

Now, if any one of the above two conditions is not satisfied then, we will get a set of compromise optimal solutions as follows: β€’ If, Condition 1 is not satisfied then, we will take the first π‘šβ€² alternatives as a compromise optimal solutions for this decision problem for maximum number of π‘šβ€² , where the number π‘šβ€² will be derived by the following equation: Μ‚ π‘šβ€² ) βˆ’ 𝐢(π‘₯ Μ‚ 𝐿) < 1 𝐢(π‘₯ π‘šβˆ’1 β€’ If, Condition 2 is not satisfied then, π‘₯𝐿 and π‘₯𝐿′ both will be the compromise optimal solutions for this decision-making problem. 6.2. Determination of the weights of the parameters Weights of the parameters play an important role in a decisionmaking problem. However in a problem, weights of the parameters may or may not be given. Moreover, an incomplete information about the weights of the parameters may be given in the problem. So, when exact weights of the parameters are not given in a problem initially, then we have to determine the exact weights of the parameters. Let, π‘Š = {π‘Š1 , π‘Š2 , … , π‘Šπ‘› } be the weighs parameters where, π‘Šπ‘— ∈ βˆ‘ [0, 1]; 𝑗 = 1, 2, … , 𝑛 and 𝑛𝑗=1 π‘Šπ‘— = 1.

By solving the above equation, we will obtain the exact weights of the parameters. Case-II. Weights of the parameters are completely unknown: If, the weights are completely unknown in a decision-making problem, then the weights of the parameters can be derived by using Eq. (6) as follows: βˆ‘π‘š Μƒ 𝑑̃𝑒𝑗 𝑠=1 𝑑CN (π‘šπ‘’π‘— , π‘₯𝑠𝑗 ) π‘Šπ‘— = βˆ‘π‘› = βˆ‘π‘› βˆ‘π‘š (9) Μƒ Μƒ 𝑗=1 𝑑𝑒𝑗 𝑗=1 𝑠=1 𝑑CN (π‘šπ‘’π‘— , π‘₯𝑠𝑗 )

Definition 13. The mean potentiality of a parameter 𝑒𝑗 associated with the complex neutrosophic soft set (𝑓𝐢̃𝑁 , 𝐸) (Table 2) (after equalization) is derived by taking arithmetic mean aggregation (as given in Definition 8) of the evaluations of all the alternatives with respect to the parameter 𝑒𝑗 . Mathematically, it is denoted by π‘šπ‘’π‘— and is defined as follows: π‘šπ‘’π‘— =

𝑑𝑠𝑗 =

𝑠=1

Μ‚ β€² ) βˆ’ 𝐢(π‘₯ Μ‚ 𝐿 ) β‰₯ 𝐷𝑄 β€’ Condition 1. Acceptable advantage: 𝐢(π‘₯ 𝐿 1 (π‘Ž π‘‘β„Žπ‘Ÿπ‘’π‘ β„Žπ‘œπ‘™π‘‘ π‘£π‘Žπ‘™π‘’π‘’) = π‘šβˆ’1 , where, π‘₯𝐿′ is the alternative placed at the second position in ranking order of the alternatives based on compromising index and π‘š is the number of alternatives. β€’ Condition 2. Acceptable stability: The alternative π‘₯𝐿 is the best π‘Š ) and solution based on the ranking order of utility measure (π‘†Μ‚π‘π‘Ÿ π‘šπ‘Žπ‘₯ ). regret measure (π‘†Μ‚π‘π‘Ÿ

Μƒ 𝐴𝑀 π‘₯2𝑗 βŠ• Μƒ 𝐴𝑀 ..βŠ• Μƒ 𝐴𝑀 π‘₯π‘šπ‘— ) (π‘₯1𝑗 βŠ• 𝑁 𝑁 𝑁

π‘š βˆ‘

6.3. Complex neutrosophic soft decision-making for multiple decision maker based problems 6.3.1. Problem description Let us consider that, π‘ˆ = {π‘₯1 , π‘₯2 , … , π‘₯π‘š } be a set of π‘š alternatives and 𝐸 = {𝑒1 , 𝑒2 , … , 𝑒𝑛 } be a set of 𝑛 corresponding parameters. Now assume that, π‘˜ decision makers, 𝐷 = {𝑑1 , 𝑑2 , … , π‘‘π‘˜ }, have been assigned to select the best alternative from π‘š alternatives over 𝑛 parameters. It is considered that, all the decision makers have given their opinions about the alternatives in terms of complex neutrosophic valued evaluations. So, we get π‘˜ complex neutrosophic soft sets, (𝑓𝐢̃𝑁 , 𝐸), (𝑓𝐢̃𝑁 , 𝐸), … , (𝑓𝐢̃𝑁 , 𝐸), where,

(6)

Then, from Definition 8 it is obtained that, ( ) Μƒ 𝐴𝑀 π‘₯2𝑗 βŠ• Μƒ 𝐴𝑀 ..βŠ• Μƒ 𝐴𝑀 π‘₯π‘šπ‘— π‘šπ‘’π‘— = π‘₯1𝑗 βŠ• 𝑁 𝑁 𝑁 1 (1 = (𝑝 + 𝑝2𝑗 + β‹― + π‘π‘šπ‘— )𝑒𝑖 π‘š (𝑒1𝑗 +𝑒2𝑗 +β‹―+π‘’π‘šπ‘— ) , π‘š 1𝑗 1 1 (π‘ž + π‘ž2𝑗 + β‹― + π‘žπ‘šπ‘— )𝑒𝑖 π‘š (𝑣1𝑗 +𝑣2𝑗 +β‹―+π‘£π‘šπ‘— ) , π‘š 1𝑗

1

9

2

π‘˜

S. Manna, T.M. Basu and S.K. Mondal

Engineering Applications of Artificial Intelligence 89 (2020) 103432

Table 5 Tabular form of the π‘˜ complex neutrosophic soft sets. (𝑓𝐢̃𝑁 , 𝐸)

… , (𝑓𝐢̃𝑁 , 𝐸) as given in Table 5. If, the weights of the parameters are π‘˜ given initially in the problem then input them (π‘Š = {π‘Š1 , π‘Š2 , ..π‘Šπ‘› }) also.

(𝑓𝐢̃𝑁 , 𝐸)

1

2

𝑒1

𝑒2

...

𝑒𝑛

𝑒1

𝑒2

...

𝑒𝑛

π‘₯1 π‘₯2

π‘₯(11,1) π‘₯(21,1)

π‘₯(12,1) π‘₯(22,1)

π‘₯(1𝑛,1) π‘₯(2𝑛,1)

π‘₯(11,2) π‘₯(21,2)

π‘₯(12,2) π‘₯(22,2)

π‘₯(π‘š1,1)

π‘₯(π‘š2,1)

π‘₯(π‘šπ‘›,1)

π‘₯(π‘š1,2)

π‘₯(π‘š2,2)

... ... ... ...

π‘₯(1𝑛,2) π‘₯(2𝑛,2)

π‘₯π‘š

... ... ... ...

Step 2. Equalization of the sense of all the parameters for handling conflicting criteria. Firstly, we have recognized the benefit parameters and cost parameters in the parameter set 𝐸. Suppose, 𝐴 is the set of benefit parameters and 𝐡 be the set cost parameters such that, 𝐴 βˆͺ 𝐡 = 𝐸 and 𝐴 ∩ 𝐡 = πœ™. Then, we have transformed the evaluations of all the alternatives with respect to each of the cost parameters into benefit sense by using complex neutrosophic complement as follows:

π‘₯(π‘šπ‘›,2)

(𝑓𝐢̃𝑁 , 𝐸) π‘˜

.. .. .. ..

𝑒1

𝑒2

...

𝑒𝑛

π‘₯(11,π‘˜) 𝑝(21,π‘˜)

π‘₯(12,π‘˜) π‘₯(22,π‘˜)

π‘₯(1𝑛,π‘˜) π‘₯(2𝑛,π‘˜)

π‘₯(π‘š1,π‘˜)

π‘₯(π‘š2,π‘˜)

... ... ... ...

βˆ€π‘™ = 1, 2, … , π‘˜ , 𝑓𝐢̃𝑁 (𝑒𝑗 ) = {π‘₯1 βˆ•π‘₯𝑐(1𝑗,𝑙) , π‘₯2 βˆ•π‘₯𝑐(2𝑗,𝑙) , … , π‘₯π‘š βˆ•π‘₯𝑐(π‘šπ‘—,𝑙) }; βˆ€π‘’π‘— ∈ 𝐡

π‘₯(π‘šπ‘›,π‘˜)

𝑙

Step 3. Construction a resultant complex neutrosophic soft set (𝐹𝐢̃𝑁 , 𝐸). Since in this problem, our aim is, select the best alternative based on the opinions of all the decision makers, therefore to take a common decision from π‘˜ decision makers, we have used complex neutrosophic soft intersection operator (given in Definition 11). Therefore, we have constructed a resultant complex neutrosophic soft set (𝐹𝐢̃𝑁 , 𝐸) from π‘˜ complex neutrosophic soft sets (𝑓𝐢̃𝑁 , 𝐸), 1 (𝑓𝐢̃𝑁 , 𝐸), … , (𝑓𝐢̃𝑁 , 𝐸) by using complex neutrosophic soft intersection π‘˜ 2 as follows,

(𝑓𝐢̃𝑁 , 𝐸) = {(𝑒1 , 𝑓𝐢̃𝑁 (𝑒1 )), (𝑒2 , 𝑓𝐢̃𝑁 (𝑒2 )), … , (𝑒𝑛 , 𝑓𝐢̃𝑁 (𝑒𝑛 ))} 𝑙 𝑙 𝑙 𝑙 = {(𝑒1 , (π‘₯1 βˆ•π‘₯(11,𝑙) , π‘₯2 βˆ•π‘₯(21,𝑙) , … , π‘₯π‘š βˆ•π‘₯(π‘š1,𝑙) )), (𝑒2 , (π‘₯1 βˆ•π‘₯(12,𝑙) , π‘₯2 βˆ•π‘₯(22,𝑙) , … , π‘₯π‘š βˆ•π‘₯(π‘š2,𝑙) )), …, (𝑒𝑛 , (π‘₯1 βˆ•π‘₯(1𝑛,𝑙) , π‘₯2 βˆ•π‘₯(2𝑛,𝑙) , … , π‘₯π‘š βˆ•π‘₯(π‘šπ‘›,𝑙) ))}. Here, π‘₯(𝑠𝑗,𝑙) = (𝑝(𝑠𝑗,𝑙) 𝑒𝑖𝑒(𝑠𝑗,𝑙) , π‘ž(𝑠𝑗,𝑙) 𝑒𝑖𝑣(𝑠𝑗,𝑙) , π‘Ÿ(𝑠𝑗,𝑙) 𝑒𝑖𝑀(𝑠𝑗,𝑙) ) represents the complex neutrosophic valued evaluation of an alternative π‘₯𝑠 ; 𝑠 = 1, 2, … , π‘š over a parameter 𝑒𝑗 ; 𝑗 = 1, 2, … , 𝑛 given by the decision maker 𝑑𝑙 ; 𝑙 = 1, 2, … , π‘˜. Tabular form of π‘˜ CNS-sets has been given in Table 5.

(𝐹𝐢̃𝑁 , 𝐸) = (𝑓𝐢̃𝑁 , 𝐸)βˆ©Μƒ 𝑁 (𝑓𝐢̃𝑁 , 𝐸)βˆ©Μƒ 𝑁 ..βˆ©Μƒ 𝑁 (𝑓𝐢̃𝑁 , 𝐸) π‘˜

2

1

β€’ Nature of the considered parameters.

Then, from Definition 11, it is concluded that, βˆ€π‘— = 1, 2, … , 𝑛

In a complex neutrosophic soft set based group decision-making problem, conflicting parameter may exist in the considered parameter set 𝐸. Therefore, we will equalize the sense of all the parameters by transforming the evaluations of all the alternatives over each of the cost parameter into benefit sense by using complex neutrosophic complement.

𝐹𝐢̃𝑁 (𝑒𝑗 ) = 𝑓𝐢̃𝑁 (𝑒𝑗 )βˆ©Μƒ 𝑁 𝑓𝐢̃𝑁 (𝑒𝑗 )βˆ©Μƒ 𝑁 ..βˆ©Μƒ 𝑁 𝑓𝐢̃𝑁 (𝑒𝑗 ) π‘˜

π‘₯2 βˆ•(π‘₯(2𝑗,1) βˆ©Μƒ 𝑁 π‘₯(2𝑗,2) βˆ©Μƒ 𝑁 ..βˆ©Μƒ 𝑁 π‘₯(2𝑗,π‘˜) ), … , π‘₯π‘š βˆ•(π‘₯(π‘šπ‘—,1) βˆ©Μƒ 𝑁 π‘₯(π‘šπ‘—,2) βˆ©Μƒ 𝑁 ..βˆ©Μƒ 𝑁 π‘₯(π‘šπ‘—,π‘˜) )} = {π‘₯1 βˆ•π‘’1𝑗 , π‘₯2 βˆ•π‘’2𝑗 , … , π‘₯π‘š βˆ•π‘’π‘šπ‘— }

β€’ Weights of the parameters.

Μ„ where, βˆ©Μƒ 𝑁 is the complex neutrosophic intersection and 𝑒𝑠𝑗 = (𝑃̄𝑠𝑗 π‘’π‘–π‘ˆπ‘ π‘— , 𝑖𝑉̄𝑠𝑗 Μ„ π‘–π‘ŠΜ„ 𝑠𝑗 Μ„ 𝑄𝑠𝑗 𝑒 , 𝑅𝑠𝑗 𝑒 ); 𝑠 = 1, 2, … , π‘š.

In this group decision-making problem, weights of the parameters may or may not be given initially in the problem. Even sometimes, the information about the weights of the parameters may be incomplete. So, when the weights of the parameters are completely unknown or incompletely known in a problem, then we will derive the weights of the parameters. If, π‘Š = {π‘Š1 , π‘Š2 , … , π‘Šπ‘› }; 0 ≀ π‘Šπ‘— ≀ 1 be the weights of the βˆ‘ parameters, then, 𝑛𝑗=1 π‘Šπ‘— = 1. Now, based on the above discussion, our goals are as follows: (𝑖) Equalize the sense of the all considered parameters for handling the conflicting criteria. (𝑖𝑖) Construct a resultant CNS-set (𝐹𝐢̃𝑁 , 𝐸) from π‘˜ CNS-sets (𝑓𝐢̃𝑁 , 𝐸); 𝑙 = 𝑙 1, 2, … , π‘˜. (𝑖𝑖𝑖) Use a normalization process for handling non commensurable parameters. (𝑖𝑣) Derive the exact weights of the parameters if they are completely unknown or incompletely known. (𝑣) Ranking the alternatives based on the opinions of all the π‘˜ decision makers.

Step 4. Derivation of the complex neutrosophic soft ideal evaluation of each of the parameters. Complex neutrosophic soft ideal evaluation 𝐹 +Μƒ (𝑒𝑗 ) of a parameter 𝐢𝑁 𝑒𝑗 over the resultant complex neutrosophic soft set (𝐹𝐢̃𝑁 , 𝐸) has been defined as follows: Μƒ π‘šπ‘Žπ‘₯ Μƒ π‘šπ‘Žπ‘₯ 𝐹 +Μƒ (𝑒𝑗 ) = 𝑒1𝑗 βˆͺΜƒ π‘šπ‘Žπ‘₯ 𝑁 𝑒2𝑗 βˆͺ𝑁 ..βˆͺ𝑁 π‘’π‘šπ‘— 𝐢𝑁

Μ„

Μ„

Μ„

= (π‘šπ‘Žπ‘₯(𝑃̄1𝑗 , 𝑃̄2𝑗 , … , π‘ƒΜ„π‘šπ‘— )π‘’π‘–π‘šπ‘Žπ‘₯(π‘ˆ1𝑗 ,π‘ˆ2𝑗 ,…,π‘ˆπ‘šπ‘— ) , π‘šπ‘Žπ‘₯(𝑄̄ 1𝑗 , 𝑄̄ 2𝑗 , … , 𝑄̄ π‘šπ‘— )𝑒

π‘–π‘šπ‘Žπ‘₯(𝑉̄1𝑗 ,𝑉̄2𝑗 ,…,π‘‰Μ„π‘šπ‘— ) Μ„

Μ„

,

Μ„

π‘šπ‘Žπ‘₯(𝑅̄ 1𝑗 , 𝑅̄ 2𝑗 , … , 𝑅̄ π‘šπ‘— )π‘’π‘–π‘šπ‘Žπ‘₯(π‘Š1𝑗 ,π‘Š2𝑗 ,…,π‘Šπ‘šπ‘— ) ) = (𝑃𝑗+ 𝑒

π‘–π‘ˆπ‘—+

, 𝑄+ 𝑗𝑒

𝑖𝑉𝑗+

, 𝑅+ 𝑗𝑒

π‘–π‘Šπ‘—+

)

Step 5. Determination of the bounded difference of 𝑒𝑠𝑗 form 𝐹 +Μƒ (𝑒𝑗 ). 𝐢𝑁 We have evaluated the complex neutrosophic bounded difference βŠ– (𝑑̄𝑠𝑗 ) of the evaluation 𝑒𝑠𝑗 of an alternative π‘₯𝑠 over the parameter 𝑒𝑗 from the associated ideal evaluation 𝐹 +Μƒ (𝑒𝑗 ) over the soft set (𝐹𝐢̃𝑁 , 𝐸) 𝐢𝑁 as follows:

6.3.2. Complex neutrosophic soft VIKOR approach for multiple decision maker (CNSVIMDM-approach) based problems In this section, we have provided a decision-making approach to get a synchronized solution from this complex neutrosophic soft set based group decision-making problem by using VIKOR method.

+ βˆ’π‘ˆ Μ„ 𝑠𝑗 )

βŠ– Μƒ 𝑁 𝑒𝑠𝑗 = (π‘šπ‘Žπ‘₯(0, 𝑃 + βˆ’ 𝑃̄𝑠𝑗 )π‘’π‘–π‘šπ‘Žπ‘₯(0,π‘ˆπ‘— 𝑑̄𝑠𝑗 = 𝐹 +Μƒ (𝑒𝑗 )βŠ– 𝑗 𝐢𝑁

π‘šπ‘Žπ‘₯(0, 𝑄+ 𝑗

βˆ’ 𝑄̄ 𝑠𝑗 )𝑒

Μ„ π‘šπ‘Žπ‘₯(0, 𝑅+ 𝑗 βˆ’ 𝑅𝑠𝑗 )𝑒

π‘–π‘šπ‘Žπ‘₯(0,𝑉𝑗+ βˆ’π‘‰Μ„π‘ π‘— )

,

,

π‘–π‘šπ‘Žπ‘₯(0,π‘Šπ‘—+ βˆ’π‘ŠΜ„ 𝑠𝑗 )

̌

̌

̌

) = (π‘ƒΜŒπ‘ π‘— π‘’π‘–π‘ˆπ‘ π‘— , π‘„ΜŒ 𝑠𝑗 𝑒𝑖𝑉𝑠𝑗 , π‘…ΜŒ 𝑠𝑗 π‘’π‘–π‘Šπ‘ π‘— )

The resultant Complex neutrosophic bounded difference soft set is Μ„βŠ– denoted by, (𝐹 βŠ– Μƒ , 𝐸) = (𝑑𝑠𝑗 )π‘šΓ—π‘› .

Algorithm II (CNSVIMDM-approach). Step 1. Input the necessary sub-

𝐢𝑁

stance. Input π‘š alternatives (π‘ˆ = {π‘₯1 , π‘₯2 , … , π‘₯π‘š }), 𝑛 parameters (𝐸 = {𝑒1 , 𝑒2 , … , 𝑒𝑛 }) and π‘˜ complex neutrosophic soft sets (𝑓𝐢̃𝑁 , 𝐸), (𝑓𝐢̃𝑁 , 𝐸), 1

2

1

= {π‘₯1 βˆ•(π‘₯(1𝑗,1) βˆ©Μƒ 𝑁 π‘₯(1𝑗,2) βˆ©Μƒ 𝑁 ..βˆ©Μƒ 𝑁 π‘₯(1𝑗,π‘˜) ),

Step 6. Evaluation of the score value of each of the entries in the soft set (𝐹 βŠ– Μƒ , 𝐸). 𝐢𝑁

2

10

S. Manna, T.M. Basu and S.K. Mondal

Engineering Applications of Artificial Intelligence 89 (2020) 103432

βŠ– We have derived the score value (π‘†Μ‚π‘π‘Ÿ ) of each of the entries 𝑑̄𝑠𝑗 in βŠ– the soft set (𝐹 Μƒ , 𝐸). The score values are presented in the score valued 𝐢𝑁 matrix (π‘†Μ‚π‘π‘Ÿ (π‘‘Μ„βŠ– ))π‘šΓ—π‘› .

Now if a decision maker evaluates the satisfaction of the parameter β€˜engineering background knowledge’ over a candidate, then, the only information about the engineering result of the candidate may not be sufficient for best manager selection. Because, in that case, additionally, the information about the β€˜technical knowledge’ of the candidate is also necessary and should be known to the decision maker. So, these two information β€˜engineering background knowledge’ and β€˜technical knowledge’ should be considered together. Now, if we will use complex neutrosophic environment, then we can handle this problem by taking β€˜engineering background knowledge’ as the amplitude term and β€˜technical knowledge’ as the phase term in the evaluation of a candidate over the parameter β€˜engineering background knowledge’. Again, for the parameter β€˜ability in budget managing ’, the information about the β€˜supervising responsibility’ of a candidate is a necessary information. Therefore, here we have taken β€˜ability in budget managing ’ as the amplitude term and β€˜supervising responsibility’ as the phase term in the evaluation of a candidate over the parameter β€˜ability in budget managing ’. Similarly, for the parameter β€˜ability to listening and understanding the customer’s comment ’, we have taken the information β€˜good behaving ability to the others’ additionally. Then, β€˜ability to listening and understanding the customer’s comment ’ has been considered as the amplitude term and ’good behaving ability to the others’ has been considered as the phase term. Lastly, for the parameter β€˜reliability’, the information about the β€˜political sensitivity’ has been considered as the phase term where, the information about the β€˜reliability’ of a candidate has been considered as the amplitude term. Now, assume that, π‘ˆ = {𝐢1 , 𝐢2 , 𝐢3 , 𝐢4 } be the set of four candidates as the initial universal set and 𝐸 = {πΈπ‘›π‘”π‘–π‘›π‘’π‘’π‘Ÿπ‘–π‘›π‘” π‘π‘Žπ‘π‘˜π‘”π‘Ÿπ‘œπ‘’π‘›π‘‘ π‘˜π‘›π‘œπ‘€π‘™π‘’π‘‘π‘”π‘’ (𝑒1 ), 𝐴𝑏𝑖𝑙𝑖𝑑𝑦 𝑖𝑛 𝐡𝑒𝑑𝑔𝑒 π‘šπ‘Žπ‘›π‘Žπ‘”π‘–π‘›π‘” (𝑒2 ), 𝐴𝑏𝑖𝑙𝑖𝑑𝑦 π‘‘π‘œ 𝑙𝑖𝑠𝑑𝑒𝑛𝑖𝑛𝑔 π‘Žπ‘›π‘‘ π‘’π‘›π‘‘π‘’π‘Ÿπ‘ π‘‘π‘Žπ‘›π‘‘π‘–π‘›π‘” π‘π‘’π‘ π‘‘π‘œπ‘šπ‘’π‘Ÿβ€² 𝑠 π‘π‘œπ‘šπ‘šπ‘’π‘›π‘‘ (𝑒3 ), π‘…π‘’π‘Žπ‘™π‘–π‘Žπ‘π‘–π‘™π‘–π‘‘π‘¦ (𝑒4 )} be the set of corresponding parameters of the elements of π‘ˆ . The evaluations of the four candidates over these four parameters have been given in a CNS-set (𝑓𝐢̃𝑁 , 𝐸) (Table 6). Now the problem is β€˜select the best candidate for the manager post who satisfies all the considered parameters with maximum evaluation level’. In Fig. 5, the graphical framework of this case study has been expressed. Solution. We have solved this manager selection problem by using our proposed Algorithm I.

𝑠𝑗

Step 7. Normalization process to deal with non commensurable parameters. + (π‘‘Μ„βŠ– )) Firstly, we have determined the maximum score value (π‘†Μ‚π‘π‘Ÿ 𝑗 over all the alternatives with respect to a parameter 𝑒𝑗 associated with βŠ– + (π‘‘Μ„βŠ– ) = π‘šπ‘Žπ‘₯π‘š 𝑆̂ (π‘‘Μ„βŠ– ). the matrix (π‘†Μ‚π‘π‘Ÿ (𝑑̄𝑠𝑗 ))π‘šΓ—π‘› as, π‘†Μ‚π‘π‘Ÿ 𝑗 𝑠=1 π‘π‘Ÿ 𝑠𝑗 βŠ– Then, we have normalized the entry π‘†Μ‚π‘π‘Ÿ (𝑑̄𝑠𝑗 ) in this matrix by the following: πœ™Μ„ 𝑠𝑗 =

βŠ– π‘†Μ‚π‘π‘Ÿ (𝑑̄𝑠𝑗 ) + Μ„βŠ– π‘†Μ‚π‘π‘Ÿ (𝑑𝑗 )

Step 8. Evaluation of the group utility value of an alternative π‘₯𝑠 (Utility measure). π‘Š (π‘₯ )) of an alternative π‘₯ is derived by The group utility value (π‘†Μ‚π‘π‘Ÿ 𝑠 𝑠 adding the weighted values of πœ™Μ„ 𝑠1 , πœ™Μ„ 𝑠2 ,.., πœ™Μ„ 𝑠𝑛 over all the parameters as follows: π‘Š π‘†Μ‚π‘π‘Ÿ (π‘₯𝑠 ) =

𝑛 βˆ‘

π‘Šπ‘— πœ™Μ„ 𝑠𝑗

(10)

𝑗=1

where, π‘Šπ‘— is the weight of the parameter 𝑒𝑗 . If π‘Šπ‘— is not given in the problem, then determine it by using Section 6.2 on the resultant complex neutrosophic soft set (𝐹𝐢̃𝑁 , 𝐸). Step 9. Evaluation of the individual regret score of an alternative π‘₯𝑠 (Regret measure). π‘šπ‘Žπ‘₯ (π‘₯ )) of an alternative π‘₯ is derived as The individual regret (π‘†Μ‚π‘π‘Ÿ 𝑠 𝑠 follows: π‘šπ‘Žπ‘₯ π‘†Μ‚π‘π‘Ÿ (π‘₯𝑠 ) = max{π‘Šπ‘— πœ™Μ„ 𝑠𝑗 }

(11)

𝑗

Μ‚ 𝑠 ) of an alternative π‘₯𝑠 . Step 10. Derivation of the compromising index 𝐢(π‘₯ The compromising index of an alternative over all the parameters is evaluated by the following equation, Μ‚ 𝑠) = πœƒ 𝐢(π‘₯

π‘Š (π‘₯ ) βˆ’ 𝑆̂ π‘Š (π‘₯ ) π‘†Μ‚π‘π‘Ÿ 𝑠 𝑠 π‘π‘Ÿ π‘Š (π‘₯ ) βˆ’ 𝑆̂ π‘Š (π‘₯ ) π‘†Μ‚π‘π‘Ÿ 𝑠 𝑠 π‘π‘Ÿ

+ (1 βˆ’ πœƒ)

π‘šπ‘Žπ‘₯ (π‘₯ ) βˆ’ 𝑆̂ π‘šπ‘Žπ‘₯ (π‘₯ ) π‘†Μ‚π‘π‘Ÿ 𝑠 𝑠 π‘π‘Ÿ

(12)

π‘šπ‘Žπ‘₯ (π‘₯ ) βˆ’ 𝑆̂ π‘šπ‘Žπ‘₯ (π‘₯ ) π‘†Μ‚π‘π‘Ÿ 𝑠 𝑠 π‘π‘Ÿ

π‘Š (π‘₯ ) = min 𝑆̂ π‘Š (π‘₯ ), 𝑆̂ π‘Š (π‘₯ ) = max 𝑆̂ π‘Š (π‘₯ ), 𝑆̂ π‘šπ‘Žπ‘₯ (π‘₯ ) = min 𝑆̂ π‘šπ‘Žπ‘₯ π‘†Μ‚π‘π‘Ÿ 𝑠 𝑠 π‘π‘Ÿ 𝑠 𝑠 𝑠 π‘π‘Ÿ 𝑠 𝑠 𝑠 π‘π‘Ÿ π‘π‘Ÿ π‘π‘Ÿ (π‘₯𝑠 ), 𝑆̂ π‘šπ‘Žπ‘₯ (π‘₯𝑠 ) = max𝑠 𝑆̂ π‘šπ‘Žπ‘₯ (π‘₯𝑠 ) where, the parameter πœƒ ∈ [0, 1] indicates π‘π‘Ÿ

Step 1,2. The corresponding CNS-set (𝑓𝐢̃𝑁 , 𝐸) has been given in Table 6. In this decision problem, all the considering parameters are benefited parameters. Therefore, equalization process is not needed.

π‘π‘Ÿ

the weight of the strategy of maximum group of utility. Step 11. Determination of a compromise optimal solution. Now, we have obtained a compromise optimal alternative by using Step 10 of Algorithm I. (CNSVISDM-approach).

Step 3. Complex neutrosophic soft ideal evaluation of each of the parameters over the CNS-set (𝑓𝐢̃𝑁 , 𝐸) is as follows: 𝑓 +Μƒ (𝑒1 ) = (0.9π‘’π‘–πœ‹ , 0.6π‘’π‘–πœ‹ , 0.8π‘’π‘–πœ‹ );

Flowchart of our proposed algorithm has been given in Fig. 4.

= 7. Case study

𝑓 +Μƒ (𝑒2 )

𝐢𝑁

=

𝐢𝑁 (0.4𝑒𝑖2πœ‹βˆ•3 , 0.5𝑒𝑖2πœ‹βˆ•3 , 0.9𝑒𝑖2πœ‹βˆ•3 );

(0.8𝑒𝑖3πœ‹βˆ•4 , 0.4𝑒𝑖3πœ‹βˆ•4 , 0.9𝑒𝑖3πœ‹βˆ•4 ); 𝑓 +Μƒ (𝑒4 ) 𝐢𝑁

=

𝑓 +Μƒ (𝑒3 )

(0.8𝑒𝑖2πœ‹ , 0.6𝑒𝑖2πœ‹ , 0.7𝑒𝑖2πœ‹ ).

𝐢𝑁

βŠ– Step 4. The bounded difference (𝑑𝑠𝑗 ) of each of the evaluations of the alternatives with respect to a parameter 𝑒𝑗 from the associated complex neutrosophic soft ideal evaluation 𝑓 +Μƒ (𝑒𝑗 ) is given in the complex

7.1. An application of our approach in manager selection of a company

𝐢𝑁

neutrosophic bounded difference soft set (𝑓 βŠ– Μƒ , 𝐸) (Table 7). 𝐢𝑁

At the present era, the number of competitors of a company is increasing rapidly in global market. Therefore, to preserve its position in global market and to increase its economic environment, proper manager selection is an important task to a company. Therefore, in this section, we have illustrated a manager selection problem of a company with the help of our proposed Algorithm I.

Step 5. By using Definition 9, the score value of each of the entries in the complex neutrosophic bounded difference soft set (𝑓 βŠ– , 𝐸) (given 𝐢̃𝑁 in Table 7) has been given in Table 8. Step 6. Normalization process. The maximum score value with respect to each of the parameters is s follows: + (𝑑 βŠ– ) = 0.74; 𝑆̂ + (𝑑 βŠ– ) = 0.72; 𝑆̂ + (𝑑 βŠ– ) = 0.66; 𝑆̂ + (𝑑 βŠ– ) = 0.77. π‘†Μ‚π‘π‘Ÿ π‘π‘Ÿ 2 π‘π‘Ÿ 3 π‘π‘Ÿ 4 1 Then, by using Eq. (2), the normalized value of each of the entries in the score valued matrix has been given in Table 9.

Example 6. Now consider that, four candidates have come to give an interview in the manager post of a company. The corresponding parameters over these four candidates are, {π‘’π‘›π‘”π‘–π‘›π‘’π‘’π‘Ÿπ‘–π‘›π‘” π‘π‘Žπ‘π‘˜π‘”π‘Ÿπ‘œπ‘’π‘›π‘‘ π‘˜π‘›π‘œπ‘€π‘™π‘’π‘‘π‘”π‘’, π‘Žπ‘π‘–π‘™π‘–π‘‘π‘¦ 𝑖𝑛 𝑏𝑒𝑑𝑔𝑒 π‘šπ‘Žπ‘›π‘Žπ‘”π‘–π‘›π‘”, π‘Žπ‘π‘–π‘™π‘–π‘‘π‘¦ π‘‘π‘œ 𝑙𝑖𝑠𝑑𝑒𝑛𝑖𝑛𝑔 π‘Žπ‘›π‘‘ π‘’π‘›π‘‘π‘’π‘Ÿπ‘ π‘‘π‘Žπ‘›π‘‘π‘–π‘›π‘” π‘‘β„Žπ‘’ π‘π‘’π‘ π‘‘π‘œπ‘šπ‘’π‘Ÿβ€² π‘ π‘π‘œπ‘šπ‘šπ‘’π‘›π‘‘, π‘Ÿπ‘’π‘Žπ‘™π‘–π‘Žπ‘π‘–π‘™π‘–π‘‘π‘¦}.

Step 7. Determination of the weights of the parameters. 11

S. Manna, T.M. Basu and S.K. Mondal

Engineering Applications of Artificial Intelligence 89 (2020) 103432

Fig. 4. Flowchart of our proposed approach.

Table 6 Tabular form of the CNS-set (𝑓𝐢̃𝑁 , 𝐸).

𝐢1 𝐢2 𝐢3 𝐢4

𝑒1

𝑒2

𝑒3

𝑒4

(0.3π‘’π‘–πœ‹βˆ•2 , 0.2π‘’π‘–πœ‹βˆ•2 , 0.7π‘’π‘–πœ‹βˆ•2 ) (0.9π‘’π‘–πœ‹ , 0.2π‘’π‘–πœ‹ , 0.1π‘’π‘–πœ‹ ) (0.6𝑒𝑖2πœ‹βˆ•3 , 0.3𝑒𝑖2πœ‹βˆ•3 , 0.5𝑒𝑖2πœ‹βˆ•3 ) (0.2π‘’π‘–πœ‹βˆ•2 , 0.6π‘’π‘–πœ‹βˆ•2 , 0.8π‘’π‘–πœ‹βˆ•2 )

(0.8𝑒𝑖2πœ‹βˆ•3 , 0.3𝑒𝑖2πœ‹βˆ•3 , 0.2𝑒𝑖2πœ‹βˆ•3 ) (0.2π‘’π‘–πœ‹βˆ•3 , 0.3π‘’π‘–πœ‹βˆ•3 , 0.9π‘’π‘–πœ‹βˆ•3 ) (0.3π‘’π‘–πœ‹βˆ•2 , 0.1π‘’π‘–πœ‹βˆ•2 , 0.8π‘’π‘–πœ‹βˆ•2 ) (0.6𝑒𝑖3πœ‹βˆ•4 , 0.4𝑒𝑖3πœ‹βˆ•4 , 0.3𝑒𝑖3πœ‹βˆ•4 )

(0.2π‘’π‘–πœ‹βˆ•2 , 0.1π‘’π‘–πœ‹βˆ•2 , 0.7π‘’π‘–πœ‹βˆ•2 ) (0.1π‘’π‘–πœ‹βˆ•6 , 0.4π‘’π‘–πœ‹βˆ•6 , 0.9π‘’π‘–πœ‹βˆ•6 ) (0.1π‘’π‘–πœ‹βˆ•4 , 0.5π‘’π‘–πœ‹βˆ•4 , 0.7π‘’π‘–πœ‹βˆ•4 ) (0.4𝑒𝑖2πœ‹βˆ•3 , 0.3𝑒𝑖2πœ‹βˆ•3 , 0.7𝑒𝑖2πœ‹βˆ•3 )

(0.8π‘’π‘–πœ‹βˆ•3 , 0.1π‘’π‘–πœ‹βˆ•3 , 0.2π‘’π‘–πœ‹βˆ•3 ) (0.3π‘’π‘–πœ‹βˆ•2 , 0.6π‘’π‘–πœ‹βˆ•2 , 0.7π‘’π‘–πœ‹βˆ•2 ) (0.1𝑒𝑖2πœ‹ , 0.5𝑒𝑖2πœ‹ , 0.7𝑒𝑖2πœ‹ ) (0.2π‘’π‘–πœ‹βˆ•2 , 0.3π‘’π‘–πœ‹βˆ•2 , 0.4π‘’π‘–πœ‹βˆ•2 )

Table 7 Tabular form of the complex neutrosophic bounded difference soft set (π‘“πΆβŠ–Μƒ , 𝐸). 𝑁

𝐢1 𝐢2 𝐢3 𝐢4

𝑒1

𝑒2

𝑒3

𝑒4

(0.6π‘’π‘–πœ‹βˆ•2 , 0.4π‘’π‘–πœ‹βˆ•2 , 0.1π‘’π‘–πœ‹βˆ•2 ) (0𝑒𝑖0 , 0.2𝑒𝑖0 , 0.7𝑒𝑖0 ) (0.3π‘’π‘–πœ‹βˆ•3 , 0.3π‘’π‘–πœ‹βˆ•3 , 0.3π‘’π‘–πœ‹βˆ•3 ) (0.7π‘’π‘–πœ‹βˆ•2 , 0π‘’π‘–πœ‹βˆ•2 , 0π‘’π‘–πœ‹βˆ•2 )

(0π‘’π‘–πœ‹βˆ•12 , 0.1π‘’π‘–πœ‹βˆ•12 , 0.7π‘’π‘–πœ‹βˆ•12 ) (0.6𝑒𝑖5πœ‹βˆ•12 , 0.1𝑒𝑖5πœ‹βˆ•12 , 0𝑒𝑖5πœ‹βˆ•12 ) (0.5π‘’π‘–πœ‹βˆ•4 , 0.3π‘’π‘–πœ‹βˆ•4 , 0.1π‘’π‘–πœ‹βˆ•4 ) (0.2𝑒𝑖0 , 0𝑒𝑖0 , 0.6𝑒𝑖0 )

(0.2π‘’π‘–πœ‹βˆ•6 , 0.4π‘’π‘–πœ‹βˆ•6 , 0.2π‘’π‘–πœ‹βˆ•6 ) (0.3π‘’π‘–πœ‹βˆ•2 , 0.1π‘’π‘–πœ‹βˆ•2 , 0π‘’π‘–πœ‹βˆ•2 ) (0.3𝑒𝑖5πœ‹βˆ•12 , 0𝑒𝑖5πœ‹βˆ•12 , 0.2𝑒𝑖5πœ‹βˆ•12 ) (0𝑒𝑖0 , 0.2𝑒𝑖0 , 0.2𝑒𝑖0 )

(0𝑒𝑖5πœ‹βˆ•3 , 0.5𝑒𝑖5πœ‹βˆ•3 , 0.5𝑒𝑖5πœ‹βˆ•3 ) (0.5𝑒𝑖3πœ‹βˆ•2 , 0𝑒𝑖3πœ‹βˆ•2 , 0𝑒𝑖3πœ‹βˆ•2 ) (0.7𝑒𝑖0 , 0.1𝑒𝑖0 , 0𝑒𝑖0 ) (0.6𝑒𝑖3πœ‹βˆ•2 , 0.3𝑒𝑖3πœ‹βˆ•2 , 0.3𝑒𝑖3πœ‹βˆ•2 )

π‘šπ‘’3 = (0.2𝑒𝑖19πœ‹βˆ•48 , 0.375𝑒𝑖19πœ‹βˆ•48 , 0.75𝑒𝑖19πœ‹βˆ•48 ); π‘šπ‘’4 = (0.35𝑒𝑖5πœ‹βˆ•6 , 0.375𝑒𝑖5πœ‹βˆ•6 , 0.5𝑒𝑖5πœ‹βˆ•6 ). 7.2 Then the complex neutrosophic distance (𝑑𝑠𝑗 ) of the evaluation π‘₯𝑠𝑗 of an alternative with respect to the parameter 𝑒𝑗 and its corresponding mean potentiality π‘šπ‘’π‘— has given in Table 10. 7.3 Now, the total complex neutrosophic distance of all the alternatives from the corresponding mean potentiality π‘šπ‘’π‘— associated with the parameter 𝑒𝑗 is as follows:

Since, in this problem, weights of the parameters are completely unknown, therefore, we have obtained the exact weights of the parameters by using Section 6.2. 7.1 By using Definition 13, the mean potentiality of each of the parameters is as follows: π‘šπ‘’1 = (0.5𝑒𝑖2πœ‹βˆ•3 , 0.375𝑒𝑖2πœ‹βˆ•3 , 0.525𝑒𝑖2πœ‹βˆ•3 ); π‘šπ‘’2 0.275𝑒𝑖9πœ‹βˆ•16 , 0.55𝑒𝑖9πœ‹βˆ•16 );

=

(0.475𝑒𝑖9πœ‹βˆ•16 ,

12

S. Manna, T.M. Basu and S.K. Mondal

Engineering Applications of Artificial Intelligence 89 (2020) 103432

Fig. 5. Modeling framework of case study 7.1. π‘Š (𝐢 ) = 0.716; 𝑆̂ π‘Š (𝐢 ) = 0.88; 𝑆̂ π‘Š (𝐢 ) = 0.933; 𝑆̂ π‘Š (𝐢 ) = 0.85. π‘†Μ‚π‘π‘Ÿ 1 2 3 4 π‘π‘Ÿ π‘π‘Ÿ π‘π‘Ÿ

Table 8 Score valued matrix.

𝐢1 𝐢2 𝐢3 𝐢4

𝑒1

𝑒2

𝑒3

𝑒4

0.64 0.52 0.59 0.74

0.53 0.71 0.66 0.60

0.59 0.66 0.65 0.60

0.36 0.62 0.77 0.54

Step 8. Then, By using Eq. (4), individual regret values of the alternatives are as follows: π‘šπ‘Žπ‘₯ (𝐢 ) = 0.225; 𝑆̂ π‘šπ‘Žπ‘₯ (𝐢 ) = 0.3; 𝑆̂ π‘šπ‘Žπ‘₯ (𝐢 ) = 0.3; 𝑆̂ π‘šπ‘Žπ‘₯ (𝐢 ) = 0.258. π‘†Μ‚π‘π‘Ÿ 4 3 2 1 π‘π‘Ÿ π‘π‘Ÿ π‘π‘Ÿ Step 9. Then, by utilizing Eq. (5), the compromising index of the alternatives are, Μ‚ 1 ) = 0; 𝐢(𝐢 Μ‚ 2 ) = 0.88; 𝐢(𝐢 Μ‚ 3 ) = 1; 𝐢(𝐢 Μ‚ 4 ) = 0.53. Here, we have 𝐢(𝐢

Table 9 Normalized score valued matrix.

𝐢1 𝐢2 𝐢3 𝐢4

taken πœƒ = 0.5

𝑒1

𝑒2

𝑒3

𝑒4

0.86 0.70 0.79 1

0.75 1 0.93 0.86

0.89 1 0.98 0.91

0.47 0.80 1 0.70

Step 10. Ranking of the alternatives: π‘Š , the ranking order is, 𝐢 > 𝐢 > 𝐢 > 𝐢 , based Now, based on π‘†Μ‚π‘π‘Ÿ 1 4 2 3 π‘šπ‘Žπ‘₯ , the ranking order is, 𝐢 > 𝐢 > 𝐢 = 𝐢 and based on 𝐢, Μ‚ the on π‘†Μ‚π‘π‘Ÿ 1 4 2 3 order is, 𝐢1 > 𝐢4 > 𝐢2 > 𝐢3 . Μ‚ 1 ) βˆ’ 𝐢(𝐢 Μ‚ 4) = Condition 1. Again, Condition 1 satisfies truly since, 𝐢(𝐢 1 0.53 βˆ’ 0 = 0.53 > 4βˆ’1 = 0.33. Condition 2. Condition 2 also satisfies truly since, 𝐢1 is the best each Μ‚ 𝑆̂ π‘Š and 𝑆̂ π‘šπ‘Žπ‘₯ . of the 𝐢, π‘π‘Ÿ π‘π‘Ÿ Therefore according to our algorithm it is concluded that, the alternative 𝐢1 is a compromise optimal solution for this decision problem. Hence, the candidate 𝐢1 is best for the manager post of this company.

Table 10 Tabular form of the matrix (𝑑𝑠𝑗 )π‘šΓ—π‘› .

𝐢1 𝐢2 𝐢3 𝐢4

𝑒1

𝑒2

𝑒3

𝑒4

0.2 0.425 0.1 0.3

0.35 0.35 0.25 0.25

0.275 0.15 0.125 0.2

0.425 0.225 0.58 0.167

7.2. Application of our approach in selecting sustainable manufacturing material of an automotive industry (Stoycheva et al., 2018)

𝑑̃𝑒1 = 1.02; 𝑑̃𝑒2 = 1.2; 𝑑̃𝑒3 = 0.75; 𝑑̃𝑒4 = 1.4. 7.4 Now, By using Eq. (9), the weights of the parameters are as

In today’s life, sustainable manufacturing in an automotive industry gains more attention because of the essentiality of balancing social, environmental and economic impacts so that, the development of the industry does not harmful to the society or environment. Now, we have used our proposed complex neutrosophic soft VIKOR approach to select the best sustainable automobile manufacturing material alternative in an automotive industry.

following, π‘Š1 = 0.2; π‘Š2 = 0.3; π‘Š3 = 0.2; π‘Š4 = 0.3. Step 8. Then, by using Eq. (3), group utility values of the alternatives are as follows: 13

S. Manna, T.M. Basu and S.K. Mondal

Engineering Applications of Artificial Intelligence 89 (2020) 103432

Table 11 Tabular form of the CNS-set (𝑓𝐢̃𝑁 , 𝐸).

𝑀1 𝑀2 𝑀3 𝑀4

𝑒1

𝑒2

𝑒3

(0.7𝑒𝑖3πœ‹βˆ•2 , 0.8𝑒𝑖3πœ‹βˆ•2 , 0.3𝑒𝑖3πœ‹βˆ•2 ) (0.1π‘’π‘–πœ‹ , 0.8π‘’π‘–πœ‹ , 0.9π‘’π‘–πœ‹ ) (0.5π‘’π‘–πœ‹βˆ•3 , 0.7π‘’π‘–πœ‹βˆ•3 , 0.6π‘’π‘–πœ‹βˆ•3 ) (0.8𝑒𝑖3πœ‹βˆ•2 , 0.6𝑒𝑖3πœ‹βˆ•2 , 0.2𝑒𝑖3πœ‹βˆ•2 )

(0.2𝑒𝑖4πœ‹βˆ•3 , 0.7𝑒𝑖4πœ‹βˆ•3 , 0.8𝑒𝑖4πœ‹βˆ•3 ) (0.9𝑒𝑖5πœ‹βˆ•3 , 0.7𝑒𝑖5πœ‹βˆ•3 , 0.2𝑒𝑖5πœ‹βˆ•3 ) (0.8𝑒𝑖3πœ‹βˆ•2 , 0.9𝑒𝑖3πœ‹βˆ•2 , 0.3𝑒𝑖3πœ‹βˆ•2 ) (0.3𝑒𝑖5πœ‹βˆ•4 , 0.6𝑒𝑖5πœ‹βˆ•4 , 0.6𝑒𝑖5πœ‹βˆ•4 )

(0.2π‘’π‘–πœ‹βˆ•2 , 0.1π‘’π‘–πœ‹βˆ•2 , 0.7π‘’π‘–πœ‹βˆ•2 ) (0.1π‘’π‘–πœ‹βˆ•6 , 0.4π‘’π‘–πœ‹βˆ•6 , 0.9π‘’π‘–πœ‹βˆ•6 ) (0.1π‘’π‘–πœ‹βˆ•4 , 0.5π‘’π‘–πœ‹βˆ•4 , 0.7π‘’π‘–πœ‹βˆ•4 ) (0.4𝑒𝑖2πœ‹βˆ•3 , 0.3𝑒𝑖2πœ‹βˆ•3 , 0.7𝑒𝑖2πœ‹βˆ•3 )

Table 12 Tabular form of the CNS-set (𝑓𝐢̃𝑁 , 𝐸).

𝑀1 𝑀2 𝑀3 𝑀4

𝑒1

𝑒2

𝑒3

(0.3π‘’π‘–πœ‹βˆ•2 , 0.2π‘’π‘–πœ‹βˆ•2 , 0.7π‘’π‘–πœ‹βˆ•2 ) (0.9π‘’π‘–πœ‹ , 0.2π‘’π‘–πœ‹ , 0.1π‘’π‘–πœ‹ ) (0.6𝑒𝑖2πœ‹βˆ•3 , 0.3𝑒𝑖2πœ‹βˆ•3 , 0.5𝑒𝑖2πœ‹βˆ•3 ) (0.2π‘’π‘–πœ‹βˆ•2 , 0.6π‘’π‘–πœ‹βˆ•2 , 0.8π‘’π‘–πœ‹βˆ•2 )

(0.8𝑒𝑖2πœ‹βˆ•3 , 0.3𝑒𝑖2πœ‹βˆ•3 , 0.2𝑒𝑖2πœ‹βˆ•3 ) (0.2π‘’π‘–πœ‹βˆ•3 , 0.3π‘’π‘–πœ‹βˆ•3 , 0.9π‘’π‘–πœ‹βˆ•3 ) (0.3π‘’π‘–πœ‹βˆ•2 , 0.1π‘’π‘–πœ‹βˆ•2 , 0.8π‘’π‘–πœ‹βˆ•2 ) (0.6𝑒𝑖3πœ‹βˆ•4 , 0.4𝑒𝑖3πœ‹βˆ•4 , 0.3𝑒𝑖3πœ‹βˆ•4 )

(0.2π‘’π‘–πœ‹βˆ•2 , 0.1π‘’π‘–πœ‹βˆ•2 , 0.7π‘’π‘–πœ‹βˆ•2 ) (0.1π‘’π‘–πœ‹βˆ•6 , 0.4π‘’π‘–πœ‹βˆ•6 , 0.9π‘’π‘–πœ‹βˆ•6 ) (0.1π‘’π‘–πœ‹βˆ•4 , 0.5π‘’π‘–πœ‹βˆ•4 , 0.7π‘’π‘–πœ‹βˆ•4 ) (0.4𝑒𝑖2πœ‹βˆ•3 , 0.3𝑒𝑖2πœ‹βˆ•3 , 0.7𝑒𝑖2πœ‹βˆ•3 )

Table 13 Tabular form of the complex neutrosophic bounded difference soft set (π‘“πΆβŠ–Μƒ , 𝐸). 𝑁

𝑀1 𝑀2 𝑀3 𝑀4

𝑒1

𝑒2

𝑒3

(0.6π‘’π‘–πœ‹βˆ•2 , 0.4π‘’π‘–πœ‹βˆ•2 , 0.1π‘’π‘–πœ‹βˆ•2 ) (0𝑒𝑖0 , 0.2𝑒𝑖0 , 0.7𝑒𝑖0 ) (0.3π‘’π‘–πœ‹βˆ•3 , 0.3π‘’π‘–πœ‹βˆ•3 , 0.3π‘’π‘–πœ‹βˆ•3 ) (0.7π‘’π‘–πœ‹βˆ•2 , 0π‘’π‘–πœ‹βˆ•2 , 0π‘’π‘–πœ‹βˆ•2 )

(0π‘’π‘–πœ‹βˆ•12 , 0.1π‘’π‘–πœ‹βˆ•12 , 0.7π‘’π‘–πœ‹βˆ•12 ) (0.6𝑒𝑖5πœ‹βˆ•12 , 0.1𝑒𝑖5πœ‹βˆ•12 , 0𝑒𝑖5πœ‹βˆ•12 ) (0.5π‘’π‘–πœ‹βˆ•4 , 0.3π‘’π‘–πœ‹βˆ•4 , 0.1π‘’π‘–πœ‹βˆ•4 ) (0.2𝑒𝑖0 , 0𝑒𝑖0 , 0.6𝑒𝑖0 )

(0.2π‘’π‘–πœ‹βˆ•6 , 0.4π‘’π‘–πœ‹βˆ•6 , 0.2π‘’π‘–πœ‹βˆ•6 ) (0.3π‘’π‘–πœ‹βˆ•2 , 0.1π‘’π‘–πœ‹βˆ•2 , 0π‘’π‘–πœ‹βˆ•2 ) (0.3𝑒𝑖5πœ‹βˆ•12 , 0𝑒𝑖5πœ‹βˆ•12 , 0.2𝑒𝑖5πœ‹βˆ•12 ) (0𝑒𝑖0 , 0.2𝑒𝑖0 , 0.2𝑒𝑖0 )

Table 14 Score valued matrix.

Example 7. Based on the Stoycheva et al. (2018), we have considered four automobile manufacturing material alternatives such as, {𝑓 π‘’π‘Ÿπ‘Ÿπ‘œπ‘’π‘  π‘šπ‘’π‘‘π‘Žπ‘™ (π‘–π‘Ÿπ‘œπ‘› π‘œπ‘Ÿ 𝑠𝑑𝑒𝑒𝑙), π‘œπ‘Ÿπ‘”π‘Žπ‘›π‘–π‘ π‘π‘œπ‘šπ‘π‘œπ‘ π‘–π‘‘π‘’, π‘π‘™π‘Žπ‘ π‘‘π‘–π‘π‘ , π‘Žπ‘™π‘’π‘šπ‘–π‘›π‘’π‘š} and three corresponding parameters including, {π‘’π‘›π‘£π‘–π‘Ÿπ‘œπ‘›π‘šπ‘’π‘›π‘‘π‘Žπ‘™ π‘–π‘šπ‘π‘Žπ‘π‘‘, π‘’π‘π‘œπ‘›π‘œπ‘šπ‘–π‘ π‘–π‘šπ‘π‘Žπ‘π‘‘, π‘ π‘œπ‘π‘–π‘Žπ‘™ π‘–π‘šπ‘π‘Žπ‘π‘‘}. Now, if the decision maker gives his/ her concentration only on the β€˜greenhouse gas emission’ of a manufacturing material alternative over the parameter β€˜environmental impact ’, then this information is not sufficient to select the best sustainable manufacturing material alternative, because additionally, the information about the β€˜energy consumption level’ associated with the parameter β€˜environmental impact ’ is also necessary for selecting the best sustainable manufacturing material. Therefore, by using complex neutrosophic environment, we have taken these two information β€˜greenhouse gas emission’ and β€˜energy consumption level’ together to evaluate the four manufacturing materials over this parameter β€˜environmental impact ’ where,β€˜greenhouse gas emission’ has been considered as the amplitude term and β€˜energy consumption level’ has been considered as the phase term. Similarly, for the parameter β€˜economic impact ’, we have taken two information together such as, β€˜product cost ’ and β€˜maintenance cost ’ where, β€˜product cost ’ has been considered as the amplitude term and β€˜maintenance cost ’ has been considered as the phase term. Lastly, for the parameter β€˜social impact ’, we have taken two information together such as, β€˜health status of adjacent people’ and β€˜safety of the adjacent people’, where, β€˜health status of adjacent people’ has been taken as amplitude term and β€˜safety of adjacent people’ has been taken as the phase term.

𝑀1 𝑀2 𝑀3 𝑀4

𝑒1

𝑒2

𝑒3

0.64 0.52 0.59 0.74

0.53 0.71 0.66 0.60

0.59 0.66 0.65 0.60

Table 15 Normalized score valued matrix.

𝑀1 𝑀2 𝑀3 𝑀4

𝑒1

𝑒2

𝑒3

0.86 0.70 0.79 1

0.75 1 0.93 0.86

0.89 1 0.98 0.91

Solution: Now, we have solved this problem by using our proposed Algorithm I. Step 1. The corresponding complex neutrosophic soft set has been given in Table 11. Step 2. Here, β€˜environmental impact ’ and β€˜economic impact ’ are two cost parameters, because over these two parameters, small evaluation indicates the goodness of a manufacturing material. Therefore, we have taken complex neutrosophic complement of all the evaluations over these two parameters. Results are given in Table 12. Step 3. The complex neutrosophic soft ideal evaluation of each of the parameters is as follows:

Now let, π‘ˆ = {𝑀1 (𝑓 π‘’π‘Ÿπ‘Ÿπ‘œπ‘’π‘  π‘šπ‘’π‘‘π‘Žπ‘™), 𝑀2 (π‘œπ‘Ÿπ‘”π‘Žπ‘›π‘–π‘ π‘π‘œπ‘šπ‘π‘œπ‘ π‘–π‘‘π‘’), 𝑀3 (π‘π‘™π‘Žπ‘ π‘‘π‘–π‘π‘ ), 𝑀4 (π‘Žπ‘™π‘’π‘šπ‘–π‘›π‘’π‘š)} be the initial universal set and 𝐸 = {𝑒1 (π‘’π‘›π‘£π‘–π‘Ÿπ‘œπ‘›π‘šπ‘’π‘›π‘‘π‘Žπ‘™ π‘–π‘šπ‘π‘Žπ‘π‘‘), 𝑒2 (π‘’π‘π‘œπ‘›π‘œπ‘šπ‘–π‘ π‘–π‘šπ‘π‘Žπ‘π‘‘), 𝑒3 (π‘ π‘œπ‘π‘–π‘Žπ‘™ π‘–π‘šπ‘π‘Žπ‘π‘‘)} be the set of parameters. The associated complex neutrosophic soft set, which represents the evaluations of the four alternatives over the three parameters, has been given in Table 11.

𝑓 +Μƒ (𝑒1 ) = (0.9π‘’π‘–πœ‹ , 0.6π‘’π‘–πœ‹ , 0.8π‘’π‘–πœ‹ ); 𝑓 +Μƒ (𝑒2 ) = (0.8𝑒𝑖 𝐢𝑁

𝑓 +Μƒ (𝑒3 ) 𝐢𝑁

= (0.4𝑒

𝑖 2πœ‹ 3

𝑖 2πœ‹ 3

, 0.5𝑒

, 0.9𝑒

𝑖 2πœ‹ 3

𝐢𝑁

3πœ‹ 4

, 0.4𝑒𝑖

3πœ‹ 4

, 0.9𝑒𝑖

3πœ‹ 4

);

).

Step 4. The bounded difference of each of the evaluations of the alternatives with respect to a parameter 𝑒𝑗 from the corresponding 𝑓 +Μƒ (𝑒𝑗 ) are given in Table 13.

Now, consider that, the information about the weights of the parameters is incompletely known.

𝐢𝑁

Step 5. The score value of each of the entries in Table 13 has been given in Table 14.

If, π‘Š = {π‘Š1 , π‘Š2 , π‘Š3 } be the weights of the three parameters, then the information is, 𝑃 = {π‘Š1 β‰₯ 0.4; 0.1 ≀ π‘Š2 ≀ 0.9; π‘Š2 βˆ’ π‘Š3 < 0.3; π‘Š3 β‰₯ 0.2}.

Step 6. Now by using linear max normalization process, the normalized evaluation of each of the entries of Table 14 has been provided in Table 15.

Now, our problem is, select the best sustainable manufacturing material alternative over the three considered parameters. 14

S. Manna, T.M. Basu and S.K. Mondal

Engineering Applications of Artificial Intelligence 89 (2020) 103432

Step 7. Now, we have determined the exact weights of the parameters by using Section 6.2. The given incomplete information about the weights of the parameters is, 𝑃 = {π‘Š1 β‰₯ 0.4; 0.1 ≀ π‘Š2 ≀ 0.9; π‘Š2 βˆ’ π‘Š3 < 0.3; π‘Š3 β‰₯ 0.2}. Then, the mean potentiality of each of the parameters is as follows: π‘šπ‘’1 = (0.5𝑒𝑖2πœ‹βˆ•3 , 0.375𝑒𝑖2πœ‹βˆ•3 , 0.525𝑒𝑖2πœ‹βˆ•3 ); π‘šπ‘’2 = (0.475𝑒𝑖9πœ‹βˆ•16 , 0.275𝑒𝑖9πœ‹βˆ•16 , 0.55𝑒𝑖9πœ‹βˆ•16 ); π‘šπ‘’3 = (0.2𝑒𝑖19πœ‹βˆ•48 , 0.375𝑒𝑖19πœ‹βˆ•48 , 0.75𝑒𝑖19πœ‹βˆ•48 ). Then the total complex neutrosophic distance of all the alternatives from the corresponding mean potentiality π‘šπ‘’π‘— associated with a parameter 𝑒𝑗 is, 𝑑̃𝑒1 = 1.02; 𝑑̃𝑒2 = 1.2; 𝑑̃𝑒3 = 0.75. Now, to obtain the weights of the parameters, we have constructed a optimization model by using the given incomplete information as follows: Μƒ π‘šπ‘Žπ‘₯ 𝐷(π‘Š ) = 1.02π‘Š1 + 1.2π‘Š2 + 0.75π‘Š3 𝑠.𝑑., π‘Š ∈ 𝑃 ; 0 ≀ π‘Š ≀ 1; 𝑗 = 1, 2, … , 𝑛; βˆ‘π‘› 𝑗 𝑗=1 π‘Šπ‘— = 1

⎫ βŽͺ βŽͺ ⎬ βŽͺ βŽͺ ⎭

(13)

Solving the above optimization model, the weights of the parameters are, π‘Š1 = 0.4; π‘Š2 = 0.4; π‘Š3 = 0.2. Step 8. The group utility values of the alternatives are, π‘Š (𝑀 ) = 0.822; 𝑆̂ π‘Š (𝑀 ) = 0.88; 𝑆̂ π‘Š (𝑀 ) = 0.884; 𝑆̂ π‘Š (𝑀 ) = π‘†Μ‚π‘π‘Ÿ 1 2 3 4 π‘π‘Ÿ π‘π‘Ÿ π‘π‘Ÿ 0.926. Step 9. The individual regret values of the alternatives are, π‘šπ‘Žπ‘₯ (𝑀 ) = 0.344; 𝑆̂ π‘šπ‘Žπ‘₯ (𝑀 ) = 0.4; 𝑆̂ π‘šπ‘Žπ‘₯ (𝑀 ) = 0.372; 𝑆̂ π‘šπ‘Žπ‘₯ (𝑀 ) = π‘†Μ‚π‘π‘Ÿ 1 2 3 4 π‘π‘Ÿ π‘π‘Ÿ π‘π‘Ÿ 0.4. Step 10. Then the compromising index of the alternatives are, Μ‚ 1 ) = 0; 𝐢(𝑀 Μ‚ 2 ) = 0.78; 𝐢(𝑀 Μ‚ 3 ) = 0.55; 𝐢(𝑀 Μ‚ 4 ) = 1. Here, we have 𝐢(𝑀 considered πœƒ = 0.5.

Fig. 6. Modeling framework of case study 7.3.

Step 11. Ranking the alternatives. π‘Š is, 𝑀 > 𝑀 > The ranking order of the alternatives based on π‘†Μ‚π‘π‘Ÿ 1 2 π‘šπ‘Žπ‘₯ is, 𝑀3 > 𝑀4 , the ranking order of the alternatives based on π‘†Μ‚π‘π‘Ÿ 𝑀1 > 𝑀3 > 𝑀2 = 𝑀4 and the ranking order of the alternatives based on 𝐢̂ is, 𝑀1 > 𝑀3 > 𝑀2 > 𝑀4 . Μ‚ 3 )βˆ’ Condition 1. Now we have seen that, condition 1 is satisfied as, 𝐢(𝑀 Μ‚ 1 ) = 0.55 > 1 = 0.33. 𝐢(𝑀 4βˆ’1 π‘Š and 𝑆̂ π‘šπ‘Žπ‘₯ . Condition 2. Again, 𝑀1 is best for both the cases of π‘†Μ‚π‘π‘Ÿ π‘π‘Ÿ Hence, 𝑀1 is a compromise optimal solution for this decisionmaking problem. Therefore, the manufacturing material alternative 𝑀1 can be selected in the automobile industry.

Example 8 (Basu et al., 2012; Das and Kar, 2014). Consider that, a patient has four symptoms such as, {π‘Žπ‘π‘‘π‘œπ‘šπ‘–π‘›π‘Žπ‘™ π‘π‘Žπ‘–π‘›, 𝑓 π‘’π‘£π‘’π‘Ÿ, β„Žπ‘’π‘Žπ‘‘π‘Žπ‘β„Žπ‘’, π‘€π‘’π‘–π‘”β„Žπ‘‘π‘™π‘œπ‘ π‘ } and the corresponding diseases related to these four symptoms are {π‘‘π‘¦π‘β„Žπ‘œπ‘–π‘‘, 𝑝𝑒𝑝𝑑𝑖𝑐 π‘’π‘™π‘π‘’π‘Ÿ, 𝑓 π‘œπ‘œπ‘‘ π‘π‘œπ‘–π‘ π‘œπ‘›π‘–π‘›π‘”, π‘Žπ‘π‘’π‘‘π‘’ π‘£π‘–π‘Ÿπ‘Žπ‘™ β„Žπ‘’π‘π‘Žπ‘‘π‘–π‘‘π‘–π‘ }. Now assume that, a set of three experts have been selected to make a common decision about β€˜which disease is more likely to appear in a patient’. Now, consider that, π‘ˆ = {π‘‘π‘¦π‘β„Žπ‘œπ‘–π‘‘ (π‘₯1 ), 𝑝𝑒𝑝𝑑𝑖𝑐 π‘’π‘™π‘π‘’π‘Ÿ (π‘₯2 ), 𝑓 π‘œπ‘œπ‘‘ π‘π‘œπ‘–π‘ π‘œπ‘›π‘–π‘›π‘” (π‘₯3 ), π‘Žπ‘π‘’π‘‘π‘’ π‘£π‘–π‘Ÿπ‘Žπ‘™β„Žπ‘’π‘π‘Žπ‘‘π‘–π‘‘π‘–π‘  (π‘₯4 )} be the universal set and 𝐸 = {π‘Žπ‘π‘‘π‘œπ‘šπ‘–π‘›π‘Žπ‘™ π‘π‘Žπ‘–π‘› (𝑒1 ), 𝑓 π‘’π‘£π‘’π‘Ÿ (𝑒2 ), β„Žπ‘’π‘Žπ‘‘π‘Žπ‘β„Žπ‘’ (𝑒3 ), π‘€π‘’π‘–π‘”β„Žπ‘‘π‘™π‘œπ‘ π‘  (𝑒4 )} be the set of corresponding parameters. The weights of the three parameters have been given initially in this group decision-making as, π‘Š1 = 0.27, π‘Š2 = 0.22, π‘Š3 = 0.17, π‘Š4 = 0.34. The three experts are, 𝐷 = {𝑑1 , 𝑑2 , 𝑑3 }. Now, three CNS-sets (𝑓𝐢̃𝑁 , 𝐸), (𝑓𝐢̃𝑁 , 𝐸), (𝑓𝐢̃𝑁 , 𝐸) provided by three experts has been given 1 2 3 in Table 16, 17, 18 respectively. In each of the membership magnitudes, amplitude term indicates the β€˜belongingness of a symptom’ and phase term indicates the β€˜time duration of the said symptom’. All the data has been collected based on 10 consecutive days. The value 2πœ‹ has been taken in the phase term instead of 10 days. In Fig. 6, the graphical framework of this decision-making problem has been given. Solution. We have solved this group decision-making problem by using our proposed Algorithm II (CNSVIMDM-approach).

7.3. An application of CNSVIMDM-approach in medical science Proper disease diagnosis of a patient is a significant task in medical science. Several researchers have proposed different mathematical models by using soft set theory to deal with disease diagnosis problems. For instance, Basu et al. (2012) proposed a fuzzy soft set based balanced solution to detect the exact disease of a patient. Then, Das and Kar (2014) proposed a group decision-making approach by using intuitionistic fuzzy soft set theory where, a set of four experts have been selected to detect the exact disease of a patient. But, in these soft set based models, they have only considered the information about the β€˜degree of belongingness of a symptom’ for disease diagnosis. But, the information about the β€˜time duration of a symptom’ to a patient is also emergent here and to be considered for proper diagnosis of the patient. Therefore, to solve these types of medical diagnosis problems, we have used our proposed complex neutrosophic soft set based approach by considering two information β€˜belongingness of a symptom’ and β€˜time duration of a symptom’ together.

Step 1. Here, all the parameters are benefit parameters. Therefore, we do not need to equalize the sense of all the parameters. Step 2. Now, we have constructed a resultant complex neutrosophic soft set (𝑓𝐢̃𝑁 , 𝐸) form these three CNS-sets (𝑓𝐢̃𝑁 , 𝐸), (𝑓𝐢̃𝑁 , 𝐸) and 1 2 (𝑓𝐢̃𝑁 , 𝐸) by using complex neutrosophic soft intersection operator. 3

15

S. Manna, T.M. Basu and S.K. Mondal

Engineering Applications of Artificial Intelligence 89 (2020) 103432

Table 16 Tabular form of the CNS-Set (𝑓𝐢̃𝑁 , 𝐸). 1

π‘₯1 π‘₯2 π‘₯3 π‘₯4

𝑒1

𝑒2

𝑒3

𝑒4

(0.4π‘’π‘–πœ‹βˆ•2 , 0.2π‘’π‘–πœ‹βˆ•2 , 0.8π‘’π‘–πœ‹βˆ•2 ) (1π‘’π‘–πœ‹ , 0.4π‘’π‘–πœ‹ , 0.3π‘’π‘–πœ‹ ) (0.5𝑒𝑖2πœ‹βˆ•3 , 0.2𝑒𝑖2πœ‹βˆ•3 , 0.6𝑒𝑖2πœ‹βˆ•3 ) (0.1π‘’π‘–πœ‹βˆ•2 , 0.5π‘’π‘–πœ‹βˆ•2 , 0.7π‘’π‘–πœ‹βˆ•2 )

(0.6𝑒𝑖2πœ‹βˆ•3 , 0.4𝑒𝑖2πœ‹βˆ•3 , 0.2𝑒𝑖2πœ‹βˆ•3 ) (0.6π‘’π‘–πœ‹βˆ•3 , 0.3π‘’π‘–πœ‹βˆ•3 , 0.1π‘’π‘–πœ‹βˆ•3 ) (0.4π‘’π‘–πœ‹βˆ•2 , 0.3π‘’π‘–πœ‹βˆ•2 , 0.7π‘’π‘–πœ‹βˆ•2 ) (0.5𝑒𝑖3πœ‹βˆ•4 , 0.2𝑒𝑖3πœ‹βˆ•4 , 0.6𝑒𝑖3πœ‹βˆ•4 )

(0.3π‘’π‘–πœ‹βˆ•2 , 0.4π‘’π‘–πœ‹βˆ•2 , 0.7π‘’π‘–πœ‹βˆ•2 ) (0.8π‘’π‘–πœ‹βˆ•6 , 0.2π‘’π‘–πœ‹βˆ•6 , 0.3π‘’π‘–πœ‹βˆ•6 ) (0.5π‘’π‘–πœ‹βˆ•4 , 0.3π‘’π‘–πœ‹βˆ•4 , 0.7π‘’π‘–πœ‹βˆ•4 ) (0.2𝑒𝑖2πœ‹βˆ•3 , 0.3𝑒𝑖2πœ‹βˆ•3 , 0.8𝑒𝑖2πœ‹βˆ•3 )

(0.7π‘’π‘–πœ‹βˆ•3 , 0.1π‘’π‘–πœ‹βˆ•3 , 0.2π‘’π‘–πœ‹βˆ•3 ) (0.8π‘’π‘–πœ‹βˆ•2 , 0.3π‘’π‘–πœ‹βˆ•2 , 0.5π‘’π‘–πœ‹βˆ•2 ) (0.1𝑒𝑖2πœ‹ , 0.5𝑒𝑖2πœ‹ , 0.7𝑒𝑖2πœ‹ ) (0.2π‘’π‘–πœ‹βˆ•2 , 0.6π‘’π‘–πœ‹βˆ•2 , 0.4π‘’π‘–πœ‹βˆ•2 )

Table 17 Tabular form of the CNS-Set (𝑓𝐢̃𝑁 , 𝐸). 2

π‘₯1 π‘₯2 π‘₯3 π‘₯4

𝑒1

𝑒2

𝑒3

𝑒4

(0.3π‘’π‘–πœ‹βˆ•2 , 0.3π‘’π‘–πœ‹βˆ•2 , 0.8π‘’π‘–πœ‹βˆ•2 ) (0.8π‘’π‘–πœ‹ , 0.4π‘’π‘–πœ‹ , 0.2π‘’π‘–πœ‹ ) (0.5𝑒𝑖2πœ‹βˆ•3 , 0.1𝑒𝑖2πœ‹βˆ•3 , 0.7𝑒𝑖2πœ‹βˆ•3 ) (0.5π‘’π‘–πœ‹βˆ•2 , 0.3π‘’π‘–πœ‹βˆ•2 , 0.6π‘’π‘–πœ‹βˆ•2 )

(0.5𝑒𝑖2πœ‹βˆ•3 , 0.1𝑒𝑖2πœ‹βˆ•3 , 0.4𝑒𝑖2πœ‹βˆ•3 ) (0.7π‘’π‘–πœ‹βˆ•3 , 0.4π‘’π‘–πœ‹βˆ•3 , 0.1π‘’π‘–πœ‹βˆ•3 ) (0.3π‘’π‘–πœ‹βˆ•2 , 0.1π‘’π‘–πœ‹βˆ•2 , 0.8π‘’π‘–πœ‹βˆ•2 ) (0.5𝑒𝑖3πœ‹βˆ•4 , 0.6𝑒𝑖3πœ‹βˆ•4 , 0.7𝑒𝑖3πœ‹βˆ•4 )

(0.2π‘’π‘–πœ‹βˆ•2 , 0.1π‘’π‘–πœ‹βˆ•2 , 0.8π‘’π‘–πœ‹βˆ•2 ) (0.5π‘’π‘–πœ‹βˆ•6 , 0.1π‘’π‘–πœ‹βˆ•6 , 0.4π‘’π‘–πœ‹βˆ•6 ) (0.1π‘’π‘–πœ‹βˆ•4 , 0.5π‘’π‘–πœ‹βˆ•4 , 0.7π‘’π‘–πœ‹βˆ•4 ) (0.4𝑒𝑖2πœ‹βˆ•3 , 0.3𝑒𝑖2πœ‹βˆ•3 , 0.6𝑒𝑖2πœ‹βˆ•3 )

(0.4π‘’π‘–πœ‹βˆ•3 , 0.1π‘’π‘–πœ‹βˆ•3 , 0.2π‘’π‘–πœ‹βˆ•3 ) (0.6π‘’π‘–πœ‹βˆ•2 , 0.3π‘’π‘–πœ‹βˆ•2 , 0.2π‘’π‘–πœ‹βˆ•2 ) (0.1𝑒𝑖2πœ‹ , 0.4𝑒𝑖2πœ‹ , 0.7𝑒𝑖2πœ‹ ) (0.2π‘’π‘–πœ‹βˆ•2 , 0.2π‘’π‘–πœ‹βˆ•2 , 0.6π‘’π‘–πœ‹βˆ•2 )

Table 18 Tabular form of the CNS-Set (𝑓𝐢̃𝑁 , 𝐸). 3

π‘₯1 π‘₯2 π‘₯3 π‘₯4

𝑒1

𝑒2

𝑒3

𝑒4

(0.2π‘’π‘–πœ‹βˆ•2 , 0.2π‘’π‘–πœ‹βˆ•2 , 0.7π‘’π‘–πœ‹βˆ•2 ) (0.9π‘’π‘–πœ‹ , 0.2π‘’π‘–πœ‹ , 0.1π‘’π‘–πœ‹ ) (0.5𝑒𝑖2πœ‹βˆ•3 , 0.3𝑒𝑖2πœ‹βˆ•3 , 0.5𝑒𝑖2πœ‹βˆ•3 ) (0.2π‘’π‘–πœ‹βˆ•2 , 0.4π‘’π‘–πœ‹βˆ•2 , 0.8π‘’π‘–πœ‹βˆ•2 )

(0.4𝑒𝑖2πœ‹βˆ•3 , 0.3𝑒𝑖2πœ‹βˆ•3 , 0.2𝑒𝑖2πœ‹βˆ•3 ) (0.7π‘’π‘–πœ‹βˆ•3 , 0.3π‘’π‘–πœ‹βˆ•3 , 0.4π‘’π‘–πœ‹βˆ•3 ) (0.3π‘’π‘–πœ‹βˆ•2 , 0.1π‘’π‘–πœ‹βˆ•2 , 0.8π‘’π‘–πœ‹βˆ•2 ) (0.5𝑒𝑖3πœ‹βˆ•4 , 0.4𝑒𝑖3πœ‹βˆ•4 , 0.3𝑒𝑖3πœ‹βˆ•4 )

(0.2π‘’π‘–πœ‹βˆ•2 , 0.1π‘’π‘–πœ‹βˆ•2 , 0.7π‘’π‘–πœ‹βˆ•2 ) (0.6π‘’π‘–πœ‹βˆ•6 , 0.4π‘’π‘–πœ‹βˆ•6 , 0.3π‘’π‘–πœ‹βˆ•6 ) (0.3π‘’π‘–πœ‹βˆ•4 , 0.4π‘’π‘–πœ‹βˆ•4 , 0.6π‘’π‘–πœ‹βˆ•4 ) (0.4𝑒𝑖2πœ‹βˆ•3 , 0.3𝑒𝑖2πœ‹βˆ•3 , 0.8𝑒𝑖2πœ‹βˆ•3 )

(0.5π‘’π‘–πœ‹βˆ•3 , 0.1π‘’π‘–πœ‹βˆ•3 , 0.2π‘’π‘–πœ‹βˆ•3 ) (0.8π‘’π‘–πœ‹βˆ•2 , 0.5π‘’π‘–πœ‹βˆ•2 , 0.3π‘’π‘–πœ‹βˆ•2 ) (0.1𝑒𝑖2πœ‹ , 0.5𝑒𝑖2πœ‹ , 0.7𝑒𝑖2πœ‹ ) (0.2π‘’π‘–πœ‹βˆ•2 , 0.3π‘’π‘–πœ‹βˆ•2 , 0.7π‘’π‘–πœ‹βˆ•2 )

Table 19 Tabular form of the resultant CNS-Set (𝑓𝐢̃𝑁 , 𝐸).

π‘₯1 π‘₯2 π‘₯3 π‘₯4

𝑒1

𝑒2

𝑒3

𝑒4

(0.2π‘’π‘–πœ‹βˆ•2 , 0.2π‘’π‘–πœ‹βˆ•2 , 0.7π‘’π‘–πœ‹βˆ•2 ) (0.8π‘’π‘–πœ‹ , 0.2π‘’π‘–πœ‹ , 0.1π‘’π‘–πœ‹ ) (0.5𝑒𝑖2πœ‹βˆ•3 , 0.1𝑒𝑖2πœ‹βˆ•3 , 0.5𝑒𝑖2πœ‹βˆ•3 ) (0.1π‘’π‘–πœ‹βˆ•2 , 0.3π‘’π‘–πœ‹βˆ•2 , 0.6π‘’π‘–πœ‹βˆ•2 )

(0.4𝑒𝑖2πœ‹βˆ•3 , 0.1𝑒𝑖2πœ‹βˆ•3 , 0.2𝑒𝑖2πœ‹βˆ•3 ) (0.6π‘’π‘–πœ‹βˆ•3 , 0.3π‘’π‘–πœ‹βˆ•3 , 0.1π‘’π‘–πœ‹βˆ•3 ) (0.3π‘’π‘–πœ‹βˆ•2 , 0.1π‘’π‘–πœ‹βˆ•2 , 0.7π‘’π‘–πœ‹βˆ•2 ) (0.5𝑒𝑖3πœ‹βˆ•4 , 0.2𝑒𝑖3πœ‹βˆ•4 , 0.3𝑒𝑖3πœ‹βˆ•4 )

(0.2π‘’π‘–πœ‹βˆ•2 , 0.1π‘’π‘–πœ‹βˆ•2 , 0.7π‘’π‘–πœ‹βˆ•2 ) (0.5π‘’π‘–πœ‹βˆ•6 , 0.1π‘’π‘–πœ‹βˆ•6 , 0.3π‘’π‘–πœ‹βˆ•6 ) (0.1π‘’π‘–πœ‹βˆ•4 , 0.3π‘’π‘–πœ‹βˆ•4 , 0.6π‘’π‘–πœ‹βˆ•4 ) (0.2𝑒𝑖2πœ‹βˆ•3 , 0.3𝑒𝑖2πœ‹βˆ•3 , 0.6𝑒𝑖2πœ‹βˆ•3 )

(0.4π‘’π‘–πœ‹βˆ•3 , 0.1π‘’π‘–πœ‹βˆ•3 , 0.2π‘’π‘–πœ‹βˆ•3 ) (0.6π‘’π‘–πœ‹βˆ•2 , 0.3π‘’π‘–πœ‹βˆ•2 , 0.2π‘’π‘–πœ‹βˆ•2 ) (0.1𝑒𝑖2πœ‹ , 0.4𝑒𝑖2πœ‹ , 0.7𝑒𝑖2πœ‹ ) (0.2π‘’π‘–πœ‹βˆ•2 , 0.2π‘’π‘–πœ‹βˆ•2 , 0.4π‘’π‘–πœ‹βˆ•2 )

Table 20 Tabular form of the soft set (π‘“πΆβŠ–Μƒ , 𝐸). 𝑁

π‘₯1 π‘₯2 π‘₯3 π‘₯4

𝑒1

𝑒2

𝑒3

𝑒4

(0.6π‘’π‘–πœ‹βˆ•2 , 0.1π‘’π‘–πœ‹βˆ•2 , 0π‘’π‘–πœ‹βˆ•2 ) (0𝑒𝑖0 , 0.1𝑒𝑖0 , 0.6𝑒𝑖0 ) (0.3π‘’π‘–πœ‹βˆ•3 , 0.2π‘’π‘–πœ‹βˆ•3 , 0.2π‘’π‘–πœ‹βˆ•3 ) (0.7π‘’π‘–πœ‹βˆ•2 , 0π‘’π‘–πœ‹βˆ•2 , 0.1π‘’π‘–πœ‹βˆ•2 )

(0.2π‘’π‘–πœ‹βˆ•12 , 0.2π‘’π‘–πœ‹βˆ•12 , 0.5π‘’π‘–πœ‹βˆ•12 ) (0𝑒𝑖5πœ‹βˆ•12 , 0𝑒𝑖5πœ‹βˆ•12 , 0.6𝑒𝑖5πœ‹βˆ•12 ) (0.3π‘’π‘–πœ‹βˆ•4 , 0.2π‘’π‘–πœ‹βˆ•4 , 0π‘’π‘–πœ‹βˆ•4 ) (0.1𝑒𝑖0 , 0.1𝑒𝑖0 , 0.4𝑒𝑖0 )

(0.3π‘’π‘–πœ‹βˆ•6 , 0.2π‘’π‘–πœ‹βˆ•6 , 0π‘’π‘–πœ‹βˆ•6 ) (0π‘’π‘–πœ‹βˆ•2 , 0.2π‘’π‘–πœ‹βˆ•2 , 0.4π‘’π‘–πœ‹βˆ•2 ) (0.4𝑒𝑖5πœ‹βˆ•12 , 0𝑒𝑖5πœ‹βˆ•12 , 0.1𝑒𝑖5πœ‹βˆ•12 ) (0.3𝑒𝑖0 , 0𝑒𝑖0 , 0.1𝑒𝑖0 )

(0.2𝑒𝑖5πœ‹βˆ•3 , 0.3𝑒𝑖5πœ‹βˆ•3 , 0.5𝑒𝑖5πœ‹βˆ•3 ) (0𝑒𝑖3πœ‹βˆ•2 , 0.1𝑒𝑖3πœ‹βˆ•2 , 0.5𝑒𝑖3πœ‹βˆ•2 ) (0.5𝑒𝑖0 , 0𝑒𝑖0 , 0𝑒𝑖0 ) (0.4𝑒𝑖3πœ‹βˆ•2 , 0.2𝑒𝑖3πœ‹βˆ•2 , 0.3𝑒𝑖3πœ‹βˆ•2 )

Table 21 Score valued matrix.

π‘₯1 π‘₯2 π‘₯3 π‘₯4

Table 22 Normalize score valued matrix.

𝑒1

𝑒2

𝑒3

𝑒4

0.71 0.55 0.62 0.72

0.58 0.53 0.66 0.60

0.67 0.52 0.68 0.7

0.43 0.44 0.75 0.52

π‘₯1 π‘₯2 π‘₯3 π‘₯4

Step 3. Then, the complex neutrosophic soft ideal evaluation of each of the parameters is as follows: =

(0.8π‘’π‘–πœ‹ , 0.3π‘’π‘–πœ‹ , 0.7π‘’π‘–πœ‹ );

𝑓 +Μƒ (𝑒2 ) 𝐢𝑁

=

𝐢𝑁

𝑒3

𝑒4

0.88 0.80 1 0.91

0.96 0.74 0.97 1

0.57 0.59 1 0.69

Step 7. The individual regret values of the alternatives over all the parameters are as follows: π‘šπ‘Žπ‘₯ (π‘₯ ) = 0.267; 𝑆̂ π‘šπ‘Žπ‘₯ (π‘₯ ) = 0.205; 𝑆̂ π‘šπ‘Žπ‘₯ (π‘₯ ) = 0.34; 𝑆̂ π‘šπ‘Žπ‘₯ (π‘₯ ) = π‘†Μ‚π‘π‘Ÿ 1 2 3 4 π‘π‘Ÿ π‘π‘Ÿ π‘π‘Ÿ 0.270.

(0.6𝑒𝑖3πœ‹βˆ•4 , 0.3𝑒𝑖3πœ‹βˆ•4 ,

𝑓 +Μƒ (𝑒3 ) = (0.5𝑒𝑖2πœ‹βˆ•3 , 0.3𝑒𝑖2πœ‹βˆ•3 , 0.7𝑒𝑖2πœ‹βˆ•3 ); 𝑓 +Μƒ (𝑒4 ) = (0.6𝑒𝑖2πœ‹ , 0.4𝑒𝑖2πœ‹ , 0.7𝑒𝑖2πœ‹ );

𝑒2

0.99 0.76 0.86 1

Step 6. The group utility values of the alternatives over all the parameters are as follows: π‘Š (π‘₯ ) = 0.818; 𝑆̂ π‘Š (π‘₯ ) = 0.708; 𝑆̂ π‘Š (π‘₯ ) = 0.957; 𝑆̂ π‘Š (π‘₯ ) = 0.875. π‘†Μ‚π‘π‘Ÿ 1 2 3 4 π‘π‘Ÿ π‘π‘Ÿ π‘π‘Ÿ

By using complex neutrosophic soft standard intersection, the resultant has been given in Table 19.

𝑓 +Μƒ (𝑒1 ) 𝐢𝑁 0.7𝑒𝑖3πœ‹βˆ•4 );

𝑒1

𝐢𝑁

Step 8. The compromising index of the alternatives over all the parameters are as follows, Μ‚ 1 ) = 0.45; 𝐢(π‘₯ Μ‚ 2 ) = 0; 𝐢(π‘₯ Μ‚ 3 ) = 1; 𝐢(π‘₯ Μ‚ 4 ) = 0.56. Here, we have 𝐢(π‘₯ considered πœƒ = 0.5

Step 4. The bounded difference of each of the evaluations of the alternatives with respect to a parameter 𝑒𝑗 from the corresponding 𝑓 +Μƒ (𝑒𝑗 ) is given in Table 20.

π‘Š , 𝑆̂ π‘šπ‘Žπ‘₯ and 𝐢̂ is Step 9. The ranking of the alternatives based on π‘†Μ‚π‘π‘Ÿ π‘π‘Ÿ same as, π‘₯2 > π‘₯1 > π‘₯4 > π‘₯3 . Μ‚ 1 ) βˆ’ 𝐢(π‘₯ Μ‚ 2 ) = 0.45 > 1 = 0.33. So, Condition 1 Condition 1. Since, 𝐢(π‘₯ 4βˆ’1 is truly satisfied.

𝐢𝑁

Step 5. Now, the score values of the entries in Table 20 have been given in Table 21. The normalize score values of Table 21 have been provided in Table 22. 16

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π‘Š and 𝑆̂ π‘šπ‘Žπ‘₯ . So, Condition 2. Again, π‘₯2 is best for both the cases π‘†Μ‚π‘π‘Ÿ π‘π‘Ÿ Condition 2 is also satisfied here. Thus, the alternative π‘₯2 is a compromise optimal solution for this decision-making problem. Hence, we can conclude that, the patient is suffered by the disease peptic ulcer (π‘₯2 ).

8. Comparative study and discussion In this article, we have proposed a new approach to evaluate the best alternative from a complex neutrosophic soft set based decisionmaking problem by using VIKOR approach. In this section, we have examined the stability of the ranking results of our proposed approach through a sensitivity analysis and a comparative analysis. In sensitivity analysis part, we have examined the robustness of our approach for different values of the parameter πœƒ which is employed in our algorithm and in comparative analysis part, we have compared our results with some well known exiting methodologies to verify the validity of our proposed approach.

Fig. 7. Sensitivity analysis based on Example 6.

8.1. Sensitivity analysis based on the parameter πœƒ In our methodology, the parameter πœƒ has been assigned as a weight of the strategy of maximum group of utility and (1βˆ’πœƒ) has been assigned as a weight of the strategy of minimum individual regret to evaluate the compromising index of an alternative, as defined in Eq. (5). Here, πœƒ can adopt any value from 0 to 1. Throughout the paper, all the results have been taken by keeping the value of πœƒ as 0.5 i.e., by giving equal importance on the both of maximum group of utility and minimum individual regret. But since, πœƒ can take any value from 0 to 1, so, we have to verify the stability of the ranking results for different values of πœƒ. Therefore, we have given a sensitivity analysis based on the parameter πœƒ. The previous illustrated examples (Examples 6, 7, 8) have been used here to conduct this process. In Figs. 7, 8, 9, the variation of compromising index values of different alternatives for different values of πœƒ have been provided based on Example 6, Example 7 and Example 8 respectively. Now, from Fig. 7 it is observed that, for any value of πœƒ from 0.1 to 1, the ranking order of the four candidates is same as, 𝐢1 > 𝐢4 > 𝐢2 > 𝐢3 and when πœƒ = 0, the ranking order of the four candidates is, 𝐢1 > 𝐢4 > 𝐢2 = 𝐢3 . So we can conclude that, the parameter πœƒ did not affect on the ranking order of the alternatives in the interval [0.1, 1]. But when πœƒ = 0, i.e., when, one focuses only on the individual evaluations of the parameters (minimum individual regret) of the candidates, then, we cannot say which one is at the third position and which one is at the fourth position between 𝐢2 and 𝐢3 . From Fig. 8 we have seen that, the ranking order of the material alternatives is same for any value of the parameter πœƒ from 0.1 to 0.9. But, when πœƒ is tending to 1, then the ranking order of 𝑀2 has been increased and the ranking order of 𝑀3 has been decreased. i.e., when one focuses only on the group utility score (utility measure) of the material alternatives over all the parameters, then the risk level is high for being that, 𝑀2 is best than 𝑀3 and when πœƒ = 0, the ranking order of the alternatives has been slightly changed as, 𝑀1 > 𝑀3 > 𝑀2 = 𝑀4 i.e., when, one focuses only on the minimum individual regret (regret measure) of the material alternatives over all the parameters then, we cannot say which one is placed at third position and which one is placed at the fourth position between 𝑀2 and 𝑀4 . Again, from Fig. 9 we have seen that, different values of πœƒ do not effect on the ranking order of the alternatives π‘₯1 , π‘₯2 , π‘₯3 , π‘₯4 . So, from this discussion it is observed that, different values of the parameter πœƒ may make an effect on the final ranking order of the alternatives in a problem. Since, in our method, decision-maker can give the importance to the strategy of maximum group of utility (utility measure) and minimum individual regret (regret measure) as he/she likes, so when, he/she changes his/her importance to the maximum

Fig. 8. Sensitivity analysis based on Example 7.

Fig. 9. Sensitivity analysis based on Example 8.

group of utility and minimum individual regret, then obviously there may exist a different ranking result of the associated alternatives. So, changing of the parameter πœƒ indicates the changing of the ranking order of the alternatives, which supports the results of the sensitivity analysis with respect to the parameter πœƒ. Therefore, from this sensitivity analysis it is concluded that, our approach is more realistic. 8.2. Comparative analysis of our proposed approach Validation or confirmation of a new approach can be tested by comparing its resulting values with the existing methods. Now, it is seen that, complex neutrosophic framework is a super set with respect to fuzzy valued framework. But since, to solve our considered soft set based decision-making problems under the complex neutrosophic framework, there exists no method in literature till now. However, in fuzzy valued framework, some methodologies such as TOPSIS method (Eraslan and Karaaslan, 2015), Kong et al. (2009) method etc. have been saved in literature. So, to compare the results obtain from our method with the results using TOPSIS and Kong et al. (2009) method, we would transform all the complex neutrosophic values of a complex 17

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Engineering Applications of Artificial Intelligence 89 (2020) 103432

Fig. 10. Comparative analysis. Table 23 Score matrix based on Table 6 (Fuzzy soft set).

𝐢1 𝐢2 𝐢3 𝐢4

Table 26 Score matrix based on Table 12 (Fuzzy soft set).

𝑒1

𝑒2

𝑒3

𝑒4

0.52 0.68 0.56 0.46

0.66 0.47 0.52 0.59

0.52 0.45 0.46 0.51

0.72 0.46 0.32 0.54

𝑀1 𝑀2 𝑀3 𝑀4

Table 24 Weighted Score matrix (Weighted fuzzy soft set).

𝐢1 𝐢2 𝐢3 𝐢4

𝑒1

𝑒2

𝑒3

𝑒4

0.104 0.136 0.112 0.092

0.198 0.141 0.156 0.177

0.104 0.09 0.092 0.102

0.216 0.138 0.096 0.162

Our approach (πœƒ = 0.5)

π‘šπ‘’π‘‘β„Žπ‘œπ‘‘

𝑒2

𝑒3

0.525 0.686 0.578 0.453

0.661 0.472 0.525 0.588

0.523 0.453 0.462 0.511

Table 27 Weighted Score matrix (Weighted fuzzy soft set).

𝑀1 𝑀2 𝑀3 𝑀4

Table 25 Comparison Table based on Example 6.

𝑇 𝑂𝑃 𝑆𝐼𝑆 π‘šπ‘’π‘‘β„Žπ‘œπ‘‘

𝑒1

𝑒1

𝑒2

𝑒3

0.21 0.274 0.231 0.181

0.264 0.189 0.21 0.235

0.105 0.091 0.092 0.102

The solution of Example 7 by using our approach has been illustrated in Section 7.2. Now, after transforming all the complex neutrosophic values (Table 12) into score values by using score function, the resultants are given in Table 26. So, Table 26 is the reflection of Table 12. In this problem, the weights of the parameters are, π‘Š1 = 0.4; π‘Š2 = 0.4; π‘Š3 = 0.2. Therefore, we have constructed weighted score matrix, by multiplying the score values of each of the alternatives over a parameter 𝑒𝑗 with the associated parameter weight π‘Šπ‘— , as given in Table 27. Now, by applying different fuzzy valued decision-making methods on Table 27, the ranking results are given in Table 28. From Table 28 we have seen that, by using our proposed approach, when, πœƒ = 0.5, i.e., when one gives equal importance on the both of maximum total group of utility and minimum individual regret, then the rank is, 𝑀1 > 𝑀3 > 𝑀2 > 𝑀4 and when, πœƒ = 1, i.e., when one gives importance only on the maximum total group of utility then, the rank as, 𝑀1 > 𝑀2 > 𝑀3 > 𝑀4 . Moreover, we have seen that, TOPSIS and Kong et al. (2009) method have given the same ranking order as, 𝑀1 > 𝑀2 > 𝑀3 > 𝑀4 . Now, the background of TOPSIS method is, distance of an alternative from ideal solution and anti ideal solution over all the parameters and the background of Kong et al. (2009) method is, the sum of the total evaluations over all the parameters (fuzzy choice value) of an alternative. So, the key idea of both the TOPSIS method and Kong et al. (2009) method is same (give the importance only on the total satisfaction over all the parameters not on the individual satisfaction of a parameter). This situation can also be controlled through our algorithm by taking πœƒ = 1. From Table 28 it is observed that, the ranking result of our method is same with TOPSIS and Kong et al. (2009) method when πœƒ = 1. So, in this situation, the ranking result of our method is consistent. But when, in a decision-making problem, the satisfaction of an individual parameter is important not the total satisfaction of all the parameters, then TOPSIS method and Kong et al. (2009) method is unsuitable whereas, in that situation, our method can solve this problem by taking πœƒ = 0. So, by taking different values of the parameter πœƒ we can get more accurate results of a decision-making problem. Therefore we can conclude that, the ranking result of our method is valid in this example.

𝐢1 > 𝐢4 > 𝐢2 > 𝐢3 √ √ √

neutrosophic soft set into some real values (fuzzy soft set) by using score function (Definition 9). Because, the score value of a complex neutrosophic number by using score function is a reflection of the complex neutrosophic number in favor of truth degree. The schematically diagram of this discussion has been given in Fig. 10. Now, in the following, we have discussed a comparative study by using some existing fuzzy valued (truth membership) methods to verify the validity of our proposed approach. β€’ Comparison based on Example 6 The solution of Example 6 through our proposed approach has been illustrated in Section 7.1. Then, after transforming all the given complex neutrosophic values (Table 6) into real values by using score function, the resultant score values are given in Table 23. So, Table 23 is the reflection of Table 6. Since, in this example, the final ranking result has been taken by using the weights of the parameters as, π‘Š1 = 0.2; π‘Š2 = 0.3; π‘Š3 = 0.2; π‘Š4 = 0.3, therefore, before applying any fuzzy valued methodology, we have constructed a weighted score matrix (Table 24) (weighted fuzzy soft set) by multiplying the score values (Table 23) of each of the alternatives over a parameter 𝑒𝑗 with its associated parameter weight π‘Šπ‘— . Now, we have applied different decision-making methods to evaluate the best alternative from Table 24. The ranking results are given in Table 25. Since, for each of the methods, the ranking of the alternatives is, 𝐢1 > 𝐢4 > 𝐢2 > 𝐢3 , therefore, this ranking order has been considered as the ideal ranking of this problem. Thus we have seen that, by using our proposed approach, the ranking result of Example 6 is consistent. β€’ Comparison based on Example 7

β€’ Comparison based on Example 8 18

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Engineering Applications of Artificial Intelligence 89 (2020) 103432

Table 28 Comparison Table based on Example 7. 𝑀1 > 𝑀3 > 𝑀2 > 𝑀4 √

Our approach (πœƒ = 0.5)

𝑀1 > 𝑀2 > 𝑀3 > 𝑀4

√

Our approach (πœƒ = 0)

√

Our approach (πœƒ = 1)

√

𝑇 𝑂𝑃 𝑆𝐼𝑆 π‘šπ‘’π‘‘β„Žπ‘œπ‘‘

√

Kong et al. (2009) π‘šπ‘’π‘‘β„Žπ‘œπ‘‘

Table 29 (Fuzzy soft set) Score matrix based on Table 16.

π‘₯1 π‘₯2 π‘₯3 π‘₯4

𝑒1

𝑒2

𝑒3

𝑒4

0.525 0.634 0.561 0.442

0.661 0.672 0.525 0.554

0.492 0.703 0.562 0.461

0.706 0.625 0.317 0.492

Table 34 Comparison Table based on Example 8.

Our approach (πœƒ = 0.5) 𝑇 𝑂𝑃 𝑆𝐼𝑆 π‘šπ‘’π‘‘β„Žπ‘œπ‘‘ Kong et al. (2009) π‘šπ‘’π‘‘β„Žπ‘œπ‘‘

Table 30 (Fuzzy soft set) Score matrix based on Table 17.

π‘₯1 π‘₯2 π‘₯3 π‘₯4

𝑒1

𝑒2

𝑒3

𝑒4

0.492 0.617 0.561 0.558

0.611 0.672 0.525 0.471

0.508 0.653 0.462 0.528

0.655 0.642 0.333 0.525

𝑒1

𝑒2

𝑒3

𝑒4

0.508 0.683 0.561 0.458

0.594 0.639 0.525 0.571

0.525 0.636 0.529 0.494

0.672 0.625 0.317 0.492

Table 32 Resultant score matrix (Fuzzy soft set).

𝑀1 𝑀2 𝑀3 𝑀4

𝑒1

𝑒2

𝑒3

𝑒4

0.492 0.617 0.561 0.442

0.594 0.639 0.525 0.471

0.492 0.636 0.462 0.461

0.655 0.625 0.317 0.492

√ √

Hence, from the above analysis it is concluded that, our proposed approach is valid and is very effective to solve decision-making problems involving a second dimension. 9. Conclusion This study presents a complex neutrosophic soft set based decisionmaking by using VIKOR method. To reach this goal, few necessary operations and properties of complex neutrosophic sets and complex neutrosophic soft sets have been introduced. The contributions of this article can be summarized as follows:

Table 33 Weighted resultant Score matrix (Weighted Fuzzy soft set).

π‘₯1 π‘₯2 π‘₯3 π‘₯4

π‘₯2 > π‘₯1 > π‘₯4 > π‘₯3 √

β€’ The domain of our considered decision-making problem is large than the existing one. β€’ There is no existing methodology to solve the real-life problems in this domain till now. β€’ In our method, two directions such as, β€˜maximization of total group of utility’ and β€˜minimization of individual regret’ have been considered together through a parameter πœƒ by which a decision maker can get an exact direction by changing the value of πœƒ according to his/her needs. But, in TOPSIS or Kong et al. method (2009), there is no such facility. β€’ Further, in our approach, we have obtained a compromise optimal solution instead of a single optimal solution of a considered complex neutrosophic soft set based decision-making problem to overcome the problems in emergency situations. But, in TOPSIS or Kong et al. (2009) method, there is no such advantage.

Table 31 (Fuzzy soft set) Score matrix based on Table 18.

π‘₯1 π‘₯2 π‘₯3 π‘₯4

𝑀1 > 𝑀3 > 𝑀2 = 𝑀4

𝑒1

𝑒2

𝑒3

𝑒4

0.133 0.166 0.151 0.119

0.131 0.140 0.116 0.104

0.084 0.108 0.078 0.078

223 212 108 167

β€’ Some new union operations (complex neutrosophic standard union, complex neutrosophic algebraic sum, complex neutrosophic bold sum), intersection operations (complex neutrosophic standard intersection, complex neutrosophic product, complex neutrosophic bold intersection) and aggregation operations (complex neutrosophic arithmetic mean aggregation, complex neutrosophic geometric mean aggregation, complex neutrosophic harmonic mean aggregation) of complex neutrosophic sets have been proposed. β€’ The above union operations and intersection operations of complex neutrosophic sets have been generalized to complex neutrosophic soft sets. β€’ Score function of a complex neutrosophic number has been initiated. β€’ Two decision-making approaches have been developed by using VIKOR method to solve single decision-maker and multiple decision-makers based decision-making problems by using complex neutrosophic soft set theory. β€’ Moreover, to show the working progress of the proposed approaches, we have applied these algorithms in company’s manager selection problem, medical diagnosis problem, sustainable manufacturing material selection problem etc.

Example 8 has been solved by our proposed approach at Section 7.3. Now, by apply score function on Tables 16, 17, 18, the score values are given in Tables 29–31. Then, by using fuzzy AND product the resultant score matrix from the three score matrices (Tables 29, 30 and 31) has been given in Table 32. Then by using the weights of the parameters (π‘Š1 = 0.27; π‘Š2 = 0.22; π‘Š3 = 0.17; π‘Š4 = 0.34), the resultant weighted score values are given in Table 33. Now, the ranking of the alternatives by using different methods has been given Table 34. From Table 34 we have seen that, the ranking order of the alternatives through every method is same. so we can conclude that, the ranking result of our proposed approach is consistent in this example. So, from the above illustration, what is the status of our method? The answer of this question is that, our proposed approach is very effective because of the following reasons. 19

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Engineering Applications of Artificial Intelligence 89 (2020) 103432

Μƒ 𝐹̃ , where, β€’ Complex fuzzy algebraic sum. denoted by, 𝐹̃𝐴 βˆ” 𝐹 𝐡 πœ‡πΉΜƒ βˆ”Μƒ 𝐹̃ (π‘₯) = (π‘ŸπΉΜƒπ΄ (π‘₯) + π‘ŸπΉΜƒπ΅ (π‘₯) βˆ’ π‘ŸπΉΜƒπ΄ (π‘₯).π‘ŸπΉΜƒπ΅ (π‘₯)) 𝐴 𝐹 𝐡 ( 𝑒𝐹̃ (π‘₯) 𝑒𝐹̃ (π‘₯) 𝑒𝐹̃ (π‘₯) 𝑒𝐹̃ (π‘₯) ) 𝐴 𝐡 𝐴 𝐡 𝑒𝑖2πœ‹ 2πœ‹ + 2πœ‹ βˆ’ 2πœ‹ . 2πœ‹ . β€’ Complex fuzzy bold intersection. denoted (by, 𝐹̃𝐴 β‹’Μƒ 𝐹 𝐹̃𝐡 , where, )

β€’ The sensitivity analysis of our proposed approach based on the parameter πœƒ has also been illustrated. β€’ Also besides, we have discussed the validity and effectiveness of our proposed approach. As further research, one can develop some algebraic structures of complex neutrosophic soft sets such as complex neutrosophic soft groups, complex neutrosophic soft rings etc. Moreover, distance measure is a useful tool that can solve many decision-making problems. So, one can propose several types of distance measures or similarity measures of complex neutrosophic soft sets. Further, single-valued neutrosophic 2-tuple linguistic environment (Wu et al., 2018), trapezoidal fuzzy environment (Wu et al., 2017, 2019a), hesitant fuzzy linguistic environment (Wu et al., 2019b,c) etc. are very effective tools to solve real-life related decision-making problems. So, one can develop our proposed approach to these effective frameworks.

πœ‡πΉΜƒπ΄ β‹’Μƒ 𝐹 𝐹̃𝐡 (π‘₯) = π‘šπ‘Žπ‘₯(0, π‘ŸπΉΜƒπ΄ (π‘₯) + π‘ŸπΉΜƒπ΅ (π‘₯) βˆ’ 1)𝑒

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Some basic operations on neutrosophic sets (Smarandache, 2003, 2005). Let, {𝑁𝐴 = (𝑇𝐴 (π‘₯), 𝐼𝐴 (π‘₯), 𝐹𝐴 (π‘₯)) ; π‘₯ ∈ π‘ˆ } and {𝑁𝐡 = (𝑇𝐡 (π‘₯), 𝐼𝐡 (π‘₯), 𝐹𝐡 (π‘₯)) ; ∈ π‘ˆ } be two neutrosophic sets over π‘ˆ . Then, some possible classes of intersection operations (βˆ©π‘ ) and union operations (βˆͺ𝑁 ) between 𝑁𝐴 and 𝑁𝐡 are as follows: = (𝑇𝐴 (π‘₯) ∩𝐹 = (𝑇𝐴 (π‘₯) ∩𝐹 = (𝑇𝐴 (π‘₯) ∩𝐹 = (𝑇𝐴 (π‘₯) ∩𝐹

𝑇𝐡 (π‘₯), 𝐼𝐴 (π‘₯) βˆͺ𝐹 𝐼𝐡 (π‘₯), 𝐹𝐴 (π‘₯) βˆͺ𝐹 𝐹𝐡 (π‘₯)); 𝑇𝐡 (π‘₯), 𝐼𝐴 (π‘₯) ∩𝐹 𝐼𝐡 (π‘₯), 𝐹𝐴 (π‘₯) βˆͺ𝐹 𝐹𝐡 (π‘₯)); 𝑇𝐡 (π‘₯), 𝐼𝐴 (π‘₯) ∩𝐹 𝐼𝐡 (π‘₯), 𝐹𝐴 (π‘₯) ∩𝐹 𝐹𝐡 (π‘₯)); 𝐼 (π‘₯)+𝐼 (π‘₯) 𝑇𝐡 (π‘₯), 𝐴 2 𝐡 , 𝐹𝐴 (π‘₯) βˆͺ𝐹 𝐹𝐡 (π‘₯));

β€’ 𝑁𝐴 βˆ©π‘ 𝑁𝐡 = (𝑇𝐴 (π‘₯) ∩𝐹 𝑇𝐡 (π‘₯), 1 βˆ’ β€’ β€’ β€’ β€’

𝑁𝐴 βˆͺ𝑁 𝑁𝐴 βˆͺ𝑁 𝑁𝐴 βˆͺ𝑁 𝑁𝐴 βˆͺ𝑁

𝑁𝐡 𝑁𝐡 𝑁𝐡 𝑁𝐡

= (𝑇𝐴 (π‘₯) βˆͺ𝐹 = (𝑇𝐴 (π‘₯) βˆͺ𝐹 = (𝑇𝐴 (π‘₯) βˆͺ𝐹 = (𝑇𝐴 (π‘₯) βˆͺ𝐹

𝐼𝐴 (π‘₯)+𝐼𝐡 (π‘₯) , 𝐹𝐴 (π‘₯) βˆͺ𝐹 2

𝐹𝐡 (π‘₯));

𝑇𝐡 (π‘₯), 𝐼𝐴 (π‘₯) ∩𝐹 𝐼𝐡 (π‘₯), 𝐹𝐴 (π‘₯) ∩𝐹 𝐹𝐡 (π‘₯)); 𝑇𝐡 (π‘₯), 𝐼𝐴 (π‘₯) βˆͺ𝐹 𝐼𝐡 (π‘₯), 𝐹𝐴 (π‘₯) ∩𝐹 𝐹𝐡 (π‘₯)); 𝑇𝐡 (π‘₯), 𝐼𝐴 (π‘₯) βˆͺ𝐹 𝐼𝐡 (π‘₯), 𝐹𝐴 (π‘₯) βˆͺ𝐹 𝐹𝐡 (π‘₯)); 𝐼 (π‘₯)+𝐼 (π‘₯) 𝑇𝐡 (π‘₯), 𝐴 2 𝐡 , 𝐹𝐴 (π‘₯) ∩𝐹 𝐹𝐡 (π‘₯));

β€’ 𝑁𝐴 βˆͺ𝑁 𝑁𝐡 = (𝑇𝐴 (π‘₯) βˆͺ𝐹 𝑇𝐡 (π‘₯), 1 βˆ’

𝐼𝐴 (π‘₯)+𝐼𝐡 (π‘₯) , 𝐹𝐴 (π‘₯) ∩𝐹 2

𝐹𝐡 (π‘₯)).

where, ∩𝐹 and βˆͺ𝐹 are the fuzzy t-norm (fuzzy intersection) and fuzzy t-conorm (fuzzy union). Appendix B Some set theoretic operations on complex fuzzy sets (Zhang et al., 2009). Let, 𝐹̃𝐴 and 𝐹̃𝐡 be two complex fuzzy sets over the universe π‘ˆ , 𝑖𝑒 (π‘₯) 𝑖𝑒 (π‘₯) where, πœ‡πΉΜƒπ΄ (π‘₯) = π‘ŸπΉΜƒπ΄ (π‘₯)𝑒 𝐹̃𝐴 and πœ‡πΉΜƒπ΅ (π‘₯) = π‘ŸπΉΜƒπ΅ (π‘₯)𝑒 𝐹̃𝐡 . Complex fuzzy bounded difference. The complex fuzzy bounded difference of 𝐹̃𝐴 and 𝐹̃𝐡 is denoted by, Μƒ 𝐹 𝐹̃𝐡 where, its membership is defined as, 𝐹̃𝐴 βŠ– ( ) πœ‡πΉΜƒπ΄ βŠ–Μƒ 𝐹 𝐹̃𝐡 (π‘₯) = π‘šπ‘Žπ‘₯(0, π‘ŸπΉΜƒπ΄ (π‘₯) βˆ’ π‘ŸπΉΜƒπ΅ (π‘₯))𝑒

π‘–π‘šπ‘Žπ‘₯ 0,𝑒𝐹̃ (π‘₯)βˆ’π‘’πΉΜƒ (π‘₯) 𝐴

𝐡

.

Μƒ β€’ Complex fuzzy standard intersection. denoted by, 𝐹̃𝐴 βˆ©Μƒ π‘šπ‘–π‘› 𝐹 𝐹𝐡 , where, π‘–π‘šπ‘–π‘›(𝑒𝐹̃ (π‘₯),𝑒𝐹̃ (π‘₯)) 𝐴 𝐡 πœ‡πΉΜƒ βˆ©Μƒ π‘šπ‘–π‘› 𝐹̃ (π‘₯) = π‘šπ‘–π‘›(π‘ŸπΉΜƒπ΄ (π‘₯), π‘ŸπΉΜƒπ΅ (π‘₯))𝑒 . 𝐴 𝐹

.

Addition of complex neutrosophic sets. For adding of two complex neutrosophic sets, 𝐢̃𝐴 and 𝐢̃𝐡 , we have used the complex neutrosophic bold sum as given in Definition 6. Now, consider π‘š complex neutrosophic sets, 𝐢̃1 , 𝐢̃2 , … , πΆΜƒπ‘š over the universal set π‘ˆ where, 𝐢̃𝑠 = (𝑝𝑠 𝑒𝑖𝑒𝑠 , π‘žπ‘  𝑒𝑖𝑣𝑠 , π‘Ÿπ‘  𝑒𝑖𝑀𝑠 ); 𝑠 = 1, 2, … , π‘š. Then, the addition of π‘š complex neutrosophic sets is denoted by, Μƒ 𝑁 𝐢̃2 βŠ• Μƒ 𝑁 ..βŠ• Μƒ 𝑁 πΆΜƒπ‘š and is defined as, 𝐢̃1 βŠ• Μƒ 𝑁 𝐢̃2 βŠ• Μƒ 𝑁 ..βŠ• Μƒ 𝑁 πΆΜƒπ‘š 𝐢̃1 βŠ• ( = π‘šπ‘–π‘›(1, 𝑝1 + 𝑝2 + β‹― + π‘π‘š )π‘’π‘–π‘šπ‘–π‘›(2πœ‹,𝑒1 +𝑒2 +β‹―+π‘’π‘š ) , π‘šπ‘–π‘›(1, π‘ž1 + π‘ž2 +) β‹― + π‘žπ‘š )π‘’π‘–π‘šπ‘–π‘›(2πœ‹,𝑣1 +𝑣2 +β‹―+π‘£π‘š ) , π‘šπ‘–π‘›(1, π‘Ÿ1 + π‘Ÿ2 + β‹― + π‘Ÿπ‘š )π‘’π‘–π‘šπ‘–π‘›(2πœ‹,𝑀1 +𝑀2 +β‹―+π‘€π‘š )

Appendix A

𝑁𝐡 𝑁𝐡 𝑁𝐡 𝑁𝐡

𝐡

Appendix C

This work is supported by the Department of Science and Technology (DST), New Delhi, India, through the letter No. DST/INSPIRE/03/2014/003326.

𝑁𝐴 βˆ©π‘ 𝑁𝐴 βˆ©π‘ 𝑁𝐴 βˆ©π‘ 𝑁𝐴 βˆ©π‘

𝐴

β€’ Complex fuzzy bold sum. denoted by, (𝐹̃𝐴 β‹“Μƒ 𝐹 𝐹̃𝐡 , where, ) 𝑖 π‘šπ‘–π‘› 2πœ‹,𝑒𝐹̃ (π‘₯)+𝑒𝐹̃ (π‘₯) 𝐴 𝐡 πœ‡πΉΜƒπ΄ β‹“Μƒ 𝐹 𝐹̃𝐡 (π‘₯) = π‘šπ‘–π‘›(1, π‘ŸπΉΜƒπ΄ (π‘₯) + π‘ŸπΉΜƒπ΅ (π‘₯))𝑒 .

Acknowledgment

β€’ β€’ β€’ β€’

𝑖 π‘šπ‘Žπ‘₯ 0,𝑒𝐹̃ (π‘₯)+𝑒𝐹̃ (π‘₯)βˆ’2πœ‹

𝐡

Μƒ β€’ Complex fuzzy standard union. denoted by, 𝐹̃𝐴 βˆͺΜƒ π‘šπ‘Žπ‘₯ 𝐹 𝐹𝐡 , where, π‘–π‘šπ‘Žπ‘₯(𝑒𝐹̃ (π‘₯),𝑒𝐹̃ (π‘₯)) 𝐴 𝐡 πœ‡πΉΜƒπ΄ βˆͺΜƒ π‘šπ‘Žπ‘₯ 𝐹̃𝐡 (π‘₯) = π‘šπ‘Žπ‘₯(π‘ŸπΉΜƒπ΄ (π‘₯), π‘ŸπΉΜƒπ΅ (π‘₯))𝑒 . 𝐹 β€’ Complex fuzzy product. denoted by, 𝐹̃𝐴 β—¦Μƒ 𝐹 𝐹̃𝐡 , where, ( 𝑒𝐹̃ (π‘₯) 𝑒𝐹̃ (π‘₯) ) 𝐴 𝐡 πœ‡πΉΜƒπ΄ β—¦Μƒ 𝐹 𝐹̃𝐡 (π‘₯) = π‘ŸπΉΜƒπ΄ (π‘₯).π‘ŸπΉΜƒπ΅ (π‘₯)𝑒𝑖2πœ‹ 2πœ‹ . 2πœ‹ . 20

S. Manna, T.M. Basu and S.K. Mondal

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