Computer Modeling in Engineering & Sciences |

DOI: 10.32604/cmes.2022.019684

ARTICLE

Interval-Valued Neutrosophic Soft Expert Set from Real Space to Complex Space

1Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi, Selangor, 43600, Malaysia

2Department of Mathematics, Faculty of Education for Pure Sciences, University of Anbar, Ramadi, 55431, Iraq

3Preparatory Year Deanship, King Faisal University, Hofuf, Al-Ahsa, 31982, Saudi Arabia

*Corresponding Author: Abd Ghafur Ahmad. Email: ghafur@ukm.edu.my

Received: 08 October 2021; Accepted: 17 December 2021

Abstract: A fuzzy system is a novel computing technique that accesses uncertain information by fuzzy representation. In the decision-making process, fuzzy system and soft computing are effective tools that are tolerant to imprecision, uncertainty, and partial truths. Evolutionary fuzzy systems have been developed with the appearance of interval fuzzy, dual fuzzy, hesitant fuzzy, neutrosophic, plithogenic representations, etc. Moreover, by capturing compound features and convey multi-dimensional data, complex numbers are utilized to generalize fuzzy and neutrosophic fuzzy sets. In this paper, a representation of neutrosophic soft expert systems based on the real and complex numbers in the interval form is proposed. The interval-valued neutrosophic soft expert set (I-VNSES) is defined, and the interval-valued complex neutrosophic soft expert set (I-VCNSES) is formally generalized from the concept of I-VNSES. For both I-VNSES and I-VCNSES, we introduce the relevant basic theoretical operations and study their properties. Based on these new concepts, a generalized algorithm is proposed and applied to handle the imbedded indeterminacy in the two-dimensional interval data. The proposed algorithm is tested on the economic factors that affected the Malaysian economy in 2020 to see which ones are the most influential. Eventually, a comparison of three current approaches is used to back up this study.

Keywords: Interval-valued neutrosophic set; complex neutrosophic set; interval-valued complex neutrosophic set; soft expert set; decision making

Fuzzy sets [1] and its extensions, such as intuitionistic fuzzy sets [2] and neutrosophic sets [3], have provided a wide range of tools that can deal with uncertainty in different types of problems. The neutrosophic set has become increasingly popular in the last two decades as a sophisticated representation of uncertain, incomplete and undetermined data. The neutrosophic set (NS) is characterized by three membership functions which represent truth, falsity, and indeterminacy, and all these three memberships take values in the non-standard interval

Recently, Al-Sharqi et al. [34] introduced interval-valued complex neutrosophic soft set (I-VCNSS) as a combination of interval complex neutrosophic set and soft set to deal with the problems involved periodicity information and varies with time in interval forms. This model is useful for handling these problems with two-dimensional characterization properties. To make this model more practical for improving new decision-making results, we will improve it into interval-valued complex neutrosophic soft expert sets (I-VCNSES) by extending interval-valued neutrosophic soft expert set from real space to complex space in order to incorporate the advantages of soft expert sets to the interval-valued complex neutrosophic soft sets. The novelty of I-VCNSES appears in its ability to provide a succinct, elegant, and comprehensive representation of two-dimensional neutrosophic information (information presented by the amplitude terms and information presented by the phase terms) as well as the adequate parameterization and the opinions of the experts, all in an interval form. In addition, the economic-related activities such as the effects of certain financial factors on the economy of a country where the period of time of the influence as a second variable and the opinion of many experts play a key role in the final decision. In this paper, we will solve the same types of decision-making problems that have been solved by CNSES and CNSER by using the interval-based membership structure while defining the I-VCNSES. I-VCNSES can describe more information range by virtue of the interval membership values, which more accurately express decision makers’ evaluation information and cause less information distortion.

1.2 Motivation and Contribution

The main contribution and motivation behind this research can be summarized below:

(1) We extend the concept of I-VNSES that is given in Section 3 to I-VCNSES to incorporate the time frame offered by phase terms, as well as the capacity to express ambiguous, indeterminate, and inconsistent data in two dimensions.

(2) We define some basic operations related to our two notions (I-VNSES and I-VCNSES), namely the complement, union, intersection, AND, and OR. In addition, prove some related properties.

(3) In terms of the application in Section 5, we propose an algorithm to solve decision-making problems in the economic field by converting our model from the complex state (I-VCNSES) to the real state (I-VNSES) and then providing in detail decision steps.

(4) In some real-life problems which the user cannot be solved by soft expert set, fuzzy soft expert set [35], intuitionistic fuzzy soft expert set [36], neutrosophic soft set expert set [37], interval-valued generalized fuzzy soft expert set [38], our first proposed interval neutrosophic soft expert set, etc. To help the user to overcome such problems, an interval-valued complex neutrosophic soft expert set has been introduced.

(5) An interval-valued complex neutrosophic soft expert set(I-VCNSES) can be viewed as follows: soft expert set (SES)

(6) Finally, the main feature in our concept (I-VCNSES) is the presence of the amplitude and phase and their memberships in the form of intervals, and this gives the user more flexibility in the decision-making process.

Based on the below flowchart, we review how to organize our manuscript briefly.

In this section, we recapitulate the concepts of interval-valued neutrosophic set (I-VNS), complex neutrosophic sets (CNS) and interval-complex neutrosophic set (I-CNS) and give an overview of the operations structures of these concepts that are relevant to the work in this paper.

Definition 2.1. [12] Let U be a space of points (objects) with generic elements in

Definition 2.2. [12] Let

(1) The complement of an I-VNS A is denoted by

(2)

(3) The union (intersection) of two INSs

where

Definition 2.3. [28] A complex neutrosophic set S defined on a universe of discourse U is characterized by three memberships that are a truth membership

Definition 2.4. [29] Let U be a space of points (objects) with generic elements in

and

where, the amplitude interval-valued terms

and

where,

Definition 2.5. [29] Let

(1) The complement of an I-VCNS N is denoted by

where,

the phase term

(2) The union (intersection) of two I-VCNSs

where,

The phase term's union (intersection) is defined in the same way as the amplitude term's union (intersection). The two symbols

3 Interval Valued Neutrosophic Soft Expert Set

In this part, we introduce the idea of an interval-valued neutrosophic soft expert set(I-VNSES) as a combination of interval-neutrosophic set(I-NS) and soft expert set (SES). Throughout this paper,

Definition 3.1. A pair

Hence

where the three-interval truth-membership

Then,

such that

Example 3.1. Suppose that one of the producing companies wanted to evaluate their products with the help of some experts. Let

The I-VNSES

Definition 3.2. Let

(1) A

(2)

This relationship is indicated by

Definition 3.3. Two I-VNSESs

Definition 3.4. An I-VNSES

Definition 3.5. An I-VNSES

Definition 3.6. An agree I-VNSES

Definition 3.7. A disagree I-VNSES

Now, we present some fundamental operations on I-VNSESs, namely the complement, union, and intersection of I-VNSESs, alongside deriving their properties and giving some numerical examples.

Definition 3.8. The complement of I-VNSES

where

where,

Example 3.2. Take the part given in Example 3.1, where

Now, by employing the I-VN-complement, we get the complement of the part that is given by

Proposition 3.1. Let

Proof. Assume that

The complement of

Thus,

This completes the proof.

Definition 3.9. The union of two I-VNSESs

where,

The union

Definition 3.10. The intersection of two I-VNSESs

where,

The intersection

Example 3.3. Consider Example 3.1, Let

By using I-VN-union, then

Proposition 3.2. If

(1)

(2)

(3)

(4)

Proof. (1) By employing Definition 3.9, we will demonstrate that

(2) The proof is similar to that of part (1).

(3) By using Definition 3.9, we want to prove that

Considering the case when

The proof of part (4) is similar to part (3).

In the following, we will introduce the definitions of AND and OR operations on IV-NSESs with a proposition on these two operations.

Definition 3.11. If

Definition 3.12. If

Proposition 3.3. If

(1)

(2)

Proof. The proof of the above two proposition is similar to the proof of Proposition 3.2 part (3) and (4).

4 Interval-Valued Complex Neutrosophic Soft Expert Sets

In this part, we establish the idea of I-VCNSES by extending I-VNSES from real space to complex space. We denote U as a universe, E is a set of parameters, X as a set of experts and O = {1 = agree, 0 = disagree} as a set of opinions, such that

Definition 4.1. A pair

The amplitude of interval-values terms

where,

Example 4.1. Suppose that an electrical appliance sales company develops two types of its products and wants to ask some experts on these products by taking into account the degree of quality and ease of use represented amplitude terms and phase terms, respectively. Suppose

Then,

In the I-VCNSES

Based on the Definition 4.1, we introduce two concepts which are the subset and equality of I-VCNSESs.

Definition 4.2. Let

(1) A

(2)

This relationship is denoted by

Definition 4.3. Two I-VCNSESs

Definition 4.4. An I-VCNSES

Definition 4.5. An I-VCNSES

Definition 4.6. An agree I-VCNSES

Definition 4.7. A disagree I-VCNSES

Next, we provide some fundamental operations on I-VCNSESs, like the complement, union, and intersection of I-VCNSESs, derive essential properties and pertinent laws to this concept as De Morgan's laws, and give some illustrative numerical examples.

Definition 4.8. The complement of an I-VCNSES

where

Similarly,

where,

Example 4.2. Take the part given in Example 4.1, where

By using the I-VCN-complement, we get the complement of the part given by

Proposition 4.1. Let

Proof. Suppose that

The complement of

Thus,

This completes the proof.

Definition 4.9. The union of two I-VCNSESs

where,

The phase term's union is defined in the same way as the amplitude term's union

Definition 4.10. The intersection of two I-VCNSESs

where,

The phase term's intersection is defined in the same way as the amplitude term's intersection

Now, we present the De Morgan's law holds for the I-VCNSES as follows.

Proposition 4.2. If

(1)

(2)

Proof. (1) Suppose that

Since

Hence,

We have,

Assume that

Hence

Therefore,

In the same method, we can get the rest terms. Thus, it follows that

The proof (2) is a similar manner proof of (1).

Proposition 4.3. If

(1)

(2)

(3)

(4)

Proof. The proof of the above four propositions is similar to the proof of proposition 3.2 depending on definitions 4.9, 4.10.

In the following, we will introduce the definitions of AND and OR operations on I-VCNSESs with a proposition on these two operations.

Definition 4.11. If

Definition 4.12. If

Proposition 4.4. If

(1)

(2)

Proof. The proof of the above two proposition is similar to the proof of proposition 4.3 part 3 and 4.

5 Decision-Making on I-VCNSES Environment

In this part, we will highlight the importance of our model in real life by solving one of the decision-making problems in the economic field.

Example 5.1. In this example, assume that we are interested in knowing the extent to which the Malaysian economy in 2020 is affected by the four economic factors. The four factors are as follows, the plunge in oil prices and commodities commercial, the slowdown in china's economy, the tax (GST) on services and goods that applied this year, and exchange rate variability in this year. Our problem is to order these four destinations in descending order from a maximum influencing the Malaysian economy to a minimum affecting the Malaysian economy in 2020. So let

In this situation, the amplitude interval terms measurement the impact degree of the above-mentioned factors on the Malaysian economy in 2020, whilst the phase interval terms represent the period of this impact. By this way, the interval-valued complex neutrosophic number

In the approximation

Then, we proceed with SVNSE-values to make the final decision using the SVNSE-method [25]. The following steps clarify the proposed algorithm:

The decision is to choose value ui as the solution to the problem. If there is more than one value with the highest ri score, then any one of those values can be chosen as the best solution.

Here, we would like to point out that this method (algorithm) is used with decision-making problems that have information that has a known weight (complete weight information). To implement these steps that were mentioned in the algorithm above, we assume that the weight vector to the amplitude terms is

Now, to convert the I-VCNSES

Hence for

By Tables 2 and 3, we get the highest numerical grade for the elements in the agree-SVNSES and disagree-SVNSES, respectively. The values of

Finally, according to the Step 8 of proposed algorithm, the maximum

6 Comparison between I-VCNSES and Other Existing Methods

In this section, we will now compare our second concept (I-VCNSES), which is an expansion of our first concept (I-VNSES) with three existing methods, which is neutrosophic soft set (NSS) [39], single-valued neutrosophic soft expert set (SVNSES) [40], and complex neutrosophic soft expert set (CNSES).

Because neutrosophic soft set lacks the phase term that indicates the time frame, so it is clear that it is unable to solve the decision-making problem given in this work (see Example 5.1), which involves two-dimensional data, i.e., the degree of the influence and the total time of the influence. Another reason is its incapacity to deal with several experts. However, the single-valued neutrosophic soft expert set can deal with more than one expert. But it is not able to deal with problems that involve two-dimensional information (amplitude terms and phase terms) like the problem presented in this work.

The complex neutrosophic soft expert set can overcome the problems which the above two concepts cannot overcome by using neutrosophic expert soft set, by virtue of the phase terms which have the ability to represent the time frame of the interaction between the variables as well as parameterization and the opinions of the experts, all in a single set. In comparison, the main distinguishing feature of our proposed is its ability to describe the grade of three complex memberships in the form of an interval that is a subset of the unit interval. Furthermore, since it is difficult for an expert to express his/her certainty by an exact real number, so it is suitable to choose an interval that expresses the certainty level. Therefore, the I-VCNSESs help in modeling the uncertainty and the consequences of any ignorance, mistakes, and confusion of experts. Therefore, it can be judged that our concept is more important and useful than the above-mentioned concepts.

6.1 Advantages and Limitations

Our proposed model has certain advantages and limitations. Firstly, our model (I-VCNSES) has the ability to provide a succinct, elegant, and comprehensive representation of two-dimensional interval neutrosophic information (amplitude terms and phase terms) as well as adequate parameterization and opinions of the experts, all in the form of an interval. Secondly, I-VCNSES includes evaluation information missing in the neutrosophic soft model and single-valued neutrosophic soft expert model, such as the time frame which is presented by the phase term. In addition, it has the added advantage of allowing the users to know the opinion of all the experts in an interval model without the necessity for any additional cumbersome operations. Thirdly, the I-VCNSES that is used in our method has the ability to handle the uncertainty information that is captured by the amplitude terms and phase terms of the complex numbers simultaneously. Fourthly, a new practical formula is employed to convert the interval-valued complex neutrosophic soft expert from the complex state to the real state, which gives decision-making with a simple computational process without the need to carry out directed operations on complex numbers. Fifthly, our model represents the information of two-dimensional in interval form, namely information of amplitude term and phase term, thus making our model more appropriate for use in real-life problems like decision-making, medical diagnosis to select the best alternative. Finally, the interval form which characterized our concept gives the user more flexibility in the real decision-making process, where the real decision we get through our concept is characterized as more trustworthy and more acceptable than the other existing concepts in which there is no attention to the interval form. Therefore, these features mentioned above are essential points that distinguish our model from existing models. Our idea, on the other hand, is incapable of handling discontinuous attribute-value sets that correspond to different attributes. Therefore, we recommend for future studies that this gap be overcome by generalizing our concept to an interval-valued complex neutrosophic hypersoft expert set.

A novel mathematical tool is created to highlight the information using time factors and to realize the opinions of all the experts in an interval model. In this article, we established the concept of IV-CNSES by extending our concept of IV-NSES from real space to complex space. The basic operations on both I-VNSES and I-VCNSES, namely subset, complement, union, intersection, AND and OR operations were defined. Subsequently, some basic algebraic properties of these operations were proven. In addition, we showed the importance of this concept in real life through a proposed new algorithm and we applied it to both models to solve a hypothetical decision-making problem related to the economic factors that affected the Malaysian economy in 2020. A comparison of our proposed model with three other existing models indicates the efficiency of our model and also it showed the superiority of our concept over these concepts with flexibility and accuracy in representing two-dimensional interval neutrosophic information. Finally, these flexible new extensions are not applied yet in many fields like computer science, social science, medical science, engineering, etc. So, in future work, we plan to combine this concept with other types of algebraic structures such as group [41,42] and ring [43,44]. We are wishful to provide our work to other MCDM models and applications for modeling vagueness and uncertainty.

Acknowledgement: We are indebted to Universiti Kebangsaan Malaysia for providing financial support and facilities for this research under the Grant TAP-K005825.

Funding Statement: Universiti Kebangsaan Malaysia Research Grant TAP-K005825.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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