A hybrid decision-making conceptual framework based on generalized information quality under neutrosophic evidence theory: a comparative analysis

Abstract

Simulating uncertainty for dealing with actual events is one of Artificial Intelligence's key difficulties and challenges. The ultimate objective for decision-makers is to manage uncertainty, particularly in indeterminate scenarios when it is not the case that the solution of the problem can be expressed with true or false values. As a result, new techniques to facilitate the interpretation of indeterminacy are currently under development. Neutrosophic logic (NL), which addresses the concept of neutralities, extends classic logic, fuzzy logic, paraconsistent logic, intuitionistic logic, and so on. The single-valued neutrosophic set (SVN) is a subclass of neutrosophic sets that has recently been presented. Solving multicriteria decision-making problems is an essential use case for SVNs to be applied. The research objectives of the article are twofold. First, we examine the potential of utilizing single-valued neutrophilic sets in a more efficient manner to address the issue of multicriteria decision-making. Within this framework, our aim is to explore and extend the concept of information quality as an uncertainty measure by comparing it to neutrosophic Dempster–Shafer (D–S) evidence theory in the context of decision-making. As a proposed solution to the aforementioned research objectives, this manuscript suggests and implements a novel conceptual framework for determining and quantifying the similarity measure between SVNSs in a multicriteria decision-making context under the principles of D–S evidence theory. Finally, illustrative case studies are given to support the logic and practicality of the suggested methodology compared to current methodologies.

1 Introduction

The process of management involves making decisions at both the strategic and operational levels; businesses and organizations encounter decision-making challenges when they are presented with a set of viable options and asked to select the option that they anticipate will yield the most advantageous outcome based on predetermined criteria. Uncertainty that is always present in the business environment adds a dimension of complexity to the decision-making process. The optimal choice in a problem alike results from the evaluation of each of the possible alternative decisions in terms of the results that will result from its possible selection and implementation.

Within this framework, the incapacity of current techniques to address the multifaceted real problems of businesses using only a single criterion resulted in the progress of multicriteria decision making. Multicriteria problems due to the presence of multiple and conflicting evaluation criteria of alternative decisions are problems with a low degree of structure. Therefore, the multitude of criteria and the complexity of the relationships between them have an effect on the system of preferences that are decided upon, which in turn is characterized by a low degree of organization. Consequently, the decisions taken belong to the categories of semi structured decisions, which creates the need to support the decision maker through the development of appropriate multicriteria models.

Multicriteria decision-making accepts the existence of a preference system that indicates the judgments of decision-makers in a set of different actions under specific evaluation criteria. In accordance with this, the decision-maker's views are considered in the formation of a value system that satisfies a set of conditions, through which the decision-maker will be led to select the best solution of solutions. The main objectives of multicriteria decision-making are (Matsatsinis 2010):

• Determine the conditions that must be met for the value system to exist.

• Support the decision maker to discover, through a given process, a value system and make the right decision.

"Similarity" is a fundamental concept that can relate to a correlation, analogy, resemblance, or comparability of various characteristics, situations, items, or even entities. The notion of similarity refers to the degree of resemblance or likeness between two or more things. It is a measure of how alike or comparable two objects, concepts, or entities are in various aspects, such as their characteristics, features, properties, or attributes (Chatterjee et al. 2019). The rational equivalent of similarity exists in mathematics and related fields as a real-valued function that measures the resemblance between items.

Fuzzy sets theory (Zadeh 1965) was the first mathematical tool to model the problem of ambiguity representation in partial form. Because of the necessity for advancement, fuzzy sets have rapidly progressed, and structures such as L-fuzzy sets, which generalize the order of structures beyond the unit interval¹ (Guo et al. 2014), and intuitionistic fuzzy sets (IFS) (Atanassov 1986) have emerged as extensions of the “classical” theory of fuzzy sets. Later, "motivated by the occurrence of paradoxism and paraconsistency in knowledge", scholar in (Smarandache 2005, 2010) originally integrated nonstandard analysis with a three-valued logic and developed the neutrosophic set (NS) as an extension of intuitionistic fuzzy sets with the addition of an indeterminacy function. The idea of a neutrosophic set is described as a set related to the degree of truth, indeterminacy, and falsity. As a remark, we could claim that neutrosophic logic stems its roots from many-valued logic, as it is actually an extension of the latter treated for dealing with approximate reasoning.

Citing the description of neutrosophic logic in (Rivieccio 2008), "The first part of neutrosophic logic is the membership or truth rate (T), the middle part is the indeterminacy rate (I), and the third part is the non-membership or falsity rate (F) of each set element." Interval neutrosophic sets (INSs) and SVNSs were introduced by scholars (Wang et al. 2005, 2010), both of which are subclasses of NSs. SVNSs and INSs are exceptionally robust methodological tools for analysing imprecise, incomplete, and uncertain data, which is common in many engineering and technological problems. Since INs and SVNSs are simple to define and because subjective judgement is considered fuzzy, these two types of neutrosophic sets are commonly employed in practice, such as in decision analysis (Zhang et al. 2014; Ye 2014), image processing (Guo and Cheng 2009; Broumi and Smarandache 2013a, b), medicine (Ma et al. 2017), fault diagnostics (Majumdar and Samanta 2014), and staff recruitment (Ji et al. 2018). Among these, decision-making is a prominent study topic for academics from numerous areas (Fatimah et al. 2019; Alcantud and Santos-Garcia 2015).

In this article, a novel hybrid methodology for measuring the similarity between SVNSs employing Dempster–Shafer (D–S) evidence theory is suggested and demonstrated in the context of multicriteria decision-making. From the literature review conducted regarding similarity measures used in neutrosophic sets, we should refer to Ye, who employed a weighted cosine similarity metric of SVNSs (Ye 2013). Then, in (Ye 2014), a multicriteria decision-making technique for SVNSs is developed based on aggregation operations and the cosine similarity measure. In their paper, Majumdar and Samanta (2014) presented a number of similarity measures based on distance metrics while proposing a new method of similarity taking into account entropy theory. For a detailed review of similarity measures used in the neutrosophic literature, interested readers are referred to (Chatterjee et al. 2019; Broumi and Smarandache 2013a, b).

When referring to information fusion, one of the prominent theories that come into mind is D–S evidence theory, which has recently received significant interest. A fundamental benefit of D–S evidence theory over ordinary probability theory is that a more qualitative fusion outcome may be achieved by adopting only a basic reasoning process, independent of previous probability. Furthermore, another important advantage of evidence theory is the ability to integrate data from multiple independent sources through the use of the D–S law of combination. Last, a significant aspect of this theory is that it may easily define information vagueness (Zadeh 1986; Jiang et al. 2017a, b).

From the above, we can understand the reason that D–S evidence theory has been widely applied in a variety of disciplines. To name a few, these include image fusion (Dong et al. 2009; Yang and Wei 2013), sensor data fusion (Jiang et al. 2016), gender discrimination (Ma et al. 2016), and device fault diagnostics (Islam et al. 2016; Jiang et al. 2017a, b), among others (Zhang et al. 2016; Deng 2016). The method of assessing similarity between SVNSs used in this study is to first convert SVNSs to the basic probability assignment (BPA), which is a key concept in D–S theory. The similarity of BPAs may then be used to compute the value of the similarity measure between SVNSs. In D–S evidence theory, the correlation coefficient or evidential distance (Jousselme et al. 2001) can be used to compute the correlation between BPAs.

Li and Deng’s (2019) correlation coefficient will be used as a blueprint in our proposed methodology to assess the similarity of the SVNSs. This correlation coefficient under generalized information quality is one of the coefficients that may efficiently reproduce the relationship between BPAs.

In summary, the problems addressed in this article are as follows:

• 1. We examine how SVNs could be employed, in a more proficient way, in the framework of multicriteria decision making because of their inherent capability of handling indeterminacy often met in such problems.

• 2. We study and expand the concept of information quality as a basic concept to measure the uncertainty of probability distribution by correlating it with neutrosophic D–S evidence theory, thus merging in an appropriate way the merits of each theory in a coherent conceptual framework.

• 3. Our work suggests a comprehensive method for calculating the similarity between SVNs, which is considered one of the most important topics in neutrosophy theory, hence believing that our proposed approach will provide new and interesting perspectives in the related research.

The key contributions and novelties of the current study are indicated below:

• 1. We suggest and establish a novel hybrid MCDM methodology based on a new similarity measure under neutrosophic evidence theory. From a more general perspective, our suggested method could be viewed as a mathematical approach to model uncertainty.

• 2. A new correlation coefficient, under generalized information quality, is formed and applied under neutrosophic evidence theory. This is achieved by proposing a new formula to quantify the information quality of BPA that will eventually enable us to better represent the conflict between evidence.

• 3. To confirm the robustness and objectivity of the suggested method, literature-based examples are examined and solved by means of our newly suggested methodology. Then, a comparative study with other similarity-based methods is undertaken.

• 4. To the best of our knowledge, no similar research article has been found in the related literature that incorporates generalized information quality theory with neutrosophic sets under evidence theory. In this way, introducing a novel hybrid multicriteria decision-making methodology is important and beneficial to the academic community and hopefully will raise more interest in the related field.

The remainder of this article is structured as follows: Sect. 2 provides an overview of the research's theoretical foundation, while we introduce our approach for measuring the similarity of SVNSs employing D–S evidence theory. Our suggested method, which quantifies the similarity between SVNSs employing evidence theory, is discussed in Sect. 3. In Sect. 4, we apply D–S evidence theory to multicriteria decision-making problems under SVNSs, whereas in Sect. 5, a comparative analysis with current neutrosophic similarity approaches is undertaken. Finally, in Sect. 6, we outline our conclusions and suggest future research work.

2 Basic concepts

In this section, which is divided by thematic Sects. 2.1–2.4, we present the theoretical foundations of our methodology, which consists of neutrosophic logic, D–S evidence theory and the notion of a correlation coefficient.

2.1 Preliminaries

2.1.1 Single-valued neutrosophic set

Definition 1 (Jiang and Shou 2017)

Consider X to be a space of points (objects) and x to be a generic element in X. A truth membership function T_A, an indeterminacy membership function I_A, and a falsity membership function F_A characterize a single-valued neutrosophic set (SVNs) A in X. Then, it holds that for each point x in X, T_A(x), I_A(x), F_A(x) $\in$ [0, 1]. A simplification of neutrosophic set A is denoted by $A = \{ x, T\left( x \right), I\left( x \right), F\left( x \right) | x \in X \} .$

Definition 2 (Jiang and Shou 2017)

Let A_j (j = 1, 2…, n) be SVNs. The simplified neutrosophic weighted arithmetic average operator is specified as:

$${\mathcal{F}}_{{\text{w}}} \left( {{\text{A}}_{{1}} ,{\text{ A}}_{{2}} ...,{\text{ A}}_{{\text{n}}} } \right) \, = \mathop \sum \limits_{j = 1}^{n} w_{j} {\text{A}}_{{\text{j}}}$$

(1)

where ${\mathcal{W}}$ = (w₁,w₂…,w_n) is the weight vector of A_j (j = 1, 2…, n), ${\mathcal{W}}_{j}$ $\in$ [0, 1] and $\sum\nolimits_{j = 1}^{n} {w_{j} }$  = 1.

Definition 3

Let A_j (j = 1, 2…, n) be SVNs. The simplified neutrosophic mean arithmetic average operator is defined as:

$${\mathcal{F}}_{ } \left( {{\text{A}}_{{1}} ,{\text{ A}}_{{2}} ...,{\text{ A}}_{{\text{n}}} } \right) \, = \frac{{\mathop \sum \nolimits_{j = 1}^{n} A_{j} }}{n}$$

(2)

2.2 Dempster–Shafer evidence theory

Definition 4 (Dempster 2008)

The frame of discernment is a definite collection of N mutually exclusive and exhaustive items in D–S evidence theory. Let ${\Theta }$ be a set denoted by

$${\Theta } = \, \{ \theta_{1} ,\theta_{2} ,\theta_{3} , \ldots \theta_{i} , \ldots \theta_{N} \}$$

(3)

Let us designate ${\mathcal{P}}$(${\Theta }$) as the power set composed of 2^N elements of ${\Theta }$

$${\mathcal{P}}(\Theta ) = \{ \emptyset ,\{ \theta_{1} \} , \ldots ,\{ \theta_{N} \} ,\{ \theta_{1} ,\theta_{2} \} , \ldots \{ \theta_{1} ,\theta_{2} , \ldots \theta_{i} \} , \ldots \Theta \}$$

(4)

Definition 5 (Dempster 2008)

A mass function is a mapping m from P(${\Theta }$) to [0,1], formally defined as:

$$m: {\mathcal{P}}\left( \Theta \right) \to \left[ {0, 1} \right],\quad {\mathcal{A}} \to m\left( {\mathcal{A}} \right)$$

(5)

which fulfils the following conditions:

$$m\left( \emptyset \right) = 0$$

(6)

and

$$\mathop \sum \limits_{A \in P\left( \Theta \right) } m\left( {\mathcal{A}} \right) = 1$$

(7)

Note: ${\mathcal{A}}$ denotes any of the elements in ${\mathcal{P}}$(${\Theta }$). The mass m (${\mathcal{A}}$) or degree of evidence represents how strongly the evidence supports ${\mathcal{A}}$ or the extent to which we believe such a claim is justified (Dempster 2008).

Since Eq. (7) resembles a similar equation for probability distributions, the function m is usually called a basic probability assignment (BPA).

Given a basic assignment m, a belief measure can be determined by the following formula (Dempster 2008):

$${\text{Bel }}({\mathcal{A}}) = \mathop \sum \limits_{{{\mathcal{B}} \subseteq {\mathcal{A}}}} m\left( {\mathcal{B}} \right)\quad {\text{for all}}\;{\mathcal{A}} \in {\mathcal{P} }\left( {\Theta } \right)$$

(8)

Definition 6 (Dempster 2008)

In the D–S evidence theory, mass functions are also known as basic probability assignments (BPAs). Assume there are two BPAs that operate in the two sets of propositions B and C denoted by m₁ and m₂. The following formula indicates how Dempster's combination rule is used to combine them:

$$m(A) = \left\{ {\begin{array}{*{20}l} 0 \hfill & {{\rm A} = \emptyset } \hfill \\ {\frac{1}{{1 - {\mathcal{K}}}} \mathop \sum \limits_{{{\text{B}} \cap {\text{C}} = {\text{A}}}} m_{1} \left( B \right)m_{2} \left( C \right)} \hfill & {{\rm A} \ne \emptyset } \hfill \\ \end{array} } \right.$$

(9)

where

$${\mathcal{K}} = \sum\limits_{{{\text{B}} \cap {\text{C}} = \emptyset }} {m_{1} \left( B \right)m_{2} \left( C \right)}$$

(10)

In Eqs. (9) and (10), ${\mathcal{K}}$ shows the conflict between the two BPAs m₁ and m₂.

Dempster's combination rule allows us to obtain aggregated evidence in the same context from two independent sources (for example, from two experts of a certain field), which are expressed by two BPAs $m_{1}$ and $m_{2}$ on some power set $P\left( X \right)$ (Klir 2006).

2.3 Correlation coefficient

To facilitate the process of measuring the similarity between two BPAs, a correlation coefficient of belief functions is proposed in (Li and Deng 2019), detailed as follows:

Definition 7 (Li and Deng 2019)

If the discernment framework is ${\Theta }$ and two BPAs are given, the correlation coefficient based on the general information quality is defined as follows.

First, a factor ${\mathcal{D} }$ is proposed, which demonstrates the relationship between the BPAs, and the formula giving the entries of the rows is as follows:

$${\mathcal{D}}\left( {{\text{A}},{\text{ B}}} \right) \, = \frac{{2^{{\left| {A \cap B} \right| }} - 1}}{{2^{{\left| {A \cup B} \right| }} - 1}}$$

(11)

where A and B only belong to nonzero focal elements in the discernment frame Θ.

To modify the BPAs, the following equation is used, which includes a factor ${\mathcal{D} }$ known as the Jaccard matrix (Li and Deng 2019):

$$\left\{ {\begin{array}{*{20}c} {m_{1}^{\prime} = m_{1} D} \\ {m_{2} ^{\prime} = m_{2} D} \\ \end{array} } \right.$$

(12)

Definition 8 (Li and Deng 2019)

Given a basic probability allocation, the generalized expression for information quality is defined as follows:

$$IQ (m_{1} ) = \mathop \sum \limits_{A \subseteq X} \left( {\frac{{m_{1} \left( A \right)}}{{2^{\left| A \right|} - 1}}} \right)^{2}$$

(13)

where m_i is a mass function defined on the frame of discernment X, and |A| is the cardinality of A.

Finally, the correlation coefficient based on generalized information quality is as follows (Jiang and Shou 2017):

$${\mathcal{C}\mathcal{I}\mathcal{Q}}\,\left( {{\text{m}}_{{1}} ,{\text{ m}}_{{2}} } \right) \, = \mathop \sum \limits_{A, B \subseteq X} \frac{{\frac{{m_{{1^{\prime}}} \left( A \right)}}{{2^{\left| A \right|} - 1}} \times \frac{{m_{{2^{\prime}}} \left( B \right)}}{{2^{\left| B \right|} - 1}} }}{{\sqrt {IQ \left( {m_{1^{\prime}} } \right)} \times \sqrt {IQ \left( {m_{2^{\prime}} } \right)} }}$$

(14)

The correlation coefficient CIQ(m₁, m₂) quantifies the importance of m₁ and m₂. A greater correlation coefficient means a greater relevance of m₁ and m₂. Two pieces of evidence's significance might be utilized to measure the conflict. The resemblance between two pieces of evidence increases as the relevance increases, and the conflict between them decreases. As a result, the correlation coefficient is inversely proportional to the measure of conflict.

The correlation coefficient under generalized information quality adheres to the following fundamental axioms (Li and Deng 2019).

• (1) ${\mathcal{C}\mathcal{I}\mathcal{Q}}$ (m₁, m₂) = ${\mathcal{C}\mathcal{I}\mathcal{Q}}$ (m₂, m1)

• (2) 0 < ${\mathcal{C}\mathcal{I}\mathcal{Q}}$ (m₁, m₂) < 1.

• (3) if m₁ = m₂${\mathcal{C}\mathcal{I}\mathcal{Q}}$ (m₁, m₂) = 1.

• (4) (4) ${\mathcal{C}\mathcal{I}\mathcal{Q}}$ (m₁, m₂) = 0 ($\cup A_{i }$)$\cap$ ($\cup A_{j }$) = $\emptyset$, $A_{i }$, where $A_{j }$ are the focal elements of m₁ and m₂, respectively.

It is interesting to note that the above properties can be proven that they logically meet the basic axioms that a neutrosophic similarity measure should satisfy, first suggested in Broumi and Smarandache (2013a, b). This observation motivated our idea to suggest the utilization of the generalized information correlation coefficient to measure similarity between SNVSs for the first time in the related literature.

Definition 9 (Li and Deng 2019)

Given two BPAs and a discerning frame, a conflict coefficient under generalized information quality is k_CIQ and is stated as follows:

$$k_{CIQ} = \, 1 \, - CIQ \, \left( {m_{1} , \, m_{2} } \right) \, = \, 1 - \mathop \sum \limits_{A, B \subseteq X} \frac{{\frac{{m_{{1^{\prime}}} \left( A \right)}}{{2^{\left| A \right|} - 1}} \times \frac{{m_{{2^{\prime}}} \left( B \right)}}{{2^{\left| B \right|} - 1}} }}{{\sqrt {IQ \left( {m_{1^{\prime}} } \right)} \times \sqrt {IQ \left( {m_{2^{\prime}} } \right)} }}$$

(15)

From the above definition, it is supposed that the greater the value of the conflict coefficient is, the higher the degree of conflict between m₁ and m₂. When k_CIQ = 1, then m₁ and m₂ are in total conflict. When k_CIQ = 0, there is no conflict between m₁ and m₂ (Li and Deng 2019).

3 Proposed similarity measure for SVNSs

The first step for the implementation of the proposed similarity measure is how to rationally convert SVNSs to BPAs. This is succeeded by the following definitions first appearing in the insightful work of scholars (Jiang and Shou 2017).

Definition 10 (Jiang and Shou 2017)

Assume neutrosophic set A is an SVNS. The relationship between SVNS and BPA can be calculated as in the next equation:

$$\begin{gathered} {\text{m}}_{{\text{A}}} \left( {\text{T}} \right) \, = {\text{ T}}_{{\text{A}}} \left( {\text{x}} \right)/\left( {{\text{T}}_{{\text{A}}} \left( {\text{x}} \right) \, + {\text{ F}}_{{\text{A}}} \left( {\text{x}} \right) \, + \, \left( {{1 } - {\text{ I}}_{{\text{A}}} \left( {\text{x}} \right)} \right)} \right) \hfill \\ {\text{m}}_{{\text{A}}} \left( {\text{F}} \right) \, = {\text{ F}}_{{\text{A}}} \left( {\text{x}} \right)/\left( {{\text{T}}_{{\text{A}}} \left( {\text{x}} \right) \, + {\text{ F}}_{{\text{A}}} \left( {\text{x}} \right) \, + \, \left( {{1 } - {\text{ I}}_{{\text{A}}} \left( {\text{x}} \right)} \right)} \right) \hfill \\ {\text{m}}_{{\text{A}}} \left( {{\text{T}},{\text{ X}}} \right) \, = \, \left( {{1 } - {\text{ I}}_{{\text{A}}} \left( {\text{x}} \right)} \right)/\left( {{\text{T}}_{{\text{A}}} \left( {\text{x}} \right) \, + {\text{ F}}_{{\text{A}}} \left( {\text{x}} \right) \, + \, \left( {{1 } - {\text{ I}}_{{\text{A}}} \left( {\text{x}} \right)} \right)} \right) \hfill \\ \end{gathered}$$

(16)

From the above definition and the meanings of T_A(x) and m_A (Tx), we can reasonably conclude that m_A(Tx) = T_A(x). The same applies to m_A (Fx) = F_A(x). A more careful thought should be given to the term of indeterminacy and its interpretation in our context. I_A(x) designates the degree of support of others, excluding trust and falsity. Thus, (1 − I_A(x)) shows the level of trust we ascribe in x and the degree of opposition to x. Therefore, m_A (Tx, Fx) = 1 − I_A(x) (Jiang and Shou 2017).

The structural decisions made in the proposed methodology, especially those related to Eqs. (15) and (16) are greatly influenced by the need to tackle the key challenges identified in this study: managing indeterminacy in MCDM, enhancing information quality assessment, and improving similarity calculations within SVNs. These structures are not only theoretically sound but also practical, providing significant value in terms of resilience, precision, and suitability for complex decision-making scenarios. Consequently, the suggested method represents a significant advancement in the field, offering new tools and perspectives for both scholars and practitioners.

It should be noted that there also exist in the literature other methods that propose how to relate SVNs with BPAs, such as in (Zou et al. 2018), where SVNs are transformed to a BPA using a mapping from SVNs to a row vector, but we adopted the method suggested in (Jiang and Shou 2017) due to its simplicity in calculations.

Subsequently, we can now present the steps needed to calculate our proposed similarity measure:

Assume that two SVNSs A and B are represented by A = {$\left\langle {{ }T_{A} \left( {\text{x}} \right),{ }I_{A} \left( {\text{x}} \right),{ }F_{A} \left( {\text{x}} \right) } \right\rangle$ | x $\in$ X} and B = {$\left\langle {{ }T_{B} \left( {\text{x}} \right),{ }I_{B} \left( {\text{x}} \right),{ }F_{B} \left( {\text{x}} \right)} \right\rangle$ | x $\in$ X}.

Step 1 Given two SVNSs A and B, two respective groups of BPAs $m_{A}$ and $m_{B}$ can be calculated according to Eq. (16). It should be noted that, depending on the nature of input data (crisp or fuzzy), appropriate normalization equations may be used (cf. Example 2 in Sect. 4).

Step 2 Calculate the correlation coefficient $r_{BPA}$ between $m_{A}$ and $m_{B}$ using Eq. (14), i.e., $r_{BPA}$ = ${\mathcal{C}\mathcal{I}\mathcal{Q}}$ (m₁, m₂).

Step 3 Finally, the similarity measure $S_{r}$(A, B) is determined by $S_{r}$(A, B) = ${\mathcal{C}\mathcal{I}\mathcal{Q}}$ (m₁, m₂).

4 Application

In this section, our proposed similarity measure between two SVNSs is utilized to illustrate its applicability to solve multicriteria decision-making problems.

Let us assume that there exists a group of alternative solutions that can be denoted as A₁, A₂, … A_m. We would like to evaluate these alternatives under criteria C₁, C₂, … C_n. The assessment procedure of option A_i under criterion C_j can be expressed in the form of SVNs as:

$$A_{i} = \left\langle {C_{j, } T_{{A_{i } }} \left( {C_{j} } \right), I_{{A_{i } }} \left( {C_{j} } \right), F_{{A_{i } }} \left( {C_{j} } \right) } \right\rangle$$

(17)

where $A_{i} \in \{ A_{1}$, $A_{2}$, … $A_{m}$}, $C_{j} \in \{ C_{1}$, $C_{2}$, … $C_{n}$} and 0 $\le T_{{A_{i } }} \left( {C_{j} } \right)$ $\le 1$, 0 $\le I_{{A_{i } }} \left( {C_{j} } \right)$ $\le 1$, 0 $\le F_{{A_{i } }} \left( {C_{j} } \right)$ $\le 1.$ The significance of each criterion is expressed as $w_{1} ,w_{2} , \ldots w_{n} ,$ ${\text{where }}\;w_{j}$ $\in$ [0,1] and $\sum\nolimits_{j = 1}^{n} {w_{j} }$ = 1.

Thus, neutrosophic decision matrix D can be formulated as:

$$D = \left[ {\begin{array}{*{20}c} {\left\langle {t_{11 , } i_{11} , f_{11} } \right\rangle \ldots \left\langle {t_{1n , } i_{1n} , f_{1n} } \right\rangle } \\ \ldots \\ {\left\langle {t_{m1 , } i_{m1} , f_{m1} } \right\rangle \ldots \left\langle {t_{mn } , i_{mn} , f_{mn} } \right\rangle } \\ \end{array} } \right]$$

(18)

Then, by Eq. (1), the simplified neutrosophic value $a_{i}$ for $A_{i}$ is $a_{i}$  =  $\left\langle {t_{i} , i_{i} , f_{i} } \right\rangle$  =  $F_{iw } \left( {a_{i1} , a_{i2} , \ldots a_{in} } \right)$ for each row in matrix D.

To determine the best solution among all available alternatives $a_{i}$ (i $\in$ 1, 2, …, m), the similarity measure between the alternative $a_{i}$ and the ideal alternative $a^{*}$ must be computed individually using Eq. (14). The ranking order of all options can be calculated using the similarity measure between each option and the best alternative $a^{*}$. The option with the highest value among all present choices is the best option.

With the goal of verifying the efficacy of our suggested similarity measure, we give two illustrative examples of multicriteria decision-making problems from the engineering field adopted from the related literature.

Example 1 (Ye 2014)

An investment firm wishes to invest money in the best business solution. There are four investment choices: (1) A1 is a car manufacturer; (2) A2 is a food manufacturer; (3) A3 is a computer manufacturer; and (4) A4 is a weapon manufacturer. The investment firm must evaluate the following three criteria: (1) C1 is the risk assessment, (2) C2 is the growth assessment, and (3) C3 is the environmental effect assessment. Consequently, the weight vector of the criterion is W = (0.35, 0.25, 0.4), and the value of W is determined by the relevance of the three criteria. We assume that the ideal alternative is $a^{*}$ = (1,0,0).

In the above example, we have four investment plans, namely, A₁, A₂, A₃ and A_4, evaluated regarding three criteria, C₁, C₂ and C₃.

Judgement given by the expert is depicted by the following neutrosophic decision matrix D:

Acording to our definitions described in Sect. 2, we proceed with the following steps:

Step 1 By applying Eq. (1), we obtain the following results:

$$a_{{1}} = \, 0.{35 }* \, (0.{4}, \, 0.{2},0.{3}) \, + \, 0.{25 }* \, \left( {0.{4}, \, 0.{2}, \, 0.{3}} \right) \, + \, 0.{4 }* \, \left( {0.{2}, \, 0.{2}, \, 0.{5}} \right) \, = \, \left( {0.{3268}, \, 0.{2}000, \, 0.{3881}} \right)$$

In the same way we get the following:

$$a_{{2}} = \, \left( {0.{5627}, \, 0.{1414}, \, 0.{2}000} \right)$$

$$a_{{3}} = \, \left( {0.{4375}, \, 0.{2416}, \, 0.{2616}} \right)$$

$$a_{{4}} = \, \left( {0.{5746}, \, 0.{1555}, \, 0.{1663}} \right)$$

Step 2 The correlation measure based on generalized information quality between each alternative ${\alpha }_{i}$ and the ideal alternative ${a}^{*}$ is determined according to calculations referred to Sect. 2.3.

First, we should define our frame of discernment. According to our analysis in Sect. 3, this is.

$X$ = $\left\{ {T_{x , } F_{x} } \right\}$, which represents support for both T_A (x) and F_A (x). Furthermore, we have $m_{1}$ ($T_{x}$) = 0.3268, $m_{1}$ ($T_{x } , F_{x}$) = 0.2000 and $m_{1}$ ($F_{x}$) = 0.3881.

Then, a Jaccard matrix D is proposed, which provides the relationship between the targets, and the entries of the matrix are given below:

Then, we can represent BPAs as vectors:

$$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{{m_{1} }} = \, \left[ {0.{3268}, \, 0.{2}, \, 0.{3881}} \right]^{{\text{T}}} \quad {\text{and}}\quad \;\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{{m_{*} }} = \, \left[ {{1}, \, 0, \, 0} \right]^{{\text{T}}}$$

$$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{{m_{{1^{\prime}}} }} = D\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{{m_{1} }} ~ = \left[ {\begin{array}{lll} 1 \hfill & 0 \hfill & {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}} \hfill \\ 0 \hfill & 1 \hfill & {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}} \hfill \\ {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}} \hfill & {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}} \hfill & 1 \hfill \\ \end{array} } \right]*~\left[ {0.3268, 0.2,0.3881} \right] = ~~\left( {\begin{array}{*{20}c} {0.5208} \\ {0.3940} \\ {0.6515} \\ \end{array} } \right)~\quad {\text{and}}\quad ~~\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{{m_{{*^{\prime}}} }} = D\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{{m_{*} }} ~ = ~\left( {\begin{array}{*{20}l} 1 \hfill \\ 0 \hfill \\ {0.5} \hfill \\ \end{array} } \right)$$

We proceed by calculating $IQ$($m_{{1^{\prime}}}$) and $IQ$($m_{{*^{\prime}}}$) by applying Eq. (13):

$$IQ (m_{{1^{\prime}}} ) = \left( {0.{52}0{8}} \right)^{{2}} + \, \left( {0.{394}0} \right)^{{2}} + \frac{{\left( {0.6515} \right)^{2} }}{{2^{2} - 1}} = 0.{271 } + \, 0.{155 } + \, 0.{141 } = \, 0.{567}$$

$$IQ (m_{{*^{\prime}}} ) = \, \left( {1} \right)^{{2}} + \, \left( 0 \right)^{{2}} + \frac{{\left( {0.5} \right)^{2} }}{{2^{2} - 1}} = {1 } + \, 0 \, + \, 0.0{83 } = { 1}.0{83}$$

$$\frac{{m_{{1^{\prime}}} \left( A \right)}}{{2^{\left| A \right|} - 1}} *\frac{{m_{{*^{\prime}}} \left( B \right)}}{{2^{\left| A \right|} - 1}} = \left( {\begin{array}{*{20}c} {0.5208} \\ {0.3940} \\ {0.6515} \\ \end{array} } \right)*\left( {\begin{array}{*{20}c} 1 \\ 0 \\ {0.5} \\ \end{array} } \right) = 0.{52}0{8 }*{ 1 } + \, 0.{394}0*0 \, + \frac{0.6515*0.5}{{2^{2} - 1}} = \, 0.{6288}$$

$${\varvec{{\mathcal{C}\mathcal{I}\mathcal{Q}}\left( {m_{{1^{\prime}}} , \, m_{{*^{\prime}}} } \right)}} \, = \mathop \sum \limits_{A, B \subseteq X} \frac{{\frac{{m_{{1^{\prime}}} \left( A \right)}}{{2^{\left| A \right|} - 1}} \times \frac{{m_{{*^{\prime}}} \left( B \right)}}{{2^{\left| B \right|} - 1}} }}{{\sqrt {IQ \left( {m_{{1^{\prime}}} } \right)} \times \sqrt {IQ \left( {m_{{*^{\prime}}} } \right)} }} = 0.6288/ \sqrt {0.567*1.083} = 0.6288/0.7836 \, = {\bf{0.802}}$$

Following the exact calculations as above, we obtain the following results:

$${\varvec{{\mathcal{C}\mathcal{I}\mathcal{Q}} \left( {m_{{2^{\prime}}} , \, m_{{*^{\prime}}} } \right) \, = \, 0.936}}$$

$${\varvec{{\mathcal{C}\mathcal{I}\mathcal{Q}} \left( {m_{{3^{\prime}}} , \, m_{{*^{\prime}}} } \right) \, = \, 0.842}}$$

$${\varvec{{\mathcal{C}\mathcal{I}\mathcal{Q}} \left( {m_{{4^{\prime}}} , \, m_{{*^{\prime}}} } \right) \, = \, 0.938}}$$

Step 3 Based on the results of the previous step, the ranking of alternative investment plans is given as

$${\varvec{A4 > A2 > A3 > A1}}$$

Given the above ranking, we can conclude that solution plan ${\varvec{A}}4$ is closest to the ideal alternative.

The aforementioned ranking is compatible with the respective results found in (Ye 2014) and (Jiang and Shou 2017) for the same case study. This example demonstrates that our method is generally operational and accurate.

Example 2 (Singh and Sekhon 1996)

(Metal stamping layout selection problem) The authors offer a strategy for picking strips using the digraph and matrix approach in this illustrative case study. They take into account the annual manufacturing of 400,000 blanks, as seen in Fig. 1. Additionally, Fig. 2 synthesizes six different strip layouts. Five relevant strip layout features are identified: cost-effective material utilization, die cost, stamp operating cost, needed production rate, and job accuracy. We assume that the ideal alternative a* = (1,0,0).

Fig. 1

Blank profile [adapted from (Elshabsery and Fattouh 2021)]

Fig. 2

Six alternative strip layouts [adopted from (Elshabsery and Fattouh 2021)]

Table 1 presents the alternative strip-layout data.

Table 1 Alternative strip-layout data [adapted from (Elshabsery and Fattouh 2021)]

The crisp decision matrix Table 1 needs to be converted into SVNS numbers by applying a vector normalization approach utilizing Eq. (18) (Elshabsery and Fattouh 2021).

$$NX_{ij = } \frac{{x_{ij} }}{{\sqrt {\mathop \sum \nolimits_{i = 1}^{m} \left( {x_{ij} } \right)^{2} } }}$$

(18)

After normalization of the above table, the matrix ($NX_{ij}$) is converted into the SVNS decision matrix. To perform this action, the relationships between normalized values of the criteria $NX_{ij }$ and SVNS number can be utilized. This is achieved by following Eq. (19) (Elshabsery and Fattouh 2021).

$${\text{SVNS}}\;number_{ij} = (T_{ij} ,I_{ij} ,F_{ij} ) = (NX_{ij} ,1 - NX_{ij} ,1 - NX_{ij} )$$

(19)

Since the attributes are of the cost and benefit types, then one can transform the SVNS decision matrix into the normalized SVNS decision matrix by transforming the cost attributes into the benefit attributes using the complement set. The complement of a neutrosophic set is denoted by and is defined by:

$${\text{X }} = \, (F_{ij} ,{ 1} - I_{ij} ,T_{ij} )$$

(20)

The normalized SVNS decision matrix is shown in Table 2.

Table 2 Normalized SVNS Decision Matrix

Then, by applying the mean arithmetic average operator given in Eq. (2), we take the following results for the alternative layouts in a simplified neutrosophic form:

$${\text{A}}_{{1}} = \, \left( {0.{49182}, \, 0.{49818}, \, 0.{5}0{818}} \right)$$

$$A_{{2}} = \, \left( {0.{51}0{86}, \, 0.{48914}, \, 0.{48914}} \right)$$

$$A_{{3}} = \, \left( {0.{53678}, \, 0.{46322}, \, 0.{45322}} \right)$$

$$A_{{4}} = \, \left( {0.{4489}0, \, 0.{5711}0, \, 0.{5511}0} \right)$$

$$A_{5} = \, \left( {0.{42792}, \, 0.{592}0{8}, \, 0.{572}0{8}} \right)$$

$$A_{6} = \, \left( {0.45288, \, 0.55712, \, 0.54712} \right)$$

Following steps 1–3 described in Sect. 3 and performing all necessary calculations, the following ranking of alternative layouts is given:

$${\varvec{A3 \, > \, A2 \, > \, A1 \, > \, A6 \, > \, A4 \, > \, A5}}$$

The above ranking emerged from the results of our proposed similarity measure:

$${\varvec{{\mathcal{C}\mathcal{I}\mathcal{Q}}}}{\varvec{\left( {A1, \, A^{*} } \right)}} \, = \, {\bf{0.73}}$$

$${\varvec{{\mathcal{C}\mathcal{I}\mathcal{Q}}}}{\varvec{\left( {A2, \, A^{*} } \right) \, = \, 0.74}}$$

$${\varvec{{\mathcal{C}\mathcal{I}\mathcal{Q}}}}{\varvec{\left( {A3, \, A^{*} } \right) \, = \, 0.76}}$$

$${\varvec{{\mathcal{C}\mathcal{I}\mathcal{Q}}}}{\varvec{\left( {A4, \, A^{*} } \right) \, = \, 0.68}}$$

$${\varvec{{\mathcal{C}\mathcal{I}\mathcal{Q}}}}{\varvec{\left( {A5, \, A^{*} } \right) \, = \, 0.66}}$$

$${\varvec{{\mathcal{C}\mathcal{I}\mathcal{Q}}}}{\varvec{\left( {A6, \, A^{*} } \right) \, = \, 0.69}}$$

It can be concluded from the above two examples that the implementation of the new method of multicriteria decision-producing problems is a logical and efficient solution according to other proposed methods found in the related literature (Elshabsery and Fattouh 2021). Further analysis of the validity and robustness of our proposed method regarding the results attained from Example 2 is given in detail in the next section.

5 Comparative analysis

The current section conducts a comparison analysis of the outcomes obtained in this study by employing several SVNS-based similarity metrics to the problem (Example 2) studied in the previous section. The findings of our study are compared to previously published results to confirm their correctness based on the same illustrative case, and finally, comments on the advantages and disadvantages indicated in the different methods are provided.

Table 3 illustrates the results obtained from several SVNS-information-based similarity measures found in (Elshabsery and Fattouh 2021).

Table 3 Rankings obtained using existing methods (Elshabsery and Fattouh 2021)

From the results obtained from Table 3, it can be easily observed that all methods, including our method, also suggest alternative A3 as the ideal solution and alternative A5 as the poorest alternative to the problem. Additionally, the remaining results in the final ranking are similarly relative.

The outcomes reveal that the two top-ranked options, A3 and A2, remain identical in almost all of the techniques used, with the exception of the WCC and Pw approaches, where alternative A1 is preferred over alternative A2. Since the vital goal of the decision-making framework is to choose the best option, changes that occur outside of the ideal solution could be considered insignificant.

By utilizing a new correlation coefficient under generalized information quality, the method reduces the impact of conflicting evidence. For example, the higher similarity score between alternatives A2 and the ideal alternative demonstrates the framework's ability to rank accurately.Moreover, unlike aggregation-based approaches (e.g., WAAO), which may hide key information, the suggested method preserves the original decision information, as evidenced by the usage of normalized SVNS matrices. This guarantees that decision-makers have more granular insights.

Some final remarks could be mentioned given the above results:

• (1) Consistent with the experimental results, the proposed method succeeds in producing the expected outcomes compared to the existing multicriteria decision-making methods for SVNs. As a result, it is inferred that the findings of this research have high application potential and can be used to solve multicriteria decision problems.

• (2) Some methods, such as CEM, WCC, WCS and WAAO, are needed to perform aggregation on the input of SVNS arguments, which could raise computational challenges and thus contribute to information loss. In contrast, other methods do not need to calculate such an aggregation and instead deal directly with the input of SVNS arguments, preserving as much of the original decision information as feasible. Despite the fact that we conduct such an operation, in step 2, we first modify all of the BPAs according to the D matrix. The size of the matrix D accounts for the true situation of each BPA, and its order is equal to the cardinality of the union of all focus elements in the system with a trust quality larger than zero. As a result, this strategy significantly minimizes the amount of computation needed (Li and Deng 2019).

• (3) Our hybrid method employs D–S evidence theory and general information quality to derive quality mixed values from different probabilistic distribution sources, where quality is related to the combined value's lack of uncertainty. This is achieved by introducing factor D, which effectively reflects the degree of relevance of BPA. As a result, our method more closely captures the BPA correlation coefficient.

5.1 Comparison of methodological features

The suggested framework, which synergistically blends generalized information quality with neutrosophic evidence theory, excels in several key areas:

Complexity: The proposed approach has a higher level of computational complexity than established methods such as WCC or WCS. This complexity is due to the use of neutrosophic logic, which naturally handles indeterminacy and partial truth values more effectively. However the utilization of neutrosophic logic allows the framework to process and integrate indeterminate information effectively. Traditional methods, which rely on binary or probabilistic logic, may oversimplify or misrepresent such data, leading to less reliable outcomes. In summary, the proposed method is the most complex due to its use of advanced theoretical frameworks, while CEM, WCC, WCS, and WAAO are simpler but less capable of handling complexity.

Flexibility: In comparison to more rigid techniques, the hybrid framework is more adaptable to a wider range of decision-making scenarios. Its ability to incorporate and accommodate varying levels of ambiguity and vagueness by integrating generalized information quality measures allows for more advanced analysis that can accommodate varying levels of ambiguity and vagueness. Overall, our proposed method and BProjw/Pw are more flexible, especially in handling complex decision scenarios. For example, in real-world engineering scenarios involving incomplete datasets, such as supply chain optimization with uncertain demand estimates, the suggested method's flexibility allows for nuanced analysis, as opposed to approaches like CEM, which rely on deterministic values.

Robustness: The incorporation of neutrosophic evidence theory enhances the proposed framework in situations with insufficient or conflicting information. This flexibility stands in contrast to the more deterministic nature of the compared approaches, which may not be as effective when faced with significant uncertainty. As demonstrated in Sect. 4, Example 2, the suggested approach consistently identifies optimal solutions despite considerable differences in input data. This emphasizes its reliability compared to methods that need considerable pre-aggregation or simplifications.

Applicability: The suggested method has the broadest application, especially in complex, multi-criteria decision-making situations. It is particularly useful in scenarios with high uncertainty, conflicting information, or when standard methodologies may oversimplify the decision-making process. Its versatility and adaptability make it an effective tool for various purposes, although it may be excessive for smaller tasks. This capability constitutes the proposed method particularly advantageous for industries such as healthcare, finance, and logistics, where decisions often involve high stakes and incomplete data.

Table 4 summarizes the methodological features of the proposed framework compared to existing methods:

Table 4 Comparison of the methodological features of proposed framework with existing SVNS-based similarity methods

A thorough review of the proposed methodology reveals several benefits:

Enhanced Decision-Making Capability: The framework's ability to integrate and interpret ambiguous and contradictory data provides a significant advantage over traditional techniques, particularly in complex decision-making scenarios. Ranking stability enables constant decision-making, which reduces the possibility of errors when inputs change. This feature is especially important for dynamic contexts, where judgements must adapt to changing conditions.

Nuanced Uncertainty Handling: By utilizing neutrosophic evidence theory, the technique adopts a more comprehensive approach to uncertainty, potentially resulting in more reliable and consistent decision outcomes. For example, the higher similarity measure between alternatives A2 and the ideal alternative reflects the framework's ability to maintain precision in ranking.

On the other hand, we could claim the following issues that need further consideration regarding our method:

Processing Demand: The increasing complexity of the method requires more processing resources, which may limit its use, especially in real-time decision-making scenarios or when working with large datasets.

Specialized Knowledge Requirement: Properly implementing the suggested technique necessitates a deep understanding of both general information theory and neutrosophic logic, which could restrict its practical application.

The methodological differences highlighted above play a crucial role in shaping the results, particularly in terms of ranking stability and decision reliability. For instance, the proposed method's ability to accommodate and process indeterminate information contributes to the generation of rankings that are more resilient to variations in input data, as demonstrated in Sect. 4.

6 Conclusions

It is well recognized that a certain theory of uncertainty is fully functional if it provides proper solutions to the following matters in question (Klir 2006): (i) it is necessary for the theory to be formalized by adhering to certain properties or axioms; (ii) a suitable calculus for the theory must be developed in such a way that the latter is properly operated; (iii) the amount of uncertainty should be measured justifiably within the theory; and (iv) methodological aspects should be established to address the various problems associated within the theory.

In response to this, we have developed a solid and sound conceptual framework that addresses the above issues by (i) utilizing belief measure as a proper fuzzy measure that satisfies axioms of (1) boundary conditions, (2) monotonicity and (3) continuity; (ii) calculus of neutrosophic theory is used dealing with concepts of uncertainty such as vagueness, imprecision, ambiguity and inconsistency; (3) in this subcategory, we adopt the theory of generalized information quality, which enables us to measure the information quality between BPAs effectively; and (4) neutrosophic sets and logic combined with evidence theory provide us with a powerful mathematical toolbox for dealing with problems that are characterized by uncertainty such as the examples illustrated in Sect. 4.

This research paper introduces a novel hybrid approach for calculating the similarity of SVNSs using D–S evidence theory, which is used in a multicriteria decision-making context. First, the background of neutrosophic sets is presented, followed by the single-valued neutrosophic set (SVNS) and D–S evidence theory. A key factor in our methodology for the similarity measure is the conversion of SVNSs to BPAs and then the utilization of a new correlation coefficient based on generalized information quality. From the numerical examples given, we can conclude that our approach can measure similarity reasonably and effectively. The application and ability of this method has been demonstrated in our comparison analysis, and it is clearly observed that the suggested methodology can generate expected results compared to the existing measurements of similarity known for simplified neutrosophic sets. From the aforesaid, we believe that our new suggested method could exhibit considerable application potential in tackling multicriteria decision-making problems.

At this point, it is important to note the cognitive demand of providing input by decision-makers, as it is crucial and affects the overall effectiveness, usability, and adoption of the decision-making method.To better understand the cognitive demands of our approach, we conducted a comparative analysis in Sect. 5 where we compared our method to other existing methodologies such as CEM, WCC, and BProj_w. Our findings suggest that while our method excels at handling contradictory information, it also imposes a greater cognitive load on decision-makers, especially during the initial stages of processing incoming data. On the other hand, approaches like WCS, which require less cognitive effort due to their straightforward aggregation process, may be more suitable in situations where decision-makers have limited knowledge or when the decision-making environment is less complex.

While the proposed decision-making framework may require more cognitive effort due to the complexity of neutrosophic logic and evidence theory, the benefits in terms of decision accuracy and resilience are significant. Future research could focus on developing decision-support technologies that streamline the input process for decision-makers, reducing cognitive strain while maintaining the quality of decision outputs.

Because our study is limited to measuring the similarity between SVNSs, as a future research plan, we could apply our method to interval-valued neutrosophic sets and further apply it to group decision problems.

Another possible and innovative future research work could include the implementation of a new method to relate SVNSs and BPAs based on the core concepts of Belief and Plausibility measures found in D–S evidence theory. This could be accomplished by obtaining a belief interval of the form [Bel {a_i}, Pls {a_i}] of a decision alternative a_i (i = 1, 2, …, N) of an MCDM problem with incomplete information. This interval is often regarded as the level of probability ignorance, i.e., the range within which the "true" probability should lie if we possess a comprehensive probabilistic model (Pearl 1988).