A Novel Decision-Making Algorithm Based on Interval-Valued Bipolar Neutrosophic Soft Sets

Abstract

Modeling uncertainty and imprecision in real-world decision-making problems often requires structures that can simultaneously capture interval-valued, bipolar, and parametric information. In this paper, we introduce the concept of an interval-valued bipolar neutrosophic soft set (IVBNSS) as a hybrid structure that combines the expressive power of interval-valued bipolar neutrosophic sets with the parametric flexibility of soft sets. Inspired by graph energy and the nuclear norm, we introduce the notion of the energy of an IVBNSS, establish its fundamental boundedness properties, and develop a novel decision-making algorithm based on this concept. The proposed algorithm is illustrated through two case studies: a sensor selection problem in industrial automation, where the goal is to identify the optimal sensor based on multiple performance criteria, including response time, environmental resistance, accuracy, and sensitivity; and a benchmark engineering problem from the literature concerning the selection of bio-inspired thin-wall structures. A comparative analysis with existing decision-making algorithms in the interval-valued bipolar neutrosophic setting confirms the validity and advantages of the proposed approach.

1. Introduction

At the end of the last century, the rapid development of artificial intelligence and intelligent decision-support systems created a growing demand for mathematical frameworks capable of representing imprecise, uncertain, and incomplete information. In response to these challenges, Zadeh introduced the theory of fuzzy sets in [1], laying the foundation for numerous extensions and hybrid models designed to handle different forms of uncertainty. As the complexity of real-world data increased, Molodtsov proposed the theory of soft sets in [2], introducing a parameterized framework that significantly enriched the representation of information and became an important basis for the development of hybrid uncertainty models. The studies in [3,4,5] further developed the theory by defining fundamental operations on soft sets, while Dalkılıç [6] and Alcantud et al. [7] contributed to the advancement of decision-making algorithms and provided comprehensive surveys of the field.

Although fuzzy sets have found numerous applications, their descriptive capability is limited by the use of a single membership function $\mu$, which cannot explicitly represent hesitation or incomplete information. To overcome this limitation, Atanassov introduced intuitionistic fuzzy sets in [8], where each element is characterized by both membership and non-membership degrees. Their hybridization with soft sets resulted in structures such as fuzzy soft sets (FSSs) and intuitionistic fuzzy soft sets (IFSSs), introduced by Maji et al. in [9,10]. These models have been successfully applied in medical reasoning [11,12] and decision-making [13], while further developments include aggregation operators proposed by Garg and Arora [14] and distance-based algorithms introduced by Selvachandran et al. [15]. Petry [16] further investigated interval-valued intuitionistic fuzzy structures and aggregation mechanisms for uncertain spatial and temporal data, demonstrating their suitability for handling interval uncertainty. Nevertheless, despite these developments, intuitionistic fuzzy models remain unable to explicitly represent indeterminacy as an independent component, which has motivated the development of more expressive uncertainty models.

Neutrosophic sets were introduced by Smarandache [17] to overcome one of the principal limitations of intuitionistic fuzzy sets, namely, the inability to represent indeterminacy as an independent component. By characterizing each element through the degrees of truth, indeterminacy, and falsity, neutrosophic sets provide a more expressive framework for modeling incomplete, inconsistent, and uncertain information. Owing to this flexibility, they have found numerous applications in similarity analysis [18], medical diagnosis [19,20], and multi-criteria decision-making.

To further improve their applicability, several extensions of neutrosophic sets have been proposed. Interval-valued neutrosophic sets [21,22] allow experts to express evaluations by intervals instead of precise values, making them more suitable for situations involving incomplete or imprecise information. Single-valued neutrosophic sets [23] simplify practical computations, while various similarity measures, including cosine similarity [24], have been developed to support decision-making tasks.

The integration of neutrosophic sets with soft sets resulted in neutrosophic soft sets (NSSs), introduced by Maji [25]. By incorporating parameters into the neutrosophic framework, NSSs significantly enhanced their applicability in multi-attribute decision-making and have since been successfully applied in medicine [26] and numerous decision-making models [27,28,29,30,31]. Subsequent studies extended the theoretical foundations through neutrosophic soft relations [32,33], neutrosophic soft matrices [34,35], and hybrid models such as neutrosophic soft rough sets [36].

A particularly important development was the introduction of interval-valued neutrosophic soft sets (IVNSSs) by Deli [37], which combine the advantages of interval-valued evaluations with the parameterized structure of soft sets. Their effectiveness has recently been demonstrated in complex engineering decision-making problems—for example, through parameter reduction techniques for bio-inspired structural design [38]. In parallel, the growing practical importance of interval-valued neutrosophic models has stimulated the development of computational tools such as the Python 3.12.4 framework proposed by Prakash et al. [39], facilitating their implementation in practical applications.

While computational tools for interval-valued bipolar neutrosophic sets (IVBNSs) are being actively developed, an important challenge remains their application to multi-attribute problems, where alternatives must be evaluated with respect to multiple independent criteria. In such situations, the absence of a parametric structure makes IVBNSs less suitable for representing attribute-oriented information. A natural way to overcome this limitation is to integrate the expressive power of interval-valued bipolar neutrosophic sets with the parameterization provided by soft set theory.

Bipolar soft sets, introduced by Shabir and Naz in [40], extend classical soft sets by incorporating a positive and a corresponding negative (NOT) attribute for each parameter, allowing simultaneous modeling of favorable and unfavorable information. Owing to this capability, bipolar soft sets have attracted considerable attention in decision-making problems [41,42,43]. Their further development has resulted in bipolar fuzzy soft sets [44] and bipolar neutrosophic soft sets [45], which provide increasingly expressive models for representing uncertain and bipolar information. These structures have been successfully applied to decision-making [45,46] as well as pattern recognition and classification problems [47].

To capture both bipolarity and interval-valued uncertainty, Deli et al. [48] introduced interval-valued bipolar neutrosophic sets (IVBNSs), while Abdel-Basset et al. [49] demonstrated their applicability to medical diagnosis. Nevertheless, despite their rich descriptive capability, IVBNSs are defined directly on the universe of alternatives and do not include a parametric component. Consequently, they cannot naturally represent situations in which each alternative must simultaneously be evaluated with respect to multiple independent attributes. This limitation becomes particularly important in multi-attribute decision-making, where parameterized information plays a central role.

Motivated by this observation, we introduce the interval-valued bipolar neutrosophic soft set (IVBNSS) as a hybrid structure that combines the interval-valued representation of uncertainty, bipolar evaluation, and the parameterized framework of soft sets. The proposed model preserves the descriptive advantages of IVBNSs while extending them with an explicit attribute-based representation, making it more suitable for complex decision-making problems involving multiple criteria.

Although several hybrid models, such as hesitant fuzzy soft sets and interval-valued hesitant fuzzy soft sets, have demonstrated their effectiveness in decision-making problems, they primarily focus on modeling hesitation in membership assessments. In contrast, many practical applications require the simultaneous representation of interval uncertainty, bipolar information, and multiple evaluation attributes. While interval-valued bipolar neutrosophic sets successfully capture interval-valued and bipolar evaluations, they do not provide a parameterized representation of attributes. Conversely, soft sets naturally organize information according to parameters but do not incorporate the expressive interval-valued bipolar neutrosophic framework. The proposed IVBNSS integrates these complementary features into a unified model, providing a richer and more flexible representation of uncertain information for complex multi-attribute decision-making problems.

Graph energy, introduced by Gutman [50], is a well-established numerical characteristic that has found numerous applications in chemistry, network analysis, and computer science. Its computation is based on the spectral properties of a matrix representation of a graph, using either eigenvalues or singular values, depending on the particular energy definition (see [51]). In particular, the sum of singular values, known as the nuclear norm, has proven to be an effective global descriptor of matrix structure and has been widely used in optimization and low-rank approximation problems [52,53].

The idea of employing energy as a global numerical characteristic has recently been extended beyond graph theory. In the fuzzy soft environment, Mudrić Staniskovski [54] introduced the concept of fuzzy soft set energy, while Alcantud et al. [55] proposed the energy of hesitant fuzzy soft sets as a tool for decision-making. Similarly, Mohamad et al. [56] defined energy, Laplacian energy, and signless Laplacian energy for bipolar single-valued neutrosophic graphs and demonstrated their applicability to renewable energy source selection.

These developments indicate that energy-based characteristics provide an effective way of summarizing the overall structure of complex mathematical objects into a single quantitative measure that can support ranking and decision-making. Motivated by this observation, and by our previous study [57], we extend this idea to interval-valued bipolar neutrosophic soft sets. Since an IVBNSS is naturally represented by a collection of matrices, singular values provide an appropriate mathematical tool for constructing a global numerical characteristic that simultaneously reflects interval-valued, bipolar, and parametric information. Based on this idea, we introduce the notion of IVBNSS energy and develop a corresponding decision-making algorithm.

The main contributions of this paper are fourfold: First, we introduce the concept of the interval-valued bipolar neutrosophic soft set (IVBNSS), a hybrid model that combines interval-valued bipolar neutrosophic representations with the parameterized framework of soft sets, providing a richer representation for complex multi-attribute information. Second, we define optimistic, pessimistic, and overall energy measures for IVBNSSs based on the singular values of appropriately constructed matrices, thereby introducing a novel numerical characteristic for these structures. Third, we develop a decision-making algorithm that exploits the proposed energy measure to rank alternatives in multi-attribute decision-making problems. Finally, we demonstrate the applicability and effectiveness of the proposed approach through practical case studies and comparative analysis with existing interval-valued bipolar neutrosophic decision-making methods.

The remainder of this paper is organized as follows: Section 2 reviews the fundamental concepts required for the proposed hybridization. Section 3 introduces interval-valued bipolar neutrosophic soft sets and establishes their basic properties. In Section 4, we define the optimistic, pessimistic, and overall energy of IVBNSSs. The proposed decision-making algorithm is presented in Section 5, while its applications are illustrated in Section 6. Section 7 provides a comparative analysis of the proposed algorithm with existing interval-valued bipolar neutrosophic decision-making approaches. Finally, Section 8 concludes the paper and discusses directions for future research.

2. Preliminaries

This section presents the definitions and properties that form the basis of the proposed hybrid structure.

2.1. Soft Sets

Unlike classical sets, such as fuzzy sets that rely on a membership function, the theory of soft sets does not impose conditions based on a membership function. Instead, it is based on a parametric approach, where each parameter is associated with a set of elements to which that parameter is assigned meaning. This property enables the modeling of data that is not sufficiently precise and does not require strict membership criteria. This structure is highly adaptable to real-world problems. For the purpose of defining a soft set, let U denote the universal set, and let E be the set of parameters, with $A\subseteq E$.

Definition 1 

([2]). A soft set $F_A$ on the universe U is defined by the set of ordered pairs

$$F_A=\left\{(x,f_A\left(x\right)):x\in E,f_A\left(x\right)\in \mathcal{P}\left(U\right)\right\},$$

where $f_A:E\to \mathcal{P}\left(U\right)$, such that $f_A\left(x\right)=\varnothing$ if $x\notin A$.

The function $f_A$ is called the approximation function of the soft set $F_A$. The set of all soft sets over U is denoted by $S\left(U\right)$.

2.2. Interval-Valued Bipolar Neutrosophic Sets

One of the main components of the desired hybridization is the interval-valued bipolar neutrosophic set. To better describe this structure, let us recall the definition of the interval-valued neutrosophic set.

Definition 2 

([21]). Let U be a universal set of elements referred to as arguments. An interval-valued neutrosophic set (IVNS), denoted by ${\mathbb{N}}_{iv}$, is defined as an object of the form

$$\begin{array}{c}\hfill {\mathbb{N}}_{iv}=\{⟨[T_L\left(u\right),T_R\left(u\right)],[I_L\left(u\right),I_R\left(u\right)],[F_L\left(u\right),F_R\left(u\right)]⟩\mid u\in U\},\end{array}$$

where $T_L,T_R,I_L,I_R,F_L,F_R:U\to [0,1]$.

The quantities $T_L\left(u\right),T_R\left(u\right),I_L\left(u\right),I_R\left(u\right),F_L\left(u\right),F_R\left(u\right)$ denote, respectively, the lower and upper bounds of the true membership degree, the lower and upper bounds of the indeterminate membership degree, and the lower and upper bounds of the false membership degree for an element $u\in U$. For convenience, the following shorter notation is often used: $T\left(u\right)=[T_L\left(u\right),T_R\left(u\right)]$, $I\left(u\right)=[I_L\left(u\right),I_R\left(u\right)]$, $F\left(u\right)=[F_L\left(u\right),F_R\left(u\right)]$, where $u\in U$. Consequently, the ordered triple of intervals $⟨T\left(u\right),I\left(u\right),F\left(u\right)⟩$ is called the interval-valued neutrosophic number for each $u\in U$, and the set of all interval-valued neutrosophic numbers over U is denoted by $IVNS\left(U\right)$.

Example 1. 

Suppose that a municipal authority needs to determine the most suitable use of a vacant urban plot. Let $U=\{u_1,u_2,u_3,u_4\}$ be a universal set of alternatives for the development of this plot, where $u_1$ represents building a bookshop, $u_2$ a chemical plant, $u_3$ a supermarket, and $u_4$ a food company. Based on an expert assessment of how suitable each option is for the given location, we define the following interval-valued neutrosophic set over U:

$$\begin{array}{cc}\hfill {\mathbb{N}}_{iv}=& \{⟨u_1,[0.4,0.5],[0.2,0.3],[0.3,0.4]⟩,\hfill \\ & ⟨u_2,[0.6,0.7],[0.1,0.2],[0.2,0.3]⟩,\hfill \\ & ⟨u_3,[0.3,0.6],[0.2,0.3],[0.3,0.4]⟩,\hfill \\ & ⟨u_4,[0.7,0.8],[0.0,0.1],[0.1,0.2]⟩\}.\hfill \end{array}$$

Here, for each alternative $u_i\in U$, the first interval denotes the truth-membership degree, i.e., the degree to which the alternative is suitable for the plot; the second interval denotes the indeterminacy-membership degree, reflecting the uncertainty of the assessment; and the third interval denotes the falsity-membership degree, i.e., the degree to which the alternative is unsuitable.

On the other hand, the bipolar aspect allows simultaneous modeling of both positive and negative information, which significantly contributes to solving more complex problems.

Definition 3 

([48]). Let U be a universal set of elements referred to as arguments. An interval-valued bipolar neutrosophic set (IVBNS), denoted by ${\mathbb{B}}_{ivn}$, is defined as an object of the form

$$\begin{array}{cc}\hfill {\mathbb{B}}_{ivn}=& \{⟨[T_L^{+}\left(u\right),T_R^{+}\left(u\right)],[I_L^{+}\left(u\right),I_R^{+}\left(u\right)],[F_L^{+}\left(u\right),F_R^{+}\left(u\right)],\hfill \\ & [T_L^{-}\left(u\right),T_R^{-}\left(u\right)],[I_L^{-}\left(u\right),I_R^{-}\left(u\right)],[F_L^{-}\left(u\right),F_R^{-}\left(u\right)]⟩\mid u\in U\},\hfill \end{array}$$

where $T_L^{+},T_R^{+},I_L^{+},I_R^{+},F_L^{+},F_R^{+}:U\to [0,1]$ and $T_L^{-},T_R^{-},I_L^{-},I_R^{-},F_L^{-},F_R^{-}:U\to [-1,0]$.

Thus, the bipolar structure enables the simultaneous analysis of contrasting aspects of the elements, thereby supporting more accurate decision-making in complex systems. The set of all interval-valued bipolar neutrosophic sets over U is denoted by $IVBNS\left(U\right)$.

Example 2. 

Consider again the problem of selecting the most suitable use of the vacant plot from Example 1. In practice, each development option, in addition to its suitability, also carries certain adverse side effects, such as pollution, noise, or increased traffic. To capture both aspects simultaneously, we define the following interval-valued bipolar neutrosophic set over $U=\{u_1,u_2,u_3,u_4\}$:

$$\begin{array}{cc}\hfill {\mathbb{B}}_{ivn}& =\{⟨u_1,[0.4,0.5],[0.2,0.3],[0.3,0.4],[-0.6,-0.4],[-0.3,-0.2],[-0.5,-0.3]⟩,\hfill \\ & ⟨u_2,[0.6,0.7],[0.1,0.2],[0.2,0.3],[-0.7,-0.5],[-0.2,-0.1],[-0.4,-0.2]⟩,\hfill \\ & ⟨u_3,[0.3,0.6],[0.2,0.3],[0.3,0.4],[-0.5,-0.3],[-0.4,-0.2],[-0.6,-0.4]⟩,\hfill \\ & ⟨u_4,[0.7,0.8],[0.0,0.1],[0.1,0.2],[-0.8,-0.6],[-0.2,-0.0],[-0.3,-0.1]⟩\}.\hfill \end{array}$$

Here, for each alternative $u_i\in U$, the first three intervals denote the positive truth-membership, indeterminacy-membership, and falsity-membership degrees, describing the suitability of the alternative for the plot. The last three intervals denote the corresponding negative degrees, describing the extent to which the alternative exhibits the adverse side effects mentioned above.

3. Interval-Valued Bipolar Neutrosophic Soft Sets

In many multi-attribute decision-making problems, it is necessary to represent several different forms of uncertainty simultaneously. Interval-valued bipolar neutrosophic sets (IVBNSs) provide a rich mathematical framework by combining interval-valued assessments with bipolar truth, indeterminacy, and falsity memberships. However, they are defined directly on the universe of alternatives and do not explicitly incorporate a parametric structure. Consequently, they are less suitable for problems in which evaluations must be organized with respect to multiple independent attributes.

On the other hand, soft sets naturally provide a parameterized representation of information, making them attractive for multi-criteria decision-making. Nevertheless, a soft set merely associates each parameter with a subset of the universe and, therefore, cannot adequately represent uncertainty, indeterminacy, interval-valued information, or bipolar evaluations. As a result, although soft sets offer a flexible parametric framework, their descriptive power is considerably more limited than that of interval-valued bipolar neutrosophic models.

Motivated by these complementary advantages and limitations, we introduce the concept of the interval-valued bipolar neutrosophic soft set (IVBNSS) as a hybrid structure that combines the expressive capability of interval-valued bipolar neutrosophic sets with the parameterized organization of soft sets. The proposed model preserves the rich interval-valued bipolar neutrosophic representation of uncertainty while simultaneously providing an explicit attribute-based framework for describing alternatives. This additional parametric layer makes IVBNSSs substantially more suitable for complex multi-attribute decision-making problems than interval-valued bipolar neutrosophic sets alone, while offering a significantly richer representation of uncertain information than classical soft sets.

To formalize this hybrid structure, let U be the initial universal set, let $IVBNS\left(U\right)$ denote the collection of all interval-valued bipolar neutrosophic sets defined on U, let E be the set of parameters (attributes), and let $A\subseteq E$.

Definition 4. 

The ordered pair $(\mathbb{B},A)$ is called an interval-valued bipolar neutrosophic soft set (IVBNSS) over U, where $\mathbb{B}:E\to IVBNS\left(U\right)$ is a mapping, for any $e\in E$, defined as

$$\begin{array}{c}\mathbb{B}\left(e\right)=\left\{\right(u,[T_L^{+}\left(u\right),T_R^{+}\left(u\right)],[I_L^{+}\left(u\right),I_R^{+}\left(u\right)],[F_L^{+}\left(u\right),F_R^{+}\left(u\right)],[T_L^{-}\left(u\right),T_R^{-}\left(u\right)],\\ [I_L^{-}\left(u\right),I_R^{-}\left(u\right)],[F_L^{-}\left(u\right),F_R^{-}\left(u\right)])\mid u\in U\}\in IVBNS\left(U\right),\end{array}$$

where $T_L^{+},T_R^{+},I_L^{+},I_R^{+},F_L^{+},F_R^{+}:U\to [0,1]$ and $T_L^{-},T_R^{-},I_L^{-},I_R^{-},F_L^{-},F_R^{-}:U\to [-1,0]$.

Instead of the notation $(\mathbb{B},A)$, we will also use the notation ${\mathbb{B}}_A$, and the intervals $[T_L^{+}\left(u\right),T_R^{+}\left(u\right)]$, $[I_L^{+}\left(u\right),I_R^{+}\left(u\right)]$, $[F_L^{+}\left(u\right),F_R^{+}\left(u\right)]$, $[T_L^{-}\left(u\right),T_R^{-}\left(u\right)]$, $[I_L^{-}\left(u\right),I_R^{-}\left(u\right)]$, $[F_L^{-}\left(u\right),F_R^{-}\left(u\right)]$ will often be denoted respectively as $T_{{\mathbb{B}}_A}^{+}\left(u\right),I_{{\mathbb{B}}_A}^{+}\left(u\right),$$F_{{\mathbb{B}}_A}^{+}\left(u\right),T_{{\mathbb{B}}_A}^{-}\left(u\right),I_{{\mathbb{B}}_A}^{-}\left(u\right),F_{{\mathbb{B}}_A}^{-}\left(u\right)$. The function $\mathbb{B}$ will be called the interval-valued bipolar neutrosophic approximating function of the interval-valued bipolar neutrosophic soft set ${\mathbb{B}}_A$. Thus, ${\mathbb{B}}_A$ can be represented as follows:

$${\mathbb{B}}_A=\{(e,\mathbb{B}\left(e\right))\mid e\in E,\mathbb{B}\left(e\right)\in IVBNS\left(U\right)\}.$$

The collection of all interval-valued bipolar neutrosophic soft sets over the universe U will be denoted by $IVBNSS\left(U\right)$. In the following, we will use the notation ${\mathbb{B}}_A,{\mathbb{B}}_B,{\mathbb{B}}_C,\dots$ for interval-valued bipolar neutrosophic soft sets. The following example illustrates this definition:

Example 3. 

One of the problems in industrial automation is the selection of the optimal sensor for a corresponding automated system. Since response time, environmental resistance, accuracy, and sensitivity represent different performance criteria, it is necessary to make a decision on the best sensor that meets the system’s requirements based on all of these parameters. Let $U=\{u_1,u_2,u_3,u_4\}$ be a set containing four alternatives, where $u_1$, $u_2$, $u_3$, and $u_4$ represent a temperature sensor, a proximity sensor, a humidity sensor, and a pressure sensor, respectively; and let $A=\{e_1,e_2,e_3,e_4\}$ be the set of parameters (attributes), where $e_1$, $e_2$, $e_3$, and $e_4$ denote response time, environmental resistance, accuracy, and sensitivity, respectively. The evaluations of these alternatives with respect to the specified attributes form an interval-valued bipolar neutrosophic soft set ${\mathbb{B}}_A$, as presented in Table 1.

Table 1. IVBNSS ${\mathbb{B}}_A$.

${\mathbb{B}}_{\mathit{A}}$	${\mathit{e}}_1$
$u_1$	$\left(\right[0.2,0.4],[0.3,0.9],[0.2,0.7],[-0.8,-0.1],[-0.8,-0.0],[-0.8,-0.5\left]\right)$
$u_2$	$\left(\right[0.9,1.0],[0.8,1.0],[0.8,0.9],[-0.9,-0.7],[-1.0,-0.8],[-0.9,-0.8\left]\right)$
$u_3$	$\left(\right[0.4,0.6],[0.2,0.4],[0.3,0.5],[-0.3,-0.1],[-0.4,-0.3],[-0.6,-0.3\left]\right)$
$u_4$	$\left(\right[0.8,1.0],[0.8,0.9],[0.7,0.9],[-1.0,-0.1],[-0.9,-0.7],[-0.9,-0.8\left]\right)$
	${\mathit{e}}_{\mathbf{2}}$
$u_1$	$\left(\right[0.1,0.8],[0.2,0.9],[0.2,1.0],[-1.0,-0.2],[-0.9,-0.0],[-0.8,-0.1\left]\right)$
$u_2$	$\left(\right[1.0,1.0],[0.8,1.0],[1.0,1.0],[-0.9,-0.8],[-1.0,-0.7],[-0.9,-0.7\left]\right)$
$u_3$	$\left(\right[0.4,0.7],[0.3,0.6],[0.5,0.6],[-0.7,-0.4],[-0.8,-0.5],[-0.7,-0.4\left]\right)$
$u_4$	$\left(\right[0.8,1.0],[0.8,0.9],[0.7,0.9],[-1.0,-0.9],[-0.9,-0.8],[-0.8,-0.6\left]\right)$
	${\mathit{e}}_{\mathbf{3}}$
$u_1$	$\left(\right[0.2,0.4],[0.3,0.8],[0.4,0.7],[-0.8,-0.1],[-0.8,-0.0],[-0.8,-0.7\left]\right)$
$u_2$	$\left(\right[1.0,1.0],[0.8,0.9],[1.0,1.0],[-0.9,-0.8],[-1.0,-0.8],[-0.9,-0.7\left]\right)$
$u_3$	$\left(\right[0.3,0.6],[0.3,0.5],[0.6,0.9],[-0.4,-0.2],[-0.9,-0.8],[-0.5,-0.3\left]\right)$
$u_4$	$\left(\right[0.7,0.9],[0.8,0.9],[0.9,1.0],[-0.8,-0.1],[-0.6,-0.4],[-0.8,-0.7\left]\right)$
	${\mathit{e}}_{\mathbf{4}}$
$u_1$	$\left(\right[0.0,0.7],[0.2,0.9],[0.2,1.0],[-0.9,-0.1],[-1.0,-0.0],[-0.8,-0.1\left]\right)$
$u_2$	$\left(\right[1.0,1.0],[0.8,0.9],[1.0,1.0],[-0.7,-0.5],[-1.0,-0.9],[-0.9,-0.7\left]\right)$
$u_3$	$\left(\right[0.4,0.6],[0.3,0.4],[0.6,0.9],[-0.4,-0.2],[-0.9,-0.7],[-0.4,-0.3\left]\right)$
$u_4$	$\left(\right[0.7,0.9],[0.7,0.9],[0.9,1.0],[-0.8,-0.0],[-0.7,-0.5],[-0.8,-0.7\left]\right)$

Fundamental Operations on Interval-Valued Bipolar Neutrosophic Soft Sets

This subsection is devoted to the fundamental relations and operations on interval-valued bipolar neutrosophic soft sets. We introduce the notions of inclusion and equality, together with the complement, union, and intersection. We also establish their fundamental properties. Let ${\mathbb{B}}_A,{\mathbb{C}}_B\in IVBNSS\left(U\right),$ where $A,B\subseteq E$. For $e\in A$ and $u\in U$, we use the representation

$$\begin{array}{cc}\hfill \mathbb{B}\left(e\right)\left(u\right)=& ⟨[T_{\mathbb{B},L}^{+}\left(e\right)\left(u\right),T_{\mathbb{B},R}^{+}\left(e\right)\left(u\right)],[I_{\mathbb{B},L}^{+}\left(e\right)\left(u\right),I_{\mathbb{B},R}^{+}\left(e\right)\left(u\right)],[F_{\mathbb{B},L}^{+}\left(e\right)\left(u\right),F_{\mathbb{B},R}^{+}\left(e\right)\left(u\right)],\hfill \\ & [T_{\mathbb{B},L}^{-}\left(e\right)\left(u\right),T_{\mathbb{B},R}^{-}\left(e\right)\left(u\right)],[I_{\mathbb{B},L}^{-}\left(e\right)\left(u\right),I_{\mathbb{B},R}^{-}\left(e\right)\left(u\right)],[F_{\mathbb{B},L}^{-}\left(e\right)\left(u\right),F_{\mathbb{B},R}^{-}\left(e\right)\left(u\right)]⟩.\hfill \end{array}$$

An analogous notation will be used for ${\mathbb{C}}_B$ and other interval-valued bipolar neutrosophic soft sets. For two closed intervals $J_1=[a_1,b_1]$ and $J_2=[a_2,b_2]$ contained either in $[0,1]$ or in $[-1,0]$, define

$$\begin{array}{cc}\hfill J_1\vee J_2& =[max\{a_1,a_2\},max\{b_1,b_2\}],\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill J_1\wedge J_2& =[min\{a_1,a_2\},min\{b_1,b_2\}].\hfill \end{array}$$

(2)

These operations preserve the ordering of the interval endpoints and, therefore, produce well-defined closed intervals.

Definition 5. 

Let ${\mathbb{B}}_A,{\mathbb{C}}_B\in IVBNSS\left(U\right)$. We say that ${\mathbb{B}}_A$ is an interval-valued bipolar neutrosophic soft subset of ${\mathbb{C}}_B$, and write ${\mathbb{B}}_A\tilde{\subseteq }{\mathbb{C}}_B,$ if $A\subseteq B$ and, for every $e\in A$ and $u\in U$, the following component-wise inequalities hold:

$$\begin{array}{cccc}\hfill T_{\mathbb{B},L}^{+}\left(e\right)\left(u\right)& \le T_{\mathbb{C},L}^{+}\left(e\right)\left(u\right), \hfill & \hfill T_{\mathbb{B},R}^{+}\left(e\right)\left(u\right)& \le T_{\mathbb{C},R}^{+}\left(e\right)\left(u\right),\hfill \\ \hfill I_{\mathbb{B},L}^{+}\left(e\right)\left(u\right)& \ge I_{\mathbb{C},L}^{+}\left(e\right)\left(u\right), \hfill & \hfill I_{\mathbb{B},R}^{+}\left(e\right)\left(u\right)& \ge I_{\mathbb{C},R}^{+}\left(e\right)\left(u\right),\hfill \\ \hfill F_{\mathbb{B},L}^{+}\left(e\right)\left(u\right)& \ge F_{\mathbb{C},L}^{+}\left(e\right)\left(u\right), \hfill & \hfill F_{\mathbb{B},R}^{+}\left(e\right)\left(u\right)& \ge F_{\mathbb{C},R}^{+}\left(e\right)\left(u\right),\hfill \\ \hfill T_{\mathbb{B},L}^{-}\left(e\right)\left(u\right)& \ge T_{\mathbb{C},L}^{-}\left(e\right)\left(u\right), \hfill & \hfill T_{\mathbb{B},R}^{-}\left(e\right)\left(u\right)& \ge T_{\mathbb{C},R}^{-}\left(e\right)\left(u\right),\hfill \\ \hfill I_{\mathbb{B},L}^{-}\left(e\right)\left(u\right)& \le I_{\mathbb{C},L}^{-}\left(e\right)\left(u\right), \hfill & \hfill I_{\mathbb{B},R}^{-}\left(e\right)\left(u\right)& \le I_{\mathbb{C},R}^{-}\left(e\right)\left(u\right),\hfill \\ \hfill F_{\mathbb{B},L}^{-}\left(e\right)\left(u\right)& \le F_{\mathbb{C},L}^{-}\left(e\right)\left(u\right), \hfill & \hfill F_{\mathbb{B},R}^{-}\left(e\right)\left(u\right)& \le F_{\mathbb{C},R}^{-}\left(e\right)\left(u\right).\hfill \end{array}$$

The reversed inequalities for the negative components reflect their bipolar interpretation and the fact that their values belong to $[-1,0]$.

Definition 6. 

Two interval-valued bipolar neutrosophic soft sets ${\mathbb{B}}_A$ and ${\mathbb{C}}_B$ are equal, written ${\mathbb{B}}_A={\mathbb{C}}_B,$ if ${\mathbb{B}}_A\tilde{\subseteq }{\mathbb{C}}_B$ and ${\mathbb{C}}_B\tilde{\subseteq }{\mathbb{B}}_A.$ Equivalently, $A=B$ and all twelve endpoint functions of ${\mathbb{B}}_A$ and ${\mathbb{C}}_B$ coincide.

Definition 7. 

Let $A\subseteq E$. The empty interval-valued bipolar neutrosophic soft set over A, denoted by ${\tilde{\varnothing }}_A$, is defined by

$${\tilde{\varnothing }}_A\left(e\right)\left(u\right)=⟨[0,0],[1,1],[1,1],[0,0],[-1,-1],[-1,-1]⟩$$

for every $e\in A$ and $u\in U$.

Definition 8. 

Let $A\subseteq E$. The universal interval-valued bipolar neutrosophic soft set over A, denoted by ${\tilde{U}}_A$, is defined by

$${\tilde{U}}_A\left(e\right)\left(u\right)=⟨[1,1],[0,0],[0,0],[-1,-1],[0,0],[0,0]⟩$$

for every $e\in A$ and $u\in U$.

The empty and universal IVBNSSs are the least and greatest elements, respectively, with respect to the relation $\tilde{\subseteq }$. In particular,

$${\tilde{\varnothing }}_A\tilde{\subseteq }{\mathbb{B}}_A\tilde{\subseteq }{\tilde{U}}_A$$

for every ${\mathbb{B}}_A\in IVBNSS\left(U\right)$.

Definition 9. 

Let ${\mathbb{B}}_A\in IVBNSS\left(U\right)$. The complement of ${\mathbb{B}}_A$, denoted by ${\mathbb{B}}_A^c$, is the IVBNSS over the same parameter set A defined, for every $e\in A$ and $u\in U$, by 

$$\begin{array}{cc}\hfill {\mathbb{B}}^c\left(e\right)\left(u\right)=& ⟨[F_{\mathbb{B},L}^{+}\left(e\right)\left(u\right),F_{\mathbb{B},R}^{+}\left(e\right)\left(u\right)],[1-I_{\mathbb{B},R}^{+}\left(e\right)\left(u\right),1-I_{\mathbb{B},L}^{+}\left(e\right)\left(u\right)],[T_{\mathbb{B},L}^{+}\left(e\right)\left(u\right),T_{\mathbb{B},R}^{+}\left(e\right)\left(u\right)],\hfill \\ & [F_{\mathbb{B},L}^{-}\left(e\right)\left(u\right),F_{\mathbb{B},R}^{-}\left(e\right)\left(u\right)],[-1-I_{\mathbb{B},R}^{-}\left(e\right)\left(u\right),-1-I_{\mathbb{B},L}^{-}\left(e\right)\left(u\right)],[T_{\mathbb{B},L}^{-}\left(e\right)\left(u\right),T_{\mathbb{B},R}^{-}\left(e\right)\left(u\right)]⟩.\hfill \end{array}$$

Definition 10. 

Let ${\mathbb{B}}_A,{\mathbb{C}}_B\in IVBNSS\left(U\right)$. Their union, denoted by ${\mathbb{B}}_A\tilde{\cup }{\mathbb{C}}_B,$ is the interval-valued bipolar neutrosophic soft set ${\mathbb{D}}_{A\cup B}$ defined by

$$\mathbb{D}\left(e\right)=\left\{\begin{array}{cc}\mathbb{B}\left(e\right),\hfill & e\in A\setminus B,\hfill \\ \mathbb{C}\left(e\right),\hfill & e\in B\setminus A,\hfill \\ \mathbb{B}\left(e\right)\bigsqcup \mathbb{C}\left(e\right),\hfill & e\in A\cap B,\hfill \end{array}\right.$$

where, for $e\in A\cap B$ and $u\in U$,

$$\begin{array}{cc}\left(\mathbb{B}\left(e\right)\bigsqcup \mathbb{C}\left(e\right)\right)\left(u\right)=\hfill & ⟨T_{\mathbb{B}}^{+}\left(e\right)\left(u\right)\vee T_{\mathbb{C}}^{+}\left(e\right)\left(u\right),I_{\mathbb{B}}^{+}\left(e\right)\left(u\right)\wedge I_{\mathbb{C}}^{+}\left(e\right)\left(u\right),F_{\mathbb{B}}^{+}\left(e\right)\left(u\right)\wedge F_{\mathbb{C}}^{+}\left(e\right)\left(u\right),\hfill \\ & T_{\mathbb{B}}^{-}\left(e\right)\left(u\right)\wedge T_{\mathbb{C}}^{-}\left(e\right)\left(u\right),I_{\mathbb{B}}^{-}\left(e\right)\left(u\right)\vee I_{\mathbb{C}}^{-}\left(e\right)\left(u\right),F_{\mathbb{B}}^{-}\left(e\right)\left(u\right)\vee F_{\mathbb{C}}^{-}\left(e\right)\left(u\right)⟩.\hfill \end{array}$$

(3)

Here, the interval operations ∨ and ∧ are defined by (1) and (2).

More explicitly, the positive truth-membership interval in (3) is

$$\begin{array}{c}\hfill T_{\mathbb{B}}^{+}\left(e\right)\left(u\right)\vee T_{\mathbb{C}}^{+}\left(e\right)\left(u\right)=\left[max\left\{T_{\mathbb{B},L}^{+}\left(e\right)\left(u\right),T_{\mathbb{C},L}^{+}\left(e\right)\left(u\right)\right\},max\left\{T_{\mathbb{B},R}^{+}\left(e\right)\left(u\right),T_{\mathbb{C},R}^{+}\left(e\right)\left(u\right)\right\}\right],\end{array}$$

and all other components are obtained analogously.

Definition 11. 

Let ${\mathbb{B}}_A,{\mathbb{C}}_B\in IVBNSS\left(U\right)$. Their intersection, denoted by ${\mathbb{B}}_A\tilde{\cap }{\mathbb{C}}_B,$ is the interval-valued bipolar neutrosophic soft set ${\mathbb{H}}_{A\cup B}$ defined by

$$\mathbb{H}\left(e\right)=\left\{\begin{array}{cc}\mathbb{B}\left(e\right),\hfill & e\in A\setminus B,\hfill \\ \mathbb{C}\left(e\right),\hfill & e\in B\setminus A,\hfill \\ \mathbb{B}\left(e\right)\sqcap \mathbb{C}\left(e\right),\hfill & e\in A\cap B,\hfill \end{array}\right.$$

where, for $e\in A\cap B$ and $u\in U$,

$$\begin{array}{cc}\hfill \left(\mathbb{B}\left(e\right)\sqcap \mathbb{C}\left(e\right)\right)\left(u\right)=& ⟨T_{\mathbb{B}}^{+}\left(e\right)\left(u\right)\wedge T_{\mathbb{C}}^{+}\left(e\right)\left(u\right),I_{\mathbb{B}}^{+}\left(e\right)\left(u\right)\vee I_{\mathbb{C}}^{+}\left(e\right)\left(u\right),F_{\mathbb{B}}^{+}\left(e\right)\left(u\right)\vee F_{\mathbb{C}}^{+}\left(e\right)\left(u\right),\hfill \\ & T_{\mathbb{B}}^{-}\left(e\right)\left(u\right)\vee T_{\mathbb{C}}^{-}\left(e\right)\left(u\right),I_{\mathbb{B}}^{-}\left(e\right)\left(u\right)\wedge I_{\mathbb{C}}^{-}\left(e\right)\left(u\right),F_{\mathbb{B}}^{-}\left(e\right)\left(u\right)\wedge F_{\mathbb{C}}^{-}\left(e\right)\left(u\right)⟩.\hfill \end{array}$$

When $A=B$, the operations $\tilde{\cup }$ and $\tilde{\cap }$ reduce to pointwise union and intersection, respectively. In the following algebraic results, all IVBNSSs are assumed to be defined over the same parameter set A.

Theorem 1. 

Let ${\mathbb{B}}_A,{\mathbb{C}}_A\in IVBNSS\left(U\right)$. Then,

(i) 

${\left({\mathbb{B}}_A^c\right)}^c={\mathbb{B}}_A;$

(ii) 

${\tilde{\varnothing }}_A^c={\tilde{U}}_A,$ and ${\tilde{U}}_A^c={\tilde{\varnothing }}_A;$

(iii) 

If ${\mathbb{B}}_A\tilde{\subseteq }{\mathbb{C}}_A$, then ${\mathbb{C}}_A^c\tilde{\subseteq }{\mathbb{B}}_A^c.$

Proof. 

For the positive truth- and falsity-membership intervals, the complement interchanges the two components. Hence, applying the complement twice restores the original intervals. For the positive indeterminacy interval $[I_L^{+},I_R^{+}]$, we have

$$\begin{array}{c}\hfill {[1-I_R^{+},1-I_L^{+}]}^c=[1-(1-I_L^{+}),1-(1-I_R^{+})]=[I_L^{+},I_R^{+}].\end{array}$$

Similarly, for a negative indeterminacy interval $[I_L^{-},I_R^{-}]$,

$$\begin{array}{c}\hfill {[-1-I_R^{-},-1-I_L^{-}]}^c=[-1-(-1-I_L^{-}),-1-(-1-I_R^{-})]=[I_L^{-},I_R^{-}].\end{array}$$

This proves (i).

Assertion (ii) follows directly by applying Definition 9 to the components of ${\tilde{\varnothing }}_A$ and ${\tilde{U}}_A$.

Finally, the functions $1-x$ on $[0,1]$ and $-1-x$ on $[-1,0]$ are decreasing. Together with the interchange of the truth- and falsity-membership components, this reverses every inequality in Definition 5. Therefore, ${\mathbb{B}}_A\tilde{\subseteq }{\mathbb{C}}_A⟹{\mathbb{C}}_A^c\tilde{\subseteq }{\mathbb{B}}_A^c,$ which proves (iii).    □

Theorem 2. 

Let ${\mathbb{B}}_A,{\mathbb{C}}_A,{\mathbb{D}}_A\in IVBNSS\left(U\right)$. Then, the following properties hold:

(i) 

${\mathbb{B}}_A\tilde{\cup }{\mathbb{B}}_A={\mathbb{B}}_A$ and ${\mathbb{B}}_A\tilde{\cap }{\mathbb{B}}_A={\mathbb{B}}_A.$

(ii) 

${\mathbb{B}}_A\tilde{\cup }{\mathbb{C}}_A={\mathbb{C}}_A\tilde{\cup }{\mathbb{B}}_A,$ and ${\mathbb{B}}_A\tilde{\cap }{\mathbb{C}}_A={\mathbb{C}}_A\tilde{\cap }{\mathbb{B}}_A.$

(iii) 

$\left({\mathbb{B}}_A\tilde{\cup }{\mathbb{C}}_A\right)\tilde{\cup }{\mathbb{D}}_A={\mathbb{B}}_A\tilde{\cup }\left({\mathbb{C}}_A\tilde{\cup }{\mathbb{D}}_A\right),$ and $\left({\mathbb{B}}_A\tilde{\cap }{\mathbb{C}}_A)\tilde{\cap }{\mathbb{D}}_A={\mathbb{B}}_A\tilde{\cap }({\mathbb{C}}_A\tilde{\cap }{\mathbb{D}}_A\right).$

(iv) 

${\mathbb{B}}_A\tilde{\cup }{\tilde{\varnothing }}_A={\mathbb{B}}_A,{\mathbb{B}}_A\tilde{\cap }{\tilde{U}}_A={\mathbb{B}}_A,$ and ${\mathbb{B}}_A\tilde{\cup }{\tilde{U}}_A={\tilde{U}}_A,{\mathbb{B}}_A\tilde{\cap }{\tilde{\varnothing }}_A={\tilde{\varnothing }}_A.$

(v) 

${\mathbb{B}}_A\tilde{\cup }\left({\mathbb{B}}_A\tilde{\cap }{\mathbb{C}}_A\right)={\mathbb{B}}_A,$ and ${\mathbb{B}}_A\tilde{\cap }\left({\mathbb{B}}_A\tilde{\cup }{\mathbb{C}}_A\right)={\mathbb{B}}_A.$

(vi) 

${\mathbb{B}}_A\tilde{\cup }\left({\mathbb{C}}_A\tilde{\cap }{\mathbb{D}}_A\right)=\left({\mathbb{B}}_A\tilde{\cup }{\mathbb{C}}_A\right)\tilde{\cap }\left({\mathbb{B}}_A\tilde{\cup }{\mathbb{D}}_A\right),$ and ${\mathbb{B}}_A\tilde{\cap }\left({\mathbb{C}}_A\tilde{\cup }{\mathbb{D}}_A\right)=\left({\mathbb{B}}_A\tilde{\cap }{\mathbb{C}}_A\right)\tilde{\cup }\left({\mathbb{B}}_A\tilde{\cap }{\mathbb{D}}_A\right).$

Proof. 

All of the assertions follow component-wise from the corresponding properties of the minimum and maximum operations.

For example, for the positive truth-membership component, the first absorption law follows from $max\{x,min\{x,y\left\}\right\}=x,$ whereas for the positive indeterminacy- and falsity-membership components it follows from $min\{x,max\{x,y\left\}\right\}=x.$ For the negative truth-membership component, the relevant identity is again $min\{x,max\{x,y\left\}\right\}=x,$ while the negative indeterminacy- and falsity-membership components satisfy $max\{x,min\{x,y\left\}\right\}=x.$ The distributive laws follow from

$$\begin{array}{cc}\hfill max\{x,min\{y,z\left\}\right\}& =min\{max\{x,y\},max\{x,z\left\}\right\},\hfill \\ \hfill min\{x,max\{y,z\left\}\right\}& =max\{min\{x,y\},min\{x,z\left\}\right\}.\hfill \end{array}$$

Since all interval operations are performed separately on the lower and upper endpoints, these scalar identities prove all stated properties.    □

Theorem 3. 

Let ${\mathbb{B}}_A,{\mathbb{C}}_A\in IVBNSS\left(U\right)$. Then,

$$\begin{array}{cc}\hfill {\left({\mathbb{B}}_A\tilde{\cup }{\mathbb{C}}_A\right)}^c& ={\mathbb{B}}_A^c\tilde{\cap }{\mathbb{C}}_A^c,\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill {\left({\mathbb{B}}_A\tilde{\cap }{\mathbb{C}}_A\right)}^c& ={\mathbb{B}}_A^c\tilde{\cup }{\mathbb{C}}_A^c.\hfill \end{array}$$

(5)

Proof. 

We prove (4); the proof of (5) is analogous. Fix $e\in A$ and $u\in U$. For the positive truth-membership interval, the complement interchanges truth and falsity. Therefore,

$$\begin{array}{c}\hfill T_{{\left(\mathbb{B}\tilde{\cup }\mathbb{C}\right)}^c}^{+}\left(e\right)\left(u\right)=F_{\mathbb{B}\tilde{\cup }\mathbb{C}}^{+}\left(e\right)\left(u\right)=F_{\mathbb{B}}^{+}\left(e\right)\left(u\right)\wedge F_{\mathbb{C}}^{+}\left(e\right)\left(u\right)=T_{{\mathbb{B}}^c}^{+}\left(e\right)\left(u\right)\wedge T_{{\mathbb{C}}^c}^{+}\left(e\right)\left(u\right)=T_{{\mathbb{B}}^c\tilde{\cap }{\mathbb{C}}^c}^{+}\left(e\right)\left(u\right).\end{array}$$

The same reasoning gives the equality of the positive falsity-membership intervals. For the positive indeterminacy component, using $1-min\{x,y\}=max\{1-x,1-y\},$ for each endpoint, we obtain

$$\begin{array}{c}\hfill I_{{\left(\mathbb{B}\tilde{\cup }\mathbb{C}\right)}^c}^{+}\left(e\right)\left(u\right)=I_{{\mathbb{B}}^c}^{+}\left(e\right)\left(u\right)\vee I_{{\mathbb{C}}^c}^{+}\left(e\right)\left(u\right)=I_{{\mathbb{B}}^c\tilde{\cap }{\mathbb{C}}^c}^{+}\left(e\right)\left(u\right).\end{array}$$

For the negative truth-membership component, we have

$$\begin{array}{c}\hfill T_{{\left(\mathbb{B}\tilde{\cup }\mathbb{C}\right)}^c}^{-}\left(e\right)\left(u\right)=F_{\mathbb{B}\tilde{\cup }\mathbb{C}}^{-}\left(e\right)\left(u\right)=F_{\mathbb{B}}^{-}\left(e\right)\left(u\right)\vee F_{\mathbb{C}}^{-}\left(e\right)\left(u\right)=T_{{\mathbb{B}}^c}^{-}\left(e\right)\left(u\right)\vee T_{{\mathbb{C}}^c}^{-}\left(e\right)\left(u\right)=T_{{\mathbb{B}}^c\tilde{\cap }{\mathbb{C}}^c}^{-}\left(e\right)\left(u\right).\end{array}$$

The negative falsity-membership component is treated identically. Consequently,

$$\begin{array}{c}\hfill I_{{\left(\mathbb{B}\tilde{\cup }\mathbb{C}\right)}^c}^{-}\left(e\right)\left(u\right)=I_{{\mathbb{B}}^c}^{-}\left(e\right)\left(u\right)\wedge I_{{\mathbb{C}}^c}^{-}\left(e\right)\left(u\right)=I_{{\mathbb{B}}^c\tilde{\cap }{\mathbb{C}}^c}^{-}\left(e\right)\left(u\right).\end{array}$$

Thus, all six interval-valued components coincide, which proves (4).    □

Theorem 4. 

Let ${\mathbb{B}}_A,{\mathbb{C}}_A\in IVBNSS\left(U\right)$. Then,

(i) 

${\mathbb{B}}_A\tilde{\subseteq }{\mathbb{B}}_A\tilde{\cup }{\mathbb{C}}_A,$ and ${\mathbb{C}}_A\tilde{\subseteq }{\mathbb{B}}_A\tilde{\cup }{\mathbb{C}}_A;$

(ii) 

${\mathbb{B}}_A\tilde{\cap }{\mathbb{C}}_A\tilde{\subseteq }{\mathbb{B}}_A,$ and ${\mathbb{B}}_A\tilde{\cap }{\mathbb{C}}_A\tilde{\subseteq }{\mathbb{C}}_A;$

(iii) 

${\mathbb{B}}_A\tilde{\subseteq }{\mathbb{C}}_A⟺{\mathbb{B}}_A\tilde{\cup }{\mathbb{C}}_A={\mathbb{C}}_A;$

(iv) 

${\mathbb{B}}_A\tilde{\subseteq }{\mathbb{C}}_A⟺{\mathbb{B}}_A\tilde{\cap }{\mathbb{C}}_A={\mathbb{B}}_A.$

Proof. 

Each statement follows directly from the defining inequalities in Definition 5 and the elementary inequalities $min\{x,y\}\le x,y\le max\{x,y\}.$ The directions of the inequalities for the negative components are already incorporated into Definition 5.    □

Our objective is to determine the optimal alternative (sensor). To develop an effective decision-making algorithm, we introduce a numerical characteristic for IVBNSSs inspired by the energy of fuzzy soft sets and hesitant fuzzy soft sets.

4. Energies of Interval-Valued Bipolar Neutrosophic Soft Sets

The main objective of this section is to define numerical characteristics for interval-valued bipolar neutrosophic soft sets, similar to what has been done for fuzzy soft sets, hesitant fuzzy soft sets, and interval-valued hesitant fuzzy soft sets. Due to the interval-valued nature of IVBNSSs, it is natural to distinguish between optimistic and pessimistic information, leading to the introduction of corresponding optimistic and pessimistic energy measures. The proposed formulation is based on the sum of singular values of appropriately constructed matrices, motivated by the notion of graph energy and the nuclear norm, both of which provide global quantitative characteristics that capture the overall structure of a matrix rather than individual entries. This makes the proposed energy particularly suitable for aggregating interval-valued bipolar neutrosophic information distributed across multiple parameters and alternatives. Although other matrix characteristics, such as the Frobenius norm, spectral radius, or individual singular values, could also be considered, the nuclear norm naturally incorporates contributions from all singular values and has been widely recognized as an effective global descriptor of matrix structure.

Since an IVBNSS is determined by six interval-valued components, it can be uniquely represented by 12 rectangular matrices. In Section 4.1, we introduce the concepts of optimistic and pessimistic energy based on these matrices, while Section 4.2 presents the overall energy of an interval-valued bipolar neutrosophic soft set. Finally, in Section 4.3, we establish a fundamental boundedness property, showing that the proposed optimistic, pessimistic, and total energies are finite and bounded.

4.1. Optimistic and Pessimistic Energies of Interval-Valued Bipolar Neutrosophic Soft Sets

Unlike “classical” bipolar neutrosophic soft sets, where the values of the functions $T^{\pm }$, $I^{\pm }$, and $F^{\pm }$ are uniquely determined for each alternative and attribute, interval-valued bipolar neutrosophic soft sets, as we have seen, provide intervals for each of these values. Therefore, it is necessary to introduce numerical characteristics for the lower bounds of these intervals as well as for their upper bounds. To define such numerical characteristics, we introduce 12 matrices for IVBNSS. These matrices form the basis for defining the optimistic and pessimistic energies of interval-valued bipolar neutrosophic soft sets.

Definition 12. 

Let $U=\{u_1,u_2,\dots ,u_n\}$ be a finite universal set (the set of alternatives); let $A=\{e_1,e_2,\dots ,e_m\}\subseteq E$, where E is a set of parameters (the set of attributes); and let ${\mathbb{B}}_A$ be an interval-valued bipolar neutrosophic soft set on $(U,E)$. The infimum and supremum value matrices of the functions $X_{{\mathbb{B}}_A}^{+}$, $X\in \{T,I,F\}$, for the interval-valued bipolar neutrosophic soft set ${\mathbb{B}}_A$ are $n\times m$ matrices of the form

$${\mathrm{\Psi }}_{{\mathbb{B}}_A}^{inf{X}^{+}}=\left[\begin{array}{cccc}inf{X}_{{\mathbb{B}}_A}^{+}\left(e_1\right)\left(u_1\right)& inf{X}_{{\mathbb{B}}_A}^{+}\left(e_2\right)\left(u_1\right)& \cdots & inf{X}_{{\mathbb{B}}_A}^{+}\left(e_m\right)\left(u_1\right)\\ inf{X}_{{\mathbb{B}}_A}^{+}\left(e_1\right)\left(u_2\right)& inf{X}_{{\mathbb{B}}_A}^{+}\left(e_2\right)\left(u_2\right)& \cdots & inf{X}_{{\mathbb{B}}_A}^{+}\left(e_m\right)\left(u_2\right)\\ ⋮& ⋮& \ddots & ⋮\\ inf{X}_{{\mathbb{B}}_A}^{+}\left(e_1\right)\left(u_n\right)& inf{X}_{{\mathbb{B}}_A}^{+}\left(e_2\right)\left(u_n\right)& \cdots & inf{X}_{{\mathbb{B}}_A}^{+}\left(e_m\right)\left(u_n\right)\end{array}\right]$$

and

$${\mathrm{\Psi }}_{{\mathbb{B}}_A}^{sup{X}^{+}}=\left[\begin{array}{cccc}sup{X}_{{\mathbb{B}}_A}^{+}\left(e_1\right)\left(u_1\right)& sup{X}_{{\mathbb{B}}_A}^{+}\left(e_2\right)\left(u_1\right)& \cdots & sup{X}_{{\mathbb{B}}_A}^{+}\left(e_m\right)\left(u_1\right)\\ sup{X}_{{\mathbb{B}}_A}^{+}\left(e_1\right)\left(u_2\right)& sup{X}_{{\mathbb{B}}_A}^{+}\left(e_2\right)\left(u_2\right)& \cdots & sup{X}_{{\mathbb{B}}_A}^{+}\left(e_m\right)\left(u_2\right)\\ ⋮& ⋮& \ddots & ⋮\\ sup{X}_{{\mathbb{B}}_A}^{+}\left(e_1\right)\left(u_n\right)& sup{X}_{{\mathbb{B}}_A}^{+}\left(e_2\right)\left(u_n\right)& \cdots & sup{X}_{{\mathbb{B}}_A}^{+}\left(e_m\right)\left(u_n\right)\end{array}\right].$$

Similarly, the infimum and supremum value matrices of the functions $X_{{\mathbb{B}}_A}^{-}$, $X\in \{T,I,F\}$, for the interval-valued bipolar neutrosophic soft set ${\mathbb{B}}_A$ are $n\times m$ matrices of the form

$${\mathrm{\Psi }}_{{\mathbb{B}}_A}^{inf{X}^{-}}=\left[\begin{array}{cccc}inf{X}_{{\mathbb{B}}_A}^{-}\left(e_1\right)\left(u_1\right)& inf{X}_{{\mathbb{B}}_A}^{-}\left(e_2\right)\left(u_1\right)& \cdots & inf{X}_{{\mathbb{B}}_A}^{-}\left(e_m\right)\left(u_1\right)\\ inf{X}_{{\mathbb{B}}_A}^{-}\left(e_1\right)\left(u_2\right)& inf{X}_{{\mathbb{B}}_A}^{-}\left(e_2\right)\left(u_2\right)& \cdots & inf{X}_{{\mathbb{B}}_A}^{-}\left(e_m\right)\left(u_2\right)\\ ⋮& ⋮& \ddots & ⋮\\ inf{X}_{{\mathbb{B}}_A}^{-}\left(e_1\right)\left(u_n\right)& inf{X}_{{\mathbb{B}}_A}^{-}\left(e_2\right)\left(u_n\right)& \cdots & inf{X}_{{\mathbb{B}}_A}^{-}\left(e_m\right)\left(u_n\right)\end{array}\right]$$

and

$${\mathrm{\Psi }}_{{\mathbb{B}}_A}^{sup{X}^{-}}=\left[\begin{array}{cccc}sup{X}_{{\mathbb{B}}_A}^{-}\left(e_1\right)\left(u_1\right)& sup{X}_{{\mathbb{B}}_A}^{-}\left(e_2\right)\left(u_1\right)& \cdots & sup{X}_{{\mathbb{B}}_A}^{-}\left(e_m\right)\left(u_1\right)\\ sup{X}_{{\mathbb{B}}_A}^{-}\left(e_1\right)\left(u_2\right)& sup{X}_{{\mathbb{B}}_A}^{-}\left(e_2\right)\left(u_2\right)& \cdots & sup{X}_{{\mathbb{B}}_A}^{-}\left(e_m\right)\left(u_2\right)\\ ⋮& ⋮& \ddots & ⋮\\ sup{X}_{{\mathbb{B}}_A}^{-}\left(e_1\right)\left(u_n\right)& sup{X}_{{\mathbb{B}}_A}^{-}\left(e_2\right)\left(u_n\right)& \cdots & sup{X}_{{\mathbb{B}}_A}^{-}\left(e_m\right)\left(u_n\right)\end{array}\right].$$

Each matrix in Definition 12 has a set of singular values, computed in the standard manner of linear algebra. We use these singular values to define the optimistic and pessimistic energies of an IVBNSS.

Definition 13. 

The optimistic energy of the interval-valued bipolar neutrosophic soft set ${\mathbb{B}}_A$, denoted by ${\mathbb{E}}^O\left({\mathbb{B}}_A\right)$, is defined as

$${\mathbb{E}}^O\left({\mathbb{B}}_A\right)={\displaystyle \sum _{i=1}^n}(({\sigma }_i^{sup{T}^{+}}+{\sigma }_i^{sup{I}^{+}}-{\sigma }_i^{inf{F}^{+}})-({\sigma }_i^{sup{T}^{-}}+{\sigma }_i^{sup{I}^{-}}-{\sigma }_i^{inf{F}^{-}})),$$

where ${\sigma }_1^{sup{T}^{+}}⩾{\sigma }_2^{sup{T}^{+}}⩾\dots ⩾{\sigma }_n^{sup{T}^{+}}⩾0$ are the singular values of the matrix ${\mathrm{\Psi }}_{{\mathbb{B}}_A}^{sup{T}^{+}}$, ${\sigma }_1^{sup{I}^{+}}⩾{\sigma }_2^{sup{I}^{+}}⩾\dots ⩾{\sigma }_n^{sup{I}^{+}}⩾0$ are the singular values of the matrix ${\mathrm{\Psi }}_{{\mathbb{B}}_A}^{sup{I}^{+}}$, ${\sigma }_1^{inf{F}^{+}}⩾{\sigma }_2^{inf{F}^{+}}⩾\dots ⩾{\sigma }_n^{inf{F}^{+}}⩾0$ are the singular values of the matrix ${\mathrm{\Psi }}_{{\mathbb{B}}_A}^{inf{F}^{+}}$, ${\sigma }_1^{sup{T}^{-}}⩾{\sigma }_2^{sup{T}^{-}}⩾\dots ⩾{\sigma }_n^{sup{T}^{-}}⩾0$ are the singular values of the matrix ${\mathrm{\Psi }}_{{\mathbb{B}}_A}^{sup{T}^{-}}$, ${\sigma }_1^{sup{I}^{-}}⩾{\sigma }_2^{sup{I}^{-}}⩾\dots ⩾{\sigma }_n^{sup{I}^{-}}⩾0$ are the singular values of the matrix ${\mathrm{\Psi }}_{{\mathbb{B}}_A}^{sup{I}^{-}}$, and ${\sigma }_1^{inf{F}^{-}}⩾{\sigma }_2^{inf{F}^{-}}⩾\dots ⩾{\sigma }_n^{inf{F}^{-}}⩾0$ are the singular values of the matrix ${\mathrm{\Psi }}_{{\mathbb{B}}_A}^{inf{F}^{-}}$, while n is the number of alternatives.

Definition 14. 

The pessimistic energy of the interval-valued bipolar neutrosophic soft set ${\mathbb{B}}_A$, denoted by ${\mathbb{E}}^P\left({\mathbb{B}}_A\right)$, is defined as

$${\mathbb{E}}^P\left({\mathbb{B}}_A\right)={\displaystyle \sum _{i=1}^n}(({\sigma }_i^{inf{T}^{+}}+{\sigma }_i^{inf{I}^{+}}-{\sigma }_i^{sup{F}^{+}})-({\sigma }_i^{inf{T}^{-}}+{\sigma }_i^{inf{I}^{-}}-{\sigma }_i^{sup{F}^{-}})),$$

where ${\sigma }_1^{inf{T}^{+}}⩾{\sigma }_2^{inf{T}^{+}}⩾\dots ⩾{\sigma }_n^{inf{T}^{+}}⩾0$ are the singular values of the matrix ${\mathrm{\Psi }}_{{\mathbb{B}}_A}^{inf{T}^{+}}$, ${\sigma }_1^{inf{I}^{+}}⩾{\sigma }_2^{inf{I}^{+}}⩾\dots ⩾{\sigma }_n^{inf{I}^{+}}⩾0$ are the singular values of the matrix ${\mathrm{\Psi }}_{{\mathbb{B}}_A}^{inf{I}^{+}}$, ${\sigma }_1^{sup{F}^{+}}⩾{\sigma }_2^{sup{F}^{+}}⩾\dots ⩾{\sigma }_n^{sup{F}^{+}}⩾0$ are the singular values of the matrix ${\mathrm{\Psi }}_{{\mathbb{B}}_A}^{sup{F}^{+}}$, ${\sigma }_1^{inf{T}^{-}}⩾{\sigma }_2^{inf{T}^{-}}⩾\dots ⩾{\sigma }_n^{inf{T}^{-}}⩾0$ are the singular values of the matrix ${\mathrm{\Psi }}_{{\mathbb{B}}_A}^{inf{T}^{-}}$, ${\sigma }_1^{inf{I}^{-}}⩾{\sigma }_2^{inf{I}^{-}}⩾\dots ⩾{\sigma }_n^{inf{I}^{-}}⩾0$ are the singular values of the matrix ${\mathrm{\Psi }}_{{\mathbb{B}}_A}^{inf{I}^{-}}$, and ${\sigma }_1^{sup{F}^{-}}⩾{\sigma }_2^{sup{F}^{-}}⩾\dots ⩾{\sigma }_n^{sup{F}^{-}}⩾0$ are the singular values of the matrix ${\mathrm{\Psi }}_{{\mathbb{B}}_A}^{sup{F}^{-}}$, while n is the number of alternatives.

In the following example, we describe how to determine the pessimistic and optimistic energy for the interval-valued neutrosophic soft set ${\mathbb{B}}_A$ given in Example 3:

Example 4. 

To illustrate the computation of the optimistic and pessimistic energies of an interval-valued bipolar neutrosophic soft set, we use Example 3 from Section 3. Based on Definition 12, we construct 12 matrices: six infimum matrices, denoted by ${\mathrm{\Psi }}_{{\mathbb{B}}_A}^{inf{X}^{+}}$ and ${\mathrm{\Psi }}_{{\mathbb{B}}_A}^{inf{X}^{-}}$, and six supremum matrices, denoted by ${\mathrm{\Psi }}_{{\mathbb{B}}_A}^{sup{X}^{+}}$ and ${\mathrm{\Psi }}_{{\mathbb{B}}_A}^{sup{X}^{-}}$, where $X\in \{T,I,F\}$:

$${\mathrm{\Psi }}_{{\mathbb{B}}_A}^{inf{T}^{+}}=\left[\begin{array}{cccc}0.2& 0.1& 0.2& 0.0\\ 0.9& 1.0& 1.0& 1.0\\ 0.4& 0.4& 0.3& 0.4\\ 0.8& 0.8& 0.7& 0.7\end{array}\right],{\mathrm{\Psi }}_{{\mathbb{B}}_A}^{inf{T}^{-}}=\left[\begin{array}{cccc}-0.8& -1.0& -0.8& -0.9\\ -0.9& -0.9& -0.9& -0.7\\ -0.3& -0.7& -0.4& -0.4\\ -1.0& -1.0& -0.8& -0.8\end{array}\right],$$

$${\mathrm{\Psi }}_{{\mathbb{B}}_A}^{inf{I}^{+}}=\left[\begin{array}{cccc}0.3& 0.2& 0.3& 0.2\\ 0.8& 0.8& 0.8& 0.8\\ 0.2& 0.3& 0.3& 0.3\\ 0.8& 0.8& 0.8& 0.7\end{array}\right],{\mathrm{\Psi }}_{{\mathbb{B}}_A}^{inf{I}^{-}}=\left[\begin{array}{cccc}-0.8& -0.9& -0.8& -1.0\\ -1.0& -1.0& -1.0& -1.0\\ -0.4& -0.8& -0.9& -0.9\\ -0.9& -0.9& -0.6& -0.7\end{array}\right],$$

$${\mathrm{\Psi }}_{{\mathbb{B}}_A}^{inf{F}^{+}}=\left[\begin{array}{cccc}0.2& 0.2& 0.4& 0.2\\ 0.8& 1.0& 1.0& 1.0\\ 0.3& 0.5& 0.6& 0.6\\ 0.7& 0.7& 0.9& 0.9\end{array}\right],{\mathrm{\Psi }}_{{\mathbb{B}}_A}^{inf{F}^{-}}=\left[\begin{array}{cccc}-0.8& -0.8& -0.8& -0.8\\ -0.9& -0.9& -0.9& -0.9\\ -0.6& -0.7& -0.5& -0.4\\ -0.9& -0.8& -0.8& -0.8\end{array}\right],$$

$${\mathrm{\Psi }}_{{\mathbb{B}}_A}^{sup{T}^{+}}=\left[\begin{array}{cccc}0.4& 0.8& 0.4& 0.7\\ 1.0& 1.0& 1.0& 1.0\\ 0.6& 0.7& 0.6& 0.6\\ 1.0& 1.0& 0.9& 0.9\end{array}\right],{\mathrm{\Psi }}_{{\mathbb{B}}_A}^{sup{T}^{-}}=\left[\begin{array}{cccc}-0.1& -0.2& -0.1& -0.1\\ -0.7& -0.8& -0.8& -0.5\\ -0.1& -0.4& -0.2& -0.2\\ -0.1& -0.9& -0.1& -0.0\end{array}\right],$$

$${\mathrm{\Psi }}_{{\mathbb{B}}_A}^{sup{I}^{+}}=\left[\begin{array}{cccc}0.9& 0.9& 0.8& 0.9\\ 1.0& 1.0& 0.9& 0.9\\ 0.4& 0.6& 0.5& 0.4\\ 0.9& 0.9& 0.9& 0.9\end{array}\right],{\mathrm{\Psi }}_{{\mathbb{B}}_A}^{sup{I}^{-}}=\left[\begin{array}{cccc}-0.0& -0.0& -0.0& -0.0\\ -0.8& -0.7& -0.8& -0.9\\ -0.3& -0.5& -0.8& -0.7\\ -0.7& -0.8& -0.4& -0.5\end{array}\right],$$

$${\mathrm{\Psi }}_{{\mathbb{B}}_A}^{sup{F}^{+}}=\left[\begin{array}{cccc}0.7& 1.0& 0.7& 1.0\\ 0.9& 1.0& 1.0& 1.0\\ 0.5& 0.6& 0.9& 0.9\\ 0.9& 0.9& 1.0& 1.0\end{array}\right],{\mathrm{\Psi }}_{{\mathbb{B}}_A}^{sup{F}^{-}}=\left[\begin{array}{cccc}-0.5& -0.1& -0.7& -0.1\\ -0.8& -0.7& -0.7& -0.7\\ -0.3& -0.4& -0.3& -0.3\\ -0.8& -0.6& -0.7& -0.7\end{array}\right].$$

Since the procedure for computing singular values is applied in the same way for all twelve matrices, we will examine it in detail only for the matrix ${\mathrm{\Psi }}_{{\mathbb{B}}_A}^{inf{T}^{+}}$, while for the remaining matrices, we will directly present the obtained results. To determine the singular values of this matrix, we first compute its transpose ${\left({\mathrm{\Psi }}_{{\mathbb{B}}_A}^{inf{T}^{+}}\right)}^T$. Since both matrices are rectangular, their product ${\mathrm{\Psi }}_{{\mathbb{B}}_A}^{inf{T}^{+}}·{\left({\mathrm{\Psi }}_{{\mathbb{B}}_A}^{inf{T}^{+}}\right)}^T$ results in a square matrix whose eigenvalues are always real and non-negative. The singular values are then obtained by computing the square roots of all eigenvalues. After performing the procedure, we obtain the following singular values for the matrix ${\mathrm{\Psi }}_{{\mathbb{B}}_A}^{inf{T}^{+}}$:

$${\sigma }_1^{inf{T}^{+}}=2.5848,{\sigma }_2^{inf{T}^{+}}=0.1807,{\sigma }_3^{inf{T}^{+}}=0.1253,{\sigma }_4^{inf{T}^{+}}=0.0188.$$

For the remaining eleven matrices, we will directly use the singular values in the further computation of the energy of the interval-valued bipolar neutrosophic soft set, since their processing follows the same method and does not require additional consideration:

$$\begin{array}{c}{\sigma }_1^{inf{T}^{-}}=3.1748,{\sigma }_2^{inf{T}^{-}}=0.2814,{\sigma }_3^{inf{T}^{-}}=0.1406,{\sigma }_4^{inf{T}^{-}}=0.1083,\\ {\sigma }_1^{inf{I}^{+}}=2.3493,{\sigma }_2^{inf{I}^{+}}=0.1232,{\sigma }_3^{inf{I}^{+}}=0.0548,{\sigma }_4^{inf{I}^{+}}=0.0505,\\ {\sigma }_1^{inf{I}^{-}}=3.4284,{\sigma }_2^{inf{I}^{-}}=0.4487,{\sigma }_3^{inf{I}^{-}}=0.1194,{\sigma }_4^{inf{I}^{-}}=0.1034,\\ {\sigma }_1^{inf{F}^{+}}=2.7425,{\sigma }_2^{inf{F}^{+}}=0.1676,{\sigma }_3^{inf{F}^{+}}=0.1369,{\sigma }_4^{inf{F}^{+}}=0.1081,\\ {\sigma }_1^{inf{F}^{-}}=3.1213,{\sigma }_2^{inf{F}^{-}}=0.2058,{\sigma }_3^{inf{F}^{-}}=0.0701,{\sigma }_4^{inf{F}^{-}}=0.0000,\\ {\sigma }_1^{sup{T}^{+}}=3.2444,{\sigma }_2^{sup{T}^{+}}=0.3266,{\sigma }_3^{sup{T}^{+}}=0.0742,{\sigma }_4^{sup{T}^{+}}=0.0381,\\ {\sigma }_1^{sup{T}^{-}}=1.6675,{\sigma }_2^{sup{T}^{-}}=0.6128,{\sigma }_3^{sup{T}^{-}}=0.1156,{\sigma }_4^{sup{T}^{-}}=0.0246,\\ {\sigma }_1^{sup{I}^{+}}=3.2904,{\sigma }_2^{sup{I}^{+}}=0.1616,{\sigma }_3^{sup{I}^{+}}=0.0685,{\sigma }_4^{sup{I}^{+}}=0.0494,\\ {\sigma }_1^{sup{I}^{-}}=2.3101,{\sigma }_2^{sup{I}^{-}}=0.4744,{\sigma }_3^{sup{I}^{-}}=0.1687,{\sigma }_4^{sup{I}^{-}}=0.0000,\\ {\sigma }_1^{sup{F}^{+}}=3.5350,{\sigma }_2^{sup{F}^{+}}=0.3144,{\sigma }_3^{sup{F}^{+}}=0.2089,{\sigma }_4^{sup{F}^{+}}=0.0401,\\ {\sigma }_1^{sup{F}^{-}}=2.2442,{\sigma }_2^{sup{F}^{-}}=0.4807,{\sigma }_3^{sup{F}^{-}}=0.1116,{\sigma }_4^{sup{F}^{-}}=0.0150.\end{array}$$

Based on Definitions 13 and 14, we can now compute the optimistic and pessimistic energy of IVBNSS ${\mathbb{B}}_A$, respectively:

$${\mathbb{E}}^O\left({\mathbb{B}}_A\right)={\displaystyle \sum _{i=1}^4}(({\sigma }_i^{sup{T}^{+}}+{\sigma }_i^{sup{I}^{+}}-{\sigma }_i^{inf{F}^{+}})-({\sigma }_i^{sup{T}^{-}}+{\sigma }_i^{sup{I}^{-}}-{\sigma }_i^{inf{F}^{-}}))=2.1219,$$

and

$${\mathbb{E}}^P\left({\mathbb{B}}_A\right)={\displaystyle \sum _{i=1}^4}(({\sigma }_i^{inf{T}^{+}}+{\sigma }_i^{inf{I}^{+}}-{\sigma }_i^{sup{F}^{+}})-({\sigma }_i^{inf{T}^{-}}+{\sigma }_i^{inf{I}^{-}}-{\sigma }_i^{sup{F}^{-}}))=-3.5645.$$

4.2. Energy of an Interval-Valued Bipolar Neutrosophic Soft Set

The values of the optimistic and pessimistic energies of an interval-valued neutrosophic soft set are used to define the energy of the observed structure.

Definition 15. 

The energy of the interval-valued bipolar neutrosophic soft set ${\mathbb{B}}_A$, denoted by $\mathbb{E}\left({\mathbb{B}}_A\right)$, is defined as $\mathbb{E}\left({\mathbb{B}}_A\right)={\displaystyle \frac{{\mathbb{E}}^O\left({\mathbb{B}}_A\right)+{\mathbb{E}}^P\left({\mathbb{B}}_A\right)}{2}}$, where ${\mathbb{E}}^O\left({\mathbb{B}}_A\right)$ and ${\mathbb{E}}^P\left({\mathbb{B}}_A\right)$ are the optimistic and pessimistic energies of the IVBNSS ${\mathbb{B}}_A$.

Example 5. 

In Example 4, the optimistic and pessimistic energy of IVBNSS ${\mathbb{B}}_A$ were determined, yielding the values ${\mathbb{E}}^O\left({\mathbb{B}}_A\right)=2.1219$ and ${\mathbb{E}}^P\left({\mathbb{B}}_A\right)=-3.5645$. Based on Definition 15, the energy of the given interval-valued bipolar neutrosophic soft set ${\mathbb{B}}_A$ can be computed as follows:

$$\mathbb{E}\left({\mathbb{B}}_A\right)={\displaystyle \frac{{\mathbb{E}}^O\left({\mathbb{B}}_A\right)+{\mathbb{E}}^P\left({\mathbb{B}}_A\right)}{2}}={\displaystyle \frac{2.1219+(-3.5645)}{2}}=-0.7213.$$

4.3. Fundamental Properties of the IVBNSS Energy

The following result shows that the proposed optimistic, pessimistic, and total energies are always finite and uniformly bounded. This is important because it guarantees that the introduced energy measures are well-defined numerical characteristics of an interval-valued bipolar neutrosophic soft set.

Theorem 5. 

Let ${\mathbb{B}}_A\in IVBNSS\left(U\right)$, where $U=\{u_1,u_2,\dots ,u_n\}$ and $A=\{e_1,e_2,\dots ,e_m\}$, and let $r=min\{n,m\}$. Then, the optimistic energy ${\mathbb{E}}^O\left({\mathbb{B}}_A\right)$, the pessimistic energy ${\mathbb{E}}^P\left({\mathbb{B}}_A\right)$, and the energy $\mathbb{E}\left({\mathbb{B}}_A\right)$ are finite real numbers. Moreover,

$$\left|{\mathbb{E}}^O\left({\mathbb{B}}_A\right)\right|\le 6\sqrt{rnm},\left|{\mathbb{E}}^P\left({\mathbb{B}}_A\right)\right|\le 6\sqrt{rnm},$$

and

$$\left|\mathbb{E}\left({\mathbb{B}}_A\right)\right|\le 6\sqrt{rnm}.$$

Proof. 

Since all matrices associated with ${\mathbb{B}}_A$ are real finite $n\times m$ matrices, their singular values are well defined and finite; moreover, because all of their entries have an absolute value of at most 1, for each such matrix M we have

$${\parallel M\parallel }_F\le \sqrt{nm}.$$

Using the standard inequality ${\parallel M\parallel }_{*}\le \sqrt{rank\left(M\right)}{\parallel M\parallel }_F$ and the fact that $rank\left(M\right)\le r=min\{n,m\}$, it follows that

$${\parallel M\parallel }_{*}\le \sqrt{rnm}.$$

By the nuclear norm representation of the optimistic energy, ${\mathbb{E}}^O\left({\mathbb{B}}_A\right)$ is a linear combination of six nuclear norms with coefficients equal to 1 or $-1$. Therefore, by the triangle inequality,

$$\left|{\mathbb{E}}^O\left({\mathbb{B}}_A\right)\right|\le {∥{\mathrm{\Psi }}_{{\mathbb{B}}_A}^{sup{T}^{+}}∥}_{*}+{∥{\mathrm{\Psi }}_{{\mathbb{B}}_A}^{sup{I}^{+}}∥}_{*}+{∥{\mathrm{\Psi }}_{{\mathbb{B}}_A}^{inf{F}^{+}}∥}_{*}+{∥{\mathrm{\Psi }}_{{\mathbb{B}}_A}^{sup{T}^{-}}∥}_{*}+{∥{\mathrm{\Psi }}_{{\mathbb{B}}_A}^{sup{I}^{-}}∥}_{*}+{∥{\mathrm{\Psi }}_{{\mathbb{B}}_A}^{inf{F}^{-}}∥}_{*}.$$

Since each of these six nuclear norms is bounded above by $\sqrt{rnm}$, it follows that

$$\left|{\mathbb{E}}^O\left({\mathbb{B}}_A\right)\right|\le 6\sqrt{rnm}.$$

The proof for the pessimistic energy is analogous. Finally, by the definition of the energy of ${\mathbb{B}}_A$, we have

$$\mathbb{E}\left({\mathbb{B}}_A\right)={\displaystyle \frac{{\mathbb{E}}^O\left({\mathbb{B}}_A\right)+{\mathbb{E}}^P\left({\mathbb{B}}_A\right)}{2}}.$$

Thus,

$$\left|\mathbb{E}\left({\mathbb{B}}_A\right)\right|\le {\displaystyle \frac{\left|{\mathbb{E}}^O\left({\mathbb{B}}_A\right)\right|+\left|{\mathbb{E}}^P\left({\mathbb{B}}_A\right)\right|}{2}}\le {\displaystyle \frac{6\sqrt{rnm}+6\sqrt{rnm}}{2}}=6\sqrt{rnm}.$$

The proof is complete.    □

Although the optimistic energy is computed from the upper interval bounds and the pessimistic energy from the corresponding lower interval bounds, there is generally no ordering between these two quantities. In particular, the inequality

$${\mathbb{E}}^O\left({\mathbb{B}}_A\right)\ge {\mathbb{E}}^P\left({\mathbb{B}}_A\right)$$

does not necessarily hold, and it is equally possible that

$${\mathbb{E}}^P\left({\mathbb{B}}_A\right)>{\mathbb{E}}^O\left({\mathbb{B}}_A\right).$$

This behavior is a direct consequence of the fact that the proposed energies are defined through the nuclear norms of the associated matrices. Although the interval endpoints satisfy the natural component-wise ordering, the nuclear norm is not monotone with respect to the entry-wise ordering of matrices. Indeed, the singular values depend on the global matrix structure rather than on individual entries. Consequently, replacing lower interval bounds with upper ones may either increase or decrease the corresponding nuclear norms; therefore, no monotonic relationship between the optimistic and pessimistic energies can be established in general.

The following example illustrates this phenomenon by presenting an IVBNSS for which the pessimistic energy exceeds the optimistic one:

Example 6. 

Consider the IVBNSS ${\mathbb{B}}_A$ given in Table 2.

Table 2. IVBNSS ${\mathbb{B}}_A$.

${\mathbb{B}}_{\mathit{A}}$	${\mathit{e}}_1$
$u_1$	$\left(\right[0.2,0.2],[0.2,0.2],[0.1,0.1],[-0.8,-0.8],[-0.7,-0.7],[-0.5,-0.5\left]\right)$
$u_2$	$\left(\right[0.4,0.4],[0.4,0.4],[0.2,0.2],[-0.7,-0.7],[-1.0,-1.0],[-0.9,-0.9\left]\right)$
$u_3$	$\left(\right[0.8,0.8],[0.8,0.8],[0.3,0.3],[-0.3,-0.3],[-0.4,-0.4],[-0.6,-0.6\left]\right)$
$u_4$	$\left(\right[0.8,0.8],[0.8,0.8],[0.8,0.8],[-0.1,-0.1],[-0.7,-0.7],[-0.8,-0.8\left]\right)$
	${\mathit{e}}_{\mathbf{2}}$
$u_1$	$\left(\right[0.6,0.6],[0.6,0.6],[0.6,0.6],[-0.8,-0.8],[-0.7,-0.7],[-0.5,-0.5\left]\right)$
$u_2$	$\left(\right[0.7,0.7],[0.7,0.7],[0.7,0.7],[-0.7,-0.7],[-1.0,-1.0],[-0.9,-0.9\left]\right)$
$u_3$	$\left(\right[0.5,0.5],[0.5,0.5],[0.6,0.6],[-0.3,-0.3],[-0.4,-0.4],[-0.6,-0.6\left]\right)$
$u_4$	$\left(\right[0.2,0.2],[0.2,0.2],[0.3,0.3],[-0.1,-0.1],[-0.7,-0.7],[-0.8,-0.8\left]\right)$
	${\mathit{e}}_{\mathbf{3}}$
$u_1$	$\left(\right[0.1,0.1],[0.2,0.2],[0.2,0.2],[-0.8,-0.8],[-0.7,-0.7],[-0.5,-0.5\left]\right)$
$u_2$	$\left(\right[0.9,0.9],[0.9,0.9],[0.4,0.4],[-0.7,-0.7],[-1.0,-1.0],[-0.9,-0.9\left]\right)$
$u_3$	$\left(\right[0.2,0.4],[0.4,0.6],[0.4,0.4],[-0.3,-0.3],[-0.4,-0.4],[-0.6,-0.6\left]\right)$
$u_4$	$\left(\right[0.5,0.5],[0.5,0.5],[0.6,0.6],[-0.1,-0.1],[-0.7,-0.7],[-0.8,-0.8\left]\right)$

By applying the proposed energy measures, we obtain

$${\mathbb{E}}^O\left({\mathbb{B}}_A\right)=1.4251,{\mathbb{E}}^P\left({\mathbb{B}}_A\right)=1.4586.$$

Hence,

$${\mathbb{E}}^P\left({\mathbb{B}}_A\right)>{\mathbb{E}}^O\left({\mathbb{B}}_A\right),$$

which confirms that, in general, the pessimistic energy may exceed the optimistic energy. This example illustrates that no monotonic relationship exists between the two energy measures.

From the decision-making perspective, the proposed energy measures provide a global quantitative assessment of the information contained in an interval-valued bipolar neutrosophic soft set. Unlike score functions, which evaluate alternatives through local aggregation of membership values, the proposed energies are derived from the singular values of the associated matrices and, therefore, capture the overall distribution and structural organization of the information across all alternatives and attributes. The optimistic and pessimistic energies reflect the upper and lower interval representations, respectively, while the total energy provides a balanced assessment of both viewpoints. Consequently, these energy measures offer an informative criterion for distinguishing alternatives within the proposed elimination-based decision-making framework.

5. Algorithm for Interval-Valued Bipolar Neutrosophic Soft Set Decision-Making

In the decision-making process, we often encounter situations where it is necessary to rank or identify the most favorable and the least favorable alternatives from a set of possible choices. Due to the complexity of real-world problems and the uncertainty in available data, traditional decision-making methods are often not sufficiently precise or flexible. Interval-valued bipolar neutrosophic soft sets (IVBNSSs) provide a powerful mathematical tool for modeling uncertainty and dual information in the decision-making process. In this section, we present an algorithm that enables efficient analysis and identification of the optimal and least favorable alternatives within the IVBNSS model. The algorithm utilizes numerical characteristics defined in the previous sections and is based on a clear sequence of steps that allow for well-founded decision-making. Below, we present the algorithm in pseudocode form and then provide a detailed explanation of its structure and its application to a dataset represented within the IVBNSS formalism.

The proposed algorithm is based on a leave-one-out principle, which evaluates each alternative through its influence on the global energy of the interval-valued bipolar neutrosophic soft set. Instead of assigning a score directly to an isolated alternative, the algorithm removes each alternative in turn and computes the energy of the corresponding reduced system. The remaining energy therefore serves as an indirect measure of the structural contribution of the omitted alternative: a smaller value indicates that removing this alternative leaves a system with lower overall energy and is interpreted as evidence of a greater contribution of the omitted alternative to the original IVBNSS. Accordingly, the alternative associated with the minimum leave-one-out energy is selected as optimal. Thus, the proposed criterion evaluates alternatives in relation to the complete parameterized decision structure rather than solely through entry-wise or local assessments.

The decision-making algorithm based on the energy of the interval-valued bipolar neutrosophic soft set enables the identification of the alternative that contributes the most to the overall system, as well as a complete ranking of all alternatives. The procedure begins with an interval-valued bipolar neutrosophic soft set ${\mathbb{B}}_A$ defined over the universal set of alternatives $U=\{u_1,u_2,\dots ,u_n\}$ and the parameter set $A=\{e_1,\dots ,e_m\}$. Each alternative $u_i$ is examined by removing it from the set and analyzing the remaining alternatives, thereby assessing its structural contribution to the system. For each alternative $u_i$, a new interval-valued bipolar neutrosophic soft set ${\mathbb{B}}_A^i$ is constructed, which includes all alternatives except for the omitted alternative $u_i$, i.e., it is defined over the universe $U\setminus \left\{u_i\right\}$. For each reduced set, the infimum and supremum matrices are formed for the truth, indeterminacy, and falsity functions, both for positive and negative values: ${\mathrm{\Psi }}_{{\mathbb{B}}_A^i}^{inf{X}^{+}}$, ${\mathrm{\Psi }}_{{\mathbb{B}}_A^i}^{sup{X}^{+}}$, ${\mathrm{\Psi }}_{{\mathbb{B}}_A^i}^{inf{X}^{-}}$, and ${\mathrm{\Psi }}_{{\mathbb{B}}_A^i}^{sup{X}^{-}}$, where $X\in \{T,I,F\}$. The singular values of these matrices ${\sigma }_{ij}^{inf{X}^{+}}$, ${\sigma }_{ij}^{sup{X}^{+}}$, ${\sigma }_{ij}^{inf{X}^{-}}$, and ${\sigma }_{ij}^{sup{X}^{-}}$ are determined, where $j=1,\dots ,k$ and $k=min(n-1,m)$. These singular values are then used to compute the optimistic energy ${\mathbb{E}}^O\left({\mathbb{B}}_A^i\right)$ and pessimistic energy ${\mathbb{E}}^P\left({\mathbb{B}}_A^i\right)$. The optimistic energy is calculated from the maximum values of the truth and indeterminacy functions and the minimum values of the falsity function, whereas the pessimistic energy is calculated from the minimum values of truth and indeterminacy and the maximum values of falsity. Based on these values, the total energy for each set ${\mathbb{B}}_A^i$ is computed as the average of the optimistic and pessimistic energy, denoted by $\mathbb{E}\left({\mathbb{B}}_A^i\right)$. Finally, the alternatives are ranked in ascending order of the total energies of the corresponding reduced IVBNSSs. Within the proposed leave-one-out interpretation, a smaller remaining energy indicates a greater structural contribution of the removed alternative to the original system. Consequently, the optimal alternative is defined by $u^{*}=arg{min}_{1\le i\le n}\mathbb{E}\left({\mathbb{B}}_A^i\right).$ If two alternatives attain the same total energy, the one with the smaller optimistic energy is preferred. This procedure provides a mathematically grounded criterion for identifying the most influential alternative under complex and uncertain conditions.

Let n denote the number of alternatives and m the number of attributes. The elimination procedure is performed once for each alternative, producing n reduced interval-valued bipolar neutrosophic soft sets. For every reduced set, twelve matrices of size $(n-1)\times m$ are constructed, corresponding to the lower and upper bounds of the truth, indeterminacy, and falsity membership functions for both positive and negative components. Since the number of matrices is constant, their construction requires $O\left(nm\right)$ time for each elimination step. The dominant computational cost arises from the computation of singular values. Computing the singular value decomposition of an $(n-1)\times m$ matrix requires $O\left(\right(n-1)mmin(n-1,m\left)\right)$, which is asymptotically equivalent to $O(nmmin(n,m\left)\right)$. As twelve singular value decompositions are performed for each eliminated alternative, the constant factor does not affect the asymptotic order. Therefore, the overall time complexity of the proposed algorithm is $O(n^{2}mmin(n,m))$. Since the algorithm exhibits polynomial-time complexity and the number of matrices processed at each elimination step is fixed, it remains computationally tractable for practical decision-making problems involving moderate and relatively large numbers of alternatives and attributes.

6. Application of the Proposed Algorithm for IVBNSSs in Practical Problems

In this section, we illustrate how the concept of energy in interval-valued bipolar neutrosophic soft sets can be used for decision-making in real-world scenarios. We apply Algorithm 1 to two case studies: the first concerns the selection of the optimal sensor for an industrial automated system, given in Example 3; while the second addresses a benchmark engineering problem from the literature—the selection of a bio-inspired thin-wall structure—which allows the results of the proposed method to be compared with previously published ones. These case studies demonstrate how a mathematical model can assist in evaluating and ranking alternatives based on multiple criteria while considering their interrelationships.

Algorithm 1 A Decision-Making Algorithm for Interval-Valued Bipolar Neutrosophic Soft Set Energy

Require: IVBNSS ${\mathbb{B}}_A$ over the universe $U=\{u_1,\dots ,u_n\}$ and the parameter set $A=\{e_1,\dots ,e_m\}$ Ensure: Ranking of alternatives; the optimal alternative $u^{*}$   1: for $i\leftarrow 1$ to n do                                                                       ▹ leave-one-out analysis   2:     Construct the reduced set ${\mathbb{B}}_A^i$ over $U\setminus \left\{u_i\right\}$   3:     for all $X\in \{T,I,F\}$ do   4:         Form the matrices ${\mathrm{\Psi }}_{{\mathbb{B}}_A^i}^{inf{X}^{+}},{\mathrm{\Psi }}_{{\mathbb{B}}_A^i}^{sup{X}^{+}},{\mathrm{\Psi }}_{{\mathbb{B}}_A^i}^{inf{X}^{-}},{\mathrm{\Psi }}_{{\mathbb{B}}_A^i}^{sup{X}^{-}}\in {\mathbb{R}}^{(n-1)\times m}$   5:         Compute their singular values ${\sigma }_{ij}^{inf{X}^{\pm }},{\sigma }_{ij}^{sup{X}^{\pm }}$, $j=1,\dots ,k$, where $k=min(n-1,m)$   6:     end for   7:     ${\displaystyle {\mathbb{E}}^O\left({\mathbb{B}}_A^i\right)\leftarrow {\displaystyle \sum _{j=1}^k}\left[\left({\sigma }_{ij}^{sup{T}^{+}}+{\sigma }_{ij}^{sup{I}^{+}}-{\sigma }_{ij}^{inf{F}^{+}}\right)-\left({\sigma }_{ij}^{sup{T}^{-}}+{\sigma }_{ij}^{sup{I}^{-}}-{\sigma }_{ij}^{inf{F}^{-}}\right)\right]}$   8:     ${\displaystyle {\mathbb{E}}^P\left({\mathbb{B}}_A^i\right)\leftarrow {\displaystyle \sum _{j=1}^k}\left[\left({\sigma }_{ij}^{inf{T}^{+}}+{\sigma }_{ij}^{inf{I}^{+}}-{\sigma }_{ij}^{sup{F}^{+}}\right)-\left({\sigma }_{ij}^{inf{T}^{-}}+{\sigma }_{ij}^{inf{I}^{-}}-{\sigma }_{ij}^{sup{F}^{-}}\right)\right]}$   9:     $\mathbb{E}\left({\mathbb{B}}_A^i\right)\leftarrow {\displaystyle \frac{{\mathbb{E}}^O\left({\mathbb{B}}_A^i\right)+{\mathbb{E}}^P\left({\mathbb{B}}_A^i\right)}{2}}$ 10: end for 11: Rank the alternatives in ascending order of $\mathbb{E}\left({\mathbb{B}}_A^i\right)$ 12: $u^{*}\leftarrow arg{min}_{1\le i\le n}\mathbb{E}\left({\mathbb{B}}_A^i\right)$                                            ▹ ties broken by the smaller ${\mathbb{E}}^O$ 13: return the ranking and $u^{*}$

6.1. Case Study 1—Sensor Selection Problem

In the observed example, the set of available sensors is given as $U=\{u_1,u_2,u_3,u_4\}$. Each of these alternatives is evaluated based on the set of parameters $A=\{e_1,e_2,e_3,e_4\}$. The data quantifying these relationships is organized in the form of IVBNSS ${\mathbb{B}}_A$, presented in Table 1. This set serves as the initial input for the decision-making algorithm.

The next step in the algorithm involves defining new IVBNSSs ${\mathbb{B}}_A^i$, where each set is constructed by removing an individual alternative from the set U, i.e., by forming the set $U\setminus \left\{u_i\right\}$. For example, Table 3 presents the set ${\mathbb{B}}_A^1$, obtained by removing the sensor $u_1$ from the initial set ${\mathbb{B}}_A$. Similarly, the sets ${\mathbb{B}}_A^2$, ${\mathbb{B}}_A^3$, and ${\mathbb{B}}_A^4$ are obtained.

Table 3. IVBNSS ${\mathbb{B}}_A^1$.

${\mathbb{B}}_{\mathit{A}}$	${\mathit{e}}_1$
$u_2$	$\left(\right[0.9,1.0],[0.8,1.0],[0.8,0.9],[-0.9,-0.7],[-1.0,-0.8],[-0.9,-0.8\left]\right)$
$u_3$	$\left(\right[0.4,0.6],[0.2,0.4],[0.3,0.5],[-0.3,-0.1],[-0.4,-0.3],[-0.6,-0.3\left]\right)$
$u_4$	$\left(\right[0.8,1.0],[0.8,0.9],[0.7,0.9],[-1.0,-0.1],[-0.9,-0.7],[-0.9,-0.8\left]\right)$
	${\mathit{e}}_{\mathbf{2}}$
$u_2$	$\left(\right[1.0,1.0],[0.8,1.0],[1.0,1.0],[-0.9,-0.8],[-1.0,-0.7],[-0.9,-0.7\left]\right)$
$u_3$	$\left(\right[0.4,0.7],[0.3,0.6],[0.5,0.6],[-0.7,-0.4],[-0.8,-0.5],[-0.7,-0.4\left]\right)$
$u_4$	$\left(\right[0.8,1.0],[0.8,0.9],[0.7,0.9],[-1.0,-0.9],[-0.9,-0.8],[-0.8,-0.6\left]\right)$
	${\mathit{e}}_{\mathbf{3}}$
$u_2$	$\left(\right[1.0,1.0],[0.8,0.9],[1.0,1.0],[-0.9,-0.8],[-1.0,-0.8],[-0.9,-0.7\left]\right)$
$u_3$	$\left(\right[0.3,0.6],[0.3,0.5],[0.6,0.9],[-0.4,-0.2],[-0.9,-0.8],[-0.5,-0.3\left]\right)$
$u_4$	$\left(\right[0.7,0.9],[0.8,0.9],[0.9,1.0],[-0.8,-0.1],[-0.6,-0.4],[-0.8,-0.7\left]\right)$
	${\mathit{e}}_{\mathbf{4}}$
$u_2$	$\left(\right[1.0,1.0],[0.8,0.9],[1.0,1.0],[-0.7,-0.5],[-1.0,-0.9],[-0.9,-0.7\left]\right)$
$u_3$	$\left(\right[0.4,0.6],[0.3,0.4],[0.6,0.9],[-0.4,-0.2],[-0.9,-0.7],[-0.4,-0.3\left]\right)$
$u_4$	$\left(\right[0.7,0.9],[0.7,0.9],[0.9,1.0],[-0.8,-0.0],[-0.7,-0.5],[-0.8,-0.7\left]\right)$

In accordance with Algorithm 1, for each IVBNSS ${\mathbb{B}}_A^i$, the corresponding infimum and supremum matrices are formed. Specifically, the following matrices are constructed: ${\mathrm{\Psi }}_{{\mathbb{B}}_A^i}^{inf{X}^{+}}$, ${\mathrm{\Psi }}_{{\mathbb{B}}_A^i}^{inf{X}^{-}}$, ${\mathrm{\Psi }}_{{\mathbb{B}}_A^i}^{sup{X}^{+}}$, and ${\mathrm{\Psi }}_{{\mathbb{B}}_A^i}^{sup{X}^{-}}$, where $X\in \{T,I,F\}.$ Below, we present the matrices corresponding to ${\mathbb{B}}_A^1$:

$${\mathrm{\Psi }}_{{\mathbb{B}}_A^1}^{inf{T}^{+}}=\left[\begin{array}{cccc}0.9& 1.0& 1.0& 1.0\\ 0.4& 0.4& 0.3& 0.4\\ 0.8& 0.8& 0.7& 0.7\end{array}\right],{\mathrm{\Psi }}_{{\mathbb{B}}_A^1}^{inf{T}^{-}}=\left[\begin{array}{cccc}-0.9& -0.9& -0.9& -0.7\\ -0.3& -0.7& -0.4& -0.4\\ -1.0& -1.0& -0.8& -0.8\end{array}\right],$$

$${\mathrm{\Psi }}_{{\mathbb{B}}_A^1}^{inf{I}^{+}}=\left[\begin{array}{cccc}0.8& 0.8& 0.8& 0.8\\ 0.2& 0.3& 0.3& 0.3\\ 0.8& 0.8& 0.8& 0.7\end{array}\right],{\mathrm{\Psi }}_{{\mathbb{B}}_A^1}^{inf{I}^{-}}=\left[\begin{array}{cccc}-1.0& -1.0& -1.0& -1.0\\ -0.4& -0.8& -0.9& -0.9\\ -0.9& -0.9& -0.6& -0.7\end{array}\right],$$

$${\mathrm{\Psi }}_{{\mathbb{B}}_A^1}^{inf{F}^{+}}=\left[\begin{array}{cccc}0.8& 1.0& 1.0& 1.0\\ 0.3& 0.5& 0.6& 0.6\\ 0.7& 0.7& 0.9& 0.9\end{array}\right],{\mathrm{\Psi }}_{{\mathbb{B}}_A^1}^{inf{F}^{-}}=\left[\begin{array}{cccc}-0.9& -0.9& -0.9& -0.9\\ -0.6& -0.7& -0.5& -0.4\\ -0.9& -0.8& -0.8& -0.8\end{array}\right],$$

$${\mathrm{\Psi }}_{{\mathbb{B}}_A^1}^{sup{T}^{+}}=\left[\begin{array}{cccc}1.0& 1.0& 1.0& 1.0\\ 0.6& 0.7& 0.6& 0.6\\ 1.0& 1.0& 0.9& 0.9\end{array}\right],{\mathrm{\Psi }}_{{\mathbb{B}}_A^1}^{sup{T}^{-}}=\left[\begin{array}{cccc}-0.7& -0.8& -0.8& -0.5\\ -0.1& -0.4& -0.2& -0.2\\ -0.1& -0.9& -0.1& -0.0\end{array}\right],$$

$${\mathrm{\Psi }}_{{\mathbb{B}}_A^1}^{sup{I}^{+}}=\left[\begin{array}{cccc}1.0& 1.0& 0.9& 0.9\\ 0.4& 0.6& 0.5& 0.4\\ 0.9& 0.9& 0.9& 0.9\end{array}\right],{\mathrm{\Psi }}_{{\mathbb{B}}_A^1}^{sup{I}^{-}}=\left[\begin{array}{cccc}-0.8& -0.7& -0.8& -0.9\\ -0.3& -0.5& -0.8& -0.7\\ -0.7& -0.8& -0.4& -0.5\end{array}\right],$$

$${\mathrm{\Psi }}_{{\mathbb{B}}_A^1}^{sup{F}^{+}}=\left[\begin{array}{cccc}0.9& 1.0& 1.0& 1.0\\ 0.5& 0.6& 0.9& 0.9\\ 0.9& 0.9& 1.0& 1.0\end{array}\right],{\mathrm{\Psi }}_{{\mathbb{B}}_A^1}^{sup{F}^{-}}=\left[\begin{array}{cccc}-0.8& -0.7& -0.7& -0.7\\ -0.3& -0.4& -0.3& -0.3\\ -0.8& -0.6& -0.7& -0.7\end{array}\right].$$

Next, the singular values of each matrix are computed and denoted by ${\sigma }_{ij}^{inf{X}^{+}}$, ${\sigma }_{ij}^{inf{X}^{-}}$, ${\sigma }_{ij}^{sup{X}^{+}}$, and ${\sigma }_{ij}^{sup{X}^{-}}$, where $X\in \{T,I,F\}$ and $j=1\dots k$, $k=min(n-1,m)$. These singular values are used to quantify the contribution of each alternative and to determine the optimistic and pessimistic energy of the set. The singular values corresponding to the matrices of the set ${\mathbb{B}}_A^1$ are

$$\begin{array}{c}{\sigma }_{11}^{inf{T}^{+}}=2.5728,{\sigma }_{12}^{inf{T}^{+}}=0.1290,{\sigma }_{13}^{inf{T}^{+}}=0.0618,\\ {\sigma }_{11}^{inf{T}^{-}}=2.6482,{\sigma }_{12}^{inf{T}^{-}}=0.2692,{\sigma }_{13}^{inf{T}^{-}}=0.1198,\\ {\sigma }_{11}^{inf{I}^{+}}=2.2952,{\sigma }_{12}^{inf{I}^{+}}=0.0967,{\sigma }_{13}^{inf{I}^{+}}=0.0509,\\ {\sigma }_{11}^{inf{I}^{-}}=2.9454,{\sigma }_{12}^{inf{I}^{-}}=0.4486,{\sigma }_{13}^{inf{I}^{-}}=0.1158,\\ {\sigma }_{11}^{inf{F}^{+}}=2.6952,{\sigma }_{12}^{inf{F}^{+}}=0.1423,{\sigma }_{13}^{inf{F}^{+}}=0.1253,\\ {\sigma }_{11}^{inf{F}^{-}}=2.6807,{\sigma }_{12}^{inf{F}^{-}}=0.1991,{\sigma }_{13}^{inf{F}^{-}}=0.0631,\\ {\sigma }_{11}^{sup{T}^{+}}=3.0299,{\sigma }_{12}^{sup{T}^{+}}=0.0784,{\sigma }_{13}^{sup{T}^{+}}=0.0595,\\ {\sigma }_{11}^{sup{T}^{-}}=1.6467,{\sigma }_{12}^{sup{T}^{-}}=0.6127,{\sigma }_{13}^{sup{T}^{-}}=0.1134,\\ {\sigma }_{11}^{sup{I}^{+}}=2.7862,{\sigma }_{12}^{sup{I}^{+}}=0.1507,{\sigma }_{13}^{sup{I}^{+}}=0.0678,\\ {\sigma }_{11}^{sup{I}^{-}}=2.3101,{\sigma }_{12}^{sup{I}^{-}}=0.4744,{\sigma }_{13}^{sup{I}^{-}}=0.1687,\\ {\sigma }_{11}^{sup{F}^{+}}=3.0965,{\sigma }_{12}^{sup{F}^{+}}=0.2628,{\sigma }_{13}^{sup{F}^{+}}=0.0539,\\ {\sigma }_{11}^{sup{F}^{-}}=2.1218,{\sigma }_{12}^{sup{F}^{-}}=0.1335,{\sigma }_{13}^{sup{F}^{-}}=0.0150.\end{array}$$

The algorithm then proceeds with computing the optimistic and pessimistic energy for each of the sets ${\mathbb{B}}_A^i$. Based on the previously mentioned singular values corresponding to the set ${\mathbb{B}}_A^1$, the optimistic and pessimistic energy of the set ${\mathbb{B}}_A^1$ is

$${\mathbb{E}}^O\left({\mathbb{B}}_A^1\right)={\displaystyle \sum _{j=1}^3}(({\sigma }_{1j}^{sup{T}^{+}}+{\sigma }_{1j}^{sup{I}^{+}}-{\sigma }_{1j}^{inf{F}^{+}})-({\sigma }_{1j}^{sup{T}^{-}}+{\sigma }_{1j}^{sup{I}^{-}}-{\sigma }_{1j}^{inf{F}^{-}}))=0.8265,$$

$${\mathbb{E}}^P\left({\mathbb{B}}_A^1\right)={\displaystyle \sum _{j=1}^3}(({\sigma }_{1j}^{inf{T}^{+}}+{\sigma }_{1j}^{inf{I}^{+}}-{\sigma }_{1j}^{sup{F}^{+}})-({\sigma }_{1j}^{inf{T}^{-}}+{\sigma }_{1j}^{inf{I}^{-}}-{\sigma }_{1j}^{sup{F}^{-}}))=-2.4833.$$

In the same manner, the optimistic and pessimistic energies are obtained for the sets ${\mathbb{B}}_A^2$, ${\mathbb{B}}_A^3$, and ${\mathbb{B}}_A^4$:

$$\begin{array}{c}{\mathbb{E}}^O\left({\mathbb{B}}_A^2\right)=2.8800,{\mathbb{E}}^P\left({\mathbb{B}}_A^2\right)=-3.7544,\\ {\mathbb{E}}^O\left({\mathbb{B}}_A^3\right)=2.2779,{\mathbb{E}}^P\left({\mathbb{B}}_A^3\right)=-2.4288,\\ {\mathbb{E}}^O\left({\mathbb{B}}_A^4\right)=2.2684,{\mathbb{E}}^P\left({\mathbb{B}}_A^4\right)=-3.3849.\end{array}$$

Finally, the total energy of each set ${\mathbb{B}}_A^i$ is determined as the average value of the optimistic and pessimistic energy:

$$\mathbb{E}\left({\mathbb{B}}_A^1\right)=-0.8284,\mathbb{E}\left({\mathbb{B}}_A^2\right)=-0.4372,\mathbb{E}\left({\mathbb{B}}_A^3\right)=-0.0755,\mathbb{E}\left({\mathbb{B}}_A^4\right)=-0.5584.$$

This step provides the final assessment of the significance of each alternative in the context of the overall system.

Finally, the alternative with the lowest total energy is determined by ranking the alternatives in ascending order of their total energies:

$$\mathbb{E}\left({\mathbb{B}}_A^1\right)<\mathbb{E}\left({\mathbb{B}}_A^4\right)<\mathbb{E}\left({\mathbb{B}}_A^2\right)<\mathbb{E}\left({\mathbb{B}}_A^3\right)$$

Based on the obtained results, it is possible to establish a linear ordering among the sensors:

$$u_1\succ u_4\succ u_2\succ u_3.$$

(6)

The alternative $u_1$, which has the lowest energy value, is considered the optimal choice, as its contribution to the system is the most significant compared to the other alternatives.

6.2. Case Study 2—Selection of Bio-Inspired Thin-Wall Structures

To further validate the effectiveness of the proposed IVBNSS energy-based decision-making approach, we consider a real engineering problem previously introduced by Zhang et al. [38]. The problem concerns the selection of an optimal bio-inspired thin-wall structure for energy absorption in high-speed railway applications.

The original study considered five bio-inspired thin-wall structural alternatives evaluated according to seven criteria: specific energy absorption (SEA), ease of production, operational life, production efficiency, initial peak crushing force (IPCF), undulation of load-carrying capacity (ULC), and lightweight level. The first four criteria are benefit-type, whereas the remaining three are cost-type. The evaluation was performed by three experts using interval-valued neutrosophic soft numbers with equal expert weights.

The original study employed three independent experts whose assessments were aggregated into a single interval-valued neutrosophic decision matrix (Table S4, Supplementary Material A in [38]). In the present study, this aggregated decision matrix is adopted as the input data. The complete input data, represented in the IVBNSS formalism, are provided in Appendix A (Table A1). Since the original model does not incorporate bipolar information, the negative membership intervals are initialized with the most general interval $[-1,0]$, covering the entire admissible domain of the negative membership functions. This initialization represents the absence of prior knowledge about the negative aspect of the evaluations and, therefore, does not bias the decision process, while preserving all information contained in the original positive assessments.

The considered benchmark problem involves five alternatives $U=\{u_1,u_2,u_3,u_4,u_5\}$, evaluated according to seven criteria $A=\{e_1,e_2,e_3,e_4,e_5,e_6,e_7\}$. The proposed IVBNSS energy-based decision-making algorithm described in the previous section is applied to the aggregated decision matrix to obtain the final ranking of the alternatives. For each alternative $u_i\in U$, a corresponding interval-valued bipolar neutrosophic soft set ${\mathbb{B}}_A^i$ over the universe $U\setminus \left\{u_i\right\}$ is constructed. The corresponding matrices are constructed, and the singular values of each matrix are determined. Based on these values, the optimistic and pessimistic energies of all alternatives are calculated. Finally, the overall energy values are obtained and used to rank the alternatives.

The total energy of each set ${\mathbb{B}}_A^i$ is determined as the average value of the optimistic and pessimistic energy:

$$\mathbb{E}\left({\mathbb{B}}_A^1\right)=2.2095,\mathbb{E}\left({\mathbb{B}}_A^2\right)=2.2967,\mathbb{E}\left({\mathbb{B}}_A^3\right)=2.4374,\mathbb{E}\left({\mathbb{B}}_A^4\right)=2.2456,\mathbb{E}\left({\mathbb{B}}_A^5\right)=1.8300.$$

Based on the obtained results, it is possible to establish a linear ordering among the structures:

$$u_5\succ u_1\succ u_4\succ u_2\succ u_3.$$

(7)

The alternative $u_5$, which has the lowest energy value, is considered the optimal choice, as its contribution to the system is the most significant compared to the other alternatives.

7. Analysis and Comparison of the Obtained Results

When creating an innovative decision-making algorithm, it is necessary to ensure that it meets certain criteria. Of course, the efficiency of each newly developed algorithm is assumed. Through the examples in this section, the superiority of the energy-based algorithm will be demonstrated in comparison to the algorithms developed by Deli et al. [48] and Abdel-Basset et al. [49].

7.1. Deli et al.’s Method and Its Limitations

This subsection focuses on the analysis of the algorithm presented in [48], which was primarily developed for decision-making in an interval-valued bipolar neutrosophic environment. However, the algorithm can also be applied in an interval-valued bipolar neutrosophic soft environment. The objective of this subsection is to identify the weaknesses of the observed algorithm. By analyzing a concrete example, the weakness of the algorithm is highlighted in its inability to identify the optimal alternative. On the other hand, using the same example, we will identify the optimal alternative by applying the energy-based algorithm, leading to the conclusion that the problem of determining the optimal alternative is resolved. Let us consider the following example:

Example 7. 

Let us consider the universal set $U=\{u_1,u_2,u_3,u_4\}$ consisting of four alternatives, and let $A=\{e_1,e_2\}$ be the set of parameters (attributes). The tabular representation of the obtained interval-valued bipolar neutrosophic soft set (IVBNSS) ${\mathbb{B}}_A$ is presented in Table 4.

Table 4. IVBNSS ${\mathbb{B}}_A$ from Example 7.

${\mathbb{B}}_{\mathit{A}}$	${\mathit{e}}_1$
$u_1$	$\left(\right[0.4,0.7],[0.2,0.6],[0.6,0.8],[-0.6,-0.4],[-0.6,-0.4],[-0.8,-0.6\left]\right)$
$u_2$	$\left(\right[0.2,0.7],[0.3,0.8],[0.1,0.3],[-0.8,-0.4],[-0.9,-0.4],[-0.8,-0.6\left]\right)$
$u_3$	$\left(\right[0.4,0.7],[0.6,0.8],[0.2,0.6],[-0.3,-0.2],[-0.8,-0.6],[-0.6,-0.4\left]\right)$
$u_4$	$\left(\right[0.2,0.7],[0.1,0.3],[0.3,0.8],[-0.4,-0.2],[-0.8,-0.6],[-0.9,-0.4\left]\right)$
	${\mathit{e}}_{\mathbf{2}}$
$u_1$	$\left(\right[0.6,0.8],[0.3,0.4],[0.2,0.6],[-0.4,-0.2],[-0.6,-0.4],[-0.8,-0.1\left]\right)$
$u_2$	$\left(\right[0.7,0.8],[0.4,0.6],[0.6,0.8],[-0.3,-0.2],[-0.6,-0.4],[-0.2,-0.1\left]\right)$
$u_3$	$\left(\right[0.6,0.8],[0.2,0.6],[0.3,0.4],[-0.8,-0.4],[-0.8,-0.1],[-0.6,-0.4\left]\right)$
$u_4$	$\left(\right[0.7,0.8],[0.6,0.8],[0.4,0.6],[-0.6,-0.4],[-0.2,-0.1],[-0.6,-0.4\left]\right)$

First, let us attempt to determine the optimal alternative by applying the algorithm developed by Deli et al. [48], assuming that each attribute is equally valuable, meaning that the attribute weights are $0.5$. By applying the given algorithm, we obtain the score values $S\left(u_i\right)$, which should provide the ranking of alternatives by priority. Since $S\left(u_1\right)=0.5765$, $S\left(u_2\right)=0.5765$, $S\left(u_3\right)=0.5765$, and $S\left(u_4\right)=0.5765$, we conclude that all alternatives are ranked equally according to the criteria of this algorithm, making it impossible to identify the optimal alternative.

On the other hand, if we apply the energy-based algorithm (Algorithm 1), we find the values

$$\mathbb{E}\left({\mathbb{B}}_A^1\right)=0.3652,\mathbb{E}\left({\mathbb{B}}_A^2\right)=0.5118,\mathbb{E}\left({\mathbb{B}}_A^3\right)=0.2084,\mathbb{E}\left({\mathbb{B}}_A^4\right)=0.0583.$$

The results obtained by applying both algorithms, and the key conclusions, are presented in Table 5.

Table 5. Comparison of the decision values, rankings, and optimal alternatives obtained by the two decision-making procedures.

Procedure	The Decision Values	Obtained Ranking	Optimal s.
Deli et al. [48]	$S\left(u_1\right)=0.5765,S\left(u_2\right)=0.5765,$	$u_1\sim u_2\sim u_3\sim u_4$	×
	$S\left(u_3\right)=0.5765,S\left(u_4\right)=0.5765,$
Method based on $\mathbb{E}$	$\mathbb{E}\left({\mathbb{B}}_A^1\right)=0.3652,\mathbb{E}\left({\mathbb{B}}_A^2\right)=0.5118,$	$u_4\succ u_3\succ u_1\succ u_2$	$u_4$
	$\mathbb{E}\left({\mathbb{B}}_A^3\right)=0.2084,\mathbb{E}\left({\mathbb{B}}_A^4\right)=0.0583,$

From Example 7, we can conclude that the main drawback of the algorithm presented in [48] is its inability to identify the optimal alternative in every case. In instances where both algorithms successfully identify the optimal alternative, the result is the same, which confirms the efficiency of the energy-based algorithm. A comparison of the results of various algorithms will be discussed in Section 7.2.

7.2. Abdel-Basset et al.’s Method and Its Limitations

This subsection analyzes the cosine similarity-based decision-making methods proposed by Abdel-Basset et al. [49]. These methods are among the most effective approaches developed for interval-valued bipolar neutrosophic environments and have demonstrated excellent performance in numerous decision-making problems. The purpose of this subsection is not to question their general effectiveness but to illustrate that there exist particular situations in which all three cosine similarity measures fail to distinguish between different alternatives. For this purpose, we construct a counterexample in which the cosine similarity measures assign identical values to all alternatives, whereas the proposed energy-based algorithm is still able to produce a unique ranking.

Example 8. 

Let $U=\{u_1,u_2,u_3\}$ be the set of alternatives and let $A=\{e_1,e_2\}$ be the set of parameters. The corresponding IVBNSS ${\mathbb{B}}_A$ is given in Table 6. Throughout this example, equal attribute weights are assumed, i.e., $w_1=w_2=0.5$.

Table 6. IVBNSS ${\mathbb{B}}_A$ from Example 8.

${\mathbb{B}}_{\mathit{A}}$	${\mathit{e}}_1$
$u_1$	$\left(\right[0.3,0.4],[0.2,0.9],[0.3,0.8],[-0.8,-0.0],[-0.9,-0.0],[-0.8,-0.6\left]\right)$
$u_2$	$\left(\right[0.3,0.4],[0.2,0.9],[0.3,0.8],[-0.8,-0.0],[-0.9,-0.0],[-0.8,-0.6\left]\right)$
$u_3$	$\left(\right[0.3,0.4],[0.2,0.9],[0.3,0.8],[-0.8,-0.0],[-0.9,-0.0],[-0.8,-0.6\left]\right)$
	${\mathit{e}}_{\mathbf{2}}$
$u_1$	$\left(\right[0.3,0.4],[0.3,0.8],[0.2,0.9],[-0.8,-0.0],[-0.9,-0.0],[-0.8,-0.6\left]\right)$
$u_2$	$\left(\right[0.3,0.4],[0.2,0.9],[0.3,0.8],[-0.8,-0.0],[-0.8,-0.6],[-0.9,-0.0\left]\right)$
$u_3$	$\left(\right[0.3,0.4],[0.2,0.9],[0.3,0.8],[-0.8,-0.0],[-0.9,-0.0],[-0.8,-0.6\left]\right)$

Applying the three cosine similarity measures proposed by Abdel-Basset et al. [49], we obtain

$$C{S}_1\left(u_1\right)=C{S}_1\left(u_2\right)=C{S}_1\left(u_3\right)=0.707107,$$

$$C{S}_2\left(u_1\right)=C{S}_2\left(u_2\right)=C{S}_2\left(u_3\right)=0.707107,$$

and

$$C{S}_3\left(u_1\right)=C{S}_3\left(u_2\right)=C{S}_3\left(u_3\right)=0.707107.$$

Hence, all three cosine similarity measures assign identical decision values to the three alternatives, yielding the ranking $u_1\sim u_2\sim u_3,$ which means that no unique optimal alternative can be identified. The inability of the cosine similarity measures to distinguish the alternatives stems from the fact that the corresponding aggregated deviation values are identical for all three alternatives, despite their different internal interval-valued bipolar neutrosophic structures.

Next, we apply the proposed energy-based algorithm. The optimistic and pessimistic energy values are

$$\begin{array}{cccc}\hfill {\mathbb{E}}^O\left({\mathbb{B}}_A^1\right)& =3.1000,\hfill & \hfill {\mathbb{E}}^P\left({\mathbb{B}}_A^1\right)& =-2.6611,\hfill \\ \hfill {\mathbb{E}}^O\left({\mathbb{B}}_A^2\right)& =3.5945,\hfill & \hfill {\mathbb{E}}^P\left({\mathbb{B}}_A^2\right)& =-2.8000,\hfill \\ \hfill {\mathbb{E}}^O\left({\mathbb{B}}_A^3\right)& =3.0945,\hfill & \hfill {\mathbb{E}}^P\left({\mathbb{B}}_A^3\right)& =-2.6611.\hfill \end{array}$$

Since the total energy is defined as the arithmetic mean of the optimistic and pessimistic energies, we obtain

$$\begin{array}{cc}\hfill \mathbb{E}\left({\mathbb{B}}_A^1\right)& ={\displaystyle \frac{3.1000+(-2.6611)}{2}}=0.2194,\hfill \\ \hfill \mathbb{E}\left({\mathbb{B}}_A^2\right)& ={\displaystyle \frac{3.5945+(-2.8000)}{2}}=0.3972,\hfill \\ \hfill \mathbb{E}\left({\mathbb{B}}_A^3\right)& ={\displaystyle \frac{3.0945+(-2.6611)}{2}}=0.2167.\hfill \end{array}$$

Therefore, $u_3\succ u_1\succ u_2,$ and the proposed energy-based algorithm uniquely identifies $u_3$ as the optimal alternative.

The results obtained by applying the three cosine similarity measures proposed by Abdel-Basset et al. [49] and the proposed energy-based method, together with the corresponding rankings and optimal solutions, are summarized in Table 7.

Table 7. Comparison of the decision values, rankings, and optimal solutions obtained by the cosine similarity measures and the proposed energy-based method.

Procedure	Decision Values	Obtained Ranking	Optimal Solution
	${CS}_1\left(u_1\right)=0.707107,$
Abdel-Basset et al. [49] (${CS}_1$)	${CS}_1\left(u_2\right)=0.707107,$	$u_1\sim u_2\sim u_3$	×
	${CS}_1\left(u_3\right)=0.707107$
	${CS}_2\left(u_1\right)=0.707107,$
Abdel-Basset et al. [49] (${CS}_2$)	${CS}_2\left(u_2\right)=0.707107,$	$u_1\sim u_2\sim u_3$	×
	${CS}_2\left(u_3\right)=0.707107$
	${CS}_3\left(u_1\right)=0.707107,$
Abdel-Basset et al. [49] (${CS}_3$)	${CS}_3\left(u_2\right)=0.707107,$	$u_1\sim u_2\sim u_3$	×
	${CS}_3\left(u_3\right)=0.707107$
	$\mathbb{E}\left({\mathbb{B}}_A^1\right)=0.2194,$
Method based on $\mathbb{E}$	$\mathbb{E}\left({\mathbb{B}}_A^2\right)=0.3972,$	$u_3\succ u_1\succ u_2$	$u_3$
	$\mathbb{E}\left({\mathbb{B}}_A^3\right)=0.2167$

Example 8 illustrates a limitation of the cosine similarity measures proposed by Abdel-Basset et al. [49]. Although the considered alternatives have different internal distributions of interval-valued bipolar neutrosophic information, all three cosine similarity measures assign identical values to them, resulting in a tied ranking and making it impossible to identify a unique optimal alternative. In contrast, the proposed energy-based method successfully distinguishes the alternatives and determines a unique optimal solution, demonstrating that the proposed energy measure is more sensitive to variations in the internal distribution of interval-valued bipolar neutrosophic information. A broader comparative analysis with other decision-making methods is presented in Section 7.3.

7.3. Comparative Analysis of the Proposed Method with Other Methods

The objective of this section is a comparative analysis of the results obtained by applying the energy-based algorithm and the results obtained by applying the algorithms developed by Deli et al. [48] and Abdel-Basset et al. [49]. To this end, let us consider the following example involving a medical diagnosis problem, as examined in [49].

Example 9. 

Let $U=\{u_1,u_2,u_3,u_4\}$ be a group of individuals with different mood and behavioral conditions described by the following attributes: $e_1$ (happiness), $e_2$ (sadness), $e_3$ (energy), and $e_4$ (sleeping rate). We aim to assess these individuals to determine whether one or more persons suffer from bipolar disorder and subsequently identify patients for treatment. To collect data on the given alternatives and attributes, an interview was conducted with a mental health expert (decision-maker). The physician performed a physical examination to ensure that these symptoms were not caused by other illnesses. After assessing the mental health of the individuals (alternatives) concerning the four conditions (attributes), the obtained data represent an IVBNSS, which is presented in Table 8.

Table 8. IVBNSS ${\mathbb{B}}_A$ from Example 9.

${\mathbb{B}}_{\mathit{A}}$	${\mathit{e}}_1$
$u_1$	$\left(\right[0.3,0.4],[0.2,0.9],[0.3,0.8],[-0.8,-0.0],[-0.9,-0.0],[-0.8,-0.6\left]\right)$
$u_2$	$\left(\right[1.0,1.0],[0.9,1.0],[1.0,1.0],[-0.9,-0.8],[-1.0,-0.9],[-0.9,-0.8\left]\right)$
$u_3$	$\left(\right[0.3,0.6],[0.2,0.3],[0.3,0.4],[-0.3,-0.2],[-0.4,-0.3],[-0.6,-0.3\left]\right)$
$u_4$	$\left(\right[0.9,1.0],[0.9,1.0],[0.8,0.9],[-1.0,-0.0],[-0.9,-0.8],[-0.8,-0.7\left]\right)$
	${\mathit{e}}_{\mathbf{2}}$
$u_1$	$\left(\right[0.0,0.8],[0.1,0.9],[0.2,1.0],[-1.0,-0.0],[-1.0,-0.0],[-0.8,-0.0\left]\right)$
$u_2$	$\left(\right[1.0,1.0],[0.9,1.0],[1.0,1.0],[-0.9,-0.8],[-1.0,-0.9],[-0.9,-0.8\left]\right)$
$u_3$	$\left(\right[0.5,0.6],[0.3,0.5],[0.4,0.6],[-0.7,-0.4],[-0.8,-0.4],[-0.7,-0.5\left]\right)$
$u_4$	$\left(\right[0.9,1.0],[0.9,1.0],[0.8,0.9],[-1.0,-0.0],[-0.9,-0.8],[-0.8,-0.7\left]\right)$
	${\mathit{e}}_{\mathbf{3}}$
$u_1$	$\left(\right[0.3,0.4],[0.2,0.9],[0.3,0.8],[-0.8,-0.0],[-0.9,-0.0],[-0.8,-0.7\left]\right)$
$u_2$	$\left(\right[1.0,1.0],[0.9,1.0],[1.0,1.0],[-0.9,-0.8],[-1.0,-0.9],[-0.9,-0.8\left]\right)$
$u_3$	$\left(\right[0.4,0.5],[0.2,0.4],[0.7,0.9],[-0.4,-0.2],[-0.9,-0.7],[-0.5,-0.4\left]\right)$
$u_4$	$\left(\right[0.8,0.9],[0.7,0.9],[0.8,0.9],[-0.9,-0.0],[-0.8,-0.7],[-0.8,-0.7\left]\right)$
	${\mathit{e}}_{\mathbf{4}}$
$u_1$	$\left(\right[0.0,0.8],[0.1,0.9],[0.2,1.0],[-1.0,-0.0],[-1.0,-0.0],[-0.8,-0.0\left]\right)$
$u_2$	$\left(\right[1.0,1.0],[0.9,1.0],[1.0,1.0],[-0.9,-0.8],[-1.0,-0.9],[-0.9,-0.8\left]\right)$
$u_3$	$\left(\right[0.4,0.5],[0.2,0.4],[0.7,0.9],[-0.4,-0.2],[-0.9,-0.7],[-0.5,-0.4\left]\right)$
$u_4$	$\left(\right[0.8,0.9],[0.7,0.9],[0.8,0.9],[-0.9,-0.0],[-0.8,-0.7],[-0.8,-0.7\left]\right)$

Abdel-Basset et al. [49] developed three algorithms based on cosine similarity measures, which were shown to be effective in determining the optimal alternative, as demonstrated in the problem described in Example 9. If we apply both the algorithm developed by Deli et al. [48] and the energy-based algorithm to the same example, we obtain sufficiently differentiated values to rank the alternatives. The values obtained using all observed algorithms are presented in Table 9.

Table 9. The table provides the values that determine the ranking of alternatives and the optimal solution.

Procedure	The Decision Values
Deli et al. [48]	$S\left(u_1\right)=0.4961,S\left(u_2\right)=0.5000,$
	$S\left(u_3\right)=0.5734,S\left(u_4\right)=0.5484,$
Abdel-Basset et al. [49] (${CS}_1$)	${CS}_1^w(u_1,I{d}_B^{*})=0.9651,C{S}_1^w(u_2,I{d}_B^{*})=0.7431,$
	${CS}_1^w(u_3,I{d}_B^{*})=0.8010,C{S}_1^w(u_4,I{d}_B^{*})=0.8314,$
Abdel-Basset et al. [49] (${CS}_2$)	${CS}_2^w(u_1,I{d}_B^{*})=0.9297,C{S}_2^w(u_2,I{d}_B^{*})=0.7071,$
	${CS}_2^w(u_3,I{d}_B^{*})=0.8704,C{S}_2^w(u_4,I{d}_B^{*})=0.7847,$
Abdel-Basset et al. [49] (${CS}_3$)	${CS}_3^w(u_1,I{d}_B^{*})=0.9297,C{S}_3^w(u_2,I{d}_B^{*})=0.7071,$
	${CS}_3^w(u_3,I{d}_B^{*})=0.8077,C{S}_3^w(u_4,I{d}_B^{*})=0.7604,$
Method based on $\mathbb{E}$	$\mathbb{E}\left({\mathbb{B}}_A^1\right)=-0.1087,\mathbb{E}\left({\mathbb{B}}_A^2\right)=0.0125,$
	$\mathbb{E}\left({\mathbb{B}}_A^3\right)=-0.0003,\mathbb{E}\left({\mathbb{B}}_A^4\right)=-0.0875.$

The values in Table 9 allow the optimal alternatives produced by the considered algorithms to be compared. In this particular example, we can conclude that the optimal solution is alternative $u_1$ when applying the algorithms developed by [49] and the energy-based algorithm, whereas applying the algorithm presented by [48] results in $u_3$ as the optimal alternative. The overall ranking of alternatives, along with the optimal solutions, is provided in Table 10. Notably, the algorithm utilizing the similarity measure $C{S}_1$ produces the same ranking of alternatives as the energy-based algorithm.

Table 10. The table with the rankings of alternatives and the optimal solution.

Procedure	Obtained Ranking	Optimal Solution
Deli et al. [48]	$u_3\succ u_4\succ u_2\succ u_1$	$u_3$
Abdel-Basset et al. [49] (${CS}_1$)	$u_1\succ u_4\succ u_3\succ u_2$	$u_1$
Abdel-Basset et al. [49] (${CS}_2$)	$u_1\succ u_3\succ u_4\succ u_2$	$u_1$
Abdel-Basset et al. [49] (${CS}_3$)	$u_1\succ u_3\succ u_4\succ u_2$	$u_1$
Method based on $\mathbb{E}$	$u_1\succ u_4\succ u_3\succ u_2$	$u_1$

The results and the resulting linear rankings reveal several characteristics of the considered algorithms. The considered characteristics include the identification of the optimal alternative, the degree to which the decision values distinguish among alternatives, and an overall insight into the distribution of the intervals $T^{+}$, $I^{+}$, $F^{+}$ on one side and the intervals $T^{-}$, $I^{-}$, $F^{-}$ on the other.

Based on the results presented in Section 7.1, Section 7.2 and Section 7.3, several important observations can be made:

• The proposed energy-based algorithm successfully identifies a unique optimal alternative even in decision-making situations where the score function of Deli et al. [48] or the cosine similarity measures proposed by Abdel-Basset et al. [49] fail to distinguish between alternatives. Moreover, in practical examples where the existing methods identify the same optimal alternative, the proposed algorithm produces consistent results, demonstrating its reliability as a decision-making tool in IVBNSS environments.
• Similar to the existing decision-making approaches, the proposed method provides finely differentiated decision values, enabling a complete linear ranking of alternatives according to their priorities.
• Unlike the compared methods, the proposed algorithm is based on optimistic and pessimistic energy values, whose arithmetic mean provides additional information about the internal distribution of interval-valued bipolar neutrosophic information. Consequently, in addition to determining the optimal alternative, the proposed method also offers insight into the balance between positive and negative interval-valued bipolar neutrosophic components, which cannot be directly inferred from the compared score- and similarity-based approaches.
• From the computational point of view, the proposed algorithm is more demanding than score- or similarity-based methods because it requires repeated singular value decompositions within the leave-one-out procedure. Nevertheless, its computational complexity remains polynomial, i.e., $O(n^{2}mmin(n,m))$, making it computationally feasible for practical decision-making problems involving moderate and relatively large numbers of alternatives and attributes.
• The illustrative examples indicate that the additional computational effort is compensated by improved discriminative capability. In particular, the proposed energy-based criterion is able to distinguish between alternatives that remain indistinguishable under representative score- and similarity-based methods. This suggests that the algorithm provides a more informative global evaluation of the underlying IVBNSS structure while preserving computational tractability.

Although the present study focuses on representative benchmark examples, the proposed framework can be naturally extended to larger decision-making problems. A systematic robustness analysis with respect to perturbations of interval-valued bipolar neutrosophic information and a broader experimental comparison with additional state-of-the-art decision-making methods constitute interesting directions for future research.

For convenience, the main characteristics and limitations of the considered decision-making methods are summarized in Table 11.

Table 11. Qualitative comparative analysis: main characteristics of the methods considered in this work, in view of the computations shown in Section 7.

Procedure	Ranking Methodology	Unique Solution	Detailed Assessment During Selection	Reflects Interval Structure
Deli et al. [48]	The score function S	No	Yes	No
Abdel-Basset et al. [49] (${CS}_1$)	Similarity measure ${CS}_1$	No	Yes	No
Abdel-Basset et al. [49] (${CS}_2$)	Similarity measure ${CS}_2$	No	Yes	No
Abdel-Basset et al. [49] (${CS}_3$)	Similarity measure ${CS}_3$	No	Yes	No
Algorithm 1	Comparison of energy values	Yes	Yes	Yes

8. Conclusions

Continuous technological development and the limitations of existing data-modeling tools create an increasing need for mathematical frameworks capable of addressing uncertainty, unreliability, and imprecision. Although interval-valued bipolar neutrosophic sets have been successfully applied, they lack an explicit parametric framework. To address this limitation, we introduced the concept of the interval-valued bipolar neutrosophic soft set as a structure resulting from the hybridization of interval-valued bipolar neutrosophic sets and soft sets. Furthermore, we established its fundamental algebraic framework by introducing the basic operations on IVBNSSs and proving their principal algebraic properties, thereby providing a solid theoretical foundation for the proposed model.

For the proposed structure, we developed a decision-making algorithm based on a new numerical characteristic of IVBNSSs. We refer to these characteristics as energies because their construction is inspired by graph energy and the nuclear norm. The proposed energy provides a global numerical characterization of the underlying IVBNSS and forms the basis of a leave-one-out decision-making strategy that evaluates the structural contribution of each alternative. The decision-making algorithm was evaluated through several examples, including an additional comparative case study, and its advantages were demonstrated through comparisons with representative score- and similarity-based methods, particularly in situations where the existing approaches fail to identify a unique optimal alternative.

The results provide a foundation for several theoretical and applied research directions. The main opportunities for further development are as follows:

• Further Algebraic Investigations: Although the fundamental algebraic operations and their basic properties have been established in this work, further research may investigate additional algebraic structures, homomorphisms, congruence relations, algebraic identities, and other theoretical aspects of interval-valued bipolar neutrosophic soft sets.
• Weighted Extensions of the IVBNSS Energy: Incorporating unequal attribute weights, e.g., via entropy-based weighting mechanisms, and analyzing their effect on the proposed energy measures represent a natural generalization of the presented framework.
• Sensitivity and Robustness Analysis: A systematic study of the behavior of the proposed energy measures under perturbations of the interval boundaries, based on the perturbation theory of singular values, would provide further insight into the robustness of the energy-based ranking.
• Algorithm Optimization for Large-Scale Systems: As decision-making algorithms become increasingly complex, optimizing algorithms for large-scale systems with high-dimensional data is of great importance.
• Multi-Criteria Decision-Making and Comparative Analysis: The application of the proposed energy-based framework to large-scale multi-criteria decision-making problems, together with systematic comparisons with additional state-of-the-art neutrosophic MADM methods such as MABAC, TODIM, TOPSIS, and VIKOR adapted to the interval-valued bipolar neutrosophic soft setting, represents another promising direction for future research.