Another View on Soft Expert Set and Its Application in Multi-Criteria Decision-Making

Abstract

The soft expert set (SES) is a useful mathematical tool for addressing uncertainty, with significant applications in decision-making. This study identifies some inconsistencies in the original SES framework. The study demonstrates that the majority of SES operations are in conflict with the foundational definition of an SES. Consequently, the definition of an SES is revised to not only resolve the existing complications but also enhance its applicability in real-world problems. In addition, a novel multi-criteria decision-making (MCDM) approach is proposed based on the revised SES framework, along with its concrete algorithm. The proposed algorithm is applied to a supply chain problem, and its results are compared with two existing SES-based decision-making approaches. The results reveal that the proposed algorithm offers a more precise ranking of decision alternatives and has greater applicability as compared to the other two methods. The findings of this study advance the theoretical understanding of the SES and provide a more robust tool for decision-makers in MCDM environments.

1. Introduction

Decision-making is a fundamental activity in human society and occurs in many aspects of daily life. It is a mental process that involves analyzing different options and selecting the one that best meets the goals or needs of the decision-makers. Early research on decision-making mainly focused on deterministic methods, assuming certainty in the process. However, due to the limited ability of humans to fully understand the objective world, uncertainty often arises in decision-making. Understanding and describing this uncertainty accurately has been a challenging task for researchers. To deal with this, several mathematical theories, such as fuzzy set theory [1], interval mathematics theory [2], rough set theory [3], and intuitionistic fuzzy set theory [4], have been developed. These theories have played a vital role in promoting research on decision-making under uncertainty.

Molodtsov [5] initiated the concept of a soft set as a new approach to dealing with uncertainties. Maji et al. [6] later defined the basic operations of soft sets and explored their basic properties. Soft set theory (SST) has been extended to create more general models such as fuzzy soft sets [7], interval-valued fuzzy soft sets [8], intuitionistic fuzzy soft sets [9], and belief interval-valued soft sets [10]. These models have been applied in various fields, including decision-making, artificial intelligence, and engineering (see [11,12,13,14,15,16,17,18,19]). However, the majority of these models are designed to handle data from a single expert and may not be suitable for situations where the decision-making process requires input from multiple experts.

To overcome this issue, Alkhazaleh et al. [20] proposed the soft expert set (SES), which considers the opinions of multiple experts within one model. They developed basic operations for SESs and explored their application to real-world decision-making problems. After their pioneering work, this concept was also combined with other mathematical structures to obtain its extended models for handling more complex decision-making scenarios. These models include vague soft expert sets [21], fuzzy soft expert sets [22], intuitionistic fuzzy soft expert sets [23], bipolar fuzzy soft expert sets [24], and fuzzy parameterized soft expert sets [25]. Various algorithms have been developed through the use of these extensions to enhance the capability of SES-based models in real-world decision-making problems.

Although SESs and their extended models have shown significant progress in recent years, there are still important gaps in the original SES framework. First, the majority of the SES operations, such as subset, union, and intersection, are in conflict with the foundational definition of the SES proposed in [20]. These complications make the SES framework less reliable, and the definition of an SES needs to be revised for better consistency. Second, most SES-based decision-making algorithms are designed for selecting a single alternative from several options. However, in many practical problems, we need the complete ranking of decision alternatives rather than selecting the optimal one. For instance, evaluation of supply chain partners is one such problem that considers different attributes and risk factors to rank the suppliers according to their service level [26,27]. In such cases, the present SES-based algorithms are not effective in providing complete rankings, which creates a gap for an improved SES-based decision-making approach.

This study is motivated by these challenges and aims to improve the SES framework and its application in decision-making. It proposes a revised definition of an SES that not only resolves the existing complications but also enhances its applicability in decision-making. It introduces a new decision-making algorithm that can provide a comprehensive ranking of alternatives. To demonstrate its practical value, the proposed algorithm is applied to a supply chain problem, and the results are compared with existing SES-based algorithms. The specific objectives of the study are as follows:

• To show, through numerical examples, the inconsistencies between the current definition of an SES and its operations.
• To propose a revised definition for an SES.
• To develop a new SES-based decision-making algorithm that provides a complete ranking of decision alternatives.
• To apply the proposed algorithm to a supply chain problem and compare its results with those of existing SES-based decision-making algorithms.

The rest of the paper is organized as follows: Section 2 provides a literature review, while Section 3 recalls some basic concepts related to soft sets and SESs. Section 4 highlights the limitations of Alkhazaleh et al.’s work in [20]. In Section 5, we provide a revised definition for SES and develop a novel algorithm for SES-based decision-making. In addition, we apply the proposed algorithm to a supply chain problem. In Section 6, we compare the proposed algorithm with two existing SES-based decision-making algorithms. Finally, we conclude our study and provide future research directions in Section 7.

2. Literature Review

Soft Set Theory (SST), introduced by Molodtsov [5], provides a mathematical framework to handle uncertainties without the constraints of traditional methods such as fuzzy or rough sets. Maji et al. [6] defined basic operations in SST and investigated several properties of these operations. Ali et al. [28] redefined some of these operations and provided some new results. Over time, SST has been extended and adapted to address decision-making problems, especially those requiring subjective expert opinions. Maji et al. [29] expressed a soft set as a binary table and discussed its applications in decision-making. Maji and Roy [30] presented a fuzzy soft set-based decision-making approach to object recognition from imprecise multi-observer data. A combined forecasting approach based on a fuzzy soft set was introduced by Xiao et al. [31]. Feng et al. [32] introduced the concept of a level soft set and provided an adjustable approach to fuzzy soft set-based decision-making. Jiang et al. [33] developed an adjustable approach to intuitionistic fuzzy soft set-based decision-making by using level soft sets. Cagman and Karatas [34] introduced a novel algorithm for intuitionistic fuzzy soft set-based decision-making. Alcantud et al. [35,36] discussed some relations between fuzzy sets and soft sets and proposed a new algorithm for fuzzy soft set-based decision-making. Fatimah et al. [37] introduced the concepts of probabilistic soft sets and dual probabilistic soft sets in decision-making.

One of the key extensions of SST is the development of soft expert set (SES) theory, first introduced by Alkhazaleh et al. [20]. This framework integrates multiple expert opinions into SST to provide structured and detailed insights for decision-making. The SES framework has undergone significant evolution to address its inherent challenges. Early research focused on defining the core properties of an SES, while subsequent studies integrated different mathematical theories to obtain its useful extensions. For instance, Enginoglu et al. [38] brought some modifications to the work of Alkhazaleh et al. [20] and proposed a new algorithm for SES-based decision-making. Hassan and Alhazaymeh [21] introduced the concept of vague soft expert sets by combining vague sets with soft expert sets. Alkhazaleh and Salleh [22] developed a new hybrid model called the fuzzy soft expert set by combining a fuzzy set with a soft expert set. Broumi and Smarandache [23] developed intuitionistic fuzzy soft expert sets as an extension of fuzzy soft expert sets. Sahin et al. [39] developed the hybrid structure of a neutrosophic soft expert set and discussed its application in decision-making. Al-Quran et al. [40] introduced the idea of a neutrosophic vague soft expert set and discussed its basic operations. Al-Qudah and Hassan [24] combined a bipolar fuzzy set with the soft expert set and developed the generalized model of a bipolar fuzzy soft expert set. Bashir and Salleh [25,41] introduced two novel hybrid models, namely fuzzy parameterized soft expert sets and possibility fuzzy soft expert sets. Qayyum et al. [42] introduced the cubic soft expert set model for decision-making. Adam and Hassan [43] proposed a multi-Q-fuzzy soft expert set model and studied an application to decision-making. In continuation of this effort, several research works have extended this important concept to many other emerging theories, which can be found in [44,45,46,47,48,49,50].

In the last five years, SES research has gained significant momentum, with numerous studies advancing its theoretical foundations and expanding its applications. For instance, Akram et al. [51] and Tchier et al. [51] developed two novel multi-criteria group decision-making techniques under the m-polar fuzzy soft expert sets and picture fuzzy soft expert sets, respectively. Ihsan et al. [52] extended SES theory by introducing a new multi-criteria decision-making (MCDM) approach based on bijective hypersoft expert sets. Akram et al. [53,54] proposed two improved MCDM techniques based on Pythagorean fuzzy N-soft expert and hesitant fuzzy soft expert information. Most recently, Ihsan et al. [55] introduced the idea of the intuitionistic fuzzy hypersoft expert set, which is both an extension and generalization of the intuitionistic fuzzy soft set and hypersoft set. They also proposed a new TOPSIS approach based on their newly defined correlation coefficient and the weighted correlation coefficient of the intuitionistic fuzzy hypersoft expert sets. These studies collectively highlight the growing versatility and impact of SES theory, especially in domains requiring collaborative decision-making frameworks under uncertainty.

3. Preliminaries

To provide a base for our upcoming discussion, this section recalls some basic concepts related to soft sets and SESs. Throughout the paper, X will denote a universal set, E will denote a parameter set, and $P\left(X\right)$ will denote the power set of X.

Definition 1

([5]). A pair $(\psi ,\Lambda )$ is called a soft set over X, where Λ is a nonempty subset of E and ψ is a mapping given by

$$\psi :\Lambda \to P\left(X\right).$$

(1)

Example 1.

Suppose that a person is interested in buying a house among the four houses $X=\{x_i:i=1,2,3,4\}$ under consideration. Let $E=\{e_j:j=1,2,3,4\}$ be the set of parameters with respect to X, where the $e_j(j=1,2,3,4)$ stand for beautiful, new, reliable, and cheap, respectively. If $\Lambda =\{e_1,e_2,e_3\}\subset E$ denotes the choice parameters of the person, then a soft set $(\psi ,\Lambda )$ over X describing “the attractiveness of the houses” can be defined by

$$(\psi ,\Lambda )=\left\{\begin{array}{c}beautifulhouses=\left\{x_1\right\},\hfill \\ newhouses=\{x_3,x_4\},\hfill \\ reliablehouses=\{x_1,x_2,x_3,x_4\}.\hfill \end{array}\right\}$$

(2)

For more illustrative examples and basic operations of soft sets, we refer [5,6]. Next, suppose that $\mathcal{Z}=E\times \mathcal{P}\times \mathcal{O}$, where E is the parameter set w.r.t. X, $\mathcal{P}$ is the set of experts (agents), and $\mathcal{O}$ is the set of opinions. The soft expert set is defined as follows.

Definition 2

([20]). A pair $(\psi ,\Lambda )$ is called a soft expert set (in short SES) over X, where $\Lambda \subseteq \mathcal{Z}$ and ψ is a mapping given by

$$\psi :\Lambda \to P\left(X\right).$$

(3)

For simplicity, $\mathcal{O}=\{0,1\}$ where 0 and 1 stand for “disagree” and “agree”, respectively. Moreover, the collection of all SESs over X will be denoted by $\mathcal{S}\tilde{\mathcal{E}}\mathcal{S}\left(X\right)$.

Example 2.

Assume that some new products were produced by a company and it wants to collect the opinions of different experts about the products. Let $X=\{x_i:i=1,2,\dots ,6\}$ represents the set of new products and $E=\{e_j:j=1,2,3\}$ represents the set parameters w.r.t. X, where each $e_j$ stands for cheap, reliable, and beautiful, respectively. Suppose that a questionnaire is distributed among the three experts $\mathcal{P}=\{p_1,p_2,p_3\}$ asking about their opinions regarding the new products. The approximations obtained from their opinions can be represented by the following SES.

$$(\psi ,\mathcal{Z})=\left\{\begin{array}{c}\psi (e_1,p_1,1)=\{x_1,x_2,x_5\},\psi (e_1,p_2,1)=\{x_2,x_3,x_5,x_6\},\psi (e_1,p_3,1)=\{x_3,x_4,x_6\},\hfill \\ \psi (e_2,p_1,1)=\{x_1,x_4,x_6\},\psi (e_2,p_2,1)=\{x_3,x_4,x_5\},\psi (e_2,p_3,1)=\{x_1,x_5\},\hfill \\ \psi (e_3,p_1,1)=\{x_1,x_4,x_5\},\psi (e_3,p_2,1)=\{x_2,x_4\},\psi (e_3,p_3,1)=\{x_1,x_3,x_5\},\hfill \\ \psi (e_1,p_1,0)=\{x_3,x_4,x_6\},\psi (e_1,p_2,0)=\{x_1,x_4\},\psi (e_1,p_3,0)=\{x_1,x_2,x_5\},\hfill \\ \psi (e_2,p_1,0)=\{x_2,x_3,x_5\},\psi (e_2,p_2,0)=\{x_1,x_2,x_6\},\psi (e_2,p_3,0)=\{x_2,x_3,x_4,x_6\},\hfill \\ \psi (e_3,p_1,0)=\{x_2,x_3,x_6\},\psi (e_3,p_2,0)=\{x_1,x_3,x_5,x_6\},\psi (e_3,p_3,0)=\{x_2,x_4,x_6\}.\hfill \end{array}\right\}$$

(4)

In the above example, we see that the first expert $p_1$ “agrees” that the “cheap” products are $\{x_1,x_2,x_5\}$, while the second and third experts “agree” that the “cheap” products are $\{x_2,x_3,x_5,x_6\}$ and $\{x_3,x_4,x_6\}$, respectively. This implies that an SES allows for the integration of multiple expert opinions for each parameter in a single model, providing greater flexibility compared to the traditional soft set approach.

Definition 3

([20]). Let $(\psi ,\Lambda )$ and $(\chi ,\Theta )\in \widetilde{\mathcal{SES}}\left(X\right)$. Then $(\psi ,\Lambda )$ is said to be a soft expert subset of $(\chi ,\Theta )$, denoted by $(\psi ,\Lambda )\tilde{\subseteq }(\chi ,\Theta )$, if the following holds:

(i) 

$\Lambda \subseteq \Theta$;

(ii) 

$\forall \alpha \in \Lambda$, $\psi \left(\alpha \right)\subseteq \chi \left(\alpha \right)$.

Two SESs $(\psi ,\Lambda )$ and $(\chi ,\Theta )\in \widetilde{\mathcal{SES}}\left(X\right)$ are called equal if $(\psi ,\Lambda )\tilde{\subseteq }(\chi ,\Theta )$ and $(\chi ,\Theta )\tilde{\subseteq }(\psi ,\Lambda )$.

Definition 4

([20]). The union of two SESs $(\psi ,\Lambda )$ and $(\chi ,\Theta )$$\in \widetilde{\mathcal{SES}}\left(X\right)$ is an SES over X, denoted by $(\phi ,{\rm Y})=(\psi ,\Lambda )\tilde{\cup }(\chi ,\Theta )$, where ${\rm Y}=\Lambda \cup \Theta$ and $\forall \alpha \in {\rm Y}$

$$\phi \left(\alpha \right)=\left\{\begin{array}{cc}\psi \left(\alpha \right),\hfill & if\alpha \in \Lambda -\Theta \hfill \\ \chi \left(\alpha \right),\hfill & if\alpha \in \Theta -\Lambda \hfill \\ \psi \left(\alpha \right)\cup \chi \left(\alpha \right),\hfill & if\alpha \in \Lambda \cap \Theta .\hfill \end{array}\right.$$

(5)

Definition 5

([20]). The intersection of two SESs $(\psi ,\Lambda )$ and $(\chi ,\Theta )\in \widetilde{\mathcal{SES}}\left(X\right)$ is an SES over X, which is denoted by $(\phi ,{\rm Y})=(\psi ,\Lambda )\tilde{\cap }(\chi ,\Theta )$, where ${\rm Y}=\Lambda \cup \Theta$ and $\forall \alpha \in {\rm Y}$

$$\phi \left(\alpha \right)=\left\{\begin{array}{cc}\psi \left(\alpha \right),\hfill & if\alpha \in \Lambda -\Theta \hfill \\ \chi \left(\alpha \right),\hfill & if\alpha \in \Theta -\Lambda \hfill \\ \psi \left(\alpha \right)\cap \chi \left(\alpha \right),\hfill & if\alpha \in \Lambda \cap \Theta .\hfill \end{array}\right.$$

(6)

Definition 6

([20]). An agree-soft expert set (agree-SES) ${(\psi ,\mathcal{Z})}_1$ is a soft expert subset of $(\psi ,\mathcal{Z})$ defined by

$${(\psi ,\mathcal{Z})}_1=\{(\alpha ,{\psi }_1\left(\alpha \right)):\alpha \in E\times \mathcal{P}\times \left\{1\right\}\}.$$

(7)

Definition 7

([20]). A disagree-soft expert set (disagree-SES) ${(\psi ,\mathcal{Z})}_0$ is a soft expert subset of $(\psi ,\mathcal{Z})$ defined by

$${(\psi ,\mathcal{Z})}_0=\{(\alpha ,{\psi }_0\left(\alpha \right)):\alpha \in E\times \mathcal{P}\times \left\{0\right\}\}.$$

(8)

Remark 1.

It follows from Definition 2 and the examples provided in [20] that for every $(\psi ,\Lambda )\in \widetilde{\mathcal{SES}}\left(X\right)$, $\psi (e_j,p_k,1)\cap \psi (e_j,p_k,0)=\varphi$ and $\psi (e_j,p_k,1)\cup \psi (e_j,p_k,0)=X$ for any fixed parameter $e_j$ and expert $p_k$ (see Example 2 and Example 3.3 in [20]). In other words, $\psi (e_j,p_k,1)$ and $\psi (e_j,p_k,0)$ satisfy the self-duality condition, i.e., $\psi (e_j,p_k,0)={\psi }^c(e_j,p_k,1)$. This shows that if an expert $p_k$ does not “agree” that $x_i$ is a cheap product (that is, $x_i\notin \psi (e_j,p_k,1)$), then it is equivalent to saying that he “disagrees” about the cheapness of $x_i$ (that is, $x_i\in \psi (e_j,p_k,0)$).

4. Analysis of Alkhazaleh et al.’s Work in [20]

It is evident from Definitions 1 and 2 that an SES is an extended version of a soft set that presents approximations from multiple experts under two opinions, such as “agree” and “disagree”. However, the SES operations as defined in [20] are similar to the soft set operations introduced in [6], which are only appropriate for single-opinion-based approximations. Consequently, these operations are insufficient to model SESs that include both agree and disagree opinions of experts for a fixed parameter $e_j$. The following proposition clarifies this point: if there are two SESs that contain the agree and disagree opinions of an expert for a fixed parameter $e_j$, then they are not comparable using the subset operation $\tilde{\subseteq }$ as defined in Definition 3.

Proposition 1.

Let $(\psi ,\Lambda )$ and $(\chi ,\Theta )$ be any two SESs that contain the agree and disagree opinions of an expert $p_k$ for a fixed parameter $e_j$, then they are non-comparable by $\tilde{\subseteq }$.

Proof. 

Assume that $(\psi ,\Lambda )$ and $(\chi ,\Theta )\in \widetilde{\mathcal{SES}}\left(X\right)$ and $\psi \left((e_j,p_k,1)\right)$ and $\psi \left((e_j,p_k,0)\right)$ simultaneously belong to $(\psi ,\Lambda )$ and $(\chi ,\Theta )$, for some fixed j and k. Assume that $(\psi ,\Lambda )\tilde{\subseteq }(\chi ,\Theta )$, then by Definition 3, $\Lambda \subseteq \Theta$ and $\forall \alpha \in \Lambda$, $\psi \left(\alpha \right)\subseteq \chi \left(\alpha \right)$. This implies that

$$\left\{\begin{array}{c}\psi \left((e_j,p_k,1)\right)\subseteq \chi \left((e_j,p_k,1)\right)\hfill \\ \psi \left((e_j,p_k,0)\right)\subseteq \chi \left((e_j,p_k,0)\right).\hfill \end{array}\right.$$

(9)

By Remark 1, $\psi \left((e_j,p_k,0)\right)={\psi }^c\left((e_j,p_k,1)\right)$ and $\chi \left((e_j,p_k,0)\right)={\chi }^c\left((e_j,p_k,1)\right)$. This implies that

$$\left\{\begin{array}{c}\psi \left((e_j,p_k,1)\right)\subseteq \chi \left((e_j,p_k,1)\right)\hfill \\ {\psi }^c\left((e_j,p_k,1)\right)\subseteq {\chi }^c\left((e_j,p_k,1)\right),\hfill \end{array}\right.$$

(10)

which is not possible. Hence, $(\psi ,\Lambda )\widetilde{⊈}(\chi ,\Theta )$. In a similar way, we can prove that $(\chi ,\Theta )\widetilde{⊈}(\psi ,\Lambda )$. Thus, $(\psi ,\Lambda )$ and $(\chi ,\Theta )$ are non-comparable by the subset operation $\tilde{\subseteq }$ as defined in Definition 3.    □

Proposition 1 implies that it is impossible to find two SESs that contain both the “agree” and “disagree” opinions of an expert for a fixed parameter $e_j$, while also being comparable under the subset operation $\tilde{\subseteq }$. This fact is further illustrated in the following.

Example 3.

Let $(\psi ,\Lambda )$ be an SES over X defined by

$$\begin{array}{cc}\hfill (\psi ,\Lambda )=& \{\psi (e_1,p_1,1)=\{x_1,x_2,x_5\},\psi (e_1,p_2,1)=\{x_2,x_3,x_5,x_6\},\psi (e_2,p_3,1)=\{x_1,x_5\},\psi (e_1,p_2,0)=\hfill \\ \hfill & \{x_1,x_4\},\psi (e_2,p_3,0)=\{x_2,x_3,x_4,x_6\}\}.\hfill \end{array}$$

Suppose that the expert’s opinions were taken once again by the company after the products had remained for one month in the market. The new SES $(\chi ,\Theta )$ is given by

$$\begin{array}{cc}\hfill (\chi ,\Theta )=& \{\chi (e_1,p_1,1)=\{x_1,x_5\},\chi (e_1,p_2,1)=\{x_2,x_3,x_5\},\chi (e_1,p_2,0)=\{x_1,x_4,x_6\},\chi (e_2,p_3,0)=\hfill \\ \hfill & \{x_2,x_3,x_4,x_6\}\left)\right\}.\hfill \end{array}$$

Now, $\Theta \subset \Lambda$ but the agree and disagree opinions of the expert $p_2$ for the parameter $e_1$ are simultaneously belong to $(\psi ,\Lambda )$ and $(\chi ,\Theta )$. Therefore, by Proposition 1, the two SESs cannot be comparable by the operation $\tilde{\subseteq }$ (i.e, $(\chi ,\Theta )\widetilde{⊈}(\psi ,\Lambda )$), which is clear from the expressions

$$\chi (e_1,p_2,1)\subseteq \psi (e_1,p_2,1)and\chi ((e_1,p_2,0)\supseteq \psi ((e_1,p_2,0).$$

In fact, practically, it is not possible to find two SESs which satisfy the condition in Proposition 1 and also comparable by the subset operation $\tilde{\subseteq }$.

Next, we provide an example to show that Definitions 4 and 5 are also not true for those SESs, which include both agree and disagree opinions of experts for a fixed parameter $e_j$.

Example 4.

Assume that $(\psi ,\Lambda )$ and $(\chi ,\Theta )$ are the same SESs as defined in Example 3. By Definitions 4, the union of $(\psi ,\Lambda )$ and $(\chi ,\Theta )$ is given by

$$\begin{array}{cc}\hfill (\phi ,{\rm Y})=(\psi ,\Lambda )\tilde{\cup }(\chi ,\Theta )=& \{\phi (e_1,p_1,1)=\{x_1,x_2,x_5\},\phi (e_1,p_2,1)=\{x_2,x_3,x_5,x_6\},\phi (e_2,p_3,1)=\hfill \\ \hfill & \{x_1,x_5\},\phi (e_1,p_2,0)=\{x_1,x_4,x_6\},\phi (e_2,p_3,0)=\{x_2,x_3,x_4,x_6\}\},\hfill \end{array}$$

where ${\rm Y}=\Lambda \cup \Theta$. Now, consider $(\phi ,{\rm Y})$ where $x_6\in \phi (e_1,p_2,1)$ and $x_6\in \phi (e_1,p_2,0)$. That is

$$\phi (e_1,p_2,1)\cap \phi (e_1,p_2,0)\ne \varphi .$$

This overlap indicates that expert $p_2$ simultaneously “agrees” and “disagrees” on $x_6$ being classified as a "cheap" product, which contradicts the basic requirements of SESs mentioned in Remark 1. Consequently, $(\phi ,{\rm Y})$ cannot be considered as an SES over X.

Next, according to Definition 5, the intersection of $(\psi ,\Lambda )$ and $(\chi ,\Theta )$ is given by

$$\begin{array}{cc}\hfill (\zeta ,{\rm Y})=(\psi ,\Lambda )\tilde{\cap }(\chi ,\Theta )=& \{\zeta (e_1,p_1,1)=\{x_1,x_5\},\zeta (e_1,p_2,1)=\{x_2,x_3,x_5\},\zeta (e_2,p_3,1)=\{x_1,x_5\},\hfill \\ & \zeta (e_1,p_2,0)=\{x_1,x_4\},\zeta (e_2,p_3,0)=\{x_2,x_3,x_4,x_6\}\},\hfill \end{array}$$

where ${\rm Y}=\Lambda \cup \Theta$. Again, if we consider $(\zeta ,{\rm Y})$, then we see that the information about $x_6$ under the expert $p_2$ and parameter $e_1$ were lost after the operation of $\tilde{\cap }$. Symbolically, we can write

$$x_6\notin \zeta (e_1,p_2,1)\cup \zeta (e_1,p_2,0).$$

But this is, again, a contradiction to the second condition of Remark 1, which states that

$$\zeta (e_1,p_2,1)\cup \zeta (e_1,p_2,0)=X.$$

Hence, $(\zeta ,{\rm Y})$ cannot be considered as an SES over X.

It follows from the above discussion that the basic operations of SESs in [20] are not compatible with the definition of an SES. This will not only affect the development of SES theory but also raise questions on the results of [20] (such as Proposition 3.14, 3.17, 3.20, and 3.21). The main reason for these complications is the structural difference between an SES and its operations. An SES is a two-fold structure that considers two contrasting opinions (agree and disagree) of experts for a set of parameters. Moreover, SES operations are defined similarly to the soft set operations that consider only a single opinion (agree) of expert(s) for a set of parameters. Using these single-opinion-based operations for two-fold structures (SESs) may result in the loss or overlap of information, as already seen in the previous examples.

To solve the problem, we have two options in hand. First, we should define new operations for SESs that can preserve the two-fold structure of SESs. Second, we should revise the definition of an SES so that it can match the current operations of SESs. The first option is not recommended because it will further complicate the process of these operations and put an extra computational burden on SES applications. Consequently, we select the second option and revise the definition of SESs in the following section.

5. Revised Definition of SES and Its Application in MCDM

In Definition 2, an SES is defined as the collection of dual approximations that consider both “agree” and “disagree” opinions of experts. However, upon closer examination, it is observed that SES-based decision-making can be effectively performed using only the “agree” approximations. Since the “disagree” component does not provide additional value in practical applications, we can remove it from the SES definition (see application section in [20]). This will not only address the existing challenges associated with SES operations but also reduce the computational complexity of SES-based decision-making.

Let $\mathcal{Z}=E\times \mathcal{P}$, where E denotes the parameter set relevant to X and $\mathcal{P}$ represents the set of available experts (agents). Based on this formulation, the revised definition of an SES is established as follows:

Definition 8.

A pair $(\psi ,\Lambda )$ is called a soft expert set (SES) over X, where $\Lambda \subseteq \mathcal{Z}$ and ψ is a mapping given by

$$\psi :\Lambda \to P\left(X\right).$$

(11)

According to Definition 8, the new representation of SES in Examples 2 is given by the following:

$$(\psi ,\mathcal{Z})=\left\{\begin{array}{c}\psi (e_1,p_1)=\{x_1,x_2,x_5\},\psi (e_1,p_2)=\{x_2,x_3,x_5,x_6\},\psi (e_1,p_3)=\{x_3,x_4,x_6\},\hfill \\ \psi (e_2,p_1)=\{x_1,x_4,x_6\},\psi (e_2,p_2)=\{x_3,x_4,x_5\},\psi (e_2,p_3)=\{x_1,x_5\},\hfill \\ \psi (e_3,p_1)=\{x_1,x_4,x_5\},\psi (e_3,p_2)=\{x_2,x_4\},\psi (e_3,p_3)=\{x_1,x_3,x_5\}.\hfill \end{array}\right\}$$

(12)

The new representation still provides the same information (expert opinions) as obtained in Examples 2. However, it is more straightforward compared to the previous definition, enhancing the efficiency and convenience of SES-based decision-making processes. Furthermore, it can also be verified that all previously defined SES operations in [20] remain consistent and compatible with the new SES definition, preserving the functionality of the SES theory.

5.1. A Novel Approach to SES-Based Decision-Making

So far, very limited algorithms have been developed for SES-based decision-making. The first algorithm was proposed by Alkhazaleh et al. [20], relying on the choice values of agree and disagree SESs. Later, Enginoğlu et al. [38] developed a new algorithm for SES-based decision-making, which focuses on the intersection of all possible parts of the agree-SES. However, these algorithms are designed for the selection of a single optimal choice and are not well-suited for ranking or selecting multiple alternatives. Consequently, an enhanced SES-based decision-making approach is needed to provide a precise ranking of all decision alternatives.

Definition 9.

The $p_k$-part of an SES $(\psi ,\mathcal{Z})$ is denoted by $p_k(\psi ,\mathcal{Z})$ and defined by

$$p_k(\psi ,\mathcal{Z})=\{(\alpha ,\psi \left(\alpha \right)):\alpha \in E\times p_k\}.$$

(13)

For example, the $p_1$-part of the SES $(\psi ,\mathcal{Z})$ as defined in (12) is given by

$$\begin{array}{c}\hfill p_1(\psi ,\mathcal{Z})=\{\psi (e_1,p_1)=\{x_1,x_2,x_5\},\psi (e_2,p_1)=\{x_1,x_4,x_6\},\psi (e_3,p_1)=\{x_1,x_4,x_5\}\}.\end{array}$$

(14)

Definition 10.

Let $p_k(\psi ,\mathcal{Z})$ be the $p_k$-part of any $(\psi ,\mathcal{Z})\in \widetilde{\mathcal{SES}}\left(X\right)$. The parametric difference from $x_i$ to $x_j$ under $p_k$ is defined by

$$x_i\stackrel{p_k}{-}{x}_j=\{e\in E:x_i\in \psi (e,p_k)and{x}_j\notin \psi (e,p_k)\}.$$

(15)

It is clear from Definition 10 that for any $p_k(\psi ,\mathcal{Z})$, we have

• $x_i\stackrel{p_k}{-}{x}_i=\varphi$,
• $x_i\stackrel{p_k}{-}{x}_j\ne x_j\stackrel{p_k}{-}{x}_i$, ∀$x_i,x_j\in X$.

Consider $p_1(\psi ,\mathcal{Z})$ in (14), the parametric difference from $x_1$ to $x_2$ is $x_1\stackrel{p_1}{-}{x}_2=\{e_2,e_3\}$ while the parametric difference from $x_2$ to $x_1$ is $x_2\stackrel{p_1}{-}{x}_1=\varphi$. In a similar way, we can compute the parametric differences of all pairs $(x_i,x_j)\in X\times X$ under $p_1$.

Definition 11.

Let $p_k(\psi ,\mathcal{Z})$ be the $p_k$-part of any $(\psi ,\mathcal{Z})\in \widetilde{\mathcal{SES}}\left(X\right)$. The difference score from $x_i$ to $x_j$ under $p_k$ is defined by

$$S_k(x_i,x_j)=\left|x_i\stackrel{p_k}{-}{x}_j\right|,$$

(16)

where the symbol $|.|$ denotes the cardinality of set. The total difference from $x_i$ to $x_j$ under $p_k$ is defined by

$$D(x_i,x_j)=\sum _k{S}_k(x_i,x_j).$$

(17)

Based on the above definitions, we propose a new algorithm for SES-based decision-making as follows (Algorithm 1):

Algorithm 1: The proposed algorithm

Step 1.       Consider an SES $(\psi ,\mathcal{Z})$; Step 2.       Find all $p_k$-parts of the $(\psi ,\mathcal{Z})$. Step 3.       Compute $x_i\stackrel{p_k}{-}{x}_j$ for all $(x_i,x_j)\in X\times X$ under each $p_k$; Step 4.       Estimate $S_k(x_i,x_j)$ and $D(x_i,x_j)$ for all $(x_i,x_j)\in X\times X$ under each $p_k$; Step 5.       Compute the score $S_i={\sum }_{x_j\in X}D(x_i,x_j),$$\forall x_i\in X$; Step 6.       Rank $x_i\in X$ according to the score $S_i$ from the largest to the smallest.

5.2. Application in MCDM

Decision-making processes often involve evaluating multiple alternatives based on various factors, which can significantly influence outcomes across different contexts [56]. Multi-Criteria Decision-Making (MCDM) is a systematic approach used to evaluate and select alternatives based on multiple criteria [57]. It facilitates decision-makers in assessing a range of options by comprehensively analyzing trade-offs between criteria such as cost, quality, performance, and risk. MCDM is widely applied in fields like business, engineering, medical science, and disaster management to support complex decision-making problems [58]. The majority of these decision-making problems involve the complete ranking of decision alternatives rather than selecting the optimal one. For instance, the evaluation of supply chain partners is one of the important MCDM problems that consider different attributes and risk factors to rank the suppliers according to their service level. Evaluation of supply chain partners is crucial for ensuring the efficiency and reliability of the overall supply chain network [59]. Factors such as supplier performance, delivery time, quality consistency, and cost competitiveness play a significant role in determining their suitability [60]. Additionally, incorporating different risk factors, such as supplier financial stability and adaptability to disruptions, ensures a more robust and resilient supply chain. Effective evaluation not only helps in identifying the best-performing suppliers but also fosters long-term partnerships that align with organizational goals [61,62]. In the following, we present a supply chain problem to demonstrate how Algorithm 1 offers a comprehensive ranking, enabling decision-makers to select the best suppliers based on their service level.

Let $X=\{x_i:i=1,2,\dots ,8\}$ represent the set of eight suppliers for an engineering project and $E=\{e_j:j=1,2,\dots ,5\}$ be the set of decision parameters, where $e_1$ = right quality, $e_2$ = right quantity, $e_3$ = right delivery time, $e_4$ = right price, and $e_5$ = right delivery place. Let $\mathcal{P}=\{p_1,p_2,p_3\}$ denote the set of three experts responsible for evaluating the suppliers based on the given decision parameters, as defined by the 5R principles. Suppose the experts evaluated the eight suppliers based on the given decision parameters. Their opinions are expressed by the SES $(\psi ,\mathcal{Z})$, whose tabular representation is given by

Table 1. Tabular form of $(\psi ,\mathcal{Z})$.

$\mathit{Z}/\mathit{X}$	${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{x}}_3$	${\mathit{x}}_4$	${\mathit{x}}_5$	${\mathit{x}}_6$	${\mathit{x}}_7$	${\mathit{x}}_8$
$(e_1,p_1)$	1	1	0	1	0	0	1	1
$(e_2,p_1)$	0	0	1	0	1	0	0	1
$(e_3,p_1)$	0	0	1	1	1	0	1	0
$(e_4,p_1)$	1	0	0	0	0	0	1	1
$(e_5,p_1)$	1	1	1	0	1	0	0	1
$(e_1,p_2)$	1	0	0	1	1	0	0	1
$(e_2,p_2)$	1	0	1	1	1	1	0	1
$(e_3,p_2)$	1	1	0	0	1	0	0	1
$(e_4,p_2)$	1	0	0	1	1	0	0	1
$(e_5,p_2)$	1	0	0	1	1	0	0	1
$(e_1,p_3)$	1	0	1	1	0	1	1	1
$(e_2,p_3)$	1	1	0	1	0	0	1	1
$(e_3,p_3)$	1	0	0	0	0	0	1	1
$(e_4,p_3)$	1	0	0	0	0	1	1	1
$(e_5,p_3)$	1	0	1	0	1	0	1	1

.

$$(\psi ,\mathcal{Z})=\left\{\begin{array}{c}(e_1,p_1)=\{x_1,x_2,x_4,x_7,x_8\},(e_1,p_2)=\{x_1,x_4,x_5,x_8\},(e_1,p_3)=\{x_1,x_3,x_4,x_6,x_7,x_8\},\hfill \\ (e_2,p_1)=\{x_3,x_5,x_8\},(e_2,p_2)=\{x_1,x_3,x_4,x_5,x_6,x_8\},(e_2,p_3)=\{x_1,x_2,x_4,x_7,x_8\},\hfill \\ (e_3,p_1)=\{x_3,x_4,x_5,x_7\},(e_3,p_2)=\{x_1,x_2,x_5,x_8\},(e_3,p_3)=\{x_1,x_7,x_8\},\hfill \\ (e_4,p_1)=\{x_1,x_7,x_8\},(e_4,p_2)=\{x_1,x_4,x_5,x_8\},(e_4,p_3)=\{x_1,x_6,x_7,x_8\},\hfill \\ (e_5,p_1)=\{x_1,x_2,x_3,x_5,x_8\},(e_5,p_2)=\{x_1,x_4,x_5,x_8\},(e_5,p_3)=\{x_1,x_3,x_5,x_7,x_8\}.\hfill \end{array}\right\}$$

(18)

Our goal is to determine a precise ranking of the eight suppliers based on the expert opinions using Algorithm 1. For this, we first compute the $p_k$-parts of the SES $(\psi ,\mathcal{Z})$ as represented by

Table 2. Tabular form of $p_1(\psi ,\mathcal{Z})$.

$(\mathit{E},{\mathit{p}}_1)/\mathit{X}$	${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{x}}_3$	${\mathit{x}}_4$	${\mathit{x}}_5$	${\mathit{x}}_6$	${\mathit{x}}_7$	${\mathit{x}}_8$
$(e_1,p_1)$	1	1	0	1	0	0	1	1
$(e_2,p_1)$	0	0	1	0	1	0	0	1
$(e_3,p_1)$	0	0	1	1	1	0	1	0
$(e_4,p_1)$	1	0	0	0	0	0	1	1
$(e_5,p_1)$	1	1	1	0	1	0	0	1

,

Table 3. Tabular form of $p_2(\psi ,\mathcal{Z})$.

$(\mathit{E},{\mathit{p}}_2)/\mathit{X}$	${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{x}}_3$	${\mathit{x}}_4$	${\mathit{x}}_5$	${\mathit{x}}_6$	${\mathit{x}}_7$	${\mathit{x}}_8$
$(e_1,p_2)$	1	0	0	1	1	0	0	1
$(e_2,p_2)$	1	0	1	1	1	1	0	1
$(e_3,p_2)$	1	1	0	0	1	0	0	1
$(e_4,p_2)$	1	0	0	1	1	0	0	1
$(e_5,p_2)$	1	0	0	1	1	0	0	1

and

Table 4. Tabular form of $p_3(\psi ,\mathcal{Z})$.

$(\mathit{E},{\mathit{p}}_3)/\mathit{X}$	${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{x}}_3$	${\mathit{x}}_4$	${\mathit{x}}_5$	${\mathit{x}}_6$	${\mathit{x}}_7$	${\mathit{x}}_8$
$(e_1,p_3)$	1	0	1	1	0	1	1	1
$(e_2,p_3)$	1	1	0	1	0	0	1	1
$(e_3,p_3)$	1	0	0	0	0	0	1	1
$(e_4,p_3)$	1	0	0	0	0	1	1	1
$(e_5,p_3)$	1	0	1	0	1	0	1	1

. From Table 2, we can easily compute the parametric difference of each pair $(x_i,x_j)\in X\times X$ under $p_1$, where $(i,j=1,2,\dots ,8)$. For example, to compute the parametric difference from $x_1$ to all other objects in X under $p_1$, we match the column of $x_1$ with all other columns in Table 2 one by one and will search those $e_t\in E$ for which $((e_t,p_1),x_1)=1$ and $((e_t,p_1),x_j)=0$, where $t=1,2,\dots ,5$ and $j\ne 1$. That is,

$$x_1\stackrel{p_1}{-}{x}_2=\left\{e_4\right\},x_1\stackrel{p_1}{-}{x}_3=\{e_1,e_4\},x_1\stackrel{p_1}{-}{x}_4=\{e_4,e_5\},x_1\stackrel{p_1}{-}{x}_5=\{e_1,e_4\},$$

$$x_1\stackrel{p_1}{-}{x}_6=\{e_1,e_4,e_5\},x_1\stackrel{p_1}{-}{x}_7=\left\{e_5\right\},x_1\stackrel{p_1}{-}{x}_8=\varphi .$$

Similarly, we can compute the parametric difference from any object to the remaining objects in X under $p_1$, $p_2$, and $p_3$.

Next, the difference score from $x_1$ to all objects in X under $p_1$ are obtained by (16) as follows:

$$S_1(x_1,x_2)=|x_1\stackrel{p_1}{-}{x}_2|=1,S_1(x_1,x_3)=|x_1\stackrel{p_1}{-}{x}_3|=2,S_1(x_1,x_4)=\left|x_1\stackrel{p_1}{-}{x}_4\right|=2,$$

$$S_1(x_1,x_5)=|x_1\stackrel{p_1}{-}{x}_5|=2,S_1(x_1,x_6)=|x_1\stackrel{p_1}{-}{x}_6|=3,S_1(x_1,x_7)=\left|x_1\stackrel{p_1}{-}{x}_7\right|=1,$$

$$S_1(x_1,x_8)=\left|x_1\stackrel{p_1}{-}{x}_8\right|=0.$$

In a similar way, we can compute the difference scores of the remaining pairs $(x_i,x_j)\in X\times X$ under $p_1$, which are listed in

Table 5. The difference score table under $p_1$.

${\mathit{S}}_1({\mathit{x}}_{\mathit{i}},{\mathit{x}}_{\mathit{j}})$	${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{x}}_3$	${\mathit{x}}_4$	${\mathit{x}}_5$	${\mathit{x}}_6$	${\mathit{x}}_7$	${\mathit{x}}_8$
$x_1$	0	1	2	2	2	3	1	0
$x_2$	0	0	1	1	1	2	1	0
$x_3$	2	2	0	2	0	3	2	1
$x_4$	1	1	1	0	1	2	0	1
$x_5$	2	2	0	2	0	3	2	1
$x_6$	0	0	0	0	0	0	0	0
$x_7$	1	2	2	1	2	3	0	1
$x_8$	1	2	2	3	2	4	2	0

. The difference scores of all pairs $(x_i,x_j)\in X\times X$ under $p_2$ and $p_3$ are also computed in the same way, which can be found in

Table 6. The difference score table under $p_2$.

${\mathit{S}}_2({\mathit{x}}_{\mathit{i}},{\mathit{x}}_{\mathit{j}})$	${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{x}}_3$	${\mathit{x}}_4$	${\mathit{x}}_5$	${\mathit{x}}_6$	${\mathit{x}}_7$	${\mathit{x}}_8$
$x_1$	0	4	4	1	0	4	5	0
$x_2$	0	0	1	1	0	1	1	0
$x_3$	0	1	0	0	0	0	1	0
$x_4$	0	4	3	0	0	3	4	0
$x_5$	0	4	4	1	0	4	5	0
$x_6$	0	1	0	0	0	0	1	0
$x_7$	0	0	1	0	0	0	0	0
$x_8$	0	4	4	1	0	4	5	0

and

Table 7. The difference score table under $p_3$.

${\mathit{S}}_3({\mathit{x}}_{\mathit{i}},{\mathit{x}}_{\mathit{j}})$	${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{x}}_3$	${\mathit{x}}_4$	${\mathit{x}}_5$	${\mathit{x}}_6$	${\mathit{x}}_7$	${\mathit{x}}_8$
$x_1$	0	4	3	3	4	3	0	0
$x_2$	0	0	1	0	1	1	0	0
$x_3$	0	2	0	1	1	1	0	0
$x_4$	0	1	1	0	2	1	0	0
$x_5$	0	1	0	1	0	1	0	0
$x_6$	0	2	1	1	2	0	0	0
$x_7$	0	4	3	3	4	3	0	0
$x_8$	0	4	3	3	4	3	0	0

, respectively.

Further, using (17), the total difference of all ordered pairs $(x_i,x_j)\in X\times X$ can be computed from Table 5, Table 6 and Table 7 as follows:

$$D(x_1,x_2)=\sum _{k=1}^3{S}_k(x_1,x_2)=1+4+4=9,D(x_1,x_3)=\sum _{k=1}^3{S}_k(x_1,x_3)=2+4+3=9$$

$$D(x_1,x_4)=\sum _{k=1}^3{S}_k(x_1,x_4)=2+1+3=6,$$

and so on, which are listed in

Table 8. The total difference table.

$\mathit{D}({\mathit{x}}_{\mathit{i}},{\mathit{x}}_{\mathit{j}})$	${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{x}}_3$	${\mathit{x}}_4$	${\mathit{x}}_5$	${\mathit{x}}_6$	${\mathit{x}}_7$	${\mathit{x}}_8$	${\mathit{S}}_{\mathit{i}}$
$x_1$	0	9	9	6	6	10	6	0	46
$x_2$	0	0	3	2	2	4	2	0	13
$x_3$	2	5	0	3	1	4	3	1	19
$x_4$	1	6	5	0	3	6	4	1	26
$x_5$	2	7	4	4	0	8	7	1	33
$x_6$	0	3	1	1	2	0	1	0	8
$x_7$	1	6	6	4	6	6	0	1	30
$x_8$	1	10	9	7	6	11	7	0	51

. From Table 8, the scores $S_i={\sum }_{x_j\in X}D(x_i,x_j)$ are computed for all $x_i\in X$, which are given in the last column of Table 8. Finally, the eight suppliers are ranked according to their scores $S_i$ from the largest to the smallest, as 

$$x_8>x_1>x_5>x_7>x_4>x_3>x_2>x_6.$$

(19)

6. Comparison with Existing Methods

This section provides a comparative analysis of the proposed algorithm with the algorithms of Alkhazaleh et al. [20] and Enginoğlu et al. [38]. Both algorithms are applied to the same supply chain problem as discussed in Section 5.2, and their results are compared with the results of the proposed algorithm (Algorithm 1).

6.1. Alkhazaleh et al.’s Algorithm

Alkhazaleh et al. [20] introduced the first algorithm for SES-based decision-making, which is based on the choice values of the agree and disagree-SESs. The specific steps of their algorithm are given in Algorithm 2.

Algorithm 2: Alkhazaleh et al.’s algorithm

Step 1.       Input the SES $(\psi ,\mathcal{Z})$; Step 2.       Find the agree and disagree-SESs ${(\psi ,\mathcal{Z})}_1$ and ${(\psi ,\mathcal{Z})}_0$; Step 3.       Find $c_i={\sum }_j{x}_{ij}$ for ${(\psi ,\mathcal{Z})}_1$; Step 4.       Find $k_i={\sum }_j{x}_{ij}$ for ${(\psi ,\mathcal{Z})}_0$; Step 5.       Compute $S_i=c_i-k_i$; Step 6.       Find r for which $S_r={max}_i{S}_i$.

Consider the SES $(\psi ,\mathcal{Z})$ as given in the supply chain problem. According to Algorithm 2, compute the agree and disagree-SESs of $(\psi ,\mathcal{Z})$ as represented by

Table 9. Tabular form of the agree-SES ${(\psi ,\mathcal{Z})}_1$.

${\mathit{Z}}_1/\mathit{X}$	${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{x}}_3$	${\mathit{x}}_4$	${\mathit{x}}_5$	${\mathit{x}}_6$	${\mathit{x}}_7$	${\mathit{x}}_8$
$(e_1,p_1,1)$	1	1	0	1	0	0	1	1
$(e_2,p_1,1)$	0	0	1	0	1	0	0	1
$(e_3,p_1,1)$	0	0	1	1	1	0	1	0
$(e_4,p_1,1)$	1	0	0	0	0	0	1	1
$(e_5,p_1,1)$	1	1	1	0	1	0	0	1
$(e_1,p_2,1)$	1	0	0	1	1	0	0	1
$(e_2,p_2,1)$	1	0	1	1	1	1	0	1
$(e_3,p_2,1)$	1	1	0	0	1	0	0	1
$(e_4,p_2,1)$	1	0	0	1	1	0	0	1
$(e_5,p_2,1)$	1	0	0	1	1	0	0	1
$(e_1,p_3,1)$	1	0	1	1	0	1	1	1
$(e_2,p_3,1)$	1	1	0	1	0	0	1	1
$(e_3,p_3,1)$	1	0	0	0	0	0	1	1
$(e_4,p_3,1)$	1	0	0	0	0	1	1	1
$(e_5,p_3,1)$	1	0	1	0	1	0	1	1

and

Table 10. Tabular form of the disagree-SES ${(\psi ,\mathcal{Z})}_0$.

${\mathit{Z}}_0/\mathit{X}$	${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{x}}_3$	${\mathit{x}}_4$	${\mathit{x}}_5$	${\mathit{x}}_6$	${\mathit{x}}_7$	${\mathit{x}}_8$
$(e_1,p_1,0)$	0	0	1	0	1	1	0	0
$(e_2,p_1,0)$	1	1	0	1	0	1	1	0
$(e_3,p_1,0)$	1	1	0	0	0	1	0	1
$(e_4,p_1,0)$	0	1	1	1	1	1	0	0
$(e_5,p_1,0)$	0	0	0	1	0	1	1	0
$(e_1,p_2,0)$	0	1	1	0	0	1	1	0
$(e_2,p_2,0)$	0	1	0	0	0	0	1	0
$(e_3,p_2,0)$	0	0	1	1	0	1	1	0
$(e_4,p_2,0)$	0	1	1	0	0	1	1	0
$(e_5,p_2,0)$	0	1	1	0	0	1	1	0
$(e_1,p_3,0)$	0	1	0	0	1	0	0	0
$(e_2,p_3,0)$	0	0	1	0	1	1	0	0
$(e_3,p_3,0)$	0	1	1	1	1	1	0	0
$(e_4,p_3,0)$	0	1	1	1	1	0	0	0
$(e_5,p_3,0)$	0	1	0	1	0	1	0	0

. From Table 9 and Table 10, the values $c_i={\sum }_j{x}_{ij}$ and $k_i={\sum }_j{x}_{ij}$ are computed for all suppliers, which are listed in the first and second columns of

Table 11. The score table for Alkhazaleh et al.’s algorithm.

${\mathit{c}}_{\mathit{i}}={\sum }_{\mathit{j}}{\mathit{x}}_{\mathit{ij}}$	${\mathit{k}}_{\mathit{i}}={\sum }_{\mathit{j}}{\mathit{x}}_{\mathit{ij}}$	${\mathit{S}}_{\mathit{i}}={\mathit{c}}_{\mathit{i}}-{\mathit{k}}_{\mathit{i}}$
$c_1=13$	$k_1=2$	$S_1=11$
$c_2=4$	$k_2=11$	$S_2=-7$
$c_3=6$	$k_3=9$	$S_3=-3$
$c_4=8$	$k_4=7$	$S_4=1$
$c_5=9$	$k_5=6$	$S_5=3$
$c_6=3$	$k_6=12$	$S_6=-9$
$c_7=8$	$k_7=7$	$S_7=1$
$c_8=14$	$k_8=1$	$S_8=13$

, respectively.

Now, the score $S_i=c_i-k_i$ for each supplier can easily be computed, which is listed in the last column of Table 11. From Table 11, we can rank the eight suppliers according to their $S_i$ values from the largest to the smallest, as 

$$x_8>x_1>x_5>x_4=x_7>x_3>x_2>x_6.$$

(20)

Since $max{S}_i=S_8$, the decision makers can, therefore, select $x_8$ as an optimal choice among the eight suppliers.

6.2. Enginoğlu et al.’s Algorithm

Enginoğlu et al. [38] investigated that the algorithm presented in [20] has some unnecessary steps, and its results are equivalent to Maji et al.’s algorithm [29] without reduction. Therefore, they developed a new algorithm for SES-based decision-making that is based on the consensus soft set obtained from the intersection of all possible parts of agree-SES. The step-by-step procedure of Enginoğlu et al.’s algorithm is labeled by Algorithm 3.

Algorithm 3: Enginoğlu et al.’s algorithm

Step 1.       Input the SES $(\psi ,\mathcal{Z})$; Step 2.       Find all parts of the agree-SES ${(\psi ,\mathcal{Z})}_1$; Step 3.       Obtain the consensus soft set by taking the intersection of all the parts                   of agree-SES; Step 4.       Compute $S_i={\sum }_j{R}_C(e_j,x_i)$, where $R_C(e_j,x_i)$ are the entries of the                   consensus soft set table; Step 5.       Find r for which $S_r={max}_i{S}_i$.

Again, consider the SES $(\psi ,\mathcal{Z})$ as defined in the supply chain problem. According to Algorithm 3, first we find all possible parts of the agree-SES ${(\psi ,\mathcal{Z})}_1$, which are already given in Table 2, Table 3 and Table 4. The consensus soft set obtained from the intersection of the three parts is represented by

Table 12. The table of the consensus soft set.

${\mathit{R}}_{\mathit{C}}$	${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{x}}_3$	${\mathit{x}}_4$	${\mathit{x}}_5$	${\mathit{x}}_6$	${\mathit{x}}_7$	${\mathit{x}}_8$
$e_1$	1	0	0	1	0	0	0	1
$e_2$	0	0	0	0	0	0	0	1
$e_3$	0	0	0	0	0	0	0	0
$e_4$	1	0	0	0	0	0	0	1
$e_5$	1	0	0	0	1	0	0	1
$S_i={\sum }_j{R}_C(e_j,x_i)$	3	0	0	1	1	0	0	4

. Finally, the scores $S_i={\sum }_j{R}_C(e_j,x_i)$ for all $x_i\in X$ are obtained by adding the entries of each column in Table 12. According to the last row of Table 12, the ranking order of the eight suppliers is given by

$$x_8>x_1>x_4=x_5>x_2=x_3=x_6=x_7.$$

(21)

According to Algorithm 3, ${max}_i{S}_i=S_8$, and hence $x_8$ is the optimal choice among the eight suppliers.

6.3. Results and Discussion

Table 13. The comparison table.

Comparison Items	Alkhazaleh et al.’s Algorithm	Enginoğlu et al.’s Algorithm	The Proposed Algorithm	Remarks
Ranking order of the eight suppliers	$x_8>x_1>x_5>x_4=x_7>x_3>x_2>x_6$	$x_8>x_1>x_4=x_5>x_2=x_3=x_6=x_7$	$x_8>x_1>x_5>x_7>x_4>x_3>x_2>x_6$	The proposed algorithm completely ranked the eight suppliers
Ranking preciseness	Medium	Low	High	The ranking order of the proposed algorithm is more precise
Parametric preference	No parametric preference	Prefers those parameters which are common in the alternatives	Prefers those parameters which are uncommon in the alternatives	The selection of parameters is different in all the three methods
Computational complexity	Low	Low	High	Estimation of difference scores require more calculations, which is why the computational complexity of the proposed algorithm is high
Applicability	Low	Low	High	The applicability of the proposed algorithm is high as we can make more selections from its ranking order
Applied situation	Best for the cases where we need to select the single best alternative	Best for the cases where we need to select the single best alternative	Best in the situations where we need to select more than one object among the given alternatives	The decision-makers can select the appropriate method according to their need and nature of the problem

presents a comparative analysis of the ranking results obtained from the proposed algorithm, Alkhazaleh et al.’s algorithm, and Enginoglu et al.’s algorithm for the supply chain problem. The ranking order of the eight suppliers clearly highlights the differences in precision and ranking capability among the three methods. The proposed algorithm successfully provides a strict and complete ranking of the suppliers, $x_8>x_1>x_5>x_7>x_4>x_3>x_2>x_6$, resolving all possible ties and differentiating between alternatives. In contrast, Alkhazaleh et al.’s method produces a partially ordered ranking ($x_8>x_1>x_5>x_4=x_7>x_3>x_2>x_6$), where $x_4$ and $x_7$ are tied, reflecting limitations in its ability to fully differentiate close alternatives. Enginoglu et al.’s algorithm further worsens this issue by generating a more imprecise ranking ($x_8>x_1>x_4=x_5>x_2=x_3=x_6=x_7$), where multiple suppliers are tied ($x_4=x_5$ and $x_2=x_3=x_6=x_7$). This indicates that Enginoglu et al.’s approach struggles to provide a clear ranking when many alternatives show similar performance.

The superior ranking precision of the proposed algorithm comes from its ability to focus on parameters or attributes that are unique to each alternative, disregarding those that are either shared by all or missing. On the other hand, Enginoglu et al.’s algorithm primarily emphasizes common attributes, which reduces its precision and results in ties when alternatives share similar attributes. Meanwhile, Alkhazaleh et al.’s method does not prioritize specific parameters and treats all attributes equally, which reduces its capacity to produce a strict ranking. Since, the proposed algorithm identifies the uncommon attributes within each part of the SES, resulting in a higher computational complexity compared to the other two algorithms. This aspect can be regarded as a significant limitation of the proposed approach.

The applicability of the three methods further highlights the advantages of the proposed algorithm. Both Alkhazaleh et al.’s and Enginoglu et al.’s methods have low applicability as they are most effective in situations requiring the selection of a single best alternative. In contrast, the proposed algorithm is highly applicable to scenarios where decision-makers need to select multiple alternatives based on a precise ranking order. This broader applicability ensures that the proposed approach can be applied to more complex decision-making problems, particularly in multi-object selection scenarios.

In light of the above considerations, decision-makers should choose the most suitable method based on their specific needs and the problem at hand. For instance, if the goal is to select only the top single alternative from a set, Alkhazaleh et al.’s and Enginoglu et al.’s algorithms are more appropriate. However, when the selection of multiple alternatives is required, the proposed algorithm is the best option due to its ability to handle such scenarios effectively.

7. Conclusions and Future Research Directions

In this study, we revisited the soft expert set (SES) framework and identified key inconsistencies between the existing SES definition and its corresponding operations as defined by Alkhazaleh et al. [20]. Through numerical examples, we demonstrated that most operations, such as soft expert subset, union, and intersection, do not align with the foundational definition of an SES. To address this gap, we proposed a revised SES definition that resolves the current complications and enhances its practical applicability in decision-making. Additionally, we developed a new SES-based multi-criteria decision-making (MCDM) algorithm, which provides a complete and precise ranking of decision alternatives. The new algorithm addresses the limitations of previous SES-based algorithms, which often focus on selecting a single alternative. The proposed algorithm was also tested on a supply chain problem, and the results demonstrated its superiority over existing approaches. It offers a more accurate and comprehensive ranking of alternatives, making it suitable for multi-object selection scenarios. In summary, the study resolves theoretical challenges within the SES framework and expands its potential for real-world decision-making.

Since most of the extended SES models, such as fuzzy soft expert set [22], intuitionistic fuzzy soft expert set [23], and bipolar fuzzy soft expert set [24] are developed in the same way as the original SES framework, they may consequently face similar complications. Therefore, it is necessary to revisit these models to address the issues identified in this study. Secondly, there is a need to extend the proposed SES-based decision-making approach to other emerging SES frameworks in order to enhance robustness across different models. Finally, the applications of an SES and its extended models in decision-making require more attention and further exploration.