Multi-Attribute Three-Way Decision Approach Based on Ideal Solutions under Interval-Valued Fuzzy Soft Environment

Abstract

The interval-valued fuzzy soft set (IVFSS) model, which combines the benefits of the soft set model with the interval-valued fuzzy set (IVFS) model, is a growing and effective mathematical tool for processing hazy data. In detail, this model is characterized by symmetry, which has the lower and upper membership degree. The study of decision-making based on IVFSS has picked up more steam recently. However, existing multi-attribute decision-making (MADM) methods can only sort alternative schemes, but are not able to classify them, which is detrimental to decision-makers’ efficient decision-making. In this paper, we propose a multi-attribute three-way decision-making (MATWDM) algorithm based on ideal solutions for IVFSS. MATWDM is extended to the IVFSS environment by incorporating the concept of the ideal solution, offering a more adaptable and comprehensive approach for addressing uncertain MADM issues. The method not only obtains the ranking results of the alternatives, but also divides them into acceptance domain, rejection domain, and delayed-decision domain, which makes the decision results more reasonable and effective, facilitating decision-makers to make better decisions. We apply the proposed three-way decision algorithm to two practical cases as diverse as mine emergency decision and Homestay selection decision. Additionally, the effectiveness and viability of the suggested method are confirmed by experimental findings.

1. Introduction

Uncertain, hazy, and suspect data from many walks of life have caused confusion and even more harmful effects in numerous fields. The enhancive needs can no longer be fully satisfied by the earlier mathematical tools. Because of this, in 1999, the eminent Russian researcher Molodtsov [1] developed soft set (SS) theory, which differs from other earlier mathematical techniques. Since the theory is unaffected by inadequate parameterization, it is frequently used in a variety of contexts, including decision-making [2,3,4], data mining [5], medical diagnosis [6], and so on. Later, many researchers began to investigate the extended models of the SS, including fuzzy SS [7], Confidence SS [8], intuitionistic fuzzy SS [9], Pythagorean fuzzy SS [10], Interval-valued intuitionistic fuzzy SS [11,12], Fermatean fuzzy soft expert set [13], complex picture fuzzy SS [14], and so on. These expanded models were applied regularly in a variety of contexts, including parameter reduction [15], medical diagnosis [16], generative adversarial networks [14], and decision-making (DM).

One of them that deserves particular notice is the IVFSS model put forth by Yang et al. [17] in 2009. Since we frequently cannot use precise data to explain the ambiguity, it is more acceptable to have interval data to provide an estimated range at that moment; and due to having the lower and upper membership degree, this model is characterized by symmetry. Yang et al. [17] first offered a score-based decision-making approach for IVFSS. Based on the novel idea of level SS, Feng [18] introduced a customizable decision-making algorithm, which is a development and enhancement over Yang et al.’s algorithm. A novel efficient DM algorithm based on IVFSS was proposed by Ma [19]. When making stochastic multi-criteria decisions and emergency decisions, the combined weights for IVFSS [20,21] were taken into account. The realization that some factors are not necessary for making decisions gave rise to the concept of parameter reduction. In [22], a novel Pearson’s Product Moment Coefficient-based parameter reduction approach for IVFSS was provided; and a parameter reduction method by means of Euclidean distance (ED) was introduced in [23].

Three-way decision-making (3WDM), first put out by [24], offers a fresh idea and approach to dealing with risk and uncertainty. The 3WDM technique entails creating three different types of decision rules that result in three-decision outcomes such as acceptance, rejection, and noncommitment [24]. The ability of 3WDM to solve issues in a way that is congruent with human cognition has grown in popularity in recent years. Rough sets have not been a restriction of the 3WDM theory thus far. Fuzzy sets [25], intuitionistic fuzzy sets [26], Pythagorean fuzzy sets [27,28], shadow sets [29], and other contexts have all made extensive use of it. Based on the data-information–knowledge-wisdom hierarchical structure model and the notion of 3WDM, ref. [30] developed a three-way computing framework and provided a three-layer conceptual model suitable for intelligent data analysis [31]. Yao also applied cognitive science to the three branches of DM theory and established the acceptance domain (AD), rejection domain (RD), and boundary domain as three separate cognitive regions in humans [32]. This led many researchers to combine the 3WDM model with conceptual analysis [33], granular computing [34], conflict analysis [35,36,37], multi-attribute decision [38,39], and concept lattice [40,41]. Later, 3WDM was expanded to the dynamic environment, leading to the development of numerous DM techniques, including sequential 3WDM [42,43], multi-class 3WDM [44], multi-granulation 3WDM [45] and multi-scale 3WDM [46]. The 3WDM theory is currently utilized frequently in medical diagnosis [47], image recognition [48], sentiment classification [49] and so on.

However, in the IVFSS environment, few documents focus on 3WDM. Alternatives cannot be classified by the current MADM methods, which is damaging to decision-makers ability to make efficient decisions. Therefore, in this study, we present a MATWDM algorithm that is based on ideal solutions for IVFSS. The study has made the following important contributions:

(1)

MATWDM is extended to IVFSS and based on the ideal solution (IS) provides a more flexible and all-encompassing approach for handling uncertain MADM problems.

(2)

In addition to obtaining the ranking results for the alternatives, the approach also divides them into domains for acceptance, rejection, and postponed decision, which improves the decision results by making them more rational and effective while helping decision-makers make better choices.

(3)

We put the proposed method under consideration in real-world scenarios, which were as diverse as mine emergency decision-making and the Homestay evaluation system.

The rest of this essay is structured as follows. Firstly, we review some fundamental definitions of IVFSS, fuzzy SS, and SS. We also go over the four current techniques’ drawbacks. In Section 3, we present a new MATWDM algorithm that is based on IS for IVFSS. In Section 4, we provide two real cases to demonstrate the viability of our approach. Finally, we wrap up the paper with a summary.

2. Preliminaries and Related Work

In this section, we merely review some fundamental ideas related to SS, fuzzy SS, IVFSS, and the MADM techniques currently in use for IVFSS.

2.1. Basic Concepts

Definition 1 

([1]). Let U be an initial non-empty universe of alternatives, E be a set of attributes in relation to alternatives in U. Let P(U) be the power set of U, and A be a subset of E. A pair (F, A) is called a SS over U, where F is a mapping given by F: E → P(U).

Definition 2 

([7]). Let E be a set of attributes and U be the initial universe set. The set of all fuzzy sets in universe U is denoted as F(U). A pair ($\tilde{F}$, E) is called a fuzzy SS over U, where $\tilde{F}$ is a mapping given by $\tilde{F}$:E→F(U).

Definition 3 

([50]). An IVFS $\widehat{X}$ on an universe U is a mapping such that $\widehat{X}:U\to Int\left(\left[0, 1\right]\right)$, where $Int\left(\left[0,1\right]\right)$ denotes the set of all closed sub-intervals of [0, 1], the set of all IVFSs on U is represented by $\tilde{\psi }\left(U\right)$ Let $\widehat{X}\in \tilde{\psi }\left(U\right)$, for every $x\in U$, where $\tilde{\psi }\left(U\right)$ denotes the set of all IVFSs on U. ${\mu }_{\widehat{x}}^{-}\left(x\right)$ and ${\mu }_{\widehat{x}}^{+}\left(x\right)$ represent the lower and upper degrees of membership x to $\widehat{X}\left(0\le {\mu }_{\widehat{x}}^{-}\left(x\right)\le {\mu }_{\widehat{x}}^{+}\left(x\right)\le 1\right)$ espectively, while ${\mu }_{\widehat{X}}\left(x\right)=\left[{\mu }_{\widehat{x}}^{-}\left(x\right),{\mu }_{\widehat{x}}^{+}\left(x\right)\right]$ is regarded as the degree of membership of an element x to $\widehat{X}$.

Definition 4 

([17]). Let U be an initial universe of alternatives and E be a set of attributes in relation to alternatives in U. A pair ($\tilde{\omega }$, E) is referred to an IVFSS over $\tilde{\psi }\left(U\right)$, where $\tilde{\omega }$ is a mapping given by $\tilde{\omega }$: E → $\tilde{\psi }\left(U\right)$.

2.2. The Existing MADM Algorithms Based on IVFSS

In this part, we review the four current MADM algorithms (Algorithms 1–4) based on IVFSS and point out the disadvantages of them.

Algorithm 1. WDBA based MADM method for IVFSS [20]
Input:	Decision matrix based on IVFSS $\left(\phi ,E\right)$.
	Step 1: Normalize the input IVFSS matrix. Step 2: Based on the values of the subjective and objective weights, calculate the total weight. Step 3: Determine the normalized IVFSS matrix’s scoring matrix. Step 4: Create the standardized matrix using the mean, average, and corrected deviation matrices. Step 5: Calculate the ideal and anti-ideal points using standardized matrices. Step 6: Substitute the weighted EDs and calculate $WE{D}_i^{+}$ and $WE{D}_i^{-}$. Step 7: Calculating each candidate’s suitability index value. Step 8: Sort the candidates based on the values of the suitability index.
Output:	The ranked objects.

Algorithm 2. CODAS based MADM method for IVFSS [20]
Input:	Decision matrix based on IVFSS $\left(\phi ,E\right)$.
	Step 1: Normalize the input IVFSS matrix. Step 2: According to the subjective and objective weights, calculate the total weight. Step 3: Determine the normalized IVFSS matrix’s scoring matrix. Step 4: Compute the weighted and normalized decision matrix. Step 5: Get the negative-ideal solution NIS and calculate the ED and the Hamming distance (HD) from the NIS. Step 6: Create the relative evaluation matrix RA. Step 7: Determine the evaluation score for every candidate. Step 8: Arrange the options in decreasing order of assessment RA values.
Output:	The ranked objects.

Algorithm 3. Similarity measure based MADM method for IVFSS [20]
Input:	Decision matrix based on IVFSS $\left(\phi ,E\right)$.
	Step 1: Normalize the input IVFSS matrix. Step 2: Calculate the total weight according to the subjective and objective weights. Step 3: Determine the similarity measure. Step 4: Arrange the options in decreasing order of similarity measure values.
Output:	The ranked objects.

Algorithm 4. A new efficient MADM method for IVFSS [19]
Input:	Decision matrix based on IVFSS $\left(\phi ,E\right)$.
	Step 1: Obtain the choice value. Step 2: Determine the total choice value. Step 3: Sort the candidates according to the total choice values.
Output:	The ranked objects.

We give the following example to demonstrate the four above methods and tend to find the disadvantages of them.

Example 1. 

Assuming that we obtain an IVFSS evaluation matrix as shown in

Table 1. An IVFSS evaluation matrix.

U	e1	e2	e3	e4	e5
h1	[0.00,0.1]	[0.8,1.0]	[0.8,0.9]	[0.8,0.9]	[0.9,0.9]
h2	[0.58,0.87]	[0.51,0.62]	[0.5,0.6]	[0.5,0.6]	[0.6,0.8]
h3	[0.61,0.81]	[0.55,0.65]	[0.3,0.4]	[0.4,0.6]	[0.7,0.8]
h4	[0.14,0.15]	[0.61,0.68]	[0.2,0.3]	[0.4,0.5]	[0.6,0.7]
h5	[0.13,0.15]	[0.43,0.49]	[0.3,0.4]	[0.4,0.6]	[0.4,0.6]

, where $U=\left\{h_1,h_2,h_3,h_4,h_5\right\}$ represents the five alternatives and $E=\left\{e_1,e_2,e_3,e_4,e_5\right\}$ represents the attributes of the alternatives. The weight given by the evaluator based on previous experience or preferences is: $w=\left\{w_1,w_2,w_3,w_4,w_5\right\}=\left\{0.35,0.15,0.15,0.15,0.2\right\}$.

The four above MADM methods for IVFSS are used for addressing the DM issue in Table 1, and the decision results are presented in

Table 2. DM results by the four existing methods for Table 1.

Algorithms	Ranking	Best Solution
Algorithm 1 [20]	h1 > h3 > h2 > h4 > h5	h1
Algorithm 2 [20]	h1 > h3 > h2 > h4 > h5	h1
Algorithm 3 [20]	h1 > h3 > h2 > h4 > h5	h1
Algorithm 4 [19]	h1 > h2 > h3 > h4 > h5	h1

.

h₁ and h₅ are the best and worst options respectively, according to Table 2. It is clear that the four existing methods are only able to rank alternative schemes but are not capable of classifying them. In some real-life applications, we not only want to know the sorting results of the candidates, but also want to know which candidates can be accepted, rejected, or delayed made decision, which is very important for decision-makers. That is, in many practical scenes, when there are too many alternatives, traditional MADM methods can only obtain the ranking results of alternative solutions and find the optimal choice but cannot directly make a decision to put alternatives into AD, RD and delayed-decision domain, which is unfavorable for decision-makers to make decisions. Therefore, in order to solve this pressing problem, we propose the following MATWDM algorithm based on IS under IVFSS environment.

3. A New MATWDM Approach Based on Ideal Solutions under IVFSS

In this part, we first suggest a few fresh definitions. Subsequently, we proposed a MATWDM algorithm based on ideal solutions in the IVFSS environment inspired by the method in [39] and introduce the implementation process of the algorithm through an example.

3.1. The Related Definitions

Definition 5. 

Given two interval-valued fuzzy soft number ${\tilde{s}}_1=\left[{\mu }_1^{-},{\mu }_1^{+}\right]$ and ${\tilde{s}}_2=\left[{\mu }_2^{-},{\mu }_2^{+}\right]$, the distance between them is defined as:

$$d\left({\tilde{s}}_1,{\tilde{s}}_2\right)=\sqrt[p]{{\displaystyle \frac{1}{2}}(|{\mu }_1^{-}-{\mu }_2^{-}{|}^p+|{\mu }_1^{+}-{\mu }_2^{+}{|}^p)}$$

(1)

When $p=1$, it denotes HD; when $p=2$, it means ED.

Definition 6. 

The distance between two alternatives A₁ and A₂ is defined as:

$$D\left(A_1,A_2\right)=\sqrt[p]{{\displaystyle \frac{1}{2n}}{\sum }_{i=1}^n\left(\left|{\mu }_1^{-}-{\mu }_2^{-}{|}^p+\right|{\mu }_1^{+}-{\mu }_2^{+}{|}^p\right)}$$

(2)

Here, n is the number of attributes. If we consider the weight, this distance can be given as:

$$D^w\left(A_1,A_2\right)=\sqrt[p]{{\displaystyle \frac{1}{2}}{\sum }_{i=1}^n{w}_i\left(\left|{\mu }_1^{-}-{\mu }_2^{-}{|}^p+\right|{\mu }_1^{+}-{\mu }_2^{+}{|}^p\right)}$$

(3)

Here $W=\left\{w_1,w_2,\mathrm{\dots },w_n\right\}$, $w_i$ is the weight of the attribute C_i and ${\sum }_{i=1}^n{w}_i=1$. Similarly, when $p=1$, it denotes HD; when $p=2$, it means ED.

Definition 7. 

Note ${\tilde{s}}_{{PIS}_i}$ and ${\tilde{s}}_{{NIS}_i}$ as the positive and negative IS on the IVFSS, respectively. On the IVFSS, three IS determination methods are proposed as follows.

(1)

Type 1 ideal solution (T1IS): using the endpoint of the value range as the IS scheme.

Positive ideal solution:

$${\tilde{S}}_{PI{S}_1}=\left\{\begin{array}{c} \left[1,1\right],C \mathrm{is} \mathrm{a} \mathrm{benefit} \mathrm{attribute}\\ \left[0,0\right],C \mathrm{is} \mathrm{a} \cos \mathrm{t} \mathrm{attribute}\end{array}\right.$$

(4)

Negative ideal solution:

$${\tilde{S}}_{NI{S}_1}=\left\{\begin{array}{c}\left[0,0\right],C \mathrm{is} \mathrm{a} \mathrm{benefit} \mathrm{attribute}\\ \left[1,1\right],C \mathrm{is} \mathrm{a} \cos \mathrm{t} \mathrm{attribute}\end{array}\right.$$

(5)

(2)

Type 2 ideal solution (T2IS): using the best and worst values as IS.

Positive ideal solution:

$${\tilde{S}}_{PI{S}_2}=\left\{\begin{array}{c}\max \widetilde{{\mathrm{s}}_{\mathrm{i}}},\mathrm{C} \mathrm{is} \mathrm{a} \mathrm{benefit} \mathrm{attribute}\\ \min \widetilde{{\mathrm{s}}_{\mathrm{i}}},\mathrm{C} \mathrm{is} \mathrm{a} \cos \mathrm{t} \mathrm{attribute}\end{array},\right.$$

(6)

Negative ideal solution:

$${\tilde{S}}_{NI{S}_2}=\left\{\begin{array}{c}\min \widetilde{{\mathrm{s}}_{\mathrm{i}}},\mathrm{C} \mathrm{is} \mathrm{a} \mathrm{benefit} \mathrm{attribute}\\ \max \widetilde{{\mathrm{s}}_{\mathrm{i}}},\mathrm{C} \mathrm{is} \mathrm{a} \cos \mathrm{t} \mathrm{attribute}\end{array}\right.,$$

(7)

where $i\in \left\{1, 2,\dots ,n\right\}$.

(3)

Type 3 ideal solution (T3IS): Selecting IS through intervals of membership and non-membership.

Positive ideal solution:

$${\tilde{S}}_{{PIS}_3}=\left\{\begin{array}{c} \left[max{\mu }_i^{-},max{\mu }_i^{+}\right],C \mathrm{is} \mathrm{a} \mathrm{benefit} \mathrm{attribute}\\ \left[min{\mu }_i^{-},min{\mu }_i^{+}\right],C \mathrm{is} \mathrm{a} \cos \mathrm{t} \mathrm{attribute}\end{array}\right.,$$

(8)

Negative ideal solution:

$${\tilde{S}}_{{NIS}_3}=\left\{\begin{array}{c} \left[min{\mu }_i^{-},min{\mu }_i^{+}\right],C \mathrm{is} \mathrm{a} \mathrm{benefit} \mathrm{attribute}\\ \left[max{\mu }_i^{-},max{\mu }_i^{+}\right],C \mathrm{is} \mathrm{a} \cos \mathrm{t} \mathrm{attribute}\end{array}\right.,$$

(9)

Definition 8. 

The threshold calculation formula is derived from the weighted distance between the IS and the evaluation matrix as follows:

$${\alpha }_i={\displaystyle \frac{\left(1-\sigma \right)D^w\left({\tilde{S}}_i,{\tilde{S}}_{PIS}\right)}{\left(1-\sigma \right)D^w\left({\tilde{S}}_i,{\tilde{S}}_{PIS}\right)+\sigma D^w\left({\tilde{S}}_i,{\tilde{S}}_{NIS}\right)}}$$

(10)

$${\beta }_i={\displaystyle \frac{\sigma D^w\left({\tilde{S}}_i,{\tilde{S}}_{PIS}\right)}{\sigma D^w\left({\tilde{S}}_i,{\tilde{S}}_{PIS}\right)+\left(1-\sigma \right)D^w\left({\tilde{S}}_i,{\tilde{S}}_{NIS}\right)}}$$

(11)

$${\gamma }_i={\displaystyle \frac{D^w\left({\tilde{S}}_i,{\tilde{S}}_{PIS}\right)}{D^w\left({\tilde{S}}_i,{\tilde{S}}_{PIS}\right)+D^w\left({\tilde{S}}_i,{\tilde{S}}_{NIS}\right)}}$$

(12)

where the risk avoidance parameter is described by means of σ and $\sigma \in \left[0, 1\right].$ The decision-makers’ risk tolerance determines the value of σ. A higher σ value may be selected if the decision-makers want findings that are certain or accurate. A smaller σ value can be set if the decision-makers prefer fuzzy or ambiguous results. Therefore, decision-makers are more resistant to hesitant decisions the higher the risk avoidance coefficient.

Determine the action that the decision-maker should take for an object $S_i$ according to the relationship between the threshold and conditional probability.

• (P) If $\mathrm{Pr}(C|S_i)\ge {\alpha }_i$, and $\mathrm{Pr}(C|S_i)\ge {\gamma }_i$, $S_i\in POS(C)$
• (B) If $\mathrm{Pr}(C|S_i)\le {\alpha }_i$, and $\mathrm{Pr}(C|S_i)\ge {\beta }_i$, $S_i\in BND(C)$
• (N) If $\mathrm{Pr}(C|S_i)\le {\beta }_i$, and $\mathrm{Pr}(C|S_i)\le {\gamma }_i$, $S_i\in NEG(C)$

Definition 9. 

According to the weighted distance value, the combined score value can be computed as

$${\delta }_i={\displaystyle \frac{D^w({\tilde{S}}_i,{\tilde{S}}_{PIS})}{D^w({\tilde{S}}_i,{\tilde{S}}_{PIS})+D^w({\tilde{S}}_i,{\tilde{S}}_{NIS})}}$$

(13)

The higher the combined score, the more likely the decision-maker will choose alternative object ${\mathrm{S}}_{\mathrm{i}}$.

3.2. The Proposed Method

Following the definitions above, we provide our algorithm (Algorithm 5) as follows:

Algorithm 5. A MATWDM algorithm based on ideal solutions for IVFSS
Input:	Decision matrix based on IVFSS $\left(\phi ,E\right)$ and the risk avoidance parameter σ.
	Step 1: Obtain the three types positive and negative ideal solutions for the input decision matrix. Step 2: Determine weighted ED (weight HD) between every interval-valued fuzzy soft number and the three types positive and negative ideal solutions for each candidate by Formula (3). Step 3: Achieve the thresholds for each object based on the above weighted ED (weight HD) by Formulas (10)–(12). Step 4: Determine which behavior to take for the object (accept, delay decision, reject), according to the relationship between threshold and conditional probability. Step 5: Compute the combined score value ${\delta }_i$ based on the weighted distance value by Formula (13). Step 6: Sort the candidates based on the combined score value.
Output:	The ranked and classified candidates.

3.3. An Example

According to the above algorithm steps, we illustrate the algorithm and its implementation process through the following Example 1.

Example 1. 

In the IVFSS evaluation matrix, $U=\left\{h_1,h_2,\dots ,h_{12}\right\}$ represents the 12 alternative objects, and $E=\left\{e_1,e_2,e_3,e_4,e_5,e_6\right\}$ represents the 6 attributes possessed by the objects. Besides, the decision-makers provide different attribute weights $w_i=\left\{w_1,w_2,w_3,w_4,w_5,w_6\right\}=\left\{0.12,0.18,0.3,0.16,0.14,0.1\right\}$. 

Table 3. IVFSS evaluation matrix.

U	e1	e2	e3	e4	e5	e6
h1	[0.81,0.92]	[0.62,0.81]	[0.62,0.87]	[0.8,0.88]	[0.75,0.85]	[0.83,0.89]
h2	[0.78,0.83]	[0.76,0.85]	[0.79,0.92]	[0.8,0.92]	[0.78,0.89]	[0.78,0.86]
h3	[0.75,0.77]	[0.72,0.78]	[0.95,0.98]	[0.85,0.92]	[0.88,1]	[0.84,0.89]
h4	[0.81,0.92]	[0.85,0.93]	[0.84,0.96]	[0.81,0.94]	[0.82,0.93]	[0.78,0.94]
h5	[0.69,0.75]	[0.76,0.86]	[0.85,0.88]	[0.82,0.95]	[0.7,0.81]	[0.7,0.85]
h6	[0.71,0.86]	[0.67,0.75]	[0.83,0.92]	[0.83,0.89]	[0.79,0.88]	[0.79,0.85]
h7	[0.75,0.79]	[0.78,0.89]	[0.95,1]	[0.77,0.86]	[0.89,0.92]	[0.89,0.93]
h8	[0.66,0.81]	[0.79,0.86]	[0.85,0.98]	[0.81,0.95]	[0.58,0.96]	[0.82,0.96]
h9	[0.57,0.76]	[0.68,0.84]	[0.67,0.87]	[0.58,0.83]	[0.54,0.85]	[0.86,0.92]
h10	[0.51,0.68]	[0.75,0.89]	[0.81,0.97]	[0.5,0.96]	[0.76,0.89]	[0.71,0.85]
h11	[0.79,0.85]	[0.81,0.89]	[0.82,1]	[0.89,0.92]	[0.86,0.93]	[0.79,0.88]
h12	[0.67,0.74]	[0.79,0.86]	[0.86,0.89]	[0.84,0.92]	[0.81,0.99]	[0.87,0.95]

shows the evaluation matrix in the form of IVFSS.

We apply the suggested approach to resolve the problem of DM in Table 3.

Step 1: Three types of positive and negative ISs are obtained for Table 3, as shown in

Table 4. Three types of ideal solutions.

	e1	e2	e3	e4	e5	e6
T1IS	PIS1	[1,1]	[1,1]	[1,1]	[1,1]	[1,1]
	NIS1	[0,0]	[0,0]	[0,0]	[0,0]	[0,0]
T2IS	PIS2	[0.81,0.92]	[0.85,0.93]	[0.95,1]	[0.89,0.92]	[0.89,0.92]
	NIS2	[0.51,0.68]	[0.62,0.81]	[0.62,0.87]	[0.5,0.96]	[0.54,0.85]
T3IS	PIS3	[0.81,0.92]	[0.85,0.93]	[0.95,1]	[0.89,0.95]	[0.89,1]
	NIS3	[0.51,0.68]	[0.62,0.75]	[0.62,0.87]	[0.5,0.83]	[0.54,0.81]

.

Step 2: According to Formula (3), calculate the weighted ED (weight HD) between the evaluation value and the three types of ISs for each object, displayed in

Table 5. Weighted ED for Example 1.

U	T1IS		T2IS		T3IS
	DPIS1	DNIS1	DPIS2	DNIS2	DPIS3	DNIS3
h1	0.235	0.795	0.166	0.143	0.171	0.143
h2	0.175	0.837	0.095	0.155	0.100	0.160
h3	0.155	0.880	0.077	0.209	0.075	0.212
h4	0.132	0.885	0.060	0.194	0.062	0.202
h5	0.198	0.820	0.116	0.149	0.123	0.156
h6	0.194	0.824	0.108	0.160	0.114	0.160
h7	0.142	0.887	0.057	0.208	0.064	0.211
h8	0.188	0.856	0.110	0.157	0.110	0.166
h9	0.281	0.757	0.200	0.068	0.204	0.063
h10	0.251	0.139	0.169	0.062	0.171	0.066
h11	0.139	0.881	0.062	0.198	0.066	0.203
h12	0.165	0.857	0.089	0.176	0.088	0.182

and

Table 6. Weighted HD for Example 1.

U	T1IS		T2IS		T3IS
	DPIS1	DNIS1	DPIS2	DNIS2	DPIS3	DNIS3
h1	0.460	0.888	0.361	0.292	0.374	0.302
h2	0.407	0.914	0.291	0.354	0.306	0.371
h3	0.353	0.936	0.234	0.414	0.229	0.422
h4	0.343	0.939	0.205	0.412	0.213	0.431
h5	0.428	0.904	0.328	0.330	0.334	0.347
h6	0.423	0.906	0.313	0.357	0.327	0.352
h7	0.342	0.940	0.189	0.428	0.211	0.432
h8	0.389	0.921	0.289	0.367	0.283	0.389
h9	0.503	0.864	0.415	0.228	0.426	0.223
h10	0.451	0.349	0.359	0.206	0.365	0.224
h11	0.349	0.924	0.206	0.380	0.224	0.395
h12	0.383	0.924	0.278	0.380	0.273	0.395

.

Step 3: According to Formulas (10), (11), and (12), calculate the thresholds corresponding to different types of ideal solutions under weighted ED (weight HD). In this decision-making problem, assuming that there is no uncertainty preference difference between attributes, the consistent risk avoidance coefficient is $\sigma =0.45$.

The following is the solution process of threshold values corresponding to different ideal solutions of $h_1$ under weighted Euclidean distance:

$${\alpha }_1={\displaystyle \frac{(1-\sigma )D^W(A_i,A_{PIS})}{(1-\sigma )D^W(A_i,A_{PIS})+\sigma D^W(A_i,A_{NIS})}}={\displaystyle \frac{(1-0.45)\times 0.235}{(1-0.45)\times 0.235+0.45\times 0.795}}=0.266$$

$${\beta }_1={\displaystyle \frac{\sigma D^W(A_i,A_{PIS})}{\sigma D^W(A_i,A_{PIS})+(1-\sigma )D^W(A_i,A_{NIS})}}={\displaystyle \frac{0.45\times 0.235}{0.45\times 0.235+(1-0.45)\times 0.795}}=0.195$$

$${\gamma }_1={\displaystyle \frac{D^W(A_i,A_{PIS})}{D^W(A_i,A_{PIS})+D^W(A_i,A_{NIS})}}={\displaystyle \frac{0.235}{0.235+0.795}}=0.228$$

$${\alpha }_2={\displaystyle \frac{(1-\sigma )D^W(A_2,A_{PIS})}{(1-\sigma )D^W(A_2,A_{PIS})+\sigma D^W(A_2,A_{NIS})}}={\displaystyle \frac{(1-0.45)\times 0.166}{(1-0.45)\times 0.166+0.45\times 0.143}}=0.587$$

$${\beta }_2={\displaystyle \frac{\sigma D^W(A_2,A_{PIS})}{\sigma D^W(A_2,A_{PIS})+(1-\sigma )D^W(A_2,A_{NIS})}}={\displaystyle \frac{0.45\times 0.166}{0.45\times 0.166+(1-0.45)\times 0.143}}=0.488$$

$${\gamma }_2={\displaystyle \frac{D^W(A_2,A_{PIS})}{D^W(A_2,A_{PIS})+D^W(A_2,A_{NIS})}}={\displaystyle \frac{0.166}{0.166+0.143}}=0.538$$

$${\alpha }_3={\displaystyle \frac{(1-\sigma )D^W(A_3,A_{PIS})}{(1-\sigma )D^W(A_3,A_{PIS})+\sigma D^W(A_3,A_{NIS})}}={\displaystyle \frac{(1-0.45)\times 0.171}{(1-0.45)\times 0.171+0.45\times 0.143}}=0.593$$

$${\beta }_3={\displaystyle \frac{\sigma D^W(A_3,A_{PIS})}{\sigma D^W(A_3,A_{PIS})+(1-\sigma )D^W(A_3,A_{NIS})}}={\displaystyle \frac{0.45\times 0.171}{0.45\times 0.171+(1-0.45)\times 0.143}}=0.494$$

$${\gamma }_3={\displaystyle \frac{D^W(A_3,A_{PIS})}{D^W(A_3,A_{PIS})+D^W(A_3,A_{NIS})}}={\displaystyle \frac{0.171}{0.171+0.143}}=0.544$$

All of thresholds for each object under weighted ED are shown in

Table 7. Thresholds under weighted ED for Example 1.

U	T1IS			T2IS			T3IS
	α1	β1	γ1	α2	β2	γ2	α3	β3	γ3
h1	0.266	0.195	0.228	0.587	0.488	0.538	0.593	0.494	0.544
h2	0.204	0.146	0.173	0.428	0.334	0.380	0.434	0.339	0.385
h3	0.177	0.126	0.150	0.311	0.232	0.270	0.303	0.225	0.262
h4	0.154	0.109	0.130	0.273	0.201	0.235	0.273	0.201	0.235
h5	0.228	0.165	0.195	0.486	0.388	0.437	0.492	0.393	0.442
h6	0.223	0.161	0.191	0.452	0.356	0.403	0.466	0.369	0.417
h7	0.164	0.116	0.138	0.251	0.183	0.215	0.270	0.199	0.233
h8	0.212	0.153	0.180	0.461	0.364	0.411	0.447	0.351	0.398
h9	0.312	0.233	0.271	0.782	0.706	0.746	0.797	0.725	0.763
h10	0.688	0.597	0.644	0.769	0.690	0.731	0.760	0.679	0.722
h11	0.162	0.114	0.136	0.277	0.204	0.239	0.284	0.210	0.245
h12	0.190	0.136	0.161	0.383	0.293	0.336	0.370	0.282	0.325

:

Similarly, all of thresholds for each object under weighted HD are shown in

Table 8. Thresholds under weighted HD for Example 1.

U	T1IS			T2IS			T3IS
	α1	β1	γ1	α2	β2	γ2	α3	β3	γ3
h1	0.388	0.298	0.341	0.602	0.503	0.553	0.602	0.503	0.553
h2	0.352	0.267	0.308	0.501	0.402	0.451	0.502	0.403	0.452
h3	0.316	0.236	0.274	0.409	0.316	0.361	0.399	0.308	0.352
h4	0.308	0.230	0.267	0.378	0.289	0.332	0.377	0.288	0.331
h5	0.367	0.279	0.322	0.548	0.448	0.498	0.540	0.441	0.490
h6	0.363	0.276	0.318	0.517	0.418	0.467	0.532	0.432	0.482
h7	0.308	0.229	0.267	0.350	0.265	0.306	0.375	0.286	0.329
h8	0.340	0.257	0.297	0.491	0.392	0.441	0.471	0.373	0.421
h9	0.416	0.323	0.368	0.690	0.599	0.646	0.700	0.610	0.657
h10	0.612	0.514	0.564	0.681	0.588	0.636	0.666	0.572	0.620
h11	0.316	0.236	0.274	0.399	0.307	0.352	0.409	0.316	0.361
h12	0.336	0.253	0.293	0.473	0.375	0.423	0.458	0.361	0.409

:

Step 4: according to the relationship between threshold value under the weighted ED and conditional probability, determine which behavior should be taken for the object (accept, delay decision, reject). Here, it is assumed that alternatives contain consistent conditional probability Pr = 0.3. The classified results are shown in

Table 9. Behaviors of each alternative object under weighted ED.

	POS(C)	BND(C)	NEG(C)
T1IS	h1, h2, h3, h4, h5, h6, h7, h8, h11, h12	h9	h10
T2IS	h4, h7, h11	h3, h12	h1, h2, h5, h6, h8, h9, h10
T3IS	h4, h7, h11	h3, h12	h1, h2, h5, h6, h8, h9, h10

.

From Table 9, for Type 2 and Type 3 ideal solution, it takes the acceptance action on alternative $h_4,h_7,h_{11}$; it takes the delayed decisions on object $h_3,h_{12}$; and object ${\mathrm{h}}_1,{\mathrm{h}}_2,{\mathrm{h}}_5,{\mathrm{h}}_6,{\mathrm{h}}_8,{\mathrm{h}}_9,{\mathrm{h}}_{10}$ are put into rejected domain.

If we compare threshold value under the weighted HD and conditional probability, the classified results are shown in

Table 10. Behaviors of each alternative object under weighted HD.

	POS(C)	BND(C)	NEG(C)
T1IS	h1, h2, h3, h4, h5, h6, h7, h8, h11, h12	h9	h10
T2IS	h4, h7, h11	h3, h8, h12	h1, h2, h5, h6, h9, h10
T3IS	h3, h4, h7	h8, h11, h12	h1, h2, h5, h6, h9, h10

(here, conditional probability is set as $Pr=0.4)$.

From Table 10, it can be seen that alternative ${\mathrm{h}}_1,{\mathrm{h}}_2,{\mathrm{h}}_3,{\mathrm{h}}_4,{\mathrm{h}}_5,{\mathrm{h}}_6,{\mathrm{h}}_7,{\mathrm{h}}_8,{\mathrm{h}}_{11},{\mathrm{h}}_{12}$ in T1IS is in the AD, ${\mathrm{h}}_9$ belongs to the boundary region, and ${\mathrm{h}}_{10}$ is in the RD.

Step 5: According to the Formula (13), based on weighted ED and HD, the combined score value ${\delta }_i$ are obtained shown in

Table 11. Combined score values under weighted ED.

U	T1IS	T2IS	T3IS
h1	0.7717	0.4623	0.4561
h2	0.8267	0.6198	0.6149
h3	0.8504	0.7301	0.7379
h4	0.8700	0.7649	0.7645
h5	0.8054	0.5633	0.5580
h6	0.8095	0.5967	0.5834
h7	0.8620	0.7851	0.7673
h8	0.8196	0.5886	0.6024
h9	0.7291	0.2537	0.2370
h10	0.3561	0.2686	0.2785
h11	0.8637	0.7614	0.7548
h12	0.8387	0.6636	0.6755

and

Table 12. Combined score value under weighted HD.

U	T1IS	T2IS	T3IS
h1	0.6587	0.4472	0.4471
h2	0.6919	0.5490	0.5480
h3	0.7261	0.6389	0.6480
h4	0.7327	0.6677	0.6687
h5	0.6784	0.5020	0.5096
h6	0.6817	0.5327	0.5183
h7	0.7334	0.6940	0.6712
h8	0.7032	0.5593	0.5790
h9	0.6319	0.3541	0.3434
h10	0.4363	0.3643	0.3798
h11	0.7256	0.6483	0.6387
h12	0.7070	0.5771	0.5914

.

Step 6: Sort the candidates based on the combined score value shown in

Table 13. Ranking results of the alternative objects under the two distance measures.

		Ranking	Best Alternative
weighted ED	T1IS	h4 > h11 > h7 > h3 > h12 > h2 > h8 > h6 > h5 > h1 > h9 > h10	h4
	T2IS	h7 > h4 > h11 > h3 > h12 > h2 > h6 > h8 > h5 > h1 > h10 > h9	h7
	T3IS	h7 > h4 > h11 > h3 > h12 > h2 > h8 > h6 > h5 > h1 > h10 > h9	h7
weighted Humming distance	T1IS	h7 > h4 > h3 > h11 > h12 > h8 > h2 > h6 > h5 > h1 > h9 > h10	h7
	T2IS	h7 > h4 > h3 > h11 > h12 > h8 > h2 > h6 > h5 > h1 > h9 > h10	h7
	T3IS	h7 > h4 > h3 > h11 > h12 > h8 > h2 > h6 > h5 > h1 > h9 > h10	h7

.

The best option, according to Table 13, is $h_7$ for both the weighted ED and the weighted HD.

By applying our proposed method, we can not only obtain the sorting result of alternative solutions, but also divide alternative solutions into AD, RD, and delayed-decision domain, which is convenient for decision-makers to make decisions.

4. Two Real-Life Applications and Comparative Analysis

In this section, we put the proposed method to use in two actual scenarios, compare it with the existing four approaches and analyze the decision results to verify the feasibility and superiority of our method.

4.1. Case One: Mine Emergency Decision [20]

One of the most dangerous risks in mining accidents is a mine explosion. The mine explosion, which also jeopardizes the safety of mine output, poses a serious threat to the safety of work and life. It is difficult to forecast an explosion disaster down to the last detail and have enough time to make necessary preparations and take emergency measures. In order to prepare for disasters and respond appropriately, emergency response plans and accident simulations are essential. The success and feasibility of the emergency preparations will directly affect how disasters develop and subsequent emergency responses. As a result, it is thought that using simulations to assess and decide on the emergency plans that have been presented is essential for the management of mine accident disasters.

Assume there are five emergency plans $U=\left\{x_1,x_2,x_3,x_4,x_5\right\}$ which are taken into account in the event of a coal mine explosion catastrophe. The expert selects the decision criteria $E=\left\{e_1,e_2,e_3,e_4,e_5\right\}$, which are the following: noxious gas concentration (denoted as e₁), casualty reduction in current events (presented as e₂), smoke and dust level (presented as e₃), viability of rescue operations (denoted as e₄), and facility damage repair (denoted as e₅). We can conclude that all criteria are benefit parameters. Assume that the expert’s prior preferences have set the following prior weights:$w=\{w_1,w_2,w_3,w_4,w_5\}=\left\{0.3,0.2,0.14,0.16,0.2\right\}$. The evaluations of emergency plans resulting from the expert’s questionnaire research and construction of an IVFSS are shown in

Table 14. IVFSS evaluation matrix for Case one [20].

U	e1	e2	e3	e4	e5
h1	[0.55,0.65]	[0.41,0.62]	[0.2,0.3]	[0.4,0.6]	[0.7,0.8]
h2	[0.58,0.67]	[0.51,0.62]	[0.2,0.3]	[0.4,0.5]	[0.6,0.8]
h3	[0.61,0.71]	[0.55,0.65]	[0.3,0.4]	[0.5,0.6]	[0.7,0.8]
h4	[0.54,0.65]	[0.61,0.68]	[0.2,0.3]	[0.4,0.5]	[0.6,0.7]
h5	[0.53,0.61]	[0.43,0.49]	[0.3,0.4]	[0.4,0.6]	[0.4,0.6]

in tabular form.

We apply our method to address the issue of achieving emergency plans for explosion accidents. Firstly, we obtain three types of positive and negative ISs which are given in

Table 15. Three ideal solutions for Case one.

		e1	e2	e3	e4	e5
T1IS	PIS1	[1,1]	[1,1]	[1,1]	[1,1]	[1,1]
	NIS1	[0.0]	[0.0]	[0.0]	[0.0]	[0.0]
T2IS	PIS2	[0.61,0.71]	[0.61,0.68]	[0.3,0.4]	[0.5,0.6]	[0.7,0.8]
	NIS2	[0.53,0.61]	[0.41,0.62]	[0.2,0.3]	[0.4,0.5]	[0.4,0.6]
T3IS	PIS3	[0.61,0.71]	[0.61,0.68]	[0.3,0.4]	[0.5,0.6]	[0.7,0.8]
	NIS3	[0.53,0.61]	[0.41,0.49]	[0.2,0.3]	[0.4,0.5]	[0.4,0.6]

.

Table 15 shows that the three viewpoints’ ideal solutions differ from one another, allowing decision-makers to select one of them as the foundation for their choices based on their preferences. Three different ISs are used to address the problem of choosing emergency plans for explosive events in order to demonstrate how the proposed model is to be used, and the outcomes of the decisions are compared. In this coal mine explosion accident problem, assuming that there is no uncertainty preference difference between attributes among decision-makers, the consistent risk avoidance coefficient is $\sigma =0.3$.

And then we compute the weighted Euclidean (Hamming) distances between the evaluated value and the three types of ISs aiming for achieving the thresholds corresponding to different types of ISs by weighted ED and HD displayed in

Table 16. Thresholds under weighted ED for Case one.

U	T1IS			T2IS			T3IS
	α1	β1	γ1		α1	β1	γ1		α1
h1	0.662	0.265	0.457	h1	0.662	0.265	0.457	h1	0.662
h2	0.662	0.265	0.457	h2	0.662	0.265	0.457	h2	0.662
h3	0.611	0.224	0.402	h3	0.611	0.224	0.402	h3	0.611
h4	0.664	0.267	0.459	h4	0.664	0.267	0.459	h4	0.664
h5	0.706	0.307	0.508	h5	0.706	0.307	0.508	h5	0.706

and

Table 17. Thresholds under weighted HD for Case one.

U	T1IS			T2IS			T3IS
	α1	β1	γ1		α1	β1	γ1		α1
h1	0.679	0.280	0.476	0.698	0.298	0.498	0.679	0.280	0.476
h2	0.680	0.280	0.476	0.700	0.300	0.500	0.681	0.282	0.478
h3	0.653	0.257	0.447	0.386	0.104	0.212	0.374	0.099	0.204
h4	0.681	0.282	0.478	0.709	0.310	0.511	0.690	0.290	0.488
h5	0.703	0.303	0.504	0.809	0.438	0.645	0.840	0.491	0.693

, respectively.

Based on the relationship between threshold value under the weighted Euclidean (Hamming) distance and the conditional probability, determine which behavior should be taken for the object (accept, delay-decision, reject). Here, it is assumed that alternatives contain consistent conditional probability $Pr=0.46$. The classified results are shown in

Table 18. The classified results under weighted Euclidean (Hamming) distance for Case one.

	Type of IS	POS(C)	BND(C)	NEG(C)
Weighted ED	T1IS	∅	h1, h2, h3, h4, h5	∅
	T2IS	h3	h1, h2, h4	h5
	T3IS	h3	h1, h2, h4	h5
Weighted HD	T1IS	∅	h1, h2, h3, h4, h5	∅
	T2IS	h3	h1, h2, h4, h5	∅
	T3IS	h3	h1, h2, h4	h5

.

It is easy to observe from Table 16 and Table 17 that each alternative threshold based on T1IS fluctuates within a small range. It leads to the results that all alternative emergency plans belong to the boundary domain and will not be accepted or rejected based on both weighted ED and weighted HD by T1IS From Table 18. Because the differences between thresholds are so small, poor decision-making may occur as a result. The fundamental cause of this issue is that when [0,0] and [1,1] are chosen as positive and negative ISs, respectively, the evaluation values of each choice are too similar and cannot adequately reflect the range of possibilities. For the other two types of ISs, due to the large range of threshold values, the differences between alternative solutions are also significant. From Table 18, it can be seen that emergency plan ${\mathrm{h}}_3$ are placed in the AD for both weighted ED and weighted HD under the ideal solution of T2IS and T3IS. This means that among all emergency plans, adopting emergency plan ${\mathrm{h}}_3$ is the most reliable. Under the weighted ED, emergency plan ${\mathrm{h}}_5$ are placed in the RD for both ideal solution of T2IS and T3IS, which means that emergency plan ${\mathrm{h}}_5$ are the least desirable. ${\mathrm{h}}_1,{\mathrm{h}}_2,{\mathrm{h}}_4$ are in the delayed-decision domain, and their decision results can be further distinguished. Under the weighted HD, the emergency plan ${\mathrm{h}}_{5 }$is placed in the RD under T3IS, which means that the emergency plan $h_{5 }$ is the least desirable. Under T2IS, ${\mathrm{h}}_1,{\mathrm{h}}_2,{\mathrm{h}}_4,{\mathrm{h}}_5$ are all in the delayed-decision domain, which can further distinguish their decision behavior.

Finally, we compute the combined score value ${\delta }_i$ based on weighted ED and weighted HD and rank the candidates based on the combined score value shown in

Table 19. Sorting results of candidate solutions for Case one.

		Methods	Type of IS	Ranking	Best Plan
Our method	Weighted Euclidean distance		T1IS	h3 > h2> h1 > h4 > h5	h3
			T2IS	h3 > h1> h2 > h4 > h5	h3
			T3IS	h3 > h1> h4 > h2 > h5	h3
	Weighted Hamming distance		T1IS	h3 > h1> h2 > h4 > h5	h3
			T2IS	h3 > h1> h2 > h4 > h5	h3
			T3IS	h3 > h1> h2 > h4 > h5	h3
Existing methods	Algorithm 1 [20]			h3 > h2> h1 > h4 > h5	h3
	Algorithm 2 [20]			h3 > h1> h4 > h2 > h5	h3
	Algorithm 3 [20]			h3 > h2> h1 > h4 > h5	h3
	Algorithm 4 [19]			h3 > h1> h2 = h4 > h5	h3

.

From Table 19, it is clear that under the weighted ED and weighted HD, the optimal choices for the three types of ideal solutions are all $h_3$, that is, in the mine explosion accident, the emergency plan $h_3$ is the optimal choice. In order to verify the feasibility of our method, we also apply the four existing methods as diverse as Algorithms 1 and 2 [20] and Algorithm 4 [19] into this case. We discover this fact that the emergency plan $h_3$ is still the optimal choice by the four existing methods, which have the same results with our method.

4.2. Case 2: Homestay Selection Decision

A real-world data set about homestays located in Siming District, Xiamen City, from the website www.agoda.com is obtained. Some college students who are about to graduate are planning a graduation trip to Xiamen. They want to find suitable homestays in Xiamen City. Visitors who have stayed in these candidate homestays evaluate them in six different categories after they check out. Cleanliness level, facilities, location, comfort level, service quality, and cost performance are the six factors listed. There are six different classifications for these occupants. For each category, guests provided an average score for a specific aspect of the homestay. Then, using the scores of each group of occupants in this aspect, we may determine the highest and lowest scores in this aspect to create an interval value. For one homestay, for instance, the highest rating provided by the six types of tenants in the category of “cost performance” is 8.8, while the lowest rating is 7.2. The score interval is then translated into [0.72, 0.88], which are sub-intervals of [0,1]. The homestay’s cost performance is at least 0.72 and reaches a maximum of 0.88 if the IVFSS model is employed to describe. That is, by normalizing the obtained minimum and maximum values as the lower and upper membership degrees of interval valued fuzzy numbers, the IVFSS are obtained. In this case, 19 alternative homestays have been identified. Here, the IVFSS for a real data set of the 19 homestays in Siming District, Xiamen City is following. U = {h₁, h₂, h₃, …, h₁₄, h₁₉}; E is a related set of six parameters, E = {e₁, e₂, e₃, e₄, e₅, e₆} = {“Cost performance”, “Comfort level”, “Service”, “Location”, “Environment and cleanliness”, “Facilities”}.

Table 20. The IVFSS for Case Two.

U	e1	e2	e3	e4	e5	e6
h1	[0.72,0.88]	[0.78,0.89]	[0.84,0.96]	[0.79,0.88]	[0.75,0.85]	[0.75,0.85]
h2	[0.78,0.91]	[0.88,0.89]	[0.93,0.97]	[0.8,0.92]	[0.78,0.89]	[0.77,0.86]
h3	[0.9,0.93]	[0.89,0.93]	[0.95,0.98]	[0.85,0.92]	[0.88,0.92]	[0.84,0.89]
h4	[0.85,0.95]	[0.83,0.95]	[0.84,0.96]	[0.81,0.94]	[0.82,0.93]	[0.78,0.94]
h5	[0.67,1]	[0.6,1]	[0.85,1]	[0.7,1]	[0.7,1]	[0.7,1]
h6	[0.79,1]	[0.75,0.88]	[0.85,1]	[0.83,1]	[0.79,0.88]	[0.75,0.85]
h7	[0.89,1]	[0.85,0.94]	[0.93,1]	[0.96,1]	[0.89,1]	[0.89,0.97]
h8	[0.67,0.96]	[0.58,0.96]	[0.92,1]	[0.5,0.95]	[0.58,0.96]	[0.58,0.96]
h9	[0.54,0.83]	[0.58,0.84]	[0.67,0.94]	[0.58,0.83]	[0.54,0.85]	[0.58,0.83]
h10	[0.25,1]	[0.75,1]	[0.5,1]	[0.5,0.96]	[0.25,1]	[0.25,0.92]
h11	[0.84,0.92]	[0.86,0.9]	[0.89,0.99]	[0.89,0.92]	[0.86,0.93]	[0.85,0.89]
h12	[0.84,1]	[0.79,1]	[0.81,1]	[0.84,1]	[0.81,1]	[0.81,1]
h13	[0.38,0.75]	[0.38,0.75]	[0.75,0.75]	[0.5,0.75]	[0.5,0.75]	[0.38,0.75]
h14	[0.67,0.8]	[0.83,0.94]	[0.67,0.85]	[0.67,0.8]	[0.67,0.75]	[0.67,0.8]
h15	[0.75,0.88]	[0.8,0.95]	[0.81,1]	[0.74,1]	[0.72,0.92]	[0.68,0.88]
h16	[0.73,0.93]	[0.83,0.96]	[0.79,0.96]	[0.67,0.93]	[0.65,0.89]	[0.65,0.93]
h17	[0.75,0.89]	[0.83,0.91]	[0.8,0.89]	[0.73,0.85]	[0.7,0.83]	[0.68,0.83]
h18	[0.71,0.89]	[0.86,0.93]	[0.88,0.96]	[0.79,0.93]	[0.75,0.89]	[0.67,0.88]
h19	[0.7,0.92]	[0.78,0.83]	[0.8,0.92]	[0.78,0.92]	[0.73,0.85]	[0.65,0.84]

below presents the IVFSS (S, E) as a tabular form of the 19 homestays in Siming District, Xiamen City.

Our approach is used to address the issue of choosing suitable homestays for travelers. First,

Table 21. Three ideal solutions for Case two.

		e1	e2	e3	e4	e5	e6
T1IS	PIS1	[1,1]	[1,1]	[1,1]	[1,1]	[1,1]	[1,1]
	NIS1	[0.0]	[0.0]	[0.0]	[0.0]	[0.0]	[0.0]
T2IS	PIS2	[0.84,1]	[0.79,1]	[0.93,1]	[0.96,1]	[0.89,1]	[0.81,1]
	NIS2	[0.25,1]	[0.38,0.75]	[0.5,1]	[0.5,0.95]	[0.25,1]	[0.25,0.92]
T3IS	PIS3	[0.89,1]	[0.89,1]	[0.95,1]	[0.96,1]	[0.89,1]	[0.89,1]
	NIS3	[0.25,0.75]	[0.38,0.75]	[0.5,0.75]	[0.5,0.83]	[0.25,0.75]	[0.25,0.75]

shows three categories of positive and negative ideal solutions that we have discovered.

Table 21 shows that the three viewpoints’ ideal solutions differ from one another, allowing decision-makers to select one of them as the foundation for their choices based on their preferences. Then, the weighted EDs between the evaluated value and the three types of ISs are calculated in order to obtain the thresholds corresponding to different types of ideal solutions under weighted ED shown in

Table 22. Thresholds under weighted ED for Case Two.

U	T1IS			T2IS			T3IS
	α1	β1	γ1	α2	β2	γ2	α3	β3	γ3
h1	0.200	0.144	0.170	0.200	0.144	0.170	0.200	0.144	0.170
h2	0.157	0.111	0.132	0.157	0.111	0.132	0.157	0.111	0.132
h3	0.109	0.076	0.091	0.109	0.076	0.091	0.109	0.076	0.091
h4	0.151	0.107	0.127	0.151	0.107	0.127	0.151	0.107	0.127
h5	0.223	0.161	0.190	0.223	0.161	0.190	0.223	0.161	0.190
h6	0.176	0.125	0.149	0.176	0.125	0.149	0.176	0.125	0.149
h7	0.087	0.060	0.072	0.087	0.060	0.072	0.087	0.060	0.072
h8	0.269	0.198	0.232	0.269	0.198	0.232	0.269	0.198	0.232
h9	0.333	0.250	0.290	0.333	0.250	0.290	0.333	0.250	0.290
h10	0.389	0.299	0.343	0.389	0.299	0.343	0.389	0.299	0.343
h11	0.126	0.088	0.105	0.126	0.088	0.105	0.126	0.088	0.105
h12	0.150	0.106	0.126	0.150	0.106	0.126	0.150	0.106	0.126
h13	0.423	0.329	0.375	0.423	0.329	0.375	0.423	0.329	0.375
h14	0.284	0.210	0.245	0.284	0.210	0.245	0.284	0.210	0.245
h15	0.199	0.143	0.169	0.199	0.143	0.169	0.199	0.143	0.169
h16	0.222	0.161	0.190	0.222	0.161	0.190	0.222	0.161	0.190
h17	0.225	0.163	0.192	0.225	0.163	0.192	0.225	0.163	0.192
h18	0.184	0.131	0.156	0.184	0.131	0.156	0.184	0.131	0.156
h19	0.224	0.162	0.191	0.224	0.162	0.191	0.224	0.162	0.191

.

We can choose the behavior as diverse as acceptance, defer decision, or rejection based on the relationship between the threshold value under the weighted ED and conditional probability, which are illustrated in

Table 23. The classified results and the related behaviors for Case two.

	BND(C)	NEG(C)	POS(C)
Type1	h1, h2, h3, h4, h5, h6, h7, h11, h12, h15, h16, h17, h18, h19	h8, h14	h9, h10, h13
Type2	h3, h4, h7, h11, h12	h1, h2, h6, h15, h18	h5, h8, h9, h10, h13, h14, h16, h17, h19
Type3	h3, h4, h7, h11, h12	h1, h2, h6, h18	h5, h8, h9, h10, h13, h14, h15, h16, h17, h19

.

From Table 23, whether it is the first type, the second type, or the third type, these candidates such as h₃, h₄, h₇, h₁₁, h₁₂ are all in the AD. That is, homestays in Siming District, Xiamen City such as h₃, h₄, h₇, h₁₁, h₁₂ are the best choice for these visitors.

Finally, we compute the combined score value ${\delta }_i$ based on weighted ED and rank the candidates shown in

Table 24. The sorting results by our methods and existing approaches for Case two.

Methods	Ranking Results	Best Choice
Type1	h7 > h3 > h11 > h12 > h4 > h2 > h6 > h18 > h15 > h1 > h16 > h5 > h19 > h17 > h8 > h14 > h9 > h10 > h13	h7
Type2	h7 > h11 > h3 > h12 > > h4 > h2 > h6 > h18 > h1 >h15 > h5 > h19 > h17 > h16 > h14 > h8 > h9 > h13 > h10	h7
Type3	h7 > h3 > h11 > h12 > h4 > h2 > h6 > h18 > h15 > h1 > h5 > h16 > h19 > h17 > h8 > h14 > h9 > h10 > h13	h7
Algorithm 1 [20]	h7 > h3 > h11 > h12 > h4 > h2 > h6 > h18 > h15 > h1 > h16 > h5 > h19 > h17 > h8 > h14 > h9 > h10 > h13	h7
Algorithm 2 [20]	h7 > h3 > h11 > h12 > h4 > h2 > h6 > h18 > h15 > h1 > h5 > h16 > h19 > h17 > h8 > h14 > h9 > h10 > h13	h7
Algorithm 3 [20]	h7 > h11 > h3 > h12 > > h4 > h2 > h6 > h18 > h1 >h15 > h5 > h19 > h17 > h16 > h14 > h8 > h9 > h13 > h10	h7
Algorithm 4 [19]	h7 > h12 > h3 > h11 > h4 > h2 > h6 > h5 > h18 > h15 > h1 > h16 > h19 > h17 > h8 > h14 > h9 > h10 > h13	h7

.

Table 24 makes it evident that for the three types of ideal solutions, the homestay h₇ is the best option. The four current approaches Algorithms 1–3 [20] and Algorithm 4 [19] are also used for this situation in order to confirm the viability of our method. The homestay h₇, is still the best option, as shown by the four other techniques that provide the same outcomes as our method, which illustrate the effectiveness of our method.

4.3. Advantage Analysis

From the above two practical cases, we can discover that our method not only obtains the ranking results of the alternatives, but also divides them into AD, RD, and delayed decision domain, which makes the decision results more reasonable and effective, facilitating decision-makers to make better decisions. However, existing methods as diverse as Algorithms 1–3 [20] and Algorithm 4 [19] only sort the alternatives to find the best choice. In some situations, we not only obtain the optimal alternative, but also intend to discover which candidates are acceptable, which ones demand a firm denial, and which ones should be delayed to making decision. For instance, in Case one, the government department and enterprises responsible for managing mining want to determine which emergency plans are feasible, and which ones should be further studied. For Case two, we find that the homestays such as $h_3,h_4,h_7,h_{11},h_{12}$ are the best choices. If we only obtain the optical choice as $h_7$ by the four existing methods, when the homestay $h_7$ is fully booked, we will not obtain the solution. By our method, if the homestay $h_7$ is fully booked, $h_3,h_4,h_{11},h_{12}$ can also be accepted. As a result, our method outperforms the four existing methods due to classification which are shown in

Table 25. Comparison results among five methods.

Methods	Algorithm 1 [20]	Algorithm 2 [20]	Algorithm 3 [20]	Algorithm 4 [19]	Our Method
sorting	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$
classification	$\times$	$\times$	$\times$	$\times$	$\surd$

.

5. Conclusions

Pioneering work on MADM based on IVFSS was made. However, there are still obvious disadvantages in the existing MADM algorithms which only sort alternative schemes, but are not able to classify them, which is detrimental to decision-makers’ efficient DM. In order to address this issue, a MATWDM algorithm based on ideal solutions for IVFSS is given in this paper. In addition to providing the ranking results of the alternatives, the approach also separates them into acceptance, rejection, and postponed decision domains. This helps decision-makers make more effective selections. Applications on two practical cases verify the effectiveness and feasibility of the proposed algorithm.

Our proposed method is not only applicable to the above two real-world examples; it also applies to many applications aiming at multi-attributes decision-making issues. The limitation of this study is only applicable to the model of interval-valued fuzzy soft sets. In the future, we will expand this idea to other fuzzy decision models and consider the possibility of developing a stochastic-fuzzy method for decision-making. On the other hand, these advantages and performances about sorting and classification can be kept on high-dimensional data or a large number of choices, while the complexity on large scale datasets will be increased when the number of candidates and attributes are added. Hence, learning how to reduce the complexity on high-dimensional datasets will be an area of focus in the future work.