An Improved TOPSIS Method Based on a New Distance Measure and Its Application to the House Selection Problem

It is difficult to choose an appropriate house for homebuyers. This is due to the difficulty of evaluating the multitude of factors, such as price, location, size, and so on. In order to help homebuyers in choosing an appropriate house, a method integrating the interval-valued Pythagorean FAHP and FTOPSIS is proposed. In the proposed approach, the evaluation criteria were determined by the experts, and the linguistic variables of interval-valued Pythagorean fuzzy numbers were used in the evaluations of the homebuyers and experts. A new distance between two IVPFNs is proposed. The weights of the evaluation criteria were determined by the interval-valued Pythagorean FAHP method by the homebuyers, and house selections were evaluated by interval-valued Pythagorean FTOPSIS method taking into account the new distance. Finally, a case study was executed to verify the feasibility of the proposed approach. The case study results reveal that the weights of criteria obtained by FAHP are not the same according to opinions of the different homebuyers.


INTRODUCTION
With the gradual improvement of people's living standards, more and more people buy houses for life in a certain city either for work or for education of children.But it is difficult for homebuyers to choose an appropriate house from the house resources by the real estate agents because they need to simultaneously consider factors such as price, value, size, and location.Some of the factors might be even contradictory.Therefore, house selection is a multi-criteria decision-making problem (MCDM).
MCDM approaches can be suitable tools to deal with the house selection problem.In the decades, researchers have proposed various methods regarding MCDM problems in fuzzy environments.For example, Prabhu and Ilangkumaran (2019) and Ahmet (2021) presented work on the analytic hierarchy process (AHP).Li (2010) and Li and Nan (2011) presented research on the technique for order preference by similarity to idea solution (TOPSIS).Other research was conducted on areas such as the relative ratio (RR) method (Li, 2009), fuzzy linear programming technique for multidimensional analysis of preference (FLINMAP) method (Li & Sun, 2007), linear programming (LP) (Yu et al., 2019), nonlinear programming approach (Li, 2011;Li & Liu, 2015), and game theory (Ye & Li, 2021;Liang et al., 2023).
The traditional AHP may not reflect the opinions of decision-makers.Therefore, new versions of AHP with fuzzy sets have been proposed.Zadeh (1965) proposed the fuzzy theory as an extension of the classical sets, and Atanassov (1986) proposed intuitionistic fuzzy sets (IFSs) as an extension of fuzzy sets.Atanassov (1989) also proposed interval-valued intuitionistic fuzzy sets (IVIFSs).However, the sum of the membership and non-membership of IVIFSs is equal to or less than one, and which may not be in line with people's way of thinking.To address this problem, Yager (2013) proposed Pythagorean fuzzy sets (PFSs) as an extension of the IFSs.Because PFSs allows the sum of membership and non-membership to exceed one, and the sum of squares to not exceed one, Pythagorean fuzzy sets theory are more powerful and flexible in solving problems involving uncertainty.Interval-valued Pythagorean fuzzy sets (IVPFSs) (Zhang, 2016), a generalization of PFSs, emerged as an effective tool to model the uncertain and imprecise information in the reallife decision evaluation process, and this method can be considered when decision-makers fail to employ crisp values, but use interval values to express their evaluation information.In the proposed method, linguistic variables of interval-valued Pythagorean fuzzy numbers (IVPFNs) are used in the evaluations by the homebuyers and experts.Hwang (1981) first proposed the TOPSIS method.Regarding the uncertainty in real situations, many studies on fuzzy extensions have been completed to enrich the theory of TOPSIS method.Different versions of TOPSIS based on fuzzy sets have been developed for considering uncertainties and vagueness in MCDM problems, such as the fuzzy TOPSIS (Dwivedi et al., 2018), the weighted fuzzy TOPSIS (Prabhu & Ilangkumaran, 2019), the intuitionistic fuzzy TOPSIS (Li & Nan, 2011), the interval-valued intuitionistic fuzzy TOPSIS (Li, 2010), and the Pythagorean fuzzy TOPSIS (Zhang & Xu, 2014).Although many studies state that methods of TOPSIS with different fuzzy sets have been applied widely in various fields, relatively little attention has been paid to the extended TOPSIS dealing with house selection problems under complex uncertainty based on IVPFSs.From the aforementioned studies, we were inspired to use weighted fuzzy TOPSIS (FTOPSIS) to rank the houses.
This study proposes a new hybrid group decision-making approach with the fuzzy AHP (FAHP) and FTOPSIS methods based on IVPFSs for the house selection problem.To choose an appropriate house for homebuyers, we propose an integrated two-stage MCDM approach.In the first stage, a panel of experts is formed to gather the opinions.The criteria of house selection are then obtained according to literature review and experts' opinions from the house perspective.IVPFSs are an extension of IFSs, and they provide more freedom to express homebuyers' judgments on the uncertainty and vagueness in house selection problems.The identified criterion weights are obtained through interval-valued Pythagorean FAHP.In the second stage, experts provide the judgment matrices of houses.Based on the judgment matrices and weights, the houses are ranked with interval-valued Pythagorean FTOPSIS according to the new distance.
In the rest of this study we include a literature review on IVPFSs, FAHP, and FTOPSIS; discuss the house selection problem; and share some basic concepts related to IVPFSs.We then present a novel distance measure of IVPFNs, and based on this new distance, propose an improved FTOPSIS method for house selection.To verify the proposed method's feasibility, we provide a numerical example of house selection.We then compare the proposed method and other methods and present our sensitivity analysis.We end the study with a conclusion and our suggestions for future areas to study.

Interval-Valued Pythagorean Fuzzy Sets
In recent decades, many studies involving PFSes theory in MCDM problems have been completed.Yager (2014) introduced a variety of aggregation operations for Pythagorean fuzzy numbers (PFNs).Peng and Yang (2016) proposed the basic concept of PFN, weighted operator, and score function.Zulqarnain et al. (2022) proposed the concept of interval Pythagorean fuzzy power-geometricgeometric Heronian mean operator.In the fuzzy sets, the problem of measurement of differences or distances is unavoidable.To manifest the distances properly, many methods of measuring distances have been proposed, and some of them have had an ideal effect on classification.Szmidt and Kacprzyk, (2000) put forward the Hamming distance, Li and Cheng (2002) proposed similarity measures of intuitionistic fuzzy sets.Li (2004) demonstrated measures of dissimilarity in intuitionistic fuzzy structures.Li and Wan (2017) introduced minimum weighted Minkowski distance power models for intuitionistic fuzzy multi-attribute decision-making (MADM).Similarly, many methods of measuring distances of PFNs were proposed and used widely in the MCDM problem.Zhang and Xu (2014) proposed a distance between PFNs, and Han et al. (2019) put forward a distance measure for linguistic Pythagorean fuzzy sets.Fei and Deng (2020) proposed a distance measure between PFNs and IVPFNs.Paul et al. (2023) proposed a new Pythagorean fuzzy-based distance operator.We propose a new distance measure of IVPFSs for house selection problem that is based on these studies.

FAHP AND FTOPSIS
The application of MCDM in various fields has attracted the interest of many scholars.In recent years, many researchers have shared results about MCDM.For example, Prabhu (2019) used FAHP and GRA-TOPSIS methods for 3D printer selection problems, and Ahmet (2021) employed the FAHP-FTOPSIS method to solve green supplier selection problems.Table 1 provides the details of these studies that are constructed in the fields of MCDM problems and methods.As Table 1 indicates, AHP, ARAS, TOPSIS, MCGP, CPT, and weighted operator methods are used frequently in MCDM problems.House selection and evaluation have been the focus of many studies concerning criteria such as price, location, house size, and transportation.Table 2 shows the details of the studies that are made for the solution and evaluation of the house selection problem.
The house selection problem has been investigated, as discussed in the Literature Review section, and shown in Tables 1 and 2. However, to the best of our knowledge, only a few studies have been completed on using the linguistic variables of IVPFSs in the house selection problem.Therefore, integrating the interval-valued Pythagorean FAHP and interval-valued Pythagorean FTOPSIS for the house selection problem is a meaningful undertaking.
In this study, we propose a group decision-making approach of FAHP and FTOPSIS methods under IVPFSs for the house selection problem.In the first stage, the criteria of the house selection problem are identified from literature review and experts.Based on the opinions of homebuyers, the weights of identified criteria are obtained by using interval-valued Pythagorean FAHP.In the second stage, houses are ranked by using interval-valued Pythagorean FTOPSIS according to the new distance measure.Figure 1 shows the flow chart of the proposed approach.The salient features of the proposed method are as follows.First, the proposed distance measure considers the membership degrees, non-membership degrees, and the degree of indeterminacy simultaneously.Hence, the new distance measure ensures the integrity of the information.Second, the weights of criteria are calculated according to homebuyers' opinions.Third, the rank of houses is calculated according to experts' opinions using the FTOPSIS method based on the new distance measure.

IVPFN
In this section, the general definition, and some basic operations of IVPFSs are introduced.
Definition 1 (Zhang, 2016): Let set X be a universe of discourse, and the interval-valued Pythagorean set (IVPFSs) P in X is defined as follows: where µ µ µ 0 1 represents the membership degree and holds that 0 ≤ ( ( )) ( ( )) µ ν 1 , and ν ν ν 0 1 represents the non-membership degree o f t h e e l e m e n t x X ∈ t o P .I n a d d i t i o n , π π π µ 1 is named the degree of indeterminacy.For the convenience, the interval-valued Pythagorean fuzzy number (IVPFNs) is defined as To compare the magnitude of two IVPFNs, Peng and Yang (2016) introduced a score function and distance.
ν ν 1 2 be two IVPFNs, a nature qu a s i -o r d e r i n g o n t h e I V P F N s i s d e f i n e d a s fo l l ows : P P ) µ µ ν ν be an IVPFNs, the score function of P is defined as shown in equation (1).
Example 1, which we present later in this paper, shows that Peng and Yang's method fails to measure IVPFS's distance in some cases.To avoid this issue, Fei and Deng (2020) developed a new distance measure.
Definition 5 (Fei & Deng, 2020): be two IVPFNs, the distance between P 1 and P 2 is defined as shown in equation (3).
This approach considers the membership and nonmembership degrees, but the degree of indeterminacy is not considered.This omission leads to information loss and thus affects the interpretation (Fei & Deng, 2020).To solve MCDM problems in a Pythagorean fuzzy environment, Zhang (2016) developed the Pythagorean fuzzy weighted averaging aggregation operator.

The Proposed Method
In this section, we propose two steps to help homebuyers choose an appropriate house from many houses.First, we calculate the weights of criteria of the house selection problem using the FAHP method.Second, we propose a new distance between IVPFNs and rank houses using the FTOPSIS method according to the proposed distance measure.

FAHP
To help homebuyers choose an appropriate house, we read a lot of literatures and consulted with five real estate experts.We determined four criteria of the house selection problem, each of which was found to include some sub-criteria.We constructed a FAHP of the house selection problem, as shown in Figure 1.
The procedures of the AHP approach in the interval-valued Pythagorean fuzzy environment are presented as follows.
Step 1: The pairwise comparison matrix A = × ( ) a ij m m is constructed based on the linguistic evaluation of house homebuyer, which adopts IVPFNs.The linguistic terms given by Karasan et al. (2018) are shown in Table 3.The difference matrix B = × ( ) B ij m m between the lower and upper values of the membership and non-membership functions is calculated using equation ( 5).
The interval multiplicative matrix S = × ( ) s ij m m is computed using equation ( 6).The determinacy value Ä = × ( ) τ ij m m of the houses is calculated using equation ( 7).
To obtain the weight matrices of houses, the determinacy degrees are multiplied with matrix S , and the weight matrix T = × ( ) t ij m m is calculated using equation (8).
The weights w i of houses are normalized using equation (9).

FTOPSIS
The careful analysis shows that the method proposed by Zhang and Xu (2016) fails to measure IVPFS's distance in some cases, and the distance proposed by Fei and Deng (2020) considers only membership and non-membership degrees, neglecting the degree of indeterminacy.This omission leads to information loss and thus affects interpretations.Based on the discussion above, a new distance measure between IVPFNs is defined and proved to satisfy all the axioms for distance.
ν ν 1 2 be two IVPFNs, then the distance between P 1 and P 2 is defined as shown in equation ( 10).

Proof (1): Because
( if and only if µ µ , if and only if P P 1 2 = . (3) Based on definition 7, d P P d P P ( , ) ( , ) is similar to the above.Now, consider the following example.
We propose an improved FTOPSIS method that is based on the new distance measure between IVPFNs.It is best to obtain through the TOPSIS method a satisfactory solution that should be as close as possible to the positive ideal solution and as far as possible to the negative ideal solution.The procedures of the FTOPSIS approach in the interval-valued Pythagorean fuzzy environment are presented as follows: The decision matrix R = × ( ( ( ))) P C x j i m n is constructed based on IVPFNs, where P C j n j ( ) ( , , , ) = 1 2 refer to the values of the criteria and housing resources, respectively.The matrix is denoted as follows: The positive ideal solution (PIS) and negative ideal solution (NIS) of criteria C j n j ( , , , ) = 1 2 of the houses x i m i ( , , , ) = 1 2 are determined using equations ( 11) and ( 12).
r + = ( ) and where r P s P j n The distance between the values of criteria of the house x i m i ( , , , ) = 1 2 and PIS are calculated using equation ( 13), and the distance between the values of criteria of the house x i m i ( , , , ) = 1 2 and NIS are calculated using equation ( 14). and where the distanced is calculated using definition 7.
In the TOPSIS method, the relative closeness RC i of all houses and the optimal house is calculated using equation ( 15).

RC d d d i
However, the relative closeness cannot achieve the aim that the optimal solution should have the shortest distance from the PIS and the farthest distance from the NIS simultaneously.To overcome this problem, first the relative ratio (RR) method, which is used in this study.The relative ratio ξ of all houses x i m ( , , , ) = 1 2 is calculated using equation ( 16).Step 5: Finally, the best ranking of houses is determined.The house with the highest revised coefficient value is the best house.

The Proposed Method
In this section, we propose a method integrating the interval-valued Pythagorean FAHP and intervalvalued Pythagorean FTOPSIS for the house selection problem.The details of this method for the house selection problem are shown in Figure 2. The procedure can be summarized as follows: 1.The criteria of house selection problem is determined based on literature review and opinions from experts. 2. The FAHP is constructed to calculate the weights of criteria based on the judgment matrices of homebuyers.3. The rank of the houses is obtained through the weighted FTOPSIS method based on the judgment matrices of the experts.

An evaluation Case of House Selection
The aim of this section is to demonstrate the effectiveness of the proposed method through a case study.Mr. Wang and Mrs. Chen, a young couple, decide to buy a house, emphasizing their consideration of money, number of rooms, and children's education.They want to buy a house at the price of 0.9 million to 1.4 million yuan.The house should be in a neighborhood with a good school, and the house should have three rooms.Based on these conditions, the housing resources x i i ( , , , ) = 1 2 8 obtained from  4. Mr. Wang and Mrs. Chen have difficulty selecting an appropriate house among this listing of houses.The proposed method can help them to make a decision because it creates a personalized ranking list according to the coefficient value of their fuzzy preferences.First, the judgment matrix is finished by Mr. Wang and Mrs. Chen and shown in Table 5.
Second, according to Table 2 and the IVPFWG operator, the interval-valued Pythagorean fuzzy matrix is obtained and shown in Table 6.
Finally, the weights of criteria are determined according to the judgment matrix of the young couple using the FAHP method.Similarly, operations are carried out within the sub-criteria.The local weights and global weights of the criteria and the sub-criteria are calculated and shown in Table 7.In Table 7, the weights ofC 11 , C 42 and C 43 are 0.341, 0.105 and 0.177, respectively, indicating that young couples pay more attention to the price of the house, the distance to workplace, and the distance to children's school.The result is consistent with their requirements.In the second stage, the judgment matrix of houses x i i ( , , , ) = 1 2 8 are finished by three experts.According to Table 2 and IVPFWG operator, the interval-valued Pythagorean fuzzy matrix is obtained.Positive and negative ideal solutions are then calculated using equations ( 11) and ( 12) and shown in Table 8.
The distances between houses x i i ( , , , ) = 1 2 8 and the positive and negative ideal solution are calculated respectively using equations ( 13) and ( 14) and shown in Table 9. Closeness coefficients of each house x i i ( , , , ) = 1 2 8 are then calculated using equation ( 16), and their rankings are obtained and shown in Table 9.The priority order of the houses is x 5 .This order suggests that the most appropriate house for the young couple is Meididadao, which is in downtown and close to children's school.

SeNSITIVITy ANALySIS
Sensitivity analysis is performed to test the results of the criteria weights.In fact, because different homebuyers have different requirements, the weights of criteria obtained by fuzzy AHP are not the ]) 0 331 0 416 0 552 0 655 same.Therefore, the results obtained are varied.Take the comparison of the second couple and the third couple as an example.Their weights are obtained using the proposed method based on their judgment matrices and shown in Table 10.
Table 11 shows that the appropriate house for the second couple is Biguiyuan with excellent property and landscape, and the appropriate one for the third couple is Mudanxincun, which is closer to a school and downtown.The rank of the houses is changed for two different homebuyers; thus, the ranking results are sensitive for the weights.

COMPARATIVe ANALySIS
We compared Ahmet's method (Ahmet, 2021) and the MCDM model proposed in this study to demonstrate the latter's effectiveness.We changed only the distance measure, using Ahmet's distance to replace the distance measure of the proposed method.Other processes remain unchanged.The results obtained are listed in Table 11.The appropriate house for second couple is x 4 (Biguiyuan) according to Ahmet's method.The result remains the same.Hence, the result of the proposed method is stable and effective.

CONCLUSION
There are many factors to be taken consideration when buying an appropriate house, and these factors create an MCDM problem.The aim of this paper is to help homebuyers to choose an appropriate house using the proposed method that integrates the interval-valued Pythagorean FAHP and interval-valued Pythagorean FTOPSIS.In this approach, first, the evaluation criteria were determined through the consultation of experts and literature review.Second, the weights of criteria were calculated using the interval-valued Pythagorean FAHP method according to the judgment matrices of homebuyers.Third, a new distance measure between IVPFNs was proposed, and the ranking of houses was evaluated using the interval-valued Pythagorean FTOPSIS method according the new distance.Finally, a case study was executed to verify the feasibility of the proposed approach.Our case study results pointed out that the weights of criteria obtained by using FAHP are not the same according to the judgment matrices of different homebuyers, and thus, the ranking results change.The limitation of this study is that various tools, such as interval-valued fuzzy Pythagorean entropy method and MCGP, can be applied for the house selection problem.

ACKNOwLeDGMeNT
This paper is supported by the Natural Science Foundation of Fujian Province (No. 2020J01384), the Education and Scientific Research Project for Young and Middle-aged Teachers of the Fujian Province Education Department (No.JAT190688), and Sanming University Introduction of .

Figure 1 .
Figure 1.The FAHP hierarchy for the house selection problem

Figure 2 .
Figure 2. Steps of the method integrating FAHP and FTOPSIS for house selection

Table 2 . A brief summary of house selection studies
Value (price), construction (size, number of rooms, number of floor levels), neighbor (pollution level, safety, landscape, recreational facilities), location (distance to downtown, distance of workplace, distance of children's school, public transportation) Ho et al. (2015) AHP Value, building quality, surrounding facility, transportation Wang (2013) AHP Location, price, design style, landscape, property service, surrounding facility, building quality, developer reputation, transportation Sun et al. (2013) Definition 2 (Zhang, 2016): Let P i

Table 4 . Information of housing resources
Meididadao 790