Housing affordability evaluation based on the third-generation prospect theory and the improved VIKOR method

Abstract

Housing affordability is an important issue and can be measured by an increasing number of indicators. Different urban settings may lead to different housing affordability criteria. This study incorporates the third-generation prospect theory and improved VIKOR method to construct a novel, comprehensive evaluation model for assessing housing affordability. The housing price to income and rent to income ratios were chosen as evaluation indicators, and the yearly median value of each indicator was taken as a dynamic reference point. The housing affordability indicators’ realistic prospect value matrix for large- and medium-sized Chinese cities were obtained for the study period’s duration. The comprehensive housing affordability prospect values were ranked using the improved VIKOR with entropy weight method. The novel proposed approach’s rationality and effectiveness were examined by comparing the original and prospect values, performing sensitivity analysis on the prospect value parameters, contrasting the ordinary and improved VIKOR methods, and comparing the proposed approach with the TOPSIS method. The results demonstrate that the proposed method can consider the decision maker’s psychological factors, endow housing affordability evaluation criteria with dynamic characteristics, overcome the problem of order reversal, and ensure the optimal compromise solution. Therefore, the proposed approach is suitable for housing affordability evaluation.

Keywords

Housing affordability third-generation prospect theory dynamic reference point prospect value VIKOR

1 Introduction

Urban economic development and the operation of the labor market are affected by housing affordability. People usually choose where they work and live based on whether the housing is affordable. However, according to the housing policy report of the Third United Nations Conference on Housing and Sustainable Urban Development, the unaffordable housing crisis, which includes high rent burdens, has become a major issue in developed and developing countries [1, 2]. A chronic shortage of affordable housing dampens the competitiveness and growth of a metropolitan area. It also contributes to the relatively poor housing quality or longer, more costly commutes borne by people in those areas [3]. Households that struggle with housing payments are more likely to suffer from adverse outcomes in health, education, and employment [4]. The Global Financial Crisis brought renewed attention to the increasing unaffordability of housing in many countries [5]. The COVID-19 pandemic heightened the focus on housing affordability because of its close relationship with public health [6].

While housing affordability has been an important issue for local governments and the public [7], accurately measuring it is exceedingly difficult and controversial [8]. Housing affordability has broad connotations and is influenced by a myriad of factors [9, 10]. Early measurement of housing affordability was grounded in the cost of living, the new urban affordability measures increasingly need to be multidimensional. The US Department of Housing and Urban Development (HUD) takes housing and ‘transportation’ costs into the Location Affordability Index. OECD states that subjective well-being can bring additional information not captured by standard objective housing affordability measures [11]. However, each indicator can only measure one aspect of housing affordability, and different indicators will lead to incompatible composite measures of housing affordability. Furthermore, housing affordability is very regional. This means that different countries, regions, and cities have different criteria for determining the boundary values of their housing affordability factors [12]. For example, it may be inappropriate to use an internationally accepted criterion of price to income ratio (PIR), to measure housing affordability in China.

To this end, the study proposes a new multiple-criteria decision-making (MCDM) approach to evaluate housing affordability. This novel approach integrates VIKOR with entropy weight and prospect theory. The third-generation prospect theory is incorporated into the calculation of housing affordability prospect values instead of the original prospect theory. The resulting, integrated prospect value considers the psychological factors regarding housing affordability from the viewpoint of the decision maker. Additionally, multiple reference points endow the housing affordability evaluation criteria in different periods and regions with dynamic characteristics. The entropy weighting method solves the problem inherent to subjective or equal weights and overcomes the problem of order reversal caused by changing the evaluation dataset. The VIKOR method’s rankings ensure that the optimal compromise solution is chosen for the housing affordability evaluation, making it easier for decision makers to resolve conflicts between housing affordability indicators.

To substantiate the significance of the novel methodology, a comparison of it and related methods was conducted using actual data and indicators. The evaluation results demonstrate the advantages of the proposed method. The most significant advantage is that the proposed method produces results that are more likely to reflect reality. The results can provide a reference for policy formulation and management practices and assist the developer and consumer choices.

The rest of this paper is organized as follows. Section 2 provides a review of some background information and related works. The housing affordability evaluation problem is defined as an MCDM problem in Section 3. The section also introduces the third-generation prospect theory and improved entropy-weighted VIKOR method (i.e., improved VIKOR), which are used in the evaluation model. In Section 4, the evaluation results of 35 large- and medium-sized cities obtained by the novel proposed approach are discussed via sensitivity and comparative analyses. Section 5 discusses the conclusions and future prospective research work.

2 Literature review

First, housing affordability evaluation are re-viewed. Next, the MCDM and VIKTOR methods are briefly discussed, the improved VIKOR method is explained. Finally, the prospect theory is discussed.

2.1 Housing affordability evaluation

Because of the diversity of indicators, some scholars have begun to use Multiple Criteria Decision Making (MCDM) to evaluate the housing affordability. Mulliner think that measurement of affordability should consider social and environmental issues, such as housing quality, location, and access to ser-vices and facilities [13]. Mulliner et al. [14] not only assess the affordability of different housing locations, but also contrast the resulting rankings of six different MCDM approaches in Liverpool, England. Said et al. assessed sustainable housing affordability of six areas in Sabah, a state of Malaysia, based on COPRAS method [15]. Haidar & Bahammam, developed a theoretical model for affordability and sustainability of housing [16].

Their work demonstrated that MCDM can effectively analyze the housing affordability problem. However, there exist many noncomprehensive, potentially conflicting housing affordability indicators that are relevant to a particular region. It may be necessary to reach compromises among all the housing affordability indicators. The priority criteria for ranking the housing affordability also need to be determine by a comprehensive approach [17].

2.2 MCDM and VIKOR

MCDM can also be effectively used by decision makers to select the best solution given multiple criteria. Some methods (e.g., AHP [18], ELECTRE [19], PROMETHEE [20], BWM [21], and DEMATEL [22]) are based on quantitative measurements and use a few attributes to compare alternatives. Other methods (e.g., TOPSIS [23], VIKOR [24], MABAC [25], and MARCOS [26]) are representative outranking methods based on a reference point or goal. Researchers normally choose which MCDM method to use based on the purpose of the study and the actual problem [27].

VIKOR is one of the best compromise methods for ranking alternatives and can solve conflicting and incommensurable problems. Instead of comparing the criteria and alternatives, it measures and evaluates each alternative independently. Therefore, the assessment can be based on raw data. Most importantly, VIKOR can balance between total and individual satisfaction via a maximum “group utility” for the “majority” and a minimum “individual regret” for the “opponent” [28]. Solutions given by TOPSIS consider the relationships between the alternatives and the positive and negative ideal points, regardless of their relative importance. This is similar to MABAC and MARCOS.

Another vital issue to consider is rank reversal. Adding or deleting alternatives may cause the rankings produced by the VIKOR method to reverse [29]. This rank reversal phenomenon was first detected in the context of AHP [30] and occurred in other MCDM methods (i.e., TOPSIS, PROMETHEE, and ELECTRE) as well [31 –35].

The entropy weighting method can improve the traditional VIKOR method. It is an objective weighting method, is mainly used to calculate the relative weights between the decision criteria based on the data obtained [36]. It takes the valid information contained in the known data into account and can resolve the rank reversal problem [29].

By extending VIKOR for fuzzy environments, Opricovic [37] allowed for some decision making with incomplete, uncertain information. Subsequently, some scholars refined it for additional fuzzy environments (e.g., hesitant fuzzy sets, interval type-2 fuzzy sets, and Pythagorean fuzzy environment [38 –41]).

Despite several enhancements, the improved VIKOR approach does not address the problem that the previous, comprehensive evaluation of housing affordability had only one judgment criterion. Since housing affordability in different regions and periods should vary, in theory [42], the feelings and expectations of policymakers and individuals about the affordability of housing in a given region may change over time. Therefore, they can make different decisions when faced with different scenarios of housing affordability losses or gains.

2.3 Prospect theory

To assess the influence of subjective factors on decision-making behavior, many scholars are, at present, turning to prospect theory. Prospect theory considers the psychological behavior of decision makers and is also often combined with other evaluation methods to construct new evaluation models. These models are then applied in various fields.

Wang et al. proposed a new group emergency decision making (GEDM) method for emergency decision making [43]. Ding and Liu [44] proposed an MCDM method based on prospect theory and VIKOR to solve emergency decision-making problems with 2-dimensional, uncertain linguistic information. Fu et al. [45] proposed a new risk prioritization method based on cumulative prospect theory and the second type of intuitionistic fuzzy VIKOR method to study railway train failure risk. Liu et al. [46] integrated fuzzy linguistic set, best-worst methods, prospect theory, and VIKOR to help new energy vehicle manufacturers select innovative suppliers. Phochanikorn and Tan [47] selected an appropriate green supplier for a particular company based on prospect theory and MCDM modeling. Environmental decision-making is another area where prospect theory has been applied [48, 49]. Based on the prospect theory of loss aversion effect, Lin et al. [50] established a dynamic price recommendation method for housing purchase by applying the fuzzy-MCDM technique.

Prospect theory has also evolved. Li and Yuan [51] tackled the fuzziness and uncertainty of decision information. They took the dynamic alternatives as the reference point and proposed a three-parameter interval grey linguistic variable decision-making method based on the projection model and prospect theory. Li and Chen [52] proposed a method to determine the prospect value based on multiple reference points.

Both the original prospect theory and the cumulative prospect theory assume that the reference points are fixed, so the influence of the reference points is not considered when calculating the probability weight. Schmidt et al. [53] proposed the third-generation prospect theory in 2008. It retains the prediction function used in the previous two generations of prospect theory and expands upon those versions. When the decision weights are designated for order dependence, uncertain reference points may be obtained. The dynamic reference point weights for each indicator are defined by the reference-dependent subjective expected utility theory method. They believed that a reference point weight should consider the corresponding probability and sized order of all results for event relative to the reference point.

3 Housing affordability evaluation model

There are multiple indicators on housing affordability, and these indicators can conflict with each other. Given the advantages of the VIKOR approach and prospect theory regarding housing affordability, it was logical to combine the third-generation prospect theory and improved VIKOR method to form a novel, comprehensive MCDM model, which can better evaluate housing affordability.

In this section, a comprehensive evaluation model is developed to analyze housing affordability in different cities. The model’s housing affordability prioritization can be treated as an MCDM problem, and the housing affordability indicator can be regarded as a decision criterion. The VIKOR accounts for the conflicts among housing affordability indicators. The entropy weighting method avoids the overweighting of expert-preferred indicators, common in subjective or equal weightings schemes, and overcomes the problem of ranking order reversal due to changes to the evaluation datasets. The integrated prospect value of this model considers the decision maker’s psychology when accessing housing affordability, and multiple reference points endow the housing affordability evaluation criteria with dynamic characteristics for different periods and regions.

Specifically, the value function and cumulative decision weights for each indicator based on the original values of the housing affordability indicators, dynamic reference points, and preference parameters for each city for different periods are determined. The integrated prospect matrix of each indicator over time for each city is obtained. Subsequently, to avoid order inversion, the integrated prospect matrix is normalized to calculate entropy weights. Then, group utility values, individual regret values, and compromise degrees are calculated according to the steps of the VIKOR method. The ranking results are obtained from the evaluation results. The general steps of this evaluation model are illustrated in Fig. 1.

Fig. 1

Evaluation flowchart.

3.1 Definition of the problem

For a housing affordability evaluation problem, it can be assumed that there exists a set of evaluated cities, A ={ a₁, a₂, a₃, …, a_m }, with m alternatives, where i = 1, 2, 3, …, m, and a_i denotes the evaluated city. Let C ={ c₁, c₂, c₃, …, c_n } be the housing affordability indicators. There are n attributes such that j = 1, 2, 3, …, n, and c_j represents the indicator attribute. The attribute value matrix of each city is X = [x_ij] _m×n, where x_ij represents the attribute value of evaluated city a_i for indicator attribute c_j. To assess the housing affordability of each indicator over a period of time, let S = (s_t|t = 1, 2, 3, … q) represent the corresponding state of each city indicator x_ij over t years.

Let V = [V_ij] be the prospective value matrix of the housing affordability indicator, where V_ij represents the corresponding prospect value of the evaluated city, a_i, in the state S of indicator attribute c_j. Let ω ={ ω₁, ω₂, ω₃, …, ω_n } be the corresponding attribute weight of the affordability indicator, where ω_j satisfies 0 ≤ ω_j ≤ 1 and $\sum_{j = 1}^{n} ω_{j} = 1$ . In this study ωis calculated by the entropy weighting method.

3.2 Calculation steps

The various steps of the algorithm are as follows:

Stage 1: Construct the prospect value matrix V.

Step 1: For each indicator for each city and each year, calculate its value function v.

The value functions of each indicator in each year are described in Equation (1): $v (f, h) = {\begin{matrix} {(f (s_{t}) - h (s_{t}))}^{α^{+}} f \geq h \\ - λ {(h (s_{d}) - f (s_{d}))}^{α^{-}} f < h \end{matrix},$ (1) where f represents the indicator value corresponding to s_t, and h represents the dynamic reference point for each indicator every year. α⁺ and α^- are the degrees of concavity and convexity, respectively, of the power function for the gain and loss of housing affordability indicators in an evaluated city. λ is the loss avoidance coefficient. This study takes α⁺ = 0.88, α^- = 0.88 and λ = 2.25 [54].

Step 2: Calculate the probability weight, W, W of each indicator for each city and year.

In accordance with the third-generation prospect theory, the median value of the annual housing affordability indicator is selected as the reference point h. Then comparing f and h under state s_t, if f ≥ h, then a weak gain state exists. Let m⁺ denote the number of weakly acquired states. Instead, if f < h, then a strong loss state exists, and the number of strong loss states is m^- = q - m⁺. Then the subscripts are reassigned. For all subjects t, d, if t > d and v (f (s_t) , h (s_t)) ≥ v (f (s_d) , h (s_d)), then the states with weak gains are indexed m⁺, …, 1, and the states with strict losses are indexed -1, …, - m^-.

The weight function expression in the third-generation prospect theory is Equation (2): $W (s_{t}; f, h) =$ (2)

${\begin{matrix} W^{+} (π_{t}) & if t = m^{+} \\ W^{+} (\sum_{d \geq t} π_{d}) - W^{+} (\sum_{d > t} π_{d}) & if 1 \leq t \leq m^{+} - 1 \\ W^{-} (\sum_{d \leq t} π_{d}) - W^{-} (\sum_{d < t} π_{d}) & if - m^{-} + 1 \leq t \leq - 1 \\ W^{-} (π_{t}) & if t = - m^{-} \end{matrix},$ where ${\begin{matrix} W^{+} (π) = \frac{p_{t}^{γ}}{{(p_{t}^{γ} + {(1 - p_{t})}^{γ})}^{1 / γ}} \\ W^{-} (π) = \frac{p_{t}^{δ}}{{(p_{t}^{δ} + {(1 - p_{t})}^{δ})}^{1 / δ}} \end{matrix}$ .

In the formula, p_t represents the probability corresponding to state s_t for the housing affordability indicator in year t. And ∑p_t = 1. Here, it is assumed that the annual housing affordability indicator has an equal influence on the comprehensive prospect value of each city. That is, the probability p_t equals $\frac{1}{q}$ each year. γ and δ influence the weight function’s curvature and the acquired risk attitude and loss risk attitude coefficient, respectively. Here, γ = 0.61 and δ = 0.69 [54].

Step 3: Calculate the prospect value, V_ij, of each indicator in each city during the research period.

Suppose F represents the set of states, s_t, with housing affordability indicator values f ≥ h, the reference point for the housing affordability indicator in the evaluated city, and G represents the set of states, s_d, with housing affordability indicator values f < h.

Then, the prospect value for each indicator in each city is Equation (3): $V (f, h) = \sum_{t = 1}^{q} v (f, h) \times W (s_{t}; f, h)$ (3) $= \sum_{t \in F} {[f (s_{t}) - h (s_{t})]}^{α^{+}} W (s_{t}; f, h)$

$- \sum_{d \in G} λ {[h (s_{d}) - f (s_{d})]}^{α^{-}} W (s_{d}; f, h) .$

Therefore, according to the third-generation prospect theory, the prospect value of each indicator for each city can be used to obtain the prospect value matrix V = [V_ij].

Stage 2: Determine the normalized prospect value matrix E and indicator attribute weight ω.

Step 4: Determine the normalized prospect value matrix E.

To eliminate the effects of different dimensions on the evaluation indicators, it is necessary to normalize the obtained original prospect value matrix. Since the selected indicators are cost-based, the standardized treatment and normalization formulas for the prospect value matrix, V_ij, are as Equation (4) and (5):

Step 5: Determine the indicator attribute weight, ω.

To reduce the objective influence of the indicator attribute weights, the weight ω of indicator attribute c_j is given according to the “entropy weighting method”. Doing so ensures that the weights can be adjusted on time as data is added, deleted, or substituted. The entropy weighting method is based on a decision matrix with alternatives and criteria. The steps of the entropy method are available in the literature [55, 56].

Stage 3: Calculate the values of S, R, Q and rank them according to the VIKOR method.

Step 6: Determine the positive ideal solution, E⁺ (PIS), and negative ideal solution, E^- (NIS), for all indicators, as illustrated in Equations (6) and (7). $E_{j}^{+} = max_{i} {E_{ij}, i = 1, 2, 3, \dots, m},$ (6) $E_{j}^{-} = min_{i} {E_{ij}, i = 1, 2, 3, \dots, m} .$ (7)

Step 7: Calculate the group utility value, S_i, and individual regret value, R_i.

Here, both the positive and negative ideal solutions are used as references to calculate the two values.

The positive ideal solution is taken as a reference, as illustrated in Equations (8) and (9). $S_{i}^{+} = \sum ω_{j} (\frac{E_{j}^{+} - E_{ij}}{E_{j}^{+} - E_{j}^{-}}),$ (8) $R_{i}^{+} = max_{j} {ω_{j} (\frac{E_{j}^{+} - E_{ij}}{E_{j}^{+} - E_{j}^{-}})} .$ (9)

The negative ideal solution is taken as a reference, as seen in Equations (10) and (11). $S_{i}^{-} = \sum ω_{j} (\frac{E_{ij} - E_{j}^{-}}{E_{j}^{+} - E_{j}^{-}}),$ (10) $R_{i}^{-} = max_{j} {ω_{j} (\frac{E_{ij} - E_{j}^{-}}{E_{j}^{+} - E_{J}^{-}})} .$ (11)

The group utility value and individual regret value of the scheme are Equations (12) and (13): $S_{i =} \frac{S_{i}^{+}}{S_{i}^{-}},$ (12) $R_{i =} \frac{R_{i}^{+}}{R_{i}^{-}} .$ (13)

Step 8: Calculate the compromise degree, Q_i.

To consider both group utility maximization and individual regret minimization, the decision making mechanism coefficient is set as u = 0.5. The compromise degree Q_i is given by Equation (14).

$\begin{matrix} Q_{i} = u \frac{S_{i} - min_{i} {S_{i}}}{max_{i} {S_{i}} - min_{i} {S_{i}}} \\ + (1 - u) \frac{R_{i} - min_{i} {R_{i}}}{max_{i} (R_{i}) - min_{i} {R_{i}}} . \end{matrix}$ (14)

Step 9: Rank the evaluation results.

According to S_i, R_i and Q_i, the evaluation results are sorted in descending order. The higher the values, the higher the ranking, and the better the result. A larger Q_i value also results in better evaluation results. When the following two conditions are met, the evaluation results can be sorted according to the Q_i value.

Advantages: $Q_{2} - Q_{1} \geq \frac{1}{m - 1}$ , where Q₁ and Q₂ are the values corresponding to the first and second evaluation results, respectively, and m is the number of evaluation results.

Acceptable decision reliability conditions: The decision results sorted by Q should be consistent with those sorted by S or R. That is, according to the decision results based on Q value ranking, the corresponding S or R value rankings should be relatively similar.

4 Empirical analysis

To test the rationality and effectiveness of the proposed approach for housing affordability evaluation, a realistic case study that includes evaluation indicators, data sources, implementation, and results was carried out. The evaluation results and discussion are presented in this section.

4.1 Evaluation indicators

In current housing affordability research in different countries, price to income ratio (PIR) and rent to income ratio (RIR) are commonly used indicators in policy decisions and academic studies [57 –59]. PIR, c₁, focuses on measuring the ability of a household to purchase housing. RIR, c₂, is mainly used to evaluate renting affordability for tenants. These indicators include both the purchase and rental aspects and vary as residential incomes and housing prices change in different locations and times. Therefore, these two major indicators were chosen as decision criteria for this study.

PIR is represented by the current market value of housing units and annual household incomes and RIR by the current market rents for housing units and monthly tenant incomes. Based on data availability, these two indicators are usually calculated, as shown in Table 1.

Table 1
Indicator descriptions

Target layer Criteria Indicator layer Method of calculation

Housing affordability c ₁ Price-to-income ratio (PIR) (Per capita housing floor area×Average population per household×Average price of housing price per unit floor area) ÷ (Average population per household×Per capita disposable income)

c ₂ Rent-to-income ratio (RIR) (Average rent per unit housing floor area×Per capita housing floor area) ÷ Per capita disposable income

Target layer	Criteria	Indicator layer	Method of calculation
Housing affordability	c ₁	Price-to-income ratio (PIR)	(Per capita housing floor area×Average population per household×Average price of housing price per unit floor area) ÷ (Average population per household×Per capita disposable income)
	c ₂	Rent-to-income ratio (RIR)	(Average rent per unit housing floor area×Per capita housing floor area) ÷ Per capita disposable income

4.2 Data sources

This study evaluated 35 large- and medium-sized cities. These cities are all provincial capitals or economically developed cities throughout China. All have large dynamic populations and high and continuously growing market demand for housing. They also have different economic development levels, growth, and resource availability. Therefore, the regional differences in housing affordability are significant. In short, the extensive data set can reflect the overall housing affordability situation in China. In this scenario, housing affordability in various regions around the country can provide an important reference for government policy making and regulation.

Calculating the PIR and RIR values in the evaluated cities required the original data (e.g., household income, housing data, and household size). The household income data is usually represented by per capita disposable income derived from the WIND Economic Database. The housing data included the average housing price per unit floor area, average rent per unit housing floor area, and per capita housing floor area. The per capita housing floor area data were obtained from the CEIC database and statistical yearbook for each city. Other necessary data were obtained from the WIND database. The household size data is usually represented by the average population per household, derived from statistical yearbooks.

China’s housing market began to develop in 2000 and has undergone tremendous changes since. It gradually stabilized in 2017 because of stringent housing policy. Therefore, this paper discusses the comprehensive housing affordability in different cities over those eight years, 2010-2017.

4.2.1 Descriptive statistics

The number of observations during the study period in the cities under study was 280. The differences between the maximum and minimum values of average housing price and rent per unit floor area were relatively large and fluctuated considerably.

The per capita disposable income and per capita housing floor area also increased over time. The average population per household changed slightly and was not significant. The gap between the housing PIR and RIR values was large. The standard deviation of the PIR was much larger than that for the RIR, indicating that the PIR’s dispersion degree is greater than that of the RIR, as seen in Table 2.

Table 2
Statistics of the variables used in the study

Variable name Number of observations Mean value Maximum value Minimum value Standard deviation

Average price of housing price per unit floor area 280 11233.534 54059.000 4562.167 7946.216

Per capita disposable income 280 31777.889 58988.000 14061.840 9453.609

Average population per household 280 2.926 4.040 2.500 0.264

Average rent per unit housing floor area 280 27.701 75.095 9.868 11.165

Per capita housing floor area 280 32.352 47.800 19.600 4.920

c₁ (PIR) 280 10.625 27.873 4.514 4.735

c₂ (RIR) 280 0.339 0.586 0.150 0.080

Variable name	Number of observations	Mean value	Maximum value	Minimum value	Standard deviation
Average price of housing price per unit floor area	280	11233.534	54059.000	4562.167	7946.216
Per capita disposable income	280	31777.889	58988.000	14061.840	9453.609
Average population per household	280	2.926	4.040	2.500	0.264
Average rent per unit housing floor area	280	27.701	75.095	9.868	11.165
Per capita housing floor area	280	32.352	47.800	19.600	4.920
c₁ (PIR)	280	10.625	27.873	4.514	4.735
c₂ (RIR)	280	0.339	0.586	0.150	0.080

4.2.2 Determine reference points h (s_t) and prospect value V_ij

For brevity, only the median PIR and RIR values used as reference points (i.e., h (s_t)) are shown in Table 3. Clearly, the reference points of the housing affordability indicators vary over the eight years.

Table 3
Reference points, h (s_t)

Criteria h (s₁) h (s₂) h (s₃) h (s₄) h (s₅) h (s₆) h (s₇) h (s₈)

2010 2011 2012 2013 2014 2015 2016 2017

Reference point of c₁ (PIR) 11.818 11.162 9.547 9.598 8.822 7.729 7.706 7.920

Reference point of c₂ (RIR) 0.531 0.507 0.500 0.516 0.526 0.518 0.488 0.279

Criteria	h (s₁)	h (s₂)	h (s₃)	h (s₄)	h (s₅)	h (s₆)	h (s₇)	h (s₈)
Reference point of c₁ (PIR)	11.818	11.162	9.547	9.598	8.822	7.729	7.706	7.920
Reference point of c₂ (RIR)	0.531	0.507	0.500	0.516	0.526	0.518	0.488	0.279

Following the third-generation prospect theory of Equation (1), the value function v of the PIR and RIR of the city a_i during the t years is calculated. That is coupled with Equation (2) to get the prospect value function of the PIR and RIR of the city a_i during the t years. Finally, using Equation (3), the comprehensive prospect value of PIR (V_i1) and RIR (V_i2) of the city a_i is obtained. The detailed procedure involved extensive calculation, so Table 4 only shows the results.

Table 4

Prospect value, V_ij

Alternative (City)	c₁ (PIR)	c₂ (RIR)	Alternative (City)	c₁ (PIR)	c₂ (RIR)
a₁ (Shanghai)	–23.933	–0.504	a₁₉ (Zhengzhou)	–0.677	–0.129
a₂ (Beijing)	–25.509	–0.525	a₂₀ (Hefei)	0.197	0.100
a₃ (Guangzhou)	–3.407	–0.076	a₂₁ (Nanchang)	–2.309	0.081
a₄ (Shenzhen)	–15.824	–0.035	a₂₂ (Harbin)	–0.756	–0.066
a₅ (Hangzhou)	–5.125	–0.149	a₂₃ (Taiyuan)	0.270	–0.077
a₆ (Nanjing)	–2.786	–0.010	a₂₄ (Changchun)	0.040	0.031
a₇ (Ningbo)	–3.542	0.040	a₂₅ (Nanning)	0.099	–0.130
a₈ (Fuzhou)	–5.986	–0.224	a₂₆ (Chongqing)	0.048	–0.062
a₉ (Xiamen)	–10.099	0.011	a₂₇ (Chengdu)	0.639	–0.034
a₁₀ (Jinan)	1.984	0.039	a₂₈ (Guiyang)	2.379	–0.018
a₁₁ (Qingdao)	1.470	0.107	a₂₉ (Kunming)	–0.300	–0.071
a₁₂ (Haikou)	–7.775	–0.071	a₃ (Xi’an)	2.057	0.065
a₁₃ (Tianjin)	–4.408	–0.191	a₃₀ (Lanzhou)	–1.112	–0.130
a₁₄ (Shijiazhuang)	0.997	0.071	a₃₁ (Xining)	1.395	–0.023
a₁₅ (Shenyang)	1.273	0.115	a₃₂ (Yinchuan)	2.219	0.169
a₁₆ (Dalian)	–5.158	–0.088	a₃₃ (Hohhot)	2.181	0.170
a₁₇ (Wuhan)	–0.037	–0.048	a₃₄ (Urumqi)	–0.033	–0.049
a₁₈ (Changsha)	3.102	0.046

4.2.3 Normalized results for the housing affordability prospect matrix and entropy weight (ω) determination

The housing affordability prospect matrix is normalized by Equations (4) and (5). The results are depicted in Fig. 2.

Fig. 2

Normalized prospect values of PIR (E_i1) and normalized prospect values of RIR (E_i2).

According to the entropy weight method in the literature of step 5 [55, 56], the entropy weights of these two indicators are ω = (0.729, 0.271).

The magnitude of the weight, ω₁ of the housing PIR is far greater than ω₂ of the housing RIR, which is consistent with most academic research and government decisions.

4.2.4 Group utility value S_i, individual regret value R_i, compromise degree Q_i and ranking

Next, Equations (6) and (7) were used to find the PIS and NIS for each indicator. The group utility value (S_i) and individual regret value (R_i) of each city were calculated according to Equations (8)–(13). Finally, Equation (14) was used to calculate the compromise degree, Q_i, and cities were sorted based on their S_i, R_i and Q_i values. Table 5 shows the specific results.

Table 5
Evaluation results

Alternative (City) S _i R _i Q _i Sorting (Q_i)

a₁ (Shanghai) 0.050 0.058 0.002 34

a₂ (Beijing) 0.000 0.000 0.000 35

a₃ (Guangzhou) 2.819 3.396 0.087 26

a₄ (Shenzhen) 0.778 0.512 0.017 33

a₅ (Hangzhou) 1.993 2.477 0.063 29

a₆ (Nanjing) 3.539 3.859 0.103 24

a₇ (Ningbo) 3.541 3.306 0.094 25

a₈ (Fuzhou) 1.595 2.148 0.053 30

a₉ (Xiamen) 1.509 1.167 0.036 32

a₁₀ (Jinan) 11.564 13.716 0.355 7

a₁₁ (Qingdao) 14.144 16.533 0.430 5

a₁₂ (Haikou) 1.695 1.630 0.046 31

a₁₃ (Tianjin) 2.011 2.810 0.068 27

a₁₄ (Shijiazhuang) 9.808 12.587 0.316 8

a₁₅ (Shenyang) 13.662 14.638 0.395 6

a₁₆ (Dalian) 2.212 2.464 0.065 28

a₁₇ (Wuhan) 5.063 7.645 0.182 14

a₁₈ (Changsha) 19.636 15.050 0.472 3

a₁₉ (Zhengzhou) 3.698 5.433 0.130 21

a₂₀ (Hefei) 8.852 8.846 0.245 10

a₂₁ (Nanchang) 4.789 4.287 0.125 23

a₂₂ (Harbin) 4.257 6.416 0.152 19

a₂₃ (Taiyuan) 4.932 6.818 0.167 17

a₂₄ (Changchun) 6.552 8.344 0.210 12

a₂₅ (Nanning) 4.169 5.583 0.138 20

a₂₆ (Chongqing) 4.936 7.189 0.173 16

a₂₇ (Chengdu) 6.026 8.380 0.205 13

a₂₈ (Guiyang) 9.908 9.705 0.272 9

a₂₉ (Kunming) 4.535 6.840 0.162 18

a₃₀ (Xi’an) 13.756 17.090 0.435 4

a₃₁ (Lanzhou) 3.459 5.323 0.126 22

a₃₂ (Xining) 7.427 9.127 0.233 11

a₃₃ (Yinchuan) 42.320 31.400 1.000 1

a₃₄ (Hohhot) 41.584 30.057 0.970 2

a₃₅ (Urumqi) 5.048 7.601 0.181 15

Alternative (City)	S _i	R _i	Q _i	Sorting (Q_i)
a₁ (Shanghai)	0.050	0.058	0.002	34
a₂ (Beijing)	0.000	0.000	0.000	35
a₃ (Guangzhou)	2.819	3.396	0.087	26
a₄ (Shenzhen)	0.778	0.512	0.017	33
a₅ (Hangzhou)	1.993	2.477	0.063	29
a₆ (Nanjing)	3.539	3.859	0.103	24
a₇ (Ningbo)	3.541	3.306	0.094	25
a₈ (Fuzhou)	1.595	2.148	0.053	30
a₉ (Xiamen)	1.509	1.167	0.036	32
a₁₀ (Jinan)	11.564	13.716	0.355	7
a₁₁ (Qingdao)	14.144	16.533	0.430	5
a₁₂ (Haikou)	1.695	1.630	0.046	31
a₁₃ (Tianjin)	2.011	2.810	0.068	27
a₁₄ (Shijiazhuang)	9.808	12.587	0.316	8
a₁₅ (Shenyang)	13.662	14.638	0.395	6
a₁₆ (Dalian)	2.212	2.464	0.065	28
a₁₇ (Wuhan)	5.063	7.645	0.182	14
a₁₈ (Changsha)	19.636	15.050	0.472	3
a₁₉ (Zhengzhou)	3.698	5.433	0.130	21
a₂₀ (Hefei)	8.852	8.846	0.245	10
a₂₁ (Nanchang)	4.789	4.287	0.125	23
a₂₂ (Harbin)	4.257	6.416	0.152	19
a₂₃ (Taiyuan)	4.932	6.818	0.167	17
a₂₄ (Changchun)	6.552	8.344	0.210	12
a₂₅ (Nanning)	4.169	5.583	0.138	20
a₂₆ (Chongqing)	4.936	7.189	0.173	16
a₂₇ (Chengdu)	6.026	8.380	0.205	13
a₂₈ (Guiyang)	9.908	9.705	0.272	9
a₂₉ (Kunming)	4.535	6.840	0.162	18
a₃₀ (Xi’an)	13.756	17.090	0.435	4
a₃₁ (Lanzhou)	3.459	5.323	0.126	22
a₃₂ (Xining)	7.427	9.127	0.233	11
a₃₃ (Yinchuan)	42.320	31.400	1.000	1
a₃₄ (Hohhot)	41.584	30.057	0.970	2
a₃₅ (Urumqi)	5.048	7.601	0.181	15

The evaluation results of the housing affordability indicators were analyzed based on the group utility value (S_i), individual regret value (R_i), and compromise degree (Q_i). The undeveloped, northwestern cities top the list, with Yinchuan and Hohhot claiming the top spots. Other cities with high rankings include Changsha in the central, Qingdao in the eastern, and Xi’an in the western region. At the bottom of the list are the developed eastern coastal cities (e.g., Beijing, Shanghai, and Shenzhen). Other cities with low rankings are Fuzhou, Haikou, and Xiamen, also in the eastern region. The evaluation results suggest that megacities have more unaffordable housing than medium-sized cities in the Midwest.

4.3 Discussions

4.3.1 Comparing the original with prospect values

First, the original value of the housing affordability indicator and the average weight were combined to get a simple evaluation result. That value was then compared with the result of prospect value V_ij with the average weight, as shown in Table 6.

Table 6
Rank results between original value and prospect value V_ij

Alternative (City) Prospect value Original value Alternative (City) Prospect value Original value Alternative (City) Prospect value Original value

a₁ (Shanghai) 34 34 a₁₃ (Tianjin) 27 27 a₂₅ (Nanning) 16 14

a₂ (Beijing) 35 35 a₁₄ (Shijiazhuang) 10 10 a₂₆ (Chongqing) 15 19

a₃ (Guangzhou) 25 26 a₁₅ (Shenyang) 8 9 a₂₇ (Chengdu) 11 12

a₄ (Shenzhen) 33 33 a₁₆ (Dalian) 28 29 a₂₈ (Guiyang) 3 2

a₅ (Hangzhou) 29 28 a₁₇ (Wuhan) 18 17 a₂₉ (Kunming) 19 16

a₆ (Nanjing) 24 24 a₁₈ (Changsha) 1 1 a₃₀ (Xi’an) 5 6

a₇ (Ningbo) 26 25 a₁₉ (Zhengzhou) 20 20 a₃₁ (Lanzhou) 22 22

a₈ (Fuzhou) 30 30 a₂₀ (Hefei) 12 11 a₃₂ (Xining) 9 8

a₉ (Xiamen) 32 32 a₂₁ (Nanchang) 23 23 a₃₃ (Yinchuan) 2 3

a₁₀ (Jinan) 6 5 a₂₂ (Harbin) 21 21 a₃₄ (Hohhot) 4 4

a₁₁ (Qingdao) 7 7 a₂₃ (Taiyuan) 13 18 a₃₅ (Urumqi) 17 13

a₁₂ (Haikou) 31 31 a₂₄ (Changchun) 14 15

Alternative (City)	Prospect value	Original value	Alternative (City)	Prospect value	Original value	Alternative (City)	Prospect value	Original value
a₁ (Shanghai)	34	34	a₁₃ (Tianjin)	27	27	a₂₅ (Nanning)	16	14
a₂ (Beijing)	35	35	a₁₄ (Shijiazhuang)	10	10	a₂₆ (Chongqing)	15	19
a₃ (Guangzhou)	25	26	a₁₅ (Shenyang)	8	9	a₂₇ (Chengdu)	11	12
a₄ (Shenzhen)	33	33	a₁₆ (Dalian)	28	29	a₂₈ (Guiyang)	3	2
a₅ (Hangzhou)	29	28	a₁₇ (Wuhan)	18	17	a₂₉ (Kunming)	19	16
a₆ (Nanjing)	24	24	a₁₈ (Changsha)	1	1	a₃₀ (Xi’an)	5	6
a₇ (Ningbo)	26	25	a₁₉ (Zhengzhou)	20	20	a₃₁ (Lanzhou)	22	22
a₈ (Fuzhou)	30	30	a₂₀ (Hefei)	12	11	a₃₂ (Xining)	9	8
a₉ (Xiamen)	32	32	a₂₁ (Nanchang)	23	23	a₃₃ (Yinchuan)	2	3
a₁₀ (Jinan)	6	5	a₂₂ (Harbin)	21	21	a₃₄ (Hohhot)	4	4
a₁₁ (Qingdao)	7	7	a₂₃ (Taiyuan)	13	18	a₃₅ (Urumqi)	17	13
a₁₂ (Haikou)	31	31	a₂₄ (Changchun)	14	15

Table 6 shows that the rankings based on the original value and prospect value in some cities vary. This can indicate that the housing affordability evaluation results were affected by the psychological factors affecting the decision maker.

4.3.2 Sensitivity analysis on the prospect value parameters

To deduce what factor had the most significant impact on the prospect value, a sensitivity analysis was conducted on the risk coefficients (α⁺ and α^-), the loss aversion coefficient (λ) of housing affordability indicator, and the risk attitude coefficients (δ and γ) of housing affordability gain and loss. The range of variation of the considered parameters was 15% and 30% higher and lower for each value. For example, α⁺ = α^- = 0.88, λ = 2.25, γ = 0.61 and δ = 0.69. The results are shown in Fig. 3. The horizontal axis represents the cities and the vertical axis the ranking changes.

Fig. 3

Ranking fluctuation diagram.

As shown in Fig. 3, the overall ranking does not fluctuate prominently. An outlier was the ranking for Taiyuan, which a change of up to 4 values. Shenyang, Xi’an, and Urumqi all saw ranking changes up to 3 values. Jinan, Qingdao, Haikou, Wuhan, Changsha, Harbin, Chongqing, and Lanzhou all had ranking changes of 2. The other cities’ rankings only changed by 1 or less. The risk coefficient and risk attitude in the value function do not really influence the results, and the values calculated from the realistic data were relatively stable. More importantly, the study’s findings suggest that the attitude of the decision makers can impact the values of the prospect theory parameters and, therefore, the evaluation results.

4.3.3 Solving the rank reversal problem in the original VIKOR method

To verify that the improved VIKOR method with entropy weight can avoid the rank inversion phenomenon, Beijing, Shanghai, Chengdu, and Harbin were removed from the data set at random. Then, the data was processed with the ordinary and improved VIKOR methods to check whether the order inversion phenomenon occurred.

When using the improved VIKOR method, the ranking results are the same as the rankings from the original scheme. Therefore, no order inversion occurred. However, when using the standard VIKOR method, the ranking results of some cities changed. Notably, the improved VIKOR method had the following ranking: Qingdao > Xi’an, Ningbo > Zhengzhou. In contrast, the ordinary scheme had the rankings: Xi’an > Qingdao, Zhengzhou > Ningbo. Therefore, it can be concluded that the improved VIKOR method with entropy weight can overcome the inversion order issue encountered in the standard VIKOR method. Thus, the proposed approach is more stable and practical, as depicted in Figs. 4 and 5.

Fig. 4

Ordinary VIKOR deletion scheme.

Fig. 5

Improved VIKOR deletion scheme.

4.3.4 Comparison of VIKOR with TOPSIS method

To prove the applicability and superiority of the novel, proposed approach, the results of the proposed approach were compared with those obtained by the TOPSIS method.

The TOPSIS method was chosen for comparison because it is commonly used for comparison in MCDM studies. Both methods, TOPSIS and the proposed method, are largely similar [60]; both use the normalized prospect value matrix and entropy weights. However, they differ in their aggregation functions and normalization methods. They also use different ranking methods. Therefore, the ranking results can reflect the differences between various schemes.

In Fig. 6, the cities are the abscissa, and the evaluation values are the ordinate. TOPSIS’s values were all remarkably close, while the proposed VIKOR method’s values were higher and varied more. Both methods produced values that were more in agreement for the cities with lower affordability rankings. The VIKOR results were greater for the high affordability ranking values. This is probably because the VIKOR method takes the group utility and individual regret into account, while the TOPSIS method does not consider the importance of relative distance.

Fig. 6

Comparison of VIKOR and TOPSIS.

According to Table 7, the proposed VIKOR method has some rankings like Zhengzhou >Lanzhou >Nanchang >Ningbo, and Xi’an >Qingdao >Shenyang. The rankings from TOPSIS include Nanchang >Ningbo >Zhengzhou >Lanzhou, and Qingdao >Shenyang >Xi’an. Comparing the rankings of both methods to actual values, the proposed method’s results were more consistent with reality in terms of the housing affordability issue.

Table 7

Ranking by VIKOR and TOPSIS

Alternative (City)	Ranking by VIKOR	Ranking by TOPSIS	Alternative (City)	Ranking by VIKOR	Ranking by TOPSIS
a₁ (Shanghai)	34	34	a₁₉ (Zhengzhou)	21	23
a₂ (Beijing)	35	35	a₂₀ (Hefei)	10	10
a₃ (Guangzhou)	26	26	a₂₁ (Nanchang)	23	14
a₄ (Shenzhen)	33	33	a₂₂ (Harbin)	19	20
a₅ (Hangzhou)	29	29	a₂₃ (Taiyuan)	17	17
a₆ (Nanjing)	24	24	a₂₄ (Changchun)	12	12
a₇ (Ningbo)	25	22	a₂₅ (Nanning)	20	21
a₈ (Fuzhou)	30	32	a₂₆ (Chongqing)	16	18
a₉ (Xiamen)	32	31	a₂₇ (Chengdu)	13	13
a₁₀ (Jinan)	7	7	a₂₈ (Guiyang)	9	9
a₁₁ (Qingdao)	5	4	a₂₉ (Kunming)	18	19
a₁₂ (Haikou)	31	30	a₃₀ (Xi’an)	4	6
a₁₃ (Tianjin)	27	28	a₃₁ (Lanzhou)	22	25
a₁₄ (Shijiazhuang)	8	8	a₃₂ (Xining)	11	11
a₁₅ (Shenyang)	6	5	a₃₃ (Yinchuan)	1	1
a₁₆ (Dalian)	28	27	a₃₄ (Hohhot)	2	2
a₁₇ (Wuhan)	14	15	a₃₅ (Urumqi)	15	16
a₁₈ (Changsha)	3	3

5 Conclusions and future work

This study uses the third-generation prospect theory and the improved VIKOR method with entropy weight to account for the psychological behaviors of decision makers and dynamic changes in indicator reference points, as well as a compromise between indicators and the order reversal problem. Using those components, a novel evaluation model of housing affordability was constructed. The model’s results show that the eastern coastal cities, led by Beijing, Shanghai, and Shenzhen, have low housing affordability, while other eastern cities (e.g., Xiamen, Haikou, and Fuzhou) also have unaffordable housing price. The cities with the best housing affordability are Yinchuan and Hohhot in the western region, followed by Changsha in the central, Xi’an in the western region, and Qingdao in the central region.

Further exploration of the rationality and effectiveness of the novel proposed method is possible by looking at the housing affordability evaluation results of those cities. Sensitivity analysis of how the parameters of prospect theory changed showed that the attitude of the decision maker has a certain influence on the housing affordability evaluation. Comparing the ordinary and improved VIKOR method revealed that the standard method encountered rank reversal when adding or deleting data. In contrast, the improved VIKOR method with entropy weight is not consistent before and after data addition or removal. Finally, by comparing with the TOPSIS method, the proposed method’s conformity to the purpose of housing affordability evaluation and its proximity to the selected city contexts is demonstrated.

Therefore, the proposed approach can evaluate the PIR and RIR situation while integrating those indicators with the prospect theory. It can consider the changes in the housing affordability criterion, allowing it also to evaluate housing affordability in a time. Local governments can use the evaluation results to develop different response mechanisms and measures (e.g., stabilizing housing prices, regulating the rental market, and increasing resident income). Residents, developers, and investors can also use the results to choose residences and make related housing decisions. This approach proposed in this paper can handle similar MCDM problems requiring decision-makers to make choices with the help of ideal reference points that may vary with their preferences or decision environment.

However, this study has some limitations that can be addressed by future research efforts. First, more housing affordability indicators can be considered for inclusion (e.g., satisfaction with housing, living condition, and government policies and strategies). Second, the qualitative or quantitative indicators can be represented with fuzzy methods, such as the hesitant fuzzy linguistic method and the intuitive fuzzy method. Third, according to the research purpose, a comparison of the different segments of the research period can be conducted with segmented analysis. For example, the analysis can look at changes before and after the implementation of a particular housing policy. Fourth, the ranking method of VIKOR can be adjusted to better suit actual decision making goals. Fifth, the theoretical parameters of the prospect value α⁺, α^-, λ, γ and δ in this study were chosen based on previous prospect theory literature. Subsequent investigation can be carried out to determine the corresponding parameters of the indicator prospect value, according to the decision making’s purpose and different decision makers’ attitude. Sixth, Due to the differences in the cultural and knowledge backgrounds of housing affordability decision makers, multi-attribute group decision making approach (MAGDM) can also be very helpful in solving this type of problem. MAGDM with multi-granular hesitant fuzzy linguistic term sets (HFLTSs) has been used considers the weight vectors of decision makers and attributes in the proposed consensus model [61, 62]. In the future, individual consensus measures and the group consensus measure [63] can help experts, policymakers, consumers and developers to reach consensus in housing affordability evaluating problems.

References

Adabre

M.A.

, Chan

A.P.

and Darko

, A scientometric analysis of the housing affordability literature, Journal of Housing and the Built Environment (2021), 1–33.

Shahar

D.B.

, Gabriel

and Oliner

S.D.

, New Research on Housing Affordability, Regional Science & Urban Economics 80 (2020), 103438.1–103438.4.

Hsieh

C.T.

and Moretti

, Housing Constraints and Spatial Misallocation, American Economic Journal: Macroeconomics 11 (2019), 1–39.

Newman

S.J.

and Holupka

C.S.

, Housing Affordability And Investments In Children, Journal of Housing Economics 24 (2013), 89–100.

Haffner

M.E.A.

and Hulse

, A fresh look at contemporary perspectives on urban housing affordability, International Journal of Urban Sciences (2019), 1226–5934.

Kang

, Choi

, Kim

, Lee

K.O.

, Lee

, Park

I.K.

, Park

and Seo

, COVID-19 impact on city and region: What’s next after lockdown, International Journal of Urban Sciences 24(3) (2020), 297–315.

Galster

and Lee

K.O.

, Housing affordability: A framing, synthesis of research and policy, and future directions, International Journal of Urban Sciences 25(2) (2020), 1–52.

Murphy

, ‘Houston, we’ve got a problem’: The political construction of a housing affordability metric in New Zealand, Housing Studies 29(7) (2014), 893–909.

Gabriel

and Painter

, Why affordability matters, Regional Science and Urban Economics 80 (2020), 103378.1–103378.6.

10.

Quigley

J.M.

and Raphael

, Is housing unaffordable? Why isn’ t it more affordable? Journal of Economic Perspectives 18(1) (2004), 191–214.

11.

Sunega

and Lux

, Subjective perception versus objective indicators of overcrowding and housing affordability, Journal of Housing and the Built Environment 31(4) (2016), 695–717.

12.

Emma

, Kieran

and Vida

, An assessment of sustainable housing affordability using a multiple criteria decision making method, Omega 41(2) (2013), 270–279.

13.

Mulliner

and Maliene

, An analysis of professional perceptions of criteria contributing to sustainable housing affordability, Sustainability 7(1) (2015), 248–270.

14.

Mulliner

, Malys

and Maliene

, Comparative analysis of MCDM methods for the assessment of sustainable housing affordability, Omega 59 (2016), 146–156.

15.

Said

, Majid

R.A.

, Alias

, Adnan

Y.M.

and Razali

M.N.

, Sustainable Housing Affordability In Sabah, Planning Malaysia 14(5) (2016), 65–76.

16.

Haidar

and Bahammam

, Developing a theoretical framework for integrated development of sustainable & affordable neighborhoods in the Kingdom of Saudi Arabia, Emirates Journal of Engineering Research 22(1) (2017), 57–84.

17.

Haidar

E.A.

and Bahammam

A.S.

, An optimal model for housing projects according to the relative importance of affordability and sustainability criteria and their implementation impact on initial cost – ScienceDirect, Sustainable Cities and Society 64(5) (2020), 102535.1–102535.22.

18.

Saaty

R.W.

, The analytic hierarchy process-what it is and how it is used, Mathematical Modelling 9(3-5) (1987), 161–176.

19.

Roy

, The outranking approach and the foundations of ELECTRE methods, Theory and Decision 31 (1991), 49–73.

20.

Vincke

J.P.

and Brans

, A preference ranking organization method the PROMETHEE method for MCDM, Management Science 31 (1985), 647–656.

21.

Rezaei

, Best-worst multi-criteria decision-making method, Omega 53 (2015), 49–57.

22.

Gabus

and Fontela

, World Problems, An Invitation to Further Thought within the Framework of DEMATEL, Battelle Geneva Research Centre, Switzerland, Geneva, 1972.

23.

Das

M.C.

, Sarkar

and Ray

, Decision making under conflicting environment: A new MCDM method, International Journal of Applied Decision Sciences 5(2) (2012), 142–162.

24.

Opricovic

, Multicriteria optimization of civil engineering systems, Faculty of Civil Engineering 37 (1998), 5–21.

25.

Pamucar

and Cirovic

, The selection of transport and handling resources in logistics centers using Multi-Attributive Border Approximation area Comparison (MABAC), Expert Systems with Applications 42(6) (2015), 3016–3028.

26.

Stević

Ž.

, Pamučar

, Puška

and Chatterjee

, Sustainable supplier selection in healthcare industries using a new MCDM method: Measurement of alternatives and ranking according to Compromise solution (MARCOS), Computers & Industrial Engineering 140 (2020), 106231.1–106231.2.

27.

Acuña-Soto

C.M.

, Liern

and Pérez-Gladish

, A VIKOR-based approach for the ranking of mathematical instructional videos, Management Decision 57(1) (2019), 501–522.

28.

Opricovic

and Tzeng

G.H.

, Compromise solution by MCDM methods: A comparative analysis of VIKOR and TOPSIS, European Journal of Operational Research 156(2) (2004), 445–455.

29.

Ceballos

, Pelta

D.A.

and Lamata.

M.T.

, Rank Reversal and the VIKOR Method: An Empirical Evaluation, International Journal of Information Technology & Decision Making 17(2) (2018), 513–525.

30.

Belton

and Gear

, On a short-coming of Saaty’s method of analytic hierarchies, Omega 11(3) (1983), 228–230.

31.

Verly

and Smet

, Some results about rank reversal instances in the PROMETHEE methods, International Journal of Multicriteria Decision Making 3(4) (2013), 325–345.

32.

Wang

and Triantaphyllou

, Ranking irregularities when evaluating alternatives by using some ELECTRE methods, Omega 36(1) (2008), 45–63.

33.

Kong

, Wei

and Gong

J.-H.

, Rank reversal and rank preservation in anp method, Journal of Discrete Mathematical Sciences and Cryptography 19(3) (2016), 821–836.

34.

Dede

, Kamalakis

and Sphicopoulos

, Convergence properties and practical estimation of the probability of rank reversal in pairwise comparisons for multi-criteria decision making problems, European Journal of Operational Research 241(2) (2015), 458–468.

35.

Garcia-Cascales

M.S.

and Lamata

M.T.

, On rank reversal and TOPSIS method, Mathematical and Computer Modelling 56(5-6) (2012), 123–132.

36.

Shannon

C.E.

, Weaver

and Wiener

, The Mathematical Theory of Communication, Physics Today 3(9) (1950), 31–32.

37.

Opricovic

, A fuzzy compromise solution for multicriteria problems, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 15(03) (2007), 363–380.

38.

Zhang

, Li

, Zhang

, Chen

and Chen

, Private-label sustainable supplier selection using a fuzzy entropy-VIKOR-based approach, Complex & Intelligent Systems 7(3) (2021), 1–18.

39.

Opricovic

, A fuzzy compromise solution for multicriteria problems, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 15(03) (2007), 363–380.

40.

Liao

H.C.

and Xu

Z.S.

, A VIKOR-based method for hesitant fuzzy multi-criteria decision making, Fuzzy Optimization and Decision Making 12(4) (2013), 373–392.

41.

Liao

H.C.

, Xu

Z.S.

and Zeng

X.J.

, Hesitant fuzzy linguistic VIKOR method and its application in qualitative multiple criteria decision making, IEEE Transactions on Fuzzy Systems 23(5) (2015), 1343–1355.

42.

Voon

Z.C.

, Hooy

C.W.

and Puah

C.H.

, An Augmented Measurement of the Housing Affordability Cycles in Malaysia, Malaysian Journal of Economic Studies 57 (2020), 133–145.

43.

Wang

, Wang

Y.M.

and Martinez

, A group decision method based on prospect theory for emergency situations, Information Sciences 418-419 (2017), 119–135.

44.

Ding

X.F.

and Liu

H.C.

, An extended prospect theory-VIKOR approach for emergency decision making with 2-dimension uncertain linguistic information, Soft Computing 23(22) (2019), 12139–12150.

45.

, Qin

, Wang

, Liu

and Jia

, An Extended FMEA Model Based on Cumulative Prospect Theory and Type-2 Intuitionistic Fuzzy VIKOR for the Railway Train Risk Prioritization, Entropy 22(12) (2020), 1418.1–1418.19.

46.

Liu

, Fan

, Tu

and Wang

, Innovative Supplier Selection from Collaboration Perspective with a Hybrid MCDM Model: A Case Study Based on NEVs Manufacturer, Symmetry 13(1) (2021), 143.1–143.18.

47.

Phochanikorn

and Tan

, An Integrated Multi-Criteria Decision-Making Model Based on Prospect Theory for Green Supplier Selection under Uncertain Environment: A Case Study of the Thailand Palm Oil Products Industry,1–, Sustainability 11(7) (2019), 1872.1–1872.22.

48.

, Zhao

, Khan

and Uthansakul

, Prospect Theory based Hesitant Fuzzy Multi-Criteria Decision Making for Low Sulphur Fuel of Maritime Transportation, Computers, Materials and Continua 66(2) (2021), 1511–1528.

49.

Zhao

, Guo

and Zhao

, Comprehensive assessment for battery energy storage systems based on fuzzy-MCDM considering risk preferences, Energy 168 (2019), 450–461.

50.

Lin

J.K.

, Lin

H.L.

, Wang

, Chang

C.H.

and Lin

C.T.

, An Evaluation Model for Property-Purchasing Plans Based on a Hybrid Multi-Criteria Decision-Making Model, Mathematics 8(6) (2020), 860.1–860.18.

51.

C.B.

and Yuan

J.H.

, A New Multi-attribute Decision-Making Method with Three-Parameter Interval Grey Linguistic Variable, International Journal of Fuzzy Systems 19(2) (2016), 292–300.

52.

and Chen

, Value determination method based on multiple reference points under a trapezoidal intuitionistic fuzzy environment, Applied Soft Computing 63 (2018), 39–49.

53.

Schmidt

, Starmer

and Sugden

, Third-generation prospect theory, Journal of Risk and Uncertainty 36(3) (2008), 203–223.

54.

Kahneman

and Tversky

, Advances in Prospect Theory: Cumulative Representation of Uncertainty, Journal of Risk and Uncertainty 5(4) (1992), 297–323.

55.

Weng

S.L.

, Weng

H.L.

, Jaaman

S.H.

and Liew

K.F.

, Performance Evaluation of Construction Companies Using Integrated Entropy–Fuzzy VIKOR Model, Entropy 23(3) (2021), 320.1–320.16.

56.

Shannon

C.E.

and Weaver

, The Mathematical Theory of Communication, The University of Illinois Press 1947.

57.

Hulchanski

J.D.

, The concept of housing affordability: Six contemporary uses of the housing expenditure-to-income ratio, Housing Studies 10(4) (1995), 471–491.

58.

McCarthy

J.M.

and Peach

R.W.

, Are home prices the next bubble? Economic Policy Review 10(3) (2004), 1–17.

59.

Kristof

, Measuring housing affordability. A case study of Flanders on the link between objective and subjective indicators, Housing Studies 3 (2021), 1–17.

60.

Kim

J.H.

and Ahn

B.S.

, Extended VIKOR method using incomplete criteria weights, Expert Systems with Applications 126 (2019), 124–132.

61.

, Zhang

and Zhong

, Consensus reaching for MAGDM with multi-granular hesitant fuzzy linguistic term sets: A minimum adjustment-based approach, Annals of Operations Research 300 (2019), 443–466.

62.

Zhang

, Gao

and Yu

, Two-sided matching decision making with multi-granular hesitant fuzzy linguistic term sets and incomplete criteria weight information, Expert Systems with Applications 168 (2020), 114311.1–114311.12.

63.

Zhang

, Gao

and Li

, Consensus reaching for social network group decision making by considering leadership and bounded confidence, Knowledge-Based Systems 204 (2020), 106240.1–106240.12.

Housing affordability evaluation based on the third-generation prospect theory and the improved VIKOR method

Abstract

Keywords

1 Introduction

2 Literature review

2.1 Housing affordability evaluation

2.2 MCDM and VIKOR

2.3 Prospect theory

3 Housing affordability evaluation model

3.2 Calculation steps

4.1 Evaluation indicators

4.2.1 Descriptive statistics

Table 3 Reference points, h (s t ) Criteria h (s1) h (s2) h (s3) h (s4) h (s5) h (s6) h (s7) h (s8) 2010 2011 2012 2013 2014 2015 2016 2017 Reference point of c1 (PIR) 11.818 11.162 9.547 9.598 8.822 7.729 7.706 7.920 Reference point of c2 (RIR) 0.531 0.507 0.500 0.516 0.526 0.518 0.488 0.279

4.3.1 Comparing the original with prospect values

References