A grey approach to site selection for nursing homes: An extension of the fuzzy analytic hierarchy process

Abstract

The construction of more nursing homes has become one of the most needed pension services in China, and the issue of site selection is one of the most important steps in their construction. The problem of site selection for nursing homes is a complex system engineering problem that involves not only economic interests but also social interests. Due to the limitations of human thinking in the evaluation process, the evaluation value of a nursing home site might be an interval grey number. Moreover, the evaluation indicator system for nursing home locations is a two-layer system that has been neglected in the literature. Therefore, the fuzzy analytical hierarchy process is extended to a new grey approach, i.e., the grey analytic hierarchy process, which can solve the evaluation problems for a two-layer indicator system under an interval grey environment. By constructing a three-point interval grey number, grey evaluation criteria are given to obtain a judgment matrix for interval grey numbers. Definitions of the initial weights, nongreyness weights and integrated weights are proposed to find the best evaluation object. Finally, the effectiveness of the method proposed by this paper is verified by comparative analyses of other grey methods.

Keywords

Nursing home site site selection grey analytic hierarchy process fuzzy analytic hierarchy process interval grey number

1 Introduction

By the end of 2019, the population in China aged 60 and above had reached 254 million, accounting for 18.1 percent of the total population. The number of disabled and semidisabled elderly people exceeded 40 million, accounting for 15.7 percent of the elderly population [1]. According to the latest projections, the population in China aged 60 and over will reach 522 million by 2050 and will account for 39.5 percent of the total population. The number of disabled and semidisabled elderly people will increase to 120 million, accounting for 23 percent of the elderly population [2]. China’s population has entered a stage of having a low birth rate, low morality rate, low growth rate and high disability rate. Under the situation of a severely declining birth rate and a rapid aging rate, the support given by the children of elderly people is not sufficiently high [3]. According to the 2018 and 2019 research reports of the “Chinese people’s livelihood survey” research group at the Development Research Center of the State Council, in recent years, the degree of the respondents’ concern about elderly people has increased significantly, and nearly 60 percent of the residents clearly stated that the current community facilities were far from meeting the needs of elderly people. The survey shows that the construction of more nursing homes has become one of the most needed pension services for elderly people in urban and rural areas [4, 5]. This has also become a focus of the government’s attention. In building a nursing home, site selection is one of the most important steps. Whether the site selection scheme is correct, from the perspective of nursing homes, is related to the sustainability of its later operation. From the perspective of the government and the public sector, site selection affects the service efficiency of public facilities and the cost-effectiveness and fairness of public attention, which is of great significance [6].

Site selection was first studied by Alfred Webber (1909). To minimize the total distance from a single warehouse to multiple customers, a Webber diagram method was proposed [7]. Modern site selection theory originated from the network site selection method proposed by Hakimi (1965). Since then, on the basis of different practical problems, scholars have begun to use decision making and models to study the classical site selection problem; these problems include coverage problems (Roth, 1968; Toregas, 1971; et al.), the P-median problem (Hakimi, 1964; Wesolowsky, 1976; et al.), and the P-center problem (Hakimi, 1964; Hedetniemi, 1981; et al.). With the deepening of research, scholars have found that the problem of site selection is a systematic engineering problem in which complex factors such as the economy and environment need to be considered, and it is a multiattribute decision analysis problem. In addition, some of these factors are difficult to measure quantitatively and involve uncertainty, so scholars have begun to apply the analytical hierarchy process (AHP), fuzzy models, grey models, etc., to the site selection problem.

Şahin et al. (2019) [8] used 6 criteria and 19 subcriteria to study the AHP of hospital location selection, which provides a reference for hospital administrators and investors in choosing hospital locations. Colak et al. (2019) [9] used geographical information system (GIS) technology to identify the construction of solar photovoltaic power stations in Malatya Province, Turkey, and then utilized the AHP method to select the best site based on effective factors. Messaoudi et al. (2019) [10] developed an integrated framework to assess the suitability of sites for hydrogen production based on solar energy site selection. The integrated framework combines multicriteria decision making (MCDM) with a GIS.

Some scholars consider that investors often evaluate investment strategies based on their own subjective preferences in terms of the numerical values of different criteria, and it is better to regard this kind of situation as a fuzzy evaluation problem. Hence, Kazemi et al. (2020) [11] used the fuzzy analytic hierarchy process (FAHP) to investigate the most suitable location for a mineral processing plant and verified its effectiveness with a case study of a gilsonite processing plant in Iran. Lin et al. (2020) [12] proposed a novel picture fuzzy multicriteria decision making (MCDM) model to solve the site selection problem for car sharing stations and verified the superiority of the proposed model through comparative analyses. Zoghi et al. (2017) [13] optimized the site selection of solar plants based on a fuzzy logic model, weighted linear combinations (WLCs) and multicriteria decision making (MCDM).

On the other hand, in recent years, some scholars have considered the limitations of human thinking that arise in the process of evaluation [14, 15]; that is, the evaluation value can be an interval grey number [16]. Although traditional grey two-point interval numbers take into account the complexity of decision making and give the preference range of the decision makers, they also amplify the uncertainty of decision making in some decision-making situations. Therefore, the concept of a three-point interval number has been proposed [17 –19]. On this basis, Li et al. (2016) [20] emphasized the importance of the “center of gravity” point and proposed a new distance measure of the three-point interval grey number. Zhang et al. (2017) [21] combined the grey system theory with the three-point interval number to determine the weight of decision-making.

When the evaluator only knows the value range of certain indicators, it is more realistic to use an interval grey number, so the grey analytic hierarchy process (GAHP) is proposed. Although the application of the GAHP to site selection analysis is just emerging, the existing literature indicates that the GAHP is superior to the AHP and fuzzy comprehensive evaluation methods in comprehensive evaluations. Jiang (2016) [22] studied the location problem of distribution centers based on the GAHP and a multiobjective optimization model. It has also been noted that GAHP analysis is more accurate than AHP analysis because it takes into account individual cognition and incomplete or uncertain information about the evaluation object. Li et al. (2019) [23] combined the AHP and DEA models and gave weight to each index of the three-point interval grey number decision-making model.

In view of the location selection problem for nursing homes, Yan et al. (2013) [24] took the location of a nursing home in Xi’an as the object, combined with public transportation, health, environment and other factors, and selected the best location through a time Petri net. Xu et al. (2017) [25] designed an evaluation index system for nursing home site selection that consists of 9 selection indexes and used the AHP to determine the weight of each index and obtain a comprehensive score for each selected address. However, there are few studies on location selection for nursing homes.

Notably, the location of nursing homes is different from traditional location selection with the goal of maximizing economic benefits. It must consider not only economic benefits but also social benefits. In the process of considering these complex problems, an interval with an unknown center of gravity rather than a numerical point is often used to express judgments in these uncertain situations [26]. Additionally, the evaluation index system for the location of nursing homes is a two-layer system, while most scholars currently consider a single-tier system. The research motivation of this paper comes from the research of Kazemi et al. [11], in which FAHP was proposed to analyze a two-layer indicator system of a new hospital site under a fuzzy environment. Hence, we extend FAHP to a new grey approach, i.e., the Grey Analytical Hierarchy Process (GAHP), which can solve the evaluation problems for a two-layer indicator system in an interval grey environment. Then, the GAHP is utilized to analyze the site selection of the nursing homes under the grey situations. The effectiveness of the method proposed by this paper is confirmed by comparison with other methods.

The paper is organized as follows: Section 2 designs the indicator system for nursing home sites. Section 3 extends the FAHP and proposes the GAHP. Section 4 presents the application of the site selection method for nursing homes in N city, J province, China. Section 5 presents the conclusions and future work.

2 The two-layer indicator system for nursing home sites

The selection of nursing home sites is affected by some common factors, such as the population status, economic development and consumption level, in traditional studies on site selection. However, for a nursing home site, decision makers pay more attention to factors such as medical conditions, commercial facilities and the environment. Integrating the related studies [24, 25], the factors influencing the selection of nursing home sites can be summarized into three categories: environment, convenience of life, and economics.

Environment: pollution, scale, and landscape.

Pollution . Elderly people have relatively high requirements for the environment, so industrial pollution, air pollution and noise pollution should be avoided.

Scale . The scale of the site needs to be considered, not only to meet the living requirements of elderly people but also to meet their catering, activity, medical care, health care and entertainment requirements.

Landscape . If the nursing home is surrounded by mountains, rivers, lakes or forests, it will increase the value of the nursing home.

Convenience of life: traffic, medical care, and commercial facilities.

Traffic . Good traffic conditions are conducive for elderly people to travel and be visited by family.

Medical care . Elderly people have a high vulnerability to disease, so they attach great importance to medical care.

Commercial facilities . Surrounding commercial facilities can add convenience to the lives of elderly people, saving their money and time.

Economic: consumption level, nursing cost, and policy support.

Consumption level . The consumption level in the area where the nursing home is located will affect the profitability of the nursing home.

Nursing cost . Elderly people need different levels of nursing care and pay different fees according to their ability to live independently.

Policy support . Preferential policies such as tax exemptions, bed subsidies, and medical treatment for elderly people are conducive to the construction of nursing homes.

According to these factors, we design the indicator system of nursing home sites, in which the target layer includes the site selection of nursing homes, the criterion layer includes the environment, convenience of life, and economy, and the indicator layer includes the pollution, scale, landscape, etc., as shown in Fig. 1.

Fig. 1

The indicator system of nursing home sites.

3 Grey analytic hierarchy process (GAHP)

3.1 Interval grey numbers & triangular fuzzy numbers

Definition 1 [27] Let M ∈ R be a fuzzy number; if the membership function μ_M : R → [0, 1] of M can be represented by $μ_{M} (x) = {\begin{matrix} \frac{x - l}{m - l}, x \in [l, m] \\ \frac{x - u}{m - u}, x \in [m, u] \\ \begin{matrix} 0, otherwise \end{matrix} \end{matrix},$ (1)

Then, M is a triangular fuzzy number and can also be represented by (l, m, u), where l ⩽ m, l ⩽ u, l, u are the lower and upper bounds of M, respectively, and m is the value when the membership function of M is equal to 1. In addition, u - l indicates the fuzzy degree.

Definition 2 Let g_⊗ ∈ R be an unknown real number within a closed interval; then, g_⊗ is an interval grey number and can be represented by $g_{\otimes} \in [a, b],$ (2) where a ⩽ b and a, b are the lower and upper bounds of g_⊗, respectively. In addition, b - a indicates the grey degree.

Liu et al. [28, 29] proposed the concept of a grey “kernel”, i.e., $\hat{g_{\otimes}} = 1 / 2 (a + b)$ . This indicates the value that is most likely to be obtained within the closed interval [a, b]. From the definitions of interval grey numbers and triangular fuzzy numbers, the similarity between them is clear. An interval grey number can also be represented by $g_{\otimes} = (a, \hat{g_{\otimes}}, b) .$ (3)

$a ⩽ \hat{g_{\otimes}}, \hat{g_{\otimes}} ⩽ b$ . This definition is based on the grey information coverage principle. For the uncertainty problems of “few data” and “poor information” that are difficult to solve, $\hat{g_{\otimes}}$ is the law of reality about how people perceive things. When the information is completely unknown, that is, when it is “black", $\hat{g_{\otimes}} = 1 / 2 (a + b)$ ; with the deepening of people’s understanding of things, new information will be added constantly, grey nucleus will gradually whiten, at this time, grey nucleus $\hat{g_{\otimes}}$ changes as additional information is added, $\hat{g_{\otimes}} \in [a, b]$ ; When the information is completely known, interval grey number degenerates into white number [28]. When the three-point interval grey number is used for evaluation, the value range of the interval grey number is guaranteed, and the most likely value of the interval grey number is highlighted, which makes up for the defect of “poor information” of the grey number and makes the evaluation result more in line with reality.

Then, we give the fundamental operations of triangular fuzzy numbers and extend them to three-point interval grey numbers.

Definition 3 [27] Denote two triangular fuzzy numbers by M₁ = (l₁, m₁, u₁) and M₂ = (l₂, m₂, u₂); then $M_{1} \oplus M_{2} = (l_{1} + l_{2}, m_{1} + m_{2}, u_{1} + u_{2})$ (4) $M_{1} \otimes M_{2} = (l_{1} l_{2}, m_{1} m_{2}, u_{1} u_{2})$ (5) $λ \otimes M_{1} = (λ l_{1}, λ m_{1}, λ u_{1})$ (6) $\frac{1}{M_{1}} = (\frac{1}{u_{1}}, \frac{1}{m_{1}}, \frac{1}{l_{1}}),$ (7) where ⊕ and ⊗ indicate the addition operator and multiplication operator of triangular fuzzy numbers, respectively.

Theorem 1. Denote two interval grey numbers by $g_{\otimes 1} = (a_{1}, \hat{g_{\otimes 1}}, b_{1})$ and $g_{\otimes 2} = (a_{2}, \hat{g_{\otimes 2}}, b_{2})$ ; then $g_{\otimes 1} + g_{\otimes 2} = (a_{1} + a_{2}, \hat{g_{\otimes 1}} + \hat{g_{\otimes 2}}, b_{1} + b_{2})$ (8) $g_{\otimes 1} \times g_{\otimes 2} = (a_{1} a_{2}, \hat{g_{\otimes 1} g_{\otimes 2}}, b_{1} b_{2})$ (9) $λ g_{\otimes 1} = (λ a_{1}, λ \hat{g_{\otimes 1}}, λ b_{1})$ (10) $\frac{1}{g_{\otimes 1}} = (\frac{1}{b_{1}}, \hat{\frac{1}{g_{\otimes 1}}}, \frac{1}{a_{1}}) .$ (11)

Proof. For Equation (8), $\begin{matrix} g_{\otimes 1} + g_{\otimes 2} = [a_{1}, b_{1}] + [a_{2}, b_{2}] = [a_{1} + a_{2}, b_{1} + b_{2}] \\ = (a_{1} + a_{2}, \frac{a_{1} + a_{2} + b_{1} + b_{2}}{2}, b_{1} + b_{2}) \\ = (a_{1} + a_{2}, \frac{a_{1} + b_{1}}{2} + \frac{a_{2} + b_{2}}{2}, b_{1} + b_{2}) \\ = (a_{1} + a_{2}, \hat{g_{\otimes 1}} + \hat{g_{\otimes 2}}, b_{1} + b_{2}) \end{matrix}$ For Equation (9), $\begin{matrix} g_{\otimes 1} \times g_{\otimes 2} = [a_{1}, b_{1}] \times [a_{2}, b_{2}] = [a_{1} a_{2}, b_{1} b_{2}] \\ = (a_{1} a_{2}, \frac{a_{1} a_{2} + b_{1} b_{2}}{2}, b_{1} b_{2}) \\ = (a_{1} a_{2}, \hat{g_{\otimes 1} g_{\otimes 2}}, b_{1} b_{2}) \end{matrix}$ For Equation (10), $\begin{matrix} λ g_{\otimes 1} = λ [a_{1}, b_{1}] = [λ a_{1}, λ b_{1}] \\ = (λ a_{1}, \frac{λ a_{1} + λ b_{1}}{2}, λ b_{1}) \\ = (λ a_{1}, \frac{a_{1} + b_{1}}{2} λ, λ b_{1}) \\ = (λ a_{1}, λ \hat{g_{\otimes 1}}, λ b_{1}) \end{matrix}$

For Equation (11), $\begin{matrix} \frac{1}{g_{\otimes 1}} = \frac{1}{[a_{1}, b_{1}]} = [\frac{1}{b_{1}}, \frac{1}{a_{1}}] \\ = (\frac{1}{b_{1}}, \frac{a_{1} + b_{1}}{2 a_{1} b_{1}}, \frac{1}{a_{1}}) \\ = (\frac{1}{b_{1}}, \hat{\frac{1}{g_{\otimes 1}}}, \frac{1}{a_{1}}) \end{matrix}$ which completes the proof.

3.2 The judgment matrix of interval grey numbers

For the comparison of two elements, we consider the greyness of human thinking and propose grey evaluation criteria, as shown in Table 1.

Table 1
Grey evaluation criteria

Relative criteria of elements A and B Relative value Description

G ₁ [1, 1] → (1, 1, 1) A is as important asB

G ₃ [2, 3] → (2, 2.5, 3) A is weakly more important thanB

G ₅ [4, 5] → (4, 4.5, 5) A is more important thanB

G ₇ [6, 7] → (6, 6.5, 7) A is obviously more important thanB

G ₉ [8, 9] → (8, 8.5, 9) A is strongly more important thanB

G₂, G₄, G₆, G₈ Median Intermediate state

Relative criteria of elements A and B	Relative value	Description
G ₁	[1, 1] → (1, 1, 1)	A is as important asB
G ₃	[2, 3] → (2, 2.5, 3)	A is weakly more important thanB
G ₅	[4, 5] → (4, 4.5, 5)	A is more important thanB
G ₇	[6, 7] → (6, 6.5, 7)	A is obviously more important thanB
G ₉	[8, 9] → (8, 8.5, 9)	A is strongly more important thanB
G₂, G₄, G₆, G₈	Median	Intermediate state

Table 1 shows that the relative value is an interval grey number, as stated in subsection 3.1. In particular, when element A is as important as element B, their relative value is [1, 1] = (1, 1, 1) =1.

Definition 4 Denote the element set of the criterion layer by X = {x₁, x₂, ⋯ , x_n} and the number of evaluators by m. According to the grey evaluation criteria, the judgment matrix of interval grey numbers can be represented as shown below.

	x ₁	x ₂	⋯	x _n
x ₁	(1, 1, 1)	$g_{\otimes 1}^{12} = (a_{1}^{12}, \hat{g_{\otimes 1}^{12}}, b_{1}^{12})$	⋯	$g_{\otimes 1}^{1 n} = (a_{1}^{1 n}, \hat{g_{\otimes 1}^{1 n}}, b_{1}^{1 n})$
		$g_{\otimes 2}^{12} = (a_{2}^{12}, \hat{g_{\otimes 2}^{12}}, b_{2}^{12})$		$g_{\otimes 2}^{1 n} = (a_{2}^{1 n}, \hat{g_{\otimes 2}^{1 n}}, b_{2}^{1 n})$
	⋯		⋯
		$g_{\otimes m}^{12} = (a_{m}^{12}, \hat{g_{\otimes m}^{12}}, b_{m}^{12})$		$g_{\otimes m}^{1 n} = (a_{m}^{1 n}, \hat{g_{\otimes m}^{1 n}}, b_{m}^{1 n})$
x ₂	$g_{\otimes 1}^{21} = 1 / g_{\otimes 1}^{12}$	(1, 1, 1)	⋯	$g_{\otimes 1}^{2 n} = (a_{1}^{2 n}, \hat{g_{\otimes 1}^{2 n}}, b_{1}^{2 n})$
	$g_{\otimes 2}^{21} = 1 / g_{\otimes 2}^{12}$			$g_{\otimes 2}^{2 n} = (a_{2}^{2 n}, \hat{g_{\otimes 2}^{2 n}}, b_{2}^{2 n})$
	⋯			⋯
	$g_{\otimes m}^{21} = 1 / g_{\otimes m}^{12}$			$g_{\otimes m}^{2 n} = (a_{m}^{2 n}, \hat{g_{\otimes m}^{2 n}}, b_{m}^{2 n})$
⋯	⋯	⋯	(1, 1, 1)	⋯
x _n	$g_{\otimes 1}^{n 1} = 1 / g_{\otimes 1}^{1 n}$	$g_{\otimes 1}^{n 2} = 1 / g_{\otimes 1}^{2 n}$	⋯	(1, 1, 1)
	$g_{\otimes 2}^{n 1} = 1 / g_{\otimes 2}^{1 n}$	$g_{\otimes 2}^{n 2} = 1 / g_{\otimes 2}^{2 n}$
	⋯	⋯
	$g_{\otimes m}^{n 1} = 1 / g_{\otimes m}^{1 n}$	$g_{\otimes m}^{n 2} = 1 / g_{\otimes m}^{2 n}$

This is a reciprocal matrix.

Definition 5 Denote the element set of the criterion layer by X = {x₁, x₂, ⋯ , x_n} and the number of evaluators by m. If the relative values of any two elements x_i, x_k ∈ Xare $g_{\otimes 1}^{ik} = (a_{1}^{ik}, \hat{g_{\otimes 1}^{ik}}, b_{1}^{ik})$ , $g_{\otimes 2}^{ik} = (a_{2}^{ik}, \hat{g_{\otimes 2}^{ik}}, b_{2}^{ik})$ , ⋯, and $g_{\otimes m}^{ik} = (a_{m}^{ik}, \hat{g_{\otimes m}^{ik}}, b_{m}^{ik})$ , then the integrated relative value of x_i, x_k can be represented by $\begin{matrix} g_{\otimes}^{ik} = (\frac{a_{1}^{ik} + a_{2}^{ik} + \dots + a_{m}^{ik}}{m}, \\ \frac{\hat{g_{\otimes 1}^{ik}} + \hat{g_{\otimes 2}^{ik}} + \dots \hat{+ g_{\otimes m}^{ik}}}{m}, \frac{b_{1}^{ik} + b_{2}^{ik} + \dots + b_{m}^{ik}}{m}) . \end{matrix}$ (12)

Definition 6 Denote the element set of the indicator layer by Y = {y₁, y₂, ⋯ , y_p}, the evaluation objects by B = {b₁, b₂, ⋯ , b_l}, and the number of evaluators by m. According to the grey evaluation criteria, the judgment matrix of the interval grey numbers for y_d ∈ Y can be represented as shown below.

	b ₁	b ₂	⋯	b _l
b ₁	(1, 1, 1)	$g_{\otimes 1}^{12 d} = (a_{1}^{12 d}, \hat{g_{\otimes 1}^{12 d}}, b_{1}^{12 d})$		$g_{\otimes 1}^{1 ld} = (a_{1}^{1 ld}, \hat{g_{\otimes 1}^{1 ld}}, b_{1}^{1 ld})$
		$g_{\otimes 2}^{12 d} = (a_{2}^{12 d}, \hat{g_{\otimes 2}^{12 d}}, b_{2}^{12 d})$		$g_{\otimes 2}^{1 ld} = (a_{2}^{1 ld}, \hat{g_{\otimes 2}^{1 ld}}, b_{2}^{1 ld})$
		⋯	⋯	⋯
		$g_{\otimes m}^{12 d} = (a_{m}^{12 d}, \hat{g_{\otimes m}^{12 d}}, b_{m}^{12 d})$		$g_{\otimes m}^{1 ld} = (a_{m}^{1 ld}, \hat{g_{\otimes m}^{1 ld}}, b_{m}^{1 ld})$
b ₂	$g_{\otimes 1}^{21 d} = 1 / g_{\otimes 1}^{12 d}$			$g_{\otimes 1}^{2 ld} = (a_{1}^{2 ld}, \hat{g_{\otimes 1}^{2 ld}}, b_{1}^{2 ld})$
	$g_{\otimes 2}^{21 d} = 1 / g_{\otimes 2}^{12 d}$	(1, 1, 1)	⋯	$g_{\otimes 2}^{2 ld} = (a_{2}^{2 ld}, \hat{g_{\otimes 2}^{2 ld}}, b_{2}^{2 ld})$
	⋯			⋯
	$g_{\otimes m}^{21 d} = 1 / g_{\otimes m}^{12 d}$			$g_{\otimes m}^{2 ld} = (a_{m}^{2 ld}, \hat{g_{\otimes m}^{2 ld}}, b_{m}^{2 ld})$
⋯	⋯	⋯	(1, 1, 1)	⋯
b _l	$g_{\otimes 1}^{l 1 d} = 1 / g_{\otimes 1}^{1 ld}$	$g_{\otimes 1}^{l 2 d} = 1 / g_{\otimes 1}^{2 ld}$
	$g_{\otimes 2}^{l 1 d} = 1 / g_{\otimes 2}^{1 ld}$	$g_{\otimes 2}^{l 2 d} = 1 / g_{\otimes 2}^{2 ld}$	⋯	(1, 1, 1)
	⋯	⋯
	$g_{\otimes m}^{l 1 d} = 1 / g_{\otimes m}^{1 ld}$	$g_{\otimes m}^{l 2 d} = 1 / g_{\otimes m}^{2 ld}$

This is a reciprocal matrix.

Definition 7 Denote the element set of the indicator layer by Y = {y₁, y₂, ⋯ , y_p}, the evaluation objects by B = {b₁, b₂, ⋯ , b_l}, and the number of evaluators by m. If the relative values of any two evaluation objects b_j, b_h ∈ B about y_d ∈ Y are $g_{\otimes 1}^{jhd} = (a_{1}^{jhd}, \hat{g_{\otimes 1}^{jhd}}, b_{1}^{jhd})$ , $g_{\otimes 2}^{jhd} = (a_{2}^{jhd}, \hat{g_{\otimes 2}^{jhd}}, b_{2}^{jhd})$ , ⋯, and $g_{\otimes m}^{jhd} = (a_{m}^{jhd}, \hat{g_{\otimes m}^{jhd}}, b_{m}^{jhd})$ , then the integrated relative value of b_j, b_h about y_d can be represented by $\begin{matrix} g_{\otimes}^{jhd} = (\frac{a_{1}^{jhd} + a_{2}^{jhd} + \dots + a_{m}^{jhd}}{m}, \\ \frac{\hat{g_{\otimes 1}^{jhd}} + \hat{g_{\otimes 2}^{jhd}} + \dots \hat{+ g_{\otimes m}^{jhd}}}{m}, \frac{b_{1}^{jhd} + b_{2}^{jhd} + \dots + b_{m}^{jhd}}{m}) . \end{matrix}$ (13)

3.3 Integrated weights

Definition 8 Suppose we have a V × V judgment matrix of interval grey numbers and that the integrated relative value of any two elements in this matrix can be denoted by a_tr (t ⩽ V, r ⩽ V); then, the initial weight of element t can be represented by $D_{t} = \sum_{r = 1}^{V} a_{tr} \div (\sum_{t = 1}^{V} \sum_{r = 1}^{V} a_{tr}) .$ (14)

If a_tr is the integrated relative value of any two elements of the criterion layer, D_t indicates the initial weight of element t of the criterion layer. If a_tr is the integrated relative value of any two evaluation objects, D_t indicates the initial weight of object t of the indicator layer.

To eliminate the greyness of the initial weights, we propose the definitions below.

Definition 9 Consider two interval grey numbers $g_{\otimes 1} = (a_{1}, \hat{g_{\otimes 1}}, b_{1})$ and $g_{\otimes 2} = (a_{2}, \hat{g_{\otimes 2}}, b_{2})$ ; the possibility of g_⊗1 > g_⊗2 can be defined by

$\begin{matrix} P (g_{\otimes 1} ⩾ g_{\otimes 2}) = \\ {\begin{matrix} 1 (\hat{g_{\otimes 1}} ⩾ \hat{g_{\otimes 2}}) \\ \frac{b_{1} - a_{2}}{\hat{g_{\otimes 2}} - a_{2} + b_{1} - \hat{g_{\otimes 1}}} (\hat{g_{\otimes 1}} ⩽ \hat{g_{\otimes 2}}, b_{1} ⩾ a_{2}) \\ 0 (otherwise) \end{matrix} . \end{matrix}$ (15)

Definition 10 Denote n interval grey numbers by g_⊗1, g_⊗2, ⋯ , g_⊗n; then, the possibility of any g_⊗i (i ⩽ n) having the largest value can be defined by

$\begin{matrix} F (g_{\otimes i}) = F (g_{\otimes i} ⩾ g_{\otimes 1}, g_{\otimes 2}, \dots, g_{\otimes n}) \\ = min P (g_{\otimes i} ⩾ g_{\otimes j}), j = 1, 2, \dots, n, j \neq i \end{matrix}$ (16)

Definition 11 Suppose that we have a V × V judgment matrix of interval grey numbers and that the possibility of element j (j ∈ V) of this matrix is denoted by F (g_⊗j); then, the nongreyness weight of element j can be represented by $w (g_{\otimes j}) = \frac{F (g_{\otimes j})}{F (g_{\otimes 1}) + F (g_{\otimes 2}) + \dots + F (g_{\otimes V})} .$ (17)

To distinguish the weights of the criterion layer and index layer, for an element i of the criterion layer, we denote its nongreyness weight by $w (g_{\otimes}^{i})$ , and for an evaluation object j about indicator d, its nongreyness weight is denoted by $w (g_{\otimes}^{jd})$ . Then, the integrated weight of evaluation object j can be represented by ${IW}_{j} = \sum_{i = 1}^{n} \sum_{d = 1, i \to d}^{p} w (g_{\otimes}^{i}) \cdot w (g_{\otimes}^{jd}),$ (18) where i → d indicates that element d of the indicator layer belongs to element i of the criterion layer. Then, we can find the best evaluation object via a comparison of the integrated weights of all the evaluation objects.

4 Case study

In this section, we present the case of the site selection of nursing homes in N City, J province, to illustrate the GAHP proposed in Section 3 to provide a scientific basis for decision-makers to formulate policies on the development of the pension industry.

We invited three experts to evaluate three alternative nursing home sites, which are denoted by B = {b₁, b₂, b₃}. According to the two-layer indicator system for nursing home sites constructed in Section 2, the elements of the criterion layer (the first layer) and indicator layer (the second layer) are denoted by X = {x₁, x₂, x₃} and Y = {y₁, y₂, ⋯ , y₉}, respectively, as shown in Fig. 2.

Fig. 2

The indicator system of the nursing home site.

In practical work, facing the complexity of the site selection problem for nursing homes, experts easily give the range of relative values of each element in the criterion layer and index layer and constitute the grey interval of the relative value of each element, namely, the grey two-point interval. It is often difficult to give the most likely value within the interval. In addition, the evaluation index system of site selection for nursing homes is a two-layer system, so we use the grey hierarchy method to solve this kind of problem. The specific operation process is as follows:

Step 1: The three experts give the relative values of each element of the grey evaluation criterion layer and the index layer to form the grey interval judgment matrix.

Step 2: According to Definition 4 and Definition 6, calculate the grey number judgment matrix of the three-point interval of the criterion layer and the index layer, respectively.

Step 3: According to Definition 5 and Definition 7, the judgment matrixes of the interval grey numbers at the three points of the criterion layer and the index layer are treated with the comprehensive relative value.

Step 4: According to Definitions 8-11, calculate the nongrey weight of the criterion layer and the index layer.

Step 5: According to Formula (18), aggregate the comprehensive weight of the site selection to determine the best site selection.

Step 6: Compare the aggregated judgment results of the grey analytic hierarchy process with other methods.

According to the operation process of the grey analytic hierarchy process, the calculation steps of nursing home site selection are as follows:

Step 1:

The relative values of the elements of the criterion layer (the first layer) are given by three experts according to the grey evaluation criteria, as shown in Table 2.

Table 2

The relative values of the elements of the criterion layer

	x ₁	x ₂	x ₃
x ₁	(1, 1)	(1, 2)	(1/4, 1/3)
		(2, 3)	(1/5, 1/4)
		(1, 2)	(1/3, 1/2)
x ₂	(1/2, 1)		(2, 3)
	(1/3, 1/2)	(1, 1)	(2, 3)
	(1/2, 1)		(1, 2)
x ₃	(3, 4)	(1/3, 1/2)
	(4, 5)	(1/3, 1/2)	(1, 1)
	(2, 3)	(1/2, 1)

The relative values of the alternative sites for the elements of the indicator layer (the second layer) are given by three experts according to the grey evaluation criteria, as shown in Tables 3.1∼3.9.

Table 3.1

The relative values of alternative sites in terms of pollution y₁

	b ₁	b ₂	b ₃
b ₁	(1, 1)	(2, 3)	(1/2, 1)
		(1, 2)	(1/2, 1)
		(2, 3)	(1/3, 1/2)
b ₂	(1/3, 1/2)	(1, 1)	(1, 2)
	(1/2, 1)		(1, 2)
	(1/3, 1/2)		(1, 2)
b ₃	(1, 2)	(1/2, 1)	(1, 1)
	(1, 2)	(1/2, 1)
	(2, 3)	(1/2, 1)

Table 3.2

The relative values of alternative sites in terms of scale y₂

	b ₁	b ₂	b ₃
b ₁	(1, 1)	(1/3, 1/2)	(4, 5)
		(1/4, 1/3)	(3, 4)
		(1/3, 1/2)	(3, 4)
b ₂	(2, 3)		(1/4, 1/3)
	(3, 4)	(1, 1)	(1/4, 1/3)
	(2, 3)		(1/4, 1/3)
b ₃	(1/5, 1/4)
	(1/4, 1/3)
	(1/4, 1/3)	(3, 4)
		(3, 4)	(1, 1)
		(3, 4)

Table 3.3

The relative values of alternative sites in terms of the landscape y₃

	b ₁	b ₂	b ₃
b ₁		(1, 2)	(1/3, 1/2)
	(1, 1)	(1, 2)	(1/4, 1/3)
		(2, 3)	(1/3, 1/2)
b ₂	(1/2, 1)		(3, 4)
	(1/2, 1)	(1, 1)	(2, 3)
	(1/3, 1/2)		(1, 2)
b ₃	(2, 3)	(1/4, 1/3)
	(3, 4)	(1/3, 1/2)	(1, 1)
	(2, 3)	(1/2, 1)

Table 3.4

The relative values of alternative sites in terms of traffic y₄

	b ₁	b ₂	b ₃
b ₁		(2, 3)	(1/2, 1)
	(1, 1)	(3, 4)	(1/3, 1/2)
		(2, 3)	(1/4, 1/3)
b ₂	(1/3, 1/2)		(2, 3)
	(1/4, 1/3)	(1, 1)	(2, 3)
	(1/3, 1/2)		(3, 4)
b ₃	(1, 2)	(1/3, 1/2)
	(2, 3)	(1/3, 1/2)	(1, 1)
	(3, 4)	(1/4, 1/3)

Table 3.5

The relative values of alternative sites in terms of medical criteria y₅

	b ₁	b ₂	b ₃
b ₁		(1/3, 1/2)	(3, 4)
	(1, 1)	(1/3, 1/2)	(4, 5)
	(1/4, 1/3)	(2, 3)
b ₂	(2, 3)		(1/4, 1/3)
	(2, 3)	(1, 1)	(1/2, 1)
	(3, 4)		(1/3, 1/2)
b ₃	(1/4, 1/3)	(3, 4)
	(1/5, 1/4)	(1, 2)	(1, 1)
	(1/3, 1/2)	(2, 3)

Table 3.6

The relative values of alternative sites in terms of commercial facilities y₆

	b ₁	b ₂	b ₃
b ₁		(3, 4)	(1/3, 1/2)
	(1, 1)	(3, 4)	(1/4, 1/3)
		(3, 4)	(1/4, 1/3)
b ₂	(1/4, 1/3)		(2, 3)
	(1/4, 1/3)	(1, 1)	(3, 4)
	(1/4, 1/3)		(4, 5)
b ₃	(2, 3)	(1/3, 1/2)
	(3, 4)	(1/4, 1/3)	(1, 1)
	(3, 4)	(1/5, 1/4)

Table 3.7

The relative values of alternative sites in terms of the consumption level y₇

	b ₁	b ₂	b ₃
b ₁		(1, 2)	(1/3, 1/2)
	(1, 1)	(2, 3)	(1/2, 1)
		(2, 3)	(1/3, 1/2)
b ₂	(1/2, 1)		(1, 2)
	(1/3, 1/2)	(1, 1)	(1, 2)
	(1/3, 1/2)		(1, 2)
b ₃	(2, 3)	(1/2, 1)
	(1, 2)	(1/2, 1)	(1, 1)
	(2, 3)	(1/2, 1)

Table 3.8

The relative values of alternative sites in terms of the nursing cost y₈

	b ₁	b ₂	b ₃
b ₁		(1/4, 1/3)	(1, 2)
	(1, 1)	(1/3, 1/2)	(2, 3)
		(1/2, 1)	(2, 3)
b ₂	(3, 4)		(1/2, 1)
	(2, 3)	(1, 1)	(1/3, 1/2)
	(1, 2)		(1/2, 1)
b ₃	(1/2, 1)	(1, 2)
	(1/3, 1/2)	(2, 3)	(1, 1)
	(1/3, 1/2)	(1, 2)

Table 3.9

The relative values of alternative sites in terms of policy support y₉

	b ₁	b ₂	b ₃
b ₁		(4, 5)	(1/4, 1/3)
	(1, 1)	(2, 3)	(1/4, 1/3)
		(2, 3)	(1/3, 1/2)
b ₂	(1/5, 1/4)		(2, 3)
	(1/3, 1/2)	(1, 1)	(3, 4)
	(1/3, 1/2)		(4, 5)
b ₃	(3, 4)	(1/3, 1/2)
	(3, 4)	(1/4, 1/3)	(1, 1)
	(2, 3)	(1/5, 1/4)

Step 2:

According to Definition 4, calculate the grey number judgment matrix of the three-point interval of the criterion layer, as shown in Table 4.

Table 4

Three-point interval grey number judgment matrix of criterion layer

	x ₁	x ₂	x ₃
x ₁		(1, 1.5, 2)	(1/4, 1/3.5, 1/3)
	(1, 1, 1)	(2, 2.5, 3)	(1/5, 1/4.5, 1/4)
		(1, 1.5, 2)	(1/3, 1/2.5, 1/2)
x ₂	(1/2, 1/1.5, 1)		(2, 2.5, 3)
	(1/3, 1/2.5, 1/2)	(1, 1, 1)	(2, 2.5, 3)
	(1/2, 1/1.5, 1)		(1, 1.5, 2)
x ₃	(3, 3.5, 4)	(1/3, 1/2.5, 1/2)
	(4, 4.5, 5)	(1/3, 1/2.5, 1/2)	(1, 1, 1)
	(2, 2.5, 3)	(1/2, 1/1.5, 1)

According to Definition 6, calculate the grey number judgment matrix of the three-point interval of the index layer, as shown in Tables 5.1∼5.2.

Table 5.1

Three-point interval grey number judgment matrix of alternative sites in terms of pollutiony₁

	b ₁	b ₂	b ₃
b ₁		(2, 2.5, 3)	(1/2, 1/1.5, 1)
	(1, 1, 1)	(1, 1.5, 2)	(1/2, 1/1.5, 1)
		(2, 2.5, 3)	(1/3, 1/2.5, 1/2)
b ₂	(1/3, 1/2.5, 1/2)		(1, 1.5, 2)
	(1/2, 1/1.5, 1)	(1, 1, 1)	(1, 1.5, 2)
	(1/3, 1/2.5, 1/2)		(1, 1.5, 2)
b ₃	(1, 1.5, 2)	(1/2, 1/1.5, 1)
	(1, 1.5, 2)	(1/2, 1/1.5, 1)	(1, 1, 1)
	(2, 2.5, 3)	(1/2, 1/1.5, 1)

Table 5.2

Three-point interval grey number judgment matrix of alternative sites in terms of scaley₂

	b ₁	b ₂	b ₃
b ₁		(1/3, 1/2.5, 1/2)	(4, 4.5, 5)
	(1, 1, 1)	(1/4, 1/3.5, 1/3)	(3, 3.5, 4)
		(1/3, 1/2.5, 1/2)	(3, 3.5, 4)
b ₂	(2, 2.5, 3)		(1/4, 1/3.5, 1/3)
	(3, 3.5, 4)	(1, 1, 1)	(1/4, 1/3.5, 1/3)
	(2, 2.5, 3)		(1/4, 1/3.5, 1/3)
b ₃	(1/5, 1/4.5, 1/4)	(3, 3.5, 4)
	(1/4, 1/3.5, 1/3)	(3, 3.5, 4)	(1, 1, 1)
	(1/4, 1/3.5, 1/3)	(3, 3.5, 4)

Due to space limitation, y₃ ∼ y₉s’ new grey number judgment matrix are omitted 1 .

Step 3:

According to Definition 5 and Definition 7, the comprehensive relative value judgment matrix of the interval grey number of the criterion layer and index layer can be obtained as follows. $G_{\otimes} (X) = [\begin{matrix} (1, 1, 1) & (1.33, 1.83, 2.33) & (0.26, 0.3, 0.36) \\ (0.44, 0.58, 0.83) & (1, 1, 1) & (1.67, 2.17, 2.67) \\ (3, 3.5, 4) & (0.39, 0.49, 0.67) & (1, 1, 1) \end{matrix}]$ $G_{\otimes} (y_{1}) = [\begin{matrix} (1, 1, 1) & (1.67, 2.17, 2.67) & (0.44, 0.58, 0.83) \\ (0.39, 0.49, 0.67) & (1, 1, 1) & (1, 1.5, 2) \\ (1.33, 1.83, 2.33) & (0.5, 0.67, 1) & (1, 1, 1) \end{matrix}]$ $G_{\otimes} (y_{2}) = [\begin{matrix} (1, 1, 1) & (0.3, 0.36, 0.44) & (3.33, 3.83, 4.33) \\ (2.33, 2.83, 3.33) & (1, 1, 1) & (0.25, 0.29, 0.33) \\ (0.23, 0.26, 0.30) & (3, 3.5, 4) & (1, 1, 1) \end{matrix}]$ $G_{\otimes} (y_{3}) = [\begin{matrix} (1, 1, 1) & (1.33, 1.83, 2.33) & (0.3, 0.36, 0.44) \\ (0.44, 0.58, 0.83) & (1, 1, 1) & (2, 2.5, 3) \\ (2.33, 2.83, 3.33) & (0.36, 0.45, 0.61) & (1, 1, 1) \end{matrix}]$ $G_{\otimes} (y_{4}) = [\begin{matrix} (1, 1, 1) & (2.33, 2.83, 3.33) & (0.36, 0.45, 0.61) \\ (0.3, 0.36, 0.44) & (1, 1, 1) & (2.33, 2.83, 3.33) \\ (2, 2.5, 3) & (0.3, 0.36, 0.44) & (1, 1, 1) \end{matrix}]$ $G_{\otimes} (y_{5}) = [\begin{matrix} (1, 1, 1) & (0.3, 0.36, 0.44) & (3, 3.5, 4) \\ (2.33, 2.83, 3.33) & (1, 1, 1) & (0.36, 0.45, 0.61) \\ (0.26, 0.3, 0.36) & (2, 2.5, 3) & (1, 1, 1) \end{matrix}]$ $G_{\otimes} (y_{6}) = [\begin{matrix} (1, 1, 1) & (3, 3.5, 4) & (0.28, 0.32, 0.39) \\ (0.25, 0.29, 0.33) & (1, 1, 1) & (3, 3.5, 4) \\ (2.67, 3.17, 3.67) & (0.26, 0.3, 0.36) & (1, 1, 1) \end{matrix}]$ $G_{\otimes} (y_{7}) = [\begin{matrix} (1, 1, 1) & (1.67, 2.17, 2.67) & (0.39, 0.49, 0.67) \\ (0.39, 0.49, 0.67) & (1, 1, 1) & (1, 1.5, 2) \\ (1.67, 2.17, 2.67) & (0.5, 0.67, 1) & (1, 1, 1) \end{matrix}]$ $G_{\otimes} (y_{8}) = [\begin{matrix} (1, 1, 1) & (0.36, 0.45, 0.61) & (1.67, 2.17, 2.67) \\ (2, 2.5, 3) & (1, 1, 1) & (0.44, 0.58, 0.83) \\ (0.39, 0.49, 0.67) & (1.33, 1.83, 2.33) & (1, 1, 1) \end{matrix}]$ $G_{\otimes} (y_{9}) = [\begin{matrix} (1, 1, 1) & (2.67, 3.17, 3.67) & (0.28, 0.32, 0.39) \\ (0.29, 0.34, 0.42) & (1, 1, 1) & (3, 3.5, 4) \\ (2.67, 3.17, 3.67) & (0.26, 0.30, 0.36) & (1, 1, 1) \end{matrix}]$

Step 4:

According to Definitions 8–11, calculate the nongrey weight of elements of the criterion layer and the index layer, as shown in Table 6.

Table 6

The nongreyness weights of the elements

Target Layer	Criterion Layer X	Weight	Indicator Layer Y	Weight
				b ₁	b ₂	b ₃
Site Selection of the Nursing Homes	Environment x₁	0.1330	Pollution y₁	0.3790	0.2733	0.3477
			Scale y₂	0.4301	0.2208	0.3491
			Landscape y₃	0.2276	0.3725	0.3999
	Convenience of Life x₂	0.3087	Traffic y₄	0.3591	0.3457	0.2952
			Medical y₅	0.4336	0.3325	0.2339
			Commercial Facilitiesy₆	0.3513	0.3472	0.3015
	Economy x₃	0.5583	Consumption Level y₇	0.3573	0.2611	0.3816
			Nursing Cost y₈	0.3261	0.3929	0.2810
			Policy Support y₉	0.3159	0.3719	0.3122

Step 5:

According to Formula (18) in Definition 11, the elements of the criterion layer (the first layer) and index layer (the second layer) are aggregated, and the comprehensive weight of the three sites is obtained by ${IW}_{j} = \sum_{i = 1}^{n} \sum_{d = 1, i \to d}^{p} w (g_{\otimes}^{i}) \cdot w (g_{\otimes}^{jd})$ , as shown in Table 7.

Table 7

The integrated weights of the three alternative sites

Chosen site	b ₁	b ₂	b ₃
Integrated Weight	1.0489	1.0046	0.9465

From Table 7, we can see that b₁ is the best site to build a nursing home in N City, while b₃ is the worst site. To verify the effectiveness and advantages of the GAHP, the grey correlation model [30], grey entropy technology [14] and the grey target method [31] are chosen to analyze the same case in this paper. The results of the four methods are shown in Table 8.

Table 8

The results of the four methods

	b ₁	b ₂	b ₃
GAHP	1.048900	1.004600	0.946500
Grey Correlation	3.089187	3.089159	3.089146
Grey Entropy	2.370990	2.370663	2.370041
Grey Target	1.778042	1.778008	1.778001

From Table 8, the result of the GAHP is the same as that of the other three methods, i.e., b₁ is the best site and b₃ is the worst site. This shows that the GAHP is an effective method to solve the evaluation problem of a two-layer indicator system in a grey situation. On the other hand, the differences in the results of the other three methods are not obvious because we can only compare the evaluation scores of the three alternative sites from the beginning of the fourth number to the right of the decimal point. The reason for this is that traditional grey methods only consider the elements of the indicator layer instead of those of the other two layers. In terms of the calculation process, the GAHP is the simplest.

5 Conclusions

The limitations of human thinking are always a factor in the process of evaluation; i.e., the evaluation values might be interval grey numbers. The evaluation indicator system for a nursing home site is a two-layer system. Hence, we propose a new grey approach to solve the evaluation problem for a two-layer indicator system under grey situations. Via a deep analysis of triangular fuzzy numbers and interval grey numbers, we find similarities between them. Then, we extend the fundamental operations of triangular fuzzy numbers to interval grey numbers and propose a judgment matrix for interval grey numbers and definitions of the initial weights, nongreyness weights and integrated weights to find the best evaluation object. Finally, a site selection case of nursing homes in N City, J Province, is considered in this paper, and the results show that the GAHP is effective in solving the evaluation problem of the two-layer indicator system in a grey situation and that it has a greater relative advantage than other grey evaluation methods.

Although this paper has made some useful explorations of the site selection of nursing homes with a new grey approach, it still has some shortcomings, such as that, depending on the number of elements of the indicator system, it may be difficult to use the GAHP to calculate and obtain the final results. Hence, we can continue to research the optimization algorithm of the GAHP by combining it with a genetic algorithm or ant colony algorithm in the future.

Footnotes

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No: 71671091; 41801119).

If necessary, please contact the corresponding author.

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