Hospital site selection using fuzzy EDAS method: case study application for districts of İstanbul

Abstract

As the Covid-19 pandemic also proved, access to health care plays a crucial role in our lives. Public officials, managers and investors should consider many criteria such as public infrastructure, environment, accessibility and demand for selecting the most appropriate site for a new hospital. Thus, the hospital site selection problem is a critical multi-criteria decision-making (MCDM) problem. This paper is a case study for İstanbul where a recent MCDM methodology, the fuzzy Evaluation based on Distance from Average Solution (EDAS) method, is applied to this problem for the first time. We used a comprehensive set of five main criteria and 17 sub-criteria found in the relevant literature regarding hospital site selection. These criteria were evaluated by three decision-makers to choose the hospital site from three districts. The recommendation of the fuzzy EDAS method was then compared to the outcome of a frequently used fuzzy MCDM method. The methods resulted in different site recommendations.

Keywords

Hospital site selection health care planning multi-criteria decision making fuzzy EDAS fuzzy TOPSIS

1 Introduction

Site selection is an important decision for health care organizations. Hospital location decisions are critical for attaining a robust public infrastructure, and hospitals should provide continuous availability of services even when emergencies and disasters strike. On the other hand, increasing population amplifies the demand for developing new health care facilities. Thus, selection criteria of a new hospital site include many factors such as transportation, market and economic conditions, and population density. The selection process is furthermore affected by the uncertainties inherent in describing and ranking available criteria and alternatives [21].

This research focuses on choosing the site for a new hospital on the Asian side of İstanbul as a case study considering the uncertainties inherent in such decisions. İstanbul is the largest city in Turkey, and also one of the largest in Europe with a population of over 17 million people. It has a total of 39 districts both on European and Asian sides. Among the total 231 hospitals in İstanbul, 159 are private hospitals, 15 are university hospitals and 37 are public hospitals. Our research utilizes the fuzzy EDAS method for this critical decision-making problem. The results are also compared with a classical methodology, the fuzzy Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) method. Fuzzy EDAS requires fewer computations compared to other MCDM methods [16]. Moreover, EDAS is efficient in solving large and complex decision-making problems [26].

This study makes the following contributions to the literature: We report the results of applying the fuzzy EDAS method to the hospital site selection problem. Since research on fuzzy EDAS is scarce, this enriches the understanding of the method. To this end, we also compare the fuzzy EDAS results with the fuzzy TOPSIS method’s outcome. TOPSIS is a popular method and is also a distance-related technique like EDAS. Interestingly, unlike the few studies that compared these methods resulting in the same recommendation, we arrive at different recommendations with the two methods. Furthermore, our criteria set is comprehensive in the sense that it includes most of the criteria used in the previous literature about hospital site selection. This work is also the first application of the fuzzy EDAS methodology to the hospital site selection problem.

The rest of the paper is organized as follows. In the rest of this section, we give a literature review; first on site selection problems, and then specifically on the use of the fuzzy EDAS methodology in selection problems. In Section 2, the definitions of the fuzzy set theory and the fuzzy EDAS methodology are given. The details of the case study for selecting an optimum site for a new hospital can be found in Section 3 where the empirical results obtained with fuzzy EDAS are presented and compared to the fuzzy TOPSIS decision. Then, Section 4 concludes the article.

1.1 Literature review

1.1.1 MCDM literature for hospital site selection

There are some studies for selecting a hospital site that use different MCDM tools other than the EDAS method. Some researchers used AHP on the problem, for example, [2 , 34]. Sahin et al. [28] investigated a decision support model for site selection of a new hospital using AHP in Muğla, Turkey. With sensitivity analysis, they also looked at the effect of changing the main criteria weights on the ranking of districts. Wu et al. [34] applied a modified Delphi method to define the evaluation criteria and sub-criteria to provide a competitive advantage. Then using AHP and sensitivity analysis, they developed an evaluation method for selecting the location of a regional hospital in Taiwan. Senvar et al. [21] suggested a new MCDM method that integrated hesitant fuzzy sets to TOPSIS. Adali and Tus [1] applied three distance-based MCDM techniques to the hospital site selection problem. The weights of this study are computed with the Criteria Importance through Inter-criteria Correlation method, and the hospital site alternatives are ranked by TOPSIS, EDAS and Combinative Distance-based Assessment methods. Some researchers combined the Analytic Network Process with Geographic Information System to determine hospital sites. Soltani and Marandi [23] used a two-step decision-making framework to select the site for a hospital in Iran using a combination of Fuzzy Analytic Network Process and Geographic Information System. Vahidni et al. [29] combined Geographic Information System analysis with Fuzzy Analytical Hierarchy Process to find the site of a new hospital in Tehran. Different from other studies, they used three methods to estimate the total weights and priorities of alternative sites. A comprehensive list of the criteria and sub-criteria used in different MCDM studies for the hospital site selection problem is given in Table 1.

Table 1
Criteria and sub-criteria used for the hospital site selection problem in the literature

Criteria Sub-criteria

Cost Cost of land [1 , 34]

Running/maintenance/labor cost [1 , 24]

Landscaping cost [10]

Environmental Factors Air pollution [1, 28]

Water and noise nuisance [1 , 28]

Density of traffic [10]

Geological Factors Earthquake, volcano, flood [1, 11]

Government Legislation [28, 34]

Policies, tax [1, 28]

Land Strategy Expansion in the future [1 , 21]

Land topography [2]

Land ownership [2]

Parking [1 , 21]

Sufficiency of infrastructure [10, 21]

Location Proximity to public transport [2, 21]

Availability of existing infrastructure [2 , 23]

Proximity to market [2]

Accessibility to hospital [1, 21]

Accessibility to center/arterial route [10 , 29]

Accessibility to hospital (staff) [10, 21]

Market Conditions Need for new hospital [1, 21]

Number of current competitors [10, 28]

Number of total beds [28]

Medical technology [28]

Potential competitors [10]

Population Age [21 , 34]

Education [2]

Economic condition [2 , 28]

Population density [1 , 34]

Criteria	Sub-criteria
Cost	Cost of land [1 , 34]
	Running/maintenance/labor cost [1 , 24]
	Landscaping cost [10]
Environmental Factors	Air pollution [1, 28]
	Water and noise nuisance [1 , 28]
	Density of traffic [10]
Geological Factors	Earthquake, volcano, flood [1, 11]
Government	Legislation [28, 34]
	Policies, tax [1, 28]
Land Strategy	Expansion in the future [1 , 21]
	Land topography [2]
	Land ownership [2]
	Parking [1 , 21]
	Sufficiency of infrastructure [10, 21]
Location	Proximity to public transport [2, 21]
	Availability of existing infrastructure [2 , 23]
	Proximity to market [2]
	Accessibility to hospital [1, 21]
	Accessibility to center/arterial route [10 , 29]
	Accessibility to hospital (staff) [10, 21]
Market Conditions	Need for new hospital [1, 21]
	Number of current competitors [10, 28]
	Number of total beds [28]
	Medical technology [28]
	Potential competitors [10]
Population	Age [21 , 34]
	Education [2]
	Economic condition [2 , 28]
	Population density [1 , 34]

1.1.2 Literature on fuzzy EDAS

Fuzzy EDAS has not been applied to the hospital site selection problem. Therefore, we provide a review of fuzzy EDAS applications to other selection problems. Jana and Pal [12] used bipolar fuzzy numbers and applied fuzzy EDAS to the selection of road construction. They introduced two aggregation operators based on the original EDAS approach with bipolar fuzzy numbers. Kas Bayraktaroglu and Kundakci [14] applied fuzzy EDAS method to select the most appropriate R&D project for their company. Rashide et al. proposed a hybrid best-worst EDAS method for ranking industrial robots. The ranking results are compared with the TOPSIS, VIKOR and Distance-Based Approach methods. A sensitivity analysis is performed to check the priority ranking consistency of the applied methods. Yanmaz et al. [30] used an extension of the EDAS method based on the use of interval-valued Pythagorean fuzzy numbers on the car selection problem. The results were compared with the interval-valued intuitionistic fuzzy sets EDAS method, and a sensitivity analysis was conducted to show the robustness of the proposed method. Another application area of this methodology is the supplier selection problem. Ecer [4] introduced a novel integrated model of the fuzzy AHP and EDAS to select a third-party logistics provider. Stevic et al. [27] used fuzzy trapezoid numbers with the fuzzy EDAS method to select the best carpenter manufacturer. A sensitivity analysis is also conducted to show the stability of the results. Polat and Bayhan [19] used the fuzzy EDAS method for solving the vary, heating, ventilating, air conditioning and the air handling unit system and also its supplier selection problem for the green multi-functional shopping center project. After they obtained results, a sensitivity analysis was carried out to show the stability of their results. Mishra et al. [18] introduced a hybrid methodology based on the Criteria Importance through Inter-criteria Correlation and EDAS methods with Fermatean fuzzy sets to select the best sustainable third-party reverse logistics providers using an improved generalized score function. They compared their results with the Fermatean fuzzy-TOPSIS and Fermatean fuzzy-Weighted Product Model to verify the robustness of the proposed methodology. Facility location selection problems are also an application area for this method. Kahraman et al. [13] extended the classical EDAS method by ordinary fuzzy sets in case of vague and incomplete data to select a waste disposal site. They introduced the interval-valued intuitionistic fuzzy EDAS method into the literature. Schitea et al. [22] applied intuitionistic fuzzy WASPAS, COPRAS and EDAS methods to select the best location for a hydrogen mobility roll-up site. The results of the proposed methods were compared with the outcome of the intuitionistic fuzzy TOPSIS method. Kutlu Gundogdu et al. [6] compared the results of hesitant fuzzy EDAS with hesitant fuzzy TOPSIS for the hospital selection problem, and found that the methods give the same ranking for the alternatives.

2 Methodology

2.1 Fuzzy set theory

The fuzzy set theory was first developed by Zadeh (1965) [31] to handle MCDM problems under fuzzy environments [13]. These applications allow decision-makers to deal with uncertainties, vagueness and impreciseness. Fuzzy numbers can have many forms such as triangular and trapezoidal. In this study, we used triangular fuzzy numbers as depicted in Figure 1. Definitions related to fuzzy sets and numbers are as follows.

Fig. 1

Triangular Fuzzy Number (TFN).

Let Ã = (a₁, a₂, a₃) and B̃ = (b₁, b₂, b₃) be Triangular Fuzzy Numbers (TFN), k̃= (k, k, k) be the crisp number.

The membership function μ_Ã(x) of Ã is defined as follows [16]:

$μ_{Ã;} (x) = {\begin{matrix} 0, & x \leq a_{1}, \\ \frac{x - a_{1}}{a_{2} - a_{1}}, & a_{1} \leq x \leq a_{2}, \\ \frac{a_{3} - x}{a_{3} - a_{2}}, & a_{2} \leq x \leq a_{3}, \\ 0, & x > a_{3} \end{matrix}$ (1) The arithmetic operations of two positive TFNs Ã and B̃ are defined as follows [16]:

Addition:

$Ã; \oplus B̃; = (a_{1}, a_{2}, a_{3}) \oplus (b_{1}, b_{2}, b_{3})$ (2) $= (a_{1} + b_{1}, a_{2} + b_{2}, a_{3} + b_{3})$ Subtraction:

$Ã; ⊖ B̃; = (a_{1}, a_{2}, a_{3}) ⊖ (b_{1}, b_{2}, b_{3})$ (3) $= (a_{1} - b_{3}, a_{2} - b_{2}, a_{3} - b_{1})$ Multiplication:

$Ã; \otimes B̃; = (a_{1}, a_{2}, a_{3}) \otimes (b_{1}, b_{2}, b_{3})$ (4) $= (a_{1} \times b_{1}, a_{2} \times b_{2}, a_{3} \times b_{3})$

$Ã; \otimes k = {\begin{matrix} (a_{1} \times k, a_{2} \times k, a_{3} \times k), & k \geq 0 \\ (a_{3} \times k, a_{2} \times k, a_{1} \times k), & k < 0 \end{matrix}$ (5) Division:

$Ã; ø B̃; = (a_{1}, a_{2}, a_{3}) ø (b_{1}, b_{2}, b_{3})$ (6) $= (a_{1} / b_{3}, a_{2} / b_{2}, a_{3} / b_{1})$

$Ã; ø k = {\begin{matrix} (a_{1} / k, a_{2} / k, a_{3} / k), & k > 0 \\ (a_{3} / k, a_{2} / k, a_{1} / k), & k < 0 \end{matrix}$ (7) The defuzified (crisp) value of a TFN Ã is defined as follows [16]:

$k (Ã;) = \frac{1}{3} (a_{1} + a_{2} + a_{3})$ (8) To find the maximum number between Ã and 0, ψ function is defined as follows [16]:

$ψ (Ã;) = {\begin{matrix} Ã;, & k ( & Ã;) > 0 \\ 0, & k ( & Ã;) \leq 0 \end{matrix}$ (9) where  0 = (0, 0, 0).

2.2 Fuzzy EDAS

The EDAS method was introduced by Keshavarz et al. [15] to deal with conflicting attributes existing in MCDM problems. This method uses the difference between all the alternatives and the average solution (AV) based on positive distance from average (PDA) and negative distance from average (NDA) measures for beneficial or non-beneficial criteria to compare candidate alternatives. Decision-makers (DMs) can express the weights of the criteria and evaluate the alternatives according to the criteria using linguistic terms.

The steps of the fuzzy EDAS method can be summarized as follows [16]:

Step 1: Create the average decision matrix (X) for aggregating the judgments of decision-makers for each alternative with respect to each criterion by the following calculations [14]: $X = [{x̃}_{ij}]_{m \times n}$ (10) ${x̃}_{ij} = \frac{1}{k} \sum_{p = 1}^{k} {x̃}_{ij}^{p}$ (11) where ${x̃}_{ij}^{p}$ denotes the performance value of alternative Ã ;_i(1 ≤ i ≤ n) with respect to criterion c_j(1 ≤ j ≤ m) dedicated by the p^th decision-maker (1 ≤ p ≤ k).

Step 2: Sum the criterion values assigned by decision-makers and divide them by the number of decision-makers to create the weighted matrix of criteria as follows: $W = [{w̃;}_{j}]_{1 \times n}$ (12) ${w̃;}_{j} = \frac{1}{k} \sum_{p = 1}^{k} {w̃;}_{i}^{p}$ (13) where ${w̃;}_{j}^{p}$ denotes the weights of criterion c_j(1 ≤ j ≤ m) determined by the p^th decision-maker (1 ≤ p ≤ k).

Step 3: Construct the matrix of average solutions as follows: $AV = [{\tilde{av}}_{j}]_{1 \times m}$ (14) ${\tilde{av}}_{j} = \frac{1}{n} \sum_{i = 1}^{n} {x̃}_{ij}$ (15) where ãv_j denotes the average solutions with respect to each criterion. The dimension of criteria weights matrix is equal to the dimension of the ãv_j matrix.

Step 4: Let B be the set of beneficial criteria and N be the set of non-beneficial (cost) criteria. In this step, the PDA and NDA matrices are calculated according to the criteria type as given: $PDA = [{\tilde{pda}}_{ij}]_{m \times n}$ (16) $NDA = [{\tilde{nda}}_{ij}]_{m \times n}$ (17) ${\tilde{pda}}_{ij} = {\begin{matrix} \frac{ψ ({x̃}_{ij} - {\tilde{av}}_{j})}{k ({\tilde{av}}_{j})}, & j \in B \\ \frac{ψ ({\tilde{av}}_{j} - {x̃}_{ij})}{k ({\tilde{av}}_{j})}, & j \in N \end{matrix}$ (18) ${\tilde{nda}}_{ij} = {\begin{matrix} \frac{ψ ({\tilde{av}}_{j} - {x̃}_{ij})}{k ({\tilde{av}}_{j})}, & j \in B \\ \frac{ψ ({x̃}_{ij} - {\tilde{av}}_{j})}{k ({\tilde{av}}_{j})}, & j \in N \end{matrix}$ (19) where p̃da and ñda represent for positive and negative distance of performance value of the i^th alternative from the average solution with respect to the j^th criterion, respectively.

Step 5: Calculate the weighted sum of positive ${\tilde{sp}}_{i}$ and negative ${\tilde{sn}}_{i}$ distances for all alternatives as shown: ${\tilde{sp}}_{i} = \sum_{i = 1}^{m} ({\tilde{w}}_{j} \otimes {\tilde{pda}}_{ij})$ (20) ${\tilde{sn}}_{i} = \sum_{i = 1}^{m} ({\tilde{w}}_{j} \otimes {\tilde{nda}}_{ij})$ (21) Step 6: Calculate the normalized weighted sum of positive ${\tilde{nsp}}_{i}$ and negative ${\tilde{nsn}}_{i}$ distances for all alternatives as shown: ${\tilde{nsp}}_{i} = \frac{{\tilde{sp}}_{i}}{\max_{i} (k ({\tilde{sp}}_{i}))}$ (22) ${\tilde{nsn}}_{i} = 1 - \frac{{\tilde{sn}}_{i}}{\max_{i} (k ({\tilde{sn}}_{i}))}$ (23) Step 7: Compute the appraisal score ${\tilde{as}}_{i}$ for all alternatives as shown:

${\tilde{as}}_{i} = \frac{1}{2} ({\tilde{nsp}}_{i} \oplus {\tilde{nsn}}_{i})$ (24)Step 8: Rank the alternatives according to the decreasing values of appraisal score ${\tilde{as}}_{i}$ . Among the candidate alternatives, the one with the highest appraisal score is selected.

A flowchart of the proposed methodology is given in Figure 2.

Fig. 2

Proposed methodology.

3 Application

3.1 The application of the fuzzy EDAS method

In this section, the fuzzy EDAS method is applied for selecting the site of a new hospital in İstanbul. Three alternative locations on the Asian side of İstanbul, namely, Beykoz, Çekmeköy and Ümraniye are considered. The reasons for choosing these particular districts include: (1) eligibility to establish a new hospital, and (2) lack of health care facilities when the total districts’ population is considered. The chosen set of criteria and sub-criteria plays an important role in hospital site selection. Many factors should be taken into account such as environmental, economical, and geological factors as well as government policies, and city land development strategies detailed in Table 1. We used five criteria and 17 sub-criteria selected faithfully to the literature. A questionnaire that includes these criteria was evaluated by considering the judgments of three decision-makers who were biomedical engineers working in the health care sector. After collecting their judgments for each criterion and sub-criterion from the decision-makers, the alternatives were compared with respect to sub-criteria illustrated in Figure 3.

Fig. 3

The hierarchical structure for hospital site selection.

Next the criteria and sub-criteria used in this study are explained in detail.

Population dynamics: Population of İstanbul has been rapidly increasing mostly due to economic migration, and thus the infrastructure needs are also increasing including in the health care sector. Population dynamics such as age structure, income level of people, population density play an important role to determine the site of a new hospital.

Cost: Labor, construction, investment and landscaping costs are important to establish a new hospital. Hospital management and the financier should consider all these costs. They scrutinize the costs with the aim of minimizing them.

Position of building: İstanbul is the most impacted city in Turkey by traffic congestion. Thus, hospital staff and people prefer to use mostly public transportation. Therefore, the distance to transportation means becomes important and the hospital should be close to the arterials and major roads. Establishment sites should be determined as locations where individuals can arrive at quickly in a reliable and economic manner.

Properties of Building: One of the most important criteria is the properties of the building. Hospital staff and people can use private transport to reach the hospital. That’s why a parking area should be considered. If needed, the hospital management can take a decision to expand the hospital. Thus, land expansion possibilities are also important for site selection. Crowded cities may have problems such as shortage of water, electricity or natural gas which could pose a serious problem for a hospital. For example, certain neighborhoods in İstanbul still do not have access to natural gas for heating. Furthermore, hospitals should be constructed for all worst-case scenarios that can happen unexpectedly such as earthquakes, fires and floods. These also need to be considered in site selection especially in a city like İstanbul which has frequent earthquakes and floods.

Market conditions: Knowledge about competitors and their activities, need of a new hospital in the area are also important to know for the site decision.

In the criterion hierarchy shown in Figure 3, “Building cost (C21)”, “Labor cost (C22)”, “Investment cost (C23)” and “Landscaping cost (C24)” are non-beneficial sub-criteria, and the remaining thirteen are beneficial sub-criteria.

The linguistic terms that decision-makers used for assigning the weights to each criterion are given in Table 3. The weights of the sub-criteria are obtained from the decision-makers and the average weighting matrix given in Table 4 is calculated using Eqs. (10) and (11). Table 5 shows the criteria weights as judged by the three decision-makers by rating alternatives concerning each criterion expressed in linguistic terms. The values of the average decision matrix and the average solution matrix are calculated using Eqs. (12) through (15), and are given in Table 6. The positive and negative distances from the average solutions are calculated by using Eqs. (16) through (19) which are listed in Tables 7 and 8. Then, the weighted sum of positive and negative distances ( ${\tilde{sp}}_{i}$ and ${\tilde{sn}}_{i}$ ) and their normalized values, the appraisal scores for all alternatives and their averages are computed by using Eqs. (20) through (24). Tables 10 and 9 present the findings of these steps. According to the calculated results, as can be seen in the last column of Table 9, A3 (Ümraniye) with average of 0.746 is the best site among all alternatives. A1 (Beykoz), on the other hand, is the worst site alternative for the new hospital. The ranking of all alternatives is found as A3 ≻ A2 ≻ A1.

Table 2

Linguistic terms and their corresponding TFNs

Terms	TFN for Weighting Criteria
Very Low (VL)	(0, 0, 0.1)
Low (L)	(0, 0.1, 0.3)
Medium Low (ML)	(0.1, 0.3, 0.5)
Medium (M)	(0.3, 0.5, 0.7)
Medium High (MH)	(0.5, 0.7, 0.9)
High (H)	(0.7, 0.9, 1)
Very High (VH)	(0.9, 1, 1)

Table 3

The weights of sub-criteria assigned by decision-makers

Criteria	DM1	DM2	DM3	Weights
C11	H	VH	MH	(0.700, 0.867, 0.967)
C12	VH	H	H	(0.767, 0.933, 1.000)
C13	VH	VH	VH	(0.900, 1.000, 1.000)
C21	H	VH	H	(0.767, 0.933, 1.000)
C22	MH	H	MH	(0.633, 0.833, 0.967)
C23	VH	VH	H	(0.833, 0.967, 1.000)
C24	MH	M	ML	(0.300, 0.500, 0.700)
C31	VH	VH	VH	(0.900, 1.000, 1.000)
C32	H	MH	H	(0.633, 0.833, 0.967)
C33	VH	H	VH	(0.833, 0.967, 1.000)
C34	MH	ML	H	(0.433, 0.633, 0.800)
C41	VH	H	H	(0.767, 0.933, 1.000)
C42	H	MH	H	(0.633, 0.833, 0.967)
C43	H	H	MH	(0.633, 0.833, 0.967)
C51	VH	H	VH	(0.833, 0.967, 1.000)
C52	H	H	MH	(0.633, 0.833, 0.967)
C53	VH	VH	H	(0.833, 0.967, 1.000)

Table 4

Decision-makers evaluations of the alternatives with respect to each criterion expressed in linguistic terms

DMs	Alt.	C11	C12	C13	C21	C22	C23	C24	C31	C32	C33	C34	C41	C42	C43	C51	C52	C53
D1	A1	MH	H	VH	VH	M	VH	ML	VH	MH	M	ML	VH	M	H	VH	H	H
	A2	MH	H	VH	H	ML	H	ML	VH	MH	MH	ML	H	MH	MH	VH	VH	H
	A3	H	VH	VH	H	ML	H	M	VH	H	M	ML	H	MH	MH	VH	VH	H
D2	A1	H	MH	H	MH	ML	H	MH	H	MH	H	MH	VH	MH	VH	VH	MH	VH
	A2	VH	H	VH	H	L	VH	H	VH	M	VH	H	VH	H	H	H	H	H
	A3	H	H	H	MH	L	VH	H	VH	MH	H	MH	VH	MH	H	VH	H	H
D3	A1	M	VH	H	H	MH	H	M	H	VH	H	VH	VH	H	MH	H	VH	MH
	A2	VH	VH	VH	MH	MH	MH	M	H	VH	VH	H	VH	H	H	H	VH	H
	A3	VH	VH	VH	MH	MH	MH	M	VH	VH	VH	H	H	MH	VH	H	H	MH

Table 5

Elements of the average decision matrix and average solution matrix

Criteria	A1	A2	A3	AV
C11	(0.500, 0.700, 0.867)	(0.767, 0.900, 0.967)	(0.767, 0.933, 1.000)	(0.678, 0.844, 0.944)
C12	(0.700, 0.867, 0.967)	(0.767, 0.933, 1.000)	(0.833, 0.967, 1.000)	(0.767, 0.922, 0.989)
C13	(0.767, 0.933,1.000)	(0.900, 1.000, 1.000)	(0.833, 0.967, 1.000)	(0.833, 0.967, 1.000)
C21	(0.700, 0.867, 0.967)	(0.633, 0.833, 0.967)	(0.567, 0.767, 0.933)	(0.633, 0.822, 0.956)
C22	(0.300, 0.500, 0.700)	(0.200, 0.367, 0.567)	(0.200, 0.367, 0.567)	(0.233, 0.411, 0.611)
C23	(0.767, 0.933, 1.000)	(0.700, 0.867, 0.967)	(0.700, 0.867, 0.967)	(0.722, 0.889, 0.978)
C24	(0.300, 0.500, 0.700)	(0.367, 0.567, 0.733)	(0.433, 0.633, 0.800)	(0.367, 0.567, 0.744)
C31	(0.767, 0.933, 1.000)	(0.833, 0.967, 1.000)	(0.900, 1.000, 1.000)	(0.833, 0.967, 1.000)
C32	(0.633, 0.800, 0.933)	(0.567, 0.733, 0.867)	(0.700, 0.867, 0.967)	(0.633, 0.800, 0.922)
C33	(0.567, 0.767, 0.900)	(0.767, 0.900, 0.967)	(0.633, 0.800, 0.900)	(0.656, 0.822, 0.922)
C34	(0.500, 0.667, 0.800)	(0.500, 0.700, 0.867)	(0.433, 0.633, 0.800)	(0.478, 0.667, 0.811)
C41	(0.900, 1.000, 1.000)	(0.833, 0.967, 1.000)	(0.767, 0.933, 1.000)	(0.833, 0.967, 1.000)
C42	(0.500, 0.700, 0.867)	(0.633, 0.833, 0.967)	(0.500, 0.700, 0.900)	(0.544, 0.744, 0.911)
C43	(0.700, 0.867, 0.967)	(0.633, 0.833, 0.967)	(0.700, 0.867, 0.967)	(0.678, 0.856, 0.967)
C51	(0.833, 0.967, 1.000)	(0.767, 0.933, 1.000)	(0.833, 0.967, 1.000)	(0.811, 0.956, 1.000)
C52	(0.700, 0.867, 0.967)	(0.700, 0.867, 0.967)	(0.767, 0.933, 1.000)	(0.722, 0.889, 0.978)
C53	(0.700, 0.867, 0.967)	(0.700, 0.900, 1.000)	(0.633, 0.833, 0.967)	(0.678, 0.867, 0.978)

Table 6

PDA values

Criteria	A1	A2	A3
C11	(0.000, 0.000, 0.000)	(-0.216, 0.068, 0.351)	(-0.216, 0.108, 0.392)
C12	(0.000, 0.000, 0.000)	(-0.249, 0.012, 0.261)	(-0.174, 0.050, 0.261)
C13	(0.000, 0.000, 0.000)	(-0.107, 0.036, 0.179)	(-0.179, 0.000, 0.179)
C21	(0.000, 0.000, 0.000)	(0.000, 0.000, 0.000)	(-0.373, 0.069, 0.484)
C22	(0.000, 0.000, 0.000)	(0.000, 0.000, 0.000)	(-0.796, 0.106, 0.982)
C23	(0.000, 0.000, 0.000)	(0.000, 0.000, 0.000)	(-0.283, 0.026, 0.322)
C24	(-0.596, 0.119, 0.795)	(0.000, 0.000, 0.000)	(0.000, 0.000, 0.000)
C31	(0.000, 0.000, 0.000)	(-0.179, 0.000, 0.179)	(-0.107, 0.036, 0.179)
C32	(-0.368, 0.000, 0.382)	(0.000, 0.000, 0.000)	(-0.283, 0.085, 0.425)
C33	(0.000, 0.000, 0.000)	(-0.194, 0.097, 0.389)	(0.000, 0.000, 0.000)
C34	(-0.477, 0.000, 0.494)	(-0.477, 0.051, 0.545)	(0.000, 0.000, 0.000)
C41	(-0.179, 0.036, 0.179)	(-0.179, 0.000, 0.179)	(0.000, 0.000, 0.000)
C42	(0.000, 0.000, 0.000)	(-0.379, 0.121, 0.576)	(0.000, 0.000, 0.000)
C43	(-0.320, 0.013, 0.347)	(-0.400, 0.000, 0.000)	(0.000, 0.013, 0.347)
C51	(-0.181, 0.012, 0.205)	(-0.253, 0.000, 0.000)	(0.000, 0.012, 0.205)
C52	(0.000, 0.000, 0.000)	(-0.322, 0.000, 0.000)	(0.000, 0.052, 0.322)
C53	(-0.330, 0.000, 0.344)	(-0.330, 0.040, 0.383)	(0.000, 0.000, 0.000)

Table 7

NDA values

Criteria	A1	A2	A3
C11	(-0.230, 0.176, 0.541)	(0.000, 0.000, 0.000)	(0.000, 0.000, 0.000)
C12	(-0.224, 0.062, 0.324)	(0.000, 0.000, 0.000)	(0.000, 0.000, 0.000)
C13	(-0.179, 0.036, 0.250)	(0.000, 0.000, 0.000)	(-0.179, 0.000, 0.179)
C21	(-0.318, 0.055, 0.415)	(-0.401, 0.014, 0.415)	(0.000, 0.000, 0.000)
C22	(-0.743, 0.212, 1.115)	(0.000, 0.000, 0.000)	(0.000, 0.000, 0.000)
C23	(-0.245, 0.052, 0.322)	(0.000, 0.000, 0.000)	(0.000, 0.000, 0.000)
C24	(0.000, 0.000, 0.000)	(0.000, 0.000, 0.000)	(-0.556, 0.119, 0.775)
C31	(-0.179, 0.036, 0.250)	(-0.179, 0.000, 0.179)	(0.000, 0.000, 0.000)
C32	(0.000, 0.000, 0.000)	(-0.297, 0.085, 0.453)	(0.000, 0.000, 0.000)
C33	(-0.306, 0.069, 0.444)	(0.000, 0.000, 0.000)	(-0.306, 0.028, 0.361)
C34	(0.000, 0.000, 0.000)	(0.000, 0.000, 0.000)	(-0.494, 0.051, 0.580)
C41	(0.000, 0.000, 0.000)	(-0.179, 0.000, 0.179)	(-0.179, 0.036, 0.250)
C42	(-0.439, 0.061, 0.561)	(0.000, 0.000, 0.000)	(-0.485, 0.061, 0.561)
C43	(0.000, 0.000, 0.000)	(-0.347, 0.027, 0.400)	(0.000, 0.000, 0.000)
C51	(0.000, 0.000, 0.000)	(-0.205, 0.024, 0.253)	(0.000, 0.000, 0.000)
C52	(-0.283, 0.026, 0.322)	(-0.283, 0.026, 0.322)	(0.000, 0.000, 0.000)
C53	(0.000, 0.000, 0.000)	(0.000, 0.000, 0.000)	(-0.344, 0.040, 0.410)

Table 8

The weighted sums of distances, the normalized values

Alternatives	${\tilde{sp}}_{i}$	${\tilde{sn}}_{i}$	${\tilde{nsp}}_{i}$	${\tilde{nsn}}_{i}$
A1	(-1.329, 0.116, 2.383)	(-2.285, 0.699, 4.458)	(-1.447, 0.126, 2.594)	(-3.656, 0.270, 3.386)
A2	(-2.288, 0.372, 2.902)	(-1.363, 0.151, 2.160)	(-2.491, 0.404, 3.158)	(-1.256, 0.843, 2.423)
A3	(-1.748, 0.490, 4.014)	(-1.527, 0.241, 2.747)	(-1.902, 0.534, 4.369)	(-1.869, 0.748, 2.594)

Table 9

The weighted normalized fuzzy decision matrix

Criteria	A1	A2	A3
C11	(0.257, 0.599, 0.856)	(0.428, 0.770, 0.856)	(0.599, 0.799, 0.856)
C12	(0.458, 0.794, 0.917)	(0.642, 0.856, 0.917)	(0.642, 0.886, 0.917)
C13	(0.688, 0.918, 0.983)	(0.885, 0.983, 0.983)	(0.688, 0.918, 0.983)
C21	(0.458, 0.529, 0.000)	(0.458, 0.550, 0.917)	(0.458, 0.598, 0.917)
C22	(0.000, 0.000, 0.000)	(0.000, 0.000, 0.000)	(0.000, 0.000, 0.000)
C23	(0.475, 0.509, 0.679)	(0.475, 0.548, 0.950)	(0.475, 0.548, 0.950)
C24	(0.056, 0.100, 0.500)	(0.050, 0.088, 0.500)	(0.050, 0.079, 0.167)
C31	(0.688, 0.918, 0.983)	(0.688, 0.951, 0.983)	(0.885, 0.983, 0.983)
C32	(0.411, 0.658, 0.822)	(0.247, 0.603, 0.822)	(0.411, 0.713, 0.822)
C33	(0.285, 0.728, 0.950)	(0.475, 0.855, 0.950)	(0.285, 0.760, 0.950)
C34	(0.063, 0.419, 0.628)	(0.063, 0.439, 0.628)	(0.063, 0.398, 0.628)
C41	(0.825, 0.917, 0.917)	(0.642, 0.886, 0.917)	(0.642, 0.856, 0.917)
C42	(0.247, 0.576, 0.822)	(0.411, 0.685, 0.822)	(0.411, 0.576, 0.740)
C43	(0.411, 0.713, 0.822)	(0.411, 0.685, 0.822)	(0.411, 0.713, 0.822)
C51	(0.665, 0.918, 0.950)	(0.665, 0.887, 0.950)	(0.665, 0.918, 0.950)
C52	(0.411, 0.713, 0.822)	(0.576, 0.795, 0.822)	(0.576, 0.767, 0.822)
C53	(0.475, 0.823, 0.950)	(0.665, 0.855, 0.950)	(0.475, 0.792, 0.950)

Table 10

The appraisal scores and ranking of fuzzy EDAS method

Alternatives	Appraisal Score	Average	Rank
A1	(-2.551, 0.198, 2.990)	0.212	3
A2	(-1.873, 0.624, 2.791)	0.514	2
A3	(-1.886, 0.641, 3.482)	0.746	1

3.2 Comparison with the fuzzy TOPSIS method

TOPSIS is a popular MCDM method first introduced by Hwang and Yoon [8]. TOPSIS is mathematically simple and flexible in the definition of the alternative set [5], and it has an intuitive and clear logic to represent the rationale of human choice [7]. Both EDAS and TOPSIS are distance-related techniques. Therefore, we compare our proposed alternative selection method with the fuzzy TOPSIS method to see how the decision will be affected. TOPSIS determines a solution with the shortest distance to the ideal solution and the greatest distance from the negative ideal solution (NIS), without considering the relative importance of these distances. While TOPSIS calculates distances to ideal solution and NIS, EDAS uses average values as reference points [9]. In addition to comparison studies of both methods in the context of the hospital site and hospital selection problem ([1] and [6]), there is some other work comparing the two methods. Keshavarz-Ghorabaee et al. [17] compared EDAS and TOPSIS via simulation of the decision data with different numbers of alternatives and criteria. They showed that the efficiency of the EDAS method is more than the TOPSIS method with respect to the defined rank reversal measures. Srivastava et al. [24] prioritized autonomous maintenance system attributes both with fuzzy EDAS and fuzzy TOPSIS approaches resulting in the same prioritization.

Next, the details of the fuzzy TOPSIS computations are given [3]:

Step 1: Construct the decision matrix and choose the appropriate linguistic variables for the alternatives with respect to criteria where ${\tilde{x}}_{ij}^{k}$ is the performance rating of alternative where ${\tilde{A}}_{i}$ with respect to criterion C_j evaluated by k^th decision-maker, and ${\tilde{x}}_{ij}^{k}$ = ( $a_{ij}^{k}$ , $b_{ij}^{k}$ , $c_{ij}^{k}$ ) and ${\tilde{w}}_{j}^{k}$ = ( $w_{j 1}^{k}$ , $w_{j 2}^{k}$ , $w_{j 3}^{k}$ ), respectively [25].

Step 2: Compute the aggregated fuzzy ratings for alternatives and fuzzy weights for criteria as shown, respectively:

$a_{ij} = min_{k} {a_{ij}^{k}}, b_{ij} = \frac{1}{K} \sum_{k = 1}^{K} b_{ij}^{k}, c_{ij} = max_{k} {c_{ij}^{k}}$ (25) $w_{j 1} = min_{k} {w_{j 1}^{k}}, w_{j 2} = \frac{1}{K} \sum_{k = 1}^{K} w_{j 2}^{k}, w_{j 3} = max_{k} {w_{j 3}^{k}}$ (26) Step 3: Construct the normalized fuzzy decision matrix $\tilde{R}$ =[ ${\tilde{r}}_{ij}$ ] as shown: ${\tilde{r}}_{ij} = (\frac{a_{ij}}{c_{j}^{*}}, \frac{b_{ij}}{c_{j}^{*}}, \frac{c_{ij}}{c_{j}^{*}}), c_{j}^{*} = max_{k} {c_{ij}}, j \in B$ (27) ${\tilde{r}}_{ij} = (\frac{a_{j}^{-}}{c_{ij}}, \frac{a_{j}^{-}}{b_{ij}}, \frac{a_{j}^{-}}{a_{ij}}), a_{j}^{-} = min_{k} {a_{ij}}, j \in C$ (28) Step 4: Compute the weighted normalized fuzzy decision matrix $\tilde{V}$ =[ ${\tilde{v}}_{ij}$ ] where ${\tilde{v}}_{ij}$ = ${\tilde{r}}_{ij}$ × {w_j.

Step 5: Determine the fuzzy positive ideal solution (FPIS) and fuzzy negative ideal solution (FNIS), respectively as shown: $A^{*} = ({\tilde{v}}_{1}^{*}, {\tilde{v}}_{2}^{*}, . . ., {\tilde{v}}_{n}^{*}), where {\tilde{v}}_{j}^{*} = max_{i} {v_{ij 3}}$ (29) $A^{-} = ({\tilde{v}}_{1}^{-}, {\tilde{v}}_{2}^{-}, . . ., {\tilde{v}}_{n}^{-}), where {\tilde{v}}_{j}^{-} = min_{i} {v_{ij 1}}$ (30) Step 6: Calculate the distance from each alternative A_i to the FPIS and FNIS, respectively as shown: $d_{i}^{*} = \sum_{j = 1}^{n} d ({\tilde{v}}_{ij}, {\tilde{v}}_{j}^{*}), d_{i}^{-} = \sum_{j = 1}^{n} d ({\tilde{v}}_{ij}, {\tilde{v}}_{j}^{-})$ (31) Step 7: Compute the closeness coefficient (CC_i) as shown: ${CC}_{i} = \frac{d_{i}^{-}}{d_{i}^{-} + d_{i}^{*}}$ (32) Step 8: Rank the alternatives according to the decreasing values of closeness coefficient CC_i. Among the candidate alternatives, the alternative with the highest closeness coefficient is selected.

The weighted normalized fuzzy decision matrix is calculated by using equations in Steps 1 through 4, and values are given in Table 11. Then, by using Eqs. (29) and (30) FPIS and FNIS are calculated which are listed in Table 12. The weighted distances to the positive and negative ideal solutions are calculated using Eq. (31) as given in Table 13. The relative closeness ratio of each alternative to the ideal solution is given in the last column of Table 13.

Table 11

The FPIS and FNIS values

Criteria	A ^*	A ^-
C11	(0.599, 0.799, 0.856)	(0.257, 0.599, 0.856)
C12	(0.642, 0.886, 0.917)	(0.458, 0.794, 0.917)
C13	(0.885, 0.983, 0.983)	(0.688, 0.918, 0.983)
C21	(0.458, 0.598, 0.917)	(0.458, 0.529, 0.000)
C22	(0.000, 0.000, 0.000)	(0.000, 0.000, 0.000)
C23	(0.475, 0.548, 0.950)	(0.475, 0.509, 0.679)
C24	(0.056, 0.100, 0.500)	(0.050, 0.079, 0.167)
C31	(0.885, 0.983, 0.983)	(0.688, 0.918, 0.983)
C32	(0.411, 0.658, 0.822)	(0.247, 0.603, 0.822)
C33	(0.475, 0.855, 0.950)	(0.285, 0.728, 0.950)
C34	(0.063, 0.439, 0.628)	(0.063, 0.398, 0.628)
C41	(0.825, 0.917, 0.917)	(0.642, 0.856, 0.917)
C42	(0.247, 0.576, 0.822)	(0.247, 0.576, 0.740)
C43	(0.411, 0.713, 0.822)	(0.411, 0.685, 0.822)
C51	(0.665, 0.918, 0.950)	(0.665, 0.887, 0.950)
C52	(0.576, 0.795, 0.822)	(0.411, 0.713, 0.822)
C53	(0.665, 0.855, 0.950)	(0.475, 0.823, 0.950)

Table 12

The weighted distances to PIS and NIS and closeness coefficient of alternatives

Alternatives	$d_{i}^{*}$	$d_{i}^{-}$	${CC}_{i}^{-}$
A1	1.783	0.516	0.225
A2	0.524	1.789	0.774
A3	0.777	1.536	0.664

Table 13

Comparison of the ranking results

Alternatives	F-EDAS	F-TOPSIS
A1	3	3
A2	2	1
A3	1	2

For comparison purposes, the same sub-criteria weights from the fuzzy EDAS method were also used in the fuzzy TOPSIS method as sub-criteria weights. The alternative site rankings of these two methods are shown in Table 13. As can be seen from the second and third columns of Table 13, the site rankings by the two methods differ from each other. The worst alternative (Alternative 1) is the same in both methods. However, the best alternative is found to be Alternative 3 with fuzzy EDAS whereas it is Alternative 2 with fuzzy TOPSIS. The main difference between the two methods is in the evaluation of the alternatives where EDAS uses distances from the average solution whereas TOPSIS considers shortest and greatest distances to the ideal and negative ideal solutions.

4 Conclusion

The existence of both quantitative and qualitative criteria and sub-criteria makes selecting the best site for a hospital a difficult MCDM problem. Using a case study from İstanbul, this research applied the fuzzy EDAS method for selecting the most appropriate hospital site from among the alternative districts. Three alternative hospital sites were considered with regard to a comprehensive set of five criteria and 17 sub-criteria selected faithfully to the relevant literature which were evaluated by three decision-makers. In the EDAS method, the evaluations of alternatives rely on a distance measure from the average solution [6]. Decision-makers evaluate their preferences by different linguistic terms which are then mapped to fuzzy numbers such as TFNs as used in this study.

In our case study, the fuzzy EDAS method resulted in a different ranking of the alternatives compared to the fuzzy TOPSIS method. It is not uncommon for different MCDM methods to produce conflicting results. It may often be up to the decision-maker to select the proper method with respect to the data s/he has [13]. It is also worth noting that Watróbski et al. [33] propose a framework which is independent of the problem domain, and provide formal guidelines for the selection of a particular MCDM method. The framework considers 56 different MCDM techniques but since EDAS is a relatively new one it has not been included in the research in [33]. As future work, integrating the EDAS method into this comparative framework could be interesting.

In the hospital site selection case study presented here, three site alternatives were considered. Including more alternatives may have provided a better understanding of how well these districts are suitable in comparison to each other for new hospitals. While these districts still cover a large geographic area, one could say the suggested alternative site works best for serving only the population in a certain part of the city rather than in a larger geographic area. Moreover, the same analysis could have yielded different suggestions with different decision-makers.

In future studies, the importance of the decision-makers’ weights can be evaluated via a sensitivity analysis. Moreover, to deal with uncertainty, new extensions of fuzzy sets such as spherical and neutrosophic sets can be used. In addition, simulation experiments can be conducted under different number of alternatives, criteria and sub-criteria to see how, in general, solutions differ when fuzzy EDAS and TOPSIS methods are applied to the hospital site selection problem.

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