Integrating Fuzzy AHP and TOPSIS Methods to Evaluate Operation Efficiency of Daycare Centers

Abstract

As the population ages and families become less able to offer care, the need for long-term care among older people increases. Evaluation of daycare centers, which provide localized long-term care services, is essential for the growth and direction of these institutions. Nevertheless, the present evaluation indexes do not adequately emphasize the significance of each index item or the actual effectiveness of an organization’s operations and management. To solve this issue, the purpose of this research was to develop an evaluation model for the operation and administration of daycare centers. Experts were consulted to collect pertinent criteria, which were further assessed using the Fuzzy Analytic Hierarchy Process (FAHP) and the Technique for Order Preference by Similarity to the Ideal Solution (TOPSIS) techniques. The results indicated that organizational operation management was the top priority, with administrative operation management and service quality management having the largest impact on productivity. Among the 10 daycare services tested, Institution 3 was judged to have the highest score. These findings shed light on the operational management effectiveness of daycare centers and give a novel evaluation methodology for gauging the efficacy of nursing management.

1. Introduction

The proportion of the population over the age of 65 years in the Organization for Economic Cooperation and Development (OECD) countries has increased from less than 9% in 1960 to 17% in 2015 and is expected to continue to rise in the coming decades, reaching about 28% by 2050 [1]. This increase is due to both declining fertility rates and increasing life expectancy, and by 2050, at least a quarter of the population in more than two-thirds of OECD countries would be over the age of 65 years [1]. In Taiwan, the older population over the age of 65 years has reached 7.10% since 1993, signaling the beginning of an aging society. In 2018, the proportion of the older population exceeded 14%, and according to the National Development Commission of the Republic of China, the proportion of the older generation aged 65 years and above would be 16.0% in 2020, increasing to 30.2% by 2040 and continuing to rise to 41.6% in 2070 [2]. In 2020, there were 119,000 people with disabilities in Taiwan [3], and with the trend of population aging, the number of people in need of long-term care is also growing rapidly.

Taiwan is facing a rapid increase in the aging population, and long-term care is also needed due to the increase in the number of people with disabilities and dementia. In 2017, the central government established a community-based long-term care service system, named the Long-Term Care Services Act, to integrate the medical care model of community care that is available, accessible, and acceptable. The Ministry of Health and Welfare actively planned the long-term care 2.0 plan to promote the establishment of a people-oriented, community-based long-term care system. Long-term care 2.0 provides a series of care services, and the modes of care services mainly include three types: home-based, community-based, and institution-based services. Referencing the development strategy of the long-term care system proposed by the World Health Organization (WHO) and the development experience of various countries compiled by the Organization for Economic Co-operation and Development (OECD), the current long-term care system in Taiwan is to accelerate the development of long-term care resources and improve long-term care needs. Accessibility to services is the primary goal. Based on the principle of aging in place, multi-objective community-based services are provided to reduce the pressure of family care [4].

Daycare centers provide localized long-term care services in the community, which enables older adults to receive care while remaining in their original community and maintains continuity of life for older adults and their families [5]. In recent years, the community care service system in Taiwan has matured, and daycare centers have also flourished due to government policies. It is important for institute owners to establish their operating characteristics and efficiency to adapt to future policy changes and operating challenges and to maintain competitiveness. According to Article 39 of the Long-term Care Service Law in Taiwan, daycare institutions are required to undergo an evaluation every four years by the competent authority, providing the public with information about long-term care options. The evaluation items include management efficiency, professional care quality and safety, environmental equipment, and the protection of individual rights and interests. Evaluating efficiency is a complex task that involves many aspects.

The Multiple Criteria Decision Making (MCDM) methods are popularly applied to determine solutions across various fields such as environmental efficiency [6,7] or public health [8]. The purpose of evaluating management efficiency is related to decisions of core value, competitive strategy, and direction. To achieve the goal, it is necessary to analyze the relevant key factors that affect operating efficiency. However, the current evaluation indicators cannot highlight the importance of each indicator item and the real operational management efficiency of the organization. To make the evaluation indicators framework more representative and that reflects the management performance, this research method is based on the Analysis Hierarchy Process (AHP) proposed by [9] and the Fuzzy Analysis Hierarchy Process (FAHP) published by [10]. In this study, the Fuzzy Analytic Hierarchy Process (FAHP) and TOPSIS were applied to evaluate the performance of daycare centers by using the evaluation criteria of daycare operation and management, calculating the relative weights and priorities, and constructing a model and scoring criteria for the performance of daycare centers.

The structure of this study is constructed as follows. After this section, Section 2 reviews relevant studies, then the methodology is presented in Section 3. Section 4 presents the findings in practical case studies. The last section, Section 5, shows the discussion and conclusion.

2. Literature Review

2.1. Daycare Services

Daycare services are community-based, long-term care programs that provide respite care to families and delay the need for residential-based institutional care for older adults with disabilities. These services may include therapeutic activities, health monitoring, socialization, medical care, transportation, and access to psychological and social resources necessary for the health and well-being of older adults. Daycare services also provide nutrition and health, social, and recreational activities for older adults during the day [11,12,13].

In Taiwan, daycare services are a critical component of community care for older adults. These services allow older adults to continue living in a familiar environment and provide support to reduce the burden of adult care and promote self-reliance. Daycare services in Taiwan offer a range of activities to meet the socialization needs of older adults, facilitate aging in place, and promote active aging. These services are primarily intended for older adults with mild to moderate disabilities or dementia and are designed to maintain and promote self-reliance, reduce social isolation, and delay functional decline. Research has shown that daycare services can improve the quality of life, social interaction, and interpersonal relationships of older adults, as well as reduce depression and stress and increase life satisfaction [14].

In recent years, there has been a significant increase in the number of community-based daycare centers for older adults in Taiwan. A variety of organizations, including non-governmental organizations, hospitals, private individuals, and corporate groups, have invested in the long-term care industry. In this competitive market, the quality of daycare management is essential for the performance, survival, social perception, and the rights and interests of care recipients. The operation and management of long-term care institutions for older adults will play a crucial role in society, and the quality of their management and performance will be a key factor in the survival of these institutions [15,16]. The operation and management of an organization are essential for its sustainability. The success of long-term care organizations is largely determined by their business philosophy, administrative system, and human resource management [15,17]. Other important factors include the availability of government resources, expansion and integration of care services, and the quality of care, as well as the utilization of institutional and community resources [16,18,19]. For example, the Mississippi Division of Medicaid in the United States has established quality assurance standards for daycare services, including standards for daycare administration, organizational operations, human resources, financial and service management, operations, and community resource planning. The evaluation system is a crucial element in the long-term sustainable development of long-term care institutions and the recognition of society. Research on the key success factors for institutional operations and evaluation has demonstrated that an evaluation system is an important tool for measuring the performance of institutions [20].

2.2. Fuzzy Analytic Hierarchy Process

In 1965, the Fuzzy Analytic Hierarchy Process (FAHP) was developed from a multi-criteria decision-making method combining Analytic Hierarchy Process (AHP) and fuzzy theory (Fuzzy Theory) [21]. FAHP is a multi-objective decision-making method comparable with the AHP method [7,22]. The method can deal with difficulties regarding quantification without deleting any unique opinions. Fuzzy AHP is widely used in solution selection and overall decision-making. The fuzzy theory solves the problems existing in traditional AHP, for example, the limitation of the application of ratio scale, the problem of correlation of decision attributes, and the problem of averages [10,23]. The Fuzzy Analytic Hierarchy Process (FAHP) was used as an operational evaluation system for daycare centers [24]. The FAHP and TOPSIS methods are integrated to establish an assessment model for the continuity of long-term care maintenance quality services in the community [25].

The main purpose of this research is to construct an evaluation model for the operation and management efficiency of daycare centers, according to the operation and management effectiveness of the daycare and evaluation benchmark according to the Ministry of Health and Welfare. A hierarchical architecture of the criteria is constructed. There are four main criteria including operation and administration (C1); organizational operation management (C2), external resource connection (C3); and employee rights and interests (C4).

Operation and administration (C1) play an important role in achieving the goal of sustainable operation for a daycare center. This includes evaluating the strategies, financial systems, and executive management of the firm. The financial stability of the organization and the efficiency of its leadership are crucial for the institution’s management quality. Good leadership is required to guarantee that the center functions efficiently and meets its objectives. Senior management should have a firm grasp of the center’s mission and values and effectively communicate them to personnel and stakeholders. Further, they should foster a healthy work environment that encourages employee development, open communication, and collaboration.

Organizational operation management (C2) involves addressing issues related to service quality and improvement. Effective organizational operation management is essential for sustainable operation, and this includes establishing a working manual, standardizing the administrative management process, and tracking and recording service quality conferences. The standardization of the administrative management process is another crucial tactic. Developing consistent procedures for operations such as budgeting, scheduling, and personnel administration is required. By standardizing these processes, businesses may eliminate errors, boost efficiency, and guarantee the effective use of resources.

External resource connection (C3) refers to the integration of resources from the external environment to support individuals in need. This can involve connecting different service resources to meet the needs of the service recipients, such as by establishing community resource links, creating essays and organization profiles, and disclosing detailed institutional service information online. Daycare centers can ensure that the needs of service users are fulfilled through a collaborative and comprehensive approach by integrating external resources, building community resource links, and releasing complete institutional service information online.

Employee rights and interests (C4) is an essential aspect of any firm, regardless of size, nature, or industry. Daycare institutions are obligated to protect the rights and interests of their workers. This is not only crucial for fostering a pleasant workplace culture, but in many jurisdictions, it is also a legal requirement. A thorough system of staff rights should be designed to clearly identify the employer’s and employees’ expectations, entitlements, and duties. This system should encompass a variety of elements, such as fair salaries and benefits, equitable opportunities, and safe working conditions. Protecting employee rights and interests is not only a moral imperative, but it also contributes to the organization’s success. Employees are more likely to be motivated, engaged, and productive if they believe they are treated fairly and with respect.

2.3. Technique for Order Preference by Similarity to an Ideal Solution

Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) is a multi-attribute decision-making method developed by [26]. It ranks all feasible solutions by measuring the distance between feasible solutions and ideal solutions to obtain the best solution. The foundation of the method is to find the best solution that is closest to the positive ideal solution and farthest away from the negative ideal solution. The positive ideal solution is to find the criterion value with the largest benefit or the smallest cost. On the other hand, the negative ideal solution is the one with the smallest benefit and the largest cost criterion among the alternatives.

2.4. Hybrid MCDMs Applications

Multiple Criteria Decision Making (MCDM) methodologies have achieved considerable acceptance in numerous industries due to their capacity to incorporate multiple criteria into a thorough analysis that supports decision-making processes. Among the many MCDM strategies described in the literature are Analytical Hierarchy Process (AHP); Technique for Ordering Preference by Similarity to an Ideal Solution (TOPSIS); Rough AHP; Complex Proportional Assessment (COPRAS); Multi-Attribute Border approximation Area Comparison (MABAC); and Ranking of Alternatives through Functional mapping of criterion sub-intervals into a Single Interval (RAFSI).

The integration of different MCDM methods provides a comprehensive decision-making model that considers the strengths of each technique and improves the accuracy and reliability of the results. Recent studies have shown the effectiveness of hybrid MCDM in various applications, such as location selection, sustainability performance measurement, hotel appraisal and selection, and railway station prioritizing. Table 1 presents some examples. Ref. [27] applied an integrated AHP–RAFSI approach to solve a location selection problem, taking into account characteristics such as accessibility, resource availability, and environmental impact. Ref. [28] emphasized the importance of the TOPSIS method to MADM in computing exponential divergence measures for Pythagorean fuzzy sets, illustrating the approach’s efficacy in uncertain decision-making procedures. Using the modified Rough AHP–MABAC technique, Ref. [29] created a multi-criteria evaluation framework for ranking Indian railway stations based on characteristics such as accessibility, cleanliness, and security. These studies illustrate the integration of many MCDM methods yields a holistic decision-making model that considers the benefits of each strategy. Improved decision-making processes have also been indicated by the creation of hybrid MCDM techniques, such as the combination of AHP and TOPSIS or Rough AHP and MABAC. Overall, the application of MCDM approaches continues to grow and provide valuable insights into complicated decision-making issues.

Access to health care equity in the health care system uses the Fuzzy Analysis Hierarchy Process (FAHP) to create a standardized weighted quality and optimal solution (TOPSIS) to rank cities in the province, and the results of the study can significantly help improve the management and overall performance of health care centers [30]. However, very few studies using hybrid MCDM models to assess performance in the health care sector, for example, Ref. [31] using fuzzy AHP and fuzzy TOPSIS method, choose the best health care insurance. Hence, to fill the research gap, this study contributes to the literature by combining fuzzy AHP and TOPSIS to evaluate the operation efficiency of daycare centers.

Table 1. Literature review of MCDM applications.

Year	References	Application Sectors	MCDM Models
2018	[29]	Multi criteria evaluation framework for prioritizing Indian railway stations	AHP–MABAC
2019	[32]	Application in web-based hotel evaluation and selection	COPRAS
2021	[33]	Sustainability performance measurement for Libyan Iron and Steel Company	Rough AHP
2021	[27]	Resolving a location selection problem	AHP–RAFSI
2022	[28]	Computing exponential divergence measures for Pythagorean fuzzy sets	Fuzzy TOPSIS
2022	[22]	Selecting optimal wind power plants	DEA and AHP

Table 1. Literature review of MCDM applications.

Year	References	Application Sectors	MCDM Models
2018	[29]	Multi criteria evaluation framework for prioritizing Indian railway stations	AHP–MABAC
2019	[32]	Application in web-based hotel evaluation and selection	COPRAS
2021	[33]	Sustainability performance measurement for Libyan Iron and Steel Company	Rough AHP
2021	[27]	Resolving a location selection problem	AHP–RAFSI
2022	[28]	Computing exponential divergence measures for Pythagorean fuzzy sets	Fuzzy TOPSIS
2022	[22]	Selecting optimal wind power plants	DEA and AHP

3. Materials and Methods

3.1. Research Design and Processes

The main purpose of this research is to construct an evaluation model for the operation and management efficiency of daycare centers. The FAHP and TOPSIS methods are used in a two-stage process. In the first stage, the fuzzy preference weights of the hierarchy were calculated using the matrix constructed by FAHP. The decision-making factors of the hierarchical structure of this study were based on the daycare evaluation structure of long-term care institutions in Article 39 of the Long-term Care Service Law passed by the government on 3 June 2017. A hierarchical structure of daycare institutions included a total of 4 main dimensions, 10 measurement levels, and 22 lane evaluation criteria. The questionnaires were reviewed and qualified by experts. In the second stage, TOPSIS was applied to select 10 daycare institutions for ranking. To make the research results reliable, the selected experts should have had professional knowledge in this research field and have had long-term academic research or more than 5 years of practical experience. The research process is shown in Figure 1.

3.1.1. Triangular Fuzzy Number

The fuzzy set theory is defined to deal with uncertain problems and used to identify the criteria for important and alternative performance [21] A triplet (a, b, c) is defined as a triangular fuzzy number (TFN) which denotes the lower, middle, and upper values, respectively, as shown in Equation (1) and Figure 2.

$$fD\left(x\right)=\left\{\begin{array}{c}\left(x-a\right)/\left(b-a\right) a\le x\le b\\ \left(x-c\right)/\left(b-c\right) b\le x\le c\\ 0 otherwise\end{array}\right.$$

(1)

where membership function $f_D$: R → [0,1].

3.1.2. Linguistic Values

The intervals of the linguistic variable were proposed by [34,35]. The criteria and the alternatives were rated by fuzzy numbers ranging from 1 to 9. In detail, the membership function of the linguistic scale of criteria and alternatives, shown in the following

Table 2. Linguistic scales of criteria’s rating for FAHP [35].

Linguistic Variable	Fuzzy Number	Triangular Fuzzy Scale			Reverse Triangular Fuzzy Number
Equal	$\tilde{1}$	1	1	1	1	1	1
Weak advantage	$\tilde{2}$	1	2	3	1/3	1/2	1
Not bad	$\tilde{3}$	2	3	4	1/4	1/3	1/2
Preferable	$\tilde{4}$	3	4	5	1/5	1/4	1/3
Good	$\tilde{5}$	4	5	6	1/6	1/5	1/4
Fairly good	$\tilde{6}$	5	6	7	1/7	1/6	1/5
Very good	$\tilde{7}$	6	7	8	1/8	1/7	1/6
Absolute	$\tilde{8}$	7	8	9	1/9	1/8	1/7
Perfect	$\tilde{9}$	8	9	10	0	1/9	1/8

and

Table 3. Linguistic scales of alternative’s rating for TOPSIS model [36].

Scale	Rating
Poor (P)	1
Medium Poor (MP)	3
Fair (F)	5
Medium Good (MG)	7
Good (G)	9
Intermediate values between the two adjacent judgments	2,4,6,8

, presents the nine-point scale for TOPSIS method [36].

The triplets, $\tilde{m}=(m_1,m_{2,}{m}_3)$ and $\tilde{n}=(n_1,n_{2,}{n}_3)$, are two triangular fuzzy numbers, and the distance between these numbers is calculated in Equation (2):

$$\mathrm{d}\left(\tilde{m},\tilde{n}\right)=\sqrt{\frac{1}{3}[{(m_1-n_1)}^2+{(m_2-n_2)}^2+{(m_3-n_3)}^2}]$$

(2)

3.2. Fuzzy Analytic Hierarchy Process (FAHP)

• Step 1: Pairwise comparison matrices of criteria

There are K experts (decision-makers). They determine the important dimensions by pair-wise comparison matrices among all the criteria of the hierarchical structure as Equation (3) matrix ($\widetilde{U^k}$):

$${\tilde{U}}^k=\left[\begin{array}{cccc}1& {\tilde{u}}_{12}^k& \cdots & {\tilde{u}}_{1n}^k\\ {\tilde{u}}_{21}^k& 1& \cdots & {\tilde{u}}_{2n}^k\\ ⋮& ⋮& \ddots & ⋮\\ {\tilde{u}}_{n1}^k& \cdots & \cdots & 1\end{array}\right]$$

(3)

where $\widetilde{u_{ij}^k}$ is the fuzzy comparison value by kth decision markers from ith to jth criterion.

• Step 2: Fuzzy geometric mean and fuzzy criteria weightage

Determining the fuzzy geometric mean and fuzzy weights of each criterion based on the geometric technique are shown in Equations (4) and (5).

$${\tilde{r}}_i={(\widetilde{u_{i1}}\otimes \cdots \otimes \widetilde{u_{ij}}\otimes \cdots \otimes \widetilde{u_{in}})}^{1/n}$$

(4)

$${\tilde{w}}_i={\tilde{r}}_i\otimes {\left[{\tilde{r}}_1\oplus \cdots \oplus {\tilde{r}}_i\oplus \cdots \oplus {\tilde{r}}_n\right]}^{-1}$$

(5)

where ${\tilde{u}}_{ij}=$ $\frac{{{\displaystyle \sum }}_{k=1}^K\widetilde{u_{ij}^k}}{K}$ is the integrated fuzzy comparison value by kth decision marker from ith to jth criterion, $\widetilde{r_i}$ is the fuzzy geometric mean of ith criterion, and $\widetilde{w_i}$ is the fuzzy weight of the ith criterion.

• Step 3: BNP value for rating weight

Calculating the best non-fuzzy performance (BNP) value [37] to analyze the rating weight of criteria is shown in Equation (6):

$$BN{P}_{wi}=\frac{\left[\left(U_{wi}-L_{wi}\right)+\left(M_{wi}-L_{wi}\right)\right]}{3}+L_{wi}$$

(6)

In which, $U_{wi},M_{wi},L_{wi}$ represent the upper, middle, and lower values, respectively, of the fuzzy weight of the ith criterion

3.3. Technique for Order Preference by Similarity to an Ideal Solution

• Step 1: Assuming that there are m alternatives and n evaluation criteria in a matrix D, Equation (7) shows that:

  $$D={\left[x_{ij}\right]}_{m\times n}=\left[\begin{array}{cccc}{x}_{11}& x_{12}& \cdots & x_{1n}\\ x_{21}& x_{22}& \cdots & x_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ x_{m1}& x_{m2}& \cdots & x_{mn}\end{array}\right]$$

  (7)

  where $x_{ij}$ represents the score of the $i$ alternative and $j$ criterion.
• Step 2: Normalization

Assuming that $r_{ij}$ is an element of the normalized decision matrix $R$, its formula and normalized decision matrix are shown in Equations (8) and (9), as follows:

$$r_{ij}=\frac{x_{ij}}{\sqrt{{{\displaystyle \sum }}_{i=1}^m{x}_{ij}^2}}i=1,2,3,\dots ,m,j=1,2,3,\dots ,n$$

(8)

$$R={\left[r_{ij}\right]}_{m\times n}=\left[\begin{array}{cccc}{r}_{11}& r_{12}& \cdots & r_{1n}\\ r_{21}& r_{22}& \cdots & r_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ r_{m1}& r_{m2}& \cdots & r_{mn}\end{array}\right]$$

(9)

• Step 3: Constructing a weighted normalized decision matrix

Let the matrix $V$ be the weighted normalized matrix in Equation (10) as follows:

$$V={\left[v_{ij}\right]}_{m\times n}=w_j\times r_{ij}=\left[\begin{array}{cccc}{w}_1{r}_{11}& w_2{r}_{12}& \cdots & w_n{r}_{1n}\\ w_1{r}_{21}& w_2{r}_{22}& \cdots & w_n{r}_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ w_1{r}_{m1}& w_2{r}_{m2}& \cdots & w_n{r}_{mn}\end{array}\right]$$

(10)

where $w=\left(w_1,w_2,w_3,\dots ,w_n\right)$ is the weights of decision criteria, ${{\displaystyle \sum }}_{j=1}^n{w}_j=1$.

• Step 4: Calculating the positive ideal solution ($A^{+}$) and the negative ideal solution ($A^{-}$)

  $$A^{+}=\left\{\left(\underset{i}{max}{v}_{ij}|j\in J\right),\left(\underset{i}{min}{v}_{ij}|j\in J^{\prime }\right)|i=1.2.3.\dots ,m\right\}=\left(v_1^{+},v_2^{+},v_3^{+},\dots ,v_j^{+},\dots ,v_n^{+}\right)$$

  (11)

  $$A^{-}=\left\{\left(\underset{i}{min}{v}_{ij}|j\in J\right),\left(\underset{i}{max}{v}_{ij}|j\in J^{\prime }\right)|i=1.2.3.\dots ,m\right\}=\left(v_1^{-},v_2^{-},v_3^{-},\dots ,v_j^{-},\dots ,v_n^{-}\right)$$

  (12)

  where J is the benefit criterion, which shows that the larger the index value, the higher the performance score of the index. J′ is the cost criterion, which shows that the smaller the index value, the higher the performance score of the index.
• Step 5: Calculating the distance between each alternative and the positive ideal solution and the negative ideal solution

Equations (13) and (14), respectively, present the distance between each alternative and the positive ideal solution ($S_i^{+}$) and the distance between each alternative and the negative ideal solution ($S_i^{-}$):

$$S_i^{+}=\sqrt{{\displaystyle \sum }_{j=1}^n{\left(v_{ij}-v_j^{+}\right)}^2},i=1,2,3,\dots ,m$$

(13)

$$S_i^{-}=\sqrt{{\displaystyle \sum }_{j=1}^n{\left(v_{ij}-v_j^{-}\right)}^2},i=1,2,3,\dots ,m$$

(14)

• Step 6: Calculating the relative closeness of each alternative to the ideal solution

The relative closeness of each alternative to the ideal solution is calculated in Equation (15) as follows:

$$C_i=\frac{S_i^{-}}{S_i^{+}+S_i^{-}}$$

(15)

where $0<C_i<1$; when the value of $C_i$ is closer to 1, it means that the solution is closer to the ideal solution.

• Step 7: Ranking of alternatives’ preference.

The optimal solution is closer to the positive ideal solution PIS $\left(S_i^{+}\right)$ and farther from negative ideal solution NIS ($S_i^{-})$ as $C_i$ approaches 1.

4. Case Application and Result Analysis

4.1. Expert Panel

In the first phase, an expert panel was assembled to evaluate the list of criteria, provide feedback to complete the hierarchical structure of criteria, and conduct pairwise comparisons. The panel of experts was selected with care based on their credentials and expertise in relevant domains. The information provided by the expert panel is briefly described in

Table 4. The expert panel’s background descriptions.

Category	Profile	No. of Respondents
Education level	Bachelor Degree	2
	Master Degree	5
	PhD Degree	3
Work experience	5–10 years	4
	10–15 years	4
	>15 years	2
Work Position	Doctor	3
	Manager	3
	Researcher	2
	Director	2
Work field	Hospitals	2
	Healthcare Institutions	2
	Daycare Centers	6

.

4.2. Criteria Description

In this study, a hierarchical structure was established based on the evaluation benchmarks of daycare institutions, and then the questionnaire was designed. There is a total of 22 evaluation criteria categorized into four main dimensions used in Table 3.

4.3. Results of Fuzzy AHP

In the analysis of FAHP, the fuzzy weights were calculated through the matrix constructed by the questionnaire, and then the fuzzy weights of 10 experts were integrated by the average method. The geometric mean of triangular fuzzy numbers was calculated, and then the fuzzy weights of each evaluation index were obtained. Then perform defuzzification was estimated for the solution fuzzy weights.

Table 5. Results of fuzzy weights from the FAHP model.

Main Criteria	Weights of Main	Sub-Criteria (Level 1)	Weights of Sub.Level1	Evaluation Criteria (Level 2)	Weights of Sub.Level2	Note
C1. Management and administration	0.3071	C1.1. Business plan formulation and implementation	0.0666	C1.1.1. Planning annual business plan	0.0172	SC1
				C1.1.2. Implementation and improvement of annual plan	0.0494	SC2
		C1.2. Financial management system	0.0971	C1.2.1. Financial statements	0.0697	SC3
				C1.2.2. Official receipt	0.0274	SC4
		C1.3. The head of the business is involved in the administration and care of quality management	0.1434	C1.3.1. Actual participation of business leaders	0.0863	SC5
				C1.3.2. Raise institutional issues	0.0571	SC6
C2. Organization operation management	0.3336	C2.1. Administrative operation and service quality management	0.2125	C2.1.1. Formulate a workbook	0.0220	SC7
				C2.1.2. Workbook content	0.0429	SC8
				C2.1.3. Administrative measures	0.0294	SC9
				C2.1.4. Service quality conference	0.0649	SC10
				C2.1.5. Meeting minutes tracking	0.0533	SC11
		C2.2. Competent authority supervision/inspection deficiencies are improved	0.1211	C2.2.1. Supervision by competent authorities	0.1211	SC12
C3. External resource connection	0.0972	C3.1. Social participation and community resource connection use	0.068	C3.1.1. Social participation and community resource connection use	0.0207	SC13
				C3.1.2. Community resource connection area	0.0473	SC14
		C3.2. Service information disclosure	0.0292	C3.2.1. Introduction or publicity of the institution	0.0101	SC15
				C3.2.2. Build a network platform	0.0191	SC16
C4. Employee rights and interests	0.2621	C4.1. Establishment and implementation of rights and interests’ system	0.1077	C4.1.1. Formulate a system of rights and interests	0.0364	SC17
				C4.1.2. Confirm system implementation status	0.0713	SC18
		C4.2. Employees undergo regular health checks	0.0536	C4.2.1. Health examination for new staff	0.0253	SC19
				C4.2.2. Health checks for service staff	0.0283	SC20
		C4.3. Pre-employment training for new recruits	0.1008	C4.3.1. Complete training within 3 months after employment	0.0657	SC21
				C4.3.2. Complete pre-employment training	0.0351	SC22

presents the global weight values and weights of sub-criteria.

According to the table, the primary evaluation criteria for long-term care institutions are management and administration, organization operation management, connectivity to external resources, and staff rights and interests. The criterion with the highest weight, 0.3336, is organization operation management, which comprises sub-criteria such as administrative operation and service quality management, competent authority monitoring, and inspection faults. With a weight of 0.3071, the design and implementation of the business plan is the second most significant factor, followed by employee rights and interests with a weight of 0.2621. It is worthwhile to note that the real participation of business executives is the most significant aspect of the organization’s operation and administrative management, followed by the raising of institutional issues. In addition, the adoption of the confirmation system for employee rights and interests is deemed essential for the human resource management of the complete long-term care facility.

In addition, the weights for external resource linkage and employee rights and interests are less than those of the other two criteria. Yet, they continue to contribute significantly to the overall performance of long-term care institutions. Among the sub-criteria of these criteria that should be prioritized are social participation, utilization of community resource connections, and health inspections for new and current employees.

Overall, the weights assigned to each sub-criteria in the table provide insightful guidance for long-term care institutions to concentrate their efforts on the areas that can have the greatest impact on their success. By recognizing the significance of each sub-criteria, institutions may manage their resources and devise plans to enhance their performance and provide better service to their citizens.

4.4. Results of TOPSIS

The fuzzy preference weight value of the criterion was obtained from the FAHP model. The TOPSIS method was used to rank the scores of the institutional evaluation. The weighted scores were normalized. Table 6 shows the positive ideal solution and negative ideal solution of 10 institutions calculated by transforming the matrix.

For example, Institution 1 obtained the distance of positive ideal solution (PIS), i.e., $S_1^{+}$= 0.145, and the distance of negative ideal solution (NIS), i.e., $S_1^{-}$= 0.003; thus, the relative closeness of Institution 1 to the ideal solution $C_1$ is calculated as below:

$$C_1=\frac{S_1^{-}}{S_1^{+}+S_1^{-}}=\frac{0.003}{0.145+0.003}=0.02$$

Table 6 displays the positive ideal solution (PIS) and the negative ideal solution (NIS) for ten institutions. The PIS reflects the highest value that an institution can attain based on all the factors analyzed, while the NIS represents the lowest value. It is evident that some institutions have a higher PIS than others, indicating that they perform better overall depending on the criteria being evaluated. For example, Institution 5 has the greatest PIS value at 0.0156, while Institution 3 has the lowest PIS value at 0.014. On the other hand, certain institutions have a higher NIS rating, suggesting that they perform poorly overall according to the criteria being evaluated. For example, Institution 3 has the highest NIS value, 0.161, while Institution 1 has the lowest, 0.003.

It is essential to note that the PIS and NIS values only provide relative comparisons between institutions based on the criteria being studied. Hence, they should be utilized in conjunction with other pertinent information, such as the exact criteria being used and the weights allocated to each criterion, in order to draw meaningful conclusions regarding the performance of each institution.

Table 6. The estimations of $S_i^{+} ; S_i^{-}; C_i$ on average and ranking.

Institution	$\mathbf{PIS}({\mathit{S}}_{\mathit{i}}^{+})$	$\mathbf{NIS}({\mathit{S}}_{\mathit{i}}^{-})$	${\mathit{C}}_{\mathit{i}}$	Ranking
Institution 1	0.145	0.003	0.020	10
Institution 2	0.038	0.142	0.789	2
Institution 3	0.014	0.161	0.920	1
Institution 4	0.048	0.09	0.652	3
Institution 5	0.156	0.013	0.077	9
Institution 6	0.063	0.131	0.675	4
Institution 7	0.131	0.039	0.229	8
Institution 8	0.079	0.101	0.561	6
Institution 9	0.081	0.098	0.547	7
Institution 10	0.071	0.102	0.590	5

Table 6. The estimations of $S_i^{+} ; S_i^{-}; C_i$ on average and ranking.

Institution	$\mathbf{PIS}({\mathit{S}}_{\mathit{i}}^{+})$	$\mathbf{NIS}({\mathit{S}}_{\mathit{i}}^{-})$	${\mathit{C}}_{\mathit{i}}$	Ranking
Institution 1	0.145	0.003	0.020	10
Institution 2	0.038	0.142	0.789	2
Institution 3	0.014	0.161	0.920	1
Institution 4	0.048	0.09	0.652	3
Institution 5	0.156	0.013	0.077	9
Institution 6	0.063	0.131	0.675	4
Institution 7	0.131	0.039	0.229	8
Institution 8	0.079	0.101	0.561	6
Institution 9	0.081	0.098	0.547	7
Institution 10	0.071	0.102	0.590	5

Because TOPSIS uses the method of relative approximation of the positive ideal solution to rank the priorities of each institution, it avoids the generation of one of the fractions that are closest to the positive ideal solution and the closest to the negative ideal solution or the farthest from a positive ideal solution and the closest to the negative ideal solution. Table 6 also shows the ranking of the institution’s performance. After calculating weights using the TOPSIS approach, the table ranks daycare institutions based on their results. Institution 3 achieved the highest score and ranked first, followed by Institution 2 and Institution 6. Instead, Institution 1 received the lowest score and was ranked last.

4.5. Sensitivity Analysis

To assess the stability of the proposed research framework and the final ranking, it is practical to do sensitivity analysis in MCDM. Due to a small difference in the relative weights of the criterion, numerous sensitivity studies are undertaken [38]. Some relative weights of the criteria are increased while others are decreased as shown in Equation (16) below:

$$w_j^{new}=w_j^{old}\pm \propto w_j^{old}$$

(16)

in which 0 < α < 1 is the percentage change of $w_j^{old}$ and the sum of all new weights ${\displaystyle \sum }_{j=1}^n{w}_j^{new}=1$. In this study, the weight of criteria was estimated by the FAHP model with linguist scale. Thus, the robustness test was conducted by assigning adjusted weights to the criteria for evaluating and selecting the most efficiency performing institution.

Table 7. Sensitive analysis.

		α = 0.1	α = 0.2	α = 0.3	α = 0.4	α = 0.5	α = 0.6	α = 0.7	α = 0.8	α = 0.9
	Original Ranking	Scenario 1 Ranking	Scenario 2 Ranking	Scenario 3 Ranking	Scenario 4 Ranking	Scenario 5 Ranking	Scenario 6 Ranking	Scenario 7 Ranking	Scenario 8 Ranking	Scenario 9 Ranking
Institution 1	10	10	10	10	10	10	10	10	5	6
Institution 2	2	2	2	2	2	2	2	2	6	7
Institution 3	1	1	1	1	1	1	1	1	3	4
Institution 4	3	3	3	3	3	3	3	4	8	8
Institution 5	9	9	9	9	9	9	9	9	4	3
Institution 6	4	4	5	4	4	4	4	5	2	5
Institution 7	8	5	8	8	5	5	8	8	1	1
Institution 8	6	6	4	6	6	6	6	7	10	9
Institution 9	7	7	7	7	7	7	7	6	9	10
Institution 10	5	8	6	5	8	8	5	3	7	2

shows that the ranking order remains consistent across most scenarios, even when we vary α from 0.1 to 0.9. The most efficient institutions are Institutions 3 and 2, while Institution 1 remains the least efficient. In Scenarios 8 and 9, however, the rankings change slightly, with Institution 7 performing best and Institutions 8 and 9 performing worst. This variation in ranks can be ascribed to the extremely variable values of α, particularly in situations where it is set to 0.8 and 0.9, which have a substantial impact on the significance of the primary criteria.

5. Discussions and Conclusions

The evaluation of daycare organizations is a crucial aspect of the development and determination of its core values, competition strategies, and directions in the operation process. However, the current evaluation index is still based on the same ratio as the distribution method, which does not effectively highlight the importance of each project index in the overall evaluation and cannot improve the operation and management efficiency of the organization. The goal of this research is to use FAHP and TOPSIS to create an evaluation model for the effectiveness of operation and management in daycare centers. The research methodology involves constructing a hierarchical structure with three layers, including four major criteria and 22 evaluation sub-criteria.

The proposed model can offer several advantages in decision-making problems. Firstly, such a methodology is an easy procedure and simple to operate. This is because AHP is one of the most prominent MCDM approaches and has numerous benefits [39]. Due to its hierarchical nature, AHP is scalable and can easily accommodate decision making problems of any size. TOPSIS is “a method for identifying an alternative that is closest to the optimal solution and farthest from the negative optimal solution in a multidimensional computing space” and it has a number of benefits [40]. Secondly, this integration has the ability to provide comprehensive ranking results and the suitability to be combined with stochastic analysis. Thirdly, using the weights objective data and smoothing tradeoffs by addressing non-linear relationships are other advantages to calculate the relative distances. However, using an expert panel is a drawback of this methodology. The outcomes may be considerably impacted by the evaluation of experts. This can be resolved in several ways. Initially, when employing the AHP, it is essential to employ a limited scale for the participant’s judgement to prevent loss of interest and distractions that could compromise the consistency of the decision. Second, a model of decision-making will be more effective if important criteria are appropriately created. Thirdly, the expertise level of the panel’s members should be carefully considered.

In this research, ten daycare and long-term care institutions were selected and evaluated using the TOPSIS method. The results of the evaluation showed that the weight values reflect the relative importance and performance of each evaluation index. An evaluation system for the efficiency of daycare center operations and management was developed using 22 evaluation sub-criteria and the distribution of evaluation index weight values. In the future, these weight values can be used to identify weaknesses in the organization and prioritize areas for improvement. The ranking of weight values can serve as a reference for evaluating management efficiency.

According to the research results, the weight analysis of all the experts in the main aspects places the highest emphasis on organization operation management. From the perspective of all the experts, improving the lack of inspection during the evaluation period through regular discussion is key to the sustainable operation of the organization. The second highest ranking criterion is business plan formulation and implementation. With good organizational operation, the person in charge of the business needs to further confirm whether the internal administrative management and operation can grasp the operator’s philosophy and attitude and move the organization towards the company’s goals. Only by advancing the management policy can long-term sustainable management be achieved. In addition, the first place in the ranking goes to the management of administrative operations and service quality. It is possible for staff to care for service items and interact with their families if a detailed operating manual and administrative management procedures are produced. This will have a positive effect on the organization. The participation of the manager in the operation and maintenance of the quality management system has the second-highest average weight ranking. Target selections can be chosen with greater precision if the institution’s leader is actively involved in the procedure. The establishment and execution of a framework for rights and interests is currently placed third in terms of the average weight it holds. Two variables that lead to high levels of job satisfaction among employees are a sense of self-actualization at work and the provision of adequate rights and interests by the organization. According to research, employee job satisfaction has a favorable effect on the achievement of the organization’s overall business objectives. Thus, this aspect must be carefully considered while assessing the viability of long-term operations.

When determining the success of a daycare organization’s administration, the oversight provided by the competent authorities is the single most important factor. This supervision serves to ensure the proper running of long-term care facilities as well as the quality of the services offered, allowing operators to make improvements to the facility’s overall operation. The participation of the organization’s leaders is the second most important factor. By actively participating in the daycare’s operations, business executives can obtain a deeper grasp of the organization’s current position and future direction. This knowledge enables them to create informed objectives and make better-informed decisions on the management and operations of the business. The implementation status of employee rights systems is the third most critical factor, according to the report (0.0713). This conclusion is supported by relevant research. By the adoption of an effective system, it is possible to motivate employees to be responsible and committed to the organization. A well-established system that protects employee rights can inspire workers to be more accountable and devoted to the firm. This incentive can result in improved organizational performance, increased employee job satisfaction, and decreased employee turnover rates.

The limitation of this study is that the relationships among sub-criteria are not concerned. If there is a correlation between criteria, the results may change. Further research can take it into account recent sophisticated MCDM models such as Decision-Making Trail and Evaluation Laboratory (DEMATEL) or Combined Compromise Solution (CoCoSo), etc.