Comparison of hospital service performances under COVID-19 pandemics for pilot regions with low vaccination rates.

It is essential to measure the quality and performance of health centers and propose policies in order for health services to continue without interruption during the pandemic period and for the continuous and proper implementation of new procedures in hospitals with COVID-19.The measurement of service quality and performance in hospitals should be provided not only for the smooth flow of health services that are vital for individuals but also for the elimination of hesitations in the treatment and vaccination processes related to COVID-19. Previously, models have been proposed by introducing some criteria to measure and evaluate hospital service performance in some extraordinary conditions, but such a study has not yet been put forward under pandemic conditions. Starting from this point, we aim to fill the gap in the literature by conducting a measurement study for hospitals in the pilot region, where COVID-19 cases are common but vaccination is observed at low rates. For this aim, the evaluation criteria are gathered under basic dimensions as in SERVPERF (Service Performance), which is a widely used tool for measuring service quality and a fuzzy multi-criteria decision analysis is proposed to measure the service performance of state hospitals for a pilot region. In the proposed methodology, the integrated methods consisting of CRITIC-TOPSIS have been extended with fermatean fuzzy sets. Expert opinions are taken via questionaries to determine hospital service performances. Based on the results obtained from the hospitals in the pilot region, the policies and strategies to be adopted by the hospitals serving under pandemic conditions worldwide to increase the service quality have been put forward. Additionally, the sensitivity of the parameters in the problem is measured, and then the validity of the obtained results is also validated. According to the results, assurance is determined as the most important main service performance factor during the pandemic period. So, the managers should develop strategies to address people's concerns about vaccines and increase people's trust in hospitals.

Introduction

Coronavirus disease (COVID-19), first detected in December 2019 in Wuhan, China, is an infectious disease (Khalilpourazari et al., 2021, Lotfi et al., 2022, Tirkolaee et al., 2022). With the COVID-19 pandemic affecting every aspect of people's lives, the burden on healthcare services has increased dramatically and medical care must be reorganized and restructured around the world. While the dynamics of health systems are trying to adapt quickly to the increasing demand due to the pandemic, they are also trying to maintain their basic services like in the other type of disasters (Baral, 2021, Ergün et al., 2021, Khalilpourazari and Pasandideh, 2021). After being declared a pandemic by the World Health Organization (WHO) due to the coronavirus, which is a very rapidly spreading pathogen, intensive care centers in hospitals exceeded their capacities, the workload of health workers increased and medical devices shortages began. For these reasons, there has been a noticeable change in the service quality of hospitals with the pandemic (Zolfani, Yazdani, Torkayesh, & Derakhti, 2020). However, regardless of the circumstances, health services must be kept at a certain level of quality due to their important role in human life (Behdioğlu et al., 2017). Hospitals, which are the most important pillars of the health sector, should do their best, considering the resources they have and the services they provide, as they are the main source of health services, especially during a pandemic like COVID-19 (Shirazi et al., 2020, Tirkolaee and Torkayesh, 2022, Torkayesh et al., 2021). Determining and improving the service quality and performance of hospitals under pandemic conditions is one of the most important actions in the fight against the virus. Hospitals with poor service quality and performance will have difficulty coping with the consequences of the COVID-19 pandemic (Gul & Yucesan, 2021). To overcome the difficulties encountered in the pandemic, some attributes have been defined to measure the quality and performance of health services; but since many healthcare standards are designed for typical situations, its use may not be appropriate in times of crisis such as the COVID-19 pandemic (Babroudi, Sabri-Laghaie, & Ghoushchi, 2021). Because, with many new applications such as vaccination against the virus, Polymerase Chain Reaction (PCR) test to determine whether the virus is in the body or treatment in separate areas for people who have caught the virus have caused changes in the service performance evaluation criteria. Therefore, in this paper, we investigate the attributes that can be used to evaluate and compare the service quality and performance of hospitals during the pandemic process. We gather the evaluation criteria under basic dimensions as in SERVPERF (Service Performance) tool, that is, we create a SERVPERF-based hierarchy, compile the criteria and conduct a case study to compare the service performances of hospitals on behalf of a selected pilot region under pandemic conditions. The multi-criteria decision-making (MCDM) approach, which is one of the most effective methods that can be used in the evaluation of many criteria encountered during the evaluation phase and that may conflict with each other, and in the comparison of more than one alternative according to all these criteria, has been utilized. MCDM is a well-known and frequently used approach, which allows predetermined alternatives to be evaluated in terms of conflicting criteria and to determine the most suitable one (Erdoğan, Kaya, Karaşan, & Çolak, 2021). Since the criteria used to evaluate service performance cannot be expressed numerically, the most reasonable way to apply is the fuzzy logic approach. By applying MCDM methods with fuzzy sets, linguistic evaluations are taken into account in the decision-making process and more precise and realistic results can be obtained while dealing with ambiguity and vagueness. For this reason, fuzzy sets are also used within the scope of this study to make the best use of the linguistic evaluations of the decision-makers in the context of MCDM.

Fuzzy logic is a useful approach for dealing with processes where nonlinear, complex, difficult to model, and information is uncertain or unclear. It uses approximate thinking instead of thinking based on exact values. Qualitative assessment and subjective judgment can be presented with fuzzy logic in the evaluation processes of complex decision-making problems. Information can be in the form of linguistic expressions (important, small, very little, etc.) in fuzzy logic. Linguistic expressions can be defined with different fuzzy sets to handle fuzziness and uncertainty in information (Wang, Peng, Zhang, Liu, & Chen, 2015). One of the main advantages of fuzzy sets over traditional sets is their ability to cope with ambiguity in linguistic variables (Kaya, Çolak, & Terzi, 2019). On top of that, traditional methods cannot handle qualitative parameters because of uncertainty in their nature; therefore, fuzzy logic should be employed (Minatour, Bonakdari, Zarghami, & Bakhshi, 2015).

Researchers and academicians broadly employed different fuzzy sets to handle fuzziness and uncertainty in decision-making problems. Zadeh presented fuzzy theory with only membership function (Zadeh, 1965). Atanassov proposed the idea of intuitionistic fuzzy (IF) sets, as an extension of traditional fuzzy sets (Atanassov, 1986). IF sets consist of both membership function and non-membership function, and their sum best be less than or equal to 1. Afterward, Yager developed the Pythagorean fuzzy (PF) sets to cope with the limitations of IF sets and model uncertainty, vagueness, and fuzziness in decision-making problems (Yager, 2013).

Nevertheless, PF sets are not suitable in some situations, when the square sum of the membership function and non-membership function is higher than 1 (Saraji, Streimikiene, & Kyriakopoulos, 2021). The aforementioned limitation motivated Senapati and Yager to introduce Fermatean Fuzzy (FF) sets into the literature to handle more fuzziness in the decision processes (Senapati & Yager, 2020). More specifically, the key characteristic of FF sets is the sum of cubes of membership function and non-membership function is less than or equal to 1 (Mishra, Rani, & Pandey, 2021). Thus, the FF sets are a more superior approach than traditional fuzzy sets, IF sets, and PF sets, and it makes FF sets more powerful to handle complex decision-making problems more effectively (Gül, 2021). Based on the aforementioned advantages of FF sets, this study adopts FF sets for the service performance evaluation. Furthermore, we propose a novel MCDM methodology using FF sets to evaluate the hospital service performances. Additionally, the indeterminacy of the decision-maker is considered in all stages of the proposed MCDM methodology, which enhances the accuracy of the methodology. Therefore, the proposed methodology can cope with complex decision-making problems effectively. In this study, we introduce a novel integrated MCDM methodology, which consists of Criteria Importance Through Intercriteria Correlation (CRITIC) and Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) under FF environment. The proposed integrated fuzzy methodology is constructed firstly to handle the problem of hospital service performance evaluation. Therefore, this study is unique because of both the proposed integrated fuzzy methodology and the application area under pandemic conditions.

The importance weights of the determined evaluation criteria to solve decision-making problems can be affected by the individual attitudes of the decision-makers (Koksalmis & Kabak, 2019). Therefore, the criteria weights determined subjectively in many studies in the literature make the reliability of the results debatable. The importance weights of the criteria are determined as a result of the judgments or evaluations of the decision-makers in subjective weighting methods. Unlike, the weights are determined as a result of the application of some mathematical models on the decision matrix, free from subjective judgments in the objective weighting methods. To overcome these issues, many objective weighting methods are introduced to the literature. These methods use existing data while objectively weighing the evaluation criteria. One of these methods mentioned above is the CRITIC method, which was developed and brought to the literature by Diakoulaki et al. (Diakoulaki, Mavrotas, & Papayannakis, 1995). The most important feature that distinguishes the CRITIC method from other methods is an objective weighting that uses the standard deviations of the criteria and the correlation between the criteria, not the subjective results obtained based on expert opinions. CRITIC method is also one of the most commonly used approaches to determine criterion weights objectively (Žižovic, Miljkovic, & Marinkovic, 2020). With this method, criteria weights are obtained from the conflict and intensity of this contrast, which forms the structure of the decision-making problem (Diakoulaki et al., 1995). In this context, correlation analysis is used to determine the contrast between criteria in the CRITIC method (Keshavarz Ghorabaee, Amiri, Zavadskas, & Antucheviciene, 2018). The CRITIC method, which was developed by considering the standard deviations of the values in the normalized decision matrix and the correlations between the criteria, is frequently preferred by researchers in recent studies (Gou and Xu, 2021, Mohamadghasemi et al., 2020, Rong et al., 2021, Singh, 2021). Criteria weights can be determined only with the help of the decision matrix, without resorting to evaluations that include subjective opinions with the CRITIC. For these reasons, CRITIC under the FF environment is adopted in this study to determine the weights of criteria.

Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) is one of the most used methodologies that an approach developed by Hwang and Yoon (Hwang & Yoon, 1981) to solve MCDM problems in many different areas (Akram et al., 2019, Biswas and Sarkar, 2019, Liu and Huang, 2020, Sarkar and Biswas, 2021). The methodology is used to determine the order of preference of alternatives and provides a more comprehensive comparison compared to other methods. The main idea behind the TOPSIS is based on determining the negative and positive ideal solutions (Ayyildiz & Taskin Gumus, 2021). In this method, the ordering of alternatives is determined by relative closeness to the ideal solution (Yildiz, Ayyildiz, Gumus, & Ozkan, 2020). The solution that minimizes the cost criterion and maximizes the benefit criterion is the positive ideal solution. The solution that maximizes the cost criterion and minimizes the benefit criterion is the negative ideal solution. In this context, as the decision point moves away from the negative ideal solution, it approaches the positive ideal solution (Wu, Lin, & Tsai, 2008). The TOPSIS method has advantages such as being understandable, simple calculation processes, allowing the best alternatives to be determined in the light of criteria, and including the importance of the criteria in the analysis (Shahroudi & Tonekaboni, 2012). In this study, TOPSIS is used to compare hospitals according to their service performances in the FF environment.

Following observations about hospital service performance have motivated this study. Hospital service performance evaluation are not considered for the COVID-19 vaccine process in the current literature. A detailed and comprehensive set of criteria to evaluate hospital service performance are not proposed in the service performance literature. Furthermore, the COVID-19 effect on hospital service performance with respect to current needs is not studied. Besides, the FF sets based performance evaluation model is not seen in the service performance literature before as far as we know. Hence, we propose SERVPERF based hospital performance evaluation model for the COVID-19 vaccine process for the first time in the literature. Although widely used in many studies, the traditional SERVPERF model which consists of five dimensions, is not suitable for analyzing up-to-date dimensions such as, environmental considerations, professional capabilities. Additionally, the COVID-19 pandemic has led to dramatic changes and great challenges, especially in the vaccine process. Therefore, analyzing both pandemic and other up-to-date dimensions is vital in terms of hospital service performance. In this study, we extend the SERVPERF model with three new dimensions: Professional capability, Environmental quality, and Pandemic conditions. By this way, a novel hospital service performance evaluation model is brought to the literature. In this study, we aim to determine the importance of service performance dimensions related to hospital service performance during the COVID-19 process, unlike the hospital service performance literature. Furthermore, we evaluate the service performance of different hospitals in the pilot region. FF sets based hybrid decision-making methodology consist of CRITIC and TOPSIS is presented to understand the effects of service performance dimensions and determine hospital service performances. This is the first study to use CRITIC and TOPSIS methods together under an FF environment. The aforementioned research gaps in the service performance literature are observed, and we aim to answer following research questions. (i) How the SERVPERF model can be improved to evaluate hospital service performance during the pandemic? (ii) Which service dimensions are effective to determine hospital service performance? (iii) Can fuzzy logic be used in hospital service performance evaluation? (iv) Can the TOPSIS method be extended with the CRITIC method under FF environment?

According to a comprehensive decision-making literature review based on fermatean fuzzy sets, there is no study that TOPSIS with CRITIC under fermatean fuzzy environment as far as we know. In this study, we present a novel hybrid decision-making solution methodology involving FF-CRITIC integrated FF-TOPSIS, which is developed to solve complex decision-making problem for the first time in the literature. In addition, using fermatean fuzzy logic for the first time to evaluate hospital service performance is one of the innovations of this study. Hence, this study stands out from the current literature in terms of both the application area and methodological novelties.

The rest of the paper is organized as follows. Section 2 presents the literature review results. Section 3 includes the proposed methodology in detail. Section 4 shows the real case analysis applied for the hospital in İstanbul, Turkey. While Section 5 includes the sensitivity analysis, Section 6 shows the validation analysis of the adopted methodology. Finally, Section 7 discusses the results and Section 8 gives the conclusion and future suggestions.

Literature review

Apart from many service quality measurement studies using the SERVPERF approach, an analytical and systematic way has been needed to determine the studies which we can emphasize the differences and which will form the basis of our study. For this purpose, the Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) approach is used while researching the relevant literature. PRISMA is a systematic approach developed by David Moher to review the literature (Moher, Liberati, Tetzlaff, Altman, & Group, 2009). The PRISMA approach has five steps in literature search: defining criteria, identifying sources, selecting literature, collecting data, and selecting data items (Santi & Putra, 2018). PRISMA approach is adopted as Systematic Literature Review Method (SLR) to minimize bias in the reviews and realize the reviews more systematic (Satria, Sensuse, & Noprisson, 2017). Studies using the SERVPERF approach, which is the method we adopt in our study, and aiming to measure service performance with this approach, are investigated. Thus, the use of fuzzy logic with the SERVPERF approach in decision-making studies is investigated, and studies that adopt fuzzy logic and the SERVPERF approach on the hospital basis in determining service performance are also examined in detail. Through studies that adopt the SERVPERF approach in determining hospital service quality and performance, criteria that can be used as evaluation criteria within the scope of this study have been determined. The literature search was performed from 20th October 2020 to 10rd February 2022 with the keywords used shown in Table 1 .

Table 1

Studies found in the literature review.

Database	Details of the Search	Number of studies
SCOPUS	(TITLE-ABS-KEY (SERVPERF) AND TITLE-ABS-KEY (“decision making”))	14
	(TITLE-ABS-KEY (SERVPERF) AND TITLE-ABS-KEY (mcdm))	6
	(TITLE-ABS-KEY (SERVPERF) AND TITLE-ABS-KEY (COVID-19))	4
	( TITLE-ABS-KEY ( SERVPERF) AND TITLE-ABS-KEY ( hospital) )	20
	( TITLE-ABS-KEY ( SERVPERF) AND TITLE-ABS-KEY ( fuzzy ) )	10
	( TITLE-ABS-KEY ( SERVPERF) AND TITLE-ABS-KEY (healthcare ) )	9

A total of 62 papers are encountered as a result of the search in the Scopus database with the keywords shown in Table 1. By specifying both inclusion and exclusion criteria after the search, papers directly related to the study are revealed. It is planned to reach similar studies that can form a background for the study and compare the inferences. Table 2 shows the inclusion and exclusion criteria for the papers encountered.

Table 2

Inclusion and exclusion criteria in the literature review.

Inclusion Criteria	Exclusion Criteria
The studies include SERVPERF implementation in multi-criteria decision-making analysis	Studies whose full text could not be reached
The studies include SERVPERF implementation in evaluating hospital performance	Studies that do not explicitly mention the method used and the results
The studies include SERVPERF implementation in evaluating hospital quality	Studies that are written in languages other than English
The studies include SERVPERF implementation for COVID-19 researches	Studies published before 2017

When the studies found as a result of the literature search are filtered with the criteria in Table 2, it is seen that 26 studies are similar to our paper that published after 2017. To present the differences between these studies with our paper and to explain their contribution to the literature, Table 3 has been created. Besides, the application in these studies and the limitations of these studies as well as the adopted methodologies are summarized, and in this way, a detailed perspective that can be used in a glance at the relevant literature is presented.

Table 3

The summary of literature review.

#	Author	Year	Aim	Method adopted	Application Area
1	Dako et al.	2017	Measuring patients' perceptions of service quality at the point of care in a PET/CT center	SERVPERF	Healthcare
2	Yalley et al.	2017	Proposing a new approach combining SERVPERF and PAKSERV to measure service quality in Ghana	GhanaQual, SERVPERF and PAKSERV	Bank and hospital
3	Carraco et al.	2018	Assessing of CRM customer somplaints through the SERVPERF scale	SERVPERF and 2-tuple model	4G TELECOMMUNICATIONS SECTOR
4	Lim et al.	2018	Proposing amodel for the relationships between hospital service quality, patient satisfaction, hospital utilization, and hospital financial performance	Factor analysis, Structural equation modeling	Hospital
5	Arumugam et al.	2018	Comparing SERVPERF and SERVQUAL models of service quality measurement tools for the healthcare industry	Review	Healthcare
6	Pedraja-Rejas et al.	2019	Evaluating the quality perception of the service provided by hospitals and family health centers	SERVPERF	Hospitals and family health centers
7	Shafei et al.	2019	Developing a scale that health care providers can use to measure health care quality and determining the best scale	SERVQUAL, weighted SERVQUAL, SERVPERF, weighted SERVPERF,factor analysis and logistic regression analysis	Hospital
8	Mıhan et al.	2019	Examining the service quality of telecommunication companies	SERVQUAL and SERVPERF	Telecommunication
9	Zehmed and Jawab	2020	Assessing the relative quality of service at the level of bus routes	Fuzzy SERVPERF and DEA	Urban Bus Transport Service
10	Psomas et al.	2020	Determining the effect of service quality of citizen service centers on citizen satisfaction	SERVPERF and Multiple linear regression analysis	Greek citizen's service centers
11	Akdere et al.	2020	Measuring the hospital service quality in Turkey and to investigate the perceived service quality levels of the patients.	SERVPERF, Logistic regression	Public Hospital
12	Liu	2020	Creating a evaluation index system for customer satisfaction by analyzing the factors affecting the high speed rail express	LSQ method, SERVPERF	High-speed rail express
13	Firlej et al.	2020	Presenting the relationship between health assessment and satisfaction with medical services in individuals with osteoarthritis	SERVPERF and Factor analysis	Rehabilitation outpatient clinics
14	Giao et al.	2020	Identifying and measuring factors influencing outpatient satisfaction in private general hospitals in Ho Chi Minh City	SERVPERF, Cronbach’s alpha analysis, Exploratory Factor analysis, and Linear regression Analysis	Private Hospital
15	Meleddu et al.	2020	Investigating the tendency of patients to consume private health services	SERVQUAL, SERVPERF, factor analysis and a partial proportional ordered logit model	Public and private healthcare services
16	Cervilheri et al.	2020	Evaluating the perceptions of professional nurses about the quality of service in an accredited hospital.	SERVPERF and Pearson’s chi-square test	Hospital
17	Subiyakto et al.	2020	Exploring the impact of service quality on the overall satisfaction of outpatients with radiology facilities.	SERVPERF and Structural equation modeling	Public Hospital
18	Frazão et al.	2021	Examining the perception and importance of biosafety actions in supermarkets	SERVPERF, Kano Model	Supermarket
19	Dzisi et al.	2021	Evaluating service quality of the paratransit minibus taxis trotro in Ghana	Modified SERVPERF	Paratransit minibus taxis trotro
20	Alp et al.	2021	Assessing the quality of occupational safety and health services to employee health and workplace productivity	SERVPERF, AHP And Fuzzy AHP	Occupational safety and health services
21	Lucadamo et al.	2021	Exploring the factors affecting patient satisfaction	SERVPERF and Principal component logistic regression	Hospital
22	Carvalho and Medeiros	2021	Examining the assessment of tourists about the services of airlines in a developing country	SERVQUAL, SERVPERF, Cluster Analysis and Structural Equation Modeling	Airlines’ service in the city of Recife
23	Campoverde Aguirre et al.	2021	Proposing a hotel service quality perception model for short-term tourists in transit	SERVPERF and Confirmatory Factor Analysis	Survey in touristic area
24	Babroudi et al.	2021	Determining the importance of SERVPERF standard criteria for healthcare services during the pandemic period, and the importance of these criteria in the prevalence of infectious diseases	SERVPERF, Z-Number theory and Fuzzy Cognitive Maps	Hospital
25	Shammot	2021	Measuring the impact of healthcare quality on patient satisfaction in public and private hospitals in Jordan	SERVPERF	Private Hospital and Public Hospital
26	Monteiro et al.	2021	Promoting the production of a virtual event presenting the results of joint international projects developed by students under conditions of social distancing	SERVPERF	Project development

When the literature research outputs are examined, it is concluded that the SERVPERF approach is frequently used in service quality and performance studies. It has been observed that successful results have been obtained in different areas such as public transportation, touristic centers, airline services, or telecommunication companies. Besides, this approach is encountered in studies of determining and weighing the criteria that affect the service performance for the patients for the health centers.

In addition, many different methods such as Fuzzy Cognitive Maps, Principal Component Logistic Regression, Kano Model and Data Envelopment Analysis (DEA) are used together with SERVPERF in the measurement of quality performance. In addition to these, it can be seen in the literature table that MCDM approaches are utilized in many papers. Among the alternative methodologies, the reason why MCDM approaches are used more can be said to enable the problem to be solved in a systematic way by putting it in a hierarchical structure under the umbrella of SERVPERF. Among the alternative methodologies, the reason why MCDM approaches are used more can be said to enable the problem to be solved in a systematic way by putting it in a hierarchical structure under the umbrella of SERVPERF. In the presence of more than one evaluation criteria that conflict with each other, the problem of evaluating service performance with SERVPERF dimensions will be easily handled together with MCDM approaches that provide the “best/suitable” solution possible.

However, as a result of the detailed literature search, no study is found for the evaluation of the service performance by considering both the activities added during the COVID-19 process and the standard services of the hospitals. Additionally, it is observed that no decision analysis is performed using the FF sets, which we utilize to model the uncertainty in the best way. Based on these two points, it can be claimed that our study is novel in the literature in terms of both application area and the proposed methodology.

Fermatean fuzzy sets

In traditional fuzzy sets, only the membership function is used to determine the membership degree of an element to the set and the non-membership degree of element to the set can be calculated by . However, using only the membership function may be insufficient to represent the fuzziness. Therefore, Atanassov introduced the IF set theory by generalizing the fuzzy set theory (Atanassov, 2016). Besides the membership function, the non-membership function is used to define an element in IF sets. The degree of both two functions can take value in the [0,1] range. Atanassov added the third function called hesitancy to complete the sum of membership and non-membership degrees to 1.

An IF number in fixed set X can be presented asin Eq. (1). IF numbers are considered as the components of the IF sets for simplicity:

and define the degree of membership and non-membership of the element to .where;

The degree of hesitancy is calculated:

Yager (2013) introduced PF sets that derived from IF sets. The sum of membership and non-membership degrees can exceed 1 in PF sets, but the sum of their squares cannot exceed 1(Ilbahar et al., 2018, Karasan et al., 2018):

A PF number in fixed set X can be presented as (Ayyildiz, Erdogan, & Taskin Gumus, 2021):

and define the degree of membership and non-membership of the element to .where;

The degree of hesitancy is calculated:

Sometimes, traditional fuzzy sets, IF sets and PF sets can be insufficient in contradictory decision-making environments. Because their sum and/or quadratic sum of membership degree and non-membership degree cannot exceed 1. This situation may limit the decision maker's evaluation. On the other hand, there are no upper limits for the sum and quadratic sum in FF sets. Besides, the sum of cubes of membership degree and non-membership degree can be between [0,1]. In this way, FF sets provide more freedom and a general perspective to decision-makers. FF numbers can be used to convert evaluations of decision-makers into mathematical expressions. FF sets are drawn the attention of many academicians to solve MCDM problems. For example, Liu et al. develop Tomada de Decisão Interativa Multicritério (TODIM) and TOPSIS methodologies in the FF environment with new distance measures (Liu, Liu, & Wang, 2019). Akram et al. use FF-TOPSIS methodology to select the most efficient sanitizer to reduce COVID-19 spread (Akram, Shahzadi, & Ahmadini, 2020). Keshavarz-Ghorabaee et al. use FF-Weighted Aggregated Sum Product Assessment (FF-WASPAS) to solve the green construction supplier selection problem (Keshavarz-Ghorabaee, Amiri, Hashemi-Tabatabaei, Zavadskas, & Kaklauskas, 2020). Mishra et al. focus on third‑party reverse logistics provider selection problem. They use FF-Criteria Importance Through Inter-criteria Correlation (FF-CRITIC) to determine the weight of criteria and use FF-Evaluation based on Distance from Average Solution (FF-EDAS) to evaluate alternatives with respect to criteria (Mishra et al., 2021). Simic et al. develop FF- Combinative Distance-based Assessment (FF-CODAS) to evaluate different tax schemes for public transit investments (Simic, Gokasar, Deveci, & Isik, 2021). Gul extends Simple Additive Weighting (SAW), Additive Ratio Assessment (ARAS), and VIseKriterijumska Optimizacija I KOmpromisno Resenje (VIKOR) in the FF environment to solve complex decision-making problems (Gül, 2021).

Preliminaries of fermatean fuzzy sets

The concept of FF sets was developed as a generalization of IF sets and PF sets by Senapati and Yager (Senapati and Yager, 2019a, Senapati and Yager, 2020). These sets can be described as an innovative approach to represent unreliable, inexact and vague information under a fuzzy environment (Garg et al., 2020, Simic et al., 2021).

A FF number in fixed set X can be presented as.(Senapati & Yager, 2019a):

and define the degree of membership and non-membership of the element to .where;

The degree of hesitancy is calculated:

Multiplication by a scalarof a FF Number (Simic et al., 2021):

The powerof a FF Number (Simic et al., 2021):

The score function of a FF Number (Senapati & Yager, 2019a).

The positive score function of a FF Number (Keshavarz-Ghorabaee et al., 2020).

Summation and multiplication of two FF numbersandare given (Senapati & Yager, 2019b):

Summation:

Multiplication:

Proposed FF-CRITIC integrated FF-TOPSIS methodology

It cannot be expected that all of the alternatives included and evaluated in the decision process have the same characteristics. The criteria weights used in MCDM problems, in which more than one conflicting criterion and objectives are handled simultaneously, are an indicator of the importance levels of the criteria. Considering that the distinctive features of these alternatives, called criteria, have different priorities of importance, one of the most important steps in the decision process is the prioritization, that is, the weighting of the criteria in this process (Wang, Jing, & Zhang, 2009).

In many MCDM problems, the determination of the best decision depends on calculating the criteria weights, which play an important role in measuring the preference values of alternatives (Bozanic et al., 2021, Mukhametzyanov, 2021). Since criterion weights expressing the relative importance of criteria in MCDM problems are initial information for the evaluation of alternatives, criteria weights should be determined according to the appropriate method from subjective or objective weighting methods. Considering the fact that the criterion importance levels may differ, subjective or objective weighting methods are used in the literature while performing the weighting process in MCDM problems (Trinkūnienė et al. 2017). Tzeng et al. classified the weighting methods as objective and subjective (Tzeng, Chen, and Wang 1998), depending on whether the weights are calculated indirectly from the outputs or obtained directly from the decision-makers. In subjective weighting methods, the criteria weights are usually based on the opinions of the decision-makers, and thus the weights obtained in this way are subjective inputs in this type of analysis. In other words, these analyzes, which are made only according to the preferences of the decision-makers, are expressed as subjective weighting methods (AHP, SMART, SIMOS Weighting Method, SWING, BWM, FUCOM or Level Based Weight Assessment (LBWA)). Since the criteria weights obtained by these methods represent the subjective evaluation of the decision-maker, the analytical results or order of the alternatives depending on the weights may be affected by the decision-maker due to the knowledge and experience of the decision-maker in the relevant field (Ahn 2011). Decision-makers and practitioners cannot easily weight the criteria because it requires experience in practice and directly affects the analysis results.

MCDM methods are considered subjective in terms of taking into account the judgments of decision-makers, and objective analysis methods in terms of being based on a mathematical algorithm. To minimize the effects of subjectivity, some objective weighting methods have been developed in the literature. These methods are used to determine the importance levels of the criteria objectively in the decision process in MCDM problems The common point of these methods (Entropy Method, CRITIC, Average Weight, Standard Deviation, etc.) is to weight the criteria by using only the available data and using mathematical programming techniques, without resorting to the subjective judgments of the decision-makers. Besides, the importance of objective weights will be more useful when aiming for an unbiased ranking of alternatives (Diakoulaki et al., 1995). In the light of this information, the CRITIC objective weighting method, which does not need to consider any of the decision maker's preferences, is combined with the TOPSIS method, since the data structure is determined by calculating the performance criteria used to model real-life situations in the decision problem discussed in the study is sufficient.

Diakoulaki et al. (1995) proposed the CRITIC as an objective weight calculation method and this method considers the alternative evaluation matrix to elicit information involved in the predetermined evaluation criteria. The objectivity in the method is due to the systematic mathematical procedure applied.This method is also an objective weighting method as it is less prone to subjective changes by a decision-maker. The information eliciting from the alternative evaluation matrix has the ability to change both the best alternative and the order of preference of the alternatives (Kremantzis, Beullens, & Klein, 2022). The main difference of this method from others is that it can reflect the interaction between criteria and define the contradictory feature and contrast strength between the evaluation indicators (Zhu et al., 2021). The CRITIC is also a more straightforward approach and can apply complex decision matrix problems (Song, Niu, & Zheng, 2021). The method needs less computational effort (Yazdani et al., 2021). Because, pairwise comparisons between criteria are not required in the CRITIC method, as the initial decision matrix is used for assigning weights, unlike any other weight determination method. Simple standard deviation and contrast intensity of criteria form the basis of this method. While determining the criteria weights in the method, the conflicts between the criteria in the structure of the problem are taken into account in the correlation analysis and thus it can be said that this method is more scientific and reasonable (Yazdani et al., 2021).

TOPSIS was developed by Hwang and Yoon (1981) as an MCDM methodology and has been applied in many different application areas with great success over decades (Biscaia, Braghini Junior, & Colmenero, 2021). TOPSIS attempts to determine the finest alternative which has the maximum distance from the negative ideal solution and the minimum distance from the positive ideal solution (Hezer, Gelmez, & Özceylan, 2021). The ideal solution in this methodology tends to promote the beneficial criteria and limit the cost criteria, while the negative ideal solution is inverse to these rules (James, Vaidya, Sodawala, & Verma, 2021). TOPSIS is an informative and very useful method to select and/or rank multiple alternatives (Elibal and Özceylan, 2022, Yorulmaz et al., 2021). The method is very commonly used by academicians and practitioners due to its user-friendliness structure and advantages from other techniques (Bertolini, Esposito, & Romagnoli, 2020). The number of alternatives doesn’t affect the calculation process, which allows faster computation time (Jati, 2012). Furthermore, the numbers of alternatives and criteria are not limited in the TOPSIS, the method can handle high numbers of alternatives and criteria. The differentiation between the cost and benefit criteria is considered in this method (Rashidi & Cullinane, 2019). TOPSIS stands out with its very reliable structure (Bertolini et al., 2020). The method uses Euclidean distance to evaluate the distance to ideal positive and negative solutions. Other similar MCDM methods, such as VIKOR, identify the solution by directly comparing it to the best and worst solution available (Jati, 2012). Due to its low complexitly, robustness and reliability, TOPSIS is utilized different areas, such as manufacturing, engineering, marketing, and management. However, TOPSIS is often criticized for its inability to adequately handle imprecision and uncertainty (Chatterjee & Stević, 2019). In the current literature, a most common trend in MCDM practices is to combine two or more methods, each filling in the gaps of the other (Velasquez & Hester, 2013). Similarly, there is a need for a fuzzy extension of the proposed method, since human judgments are often based on uncertainty, subjectivity, and fuzziness (Bertolini et al., 2020). Therefore, in this study, we propose a fermatean fuzzy-based approach that uses TOPSIS and CRITIC methods together, due to its compatibility with the structure of the hospital performance evaluation problem.

In this study, we introduce a novel integrated MCDM methodology, which consists of CRITIC and TOPSIS under the FF environment. The steps of the proposed methodology are presented in this section. Firstly, the FF-CRITIC proposed by Diakoulaki et al. (1995) and extended for FF sets by Mishra et al. (2021) is employed to determine criteria weights objectively, then FF-TOPSIS introduced to literature by Hwang and Yoon (1981) and extended for FF sets by Senapati and Yager (2020) is utilized to evaluate and determine hospital service performance based on experts’ opinions. The flowchart of our suggested integrated fuzzy methodology can be shown in Fig. 1 .

Fig. 1

Flowchart of the methodology.

Firstly, to perform integrated MCDM methodology, experts are evaluated the alternatives via linguistic terms given in Table 4 (Saraji et al., 2021).

Table 4

Linguistic terms for evaluating alternatives.

Linguistic Terms	FF Numbers
	μ	v
EL-Extremely Low	0.1	0.9
VL-Very Low	0.1	0.75
L-Low	0.25	0.6
ML-Medium Low	0.4	0.5
M−Medium	0.5	0.4
MH-Medium High	0.6	0.3
H-High	0.7	0.2
VH-Very High	0.8	0.1
EH-Extremely High	0.9	0.1

Step 1: Experts are evaluated using linguistic terms given in Table 5 and weights for each expert are determined.

Table 5

Linguistic terms for evaluating experts.

Linguistic Term	FF NumbersFF Numbers
	μ	v
Absolutely Skilled-AS	0.95	0.10
Very Skilled-VS	0.75	0.30
More Skilled-MS	0.55	0.50
Skilled-S	0.30	0.75
Less Skilled-LS	0.10	0.95

Then, the following equation is used to determine the weight of each expert. Let be the number of experts, and be the corresponding FF number to determine the weight of expert based on the evaluations.

Step 2: Construct a decision matrix by taking opinions from experts about each criterion using linguistic terms given in Table 4. Let be the evaluation of alternative with respect to criterion by expert . Let and be the number of alternatives and criteria, respectively.

Step 3: Expert opinions are aggregated to construct the FF decision matrix.

Step 4: Positive score of each criterion is calculated.

Step 5: Maximum and minimum values of are determined for each criterion.

Step 6: The score matrix is normalized.

if is a beneficial criterion:

if is a cost criterion:

Step 7: Standard deviations of criteria are estimated.

where;

Step 8: Correlations between criteria are calculated.

Step 9: The quantity of information in each criterion is evaluated.

Step 10: The weights of criteria are calculated.

Step 11: FF Positive and FF Negative Ideal Solutions are calculated (FF-PIS and FF-NIS) for aggregated decision matrix using score function:

Step 12: The weighted Euclidean distances from FF-PIS and FF-NIS are calculated for each alternative:

Step 13: The closeness coefficient for each alternative is computed:

Step 14: The alternatives are ranked from the highest to the lowest.

Real case application

In Istanbul, Turkey's most populous city, the number of COVID-19 cases is at its highest since the beginning of the pandemic. Most of confirmed COVID-19 cases in Turkey is seen in İstanbul during the COVID-19 pandemic (Republic of Turkey Ministry of Health. (2021), 2021). In this context, the service performance level of the hospitals in this city is critical during the pandemic. In this city, which was built on the Anatolian and European sides, evaluations about cultural, economic, transportation and health are usually performed based on two separate sides. In this study, a case study is conducted based on the Anatolian side state hospitals to focus the hospital in detail.

Within the scope of the study, many criteria are taken into consideration for the selection of hospitals whose service performance will be evaluated. Hospitals with the highest number of COVID-19 patients and low vaccine density, as well as those with the busiest usual activities, are considered. Additionally, the level of knowledge of the experts about the hospitals is another criterion taken into account in determining the hospitals to be included in the real-case analysis. There are more than 20 state hospitals on the Anatolian side. In this study, eight hospitals in the Anatolian side of İstanbul are evaluated in the real case application. Because of privacy, hospital names are abbreviated as H-1, H-2,…, H-8. Fig. 2 shows the state hospitals in the Anatolian side of İstanbul.

Fig. 2

State hospitals in the Anatolian side of İstanbul.

To determine the evaluation criteria, the literature is searched in detail and the opinions of experts are consulted. Many criteria previously used to evaluate hospital service quality and performance are examined in detail, and it is discussed with experts that criteria would be valid during the COVID-19 process. Additionally, it is questioned whether there are different criteria that the experts suggested adding in this process. In the end, the criteria to be used within the scope of this study are determined as follows by blending expert opinions and literature research.

Three experts with different experiences have been consulted to evaluate the performance of hospitals during the pandemic. One expert is an academician studying healthcare management, one expert is a hospital manager and the last one works as a hospital inspector. To determine the weight of experts, they are evaluated according to their experiences using Table 5. Three experts are evaluated as MS, S, and VS respectively. Then, the weights of experts (E-1, E-2, E-3) are determined by Eq. (16), which are given as 0.342, 0.199, and 0.459. Firstly, experts evaluate hospitals according to main criteria to determine the criteria weights by linguistic terms given in Table 4. Table 7 presents the hospital evaluation according to the main criteria by each expert.

Table 7

Main criteria evaluation.

Expert-1	H-1	H-2	H-3	H-4	H-5	H-6	H-7	H-8
Tangible	MH	M	M	MH	MH	M	VH	M
Reliability	M	M	M	MH	MH	ML	ML	M
Responsiveness	MH	MH	H	MH	MH	VH	H	H
Assurance	M	M	ML	MH	M	MH	MH	MH
Empathy	MH	MH	MH	MH	H	H	MH	H
Professional capability	MH	M	H	MH	H	ML	ML	M
Environmental quality	H	H	MH	H	H	H	H	H
Pandemic conditions	H	H	H	MH	H	M	MH	MH
Expert-2
Tangible	MH	MH	H	MH	MH	H	M	MH
Reliability	M	M	MH	ML	M	H	M	MH
Responsiveness	MH	H	H	M	MH	VH	H	H
Assurance	M	H	H	MH	MH	VH	M	H
Empathy	MH	M	M	L	M	H	L	M
Professional capability	MH	MH	H	MH	H	H	MH	H
Environmental quality	H	H	H	M	H	VH	MH	H
Pandemic conditions	H	M	MH	ML	ML	MH	ML	M
Expert-3
Tangible	M	M	MH	M	MH	ML	H	M
Reliability	ML	M	M	MH	MH	M	ML	M
Responsiveness	MH	MH	MH	MH	MH	H	H	H
Assurance	MH	ML	ML	M	MH	M	M	M
Empathy	M	M	MH	MH	H	M	MH	MH
Professional capability	MH	MH	M	M	MH	ML	M	M
Environmental quality	M	MH	M	H	H	MH	H	MH
Pandemic conditions	MH	MH	M	MH	MH	MH	H	MH

Expert evaluations are converted to FF numbers and then aggregated based on experts’ weights by Equation (18), as given in Table 8 .

Table 8

Aggregated main criteria evaluation matrix.

	H-1		H-2		H-3		H-4
	μ	v	μ	v	μ	v	μ	v
Tangible	0.560	0.342	0.524	0.378	0.598	0.305	0.560	0.342
Reliability	0.460	0.443	0.500	0.400	0.524	0.378	0.573	0.332
Responsiveness	0.600	0.300	0.624	0.277	0.660	0.241	0.584	0.318
Assurance	0.552	0.351	0.528	0.386	0.504	0.417	0.560	0.342
Empathy	0.560	0.342	0.540	0.362	0.584	0.318	0.564	0.344
Professional capability	0.600	0.300	0.571	0.331	0.631	0.275	0.560	0.342
Environmental quality	0.631	0.275	0.660	0.241	0.589	0.316	0.673	0.230
Pandemic conditions	0.660	0.241	0.626	0.276	0.607	0.298	0.573	0.332
	H-5		H-6		H-7		H-8
	μ	v	μ	v	μ	v	μ	v
Tangible	0.600	0.300	0.528	0.386	0.719	0.181	0.524	0.378
Reliability	0.584	0.318	0.536	0.376	0.424	0.478	0.524	0.378
Responsiveness	0.600	0.300	0.761	0.137	0.700	0.200	0.700	0.200
Assurance	0.571	0.331	0.630	0.275	0.540	0.362	0.589	0.316
Empathy	0.673	0.230	0.631	0.275	0.564	0.344	0.626	0.276
Professional capability	0.660	0.241	0.504	0.417	0.499	0.408	0.558	0.348
Environmental quality	0.700	0.200	0.690	0.210	0.684	0.217	0.660	0.241
Pandemic conditions	0.618	0.289	0.571	0.331	0.631	0.276	0.584	0.318

Steps of 4 and 10 in the proposed methodology are applied to determine main criteria weights as given in Table 9 .

Table 9

Main criteria weights.

Main Criterion	Weight
Tangible	0.108
Reliability	0.016
Responsiveness	0.032
Assurance	0.382
Empathy	0.117
Professional capability	0.098
Environmental quality	0.160
Pandemic conditions	0.086

When the criteria weights are examined, the most important criterion in the decision process is determined as “assurance” which refers to employees' knowledge and ability to develop confidence with patients (Babroudi et al., 2021), and the second important one is “environmental quality”. The least important criterion is found as “reliability”. Considering that confidence in vaccine and treatment approaches is often discussed during the pandemic, and people's hesitations about hospital visits are taken into account, it is quite reasonable that the most important criterion in the decision process is found as assurance. The level of success of the hospital in medical operations, the trust provided by the staff to the patient during the procedures, and the attitude of the hospital staff to the patients in pandemic cures will significantly affect hospital service performance and quality, especially during the pandemic process. Reliability-based attributes such as the punctuality of the personnel or the data privacy appeared as the least important factors in the evaluation process. Additionally, in a pandemic period when the intensive care units are working with above-normal capacity and people flock to the hospitals, it may be considered usual to detect this attribute in the last place. When the results obtained for the criteria weights are discussed with the decision-makers, they state that the results obtained are acceptable for the importance ranking of evaluation attributes.

After the criteria weights are determined, the alternative evaluation according to sub-criteria is performed for each main criterion by experts. Table 10 presents the sub-criteria evaluation matrix.

Table 10

Evaluation of experts for sub-criteria.

EXPERT-1	C11	C12	C13	C14	C15	C16	C21	C22	C23	C24	C25	C26	C31	C32	C33	C34	C35	C41	C42	C43	C44
H-1	ML	MH	M	H	ML	VH	M	MH	M	MH	M	M	MH	MH	M	ML	L	M	H	M	M
H-2	M	ML	M	H	ML	M	ML	MH	M	MH	ML	M	M	MH	M	VH	L	M	VH	M	M
H-3	M	H	M	H	M	H	M	H	H	VH	MH	M	MH	H	H	MH	H	H	M	MH	MH
H-4	MH	M	M	ML	L	VH	ML	ML	M	MH	ML	M	M	ML	M	L	ML	M	H	M	M
H-5	H	H	ML	ML	MH	VH	M	M	MH	H	MH	ML	MH	H	MH	ML	MH	MH	MH	MH	M
H-6	VH	MH	M	M	MH	MH	M	MH	M	MH	M	M	H	MH	M	H	H	M	VH	M	M
H-7	H	H	MH	VH	ML	H	H	M	VH	EH	H	MH	MH	M	MH	M	VH	VH	VH	H	ML
H-8	H	H	MH	VH	H	VH	H	H	H	VH	MH	M	H	H	H	M	VH	H	MH	MH	MH
	C51	C52	C53	C61	C62	C63	C64	C71	C72	C73	C74	C81	C82	C83	C84	C85	C86	C87	C88
H-1	MH	M	MH	M	M	ML	M	VH	VH	H	H	ML	M	M	M	M	M	H	H
H-2	MH	M	MH	M	ML	L	M	M	ML	L	ML	L	M	M	ML	M	ML	H	M
H-3	H	M	H	VH	H	EH	MH	VH	VH	H	M	VH	MH	MH	H	MH	MH	H	VH
H-4	MH	ML	MH	MH	M	MH	M	H	MH	ML	M	MH	M	M	MH	M	M	H	VH
H-5	H	MH	VH	H	M	VH	H	M	M	MH	MH	H	MH	M	M	H	H	H	VH
H-6	H	H	MH	ML	ML	VL	M	M	M	L	M	L	MH	M	ML	M	H	H	MH
H-7	MH	H	VH	MH	MH	H	H	H	MH	MH	H	VH	VH	M	MH	H	H	VH	VH
H-8	VH	H	H	M	M	M	MH	MH	M	M	MH	M	H	MH	M	MH	H	H	VH
EXPERT-2	C11	C12	C13	C14	C15	C16	C21	C22	C23	C24	C25	C26	C31	C32	C33	C34	C35	C41	C42	C43	C44
H-1	ML	MH	M	H	ML	VH	ML	H	M	H	H	VH	H	H	M	H	H	MH	M	M	MH
H-2	M	ML	M	H	ML	M	ML	L	ML	M	H	H	MH	MH	ML	MH	MH	M	L	ML	L
H-3	M	H	M	H	M	H	H	H	ML	H	VH	H	H	H	H	H	H	M	ML	M	MH
H-4	MH	M	M	ML	L	VH	M	H	H	M	VH	H	VH	H	ML	H	H	MH	M	MH	MH
H-5	H	H	ML	ML	MH	VH	ML	ML	M	L	MH	VH	M	MH	L	M	M	VL	VL	L	VL
H-6	VH	MH	M	M	MH	MH	M	M	M	M	H	H	H	H	M	MH	H	M	M	H	MH
H-7	H	H	MH	VH	ML	H	H	M	H	M	VH	VH	H	H	ML	H	H	M	ML	MH	MH
H-8	H	H	MH	VH	H	VH	ML	M	L	L	MH	H	M	H	M	M	MH	L	L	M	L
	C51	C52	C53	C61	C62	C63	C64	C71	C72	C73	C74	C81	C82	C83	C84	C85	C86	C87	C88
H-1	H	M	MH	H	MH	H	H	H	MH	M	L	MH	MH	M	H	H	MH	H	MH
H-2	H	ML	M	MH	H	VH	VH	H	H	VL	M	H	M	MH	MH	H	H	M	MH
H-3	VH	M	MH	VH	VH	H	H	VH	H	M	MH	VH	H	M	H	H	VH	MH	H
H-4	VH	M	M	H	VH	EH	VH	EH	VH	VL	L	H	MH	MH	VH	VH	VH	MH	H
H-5	H	ML	L	M	H	VH	MH	ML	L	VL	L	M	M	ML	H	MH	MH	H	L
H-6	VH	MH	MH	MH	H	H	VH	H	MH	MH	M	H	MH	MH	MH	H	VH	H	MH
H-7	VH	M	H	H	VH	VH	VH	H	H	M	MH	H	H	H	H	MH	MH	H	H
H-8	H	L	L	M	H	MH	H	M	MH	ML	L	M	M	M	MH	H	H	M	L
EXPERT-3	C11	C12	C13	C14	C15	C16	C21	C22	C23	C24	C25	C26	C31	C32	C33	C34	C35	C41	C42	C43	C44
H-1	M	H	ML	MH	L	MH	L	H	ML	H	ML	ML	H	VH	MH	L	M	ML	VH	ML	ML
H-2	ML	M	L	M	M	ML	M	M	MH	MH	L	L	ML	M	ML	H	ML	MH	H	MH	MH
H-3	MH	MH	M	M	H	MH	ML	MH	MH	H	H	MH	H	MH	MH	M	MH	MH	MH	M	M
H-4	H	ML	MH	L	MH	H	MH	L	L	ML	M	ML	MH	L	ML	ML	M	ML	MH	ML	ML
H-5	MH	VH	L	VL	H	MH	H	MH	H	MH	ML	L	ML	MH	H	MH	M	H	H	M	MH
H-6	H	M	MH	MH	M	ML	MH	H	ML	ML	H	MH	M	M	L	MH	MH	L	H	ML	MH
H-7	M	ML	H	H	L	M	MH	ML	H	H	MH	H	ML	ML	H	ML	MH	H	MH	VH	H
H-8	ML	M	ML	MH	VH	H	M	MH	VH	MH	ML	MH	M	MH	MH	L	M	MH	H	ML	H
	C51	C52	C53	C61	C62	C63	C64	C71	C72	C73	C74	C81	C82	C83	C84	C85	C86	C87	C88
H-1	H	ML	H	MH	ML	L	ML	H	H	MH	MH	L	ML	MH	ML	MH	ML	MH	MH
H-2	MH	MH	MH	ML	M	ML	MH	ML	L	ML	L	ML	MH	ML	M	ML	M	MH	ML
H-3	MH	MH	MH	H	MH	H	H	H	H	MH	MH	H	M	H	MH	H	H	MH	H
H-4	H	M	MH	M	ML	M	ML	MH	H	L	ML	H	MH	ML	H	ML	ML	MH	L
H-5	MH	H	H	MH	L	H	MH	ML	ML	H	H	MH	H	L	ML	MH	MH	H	H
H-6	MH	MH	MH	L	MH	EL	ML	M	MH	ML	MH	VL	MH	MH	L	ML	H	M	M
H-7	H	VH	H	H	M	MH	MH	MH	H	H	MH	H	H	ML	H	MH	MH	H	H
H-8	H	MH	MH	ML	H	ML	H	H	MH	MH	H	MH	MH	M	ML	H	MH	M	MH

The weight calculation process is repeated to determine the local weights of each sub-criterion. The local weights of each sub-criterion are multiplied with their related main criterion weight to find the final weight of the related sub-criterion. So, the final criteria weights are determined as given in Table 11 .

Table 11

Sub-criteria weights.

Sub-Criteria	Main Criteria Weight	Local Weight	Final Weight	Ranking	Sub-Criteria	Main Criteria Weight	Local Weight	Final Weight	Ranking
C11	C1: 0.108	0.079	0.0085	25	C51	C5: 0.117	0.433	0.0509	7
C12		0.0196	0.0021	36	C52		0.097	0.0115	23
C13		0.2612	0.0281	11	C53		0.4695	0.0551	5
C14		0.1603	0.0173	17	C61	C6: 0.098	0.0847	0.0083	26
C15		0.132	0.0142	21	C62		0.4187	0.0411	8
C16		0.3478	0.0375	9	C63		0.1762	0.0173	16
C21	C2: 0.016	0.1842	0.003	34	C64		0.3205	0.0315	10
C22		0.0421	0.0007	40	C71	C7: 0.16	0.0997	0.016	18
C23		0.0933	0.0015	38	C72		0.128	0.0205	13
C24		0.073	0.0012	39	C73		0.3427	0.0549	6
C25		0.2765	0.0045	31	C74		0.4296	0.0688	4
C26		0.3308	0.0053	29	C81	C8: 0.086	0.0923	0.0079	27
C31	C3: 0.032	0.2738	0.0088	24	C82		0.0369	0.0032	32
C32		0.0975	0.0031	33	C83		0.0307	0.0026	35
C33		0.0567	0.0018	37	C84		0.1672	0.0143	20
C34		0.4072	0.0131	22	C85		0.1767	0.0151	19
C35		0.1648	0.0053	30	C86		0.0724	0.0062	28
C41	C4: 0.382	0.0576	0.022	12	C87		0.2069	0.0177	15
C42		0.354	0.1354	2	C88		0.2169	0.0186	14
C43		0.395	0.1511	1
C44		0.1934	0.074	3

When the sub-criteria weights are examined, the most important criterion in the decision process is determined as “C41:Support of hospital management to employees” which refers to the supports provided by the hospital management to the hospital staff. These supports can be financial support, training and psychological counseling services that can help him cope with difficult situations such as pandemics. “C42:Politeness of employees” is determined as the second most important sub-criterion. Effective communication between the patient and the health personnel will both affect the patient's health and positively affect the motivation of the health personnel. All people who receive service expect attention. It is possible to say that relevant hospital staff has a great influence on the patient. Individuals who come to the hospital for service experience psychological timidity, fear and uneasiness when they first arrive at the hospital. They need attention to get rid of such negative thoughts. The interest that the hospital staff will show towards the sick individuals relaxes them and can relieve their uneasiness.

To evaluate hospital performance, expert opinions given in Table 9 are aggregated by Equation (18) as given in Table 12 .

Table 12

Aggregated expert opinions.

	C11		C12		C13		C14		C15		C16		C21		C22		C23		C24		C25		C26		C31		C32
	μ	v	μ	v	μ	v	μ	v	μ	v	μ	v	μ	v	μ	v	μ	v	μ	v	μ	v	μ	v	μ	v	μ	v
H-1	0.452	0.451	0.652	0.249	0.460	0.443	0.660	0.241	0.348	0.544	0.733	0.166	0.380	0.520	0.684	0.217	0.460	0.443	0.684	0.217	0.567	0.349	0.639	0.276	0.684	0.217	0.741	0.158
H-2	0.460	0.443	0.452	0.451	0.424	0.482	0.631	0.275	0.452	0.451	0.460	0.443	0.452	0.451	0.477	0.434	0.530	0.378	0.571	0.331	0.533	0.397	0.546	0.380	0.508	0.402	0.560	0.342
H-3	0.552	0.351	0.660	0.241	0.500	0.400	0.631	0.275	0.615	0.291	0.660	0.241	0.567	0.349	0.660	0.241	0.581	0.330	0.725	0.174	0.728	0.171	0.626	0.276	0.684	0.217	0.660	0.241
H-4	0.652	0.249	0.460	0.443	0.552	0.351	0.348	0.544	0.484	0.437	0.761	0.137	0.539	0.366	0.533	0.397	0.546	0.380	0.489	0.418	0.649	0.260	0.567	0.349	0.683	0.218	0.533	0.397
H-5	0.660	0.241	0.753	0.146	0.348	0.544	0.329	0.602	0.652	0.249	0.733	0.166	0.599	0.314	0.530	0.378	0.629	0.275	0.566	0.351	0.531	0.379	0.616	0.313	0.489	0.418	0.624	0.277
H-6	0.761	0.137	0.560	0.342	0.552	0.351	0.552	0.351	0.560	0.342	0.531	0.379	0.552	0.351	0.629	0.275	0.460	0.443	0.489	0.418	0.673	0.230	0.626	0.276	0.631	0.275	0.607	0.298
H-7	0.631	0.275	0.610	0.304	0.652	0.249	0.761	0.137	0.348	0.544	0.631	0.275	0.660	0.241	0.460	0.443	0.725	0.174	0.732	0.221	0.709	0.190	0.728	0.171	0.584	0.330	0.567	0.349
H-8	0.610	0.304	0.631	0.275	0.531	0.379	0.733	0.166	0.753	0.146	0.761	0.137	0.536	0.376	0.598	0.305	0.699	0.212	0.612	0.306	0.531	0.379	0.626	0.276	0.558	0.348	0.660	0.241
	C33		C34		C35		C41		C42		C43		C44		C51		C52		C53		C61		C62		C63		C64
H-1	0.552	0.351	0.533	0.397	0.573	0.342	0.508	0.402	0.716	0.185	0.460	0.443	0.508	0.402	0.684	0.217	0.460	0.443	0.652	0.249	0.626	0.276	0.508	0.402	0.533	0.397	0.567	0.349
H-2	0.424	0.478	0.699	0.200	0.479	0.435	0.552	0.351	0.661	0.254	0.530	0.378	0.511	0.403	0.640	0.261	0.530	0.378	0.571	0.331	0.508	0.402	0.581	0.330	0.624	0.299	0.683	0.218
H-3	0.660	0.241	0.607	0.298	0.660	0.241	0.598	0.305	0.530	0.378	0.524	0.378	0.560	0.342	0.709	0.190	0.552	0.351	0.624	0.277	0.761	0.137	0.709	0.190	0.766	0.174	0.684	0.217
H-4	0.424	0.478	0.546	0.379	0.581	0.330	0.508	0.402	0.598	0.305	0.508	0.402	0.508	0.402	0.728	0.171	0.484	0.418	0.571	0.331	0.607	0.298	0.639	0.276	0.753	0.235	0.639	0.276
H-5	0.602	0.316	0.539	0.366	0.524	0.378	0.599	0.341	0.599	0.341	0.477	0.434	0.506	0.435	0.660	0.241	0.614	0.297	0.661	0.254	0.598	0.305	0.546	0.380	0.761	0.137	0.624	0.277
H-6	0.424	0.482	0.624	0.277	0.660	0.241	0.424	0.482	0.682	0.221	0.567	0.349	0.584	0.318	0.709	0.190	0.624	0.277	0.600	0.300	0.462	0.456	0.618	0.289	0.512	0.519	0.639	0.276
H-7	0.614	0.297	0.567	0.349	0.690	0.210	0.682	0.221	0.623	0.287	0.731	0.167	0.631	0.276	0.728	0.171	0.716	0.185	0.725	0.174	0.684	0.217	0.669	0.235	0.709	0.190	0.709	0.190
H-8	0.598	0.305	0.424	0.482	0.630	0.275	0.566	0.351	0.602	0.316	0.489	0.418	0.602	0.316	0.725	0.174	0.566	0.351	0.566	0.351	0.460	0.443	0.673	0.230	0.508	0.402	0.684	0.217
	C71		C72		C73		C74		C81		C82		C83		C84		C85		C86		C87		C88
H-1	0.725	0.174	0.699	0.200	0.598	0.305	0.566	0.351	0.462	0.456	0.508	0.402	0.552	0.351	0.567	0.349	0.626	0.276	0.508	0.402	0.660	0.241	0.624	0.277
H-2	0.567	0.349	0.533	0.397	0.322	0.596	0.400	0.504	0.546	0.379	0.552	0.351	0.508	0.402	0.526	0.379	0.567	0.349	0.581	0.330	0.598	0.305	0.508	0.402
H-3	0.761	0.137	0.725	0.174	0.598	0.305	0.584	0.318	0.761	0.137	0.607	0.298	0.629	0.275	0.660	0.241	0.684	0.217	0.728	0.171	0.624	0.277	0.725	0.174
H-4	0.780	0.190	0.728	0.171	0.274	0.625	0.392	0.509	0.684	0.217	0.584	0.318	0.508	0.402	0.728	0.171	0.639	0.276	0.639	0.276	0.624	0.277	0.634	0.288
H-5	0.424	0.478	0.392	0.509	0.599	0.341	0.602	0.316	0.598	0.305	0.629	0.275	0.380	0.520	0.567	0.349	0.624	0.277	0.624	0.277	0.700	0.200	0.661	0.254
H-6	0.591	0.315	0.584	0.318	0.479	0.435	0.552	0.351	0.516	0.456	0.600	0.300	0.584	0.318	0.462	0.456	0.567	0.349	0.741	0.158	0.631	0.275	0.560	0.342
H-7	0.660	0.241	0.684	0.217	0.629	0.275	0.624	0.277	0.725	0.174	0.725	0.174	0.567	0.349	0.684	0.217	0.624	0.277	0.624	0.277	0.725	0.174	0.725	0.174
H-8	0.629	0.275	0.584	0.318	0.530	0.378	0.602	0.316	0.552	0.351	0.598	0.305	0.524	0.378	0.508	0.402	0.684	0.217	0.660	0.241	0.558	0.348	0.612	0.306

Then, FF-PIS and FF-NIS are determined based on positive score values for each sub-criterion by Equations (29), (30). Table 13 and Table 14 show the FF-PIS and FF-NIS for each sub-criterion, respectively.

Table 13

Positive ideal solutions.

	μ	v	π		μ	v	π		μ	v	π		μ	v	π
C11	0.761	0.137	0.823	C25	0.728	0.171	0.848	C44	0.631	0.276	0.900	C73	0.629	0.275	0.900
C12	0.753	0.146	0.829	C26	0.728	0.171	0.848	C51	0.728	0.171	0.848	C74	0.624	0.277	0.903
C13	0.652	0.249	0.891	C31	0.684	0.217	0.875	C52	0.716	0.185	0.855	C81	0.761	0.137	0.823
C14	0.761	0.137	0.823	C32	0.741	0.158	0.839	C53	0.725	0.174	0.850	C82	0.725	0.174	0.850
C15	0.753	0.146	0.829	C33	0.660	0.241	0.887	C61	0.761	0.137	0.823	C83	0.629	0.275	0.900
C16	0.761	0.137	0.823	C34	0.699	0.200	0.866	C62	0.709	0.190	0.860	C84	0.728	0.171	0.848
C21	0.660	0.241	0.887	C35	0.690	0.210	0.872	C63	0.766	0.137	0.818	C85	0.684	0.217	0.875
C22	0.684	0.217	0.875	C41	0.682	0.221	0.876	C64	0.709	0.190	0.860	C86	0.741	0.158	0.838
C23	0.725	0.174	0.850	C42	0.716	0.185	0.855	C71	0.780	0.137	0.806	C87	0.725	0.174	0.850
C24	0.732	0.221	0.842	C43	0.731	0.167	0.845	C72	0.728	0.171	0.848	C88	0.725	0.174	0.850

Table 14

Negative ideal solutions.

	μ	v	π		μ	v	π		μ	v	π		μ	v	π
C11	0.452	0.451	0.934	C25	0.531	0.397	0.924	C44	0.506	0.435	0.924	C73	0.274	0.625	0.903
C12	0.452	0.451	0.934	C26	0.546	0.380	0.921	C51	0.640	0.261	0.896	C74	0.392	0.509	0.931
C13	0.348	0.544	0.927	C31	0.489	0.418	0.932	C52	0.460	0.443	0.934	C81	0.462	0.456	0.931
C14	0.329	0.602	0.907	C32	0.533	0.397	0.923	C53	0.566	0.351	0.919	C82	0.508	0.402	0.930
C15	0.348	0.544	0.927	C33	0.424	0.482	0.933	C61	0.460	0.456	0.931	C83	0.380	0.520	0.930
C16	0.460	0.443	0.934	C34	0.424	0.482	0.933	C62	0.508	0.402	0.930	C84	0.462	0.456	0.931
C21	0.380	0.520	0.930	C35	0.479	0.435	0.931	C63	0.508	0.519	0.900	C85	0.567	0.349	0.919
C22	0.460	0.443	0.934	C41	0.424	0.482	0.933	C64	0.567	0.349	0.919	C86	0.508	0.402	0.930
C23	0.460	0.443	0.934	C42	0.530	0.378	0.927	C71	0.424	0.478	0.934	C87	0.558	0.348	0.922
C24	0.489	0.418	0.932	C43	0.460	0.443	0.934	C72	0.392	0.509	0.931	C88	0.508	0.402	0.930

Distances from FF-PIS and FF-NIS are calculated by Equations (32), (33). Lastly, FF-TOPSIS scores for hospitals are determined by Equation (34). The final scores and the rankings of hospitals are given in Table 15 .

Table 15

Final scores of hospitals.

	DS_i,S^-	DS_i,S^+	Score	Ranking
H-1	0.0937	0.1246	0.4292	4
H-2	0.0526	0.1661	0.2405	8
H-3	0.1075	0.1105	0.4931	2
H-4	0.0777	0.1401	0.3567	6
H-5	0.0754	0.1392	0.3516	7
H-6	0.0909	0.1288	0.4137	5
H-7	0.1685	0.0459	0.7858	1
H-8	0.0953	0.1236	0.4354	3

According to the final scores, the hospital with the highest service performance during the COVID-19 pandemic is the H-7 hospital. This hospital is followed by H-3 and H-8 hospitals, respectively. The hospital with the lowest service performance is H-2. H-7 hospital is located in a district with less population compared to other hospitals evaluated on the Anatolian side. Despite this, it is a hospital that also serves the surrounding districts and is located in an easily accessible area. Additionally, when the results are shared with the experts, they explicate that the hospital H-7 is successful in ordinary medical procedures and had a high vaccination rate. Therefore, it can be argued that the ranking obtained based on these facts is reasonable and consistent.

Sensitivity analysis

Sensitivity analysis is performed to evaluate the effectiveness of the proposed integrated fuzzy methodology due to the change in distance measures used in the hospital performance evaluation calculations. In the proposed methodology, Euclidean distance is used to calculate distance from the ideal solution. To evaluate the effect of different measures, Hamming distance is used in the sensitivity analysis. The Hamming distance of two FF numbers and is calculated as given in Equation (34).

In this way, Step 12 of the proposed methodology is changed with the Hamming distance as formulas given below.

Then, the Hamming distances from the FF-PIS and FF-NIS are calculated using Equation (35) and Equation (36). FF-TOPSIS scores for hospitals are determined by Equation (34) again. The final scores and ranking of hospitals according to Hamming distances are given in Table 16 .

Table 16

Final scores of hospitals according to Hamming distance.

	DS_i,S^-	DS_i,S^+	Score	Ranking
H-1	0.2578	0.3254	0.4420	5
H-2	0.2035	0.4133	0.3300	8
H-3	0.3047	0.3046	0.5001	2
H-4	0.2466	0.3547	0.4101	6
H-5	0.2401	0.3700	0.3935	7
H-6	0.2990	0.3540	0.4578	4
H-7	0.4219	0.1323	0.7613	1
H-8	0.2915	0.3295	0.4693	3

As shown in Table 16, the top three hospitals with the best performance are the same as the Euclidean distance. So, it can be said that H-7, H-8, and H-3 are determined as the best hospitals among the evaluated eight hospitals. Furthermore, H-7 is determined as the best hospital during the pandemic process, again. The comparison of the final ranking of hospitals for different distance measures is presented in Fig. 3 .

Fig. 3

Ranking of the hospitals for both Euclidean and Hamming distances.

As shown in Fig. 3, distance measures affect the results, therefore accurately determining distance measures to evaluate alternatives can provide more robust results. The performance scores and final ranking of the hospitals are changing, while the distance measure is changing. For example, the used distance measures affect the final ranking of H-6. The final ranking of H-6 becomes four while using the Hamming distance, while it is determined as the fifth-best hospital.

Validation analysis

A comparative analysis for the proposed integrated FF-based MCDM methodology for service performance evaluation of state hospitals is performed to evaluate the validity and reliability. For this purpose, one state of the art FF-based MCDM methodology, namely, FF-CODAS (Simic et al., 2021), and one state of the art TOPSIS methodology in the Pythagorean fuzzy environment, namely, PF-TOPSIS (Bakioglu and Atahan, 2021, Yildiz et al., 2020), are employed to determine the ranking of hospitals. The results and efficiency of the proposed integrated methodology are analyzed by comparing the results of FF-CODAS and PF-TOPSIS with the results of the proposed method.

FF-Codas

Steps 1–10: The same as in the proposed methodology.

Step 11: FF Negative Ideal Solutions are determined (FF-NIS) for each criterion based on aggregated decision matrix (Z) by Equation (30).

Step 12: The weighted Euclidean and weighted Hamming distances from FF-NIS are calculated for each hospital by Equation (32) and Equation (36), respectively. Table 17 presents the weighted distances for each hospital.

Table 17

Distances of hospitals.

	H-1	H-2	H-3	H-4	H-5	H-6	H-7	H-8
Weighted Euclidean	0.0937	0.0526	0.1075	0.0777	0.0754	0.0909	0.1685	0.0953
Weighted Hamming	0.2578	0.2035	0.3047	0.2466	0.2401	0.2990	0.4219	0.2915

Step 13: The relative assessment matrix is constructed by Equation (37).

is a threshold function to determine the equality of weighted Euclidean distance of two hospitals.

is the threshold parameter and it is set to 0.04 in this study as used in the literature (Simic et al., 2021). Then matrix is constructed based on Table 16 and Equation (37) as given in Table 18 .

Table 18

The relative assessment matrix.

	H-1	H-2	H-3	H-4	H-5	H-6	H-7	H-8
H-1	0	0.0953	−0.0138	0.0160	0.0182	0.0027	−0.2389	−0.0016
H-2	−0.0953	0	−0.1560	−0.0251	−0.0229	−0.0383	−0.3342	−0.1306
H-3	0.0138	0.1560	0	0.0298	0.0320	0.0165	−0.1782	0.0122
H-4	−0.0160	0.0251	−0.0298	0	0.0022	−0.0132	−0.2661	−0.0176
H-5	−0.0182	0.0229	−0.0320	−0.0022	0	−0.0155	−0.2748	−0.0198
H-6	−0.0027	0.0383	−0.0165	0.0132	0.0155	0	−0.2004	−0.0043
H-7	0.2389	0.3342	0.1782	0.2661	0.2748	0.2004	0	0.2036
H-8	0.0016	0.1306	−0.0122	0.0176	0.0198	0.0043	−0.2036	0

Step 14: Assessment scores are calculated using Equation (39) and hospitals are ranked based on assessment scores.

The assessment scores and ranking of hospitals are given in Table 19 .

Table 19

Ranking of hospitals by FF-CODAS.

	H-1	H-2	H-3	H-4	H-5	H-6	H-7	H-8
B_i	−0.1219	−0.8025	0.0821	−0.3153	−0.3398	−0.1570	1.6962	−0.0418
Ranking	4	8	2	6	7	5	1	3

According to FF-CODAS application results, the rankings of the hospitals with respect to the determined service quality criteria during the pandemic are H-7 > H-3 > H-8 > H-1 > H-6 > H-4 > H-5 > H-2, which is the same as the results of the proposed methodology.

PF-Topsis

Step 1: Construct a decision matrix by taking opinions from experts about each criterion using linguistic terms given in Table 19.

Step 2: Expert opinions are aggregated by the Pythagorean Fuzzy Weighted Averaging (PFWA) operator given in Equation (40). In this step, the same expert weights are used determined in Step 1 of the proposed methodology.

Step 3: PF-Positive and PF Negative Ideal Solutions are determined (PF-PIS and PF-NIS) for aggregated decision matrix using Equation (41), (42): where the score function of PF number is identified as follows:

Step 4: The weighted Euclidean distances from PF-PIS and PF-NIS are calculated for each alternative. In the PF-TOPSIS, the same criteria weights are used, which are determined by FF-CRITIC.

PF-PIS and PF-NIS are determined based on score values for each sub-criterion. Table 20 and Table 21 show the PF-PIS and PF-NIS for each sub-criterion, respectively.

Table 20

Positive ideal solutions for PF-TOPSIS application.

	μ	v	π		μ	v	π		μ	v	π		μ	v	π
C11	0.760	0.507	0.406	C25	0.726	0.558	0.403	C44	0.625	0.684	0.375	C73	0.354	0.685	0.637
C12	0.752	0.520	0.405	C26	0.726	0.558	0.403	C51	0.726	0.558	0.403	C74	0.354	0.687	0.635
C13	0.651	0.657	0.380	C31	0.683	0.617	0.391	C52	0.354	0.574	0.738	C81	0.429	0.507	0.747
C14	0.760	0.507	0.406	C32	0.738	0.538	0.406	C53	0.429	0.564	0.706	C82	0.429	0.564	0.706
C15	0.752	0.520	0.405	C33	0.659	0.648	0.382	C61	0.429	0.507	0.747	C83	0.291	0.685	0.668
C16	0.760	0.507	0.406	C34	0.697	0.598	0.397	C62	0.354	0.583	0.731	C84	0.354	0.558	0.751
C21	0.659	0.648	0.382	C35	0.687	0.609	0.396	C63	1.000	0.000	0.000	C85	0.354	0.620	0.700
C22	0.683	0.620	0.386	C41	0.676	0.622	0.394	C64	0.354	0.583	0.731	C86	0.354	0.540	0.764
C23	0.724	0.564	0.397	C42	0.710	0.574	0.408	C71	0.429	0.000	0.903	C87	0.429	0.564	0.706
C24	1.000	0.000	0.000	C43	0.728	0.551	0.407	C72	0.429	0.558	0.710	C88	0.429	0.564	0.706

Table 21

Negative ideal solutions for PF-TOPSIS application.

	μ	v	π		μ	v	π		μ	v	π		μ	v	π
C11	0.450	0.837	0.311	C25	0.505	0.786	0.356	C44	0.483	0.809	0.335	C73	0.113	0.926	0.359
C12	0.450	0.837	0.311	C26	0.521	0.773	0.361	C51	0.639	0.670	0.378	C74	0.185	0.872	0.453
C13	0.341	0.893	0.294	C31	0.484	0.812	0.326	C52	0.185	0.831	0.524	C81	0.113	0.833	0.542
C14	0.307	0.915	0.263	C32	0.505	0.786	0.356	C53	0.291	0.750	0.593	C82	0.236	0.798	0.554
C15	0.341	0.893	0.294	C33	0.411	0.856	0.314	C61	0.185	0.833	0.522	C83	0.236	0.878	0.417
C16	0.458	0.831	0.314	C34	0.411	0.853	0.321	C62	0.185	0.798	0.574	C84	0.185	0.833	0.522
C21	0.369	0.878	0.305	C35	0.467	0.821	0.330	C63	0.045	0.831	0.555	C85	0.236	0.753	0.614
C22	0.458	0.831	0.314	C41	0.411	0.853	0.321	C64	0.236	0.753	0.614	C86	0.185	0.798	0.574
C23	0.458	0.831	0.314	C42	0.524	0.779	0.343	C71	0.236	0.856	0.461	C87	0.354	0.755	0.551
C24	0.484	0.812	0.326	C43	0.458	0.831	0.314	C72	0.185	0.872	0.453	C88	0.236	0.798	0.554

Step 13: The closeness coefficient for each alternative is computed and the alternatives are ranked from the highest to the lowest:

Distances from PF-PIS and PF-NIS are calculated. Lastly, PF-TOPSIS scores for hospitals are determined by. The final scores and the rankings of hospitals are given in Table 22 .

Table 22

Final scores of hospitals.

	Dx_i,x^-	Dx_i,x^+	Score	Ranking
H-1	0.1457	0.1920	0.4315	5
H-2	0.0849	0.2547	0.2500	8
H-3	0.1856	0.1514	0.5508	2
H-4	0.1307	0.2215	0.3711	6
H-5	0.1188	0.2218	0.3488	7
H-6	0.1469	0.1885	0.4380	4
H-7	0.2645	0.0775	0.7735	1
H-8	0.1522	0.1844	0.4522	3

According to PF-TOPSIS application results, the rankings of the hospitals with respect to the determined service performance criteria during the pandemic are H-7 > H-3 > H-8 > H-6 > H-1 > H-4 > H-5 > H-2, which agrees with the results of the proposed methodology, except H-6 and H-1. The rankings of these two hospitals have changed among themselves. Namely, H-6 becomes the 4th best hospital in the PF-TOPSIS application, while it is determined as the 5th best hospital in the proposed methodology. The rankings of hospitals according to the three different methods are presented in Table 23 .

Table 23

Results of the comparative analysis.

	H-1	H-2	H-3	H-4	H-5	H-6	H-7	H-8
Proposed Methodology	4	8	2	6	7	5	1	3
FF-CODAS	4	8	2	6	7	5	1	3
PF-TOPSIS	5	8	2	6	7	4	1	3

As shown in Table 22, the results of the three different methods are almost the same, and H-7 is the best hospital based on its service performance with respect to the determined criteria during the pandemic process. Therefore, the proposed integrated methodology is determined to be effective in solving the MCDM problems with many conflicting criteria.

Discussion

To be successful in the health sector and to meet the expectations of the patients, being able to provide a good health service quality is the most important factor. Clarifying the attributes of quality and performance in health services is more tricky than in other services as customers and their lives are assessed (Tuzkaya, Sennaroglu, Kalender, & Mutlu, 2019). With the rapid spread of the pandemic worldwide, hospitals faced difficulties in their resources and capacities after many new occupations such as treating and testing infected patients, as well as vaccination. Along with the increasing workload of hospitals, the decrease in service quality and performance has also negatively affected the COVID-19 vaccine process. Apart from those who reject the vaccine outright, the unreliability of the hospital and its components also affects both the vaccine and the treatment process. In other words, it can be claimed that the decreases in the quality and performance of hospitals increase the hesitation of the COVID-19 vaccine. At this point, it is essential to increase the quality and performance of hospital services to control the pandemic and increase the confidence in the vaccine, and finally to ensure community immunity by getting more people vaccinated.

As of February 13, 2022, the rate of citizens who have had their first dose of vaccine in Turkey is 92.73%, the rate of the second dose vaccination across the country is 84.77%, and the rate of third dose only 42.13% (T.C. Sağlık Bakanlığı, 2021). The high vaccination rates across the country, unfortunately, could not be observed in big cities with a high number of COVID-19 cases. Especially in the city of Istanbul, which is the most populated city in the country, vaccination rates are below the country average on a regional basis. At this point, the regions on the Anatolian side of the city are leading with low vaccination rates. Table 24 shows the vaccination rates for the districts of Istanbul on the Anatolian side.

Table 24

Vaccination Rates for districts in the Anatolian side of İstanbul.

District	Vaccination Rate (%)	District	Vaccination Rate (%)
Kadıköy	85	Beykoz	75
Adalar	80	Tuzla	73
Maltepe	78	Çekmeköy	72
Kartal	76	Ümraniye	71
Şile	76	Pendik	71
Ataşehir	75	Sancaktepe	68
Üsküdar	75	Sultanbeyli	61

This study is conducted to investigate the fact that vaccination rates are so low in a big metropolis like Istanbul compared with the country average and to examine the relationship between this low vaccination rate and hospital quality and performance with case analysis. For this purpose, attributes affecting hospital service performance during the pandemic period are investigated and a detailed hierarchy of evaluation criteria is revealed. To put the criteria hierarchy on a systematic basis, the approaches used previously to examine service quality and performance are examined. At this point, the SERVPERF approach, which is successfully used in measuring the service performance of both hospitals and other different service institutions, is used. The evaluation criteria determined because of expert opinions and detailed literature research are set hierarchically under the dimensions of the SERVPERF approach, and the decision-making structure to be used in the solution of the problem is created.

The CRITIC method is adopted to determine the criteria weights by the objective way. Besides, a fuzzy logic-based approach has been used to concurrently evaluate the criteria that cannot be expressed numerically and that are in conflict with each other. As a result of fuzzy-based multi-criteria decision analysis, the “assurance” criterion, which is defined as the knowledge and courtesy of the employees and their ability to create a sense of trust in the customers, has emerged as the most important criterion in the decision process. Assurance criteria consist of the patients' level of feeling safe, politeness of employees, support of hospital management to employees, support service, and the success rate of operation. It is quite consistent to determine the assurance criterion as the most important criterion, as the confidence of hospitals during the pandemic period is lost. The conclusion that can be made at this point is that both hospital staff and management should focus on the sub-dimensions of assurance in improving service performance. Especially in improving hospital service performance, an internal environment where patients feel confident and safe should be procured by the staff; at this point, all the support of the hospital staff during this chaotic and tiring pandemic process must be provided by the management.

The second most important criterion is found as “environmental quality”. Managers aiming to improve hospital service performance should create a hospital environment suitable for citizens who reach the hospital to ensure easy access to the hospital for severe COVID-19 patients and to get vaccinated at any time during the day. At this point, the hospital management should adopt priority policies regarding efforts to increase the environmental quality by getting support from the municipality of the district where it is located. At this point, the hospital management should adopt policies regarding easy access and transport to the hospital, with the support of the municipality of the district where it is located.

The proposed SERVPERF embedded service performance evaluation model consists of eight different dimensions and their inner levels. Since the main and inner levels of performance evaluation dimension may include qualitative and quantitative information with conflicts, evaluating hospital performances using MCDM approaches enables meaningful analysis. Additionally, fuzzy logic can be applied to deal with the uncertainties inherent in the specified criteria. Along with the proposed methodology in this study, an integrated fuzzy decision-making model based on FF sets, which can transform into numerical expressions by modeling uncertain expressions in the expert opinions and data, is presented in the literature. This proposed integrated methodology consists of CRITIC and TOPSIS methods under a FF environment that can be used in different complex decision-making problems as used in the hospital service performance evaluation in this study.

In decision-making problems involving criteria in such a detailed hierarchical structure, it is inevitable to assume that the effects of the criteria on the alternatives are not equal. In this context, it is necessary to determine the criteria weights. While determining the criteria weights, methods based on expert opinions can be preferred. However, expert opinions can sometimes be subjective and based on individual characteristics. Therefore, in this study, the FF-CRITIC methodology is used and criteria weights are determined in an objective way. When the decision-making-based studies in the literature are reviewed, one of the most effective techniques for evaluating alternatives can be seen as distance-based evaluation. In this context, the service performances of the hospitals are determined by applying the TOPSIS method, which is calculated by considering both the best and the worst solutions, in a FF environment.

In theory, an integrated MCDM framework based on SERVPERF is proposed, which can be used as a generic yet comprehensive and rigorous decision-making tool to evaluate hospital service performance. Meanwhile, this study proposes a conceptual model by collecting 40 relevant sub-criteria and classifying them into eight SERVPERF-based service performance criteria. Therefore, the proposed framework can be considered a benchmarking approach to develop performance evaluation applications in various systems, modified if necessary. In other words, the performance level of the alternatives will be obtained by determining the main criteria, collecting the relevant sub-criteria, and applying the proposed integrated decision-making method. These main and sub-criteria can be changed and/or eliminated according to the objectives of the problem. This study has some theoretical implications. To be specific, (i) The proposed SERVPERF based hierarchy shown in Table 6 can be used to collect relevant criteria for performance evaluation in other industries; (ii) The FF membership function and the corresponding linguistic terms shown in Table 4, Table 5 can be used to consider uncertainty in other service performance evaluations; (iii) The proposed FF-CRITIC approach facilitates comprehensive weighting of criteria in a hierarchical structure and thus is used to obtain objective weighting by combining the judgments of experts with the objective characteristics of them; (iv) The TOPSIS method under FF environment is proposed to determine the service performance evaluation of public hospitals in Turkey, which would enhance the reliability of the performance evaluation scores since it overcomes the limitation of the traditional fuzzy sets in respect of ignoring the hesitancy in information.

Table 6

Evaluation criteria.

Criteria	Sub-criteria
C1: Tangible	C11: Up-to-date equipment ownership
	C12: Visual appeal of physical facilities
	C13: Employees dressing and cleaning
	C14: Alignment with the appearance of physical facilities and the type of service
	C15: Heating and air conditioning
	C16: Having laboratory, imaging and diagnostic equipment and facilities
C2: Reliability	C21: Employee scheduling ability
	C22: The understanding of the employees
	C23: Reliability of the employees
	C24: Employee punctuality
	C25: The precision of record-keeping
	C26: Privacy of patient information
C3:Responsiveness	C31: Employees' willingness to help patients
	C32: Caring attitude of medical staff
	C33: Low waiting time for services
	C34: Appropriate treatment costs
	C35: Providing adequate information regarding diseases, their treatments and consequences
	C36: Immediate treatment in case of emergency
C4: Assurance	C41: The patients' level of feeling safe
	C42: Politeness of employees
	C43: Support of hospital management to employees
	C44: Support service
	C45: The success rate of operation
C5: Empathy	C51: The level of care that patients expect
	C52: The level of observance of the patient's interests by the employees
	C53: Directing the care staff according to the interest of the patients
	C54: Understanding the specific needs of patients for staff
C6: Professionalcapability	C61: Number of nurses
	C62: Number of hospital attendants
	C63: Number of experienced doctors and specialists in different medical fields
	C64: Professional qualification
C7:Environmental quality	C71: Convenient access to the hospital
	C72: Transportation
	C73: Parking lot
	C74: Recreation and accommodation facilities
C8: Pandemic conditions	C81: Feasibility of placing patients in single rooms with adequate ventilation
	C82: Obeying distance rule between beds
	C83: Follow-up of equipment which is disposable
	C84: Possibility of assigning a healthcare team to look after cases to reduce the risk of contamination
	C85: Availability of adequate personal protective equipment
	C86: Ability to follow safe routine procedures and manage medical waste according to infection prevention and control (IPC) guidelines
	C87: Existence of a policy to monitor and manage personnel suspected or infected with COVID-19
	C88:Availability of necessary laboratory tests at all times

As for the limitations of the study, not being able to consider all the hospitals in the city could be our first drawback. In addition, the second most important limitation is the inability to find more experts with sufficient competence in the evaluation of criteria and alternatives. However, beyond all these limitations, the service performance comparison study for the important hospitals serving in Istanbul has been successfully carried out with a detailed and systematic hierarchy under the pandemic process.

Conclusion and future suggestions

With the impact of COVID-19 on the whole world, the demand for health services has increased and hospitals in many countries are trying to provide services beyond their capacity. It has been observed that service quality and performance in hospitals have decreased with increasing workload, insufficient medical equipment, and the addition of new practices such as vaccination. For these reasons, hospitals must try maintaining the quality of service under current conditions and challenges during the COVID-19 pandemic. At this point, the first step to be taken will be to determine the criteria that affect the hospital service and quality during the pandemic process. In this study, we focus on the service performance measurement of hospitals during the pandemic period and investigate which criteria should be considered in this process, and conduct a case study for hospitals in a particular region. To gather the evaluation criteria under certain headings and to create a decision hierarchy, the SERVPERF tool is applied. The multi-criteria decision-making (MCDM) approach is adopted to use in the evaluation process of many conflicting criteria in the most effective way. Since the evaluation of the criteria and scoring of the criteria according to the alternatives in the decision analysis stage cannot be expressed numerically, fuzzy logic is used. Extended versions of the ordinary fuzzy sets are used to best reflect the uncertainty and vagueness arising from the criteria that cannot be expressed numerically in the decision process. For this purpose, an integrated decision-making methodology based on FF sets is proposed to evaluate hospital service performance. The criteria weights are calculated by applying the FF CRITIC method and the FF TOPSIS approach is used to compare the alternatives. The main criterion with the highest weight among the evaluation attributes is calculated as “assurance” and the least important main criterion is “reliability”. In the real-case study, state hospitals located on the Anatolian side of Istanbul, Turkey's most populous city, are considered as alternatives. An expert group consisting of three people is consulted in weighting the criteria and evaluating the alternatives. As a result of the proposed fuzzy-based integrated methodology, the best and worst hospitals are determined in terms of service performance during the pandemic process. After the results are obtained, sensitivity analysis is applied to question the effect in the ranking depending on the changes in the parameters. To validate the results, a comparative analysis is performed using different MCDM methods and different fuzzy extensions. In the results of the comparison analysis, a change is observed in the places of the two alternatives in the ranking obtained by using the different fuzzy set extension, but there is no difference in the places of the first three and the last three alternatives for this analysis. As a result of the comparative analysis, it has been revealed that the proposed methodology is reliable, and results are validated.

According to the results, it can be said that “assurance” is the most important criterion for the vaccination process during the pandemic. It can be expressed as “the knowledge and courtesy of hospital staff and their ability to instill a sense of trust in patients”. The fact that the person who will be vaccinated feels psychologically dependent on the service providers, wants to rely on the knowledge and experience of the healthcare professionals who receive service during the treatment, and the need for the employees to be courteous increases the importance of the “assurance” dimension in hospitals. As it is seen, the fact that the service provided in hospitals is directly related to human life and the need to provide the service in an accurate and reliable manner causes the “assurance” dimension to be evaluated as the most important service quality dimension by the patients. This criterion was also found to be the highest perceived quality dimension in the paper of (Akdere, Top, & Tekingündüz, 2020). In this study, in which hospital performances were analyzed using the SERVPERF approach before the pandemic conditions, the “level of being knowledgable of the personnel” under the “assurance” criterion. In our study, the sub-criteria considered in the first places are “hospital support” and “politeness of employees”. At this point, our determination of a separate main criterion for the qualifications of hospital staff may cause differences in the findings obtained. However, the “tangibles” criterion, which was in the last order in the study of Akdere et al. (Akdere et al., 2020), has also been found in the last places in the ranking in our study. However, unlike our findings, in the study of Babroudi et al. (Babroudi et al., 2021), in which only the criteria were evaluated, the “assurance” criterion was determined in the 4th place in the order of importance. In fact, we assert that taking the hierarchy of criteria in this study, which was also carried out considering the pandemic conditions, from a study that was put forward many years ago (in 1992), may cause the differences in findings. In addition, since the vaccination process has also been taken into account in our study, we think that it is reasonable to arise a new perspective and thus obtain different results, by leaving only the limitations of preventing transmission and being treated in pandemic conditions. In short, we can argue that our results are similar to many studies evaluating hospital performance criteria, but offer some different results due to the handling of extraordinary conditions such as a pandemic or even a specific process such as vaccination.

The contributions of this study to the current service performance and MCDM literature can be specified as follows: (i) The SERVPERF model is adapted to hospital service performance evaluation during the COVID-19 vaccine process; (ii) Novel dimensions are added into SERVPERF model considering the current needs; (iii) The most important service performance dimensions for vaccine process are determined under the fuzzy environment; (iv) TOPSIS method is integrated with CRITIC under FF environment for the first time in the literature; (v) A real-life application in İstanbul is performed and presented to show the applicability and reliability of the proposed methodology; (vi) Eight different hospitals are evaluated according to the determined service performance dimensions and their performance are analyzed; (vii) The proposed methodology is intended to be used by the private and public hospital to improve their service performances.

Although this study has contributed to the decision-making literature and hospital service performance evaluation concept, still has some research limitations. As for the limitations, not being able to consider all the hospitals in the city could be our first drawback. In addition, the second most important limitation is the inability to find more experts with sufficient competence in the evaluation of criteria and alternatives. However, beyond all these limitations, the service performance comparison study for the important hospitals serving in Istanbul has been successfully carried out with a detailed and systematic hierarchy under the pandemic process.

As suggestions for future studies, different fuzzy-based MCDM methodologies can be applied in the hospital service quality evaluation process and the results obtained can be compared with this paper’s results. The proposed decision-making framework for service performance evaluation for the hospitals is limited to the identified criteria, but the number of criteria can be increased or decreased according to the problem’s situation/requirement.

CRediT authorship contribution statement

Melike Erdogan: Methodology, Supervision, Validation, Writing – review & editing. Ertugrul Ayyildiz: Writing – original draft, Methodology.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.