1 Introduction
The healthcare system is an indispensable pillar of any country’s economic growth. Diverse healthcare needs, increasing public expectations, and more ambitious health objectives are placing greater demands on health systems to achieve improved health outcomes (Rehman et al., 2023). Delivering health services that align with patients’ needs and expectations enhances organizations’ ability to remain competitive in today’s healthcare sector. Consequently, enhancing healthcare quality has become a central objective for healthcare organizations worldwide, as it supports service improvement, financial sustainability, and patient satisfaction regarding the standard of care (Darzi et al., 2023). Moreover, enhanced patient care and support services within hospitals, beyond the treatment of medical conditions, significantly enhance patients’ trust in hospital management while keeping costs under control. In recent years, this sector has undergone transformative changes, driven by technological advancements, data ecosystems, evolving patient expectations, and increased regulatory pressures (Khan et al., 2023). Ensuring patient satisfaction while delivering accurate diagnoses represents a key challenge that hospitals must address in their efforts to improve healthcare service quality.
Since healthcare organizations are directly responsible for saving lives, the ongoing demand for improved outcomes and high-quality healthcare services is critically important for developing effective organizational policies for reducing premature mortality and preventable patient harm. Healthcare sectors, as one of the leading service sectors, recognize that the service quality of hospitals is particularly relevant to the promotion of the healthcare market in today’s competitive environment (Ferreira et al., 2023; Zan et al., 2026). The survivability of healthcare institutions depends heavily on clients utilizing the services and determining that they adequately satisfy their immediate needs (Sanıl and Eminer, 2021). In developing countries, customers of the healthcare sector are increasingly aware of their right to receive quality healthcare (Darzi et al., 2023). Therefore, delivering quality services within healthcare organizations has become an increasingly important priority in the nation’s booming health industry. According to Cui and Tan (2025), healthcare providers need to deliver high-quality services that offer improved customer retention, higher customer satisfaction, enlarged market shares, and increased profit margins. Additionally, enhancing service quality while reducing costs provides healthcare institutions with a competitive advantage in distinguishing themselves from their competitors (Khan et al., 2023; Amit et al., 2026).
Patient satisfaction is a key indicator in assessing hospital quality, as it reflects how effectively healthcare staff meet patients’ needs and expectations. This satisfaction is largely shaped by patients’ perceptions of the quality of care and services they receive (Farrokhi et al., 2023). High levels of hospital service quality enhance patient satisfaction, thereby increasing patients’ willingness to establish and maintain long-term relationships with healthcare providers (Zehra et al., 2025). Patient-perceived healthcare quality, which covers dimensions, namely patient relationship management, patient management system, and patient-physician relationship, has been recognized as a significant factor in measuring patient satisfaction (Rehman et al., 2023; Mini et al., 2025). Assessment of patient-perceived hospital service quality is taken into account as a complex Multi-Criteria Decision-Making (MCDM) problem, as it involves diverse criteria and uncertain information.
In real-life decision-making situations, the DMs may encounter problems in articulating their preferences as a numerical value (Alkan and Kahraman, 2025). To solve these problems, Zadeh (1965) proposed the theoretical notion of Fuzzy Set (FS) to capture uncertainty arising from human subjectivity. It represents the degree to which an object belongs to a set through membership functions defined on a continuous scale from 0 to 1. To address the limitations of FS theory, Atanassov (1986) presented the Intuitionistic Fuzzy Set (IFS), which is denoted by the Membership Degree (MD) ‘μ’ and Non-Membership Degree (NMD) ‘ν’, with constraint $0\leqslant \mu +\nu \leqslant 1$. Further, Yager (2014) put forward the notion of Pythagorean Fuzzy Set (PFS), which extends the theory of IFS by replacing the linear constraint $0\leqslant \mu +\nu \leqslant 1$ subject to $0\leqslant {\mu ^{2}}+{\nu ^{2}}\leqslant 1$. Accordingly, PFS offers decision experts a broader range for expressing fuzzy information.
Afterward, Peng and Yang (2016) defined the Interval-Valued Pythagorean Fuzzy Set (IVPFS) with its own mathematical procedures. IVPFS extends the concept of PFS by expressing membership degrees as intervals instead of precise values; therefore, it can tackle a wider range of uncertain information. Considering the effectiveness of IVPFSs, there has been an increasing interest among scholars in utilizing them for solving real-life problems. For instance, Kose et al. (2022) proposed a hybrid MCDM framework for ergonomic assessment of the setup process under a single-minute exchange of dies. In this regard, they integrated the Analytic Hierarchy Process (AHP) method and TOPSIS within the context of IVPFSs and applied it to a manufacturing problem. Alhamzi et al. (2023) presented a score function for ordering Interval-Valued Pythagorean Fuzzy Numbers (IVPFNs). Additionally, they proposed a set of Dombi operators for aggregating IVPFNs into a single IVPFN. Within the context of IVPFSs, Rani et al. (2023) developed a hybrid Operational Competitiveness Rating Analysis (OCRA) approach based on a new score function, interaction operators, and cross-entropy, and employed to a case study focused on metaverse integration options in the transportation sharing economy. On the basis of the proposed membership score, Alrasheedi and Jeevaraj (2023) defined a total ordering principle on IVPFNs to deal with MCDM problems. Mishra et al. (2023) described the degree of fuzziness of IVPFS using a new entropy measure. Further, they aggregated a set of IVPFNs into an IVPFN with the use of fairly aggregation operator. Moreover, they introduced a hybrid Multi-Attribute Ideal-Real Comparative Analysis (IVPF-MAIRCA) method for assessing the blockchain-based healthcare supply chain.
Recently, Arisoy et al. (2025) proposed four score functions for distinguishing different IVPFNs with some interesting mathematical axioms and presented a hybrid Multi-Criteria Group Decision-Making (MCGDM) model for assessing the best agricultural logistics system in developing countries. Extending the Decision-Making Trial and Evaluation Laboratory (DEMATEL) approach in the IVPFS environment, Alam et al. (2025) offered an integrated MCDM framework to mitigate traffic congestion, which helps policymakers in long-term urban planning and environmental sustainability. Based on triangular divergence, Liu et al. (2025) suggested a distance measure on IVPFSs and developed a collective IVPF distance-based extended inferior ratio approach for solving the green supplier selection problem. With the use of an integrated IVPF-MCDM problem, Oh et al. (2025) evaluated five hydrogen storage technologies and identified underground storage as the optimal solution for large-scale hydrogen storage under Saudi Arabia’s specific conditions.
As one of the most recent MCDM techniques, the Compromise Ranking of Alternatives from Distance to Ideal Solution (CRADIS) (Puška et al., 2022) prioritizes alternatives considering their closeness to an ideal solution. The approach combines normalized and weighted distance calculations to derive a compromise ranking for each option. Recently, many studies on extended CRADIS methods with different extended versions of FS theory have been conducted to rank the alternatives under uncertain environments. For instance, Yuan et al. (2023) offered a hybrid picture fuzzy Jensen-Shannon divergence-based CRADIS approach and applied it to rank six alternatives across four evaluation criteria, including resource consumption, carbon emission, green production, and green technology. Demir and Moslem (2024) integrated the fuzzy CRADIS method with a fuzzy preference selection index for evaluating the most appropriate medical waste treatment techniques, considering technological, economic, environmental, and social factors. By combining the CRADIS technique with the entropy method, Anjum et al. (2025) introduced a combined intuitionistic fuzzy MCDM framework to optimize a range of energy-efficient solutions in smart manufacturing. Işık and Adalar (2025) suggested a fuzzy CRADIS model for assessing and ranking the environmental, social, and governance performance of non-life insurance firms in Turkey. Thus far, the literature does not report any extensions of the CRADIS method under the setting of IVPFSs.
Considering the effectiveness of the CRADIS method and IVPFSs, this research offers a hybrid decision-making methodology and employs it to evaluate hospital service quality. In this framework, we first derive the DMs’ weights through the entropy method and incorporate them into the IVPF-weighted averaging operator; we aggregate the individual IVPF opinions of each expert to appraise the quality of service of hospitals. Next, we derive the weights of criteria with the help of an integrated objective-subjective model, consisting of Weights by ENvelope and SLOpe (WENSLO) and RANking COMparison (RANCOM), respectively. Based on the combination of the above steps, the algorithm of CRADIS is to determine alternatives’ ranking in the context of IVPF information. Consequently, a novel entropy measure, a modified score function, and an improved distance measure are developed in the context of IVPFSs, followed by comparative analyses. The suggested entropy technique is used to derive the DMs’ weights, while the score function is employed to normalize the input information during the computation of criteria weights through the WENSLO technique. On the flip side, the suggested distance formula is employed to find the closeness of each option to the IVPF-ideal and IVPF anti-ideal solutions via the CRADIS technique.
The originality of this work is primarily reflected in the following:
-
i. A novel IVPF-entropy measure is offered in this work for measuring the degree of fuzziness of an IVPFS. Further, the DMs’ weights are obtained based on the suggested entropy measure.
-
ii. To compare different IVPFNs, this research proposes an enhanced IVPF-score function, which relaxes the weaknesses of existing IVPF-score functions. Moreover, the proposed score formula is utilized to find the normalized decision matrix in the WENSLO model for criteria weighting.
-
iii. To distinguish the IVPFSs, a modified distance measure is proposed for IVPFSs, followed by comparison with existing IVPF-score functions. It is further utilized to find the dissimilarity degrees between alternatives and ideal solutions.
-
iv. To treat the MCGDM problems, a hybrid IVPF-CRADIS method is developed by merging its algorithm with the entropy method for DMs’ weights and an integrated WENSLO-RANCOM model for criteria weighting within the setting of IVPFSs.
-
v. To demonstrate the efficiency of the developed methodology, it is implemented in a case study of a hospital service quality evaluation problem under an IVPF environment.
The rest of the manuscript is structured as follows. Section 2 reviews the literature of hospital service quality assessment. Section 3 first recalls the concepts related to IVPFSs and further introduces the novel entropy measure, a modified score function, and an improved distance measure in the context of IVPFSs, followed by comparative discussions. Section 4 elaborates the application steps of an incorporated IVPF-WENSLO-RANCOM-CRADIS methodology for treating MCGDM problems. Section 5 implements the developed methodology to a hospital service quality selection problem. Section 6 presents the sensitivity and comparative analyses to reveal the stability and robustness of the decision support framework, respectively. Section 7 concludes the research and recommends several directions for future work.
2 Literature Survey
This section explores the existing works on service quality of health organizations using different decision-making methods. Akdag et al. (2014) presented an MCDM framework to evaluate service quality of Turkish hospitals. In this regard, they combined AHP, TOPSIS, and Yager’s min-max approach within the setting of fuzzy information and later implemented it in an empirical study of the healthcare service quality evaluation problem of four B-class hospitals of Istanbul. Employing a fuzzy AHP-TOPSIS model, Shafii et al. (2016) evaluated the service quality of teaching hospitals in Yazd, Iran, considering six dimensions of assurance, responsiveness, tangibles, security, patient orientation, and health communication. Liu et al. (2019) presented a Grey Relational Analysis (GRA) for appraising the healthcare service quality of hospitals. Based on the obtained outcomes, they determined that the GRA decreases the latency in the assessment process and increases the efficacy range of tools used for healthcare services. Utilizing the Pythagorean fuzzy AHP model, Yucesan and Gul (2020) specified the weights of service quality assessment factors of 32 hospitals. Based on this, they determined the priorities of healthcare institutions through the Pythagorean fuzzy TOPSIS method and offered the opportunity to make evaluations in view of DMs’ subjective opinions.
Babroudi et al. (2021) measured the customer perceptions of hospital service quality. For this purpose, they suggested an incorporated framework model that integrates Z-number with fuzzy cognitive map to assess healthcare service quality. Sivakumar et al. (2022) integrated a multi-objective linear programming model with BWM and Multi-Actor Multi-Criteria Assessment (MAMCA) for enhancing the patient service quality in the intensive care unit. Their findings provided an effective solution that can be utilized to advance the quality of services for intensive care unit patients during the COVID-19 pandemic. By combining the fuzzy rule-based system and TOPSIS model, Karasan et al. (2022) suggested a Pythagorean fuzzy-based MCDM tool to assess the service quality performance of ten clinics in a private hospital. With a focus on patient-centred services, Khan et al. (2023) identified the quality of services of four hospitals during the recent pandemic. For this purpose, they used a fuzzy BWM and implemented it in a case study of a service quality factors assessment problem, taking into account the patients’ perceptions and expectations of the services.
Considering a variety of criteria, Li et al. (2023) presented a fuzzy rough copula Bayesian network to assess the service quality of hospitals under an uncertain environment, which deals with the subjective uncertainty of DMs’ opinions. Using fuzzy logic and decision modelling, Rehman et al. (2023) assessed hospital service quality considering numerous criteria and dimensions based on patients’ viewpoints. Their findings suggested that the dynamic nature of hospitals and varying patient expectations require periodic service quality evaluations. To evaluate the quality of healthcare services, Alshouha et al. (2024) presented a fuzzy recommendation procedure based on the Quality Function Deployment (QFD) approach. By means of patient opinions and preferences, their findings indicated a step toward a more adapted and operative healthcare structure selection using fuzzy information.
Accounting for smart medical systems, Wang and Shao (2024) integrated the BWM and VIKOR models for assessing the level of medical services offered by hospitals. Cui and Tan (2025) suggested a two-phase fuzzy MCDM model for periodic service quality enhancements in hospitals, in which fuzzy logic was utilized to enhance the accuracy of decisions in hospital quality management. Their method aimed to provide a novel and inspiring mode to advance the effectiveness and ease of healthcare services for patients. Through a comprehensive literature review, Datt et al. (2025) identified the principal quality dimensions within the healthcare services domain. Furthermore, they developed an evaluation model for healthcare centres that highlights their capacity to enhance service quality and increase patient satisfaction.
In accordance with the existing literature, we identify that the previous decision-making models ignore the significance of decision experts’ opinions during the assessment of hospital service quality. Additionally, these studies overlook the importance of multiple experts in making decisions on the quality of healthcare services. An additional research gap concerns the insufficient consideration of service quality indicators from both objective and subjective perspectives. To handle these gaps, this work suggests a fuzzy integrated decision support framework by incorporating the entropy method for DMs weighting and an integrated WENSLO-RANCOM model for criteria weighting with the steps of the CRADIS approach under the setting of IVPFSs.
3 Proposed Entropy, Score, and Distance Formulae in the Context of IVPFSs
This section begins with a brief overview of the basic notions of IVPFSs, followed by the development of an improved score function, entropy measure, and distance measure.
3.1 Preliminaries
Definition 1.
Peng and Yang (2016) presented a mathematical definition of an IVPFS P on a fixed universal set $L=\{{l_{1}},{l_{2}},\dots ,{l_{r}}\},$ and given as
where $0\leqslant {\mu _{B}^{-}}({l_{i}})\leqslant {\mu _{B}^{+}}({l_{i}})\leqslant 1$, $0\leqslant {\nu _{B}^{-}}({l_{i}})\leqslant {\nu _{B}^{+}}({l_{i}})\leqslant 1$ and $0\leqslant {({\mu _{B}^{+}}({l_{i}}))^{2}}+{({\nu _{B}^{+}}({l_{i}}))^{2}}\leqslant 1$. Here, ${\mu _{B}}({l_{i}})=[{\mu _{B}^{-}}({l_{i}}),{\mu _{B}^{+}}({l_{i}})]$ and ${\nu _{B}}({l_{i}})=[{\nu _{B}^{-}}({l_{i}}),{\nu _{B}^{+}}({l_{i}})]$ stand for the interval-valued MD and NMD of an element ${l_{i}}$ in B, respectively. The function ${\pi _{B}}({l_{i}})=[{\pi _{B}^{-}}({l_{i}}),{\pi _{B}^{+}}({l_{i}})]$ denotes the degree of hesitancy/indeterminacy ${l_{i}}$ to B, where ${\pi _{B}^{-}}({l_{i}})=\sqrt{1-{({\mu _{B}^{+}}({l_{i}}))^{2}}-{({\nu _{B}^{+}}({l_{i}}))^{2}}}$ and ${\pi _{B}^{+}}({l_{i}})=\sqrt{1-{({\mu _{B}^{-}}({l_{i}}))^{2}}-{({\nu _{B}^{-}}({l_{i}}))^{2}}}$. Peng and Yang (2016) defined the term $([{\mu _{B}^{-}}({l_{i}}),{\mu _{B}^{+}}({l_{i}})],[{\nu _{B}^{-}}({l_{i}}),{\nu _{B}^{+}}({l_{i}})])$ as an interval-valued Pythagorean fuzzy value (IVPFV)/interval-valued Pythagorean fuzzy number (IVPFN) and displayed in this work as $\delta =([{\mu ^{-}},{\mu ^{+}}],[{\nu ^{-}},{\nu ^{+}}])$, which satisfies $0\leqslant ({\mu ^{+}})+({\nu ^{+}})\leqslant 1$.
Definition 2.
Let ${\delta _{1}}=([{\mu _{1}^{-}},{\mu _{1}^{+}}],[{\nu _{1}^{-}},{\nu _{1}^{+}}])$ and ${\delta _{2}}=([{\mu _{2}^{-}},{\mu _{2}^{+}}],[{\nu _{2}^{-}},{\nu _{2}^{+}}])$ be two IVPFNs. The main operations of IVPFNs are defined as in Peng and Yang (2016):
-
(i) $\begin{array}{l}{\delta _{1}}\oplus {\delta _{2}}\\ {} \hspace{10.0pt}=\displaystyle \Bigg(\Big[\sqrt{{\Big({\mu _{1}^{-}}\Big)^{2}}+{\Big({\mu _{2}^{-}}\Big)^{2}}-{\Big({\mu _{1}^{-}}\Big)^{2}}{\Big({\mu _{2}^{-}}\Big)^{2}}},\sqrt{{\Big({\mu _{1}^{+}}\Big)^{2}}+{\Big({\mu _{2}^{+}}\Big)^{2}}-{\Big({\mu _{1}^{+}}\Big)^{2}}{\Big({\mu _{2}^{+}}\Big)^{2}}}\Big],\Big[{\nu _{1}^{-}}{\nu _{2}^{-}},{\nu _{1}^{+}}{\nu _{2}^{+}}\Big]\Bigg),\end{array}$
-
(ii) $\begin{array}{l}{\delta _{1}}\otimes {\delta _{2}}\\ {} \hspace{10.0pt}=\displaystyle \Bigg(\Big[{\mu _{1}^{-}}{\mu _{2}^{-}},{\mu _{1}^{+}}{\mu _{2}^{+}}\Big],\Bigg[\sqrt{{\Big({\nu _{1}^{-}}\Big)^{2}}+{\Big({\nu _{2}^{-}}\Big)^{2}}-{\Big({\nu _{1}^{-}}\Big)^{2}}{\Big({\nu _{2}^{-}}\Big)^{2}}},\sqrt{{\Big({\nu _{1}^{+}}\Big)^{2}}+{\Big({\nu _{2}^{+}}\Big)^{2}}-{\Big({\nu _{1}^{+}}\Big)^{2}}{\Big({\nu _{2}^{+}}\Big)^{2}}}\Bigg]\Bigg),\end{array}$
-
(iii) $\begin{array}{l}\lambda {\delta _{i}}=\left(\left[\sqrt{1-{\left(1-{\left({\mu _{i}^{-}}\right)^{2}}\right)^{\lambda }}}\sqrt{1-{\left(1-{\left({\mu _{i}^{+}}\right)^{2}}\right)^{\lambda }}}\right],\left[{\left({\nu _{i}^{-}}\right)^{\lambda }},{\left({\nu _{i}^{+}}\right)^{\lambda }}\right]\right)\\ {} \hspace{10.0pt}(\lambda \gt 0;i=1,2),\end{array}$
-
(iv) $\begin{array}{l}{({\delta _{i}})^{\lambda }}=\left(\left[{\left({\mu _{i}^{-}}\right)^{\lambda }},{\left({\mu _{i}^{+}}\right)^{\lambda }}\right],\left[\sqrt{1-{\left(1-{\left({\nu _{i}^{-}}\right)^{2}}\right)^{\lambda }}},\sqrt{1-{\left(1-{\left({\nu _{i}^{+}}\right)^{2}}\right)^{\lambda }}}\right]\right)\\ {} \hspace{10.0pt}(\lambda \gt 0;i=1,2),\end{array}$
-
(v) ${({\delta _{i}})^{c}}=\left(\left[{\nu _{i}^{-}},{\nu _{i}^{+}}\right],\left[{\mu _{i}^{-}},{\mu _{i}^{+}}\right]\right)$ $(i=1,2)$.
Definition 3.
For an IVPFN $\delta =([{\mu ^{-}},{\mu ^{+}}],[{\nu ^{-}},{\nu ^{+}}])$, Peng and Yang (2016) defined the score function as
With the help of this setting, Peng and Yang (2016) presented a comparison rule for two IVPFNs ${\delta _{1}}=([{\mu _{1}^{-}},{\mu _{1}^{+}}],[{\nu _{1}^{-}},{\nu _{1}^{+}}])$ and ${\delta _{2}}=([{\mu _{2}^{-}},{\mu _{2}^{+}}],[{\nu _{2}^{-}},{\nu _{2}^{+}}])$, given as
-
∙ If ${S_{P}}({\delta _{1}})\lt {S_{P}}({\delta _{2}})$, then ${\delta _{1}}$ is considered smaller than ${\delta _{2}}$, and denoted as ${\delta _{1}}\lt {\delta _{2}}$,
-
∙ If ${S_{P}}({\delta _{1}})\gt {S_{P}}({\delta _{2}})$, then ${\delta _{1}}$ is considered larger than ${\delta _{2}}$, and denoted as ${\delta _{1}}\gt {\delta _{2}}$,
-
∙ If ${S_{P}}({\delta _{1}})={S_{P}}({\delta _{2}})$, then ${\theta _{1}}$ is considered similar as ${\theta _{2}}$, and denoted as ${\delta _{1}}={\delta _{2}}$.
For any two IVPFNs ${\delta _{1}}=([0.3,0.4],[0.3,0.4])$ and ${\delta _{2}}=([0.4,0.5],[0.4,0.5])$, the score function defined by Eq. (2) gives ${S_{P}}({\delta _{1}})=0$ and ${S_{P}}({\delta _{2}})=0$. In this context, an expert cannot discriminate the difference between two IVPFNs. To this aim, Peng and Yang (2016) defined the accuracy function, given by
Definition 4.
Let ${\delta _{i}}$ $(i=1,2,\dots ,r)$ be a set of IVPFNs and $\kappa ={({\kappa _{1}},{\kappa _{2}},\dots ,{\kappa _{r}})^{T}}$ be the weight vector of ${\delta _{i}}$ $(i=1,2,\dots ,r)$, where ${\kappa _{i}}\in [0,1]$ and ${\sum _{i=1}^{r}}{\kappa _{i}}=1$. An Interval-Valued Pythagorean Fuzzy Weighed Averaging (IVPFWA) operator is then defined as in Peng and Yang (2016):
(4)
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \textit{IVPFWA}({\delta _{1}},{\delta _{2}},\dots ,{\delta _{r}})={\underset{i=1}{\overset{r}{\bigoplus }}}({\kappa _{i}}{\delta _{i}})\\ {} & \displaystyle \hspace{10.0pt}=\left(\left[1-{\prod \limits_{i=1}^{r}}{\left(1-{\left({\mu _{{\delta _{i}}}^{-}}\right)^{2}}\right)^{{\kappa _{i}}}},1-{\prod \limits_{i=1}^{r}}{\left(1-{\left({\mu _{{\delta _{i}}}^{+}}\right)^{2}}\right)^{{\kappa _{i}}}}\right],\left[{\prod \limits_{i=1}^{r}}{\left({\nu _{{\delta _{i}}}^{-}}\right)^{2{\kappa _{i}}}},{\prod \limits_{i=1}^{r}}{\left({\nu _{{\delta _{i}}}^{+}}\right)^{2{\kappa _{i}}}}\right]\right).\end{array}\]Additionally, Peng and Yang (2016) defined an Interval-Valued Pythagorean Fuzzy Weighed Geometric (IVPFWG) operator, given as
(5)
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \textit{IVPFWG}({\delta _{1}},{\delta _{2}},\dots ,{\delta _{r}})={\underset{i=1}{\overset{r}{\bigotimes }}}\left({\delta _{i}^{{\kappa _{i}}}}\right)\\ {} & \displaystyle \hspace{10.0pt}=\left(\left[{\prod \limits_{i=1}^{r}}{\left({\mu _{{\delta _{i}}}^{-}}\right)^{2{\kappa _{i}}}},{\prod \limits_{i=1}^{r}}{\left({\mu _{{\delta _{i}}}^{+}}\right)^{2{\kappa _{i}}}}\right],\left[1-{\prod \limits_{i=1}^{r}}{\left(1-{\left({\nu _{{\delta _{i}}}^{-}}\right)^{2}}\right)^{{\kappa _{i}}}},1-{\prod \limits_{i=1}^{r}}{\left(1-{\left({\nu _{{\delta _{i}}}^{+}}\right)^{2}}\right)^{{\kappa _{i}}}}\right]\right).\end{array}\]Definition 5.
Let $B\in \textit{IVPFS}(L)$, then a real-valued function $E:\textit{IVPFS}(L)\to [0,1]$ is called an IVPF-entropy if it satisfies the following axioms (Peng and Li, 2019):
-
(a1) $E(B)=0$ iff B is a crisp set,
-
(a2) $E(B)=1$ iff ${\mu _{B}^{-}}({l_{i}})={\nu _{B}^{-}}({l_{i}})$ and ${\mu _{B}^{+}}({l_{i}})={\nu _{B}^{+}}({l_{i}})$, for each ${l_{i}}\in L$,
-
(a3) $E(B)=E({B^{c}})$,
-
(a4) $E(B)\leqslant E(N)$ if $B\subseteq N$ with ${\mu _{N}^{-}}({l_{i}})\leqslant {\nu _{N}^{-}}({l_{i}})$ and ${\mu _{N}^{+}}({l_{i}})\leqslant {\nu _{N}^{+}}({l_{i}})$, for each ${l_{i}}\in L$, or $B\supseteq N$ with ${\mu _{N}^{-}}({l_{i}})\geqslant {\nu _{N}^{-}}({l_{i}})$ and ${\mu _{N}^{+}}({l_{i}})\geqslant {\nu _{N}^{+}}({l_{i}})$, for each ${l_{i}}\in L$, where $N\in \textit{IVPFS}(L)$.
Definition 6.
Let $B,N\in \textit{IVPFSs}(L)$. Then a real-valued mapping $d:\textit{IVPFSs}(L)\times \textit{IVPFSs}(L)\to [0,1]$ is a real-valued function, satisfying the following requirements (Zhang, 2016):
3.2 Proposed Entropy Measure for IVPFS
The concept of entropy plays a fundamental role in measuring the degree of fuzziness and uncertainty within FS theory and its extensions (Saha et al., 2025). As an important research topic, entropy has received significant scholarly attention over the past several decades. For IVPFS, Peng and Li (2019) firstly proposed an IVPF-entropy and its applications. Besides, some scholars put forward different entropy formulae for IVPFS (e.g., Moazzamigodarzi et al., 2025). Compared with FS, IFS, and PFS, there are not as many studies on the entropy of IVPFS. Unlike existing studies, this paper not only gives the formula of IVPF-entropy, but also presents the use of the suggested measure in determining experts’ weights.
Theorem 1.
For $B\in \textit{IVPFS}(L)$, the function
(6)
\[\begin{array}{c}\displaystyle E\displaystyle (B\displaystyle )\displaystyle =\displaystyle \frac{1}{r}{\displaystyle \displaystyle \sum \limits_{i=1}^{r}}\left(\frac{2+{({\pi _{B}^{-}}({l_{i}}))^{2}}+{({\pi _{B}^{+}}({l_{i}}))^{2}}-\mathrm{|}{({\mu _{B}^{-}}({l_{i}}))^{2}}-{({\nu _{B}^{-}}({l_{i}}))^{2}}\mathrm{|}-\mathrm{|}{({\mu _{B}^{+}}({l_{i}}))^{2}}-{({\nu _{B}^{+}}({l_{i}}))^{2}}\mathrm{|}}{2+{({\pi _{B}^{-}}({l_{i}}))^{2}}+{({\pi _{B}^{+}}({l_{i}}))^{2}}}\right)\\ {} \end{array}\]
is an entropy measure for IVPFS ‘B’.
Proof.
(a${_{1}}$) Let B be a crisp set. It implies ${\mu _{B}^{-}}({l_{i}})=1={\mu _{B}^{+}}({l_{i}})$, ${\nu _{B}^{-}}({l_{i}})=0={\nu _{B}^{+}}({l_{i}})$ or ${\mu _{B}^{-}}({l_{i}})=0={\mu _{B}^{+}}({l_{i}})$, ${\nu _{B}^{-}}({l_{i}})=1={\nu _{B}^{+}}({l_{i}})$, and therefore, ${\pi _{B}^{-}}({l_{i}})=0={\pi _{B}^{+}}({l_{i}})$, $\forall {l_{i}}\in L$. In this case, Eq. (6) becomes $E(B)=0$.
On the other hand, assume that $E(P)=0$, then from Eq. (6), we have
(7)
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \displaystyle \frac{2+{\Big({\pi _{B}^{-}}({l_{i}})\Big)^{2}}+{\Big({\pi _{B}^{+}}({l_{i}})\Big)^{2}}-\big|{({\mu _{B}^{-}}({l_{i}}))^{2}}-{({\nu _{B}^{-}}({l_{i}}))^{2}}\big|-\big|{({\mu _{B}^{+}}({l_{i}}))^{2}}-{({\nu _{B}^{+}}({l_{i}}))^{2}}\big|}{2+{\Big({\pi _{B}^{-}}({l_{i}})\Big)^{2}}+{\Big({\pi _{B}^{+}}({l_{i}})\Big)^{2}}}=0\hspace{8.7pt}\mathrm{\forall }{l_{i}}\in L,\\ {} & \displaystyle \Rightarrow \hspace{10.0pt}\Big|{\left({\mu _{B}^{-}}({l_{i}})\right)^{2}}-{\left({\nu _{B}^{-}}({l_{i}})\right)^{2}}\Big|+\Big|{\left({\mu _{B}^{+}}({l_{i}})\right)^{2}}-{\left({\nu _{B}^{+}}({l_{i}})\right)^{2}}\Big|\\ {} & \displaystyle \hspace{30.43pt}=2+{\left({\pi _{B}^{-}}({l_{i}})\right)^{2}}+{\left({\pi _{B}^{+}}({l_{i}})\right)^{2}},\hspace{10.0pt}\forall {l_{i}}\in L.\end{array}\]As $B\in \textit{IVPFS}(L)$, then $0\leqslant {\mu _{B}^{-}}({l_{i}})\leqslant {\mu _{B}^{+}}({l_{i}})\leqslant 1,0\leqslant {\nu _{B}^{-}}({l_{i}})\leqslant {\nu _{B}^{+}}({l_{i}})\leqslant 1$ and $0\leqslant {({\mu _{B}^{+}}({l_{i}}))^{2}}+{({\nu _{B}^{+}}({l_{i}}))^{2}}\leqslant 1$. Therefore, the only possible solution of Eq. (7) is ${\mu _{B}^{-}}({l_{i}})=1={\mu _{B}^{+}}({l_{i}})$, ${\nu _{B}^{-}}({l_{i}})=0={\nu _{B}^{+}}({l_{i}})$ or ${\mu _{B}^{-}}({l_{i}})=0={\mu _{B}^{+}}({l_{i}})$, ${\nu _{B}^{-}}({l_{i}})=1={\nu _{B}^{+}}({l_{i}})$ and ${\pi _{B}^{-}}({l_{i}})=0={\pi _{B}^{+}}({l_{i}})$, $\forall {l_{i}}\in L$. It implies that ‘B’ is a crisp set.
(a${_{2}}$) Let ${\mu _{B}^{-}}({l_{i}})={\nu _{B}^{-}}({l_{i}})$ and ${\mu _{B}^{+}}({l_{i}})={\nu _{B}^{+}}({l_{i}})$, $\forall {l_{i}}\in L$. Then, from Eq. (6), we get $E(B)=1$. On the other hand, assume that $E(B)=1$, then from Eq. (6), we have
(8)
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \displaystyle \frac{2+{\Big({\pi _{B}^{-}}({l_{i}})\Big)^{2}}+{\Big({\pi _{B}^{+}}({l_{i}})\Big)^{2}}-\big|{({\mu _{B}^{-}}({l_{i}}))^{2}}-{({\nu _{B}^{-}}({l_{i}}))^{2}}\big|-\big|{({\mu _{B}^{+}}({l_{i}}))^{2}}-{({\nu _{B}^{+}}({l_{i}}))^{2}}\big|}{2+{\Big({\pi _{B}^{-}}({l_{i}})\Big)^{2}}+{\Big({\pi _{B}^{+}}({l_{i}})\Big)^{2}}}=1,\hspace{8.7pt}\mathrm{\forall }{l_{i}}\in L,\\ {} & \displaystyle \Rightarrow \hspace{10.0pt}\Big|{\left({\mu _{B}^{-}}({l_{i}})\right)^{2}}-{\left({\nu _{B}^{-}}({l_{i}})\right)^{2}}\Big|+\Big|{\left({\mu _{B}^{+}}({l_{i}})\right)^{2}}-{\left({\nu _{B}^{+}}({l_{i}})\right)^{2}}\Big|=0,\hspace{10.0pt}\forall {l_{i}}\in L.\end{array}\]Thus, the possible solution of Eq. (8) is ${\mu _{B}^{-}}({l_{i}})={\nu _{B}^{-}}({l_{i}})$ and ${\mu _{B}^{+}}({l_{i}})={\nu _{B}^{+}}({l_{i}})$, for each ${l_{i}}\in L$.
(a${_{3}}$) Let $B\in \textit{IVPFS}(L)$, where $B=\{({l_{i}},([{\mu _{B}^{-}}({l_{i}}),{\mu _{B}^{+}}({l_{i}})],[{\nu _{B}^{-}}({l_{i}}),{\nu _{B}^{+}}({l_{i}})])):{l_{i}}\in L\}$. Then the complement of B is denoted as ${B^{c}}=\{({l_{i}},([{\nu _{B}^{-}}({l_{i}}),{\nu _{B}^{+}}({l_{i}})],[{\mu _{B}^{-}}({l_{i}}),{\mu _{B}^{+}}({l_{i}})])):{l_{i}}\in L\}$. Thus, from Eq. (6), we have $E(B)=E({B^{c}})$.
(a4) For $B,N\in \textit{IVPFSs}(L)$, let $B\subseteq N$ with ${\mu _{N}^{-}}({l_{i}})\leqslant {\nu _{N}^{-}}({l_{i}})$ and ${\mu _{N}^{+}}({l_{i}})\leqslant {\nu _{N}^{+}}({l_{i}})$, for each ${l_{i}}\in L$. It follows that ${\mu _{B}^{-}}({l_{i}})\leqslant {\mu _{N}^{-}}({l_{i}})\leqslant {\nu _{N}^{-}}({l_{i}})\leqslant {\nu _{B}^{-}}({l_{i}})$ and ${\mu _{B}^{+}}({l_{i}})\leqslant {\mu _{N}^{+}}({l_{i}})\leqslant {\nu _{N}^{+}}({l_{i}})\leqslant {\nu _{B}^{+}}({l_{i}})$, for each ${l_{i}}\in L$.
Since $|{({\mu _{B}^{-}}({l_{i}}))^{2}}-{({\nu _{B}^{-}}({l_{i}}))^{2}}|\geqslant |{({\mu _{N}^{-}}({l_{i}}))^{2}}-{({\nu _{N}^{-}}({l_{i}}))^{2}}|$ and $|{({\mu _{B}^{+}}({l_{i}}))^{2}}-{({\nu _{B}^{+}}({l_{i}}))^{2}}|\geqslant |{({\mu _{N}^{+}}({l_{i}}))^{2}}-{({\nu _{N}^{+}}({l_{i}}))^{2}}|$, it implies that $-|{({\mu _{B}^{-}}({l_{i}}))^{2}}-{({\nu _{B}^{-}}({l_{i}}))^{2}}|\leqslant -|{({\mu _{N}^{-}}({l_{i}}))^{2}}-{({\nu _{N}^{-}}({l_{i}}))^{2}}|$ and $-|{({\mu _{B}^{+}}({l_{i}}))^{2}}-{({\nu _{B}^{+}}({l_{i}}))^{2}}|\leqslant -|{({\mu _{N}^{+}}({l_{i}}))^{2}}-{({\nu _{N}^{+}}({l_{i}}))^{2}}|$, and therefore, $-|{({\mu _{B}^{-}}({l_{i}}))^{2}}-{({\nu _{B}^{-}}({l_{i}}))^{2}}|-|{({\mu _{B}^{+}}({l_{i}}))^{2}}-{({\nu _{B}^{+}}({l_{i}}))^{2}}|$ is also less than or equal to $-|{({\mu _{N}^{-}}({l_{i}}))^{2}}-{({\nu _{N}^{-}}({l_{i}}))^{2}}|-|{({\mu _{N}^{+}}({l_{i}}))^{2}}-{({\nu _{N}^{+}}({l_{i}}))^{2}}|$,
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \Rightarrow \left(\begin{array}{l}2+\left(1-{\left({\mu _{B}^{+}}({l_{i}})\right)^{2}}-{\left({\nu _{B}^{+}}({l_{i}})\right)^{2}}\right)+\left(1-{\left({\mu _{B}^{-}}({l_{i}})\right)^{2}}-{\left({\nu _{B}^{-}}({l_{i}})\right)^{2}}\right)\\ {} -\Big|{\left({\mu _{B}^{-}}({l_{i}})\right)^{2}}-{\left({\nu _{B}^{-}}({l_{i}})\right)^{2}}\Big|-\Big|{\left({\mu _{B}^{+}}({l_{i}})\right)^{2}}-{\left({\nu _{B}^{+}}({l_{i}})\right)^{2}}\Big|\end{array}\right)\hspace{10.0pt}\\ {} & \displaystyle \leqslant \left(\begin{array}{l}2+\left(1-{\left({\mu _{N}^{+}}({l_{i}})\right)^{2}}-{\left({\nu _{N}^{+}}({l_{i}})\right)^{2}}\right)+\left(1-{\left({\mu _{N}^{-}}({l_{i}})\right)^{2}}-{\left({\nu _{N}^{-}}({l_{i}})\right)^{2}}\right)\\ {} -\Big|{\left({\mu _{N}^{-}}({l_{i}})\right)^{2}}-{\left({\nu _{N}^{-}}({l_{i}})\right)^{2}}\Big|-\Big|{\left({\mu _{N}^{+}}({l_{i}})\right)^{2}}-{\left({\nu _{N}^{+}}({l_{i}})\right)^{2}}\Big|\end{array}\right),\hspace{10.0pt}\forall {l_{i}}\in L,\\ {} & \displaystyle \Rightarrow \displaystyle \Big(2+{\Big({\pi _{B}^{-}}({l_{i}})\Big)^{2}}+{\Big({\pi _{B}^{+}}({l_{i}})\Big)^{2}}-|{\Big({\mu _{B}^{-}}({l_{i}})\Big)^{2}}-{\Big({\nu _{B}^{-}}({l_{i}})\Big)^{2}}|-|{\Big({\mu _{B}^{+}}({l_{i}})\Big)^{2}}-{\Big({\nu _{B}^{+}}({l_{i}})\Big)^{2}}|\Big)\\ {} & \displaystyle \leqslant \displaystyle \Big(2+{\Big({\pi _{N}^{-}}({l_{i}})\Big)^{2}}+{\Big({\pi _{N}^{+}}({l_{i}})\Big)^{2}}-|{\Big({\mu _{N}^{-}}({l_{i}})\Big)^{2}}-{\Big({\nu _{N}^{-}}({l_{i}})\Big)^{2}}|-|{\Big({\mu _{N}^{+}}({l_{i}})\Big)^{2}}-{\Big({\nu _{N}^{+}}({l_{i}})\Big)^{2}}|\Big),\\ {} & \displaystyle \hspace{10.0pt}\forall {l_{i}}\in L.\end{array}\]
It will also imply that
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \frac{(2+{({\pi _{B}^{-}}({l_{i}}))^{2}}+{({\pi _{B}^{+}}({l_{i}}))^{2}}-|{({\mu _{B}^{-}}({l_{i}}))^{2}}-{({\nu _{B}^{-}}({l_{i}}))^{2}}|-|{({\mu _{B}^{+}}({l_{i}}))^{2}}-{({\nu _{B}^{+}}({l_{i}}))^{2}}|)}{2+{({\pi _{B}^{-}}({l_{i}}))^{2}}+{({\pi _{B}^{+}}({l_{i}}))^{2}}}\\ {} & \displaystyle \leqslant \displaystyle \frac{(2+{({\pi _{N}^{-}}({l_{i}}))^{2}}+{({\pi _{N}^{+}}({l_{i}}))^{2}}-\mathrm{|}{({\mu _{N}^{-}}({l_{i}}))^{2}}-{({\nu _{N}^{-}}({l_{i}}))^{2}}\mathrm{|}-\mathrm{|}{({\mu _{N}^{+}}({l_{i}}))^{2}}-{({\nu _{N}^{+}}({l_{i}}))^{2}}\mathrm{|})}{2+{({\pi _{N}^{-}}({l_{i}}))^{2}}+{({\pi _{N}^{+}}({l_{i}}))^{2}}},\\ {} & \displaystyle \hspace{10.0pt}\forall {l_{i}}\in L.\end{array}\]
Thus, $E(B)\leqslant E(N)$. Similarly, for $B\supseteq N$ with ${\mu _{N}^{-}}({l_{i}})\geqslant {\nu _{N}^{-}}({l_{i}})$ and ${\mu _{N}^{+}}({l_{i}})\geqslant {\nu _{N}^{+}}({l_{i}})$, for each ${l_{i}}\in L$, it is demonstrated that $E(B)\leqslant E(N)$. □
In the following, the developed IVPF-entropy is evaluated against existing IVPF-entropy measures to reveal its effectiveness (Peng and Li, 2019; Mishra et al., 2023; Rani et al., 2023; Moazzamigodarzi et al., 2025). Table S1 (Supplementary Material) reviews the formulations of these measures.
Example 1 (Hung and Yang (2006)).
Consider an IVPFS $B=\{({l_{i}},([{\mu _{B}^{-}}({l_{i}}),{\mu _{B}^{+}}({l_{i}})],[{\nu _{B}^{-}}({l_{i}}),{\nu _{B}^{+}}({l_{i}})])):{l_{i}}\in L\}$. For any $m\in {\mathbb{R}^{+}}$, define the IVPFS ${B^{m}}$ as follows:
The IVPFS B is denoted on the universal set $L=\{6,7,8,9,10\}$:
Similarly, we can define the following IVPFSs:
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle {B^{2}}=\displaystyle \Bigg\{\begin{array}{l}\langle 6,[0\mathrm{.}01,0\mathrm{.}04],[0\mathrm{.}84,0\mathrm{.}91]\rangle ,\langle 7,[0\mathrm{.}09,0\mathrm{.}25],[0\mathrm{.}64,0\mathrm{.}75]\rangle ,\\ {} \langle 8,[0\mathrm{.}36,0\mathrm{.}49],[0\mathrm{.}19,0\mathrm{.}36]\rangle ,\langle 9,[0\mathrm{.}64,0\mathrm{.}81],[0,0\mathrm{.}19]\rangle ,\langle 10,[1,1],[0,0]\rangle \end{array}\Bigg\},\\ {} & \displaystyle {B^{3}}=\displaystyle \Bigg\{\begin{array}{l}\langle 6,[0\mathrm{.}0010,0\mathrm{.}0080],[0\mathrm{.}9360,0\mathrm{.}9730]\rangle ,\langle 7,[0\mathrm{.}0270,0\mathrm{.}1250],[0\mathrm{.}7840,0\mathrm{.}8750]\rangle ,\\ {} \langle 8,[0\mathrm{.}2160,0\mathrm{.}3430],[0\mathrm{.}2710,0\mathrm{.}4880]\rangle ,\langle 9,[0\mathrm{.}5120,0\mathrm{.}7290],[0,0\mathrm{.}2710]\rangle ,\\ {} \langle 10,[1,1],[0,0]\rangle \end{array}\hspace{-1.55pt}\Bigg\},\\ {} & \displaystyle {B^{4}}=\displaystyle \Bigg\{\begin{array}{l}\langle 6,[0\mathrm{.}0001,0\mathrm{.}0016],[0\mathrm{.}9744,0\mathrm{.}9919]\rangle ,\langle 7,[0\mathrm{.}0081,0\mathrm{.}0625],[0\mathrm{.}8704,0\mathrm{.}9375]\rangle ,\\ {} \langle 8,[0\mathrm{.}1296,0\mathrm{.}2401],[0\mathrm{.}3439,0\mathrm{.}5904]\rangle ,\langle 9,[0\mathrm{.}4096,0\mathrm{.}6561],[0,0\mathrm{.}3439]\rangle ,\\ {} \langle 10,[1,1],[0,0]\rangle \end{array}\hspace{-1.55pt}\Bigg\}\mathrm{.}\end{array}\]
To capture the semantics of linguistic expressions, IVPFSs are adopted to model the effectiveness of structural linguistic variables. In line with the classification framework introduced by De et al. (2000), the linguistic term “LARGE” serves as the baseline reference. Based on this foundation, a hierarchy of modified linguistic intensifiers is defined, including “More or Less LARGE” (B), “Very LARGE” (${B^{2}}$), “Quite Very LARGE” (${B^{3}}$), and “Very Very LARGE” (${B^{4}}$), to represent increasing degrees of emphasis.
In accordance with the underlying mathematical properties, the entropy measures defined for interval-valued Pythagorean fuzzy sets must satisfy the criteria established by Hung and Yang (2006), as summarized in Table 1; i.e. $E(B)\gt E({B^{2}})\gt E({B^{3}})\gt E({B^{4}})$. It is evident that the proposed entropy measure demonstrates sound behaviour and compares favourably with existing interval-valued Pythagorean fuzzy entropy measures.
Table 1
Evaluation of the proposed and existing entropy measures for IVPFSs.
| ${E_{1}}$ (.) | ${E_{2}}$ (.) | ${E_{3}}$ (.) | ${E_{4}}$ (.) | ${E_{5}}$ (.) | E (.) | |
| B | 0.478 | 0.489 | 0.524 | 0.508 | 0.637 | 0.579 |
| ${B^{2}}$ | 0.424 | 0.433 | 0.472 | 0.364 | 0.593 | 0.532 |
| ${B^{3}}$ | 0.380 | 0.394 | 0.424 | 0.254 | 0.519 | 0.473 |
| ${B^{4}}$ | 0.339 | 0.356 | 0.382 | 0.191 | 0.464 | 0.423 |
3.3 An Improved Score Function for IVPFN
The notion of a score function was proposed by Peng and Yang (2016), which can derive a crisp value directly from a single IVPFV. Further, the researchers have paid considerable emphasis on studying the score function and its properties for comparing the IVPFVs. However, there are some situations where existing IVPF-score functions (e.g. Alrasheedi and Jeevaraj, 2023; Rani et al., 2023) produce the same score values. To overcome the limitations of existing IVPF-score functions, a modified score function for an IVPFV is offered.
Let $\delta =([{\mu ^{-}},{\mu ^{+}}],[{\nu ^{-}},{\nu ^{+}}])$ be an IVPFN. Then a modified score function is proposed for ‘δ’ and given as
(10)
\[\begin{array}{r@{\hskip0pt}l}\displaystyle S(\delta )& \displaystyle =\Bigl\{{\left({\mu ^{-}}\right)^{2}}+{\left({\mu ^{+}}\right)^{2}}+\left(1-{\left({\nu ^{-}}\right)^{2}}\right)+\left(1-{\left({\nu ^{+}}\right)^{2}}\right)+{\left({\mu ^{-}}\right)^{2}}{\left({\mu ^{+}}\right)^{2}}\\ {} & \displaystyle \hspace{10.0pt}-\sqrt{\left(1-{\left({\nu ^{-}}\right)^{2}}\right)\left(1-{\left({\nu ^{+}}\right)^{2}}\right)}\Bigr\}/4.\end{array}\]Proof.
Similarly, we differentiate Eq. (10) partially to ${\nu ^{-}}$ and ${\nu ^{+}}$, respectively:
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \frac{\partial S(\delta )}{\partial {\nu ^{-}}}=-\frac{{\nu ^{-}}}{2}\Bigg(1-\frac{(1-{({\nu ^{+}})^{2}})}{\sqrt{(1-{({\nu ^{-}})^{2}})(1-{({\nu ^{-}})^{2}})}}\Bigg)\hspace{10.0pt}\text{and}\hspace{10.0pt}\\ {} & \displaystyle \frac{\partial S(\delta )}{\partial {\nu ^{+}}}=-\frac{{\nu ^{+}}}{2}\Bigg(1-\frac{(1-{({\nu ^{-}})^{2}})}{\sqrt{(1-{({\nu ^{-}})^{2}})(1-{({\nu ^{-}})^{2}})}}\Bigg).\end{array}\]
Since $0\leqslant {\mu ^{-}}\leqslant {\mu ^{+}}\leqslant 1$ and $0\leqslant {\nu ^{-}}\leqslant {\nu ^{+}}\leqslant 1$, therefore, $\frac{\partial S(\delta )}{\partial {\mu ^{-}}}\geqslant 0$, $\frac{\partial S(\delta )}{\partial {\mu ^{+}}}\geqslant 0$, $\frac{\partial S(\delta )}{\partial {\nu ^{-}}}\leqslant 0$ and $\frac{\partial S(\delta )}{\partial {\nu ^{+}}}\leqslant 0$. Consequently, we obtain that the score function $S(\delta )$ raises continuously over ${\mu ^{-}}$ and ${\mu ^{+}}$, and decreases continuously over ${\nu ^{-}}$ and ${\nu ^{+}}$. □
Proof.
According to Property 1, we obtain that the score function $S(\delta )$ raises continuously over ${\mu ^{-}}$ and ${\mu ^{+}}$, and decreases continuously over ${\nu ^{-}}$ and ${\nu ^{+}}$. Therefore, $S(\delta )$ can have the maximum value ‘1’ if and only if $\delta =([1,1],[0,0])$ and $S(\delta )$ can have the minimum value ‘0’ if and only if $\delta =([0,0],[1,1])$. □
Property 3.
Let ${\delta _{1}}=([{\mu _{1}^{-}},{\mu _{1}^{+}}],[{\nu _{1}^{-}},{\nu _{1}^{+}}])$ and ${\delta _{2}}=([{\mu _{2}^{-}},{\mu _{2}^{+}}],[{\nu _{2}^{-}},{\nu _{2}^{+}}])$ be two IVPFNs. If ${\mu _{1}^{-}}\gt {\mu _{2}^{-}}$, ${\mu _{1}^{+}}\gt {\mu _{2}^{+}}$, ${\nu _{1}^{-}}\lt {\nu _{2}^{-}}$ and ${\nu _{1}^{+}}\lt {\nu _{2}^{+}}$, then $S({\delta _{1}})\gt S({\delta _{2}})$.
Proof.
From Property 1, we obtain that the score function $S(\delta )$ increases continuously over ${\mu ^{-}}$ and ${\mu ^{+}}$, and decreases continuously over ${\nu ^{-}}$ and ${\nu ^{+}}$. Hence, if ${\mu _{1}^{-}}\gt {\mu _{2}^{-}},{\mu _{1}^{+}}\gt {\mu _{2}^{+}}$, ${\nu _{1}^{-}}\lt {\nu _{2}^{-}}$ and ${\nu _{1}^{+}}\lt {\nu _{2}^{+}}$, then $S({\delta _{1}})\gt S({\delta _{2}})$. □
To appraise the feasibility of the developed score function $S(\delta )$ for comparing IVPFNs, Table 2 presents a comparison between the results achieved using the proposed score function and those obtained from the score functions developed by Peng and Yang (2016), Garg (2016), Garg (2017), Peng and Li (2019), Alrasheedi and Jeevaraj (2023) and Rani et al. (2023).
-
I. To compare three different IVPFNs ${\delta _{1}}=([0.4,0.7],[0.4,0.7])$, ${\delta _{2}}=([0.3,0.4],[0.3,0.4])$, and ${\delta _{8}}=([0,0.7],[0,0.7])$, the score function by Peng and Yang (2016) presents the same result, ‘0’. Additionally, Peng & Yang’s score function is not able to distinguish the IVPFNs ${\delta _{3}}=([0,0.5],[0.1,0.7])$ and ${\delta _{4}}=([0.3,0.4],[0.5,0.5])$ as ${S_{1}}({\delta _{3}})=-0.125={S_{1}}({\delta _{4}})$.
-
II. Similarly, Garg’s score formula (Garg, 2016) is unable to compare two different IVPFNs, ${\delta _{5}}=([0,1],[0,0])$ and ${\delta _{6}}=([0,0],[1,1])$, as ${S_{2}}({\delta _{5}})=0={S_{2}}({\delta _{6}})$.
-
III. For ${\delta _{1}}=([0.4,0.7],[0.4,0.7])$ and ${\delta _{2}}=([0.3,0.4],[0.3,0.4])$, we get ${S_{3}}({\delta _{1}})=0={S_{3}}({\delta _{2}})$, indicating that Garg’s score formula ${S_{3}}(.)$ (Garg, 2017) cannot distinguish ${\delta _{1}}$ and ${\delta _{2}}$.
-
IV. The score function is used to convert IVPFN into a crisp number, but the score formula ${S_{4}}(.)$ offered by Peng and Li (2019) is unable to convert an IVPFN ${\delta _{6}}=([0,0],[1,1])$ into a crisp value and provides an unreasonable result; i.e. “division-by-zero problem”. For the same IVPFN ${\delta _{6}}=([0,0],[1,1])$, the result obtained by the score function offered by Alrasheedi and Jeevaraj (2023) is not consistent with the other score results.
-
V. For any two IVPFNs ${\delta _{3}}=([0,0.5],[0.1,0.7])$ and ${\delta _{7}}=([0.7,0.7]$, $[\sqrt{0.495},\sqrt{0.51}])$, we apply the score function ${S_{6}}(.)$ and obtain the result, as ${S_{6}}({\delta _{3}})=0.124={S_{6}}({\delta _{7}})$, it means that the score function offered by Rani et al. (2023) fails to distinguish these IVPFNs.
According to Table 2, it can be observed that the suggested score function $S(\delta )$ can efficiently avoid the inadequacies of existing IVPF-score functions (Peng and Yang, 2016; Garg, 2016; 2017; Peng and Li, 2019; Alrasheedi and Jeevaraj, 2023; Rani et al., 2023). The proposed score function is capable of distinguishing between two IVPFNs even in cases where the existing score functions fail to operate.
Table 2
Comparative results for the IVPF-score function.
| Score formulae | ${\delta _{1}}=([0\mathrm{.}4,0\mathrm{.}7],[0\mathrm{.}4,0\mathrm{.}7])$ | ${\delta _{2}}=([0\mathrm{.}3,0\mathrm{.}4],[0\mathrm{.}3,0\mathrm{.}4])$ | ${\delta _{3}}=([0,0\mathrm{.}5],[0\mathrm{.}1,0\mathrm{.}7])$ | ${\delta _{4}}=([0\mathrm{.}3,0\mathrm{.}4],[0\mathrm{.}5,0\mathrm{.}5])$ | ${\delta _{5}}=([0,1],[0,0])$ | ${\delta _{6}}=([0,0],[1,1])$ | ${\delta _{7}}=([0\mathrm{.}7,0\mathrm{.}7],[\sqrt{0\mathrm{.}495},\sqrt{0\mathrm{.}51}])$ | ${\delta _{8}}=([0,0\mathrm{.}7],[0,0\mathrm{.}7])$ |
| Peng and Yang (2016) ${S_{1}}(\delta )=\frac{1}{2}({({\mu ^{-}})^{2}}+{({\mu ^{+}})^{2}}-{({\nu ^{-}})^{2}}-{({\nu ^{+}})^{2}})$ | ${0^{\ast }}$ | 0 | $-0\mathrm{.}125$ | $-0\mathrm{.}125$ | 0.5 | $-1$ | $-0\mathrm{.}013$ | 0 |
| Garg (2016) ${S_{2}}(\delta )=\frac{({({\mu ^{-}})^{2}}-\sqrt{1-{({\mu ^{-}})^{2}}-{({\nu ^{-}})^{2}}})+({({\mu ^{+}})^{2}}-\sqrt{1-{({\mu ^{+}})^{2}}-{({\nu ^{+}})^{2}}})}{2}$ | $-0\mathrm{.}158$ | $-0\mathrm{.}74$ | $-0\mathrm{.}627$ | $-0\mathrm{.}665$ | 0 | 0 | 0.429 | $-0\mathrm{.}326$ |
| Garg (2017) ${S_{3}}(\delta )=\frac{1}{2}\Bigg(\begin{array}{l}({({\mu ^{-}})^{2}}-{({\nu ^{-}})^{2}})(1+\sqrt{1-{({\mu ^{-}})^{2}}-{({\nu ^{-}})^{2}}})\\ {} +({({\mu ^{-}})^{2}}-{({\nu ^{-}})^{2}})(1+\sqrt{1-{({\mu ^{+}})^{2}}-{({\nu ^{+}})^{2}}})\end{array}\Bigg)$ | 0 | 0 | $-0\mathrm{.}191$ | $-0\mathrm{.}225$ | 0.5 | $-1$ | $-0\mathrm{.}013$ | 0 |
| Peng and Li (2019) ${S_{4}}(\delta )=\frac{({({\mu ^{-}})^{2}}+{({\mu ^{+}})^{2}}-{({\nu ^{-}})^{2}}-{({\nu ^{+}})^{2}})}{2}+\frac{({({\mu ^{-}})^{2}}-{({\mu ^{-}})^{2}}{({\nu ^{+}})^{2}}-{({\mu ^{-}})^{4}})}{2({({\nu ^{-}})^{2}}-{({\mu ^{+}})^{2}}{({\nu ^{-}})^{2}}-{({\nu ^{-}})^{4}}+1)}$ | 0.052 | 0.035 | $-0\mathrm{.}125$ | $-0\mathrm{.}099$ | 0.5 | NaN${^{\ast \ast }}$ | $-0\mathrm{.}009$ | 0 |
| Alrasheedi and Jeevaraj (2023) ${S_{5}}(\delta )=\frac{1}{3}({({\mu ^{-}})^{2}}+{({\mu ^{+}})^{2}}-{({\nu ^{-}})^{2}}-{({\nu ^{+}})^{2}}+{({\mu ^{-}})^{2}}{({\mu ^{+}})^{2}}+{({\nu ^{-}})^{2}}{({\nu ^{+}})^{2}})$ | 0.052 | 0.01 | $-0\mathrm{.}082$ | $-0\mathrm{.}058$ | 0.333 | $-0\mathrm{.}333$ | 0.156 | 0 |
| Rani et al. (2023) ${S_{6}}(\delta )=\frac{{({\mu ^{-}})^{2}}(1-{({\nu ^{+}})^{2}})+{({\mu ^{+}})^{2}}(1-{({\nu ^{-}})^{2}})-{({\mu ^{-}})^{2}}{({\mu ^{+}})^{2}}}{2}$ | 0.207 | 0.103 | 0.124 | 0.087 | 0.5 | 0 | 0.124 | 0.245 |
| Proposed IVPF-score function | 0.356 | 0.285 | 0.26 | 0.254 | 0.5 | 0 | 0.429 | 0.321 |
3.4 Proposed Distance Measure for IVPFSs
The distance measure concept is a mathematical term that calculates the dissimilarity score between two objects/entities.
Theorem 2.
For $B,N\in \textit{IVPFSs}(L)$, the function ‘d’, defined by
(11)
\[d(B,N)=\frac{1}{r}{\sum \limits_{i=1}^{r}}\sqrt{\begin{array}{l}\Bigl[{\left(\begin{array}{l}\frac{{({\mu _{B}^{-}}({l_{i}}))^{2}}+{({\mu _{B}^{+}}({l_{i}}))^{2}}-{({\nu _{B}^{-}}({l_{i}}))^{2}}-{({\nu _{B}^{+}}({l_{i}}))^{2}}}{4}\\ {} -\frac{{({\mu _{N}^{-}}({l_{i}}))^{2}}+{({\mu _{N}^{+}}({l_{i}}))^{2}}-{({\nu _{N}^{-}}({l_{i}}))^{2}}-{({\nu _{N}^{+}}({l_{i}}))^{2}}}{4}\end{array}\right)^{2}}\\ {} +\frac{1}{3}{\left(\begin{array}{l}\frac{{({\mu _{B}^{-}}({l_{i}}))^{2}}+{({\mu _{B}^{+}}({l_{i}}))^{2}}+{({\nu _{B}^{-}}({l_{i}}))^{2}}+{({\nu _{B}^{+}}({l_{i}}))^{2}}}{4}\\ {} -\frac{{({\mu _{N}^{-}}({l_{i}}))^{2}}+{({\mu _{N}^{+}}({l_{i}}))^{2}}+{({\nu _{N}^{-}}({l_{i}}))^{2}}+{({\nu _{N}^{+}}({l_{i}}))^{2}}}{4}\end{array}\right)^{2}}\Bigr]\end{array}}\]is a distance measure for IVPFSs.
Proof.
(b1) For any two IVPFSs P and Q, we have $0\leqslant {\mu _{B}^{-}}({l_{i}})\leqslant {\mu _{B}^{+}}({l_{i}})\leqslant 1,0\leqslant {\mu _{N}^{-}}({l_{i}})\leqslant {\mu _{N}^{+}}({l_{i}})\leqslant 1$, $0\leqslant {\nu _{B}^{-}}({l_{i}})\leqslant {\nu _{B}^{+}}({l_{i}})\leqslant 1$ and $0\leqslant {\nu _{N}^{-}}({l_{i}})\leqslant {\nu _{N}^{+}}({l_{i}})\leqslant 1,\forall {l_{i}}\in L$. Here, we can observe that each term of Eq. (11) is non-negative; and then $d(B,N)\geqslant 0$. Moreover, for any two IVPFSs $B=([1,1],[0,0])$ and $N=([0,0],[1,1])$ in the fixed universal set L, then we get $d(B,N)\leqslant 1$.
(b2) Suppose that $d(B,N)=0$, and based on Eq. (11):
(12)
\[{\left(\begin{array}{l}\frac{{({\mu _{B}^{-}}({l_{i}}))^{2}}+{({\mu _{B}^{+}}({l_{i}}))^{2}}-{({\nu _{B}^{-}}({l_{i}}))^{2}}-{({\nu _{B}^{+}}({l_{i}}))^{2}}}{4}\\ {} -\frac{{({\mu _{N}^{-}}({l_{i}}))^{2}}+{({\mu _{N}^{+}}({l_{i}}))^{2}}-{({\nu _{N}^{-}}({l_{i}}))^{2}}-{({\nu _{N}^{+}}({l_{i}}))^{2}}}{4}\end{array}\right)^{2}}=0,\]and
(13)
\[{\left(\begin{array}{l}\frac{{({\mu _{B}^{-}}({l_{i}}))^{2}}+{({\mu _{B}^{+}}({l_{i}}))^{2}}+{({\nu _{B}^{-}}({l_{i}}))^{2}}+{({\nu _{B}^{+}}({l_{i}}))^{2}}}{4}\\ {} -\frac{{({\mu _{N}^{-}}({l_{i}}))^{2}}+{({\mu _{N}^{+}}({l_{i}}))^{2}}+{({\nu _{N}^{-}}({l_{i}}))^{2}}+{({\nu _{N}^{+}}({l_{i}}))^{2}}}{4}\end{array}\right)^{2}}=0.\]Solving Eqs. (12) and (13), we get ${\mu _{B}^{-}}({l_{i}})={\mu _{N}^{-}}({l_{i}}),{\mu _{B}^{+}}({l_{i}})={\mu _{N}^{+}}({l_{i}}),{\nu _{B}^{-}}({l_{i}})={\nu _{N}^{-}}({l_{i}})$ and ${\nu _{B}^{+}}({l_{i}})={\nu _{N}^{+}}({l_{i}})$, $\forall {l_{i}}\in L$. Hence, B = N.
On the contrary, consider $B=N$, then ${\mu _{B}^{-}}({l_{i}})={\mu _{N}^{-}}({l_{i}})$, ${\mu _{B}^{+}}({l_{i}})={\mu _{N}^{+}}({l_{i}})$, ${\nu _{B}^{-}}({l_{i}})={\nu _{N}^{-}}({l_{i}})$ and ${\nu _{B}^{+}}({l_{i}})={\nu _{N}^{+}}({l_{i}})$, $\forall {l_{i}}\in L$. Then, from Eq. (11), it is obvious that $d(B,N)=0$.
(b3) Since the result follows directly from Eq. (11), the proof is omitted.
(b4) For any three IVPFSs B, N and G, let $B\subseteq N\subseteq G$, we have ${\mu _{B}^{-}}({l_{i}})\leqslant {\mu _{N}^{-}}({l_{i}})\leqslant {\mu _{G}^{-}}({l_{i}})$, ${\mu _{B}^{+}}({l_{i}})\leqslant {\mu _{N}^{+}}({l_{i}})\leqslant {\mu _{G}^{+}}({l_{i}})$, ${\nu _{G}^{-}}({l_{i}})\leqslant {\nu _{N}^{-}}({l_{i}})\leqslant {\nu _{B}^{-}}({l_{i}})$ and ${\nu _{G}^{+}}({l_{i}})\leqslant {\nu _{N}^{+}}({l_{i}})\leqslant {\nu _{B}^{+}}({l_{i}})$, $\forall {l_{i}}\in L$.
Consider a function $f({z_{1}},{z_{2}},{z_{3}},{z_{4}})$, where
(14)
\[f({z_{1}},{z_{2}},{z_{3}},{z_{4}})=\left(\begin{array}{l}{\left(\left({z_{1}^{2}}+{z_{2}^{2}}-{z_{3}^{2}}-{z_{4}^{2}}\right)-\left({\alpha _{1}^{2}}+{\alpha _{2}^{2}}-{\alpha _{3}^{2}}-{\alpha _{4}^{2}}\right)\right)^{2}}\\ {} +\frac{1}{3}{\left(\left({z_{1}^{2}}+{z_{2}^{2}}+{z_{3}^{2}}+{z_{4}^{2}}\right)-\left({\alpha _{1}^{2}}+{\alpha _{2}^{2}}+{\alpha _{3}^{2}}+{\alpha _{4}^{2}}\right)\right)^{2}}\end{array}\right),\]where $0\leqslant {z_{1}},{z_{2}},{z_{3}},{z_{4}}\leqslant 1$ and $0\leqslant {\alpha _{1}},{\alpha _{2}},{\alpha _{3}},{\alpha _{4}}\leqslant 1$.
Differentiate $f({z_{1}},{z_{2}},{z_{3}},{z_{4}})$ partially with respect to ${z_{1}}$, ${z_{2}}$, ${z_{3}}$ and ${z_{4}}$, respectively, we have
(15)
\[\frac{\partial f({z_{1}},{z_{2}},{z_{3}},{z_{4}})}{\partial {z_{1}}}=\frac{8}{3}{z_{1}}\left(2\left({z_{1}^{2}}+{z_{2}^{2}}\right)-\left({z_{3}^{2}}+{z_{4}^{2}}\right)-2\left({\alpha _{1}^{2}}+{\alpha _{2}^{2}}\right)+\left({\alpha _{3}^{2}}+{\alpha _{4}^{2}}\right)\right),\](16)
\[\frac{\partial f({z_{1}},{z_{2}},{z_{3}},{z_{4}})}{\partial {z_{2}}}=\frac{8}{3}{z_{2}}\left(2\left({z_{1}^{2}}+{z_{2}^{2}}\right)-\left({z_{3}^{2}}+{z_{4}^{2}}\right)-2\left({\alpha _{1}^{2}}+{\alpha _{2}^{2}}\right)+\left({\alpha _{3}^{2}}+{\alpha _{4}^{2}}\right)\right),\]For the conditions $0\leqslant {\alpha _{1}}\leqslant {z_{1}}\leqslant 1$, $0\leqslant {\alpha _{2}}\leqslant {z_{2}}\leqslant 1$, $0\leqslant {z_{3}}\leqslant {\alpha _{3}}\leqslant 1$ and $0\leqslant {z_{4}}\leqslant {\alpha _{4}}\leqslant 1$, we have $\frac{\partial f({z_{1}},{z_{2}},{z_{3}},{z_{4}})}{\partial {z_{1}}}\geqslant 0$, $\frac{\partial f({z_{1}},{z_{2}},{z_{3}},{z_{4}})}{\partial {z_{2}}}\geqslant 0$, $\frac{\partial f({z_{1}},{z_{2}},{z_{3}},{z_{4}})}{\partial {z_{3}}}\leqslant 0$ and $\frac{\partial f({z_{1}},{z_{2}},{z_{3}},{z_{4}})}{\partial {z_{4}}}\leqslant 0$, respectively. Here, we can observe that the function $f({z_{1}},{z_{2}},{z_{3}},{z_{4}})$ is increasing with respect to ${z_{1}}$ and ${z_{2}}$ for $0\leqslant {\alpha _{1}}\leqslant {z_{1}}\leqslant 1$ and $0\leqslant {\alpha _{2}}\leqslant {z_{2}}\leqslant 1$, respectively, and decreasing with respect to ${z_{3}}$ and ${z_{4}}$ for $0\leqslant {z_{3}}\leqslant {\alpha _{3}}\leqslant 1$ and $0\leqslant {z_{4}}\leqslant {\alpha _{4}}\leqslant 1$, respectively. Now, let us assume three IVPFSs B, N and G such that $B\subseteq N\subseteq G$, then ${\mu _{B}^{-}}({l_{i}})\leqslant {\mu _{N}^{-}}({l_{i}})\leqslant {\mu _{G}^{-}}({l_{i}})$, ${\mu _{B}^{+}}({l_{i}})\leqslant {\mu _{N}^{+}}({l_{i}})\leqslant {\mu _{G}^{+}}({l_{i}})$, ${\nu _{G}^{-}}({l_{i}})\leqslant {\nu _{N}^{-}}({l_{i}})\leqslant {\nu _{B}^{-}}({l_{i}})$ and ${\nu _{G}^{+}}({l_{i}})\leqslant {\nu _{N}^{+}}({l_{i}})\leqslant {\nu _{B}^{+}}({l_{i}}),\forall {l_{i}}\in L$. From the monotonicity of $f({z_{1}},{z_{2}},{z_{3}},{z_{4}})$, we can have $d(B,G)\geqslant d(B,N)$. By a similar argument, we can demonstrate that $d(B,G)\geqslant d(N,G)$.
Theoretically, it is clear that the proposed function, given in Eq. (11), satisfies all the required characteristics of a distance measure under the IVPF environment. To prove the superiority of the introduced measure, a comparison with several existing IVPF distance measures is made. To this aim, we first recall these IVPF distance measures, given as follows:
-
∙ IVPF distance by Zhang (2016)
(19)
\[\begin{array}{c}\displaystyle {d_{1}}\displaystyle (B\displaystyle ,N\displaystyle )\displaystyle =\displaystyle \frac{1}{4r}{\sum \limits_{i=1}^{r}}\left(\begin{array}{l}|{({\mu _{B}^{-}}({l_{i}}))^{2}}-{({\mu _{N}^{-}}({l_{i}}))^{2}}|+|{({\mu _{B}^{+}}({l_{i}}))^{2}}-{({\mu _{N}^{+}}({l_{i}}))^{2}}|\\ {} +|{({\nu _{B}^{-}}({l_{i}}))^{2}}-{({\nu _{N}^{-}}({l_{i}}))^{2}}|+|{({\nu _{B}^{+}}({l_{i}}))^{2}}-{({\nu _{N}^{+}}({l_{i}}))^{2}}|\\ {} +|{({\pi _{B}^{-}}({l_{i}}))^{2}}-{({\pi _{N}^{-}}({l_{i}}))^{2}}|+|{({\pi _{B}^{+}}({l_{i}}))^{2}}-{({\pi _{N}^{+}}({l_{i}}))^{2}}|\end{array}\right)\displaystyle .\\ {} \end{array}\] -
∙ IVPF distance by Kumar et al. (2020)
(20)
\[\begin{array}{c}\displaystyle {d_{2}}\displaystyle (B\displaystyle ,N\displaystyle )\displaystyle =\sqrt[p]{\displaystyle \frac{1}{4r}{\sum \limits_{i=1}^{r}}\left(\begin{array}{l}{|{({\mu _{B}^{-}}({l_{i}}))^{2}}-{({\mu _{N}^{-}}({l_{i}}))^{2}}|^{p}}+{|{({\mu _{B}^{+}}({l_{i}}))^{2}}-{({\mu _{N}^{+}}({l_{i}}))^{2}}|^{p}}\\ {} +{|{({\nu _{B}^{-}}({l_{i}}))^{2}}-{({\nu _{N}^{-}}({l_{i}}))^{2}}|^{p}}+{|{({\nu _{B}^{+}}({l_{i}}))^{2}}-{({\nu _{N}^{+}}({l_{i}}))^{2}}|^{p}}\\ {} +{|{({\pi _{B}^{-}}({l_{i}}))^{2}}-{({\pi _{N}^{-}}({l_{i}}))^{2}}|^{p}}+{|{({\pi _{B}^{+}}({l_{i}}))^{2}}-{({\pi _{N}^{+}}({l_{i}}))^{2}}|^{p}}\end{array}\right)}\displaystyle .\\ {} \end{array}\] -
∙ IVPF distance by Liu et al. (2025)
(21)
\[{d_{3}}(B,N)=\sqrt{\frac{1}{4r}{\sum \limits_{i=1}^{r}}\left(\begin{array}{l}\frac{{({({\mu _{B}^{-}}({l_{i}}))^{2}}-{({\mu _{N}^{-}}({l_{i}}))^{2}})^{2}}}{{({\mu _{B}^{-}}({l_{i}}))^{2}}+{({\mu _{N}^{-}}({l_{i}}))^{2}}}+\frac{{({({\mu _{B}^{+}}({l_{i}}))^{2}}-{({\mu _{N}^{+}}({l_{i}}))^{2}})^{2}}}{{({\mu _{B}^{+}}({l_{i}}))^{2}}+{({\mu _{N}^{+}}({l_{i}}))^{2}}}\\ {} +\frac{{({({\nu _{B}^{-}}({l_{i}}))^{2}}-{({\nu _{N}^{-}}({l_{i}}))^{2}})^{2}}}{{({\nu _{B}^{-}}({l_{i}}))^{2}}+{({\nu _{N}^{-}}({l_{i}}))^{2}}}+\frac{{({({\nu _{B}^{+}}({l_{i}}))^{2}}-{({\nu _{N}^{+}}({l_{i}}))^{2}})^{2}}}{{({\nu _{B}^{+}}({l_{i}}))^{2}}+{({\nu _{N}^{+}}({l_{i}}))^{2}}}\end{array}\right)}.\] -
∙ IVPF distance by Yu et al. (2025)
(22)
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle {d_{4}}(B,N)\\ {} & \displaystyle \hspace{10.0pt}=\displaystyle 1-{\log _{2}}\left(2-\sqrt[q]{\displaystyle \frac{1}{4r}{\displaystyle \displaystyle \sum \limits_{i=1}^{r}}\Bigg(\begin{array}{l}\mathrm{|}{({\mu _{B}^{-}}({l_{i}}))^{2}}-{({\mu _{N}^{-}}({l_{i}}))^{2}}{\mathrm{|}^{q}}+\mathrm{|}{({\mu _{B}^{+}}({l_{i}}))^{2}}-{({\mu _{N}^{+}}({l_{i}}))^{2}}{\mathrm{|}^{q}}\\ {} +\mathrm{|}{({\nu _{B}^{-}}({l_{i}}))^{2}}-{({\nu _{N}^{-}}({l_{i}}))^{2}}{\mathrm{|}^{q}}+\mathrm{|}{({\nu _{B}^{+}}({l_{i}}))^{2}}-{({\nu _{N}^{+}}({l_{i}}))^{2}}{\mathrm{|}^{q}}\end{array}\Bigg)}\right),\end{array}\](23)
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle {d_{5}}(B,N)\\ {} & \displaystyle \hspace{10.0pt}=\displaystyle 1-{\log _{2}}\left(2-\sqrt[q]{\displaystyle \frac{1}{4r}{\displaystyle \displaystyle \sum \limits_{i=1}^{r}}\left(\begin{array}{l}\mathrm{|}{({\mu _{B}^{-}}({l_{i}}))^{2}}-{({\mu _{N}^{-}}({l_{i}}))^{2}}{\mathrm{|}^{q}}+\mathrm{|}{({\mu _{B}^{+}}({l_{i}}))^{2}}-{({\mu _{N}^{+}}({l_{i}}))^{2}}{\mathrm{|}^{q}}\\ {} +\mathrm{|}{({\nu _{B}^{-}}({l_{i}}))^{2}}-{({\nu _{N}^{-}}({l_{i}}))^{2}}{\mathrm{|}^{q}}+\mathrm{|}{({\nu _{B}^{+}}({l_{i}}))^{2}}-{({\nu _{N}^{+}}({l_{i}}))^{2}}{\mathrm{|}^{q}}\\ {} +\mathrm{|}{({\pi _{B}^{-}}({l_{i}}))^{2}}-{({\pi _{N}^{-}}({l_{i}}))^{2}}{\mathrm{|}^{q}}+\mathrm{|}{({\pi _{B}^{+}}({l_{i}}))^{2}}-{({\pi _{N}^{+}}({l_{i}}))^{2}}{\mathrm{|}^{q}}\end{array}\right)}\right)\mathrm{.}\end{array}\] -
∙ IVPF distance by Moazzamigodarzi et al. (2025)
(24)
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle {d_{6}}(B,N)\\ {} & \displaystyle \hspace{10.0pt}=\displaystyle \frac{1}{4r({e^{2}}-1)}{\displaystyle \displaystyle \sum \limits_{i=1}^{r}}\Bigl[\Big({\Big({\mu _{B}^{-}}({l_{i}})\Big)^{2}}-{\Big({\mu _{N}^{-}}({l_{i}})\Big)^{2}}\Big)\Bigg({e^{\Big(\frac{4{({\mu _{B}^{-}}({l_{i}}))^{2}}}{1+{({\mu _{B}^{-}}({l_{i}}))^{2}}+{({\mu _{N}^{-}}({l_{i}}))^{2}}}\Big)}}-{e^{\Big(\frac{4{({\mu _{N}^{-}}({l_{i}}))^{2}}}{1+{({\mu _{B}^{-}}({l_{i}}))^{2}}+{({\mu _{N}^{-}}({l_{i}}))^{2}}}\Big)}}\Bigg)\\ {} & \displaystyle \hspace{20.0pt}+\left({\left({\mu _{B}^{+}}({l_{i}})\right)^{2}}-{\left({\mu _{N}^{+}}({l_{i}})\right)^{2}}\right)\left({e^{\left(\frac{4{({\mu _{B}^{+}}({l_{i}}))^{2}}}{1+{({\mu _{B}^{+}}({l_{i}}))^{2}}+{({\mu _{N}^{+}}({l_{i}}))^{2}}}\right)}}-{e^{\left(\frac{4{({\mu _{N}^{+}}({l_{i}}))^{2}}}{1+{({\mu _{B}^{+}}({l_{i}}))^{2}}+{({\mu _{N}^{+}}({l_{i}}))^{2}}}\right)}}\right)\\ {} & \displaystyle \hspace{20.0pt}+\left({\left({\nu _{B}^{-}}({l_{i}})\right)^{2}}-{\left({\nu _{N}^{-}}({l_{i}})\right)^{2}}\right)\left({e^{\left(\frac{4{({\nu _{B}^{-}}({l_{i}}))^{2}}}{1+{({\nu _{B}^{-}}({l_{i}}))^{2}}+{({\nu _{N}^{-}}({l_{i}}))^{2}}}\right)}}-{e^{\left(\frac{4{({\nu _{N}^{-}}({l_{i}}))^{2}}}{1+{({\nu _{B}^{-}}({l_{i}}))^{2}}+{({\nu _{N}^{-}}({l_{i}}))^{2}}}\right)}}\right)\\ {} & \displaystyle \hspace{20.0pt}+\left({\left({\nu _{B}^{+}}({l_{i}})\right)^{2}}-{\left({\nu _{N}^{+}}({l_{i}})\right)^{2}}\right)\left({e^{\left(\frac{4{({\nu _{B}^{+}}({l_{i}}))^{2}}}{1+{({\nu _{B}^{+}}({l_{i}}))^{2}}+{({\nu _{N}^{+}}({l_{i}}))^{2}}}\right)}}-{e^{\left(\frac{4{({\nu _{N}^{+}}({l_{i}}))^{2}}}{1+{({\nu _{B}^{+}}({l_{i}}))^{2}}+{({\nu _{N}^{+}}({l_{i}}))^{2}}}\right)}}\right)\Bigr].\end{array}\]
Table 3
Comparison of outcomes of developed and existing IVPF distance measures.
| Set-1 | Set-2 | Set-3 | Set-4 | Set-5 | Set-6 | |
| B | $([0\mathrm{.}25,0\mathrm{.}35],[0\mathrm{.}25,0\mathrm{.}35])$, | $([0\mathrm{.}25,0\mathrm{.}35],[0\mathrm{.}35,0\mathrm{.}45])$ | $([1,1],[0,0])$ | $([0\mathrm{.}3,0\mathrm{.}3],[0\mathrm{.}3,0\mathrm{.}3])$ | $([0\mathrm{.}1,0\mathrm{.}1],[0\mathrm{.}2,0\mathrm{.}2])$ | $([1,1],[0,0])$ |
| N | $([0\mathrm{.}35,0\mathrm{.}45],[0\mathrm{.}35,0\mathrm{.}45])$ | $([0\mathrm{.}35,0\mathrm{.}45],[0\mathrm{.}25,0\mathrm{.}35])$ | $([0,0],[0,0])$ | $([0,0],[0,0])$ | $([0\mathrm{.}3,0\mathrm{.}3],[0\mathrm{.}4,0\mathrm{.}4])$ | $([0,0],[1,1])$ |
| ${d_{1}}(B,N)$ | 0.14 | 0.07 | 1${^{\textbf{*}}}$ | 0.18 | 0.2 | 1 |
| ${d_{2}}(B,N)$ | 0.01 | 0.01 | 0.8409 | 0.09 | 0.04 | 1 |
| ${d_{3}}(B,N)$ | 0.56 | 0.28 | 4 | 0.72 | 0.8 | 4 |
| ${d_{4}}(B,N)$ | 0.052 | 0.052 | 0.729 | 0.066 | 0.077 | 1 |
| ${d_{5}}(B,N)$ | 0.0038 | 0.0038 | 0.5 | 0.0055 | 0.0075 | 1 |
| ${d_{6}}(B,N)$ | 0.14 | 0.07 | 1 | 0.18 | 0.2 | 1 |
| Proposed IVPF distance measure | 0.0404 | 0.07 | 0.5774 | 0.052 | 0.0611 | 1 |
Table 3 presents the required comparative outcomes of suggested and existing measures. It is observed that the previously developed IVPF distances: i.e. ${d_{2}}(B,N)$, ${d_{4}}(B,N)$ and ${d_{5}}(B,N)$, produced the same results for two various sets of IVPFSs, given by Set-1 and Set-2. Similarly, the distance formulae ${d_{1}}(B,N)$, ${d_{3}}(B,N)$ and ${d_{6}}(B,N)$ failed to distinguish the two sets of IVPFSs, given by Set-3 and Set-6. In addition, the property of the distance measure in Theorem 2 is not satisfied using ${d_{3}}(B,N)$ in Case 6. As $B=([1,1],[0,0])$ and $N=([0,0],[1,1])$ are the maximum and the minimum IVPFNs, therefore, it is expected that the distance between B and N is the maximum distance, arriving at ‘1’, while the IVPF distance suggested by Liu et al. (2025) obtains the result as ‘4’. The proposed IVPF distance does not exhibit any counter-intuitive behaviour and demonstrates a strong performance compared to others. □
4 The IVPF-WENSLO-RANCOM-CRADIS Model
With the help of proposed measures in Section 3, this section develops a novel fuzzy integrated decision support framework under the setting of IVPFSs and named “IVPF-WENSLO-RANCOM-CRADIS”. Figure 1 presents the flowchart that illustrates the execution steps of the developed framework.
4.1 Aggregate the Individual Decision Matrices
This procedure involves the following steps:
-
1. Create the Linguistic Assessment Decision matrix (LADM) on the basis of the opinions of a set of experts.
-
2. Determine the significance value of each expert.
-
3. Utilize the experts’ weights in the IVPFWA operator and find the aggregated opinion about the performance of each alternative over each criterion.
| Stage 1: Process for aggregating the individual decision opinions | |
| No. of steps | Description of the procedure |
| Step 1. | Create the LADM |
| Suppose that a MCGDM problem involves a finite set of ‘r’ alternatives $A=\{{A_{1}},{A_{2}},\dots ,{A_{s}}\}$ and they are to be ranked over a set of ‘t’ criteria $V=\{{v_{1}},{v_{2}},\dots ,{v_{t}}\}$. Let us invite a set of ‘q’ DMs $G=\{{g_{1}},{g_{2}},\dots ,{g_{q}}\}$ to give the linguistic assessment ratings of alternatives concerning each criterion. Consequently, a LADM is formed as $Y={({Y_{\mathit{ij}}^{(k)}})_{s\times t}}$, where ${Y_{\mathit{ij}}^{(k)}}$ stands for the linguistic assessment rating of ith alternative over jth criterion, provided by kth DM. Using the Likert’s scale, the LADM is transformed into IVPF decision matrix (IVPFDM) $F={({\upsilon _{\mathit{ij}}^{(k)}})_{s\times t}}$. | |
| Step 2. | Derive the DMs’ significance values |
Let $\omega ={({\omega _{1}},{\omega _{2}},\dots ,{\omega _{q}})^{T}}$ be the weight vector of q DMs $G=\{{g_{1}},{g_{2}},\dots ,{g_{q}}\}$ with $0\leqslant {\omega _{j}}\leqslant 1$ and ${\omega _{1}}+{\omega _{2}}+\cdots +{\omega _{q}}=1$. To find the numeric value of ω, we first create the average IVPFDM $\stackrel{ˉ}{F}={({\stackrel{ˉ}{\upsilon }_{\mathit{ij}}^{(k)}})_{s\times q}}$ using IVPFWA operator for each expert and then find the entropy of each DE over different alternatives through proposed entropy (6). Finally, the formula for finding ${\omega _{k}}$ is given as
(25)\[{\omega _{k}}=\frac{1-{\textstyle\textstyle\sum _{i=1}^{s}}(\stackrel{ˉ}{E}({\upsilon _{\mathit{ik}}}))}{q-{\textstyle\textstyle\sum _{k=1}^{q}}{\textstyle\textstyle\sum _{i=1}^{s}}(\stackrel{ˉ}{E}({\upsilon _{\mathit{ik}}}))}\hspace{8.0pt}(k=1,2,\dots ,q),\] |
|
| Step 3. | Create the IVPF-Aggregated Decision matrix (IVPFADM) |
It is important to note that, in the development of IVPFADM, individual viewpoints must be aggregated into a collective opinion to facilitate group decision-making. To construct an IVPFADM ${F_{A}}={({\tilde{\upsilon }_{\mathit{ij}}})_{s\times t}}$, the individual opinions of DMs about each alternative are aggregated through the use of experts’ weights into the IVPFWA operator, where ${\tilde{\upsilon }_{\mathit{ij}}}$ is an IVPFN, denotes an assessment value of ith alternative by means of jth criterion
(26)\[{\tilde{\upsilon }_{\mathit{ij}}}=\displaystyle \Bigg(\Bigg[1-{\displaystyle \prod \limits_{k=1}^{q}}{(1-{({\mu _{{\upsilon _{\mathit{ij}}^{k}}}^{-}})^{2}})^{{\omega _{k}}}},1-{\displaystyle \prod \limits_{k=1}^{q}}{(1-{({\mu _{{\upsilon _{\mathit{ij}}^{k}}}^{+}})^{2}})^{{\omega _{k}}}}\Bigg],\Bigg[{\displaystyle \prod \limits_{k=1}^{q}}{({\nu _{{\upsilon _{\mathit{ij}}^{k}}}^{-}})^{2{\omega _{k}}}},{\displaystyle \prod \limits_{\mathit{ik}=1}^{q}}{({\nu _{{\upsilon _{\mathit{ij}}^{k}}}^{+}})^{2{\omega _{k}}}}\Bigg]\Bigg)\mathrm{.}\] |
|
4.2 Integrated Weighting Model for Criteria Weights Determination
Let $w={({w_{1}},{w_{2}},\dots ,{w_{t}})^{T}}$ be the weight vector of the criteria set $V=\{{v_{1}},{v_{2}},\dots ,{v_{t}}\}$ with ${w_{j}}\in [0,1]$ and ${\sum _{j=1}^{t}}{w_{j}}=1$. To specify the numeric value of w, we incorporate the IVPF-WENSLO and the IVPF-RANCOM models for objective and subjective weights of criteria, on the basis of the following steps:
| Stage 2: Procedure to determine the criteria weights | |
| No. of steps | Description of the procedure |
| Step 1. | IVPF-WENSLO model for attaining the objective weight of each criterion |
| Sub-step 1.1: Find the normalized decision matrix. | |
| In this method, we first create the IVPF-score decision matrix and further normalize it as $N={({n_{\mathit{ij}}})_{s\times t}}$, where ${n_{\mathit{ij}}}=\frac{S({\tilde{\upsilon }_{\mathit{ij}}})}{{\textstyle\sum _{i=1}^{s}}S({\tilde{\upsilon }_{\mathit{ij}}})}$ and $S({\tilde{\upsilon }_{\mathit{ij}}})$ denotes the score value of an element ‘${\tilde{\upsilon }_{\mathit{ij}}}$’ of an IVPFADM, $i=1,2,\dots ,s$ and $j=1,2,\dots ,t$. | |
| Sub-step 1.2: Specify the class interval of criterion. | |
| With the use of Sturges’ rule (Pamucar et al., 2023), the class interval of each criterion is obtained via Eq. (27): | |
| Sub-step 1.3: Estimate the criterion slope. | |
| In line with Eq. (27), the slope of each criterion is obtained by means of Eq. (28): | |
| Sub-step 1.4: Compute the criterion envelope. | |
| Considering the class interval of the criterion, the envelope is achieved as the total Euclidean distance across consecutive normalized values and calculated through Eq. (29): | |
| Sub-step 1.5: Compute the ratio of envelope-slope. | |
| Taking into account the criterion slope ‘$\tan {\phi _{j}}$’ and criterion envelope ‘${O_{j}}$’, the envelope-slope ratio of each criterion is determined with the help of Eq. (30): | |
| Sub-step 1.6: Obtain the objective weight ‘${w_{j}^{o}}$’ of jth criterion using Eq. (31): | |
| Step 2. | IVPF-RANCOM model for finding the subjective weight of each criterion. |
| Sub-step 2.1: Create the aggregated column matrix. | |
| The DMs offer their linguistic opinions regarding the performance of each criterion and further converted into IVPFNs. Consequently, construct the aggregated column matrix ${C_{A}}={({c_{j}})_{1\times t}}$, where ${c_{j}}$ is an aggregated IVPFN, calculated through IVPFWA operator. | |
| Sub-step 2.2: Arrange the criteria positions. | |
| Obtain the score value of each element of aggregated column matrix and then arrange the criteria positions accordingly. | |
| Sub-step 2.3: Create the RANCOM matrix. | |
Considering the pairwise comparison of criteria preferences, create the RANCOM matrix $M={({\beta _{\mathit{lj}}})_{t\times t}}$, where
(32)\[M=\hspace{2.22pt}\hspace{9.8pt}\left[\hspace{-18.55pt}\begin{array}{r@{\hskip4.0pt}c@{\hskip4.0pt}c@{\hskip4.0pt}c@{\hskip4.0pt}c}& & & & \\ {} & \hspace{1.83pt}{v_{1}}& {v_{2}}& \dots & {v_{t}}\\ {} {v_{1}}\hfill & \hspace{1.83pt}{\beta _{11}}& {\beta _{11}}& \cdots \hspace{1.33pt}& {\beta _{1t}}\\ {} {v_{2}}\hfill & \hspace{1.83pt}{\beta _{21}}& {\beta _{22}}& \cdots \hspace{1.33pt}& {\beta _{2t}}\\ {} \hfill \vdots \hfill & \hspace{1.83pt}\vdots & \vdots & \ddots & \vdots \\ {} {v_{t}}\hfill & \hspace{1.83pt}{\beta _{t1}}& {\beta _{t2}}& \cdots \hspace{1.33pt}& {\beta _{\mathit{tt}}}\\ {} & & & & \end{array}\hspace{1.33pt}\right]\hspace{8.0pt}\text{and}\hspace{8.0pt}{\beta _{\mathit{lj}}}=\left\{\begin{array}{l}1,\text{if}\hspace{2.5pt}S({c_{j}})\lt S({c_{l}}),\\ {} 0\mathrm{.}5,\text{if}\hspace{2.5pt}S({c_{j}})=S({c_{l}}),\\ {} 0,\text{if}\hspace{2.5pt}S({c_{j}})\gt S({c_{l}})\mathrm{.}\end{array}\right.\] |
|
| Sub-step 2.4: Find the summed weight of jth criterion using Eq. (33): | |
| Sub-step 2.5: Obtain the subjective weight ‘${w_{j}^{s}}$’ of jth criterion through Eq. (34): | |
| Step 3. | Criteria weights through IVPF-WENSLO-RANCOM model. |
| Combining the objective and subjective weights of criteria, derive the combined weight of each criterion using Eq. (35): where $\varsigma \in [0,1]$ stands for the decision-precision coefficient. |
4.3 Integrated CRADIS Model for Alternatives’ Ranking
The suggested CRADIS model is implemented through the following steps:
-
1. Find the IVPF ideal and IVPF anti-ideal solutions.
-
2. Obtain the distances from each alternative to the IVPF ideal and anti-ideal solutions.
-
3. Determine the utility functions.
-
4. Find the collective measure of each alternative.
-
5. Rank the alternatives as per the values of the collective measure.
| Stage3:ProcedureofanIVPFextensionofCRADISmethod | |
| No.ofsteps | Descriptionoftheprocedure |
| Step1. | DeterminetheIVPF-idealsolutionandIVPFanti-idealsolution |
WiththeuseofEq.(36)andEq.(37),theidealandanti-idealsolutionsarecomputedrespectively,intermsofIVPFNs:
(36)\[t{\alpha _{j}^{+}}=\left\{\begin{array}{l}([\underset{i=1,2,\dots ,s}{\max }{\mu _{\mathit{ij}}^{-}},\underset{i=1,2,\dots ,s}{\min }{\nu _{\mathit{ij}}^{-}}],[\underset{i=1,2,\dots ,s}{\max }{\mu _{\mathit{ij}}^{+}},\underset{i=1,2,\dots ,s}{\min }{\nu _{\mathit{ij}}^{+}}]),j\in {v_{b}},\\ {} ([\underset{i=1,2,\dots ,s}{\min }{\mu _{\mathit{ij}}^{-}},\underset{i=1,2,\dots ,s}{\max }{\nu _{\mathit{ij}}^{-}}],[\underset{i=1,2,\dots ,s}{\min }{\mu _{\mathit{ij}}^{+}},\underset{i=1,2,\dots ,s}{\max }{\nu _{\mathit{ij}}^{+}}]),j\in {v_{n}},\end{array}\right.\](37)\[{\alpha _{j}^{-}}=\left\{\begin{array}{l}([\underset{i=1,2,\dots ,s}{\min }{\mu _{\mathit{ij}}^{-}},\underset{i=1,2,\dots ,s}{\max }{\nu _{\mathit{ij}}^{-}}],[\underset{i=1,2,\dots ,s}{\min }{\mu _{\mathit{ij}}^{+}},\underset{i=1,2,\dots ,s}{\max }{\nu _{\mathit{ij}}^{+}}]),j\in {v_{b}},\\ {} ([\underset{i=1,2,\dots ,s}{\max }{\mu _{\mathit{ij}}^{-}},\underset{i=1,2,\dots ,s}{\min }{\nu _{\mathit{ij}}^{-}}],[\underset{i=1,2,\dots ,s}{\max }{\mu _{\mathit{ij}}^{+}},\underset{i=1,2,\dots ,s}{\min }{\nu _{\mathit{ij}}^{+}}]),j\in {v_{n}}\mathrm{.}\end{array}\right.\] |
|
| Step 2. | Compute the distances$({d_{j}^{+}})$ and $({d_{j}^{-}})$ |
In line with the proposed distance measure, calculate the deviation relation between each alternative and IVPF-ideal solution using Eq. (38):
(38)\[{d_{j}^{+}}={\displaystyle \sum \limits_{j=1}^{t}}{w_{j}}\sqrt{\begin{array}{l}\Bigl[{\left(\frac{{({\mu _{\mathit{ij}}^{-}})^{2}}+{({\mu _{\mathit{ij}}^{+}})^{2}}-{({\nu _{\mathit{ij}}^{-}})^{2}}-{({\nu _{\mathit{ij}}^{+}})^{2}}}{4}-\frac{{({\mu _{{j_{\mathit{IS}}}}^{-}})^{2}}+{({\mu _{{j_{\mathit{IS}}}}^{+}})^{2}}-{({\nu _{{j_{\mathit{IS}}}}^{-}})^{2}}-{({\nu _{{j_{\mathit{IS}}}}^{+}})^{2}}}{4}\right)^{2}}\\ {} +\frac{1}{3}{\left(\frac{{({\mu _{\mathit{ij}}^{-}})^{2}}+{({\mu _{\mathit{ij}}^{-}})^{2}}+{({\nu _{\mathit{ij}}^{-}})^{2}}+{({\nu _{\mathit{ij}}^{-}})^{2}}}{4}-\frac{{({\mu _{{j_{\mathit{IS}}}}^{-}})^{2}}+{({\mu _{{j_{\mathit{IS}}}}^{-}})^{2}}+{({\nu _{{j_{\mathit{IS}}}}^{-}})^{2}}+{({\nu _{{j_{\mathit{IS}}}}^{-}})^{2}}}{4}\right)^{2}}\Bigr]\mathrm{.}\end{array}}\](39)\[{d_{j}^{-}}=\displaystyle {\displaystyle \sum \limits_{j=1}^{t}}{w_{j}}\sqrt{\begin{array}{l}\Bigl[{\left(\frac{{({\mu _{\mathit{ij}}^{-}})^{2}}+{({\mu _{\mathit{ij}}^{+}})^{2}}-{({\nu _{\mathit{ij}}^{-}})^{2}}-{({\nu _{\mathit{ij}}^{+}})^{2}}}{4}-\frac{{({\mu _{{j_{\mathit{AIS}}}}^{-}})^{2}}+{({\mu _{{j_{\mathit{AIS}}}}^{+}})^{2}}-{({\nu _{{j_{\mathit{AIS}}}}^{-}})^{2}}-{({\nu _{{j_{\mathit{AIS}}}}^{+}})^{2}}}{4}\right)^{2}}\\ {} +\frac{1}{3}{\left(\frac{{({\mu _{\mathit{ij}}^{-}})^{2}}+{({\mu _{\mathit{ij}}^{-}})^{2}}+{({\nu _{\mathit{ij}}^{-}})^{2}}+{({\nu _{\mathit{ij}}^{-}})^{2}}}{4}-\frac{{({\mu _{{j_{\mathit{AIS}}}}^{-}})^{2}}+{({\mu _{{j_{\mathit{AIS}}}}^{-}})^{2}}+{({\nu _{{j_{\mathit{AIS}}}}^{-}})^{2}}+{({\nu _{{j_{\mathit{AIS}}}}^{-}})^{2}}}{4}\right)^{2}}\Bigr]\mathrm{.}\end{array}}\] |
|
| Step 3. | Find the utility functions |
| Considering the deviation relations from Step 2, estimate the utility functions (${U_{i}^{+}}$) and (${U_{i}^{-}}$) from Eqs. (40) and (41): | |
| Step 4. | Calculate the collective measure |
| Taking into account the utility functions (${U_{i}^{+}}$) and (${U_{i}^{-}}$), find the collective measure of each alternative through Eq. (42), where ‘$\lambda \in [0,1]$’ stands for the strategy coefficient: | |
| Step 5. | Determine the alternatives’ ranking |
| On the basis of the determined collective measure of each alternative, estimate the ranking order of alternatives. | |
5 Case Study: Hospital Service Quality Assessment
To demonstrate the applicability of the proposed method, it was applied to assess the service quality of five hospitals in Patna district, Bihar, India. Patna serves as the capital and largest city of Bihar. It is also the 19th most populous city in the country and caters to a very large population. Since the management of the considered hospitals did not agree to reveal their identity, we referred to the name of the hospitals as Hospital-1 (${A_{1}}$), Hospi tal-2 (${A_{2}}$), Hospital-3 (${A_{3}}$), Hospital-4 (${A_{4}}$), and Hospital-5 (${A_{5}}$). In order to determine the criteria, a comprehensive literature survey was conducted with the keywords “healthcare service quality”, “patient satisfaction”, “patient-perceived”, and “multi-criteria decision-making”. Table 4 presents the list of selected criteria with their sources. Next, a group of four experts is asked to confirm the validity of selected service quality dimensions and criteria for evaluating the hospitals. This group includes experts from healthcare administration, hospital staff from the emergency department, ward service provider, and quality consultants, and details of DEs are presented in Table 5. In the following, we present the step-by-step procedure for assessing the quality of service at the five hospitals of Patna.
Table 4
Details of the criteria for assessing the quality of service of five hospitals in Patna.
| Service quality dimensions | Criteria | References |
| Patient Management System | Hospital location (${v_{1}}$) | (Rehman et al., 2023; Alshouha et al., 2024) |
| Emergency parking (${v_{2}}$) | (Aaronson et al., 2015; Rehman et al., 2023) | |
| Public transport (${v_{3}}$) | (Yucesan and Gul, 2020; Rehman et al., 2023) | |
| Ambulance service (${v_{4}}$) | (Yucesan and Gul, 2020; Rehman et al., 2023) | |
| Operating hour (${v_{5}}$) | (Kondasani and Panda, 2015; Yucesan and Gul, 2020) | |
| Patient relationship management | Appointment system (${v_{6}}$) | (Kondasani and Panda, 2015; Rehman et al., 2023) |
| Registration process (${v_{7}}$) | (Amin and Nasharuddin, 2013; Rehman et al., 2023) | |
| Walk-in facility (${v_{8}}$) | (Aliman and Mohamad, 2013; Rehman et al., 2023) | |
| Prompt response (${v_{9}}$) | (Rehman et al., 2023) | |
| Effective communication (${v_{10}}$) | (Rehman et al., 2023; Alshouha et al., 2024) | |
| Managerial dependence (${v_{11}}$) | (Aliman and Mohamad, 2013; Rehman et al., 2023) | |
| Reception assistance (${v_{12}}$) | (Rehman et al., 2023) | |
| Patient-physician relationship | Waiting time (${v_{13}}$) | (Aliman and Mohamad, 2013; Rehman et al., 2023) |
| On-demand doctors (${v_{14}}$) | (Aliman and Mohamad, 2013; Rehman et al., 2023) | |
| Personalized care (${v_{15}}$) | (Yucesan and Gul, 2020; Rehman et al., 2023) | |
| Medical examination (${v_{16}}$) | (Rehman et al., 2023) | |
| Medication expertise (${v_{17}}$) | (Amin and Nasharuddin, 2013; Rehman et al., 2023) | |
| Trustworthy care (${v_{18}}$) | (Aliman and Mohamad, 2013; Rehman et al., 2023) |
Table 5
Information about the members of the DEs’ committee.
| DM | Professional position | Experience | Primary perspective in the evaluation |
| ${g_{1}}$ | Senior Medical Superintendent / Hospital Administrator | 15–20 years | Hospital administration, management, patient-service delivery, and organizational operations |
| ${g_{2}}$ | Senior Physician / Consultant | 12–18 years | Clinical service delivery, physician-patient interaction, and clinical aspects of service quality |
| ${g_{3}}$ | Hospital Quality Manager / NABH Quality Professional | 10–15 years | Healthcare quality management, quality assurance, accreditation, and service quality monitoring |
| ${g_{4}}$ | Public Health / Health Services Management Professor | 12–18 years | Public health, health-services management, healthcare-system evaluation, and academic expertise |
Stage 1: Process for aggregating the individual decision opinions
Step 1: With the use of Table S2, a group of four experts is provided with their linguistic opinions about the performance of each hospital in respect of each evaluation criterion. Thus, a linguistic assessment decision matrix is formed in Table S3.
Step 2: To find the experts’ weights; initially, an average IVPFDM $\stackrel{ˉ}{F}={({\stackrel{ˉ}{\upsilon }_{\mathit{ij}}^{(k)}})_{5\times 4}}$ is constructed in Table 6, and then an entropy measure of each element of the average IVPFDM is computed and shown in Table 7. Further, the computed entropy value is normalized and presented in Table 7. Utilizing the normalized entropy values in Eq. (25), the weight of each expert is calculated and outlined in Table 7.
Table 6
Average IVPF decision matrix for finding the experts’ weights.
| Alternatives | ${g_{1}}$ | ${g_{2}}$ | ${g_{3}}$ | ${g_{4}}$ |
| ${A_{1}}$ | $([0\mathrm{.}633,0\mathrm{.}751],[0\mathrm{.}412,0\mathrm{.}523])$ | $([0\mathrm{.}579,0\mathrm{.}717],[0\mathrm{.}472,0\mathrm{.}590])$ | $([0\mathrm{.}511,0\mathrm{.}625],[0\mathrm{.}527,0\mathrm{.}629])$ | $([0\mathrm{.}472,0\mathrm{.}589],[0\mathrm{.}565,0\mathrm{.}660])$ |
| ${A_{2}}$ | $([0\mathrm{.}628,0\mathrm{.}754],[0\mathrm{.}415,0\mathrm{.}534])$ | $([0\mathrm{.}606,0\mathrm{.}738],[0\mathrm{.}447,0\mathrm{.}554])$ | $([0\mathrm{.}645,0\mathrm{.}774],[0\mathrm{.}390,0\mathrm{.}530])$ | $([0\mathrm{.}500,0\mathrm{.}619],[0\mathrm{.}535,0\mathrm{.}632])$ |
| ${A_{3}}$ | $([0\mathrm{.}492,0\mathrm{.}622],[0\mathrm{.}543,0\mathrm{.}652])$ | $([0\mathrm{.}545,0\mathrm{.}677],[0\mathrm{.}487,0\mathrm{.}608])$ | $([0\mathrm{.}494,0\mathrm{.}603],[0\mathrm{.}523,0\mathrm{.}647])$ | $([0\mathrm{.}486,0\mathrm{.}627],[0\mathrm{.}559,0\mathrm{.}657])$ |
| ${A_{4}}$ | $([0\mathrm{.}638,0\mathrm{.}762],[0\mathrm{.}397,0\mathrm{.}523])$ | $([0\mathrm{.}534,0\mathrm{.}669],[0\mathrm{.}506,0\mathrm{.}625])$ | $([0\mathrm{.}460,0\mathrm{.}583],[0\mathrm{.}570,0\mathrm{.}674])$ | $([0\mathrm{.}495,0\mathrm{.}631],[0\mathrm{.}548,0\mathrm{.}652])$ |
| ${A_{5}}$ | $([0\mathrm{.}603,0\mathrm{.}722],[0\mathrm{.}436,0\mathrm{.}560])$ | $([0\mathrm{.}571,0\mathrm{.}701],[0\mathrm{.}480,0\mathrm{.}584])$ | $([0\mathrm{.}530,0\mathrm{.}652],[0\mathrm{.}509,0\mathrm{.}611])$ | $([0\mathrm{.}521,0\mathrm{.}664],[0\mathrm{.}522,0\mathrm{.}625])$ |
Table 7
Weight of DEs based on the proposed IVPF-entropy.
| $E({\stackrel{ˉ}{\upsilon }_{\mathit{ik}}})$ | ${g_{1}}$ | ${g_{2}}$ | ${g_{3}}$ | ${g_{4}}$ |
| ${A_{1}}$ | 0.799 | 0.892 | 0.992 | 0.931 |
| ${A_{2}}$ | 0.804 | 0.843 | 0.772 | 0.98 |
| ${A_{3}}$ | 0.966 | 0.944 | 0.969 | 0.956 |
| ${A_{4}}$ | 0.785 | 0.967 | 0.915 | 0.969 |
| ${A_{5}}$ | 0.854 | 0.906 | 0.972 | 0.98 |
| $\stackrel{ˉ}{E}({\stackrel{ˉ}{\upsilon }_{\mathit{ij}}})$ | ${g_{1}}$ | ${g_{2}}$ | ${g_{3}}$ | ${g_{4}}$ |
| ${A_{1}}$ | 0.827 | 0.833 | 1 | 0.813 |
| ${A_{2}}$ | 0.923 | 0.871 | 0.976 | 1 |
| ${A_{3}}$ | 1 | 0.778 | 0.977 | 0.922 |
| ${A_{4}}$ | 0.95 | 1 | 0.976 | 0.988 |
| ${A_{5}}$ | 0.827 | 0.833 | 1 | 0.813 |
| ${\omega _{k}}$ | 0.229 | 0.253 | 0.250 | 0.267 |
Step 3: Using the DMs’ weights in the IVPFWA operator, an IVPFADM is created via Eq. (26) and represented in Table 8, where each element denotes the collective IVPF-assessment rating of hospitals with respect to shortlisted evaluation criteria.
Table 8
Aggregated decision matrix for assessing the quality of service of five hospitals.
| Criteria | Alternatives | ||||
| ${A_{1}}$ | ${A_{2}}$ | ${A_{3}}$ | ${A_{4}}$ | ${A_{5}}$ | |
| ${v_{1}}$ | $([0\mathrm{.}454,0\mathrm{.}585],[0\mathrm{.}547,0\mathrm{.}648])$ | $([0\mathrm{.}617,0\mathrm{.}769],[0\mathrm{.}425,0\mathrm{.}525])$ | $([0\mathrm{.}433,0\mathrm{.}564],[0\mathrm{.}573,0\mathrm{.}673])$ | $([0\mathrm{.}501,0\mathrm{.}638],[0\mathrm{.}522,0\mathrm{.}623])$ | $([0\mathrm{.}573,0\mathrm{.}722],[0\mathrm{.}464,0\mathrm{.}565])$ |
| ${v_{2}}$ | $([0\mathrm{.}667,0\mathrm{.}773],[0\mathrm{.}374,0\mathrm{.}468])$ | $([0\mathrm{.}730,0\mathrm{.}836],[0\mathrm{.}303,0\mathrm{.}391])$ | $([0\mathrm{.}529,0\mathrm{.}675],[0\mathrm{.}493,0\mathrm{.}594])$ | $([0\mathrm{.}713,0\mathrm{.}837],[0\mathrm{.}304,0\mathrm{.}441])$ | $([0\mathrm{.}509,0\mathrm{.}648],[0\mathrm{.}516,0\mathrm{.}617])$ |
| ${v_{3}}$ | $([0\mathrm{.}649,0\mathrm{.}785],[0\mathrm{.}390,0\mathrm{.}518])$ | $([0\mathrm{.}741,0\mathrm{.}852],[0\mathrm{.}268,0\mathrm{.}421])$ | $([0\mathrm{.}511,0\mathrm{.}650],[0\mathrm{.}514,0\mathrm{.}615])$ | $([0\mathrm{.}489,0\mathrm{.}629],[0\mathrm{.}541,0\mathrm{.}643])$ | $([0\mathrm{.}447,0\mathrm{.}580],[0\mathrm{.}599,0\mathrm{.}692])$ |
| ${v_{4}}$ | $([0\mathrm{.}661,0\mathrm{.}795],[0\mathrm{.}371,0\mathrm{.}497])$ | $([0\mathrm{.}628,0\mathrm{.}761],[0\mathrm{.}397,0\mathrm{.}524])$ | $([0\mathrm{.}590,0\mathrm{.}723],[0\mathrm{.}424,0\mathrm{.}551])$ | $([0\mathrm{.}494,0\mathrm{.}635],[0\mathrm{.}537,0\mathrm{.}639])$ | $([0\mathrm{.}730,0\mathrm{.}841],[0\mathrm{.}299,0\mathrm{.}387])$ |
| ${v_{5}}$ | $([0\mathrm{.}527,0\mathrm{.}673],[0\mathrm{.}495,0\mathrm{.}595])$ | $([0\mathrm{.}701,0\mathrm{.}823],[0\mathrm{.}320,0\mathrm{.}461])$ | $([0\mathrm{.}661,0\mathrm{.}795],[0\mathrm{.}371,0\mathrm{.}497])$ | $([0\mathrm{.}540,0\mathrm{.}670],[0\mathrm{.}496,0\mathrm{.}622])$ | $([0\mathrm{.}656,0\mathrm{.}791],[0\mathrm{.}378,0\mathrm{.}502])$ |
| ${v_{6}}$ | $([0\mathrm{.}797,0\mathrm{.}889],[0\mathrm{.}209,0\mathrm{.}319])$ | $([0\mathrm{.}705,0\mathrm{.}826],[0\mathrm{.}315,0\mathrm{.}457])$ | $([0\mathrm{.}599,0\mathrm{.}732],[0\mathrm{.}415,0\mathrm{.}543])$ | $([0\mathrm{.}525,0\mathrm{.}671],[0\mathrm{.}497,0\mathrm{.}597])$ | $([0\mathrm{.}706,0\mathrm{.}817],[0\mathrm{.}318,0\mathrm{.}407])$ |
| ${v_{7}}$ | $([0\mathrm{.}489,0\mathrm{.}629],[0\mathrm{.}541,0\mathrm{.}643])$ | $([0\mathrm{.}723,0\mathrm{.}837],[0\mathrm{.}308,0\mathrm{.}397])$ | $([0\mathrm{.}581,0\mathrm{.}706],[0\mathrm{.}437,0\mathrm{.}568])$ | $([0\mathrm{.}688,0\mathrm{.}798],[0\mathrm{.}342,0\mathrm{.}433])$ | $([0\mathrm{.}556,0\mathrm{.}699],[0\mathrm{.}486,0\mathrm{.}587])$ |
| ${v_{8}}$ | $([0\mathrm{.}701,0\mathrm{.}823],[0\mathrm{.}320,0\mathrm{.}461])$ | $([0\mathrm{.}756,0\mathrm{.}855],[0\mathrm{.}264,0\mathrm{.}369])$ | $([0\mathrm{.}568,0\mathrm{.}715],[0\mathrm{.}469,0\mathrm{.}570])$ | $([0\mathrm{.}635,0\mathrm{.}768],[0\mathrm{.}391,0\mathrm{.}518])$ | $([0\mathrm{.}665,0\mathrm{.}783],[0\mathrm{.}353,0\mathrm{.}501])$ |
| ${v_{9}}$ | $([0\mathrm{.}236,0\mathrm{.}321],[0\mathrm{.}778,0\mathrm{.}844])$ | $([0\mathrm{.}276,0\mathrm{.}371],[0\mathrm{.}732,0\mathrm{.}810])$ | $([0\mathrm{.}299,0\mathrm{.}269],[0\mathrm{.}691,0\mathrm{.}831])$ | $([0\mathrm{.}239,0\mathrm{.}335],[0\mathrm{.}762,0\mathrm{.}838])$ | $([0\mathrm{.}210,0\mathrm{.}307],[0\mathrm{.}788,0\mathrm{.}850])$ |
| ${v_{10}}$ | $([0\mathrm{.}211,0\mathrm{.}307],[0\mathrm{.}788,0\mathrm{.}850])$ | $([0\mathrm{.}222,0\mathrm{.}315],[0\mathrm{.}774,0\mathrm{.}851])$ | $([0\mathrm{.}313,0\mathrm{.}412],[0\mathrm{.}693,0\mathrm{.}783])$ | $([0\mathrm{.}245,0\mathrm{.}337],[0\mathrm{.}763,0\mathrm{.}830])$ | $([0\mathrm{.}258,0\mathrm{.}357],[0\mathrm{.}746,0\mathrm{.}824])$ |
| ${v_{11}}$ | $([0\mathrm{.}274,0\mathrm{.}369],[0\mathrm{.}734,0\mathrm{.}823])$ | $([0\mathrm{.}199,0\mathrm{.}293],[0\mathrm{.}795,0\mathrm{.}860])$ | $([0\mathrm{.}243,0\mathrm{.}340],[0\mathrm{.}758,0\mathrm{.}835])$ | $([0\mathrm{.}188,0\mathrm{.}274],[0\mathrm{.}811,0\mathrm{.}875])$ | $([0\mathrm{.}191,0\mathrm{.}285],[0\mathrm{.}801,0\mathrm{.}864])$ |
| ${v_{12}}$ | $([0\mathrm{.}215,0\mathrm{.}312],[0\mathrm{.}784,0\mathrm{.}849])$ | $([0\mathrm{.}197,0\mathrm{.}291],[0\mathrm{.}796,0\mathrm{.}861])$ | $([0\mathrm{.}215,0\mathrm{.}312],[0\mathrm{.}784,0\mathrm{.}849])$ | $([0\mathrm{.}296,0\mathrm{.}392],[0\mathrm{.}709,0\mathrm{.}799])$ | $([0\mathrm{.}306,0\mathrm{.}405],[0\mathrm{.}700,0\mathrm{.}789])$ |
| ${v_{13}}$ | $([0\mathrm{.}215,0\mathrm{.}312],[0\mathrm{.}784,0\mathrm{.}848])$ | $([0\mathrm{.}257,0\mathrm{.}356],[0\mathrm{.}747,0\mathrm{.}824])$ | $([0\mathrm{.}296,0\mathrm{.}392],[0\mathrm{.}709,0\mathrm{.}799])$ | $([0\mathrm{.}290,0\mathrm{.}388],[0\mathrm{.}720,0\mathrm{.}797])$ | $([0\mathrm{.}298,0\mathrm{.}394],[0\mathrm{.}708,0\mathrm{.}797])$ |
| ${v_{14}}$ | $([0\mathrm{.}243,0\mathrm{.}340],[0\mathrm{.}758,0\mathrm{.}835])$ | $([0\mathrm{.}293,0\mathrm{.}391],[0\mathrm{.}716,0\mathrm{.}795])$ | $([0\mathrm{.}292,0\mathrm{.}390],[0\mathrm{.}718,0\mathrm{.}796])$ | $([0\mathrm{.}310,0\mathrm{.}408],[0\mathrm{.}697,0\mathrm{.}786])$ | $([0\mathrm{.}278,0\mathrm{.}374],[0\mathrm{.}730,0\mathrm{.}808])$ |
| ${v_{15}}$ | $([0\mathrm{.}523,0\mathrm{.}668],[0\mathrm{.}497,0\mathrm{.}598])$ | $([0\mathrm{.}638,0\mathrm{.}777],[0\mathrm{.}383,0\mathrm{.}506])$ | $([0\mathrm{.}527,0\mathrm{.}673],[0\mathrm{.}495,0\mathrm{.}595])$ | $([0\mathrm{.}712,0\mathrm{.}837],[0\mathrm{.}304,0\mathrm{.}442])$ | $([0\mathrm{.}590,0\mathrm{.}722],[0\mathrm{.}424,0\mathrm{.}551])$ |
| ${v_{16}}$ | $([0\mathrm{.}676,0\mathrm{.}781],[0\mathrm{.}359,0\mathrm{.}451])$ | $([0\mathrm{.}675,0\mathrm{.}811],[0\mathrm{.}354,0\mathrm{.}477])$ | $([0\mathrm{.}622,0\mathrm{.}743],[0\mathrm{.}389,0\mathrm{.}516])$ | $([0\mathrm{.}584,0\mathrm{.}737],[0\mathrm{.}448,0\mathrm{.}548])$ | $([0\mathrm{.}639,0\mathrm{.}772],[0\mathrm{.}386,0\mathrm{.}514])$ |
| ${v_{17}}$ | $([0\mathrm{.}412,0\mathrm{.}531],[0\mathrm{.}592,0\mathrm{.}693])$ | $([0\mathrm{.}489,0\mathrm{.}628],[0\mathrm{.}541,0\mathrm{.}643])$ | $([0\mathrm{.}626,0\mathrm{.}761],[0\mathrm{.}424,0\mathrm{.}551])$ | $([0\mathrm{.}668,0\mathrm{.}792],[0\mathrm{.}357,0\mathrm{.}501])$ | $([0\mathrm{.}688,0\mathrm{.}798],[0\mathrm{.}342,0\mathrm{.}433])$ |
| ${v_{18}}$ | $([0\mathrm{.}761,0\mathrm{.}863],[0\mathrm{.}256,0\mathrm{.}358])$ | $([0\mathrm{.}711,0\mathrm{.}836],[0\mathrm{.}305,0\mathrm{.}442])$ | $([0\mathrm{.}585,0\mathrm{.}711],[0\mathrm{.}433,0\mathrm{.}564])$ | $([0\mathrm{.}665,0\mathrm{.}783],[0\mathrm{.}353,0\mathrm{.}501])$ | $([0\mathrm{.}735,0\mathrm{.}857],[0\mathrm{.}287,0\mathrm{.}421])$ |
Stage 2: Procedure to determine the criteria weights
Step 1: To determine the objective weight through the IVPF-WENSLO approach, we need to first find the IVPF decision matrix through Eq. (10). Further, the IVPF-score decision matrix is normalized, and the results are presented in Table 9.
Table 9
IVPF-score and normalized IVPF decision matrix for assessing the quality of service of five hospitals.
| Criteria | Alternatives | |||||||||
| ${A_{1}}$ | ${A_{2}}$ | ${A_{3}}$ | ${A_{4}}$ | ${A_{5}}$ | ${A_{1}}$ | ${A_{2}}$ | ${A_{3}}$ | ${A_{4}}$ | ${A_{5}}$ | |
| ${v_{1}}$ | 0.316 | 0.493 | 0.295 | 0.358 | 0.438 | 0.166 | 0.259 | 0.155 | 0.188 | 0.231 |
| ${v_{2}}$ | 0.533 | 0.621 | 0.392 | 0.606 | 0.366 | 0.212 | 0.247 | 0.156 | 0.241 | 0.146 |
| ${v_{3}}$ | 0.522 | 0.637 | 0.369 | 0.345 | 0.297 | 0.241 | 0.294 | 0.17 | 0.159 | 0.137 |
| ${v_{4}}$ | 0.539 | 0.497 | 0.453 | 0.35 | 0.624 | 0.219 | 0.202 | 0.184 | 0.142 | 0.253 |
| ${v_{5}}$ | 0.39 | 0.586 | 0.539 | 0.39 | 0.532 | 0.16 | 0.241 | 0.221 | 0.16 | 0.218 |
| ${v_{6}}$ | 0.714 | 0.591 | 0.464 | 0.387 | 0.592 | 0.26 | 0.215 | 0.169 | 0.141 | 0.215 |
| ${v_{7}}$ | 0.345 | 0.615 | 0.438 | 0.565 | 0.415 | 0.145 | 0.259 | 0.184 | 0.238 | 0.174 |
| ${v_{8}}$ | 0.586 | 0.655 | 0.432 | 0.505 | 0.535 | 0.216 | 0.241 | 0.159 | 0.186 | 0.197 |
| ${v_{9}}$ | 0.128 | 0.158 | 0.15 | 0.135 | 0.119 | 0.185 | 0.23 | 0.217 | 0.196 | 0.172 |
| ${v_{10}}$ | 0.119 | 0.124 | 0.186 | 0.137 | 0.147 | 0.167 | 0.174 | 0.261 | 0.192 | 0.206 |
| ${v_{11}}$ | 0.155 | 0.112 | 0.138 | 0.102 | 0.108 | 0.252 | 0.182 | 0.224 | 0.166 | 0.176 |
| ${v_{12}}$ | 0.121 | 0.111 | 0.121 | 0.172 | 0.18 | 0.172 | 0.157 | 0.172 | 0.244 | 0.255 |
| ${v_{13}}$ | 0.121 | 0.147 | 0.172 | 0.169 | 0.174 | 0.155 | 0.188 | 0.22 | 0.215 | 0.222 |
| ${v_{14}}$ | 0.138 | 0.171 | 0.17 | 0.183 | 0.16 | 0.168 | 0.208 | 0.207 | 0.223 | 0.195 |
| ${v_{15}}$ | 0.386 | 0.514 | 0.39 | 0.605 | 0.453 | 0.164 | 0.219 | 0.166 | 0.258 | 0.193 |
| ${v_{16}}$ | 0.545 | 0.559 | 0.486 | 0.455 | 0.511 | 0.213 | 0.219 | 0.19 | 0.178 | 0.2 |
| ${v_{17}}$ | 0.272 | 0.345 | 0.489 | 0.542 | 0.565 | 0.123 | 0.156 | 0.221 | 0.245 | 0.255 |
| ${v_{18}}$ | 0.665 | 0.604 | 0.443 | 0.535 | 0.636 | 0.231 | 0.21 | 0.154 | 0.186 | 0.22 |
Considering Sturges’ rule (Pamucar et al., 2023) given in Eq. (27), we determine the class interval ($\Delta {n_{j}}$) of each criterion and given as $\Delta {n_{1}}=0.016$, $\Delta {n_{2}}=0.016$, $\Delta {n_{3}}=0.025$, $\Delta {n_{4}}=0.017$, $\Delta {n_{5}}=0.013$, $\Delta {n_{6}}=0.019$, $\Delta {n_{7}}=0.018$, $\Delta {n_{8}}=0.013$, $\Delta {n_{9}}=0.009$, $\Delta {n_{10}}=0.015$, $\Delta {n_{11}}=0.014$, $\Delta {n_{12}}=0.015$, $\Delta {n_{13}}=0.011$, $\Delta {n_{14}}=0.009$, $\Delta {n_{15}}=0.015$, $\Delta {n_{16}}=0.006$, $\Delta {n_{17}}=0.021$, and $\Delta {n_{18}}=0.012$. In the next step, we compute the slope value ($\tan {\zeta _{j}}$) of each criterion using Eq. (28), and presented as $\tan {\zeta _{1}}=15.25$, $\tan {\zeta _{2}}=15.757$, $\tan {\zeta _{3}}=10.135$, $\tan {\zeta _{4}}=14.291$, $\tan {\zeta _{5}}=19.708$, $\tan {\zeta _{6}}=13.397$, $\tan {\zeta _{7}}=13.975$, $\tan {\zeta _{8}}=19.358$, $\tan {\zeta _{9}}=27.703$, $\tan {\zeta _{10}}=16.928$, $\tan {\zeta _{11}}=18.315$, $\tan {\zeta _{12}}=16.223$, $\tan {\zeta _{13}}=23.747$, $\tan {\zeta _{14}}=28.823$, $\tan {\zeta _{15}}=17.019$, $\tan {\zeta _{16}}=39.079$, $\tan {\zeta _{17}}=12.031$, and $\tan {\zeta _{18}}=20.67$.
Based on Eq. (29), the envelope value (${O_{j}}$) of each criterion is obtained as ${O_{1}}=0.151$, ${O_{2}}=0.161$, ${O_{3}}=0.139$, ${O_{4}}=0.123$, ${O_{5}}=0.119$, ${O_{6}}=0.104$, ${O_{7}}=0.16$, ${O_{8}}=0.092$, ${O_{9}}=0.057$, ${O_{10}}=0.112$, ${O_{11}}=0.102$, ${O_{12}}=0.078$, ${O_{13}}=0.048$, ${O_{14}}=0.052$, ${O_{15}}=0.136$, ${O_{16}}=0.039$, ${O_{17}}=0.08$ and ${O_{18}}=0.077$. Further, the ratio of envelope-slope for each criterion is calculated via Eq. (30) and given as ${\alpha _{1}}=0.0099$, ${\alpha _{2}}=0.0102$, ${\alpha _{3}}=0.0137$, ${\alpha _{4}}=0.0086$, ${\alpha _{5}}=0.0061$, ${\alpha _{6}}=0.0077$, ${\alpha _{7}}=0.0115$, ${\alpha _{8}}=0.0047$, ${\alpha _{9}}=0.002$, ${\alpha _{10}}=0.0066$, ${\alpha _{11}}=0.0055$, ${\alpha _{12}}=0.0048$, ${\alpha _{13}}=0.002$, ${\alpha _{14}}=0.0018$, ${\alpha _{15}}=0.008$, ${\alpha _{16}}=0.001$, ${\alpha _{17}}=0.0067$ and ${\alpha _{18}}=0.0037$. Finally, we derive the objective weight of each criterion with the help of Eq. (31), and thus, we get ${w_{1}^{o}}=0.0862$, ${w_{2}^{o}}=0.0892$, ${w_{3}^{o}}=0.1195$, ${w_{4}^{o}}=0.0749$, ${w_{5}^{o}}=0.0528$, ${w_{6}^{o}}=0.0676$, ${w_{7}^{o}}=0.1$, ${w_{8}^{o}}=0.0413$, ${w_{9}^{o}}=0.0179$, ${w_{10}^{o}}=0.0578$, ${w_{11}^{o}}=0.0484$, ${w_{12}^{o}}=0.0417$, ${w_{13}^{o}}=0.0177$, ${w_{14}^{o}}=0.0159$, ${w_{15}^{o}}=0.0698$, ${w_{16}^{o}}=0.0087$, ${w_{17}^{o}}=0.0582$, and ${w_{18}^{o}}=0.0325$.
Step 2: To derive the subjective weight through the IVPF-RANCOM model, we need to collect the DMs’ linguistic opinions regarding the performance of each criterion. Consequently, an aggregated column matrix is constructed through the IVPFWA operator (Eq. (4)), and their score values are further determined, given in Table 10. In accordance with the score values of the criteria, the rank of each criterion is attained. The last two columns of Table 10 present the score value and rank of each criterion, respectively.
Table 10
Significance ratings of each criterion during the assessment of the quality of service of five hospitals.
| Criteria | ${g_{1}}$ | ${g_{2}}$ | ${g_{3}}$ | ${g_{4}}$ | Aggregated IVPFNs | Score values | Rank |
| ${v_{1}}$ | M | VH | MH | M | $([0\mathrm{.}584,0\mathrm{.}710],[0\mathrm{.}434,0\mathrm{.}565])$ | 0.441 | 7 |
| ${v_{2}}$ | ML | M | MH | H | $([0\mathrm{.}498,0\mathrm{.}638],[0\mathrm{.}533,0\mathrm{.}635])$ | 0.354 | 10 |
| ${v_{3}}$ | VH | MH | H | M | $([0\mathrm{.}623,0\mathrm{.}757],[0\mathrm{.}402,0\mathrm{.}528])$ | 0.492 | 5 |
| ${v_{4}}$ | M | H | MH | VH | $([0\mathrm{.}639,0\mathrm{.}772],[0\mathrm{.}386,0\mathrm{.}514])$ | 0.511 | 3 |
| ${v_{5}}$ | VH | ML | H | MH | $([0\mathrm{.}614,0\mathrm{.}750],[0\mathrm{.}417,0\mathrm{.}545])$ | 0.48 | 6 |
| ${v_{6}}$ | M | L | MH | M | $([0\mathrm{.}394,0\mathrm{.}512],[0\mathrm{.}617,0\mathrm{.}708])$ | 0.255 | 12 |
| ${v_{7}}$ | ML | M | H | L | $([0\mathrm{.}439,0\mathrm{.}570],[0\mathrm{.}607,0\mathrm{.}699])$ | 0.289 | 11 |
| ${v_{8}}$ | MH | M | L | ML | $([0\mathrm{.}368,0\mathrm{.}486],[0\mathrm{.}644,0\mathrm{.}735])$ | 0.232 | 13 |
| ${v_{9}}$ | ML | M | ML | ML | $([0\mathrm{.}329,0\mathrm{.}429],[0\mathrm{.}673,0\mathrm{.}773])$ | 0.198 | 17 |
| ${v_{10}}$ | M | L | ML | M | $([0\mathrm{.}337,0\mathrm{.}436],[0\mathrm{.}671,0\mathrm{.}760])$ | 0.204 | 16 |
| ${v_{11}}$ | L | VL | MH | ML | $([0\mathrm{.}321,0\mathrm{.}441],[0\mathrm{.}697,0\mathrm{.}778])$ | 0.194 | 18 |
| ${v_{12}}$ | VL | M | MH | L | $([0\mathrm{.}348,0\mathrm{.}466],[0\mathrm{.}671,0\mathrm{.}752])$ | 0.215 | 14 |
| ${v_{13}}$ | MH | ML | L | ML | $([0\mathrm{.}343,0\mathrm{.}462],[0\mathrm{.}670,0\mathrm{.}760])$ | 0.212 | 15 |
| ${v_{14}}$ | H | H | ML | MH | $([0\mathrm{.}553,0\mathrm{.}704],[0\mathrm{.}488,0\mathrm{.}590])$ | 0.415 | 8 |
| ${v_{15}}$ | MH | H | M | M | $([0\mathrm{.}506,0\mathrm{.}643],[0\mathrm{.}519,0\mathrm{.}620])$ | 0.363 | 9 |
| ${v_{16}}$ | H | H | M | VH | $([0\mathrm{.}664,0\mathrm{.}797],[0\mathrm{.}368,0\mathrm{.}494])$ | 0.542 | 1 |
| ${v_{17}}$ | VH | H | M | H | $([0\mathrm{.}656,0\mathrm{.}792],[0\mathrm{.}378,0\mathrm{.}501])$ | 0.533 | 2 |
| ${v_{18}}$ | H | M | MH | VH | $([0\mathrm{.}635,0\mathrm{.}767],[0\mathrm{.}389,0\mathrm{.}518])$ | 0.505 | 4 |
Taking into account the pairwise comparison of criteria preferences, the RANCOM matrix is created with the utilization of Eq. (32); consequently, the summed weight of each criterion is found through Eq. (33). Table 11 outlines the computational steps required to construct the RANCOM matrix and to calculate the summed criteria weights. In addition, the subjective weights of the criteria are obtained via Eq. (34) as ${w_{1}^{s}}=0.071$, ${w_{2}^{s}}=0.0525$, ${w_{3}^{s}}=0.0833$, ${w_{4}^{s}}=0.0957$, ${w_{5}^{s}}=0.0772$, ${w_{6}^{s}}=0.0401$, ${w_{7}^{s}}=0.0463$, ${w_{8}^{s}}=0.034$, ${w_{9}^{s}}=0.0093$, ${w_{10}^{s}}=0.0154$, ${w_{11}^{s}}=0.0031$, ${w_{12}^{s}}=0.0278$, ${w_{13}^{s}}=0.0216$, ${w_{14}^{s}}=0.0648$, ${w_{15}^{s}}=0.0586$, ${w_{16}^{s}}=0.108$, ${w_{17}^{s}}=0.1019$, and ${w_{18}^{s}}=0.0895$.
Table 11
Results of the IVPF-RANCOM model for assessing hospital service quality.
| Criteria | RANCOM matrix | ${\sigma _{j}}$ | |||||||||||||||||
| ${v_{1}}$ | ${v_{2}}$ | ${v_{3}}$ | ${v_{4}}$ | ${v_{5}}$ | ${v_{6}}$ | ${v_{7}}$ | ${v_{8}}$ | ${v_{9}}$ | ${v_{10}}$ | ${v_{11}}$ | ${v_{12}}$ | ${v_{13}}$ | ${v_{14}}$ | ${v_{15}}$ | ${v_{16}}$ | ${v_{17}}$ | ${v_{18}}$ | ||
| ${v_{1}}$ | 0.5 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 11.5 |
| ${v_{2}}$ | 0 | 0.5 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 8.5 |
| ${v_{3}}$ | 1 | 1 | 0.5 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 13.5 |
| ${v_{4}}$ | 1 | 1 | 1 | 0.5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 15.5 |
| ${v_{5}}$ | 1 | 1 | 0 | 0 | 0.5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 12.5 |
| ${v_{6}}$ | 0 | 0 | 0 | 0 | 0 | 0.5 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 6.5 |
| ${v_{7}}$ | 0 | 0 | 0 | 0 | 0 | 1 | 0.5 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 7.5 |
| ${v_{8}}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.5 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 5.5 |
| ${v_{9}}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.5 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1.5 |
| ${v_{10}}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0.5 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2.5 |
| ${v_{11}}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.5 |
| ${v_{12}}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0.5 | 1 | 0 | 0 | 0 | 0 | 0 | 4.5 |
| ${v_{13}}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0.5 | 0 | 0 | 0 | 0 | 0 | 3.5 |
| ${v_{14}}$ | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0.5 | 1 | 0 | 0 | 0 | 10.5 |
| ${v_{15}}$ | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0.5 | 0 | 0 | 0 | 9.5 |
| ${v_{16}}$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0.5 | 1 | 1 | 17.5 |
| ${v_{17}}$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0.5 | 1 | 16.5 |
| ${v_{18}}$ | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0.5 | 14.5 |
Step 3: Applying Eq. (35), the combined weight of each criterion is calculated as the product of the “sum of ς” and “objective weight”, as well as the product of the “complement of ς; i.e. $(1-\varsigma )$” and “subjective weight of the criterion”. For $\varsigma =0.5$, the combined weight of the criterion is achieved via Eq. (35) and shown as ${w_{1}}=0.0786$, ${w_{2}}=0.0708$, ${w_{3}}=0.1014$, ${w_{4}}=0.0853$, ${w_{5}}=0.065$, ${w_{6}}=0.0538$, ${w_{7}}=0.0732$, ${w_{8}}=0.0377$, ${w_{9}}=0.0136$, ${w_{10}}=0.0366$, ${w_{11}}=0.0257$, ${w_{12}}=0.0348$, ${w_{13}}=0.0196$, ${w_{14}}=0.0403$, ${w_{15}}=0.0642$, ${w_{16}}=0.0583$, ${w_{17}}=0.0801$ and ${w_{18}}=0.061$. Figure 2 presents the graphical representation of the criteria weighting values.
Stage 3: Procedure for an IVPF extension of the CRADIS
Step 1: Considering Table 7, we determine the IVPF-ideal solution using Eq. (36) and given as ${\alpha _{j}^{+}}=$ {([0.617, 0.769], [0.425, 0.525]), ([0.730, 0.836], [0.303, 0.391]), ([0.741, 0.852], [0.268, 0.421]), ([0.730, 0.841], [0.299, 0.387]), ([0.701, 0.823], [0.320, 0.461]), ([0.797, 0.889], [0.209, 0.319]), ([0.723, 0.837], [0.308, 0.397]), ([0.756, 0.855], [0.264, 0.369]), ([0.276, 0.371], [0.732, 0.810]), ([0.313, 0.412], [0.693, 0.783]), ([0.274, 0.369], [0.734, 0.823]), ([0.306, 0.405], [0.7, 0.789]), ([0.298, 0.394], [0.708, 0.797]), ([0.31, 0.408], [0.697, 0.786]), ([0.712, 0.837], [0.304, 0.442]), ([0.675, 0.811], [0.354, 0.477]), ([0.688, 0.798], [0.342, 0.433]), ([0.761, 0.863], [0.256, 0.358])}, where $j=1,2,\dots ,18$.
Next, we calculate the IVPF anti-ideal solution through Eq. (37) and given as ${\alpha _{j}^{-}}=$ {([0.433, 0.564], [0.573, 0.673]), ([0.509, 0.648], [0.516, 0.617]), ([0.447, 0.580], [0.599, 0.692]), ([0.494, 0.635], [0.537, 0.639]), ([0.527, 0.673], [0.495, 0.595]), ([0.525, 0.671], [0.497, 0.597]), ([0.489, 0.629], [0.541, 0.643]), ([0.568, 0.715], [0.469, 0.57]), ([0.21, 0.307], [0.788, 0.85]), ([0.211, 0.307], [0.788, 0.85]), ([0.188, 0.274], [0.811, 0.875]), ([0.197, 0.291], [0.796, 0.861]), ([0.215, 0.312], [0.784, 0.848]), ([0.243, 0.34], [0.758, 0.835]), ([0.523, 0.668], [0.497, 0.598]), ([0.584, 0.737], [0.448, 0.548]), ([0.412, 0.531], [0.592, 0.693]), ([0.585, 0.711], [0.433, 0.564])}, where $j=1,2,\dots ,18$.
Step 2: With the utilization of the proposed distance measure, the deviation relation between each alternative and the IVPF-ideal solution is attained through Eq. (38), and given as ${d_{1}^{+}}$ = 0.1174, ${d_{2}^{+}}=0.0506$, ${d_{3}^{+}}=0.143$, ${d_{4}^{+}}=0.1185$, and ${d_{5}^{+}}=0.1005$. Similarly, the deviation relation between each alternative and the IVPF anti-ideal solution is computed via Eq. (39) and presented as ${d_{1}^{-}}=0.0937$, ${d_{2}^{-}}=0.1604$, ${d_{3}^{-}}=0.0697$, ${d_{4}^{-}}=0.0946$, and ${d_{5}^{-}}=0.1119$.
Step 3: Considering the deviation relations of above step, we compute the utility functions ${U_{i}^{+}}$ and ${U_{i}^{-}}$ for each alternative ${A_{i}}$ using Eq. (41) and Eq. (42), where $i=1,2,\dots ,5$, and presented as ${U_{1}^{+}}=0.431$, ${U_{2}^{+}}=1.0$, ${U_{3}^{+}}=0.354$, ${U_{4}^{+}}=0.427$, ${U_{5}^{+}}=0.504$, ${U_{1}^{-}}=0.584$, ${U_{2}^{-}}=1.0$, ${U_{3}^{-}}=0.434$, ${U_{4}^{-}}=0.59$, and ${U_{5}^{-}}=0.698$.
Step 4: Using Eq. (42), we obtain the collective measure (for strategy coefficient λ = 0.5) for each alternative ${A_{i}}$, where i = 1, 2, ..., 5, ${U_{1}}=0.5077$, ${U_{2}}=1.0$, ${U_{3}}=0.3942$, ${U_{4}}=0.5085$, and ${U_{5}}=0.6007$.
Step 5: The collective measure of each option allows for deriving the prioritization among options, which is ${A_{2}}\succ {A_{5}}\succ {A_{4}}\succ {A_{1}}\succ {A_{3}}$. Thus, the option “Hospital-2 (${A_{2}}$)” has the maximum preference in terms of better service quality.
6 Results Validation
In this section, we conduct sensitivity and comparative tests to confirm the stability and robustness of the outcomes.
6.1 Sensitivity Analysis
This subsection considers the following two cases to examine the influence of fluctuations in the criteria weighting parameter (ς) and strategy coefficient (λ) on the final ranking output of the IVPF-WENSLO-RANCOM-CRADIS framework. The first case underlines the rationale for carrying out sensitivity analysis on the criteria weights when ranking the hospitals. The second case spotlights the rationale for carrying out sensitivity analysis with respect to the strategy coefficient when determining the hospital alternatives’ ranking.
Case I (Analysing the influence of varying values of the criteria weighting parameter $\boldsymbol{\varsigma }$): In this case, we investigate the degrees of collective measure over the varying values of the criteria weighting parameter (ς). To this aim, we consider the different values of the criteria weighting parameter (ς), ranging from $\varsigma =0$ to $\varsigma =1$. In accordance with the diverse values of ς, the collective measure of each alternative hospital is determined and shown in Table 12. Moreover, Fig. 3 demonstrates the results of the sensitivity investigation over the criteria weighting parameter (ς). From Table 12 and Fig. 3, we can observe that the ranking result varies for $\varsigma =0$ to $\varsigma =0.4$, while there is no change in the ranking of these hospitals for $\varsigma =0.5$ to $\varsigma =1$. Thus, the option “Hospital-2 (${A_{2}}$)” has the maximum preference for different values of ς.
Table 12
Results of the sensitivity analysis over varying values of the coefficient (ς).
| ${A_{1}}$ | ${A_{2}}$ | ${A_{3}}$ | ${A_{4}}$ | ${A_{5}}$ | Ranking order | |
| $\varsigma =0\mathrm{.}0$ (IVPF-RANCOM) | 0.5343 | 1 | 0.427 | 0.5231 | 0.7188 | ${A_{2}}\succ {A_{5}}\succ {A_{1}}\succ {A_{4}}\succ {A_{3}}$ |
| $\varsigma =0\mathrm{.}1$ | 0.5288 | 1 | 0.4202 | 0.5201 | 0.6928 | ${A_{2}}\succ {A_{5}}\succ {A_{1}}\succ {A_{4}}\succ {A_{3}}$ |
| $\varsigma =0\mathrm{.}2$ | 0.5234 | 1 | 0.4135 | 0.5172 | 0.668 | ${A_{2}}\succ {A_{5}}\succ {A_{1}}\succ {A_{4}}\succ {A_{3}}$ |
| $\varsigma =0\mathrm{.}3$ | 0.5181 | 1 | 0.4069 | 0.5143 | 0.6445 | ${A_{2}}\succ {A_{5}}\succ {A_{1}}\succ {A_{4}}\succ {A_{3}}$ |
| $\varsigma =0\mathrm{.}4$ | 0.5129 | 1 | 0.4005 | 0.5113 | 0.6221 | ${A_{2}}\succ {A_{5}}\succ {A_{1}}\succ {A_{4}}\succ {A_{3}}$ |
| $\varsigma =0\mathrm{.}5$ (IVPF-WENSLO-RANCOM) | 0.5077 | 1 | 0.3942 | 0.5085 | 0.6007 | ${A_{2}}\succ {A_{5}}\succ {A_{4}}\succ {A_{1}}\succ {A_{3}}$ |
| $\varsigma =0\mathrm{.}6$ | 0.5027 | 1 | 0.388 | 0.5056 | 0.5802 | ${A_{2}}\succ {A_{5}}\succ {A_{4}}\succ {A_{1}}\succ {A_{3}}$ |
| $\varsigma =0\mathrm{.}7$ | 0.4977 | 1 | 0.382 | 0.5028 | 0.5605 | ${A_{2}}\succ {A_{5}}\succ {A_{4}}\succ {A_{1}}\succ {A_{3}}$ |
| $\varsigma =0\mathrm{.}8$ | 0.4928 | 1 | 0.376 | 0.4999 | 0.5417 | ${A_{2}}\succ {A_{5}}\succ {A_{4}}\succ {A_{1}}\succ {A_{3}}$ |
| $\varsigma =0\mathrm{.}9$ | 0.488 | 1 | 0.3702 | 0.4971 | 0.5236 | ${A_{2}}\succ {A_{5}}\succ {A_{4}}\succ {A_{1}}\succ {A_{3}}$ |
| $\varsigma =1\mathrm{.}0$ (IVPF-WENSLO) | 0.4833 | 1 | 0.3645 | 0.4944 | 0.5061 | ${A_{2}}\succ {A_{5}}\succ {A_{4}}\succ {A_{1}}\succ {A_{3}}$ |
Case II (Examining the impact of the strategy coefficient $\boldsymbol{\lambda }$): In this case, we observe the effect of the possible variations in λ values on the assessment of service quality of hospitals. To this end, we consider various values of the strategy parameter (λ) between $\lambda =0$ to $\lambda =1$, with a step size of ‘0.1’. Table 13 and Fig. 4 present different scenarios taking into account $\lambda =0.0$ to $\lambda =1.0$. From Table 13 and Fig. 4, we can find that there is no change in the ranking of these hospitals for $\lambda =0$ to $\lambda =0.5$, while the ranking result varies for $\lambda =0.6$ to $\lambda =1$. However, the option “Hospital-2 (${A_{2}}$)” has the maximum preference for different values of the strategy parameter (λ).
Based on these two cases of sensitivity analysis, we observe that the potential scores of the alternatives’ collective measures display variations, but it should be noted that the “Hospital-2 (${A_{2}}$)” has the maximum preference among the others in terms of service quality assessment of hospitals across both scenarios. This proves that the proposed MCGDM methodology has good robustness.
Table 13
Performance degree of options over strategy parameter (λ).
| ${A_{1}}$ | ${A_{2}}$ | ${A_{3}}$ | ${A_{4}}$ | ${A_{5}}$ | Ranking order | |
| $\lambda =0\mathrm{.}0$ (Utility measure by IVPFAIS) | 0.584 | 1.0 | 0.434 | 0.59 | 0.698 | ${A_{2}}\succ {A_{5}}\succ {A_{4}}\succ {A_{1}}\succ {A_{3}}$ |
| $\lambda =0\mathrm{.}1$ | 0.5687 | 1.0 | 0.426 | 0.5737 | 0.6786 | ${A_{2}}\succ {A_{5}}\succ {A_{4}}\succ {A_{1}}\succ {A_{3}}$ |
| $\lambda =0\mathrm{.}2$ | 0.5534 | 1.0 | 0.418 | 0.5574 | 0.6592 | ${A_{2}}\succ {A_{5}}\succ {A_{4}}\succ {A_{1}}\succ {A_{3}}$ |
| $\lambda =0\mathrm{.}3$ | 0.5381 | 1.0 | 0.41 | 0.5411 | 0.6398 | ${A_{2}}\succ {A_{5}}\succ {A_{4}}\succ {A_{1}}\succ {A_{3}}$ |
| $\lambda =0\mathrm{.}4$ | 0.5228 | 1.0 | 0.402 | 0.5248 | 0.6204 | ${A_{2}}\succ {A_{5}}\succ {A_{4}}\succ {A_{1}}\succ {A_{3}}$ |
| $\lambda =0\mathrm{.}5$ (Collective measure) | 0.5075 | 1.0 | 0.394 | 0.5085 | 0.601 | ${A_{2}}\succ {A_{5}}\succ {A_{4}}\succ {A_{1}}\succ {A_{3}}$ |
| $\lambda =0\mathrm{.}6$ | 0.4922 | 1.0 | 0.386 | 0.4922 | 0.5816 | ${A_{2}}\succ {A_{5}}\succ {A_{4}}\approx {A_{1}}\succ {A_{3}}$ |
| $\lambda =0\mathrm{.}7$ | 0.4769 | 1.0 | 0.378 | 0.4759 | 0.5622 | ${A_{2}}\succ {A_{5}}\succ {A_{1}}\succ {A_{4}}\succ {A_{3}}$ |
| $\lambda =0\mathrm{.}8$ | 0.4616 | 1.0 | 0.37 | 0.4596 | 0.5428 | ${A_{2}}\succ {A_{5}}\succ {A_{1}}\succ {A_{4}}\succ {A_{3}}$ |
| $\lambda =0\mathrm{.}9$ | 0.4463 | 1.0 | 0.362 | 0.4433 | 0.5234 | ${A_{2}}\succ {A_{5}}\succ {A_{1}}\succ {A_{4}}\succ {A_{3}}$ |
| $\lambda =1\mathrm{.}0$ (Utility measure by IVPFIS) | 0.431 | 1.0 | 0.354 | 0.427 | 0.504 | ${A_{2}}\succ {A_{5}}\succ {A_{1}}\succ {A_{4}}\succ {A_{3}}$ |
6.2 Practical Implications of the Study Based on Variation of Ranking
The sensitivity assessment reveals that the developed IVPF-based MCDM framework offers stable outcomes over diverse parameter settings such as criteria weighting parameter (ς) and strategy coefficient (λ). However, the experimental changes in the relative places of some hospitals over specific parameter arrangements also offer valuable real-world material for healthcare managers. A change in preference does not inevitably specify a weakness of the developed approach; rather, it reflects the diverse priorities allocated to the assessment criteria through the criteria weighting parameter (ς) and strategy coefficient (λ).
From a practical viewpoint, hospitals that sustain a consistently higher prioritization across the majority of parameter configurations can be viewed as indicating relatively sta ble performance across diverse interpretations of service quality significance. Such stability indicates that their overall performance is not strongly reliant on a specific parameter condition. On the other hand, hospitals whose positions change noticeably with the criteria weighting parameter (ς) and strategy coefficient (λ) should be interpreted with greater consideration because their relative outcome is more sensitive to changes in the importance or treatment of the assessment criteria in terms of better service quality.
This result is mainly appropriate for hospital administrators and healthcare policymakers. If any hospital changes from a higher to a lower preference when the significance rating of a specific service quality criterion is increased, this may indicate that the hospital has comparatively weaker performance on that criterion. Such an outcome can support administrators in recognizing areas needing targeted improvement rather than relying solely on the overall prioritization. For instance, variation in ranking associated with patient-management, patient-relationship, or patient-physician relationship criteria can be utilized to recognize which features of service quality have a greater influence on the hospital’s overall position.
Thus, the sensitivity assessment can assist not only as a validation mechanism for the robustness of the developed approach but also as a managerial analytical procedure. Stable rankings offer confidence in the steadiness of the decision results, while changes in preference orders highlight hospitals and service quality dimensions that permit further investigation. In real-world applications, hospital DEs can utilize these outcomes to discriminate between structural performance differences and priority-dependent performance differences, by this means supporting more focused quality-improvement interventions.
6.3 Comparative Analysis
Here, a comparative analysis is performed to assess the robustness of the developed methodology over the existing IVPF information-based MCDM methods. To this end, we selected the previously developed methods from distance and score function-based perspectives and applied them to the aforesaid case study to find a suitable hospital in terms of 18 service criteria. For this purpose, some of the well-known PF information-based decision-making methods, including an extension of the TOPSIS method under the IVPF environment (Kose et al., 2022), score function-based IVPF-TOPSIS (Oh et al., 2025), IVPF-VIKOR (Yu et al., 2025), IVPF distance-based extended inferior ratio method (Liu et al., 2025), are selected and applied to the aforesaid case study. Details of the comparative analysis are given in Supplementary Material. Finally, Fig. 5 demonstrates the ranking orders of hospitals attained by the suggested and previous models.
The main advantages of the proposed IVPF-WENSLO-RANCOM-CRADIS model are presented as follows:
-
i. This method uses a novel entropy to obtain the significance values of DMs, as the opinions of DMs are very important in the real-life hospital service quality assessment problem, while existing studies related to healthcare service quality evaluation ignore the DMs’ weights.
-
ii. Using the WENSLO and RANCOM models, the proposed method combines the objective and subjective weights of criteria during the evaluation of hospital service quality. This means that the present method not only considers the quantitative data for determining the criterion’s objective weight through the IVPF-WENSLO model but also involves the experts’ preferences during the computation of the subjective weight of the criterion via the IVPF-RANCOM model.
-
iv. The proposed method combines the DMs and criteria weighting tools with the IVPF extension of the CRADIS model, in which the deviation relation from the ideal solutions to the alternatives is computed through a novel distance measure. Thus, the results obtained by the developed IVPF-WENSLO-RANCOM-CRADIS model are more effective and robust than existing methods (Kose et al., 2022; Oh et al., 2025; Yu et al., 2025; Liu et al., 2025).
7 Conclusions and Recommendations
Given the increasing importance of evaluating healthcare service quality, this work proposed a hybrid decision support framework within the context of IVPFSs, as such types of real-life problems contain uncertain and imprecise information. Before developing the algorithm, several formulas were introduced, including entropy measures, score functions, and distance measures. These formulae demonstrated significant advantages in overcoming the limitations of existing entropy, score function, and distance measures within the IVPF environment. Furthermore, we explored the application of entropy, score function, and distance measure in the development of the IVPF-WENSLO-RANCOM-CRADIS model. The DMs’ weights were calculated through an entropy-based model, while the criteria weights were determined by an integrated WENSLO-RANCOM model. The combined criteria weights identify public transport, ambulance service, and medication expertise as the leading priorities in the evaluated setting, suggesting that hospital managers and policymakers should consider accessibility, emergency transport, and medication-related expertise when planning service improvements and allocating resources. Based on these steps, the alternatives’ ranking was determined via an IVPF extension of the CRADIS model. Moreover, this methodology was implemented in a case study of healthcare service quality evaluation of five hospitals in Patna, the most populous city of Bihar (India), which proves its applicability and effectiveness. The insights gained from this work can support policymakers in designing service quality policies that rank healthcare services based on performance. Furthermore, sensitivity and comparative analyses were executed to demonstrate the stability and robustness of the developed methodology, respectively. The method simultaneously ranked hospitals according to 18 service criteria and calculated the combined objective-subjective weights of the criteria through the IVPF-WENSLO-RANCOM model together with DMs’ weighting via the IVPF-entropy-based model. Moreover, there were obvious advantages of the suggested methodology in comparison with existing fuzzy MCDM methods related to healthcare service quality evaluation problems.
Nevertheless, this study has some limitations. First, there is no discussion about the interrelationships of criteria. Second, the considered criteria and DMs are very limited in this study. Third, it is necessary to separately appraise the quality of service of private and public hospitals. In future studies, we aim to overcome the limitations identified in this work. Specifically, machine learning and optimization methods may be combined with fuzzy information to render a more comprehensive assessment of hospitals’ overall service quality.
