1 Introduction
Advancements in science and technology have highlighted the necessity of managing uncertainty in decision-making processes, a challenge that has become increasingly critical in today’s complex and data-driven world. As the volume and complexity of data continue to grow, so does the need for tools and methodologies that can effectively handle ambiguity and incomplete information. Traditional binary logic, which operates on clear-cut true or false values, often falls short in dealing with the nuances of real-world scenarios where data is seldom straightforward. The introduction of fuzzy sets (FSs) by Zadeh (1965) has played a crucial role in addressing data ambiguity. Alongside FSs, hesitant FSs (HF) (Torra, 2010) have emerged, allowing for more flexible membership degrees (MDs) to consider various potential inputs, as perceived in the work of Rodriguez et al. (2014). This development not only reduces subjective randomness but also aids in expressing expert preferences and accommodating occurrence probabilities. Building on this foundation, Xu and Zhou (2016) introduced the probabilistic HFSs (PHFSs), which assign occurrence probabilities to elements based on systematic reviews. Integrating PHF into decision-making frameworks, as demonstrated by Li and Wang (2017), has led to the establishment of novel decision models such as those incorporating preference ranking organization method for enrichment evaluation (PROMETHEE) and (qualitative flexible multiple criteria method (QUALIFLEX). Further advancements include distance measures of PHFSs by Ding et al. (2017), and development of a density function for investor assessments by Li et al. (2019). Reference ideal-based algorithms by He and Xu (2019) provide a means to evaluate projects, linking ideal values with PHF information (PHFI). Additionally, Liu et al. (2020) combined PHFI with regret theory and entropy measures for venture capital evaluations. Li et al. (2020) introduced an approach based on Organization Rangement EtSynthese De Donnes Relationnelles (ORESTE) employing PHFI. Lin et al. (2020a) used a PHFI algorithm for consistency testing in investment projects. Jin et al. (2020) suggested a preference relation-based measure under PHFI, and applied to logistical selection. To address rational CO2 storage location selection, Guo et al. (2020) proposed Tomada de Decisão Interativa Multicritério (TODIM) method incorporating Choquet Integrals under PHFI. Lin et al. (2020b) proposed PHF-Multi-Objective Optimization by Ratio Analysis plus Full Multiplicative Form (MULTIMOORA) method, while Liu et al. (2021) defined cross-efficiency model of Data Envelopment Analysis (DEA) using PHF preference relations. Krishankumar et al. (2022) developed PHF-Complex Proportional Assessment (COPRAS) method, while Liao et al. (2022a) addressed a supplier selection problem based on PHF-CODAS (Combinative Distance based Assessment) model. Liao et al. (2022b) employed prospect theory based TODIM method under PHF setting, and Qi (2023) used PHF-Technique for Order of Preference by Similarity to Ideal Solution (PHF-TOPSIS) for quality assessment of public charging services. Jaisankar et al. (2023) used a hybrid PHF decision-making approach for assessment of plastic disposal technologies, while Liu et al. (2023) developed a modified Measurement of Alternatives and Ranking according to Compromise Solution (MARCOS) method incorporating PHFI.
1.1 Research Gaps and Motivations
MCDM is an important sub-set of decision theory, focusing on selecting the optimal alternative from a diverse set. The dynamic and ever-evolving socio-economic environment has significantly increased the complexity of real-world decision-making problems. This complexity arises from the need to consider various, often conflicting, criteria that impact the outcomes of decisions. Over recent decades, various methods have been developed to address the complexities of MCDM problems. These methods aim to enhance both the accuracy and efficiency of MCDM processes, each with distinct advantages and limitations. Some methods offer computational efficiency and ease of implementation, making them suitable for applications with limited resources; however, they may be less effective in managing uncertainty or imprecise data. On the other hand, certain methods are well-suited for handling uncertainty and providing stable solutions under variable conditions, though they often require increased computational resources and complexity.
The significance of each criterion in any decision-making process can vary based on the context, the specific decision to be made, and the stakeholders involved. Criteria weights represent the relative importance of each criterion in the decision-making process. Hence, determining the criteria weights must be done logically and systematically. The pertinent existing methods (Lin et al., 2020b; Krishankumar et al., 2022; Liao et al., 2022a; Qi, 2023; Liu et al., 2023a), often fail to derive these weights systematically, leading to subjectivity and inaccuracies. The difficulty in accurately weighting criteria arises from the complex nature of decision-making environments and varied perspectives of stakeholders. Thus, developing a comprehensive framework for determining criteria weights remains a challenging problem. Multi-criteria group decision-making (MCGDM) addresses this problem by integrating diverse expert perspectives and effectively managing trade-offs among conflicting criteria. By involving experts from various fields, MCGDM ensures that decisions are informed and balanced, enabling proper prioritization of the criteria. MCGDM models systematically evaluate trade-offs, providing clarity on compromises and optimizing outcomes. The process promotes transparency, consistency, and accountability, incorporating both objective data and subjective opinions. MCGDM also enhances decision quality by considering all relevant criteria, fostering collaboration, and increasing stakeholder trust. It helps decision-makers to understand the compromises involved in choosing between alternatives, ensuring that the final decision aligns with the overarching objectives. Furthermore, considering all relevant criteria ensures that no important aspects are overlooked, thereby improving the overall quality of the decision. However, due to their varied experiences and backgrounds, decision-makers often exhibit significant differences in evaluation, making consensus challenging. Existing methods like PHF-COPRAS (Krishankumar et al., 2022), PHF-TODIM (Liao et al., 2022b), PHF-MARCOS (Liu et al., 2023a), and PHF-TOPSIS (Qi, 2023), may fail to fully capture the ambiguity in expert judgments, especially when biases are present. Therefore, it becomes necessary to implement consensus-building strategies to increase agreement among decision-makers, a factor that previous studies with PHFI have not adequately addressed.
One critical aspect of improving consensus and enhancing decision quality is the effective aggregation of diverse expert inputs. Combining various input data into a unified entity often requires aggregation operators (AOs), which have proven highly effective in data processing, decision-making, pattern recognition, data analytics, and neural networks. While AOs like Archimedean AOs, Hamacher AOs, Einstein AOs, and Dombi AOs have been employed for aggregating PHFI, the potential of Aczel-Alsina AOs has been increasingly recognized in this context (Senapati et al., 2022a, 2023a, 2023b). These studies suggested that Aczel-Alsina AOs offer a promising approach to address these aggregation challenges, providing significantly accurate results in MCGDM environment, thus making Aczel-Alsina AOs a valuable tool for improving both consensus-building process and overall decision-making framework. The existing works on Aczel-Alsina AOs are summarized in Table 1.
Table 1
Existing works on Aczel-Alsina AOs.
| Reference | AO | Application |
| Senapati et al. (2022a) | Interval-valued Pythagorean fuzzy Aczel-Alsina AOs | Selection of an emerging IT software company |
| Mahmood et al. (2022) | Complex intuitionistic fuzzy Aczel-Alsina AOs | Selection of an advertising administrator |
| Ali and Naeem (2022) | Complex q-rung orthopair fuzzy Aczel-Alsina AOs | Selection of the most impactful sector effecting the Stock Exchange |
| Li et al. (2023) | Neutrosophic multi-valued Aczel-Alsina AOs | Selection of service robots |
| Wang et al. (2023) | T-spherical fuzzy Aczel-Alsina Hamy Mean AOs | Assessment of investment company plans |
| Chen et al. (2023) | Complex Fermatean fuzzy Aczel-Alsina AOs | Assessment of different bands for solar panel system |
| Athar Farid and Riaz (2023) | q-rung orthopair fuzzy Aczel-Alsina AOs | Green supplier selection |
| Liu et al. (2023) | Complex intuitionistic fuzzy Aczel-Alsina prioritized AOs | Assessment of business alternatives |
| Senapati (2024) | Single valued neutrosophic Aczel-Alsina AOs | Assessment of investment opportunities |
| Gula et al. (2024) | Aczel-Alsina linear Diophantine fuzzy AOs | Selection of weather forecasting techniques |
The continuous improvement and adaptation of MCDM methods reflect ongoing efforts to bridge theoretical progress with practical application. Efforts to make these methods more responsive to emerging needs, including large-scale data processing and adaptable decision-making in dynamic environments, are steadily advancing. Balancing theoretical rigor with practical usefulness highlights the essential role of MCDM frameworks in modern applications, where solutions must combine accuracy with adaptability to meet the complexities of real-world problems. Brauers and Zavadskas (2006) proposed MOORA model, a well-known and effective MCDM method that combines reference point (RP) and ratio system (RS) models. MULTIMOORA (Brauers and Zavadskas, 2010), an extension of MOORA, was developed based on RS, RP and full multiplicative form (FMF) models. In recent years, under various fuzzy contexts MULTIMOORA method have been utilized for purchasing rental space (Stanujkic et al., 2019), selection of technology for food waste treatment (Rani et al., 2021), charging station selection for electric vehicles (Rani and Mishra, 2021), CNC machine tool selection (Sahin and Aydemir, 2022), solid waste disposal method selection (Mishra et al., 2023), failure mode and effects analysis (Yu et al., 2023), green supplier selection (Gai et al., 2023), welding process selection (Saluja and Singh, 2023), offshore wind power station site selection (Zhou et al., 2024), crop disease detection (Zhang et al., 2024), sustainable supplier selection (Vaezi et al., 2024), car selection through online reviews (Liu et al., 2024), business strategies evaluation (Ghaemi-Zadeh and Eghbali-Zarch, 2024), sustainability of urban mobility evaluation (Yucesan et al., 2024). Consensus-reaching mechanisms for structured group decision-making have not yet been incorporated into MULTIMOORA method. With increasing environmental concerns and regulatory pressures, the importance of sustainable practices in operations and supply chains is being widely acknowledged. Lean, agile, resilient, green, and sustainable approaches are being integrated throughout supply chain and manufacturing processes to address uncertainties and support competitiveness. The selection of manufacturing outsourcing vendors (MOVs) significantly impacts operational efficiency, environmental performance, and resilience to disruptions. Although promising, the PHF-based MULTIMOORA method has not been applied to MOV selection, marking an area with potential for further development.
1.2 Contributions
Nowadays, decision makers are increasingly seeking systematic approaches to determine optimal actions. Despite the wealth of literature on PHFSs, there is a notable dearth of research leveraging the complete potential of PHFI, Aczel-Alsina AOs, the consensus-building process, and MULTIMOORA method to address decision-making complexities. This study fills critical gaps in the literature by making significant contributions in these areas.
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a) The research introduces two novel PHF Aczel-Alsina Aggregation Operators (PHFAAWA and PHFAAWG) designed specifically to tackle the intricate challenges of group decision-making utilizing PHFI. These adaptable weighted operators provide decision-makers with a resilient tool to efficiently amalgamate varied perspectives within a group context.
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b) A comprehensive consensus-based MULTIMOORA model is presented in this research, specifically designed to assess MOVs. This model incorporates an advanced optimization process to calculate criteria weights, ensuring the objectivity and robustness of the assessment process. By utilizing this model, decision-makers are equipped to make well-informed decisions when selecting manufacturing outsourcing vendors.
1.3 Structural Overview of the Paper
This paper follows a structured framework: Section 2 provides a comprehensive overview of essential concepts necessary for understanding the subsequent discussions. Section 3 examines the details of Aczel-Alsina operations between PHF elements, explaining the definitions and characteristics while introducing PHFAAWA and PHFAAWG AOs. Building upon this foundation, Section 4 elucidates the consensus-based PHF decision support paradigm. The practical application of these methodologies is exemplified in Section 5, which presents a detailed case study and the corresponding solution. Section 6 provides comprehensive discussions on sensitivity analysis, and comparative study. Finally, Section 7 encapsulates the research outcomes, offering insightful recommendations for future investigations in the field.
2 Preliminaries
Definition 1 (Xu and Zhou, 2016).
A PHF set ℵ on a set U is defined as: $\mathrm{\aleph }=\{\langle \alpha ,\varpi (\alpha )\rangle :\alpha \in U\}$ where $\varpi (\alpha )={\textstyle\bigcup _{t}}\{{\delta _{\alpha }^{(t)}}({p^{(t)}})\}$, and $0\leqslant {\textstyle\sum _{t}}{p^{(t)}}\leqslant 1$ (${P^{(q)}}$ being the probability of ${\delta _{\alpha }^{(t)}}\in [0,1]$). If U is singleton, then ℵ reduces to a PHF element (PHFE) and we write $\mathrm{\aleph }={\textstyle\bigcup _{t}}\{{\delta ^{(t)}}({p^{(t)}})\}$ and $s(\mathrm{\aleph })={\textstyle\sum _{t}}({\delta ^{(t)}}\times {p^{(t)}})$ to denote it’s a score value. For two PHFEs ℵ and ℜ, $s(\mathrm{\aleph })\gt s(\mathrm{\Re })\Rightarrow \mathrm{\aleph }\succ \mathrm{\Re }$.
Definition 2 (Senapati et al., 2022b).
3 Aczel-Alsina-Operations Between PHFEs and Associated Weighted Operators
Let ${\mathrm{\aleph }_{n}}=\langle {\textstyle\bigcup _{t}}\{{\delta _{n}^{(t)}}({p_{n}^{(t)}})\}\rangle $ $(n=1,2,\dots ,\ell )$ be a collection of PHFEs.
Definition 3.
For $\sigma \gt 0$, we define:
(3)
\[\begin{aligned}{}(\mathrm{i})\hspace{2.5pt}& {\mathrm{\aleph }_{1}}\oplus {\mathrm{\aleph }_{2}}=\Bigg\langle \bigcup \limits_{t}\Bigg\{1-\exp \Bigg[-{\Bigg({\sum \limits_{n=1}^{2}}{\big(-\ln \big(1-{\delta _{n}^{(t)}}\big)\big)^{\theta }}\Bigg)^{\frac{1}{\theta }}}\Bigg]\Bigg\}\big({p_{{^{1}}}^{(t)}}{p_{{^{2}}}^{(t)}}\big)\Bigg\rangle ,\end{aligned}\](4)
\[\begin{aligned}{}(\mathrm{ii})\hspace{2.5pt}& {\mathrm{\aleph }_{1}}\otimes {\mathrm{\aleph }_{2}}=\Bigg\langle \bigcup \limits_{t}\Bigg\{\exp \Bigg[-{\Bigg({\sum \limits_{n=1}^{2}}{\big(-\ln \big({\delta _{n}^{(t)}}\big)\big)^{\theta }}\Bigg)^{\frac{1}{\theta }}}\Bigg]\Bigg\}\big({p_{{^{1}}}^{(t)}}{p_{{^{2}}}^{(t)}}\big)\Bigg\rangle ,\end{aligned}\]Theorem 1.
Let ${\mathrm{\aleph }_{n}}=\langle {\textstyle\bigcup _{t}}\{{\delta _{n}^{(t)}}({p_{n}^{(t)}})\}\rangle $ $(n=1,2)$ be a collection of PHFEs. Then
\[\begin{aligned}{}(\mathrm{i})\hspace{2.5pt}& {\mathrm{\aleph }_{1}}\oplus {\mathrm{\aleph }_{2}}={\mathrm{\aleph }_{2}}\oplus {\mathrm{\aleph }_{1}},\\ {} (\mathrm{ii})\hspace{2.5pt}& {\mathrm{\aleph }_{1}}\otimes {\mathrm{\aleph }_{2}}={\mathrm{\aleph }_{2}}\otimes {\mathrm{\aleph }_{1}},\\ {} (\mathrm{iii})\hspace{2.5pt}& {\sigma _{1}}({\mathrm{\aleph }_{1}}\oplus {\mathrm{\aleph }_{2}})=({\sigma _{1}}{\mathrm{\aleph }_{1}})\oplus ({\sigma _{2}}{\mathrm{\aleph }_{2}}),\\ {} (\mathrm{iv})\hspace{2.5pt}& {({\mathrm{\aleph }_{1}}\otimes {\mathrm{\aleph }_{2}})^{{\sigma _{1}}}}=\big({\mathrm{\aleph }_{1}^{{\sigma _{1}}}}\big)\oplus \big({\mathrm{\aleph }_{2}^{{\sigma _{1}}}}\big),\\ {} (\mathrm{v})\hspace{2.5pt}& ({\sigma _{1}}+{\sigma _{2}}){\mathrm{\aleph }_{1}}=({\sigma _{1}}{\mathrm{\aleph }_{1}})\oplus ({\sigma _{2}}{\mathrm{\aleph }_{2}}),\\ {} (\mathrm{vi})\hspace{2.5pt}& {\mathrm{\aleph }_{1}^{{\sigma _{1}}+{\sigma _{2}}}}=\big({\mathrm{\aleph }_{1}^{{\sigma _{1}}}}\big)\otimes \big({\mathrm{\aleph }_{2}^{{\sigma _{2}}}}\big).\end{aligned}\]
Definition 4.
Suppose ${W_{n}}$ denotes the weight of ${\mathrm{\aleph }_{n}}$ such that $0\leqslant {W_{n}}\leqslant 1$ and ${\textstyle\sum _{n=1}^{\ell }}{W_{n}}=1$. Then the PHF-AA weighted averaging (PHFAAWA) operator is presented as:
Theorem 2.
$\textit{PHFAAWA}({\mathrm{\aleph }_{1}},{\mathrm{\aleph }_{2}},\dots ,{\mathrm{\aleph }_{\ell }})$ can be expressed as a PHFE and
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\[\begin{aligned}{}& \textit{PHFAAWA}({\mathrm{\aleph }_{1}},{\mathrm{\aleph }_{2}},\dots ,{\mathrm{\aleph }_{\ell }})\\ {} & \hspace{1em}=\Bigg\langle \bigcup \limits_{t}\Bigg\{1-\exp \Bigg[-{\Bigg({\sum \limits_{n=1}^{\ell }}{W_{n}}{\big(-\ln \big(1-{\delta _{n}^{(t)}}\big)\big)^{\theta }}\Bigg)^{\frac{1}{\theta }}}\Bigg]\Bigg\}\Bigg({\prod \limits_{n=1}^{\ell }}{p_{{^{n}}}^{(t)}}\Bigg)\Bigg\rangle .\end{aligned}\]Several properties of the PHFAAWA operator are presented below.
Theorem 3.
If ${\mathrm{\aleph }_{0}}$ ($\ne {\mathrm{\aleph }_{n}}$ for any n) is a PHFE, then $\textit{PHFAAWA}({\mathrm{\aleph }_{0}}\oplus {\mathrm{\aleph }_{1}},{\mathrm{\aleph }_{0}}\oplus {\mathrm{\aleph }_{2}},\dots ,{\mathrm{\aleph }_{0}}\oplus {\mathrm{\aleph }_{\ell }})={\mathrm{\aleph }_{0}}\oplus \textit{PHFAAWA}({\mathrm{\aleph }_{1}},{\mathrm{\aleph }_{2}},\dots ,{\mathrm{\aleph }_{\ell }})$.
Theorem 4.
If ${\mathrm{\aleph }_{0}}$ ($\ne {\mathrm{\aleph }_{n}}$ for any n) is a PHFE, then $\textit{PHFAAWA}({\mathrm{\aleph }_{1}},{\mathrm{\aleph }_{2}},\dots ,{\mathrm{\aleph }_{\ell }})={\mathrm{\aleph }_{0}}$.
Theorem 5.
If ${({\mathrm{\aleph }_{n}})^{-}}=\big\langle {\textstyle\bigcup _{t}}\{{\min _{n}}{\delta _{n}^{(t)}}({p_{{^{n}}}^{(t)}})\}\big\rangle $ and ${({\mathrm{\aleph }_{n}})^{+}}=\langle {\textstyle\bigcup _{t}}\{{\max _{n}}{\delta _{n}^{(t)}}({p_{{^{n}}}^{(t)}})\}\rangle $, then ${({\mathrm{\aleph }_{n}})^{-}}\prec \textit{PHFAAWA}({\mathrm{\aleph }_{1}},{\mathrm{\aleph }_{2}},\dots ,{\mathrm{\aleph }_{\ell }})\prec {({\mathrm{\aleph }_{n}})^{+}}$.
Theorem 6.
If ${\mathrm{\Re }_{n}}=\big\langle {\textstyle\bigcup _{t}}\{{\delta ^{\prime \hspace{0.1667em}(t)}_{n}}({p_{n}^{(t)}})\}\big\rangle $ $(n=1,2,\dots ,\ell )$ be another collection of PHFEs with ${\delta _{n}^{(t)}}\leqslant {\delta ^{\prime \hspace{0.1667em}(t)}_{n}}$ $\forall n$, then $\textit{PHFAAWA}({\mathrm{\aleph }_{1}},{\mathrm{\aleph }_{2}},\dots ,{\mathrm{\aleph }_{\ell }})\prec \textit{PHFAAWA}({\mathrm{\Re }_{1}},{\mathrm{\Re }_{2}},\dots ,{\mathrm{\Re }_{\ell }})$.
Definition 5.
Suppose ${W_{n}}$ denotes the weight of ${\mathrm{\aleph }_{n}}$ such that $0\leqslant {W_{n}}\leqslant 1$ and ${\textstyle\sum _{n=1}^{\ell }}{W_{n}}=1$. Then the PHF-AA weighted geometric (PHFAAWG) operator is presented as:
Theorem 7.
$\textit{PHFAAWG}({\mathrm{\aleph }_{1}},{\mathrm{\aleph }_{2}},\dots ,{\mathrm{\aleph }_{\ell }})$ can be expressed as a PHFE and
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\[\begin{aligned}{}& \textit{PHFAAWG}({\mathrm{\aleph }_{1}},{\mathrm{\aleph }_{2}},\dots ,{\mathrm{\aleph }_{\ell }})\\ {} & \hspace{1em}=\Bigg\langle \bigcup \limits_{t}\Bigg\{\exp \Bigg[-{\Bigg({\sum \limits_{n=1}^{\ell }}{W_{n}}{\big(-\ln \big({\delta _{n}^{(t)}}\big)\big)^{\theta }}\Bigg)^{\frac{1}{\theta }}}\Bigg]\Bigg\}\Bigg({\prod \limits_{n=1}^{\ell }}{p_{{^{n}}}^{(t)}}\Bigg)\Bigg\rangle .\end{aligned}\]Some properties of PHFAAWG operator are presented as follows:
Theorem 8.
If ${\mathrm{\aleph }_{0}}$ ($\ne {\mathrm{\aleph }_{n}}$ for any n) is a PHFE, then $\textit{PHFAAWG}({\mathrm{\aleph }_{0}}\otimes {\mathrm{\aleph }_{1}},{\mathrm{\aleph }_{0}}\otimes {\mathrm{\aleph }_{2}},\dots ,{\mathrm{\aleph }_{0}}\otimes {\mathrm{\aleph }_{\ell }})={\mathrm{\aleph }_{0}}\otimes \textit{PHFAAWG}({\mathrm{\aleph }_{1}},{\mathrm{\aleph }_{2}},\dots ,{\mathrm{\aleph }_{\ell }})$.
Theorem 9.
If ${\mathrm{\aleph }_{0}}$ ($\ne {\mathrm{\aleph }_{n}}$ for any n) is a PHFE, then $\textit{PHFAAWG}({\mathrm{\aleph }_{1}},{\mathrm{\aleph }_{2}},\dots ,{\mathrm{\aleph }_{\ell }})={\mathrm{\aleph }_{0}}$.
Theorem 10.
If ${({\mathrm{\aleph }_{n}})^{-}}=\big\langle {\textstyle\bigcup _{t}}\{{\min _{n}}{\delta _{n}^{(t)}}({p_{{^{n}}}^{(t)}})\}\big\rangle $ and ${({\mathrm{\aleph }_{n}})^{+}}=\big\langle {\textstyle\bigcup _{t}}\{{\max _{n}}{\delta _{n}^{(t)}}({p_{{^{n}}}^{(t)}})\}\big\rangle $, then
Theorem 11.
If ${\mathrm{\Re }_{n}}=\big\langle {\textstyle\bigcup _{t}}\{{\delta ^{\prime \hspace{0.1667em}(t)}_{n}}({p_{n}^{(t)}})\}\big\rangle $ $(n=1,2,\dots ,\ell )$ be another collection of PHFEs with ${\delta _{n}^{(t)}}\leqslant {\delta ^{\prime \hspace{0.1667em}(t)}_{n}}$ $\forall n$, then $\textit{PHFAAWG}({\mathrm{\aleph }_{1}},{\mathrm{\aleph }_{2}},\dots ,{\mathrm{\aleph }_{\ell }})\prec \textit{PHFAAWG}({\mathrm{\Re }_{1}},{\mathrm{\Re }_{2}},\dots ,{\mathrm{\Re }_{\ell }})$.
4 Group Decision-Making Methodology
Let’s consider that the alternatives ${N_{1}},{N_{2}},\dots ,{N_{n}}$ and factors (criteria) ${F_{1}},{F_{2}},\dots ,{F_{q}}$ are linked to a group assessment scenario, where each alternative is evaluated by the experts (specialists) ${E_{1}},{E_{2}},\dots ,{E_{l}}$ under the PHF framework. The initial results, analysed by the experts, are depicted as the PHF decision matrices, as given below:
The proposed framework includes the following steps.
Step 1: Use PHFAAWA or PHFAAWG operator to get the aggregated PHF matrix (A-PHF-M).
A-PHF-M is ${[{\mathrm{\aleph }^{rs}}]_{n\times q}}={\big[\big\langle {\textstyle\bigcup _{b}}\{{\delta _{rs}^{(b)}}({p_{rs}^{(b)}})\}\big\rangle \big]_{n\times q}}$ where:
OR
(${\varpi _{m}}$ being the weight of ${E_{m}}$).
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\[ {\mathrm{\aleph }^{rs}}=\textit{PHFAAWA}\big({\mathrm{\aleph }_{1}^{rs}},{\mathrm{\aleph }_{2}^{rs}},\dots ,{\mathrm{\aleph }_{l}^{rs}}\big)={\underset{m=1}{\overset{l}{\bigoplus }}}\big({\varpi _{m}}{\mathrm{\aleph }_{m}^{rs}}\big)\](13)
\[ {\mathrm{\aleph }^{rs}}=\textit{PHFAAWG}\big({\mathrm{\aleph }_{1}^{rs}},{\mathrm{\aleph }_{2}^{rs}},\dots ,{\mathrm{\aleph }_{l}^{rs}}\big)={\underset{m=1}{\overset{l}{\bigotimes }}}{\big({\mathrm{\aleph }_{m}^{rs}}\big)^{{\varpi _{m}}}}\]
Step 2: Calculate the “degree of consensus” (DC) for each expert.
Let us consider the following fuzzy matrices.
The correlation measure (CM) ${\Omega _{s}^{(m)}}$ for ${E_{m}}$ under ${H_{s}}$ can be defined as:
where
(16)
\[ {\Omega _{s}^{(m)}}=\frac{{\textstyle\textstyle\sum _{r=1}^{p}}\big[\big(\frac{{D_{rs}^{(m)}}}{{D_{s}^{(m)}}}-\frac{1}{p}{\textstyle\textstyle\sum _{r=1}^{p}}\frac{{D_{ij}^{(m)}}}{{D_{s}^{(m)}}}\big)\times \big(\frac{{D_{rs}}}{{D_{s}}}-\frac{1}{p}{\textstyle\textstyle\sum _{r=1}^{p}}\frac{{D_{rs}}}{{D_{s}}}\big)\big]}{\sqrt{{\textstyle\textstyle\sum _{r=1}^{p}}{\big(\frac{{D_{ij}^{(m)}}}{{D_{s}^{(m)}}}-\frac{1}{p}{\textstyle\textstyle\sum _{r=1}^{p}}\frac{{D_{rs}^{(m)}}}{{D_{s}^{(m)}}}\big)^{2}}}\times \sqrt{{\textstyle\textstyle\sum _{r=1}^{p}}{\big(\frac{{D_{rs}}}{{D_{s}}}-\frac{1}{p}{\textstyle\textstyle\sum _{r=1}^{p}}\frac{{D_{rs}}}{{D_{s}}}\big)^{2}}}},\]
\[\begin{aligned}{}& {\bar{\zeta }_{m}^{(rs)}}=\Big\langle \underset{r}{\max }{\Delta _{m}^{rs}}\Big\rangle ,\hspace{1em}{\underline{\zeta }_{m}^{(rs)}}=\Big\langle \underset{r}{\min }{\Delta _{m}^{rs}}\Big\rangle ,\hspace{1em}{\bar{\zeta }^{(rs)}}=\Big\langle \underset{r}{\max }{\Delta ^{rs}}\Big\rangle ,\hspace{1em}{\underline{\zeta }^{(rs)}}=\Big\langle \underset{r}{\min }{\Delta ^{rs}}\Big\rangle ,\\ {} & {D_{rs}^{(m)}}=\mathrm{Distance}\big({\zeta _{m}^{(rs)}},{\bar{\zeta }_{m}^{(rs)}}\big)=\big|{\zeta _{m}^{(rs)}}-{\bar{\zeta }_{m}^{(rs)}}\big|,\\ {} & {D_{s}^{(m)}}=\mathrm{Distance}\big({\bar{\zeta }_{m}^{(rs)}},{\underline{\zeta }_{m}^{(rs)}}\big)=\big|{\bar{\zeta }_{m}^{(rs)}}-{\underline{\zeta }_{m}^{(rs)}}\big|,\\ {} & {D_{rs}}=\mathrm{Distance}\big({\bar{\zeta }^{(rs)}},{\zeta ^{(rs)}}\big)=\big|{\bar{\zeta }^{(rs)}}-{\zeta ^{(rs)}}\big|,\\ {} & Dis{t_{s}}=\mathrm{Distance}\big({\bar{\zeta }^{(rs)}},{\underline{\zeta }^{(rs)}}\big)=\big|{\bar{\zeta }^{(rs)}}-{\underline{\zeta }^{(rs)}}\big|.\end{aligned}\]
Next, CD ${\vartheta ^{(m)}}$ of ${E_{m}}$ can be defined as:
Obviously ${\vartheta ^{(m)}}\in [0,1]$. If ϑ denotes the predefined DC for each expert, then ${\vartheta ^{(m)}}\geqslant \vartheta $ must be fulfilled. For ${\vartheta ^{(m)}}\lt \vartheta $, then we have to consider other values of θ till ${\vartheta ^{(m)}}\geqslant \vartheta $ is achieved for all ${E_{m}}$.
Step 3: Calculation of criteria weights.
The divergence measure, as given in Eq. (18), exhibits the difference between rth alternative and other alternatives under sth criterion.
where $C({\Delta ^{rs}},{\Delta ^{zs}})$ which corresponds the cross-entropy measure between ${\Delta ^{rs}}$ and ${\Delta ^{zs}}$ is defined as:
The overall divergence corresponding to sth criterion is calculated as:
Hence, upon solving the optimization model, criteria weights can be determined.
where ${\beta _{s}}$ represents the weight of sth criterion and Ξ represents the set of partial information regarding criteria weights.
(18)
\[ {\textit{DIV}_{rs}}=\frac{1}{n-1}{\sum \limits_{z=1}^{n}}C\big({\Delta ^{rs}},{\Delta ^{zs}}\big).\](19)
\[\begin{aligned}{}C\big({\Delta ^{rs}},{\Delta ^{zs}}\big)& ={\Delta ^{rs}}\times \ln \bigg(\frac{2{\Delta ^{rs}}}{{\Delta ^{rs}}+{\Delta ^{zs}}}\bigg)+{\Delta ^{zs}}\times \ln \bigg(\frac{2{\Delta ^{zs}}}{{\Delta ^{rs}}+{\Delta ^{zs}}}\bigg)\\ {} & \hspace{1em}+\big(1-{\Delta ^{rs}}\big)\times \ln \bigg(\frac{1-{\Delta ^{rs}}}{1-\frac{1}{2}({\Delta ^{rs}}+{\Delta ^{zs}})}\bigg)\\ {} & \hspace{1em}+\big(1-{\Delta ^{zs}}\big)\times \ln \bigg(\frac{1-{\Delta ^{zs}}}{1-\frac{1}{2}({\Delta ^{rs}}+{\Delta ^{zs}})}\bigg).\end{aligned}\](20)
\[ {\textit{DIV}_{s}}=\frac{1}{n-1}{\sum \limits_{r=1}^{n}}{\sum \limits_{z=1}^{n}}C\big({\Delta ^{rs}},{\Delta ^{zs}}\big).\](21)
\[ \begin{aligned}{}& \text{Max}\chi ={\sum \limits_{s=1}^{q}}{\beta _{s}}\frac{1}{n-1}{\sum \limits_{r=1}^{n}}{\sum \limits_{z=1}^{n}}C\big({\Delta ^{rs}},{\Delta ^{zs}}\big),\\ {} & \text{Subject to:}\hspace{2.5pt}{\beta _{s}}\in \Xi ,{\sum \limits_{s=1}^{q}}{\beta _{s}}=1,\hspace{1em}{\beta _{s}}\geqslant 0\forall s,\end{aligned}\]
Step 4: Derive priority order of the alternatives using MULTIMOORA.
Step 4.1: Determine the significance values (SVs) of alternatives by “ratio system” (RS) technique.
Suppose ${\zeta _{r}^{+}}$ and ${\zeta _{r}^{-}}$ denote the level of significance of ${N_{r}}$ in relation to benefit and cost attributes respectively. They can be calculated by:
Determine the SV for ${N_{r}}$ utilizing Eq. (23):
(22)
\[ {\zeta _{r}^{+}}=\underset{s\in \mathrm{Benefit}}{\bigoplus }\big({\beta _{s}}{\Delta ^{rs}}\big),\hspace{2em}{\zeta _{r}^{-}}=\underset{s\in \mathrm{Cost}}{\bigoplus }\big({\beta _{s}}{\Delta ^{rs}}\big).\]
Step 4.2: Calculate the overall SVs (OSVs) of alternatives by “ratio point” (RP) model.
The OSVs (${\vartheta _{s}}$) of RP model for the sth criterion (${F_{s}}$) are obtained using Eq. (24).
Use Eq. (25) to calculate the weighted distances (WDs) for all alternatives.
Determine the maximum WD based on Eq. (26).
(24)
\[ {\vartheta _{s}}=\left\{\begin{array}{l@{\hskip4.0pt}l}{\max _{r}}{\Delta ^{rs}},\hspace{1em}& \text{for}\hspace{2.5pt}s\in \mathrm{Benefit},\\ {} {\min _{r}}{\Delta ^{rs}},\hspace{1em}& \text{for}\hspace{2.5pt}s\in \mathrm{Cost}.\end{array}\right.\]
Step 4.3: Calculate SVs of the alternatives by FMF model.
Suppose ${\alpha _{r}^{+}}$ and ${\alpha _{r}^{+}}$ denote the level of significance of ${N_{r}}$ related to benefit and cost attributes respectively. We calculate them as:
Determine the SV of ${N_{r}}$ using Eq. (28).
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\[ {\alpha _{r}^{+}}=\underset{s\in \mathrm{Benefit}}{\bigotimes }{\big({\Delta ^{rs}}\big)^{{\beta _{s}}}},\hspace{2em}{\alpha _{r}^{-}}=\underset{s\in \mathrm{Cost}}{\bigotimes }{\big({\Delta ^{rs}}\big)^{{\beta _{s}}}}.\]
Step 4.4: Calculate the final priority values (FPVs) of alternatives.
The FPV of ${N_{r}}$ by improved Borda Rule, is calculated as follows:
Here, ${\tilde{\zeta }_{r}}$, ${\tilde{\eta }_{r}}$, ${\tilde{\tau }_{r}}$ are the scores (normalized) of ${B_{r}}$ by RS, RP and FMF models, respectively. Alternatives are ranked based on their FPVs.
(29)
\[\begin{aligned}{}B({N_{r}})& =\frac{2}{n(n+1)}\big[{\tilde{\zeta }_{r}}\times \big(n-rank({\tilde{\zeta }_{r}})+1\big)-{\tilde{\eta }_{r}}\times rank({\tilde{\eta }_{r}})\\ {} & \hspace{1em}+{\tilde{\tau }_{r}}\times \big(n-rank({\tilde{\tau }_{r}})+1\big)\big]\hspace{1em}(r=1,2,\dots ,n).\end{aligned}\]5 Case Study
5.1 Problem Description
One of the main forces for macroeconomic expansion and technological improvement in India is the automobile sector. By 2026, volume-wise, India is predicted to become the third-largest vehicle market in the world (Miglani, 2019). Approximately 22.70 million vehicles, including passenger, commercial, three- and four-wheeler vehicles, were produced in India during the 2020–2021 fiscal year (Luthra et al., 2016). By 2026, the Indian automobile sector is projected to generate $300 billion in revenue (Mathivathanan et al., 2018). The automobile sector in India generates around 37 million direct and indirect jobs, accounts for 49% of all manufacturing output, and contributes 7.1% of the country’s GDP (Borkhade et al., 2022).
In this paper, a well-known Indian producer of commercial vehicles (referred to as Company Z), which has a robust domestic and global market and has been in operation for the previous forty years, is examined. The business holds an ISO 9001 accreditation. As one of the leading car manufacturers in India, Company Z is well-known for its excellent performance in terms of mileage, payload, strength and durability, luxury and comfort, convenience of warranty and servicing, and so on. The corporation has been working to achieve sustainability and agility in their operations. The case company has therefore put outsourcing rules into place and is searching for a trustworthy outsourcing vendor.
To identify and assess agile and sustainability criteria, a committee comprising three experts was formed. The committee chose six agile criteria, namely production flexibility and capability (F1), service level and lead time (F2), multi-skilled and flexible workforce (F3), collaboration with partners (F4), customer driven innovation (F5), delivery and sourcing flexibility (F6), and six sustainability criteria, namely: product price (F7), resource consumption (F8), green product (F9), green manufacturing process (F10), workers’ occupational health and safety (F11), social welfare and community development (F12). A brief description about the criteria is presented in Table 2.
Table 2
Significance of the considered MOV selection criteria.
| Category | Criteria | Significance |
| Agile | Production flexibility and capability (F1) | Ability of the vendor to adapt their production process to meet changing demands, enabling faster product variations or customized orders. |
| Service level and lead time (F2) | The time it takes to fulfil an order and deliver the product, crucial for meeting customer expectations and maintaining competitiveness in dynamic markets. | |
| Multi-skilled and flexible workforce (F3) | The versatility of the workforce in handling various tasks, ensuring smooth transitions in production processes without delays or quality compromise. | |
| Collaboration with partners (F4) | The extent of cooperation with supply chain partners to improve efficiency, transparency, and responsiveness. | |
| Customer-driven innovation (F5) | Innovation driven by customer needs, which helps in creating customized or advanced products that meet market demands. | |
| Delivery and sourcing flexibility (F6) | Ability to adjust delivery schedules and sourcing strategies according to demand fluctuations, reducing risks and delays in the supply chain. | |
| Sustainability | Product price (F7) | The cost at which the product is sold, influencing both affordability and the vendor’s ability to maintain competitive pricing in the market. |
| Resource consumption (F8) | The efficiency with which the vendor uses resources, such as energy and raw materials, to minimize waste and environmental impact. | |
| Green product (F9) | A product designed with environmental considerations, using sustainable materials and processes to minimize environmental impact. | |
| Green manufacturing process (F10) | The adoption of eco-friendly manufacturing processes that reduce environmental harm, such as energy consumption and emissions. | |
| Workers’ occupational health and safety (F11) | Focus on the health, safety, and well-being of workers in the production environment, ensuring a safe working atmosphere and compliance with regulations. | |
| Social welfare and community development (F12) | Contributions to societal well-being, including community engagement, promoting social equity, and supporting local development. |
The initial evaluations provided by the three experts for the outsourcing vendors Altras (${N_{1}}$), Bertrandt (${N_{2}}$), Tata technology (${N_{3}}$) and EDAG (${N_{4}}$) are presented in Tables 3, 4, and 5 respectively.
Table 3
Primary assessments by the 1st expert.
| MOV | ${\boldsymbol{F}_{\textbf{1}}}$ | ${\boldsymbol{F}_{\textbf{2}}}$ | ${\boldsymbol{F}_{\textbf{3}}}$ | ${\boldsymbol{F}_{\textbf{4}}}$ |
| ${\boldsymbol{N}_{\textbf{1}}}$ | $\langle 0.6(0.5),0.8(0.5)\rangle $ | $\langle 0.8(0.5),0.9(0.5)\rangle $ | $\langle 0.4(0.6),0.5(0.4)\rangle $ | $\langle 0.6(0.5),0.8(0.5)\rangle $ |
| ${\boldsymbol{N}_{\textbf{2}}}$ | $\langle 0.8(1)\rangle $ | $\langle 0.7(1)\rangle $ | $\langle 0.6(1)\rangle $ | $\langle 0.8(1)\rangle $ |
| ${\boldsymbol{N}_{\textbf{3}}}$ | $\langle 0.5(1)\rangle $ | $\langle 0.6(1)\rangle $ | $\langle 0.3(1)\rangle $ | $\langle 0.4(1)\rangle $ |
| ${\boldsymbol{N}_{\textbf{4}}}$ | $\langle 0.1(0.5),0.3(0.5)\rangle $ | $\langle 0.5(0.5),0.6(0.5)\rangle $ | $\langle 0.1(0.5),0.2(0.5)\rangle $ | $\langle 0.2(0.5),0.3(0.5)\rangle $ |
| ${\boldsymbol{F}_{\textbf{5}}}$ | ${\boldsymbol{F}_{\textbf{6}}}$ | ${\boldsymbol{F}_{\textbf{7}}}$ | ${\boldsymbol{F}_{\textbf{8}}}$ | |
| ${\boldsymbol{N}_{\textbf{1}}}$ | $\langle 0.3(0.5),0.5(0.5)\rangle $ | $\langle 0.5(0.5),0.7(0.5)\rangle $ | $\langle 0.6(0.5),0.8(0.5)\rangle $ | $\langle 0.5(0.5),0.6(0.5)\rangle $ |
| ${\boldsymbol{N}_{\textbf{2}}}$ | $\langle 0.6(1)\rangle $ | $\langle 0.7(1)\rangle $ | $\langle 0.9(1)\rangle $ | $\langle 0.8(1)\rangle $ |
| ${\boldsymbol{N}_{\textbf{3}}}$ | $\langle 0.3(1)\rangle $ | $\langle 0.5(1)\rangle $ | $\langle 0.7(1)\rangle $ | $\langle 0.5(1)\rangle $ |
| ${\boldsymbol{N}_{\textbf{4}}}$ | $\langle 0.3(0.5),0.5(0.5)\rangle $ | $\langle 0.2(0.6),0.4(0.4)\rangle $ | $\langle 0.5(0.5),0.6(0.5)\rangle $ | $\langle 0.3(0.4),0.4(0.6)\rangle $ |
| ${\boldsymbol{F}_{\textbf{9}}}$ | ${\boldsymbol{F}_{\textbf{10}}}$ | ${\boldsymbol{F}_{\textbf{11}}}$ | ${\boldsymbol{F}_{\textbf{12}}}$ | |
| ${\boldsymbol{N}_{\textbf{1}}}$ | $\langle 0.3(0.4),0.6(0.6)\rangle $ | $\langle 0.5(0.5),0.7(0.5)\rangle $ | $\langle 0.6(0.5),0.9(0.5)\rangle $ | $\langle 0.5(0.5),0.6(0.5)\rangle $ |
| ${\boldsymbol{N}_{\textbf{2}}}$ | $\langle 0.6(1)\rangle $ | $\langle 0.9(1)\rangle $ | $\langle 0.9(1)\rangle $ | $\langle 0.7(1)\rangle $ |
| ${\boldsymbol{N}_{\textbf{3}}}$ | $\langle 0.1(1)\rangle $ | $\langle 0.3(1)\rangle $ | $\langle 0.5(1)\rangle $ | $\langle 0.3(1)\rangle $ |
| ${\boldsymbol{N}_{\textbf{4}}}$ | $\langle 0.2(0.5),0.4(0.5)\rangle $ | $\langle 0.3(0.5),0.5(0.5)\rangle $ | $\langle 0.3(0.5),0.5(0.5)\rangle $ | $\langle 0.2(0.5),0.3(0.5)\rangle $ |
Table 4
Primary assessments by the 2nd expert.
| MOV | ${\boldsymbol{F}_{\textbf{1}}}$ | ${\boldsymbol{F}_{\textbf{2}}}$ | ${\boldsymbol{F}_{\textbf{3}}}$ | ${\boldsymbol{F}_{\textbf{4}}}$ |
| ${\boldsymbol{N}_{\textbf{1}}}$ | $\langle 0.6(0.5),0.8(0.5)\rangle $ | $\langle 0.6(0.5),0.8(0.5)\rangle $ | $\langle 0.5(0.6),0.8(0.4)\rangle $ | $\langle 0.6(0.5),0.7(0.5)\rangle $ |
| ${\boldsymbol{N}_{\textbf{2}}}$ | $\langle 0.7(1)\rangle $ | $\langle 0.9(1)\rangle $ | $\langle 0.9(1)\rangle $ | $\langle 0.8(1)\rangle $ |
| ${\boldsymbol{N}_{\textbf{3}}}$ | $\langle 0.7(1)\rangle $ | $\langle 0.5(1)\rangle $ | $\langle 0.2(1)\rangle $ | $\langle 0.6(1)\rangle $ |
| ${\boldsymbol{N}_{\textbf{4}}}$ | $\langle 0.5(0.5),0.6(0.5)\rangle $ | $\langle 0.1(0.5),0.3(0.5)\rangle $ | $\langle 0.3(0.5),0.5(0.5)\rangle $ | $\langle 0.5(0.5),0.6(0.5)\rangle $ |
| ${\boldsymbol{F}_{\textbf{5}}}$ | ${\boldsymbol{F}_{\textbf{6}}}$ | ${\boldsymbol{F}_{\textbf{7}}}$ | ${\boldsymbol{F}_{\textbf{8}}}$ | |
| ${\boldsymbol{N}_{\textbf{1}}}$ | $\langle 0.3(0.5),0.5(0.5)\rangle $ | $\langle 0.5(0.5),0.8(0.5)\rangle $ | $\langle 0.4(0.5),0.6(0.5)\rangle $ | $\langle 0.5(0.5),0.7(0.5)\rangle $ |
| ${\boldsymbol{N}_{\textbf{2}}}$ | $\langle 0.5(1)\rangle $ | $\langle 0.6(1)\rangle $ | $\langle 0.8(1)\rangle $ | $\langle 0.7(1)\rangle $ |
| ${\boldsymbol{N}_{\textbf{3}}}$ | $\langle 0.3(1)\rangle $ | $\langle 0.2(1)\rangle $ | $\langle 0.6(1)\rangle $ | $\langle 0.5(1)\rangle $ |
| ${\boldsymbol{N}_{\textbf{4}}}$ | $\langle 0.4(0.5),0.5(0.5)\rangle $ | $\langle 0.1(0.6),0.3(0.4)\rangle $ | $\langle 0.4(0.5),0.6(0.5)\rangle $ | $\langle 0.2(0.4),0.3(0.6)\rangle $ |
| ${\boldsymbol{F}_{\textbf{9}}}$ | ${\boldsymbol{F}_{\textbf{10}}}$ | ${\boldsymbol{F}_{\textbf{11}}}$ | ${\boldsymbol{F}_{\textbf{12}}}$ | |
| ${\boldsymbol{N}_{\textbf{1}}}$ | $\langle 0.5(0.4),0.6(0.6)\rangle $ | $\langle 0.8(0.5),0.9(0.5)\rangle $ | $\langle 0.6(0.5),0.8(0.5)\rangle $ | $\langle 0.6(0.5),0.7(0.5)\rangle $ |
| ${\boldsymbol{N}_{\textbf{2}}}$ | $\langle 0.5(1)\rangle $ | $\langle 0.6(1)\rangle $ | $\langle 0.4(1)\rangle $ | $\langle 0.6(1)\rangle $ |
| ${\boldsymbol{N}_{\textbf{3}}}$ | $\langle 0.5(1)\rangle $ | $\langle 0.3(1)\rangle $ | $\langle 0.7(1)\rangle $ | $\langle 0.4(1)\rangle $ |
| ${\boldsymbol{N}_{\textbf{4}}}$ | $\langle 0.6(0.5),0.9(0.5)\rangle $ | $\langle 0.3(0.5),0.5(0.5)\rangle $ | $\langle 0.4(0.5),0.7(0.5)\rangle $ | $\langle 0.3(0.5),0.4(0.5)\rangle $ |
Table 5
Primary assessments by the 3rd expert.
| MOV | ${\boldsymbol{F}_{\textbf{1}}}$ | ${\boldsymbol{F}_{\textbf{2}}}$ | ${\boldsymbol{F}_{\textbf{3}}}$ | ${\boldsymbol{F}_{\textbf{4}}}$ |
| ${\boldsymbol{N}_{\textbf{1}}}$ | $\langle 0.7(0.5),0.8(0.5)\rangle $ | $\langle 0.5(0.5),0.6(0.5)\rangle $ | $\langle 0.3(0.6),0.6(0.4)\rangle $ | $\langle 0.6(0.5),0.8(0.5)\rangle $ |
| ${\boldsymbol{N}_{\textbf{2}}}$ | $\langle 0.6(1)\rangle $ | $\langle 0.9(1)\rangle $ | $\langle 0.8(1)\rangle $ | $\langle 0.5(1)\rangle $ |
| ${\boldsymbol{N}_{\textbf{3}}}$ | $\langle 0.2(1)\rangle $ | $\langle 0.5(1)\rangle $ | $\langle 0.5(1)\rangle $ | $\langle 0.7(1)\rangle $ |
| ${\boldsymbol{N}_{\textbf{4}}}$ | $\langle 0.3(0.5),0.4(0.5)\rangle $ | $\langle 0.3(0.5),0.5(0.5)\rangle $ | $\langle 0.5(0.5),0.7(0.5)\rangle $ | $\langle 0.4(0.5),0.6(0.5)\rangle $ |
| ${\boldsymbol{F}_{\textbf{5}}}$ | ${\boldsymbol{F}_{\textbf{6}}}$ | ${\boldsymbol{F}_{\textbf{7}}}$ | ${\boldsymbol{F}_{\textbf{8}}}$ | |
| ${\boldsymbol{N}_{\textbf{1}}}$ | $\langle 0.3(0.5),0.5(0.5)\rangle $ | $\langle 0.5(0.5),0.6(0.5)\rangle $ | $\langle 0.6(0.5),0.8(0.5)\rangle $ | $\langle 0.6(0.5),0.8(0.5)\rangle $ |
| ${\boldsymbol{N}_{\textbf{2}}}$ | $\langle 0.6(1)\rangle $ | $\langle 0.8(1)\rangle $ | $\langle 0.9(1)\rangle $ | $\langle 0.7(1)\rangle $ |
| ${\boldsymbol{N}_{\textbf{3}}}$ | $\langle 0.2(1)\rangle $ | $\langle 0.5(1)\rangle $ | $\langle 0.6(1)\rangle $ | $\langle 0.3(1)\rangle $ |
| ${\boldsymbol{N}_{\textbf{4}}}$ | $\langle 0.1(0.5),0.2(0.5)\rangle $ | $\langle 0.3(0.6),0.5(0.4)\rangle $ | $\langle 0.5(0.5),0.7(0.5)\rangle $ | $\langle 0.1(0.4),0.3(0.6)\rangle $ |
| ${\boldsymbol{F}_{\textbf{9}}}$ | ${\boldsymbol{F}_{\textbf{10}}}$ | ${\boldsymbol{F}_{\textbf{11}}}$ | ${\boldsymbol{F}_{\textbf{12}}}$ | |
| ${\boldsymbol{N}_{\textbf{1}}}$ | $\langle 0.3(0.4),0.5(0.6)\rangle $ | $\langle 0.6(0.5),0.8(0.5)\rangle $ | $\langle 0.8(0.5),0.9(0.5)\rangle $ | $\langle 0.4(0.5),0.5(0.5)\rangle $ |
| ${\boldsymbol{N}_{\textbf{2}}}$ | $\langle 0.6(1)\rangle $ | $\langle 0.9(1)\rangle $ | $\langle 0.7(1)\rangle $ | $\langle 0.8(1)\rangle $ |
| ${\boldsymbol{N}_{\textbf{3}}}$ | $\langle 0.1(1)\rangle $ | $\langle 0.5(1)\rangle $ | $\langle 0.5(1)\rangle $ | $\langle 0.4(1)\rangle $ |
| ${\boldsymbol{N}_{\textbf{4}}}$ | $\langle 0.2(0.5),0.4(0.5)\rangle $ | $\langle 0.4(0.5),0.5(0.5)\rangle $ | $\langle 0.2(0.5),0.3(0.5)\rangle $ | $\langle 0.1(0.5),0.3(0.5)\rangle $ |
5.2 Problem Solution
PHFAAWA operator, as shown in Eq. (13), is employed to derive the aggregated PHF matrix for $\theta =2$, with expert weights of 0.35, 0.40, and 0.25. The initial PHF assessment matrices are converted to fuzzy matrices based on Eq. (14). The converted aggregated matrix (for $\theta =2$) is formed employing Eq. (15), as shown in Table 6. Suppose the minimum DC is $\vartheta =0.80$. DC value for each expert are calculated using Eqs. (16) and (17) for different values of Aczel-Alsina parameter θ have been given in Fig. 1. Since ${\vartheta ^{(3)}}=0.7930$ for $\theta =3$, ${\vartheta ^{(3)}}=0.7857$ for $\theta =4$, and ${\vartheta ^{(3)}}=0.7815$ for $\theta =5$, so these values of θ cannot be considered. We can choose either $\theta =1$ or $\theta =2$. Let us take $\theta =2$. Then the DC are: ${\vartheta ^{(1)}}=0.9118,{\vartheta ^{(2)}}=0.8186$, and ${\vartheta ^{(3)}}=0.8056$. Since ${\min _{m}}{\vartheta ^{(m)}}=0.8056$, we are done with the consensus reaching among experts.
Next, based on the Eqs. (18)–(21), we have the following optimization model in linear form:
\[\begin{aligned}{}\text{Max}\hspace{2.5pt}Z& =0.1314{\beta _{1}}+0.3583{\beta _{2}}+0.3702{\beta _{3}}+0.1167{\beta _{4}}+0.1139{\beta _{5}}+0.2844{\beta _{6}}\\ {} & \hspace{1em}+0.1858{\beta _{7}}+0.3089{\beta _{8}}+0.1050{\beta _{9}}+0.5059{\beta _{10}}+0.2100{\beta _{11}}+0.3157{\beta _{12}}\end{aligned}\]
Subject to: ${\beta _{1}}+{\beta _{2}}+\cdots +{\beta _{12}}=1$ and ${\beta _{1}},{\beta _{2}},\dots ,{\beta _{12}}\geqslant 0$.Table 6
Converted fuzzy matrix (for $\theta =2$).
| MOV | ${\boldsymbol{F}_{\textbf{1}}}$ | ${\boldsymbol{F}_{\textbf{2}}}$ | ${\boldsymbol{F}_{\textbf{3}}}$ | ${\boldsymbol{F}_{\textbf{4}}}$ | ${\boldsymbol{F}_{\textbf{5}}}$ | ${\boldsymbol{F}_{\textbf{6}}}$ |
| ${\boldsymbol{N}_{\textbf{1}}}$ | 0.7423 | 0.7677 | 0.5530 | 0.697 | 0.4158 | 0.6348 |
| ${\boldsymbol{N}_{\textbf{2}}}$ | 0.7281 | 0.8631 | 0.8262 | 0.7622 | 0.5658 | 0.7051 |
| ${\boldsymbol{N}_{\textbf{3}}}$ | 0.5865 | 0.5409 | 0.3492 | 0.5888 | 0.2799 | 0.4260 |
| ${\boldsymbol{N}_{\textbf{4}}}$ | 0.4809 | 0.4354 | 0.4469 | 0.4998 | 0.3956 | 0.3001 |
| ${\boldsymbol{F}_{\textbf{7}}}$ | ${\boldsymbol{F}_{\textbf{8}}}$ | ${\boldsymbol{F}_{\textbf{9}}}$ | ${\boldsymbol{F}_{\textbf{10}}}$ | ${\boldsymbol{F}_{\textbf{11}}}$ | ${\boldsymbol{F}_{\textbf{12}}}$ | |
| ${\boldsymbol{N}_{\textbf{1}}}$ | 0.6624 | 0.6322 | 0.5200 | 0.7763 | 0.7962 | 0.5841 |
| ${\boldsymbol{N}_{\textbf{2}}}$ | 0.8717 | 0.7433 | 0.5658 | 0.8467 | 0.7822 | 0.7051 |
| ${\boldsymbol{N}_{\textbf{3}}}$ | 0.6416 | 0.4654 | 0.3598 | 0.3714 | 0.6061 | 0.3705 |
| ${\boldsymbol{N}_{\textbf{4}}}$ | 0.5590 | 0.3014 | 0.6269 | 0.4260 | 0.4735 | 0.2748 |
Suppose that the partial weights information for the criteria are:
\[\begin{aligned}{}\Xi & =\{0.03\leqslant {\beta _{1}}\leqslant 0.07,0.05\leqslant {\beta _{2}}\leqslant 0.07,0.15\leqslant {\beta _{3}}\leqslant 0.25,0.05\leqslant {\beta _{4}}\leqslant 0.08,\\ {} & \hspace{2em}0.05\leqslant {\beta _{5}}\leqslant 0.08,0.08\leqslant {\beta _{6}}\leqslant 0.15,0.10\leqslant {\beta _{7}}\leqslant 0.15,0.20\leqslant {\beta _{8}}\leqslant 0.25,\\ {} & \hspace{2em}0.08\leqslant {\beta _{9}}\leqslant 0.15,0.15\leqslant {\beta _{10}}\leqslant 0.25,0.03\leqslant {\beta _{11}}\leqslant 0.08,0.02\leqslant {\beta _{12}}\leqslant 0.08\}.\end{aligned}\]
This leads to the following solution:
${\beta _{1}}=0.03$, ${\beta _{2}}=0.05$, ${\beta _{3}}=0.15$, ${\beta _{4}}=0.05$, ${\beta _{5}}=0.05$, ${\beta _{6}}=0.08$, ${\beta _{7}}=0.10$, ${\beta _{8}}=0.20$, ${\beta _{9}}=0.09$, ${\beta _{10}}=0.15$, ${\beta _{11}}=0.03$, ${\beta _{12}}=0.02$ with $\text{Max}Z=94.7805$.
SVs of the alternatives are calculated using RS approach, as described in Eqs. (22) and (23), and are presented in Table 7.
Table 7
SVs of MOVs by RS approach.
| MOV | ${\zeta _{r}^{+}}$ | ${\zeta _{r}^{-}}$ | ${\zeta _{r}}$ |
| ${\boldsymbol{N}_{\textbf{1}}}$ | 0.448841 | 0.192678 | 0.256163 |
| ${\boldsymbol{N}_{\textbf{2}}}$ | 0.527224 | 0.235825 | 0.291398 |
| ${\boldsymbol{N}_{\textbf{3}}}$ | 0.288221 | 0.157240 | 0.130980 |
| ${\boldsymbol{N}_{\textbf{4}}}$ | 0.312025 | 0.116175 | 0.195850 |
${\vartheta _{1}}=0.742344787$, ${\vartheta _{2}}=0.863080243$, ${\vartheta _{3}}=0.826210559$, ${\vartheta _{4}}=0.762183501$, ${\vartheta _{5}}=0.565789937$, ${\vartheta _{6}}=0.705054017$, ${\vartheta _{7}}=0.558990445$, ${\vartheta _{8}}=0.301380812$, ${\vartheta _{9}}=0.626947861$, ${\vartheta _{10}}=0.846700199$, ${\vartheta _{11}}=0.796187962$, ${\vartheta _{12}}=0.705054017$.
Table 8
Distance from each alternative to the RPs.
| Criteria | ${\boldsymbol{N}_{\textbf{1}}}$ | ${\boldsymbol{N}_{\textbf{2}}}$ | ${\boldsymbol{N}_{\textbf{3}}}$ | ${\boldsymbol{N}_{\textbf{4}}}$ |
| ${\boldsymbol{F}_{\textbf{1}}}$ | 0 | 0.000426 | 0.004675 | 0.007843 |
| ${\boldsymbol{F}_{\textbf{2}}}$ | 0.004768 | 0 | 0.016107 | 0.021383 |
| ${\boldsymbol{F}_{\textbf{3}}}$ | 0.040975 | 0 | 0.071546 | 0.056903 |
| ${\boldsymbol{F}_{\textbf{4}}}$ | 0.003258 | 0 | 0.008671 | 0.013119 |
| ${\boldsymbol{F}_{\textbf{5}}}$ | 0.007500 | 0 | 0.014292 | 0.008509 |
| ${\boldsymbol{F}_{\textbf{6}}}$ | 0.005623 | 0 | 0.022323 | 0.032399 |
| ${\boldsymbol{F}_{\textbf{7}}}$ | 0.010341 | 0.0312736 | 0.008264 | 0 |
| ${\boldsymbol{F}_{\textbf{8}}}$ | 0.066161 | 0.088376 | 0.032801 | 0 |
| ${\boldsymbol{F}_{\textbf{9}}}$ | 0.009625 | 0.005504 | 0.024046 | 0 |
| ${\boldsymbol{F}_{\textbf{10}}}$ | 0.010562 | 0 | 0.071296 | 0.063105 |
| ${\boldsymbol{F}_{\textbf{11}}}$ | 0 | 0.000421 | 0.005701 | 0.009680 |
| ${\boldsymbol{F}_{\textbf{12}}}$ | 0.002419 | 0 | 0.006692 | 0.008605 |
Finally, using Eq. (26) of RP model, the maximum distances of the alternatives are calculated, as given below:
\[ {\eta _{1}}=0.06616,\hspace{1em}{\eta _{2}}=0.08837,\hspace{1em}{\eta _{3}}=0.07154,\hspace{1em}{\eta _{4}}=0.06310.\]
Next, Eqs. (27)–(28) of FMF model are used to compute the SVs of alternatives, as given in Table 9.Table 9
Outcomes by FMF model.
| MOV | ${\alpha _{r}^{+}}$ | ${\alpha _{r}^{-}}$ | ${\tau _{r}}$ |
| ${\boldsymbol{N}_{\textbf{1}}}$ | 0.723450 | 0.875551 | 0.826279 |
| ${\boldsymbol{N}_{\textbf{2}}}$ | 0.813743 | 0.929537 | 0.875428 |
| ${\boldsymbol{N}_{\textbf{3}}}$ | 0.528133 | 0.820902 | 0.643356 |
| ${\boldsymbol{N}_{\textbf{4}}}$ | 0.559902 | 0.742272 | 0.754308 |
Table 10
Final priority values of the MOV alternatives.
| RS model | RP model | FMF model | $B({N_{r}})$ | ||||
| MOV | ${\tilde{\zeta }_{r}}$ | Rank | ${\tilde{\eta }_{r}}$ | Rank | ${\tilde{\tau }_{r}}$ | Rank | |
| ${\boldsymbol{N}_{\textbf{1}}}$ | 0.5643 | 2 | 0.4534 | 2 | 0.5298 | 2 | 0.2376 |
| ${\boldsymbol{N}_{\textbf{2}}}$ | 0.642 | 1 | 0.6057 | 4 | 0.5614 | 1 | 0.2390 |
| ${\boldsymbol{N}_{\textbf{3}}}$ | 0.2886 | 4 | 0.4904 | 3 | 0.4125 | 4 | −0.077 |
| ${\boldsymbol{N}_{\textbf{4}}}$ | 0.4315 | 3 | 0.4325 | 1 | 0.4837 | 3 | 0.1398 |
Hence, the priority order of the vendors is: ${N_{2}}\succ {N_{1}}\succ {N_{4}}\succ {N_{3}}$ where the sign “≻” signifies “superior to”. Therefore, the most suitable vendor is ${N_{2}}$ (Bertrandt).
6 Discussions
The entire discussion is classified into two parts: (A) sensitivity analysis of criteria weights, (B) comparison of the proposed approach with the extant methods, and (C) managerial implications.
6.1 Sensitivity Analysis of Criteria Weights
In this section, sensitivity analysis is conducted to understand the effect of criteria weights on the ranking order. This is achieved using 24 different weight sets (CWS1, CWS2, CWS3, $\dots \hspace{0.1667em}$, CWS24), as depicted in Fig. 2, formed by considering twenty-four arbitrary combinations of the criteria weights. Especially, this is valuable in achieving a broader scope of criteria weights for taking a look at the performance of the created model. The final priority scores of alternatives are shown in Fig. 3. The positioning places of alternatives along with the Spearman’s rank correlation coefficient (SRCC) values (Saha et al., 2023) have been calculated for those 24 weight sets and depicted in Fig. 4. The average SRCC value is calculated as ‘0.8916’, indicating a strong correlation (Saha et al., 2023) between the ranking positions. Therefore, the priority order of the alternatives obtained using the proposed framework is reliable.
6.2 Comparative Study
This section aims to compare the proposed model with existing ones, specifically PHF-COPRAS (Krishankumar et al., 2022), PHF-MARCOS (Liu et al., 2023b), and PHF-TOPSIS (Qi, 2023). These tools have been applied to the case study considered by us. Outcomes are presented in Fig. 5. According to Fig. 5, the priority order derived by each technique is ${N_{2}}\succ {N_{1}}\succ {N_{4}}\succ {N_{3}}$ which matches perfectly the results from our suggested strategy.
The main benefits of the proposed framework are:
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1. Existing methods, especially those relying on t-norm and t-conorm operators, often fail to handle bias or disagreement among experts effectively. Aczel-Alsina AOs integrated in the proposed framework include an adjustable parameter, allowing them to better account for expert biases and preferences, which leads to a more robust consensus among the group. This makes the proposed method more effective in situations where expert opinions are uncertain or divergent.
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2. The existing methodologies (Krishankumar et al., 2022; Liu et al., 2023b; Qi, 2023) might result in information loss when determining criteria weights due to the absence of optimization models. Consequently, the accuracy of the inputs cannot be thoroughly evaluated by current methods (Krishankumar et al., 2022; Liu et al., 2023b; Qi, 2023). In contrast, the proposed framework uses a cross-entropy based model to determine the criteria weights, which enhances the accuracy and relevance of the weights assigned to each criterion. This model helps to reduce uncertainty in the weight assignment process, which is often a challenge in other methods that rely on fixed or subjective weight distributions.
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4. PHF-MULTIMOORA is more efficient compared to PHF-COPRAS (Krishankumar et al., 2022), PHF-MARCOS (Liu et al., 2023b), and PHF-TOPSIS (Qi, 2023) as “ratio system”, “reference point” and “full multiplicative form” are included in that. Moreover, PHF-MULTIMOORA approach is simple, highly robust, and has less computation time in comparison to PHF-COPRAS (Krishankumar et al., 2022), PHF-MARCOS Liu et al. (2023b), and PHF-TOPSIS (Qi, 2023).
6.3 Adaptability and Versatility of Proposed Framework Across Industries
The integration of PHF sets allows the framework to handle uncertainty and vagueness in expert opinions, making it suitable for a variety of decision-making contexts across industries. PHF sets enable the incorporation of multiple conflicting and uncertain evaluations, which is crucial in sectors like healthcare, where decisions often rely on subjective judgments about treatments, resource allocation, and medical technologies. Aczel-Alsina aggregation operations, with their adjustable parameters, provide the flexibility to modify the aggregation process based on the specific requirements of different industries, allowing for fine-tuning of the consensus mechanism to match the decision-making environment. For example, in the context of sustainable supplier selection, the framework can be employed to aggregate diverse criteria such as environmental impact, cost-effectiveness, and compliance with sustainability standards. By adjusting the parameters of the aggregation operations, different weightings can be applied to these criteria depending on the sustainability goals of the industry. Similarly, in financial risk assessment, MULTIMOORA method, in combination with PHF aggregation, can be used to rank investment portfolios based on criteria like return potential, risk exposure, and liquidity, offering a comprehensive evaluation framework under uncertainty. The adopted cross-entropy-based criteria weighting model ensures that the weight distribution reflects the relative importance of each criterion, which is particularly useful in industries like transportation, where fuel efficiency, safety, and cost criteria may vary in significance based on the region or operational conditions. The versatility of the framework is further enhanced by its multi-objective optimization approach, which can be applied to a wide range of problems like urban infrastructure development or public policy formulation. These technical aspects make the framework highly adaptable, capable of addressing the specific needs of different industries while maintaining its reliability and accuracy in aggregating expert judgments.
7 Conclusions
The shortfalls in existing methods for managing uncertainty in decision-making are addressed using PHF sets to systematically handle uncertainty. Traditional aggregation operations for PHF sets often lack adaptability, which has been improved by introducing PHF Aczel-Alsina weighted averaging and geometric operations, providing greater flexibility in aggregating uncertain information. A consensus-building approach is proposed to identify the best alternative within a PHF environment, effectively mitigating the influence of biased expert perspectives. The proposed framework also incorporates a cross-entropy based model to determine criteria weights, ensuring that the weighting process accurately reflects the real-world importance of each criterion. MULTIMOORA method is then applied to establish the priority orders of alternatives, offering a comprehensive approach to decision-making. This combination of techniques improves both the reliability and objectivity of the decision process. A detailed case study on MOV selection demonstrates the practical applicability of the framework, showing how it can be used to make informed and balanced decisions in real-world scenarios. Sensitivity analysis and comparative studies further validate the effectiveness of the proposed framework in managing the uncertainties in PHF-based decision-making. These analyses confirm that the framework performs well in adjusting to different levels of uncertainty, thus ensuring robust and consistent results.
The framework may be extended in future research by incorporating additional criteria. New factors can be added or existing ones modified depending on specific industry or business requirements to better reflect the evolving needs of decision-making environments. The integration of dynamic or time-varying uncertainty models within PHF sets can also be explored to handle uncertainties that evolve over time. The proposed framework can be further extended to other MCGDM problems across various sectors like supply chain management, energy planning, or healthcare, where uncertainty and hesitation play significant roles in decision outcomes. The adaptability of PHF Aczel-Alsina operations in these diverse contexts should be investigated in future studies. The framework can also be expanded to include machine learning or artificial intelligence-based techniques for dynamic consensus-building, enabling the system to learn from previous decision outcomes and improve the aggregation and consensus processes over time, refining the decision-making process in environments with large amounts of data and complex decision criteria.