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.