Abstract
Sustainable energy planning is a critical challenge, particularly in regions with complex decision-making environments and uncertain data. The selection of an optimal energy source requires robust methodologies that can effectively handle multi-criteria decision-making (MCDM) under uncertainty. This study explores the application of the multiple objective optimization on the basis of ratio analysis plus full multiplicative form (MULTIMOORA) method by incorporating the concept of the complex linear Diophantine fuzzy (\(C_pLD_yF\)) set. The \(C_pLD_yF\) set extends the conventional linear Diophantine fuzzy set by introducing a phase component, thereby enhancing the system’s adaptability. To examine the interrelationships among multiple \(C_pLD_yF\) numbers, we develop the \(C_pLD_yF\) Dombi prioritized averaging (\(C_pLD_yFDPA\)) and the \(C_pLD_yF\) Dombi prioritized geometric (\(C_pLD_yFDPG\)) aggregation operators, along with their weighted versions, based on the proposed Dombi operational laws. The fundamental properties of these aggregation operators are systematically analyzed. The developed \(C_pLD_yF\) operators are then integrated into the MULTIMOORA method to address MCDM problems. To illustrate the practical effectiveness of the proposed framework, a case study is conducted to determine a sustainable energy source for Gwadar, Pakistan, utilizing \(C_pLD_yF\) information. Furthermore, comparative analyses are performed against existing methodologies to validate the applicability and accuracy of the proposed approach.
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Introduction
The concept of fuzzy sets (FS) was first introduced by Zadeh1 in 1965 as a mathematical tool to handle imprecision and uncertainty in real-world problems. FS employs a membership degree (MD) within the interval \([0,1]\) to describe the extent to which an element belongs to a given set. Over the years, researchers have explored various extensions of FS, including modifications of its codomain. This led to the development of the complex FS (CFS), proposed in 2002 by2, which extends the FS domain from \([0,1]\) to the unit disc in the complex plane. Unlike traditional FS, CFS allows membership degrees to take complex values, thus providing a richer framework for modeling uncertainty. The field of FS has witnessed significant advancements, including the introduction of the intuitionistic FS (IFS) by3, which incorporates both MD and a non-membership degree (NMD), subject to the constraint that their sum lies within the interval \([0,1]\). IFS enhances decision-making models by allowing partial acceptance and rejection, better reflecting human reasoning. Several researchers have explored IFS applications across diverse domains4,5,6. A further extension, the complex intuitionistic FS (CIFS), was introduced in 2012 by7, incorporating complex-valued MD and NMD within the unit circle of the complex plane. CIFS has been applied in multi-criteria decision-making (MCDM), leading to the development of various CIF aggregation operators (AOs)8,9 and correlation measures10.
In 2013, Yager11 proposed the Pythagorean fuzzy set (PyFS), which generalizes IFS by relaxing the constraint on MD and NMD to \((MD^2 + NMD^2) \in [0,1]\). This extension provides greater flexibility in representing uncertainty and has proven effective in handling complex decision-making scenarios. The integration of PyFS with complex numbers was first explored in12, leading to further refinements and applications in various distance measures13. Subsequently, Yager14 introduced the q-rung orthopair FS (q-ROFS) in 2017, further generalizing PyFS by imposing the condition \((MD^q + NMD^q) \in [0,1]\), which enhanced flexibility for decision-making under uncertainty. To address two-dimensional scenarios within the q-rung orthopair fuzzy environment, Liu et al.15 extended this concept to complex q-rung orthopair fuzzy sets (Cq-ROFS). Several studies have demonstrated the effectiveness of Cq-ROFS across multiple domains16,17,18.
Despite the advancements in FS-based methodologies, traditional sets such as IFS, PyFS, and q-ROFS lack the structural capacity to incorporate reference parameters (RPs)—a vital component in many real-world decision-making problems involving dual-dimensional or context-sensitive information. To overcome this limitation, Riaz and Hashmi19 introduced the linear Diophantine FS (\(LD_yFS\)), which integrates reference parameters into the FS structure. This model was later generalized by Kamacı in 2021 through the introduction of Complex \(LD_yFS\) (\(C_pLD_yFS\))20, which allows the representation of fuzziness using both amplitude and phase components, significantly improving the modeling of time-dependent, uncertain, or cyclic phenomena. In real-world decision-making—especially in sustainable energy planning, medical diagnosis, or climate analysis—the information being processed is often not only vague but also cyclic or periodic, with embedded uncertainty. For example, in climate change modeling, seasonal temperature cycles exhibit regular periodicity, while global warming introduces unpredictable variations. Similarly, during the COVID-19 pandemic, symptoms like fever and cough may appear in regular patterns, but the timing and severity differ across individuals due to underlying conditions or viral mutations. In both cases, traditional fuzzy sets may cause information loss by failing to reflect the dual-dimensional (amplitude + phase) nature of the data. The \(C_pLD_yFS\) structure excels in such contexts by encoding both the magnitude of belief (amplitude) and frequency or temporal distribution (phase) of assessment values. For instance, a patient experiencing moderate fever for 6 out of 14 days would be represented not just by the average severity (amplitude), but also by the frequency of occurrence (phase) — enabling more nuanced decision support in medical triage systems. Likewise, in energy policy planning, consider subjective inputs from stakeholders like environmental experts and economists. One may emphasize carbon reduction (with high membership and low non-membership), while the other may have conditional beliefs depending on fluctuating market data. The \(C_pLD_yFS\) allows encoding these complex, condition-based preferences while preserving relational structure through reference parameters. Such encoding is especially useful when integrating subjective (expert opinion) and objective (sensor data or time-series analysis) information into a single consistent framework.
Given the above capabilities, \(C_pLD_yFS\) has been recognized as a powerful tool in complex decision-making scenarios21,22,23. Research on the theoretical and practical applications of \(C_pLD_yFS\) is still in its early stages of development. Further investigation is required to comprehensively understand the capabilities of CpLDyFS and to enhance its theoretical framework. The better understanding and use of \(C_pLD_yF\) will lead to the creation of better tools for making decisions, especially in areas where two-dimensional data needs to be managed.
In complex decision-making environments, selecting the best alternative often involves multiple conflicting criteria that require careful aggregation24,25,26. MCDM provides a structured framework for handling such problems by integrating diverse information and expert evaluations27. Traditional aggregation methods, however, typically assume equal importance among criteria, which does not align with real-world scenarios where some factors inherently hold greater significance than others. Prioritization is often crucial in these cases to ensure that more critical attributes exert a dominant influence on the decision while maintaining fairness in the evaluation process28. A classic example illustrating the need for prioritization can be found in residential property selection. When choosing a home, factors such as utility access, location, and cost must be considered. A house without access to essential services like electricity and water cannot be considered livable, even if it is in an excellent location at a low price. Similarly, a well-located property is meaningless if it lacks fundamental amenities. This ranking of priorities, where utility access is the most critical factor, followed by location, and finally cost, demands an aggregation method that respects the inherent hierarchical structure among criteria. Traditional AOs fail to capture such prioritization, leading to suboptimal and misleading results29.
AOs play a crucial role in MCDM, and numerous models have been introduced to enhance decision-making. Dombi operators, introduced by Dombi30, have gained significant attention due to their parametric flexibility, allowing decision-makers to fine-tune aggregation based on varying degrees of importance. These operators have been widely applied in IFS31, PyFS32, and hesitant fuzzy environments33, demonstrating their effectiveness in handling uncertainty. Similarly, prioritized AOs, introduced by Yager28, ensure that higher-priority factors exert greater influence over the decision process. Additionally, Bonferroni mean (BM) operators34 have been explored for modeling inter-criteria relationships, providing an interactive way to aggregate values. However, traditional BM operators suffer from non-reducibility, leading to biased results, a problem that was later addressed by the normalized weighted Bonferroni mean operator, which enhances flexibility in data aggregation35. Despite these advancements, no existing research has explored the integration of Dombi operators with prioritized AOs in the context of \(C_pLD_yF\) environments. While prioritization techniques have been applied in various fuzzy models36,37, they have not been systematically combined with Dombi operators to develop a structured decision-making framework that captures both priority-based relationships and inter-criteria interactions. This gap highlights the need for a new class of prioritized Dombi AOs that can accommodate hierarchical decision structures while leveraging the parametric strength of Dombi operations.
The multi-objective optimization by ratio analysis (MOORA) method, initially introduced by38, is a widely recognized multi-criteria decision analysis tool that employs two aggregation structures: the ratio system model (RSM) and the reference point odel (RPM). To further enhance its decision-making capabilities,39 extended MOORA into the multi-objective optimization on the basis of ratio analysis plus full multiplicative form (MULTIMOORA) framework by incorporating the full multiplicative form alongside RSM and RPM, thereby improving the comprehensiveness of the model. This extension demonstrated notable advantages over conventional MCDM approaches, offering greater stability, reduced computational complexity, and enhanced robustness40. Over time, MULTIMOORA has been extended to various fuzzy and uncertain environments to better handle complex decision-making problems. Zavadskas et al.41 proposed a hybrid interval-valued intuitionistic fuzzy (IVIF) MULTIMOORA approach to model uncertainty while disregarding decision-maker expertise and criteria weights. Zhang et al.42 further adapted the classical MULTIMOORA method to an intuitionistic fuzzy environment, applying it to assess multi-criteria energy storage technologies. Additionally, the authors in43 integrated the best-worst method (BWM) with fuzzy MULTIMOORA for ranking logistics service providers, showcasing its effectiveness in real-world applications.
Recent studies have continued to enhance MULTIMOORA by incorporating advanced fuzzy set theories and optimization techniques. Rani and Mishra44 introduced an integrated MULTIMOORA model based on the maximizing deviation model and Fermatean FSs for improved uncertainty management. Similarly, Wang et al.45 proposed an integrated MULTIMOORA model, demonstrating its applicability in complex decision-making scenarios through a real-world case study. Further, Shang et al.46 originated a Shannon entropy-based MULTIMOORA model combined with BWM and fuzzy information, which was successfully implemented for sustainable supplier evaluation. Garg and Rani47 designed an integrated MULTIMOORA framework leveraging IFS, particle swarm optimization, and AOs to analyze solid waste management techniques, thereby accounting for uncertainty in decision-making. Yu et al.48 employed MULTIMOORA with interval asymmetric rough cloud theory to develop an advanced failure mode and effects assessment model, applying it to risk assessment in floating production storage and offloading single-point mooring systems. Despite these advancements, the extension of MULTIMOORA to the \(C_pLD_yF\) environment remains unexplored, even though it provides a more comprehensive framework for managing uncertainty and imprecision. Bridging this gap is crucial, as integrating Dombi-prioritized AOs into the MULTIMOORA framework can significantly enhance decision-making precision, ensuring more reliable and structured evaluations in complex uncertain scenarios.
Based on the analysis of the existing literature, several research gaps have been identified and are outlined as follows.
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i)
Information AOs serve as a fundamental component in decision-making frameworks, facilitating the fusion of multiple individual inputs into a single representative value. However, existing \(C_pLD_yF\) AOs21,22,23 lack the ability to effectively incorporate priority relationships among multiple criteria, which is crucial in hierarchical decision-making scenarios. To overcome this limitation, it is essential to establish an advanced aggregation mechanism that integrates Dombi t-norm and t-conorm, ensuring a more structured and priority-sensitive decision-making approach.
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ii)
MULTIMOORA is a highly versatile decision-making approach, as it integrates three robust methodologies: the ratio system, the reference point approach, and the full multiplicative form, ensuring a comprehensive ranking mechanism. While the classical MULTIMOORA method has been successfully extended to various fuzzy environments45,46,47,48, both its original and extended versions remain inadequate for addressing selection problems within the \(C_pLD_yF\) environment.
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iii)
Although numerous decision-making methods have been effectively applied in sustainable energy planning (see Sect. 5.1), no research has yet investigated a \(C_pLD_yF\)-based MCDM model for selecting the most suitable sustainable energy alternatives.
To fill these gaps, this study presents the following key contributions.
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i)
This study introduces Prioritized Dombi AOs under the \(C_pLD_yF\) framework, combining the flexibility of Dombi operators with the ability to enforce prioritized relationships in decision-making. The proposed operators aim to enhance decision accuracy and robustness by integrating Dombi operations with prioritization-based aggregation techniques, ensuring that criteria are aggregated not only based on their values but also weighted according to their relative importance.
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ii)
Building on the proposed operators, this study extends the classical MULTIMOORA method to the \(C_pLD_yF\) framework, enabling more precise and effective handling of complex decision-making problems.
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iii)
The framed approach is implemented in a case study focused on selecting the most appropriate sustainable energy alternatives, which constitutes an MCDM problem. By utilizing \(C_pLD_yF\) information in the evaluation of each criterion, the integrity of the decision-making process is preserved, ensuring a more precise and reliable assessment of sustainable energy options.
The reminder of the manuscript is arranged as folllows: Sect. 2 provides a concise overview of key terms relevant to the proposed investigation. In Sect. 3, we introduce innovative operational regulations by using Prioritized aggregation operators and Dombi TN and TCN inside the framework of \(C_pLD_yFNs\), with a strong focus on practical application. Section 4 outlines the suggested approach of \(C_pLD_yFN\) aggregations using Dombi Prioritized aggregation operators. In Sect. 5, we analyze a real-life scenario to demonstrate the suitability of the proposed technique and choose the most favourable option. Section 6 incorporates a sensitivity analysis and comparison analysis to demonstrate the soundness and effectiveness of the devised strategy. Some conclusions are presented in Sect. 7.
Preliminaries
This section summarizes the concepts of LDFS, \(C_pLD_yFS\), operational laws of \(C_pLD_yFS\), Dombi t-norm, Dombi t-conorm and Prioritized AOs.
Definition 1
19 Let W be a universe of discourse. A linear Diophantine fuzzy set (LDFS) L on W is defined as an object of the form:
where \({m_{s}, \; n_{s}, \; \alpha _{s}, \; \beta _{s}} \in [0,1]\) represent membership grades (MG), non-membership grades (NMG) and reference parameters for \({s} \; \in \; {W}\), respectively. These parameters should satisfy the conditions \({0\le \alpha _s +\beta _s \le 1}\) and \({0\le \alpha _s m_s+\beta _s n_s\le 1}\).
Definition 2
20 Let W be a universe of discourse. A complex linear Diophantine fuzzy set \((C_pLD_yFS)\) C on W is defined as an object of the form:
where \({m(s)e^{i2\pi (\theta (s))}}\) and \({ne^{i2\pi \phi (s)}}\) represent the complex-valued MG and NMG, respectively, with \({m(s)}, {n(s)}, {\alpha (s)}, {\beta (s)}, {\theta (s)}, {\phi (s)}, {\delta (s)}, {\rho (s)} \; \in \; [0,1].\) These quantities should meet the following restrictions: \(0\le {\alpha +\beta } \le 1\) , \(0\le {\alpha m+\beta n}\le 1\) , \(0\le {\delta +\rho }\le 1\), and \(0\le {\theta \delta +\phi \rho }\le 1\).
The hesitant part can be evaluated by \({H=1-\{(\alpha m+\beta n))e^{i2\pi (1-(\delta \theta +\rho \phi )}\}}\). For simplicity, we denote any complex linear Diophantine number \((C_pLD_yFN)\) as:
where \({C_{i}\in C}\). The set of all \(C_pLD_yFS\) over the universe of discourse W is denoted by \((C_pLD_yFS(W))\).
Definition 3
The score and accuracy values of a \(C_pLD_yFN\) \({C}=\left\langle m e^{i 2 \pi \theta }, n e^{i 2 \pi \phi } \right\rangle , \left\langle \alpha e^{i 2 \pi \delta }, \beta e^{i 2 \pi \rho }\right\rangle\) are defined in Equations (4) and (5) as follows:
where \(\ddot{S_{r}}(C ) \; \in \; [0,1]\), and
where \({\ddot{A_{r}}(C_{i})} \; \in \; [0,1]\).
Based on the score and accuracy functions, any two \(C_pLD_yFNs\) \({C_{i}}(i=1,2)\) can be easily compared as follows:
-
1.
If \({\ddot{S_{r}}(C_{1})} > {\ddot{S_{r}}(C_{2})}\), then \({C_{1}} > {C_{2}};\)
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2.
If \({\ddot{S_{r}}(C_{1})} < {\ddot{S_{r}}(C_{2})}\), then \({C_{1}} < {C_{2}};\)
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3.
If \({\ddot{S_{r}}(C_{1})} = {\ddot{S_{r}}(C_{2})}\), then
-
(i)
If \({\ddot{A_{r}}(C_{1})} > {\ddot{A_{r}}(C_{2})}\), then \({C_{1}} > {C_{2}};\)
-
(ii)
If \({\ddot{A_{r}}(C_{1})} <{\ddot{A_{r}}(C_{2})}\), then \({C_{1}} < {C_{2}};\)
-
(iii)
If \({\ddot{A_{r}}(C_{1})} = {\ddot{A_{r}}(C_{2})}\), then \({C_{1}} = {C_{2}}.\)
Definition 4
20 The basic operational rules for two \(C_pLD_yFNs\), \(C_{1} =\left( \left\langle m_{1}e^{i 2 \pi \left( \theta _{1} \right) }, n_{1}e^{i 2 \pi \left( \phi _{n_1} \right) } \right\rangle \right.,\)\(\left.\left\langle \alpha _{1} e^{i 2 \pi \left( \delta _{1} \right) }, \beta _{1}e^{i 2 \pi \left( \rho _{1} \right) } \right\rangle \right)\) and \(C_{2} = \left( \left\langle m_{2}e^{i 2 \pi \left( \theta _{2} \right) }, n_{2}e^{i 2 \pi \left( \phi _{n_2} \right) } \right\rangle\right.,\)\(\left. \left\langle \alpha _{2}e^{i 2 \pi \left( \delta _{2} \right) }, \beta _{2}e^{i 2 \pi \left( \rho _{2} \right) } \right\rangle \right)\) are given as follows:
-
1.
\(C_{1} \oplus C_{2} = \left( \begin{array}{c} \left\langle (m_{1} + m_{2} - m_{1}m_{2})e^{i 2 \pi \left( \theta _{1} + \theta _{2} - \theta _{1}\theta _{2}\right) }, (n_{1}n_{2})e^{i 2 \pi \left( \phi _{1}\phi _{2}\right) } \right\rangle , \\ \left\langle (\alpha _{1} + \alpha _{2} - \alpha _{1}\alpha _{2})e^{i 2 \pi \left( \delta _{1} + \delta _{2} - \delta _{1}\delta _{2}\right) }, (\beta _{1}\beta _{2})e^{i 2 \pi \left( \rho _{1}\rho _{2}\right) } \right\rangle \end{array}\right) .\)
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2.
\(C_{1} \otimes C_{2} = \left( \begin{array}{c} \left\langle (m_{1}m_{2})e^{i 2 \pi \left( \theta _{1}\theta _{2}\right) }, (n_{1}+n_{2}-n_{1}n_{2})e^{i 2 \pi \left( \phi _{1}+\phi _{2}-\phi _{1}\phi _{2}\right) } \right\rangle , \\ \left\langle (\alpha _{1}\alpha _{2})e^{i 2 \pi \left( \delta _{1}\delta _{2}\right) }, (\beta _{1}+\beta _{2}-\beta _{1}\beta _{2})e^{i 2 \pi \left( \rho _{1}+\rho _{2}-\rho _{1}\rho _{2}\right) } \right\rangle \end{array}\right) .\)
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3.
\({ {\pounds }{C_1}}=\left( \begin{array}{c} \left\langle (1-(1-m_{1})^{{\pounds }})e^{i 2 \pi \left( 1-(1-\theta _{1})^{{\pounds }}\right) }, (n_{1})^{{\pounds }}e^{i 2 \pi \left( (\phi _{1})^{{\pounds }}\right) } \right\rangle , \\ \left\langle (1-(1-\alpha _{1}))^{{\pounds }}e^{i 2 \pi \left( 1-(1-\delta _{1})^{{\pounds }}\right) }, (\beta _{1})^{{\pounds }}e^{i 2 \pi \left( (\rho _{1})^{{\pounds }}\right) }\right\rangle \end{array}\right)\) where \({\pounds } \; \ge \; 0.\)
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4.
\({{C_1}^{{\pounds }}}=\left( \begin{array}{c} \left\langle (m_{1})^{{\pounds }}e^{i 2 \pi \left( (\theta _{1})^{{\pounds }}\right) }, (1-(1-n_{1}))^{{\pounds }}e^{i 2 \pi \left( 1-(1-\phi _{1})^{{\pounds }}\right) } \right\rangle , \\ \left\langle (\alpha _{1})^{{\pounds }}e^{i 2 \pi \left( (\delta _{1})^{{\pounds }}\right) }, (1-(1-\beta _{1}))^{{\pounds }}e^{i 2 \pi \left( 1-(1-\rho _{1})^{{\pounds }}\right) } \right\rangle \end{array}\right)\) where \({\pounds } \; \ge \; 0.\)
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5.
\(C_{1}^{c}=\left( \langle n_{1}e^{i 2 \pi \left( \phi _{1}\right) }, m_{1}e^{i 2 \pi \left( \theta _{1}\right) } \rangle , \langle \beta _{1}e^{i 2 \pi \left( \rho _{1}\right) }, \alpha _{1}e^{i 2 \pi \left( \delta _{1}\right) } \rangle \right) .\)
Definition 5
30 Let \(\nu\) and \(\mu\) be any two real numbers. The Dombi \(T_{DN}\) and \(T_{DCN}\) are defined as follows:
where \(k \ge 1\) and \((\nu , \mu ) \in [0, 1] \times [0, 1]\).
Definition 6
28 Let \(S = {\{S_1, S_2, \dots , S_t}\}\) denote a set of characteristics. Assume that the hierarchy of characteristics is \(S_1> S_2> S_3> \dots > S_t\).
If i is less than j, then \(S_i\) has a greater priority than \(S_j\). The value of \(S_{i}(\nu )\) represents the performance of any alternative \(\nu\) about attribute \(S_i\), where \(S_{i}(x)\) is within the range [0, 1]. This equality pertains to the PA.
where \(\zeta _{i} = \frac{\xi _{i}}{\sum _{i=1}^{t} \xi _{i}}\), \(\xi _{i} = \prod _{j=1}^{i-1} S_{j}(\nu )\) for \(i = 2, 3, \dots , t\), and \(\xi _{1}=1.\). Crisp values were used in Omar and Fayek49. Jana et al.50 and Khan et al.51 used prioritized aggregation operators on bipolar fuzzy sets and PFSs. This work used a more generalized fuzzy set known as \(C_pLD_yFS\).
Complex linear Diophantine Fuzzy dombi prioitized aggregation operators
In this section, we begin by introducing Dombi operations for \(C_pLD_yFNs\). Building on these operations, we define several AOs for \(C_pLD_yFNs\) and examine their characteristics.
Definition 7
Let \({C_{1}=(\langle m_{1}e^{i2\pi (\theta _{1})},n_{1}e^{i2\pi \phi _{1}}\rangle ,\langle \alpha _{1}e^{i2\pi (\delta _{1})},\beta _{1}e^{i2\pi (\rho _{1})}\rangle )}\) \({C_{2}}\) = \((\langle m_{2}e^{i2\pi (\theta _{2})},n_{2}e^{i2\pi \phi _{2}}\rangle ,\)\(\langle \alpha _{2}e^{i2\pi (\delta _{2})},\beta _{2}e^{i2\pi (\rho _{2})}\rangle )\) be any two \(C_pLD_yFNs\) and \({{\pounds }\ge 0}\). Then the Dombi operational laws on them are listed as follows:
1. \({C}_1 \oplus {C}_2 = \left( \begin{array}{c} \left\langle { 1 - \frac{1}{1 + \left( \left( \frac{m_{1}}{1 - m_{1}} \right) ^k + \left( \frac{m_{2}}{1 - m_{2}} \right) ^k \right) ^{\frac{1}{k}}}} e^{i 2 \pi \left( { 1 - \frac{1}{1 + \left( \left( \frac{\theta _{1}}{1 - \theta _{1}} \right) ^k + \left( \frac{\theta _{2}}{1 - \theta _{2}} \right) ^k \right) ^{\frac{1}{k}}}} \right) }, \right. \\ \left. {\frac{1}{1 + \left( \left( \frac{1 - n_1}{n_1} \right) ^k + \left( \frac{1 - n_2}{n_2} \right) ^k \right) ^{\frac{1}{k}}}} e^{i 2 \pi \left( {\frac{1}{1 + \left( \left( \frac{1 - \phi _1}{\phi _1} \right) ^k + \left( \frac{1 - \phi _2}{\phi _2} \right) ^k \right) ^{\frac{1}{k}}}} \right) } \right\rangle , \\ \left\langle { 1 - \frac{1}{1 + \left( \left( \frac{\alpha _{1}}{1 - \alpha _{1}} \right) ^k + \left( \frac{\alpha _{2}}{1 - \alpha _{2}} \right) ^k \right) ^{\frac{1}{k}}}} e^{i 2 \pi \left( { 1 - \frac{1}{1 + \left( \left( \frac{\delta _{1}}{1 - \delta _{1}} \right) ^k + \left( \frac{\delta _{2}}{1 - \delta _{2}} \right) ^k \right) ^{\frac{1}{k}}}} \right) }, \right. \\ \left. {\frac{1}{1 + \left( \left( \frac{1 - \beta _1}{\beta _1} \right) ^k + \left( \frac{1 - \beta _2}{\beta _2} \right) ^k \right) ^{\frac{1}{k}}}} e^{i 2 \pi \left( {\frac{1}{1 + \left( \left( \frac{1 - \rho _1}{\rho _1} \right) ^k + \left( \frac{1 - \rho _2}{\rho _2} \right) ^k \right) ^{\frac{1}{k}}}} \right) } \right\rangle \end{array} \right) ;\)
2. \({C_1 \otimes C_2} = \left( \begin{array}{c} \left\langle {\frac{1}{1 + \left( \left( \frac{1 - m_1}{m_1} \right) ^k + \left( \frac{1 - m_2}{m_2} \right) ^k \right) ^{\frac{1}{k}}}} e^{i 2 \pi \left( {\frac{1}{1 + \left( \left( \frac{1 - \theta _1}{\theta _1} \right) ^k + \left( \frac{1 - \theta _2}{\theta _2} \right) ^k \right) ^{\frac{1}{k}}}} \right) }, \right. \\ \left. { 1 - \frac{1}{1 + \left( \left( \frac{n_{1}}{1 - n_{1}} \right) ^k + \left( \frac{n_{2}}{1 - n_{2}} \right) ^k \right) ^{\frac{1}{k}}}} e^{i 2 \pi \left( { 1 - \frac{1}{1 + \left( \left( \frac{\phi _{1}}{1 - \phi _{1}} \right) ^k + \left( \frac{\phi _{2}}{1 - \phi _{2}} \right) ^k \right) ^{\frac{1}{k}}}} \right) } \right\rangle , \\ \left\langle {\frac{1}{1 + \left( \left( \frac{1 - \alpha _1}{m_1} \right) ^k + \left( \frac{1 - \alpha _2}{\alpha _2} \right) ^k \right) ^{\frac{1}{k}}}} e^{i 2 \pi \left( {\frac{1}{1 + \left( \left( \frac{1 - \delta _1}{\delta _1} \right) ^k + \left( \frac{1 - \delta _2}{\delta _2} \right) ^k \right) ^{\frac{1}{k}}}} \right) }, \right. \\ \left. { 1 - \frac{1}{1 + \left( \left( \frac{\beta _{1}}{1 - \beta _{1}} \right) ^k + \left( \frac{\beta _{2}}{1 - \beta _{2}} \right) ^k \right) ^{\frac{1}{k}}}} e^{i 2 \pi \left( { 1 - \frac{1}{1 + \left( \left( \frac{\rho _{1}}{1 - \rho _{1}} \right) ^k + \left( \frac{\rho _{2}}{1 - \rho _{2}} \right) ^k \right) ^{\frac{1}{k}}}} \right) } \right\rangle \end{array} \right) ;\)
3. \({{C_1}^{{\pounds }}} = \left( \begin{array}{c} \left\langle {\frac{1}{1 + \left( {\pounds } \left( \frac{1 - m_1}{m_1} \right) ^k \right) ^{\frac{1}{k}}}} e^{i 2 \pi \left( {\frac{1}{1 + \left( {\pounds } \left( \frac{1 - \theta _1}{\theta _1} \right) ^k \right) ^{\frac{1}{k}}}} \right) } , { 1 - \frac{1}{1 + \left( {\pounds } \left( \frac{n_{1}}{1 - n_{1}} \right) ^k \right) ^{\frac{1}{k}}}} e^{i 2 \pi \left( { 1 - \frac{1}{1 + \left( {\pounds } \left( \frac{\phi _{1}}{1 - \phi _{1}} \right) ^k \right) ^{\frac{1}{k}}}} \right) } \right\rangle , \\ \left\langle {\frac{1}{1 + \left( {\pounds } \left( \frac{1 - \alpha _1}{\alpha _1} \right) ^k \right) ^{\frac{1}{k}}}} e^{i 2 \pi \left( {\frac{1}{1 + \left( {\pounds } \left( \frac{1 - \delta _1}{\delta _1} \right) ^k \right) ^{\frac{1}{k}}}} \right) } , { 1 - \frac{1}{1 + \left( {\pounds } \left( \frac{\beta _{1}}{1 - \beta _{1}} \right) ^k \right) ^{\frac{1}{k}}}} e^{i 2 \pi \left( { 1 - \frac{1}{1 + \left( {\pounds } \left( \frac{\rho _{1}}{1 - \rho _{1}} \right) ^k \right) ^{\frac{1}{k}}}} \right) } \right\rangle \end{array} \right) ;\)
4. \({{\pounds }{C_1}} = \left( \begin{array}{c} \left\langle {1 - \frac{1}{1 + \left( {\pounds } \left( \frac{m_1}{1 - m_1} \right) ^k \right) ^{\frac{1}{k}}}} e^{i 2 \pi \left( {1 - \frac{1}{1 + \left( {\pounds } \left( \frac{\theta _1}{1 - \theta _1} \right) ^k \right) ^{\frac{1}{k}}}} \right) }, { \frac{1}{1 + \left( {\pounds } \left( \frac{1 - n_{1}}{n_{1}} \right) ^k \right) ^{\frac{1}{k}}}} e^{i 2 \pi \left( { \frac{1}{1 + \left( {\pounds } \left( \frac{1 - \phi _{1}}{\phi _{1}} \right) ^k \right) ^{\frac{1}{k}}}} \right) } \right\rangle , \\ \left\langle {1 - \frac{1}{1 + \left( {\pounds } \left( \frac{\alpha _1}{1 - \alpha _1} \right) ^k \right) ^{\frac{1}{k}}}} e^{i 2 \pi \left( {1 - \frac{1}{1 + \left( {\pounds } \left( \frac{\delta _1}{1 - \delta _1} \right) ^k \right) ^{\frac{1}{k}}}} \right) }, { \frac{1}{1 + \left( {\pounds } \left( \frac{1 - \beta _{1}}{\beta _{1}} \right) ^k \right) ^{\frac{1}{k}}}} e^{i 2 \pi \left( { \frac{1}{1 + \left( {\pounds } \left( \frac{1 - \rho _{1}}{\rho _{1}} \right) ^k \right) ^{\frac{1}{k}}}} \right) } \right\rangle \end{array} \right) .\)
Utilizing these operations, a range of AOs is explored as follows:
Definition 8
Let \({C_{i}=(\langle m_{i}e^{i2\pi (\theta _{i})},n_{i}e^{i2\pi \phi _{i}}\rangle ,\langle \alpha _{i}e^{i2\pi (\delta _{i})},\beta _{i}e^{i2\pi (\rho _{i})}\rangle )}(i=1,2,3,...,n)\) be a collection of n \(C_pLD_yFNs\). Then the \(C_pLD_yFDPA\) is given as follows:
Here \(Q_{i}=(\frac{\xi _{i}}{\sum _{i=1}^{n} \xi _{i}})\) where
Theorem 1
Let \({C_{i}=(\langle m_{i}e^{i2\pi (\theta _{i})},n_{i}e^{i2\pi \phi _{i}}\rangle ,\langle \alpha _{i}e^{i2\pi (\delta _{i})},\beta _{i}e^{i2\pi (\rho _{i})}\rangle )}(i=1,2,3,...,n)\) be a collection of n \(C_pLD_yFNs\). Then, the extended form of Equation (9) is given as
Proof
For \(i=2\) we have \(C_pLD_yFDPA{(C_1,C_2)=Q_{1}C_{1}\bigoplus Q_{2}C_{2}}\), where
and
Thus
Hence Equation (10) is true for \(i=2\).
Suppose Equation (10) is true for \(i=1,2,3,...,n\) that is
Now we need to show that Equation (10) is true for \(i=n+1\).
Hence Equation (10) is true for \(i=n+1\). \(\square\)
Theorem 2
(Idempotency) Let \({C_{i}=(\langle m_{i}e^{i2\pi (\theta _{i})},n_{i}e^{i2\pi \phi _{i}}\rangle ,\langle \alpha _{i}e^{i2\pi (\delta _{i})},\beta _{i}e^{i2\pi (\rho _{i})}\rangle )}\) \((i=1,2,3,...,n)\) be a collection of n \(C_pLD_yFNs\) where \(\xi _{i} = \prod _{j=1}^{i-1} S(C_j) \quad \text {for}\; \quad i = 2, 3, \dots , n, \; and\)\(for\, i=1 \quad \xi _{1} = 1. \; S(C_j) \; represent \; score \; function \; value.\) If \(C_i=C_s\) then \(C_pLD_yFDPA(C_1,C_2,C_3,...,C_i)=C_s\).
Proof
Since \(C_i=C_s\) \(\forall (i=1,2,3,...n)\), so according to Theorem 1, we have
\(for \; all\; i\)
\(\square\)
Theorem 3
(Monotonicity) Let \(C_{i}=(\langle m_{i}e^{i2\pi (\theta _{i})},n_{i}e^{i2\pi \phi _{i}}\rangle ,\)\(\langle \alpha _{i}e^{i2\pi (\delta _{i})},\beta _{i}e^{i2\pi (\rho _{i})}\rangle )\) and \({C_{j}}\) = \(\left( \langle m_{j}e^{i2\pi (\theta _{j})},\right.\)\(n_{j}e^{i2\pi \phi _{j}}\rangle ,\)\(\left. \langle \alpha _{j}e^{i2\pi (\delta _{j})},\beta _{j}e^{i2\pi (\rho _{j})}\rangle \right)\) \((i,j=1,2,3,...,n)\) be two collections of n \(C_pLD_yFNs\). If \(m_{i} \le m_{j}, n_{i} \ge n_{j}, \theta _{i} \le \theta _{j}, \phi _{i} \ge \phi _{j}, {\alpha _{i}} \le {\alpha _{j}} , \delta _{i} \le \delta _{j}, \rho _{i}\ge \rho _{j} \; and \; {\beta _{i}} \ge {\beta _{j}}\) , then
where \(\xi _{i} = \prod _{j=1}^{i-1} S(C_j), \xi _{j} = \prod _{i=1}^{j-1} S(C_i) \quad \text {for} \quad i = 2, 3, \dots , t,\)\(\xi _{1} =1. \; S(C_i) \;and\; S(C_j) \; represent \; score \; function \; values.\)
Proof
Since \(m_{i} \le m_{j}, n_{i} \ge n_{j}, \theta _{i} \le \theta _{j}, \phi _{i} \ge \phi _{j}, {\alpha _{i}} \le {\alpha _{j}} , \delta _{i} \le \delta _{j}, \rho _{i}\ge \rho _{j} \; and \; {\beta _{i}} \ge {\beta _{j}} \; \forall i\). Taking into consideration these observations, we get the following inequalities,
which gives
this implies that \(C_pLD_yFDPA(C_{i}) \le C_pLD_yFDPA(C_{j})\) which completes the proof. \(\square\)
Theorem 4
(Boundedness) Let \({C_{i}=(\langle m_{i}e^{i2\pi (\theta _{i})},n_{i}e^{i2\pi \phi _{i}}\rangle ,\langle \alpha _{i}e^{i2\pi (\delta _{i})},\beta _{i}e^{i2\pi (\rho _{i})}\rangle )}(i=1,2,3,...,n)\) be a collection of n \(C_pLD_yFNs\). Then \(C_{i}^{-} \le C_pLD_yFDPA(C_i) \le C_{i}^{+}\), where \(C_{i}^{-}=\left\langle \left( \underset{1\le i\le n}{\min }m_{i}, \underset{1\le i\le n}{\min }\theta _{i} \right) , \left( \underset{1\le i\le n}{\max }n_{i}, \underset{1\le i\le n}{\max }\phi _{i} \right) \right\rangle ,\)\(\left\langle \left( \underset{1\le i\le n}{\min }\alpha _{i},\underset{1\le i\le n}{\min }\delta _{i} \right) , \left( \underset{1\le i\le n}{\max }\beta _{i}, \underset{1\le i\le n}{\max }\rho _{i} \right) \right\rangle\) and \(C_{i}^{+}=\left\langle \left( \underset{1\le i\le n}{\max }m_{i}, \underset{1\le i\le n}{\max }\theta _{i} \right) , \left( \underset{1\le i\le n}{\min }n_{i}, \underset{1\le i\le n}{\min }\phi _{i} \right) \right\rangle ,\)\(\left\langle \left( \underset{1\le i\le n}{\max }\alpha _{i},\underset{1\le i\le n}{\max }\delta _{i} \right) , \left( \underset{1\le i\le n}{\min }\beta _{i}, \underset{1\le i\le n}{\min }\rho _{i} \right) \right\rangle .\)
Proof
The proof is straightforward. \(\square\)
Definition 9
Let \({C_{i}=(\langle m_{i}e^{i2\pi (\theta _{i})},n_{i}e^{i2\pi \phi _{i}}\rangle ,\langle \alpha _{i}e^{i2\pi (\delta _{i})},\beta _{i}e^{i2\pi (\rho _{i})}\rangle )}(i=1,2,3,...,n)\) be a collection of n \(C_pLD_yFNs\). Then the \(C_pLD_yFDPWA\) is given as follows:
where
In Equation (12) \(w=(w_1,w_2,w_3,...,w_n)\) represents weight vector such that \(w_i \; \in \; [0,1] \forall i\) and \(\sum \limits _{i=1}^{n}{w_i}=1\).
Theorem 5
Let \({C_{i}=(\langle m_{i}e^{i2\pi (\theta _{i})},n_{i}e^{i2\pi \phi _{i}}\rangle ,\langle \alpha _{i}e^{i2\pi (\delta _{i})},\beta _{i}e^{i2\pi (\rho _{i})}\rangle )}(i=1,2,3,...,n)\) be a collection of n \(C_pLD_yFNs\). Then, the extended form of Equation (12) is given as
Proof
The proof is omitted as it is similar to Theorem 1. \(\square\)
Theorem 6
(Idempotency) Let \({C_{i}=(\langle m_{i}e^{i2\pi (\theta _{i})},n_{i}e^{i2\pi \phi _{i}}\rangle ,\langle \alpha _{i}e^{i2\pi (\delta _{i})},\beta _{i}e^{i2\pi (\rho _{i})}\rangle )}(i=1,2,3,...,n)\) be a collection of n \(C_pLD_yFNs\). If \(C_i=C_s\) then \(C_pLD_yFDPWA(C_1,C_2,C_3,...,C_i)=C_s\).
Proof
The proof is omitted as it is similar to Theorem 2. \(\square\)
Theorem 7
(Monotonicity) Let \(C_{i}\) = \((\langle m_{i}e^{i2\pi (\theta _{i})},n_{i}e^{i2\pi \phi _{i}}\rangle ,\)\(\langle \alpha _{i}e^{i2\pi (\delta _{i})},\beta _{i}e^{i2\pi (\rho _{i})}\rangle )\) and \(C_{j}\) = \(\left( \langle m_{j}e^{i2\pi (\theta _{j})},n_{j}e^{i2\pi \phi _{j}}\rangle \right.,\)\(\left.\langle \alpha _{j}e^{i2\pi (\delta _{j})},\beta _{j}e^{i2\pi (\rho _{j})}\rangle \right)\) \((i,j=1,2,3,...,n)\) be two collections of n \(C_pLD_yFNs\). If \(m_{i} \le m_{j}, n_{i} \ge n_{j}, \theta _{i} \le \theta _{j}, \phi _{i} \ge \phi _{j}, {\alpha _{i}} \le {\alpha _{j}} , \delta _{i} \le \delta _{j}, \rho _{i}\ge \rho _{j} \; and \; {\beta _{i}} \ge {\beta _{j}}\), then
Proof
The proof is omitted as it is similar to Theorem 3. \(\square\)
Theorem 8
(Boundedness) Let \({C_{i}=(\langle m_{i}e^{i2\pi (\theta _{i})},n_{i}e^{i2\pi \phi _{i}}\rangle ,\langle \alpha _{i}e^{i2\pi (\delta _{i})},\beta _{i}e^{i2\pi (\rho _{i})}\rangle )}(i=1,2,3,...,n)\)be a collection of n \(C_pLD_yFNs\). Then \(C_{i}^{-} \le C_pLD_yFDPWA(C_i) \le C_{i}^{+}\), where \(C_{i}^{-}=\left\langle \left( \underset{1\le i\le n}{\min }m_{i}, \underset{1\le i\le n}{\min }\theta _{i} \right) , \left( \underset{1\le i\le n}{\max }n_{i}, \underset{1\le i\le n}{\max }\phi _{i} \right) \right\rangle ,\)\(\left\langle \left( \underset{1\le i\le n}{\min }\alpha _{i},\underset{1\le i\le n}{\min }\delta _{i} \right) , \left( \underset{1\le i\le n}{\max }\beta _{i}, \underset{1\le i\le n}{\max }\rho _{i} \right) \right\rangle\) and \(C_{i}^{+}=\left\langle \left( \underset{1\le i\le n}{\max }m_{i}, \underset{1\le i\le n}{\max }\theta _{i} \right) , \left( \underset{1\le i\le n}{\min }n_{i}, \underset{1\le i\le n}{\min }\phi _{i} \right) \right\rangle ,\)\(\left\langle \left( \underset{1\le i\le n}{\max }\alpha _{i},\underset{1\le i\le n}{\max }\delta _{i} \right) , \left( \underset{1\le i\le n}{\min }\beta _{i}, \underset{1\le i\le n}{\min }\rho _{i} \right) \right\rangle\).
Proof
The proof is straightforward. \(\square\)
Definition 10
Let \({C_{i}=(\langle m_{i}e^{i2\pi (\theta _{i})},n_{i}e^{i2\pi \phi _{i}}\rangle ,\langle \alpha _{i}e^{i2\pi (\delta _{i})},\beta _{i}e^{i2\pi (\rho _{i})}\rangle )}(i=1,2,3,...,n)\) be a collection of n \(C_pLD_yFNs\). Then the \(C_pLD_yFPEG\) is given as follows:
where \(Q_{i}=(\frac{\xi _{i}}{\sum _{i=1}^{n} \xi _{i}}).\)
Theorem 9
Let \({C_{i}=(\langle m_{i}e^{i2\pi (\theta _{i})},n_{i}e^{i2\pi \phi _{i}}\rangle ,\langle \alpha _{i}e^{i2\pi (\delta _{i})},\beta _{i}e^{i2\pi (\rho _{i})}\rangle )}(i=1,2,3,...,n)\)be a collection of n \(C_pLD_yFNs\). Then, the extended form of Equation (15) is given as
Proof
For \(i=2\) we have \(C_pLD_yFDPG(C_1,C_2)={C_{1}}^{Q_{1}}\bigotimes {C_{2}}^{Q_{2}}\) where
and
Hence Equation (16) is true for \(i=2\). Suppose Equation (16) is true for \(i=n\).
Now we need to prove that Equation (16) is true for \(i=n+1\).
Hence the theorem is true \(\forall\) i. \(\square\)
Theorem 10
(Idempotency) Let \({C_{i}=(\langle m_{i}e^{i2\pi (\theta _{i})},n_{i}e^{i2\pi \phi _{i}}\rangle ,\langle \alpha _{i}e^{i2\pi (\delta _{i})},\beta _{i}e^{i2\pi (\rho _{i})}\rangle )}(i=1,2,3,...,n)\) be a collection of n \(C_pLD_yFNs\), where \(\xi _{i} = \prod _{j=1}^{i-1} S(C_j) \quad \text {for} \quad i = 2, 3, \dots , n, \; and \; for\; i=1 \quad \xi _{1} = 1. \; S(C_j) \; represents \; score \; function \; value.\) If \(C_i=C_s\) then \(C_pLD_yFDPG(C_1,C_2,C_3,...,C_i)=C_s\).
Proof
Since \(C_i=C_s\) \(\forall (i=1,2,3,...n)\), so according to Theorem 9, we have
Hence proved. \(\square\)
Theorem 11
(Monotonicity) Let \({C_{i}=(\langle m_{i}e^{i2\pi (\theta _{i})},n_{i}e^{i2\pi \phi _{i}}\rangle ,\langle \alpha _{i}e^{i2\pi (\delta _{i})},\beta _{i}e^{i2\pi (\rho _{i})}\rangle )}\) and \({C_{j}=\left( \langle m_{j}e^{i2\pi (\theta _{j})},n_{j}e^{i2\pi \phi _{j}}\rangle ,\langle \alpha _{j}e^{i2\pi (\delta _{j})},\beta _{j}e^{i2\pi (\rho _{j})}\rangle \right) }\) \((i,j=1,2,3,...,n)\) be two collections of n \(C_pLD_yFNs\). If \(m_{i} \le m_{j}, n_{i} \ge n_{j}, \theta _{i} \le \theta _{j}, \phi _{i} \ge \phi _{j}, {\alpha _{i}} \le {\alpha _{j}} , \delta _{i} \le \delta _{j}, \rho _{i}\ge \rho _{j} \; and \; {\beta _{i}} \ge {\beta _{j}}\) , then
Proof
Since \(m_{i} \le m_{j}, n_{i} \ge n_{j}, \theta _{i} \le \theta _{j}, \phi _{i} \ge \phi _{j}, {\alpha _{i}} \le {\alpha _{j}} , \delta _{i} \le \delta _{j}, \rho _{i}\ge \rho _{j} \; and \; {\beta _{i}} \ge {\beta _{j}} \; \forall i\). Taking into consideration these observations, we get the following inequalities,
which gives
This implies that \(C_pLD_yFDPG(C_{i}) \le C_pLD_yFDPG(C_{j})\), which completes the proof. \(\square\)
Theorem 12
(Boundedness) Let \({C_{i}=(\langle m_{i}e^{i2\pi (\theta _{i})},n_{i}e^{i2\pi \phi _{i}}\rangle ,\langle \alpha _{i}e^{i2\pi (\delta _{i})},\beta _{i}e^{i2\pi (\rho _{i})}\rangle )}(i=1,2,3,...,n)\) be a collection of n \(C_pLD_yFNs\). Then \(C_{i}^{-} \le C_pLD_yFDPG(C_i) \le C_{i}^{+}\), where \(C_{i}^{-}=\left\langle \left( \underset{1\le i\le n}{\min }m_{i}, \underset{1\le i\le n}{\min }\theta _{i} \right) , \left( \underset{1\le i\le n}{\max }n_{i}, \underset{1\le i\le n}{\max }\phi _{i} \right) \right\rangle ,\)\(\left\langle \left( \underset{1\le i\le n}{\min }\alpha _{i},\underset{1\le i\le n}{\min }\delta _{i} \right) , \left( \underset{1\le i\le n}{\max }\beta _{i}, \underset{1\le i\le n}{\max }\rho _{i} \right) \right\rangle\) and \(C_{i}^{+}=\left\langle \left( \underset{1\le i\le n}{\max }m_{i}, \underset{1\le i\le n}{\max }\theta _{i} \right) , \left( \underset{1\le i\le n}{\min }n_{i}, \underset{1\le i\le n}{\min }\phi _{i} \right) \right\rangle ,\)\(\left\langle \left( \underset{1\le i\le n}{\max }\alpha _{i},\underset{1\le i\le n}{\max }\delta _{i} \right) , \left( \underset{1\le i\le n}{\min }\beta _{i}, \underset{1\le i\le n}{\min }\rho _{i} \right) \right\rangle\).
Proof
The proof is straightforward. \(\square\)
Definition 11
Let \({C_{i}=(\langle m_{i}e^{i2\pi (\theta _{i})},n_{i}e^{i2\pi \phi _{i}}\rangle ,\langle \alpha _{i}e^{i2\pi (\delta _{i})},\beta _{i}e^{i2\pi (\rho _{i})}\rangle )}(i=1,2,3,...,n)\) be a collection of n \(C_pLD_yFNs\). Then the \(C_pLD_yFDPWG\) is given as follows
where \(Q_{wi}=(\frac{{w_i}\xi _{i}}{\sum _{i=1}^{n} \xi _{i}})\). Here \(w=(w_1,w_2,w_3,...,w_n)\) represents weight vector such that \(w_i \; \in \; [0,1] \forall i\) and \(\sum \limits _{i=1}^{n}{w_i}=1\).
Theorem 13
Let \({C_{i}=(\langle m_{i}e^{i2\pi (\theta _{i})},n_{i}e^{i2\pi \phi _{i}}\rangle ,\langle \alpha _{i}e^{i2\pi (\delta _{i})},\beta _{i}e^{i2\pi (\rho _{i})}\rangle )}(i=1,2,3,...,n)\) be a collection of n \(C_pLD_yFNs\). Then, the extended form of Equation (18) is given as
Proof
The proof is omitted as it is similar to Theorem 9. \(\square\)
Theorem 14
(Idempotency) Let \({C_{i}=(\langle m_{i}e^{i2\pi (\theta _{i})},n_{i}e^{i2\pi \phi _{i}}\rangle ,\langle \alpha _{i}e^{i2\pi (\delta _{i})},\beta _{i}e^{i2\pi (\rho _{i})}\rangle )}(i=1,2,3,...,n)\) be a collection of n \(C_pLD_yFNs\). If \(C_i=C_s\) then \(C_pLD_yFDPWG(C_1,C_2,C_3,...,C_i)=C_s\).
Proof
The proof is omitted as it is similar to Theorem 10. \(\square\)
Theorem 15
(Monotonicity) Let \({C_{i}=(\langle m_{i}e^{i2\pi (\theta _{i})},n_{i}e^{i2\pi \phi _{i}}\rangle ,\langle \alpha _{i}e^{i2\pi (\delta _{i})},\beta _{i}e^{i2\pi (\rho _{i})}\rangle )}\) and \({C_{j}=\left( \langle m_{j}e^{i2\pi (\theta _{j})},n_{j}e^{i2\pi \phi _{j}}\rangle ,\langle \alpha _{j}e^{i2\pi (\delta _{j})},\beta _{j}e^{i2\pi (\rho _{j})}\rangle \right) }\) \((i,j=1,2,3,...,n)\) be two collections of n \(C_pLD_yFNs\). If \(m_{i} \le m_{j}, n_{i} \ge n_{j}, \theta _{i} \le \theta _{j}, \phi _{i} \ge \phi _{j}, {\alpha _{i}} \le {\alpha _{j}} , \delta _{i} \le \delta _{j}, \rho _{i}\ge \rho _{j} \; and \; {\beta _{i}} \ge {\beta _{j}}\) , then
Proof
The proof is omitted as it is similar to Theorem 11. \(\square\)
Theorem 16
(Boundedness) Let \({C_{i}=(\langle m_{i}e^{i2\pi (\theta _{i})},n_{i}e^{i2\pi \phi _{i}}\rangle ,\langle \alpha _{i}e^{i2\pi (\delta _{i})},\beta _{i}e^{i2\pi (\rho _{i})}\rangle )}(i=1,2,3,...,n)\) be a collection of n \(C_pLD_yFNs\). Then \(C_{i}^{-} \le C_pLD_yFDPWG(C_i) \le C_{i}^{+}\), where \(C_{i}^{-}=\left\langle \left( \underset{1\le i\le n}{\min }m_{i}, \underset{1\le i\le n}{\min }\theta _{i} \right) , \left( \underset{1\le i\le n}{\max }n_{i}, \underset{1\le i\le n}{\max }\phi _{i} \right) \right\rangle,\)\(\left\langle \left( \underset{1\le i\le n}{\min }\alpha _{i},\underset{1\le i\le n}{\min }\delta _{i} \right) , \left( \underset{1\le i\le n}{\max }\beta _{i}, \underset{1\le i\le n}{\max }\rho _{i} \right) \right\rangle\) and \(C_{i}^{+}=\left\langle \left( \underset{1\le i\le n}{\max }m_{i}, \underset{1\le i\le n}{\max }\theta _{i} \right) , \left( \underset{1\le i\le n}{\min }n_{i}, \underset{1\le i\le n}{\min }\phi _{i} \right) \right\rangle,\)\(\left\langle \left( \underset{1\le i\le n}{\max }\alpha _{i},\underset{1\le i\le n}{\max }\delta _{i} \right) , \left( \underset{1\le i\le n}{\min }\beta _{i}, \underset{1\le i\le n}{\min }\rho _{i} \right) \right\rangle\).
Proof
The proof is straightforward. \(\square\)
Proposed methodology
This section outlines the methodology of the proposed \(C_pLD_yF\) model, which is based on the diagnosed theoretical framework.
Let \({\daleth } = {\{\daleth }_1, {\daleth }_2, {\daleth }_3, \dots , {\daleth }_m\}\) represent a set of alternatives, and \(D = {\{D_1, D_2, D_3, \dots , D_n}\}\) denote a set of criteria. The decision maker or expert provides evaluation values for each alternative based on the specified criteria, resulting in the formation of the \(C_pLD_yF\) decision matrix \({\mathfrak {M}} = \left( {C_{ij}}\right) _{m \times n}\), where \(C_{ij}\) represents the \(C_pLD_yFN\)—the evaluation value assigned by the expert. This matrix encapsulates the evaluations for each alternative across all criteria, serving as a key component in the decision-making process. Assume that characteristics are prioritized as follows: \(D_1> D_2>\ldots > D_t\). \(D_i\) has a greater priority than \(D_j\) when \(i < j\). The MULTIMOORA technique52 was adapted to \(C_pLD_yFS\) in the current study.
The MULTIMOORA technique combines multiple Moora methods and a multiplicative utility function. According to Bauers and Zavadskas52, there are three methods: the ratio system, the reference point approach, and the complete multiplicative utility function.
The proposed aggregation operators are utilized in the ratio system and complete multiplicative form areas of the MULTIMORA approach. The prioritized aggregation and Dombi operators have excellent parameter flexibility, allowing them to express the properties better.
The following procedures can be used to describe the MULTIMOORA MCDM using \(C_pLD_yFNs\):
Step 1: Normalization of the criteria values is not required when all criteria \({{\daleth }i}\), \((i = 1, 2, 3, \dots , m)\) are of the same type. However, in DM problems, criteria generally fall into two categories: Benefit (where higher values are preferred) and Cost (where lower values are preferred). For cost-type criteria, it is necessary to convert them into benefit-type values. In this case, the decision matrix \({\mathfrak {M}} = {(C_{ij})_{m \times n}}\) can be transformed into a normalized \(C_pLD_yF\) matrix, represented as \(\mathfrak {{\widetilde{M}}} = \widetilde{(C_{ij})}_{m \times n}\). The normalization process can be achieved using the following formula.
where BT and CT represent benefit-type and cost-type, respectively.
Step 2: In this step, the \(C_pLD_yFDPWA\) operator is applied to the ratio system for MULTIMOORA \(C_pLD_yFS\). The \(C_pLD_yFS\) utility can be computed as follows.
where \(Q_{wij}=(\frac{{w_i}\xi _{ij}}{\sum _{i=1}^{n}\sum _{j=1}^{n} \xi _{ij}}).\) Here \(w=(w_1,w_2,w_3,...,w_n)\) represents weight vector such that \(w_i \; \in \; [0,1] \forall i\) and \(\sum \limits _{i=1}^{n}{w_i}=1\).
Step 3: The score function is used to calculate the values derived in step 2.
where \(({\ddot{S}}_{i})^{RS} \in [0,1]\) \(i=1, 2, 3, \dots t.\)
Further, the derived values are then normalized as follows:
Ultimately, the greatest utility value for \(\overline{({\ddot{S}}_{i})}^{RS} \in [0, 1]\) is determined.
Step 4: This phase employs the reference point approach. The reference point is established, and the Chebyshev distance is computed for each alternative. The reference point derived from the inputs are as follows.
Subsequently, we shall determine the distances between the reference point and the alternatives. The Hamming distance is employed as a distance measurement53,54.
Let \(C_{1}=(\langle m_{1}e^{i2\pi (\theta _{1})},n_{1}e^{i2\pi \phi _{1}}\rangle ,\)\(\langle \alpha _{1}e^{i2\pi (\delta _{1})},\beta _{1}e^{i2\pi (\rho _{1})}\rangle )\) \({C_{2}}=(\langle m_{2}e^{i2\pi (\theta _{2})},n_{2}e^{i2\pi \phi _{2}}\rangle,\)\(\langle \alpha _{2}e^{i2\pi (\delta _{2})},\beta _{2}e^{i2\pi (\rho _{2})}\rangle )\) be any two \(C_pLD_yFNs\). Then hamming distance can be calculated as
Additionally, the maximum Chebyshev distance from the reference point is determined for each alternative as \(({\ddot{S}}_{i})^{RP}= \underset{j}{\max }d_{ij}\).
The reference point employs a non-compensatory approach. A lower \(({\ddot{S}}_{i})^{RP}\) value indicates greater utility. In the subsequent step, normalized utility scores can be computed.
Step 5: The calculation of the normalized utility scores is presented here.
Ultimately, the maximal utility value for \(({\ddot{S}}_{i})^{RP} \in [0, 1]\) is determined.
Step 6: In this step, the \(C_pLD_yFDPWG\) operator is applied to the multiplicative function for MULTIMOORA \(C_pLD_yFS\). The \(C_pLD_yFS\) utility can be computed as follows.
where \(Q_{wij}=(\frac{{w_i}\xi _{ij}}{\sum _{i=1}^{n}\sum _{j=1}^{n} \xi _{ij}}).\) Here \(w=(w_1,w_2,w_3,...,w_n)\) represents weight vector such that \(w_i \; \in \; [0,1] \forall i\) and \(\sum \limits _{i=1}^{n}{w_i}=1\).
Step 7: The score function is used to calculate the values derived in Step 6.
where \(({\ddot{S}}_{i})^{MF} \in [-1,1]\) \(i=1, 2, 3, \dots t.\)
Further, the obtained values are then normalized as followsL
Ultimately, the greatest utility value for \(\overline{({\ddot{S}}_{i})}^{MF} \in [0, 1]\) is determined.
Step 8: Considering the ranking values \(\overline{({\ddot{S}}_{i})}^{RS}, \overline{({\ddot{S}}_{i})}^{RP}, \overline{({\ddot{S}}_{i})}^{MF}\) for each alternative \(\overline{{\ddot{S}}_{i}} \ (i = 1, 2,..., m)\) over three distinct subsystems given in Equations (24), (28), and (31), we utilize the enhanced Borda rule55 to amalgamate these rankings into a conclusive ranking.
The following formula is used to determine the alternatives’ final ranking results:
The classification order of each alternative in the ratio system, reference point, and complete multiplicative form is denoted by \({\bar{O}}\overline{({\ddot{S}}_{i})}^{RS},\) \({\bar{O}}\overline{({\ddot{S}}_{i})}^{RP},\) and \({\bar{O}}\overline{({\ddot{S}}_{i})}^{MF}\), respectively. The alternative that achieves a higher final ranking result is deemed preferable. Consequently, the alternatives should be ranked in descending order according to their ultimate ranking results.
The flowchart of the proposed MULTIMOORA method is shown in Fig. 1
Flowchart of the proposed method.
Case study
This section presents practical examples of sustainable energy alternative selection, adapted from56, to demonstrate the reliability and consistency of the proposed method.
Problem description
The majority of the world’s energy comes from fossil fuels. The supply of fossil fuels has steadily decreased throughout the years due to their finite nature. There was an all-time high in the reliance of energy production on fossil fuels. Future decades will see shortages due to the continued use of fossil fuels at current rates. Did reliance bring about the degradation of the environment? The excessive use of fossil fuels has led to an alarming rise in atmospheric \(CO_{2}\) concentrations. This has affected the air and water, damaging the environment. The primary contributors to acid rain are coal and nuclear power stations. Contaminated air adversely affects our bodies, and health risks are associated. A multitude of medical diseases has been recognized and recorded. Energy policymakers have sought new methods to enhance air quality and mitigate emissions in response to these urgent issues. Consequently, it is and continues to be imperative to utilize renewable energy sources57. Hazardous smog, particulate materials, haze, and smoke are all too common in Pakistan, so the country ranks so low in pollution levels. The primary contributors to this pollution come from fossil fuels. The numbers of Pakistan’s Air Quality Index (AQI) are problematic58. The energy industry is Pakistan’s primary source of emissions; by regulating and decreasing the usage of fossil fuels, the country can improve its AQI ranking. Up until March 2020, 35,975 MWh of power will be generated in Pakistan. In the fiscal years 2019 and 2020, Pakistan’s share of renewable energy was a meager 2057 GWh. Pakistan has significant unrealized potential for renewable energy sources such as solar, wind, geothermal, and tidal power. To address clean and green energy challenges, Pakistan must examine these resources. Renewable energy sources have replaced fossil fuels to combat climate change. Using renewable resources can help to address electricity supply and demand challenges. Renewable energy sources (RE) meet environmental challenges since they produce clean, eco-friendly electricity59. Pakistan has great possibilities for renewable energy sources, including tidal, solar, and wind. These supplies can meet the national energy demand. Pakistan is striving to establish targets for the generation of renewable energy. Recently, federal-level renewable energy policy projects have attracted attention. Installation of RE plants can benefit greatly from energy policies. While starting is mostly hampered by finance, Pakistan is highly determined to reach RE-generating goals60.
This study is on the energy scenario of Gwadar City, which is located on the southwest coast of the Arabian Sea region of Balochistan, and how it generates, demands, and consumes energy. Coastal winds and sunshine are very concentrated in this area.Today’s Power generation depends on wind and coal, both near Gwadar. Power units with comparatively small capacities have been installed. Coal-fired power plants are also responsible for regional energy production. This research aims to comprehensively review renewable energy (RE), which may encompass wind, solar, tidal, and coal power plants. Enquiring as to how the Gwadar area’s electricity needs might be met by renewable energy sources.Using the country’s plentiful natural resources can solve its energy problem. Pollutants from coal power stations hurt the environment. Renewable energy sources such as solar, wind, and tidal power can potentially displace coal-fired power plants in the long run. The Gwadar region has the potential to produce 1800-2100 KW of electricity from direct solar due to the 3000 hours of sunshine it receives annually61. Consequently, Pakistan benefits from a notably higher average solar irradiance than numerous other nations. This characteristic renders the nation exceptionally well-suited for the production of photovoltaic energy. Authorizations have been issued to various non-governmental energy producers to convert kinetic wind energy into electrical power. The currently functioning wind energy conversion systems are situated within the geographical area of Gwadar. Our research findings suggest that integrating water-driven tidal power stations alongside wind turbine arrays can potentially increase the total output of electrical energy. This combined strategy provides a superior level of operational efficiency. Developing a tidal energy generation plant in the Gwadar coastal region is practically achievable. This installation has the potential to generate sufficient electrical power to meet the requirements of rural populations within the Balochistan province. Tidal electricity generation is also ecologically benign, as it significantly diminishes the release of noxious substances into the atmosphere and surrounding ecosystem. Fossilized carbonaceous material is being utilized to generate electricity, with collaborative support from the People’s Republic of China. Fossil fuel-based energy projects are being executed through a collaborative partnership under the China-Pakistan Economic Corridor (CPEC) framework. While coal-fired power stations have implemented advanced emission control technologies, particulate and gaseous byproducts are inevitably generated during electrical power production. We have not achieved compliance with established air quality index (AQI) thresholds. Therefore, we have undertaken a detailed comparative evaluation of various electricity generation modalities. Based on the empirical evidence, it is evident that Pakistan can produce electrical energy without reliance on fossil fuel consumption across all sectors. Renewable energy (RE) resources can adequately meet the nation’s energy requirements. Consequently, environmental pollution can be mitigated, and a substantial reduction in emission levels can be achieved. The adoption and establishment of renewable energy systems are anticipated to yield positive effects on the population’s well-being. Nevertheless, a continuing economic obstacle demands the application of carefully planned state policy actions. Moreover, strong administrative structures are essential to encourage and promote initiatives that generate environmentally sound energy.
Tidal energy
Gwadar, Pakistan, possesses significant potential for tidal energy generation along its coastal regions. A study has projected the feasibility of installing tidal power plants along the Gwadar coastline62. Further investigation into the potential of tidal electricity in coastal areas and the deployment of tidal energy plants to generate affordable and clean energy is essential. The fluctuations between high and low tides of water produce tidal energy. Tidal turbines are implemented in the barrage to generate clean electricity from the water waves. The energy is both clean and cost-effective63. Reducing pollutants is the primary benefit of tidal energy. The amount cut down environmental pollutants, including carbon dioxide and greenhouse gas emissions. Compared to a coal-powered power station, a tidal power plant can avoid emitting around 1000 g of carbon dioxide into the atmosphere. In addition, it challenges others to reduce emissions effectively. It helps lower nitrous oxide \((N_{2}O)\) emissions, greenhouse gases, and methane \((CH_{4})\). Unlike other sources of air pollution, it does not release particle matter64. The design and installation of tidal power plants are less complicated and more predictable. Compared with solar and wind power, tidal energy is more efficient and lasts longer. Thus, tidal energy is competitive with other forms of renewable power. Commercial tidal energy generating plants are present in France (240 MW), Russia (0.4 MW), Canada (20 MW), and China (5 MW). A 254 MW tidal power facility was recently established in South Korea65. Pakistan still relies heavily on electricity generated by oil. Oil releases various pollutants into the atmosphere, making it less eco-friendly. The power sector, which now relies on oil, has enormous potential to become more sustainable and environmentally friendly. By tapping into its renewable natural resource reserves, Pakistan has the potential to meet its energy needs and possibly export excess power to its neighboring countries.
Wind energy
According to resources used to evaluate renewable energy, wind corridors can be found in several parts of Balochistan in the southern, northwestern, and central regions. The wind potential evaluation for Balochistan is tabulated in Table 1. As stated in Table 2, Independent Power Projects (IPPs) have been working on wind energy generation projects with a capacity of 50 MW each, having obtained the necessary licenses for wind power. The potential of wind energy in coastal locations has been explained in multiple submitted feasibility assessments66. The installation of wind power generating facilities depends on the fulfillment of several critical criteria. Wind speed can serve as a proxy for the potential and characteristics of other relevant metrics. The average wind speed measures the effectiveness and potential of a wind farm. Second, there must be convenient transportation options and third; transmission lines must connect the generation grid to the distribution grid. Among Pakistan’s provinces, Balochistan offers the most promise for wind power. This province’s renewable and non-renewable energy resources are abundant and of excellent quality. The aggregate potential of our indigenous resources has yet to undergo thorough quantification or appraisal. In many geographical locales, the necessary infrastructural foundations for the exploitation of the region’s intrinsic resources are deficient. Notwithstanding, an adequate number of locations are available to sustain modest wind energy projects, provided they operate within the established wind speed criteria. A project of this kind can potentially meet the neighborhood’s electrical needs67.
While numerous regions remain underdeveloped, they are being encompassed strategically into the Infrastructure initiatives undertaken by the government of Pakistan. Establishing a seaport will serve as a central nexus for traders and has the potential to draw in new energy initiatives. It has the potential to serve as the central nexus for trade activities within Central Asia and China. Consequently, the absence of adequate infrastructure represents the primary challenge that must be addressed. A transmission network must be established to facilitate the integration of renewable energy resources. The construction of roadway transportation networks is essential to facilitate unobstructed access to remote regions. Pakistan possesses considerable potential for harnessing wind energy. The estimated potential for installable wind energy in Pakistan is 346,000 MW68. An exhaustive evaluation of wind energy across different regions of Pakistan is essential to underscore its potential for investment opportunities for both the public and private sectors. The overall assessment is projected by the three principal entities, including the Pakistan Meteorological Department (PMD), the National Renewable Energy Laboratory (NREL), and the Alternative Energy Development Board (AEDB). It has been reported that wind energy generation can meet national electricity demand. Wind energy utilization continues to progress, hindered by the scarcity of accessible resources and the requisite infrastructure within the nation. Pakistan possesses significant potential in exploiting wind energy resources. Reports indicate that Pakistan possesses an estimated potential of 346,000 MW for wind energy. The coastal regions of Gwadar present promising opportunities for the development of wind power initiatives. The implementation has the potential to meet the electricity requirements of the Gwadar region. Wind energy presents a sustainable and economically viable solution for our energy needs. Its availability can be significantly enhanced through the installation of wind turbines at various sites.
Significant challenges that will necessitate financial resources include the initial installation cost as well as transmission and distribution infrastructure. The pros and cons of generating power from wind turbines in Gwadar’s coastal regions are detailed in this paper. It offers a preliminary evaluation of the resource’s social, environmental, and economic viability in comparison to other renewable power options. The results will be useful for the government as they reevaluate their policies regarding traditional energy sources60. Since wind power doesn’t pollute the air, water, or land, it has a net beneficial effect on the environment. From a societal, political, and economic perspective, wind power is competitive with other renewable sources. In the majority of industrialized nations, this green energy source is financially viable. The environmental effects of wind power plants are minimal. Activities involving humans and industry are mostly unaffected. Some land-dwelling species may experience slight adverse effects. There have been attempts to lessen the severity of these unfavorable effects69.
Solar energy
Pakistan is a land of abundant sunlight, with three thousand hours of sunshine per year. This natural resource translates to an estimated electrical potential of solar power ranging from 1900 to 2200 MW. Table 3 showcases the operating solar power installations, a testament to the country’s progress in harnessing solar energy. The average yearly global solar irradiation is 3 hours, or 1.9 to 2.3 megawatt hours per square meter, further underlining the country’s promising solar energy potential70. The arid terrain of Gwadar benefits from an abundance of solar radiation. Its capacity for harnessing solar power is exceptionally high, exceeding 70 percent. This renewable resource presents a viable solution for fulfilling the energy demands of outlying settlements. As documented in reference71, the typical domestic energy consumption in Gwadar approximates 100 watts. The availability of sun exposure makes solar electricity a cost-beneficial option in the long run. The rural communities of Gwadar could profit economically, socially, and environmentally from switching to solar-generated power to meet their energy needs. Lower costs for electrical transfer and allocation can be achieved in the Gwadar area by meeting power needs with solar photovoltaic systems. Societal support for photovoltaic energy extends to moderately wealthy families in Gwadar since it meets the power needs of economically disadvantaged homes. In 2006, the government issued the National Sustainable Energy Directive, which outlined the ways in which renewable energy might be used. This regulatory framework emphasizes the nationwide integration of renewable energy resources72.
Solar energy is the process of absorbing the sun’s rays. It produces zero emissions of smoke, particles, or greenhouse gases (GHG). If we want to reduce emissions and promote green energy, solar power is a great example to follow. Solar energy generation technologies can satisfy the world’s commitments to climate change and warming. As shown in Table 4, it cuts emissions by 20,000 metric tons annually from fossil fuel combustion.
Solar power is more cost-effective to install than other energy sources, whether renewable or not. Fuels like gas, coal, and oil, which can be quite pricey, are unnecessary. Less money is needed to run solar power than conventional power plants. You won’t even notice the transportation cost compared to other commodities transported to factories. Solar panels have an expected lifetime of 25–30 years and can be extended to 40 years if needed73. Sunlight is abundant in Balochistan, which is a gift. The Gwadar region’s electrical requirement might be met by it. You can generate solar power with either an on-grid or off-grid setup. The extra demand can be met by on-grid solar facilities, which are linked to the national grid. Solar power’s main benefit is decentralization. Communities and societies at large can benefit from it because transmission and distribution networks are unnecessary. The additional infrastructure expense is so saved. Decentralization is still a feasible option despite its limitations and expensive initial installation costs. Local communities would benefit from the creation of jobs in this sector, which involves installing, maintaining, and monitoring panels. There will be less need for jobless benefits in that area if these positions are filled74.
Coal energy
Coal, in contrast to other renewable energy sources, is a very accessible energy source. Unlike solar power, which is affected by darkness, and hydropower by water constraints, this one is unaffected because it is available without delay. This energy source is currently being used extensively around the globe, including in Pakistan. This choice is the most economical when considering its accessibility and usability. Table 5 indicates an analysis of coal’s operational efficacy inside Pakistan’s energy infrastructure. Coal faces challenges in sustaining competitive equivalence with the economic affordability of current electricity sources. Conversely, in a comparative assessment against alternative renewable energy modalities, coal demonstrates a minimized cost structure. It warrants emphasis that the methodology of power generation predicated on coal combustion diverges fundamentally from those employed by other renewable energy paradigms. Specifically, coal combustion for energy release is characterized by substantial smoke emissions. This process inherently generates pollutants and smoke, which are environmental considerations. Conversely, coal byproducts can be repurposed for diverse industrial applications. Furthermore, the extraction of heat from coal is a comparatively straightforward procedure when juxtaposed with the intricacies of electricity generation from other energy sources75.
The energy sector relies on labor and so requires a larger workforce to function. It generates a huge amount of direct jobs, which is good for the community and society at large. The building industry generates a lot of new jobs, including drivers, cleaners, and loaders. Monetary incentives exist for technical engineering positions. Gwadar must attain electrical self-sufficiency to satisfy the energy requirements of its inhabitants. In the Makran region, domestic sources provide thirty megawatts (MW), with an additional eight and a half megawatts (MW) imported from the Gwadar Free Zone. The majority of this power is derived from imports from Iran. The anticipated rise in electricity demand in the Gwadar region to 778 MW by 2030 is significant76. Coal energy initiatives are set to be established in Gwadar, which is part of the China-Pakistan Economic Corridor. The newly established coal plant boasts a capacity of 300 MW. It would address the deficiencies in the Gwadar region. The generation of electricity through coal continues to pose a significant risk to our climate. Coal power stations have many major negative effects on the environment. They release many contaminants into the air, including sulfur dioxide, nitrogen oxides, and carbon oxides. Plants also release particulate matter into the air. Plants’ hot water runoff is a major contributor to soil erosion and a threat to aquatic habitats. There are also a number of health effects. Coal is also a source of radiation, which can cause cancer. Along with mutations in cells, they are the primary cause of cancer. Those who reside near coal power plants are reportedly more likely to develop asthma, lung cancer, and a host of other health problems. The effects on people’s health are devastating. Liquid heat from coal-fired power stations is extremely harmful to plant life. The byproducts generated by coal power stations are having a detrimental impact on the natural environment. The mineral deposits that supply groundwater have also suffered damage as a result. The species possesses the capacity to disrupt the intricate equilibrium that nature has meticulously cultivated over time. It eradicates the indigenous species within the ecosystem. The existence of power plants exerts detrimental effects on local species. The extensive exploitation has significantly transformed the geographic distribution of the species77. The occurrence of acid rain, mainly linked to this, poses a significant threat to the community’s infrastructure. A notable correlation can be observed between emissions and detrimental impacts on human and environmental health. Evidence from scientific studies indicates that coal is the main factor behind the observed increase in temperature78. Innovative coal technology has been implemented to manage emissions effectively. Nonetheless, long-term ecological sustainability remains inadequate. The construction of coal power plants serves as a primary contributor to vibrations, noise, and both water and air pollution. Consequently, it lacks societal endorsement. It also disrupts the existence of the local populace. Its impurity distorts the sites of cultural legacy. The implementation of modern renewable energy policies for Pakistan puts political approval in danger. Furthermore, coal power plants in Gwadar are unfit for inhabited areas and humans. The extraction of coal will affect society, the economy, and the surroundings. Installation of clean plants helps to restore damage caused to the environment and waterways. Affected individuals can be moved to their local locations, and compensation can also be changed in some sort57. Coal clearly has both advantages and disadvantages. Modern construction has benefited greatly from the great contribution coal makes. Though it cannot be disregarded, technological development will help satisfy the demand for clean energy. Thus, it is necessary to implement the advanced stages to reduce coal’s negative effects on the environment and mankind. Areas located along coastlines are particularly well-suited for generating and using wind-based electricity. This is because these regions typically experience a high volume of readily available wind resources. Among environmentally friendly energy options, wind power stands out as a relatively economical choice. Wind power production frequently exceeds the operational limits of solar, tidal, and coal-based energy facilities. The China-Pakistan Economic Corridor (CPEC) reflects the strategic goals and aspirations of both the Chinese and Pakistani governments. It is anticipated to develop into a prominent commercial hub, largely due to the extensive deployment of wind energy infrastructure. Infrastructure construction has already commenced. The generation of wind energy in coastal locations has a less detrimental impact on the ecosystem. The location of wind turbines will not compromise the adjacent activities. It has the potential to have a somewhat detrimental effect on coastal regions. Wind turbines generate clean electricity while reducing pollution because they are eco-friendly. It doesn’t pollute the air, land, or water with harmful gases or carbon dioxide. It follows the rules set out by environmental policy. The nearby area’s air quality is improved. It will enhance matters pertaining to health. Positioned in the coastal regions is Gwadar. People are living in close quarters to the shore. They are used to human habitation. There are no societal barriers associated with wind electricity production. But it will meet their basic energy needs, which will improve the living conditions for everyone in the neighborhood. The establishment of numerous wind energy initiatives will assuredly enhance employment possibilities for the regional population. As a greater number of work positions become accessible, the general quality of life for the local inhabitants will be elevated. Furthermore, an expanded range of commercial exchange opportunities with residents will be provided.
We rigorously evaluate four energy sources. The output is written as Tidal energy \((\beth _1)\), Wind energy \((\beth _2)\), Solar energy \((\beth _3)\), and Coal energy \((\beth _4)\). These alternatives are assessed using four criteria. These are Impact on environment \((E_{1})\), Cost \((E_{2})\), Political acceptance \((E_{3})\), and Job creation and social acceptance \((E_{4})\) are the four characteristics that will be considered. All the criteria are benefit-type. The characteristics’ prioritization relation is defined as: \(E_{1}> E_{2}> E_{3} > E_{4}\). Decision makers utilized \(C_pLD_yFNs\) when creating the decision matrix. The weight vector of the attributes is denoted by \(w=(0.4,0.3,0.2,0.1)^{T}\).
Numerical example
Let \({\beth }={\{{\beth }_1,{\beth }_2,{\beth }_3,{\beth }_4}\}\) represent a set of alternatives, and \(E={\{E_1,E_2,E_3,E_4}\}\) represent a set of objectives. The decision-maker’s or expert’s evaluation values for each option would be determined using the interpreted criteria to create the decision matrix \(C_pLD_yFN's\) as \(\mathfrak {M={(d_{ij})_{m\times n}}}\). \(C= {\{s, \langle m(s)e^{i2\pi \theta (s)},n(s)e^{i2\pi \phi (s)}\rangle ,\langle \alpha (s)e^{i2\pi \delta (s)},\beta (s)e^{i2\pi \rho (s)}\rangle ,s \in W\}}\). \(C_yLD_yFNs\) represents all elements of the decision matrix. With the help of Equation 3, we can use \(C_pLD_yFN's\) as follows: \(C_{i} = \left\langle \left( m_{i}, \theta _{i} \right) , \left( n_{i}, \phi _{n_i} \right) \right\rangle\), This is for our convenience. Table 1 will display the evaluation values that the decision-makers or experts have assigned to each alternative.
Step 1: There is no need to standardize the data because every criterion falls within the benefit type. Therefore, we first use the DM technique, which includes the \(C_pLD_yDPWA\) and \(C_pLD_yFDPWG\) operators and reference point approach, to resolve the data shown in Table 6.
Step 2: The prioritized matrix for k = 2 is computed as shown in Table 7.
Based on Equation (22), we get \(\beth _1=(\langle (0.6458,0.7224),(0.0217,0.1417)\rangle ,\langle (0.3794,0.1487),(0.3877,0.5720)\rangle ),\) \(\beth _2=(\langle (0.6669,0.8184),(0.6349,0.6868)\rangle ,\langle (0.1554,0.1721),(0.4409,0.7339)\rangle ),\) \(\beth _3=(\langle (0.5268,0.2769),(0.4080,0.8055)\rangle ,\langle (0.0644,0.1515),(0.0207,0.1355)\rangle ),\) \(\beth _4=(\langle (0.9667,0.4558),(0.4439,0.0927)\rangle ,\langle (0.6086,0.5952),(0.1156,0.2868)\rangle ).\)
Step 3: The following are the score values of the combined result generated by \(C_pLD_yFDPWA\) using Equation (23): \({\ddot{S}}(\beth _1)=0.5967, {\ddot{S}}(\beth _2)=0.4145, {\ddot{S}}(\beth _3)=0.4562, {\ddot{S}}(\beth _4)=0.7109.\)
Equation (24) can be used to normalisze crisp values of ratio system \(\overline{({\ddot{S}}_{i})}^{RS}=(0.8394,0.5831,0.6417,1)\).
The ranking outcome of the ratio system for MULTIMOORA \(C_pLD_yFDPWA\) as follows: \(\beth _4> \beth _1> \beth _3 > \beth _2.\)
Step 4: Equations (25) and (26) are used to determine the reference point and calculate the Chebyshev distance (shown in Table 8) for each alternative respectively.
\(\wp _{1}=(\langle (0.84,0.90),(0.01,0.04)\rangle ,\langle (0.43,0.71),(0.01,0.07)\rangle ),\) \(\wp _{2}=(\langle (0.99,0.72),(0.26,0.73)\rangle ,\langle (0.84,0.51),(0.07,0.23)\rangle ),\) \(\wp _{3}=(\langle (0.90,0.94),(0.32,0.41)\rangle ,\langle (0.24,0.84),(0.03,0.15)\rangle ),\) \(\wp _{4}=(\langle (0.55,0.73),(0.14,0.18)\rangle ,\langle (0.55,0.36),(0.05,0.39)\rangle ).\)
The maximum Chebyshev distance for each alternative is determined as: \({({\ddot{S}}_{i})}^{RP}=(0.3062,0.4475,0.3888,0.2050).\)
Step 5: Equation (28) can be used to normalisze crisp values of reference point: \(\overline{({\ddot{S}}_{i})}^{RP}=(0.6702,0.4585,0.5272,1).\)
The ranking outcome of the refrence point for MULTIMOORA \(C_pLD_yFS\) as follows: \(\beth _4> \beth _1> \beth _3 > \beth _2.\)
Step 6: Based on Equation (29), we get
Step 7: The following are the score values of the combined result generated by \(C_pLD_yFDPWG\) using Equation (30): \({\ddot{S}}(\beth _1)=0.5450 , {\ddot{S}}(\beth _2)=0.4564 , {\ddot{S}}(\beth _3)=0.5051 , {\ddot{S}}(\beth _4)=0.6892.\)
According to Equation (31), the above value are normalized as follows: \(\overline{({\ddot{S}}_{i})}^{MF}=(0.7908,0.6692,0.7329,1).\)
The ranking outcome of the multiplicative function is as follows: \(\beth _4> \beth _1> \beth _3 > \beth _2\).
Step 8: Finally, based on Table 9, Equation (32) presents the final ranking of alternatives: \(\beth _4> \beth _1> \beth _3 > \beth _2\).
Sensitivity and comparative analysis
In order to assess the stability and reliability of the proposed approach, a sensitivity and comparative analysis is conducted, juxtaposing its results with those obtained from earlier methodologies.
Sensitivity analysis
This subsection analyzes the ranking outcomes of the developed model based on sensitivity. The sensitivity analysis aims to examine the impact of various configurations for the different values of parameter k in Example 5.2. Tables 10 and 11 examines the impact of a variation in the value of k. Tables 10 and 11 demonstrates that the best alternative and the worst element are identical in every instance of \(C_pLD_yFDPWA\) and \(C_pLD_yFDPWG\), respectively.
Example 5.2 compares the exceptional case with k = 2. We chose this unique scenario to compare with other methods and more accurately represent uncertainties. Let us analyze in depth the impact of the k parameter on the suggested methodologies. Tables 10 and 11 present the rankings derived from the parameter effect results for MULTIMOORA \(C_pLD_yFS\). In the reference point method, the absence of an aggregation approach results in variations only based on the ideal point and Hamming distance. Therefore, we do not discuss reference point approaches in this section. Tables 10 and 11 illustrate the impact of the k parameter. The most appropriate choice remains unchanged based on the k values, as indicated by Tables 10 and 11. Applying prioritized aggregation with Dombi TN and TCN yielded the best candidate, \(\beth _4\). As we can see, the candidates’ ordering is the same for k = 1, 2, 3, 5, 7, 10, 15, 20 values, and \(\beth _4\) is the best candidate regardless of the aggregating operators applied. Dombi TN offers us a trustworthy and versatile rating based on Tables 10 and 11.
Comparative analysis
This section aims to evaluate the comparability of the proposed model against established methodologies. This comparison highlights the effectiveness of the method and the logical reasoning presented in this paper. We compare the new method to others that have already been used, such as \(q-LDFWA\) and \(q-LDFWG\)79, LDFWA80 and LDFWG80, \(C_pLD_yFS\), the Cosine similarity measures created by Huseyin Kamaci20 and \(C_pLD_yFS\) in MCDM as suggested by Zia et al.21. Tables 12 present the rankings of the options derived from the established methodology and the present techniques.
The information in Table 12 shows that there are some variations in the alternatives’ rating order. But it’s important to remember that \({\beth }_4\) is the best choice. Therefore, as the previous discussion has shown, the outcomes of the intended approach are highly reliable.
In Table 12 for LDFS shows the tiny transformation, as \({\beth }_1\) rather than \({\beth }_4\) is the ideal object. The change is due to the fact that complex portions of life are excluded from the data, which causes a little fluctuation after the data is aggregated. For this reason, \({\beth }_1\), not \({\beth }_2\), is the ideal option. It’s crucial to remember that the best solutions do not change when we compare the outcomes from \(C_pLD_yFS\), giving our conclusions a more comprehensive context.
Riaz and Hashmi19 developed LDFS, which demonstrate better flexibility and reliability compared to concepts such as IFSs, PyFSs, and q-ROFSs. This is due to the incorporation of reference or control elements that possess membership and non-membership functions. The LDFS offers decision makers the freedom for expressing their opinions freely within the interval [0, 1], while the \(C_pLD_yFS\) developed by Huseyin Kamacı20 and utilized by Zia et al.21 allows decision makers to express their opinions with greater precision, as its range is the unit circle in the complex plane. In practical situations, memberships, non-memberships, and reference values often assume complex forms, which are effectively managed by \(C_pLD_yFS\). \(C_pLD_yFS\) is generally an extension of LDFS characterized by a zero-complex component, permitting it to address real-world issues with higher levels of precision and accuracy.
The suggested methodology surpasses Riaz and Hashmi’s LDFS19, as it can handle complex membership, non-membership, and reference parameters. Our methodology successfully handles these characteristics, which are commonly encountered in decision-making circumstances. It is superior to the Kamacai technique20, which does not employ criteria weighting and relies on algebraic aggregation. As a result, bias may exist. Furthermore, our suggested approach prevails over that of Zia et al.21, Faisal et al.79, and others. Zia21 uses algebraic aggregation operators of \(C_pLD_yFS\) with known weight vectors, whereas Faisal et al.79 use Dombi aggregation operators with known weight vectors and zero complex part.
As a result, the originality of their model is less than that of our suggested model, which utilizes Dombi aggregation operations that are several generations old. Furthermore, we employed the MULTIMOORA technique, an innovative way in our approach, to determine the ranking of the alternatives. This method reduced the probability of bias and provided a precise ranking. This marked the first-ever application of the MULTIMOORA method for \(C_pLD_yFS\). In contrast to alternative methodologies, our approach yields an unbiased ranking, demonstrating the originality of our model.
\(\beth _4\) represents the optimal candidate within the proposed MULTIMOORA framework. The proposed methods incorporate aggregation techniques and operators combined with the MULTIMOORA method. Additionally, the other existing operators do not take into consideration that particular attributes may hold a higher priority than others in specific situations.
The suggested model has various benefits, some of which are detailed below.
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By integrating Dombi-prioritized aggregation operators into the suggested structure, we can manage preferences among criteria and give more weight to the most important ones when making decisions. When the criteria are not all equally significant, this prioritization becomes crucial because it allows us to mold the decision-making process based on our priorities.
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Unlike traditional ranking approaches, the extended MULTIMOORA method we applied reduces the likelihood of subjective bias. It does so by balancing three decision perspectives—ratio system, reference point, and full multiplicative form—thereby ensuring a more holistic and accurate ranking of alternatives.
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The proposed framework uses complex linear Diophantine equations as constraints. When decision variables do not have to represent real-valued MD and NMD, these constraints, complex-based MD and NMD, can be quite helpful. For situations involving complex issues in the real world, this quality is crucial.
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Dombi operators provide significant flexibility in handling the aggregation operators of different fuzzy sets, serving as a powerful instrument for aggregating disparate information sources with various degrees of certainty and significance. Complex decision-making environments require this versatility to integrate various information types and degrees of uncertainty into a cohesive decision.
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To deal with uncertainty in MCDM problems, a hybrid \(C_pLD_yF\) model was created. This model combines the MULTIMOORA method with Dombi Prioritized AOs. Other than a few other models, the hybrid model has a wider range of applications and can better handle the complexity and uncertainty of the decision-making problem.
Concluding remarks
In this study, we introduced a novel decision-making framework by integrating \(C_pLD_yF\) operators with the MULTIMOORA method to enhance the accuracy and reliability of MCDM under uncertainty. The proposed approach extends traditional fuzzy decision models by incorporating Dombi-prioritized AOs, allowing for a more nuanced representation of complex decision environments. By developing and systematically analyzing new \(C_pLD_yFS\) AOs—such as \(C_pLD_yFDPA\), \(C_pLD_yFDPG\), \(C_pLD_yFDPWA\), and \(C_pLD_yFDPWG\)—this study contributes to advancing MCDM methodologies. The integration of these operators within the MULTIMOORA framework facilitates an improved decision-making process by effectively handling interdependencies among criteria, prioritization of attributes, and uncertain data. To validate the practical applicability of the proposed framework, we conducted a case study focusing on sustainable energy planning for Gwadar, Pakistan. The results demonstrate that the \(C_pLD_yF\)-based MULTIMOORA approach provides a more comprehensive, precise, and adaptable evaluation of sustainable energy alternatives compared to existing methodologies. Additionally, sensitivity and comparative analyses confirm the robustness and reliability of the developed framework.
However, the proposed model possesses certain limitations that warrant consideration. First, the computational complexity of the method may increase significantly due to the integration of complex fuzzy operators and constraints. Second, the criteria weights are assumed to be known in advance, which may not always align with real-world decision-making contexts. Third, the case study employed a limited number of criteria, which might not capture the full scope of influencing factors in large-scale decision problems.
Future research can address these limitations in several ways. One direction is to integrate subjective or objective weight derivation models into the proposed framework. Additionally, the number of criteria can be expanded to cover broader aspects of decision problems. The model can also be adapted to other domains beyond energy planning, enhancing its versatility.
Data availability
All the data generated or analysed during this study are included in this published article.
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Mustafa, A.W., Bashir, Z., Ali, J. et al. Complex linear Diophantine fuzzy Dombi prioritized operators-based MULTIMOORA approach with applications to sustainable energy planning. Sci Rep 15, 29051 (2025). https://doi.org/10.1038/s41598-025-05710-w
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DOI: https://doi.org/10.1038/s41598-025-05710-w
