A comprehensive review of the existing literature has been conducted to identify and define the problem statement surrounding control strategies in aquaponic systems. Goddek et al. (2015) highlight the various challenges involved in implementing effective control strategies within aquaponic systems, emphasizing the unique complexities of these environments. Similarly, Yep and Zheng (2019) provide a review that addresses the challenges faced when applying control systems in aquaponics, noting the intricate nature of the system and the difficulties in achieving effective regulation. Okomoda et al. (2023) offer an in-depth discussion of the challenges associated with the adoption of aquaponics, focusing on the inherent complexities such as nonlinear behaviour, the Multi-Input Multi-Output (MIMO) characteristics of the system, and other system-specific intricacies. Additionally, Ng and Mahkeswaran (2024) examine the technological barriers in aquaponic systems, which further complicate the implementation of control strategies, thus limiting their potential for optimization. Through this extensive analysis, it became evident that, given the complexities and challenges identified in the literature, there is a pressing need to explore and identify suitable control approaches that can address the unique needs of large-scale aquaponic systems. The ultimate objective of the study is to enhance the production efficiency, profitability, and sustainability of aquaponic systems—goals that cannot be achieved without the effective implementation of control strategies. Therefore, addressing this issue and selecting the most appropriate control strategy for large-scale applications is the central focus of our research.

In recent years, there has been a notable increase in proposals for control mechanisms designed specifically for aquaponic systems. These mechanisms employ a variety of strategies, including ON-OFF control, Rule-based control, Open-loop control, PID (Proportional Integral Derivative), MPC (Model Predictive Control), and PLC (Programmable logic control), each offering unique advantages and applications. However, a significant portion of these proposals focuses on small-scale aquaponic setups, such as kitchen gardens, indoor aquaponic farming, and balcony gardening. In response to the growing interest in small-scale aquaponics, researchers have begun integrating Artificial Intelligence (AI) and Internet of Things (IoT) technologies with traditional ON-OFF control methods. For instance, Vernandhes et al. (2017) introduced a smart aquaponics monitoring and control system utilizing a sensor network for water quality parameters, managed through a microcontroller. Similarly, Dutta et al. (2018) and Zamora-Izquierdo et al. (2019) integrated IoT technology and sensor networks to regulate water quality parameters. Khaoula et al. (2021) implemented an IoT-based solution for monitoring and controlling water quality and environmental parameters using sensors for water level, temperature, and CO₂, along with actuators. Additionally, Channa et al. (2024) explored the integration of Artificial Intelligence and IoT in a smart aquaponics system to monitor and control essential parameters using Rule-based Control approach. While, much of the focus remains on control mechanisms for small-scale aquaponic systems, there is also emerging interest in applying machine learning techniques for aquaponic setups. Debroy and Seban (2022a, 2022b) proposed prediction methods for fish weight estimation using Artificial Neural Network (ANN) and its hybrid with fuzzy logic (ANFIS), as well as ANN models for predicting tomato biomass in aquaponic systems, respectively. Eneh et al. (2023) presented a yield prediction method for aquaponic systems employing various machine learning algorithms. Furthermore, Rajendiran and Rethnaraj (2024) discussed a study on IoT-integrated Machine Learning-based Indoor Aquaponics farming.

Recent studies have focused on employing PID (Proportional Integral Derivative) control strategies in aquaponics to efficiently regulate specific water quality parameters. For example, Alipon et al. (2021) introduced a design for an automated fertigation system that monitors photoperiod and nutrient consumption, employing a Proportional-Integral-Derivative (PID) system. Li et al. (2022) applied PID control to regulate dissolved oxygen concentration in aquaponic recirculating water. Kim et al. (2023) presented a Dissolved Oxygen (DO) management system for aquaponic systems using PI and PID controllers. Kannabiran et al. (2024) suggested that a PI controller demonstrated robustness in maintaining pH levels within the desired range under varying operating conditions. Wei et al. (2019) proposed a laboratory-based aquaponic system utilizing PLC (Programmable Logic Controller) and LabVIEW. Another study by Selvalakshmi et al. (2023) developed a PLC-based approach for a small-scale aquaponic system. Chahid et al. (2021) conducted a comparative analysis of four Model Predictive Control (MPC) strategies for fish growth reference tracking using a representative bioenergetic growth model in precision aquaculture. Ding et al. (2018) explored the opportunities and challenges associated with implementing MPC in aquaponic systems. Lin et al. (2020) proposes the use of open-loop control and Model Predictive Control (MPC) for managing greenhouse parameters. Similarly, Debroy et al. (2024a) present an MPC-based strategy for controlling aquaponic greenhouse parameters, and they also provide a comparison of this approach with a traditional PI controller. Another publication by Debroy et al. (2024c) presents a similar MPC-based strategy, but for controlling water quality parameters in aquaponic systems, and again, they compare this method with a conventional PI controller. In a recent study by Debroy et al. (2024b), the authors employed Multi-Criteria Decision Making (MCDM) techniques to identify the most suitable water quality parameter for an aquaponic system. Their research aimed to evaluate and prioritize various water quality indicators by considering multiple criteria, ultimately selecting the one most critical for ensuring the optimal functioning and sustainability of aquaponics. The application of MCDM in this context provides a structured approach to decision-making, helping to balance and assess the different factors that influence water quality in aquaponic environments.

The existing literature clearly demonstrates the profound impact of integrating aquaculture and hydroponics within aquaponics systems on the reliance of fish growth and yield on water quality parameters. Maintaining these parameters at optimal levels is imperative for the success and productivity of aquaponic systems. However, previous research has predominantly focused on monitoring and controlling small-scale smart aquaponics setups, often employing various control approaches. While some studies have ventured into incorporating IoT-based machine learning methods to predict yields and manage system parameters, the diverse array of control approaches utilized poses a significant challenge in determining the most suitable one for broader application in aquaponic systems. This variability in control strategies further complicates the selection process, particularly when considering implementation in larger-scale aquaponic operations. Moreover, the predominant emphasis on small-scale setups in past research exacerbates the difficulty in identifying the ideal control approach for industrial-scale aquaponics. The lack of specific research tailored to the unique requirements and complexities of large-scale systems underscores a notable gap in the current literature. Consequently, a critical research question arises: Which control strategy is most pertinent for implementation in large-scale aquaponic systems?

Addressing this research gap is paramount for advancing our understanding of optimal control strategies tailored to industrial-scale aquaponics. Such advancements are vital for improving the sustainability, productivity, and viability of large-scale aquaponic operations in real-world applications. Therefore, bridging this gap is essential for driving progress in the field and ensuring the successful integration of aquaponics into broader agricultural practices.

Aquaponics stands as a sustainable farming method, marrying aquaculture with hydroponics to create a harmonious ecosystem. In this system, fish waste provides nutrients for plants, while plants filter and purify the water for the fish. Given the intricate balance required, understanding water quality parameters is paramount, as they directly impact the health and growth of both fish and plants. However, these parameters are susceptible to fluctuations due to external factors and are interrelated, necessitating a careful balance. To tackle these challenges, a robust control strategy is imperative. Moreover, with aquaponics poised for larger-scale adoption, selecting the optimal control approach is crucial, given its promising future prospects. Therefore, the primary objective of this study is to determine the most effective control strategy tailored specifically for industrial-scale aquaponic systems. The overarching goal is to optimize production, enhance system stability, and maximize profitability within these large-scale operations. By identifying the most effective control strategy tailored to the unique requirements of industrial-scale aquaponic systems, this study aims to drive advancements in aquaponic technology and contribute to the sustainable development of agriculture. Ultimately, the findings of this research have the potential to significantly impact the future of food production by enabling the scalable and profitable implementation of aquaponic systems on a large scale. Advancing aquaponic systems through effective control measures holds immense promise in addressing the looming challenges of global food demands and hunger. It can significantly contribute to enhancing food security worldwide by providing a sustainable and efficient method of agricultural production. Thus, the elaboration of this study underscores its potential to revolutionize food production practices and address critical global challenges.

This study aims to introduce a novel hybrid Multiple Criteria Decision Making (MCDM) model tailored to identify the most effective control approach for large-scale aquaponic systems. The proposed approach, named OPA-IF-Neutrosophic-TOPSIS under SVNS Environment, integrates various decision-making techniques into a cohesive framework—a concept not yet explored in existing literature. In this hybrid model, criteria weights are determined using the Intuitionistic Fuzzy ordinal priority Approach (OPA-IF). Subsequently, the ranking of alternatives is refined through the use of TOPSIS within a Neutrosophic fuzzy environment. This comprehensive methodology provides a fresh perspective on optimizing decision-making processes in aquaponic systems by synergistically leveraging diverse analytical tools.

In 2020, Ataei et al. introduced the MCDM method known as OPA (Ordinal Priority Approach), representing a departure from traditional pairwise comparisons. Building upon this, Mahmoudi et al. (2022) developed OPA-F (Fuzzy ordinal priority Approach) in 2022. This approach eliminates the need for pairwise comparisons, automatically estimates attribute weights, and integrates observations without averaging them. However, traditional fuzzy sets face challenges in precisely determining membership mappings, particularly under specific circumstances (Chiao, 2016). To address this limitation, intuitionistic fuzzy sets (IFSs) were introduced. IFSs specify both membership and non-membership degrees of elements within a fuzzy set, thereby accommodating ambiguity levels (Jin et al., 2016; Wan et al., 2016). In 2024, Majumder and Salomon introduced the Intuitionistic Fuzzy Ordinal Priority Approach (OPA-IF) to better handle uncertainty in decision-making. This method extends the traditional OPA and OPA-F by using triangular intuitionistic fuzzy sets (TIFS) instead of standard fuzzy sets, addressing challenges in determining exact membership values. Unlike OPA-F, OPA-IF relies on ranks rather than weights for criteria, offering a more flexible approach. The study combines OPA-IF with the OPA-F method to improve Multi-Criteria Decision-Making (MCDM) in aquaponic systems, effectively managing ambiguity and optimizing decision outcomes.

In this study, the TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution) method, as delineated in the primary reference (Hwang et al., 1981), is utilized to ascertain the optimal option. The process of assigning criteria weights is guided by TrF-FOCUM (Majumder, 2023), facilitating this determination. The rationale for opting for Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) stems from its user-friendly interface and its adaptability to meet both qualitative and quantitative requirements. By evaluating each option based on its best and worst outcomes, TOPSIS contributes to a more robust ranking outcome. Furthermore, it can incorporate cost-benefit considerations, rendering it suitable for scenarios where the interaction between performance and cost is significant. While previous research has extensively delved into fuzzy and intuitionistic fuzzy Multiple Criteria Decision Making (MCDM) problems, the growing recognition of ambiguity’s role in MCDM complexities highlights the need to incorporate Neutrosophic sets. Neutrosophic sets are adept at addressing environments characterized by uncertainty, indeterminacy, and inconsistency within the MCDM methodology. Despite the attention dedicated to challenges posed by fuzzy and intuitionistic fuzzy MCDM, integrating indeterminacy into the realm of MCDM complexities is deemed crucial. In 2023, Neutrosophic-TOPSIS was developed by Pramanik et al. (2023) to determine alternative rankings, marking a significant advancement in tackling the intricacies of decision-making under uncertainty.

The hybrid approach that integrates the Neutrosophic-TOPSIS strategy for assessing alternative levels within a Single Valued Neutrosophic (SVN) framework with the Intuitionistic Fuzzy-Ordinary priority approach (OPA-IF) for evaluating criteria levels offers numerous advantages: I.

This method adeptly manages uncertainty within SVN contexts by leveraging both OPA-IF and Neutrosophic logic. Neutrosophic logic addresses uncertainty at the alternative level, while OPA-IF tackles uncertainty at the criteria level, ensuring comprehensive treatment of ambiguity throughout decision-making.

Fuzzy Sets (FS), as introduced by Zadeh (1965), along with Intuitionistic Fuzzy Sets (IFS), pioneered by Atanassov and Stoeva (1986), or their extensions, are commonly employed to manage information characterized by incompleteness and imprecision. Mardani et al. (2015) offer an outline of various fuzzy Multiple Criteria Decision Making (MCDM) methodologies. Within this framework, several fundamental notions are utilized in constructing OPA-IF. Definition 1.

Assuming Z ≠ { } is a set, the intuitionistic fuzzy set in Z has the property Y given by, Y ˜ = { ⟨ y , α Y ¯ ( y ) , β Y ¯ ( y ) ⟩ ; y ∈ Y } as long as, α Y ¯ : Z → { 0 , 1 } ∪ ( 0 , 1 ) and β Y ¯ : Z → { 0 , 1 } ∪ ( 0 , 1 ) meet the condition α Y ¯ ( y ) + β Y ¯ ( y ) ∈ { 0 , 1 } ∪ ( 0 , 1 ).

Smarandache (1998) laid the foundation for Neutrosophic Sets in 1998, which was later built upon by Wang et al. (2010) with the introduction of Single-Valued Neutrosophic Sets (SVNS). This concept aimed to address situations marked by uncertainty and incomplete data.

The following definition outlines an SVNS Θ defined over a specified set G: Θ = { ( n , P m ( n ) , Q m ( n ) , S m ( n ) ) : n ∈ G } , where P m : R → { 0 , 1 } ∪ ( 0 , 1 ), Q m : R → { 0 , 1 } ∪ ( 0 , 1 ), S m : R → { 0 , 1 } ∪ ( 0 , 1 ) and so 0 ⩽ P m ( n ) + Q m ( n ) + S m ( n ) ⩽ 3. If an SVNS Θ over a given set G, we refer to the triplet ( P m ( n ) , Q m ( n ) , S m ( n ) ) as a Single-Valued Neutrosophic Number (SVNN).

Mandal and Basu (2019) proposed a new scoring function designed to tackle Multiple Attribute Decision Making (MADM) challenges within the SVNS framework. The scoring process involves the following steps: (i)

Consider a three-dimensional space with the origin represented as Γ. Within this space, let denote a specific point Π = ( i θ , j θ , k θ ), referred to as an SVNN. Perform a translation of this point into Π to arrive at Δ = ( i ϖ , j ϖ , k ϖ ). Here i ϖ = i θ + ζ , j ϖ = j θ + ζ , k ϖ = k θ + ζ , where ζ > 0, each representing k ϖ , a real number that remains distinct and unchanging throughout the specific problem, play a crucial role. Now, let’s consider another point, Δ / = ( i ϖ , − j ϖ , − k ϖ ), resulting from reflecting Λ = ( i ϖ , j ϖ , k ϖ ) across the x-axis, acting as a mirror.

The process of obtaining attribute weights in the OPA-IF model involves solving the linear optimization model (1) (refer to equation (1)) for attributes in the following way:

Step 1: Let E = { λ e : e = 1 ( 1 ) f } be the set of experts and C = { ϕ a : a = 1 ( 1 ) b } be the set of Attributes. The profit function is denoted by ζ ˜, also the decision variable δ ˜ e a r denote fuzzy weight of ath. Attributes by eth expert at rth rank. ℜ ˜ e a r denote the linguistic-based measures of significance of ath Attributes by eth expert at rth rank from the Table 1. Equation (1) presents the mathematical model in a linear form. (1) Max ζ ˜ Subject to ℜ ˜ e a r ( δ ˜ e a r − δ ˜ e a r + 1 ) ⩾ ζ , ∀ e , a , γ Subject to ℜ ˜ e a r δ ˜ e a ρ ⩾ ξ ˜ , ∀ e , a Subject to ∑ e = 1 f ∑ a = 1 ρ δ ˜ e a = ( 1 , 1 , 1 ; 1 , 1 , 1 ) , Subject to δ e a / v ⩽ δ e a v ⩽ δ e a u = δ e a / u ⩽ δ e a t ⩽ δ e a / t , ∀ e , a Subject to δ e a / v ⩾ 0 , ∀ e , a

Step 2: Once the optimization problem outlined in equation (1) has been addressed, the ranking can be computed using an appropriate defuzzification formula as described in equation (2): (2) W a = [ ( δ a v + 2 δ a u + δ a t ) + ( δ a / v + 2 δ a / u + δ a / t ) 8 ] , ∀ a , where ( δ a v , δ a u , δ a t ; δ a / v , δ a / u , δ a / t ) represents the optimal fuzzy weight of ath Attributes.

Consider the set of alternative B = { ε d : d = 1 ( 1 ) λ }, d ⩾ 1 and C = { ϕ a : a = 1 ( 1 ) b }, a ⩾ 2 be the set of attributes with weights w ( α a ∗ ), a = 1 ( 1 ) b, respectively.

Decision-makers assign ratings to the ε d , d = 1 ( 1 ) λ alternatives based on the attributes ϕ a , a = 1 ( 1 ) b, which are represented using an SVNN. Let’s assume the rating for the ath attribute concerning the dth alternative is presented as follows: ε d ∗ = ( κ a , O ε κ ( ϕ a ) , Q ε κ ( ϕ a ) , S ε κ ( ϕ a ) ) , d = 1 ( 1 ) λ , where 0 ⩽ O ε κ ( Z φ ) + Q ε κ ( Z ϕ ) + S ε κ ( Z φ ) ⩽ 3. Here, ( O d a , Q d a , S d a ) is denoted as an SVNN. ε d a ∗ , ( d = 1 ( 1 ) λ and a = 1 ( 1 ) b), where a represents the number of attributes and d represents the number of alternatives. The decision matrix is determined based on the ratings as, Φ ∗ = [ ε d a ∗ ] λ × δ .

The TOPSIS method is encapsulated in the following manner:

Step 1: The score-matrix Φ = [ ε d a ] λ × δ , ( d = 1 ( 1 ) λ and a = 1 ( 1 ) b)is acquired from the decision matrix Φ ∗ = [ ε d a ∗ ] λ × δ utilizing the following described in preliminary section: i.e. ε d a = L 1 ( [ ε d a ∗ ] λ × δ ).

Step 2: Determination of normalized decision matrix F = [ τ d a ] λ × δ , where, (3) τ d a = ε d a ∑ a = 1 b ε d a , d = 1 ( 1 ) λ .

Step 3: Calculation of the weighted normalized decision matrix Ψ = [ υ d a ] λ × δ , where, υ d a = w ( α ˜ a ∗ ) λ d a , d = 1 ( 1 ) λ and a = 1 ( 1 ) b.

Step 4: Determination of the Neutrosophic Positive Ideal Solution (NPIS) and Neutrosophic Negative Ideal Solution (NNIS), denoted by μ + and μ − , respectively, μ + = { υ 1 + , υ 2 + , … , υ n + } , where υ θ + = max θ υ d θ , θ = 1 ( 1 ) n , μ − = { υ 1 − , υ 2 − , … , υ n − } , where υ θ − = max θ υ d θ , θ = 1 ( 1 ) n .

Step 5: Computation of the distance of each alternative from both the NPIS and NNIS using the equations (4) and (5) provided below: (4) ∂ d + = ∑ θ = 1 δ ( υ d θ − μ + ) 2 , d = 1 ( 1 ) λ , (5) ∂ d − = ∑ θ = 1 δ ( υ d θ − μ − ) 2 , d = 1 ( 1 ) λ .

Step 6: Evaluation of the performance score for each alternative using the equation (6): (6) ℘ d = ∂ d − / ( ∂ d + + ℏ d − ) , d = 1 ( 1 ) λ .

Step 7: Arrangement of the alternatives based on their performance scores, with the alternative having the highest score receiving the top ranking, and the one with the lowest score being allocated the lowest ranking.

The objective of the upcoming section is to assess the priority values (PV) of both criteria and alternatives. This phase entails three key components: identifying factors, employing OPA-IF, and utilizing the Neutrosophic-TOPSIS strategy within the SVNS environment. The decision hierarchy for the matter is illustrated in Fig. 3.

Step 1: Selection of Factors:

A comprehensive examination of pertinent literature is conducted to select criteria and alternatives, followed by assembling a panel comprising specialists and stakeholders. Table 2 and 3 presents all the identified criteria and alternatives being investigated.

Step 2: Application of OPA-IF (Model-I):

In the present study, opinions from three experts were taken. Based on these expert opinions, Capability to Handle MIMO Systems ( ϕ 3 ) and Capability to Handle Complexity ( ϕ 2 ) were identified as being more highly and moderately highly responsible for efficiency, profitability, and sustainability of aquaponic systems, respectively; while Capability to Handle Large-Scale Systems ( ϕ 1 ) and Capability to Handle Non-Linear Systems ( ϕ 4 ) were categorized as having high and medium responsibility, respectively. It was clearly indicated by the experts that for Capability to Handle MIMO Systems and all other criteria, higher values are preferable. The fuzzy priority value of criteria can be determined using OPA-IF. Fuzzy weights for attributes can be estimated using equation (7). (7) Max ζ ˜ Subject to ( 6 , 7 , 8 ; 5 , 7 , 9 ) ( δ ˜ 3 − δ ˜ 1 ) ⩾ ξ ˜ Subject to ( 4 , 5 , 6 ; 3 , 5 , 7 ) ( δ ˜ 1 − δ ˜ 4 ) ⩾ ξ ˜ Subject to ( 2 , 3 , 4 ; 1 , 3 , 5 ) ( δ ˜ 4 − δ ˜ 2 ) ⩾ ξ ˜ Subject to ( 1 , 1 , 1 ; 1 , 1 , 1 ) δ ˜ 2 ⩾ ξ ˜ Subject to δ ˜ 1 + δ ˜ 2 + δ ˜ 3 + δ ˜ 4 = ( 1 , 1 , 1 ; 1 , 1 , 1 ) Subject to δ i / v ⩽ δ i v ⩽ δ i u = δ i / u ⩽ δ i t ⩽ δ i / t , ∀ i = 1 ( 1 ) 4 Subject to δ i / v ⩾ 0 , ∀ i = 1 ( 1 ) 4

Step 3: Neutrosophic-TOPSIS Strategy under SVNS Environment (Model II): The decision maker utilizes SVNNs to assess alternatives according to their attributes, leading to the generation of Decision Matrix as Matrix-1.

Matrix-1: Decision Matrix:

Matrix Φ is derived by shifting the values of each entry. Each entry in Matrix Φ is incremented by 0.01 across all components, thereby producing Matrix-2.

Matrix-2: Translation of Φ: The next step involves generating the score matrix using the score function. Matrix-3 represents the score matrix denoted as Φ ∗ . The score value is given by, L 1 ( 0.91 , 0.71 , 0.51 ) = 0.91 × 0.91 + 0.71 × ( − 0.71 ) + 0.51 × ( − 0.51 ) 0.91 2 + 0.71 2 + 0.51 2 0.91 2 + ( − 0.71 ) 2 + ( − 0.51 ) 2 = 0.04013.

Matrix-3: Score Matrix: The Normalized Decision Matrix is determined by using equation (4) on matrix Φ ∗ . As shown in Matrix-4, the Normalized Decision Matrix is denoted by F. τ 11 = 0.04013 0.04013 2 + ( − 0.007819 ) 2 + 0.018551 2 + 0.100399 2 + 0.312247 2 + 0.096882 2 = 0.116342.

Matrix-4: Decision Matrix with Normalization:

Once the criteria crisp weights are determined by solving the equation (7), the weighted normalized decision matrix is computed. This process entails multiplying each criteria weight by the corresponding element in its respective row of Matrix. Denoted as matrix Ψ, it is illustrated by Matrix-5.

Matrix-5: Weighted Normalized Decision Matrix: Subsequently, ascertain NPIS and NNIS using the formulas υ θ + = max θ υ d θ , θ = 1 ( 1 ) 6 and υ θ − = min θ υ d θ , θ = 1 ( 1 ) 6 values, respectively. υ 1 − = min { 0.032543316 , − 0.006341469 , 0.015044118 , 0.081417212 , 0.253212567 , 0.078565011 } = − 0.006341469. So, μ + = { υ 1 + , υ 2 + , υ 3 + , υ 4 + } = { 0.253212567 , 0.136880078 , 0.194153626 , 0.12406148 } , μ − = { υ 1 − , υ 2 − , υ 3 − , υ 4 − } = { − 0.006341469 , − 0.03340555 , − 0.045233224 , μ − = 0.044424922 } , τ 1 + = max { 0.032543316 , − 0.006341469 , 0.015044118 , 0.081417212 , 0.253212567 , 0.078565011 } τ 1 + = 0.253212567. Next, the distances between each alternative using the NPIS and NNIS are computed using the equations (4) and (5), respectively. Table 4 displays the distances between alternatives calculated using NPIS and NNIS. ∂ 1 + = ( υ 11 − μ 1 + ) 2 + ( υ 12 − μ 2 + ) 2 + ( υ 13 − μ 3 + ) 2 + ( υ 14 − μ 4 + ) 2 = ( 0.032543316 − 0.253212567 ) 2 + ( 0.018933394 − 0.136880078 ) 2 + ( 0.034162112 − 0.194153626 ) 2 + ( 0.044424922 − 0.12406148 ) 2 = 0.30748269 , ∂ 1 − = ( υ 11 − μ 1 − ) 2 + ( υ 12 − μ 2 − ) 2 + ( υ 13 − μ 3 − ) 2 + ( υ 14 − μ 4 − ) 2 = ( 0.032543316 − ( − 0.006341469 ) ) 2 + ( 0.018933394 − ( − 0.03340555 ) ) 2 + ( 0.034162112 − ( − 0.045233224 ) ) 2 + ( 0.044424922 − 0.044424922 ) 2 = 0.102737582.

The performance score of each alternative is computed using equation (6). Table 5 presents the performance scores for all alternatives. Arrange the alternatives in ascending order according to their performance scores and assign ranks accordingly. ℘ 1 = ∂ 1 − / ( ∂ 1 + + ∂ 1 − ) = 0.102737582 / ( 0.30748269 + 0.102737582 ) = 0.250445.

Experts have been consulted to identify the vectors appropriate for utilization in the BWM-Neutrosophic-TOPSIS Strategy under SVNS Environment, along with determining the most and least significant aspects. Through expert consensus, it has been established that ϕ 3 holds the highest significance, while ϕ 2 is deemed to be the least significant criterion. The best-to-others vector is presented in Table 6, and the worst-to-others vector is outlined in Table 7.

The nonlinear mathematical model (8) (refer to equation (8)) can also be used to determine the weight of each criterion. (8) min ℏ s.t. | D 3 D 1 − 2 | < ℏ , | D 3 D 2 − 4 | < ℏ , | D 3 D 4 − 3 | < ℏ , | D 1 D 2 − 4 | < ℏ , | D 1 D 2 − 3 | < ℏ , | D 3 D 2 − 4 | < ℏ , | D 4 D 2 − 2 | < ℏ , ∑ j = 1 4 D j = 1 , D j ⩾ 0 , for all j = 1 ( 1 ) 4 .

Utilizing the equation (3) on matrix Φ ∗ , the Normalized Decision Matrix can be determined. As shown in Matrix-6, the Normalized Decision Matrix decision matrix is denoted by F.

Matrix-6: Normalized Decision Matrix:

Once the criteria weights are determined by the equation (8), the subsequent step involves computing the weighted normalized decision matrix. This matrix is generated by multiplying each criterion weight by the corresponding element in the respective row of the associated matrix F. The resultant weighted normalized decision matrix (Matrix-7) is represented by matrix Ψ.

Matrix-7: Weighted Normalized Decision Matrix:

Next step is to identify NPIS and NNIS using the formulas υ θ + = max θ υ d θ , θ = 1 ( 1 ) 6 and υ θ − = min θ υ d θ , θ = 1 ( 1 ) 6 values, respectively.

So, μ + = { υ 1 + , υ 2 + , υ 3 + , υ 4 + } = { 0.234112482 , 0.072317184 , 0.315234293 , 0.89963446 } , μ − = { υ 1 − , υ 2 − , υ 3 − , υ 4 − } μ − = { − 0.005863125 , − 0.017648991 , − 0.073442168 , 0.032214827 } . The distance of each alternative from both NPIS and NNIS is calculated using equations (4) and (5), respectively. Table 8 presents the distances for each alternative’s weights from NPIS and NNIS.

The evaluation score for each option is calculated utilizing equation (6). Fig. 4 displays the performance scores for each alternative according to both the proposed method and the current method.

The rankings produced by the two methods can be compared using the Spearman correlation coefficient, which measures the linear relationship between two variables. This coefficient ranges from −1 to 1: −1 indicates no linear correlation, 0 signifies no linear correlation, and 1 indicates a perfect linear correlation. To evaluate the association between variables on interval scales, Pearson’s correlation coefficient (Sedgwick, 2012) can be employed, as demonstrated in equation (9): (9) χ ( ϖ , ι ) = cov ( ϖ , ι ) η ϖ η ι . σ, as well as ξ, is a covariant of cov ( σ , ξ ). SD is represented by σ, as well as ξ, in both η σ , as well as η ξ .

The Pearson correlation coefficient is important for assessing the absence of a perfect correlation between two variables when it deviates from a value of 1, as given in equation (10): (10) F 0 : − ∞ < χ ⩽ 0 , F 1 : 0 < χ < ∞ .

Pearson correlation coefficient alongside Student’s t-distribution with degrees of freedom λ – − 1, is presented by equation (11). (11) t = χ ( λ – − 2 1 − χ 2 ) 1 2 . The null hypothesis should be rejected if t (equation (11)) suppresses t α ( λ – − 2 ). The Pearson correlation coefficient χ falls within the range of λ –.

Both methods generate rankings, which are then assessed using the Spearman correlation coefficient. If the rankings are identical, resulting in a Spearman correlation coefficient of 1, further hypothesis testing is unnecessary. However, if the rankings differ, a hypothesis test can be conducted to validate the Spearman correlation coefficients, as described in equation (9). To compare the proposed approach with the BWM-TOPSIS strategy under the SVNS environment weights, the analysis will employ the Pearson correlation coefficient, as specified in Table 9. It’s worth noting that there is a noticeable correlation between the proposed PV models and the currently established PV models, as corroborated by Ataei et al. (2020). The analysis findings suggest that t = χ ( λ – − 2 1 − χ 2 ) 1 2 should outperform t 0.05 ( λ – − 2 ). A hypothesis test is carried out to confirm a positive correlation between the attributes of the existing and proposed methods.

The objective is to understand how changing the weight coefficients impacts various scenarios, each defined by a unique set of parameters. To achieve this, sensitivity analysis is utilized. This analytical method allows us to evaluate the primary criterion, as defined by equation (12), and to assess how sensitive the criterion PVs ( N ( K ˜ ℜ )) are to changes in these coefficients. Furthermore, we delve into the progression of the leading criterion to gain insights into how it evolves over time or under different conditions. (12) K ˜ ℜ ′ = K ˜ ℜ ( 1 − r K ˜ ε 1 − K ˜ ε ) , ℜ = 1 ( 1 ) 4 . When K ℜ ′ represents the initial value of the criterion, denoted K ˜ ℜ ( ℜ = 1 ( 1 ) Ξ ); K ˜ ε , it signifies the criterion’s starting point. r ℜ ∈ ( 0 , 1 ) ∪ { 0 , 1 }, on the other hand, represents the adjusted value.

For this study, 25 unique scenarios are generated using equation (12). In these scenarios, the variable r is capable of assuming random values between 0 and 1. Figures 5 and 6 depict the results of sensitivity analyses conducted separately for each alternative of ∂ d + and ∂ d − . Figure 7 presents an overview of the sensitivity analysis results. Notably, according to Fig. 7, the “Model Predictive Control (MPC) strategy” emerges as the most sensitive parameter across all cases.