Selecting Optimal Cybersecurity Measures for Power Systems Using Complex Intuitionistic Fuzzy Rough Frank Aggregation Operators

Technical Summary: Selecting Optimal Cybersecurity Measures for Power Systems Using Complex Intuitionistic Fuzzy Rough Frank Aggregation Operators

Problem Statement
Modern power systems have evolved into complex cyber-physical systems due to the integration of cloud computing, the Internet of Things (IoT), and 5G communications. While these technologies enhance efficiency and grid flexibility, they significantly expand the attack surface, making power grids vulnerable to sophisticated cyberattacks (e.g., Stuxnet, Ukraine power outages). Selecting the optimal cybersecurity (CS) measure for such critical infrastructure is a Multi-Criteria Decision-Making (MCDM) problem characterized by high uncertainty, imprecision, and conflicting criteria (e.g., cost vs. security effectiveness). Conventional decision-making methods often fail to adequately model the dual nature of this uncertainty: the vagueness of expert judgments and the roughness (boundary ambiguity) inherent in incomplete data. Furthermore, existing aggregation operators often lack the flexibility to handle the complex, oscillatory nature of decision data in dynamic environments.

Methodology
The paper proposes a novel MCDM framework based on Complex Intuitionistic Fuzzy Rough Sets (CIFRSs) integrated with Frank t-norm and t-conorm operations. The methodology proceeds as follows:

1. Theoretical Foundation: The authors establish the mathematical definitions for CIFRSs, which combine Complex Intuitionistic Fuzzy Sets (handling membership, non-membership, and hesitancy with phase terms) and Rough Sets (handling lower and upper approximations).
2. Operational Laws: New operational laws are developed for Complex Intuitionistic Fuzzy Rough Numbers (CIFRNs) using Frank operations. These include Frank sum, product, and scalar multiplication, which offer tunable flexibility through a parameter $\lambda$ (where $\lambda \neq 1$).
3. Aggregation Operators (AOs): Two specific aggregation operators are introduced:
   • CIFR Frank Weighted Average (CIFRFWA): Aggregates inputs using a weighted averaging approach based on Frank operations.
   • CIFR Frank Weighted Geometric (CIFRFWG): Aggregates inputs using a weighted geometric approach.
   • The paper provides rigorous proofs for the idempotency, boundedness, and monotonicity of these operators.
4. Decision Algorithm: A step-by-step MCDM algorithm is formulated:
   • Construction of a CIFR decision matrix.
   • Normalization of criteria (benefit vs. cost types).
   • Aggregation of alternatives using CIFRFWA or CIFRFWG operators.
   • Calculation of score and accuracy functions to rank alternatives.

Case Study and Results
To validate the proposed approach, the authors conducted a case study on selecting the optimal CS measure for a nationwide power grid.

• Alternatives: Five measures were evaluated: Firewalls and Network Segmentation (FNS), Intrusion Detection and Prevention Systems (IDPS), Access Control Mechanisms (ACMs), Cryptographic Technologies (CTs), and Security Information and Event Management (SIEM).
• Criteria: Evaluation was based on five criteria: Effectiveness, Scalability, Ease of Integration, Response Time, and Compliance, with assigned weights (0.4, 0.3, 0.1, 0.1, 0.1).
• Findings:
  • Using the CIFRFWA operator, SIEM was ranked first (Score: 0.61647), followed by FNS, IDPS, ACM, and CT.
  • Using the CIFRFWG operator, SIEM was again ranked first (Score: 0.58809), though the relative ordering of the remaining alternatives shifted slightly (FNS second, ACM third).
  • In both scenarios, SIEM emerged as the most robust solution, demonstrating the model's ability to handle complex, uncertain data to produce a consistent optimal choice.

Key Contributions

1. Novel Aggregation Operators: The paper introduces the first application of Frank operational laws within the CIFRS framework, creating the CIFRFWA and CIFRFWG operators.
2. Enhanced Uncertainty Modeling: By integrating complex intuitionistic fuzzy sets with rough set approximations, the model captures both the phase-based complexity of data and the boundary roughness of expert knowledge, offering a more comprehensive representation of uncertainty than standard fuzzy or rough sets alone.
3. Tunable Flexibility: The use of Frank operators allows for adjustable aggregation behavior via the parameter $\lambda$, providing greater adaptability to different decision-making contexts compared to fixed operators (e.g., algebraic or Einstein t-norms).
4. Practical Application: The study provides a concrete, validated framework for selecting cybersecurity measures in power systems, a domain where decision errors can have severe national security and economic consequences.

Significance and Claims
The authors claim that their proposed CIFR-MCDM technique offers superior flexibility, accuracy, and robustness compared to existing methods. Through a comparative analysis with approaches by Garg and Rani, Fang et al., Yahya et al., Hussain et al., Khan et al., and Ahmmad et al., the paper argues that:

• Unlike methods using only fuzzy rough sets, the proposed approach incorporates intuitionistic uncertainty (hesitancy) and complex-valued data (phase terms), making it suitable for oscillatory and dynamic decision scenarios.
• Unlike methods using fixed aggregation operators, the Frank-based operators provide tunable strictness, allowing decision-makers to better adapt to the specific nature of the risk.
• The integration of rough set approximations (lower and upper bounds) enhances the discrimination power among alternatives, leading to more stable rankings.

The paper concludes that this framework serves as a reliable decision-support tool for cybersecurity experts, enabling more systematic and resilient selection of protective measures for modern power infrastructure. The authors note that future work will involve exploring other aggregation operators within this framework and applying the model to larger-scale empirical datasets.