Dynamic Modeling and Multi-Objective Optimization of Takaful Insurance System

This paper proposes a novel dynamical systems approach to model and optimize Takaful insurance operations, contributing to the growing body of research in Islamic finance. To the best of our knowledge, this is the first study to formalize the interactions between the three core components of Takaful—participants, claims, and the mutual fund—within a continuous-time dynamical framework. The model integrates key operational parameters such as enrollment and attrition rates, claim frequency, contribution levels, and profit-and-loss sharing mechanisms. We first establish the mathematical well-posedness of the system, proving existence, uniqueness, positivity, and boundedness of solutions, followed by a stability analysis of equilibrium points supported by numerical simulations. Building on this foundation, we formulate a multi-objective optimization problem to address the strategic goals of Takaful operators: maximizing participant retention, minimizing claim incidence, and ensuring fund stability. The problem consists in determining the optimal values for the attrition rate, claim occurrence rate, and average contribution, subject to realistic operational constraints. We solve this problem using an integrated NSGA-II and entropy weighting approach, enabling robust trade-off analysis between conflicting objectives. The proposed methodology offers practitioners a quantitative decision-support tool for enhancing membership strategies and risk management while maintaining financial sustainability in accordance with Sharia-compliant principles.