Discrete-Time Surplus Process Model with Multiple Threshold Levels for the Productive Waqf-Based Crab Cracker Industry: Theory and Simulation

Abstract

This study develops a discrete-time surplus process model with multiple threshold levels in productive waqf-based crab cracker industry through theoretical and simulation approaches. This study is a mathematical reinforcement of the productive waqf integration scheme with fishermen's welfare benefits previously published by the author, where the crab cracker industry acts as a surplus generator used to finance benefits. The sustainability of the scheme depends on the stability of the surplus dynamics over time, so a structured risk modeling framework is needed to control potential fluctuations and deficit risks. Empirical data on crab catches are modeled in a frequency and severity framework, where the number of fishermen per day represents the frequency, while the value of crab purchases represents the severity. The probability distributions of both components are tested statistically and then used to form compound claims in the discrete-time surplus model. This model introduces multiple threshold levels as a control mechanism in responding to changes in surplus conditions. Furthermore, a theorem related to model characteristics is formulated and proven both analytically and simulation. Theoretically, this study extends the development of the discrete-time surplus model, while practically providing a quantitative basis for maintaining the stability of surpluses againts stochastically uncertainty.