It is highly interesting to explore the implications of Markov chain stability because its importance in engineering, dynamic systems, energy subject and decision-making issues. This paper deals with a method to assess the stability of Markov chains, finishing our notes in finite fuzzy Markov chains. The optimization of KKT was evaluated in two approaches: a. For one thing, optimized KKT methods substitute the random transition probability with a fuzzy value in the transition matrix. The problem illustrates the efficacy of method in proving the unique properties for power bounds of a 3 × 3 matrix depending where on this it lies. This transition matrix defines a fuzzy link within an essentially constrained state space. The second attempt was to model the problem of computing dependability vector by the proposed optimization technique. Additional components were successfully incorporated criteria into fuzzy 3 × 3 transition matrices via KKT optimization techniques. This enhanced the adaptability of the behavior. The model determined that ability constituted an optimization problem. This applicable feature involves the collection of data, namely the exchange rate of the Iraqi dinar against the US dollar as established by the Central Bank of Iraq for the period from October 1, 2023, to December 1, 2023. Data was evaluated, results were derived to fulfill the study objective, and suggestions were formulated based on the findings.
Jameel et al. (2026) studied this question.