Numerical investigation of non-probabilistic systems using Inner Outer Direct Search optimization technique

Fuzzy systems of equations often appear while modeling physical systems with imprecisely defined parameters. Many mathematical methods are available to investigate them, but handling them is challenging due to the computational complexity and difficult implementation. As such, in this paper, the Inner-Outer Direct Search (IODS) optimization technique is extended in the fuzzy environment to solve a fuzzy system of nonlinear equations. The main purpose of the extension is to study the system variables in the presence of fuzzy information. To manage fuzziness, a fuzzy parametric form is employed in the uncertain system and controls the search process toward the optimal solution. The proposed approach of fuzzy IODS converts the fuzzy system of nonlinear equations to an unconstrained fuzzy optimization problem. Then, the unconstrained fuzzy optimization problem is studied through the IODS technique. To solve the unconstrained fuzzy optimization problem, the fuzzy objective function is minimized with the help of exploratory and pattern search approaches. These searches are performed with inner and outer computations. Then, the obtained united solution provides the desired solution which minimizes the objective function. From the same the uncertain system, variables are derived. To verify the solution and proposed algorithm, convergence analysis is performed. Three case studies are considered with only fuzzy and fully fuzzy systems, and various cases are discussed. A comparison with other methods is made to test the efficacy of the method. The proposed algorithm is coded with the help of MATLAB software, and the results are analyzed graphically. Finally, the simple procedure and computationally efficient approach may help to implement the same in many engineering and science problems that can be modeled into systems of equations.