Optimization of adaptive neuro–fuzzy inference system (ANFIS) parameters via Box-Behnken experimental design approach: The prediction of chromium adsorption

Abstract

Prediction of adsorption via Adaptive Neuro–Fuzzy Inference System (ANFIS) can save the cost and time in practical applications. Chromium (VI) adsorption data obtained at different temperature, activated carbon dosage and pH values were evaluated by using MATLAB ANFIS. In order to achieve prediction of adsorption via ANFIS with acceptable error values, optimum membership function (MF) and optimum number of MF were determined by using Box-Behnken experimental design (BBD) method. In order to determine the optimum number of MF for each input, all combinations given in BBD matrix were examined via ANFIS, then, regression models for each MFs were developed between the root mean square error (RMSE) and MF numbers of each input. The most used five membership functions (triangular, trapezoidal, generalized bell shaped, Gaussian, Gaussian 2) were investigated. According to the analysis of variance (ANOVA), regression models developed for the test data with triangular and trapezoidal membership functions were significant in the 95 % confidence level. Predictions were employed via ANFIS by using optimum MF numbers of each inputs (6, 6, 3 for triangular MF and 8, 8, 2 for trapezoidal MF). Consequently, the best Cr(VI) adsorption percentage prediction (RMSE = 1.9084 and R2 = 0.992) was obtained by using triangular membership function with optimum MF numbers. Response surface plots, which gives the relationship between MF numbers and RMSE values for triangular MF were also evaluated. In this study, it was demonstrated that MF type and numbers, which are crucial for good prediction via ANFIS grid partition method, can be determined optimally by applying experimental design methodology.

Keywords

• Adaptive Neuro–Fuzzy Inference System (ANFIS)
• Chromium adsorption
• Prediction
• Membership function
• Experimental design
• Optimization