Development of committee machine models for multiple response optimization problems

Multiple response optimization (MRO) problems need to optimize several response variables simultaneously. Three phases are considered to solve MRO problems and they include design of experiments, modeling, and optimization. Committee Machine (CM) is a group of some experts such as some Artificial Neural Networks (ANNs) or other mathematical models that can be used in modeling phase of MRO problem solving. There are several methods in two categories to solve MRO problems. The first category includes premier methods such as Response Surface Methodology (RSM) and Taguchi method and the second one includes newer methods like hybrid methods of ANNs and Genetic Algorithm (GA). Each of these methods has deficiencies. RSM uses quadratic polynomial in modeling phase and it is not an accurate modeling tool because all problems including curvilinearity do not fit and compatible to a quadratic polynomial. Responses of Taguchi are restricted to be selected from defined levels of input variables and newer hybrid methods use only one ANN in modeling. As usual, to evaluate the responses of each methodology,Global Desirability (GD) function is used as performance metrics. Higher GD for responses means superior performance for methodology and in MRO it is needed to introduce the new methods to obtain responses with more accuracy and higher Global Desirability. Methodology of the current research is an overall methodology that includes five methodologies. Four methodologies are to make four different CM models to solve MRO problems. The fifth methodology proposes the final algorithm which uses four CM models together to solve MRO problems. Accordingly, GD is computed for all design points of experiments. Then, eight different models are made as CM experts including seven well-known ANNs and one multi-linear regression (MLR) model based on experiments design points. CM models are in four types of categories including Sequential Combination Model (SCM), Optimum Combination Model (OCM), Point Approach Model (PAM), and Mixed Combination Model (MCM). Depending on each CM model, two to eight numbers of experts will participate. The weights of all CM models are obtained by GA with object minimum Root Mean Squire Error (RMSE). This is the modeling phase. Then, in the optimization phase, GA searches best responses for each Committee Machine model with object maximum Global Desirability. For each response, the nearest experiment point number is determined according to x's and y's Euclidean distances. Five MRO case studies were selected from the literature and one real case study was selected from industry. The results showed optimum points can happen in all CM groups. The results showed that MCMs have less difference in GD between model and real responses in comparison with SCMs. However, SCMs may yield good responses. So,it is necessary to get all MCMs and SCMs responses. Furthermore, each response that is nearest to the point number for x's and y's is the same as experiments point number with highest GD and has a difference of less than 3%. Consequently, current algorithm helps to find a set of accurate responses. Implementation of the current algorithm on case 1, a wire-bonding problem, shows response surface methodology yields Global Desirability equal to 0.31, while neural networks model offers GD equal to 0.42, and usual Committee Machine has obtained GD of 0.48 and finally current algorithm offers Global Desirability equal to 0.52. The results represent Global Desirability of proposed algorithm is equal or higher than GD of case studies. Final conclusion shows that comparison between the results of Committee Machine and its experts in case studies indicate that Global Desirability of CM is equal to or higher than its experts, and proposed algorithm can be applied to solve different MRO problems in industry and scientific areas.

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