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Robust Parameter Design and Economical Multi-Objective Optimization on Characterizing Rubber for Shoe Soles•

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Robust Parameter Design and Economical Multi-Objective Optimization on Characterizing Rubber for Shoe Soles• Robust parameter design and economical multi-objective optimization on characterizing rubber for shoe soles• Armando Mares-Castro Tecnológico Nacional de México / ITS de Purísima del Rincón, Guardarrayas Purísima del Rincón, Guanajuato. México. [email protected] Received: August 3rd, 2020. Received in revised form: January 14th, 2021. Accepted: January 26th, 2021. Abstract Within Taguchi methods, robust parameter design is a widely used tool for quality improvement in processes and products. The loss function is another quality improvement technique with a focus on cost reduction. Traditional Taguchi methods focus on process improvement or optimization with a unique quality characteristic. Analytical approaches for optimizing processes with multiple quality characteristics have been presented in the literature. In this investigation, a case of analysis for two quality characteristics in rubber for shoe sole is presented. A methodology supported in robust parameter design in combined array is used in order to obtain optimal levels in the vulcanization process. Optimization techniques based on the loss function and the use of restricted nonlinear optimization with genetic algorithms are proposed. Keywords: multiobjective optimization; robust parameter design; loss function; genetic algorithms; rubber for shoe sole; vulcanization. Diseño robusto de parámetros y optimización multi-objetivo económica en la caracterización de hule para suela de calzado Resumen Dentro de los métodos Taguchi, el diseño robusto de parámetros es una herramienta ampliamente utilizada para la mejora de calidad en procesos y productos. La función de pérdida es otra técnica de mejora calidad con un enfoque en la reducción de costos. Los métodos Taguchi tradicionales se enfocan en la mejora u optimización de procesos con una característica de calidad única. En la literatura se han presentado enfoques de análisis para la optimización de procesos con múltiples características de calidad. En esta investigación de presenta un caso de análisis para dos características de calidad en el hule para suela de calzado mediante el uso del diseño robusto de parámetros en arreglo combinado, con la finalidad de obtener los niveles óptimos en el proceso de vulcanización. Se proponen técnicas de optimización basadas en la función de pérdida y el uso optimización no lineal restringida basada en algoritmos genéticos. Palabras clave: optimización multiobjetivo; diseño robusto de parámetros; función de pérdida; hule para suela de calzado; vulcanización. 1. Introduction one in which is minimized the effect of noise factors in the process. Noise is something difficult to control in the process, Quality engineering involves management in the stages of there is a need to reduce the variation in critical quality product design, manufacturing process design, manufacturing, and characteristics. customer use. Reduction of the effect of noise factors and part-by- In Taguchi's original proposal, a double experimental part variation in manufacturing should be considered within the arrangement is used. There is an arrangement for noise factors design phases. The main objective of robust parameter design called “internal arrangement” and an arrangement for noise (RPD) is obtaining the optimum levels in the control factors, factors called “external array” in which noise factors are with which the average performance of the process is integrated, these factors are expensive or difficult to control. The improved. At the same time, the variation around the mean combination of values for each factor is defined in rows and of the process is reduced [1]. The definition of robustness consists of a treatment. The experimental region is defined by indicates that the solution obtained by means of the RPD is the extreme values of the control factors under study. How to cite: Mares-Castro, A., Robust parameter design and economical multi-objective optimization on characterizing rubber for shoe soles. DYNA, 88(216), pp. 160-169, January - March, 2021 © The author; licensee Universidad Nacional de Colombia. Revista DYNA, 88(216), pp. 160-169, January - March, 2021, ISSN 0012-7353 DOI: http://doi.org/10.15446/dyna.v88n216.89676 Mares-Castro / Revista DYNA, 88(216), pp. 160-169, January - March, 2021 The RPD methodology has evolved from the advice of areas Table 1. of opportunity by several western statistical authors [2]. One of the proposals to streamline the experimental effort is based on the use of an experimental matrix in which both the control and noise ⋯ ⋯ 11 1 11 1 11 factors are integrated, called the "combined array", which can ⋯ ⋯ reduce the number of experimental runs. The use of the combined Source: ∙The authors⋯ ∙ ∙ ⋯ ∙ ∙ array allows working on the response surface scheme with 1 ⋯ 1 ⋯ 1 several designs, such as factorials, fractional factorials, Box- Behnken, Box Draper designs, among others [3]. Data generated by this experimental strategy allows the Optimization procedures for the double orthogonal array and generation of a model that is based on the control and noise the combined array are found in the literature. In these proposals, factors obtained, the expression can be represented as eq. (1): an approach based of response surface models for the mean and models for the variation (variance, standard deviation, signal to = + + + + + (1) noise ratio SNR) is managed. Among the optimization proposals ′ ′ ′ ′ 0 for the RPD is that of Kim & Lin [4] which is an approach to Where =( 1, … , ) k control factors, = ( , … , ) diffuse modeling. Kim & Cho [5] presented a proposal based on q noise factors,′ the constant. Parameter vectors, = 0 1 an RPD model that ponders the point of view of the engineer at ( , … , ), =( 1,…, ), = ( , … , , , … , ) the product design stage. Del Castillo & Montgomery [6] used and = ( ,′… , , … , ,… , ) are the second order′ the nonlinear programming problem for RPD optimization. parameter1 matrix , y ~ (0, ). We11 have1 the 1following 11 1 1 Köksoy & Doganaksoy [7] use a mean squared error (MSE) assumptions for noise factors: 2 ( ) = 0, ( ) = and reduction approach and joint optimization of the mean and ( ) = . variance. Copeland & Nelson [8] used quadratic nonlinear The expressions2 for the mean (conditional expectation) programming techniques for the direct minimization of the and variance are shown in eq. (2) and eq. (3) variance and set the target to a target value as a constraint. Lin & Tu [9] presented a definition of the MSE that allows moving a ( ) = + + (2) little away the mean of the Target value, in addition to the ′ ′ 0 minimization of the variance. This approach allows obtaining ( ) = ( + ) ( + ) + (3) better optimization solutions. All of the above methods are based ′ ′ ′ ′ 2 on the analysis of a single process response. The model in eq. (1) is fitted by the least squares’ method. In this research, a proposal of analysis for a sole vulcanizing Subsequently, eq. (2-3) are estimated according to the process is presented. The two quality characteristics are analyzed required terms. An important detail is that the model in eq. based on tests of tension and hardness in the material. The (1) must contain control x noise interactions, in order to methodology involves the use of the combined array in the determine the eq model. (3). Terms + are the partial experiment and the use of optimization techniques based on derivative with respect to of the individual′ elements and restricted nonlinear optimization by genetic algorithms. The interactions that contain in eq. (1). objective function of the optimization problem involves the use of the Taguchi loss function. In order to use an economic 2.2 Quadratic loss function approach to reduce the expected cost due to costs of non- confirming in the product. The main contribution of this work lies This model was introduced by Dr. Taguchi to provide a in the analysis of the multi-objective problem in the RPD and the monetary estimate of the loss that the manufacturer causes to use of a loss function-based scheme. Likewise, there are no customer when the performance of their product differs from the analyzes in the literature to optimize parameters of the vulcanized target value. That is, it is used in cases for quality characteristics process in rubber for shoe soles. Due to the high competitiveness of the type “Target is the best”. Let the quality characteristic and quality demands on the part of footwear users, it is important of a product and the target value for . According to the to carry out studies to establish standards for the evaluation and quadratic loss function, its equation is represented in eq. (4). improvement of quality in the materials used in the product to achieve optimal performance. The objective can be achieved by [ ] = [ ] (4) using techniques for process optimization. 2 [ ] is given by a Taylor series − expansion around = 2. Generalization and approach of the proposal neglecting the cubic terms hereafter. Constant is known as the coefficient of quality loss for the quality characteristic, 2.1 Combined array, models for mean and variance it is defined as eq. (5). Welch et al [10], presented an alternative to the double = (5) orthogonal array of Taguchi. In this strategy, the control and 0 2 noise factors are combined in a single experimental matrix. The ∆0 main advantage of this scheme is that the number of runs or In which is the cost r los to repair a product when experimental treatments is significantly reduced, remembering quality
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