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Robust Design and Two-Step Optimization Lihui Shi

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Outline Introduction of Taguchi Basic Concepts and Tools in Robust Design Signal-to-Noise Ratio Static Robust Design & Two-Step Optimization Dynamic Robust Design & Two-Step Optimization Reference

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Introduction Genichi Taguchi ( 田口 玄一 ) From 1950s developed a methodology to improve the quality of products. to improve the quality of products. Much of his work was carried out in isolation from the mainstream of isolation from the mainstream of Western statistics. Western statistics. Unknown outside of Japan. Introduced into US in 1980. Taguchi’s method. Controversial among statisticians, but many concepts introduced by him have been accepted.

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Basic Block Diagram, Concepts and Tools Quality characteristics Quadratic Loss function Design of Experiments (DOE) (DOE) Signal-to-noise ratio (SN ratio) (SN ratio) Orthogonal arrays Linear graph Basic question: How to choose the levels of the control factors to make the output on target and the process robust again noise factors?

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Quadratic Loss Function Y: output t: target value Objective: Minimize the average loss =(d,a), design parameters. =(d,a), design parameters.

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Signal-to-Noise (SN) Ratio SN ratio h is defined as Question 1: Why use the log transformation? Box (1988), (1987 discussion): The standard deviation will be independent of the mean, so the design factors will separate into some that affect the variation and some others that affect the mean without changing the variation. Question 2: Why use the ratio instead of the standard deviation? Phadke: Frequently, as the mean decreases the standard deviation also decreases and vice versa.

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Various Factors Among many applications, Taguchi has empirically found that the two stage optimization procedure involving the SN ratio indeed gives the parameter level combination where the standard deviation is minimum while keeping the mean on target. Control factors d: a significant effect on SN ratio. Adjustment factor (scaling factor) a: significant effect on mean, but no effect on SN ratio. Other factors: have no effect on SN ratio and mean. d and a all both set of factors, and use =(d,a), design parameters.

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Static Robust Design When the target is fixed, then the signal factor is trivial, or absent. Objective: Minimize the variance, and keep the mean on target. It is a constrained optimization problem.

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Two-Step Optimization It is equivalent to: Maximize h, and keep the mean on target. Use the two-step optimization method: 1. Choose d to maximize h (no worry about mean): 2. Adjust the mean on target by using a: It is an unconstrained optimization problem. Much easier now!!!

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Dynamic Robust Design Also called: robust design in signal-response system. A signal factor is selected from the set of control factors, and is changed continuously depending on the customer’s intent, to meet his requirements. Aim: make the signal-response relationship insensitive to the noise variation, by choosing the appropriate levels of the control factors. Two types of systems: 1. measurement system 1. measurement system 2. multiple-target system 2. multiple-target system

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Multiple Target System Linear relationship between the signal and response: SN ratio is given by Nonlinear: Performance measure

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Optimization Objective: Minimize the PM. The system requires that the value of M be between ML and MH. Let (t1,t2) be the range of t, and W(t) be the probability density function. h(Z,t) is the solution of M from f(Z,M)=t.

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Two-Step Optimization A special form of f(Z,M): Optimization: It is equivalent to the two-step optimization: 1. Choose Z to maximize h. 2. Adjust to the desired range, by using the adjustment factor a.

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Reference Box, G. E. P., Signal-to-noise ratios, performance criteria, and transformations (with discussion), Technometrics, 30 (1988), 1-40. Nair, V. N., Taguchi’s parameter design: A panel discussion, Technometrics, 34 (1992), 127-161. Phadke, M. S., Quality engineering using robust design, (1989), Prentice-Hall, New Jersey. Roshan Joseph, V. and C.F.Jeff Wu, Robust parameter design of multiple-target systems, Technometrics, 44 (2002), 338-346. Le¡äon, R., Shoemaker, A. C. and Kacker, R. N., Performance measures independent of adjustment: An explanation and extension of Taguchi’s signal-to-noise ratios (with discussion)," Technometrics, 29, (1987), 253- 285.

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