Joint optimization of independent multiple responses

Erdbrügge, Martina; Kuhnt, Sonja; Rudak, Nikolaus

doi:10.1002/qre.1229

Cited by 5 publications

(11 citation statements)

References 20 publications

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“…If the cost matrix C is diagonal, the risk function reduces to

\begin{matrix} R (x) & = \sum_{j = 1}^{p} c_{jj} (σ_{j}^{2} (x) + {(μ_{j} (x) - τ_{j})}^{2}) \\ = : \sum_{j = 1}^{p} c_{jj} r_{j} (x) \end{matrix}

with c j j =[ C ] j j and

r_{j} (x) = σ_{j}^{2} (x) + {(μ_{j} (x) - τ_{j})}^{2}

. In this case, minimization of the risk function leads to a Pareto optimal point for the vector of single risks ( r 1 ( x ),…, r p ( x ))′ in the sense of Definition . According to Definition , a point is Pareto optimal if no coordinatewise improvement is possible.…”

Section: Jop Methodsmentioning

confidence: 99%

“…Erdbrügge et al propose to use a sequence of diagonal cost matrices of the form

C_{t} = A_{y}^{^{'}} W_{t} A_{y}

for t = 1,…, m with

A_{y} = diag ((), {([], \frac{1}{n} \sum_{i = 1}^{n} \overset{E}{false} (Y_{j} | x_{i}))}_{j = 1, \dots, p}^{- 1}) \in {double-struckR}^{p \times p}

where x 1 ,…, x n are the different settings of x in the used experimental design.…”

Section: Jop Methodsmentioning

confidence: 99%

“…The joint optimization (JOP) method solves the robust parameter design problem for multiple responses by minimizing an estimated risk function, which so far is based on double generalized linear models. They allow the mean as well as the variance of the individual response variables to depend on the input factors .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Simultaneous Optimization of Multiple Correlated Responses with Application to a Thermal Spraying Process

Rudak

Hussong

Kuhnt

2015

Quality & Reliability Eng

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In industrial applications, it is often desired to find settings of process parameters, which lead to pre-specified target values of multiple quality characteristics with minimal variance. One approach to solve this problem is to minimize an estimated risk function depending on a cost matrix. The joint optimization (JOP) method follows this general strategy using a sequence of diagonal cost matrices and requires estimated models for the expectation and the variance of the responses. However, if the quality characteristics might be correlated, this should be considered at the model or optimization stage in order to find a realistic solution. In this contribution, we extend the JOP method to the simultaneous optimization of correlated multiple responses. We also introduce a new approach for the choice of non-diagonal cost matrices. The resulting JOP method for correlated responses is illustrated on an application arising in the field of thermal spraying.

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“…If the cost matrix C is diagonal, the risk function reduces to

\begin{matrix} R (x) & = \sum_{j = 1}^{p} c_{jj} (σ_{j}^{2} (x) + {(μ_{j} (x) - τ_{j})}^{2}) \\ = : \sum_{j = 1}^{p} c_{jj} r_{j} (x) \end{matrix}

with c j j =[ C ] j j and

r_{j} (x) = σ_{j}^{2} (x) + {(μ_{j} (x) - τ_{j})}^{2}

Section: Jop Methodsmentioning

confidence: 99%

“…Erdbrügge et al propose to use a sequence of diagonal cost matrices of the form

C_{t} = A_{y}^{^{'}} W_{t} A_{y}

for t = 1,…, m with

A_{y} = diag ((), {([], \frac{1}{n} \sum_{i = 1}^{n} \overset{E}{false} (Y_{j} | x_{i}))}_{j = 1, \dots, p}^{- 1}) \in {double-struckR}^{p \times p}

where x 1 ,…, x n are the different settings of x in the used experimental design.…”

Section: Jop Methodsmentioning

confidence: 99%

See 1 more Smart Citation

Simultaneous Optimization of Multiple Correlated Responses with Application to a Thermal Spraying Process

Rudak

Hussong

Kuhnt

2015

Quality & Reliability Eng

Self Cite

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“…The matrix W could be constructed using the inverse of the constant covariance matrix for responses

Σ^{- 1}

K {boldΣ}^{- 1}

, where K typically is a diagonal matrix reflecting the relative importance of each response . The choice of W is important and sometimes subjective because it conveys process economics or correlations …”

Section: Review Of Multiobjective Approachesmentioning

confidence: 99%

“…62,63 The choice of W is important and sometimes subjective because it conveys process economics or correlations. 64 Pignatiello 65 used the expected loss function. The expected value of the quadratic form MQL is given by…”

Section: Loss Functionmentioning

confidence: 99%

An Overview of Optimization Formulations for Multiresponse Surface Problems

Ardakani

Wulff

2012

Quality & Reliability Eng

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Many industrial applications involve more than one quality characteristic. For example, in robust parameter design, the quality characteristics include the process mean and process variance. Such applications lead to multiresponse surface problems in which it is necessary to determine optimal operating conditions according to some specified optimization criterion involving the quality characteristics. The purpose of this article is to address this problem from a multiobjective decision‐making framework. The foremost approaches in multiresponse optimization are categorized and integrated. Guidelines are presented to help select appropriate formulations. Moreover, the applicability and computational aspects of the methods in various decision‐making contexts are discussed. Numerical examples are also provided. Copyright © 2012 John Wiley & Sons, Ltd.

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