Using POE with Tolerance Intervals to Define Design Space

Transcription

1 Using POE with Tolerance Intervals to Define Design Space Patrick J. Whitcomb (Speaker) Stat-Ease, Inc. Mark J. Anderson, PE, CQE Stat-Ease, Inc. Minneapolis, MN USA Presented to: Chemometrics for Online Process Analytics (COPA) Session IFPAC Annual Meeting, Jan , 215, Arlington, VA IFPAC Last Year s Talk Managing Uncertainty in Design Space Patrick J. Whitcomb Stat-Ease, Inc. Minneapolis, MN USA Mark J. Anderson, PE, CQE (Speaker) Stat-Ease, Inc. mark@statease.com Presented to: Chemometrics for Online Process Analytics (COPA) Session IFPAC Annual Meeting, Jan , 214, Arlington, VA IFPAC

2 Managing Uncertainty in Design Space Objective: Define an operating window in process factor space where we have 95% confidence that 99% of the population meet (or exceed) specifications. Use empirical DOE to model the responses as functions of the process factors. Use a tolerance interval to back off (provide a buffer) from the specifications. Size the DOE for required half-width of tolerance interval IFPAC New Idea: Accounting for Factor Variation via POE Propagation of error (POE) is well-established* tool for reducing variation transmitted to the response from variation in control factors. Pictured here is the impact of standard deviation in the set-point for granulation and lubrification increasing from.5 to.333 minutes. Overlay Plot Overlay Plot B : L u b r i f i c a t i o n 4. Friability:.5 Friability TI:.5 Dissolution TI: 7 Dissolution: 7 Hardness TI: 1. B : L u b r i f i c a t i o n 4. Friability:.5 Dissolution: 7 Friability TI*:.5 Dissolution TI*: 7 Hardness TI*: 1. Hardness: 1. Hardness: A: Granulation A: Granulation * Matthias Otto, Chemometrics, 2nd Edition, (27) J. Wiley, pp IFPAC

3 Agenda Managing Uncertainty in Design Space: Size design to estimate tolerance intervals. Run DOE and fit appropriate models. Using POE with Tolerance Intervals to Define Design Space: Use POE to approximate long term variation. Optimize using C pk. Use tolerance intervals to back off specifications. IFPAC Illustrative Example Tableting Process This case study illustrates how response surface DOE and tolerance intervals can be used to set an operating window where specifications are consistently met. An optimal design is run on two process parameters (granulation time and lubrification time) in a tableting process. Three responses (dissolution, friability and hardness) are measured. The specifications are: Dissolution 75% Friability.5% Hardness 1 kp IFPAC

4 B: Lubrification 2/22/211 Initial Operating Window Specifications as Bounds y 1 = Dissolution % (specification 75%) Specifications on Graphical Overlay y 2 = Friability % (specification.5%) Friability:.5 y 3 = Hardness kp (specification 1 kp) 4. Dissolution: 7 Hardness: A: Granulation IFPAC Uncertainty is a BIG Problem Specifications as Bounds If the tableting process is operated on a boundary, then 5% of tablets produced fall outside the specification. Specification IFPAC

5 Back off from Specification One-Sided 95% tolerance interval (TI) One-sided TI calculation for DOE TI yˆ d or TI yˆ d d is the offset for the tolerance interval d 95% TI Specification new boundary IFPAC DOE with Tolerance Intervals Sizing for Precision Requirements Define the precision required as the half-width of a Tolerance Interval that contains 99% of the population with 95% confidence: Response Desired d s Lower bound on d* Size the DOE to the smallest (d/s) ratio. d/s dissolution friability hardness *(1.2)(K 1 )(s) is a lower bound on d. Since the desired d is higher than the lower bound on d, we can achieve a precise enough design. IFPAC

6 B: Lubrification 2/22/211 Sizing for Precision Requirements Tolerance Intervals (using optimal defaults) 16 runs (6 model, 5 lack of fit, 5 replicates) 2.7 Std Error of Design B : L u b r i f i c a t i o n A: Granulation A: Granulation IFPAC Sizing for Precision Requirements Sizing for Tolerance Intervals 16 runs (6 model, 5 lack of fit, 5 replicates) FDS = % Design-Expert Software Min Tolerance: Avg Tolerance: Max Tolerance: Constrained Points = 5 t(.5,1) = P =.99 Tolerance = FDS Graph T I M u lt ip lie r d = 9 s = 2 a =.5 P= Fraction of Design Space IFPAC

7 Functional versus Verification Design Confidence vs. Tolerance Intervals Take home message Sizing for verification using tolerance intervals requires a larger DOE (sample size) than sizing for functional properties using confidence intervals. Next step Increase the size of the tableting process DOE by seven additional model points. IFPAC Sizing for Precision Requirements Sizing for Tolerance Intervals Re-build the design with increasing the number of model points with 7 Additional model points for a total of 23 runs: (6+7=13 model, 5 lack of fit and 5 replicate points) IFPAC

8 Ti Multiplier 2/22/211 Sizing for Precision Requirements Sizing for Tolerance Intervals 23 runs (13 model, 5 lack of fit, 5 replicates) FDS = 93% 4. FDS Graph Tolerance One-sided d = 9 s = 2 a =.5 P=.99 T I M u l t i p l i e r 1. Much better! Fraction of Design Space Fraction of Design Space IFPAC Tableting Process Results Final design: 23 runs 13 model points 5 lack of fit points 5 replicates IFPAC

9 Agenda Managing Uncertainty in Design Space: Size design to estimate tolerance intervals. Run DOE and fit appropriate models. Using POE with Tolerance Intervals to Define Design Space: Use POE to approximate long term variation. Optimize using C pk. Use tolerance intervals to back off specifications. IFPAC Tableting Process Results Quadratic Models IFPAC

10 Final Operating Window Good models are essential! Be sure the fitted surface adequately represents your process before you use it for optimization. Check for: 1. A significant model: Large F-value with p < Insignificant lack-of-fit: F-value near one with p > Well behaved residuals. 4. Adequate precision > 4: Want signal to noise ratio large enough that we can successfully navigate the design space. IFPAC Tableting Process Results Response Significant Model Insignificant lack-of-fit Well behaved residuals Adequate precision Dissolution p <.1 p=.6885 Yes Friability p <.1 p=.574 Yes 48.8 Hardness p <.1 p=.5848 Yes 46.4 IFPAC

11 Design Space via Tolerance Intervals B : L u b r i f i c a t i o n ( m i n u t e s ) 4. Friability:.5 Dissolution: 75 Specifications B : L u b r i f i c a t i o n ( m i n u t e s ) 4. Tolerance Intervals Friability:.5 Friability TI:.5 Dissolution TI: 75 Hardness TI: 1 Dissolution: 75 Hardness: 1 Hardness: A: Granulation (minutes) A: Granulation (minutes) IFPAC Agenda Managing Uncertainty in Design Space: Size design to estimate tolerance intervals. Run DOE and fit appropriate models. Using POE with Tolerance Intervals to Define Design Space: Use POE to approximate long term variation. Optimize using C pk. Use tolerance intervals to back off specifications. IFPAC

12 Realistic Tolerances Approximating Long Term Variance Problem: Extra attention is paid during a DOE observer effect. DOE is run over a limited time period only captures short term variation. DOE may be run on smaller scale equipment that is more easily controlled. etc. Solution: Use POE to approximate long term variation. IFPAC Propagation of Error (POE) Transmitted Variation Objective: Reduce the variation transmitted to the response from variation in control factors. IFPAC

13 Propagation of Error How it works Once a relationship has been established between a factor and a response, the variation in the output can be: 1. Dependent on the level of the control factor 2. Independent of the level of the control factor See pictures on next two pages RDTA section 3 25 Propagation of error Dependent The transmitted variation is dependent on the level of the control factor. Therefore, set the level of the control factor to reduce variation transmitted to the response from variation (lack-of-control) of the control factor. Effect of Input on Response A Control Factor B RDTA section

14 Response 2/22/211 Propagation of error Independent The transmitted variation is independent of the level of the control factor. Therefore, set the level of the control factor to center the process mean on target. Effect of Input on Response A Control Factor B RDTA section 3 27 Typical Application of POE on target with reduced variation The control factors are used in a two step process: the more curved factors to decrease variation the more linear factors to move back on target. In practice, numerical optimization can be used to simultaneously obtain all the goals. Target RDTA section

15 Propagation of error Transmitted Variation First order: f Yˆ xi e i xi 2 ŷ f x,...,x 1 k Statistical detail Second order: k k 2 k 2 2 f 2 1 f 4 f Yˆ ii 2 ii ii jj e i 1 xi 2 i 1 x i j x i i x j ŷ f x,...,x k 2 1 f 2 1 k 2 ii 2 i 1 x i MS from the ANOVA and POE 2 2 e residual Yˆ IFPAC Propagation of Error Goal: Minimize propagated error (POE) The amount of variation transmitted to the response (POE) depends on: The response model (need second order terms), The lack of control of the control factors (you enter these standard deviations), plus the normal process variation (obtained from the ANOVA or manually entered). POE is expressed as a standard deviation. IFPAC

16 Propagation of Error Approximating Long Term Variance (page 1 of 2) Approximating Long Term Variance: Enter the standard deviation of the input factors around their set point. For the tableting example the long term standard deviations (from plant data) are about 1 seconds, or.17 minutes Enter estimate of unexplained variation in plant: Dissolution 1.7% Friability.4% Hardness.2 kp IFPAC Propagation of Error Approximating Long Term Variance (page 2 of 2) IFPAC

17 Agenda Managing Uncertainty in Design Space: Size design to estimate tolerance intervals. Run DOE and fit appropriate models. Using POE with Tolerance Intervals to Define Design Space: Use POE to approximate long term variation. Optimize using C pk. Use tolerance intervals to back off specifications. IFPAC Optimize with respect to Specifications Maximize C pk (page 1 of 2) Dissolution 75% IFPAC

18 Optimize with respect to Specifications Maximize C pk (page 2 of 2) Friability.5% Hardness 1 kp IFPAC Numerical Optimization Solution B : L u b r i f i c a t i o n ( m i n u t e s ) Desirability Prediction IFPAC 215 A: Granulation (minutes) 36 18

19 B: Lubrification 2/22/211 Agenda Managing Uncertainty in Design Space: Size design to estimate tolerance intervals. Run DOE and fit appropriate models. Using POE with Tolerance Intervals to Define Design Space: Use POE to approximate long term variation. Optimize using C pk. Use tolerance intervals to back off specifications. IFPAC Operating Window Specifications y 1 = Dissolution % (specification 75%) Specifications y 2 = Friability % (specification.5%) Friability:.5 y 3 = Hardness kp (specification 1 kp) 4. Dissolution: 7 Hardness: A: Granulation IFPAC

20 Operating Window Tolerance Intervals as Bounds Design-Expert Software Factor Coding: Actual Overlay Plot Dissolution TI Low* Friability TI High* Hardness TI Low* Design Points X1 = A: Granulation X2 = B: Lubrification * Intervals adjusted for variation in the factors Tolerance Intervals (a = 5%, P = 99%) Overlay Plot B : L u b r i f i c a t i o n ( m i n u t e s ) Dissolution: 9.25 TI Low*: Friability: TI High*: Friability: Hardness: TI Low*: X X2 Friability TI*: Dissolution: 75 2 Dissolution TI*: 75 Hardness TI*: 1 3 Hardness: IFPAC 215 A: Granulation (minutes) 39 Final Operating Window Approximating Long Term Variance Tolerance Intervals w/o POE Overlay Plot Tolerance Intervals with POE Overlay Plot B : L u b r i f i c a t i o n 4. Friability:.5 Friability TI:.5 Dissolution TI: 7 Dissolution: 7 Hardness TI: 1. B : L u b r i f i c a t i o n 4. Friability:.5 Dissolution: 7 Friability TI*:.5 Dissolution TI*: 7 Hardness TI*: 1. Hardness: 1. Hardness: A: Granulation A: Granulation IFPAC

21 References 1. Alyaa R. Zahran, Christine M. Anderson-Cook and Raymond H. Myers, Fraction of Design Space to Assess Prediction, Journal of Quality Technology, Vol. 35, No. 4, October Heidi B. Goldfarb, Christine M. Anderson-Cook, Connie M. Borror and Douglas C. Montgomery, Fraction of Design Space plots for Assessing Mixture and Mixture- Process Designs, Journal of Quality Technology, Vol. 36, No. 2, October Steven DE Gryze, Ivan Langhans and Martina Vandebroek, Using the Intervals for Prediction: A Tutorial on Tolerance Intervals for Ordinary Least-Squares Regression, Chemometrics and Intelligent Laboratory Systems, 87 (27) Pat Whitcomb (29), Sizing Mixture (RSM) Designs on the Chemical and Process Industries web site at: //asq.org/cpi/, 5. Gerald J. Hahn and William Q. Meeker, Statistical Intervals; A Guide for Practitioners, (1991) John Wiley and Sons, Inc, 6. Gary Oehlert and Patrick Whitcomb (21), Sizing Fixed Effects for Computing Power in Experimental Designs, Quality and Reliability Engineering International, July 27, 21. IFPAC Using POE with Tolerance Intervals to Define Design Space Pat Whitcomb pat@statease.com 215 IFPAC Conference Thank you for attending! IFPAC