Batch Startups Using Multivariate Statistics and Optimization

1. Batch Startups Using Multivariate Statistics and Optimization Susan L. Albin Di Xu Rutgers University supported by NSF/Industry-University Cooperative Center for Quality and Reliability Engineering IBM, January 2003

3. Startup Stage Accounts for Up to 50% of Batch Time

4. Goal: Decrease Mean and Variance of Batch Startup Time create capacity without adding machines, personnel, or space improve production planning reduce scrap ease bottleneck at off-line testing

5. Multiple Input and Output Variables in Batch Processes Process variables, X temperature, pressure, speeds Product variables, Y diameter, tensile strength, elongation Correlations among all variables

6. Traditional Batch Startup Procedure � One Variable at-a-time

7. Consequences of Monitoring Multiple Process Variables One-at-a-Time X1 and X2 correlated

8. Why Different Settings for Different Batches? Long time between batches Uncontrollable variables change batch-to-batch Raw material changes Environment changes Maintenance levels Uncontrollable variables often unknown Different system Not easily measured by sensor

9. Batch Startup: Characterize Good Baseline Data Process & product variables Multivariate statistical Model For New Batch - Start at baseline average If product not ok, select new setting Consistent with Model Taking into account operator/engineering advice

10. Partial Least Squares (PLS) Characterizes Process & Product Variables in Baseline

11. Construct PLS Component T�s T1 = w1X1 + w2X2 + w3X3 + ������ U1 = c1Y1 + c2Y2 + c3Y3 + ������ Find w�s and c�s (normalized): Max Cov(T1 , U1) Find w�s and c�s: Max Cov(T2 , U2) s.t. T2 ? T1

12. Comparison of Principal Components Analysis & PLS Both Reduce dimension of data Components are linear combinations of the X�s BUT PLS components consider the Y�s X�s that are correlated with Y�s emphasized in PLS components

13. Measure Distance Between Current Process & Baseline: Squared Prediction Error SPE

14. Calculate SPE

15. A Filament Extrusion Process Conveying screw pushes solid raw material down length of enclosed barrel Melting occurs due to shear stresses, increased pressure and externally added heat Semi-molten extrudate pushed through die, producing desired filament shape Stretching and re-heating steps control molecular properties e.g. diameter and tensile strength Finished product wound onto take-up spools, each batch producing dozens

16. Process & Product Variables Input: 25 On-line Process Variables ex: temperatures, pressures, speeds observations every few minutes Output: 12 Off-line Product Vars ex: diameters, tensile strength observations every few hours delay of an hour or more

17. Develop PLS Model on Baseline Data(17 batches, 114 observations) 5 PLS components account for 98% cov (Xs, Ys) 84% var(Xs) 29% var(Ys) Could use fewer - 3 comps acct for 91% cov (Xs, Ys) 70% var(Xs) 22% var(Ys) 1Geladi, P. and Kowalski, B.R., (1986)2Lindberg, W., Persson, J., and Wold, S. (1983)3Wold, S., (1978)

18. Graph of SPE for Baseline Data with Control Limit

19. Ad Hoc Use of PLS to Find Adjustment: Decompose SPE

20. Improving on the Ad Hoc Decomposition Method Decomposing SPE suggests which variable to adjust Does not give how much to adjust what related variables need adjustment New methodology combines optimization & multivariate statistics gives which variables to adjust and how much

21. Operator-Assisted Batch Startup

22. Operator Interfaces with Startup Algorithm in Several Modes Operator gives the variable to adjust algorithm gives setting and other process settings Operator gives several possible variables algorithm helps choose Operator unaware adjustment needed without prompt, algorithm suggests adjustment

23. Relationship Between Process Settings and Variables Process variables are a linear function of process settings

24. Mathematical Optimization: Determine Adjusted Process Vars xa & Settings sa Minimize SPE(xa) Subject to:

25. Objective Function Given current process settings sc variables xc Find adjusted settings settings sa Minimize SPE(xa) distance from adjusted variables to baseline

26. Constraint: Follow the Operator�s Recommendation ex: adjust setting 23 to a new value u ex: adjust setting 23 to a new value exceeding the current setting

27. Constraints: Limit Size of Adjustments & No. of Variables Adjusted Introduce one integer variable zi for each possible adjustment Limit size of each adjustment Limit number of variables adjusted, typically 2 or 3

28. Constraint: PLS Components Should Be Within Reasonable Range Compute PLS components, Ts, after adjustment Ts should be in a reasonable range

29. Mixed Integer Quadratic Program Objective function: convex quadratic Mixed decision variables 0-1 variables in constraint limiting no. of adjustments continuous process settings Linear constraints Solve with Bender�s Algorithm or Search

30. About SPE B contains weights to compute PLS components, t, from process variables x loadings to computefrom PLS components t

31. Example: Operator Considers Two possibilities and Algorithm Helps to Select Historical t=40: adjust v7 t=60: adjust v4, v5, v6 t=210: adjust v5, v6 t=240: adjust v5, v6 t=330: adjust v5, v6 t=360: adjust v7 (start) & production With algorithm t=40: input v4 OR v7 output v4, v5, v6 t=50: production! Startup reduced 86% from 360 to 50 minutes

32. Example cont: Two possible adjustments at t=40 Adjust v7 SPE 13.8 plus other adjustments Adjust v4 SPE 8.3 also adjust v5 & v6 Select second choice with min SPE

33. Uncontrollable Variables Contribute to Batch-to-Batch Variability Uncontrollable variables are random variables New values for each batch You can measure them You can control them within specifications You cannot set them Examples � raw material characteristics, environmental and maintenance variables

34. Select Better Settings by Accounting for Uncontrollable Variables

35. Objective Given means and variances for uncontrollable variables Identify optimal settings quickly Predict whether likely to produce successful outputs

36. Extend SPE to Include Uncontrollable Variables Original Divide x into two groups

37. Optimization Objective Function Min Expected Value of SPE Select new settings xS xu are random variables mean vector & variance matrix known

38. Mathematical Optimization: Choose Settings xS to Minimize ESPE Subject to:

39. Settings depend on mean xu - min ESPE depends on mean and variances

40. Predicting if this Batch is Likely to Work Well Find mean and variance for uncontrollable variables Solve for optimal settings If min ESPE exceeds threshold from baseline data, optimal settings are unlikely to produce successful outputs

41. Polystyrene Extrusion Simulation: Baseline of 260 Good Batches 4 uncontrollable raw material vars density, specific heat, thermal conductivity, power law index 3 process settings flow rate, screw speed, barrel temp 8 outputs - extruder performance req axial length, bulk temp, pressure at screw tip & die entrance, max shear rate in channel & die, specific mechanical energy, ave residence time

42. Comparison of Success Rates: Ave Baseline vs. Min ESPE Settings

43. Raw Material Sample Estimates May Be Uncertain High variability in some materials food, oil, bulk chemicals Measurement error lab-to-lab and other testing errors Sampling problems how to sample from a large lot of bulk chemical Constraints on time/money small samples

44. Sample Estimates of Input Variables Form Joint Confidence Interval

45. ESPE Between Baseline and Uncontrollables Vars & Settings

46. Compute Confidence Interval for ESPE

47. Sequential Sampling Algorithm to Determine Whether to Process Batch Compare ESPE CI to 90th percentile of SPE�s in baseline control limit

48. If We Proceed with Batch, Select Settings Use point estimates of uncontrollable variables mean and variance, find settings to min ESPE More conservative � Use minimax optimization to minimize �worst case� ESPE over the CI of the uncontrollable variables

49. Summary: Batch Startups Using Multivariate Statistics and Optimization Uncontrollable variables contribute to batch-to-batch variability no info on uncontrollables means and variances estimates of means and variances Feedforward info on uncontrollables to select optimal batch settings (or quit batch)

50. Summary: Batch Startups Using Multivariate Statistics and Optimization PLS baseline model characterizes uncontrollable variables, settings & process output Math program finds settings Objective: min distance from baseline PLS model to current process Constraints: consistent with PLS model, operator suggestions, & engineering considerations Synthesis of multivariate statistics and mathematical programming

51. Continuing Research Monitoring Batch-to-Batch and Within Batch Variance during the production stage Robust optimization - takes into account that the objective function contains parameter estimates with confidence intervals