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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

Outline of Talk

- Batch processes start-ups
- Multivariate models for process and product

variables - Optimization algorithm, operator assisted, to

reduce batch startup time - Optimizing startup by accounting for

uncontrollable variables raw materials,

environmental variables - Sequential sampling method to estimate

uncontrollable variable parameters

Startup Stage Accounts for Up to 50 of Batch

Time

Long time interval

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

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

Traditional Batch Startup Procedure One

Variable at-a-time

Adjust Process Variables, Take process

measurement

no

in spec?

real-time

Take Product Sample

no

in spec?

with delay

Production

25-D Hypercube

Consequences of Monitoring Multiple Process

Variables One-at-a-Time

- X1 and X2 correlated

X2

good combination

USL

LSL

X1

USL

LSL

bad combination

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

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

Partial Least Squares (PLS) Characterizes Process

Product Variables in Baseline

- Construct PLS component Ts
- Each T is linear combination of Xs
- T1 w11X1 w12X2 w13X3
- T2 w21X1 w22X2 w23X3
- PLS components are independent
- Data reduction 3 5 components contain

sufficient information in data

Construct PLS Component Ts

- T1 w1X1 w2X2 w3X3
- U1 c1Y1 c2Y2 c3Y3
- Find ws and cs (normalized)
- Max Cov(T1 , U1)
- Find ws and cs
- Max Cov(T2 , U2)
- s.t. T2 ? T1

Comparison of Principal Components Analysis PLS

- Both
- Reduce dimension of data
- Components are linear combinations of the Xs
- BUT PLS components consider the Ys
- Xs that are correlated with Ys emphasized in

PLS components

Measure Distance Between Current Process

Baseline Squared Prediction Error SPE

Current process

X2

SPE

PLS model

X1

Baseline data

Calculate SPE

SPE is sum over all process variables

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

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

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)

Graph of SPE for Baseline Data with Control Limit

SPE

Baseline production data 17 batches covering 114

observation points

- If observations Normalthen calculate control

limit - Control limit used to assessstartup

Ad Hoc Use of PLS to Find Adjustment Decompose

SPE

SPE

Startup Batch

2

Time

Contribution of variable i

Variable i

1 2 3 4 5 6 7 8 9..

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

Operator-Assisted Batch Startup

Begin Startup

OK?

Yes

Production

No

Operator may inputprocess variable to adjust

Algorithm recommends adjustment

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

Relationship Between Process Settings and

Variables

- Process variables are a linear function of

process settings

Process Variables X

Setpoints S

Linearmodel

Mathematical Optimization Determine Adjusted

Process Vars xa Settings sa

- Minimize SPE(xa)
- Subject to

Objective Function

- Given current process
- settings sc
- variables xc
- Find adjusted settings
- settings sa
- Minimize SPE(xa)
- distance from adjusted variables to baseline

Predicted from PLS components

Constraint Follow the Operators Recommendation

- ex adjust setting 23 to a new value u
- ex adjust setting 23 to a new value exceeding

the current setting

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

Constraint PLS Components Should Be Within

Reasonable Range

- Compute PLS components, Ts, after adjustment
- Ts should be in a reasonable range

X2

T1w1X1 w2X2

X1

Baseline data

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 Benders Algorithm or Search

Derive SPE as xBx prove B is postive semi definte

About SPE

- B contains
- weights to compute PLS components, t, from

process variables x - loadings to computefrom PLS components t

Example Operator Considers Two possibilities and

Algorithm Helps to Select

- Historical
- t40 adjust v7
- t60 adjust v4, v5, v6
- t210 adjust v5, v6
- t240 adjust v5, v6
- t330 adjust v5, v6
- t360 adjust v7 (start) production
- With algorithm
- t40 input v4 OR v7 output v4, v5, v6
- t50 production!
- Startup reduced 86 from 360 to 50 minutes

Example cont Two possible adjustments at t40

- Adjust v7
- SPE 13.8
- plus other adjustments
- Adjust v4
- SPE 8.3
- also adjust v5 v6
- Select second choice with min SPE

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

Select Better Settings by Accounting for

Uncontrollable Variables

Input raw material, environmental, output stage

n-1)

Process Settings

PROCESS

Feedforward control to reducebatch-to-batch varia

tion

Output

Objective

- Given means and variances for uncontrollable

variables - Identify optimal settings quickly
- Predict whether likely to produce successful

outputs

Extend SPE to Include Uncontrollable Variables

- Original
- Divide x into two groups

Process settings

Uncontrollable variables (random variables)

Optimization Objective Function

- Min Expected Value of SPE
- Select new settings xS
- xu are random variables
- mean vector variance matrix known

Mathematical Optimization Choose Settings xS

to Minimize ESPE

- Subject to

Find xS

Defn of PLScomps

PLS comps in baseline range

Settings withinlimits

Settings depend on mean xu - min ESPE depends on

mean and variances

Best settings only depend on mean of

uncontrollable variables

Min ESPE depends on both means and variances of

uncontrollable variables

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

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

Comparison of Success Rates Ave Baseline vs. Min

ESPE Settings

- 100 scenarios
- uncontrollable variables taken from
- join normal with mean var known
- settings from optimization

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

Sample Estimates of Input Variables Form Joint

Confidence Interval

?1 and ?2 are means of two inputs

?2

Conf Interval

Point Estimate

?1

ConfInterval

Yellow Box is CI for Inputs

ESPE Between Baseline and Uncontrollables Vars

Settings

Current samplelarge ?

X2

PLS model

X1

Baseline data

- CI around current uncontrollables
- ESPE is distance averaged over CI
- ESPE large if CI or distance is large

Compute Confidence Interval for ESPE

Yellow CI on uncontrollable variable means

?2

Max ESPE

Min ESPE

Find ESPE under optimal settings(math program)

Confidence Intervalfor ESPE

Sequential Sampling Algorithm to Determine

Whether to Process Batch

- Compare ESPE CI to 90th percentile of SPEs in

baseline control limit

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

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)

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

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

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