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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

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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

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Startup Stage Accounts for Up to 50% of Batch Time Startup stage Production stage Batch 1 Startup stage Production stage Batch 2 Long time interval

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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

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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

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Traditional Batch Startup Procedure – One Variable at-a-time with delay Production Adjust Process Variables, Take process measurement Take Product Sample no in spec? real-time 25-D Hypercube

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USL X2X2 X1X1 good combination bad combination USL LSL Consequences of Monitoring Multiple Process Variables One-at-a-Time X 1 and X 2 correlated

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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

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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

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Partial Least Squares (PLS) Characterizes Process & Product Variables in Baseline Baseline: Good Production Data Input XsOutput Ys Construct PLS component Ts Each T is linear combination of Xs T 1 = w 11 X 1 + w 12 X 2 + w 13 X 3 + ··· T 2 = w 21 X 1 + w 22 X 2 + w 23 X 3 + ··· PLS components are independent Data reduction: 3 ~ 5 components contain sufficient information in data

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Construct PLS Component Ts T 1 = w 1 X 1 + w 2 X 2 + w 3 X 3 + ······ U 1 = c 1 Y 1 + c 2 Y 2 + c 3 Y 3 + ······ Find ws and cs (normalized): Max Cov(T 1, U 1 ) Find ws and cs: Max Cov(T 2, U 2 ) s.t. T 2 T 1

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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

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Measure Distance Between Current Process & Baseline: Squared Prediction Error SPE SPE X1X1 X2X2 Current process Baseline data PLS model

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Calculate SPE PLS baseline model x1x1 x2x2 xkxk T1T1 T2T2 T3T3 Ts predict Xs Regress X on Ts SPE is sum over all process variables

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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

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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

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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) – 1 Geladi, P. and Kowalski, B.R., (1986) 2 Lindberg, W., Persson, J., and Wold, S. (1983) 3 Wold, S., (1978)

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Graph of SPE for Baseline Data with Control Limit Baseline production data: 17 batches covering 114 observation points SPE If observations ~Normal then calculate control limit Control limit used to assess startup

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Ad Hoc Use of PLS to Find Adjustment: Decompose SPE SPE 2 Variable i Contribution of variable i: 1 2 3 4 5 6 7 8 9….. Startup Batch Time

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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

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Operator-Assisted Batch Startup Begin Startup OK? Production No Yes Operator may input process variable to adjust Algorithm recommends adjustment

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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

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Relationship Between Process Settings and Variables Process variables are a linear function of process settings Process Variables X Linear model Setpoints S

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Mathematical Optimization: Determine Adjusted Process Vars x a & Settings s a Minimize SPE(x a ) Subject to:

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Objective Function Given current process –settings s c –variables x c Find adjusted settings –settings s a Minimize SPE(x a ) –distance from adjusted variables to baseline Predicted from PLS components

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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

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

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Constraint: PLS Components Should Be Within Reasonable Range Compute PLS components, Ts, after adjustment Ts should be in a reasonable range X1X1 X2X2 Baseline data T 1 =w 1 X 1 + w 2 X 2

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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

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

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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

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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

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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

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Select Better Settings by Accounting for Uncontrollable Variables Input raw material, environmental, output stage n-1) Process Settings Output PROCESS Feedforward control to reduce batch-to-batch variation

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

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Extend SPE to Include Uncontrollable Variables Original Divide x into two groups Process settings Uncontrollable variables (random variables)

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Optimization Objective Function Min Expected Value of SPE Select new settings x S x u are random variables –mean vector & variance matrix known

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Mathematical Optimization: Choose Settings x S to Minimize ESPE Subject to: Defn of PLS comps PLS comps in baseline range Settings within limits Find x S

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Settings depend on mean x u - min ESPE depends on mean and variances Min ESPE depends on both means and variances of uncontrollable variables Best settings only depend on mean of uncontrollable variables

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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

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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

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Comparison of Success Rates: Ave Baseline vs. Min ESPE Settings Variance of x s ave baseline settings min ESPE settings reasonable57/10093/100 large61/100Quit batch! Min ESPE > 95 %tile 100 scenarios uncontrollable variables taken from join normal with mean & var known settings from optimization

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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

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Sample Estimates of Input Variables Form Joint Confidence Interval Conf Interval 1 2 Point Estimate 1 and 2 are means of two inputs Yellow Box is CI for Inputs

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ESPE Between Baseline and Uncontrollables Vars & Settings X1X1 Current sample large Baseline data PLS model X2X2 CI around current uncontrollables ESPE is distance averaged over CI ESPE large if CI or distance is large

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Compute Confidence Interval for ESPE Find ESPE under optimal settings (math program) Max ESPE Min ESPE Yellow CI on uncontrollable variable means 2 ESPE Confidence Interval for ESPE

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Sequential Sampling Algorithm to Determine Whether to Process Batch Compare ESPE CI to 90th percentile of SPEs in baseline control limit Sample more Control limit ESPE Input infeasible Proceed ESPE

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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

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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)

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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

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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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