2.7

Empirical Process Optimization D. W. HOYTE

(1995)

B. G. LIPTÁK

Suppliers of Optimization Software:

(2005)

DMC; Gensym; Setpoint; Statistical Programs; Ultramax ($1000 to $20,000)

The optimization of control loops is discussed in Section 2.20, and the optimization of such unit operations as chemical reaction or distillation is covered in Chapter 8. Here the subject is the optimization of empirical processes, but before that discussion, it is appropriate to say a few words about optimization in general. LEVELS OF OPTIMIZATION The highest level of optimization is business- or enterprisewide optimization, which includes not only the manufacturing process, but also the optimization of the raw material supply chain and of the packaging and product distribution chain. This is a higher and more important level of optimization than process optimization alone because it requires the simultaneous consideration of all three areas of optimization and requires the finding of enterprise-wide operation strategies that will keep all three areas within their optimum areas of operation (Figure 2.7a). Plant-wide optimization also involves more than the optimization of the unit processes because it must also consider Process

Distribution intensive Supplier

Sourcing intensive Manufacturing intensive

Distributor/ consumer

Batch Processes

Discrete

FIG. 2.7a Enterprise-wide optimization requires the optimization of not only the manufacturing plant but also the raw material supply chain and the product distribution chain.

documentation, maintenance, scheduling, and quality management considerations. Plant-wide optimization involves the resolution of the sometimes conflicting objectives of the many unit operations and the envelope strategies required to optimize the entire plant. At the unit operations level, it goes without saying that multivariable optimization cannot be achieved when individual processing equipment is defective or when control loops are not properly tuned. In addition, it is important that measurements be sampled fast enough and that loops be tuned for sufficiently fast rates of recovery. Loop cycling must also be eliminated, which usually requires the elimination or correction for interactions between loops. When no mathematical model can describe a process and therefore the process can only be optimized experimentally, empirical optimization is required. This is the subject of this section. EMPIRICAL OPTIMIZATION The performance of some processes can be described by theoretical mathematical models. In these cases, mathematical techniques are available to determine the best operating conditions to satisfy selected performance criteria and 1 thereby optimize the process. There are, however, many processes for which no adequate mathematical model exists. These are known as empirical processes, and they can only be optimized experimentally. This section discusses approaches to finding the optimum process settings by experiment. The discussion concentrates on the Ultramax method, which is appropriate to a wide variety of processes and for which excellent user-friendly computer tools are available. Optimization The term “optimize” is used here as it refers to a measurable criterion or quality of the process (Q) that the user wishes to maximize or minimize. For example: 1. Plant production rate is to be maximized. 2. Cost-per-ton of output, calculated from many measurements, is to be minimized. 157

© 2006 by Béla Lipták

Control Theory

3. Flavor of a product, rated 1 to 10 by a group of tasters, is to be maximized. 4. Height/width ratio of a chromatograph peak is to be maximized. Q is sometimes referred to as the target variable or the objective function. Several contending criteria may be given weights to produce a single measurement of process perfor2 mance. To simplify the discussion, the examples here will be limited to maximizing the value of Q. Optimizing the process may include maximizing Q while other measured results of the process (R, S, etc.) remain within specified limits (e.g., maximizing plant production rate in making butter, paper, or sulfuric acid must be subject to a specified limit on water content in each product).

Information Cycle Times Some processes have inherently long information cycle times; resistance to weathering of different paint formulations or increase in farmland productivity from different mixtures of synthetic fertilizers are examples. In both cases, the search for an optimum would take an unacceptably long time if the results from one test were awaited before deciding the conditions for the next test. Therefore, multiple formulations of paints and fertilizers must be planned and then tested in parallel at the same time. This type of planning, and the subsequent analysis of the results, is often called parallel design of experiments.

Parallel Design Taking a specific example: A material is baked for time duration (D), at an oven temperature (T ), to result in a quality (Q), which is to be maximized. This process could be, for example, the case-hardening of steel, the polymerization of resin, or the diffusion of ions in silicon chips. Only D and T are manipulated to affect Q. Processing is performed with a selected array of D and T values, and the corresponding values of Q are measured. The optimum can be found graphically by drawing contours of Q on a graph showing the points (D1, T1), (D2, T2), and so on. Optimum operating conditions for the process are the values of D and T at the top of the “hill.” The contours of Q are sometimes called the response surface of the process and can be fitted to the data points 3 mathematically, instead of sketching contours by eye. This has clear advantages when there are many manipulated variables in the process. The math is simplified by selecting the array of experimental points to suit it. Also, the analysis is simplified by limiting the equation for the response surface to the second order. Note that while statistical texts often refer to D and T as the controlled variables, an engineer will usually refer to D and T 4 as manipulated variables and to Q as the controlled variable. Some commercially available computer programs for parallel designs of experiments are in the nature of a collection of routines that are useful to an experienced industrial statistician.

© 2006 by Béla Lipták

The practicing engineer is likely to work faster with a user5 friendly, menu-driven package such as Catalyst or Discover. Sequential Design Many processes have short information cycle times. In such cases, the results from using particular process settings become available in time to help in deciding the settings for the next test. This method is often called sequential design of experiments, or hill climbing. Using the previous example, three different values for the settings D and T are evaluated in this process. The slope of the response surface defined by the values of Q1, Q2, and Q3 shows the direction “up the hill” to increase Q. The next values of D and T are selected to proceed in this direction. Repeating this procedure, the experimenter can “hill-climb” step by step, toward the maximum value for Q. If the process has many independent variables and also has some noise that is affecting the results, it can be difficult to determine how each variable is affecting the quality of the product (Q). The Simplex search procedure works (finds the maximum value of Q) all the time, with one more process result than the number of independent variables. In the example used here, it would require three process results. Figure 2.7b illustrates the method. To find the next settings for the process, a line is drawn from the settings with the lowest result, through the midpoint of the opposite face, to an equal distance beyond. In this example, the search reaches optimum with 20 data points. The Simplex search was invented by William Spendley at ICI Billingham, England to automate the evolutionary operation (EVOP) procedures of G. P. Box and others, and the methods are amenable to implementation using paper and 6,7 pencil or a desk calculator. Reference 8 provides a practical treatment of the subject in which the authors review related computer programs.

14 Duration (hours)

158

13 20

4 5

3

2 4

1 3

Temperature (deg F)

2

1: Lowest result. Project to #4 Via mid point, O.

FIG. 2.7b SIMPLEX search for a maximum of quality, Q. Contours of Q are shown, though usually unknown. Inset shows how the next process settings are found. Direction changes where #14 has lower Q than #13.

2.7 Empirical Process Optimization

Main menu of ultramax functions. F Formulate problem… E Enter run data… L Build models… A Advice… W What-if analysis… D Data management… H Historical data report… P Plot data… Q Quit ultramax. function--> −

Problem formulation: find duration and temperature settings to give max quality subject to limits on volume VAR # Name 1 2 3 4

Temperature Duration Volume Quality

Units

Type

Deg. F Hours Cu.In. Score

1 1 5 6

Prior Region LO HI 450. 6. 30. 5.

500. 7. 31. 6.

Constraints LO HI

Providing Process Data 30.

31.

FIG. 2.7c Main menu and a Problem Formulation for the Ultramax program. In the formulation, “Type” designates treatment of the variable. Note that all Ultramax reports have been simplified here to highlight concepts. (Courtesy of Ultramax Corp., Cincinnati, OH.)

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1. Q is only acceptable when V is ranged from 30 to 31 cu. in. 2. TYPE code specifies how the program is to treat each variable, as follows: Types 0 and 4 are for data-recording only. (Types 0, 2, and 4 are not used in this example.) Type 1 is an independent (manipulated) variable. Type 2 is an independent variable that affects the target but cannot be manipulated (e.g., the time of day, the price of crude oil, a process disturbance).

After the formulation of the problem, the menu function E (Enter Data) is used next. The user can enter, at one time, one or many data points: for example, sets of measurements of D, T, V, and Q from the process. Process data can be keyed in or can be read in from a file. Although users often have extensive data to enter from prior process runs, they may wish to obtain and enter data points one at a time after the initial loading in order to obtain advice from Ultramax on the recommended process settings to use next. This way the optimum will be reached with the fewest runs of the process. The search up through run #14, following the advice of Ultramax, is shown in Figure 2.7d. This starts with the same three data points used in Figure 2.7b and explores the contours

H

Bayesian Methods Bayesian statistics are designed to match the human decision-making process in situations where few data are available. They are used to help in selecting the next process settings in a sequential hill-climbing program, 9 Ultramax, designed for the PC or mainframe computers. Ultramax is a menu-driven, engineer-oriented package. The following illustration uses the baking process mentioned earlier, but with the addition of a constraint: volume (V) per unit of product. Here, volume and quality are both affected by duration and temperature of baking; the goal is to find settings of D and T to give maximum Q while holding V between specified limits. Figure 2.7c shows the Main Menu of Ultramax. Keying F prompts the user to specify the “Problem Formulation” shown there. (Note that all examples shown have been simplified to clarify the concepts.) Salient points of this formulation are as follows:

© 2006 by Béla Lipták

Type 5 is a dependent (results) variable with constraints. Type 6 is the target variable. 3. One Type 6 target variable must be specified in the formulation; several variables of other types may be specified. The standard version of Ultramax can deal with ten Type 1 variables, and all can be changed at will in each step of the search. 4. The “Prior Region” specifies the range where the user expects the optimum to lie and wishes the search to start. Note: A parallel design of experiments for two independent variables requires a dozen data points in this region and in every further region selected for exploration.

Duration (Hours)

*** Enter

159

6

300 350 400 Temperature (Deg F)

450

500

FIG. 2.7d Ultramax search for process settings to give maximum quality (Q) without violating limits on volume (V). The contours are not known before the search.

160

Control Theory

of Q as it seeks the maximum. At the same time, the search is recognizing the contours of V and locating the barrier of V = 31 cu. in. By run #14, the search is close to the optimum settings for duration and temperature in this process. (The contours are shown for illustration only; they are not, of course, known while the search is being made.) Search Procedure After process data have been entered, menu function L (Build Model), is used to fit a second-order model to the data points. This is illustrated in the “Coefficients Report for Quality,” which appears after the data for run #13 have been entered (see Figure 2.7e). (A similar report obtained for the model of V is not shown here.) The user seeks process settings that give the largest value of Q and is only interested in fitting the model in that region. The data points used earlier in the search may detract from that fit; if they do, Ultramax automatically discards them. The actual points used in the model are listed at the bottom of Figure 2.7e together with their errors. “Minimum Prediction Error” at the top of Figure 2.7e gives a single value for goodness-of-fit.

Model coefficients for variable #4: quality. problem: maximum quality subject to limits on volume Model type: local fixed-point-centered quadratic Minimum prediction error = 0.26 score Model Equation: F(x) = + 12.28 − 0.0392 + 1.139 − 0.00006016 + 0.009428 − 0.289 Key F(x) X(1) X(2)

Variable Quality Temperature Duration

∗

X(1) X(2) ∗ X(1)∗∗2 ∗ X(2)∗X(1) ∗ X(2)∗∗2

Optimum 14

Advice 15

Advice 16

Advice 17

Process Settings 1 Temperature Deg. F 2 Duration Hours

303 7.44

321 7.52

324 7.86

359 9.27

Expected Results 3 Volume Cu.In. Expected Error 4 Quality Score Expected Error

31.00 0.19 8.6 0.6

30.91 0.19 8.5 0.6

30.81 0.19 8.4 0.6

30.33 0.39 7.6 1.2

1.1

1.0

1.0

2.8

Advice #

Standard distance (“Boldness”)

FIG. 2.7f Example of an “Advice Report” showing Advices for the next settings to run the process after run #13. “Optimum” gives the best process settings based on present data. “Standard Distance” is the estimated risk in the results from each Advice (similar to confidence limits). Maximum Standard Distance has been set by the user at 3.0. (Courtesy of Ultramax Corp., Cincinnati, OH.)

After the model-building step, the menu function A (Generate Advice) gives a report for a selected number of “Advices,” as shown in Figure 2.7f. The column headed “Optimum” gives the best calculated settings for the process, based on the data at that stage. The user can then follow Advices for the next process settings to explore further for the optimum, in the priority order that Ultramax sees as the need for information. The user may choose to select several or only one of the Advices for the next process settings, or the user can use settings to explore the process based solely on the user’s own knowledge and intuition about the process. Ultramax will use any data points it is given.

∗

Units Score Deg F Hours

Type 6 1 1

Datapoints Used and the Errors of the Model: Run #

Actual quality

Model quality

Error

13 12 11 10 9 8 7 6

8.00 8.10 8.00 8.76 8.62 8.09 7.78 6.71

7.97 8.06 8.19 8.43 8.70 8.16 7.70 6.74

−0.03 −0.04 0.19 −0.33 0.08 0.07 −0.08 0.03

FIG. 2.7e Example of a Model Coefficients Report: Run #13. Goodness-of-fit is shown by minimum prediction error and list of model errors. Ultramax fits the model in the region of the maximum quality, as desired by the user. It has discarded outlying data points numbers 1 to 5. (Courtesy of Ultramax Corp., Cincinnati, OH.)

© 2006 by Béla Lipták

Advice Report: Seeking Maximum Quality Subject to Limits on Volume

Finding the Optimum

Sequential cycles consist of:

Data Entry Build Model Get Advices Run Process Data Entry and will initially lead to data points in the general region of the process maximum. In this phase of the search, the size of steps made in changing the process settings is limited by a “Boldness-of-Search” parameter that the user specifies. Thus the user can choose either to search quickly or to make smaller and more cautious changes. This is somewhat similar to tuning one of the process control loops. The “boldness” of each Advice is ranged from 0 to 3 and is shown as “Standard Distance” in Figure 2.7f. The search continues until the distinction between the best process settings is limited only by the repeatability (noise) of the process. At this point, changes in the “Optimum” value from one process-run to another for each manipulated variable (Figure 2.7f ) will bottom out.

2.7 Empirical Process Optimization

Process Sensitivity The model can be explored with menu function W (What-If Analysis). This evaluates Q and V from the models for any desired settings of D and T, showing the sensitivities of Q and V to changes in each independent variable. Ultramax may be called periodically by a process control program such as The Fix or Onspec (see References 10 and 11) to calculate new values for the setpoints of manipulated variables. The program will perform a reasonableness check, then transmit these setpoints to its own process-control algorithm or to hardware controller setpoints.

6.

7. 8.

9.

10.

Process Repeatability Where the repeatability of the process is not adequate for analyzing for optimum, the following steps can be considered: 1. Average many measurements of a variable to reduce “noise.” 2. Improve repeatability of the measuring devices through recalibration and other means. 3. Seek further variables that affect the process and are not yet in the model. 4. Do a time-series analysis to investigate interactions 12 and delayed actions. 13

Programs are available to make this analysis. A userfriendly interface was written for them, but it is not now 14 available in the United States. However, some controlsystem contractors, such as the ACCORD system, offer time15,16 series analysis.

11. 12. 13.

14. 15.

16.

17.

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Spendley, W., et al., “Sequential Application of Simplex Designs in Optimization and Evolutionary Operation,” Technometrics, November 1962, pp. 441–461. Box, G. E. P., “Evolutionary Operation: A Method for Increasing Industrial Productivity,” Applied Statistics, No. 6, 1957, pp. 3–23. Deming, S. N., et al., Sequential Simplex Optimization: A Technique for Improving Quality and Productivity in Research, Development and Manufacturing, Boca Raton, FL: CRC Press, 1991. Moreno, C. W., “Self-Learning Optimization Control Software,” Instrument Society of America, ROBEXS ’86 Conference, Houston, TX, June 1986. (Available from Ultramax Corp., 1251 Kemper Meadow Drive, Cincinnati, OH 45240.) The Fix, from Intellution, Inc., 315 Norwood Park South, Norwood, MA 02062. Onspec, from Heuristics, Inc., 9845 Horn Road, Mail Stop 170, Sacramento, CA 95827. Findley, D. F., Applied Time Series Analysis, II, New York: Academic Press, 1981. Akaike, H., et al., “Time Series Analysis and Control: TIMSAC-74, -78 and -84 Program Packages,” Computer Science Monographs, Nos. 5, 6, 11, 22, and 23, Institute of Statistical Mathematics, 4–6–7 Minami-Azabu, Minato-ku, Tokyo, Japan 106, March 1985. Schneider, P., private communication, January 1992. Nakamura, H., and Uchida, M., “Implementation of Optimum Regulator at Supercritical Thermal Power Plants,” Conference on Power Plant Controls and Automation, Miami, Florida, February 7–9, 1989. Proceedings available from Electric Power Research Institute, 3412 Hillview Avenue, Palo Alto, CA 94304. Nakamura, H., et al., “Optimum Regulator Implementation by Microprocessor-Based Distributed Control System,” Conference on Power Plant Controls and Automation, Miami, Florida, February 7–9, 1989. Proceedings available from Electric Power Research Institute, 3412 Hillview Avenue, Palo Alto, California 94304. Brown, J. M., University of Manchester Institute of Science and Technology, private communications, 1986–1989.

CONCLUSIONS Bibliography The techniques of optimizing an empirical process owe much 17 to the insights and work of the late James M. Brown. These methods were once the sole preserve of the specialist statisticians but are now available in forms that are suitable for use by the practicing engineer. Industrial user companies seldom advertise the successes they are achieving by the use of these tools, but information and assistance are readily available from the suppliers of optimization software. References 1. 2.

3. 4.

5.

Reklaitis, G. V., et al., Engineering Optimization: Methods and Applications, New York, Wiley, 1983. Derringer, G., and Suich, R., “Simultaneous Optimization of Several Response Variables,” Journal of Quality Technology, October 1980, pp. 214–219. Myers, R. H., “Response Surface Methodology,” Virginia Polytechnic Institute and State University, Blacksburg, 1976. Cubberly, W. H. (Ed.), Comprehensive Dictionary of Instrumentation and Control: Controlled Variable, Research Triangle Park, NC: Instrument Society of America, 1988. Catalyst and Discover, from BBN Software Products, 150 Cambridge Park Drive, Cambridge, MA 02140.

© 2006 by Béla Lipták

ARC Report, “Batch Process Automation Strategies,” ARC Advisory Group, October 1999. Box, G. E. P., et al., Statistics for Experimenters, New York: Wiley, 1978. Draper, N. R., and Smith, H., Applied Regression Analysis, New York: Wiley, 1981. Ghosh, A., “Maximizing the Potential of Batch Process Control,” presented at WBF Conference, Brussels, October 2, 2000. Luyben, W. L., et al., Plantwide Process Control, New York: McGraw-Hill, 1998. Nachtsheim, C. J., “Tools for Computer-Aided Design of Experiments,” Journal of Quality Technology, Vol. 19, No. 3, July 1987, pp. 132–159. Ruel, M., “Stiction: The Hidden Menace,” Control Magazine, November 2000. Savas, E. S., Computer Control of Industrial Processes, New York: McGrawHill, 1965, pp. 55–77. Shinskey, F. G., Process Control Systems, 4th ed., New York: McGraw-Hill, 1996. Smoak, R. A., “Control Design Methodology for Power Plants,” EPRI Conference, 1989. Storm, M. E., “Sequential Design of Experiments with Physically Based Models,” Master’s Thesis, Dept. Mech. Engineering, MIT, April 1989. Wong, K. Y., et al., “Computer Control of the Clarksville Cement Plant by State Space Design Method,” Cement Industry Conference of IEEE in St. Louis, Missouri, May 1968.