Pro-active Statistical Process Control	June 1999
Dr. Dennis Ridley, Operations Research & Statistic Professor. dridley@scri.fsu.edu

 ABSTRACT
     This case teaches the state of the art  in pro-active model based statistical process control.  It  changes the way this type of problem is viewed. Steel Maker Inc., is a large producer of iron and steel ingots. Ingots are shipped to various fabrication plants. Special cause and common cause statistical process control charts (SPC) have been successfully implemented. However, human operators are often preoccupied and miss current out of control signals, as well as fail to diagnose predictable upcoming out of control conditions. The case shows how to improve on past efforts by coupling the  "Moving Window Spectral Method (MWS)" [1] with a diagnostic Neural Network. The result is dependable action on out of control signals, and impending failure, and advanced action to reduce same. The MWS & Neural Network method is the only method that will perform  this  analysis automatically.

CASE
     The production management planning meeting of a large corporation was convened to review the quality control procedures. One of the procedures in place is statistical process control. In that procedure, a time series model decomposes process measurements into two components, namely random residuals and systematic fitted values. The residuals are plotted on a special cause process control chart. The fitted values are plotted on a common cause process control chart. Both charts are marked with 3 sigma upper and lower control limits. If any measurements fall outside the 3 sigma limits of the special cause chart, the process is stopped, the cause of the breach is determined, the cause is removed and the process restarted. If any measurements fall outside the 3 sigma limits of the common cause chart, failure is assumed to be imminent. If the common cause is identified, it is corrected at the next scheduled maintenance shutdown.
     Stafford (director of production management) opened the meeting by thanking and congratulating everyone on their fine efforts in implementing the new time series model based dual chart process control system. “However” he continued, “due to  operator preoccupation with various other details, current out of control signals are often missed.  Also, upcoming out of control conditions which appear to be predictable are going undiagnosed.”
	“I could have told you this would,” interjected Sean (chemical engineer), “the controllers always had enough to do. Now we are asking them to read two charts instead of one.  Did you guys honestly believe that they could diagnose a chemical process from just looking at charts. They have no engineering training..” Stafford reassured Sean. “You make a valid point. There hasn’t been any additional training, much less a professional education. However, the new dual chart system has made significant improvements throughout the plant, more so than in the phosphorus process. Your only experience with it has been on the phosphorus process. To help you gain some additional insights, I am reassigning you to other processes over the next two weeks, after which time we will convene again to discuss this matter. In the mean time I urge the rest of you to give this matter serious thought. We must find an answer.”
     The next production planning meeting was convened two weeks later, right on time, except Andrew (electrical engineer) and Jon (computer scientist) were missing. “O.K., lets get started,” said Stafford, “we can’t wait for Andrew and Jon. The purpose of this meeting is to discuss the production personnel, so I guess we can get by without the computer geniuses today. Well guys, what do we have.” “If you are correct,” said Yolette (cost accountant), there continues to be huge money saving potential if we can improve quality in the phosphorus process.” “The level of training and experience in the phosphorus process exceeds that in other processes,” said Peter (personnel director). “Oh,” interrupted Stafford, “you may recall from our last meeting that Peter is  our new personnel director. Sean did say that the people operating the phosphorus process may require additional training.” “Stress levels continue to rise,” Shannon (medical) warned. “Well what I was about to say” Peter continued, “is that the operators have limited academic schooling, and can only absorb so much on the job training.” “Great, just when our competitors are gaining on us” said Jennifer (director of marketing) “we have a dead end situation here. Well, I suppose our competition will face the same problems. Still, I am  trying to win back some of our old customers. We should try training the operators anyway.”
     “Maybe you won’t have to train the untrainable,” Andrew and Jon busted into the room. “You’re late,” said Stafford, “and what do you mean anyway.” “Well we don’t think anybody can really keep up with the dynamics of the common cause chart,” said Andrew. “I agree,” replied Sean. “I did my tour of duty as prescribed, and it’s not what you think. Everywhere in the plant the operators are so focused on diagnosing the common cause chart, they are falling behind in other work.” “The accident rate has risen too,” nodded Shannon. “I wondered why we were spending so much on medical supplies,” chimed Kingsley (purchasing director). “What is unique about the phosphorus,” said Sean, “is that it is more rapid than other processes. That does not provide for sufficient time to diagnose a related cause.” “In addition,” said Andrew, “the correlation structure depicted is extremely subtle, and requires a keen mind and eye.”  “Well,” said Stafford, “in that case we will simply change the policy. This is a case where moving decision down to the lowest possible level has over shot its target. From now, at the slightest indication of a pattern in the common cause chart, the operator must alert an engineer. The engineer can take it from there.” “I’m afraid that it is not that simple,” said Jon. “The desired reaction time is just not humanly possible, engineer or no engineer!” “Great,” Jennifer sighed, “there goes my bonus.”
     “Wait just a minute. Maybe, just maybe ....” Stafford was beaming. “When our friend, the professor ....Dennis was here to consult on the new dual chart system, I believe he mentioned something about closing the loop electronically.” “That’s right,” said Yolette. “He also mentioned an industrial engineer. His name is Felipe. Felipe believes that a Neural Network can diagnose the common cause chart of fitted values from the Moving Window Spectral (MWS),  time series model, and prescribe the corrective action.” “In that case we could take the output from the Neural Network, and use it to adjust the processes electronically. That is what Dennis meant by closing the loop. The MWS method is particularly effective when there are several periodic cycles, in addition to trend and randomness, in the data. The periodic cycles are deceptive to the human brain, and are very difficult if not impossible to sort out, even by an expert.” “Actually,” remarked Jon, “this type of approach, the MWS & Neural Network method,   is the only feasible way to do it. We can feed  the fitted values from computer program FOURCAST (which we have been using for sales time series analysis and forecasting, and for the new dual chart system), directly to the Neural Network, as shown in Figure 1. The antithetic time series bias reduction feature is applicable.”
     “Well done guys,” Stafford was relieved. “We will dedicate a PC to this task, and simply feed the numbers from the phosphorus process into FOURCAST and the Neural Network program. At a very minimum we can make decisions based on the diagnosis from the Neural Network while we are waiting for the new equipment required to  close the loop automatically. O.K. lets do it!”
     SPC - Statistical Process Control. PC - Personal computer
 
                                   DATA
PHOS (read left to right then down)
0.0432500 0.0410000 0.0422500 0.0388000 0.0422500 0.0412000 0.0377500 0.0367500 0.0388000 0.0377500 0.0388000 0.0357000 0.0387500 0.0422000 0.0410500 0.0410500 0.0388500 0.0400000 0.0422500 0.0422000 0.0400000 0.0422000 0.0357000 0.0346000 0.0422500 0.0411000 0.0367500 0.0357500 0.0388500 0.0432500 0.0445000 0.0399500 0.0400000 0.0375000 0.0388000 0.0367500 0.0378000 0.0378000 0.0378000 0.0422500 0.0400000 0.0390000 0.0367500 0.0390000 0.0400000 0.0400000 0.0390000 0.0432500 0.0390000 0.0378000 0.0357500 0.0378000 0.0411000 0.0422500 0.0377500 0.0388500 0.0412000 0.0432500 0.0454000 0.0465000 0.0443000 0.0455000 0.0411000 0.0465000 0.0422000 0.0443000 0.0422000 0.0422000 0.0389500 0.0389500 0.0422000 0.0400000 0.0455000 0.0482000 0.0465000 0.0422500 0.0442500 0.0445000 0.0477000 0.0432500 0.0432500 0.0487500 0.0476500 0.0487500 0.0507500 0.0476500 0.0453500 0.0432500 0.0453500 0.0454000 0.0475000 0.0465000 0.0455000 0.0432500 0.0432500 0.0475000 0.0378500 0.0367500 0.0475000 0.0497500 0.0518000 0.0497500 0.0482500 0.0507500 0.0530000 0.0432500 0.0476000 0.0453500 0.0442500 0.0476000 0.0476000 0.0497000 0.0487500 0.0466000 0.0443500 0.0432500 0.0443500 0.0432500 0.0412500 0.0432500 0.0443500 0.0432500 0.0443500 0.0443750 0.0444000 0.0422000 0.0443500 0.0455000 0.0465000 0.0455000 0.0465000 0.0477000 0.0422500 0.0422000 0.0400000 0.0412000 0.0412500 0.0432500 0.0432500 0.0445000 0.0455000 0.0445000 0.0412000 0.0465000 0.0455000 0.0465000 0.0497000 0.0487500 0.0453000 0.0420500 0.0432500 0.0420500 0.0454000 0.0465000 0.0432500 0.0422000 0.0477000 0.0477500 0.0487000 0.0508500 
Notes on Input Data
PHOS - Percentage of phosphorus in iron

Question

Develop a model (software is provided [1] based on theory in [2],[5],[6],[7],[8]) that  fits the above data (provided on the internet [3]). Management is interested in identifying and understanding the correlation and random structures of the phosphorus process. Incorporate the concepts discussed in [4]. Use the schematic diagram in Figure 1 to illustrate how management can incorporate the time series model and neural network to provide a pro-active diagnostic quality control planning tool.  Incorporate the concepts discussed in [5],[6],[7] ,[8].

Key Steps

Data, Plot, Transform, Model, Inverse Transform, Residuals, Fitted values [9][10][11].

Key Related Ideas

Transformation[9][10][11], Spectrum, Cycles, Frequency[2], Time Series, Trend, Regression, Historical, Parameters, Statistical inference, 3 sigma limits, Upper control limit, Lower control limit,  Statistical process control, Special cause chart, Common cause chart, t-test[1][2], F-test[1], Decomposition, Analysis of Variance (ANOVA)[12],[13], Stationary,  Mean, Variance,  Probability distribution, Mean absolute percentage error, Frequency response, Impulse response, Moving window[14], Antithetic time series analysis[15],[16], Neural Network [5], [6], [7], [8].  

APPENDIX
Courses & Levels
This case is suitable for  graduate & senior undergraduate courses in production & operations management, management science, operations research, econometrics, systems engineering, industrial engineering and engineering management.

Scope
	Steel Maker Inc. business activities includes research, design and the production of various grades of iron products. Quality control is an integral part their production operation. They  face local and significant foreign competition from other producers.  They have in place, a system for collecting data on the various processes and storing that data in their computers. This data is examined by various means to determine progress as it relates to productivity, quality and cost.
	Steel Maker Inc. wants to explore new methodologies designed to improve their quality and productivity, and to reduce cost. The case is intended to show that the way to take past efforts to decompose the process data into random independent and correlated components, to a higher level, is to integrate automatic on line diagnosis and control. The case incorporates the state of the art "Moving Window Spectral Method (MWS) & neural network analysis." It is the only method that will perform this analysis automatically. The MWS approach may be classified as a frequency domain method. The computer program FOURCAST [1] carries out the procedure, so that neither the instructor nor the student is required to do any programming. The program and the data are provided on the internet. No keyboard data entry is required. The student may therefore focus on problem definition, and on applying the method to the data.

Discussion Question and Answers
     The question that was  asked in the case presentation is directed, yet open ended. As a senior undergraduate student, there are several concepts wherein you may apply your statistical training. In doing so it is unlikely that all facets of the Moving Window Spectral Methodology will be utilized. The case will however provide useful exposure to the methodology, particularly the important concepts of decomposition, random independence and correlated structures.                                                                       
     As a graduate student majoring in quantitative methods, you are expected to answer the question in great depth, with thoughtfulness and specificity. This includes hypothesis testing (e.g. statistical & other inference)  regarding  the characteristics of the data, potential models, analysis of variance, and  parameter estimates. Use is to be made of the frequency domain, as well as the time domain characteristics of the data. You must also discuss the relevance, properties and expectations of the neural network.

The Moving Window-Spectral Model
     Let Y(t), t=1,2,.....,n represent a time series.  In order to estimate a forecasting model, a moving window of length T is defined in the time domain (1). The moving window is used to generate sequences of observations in the time domain, each sequence containing an observation  of length T on the input and output processes and for each frequency in the frequency domain (2) as follows [1][2].
     The general  discrete time domain AR(T) model relating the model variables is given by
                                       N    T
       	            Yi(t) =      Yj(t-k) Bij(k)+Ei(t)                            (1)
                                       j=1 k=1
where N = the number of variables in the model 
Bij(k)  =  system parameter,  coefficient  of  Yj in equation i,  lagged k time periods.
Ei(t) = the unobservable error term in  the ith equation.
 i = 1,2,3,...,N.     t=T+1,T+2,...,n.
Transforming the time domain model to the frequency domain yields
                                      n
                        Ymi(w)=    Ym-1,j(w) Bij(w)+Emij(w)                         (2)
                                      j=1 
for  m=2,3,..,n-T+1
and where the frequency   -w
where Y(w) is the frequency domain counter part of Y(t) and B(w) is the spectral density function of the impulse response function B(t).

Neural Network Model
See [5],[6],[7],[8].

ACKNOWLEDGMENT
The neural network  for this case was suggested by Felipe Llaugel, Universidad Dominicana O&M.

REFERENCES
[1] Ridley, A.D.  "FOURCAST - Multivariate Spectral Time Series Analysis & Forecasting (MWS),"  computer program and practice sample case study data (version 1999.5 or later) available in micro computer (pc) laboratory. Otherwise ask your professor to obtain (and install) the educational or full professional version from Engineering Management Consultants, P.O.Box 12518, Centerville Road, Tallahassee Fl 32317.
Tel (850) 668 0635. Fax (850) 668 5937 www.fourcast.net
 [2]  _____________ "The Global Univariate Moving Window Spectral Method," Technical Report FSU-SCRI-94-17, Supercomputer Computations Research Institute, Florida State University, Tallahassee Fl 32306.      1994.    Tel (850) 644 1010.
[3]  ______ (1997) "An Internet Learning Center for Time Series Analysis and Forecasting." http://www.polaris.net/~fourcast/
[4] Alwan, L. C.  and D. Radson, “Implementation issues of  time-series based statistical	process control, Production and Operations Management, 4, 3, p263-276, 1995.
[5] Chen, C., “A Multi-layer Neural Network Model for Detecting Changes in Process Mean,” Computers and Industrial Engineering, Vol. 28, p51-61, 1995.
[6] Ramirez-Beltran, N.D., and F. Llaugel “The Ratio Control Chart and Pattern Recognition,” ASQC 48th Annual Quality Control Proceedings, p583-590, 1994.
[7] Hwarng, H.B., and N.F. Hubele. “X-bar Control Chart Pattern Identification Through Efficient Off-line Neural Network Training,” IIE Transactions, Vol. 25, No. 3, p27-39, 1993.
[8] ___________ “Back-propagation Pattern Recognizers for X Control Chart: methodology and performance,” Computers and Industrial Engineering, Vol.24, p219-235, 1993.
[9] Ridley, A.D. "A Model-Free Power Transformation to Homoscedasticity," International Journal of Production Economics,36, p191-202, 1994.
[10] ____________  "Variance Stabilization: A Direct Yet Robust Method," Review of Business, Vol.15,no 1, p28-30, Summer/Fall 1993.
[11]   ____________  “Counterbalancing using unequal weights,” Production Planning & Control, Vol. 5, no 2, p193-198, 1994.
[12]  ____________ "Price Forecasting and the Financial Markets," Technical Analysis of Stocks & Commodities, Vol. 2, p192-196, 1984.
[13]  ____________"Spectral Analysis and the Financial Markets," Technical Analysis of Stocks & Commodities, Vol.11, p84-87, January 1993.
[14] ____________"Optimal Window Length for MWS forecasting,"  Technical Analysis of Stocks & Commodities, Vol.12, p70-75, March 1994.
[15] ____________  “Combining Global Antithetic Forecasts,” International Transactions in Operational Research, Vol. 2, No.4, p387-398, 1995.
[16]  ____________  “Optimal Weights for Combining Antithetic Forecasts,” Computers and Industrial Engineering, Vol. 32, No. 2, p371-381, 1997.
[17] _______and Llaugel “Feedforward Process  Control: an mws model,” Refereed proceedings of the 10th annual Production & Operation Management conference, Mar. 20-23, 1999, Charleston, South Carolina. www.poms.org 

RESIDUALS

Neural Network

PRODUCTION
 PROCESS

FITTED
VALUES

special
cause
chart

common
cause
chart

forecast
chart

FORECAST
Estimated 
current
output

Moving Window Spectral
antithetic time series analysis & forecasting
Model

Feedback controller
(trim)

Current
output

Historical
output

Figure 1. Moving Window Spectral Neural Network Feedforward Process Control.

........................

Indirect Feedforward controller
(systematic predictable drift)

Controlled output

TIME

Manipulated
 input

X|

Dynamic compensation

Direct Feedforward controller
(random unpredictable error)

Unmeasured disturbances

Measured disturbances