Common cause Special cause Statistical Process Control June 1999
Dr. Dennis Ridley,  Operations Research & Statistics Professor. dridley@scri.fsu.edu

 ABSTRACT
     This case teaches the state of the art  in 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. Traditional statistical process control charts (SPC) have been generating false signals. The case shows how to improve on past efforts by using the  "Moving Window Spectral Method (MWS)" [1] to create separate common cause and special cause SPC charts. The result is accurate out of control signals, including advanced warning of impending failure. The MWS 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, process measurements are plotted on a process control chart. The chart is marked with 3 sigma upper and lower control limits. If any measurements fall outside the 3 sigma limits, the process is stopped, the cause of the breach is determined, the cause is removed and the process restarted. Ever since its inception, the phosphorus process has been a problem.
     Stafford (director of production management) looked puzzled as he explained the problem as follows. “There have been numerous false positives, when the phosphorus process was stopped and no cause  could be determined.” “Also,” he continued, “it appears that there are about as many false negatives, since we have received customer complaints even when there are no out of control signals.”
     “That’s what I have been telling you,” interjected Sean (chemical engineer), “statistical process control is just a guessing game. There are so many variables, how do we ever know if we are looking at the right ones anyway. Did you guys honestly believe that this system could really work? Just  get me the new smelter that I have been asking for. That’ll fix it, you’ll see.” Stafford reassured Sean. “I understand your frustration Sean, but all of your experience is associated with the phosphorus process. We have many other processes throughout the plant where the statistical process control charts are fairly reliable, and while not perfect, they have improved our quality significantly. I have to admit that we have much to learn about the phosphorus process. To help you understand the potential value of control charts I am reassigning you to other processes elsewhere in the plant for the next two weeks, after which time we will reconvene to discuss this matter further. 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 Sean was missing. “O.K., lets get started,” said Stafford, “we can’t wait for Sean. The purpose of this meeting is control charts, and we know that he is agnostic on that subject anyway. Well guys, what do we have.” “You were right Stafford,” said Yolette (cost accountant), it is as you said, we have a problem in the phosphorus process, and it is causing us to lose money.” “We are following all of the same practices as we do in all the other processes,” said Peter (personnel director). “Oh,” interrupted Stafford, “let me introduce Peter, our personnel director. We can’t rule out the possibility that the people operating the phosphorus process require additional training.” “Stress levels are up,” Shannon (medical) warned. “Well what I was about to say,” continued Peter, “is that the only place to train them is right where they are working now. I tell you, they are doing exactly what we ask them to do. Most of them have prior experience on other processes which had fewer problems.” “Good point,” said Jennifer (director of marketing). “However, I am running out of excuses, we could lose some major contracts, so I’m in favor of trying anything.” “Well maybe you won’t have to try just anything, maybe management can try following its own rules for a change,” Sean busted into the room. “You’re late,” said Stafford, “and once again, I am not purchasing any new equipment until you can give good cause that that will solve the problem! Furthermore, the equipment, while not state of the art, is not old.” “Who said anything about equipment?” replied Sean. “I did my tour of duty as you ordered, and things are worse than I thought. It’s a wonder that any of the process control charts work at all.” “What do you mean?” asked Andrew (electrical engineer). “I designed the charting equipment myself.”  “Yeah what exactly do you mean?” chimed Kingsley (purchasing director), “all parts came from reputable suppliers.” “Calm down, the problem is not equipment” explained Sean, “the assumption has always been that we are plotting independent observations from each of the various processes. In fact, that is management policy. Yet in every process that I observed, the charts show that the data being plotted are obviously correlated, phosphorus being the worst of the bunch.” “He’s right,” said Jon (computer scientist). “But that is the way the data is. We can’t change that.”
     “Wait just a minute. Maybe, just maybe we can.” Stafford was beaming. “There is this guy, a professor, I believe his name is Layth. He published a paper that addressed this issue. He recommended that two charts be used. One chart should contain only the random independent part of the data, the other should contain only the correlated part of the data.” “That’s right,” said Yolette. “There is also that other professor. His name is Dennis. He invented a statistical procedure to identify and separate out the two components. It is called the Moving Window Spectral Method (MWS), and is particularly effective when there are several periodic cycles, in addition to trend and randomness, in the data.” “Actually,” remarked Jon, “the MWS method is the basis for the computer program FOURCAST, which we have been using for sales time series analysis and forecasting. We can simply use FOURCAST to identify the correlation structure of the phosphorus data. Better yet, FOURCAST  will even create the two charts (special cause and common cause), referred to by Layth. 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. We can then examine the charts produced by FOURCAST, and make our quality control decisions based on that. 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]) 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. How can management incorporate this model as a quality control planning tool. Incorporate the concepts discussed in [4].  

Key Steps

Data, Plot, Transform, Model, Inverse Transform, Residuals, Fitted values [5][6][7].

Key Related Ideas

Transformation[5][6][7], 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)[8,9], Stationary,  Mean, Variance,  Probability distribution, Mean absolute percentage error, Frequency response, Impulse response, Moving window[10], Antithetic time series analysis[11,12].  

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 of 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 improve on past efforts is to decompose the process data into random independent and correlated components, and analyze each component  separately. The case incorporates the state of the art "Moving Window Spectral Method (MWS)." It is the only method that will perform this analysis automatically. This approach may be classified as a frequency domain method. The computer program FOURCAST [1] carries out this 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 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. By making use of  technical  tools (e.g. statistical & other inference) many observations regarding  the characteristics of the data, potential models, analysis of variance, and  parameter estimates are possible. Use is to be made of the frequency domain, as well as the time domain characteristics of the data.

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).

ACKNOWLEDGMENT
The data for this case was suggested by Professor Layth Alwan.

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.
(request from: mimi@scri.fsu.edu)

[3]  ____________ (1997) “An Internet Learning Center for Time Series Analysis and Forecasting.” www.fourcast.net

[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] Ridley, A.D. "A Model-Free Power Transformation to Homoscedasticity," International Journal of Production Economics,36, p191-202, 1994.

[6] ____________  "Variance Stabilization: A Direct Yet Robust Method," Review of Business, Vol.15,no 1, p28-30, Summer/Fall 1993.

[7] ____________   “Counterbalancing using unequal weights,” Production Planning & Control, Vol. 5, no 2, p193-198, 1994.

[8] ____________   "Price Forecasting and the Financial Markets," Technical Analysis of Stocks & Commodities, Vol. 2, p192-196, 1984.

[9] ____________  "Spectral Analysis and the Financial Markets," Technical Analysis of Stocks & Commodities, Vol.11, p84-87, January 1993.

[10] ___________"Optimal Window Length for MWS forecasting,"  Technical Analysis of Stocks & Commodities, Vol.12, p70-75, March 1994.

[11] ____________ “Combining Global Antithetic Forecasts,” International Transactions in Operational Research, Vol. 2, No.4, p387-398, 1995.

[12] _____________ “Optimal Weights for Combining Antithetic Forecasts,” Computers and Industrial Engineering, Vol. 32, No. 2, p371-381, 1997.

End of Case