tivariate control chart is often preferable statistical control occurred for both the MDF and OSB samples. Multivariate control charts ofHotelling's T 2statistic ...
COMPOSITES AND MANUFACTURED PRODUCTS
MULTIVARIATE CONTROL CHARTS OF MDF AND OSB VERTICAL DENSITY PROFILE ATTRIBUTES TIMOTHY M. YOUNG ? PAUL M. WlNISTORFER ? SIQUN
WANG _
ABSTRACT
trol charts needed also increases. Often a
The vertical density profile of wood composite panels is influenced by many variables in the manufacturing process and is an important product attribute for composite panel end-users. Monitoring the quality of product attributes, such as the vertical density profile (VDP), is a critical step in ensuring the delivery of product value to the end-user and for maintaining the competitive position of a firm. In this study, multivariate control charting procedures using Hotelling's T 2 statistic for correlated VDP variables were comparedwithtmivariate control charts derived fromthe same VDP variables. Comparisons were based on samples obtained from typical production runs of a manufacturer of medium density fiberboard (MDF) and a manufacturer &oriented strandboard (OSB). The MDF samples were taken from 3/4-inch stock and the OSB samples were taken from 23/32-inch stock. Statistical process control (SPC) is intended
single batch, or even a single item, will have several related attributes, as is the case with vertical density profile (VDP)
to prevent the manufacture of defective product that may otherwise occur using traditional quality-control procedures. The Shewhart control chart is the primary tool of SPC, which separates variation as either "special cause" or "random." Even though Shewhart control charts provide a sound method for detecting problems in manufacturing processes that may otherwise go undetected, such charts are univariate in nature and have limitations for both uncorrelated and correlated variables. The typical analysis of univariate control charts for correlated VDP variables revealed that false signals of statistical control occurred for both the MDF and OSB samples. Multivariate control charts ofHotelling's T 2 statistic provided a more robust (less false-signals) method for control charting of correlated variables,
Statistical process control (SPC) has evolved over the last five decades to include a wide range of statistical methods to improve process productivity and product quality. While the Shewhart control chart (17) is one of the most prominent industrial statistical techniques, other techniques such as design of experiments, evolutionary operation (EVOP), response surface analysis, time series analysis, and multivariate analysis have become extremely useful to all manufacturing industries (2,3,5,6,12,20). Shewhart control charts are univariate in nature and provide useful insight into FOREST PRODUCTS JOURNAL
the type of variability for any individual process variable or product attribute, i.e., quantifying variation as "special cause" or "random." However, as the number of process variables and product attributes increase, the number of univariate con-
data for wood composite panels. Separate Shewhart univariate control charts provide the greatest insight into each variable, but taken independently they do not provide a characterization of how close a batch or item may be to a collective target value (21). Multivariate control charts are needed for controlling and understanding the multivariate mean of a process (16). As Jackson (9) notes, '_hen two or more variables are being controlled simultaneously, the control region should take into account the relationships between (among) them." If variables are not independent, a multivariate control chart is often preferable for joint monitoring of all variables in order to control the probability of a false alarm(14).Independent controlcharts for related variables may lead to false signals of statistical control (9). As Nedumaran and Pignatiello (14) note, ifc_ denotes the probability of a false alarm (the probability of a Type I elxor) for any one subgroup on each of the p individual univariate control charts, and the vailables being monitored are independent, the overall probability of a false alarm for
The authors are, respectively, Research Assistant Professor, Professor and Director, and Post Doctoral Research Associate, The Univ. of Tennessee, Tennessee Forest Prod. Center, 274 Ellington Plant Sci. Bldg., Knoxville, TN 37901-1071. This work was supported by the Tennessee Agri. Expt. Sta., project MS #70, and by the USDA NRI award #95-37103-2104. This paper was received for publication in June 1998. Reprint No. 8825. I"Forest Products Society Member. ©Forest Products Society 1999. Forest Prod. J. 49(5):79-86.
VOL. 49, NO. 5
79
any one multivariate subgroup on all of thep univariate charts is o( = 1- ( 1- o_)p. For example, if five separate univariate control charts are used to monitor five
wood composite panels commonly monitor product quality by analyzing the overall shape of the VDP and associated key attributes such as overall density,
where p is the number of variables, n is
independent variables and the probability of a false alarm is 0.0027 for each chart, then the overall probability of a false alarm is 0.0134, approximately five times the false alarm probability ofincorrectly deciding that the process is out of control (14).
top-face density, bottom-face density, and core density, or by ratio relationships among these attributes. Some producers monitor product quality by univariate control charting of these attributes. The authors are not aware of any application of multivariatecontrol charting of VDP
the sample size, and F(p,n-p) refers to the F distribution. HOTELLING'S T 2 STATISTIC AND CONTROL LIMITS FOR SUBGROUPED DATA In practice, the population mean g is generally unknown, so it is necessary to
Multivariate procedures for control charts are based heavily upon Hotelling's T 2distribution, which was introduced by Hotelling (8). Even though other multivariate charting techniques exist, such as Standardized Euclidean Distance (Dp) and Composite Multivariate Quality Control (CMQC), Hotelling's T z statistic has been widely accepted for multivariate control charting (4,16,21).
estimate tx analogous to the way it is estimated when an X-bar chart 2 is used, i.e., for k subgroups of size n, tx is estimated by x. For a multivariate control chart using subgrouped data, one would plot:
VERTICAL DENSITY PROFILE For over three decades, researchers and industrial producers in the wood products industry have noted the formation of a density gradient through the thickness of flat-pressed wood panel products as a result of hot-pressing (19,23). A density gradient through the panel thickness is typically reflected by the presence of a higher-density face
attributes, As Jackson (9) emphasized, multivariate control charts were designed for use at the shop level where it was believed that an approximate multivariate tool is better than none at all. He also noted that considerablymore insight into the nature of the process can be gained from the use of multivariate control charts. Multivariate control charting of VDP product attributes may provide a more accurate method of monitoring product quality than univariate control charting by reducing false signals of statistical control and ensuring that the product mean is on target. Such techniques are critical for an improved understanding of product variability and may enhance overall panel performance (26). METHODS HOTELLING'S T 2 STATISTIC
layer and a lower-density core layer within the panel (25). VDP will be the nomenclature used throughout this paper to describe this density gradient, VDP attributes, such as density in the face and core layers, are of interest to
Hotelling's T 2 distribution, introduced by Hotelling (8), is a multivariate analogue of the univariate Student's t distribution (t). Recall that for a random sample of size n from a normal distribution with mean Ixand variance o-2:
UCL = (knp - np (kn -- kp k -p + +1)p) x
panel producers because of the effect such attributes have on surface quality, profiling characteristics, fastener per-
2_g t - s / _-
formance, bending The properties, internal bond strength. ability toand measure and analyze the VDP of pressed-wood panels has improved in recent years and most wood panel producers and re-
has a Student's t distribution - 1 degrees of freedom.1 It followswith that nwhen interest involves symmetric relationships oft:
CL = (knp - kp -np +p) x (kn - k - p + 1) F.50(p,_- k-p - 1)
searchers currently have access to some type of automated, nondestructive gamma or x-ray densitometry measurement device (11,23,24). Producers of
t2 _ ( _ _ Ix)2 s2/n
[2]
There is no lower control limit for the multivariate control chart using Hotelling's T 2 statistic. If any of the Tf
=n(Y-_t)(s2)-l(Y-ix)
[3]
[1]
may also be used. When Equation [3] is generalized top variables, it becomes: 1Note, _ mean,
is the samplemean,g is the population deviation, and s z is the
s is the standard
samplevariance,
2 An "X-bar
chart"
is a Shewhart
univariate
control
chart where the averagesfor each subgroupare plotted in a time-ordered sequence starting with the
oldestdate.The centerlineof the controlchartis the average
of all the subgroup
averages;
the upper
and lower control limits are approximations of 3sigma limits (22).
80
[4]
[5]
7"/2= n ( x (j) -x )' Sp- l( x (/) -x ) [6] for thejth subgroup (j = 1, 2,... k), where Y(.)jdenotes, a vector withp elements that contains the subgroup averages for each of the p characteristics for the jth subgroup. Sp-1 is the inverse of the ']9ooled" variance-covariance matrix S,P which is • , . obtained by averaging the sutsgroup vanance-covariance matrices over the k subgroups (see ref. 16, pp. 215-223). Each of the k values of Equation [6] would be compared with the upper control limit (UCL):
F.997(/,, kn- k-p- t) [7] Note that 1 - ct = 0.997 follows control chart convention that the upper and lower control limits are approximately plus or minus three standard deviations from the
central line. The central line (CL) for this subgrouped data is:
[8]
values exceed the UCL from Equation [7], the corresponding subgroup(s) would be investigated and the process adjusted. See Ryan (ref. 16, pp. 219-220)
where Yand Ixare vectors of size p, S-1 is
for modifications Hotelling's T 2statistic for monitoring totilture production. Since this is multivariate data, it is nec-
the inverse of the sample variance-covariance matrix S, and n is the sample size upon which each 2i' i = 1, 2, ..., p, is based. When _t is equal to go for the null hypothesis, then T 2 is distributed as:
essary to determine which quality attribute(s) is causing the "out-of-control"signal. Since this signal may not occur on univariate control charts, it is advisable to analyze the degree of correlation of the
T 2=
n( _ - g)' S-1( y _ IX)
p (n - 1) x F(p n- p ,n-p)
MAY
1999
TOII Face Den&v
attributes and the sign of the correlation. A comparison of the correlations, univariate control charts, and multivariate control charts may be necessary to determine the cause of the out-of-control signal. Alt (l), Hayter and Tsui (7), Jackson (9), and Nedumaran and Pigatiello (14) have each developed methods for identifying the source of out-of-control signals for multivariate control charts.
66 66
Girded Areas signal of “Run-of-Seven” indicating a shift in process average
\
I UCL
64 62
“s: p 60
Average
= 56 56
HOTELLING'S T2 STATISTIC ANDCONTROLLIMITS FOR INDIVIDUALOBSERVATIONS Core Density
A multivariate analogueof the “X Individual charV3 waspresentedby Jackson (9). As Ryan (16) discussed,if pnindividual multivariate observations are to be usedfor estimatingthe meanvector and variance-covariancematrix (the estimates denotedby x, andS,), for eachobservation, x, would be usedin computing: T2 = (x -Z,,J’S,,-1(x -X,)
Average
[9]
andcomparingit againstthe UCL: ucL=p(m
+ l)(m - 1) x m2-mp mm-p) F.99~3
Bottom
WI 66 66 64
where, as before, p denotesthe number of variables. The central line (CL) for individual observations using this methodis:
m -P)
= 58 56
WI
MEDIUM DENSITY FIBERBOARD DATASET
A data set for medium density fiberboard(MDF) wasobtainedfrom a manufacturer in the southeastern United States.This manufacturerusesa furnish consisting primarily of a mixture of southernpines(Pinustaeda,Pinuselliot-
An “X-Individuals chart” is a Shewhart univariate control chart where individual measurements for a subgroup size of n = 1 are plotted in a time-ordered sequence starting with the oldest date. The center line of the control chart is the average of all the individual measurements; the upper and lower control limits are the average k 2.66 x mR, where mR is the average ofthe moving ranges of the individual neighboring measurements, i.e., tij= 1xi -xi- 1 I
(22). FORESTPRODUCTSJOURNAL
Average
54 52
It is important to note that Ryan (16) indicatedthat this method is most applicablefor controlling future observations and is robust for large samplesizesof m > 75.
3
UCL
5 62 2 60
cL =p(m + l)(m - 1) x m2-mp FSa3,
Face Density
Sample
Figure 1. - Univariate for MDF samples.
control charts for top face, bottom face, and cores densities
tii, and Pinus palustvis). Seventy-eight samples(each of subgroupsize n = 1) were obtained from time-ordered production runs of sanded3/4-inch MDF. Each samplerepresentsone of the facility’s typical destructive laboratory tests of a 2- by 2-inch sampleblock from a finished4- by g-foot panel, andwaspart of the typical production of the manufacturer. The samplingschemeof the manufacturer was not altered for this study. Each sampleis representativeof a batch of 3/4-inch MDF for a customerorder. Somegroupsof samplesweretakenfrom a sequenceof 2 or 3 days of consecutive productionof 3/4-inch MDF. Every point does not representa consecutivetime sequence.The largesttime-gapbetween
Vcx.49,
NO. 5
samples was a 3-day period during which the 3/4-inch MDF product was not manufactured. An x-ray densitometerowned by the panelmanufacturerwasusedfor the density profile analysisof the sampleblocks (15). The top-face and bottom-face densitieswere taken asthe averagedensity over a 0.020-inch zone from the sand-off point of the sampleblock. The core density wastaken asthe averagedensityover a0.030-inchzone in the centerof the 3/4inch sampleblock. The scan resolution of the x-ray device was 0.002 inch, so each face density measurementwas the averageof 10measurementpositionsand each core density measurementwas the averageof 15 measurementpositions. 81
Average Weight UCL
39 38 E $7 36
Average
Figure 2. - Univariate MDF samples.
ORIENTED DATA
control
charts for
STRANDBOARD
SET
The data set for oriented strandboard (OSB) wasobtainedfrom a manufacturer in the southeasternUnited States.Thirty subgroupedsamples(each of subgroup size n = 2) were obtained from timeorderedproduction runsof sanded23/32inch OSB. Each samplerepresentedan
4 “Run Rules” are basic tools for detecting patterns in the running data record. The most common is the run about the central line. In most cases, runs in the data record are evidence of a sustained shift in the process average even when no points fall outside the control limits (22).
82
RESULTS MULTIVARIATE CHARTS OF INDIVIDUAL
AND
DISCUSSION
CONTROL MDF OBSERVATIONS
X-Individual control chartswere generated for the 78 MDF samplesfor topAverage face density, core density, bottom-face density, averageweight, averagedensity, and average thickness (Figs. 1 and 2). The purpose of control charts, as intended by Shewhart (17), is to reduce processand product variation by adjustment and improvement of the process. “Out-of-Control” Signal Density The six univariate control chartsin FigI ures 1 and 2 had only oneout-of-control point for average density (sampleNo. 74). This samplehad a density greater than the UCL. In a univariate control chart setting, this is a signal that this specialcausesourceof variation should be investigated and corrective action(s) shouldbe taken to eliminatethis special causesourceof variation from occurring in the future. The univariate control chartsalsohave signalsof “runs” in the datawhich indicate the process average is not stable. Using standard “Run Rules”4 where a shift in the averageis indicated by seven consecutive points above or below the center line, or seven consecutivepoints increasingor decreasing,it appearsthat the averagesof this data set for top-face density,bottom-face density,andaverage density were not stable. The correlation matrix of thesevariables (Table 1) showed that average weight and averagedensity had a strong positive correlation(Y= 0.8350), top-face average weight, density, and thicknes for and bottom-face density had a strong positive correlation (Y= 0.7236), average density and bottom-face density had a positive correlation (r = 0.5 126),average density and top-face density had a posiaverageof two destructivelaboratorytests tive correlation (r = 0.4896), and core of 2- by 2-inch sampleblocks from one density and average thickness had a finished panel. All sampleswere timenegative correlation (u = -0.4141). The signs and strength of the correlation orderedovermultiple daysof production. An x-ray densitometry measurement coefficients for these variables follow device (not owned by the manufacturer)’ conventional wisdom related to the manufacture of medium density fiberwas used for the density analysisof the board (13,18). sampleblocks. The scanresolution was Wheeler (2l), Ryan (16), and Jackson 0.002 inch.Thetop-face andbottom-face and Morris (lo), have all shownthat indensitieswere taken asthe averagedendependent control charts of correlated sity over a 0.020-inch zone, starting variables can lead to misleadingresults 0.040 inch from each face of the 23/32or false signals of statistical control. inch sampleblock. The core density was False signals of statistical control can taken asthe averagedensity over a 0.030leadto anadjustmentofthe processwhen inch zone in the center of the sampleof no adjustmentis necessary,which. may lead to inducing more variation in the the 23/32-&h sampleblock. MAY
1999
TABLE1. -- Correlationmatrixof criticalMDFvariables(n = 78). Variable Bottom-face density Top-facedensity Core density Average weight Average density Average thickness
Bottom-face density 1.0000 0.7236 0.0205 0.3397 0.5126 -0.2035
Top-face density 0.7236 1.0000 -0.0194 0.3876 0.4896 -0.1617
process and product by over-adjustment (5,6)). False signals of statistical control can also lead to failure to adjust the process when an adjustment is necessary. Thissituation wouldalsolimitpotential (see Deming's efforts improvement Funnelbecause Experiments out-ofcontrol points or special cause variations would not be detected when they actually occurred. The consequences of this situation may also lead to significant financial losses for a producer if defective or low-grade product is shipped to the customer. Multivariate control charts of these same variables and data sets indicated that false signals of statistical control were present when only univariate control charts were used. A multivariate control chart of Hotelling's T 2 statistic for top-face, bottom-face, and core densities indicated that only one out-of-control point occurredatsampleNo. 50(Fig. 3). Thiswouldsignalthatthe productassociated with this sample should not be shipped to the customer without further analysis. Sample No. 50 had a core density near the upper control limit for the univariate control chart. Recall that core density and average thickness were negatively correlated (r = -0.4141), and this may have a combined effect on the VDP that is not apparent from an analysis of individual control charts. These out-ofcontrol attributes would require more detailed investigation by the producer, When a multivariate control chart of Hotelling's T 2statistic was produced for all six variables (average thickness, average weight, top-face, bottom-face, core, and average densities), two out-of-control signals occurred at samples No. 68 and No. 74 (Fig. 4). The out-of-control point signal for sample No. 50 for the three variables top-face, bottom-face, and core densitiesis no longer out-of-control, thus indicating that one or more variables correlated with the face and core densities were omitted from the original multivariFOREST PRODUCTS JOURNAL
Core density 0.0205 -0.0194 1.0000 0.0590 0.1299 -0.414 !
Average weight 0.3397 0.3876 0.0590 1.0000 0.8350 0.2423
12 o 8 _. 6 69 % 4 2 10 0 __]_4---------_ _o _ _ _
Average density 0.5126 0.4896 0.1299 0.8350 1.0000 0.0450
Average thickness -0.2035 -0.1617 -0.4141 0.2423 0.0450 1.0000
"out-of-control" Signal
_
_
_ ._ _ Sample
Yo _
_
_
_
CL i UCL g'.
Figure 3. -- Multivariate control chart using Hotelling's T 2 statistic (top-face, bottom-face, and core densities for medium density fiberboard). 20 "Out-of
o 15
.
..... "Signals _ _
UCL
*_ 10 % 5
Ck
o ............................................................................. _ _ _ TM ,,, _o ,, ,, _o _o _o © _ sample
_-
Figure 4. -- Multivariate control chart using Hotelling's T 2 statistic (average weight, average thickness, average density, to0-face density, bottom-face density, and core density for medium density fiberboard).
ate control chart. It is important to note in Figure 4 that the variation around the center line appears to be increasing for samples greater than sample No. 66. This may signal that these product attributes were moving farther away from a collectire mean or target, and the producer should focus attention on reasons for this drift. The drift may also result in a loss of product value to the customer, Hotelling's T 2 multivariate control charts may reduce the risk of false alarms of out-of-control situations, but may not be adequate as complete replacements for univariate control charts. Univariate control charts provide data with the unit of measurement for a particular product attribute or process variable. For example, if the process average for MDF core
VOL.49, NO. 5
density is 47 pcf, the univariate control chart would display sample data around this process average. Hotelling's T 2multivariate control charts do not provide a unit of measurement for individual sample points that can be easily related to a process average or target. In many instances, the users ofHotelling's T 2multivariate control charts may also need corresponding univariate control charts if the unit of measurement is an important feature for the product attribute or process variable. MULTIVARIATECONTROL CHARTS OF OSB SUBGROUPED OBSERVATIONS Thirty samples, where each sample has a subgroup size of n - 2, were analyzed for OSB. A univariate X-bar con83
trol chart of top-face density revealed out of control points at samples No. 4, No. 17, and No. 28 (Fig. 5). Samples No. 6, No. 14, No. 17, and sample No. 30 are out-of-control for core density, and sample No. 17 is out-of-control for bottomface density (Fig. 5). Top-face density and bottom-face density had a strong positive correlation (r = 0.8610, Table
2). Bottom-face density and core density had a positive correlation (r = 0.6594). Top-face density and core density had a positive correlation (r = 0.5041). The sign of the correlation coefficient derived from the pooled covariance matrix forthe face densities and core density was surprising, given that many manufacturers find this correlation to be negative. One
Top Face Density
55.00
50.00
"Out-of-Control"Signal _
45.00 7_ 40.00 "_ __ 35.00 30.00 ,,out-of-co°fro," s_g,a,, 25.00 20.00 ............................. _ ._ _. o, _ _ ._ _- = sample
NO. 9, No. 17, No. 27, and No. 28 (Fig. Average LCL UCL
_
_
_
_
_
Core Density 55.00
50.00 45.00
"Out-af-Contml"_ Signal
40.00
_
30.00 35.00 __._.. 25.00 20.00
_.o_
Ck
_
_
UCL
LCL Average
..out-_-co,_,' s_=.
.............................
......
_, _, _. _ Sample
_
_
_
_
_
"Out-of-Control" sig..,--._
UCL
45.00 Average 4o.o0, 50.00 _a_35-00 30.00 25.00 20.00 ............................. _' " _" = .....
LCL
_
_
_
_
corresponded with the univariate control chartsoutofcontrol signals, butthenml6). Samples No. 6, No. 17, and No. 28 tivariate control chart of Hotelling's T 2 portrayed a much stronger out-of-control signal forsample No.28.Sample No.9is outof controlforthemultivariate control chartbutnotouto:_'control foranyofthe univariate control charts; therefore, a false of statistical occurred when signal univariate control control charts were used as the only charting method. The multivariate control chart indicated a severe deviationfrom the collective product mean of the attributes of top-face, bottomNo.9 andNo.28revealed largedeviaface, and core densities. Both samples dons from the collective product mean. If therewereno othervariablescorrelated with top-face, bottom-face, and core densities, multivariate control charts of additional variables would indicate a
Bottom Face Density 55.00
explanation for the unexpected positive correlation between face and core densities is that the average density for the OSB samples varied from 32 pcfto 41 pcf. This large range of density variation may mask face andcore density relationships because both densities are moving in the same direction as overall panel density changes. A multivariate control chart using Hotelling's T 2 statistic of top-face, bottom-face, and core densities revealed five out-of-control points, i.e., samples No. 6,
_
Sample
Figure 5. _ Univariate control charts for top face, bottom face, and core OSB densities,
similar pattem of statistical control as in Figure 6. However, uncorrelated variablesmaynotbein a stateofstatistical control. X-bar univariate control charts of average density had out-of-control signals for samples No. 4, No. 6, and No. 30, and average thickness of the OSB sampies had out-of-control signals at sampies No. 10, No. 117and No. 21 (Fig. 7). Averagedensitywas highlypositively correlated with core density (r = 0.8998), bottom-face density (r = 0.8346), and topface density (r = 0.7087). When a multivariate control chart using Hotelling's T 2
TABLE 2. -- Correlation matrix of critical OSB variables (n = 30). Bottom-face density
Top-face density
Core density
Average density
Average thickness
Bottom-face density
1.0000
0.8610
0.6594
0.8346
0.1406
Top-face density Coredensity
0.8610 0.6594
1.0000 0.5041
0.5041 1.0000
0.7087 0.8998
0.3194 0.0824
Average density
0.8346
0.7087
0.8998
1.0000
0.0441
Average thickness
0.1406
0.3194
0.0824
0.0441
1.0000
Variable
84
MAY 1999
18.00 |
statistic was generated for top-face den-
1600•
average density, and average thickness, sity, bottom-face density, core density, two out-of-control signals occurred (Fig.
::i:: t "_ I
thatthe OSB panels associated with these samples were not part of the typical processvariation andthatsuchspecial cause 6). SamplesNo. variation should require 9 and No. further 28 examinaindicated tins. Adjustments to the process should be made for these samples and corrective action(s) should be developed to ensure that these out-of-control instances do not occur in the future,
,___----7..o=-o,-co_,ro,.. _,go_,° __ _
/
__
_/_ UCL
% 6._
i Ck
_ ii!]_i.._ ....
. . = .....
"" _
_
_
_
Sample Figure 6. -- Multivariate control chart using Hotelling's T2 statistic (top-face, hottom-face, and core densities for oriented strand board samples).
CONCLUSIONS
As competition
among producers of Average Density
wood composite panels increases and customers of these products demand higher quality, it will be imperative for successful businesses of the future to focus on continuous improvement ofproc-
55.00 50.00 "out-of-con_ror'
45.00
esses and manufactured products. Multivariate control chart procedures using Hotelling's T 2 statistic can provide SPC
Signal
_._
7_ 40.00 ___._. _ 35.00 ---,_ _o/ 30.00 _p
,O,
_
,h(
-
..
-
_
_
UCL Average LCL
25.00 "out-of-contror'signalsV
monitoring systems that prevent the manufacture and shipment of defective or low-grade product. Multivariate control chart procedures of correlated variables are more robust (produce less false signals) for detecting out-of-control points or special cause variation than Shewhart univariate control charts of
20.00
.............................
_
., _- = Sample
......
_
_
Average Thickness 0.740 0.730
variate control charts may not be replacemerits for univariate control charts, the use of multivariate control charts in con-
_, 0.720
junction with univariate control charts can provide practitioners with powerful process improvement tools.
0.7o0
"Out-of-Control" Signals
thes----b,s.n ouhm
The multivariate control charting procedure presented in this paper for VDPs of wood composite panels represents an improved method for monitoring product quality. A more robust statistical control method, such as multivariate control charting, also reduces the risk of financiallossdueto excessive variation or customer claims. A realistic limitation for the applicatins of this analysis technique by manufacturers is the lack of availability of this technique in commercial SPC software. However, the analysis for this paper using the variance-covariance matrix,its inverse, and the matrix algebra necessary to compute the Hotelling's T 2 statistic were done on personal computers using MicrosofP Excel and Mathcad_.The atgorithm of Hotelling's T 2 multivariate control chart is available on the commerFOREST
PRODUCTS
JOURNAL
VOL.
.l= 0.710
Average
LCL
..
0.690
Sample
Figure 7. -- Univariate control charts for average OSB density and thickness.
18 16 14 .o12 _ 10 _ 8 % 64 2 0 .............................
....
""
UCL
CL
Sample Figure 8. -- Multivariate control chart using Hotelling's T 2 statistic (average thickhess, average density, top-face density, bottom-face density, and (:ore density for oriented strand board samples). 49,
NO.
5
85
• cially available
software
SAS © and can
be easily encoded into the newer humanmachine interface software packages such as Wonderware © or Intellusions ©. LITERATURE CITED 1. Alt, F.B. 1985. Multivariate quality control. In: Encyclopedia of Statistical Sciences 6. S. Kotz and N.L. Johnson, eds. John Wiley and Sons, New York. 2. Box, G.E.P. and N.R. Draper. 1969. Evolutionary Operation: A Statistical Method for Process Improvement. JohnWiley and Sons, NewYork.459pp. 3. , W.G. Hunter, and J.S. Hunter. 1978. Statistics for Experimenters. John Wiley and Sons,New York. 653pp. 4. Caudill, S.P., S.J. Smith, J.L. Pirkle, and D.L. Ashley. 1992. Performance characteristics of a composite multivariate quality control system. Analytical Chemistry 64(13): 1390-1395. 5. Deming, W.E. 1986. Out of the Crisis. Massachusetts Inst. of Technology, Center for Advanced Engineering Study. Cambridge, Mass. 507 pp. 6. . 1993. The New Economics. Massachusetts Inst. of Technology, Center for Advanced Engineering Study. Cambridge, Mass. 507 pp. 7. Hayter, A.J. and K.L. Tsui. 1994. Identifieation and quantification in multivariate quality control problems. J. of Quality Technology 26:197-208. 8. Hotelling, H. 1947. Multivariate quality control. In: Techniques of Statistical Analy-
86
sis, C. Eisenhart, M.W. Hastay, W.A. Wallis, eds. McGraw-Hill, New and York. pp. 113-185. 9. Jackson, J.E. 1959. Quality control methods for several related variables. Technometrics 1(4):359-377. 10. Jackson, J.E. and R.H. Morris. 1957. An application of multivariate quality control to photographic processing. J. of the American Statistical Assoc. 52(6): 186-199. 11. Laufenberg, T.L. 1986. Using gammaradiation to measure density gradients in recon-
Michigan 104-121. State Univ., Quart. Bull. 45(1): 19. and G.E.Woodson.1987.Fiberboard Manufacturing Practices in the United States. Proc. No. 7360. Forest Prod. Soc., Madison, Wis. 20. Taguchi, G. 1993. Taguchi on Robust Tech-
stituted wood products. Forest Prod. J. 36(2):59-62. 12. MacGregor, J.F. 1988. On-line statistical process control. Chemical Engineering Progress 84(10):21-31. 13. Maloney, T.M. 1993. Modern Particleboard and Dry-Process Fiberboard Manufacturing. Miller Freeman, Inc., San Francisco, Calif. 681pp. 14. Nedumaran, G. and J.J. Pignatiello, Jr. 1998. Diagnosing signals from T_ and X multivariate control charts. Quality Engineering 10(4):657-667. 15. Quintek Measurement Systems, Inc. Model QDP-01X. 201 Center Park Dr., Suite 1140, Knoxville, Tenn. 37922. 16. Ryan, T.P. 1989. Statistical Methods for Quality Improvement. John Wiley and Sons, New York. 446 pp. 17. Shewhart, W.A. 1931. Economic Control of Quality of Manufactured Product. D. Van NostrandCo.,NewYork.501pp. 18. Suchsland, O. 1962. The density distribution of flakeboards. Michigan Agri. Expt. Sta.,
Knoxville, Tenn. 470 pp. 22. andD.S.Chambers. 1992. Understanding Statistical Process Control. SPC Press Inc., Knoxville, Tenn. 406 pp. 23. Winistorfer,P.M. 1992.Pressing strategies.
nology Development. The Am. Soc. of Mechanical Engineers, New York. 136 pp. 21. Wheeler, D.J. 1995:. Advanced Topics in Statistical Process Control. SPC Press, Inc.,
In: Proc. of the NPA Pressline Technology Seminars, Charlotte, N.C. Partieleboard Composite Panel Assoc. (formerly National Assoc.),SilverSprings., Md.pp 19-25. 24.
, W.C. Davis and W.W. Moschler. 1986. A direct scanning densitometerto measuredensityprofilesin wood composite products. ForestProd. J. 36(11/12): 82-86. 25. _ T.M. Young, and E. Walker. 1996. Modeling and comparing vertical density profiles. Wood and Fiber Sci. 28(1): 133-141. 26. Young, T.M. and F.M. Guess. 1994. Reliabilityprocessesand corporate structures. Microelectronics and Reliability 34(6): 1107-1119.
MAY 1999
