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POISSON GWMA CONTROL CHART FOR ENHANCING SUPPLY
CHAIN QUALITY
Tse-Chieh Lin,
Department of Industrial Management, Lunghwa University of Science and Technology
No. 300, Sec. 1, Wanshou Rd., Guishan, Taoyuan County 33306, Taiwan ROC
tsejes.lin@msa.hinet.net
Mei-Chun Su
Graduate School of Industrial Management, MingChi University of Technology
No. 84, Gongjhuan RD., Taishan Township, Taipei County 243, Taiwan ROC
maruko4151@yahoo.com.tw
Shih-Hung Tai,
Department of Industrial Management, Lunghwa University of Science and Technology
No. 300, Sec. 1, Wanshou Rd., Guishan, Taoyuan County 33306, Taiwan ROC
tai3662@mail.lhu.edu.tw
ABSTRACT
Quality has always been of priority concern in supply management. Selecting the most
appropriate control chart is important. This study presents a Poisson generally weighted
moving average control chart (Poisson GWMA control chart) for monitoring Poisson
counts. The Poisson GWMA control chart is superior to the c chart as measured by average
run length (ARL). Managers use information technology throughout the entire supply
chain through to customer relationships. This study uses the Visual Basic for Application
(VBA) program language to design computer-aided software for the Poisson GWMA
control chart to provide enterprises be promoting the reference for supply chain quality
system.
Keyword: Supply chain, Generally weighted moving average, c chart, Average run length
INTRODUCTION
In today fast-paced global economy, time-to-market constraints demand that business
produce products with increased speed, and quickly delivery high-quality products to
customers. Total quality management (TQM) is a philosophy and management system that
focuses on customer satisfaction. In TQM, a customer can be internal or external, and is
anyone in the supply chain who receives materials from a previous stage in the chain.
Quality, quantity, delivery, price, and service are the five most common supply
requirements. Quality has always been a primary concern in supply management. In the
total quality management context, the definition of was expanded to represent a
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combination of corporate philosophy and quality tools directed toward satisfying customer
needs (Leeders et al., 2006). Monczka et al. (2005) proposed that supplier quality is the
ability to meet or exceed current and future customer expectations or requirements within
critical performance areas on a continual basis. Managing quality in supply chain activities
focuses on how efficiency and profitability can be enhanced in relationships between
buyers and sellers by continuous quality improvement. Monitoring quality carefully
throughout the duration of contractual relationships is important as quality is continuously
adjusted through interactions; in supply relationships, quality is often developed jointly
(Jenster, 2005).
Managing quality using statistical process control techniques involves sampling
processes and using the obtained data to establish statistical performance criteria and using
control charts to monitor and improve processes. In determining whether a process is stable,
suppliers must determine whether products meet buyer specifications. When a process in
“in control,” a supplier can predict the future distributions based on the process mean.
When using repetitive operations, a control chart is an invaluable tool. Control charts are
specialized run charts that assist organizations in tracking changes to primary measures
over time. By using control charts, organizations can quickly determine whether a process
is “in control” and take action when required. Control charts of the type that have been very
popular in manufacturing quality control can be applied in logistics performance control to
improve tracking of costs, customer service, or productivity ratios over time and to
pinpoint when adverse trends occur. When sufficient data are available, statistical
procedures can be utilized to generate signals regarding when corrective action should
implemented. Ballou (2004) proposed that control charts provide a graphic picture of
performance as well as facilitating the comparison of performance measures over multiple
consecutive periods. Numerous studies have focused on increasing the ability of control
charts to detect process shifts. In numerous manufacturing industry applications, Shewhart
control charts are utilized for statistical process control as they are simple to plot and easily
interpreted.
Attribute data are based on counts, or the number of times a particular event occurs.
Attributes charts have many applications in logistics management, including counting the
number of employee errors on order entry, numbers of errors workers performing during
picking tasks, number of returns to total orders, number of damage and loss claims, number
of quality complaints, number of warranty claims, stockout percentages, and percent of
deliveries made on time. Most of these problems assume that sampling assumptions are
either binomial or Poisson distributions. The c chart is likely the simplest control chart for
monitoring Poisson counts. When small shifts in nonconformities of the process mean
results from assignable causes, classical c charts are relatively inefficient in detecting small
shifts in nonconformities of the process mean. Nakagawa and Osaki (1975) first introduced
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the discrete Weibull distribution. Sheu and Lin (2004) who applied the probability property
of a discrete Weibull distribution for weighting observations introduced generally weighted
moving average (GWMA) control charts to improve the detection abilities of control charts.
This study proposes GWMA chart, called the Poisson GWMA control chart, for monitoring
the mean of a Poisson distribution. This study utilizes VBA program language to design
computer-aided software for related control skills and provide an enterprise be promoting
the reference of the supply chain quality system.
THE POISSON GWMA CONTROL CHART
Suppose that the sequence of independent samples that include events A and B are
complementary and mutually exclusive. Let M count the number of samples until the first
occurrence of event A since the previous occurrence of event A. Let P j P ( M j ) . That
is, P j is the probability that the event A does not occur in the first j samples. Therefore,
P (M m) 1
m 1
The P satisfies 1 P 0 P 1 0 . Let P ( M
j j ) P j 1 P j . Then, P (M 1) ,
P (M 2 ) , …, P ( M j ) can be regarded as the weights in the weighted moving average
and the weights of the current sample, the second updated sample,…, and the remote
sample, respectively.
Let Y j denote the generally weighted moving average in the plotted test statistics at time
j (j 1, 2 ,3, ) , and C
represents an observation at the time j ; C j , j 1, 2 ,3, are
j
assumed to be independent identically distributed Poisson random variables with mean c ,
where c is the mean number of defects per inspection unit. When the process is in
control, c c 0 , where c 0 is the target value of a mean number of defects per inspection unit.
The Y 0 represents the starting value set by a practitioner. Setting Y 0 c 0 is convenient.
For the Poisson GWMA control chart, the sample statistic is a weighted average of the
current observation C j and all previous observations, with the current observation most
heavily weighted. Consequently, Y j can be configured as
Yj P (M 1) C j
P (M 2 )C j 1
P (M j )C 1 P (M j )c 0
(1)
(P 0 P 1 )C j
(P1 P 2 )C j 1
(P j 1 P j )C 1 P jc0
A discrete analogue of the continuous Weibull, developed by Nakagawa and Osaki
(1975), is called the discrete Weibull distribution. They considered the discrete distribution
j
{P j } j 0
and defined P i ( j : q, ) (q ) for j 0 ,1, 2 ,3,... , 0, and 0 q 1 . The
i j
( j 1) j
probability mass function (pmf) is p ( j : q , ) q q , j 1, 2 ,3,..., and the
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j
1 q , j 1 , 2 ,3 ...
cumulative density function (cdf) is F ( j ; q , )
0 , j 1
j
Let P j q , where design parameter q is constant ( 0 q 1) , ( j 0 ,1, 2 ,3, ) and is
adjustment parameter determined by a practitioner. Thus, the expected value of Eq. (1) can
then be calculated as
0 1 1 2 ( j 1) j j
E (Y j ) E [( q q )C j
(q q )C j 1
(q q )C 1 q c0 ]
(2)
c0
Variance is,
0 1 2 1 2 2 ( j 1) j 2
Var (Y j ) [( q q ) (q q ) (q q ) ]Var ( C )
0 1 2 1 2 2 ( j 1) j 2
[( q q ) (q q ) (q q ) ]c 0 (3)
Q j c0
0 1 2 1 2 2 ( j 1) j 2
where, Q j (q q ) (q q ) (q q ) .
Let L denote the multiplier to determine the width of control limits; the control limits
can be written as
UCL c0 L Q jc0
CL c0 (4)
LCL c0 L Q jc0
When statistics Y j falls outside the range of control limits, the process is out of control
and appropriate actions should be undertaken.
The sensitivity of a control chart has frequently been summarized utilizing the mean of
the run length distribution, known as the ARL. The ARL is defined as the expected number
of points requiring plotting before an out-of-control signal is generated. When the process
is in control, the ARL (ARL0) should be sufficiently large to preclude false alarms. When
the process is out of control, the ARL (ARL1) should be sufficiently small to detect shifts
rapidly. However, the ARL can be obtained through numerical analysis and computer
simulation (Crowder (1987), Roberts (1959), Robinson and Ho (1978), White and Keats
(1996)). Table 1 shows the ARLs for various values of c when the in-control process mean
is c 0 =30. The Poisson-GWMA control chart is superior to the Shewhart c chart based on
ARL.
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Table 1. Comparison of ARLs of the Poisson GWMA control chart and Shewhart c
control chart
C chart Poisson GWMA control Chart q=0.90
Contr
ol
L=3.00 α =1.00 α =0.90 α =0.80 α =0.75 α =0.50
chart L=2.691 L=2.708 L=2.731 L=2.750 L=2.885
c
25 153.31 8.92 8.91 9.01 9.26 11.53
26 251.88 13.30 13.16 13.20 13.46 16.66
27 393.43 22.33 21.65 21.39 21.69 26.64
28 517.96 47.34 44.33 42.50 42.61 50.18
29 495.13 145.17 135.29 125.73 123.13 132.06
30 349.94 350.13 349.78 350.20 349.86 350.52
31 215.17 117.59 109.72 101.32 98.46 95.99
32 129.20 41.24 38.87 36.90 36.59 38.95
33 79.21 20.55 19.77 19.36 19.44 21.68
34 50.22 12.52 12.28 12.23 12.37 14.03
35 33.00 8.63 8.55 8.52 8.65 9.88
The optimum design of a Poisson GWMA control chart is the choice of charting
parameter (q, α, and L) that satisfies specific criteria, which can be either economical or
statistical. A simple three-step procedure is developed to design a Poisson-GWMA chart
for detecting small shifts in Poisson counts. The following steps are recommended:
Step 1. Select an acceptable ARL0.
Step 2. The magnitude of the shift in the process must be determined quickly,
that is, the magnitude with a small ARL1, and then select the q, α and L
that produces a minimum ARL1 for the size shift, and then satisfy the
ARL0 constraint from Step 1.
Step 3. Use the values of q, α and L obtained in Step 2 to determine the control
limits and plot the Poisson GWMA control chart.
POISSON GWMA CONTROL CHART SOFTWARE
Microsoft VBA is easy to use. Any Visual Basic–like language can be used as a scripting
language to program and automate numerous applications (Albright, 2001). Developers
using applications programmed with VBA can automate and extend the functionality of
those applications, reducing the development cycle for custom business solutions. This
study uses the VBA to design computer-aided software for the Poisson GWMA control
chart and provide an enterprise with a reference for the supply chain quality system.
Users can select the sample size when running the Poisson GWMA chart (Fig. 1), and
the system has the following five buttons (Fig. 2):
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1. Clear data.
2. Calculate Poisson GWMA statistics.
3. Calculate the control limit.
4. Display Out-of-control signals.
5. Return to the main screen.
Fig. 1 Poisson-GWMA control chart main screen
Data is cleared when the clear button is pushed. Calculate the Poisson-GWMA Yj
button for the Poisson GWMA control chart being plotted. Input q, α, L, and target mean
c 0 , the system then calculates the Poisson-GWMA statistics. When the calculate control
limits button is selected, the system displays the control limits. Finally, when the display
out-of-signals button is selected, the system displays the Poisson-GWMA Yj in a red
boldface font. The system directly presents the Poisson-GWMA control chart in real time.
Fig.2 Poisson GWMA control button function
A delivery complaints process example is utilized to demonstrate operation of the
Poisson GWMA control scheme and software application. For the quality complaint
characteristic being monitored, C j is an independent identically distributed Poisson
random variable and the target value for c 0 is 30 times per day. The parameters of the c
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chart are set to c 0 =30 and L=3, yielding Control limits of (LCL, UCL)=(14,46). The
parameters of the Poisson GWMA control chart are set to q=0.9,α =0.8, and L=2.731,
yielding approximate control limits of (LCL, UCL)=(27.27, 32.73). Consider an upward
shift in the delivery complaints process mean to c=35 after a new person delivered from
15th observation. Table 2 presents delivery complaints process data. The c chart displays
the out-of-control signal at the 28th observation. However, a signal is first generated at the
23th observation when the Poisson-GWMA statistic is Y23=32.858.
Table 2. Delivery complaints process example data for the c and Poisson GWMA control
scheme
No. C c UCL c CL c LCL Yj P-G UCL P-G CL P-G LCL
1 33 46 30 14 30.300 31.496 30.000 28.504
2 32 46 30 14 30.403 31.806 30.000 28.194
3 32 46 30 14 30.505 31.994 30.000 28.006
4 33 46 30 14 30.696 32.126 30.000 27.874
5 26 46 30 14 30.146 32.225 30.000 27.775
6 31 46 30 14 30.305 32.303 30.000 27.697
7 36 46 30 14 30.866 32.366 30.000 27.634
8 32 46 30 14 30.848 32.418 30.000 27.582
9 31 46 30 14 30.798 32.461 30.000 27.539
10 24 46 30 14 30.083 32.497 30.000 27.503
11 26 46 30 14 29.805 32.528 30.000 27.472
12 37 46 30 14 30.645 32.555 30.000 27.445
13 26 46 30 14 30.059 32.578 30.000 27.422
14 35 46 30 14 30.632 32.598 30.000 27.402
15 26 46 30 14 30.076 32.615 30.000 27.385
16 39 46 30 14 31.049 32.631 30.000 27.369
17 41 46 30 14 31.862 32.644 30.000 27.356
18 38 46 30 14 32.203 32.656 30.000 27.344
19 37 46 30 14 32.452 32.667 30.000 27.333
20 36 46 30 14 32.603 32.676 30.000 27.324
21 35 46 30 14 32.670 32.684 30.000 27.316
22 32 46 30 14 32.464 32.692 30.000 27.308
23 37 46 30 14 32.858 32.698 30.000 27.302
24 42 46 30 14 33.622 32.704 30.000 27.296
25 37 46 30 14 33.686 32.710 30.000 27.290
26 36 46 30 14 33.733 32.715 30.000 27.285
27 29 46 30 14 33.114 32.719 30.000 27.281
28 47 46 30 14 34.528 32.723 30.000 27.277
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29 31 46 30 14 33.836 32.726 30.000 27.274
30 42 46 30 14 34.611 32.730 30.000 27.270
Values q=0.9, α =0.8, L=2.731, and target mean=30 are then input and the system
calculates the Poisson GWMA statistics (Figs. 3 and 4). Completing all steps results in
display of the Poisson GWMA chart and out-of-control; the out-of-control is displayed in
red.
Fig. 3 Poisson-GWMA control chart of example
Conclusion
Numerous companies utilized software and hardware to assess driftage. Information
technology is typically used throughout the whole supply chain through to customer
relationships. This study utilizes Visual Basic for system applications to monitor the
process. Selecting the most appropriate control chart is important. The traditional Shewhart
chart is suitable for detecting large shifts in the process mean. However, the proposed
Poisson GWMA control chart for monitoring Poisson counts is superior to Shewhart c
charts in terms of ARL considerations. The Poisson GWMA control scheme, with
time-varying control limits that increase the sensitivity at start-up shifts, performs better
when detecting small shifts in the process mean.
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