Quality Technology &
Quantitative Management
Vol. 12, No. 1, pp. 53-67, 2015
QTQM
© ICAQM 2015
One-Sided Shewhart-type Charts for Monitoring the
Coefficient of Variation in Short Production Runs
Philippe Castagliola1,*, Asma Amdouni2, Hassen Taleb3 and Giovanni Celano4
1
LUNAM Université, Université de Nantes & IRCCyN UMR CNRS 6597, Nantes, France
2
Institut Supérieur de Gestion, Université de Tunis, Tunisie
3
Higher Institute of Business Administration of Gafsa, University of Gafsa, Tunisia
4
Universitá di Catania, Catania, Italy
(Received September 2013, accepted February 2014)
______________________________________________________________________
Abstract: Monitoring the coefficient of variation is an effective approach to Statistical Process Control
when the process mean and standard deviation are not constant but their ratio is constant. Until now,
research has not investigated the monitoring of the coefficient of variation for short production runs.
Viewed under this perspective, this paper proposes a new method to monitor the coefficient of variation
for a finite horizon production by means of one-sided Shewhart-type charts. Tables are provided for the
statistical properties of the proposed charts when the shift size is deterministic. Two illustrative examples
are given in order to illustrate the use of these charts on real data.
Keywords: Coefficient of variation, shewhart-type chart, short production runs, truncated run length.
______________________________________________________________________
1. Introduction
Q
uality is one of the most important consumer decision factors. It has become one of
the main strategies to increase the productivity of industrial and service organizations.
One of the fundamental principles of the SPC (Statistical Process Control) is that a
normally distributed process cannot be claimed to be in-control until it has a constant mean
and variance. This implies that a shift in the mean and / or the standard deviation makes
the process out-of-control. However, control charting techniques were recently extended to
various sectors such as health, education, finance and various societal applications where
the mean and the standard deviation may not be constant all the time but the process is
operating in-control. In this case, it is natural to explore the use of the coefficient of
variation (CV, in short)  which is a normalized measure of dispersion of a probability
distribution that is defined as the ratio of the standard deviation  to the mean  .
Generally, it is widely used to compare data sets having different units or widely different
means.
The CV has several applications. For instance, the CV is commonly used in renewal
theory, queuing theory, and reliability theory. In the field of finance (see Sharpe [24]), it is
interpreted as a measure of the risk faced by investors, by relating the volatility of the
return on an asset to the expected value of the return. It is also adopted in chemical and
biological assay quality control to validate results (see Reed et al. [22]). It can also be used
in the fields of materials engineering and manufacturing where some quality characteristics
related to the physical properties of products constituted by metal alloys or composite
*
Corresponding author. E-mail: [email protected]
54
Castagliola, Amdouni, Taleb and Celano
materials often have a standard-deviation which is proportional to their population mean.
These properties are usually related to the way atoms of a metal diffuse into another. Tool
cutting life and several properties of sintered materials are typical examples from this
setting.
As pioneers in this field, Kang et al. [14] developed a control chart for monitoring the
CV using rational subgroups and applied it to a clinical chemistry-control process in order
to show that the CV is a potentially attractive tool in quality improvement, where neither
the process mean nor the variance are constant. In Kang et al. [14]’s paper, the CV is
monitored through a Shewhart-type chart, making this chart sensitive to large shifts in the
CV but not very sensitive to small to moderate shifts. For this reason, several authors have
tried to improve this chart, suggesting and investigating various more advanced approaches
dedicated to the monitoring of the CV. Hong et al. [11], were the first to propose an
EWMA-CV (Exponentially Weighted Moving Average) control chart in order to improve
the CV chart proposed by Kang et al. [14] and detect small shifts more efficiently. More
recently, Castagliola et al. [6] suggested a new method to monitor the CV by means of two
one-sided EWMA charts of the CV squared. Calzada and Scariano [2] suggested a
synthetic control chart (denoted SynCV) for monitoring the coefficient of variation and
Castagliola et al. [3-4] proposed alternative approaches to monitor the CV based on Run
Rules and Variable Sampling Interval strategies.
The research presented above are aimed to monitor the CV over a production horizon
considered as infinite. But, there are many situations in which the production horizon is
very short, i.e. a few hours or a few days, and is considered as finite. Traditional SPC
techniques have been designed to keep high volume production along an infinite horizon
production, while the new industry is moving towards specialization and diversity of
products and flexible manufacturing, which gives an increased importance for short
production run. Examples of processes with finite production horizon in the automotive
industry include short production runs of mechanical components within flexible
manufacturing cells and, in the semiconductor industry, the assembly of electronic boards.
Also, there exist high volume manufacturing processes with a low inspection rate, where
the rolling horizon scheduled for the production of a product code can be so short that the
number of allowed inspections should be limited to a few. As an example, in several
beverage industries bottling soft drinks, set-ups can occur every 24-48 hours to change
brand and packaging, according to the production planning decisions. The measurement of
the percentage of carbonation is an important quality characteristic of these processes.
Although the production rate is high, (thousands of bottles/cans per hour), the
measurement test used to get the correct value of percentage of carbonation inside one
bottle requires several minutes to be completed. For this reason, a limited number of
bottles/cans of the same brand and package can be inspected every eight hours, thus
limiting to 20-30 the number of scheduled inspections between two consecutive set-ups.
The design of control charts for processes with limited production horizon H is a
challenge tackled by practitioners and scholars. Control charts specifically designed for
finite production run processes have been first introduced by Ladany [15] who presents a
methodology for the economic optimization of a p -chart for short runs. Later, Bedi [16]
extended this work to allow time H to be a decision variable. More recently, the design of
Shewhart-type X control charts for short runs have been discussed in Montgomery [8, 9].
Bayesian type control charts for monitoring the sample mean during a short run have been
proposed by Calabrese [1], Tagaras [25], Tagaras and Nikolaidis [26] and Nenes and
Tagaras [20]. The statistical measures of performance of the Fixed Sampling Rate (FSR)
One-Sided Shewhart-type Charts for Monitoring the Coefficient of Variation
55
Shewhart and EWMA t and X control charts, monitoring the process mean in short
horizon processes, have been investigated by Celano et al. [7]. Nenes and Tagaras [19, 21]
investigated the performance of the CUSUM control chart, again under the assumption of
a finite run. Very recently, Castagliola et al. [5] investigated the statistical properties of VSS
(Variable Sample Size) Shewhart control charts monitoring the mean in a short production
run context.
If several X type control charts have been proposed for short run processes, as far as
we know, no research has been done concerning the monitoring of the CV in a short run
context. Consequently, the purpose of this paper is to fill this gap by proposing two
one-sided Shewhart-type CV charts for monitoring the coefficient of variation (i.e. an
upward (downward) Shewhart CV chart aiming at detecting an increase (decrease) in the
CV) for short run processes and by investigating their truncated run length properties. The
choice of two separate one-sided Shewhart-type control charts instead of a single two-sided
Shewhart-type control chart is motivated by the fact that the former are more efficient than
the later to detect shifts in an a priori known direction for an asymmetric statistic like the
sample CV.
The remainder of the paper is organized as follows. In Section 2, the main distribution
properties of the sample CV are discussed. In Section 3, two one-sided Shewhart-type CV
charts for short production runs are introduced. The truncated run length properties
TARL (average of the truncated run length), TSDRL (standard deviation of the
truncated run length), TRL05 (interpolated 05 -quantile, i.e. the median, of the truncated
run length distribution) and TRL095 (interpolated 095 -quantile of the truncated run
length distribution) are presented in Section 4 as measures of performance. Section 5
reports the results of the numerical analysis. Two illustrative examples are presented in
Section 6 in order to show the implementation of a one-sided Shewhart chart monitoring
the CV in a short run context. Finally, conclusions and future research directions in Section
7 complete the paper.
2. Properties of the (Sample) Coefficient of Variation
Let X be a positive random variable and let   E ( X )  0 and    ( X ) be the
mean and standard deviation of X respectively. By definition, the CV  of the random
variable X is defined as




(1)
Now, let us assume that {X 1 … X n } is a sample of n normal i.i.d. N (   ) random
variables. Let X and S be the sample mean and the sample standard-deviation of
X1 … X n , i.e.,
X
1 n
 Xi 
n i 1
(2)
and
S
The sample CV ˆ is defined as
1 n
2
 (Xi  X ) 
n  1 i 1
(3)
56
Castagliola, Amdouni, Taleb and Celano
ˆ 
S

X
(4)
By definition, ˆ is defined on (0 ) . The distributional properties of the sample CV ˆ
have been studied by McKay [18], Hendricks and Robey [10], Iglewicz et al. [13], Iglewicz
and Myers [12], Warren [30], Vangel [28] and Reh and Scheffler [23]. Among these authors,
Iglewicz et al. [13] noticed that n / ˆ follows a noncentral t distribution with n  1
degrees of freedom and noncentrality parameter n / ˆ . Based on this property, it is easy
to derive the c.d.f. (cumulative distribution function) Fˆ ( x  n  ) of ˆ as
 n
n
Fˆ ( x  n  )  1  Ft 
n  1

 x




(5)
where Ft () is the c.d.f. of the noncentral t distribution with n  1 degrees of freedom
and noncentrality parameter
n / ˆ . Inverting Fˆ ( x  n  ) gives the inverse c.d.f.
Fˆ1 (  n  ) of ˆ as
Fˆ1 (  n  ) 
Ft
1

n
1   n  1
n



(6)
where Ft 1 () is the inverse c.d.f. of the non-central t distribution. To get confidence
intervals or hypothesis tests involving the coefficient of variation, the reader can refer to
Tian [27], Verrill and Johnson [29] and Mahmoudvand and Hassani [17].
3. One-sided Shewhart-type CV Charts for Short Production Runs
A manufacturing process is scheduled to produce a small lot of N parts during a
production horizon having finite length H . Let I be the number of scheduled
inspections within the production horizon H . The interval between two consecutive
inspections, i.e. the sampling frequency is h  H / ( I  1) hours since no inspection takes
place at the end of the run. Let us suppose that we observe subgroups {X i 1  X i 2 … X i n }
of size n , at time i  1 2… I . We assume that there is independence within and between
these subgroups and we also assume that each random variable X i  j follows a normal
( i  i ) distribution where parameters i and  i are constrained by the relation
 i   i / i   0 when the process is in-control. This implies that from one subgroup to

another, the values of i and  i may change, but the coefficient of variation  i  i
i
must be equal to some predefined in-control value  0   0 / 0 common to all the
subgroups, where 0 is the in-control mean and  0 is the in-control standard deviation.
In this paper, we propose two separate one-sided Shewhart-type control charts for
monitoring the CV for short production runs:

a downward Shewhart-type chart (denoted as the SH   chart) aiming at
detecting a decrease in the CV, with the following control limits
LCL  0 (ˆ )  K  0 (ˆ ),
(7)
UCL  ,

an upward Shewhart-type chart (denoted as the SH   chart) aiming at detecting
an increase in the CV, with the following control limits
One-Sided Shewhart-type Charts for Monitoring the Coefficient of Variation
57
LCL  0,
UCL  0 (ˆ )  K  0 (ˆ ),
(8)
where K   0 and K   0 are the control limit parameters and where 0 (ˆ ) and
 0 (ˆ ) are the mean and standard deviation of the sample coefficient of variation ˆ when
the process is in-control, i.e.  i   0 . Since there is no closed form for 0 (ˆ ) and  0 (ˆ ) ,
the following approximations proposed by Reh and Scheffler [23] can be used

1
1



0 (ˆ )   0  1    02   
 n
4
3 04 7 02 19  
1  4  02 7  1 
6








3
15



 
0
0
4 32  n 3 
4
32 128  
n2 
(9)
1 2
7 4 3 2 3  
1 1 
3 1 
1
 0 (ˆ )   0    02    2  8 04   02    3  69 06  0  0    
2 n 
8 n 
2
4
16  
n
(10)
4. Truncated Run Length Properties
The production run ends after a fixed rolling horizon H coinciding with the
production lot cycle time. For this reason, the computation of the statistical measures of
performance of the investigated control chart must be a function of the finite number I
of scheduled inspections and should account for the fact that the run may end without any
signal issued by the chart. The short run measures of statistical performance of a control
chart have been originally proposed by Nenes and Tagaras [21] who assume that the
truncated run length TRL of the short run chart is defined for   1 2… I  1 and the
p.m.f. (probability mass function) fTRL () is equal to
 1
(1   )  if   1 2… I 
fTRL ()   I
if   I  1
 
where  is the Type II error, i.e.
1  Fˆ ( LCL  n  1 ) for the SH  chart

 Fˆ (UCL  n  1 ) for the SH chart
 
where  1   0 is an out-of-control value for the CV. Values of   (01) correspond to a
decrease of the nominal coefficient of variation, while values of   1 correspond to an
increase of the nominal coefficient of variation. The c.d.f. FTRL () of TRL is
1   
FTRL ()  
 1
if 12…,I
.
if   I  1
The measure of performance proposed by Nenes and Tagaras [21] for finite runs in
place of the “classical” ARL for infinite runs is the average of the truncated run length
TARL  E (TRL ) , i.e.
I
1   I 1
TARL  (1   )    1  ( I  1)  I 

1 
 1
In order to gain more insight concerning the variability of the TRL , we have also
decided to introduce in this paper two other measures of performance for short runs: the
standard deviation of the truncated run length TSDRL and the interpolated r -quantile
of the truncated run length distribution TRLr . In order to derive TSDRL , we firstly need
to define TRL 2  E (TRL2 ) as the central moment of order 2 of the truncated run length
58
Castagliola, Amdouni, Taleb and Celano
TRL , i.e.
I
TRL 2  (1   )  2   1  ( I  1) 2  I 
 1
After some tedious sum manipulations, it can be proven that
TRL 2 
2 I  I  2  2 I  I 1   I  2  3 I 1    1

(1   ) 2
The TSDRL can thus be obtained using TSDRL  E (TRL2 )  TARL2 , i.e.
2
2 I  I  2  2 I  I 1   I  2  3 I 1    1  1   I 1 

TSDRL 
 
(1   ) 2
 1  
After some easy simplifications, it remains
TSDRL 
 (1   2 I 1 )  (1   )  I 1 (1  2 I )
.
1 
(11)
Concerning the definition of the interpolated r -quantile TRLr for the truncated run
length distribution, since fTRL (1)  1   , r  [1   1] . For r  [1   1   I ] , we can find
the value of TRLr by solving the equation FTRL (TRLr )  r  1   TRLr , i.e.
TRLr  ln(1  r ) / ln(  ) . When r  (1   I 1] , we suggest to linearly interpolate TRLr
using the points (1   I  I ) and (1 I  1) , i.e. TRLr  I  1  (1  r ) /  I . If we summarize,
for r  [1    1] , we can define the interpolated r -quantile TRLr as
 ln(1  r )
if r  [1   1   I ]
 ln(  )

.
TRLr  
 I  1  1  r if r  (1   I 1]

I
In the Numerical Analysis Section, we will exclusively focus on two particular r -quantiles
TRLr : TRL05 (i.e. the median of the truncated run length distribution) and TRL095 .
5. Numerical Analysis
In this Section we evaluate the statistical performance of the SH   and SH   charts
for short runs. The statistical performance is computed by considering an assignable cause
occurring immediately after the start-up of the short run. Table 1 presents the chart
parameters K  (assuming a SH   chart) and K  (assuming a SH   chart). These
values are designed such that TARL  TARL0  I (as suggested in Nenes and Tagaras [21])
when the process is functioning at the nominal / in-control coefficient of variation    0 .
Table 1 also presents TARL and TSDRL values (first row of each block), TRL05 and
TRL095 values (second row of each block) for I  10 ,  0  {005 01 015 02} ,
n  {5 7 10 15} ,   {05 , 065 , 08 , 09} (assuming a SH   chart) and   {11 ,
125 , 15 , 2} (assuming a SH   chart). Some values of TRL05 for which 05  1  
are not computable and are denoted as “   ” in Table 1. Table 2 (Table 3) has a similar
structure as for Table 1 but for I  30 ( I  50 ) inspections. As an example, in Table 1, for
n  5 and  0  005 , the chart parameter corresponding to the SH   chart is
K   2272 (this value is such that TARL0  10 ) and, for an increase of 25% (i.e.   125 )
of  0  005 we have TARL  656 , TSDRL  377 , TRL05  593 and TRL095  1084 .
One-Sided Shewhart-type Charts for Monitoring the Coefficient of Variation

59


Table 1. Chart parameters K (assuming a SH  chart) and K (assuming a
SH   chart), TARL and TSDRL values (first row of each block), TRL05 and
TRL095 values (second row of each block) for I  10 ,  0  {005 01 015 02} ,
n  {5 71015} ,   {05 , 065 , 08 , 09} (assuming a SH   chart) and
  {11 , 125 , 15 , 2} (assuming a SH   chart).
I  10 n  5
I  10 n  7
 0  005

 0 01
 0  015
 0 02



K 1801 K 1793 K 1779 K  1759
050 447324
258197
259198
3011050 3021051 3051052 3091053
142614
560363
720376
452326
 0 01
 0  015
 0  0 2



K 1868 K 1860 K 1847 K  1829
456328
065 719376
449325
 0  005
723376
725376
7401087 7431087 7491087 7571088
080 890329
890329
891329
892328
262200
265203
143619
145627
148638
562364
566364
570366
4331075 4361076 4411076 4481077
829354
830354
832354
834 353
10221092 10221092 10221092 10231092 10091091 10091091 10091091 10101091
090 956287

110
957287
957287
957286
936302
936302
937301
938301
10331093 10331093 10331093 10331093 10301093 10301093 10301093 10301093
K   2272 K   2283 K   2302 K  2327 K  2231 K  2241 K  2257 K  2279
881333
882333
883332
885332
856344
857344
859343
862342
10201092 10211092 10211092 10211092 10151092 10151092 10161092 10161092
125
656377
658377
662377
668377
5931084 5981084 6061084 6171085
583368
586369
591370
599371
4661078 4711078 4791079 4911079
150 366280
369282
375285
383290
287222
290224
296229
304 236
226977
229990
234 1003 2411011
164 707
166718
170737
177764
200 177116
359
179119
366
182122
377
187128
393
142077
247
I  10 n  10
 0  005

285
 0  015
 0  0 2
155092
156094
158096
 0  005
 0 01
 0  015
 0  0 2



K 1950 K 1945 K 1935 K  1921
110034
111035
111035
112037
288
293
299
127
129
131
134
387293
393296
226167
228169
231172
236176
2381008 2411011 244 1015 2491019
119515
121521
123531
126545
609372
612373
616373
622374
080 741374
743374
745373
749373
8021088 8071088 8161088 8281088
090 907320
907320
908319
910318
5071080 5121081 5201081 5301082
862342
863342
865341
867340
10251093 10251093 10261093 10261093 10161092 10171092 10171092 10181092
K   2200 K   2208 K  2221 K  2240 K   2173 K   2180 K  2191 K  2207
110 824 356
826356
828355
831354
10071091 10081091 10081091 10091091
125
151087
275
383290
065 380288

147083
262
I  10 n  15
 0 01
K  1914 K  1907 K  1896 K  1880
050 154 091
144 079
252
497345
501346
508348
517351
778368
780368
783367
788366
9251089 9331089 9461090 9631090
396298
400300
407304
417309
354 1065 3591066 3671067 3771069
2521022 2561025 2621030 2711036
220161
223164
228169
166104
114 495
117504
120519
125541
324
331
342
358
200 120048
121050
123053
125056
106025
107027
108029
109031
170
177
187
105
108
113
120
150
166
235175
168107
171111
176116
60
Castagliola, Amdouni, Taleb and Celano


Table 2. Chart parameters K (assuming a SH  chart) and K  (assuming a
SH   chart), TARL and TSDRL values (first row of each block), TRL05 and
TRL095 values (second row of each block) for I  30 ,  0  {005 01 015 02} ,
n  {5 7 10 15} ,   {05 , 065 , 08 , 09} (assuming a SH   chart) and
  {11 , 125 , 15 , 2} (assuming a SH   chart).
I  30 n  5
I  30 n  7
 0  005
 0  0 1
 0  015
 0  0 2



K 2211 K 2198 K 2178 K  2149
 0  0 1
 0  015
 0  0 2



K 2370 K 2357 K 2337 K  2308
050 20191092 2024 1091 20321090 20431089
1143918 1152922 1165928 1184 937

22243087 22383087 22613087 22933088
 0  005
8193037 8273038
8403041 8583044
065 2618881 2620879 2624 877 2628874
22471050 22531048 22611046 22731043
30293093 30293093 30293093 30303093
30013090 30013090 30023090 30033090
080 2869650 2870649
2871648 2872646
30413094 30423094 30423094 30423094
090 2951532

2951532
2951531
2952531
30453094 30453094 30453094 30453094
K  3246 K  3278 K  3334 K  3415
110 2759768
2761767
2774 755
2775753
2778751
2781747
30373094 30373094 30373094 30383094
2923576 2923576
2924 574
2925573
3044 3094 30443094 30443094 3044 3094
K  3170 K  3196 K  3242 K  3308
2764 764 2768760
2705816 2708814 2712810 2718805
30373094 30373094 30373094 30373094
3034 3093 30343093 30343093 30353093
125 20351090 20441089 20591087 20811084
17871095 18001096 18221097 18521098
22703088 22973088 23423088 24093088
16833083 17083083 17523084 18143084
150
855744
868753
605542
617553
637571
666596
5712469
5812513 5992588 624 2699
890768
921790
3851664
3931701
4081763
4291855
200 263207
268212
277221
290235
191132
195136
202143
211153
145627
149642
155669
164 708
405
417
101438
108468
 0  005
 0  0 1
 0  015
 0  0 2
 0  005
 0  0 1
452397
457402
465410
476422
182123
184 124
2771198
2811213
2861238
294 1273
I  30 n  10

050
I  30 n  15
K  2484 K  2472 K  2453 K  2426
065 15991072 16091074 16261077 16491080
13623077 13773077 14033078 14393079
080 2604 890 2608888 2613884
187127
191132
377
382
391
404
786693
796701
812713
835730
5182239 5262272 5382327 5562403
2621879
22731042 22811040 22921037 23081032
30283093 30293093 30293093 30293093
30033090 30043090 30053091 30073091
090 2880636

 0  015
 0  0 2



K 2573 K 2562 K 2545 K  2521
2881635 2882633 2884 631
30423094 30423094 30423094 30423094
K  3111 K  3132 K  3169 K  3222
110 2630872 2634 870
2805723
2807721 2809718 2813714
30393094 30393094 30393094 30393094
K  3060 K  3077 K  3106 K  3147
2641865 2650859
2515943 2520940 2530935 2543928
30303093 30303093 30303093 30313093
30233092 30233092 30243092 30253092
125 14781045 14951049 15221056 15601064
1094 893 1112902 1142917 1183937
11923071 1214 3072 12503074 13033075
774 3026 7903030 8183036 8583044
150
200
411357
420366
436382
458404
268212
274 218
284 228
299243
2491076
2551103
2661150
2821218
145081
147084
152089
158096
148641
152659
160690
170734
116043
117045
120049
123053
256
264
279
299
152
157
166
179
One-Sided Shewhart-type Charts for Monitoring the Coefficient of Variation

61


Table 3. Chart parameters K (assuming a SH  chart) and K (assuming a
SH   chart), TARL and TSDRL values (first row of each block), TRL05 and
TRL095 values (second row of each block) for I  50 ,  0  {005 01 015 02} ,
n  {5 7 10 15} ,   {05 , 065 , 08 , 09} (assuming a SH   chart) and
  {11 , 125 , 15 , 2} (assuming a SH   chart).
I  50 n  5
I  50 n  7
 0  005
 0 01
 0  015
 0  0 2



K 2333 K 2320 K 2297 K  2266
 0 01
 0  015
 0  0 2



K 2537 K 2523 K 2500 K  2468
050 38571685 38631683 38741679 38881673
24181731 24331735 24581741 24921749

 0  005
50105091 50105091 50115091 50125091
19785071 19985072 20325072 20785074
065 45901219 45931217 45961213 46021208
4124 1566 41311562 41421556 41571548
50385094 50385094 50385094 50385094
50225092 50235092 50235092 50245092
080 4864 863 4865862 4866860 4868857
47571024 47591022 47621018 47661013
50455094 50455095 50455095 50455095
50425094 50425094 50435094 50435094
090 4950697 4950697 4950696

4951695
50475095 50475095 50475095 50475095
K  3642 K  3687 K  3764 K  3879
4920761 4920760
4921758
4922756
50465095 50465095 50465095 50465095
K  3552 K  3588 K  3650 K  3741
110 46821119 46851116 46891111 46951104
46081201 46111198 46181191 46271181
50415094 50415094 50415094 50415094
50395094 50395094 50395094 50395094
125 34871783 35031781 35311775 35691767
30601818 30831818 31231818 31781816
42375089 42935089 43915089 45365089
30635084 31155085 32055085 33385086
150 12861163 13091180 13491209 14071251
8723769 8903845 9203978 9664175
871811
891829
926861
977907
5702462 584 2524 6092630 6452787
200 320266
328273
340286
359305
223166
229171
238181
252195
185801
190823
199861
213919
117504
121521
127549
137592
 0  005
 0 01
 0  015
 0  0 2
 0  005
 0 01
884 823
895833
914 850
940874
273217
276220
282226
290234
152656
154 666
158683
164 707
I  50 n  10

050
I  50 n  15
K  2686 K  2673 K  2650 K  2619
5792502 5872536 6002592 6182672
 0  015
 0  0 2



K 2803 K 2790 K 2770 K  2741
065 31391818 31561817 31831816 32191814
15581351 15791364 1614 1385 16611413
32435085 32835086 33485086 34405086
10894705 11064781 11354907 11765005
080 45531256 45571252 4564 1246 45731237
41081574 41181569 4134 1560 41561549
50375094 50375094 50375094 50385094
50225092 50225092 50235092 50245092
090 4871851 4872849 4874 846 4876842
4784 988 4786985 4789981 4793975
50455095 50455095 50455095 50455095
K  3481 K  3510 K  3560 K  3631
50435094 50435094 50435094 50435094
K  3419 K  3442 K  3481 K  3536
110 45031303 45081297 45181289 45311277
43351436 43431430 43581419 43771405

50355094 50365094 50365094 50365094
50305093 50305093 50315093 50325093
125 25101753 25401760 25921771 26631783
18101495 1844 1512 19001540 19791576
21035074 21465075 22215076 23295078
13105030 13425034 13965040 1474 5048
150 564 512
579526
604 552
641588
3551536
345291
354 300
370316
393339
3661580
3831656
4091768
203875
209903
220951
2361020
200 161099
165103
171110
180119
123053
125055
128060
132066
309
321
340
368
178
185
196
213
62
Castagliola, Amdouni, Taleb and Celano
The following conclusions can be drawn from Tables 1–3:

For fixed values of I , n and  0 , when the shift size  decreases (i.e.   1 ) or
increases (i.e.   1 ), the values of TARL , TSDRL , TRL05 and TRL095 decrease.
This is a common property of many control charts. It is also worth to note that the
TARL is always unbiased, i.e. when  decreases or increases, TARL  TARL0 .

For fixed values of I ,  0 and  , when the sample size n increases, the values of
TARL , TRL05 and TRL095 tend to decrease. For example, in Table 3, if I  50 ,
 0  005 and   11 , we have TARL  {4682 , 4608 , 4503 , 4335} ,
TRL05  {5041 , 5039 , 5035 , 5030} and TRL095  {5094 , 5094 , 5094 ,
5093} when n  {5 71015} . Concerning TSDRL the situation is a bit more
contrasted. For example, in Table 3, if I  50 ,  0  005 and   11 , we have
TSDRL  {1119 , 1201 , 1303 , 1436} when n  {5 71015} , i.e. the TSDRL
values increase when n increases. On the other side, in Table 3, if I  50 ,
 0  005 and   15 , we have TSDRL  {1163 , 811 , 512 , 291} when
n  {5 71015} , i.e. the TSDRL values decrease when n increases.

For fixed values
TARL , TSDRL ,
if I  50 , n  5
TSDRL  {697 ,
TRL095  {5095 ,

The SH   chart is more sensitive than the SH   chart to shifts in the CV. For
example, for I  30 ,  0  005 , n  {5 71015} ,   05 we have TARL  {2019 ,
1143 , 452 , 182} . When   20 , we have TARL  {263 , 191 , 145 , 116} .
This difference is due on an asymmetrical change of shape from the in-control to the
out-of-control distribution of the CV.
of I , n and  close to 1, when  0 increases, the values of
TRL05 and TRL095 tend to be constant. For example, in Table 3,
and   09 , we have TARL  {4950 , 4950 , 4950 , 4951} ,
697 , 696 , 695} , TRL05  {5047 , 5047 , 5047 , 5047} and
5095 , 5095 , 5095} when  0  {005 01 015 02} .
6. Illustrative Examples
Example #1
The following example has been introduced in Castagliola et al. [6] for the
implementation of an EWMA chart monitoring the CV in a long production run context
and will be adapted, in this paper, to a short production run context. For more information
concerning this example, refer to Castagliola et al. [6]. This example considers actual data
from a sintering process kindly provided by an Italian company that manufactures sintered
mechanical parts. The process manufactures parts which are required to guarantee a
pressure test drop time T pd from 2 bar to 15 bar larger than 30 sec as a quality
characteristic related to the pore shrinkage. Using molten copper to fill pores during the
sintering process allows the drop time to be significantly extended. In fact, the larger the
quantity QC of molten copper absorbed within the sintered compact during cooling, the
larger is the expected pressure drop time T pd . A preliminary regression study has
demonstrated the presence of a constant proportionality  pd   pd   pd between the
standard deviation of the pressure drop time and its mean. To perform SPC by means of
control charts the quality practitioner decided to monitor the coefficient of variation
 pd   pd  pd in order to detect changes in the process variability. Based on past
production of batches of the same part codes, this company has obtained enough Phase I
data to get a reliable estimate for the in-control CV  0  0417 . The process engineer has
decided to implement a SH   chart in order to monitor the CV (and to potentially detect
unexpected increase in the CV) for a short run production of H  21 hours calling for
One-Sided Shewhart-type Charts for Monitoring the Coefficient of Variation
63
I  20 inspections, i.e. an inspection every hour. Assuming a sample size n  5 during
the short run production allows the process engineer to compute K   3575 . Using (9)
and (10) gives 0 (ˆ )  0407 and  0 (ˆ )  0173 and the upper control limit of the SH  
chart is
UCL  0417  3575  0173  1035 .
Table 4. Sintering process example values of X i , Si and ˆ i ,
i  1… 20 , obtained during a short run of H  21 hours.
i
Xi
Si
ˆ i
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
906.4
805.1
1187.2
663.4
1012.1
863.2
1561.0
697.1
1024.6
355.3
485.6
1224.3
1365.0
704.0
1584.7
1130.0
824.7
921.2
870.3
1068.3
476.0
493.9
1105.9
304.8
367.4
350.4
1652.2
253.2
120.9
235.2
106.5
915.4
1051.6
449.7
1050.8
680.6
393.5
391.6
730.0
150.8
0.525
0.614
0.932
0.459
0.363
0.406
1.058
0.363
0.118
0.662
0.219
0.748
0.770
0.639
0.663
0.602
0.477
0.425
0.839
0.141
Figure 1. SH   chart corresponding to sintering process data in Table 4.
Table 4 presents the values of X i , Si and ˆ i , i  1… 20 , obtained during a short
run of H  21 hours and taken from the process after the occurrence of a special cause.
The corresponding SH   chart is plotted in Figure 1. As it can be noticed, the SH  
chart actually detects a single out-of-control situation (in bold in Table 5) for sample 7,
64
Castagliola, Amdouni, Taleb and Celano
confirming the occurrence of a special cause as it was expected by the engineers. After the
out-of-control triggering from the chart and the end of the corrective actions, the process
continues to operate in control (samples 8–20).
Example #2
The following example has been introduced in Castagliola et al. [3] for the
implementation of a VSI (Variable Sampling Interval) control chart monitoring the CV in a
long production run context and will be adapted, in this paper, for a short production run
context. For more information concerning this example, refer to Castagliola et al. [3]. This
example considers actual data from a die casting hot chamber process kindly provided by a
Tunisian company manufacturing zinc alloy (ZAMAK) parts for the sanitary sector. The
quality characteristic X of interest is the weight (in grams) of scrap zinc alloy material to
be removed between the molding process and the continuous plating surface treatment. A
preliminary regression study has demonstrated the presence of a constant proportionality
     between the standard-deviation  and the mean  of the weight of scrap
alloy. The in-control CV  0 has been estimated to  0  000975 and rounded to
 0  001 for simplicity. The process engineer has decided to implement a SH   chart in
order to monitor the CV (and to potentially detect unexpected increase in the CV) for a
short run production of H  31 hours calling for I  30 inspections, i.e. an inspection
every hour. Assuming a sample size n  5 during the short run production allows the
process engineer to select K   3278 in Table 2. Using (9) and (10) gives 0 (ˆ )  00094
and  0 (ˆ )  000341 and the upper control limit of the SH   chart is
UCL  00094  3278  000341  002058 .
Table 5 presents the values of X i , Si and ˆ i , i  1… 30 , obtained during a short
run of H  31 hours and taken from the process after the occurrence of a special cause.
The corresponding SH   chart is plotted in Figure 2. As it can be noticed, the SH  
chart actually detects 2 out-of-control situations (in bold in Table 5), for samples 18 and 19,
confirming the occurrence of a special cause as it was expected by the engineers. After the
out-of-control triggering from the chart and the end of the corrective actions, the process
continues to operate in control (samples 20–30).
Table 5. Die casting process values of X i , Si and ˆ i , i  1… 30 , obtained
during a short run of H  31 hours.
i
Xi
Si
ˆ i
i
Xi
Si
ˆ i
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
396.4
393.2
404.6
396.0
301.4
295.4
293.2
297.4
642.8
640.2
650.4
647.8
646.0
549.8
522.6
4.037
1.923
3.049
2.449
3.049
1.816
1.788
2.190
2.280
1.095
3.435
1.643
2.345
3.114
10.310
0.0102
0.0049
0.0075
0.0062
0.0101
0.0061
0.0061
0.0074
0.0035
0.0017
0.0053
0.0025
0.0036
0.0057
0.0197
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
519.8
518.8
515.4
550.4
529.0
526.8
529.2
521.8
534.0
525.0
533.0
287.8
287.2
289.8
288.4
7.259
8.927
11.760
15.678
10.440
9.602
7.949
7.981
7.681
5.656
5.522
3.114
3.271
1.095
3.049
0.0140
0.0172
0.0228
0.0285
0.0197
0.0182
0.0150
0.0153
0.0144
0.0108
0.0104
0.0108
0.0114
0.0038
0.0106
One-Sided Shewhart-type Charts for Monitoring the Coefficient of Variation
65
Figure 2. SH   chart corresponding to die casting data in Table 5.
7. Conclusions
In this paper, we have proposed two separate one-sided Shewhart-type control charts
monitoring the coefficient of variation in a short production run context: a downward
(upward) Shewhart-type chart denoted as SH   (SH   ) aiming at detecting a shift
decreasing (increasing) the in-control CV  0 . In order to help the quality practitioner to
implement these control charts, we have provided tables presenting the control limit
parameters K  and K  and the corresponding measures of performance ( TARL ,
TSDRL , TRL05 and TRL095 ). The main conclusions that can be derived from these
tables are a) as many control charts, for fixed values of I , n and  0 , when the shift size
 decreases (i.e.   1 ) or increases (i.e.   1 ), the values of TARL , TSDRL , TRL05
and TRL095 decrease, b) for fixed values of I ,  0 and  , when the sample size n
increases, the values of TARL , TRL05 and TRL095 tend to decrease, c) for fixed values
of I , n and  close to 1, when  0 increases, the values of TARL , TSDRL , TRL05
and TRL095 tend to be constant. Since monitoring the CV in a short production run is a
new subject of SPC research, there is room for many extensions like, for instance, the
implementation of run rules, the use of adaptive strategies like the VSI (Variable Sampling
Interval), VSS (Variable Sampling Size) or DS (Double Sampling) and the design of
advanced schemes like EWMA or CUSUM.
Acknowledgements
This work is partially funded by the EGIDE Utique PHC program 13G1109/
28771XE and the University of Gafsa.
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Authors’ Biographies:
Philippe Castagliola is graduated (Ph.D. 1991) from the UTC (Universite de Technologie
de Compiegne, France). He is currently professor at the Universite de Nantes, Institut
Universitaire de Technologie de Nantes, France, and he is also a member of the IRCCyN
(Institut de Recherche en Communications et Cybernetique de Nantes), UMR CNRS 6597.
He is associate editor for the International Journal of Reliability, Quality and Safety
Engineering. His research activity includes developments of new SPC techniques (non
normal control charts, optimized EWMA type control charts, control charts with estimated
parameters, multivariate capability indices, monitoring of batch processes,...).
Asma Amdouni is graduated from the Universite de Tunis, Institut Superieur de Gestion,
Tunisie. She is currently a Ph.D. student at the Universite de Nantes and at the Universite
de Tunis. She is a member of the IRCCyN (Institut de Recherche en Communications et
Cybernetique de Nantes), and LARODEC (Laboratoire de Recherche Operationnelle, de
Decision et de Controle de Processus). His thesis research is focused on Control Chart
Specification using Coefficient of Variation in Short Production Runs.
Hassen Taleb is graduated (Ph.D. 2006) from the university of Tunis, Tunisia. He is
currently associate professor at the University of Gafsa, and he is also the Director of the
"Institut Superieur d'Administration des Entreprises'' de Gafsa, Tunisia. He is also a
member of the LARODEC (Laboratory of Operational Research, Decision Making and
Process Control. His research activity includes SPC developments and applications (control
charts for multinomial processes, fuzzy control charts, self-starting control chart, DOE, ...).
Giovanni Celano has a Master Degree in Mechanical Engineering and holds a Ph.D. in
Manufacturing Engineering from the University of Palermo (Italy). Currently, he is
assistant professor at the University of Catania (Italy), where he teaches Quality
Management. His research is mainly focused on Statistical Quality Control. He is currently
Associate Editor of the Quality Technology and Quantitative Management journal and the
Journal of Industrial Engineering. He is member of the Associazione Italiana di
Tecnologia Meccanica (AITeM) and the European Network of Business and Industrial
Statistics (ENBIS).