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Quality Engineering

ISSN: 0898-2112 (Print) 1532-4222 (Online) Journal homepage: https://www.tandfonline.com/loi/lqen20

Optimal design of cause selecting control chartsfor monitoring the processes of coating fireextinguishers: A case study

Salih O. Duffuaa, Ahmed M. Attia & Ahmed M. Ghaithan

To cite this article: Salih O. Duffuaa, Ahmed M. Attia & Ahmed M. Ghaithan (2018): Optimaldesign of cause selecting control charts for monitoring the processes of coating fire extinguishers: Acase study, Quality Engineering, DOI: 10.1080/08982112.2018.1450510

To link to this article: https://doi.org/10.1080/08982112.2018.1450510

Accepted author version posted online: 11Apr 2018.Published online: 07 Dec 2018.

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CASE STUDY

Optimal design of cause selecting control charts for monitoring theprocesses of coating fire extinguishers: A case study

Salih O. Duffuaa, Ahmed M. Attia, and Ahmed M. Ghaithan

Department of Systems Engineering, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia

ABSTRACTThe purpose of this article is to develop a mathematical optimization model for the optimaldesign of a cause selecting chart (CSC) for a real manufacturing case. The �X and CSC chartsare designed and used for monitoring two dependent finishing processes for manufacturingfire extinguishers; an important equipment for safety. An algorithm is developed to obtainthe design parameters for both processes and sensitivity analysis indicated that theobtained design parameters are practical. The applicability of the model has been demon-strated on other cases from the literature.

KEYWORDScoating process; controlcharts; dependentprocesses; design; optimal

Introduction

Items are usually processed through a sequential man-ufacturing processes, where the quality characteristicof each process affected by the accuracy of the preced-ing processes. As an example, Figure 1 shows manu-facturing processes for an item that is manufacturedin several stages. In each process, a specific qualitycharacteristics need to be monitored using a suitablecontrol chart. If an assignable cause occurs in anystage of the manufacturing process, it may affect someor all of the quality characteristics in the downstreamstages but none of the quality characteristicspreceding it.

The Shewhart control charts can be used tomonitor each process separately, based on theassumption of independency between processes. Toovercome dependency problem, a Hotelling T2

chart may be employed which is an analog to theShewhart �X control chart but for monitoring andcontrolling sequential processes. However, theHotelling T2 chart has a disadvantage; when anassignable cause occurs in any of the processes; itis usually hard to specify which one of theseprocesses is out-of-control. Another type of con-trol charts used for monitoring and controllingsequential processes are regression adjusted con-trols chart Hawkins (1991) and cause selecting

charts Zhang (1984). The two charts are con-structed for a quality characteristic of interestafter adjusting for the effect of some associatedcharacteristic(s) (i.e. the residuals are obtained andused for monitoring the quality characteristic ofinterest). A new type of control chart called auxil-iary variables chart is proposed by Riaz (2008a).The idea of auxiliary-variables based control chartsis by based on observing auxiliary variables alongwith variable of interest. Then, the information onthe relationship between auxiliary variables andvariable of interest is used to improve the preci-sion with which parameters are estimated.

In this study, cause selecting control chart (CSC) isutilized to monitor the process of coating fire extin-guisher. The prime motivation for this article is oftwo folds. The first fold is to generalize the existingmodels for obtaining the optimal design parametersfor the CSC by having a general value n for the sam-ple size in contrast to the existing models that fix thesample size to be 1. This generalization has implica-tions on the algorithm that is used to obtain the opti-mal design parameters. The second fold is to improvethe quality monitoring and control at the plant con-sidered in this article where two dependent processesare used in coating fire extinguishers and help thequality manager in the plant to move away frominspection which is the current practice. In addition

CONTACT Salih O. Duffuaa duffuaa@kfupm.edu.sa Department of Systems Engineering, King Fahd University of Petroleum and Minerals, Al-MoroojCourt, Nakhail Street, #8913, Dhahran 31261, Saudi Arabia.Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/lqen.� 2018 Taylor & Francis Group, LLC

QUALITY ENGINEERINGhttps://doi.org/10.1080/08982112.2018.1450510

demonstrate to practitioners in other industries theapplication of CSC charts.

Therefore, this article presents a real case for twodependent coating processes in a plant that manufac-tures fire extinguishers. The plant-quality managerwants to adopt statistical processes control (SPC) forthese processes and move away from its currentdependency on inspection. The featured tool of SPC iscontrol charts. The most suitable control chart for thispurpose is the CSC, which can be used to monitortwo dependent processes, employing �X and CSCcharts for first and second processes, respectively. Thepurpose of this article is to determine the optimaldesign parameters of both charts (n, h, k1, k2); where nsample size, h sampling frequency, K1 width of firstchart, and K2 width of the second chart. The detail ofthe real case study is provided in section 2. Real dataare collected from the plant and the design parametersfor both control charts have been determined. Analgorithm based on direct search in multidimensionalis proposed and implemented using MATLAB to findthe �X and the CSC optimal design parameters. Moredetails about the algorithm are provided in the resultsand interpretation section. The results and the sensi-tivity analysis indicated that the optimal values of thecharts design parameters (n, h, k1, k2) are insensitive tochanges in the input parameters. This is an indicationof stability and hence the optimal design parametersare robust.

To demonstrate the applicability of the model andthe algorithm in this article data has been collectedfrom published literature for loan-granting processesin banks and processes for the control of outgoingquality of cotton yarn. The data are run using themodel and algorithm and the results compared withthe results in the literature.

Next the statement of the problem is provided andfollowed by the literature review. After the literaturereview, the model development is presented andtheoptimal design parameters are obtained. Then anextensive sensitivity analysis is carried out and otherpractical examples are used to demonstrate the applic-ability of themodel and the algorithm. Finally, the art-icle is concluded.

Problem statements

The real case in this section that fits the above generaldescription is motivated by the case presented inDuffuaa, Al-Turki, and Kolus (2009). The real casestudy is obtained from a plant that manufactures fire-fighting extinguishers which are important safetyequipment used in putting fires away. Manufacturingfirefighting extinguishers (cylinders) pass through sev-eral processes before it becomes ready for the finalstage. The final stage consists of coating the cylinderswith zinc phosphate at process 1 and then coatingthem with powder paint at process 2. The qualitycharacteristics of interest are the thickness of the zinccoating after process 1 and the total thickness afterprocess 2. The total thickness is the sum of zinc phos-phate and powder paint thicknesses. In Duffuaa, Al-Turki, and Kolus (2009), the case is used in the con-text of process targeting which is totally different thanthe way dealt with in this article.

The plant is using inspection to determine acceptedfire extinguishers. They have a high reject and reworkwhich is costly for the plant. The reject rate reaches20 percent of which amount to 20 percent rework.Depending on inspection does not improve the coat-ing and painting processes. The plant decided toadopt statistical process control (SPC) mainly use con-trol charts for monitoring and improving both proc-esses. SPC will enable the quality control departmentto improve the processes. The focus of this article isthe design of the control charts for both processes.The description above indicates that the two processesare dependent since the second quality characteristicvariability is determined by both processes. Hence, theclassical Shewhart control charts are not adequate formonitoring the two processes. An alternative to theShewhart control charts is the CSC proposed for mon-itoring-dependent processes.

Consequently, this type of problems can be formu-lated in general terms as follows: a product is proc-essed by two manufacturing processes in series; eachprocess has a quality characteristic. The first qualitycharacteristic of the product is a random variable X1

and is determined by the first process. The second(final) quality characteristic of the product is denotedby Y and is determined by both processes, whereY¼X1þX2 is a random variable and X2 is deter-mined by the second process. Two control charts areused to monitor and control the two processes. The �Xchart monitors the first quality characteristic X1 anddetects assignable causes for the first process. TheCSC monitors the second quality characteristic value

Figure 1. A sequential production line

2 S. O. DUFFUAA ET AL.

which is determined by both processes. Figure 2depicts the described production system.

One or both of the above two dependent manufac-turing processes may be influenced by an assignablecause. It is desired to determine the optimal designparameters (n, h, k1, k2) for �X chart used to monitorprocess 1 and CSC that monitors the final qualitycharacteristics that is influenced by both processes.The optimal design parameters are determined usingan economic-statistical model in section 4. The modelminimizes the total cost consisting of the cost of qual-ity, sampling, testing, and cost of false alarms, whilesatisfying statistical constraints.

Some data are collected to estimate the parameters ofthe random variables X1 andY that represent the thick-ness of the zinc and the total coating that includes thezinc plus paint coatings. A sample of 100 observations(Xi, Yi) is collected from the fire extinguishers factory.The average (mean) of the zinc and total thicknesses are22.2 mm and 126 mm, respectively. The standard devia-tions of both thicknesses are estimated to be 5.13 and11.14 mm2, respectively. The interarrival for the assign-able cause for the first process is assumed exponentialwith an estimated mean time 100 hours and the interar-rival for the assignable cause for second process isassumed to be exponential with estimated mean150 hours. The mean times for the interarrival times forthe assignable causes were based on the experience ofthe quality manager. These data are utilized in themodel in the results and interpretation section.

Literature review

The literature review of CSC is divided into threeparts. The first part covers normal measured output

observations, the second part is based on auto-correlatedobservations, and the review for adaptive CSC ispresented in the third part. Furthermore, Asadzadeh etal. (2009) provide a comprehensive literature reviewon the CSC application in monitoring-dependentprocesses.

CSC based on normal measured outputobservations

Yang S.F (1997) was the first to propose an economicCSC model utilizing the renewal theory for twodependent processes. The model minimized the totalcost of sampling, testing, and residual costs, while sat-isfying statistical constraints. The proposed model wasderived under the assumption of single assignablecause occurs in both processes and the utility of themodel was verified using numerical example. Yang(1998) generalized the same model for multistage-dependent processes. The model was applied for morethan two processes, however, it assumed only oneassignable cause occurs in each process. Yang C.M(1999) developed an economic statistical CSC tomonitor and control two dependent processes. Toshow the CSC design procedure, a practical applica-tion from the banking industry was solved.

Yang (2005) formulated X chart and CSC chartseconomic model for the case of over adjustment fortwo dependent processes. The proposed model wasapplied to manufacturing processes of cotton yarn.Then, Yang and Yang (2006) extended Yang (2005)model for the over adjustment of two dependentprocesses and considered a joint design of �X � S2

control charts to control both shifts in means andvariability. The validity of the proposed model was

Figure 2. A two dependent manufacturing processes in series

QUALITY ENGINEERING 3

investigated using a real case study about thin goldenfilms. All the above models assumed exponentialarrival time for the assignable causes and normal out-put observations.

Yang (2003) developed an economic statistical CSCmodel for two dependent processes when the assign-able cause follows an increasing failure rate repre-sented by Weibull distribution. The utility of theproposed model was investigated using the same prac-tical example used by Yang (2005). A sensitivity ana-lysis was presented to show the effect of input data onthe design parameters. Yang and Chen (2003) modi-fied Yang (1999) work to the case of two failuremechanisms under Weibull shock models and increas-ing failure rates. The model was applied to the samereal case done by Yang (2005). Yang (2006) extendedYang (1999) model to account for over adjustment oftwo dependent processes. The model was applied to apractical application from bank industry. It is worthnoting that, the models in this paragraph made it pos-sible to address mechanical failures that usually resultsfrom degradation and fatigue.

Several studies have been proposed and utilizedcontrol chart called regression adjusted control chartand auxiliary-variable-based control chart for control-ling and monitoring sequential steps and multivariatedata. Hawkins (1991) proposed the first regressionadjusted control chart based on residuals from theregression of each variable on all others. The modelwas constructed based on the assumption of if anassignable cause affect one of the quality characteris-tics; it will not affect the others. While, Hawkins(1993) proposed another model-based regressionadjusted control chart by relaxing the assumption ofHawkins (1991) to affect some or all of the qualitycharacteristics following it. Hauck, Runger, andMontgomery (1999) extended the Hawkins (1993)work for multivariate cases. Zeng and Zhou (2007)proposed a regression adjusted chart for monitoringpropagation of variance for multistage processes.

Riaz (2008a) proposed an auxiliary variable controlchart for monitoring variance of a quality characteris-tic by using information on an auxiliary characteristic.Next, Riaz (2008b) proposed an auxiliary variable con-trol chart for monitoring of mean of a quality charac-teristic of interest by using information on anauxiliary characteristic.

CSC based on auto-correlated observations

Traditionally sample observations are assumed to beindependent; however, samples taken from production

lines or output of industrial processes are usuallyautocorrelated. Several articles on the topic of CSCappeared in the literature using correlated data.Shu and Tsung (2003) studied the influence ofautocorrelated observations on the detecting power ofthe cause selecting chart. A mathematical model wasdeveloped without any practical application or evennumerical example. Yang and Yang (2005) investi-gated the effect of observations and instrument meas-urement errors on the performance of the CSC. Theperformance of the model was investigated using areal case study about thin golden films. Yang andYang (2006) extended the preceding model to con-sider the case when the observations of the first stepare modeled as one step autoregressive model AR (1)without measurement error. Furthermore, the utilityof the model was investigated using the same real casestudy done by Yang and Yang (2005). The advantageof the models in this section made it possible toaddress autocorrelated data that arise in manyreal situations.

Adaptive CSC chart

Often, SPC utilizes sample of fixed size collected atfixed sampling intervals and then charted under fixedcontrol limits. Therefore, an adaptive control chart isdesigned under variable parameters which make itmore sensitive to detect small assignable causes. Yangand Su (2007a) presented �X chart and CSC for twodependent processes under the assumptions of vari-able sampling interval (VSI), fixed sample size andfixed control limits. They utilized Markov chainapproach to measure the power of the constructedcharts by controlling adjusted average time to signal(AATS). The model was verified using a real casestudy about the braking system of automobiles. YangS.F and Su (2007b) extended the work and developed�X control chart and CSC with both VSI and variablesample size (VSS) for two dependent processes.

Yang and Yu (2009) developed an exponentialweighted moving average (EWMA) chart and CSCwith VSI for two dependent processes under incorrectadjustment. They argued that the quality of items canbe affected by improper adjustment of the processwhich leads to process shifts. The utility of the modelwas studied using the same real example done byYang S.F and Su (2007a). Also, Yang (2010) con-structed a CSC under the assumptions of fixed samplesize and fixed control limits to monitor the mean andvariability of two dependent processes under incorrectadjustment. The Markov chain approach was utilized

4 S. O. DUFFUAA ET AL.

Table1.

Asummarized

overview

oftheliterature

Reference

Type

ofchart

Failure

mechanism

Adaptive

parameters

Auto-correlatio

nsAlgo

rithm

/solutio

nmetho

dAp

plications

Weibu

llshock

Expo

nentialsho

ck

NoorossanaandShekary(2012)

� X&CSC

��

Tail-light

manufacturin

gNoorossanaandShekary(2015)

� X&CSC

��

ShuandTsun

g(2003)

� X&CSC

�Yang

(1999)

� X&CSC

�Simplegrid

search

metho

dBank

indu

stry

Yang

(2006)

� X&CSC

�Simplegrid

search

metho

dBank

indu

stry

Yang

1997)

� X&CSC

�Simplegrid

search

metho

dAn

exam

ple

Yang

(1998)

� X&CSC

�Simplegrid

search

metho

dCo

tton

manufacturin

gYang

(2003)

� X&CSC

�Speciala

lgorith

mCo

tton

manufacturin

gYang

(2005)

� X&CSC

�Simplegrid

search

metho

dCo

tton

manufacturin

gYang

(2010)

� X&CSC

��

Metallic

film

ofthecom-

puterconn

ectors

Yang

andYang

(2006)

� X&CSC

��

Thin

golden

films

Yang

andCh

en(2011)

� X&CSC

��

Metallic

film

ofthecom-

puteron

nectors

Yang

andCh

en(2003)

� X&CSC

�Speciala

lgorith

mCo

tton

manufacturin

gYang

andSu

(2006)

� X&CSC

��

Cotton

manufacturin

gYang

andSu

(2007a)

� X&CSC

��

Speciala

lgorith

mAu

tomob

ilebrakingsystem

Yang

andSu

(2007b)

� X&CSC

��

Speciala

lgorith

mAu

tomob

ilebrakingsystem

Yang

andYang

C.M

(2005)

� X&CSC

��

Simplegrid

search

metho

dThin

golden

films

Yang

andYang

(2006)

� X-S2&CSC

��

Simplegrid

search

metho

dThin

golden

films

Yang

andYu

(2009)

EWMA&

CSC

�Speciala

lgorith

mAu

tomob

ilebrakingsystem

Our

paper

� X&CSC

��

Simplegrid

search

metho

d,Implem

entedusingMatlab.

CoatingFire

Extin

guishers

QUALITY ENGINEERING 5

to study the power of VSI charts by calculating theAATS. The CSC chart used to monitor the thicknessof connectors film used for computer system. Yangand Chen (2011) monitored the means and the var-iances of two dependent processes using VSI controlchart. The model compared the ability of detectingthe shift of both VSI and conventional control charts.

Noorossana and Shekary (2011) utilized an adaptiveVSS and VSI CSC to monitor a quality characteristicof two dependent industrial processes. The VSS andVSI CSC control chart was tested for monitoring plas-tic injection. The Markov chain was employed tomeasure the performance of the VSS and VSI CSCchart by calculating AATS. Recently, Noorossana and

Shekary (2012) extended the work done byNoorossana and Shekary (2011) to the case of VSS,VSI, and variable control limit in a single model.Table 1 summarizes literatures in terms of type ofcontrol chart, type of failure mechanism, adaptive orcorrelated parameters, solution methods, andapplications.

Model development

In this section, the economic-statistical model is con-structed. The notations used to develop the model aredefined followed by the model assumptions. Then themodel development is presented.

Table 2. NotationsSymbols Description

X1 : The random variable representing the quality characteristic of the zinc-coating process (first process) which is normally distributedas X1�N(l1,r

21).

X2 : The random variable representing the quality characteristic of the paint-coating process (second process) which is normally distributed as X2�N(m2, r

22).

Y : The random variable representing the overall quality characteristic of the two processes which is normally distributed as Y3�N(l3, r23), where

r23 ¼r12=n þ r22=n, and random variable Y related to variables X1and X2 by simple linear regression E(YijX1i,X2i)¼l1þl2¼l3

�X chart : The shewhart control chart with center line, upper control limit, and lower control limit set at l1; l1 þ k1r1=ffiffiffin

pand l1 � k1r1=

ffiffiffin

p,

respectively.Where k1is the number of standard deviation above or below the center line of �X chart.CSC : Cause-selecting Control chart with Zi ¼ ðYi � l3Þ=ðr3=

ffiffiffin

p Þ are the values of Yiadjusted for the effects of Xi. Thus the Zi0sare independentN(0,1) random variables. The center line, upper control limit, and lower control limit for the cause-selecting control chart are 0, K2, and–K2, respectively, where K2is the number of standarddeviations above or below the centre line of the cause-selecting chart.

A1 : The assignable cause that affects the zinc-coating process and it cannot be controlled at the paint-coating process. Due toA1, the mean ofthe zinc-coating process shifts from l1 to m11¼ l1 þ d1r1=

ffiffiffin

pandthe mean of the paint-coating process, Y given X1 and X2 shifts from

l3 to l31. The variances of X1 and Y given X1 and X2 are unchanged.A2 : The assignable cause which affects the paint-coating process but not zinc-coating process. Due to A2, the mean of the paint-coating process,

Y given X1 and X2 shifts from l3 to m32¼ l3 þ d2r2=ffiffiffin

pand the variances of X1is unchanged.

A1,A2 : Assignable causes occur in the zinc-coating and paint-coating processes simultaneously. Due to A1 and A2, the mean of the zinc-coating pro-cess shift from m1 to m11¼ l1 þ d1r1=

ffiffiffin

pand the mean of the paint-coating process, Y given X1 and X2 shifts from m3

to m33¼ l31 þ d2r3=ffiffiffin

pa : The probability of false alarm occurring in at least one of the control charts

a¼ a1þ a2 – a1a2a1 : The probability of false alarm occurring in �X control chart,where a1¼ 2(1�U(k1))a2 : The probability that the CSC has a false alarm, where, a2¼ 2(1�U(k2))b01 : The probability of no alarm for the two control charts given that the paint-coating process is out of control, where b01 ¼ (1� a1)b2 and b2is

the probability of no alarm for CSC given that the paint-coating process is out of control. b2¼Uðk2 � d2ffiffiffin

p Þ � Uð�k2 � d2ffiffiffin

p Þ.b10 : The probability of no alarm for the two control charts given that the zinc-coating process is out of control, where b10 ¼b1(1� a2) and b1is

the probability of no alarm for �X chart given that the zinc-coating process is out of control.b1 ¼ Uðk1 � d1

ffiffiffin

p Þ � Uð�k1 � d1ffiffiffin

p Þ:b11 : The probability of no alarm for the two control charts given that the zinc-coating process and the paint-coating process are all out of control,

where b11 ¼b1b2C0 : The quality cost per hour when the zinc-coating and paint-coating processes are in control.C1 : The quality cost per hour when an assignable cause A1 affects the zinc-coating process.C2 : The quality cost per hour when an assignable cause A2 affects the paint-coating process.C12 : The quality cost per hour when assignable causes A1andA2 affect the zinc-coating and paint-coating processes, respectively.Csr : The cost of searching and repairing the whole coating process due to at least one true alarm for the two charts.Cf : The cost of searching and repairing the whole coating process due to at least one false alarm for the two charts.b : The cost of sampling and testing, b> 0.TAi : Inter arrival time between the occurrences of assignable causeAi, where i¼ 1, 2.Tsr : The time of searching and repairing a true alarm.Tf : The time of searching for a false alarm.ki : Exponential distribution parameters i¼ 1, 2.si : The expected arrival time of the assignable cause Aigiven that it occurred in the first sampling and testing interval, i¼ 1, 2. That is,

¼ð1� e�ki h � ki h e�ki hÞ=kið1� e�ki hÞ; i ¼ 1; 2sis(i) : The expected arrival time of the ith arrived assignable cause given that A1and A2 occurred between (n)h and (nþ1)h sampling and testing

interval, i¼ 1, 2. Where,s(1) ¼ e�ðk1þk2 Þh½

�hþ 1=k1 þ 1=k2 � 1=ðk1 þ k2Þ

�� e�k1 h=k2 � e�k2 h=k1 þ 1=ðk1 þ k2 Þ�=ð1� e�k1 hÞð1� e�k2 hÞ

s(2) ¼ e�ðk1þk2 Þh½�hþ 1=ðk1 þ k2 Þ

�� e�k1 h ðhþ 1=k1 Þ � e�k2 hðhþ 1=k2 Þ þ 1=k1 þ 1=k2 � 1=ðk1 þ k2 Þ�=ð1� e�k1 h Þð1� e�k2 h Þ

6 S. O. DUFFUAA ET AL.

Notations

To develop the model, the same notations used in theliterature are adopted and presented in Table 2.

Assumptions

The following assumptions are made to formulate themodel for the problem stated in section 2:

1. Interarrival timeTAi between the occurrences ofassignable causesAi , i¼ 1, 2 for the zinc-coatingprocess and the paint coating follow an exponen-tial distribution with parameter ki, i¼ 1, 2.

2. Sampling and inspection of fire extinguishers inaddition to the CSC charting times are assumedto be negligible.

3. The processes of zinc and paint coatings are haltedduring searching and repairing of assignable causes.That is, the coating process is discontinuous.

Table 3. Possible states.

StateZinc coating

process in control?Paint coating

process in control?

At least oneassignable cause

for the two processes?

1 Yes Yes No2 Yes Yes Yes3 No Yes No4 No Yes Yes5 Yes No No6 Yes No Yes7 No No No8 No No Yes

Table 4. Expected residual times with correspondingprobabilities.State Probability Expected residual cycle time

1 P1 ¼ e�k1he�k2hð1� aÞ R1 ¼ E(T)2 P2 ¼ e�k1he�k2ha R2 ¼ Tf þ E(T)3 P3 ¼ ð1� e�k1hÞe�k2hb10 R3 ¼ h=(1�b10) þ Tsr4 P4 ¼ ð1� e�k1hÞe�k2hð1� b10Þ R4 ¼ Tsr5 P5 ¼ e�k1hð1� e�k2hÞb01 R5 ¼ h=(1�b01) þ Tsr6 P6 ¼ e�k1hð1� e�k2hÞð1� b01Þ R6 ¼ Tsr7 P7 ¼ ð1� e�k1hÞð1� e�k2hÞb11 R7 ¼ h=(1�b11) þ Tsr8 P8 ¼ ð1� e�k1hÞð1� e�k2hÞð1� b11Þ R8 ¼ Tsr

Table 5. Costs of sampling and testing with their correspond-ing expected residual costs.State Cost of sampling and testing þ Expected residual cost

1 R01 ¼ bn þ C0h þ E(C)2 R02 ¼ bn þ C0h þ Cfþ E(C)3 R03 ¼ bn þ C0s1 þ C1(h� s1) þ h C1=(1�b10) þ Csr4 R04 ¼ bn þ C0s1 þ C1(h� s1) þ Csr5 R05 ¼ bn þ C0s2 þ C2(h� s2) þ h C2=(1�b01) þ Csr6 R06 ¼ bn þ C0s2 þ C2(h� s2) þ Csr7 R07 ¼ bn þ C0s(1) þ C12(h� s(2)) þ (s(2)� s(1))

( C1k1þ C2k2)=(k1þ k2)þ h C12=(1�b11) þ Csr

8 R08 ¼ bn þ C0s(1) þ C12(h� s(2)) þ (s(2)� s(1))( C1k1þ C2k2)=(k1þ k2)

þ Csr

Table 6. Input data to the model.Symbol Value Symbol Value

l1 ¼ 22.2 mm Csr ¼ 40 SRr21 ¼ 5.13 mm2 Cf¼ 30 SRl2 ¼ 126 mm C0 ¼ 25 SRr22 ¼ 11.14 mm2 C1 ¼ 30 SRTsr ¼ 0.5 day C2¼ 50 SRTf ¼ 1 day C12 ¼ 60 SR

b ¼ 20 SR

SR : Saudi Riyal, mm: millimeter

Figure 3. Flowchart of the systematic search method

QUALITY ENGINEERING 7

4. A quality cycle is defined as the time between thestart of successive in-control periods. In general,the quality cycle consists of the followings: (a) thetime until the occurrence of alarm, (b) the timeuntil taking the next sample, (c) analyzing andcharting times, (d) the time until an out of con-trol signal, and (e) time of discovering and repair-ing the process.

Model formulation

The development of the model is based on eight statesshown in Table 3. The expected residual time withcorresponding probabilities, and sampling and testingcosts with corresponding expected residual costs foreach possible state are formulated in Table 4 andTable 5, respectively, as given in Yang S.F (1997). Themodel minimizes the expected cost per unit timeE(C)/E(T), where E(C) is the expected total cost andE(T) is the expected cycle time.

The cycle time was divided into three parts: the in-control period, the time to find a true alarm while theprocess is out of control, and the time to find andrepair the assignable cause. The expected time equa-tion was given as:

E Tð Þ ¼ hþ P1E Tð Þ þ P1 E Tð Þ þ Tf� �þ

X8i¼3

PiRi

where Pi and Ri are defined in Table 4 below.The expected cycle cost is composed of three parts:

the cost of first sampling and testing, the costincurred from the time that the process is influencedby an assignable cause until assignable cause isrepaired. The elements of the expected cost are

provided in Table 5. The expected cost equation wasgiven as:

E Cð Þ ¼ P1 bnþ C0hð Þ þ E Cð Þ� �þ P2 bnþ C0hð Þ½

þE Cð Þ þ Cf � þX8i¼3

PiR0i

where R0i and expected residual costs are defined in

Table 5.Eventually, the general form of the economic-

statistical model of the CSC can be represented as fol-lows:

min f n; h; k1; k2ð Þs:t: a � a0;

where the objective function for the proposed modelis defined as minimization of expected cost perexpected time f(n, h, k1, k2)¼E(C)=E(T).

Results and interpretation

Based on communication with the production super-visor, realistic data representing the two processes aresummarized in Table 6. Input data to the model:Input data to the model and the following processesparameters are assumed: d1¼ 2�5, d2¼ 2�5, k1¼ 1/100, k2¼ 1/150. They also added that the constrainton type I error (probability of false alarm) for the twocontrol charts should not exceed 0.1, that is (a � 0.1).

The optimal design parameters for both controlcharts are determined using a systematic searchmethod under different values of shifts. The flowchartof the systematic search method is shown in Figure 3.Table 7 presents the expected cost per unit time andthe optimal design parameters for the CSC for differ-ent values of shifts. The expected cost per expectedcycle time E(C)/E(T) decreases as the shifts in bothprocesses increases. Whereas, the optimal value ofsampling interval h

�is constant for different values of

shifts. Type I (a�) and type II errors (b10

�, b01

�, and,

b11�) decreases as the shifts size increases. As the shift

size increases, the probability of detecting a processmean shift increases which is in line with commonsense. The results in Table 7 are consistent withexpected control charts behavior and provide confi-dence in the model results.

Table 7. Optimal values of charts design parameters for different values of shiftsd1 d2 a b10 b01 b11 1�b10 1�b01 n h k1 k2 E(C)/E(T)

2.5 2.5 0.10 0.49 0.21 0.11 0.46 0.79 1 8.0 2.6 1.7 29.113.0 3.0 0.10 0.31 0.10 0.03 0.66 0.90 1 8.0 2.6 1.7 28.923.5 3.5 0.10 0.09 0.04 0.00 0.90 0.96 1 8.0 2.2 1.8 28.82

Figure 4. E(C)/E(T) versus shifts on the paint coating process

8 S. O. DUFFUAA ET AL.

Table 7 shows if shifts of 2.5 for both charts, theoptimal design parameters and objective function val-ues are h� ¼ 8 days, k�1 ¼ 2.6, k�2 ¼ 1.7, and E(C)/E(T) ¼ 29.11. By substituting the optimal designparameters in center line and control limits of the �Xchart formulas; l1; l17; k1r1=

ffiffiffin

p, respectively, CL =

22.2, UCL = 28.089, and LCL = 16.311. While, thecenter line and the control limits of the CSC are CL =0, UCL = 4.033 k2 = 6.856, and LCL= �4.033 k2= �6.856.

To monitor the processes of coating of fire extin-guishers, every 8 days a sample of size one (Xi, Yi) ismeasured and tested. There are four possible

outcomes for the zinc-coating and painting processes.In state 1, the observation (Xi, Yi) falls inside the con-trol limits of both the �X chart and CSC with probabil-ity of less than or equal to 0.1 (type I error), that isthe two processes are in-control and the next observa-tion will be sampled after 8 days. In state 2, theobservation(Xi, Yi) falls inside the control limits of the�X chart with probability of less than a (type I error)and outside of the CSC control limits, that is thewhole painting process should be stopped to searchand repair the second process and remove the assign-able causeA2. After repairing the out-of-control paint-ing process, the coating process starts and the next

Table 8. Optimal values for different values of d1d1 d2 a b10 b01 b11 1�b10 1�b01 n h k1 k2 E(C)/E(T)

1.0 1.0 0.09 0.91 0.75 0.75 0.00 0.25 1 7.0 4.0 1.7 31.051.5 0.10 0.86 0.57 0.55 0.05 0.42 1 8.0 2.6 1.7 30.112.0 0.10 0.90 0.38 0.40 0.05 0.62 1 8.0 2.6 1.7 29.602.5 0.10 0.82 0.24 0.21 0.12 0.76 1 8.0 2.2 1.8 29.323.0 0.10 0.82 0.11 0.10 0.12 0.88 1 8.0 2.2 1.8 29.173.5 0.10 0.80 0.08 0.10 0.19 0.92 1 8.0 1.9 2.1 29.09

1.5 1.0 0.10 0.91 0.75 0.75 0.01 0.30 1 7.0 4.0 1.7 31.061.5 0.10 0.79 0.57 0.50 0.14 0.42 1 8.0 2.6 1.7 30.112.0 0.10 0.80 0.38 0.30 0.14 0.62 1 8.0 2.6 1.7 29.542.5 0.10 0.70 0.24 0.18 0.24 0.76 1 8.0 2.2 1.8 29.233.0 0.10 0.70 0.11 0.09 0.24 0.88 1 8.0 2.2 1.8 29.063.5 0.10 0.60 0.10 0.06 0.38 0.90 1 8.0 1.8 2.2 28.98

2.0 1.0 0.09 0.89 0.80 0.70 0.02 0.25 1 7.0 4.0 1.7 31.111.5 0.10 0.70 0.57 0.40 0.27 0.42 1 8.0 2.6 1.7 30.102.0 0.10 0.66 0.38 0.28 0.27 0.62 1 8.0 2.6 1.7 29.472.5 0.10 0.54 0.24 0.14 0.42 0.76 1 8.0 2.2 1.8 29.163.0 0.10 0.54 0.11 0.07 0.42 0.88 1 8.0 2.2 1.8 28.983.5 0.10 0.54 0.04 0.00 0.42 0.96 1 8.0 2.2 1.8 28.91

2.5 1.0 0.09 0.85 0.75 0.70 0.07 0.25 1 6.9 4.0 1.7 31.201.5 0.10 0.49 0.57 0.31 0.46 0.42 1 8.0 2.6 1.7 30.072.0 0.10 0.49 0.38 0.21 0.46 0.62 1 8.0 2.6 1.7 29.432.5 0.10 0.49 0.21 0.11 0.46 0.79 1 8.0 2.6 1.7 29.113.0 0.10 0.35 0.11 0.04 0.62 0.88 1 8.0 2.2 1.8 28.943.5 0.10 0.35 0.04 0.02 0.62 0.96 1 8.0 2.2 1.8 28.86

3 1.0 0.09 0.77 0.75 0.63 0.16 0.25 1 6.8 4.0 1.7 31.301.5 0.10 0.31 0.57 0.20 0.66 0.42 1 8.0 2.6 1.7 30.062.0 0.10 0.31 0.38 0.13 0.66 0.62 1 8.0 2.6 1.7 29.402.5 0.10 0.31 0.21 0.07 0.66 0.79 1 8.0 2.6 1.7 29.073.0 0.10 0.31 0.10 0.03 0.66 0.90 1 8.0 2.6 1.7 28.913.5 0.10 0.2 0.04 0.01 0.79 0.96 1 8.0 2.2 1.8 28.83

3.5 1.0 0.10 0.17 0.75 0.14 0.82 0.25 1 7.0 2.6 1.7 31.321.5 0.10 0.17 0.57 0.11 0.82 0.42 1 8.0 2.6 1.7 30.042.0 0.10 0.17 0.38 0.07 0.82 0.62 1 8.0 2.6 1.7 29.382.5 0.10 0.17 0.21 0.04 0.82 0.79 1 8.0 2.6 1.7 29.053.0 0.10 0.17 0.10 0.02 0.82 0.90 1 8.0 2.6 1.7 28.893.5 0.10 0.09 0.04 0.00 0.90 0.96 1 8.0 2.2 1.8 28.82

Figure 6. E(C)/E(T) versus shifts on the zinc coating processFigure 5. Power of CSC chart versus shifts on the paint coat-ing process

QUALITY ENGINEERING 9

observation (Xi, Yi) will be taken after 8 days. In state3, the observation (Xi, Yi) falls outside the controllimits of the �X chart and inside of the CSC controllimits, therefore, the zinc-coating process should bestopped to search and repair assignable cause A1.After repairing the out-of-control zinc-coating pro-cess, the process starts and the next observation (Xi,Yi) will be taken after 8 days. Finally, in state 4, theobservation (Xi, Yi) falls outside the control limits ofboth �X chart and CSC, that is both processes are outof control and should be stopped to search and repairassignable causes A1 and A2. After repairing the twoprocesses, the production starts and continues. Themonitoring continues and the next observation (Xi,Yi) will be taken after 8 days.

At a shift of 2.5 for processes, the �X chart and CSChas test powers of 0.51, 0.79, and 0.98 when the zinc-

coating process is out-of-control, the painting processis out-of-control, and both processes are out-of-control, respectively. Consequently, it is noted that theproposed control charts have high ability or power todetect out of control state when both processes areout-of-control.

Sensitivity analysis

The effect of the shifts magnitude on the ability ofboth control charts to detect the out-of-control state,E(C)/E(T), and the optimal design parameters areillustrated in Tables 8 and 9. The obtained results givemeaningful indications which are valid at least for therange of the input parameters.

Table 8 shows the optimal control charts designparameters and expected cost per expected cycle timeunder different values of d2 at fixed values of d1. Theupper and lower control limits of the CSC (K2)increases as the size of the shift increases (d2).Figure 4 plots the E(C)/E(T) (the expected cost perunit time) versus different values of d2 and fixed val-ues of d1. The figure shows that E(C)/E(T) slightlydecreases as d2 increases because as the shift increasethe ability of the chart to detect the out of controlstate increases and this lead to avoiding the produc-tion of defective items which is usually more

Table 9. Optimal values for different values ofd2d1 d2 a b10 b01 b11 1�b10 1�b01 n h k1 k2 E(C)/E(T)

1.0 1.0 0.09 0.91 0.75 0.75 0.00 0.25 1 7.0 4.0 1.7 31.051.5 0.10 0.91 0.75 0.75 0.01 0.30 1 7.0 4.0 1.7 31.062.0 0.09 0.89 0.80 0.70 0.02 0.25 1 7.0 4.0 1.7 31.112.5 0.09 0.85 0.75 0.70 0.07 0.25 1 6.9 4.0 1.7 31.203.0 0.09 0.77 0.75 0.63 0.16 0.25 1 6.8 4.0 1.7 31.303.5 0.10 0.17 0.75 0.14 0.82 0.25 1 7.0 2.6 1.7 31.321.0 1.5 0.10 0.86 0.57 0.55 0.05 0.42 1 8.0 2.6 1.7 30.111.5 0.10 0.79 0.57 0.50 0.14 0.42 1 8.0 2.6 1.7 30.112.0 0.10 0.70 0.57 0.40 0.27 0.42 1 8.0 2.6 1.7 30.12.5 0.10 0.49 0.57 0.31 0.46 0.42 1 8.0 2.6 1.7 30.073.0 0.10 0.31 0.57 0.20 0.66 0.42 1 8.0 2.6 1.7 30.063.5 0.10 0.17 0.57 0.11 0.82 0.42 1 8.0 2.6 1.7 30.041.0 2.0 0.10 0.90 0.38 0.40 0.05 0.62 1 8.0 2.6 1.7 29.601.5 0.10 0.80 0.38 0.30 0.14 0.62 1 8.0 2.6 1.7 29.542.0 0.10 0.66 0.38 0.28 0.27 0.62 1 8.0 2.6 1.7 29.472.5 0.10 0.49 0.38 0.21 0.46 0.62 1 8.0 2.6 1.7 29.433.0 0.10 0.31 0.38 0.13 0.66 0.62 1 8.0 2.6 1.7 29.403.5 0.10 0.17 0.38 0.07 0.82 0.62 1 8.0 2.6 1.7 29.381.0 2.5 0.10 0.82 0.24 0.21 0.12 0.76 1 8.0 2.2 1.8 29.321.5 0.10 0.70 0.24 0.18 0.24 0.76 1 8.0 2.2 1.8 29.232.0 0.10 0.54 0.24 0.14 0.42 0.76 1 8.0 2.2 1.8 29.162.5 0.10 0.49 0.21 0.11 0.46 0.79 1 8.0 2.6 1.7 29.113.0 0.10 0.31 0.21 0.07 0.66 0.79 1 8.0 2.6 1.7 29.073.5 0.10 0.17 0.21 0.04 0.82 0.79 1 8.0 2.6 1.7 29.051.0 3.0 0.10 0.82 0.11 0.10 0.12 0.88 1 8.0 2.2 1.8 29.171.5 0.10 0.70 0.11 0.09 0.24 0.88 1 8.0 2.2 1.8 29.062.0 0.10 0.54 0.11 0.07 0.42 0.88 1 8.0 2.2 1.8 28.982.5 0.10 0.35 0.11 0.04 0.62 0.88 1 8.0 2.2 1.8 28.943.0 0.10 0.31 0.1 0.03 0.66 0.9 1 8.0 2.6 1.7 28.913.5 0.10 0.17 0.1 0.02 0.82 0.9 1 8.0 2.6 1.7 28.89

Figure 7. Power of X � chart versus shifts on the zinc coat-ing process

10 S. O. DUFFUAA ET AL.

expensive than other costs considered. On the otherhand, Figure 5 shows the power of the CSC for differ-ent values of d2 and fixed values of d1. The figureindicates that the power of the CSC increases as d2increases because when the shift size increases theability of control chart to discover and detect this shiftis enhanced. The sample size, sampling interval, and

upper and lower control limits for �X chart do notchange in Table 8.

Table 9 shows the optimal control charts designparameters and expected cost per unit time under dif-ferent values of d1for fixed values of d2. For �X chartthe upper and lower control limits (depends onK1)increase slightly as d1 increases.

Figure 6 plots E(C)/E(T) versus shifts in the zinc-coating process d1 at fixed values of d2. The figureindicates that E(C)/E(T) decreases slightly as d1increases because the power of the first chart increaseswhich leads to early detection of the out of controlstate and thus avoiding the production of defectiveitems. Figure 7 plots the power of �X chart against theshift in the zinc-coating process d1 for fixed values ofd2. It is worth noting that as the shift in the zinc-coat-ing process (d1) exceeds 3 the power of the chart isthe same which reaches a high value under differentvalues of d2. Figure 7 illustrates that the probability of

Table 11. Optimal values of charts design parameters for different values of shiftsd1 d2 a b10 b01 b11 1�b10 1�b01 n h k1 k2 E(C)/E(T)

-1 -2.5 0.05 0.93 0.31 0.3 0.03 0.69 1 2 2.9 2 137.22-1 -2.5 0.05 0.83 0.07 0.07 0.14 0.92 2 3 2.5 2.1 145.54-1 -2.5 0.05 0.7 0.02 0.01 0.29 0.98 3 4 2.3 2.2 161.34-1 -2.5 0.05 0.57 0 0 0.42 1 4 5 2.2 2.3 178.58-1 -2.5 0.05 0.44 0 0 0.55 1 5 6 2.1 2.5 195.42-1 -2.5 0.04 0.36 0 0 0.64 1 6 6 2.1 2.8 211.33-1 -2.5 0.03 0.33 0 0 0.67 1 7 7 2.2 3 225.40-1 -2.5 0.02 0.3 0 0 0.7 1 8 7 2.3 3.3 239.30-1 -2.5 0.02 0.27 0 0 0.73 1 9 8 2.4 3.5 251.51-1 -2.5 0.01 0.25 0 0 0.75 1 10 8 2.5 3.7 263.77

Table 12. Input data to the model of cotton yarn produc-tion exampleSymbol Value Symbol Value

d1 ¼ 3.00 Csr ¼ $ 1000/hourk1 ¼ 1/446 Cf ¼ $ 400/hourd2 ¼ 3.00 C0 ¼ $ 5k2 ¼ 1/893 C1 ¼ $ 20Tsr ¼ 0.40 hour C2 ¼ $ 30Tf ¼ 0.10 hour C12 ¼ $ 40

b ¼ $ 25

Figure 8. Power of the charts versus sample size

Figure 9. E(C)/E(T) versus sample size

Table 10. Input data to the model of bank industry exampleSymbol Value Symbol Value

d1 ¼ -1.00 Csr ¼ $ 500k 1 ¼ 1/358 Cf ¼ $ 125d 2 ¼ -2.50 C0 ¼ $ 0k2 ¼ 1/31 C1 ¼ $ 300Tsr ¼ 0.25 day C2 ¼ $ 1000Tf ¼ 0.06 day C12 ¼ $ 1300

b ¼ $ 100

QUALITY ENGINEERING 11

detecting a process mean shift of the Xchart increasesas d1increases because when shift size increases, theability of the control chart to detect the shift isenhanced. Again the sample size, sampling interval,and upper and lower control limits for CSC chart donot change in Table 9.

Under low values of shifts both charts have a weekability to detect the out-of-control condition. Toimprove the power of the chart large sample size haveto be taken, Figure 8. As a result of increasing samplesize the expected cost per unit time increases,Figure 9. Consequently, for practical use of X chartsand CSC production supervisor have to increase thesample size if the process shift less than 2 at a highercost than that associated with the optimum, if he/shecan afford it.

Other applications

The purpose and the focus of the article is the realcase study of coating fire extinguishers provided inthe statement of the problem. This section presentstwo practical examples to demonstrate the applicabil-ity of the model and the algorithm. The first exampleis from the banking industry and the second fromthreading industry, representing the application incontrolling service quality and product quality,respectively.

Yang (1999) presented an example concerningmonitoring the quality of a loan granting by a banksystem (i.e., a major bank in Taiwan). The loan appli-cant has to pass through two dependent processes:open a savings account (loan application and creditcheck list, open savings account, credit investigation,mortgage loan, appraisal, and approval) and loanagreement (document requirement, make contracts,and loan delivery). In this case, X1 is the open savingaccount, X2 is the loan agreement, and Y is the qualityof all processes (i.e., the overall quality). Where, Yi

has a linear relation with X1i, where Yi¼ a0þ a1X1i.Yang (1999) considered the case of two types ofassignable causes that may occur on the second

process. This results in 16 different possible states forthe quality systems.

The relevant data for the model is obtained fromYang (1999) and provided in Table 10. The data col-lected by conducting a customer satisfaction question-naire, and eighty-six observations randomly selectedto estimate the data. Optimal design parameters hasbeen obtained by applying the algorithm described inFigure 3 using the following parameters: a � 0.05, 1� n � 10 step size 1, 0 � h � 30 step size 1, and 0 �k1, k2 � 4 step size 0.1. Table 11 shows the expectedcost per unit time and the optimal design parametersfor the CSC for different values of shifts.

Practically, these outcomes show that, if the stake-holder objective is to minimize the cost of controllingquality. He/she has to take one sample every 2 days,and still the statistical constraint of type I error is sat-isfied. But, in this case, the power of the charts is low.On the other hand, if the objective of the stakeholderis to increase the power of the charts, that is, if his/her objective is to detect true alarms, even if the costincreases, the sample size should be increased andsampling interval should be decreased.

The results provided by Yang (1999) are n¼ 1,h*¼ 30.0, k1

*¼ 3.0, k2*¼ 3.0, and E(V )¼ 856.882.

The results of Yang (1999) have wider control chartswith longer sampling frequency when compared tothe results in this article. The results in this article aren¼ 1, h*¼ 2.0, k1

*¼ 2.9, k2*¼ 2.0, and E(V )¼ 137.22

as shown in Table 11. The design parameters in thisarticle has a higher probability of detecting out ofcontrol states since the design parameters for thechart has narrower control chart with smaller sam-pling frequency.

Another example proposed by Yang and Chen(2003) has been used to demonstrate the practicabilityof the algorithm and consistency of the results. Theobjective is to monitor the outgoing quality of cottonyarn; where, the first quality characteristic to be moni-tored is the fiber length and the second is the skeinstrength. They assumed that, the two processes shiftsfrom in-control state to out-of-control state under a

Table 13. Optimal values of charts design parameters for different values of shiftsd1 d2 a b10 b01 b11 1�b10 1�b01 n h k1 k2 E(C)/E(T)

3 3 0 0.54 0.54 0.29 0.46 0.46 1 8 3.1 3.1 12.203 3 0 0.26 0.23 0.06 0.74 0.77 2 8 3.6 3.5 14.903 3 0 0.12 0.1 0.01 0.88 0.9 3 8 4 3.9 17.943 3 0 0.02 0.02 0 0.98 0.98 4 8 4 4 21.043 3 0 0 0 0 1 1 5 8 4 4 24.163 3 0 0 0 0 1 1 6 8 4 4 27.283 3 0 0 0 0 1 1 7 8 4 4 30.403 3 0 0 0 0 1 1 8 8 4 4 33.523 3 0 0 0 0 1 1 9 8 4 4 36.643 3 0 0 0 0 1 1 10 8 4 4 39.77

12 S. O. DUFFUAA ET AL.

Weibull shock model with increasing failure rate.Consequently, the sampling intervals are decreasingfrom sample to sample, using the equation proposedby Banerjee and Rahim (1988) to calculate hi from h1.

Table 12 lists the relevant data based on theassumption that the failure occurs to an exponentialdistribution. The specification of the systematic searchprocedure used to specify the optimum design param-eters are a � 0.05, 1 � n � 10 , 0 � h �8 step size 0.1, and 0 � k1, k2 � 4 step size 0.1. Theresult presented in Yang and Chen (2003) are n¼ 1,h1

*¼ 1.478, k1*¼ 2.255, k2

*¼ 1.764, and E(V)¼ 4945.93 and the results of this article given inTable 13 are: n¼ 1, h*¼ 8.0, k1

*¼ 3.1, k2*¼ 3.1, and

E(V )¼ 12.20. In this case the control charts designedin this article are wider than the results in Yang andChen (2003). This is expected because our model isbased on exponential and Yang and Chen model onWeibull, since Weibull has an increasing hazard rate.

The case study and the above two examples dem-onstrate the versatility and applicability of the modeland the algorithm in this article.

Conclusion

The CSC chart is a SPC tool for monitoring- and con-trolling-dependent processes in series. It is used inconjunction with �X control charts to distinguish theassignable causes. A mathematical model has beenformulated and tailored to determine the optimaldesign parameters for both �X and CSC control charts.These charts are used to monitor two dependent proc-esses for coating fire extinguishers as a case studyfrom industry. The sensitivity analysis indicated thatthe charts design parameters are not sensitive tochanges in input parameters, which is an indication ofstability and robustness. However, the obtained resultsshowed more sensitivity to the shift in the paint-coat-ing process than the first one. This is due to the factthat the paint-coating process is impacting vividly thefinal product-quality characteristics in terms of cost.

Additional examples are provided to demonstratethe applicability of the model and the algorithm. Theexamples are from the banking and threading indus-tries. The model could be extended to the case ofmultiple assignable causes. Also, instead of fixedparameters; a model could be modified for the case ofVSS, VSI, and variable control limit control charts.Another interesting research direction is by comparingthe results with EMWA chart and CSC model thatproposed by Yang and Yu (2009). Robust economicdesign of CSC is a promising research area and still

not addressed in the literature. The main limitationsof the proposed approach as the number of stagesincrease the problem became complicated due to thedependency between quality characteristics and insome cases variable sample size should be taken ateach stage which increases the number of deci-sion variables.

About the authors

Dr. Salih O. Duffuaa is a Professor of Industrial andSystems Engineering at the Department of SystemsEngineering at King Fahd University of Petroleum andMinerals, Dhahran, Saudi Arabia. He received his B.S.and a higher Diploma from the University ofKhartoum, Sudan and Ph.D. in Operations Researchfrom the University of Texas at Austin, USA. Hisresearch interests are in the areas of Operationsresearch, quality and maintenance systems optimiza-tion, facility and strategic planning. He authored abook on maintenance planning and control publishedby John Wiley and Sons and 2nd edition of the samebook published by Springer. He edited two books onmaintenance optimization and management and he isthe Editor of the Journal of Quality in MaintenanceEngineering.

Ahmed M. Attia is a Ph.D. candidate and aLecturer of Industrial and Systems Engineering at theDepartment of Systems Engineering at King FahdUniversity of Petroleum and Minerals, Dhahran, SaudiArabia. He received his B.Sc. and M.Sc. in IndustrialEngineering from Zagazig University, Egypt. Hisresearch interests are in the areas of quality control,inventory control, supply chain management, opera-tions research, and optimization.

Ahmed M. Ghaithan is a Ph.D. candidate at theDepartment of Systems Engineering at King FahdUniversity of Petroleum and Minerals, Dhahran, SaudiArabia. He received his B.Sc. in Mechanical Engineeringfrom Hashemite University, Jordan. Then he got hisM.Sc. in Industrial Engineering from the JordanUniversity of Science and Technology, Jordan. Hisresearch interests include Modeling and Optimizationof Supply Chain, Inventory and Production Control,and Optimization of Quality Systems.

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