Research ArticleWeibull-Exponential Distribution and Its Application inMonitoring Industrial Process

Muhammad Bilal12 Muhammad Mohsin2 and Muhammad Aslam 3

1Department of Statistics and Computer Science University of Veterinary and Animal Sciences Lahore Pakistan2Department of Statistics COMSATS University Islamabad Lahore Campus Lahore 54000 Pakistan3Department of Statistics Faculty of Science King Abdulaziz University Jeddah 21551 Saudi Arabia

Correspondence should be addressed to Muhammad Aslam aslam_ravianhotmailcom

Received 4 December 2020 Revised 12 February 2021 Accepted 6 March 2021 Published 26 March 2021

Academic Editor Hussein Abulkasim

Copyright copy 2021 Muhammad Bilal et al )is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

)is paper presents a new Weibull family of distributions )e compatibility of the newly developed class is justified through itsapplication in the field of quality control using Weibull-exponential distribution a special case of the proposed family In thispaper an attribute control chart using Weibull-exponential distribution is developed )e estimations of the model parametersand the proposed chart parameters are performed through the methods of maximum likelihood and average run-length )esignificance of the proposed model is demonstrated using a simulation study and real-life problems)e results of the monitoringprocess and quick detection are compared with exponential distribution

1 Introduction

)e Weibull (W) distribution is extensively used to modelfailure time data Numerous distributions have been de-veloped and applied to describe various phenomena inengineering and quality control Several generalizations ofthe family of distribution have been studied in the last twodecades Eugene et al [1] proposed a new beta generatedfamily of distributions for skewed and bimodal data Manypractitioners discussed a technique to generate differentdistributions Nadarajah and Kotz [2] Nadarajah and Gupta[3] Nadarajah and Kotz [4] and Nadarajah et al [5]Alzaatreh et al [6] generalized the technique of Eugene et al[1] to develop distributions defined over any domain A lotof work is available in the recent literature regardingAlzaatreh et al [6] Cordeiro et al [7] Afify et al [8]Alizadeh et al [9] Afify et al [8 10] and Nofal et al [11]Chahkandi and Ganjali [12] established an exponentialpower series (EPS) family of distributions which included anew mixture of an exponential and binomial distribution(see Barreto-Souza et al [13]) Lu and Shi [14] introducedthe W-geometric (WG) and W-Poisson (WP) distributions

were expanded as exponential geometric and exponentialPoisson distributions respectively A comprehensive ac-count of statistical methods can be traced in quality controlsince its commencement In 1924 Walter A Shewhartdeveloped the concept of statistical quality control [15] Inearly days the control charts were extensively applied tomonitor production processes only but nowadays thesecontrol charts are applied in various fields like health care[16] education [17] coal monitoring [18] nuclear engi-neering [17] and veterinary medicine [19 20] Roberts [21]depicted a graphical technique to generate geometricmoving average by selecting weight for most recent valuesBrook and Evans [22] suggested the technique of adopting aMarkov chain approach By using this method they attaineda transition probability matrix along with moments per-centage points of run-length distribution and exact prob-ability of run-length Lucas [23] developed a scheme bycombining the Shewhart control chart with the cumulativesum control chart (CUSM) which detected the small shiftsrapidly Borror et al [24] suggested the Markov chain ap-proach to assess the average run-length (ARL) Khoo [25]designed a new approach to monitor the fraction of

HindawiMathematical Problems in EngineeringVolume 2021 Article ID 6650237 13 pageshttpsdoiorg10115520216650237

nonconforming products )ey constructed a moving av-erage (MA) control chart for a fraction of nonconformingusing the ARL profile of simulation results andmathematicalcalculations

Sukparungsee and Novikov [26] devolved analytical ap-proximation of exponentially weighted moving average(EWMA) using a martingale procedure for the expectation ofexit times Areepong and Novikov [27] derived ARL andaverage delay time (AD) utilizing the martingale approachNoorossana et al [28] worked on different methods to monitorrare health events utilizing the ZIB-EWMA chart Santiago andSmith [29] put forward the idea of the probabilistic-basedprocedure to create a t-chart to observe the stability of theprocess )ey yielded exponential distribution and definedsupplementary rules like ARL and shift detection Ho andQuinino [30] used the ldquonprdquo control chart to monitor theprocess variability instead of the mean chart )ey exploredthat the ldquonprdquo control chart was better in performance than R(range chart) and S2 control chart Aslam et al [31] imple-mented an ordinary sampling plan and group sampling plan)ey developed sampling plans for the multistage processwhich depended upon time-truncated tests and gave nonlinearoptimized solutions to decrease the average sample numberRakitzis andCastagliola [32] studied the Shewhart control chartfor zero-inflated probability models )ey suggested pragmaticrules for the statistical design of inspected charts for a fixedsample size Azam et al [33] designed a control chart under arepetitive sampling technique using an exponential distribu-tion )ey analyzed two existing control charts with the newproposed chart using average run-length Azam et al [34]unfolded an attribute chart for the W distribution using thediscriminant limits)ey established a control chart to estimateparameters in light of ARL Wu and Jiao [35] performed at-tribute inspection to track the mean value of variable char-acteristic for normal distribution )is chart produced an out-of-control signal if the interval between the two suspectsamples is lower than a control limit Kim and Lee [36] used afixed sampling interval scheme (FSI) for the detection of smallshifts to moderate shifts by increasing fractional parameters forzero-inflated binomial process )e effectiveness of the FSIscheme over the VSI (variable sampling interval) scheme wasjustified through a numerical study For further achievementsto the statistical process monitoring literature on the Shewhart-CUSUM and Shewhart-EWMA for instance Aslam et al [37]improved the GWMA (generally weighted moving average)monitoring scheme for the detection of large shift in theprocess For this purpose they proposed the phase-II com-posite Shewhart-GWMA scheme using the MannndashWhitney Ustatistic Shah et al [38] proposed a new monitoring schemehas been developed for time between events under exponentialand gamma distribution Shaheen et al [39] developed acontrol chart using the lognormal distribution to monitor thevariation under repetitive sampling scheme Aslam et al [40]developed and designed the modification of multiple depen-dent statesrsquo sampling plan for satisfying a mean lifetime of theproducts under Birnbaum-Saunders andWeibull distributions

In modeling of real-data problems the common failuretime distribution needs not to be the same but can be amixture of different lifetime distributions Each of these

distinct lifetime distributions can represent a different typeof failure cause for the population like weather changedepending on different parameters such as temperaturehumidity wind flow and precipitation Each parameterfollows different distributions which together model theweather change In this article a new class of Weibull-Gdistributions is derived by using the method given byAlzaatreh et al [6] In fact we follow a certain methodologypresented in equations (1) and (2) According to thismethodology we need H (g(x)) and r (t) and for it we useexponential and Weibull distributions )e exponential is aspecial case of Weibull distribution but when we use thegiven methodology the resulting distribution appears as thecombination of exponential and Weibull distribution In-deed it is the combination of these two distributions not themixture of distributions )e benefit of using the proposeddistribution is that it provides an extraparameter whichcontrols the kurtosis and provides better efficiency indetecting early shift )is can be seen from the comparisonof proposed distribution with mere simple exponential andsimple Weibull distributions )us the four parameterdistributions are more useful to explain several real-lifephenomena much better than two or five parameters (seeBerk et al [41]) In quality control this feature helps todetect out-of-control process much early

We develop an attribute control chart scheme usingWEx distribution )e parameter estimation of the WExdistribution is carried out through the maximum likelihoodestimation method )e designed scheme is used to monitorthe process out-of-control and found more suitable thanexisting schemes We compare the results of the simulateddata and real data of urinary tract infection in the health caredepartment using the proposed scheme with the existingmethod based on ARL and control charts

2 Weibull-G Class of Distributions

In this section we derive the cumulative distributionfunction (cdf) and the probability density function (pdf) oftheWeibull-G class of distributions For this purpose we usethe method of Alzaatreh et al [6] )e cdf of the proposedmethod is given as

F(y) 1113946H(G(x))

ar(t)dt (1)

and the pdf is given as follows

f(x) ddx

H(G(x))1113896 1113897r(H(G(x))) (2)

where r(t) is the pdf of the parent distribution and H()

should satisfy the following conditions

(i) H(G(x)) isin [a b](ii) H(G(x)) is differentiable and monotonically

nondecreasing(iii) H(G(x))⟶ a as x⟶ minus infin and

H(G(x))⟶ b as x⟶infin

2 Mathematical Problems in Engineering

Using H(G(x)) minus log[(1 minus G(x))α] and r(t) of Wei-bull distribution the cdf of the WG class of distributions isobtained as

F(x) 1113946minus log (1minus G(x))α[ ]

a

c

c

t

c1113888 1113889

cminus 1

exp minust

c1113888 1113889

c

1113888 1113889dt

1 minus exp minusminus log (1 minus G(x))α( )

c1113888 1113889

c

1113888 1113889

(3)

and pdf is obtained by differentiating (3) as

f(x) αc

c1113888 1113889

minus log [1 minus G(x)]α

c1113890 1113891

cminus 1

middot eminus minus log [1minus G(x)]αc( )[ ]

c g(x)

1 minus G(x)

(4)

Now several new distributions can be generated for thedifferent values of G(x)

3 Weibull-Exponential Distribution

In this section we derive Weibull-exponential distributionusing the proposed class of distributions

If x is an exponential distribution with cdf and pdf thenit is

G(x) 1 minus eminus λx

(5)

g(x) λeminus λx

(6)

respectively )e cdf and the pdf of the WEx distribution isgenerated by inserting (5) and (6) in (3) as

F(x) 1 minus eminus (αλxc)c

f(x) cα λc

1113888 1113889

c

xcminus 1

eminus (αλxc)c

xgt 0 αgt 0 c λgt 0

(7)

Some special cases of the proposed distribution are givenin Appendix (Table 1)

4 The Proposed Control Chart

)is article proposes the new scheme of attribute charac-teristics control charts for the manufacturing process usingWEx distribution For a fixed sampling interval where n isconsidered as a sample for each subgroup we discuss thelifetime of the product when random variable follows theproposed model If the selected unit falls within controllimits we accept it otherwise it would be rejected)erefore we present the lower control limit (LCL) andupper control limit (UCL) as follows

)e LCL and UCL are defined as

Pr Xlt LCL|c0( 1113857 PL

Pr XltUCL|c0( 1113857 PU(8)

respectively where c0 is the shape parameter )e LCL andUCL for the control process by following Azam et al [34] aregiven by

LCL c

αλln

11 minus PL

1113890 1113891

(1c)

(9)

UCL c

αλln

11 minus PU

1113890 1113891

(1c)

(10)

Ho and Quinino [30] presented the derivation of controlcharts

Firstly a sample is chosen at random from each sub-group and its quality characteristics X are measured )eselected unit is accepted if it lies within defined limits andotherwise declared as rejected

Secondly the process is in control if a item is accepted atfirst and out-of-control if b item is rejected first

When the process is in control the probability that theprocess is out-of-control is defined as

P0 P LCLltXltUCL|c c0( 1113857 1 minus PU + PL (11)

Moreover for the in-control process the probability ofthe out-of-control process is demonstrated through controlcharts is given as

P0out 1 minus 1113944

a+bminus 1

xa

x minus 1

a minus 11113888 1113889P

ao 1 minus P0( 1113857

xminus a (12)

Furthermore the shape parameter is converted to c1)en the probability of the rejected item is derived as

P1 P LCLltXltUCL|c c1( 1113857 (13)

)e probability that the process is found to be out-of-control due to the shifted process using control chart isobtained as

P1out 1 minus 1113944

a+bminus 1

xa

x minus 1

a minus 11113888 1113889P

a1 1 minus P1( 1113857

xminus a (14)

Now the shape parameter is considered for the shiftedprocess which is given as follows

c flowast c1 (15)

where f is a shifted constant and lies between 025 and 1)erefore (11) is formed as

Table 1 Special cases of the proposed distribution

Special cases Parameters DistributionCase 1 (αλc) θ f(x) sim W(θ c)

Case 2 If c 1 λ 1 α 1 f(x) sim Exp(c)

Case 3 If c 1 λ 1 α 1 f(x) sim W(c)

Case 4 If c 1 α 1 f(x) sim Exp(λ c)

Case 5 If c 2 f(x) sim Rayleigh(α λ c)

Mathematical Problems in Engineering 3

P1 expαλUCL

c1113888 1113889

fc1

⎡⎣ ⎤⎦ + 1 minus expαλLCL

c1113888 1113889

fc1

⎡⎣ ⎤⎦

P1 1 + exp minusαλclowast

c

α λ1113888 1113889 ln

11 minus PU

1113888 1113889

1c1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

fc1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦minus exp minus

αλclowast

c

α λ1113888 1113889 ln

11 minus PL

1113888 1113889

1c1( )⎛⎝ ⎞⎠

fc1

⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦

P1 1 + exp minus ln1

1 minus PU

1113888 1113889

f

⎡⎣ ⎤⎦ minus exp minus ln1

1 minus PL

1113888 1113889

f

⎡⎣ ⎤⎦

(16)

)eARL is evaluated using P0 and P1 which are obtainedthrough the parameters of the proposed distribution Notehere that the equations ofP0 andP1 are independent of scaleparameters )erefore we fixed only the shape parameter

Now the ARL of in-control and ARL of out-of-controlhave been calculated as

ARL0 1

P0out

andARL1 1

P1out

(17)

5 ARL Behavior of the Newly ProposedControl Chart

In this section we discuss the ARL behavior of thesuggested control chart A simulation study is conductedin the R-language program using the above-estimatedequations of probabilities In simulation study 10000samples are generated from the well-known exponentialWeibull and the proposed WEx distributions )e MLestimates of the parameters of the suggested distributionare obtained by using SANN (simulated annealing) givenin Belisle [42] SANN is an optimization technique whichis more recent and flexible as compared to New-tonndashRaphson )e SANN method is available inR-package maxLik Henningsen and Toomet [43] Wefixed the shape parameters to find the ARLs

)ese ML estimates are used to find ARLs forARL0 200 300 and 370 Furthermore ARLvalues areexplained by calculating for different drift levels It is im-portant to mention that the shift unit is equal to the in-control ARL)e performance of ARL can be understood byreading the R-code program Simulated data of 10000samples are generated by choosing the random values of a bandn For in-control processes the ARL is calculated byusing these given random values For the selection of a and b

parameters see details in Ho and Quinino [30] Moreoverwe compare the exponential and Weibull distribution withthe proposed WEx distribution for the efficiency of ARL

6 Comparative Study

In this section we compare the performance of theproposed WEx distribution with the exponential distri-bution under ARL In Table 2 we report the estimatedvalues of the proposed model parameters Various values

of the parameters of exponential Weibull and theproposed WEx distributions are placed in Tables 3ndash5 forvarious combinations of a and b atARL0 200 300 and 370 From Table 3 it is seen thatvarious values of a and b at f 1 ie there is no shift inthe values of ARLs of the exponential Weibull and theproposed WEx distributions are the same ie 2019830333 and 37199 at ARL0 200 300 and 370 On theother hand it is observed that at a different shift levelf 1 to 025 the values of ARL decrease for the expo-nential and the proposed models

One can clearly observe in Tables 3ndash5 that the pro-posed model suppressed the exponential and Weibulldistribution in terms of shift detection From Table 3 forinstance the value of ARL is 1704 for the WEx distri-bution and 655 for the exponential distribution whenf 085 a 9 b 20 andARL0 200 FurthermoreTables 4 and 5 also represent the values of ARLs of theexponential and the proposed models at ARL0 200 300and 370 with PL 01 andPU 09 and PL 02 andPU

08 for various combinations of a and b at different levelsof shifts It can be noticed that the values of ARLs de-crease for both models when the values of a and b de-crease By comparing both models it is concluded thatthe proposed model gives quick detection of out-of-control process at different levels of shifts Tables 4 and 5are presented in Appendix

Figures 1ndash9 demonstrate the graphical representations ofthe performance of ARLs at 200 300 and 370 for severalcombinations of aampb and PLampPU One can see fromFigures 1ndash9 as shift level f decreases the curves of ARLs theexponential and the proposed WEx distributions also de-crease Also the ARL curves show that the behavior of theproposed model is better than that of the compared modelFigures 4ndash9 are also presented in Appendix

7 Real-Life Example

In this section the proposed control chart is imple-mented in real-life data set from the health sector )edata of urinary tract infection (UTI) are taken from thehospital system )ese data are attained from Azam et al[34] which was initially used by Santiago and Smith [29]Data are collected from a hospital with a high risk ofurinary tract infections particularly to identify risk rate

4 Mathematical Problems in Engineering

and the frequency of the UTI patients being dischargedIn this example male patients are focused )e data showthe number of days between admissions and discharge ofthe patients having UTIs )e simulated data for theabove scenario are generated from the proposed WExdistribution with PL 01 PU 09 a 29 and b 59Firstly we generate 40 observations for the in-controlprocess and then the next 40 observations for the shiftedprocess withf 090 as presented in Tables 6 and 7 Inaddition we estimate the values of LCL and UCL as003206 and 095051 respectively and display in Fig-ure 10 Figures 10 and 11 show the LCL and UCL for in-control and out-of-control data It is clearly shown that

40th value is detected as out-of-control for the proposedmodel

8 Comparisons of the Exponential and theWeibull-Exponential Distribution UsingReal-Life Example

In this section we compare the control charts of theexponential and the proposed WEx distributions usingreal data of UTI patients It can be easily seen in Figure 12that the control chart of the exponential distribution isin-control and no value is detected as out-of-control

Table 2 Maximum likelihood estimation of the proposed WEx distribution parameterMaximum likelihood estimationSANN maximization 10000 iterationsReturn code 0 successful convergenceLog-likelihood minus 4288574Parameter estimates Estimate Std error t value P valueα 274 006969 393169 2e minus 16C 1328 44492 029484 0653c 701 71518 09801 0613λ 488 113607 04293 0697

Table 3 Comparison of ARL (SDRL) using exponential and Weibull distributions and WEx distribution with pL 005 and pU 095

f

ARL0 200 ARL0 300 ARL0 370a 9 b 20 a 7 b 10 a 13 b 37

Exp Weibull WEx Exp Weibull WEx Exp Weibull WEx

1 20198(20148)

20198(20148)

20198(20148)

30333(30283)

30333(30283)

30333(30283)

37199(37149)

37199(37149)

37199(37149)

099 18971(18921)

22804(22754)

16750(16699)

28791(28741)

33552(33502)

25948(25898)

34146(34096)

43925(43875)

28813(28762)

095 14493(14443)

35189(35139) 8172 (8122) 22987

(22937)48020(47970)

14165(14115)

23666(23616)

79829(79779)

1094(10890)

09 9943(9893)

51359(51309)

3581(3530)

16732(16682)

(65478)(65428)

6945(6895)

14227(14177)

134895(134845) 3691 (3641)

085 655 (6499) 58654(58604)

1704(1654)

11723(11673)

72978(72928) 3581 (3530) 8149

(8099)162317(162266) 1437 (1386)

08 417 (4120) 50734(50684) 884 (832) 7935

(7885)64825(64775) 1944 (1894) 4500

(4449)132614(132564) 649 (597)

075 2588(2538)

33790(33740) 501 (448) 5214 (5163) 46442

(46392) 1114 (1063) 2432(2382)

75476(75426) 341 (287)

07 1583(1532)

18337(18287) 311 (256) 3347

(3297)27986(27936) 674 (622) 1311 (1260) 32598

(32547) 209 (151)

065 966 (915) 8714 (8664) 212 (154) 2115 (2065) 14960(14910) 433 (379) 721 (669) 11920

(11870) 148 (0838)

06 597 (545) 3870(3820) 158 (0953) 1328 (1277) 7435 (7384) 294 (239) 414 (360) 4083 (4033) 119 (0476)

055 380 (326) 1689 (1639) 128 (0599) 837 (785) 3552 (3502) 213 (155) 255 (199) 1422 (1371) 106 (0259)05 253 (197) 754 (702) 112 (0371) 535 (482) 1670 (1619) 164 (103) 172 (112) 540 (487) 102 (113)

025 101 (0098) 1001(00296) 1 (0005) 115 (0409) 104 (0211) 1 (0082) 1 (0012) 100 (00013) 1 (000004)

Mathematical Problems in Engineering 5

Table 4 Comparison of ARL (SDRL) using exponential and Weibull distributions and WEx distribution with pL 01 and pU 09

f

ARL0 200 ARL0 300 ARL0 370a 29 b 59 a 23 b 40 a 34 b 70

Exp Weibull WEx Exp Weibull WEx Exp Weibull WEx

1 20001(19951)

20001(19951)

20001(19951)

30499(30449)

30499(30449)

30499(30449)

37491(37441)

37491(37441)

37491(37441)

099 18127(18077)

24246(24196)

1501(14960)

27928(27878)

36215(36165)

23577(23527)

33449(33399)

46870(46820)

26878(26828)

095 11961(11911)

49423(49376)

5155(5105)

19214(19164)

68054(68003) 8912 (8862) 20662

(20612)107188(107138)

78115(7761)

09 6805(6755)

96705(96652)

1629(1578)

11504(11454)

122584(122534) 3013 (2962) 10763

(10713)234044(233994)

2078(2028)

085 373 (3679) 129441(129391) 635 (583) 6601

(6551)158035(157985) 1185 (1134) 5379

(5329)328659(328609) 712 (660)

08 2002(1951)

106293(106243) 306 (251) 3671

(3620)133121(133071) 545 (492) 2631

(2581)261275(261225) 315 (260)

075 1074(1023)

52202(52152) 182 (122) 2008

(1957)71411(71361) 294 (239) 1291

(1240)114225(114175) 179 (118)

07 590 (538) 16663(16613) 131 (0633) 1100

(1048)25897(25847) 186 (126) 655 (603) 30337

(30287) 127 (0583)

065 342 (288) 4137 (4087) 11 (0328) 616 (564) 7269 (7219) 136 (0702) 356 (302) 6061 (6011) 108 (0284)06 215 (157) 1009 (958) 102 (0155) 362 (308) 1888 (1838) 114 (0392) 214 (156) 1203 (1152) 101 (0122)

055 151 (0872) 307 (253) 1 (0061) 229 (172) 547 (495) 104 (0206) 146(08244) 316 (261) 1 (0041)

05 119 (0476) 142 (0770) 1 (0018) 159(09692) 210 (1520) 101 (0095) 116

(04275) 138 (07208) 1 (0010)

025 1 (00004) 100 (---) 1 (0000) 1 (00079) 100(0000001) 1 (000002) 1 (00001) 100 (000000) 1 (0000)

Table 5 Comparison of ARLs (SDRL) using exponential and Weibull distribution and WEx distribution with pL 02 and pU 08

f

ARL0 200 ARL0 300 ARL0 370a 63 b 57 a 54 b 45 a 42 b 31

Exp Weibull WEx Exp Weibull WEx Exp Weibull WEx

1 20021(19971)

20021(19971)

20021(11971)

30051(30001)

30051(30001)

30051(30001)

37082(37032)

37082(37032)

37082(37032)

099 18213(18163)

24105(24055)

15262(15212)

27470(27420)

35829(35778)

2322(23170)

34273(34223)

43255(43205)

29564(29514)

095 1223(12180)

48380(48330)

5517(5467)

18800(18750)

69067(69017)

8745(8695)

24541(24491)

76586(76535)

12417(12367)

09 7144(7094)

96152(96102)

1814(1763)

11225(11175)

131293(131243)

2935(2884)

15525(15475)

133346(133296)

4598(4548)

085 403 (3980) 135757(135707) 718 (666) 6444

(6394)180998(180948) 1145 (1094) 9433

(9383)175673(175623)

1896(1845)

08 2224(2174)

118192(118142) 344 (290) 3594

(3544)159117(159067) 524 (471) 5543

(5492)157292(157242) 876 (824)

075 1222(1171)

57896(57846) 200 (141) 1974

(1923)81731(81681) 282 (227) 3179

(3128)88602(88552) 456 (403)

07 682 (630) 16228(16177) 139 (0740) 1086

(1034)24616(24566) 179 (119) 1800

(1749)31123(31073) 269 (213)

065 396 (342) 3111 (3060) 114 (0393) 610 (558) 5003 (4952) 132 (0654) 1022 (970) 7498 (7448) 180 (1202)06 245 (189) 583 (531) 104 (0195) 359 (305) 922 (871) 112 (0358) 592 (539) 1537 (1486) 136 (0702)055 167 (106) 171 (1098) 101 (0083) 227 (170) 233 (175) 103 (0182) 357 (303) 368 (314) 115 (0409)05 127 (0589) 105 (02310) 1 (0028) 158 (0954) 115 (04118) 101 (0079) 229 (172) 144 (07909) 105 (0225)025 1 (00009) 100 (000000) 1 (00000) 1 (0006) 100 (000000) 1 (000001) 1 (0042) 100 (----) 1 (00004)

6 Mathematical Problems in Engineering

00 02 04 06 08 10

600

500

400

300

200

100

0

ARL

s

ARL of exponentialARL of weibullARL of weibull-exponential

Comparison of ARLs

f

Figure 1 ARLs of exponential Weibull and proposed WEx at 200with pL 005 and pU 095

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

800

600

400

200

0

ARL

s

Comparison of ARLs

Figure 2 ARLs of exponential Weibull and proposed WEx at 300with pL 005 and pU 095

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

1500

1000

500

0

ARL

s

Comparison of ARLs

Figure 3 ARLs of exponential Weibull and proposed WEx at 370with pL 005 and pU 095

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

1200

1000

800

600

400

200

0

ARL

s

Comparison of ARLs

Figure 4 ARLs of exponential Weibull and proposed WEx at 200with pL 01 and pU 09

Mathematical Problems in Engineering 7

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

1500

1000

500

0

ARL

s

Comparison of ARLs

Figure 5 ARLs of exponential Weibull and proposed WEx at 300with pL 01 and pU 09

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

3000

2500

2000

1500

1000

500

0

ARL

s

Comparison of ARLs

Figure 6 ARLs of exponential Weibull and proposed WEx at 370with PL 01 an dPU 09

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

1400

1200

1000

800

600

400

200

0

ARL

s

Comparison of ARLs

Figure 7 ARLs of exponential Weibull and proposed WEx at 200with pL 02 and pU 08

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

1500

1000

500

0

ARL

s

Comparison of ARLs

Figure 8 ARLs of exponential Weibull and proposed WEx at 300with pL 02 and pU 08

8 Mathematical Problems in Engineering

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

1500

1000

500

0

ARL

s

Comparison of ARLs

Figure 9 ARLs of exponential Weibull and proposed WEx at 370 with pL 02 and pU 08

Table 6 Simulated in-control data0220499 0467107 0165884 0478009 0186779 00584930115906 0088989 0450255 0287988 0867485 025340604247 0714495 0356458 026482 0503415 05297730278757 008675 0204802 0181542 0417945 01030820269525 0543777 0096818 004666 0070134 0177020166788 0632749 0482233 0208581 0354741 02424770592133 0837021 0513781 0086579

Table 7 Simulated shifted data0650604 0144811 0585181 01867 0613353 05015330092963 0733554 0192515 0154301 0913754 05739580462511 0190089 058542 0297584 0162154 00806710066036 0746633 0545823 0061 0198106 0921250057984 0692892 026246 0463523 0204097 03307340454024 0765241 051748 0676308 0341994 04709630289353 0391563 0502378 0011261

0 10 20 30 40Sample number

10

08

06

04

02

00

Sam

ple v

alue

s

UDL = 095051

LDL = 003206

Figure 10 Plot for simulated in-control data

Mathematical Problems in Engineering 9

0 10 20 30 40Sample number

10

08

06

04

02

00

Sam

ple v

alue

s

UDL = 095051

LDL = 003206

Figure 11 Plot for simulated shift data with f 09 pL 01 pU 09 andARL0 200

0 10 20 30 5040Sample number

15

10

05

00

Sam

ple v

alue

s

UDL = 13544

LDL = 006197

Figure 12 Plot of UTI using exponential distribution control chart at pL 01 andpU 09

0 10 20 30 5040Sample number

15

10

05

00

Sam

ple v

alue

s UDL = 098102

LDL = 009615

Figure 13 Plot of UTI using proposed (WEx) control chart at pL 01 andpU 09

0 10 20 30 5040Sample number

15

10

05

00

Val

ues o

f T =

UTI

UDL = 15174

LDL = 03246

Figure 14 Plot of UTI using Weibull distribution control chart at pL 01 andpU 09

10 Mathematical Problems in Engineering

whereas the control chart of the proposed distributionwhich can be seen in Figure 13 that detected that theprocess is out-of-control at the 40th value It may beconcluded that our proposed distribution using real dataset of UTI shows better performance to detect the out-of-control process quickly

9 Conclusion

In this paper we develop a newWeibull class of distributionsusing the distribution generator technique )e newlyproposed WEx distribution is applied in the attributecharacteristics control chart for monitoring themanufacturing process Furthermore we compare theproposed WEx distribution with exponential distribution tomonitor the quick detection of the item in the out-of-controlprocess We conclude that under various shift levels ourproposed model is more efficient than the exponentialdistribution A simulation study is also conducted to knownthe behavior of ARL using the proposed WEx Weibull andexponential distributions )e implementation of the pro-posed model in a real-life example from the health caredepartment and the comparison of the two models endorsethe scope of WEx distribution We hope that this proposedcontrol chart can also be used for other accepting samplingschemes and other lifetime models (Figure 14)

Notations

W WeibullWEx Weibull-exponentialR Range chartS2 chart Variance chartW-G Weibull-generalizedPL PU PercentilesUDL Upper discriminant limitLDL Lower discriminant limita Accepted itemb Rejected itemf Shift levelα c c λ Parameters of Weibull-exponential distributionARL Average run-lengthP0out Probability of out-of-control process

P1out Probability of out-of-control process of shifted

processP0 Probability of rejected item when process is in-

controlP1 Probability of rejected item when process is shiftedUTI Urinary tract infection

Appendix

A R-Code

Derivation of Equation (9) (Algorithms 1 and 2)

Pr Xlt LCL|c0( 1113857 PL

1 minus exp minusαλx

c1113888 1113889

c

1113888 1113889 PL

1 minus PL exp minusαλx

c1113888 1113889

c

1113888 1113889

(A1)

Taking ln on both sides

ln 1 minus PL( 1113857 ln exp minusαλx

c1113888 1113889

c

1113888 1113889

minus ln 1 minus PL( 1113857 αλx

c1113888 1113889

c

minus ln 1 minus PL( 11138571113858 1113859(1c)

αλx

c1113888 1113889

ln1

1 minus PL

1113888 11138891113890 1113891

(1c)

αλx

c1113888 1113889

c

α λln

11 minus PL

1113888 11138891113890 1113891

(1c)

x

(A2)

)e LCL is (cαλ)[ln(11 minus PL)](1c)

Input different values of shift (f )f1 log(1(1 minus pU)) f2 log(1(1minus pL))p1 1 + exp(minus f1lowast f ) minus exp(minus f2lowastf )u a+ b minus 1 l a minus 1beta sum(dbinom(lu n p1))ARL1 1beta SDRL sqrt((1minus beta)((beta)2))Output ARL for different values of shift

ALGORITHM 2 Computation of ARL for different shifts

Input sample sizes (ss) PL PU and r0P0 1 minus PU+PLa sample(1 70 1 T) b sample(1 70 1 T)u a+ b minus 1 l a minus 1 n a+ b minus 1Pout0 sum(dbinom(lu n p0))ARL0[i] 1Pout0ARL0 [i] 1Pout0 aa[i] a bb[i] b nn[i] nImpose condition ARL0 ge r0Output obtain corresponding values of a b n and ARL0

ALGORITHM 1 Computation of a b n and ARL0

Mathematical Problems in Engineering 11

Derivation of Equation (10)

Pr XltUCL|c0( 1113857 PU

1 minus exp minusαλx

c1113888 1113889

c

1113888 1113889 PU

1 minus PU exp minusαλx

c1113888 1113889

c

1113888 1113889

(A3)

Taking ln on both sides

ln 1 minus PU( 1113857 ln exp minusαλx

c1113888 1113889

c

1113888 1113889

minus ln 1 minus PU( 1113857 αλx

c1113888 1113889

c

minus ln 1 minus Pu( 11138571113858 1113859(1c)

αλx

c1113888 1113889

ln1

1 minus PU

1113888 11138891113890 1113891

(1c)

αλx

c1113888 1113889

c

αλln

11 minus PU

1113888 11138891113890 1113891

(1c)

x

(A4)

)e UCL is (cαλ)[ln(11 minus PL)](1c)Equation (12) DerivationFor the in-control process the probability of the out-of-

control process is demonstrated through control charts isgiven as

P0out 1 minus 1113944

a+bminus 1

xa

x minus 1

a minus 11113888 1113889P

ao 1 minus P0( 1113857

xminus a (A5)

)e authors compute the probability of out-of-controlprocess as P(Xge b)

Now

P(Xge b) 1 minus P(Xlt b) (A6)

Since ldquoXrdquo follows the negative binomial distributionthen

P(Xlt b) 1113944

a+bminus 1

xa

x minus 1

a minus 11113888 1113889p

a0 1 minus p0( 1113857

xminus a (A7)

Equation (14) Derivation)e probability that the process is found out-of-control

due to the shifted process using control chart is obtained as

P1out 1 minus 1113944

a+bminus 1

xa

x minus 1

a minus 11113888 1113889P

a1 1 minus P1( 1113857

xminus a (A8)

)e authors compute the probability of out-of-controlprocess as P(Xge b)

Now

P(Xge b) 1 minus P(Xlt b) (A9)

Since ldquoXrdquo follows the negative binomial distributionthen

P(Xlt b) 1113944a+bminus 1

xa

x minus 1

a minus 11113888 1113889p

a1 1 minus p1( 1113857

xminus a (A10)

Data Availability

)e data used to support the findings of this study are givenin the paper

Disclosure

No funds were obtained for this paper

Conflicts of Interest

)e authors declare that there are no conflicts of interest

References

[1] N Eugene C Lee and F Famoye ldquoBeta-normal distributionand its applicationsrdquo Communications in Statistics-5eoryand Methods vol 31 no 4 pp 497ndash512 2002

[2] S Nadarajah and S Kotz ldquo)e beta Gumbel distributionrdquoMathematical Problems in Engineering vol 2004 no 4pp 323ndash332 2004

[3] S Nadarajah and A K Gupta ldquo)e beta Frechet distributionrdquoFar East Journal of 5eoretical Statistics vol 14 no 1pp 15ndash24 2004

[4] S Nadarajah and S Kotz ldquo)e beta exponential distributionrdquoReliability Engineering amp System Safety vol 91 no 6pp 689ndash697 2006

[5] S Nadarajah G M Cordeiro and E M M Ortega ldquo)ezografos-balakrishnan-GFamily of distributions mathemati-cal properties and applicationsrdquo Communications in Statistics- 5eory and Methods vol 44 no 1 pp 186ndash215 2015

[6] A Alzaatreh C Lee and F Famoye ldquoA new method forgenerating families of continuous distributionsrdquo Metronvol 71 no 1 pp 63ndash79 2013

[7] G M Cordeiro E M M Ortega B V Popovic andR R Pescim ldquo)e Lomax generator of distributions prop-erties minification process and regression modelrdquo AppliedMathematics and Computation vol 247 pp 465ndash486 2014

[8] A Z Afify M Alizadeh H M Yousof G Aryal andM Ahmad ldquo)e transmuted geometric-g family OF distri-butions THEORY and applicationsrdquo Pakistan Journal ofStatistics vol 32 no 2 pp 139ndash160 2016

[9] M Alizadeh M H Tahir G M Cordeiro M MansoorM Zubair and G G Hamedani ldquo)e Kumaraswamy Mar-shal-Olkin family of distributionsrdquo Journal of the EgyptianMathematical Society vol 23 no 3 pp 546ndash557 2015

[10] A Z Afify G M Cordeiro H M Yousof A Alzaatreh andZ M Nofal ldquo)e Kumaraswamy transmuted-G family ofdistributions properties and applicationsrdquo Journal of DataScience vol 14 no 2 pp 245ndash270 2016

[11] Z M Nofal A Z Afify H M Yousof and G M Cordeiroldquo)e generalized transmuted-G family of distributionsrdquo

12 Mathematical Problems in Engineering

Communications in Statistics-5eory and Methods vol 46no 8 pp 4119ndash4136 2017

[12] M Chahkandi and M Ganjali ldquoOn some lifetime distribu-tions with decreasing failure raterdquo Computational Statisticsand Data Analysis vol 53 no 12 pp 4433ndash4440 2009

[13] W Barreto-Souza A H S Santos and G M Cordeiro ldquo)ebeta generalized exponential distributionrdquo Journal of Statis-tical Computation and Simulation vol 80 no 2 pp 159ndash1722010

[14] W Lu and D Shi ldquoA new compounding life distribution theWeibull-Poisson distributionrdquo Journal of Applied Statisticsvol 39 no 1 pp 21ndash38 2012

[15] D C Montgomery Statistical Quality Control Vol 7 WileyNew York NY USA 2009

[16] W H Woodall ldquo)e use of control charts in health-care andpublic-health surveillancerdquo Journal of Quality Technologyvol 38 no 2 pp 89ndash104 2006

[17] S-L Hwang J-T Lin G-F Liang Y-J Yau T-C Yenn andC-C Hsu ldquoApplication control chart concepts of designing apre-alarm system in the nuclear power plant control roomrdquoNuclear Engineering and Design vol 238 no 12 pp 3522ndash3527 2008

[18] Z Wang and R Liang ldquoDiscuss on applying SPC to qualitymanagement in university educationrdquo in Proceedings of the9th International Conference for Young Computer Scientistspp 2372ndash2375 Hunan China November 2008

[19] M Pastell and H Madsen ldquoApplication of CUSUM charts todetect lameness in a milking robotrdquo Expert Systems withApplications vol 35 no 4 pp 2032ndash2040 2008

[20] A Ivanova C Xiao and Y Tymofyeyev ldquoTwo-stage designsfor phase 2 dose-finding trialsrdquo Statistics in Medicine vol 31no 24 pp 2872ndash2881 2012

[21] S W Roberts ldquoControl chart tests based on geometricmoving averagesrdquo Technometrics vol 1 no 3 pp 239ndash2501959

[22] D Brook and D A Evans ldquoAn approach to the probabilitydistribution of CUSUM run lengthrdquo Biometrika vol 59 no 3pp 539ndash549 1972

[23] J M Lucas ldquoCombined Shewhart-CUSUM quality controlschemesrdquo Journal of Quality Technology vol 14 no 2pp 51ndash59 1982

[24] C M Borror J B Keats and D C Montgomery ldquoRobustnessof the time between events CUSUMrdquo International Journal ofProduction Research vol 41 no 15 pp 3435ndash3444 2003

[25] M B C Khoo ldquoA moving average control chart for moni-toring the fraction non-conformingrdquo Quality and ReliabilityEngineering International vol 20 no 6 pp 617ndash635 2004

[26] S Sukparungsee and A A Novikov ldquoAnalytical approxi-mations for detection of a change-point in case of light-taileddistributionsrdquo Journal of Quality Measurement and Analysisvol 4 no 2 pp 49ndash56 2008

[27] Y Areepong and A A Novikov ldquoMartingale approach toEWMA control chart for changes in Exponential distribu-tionrdquo Journal of Quality Measurement and Analysis vol 4pp 197ndash203 2008

[28] R Noorossana A A Fatahi P Dokouhaki andM Babakhani ldquoZIB-EWMA control chart for monitoringrare health eventsrdquo Journal of Mechanics in Medicine andBiology vol 11 no 04 pp 881ndash895 2011

[29] E Santiago and J Smith ldquoControl charts based on the ex-ponential distribution adapting runs rules for the tChartrdquoQuality Engineering vol 25 no 2 pp 85ndash96 2013

[30] L L Ho and R C Quinino ldquoAn attribute control chart formonitoring the variability of a processrdquo International Journalof Production Economics vol 145 no 1 pp 263ndash267 2013

[31] M Aslam M Azam and C-H Jun ldquoAcceptance samplingplans for multi-stage process based on time-truncated test forWeibull distributionrdquo 5e International Journal of AdvancedManufacturing Technology vol 79 no 9ndash12 pp 1779ndash17852015

[32] A C Rakitzis and P Castagliola ldquo)e effect of parameterestimation on the performance of one-sided Shewhart controlcharts for zero-inflated processesrdquo Communications in Sta-tistics - 5eory and Methods vol 45 no 14 pp 4194ndash42142016

[33] M Azam M N Aslam and C-H Jun ldquoAn EWMA controlchart for the exponential distribution using repetitive sam-pling planrdquo 2017

[34] M Azam L Ahmad M Aslam and C-H Jun ldquoAn attributecontrol chart using discriminant limits for monitoring processunder the Weibull distributionrdquo Production Engineeringvol 12 no 5 pp 659ndash665 2018

[35] Z Wu and J Jiao ldquoA control chart for monitoring processmean based on attribute inspectionrdquo International Journal ofProduction Research vol 46 no 15 pp 4331ndash4347 2008

[36] H Kim and S Lee ldquoOn the VSI CUSUM chart for countprocesses and its implementation with R package attrCU-SUMrdquo Industrial Engineering ampManagement Systems vol 17no 1 pp 91ndash101 2018

[37] K Mabude J C Malela-Majika M Aslam Z L Chong andS C Shongwe ldquoDistribution-free composite Shewhart-GWMA Mann-Whitney charts for monitoring the processlocationrdquo Quality and Reliability Engineering International2020

[38] M T Shah M Azam M Aslam and U Sherazi ldquoTimebetween events control charts for gamma distributionrdquoQuality and Reliability Engineering International vol 37no 2 pp 785ndash803 2020

[39] U Shaheen M Azam and M Aslam ldquoA control chart formonitoring the lognormal process variation using repetitivesamplingrdquo Quality and Reliability Engineering Internationalvol 36 no 3 pp 1028ndash1047 2020

[40] M Aslam P Jeyadurga S Balamurali M Azam and A AL-Marshadi ldquoEconomic determination of modified multipledependent state sampling plan under some lifetime distri-butionsrdquo Journal of Mathematics vol 2021 Article ID7470196 2021

[41] R H Berk N L Johnson S Kotz and N BalakrishnanldquoContinuous univariate distributionsrdquo Technometrics vol 2p 752 1996

[42] C J P Belisle ldquoConvergence theorems for a class of simulatedannealing algorithms on Rdrdquo Journal of Applied Probabilityvol 29 no 4 pp 885ndash895 1992

[43] A Henningsen and O Toomet ldquoMaxlik a package formaximum likelihood estimation in Rrdquo Computational Sta-tistics vol 26 pp 443ndash458 2011

Mathematical Problems in Engineering 13

Using H(G(x)) minus log[(1 minus G(x))α] and r(t) of Wei-bull distribution the cdf of the WG class of distributions isobtained as

F(x) 1113946minus log (1minus G(x))α[ ]

a

c

c

t

c1113888 1113889

cminus 1

exp minust

c1113888 1113889

c

1113888 1113889dt

1 minus exp minusminus log (1 minus G(x))α( )

c1113888 1113889

c

1113888 1113889

(3)

and pdf is obtained by differentiating (3) as

f(x) αc

c1113888 1113889

minus log [1 minus G(x)]α

c1113890 1113891

cminus 1

middot eminus minus log [1minus G(x)]αc( )[ ]

c g(x)

1 minus G(x)

(4)

Now several new distributions can be generated for thedifferent values of G(x)

3 Weibull-Exponential Distribution

In this section we derive Weibull-exponential distributionusing the proposed class of distributions

If x is an exponential distribution with cdf and pdf thenit is

G(x) 1 minus eminus λx

(5)

g(x) λeminus λx

(6)

respectively )e cdf and the pdf of the WEx distribution isgenerated by inserting (5) and (6) in (3) as

F(x) 1 minus eminus (αλxc)c

f(x) cα λc

1113888 1113889

c

xcminus 1

eminus (αλxc)c

xgt 0 αgt 0 c λgt 0

(7)

Some special cases of the proposed distribution are givenin Appendix (Table 1)

4 The Proposed Control Chart

)is article proposes the new scheme of attribute charac-teristics control charts for the manufacturing process usingWEx distribution For a fixed sampling interval where n isconsidered as a sample for each subgroup we discuss thelifetime of the product when random variable follows theproposed model If the selected unit falls within controllimits we accept it otherwise it would be rejected)erefore we present the lower control limit (LCL) andupper control limit (UCL) as follows

)e LCL and UCL are defined as

Pr Xlt LCL|c0( 1113857 PL

Pr XltUCL|c0( 1113857 PU(8)

respectively where c0 is the shape parameter )e LCL andUCL for the control process by following Azam et al [34] aregiven by

LCL c

αλln

11 minus PL

1113890 1113891

(1c)

(9)

UCL c

αλln

11 minus PU

1113890 1113891

(1c)

(10)

Ho and Quinino [30] presented the derivation of controlcharts

Firstly a sample is chosen at random from each sub-group and its quality characteristics X are measured )eselected unit is accepted if it lies within defined limits andotherwise declared as rejected

Secondly the process is in control if a item is accepted atfirst and out-of-control if b item is rejected first

When the process is in control the probability that theprocess is out-of-control is defined as

P0 P LCLltXltUCL|c c0( 1113857 1 minus PU + PL (11)

Moreover for the in-control process the probability ofthe out-of-control process is demonstrated through controlcharts is given as

P0out 1 minus 1113944

a+bminus 1

xa

x minus 1

a minus 11113888 1113889P

ao 1 minus P0( 1113857

xminus a (12)

Furthermore the shape parameter is converted to c1)en the probability of the rejected item is derived as

P1 P LCLltXltUCL|c c1( 1113857 (13)

)e probability that the process is found to be out-of-control due to the shifted process using control chart isobtained as

P1out 1 minus 1113944

a+bminus 1

xa

x minus 1

a minus 11113888 1113889P

a1 1 minus P1( 1113857

xminus a (14)

Now the shape parameter is considered for the shiftedprocess which is given as follows

c flowast c1 (15)

where f is a shifted constant and lies between 025 and 1)erefore (11) is formed as

Table 1 Special cases of the proposed distribution

Special cases Parameters DistributionCase 1 (αλc) θ f(x) sim W(θ c)

Case 2 If c 1 λ 1 α 1 f(x) sim Exp(c)

Case 3 If c 1 λ 1 α 1 f(x) sim W(c)

Case 4 If c 1 α 1 f(x) sim Exp(λ c)

Case 5 If c 2 f(x) sim Rayleigh(α λ c)

Mathematical Problems in Engineering 3

P1 expαλUCL

c1113888 1113889

fc1

⎡⎣ ⎤⎦ + 1 minus expαλLCL

c1113888 1113889

fc1

⎡⎣ ⎤⎦

P1 1 + exp minusαλclowast

c

α λ1113888 1113889 ln

11 minus PU

1113888 1113889

1c1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

fc1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦minus exp minus

αλclowast

c

α λ1113888 1113889 ln

11 minus PL

1113888 1113889

1c1( )⎛⎝ ⎞⎠

fc1

⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦

P1 1 + exp minus ln1

1 minus PU

1113888 1113889

f

⎡⎣ ⎤⎦ minus exp minus ln1

1 minus PL

1113888 1113889

f

⎡⎣ ⎤⎦

(16)

)eARL is evaluated using P0 and P1 which are obtainedthrough the parameters of the proposed distribution Notehere that the equations ofP0 andP1 are independent of scaleparameters )erefore we fixed only the shape parameter

Now the ARL of in-control and ARL of out-of-controlhave been calculated as

ARL0 1

P0out

andARL1 1

P1out

(17)

5 ARL Behavior of the Newly ProposedControl Chart

In this section we discuss the ARL behavior of thesuggested control chart A simulation study is conductedin the R-language program using the above-estimatedequations of probabilities In simulation study 10000samples are generated from the well-known exponentialWeibull and the proposed WEx distributions )e MLestimates of the parameters of the suggested distributionare obtained by using SANN (simulated annealing) givenin Belisle [42] SANN is an optimization technique whichis more recent and flexible as compared to New-tonndashRaphson )e SANN method is available inR-package maxLik Henningsen and Toomet [43] Wefixed the shape parameters to find the ARLs

)ese ML estimates are used to find ARLs forARL0 200 300 and 370 Furthermore ARLvalues areexplained by calculating for different drift levels It is im-portant to mention that the shift unit is equal to the in-control ARL)e performance of ARL can be understood byreading the R-code program Simulated data of 10000samples are generated by choosing the random values of a bandn For in-control processes the ARL is calculated byusing these given random values For the selection of a and b

parameters see details in Ho and Quinino [30] Moreoverwe compare the exponential and Weibull distribution withthe proposed WEx distribution for the efficiency of ARL

6 Comparative Study

In this section we compare the performance of theproposed WEx distribution with the exponential distri-bution under ARL In Table 2 we report the estimatedvalues of the proposed model parameters Various values

of the parameters of exponential Weibull and theproposed WEx distributions are placed in Tables 3ndash5 forvarious combinations of a and b atARL0 200 300 and 370 From Table 3 it is seen thatvarious values of a and b at f 1 ie there is no shift inthe values of ARLs of the exponential Weibull and theproposed WEx distributions are the same ie 2019830333 and 37199 at ARL0 200 300 and 370 On theother hand it is observed that at a different shift levelf 1 to 025 the values of ARL decrease for the expo-nential and the proposed models

One can clearly observe in Tables 3ndash5 that the pro-posed model suppressed the exponential and Weibulldistribution in terms of shift detection From Table 3 forinstance the value of ARL is 1704 for the WEx distri-bution and 655 for the exponential distribution whenf 085 a 9 b 20 andARL0 200 FurthermoreTables 4 and 5 also represent the values of ARLs of theexponential and the proposed models at ARL0 200 300and 370 with PL 01 andPU 09 and PL 02 andPU

08 for various combinations of a and b at different levelsof shifts It can be noticed that the values of ARLs de-crease for both models when the values of a and b de-crease By comparing both models it is concluded thatthe proposed model gives quick detection of out-of-control process at different levels of shifts Tables 4 and 5are presented in Appendix

Figures 1ndash9 demonstrate the graphical representations ofthe performance of ARLs at 200 300 and 370 for severalcombinations of aampb and PLampPU One can see fromFigures 1ndash9 as shift level f decreases the curves of ARLs theexponential and the proposed WEx distributions also de-crease Also the ARL curves show that the behavior of theproposed model is better than that of the compared modelFigures 4ndash9 are also presented in Appendix

7 Real-Life Example

In this section the proposed control chart is imple-mented in real-life data set from the health sector )edata of urinary tract infection (UTI) are taken from thehospital system )ese data are attained from Azam et al[34] which was initially used by Santiago and Smith [29]Data are collected from a hospital with a high risk ofurinary tract infections particularly to identify risk rate

4 Mathematical Problems in Engineering

and the frequency of the UTI patients being dischargedIn this example male patients are focused )e data showthe number of days between admissions and discharge ofthe patients having UTIs )e simulated data for theabove scenario are generated from the proposed WExdistribution with PL 01 PU 09 a 29 and b 59Firstly we generate 40 observations for the in-controlprocess and then the next 40 observations for the shiftedprocess withf 090 as presented in Tables 6 and 7 Inaddition we estimate the values of LCL and UCL as003206 and 095051 respectively and display in Fig-ure 10 Figures 10 and 11 show the LCL and UCL for in-control and out-of-control data It is clearly shown that

40th value is detected as out-of-control for the proposedmodel

8 Comparisons of the Exponential and theWeibull-Exponential Distribution UsingReal-Life Example

In this section we compare the control charts of theexponential and the proposed WEx distributions usingreal data of UTI patients It can be easily seen in Figure 12that the control chart of the exponential distribution isin-control and no value is detected as out-of-control

Table 2 Maximum likelihood estimation of the proposed WEx distribution parameterMaximum likelihood estimationSANN maximization 10000 iterationsReturn code 0 successful convergenceLog-likelihood minus 4288574Parameter estimates Estimate Std error t value P valueα 274 006969 393169 2e minus 16C 1328 44492 029484 0653c 701 71518 09801 0613λ 488 113607 04293 0697

Table 3 Comparison of ARL (SDRL) using exponential and Weibull distributions and WEx distribution with pL 005 and pU 095

f

ARL0 200 ARL0 300 ARL0 370a 9 b 20 a 7 b 10 a 13 b 37

Exp Weibull WEx Exp Weibull WEx Exp Weibull WEx

1 20198(20148)

20198(20148)

20198(20148)

30333(30283)

30333(30283)

30333(30283)

37199(37149)

37199(37149)

37199(37149)

099 18971(18921)

22804(22754)

16750(16699)

28791(28741)

33552(33502)

25948(25898)

34146(34096)

43925(43875)

28813(28762)

095 14493(14443)

35189(35139) 8172 (8122) 22987

(22937)48020(47970)

14165(14115)

23666(23616)

79829(79779)

1094(10890)

09 9943(9893)

51359(51309)

3581(3530)

16732(16682)

(65478)(65428)

6945(6895)

14227(14177)

134895(134845) 3691 (3641)

085 655 (6499) 58654(58604)

1704(1654)

11723(11673)

72978(72928) 3581 (3530) 8149

(8099)162317(162266) 1437 (1386)

08 417 (4120) 50734(50684) 884 (832) 7935

(7885)64825(64775) 1944 (1894) 4500

(4449)132614(132564) 649 (597)

075 2588(2538)

33790(33740) 501 (448) 5214 (5163) 46442

(46392) 1114 (1063) 2432(2382)

75476(75426) 341 (287)

07 1583(1532)

18337(18287) 311 (256) 3347

(3297)27986(27936) 674 (622) 1311 (1260) 32598

(32547) 209 (151)

065 966 (915) 8714 (8664) 212 (154) 2115 (2065) 14960(14910) 433 (379) 721 (669) 11920

(11870) 148 (0838)

06 597 (545) 3870(3820) 158 (0953) 1328 (1277) 7435 (7384) 294 (239) 414 (360) 4083 (4033) 119 (0476)

055 380 (326) 1689 (1639) 128 (0599) 837 (785) 3552 (3502) 213 (155) 255 (199) 1422 (1371) 106 (0259)05 253 (197) 754 (702) 112 (0371) 535 (482) 1670 (1619) 164 (103) 172 (112) 540 (487) 102 (113)

025 101 (0098) 1001(00296) 1 (0005) 115 (0409) 104 (0211) 1 (0082) 1 (0012) 100 (00013) 1 (000004)

Mathematical Problems in Engineering 5

Table 4 Comparison of ARL (SDRL) using exponential and Weibull distributions and WEx distribution with pL 01 and pU 09

f

ARL0 200 ARL0 300 ARL0 370a 29 b 59 a 23 b 40 a 34 b 70

Exp Weibull WEx Exp Weibull WEx Exp Weibull WEx

1 20001(19951)

20001(19951)

20001(19951)

30499(30449)

30499(30449)

30499(30449)

37491(37441)

37491(37441)

37491(37441)

099 18127(18077)

24246(24196)

1501(14960)

27928(27878)

36215(36165)

23577(23527)

33449(33399)

46870(46820)

26878(26828)

095 11961(11911)

49423(49376)

5155(5105)

19214(19164)

68054(68003) 8912 (8862) 20662

(20612)107188(107138)

78115(7761)

09 6805(6755)

96705(96652)

1629(1578)

11504(11454)

122584(122534) 3013 (2962) 10763

(10713)234044(233994)

2078(2028)

085 373 (3679) 129441(129391) 635 (583) 6601

(6551)158035(157985) 1185 (1134) 5379

(5329)328659(328609) 712 (660)

08 2002(1951)

106293(106243) 306 (251) 3671

(3620)133121(133071) 545 (492) 2631

(2581)261275(261225) 315 (260)

075 1074(1023)

52202(52152) 182 (122) 2008

(1957)71411(71361) 294 (239) 1291

(1240)114225(114175) 179 (118)

07 590 (538) 16663(16613) 131 (0633) 1100

(1048)25897(25847) 186 (126) 655 (603) 30337

(30287) 127 (0583)

065 342 (288) 4137 (4087) 11 (0328) 616 (564) 7269 (7219) 136 (0702) 356 (302) 6061 (6011) 108 (0284)06 215 (157) 1009 (958) 102 (0155) 362 (308) 1888 (1838) 114 (0392) 214 (156) 1203 (1152) 101 (0122)

055 151 (0872) 307 (253) 1 (0061) 229 (172) 547 (495) 104 (0206) 146(08244) 316 (261) 1 (0041)

05 119 (0476) 142 (0770) 1 (0018) 159(09692) 210 (1520) 101 (0095) 116

(04275) 138 (07208) 1 (0010)

025 1 (00004) 100 (---) 1 (0000) 1 (00079) 100(0000001) 1 (000002) 1 (00001) 100 (000000) 1 (0000)

Table 5 Comparison of ARLs (SDRL) using exponential and Weibull distribution and WEx distribution with pL 02 and pU 08

f

ARL0 200 ARL0 300 ARL0 370a 63 b 57 a 54 b 45 a 42 b 31

Exp Weibull WEx Exp Weibull WEx Exp Weibull WEx

1 20021(19971)

20021(19971)

20021(11971)

30051(30001)

30051(30001)

30051(30001)

37082(37032)

37082(37032)

37082(37032)

099 18213(18163)

24105(24055)

15262(15212)

27470(27420)

35829(35778)

2322(23170)

34273(34223)

43255(43205)

29564(29514)

095 1223(12180)

48380(48330)

5517(5467)

18800(18750)

69067(69017)

8745(8695)

24541(24491)

76586(76535)

12417(12367)

09 7144(7094)

96152(96102)

1814(1763)

11225(11175)

131293(131243)

2935(2884)

15525(15475)

133346(133296)

4598(4548)

085 403 (3980) 135757(135707) 718 (666) 6444

(6394)180998(180948) 1145 (1094) 9433

(9383)175673(175623)

1896(1845)

08 2224(2174)

118192(118142) 344 (290) 3594

(3544)159117(159067) 524 (471) 5543

(5492)157292(157242) 876 (824)

075 1222(1171)

57896(57846) 200 (141) 1974

(1923)81731(81681) 282 (227) 3179

(3128)88602(88552) 456 (403)

07 682 (630) 16228(16177) 139 (0740) 1086

(1034)24616(24566) 179 (119) 1800

(1749)31123(31073) 269 (213)

065 396 (342) 3111 (3060) 114 (0393) 610 (558) 5003 (4952) 132 (0654) 1022 (970) 7498 (7448) 180 (1202)06 245 (189) 583 (531) 104 (0195) 359 (305) 922 (871) 112 (0358) 592 (539) 1537 (1486) 136 (0702)055 167 (106) 171 (1098) 101 (0083) 227 (170) 233 (175) 103 (0182) 357 (303) 368 (314) 115 (0409)05 127 (0589) 105 (02310) 1 (0028) 158 (0954) 115 (04118) 101 (0079) 229 (172) 144 (07909) 105 (0225)025 1 (00009) 100 (000000) 1 (00000) 1 (0006) 100 (000000) 1 (000001) 1 (0042) 100 (----) 1 (00004)

6 Mathematical Problems in Engineering

00 02 04 06 08 10

600

500

400

300

200

100

0

ARL

s

ARL of exponentialARL of weibullARL of weibull-exponential

Comparison of ARLs

f

Figure 1 ARLs of exponential Weibull and proposed WEx at 200with pL 005 and pU 095

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

800

600

400

200

0

ARL

s

Comparison of ARLs

Figure 2 ARLs of exponential Weibull and proposed WEx at 300with pL 005 and pU 095

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

1500

1000

500

0

ARL

s

Comparison of ARLs

Figure 3 ARLs of exponential Weibull and proposed WEx at 370with pL 005 and pU 095

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

1200

1000

800

600

400

200

0

ARL

s

Comparison of ARLs

Figure 4 ARLs of exponential Weibull and proposed WEx at 200with pL 01 and pU 09

Mathematical Problems in Engineering 7

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

1500

1000

500

0

ARL

s

Comparison of ARLs

Figure 5 ARLs of exponential Weibull and proposed WEx at 300with pL 01 and pU 09

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

3000

2500

2000

1500

1000

500

0

ARL

s

Comparison of ARLs

Figure 6 ARLs of exponential Weibull and proposed WEx at 370with PL 01 an dPU 09

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

1400

1200

1000

800

600

400

200

0

ARL

s

Comparison of ARLs

Figure 7 ARLs of exponential Weibull and proposed WEx at 200with pL 02 and pU 08

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

1500

1000

500

0

ARL

s

Comparison of ARLs

Figure 8 ARLs of exponential Weibull and proposed WEx at 300with pL 02 and pU 08

8 Mathematical Problems in Engineering

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

1500

1000

500

0

ARL

s

Comparison of ARLs

Figure 9 ARLs of exponential Weibull and proposed WEx at 370 with pL 02 and pU 08

Table 6 Simulated in-control data0220499 0467107 0165884 0478009 0186779 00584930115906 0088989 0450255 0287988 0867485 025340604247 0714495 0356458 026482 0503415 05297730278757 008675 0204802 0181542 0417945 01030820269525 0543777 0096818 004666 0070134 0177020166788 0632749 0482233 0208581 0354741 02424770592133 0837021 0513781 0086579

Table 7 Simulated shifted data0650604 0144811 0585181 01867 0613353 05015330092963 0733554 0192515 0154301 0913754 05739580462511 0190089 058542 0297584 0162154 00806710066036 0746633 0545823 0061 0198106 0921250057984 0692892 026246 0463523 0204097 03307340454024 0765241 051748 0676308 0341994 04709630289353 0391563 0502378 0011261

0 10 20 30 40Sample number

10

08

06

04

02

00

Sam

ple v

alue

s

UDL = 095051

LDL = 003206

Figure 10 Plot for simulated in-control data

Mathematical Problems in Engineering 9

0 10 20 30 40Sample number

10

08

06

04

02

00

Sam

ple v

alue

s

UDL = 095051

LDL = 003206

Figure 11 Plot for simulated shift data with f 09 pL 01 pU 09 andARL0 200

0 10 20 30 5040Sample number

15

10

05

00

Sam

ple v

alue

s

UDL = 13544

LDL = 006197

Figure 12 Plot of UTI using exponential distribution control chart at pL 01 andpU 09

0 10 20 30 5040Sample number

15

10

05

00

Sam

ple v

alue

s UDL = 098102

LDL = 009615

Figure 13 Plot of UTI using proposed (WEx) control chart at pL 01 andpU 09

0 10 20 30 5040Sample number

15

10

05

00

Val

ues o

f T =

UTI

UDL = 15174

LDL = 03246

Figure 14 Plot of UTI using Weibull distribution control chart at pL 01 andpU 09

10 Mathematical Problems in Engineering

whereas the control chart of the proposed distributionwhich can be seen in Figure 13 that detected that theprocess is out-of-control at the 40th value It may beconcluded that our proposed distribution using real dataset of UTI shows better performance to detect the out-of-control process quickly

9 Conclusion

In this paper we develop a newWeibull class of distributionsusing the distribution generator technique )e newlyproposed WEx distribution is applied in the attributecharacteristics control chart for monitoring themanufacturing process Furthermore we compare theproposed WEx distribution with exponential distribution tomonitor the quick detection of the item in the out-of-controlprocess We conclude that under various shift levels ourproposed model is more efficient than the exponentialdistribution A simulation study is also conducted to knownthe behavior of ARL using the proposed WEx Weibull andexponential distributions )e implementation of the pro-posed model in a real-life example from the health caredepartment and the comparison of the two models endorsethe scope of WEx distribution We hope that this proposedcontrol chart can also be used for other accepting samplingschemes and other lifetime models (Figure 14)

Notations

W WeibullWEx Weibull-exponentialR Range chartS2 chart Variance chartW-G Weibull-generalizedPL PU PercentilesUDL Upper discriminant limitLDL Lower discriminant limita Accepted itemb Rejected itemf Shift levelα c c λ Parameters of Weibull-exponential distributionARL Average run-lengthP0out Probability of out-of-control process

P1out Probability of out-of-control process of shifted

processP0 Probability of rejected item when process is in-

controlP1 Probability of rejected item when process is shiftedUTI Urinary tract infection

Appendix

A R-Code

Derivation of Equation (9) (Algorithms 1 and 2)

Pr Xlt LCL|c0( 1113857 PL

1 minus exp minusαλx

c1113888 1113889

c

1113888 1113889 PL

1 minus PL exp minusαλx

c1113888 1113889

c

1113888 1113889

(A1)

Taking ln on both sides

ln 1 minus PL( 1113857 ln exp minusαλx

c1113888 1113889

c

1113888 1113889

minus ln 1 minus PL( 1113857 αλx

c1113888 1113889

c

minus ln 1 minus PL( 11138571113858 1113859(1c)

αλx

c1113888 1113889

ln1

1 minus PL

1113888 11138891113890 1113891

(1c)

αλx

c1113888 1113889

c

α λln

11 minus PL

1113888 11138891113890 1113891

(1c)

x

(A2)

)e LCL is (cαλ)[ln(11 minus PL)](1c)

Input different values of shift (f )f1 log(1(1 minus pU)) f2 log(1(1minus pL))p1 1 + exp(minus f1lowast f ) minus exp(minus f2lowastf )u a+ b minus 1 l a minus 1beta sum(dbinom(lu n p1))ARL1 1beta SDRL sqrt((1minus beta)((beta)2))Output ARL for different values of shift

ALGORITHM 2 Computation of ARL for different shifts

Input sample sizes (ss) PL PU and r0P0 1 minus PU+PLa sample(1 70 1 T) b sample(1 70 1 T)u a+ b minus 1 l a minus 1 n a+ b minus 1Pout0 sum(dbinom(lu n p0))ARL0[i] 1Pout0ARL0 [i] 1Pout0 aa[i] a bb[i] b nn[i] nImpose condition ARL0 ge r0Output obtain corresponding values of a b n and ARL0

ALGORITHM 1 Computation of a b n and ARL0

Mathematical Problems in Engineering 11

Derivation of Equation (10)

Pr XltUCL|c0( 1113857 PU

1 minus exp minusαλx

c1113888 1113889

c

1113888 1113889 PU

1 minus PU exp minusαλx

c1113888 1113889

c

1113888 1113889

(A3)

Taking ln on both sides

ln 1 minus PU( 1113857 ln exp minusαλx

c1113888 1113889

c

1113888 1113889

minus ln 1 minus PU( 1113857 αλx

c1113888 1113889

c

minus ln 1 minus Pu( 11138571113858 1113859(1c)

αλx

c1113888 1113889

ln1

1 minus PU

1113888 11138891113890 1113891

(1c)

αλx

c1113888 1113889

c

αλln

11 minus PU

1113888 11138891113890 1113891

(1c)

x

(A4)

)e UCL is (cαλ)[ln(11 minus PL)](1c)Equation (12) DerivationFor the in-control process the probability of the out-of-

control process is demonstrated through control charts isgiven as

P0out 1 minus 1113944

a+bminus 1

xa

x minus 1

a minus 11113888 1113889P

ao 1 minus P0( 1113857

xminus a (A5)

)e authors compute the probability of out-of-controlprocess as P(Xge b)

Now

P(Xge b) 1 minus P(Xlt b) (A6)

Since ldquoXrdquo follows the negative binomial distributionthen

P(Xlt b) 1113944

a+bminus 1

xa

x minus 1

a minus 11113888 1113889p

a0 1 minus p0( 1113857

xminus a (A7)

Equation (14) Derivation)e probability that the process is found out-of-control

due to the shifted process using control chart is obtained as

P1out 1 minus 1113944

a+bminus 1

xa

x minus 1

a minus 11113888 1113889P

a1 1 minus P1( 1113857

xminus a (A8)

)e authors compute the probability of out-of-controlprocess as P(Xge b)

Now

P(Xge b) 1 minus P(Xlt b) (A9)

Since ldquoXrdquo follows the negative binomial distributionthen

P(Xlt b) 1113944a+bminus 1

xa

x minus 1

a minus 11113888 1113889p

a1 1 minus p1( 1113857

xminus a (A10)

Data Availability

)e data used to support the findings of this study are givenin the paper

Disclosure

No funds were obtained for this paper

Conflicts of Interest

)e authors declare that there are no conflicts of interest

References

[1] N Eugene C Lee and F Famoye ldquoBeta-normal distributionand its applicationsrdquo Communications in Statistics-5eoryand Methods vol 31 no 4 pp 497ndash512 2002

[2] S Nadarajah and S Kotz ldquo)e beta Gumbel distributionrdquoMathematical Problems in Engineering vol 2004 no 4pp 323ndash332 2004

[3] S Nadarajah and A K Gupta ldquo)e beta Frechet distributionrdquoFar East Journal of 5eoretical Statistics vol 14 no 1pp 15ndash24 2004

[4] S Nadarajah and S Kotz ldquo)e beta exponential distributionrdquoReliability Engineering amp System Safety vol 91 no 6pp 689ndash697 2006

[5] S Nadarajah G M Cordeiro and E M M Ortega ldquo)ezografos-balakrishnan-GFamily of distributions mathemati-cal properties and applicationsrdquo Communications in Statistics- 5eory and Methods vol 44 no 1 pp 186ndash215 2015

[6] A Alzaatreh C Lee and F Famoye ldquoA new method forgenerating families of continuous distributionsrdquo Metronvol 71 no 1 pp 63ndash79 2013

[7] G M Cordeiro E M M Ortega B V Popovic andR R Pescim ldquo)e Lomax generator of distributions prop-erties minification process and regression modelrdquo AppliedMathematics and Computation vol 247 pp 465ndash486 2014

[8] A Z Afify M Alizadeh H M Yousof G Aryal andM Ahmad ldquo)e transmuted geometric-g family OF distri-butions THEORY and applicationsrdquo Pakistan Journal ofStatistics vol 32 no 2 pp 139ndash160 2016

[9] M Alizadeh M H Tahir G M Cordeiro M MansoorM Zubair and G G Hamedani ldquo)e Kumaraswamy Mar-shal-Olkin family of distributionsrdquo Journal of the EgyptianMathematical Society vol 23 no 3 pp 546ndash557 2015

[10] A Z Afify G M Cordeiro H M Yousof A Alzaatreh andZ M Nofal ldquo)e Kumaraswamy transmuted-G family ofdistributions properties and applicationsrdquo Journal of DataScience vol 14 no 2 pp 245ndash270 2016

[11] Z M Nofal A Z Afify H M Yousof and G M Cordeiroldquo)e generalized transmuted-G family of distributionsrdquo

12 Mathematical Problems in Engineering

Communications in Statistics-5eory and Methods vol 46no 8 pp 4119ndash4136 2017

[12] M Chahkandi and M Ganjali ldquoOn some lifetime distribu-tions with decreasing failure raterdquo Computational Statisticsand Data Analysis vol 53 no 12 pp 4433ndash4440 2009

[13] W Barreto-Souza A H S Santos and G M Cordeiro ldquo)ebeta generalized exponential distributionrdquo Journal of Statis-tical Computation and Simulation vol 80 no 2 pp 159ndash1722010

[14] W Lu and D Shi ldquoA new compounding life distribution theWeibull-Poisson distributionrdquo Journal of Applied Statisticsvol 39 no 1 pp 21ndash38 2012

[15] D C Montgomery Statistical Quality Control Vol 7 WileyNew York NY USA 2009

[16] W H Woodall ldquo)e use of control charts in health-care andpublic-health surveillancerdquo Journal of Quality Technologyvol 38 no 2 pp 89ndash104 2006

[17] S-L Hwang J-T Lin G-F Liang Y-J Yau T-C Yenn andC-C Hsu ldquoApplication control chart concepts of designing apre-alarm system in the nuclear power plant control roomrdquoNuclear Engineering and Design vol 238 no 12 pp 3522ndash3527 2008

[18] Z Wang and R Liang ldquoDiscuss on applying SPC to qualitymanagement in university educationrdquo in Proceedings of the9th International Conference for Young Computer Scientistspp 2372ndash2375 Hunan China November 2008

[19] M Pastell and H Madsen ldquoApplication of CUSUM charts todetect lameness in a milking robotrdquo Expert Systems withApplications vol 35 no 4 pp 2032ndash2040 2008

[20] A Ivanova C Xiao and Y Tymofyeyev ldquoTwo-stage designsfor phase 2 dose-finding trialsrdquo Statistics in Medicine vol 31no 24 pp 2872ndash2881 2012

[21] S W Roberts ldquoControl chart tests based on geometricmoving averagesrdquo Technometrics vol 1 no 3 pp 239ndash2501959

[22] D Brook and D A Evans ldquoAn approach to the probabilitydistribution of CUSUM run lengthrdquo Biometrika vol 59 no 3pp 539ndash549 1972

[23] J M Lucas ldquoCombined Shewhart-CUSUM quality controlschemesrdquo Journal of Quality Technology vol 14 no 2pp 51ndash59 1982

[24] C M Borror J B Keats and D C Montgomery ldquoRobustnessof the time between events CUSUMrdquo International Journal ofProduction Research vol 41 no 15 pp 3435ndash3444 2003

[25] M B C Khoo ldquoA moving average control chart for moni-toring the fraction non-conformingrdquo Quality and ReliabilityEngineering International vol 20 no 6 pp 617ndash635 2004

[26] S Sukparungsee and A A Novikov ldquoAnalytical approxi-mations for detection of a change-point in case of light-taileddistributionsrdquo Journal of Quality Measurement and Analysisvol 4 no 2 pp 49ndash56 2008

[27] Y Areepong and A A Novikov ldquoMartingale approach toEWMA control chart for changes in Exponential distribu-tionrdquo Journal of Quality Measurement and Analysis vol 4pp 197ndash203 2008

[28] R Noorossana A A Fatahi P Dokouhaki andM Babakhani ldquoZIB-EWMA control chart for monitoringrare health eventsrdquo Journal of Mechanics in Medicine andBiology vol 11 no 04 pp 881ndash895 2011

[29] E Santiago and J Smith ldquoControl charts based on the ex-ponential distribution adapting runs rules for the tChartrdquoQuality Engineering vol 25 no 2 pp 85ndash96 2013

[30] L L Ho and R C Quinino ldquoAn attribute control chart formonitoring the variability of a processrdquo International Journalof Production Economics vol 145 no 1 pp 263ndash267 2013

[31] M Aslam M Azam and C-H Jun ldquoAcceptance samplingplans for multi-stage process based on time-truncated test forWeibull distributionrdquo 5e International Journal of AdvancedManufacturing Technology vol 79 no 9ndash12 pp 1779ndash17852015

[32] A C Rakitzis and P Castagliola ldquo)e effect of parameterestimation on the performance of one-sided Shewhart controlcharts for zero-inflated processesrdquo Communications in Sta-tistics - 5eory and Methods vol 45 no 14 pp 4194ndash42142016

[33] M Azam M N Aslam and C-H Jun ldquoAn EWMA controlchart for the exponential distribution using repetitive sam-pling planrdquo 2017

[34] M Azam L Ahmad M Aslam and C-H Jun ldquoAn attributecontrol chart using discriminant limits for monitoring processunder the Weibull distributionrdquo Production Engineeringvol 12 no 5 pp 659ndash665 2018

[35] Z Wu and J Jiao ldquoA control chart for monitoring processmean based on attribute inspectionrdquo International Journal ofProduction Research vol 46 no 15 pp 4331ndash4347 2008

[36] H Kim and S Lee ldquoOn the VSI CUSUM chart for countprocesses and its implementation with R package attrCU-SUMrdquo Industrial Engineering ampManagement Systems vol 17no 1 pp 91ndash101 2018

[37] K Mabude J C Malela-Majika M Aslam Z L Chong andS C Shongwe ldquoDistribution-free composite Shewhart-GWMA Mann-Whitney charts for monitoring the processlocationrdquo Quality and Reliability Engineering International2020

[38] M T Shah M Azam M Aslam and U Sherazi ldquoTimebetween events control charts for gamma distributionrdquoQuality and Reliability Engineering International vol 37no 2 pp 785ndash803 2020

[39] U Shaheen M Azam and M Aslam ldquoA control chart formonitoring the lognormal process variation using repetitivesamplingrdquo Quality and Reliability Engineering Internationalvol 36 no 3 pp 1028ndash1047 2020

[40] M Aslam P Jeyadurga S Balamurali M Azam and A AL-Marshadi ldquoEconomic determination of modified multipledependent state sampling plan under some lifetime distri-butionsrdquo Journal of Mathematics vol 2021 Article ID7470196 2021

[41] R H Berk N L Johnson S Kotz and N BalakrishnanldquoContinuous univariate distributionsrdquo Technometrics vol 2p 752 1996

[42] C J P Belisle ldquoConvergence theorems for a class of simulatedannealing algorithms on Rdrdquo Journal of Applied Probabilityvol 29 no 4 pp 885ndash895 1992

[43] A Henningsen and O Toomet ldquoMaxlik a package formaximum likelihood estimation in Rrdquo Computational Sta-tistics vol 26 pp 443ndash458 2011

Mathematical Problems in Engineering 13

P1 expαλUCL

c1113888 1113889

fc1

⎡⎣ ⎤⎦ + 1 minus expαλLCL

c1113888 1113889

fc1

⎡⎣ ⎤⎦

P1 1 + exp minusαλclowast

c

α λ1113888 1113889 ln

11 minus PU

1113888 1113889

1c1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

fc1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦minus exp minus

αλclowast

c

α λ1113888 1113889 ln

11 minus PL

1113888 1113889

1c1( )⎛⎝ ⎞⎠

fc1

⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦

P1 1 + exp minus ln1

1 minus PU

1113888 1113889

f

⎡⎣ ⎤⎦ minus exp minus ln1

1 minus PL

1113888 1113889

f

⎡⎣ ⎤⎦

(16)

)eARL is evaluated using P0 and P1 which are obtainedthrough the parameters of the proposed distribution Notehere that the equations ofP0 andP1 are independent of scaleparameters )erefore we fixed only the shape parameter

Now the ARL of in-control and ARL of out-of-controlhave been calculated as

ARL0 1

P0out

andARL1 1

P1out

(17)

5 ARL Behavior of the Newly ProposedControl Chart

In this section we discuss the ARL behavior of thesuggested control chart A simulation study is conductedin the R-language program using the above-estimatedequations of probabilities In simulation study 10000samples are generated from the well-known exponentialWeibull and the proposed WEx distributions )e MLestimates of the parameters of the suggested distributionare obtained by using SANN (simulated annealing) givenin Belisle [42] SANN is an optimization technique whichis more recent and flexible as compared to New-tonndashRaphson )e SANN method is available inR-package maxLik Henningsen and Toomet [43] Wefixed the shape parameters to find the ARLs

)ese ML estimates are used to find ARLs forARL0 200 300 and 370 Furthermore ARLvalues areexplained by calculating for different drift levels It is im-portant to mention that the shift unit is equal to the in-control ARL)e performance of ARL can be understood byreading the R-code program Simulated data of 10000samples are generated by choosing the random values of a bandn For in-control processes the ARL is calculated byusing these given random values For the selection of a and b

parameters see details in Ho and Quinino [30] Moreoverwe compare the exponential and Weibull distribution withthe proposed WEx distribution for the efficiency of ARL

6 Comparative Study

In this section we compare the performance of theproposed WEx distribution with the exponential distri-bution under ARL In Table 2 we report the estimatedvalues of the proposed model parameters Various values

of the parameters of exponential Weibull and theproposed WEx distributions are placed in Tables 3ndash5 forvarious combinations of a and b atARL0 200 300 and 370 From Table 3 it is seen thatvarious values of a and b at f 1 ie there is no shift inthe values of ARLs of the exponential Weibull and theproposed WEx distributions are the same ie 2019830333 and 37199 at ARL0 200 300 and 370 On theother hand it is observed that at a different shift levelf 1 to 025 the values of ARL decrease for the expo-nential and the proposed models

One can clearly observe in Tables 3ndash5 that the pro-posed model suppressed the exponential and Weibulldistribution in terms of shift detection From Table 3 forinstance the value of ARL is 1704 for the WEx distri-bution and 655 for the exponential distribution whenf 085 a 9 b 20 andARL0 200 FurthermoreTables 4 and 5 also represent the values of ARLs of theexponential and the proposed models at ARL0 200 300and 370 with PL 01 andPU 09 and PL 02 andPU

08 for various combinations of a and b at different levelsof shifts It can be noticed that the values of ARLs de-crease for both models when the values of a and b de-crease By comparing both models it is concluded thatthe proposed model gives quick detection of out-of-control process at different levels of shifts Tables 4 and 5are presented in Appendix

Figures 1ndash9 demonstrate the graphical representations ofthe performance of ARLs at 200 300 and 370 for severalcombinations of aampb and PLampPU One can see fromFigures 1ndash9 as shift level f decreases the curves of ARLs theexponential and the proposed WEx distributions also de-crease Also the ARL curves show that the behavior of theproposed model is better than that of the compared modelFigures 4ndash9 are also presented in Appendix

7 Real-Life Example

In this section the proposed control chart is imple-mented in real-life data set from the health sector )edata of urinary tract infection (UTI) are taken from thehospital system )ese data are attained from Azam et al[34] which was initially used by Santiago and Smith [29]Data are collected from a hospital with a high risk ofurinary tract infections particularly to identify risk rate

4 Mathematical Problems in Engineering

and the frequency of the UTI patients being dischargedIn this example male patients are focused )e data showthe number of days between admissions and discharge ofthe patients having UTIs )e simulated data for theabove scenario are generated from the proposed WExdistribution with PL 01 PU 09 a 29 and b 59Firstly we generate 40 observations for the in-controlprocess and then the next 40 observations for the shiftedprocess withf 090 as presented in Tables 6 and 7 Inaddition we estimate the values of LCL and UCL as003206 and 095051 respectively and display in Fig-ure 10 Figures 10 and 11 show the LCL and UCL for in-control and out-of-control data It is clearly shown that

40th value is detected as out-of-control for the proposedmodel

8 Comparisons of the Exponential and theWeibull-Exponential Distribution UsingReal-Life Example

In this section we compare the control charts of theexponential and the proposed WEx distributions usingreal data of UTI patients It can be easily seen in Figure 12that the control chart of the exponential distribution isin-control and no value is detected as out-of-control

Table 2 Maximum likelihood estimation of the proposed WEx distribution parameterMaximum likelihood estimationSANN maximization 10000 iterationsReturn code 0 successful convergenceLog-likelihood minus 4288574Parameter estimates Estimate Std error t value P valueα 274 006969 393169 2e minus 16C 1328 44492 029484 0653c 701 71518 09801 0613λ 488 113607 04293 0697

Table 3 Comparison of ARL (SDRL) using exponential and Weibull distributions and WEx distribution with pL 005 and pU 095

f

ARL0 200 ARL0 300 ARL0 370a 9 b 20 a 7 b 10 a 13 b 37

Exp Weibull WEx Exp Weibull WEx Exp Weibull WEx

1 20198(20148)

20198(20148)

20198(20148)

30333(30283)

30333(30283)

30333(30283)

37199(37149)

37199(37149)

37199(37149)

099 18971(18921)

22804(22754)

16750(16699)

28791(28741)

33552(33502)

25948(25898)

34146(34096)

43925(43875)

28813(28762)

095 14493(14443)

35189(35139) 8172 (8122) 22987

(22937)48020(47970)

14165(14115)

23666(23616)

79829(79779)

1094(10890)

09 9943(9893)

51359(51309)

3581(3530)

16732(16682)

(65478)(65428)

6945(6895)

14227(14177)

134895(134845) 3691 (3641)

085 655 (6499) 58654(58604)

1704(1654)

11723(11673)

72978(72928) 3581 (3530) 8149

(8099)162317(162266) 1437 (1386)

08 417 (4120) 50734(50684) 884 (832) 7935

(7885)64825(64775) 1944 (1894) 4500

(4449)132614(132564) 649 (597)

075 2588(2538)

33790(33740) 501 (448) 5214 (5163) 46442

(46392) 1114 (1063) 2432(2382)

75476(75426) 341 (287)

07 1583(1532)

18337(18287) 311 (256) 3347

(3297)27986(27936) 674 (622) 1311 (1260) 32598

(32547) 209 (151)

065 966 (915) 8714 (8664) 212 (154) 2115 (2065) 14960(14910) 433 (379) 721 (669) 11920

(11870) 148 (0838)

06 597 (545) 3870(3820) 158 (0953) 1328 (1277) 7435 (7384) 294 (239) 414 (360) 4083 (4033) 119 (0476)

055 380 (326) 1689 (1639) 128 (0599) 837 (785) 3552 (3502) 213 (155) 255 (199) 1422 (1371) 106 (0259)05 253 (197) 754 (702) 112 (0371) 535 (482) 1670 (1619) 164 (103) 172 (112) 540 (487) 102 (113)

025 101 (0098) 1001(00296) 1 (0005) 115 (0409) 104 (0211) 1 (0082) 1 (0012) 100 (00013) 1 (000004)

Mathematical Problems in Engineering 5

Table 4 Comparison of ARL (SDRL) using exponential and Weibull distributions and WEx distribution with pL 01 and pU 09

f

ARL0 200 ARL0 300 ARL0 370a 29 b 59 a 23 b 40 a 34 b 70

Exp Weibull WEx Exp Weibull WEx Exp Weibull WEx

1 20001(19951)

20001(19951)

20001(19951)

30499(30449)

30499(30449)

30499(30449)

37491(37441)

37491(37441)

37491(37441)

099 18127(18077)

24246(24196)

1501(14960)

27928(27878)

36215(36165)

23577(23527)

33449(33399)

46870(46820)

26878(26828)

095 11961(11911)

49423(49376)

5155(5105)

19214(19164)

68054(68003) 8912 (8862) 20662

(20612)107188(107138)

78115(7761)

09 6805(6755)

96705(96652)

1629(1578)

11504(11454)

122584(122534) 3013 (2962) 10763

(10713)234044(233994)

2078(2028)

085 373 (3679) 129441(129391) 635 (583) 6601

(6551)158035(157985) 1185 (1134) 5379

(5329)328659(328609) 712 (660)

08 2002(1951)

106293(106243) 306 (251) 3671

(3620)133121(133071) 545 (492) 2631

(2581)261275(261225) 315 (260)

075 1074(1023)

52202(52152) 182 (122) 2008

(1957)71411(71361) 294 (239) 1291

(1240)114225(114175) 179 (118)

07 590 (538) 16663(16613) 131 (0633) 1100

(1048)25897(25847) 186 (126) 655 (603) 30337

(30287) 127 (0583)

065 342 (288) 4137 (4087) 11 (0328) 616 (564) 7269 (7219) 136 (0702) 356 (302) 6061 (6011) 108 (0284)06 215 (157) 1009 (958) 102 (0155) 362 (308) 1888 (1838) 114 (0392) 214 (156) 1203 (1152) 101 (0122)

055 151 (0872) 307 (253) 1 (0061) 229 (172) 547 (495) 104 (0206) 146(08244) 316 (261) 1 (0041)

05 119 (0476) 142 (0770) 1 (0018) 159(09692) 210 (1520) 101 (0095) 116

(04275) 138 (07208) 1 (0010)

025 1 (00004) 100 (---) 1 (0000) 1 (00079) 100(0000001) 1 (000002) 1 (00001) 100 (000000) 1 (0000)

Table 5 Comparison of ARLs (SDRL) using exponential and Weibull distribution and WEx distribution with pL 02 and pU 08

f

ARL0 200 ARL0 300 ARL0 370a 63 b 57 a 54 b 45 a 42 b 31

Exp Weibull WEx Exp Weibull WEx Exp Weibull WEx

1 20021(19971)

20021(19971)

20021(11971)

30051(30001)

30051(30001)

30051(30001)

37082(37032)

37082(37032)

37082(37032)

099 18213(18163)

24105(24055)

15262(15212)

27470(27420)

35829(35778)

2322(23170)

34273(34223)

43255(43205)

29564(29514)

095 1223(12180)

48380(48330)

5517(5467)

18800(18750)

69067(69017)

8745(8695)

24541(24491)

76586(76535)

12417(12367)

09 7144(7094)

96152(96102)

1814(1763)

11225(11175)

131293(131243)

2935(2884)

15525(15475)

133346(133296)

4598(4548)

085 403 (3980) 135757(135707) 718 (666) 6444

(6394)180998(180948) 1145 (1094) 9433

(9383)175673(175623)

1896(1845)

08 2224(2174)

118192(118142) 344 (290) 3594

(3544)159117(159067) 524 (471) 5543

(5492)157292(157242) 876 (824)

075 1222(1171)

57896(57846) 200 (141) 1974

(1923)81731(81681) 282 (227) 3179

(3128)88602(88552) 456 (403)

07 682 (630) 16228(16177) 139 (0740) 1086

(1034)24616(24566) 179 (119) 1800

(1749)31123(31073) 269 (213)

065 396 (342) 3111 (3060) 114 (0393) 610 (558) 5003 (4952) 132 (0654) 1022 (970) 7498 (7448) 180 (1202)06 245 (189) 583 (531) 104 (0195) 359 (305) 922 (871) 112 (0358) 592 (539) 1537 (1486) 136 (0702)055 167 (106) 171 (1098) 101 (0083) 227 (170) 233 (175) 103 (0182) 357 (303) 368 (314) 115 (0409)05 127 (0589) 105 (02310) 1 (0028) 158 (0954) 115 (04118) 101 (0079) 229 (172) 144 (07909) 105 (0225)025 1 (00009) 100 (000000) 1 (00000) 1 (0006) 100 (000000) 1 (000001) 1 (0042) 100 (----) 1 (00004)

6 Mathematical Problems in Engineering

00 02 04 06 08 10

600

500

400

300

200

100

0

ARL

s

ARL of exponentialARL of weibullARL of weibull-exponential

Comparison of ARLs

f

Figure 1 ARLs of exponential Weibull and proposed WEx at 200with pL 005 and pU 095

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

800

600

400

200

0

ARL

s

Comparison of ARLs

Figure 2 ARLs of exponential Weibull and proposed WEx at 300with pL 005 and pU 095

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

1500

1000

500

0

ARL

s

Comparison of ARLs

Figure 3 ARLs of exponential Weibull and proposed WEx at 370with pL 005 and pU 095

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

1200

1000

800

600

400

200

0

ARL

s

Comparison of ARLs

Figure 4 ARLs of exponential Weibull and proposed WEx at 200with pL 01 and pU 09

Mathematical Problems in Engineering 7

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

1500

1000

500

0

ARL

s

Comparison of ARLs

Figure 5 ARLs of exponential Weibull and proposed WEx at 300with pL 01 and pU 09

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

3000

2500

2000

1500

1000

500

0

ARL

s

Comparison of ARLs

Figure 6 ARLs of exponential Weibull and proposed WEx at 370with PL 01 an dPU 09

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

1400

1200

1000

800

600

400

200

0

ARL

s

Comparison of ARLs

Figure 7 ARLs of exponential Weibull and proposed WEx at 200with pL 02 and pU 08

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

1500

1000

500

0

ARL

s

Comparison of ARLs

Figure 8 ARLs of exponential Weibull and proposed WEx at 300with pL 02 and pU 08

8 Mathematical Problems in Engineering

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

1500

1000

500

0

ARL

s

Comparison of ARLs

Figure 9 ARLs of exponential Weibull and proposed WEx at 370 with pL 02 and pU 08

Table 6 Simulated in-control data0220499 0467107 0165884 0478009 0186779 00584930115906 0088989 0450255 0287988 0867485 025340604247 0714495 0356458 026482 0503415 05297730278757 008675 0204802 0181542 0417945 01030820269525 0543777 0096818 004666 0070134 0177020166788 0632749 0482233 0208581 0354741 02424770592133 0837021 0513781 0086579

Table 7 Simulated shifted data0650604 0144811 0585181 01867 0613353 05015330092963 0733554 0192515 0154301 0913754 05739580462511 0190089 058542 0297584 0162154 00806710066036 0746633 0545823 0061 0198106 0921250057984 0692892 026246 0463523 0204097 03307340454024 0765241 051748 0676308 0341994 04709630289353 0391563 0502378 0011261

0 10 20 30 40Sample number

10

08

06

04

02

00

Sam

ple v

alue

s

UDL = 095051

LDL = 003206

Figure 10 Plot for simulated in-control data

Mathematical Problems in Engineering 9

0 10 20 30 40Sample number

10

08

06

04

02

00

Sam

ple v

alue

s

UDL = 095051

LDL = 003206

Figure 11 Plot for simulated shift data with f 09 pL 01 pU 09 andARL0 200

0 10 20 30 5040Sample number

15

10

05

00

Sam

ple v

alue

s

UDL = 13544

LDL = 006197

Figure 12 Plot of UTI using exponential distribution control chart at pL 01 andpU 09

0 10 20 30 5040Sample number

15

10

05

00

Sam

ple v

alue

s UDL = 098102

LDL = 009615

Figure 13 Plot of UTI using proposed (WEx) control chart at pL 01 andpU 09

0 10 20 30 5040Sample number

15

10

05

00

Val

ues o

f T =

UTI

UDL = 15174

LDL = 03246

Figure 14 Plot of UTI using Weibull distribution control chart at pL 01 andpU 09

10 Mathematical Problems in Engineering

whereas the control chart of the proposed distributionwhich can be seen in Figure 13 that detected that theprocess is out-of-control at the 40th value It may beconcluded that our proposed distribution using real dataset of UTI shows better performance to detect the out-of-control process quickly

9 Conclusion

In this paper we develop a newWeibull class of distributionsusing the distribution generator technique )e newlyproposed WEx distribution is applied in the attributecharacteristics control chart for monitoring themanufacturing process Furthermore we compare theproposed WEx distribution with exponential distribution tomonitor the quick detection of the item in the out-of-controlprocess We conclude that under various shift levels ourproposed model is more efficient than the exponentialdistribution A simulation study is also conducted to knownthe behavior of ARL using the proposed WEx Weibull andexponential distributions )e implementation of the pro-posed model in a real-life example from the health caredepartment and the comparison of the two models endorsethe scope of WEx distribution We hope that this proposedcontrol chart can also be used for other accepting samplingschemes and other lifetime models (Figure 14)

Notations

W WeibullWEx Weibull-exponentialR Range chartS2 chart Variance chartW-G Weibull-generalizedPL PU PercentilesUDL Upper discriminant limitLDL Lower discriminant limita Accepted itemb Rejected itemf Shift levelα c c λ Parameters of Weibull-exponential distributionARL Average run-lengthP0out Probability of out-of-control process

P1out Probability of out-of-control process of shifted

processP0 Probability of rejected item when process is in-

controlP1 Probability of rejected item when process is shiftedUTI Urinary tract infection

Appendix

A R-Code

Derivation of Equation (9) (Algorithms 1 and 2)

Pr Xlt LCL|c0( 1113857 PL

1 minus exp minusαλx

c1113888 1113889

c

1113888 1113889 PL

1 minus PL exp minusαλx

c1113888 1113889

c

1113888 1113889

(A1)

Taking ln on both sides

ln 1 minus PL( 1113857 ln exp minusαλx

c1113888 1113889

c

1113888 1113889

minus ln 1 minus PL( 1113857 αλx

c1113888 1113889

c

minus ln 1 minus PL( 11138571113858 1113859(1c)

αλx

c1113888 1113889

ln1

1 minus PL

1113888 11138891113890 1113891

(1c)

αλx

c1113888 1113889

c

α λln

11 minus PL

1113888 11138891113890 1113891

(1c)

x

(A2)

)e LCL is (cαλ)[ln(11 minus PL)](1c)

Input different values of shift (f )f1 log(1(1 minus pU)) f2 log(1(1minus pL))p1 1 + exp(minus f1lowast f ) minus exp(minus f2lowastf )u a+ b minus 1 l a minus 1beta sum(dbinom(lu n p1))ARL1 1beta SDRL sqrt((1minus beta)((beta)2))Output ARL for different values of shift

ALGORITHM 2 Computation of ARL for different shifts

Input sample sizes (ss) PL PU and r0P0 1 minus PU+PLa sample(1 70 1 T) b sample(1 70 1 T)u a+ b minus 1 l a minus 1 n a+ b minus 1Pout0 sum(dbinom(lu n p0))ARL0[i] 1Pout0ARL0 [i] 1Pout0 aa[i] a bb[i] b nn[i] nImpose condition ARL0 ge r0Output obtain corresponding values of a b n and ARL0

ALGORITHM 1 Computation of a b n and ARL0

Mathematical Problems in Engineering 11

Derivation of Equation (10)

Pr XltUCL|c0( 1113857 PU

1 minus exp minusαλx

c1113888 1113889

c

1113888 1113889 PU

1 minus PU exp minusαλx

c1113888 1113889

c

1113888 1113889

(A3)

Taking ln on both sides

ln 1 minus PU( 1113857 ln exp minusαλx

c1113888 1113889

c

1113888 1113889

minus ln 1 minus PU( 1113857 αλx

c1113888 1113889

c

minus ln 1 minus Pu( 11138571113858 1113859(1c)

αλx

c1113888 1113889

ln1

1 minus PU

1113888 11138891113890 1113891

(1c)

αλx

c1113888 1113889

c

αλln

11 minus PU

1113888 11138891113890 1113891

(1c)

x

(A4)

)e UCL is (cαλ)[ln(11 minus PL)](1c)Equation (12) DerivationFor the in-control process the probability of the out-of-

control process is demonstrated through control charts isgiven as

P0out 1 minus 1113944

a+bminus 1

xa

x minus 1

a minus 11113888 1113889P

ao 1 minus P0( 1113857

xminus a (A5)

)e authors compute the probability of out-of-controlprocess as P(Xge b)

Now

P(Xge b) 1 minus P(Xlt b) (A6)

Since ldquoXrdquo follows the negative binomial distributionthen

P(Xlt b) 1113944

a+bminus 1

xa

x minus 1

a minus 11113888 1113889p

a0 1 minus p0( 1113857

xminus a (A7)

Equation (14) Derivation)e probability that the process is found out-of-control

due to the shifted process using control chart is obtained as

P1out 1 minus 1113944

a+bminus 1

xa

x minus 1

a minus 11113888 1113889P

a1 1 minus P1( 1113857

xminus a (A8)

)e authors compute the probability of out-of-controlprocess as P(Xge b)

Now

P(Xge b) 1 minus P(Xlt b) (A9)

Since ldquoXrdquo follows the negative binomial distributionthen

P(Xlt b) 1113944a+bminus 1

xa

x minus 1

a minus 11113888 1113889p

a1 1 minus p1( 1113857

xminus a (A10)

Data Availability

)e data used to support the findings of this study are givenin the paper

Disclosure

No funds were obtained for this paper

Conflicts of Interest

)e authors declare that there are no conflicts of interest

References

[1] N Eugene C Lee and F Famoye ldquoBeta-normal distributionand its applicationsrdquo Communications in Statistics-5eoryand Methods vol 31 no 4 pp 497ndash512 2002

[2] S Nadarajah and S Kotz ldquo)e beta Gumbel distributionrdquoMathematical Problems in Engineering vol 2004 no 4pp 323ndash332 2004

[3] S Nadarajah and A K Gupta ldquo)e beta Frechet distributionrdquoFar East Journal of 5eoretical Statistics vol 14 no 1pp 15ndash24 2004

[4] S Nadarajah and S Kotz ldquo)e beta exponential distributionrdquoReliability Engineering amp System Safety vol 91 no 6pp 689ndash697 2006

[5] S Nadarajah G M Cordeiro and E M M Ortega ldquo)ezografos-balakrishnan-GFamily of distributions mathemati-cal properties and applicationsrdquo Communications in Statistics- 5eory and Methods vol 44 no 1 pp 186ndash215 2015

[6] A Alzaatreh C Lee and F Famoye ldquoA new method forgenerating families of continuous distributionsrdquo Metronvol 71 no 1 pp 63ndash79 2013

[7] G M Cordeiro E M M Ortega B V Popovic andR R Pescim ldquo)e Lomax generator of distributions prop-erties minification process and regression modelrdquo AppliedMathematics and Computation vol 247 pp 465ndash486 2014

[8] A Z Afify M Alizadeh H M Yousof G Aryal andM Ahmad ldquo)e transmuted geometric-g family OF distri-butions THEORY and applicationsrdquo Pakistan Journal ofStatistics vol 32 no 2 pp 139ndash160 2016

[9] M Alizadeh M H Tahir G M Cordeiro M MansoorM Zubair and G G Hamedani ldquo)e Kumaraswamy Mar-shal-Olkin family of distributionsrdquo Journal of the EgyptianMathematical Society vol 23 no 3 pp 546ndash557 2015

[10] A Z Afify G M Cordeiro H M Yousof A Alzaatreh andZ M Nofal ldquo)e Kumaraswamy transmuted-G family ofdistributions properties and applicationsrdquo Journal of DataScience vol 14 no 2 pp 245ndash270 2016

[11] Z M Nofal A Z Afify H M Yousof and G M Cordeiroldquo)e generalized transmuted-G family of distributionsrdquo

12 Mathematical Problems in Engineering

Communications in Statistics-5eory and Methods vol 46no 8 pp 4119ndash4136 2017

[12] M Chahkandi and M Ganjali ldquoOn some lifetime distribu-tions with decreasing failure raterdquo Computational Statisticsand Data Analysis vol 53 no 12 pp 4433ndash4440 2009

[13] W Barreto-Souza A H S Santos and G M Cordeiro ldquo)ebeta generalized exponential distributionrdquo Journal of Statis-tical Computation and Simulation vol 80 no 2 pp 159ndash1722010

[14] W Lu and D Shi ldquoA new compounding life distribution theWeibull-Poisson distributionrdquo Journal of Applied Statisticsvol 39 no 1 pp 21ndash38 2012

[15] D C Montgomery Statistical Quality Control Vol 7 WileyNew York NY USA 2009

[16] W H Woodall ldquo)e use of control charts in health-care andpublic-health surveillancerdquo Journal of Quality Technologyvol 38 no 2 pp 89ndash104 2006

[17] S-L Hwang J-T Lin G-F Liang Y-J Yau T-C Yenn andC-C Hsu ldquoApplication control chart concepts of designing apre-alarm system in the nuclear power plant control roomrdquoNuclear Engineering and Design vol 238 no 12 pp 3522ndash3527 2008

[18] Z Wang and R Liang ldquoDiscuss on applying SPC to qualitymanagement in university educationrdquo in Proceedings of the9th International Conference for Young Computer Scientistspp 2372ndash2375 Hunan China November 2008

[19] M Pastell and H Madsen ldquoApplication of CUSUM charts todetect lameness in a milking robotrdquo Expert Systems withApplications vol 35 no 4 pp 2032ndash2040 2008

[20] A Ivanova C Xiao and Y Tymofyeyev ldquoTwo-stage designsfor phase 2 dose-finding trialsrdquo Statistics in Medicine vol 31no 24 pp 2872ndash2881 2012

[21] S W Roberts ldquoControl chart tests based on geometricmoving averagesrdquo Technometrics vol 1 no 3 pp 239ndash2501959

[22] D Brook and D A Evans ldquoAn approach to the probabilitydistribution of CUSUM run lengthrdquo Biometrika vol 59 no 3pp 539ndash549 1972

[23] J M Lucas ldquoCombined Shewhart-CUSUM quality controlschemesrdquo Journal of Quality Technology vol 14 no 2pp 51ndash59 1982

[24] C M Borror J B Keats and D C Montgomery ldquoRobustnessof the time between events CUSUMrdquo International Journal ofProduction Research vol 41 no 15 pp 3435ndash3444 2003

[25] M B C Khoo ldquoA moving average control chart for moni-toring the fraction non-conformingrdquo Quality and ReliabilityEngineering International vol 20 no 6 pp 617ndash635 2004

[26] S Sukparungsee and A A Novikov ldquoAnalytical approxi-mations for detection of a change-point in case of light-taileddistributionsrdquo Journal of Quality Measurement and Analysisvol 4 no 2 pp 49ndash56 2008

[27] Y Areepong and A A Novikov ldquoMartingale approach toEWMA control chart for changes in Exponential distribu-tionrdquo Journal of Quality Measurement and Analysis vol 4pp 197ndash203 2008

[28] R Noorossana A A Fatahi P Dokouhaki andM Babakhani ldquoZIB-EWMA control chart for monitoringrare health eventsrdquo Journal of Mechanics in Medicine andBiology vol 11 no 04 pp 881ndash895 2011

[29] E Santiago and J Smith ldquoControl charts based on the ex-ponential distribution adapting runs rules for the tChartrdquoQuality Engineering vol 25 no 2 pp 85ndash96 2013

[30] L L Ho and R C Quinino ldquoAn attribute control chart formonitoring the variability of a processrdquo International Journalof Production Economics vol 145 no 1 pp 263ndash267 2013

[31] M Aslam M Azam and C-H Jun ldquoAcceptance samplingplans for multi-stage process based on time-truncated test forWeibull distributionrdquo 5e International Journal of AdvancedManufacturing Technology vol 79 no 9ndash12 pp 1779ndash17852015

[32] A C Rakitzis and P Castagliola ldquo)e effect of parameterestimation on the performance of one-sided Shewhart controlcharts for zero-inflated processesrdquo Communications in Sta-tistics - 5eory and Methods vol 45 no 14 pp 4194ndash42142016

[33] M Azam M N Aslam and C-H Jun ldquoAn EWMA controlchart for the exponential distribution using repetitive sam-pling planrdquo 2017

[34] M Azam L Ahmad M Aslam and C-H Jun ldquoAn attributecontrol chart using discriminant limits for monitoring processunder the Weibull distributionrdquo Production Engineeringvol 12 no 5 pp 659ndash665 2018

[35] Z Wu and J Jiao ldquoA control chart for monitoring processmean based on attribute inspectionrdquo International Journal ofProduction Research vol 46 no 15 pp 4331ndash4347 2008

[36] H Kim and S Lee ldquoOn the VSI CUSUM chart for countprocesses and its implementation with R package attrCU-SUMrdquo Industrial Engineering ampManagement Systems vol 17no 1 pp 91ndash101 2018

[37] K Mabude J C Malela-Majika M Aslam Z L Chong andS C Shongwe ldquoDistribution-free composite Shewhart-GWMA Mann-Whitney charts for monitoring the processlocationrdquo Quality and Reliability Engineering International2020

[38] M T Shah M Azam M Aslam and U Sherazi ldquoTimebetween events control charts for gamma distributionrdquoQuality and Reliability Engineering International vol 37no 2 pp 785ndash803 2020

[39] U Shaheen M Azam and M Aslam ldquoA control chart formonitoring the lognormal process variation using repetitivesamplingrdquo Quality and Reliability Engineering Internationalvol 36 no 3 pp 1028ndash1047 2020

[40] M Aslam P Jeyadurga S Balamurali M Azam and A AL-Marshadi ldquoEconomic determination of modified multipledependent state sampling plan under some lifetime distri-butionsrdquo Journal of Mathematics vol 2021 Article ID7470196 2021

[41] R H Berk N L Johnson S Kotz and N BalakrishnanldquoContinuous univariate distributionsrdquo Technometrics vol 2p 752 1996

[42] C J P Belisle ldquoConvergence theorems for a class of simulatedannealing algorithms on Rdrdquo Journal of Applied Probabilityvol 29 no 4 pp 885ndash895 1992

[43] A Henningsen and O Toomet ldquoMaxlik a package formaximum likelihood estimation in Rrdquo Computational Sta-tistics vol 26 pp 443ndash458 2011

Mathematical Problems in Engineering 13

and the frequency of the UTI patients being dischargedIn this example male patients are focused )e data showthe number of days between admissions and discharge ofthe patients having UTIs )e simulated data for theabove scenario are generated from the proposed WExdistribution with PL 01 PU 09 a 29 and b 59Firstly we generate 40 observations for the in-controlprocess and then the next 40 observations for the shiftedprocess withf 090 as presented in Tables 6 and 7 Inaddition we estimate the values of LCL and UCL as003206 and 095051 respectively and display in Fig-ure 10 Figures 10 and 11 show the LCL and UCL for in-control and out-of-control data It is clearly shown that

40th value is detected as out-of-control for the proposedmodel

8 Comparisons of the Exponential and theWeibull-Exponential Distribution UsingReal-Life Example

In this section we compare the control charts of theexponential and the proposed WEx distributions usingreal data of UTI patients It can be easily seen in Figure 12that the control chart of the exponential distribution isin-control and no value is detected as out-of-control

Table 2 Maximum likelihood estimation of the proposed WEx distribution parameterMaximum likelihood estimationSANN maximization 10000 iterationsReturn code 0 successful convergenceLog-likelihood minus 4288574Parameter estimates Estimate Std error t value P valueα 274 006969 393169 2e minus 16C 1328 44492 029484 0653c 701 71518 09801 0613λ 488 113607 04293 0697

Table 3 Comparison of ARL (SDRL) using exponential and Weibull distributions and WEx distribution with pL 005 and pU 095

f

ARL0 200 ARL0 300 ARL0 370a 9 b 20 a 7 b 10 a 13 b 37

Exp Weibull WEx Exp Weibull WEx Exp Weibull WEx

1 20198(20148)

20198(20148)

20198(20148)

30333(30283)

30333(30283)

30333(30283)

37199(37149)

37199(37149)

37199(37149)

099 18971(18921)

22804(22754)

16750(16699)

28791(28741)

33552(33502)

25948(25898)

34146(34096)

43925(43875)

28813(28762)

095 14493(14443)

35189(35139) 8172 (8122) 22987

(22937)48020(47970)

14165(14115)

23666(23616)

79829(79779)

1094(10890)

09 9943(9893)

51359(51309)

3581(3530)

16732(16682)

(65478)(65428)

6945(6895)

14227(14177)

134895(134845) 3691 (3641)

085 655 (6499) 58654(58604)

1704(1654)

11723(11673)

72978(72928) 3581 (3530) 8149

(8099)162317(162266) 1437 (1386)

08 417 (4120) 50734(50684) 884 (832) 7935

(7885)64825(64775) 1944 (1894) 4500

(4449)132614(132564) 649 (597)

075 2588(2538)

33790(33740) 501 (448) 5214 (5163) 46442

(46392) 1114 (1063) 2432(2382)

75476(75426) 341 (287)

07 1583(1532)

18337(18287) 311 (256) 3347

(3297)27986(27936) 674 (622) 1311 (1260) 32598

(32547) 209 (151)

065 966 (915) 8714 (8664) 212 (154) 2115 (2065) 14960(14910) 433 (379) 721 (669) 11920

(11870) 148 (0838)

06 597 (545) 3870(3820) 158 (0953) 1328 (1277) 7435 (7384) 294 (239) 414 (360) 4083 (4033) 119 (0476)

055 380 (326) 1689 (1639) 128 (0599) 837 (785) 3552 (3502) 213 (155) 255 (199) 1422 (1371) 106 (0259)05 253 (197) 754 (702) 112 (0371) 535 (482) 1670 (1619) 164 (103) 172 (112) 540 (487) 102 (113)

025 101 (0098) 1001(00296) 1 (0005) 115 (0409) 104 (0211) 1 (0082) 1 (0012) 100 (00013) 1 (000004)

Mathematical Problems in Engineering 5

Table 4 Comparison of ARL (SDRL) using exponential and Weibull distributions and WEx distribution with pL 01 and pU 09

f

ARL0 200 ARL0 300 ARL0 370a 29 b 59 a 23 b 40 a 34 b 70

Exp Weibull WEx Exp Weibull WEx Exp Weibull WEx

1 20001(19951)

20001(19951)

20001(19951)

30499(30449)

30499(30449)

30499(30449)

37491(37441)

37491(37441)

37491(37441)

099 18127(18077)

24246(24196)

1501(14960)

27928(27878)

36215(36165)

23577(23527)

33449(33399)

46870(46820)

26878(26828)

095 11961(11911)

49423(49376)

5155(5105)

19214(19164)

68054(68003) 8912 (8862) 20662

(20612)107188(107138)

78115(7761)

09 6805(6755)

96705(96652)

1629(1578)

11504(11454)

122584(122534) 3013 (2962) 10763

(10713)234044(233994)

2078(2028)

085 373 (3679) 129441(129391) 635 (583) 6601

(6551)158035(157985) 1185 (1134) 5379

(5329)328659(328609) 712 (660)

08 2002(1951)

106293(106243) 306 (251) 3671

(3620)133121(133071) 545 (492) 2631

(2581)261275(261225) 315 (260)

075 1074(1023)

52202(52152) 182 (122) 2008

(1957)71411(71361) 294 (239) 1291

(1240)114225(114175) 179 (118)

07 590 (538) 16663(16613) 131 (0633) 1100

(1048)25897(25847) 186 (126) 655 (603) 30337

(30287) 127 (0583)

065 342 (288) 4137 (4087) 11 (0328) 616 (564) 7269 (7219) 136 (0702) 356 (302) 6061 (6011) 108 (0284)06 215 (157) 1009 (958) 102 (0155) 362 (308) 1888 (1838) 114 (0392) 214 (156) 1203 (1152) 101 (0122)

055 151 (0872) 307 (253) 1 (0061) 229 (172) 547 (495) 104 (0206) 146(08244) 316 (261) 1 (0041)

05 119 (0476) 142 (0770) 1 (0018) 159(09692) 210 (1520) 101 (0095) 116

(04275) 138 (07208) 1 (0010)

025 1 (00004) 100 (---) 1 (0000) 1 (00079) 100(0000001) 1 (000002) 1 (00001) 100 (000000) 1 (0000)

Table 5 Comparison of ARLs (SDRL) using exponential and Weibull distribution and WEx distribution with pL 02 and pU 08

f

ARL0 200 ARL0 300 ARL0 370a 63 b 57 a 54 b 45 a 42 b 31

Exp Weibull WEx Exp Weibull WEx Exp Weibull WEx

1 20021(19971)

20021(19971)

20021(11971)

30051(30001)

30051(30001)

30051(30001)

37082(37032)

37082(37032)

37082(37032)

099 18213(18163)

24105(24055)

15262(15212)

27470(27420)

35829(35778)

2322(23170)

34273(34223)

43255(43205)

29564(29514)

095 1223(12180)

48380(48330)

5517(5467)

18800(18750)

69067(69017)

8745(8695)

24541(24491)

76586(76535)

12417(12367)

09 7144(7094)

96152(96102)

1814(1763)

11225(11175)

131293(131243)

2935(2884)

15525(15475)

133346(133296)

4598(4548)

085 403 (3980) 135757(135707) 718 (666) 6444

(6394)180998(180948) 1145 (1094) 9433

(9383)175673(175623)

1896(1845)

08 2224(2174)

118192(118142) 344 (290) 3594

(3544)159117(159067) 524 (471) 5543

(5492)157292(157242) 876 (824)

075 1222(1171)

57896(57846) 200 (141) 1974

(1923)81731(81681) 282 (227) 3179

(3128)88602(88552) 456 (403)

07 682 (630) 16228(16177) 139 (0740) 1086

(1034)24616(24566) 179 (119) 1800

(1749)31123(31073) 269 (213)

065 396 (342) 3111 (3060) 114 (0393) 610 (558) 5003 (4952) 132 (0654) 1022 (970) 7498 (7448) 180 (1202)06 245 (189) 583 (531) 104 (0195) 359 (305) 922 (871) 112 (0358) 592 (539) 1537 (1486) 136 (0702)055 167 (106) 171 (1098) 101 (0083) 227 (170) 233 (175) 103 (0182) 357 (303) 368 (314) 115 (0409)05 127 (0589) 105 (02310) 1 (0028) 158 (0954) 115 (04118) 101 (0079) 229 (172) 144 (07909) 105 (0225)025 1 (00009) 100 (000000) 1 (00000) 1 (0006) 100 (000000) 1 (000001) 1 (0042) 100 (----) 1 (00004)

6 Mathematical Problems in Engineering

00 02 04 06 08 10

600

500

400

300

200

100

0

ARL

s

ARL of exponentialARL of weibullARL of weibull-exponential

Comparison of ARLs

f

Figure 1 ARLs of exponential Weibull and proposed WEx at 200with pL 005 and pU 095

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

800

600

400

200

0

ARL

s

Comparison of ARLs

Figure 2 ARLs of exponential Weibull and proposed WEx at 300with pL 005 and pU 095

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

1500

1000

500

0

ARL

s

Comparison of ARLs

Figure 3 ARLs of exponential Weibull and proposed WEx at 370with pL 005 and pU 095

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

1200

1000

800

600

400

200

0

ARL

s

Comparison of ARLs

Figure 4 ARLs of exponential Weibull and proposed WEx at 200with pL 01 and pU 09

Mathematical Problems in Engineering 7

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

1500

1000

500

0

ARL

s

Comparison of ARLs

Figure 5 ARLs of exponential Weibull and proposed WEx at 300with pL 01 and pU 09

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

3000

2500

2000

1500

1000

500

0

ARL

s

Comparison of ARLs

Figure 6 ARLs of exponential Weibull and proposed WEx at 370with PL 01 an dPU 09

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

1400

1200

1000

800

600

400

200

0

ARL

s

Comparison of ARLs

Figure 7 ARLs of exponential Weibull and proposed WEx at 200with pL 02 and pU 08

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

1500

1000

500

0

ARL

s

Comparison of ARLs

Figure 8 ARLs of exponential Weibull and proposed WEx at 300with pL 02 and pU 08

8 Mathematical Problems in Engineering

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

1500

1000

500

0

ARL

s

Comparison of ARLs

Figure 9 ARLs of exponential Weibull and proposed WEx at 370 with pL 02 and pU 08

Table 6 Simulated in-control data0220499 0467107 0165884 0478009 0186779 00584930115906 0088989 0450255 0287988 0867485 025340604247 0714495 0356458 026482 0503415 05297730278757 008675 0204802 0181542 0417945 01030820269525 0543777 0096818 004666 0070134 0177020166788 0632749 0482233 0208581 0354741 02424770592133 0837021 0513781 0086579

Table 7 Simulated shifted data0650604 0144811 0585181 01867 0613353 05015330092963 0733554 0192515 0154301 0913754 05739580462511 0190089 058542 0297584 0162154 00806710066036 0746633 0545823 0061 0198106 0921250057984 0692892 026246 0463523 0204097 03307340454024 0765241 051748 0676308 0341994 04709630289353 0391563 0502378 0011261

0 10 20 30 40Sample number

10

08

06

04

02

00

Sam

ple v

alue

s

UDL = 095051

LDL = 003206

Figure 10 Plot for simulated in-control data

Mathematical Problems in Engineering 9

0 10 20 30 40Sample number

10

08

06

04

02

00

Sam

ple v

alue

s

UDL = 095051

LDL = 003206

Figure 11 Plot for simulated shift data with f 09 pL 01 pU 09 andARL0 200

0 10 20 30 5040Sample number

15

10

05

00

Sam

ple v

alue

s

UDL = 13544

LDL = 006197

Figure 12 Plot of UTI using exponential distribution control chart at pL 01 andpU 09

0 10 20 30 5040Sample number

15

10

05

00

Sam

ple v

alue

s UDL = 098102

LDL = 009615

Figure 13 Plot of UTI using proposed (WEx) control chart at pL 01 andpU 09

0 10 20 30 5040Sample number

15

10

05

00

Val

ues o

f T =

UTI

UDL = 15174

LDL = 03246

Figure 14 Plot of UTI using Weibull distribution control chart at pL 01 andpU 09

10 Mathematical Problems in Engineering

whereas the control chart of the proposed distributionwhich can be seen in Figure 13 that detected that theprocess is out-of-control at the 40th value It may beconcluded that our proposed distribution using real dataset of UTI shows better performance to detect the out-of-control process quickly

9 Conclusion

In this paper we develop a newWeibull class of distributionsusing the distribution generator technique )e newlyproposed WEx distribution is applied in the attributecharacteristics control chart for monitoring themanufacturing process Furthermore we compare theproposed WEx distribution with exponential distribution tomonitor the quick detection of the item in the out-of-controlprocess We conclude that under various shift levels ourproposed model is more efficient than the exponentialdistribution A simulation study is also conducted to knownthe behavior of ARL using the proposed WEx Weibull andexponential distributions )e implementation of the pro-posed model in a real-life example from the health caredepartment and the comparison of the two models endorsethe scope of WEx distribution We hope that this proposedcontrol chart can also be used for other accepting samplingschemes and other lifetime models (Figure 14)

Notations

W WeibullWEx Weibull-exponentialR Range chartS2 chart Variance chartW-G Weibull-generalizedPL PU PercentilesUDL Upper discriminant limitLDL Lower discriminant limita Accepted itemb Rejected itemf Shift levelα c c λ Parameters of Weibull-exponential distributionARL Average run-lengthP0out Probability of out-of-control process

P1out Probability of out-of-control process of shifted

processP0 Probability of rejected item when process is in-

controlP1 Probability of rejected item when process is shiftedUTI Urinary tract infection

Appendix

A R-Code

Derivation of Equation (9) (Algorithms 1 and 2)

Pr Xlt LCL|c0( 1113857 PL

1 minus exp minusαλx

c1113888 1113889

c

1113888 1113889 PL

1 minus PL exp minusαλx

c1113888 1113889

c

1113888 1113889

(A1)

Taking ln on both sides

ln 1 minus PL( 1113857 ln exp minusαλx

c1113888 1113889

c

1113888 1113889

minus ln 1 minus PL( 1113857 αλx

c1113888 1113889

c

minus ln 1 minus PL( 11138571113858 1113859(1c)

αλx

c1113888 1113889

ln1

1 minus PL

1113888 11138891113890 1113891

(1c)

αλx

c1113888 1113889

c

α λln

11 minus PL

1113888 11138891113890 1113891

(1c)

x

(A2)

)e LCL is (cαλ)[ln(11 minus PL)](1c)

Input different values of shift (f )f1 log(1(1 minus pU)) f2 log(1(1minus pL))p1 1 + exp(minus f1lowast f ) minus exp(minus f2lowastf )u a+ b minus 1 l a minus 1beta sum(dbinom(lu n p1))ARL1 1beta SDRL sqrt((1minus beta)((beta)2))Output ARL for different values of shift

ALGORITHM 2 Computation of ARL for different shifts

Input sample sizes (ss) PL PU and r0P0 1 minus PU+PLa sample(1 70 1 T) b sample(1 70 1 T)u a+ b minus 1 l a minus 1 n a+ b minus 1Pout0 sum(dbinom(lu n p0))ARL0[i] 1Pout0ARL0 [i] 1Pout0 aa[i] a bb[i] b nn[i] nImpose condition ARL0 ge r0Output obtain corresponding values of a b n and ARL0

ALGORITHM 1 Computation of a b n and ARL0

Mathematical Problems in Engineering 11

Derivation of Equation (10)

Pr XltUCL|c0( 1113857 PU

1 minus exp minusαλx

c1113888 1113889

c

1113888 1113889 PU

1 minus PU exp minusαλx

c1113888 1113889

c

1113888 1113889

(A3)

Taking ln on both sides

ln 1 minus PU( 1113857 ln exp minusαλx

c1113888 1113889

c

1113888 1113889

minus ln 1 minus PU( 1113857 αλx

c1113888 1113889

c

minus ln 1 minus Pu( 11138571113858 1113859(1c)

αλx

c1113888 1113889

ln1

1 minus PU

1113888 11138891113890 1113891

(1c)

αλx

c1113888 1113889

c

αλln

11 minus PU

1113888 11138891113890 1113891

(1c)

x

(A4)

)e UCL is (cαλ)[ln(11 minus PL)](1c)Equation (12) DerivationFor the in-control process the probability of the out-of-

control process is demonstrated through control charts isgiven as

P0out 1 minus 1113944

a+bminus 1

xa

x minus 1

a minus 11113888 1113889P

ao 1 minus P0( 1113857

xminus a (A5)

)e authors compute the probability of out-of-controlprocess as P(Xge b)

Now

P(Xge b) 1 minus P(Xlt b) (A6)

Since ldquoXrdquo follows the negative binomial distributionthen

P(Xlt b) 1113944

a+bminus 1

xa

x minus 1

a minus 11113888 1113889p

a0 1 minus p0( 1113857

xminus a (A7)

Equation (14) Derivation)e probability that the process is found out-of-control

due to the shifted process using control chart is obtained as

P1out 1 minus 1113944

a+bminus 1

xa

x minus 1

a minus 11113888 1113889P

a1 1 minus P1( 1113857

xminus a (A8)

)e authors compute the probability of out-of-controlprocess as P(Xge b)

Now

P(Xge b) 1 minus P(Xlt b) (A9)

Since ldquoXrdquo follows the negative binomial distributionthen

P(Xlt b) 1113944a+bminus 1

xa

x minus 1

a minus 11113888 1113889p

a1 1 minus p1( 1113857

xminus a (A10)

Data Availability

)e data used to support the findings of this study are givenin the paper

Disclosure

No funds were obtained for this paper

Conflicts of Interest

)e authors declare that there are no conflicts of interest

References

[1] N Eugene C Lee and F Famoye ldquoBeta-normal distributionand its applicationsrdquo Communications in Statistics-5eoryand Methods vol 31 no 4 pp 497ndash512 2002

[2] S Nadarajah and S Kotz ldquo)e beta Gumbel distributionrdquoMathematical Problems in Engineering vol 2004 no 4pp 323ndash332 2004

[3] S Nadarajah and A K Gupta ldquo)e beta Frechet distributionrdquoFar East Journal of 5eoretical Statistics vol 14 no 1pp 15ndash24 2004

[4] S Nadarajah and S Kotz ldquo)e beta exponential distributionrdquoReliability Engineering amp System Safety vol 91 no 6pp 689ndash697 2006

[5] S Nadarajah G M Cordeiro and E M M Ortega ldquo)ezografos-balakrishnan-GFamily of distributions mathemati-cal properties and applicationsrdquo Communications in Statistics- 5eory and Methods vol 44 no 1 pp 186ndash215 2015

[6] A Alzaatreh C Lee and F Famoye ldquoA new method forgenerating families of continuous distributionsrdquo Metronvol 71 no 1 pp 63ndash79 2013

[7] G M Cordeiro E M M Ortega B V Popovic andR R Pescim ldquo)e Lomax generator of distributions prop-erties minification process and regression modelrdquo AppliedMathematics and Computation vol 247 pp 465ndash486 2014

[8] A Z Afify M Alizadeh H M Yousof G Aryal andM Ahmad ldquo)e transmuted geometric-g family OF distri-butions THEORY and applicationsrdquo Pakistan Journal ofStatistics vol 32 no 2 pp 139ndash160 2016

[9] M Alizadeh M H Tahir G M Cordeiro M MansoorM Zubair and G G Hamedani ldquo)e Kumaraswamy Mar-shal-Olkin family of distributionsrdquo Journal of the EgyptianMathematical Society vol 23 no 3 pp 546ndash557 2015

[10] A Z Afify G M Cordeiro H M Yousof A Alzaatreh andZ M Nofal ldquo)e Kumaraswamy transmuted-G family ofdistributions properties and applicationsrdquo Journal of DataScience vol 14 no 2 pp 245ndash270 2016

[11] Z M Nofal A Z Afify H M Yousof and G M Cordeiroldquo)e generalized transmuted-G family of distributionsrdquo

12 Mathematical Problems in Engineering

Communications in Statistics-5eory and Methods vol 46no 8 pp 4119ndash4136 2017

[12] M Chahkandi and M Ganjali ldquoOn some lifetime distribu-tions with decreasing failure raterdquo Computational Statisticsand Data Analysis vol 53 no 12 pp 4433ndash4440 2009

[13] W Barreto-Souza A H S Santos and G M Cordeiro ldquo)ebeta generalized exponential distributionrdquo Journal of Statis-tical Computation and Simulation vol 80 no 2 pp 159ndash1722010

[14] W Lu and D Shi ldquoA new compounding life distribution theWeibull-Poisson distributionrdquo Journal of Applied Statisticsvol 39 no 1 pp 21ndash38 2012

[15] D C Montgomery Statistical Quality Control Vol 7 WileyNew York NY USA 2009

[16] W H Woodall ldquo)e use of control charts in health-care andpublic-health surveillancerdquo Journal of Quality Technologyvol 38 no 2 pp 89ndash104 2006

[17] S-L Hwang J-T Lin G-F Liang Y-J Yau T-C Yenn andC-C Hsu ldquoApplication control chart concepts of designing apre-alarm system in the nuclear power plant control roomrdquoNuclear Engineering and Design vol 238 no 12 pp 3522ndash3527 2008

[18] Z Wang and R Liang ldquoDiscuss on applying SPC to qualitymanagement in university educationrdquo in Proceedings of the9th International Conference for Young Computer Scientistspp 2372ndash2375 Hunan China November 2008

[19] M Pastell and H Madsen ldquoApplication of CUSUM charts todetect lameness in a milking robotrdquo Expert Systems withApplications vol 35 no 4 pp 2032ndash2040 2008

[20] A Ivanova C Xiao and Y Tymofyeyev ldquoTwo-stage designsfor phase 2 dose-finding trialsrdquo Statistics in Medicine vol 31no 24 pp 2872ndash2881 2012

[21] S W Roberts ldquoControl chart tests based on geometricmoving averagesrdquo Technometrics vol 1 no 3 pp 239ndash2501959

[22] D Brook and D A Evans ldquoAn approach to the probabilitydistribution of CUSUM run lengthrdquo Biometrika vol 59 no 3pp 539ndash549 1972

[23] J M Lucas ldquoCombined Shewhart-CUSUM quality controlschemesrdquo Journal of Quality Technology vol 14 no 2pp 51ndash59 1982

[24] C M Borror J B Keats and D C Montgomery ldquoRobustnessof the time between events CUSUMrdquo International Journal ofProduction Research vol 41 no 15 pp 3435ndash3444 2003

[25] M B C Khoo ldquoA moving average control chart for moni-toring the fraction non-conformingrdquo Quality and ReliabilityEngineering International vol 20 no 6 pp 617ndash635 2004

[26] S Sukparungsee and A A Novikov ldquoAnalytical approxi-mations for detection of a change-point in case of light-taileddistributionsrdquo Journal of Quality Measurement and Analysisvol 4 no 2 pp 49ndash56 2008

[27] Y Areepong and A A Novikov ldquoMartingale approach toEWMA control chart for changes in Exponential distribu-tionrdquo Journal of Quality Measurement and Analysis vol 4pp 197ndash203 2008

[28] R Noorossana A A Fatahi P Dokouhaki andM Babakhani ldquoZIB-EWMA control chart for monitoringrare health eventsrdquo Journal of Mechanics in Medicine andBiology vol 11 no 04 pp 881ndash895 2011

[29] E Santiago and J Smith ldquoControl charts based on the ex-ponential distribution adapting runs rules for the tChartrdquoQuality Engineering vol 25 no 2 pp 85ndash96 2013

[30] L L Ho and R C Quinino ldquoAn attribute control chart formonitoring the variability of a processrdquo International Journalof Production Economics vol 145 no 1 pp 263ndash267 2013

[31] M Aslam M Azam and C-H Jun ldquoAcceptance samplingplans for multi-stage process based on time-truncated test forWeibull distributionrdquo 5e International Journal of AdvancedManufacturing Technology vol 79 no 9ndash12 pp 1779ndash17852015

[32] A C Rakitzis and P Castagliola ldquo)e effect of parameterestimation on the performance of one-sided Shewhart controlcharts for zero-inflated processesrdquo Communications in Sta-tistics - 5eory and Methods vol 45 no 14 pp 4194ndash42142016

[33] M Azam M N Aslam and C-H Jun ldquoAn EWMA controlchart for the exponential distribution using repetitive sam-pling planrdquo 2017

[34] M Azam L Ahmad M Aslam and C-H Jun ldquoAn attributecontrol chart using discriminant limits for monitoring processunder the Weibull distributionrdquo Production Engineeringvol 12 no 5 pp 659ndash665 2018

[35] Z Wu and J Jiao ldquoA control chart for monitoring processmean based on attribute inspectionrdquo International Journal ofProduction Research vol 46 no 15 pp 4331ndash4347 2008

[36] H Kim and S Lee ldquoOn the VSI CUSUM chart for countprocesses and its implementation with R package attrCU-SUMrdquo Industrial Engineering ampManagement Systems vol 17no 1 pp 91ndash101 2018

[37] K Mabude J C Malela-Majika M Aslam Z L Chong andS C Shongwe ldquoDistribution-free composite Shewhart-GWMA Mann-Whitney charts for monitoring the processlocationrdquo Quality and Reliability Engineering International2020

[38] M T Shah M Azam M Aslam and U Sherazi ldquoTimebetween events control charts for gamma distributionrdquoQuality and Reliability Engineering International vol 37no 2 pp 785ndash803 2020

[39] U Shaheen M Azam and M Aslam ldquoA control chart formonitoring the lognormal process variation using repetitivesamplingrdquo Quality and Reliability Engineering Internationalvol 36 no 3 pp 1028ndash1047 2020

[40] M Aslam P Jeyadurga S Balamurali M Azam and A AL-Marshadi ldquoEconomic determination of modified multipledependent state sampling plan under some lifetime distri-butionsrdquo Journal of Mathematics vol 2021 Article ID7470196 2021

[41] R H Berk N L Johnson S Kotz and N BalakrishnanldquoContinuous univariate distributionsrdquo Technometrics vol 2p 752 1996

[42] C J P Belisle ldquoConvergence theorems for a class of simulatedannealing algorithms on Rdrdquo Journal of Applied Probabilityvol 29 no 4 pp 885ndash895 1992

[43] A Henningsen and O Toomet ldquoMaxlik a package formaximum likelihood estimation in Rrdquo Computational Sta-tistics vol 26 pp 443ndash458 2011

Mathematical Problems in Engineering 13

Table 4 Comparison of ARL (SDRL) using exponential and Weibull distributions and WEx distribution with pL 01 and pU 09

f

ARL0 200 ARL0 300 ARL0 370a 29 b 59 a 23 b 40 a 34 b 70

Exp Weibull WEx Exp Weibull WEx Exp Weibull WEx

1 20001(19951)

20001(19951)

20001(19951)

30499(30449)

30499(30449)

30499(30449)

37491(37441)

37491(37441)

37491(37441)

099 18127(18077)

24246(24196)

1501(14960)

27928(27878)

36215(36165)

23577(23527)

33449(33399)

46870(46820)

26878(26828)

095 11961(11911)

49423(49376)

5155(5105)

19214(19164)

68054(68003) 8912 (8862) 20662

(20612)107188(107138)

78115(7761)

09 6805(6755)

96705(96652)

1629(1578)

11504(11454)

122584(122534) 3013 (2962) 10763

(10713)234044(233994)

2078(2028)

085 373 (3679) 129441(129391) 635 (583) 6601

(6551)158035(157985) 1185 (1134) 5379

(5329)328659(328609) 712 (660)

08 2002(1951)

106293(106243) 306 (251) 3671

(3620)133121(133071) 545 (492) 2631

(2581)261275(261225) 315 (260)

075 1074(1023)

52202(52152) 182 (122) 2008

(1957)71411(71361) 294 (239) 1291

(1240)114225(114175) 179 (118)

07 590 (538) 16663(16613) 131 (0633) 1100

(1048)25897(25847) 186 (126) 655 (603) 30337

(30287) 127 (0583)

065 342 (288) 4137 (4087) 11 (0328) 616 (564) 7269 (7219) 136 (0702) 356 (302) 6061 (6011) 108 (0284)06 215 (157) 1009 (958) 102 (0155) 362 (308) 1888 (1838) 114 (0392) 214 (156) 1203 (1152) 101 (0122)

055 151 (0872) 307 (253) 1 (0061) 229 (172) 547 (495) 104 (0206) 146(08244) 316 (261) 1 (0041)

05 119 (0476) 142 (0770) 1 (0018) 159(09692) 210 (1520) 101 (0095) 116

(04275) 138 (07208) 1 (0010)

025 1 (00004) 100 (---) 1 (0000) 1 (00079) 100(0000001) 1 (000002) 1 (00001) 100 (000000) 1 (0000)

Table 5 Comparison of ARLs (SDRL) using exponential and Weibull distribution and WEx distribution with pL 02 and pU 08

f

ARL0 200 ARL0 300 ARL0 370a 63 b 57 a 54 b 45 a 42 b 31

Exp Weibull WEx Exp Weibull WEx Exp Weibull WEx

1 20021(19971)

20021(19971)

20021(11971)

30051(30001)

30051(30001)

30051(30001)

37082(37032)

37082(37032)

37082(37032)

099 18213(18163)

24105(24055)

15262(15212)

27470(27420)

35829(35778)

2322(23170)

34273(34223)

43255(43205)

29564(29514)

095 1223(12180)

48380(48330)

5517(5467)

18800(18750)

69067(69017)

8745(8695)

24541(24491)

76586(76535)

12417(12367)

09 7144(7094)

96152(96102)

1814(1763)

11225(11175)

131293(131243)

2935(2884)

15525(15475)

133346(133296)

4598(4548)

085 403 (3980) 135757(135707) 718 (666) 6444

(6394)180998(180948) 1145 (1094) 9433

(9383)175673(175623)

1896(1845)

08 2224(2174)

118192(118142) 344 (290) 3594

(3544)159117(159067) 524 (471) 5543

(5492)157292(157242) 876 (824)

075 1222(1171)

57896(57846) 200 (141) 1974

(1923)81731(81681) 282 (227) 3179

(3128)88602(88552) 456 (403)

07 682 (630) 16228(16177) 139 (0740) 1086

(1034)24616(24566) 179 (119) 1800

(1749)31123(31073) 269 (213)

065 396 (342) 3111 (3060) 114 (0393) 610 (558) 5003 (4952) 132 (0654) 1022 (970) 7498 (7448) 180 (1202)06 245 (189) 583 (531) 104 (0195) 359 (305) 922 (871) 112 (0358) 592 (539) 1537 (1486) 136 (0702)055 167 (106) 171 (1098) 101 (0083) 227 (170) 233 (175) 103 (0182) 357 (303) 368 (314) 115 (0409)05 127 (0589) 105 (02310) 1 (0028) 158 (0954) 115 (04118) 101 (0079) 229 (172) 144 (07909) 105 (0225)025 1 (00009) 100 (000000) 1 (00000) 1 (0006) 100 (000000) 1 (000001) 1 (0042) 100 (----) 1 (00004)

6 Mathematical Problems in Engineering

00 02 04 06 08 10

600

500

400

300

200

100

0

ARL

s

ARL of exponentialARL of weibullARL of weibull-exponential

Comparison of ARLs

f

Figure 1 ARLs of exponential Weibull and proposed WEx at 200with pL 005 and pU 095

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

800

600

400

200

0

ARL

s

Comparison of ARLs

Figure 2 ARLs of exponential Weibull and proposed WEx at 300with pL 005 and pU 095

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

1500

1000

500

0

ARL

s

Comparison of ARLs

Figure 3 ARLs of exponential Weibull and proposed WEx at 370with pL 005 and pU 095

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

1200

1000

800

600

400

200

0

ARL

s

Comparison of ARLs

Figure 4 ARLs of exponential Weibull and proposed WEx at 200with pL 01 and pU 09

Mathematical Problems in Engineering 7

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

1500

1000

500

0

ARL

s

Comparison of ARLs

Figure 5 ARLs of exponential Weibull and proposed WEx at 300with pL 01 and pU 09

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

3000

2500

2000

1500

1000

500

0

ARL

s

Comparison of ARLs

Figure 6 ARLs of exponential Weibull and proposed WEx at 370with PL 01 an dPU 09

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

1400

1200

1000

800

600

400

200

0

ARL

s

Comparison of ARLs

Figure 7 ARLs of exponential Weibull and proposed WEx at 200with pL 02 and pU 08

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

1500

1000

500

0

ARL

s

Comparison of ARLs

Figure 8 ARLs of exponential Weibull and proposed WEx at 300with pL 02 and pU 08

8 Mathematical Problems in Engineering

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

1500

1000

500

0

ARL

s

Comparison of ARLs

Figure 9 ARLs of exponential Weibull and proposed WEx at 370 with pL 02 and pU 08

Table 6 Simulated in-control data0220499 0467107 0165884 0478009 0186779 00584930115906 0088989 0450255 0287988 0867485 025340604247 0714495 0356458 026482 0503415 05297730278757 008675 0204802 0181542 0417945 01030820269525 0543777 0096818 004666 0070134 0177020166788 0632749 0482233 0208581 0354741 02424770592133 0837021 0513781 0086579

Table 7 Simulated shifted data0650604 0144811 0585181 01867 0613353 05015330092963 0733554 0192515 0154301 0913754 05739580462511 0190089 058542 0297584 0162154 00806710066036 0746633 0545823 0061 0198106 0921250057984 0692892 026246 0463523 0204097 03307340454024 0765241 051748 0676308 0341994 04709630289353 0391563 0502378 0011261

0 10 20 30 40Sample number

10

08

06

04

02

00

Sam

ple v

alue

s

UDL = 095051

LDL = 003206

Figure 10 Plot for simulated in-control data

Mathematical Problems in Engineering 9

0 10 20 30 40Sample number

10

08

06

04

02

00

Sam

ple v

alue

s

UDL = 095051

LDL = 003206

Figure 11 Plot for simulated shift data with f 09 pL 01 pU 09 andARL0 200

0 10 20 30 5040Sample number

15

10

05

00

Sam

ple v

alue

s

UDL = 13544

LDL = 006197

Figure 12 Plot of UTI using exponential distribution control chart at pL 01 andpU 09

0 10 20 30 5040Sample number

15

10

05

00

Sam

ple v

alue

s UDL = 098102

LDL = 009615

Figure 13 Plot of UTI using proposed (WEx) control chart at pL 01 andpU 09

0 10 20 30 5040Sample number

15

10

05

00

Val

ues o

f T =

UTI

UDL = 15174

LDL = 03246

Figure 14 Plot of UTI using Weibull distribution control chart at pL 01 andpU 09

10 Mathematical Problems in Engineering

whereas the control chart of the proposed distributionwhich can be seen in Figure 13 that detected that theprocess is out-of-control at the 40th value It may beconcluded that our proposed distribution using real dataset of UTI shows better performance to detect the out-of-control process quickly

9 Conclusion

In this paper we develop a newWeibull class of distributionsusing the distribution generator technique )e newlyproposed WEx distribution is applied in the attributecharacteristics control chart for monitoring themanufacturing process Furthermore we compare theproposed WEx distribution with exponential distribution tomonitor the quick detection of the item in the out-of-controlprocess We conclude that under various shift levels ourproposed model is more efficient than the exponentialdistribution A simulation study is also conducted to knownthe behavior of ARL using the proposed WEx Weibull andexponential distributions )e implementation of the pro-posed model in a real-life example from the health caredepartment and the comparison of the two models endorsethe scope of WEx distribution We hope that this proposedcontrol chart can also be used for other accepting samplingschemes and other lifetime models (Figure 14)

Notations

W WeibullWEx Weibull-exponentialR Range chartS2 chart Variance chartW-G Weibull-generalizedPL PU PercentilesUDL Upper discriminant limitLDL Lower discriminant limita Accepted itemb Rejected itemf Shift levelα c c λ Parameters of Weibull-exponential distributionARL Average run-lengthP0out Probability of out-of-control process

P1out Probability of out-of-control process of shifted

processP0 Probability of rejected item when process is in-

controlP1 Probability of rejected item when process is shiftedUTI Urinary tract infection

Appendix

A R-Code

Derivation of Equation (9) (Algorithms 1 and 2)

Pr Xlt LCL|c0( 1113857 PL

1 minus exp minusαλx

c1113888 1113889

c

1113888 1113889 PL

1 minus PL exp minusαλx

c1113888 1113889

c

1113888 1113889

(A1)

Taking ln on both sides

ln 1 minus PL( 1113857 ln exp minusαλx

c1113888 1113889

c

1113888 1113889

minus ln 1 minus PL( 1113857 αλx

c1113888 1113889

c

minus ln 1 minus PL( 11138571113858 1113859(1c)

αλx

c1113888 1113889

ln1

1 minus PL

1113888 11138891113890 1113891

(1c)

αλx

c1113888 1113889

c

α λln

11 minus PL

1113888 11138891113890 1113891

(1c)

x

(A2)

)e LCL is (cαλ)[ln(11 minus PL)](1c)

Input different values of shift (f )f1 log(1(1 minus pU)) f2 log(1(1minus pL))p1 1 + exp(minus f1lowast f ) minus exp(minus f2lowastf )u a+ b minus 1 l a minus 1beta sum(dbinom(lu n p1))ARL1 1beta SDRL sqrt((1minus beta)((beta)2))Output ARL for different values of shift

ALGORITHM 2 Computation of ARL for different shifts

Input sample sizes (ss) PL PU and r0P0 1 minus PU+PLa sample(1 70 1 T) b sample(1 70 1 T)u a+ b minus 1 l a minus 1 n a+ b minus 1Pout0 sum(dbinom(lu n p0))ARL0[i] 1Pout0ARL0 [i] 1Pout0 aa[i] a bb[i] b nn[i] nImpose condition ARL0 ge r0Output obtain corresponding values of a b n and ARL0

ALGORITHM 1 Computation of a b n and ARL0

Mathematical Problems in Engineering 11

Derivation of Equation (10)

Pr XltUCL|c0( 1113857 PU

1 minus exp minusαλx

c1113888 1113889

c

1113888 1113889 PU

1 minus PU exp minusαλx

c1113888 1113889

c

1113888 1113889

(A3)

Taking ln on both sides

ln 1 minus PU( 1113857 ln exp minusαλx

c1113888 1113889

c

1113888 1113889

minus ln 1 minus PU( 1113857 αλx

c1113888 1113889

c

minus ln 1 minus Pu( 11138571113858 1113859(1c)

αλx

c1113888 1113889

ln1

1 minus PU

1113888 11138891113890 1113891

(1c)

αλx

c1113888 1113889

c

αλln

11 minus PU

1113888 11138891113890 1113891

(1c)

x

(A4)

)e UCL is (cαλ)[ln(11 minus PL)](1c)Equation (12) DerivationFor the in-control process the probability of the out-of-

control process is demonstrated through control charts isgiven as

P0out 1 minus 1113944

a+bminus 1

xa

x minus 1

a minus 11113888 1113889P

ao 1 minus P0( 1113857

xminus a (A5)

)e authors compute the probability of out-of-controlprocess as P(Xge b)

Now

P(Xge b) 1 minus P(Xlt b) (A6)

Since ldquoXrdquo follows the negative binomial distributionthen

P(Xlt b) 1113944

a+bminus 1

xa

x minus 1

a minus 11113888 1113889p

a0 1 minus p0( 1113857

xminus a (A7)

Equation (14) Derivation)e probability that the process is found out-of-control

due to the shifted process using control chart is obtained as

P1out 1 minus 1113944

a+bminus 1

xa

x minus 1

a minus 11113888 1113889P

a1 1 minus P1( 1113857

xminus a (A8)

)e authors compute the probability of out-of-controlprocess as P(Xge b)

Now

P(Xge b) 1 minus P(Xlt b) (A9)

Since ldquoXrdquo follows the negative binomial distributionthen

P(Xlt b) 1113944a+bminus 1

xa

x minus 1

a minus 11113888 1113889p

a1 1 minus p1( 1113857

xminus a (A10)

Data Availability

)e data used to support the findings of this study are givenin the paper

Disclosure

No funds were obtained for this paper

Conflicts of Interest

)e authors declare that there are no conflicts of interest

References

[1] N Eugene C Lee and F Famoye ldquoBeta-normal distributionand its applicationsrdquo Communications in Statistics-5eoryand Methods vol 31 no 4 pp 497ndash512 2002

[2] S Nadarajah and S Kotz ldquo)e beta Gumbel distributionrdquoMathematical Problems in Engineering vol 2004 no 4pp 323ndash332 2004

[3] S Nadarajah and A K Gupta ldquo)e beta Frechet distributionrdquoFar East Journal of 5eoretical Statistics vol 14 no 1pp 15ndash24 2004

[4] S Nadarajah and S Kotz ldquo)e beta exponential distributionrdquoReliability Engineering amp System Safety vol 91 no 6pp 689ndash697 2006

[5] S Nadarajah G M Cordeiro and E M M Ortega ldquo)ezografos-balakrishnan-GFamily of distributions mathemati-cal properties and applicationsrdquo Communications in Statistics- 5eory and Methods vol 44 no 1 pp 186ndash215 2015

[6] A Alzaatreh C Lee and F Famoye ldquoA new method forgenerating families of continuous distributionsrdquo Metronvol 71 no 1 pp 63ndash79 2013

[7] G M Cordeiro E M M Ortega B V Popovic andR R Pescim ldquo)e Lomax generator of distributions prop-erties minification process and regression modelrdquo AppliedMathematics and Computation vol 247 pp 465ndash486 2014

[8] A Z Afify M Alizadeh H M Yousof G Aryal andM Ahmad ldquo)e transmuted geometric-g family OF distri-butions THEORY and applicationsrdquo Pakistan Journal ofStatistics vol 32 no 2 pp 139ndash160 2016

[9] M Alizadeh M H Tahir G M Cordeiro M MansoorM Zubair and G G Hamedani ldquo)e Kumaraswamy Mar-shal-Olkin family of distributionsrdquo Journal of the EgyptianMathematical Society vol 23 no 3 pp 546ndash557 2015

[10] A Z Afify G M Cordeiro H M Yousof A Alzaatreh andZ M Nofal ldquo)e Kumaraswamy transmuted-G family ofdistributions properties and applicationsrdquo Journal of DataScience vol 14 no 2 pp 245ndash270 2016

[11] Z M Nofal A Z Afify H M Yousof and G M Cordeiroldquo)e generalized transmuted-G family of distributionsrdquo

12 Mathematical Problems in Engineering

Communications in Statistics-5eory and Methods vol 46no 8 pp 4119ndash4136 2017

[12] M Chahkandi and M Ganjali ldquoOn some lifetime distribu-tions with decreasing failure raterdquo Computational Statisticsand Data Analysis vol 53 no 12 pp 4433ndash4440 2009

[13] W Barreto-Souza A H S Santos and G M Cordeiro ldquo)ebeta generalized exponential distributionrdquo Journal of Statis-tical Computation and Simulation vol 80 no 2 pp 159ndash1722010

[14] W Lu and D Shi ldquoA new compounding life distribution theWeibull-Poisson distributionrdquo Journal of Applied Statisticsvol 39 no 1 pp 21ndash38 2012

[15] D C Montgomery Statistical Quality Control Vol 7 WileyNew York NY USA 2009

[16] W H Woodall ldquo)e use of control charts in health-care andpublic-health surveillancerdquo Journal of Quality Technologyvol 38 no 2 pp 89ndash104 2006

[17] S-L Hwang J-T Lin G-F Liang Y-J Yau T-C Yenn andC-C Hsu ldquoApplication control chart concepts of designing apre-alarm system in the nuclear power plant control roomrdquoNuclear Engineering and Design vol 238 no 12 pp 3522ndash3527 2008

[18] Z Wang and R Liang ldquoDiscuss on applying SPC to qualitymanagement in university educationrdquo in Proceedings of the9th International Conference for Young Computer Scientistspp 2372ndash2375 Hunan China November 2008

[19] M Pastell and H Madsen ldquoApplication of CUSUM charts todetect lameness in a milking robotrdquo Expert Systems withApplications vol 35 no 4 pp 2032ndash2040 2008

[20] A Ivanova C Xiao and Y Tymofyeyev ldquoTwo-stage designsfor phase 2 dose-finding trialsrdquo Statistics in Medicine vol 31no 24 pp 2872ndash2881 2012

[21] S W Roberts ldquoControl chart tests based on geometricmoving averagesrdquo Technometrics vol 1 no 3 pp 239ndash2501959

[22] D Brook and D A Evans ldquoAn approach to the probabilitydistribution of CUSUM run lengthrdquo Biometrika vol 59 no 3pp 539ndash549 1972

[23] J M Lucas ldquoCombined Shewhart-CUSUM quality controlschemesrdquo Journal of Quality Technology vol 14 no 2pp 51ndash59 1982

[24] C M Borror J B Keats and D C Montgomery ldquoRobustnessof the time between events CUSUMrdquo International Journal ofProduction Research vol 41 no 15 pp 3435ndash3444 2003

[25] M B C Khoo ldquoA moving average control chart for moni-toring the fraction non-conformingrdquo Quality and ReliabilityEngineering International vol 20 no 6 pp 617ndash635 2004

[26] S Sukparungsee and A A Novikov ldquoAnalytical approxi-mations for detection of a change-point in case of light-taileddistributionsrdquo Journal of Quality Measurement and Analysisvol 4 no 2 pp 49ndash56 2008

[27] Y Areepong and A A Novikov ldquoMartingale approach toEWMA control chart for changes in Exponential distribu-tionrdquo Journal of Quality Measurement and Analysis vol 4pp 197ndash203 2008

[28] R Noorossana A A Fatahi P Dokouhaki andM Babakhani ldquoZIB-EWMA control chart for monitoringrare health eventsrdquo Journal of Mechanics in Medicine andBiology vol 11 no 04 pp 881ndash895 2011

[29] E Santiago and J Smith ldquoControl charts based on the ex-ponential distribution adapting runs rules for the tChartrdquoQuality Engineering vol 25 no 2 pp 85ndash96 2013

[30] L L Ho and R C Quinino ldquoAn attribute control chart formonitoring the variability of a processrdquo International Journalof Production Economics vol 145 no 1 pp 263ndash267 2013

[31] M Aslam M Azam and C-H Jun ldquoAcceptance samplingplans for multi-stage process based on time-truncated test forWeibull distributionrdquo 5e International Journal of AdvancedManufacturing Technology vol 79 no 9ndash12 pp 1779ndash17852015

[32] A C Rakitzis and P Castagliola ldquo)e effect of parameterestimation on the performance of one-sided Shewhart controlcharts for zero-inflated processesrdquo Communications in Sta-tistics - 5eory and Methods vol 45 no 14 pp 4194ndash42142016

[33] M Azam M N Aslam and C-H Jun ldquoAn EWMA controlchart for the exponential distribution using repetitive sam-pling planrdquo 2017

[34] M Azam L Ahmad M Aslam and C-H Jun ldquoAn attributecontrol chart using discriminant limits for monitoring processunder the Weibull distributionrdquo Production Engineeringvol 12 no 5 pp 659ndash665 2018

[35] Z Wu and J Jiao ldquoA control chart for monitoring processmean based on attribute inspectionrdquo International Journal ofProduction Research vol 46 no 15 pp 4331ndash4347 2008

[36] H Kim and S Lee ldquoOn the VSI CUSUM chart for countprocesses and its implementation with R package attrCU-SUMrdquo Industrial Engineering ampManagement Systems vol 17no 1 pp 91ndash101 2018

[37] K Mabude J C Malela-Majika M Aslam Z L Chong andS C Shongwe ldquoDistribution-free composite Shewhart-GWMA Mann-Whitney charts for monitoring the processlocationrdquo Quality and Reliability Engineering International2020

[38] M T Shah M Azam M Aslam and U Sherazi ldquoTimebetween events control charts for gamma distributionrdquoQuality and Reliability Engineering International vol 37no 2 pp 785ndash803 2020

[39] U Shaheen M Azam and M Aslam ldquoA control chart formonitoring the lognormal process variation using repetitivesamplingrdquo Quality and Reliability Engineering Internationalvol 36 no 3 pp 1028ndash1047 2020

[40] M Aslam P Jeyadurga S Balamurali M Azam and A AL-Marshadi ldquoEconomic determination of modified multipledependent state sampling plan under some lifetime distri-butionsrdquo Journal of Mathematics vol 2021 Article ID7470196 2021

[41] R H Berk N L Johnson S Kotz and N BalakrishnanldquoContinuous univariate distributionsrdquo Technometrics vol 2p 752 1996

[42] C J P Belisle ldquoConvergence theorems for a class of simulatedannealing algorithms on Rdrdquo Journal of Applied Probabilityvol 29 no 4 pp 885ndash895 1992

[43] A Henningsen and O Toomet ldquoMaxlik a package formaximum likelihood estimation in Rrdquo Computational Sta-tistics vol 26 pp 443ndash458 2011

Mathematical Problems in Engineering 13

00 02 04 06 08 10

600

500

400

300

200

100

0

ARL

s

ARL of exponentialARL of weibullARL of weibull-exponential

Comparison of ARLs

f

Figure 1 ARLs of exponential Weibull and proposed WEx at 200with pL 005 and pU 095

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

800

600

400

200

0

ARL

s

Comparison of ARLs

Figure 2 ARLs of exponential Weibull and proposed WEx at 300with pL 005 and pU 095

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

1500

1000

500

0

ARL

s

Comparison of ARLs

Figure 3 ARLs of exponential Weibull and proposed WEx at 370with pL 005 and pU 095

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

1200

1000

800

600

400

200

0

ARL

s

Comparison of ARLs

Figure 4 ARLs of exponential Weibull and proposed WEx at 200with pL 01 and pU 09

Mathematical Problems in Engineering 7

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

1500

1000

500

0

ARL

s

Comparison of ARLs

Figure 5 ARLs of exponential Weibull and proposed WEx at 300with pL 01 and pU 09

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

3000

2500

2000

1500

1000

500

0

ARL

s

Comparison of ARLs

Figure 6 ARLs of exponential Weibull and proposed WEx at 370with PL 01 an dPU 09

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

1400

1200

1000

800

600

400

200

0

ARL

s

Comparison of ARLs

Figure 7 ARLs of exponential Weibull and proposed WEx at 200with pL 02 and pU 08

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

1500

1000

500

0

ARL

s

Comparison of ARLs

Figure 8 ARLs of exponential Weibull and proposed WEx at 300with pL 02 and pU 08

8 Mathematical Problems in Engineering

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

1500

1000

500

0

ARL

s

Comparison of ARLs

Figure 9 ARLs of exponential Weibull and proposed WEx at 370 with pL 02 and pU 08

Table 6 Simulated in-control data0220499 0467107 0165884 0478009 0186779 00584930115906 0088989 0450255 0287988 0867485 025340604247 0714495 0356458 026482 0503415 05297730278757 008675 0204802 0181542 0417945 01030820269525 0543777 0096818 004666 0070134 0177020166788 0632749 0482233 0208581 0354741 02424770592133 0837021 0513781 0086579

Table 7 Simulated shifted data0650604 0144811 0585181 01867 0613353 05015330092963 0733554 0192515 0154301 0913754 05739580462511 0190089 058542 0297584 0162154 00806710066036 0746633 0545823 0061 0198106 0921250057984 0692892 026246 0463523 0204097 03307340454024 0765241 051748 0676308 0341994 04709630289353 0391563 0502378 0011261

0 10 20 30 40Sample number

10

08

06

04

02

00

Sam

ple v

alue

s

UDL = 095051

LDL = 003206

Figure 10 Plot for simulated in-control data

Mathematical Problems in Engineering 9

0 10 20 30 40Sample number

10

08

06

04

02

00

Sam

ple v

alue

s

UDL = 095051

LDL = 003206

Figure 11 Plot for simulated shift data with f 09 pL 01 pU 09 andARL0 200

0 10 20 30 5040Sample number

15

10

05

00

Sam

ple v

alue

s

UDL = 13544

LDL = 006197

Figure 12 Plot of UTI using exponential distribution control chart at pL 01 andpU 09

0 10 20 30 5040Sample number

15

10

05

00

Sam

ple v

alue

s UDL = 098102

LDL = 009615

Figure 13 Plot of UTI using proposed (WEx) control chart at pL 01 andpU 09

0 10 20 30 5040Sample number

15

10

05

00

Val

ues o

f T =

UTI

UDL = 15174

LDL = 03246

Figure 14 Plot of UTI using Weibull distribution control chart at pL 01 andpU 09

10 Mathematical Problems in Engineering

whereas the control chart of the proposed distributionwhich can be seen in Figure 13 that detected that theprocess is out-of-control at the 40th value It may beconcluded that our proposed distribution using real dataset of UTI shows better performance to detect the out-of-control process quickly

9 Conclusion

In this paper we develop a newWeibull class of distributionsusing the distribution generator technique )e newlyproposed WEx distribution is applied in the attributecharacteristics control chart for monitoring themanufacturing process Furthermore we compare theproposed WEx distribution with exponential distribution tomonitor the quick detection of the item in the out-of-controlprocess We conclude that under various shift levels ourproposed model is more efficient than the exponentialdistribution A simulation study is also conducted to knownthe behavior of ARL using the proposed WEx Weibull andexponential distributions )e implementation of the pro-posed model in a real-life example from the health caredepartment and the comparison of the two models endorsethe scope of WEx distribution We hope that this proposedcontrol chart can also be used for other accepting samplingschemes and other lifetime models (Figure 14)

Notations

W WeibullWEx Weibull-exponentialR Range chartS2 chart Variance chartW-G Weibull-generalizedPL PU PercentilesUDL Upper discriminant limitLDL Lower discriminant limita Accepted itemb Rejected itemf Shift levelα c c λ Parameters of Weibull-exponential distributionARL Average run-lengthP0out Probability of out-of-control process

P1out Probability of out-of-control process of shifted

processP0 Probability of rejected item when process is in-

controlP1 Probability of rejected item when process is shiftedUTI Urinary tract infection

Appendix

A R-Code

Derivation of Equation (9) (Algorithms 1 and 2)

Pr Xlt LCL|c0( 1113857 PL

1 minus exp minusαλx

c1113888 1113889

c

1113888 1113889 PL

1 minus PL exp minusαλx

c1113888 1113889

c

1113888 1113889

(A1)

Taking ln on both sides

ln 1 minus PL( 1113857 ln exp minusαλx

c1113888 1113889

c

1113888 1113889

minus ln 1 minus PL( 1113857 αλx

c1113888 1113889

c

minus ln 1 minus PL( 11138571113858 1113859(1c)

αλx

c1113888 1113889

ln1

1 minus PL

1113888 11138891113890 1113891

(1c)

αλx

c1113888 1113889

c

α λln

11 minus PL

1113888 11138891113890 1113891

(1c)

x

(A2)

)e LCL is (cαλ)[ln(11 minus PL)](1c)

Input different values of shift (f )f1 log(1(1 minus pU)) f2 log(1(1minus pL))p1 1 + exp(minus f1lowast f ) minus exp(minus f2lowastf )u a+ b minus 1 l a minus 1beta sum(dbinom(lu n p1))ARL1 1beta SDRL sqrt((1minus beta)((beta)2))Output ARL for different values of shift

ALGORITHM 2 Computation of ARL for different shifts

Input sample sizes (ss) PL PU and r0P0 1 minus PU+PLa sample(1 70 1 T) b sample(1 70 1 T)u a+ b minus 1 l a minus 1 n a+ b minus 1Pout0 sum(dbinom(lu n p0))ARL0[i] 1Pout0ARL0 [i] 1Pout0 aa[i] a bb[i] b nn[i] nImpose condition ARL0 ge r0Output obtain corresponding values of a b n and ARL0

ALGORITHM 1 Computation of a b n and ARL0

Mathematical Problems in Engineering 11

Derivation of Equation (10)

Pr XltUCL|c0( 1113857 PU

1 minus exp minusαλx

c1113888 1113889

c

1113888 1113889 PU

1 minus PU exp minusαλx

c1113888 1113889

c

1113888 1113889

(A3)

Taking ln on both sides

ln 1 minus PU( 1113857 ln exp minusαλx

c1113888 1113889

c

1113888 1113889

minus ln 1 minus PU( 1113857 αλx

c1113888 1113889

c

minus ln 1 minus Pu( 11138571113858 1113859(1c)

αλx

c1113888 1113889

ln1

1 minus PU

1113888 11138891113890 1113891

(1c)

αλx

c1113888 1113889

c

αλln

11 minus PU

1113888 11138891113890 1113891

(1c)

x

(A4)

)e UCL is (cαλ)[ln(11 minus PL)](1c)Equation (12) DerivationFor the in-control process the probability of the out-of-

control process is demonstrated through control charts isgiven as

P0out 1 minus 1113944

a+bminus 1

xa

x minus 1

a minus 11113888 1113889P

ao 1 minus P0( 1113857

xminus a (A5)

)e authors compute the probability of out-of-controlprocess as P(Xge b)

Now

P(Xge b) 1 minus P(Xlt b) (A6)

Since ldquoXrdquo follows the negative binomial distributionthen

P(Xlt b) 1113944

a+bminus 1

xa

x minus 1

a minus 11113888 1113889p

a0 1 minus p0( 1113857

xminus a (A7)

Equation (14) Derivation)e probability that the process is found out-of-control

due to the shifted process using control chart is obtained as

P1out 1 minus 1113944

a+bminus 1

xa

x minus 1

a minus 11113888 1113889P

a1 1 minus P1( 1113857

xminus a (A8)

)e authors compute the probability of out-of-controlprocess as P(Xge b)

Now

P(Xge b) 1 minus P(Xlt b) (A9)

Since ldquoXrdquo follows the negative binomial distributionthen

P(Xlt b) 1113944a+bminus 1

xa

x minus 1

a minus 11113888 1113889p

a1 1 minus p1( 1113857

xminus a (A10)

Data Availability

)e data used to support the findings of this study are givenin the paper

Disclosure

No funds were obtained for this paper

Conflicts of Interest

)e authors declare that there are no conflicts of interest

References

[1] N Eugene C Lee and F Famoye ldquoBeta-normal distributionand its applicationsrdquo Communications in Statistics-5eoryand Methods vol 31 no 4 pp 497ndash512 2002

[2] S Nadarajah and S Kotz ldquo)e beta Gumbel distributionrdquoMathematical Problems in Engineering vol 2004 no 4pp 323ndash332 2004

[3] S Nadarajah and A K Gupta ldquo)e beta Frechet distributionrdquoFar East Journal of 5eoretical Statistics vol 14 no 1pp 15ndash24 2004

[4] S Nadarajah and S Kotz ldquo)e beta exponential distributionrdquoReliability Engineering amp System Safety vol 91 no 6pp 689ndash697 2006

[5] S Nadarajah G M Cordeiro and E M M Ortega ldquo)ezografos-balakrishnan-GFamily of distributions mathemati-cal properties and applicationsrdquo Communications in Statistics- 5eory and Methods vol 44 no 1 pp 186ndash215 2015

[6] A Alzaatreh C Lee and F Famoye ldquoA new method forgenerating families of continuous distributionsrdquo Metronvol 71 no 1 pp 63ndash79 2013

[7] G M Cordeiro E M M Ortega B V Popovic andR R Pescim ldquo)e Lomax generator of distributions prop-erties minification process and regression modelrdquo AppliedMathematics and Computation vol 247 pp 465ndash486 2014

[8] A Z Afify M Alizadeh H M Yousof G Aryal andM Ahmad ldquo)e transmuted geometric-g family OF distri-butions THEORY and applicationsrdquo Pakistan Journal ofStatistics vol 32 no 2 pp 139ndash160 2016

[9] M Alizadeh M H Tahir G M Cordeiro M MansoorM Zubair and G G Hamedani ldquo)e Kumaraswamy Mar-shal-Olkin family of distributionsrdquo Journal of the EgyptianMathematical Society vol 23 no 3 pp 546ndash557 2015

[10] A Z Afify G M Cordeiro H M Yousof A Alzaatreh andZ M Nofal ldquo)e Kumaraswamy transmuted-G family ofdistributions properties and applicationsrdquo Journal of DataScience vol 14 no 2 pp 245ndash270 2016

[11] Z M Nofal A Z Afify H M Yousof and G M Cordeiroldquo)e generalized transmuted-G family of distributionsrdquo

12 Mathematical Problems in Engineering

Communications in Statistics-5eory and Methods vol 46no 8 pp 4119ndash4136 2017

[12] M Chahkandi and M Ganjali ldquoOn some lifetime distribu-tions with decreasing failure raterdquo Computational Statisticsand Data Analysis vol 53 no 12 pp 4433ndash4440 2009

[13] W Barreto-Souza A H S Santos and G M Cordeiro ldquo)ebeta generalized exponential distributionrdquo Journal of Statis-tical Computation and Simulation vol 80 no 2 pp 159ndash1722010

[14] W Lu and D Shi ldquoA new compounding life distribution theWeibull-Poisson distributionrdquo Journal of Applied Statisticsvol 39 no 1 pp 21ndash38 2012

[15] D C Montgomery Statistical Quality Control Vol 7 WileyNew York NY USA 2009

[16] W H Woodall ldquo)e use of control charts in health-care andpublic-health surveillancerdquo Journal of Quality Technologyvol 38 no 2 pp 89ndash104 2006

[17] S-L Hwang J-T Lin G-F Liang Y-J Yau T-C Yenn andC-C Hsu ldquoApplication control chart concepts of designing apre-alarm system in the nuclear power plant control roomrdquoNuclear Engineering and Design vol 238 no 12 pp 3522ndash3527 2008

[18] Z Wang and R Liang ldquoDiscuss on applying SPC to qualitymanagement in university educationrdquo in Proceedings of the9th International Conference for Young Computer Scientistspp 2372ndash2375 Hunan China November 2008

[19] M Pastell and H Madsen ldquoApplication of CUSUM charts todetect lameness in a milking robotrdquo Expert Systems withApplications vol 35 no 4 pp 2032ndash2040 2008

[20] A Ivanova C Xiao and Y Tymofyeyev ldquoTwo-stage designsfor phase 2 dose-finding trialsrdquo Statistics in Medicine vol 31no 24 pp 2872ndash2881 2012

[21] S W Roberts ldquoControl chart tests based on geometricmoving averagesrdquo Technometrics vol 1 no 3 pp 239ndash2501959

[22] D Brook and D A Evans ldquoAn approach to the probabilitydistribution of CUSUM run lengthrdquo Biometrika vol 59 no 3pp 539ndash549 1972

[23] J M Lucas ldquoCombined Shewhart-CUSUM quality controlschemesrdquo Journal of Quality Technology vol 14 no 2pp 51ndash59 1982

[24] C M Borror J B Keats and D C Montgomery ldquoRobustnessof the time between events CUSUMrdquo International Journal ofProduction Research vol 41 no 15 pp 3435ndash3444 2003

[25] M B C Khoo ldquoA moving average control chart for moni-toring the fraction non-conformingrdquo Quality and ReliabilityEngineering International vol 20 no 6 pp 617ndash635 2004

[26] S Sukparungsee and A A Novikov ldquoAnalytical approxi-mations for detection of a change-point in case of light-taileddistributionsrdquo Journal of Quality Measurement and Analysisvol 4 no 2 pp 49ndash56 2008

[27] Y Areepong and A A Novikov ldquoMartingale approach toEWMA control chart for changes in Exponential distribu-tionrdquo Journal of Quality Measurement and Analysis vol 4pp 197ndash203 2008

[28] R Noorossana A A Fatahi P Dokouhaki andM Babakhani ldquoZIB-EWMA control chart for monitoringrare health eventsrdquo Journal of Mechanics in Medicine andBiology vol 11 no 04 pp 881ndash895 2011

[29] E Santiago and J Smith ldquoControl charts based on the ex-ponential distribution adapting runs rules for the tChartrdquoQuality Engineering vol 25 no 2 pp 85ndash96 2013

[30] L L Ho and R C Quinino ldquoAn attribute control chart formonitoring the variability of a processrdquo International Journalof Production Economics vol 145 no 1 pp 263ndash267 2013

[31] M Aslam M Azam and C-H Jun ldquoAcceptance samplingplans for multi-stage process based on time-truncated test forWeibull distributionrdquo 5e International Journal of AdvancedManufacturing Technology vol 79 no 9ndash12 pp 1779ndash17852015

[32] A C Rakitzis and P Castagliola ldquo)e effect of parameterestimation on the performance of one-sided Shewhart controlcharts for zero-inflated processesrdquo Communications in Sta-tistics - 5eory and Methods vol 45 no 14 pp 4194ndash42142016

[33] M Azam M N Aslam and C-H Jun ldquoAn EWMA controlchart for the exponential distribution using repetitive sam-pling planrdquo 2017

[34] M Azam L Ahmad M Aslam and C-H Jun ldquoAn attributecontrol chart using discriminant limits for monitoring processunder the Weibull distributionrdquo Production Engineeringvol 12 no 5 pp 659ndash665 2018

[35] Z Wu and J Jiao ldquoA control chart for monitoring processmean based on attribute inspectionrdquo International Journal ofProduction Research vol 46 no 15 pp 4331ndash4347 2008

[36] H Kim and S Lee ldquoOn the VSI CUSUM chart for countprocesses and its implementation with R package attrCU-SUMrdquo Industrial Engineering ampManagement Systems vol 17no 1 pp 91ndash101 2018

[37] K Mabude J C Malela-Majika M Aslam Z L Chong andS C Shongwe ldquoDistribution-free composite Shewhart-GWMA Mann-Whitney charts for monitoring the processlocationrdquo Quality and Reliability Engineering International2020

[38] M T Shah M Azam M Aslam and U Sherazi ldquoTimebetween events control charts for gamma distributionrdquoQuality and Reliability Engineering International vol 37no 2 pp 785ndash803 2020

[39] U Shaheen M Azam and M Aslam ldquoA control chart formonitoring the lognormal process variation using repetitivesamplingrdquo Quality and Reliability Engineering Internationalvol 36 no 3 pp 1028ndash1047 2020

[40] M Aslam P Jeyadurga S Balamurali M Azam and A AL-Marshadi ldquoEconomic determination of modified multipledependent state sampling plan under some lifetime distri-butionsrdquo Journal of Mathematics vol 2021 Article ID7470196 2021

[41] R H Berk N L Johnson S Kotz and N BalakrishnanldquoContinuous univariate distributionsrdquo Technometrics vol 2p 752 1996

[42] C J P Belisle ldquoConvergence theorems for a class of simulatedannealing algorithms on Rdrdquo Journal of Applied Probabilityvol 29 no 4 pp 885ndash895 1992

[43] A Henningsen and O Toomet ldquoMaxlik a package formaximum likelihood estimation in Rrdquo Computational Sta-tistics vol 26 pp 443ndash458 2011

Mathematical Problems in Engineering 13

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

1500

1000

500

0

ARL

s

Comparison of ARLs

Figure 5 ARLs of exponential Weibull and proposed WEx at 300with pL 01 and pU 09

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

3000

2500

2000

1500

1000

500

0

ARL

s

Comparison of ARLs

Figure 6 ARLs of exponential Weibull and proposed WEx at 370with PL 01 an dPU 09

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

1400

1200

1000

800

600

400

200

0

ARL

s

Comparison of ARLs

Figure 7 ARLs of exponential Weibull and proposed WEx at 200with pL 02 and pU 08

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

1500

1000

500

0

ARL

s

Comparison of ARLs

Figure 8 ARLs of exponential Weibull and proposed WEx at 300with pL 02 and pU 08

8 Mathematical Problems in Engineering

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

1500

1000

500

0

ARL

s

Comparison of ARLs

Figure 9 ARLs of exponential Weibull and proposed WEx at 370 with pL 02 and pU 08

Table 6 Simulated in-control data0220499 0467107 0165884 0478009 0186779 00584930115906 0088989 0450255 0287988 0867485 025340604247 0714495 0356458 026482 0503415 05297730278757 008675 0204802 0181542 0417945 01030820269525 0543777 0096818 004666 0070134 0177020166788 0632749 0482233 0208581 0354741 02424770592133 0837021 0513781 0086579

Table 7 Simulated shifted data0650604 0144811 0585181 01867 0613353 05015330092963 0733554 0192515 0154301 0913754 05739580462511 0190089 058542 0297584 0162154 00806710066036 0746633 0545823 0061 0198106 0921250057984 0692892 026246 0463523 0204097 03307340454024 0765241 051748 0676308 0341994 04709630289353 0391563 0502378 0011261

0 10 20 30 40Sample number

10

08

06

04

02

00

Sam

ple v

alue

s

UDL = 095051

LDL = 003206

Figure 10 Plot for simulated in-control data

Mathematical Problems in Engineering 9

0 10 20 30 40Sample number

10

08

06

04

02

00

Sam

ple v

alue

s

UDL = 095051

LDL = 003206

Figure 11 Plot for simulated shift data with f 09 pL 01 pU 09 andARL0 200

0 10 20 30 5040Sample number

15

10

05

00

Sam

ple v

alue

s

UDL = 13544

LDL = 006197

Figure 12 Plot of UTI using exponential distribution control chart at pL 01 andpU 09

0 10 20 30 5040Sample number

15

10

05

00

Sam

ple v

alue

s UDL = 098102

LDL = 009615

Figure 13 Plot of UTI using proposed (WEx) control chart at pL 01 andpU 09

0 10 20 30 5040Sample number

15

10

05

00

Val

ues o

f T =

UTI

UDL = 15174

LDL = 03246

Figure 14 Plot of UTI using Weibull distribution control chart at pL 01 andpU 09

10 Mathematical Problems in Engineering

whereas the control chart of the proposed distributionwhich can be seen in Figure 13 that detected that theprocess is out-of-control at the 40th value It may beconcluded that our proposed distribution using real dataset of UTI shows better performance to detect the out-of-control process quickly

9 Conclusion

In this paper we develop a newWeibull class of distributionsusing the distribution generator technique )e newlyproposed WEx distribution is applied in the attributecharacteristics control chart for monitoring themanufacturing process Furthermore we compare theproposed WEx distribution with exponential distribution tomonitor the quick detection of the item in the out-of-controlprocess We conclude that under various shift levels ourproposed model is more efficient than the exponentialdistribution A simulation study is also conducted to knownthe behavior of ARL using the proposed WEx Weibull andexponential distributions )e implementation of the pro-posed model in a real-life example from the health caredepartment and the comparison of the two models endorsethe scope of WEx distribution We hope that this proposedcontrol chart can also be used for other accepting samplingschemes and other lifetime models (Figure 14)

Notations

W WeibullWEx Weibull-exponentialR Range chartS2 chart Variance chartW-G Weibull-generalizedPL PU PercentilesUDL Upper discriminant limitLDL Lower discriminant limita Accepted itemb Rejected itemf Shift levelα c c λ Parameters of Weibull-exponential distributionARL Average run-lengthP0out Probability of out-of-control process

P1out Probability of out-of-control process of shifted

processP0 Probability of rejected item when process is in-

controlP1 Probability of rejected item when process is shiftedUTI Urinary tract infection

Appendix

A R-Code

Derivation of Equation (9) (Algorithms 1 and 2)

Pr Xlt LCL|c0( 1113857 PL

1 minus exp minusαλx

c1113888 1113889

c

1113888 1113889 PL

1 minus PL exp minusαλx

c1113888 1113889

c

1113888 1113889

(A1)

Taking ln on both sides

ln 1 minus PL( 1113857 ln exp minusαλx

c1113888 1113889

c

1113888 1113889

minus ln 1 minus PL( 1113857 αλx

c1113888 1113889

c

minus ln 1 minus PL( 11138571113858 1113859(1c)

αλx

c1113888 1113889

ln1

1 minus PL

1113888 11138891113890 1113891

(1c)

αλx

c1113888 1113889

c

α λln

11 minus PL

1113888 11138891113890 1113891

(1c)

x

(A2)

)e LCL is (cαλ)[ln(11 minus PL)](1c)

Input different values of shift (f )f1 log(1(1 minus pU)) f2 log(1(1minus pL))p1 1 + exp(minus f1lowast f ) minus exp(minus f2lowastf )u a+ b minus 1 l a minus 1beta sum(dbinom(lu n p1))ARL1 1beta SDRL sqrt((1minus beta)((beta)2))Output ARL for different values of shift

ALGORITHM 2 Computation of ARL for different shifts

Input sample sizes (ss) PL PU and r0P0 1 minus PU+PLa sample(1 70 1 T) b sample(1 70 1 T)u a+ b minus 1 l a minus 1 n a+ b minus 1Pout0 sum(dbinom(lu n p0))ARL0[i] 1Pout0ARL0 [i] 1Pout0 aa[i] a bb[i] b nn[i] nImpose condition ARL0 ge r0Output obtain corresponding values of a b n and ARL0

ALGORITHM 1 Computation of a b n and ARL0

Mathematical Problems in Engineering 11

Derivation of Equation (10)

Pr XltUCL|c0( 1113857 PU

1 minus exp minusαλx

c1113888 1113889

c

1113888 1113889 PU

1 minus PU exp minusαλx

c1113888 1113889

c

1113888 1113889

(A3)

Taking ln on both sides

ln 1 minus PU( 1113857 ln exp minusαλx

c1113888 1113889

c

1113888 1113889

minus ln 1 minus PU( 1113857 αλx

c1113888 1113889

c

minus ln 1 minus Pu( 11138571113858 1113859(1c)

αλx

c1113888 1113889

ln1

1 minus PU

1113888 11138891113890 1113891

(1c)

αλx

c1113888 1113889

c

αλln

11 minus PU

1113888 11138891113890 1113891

(1c)

x

(A4)

)e UCL is (cαλ)[ln(11 minus PL)](1c)Equation (12) DerivationFor the in-control process the probability of the out-of-

control process is demonstrated through control charts isgiven as

P0out 1 minus 1113944

a+bminus 1

xa

x minus 1

a minus 11113888 1113889P

ao 1 minus P0( 1113857

xminus a (A5)

)e authors compute the probability of out-of-controlprocess as P(Xge b)

Now

P(Xge b) 1 minus P(Xlt b) (A6)

Since ldquoXrdquo follows the negative binomial distributionthen

P(Xlt b) 1113944

a+bminus 1

xa

x minus 1

a minus 11113888 1113889p

a0 1 minus p0( 1113857

xminus a (A7)

Equation (14) Derivation)e probability that the process is found out-of-control

due to the shifted process using control chart is obtained as

P1out 1 minus 1113944

a+bminus 1

xa

x minus 1

a minus 11113888 1113889P

a1 1 minus P1( 1113857

xminus a (A8)

)e authors compute the probability of out-of-controlprocess as P(Xge b)

Now

P(Xge b) 1 minus P(Xlt b) (A9)

Since ldquoXrdquo follows the negative binomial distributionthen

P(Xlt b) 1113944a+bminus 1

xa

x minus 1

a minus 11113888 1113889p

a1 1 minus p1( 1113857

xminus a (A10)

Data Availability

)e data used to support the findings of this study are givenin the paper

Disclosure

No funds were obtained for this paper

Conflicts of Interest

)e authors declare that there are no conflicts of interest

References

[1] N Eugene C Lee and F Famoye ldquoBeta-normal distributionand its applicationsrdquo Communications in Statistics-5eoryand Methods vol 31 no 4 pp 497ndash512 2002

[2] S Nadarajah and S Kotz ldquo)e beta Gumbel distributionrdquoMathematical Problems in Engineering vol 2004 no 4pp 323ndash332 2004

[3] S Nadarajah and A K Gupta ldquo)e beta Frechet distributionrdquoFar East Journal of 5eoretical Statistics vol 14 no 1pp 15ndash24 2004

[4] S Nadarajah and S Kotz ldquo)e beta exponential distributionrdquoReliability Engineering amp System Safety vol 91 no 6pp 689ndash697 2006

[5] S Nadarajah G M Cordeiro and E M M Ortega ldquo)ezografos-balakrishnan-GFamily of distributions mathemati-cal properties and applicationsrdquo Communications in Statistics- 5eory and Methods vol 44 no 1 pp 186ndash215 2015

[6] A Alzaatreh C Lee and F Famoye ldquoA new method forgenerating families of continuous distributionsrdquo Metronvol 71 no 1 pp 63ndash79 2013

[7] G M Cordeiro E M M Ortega B V Popovic andR R Pescim ldquo)e Lomax generator of distributions prop-erties minification process and regression modelrdquo AppliedMathematics and Computation vol 247 pp 465ndash486 2014

[8] A Z Afify M Alizadeh H M Yousof G Aryal andM Ahmad ldquo)e transmuted geometric-g family OF distri-butions THEORY and applicationsrdquo Pakistan Journal ofStatistics vol 32 no 2 pp 139ndash160 2016

[9] M Alizadeh M H Tahir G M Cordeiro M MansoorM Zubair and G G Hamedani ldquo)e Kumaraswamy Mar-shal-Olkin family of distributionsrdquo Journal of the EgyptianMathematical Society vol 23 no 3 pp 546ndash557 2015

[10] A Z Afify G M Cordeiro H M Yousof A Alzaatreh andZ M Nofal ldquo)e Kumaraswamy transmuted-G family ofdistributions properties and applicationsrdquo Journal of DataScience vol 14 no 2 pp 245ndash270 2016

[11] Z M Nofal A Z Afify H M Yousof and G M Cordeiroldquo)e generalized transmuted-G family of distributionsrdquo

12 Mathematical Problems in Engineering

Communications in Statistics-5eory and Methods vol 46no 8 pp 4119ndash4136 2017

[12] M Chahkandi and M Ganjali ldquoOn some lifetime distribu-tions with decreasing failure raterdquo Computational Statisticsand Data Analysis vol 53 no 12 pp 4433ndash4440 2009

[13] W Barreto-Souza A H S Santos and G M Cordeiro ldquo)ebeta generalized exponential distributionrdquo Journal of Statis-tical Computation and Simulation vol 80 no 2 pp 159ndash1722010

[14] W Lu and D Shi ldquoA new compounding life distribution theWeibull-Poisson distributionrdquo Journal of Applied Statisticsvol 39 no 1 pp 21ndash38 2012

[15] D C Montgomery Statistical Quality Control Vol 7 WileyNew York NY USA 2009

[16] W H Woodall ldquo)e use of control charts in health-care andpublic-health surveillancerdquo Journal of Quality Technologyvol 38 no 2 pp 89ndash104 2006

[17] S-L Hwang J-T Lin G-F Liang Y-J Yau T-C Yenn andC-C Hsu ldquoApplication control chart concepts of designing apre-alarm system in the nuclear power plant control roomrdquoNuclear Engineering and Design vol 238 no 12 pp 3522ndash3527 2008

[18] Z Wang and R Liang ldquoDiscuss on applying SPC to qualitymanagement in university educationrdquo in Proceedings of the9th International Conference for Young Computer Scientistspp 2372ndash2375 Hunan China November 2008

[19] M Pastell and H Madsen ldquoApplication of CUSUM charts todetect lameness in a milking robotrdquo Expert Systems withApplications vol 35 no 4 pp 2032ndash2040 2008

[20] A Ivanova C Xiao and Y Tymofyeyev ldquoTwo-stage designsfor phase 2 dose-finding trialsrdquo Statistics in Medicine vol 31no 24 pp 2872ndash2881 2012

[21] S W Roberts ldquoControl chart tests based on geometricmoving averagesrdquo Technometrics vol 1 no 3 pp 239ndash2501959

[22] D Brook and D A Evans ldquoAn approach to the probabilitydistribution of CUSUM run lengthrdquo Biometrika vol 59 no 3pp 539ndash549 1972

[23] J M Lucas ldquoCombined Shewhart-CUSUM quality controlschemesrdquo Journal of Quality Technology vol 14 no 2pp 51ndash59 1982

[24] C M Borror J B Keats and D C Montgomery ldquoRobustnessof the time between events CUSUMrdquo International Journal ofProduction Research vol 41 no 15 pp 3435ndash3444 2003

[25] M B C Khoo ldquoA moving average control chart for moni-toring the fraction non-conformingrdquo Quality and ReliabilityEngineering International vol 20 no 6 pp 617ndash635 2004

[26] S Sukparungsee and A A Novikov ldquoAnalytical approxi-mations for detection of a change-point in case of light-taileddistributionsrdquo Journal of Quality Measurement and Analysisvol 4 no 2 pp 49ndash56 2008

[27] Y Areepong and A A Novikov ldquoMartingale approach toEWMA control chart for changes in Exponential distribu-tionrdquo Journal of Quality Measurement and Analysis vol 4pp 197ndash203 2008

[28] R Noorossana A A Fatahi P Dokouhaki andM Babakhani ldquoZIB-EWMA control chart for monitoringrare health eventsrdquo Journal of Mechanics in Medicine andBiology vol 11 no 04 pp 881ndash895 2011

[29] E Santiago and J Smith ldquoControl charts based on the ex-ponential distribution adapting runs rules for the tChartrdquoQuality Engineering vol 25 no 2 pp 85ndash96 2013

[30] L L Ho and R C Quinino ldquoAn attribute control chart formonitoring the variability of a processrdquo International Journalof Production Economics vol 145 no 1 pp 263ndash267 2013

[31] M Aslam M Azam and C-H Jun ldquoAcceptance samplingplans for multi-stage process based on time-truncated test forWeibull distributionrdquo 5e International Journal of AdvancedManufacturing Technology vol 79 no 9ndash12 pp 1779ndash17852015

[32] A C Rakitzis and P Castagliola ldquo)e effect of parameterestimation on the performance of one-sided Shewhart controlcharts for zero-inflated processesrdquo Communications in Sta-tistics - 5eory and Methods vol 45 no 14 pp 4194ndash42142016

[33] M Azam M N Aslam and C-H Jun ldquoAn EWMA controlchart for the exponential distribution using repetitive sam-pling planrdquo 2017

[34] M Azam L Ahmad M Aslam and C-H Jun ldquoAn attributecontrol chart using discriminant limits for monitoring processunder the Weibull distributionrdquo Production Engineeringvol 12 no 5 pp 659ndash665 2018

[35] Z Wu and J Jiao ldquoA control chart for monitoring processmean based on attribute inspectionrdquo International Journal ofProduction Research vol 46 no 15 pp 4331ndash4347 2008

[36] H Kim and S Lee ldquoOn the VSI CUSUM chart for countprocesses and its implementation with R package attrCU-SUMrdquo Industrial Engineering ampManagement Systems vol 17no 1 pp 91ndash101 2018

[37] K Mabude J C Malela-Majika M Aslam Z L Chong andS C Shongwe ldquoDistribution-free composite Shewhart-GWMA Mann-Whitney charts for monitoring the processlocationrdquo Quality and Reliability Engineering International2020

[38] M T Shah M Azam M Aslam and U Sherazi ldquoTimebetween events control charts for gamma distributionrdquoQuality and Reliability Engineering International vol 37no 2 pp 785ndash803 2020

[39] U Shaheen M Azam and M Aslam ldquoA control chart formonitoring the lognormal process variation using repetitivesamplingrdquo Quality and Reliability Engineering Internationalvol 36 no 3 pp 1028ndash1047 2020

[40] M Aslam P Jeyadurga S Balamurali M Azam and A AL-Marshadi ldquoEconomic determination of modified multipledependent state sampling plan under some lifetime distri-butionsrdquo Journal of Mathematics vol 2021 Article ID7470196 2021

[41] R H Berk N L Johnson S Kotz and N BalakrishnanldquoContinuous univariate distributionsrdquo Technometrics vol 2p 752 1996

[42] C J P Belisle ldquoConvergence theorems for a class of simulatedannealing algorithms on Rdrdquo Journal of Applied Probabilityvol 29 no 4 pp 885ndash895 1992

[43] A Henningsen and O Toomet ldquoMaxlik a package formaximum likelihood estimation in Rrdquo Computational Sta-tistics vol 26 pp 443ndash458 2011

Mathematical Problems in Engineering 13

00 02 04 06 08 10

ARL of exponentialARL of weibullARL of weibull-exponential

f

1500

1000

500

0

ARL

s

Comparison of ARLs

Figure 9 ARLs of exponential Weibull and proposed WEx at 370 with pL 02 and pU 08

Table 6 Simulated in-control data0220499 0467107 0165884 0478009 0186779 00584930115906 0088989 0450255 0287988 0867485 025340604247 0714495 0356458 026482 0503415 05297730278757 008675 0204802 0181542 0417945 01030820269525 0543777 0096818 004666 0070134 0177020166788 0632749 0482233 0208581 0354741 02424770592133 0837021 0513781 0086579

Table 7 Simulated shifted data0650604 0144811 0585181 01867 0613353 05015330092963 0733554 0192515 0154301 0913754 05739580462511 0190089 058542 0297584 0162154 00806710066036 0746633 0545823 0061 0198106 0921250057984 0692892 026246 0463523 0204097 03307340454024 0765241 051748 0676308 0341994 04709630289353 0391563 0502378 0011261

0 10 20 30 40Sample number

10

08

06

04

02

00

Sam

ple v

alue

s

UDL = 095051

LDL = 003206

Figure 10 Plot for simulated in-control data

Mathematical Problems in Engineering 9

0 10 20 30 40Sample number

10

08

06

04

02

00

Sam

ple v

alue

s

UDL = 095051

LDL = 003206

Figure 11 Plot for simulated shift data with f 09 pL 01 pU 09 andARL0 200

0 10 20 30 5040Sample number

15

10

05

00

Sam

ple v

alue

s

UDL = 13544

LDL = 006197

Figure 12 Plot of UTI using exponential distribution control chart at pL 01 andpU 09

0 10 20 30 5040Sample number

15

10

05

00

Sam

ple v

alue

s UDL = 098102

LDL = 009615

Figure 13 Plot of UTI using proposed (WEx) control chart at pL 01 andpU 09

0 10 20 30 5040Sample number

15

10

05

00

Val

ues o

f T =

UTI

UDL = 15174

LDL = 03246

Figure 14 Plot of UTI using Weibull distribution control chart at pL 01 andpU 09

10 Mathematical Problems in Engineering

whereas the control chart of the proposed distributionwhich can be seen in Figure 13 that detected that theprocess is out-of-control at the 40th value It may beconcluded that our proposed distribution using real dataset of UTI shows better performance to detect the out-of-control process quickly

9 Conclusion

In this paper we develop a newWeibull class of distributionsusing the distribution generator technique )e newlyproposed WEx distribution is applied in the attributecharacteristics control chart for monitoring themanufacturing process Furthermore we compare theproposed WEx distribution with exponential distribution tomonitor the quick detection of the item in the out-of-controlprocess We conclude that under various shift levels ourproposed model is more efficient than the exponentialdistribution A simulation study is also conducted to knownthe behavior of ARL using the proposed WEx Weibull andexponential distributions )e implementation of the pro-posed model in a real-life example from the health caredepartment and the comparison of the two models endorsethe scope of WEx distribution We hope that this proposedcontrol chart can also be used for other accepting samplingschemes and other lifetime models (Figure 14)

Notations

W WeibullWEx Weibull-exponentialR Range chartS2 chart Variance chartW-G Weibull-generalizedPL PU PercentilesUDL Upper discriminant limitLDL Lower discriminant limita Accepted itemb Rejected itemf Shift levelα c c λ Parameters of Weibull-exponential distributionARL Average run-lengthP0out Probability of out-of-control process

P1out Probability of out-of-control process of shifted

processP0 Probability of rejected item when process is in-

controlP1 Probability of rejected item when process is shiftedUTI Urinary tract infection

Appendix

A R-Code

Derivation of Equation (9) (Algorithms 1 and 2)

Pr Xlt LCL|c0( 1113857 PL

1 minus exp minusαλx

c1113888 1113889

c

1113888 1113889 PL

1 minus PL exp minusαλx

c1113888 1113889

c

1113888 1113889

(A1)

Taking ln on both sides

ln 1 minus PL( 1113857 ln exp minusαλx

c1113888 1113889

c

1113888 1113889

minus ln 1 minus PL( 1113857 αλx

c1113888 1113889

c

minus ln 1 minus PL( 11138571113858 1113859(1c)

αλx

c1113888 1113889

ln1

1 minus PL

1113888 11138891113890 1113891

(1c)

αλx

c1113888 1113889

c

α λln

11 minus PL

1113888 11138891113890 1113891

(1c)

x

(A2)

)e LCL is (cαλ)[ln(11 minus PL)](1c)

Input different values of shift (f )f1 log(1(1 minus pU)) f2 log(1(1minus pL))p1 1 + exp(minus f1lowast f ) minus exp(minus f2lowastf )u a+ b minus 1 l a minus 1beta sum(dbinom(lu n p1))ARL1 1beta SDRL sqrt((1minus beta)((beta)2))Output ARL for different values of shift

ALGORITHM 2 Computation of ARL for different shifts

Input sample sizes (ss) PL PU and r0P0 1 minus PU+PLa sample(1 70 1 T) b sample(1 70 1 T)u a+ b minus 1 l a minus 1 n a+ b minus 1Pout0 sum(dbinom(lu n p0))ARL0[i] 1Pout0ARL0 [i] 1Pout0 aa[i] a bb[i] b nn[i] nImpose condition ARL0 ge r0Output obtain corresponding values of a b n and ARL0

ALGORITHM 1 Computation of a b n and ARL0

Mathematical Problems in Engineering 11

Derivation of Equation (10)

Pr XltUCL|c0( 1113857 PU

1 minus exp minusαλx

c1113888 1113889

c

1113888 1113889 PU

1 minus PU exp minusαλx

c1113888 1113889

c

1113888 1113889

(A3)

Taking ln on both sides

ln 1 minus PU( 1113857 ln exp minusαλx

c1113888 1113889

c

1113888 1113889

minus ln 1 minus PU( 1113857 αλx

c1113888 1113889

c

minus ln 1 minus Pu( 11138571113858 1113859(1c)

αλx

c1113888 1113889

ln1

1 minus PU

1113888 11138891113890 1113891

(1c)

αλx

c1113888 1113889

c

αλln

11 minus PU

1113888 11138891113890 1113891

(1c)

x

(A4)

)e UCL is (cαλ)[ln(11 minus PL)](1c)Equation (12) DerivationFor the in-control process the probability of the out-of-

control process is demonstrated through control charts isgiven as

P0out 1 minus 1113944

a+bminus 1

xa

x minus 1

a minus 11113888 1113889P

ao 1 minus P0( 1113857

xminus a (A5)

)e authors compute the probability of out-of-controlprocess as P(Xge b)

Now

P(Xge b) 1 minus P(Xlt b) (A6)

Since ldquoXrdquo follows the negative binomial distributionthen

P(Xlt b) 1113944

a+bminus 1

xa

x minus 1

a minus 11113888 1113889p

a0 1 minus p0( 1113857

xminus a (A7)

Equation (14) Derivation)e probability that the process is found out-of-control

due to the shifted process using control chart is obtained as

P1out 1 minus 1113944

a+bminus 1

xa

x minus 1

a minus 11113888 1113889P

a1 1 minus P1( 1113857

xminus a (A8)

)e authors compute the probability of out-of-controlprocess as P(Xge b)

Now

P(Xge b) 1 minus P(Xlt b) (A9)

Since ldquoXrdquo follows the negative binomial distributionthen

P(Xlt b) 1113944a+bminus 1

xa

x minus 1

a minus 11113888 1113889p

a1 1 minus p1( 1113857

xminus a (A10)

Data Availability

)e data used to support the findings of this study are givenin the paper

Disclosure

No funds were obtained for this paper

Conflicts of Interest

)e authors declare that there are no conflicts of interest

References

[1] N Eugene C Lee and F Famoye ldquoBeta-normal distributionand its applicationsrdquo Communications in Statistics-5eoryand Methods vol 31 no 4 pp 497ndash512 2002

[2] S Nadarajah and S Kotz ldquo)e beta Gumbel distributionrdquoMathematical Problems in Engineering vol 2004 no 4pp 323ndash332 2004

[3] S Nadarajah and A K Gupta ldquo)e beta Frechet distributionrdquoFar East Journal of 5eoretical Statistics vol 14 no 1pp 15ndash24 2004

[4] S Nadarajah and S Kotz ldquo)e beta exponential distributionrdquoReliability Engineering amp System Safety vol 91 no 6pp 689ndash697 2006

[5] S Nadarajah G M Cordeiro and E M M Ortega ldquo)ezografos-balakrishnan-GFamily of distributions mathemati-cal properties and applicationsrdquo Communications in Statistics- 5eory and Methods vol 44 no 1 pp 186ndash215 2015

[6] A Alzaatreh C Lee and F Famoye ldquoA new method forgenerating families of continuous distributionsrdquo Metronvol 71 no 1 pp 63ndash79 2013

[7] G M Cordeiro E M M Ortega B V Popovic andR R Pescim ldquo)e Lomax generator of distributions prop-erties minification process and regression modelrdquo AppliedMathematics and Computation vol 247 pp 465ndash486 2014

[8] A Z Afify M Alizadeh H M Yousof G Aryal andM Ahmad ldquo)e transmuted geometric-g family OF distri-butions THEORY and applicationsrdquo Pakistan Journal ofStatistics vol 32 no 2 pp 139ndash160 2016

[9] M Alizadeh M H Tahir G M Cordeiro M MansoorM Zubair and G G Hamedani ldquo)e Kumaraswamy Mar-shal-Olkin family of distributionsrdquo Journal of the EgyptianMathematical Society vol 23 no 3 pp 546ndash557 2015

[10] A Z Afify G M Cordeiro H M Yousof A Alzaatreh andZ M Nofal ldquo)e Kumaraswamy transmuted-G family ofdistributions properties and applicationsrdquo Journal of DataScience vol 14 no 2 pp 245ndash270 2016

[11] Z M Nofal A Z Afify H M Yousof and G M Cordeiroldquo)e generalized transmuted-G family of distributionsrdquo

12 Mathematical Problems in Engineering

Communications in Statistics-5eory and Methods vol 46no 8 pp 4119ndash4136 2017

[12] M Chahkandi and M Ganjali ldquoOn some lifetime distribu-tions with decreasing failure raterdquo Computational Statisticsand Data Analysis vol 53 no 12 pp 4433ndash4440 2009

[13] W Barreto-Souza A H S Santos and G M Cordeiro ldquo)ebeta generalized exponential distributionrdquo Journal of Statis-tical Computation and Simulation vol 80 no 2 pp 159ndash1722010

[14] W Lu and D Shi ldquoA new compounding life distribution theWeibull-Poisson distributionrdquo Journal of Applied Statisticsvol 39 no 1 pp 21ndash38 2012

[15] D C Montgomery Statistical Quality Control Vol 7 WileyNew York NY USA 2009

[16] W H Woodall ldquo)e use of control charts in health-care andpublic-health surveillancerdquo Journal of Quality Technologyvol 38 no 2 pp 89ndash104 2006

[17] S-L Hwang J-T Lin G-F Liang Y-J Yau T-C Yenn andC-C Hsu ldquoApplication control chart concepts of designing apre-alarm system in the nuclear power plant control roomrdquoNuclear Engineering and Design vol 238 no 12 pp 3522ndash3527 2008

[18] Z Wang and R Liang ldquoDiscuss on applying SPC to qualitymanagement in university educationrdquo in Proceedings of the9th International Conference for Young Computer Scientistspp 2372ndash2375 Hunan China November 2008

[19] M Pastell and H Madsen ldquoApplication of CUSUM charts todetect lameness in a milking robotrdquo Expert Systems withApplications vol 35 no 4 pp 2032ndash2040 2008

[20] A Ivanova C Xiao and Y Tymofyeyev ldquoTwo-stage designsfor phase 2 dose-finding trialsrdquo Statistics in Medicine vol 31no 24 pp 2872ndash2881 2012

[21] S W Roberts ldquoControl chart tests based on geometricmoving averagesrdquo Technometrics vol 1 no 3 pp 239ndash2501959

[22] D Brook and D A Evans ldquoAn approach to the probabilitydistribution of CUSUM run lengthrdquo Biometrika vol 59 no 3pp 539ndash549 1972

[23] J M Lucas ldquoCombined Shewhart-CUSUM quality controlschemesrdquo Journal of Quality Technology vol 14 no 2pp 51ndash59 1982

[24] C M Borror J B Keats and D C Montgomery ldquoRobustnessof the time between events CUSUMrdquo International Journal ofProduction Research vol 41 no 15 pp 3435ndash3444 2003

[25] M B C Khoo ldquoA moving average control chart for moni-toring the fraction non-conformingrdquo Quality and ReliabilityEngineering International vol 20 no 6 pp 617ndash635 2004

[26] S Sukparungsee and A A Novikov ldquoAnalytical approxi-mations for detection of a change-point in case of light-taileddistributionsrdquo Journal of Quality Measurement and Analysisvol 4 no 2 pp 49ndash56 2008

[27] Y Areepong and A A Novikov ldquoMartingale approach toEWMA control chart for changes in Exponential distribu-tionrdquo Journal of Quality Measurement and Analysis vol 4pp 197ndash203 2008

[28] R Noorossana A A Fatahi P Dokouhaki andM Babakhani ldquoZIB-EWMA control chart for monitoringrare health eventsrdquo Journal of Mechanics in Medicine andBiology vol 11 no 04 pp 881ndash895 2011

[29] E Santiago and J Smith ldquoControl charts based on the ex-ponential distribution adapting runs rules for the tChartrdquoQuality Engineering vol 25 no 2 pp 85ndash96 2013

[30] L L Ho and R C Quinino ldquoAn attribute control chart formonitoring the variability of a processrdquo International Journalof Production Economics vol 145 no 1 pp 263ndash267 2013

[31] M Aslam M Azam and C-H Jun ldquoAcceptance samplingplans for multi-stage process based on time-truncated test forWeibull distributionrdquo 5e International Journal of AdvancedManufacturing Technology vol 79 no 9ndash12 pp 1779ndash17852015

[32] A C Rakitzis and P Castagliola ldquo)e effect of parameterestimation on the performance of one-sided Shewhart controlcharts for zero-inflated processesrdquo Communications in Sta-tistics - 5eory and Methods vol 45 no 14 pp 4194ndash42142016

[33] M Azam M N Aslam and C-H Jun ldquoAn EWMA controlchart for the exponential distribution using repetitive sam-pling planrdquo 2017

[34] M Azam L Ahmad M Aslam and C-H Jun ldquoAn attributecontrol chart using discriminant limits for monitoring processunder the Weibull distributionrdquo Production Engineeringvol 12 no 5 pp 659ndash665 2018

[35] Z Wu and J Jiao ldquoA control chart for monitoring processmean based on attribute inspectionrdquo International Journal ofProduction Research vol 46 no 15 pp 4331ndash4347 2008

[36] H Kim and S Lee ldquoOn the VSI CUSUM chart for countprocesses and its implementation with R package attrCU-SUMrdquo Industrial Engineering ampManagement Systems vol 17no 1 pp 91ndash101 2018

[37] K Mabude J C Malela-Majika M Aslam Z L Chong andS C Shongwe ldquoDistribution-free composite Shewhart-GWMA Mann-Whitney charts for monitoring the processlocationrdquo Quality and Reliability Engineering International2020

[38] M T Shah M Azam M Aslam and U Sherazi ldquoTimebetween events control charts for gamma distributionrdquoQuality and Reliability Engineering International vol 37no 2 pp 785ndash803 2020

[39] U Shaheen M Azam and M Aslam ldquoA control chart formonitoring the lognormal process variation using repetitivesamplingrdquo Quality and Reliability Engineering Internationalvol 36 no 3 pp 1028ndash1047 2020

[40] M Aslam P Jeyadurga S Balamurali M Azam and A AL-Marshadi ldquoEconomic determination of modified multipledependent state sampling plan under some lifetime distri-butionsrdquo Journal of Mathematics vol 2021 Article ID7470196 2021

[41] R H Berk N L Johnson S Kotz and N BalakrishnanldquoContinuous univariate distributionsrdquo Technometrics vol 2p 752 1996

[42] C J P Belisle ldquoConvergence theorems for a class of simulatedannealing algorithms on Rdrdquo Journal of Applied Probabilityvol 29 no 4 pp 885ndash895 1992

[43] A Henningsen and O Toomet ldquoMaxlik a package formaximum likelihood estimation in Rrdquo Computational Sta-tistics vol 26 pp 443ndash458 2011

Mathematical Problems in Engineering 13

0 10 20 30 40Sample number

10

08

06

04

02

00

Sam

ple v

alue

s

UDL = 095051

LDL = 003206

Figure 11 Plot for simulated shift data with f 09 pL 01 pU 09 andARL0 200

0 10 20 30 5040Sample number

15

10

05

00

Sam

ple v

alue

s

UDL = 13544

LDL = 006197

Figure 12 Plot of UTI using exponential distribution control chart at pL 01 andpU 09

0 10 20 30 5040Sample number

15

10

05

00

Sam

ple v

alue

s UDL = 098102

LDL = 009615

Figure 13 Plot of UTI using proposed (WEx) control chart at pL 01 andpU 09

0 10 20 30 5040Sample number

15

10

05

00

Val

ues o

f T =

UTI

UDL = 15174

LDL = 03246

Figure 14 Plot of UTI using Weibull distribution control chart at pL 01 andpU 09

10 Mathematical Problems in Engineering

whereas the control chart of the proposed distributionwhich can be seen in Figure 13 that detected that theprocess is out-of-control at the 40th value It may beconcluded that our proposed distribution using real dataset of UTI shows better performance to detect the out-of-control process quickly

9 Conclusion

In this paper we develop a newWeibull class of distributionsusing the distribution generator technique )e newlyproposed WEx distribution is applied in the attributecharacteristics control chart for monitoring themanufacturing process Furthermore we compare theproposed WEx distribution with exponential distribution tomonitor the quick detection of the item in the out-of-controlprocess We conclude that under various shift levels ourproposed model is more efficient than the exponentialdistribution A simulation study is also conducted to knownthe behavior of ARL using the proposed WEx Weibull andexponential distributions )e implementation of the pro-posed model in a real-life example from the health caredepartment and the comparison of the two models endorsethe scope of WEx distribution We hope that this proposedcontrol chart can also be used for other accepting samplingschemes and other lifetime models (Figure 14)

Notations

W WeibullWEx Weibull-exponentialR Range chartS2 chart Variance chartW-G Weibull-generalizedPL PU PercentilesUDL Upper discriminant limitLDL Lower discriminant limita Accepted itemb Rejected itemf Shift levelα c c λ Parameters of Weibull-exponential distributionARL Average run-lengthP0out Probability of out-of-control process

P1out Probability of out-of-control process of shifted

processP0 Probability of rejected item when process is in-

controlP1 Probability of rejected item when process is shiftedUTI Urinary tract infection

Appendix

A R-Code

Derivation of Equation (9) (Algorithms 1 and 2)

Pr Xlt LCL|c0( 1113857 PL

1 minus exp minusαλx

c1113888 1113889

c

1113888 1113889 PL

1 minus PL exp minusαλx

c1113888 1113889

c

1113888 1113889

(A1)

Taking ln on both sides

ln 1 minus PL( 1113857 ln exp minusαλx

c1113888 1113889

c

1113888 1113889

minus ln 1 minus PL( 1113857 αλx

c1113888 1113889

c

minus ln 1 minus PL( 11138571113858 1113859(1c)

αλx

c1113888 1113889

ln1

1 minus PL

1113888 11138891113890 1113891

(1c)

αλx

c1113888 1113889

c

α λln

11 minus PL

1113888 11138891113890 1113891

(1c)

x

(A2)

)e LCL is (cαλ)[ln(11 minus PL)](1c)

Input different values of shift (f )f1 log(1(1 minus pU)) f2 log(1(1minus pL))p1 1 + exp(minus f1lowast f ) minus exp(minus f2lowastf )u a+ b minus 1 l a minus 1beta sum(dbinom(lu n p1))ARL1 1beta SDRL sqrt((1minus beta)((beta)2))Output ARL for different values of shift

ALGORITHM 2 Computation of ARL for different shifts

Input sample sizes (ss) PL PU and r0P0 1 minus PU+PLa sample(1 70 1 T) b sample(1 70 1 T)u a+ b minus 1 l a minus 1 n a+ b minus 1Pout0 sum(dbinom(lu n p0))ARL0[i] 1Pout0ARL0 [i] 1Pout0 aa[i] a bb[i] b nn[i] nImpose condition ARL0 ge r0Output obtain corresponding values of a b n and ARL0

ALGORITHM 1 Computation of a b n and ARL0

Mathematical Problems in Engineering 11

Derivation of Equation (10)

Pr XltUCL|c0( 1113857 PU

1 minus exp minusαλx

c1113888 1113889

c

1113888 1113889 PU

1 minus PU exp minusαλx

c1113888 1113889

c

1113888 1113889

(A3)

Taking ln on both sides

ln 1 minus PU( 1113857 ln exp minusαλx

c1113888 1113889

c

1113888 1113889

minus ln 1 minus PU( 1113857 αλx

c1113888 1113889

c

minus ln 1 minus Pu( 11138571113858 1113859(1c)

αλx

c1113888 1113889

ln1

1 minus PU

1113888 11138891113890 1113891

(1c)

αλx

c1113888 1113889

c

αλln

11 minus PU

1113888 11138891113890 1113891

(1c)

x

(A4)

)e UCL is (cαλ)[ln(11 minus PL)](1c)Equation (12) DerivationFor the in-control process the probability of the out-of-

control process is demonstrated through control charts isgiven as

P0out 1 minus 1113944

a+bminus 1

xa

x minus 1

a minus 11113888 1113889P

ao 1 minus P0( 1113857

xminus a (A5)

)e authors compute the probability of out-of-controlprocess as P(Xge b)

Now

P(Xge b) 1 minus P(Xlt b) (A6)

Since ldquoXrdquo follows the negative binomial distributionthen

P(Xlt b) 1113944

a+bminus 1

xa

x minus 1

a minus 11113888 1113889p

a0 1 minus p0( 1113857

xminus a (A7)

Equation (14) Derivation)e probability that the process is found out-of-control

due to the shifted process using control chart is obtained as

P1out 1 minus 1113944

a+bminus 1

xa

x minus 1

a minus 11113888 1113889P

a1 1 minus P1( 1113857

xminus a (A8)

)e authors compute the probability of out-of-controlprocess as P(Xge b)

Now

P(Xge b) 1 minus P(Xlt b) (A9)

Since ldquoXrdquo follows the negative binomial distributionthen

P(Xlt b) 1113944a+bminus 1

xa

x minus 1

a minus 11113888 1113889p

a1 1 minus p1( 1113857

xminus a (A10)

Data Availability

)e data used to support the findings of this study are givenin the paper

Disclosure

No funds were obtained for this paper

Conflicts of Interest

)e authors declare that there are no conflicts of interest

References

[1] N Eugene C Lee and F Famoye ldquoBeta-normal distributionand its applicationsrdquo Communications in Statistics-5eoryand Methods vol 31 no 4 pp 497ndash512 2002

[2] S Nadarajah and S Kotz ldquo)e beta Gumbel distributionrdquoMathematical Problems in Engineering vol 2004 no 4pp 323ndash332 2004

[3] S Nadarajah and A K Gupta ldquo)e beta Frechet distributionrdquoFar East Journal of 5eoretical Statistics vol 14 no 1pp 15ndash24 2004

[4] S Nadarajah and S Kotz ldquo)e beta exponential distributionrdquoReliability Engineering amp System Safety vol 91 no 6pp 689ndash697 2006

[5] S Nadarajah G M Cordeiro and E M M Ortega ldquo)ezografos-balakrishnan-GFamily of distributions mathemati-cal properties and applicationsrdquo Communications in Statistics- 5eory and Methods vol 44 no 1 pp 186ndash215 2015

[6] A Alzaatreh C Lee and F Famoye ldquoA new method forgenerating families of continuous distributionsrdquo Metronvol 71 no 1 pp 63ndash79 2013

[7] G M Cordeiro E M M Ortega B V Popovic andR R Pescim ldquo)e Lomax generator of distributions prop-erties minification process and regression modelrdquo AppliedMathematics and Computation vol 247 pp 465ndash486 2014

[8] A Z Afify M Alizadeh H M Yousof G Aryal andM Ahmad ldquo)e transmuted geometric-g family OF distri-butions THEORY and applicationsrdquo Pakistan Journal ofStatistics vol 32 no 2 pp 139ndash160 2016

[9] M Alizadeh M H Tahir G M Cordeiro M MansoorM Zubair and G G Hamedani ldquo)e Kumaraswamy Mar-shal-Olkin family of distributionsrdquo Journal of the EgyptianMathematical Society vol 23 no 3 pp 546ndash557 2015

[10] A Z Afify G M Cordeiro H M Yousof A Alzaatreh andZ M Nofal ldquo)e Kumaraswamy transmuted-G family ofdistributions properties and applicationsrdquo Journal of DataScience vol 14 no 2 pp 245ndash270 2016

[11] Z M Nofal A Z Afify H M Yousof and G M Cordeiroldquo)e generalized transmuted-G family of distributionsrdquo

12 Mathematical Problems in Engineering

Communications in Statistics-5eory and Methods vol 46no 8 pp 4119ndash4136 2017

[12] M Chahkandi and M Ganjali ldquoOn some lifetime distribu-tions with decreasing failure raterdquo Computational Statisticsand Data Analysis vol 53 no 12 pp 4433ndash4440 2009

[13] W Barreto-Souza A H S Santos and G M Cordeiro ldquo)ebeta generalized exponential distributionrdquo Journal of Statis-tical Computation and Simulation vol 80 no 2 pp 159ndash1722010

[14] W Lu and D Shi ldquoA new compounding life distribution theWeibull-Poisson distributionrdquo Journal of Applied Statisticsvol 39 no 1 pp 21ndash38 2012

[15] D C Montgomery Statistical Quality Control Vol 7 WileyNew York NY USA 2009

[16] W H Woodall ldquo)e use of control charts in health-care andpublic-health surveillancerdquo Journal of Quality Technologyvol 38 no 2 pp 89ndash104 2006

[17] S-L Hwang J-T Lin G-F Liang Y-J Yau T-C Yenn andC-C Hsu ldquoApplication control chart concepts of designing apre-alarm system in the nuclear power plant control roomrdquoNuclear Engineering and Design vol 238 no 12 pp 3522ndash3527 2008

[18] Z Wang and R Liang ldquoDiscuss on applying SPC to qualitymanagement in university educationrdquo in Proceedings of the9th International Conference for Young Computer Scientistspp 2372ndash2375 Hunan China November 2008

[19] M Pastell and H Madsen ldquoApplication of CUSUM charts todetect lameness in a milking robotrdquo Expert Systems withApplications vol 35 no 4 pp 2032ndash2040 2008

[20] A Ivanova C Xiao and Y Tymofyeyev ldquoTwo-stage designsfor phase 2 dose-finding trialsrdquo Statistics in Medicine vol 31no 24 pp 2872ndash2881 2012

[21] S W Roberts ldquoControl chart tests based on geometricmoving averagesrdquo Technometrics vol 1 no 3 pp 239ndash2501959

[22] D Brook and D A Evans ldquoAn approach to the probabilitydistribution of CUSUM run lengthrdquo Biometrika vol 59 no 3pp 539ndash549 1972

[23] J M Lucas ldquoCombined Shewhart-CUSUM quality controlschemesrdquo Journal of Quality Technology vol 14 no 2pp 51ndash59 1982

[24] C M Borror J B Keats and D C Montgomery ldquoRobustnessof the time between events CUSUMrdquo International Journal ofProduction Research vol 41 no 15 pp 3435ndash3444 2003

[25] M B C Khoo ldquoA moving average control chart for moni-toring the fraction non-conformingrdquo Quality and ReliabilityEngineering International vol 20 no 6 pp 617ndash635 2004

[26] S Sukparungsee and A A Novikov ldquoAnalytical approxi-mations for detection of a change-point in case of light-taileddistributionsrdquo Journal of Quality Measurement and Analysisvol 4 no 2 pp 49ndash56 2008

[27] Y Areepong and A A Novikov ldquoMartingale approach toEWMA control chart for changes in Exponential distribu-tionrdquo Journal of Quality Measurement and Analysis vol 4pp 197ndash203 2008

[28] R Noorossana A A Fatahi P Dokouhaki andM Babakhani ldquoZIB-EWMA control chart for monitoringrare health eventsrdquo Journal of Mechanics in Medicine andBiology vol 11 no 04 pp 881ndash895 2011

[29] E Santiago and J Smith ldquoControl charts based on the ex-ponential distribution adapting runs rules for the tChartrdquoQuality Engineering vol 25 no 2 pp 85ndash96 2013

[30] L L Ho and R C Quinino ldquoAn attribute control chart formonitoring the variability of a processrdquo International Journalof Production Economics vol 145 no 1 pp 263ndash267 2013

[31] M Aslam M Azam and C-H Jun ldquoAcceptance samplingplans for multi-stage process based on time-truncated test forWeibull distributionrdquo 5e International Journal of AdvancedManufacturing Technology vol 79 no 9ndash12 pp 1779ndash17852015

[32] A C Rakitzis and P Castagliola ldquo)e effect of parameterestimation on the performance of one-sided Shewhart controlcharts for zero-inflated processesrdquo Communications in Sta-tistics - 5eory and Methods vol 45 no 14 pp 4194ndash42142016

[33] M Azam M N Aslam and C-H Jun ldquoAn EWMA controlchart for the exponential distribution using repetitive sam-pling planrdquo 2017

[34] M Azam L Ahmad M Aslam and C-H Jun ldquoAn attributecontrol chart using discriminant limits for monitoring processunder the Weibull distributionrdquo Production Engineeringvol 12 no 5 pp 659ndash665 2018

[35] Z Wu and J Jiao ldquoA control chart for monitoring processmean based on attribute inspectionrdquo International Journal ofProduction Research vol 46 no 15 pp 4331ndash4347 2008

[36] H Kim and S Lee ldquoOn the VSI CUSUM chart for countprocesses and its implementation with R package attrCU-SUMrdquo Industrial Engineering ampManagement Systems vol 17no 1 pp 91ndash101 2018

[37] K Mabude J C Malela-Majika M Aslam Z L Chong andS C Shongwe ldquoDistribution-free composite Shewhart-GWMA Mann-Whitney charts for monitoring the processlocationrdquo Quality and Reliability Engineering International2020

[38] M T Shah M Azam M Aslam and U Sherazi ldquoTimebetween events control charts for gamma distributionrdquoQuality and Reliability Engineering International vol 37no 2 pp 785ndash803 2020

[39] U Shaheen M Azam and M Aslam ldquoA control chart formonitoring the lognormal process variation using repetitivesamplingrdquo Quality and Reliability Engineering Internationalvol 36 no 3 pp 1028ndash1047 2020

[40] M Aslam P Jeyadurga S Balamurali M Azam and A AL-Marshadi ldquoEconomic determination of modified multipledependent state sampling plan under some lifetime distri-butionsrdquo Journal of Mathematics vol 2021 Article ID7470196 2021

[41] R H Berk N L Johnson S Kotz and N BalakrishnanldquoContinuous univariate distributionsrdquo Technometrics vol 2p 752 1996

[42] C J P Belisle ldquoConvergence theorems for a class of simulatedannealing algorithms on Rdrdquo Journal of Applied Probabilityvol 29 no 4 pp 885ndash895 1992

[43] A Henningsen and O Toomet ldquoMaxlik a package formaximum likelihood estimation in Rrdquo Computational Sta-tistics vol 26 pp 443ndash458 2011

Mathematical Problems in Engineering 13

whereas the control chart of the proposed distributionwhich can be seen in Figure 13 that detected that theprocess is out-of-control at the 40th value It may beconcluded that our proposed distribution using real dataset of UTI shows better performance to detect the out-of-control process quickly

9 Conclusion

In this paper we develop a newWeibull class of distributionsusing the distribution generator technique )e newlyproposed WEx distribution is applied in the attributecharacteristics control chart for monitoring themanufacturing process Furthermore we compare theproposed WEx distribution with exponential distribution tomonitor the quick detection of the item in the out-of-controlprocess We conclude that under various shift levels ourproposed model is more efficient than the exponentialdistribution A simulation study is also conducted to knownthe behavior of ARL using the proposed WEx Weibull andexponential distributions )e implementation of the pro-posed model in a real-life example from the health caredepartment and the comparison of the two models endorsethe scope of WEx distribution We hope that this proposedcontrol chart can also be used for other accepting samplingschemes and other lifetime models (Figure 14)

Notations

W WeibullWEx Weibull-exponentialR Range chartS2 chart Variance chartW-G Weibull-generalizedPL PU PercentilesUDL Upper discriminant limitLDL Lower discriminant limita Accepted itemb Rejected itemf Shift levelα c c λ Parameters of Weibull-exponential distributionARL Average run-lengthP0out Probability of out-of-control process

P1out Probability of out-of-control process of shifted

processP0 Probability of rejected item when process is in-

controlP1 Probability of rejected item when process is shiftedUTI Urinary tract infection

Appendix

A R-Code

Derivation of Equation (9) (Algorithms 1 and 2)

Pr Xlt LCL|c0( 1113857 PL

1 minus exp minusαλx

c1113888 1113889

c

1113888 1113889 PL

1 minus PL exp minusαλx

c1113888 1113889

c

1113888 1113889

(A1)

Taking ln on both sides

ln 1 minus PL( 1113857 ln exp minusαλx

c1113888 1113889

c

1113888 1113889

minus ln 1 minus PL( 1113857 αλx

c1113888 1113889

c

minus ln 1 minus PL( 11138571113858 1113859(1c)

αλx

c1113888 1113889

ln1

1 minus PL

1113888 11138891113890 1113891

(1c)

αλx

c1113888 1113889

c

α λln

11 minus PL

1113888 11138891113890 1113891

(1c)

x

(A2)

)e LCL is (cαλ)[ln(11 minus PL)](1c)

Input different values of shift (f )f1 log(1(1 minus pU)) f2 log(1(1minus pL))p1 1 + exp(minus f1lowast f ) minus exp(minus f2lowastf )u a+ b minus 1 l a minus 1beta sum(dbinom(lu n p1))ARL1 1beta SDRL sqrt((1minus beta)((beta)2))Output ARL for different values of shift

ALGORITHM 2 Computation of ARL for different shifts

Input sample sizes (ss) PL PU and r0P0 1 minus PU+PLa sample(1 70 1 T) b sample(1 70 1 T)u a+ b minus 1 l a minus 1 n a+ b minus 1Pout0 sum(dbinom(lu n p0))ARL0[i] 1Pout0ARL0 [i] 1Pout0 aa[i] a bb[i] b nn[i] nImpose condition ARL0 ge r0Output obtain corresponding values of a b n and ARL0

ALGORITHM 1 Computation of a b n and ARL0

Mathematical Problems in Engineering 11

Derivation of Equation (10)

Pr XltUCL|c0( 1113857 PU

1 minus exp minusαλx

c1113888 1113889

c

1113888 1113889 PU

1 minus PU exp minusαλx

c1113888 1113889

c

1113888 1113889

(A3)

Taking ln on both sides

ln 1 minus PU( 1113857 ln exp minusαλx

c1113888 1113889

c

1113888 1113889

minus ln 1 minus PU( 1113857 αλx

c1113888 1113889

c

minus ln 1 minus Pu( 11138571113858 1113859(1c)

αλx

c1113888 1113889

ln1

1 minus PU

1113888 11138891113890 1113891

(1c)

αλx

c1113888 1113889

c

αλln

11 minus PU

1113888 11138891113890 1113891

(1c)

x

(A4)

)e UCL is (cαλ)[ln(11 minus PL)](1c)Equation (12) DerivationFor the in-control process the probability of the out-of-

control process is demonstrated through control charts isgiven as

P0out 1 minus 1113944

a+bminus 1

xa

x minus 1

a minus 11113888 1113889P

ao 1 minus P0( 1113857

xminus a (A5)

)e authors compute the probability of out-of-controlprocess as P(Xge b)

Now

P(Xge b) 1 minus P(Xlt b) (A6)

Since ldquoXrdquo follows the negative binomial distributionthen

P(Xlt b) 1113944

a+bminus 1

xa

x minus 1

a minus 11113888 1113889p

a0 1 minus p0( 1113857

xminus a (A7)

Equation (14) Derivation)e probability that the process is found out-of-control

due to the shifted process using control chart is obtained as

P1out 1 minus 1113944

a+bminus 1

xa

x minus 1

a minus 11113888 1113889P

a1 1 minus P1( 1113857

xminus a (A8)

)e authors compute the probability of out-of-controlprocess as P(Xge b)

Now

P(Xge b) 1 minus P(Xlt b) (A9)

Since ldquoXrdquo follows the negative binomial distributionthen

P(Xlt b) 1113944a+bminus 1

xa

x minus 1

a minus 11113888 1113889p

a1 1 minus p1( 1113857

xminus a (A10)

Data Availability

)e data used to support the findings of this study are givenin the paper

Disclosure

No funds were obtained for this paper

Conflicts of Interest

)e authors declare that there are no conflicts of interest

References

[1] N Eugene C Lee and F Famoye ldquoBeta-normal distributionand its applicationsrdquo Communications in Statistics-5eoryand Methods vol 31 no 4 pp 497ndash512 2002

[2] S Nadarajah and S Kotz ldquo)e beta Gumbel distributionrdquoMathematical Problems in Engineering vol 2004 no 4pp 323ndash332 2004

[3] S Nadarajah and A K Gupta ldquo)e beta Frechet distributionrdquoFar East Journal of 5eoretical Statistics vol 14 no 1pp 15ndash24 2004

[4] S Nadarajah and S Kotz ldquo)e beta exponential distributionrdquoReliability Engineering amp System Safety vol 91 no 6pp 689ndash697 2006

[5] S Nadarajah G M Cordeiro and E M M Ortega ldquo)ezografos-balakrishnan-GFamily of distributions mathemati-cal properties and applicationsrdquo Communications in Statistics- 5eory and Methods vol 44 no 1 pp 186ndash215 2015

[6] A Alzaatreh C Lee and F Famoye ldquoA new method forgenerating families of continuous distributionsrdquo Metronvol 71 no 1 pp 63ndash79 2013

[7] G M Cordeiro E M M Ortega B V Popovic andR R Pescim ldquo)e Lomax generator of distributions prop-erties minification process and regression modelrdquo AppliedMathematics and Computation vol 247 pp 465ndash486 2014

[8] A Z Afify M Alizadeh H M Yousof G Aryal andM Ahmad ldquo)e transmuted geometric-g family OF distri-butions THEORY and applicationsrdquo Pakistan Journal ofStatistics vol 32 no 2 pp 139ndash160 2016

[9] M Alizadeh M H Tahir G M Cordeiro M MansoorM Zubair and G G Hamedani ldquo)e Kumaraswamy Mar-shal-Olkin family of distributionsrdquo Journal of the EgyptianMathematical Society vol 23 no 3 pp 546ndash557 2015

[10] A Z Afify G M Cordeiro H M Yousof A Alzaatreh andZ M Nofal ldquo)e Kumaraswamy transmuted-G family ofdistributions properties and applicationsrdquo Journal of DataScience vol 14 no 2 pp 245ndash270 2016

[11] Z M Nofal A Z Afify H M Yousof and G M Cordeiroldquo)e generalized transmuted-G family of distributionsrdquo

12 Mathematical Problems in Engineering

Communications in Statistics-5eory and Methods vol 46no 8 pp 4119ndash4136 2017

[12] M Chahkandi and M Ganjali ldquoOn some lifetime distribu-tions with decreasing failure raterdquo Computational Statisticsand Data Analysis vol 53 no 12 pp 4433ndash4440 2009

[13] W Barreto-Souza A H S Santos and G M Cordeiro ldquo)ebeta generalized exponential distributionrdquo Journal of Statis-tical Computation and Simulation vol 80 no 2 pp 159ndash1722010

[14] W Lu and D Shi ldquoA new compounding life distribution theWeibull-Poisson distributionrdquo Journal of Applied Statisticsvol 39 no 1 pp 21ndash38 2012

[15] D C Montgomery Statistical Quality Control Vol 7 WileyNew York NY USA 2009

[16] W H Woodall ldquo)e use of control charts in health-care andpublic-health surveillancerdquo Journal of Quality Technologyvol 38 no 2 pp 89ndash104 2006

[17] S-L Hwang J-T Lin G-F Liang Y-J Yau T-C Yenn andC-C Hsu ldquoApplication control chart concepts of designing apre-alarm system in the nuclear power plant control roomrdquoNuclear Engineering and Design vol 238 no 12 pp 3522ndash3527 2008

[18] Z Wang and R Liang ldquoDiscuss on applying SPC to qualitymanagement in university educationrdquo in Proceedings of the9th International Conference for Young Computer Scientistspp 2372ndash2375 Hunan China November 2008

[19] M Pastell and H Madsen ldquoApplication of CUSUM charts todetect lameness in a milking robotrdquo Expert Systems withApplications vol 35 no 4 pp 2032ndash2040 2008

[20] A Ivanova C Xiao and Y Tymofyeyev ldquoTwo-stage designsfor phase 2 dose-finding trialsrdquo Statistics in Medicine vol 31no 24 pp 2872ndash2881 2012

[21] S W Roberts ldquoControl chart tests based on geometricmoving averagesrdquo Technometrics vol 1 no 3 pp 239ndash2501959

[22] D Brook and D A Evans ldquoAn approach to the probabilitydistribution of CUSUM run lengthrdquo Biometrika vol 59 no 3pp 539ndash549 1972

[23] J M Lucas ldquoCombined Shewhart-CUSUM quality controlschemesrdquo Journal of Quality Technology vol 14 no 2pp 51ndash59 1982

[24] C M Borror J B Keats and D C Montgomery ldquoRobustnessof the time between events CUSUMrdquo International Journal ofProduction Research vol 41 no 15 pp 3435ndash3444 2003

[25] M B C Khoo ldquoA moving average control chart for moni-toring the fraction non-conformingrdquo Quality and ReliabilityEngineering International vol 20 no 6 pp 617ndash635 2004

[26] S Sukparungsee and A A Novikov ldquoAnalytical approxi-mations for detection of a change-point in case of light-taileddistributionsrdquo Journal of Quality Measurement and Analysisvol 4 no 2 pp 49ndash56 2008

[27] Y Areepong and A A Novikov ldquoMartingale approach toEWMA control chart for changes in Exponential distribu-tionrdquo Journal of Quality Measurement and Analysis vol 4pp 197ndash203 2008

[28] R Noorossana A A Fatahi P Dokouhaki andM Babakhani ldquoZIB-EWMA control chart for monitoringrare health eventsrdquo Journal of Mechanics in Medicine andBiology vol 11 no 04 pp 881ndash895 2011

[29] E Santiago and J Smith ldquoControl charts based on the ex-ponential distribution adapting runs rules for the tChartrdquoQuality Engineering vol 25 no 2 pp 85ndash96 2013

[30] L L Ho and R C Quinino ldquoAn attribute control chart formonitoring the variability of a processrdquo International Journalof Production Economics vol 145 no 1 pp 263ndash267 2013

[31] M Aslam M Azam and C-H Jun ldquoAcceptance samplingplans for multi-stage process based on time-truncated test forWeibull distributionrdquo 5e International Journal of AdvancedManufacturing Technology vol 79 no 9ndash12 pp 1779ndash17852015

[32] A C Rakitzis and P Castagliola ldquo)e effect of parameterestimation on the performance of one-sided Shewhart controlcharts for zero-inflated processesrdquo Communications in Sta-tistics - 5eory and Methods vol 45 no 14 pp 4194ndash42142016

[33] M Azam M N Aslam and C-H Jun ldquoAn EWMA controlchart for the exponential distribution using repetitive sam-pling planrdquo 2017

[34] M Azam L Ahmad M Aslam and C-H Jun ldquoAn attributecontrol chart using discriminant limits for monitoring processunder the Weibull distributionrdquo Production Engineeringvol 12 no 5 pp 659ndash665 2018

[35] Z Wu and J Jiao ldquoA control chart for monitoring processmean based on attribute inspectionrdquo International Journal ofProduction Research vol 46 no 15 pp 4331ndash4347 2008

[36] H Kim and S Lee ldquoOn the VSI CUSUM chart for countprocesses and its implementation with R package attrCU-SUMrdquo Industrial Engineering ampManagement Systems vol 17no 1 pp 91ndash101 2018

[37] K Mabude J C Malela-Majika M Aslam Z L Chong andS C Shongwe ldquoDistribution-free composite Shewhart-GWMA Mann-Whitney charts for monitoring the processlocationrdquo Quality and Reliability Engineering International2020

[38] M T Shah M Azam M Aslam and U Sherazi ldquoTimebetween events control charts for gamma distributionrdquoQuality and Reliability Engineering International vol 37no 2 pp 785ndash803 2020

[39] U Shaheen M Azam and M Aslam ldquoA control chart formonitoring the lognormal process variation using repetitivesamplingrdquo Quality and Reliability Engineering Internationalvol 36 no 3 pp 1028ndash1047 2020

[40] M Aslam P Jeyadurga S Balamurali M Azam and A AL-Marshadi ldquoEconomic determination of modified multipledependent state sampling plan under some lifetime distri-butionsrdquo Journal of Mathematics vol 2021 Article ID7470196 2021

[41] R H Berk N L Johnson S Kotz and N BalakrishnanldquoContinuous univariate distributionsrdquo Technometrics vol 2p 752 1996

[42] C J P Belisle ldquoConvergence theorems for a class of simulatedannealing algorithms on Rdrdquo Journal of Applied Probabilityvol 29 no 4 pp 885ndash895 1992

[43] A Henningsen and O Toomet ldquoMaxlik a package formaximum likelihood estimation in Rrdquo Computational Sta-tistics vol 26 pp 443ndash458 2011

Mathematical Problems in Engineering 13

Derivation of Equation (10)

Pr XltUCL|c0( 1113857 PU

1 minus exp minusαλx

c1113888 1113889

c

1113888 1113889 PU

1 minus PU exp minusαλx

c1113888 1113889

c

1113888 1113889

(A3)

Taking ln on both sides

ln 1 minus PU( 1113857 ln exp minusαλx

c1113888 1113889

c

1113888 1113889

minus ln 1 minus PU( 1113857 αλx

c1113888 1113889

c

minus ln 1 minus Pu( 11138571113858 1113859(1c)

αλx

c1113888 1113889

ln1

1 minus PU

1113888 11138891113890 1113891

(1c)

αλx

c1113888 1113889

c

αλln

11 minus PU

1113888 11138891113890 1113891

(1c)

x

(A4)

)e UCL is (cαλ)[ln(11 minus PL)](1c)Equation (12) DerivationFor the in-control process the probability of the out-of-

control process is demonstrated through control charts isgiven as

P0out 1 minus 1113944

a+bminus 1

xa

x minus 1

a minus 11113888 1113889P

ao 1 minus P0( 1113857

xminus a (A5)

)e authors compute the probability of out-of-controlprocess as P(Xge b)

Now

P(Xge b) 1 minus P(Xlt b) (A6)

Since ldquoXrdquo follows the negative binomial distributionthen

P(Xlt b) 1113944

a+bminus 1

xa

x minus 1

a minus 11113888 1113889p

a0 1 minus p0( 1113857

xminus a (A7)

Equation (14) Derivation)e probability that the process is found out-of-control

due to the shifted process using control chart is obtained as

P1out 1 minus 1113944

a+bminus 1

xa

x minus 1

a minus 11113888 1113889P

a1 1 minus P1( 1113857

xminus a (A8)

)e authors compute the probability of out-of-controlprocess as P(Xge b)

Now

P(Xge b) 1 minus P(Xlt b) (A9)

Since ldquoXrdquo follows the negative binomial distributionthen

P(Xlt b) 1113944a+bminus 1

xa

x minus 1

a minus 11113888 1113889p

a1 1 minus p1( 1113857

xminus a (A10)

Data Availability

)e data used to support the findings of this study are givenin the paper

Disclosure

No funds were obtained for this paper

Conflicts of Interest

)e authors declare that there are no conflicts of interest

References

[1] N Eugene C Lee and F Famoye ldquoBeta-normal distributionand its applicationsrdquo Communications in Statistics-5eoryand Methods vol 31 no 4 pp 497ndash512 2002

[2] S Nadarajah and S Kotz ldquo)e beta Gumbel distributionrdquoMathematical Problems in Engineering vol 2004 no 4pp 323ndash332 2004

[3] S Nadarajah and A K Gupta ldquo)e beta Frechet distributionrdquoFar East Journal of 5eoretical Statistics vol 14 no 1pp 15ndash24 2004

[4] S Nadarajah and S Kotz ldquo)e beta exponential distributionrdquoReliability Engineering amp System Safety vol 91 no 6pp 689ndash697 2006

[5] S Nadarajah G M Cordeiro and E M M Ortega ldquo)ezografos-balakrishnan-GFamily of distributions mathemati-cal properties and applicationsrdquo Communications in Statistics- 5eory and Methods vol 44 no 1 pp 186ndash215 2015

[6] A Alzaatreh C Lee and F Famoye ldquoA new method forgenerating families of continuous distributionsrdquo Metronvol 71 no 1 pp 63ndash79 2013

[7] G M Cordeiro E M M Ortega B V Popovic andR R Pescim ldquo)e Lomax generator of distributions prop-erties minification process and regression modelrdquo AppliedMathematics and Computation vol 247 pp 465ndash486 2014

[8] A Z Afify M Alizadeh H M Yousof G Aryal andM Ahmad ldquo)e transmuted geometric-g family OF distri-butions THEORY and applicationsrdquo Pakistan Journal ofStatistics vol 32 no 2 pp 139ndash160 2016

[9] M Alizadeh M H Tahir G M Cordeiro M MansoorM Zubair and G G Hamedani ldquo)e Kumaraswamy Mar-shal-Olkin family of distributionsrdquo Journal of the EgyptianMathematical Society vol 23 no 3 pp 546ndash557 2015

[10] A Z Afify G M Cordeiro H M Yousof A Alzaatreh andZ M Nofal ldquo)e Kumaraswamy transmuted-G family ofdistributions properties and applicationsrdquo Journal of DataScience vol 14 no 2 pp 245ndash270 2016

[11] Z M Nofal A Z Afify H M Yousof and G M Cordeiroldquo)e generalized transmuted-G family of distributionsrdquo

12 Mathematical Problems in Engineering

Communications in Statistics-5eory and Methods vol 46no 8 pp 4119ndash4136 2017

[12] M Chahkandi and M Ganjali ldquoOn some lifetime distribu-tions with decreasing failure raterdquo Computational Statisticsand Data Analysis vol 53 no 12 pp 4433ndash4440 2009

[13] W Barreto-Souza A H S Santos and G M Cordeiro ldquo)ebeta generalized exponential distributionrdquo Journal of Statis-tical Computation and Simulation vol 80 no 2 pp 159ndash1722010

[14] W Lu and D Shi ldquoA new compounding life distribution theWeibull-Poisson distributionrdquo Journal of Applied Statisticsvol 39 no 1 pp 21ndash38 2012

[15] D C Montgomery Statistical Quality Control Vol 7 WileyNew York NY USA 2009

[16] W H Woodall ldquo)e use of control charts in health-care andpublic-health surveillancerdquo Journal of Quality Technologyvol 38 no 2 pp 89ndash104 2006

[17] S-L Hwang J-T Lin G-F Liang Y-J Yau T-C Yenn andC-C Hsu ldquoApplication control chart concepts of designing apre-alarm system in the nuclear power plant control roomrdquoNuclear Engineering and Design vol 238 no 12 pp 3522ndash3527 2008

[18] Z Wang and R Liang ldquoDiscuss on applying SPC to qualitymanagement in university educationrdquo in Proceedings of the9th International Conference for Young Computer Scientistspp 2372ndash2375 Hunan China November 2008

[19] M Pastell and H Madsen ldquoApplication of CUSUM charts todetect lameness in a milking robotrdquo Expert Systems withApplications vol 35 no 4 pp 2032ndash2040 2008

[20] A Ivanova C Xiao and Y Tymofyeyev ldquoTwo-stage designsfor phase 2 dose-finding trialsrdquo Statistics in Medicine vol 31no 24 pp 2872ndash2881 2012

[21] S W Roberts ldquoControl chart tests based on geometricmoving averagesrdquo Technometrics vol 1 no 3 pp 239ndash2501959

[22] D Brook and D A Evans ldquoAn approach to the probabilitydistribution of CUSUM run lengthrdquo Biometrika vol 59 no 3pp 539ndash549 1972

[23] J M Lucas ldquoCombined Shewhart-CUSUM quality controlschemesrdquo Journal of Quality Technology vol 14 no 2pp 51ndash59 1982

[24] C M Borror J B Keats and D C Montgomery ldquoRobustnessof the time between events CUSUMrdquo International Journal ofProduction Research vol 41 no 15 pp 3435ndash3444 2003

[25] M B C Khoo ldquoA moving average control chart for moni-toring the fraction non-conformingrdquo Quality and ReliabilityEngineering International vol 20 no 6 pp 617ndash635 2004

[26] S Sukparungsee and A A Novikov ldquoAnalytical approxi-mations for detection of a change-point in case of light-taileddistributionsrdquo Journal of Quality Measurement and Analysisvol 4 no 2 pp 49ndash56 2008

[27] Y Areepong and A A Novikov ldquoMartingale approach toEWMA control chart for changes in Exponential distribu-tionrdquo Journal of Quality Measurement and Analysis vol 4pp 197ndash203 2008

[28] R Noorossana A A Fatahi P Dokouhaki andM Babakhani ldquoZIB-EWMA control chart for monitoringrare health eventsrdquo Journal of Mechanics in Medicine andBiology vol 11 no 04 pp 881ndash895 2011

[29] E Santiago and J Smith ldquoControl charts based on the ex-ponential distribution adapting runs rules for the tChartrdquoQuality Engineering vol 25 no 2 pp 85ndash96 2013

[30] L L Ho and R C Quinino ldquoAn attribute control chart formonitoring the variability of a processrdquo International Journalof Production Economics vol 145 no 1 pp 263ndash267 2013

[31] M Aslam M Azam and C-H Jun ldquoAcceptance samplingplans for multi-stage process based on time-truncated test forWeibull distributionrdquo 5e International Journal of AdvancedManufacturing Technology vol 79 no 9ndash12 pp 1779ndash17852015

[32] A C Rakitzis and P Castagliola ldquo)e effect of parameterestimation on the performance of one-sided Shewhart controlcharts for zero-inflated processesrdquo Communications in Sta-tistics - 5eory and Methods vol 45 no 14 pp 4194ndash42142016

[33] M Azam M N Aslam and C-H Jun ldquoAn EWMA controlchart for the exponential distribution using repetitive sam-pling planrdquo 2017

[34] M Azam L Ahmad M Aslam and C-H Jun ldquoAn attributecontrol chart using discriminant limits for monitoring processunder the Weibull distributionrdquo Production Engineeringvol 12 no 5 pp 659ndash665 2018

[35] Z Wu and J Jiao ldquoA control chart for monitoring processmean based on attribute inspectionrdquo International Journal ofProduction Research vol 46 no 15 pp 4331ndash4347 2008

[36] H Kim and S Lee ldquoOn the VSI CUSUM chart for countprocesses and its implementation with R package attrCU-SUMrdquo Industrial Engineering ampManagement Systems vol 17no 1 pp 91ndash101 2018

[37] K Mabude J C Malela-Majika M Aslam Z L Chong andS C Shongwe ldquoDistribution-free composite Shewhart-GWMA Mann-Whitney charts for monitoring the processlocationrdquo Quality and Reliability Engineering International2020

[38] M T Shah M Azam M Aslam and U Sherazi ldquoTimebetween events control charts for gamma distributionrdquoQuality and Reliability Engineering International vol 37no 2 pp 785ndash803 2020

[39] U Shaheen M Azam and M Aslam ldquoA control chart formonitoring the lognormal process variation using repetitivesamplingrdquo Quality and Reliability Engineering Internationalvol 36 no 3 pp 1028ndash1047 2020

[40] M Aslam P Jeyadurga S Balamurali M Azam and A AL-Marshadi ldquoEconomic determination of modified multipledependent state sampling plan under some lifetime distri-butionsrdquo Journal of Mathematics vol 2021 Article ID7470196 2021

[41] R H Berk N L Johnson S Kotz and N BalakrishnanldquoContinuous univariate distributionsrdquo Technometrics vol 2p 752 1996

[42] C J P Belisle ldquoConvergence theorems for a class of simulatedannealing algorithms on Rdrdquo Journal of Applied Probabilityvol 29 no 4 pp 885ndash895 1992

[43] A Henningsen and O Toomet ldquoMaxlik a package formaximum likelihood estimation in Rrdquo Computational Sta-tistics vol 26 pp 443ndash458 2011

Mathematical Problems in Engineering 13

Communications in Statistics-5eory and Methods vol 46no 8 pp 4119ndash4136 2017

[12] M Chahkandi and M Ganjali ldquoOn some lifetime distribu-tions with decreasing failure raterdquo Computational Statisticsand Data Analysis vol 53 no 12 pp 4433ndash4440 2009

[13] W Barreto-Souza A H S Santos and G M Cordeiro ldquo)ebeta generalized exponential distributionrdquo Journal of Statis-tical Computation and Simulation vol 80 no 2 pp 159ndash1722010

[14] W Lu and D Shi ldquoA new compounding life distribution theWeibull-Poisson distributionrdquo Journal of Applied Statisticsvol 39 no 1 pp 21ndash38 2012

[15] D C Montgomery Statistical Quality Control Vol 7 WileyNew York NY USA 2009

[16] W H Woodall ldquo)e use of control charts in health-care andpublic-health surveillancerdquo Journal of Quality Technologyvol 38 no 2 pp 89ndash104 2006

[17] S-L Hwang J-T Lin G-F Liang Y-J Yau T-C Yenn andC-C Hsu ldquoApplication control chart concepts of designing apre-alarm system in the nuclear power plant control roomrdquoNuclear Engineering and Design vol 238 no 12 pp 3522ndash3527 2008

[18] Z Wang and R Liang ldquoDiscuss on applying SPC to qualitymanagement in university educationrdquo in Proceedings of the9th International Conference for Young Computer Scientistspp 2372ndash2375 Hunan China November 2008

[19] M Pastell and H Madsen ldquoApplication of CUSUM charts todetect lameness in a milking robotrdquo Expert Systems withApplications vol 35 no 4 pp 2032ndash2040 2008

[20] A Ivanova C Xiao and Y Tymofyeyev ldquoTwo-stage designsfor phase 2 dose-finding trialsrdquo Statistics in Medicine vol 31no 24 pp 2872ndash2881 2012

[21] S W Roberts ldquoControl chart tests based on geometricmoving averagesrdquo Technometrics vol 1 no 3 pp 239ndash2501959

[22] D Brook and D A Evans ldquoAn approach to the probabilitydistribution of CUSUM run lengthrdquo Biometrika vol 59 no 3pp 539ndash549 1972

[23] J M Lucas ldquoCombined Shewhart-CUSUM quality controlschemesrdquo Journal of Quality Technology vol 14 no 2pp 51ndash59 1982

[24] C M Borror J B Keats and D C Montgomery ldquoRobustnessof the time between events CUSUMrdquo International Journal ofProduction Research vol 41 no 15 pp 3435ndash3444 2003

[25] M B C Khoo ldquoA moving average control chart for moni-toring the fraction non-conformingrdquo Quality and ReliabilityEngineering International vol 20 no 6 pp 617ndash635 2004

[26] S Sukparungsee and A A Novikov ldquoAnalytical approxi-mations for detection of a change-point in case of light-taileddistributionsrdquo Journal of Quality Measurement and Analysisvol 4 no 2 pp 49ndash56 2008

[27] Y Areepong and A A Novikov ldquoMartingale approach toEWMA control chart for changes in Exponential distribu-tionrdquo Journal of Quality Measurement and Analysis vol 4pp 197ndash203 2008

[28] R Noorossana A A Fatahi P Dokouhaki andM Babakhani ldquoZIB-EWMA control chart for monitoringrare health eventsrdquo Journal of Mechanics in Medicine andBiology vol 11 no 04 pp 881ndash895 2011

[29] E Santiago and J Smith ldquoControl charts based on the ex-ponential distribution adapting runs rules for the tChartrdquoQuality Engineering vol 25 no 2 pp 85ndash96 2013

[30] L L Ho and R C Quinino ldquoAn attribute control chart formonitoring the variability of a processrdquo International Journalof Production Economics vol 145 no 1 pp 263ndash267 2013

[31] M Aslam M Azam and C-H Jun ldquoAcceptance samplingplans for multi-stage process based on time-truncated test forWeibull distributionrdquo 5e International Journal of AdvancedManufacturing Technology vol 79 no 9ndash12 pp 1779ndash17852015

[32] A C Rakitzis and P Castagliola ldquo)e effect of parameterestimation on the performance of one-sided Shewhart controlcharts for zero-inflated processesrdquo Communications in Sta-tistics - 5eory and Methods vol 45 no 14 pp 4194ndash42142016

[33] M Azam M N Aslam and C-H Jun ldquoAn EWMA controlchart for the exponential distribution using repetitive sam-pling planrdquo 2017

[34] M Azam L Ahmad M Aslam and C-H Jun ldquoAn attributecontrol chart using discriminant limits for monitoring processunder the Weibull distributionrdquo Production Engineeringvol 12 no 5 pp 659ndash665 2018

[35] Z Wu and J Jiao ldquoA control chart for monitoring processmean based on attribute inspectionrdquo International Journal ofProduction Research vol 46 no 15 pp 4331ndash4347 2008

[36] H Kim and S Lee ldquoOn the VSI CUSUM chart for countprocesses and its implementation with R package attrCU-SUMrdquo Industrial Engineering ampManagement Systems vol 17no 1 pp 91ndash101 2018

[37] K Mabude J C Malela-Majika M Aslam Z L Chong andS C Shongwe ldquoDistribution-free composite Shewhart-GWMA Mann-Whitney charts for monitoring the processlocationrdquo Quality and Reliability Engineering International2020

[38] M T Shah M Azam M Aslam and U Sherazi ldquoTimebetween events control charts for gamma distributionrdquoQuality and Reliability Engineering International vol 37no 2 pp 785ndash803 2020

[39] U Shaheen M Azam and M Aslam ldquoA control chart formonitoring the lognormal process variation using repetitivesamplingrdquo Quality and Reliability Engineering Internationalvol 36 no 3 pp 1028ndash1047 2020

[40] M Aslam P Jeyadurga S Balamurali M Azam and A AL-Marshadi ldquoEconomic determination of modified multipledependent state sampling plan under some lifetime distri-butionsrdquo Journal of Mathematics vol 2021 Article ID7470196 2021

[41] R H Berk N L Johnson S Kotz and N BalakrishnanldquoContinuous univariate distributionsrdquo Technometrics vol 2p 752 1996

[42] C J P Belisle ldquoConvergence theorems for a class of simulatedannealing algorithms on Rdrdquo Journal of Applied Probabilityvol 29 no 4 pp 885ndash895 1992

[43] A Henningsen and O Toomet ldquoMaxlik a package formaximum likelihood estimation in Rrdquo Computational Sta-tistics vol 26 pp 443ndash458 2011

Mathematical Problems in Engineering 13