Submit manuscript...
eISSN: 2378-315X

Biometrics & Biostatistics International Journal

Research Article Volume 9 Issue 2

Time truncated control chart using log logistic distribution

Gadde Srinivasa Rao, Edwin Paul

Department of Mathematics and Statistics, The University of Dodoma, Tanzania

Correspondence: G. Srinivasa Rao, Department of mathematics and statistics, The University of Dodoma, Dodoma, Tanzania, PO. Box: 259, Tanzania

Received: March 29, 2020 | Published: April 30, 2020

Citation: Rao GS, Paul E. Time truncated control chart using log logistic distribution. Biom Biostat Int J. 2020;9(2):76-82. DOI: 10.15406/bbij.2020.09.00303

Download PDF

Abstract

In this article, a log logistic distribution considered to develop an attribute control chart for time truncated life tests with known or unknown shape parameter. The performance of the proposed chart is evaluated in terms of average run length (ARL) using the Monte Carlo simulation. The extensive tables are provided for the industrial use for various values of shape parameter, sample size, specified ARL and shift constants. The advantages of the proposed control chart are discussed over the existing truncated life test control charts. The performance of the proposed control chart is also studied using the simulated data sets for industrial purpose.

Keywords: Log logistic distribution, attribute control chart, truncated life test, average run length, simulation

Introduction

Control charts are considered as important tools when producer wants to produce goods or services of high–quality. These charts help producers to manufacture products based on specified limits by monitoring the quality beforehand.1 There are a number of control charts developed to monitor production process in different situations. One of the major characteristics of many control charts is that the production process should follow normal distribution. Ouyang et al.2 and Pearn and Wu3 they mentioned efficiency of process capability (PC) based on the production process which follows normally distributed processes. According to Aslam and Jun1 there are also other control charts which are developed based on non–normal distributions which are being used when the production process follows other distributions rather than normal. Rao4 developed a control chart for time truncated life tests using exponentiated half logistic distribution and Rao et al.5 constructed attribute control charts for the Dagum distribution under truncated life tests.

If a quality characteristics of the production process does not follow normal distribution and the experimenter developed a control chart based on the assumption that it follows normal distribution, it will led to a wrong result. A number of non–normal control chart have being developed by Al–Oraini and Rahim.,6; Amin et al.,7; Lin and Chou., 8; McCracken and Chakraborti,9; Ahmad et al.,10). On the other hand control charts are divided into variables or measurements and attributes.11 Control charts for variables monitor characteristics that can be measured and have a continuous scale whereas control charts for attributes are used to measure quality characteristics that are counted rather than measured, such as a fraction defective or nonconformities per unit of product.

One among control charts for attributes include np chart based on the study conducted by Rodrigues et al.12 and Epprecht et al.13 considered usual Shewhart np control charts are used for monitoring the number of non–conforming products rather than proportion as in p–charts. Several authors have conducted studies on how control charts for attributes are being used in several situations such as Epprecht et al.,13 Costa and Rahim,14 Hsu15,16, Wu et al.,17 Wu and Wang18 and Barbosa and Joekes.19 In a study conducted by Kantam and Rosaiah20 suggested sampling plan on life tests while the failure density model is half logistic distribution. Furthermore Kantam et al.21 considered acceptance sampling mainly on life tests while the failure density model of the products was a log–logistic distribution. It is also noted that Kantam et al.22 studied an economic reliability test plan using log–logistic distribution.

Based on the literature review made, it is noted that there is no study on log–logistic distribution based on truncated life tests. In this study we have developed attribute control chart of log–logistic distribution by incorporating the truncated life test.

Design of the control chart

Assume the failure time of a product follows a log logistic distribution in which its cumulative distribution function (cdf) is

F( t,σ,β )= ( t/σ ) β 1+ ( t/σ ) β ;t0,σ>0,β>1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAeadaqada qaaiaadshacaGGSaGaaGPaVlabeo8aZjaacYcacqaHYoGyaiaawIca caGLPaaacqGH9aqpdaWcaaqaamaabmaabaWaaSGbaeaacaWG0baaba Gaeq4WdmhaaaGaayjkaiaawMcaamaaCaaaleqabaGaeqOSdigaaaGc baGaaGymaiabgUcaRmaabmaabaWaaSGbaeaacaWG0baabaGaeq4Wdm haaaGaayjkaiaawMcaamaaCaaaleqabaGaeqOSdigaaOGaaGPaVdaa caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caGG7aGaaGPaVl aadshacqGHLjYScaaIWaGaaiilaiaaykW7caaMc8Uaeq4WdmNaeyOp a4JaaGimaiaacYcacaaMc8UaaGPaVlabek7aIjabg6da+iaaigdaaa a@6EC8@         (1)

whereas the pdf is given by

f( t;σ,β )= β σ ( t/σ ) β1 [ 1+ ( t/σ ) β ] 2 ;t>0,σ>0,β>1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgadaqada qaaiaadshacaGG7aGaeq4WdmNaaiilaiabek7aIbGaayjkaiaawMca aiabg2da9maalaaabaGaeqOSdigabaGaeq4WdmhaamaalaaabaWaae WaaeaadaWcgaqaaiaadshaaeaacqaHdpWCaaaacaGLOaGaayzkaaWa aWbaaSqabeaacqaHYoGycqGHsislcaaIXaaaaaGcbaWaamWaaeaaca aIXaGaey4kaSYaaeWaaeaadaWcgaqaaiaadshaaeaacqaHdpWCaaaa caGLOaGaayzkaaWaaWbaaSqabeaacqaHYoGyaaaakiaawUfacaGLDb aadaahaaWcbeqaaiaaikdaaaaaaOGaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8Uaai4oaiaadshacqGH+aGpcaaIWaGaaiilaiaayk W7caaMc8Uaeq4WdmNaeyOpa4JaaGimaiaacYcacaaMc8UaaGPaVlab ek7aIjabg6da+iaaigdaaaa@7199@ (2)

where β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabek7aIbaa@38AF@ is the shape parameter. The mean life of a product for a log–logistic distribution is given as:

μ=σΓ( 1+ 1 β )Γ( 1 1 β ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeY7aTjabg2 da9iabeo8aZjaaykW7cqqHtoWrdaqadaqaaiaaigdacqGHRaWkdaWc aaqaaiaaigdaaeaacqaHYoGyaaaacaGLOaGaayzkaaGaeu4KdC0aae WaaeaacaaIXaGaeyOeI0YaaSaaaeaacaaIXaaabaGaeqOSdigaaaGa ayjkaiaawMcaaaaa@4B17@ (3)

Let μ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeY7aTnaaBa aaleaacaaIWaaabeaaaaa@39AA@ be the center mean life when the production process is in control. The aim is to design a control chart for monitoring shift from center line by counting the number of failed items in a specified truncated time t 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadshadaWgaa WcbaGaaGimaaqabaaaaa@38ED@ . The probability that a product fails by time t 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadshadaWgaa WcbaGaaGimaaqabaaaaa@38ED@ is given by

p= ( t 0 /σ ) β 1+ ( t 0 /σ ) β 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadchacqGH9a qpdaWcaaqaamaabmaabaWaaSGbaeaacaWG0bWaaSbaaSqaaiaaicda aeqaaaGcbaGaeq4WdmhaaaGaayjkaiaawMcaamaaCaaaleqabaGaeq OSdigaaaGcbaGaaGymaiabgUcaRmaabmaabaWaaSGbaeaacaWG0bWa aSbaaSqaaiaaicdaaeqaaaGcbaGaeq4WdmhaaaGaayjkaiaawMcaam aaCaaaleqabaGaeqOSdigaaaaakmaaBaaaleaacaaIWaaabeaaaaa@49E2@ (4)

By specifying the truncation time t 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadshadaWgaa WcbaGaaGimaaqabaaaaa@38ED@ using a multiple in–control process mean in the cause of t 0 =a μ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadshadaWgaa WcbaGaaGimaaqabaGccqGH9aqpcaWGHbGaeqiVd02aaSbaaSqaaiaa icdaaeqaaaaa@3D7F@ where a is constant (termed as a truncated time constant) then the equation (4) above can be rewritten as

p 0 = ( a η 0 ) β 0 1+ ( a η 0 ) β 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadchadaWgaa WcbaGaaGimaaqabaGccqGH9aqpdaWcaaqaamaabmaabaGaamyyaiab eE7aOnaaBaaaleaacaaIWaaabeaaaOGaayjkaiaawMcaamaaCaaale qabaGaeqOSdi2aaSbaaWqaaiaaicdaaeqaaaaaaOqaaiaaigdacqGH RaWkdaqadaqaaiaadggacqaH3oaAdaWgaaWcbaGaaGimaaqabaaaki aawIcacaGLPaaadaahaaWcbeqaaiabek7aInaaBaaameaacaaIWaaa beaaaaaaaaaa@4B30@                                                                                                                   (5)

where μ 0 = σ 0 η 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeY7aTnaaBa aaleaacaaIWaaabeaakiabg2da9iabeo8aZnaaBaaaleaacaaIWaaa beaakiabeE7aOnaaBaaaleaacaaIWaaabeaaaaa@3FFF@ and hence σ 0 = μ 0 / η 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo8aZnaaBa aaleaacaaIWaaabeaakiabg2da9maalyaabaGaeqiVd02aaSbaaSqa aiaaicdaaeqaaaGcbaGaeq4TdG2aaSbaaSqaaiaaicdaaeqaaaaaaa a@4015@ with η 0 =Γ( 1+ 1 β 0 )Γ( 1 1 β 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeE7aOnaaBa aaleaacaaIWaaabeaakiabg2da9iabfo5ahnaabmaabaGaaGymaiab gUcaRmaalaaabaGaaGymaaqaaiabek7aInaaBaaaleaacaaIWaaabe aaaaaakiaawIcacaGLPaaacqqHtoWrdaqadaqaaiaaigdacqGHsisl daWcaaqaaiaaigdaaeaacqaHYoGydaWgaaWcbaGaaGimaaqabaaaaa GccaGLOaGaayzkaaaaaa@4A8F@ .

We have therefore proposed np control chart for a log–logistic distribution based on the number of failed items for each subgroup:

Step 1: Taking a sample of size n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad6gaaaa@3801@ from a certain production process. If we count the number of failures by considering the specified time t 0 =a μ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadshadaWgaa WcbaGaaGimaaqabaGccqGH9aqpcaWGHbGaeqiVd02aaSbaaSqaaiaa icdaaeqaaaaa@3D7F@ whereby a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadggaaaa@37F4@ is a constant and μ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeY7aTnaaBa aaleaacaaIWaaabeaaaaa@39AA@ is the target mean when the production process is in control.

Step 2: Conclude the process is out of control when D>UCL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadseacqGH+a GpcaWGvbGaam4qaiaadYeaaaa@3B52@ or D<LCL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadseacqGH8a apcaWGmbGaam4qaiaadYeaaaa@3B45@ and the production process said to in control when LCLDUCL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadYeacaWGdb GaamitaiabgsMiJkaadseacqGHKjYOcaWGvbGaam4qaiaadYeaaaa@401E@ .

It should be noted that the above chart is np MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad6gacaWGWb aaaa@38F6@ chart because number of failures has being used instead of proportion of failures. When the production process is in control the random variable D MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadseaaaa@37D7@ follows a binomial distribution with parameters n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad6gaaaa@3801@ and p 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadchadaWgaa WcbaGaaGimaaqabaaaaa@38E9@ . Thus the upper and lower control limits for the suggested np MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad6gacaWGWb aaaa@38F6@ chart will be:

UCL=n p 0 +L n p 0 ( 1 p 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwfacaWGdb Gaamitaiabg2da9iaad6gacaWGWbWaaSbaaSqaaiaaicdaaeqaaOGa ey4kaSIaamitamaakaaabaGaamOBaiaadchadaWgaaWcbaGaaGimaa qabaGcdaqadaqaaiaaigdacqGHsislcaWGWbWaaSbaaSqaaiaaicda aeqaaaGccaGLOaGaayzkaaaaleqaaaaa@471B@                                                                                                                                (6a)

LCL=max[ 0,n p 0 L n p 0 ( 1 p 0 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadYeacaWGdb Gaamitaiabg2da9iGac2gacaGGHbGaaiiEamaadmaabaGaaGimaiaa cYcacaaMc8UaamOBaiaadchadaWgaaWcbaGaaGimaaqabaGccqGHsi slcaWGmbWaaOaaaeaacaWGUbGaamiCamaaBaaaleaacaaIWaaabeaa kmaabmaabaGaaGymaiabgkHiTiaadchadaWgaaWcbaGaaGimaaqaba aakiaawIcacaGLPaaaaSqabaaakiaawUfacaGLDbaaaaa@4EE2@                                                                                                                   (6b)

Where L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadYeaaaa@37DF@ is the control constant or coefficient of the control limits to be determined. The probability p 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadchadaWgaa WcbaGaaGimaaqabaaaaa@38E9@ should be estimated when the process is in control from a preliminary sample, since its unknown. Thus, the UCL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwfacaWGdb Gaamitaaaa@3981@ and LCL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadYeacaWGdb Gaamitaaaa@3978@ limits to be used in practice will be:

UCL= D ¯ +L D ¯ ( 1 D ¯ /n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwfacaWGdb Gaamitaiabg2da9maanaaabaGaamiraaaacqGHRaWkcaWGmbWaaOaa aeaadaqdaaqaaiaadseaaaWaaeWaaeaacaaIXaGaeyOeI0YaaSGbae aadaqdaaqaaiaadseaaaaabaGaamOBaaaaaiaawIcacaGLPaaaaSqa baaaaa@431D@                                                                                                                                    (7a)

LCL=max[ 0, D ¯ L D ¯ ( 1 D ¯ /n ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadYeacaWGdb Gaamitaiabg2da9iGac2gacaGGHbGaaiiEamaadmaabaGaaGimaiaa cYcacaaMc8UaaGPaVpaanaaabaGaamiraaaacqGHsislcaWGmbWaaO aaaeaadaqdaaqaaiaadseaaaWaaeWaaeaacaaIXaGaeyOeI0YaaSGb aeaadaqdaaqaaiaadseaaaaabaGaamOBaaaaaiaawIcacaGLPaaaaS qabaaakiaawUfacaGLDbaaaaa@4C6F@                                                                                                                     (7b)

Where D ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaanaaabaGaam iraaaaaaa@37E8@ is the average number of failures in a batch or lot over a preliminary sample.

It should be noted that in this study we have considered control limits in the form of equation [6a&6b]. The aim is to investigate of the new control chart proposed by using average run length. The proportion that production process is declared to be in control when in fact it is real in control is given by

P in 0 =P( LCLDUCL| p 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadcfadaqhaa WcbaGaamyAaiaad6gaaeaacaaIWaaaaOGaeyypa0Jaamiuamaabmaa baGaamitaiaadoeacaWGmbGaeyizImQaamiraiabgsMiJkaadwfaca WGdbGaamitamaaeeaabaGaamiCamaaBaaaleaacaaIWaaabeaaaOGa ay5bSdaacaGLOaGaayzkaaaaaa@4AA2@   

= d=LCL+1 UCL ( n d ) P 0 d ( 1 P 0 ) nd MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabg2da9maaqa habaWaaeWaaqaabeqaaiaad6gaaeaacaWGKbaaaiaawIcacaGLPaaa aSqaaiaadsgacqGH9aqpcaWGmbGaam4qaiaadYeacqGHRaWkcaaIXa aabaGaamyvaiaadoeacaWGmbaaniabggHiLdGccaWGqbWaa0baaSqa aiaaicdaaeaacaWGKbaaaOWaaeWaaeaacaaIXaGaeyOeI0Iaamiuam aaBaaaleaacaaIWaaabeaaaOGaayjkaiaawMcaamaaCaaaleqabaGa amOBaiabgkHiTiaadsgaaaaaaa@50D0@

 = d=[ LCL ]+1 [ UCL ] ( n d ) ( ( a η 0 ) β 0 1+ ( a η 0 ) β 0 ) d { 1 ( a η 0 ) β 0 1+ ( a η 0 ) β 0 } nd MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaaqahabaWaae Waaqaabeqaaiaad6gaaeaacaWGKbaaaiaawIcacaGLPaaaaSqaaiaa dsgacqGH9aqpdaWadaqaaiaadYeacaWGdbGaamitaaGaay5waiaaw2 faaiabgUcaRiaaigdaaeaadaWadaqaaiaadwfacaWGdbGaamitaaGa ay5waiaaw2faaaqdcqGHris5aOWaaeWaaeaadaWcaaqaamaabmaaba GaamyyaiabeE7aOnaaBaaaleaacaaIWaaabeaaaOGaayjkaiaawMca amaaCaaaleqabaGaeqOSdi2aaSbaaWqaaiaaicdaaeqaaaaaaOqaai aaigdacqGHRaWkdaqadaqaaiaadggacqaH3oaAdaWgaaWcbaGaaGim aaqabaaakiaawIcacaGLPaaadaahaaWcbeqaaiabek7aInaaBaaame aacaaIWaaabeaaaaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaWG KbaaaOWaaiWaaeaacaaIXaGaeyOeI0YaaSaaaeaadaqadaqaaiaadg gacqaH3oaAdaWgaaWcbaGaaGimaaqabaaakiaawIcacaGLPaaadaah aaWcbeqaaiabek7aInaaBaaameaacaaIWaaabeaaaaaakeaacaaIXa Gaey4kaSYaaeWaaeaacaWGHbGaeq4TdG2aaSbaaSqaaiaaicdaaeqa aaGccaGLOaGaayzkaaWaaWbaaSqabeaacqaHYoGydaWgaaadbaGaaG imaaqabaaaaaaaaOGaay5Eaiaaw2haamaaCaaaleqabaGaamOBaiab gkHiTiaadsgaaaaaaa@750D@                                                                                              (8)

During evaluation of the sum above it should be noted that the value of d MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadsgaaaa@37F7@ should be 0 if LCL=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadYeacaWGdb Gaamitaiabg2da9iaaicdaaaa@3B38@ . Average run length (ARL) is usually used to evaluate the performance of the control chart. When the process is in control ARL is always given by

. AR L 0 = 1 1 P in 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeacaWGsb GaamitamaaBaaaleaacaaIWaaabeaakiabg2da9maalaaabaGaaGym aaqaaiaaigdacqGHsislcaWGqbWaa0baaSqaaiaadMgacaWGUbaaba GaaGimaaaaaaaaaa@4182@                                                                                                                                                      (9)

ARL when scale parameter is shifted

The process is declared to be out–of–control when the process is shifted to a new scale parameter σ 1 =c σ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo8aZnaaBa aaleaacaaIXaaabeaakiabg2da9iaadogacqaHdpWCdaWgaaWcbaGa aGimaaqabaaaaa@3E59@ , where c is a shift constant. In this case, the probability that an item is failed before the experiment time t 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadshalmaaBa aabaqcLbmacaaIWaaaleqaaaaa@3A26@ denoted by p1, is obtained by

p 1 =F( t 0 ; β 0 , σ 1 )=( ( t 0 / σ 1 ) β 0 1+ ( t 0 / σ 1 ) β 0 )=( ( a η 0 /c ) β 0 1+ ( a η 0 /c ) β 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadchadaWgaa WcbaGaaGymaaqabaGccqGH9aqpcaWGgbGaaiikaiaadshadaWgaaWc baGaaGimaaqabaGccaGG7aGaeqOSdi2aaSbaaSqaaiaaicdaaeqaaO GaaiilaiaaykW7cqaHdpWCdaWgaaWcbaGaaGymaaqabaGccaGGPaGa eyypa0ZaaeWaaeaadaWcaaqaamaabmaabaWaaSGbaeaacaWG0bWaaS baaSqaaiaaicdaaeqaaaGcbaGaeq4Wdm3aaSbaaSqaaiaaigdaaeqa aaaaaOGaayjkaiaawMcaamaaCaaaleqabaGaeqOSdi2aaSbaaWqaai aaicdaaeqaaaaaaOqaaiaaigdacqGHRaWkdaqadaqaamaalyaabaGa amiDamaaBaaaleaacaaIWaaabeaaaOqaaiabeo8aZnaaBaaaleaaca aIXaaabeaaaaaakiaawIcacaGLPaaadaahaaWcbeqaaiabek7aInaa BaaameaacaaIWaaabeaaaaaaaaGccaGLOaGaayzkaaGaeyypa0Zaae WaaeaadaWcaaqaamaabmaabaWaaSGbaeaacaWGHbGaeq4TdG2aaSba aSqaaiaaicdaaeqaaaGcbaGaam4yaaaaaiaawIcacaGLPaaadaahaa Wcbeqaaiabek7aInaaBaaameaacaaIWaaabeaaaaaakeaacaaIXaGa ey4kaSYaaeWaaeaadaWcgaqaaiaadggacqaH3oaAdaWgaaWcbaGaaG imaaqabaaakeaacaWGJbaaaaGaayjkaiaawMcaamaaCaaaleqabaGa eqOSdi2aaSbaaWqaaiaaicdaaeqaaaaaaaaakiaawIcacaGLPaaaaa a@7244@ . (10)

It should be noted that mean μ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeY7aTnaaBa aaleaacaaIWaaabeaaaaa@39AA@ corresponds to the probability p 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadchadaWgaa WcbaGaaGimaaqabaaaaa@38E9@ while when the process is out of control mean μ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeY7aTnaaBa aaleaacaaIXaaabeaaaaa@39AB@ corresponds to the probability p 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadchadaWgaa WcbaGaaGymaaqabaaaaa@38EA@ . Therefore the probability that, the process is said to be in control when the mean has shifted to μ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeY7aTnaaBa aaleaacaaIXaaabeaaaaa@39AB@ is given by

P in 1 =( LCLDUCL| p 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadcfadaqhaa WcbaGaamyAaiaad6gaaeaacaaIXaaaaOGaeyypa0ZaaeWaaeaacaWG mbGaam4qaiaadYeacqGHKjYOcaWGebGaeyizImQaamyvaiaadoeaca WGmbWaaqqaaeaacaWGWbWaaSbaaSqaaiaaigdaaeqaaaGccaGLhWoa aiaawIcacaGLPaaaaaa@49CF@

= d=LCL+1 UCL ( n d ) P 1 d ( 1 p 1 ) nd MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabg2da9maaqa habaWaaeWaaqaabeqaaiaad6gaaeaacaWGKbaaaiaawIcacaGLPaaa aSqaaiaadsgacqGH9aqpcaWGmbGaam4qaiaadYeacqGHRaWkcaaIXa aabaGaamyvaiaadoeacaWGmbaaniabggHiLdGccaWGqbWaa0baaSqa aiaaigdaaeaacaWGKbaaaOWaaeWaaeaacaaIXaGaeyOeI0IaamiCam aaBaaaleaacaaIXaaabeaaaOGaayjkaiaawMcaamaaCaaaleqabaGa amOBaiabgkHiTiaadsgaaaaaaa@50F2@                                                                                                                                   

d=[ LCL ]+1 [ UCL ] ( n d ) ( ( a η 0 /c ) β 0 1+ ( a η 0 /c ) β 0 ) d { 1 ( a η 0 /c ) β 0 1+ ( a η 0 /c ) β 0 } nd MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaaqahabaWaae Waaqaabeqaaiaad6gaaeaacaWGKbaaaiaawIcacaGLPaaaaSqaaiaa dsgacqGH9aqpdaWadaqaaiaadYeacaWGdbGaamitaaGaay5waiaaw2 faaiabgUcaRiaaigdaaeaadaWadaqaaiaadwfacaWGdbGaamitaaGa ay5waiaaw2faaaqdcqGHris5aOWaaeWaaeaadaWcaaqaamaabmaaba WaaSGbaeaacaWGHbGaeq4TdG2aaSbaaSqaaiaaicdaaeqaaaGcbaGa am4yaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiabek7aInaaBaaame aacaaIWaaabeaaaaaakeaacaaIXaGaey4kaSYaaeWaaeaadaWcgaqa aiaadggacqaH3oaAdaWgaaWcbaGaaGimaaqabaaakeaacaWGJbaaaa GaayjkaiaawMcaamaaCaaaleqabaGaeqOSdi2aaSbaaWqaaiaaicda aeqaaaaaaaaakiaawIcacaGLPaaadaahaaWcbeqaaiaadsgaaaGcda GadaqaaiaaigdacqGHsisldaWcaaqaamaabmaabaWaaSGbaeaacaWG HbGaeq4TdG2aaSbaaSqaaiaaicdaaeqaaaGcbaGaam4yaaaaaiaawI cacaGLPaaadaahaaWcbeqaaiabek7aInaaBaaameaacaaIWaaabeaa aaaakeaacaaIXaGaey4kaSYaaeWaaeaadaWcgaqaaiaadggacqaH3o aAdaWgaaWcbaGaaGimaaqabaaakeaacaWGJbaaaaGaayjkaiaawMca amaaCaaaleqabaGaeqOSdi2aaSbaaWqaaiaaicdaaeqaaaaaaaaaki aawUhacaGL9baadaahaaWcbeqaaiaad6gacqGHsislcaWGKbaaaaaa @7905@ (11)

and when the process is out of control ARL is given by

. AR L 1 = 1 1 P in 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeacaWGsb GaamitamaaBaaaleaacaaIXaaabeaakiabg2da9maalaaabaGaaGym aaqaaiaaigdacqGHsislcaWGqbWaa0baaSqaaiaadMgacaWGUbaaba GaaGymaaaaaaaaaa@4184@                                                                                                                                                       (12)

We used the following algorithm to complete the tables for the proposed control chart.

  1. Specify the values of ARL, say r0 and shape parameters β 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabek7aInaaBa aaleaacaaIWaaabeaaaaa@3995@ .
  2. Determine the values of control chart parameters and sample size n for which the ARL0 from Equation (9) is close to r0.
  3. Use the values of control chart parameters obtained in step 2 to find ARL1according to shift constant c using Equation (12).

We determined the control chart parameters and ARL1 for various values of β 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabek7aInaaBa aaleaacaaIWaaabeaaaaa@3995@ , r0 and n and placed in Tables 1–4.

β 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHYoGydaWgaa WcbaGaaGimaaqabaaaaa@3A4A@

1.5

2

2.5

3

n

25

25

26

26

LCL

9

9

5

7

UCL

22

22

19

21

a

0.6296

0.8727

0.7238

0.8845

L

2.8072

2.8175

2.8582

2.8815

c

ARL1

ARL1

ARL1

ARL1

1.00

200.01

200.05

200.03

200.03

0.95

188.09

174.69

157.04

174.87

0.90

140.55

109.30

73.46

72.63

0.85

92.64

61.31

31.91

28.49

0.80

58.48

34.27

14.69

12.34

0.75

36.71

19.69

7.37

6.04

0.70

23.26

11.75

4.09

3.37

0.65

14.97

7.31

2.52

2.14

0.60

9.83

4.76

1.74

1.54

0.55

6.61

3.26

1.34

1.24

0.50

4.56

2.36

1.14

1.10

0.40

2.40

1.45

1.01

1.01

0.30

1.48

1.12

1.00

1.00

0.20

1.11

1.02

1.00

1.00

0.10

1.01

1.00

1.00

1.00

Table 1 ARLs for the proposed chart for r0=200 when scale parameter is shifted

β 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHYoGydaWgaa WcbaGaaGimaaqabaaaaa@3A4A@

1.5

2

2.5

3

n

23

25

23

25

LCL

7

4

2

4

UCL

20

18

15

18

a

0.5723

0.5732

0.6126

0.7711

L

2.8991

2.8934

2.8183

2.8923

c

ARL1

ARL1

ARL1

ARL1

1.00

250.01

250.05

250.04

250.03

0.95

235.18

221.78

196.08

176.32

0.90

175.36

128.02

94.33

67.43

0.85

114.89

63.86

41.05

25.45

0.80

71.86

31.74

18.57

10.70

0.75

44.60

16.38

9.04

5.14

0.70

27.90

8.91

4.82

2.85

0.65

17.70

5.16

2.85

1.83

0.60

11.43

3.21

1.88

1.35

0.55

7.55

2.17

1.40

1.14

0.50

5.12

1.59

1.16

1.04

0.40

2.59

1.12

1.01

1.00

0.30

1.55

1.01

1.00

1.00

0.20

1.13

1.00

1.00

1.00

0.10

1.01

1.00

1.00

1.00

Table 2 ARLs for the proposed chart for r0=250 when scale parameter is shifted

β 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHYoGydaWgaa WcbaGaaGimaaqabaaaaa@3A4A@

1.5

2

2.5

3

n

28

24

27

24

LCL

8

4

6

4

UCL

23

18

21

18

a

0.5222

0.5979

0.7655

0.7931

L

2.9063

2.9645

2.9613

2.9635

c

ARL1

ARL1

ARL1

ARL1

1.00

300.00

300.04

300.02

300.02

0.95

228.91

268.57

281.24

214.41

0.90

144.20

156.30

131.34

82.71

0.85

84.94

78.36

53.82

31.28

0.80

49.58

39.02

23.25

13.08

0.75

29.27

20.10

10.95

6.19

0.70

17.64

10.86

5.69

3.35

0.65

10.90

6.22

3.29

2.09

0.60

6.94

3.81

2.13

1.49

0.55

4.58

2.51

1.54

1.20

0.50

3.15

1.79

1.24

1.07

0.40

1.73

1.18

1.03

1.00

0.30

1.19

1.02

1.00

1.00

0.20

1.02

1.00

1.00

1.00

0.10

1.00

1.00

1.00

1.00

Table 3 ARLs for the proposed chart for r0=300 when scale parameter is shifted

β 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHYoGydaWgaa WcbaGaaGimaaqabaaaaa@3A4A@

1.5

2

2.5

3

n

13

30

17

23

LCL

2

5

0

5

UCL

12

21

11

19

a

0.5651

0.5603

0.552

0.8671

L

2.8593

2.9793

2.9876

2.9981

c

ARL1

ARL1

ARL1

ARL1

1.00

370.03

370.05

370.00

370.05

0.95

314.20

323.07

316.82

329.65

0.90

240.56

166.74

175.21

136.60

0.85

173.40

74.71

82.38

52.22

0.80

121.40

33.99

38.34

21.69

0.75

83.98

16.31

18.53

10.02

0.70

57.90

8.37

9.48

5.20

0.65

39.94

4.66

5.21

3.04

0.60

27.64

2.84

3.11

2.01

0.55

19.22

1.91

2.05

1.49

0.50

13.45

1.43

1.50

1.22

0.40

6.76

1.06

1.08

1.03

0.30

3.57

1.00

1.00

1.00

0.20

2.03

1.00

1.00

1.00

0.10

1.29

1.00

1.00

1.00

Table 4 ARLs for the proposed chart for r0=370 when scale parameter is shifted

From these tables, we note that a rapidly decreasing trend in ARLs as the shift constant decreases. Same is observed for various parametric combinations that we considered in this article.

ARL when shape parameter is shifted

In this section, we will present the designing of the proposed chart when the shape is shifted due to some extraneous factors. Let us assume that the shape parameter is shifted to β 1 =f β 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdi2aaS baaSqaaiaaigdaaeqaaOGaeyypa0JaamOzaiabek7aInaaBaaaleaa caaIWaaabeaaaaa@3D00@ for a shift constant f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGMbaaaa@366E@ . In this case, the probability that an item is failed before the experiment time t 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDamaaBa aaleaacaaIWaaabeaaaaa@37D5@ , denoted by p 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGWbWaaSbaaS qaaiaaikdaaeqaaaaa@3760@ , is obtained by

p 2 =F( t 0 ; β 1 , σ 0 )=( ( t 0 / σ 0 ) β 1 1+ ( t 0 / σ 0 ) β 1 )=( ( a η 1 ) β 1 1+ ( a η 1 ) β 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadchadaWgaa WcbaGaaGOmaaqabaGccqGH9aqpcaWGgbGaaiikaiaadshadaWgaaWc baGaaGimaaqabaGccaGG7aGaeqOSdi2aaSbaaSqaaiaaigdaaeqaaO GaaiilaiaaykW7cqaHdpWCdaWgaaWcbaGaaGimaaqabaGccaGGPaGa eyypa0ZaaeWaaeaadaWcaaqaamaabmaabaWaaSGbaeaacaWG0bWaaS baaSqaaiaaicdaaeqaaaGcbaGaeq4Wdm3aaSbaaSqaaiaaicdaaeqa aaaaaOGaayjkaiaawMcaamaaCaaaleqabaGaeqOSdi2aaSbaaWqaai aaigdaaeqaaaaaaOqaaiaaigdacqGHRaWkdaqadaqaamaalyaabaGa amiDamaaBaaaleaacaaIWaaabeaaaOqaaiabeo8aZnaaBaaaleaaca aIWaaabeaaaaaakiaawIcacaGLPaaadaahaaWcbeqaaiabek7aInaa BaaameaacaaIXaaabeaaaaaaaaGccaGLOaGaayzkaaGaeyypa0Zaae WaaeaadaWcaaqaamaabmaabaGaamyyaiabeE7aOnaaBaaaleaacaaI XaaabeaaaOGaayjkaiaawMcaamaaCaaaleqabaGaeqOSdi2aaSbaaW qaaiaaigdaaeqaaaaaaOqaaiaaigdacqGHRaWkdaqadaqaaiaadgga cqaH3oaAdaWgaaWcbaGaaGymaaqabaaakiaawIcacaGLPaaadaahaa Wcbeqaaiabek7aInaaBaaameaacaaIXaaabeaaaaaaaaGccaGLOaGa ayzkaaaaaa@704D@ (13)

where η 1 =Γ( 1+ 1 β 1 )Γ( 1 1 β 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeE7aOnaaBa aaleaacaaIXaaabeaakiabg2da9iabfo5ahnaabmaabaGaaGymaiab gUcaRmaalaaabaGaaGymaaqaaiabek7aInaaBaaaleaacaaIXaaabe aaaaaakiaawIcacaGLPaaacqqHtoWrdaqadaqaaiaaigdacqGHsisl daWcaaqaaiaaigdaaeaacqaHYoGydaWgaaWcbaGaaGymaaqabaaaaa GccaGLOaGaayzkaaaaaa@4A92@ .

The probability of in control for the shifted process is given as follows:

P in 2 =P( LCLDUCL| p 2 )= d=[ LCL ]+1 [ UCL ] ( n d ) ( ( a η 1 ) β 1 1+ ( a η 1 ) β 1 ) d { 1 ( a η 1 ) β 1 1+ ( a η 1 ) β 1 } nd MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadcfadaqhaa WcbaGaamyAaiaad6gaaeaacaaIYaaaaOGaeyypa0Jaamiuamaabmaa baGaamitaiaadoeacaWGmbGaeyizImQaamiraiabgsMiJkaadwfaca WGdbGaamitaiaacYhacaWGWbWaaSbaaSqaaiaaikdaaeqaaaGccaGL OaGaayzkaaGaeyypa0ZaaabCaeaadaqadaabaeqabaGaamOBaaqaai aadsgaaaGaayjkaiaawMcaaaWcbaGaamizaiabg2da9maadmaabaGa amitaiaadoeacaWGmbaacaGLBbGaayzxaaGaey4kaSIaaGymaaqaam aadmaabaGaamyvaiaadoeacaWGmbaacaGLBbGaayzxaaaaniabggHi LdGcdaqadaqaamaalaaabaWaaeWaaeaacaWGHbGaeq4TdG2aaSbaaS qaaiaaigdaaeqaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacqaHYoGy daWgaaadbaGaaGymaaqabaaaaaGcbaGaaGymaiabgUcaRmaabmaaba GaamyyaiabeE7aOnaaBaaaleaacaaIXaaabeaaaOGaayjkaiaawMca amaaCaaaleqabaGaeqOSdi2aaSbaaWqaaiaaigdaaeqaaaaaaaaaki aawIcacaGLPaaadaahaaWcbeqaaiaadsgaaaGcdaGadaqaaiaaigda cqGHsisldaWcaaqaamaabmaabaGaamyyaiabeE7aOnaaBaaaleaaca aIXaaabeaaaOGaayjkaiaawMcaamaaCaaaleqabaGaeqOSdi2aaSba aWqaaiaaigdaaeqaaaaaaOqaaiaaigdacqGHRaWkdaqadaqaaiaadg gacqaH3oaAdaWgaaWcbaGaaGymaaqabaaakiaawIcacaGLPaaadaah aaWcbeqaaiabek7aInaaBaaameaacaaIXaaabeaaaaaaaaGccaGL7b GaayzFaaWaaWbaaSqabeaacaWGUbGaeyOeI0Iaamizaaaaaaa@891F@ (14)

The efficiency of the control chart is measured using the ARL. The ARL for the shifted process is given as follows:

AR L 2 = 1 1 p in 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeacaWGsb GaamitamaaBaaaleaacaaIYaaabeaakiabg2da9maalaaabaGaaGym aaqaaiaaigdacqGHsislcaWGWbWaa0baaSqaaiaadMgacaWGUbaaba GaaGOmaaaaaaaaaa@41A6@ (15)

We determined the control chart parameters and ARL2 for various values of β 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabek7aInaaBa aaleaacaaIWaaabeaaaaa@3995@ , r0 and n and placed in Tables 5–8.

n

36

21

26

17

18

LCL

12

5

9

5

6

UCL

29

18

23

16

17

a

0.7633

0.7253

0.8466

0.8787

0.9422

L

2.971

3.02

3.017

2.987

2.866

f

ARL1

ARL1

ARL1

ARL1

ARL1

1.0

300.05

300.03

300.05

300.02

300.00

1.1

144.00

148.18

153.29

171.87

200.53

1.2

53.97

59.03

75.19

99.86

127.77

1.3

24.26

30.12

41.38

62.82

85.67

1.4

12.83

17.33

25.26

42.47

60.98

1.5

7.69

10.95

16.72

30.41

45.65

1.6

5.08

7.45

11.79

22.78

35.59

1.7

3.63

5.38

8.74

17.69

28.65

1.8

2.76

4.09

6.74

14.14

23.68

1.9

2.21

3.24

5.38

11.57

19.98

2.0

1.85

2.66

4.41

9.67

17.16

2.5

1.16

1.44

2.20

4.84

9.58

3.0

1.02

1.12

1.49

3.04

6.38

3.5

1.00

1.03

1.21

2.18

4.70

Table 5 ARLs for the proposed chart for r0=300 when shape parameter is shifted when=2

n

31

24

24

35

18

LCL

1

1

4

10

3

UCL

15

14

18

27

15

a

0.5878

0.6459

0.8092

0.8872

0.8771

L

2.9290

2.8670

3.0190

2.9440

2.9370

f

ARL1

ARL1

ARL1

ARL1

ARL1

1.0

300.02

300.02

300.03

300.02

300.02

1.1

242.63

247.56

252.66

257.17

294.45

1.2

85.33

106.27

143.13

154.28

223.16

1.3

34.59

48.32

78.67

89.34

154.51

1.4

16.71

24.92

45.88

54.44

106.55

1.5

9.31

14.34

28.55

35.18

75.23

1.6

5.82

9.04

18.82

23.94

54.66

1.7

3.99

6.14

13.04

17.03

40.82

1.8

2.96

4.45

9.42

12.57

31.24

1.9

2.33

3.40

7.07

9.58

24.43

2.0

1.92

2.72

5.48

7.51

19.48

2.5

1.18

1.41

2.25

3.05

7.83

3.0

1.04

1.12

1.42

1.79

4.11

3.5

1.01

1.03

1.14

1.32

2.59

Table 6 ARLs for the proposed chart for r0=300 when shape parameter is shifted when=3

n

17

20

24

22

17

LCL

0

4

8

8

6

UCL

11

17

22

21

17

a

0.4291

0.6909

0.8438

0.9213

0.9867

L

3.4100

3.0580

3.0960

3.1400

3.1020

f

ARL1

ARL1

ARL1

ARL1

ARL1

1.0

370.00

370.05

370.01

370.03

370.04

1.1

157.05

157.01

157.87

178.64

202.24

1.2

59.85

66.56

76.70

98.85

127.82

1.3

27.73

32.80

42.70

61.36

89.08

1.4

15.06

18.34

26.37

41.48

66.52

1.5

9.23

11.32

17.61

29.88

52.26

1.6

6.22

7.57

12.50

22.59

42.65

1.7

4.51

5.39

9.30

17.73

35.85

1.8

3.48

4.05

7.20

14.34

30.85

1.9

2.80

3.18

5.75

11.88

27.05

2.0

2.35

2.60

4.72

10.05

24.08

2.5

1.41

1.40

2.34

5.32

15.72

3.0

1.15

1.11

1.57

3.47

11.91

3.5

1.06

1.03

1.25

2.55

9.75

Table 7 ARLs for the proposed chart for r0=370 when shape parameter is shifted when=2

n

42

30

29

34

35

LCL

3

5

7

10

12

UCL

20

21

23

27

29

a

0.5997

0.7595

0.8493

0.8982

0.9521

L

2.966

2.979

3.002

3.077

3.061

f

ARL1

ARL1

ARL1

ARL1

ARL1

1.0

370.03

370.04

370.01

370.04

370.03

1.1

183.21

184.26

194.68

261.96

275.94

1.2

52.23

78.73

100.45

151.78

182.43

1.3

19.26

37.66

56.16

89.61

122.16

1.4

8.90

20.24

33.98

56.15

85.37

1.5

4.93

11.99

21.96

37.24

62.27

1.6

3.14

7.71

14.99

25.93

47.12

1.7

2.24

5.31

10.72

18.80

36.76

1.8

1.75

3.88

7.97

14.10

29.42

1.9

1.46

2.99

6.13

10.89

24.05

2.0

1.29

2.40

4.86

8.63

20.01

2.5

1.02

1.28

2.14

3.58

9.67

3.0

1.00

1.05

1.39

2.06

5.74

3.5

1.00

1.01

1.13

1.47

3.85

Table 8 ARLs for the proposed chart for r0=370 when shape parameter is shifted when=3

From these tables, we noticed that the decreasing trend in ARLs as the shift constant f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgaaaa@37F9@ increases. We observed the same trend for other combination of parameters we considered.

Industrial application of proposed chart

The industrial application of the proposed control chart can be implemented as follows: Presume that the lifetime of an electronic equipment follows the log logistic distribution with shape parameter β 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabek7aInaaBa aaleaacaaIWaaabeaaaaa@3995@ =3. Assume that the target average life of electronic equipment is μ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeY7aTnaaBa aaleaacaaIWaaabeaaaaa@39AA@ = 1000 hours and r0= 370. Then from Equation (5) we have p0 = 0.5355. Also, from Table 4 we obtain the sample size of n = 23, a = 0.8671, L = 2.9981, LCL = 5 and UCL = 19. Thus the inspection time t 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadshadaWgaa WcbaGaaGimaaqabaaaaa@38ED@ =867 hours. Therefore, the proposed control chart works as follows:

Step 1. Take a sample of size 23 at each subgroup and put them on the life test during 867 hours. Plot the number of failed items (D) during the test.

Step 2. Declare the process as in–control if 5≤ D ≤ 19 otherwise process as out–of–control.

Simulation study

To demonstrate the performance of the proposed control chart methodology, the following steps depicts the generation of data using simulation from log logistic distribution and constructing the control chart:

Step 1: Choose a subgroup sample size n.

Step 2: Generate log logistic random variable X of size n with scale parameter σ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo8aZbaa@38D1@ =1, shape parameters β 0 =2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabek7aInaaBa aaleaacaaIWaaabeaakiabg2da9iaaikdaaaa@3B61@ .

Step 3: Obtain the chart statistic, the number of failures D for each subgroup.

Step 4: Repeat step 1 to step 3 until desired number of sample groups (m=20) are attained.

Step 5: Construct the control limits for the proposed chart.

Step 6: Plot all statistics D against their sample groups.

For this design, the first 20 samples of subgroup size 24 are generated from log logistic distribution with in–control parameters σ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo8aZbaa@38D1@ =1, β 0 =2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabek7aInaaBa aaleaacaaIWaaabeaakiabg2da9iaaikdaaaa@3B61@ and the second set of the 20 samples of subgroup size 24 are from log logistic distribution with parameters σ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo8aZbaa@38D1@ =0.75, β 0 =2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabek7aInaaBa aaleaacaaIWaaabeaakiabg2da9iaaikdaaaa@3B61@ (i.e. out–of–control situation having a shift of c=0.75). From Table 3, when ARL at 300 and in–control parameters we find control chart coefficient L is 2.9645, a is 0.5979 and n is 24. The life test termination time be t0=0.5979x0.46868=0.2802.

The values of D for in–control and shifted cases for each subgroup are reported as follows:

In–control: 2, 1, 1, 3, 2, 2, 1, 1, 2, 1, 2, 3, 1, 2, 3, 1, 2, 1, 3, 2.

 Shifted : 2, 2, 3, 2, 4, 6, 1, 2, 3, 5, 5, 3, 3, 2, 1, 3, 7, 2, 3, 2.

The average of failure D ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqadseagaqeaa aa@37EF@ is 1.6, using Eq. (7) the control limits for the proposed control chart is found as UCL=5.2227 and LCL=0. Figure 1 is the proposed control chart based on simulated data, along with chart statistics and control limits, can detect an out–of–control process after 26 samples, i.e. after the 6th sample of the shifted process as ARL1 is estimated to be 20.1 for a shift of 0.75. Thus the proposed chart efficiently detects the shift in the process.

Figure 1 The proposed control chart for simulated data.

Conclusion

This paper proposed a new np control chart assuming that the lifetime of the manufactured goods follows to log logistic distribution. The chart constants and extensive tables are given for the industrial use. Through simulated data the methodology has explained. We notice the decreasing trend in ARLs values as the shift constant increases. The recommended control chart can be used in the electronic appliances industries for mentoring of non–conforming products. The proposed control chart can be extended for some other non normal distributions as a future research.

Acknowledgments

The authors would like to thank the editor and the anonymous reviewers for their precious comments and suggestions which helped to improve the quality of the manuscript

Conflicts of interest

No potential conflict of interest was reported by the authors.

References

  1. Aslam, M, Jun CH. Attribute control charts for the weibull distribution under truncated life tests. Quality Engineering. 2015;27(3):283–288.
  2. Ouyang L–Y, Hsu C–H, Yang C–M. A new process capability analysis chart approach on the chip resistor quality management. Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture, 2013;227(7):1075–1082.
  3. Pearn WL, Wu C. Production quality and yield assurance for processes with multiple independent characteristics. European Journal of Operational Research. 2005;173(2):637–647.
  4. Rao GS. A control chart for time truncated life tests using exponentiated half logistic distribution. Applied Mathematics & Information Sciences: An International Journal. 2018;12(1):125–131.
  5. Rao GS, Fulment AK, Josephat PK. Attribute control charts for the Dagum distribution under truncated life tests. Life Cycle Reliability and Safety Engineering. 2019;8(4):329–335.
  6. Al–Oraini HA, Rahim MA. Economic statistical design of control charts X for systems with gamma (λ, 2) in–control times. Journal of Applied Statistics. 2003;30:397–409.
  7. Amin R, Reynolds Jr. MR, Bakir ST. Nonparametric quality control charts based on the sign statistic. Communications in Statistics. 1995;24(6):1597–1623.
  8. Lin YC, Chou CY. Non–normality and the variable parameters X bar control charts. European Journal of Operational Research. 2007;176:361–373.
  9. McCracken AK, Chakraborti S. Control charts for joint monitoring of mean and variance: An overview. Quality Technology and Quantitative Management. 2013;10(1):17–36.
  10. Ahmad S, Riaz M, Abbasi SA. et al. On efficient median control charting. Journal of the Chinese Institute of Engineers. 2014; 37(3):358–375.
  11. Montgomery DC. Introduction to Statistical Quality Control, John Wiley & Sons, Sixth Edition. 2009.
  12. Rodrigues AD, Epprecht EK, De Magalhaes MS. Double–sampling control charts for attributes. Journal of Applied Statistics. 2011;38(1):87–112.
  13. Epprecht EK, Costa AFB, Mendes FCT. Adaptive control charts for attributes. IIE Transactions. 2003;35(6):567–582.
  14. Costa AFB, Rahim MA. Joint X bar and R charts with two stage samplings. Quality and Reliability Engineering International. 2004; 20(7):699–708.
  15. Hsu LF. Note on Design of Double– and Triple–Sampling Control Charts Using Genetic Algorithms. International Journal Production Research. 2004;42(5):1043–1047.
  16. Hsu LF. Note on Construction of Double Sampling s–Control Charts for Agile Manufacturing. Quality and Reliability Engineering International. 2007;23(2):269–272.
  17. Wu Z, Luo H, Zhang X. Optimal np control chart with curtailment. European Journal of Operational Research. 2006;174(3): 1723–1741.
  18. Wu Z, Wang Q. An np control chart using double inspections. Journal of Applied Statistics. 2007;34(7):843–855.
  19. Barbosa EP, Joekes S. An improved attribute control chart for monitoring non–conforming proportion in high quality processes. Control Engineering Practice. 2013;21(4):407–412.
  20. Kantam RRL, Rosaiah K. Half logistic distribution in acceptance sampling based on life tests. Indian Association for Productivity Quality & Reliability Transactions. 1998;23(2):117–125.
  21. Kantam, RRL, Rosaiah K, Rao GS. Acceptance sampling based on life tests: log–logistic model. Journal of Applied Statistics. 2001; 28(1):121–128.
  22. Kantam RRL, Rao GS, Sriram B. An economic reliability test plan: Log–Logistic Distribution. Journal of Applied Statistics. 2006;33(3):291–296.
Creative Commons Attribution License

©2020 Rao, et al . This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and build upon your work non-commercially.