# Attribute Control Charts Based on TLT for Length-Biased Weighted Lomax Distribution

Authors: Srinivasa Rao Gadde; Amer Ibrahim Al-Omari
Journal: Journal of Mathematics
 (2022)

---

## Abstract

The length-biased weighted Lomax distribution (LBWD) is a novel continuous of two parameters lifetime distribution. In this article, we introduced an attribute control chart (CC) for the lifetime of a product that follows the LBWD in terms of the number of failure items before a fixed time period is investigated. The performance of the suggested charts is investigated using in term of the average run length (ARL). The necessary tables of shift sizes and various sample sizes are offered for numerous values of the distribution parameters as well as specified ARL and shift constants. Some numerical examples are discussed for various scheme parameters to study the performance of the new LBWD attribute control charts.


---

## Body

## 1. Introduction

Quality control (QC) is a substantially continuous process in different fields to introduce either an excellent service for the customers or an improved product. Quality control charts or control charts for abbreviation is a tool used in the manufacturing to maintain the feature of the product in terms of its characteristics as its hardness, colors, size, and age, etc.In the control charts technique, there are two limits, the first one is the upper control limit (UCL) and the other is Lower control limit (LCL), which are set to test whether the control status is falling in the range between the two borders. The mean value for in control process is usually represented by the central line (CL). The control status is in a stable situation if it is within the limits, otherwise out of control, if it is out of the range.There are various valuable CC to keep the production under control, incorporating lower and upper control limits. Two well-known types of the control charts which are CC for attributes and the CC for variables. The variables CC provide good information regard the process and comprise minimum sample sizes because it employs quantitative data. The use of attribute CC is more flexible as compared to the variables chart according to its simple evaluation. Different types of the attribute CC are well known in the literature as thenp chart, u chart and the c chart.Chen et al. [1] suggested exponentially weighted moving average chart (EWMA) for observing variance and mean simultaneously. Bakir [2] investigated distribution-free Shewhart QC chart in terms of signed-ranks. Nichols and Padgett [3] investigated bootstrap CC for Weibull distribution. Wu and Jiao [4] studied a control CC for checking process of mean in terms attribute assessment. Wu et al. [5] recommended np CC to screen a process average by attribute assessment. Haq and Al-Omari [6] suggested a Shewhart CC for observing process of mean. Aslam and Jun [7] constructed attribute control charts for the Weibull distribution. Aslam et al. [8] investigated the CC for the truncated life tests (TLT) time for Pareto model of the 2nd kind. Aslam [9] developed attribute control chart for the Weibull distribution using multiple dependent state sampling. Nanthakumar and Kavitha [10] proposed attribute CC for the inverse Rayleigh model in terms of type-I censoring. Zhou et al. [11] considered joint-adaptive np CC with multiple dependent state sampling method. Quinino et al. [12] offered attribute inspection CC for process mean monitoring. Khan et al. [13] developed a variable control chart based on Weibull distribution. Balamurali and Jeyadurga [14] recommended an attribute np CC for observing mean life by means of multiple deferred state sampling under TLT. Quinino et al. [15] suggested a CC for monitoring the process of mean under scrutinizing attributes using control limits of the commonly X-bar chart. Adeoti and Rao [16] developed moving average CC for the Rayleigh and inverse Rayleigh models in means of time truncated life test. Adeoti and Ogundipe [17]. Studied control chart for the generalized exponential distribution.To best of knowledge this work is the first which proposed an attribute CC for the lifetime of a product that tracks the length-biased weighted Lomax distribution. In this article, new attribute CC under TLT are proposed using the length-biased weighted Lomax model. The coming part of this paper is prepared as follows. In Section2, we define the LBWLD with its main properties. The advised CC is defined in detail in Section 3. Numerical illustrations are assumed in Section 4. Industrial application is demonstrated in Section 5. Monitoring of the proposed CC is carried out in Section 6. The paper is ended in Section 7 with some conclusions.


## 2. The LBWLD

Most of the industrial data reported are sometimes cannot be treated as random sample from the parent distribution for instance these may happen due to some events not possible to observe, smash up to the original data. As an outcome, weighted models are formulated in such difficulties to record the data corresponding to some weighted functions. In most of these situations, the statistical distribution that takes place in industrial application could be recorded with unequal probability. More commonly, when the sampling apparatus, chooses items with probability proportional to some appraise of the item size, consequential distribution is known as size-biased. These types of various situations are well addressed by Rao [18]. The weighted distributions are more flexible model for modeling data take place more frequently in textile industry, reliability, packing industry, biomedical sciences, lifetime analysis, ecology and industrial production process for more details refer Patil et al. [19].In 1974 Lomax suggested the Lomax distribution to be a well-known distribution in the literature and commonly used in life testing and reliability. Due to the importance of the Lomax distribution many researchers generalized some new models related to it, for example Abdul-Moniem and Abdel-Hameed [20] proposed the exponentiated Lomax distribution. Abdul-Moniem and Diab [21] offered length-biased weighted exponentiated Lomax distribution. Al-Omari et al. [22] recommended acceptance sampling plans with TLT for the LBWLD. Kilany [23] proposed the weighted Lomax distribution. Ahmad et al. [24] generalized the Lomax distribution using the weighted distribution generation idea and suggested the LBWLD with a probability density function(1)fLBWLDx,α,λ=αα−1λ2x1+xλ−α+1,x>0,λ,α>0,where α and are the shape and scale parameters, and the corresponding cumulative distribution function (cdf) is(2)FLBWLDx,α,λ=1−1+xλ−α1+αxλ,x>0,λ,α>0.Figure1 represents the pdf of the LBWLD for selected parameters values. It is reflected that the distribution is symmetric to the right which is suitable for mode king some real life data.Figure 1
The pdf Plots of the LBWLD withl = 9.1 and at different α.The reliability, hazard, and cumulative hazard functions of the LBWLD, respectively, are(3)RLBWLDx,α,λ=1+xλ−α1+αxλ,HLBWLDx,α,λ=fLBWLDx,α,λSLBWLDx,α,λ=αα−1λ2x1+xλ−11+αxλ−1,RHLBWLDx,α,λ=fLBWLDx,α,λFLBWLDx,α,λ=αα−1/λ2x1+x/λ−α+11−1+x/λ−α1+αx/λ.Figures2 and 3 include the reliability and hazard functions of the LBWLD for various values of the distribution parameters.Figure 2
Plots of the reliability function of LBWLD for selected parameters.Figure 3
Plots of the hazard function of LBWLD for selected parameters.It can be seen that the LBWLD has a decreasing reliability function in time for the parameters considered in this study. But the hazard function is increasing up to some times then start decreasing. These various flexibility behaviors of the reliability and hazard functions of the LBWLD give the distribution more superiety to be considered in different real life situations.The mean of the LBWLD distribution is(4)EXLBWLD=μ=2λα−2.For more about the LBWLD and its modifications, see Ahmad et al. [24].


## 3. The Suggested Control Chart

We suggest a newnp CC based on time TLT as proposed by Aslam and Jun [7]:Step 1: choose a sample of sizem from the lot process and testing them. The number of failures dented by (D) is counted by the identified inspection time t0=βμ0 where μ0 is the quality parameter assuming that the process is in control and β is a constant.Step 2: assert that the process as out of control whenD>UCL or D<LCL, while it is in control


### 3.1. The Process Is In-Control

It is well known that the number of defectives is binomially distributed with parametersp0 and n. The control limits for np charts are defined as(5)UCL=np0+knp01−p0,LCL=max0,np0+knp01−p0,where p0 is the probability of unsuccessful item earlier the experiment time t0 when the process is in control, and k is the control limit constant to be identified. However, we can say the process is in control when μ0=μ (or the distribution parameters α=α0 and λ=λ0). Therefore, from (2), the probability p0 is obtained as follows:(6)p0=FLBWLDt0;λ0,α0=1−1+2βα0−2−α01+2α0βα0−2.LetD¯ is the mean of failures over the subgroups from the sample. If the p0 is unknown, the control limits for practical application can be defined as(7)UCL=D¯+kD¯1−D¯n,(8)LCL=max0,D¯−kD¯1−D¯n.The probability of announcing that the process is in control for the suggested CC is defined as(9)pIC0=PLCL≤D≤UCL|p0=∑z=LCL+1UCLnzp0z1−p0n−z=∑z=LCL+1UCLnz1−1+2βα0−2−α01+2α0βα0−2z1+2βα0−2−α01+2α0βα0−2n−z.The performance of the suggested control chart can be investigated by its average control length (ARL) which is in the case of in-control process is defined as:(10)ARL0=11−pIC0.


### 3.2. Out-of-Control ARLs When Scale Parameter Is Shifted

The process is out-of-control if the process scale parameter is shifted fromλ1=cλ0, where c is a shift constant. Hence, the probability that an item is failed earlier the experiment time t0 denoted by p1 is(11)p1=FLBWLDt0;λ1,α0=1−1+2βcα0−2−α01+2α0βcα0−2.The probability that the process is in-control for the shifted process is given by(12)pIC1=PLCL≤D≤UCL|p1=∑z=LCL+1UCLnz1−1+2βcα0−2−α01+2α0βcα0−2z1+2βcα0−2−α01+2α0βcα0−2n−z.In this case, the ARL for the shifted process is(13)ARL1=11−pIC1.


### 3.3. Out-of-Control ARLs When Shape Parameter Is Shifted

The offered chart is investigated here when the shape is shifted due to some extrinsic factors. Assume that the shape parameter is shifted toα1=dα0 where d is shift constant. Therefore, the probability that an item is failed earlier the experiment time t0, denoted by p2, is(14)p2=FLBWLDt0;λ0,α1=1−1+2βdα0−2−dα01+2dα0βdα0−2.The probability that the process declared as in-control when shape parameter shifted can be written as(15)pIC2=PLCL≤D≤UCL|p2=∑z=LCL+1UCLnz1−1+2βdα0−2−dα01+2dα0βdα0−2z1+2βdα0−2−dα01+2dα0βdα0−2n−z.The ARL for the shifted process when the shape parameter is shifted as follows:(16)ARL1=11−pIC2.


### 3.4. Out-of-Control When Both Scale and Shape Parameters Are Shifted

In this subsection, we will develop the designing of the planned chart when both the scale and shape parameters are shifted due to some extrinsic factors. Let us suppose that the process scale parameter is shifted fromλ1=cλ0, and the shape parameter is shifted to α1=dα0. In this case, the probability that an item is failed before the experiment time t0, denoted by p3 is(17)p3=FLBWLDt0;λ1,α1=1−1+2βcdα0−2−dα01+2dα0βcdα0−2.When both parameters are shifted, then the probability that the process is declared as in-control is(18)pIC3=PLCL≤D≤UCL|p3=∑z=.LCL+1UCLnz1−1+2βcdα0−2−dα01+2dα0βcdα0−2z1+2βcdα0−2−dα01+2dα0βcdα0−2n−z.The ARL for the shifted process when both the scale and shape are shifted as follows:(19)ARL1=11−pIC3.We used the following algorithm to obtain the tables for the offered control chart.(1)
Determine the values of ARL, sayr0, and shape parameters α0, respectively(2)
Obtain the standards of control chart parametersk, β and n (sample size) under which the ARL0 given in equation (9) is close to r0, i.e., ARL0≥r0(3)
After obtaining the values in Step 2, determine theARL1 according to shift constant c based on equation (12), shift constant d by equation (15), and both shift constants c and d by equation (18)We determined the control chart parameters andARL1 for various values of α0, r0 and n and given in Tables 1–4 for shifting of scale parameter; Tables 5–7 for shifting of shape parameter; Tables 8–11 for shifting of both scale and shape parameters.Table 1
ARL values of the proposed chart forr0=200 when scale parameter is shifted.
α2.533.54β0.56500.63100.72000.8720K2.1312.1102.7302.626N561917LCL0055UCL561716CARL1ARL1ARL1ARL11.00200.04200.00200.00200.031.05161.64155.09132.46129.921.10132.52122.6387.7285.451.15110.0798.6759.6257.941.2092.5080.6341.8340.631.2578.5766.8230.3029.431.3067.3956.0822.6021.961.3558.3247.6217.3116.841.4050.8940.8513.5913.221.4544.7335.3810.9010.611.5039.5930.918.918.691.6031.6024.166.286.131.7025.7919.394.694.591.8021.4415.933.673.601.9018.1313.352.992.942.0015.5411.382.522.482.508.516.191.481.473.005.614.151.191.183.504.153.141.081.074.003.302.561.031.03Table 2
ARL values of the proposed chart forr0=250 when scale parameter is shifted.
α2.533.54β0.68100.97501.17101.0830K2.87102.6302.5782.636N20191830LCL88814UCL20191828CARL1ARL1ARL11.00250.01250.02250.03250.021.05164.69158.44154.83144.971.10112.36104.60100.2583.321.1579.1171.6267.5449.971.2057.3050.6847.1531.561.2542.5936.9333.9920.951.3032.3927.6425.2114.551.3525.1621.1919.2010.521.4019.9316.6114.967.891.4516.0613.2811.916.111.5013.1610.819.674.871.609.227.516.713.341.706.795.524.932.491.805.214.253.811.981.904.153.413.061.662.003.412.822.551.452.501.791.561.461.073.001.311.201.161.013.501.131.081.061.004.001.061.031.021.00Table 3
ARL values of the proposed chart forr0=300 when scale parameter is shifted.
α2.533.54β0.82900.93200.99800.9800K2.80902.6782.9982.866N25253443LCL12111620UCL24243238CARL1ARL1ARL1ARL11.00300.04300.03300.05300.011.05228.72185.38158.21152.741.10162.83116.3688.0477.901.15114.0275.5352.0642.541.2080.5450.8632.6025.001.2557.9735.4721.4915.711.3042.6525.5514.8310.481.3532.0718.9610.677.371.4024.6414.457.965.431.4519.3011.286.144.171.5015.409.014.883.321.6010.306.093.322.311.707.304.402.461.771.805.433.361.961.471.904.222.691.641.292.003.392.241.431.182.501.691.311.071.013.001.241.091.011.003.501.091.031.001.004.001.031.011.001.00Table 4
ARL values of the proposed chart forr0=370 when scale parameter is shifted.
α2.533.54β0.88800.69200.89101.0940K2.76202.96702.86702.7780N24474132LCL12191815UCL24393630CARL1ARL1ARL1ARL11.00370.04370.03370.04370.001.05235.64201.67200.59199.721.10155.73106.48105.54104.851.15106.4359.2458.3156.531.2074.9635.1234.1933.401.2554.2722.1221.1720.391.3040.2814.7013.7513.181.3530.5910.269.298.141.4023.717.476.505.921.4518.745.664.683.791.5015.074.433.453.331.6010.212.971.991.861.707.312.191.211.201.805.481.751.171.141.904.281.481.141.112.003.461.311.131.112.501.731.131.011.013.001.261.111.011.013.501.101.071.011.004.001.041.001.001.00Table 5
The ARLs of the proposed chart forr0=370 when the shape parameter is shifted from α0=2.5.
N202429817LCL5121315UCL182427816β0.4590.8880.7160.5860.609K3.0633.1182.9272.7582.854DARL1ARL1ARL1ARL1ARL11.00370.02370.04370.04370.01370.051.05142.49153.10174.87193.86234.571.1067.5281.7082.92120.75129.661.1538.6651.0746.5084.0078.721.2025.2235.4729.7262.9952.751.2518.0226.5220.8649.8438.171.3013.7520.9115.6841.0429.261.3511.0117.1712.4034.8323.441.409.1514.5410.1930.2719.441.457.8312.618.6426.8016.551.506.8511.157.5024.1014.411.605.539.115.9620.1911.471.704.697.785.0017.529.591.804.116.854.3415.618.301.903.706.163.8814.177.372.003.395.653.5313.066.672.502.584.252.649.944.843.002.243.642.268.514.063.502.063.302.067.713.644.001.943.081.947.193.37Table 6
The ARLs of the proposed chart forr0=370 when the shape parameter is shifted from α0=3.0.
N4738473848LCL1919272229UCL3935453846β0.6920.9711.0971.1921.196K3.0432.9983.1283.1853.172DARL1ARL1ARL1ARL1ARL11.00370.03370.03370.01370.01370.051.05201.38261.33220.92238.62236.321.10114.87181.04146.32168.95163.331.1572.19130.39104.81127.86120.701.2049.2898.3479.62101.6493.971.2535.8977.2763.2683.8576.181.3027.4962.8052.0471.2163.741.3521.9252.4944.0161.8754.701.4018.0444.8838.0554.7547.901.4515.2439.1133.5049.1942.641.5013.1534.6229.9344.7438.481.6010.2928.1624.7738.1332.381.708.4623.8221.2433.4928.161.807.2220.7318.7230.0825.091.906.3418.4416.8327.4822.782.005.6816.6915.3725.4520.982.503.9711.9111.3319.6115.913.003.279.819.5216.8913.583.502.898.658.5015.3312.264.002.667.927.8614.3211.42Table 7
The ARLs of the proposed chart forr0=370 when the shape parameter is shifted from α0=3.5.
N3446562741LCL614201018UCL2334422536β0.4950.6350.6890.8570.891K3.0773.0463.0023.0823.098DARL1ARL1ARL1ARL1ARL11.00370.04370.03370.05370.02370.041.05345.07335.32341.71270.01272.551.10274.74256.29258.19205.60202.801.15209.75188.51186.06162.67155.601.20161.52140.80136.24132.90123.331.25127.38108.33103.05111.49100.651.30103.1085.9180.5695.6084.221.3585.4870.0364.8883.4671.971.4072.3858.4453.6073.9862.611.4562.4049.7645.2566.4255.291.5054.6443.1138.9160.2949.461.6043.5333.7530.0951.0140.831.7036.1027.6024.3944.3834.841.8030.8723.3420.4739.4730.491.9027.0320.2617.6635.6927.212.0024.1217.9415.5732.7224.672.5016.3211.8710.1624.1517.573.0013.009.357.9620.1514.383.5011.218.016.7917.8612.604.0010.107.196.0916.3911.47Table 8
The ARLs of the proposed chart forr0=370 when both scale and shape parameter are shifted when α0=2.5.
Shape parameter shifted (d)Scale parameter shifted (c) at β=0.609, k = 2.854, n = 17, LCL = 5 UCL = 161.01.11.21.31.41.51.61.71.81.92.01.0370.05258.43144.7681.4348.6430.9720.8714.7810.928.376.621.1129.6663.0134.0120.2513.089.056.625.074.043.332.821.252.7526.6315.249.656.624.853.753.022.532.181.921.329.2615.459.276.154.413.372.702.261.951.731.571.419.4410.656.644.563.382.662.201.891.671.511.401.514.418.145.233.692.812.271.921.671.511.381.301.611.476.654.373.162.462.021.741.541.401.301.231.79.595.683.812.812.221.851.611.451.331.251.191.88.305.013.422.562.051.731.531.381.281.211.161.97.374.523.132.371.931.651.461.331.251.181.142.06.674.152.912.231.831.581.411.301.221.161.12Table 9
The ARLs of the proposed chart forr0=370 when both scale and shape parameter are shifted when α0=3.0.
Shape parameter shifted (d)Scale parameter shifted (c) at β=0.971, k = 2.998, n = 38, LCL = 19, UCL = 351.01.11.21.31.41.51.61.71.81.92.01.0370.03139.4551.0422.2911.426.684.353.092.361.911.621.1181.0456.3321.7710.335.783.682.612.001.651.421.281.298.3431.0212.716.433.842.611.961.591.371.241.151.362.8020.488.804.692.952.101.651.391.241.151.091.444.8815.116.763.752.461.821.481.281.171.101.061.534.6211.985.543.182.151.641.371.211.121.071.041.628.169.984.752.811.951.521.291.171.091.051.031.723.828.624.202.541.811.441.241.141.071.041.021.820.737.643.802.351.701.381.211.111.061.031.021.918.446.913.502.201.621.331.181.101.051.031.012.016.696.343.262.091.561.291.161.081.041.021.01Table 10
The ARLs of the proposed chart forr0=370 when both scale and shape shifted parameter are shifted when α0=3.5.
Shape parameter shifted (d)Scale parameter shifted (c) at β=0.857, k = 3.082, n = 27, LCL = 10, UCL = 251.01.11.21.31.41.51.61.71.81.92.01.0370.02121.2447.6322.3612.147.404.963.582.762.231.891.1205.6067.4327.6713.657.794.993.512.652.131.791.561.2132.9044.6118.989.745.783.842.802.191.811.561.391.395.6032.8514.417.634.673.202.391.921.621.431.301.473.9825.9711.686.343.982.802.141.751.501.341.241.560.2921.559.905.503.522.521.961.631.421.281.191.651.0118.538.664.903.202.331.841.551.361.241.161.744.3816.357.774.462.962.181.741.481.321.211.141.839.4714.727.094.132.772.071.671.431.281.191.131.935.6913.456.563.862.621.981.611.391.261.171.112.032.7212.456.133.652.501.911.571.361.241.151.10Table 11
ARLs for the proposed chart forr0=370 when both scale and shape parameter are shifted when α0=4.0.
Shape parameter shifted (d)Scale parameter shifted (c) atβ=0.680, k = 3.096, n = 46, LCL = 14, UCL = 341.01.11.21.31.41.51.61.71.81.92.01.0370.03134.2640.2115.717.664.442.952.171.731.471.301.1275.1370.8322.739.665.103.182.251.751.461.291.181.2179.4144.4015.236.913.882.551.891.531.321.201.121.3123.1531.2811.345.433.192.191.681.401.241.151.091.490.5123.859.064.522.771.961.551.321.191.111.071.570.4019.237.593.932.481.801.461.261.151.091.051.657.2216.156.583.512.271.691.391.221.131.071.041.748.1113.985.863.202.121.611.341.191.111.061.031.841.5412.385.322.972.001.541.301.171.101.051.031.936.6211.164.902.791.911.491.271.151.081.051.032.032.8410.214.572.641.841.451.251.141.081.041.02From the tables, we noticed the following:(1)
There is a rapid decreasing tendency in ARL1 as the shift constants (either c, or d, or both) decreases.(2)
From Tables1–4, it is clear that when the shape parameter α increases ARL1 values are decreases for other chart parameters fixed. Also, if the constant multiplier β increases, the ARL1 values decreases when other chart parameters are fixed.(3)
From Tables5–7, it is understandable that when the shape parameter α increases, the ARL1 values are also increases for other chart parameters fixed.(4)
From Tables8–11, it is interesting to observe that when both scale and shape parameters are shifted, the ARL1 values decrease. More specifically, for shifting of scale parameter, ARL1 values rapid decreasing, whereas when the shape parameter is shifted, the ARL1 values decrease slow tendency. In addition, when the shape parameter α increases, the ARL1 values are also increases for other chart parameters fixed.


## 3.1. The Process Is In-Control

It is well known that the number of defectives is binomially distributed with parametersp0 and n. The control limits for np charts are defined as(5)UCL=np0+knp01−p0,LCL=max0,np0+knp01−p0,where p0 is the probability of unsuccessful item earlier the experiment time t0 when the process is in control, and k is the control limit constant to be identified. However, we can say the process is in control when μ0=μ (or the distribution parameters α=α0 and λ=λ0). Therefore, from (2), the probability p0 is obtained as follows:(6)p0=FLBWLDt0;λ0,α0=1−1+2βα0−2−α01+2α0βα0−2.LetD¯ is the mean of failures over the subgroups from the sample. If the p0 is unknown, the control limits for practical application can be defined as(7)UCL=D¯+kD¯1−D¯n,(8)LCL=max0,D¯−kD¯1−D¯n.The probability of announcing that the process is in control for the suggested CC is defined as(9)pIC0=PLCL≤D≤UCL|p0=∑z=LCL+1UCLnzp0z1−p0n−z=∑z=LCL+1UCLnz1−1+2βα0−2−α01+2α0βα0−2z1+2βα0−2−α01+2α0βα0−2n−z.The performance of the suggested control chart can be investigated by its average control length (ARL) which is in the case of in-control process is defined as:(10)ARL0=11−pIC0.


## 3.2. Out-of-Control ARLs When Scale Parameter Is Shifted

The process is out-of-control if the process scale parameter is shifted fromλ1=cλ0, where c is a shift constant. Hence, the probability that an item is failed earlier the experiment time t0 denoted by p1 is(11)p1=FLBWLDt0;λ1,α0=1−1+2βcα0−2−α01+2α0βcα0−2.The probability that the process is in-control for the shifted process is given by(12)pIC1=PLCL≤D≤UCL|p1=∑z=LCL+1UCLnz1−1+2βcα0−2−α01+2α0βcα0−2z1+2βcα0−2−α01+2α0βcα0−2n−z.In this case, the ARL for the shifted process is(13)ARL1=11−pIC1.


## 3.3. Out-of-Control ARLs When Shape Parameter Is Shifted

The offered chart is investigated here when the shape is shifted due to some extrinsic factors. Assume that the shape parameter is shifted toα1=dα0 where d is shift constant. Therefore, the probability that an item is failed earlier the experiment time t0, denoted by p2, is(14)p2=FLBWLDt0;λ0,α1=1−1+2βdα0−2−dα01+2dα0βdα0−2.The probability that the process declared as in-control when shape parameter shifted can be written as(15)pIC2=PLCL≤D≤UCL|p2=∑z=LCL+1UCLnz1−1+2βdα0−2−dα01+2dα0βdα0−2z1+2βdα0−2−dα01+2dα0βdα0−2n−z.The ARL for the shifted process when the shape parameter is shifted as follows:(16)ARL1=11−pIC2.


## 3.4. Out-of-Control When Both Scale and Shape Parameters Are Shifted

In this subsection, we will develop the designing of the planned chart when both the scale and shape parameters are shifted due to some extrinsic factors. Let us suppose that the process scale parameter is shifted fromλ1=cλ0, and the shape parameter is shifted to α1=dα0. In this case, the probability that an item is failed before the experiment time t0, denoted by p3 is(17)p3=FLBWLDt0;λ1,α1=1−1+2βcdα0−2−dα01+2dα0βcdα0−2.When both parameters are shifted, then the probability that the process is declared as in-control is(18)pIC3=PLCL≤D≤UCL|p3=∑z=.LCL+1UCLnz1−1+2βcdα0−2−dα01+2dα0βcdα0−2z1+2βcdα0−2−dα01+2dα0βcdα0−2n−z.The ARL for the shifted process when both the scale and shape are shifted as follows:(19)ARL1=11−pIC3.We used the following algorithm to obtain the tables for the offered control chart.(1)
Determine the values of ARL, sayr0, and shape parameters α0, respectively(2)
Obtain the standards of control chart parametersk, β and n (sample size) under which the ARL0 given in equation (9) is close to r0, i.e., ARL0≥r0(3)
After obtaining the values in Step 2, determine theARL1 according to shift constant c based on equation (12), shift constant d by equation (15), and both shift constants c and d by equation (18)We determined the control chart parameters andARL1 for various values of α0, r0 and n and given in Tables 1–4 for shifting of scale parameter; Tables 5–7 for shifting of shape parameter; Tables 8–11 for shifting of both scale and shape parameters.Table 1
ARL values of the proposed chart forr0=200 when scale parameter is shifted.
α2.533.54β0.56500.63100.72000.8720K2.1312.1102.7302.626N561917LCL0055UCL561716CARL1ARL1ARL1ARL11.00200.04200.00200.00200.031.05161.64155.09132.46129.921.10132.52122.6387.7285.451.15110.0798.6759.6257.941.2092.5080.6341.8340.631.2578.5766.8230.3029.431.3067.3956.0822.6021.961.3558.3247.6217.3116.841.4050.8940.8513.5913.221.4544.7335.3810.9010.611.5039.5930.918.918.691.6031.6024.166.286.131.7025.7919.394.694.591.8021.4415.933.673.601.9018.1313.352.992.942.0015.5411.382.522.482.508.516.191.481.473.005.614.151.191.183.504.153.141.081.074.003.302.561.031.03Table 2
ARL values of the proposed chart forr0=250 when scale parameter is shifted.
α2.533.54β0.68100.97501.17101.0830K2.87102.6302.5782.636N20191830LCL88814UCL20191828CARL1ARL1ARL11.00250.01250.02250.03250.021.05164.69158.44154.83144.971.10112.36104.60100.2583.321.1579.1171.6267.5449.971.2057.3050.6847.1531.561.2542.5936.9333.9920.951.3032.3927.6425.2114.551.3525.1621.1919.2010.521.4019.9316.6114.967.891.4516.0613.2811.916.111.5013.1610.819.674.871.609.227.516.713.341.706.795.524.932.491.805.214.253.811.981.904.153.413.061.662.003.412.822.551.452.501.791.561.461.073.001.311.201.161.013.501.131.081.061.004.001.061.031.021.00Table 3
ARL values of the proposed chart forr0=300 when scale parameter is shifted.
α2.533.54β0.82900.93200.99800.9800K2.80902.6782.9982.866N25253443LCL12111620UCL24243238CARL1ARL1ARL1ARL11.00300.04300.03300.05300.011.05228.72185.38158.21152.741.10162.83116.3688.0477.901.15114.0275.5352.0642.541.2080.5450.8632.6025.001.2557.9735.4721.4915.711.3042.6525.5514.8310.481.3532.0718.9610.677.371.4024.6414.457.965.431.4519.3011.286.144.171.5015.409.014.883.321.6010.306.093.322.311.707.304.402.461.771.805.433.361.961.471.904.222.691.641.292.003.392.241.431.182.501.691.311.071.013.001.241.091.011.003.501.091.031.001.004.001.031.011.001.00Table 4
ARL values of the proposed chart forr0=370 when scale parameter is shifted.
α2.533.54β0.88800.69200.89101.0940K2.76202.96702.86702.7780N24474132LCL12191815UCL24393630CARL1ARL1ARL1ARL11.00370.04370.03370.04370.001.05235.64201.67200.59199.721.10155.73106.48105.54104.851.15106.4359.2458.3156.531.2074.9635.1234.1933.401.2554.2722.1221.1720.391.3040.2814.7013.7513.181.3530.5910.269.298.141.4023.717.476.505.921.4518.745.664.683.791.5015.074.433.453.331.6010.212.971.991.861.707.312.191.211.201.805.481.751.171.141.904.281.481.141.112.003.461.311.131.112.501.731.131.011.013.001.261.111.011.013.501.101.071.011.004.001.041.001.001.00Table 5
The ARLs of the proposed chart forr0=370 when the shape parameter is shifted from α0=2.5.
N202429817LCL5121315UCL182427816β0.4590.8880.7160.5860.609K3.0633.1182.9272.7582.854DARL1ARL1ARL1ARL1ARL11.00370.02370.04370.04370.01370.051.05142.49153.10174.87193.86234.571.1067.5281.7082.92120.75129.661.1538.6651.0746.5084.0078.721.2025.2235.4729.7262.9952.751.2518.0226.5220.8649.8438.171.3013.7520.9115.6841.0429.261.3511.0117.1712.4034.8323.441.409.1514.5410.1930.2719.441.457.8312.618.6426.8016.551.506.8511.157.5024.1014.411.605.539.115.9620.1911.471.704.697.785.0017.529.591.804.116.854.3415.618.301.903.706.163.8814.177.372.003.395.653.5313.066.672.502.584.252.649.944.843.002.243.642.268.514.063.502.063.302.067.713.644.001.943.081.947.193.37Table 6
The ARLs of the proposed chart forr0=370 when the shape parameter is shifted from α0=3.0.
N4738473848LCL1919272229UCL3935453846β0.6920.9711.0971.1921.196K3.0432.9983.1283.1853.172DARL1ARL1ARL1ARL1ARL11.00370.03370.03370.01370.01370.051.05201.38261.33220.92238.62236.321.10114.87181.04146.32168.95163.331.1572.19130.39104.81127.86120.701.2049.2898.3479.62101.6493.971.2535.8977.2763.2683.8576.181.3027.4962.8052.0471.2163.741.3521.9252.4944.0161.8754.701.4018.0444.8838.0554.7547.901.4515.2439.1133.5049.1942.641.5013.1534.6229.9344.7438.481.6010.2928.1624.7738.1332.381.708.4623.8221.2433.4928.161.807.2220.7318.7230.0825.091.906.3418.4416.8327.4822.782.005.6816.6915.3725.4520.982.503.9711.9111.3319.6115.913.003.279.819.5216.8913.583.502.898.658.5015.3312.264.002.667.927.8614.3211.42Table 7
The ARLs of the proposed chart forr0=370 when the shape parameter is shifted from α0=3.5.
N3446562741LCL614201018UCL2334422536β0.4950.6350.6890.8570.891K3.0773.0463.0023.0823.098DARL1ARL1ARL1ARL1ARL11.00370.04370.03370.05370.02370.041.05345.07335.32341.71270.01272.551.10274.74256.29258.19205.60202.801.15209.75188.51186.06162.67155.601.20161.52140.80136.24132.90123.331.25127.38108.33103.05111.49100.651.30103.1085.9180.5695.6084.221.3585.4870.0364.8883.4671.971.4072.3858.4453.6073.9862.611.4562.4049.7645.2566.4255.291.5054.6443.1138.9160.2949.461.6043.5333.7530.0951.0140.831.7036.1027.6024.3944.3834.841.8030.8723.3420.4739.4730.491.9027.0320.2617.6635.6927.212.0024.1217.9415.5732.7224.672.5016.3211.8710.1624.1517.573.0013.009.357.9620.1514.383.5011.218.016.7917.8612.604.0010.107.196.0916.3911.47Table 8
The ARLs of the proposed chart forr0=370 when both scale and shape parameter are shifted when α0=2.5.
Shape parameter shifted (d)Scale parameter shifted (c) at β=0.609, k = 2.854, n = 17, LCL = 5 UCL = 161.01.11.21.31.41.51.61.71.81.92.01.0370.05258.43144.7681.4348.6430.9720.8714.7810.928.376.621.1129.6663.0134.0120.2513.089.056.625.074.043.332.821.252.7526.6315.249.656.624.853.753.022.532.181.921.329.2615.459.276.154.413.372.702.261.951.731.571.419.4410.656.644.563.382.662.201.891.671.511.401.514.418.145.233.692.812.271.921.671.511.381.301.611.476.654.373.162.462.021.741.541.401.301.231.79.595.683.812.812.221.851.611.451.331.251.191.88.305.013.422.562.051.731.531.381.281.211.161.97.374.523.132.371.931.651.461.331.251.181.142.06.674.152.912.231.831.581.411.301.221.161.12Table 9
The ARLs of the proposed chart forr0=370 when both scale and shape parameter are shifted when α0=3.0.
Shape parameter shifted (d)Scale parameter shifted (c) at β=0.971, k = 2.998, n = 38, LCL = 19, UCL = 351.01.11.21.31.41.51.61.71.81.92.01.0370.03139.4551.0422.2911.426.684.353.092.361.911.621.1181.0456.3321.7710.335.783.682.612.001.651.421.281.298.3431.0212.716.433.842.611.961.591.371.241.151.362.8020.488.804.692.952.101.651.391.241.151.091.444.8815.116.763.752.461.821.481.281.171.101.061.534.6211.985.543.182.151.641.371.211.121.071.041.628.169.984.752.811.951.521.291.171.091.051.031.723.828.624.202.541.811.441.241.141.071.041.021.820.737.643.802.351.701.381.211.111.061.031.021.918.446.913.502.201.621.331.181.101.051.031.012.016.696.343.262.091.561.291.161.081.041.021.01Table 10
The ARLs of the proposed chart forr0=370 when both scale and shape shifted parameter are shifted when α0=3.5.
Shape parameter shifted (d)Scale parameter shifted (c) at β=0.857, k = 3.082, n = 27, LCL = 10, UCL = 251.01.11.21.31.41.51.61.71.81.92.01.0370.02121.2447.6322.3612.147.404.963.582.762.231.891.1205.6067.4327.6713.657.794.993.512.652.131.791.561.2132.9044.6118.989.745.783.842.802.191.811.561.391.395.6032.8514.417.634.673.202.391.921.621.431.301.473.9825.9711.686.343.982.802.141.751.501.341.241.560.2921.559.905.503.522.521.961.631.421.281.191.651.0118.538.664.903.202.331.841.551.361.241.161.744.3816.357.774.462.962.181.741.481.321.211.141.839.4714.727.094.132.772.071.671.431.281.191.131.935.6913.456.563.862.621.981.611.391.261.171.112.032.7212.456.133.652.501.911.571.361.241.151.10Table 11
ARLs for the proposed chart forr0=370 when both scale and shape parameter are shifted when α0=4.0.
Shape parameter shifted (d)Scale parameter shifted (c) atβ=0.680, k = 3.096, n = 46, LCL = 14, UCL = 341.01.11.21.31.41.51.61.71.81.92.01.0370.03134.2640.2115.717.664.442.952.171.731.471.301.1275.1370.8322.739.665.103.182.251.751.461.291.181.2179.4144.4015.236.913.882.551.891.531.321.201.121.3123.1531.2811.345.433.192.191.681.401.241.151.091.490.5123.859.064.522.771.961.551.321.191.111.071.570.4019.237.593.932.481.801.461.261.151.091.051.657.2216.156.583.512.271.691.391.221.131.071.041.748.1113.985.863.202.121.611.341.191.111.061.031.841.5412.385.322.972.001.541.301.171.101.051.031.936.6211.164.902.791.911.491.271.151.081.051.032.032.8410.214.572.641.841.451.251.141.081.041.02From the tables, we noticed the following:(1)
There is a rapid decreasing tendency in ARL1 as the shift constants (either c, or d, or both) decreases.(2)
From Tables1–4, it is clear that when the shape parameter α increases ARL1 values are decreases for other chart parameters fixed. Also, if the constant multiplier β increases, the ARL1 values decreases when other chart parameters are fixed.(3)
From Tables5–7, it is understandable that when the shape parameter α increases, the ARL1 values are also increases for other chart parameters fixed.(4)
From Tables8–11, it is interesting to observe that when both scale and shape parameters are shifted, the ARL1 values decrease. More specifically, for shifting of scale parameter, ARL1 values rapid decreasing, whereas when the shape parameter is shifted, the ARL1 values decrease slow tendency. In addition, when the shape parameter α increases, the ARL1 values are also increases for other chart parameters fixed.


## 4. Illustration of the Suggested Control Chart

The industrialization purpose of the developed CC can be enforced as follows: suppose that the lifetime of the item follows the length-biased weighted Lomax distribution with shape parameterα0=4. Assume that the average target lifetime of the item is μ0=1000 hours and r0=370. Hence, from (6) we get p0=0.72039. In addition, from Table 4 we get n = 32, β=1.094, k = 2.778, LCL = 15 and UCL = 30. Thus the examination time t0=1094 hours. Hence, the developed control chart implemented as follows:Step 1. Take a random sample of size 32 at each subgroup and deposit them for the life testing examination during 1094 hours. Obtain the number D (of failed items) during the examination time.Step 2. Declare the process as in-control if 15 ≤ D ≤ 30; otherwise, the process can be considered as out-of-control.


## 5. Industrial Application

The suggested CC is illustrated with a real data application in this section; the set is taken from Murthy et al. [25]. The data set is related to the time between failures for repairable item for ready reference the data is given below:1.43, 0.11, 0.71, 0.77, 2.63, 1.49, 3.46, 2.46, 0.59, 0.74, 1.23, 0.94, 4.36, 0.40, 1.74, 4.73, 2.23, 0.45, 0.70, 1.06, 1.46, 0.30, 1.82, 2.37, 0.63, 1.23, 1.24, 1.97, 1.86 and 1.17It is establish that the time between failures for repairable item data comes from the LBWLD with shape parameterα^=84.0881 and scale parameter λ^=63.3237 and the maximum distance between the real time data and the fitted of LBWLD is found from the Kolmogorov-Smirnov test as 0.0653 and also the p-value is 0.9995. The demonstration of the goodness of fit for the given model is shown in Figure 4, the empirical and theoretical cdfs and P-P plots for the LBWLD for the time between failures for repairable item data. Table 12 is presented the comparison of the proposed control and existing Shewhart type control chart for real example data estimated parameters. From Table 12 it is noticed that offered CC shows smaller ARL1 values as compared with Shewhart type control chart.Figure 4
The empirical and theoretical cdf and P-P plots for the LBWLD for the time between failures for repairable item.Table 12
ARLs comparison for the proposed and Schewhart charts forr0=370 for real example estimated parameters.
ARLscPropSchewhart1.00370.02370.541.05297.57301.171.10217.23246.431.15155.84182.641.20113.42143.521.2584.52112.121.3064.5691.261.3550.4979.561.4040.3252.111.4532.8241.271.5027.1832.321.6019.4624.681.7014.6418.121.8011.4514.221.909.2612.062.007.699.232.504.017.513.002.744.903.502.153.754.001.822.22Here,n = 13, α^=84.0881, β=0.66, and r0=370. The value of p0 is 0.38496 using (6). The value of μ=1.5428 by (4) is obtained as 1.5428 for duration of test t0=βμ0=0.661.5428=1.0182. The control limits proposed chart are LCL = 0 and UCL = 4.3308 for the parameters k = 2.391. The proposed control chart for time between failures for repairable item data is depicted in Figure 5 and existing Shewhart type control chart for k = 4.1515, LCL = 0.0 and UCL = 6.1549 is presented in Figure 6. From Figure 5, it is noticed that the proposed chart detect the first out-of-control signal at sample 13, whereas from Figure 6 the existing Shewhart type control chart fail to detect the out-of-control signal. Thus the proposed control chart shows 2 out-of-control signal but the existing Shewhart type control chart fail to detect. Hence, the proposed chart is better to monitor the quality of the products. Finally, the suggested CC is more efficient than the some existing control charts.Figure 5
The proposed control chart for real data.Figure 6
The Shewhart type control chart for real data.


## 6. Monitoring of the Proposed Control Chart

To investigate the performance of the developed control chart, the following simulation technique adopted to the generation of data from length-biased weighted Lomax distribution and building the control chart:Step 1: select a subgroup sample sizenStep 2: generate length-biased weighted Lomax random variableT of size n with scale parameter λ=1, shape parameter α0=2.5Step 3: find the chart data point, the values ofD for each subgroupStep 4: replicate step 1 to step 3 until the desired number of sample groups (m = 15) is attainedStep 5: make the proposed chart control limits using (7) and (8)Step 6: draw all chart data pointsD against their subgroups numbersTo build the new control chart, generate the first 15 samples of subgroup size 17 are generated from length-biased weighted Lomax distribution with in-control scale parameterλ0=1, shape parameter α0=2.5. The second set of the 15 samples of subgroup size 17 are from length-biased weighted Lomax distribution with scale parameter λ1=1.2, shape parameter α1=2.7 (i.e., both scale and shape parameters are shifted 0.2, i.e., c = 0.2 and d = 0.2). Table 8 shows that when ARL at 370 and in-control parameters we find control chart constant k is 2.854, β is 0.609, and n is 17. The life test termination time is t0 = 0.609 × 4.0 = 2.436. The chart data points D, along with sample values, are displayed in Table 13.Table 13
The chart data points for simulated data.
Sample1234567891011121314151617D111.410.0751.470.7657.551.721.562.311.387.030.480.8410.3210.0412.412.860.82922.672.814.501.411.8917.6511.22325.441.764.0039.622.335.574.944.091.467.165340.829.642.6612.446.3210.130.6611.411.423.270.791.463.421.917.6319.8410.16549.4430.249.7314.393.393.2115.0221.440.4645.981.167.7846.926.440.3011.655.303540.322.100.2727.593.734.751.2919.5710.932.671.506.901.612.370.730.3042.778638.939.8035.421.624.2610.301.2811.800.217.329.139.7311.077.894.3434.360.964718.093.7015.454.073.7539.9425.616.490.9326.194.057.059.1528.525.031.0312.86281.49113.4114.895.9210.9259.601.465.001.811.834.496.8610.6212.010.442.381.00798.461.571.633.1810.1016.9217.3312.004.1834.8518.994.104.6712.140.765.276.6731016.256.204.322.893.8421.452.3419.782.991.413.661.232.4916.3120.251.414.254116.825.674.249.813.4711.450.7617.144.611.2637.0057.700.631.6770.230.141.986122.3962.754.230.743.1111.024.6817.110.0313.7866.305.792.632.3022.716.901.555136.014.7211.9794.9730.2611.3216.595.5322.861.322.0611.514.7833.212.6829.839.142143.063.4024.730.9210.416.017.9413.926.4137.521.845.392.9344.799.3832.720.7131518.8720.201.2329.6714.6530.8528.612.3585.561.9230.756.980.8612.4311.850.2712.6751619.227.214.558.53141.450.115.230.664.254.0914.25147.5838.9992.492.951.1345.3931713.6868.910.742.594.812.797.925.655.484.680.929.3890.141.485.1753.006.023180.911.593.693.170.3027.507.693.614.973.298.763.426.367.68101.685.8931.9231911.190.890.572.011.4797.1848.5345.544.173.891.626.870.1419.876.5423.087.6462016.380.968.774.634.202.340.571.396.200.571.171.261.201.451.131.870.81122184.151.304.475.286.131.918.262.360.8212.944.7311.606.1011.440.251.259.876222.290.7525.799.376.105.196.711.49194.672.151.4518.502.1862.065.661.201.318235.3513.121.1028.17118.760.5510.809.241.922.301.691.555.618.3410.362.0833.337240.141.620.200.370.297.776.312.047.943.224.161.875.6119.492.274.611.7492510.2574.958.245.115.511.965.812.152.7457.953.6112.561.852.655.204.866.263266.1217.111.108.272.153.490.561.008.230.1712.463.394.2917.747.044.0211.1252736.485.8932.805.092.773.680.454.997.699.631.546.087.7540.423.163.4215.6522833.7928.420.642.729.8614.395.521.058.142.211.030.272.942.564.809.6843.9552923.126.165.4917.1854.560.771.1425.2148.200.9312.1716.161.367.269.8412.3216.2043026.236.638.2633.746.644.2394.262.432.8010.4125.140.5110.570.5536.190.203.054The average of failureD¯ is 4.7333, proposed chart control limits using (7) and (8) is obtained as UCL = 10.0077 and LCL = 0. The suggested control chart based on simulated data is depicted in Figure 7. It shows that an out-of-control for the process is obtained after 20th samples, i.e., after the 5th sample, when process shifted. Whereas when both scale and shape parameters are shifted to 0.2, Table 8 shows that the estimated ARL1 is 15.24. Therefore, the developed control chart resourcefully detects when the process is shifted.Figure 7
Control chart for simulated data.


## 7. Conclusions

The proposed article developed for thenp control chart when quality characteristics of life testing items are follows to length-biased weighted Lomax distribution. The chart coefficients and widespread tables are provided to use the practitioners in the industry. The proposed control chart design is demonstrated through simulated data. We observe the decreasing trend in the values of ARLs as the shift constant increases. The developed control chart can be used in the packing, electronic manufacturing industries to investigate the nonconforming products. The future research can be considered as an extension of proposed control chart, like repetitive control charts, multiple dependent sampling control charts. The proposed study can be extended for neutrosophic statistics [9, 26, 27].



---

*Source: 3091850-2022-04-22.xml*