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ISSN : 1598-7248 (Print)
ISSN : 2234-6473 (Online)
Industrial Engineering & Management Systems Vol.16 No.1 pp.109-117
DOI : https://doi.org/10.7232/iems.2017.16.1.109

A Control Chart for Gamma Distribution using Multiple Dependent State Sampling

Muhammad Aslam*, Osama-H. Arif, Chi-Hyuck Jun
Department of Statistics, Faculty of Sciences, King Abdulaziz University, Jeddah, Saudi Arabia
Department of Statistics, Faculty of Sciences, King Abdulaziz University, Jeddah, Saudi Arabia
Department of Industrial and Management Engineering, POSTECH
Corresponding author : chjun@postech.ac.kr
May 18, 2016 August 31, 2016 September 27, 2016

ABSTRACT

In this article, a control chart based on multiple dependent (or deferred) state sampling for the gamma distributed quality characteristic is proposed using the gamma to normal transformation. The proposed control chart has two pairs of control limits, which can be determined by considering the in-control average run length (ARL). The shift in the scale parameter of a gamma distribution is considered and the out-of-control ARL is evaluated. The performance of the proposed chart has been shown for different levels of the parameters of the proposed control chart. It is also shown that the proposed chart is better than the Shewhart chart in terms of ARLs. A case study with a real data has been included for the practical usage of the proposed scheme.


초록


    1.INTRODUCTION

    A control chart is an important tool of statistical process control for monitoring and improving the quality of products of any manufacturing process. The idea of control chart was rooted by Shewhart A. Walter during 1920s in Bell Telephone Laboratories. Several modifications have been introduced since its existence but the basic idea of plotting the statistic on the graph of lower and upper limits remains unchanged. It becomes necessary for quality engineers to evaluate the control chart in use whether it has the ability of early detection of the out-ofcontrol process. Early and quick detection of the assignable causes of the on-line process is the prime purpose behind constructing the control charts.

    The concept of multiple dependent (or deferred) state (MDS) sampling was initiated by (Wortham and Baker, 1976). Balamurali and Jun (2007) presented a variable acceptance sampling plan using the MDS scheme and concluded that this sampling scheme is better in risk protection to the manufacturer and the consumer as compared to the conventional single and double sampling plans. (Aslam et al., 2015) studied the MDS schemes in the area of acceptance sampling plans and argued that the MDS sampling performs better than the conventional single sampling plans in terms of average sample number. Under the MDS scheme the decision about the in-control or the out-of-control process is made considering the results of the previous samples. If we select a sample from the on line process and posted it on the control chart, then it may fall in any of three mutually exclusive states i.e., in-control state, out-of-control state or the state in which the decision depends on the previous samples. The MDS sampling have been studied by many authors including among others (Soundararajan and Vijayaraghavan, 1990).

    Most of the control charts have been studied assuming that the specific quality characteristics of the manufacturing process follow the normal distribution. But there are situations when the specific quality characteristic does not follow the normal distribution. According to Santiago and Smith (2013) the data not collected in subgroups or a skewed data may not produce good results under the normal distribution. Schilling and Nelson (1976) and Stoumbos and Reynolds Jr (2000) have suggested alternative methods when the quality characteristic of interest follows a skewed distribution. Santiago and Smith (2013) proposed control chart for exponential distribution and named it as t-chart. Santiago and Smith (2013) used transformation given by Johnson and Kotz (1970) and Nelson (1994). Mohammed (2004) and Mohammed and Laney (2006) applied t-chart in healthcare. Aslam et al. (2016) proposed t-chart using process capability index. For a skewed distributed quality characteristic, a popularly used distribution to study the phenomena is a gamma distribution. The gamma distribution is frequently used in modeling the waiting time of the life events (Hogg and Craig, 1970; Aksoy, 2000). Al-Oraini and Rahim (2002) worked for economical X-bar chart for gamma distribution. Jearkpaporn et al. (2003) designed control chart for gamma distribution using generalized linear model. Sheu and Lin (2003) used the gamma distribution to study a small shift in the process. Aslam et al. (2014) used the Wilson- Hilferty transformation to propose a control chart for an exponential distribution. Zhang et al. (2007) proposed control chart for gamma distribution.

    By exploring the literature and according to the best of author knowledge, there is no work on the designing the control chart for a gamma distribution using MDS sampling. Therefore, this study proposes a new control chart for a gamma distribution using MDS sampling. The rest of the paper is organized as follows: In Section 2 the design of the proposed control chart has been explained. Section 3 explains the performance evaluation of the proposed chart in terms of the average run lengths. In Section 4 the comparison of the proposed chart with the Shewhart chart has been described and a simulation study is performed to demonstrate the merit of the proposed control chart. A case study with a real data is also added in this section. In Section 5 some conclusions and findings have been explained.

    2.DESIGN OF PROPOSED CONTROL CHART

    The proposed control chart utilizes a gamma to normal approximation under the Wilson-Hilferty transformation. Let T be a random variable from a gamma distribution with shape parameter ‘a’ and scale parameter ‘b’. The cumulative distribution function (cdf) of the gamma distribution is given by

    P ( T t ) =1- j=1 a-1 e - t b ( t b ) j j!
    (1)

    The Wilson and Hilferty (1931) suggested that the transformed variable of T = T1/3 is distributed approximately as normal with mean

    μ T * = b 1/3 Γ ( a+1/3 ) Γ ( a )
    (2)

    and variance

    σ T * = b 2/3 Γ ( a+2/3 ) Γ ( a ) μ T * 2
    (3)

    This suggests that T is symmetric in distribution, so a control chart can be designed with the usual upper control limit (UCL) and lower control limit (LCL). Therefore, we propose the following steps for the development of the control chart for a gamma distributed quality characteristic:

    • Step 1:  Select an item randomly and measure its quality characteristic T. Then, calculate T:

      T * =T 1/3

    • Step 2:  Declare the process as in-control if LCL2 ≤ T* ≤ UCL2 . Declare the process to be out-of-control if T* ≥ UCL1 or T* ≤ LCL1. Otherwise, go to Step-3.

    • Step-3:  Declare the process is in-control if i proceeding subgroups have been declared as in-control. Otherwise, declare the process to be out-of-control.

    The proposed control chart is based on two pairs of control limits, that is the outer control limits of (LCL1, UCL1) and the inner control limits of (LCL2, UCL2) as well as the parameter i. The outer control limits are given by

    LCL 1 = μ T *  -  k 1 σ T * = b 0 1 / 3 Γ ( a+1/3 ) Γ ( a ) k 1 b 0 2/3 Γ ( a+2/3 ) Γ ( a ) μ T * 2
    (4a)

    UCL 1 = μ T *  +  k 1 σ T * = b 0 1 / 3 Γ(a+1 Γ ( a ) +k 1 b 0 2/3 Γ ( a+2/3 ) Γ ( a ) μ T * 2
    (4b)

    Also, the inner control limits are given by

    LCL 2 = μ T * k 2 σ T * = b 0 1 / 3 Γ ( a+1/3 ) Γ ( a ) -k 2 b 0 2/3 Γ ( a+2/3 ) Γ ( a ) μ T * 2
    (5a)

    UCL 2 = μ T * + k 2 σ T * = b 0 1 / 3 Γ ( a+1/3 ) Γ ( a ) +k 2 b 0 2/3 Γ ( a+2/3 ) Γ ( a ) μ T * 2
    (5b)

    In above, k1 and k2 are control coefficient to be determined by considering the in-control ARLs while b0 is the scale parameter when the process is in control. The proposed plan reduces to the traditional Shewhart control chart when the control coefficients k1=k2=k and i = 1 .

    The control limits can also be written as follows

    LCL 1 =b 0 1/3 LL 1 UCL 1 =b 0 1/3 UL 1 LCL 2 =b 0 1/3 LL 2

    and

    UCL 2 =b 0 1/3 UL 2

    where

    LL 1 = [ Γ ( a+1/3 ) Γ ( a )  -  k 1 Γ ( a+2/3 ) Γ ( a ) - ( Γ ( a+1/3 ) Γ ( a ) ) 2 ]
    (6a)

    UL 1 = [ Γ ( a+1/3 ) Γ ( a )  +  k 1 Γ ( a+2/3 ) Γ ( a ) ​ -  ( Γ ( a+1/3 ) Γ ( a ) ) 2 ]
    (6b)

    LL 2 = [ Γ ( a+1/3 ) Γ ( a ) k 2 Γ ( a+2/3 ) Γ ( a ) ​ -  ( Γ ( a+1/3 ) Γ ( a ) ) 2 ]
    (7a)

    and

    UL 2 = [ Γ ( a+1/3 ) Γ ( a ) + k 2 Γ ( a+2/3 ) Γ ( a ) ​ -  ( Γ ( a+1/3 ) Γ ( a ) ) 2 ]
    (7b)

    The probability of declaring as in-control for the proposed control chart when the process is actually in control is given as follows

    P in 0  = P ( LCL 2 T * UCL 2 | b 0 )  + { P ( LCL 1  <  T *  <  LCL 2 | b 0 )  + P ( UCL 2  <  T *  <  UCL 1 | b 0 ) } { P ( LCL 2  <  T *  <  UCL 2 | b 0 ) } i
    (8)

    Here,

    P ( LCL 2 T * UCL 2 | b 0 )  = P ( T * < UCL 2 | b 0 ) -P ( T * < L CL 2 | b 0 ) P ( LCL 2 T * UCL 2 | b 0 )  =  j=1 a-1 e -UL 2 3 ( UL 2 3 ) j j! j=1 a-1 e -LL 2 3 ( LL 2 3 ) j j! P ( LCL 1 < T * < LCL 2 | b 0 )  =  j=1 a-1 e -LL 2 3 ( UL 2 3 ) j j! j=1 a-1 e -LL 1 3 ( LL 1 3 ) j j! P ( UCL 1 < T * < UCL 1 | b 0 )  =  j=1 a-1 e -UL 1 3 ( UL 1 3 ) j j! j=1 a-1 e -UL 2 3 ( UL 2 3 ) j j!

    The average run length (ARL) for the in-control process is given as follows

    ARL 0 = 1 1-P in 0
    (9)

    Now, we will work for the shifted process. We assumed that the scale parameter of the gamma distribution is shifted from b0 to b1 when the process is shifted. Let us assume that b1 = cb0 , where b1 is the shifted scale parameter of the gamma distribution and c is the shift constant. Then, the probability of declaring in-control when the process is shifted is given by P in 1  = P ( LCL 2 T * UCL 2 | b 1 )  +  { P ( LCL 1 < T * < LCL 2 | b 1 )  + P ( UCL 2  <  T * < UCL 1 | b 1 ) } { P ( LCL 2 T * UCL 2 | b 1 ) } i  

    Here,

    P ( LCL 2 T * UCL 2 | b 1 )  = P ( T * < UCL 2 | b 1 ) -P ( T * < LCL 2 | b 1 ) P ( LCL 2 T * UCL 2 | b 1 )  =  j=1 a-1 e -UL 2 3 c ( UL 2 3 c ) j j! j=1 a-1 e LL 2 3 c ( LL 2 3 c ) j j! P ( LCL 1 < T * < LCL 2 | b 1 )  =  j=1 a-1 e -LL 2 3 c ( UL 2 3 c ) j j! j=1 a-1 e LL 1 3 c ( LL 1 3 c ) j j! P ( UCL 2 < T * < UCL 1 | b 1 )  =  j=1 a-1 e -UL 1 3 c ( UL 1 3 c ) j j! j=1 a-1 e - UL 2 3 c ( UL 2 3 c ) j j!

    The ARL for the shifted process ARL1 is given as follows

    ARL 1 = 1 1-P in 1

    3.PERFORMANCE EVALUATION OF THE PROPOSED CHART

    The performance indicator of any control chart can be best examined and evaluated by the average run length (ARL). Traditionally, the ARL is defined as the average number of samples before the process shows an out-ofcontrol signal (Montgomery, 2007). A greater value of ALR is required when the process is stable and a smaller value is desirable when the process is shifted or out of control. The simulation approach has been used for estimating the ARL with the help of the R-language software. This simulation approach is commonly used when the exact form of the mean and other measures of the proposed process is not available. Many researchers have used the simulation approach for the effectiveness of control charts including among others Santos (2009), Abbasi and Miller (2013), Ahmad et al. (2013), Ahmad et al. (2013), Chananet et al. (2014), Shu et al. (2014), Aslam et al. (2015), Azam et al. (2015) and Aslam (2016).

    The ARL1 values of the shifted process for r0 = 200, 300 and 370, for different shift levels c and five values of the shape parameter a = 1, 2, 5, 10 and 20 are given in Table 1-Table 6. Table 1-Table 3 are for I = 2 and Table 4-Table 6 are for i = 3. As mentioned earlier that the shift occurs in the scale parameter as b1 = cb0 when all other settings are held constants, the decreasing pattern of the ARL1 shows the performance of the proposed chart. From Table 1~Table 6, we note following trends in control chart parameters. Table 2-Table 5

    • 1. For all other same parameters, as r0 increases from 200 to 370, the values of ARL1 increases.

    • 2. For all other same parameters, as a increases from 2 to 10, the values of ARL1 decreases.

    • 3. For all other same parameters, as i increases from 2 to 3, the values of ARL1 decreases.

    4.ADVANTGES OF PROPOSED CHART

    As mentioned earlier the proposed control chart is equal to the Shewhart chart when the two control constants are equal (k1 = k2) and i = 1. Tables 7 and 8 have been generated for the ARL1 comparison of the proposed control chart with the Shewhart chart. The efficiency of the proposed chart can be observed by the decreasing pattern of the ARL1 values; for instance, the ARL1 of the proposed chart is 147.04 for a = 20 and c = 1.05 whereas the same shift is detected after 170.71 samples on the average for the existing chart as mentioned in Table 7. The efficiency of the proposed chart is checked for all the possible combinations of the different sittings of r0 = 200, 300 and 370, a = 1, 2, 5, 10 and 20 and shift levels c = 1, 1.01, 1.02, 1.03, 1.04, 1.05, 1.1, 1.15, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0, 2.5 and 3.

    4.1.Simulation Study

    In this section, we will demonstrate the efficiency of the proposed control chart. For this purpose, we will use the simulated data from the gamma distribution. The data is generated and placed in Table 9. The first 20 observations have been generated for in-control process using gamma distribution with a = 2 and b0 = 1. Next 30 observations have been generated from a shifted parameter of b1 = 1.5. Figure 1 shows the proposed control chart with r0 = 370 for the simulated data, which indicates the out-of-control process at 49th (or 29th subgroup after the actual process shift) subgroup.

    The Shewhart chart for this data is shown in Figure 2. From this figure, it can be read that all values are within the control limits which indicates that the process is in control. So, we can say that the proposed control chart performs better to detect a shifted process than the Shewhart chart.

    4.2.Industrial Example

    In this section, the proposed control chart is applied to monitoring of urinary tract infections (UTIs) at a large hospital. The data represents the duration of male UTIs patient at a hospital. Similar data was used by Santiago and Smith (2013). The data is known to follow the gamma distribution with shape a = 2. The UTIs data is reported in Table 10.

    The control limits of the proposed control chart for UTIs data are given in Figure 3. It can be seen from Figure 3 that the process is in-control although some points are close to UCL2.

    5.CONCLUDING REMARKS

    The control chart for the efficient monitoring of the production process has been developed for the multiple dependent state sampling scheme under the gamma distribution. The control chart coefficients have been estimated for various target in-control ARLs. Numerical tables have been constructed for the ARL0 and ARL1 values. The proposed chart is found to be comparatively effective for the monitoring of process shifts from the ARL comparison. It has been observed from a simulation study that the proposed scheme is effective for the quick response of the shifted process. A real example is added to explain the application of the proposed chart to a healthcare area. This example shows that the proposed control chart can be also used in health monitoring. The proposed scheme can be extended for other non-normal distributions as a future research.

    ACKNOWLEDGMENTS

    This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia. The authors, therefore, acknowledge with thanks DSR technical and financial support. The work by Chi-Hyuck Jun is a part of the project entitled ‘Development of TCS System on ECO-Ship Technology’, funded by the Ministry of Oceans and Fisheries, Korea.

    Figure

    IEMS-16-109_F1.gif

    Proposed control chart for the simulated data.

    IEMS-16-109_F2.gif

    Shewhart control chart for the simulated data.

    IEMS-16-109_F3.gif

    The control chart for UTIs data.

    Table

    The ARL1 values for the proposed chart with i = 2 when r0 = 200

    The ARL1 values for the proposed chart with i = 2 when r0 = 300

    The ARL1 values for the proposed chart with i = 2 when r0 = 370

    The ARL1 values for the proposed chart with i = 3 when r0 = 200

    The ARL1 values for the proposed chart with i = 3 when r0 = 300

    The ARL1 values for the proposed chart with i = 3 when r0 = 370

    The Comparison of ARL1 values for proposed chart with i = 2 and Shewhart Chart when r0 = 370

    The Comparison of ARL1 values for proposed chart with i = 3 and Shewhart Chart when r0 = 370

    Simulated data

    The UTIs data

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