Optimization of CASP-CUSUM schemes based on truncated Frechet distribution using gauss-legendre integration method

Vardeman. S, Di-ou Ray 1985 [3] introduced CUSUM control charts with the restriction that the quality values were exponentially distributed. Moreover, the phenomenon under study is the occurrence of rare events and the arrival times for a homogenous poison process, which are identically distributed exponential random variables. Kakoty. S. Chakraborthy A.B. 1990 [5] presented CASP-CUSUM charts under the assumption that the variable under study follows a Truncated Normal distribution. Truncated distributions are commonly employed in many practical phenomena where there is an upper and lower limit to the variable under study. The sorting procedure, for example, eliminates items that are above or below specified tolerance limits in the production engineering items. Any continuous variable should be first approximated as an exponential variable. Muhammad Riaz, Nasir Abbas, and Ronald J.M.M Does propose two Runs rule schemes for the CUSUM Charts 2011 [9] . A comparison of CUSUM and EWMA charts is made with the usual CUSUM and weighted CUSUM, with the first initial CUSUM as compared with the usual EWMA schemes. Using this comparison, it was concluded that the proposed schemes perform better for small and moderate shifts. Akhtar P.Md and Sarma K.L.A.P 2004 [8] presented an optimization of CASP-CUSUM Schemes based on truncated two parametric Gamma distributions and evaluated L 0 , L' 0 and Probability of Acceptance, as well as optimized CASP-CUSUM Schemes based on numerical results. It is proposed in this paper that CASP-CUSUM Charts be used when the variable under study follows a Truncated Frechet Distribution. Therefore, it is more important to investigate some of the interesting characteristics of the distribution.

Frechet distribution

Extreme value theory plays a crucial role in statistical analysis. Generally, extreme data are described by generalized extreme value (GEV) distributions Gumbel, Weibull, and Fréchet distributions are all special cases of GEV distributions.

Definition:

A random variable with a non-negative Frechet distribution is said to have the P.D.F is given by

Where α, β, x ≥ γ …… (1.1.1)

Truncated frechet distribution

It is the ratio of the probability density function of the Frechet distribution to the corresponding cumulative distribution function at B.

Where 'B' is the upper truncated point of the Frechet Distribution.

A Description of the Plan and the O.C Curve of Type-C

Beattie [2] proposed a method for the construction of continuous acceptance sampling plans. It consists of a chosen decision interval, "Return interval" with the length h', above which a decision line is taken. Sum

is distributed independently, and k 1 is the reference value, plotted on the chart. If the sum lies in the area of the normal chart, the product is accepted and if it lies in the return chart, the product is rejected, subject to the following assumptions. 1. When the recently plotted point on the chart touches the decision line, then the next point is to be plotted at maximum, i.e., h+h'. 2. Once the decision line is reached or crossed from above, the next point on the chart should be plotted from the baseline.

A network or a change of specification may be used instead of outright rejection when the CUSUM falls in the return chart. The procedure is summarized below.

When S m < 0, return cumulative to zero.

3. When h <S m <h+h' the product is rejected: when S m crosses h, i.e., when S m >h+h' and Continues rejecting the product until S m >h+h' return cumulative toh+h'. The Type -C, OC function is determined by incoming item quality and the sampling rate in acceptance and rejection regions is equal. Therefore, the probability of acceptance P A can be calculated as follows:

Where, 𝐿 0 = Average Run Length in acceptance zone and 𝐿 0 ′ = Average Run Length in rejection zone.

Page E.S [1] has introduced the formulae for 𝐿 0 and 𝐿 0 ′ as

Where P 0 = Probability for the test starting from zero on the normal chart, N 0 = ASN for the test starting from zero on the normal chart, 𝑃 0 ′ = Probability for the test on the return chart and 𝑁 0 ′ = ASN for the test on the return chart He further obtained integral equations for the quantities P 0 , N 0 , and 𝑁 0 ′ , 𝑃 0 ′ as follows:

And z is the distance of the starting of the test in the normal chart from zero.

Determination of ARLs and PA

The Gauss-Legendre integration method helps us determine the integral equations that are related to the probability of acceptance and the average sample number values, which help estimate the value of the probability of acceptance and the average run length of the distribution. We developed computer programs to solve equations (2.4), (2.5), (2.6) and (2.7) and we obtained the following results in Tables (3.1) to (3.28). Table 3.1 to Table 3.28 represents the values of Average Run Length and Type-C OC Cueves.

References

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