Gompertz Log-Logistic Distribution: A Case Study of CASP-CUSUM Schemes

Abstract

The acceptance sampling technique is one of the oldest methods of quality control and relates to inspection and decision-making regarding lots of goods. In this way, many optimal techniques were developed to amplify and manage the quality of the products. Consistent with the supposition that the quality characteristic variable is scattered according to certain probability laws. Based on this assumption, we optimized CASP-CUSUM schemes for continuous variables which used a Truncated Gompertz Log-Logistic distribution used in Statistical Quality Control and Reliability analysis. In particular, the distribution is intended to estimate the optimal truncated point and the probability of acceptance of lots. The operating characteristics and average run length values are shown. The results are illustrated with figures.

Introduction

I. INTRODUCTION

Quality has become one of the most important factors for consumers to choose competitive goods and services. Consumers can be individuals, industrial organizations, retail stores, banks, financial institutions or military defense projects, this phenomenon is everywhere. Therefore, implementing and improving features are key drivers of business performance.

  Acceptance sampling procedures are used for statistical quality control. It is part of the continuity of operations management and quality of service. A quality management system is fundamental for industrial and commercial purposes. They are very vigilant about the quality of their products, so when consumers come to buy them, they can accept them without any problem.

 Acceptance sampling is more likely to be used in situations where the test is destructive, 100% inspection would be extremely expensive, or 100% inspection is not technically feasible or would take so long that the production planning would be grossly artificial.  

CUSUM charts are more effective than Shewhart charts for process monitoring because they can more quickly detect small disturbances in the middle, so CUSUM charts are  increasingly popular with researchers.

Later on, Vance12 provides a computer program for evaluating Average Run Length, Hawkins9 gives consistency simple at the very accurate estimated equation for the evaluation of Average Run Length. Numerous Markov chain approaches have been used for the computation of A.R.L by Ewan and Kemp7.  The integral equation is encountered in a variety of applications from many fields including continuum mechanics, mathematical economics, queuing theory, potential theory, geophysics, electricity and magnetism, organization, optimal control systems, communication theory, population genetics, medicine and so on. 

The integral equation was provided by Page14 and was used to approximate the ARLs of control chart by assuming a small shift in the mean.  A computer program based on the integral equation procedure was given by Vance12.  Goel and Wu8 provided a nomogram for the fortitude of chart parameters of a CUSUM control chart.  Lashkari and Rahim11 and Chung5 reported the economical design of CUSUM control charts.

Sarma and Akhtar2 studied Continuous acceptance sampling plans based on the truncated negative exponential distribution for Optimizing CASP-CUSUM schemes by solving the integral equation using Gauss-Chebyshev integration method with facilitate of computer program.  Lastly, the obtained results were compared at different values of the parameters.

 Sainath et.al15 Considered Continuous acceptance sampling plan Cumulative sum to determine ARL values through truncated two parameters Burr distribution.  To determine Optimum CASP-CUSUM values, a computer program is generated to solve the integral equations.  By executing the computer program is generated to solve the integral equations.  By executing the computer program, thus obtained ARL values for CASP-CUSUM schemes. 

Type-C OC curve values and ARL values are compared at different values of the parameters of the underlying probability distribution.  They also determined an optimal CASP-CUSUM scheme at which the probability of acceptance is utmost.

Venkatesulu.G and Mohammed Akhtar.P determined Truncated Lomax Distribution17 and Truncated Gompertz Distribution16 and its Optimization of CASP-CUSUM Schemes by altering the values of the parameters and in conclusion critical comparisons have based on the obtained numerical fallout.

In the present paper, it is unwavering CASP-CUSUM Chart when the variable under study follows Truncated Gompertz Log-Logistic Distribution. Thus it is more worthwhile to study some fascinating characteristics of this distribution.

A. Gompertz Log-Logistic Distribution

Gompertz13 introduced the Gompertz distribution to describe human transience and establish actuarial tables.  It is well-known human lifetime model and has many applications for instance; biology, gerontology, and marketing science.  The Log-Logistic distribution (branded as the Fisk distribution in economics) possesses a rather flexible functional form.

The log-logistic distribution is amongst the class of survival time parametric models where the hazard rate initially increases and then decreases and at times can be hump-shaped. The log-logistic distribution can be used as an appropriate substitute for Weibull distribution. It is, actually a mixture of Gompertz distribution with the value of the mean and the variance coincide equal to one. The log-logistic distribution as life testing model has its own standing; it is increasing failure rate (IFR) model and also is viewed as a weighted exponential distribution.

Definition: The non-negative random variable X is said to have Gompertz log-logistic distribution if its P.D.F is given by

References

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Copyright

Copyright © 2023 A Dhanunjaya, K Pushpanjali, G Venkatesulu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.