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International Journal of Mathematics And its Applications
Time Truncated Special Purpose Double Sampling Plan for Weighted Exponential Distribution
A. R. Sudamani Ramaswamy1 and P. Ponmani1,∗
1 Department of Mathematics, Avinashilingam University, Coimbatore, Tamilnadu, India.
Abstract: Special purpose Double Sampling Plan DSP (0,1) is developed for a truncated life test when the life time of an item follows weighted exponential distribution. The minimum sample sizes are determined when the consumers risk and the test termination time are specified. Tables are constructed and examples are provided. The Operating Characteristic values are also provided in the table.
Keywords: Double sampling plan, Probability of acceptance, Consumers risk, Truncated life test, Operating Characteristic curve, Weighted exponential distribution.
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JS Publication. Accepted on: 28.04.2018
1. Introduction
Statistical quality control is systematic as compared to guess-work of haphazard process inspection. It plays an important role in total quality control. Whenever a statistical technique is employed to control, improve and maintain the quality or to solve quality problem it is termed as Statistical Quality Control. Acceptance sampling is an inspecting procedure applied in statistical quality control. Acceptance sampling is a part of operations management and service quality maintenance. Life test sampling plan is a technique, which consist of decision making based on sampling inspection of batch of products by experiments for examining the continuous utility of the products for the specified function. In a truncated life test, the units are randomly selected from a lot of products and are subjected to a set of test procedures, where the number of failures is recorded until the pre-specified time. If the number of observed failures at the end of fixed time is not greater than the specified acceptance number, then the lot will be accepted. For such a truncated life test and the associated decision rule we are interested in obtaining the smallest sample size to arrive at a decision where the life time of an item follows weighted exponential distribution. Two risks are always attached to an acceptance sampling. The probability of rejecting the good lot is known as the type-1 error (Producers risk) and it is denoted by α. The probability of accepting the bad lot is known as the type-2 error (Consumers risk) and it is denoted by β.
Dodge and Romig (1959)have studied the use of DSP(0,1) plan to product characteristics involving costly and destructive testing. Sudamani Ramaswamy and Sutharani (2013) discussed designing double acceptance sampling plans based on truncated life tests in Rayleigh distribution using minimum angle method. Sudamani Ramaswamy and Jayasri (2014) discussed time truncated special purpose double sampling plan for selected distributions. Sudamani Ramaswamy and Sowmya (2016) discussed time truncated special purpose double sampling DSP(0,1) for compound Rayleigh distribution.
∗ E-mail: [email protected] (Research Scholar)
Wenhao Gui and Aslam (2016) discussed time truncated single sampling plan for weighted exponential distribution. The purpose of this paper is to study special purpose double acceptance sampling plan when the lifetime follows Weighted exponential distribution. The results are analyzed with the help of tables and examples.
2. Weighted Exponential Distribution
Gupta and Kundu (2009) proposed a new class of weighted exponential distributions using the idea of Azzalini (1985).
Makhdoom (2012) investicated the statistical inference for reliability and stress-strength for weighted exponential distribu- tion. Zamani and Ismail (2010) mixed the distributions of the poisson and the weighted exponential and applied it on claim count data. The PDF of the weighted exponential distribution is given by
f (t; α; λ) =α + 1
α λe−λt(1 − e−αλt), t > 0
where α > 0 is the shape parameter and λ > 0 is the scale parameter. We denote it by writing T ∼ W E (α, λ). The PDF is always unimodal with log(1 + α)/(αλ). As α −→ ∞, the weighted exponential distribution T converges to Exponential distribution Exp(λ). As α −→ 0, the weighted exponential distribution T converges to a gamma distribution Gamma (2, λ).
The corresponding cumulative distribution function of T is
f (t; α; λ) =α + 1 α
e−(α+1)λt− 1
α + 1 − e−λt+ 1
, t > 0
3. Concept of DSP(0,1)
From Cameron (1952) table, one can observe a jump between the operating ratios of Single Sampling Plan with c = 0 and c = 1 and slow reduction of operating ratios for other values of c. It may also be seen that, in between the Operating Characteristic (OC) curves of single sampling plans with c = 0 and c = 1 plans, there is a vast gap to be filled which leads one to assess the possibility of designing plans having OC curves lying between the OC curves of c = 0 and c = 1 plans. To overcome such situation Craig (1981) have proposed double sampling plan with acceptance numbers 0 and 1 and rejection number 2. Vijayaraghavan (1990) has presented tables for the selection of DSP-(0,1) plan for attributes under Poisson and Binomial conditions of sampling.
4. Operating Procedure of Double Sampling Plan of Type DSP(0,1)
According to Hald (1981), the operating procedure of DSP-(0,1) is as follows
(1). From a lot, select a sample of size n1, and observe the number of defectives, d1.
(2). If d1= 0, accept the lot; If d1> 1, reject the lot;
(3). If d1= 1, select a second sample of size n2 and observe d2; If d2= 0, accept the lot, otherwise reject the lot.
4.1. Operating procedure of double sampling plan of type DSP(0,1) for the life tests
(1). From a lot, select a sample of size n1, and observe the number of defectives d1, during the time t0.
(2). If d1= 0, accept the lot; If d1> 1, reject the lot;
(3). If d1= 1, select a second sample of size n2and observe d2, during the time t0. If d2= 0, accept the lot, otherwise reject the lot.
The operating procedure is presented in the form of flow chart in Figure 1.
Figure 1. Operating procedure of DSP(0,1) plan for the life tests in the form of flow chart.
5. Operating Characteristic Curve
Every sampling plan is associated with an operating characteristic curve, familiarly known as OC curve of the plan. This curve when referred to two axes, the axis of p-proportion nonconforming of the material offered for inspection and the axis of Pa(p)-probability of acceptance of a lot or process, is the locus of (p, Pa(p)). The OC curve gives the practical performance of a sampling plan.
6. Design of the Sampling Plan
Suppose that life times of the submitted products follow a weighted exponential distribution defined by equation number (1). The life test terminates at a preassigned time t0 and the number of failures during this time interval [0, t] are recorded.
The decision to accept the lot occurs if and only if the number of failures at the end of the time point t0 does not exceed the acceptance number c. The lot size is assumed to be infinitely large so that the theory of binomial distribution can be applied. The acceptance or rejection of the lot are equivalent to the acceptance or rejection of the hypothesis H0: µ ≥ µ0.
For fixed α, µ ≥ µ0 ⇔ λ ≤ λ0, where λ0 = α + 2/(α + 1)µ0. One can note that λ0 also depends on the shape parameter α. It is assumed that α is known in this article. For the sake of convenience, set the termination time as a multiple of the specified mean lifetime, that is, t0 = dµ0, where d is a positive constant. The proposed acceptance sampling plan is characterized by (n, k), which consists of
(1). the number of items n to be drawn from the lot,
(2). the acceptance number k,
(3). the ratio t0= dµ0, where µ0 corresponds to the specified mean lifetime and t0 is the preassigned testing time.
Let P∗be the consumer’s confidence level in the sense that the chance of rejecting a bad lot (having mean lifetime µ < µ0) is at least P∗. The consumer’s risk, the probability of accepting a bad lot, is fixed not to exceed 1P∗. The DSP-(0,1) is composed of parameters n1 and n2 if t0 is specified. The probability of acceptance can be regarded as a function of the deviation of specified average from the true average. This function is called Operating Characteristic (OC) function of the sampling plan. Once the minimum sample size is obtained one may be interested to find the probability of acceptance of a lot when the quality of the product is good enough. It is assumed that the lot size is large enough to use the binomial distribution to find the probability of acceptance. The probability of acceptance for the sampling plan is calculated as follows
Pa(p) = (1 − p)n1+ n1p(1 − p)n1+n2−1 (1)
Where n1= n and n2= kn. The time determination ratio t/µ0are fixed as 0.4, 0.6, 0.8, 1.0, 1.5, 2.0, 2.5, 3.0, the consumers risk β as 0.25, 0.10, 0.05, 0.01 and the mean ratio µ/µ0 are fixed as 2, 4, 6, 8, 10, 12. These choices of t/µ0 are consistent with Gupta and Groll, Gupta, Kantam et al , Baklizi and EI Masri, Balakrishnan. For various time termination ratios and mean ratios, the design parameter values n1 = n and n2 = kn are obtained and presented in Table 1. The probability of acceptance for DSP(0,1) sampling plan are also calculated and presented in Table 2.
P∗ K d = t0/µ0
0.4 0.6 0.8 1.0 1.5 2.0 2.5 3.0
0.75
1 7 3 2 1 1 1 1 1
2 5 2 1 1 1 1 1 1
3 4 2 1 1 1 1 1 1
4 4 2 1 1 1 1 1 1
5 4 2 1 1 1 1 1 1
0.90
1 10 4 3 2 1 1 1 1
2 7 3 2 1 1 1 1 1
3 6 3 2 1 1 1 1 1
4 6 3 2 1 1 1 1 1
5 6 3 2 1 1 1 1 1
0.95
1 12 5 3 2 1 1 1 1
2 8 4 2 2 1 1 1 1
3 8 4 2 2 1 1 1 1
4 8 4 2 2 1 1 1 1
5 8 4 2 2 1 1 1 1
0.99
1 17 8 4 3 1 1 1 1
2 12 6 3 2 1 1 1 1
3 12 6 3 2 1 1 1 1
4 12 6 3 2 1 1 1 1
5 12 6 3 2 1 1 1 1
Table 1. Minimum sample size for DSP (0,1) plan when the life time of the item follows Weighted exponential distribution (α = 2)
p∗ n d = t0/µ0 µ/µ0
2 4 6 8 10 12
0.75
5 0.4 0.800080 0.974465 0.993830 0.997862 0.999079 0.999541 2 0.6 0.863344 0.982595 0.995703 0.998487 0.999341 0.999669 1 0.8 0.913127 0.988952 0.997221 0.999007 0.999563 0.999778 1 1 0.849279 0.977475 0.993988 0.997787 0.999007 0.999491 1 1.5 0.658275 0.926792 0.977475 0.991071 0.995810 0.997787 1 2 0.477798 0.849279 0.947096 0.977475 0.988952 0.993988 1 2.5 0.336374 0.756061 0.903449 0.956006 0.977475 0.987378 1 3 0.234258 0.658275 0.849279 0.926792 0.960938 0.977475
0.90
5 0.4 0.685737 0.953023 0.988196 0.995849 0.998198 0.999099 2 0.6 0.748962 0.962477 0.990355 0.996551 0.998485 0.999236 1 0.8 0.733676 0.956394 0.988282 0.995704 0.998083 0.999023 1 1 0.849279 0.977476 0.993989 0.997788 0.999008 0.999492 1 1.5 0.658275 0.926793 0.977476 0.991071 0.995810 0.997788 1 2 0.477799 0.849279 0.947096 0.977476 0.988952 0.993989 1 2.5 0.336374 0.756062 0.903449 0.956007 0.977476 0.987379 1 3 0.234259 0.658275 0.849279 0.926793 0.960938 0.977476
0.95
5 0.4 0.631367 0.940662 0.984800 0.994615 0.997654 0.998825 2 0.6 0.637560 0.937191 0.983202 0.993898 0.997299 0.998632 1 0.8 0.733676 0.956394 0.988282 0.995704 0.998083 0.999023 1 1 0.595231 0.916107 0.975427 0.990604 0.995704 0.997775 1 1.5 0.658275 0.926793 0.977476 0.991071 0.995810 0.997788 1 2 0.477799 0.849279 0.947096 0.977476 0.988952 0.993989 1 2.5 0.336374 0.756062 0.903449 0.956007 0.977476 0.987379 1 3 0.234259 0.658275 0.849279 0.926793 0.960938 0.977476
0.99
5 0.4 0.446134 0.883602 0.967871 0.988278 0.994820 0.997384 2 0.6 0.450911 0.876485 0.964346 0.986644 0.993998 0.996933 1 0.8 0.562689 0.910495 0.974364 0.990355 0.995640 0.997761 1 1 0.595231 0.916107 0.975427 0.990604 0.995704 0.997775 1 1.5 0.658275 0.926793 0.977476 0.991071 0.995810 0.997788 1 2 0.477799 0.849279 0.947096 0.977476 0.988952 0.993989 1 2.5 0.336374 0.756062 0.904490 0.956007 0.977476 0.987379 1 3 0.234259 0.658275 0.849279 0.926793 0.960938 0.977476
Table 2. Probability of acceptance for DSP (0,1) plan when the life time of item follows weighted exponential distribution (k = 2)
Figure 2.
7. Conclusion
In this paper, special purpose double acceptance sampling plan for truncated life tests by using weighted exponential distribution is presented. The minimum sample size and the acceptance number are calculated, for various values of the mean ratios and different experiment times assuming that the lifetime of an item follows weighted exponential distributions.
The operating characteristic curve values and the associated consumers risks are also provided. The operating characteristic values of weighted exponential distribution increases disproportionately and reaches the maximum value 1 when µ/ µ0 is greater than 2. Obviously, from the Figure 2 one can conclude that the operating characteristic values increases when the quality improves and can be used conveniently in practical situations to save the time and cost of life rest experiments. One can see that the probability of acceptance increases when µ/ µ0 increases and it reaches the maximum value 1 when µ/ µ0is greater than 2.
References
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