Time Truncated Chain Sampling Plans for Inverse Rayleigh Distribution

International Journal of Engineering Inventions
e-ISSN: 2278-7461, p-ISSN: 2319-6491
Volume 3, Issue 6 (January 2014) PP: 41-45

Time Truncated Chain Sampling Plans for Inverse Rayleigh
Distribution
Dr. A. R. Sudamani Ramaswamy1, S. Jayasri2
1

Associate Professor, Department of Mathematics, Avinashilingam University, Coimbatore
2
Assistant Professor, Department of Mathematics, CIT, Coimbatore

Abstract: A Chain sampling plan is developed based on truncated lifetimes when the lifetime of an item
follows a inverse Rayleigh distribution. The design parameters such as the minimum sample size and the
acceptance number are obtained by satisfying the producer’s and consumer’s risks at the specified quality
levels, under the assumption that the termination time and the number of items are pre-fixed. The results are
illustrated by an example.
Keywords: Chain sampling plan, Consumer’s risk, Inverse Rayleigh distribution, Operating characteristics,
Producer’s risk, truncated life test.

I.

Introduction

Acceptance sampling plans are used for quality control purposes in firms and industries. Keeping in
view two basic factors of time and cost, it is not possible to check the life time of each and every electronic
device. Whenever a sampling inspection is considered, the lot is either accepted or rejected along with
associated producer and consumer’s risk. In a time truncated sampling plan suppose n units are placed in a life
test and the experiment is stopped at a predetermined time T, where the number of failures is recorded until the
pre specified time. If the number of failures observed is not greater than the specified acceptance number, then
the lot will be accepted. Sampling inspection in which the criteria for acceptance and nonacceptance of the lot
depend in part on the results of the inspection of immediately preceding lots is adopted in Chain Sampling Plan
and so the use of chain sampling plan is often recommended when an extremely high quality is essential. Using
life tests one may find the probability of acceptance, minimum sample size put on test and the minimum ratio of
true average life to the specified average life or quality level subject to the consumer’s risk. These life tests are
studied by many authors using the different statistical distributions, More recently, Aslam and Jun (2009)
proposed the group acceptance sampling plan based on the truncated life test when the lifetime of an item
follows the inverse Rayleigh and Log-logistic distribution, Kantam R.R. L., Rosaiah K. and Srinivasa Rao G.
(2001) discussed acceptance sampling based on life tests with Log-logistic models. Rosaiah K. and Kantam
R.R.L. (2005) discussed acceptance sampling based on the inverse Rayleigh distribution. Gupta and
Groll(1961), Baklizi and EI Masri(2004), and Tsai, Tzong and Shou(2006), Balakrishnan, Victor Leiva & Lopez
(2007). All these authors developed the sampling plans for life tests using single acceptance sampling.
The
purpose of this study is to find the probability of acceptance for chain sampling plan assuming the experiment is
truncated at preassigned time and lifetime follows a Inverse Rayleigh distribution.

II.
n
λ
T
β
p*
d
i
Pa(p)

-

Glossary of Symbols

Sample size
Shape parameter
Prefixed time
Consumer’s risk
Minimum probability
Number of defectives
Acceptance criteria
Probability of Acceptance

III.

Inverse Rayleigh Distribution

The probability density function g(z) and cumulative distribution function G(z) of the Rayleigh
distribution are given by

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Time Truncated Chain Sampling Plans for Inverse Rayleigh Distribution
g ( z) 

2
 1
exp  2 
z3
z 

for z > 0

for z > 0

 1
G( Z )  exp  2 
z 

------------------(1)

-------------------(2)

If a scale parameter λ > 0 is introduced, the distribution function and the density function of Inverse Rayleigh
distribution, respectively are given by

  2 
22
exp  2 
 t 
t3


2
 
F (t ,  )  exp  2 
 t 


f (t ,  ) 

for t > 0,λ > 0

---------------------(3)

for t > 0, λ > 0

----------------------(4)

The failure rate of a single parameter inverse Rayleigh distribution is increasing for t < 1.0694543
decreasing for t > 1.0694543





and

as shown by Mukherjee and Saran (1984). The mean of the inverse Rayleigh

distribution is 2  .Voda (1972) studied the properties of the maximum likelihood estimator of λ,
determined tolerance limits and confidence intervals of minimum length and discussed problems of hypothesis
testing. Mukherjee and Maiti (1997) considered quantile estimation in Inverse Rayleigh distribution

IV.

Chain Sampling Plan

Chain Sampling Plan (ChSP-1) proposed by Dodge (1955) making use of cumulative results of several samples
help to overcome the shortcomings of the Single Sampling Plan. The distinguish feature is that the current lot
under inspection can also be accepted if one defective unit is observed in the sample provided that no other
defective units were found in the samples from the immediately preceeding i lots, i.e. the chain It avoids
rejection of a lot on the basis of a single nonconfirming unit and improves the poor discrimination between good
and bad quality. The conditions for application and operating procedure of chsp-1 are as follows
4.1 Conditions for application of ChSP -1:
The cost of destructiveness of testing is such that a relatively small sample size is necessary, although
other factors make a large sample desirable.
1) The product to be inspected comprises a series of successive lots produced by a continuing process.
2) Normally lots are expected to be of essentially the same quality.
3) The consumer has faith in the integrity of the producer.
4.2 Operating Procedure
The plan is implemented in the following way:
1) For each lot, select a sample of n units and test each unit for conformance to the specified requirements.
2) Accept the lot if d (the observed number of defectives) is zero in the sample of n units, and reject if d > 1.
3) Accept the lot if d is equal to 1 and if no defectives are found in the immediately preceeding i samples of size
n.
The Chain sampling Plan is characterized by the parameters n and i. We are interested in determining
the sample size required for in the case of Inverse Rayleigh distribution and various values of acceptance
number i.
The probability (α) of rejecting a good lot is called the producer’s risk, whereas the probability(β) of
accepting a bad lot is known as the consumer’s risk. Often the consumer risk is expressed by the consumer’s
confidence level. If the confidence level is p* then the consumer’s risk will be β = 1- p*.We will determine the
sample size so that the consumer’s risk does not exceed a given value β. The probability of acceptance in the
case of chain sampling plan is given by
n

Pa(p) = (1-p) + np(1-p)
Where

p=

n 1

(1-p)

ni

  2 
F (t ,  )  exp  2 
 t 



In Table 1 we present the minimum values of n, satisfying equation for p* = 0.75, 0.90, 0.95, 0.99 and for T/
λ 0 = 0.628,0.942,1.257,1.571,2.356,3.141,4.712. These choices are consistent with Gupta and Groll (1961),
Gupta (1962),Kantam et al (2001), Baklizi and EI Masri (2004), Balakrishnan et Al (2007).
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V.

Operating Characteristic Function

If the true but unknown life product deviates from the specified life of the product it should result in a
considerable change in the probability of acceptance of the lot based on the sampling plan. Hence 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. For a fixed T, p is a decreasing function of λ ≥ λ 0 .For a fixed i the operating
chartacteristic function values as a function of λ/ λ 0 are presented in Table 2 for different values of p* and for
the chsp-1 and the corresponding OC curves are also drawn.

VI.

Description Of Tables And An Example

Assume that the life time distribution is an Inverse Rayleigh distribution and that the experimenter is
interested in knowing that the true mean life is atleast 1000 hours with confidence 0.99. It is assumed that the
maximum affordable time is 767 hours and T/ λ 0 = 0.942, from the table 1, we obtain n = 11. Therefore, for this
sampling plan we accept the product if not more than 1 item fail out of 11 items and if no defectives are found
in the immediately preceeding i samples before T = 767 units of time, with the assurance that the true median
life is atleast 1000 with probability 0.99.
For the sampling plan(n = 11, i= 2, T/ λ 0 = 0.942) and confidence level p* = 0.99 under inverse Rayleigh
distribution the values of the operating characteristic function from Table 2 as follows

 / 0

2

4

6

8

10

12

L(p)

0.965196

1

1

1

1

1

Table 1:
Minimum sample size for the proposed plan in case of Inverse Rayleigh distribution with probability p* and the
corresponding acceptance number i .
p*
i
T/ λ
0

0.628
0.75

0.90

0.95

0.99

1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4

0.942

12
18
18
17
17
17
31
29
28
28
28
28
38
37
37
37
37
37
57
56
56
56

5
4
4
4
4
4
7
6
6
6
6
6
9
8
8
8
8
8
12
12
12
12

1.257

3
2
2
2
2
2
4
4
4
4
4
4
5
4
4
4
4
4
7
7
7
7

1.571

2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
3
3
5
5
5
5

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2.356

2
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
3
3
3
3

3.141

1
1
1
1
1
1
2
2
2
1
1
1
2
2
2
2
2
2
3
2
2
2

3.927

1
1
1
1
1
1
2
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2

4.712

1
1
1
1
1
1
1
1
1
1
1
1
2
1
1
1
1
1
2
2
2
2
Page | 43

5
6

56
56

12
12

7
7

5
5

3
3

2
2

2
2

2
2

Table 2: Operating Characteristic values for the time truncated chain sampling plan (n, i, T// λ 0 ) for a given
p*, when i = 2 .
0
p*
n
 /
T/λ
0

m

2
0.75

0.90

0.95

0.99

8
4
2
2
1
1
1
1
12
6
3
2
1
1
1
1
16
7
4
3
2
1
1
1
24
11
6
4
2
1
1
1

0.628
0.942
1.257
1.571
2.356
3.141
3.927
4.712
0.628
0.942
1.257
1.571
2.356
3.141
3.927
4.712
0.628
0.942
1.257
1.571
2.356
3.141
3.927
4.712
0.628
0.942
1.257
1.571
2.356
3.141
3.927
4.712

4

6

8

10

12

0.952362

1
1
1

0.775024

0.999979

1
1
1
1

0.641846

0.993905

0.998475

1
1
1
1
1

0.407383

0.929656

0.973981

0.999995

1
1
1
1
1
1

0.268744

0.793386

0.903134

0.999507

0.999995

1
1
1
1
1
1
1

0.187559

0.641846

0.802379

0.993905

0.999756

0.999995

0.999996769
0.845694

1
1
1

0.618109

0.999951

1
1
1
1

0.298488

0.975102

0.999979

1
1
1
1
1

0.116585

0.775383

0.99425

0.999979

1
1
1
1
1
1

0.268744

0.793386

0.982143

0.999507

0.999995

1
1
1
1
1
1
1

0.187559

0.641846

0.92961

0.993905

0.999756

0.999995

0.999994736
0.845694

1
1
1

0.618109

0.999951

1
1
1
1

0.298488

0.975102

0.999979

1
1
1
1
1

0.116585

0.775383

0.99425

0.999979

1
1
1
1
1
1

0.05316

0.496411

0.932054

0.997841

0.999995

1
1
1
1
1
1
1

0.187559

0.641846

0.92961

0.993905

0.999756

0.999995

0.999987934
0.665948

1
1
1

0.377526

0.999861

1
1
1
1

0.142502

0.947188

0.999951

1
1
1
1
1

0.116585

0.775383

0.99425

0.999979

1
1
1
1
1
1

0.05316

0.496411

0.932054

0.997841

0.999979

1
1
1
1
1
1
1

0.027382

0.298488

0.775263

0.975102

0.998925

0.999979

0.99999876
0.995662

0.990437

0.983483

0.965196

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FIGURE 1:

VII.

Conclusions

In this paper, chain sampling plan for the truncated life test was proposed in the case of Inverse
Rayleigh distribution. The construction yields the minimum sample size required to test the items to decide
upon whether a submitted lot is good having more median life or not.The operating characteristics values of the
plan against a specified producer’s risk are also presented. From the figure 1 we can see that the probability of
acceptance increases when  / 0 increases and it reaches the maximum value 1 when  / 0 is greater than 2 .

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Time Truncated Chain Sampling Plans for Inverse Rayleigh Distribution

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