1. Introduction

Acceptance sampling procedures play an important role in improving the quality. The basic aim of all companies in this world is to improve the quality of their products. The high quality product has the high probability of acceptance. In a time- truncated sampling plan, a random sample is selected from a lot of products and put on the test 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. Two risks are always attached to an acceptance sampling. The probability of rejecting the good lot is known as the type – 1 error (producer’s risk) and it is denoted by α. The probability of accepting the bad lot is known as the type – 2 error (consumer’s risk) and it is denoted by β. An acceptance sampling plan should be designed so that both risks are smaller than the required values. An acceptance sampling plan involves quality contracting on product orders between the producer’s risk and consumer’s risk.

These life tests are discussed by many authors Goode and Kao (1961).[1] Ayman Baklizi (2003),[2] Baklizi A,,EI Qader, and EI Masri (2004).[3] Rosaiah and Kantam (2005) [4] Tsai, Tzong and Shuo (2006).[5] and Mohammad Aslam [6] have designed double acceptance sampling plan based on truncated life tests in Rayliegh distribution. Srinivasa Rao[8] have designed double acceptance sampling plan based on truncated life tests for the Marshall – Olkin extended exponential distribution.

The intent of this paper is to design double sampling plans for truncated life tests using minimum angle method, when life times of the items follows Rayleigh distribution.

It is known that the double acceptance sampling plan (DASP) is more efficient than the single sampling plan in terms of the sample size required. Further, a DASP is expected to reduce the producer’s risk when specifying the consumer’s risk.

2. Operating Procedure for Double Sampling Plan

1) From a lot, take a first sample of size n₁ and observe the number of nonconforming units, d_1.

2) If d₁ ≤ c₁, accept the lot; if d₁ ≥ c₂, reject the lot. If c₁ < d₁ < c₂ take a second sample of size n₂ and observe the number of nonconforming units, d₂.

3) If d₁ + d₂ ≤ c₂, accept the lot; otherwise reject the lot.

Thus the double sampling plan is characterized by the parameters n₁, n₂, c₁, c₂, and designated as DASP – (n₁, n₂, c₁, c₂).

3. Double Sampling Plans in Life Tests

We propose the following Double sampling plan procedure based on a truncated life test:

1. Draw the first sample of size n₁ and put them on test during time t₀

2. Accept the lot if there are no more than c₁ failures. Reject the lot and terminate the test if there are more than c₂ failures.

3. If the number of failures is between c₁ and c₂, then draw the second sample of size n₂ and put them on test during time t₀.

4. Accept the lot if the total number of failures not more than c₂ during the time t_{0 .}

DECISION FLOWCHART FOR DOUBLE SAMPLING PLAN IN LIFE TESTS.

The DASP is composed of four parameters of (n₁, n₂, c₁, c₂) if t₀ is specified. Here n₁ and n₂ are sample sizes of the first and second sample, whereas c₁ and c₂ are the acceptance numbers associated with the first and the second sample, respectively. Let λ be the unknown average life and λ₀ be the specified average life. A lot is considered to be good if the true unknown average life is more than the specified average life.

We assume that the lot size is large enough to use the binomial distribution to find the probability of acceptance of the lot. In this paper we have considered c₁= 0 and c₂= 2, ie. DASP (c₁= 0 and c₂= 2).

Where, P = F (t, λ) = F ( ) is the function CDF of the Rayleigh distribution and the Cumulative Distribution Function (CDF) is,

F(t,λ) = t >0, λ >0

Then,

(1)

For different time ratio t/λ₀ = 0.628, 0.942, 1.257, 1.571, 2.356, 3.141 and different mean ratios λ/λ₀ = 4 ,6, 8, 10 and 12, the cumulative distribution function of the Rayleigh distribution is calculated using the formula (1).The probability of acceptance L(p₁) and L(p₂) of DASP are calculated using above values. The probability of acceptance for DASP is given by P (A) = P (no failure occur in sample 1) + P (1 failure occur in sample 1 and 0, 1 failure occur in sample 2) + P (2 failures occur in sample 1 and 0 failure occurs in sample 2).

The probability of acceptance DASP (n₁, n₂, 0,2) is given in formula( 2)

(2)

Figure 1. Minimal angle for given P1 and P2

4. Operating Characteristics Function

The probability of acceptance can be regarded as a function of the deviation of the unknown average life λ₀ from its specified average life λ_. This function is called Operating Characteristic (OC) function of the sampling plan. For different time ratio t/λ₀ = 0.628, 0.942, 1.257, 1.571,2.356 and 3.141 the parameters n₁ and n₂ are determined using minimum angle method.

Notation:

5. Minimum Angle Method

The practical performance of a sampling plan is revealed by it operating characteristic curve. Norman Bush et. al.[7] have used different techniques involving comparison of some portion of the OC curve to that of the ideal curve. The approach of minimum angle method by considering the tangent of the angle between the lines joining the points Acceptable Quality Level (AQL, 1- α) and Limiting Quality Level (LQL β) is shown in Figure 1, where p₁ = AQL, p₂ = LQL. By employing this method one can get a better discriminating plan with the minimum angle. Tangent of angle made by lines AB and AC is

(3)

The smaller the value of this tan θ, closer is the angle θ approaching zero and the chord AB approaching AC, the ideal condition through (AQL, 1-α). This criterion minimizes simultaneously the consumer’s and producer’s risks. Thus both the producer and consumer favor the plans evolved by the criterion.

In this paper we design parameters of the double acceptance sampling plan based on truncated life tests for Rayleigh distribution, using minimum angle method. The minimum angle method of the double sampling plan under Rayleigh distribution for truncated life test is given below. Let us assume mean ratio λ/λ₀ (4,6,8,10,12) and the consumer’s risk β ≤ .10 and producer’s risk α = 0.05 are specified. The probability of acceptance L (p₁) and L (p₂) is placed in Table 1 to Table 5 for c₁ = 0 and c₂ = 2 and the time ratios t/λ₀ = 0.628, 0.942, 1.257, 1.571,2.356, 3.141.

From the Tables 1, 2, 3, 4 and 5 it can be noted that from the given values for fixed mean ratio and various time ratios. We select the parameters corresponding to minimum angle.

Table 1. Minimum Angle Double Sampling Plan Under Rayleigh Distribution for C1 = 0 & C2 = 2

Table 2. Minimum Angle Double Sampling Plan Under Rayleigh Distribution for C1 = 0 & C2 = 2

Table 3. Minimum Angle Double Sampling Plan Under Rayleigh Distribution for C1 = 0 & C2 = 2

Table 4. Minimum Angle Double Sampling Plan Under Rayleigh Distribution for C1 = 0 & C2 = 2

Table 5. Minimum Angle Double Sampling Plan Under Rayleigh Distribution for C1 = 0 & C2 = 2

6. Designing DSP Based on Truncated Life Tests for the Rayleigh Distribution Using Minimum Angle Method

First let us fix the value of time ratio t/λ₀ and mean ratio λ/λ₀ corresponding to c₁ = 0 and c₂ = 2. Where the mean ratio λ/λ₀ = 2,4,6,8,10 and 12 be the acceptable reliability level (ARL) at the producer’s risk and the mean ratio λ/λ₀ which is equal to 1, be the Lot Tolerance Reliability Level (LTRL) at the consumer’s risk.

♦ The parameters n₁ & n ₂ can be obtained from the table along with producers and consumers risk.

♦ First select the time ratio the t/λ₀

♦ Select the parameter of the sampling plan corresponding to smallest value of θ.

♦ Construction of Tables

The Tables are constructed using OC function for Double sampling plans under Rayleigh distribution is given by the equations (1), (2) & (3). Using the above values the minimum angle tan θ is calculated using the equation (3). For various time ratios t/λ₀ and mean ratios λ/λ₀ the parameter values n₁ and n₂ are obtained by DASP under Rayleigh Distribution for c₁ = 0 and c₂ = 2 and are presented in Table 1 to Table 5. Numerical value in these tables reveals the following facts.

For given mean ratio and time ratio c₁ = 0 and c₂ = 2, values in Tables 1-5 can be used to select the parameters of Double sampling plan under Rayleigh distribution for certain specified values of AQL and LQL. The parameters n₁, n₂ and θ can be obtained from the selected table corresponding to λ/λ₀ along with producer’s risk and consumer’s risk.

Example 1: Suppose one wants to design Double sampling plan under Rayleigh distribution for given α = .02, β = .01, λ/λ₀ = 4, given t/λ_{0 =} 0.628, c₁ = 0, c2 = 2. From Table 1, one can observe that the minimum angle is θ = 45.3166° it corresponds to n₁= 22 n₂=28. Thus the required sampling plan has parameters (22, 28, 0, 2).

Example 2: For given λ/λ₀ = 6, α = 0.007, β = 0.001 from Table 2, one can observe the minimum angle is θ = 44.91877^0. It corresponds to n₁ = 35 n₂ = 40. Thus the required sample plan has parameters (35, 40, 0, 2).

Example 3: For givenλ/λ₀ = 8, α = 0.001, β = 0.002 from Table 3, one can observe the minimum angle is θ = 44.91254^°. It corresponds to n₁ = 32 n₂ = 37. Thus the required sample plan has parameters (32, 37, 0, 2).

Example 4: For given λ/λ₀ = 10, α = 0.001, β = 0.009 from Table 4, one can observe the minimum angle is θ = 45.15134^0. It corresponds to n₁ = 25 n₂ = 27. Thus the required sample plan has parameters (25, 27, 0, 2).

Example 5: For given λ/λ₀ = 12, α = 0.0002, β = 0.071 from Table 5, one can observe the minimum

angle is θ = 44.99549^0. It corresponds to n₁ = 35 n₂ = 39. Thus the required sample plan has parameters (35, 39, 0, 2).

7. Conclusions

In this paper designing of double sampling plan for truncated life tests by using minimum angle method is presented. It is assumed that life times of the items follow Rayleigh distribution. It can be seen that by applying minimum angle method there is a great reduction in the sample sizes and at the same time this criterion minimizes simultaneously the consumer’s and producer’s risk. This minimum angle method plan provides better discrimination of accepting good lots