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Quick Switching System with Different Reference Plans

Vennila J*, Devaraj Arumainayagam S

KMCH College of Pharmacy, Coimbatore – 641 048, Tamil Nadu, India

Abstract: A new sampling inspection system with normal single sampling plan and tightened double sampling plan is introduced. Advantages of this system are highlighted.

Keywords: sampling plan, sampling system, single sampling plan, double sampling plan

Citation: Vennila J, Devaraj Arumainayagam S, Quick Switching System with Different Reference Plans. American Journal of Applied Mathematics and Statistics, 6 (4). 141-148. January 2018. doi:10.12691/ajams-6-4-4

1. Introduction

Sampling system is a procedure used to accept or reject the lots in the acceptance sampling [1]. Dodge [2] proposed a new sampling inspection system called “Quick Switching System (QSS-1)”. It is operated as follows:

1. Adopt a normal plan (N) and a tightened plan (T), where plan T has a tighter OC Curve than plan N.

2. Use plan N for the first lot. It is also possible to start with plan T when addition protection in the first lot production is desired. The OC curve properties are the same for the both plans.

3. For each is lot inspected; if the lot is accepted, then use plan N for the next lot; and if the lot is rejected, then use plan T for the next lot.

Rombaski [3] studied the QSS-1 with single sampling plan as a reference plan. After comparing the switching rules of many systems, Rombaski [3] made certain modifications on the switching rules of QSS-1. The resulting systems are QSS-2 and QSS-3. These systems are having an operating characteristic curve (OC) which is more discriminating than the corresponding OC curves of normal and tightened plans. Also the sample size required for QSS is much lower than any comparable equivalent sampling plan and system.

2. Literature Review

Soundarajan and Arumainayagam [4,5,6,7,8,9,10,11] have analyzed QSS-r, r=1,2,3 with single sampling plan, double sampling plan, chain sampling plan and repetitive group sampling plan as reference plan. Arumainayagam and Uma [12,13,14,15] studied QSS-r, r=1, 2, 3 with three stage multiple sampling plan as a reference plan. Suresh and Jeyalakshmi [16,17,18] used multiple deferred sampling plans (MDS) as reference plan in QSS-1. Suresh and Kaviyarasu [19,20] have analyzed QSS-r with conditional repetitive group (CRGS) sampling plan, two stage CRGS plan and multiple repetitive groups sampling plan as a reference plans.

Arumainaygam and Vennila [21] have applied two different types of reference plans in QSS-1 and find that the resulting system is more advantages than the system using same reference plan for normal and tightened inspection. This paper extends this method to a system involving the switching roles of QSS-3.

3. Methodology

3.1. Quick Switching Single Double Sampling System (QSSDSS-3)

The new system incorporate the switching rules of QSS-3 with single sampling plan as normal plan and double sampling plan as the tightened plan. This system is called quick switching single double sampling system designated as QSSDSS – 3 (n; c; c₁, c₂). Here n and c are the parameters of the normal plan and n (n=n₁=n₂), c₁ and c2 are the parameters of the tightened plan. The sample sizes of the normal plan and tightened plan are equal where as c1 < c2 < c.

3.2. Conditions for applications

a) The production is stable so that outcome on current and preceding lots are broadly investigative of a continuing process and proposed lots are expected to be effectively of the same quality.

b) Lots are submitted substantially in order of their production.

c) Inspection by attributes is considered with quality defined as fraction nonconforming p.

3.3. Operating Procedure of QSSDSS-3 (n; c; c₁, c₂)

Step 1: From a lot, take a random sample of size n at normal inspection level and count the number of defectives x.

i) If x ≤ c, accept the lot and repeat step 1 for the next lot.

ii) If x > c, reject the lot and go to step 2.

Step 2: From the next lot, take a random sample of size n at tightened inspection level and count the number of defectives x₁

i) If x₁ ≤ c₁, then accept the lot and repeat step 1 for the next lot

ii) If x₁ > c₂, then reject the lot and continue step 2 for the next lot.

iii) If c₁ < x₁ ≤ c₂, then take a second random sample of size n form the same lot, and count the number of defectives x₂.

iv) If x₁+x₂ ≤ c₂, then accept the lot and repeat step 1 for the next lot

v) If x₁+x₂ > c₂, then reject the lot and repeat step 2 for the next lot.

3.4. Operating Characteristic Function

Based on Romboski [3], the OC function of QSSDSS -3 (n; c; c₁, c₂) is given below

[picture] | (1)

Where

PN = Probability of acceptance of normal single sampling plan

PT = Probability of acceptance of tightened double sampling plan

4. Findings and Discussion

4.1. Properties of OC Curve

The properties of OC curve of QSSDSS-3 are given below

1. Figure 1 – Figure 3 give the normal, tightened and composite OC curve of QSSDSS - 3. The composite OC curve lies between normal and tightened OC curves. For good quality, the normal plan has more probability being applied in the system and hence it is closer to the composite OC curve. From these curves; it is observed that the relationship of the switching systems of composite OC curve is a weighted average of the normal and tightened OC curves. Incorporating the advantages of normal and tightened OC curve the composite OC curve attains more desirable form than the two OC curves.

Figure 1. Normal, Tightened and Composite OC curves of QSSDSS – 3 (n; c; c₁, c₂), QSSDSS – 3 (100, 2, 0, 1)

Figure 2. Normal, Tightened and Composite OC curves of QSSDSS – 3 (n; c; c₁, c₂), QSSDSS – 3 (150, 5, 0, 3)

Figure 3. Normal, Tightened and Composite OC curves of QSSDSS – 3 (n; c; c₁, c₂), QSSDSS – 3 (175, 4, 1, 3)

Figure 4. Composite OC curves of (QSSDSS-3) (n; c; c₁,c₂); 1.QSSDSS-3 (225, 5, 0, 4); 2.QSSDSS-3 (225, 5, 0, 3); 3.QSSDSS-3 (225, 5, 0, 2); 4. QSSDSS-3 (225, 5, 0, 1)

2. Figure 4 give a set of composite OC curves of the QSSDSS. In these curves, the normal plan is fixed and in the tightened plan acceptance number is allowed to decrease. That is tightening is made severe. It is observed that as the value of p increases, the OC curve approaches to the shape of an ideal OC curve.

4.2. Designing the System

4.2.1. Designing Systems Given p₁, p₂, α and β

Table 3 can be used to design QSSDSS – 3 (n, c, c₁, c₂) for given p₁, p₂, α and β by the following steps:

1. Find the value of p₂/p₁.

2. Determine the value of p₂/p₁ in Table 4 in the column for appropriate α and β that is closer to the computed p₂/p₁.

3. Find the values of c, c₁ and c₂ corresponding to the ratio located.

4. Corresponding to the selected c, c₁ and c₂, from Table 3, find the value of np₁

5. The sample size of the system is found by dividing np₁ by p1.

Example:

To obtain QSSDSS – 3 (n, c, c₁, c₂) for the given values of p₁ = 0.03, α = 0.05, p₂ = 0.04 and β = 0.10, the following steps are to be followed:

1. Compute p₂/p1 = 0.04 / 0.03 = 1.33

2. The value of p₂/p₁ which is nearly equal to 1.3 in Table 4 under the column of α = 0.05 and β = 0.10 is 1.3429

3. The value of c, c₁ and c₂ corresponding to 1.3429 are c = 10, c₁= 0, c₂ = 2.

4. For c = 10, c₁= 0, c₂ = 2, value of np₁ obtained from Table 3 is 2.8306.

5. The sample size is determined as n = np₁ / p₁ = 2.8306 / 0.03 = 94.

6. The designed system is QSSDSS – 3 (94; 10, 0, 2).

4.2.2. Calculating AOQL for the given system

Table 5 provides the npm and nAOQL values for QSSDSS – 3 (n, c, c₁, c₂). This table can be used to determine npm and nAOQL of a system.

Example:

Determine the pm and AOQL of QSSDSS – 3 (94, 4, 0, 2). From Table 5, corresponding to c = 10, c₁ = 0 and c₂ = 2, nAOQL = 4.3333 and npm = 4.7960. so AOQL = nAOQL / n = 4.3333 / 94 = 0.046% and pm = npm / n = 4.7960 / 94 = 0.05%.

4.2.3. Designing Systems Given AQL and AOQL

Table 5 can be used to design QSSDSS - 3 (n, c, c₁, c₂) for specified values of AQL and AOQL.

Example

To determine a QSSDSS - 3 (n; c; c₁, c₂), having AQL (α = 0.05) = 0.06 and AOQL = 0.08, compute AOQL/p₁ = 0.08 / 0.06 = 1.3. From Table 5, under the column headed AOQL/p₁, value closer to the desired value is 1.3119, which corresponds to a value of c = 9, c₁ = 0 and c₂ = 3. Corresponding to these parameters, value of np₁ obtained from Table 5 is 3.0601. The normal single sampling plan sample size is obtained by n = np1 / p₁ = 3.0601 / 0.06 = 51. The designed system is QSSDSS - 3 (51; 9 ;0 ,3).

4.2.4. Designing Systems Given Indifference Quality Level (IQL)

Table 5 can be used to design QSSDSS – 3 (n; c; c₁, c₂) indexed by point of control and point of control (h₀) Hamaker [22]

Example

If one wants to design a QSSDSS-3 having p₀ = 0.03 and h₀ = 2.6, from Table 3, under the column headed h₀ find the value which is closer to the desired value. The value is 2.5782 which has associated with it a value of c = 3, c₁ = 0 and c₂ = 2. Corresponding to these parameters, value of np₀ is 1.9059. The sample size of normal double sampling plan is obtained as n = np₀ / n = 1.9059 / 0.03 = 63.53 ≈ 64. The designed system is QSSDSS - 3 (64; 3; 0, 2).

4.2.5. Conversion of One Set of Parameters to Other

For QSSDSS – 3 (n; c; c₁, c₂), if p₁=0.02, p₂=0.04, α=0.05 and β=0.10, the system satisfying the requirements can be obtained from Table 5 as n = 94, c = 10, c₁ = 0 and c2 = 2. Corresponding to c = 10, c₁ = 0 and c2 = 2, from Tables 3, one can get the following:

np₁ = 2.8306, npm = 4.7960, nAOQL = 4.3330

np₀ = 3.3278 and h₀ = 4.8118.

So, AOQL = nAOQL/n = 4.3330/94 = 0.0461

p₀ = np₀ / n = 3.3278/94 = 0.0354.

So, when p₁=0.02, α = 0.05, p₂ = 0.04 and β = 0.10, the other similar sets of parameters are given by

1. p₁ = 0.02 (α = 0.05) and AOQL = 0.0461

2. p₀ = 0.0354 and h₀ = 4.8118.

4.3. Plotting the OC Curve

Table 1 can be used to plot the operating characteristic curve of the given QSSDSS – 3 (n; c; c₁, c₂). This can be done by dividing each entry for given acceptance numbers, by the value of n. The result of each division is the proportion nonconforming p for which the proportion of lots expected to be accepted Pa (p) is shown in the corresponding column heading.

Example

For QSSDSS – 3 (60; 3; 0, 1), division of entries opposite to c=3, c₁=0 and c₂=1 in Table 3 by 100 leads to the following Table-A for plotting the OC curve of QSSDSS-3.

Table 1. Values for OC curve of (QSSDSS - 3) (n; c; c₁; c₂) (n=n₁=n₂) (60, 3, 0, 1)

5. Conclusion

5.1. Comparison

The last three columns of below Table 2 respectively give the values obtained by dividing the np0.95 values of SSP (n; cN, cT) (values are taken from Cameron J.M., [23]), DSP (n; cN, cT) (Duncun A.J., [24]), (US Chemical Corps Agency [25]) and QSS-3 (n; cN, cT) (Romboski L.D. [3] respectively) by the np0.95 value of the matched QSSDSS-3 (n; c, c₁; c₂). Six single sampling plans, double sampling plan, and QSS – 3 their equivalent QSSDSS-3 (n, c, c1, c2) values are presented in the below table. From this, one can observe the following, On comparing the np0.95 values of different systems included in various sets, it is observed that the sample size of QSSDSS-3 (n; c, c₁; c₂) requires lesser sample sizes than those of the corresponding matched SSP (n; cN, cT), DSP (n; cN, cT) and QSS-3 (n; cN, cT). However, in some cases, QSSDSS-3 (n; c, c₁; c₂) requires a smaller sample size than that of QSS-3 (n; cN, cT).

Table 2. Single, Double and QSSDSS – 3 Comparison Values

Table 3. Values of np tabulated against c, c₁ & c₂ for give Pa(p) for QSSDSS – 3 (n, c, c₁, c₂)

Table 4. Values of p₂/p1 tabulated against c, c₁&c₂ for given α and β for QSSDSS-3 (n,c,c₁,c₂)

Table 5. Parametric values for QSSDSS-3 (n,c,c₁,c₂)

5.2. Construction of Table 3, Table 4 and Table 5

Under the assumption of poisson model, the OC function of QSSDSS -3 (n; c; c₁, c₂) is given by

[picture] | (1)

Under the assumption of the poisson model Pa(p) is solved for using the equation (1) in MATLAB program and we get the different np values. The values are presented in Table 3. Using Table 3, for assumed values of α and β, the operating ratio p₂/p₁ is calculated and the values are presented in Table 4. Using the np₀ values given in Table 3, the relative slope h₀ is evaluated and these values are provided in Table 5 for specified values of QSSDSS – 3 (n, c, c₁, c₂).

References

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