Repetitive Deferred Variables Sampling Plan Indexed By Six Sigma Quality Levels

Repetitive Deferred Variables Sampling Plan Indexed By Six Sigma Quality Levels

D. Senthilkumar*, A. Prakash and B. Esha Raffie

Department of Statistics, PSG College of Arts & Science, Coimbatore-614014, TN, India.

Abstract- This article proposes a repetitive deferred variable sampling plan (RDVSP) indexed by six sigma quality levels for the inspection of normally distributed quality characteristics. In this plan, the acceptance or rejection of a lot in the deferred state is dependent on the inspection results of the preceding or succeeding lots under repetitive group sampling (RGS) inspection is designing plan indexed by Six Sigma AQL (SSAQL, 1-) and Six Sigma LQL (SSLQL, ) is indicated. Sampling plan tables are constructed for the selection of parameters indexed by SSAQL and SSLQL in the case of known and unknown standard deviation and sigma values.

Keywords – Acceptable quality level, sampling plan, consumers risk, limiting quality level, producers risk. Repetitive group sampling, Deferred sampling, variables sampling plan, Six Sigma AQL and Six Sigma LQL.

Acceptance sampling was mainly designed to decide whether to accept or reject a lot on the basis of information provided by the sample taken from the particular lot. Acceptance sampling plan may be classified by attributes and variables. Acceptance sampling plan for attributes means that items will be judged as defective/bad or non-defective/well. Further, a sampling plan may be either type are Acceptance Rejection type or Acceptance -Rectification type. In an acceptance-rejection sampling inspection plan, lots are either accepted or rejected by the sample. In an acceptance rectification sampling plan if we do not accept on the basis of the sample, we take recourse to 100% inspection and in either case replace all defectives by non-defectives.

The RDS plan has been developed by Sankar and Mahopatra (1991) and this plan is essentially an extension of the Multiple deferred sampling plan MDS(c1, c2) which was proposed by Rambert and Vaerst (1981). In this plan, the acceptance or rejection of a lot in deferred state is dependent on the inspection results of the preceding or succeeding lots under repetitive group sampling (RGS) inspection. RGS is the particular case of RDS plan. Wortham and Baker (1976) have developed multiple deferred state sampling (MDS) plans and also provided tables for construction of plans. Suresh (1992) has proposed procedures to select deferred state sampling plan indexed through AQL and LQL. Suresh (1993) has proposed procedures to select Multiple Deferred State Plan of type MDS and MDS1 indexed through producer and consumer quality levels

considering filter and incentive effects. Senthulkumar et al

• Production is steady so that results on current preceding and succeeding lots are broadly indicative of a continuing process.

• Lots are submitted substantially in the order of production.

• Normally lots are expected to be essentially of the same quality.

• Inspection is by variables with quality defined as the fraction non – conforming.

  Basic Assumptions

• The quality characteristic is represented by a random variable X measurable on a continuous scale.

• Distribution of X is normal with mean and standard deviation

• An upper limit U has been specified and a product is qualified as defective when X>U.

  [When the lower limit L is specified, the product is a defective one if X < L]

• The purpose of inspection is to control the fraction defective p in the lot inspected.

When the conditions listed above are satisfied the fraction defective in a lot will be defined by

P=1-F (v) with v = (U-µ) / and

2

F(y) = y 1 exp(z2/ 2)dz (1)

Where Z -N(0,1). It is to be recalled, here, that the criterion for the -method variable plan is to accept the lot if X + k1 U or X k1 L, when the upper specification limit, U or the lower specification limit, L is specified.

Operating Procedure

Step 1: From each submitted lot take a random sample of size n

n

U X X

(2015) have studied Repetitive Deferred Variables Sampling (RDVS) plan. The resulting plan would be designated as RDVSP and would be applied under the following conditions.

(x1,x2,x3) say and Compute v =

o Where, X = i=1 i

n

11 21 11 21 11 11

Step 2: Accept the lot of v k1

and reject the lot if v < k2

P ( p )

(w ) 1(w ) (w )i (w ) (w ) (w )i

1

7

21 11

a 1 1(w ) (w )i

(w ) 1(w ) (w )i (w ) (w ) (w )i

Step 3: If k2 v < k1 accept the lot provided i proceeding or Pa( p )

2

succeeding lots are accepted under RGS inspection plan.

12

22 12 22 12

22 12

1(w ) (w )i

12

8

Otherwise reject the lot.

Thus, the RDVS plan has the parameters of the sample size n and the acceptable criterion k1 and k2. The RDS plan for variables is simply designated as RDVSP (n; k1, k2).

Operating Characteristic Function

According to Sankar and Mohoputra (1991) the OC function of RDS is given

Pa(1Pc)i+ PcPi

Here w11 is the value of w1 at p=p1, w21 is the value of w2 at p=p1, w12 is the value of w1 at p=p2 and w22 is the value of w2 at p=p2, That is,

w11 = ( Zp1 k1 ) , w 21 = ( Zp2 k2 ) , w12 = ( Zp1 k1 ) , w22 = ( Zp2 k2 )

And Zp is the standard normal variate having upper tail probability of p. By fixing the probability of acceptance of the

Pa(p) ==

a

(1Pc)i

(2)

lot, Pa(p) as 99.99% with normal distribution, where the value of v1 at SSAQL and the value of v2 at SSLQL. For example, if

Under the assumption of normal approximation to the non-

central t distributes the values of Pa and Pc are respectively given by,

p1 and p2 are prescribed, then the corresponding value of v1 and v2 will be fixed and if Pa(p1) and Pa(p2) are fixed at 99.99966% and 0.000068% respectively, then we have

(w ) 1(w ) (w )i (w ) (w ) (w )i

P ( p )

11

21 11

21 11

11

0.9999966

9

a 1 1(w ) (w )i

P = (w ) = P [ v k

/ p ] (3)

21 11

a 1 r 1

and

Pc = [(w2) – (w1) ] = Pr [ k2 v < k1 / p] (4)

(w ) 1(w ) (w )i (w ) (w ) (w )i

P ( p )

12

22 12 22 12

12

0.0000068

10

Where, P

= P [ v k

a 2

] is the probability of

1(w ) (w )i

a r 1

22 12

accepting a lot based on under VRGS plan with parameters ( n , k1) and Pc = Pr [ k2 < v < k1a / p ] is the probability of repeating the sample a lot based on under VRGS plan with parameters ( n, k1 , k2, i).

When the expression of Pa and Pc are substituted in equation (2), the OC function of known RDVS ( n; k1 , k2 ) would become

P {v k / p}1 P {k v k / p}i P {k v k / p}P {v k / pi

For given SSAQL and SSLQL, the parametric values of the RDVS plan namely k1, k2 and the sample size n are determined by using a computer search.

Selection of known RDVS Plan Indexed by SSAQL and SSLQL Example

Table 1 can be used to determine RDVS (n; k1, k2, i) for specified values of SSAQL and SSLQL. For example, if it is desired to have a RDVS (n; k1, k2, i for given p1=0.00005

p2=0.00006 and i=1, =0.00034%, =0.00068%, Table 1 given

( p) r

1

r 2

1

r 2

1 r

1

(5)

r 2 1

Pa 1 P {k v k / p}i

Where the first term in the right hand side represents the probability of accepting a lot based on a RGS and the second term is the probability of accepting a lot based on the states of the preceding lots. The probability of acceptance of the lot can be written as

n=417, k1=4.156, k2=4.150 as desired plan parameters with sigma level is 3.8. For the above example, the plan is operated as follows. Take a random sample of size 417 and compute, v

= U X/. Accept the lot if v4.153 and reject the lot if v<4.150. If 4.150v<4.156 accept the lot provided i proceeding or succeeding lots are accepted under RGS inspection plan.

( ) 1( ) ( )

w

w w i

( ) ( )

( )

w w

w i

where

Explanation

( p) 1

2 1

2 1 1

(6)

In screw manufacturing company if the manufacturer

2 1

Pa 1(w ) (w )i

w1 = (U – k1-µ) / = (v k1)

w2 = (U k2-µ) / = (v k2)

and v = ( U – µ) /

If SSAQL, SSLQL, the producers risk () and the consumers risk () are prescribed then we have

fixes the incoming quality p1=0.00005 and p2=0.00006 (5 to 6 defects out of one lack). Then take a sample of size 417 screw from a given lot. Then the lot has accepted if v4.156 and reject the lot if v<4.150. If 4.150v<4.156 accept the lot provided i proceeding or succeeding lots are accepted under RGS inspection plan.

RDVS with unknown variables plan as the reference plan

If the population standard deviation is unknown, then it is estimated from the sample standard deviation S (n-1 as the divisor). If the sample size of the unknown sigma variables

system (S-method) are ns the acceptance constants are k1s and k2s, then the operating procedure is as follows:

Step 1: From each submitted lot take a random sample of size

The design parameters of the unknown sigma variables RDS plan, namely n, i, k1s, k2s can be determined by solving the following optimization problem when SSAQL and SSLQL are

specified.

n(x1, x2,.xn) say and compute v =

U X

i

Where, = =1

Plotting the OC Curve

Step 2 : Accept the lot if v k1s and reject the lot if v < k2s (K1s>k2s)

Step 3 : If k2s < v < k1s accept the lot provided i proceeding or succeeding lots are accepted under RGS inspection plan. Otherwise reject the lot.

0

1E-20

Probability of Acceptance (pa(p))

Thus, the proposed unknown sigma variables RDVS plan is characterized by four parameters, namely, ns, i, k1s, If k1s and k2s. If k1s = k2s then the proposed plan reduces to the RGS variables sampling plan with unknown standard deviation. The determination of parameters of the unknown sigma plan is slightly different from the unknown sigma case. It is known that X+k2sS for a large sample size approximately follows (see Hamaker(1979), Duncan,(1986))

The OC curve for the repetitive deferred variables sampling plan with i=2, n=373, k1=4.367, k2=4.359

1.2

1

0.8

0.6

0.4

0.2

+

~ ( + 2 + 2 2)

2E-05

1E-05

-1E-05

1

1

1 2

Fraction Defective (p)

Therefore, the probability of accepting a lot based on a single sample is given approximately by,

P{X U k

| p}

U- k1s s-

Figure 1: The OC curve of Repetitive deferred (n; k1, k2, i) variables sampling plans where n = 373, k1 = 4.367, k2 = 4.359, i=2.

1s / n

1+ k2 / 2

s 1s

1 2k / 2

p 1s

= z k ns

(11)

Behavior of OC curve

Figure 1 shows that the OC curve of Repetitive Deferred (n; k1, k2, i) variables Sampling plan with i=1,

1s

Analogously to equation (6), the lot acceptance probability for the sigma unknown case is given by

(w ) 1(w ) (w )i (w ) (w ) (w )i

n=373, k1=4.367, k2=4.359.

Figure 2 shows that the comparison of OC curves of the above three nearly equivalent variables RDVS plans having different values of i. It can be seen that the RDVS plan with a

larger value of i seems to be closer to the ideal OC curve in that

( p)

1s

2s 1s 2s

1s

1s

(12)

2s 1s

Pa 1(w ) (w )i

Where,

(w2s) = z k ns

the probability of acceptance at SSAQL, or smaller is increased and the slope at levels between SSAQL and SSLQL becomes sharper although it will require larger sample size. The decision upon choice of the value of i when implementing the plan could be made by considering the sample size and its OC curve.

p

(w1s) = zp

2s

k1s

1 k2 / 2

2s

ns

1s

1 k2 / 2

Here w11s the value of w1s at p=p1, w21s is the value of w2s at p=p1, w12s is the value of w1s at p=p2 and w22s is the value of w2s at p=p2. That is , If (SSAQL, 1-) and (SSLQL, ) are prescribed, then we require

(w ) 1(w ) (w )i (w ) (w ) (w )i

and

P ( p )

11

21 11

21 11

11

1

13

21 11

a 1 1(w ) (w )i

(w ) 1(w ) (w )i (w ) (w ) (w )i

P ( p )

12

22 12

22 12

12

14

22 12

a 2 1(w ) (w )i

1.2

1

0.8

0.6

0.4

0.2

1

2

3

0

0

0.00001 0.00002 0.00003

Figure 2: shows the comparison of OC curves of 1) i=1, n=386, k1=4.375, k2 = 4.348;2) i=2, n=373, k1=4.367, k2=4.359; 3) i=3, n=362, k1=4.362, k2=4.354;

Construction of table

The OC function of the variables RDS sampling plan, which gives the proportion of lots are expected to be accepted for a given lot quality p, is obtained by

CONCLUSION

Acceptance sampling is the technique which deals with procedures in which decision to accept or reject lots or process based on the examination of samples. The percent work mainly relates to the construction and selection of tables for Six Sigma Repetitive Deferred Variables Sampling (SSRDVS n, k1, k2,

i) plan through the SSAQL and SSLQL. Tables are provided

here which tailor made, handy and ready-made use to the industrial shop-floor condition.

REFERENCES

1. A.L. Christina, Contribution to the study of Design and Analysis of Suspension System and Some other Sampling Plans, Ph.D. Thesis, Bharathiar University, Tamil Nadu, India, 1995.

2. A.W. Wortham, and R.C. Baker, Multiple Deferred State Sampling Inspection. Int. J. Prod. Res. 14(6), pp.719-731, 1976.

3. D. Senthilkumar, S.R. Ramya and B. Esha Raffie Construction and Selection of Repetitive Deferred Variables Sampling (RDVS) Plan Indexed by Quality Levels, Journal of Academia and Industrial Research (JAIR), Volume 3, Issue 10, pp.497-503. 2015

4. G. Sankar, and B.N. Mahopatra, GERT analysis of repetitive deferred sampling plans. IAPQR Trans. 16(2): pp. 17-25, 1991.

5. K.K. Suresh, A study on Acceptance Sampling using Acceptable and Limiting Quality Levels, Ph.D. Thesis, Bharathiar University, Tamil Nadu, 1993.

6. K.K. Suresh, and R. Saminathan, Construction and selection of Repetitive Deferred Sampling plan through AQL and LQL, Int. J. Pure

   ( p) r 1 r 2 1 r 2 1 r 1

   P {v k / p}1 P {k v k / p}i P {k v k / p}P {v k / pi

   r 2 1

   Pa 1 P {k v k / p}i

   (15)

   Where

   Appl. Math. 65(3): pp.257-264, 2010.

7. R. Vedaldi, A new criterion for the construction of single sampling inspection plans by attributes. Rivista di Statistica Applicata. 19(3), pp.235-244. 1986.

   the first term in the right hand side represents the probability of

   accepting a lot based on a RGS and the second term is the probability of accepting a lot based on the states of the preceding lots. The probability of acceptance of the lot can be written as

8. Rombert and Vaerst. A procedure to construct Multiple deferred state sampling plan. Meth. Operat. Res. 37: pp. 477-485. 1981.

9. S. Balamurali, and C.H. Jun, Multiple dependent state sampling plans for lot acceptance based on measurement data. Euro. J. Operat. Res. 180: pp.1221-1230. 2006.

10. V. Soundararajan, and V. Ramasamy, Procedures and tables for

P (p) =

(1)[1(2)+(1)]+[ (2)(1)][(1)]

a [1(2)+(1)]

(16)

construction and selection of Repetitive Group Sampling (RGS) plans. QR J. 13(13), pp.14-21, 1986.

Where w1 = ( U – k1-µ ) / = ( v k1)

w2 = ( U k2-µ ) / = ( v k2)

and v = ( U – µ) /

If SSAQL, SSLQL, the producers risk () and the consumers risk () are prescribed then we have

P (p )=(w11)[1(w21)+(w11)]i+[ (w21)(w11)][(w11)]i=1- (17)

a 1 [1(w21)+(w12)]i

P (p ) = (w12)[1(w22)+(w12)]i+[ (w22)(w12)][(w12)]i = (18)

a 2 [1(w22)+(w12)]i

For given SSAQL and SSLQL, the design parameters of the variables MDS plan, namely n, m, ka and kr may be determined by satisfying the required producer and consumer conditions in equations (9) and (10). Optimization problem to determine the parameters is formulated as follows:

Values of the parameters (n, k1, and k2) when i=1 and when sigma is known and unknown are tabulated in Table

1. Also i=2 and i=3, the value of the parameters are tabulated in Table 2 and Table 3

Table 1: Variables RDS sapling plans for i=1 indexed by SSAQL and SSLQL

Table 1 (continued )

Table 2: Variables RDS sapling plans for i=2 indexed by SSAQL and SSLQL

Table 2 (continued )

Table 3: Variables RDS sampling plans for i=3 indexed by SSAQL and SSLQL

Table 3 (continued )