# Construction of chain sampling plan -1 indexed through convex combination of AOQL and MAAOQ.

1. IntroductionDodge (1955) developed ChSP-1 to improve the single sampling plans having c=0 without increasing the sample size. Clark (1960) has given a procedure for the selection of ChSP-1. Fred Frishman(1960) developed two types of chain sampling plans which are extension of ChSP-1 plan in which rejection number greater than 1 is included. Soundararajan (1978a, 1978b) constructed tables for selection of ChSP-1. Soundararajan and Govindaraju (1982) have made contribution in designing ChSP-1 plans. Soundararajan and Doraiswamy (1984) have constructed ChSP-1 indexed by inflection point and Point of Control

Mandelson (1962) explained a system of sampling plans indexed through Maximum Allowable Percent Defective. Mayer(1967) suggested that the quality standard 'p' can be considered as a quality level, along with certain other conditions to specify an operating characteristic curve later studied by Soundararajan (1975) as the quality level corresponding to the inflection point of the OC curve. One of the desirable properties of an OC curve is that the decrease of the probability of acceptance should be slower for lesser values of 'p' (good quality level) and steeper for larger values of 'p' (bad quality level), which provides a better overall discrimination. If [p.sup.*] is considered as a standard quality measure then the property of a desirable OC curve is exactly followed. In the literature, the AOQL is defined as the worst average quality that the consumer will receive in the long run, when defective items are replaced by non-defective items. Dodge and Romig (1959) have proposed a procedure for the selection of a single sampling plan indexed through AOQL. Maximum allowable average Outgoing quality (MAAOQ) is the average outgoing quality at the inflection point. The use of MAAOQ for deriving sampling plan was justified by Suresh and Ramkumar(1996). Radhakrishnan (2002) studied various sampling plans indexed through MAAOQ and established that the sample size is less in the inspection when MAAOQ is used as an indexed parameter than AOQL. Sekkhizhar (2007) constructed sampling plans using MAAOQ with Intervened Random effect Poisson Distribution. Sampath kumar(2007) constructed Mixed Sampling plan with ChSP-1 as attribute plan indexed through AOQL, MAPD and MAAOQ. Radhakrishnan and Mallika(2008,2009) constructed single sampling plan indexed through [AOQ.sub.cc] which is the convex combination of AOQL and MAAOQ, which may safeguard the interests of both producer as well as consumer by properly choosing a right combination using the gain parameter [lambda].

2. Definitions

2.1 Definition of AOQL

The AOQL is defined as the worst average quality that the consumer will receive in the end, when defective items are replaced by non-defective items. It is obtained by maximizing Average Outgoing Quality (AOQ), AOQ= p.[P.sub.a](p)

2.2 Definition of MAPD

The MAPD is the value of fraction defective (p=[p.sup.*]) at which [d.sup.2][P.sub.a](p)/ d[p.sup.2] = 0 and [d.sup.3][P.sub.a](p)/d[p.sup.3] [not equal to], 0

It is also the inflection point of the operating characteristic (OC) curve and [P.sub.a](p) is the probability of acceptance at the quality level p fraction defective

2.3 Definition of MAAOQ

The MAAOQ is defined as average outgoing quality at MAPD, we have MAAOQ = AOQ at p = [p.sup.*].

3. Operating procedure of ChSP-1.

The Operating procedure for ChSP-1 is as follows

Step 1: Select a random sample of size 'n' from a lot of size 'N'

Step 2: Inspect all the articles included in the sample. Let 'd' be the number of defectives in the sample.

Step 3: If d = 0, accept the lot

Step 4: If d [greater than or equal to] 2, reject the lot

Step 5: Accept the lot if d = 1 and if no defective units are found in the immediately preceding i samples of size 'n'

4. Glossary of Symbols

N -lot size

n -sample size

d -number of non-conformities

[lambda] -gain parameter, 0<[lambda]<1

p -submitted lot quality of lot or process

[p.sup.*] -Maximum allowable percent defective

[P.sub.a](p) -probability of acceptance for given quality p

[AOQ.sub.cc] -convex combination of AOQL and MAAOQ

i - Number of previous samples from which the decision of lot is made.

5. Operating characteristic function of ChSP-1

Under Poisson model, the OC function of ChSP-1 plan is given by

[P.sub.a](p) = [P.sub.0,n] + [P.sub.1,n][([P.sub.0,n]).sup.i]

6. Construction of ChSP-1 indexed through [AOQ.sub.cc]

The general procedure for designing a ChSP-1 indexed through a parameter which is a convex combination of AOQL and MAAOQ using Poisson distribution as base line distribution is given below

Step 1: Determine n[p.sup.*], nMAAOQ and nAOQL for ChSP-1 for various values of i, find [R.sub.1] = nAOQL/nMAPD and [R.sub.2] = nMAAOQ/nMAPD

Step 2: Find n[AOQ.sub.cc] = [lambda]nAOQL+ (1 - [lambda])nMAAOQ for various values of [lambda] and find [R.sub.3] = n[AOQ.sub.cc]/nMAPD

Step 3: Present the results of Step 1 and Step 2 for various values of [lambda] (from 0.1 to 0.9) in Table 1

7. Selection of the plan

Table 1 is used to construct the plan when the MAPD and [AOQ.sub.cc] are specified. One can find the ratio [R.sub.3] = [AOQ.sub.cc]/MAPD and locate the value in Table 1 under the column [R.sub.3] (for a fixed values of [lambda]) and the corresponding values of i and n[p.sup.*] are noted. The value of n is determined using n = n[p.sup.*]/MAPD and hence the parameters, n and i are determined

Example 1: For a specified AOQL = 0.0060 and MAPD =0.0067 compute the ratio [R.sub.1] = AOQL/MAPD = 0.8955 which is nearer to the value 0.8950 and the associated values of i and n from the Table 1 are i=1 and n= n[p.sup.*]/MAPD = 84. Thus the sampling plan for specified AOQL = 0.0060 is n = 84 and i = 1

For a specified MAAOQ = 0.0059 and MAPD =0.0067 compute the ratio [R.sub.2] = MAAOQ/MAPD = 0.8806 which is nearer to the value 0.8786 and the associated values of i and n from the Table 1 are i=8 n= n[p.sup.*]/MAPD = 26. Thus the sampling plan for specified MAAOQ = 0.0059 is n = 26 an i = 8

For a specified value of AOQL = 0.0060, MAAOQ = 0.0059, MAPD =0.0067 and [lambda] = 0.2 the value of [AOQ.sub.cc] = 0.00592. Find the ratio [R.sub.3] = [AOQ.sub.cc]/MAPD = 0.8896 and locate the value in [R.sub.3] corresponding to [lambda]=0.2 which is nearer to the ratio 0.8688 and the value of i and n from the Table 1 are i=2 and n = n[p.sup.*]/MAPD = 62. Thus the sampling plan for specified [AOQ.sub.cc] = 0.00592 is n = 62 and i = 2.

Explanation: In a Tube light manufacturing company, if the producer fixes the quality level MAPD=0.0067 (67 non-conformities out of 10,000) and the consumer fixes the quality level [AOQ.sub.cc] = 0.00592 (592 non-conformities out of 1,00,000) select a sample of 62 tube lights from a lot and count the number of non-conformities(d). If d=0 accept the lot and if d [greater than or equal to] 2 reject the lot and inform the management for improving the quality of the product. If d=1 accept the lot if no non-conformities are found in the immediately preceding 2 samples.

The Operating characteristic curve for the plans given in the example 1 are presented in figure 1 and their AOQ curves are presented in figure 2.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

Conclusion

In this paper a procedure for the construction and selection of ChSP-1 indexed through gain parameter [lambda] which is a convex combination of AOQL and MAAOQ is stated. Tables are also constructed for the easy selection of the plans when the indexing parameters and gain parameter are known. The engineers after knowing the minds of both producer and consumer can search for the quality level AOQcc and select the appropriate plan. It is also concluded from the study that the sample size required for the ChSP-1 indexed through(MAPD, AOQcc) is lesser than the sample size indexed through (MAPD, AOQL) which also satisfies the interests of both the producer as well as the consumer, which can be understood from the OC and AOQ curves. Readymade tables are also provided in this paper for the engineers to take quick decisions on the nature of the sampling plan when the quality level of the producer and consumer are known. This study can be extended for constructing other sampling plans and efficiency of these plans can also be compared with the plans indexed through other parameters.

References

[1] Dodge, H.F. and Roming, H. G., 1959, "Sampling Inspecting Tables-Single Double Sampling", 2nd edition, John Wiley and sons, New York.

[2] Dodge, H.F., 1955, "Chain Sampling Inspection Plan", Industrial Quality Control, 11(4), pp.10-13

[3] Clark.C.R., 1960, "OC curves for ChSp-1, Chain Sampling Plans", Industrial Quality Control 17 (4) pp.10-12

[4] Frishman, Fred., 1960, "An extended Chain Sampling Plan", Industrial Quality Control 17 (1) pp.10.12

[5] Mandelson, J., 1962, The Statistician, "The Engineer and sampling plans", Industrial Quality control 19 (5), pp.12-15

[6] Mayer.P.L., 1967, "A note on sum of Poisson Probabilities and an application", Annals of Institute of Statistical Mathematics, 19,537-542

[7] Radhakrishnan, R., 2002, "Contribution to the study on selection of certain acceptance sampling plans", Ph.D thesis, Bharathiar University, Coimbatore, India.

[8] Radhakrishnan, R. and Mallika, M., 2008, "Designing of Sampling plans indexed through the convex combination of AOQL and MAAOQ", Published as a proceedings of the National level Conference on IT and Business Intelligence organized by Institute of Management Technology, Nagpur, Maharashtra, India

[9] Radhakrishnan, R. and Mallika, M., 2009 "Construction of Sampling Plans indexed through AOQcc", International Journal of Statistics and Systems(Accepted for Publication)

[10] Sampathkumar, R., 2007, "Construction and selection of mixed variables- attributes sampling plans", Ph.D Thesis, Bharathiar University, Coimbatore, India.

[11] Sekkizhar, J., 2007, "Designing of sampling plans using Intervened Random effect Poisson Distribution", Ph.D Thesis, Bharathiar University, Coimbatore, India.

[12] Soundararajan, V., 1978 a, "Procedures and Tables for construction and selection of Chain Sampling plans(ChSp-1) Part-I", 'Journal of Quality Technology, 10(2) pp-56-60

[13] Soundararajan, V., 1978b, "Procedures and Tables for construction and selection of Chain Sampling plans (ChSP-1)", Part II, 'Journal of Quality Technology, 10(3) pp.99-103

[14] Soundararajan, V., 1975, "Maximum Allowable percent defective (MAPD) Single sampling Inspection by Attributes plans", Journal of Quality Technology, 7 (4) pp.173-177

[15] Soundararajan. V., and Doraiswamy, P., 1984, "Chain Sampling Plans(ChSp-1) indexed by Inflection point and point of control", 9(3) pp.121-123

[16] Soundararajan, V., and Govindaraju, K., 1982, "A note on Designing Chain Sampling Plan (ChSP-1)", The QR Journal, 9(3) pp. 121-123

[17] Suresh K.K., and Ramkumar T.B., 1996, "Selection of a sampling plan indexed with Maximum Allowable Average Outgoing Quality", Journal of Applied Statistics, 23 (6) pp.643-652.

(1) R. Radhakrishnan and (2) M. Mallika

(1) Reader in Statistics (2) Research Scholar (1,2), P.S.G. College of Arts and Science, Coimbatore, Tamil Nadu, India.

(1) Email: rkrishnan_cbe@yahoo.com

Table 1: Characteristics of Chain sampling plan - 1. n i [np.sup.*] AOQL [R.sub.1] 1 0.562 0.503 0.8950 2 0.413 0.42 1.0169 3 0.331 0.388 1.1722 4 0.278 0.375 1.3489 5 0.241 0.371 1.5394 6 0.213 0.369 1.7324 7 0.191 0.368 1.9267 8 0.173 0.368 2.1272 9 0.158 0.368 2.3291 10 0.146 0.368 2.5205 n i [np.sup.*] MAAOQ [R.sub.2] 1 0.562 0.423 0.7527 2 0.413 0.323 0.7821 3 0.331 0.267 0.8066 4 0.278 0.23 0.8273 5 0.241 0.203 0.8423 6 0.213 0.182 0.8545 7 0.191 0.165 0.8639 8 0.173 0.152 0.8786 9 0.158 0.14 0.8861 10 0.146 0.131 0.8973 [lambda]=0.1 i [np.sup.*] [AOQ.sub.cc] [R.sub.3] 1 0.562 0.4387 0.7806 2 0.413 0.3421 0.8282 3 0.331 0.2907 0.8782 4 0.278 0.2577 0.9269 5 0.241 0.2332 0.9677 6 0.213 0.2149 1.0091 7 0.191 0.1996 1.0452 8 0.173 0.1871 1.0814 9 0.158 0.1771 1.1209 10 0.146 0.1685 1.1542 [lambda]=0.2 i [np.sup.*] [AOQ.sub.cc] [R.sub.3] 1 0.562 0.4525 0.8052 2 0.413 0.3588 0.8688 3 0.331 0.3112 0.9401 4 0.278 0.2816 1.0131 5 0.241 0.2600 1.0789 6 0.213 0.2438 1.1444 7 0.191 0.2302 1.2053 8 0.173 0.2193 1.2675 9 0.158 0.2105 1.3326 10 0.146 0.2031 1.3909 [lambda]=0.3 i [np.sup.*] [AOQ.sub.cc] [R.sub.3] 1 0.562 0.4646 0.8267 2 0.413 0.3734 0.9041 3 0.331 0.3291 0.9941 4 0.278 0.3027 1.0888 5 0.241 0.2837 1.1773 6 0.213 0.2696 1.2660 7 0.191 0.2581 1.3514 8 0.173 0.2488 1.4380 9 0.158 0.2416 1.5292 10 0.146 0.2355 1.6131 [lambda]=0.4 i [np.sup.*] [AOQ.sub.cc] [R.sub.3] 1 0.562 0.4750 0.8452 2 0.413 0.3858 0.9343 3 0.331 0.3445 1.0406 4 0.278 0.3210 1.1546 5 0.241 0.3047 1.2645 6 0.213 0.2930 1.3754 7 0.191 0.2835 1.4843 8 0.173 0.2761 1.5960 9 0.158 0.2705 1.7120 10 0.146 0.2658 1.8207 [lambda]=0.5 i [np.sup.*] [AOQ.sub.cc] [R.sub.3] 1 0.562 0.4837 0.8607 2 0.413 0.3964 0.9597 3 0.331 0.3575 1.0801 4 0.278 0.3367 1.2112 5 0.241 0.3232 1.3411 6 0.213 0.3138 1.4731 7 0.191 0.3065 1.6045 8 0.173 0.3009 1.7393 9 0.158 0.2969 1.8790 10 0.146 0.2936 2.0111 [lambda]=0.6 i [np.sup.*] [AOQ.sub.cc] [R.sub.3] 1 0.562 0.4908 0.8733 2 0.413 0.4050 0.9806 3 0.331 0.3683 1.1126 4 0.278 0.3499 1.2588 5 0.241 0.3390 1.4065 6 0.213 0.3318 1.5580 7 0.191 0.3267 1.7103 8 0.173 0.3228 1.8661 9 0.158 0.3203 2.0270 10 0.146 0.3182 2.1794 [lambda]=0.7 i [np.sup.*] [AOQ.sub.cc] [R.sub.3] 1 0.562 0.4962 0.8829 2 0.413 0.4117 0.9967 3 0.331 0.3768 1.1382 4 0.278 0.3606 1.2971 5 0.241 0.3520 1.4605 6 0.213 0.3470 1.6291 7 0.191 0.3436 1.7991 8 0.173 0.3413 1.9728 9 0.158 0.3398 2.1509 10 0.146 0.3387 2.3198 [lambda]=0.8 i [np.sup.*] [AOQ.sub.cc] [R.sub.3] 1 0.562 0.5001 0.8899 2 0.413 0.4165 1.0084 3 0.331 0.3829 1.1569 4 0.278 0.3686 1.3257 5 0.241 0.3619 1.5017 6 0.213 0.3586 1.6837 7 0.191 0.3567 1.8676 8 0.173 0.3555 2.0552 9 0.158 0.3549 2.2461 10 0.146 0.3543 2.4269 [lambda]=0.9 i [np.sup.*] [AOQ.sub.cc] [R.sub.3] 1 0.562 0.5024 0.8940 2 0.413 0.4194 1.0154 3 0.331 0.3868 1.1684 4 0.278 0.3735 1.3436 5 0.241 0.3683 1.5282 6 0.213 0.3662 1.7192 7 0.191 0.3652 1.9122 8 0.173 0.3647 2.1082 9 0.158 0.3646 2.3073 10 0.146 0.3644 2.4956

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Title Annotation: | Maximum allowable average Outgoing quality |
---|---|

Author: | Radhakrishnan, R.; Mallika, M. |

Publication: | Global Journal of Pure and Applied Mathematics |

Geographic Code: | 9INDI |

Date: | Apr 1, 2009 |

Words: | 2824 |

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