Designing of mixed sampling plan with chain sampling plan (ChSP-1) plan as attribute plan using MAPD and discriminant tangential angles

Introduction

A variety of plans and procedures have been developed for special sampling situation involving both measurements and attributes. Each is tailored to do a specific job under prescribed circumstances. They range from a simplified variables approach to a more technically complicated combination of variables and attribute sampling called Mixed sampling plans. Mixed sampling plans are of two types, namely independent and dependent plans. Independent mixed plans do not incorporate first sample results in the assessment of the second sample. Dependent mixed plans combine the results of the first and second samples in making a decision if a second sample is necessary. Mixed sampling plans consist of two stages of rather different nature. During the first stage the given lot is considered as a sample from the respective production process a criterion by variables is used to check process quality. If process quality is judged to be sufficiently good, the lot is accepted, otherwise the second stage of the sampling plan is entered and lot quality is checked directly by means of an attribute sampling plan. MAPD is a key measure assessing to what degree the inflection point empowers the OC curve to discriminate between good and bad lots. The concept of MAPD (p*) was introduced by Mayer (1967) [2] and further studied by Soundararajan (1975) [7] is the quality level corresponding to the inflection point on the OC curve. Mixed Sampling Plan (MSP) was first developed by Schilling (1967) [6] for the case of single sided specifications, standard deviation known by assuming an underlying normal distribution for measurements. Using Schilling's procedure, Devaarul (2003) has constructed tables for mixed sampling plans (independent case) having various sampling plans as attribute plans. Radhakrishnan and Sampath Kumar (2006) [3] have made contributions to mixed sampling plan with chain sampling plan as attribute plan for independent case. Sampath Kumar (2007) [8] has constructed mixed sampling plan with various sampling plans as attribute plans using Poisson distribution as a base line distribution. A tangent is drawn at the inflection point of an OC curve, where pT is the tangent intercept to the p-axis and L(pT) is the tangent intercept to the L(p) axis. The discriminant 'D' is defined as the distance between L(p*) and L(pT) in the L(p) axis ie., D = L(pT) -L(p*).

When D is large there is a high degree of discrimination power and when D is small the discrimination decreases. A procedure for designing a SSP indexed through MAPD and discriminant distance is developed by Ramkumar and Manjula (2007) [4] . They discuss a unique sampling plan at fixed MAPD and tangential distance and this distance is a powerful tool to identify the OC curve at the designed MAPD. Ramkumar (2009) [5] has designed a SSP indexed through MAPD and tangential distance (discriminant). In this paper mixed sampling plan (independent case) with ChSP-1 plan as attribute plan is constructed using Poisson distribution as a base line distribution. The plans indexed through MAPD and tangential angles are constructed by fixing the values c and *    . Tables are presented for easy selection of the plans based on MAPD with discriminant and tangential angles.

Glossary of symbols

The symbols used in this paper are as follows. P: Submitted quality of lot or process. P*: Maximum allowable percent defective (MAPD): Probability of acceptance for lot quality 'pj'

Operating procedure of mixed sampling plan having chsp-1 plan as attribute plan

Using Schilling (1967) [6] the operating procedure of Mixed Sampling plan using ChSP-1 as attribute plan with upper specification limit (U) and standard deviation (σ) is given as follows.

1. Determine the parameters of the mixed sampling plan n1, n2, k and c. 2. Take a random sample of size n1 from the lot. The OC function of the mixed sampling plan, suggested by Schilling (1967) [6] for the single sampling plan is

If the sample average

Equation ( 1) can be expressed as . ) 1 (

Construction of mixed sampling plan having chsp-1 plan as attribute plan

The detailed procedure adopted in this paper for the construction of mixed sampling plan having ChSP-1 plan as attribute plan indexed through MAPD is given below.  Assume that the mixed sampling plan is independent  Decide the sample size n1 (for variable sampling plan) to be used  Calculate the acceptance limit for the variable sampling plan as

where z(t) is the standard normal variate corresponding to 't' such that .

 Determine the sample average X . If a sample average

, take a second stage sample of size 'n2' using attribute sampling plan.  Split the probability of acceptance *



Using the above procedure tables can be constructed to facilitate easy selection of mixed sampling plan by using ChSP-1 plan as attribute plan indexed through MAPD.

Designing of chsp-1 with mapd and tangential angles

The equation of the tangent to the OC curve at the inflection point is given by Soundararajan (1975) [7] is (5.1) The tangent intercept to the L(p) axis is given by (5.2)

Which reduces to

Then the discriminant (5.3) 2.

Selection of the plan

Construction of OC curves

The ChSP-1 plan for various Angles with fixed D and c are constructed and are presented in Table 1. The OC curves for the sampling plans are also drawn and are presented in Figure 2.

Conclusion

In this paper designing of mixed sampling plan with ChSP-1 plan as attribute sampling using MAPD and discriminant tangential angles with Poisson distribution as baseline distribution is presented. It is concluded from this study when angle of the tangent decreases, the discriminating power is increasing. Suitable tables are also provided for the easy selection of the plans for the engineers who are working on the floor of the assembly. Different sampling plans can also be constructed by changing the first stage probabilities ) ( j   .

References

1. Mayer PL. A note on sum of poisson probabilities and an application. Industrial Quality Control. 1967;19(15):1215.
2. Radhakrishnan R, Sampath Kumar R. Construction of mixed sampling plan indexed through MAPD and AQL with chain sampling plan as attribute plan. STARS, Interdisciplinary Journal. 2006;7(1):14-22.
3. Ramkumar TB, Manjula K. Design of single sampling Plan with MAPD and discriminant tangential distance. ISPS conference Thirupati, January 2007.
4. Ramkumar TB. Design of single sampling plan by discriminant at MAPD. Prob Stat Forum. 2009;2:104114.
5. Soundararajan V. Maximum allowable percent defective (MAPD) single sampling inspection by attributes plam, Journal of Quality Technology. 1975;7(4):173-177.