Author Topic: Selection Of Mixed Sampling Plan With CSP-1 (C=2) Plan As Attribute Plan Indexed  (Read 3196 times)

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Author : R. Sampath Kumar, S. Sumithra and R. Radhakrishnan
International Journal of Scientific & Engineering Research Volume 3, Issue 1, January-2012
ISSN 2229-5518
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Abstract - In this paper a procedure for the construction and selection of the independent mixed sampling plan using MAPD and MAAOQ as quality standards with Continuous Sampling plan of the type CSP-1 (c=2) as attribute plan is presented. Tables are constructed for the selection of parameters of the plan when MAPD and MAAOQ are given. Practical applications of the sampling plan are also discussed with suitable example.

Key words and Phrases: Maximum allowable percent defective, maximum allowable average outgoing quality, Operating characteristic.
AMS (2000) Subject classification Number: Primary: 62P30, Secondary: 62D05.

1. 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 Plan (MSP) was first developed by Schilling (1967) for the case of single sided specifications, standard deviation known by assuming an underlying normal distribution for measurements. Dodge (1943) provided the concept of continuous sampling inspection and introduced the first continuous sampling plan. Dodge (1947) outlined several sampling plans for continuous production, originally referred to as the random – order and later designated as CSP-1 plan by Dodge and Torrey (1951). The desirability of developing a set of sampling plans indexed with p* has been explained by, Soundararajan (1975), Kandasamy (1993) studied in designing of various types of continuous sampling plans. Suresh and Ramkumar (1996)discussed about the use of MAAOQ for the selection of sampling plans. Radhakrishnan (2002) constructed various continuous sampling plans indexed through MAAOQ and mentioned its advantage over AOQL. Devaarul (2003), Sampath Kumar (2007), Radhakrishnan and Sampath Kumar (2006, 2007, 2009), Radhakrishnan et.al (2010) have made contributions to mixed sampling plans for independent case. Radhakrishnan et.al and (2009) studied mixed sampling plan for dependent case.
 In this paper, using the operating procedure of mixed sampling plan (independent case) with CSP-1 (c=2) as attribute plan, tables are constructed for the mixed sampling plan indexed through MAPD and MAAOQ. Suitable suggestions are also provided for the future.

2. 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’
         :  probability of acceptance assigned to first stage for       
                 percent defective ‘pj’
       :  probability of acceptance assigned to second stage
                  for percent defective ‘pj’
       k    :  variable factor such that a lot is accepted
                 if
       f     :  the rate of inspection(=1/n)
        i     :  number of consecutive units are found conforming
       n1     :  sample size for the variable sampling plan
       n2     :  sample size for the attribute sampling plan = (1/f)
                 units
3. Formulation of Mixed Sampling Plan with CSP-1 (c=2) as Attribute Plan
        The development of mixed sampling plans and the subsequent discussions are limited only to the lower specification limit ‘L’. By symmetry a parallel discussion can be used for upper specification limits also It is suggested that the mixed sampling plan with CSP–1(c=2) in the case of single sided specification (L), standard deviation (σ) known can be formulated by the parameters i, n1, n2 and k.

      Mixed sampling procedure suggested by Schilling (1967) is slightly modified and presented in this paper and this procedure is to be adopted separately for each time period fixed by the manufacturer. By giving the values for the parameters an independent plan for single sided specification, σ known would be carried out as follows:
   Determine the parameters with reference to ASN and OC curves
   Take a random sample of size n1 from the lot assumed to be large during the time period‘t’ (may be an hour / a shift / a day / a week …). [This is the modification suggested in this paper over Schilling (1967)]
   If a sample average ,  accept the lot
   If the sample average , apply the operating procedure of CSP-1 (c=2)
Operating Procedure of CSP-1(c=2) plan
Step 1: At the outset, inspect 100% of the units consecutively in the order of production until i successive conforming units are found.
Step 2: When i units in succession are found conforming discontinue 100% inspection and inspect f (=1/n) units until total of (c+ 1) sampled units are found nonconforming.
Step 3: When the number of nonconforming sampled units reaches (c+1), revert to 100% inspection as in step 1.
Step 4: Correct or replace all nonconforming units found with conforming units.   
Step 5: At the end of the time t switch back to variable sampling plan for the units produced.

4. Construction and Selection of Mixed Sampling Plan having CSP-1 (c=2) as attribute plan indexed through MAPD and MAAOQ
              Maximum allowable percent defective (MAPD) is the quality level that corresponds to the point of inflection of the OC curve. It is the quality level at which the second order derivative of the OC function Pa(p) with respect to p is zero ans the third order not equal to zero. When some specific value for a characteristic or group of characteristics is designated, the continuous sampling plan will have a tendency to accept product during periods of sampling if the submitted quality is upto MAPD and if the submitted quality is beyond MAPD, the sampling plan will have a tendency to submit the product for screening. The inflection point (p*) is obtained by using d2Pa(p)/dp2 = 0 and d3Pa(p)/dp3 ≠ 0.

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