R.Radhakrishnan, K. Esther Jenitha

Transcription

1 Selection of Mixed Sampling Plan Indexed Through AOQ cc with Conditional Double Sampling Plan as Attribute Plan Abstract - In this paper a procedure for the selection of Mixed Sampling Plan (MSP) indexed through AOQ cc which is a convex combination of consumer oriented concept, AOQL (Average Outgoing Quality Limit) and producer oriented concept, MAAOQ (Maximum Allowable Average Outgoing Quality) with Conditional Double Sampling Plan (CDSP) as attribute plan is given. This plan may safeguard the interests of both producer as well as consumer by properly choosing a right combination using the gain parameter (0<<1). Tables are also constructed for the easy selection of the plan. Index Terms - Operating Characteristic Curve, Average Outgoing Quality Limit, Maximum Allowable Percent Defective, Maximum Allowable Average Outgoing Quality, Poisson Distribution, Convex Combination. I. INTRODUCTION The AOQL is the average quality that the consumer will receive in the long run when the defective items are replaced by non-defective items. The construction of sampling plans based on AOQL is largely consumer oriented and MAAOQ is the average outgoing quality at the inflection point is a producer oriented, which is the average outgoing quality at MAPD. The advantage of using MAAOQ for designing a sampling plan instead of AOQL is that it reduces the sample size to be inspected which reduces the cost of inspection and results reduction in total cost. The use of MAAOQ for deriving sampling plans was justified by Suresh and Ram Kumar (1996). Radhakrishnan (2002) studied various sampling plans indexed through MAPD and MAAOQ. Radhakrishnan and Mallika (2008) constructed single sampling plans indexed through AOQ cc. Radhakrishnan and Esther Jenitha (2011a, 2011b, 2011c, 2011d) constructed continuous sampling plan of the type CSP, CSP-3, CSP V (i-2x), Continuous Sampling Plan of the Type T CSP-3 plan indexed through AOQ cc which is the convex combination of AOQL and MAAOQ.In this paper an attempt is made to introduce AOQ cc in MSP using CDSP as an attribute plan. This plan may safeguard the interests of both producer as well as consumer by choosing a right combination using the gain parameter λ. Sampling plans indexed through p* (MAPD) which is the quality level corresponding to the inflection point of the operating characteristic (OC) curve has been explained by Mandelson (1962), Mayer (1967) and further studied by Soundararajan (1975). The MAPD is the value of fraction defective (p= p * ) at which d 2 P a (p)/ dp 2 = 0, for p= p * d 2 P a (p)/dp 2 < 0, for p< p * R.Radhakrishnan, K. Esther Jenitha 311 and d 2 P a (p)/dp 2 > 0, for p> p * A variety of plans and procedures have been developed for special sampling situations 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 attributes sampling in a so-called mixed sampling plans. Mixed sampling plan is a two stage sampling procedure involving variables inspection in the first stage and attributes inspection in the second stage if the variables inspection of the first sample does not lead to acceptance. Use of variables on the first sample with attributes on the second sample combines the economy of variables for quick acceptance on the first sample with the broad nonparametric protection of attributes sampling when a questionable lot requires a second sample. The mixed sampling plans are initially introduced by Dodge (1932) and later developed by Bowker and Goode (1952). Schilling (1967) has given a method for determining the operating characteristics for mixed variables-attributes sampling plans. Using Schilling s procedure, Devaarul (2003) has constructed mixed sampling plan. Sampath Kumar (2007) constructed mixed sampling plan with DSP plan as attribute plans indexed through the parameters MAPD, AOQL and MAAOQ. Radhakrishnan and Saravanan (2011) Constructed Dependent Mixed Sampling Plan using Double Sampling Plan of the type DSP (0, 1) indexed through IQL. II. GLOSSARY OF SYMBOLS The symbols used in this paper are as follows: p: submitted quality of lot or process p j : submitted quality of lot or process j P a (p): probability of acceptance for given quality p p * : maximum allowable percent defective (MAPD) p m : the product quality at which AOQ is maximum p t : the point at which the inflection tangent of the OC curve cuts the p axis h * : relative slope at p * c : attributes acceptance number c 1 :first attributes acceptance number c 2 : second attributes acceptance number c 3 :third attributes acceptance number d : number of defectives in the sample d j : number of defectives in the j th sample (j = 1,2,3, ) n 1 : sample size for variable sampling plan n 1,2 : first sample size for attribute sampling plan n 2,2 : second sample size for attribute sampling plan j : probability of acceptance for lot quality p j

2 for Variate corresponding to t such that t β j ': probability of acceptance assigned to first stage percent defective p j β j " :probability of acceptance assigned to second stage for percent defective pj z (j): z value for the j th ordered observation z(p) : standard normal deviate k: variable factor such that a lot is accepted if X A = U- kσ III. FORMULATION OF MSP WITH CDSP Procedure: Independent Plan Determine the parameters of the mixed plan n 1, n 1, 2, n 2, 2, k, c 1, c 2 and c 3. Take a random sample of n 1 from the lot. If the sample average X A = U - k, accept the lot. If the sample average X > A = U - k, take another sample of size n 1, 2 and count the number of defectives d 1 therein. If the number of defectives d 1 c 1, accept the lot. If the number of defectives d 1 >c 3, reject the lot. If c 1 +1 < d 1 c 3, take a second sample of size n 2,2 from the remaining lot and find the number of defectives d 2. If d 2 c 2 or d1 + d 2 c 3 accept the lot, otherwise reject the lot. = z() t 1 2 u 2 /2 e du Determine the sample average X. If a sample average X > A = U - k, take a second stage sample of size n 2 using attribute sampling plan. Now determine β * ", the probability of acceptance assigned to the attributes plan associated with the second stage sample as β j " = (β j β j ') / (1-β j '). Determine the appropriate second stage sample of size n 2 and c from Pa (p) = β j " for p= p j. Using the above procedure tables can be constructed to facilitate easy selection of mixed sampling plan with any attribute plan indexed through p *, AOQL, MAAOQ and AOQ cc. VI. CONSTRUCTION OF TABLES The probability of acceptance for CDSP under Poisson model is given by P a (p)= +[ c c1 r0 -n1,2p r 1,2 e (n p) r! c n1,2 p 2 e ( n1, 2 k 1 1 k! p) k c3 k 2,2 r e ( n2,2 p) { }] = r! r0 n p IV. CONDITIONS FOR APPLICATIONS (i) Production process should be steady and continuous. (ii) Lots are submitted substantially in the order of their production. (iii) Inspection is by variable in the first stage and attribute in the second stage with quality defined as the fraction defective. V. CONSTRUCTION OF MSP WITH CDSP AS ATTRIBUTE PLAN INDEXED THROUGH MAPD The procedure for the construction of mixed variables attributes sampling plans is provided by Schilling (1967) for a given n 1 and a point p j on the OC curve. A modified procedure for the construction of mixed variables attributes sampling plan for a given MAPD and n 1 is given below. Assume that the mixed sampling plan is independent. Split the probability of acceptance (β j ) determining the probability of acceptance that will be assigned to the first stage. Let it be β j '. Decide the sample size n 1 (for variable sampling plan) to be used.. Calculate the acceptance limit for the variable sampling plan as A = U - k = U [z (p j ) + {z (β j ')/ n 1 }], where z (t) is the standard normal β * " for p = p *. For n 1,2 =n 2,2 = n (say), the inflection point (p * ) is obtained by using [d 2 Pa(p)/dp 2 ]=0 and [d 3 Pa(p)/dp 3 ] 0. The p dpa( p) relative slope of the OC curve h * = Pa( p) dp at p = p *. The inflection tangent of the OC curve cuts the p axis at p t =p * + (p * /h * ). The values of h *, np *, np t and R=p t /p * are calculated for a specified β * '=0.35 using visual basic program and presented in Table 1. The general procedure for designing a CDSP 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 2 MAAOQ and n 2 AOQL for CDSP for various combinations of c 1, c 2, c 3 and n 2 p *. Find R 1 =n 2 AOQL/ n 2 p * and R 2 =n 2 MAAOQ/ n 2 p * Step 2. Find n 2 AOQcc= n 2 ΑOQL+ (1- ) n 2 MAAOQ and R3 = n 2 AOQcc/ n 2 p * for selected values of. Step 3. Present the results of Step1and Step 2 in Table 1. VII. SELECTION OF THE PLAN Table 1 is used to construct the plans when MAPD (p * ) and tangent intercept (p t ) are given. For any given values of c 1, c 2, c 3, p t and p *, one can find the ratio R=p t /p *. Corresponding to the value of c 1, c 2 and c 3 find the value of R in Table 1. From this c 1, c 2 and c 3 values one can determine the value of n using n=np * /p *. Example: (i) For a specified AOQL = and p * = compute the ratio R 1 = AOQL/ p * = which is 312

3 associated with c 1 =9, c 2 =13 and c 3 =23 in Table 1 and n 2 =n 2 p * / p * =13.069/ = Thus n 1, 2 =2376, n 2, 2 =2376, c 1 =9, c 2 =13 and c 3 =23are the parameters selected for the MSP with CDSP as attribute plan for a specified p * =0.0055, AOQL= (ii) For a specified MAAOQ = and p * = compute the ratio R 2 = MAAOQ/ p * = which is associated with c 1 =2,c 2 =6 and c 3 =8 in Table 1 and n 2 =n 2 p * / p * =4.867/ = 885. Thus n 1, 2 =885, n 2, 2 =885, c 1 =2,c 2 =6 and c 3 =8 are the parameters selected for the MSP with CDSP as attribute plan for a specified p * =0.0055, MAAOQ= (iii) For a specified value of AOQL = , MAAOQ = and p * = and =0.2, AOQ cc = compute R 3 = AOQ cc / p * = with c 1 =5, c 2 =8 and c 3 =12 in Table 1 and n 2 =n 2 p * / p * =7.3230/ = Thus n 1, 2 =1331, n 2, 2 =1331, c 1 =5, c 2 =8 and c 3 =12 are the parameters selected for the MSP with CDSP as attribute plan for a specified p * =0.0055, AOQL = , MAAOQ= Practical application: Suppose the plan with n 1 = 10, k = 1.5 is to be applied to the lot-by-lot acceptance inspection of shafts from a production line, the characteristic to be inspected is the shafts diameters in mm for which there is a specified upper limit (U) of 45mm with a known standard deviation ( ) of mm. In this example, U=45 mm, = mm and k = 1.5. Now, in applying the variable inspection first, take a random sample of size n 1 =10 from the lot. Record the sample results and find X. If X A = U k = mm, accept the lot otherwise take a random sample of size n 2 =1331 and apply attribute inspection. number of defectives d 1 >12, reject the lot. If 6 < d 1 12, take a second sample of size 1331 from the remaining lot and find the number of defectives d 2. If d 2 8 or d 1 + d 2 12 accept the lot, otherwise reject the lot and inform the management for corrective action. The OC and AOQ curves for the above Example are represented in Figure 1 and Figure 2 respectively. Fig 2:AOQ curves for n 1,2 =2376, n 2,2 =2376, c 1 =9,c 2 =13, c 3 =23 (AOQL) n 1,2 =885, n 2,2 =885,c 1 =2,c 2 =6, c 3 =8 (MAAOQ) n 1,2 =1331, n 2,2 =1331, c 1 =5, c 2 =8, c 3 =12 (AOQ cc ) VIII. CONCLUSION In this paper an attempt is made to construct mixed sampling plans indexed through AOQ cc, a convex combination of AOQL, consumer s preference and MAAOQ, producer s preference with the gain parameter λ using Conditional Double sampling plan as attribute plan. naoq cc values are also presented for the selected values of λ in Table1. These sampling plans will safeguard the interest of producer as well as consumer by choosing a right combination of gain parameter λ. These plans will help the floor engineers to suggest the quality level AOQ cc after knowing the interest of both producer and consumer which can be understood from OC and AOQ curves. This work can be extended for mixed sampling plan using other plans as attribute plan. Fig 1:OC curves for n 1,2 =2376, n 2,2 =2376, c 1 =9,c 2 =13, c 3 =23 (AOQL) n 1,2 =885, n 2,2 =885,c 1 =2,c 2 =6, c 3 =8 (MAAOQ) n 1,2 =1331, n 2,2 =1331, c 1 =5, c 2 =8, c 3 =12 (AOQ cc ) Under attribute inspection, by taking Conditional Double Sampling Plan as attribute plan, if the manufacturer fixes the values β * '=0.35, p * = (55 non-conformity shafts out of 10000) and the consumer fixes the quality level AOQ cc = (23 non-conformity shafts out of shafts) then select a sample of 1331 shafts and count the number of non-conformities (d 1 ). If the number of defectives d 1 5, accept the lot. If the REFERENCES [1]. A. H. Bowker, and H.P.Goode. Sampling Inspection by Variables, McGraw Hill, New York, [2]. S. Devarrul, Certain Studies Relating to Mixed Sampling Plans and Reliability Based Sampling Plans, PhD thesis, Bharathiar University, Coimbatore, India, [3]. Dodge, Statistical control in sampling Inspection, American Mechanist, Oct 1932: Nov 1932,: [4]. J. Mandelson, The Statistician, The Engineer and sampling plans, Industrial Quality control, 19 (5): 12-15, [5]. P.L. Mayer. A note on sum of Poisson Probabilities and an application, Annals of Institute of Statistical Mathematics, 19: ,

4 [6]. R. Radhakrishnan, Contribution to the study on selection of certain acceptance sampling plans, Ph.D thesis, Bharathiar University, Coimbatore, India, [7]. R. Radhakrishnan, and M. Mallika, 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, India Excel publishers pp : 45-52, [8]. R. Radhakrishnan, and K. Esther Jenitha, Construction of Continuous Sampling Plan Indexed through AOQ cc, International Journal of Statistics & Systems, Volume 6, Number, PP , 2011a. [9]. R. Radhakrishnan, and K. Esther Jenitha, Construction of CSP 3 indexed through AOQ cc, Global Journal of Mathematical Sciences: Theory & Practical (GJMS), Volume 3, Number 1, PP , 2011b. [10]. R. Radhakrishnan, and K. Esther Jenitha, Construction of CSP of the type CSP V (i=2x) indexed through the convex combination of AOQL & MAAOQ, Research in Management: A Contemporary Approach (VIIMS Conference Proceedings). PP , 2011c. [11]. R. Radhakrishnan, and K. Esther Jenitha,. Construction of CSP of the type Tightened CSP-3 indexed through the convex combination of AOQL & MAAOQ, International Journal of Mathematical Sciences & Applications, Volume 1, Number 2, PP , 2011d. [12]. R. Radhakrishnan, and P.G. Saravanan, Construction & Selection of Dependent Mixed Sampling Plan using Double Sampling Plan of the type DSP(0,1) indexed through IQL, Published as a proceedings of the International Conference on Mathematics & its Applications organized by Department of Mathematics, Avinashilingam University, Coimbatore, [13]. R. Sampath Kumar, Construction and selection of mixed variables-attributes sampling plans, Ph.D thesis, Bharathiar University, Coimbatore, India, [14]. E.G. Schilling, A general method for determining the operating characteristics of mixed variables, Attribute sampling Plans single side specifications, S.D known, Ph.D thesis Rutgers, The State University, New Brunswick, New Jersy, [15]. V. Soundararajan. Maximum allowable percent defective (MAPD) single sampling inspection by attribute plan, Journal of Quality Technology, 7(4): , [16]. K.K Suresh. and T.B. Ramkumar, Selection of a sampling plan indexed with maximum allowable average outgoing quality, Journal of Applied Statistics, 23(6): , AUTHOR BIOGRAPHY Dr. Radhakrishnan is having a bachelor degree, post graduate degree in Statistics and also possessing research degrees such as MPhil and PhD and also has additional qualification of post graduate degree in Business Administration. He has 34 years of teaching experience in teaching theoretical and applied Statistics, presented more than 160 papers in National and International seminars & conferences and published more than 119 articles in reputed National and International journals. He is also a quality auditor for ISO certifications and certified Six Sigma Black Belt also. He gains sufficient knowledge in Six Sigma methodologies and he is conducting training programme for Six Sigma black belts. He has visited countries like Sri Lanka and China for presenting his research contributions. Esther jenitha is having is having postgraduate and MPhil degrees in Statistics. She is an Assistant Professor in Statistics with more than five years of teaching experience and doing research under the guidance of Dr. R. Radhakrishnan. She has presented 4 papers in the National and International seminars & conferences and published 8 articles in reputed National and International journals. 314

5 C1 C2 C3 n 2 p* naoql R 1 nmaaoq R 2 APPENDIX Table 1: Parameters of Mixed Sampling plan (CDSP) for β * ' = 0.35, β m ' = 0.35 =0.2 =0.4 =0.6 naoq cc R 3 naoq cc R 3 naoq cc R