Minimum Average Fraction Inspected for Combined Continuous Lot by Lot Acceptance Sampling Plan

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1 c Heldermann Verlag Economic Quality Control ISSN Vol 19 (2004), No. 1, Minimum Average Fraction Inspected for Combined Continuous Lot by Lot Acceptance Sampling Plan Chung-Ho Chen Abstract: This article investigates the problem of minimizing the average fraction inspected (AFI) for a combined continuous lot by lot acceptance sampling plan. A solution procedure is developed to find sub-optimal sampling plan parameters that will meet the average outgoing quality limit (AOQL) requirement, while guaranteeing a small AFI when the process average (>AOQL) is known. Key Words: Continuous Sampling Plan, Average Outgoing Quality (AOQ), Average Outgoing Quality Limit (AOQL), Average Fraction Inspected (AFI), Lot by Lot Inspection Plan. 1 Introduction Type I continuous sampling plans (CSP-1) were introduced by Dodge in The operating procedure of a CSP-1 plan is as follows: Inspect every item until i successive units are found to be free of defects, then inspect at a frequency f. When a defective item is found, revert to 100% inspection and continue again until i successive items are found free of defects. The average outgoing quality limit (AOQL) is widely used as primary performance inde for CSP-1 plans and other types of continuous sampling plans. Hence, the sampling plan parameters are determined so as to meet the AOQL requirement and at the same time guarantee minimum inspection cost. The last aim is generally met by minimizing the long run inspection cost given by the average fraction inspected (AFI). Govindaraju (1989) and Wasserman (1990) matched CSP-1 plans and single sampling plans with respect to their properties of outgoing quality and long-run proportion accepted, respectively. Pesotchinsky (1987), Bebbington and Govindaraju (1998), and Govindaraju and Bebbington (2000) proposed the combined continuous lot by lot inspection plan. From Govindaraju and Bebbington (2000, p. 725), the operating procedure of the combined continuous lot by lot inspection plan is outlined below: Step 1: Inspect 100% of the units consecutively until i units in succession are found to be clear of defects.

2 116 Chung-Ho Chen Step 2: Once clearance takes place as in Step 1, start forming lots of desired size. Apply any lot by lot sampling inspection plan, such as a single sampling plan and inspect the lots at a frequency of f. Step 3: If a lot is rejected, go to Step 1; otherwise, continue with lot by lot inspection. Previously researchers (Resnikoff (1960) and Ghosh (1988)) addressed the problems of achieving the minimum AFI for the CSP-1 plan. In this paper, we investigate the problem of minimizing the AFI for a combined continuous lot by lot acceptance sampling plan. A procedure is developed to find a sub-optimal solution that will meet the AOQL requirement, while come close to the minimum AFI when the process average (>AOQL) is known. 2 Average Outgoing Quality and Average Fraction Inspected for Combined Continuous Lot by Lot Acceptance Sampling Plans In Govindaraju and Bebbington (2000, p. 728), the average outgoing quality (AOQ) and AFI functions for the combined continuous lot by lot acceptance sampling plan are given. AOQ = p 2 q i (1 f) (1 P a )(1 q i )+pq i (1) AFI = (1 P a)(1 q i )+fpq i (1 P a )(1 q i )+pq i (2) where i = clearance number f = lot inspection frequency p = process defective probability q = 1 p P a = probability of accepting an inspected lot with fraction defective p The problem investigated here is to determine a single sampling plan (n, c) and an continuous sampling plan (i, f) so that a prescribed AOQL is guaranteed with minimum AFI. 3 Minimum AFI for a Combined Continuous Lot by Lot Acceptance Sampling Plan Dodge (1943, p. 271) presented the AOQ and AFI functions for a CSP-1 plan given by (i, f).

3 Minimum Average Fraction Inspected for Acceptance Sampling Plan 117 AOQ = AFI = (1 f)pq i f +(1 f)q i f f +(1 f)q i (3) (4) Ghosh (1988, p. 419) showed that the minimum AFI for a CSP-1 plan occurs when the quality level that yields the specified AOQL is equal to process average p. By means of a comparison we notice that by setting 1 P a = pf (5) the epressions on the right hand side of (1) and (2) are transformed into the right hand sides of (3) and (4). Hence, we assume (5) which allows the development of a recursive minimization procedure. Of course, the result will be only sub-optimal, but it is to be epected that it is close to the optimum. Assuming (5) allows to adopt Ghosh s (1988) method for minimizing the AFI for a CSP-1 plan and obtain the parameters (i,f ). Net we use (5) to determine the corresponding optimal parameters (n,c ) for a single sampling plan. The detailed solution procedure of minimizing AFI for the combined continuous lot by lot acceptance sampling plan under the specified AOQL, p L, is as follows: Step 1: Estimate the process average p. Step 2: Determine the optimum continuous sampling plan (i,f ). From Ghosh (1988, p. 419), the optimal clearance number is given by 1 p i = p p L (6) and the corresponding optimal sampling frequency by f = (1 p) i+1 ip L +(1 p) i+1 (7) Step 3: Determine the optimum lot by lot sampling plan (n,c ). From (5) we have for a single sampling plan (n, c) with sufficiently large lot size, and the optimum sampling frequency f : P a = c =0 ( ) n p (1 p) n =1 pf (8) For a given acceptance number c, P a is a decreasing function in the sample size n and for given n an increasing function of c. Ifc is too small, then there is no solution of (8). Hence we search for the minimum sample size for which (8) is met.

4 118 Chung-Ho Chen The above algorithm has one major drawback. Because of the discrete character of the acceptance number and sample size and hence of P a, it is generally not possible to find (n, c) being a solution of (8). In order to overcome this difficulty, we replace the equality (8) by the following requirement for (n,c ): c ( ) n 1 p (1 p) n 1 > 1 pf (9) =0 c ( ) n p (1 p) n 1 pf (10) =0 n minimum (11) With (9), (10) and (11) there is always a unique solution (i,f,n,c ) of the problem with c =1. 4 Numerical eample Assume the situation of a given process with defective probability of p = 400 PPM 4 and a requited outgoing quality level of AOQL = 200 PPM. A combined continuous lot by lot acceptance sampling plan with minimum AFI shall be used for meeting the AOQL-requirement. The above given algorithm yields the following values: i = = (12) f ( ) 4998 = = ( ) 4998 (13) pf = (14) For c = 1, the desired sample size is found to be n = 25. Hence the sub-optimal combined continuous lot by lot sampling plan is given by: (i = 4998,f = ,c =1,n = 25) (15) 5 Conclusions Govindaraju and Bebbington (2000, p. 728) pointed out that the combined continuous lot by lot acceptance sampling plan is particularly useful for high quality production 4 parts per million

5 Minimum Average Fraction Inspected for Acceptance Sampling Plan 119 processes, where the fraction non-conforming is very low, of the order of 100 PPM. In this paper, we present an easy solution procedure for determining a continuous lot by lot single sampling plan under a specified AOQL value yielding a solution which has an AFI close to the minimum. Further investigations aim at quantified statements about the quality of the obtained solutions with respect to the closeness to the optimal ones. References [1] Bebbinton, M. and Govindaraju, K. (1998): On Pesotchinsky s Scheme for very low fraction nonconforming. Journal of Quality Technology 30, [2] Dodge, H.F.,(1943): A sampling inspection plan for continuous production, Annals of Mathematical Statistics 14, [3] Ghosh, D.T. (1988): The continuous sampling plan that minimises the amount of inspection. Sankhya 50, B, [4] Govindaraju, K. (1989): Procedures and tables for the selection of CSP-1 plans. Journal of Quality Technology 21, [5] Govindaraju, K. and Bebbington, M. (2000): Combined continuous lot by lot acceptance sampling plan. Journal of Applied Statistics 27, [6] Pesotchinsky, L. (1987): Plans for low fraction nonconforming. Journal of Quality Technology 19, [7] Resnikoff, G.J. (1960): Minimum average fraction inspected for a continuous sampling plan. Journal of Industrial Engineering 11, [8] Wasserman, G.S. (1990): Matching performance of continuous sampling plans with single sampling plans. Communications in Statistics-Simulation and Computation 19, Chung-Ho Chen Department of Industrial Management Southern Taiwan University of Technology 1 Nan-Tai Street, Yung-Kang City Tainan 710 Taiwan

Joint Design of Economic Manufacturing Quantity, Sampling Plan and Specification Limits

c Heldermann Verlag Economic Quality Control ISSN 0940-5151 Vol 17 (2002), No. 2, 145 153 Joint Design of Economic Manufacturing Quantity, Sampling Plan and Specification Limits Chung-Ho Chen and Chao-Yu