Minimum Risk Acceptance Sampling Plans: A Review

Transcription

1 c Heldermann Verlag Economic Quality Control ISSN Vol 19 (2004), No. 1, Minimum Risk Acceptance Sampling Plans: A Review K. Subramani Abstract: Since the first acceptance sampling plans have been developed almost 80 years ago, a number of selection principles have emerged. The majority of these principles is characterized by the fact that they look upon producer and consumer as two opposing parties. However, in many occasions, e.g., in final inspection, producer and consumer represent the same party and, therefore, the used sampling plan should not make an attempt to discriminate between their interests. In this case the interest is to avoid wrong decisions, i.e., reject product of sufficient quality and accept product of insufficient quality. Thus, the natural objective in these cases is to use the overall risk for a wrong decision as optimization criteria. This is the case with so-called minimum risks sampling plans, which are reviewed in this paper so as to make them better known to those responsible for quality control. 1 Introduction The statistical performance of a sampling inspection plan is revealed by its operating characteristic (OC) curve P a. In the non-bayesian risk-based design of acceptance sampling plans, two points on the OC curve are usually specified namely (, 1 α) and(p 2,β) where = acceptable quality level, p 2 = limiting quality level, α= producer s risk, and β = consumer s risk. The sampling plan parameters (sample size n and acceptance number Ac) are determined so as to meet the two point requirement with minimum sample size. A number of published tables in various text books such as Duncan (1986), Schilling (1982), Montgomery (1991) etc. provide the sample size and acceptance number for given operating ratios p 2 referring to fractional parts and p 2 that are not usually observed in practice. This is because these tables have been constructed fixing the parameters (n, Ac) of the plans which are integers. The conditions of fixed risks are often changed to P a ( ) 1 α and P a (p 2 ) β, where P a (p) is the probability of acceptance for given lot quality p. Unfortunately, it is not possible to simultaneously minimize the producer s and consumer s risks without letting n get unreasonable large. However, it is possible minimize the sum of producer s and consumer s risks.

2 122 K. Subramani 2 Minimum Risk Plans Golub introduced the concept of minimizing the sum of the producer s and consumer s risks for single sampling attribute plan when the sample size is fixed. For a fixed sample size n, he developed a method of obtaining the acceptance number Ac which minimizes the sum of producer s and consumer s risks or equivalently which maximizes the sum of probabilities of classifying the lot of quality as acceptable and classifying the lot p 2 quality as not acceptable. Mathematically, the expression which is to be maximized may be written as P a ( )+(1 P a (p 2 )) (1) where P a ( ) represents the probability of accepting the lots of quality and 1 P a (p 2 ) represents the probability of rejecting lots of quality p 2. Alternatively we can minimize the sum of producer s and consumer s risks, 1 P a ( )+P a (p 2 ). Golub (1953) has shown that, for fixed sample size n, the value of the acceptance number Ac which minimize the sum of risks is given by the integer nearest to n 1 log p 2 log +1 (2) 2 log(1 ) log(1 p 2 ) Moreover, Golub has developed tables providing the values of Ac for different values of n based on the binomial model. He also extended his procedure for classifying a lot into different categories when the sample size is fixed. Soundararajan (1981) modified this approach to single sampling plan by assuming the Poisson model for the OC curve. One shortcoming of Golub s approach is that the sampling plan guaranteeing the minimum sum of risks may have unacceptably large individual producer s and consumer s risks. During the 1990s, in a series of papers (see Table 19) Govindaraju and Subramani developed a new method for determining minimum risk plans which do not exhibit the weakness of uncontrolled individual risks. They designed and constructed tables based on the Poisson model which combine Golub s approach with that of selecting a plan for given, p 2, α and β without assuming that the sample size n is known. Further, this approach results in rounded values of the operating ratios p 2 /. 3 Selection of Minimum Risk Plans The tables constructed by Govindaraju and Subramani may be used to select various sampling plans for given and p 2 with minimum sum of producer s and consumer s risks. For the plans given in the tables, the producer s and consumer s risks will be at most 10 percent each. Against the fixed value of the operating ratio p 2 /, the tables give the parameters of the respective sampling plans and the associated producer s and consumer s risks in the body of the table against the product of sample size and acceptable

3 Minimum Risk Acceptance Sampling Plans: A Review 123 quality level (n ). The structure of a typical table giving minimum risk plans is shown in Table 2 5 for a double sampling plan (n 1,Ac 1,Re 1 ; n 2,Ac 2,Re 2 ). Table 1: Papers on Minimum Risk Attribute Plans by K.Govindaraju and K.Subramani Title Publication Selection of Single Sampling Attribute Plans Communications in Statistics - for Given Acceptable Quality Level and Simulation and Computation Limiting Quality Level Involving Minimum Risks Vol. 19 (1990), Selection of Multiple Deferred State MDS-1 Journal of Applied Statistics Sampling Plans for Given Acceptable Quality Vol. 17 (1990), Level and Limiting Quality Level Involving Minimum Risks Selection of Single Sampling Quick Switching International Journal of Quality System for Given Acceptable Quality Level and Reliability Management and Limiting Quality Level Vol. 18 (1991), Selection of Double Sampling Attributes Communications in Statistics - Plan for Given Acceptable Quality Level Simulation and Computation and Limiting Quality Level Vol. 21 (1992), Selection of Tightened-Normal-Tightened Journal of Applied Statistics System for Given Values of the Acceptable Vol. 19 (1992), Quality Level and Limiting Quality Level Selection of Chain Sampling Plans ChSP-1 American Journal of Mathematical and ChSP-(0,1) for Given Acceptable Quality and Management Sciences Level and Limiting Quality Level Vol.13 (1993), Selection of Multiple Deferred (Dependent) State Journal of Applied Statistics Sampling Plans for Given Acceptable Vol. 20 (1993), Quality Level and Limiting Quality Level Selection of Modified Quick Switching Journal of Applied Statistics Systems for Given Acceptable and Limiting Vol. 20 (1993), Quality Levels A double sampling plan may determined for given, p 2, α and β by means of the following four steps and Table Compute the operating ratio p Enter the table in the column headed by p 2 of just smaller than the computed ratio. and select the row with an entry equal 3. Next, proceeding from left to right in the row identified in Step 2, the relevant column is determined. It is obtained when the tabulated producer s and consumer s risks are equal to or just less than the desired values. 5 From Govindaraju and Subramani (1992)

4 124 K. Subramani 4. The sample size n is obtained as n = n, where the value of n isgiveninthe heading of the column identified in Step 3. ( ) p Each entry for a pair 2,n consists of three pairs of numbers, which are displayed in three rows. The pair in the first row gives the optimal acceptance numbers Ac 1 and Ac 2, the second row contains the rejection numbers Re 1 and Re 2, while the third row gives the producer s and consumer s risks in percent. Table 2: Parameters of double sampling plan for given p 2 and n, p 2 n ,1 0,2 0,3 0,3 0,5 0,6 0,9 0,12 0,15 3,19 2,21 1,23 0, 28 10,40 2,2 3,3 4,4 4,4 6,6 7,7 9,10 11,13 13,16 15,20 16,22 17,24 26,29 26,41 3,6 1,3.2,2.3,.4 0,.2 0,0 0,0 0,0 0,0 0,0 0,0 0,0 0,0 0, ,1 0,2 0,2 0,3 0,4 0,5 0,8 0,11 0,15 3,19 2,21 1,23 0,26 7,40 2,2 3,3 3,3 4,4 5,5 6,6 9,9 11,12 13,16 15,20 16,22 17,24 20, 27 26,41 3,9 1,6 1,2.3,1.1,.3.1,.1 0,0 0,0 0,0 0,0 0,0 0,0 0,0 0, ,2 0,2 0,3 0,4 0,5 0,7 0,10 0,15 3,19 2,21 1,23 0,26 7,32 3,3 3,3 4,4 5,5 6,6 8,8 11,11 13,16 15,20 16,22 17,24 20,27 23,33 1,9 1,4.3,3.1,1.1,.3 0,0 0,0 0,0 0,0 0,0 0,0 0,0 0, ,2 0,2 0,3 0,4 0,6 0,8 0,13 2,17 2,21 1,23 0,26 7,32 3,3 3,3 4,4 5,5 7,7 9,9 13,14 15,16 16,22 17,24 20,27 23,33 1,9 2,4 1,2.4,1.1,.1 0,0 0,0 0,0 0,0 0,0 0,0 0, ,2 0,3 0,4 0,6 0,8 0,12 1,16 2,20 1,23 0,26 7,32 3,3 4,4 5,5 7,7 9,9 12,13 14,17 16,21 17,24 20,27 23,33 2,6 1,3.4,2.1,.3 0,.1 0,0 0,0 0,0 0,0 0,0 0, ,2 0,3 0,3 0,5 0,7 0,11 1,15 0,19 1,23 0,26 7,32 3,3 4,4 4,4 6,6 8,8 12,12 14,16 16,20 17,24 20,27 23,33 2,9 1,5 2,2.4,.3.1,.1 0,0 0,0 0,0 0,0 0,0 0, ,3 0,3 0,5 0,7 0,11 1,15 2,18 0,22 0,26 7,32 4,4 4,4 6,6 8,8 12,12 14,16 16,19 17,23 20,27 23,33 1,7 2,2.4,1.1,.2 0,0 0,0 0,0 0,0 0,0 0, ,2 0,3 0,5 0,7 0,10 0,14 2,18 4,21 0,26 7,32 3,3 4,4 6,6 8,8 11,11 14,15 16,19 18,22 20,27 23,33 4,4 2,3.4,1.1,.3 0,0 0,0 0,0 0,0 0,0 0, ,2 0,3 0,5 0,7 0,10 0,14 0,17 0,21 0,26 7,32 3,3 4,4 6,6 8,8 11,11 14,15 16,18 17,22 20,22 23,33 4,6 2,4.4,1.1,1 0,0 0,0 0,0 0,0 0,0 0, ,2 0,3 0,5 0,6 0,10 0,14 0,16 1,20 0,26 7,32 3,3 4,4 6,6 7,7 11,11 13,14 15,17 17,21 20,27 23,33 4,7 2,5.4,2.5,.4 0,.1 0,0 0,0 0,0 0,0 0, ,2 0,3 0,4 0,6 0,9 0,12 0,16 2,19 2,25 8,31 3,3 4,4 5,5 7,7 10,10 13,13 15,17 17,20 20,26 22,32 4,9 2,7 2,2.5,1.1,.1 0,0 0,0 0,0 0,0 0,0

5 Minimum Risk Acceptance Sampling Plans: A Review 125 Example: Consider a situation given by =.01, p 2 =.075 with α =5%andβ =10%. Using tables given in Duncan [2] the corresponding tabulated operating ratio is found to be 7.54 with n =.52 (k = 1). The sampling plan has the following parameter values: n 1 = n 2 =52,Ac 1 =1,Ac 2 =2,Re 1 =3andRe 2 =3 (3) The actual producer s and consumer s risks are given by α =.0485 and β =.1023 against the desired values α =.05 and β =.1. For the same conditions, one obtains the following plan from Table 2. n 1 = n 2 =40,Ac 1 =0,Ac 2 =2,Re 1 =3andRe 2 =3 (4) There are similar tables in the papers listed in Table 1 for selecting other attribute sampling plans. 4 Advantages of Minimum Risk Plans There are several advantages for using minimum risk plans: 1. The plans tabulated refer to rather realistic operating ratios which are often encountered in practice. 2. The OC curves of minimum risk plans have a better shoulder (i.e. the plans are better discriminating between lots of good and poor quality). 3. When the producer and the consumer belong to the same company or interest, the sum of risks may be minimized rather than fixing them at given levels. The main guiding principle in constructing the tables has been stressed by Harold F. Dodge (1973) while receiving the 1972 E.L. Grant Award, namely If you want a method or system used, keep it simple. The tables presented in the papers listed above are simple to use and hence facilitate an easy selection of various special purpose plans for use in shop-floor situations. Acknowledgement: The author is grateful to Dr. K. Govindaraju, Institute of Inf. Sci. & Tech., Massey University, New Zealand for his suggestions and to the Kumaraguru College of Technology, Coimbatore for its valuable support rendered. References [1] Dodge, H.F. (1973): Keep it Simple. Quality Progress 6,

6 126 K. Subramani [2] Duncan, A.J. (1986): Quality Control and Industrial Statistics. 5th Edition, Homewood, Illinois. [3] Golub, A. (1953): Designing Single-Sampling Inspection Plans when the Sample size is Fixed. Journal of the American Statistical Association 48, [4] Govindaraju, K and Subramani, K (1992): Selection of Double Sampling Attributes Plan for Given Acceptable Quality Level and Limiting Quality Level. Communications in Statistics - Simulation and Computation 21, [5] Montgomery, D.C. (1991): Introduction to Statistical Quality Control. Second Edition, John Wiley & Sons, Inc., New York. [6] Schilling, E.G. (1992): Acceptance Sampling in Quality Control. Marcel Dekker, Inc., New York. [7] Soundararajan, V. (1981) : Sampling Inspection Plans When the Sample Size is Fixed. Journal of Madras University, Section B, 44, K. Subramani Department of Mathematics Kumaraguru College of Technology Coimbatore Tamil Nadu, India