The Design of Two-Level Continuous Sampling Plan MCSP-2-C

Transcription

1 Applied Mathematical Sciences, Vol. 6, 2012, no. 90, The esign of Two-Level Continuous Sampling Plan MCSP-2-C Pannarat Guayjarernpanishk epartment of Applied Statistics, Faculty of Applied Science King Mongkut s University of Technology North Bangkok 10800, Thailand pannarat @gmail.com Tidadeaw Mayureesawan epartment of Applied Statistics, Faculty of Applied Science King Mongkut s University of Technology North Bangkok 10800, Thailand tidadeaw@yahoo.com Abstract In this paper, the concept two-level continuous sampling plan has been developed from the single-level continuous sampling plan MCSP- C.The new plan is designated as the MCSP-2-C that is defined by five parameters: i (the clearance number), m (the number of conforming units to be found before allowing c non-conforming units in the sampling inspection), c (the acceptance number), f 1 (the sampling frequency at level 1) and f 2 (the sampling frequency at level 2). MCSP-2-C computes four performance measures: average fraction inspected (AFI ), average outgoing quality (AOQ), average outgoing quality limit (AOQL) and average fraction of total produced accepted on sampling basis (P a (p)), for given values of the parameters and incoming fractions of non-conforming units actually produced on line (p). The validity and accuracy of performance measure formulas have been tested by extensive simulations. The formulas of performance measures, AFI and AOQ are valid and accurate for all sets of i, m, c, p and r (r =1/f 1 and r =2/f 2 ) values. Keywords: Continuous Sampling Plan, Average Fraction Inspected, Average Outgoing Quality 1 Introduction A continuous sampling plan (CSP) is a plan of sampling inspection for a product consisting for individual units that are manufactured in quantity by an

2 4484 P. Guayjarernpanishk and T. Mayureesawan essentially continuous process. The original continuous sampling plan was the single-level continuous sampling plan known as CSP-1, and described by odge [6]. This plan is the simplest and most common, and an inspection procedure in CSP-1 requires periods of both 100% inspection and sampling inspection. The other continuous sampling plans have been developed by odge and Torrey [7], namely CSP-2 and CSP-3, Lieberman and Solomon [5], namely CSP-M, and Govindaraju and Kandnsamy [8], namely CSP-C. Reviews of these continuous sampling plans are now available in many statistical quality control textbooks such as Grant [4], Stephens [10], and Montgomery [2]. Balamurali and Subramani [11] have developed a new single level continuous sampling plan based on the CSP-C and the resultant plan is designated as the MCSP-C. In this plan, sampling inspection is continued until the event of c+1 non-conforming units if the first m sampled units have been found to conform during the sampling phase. So MCSP-C imports an additional requirement that, after invoking sampling inspection, the event of a non-conforming unit before finding m sampled units, and then revert immediately to 100% inspection. In this paper, we developed the MCSP-C as a single level continuous sampling plan with two sampling inspection levels to reduce inspection or extended restart 100% inspection on processes designated as MCSP-2-C. The new plan introduces a sampling inspection phase that is characterized by two sampling inspection rates, f 1 and f 2. The paper describes the following: 1. The design of a two level continuous sampling plan which was developed from the one level continuous sampling plan the Modified CSP-C (MCSP- C) and the resultant plan is designated as MCSP-2-C. 2. The development of the formulas for performance measures in MCSP- 2-C carried out using a Markov Chain, such as the average fraction inspected (AFI ), the average outgoing quality (AOQ), the average outgoing quality limit (AOQL) and the average fraction of total produced accepted on sampling basis (P a (p)). 3. Tests of the validity and accuracy of the formulas for the performance measures by comparison of the values computed from the formulas with values obtained through extensive simulations. 2 Materials and Methods The Operating Procedure of the MCSP-2-C The MCSP-2-C uses five parameters for inspection of the units being produced on the production line,namely three positive integers i, m and c, and two fractions f 1 and f 2, which are defined by: i = the clearance number, m = the number of conforming units to be found before allowing c non-

3 The design of two-level continuous sampling plan MCSP-2-C 4485 conforming units in the sampling inspection, c = the acceptance number, f 1 = the sampling frequency at level 1 or f 1 =1/r, f 2 = the sampling frequency at level 2 or f 2 =2/f 1. The procedure for inspection of the MCSP-2-C is as follows: i. Start with 100% inspection of units from the flow of production and continue the inspection until i successive conforming units are found, discontinue 100% inspection and start sampling inspection at level 1. ii. uring the sampling inspection at level 1, inspect only a fraction f 1 of the units, selecting individual units one at a time in the order of production in such a way as to ensure an unbiased sample. iii. If c non-conforming units are found after m sampled units have been found to conform then continue the sampling at rate f 1 until a total of c+1 non-conforming sampled units have been found, and then revert immediately to 100% inspection. iv. If a non-conforming unit is found within m sampled conforming units then start sampling inspection at level 2, inspect only a fraction f 2 until a total of c+1 non-conforming sampled units have been found then revert to a 100% inspection for succeeding units and return to step i. v. Replace or correct all the non-conforming units found with conforming units. The formulations of the MCSP-2-C are derived using a Markov Chain model, assuming that the production process is under statistical control. Such appearances are discussed in divided sections. The MCSP-2-C Procedure as a Markov Chain Let [X t ] (t = 1,2,) denote a discrete-parameter Markov Chain with finite state space (S j ), j = 1,, i+3m+6c+7. The states of the process are defined, in a same way Roberts (1965) and Lasater (1970), as follows: S 1 = A 0 = Non-conforming unit is found on 100% inspection. S j+1 = A j (j =1, 2,...,i) = j consecutive conforming units found during 100% inspection following its commencement. S i+3j+2 = f 1 Id j+1 (j =0, 1,..., m-1) = Sampling inspection at level 1 is in effect, the j +1 units submitted for inspection and only unit j +1 was found to be non-conforming. S i+3j+3 = f 1 In j+1 (j =0, 1,..., m-1) = Sampling inspection at level 1 is in effect and the j +1 units submitted for inspection were all found to be conforming. S i+3j+4 = f 1 N j+1 (j =0, 1,..., m-1)

4 4486 P. Guayjarernpanishk and T. Mayureesawan = Sampling inspection at level 1 is in effect and the j units submitted for inspection were all found to be conforming but the last unit was not selected for inspection. S i+3m+3j+2 = f 1 SIn j (j =0, 1,..., c) = Sampling inspection at level 1 is in effect and the last unit submitted for inspection and found to be conforming and the number of nonconforming units found during this sampling phase is j. S i+3m+3j+3 = f 1 SId j (j =0, 1,..., c) = Sampling inspection at level 1 is in effect and the last unit submitted for inspection and found to be non-conforming and the number of non-conforming units found during this sampling phase is j. S i+3m+3j+4 = f 1 SN j (j =0, 1,..., c) = Sampling inspection at level 1 is in effect and the last unit was not inspected and the number of non-conforming units found during this sampling phase is j. S i+3m+3c+3j+5 = f 2 SIn j (j =0, 1,..., c) = Sampling inspection at level 2 is in effect and the last unit submitted for inspection and found to be conforming and the number of nonconforming units found during this sampling phase is j. S i+3m+3c+3j+6 = f 2 SId j (j =0, 1,..., c) = Sampling inspection at level 2 is in effect and the last unit submitted for inspection and found to be non-conforming and the number of non-conforming units found during this sampling phase is j. S i+3m+3c+3j+7 = f 2 SN j (j =0, 1,..., c) = Sampling inspection at level 2 is in effect and the last unit was not inspected and the number of non-conforming units found during this sampling phase is j. The set (i+3m+6c+7) states defined above completely describe the mutually exclusive phases of inspection for the MCSP-2-C procedure. A flow chart showing the description of the process by means of states and transition is given in Figure 1 and the one-step transition probability is given in Table 4 in the Appendix. The transition probability matrix reveals that the process is a discrete-parameter, finite, recurrent, irreducible, aperiodic (FRIA) Markov Chain (see Karlin, 1966 and Lasater, 1970). The Performance Measures of the MCSP-2-C Letting p be the incoming fractions of non-conforming units actually produced on line, the following performance measures may be obtained: The average number of units inspected in a 100% screening sequence following the finding of a non-conforming unit, u: u = 1 qi pq i (1)

5 The design of two-level continuous sampling plan MCSP-2-C 4487 The average number of units passed under the sampling inspection, v: v = f 2(1 + cq i ) + (c + 1)f 1 (1 q m ) pf 1 f 2 (2) The average cycle length, which is the average number of units, passed on each cycle, ACL: ACL = f 1f 2 (1 q i ) + q i f 2 (1 + cq m ) + (c + 1)q i f 1 (1 q m ) pq i f 1 f 2 (3) The average fraction inspected, AFI : AF I = f 1 f 2 (1 + (c + 1)q i q i+m ) f 1 f 2 (1 q i ) + q i f 2 (1 + cq m ) + q i f 1 (c + 1)(1 q m ) The average outgoing quality, AOQ: AOQ = pqi (1 q m )(1 f 1 )f 2 + (c + 1)q m (1 f 1 )f 2 + f 1 (1 q m )(1 f 2 ) f 1 f 2 (1 q i ) + q i f 2 (1 + cq m ) + q i f 1 (c + 1)(1 q m ) The average outgoing quality limit, AOQL that is the maximum of AOQ for all values of p, may be found graphically using the AOQ function given in equation (5). The average fraction of the total produced accepted on sampling basis, P a (p): P a (p) = q i f 2 (1 + cq m ) + (c + 1)f 1 (1 q m ) f 1 f 2 (1 q i ) + q i f 2 (1 + cq m ) + q i f 1 (c + 1)(1 q m ) A detailed derivation of these measures, based on Markov Chain formulation of the plan, is presented in the Appendix. (4) (6) (5) Test of the Validity and Accuracy of Performance Measures for the MCSP-2-C In this section, the test of the validity and accuracy of performance measures for MCSP-2-C are given. We have compared the results from our formulas given in equations (4) and (5) with the values for the performance measures from extensive simulations. We tested six different levels for the incoming fractions p of non-conforming units actually produced on line: 0.005, 0.008, 0.01, 0.02, 0.05 and For each p we selected i = 10, 15, 20, 30, 40 and 50, values of m = i, values of r = 4 and 10 and values of c = 2 and 3. For each set of values of the parameters p, i, m, r and c, we carried out a simulation to compute the fraction of units inspected, and the fraction of outgoing non-conforming units. The simulation was repeated 200 times and the values of AFI and AOQ were calculated. An AFI formula is accepted as a valid and accurate formula if the AFI

6 4488 P. Guayjarernpanishk and T. Mayureesawan values from the formula were contained in the 95% confidence interval of AFI values from the simulation. In testing the validity and accuracy of an AOQ formula, a formula is accepted as a valid and accurate formula if the AOQ values from the formula were contained in the 95% confidence interval of AOQ values from the simulation. We compared the validity and accuracy of formulas for each set of values of p, i, m, r and c. The results are presented in the next section. Table 1. The AFI-F vales, AFI-S vales, P-AFI vales, AOQ-F vales, AOQ-S vales and P-AOQ vales for p = Note: i r c AFI-F AFI-S P-AFI AOQ-F AOQ-S P-AOQ AF I F = the AFI vales from the formula, AF I S = the AFI vales from the simulation, P AF I = the percentage of AF I F contained in the 95% confidence interval of AF I S, AOQ F = the AOQ vales from the formula, AOQ S = the AOQ vales from the simulation, P AOQ = the percentage of AOQ F contained in the 95% confidence interval of AOQ S.

7 The design of two-level continuous sampling plan MCSP-2-C 4489 Table 2. The AFI-F vales, AFI-S vales, P-AFI vales, AOQ-F vales, AOQ-S vales and P-AOQ vales for p =0.01. i r c AFI-F AFI-S P-AFI AOQ-F AOQ-S P-AOQ Table 3. The AFI-F vales, AFI-S vales, P-AFI vales, AOQ-F vales, AOQ-S vales and P-AOQ vales for p =0.02. i r c AFI-F AFI-S P-AFI AOQ-F AOQ-S P-AOQ

8 4490 P. Guayjarernpanishk and T. Mayureesawan 3 Results In this section a summary is given of some of the results that were obtained from the simulations. In Table 1 to 3, we show that for all sets of i, r and c when p = 0.005, p=0.01 and p = 0.02, we found that the percentage of AFI values from the formula which is contained in the 95% confidence interval of AFI values from the simulation are 100% and the percentage of AOQ values from the formula which is contained in the 95% confidence interval of AOQ values from the simulation are 100% for all sets of p, i, m, r and c. Our simulations indicated that the AFI and AOQ formulas are valid and accurate. 4 iscussions and Conclusions In this paper, the MCSP-2-C plan differs from the original MCSP-C plan in that a two level continuous sampling plan has been considered if a nonconforming unit is found within msampled conforming units during the sampling inspection at level 1. Formulas for the average fraction inspected (AFI ), the average outgoing quality (AOQ), the average outgoing quality limit (AOQL) and the average fraction of total produced accepted on sampling basis (P a (p)) has been developed. The validity and accuracy of AFI and AOQ for MCSP-2-C has been tested by extensive simulations for a range of values of p (probability of nonconforming units), i (clearance number), c (acceptance number), m (number of conforming units to be found before allowing c non-conforming units in the sampling inspection) and r (number of units produced between inspections in a fractional inspection). The formula values are contained in the 95% confidence interval of the simulation values in all simulations. One advantage of the MCSP-2-C plan is that it reduces inspections or extended restart of 100% inspection on process. In contrast, the new plan is a more complicated inspection procedure than the MCSP-C plan. Acknowledgements The authors would like to thank for the Graduate Colleges, King Mongkut s University of Technology North Bangkok for the financial support during this research. References [1] Chi-Hyuck Jun and etc, Evaluation and design of two level Continuous Sampling plans, Tamkang Journal of Science and Engineering, 9(2006),

9 The design of two-level continuous sampling plan MCSP-2-C [2].C. Montgomery, Introduction to Statistical Quality Control 5th Edition, John Wiley and Sons, Inc, United States, [3] E. G. Schilling, Acceptance Sampling in Quality Control, New York: Marcel ekker, [4] E. L. Grant and R. S. Leavenworth, Statistical Quality Control, 6th Edition, McGraw-Hill,New York,1988. [5] G. J. Lieberman and H. Solomon, Multi-level continuous sampling plans, Annals of Mathematical Statistics, 26(1955), [6] H. F. odge, A Sampling inspection plan for continuous production, Annals of Mathematical Statistics, 14(1943), [7] H. F. odge and M.N. Torrey, Additional continuous sampling inspection plans, Industrial Control, 7(1951), [8] K. Govindaraju and C. Kandasamy, esign of generalized CSP-C Continuous Sampling inspection plan, Journal of Applied Statistics, 27(2000), [9] K. S. Stephens, How to Perform Continuous Sampling (CSP). Vol. 2, ASQC Basic References in Quality Control, Statistical Techniques, Milwaukee, WI: American Society for Quality Control, [10] K. S. Stephens, The Handbook of Applied Acceptance Sampling Plans, Procedures, and Principles, Milwaukee: American Society for Quality, [11] S. Balamurali and K. Subramali, Modified CSP-C Continuous Sampling plan for Consumer Protection, Journal of Applied Statistics, 31(2004), [12] S. Karlin, A First Course in Stochastic Process, New York: Academic Press, Appendix Glossary of symbols S j = the j th state of the process, P (S j ) = the steady-state probability for the state S j, p ij = the probability that the process transits from state S i to S j in one step.

10 4492 P. Guayjarernpanishk and T. Mayureesawan erivation of Performance Measures of the MCSP-2-C plan The formulation of the MCSP-2-C using the Markov Chain development is similar to Stephens (1995a). Let [X t ] (t = 1,2,) denote a discrete-parameter Markov Chain with finite state space (S j ), j = 1,, i+3m+6c+7. The states of the process are defined, in a way similar to that of Roberts (1965). These steady-state probabilities P (S j ) satisfy the following conditions: and P (S j ) = i+3m+6c+7 k=1 P (S j ) 0, j = 1, 2,..., i + 3m + 6c + 7 (7) P (S k )p ij, j = 1, 2,..., i + 3m + 6c + 7 (8) P (S j ) = 1, j = 1, 2,..., i + 3m + 6c + 7 (9) j Figure 1: Flow chart showing states and transition of the MCSP-2-C procedure. From conditions (8) and (9), we acquire the following: i 1 P (A 0 ) = p[p (A 0 ) + P (A j ) + P (f 1 SId c+1 ) + P (f 2 SId c+1 )] (10) j=1

11 The design of two-level continuous sampling plan MCSP-2-C 4493 Table 4. One-step transition probability matrix of the MCSP-2-C plan. step A0 A1... Ai f1id1 f1in1 f1n1... f1sin0 f1sid1... f1sidc+1 f1snc f2sin0 f2sid1 f2sn0... f2snc A0 p q A1 p Ai f1p f1q 1 f f1id f2p f2q 1 f f1in f1n f1p f1q 1 f f1idm f2p f2q 1 f f1inm f1p f1q f1nm f1sin f1p f1q f1sid f1sn f1p f1q f1sinc f1q 1 f f1sidc+1 p q f1snc f1q 1 f f2sin f2p f2q 1 f f2sid f2sn f2p f2q 1 f f2sinc f2 f2sidc+1 p q f2snc f2

12 4494 P. Guayjarernpanishk and T. Mayureesawan P (A j ) = q j [P (A 0 ) + P (f 1 SId c+1 ) + P (f 2 SId c+1 )], j = 1, 2,..., i (11) P (f 1 N j ) = (1 f 1)q i+j 1 P (A 0 ) f 1 (1 q i ), j = 1, 2,..., m (12) P (f 1 Id j ) = pqi+j 1 P (A 0 ) 1 q i, j = 1, 2,..., m (13) P (f 1 In j ) = qi+j P (A 0 ) 1 q i, j = 1, 2,..., m (14) P (f 1 SId j+1 ) = qi+m P (A 0 ) 1 q i, j = 0, 1,..., c (15) P (f 1 SIn j ) = qi+m+1 P (A 0 ) p(1 q i ) P (f 1 SN j ) = (1 f 1)q i+m P (A 0 ) pf 1 (1 q i ), j = 0, 1,..., c (16), j = 0, 1,..., c (17) P (f 2 SId j+1 ) = qi (1 q m )P (A 0 ) 1 q i, j = 0, 1,..., c (18) P (f 2 SIn j ) = qi+1 (1 q m )P (A 0 ) p(1 q i ), j = 0, 1,..., c (19) P (f 2 SN j ) = (1 f 2)q i (1 q m )P (A 0 ), j = 0, 1,..., c (20) pf 2 (1 q i ) 1 = i m P (A 0 ) + [P (f 1 In j ) + P (f 1 Id j ) + P (f 1 N j )] j=0 j=1 c + [P (f 1 SIn j ) + P (f 1 SId j+1 ) + P (f 1 SN j )] j=0 c + [P (f 2 SIn j ) + P (f 2 SId j+1 ) + P (f 2 SN j )] (21) j=0 By (10) to (21), (10) can be written as P (A 0 ) = pf 1f 2 (1 q i ) where =f 1 f 2 (1 q i ) + q i f 2 (1 + cq m ) + (c + 1)q i f 1 (1 q m ). The steady-state probabilities can be written as follows: P (A j ) = pqi f 1 f 2 P (f 1 N j ) = pqi+j 1 (1 f 1 )f 2, j = 1, 2,..., i, j = 1, 2,..., m

13 The design of two-level continuous sampling plan MCSP-2-C 4495 P (f 1 Id j ) = p2 q i+j 1 f 1 f 2 P (f 1 In j ) = pqi+j f 1 f 2 P (f 1 SId j+1 ) = pqi+m f 1 f 2 P (f 1 SIn j ) = qi+m+1 f 1 f 2 P (f 1 SN j ) = qi+m (1 f 1 )f 2 P (f 2 SId j+1 ) = pqi f 1 f 2 (1 q m ) P (f 2 SIn j ) = qi+1 f 1 f 2 (1 q m ) P (f 2 SN j ) = qi f 1 (1 q m )(1 f 2 ), j = 1, 2,..., m, j = 1, 2,..., m, j = 0, 1,..., c, j = 0, 1,..., c, j = 0, 1,..., c, j = 0, 1,..., c, j = 0, 1,..., c, j = 0, 1,..., c i Let P (100) = P (A 0 ) + P (A j ) j=1 m c P (F N) = P (f 1 N j ) + [P (f 1 SN j ) + P (f 2 SN j )] j=1 j=0 m and P (S) = [P (f 1 In j ) + P (f 1 Id j ) + P (f 1 N j )] j=1 c + [P (f 1 SIn j ) + P (f 1 SId j+1 ) + P (f 1 SN j )] j=0 then u = P (100) P (A i ) v = P (S) P (A i ) ACL = u + v AF I = 1 P (F N) AOQ = pp (F N) P a (p) = 1 P (100) Received: April, 2012