Construction of Continuous Sampling Plan of the Type Tightened CSP-3 Indexed Through. the Convex Combination of AOQL and MAAOQ.

Transcription

1 International Journal of Mathematical Sciences and Applications Vol. 1 No. 2 (May, 2011) Copyright Mind Reader Publications Construction of Continuous Sampling Plan of the Type Tightened CSP-3 Indexed Through the Convex Combination of AOQL and MAAOQ Radhakrishnan. R 1 and Esther Jenitha.K 2 1 Associate Professor, Department of Statistics, P.S.G. College of Arts and Science, Coimbatore, India Assistant Professor, RVS Institute of Management Studies, Coimbatore, India Abstract Continuous Sampling is an acceptance-rectification inspection procedure that involves alternative periods of 100% inspection and sampling inspection. If the consumer wishes to increase the amount of inspection performed whenever the quality of submitted material is unsatisfactory the tightened inspection may be utilized. In this paper a procedure for constructing Tightened Continuous Sampling Plan-3 indexed through the Convex Combination of Average Outgoing Quality Limit (AOQL) and Maximum Allowable Average Outgoing Quality (MAAOQ) is presented. This plan may safeguard the interests of both producer as well as consumer by properly choosing a right combination using the gain parameter. A table is also provided for the easy selection of the plan when = 0.2, = 0.7 and MAPD = Keywords Operating Characteristic curve, Average Outgoing Quality Limit, Maximum Allowable Percent Defective, Maximum Allowable Average Outgoing Quality, Continuous Sampling Plan, Convex combination. AMS (2000) Subject Classification: 62P Introduction In the literature, the average outgoing quality limit (AOQL) is defined as the worst

2 728 Radhakrishnan. R average quality that the consumer will receive in the long run, when the defective items are replaced by non-defective items. The proportion defective corresponding to the inflection point of the OC curve is interpreted as the maximum allowable percent defective (MAPD = p*). Sampling plans indexed through p* 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 construction of sampling plans based on AOQL is largely consumer oriented and the MAAOQ is the average outgoing quality at the inflection point is a producer oriented, which is the average outgoing quality at p*. The advantage of using MAAOQ for designing a sampling plan instead of AOQL is that it reduces the sample size to be inspected which directly reduces the inspection cost and indirectly reduces the total cost. The use of MAAOQ for designing sampling plans was justified by Suresh and Ramkumar (1996). Fordice (1972) constructed a tightened multi-level continuous sampling plan- T. Liberman and Solomon (1955) studied multilevel continuous sampling plan. Kandasamy (1993) constructed the sampling plan of the type CSP-T indexed through AOQL. Radhakrishnan (2002) studied various sampling plans including continuous sampling plans indexed through MAPD and MAAOQ. Sampathkumar (2007) constructed mixed sampling plan indexed through AOQL, MAPD, MAAOQ and emphasized the superiority of MAAOQ. Radhakrishnan and Mallika (2008, 2009a, 2009b, 2009c, 2010a, 2010b) constructed single, Double, ChSP-1(Chain Sampling Plan-1) and ChSP-2(Chain Sampling Plan-2)and Link sampling plans indexed through AOQ cc which is the convex combination of AOQL and MAAOQ with gain parameter (0< <1). Radhakrishnan and Esther Jenitha (2011) constructed continuous sampling plan indexed through AOQ cc.. In this paper a procedure for the construction of Tightened CSP-3 plan indexed through AOQ cc. This plan may safeguard the interests of both producer as well as consumer by choosing

3 Construction of Continuous Sampling 729 a right combination using the gain parameter (0< <1). 2. Definition of AOQ cc AOQ cc is the convex combination of AOQL and MAAOQ with gain parameter suggested by Radhakrishnan and Mallika (2009) as AOQ cc = AOQL+ (1- ) MAAOQ 3. Operating Procedure of Tightened CSP-3 Plan The operating procedure of Tightened Continuous Sampling Plan -3 is as follows Step l: Inspect units consecutively in the order of production. This 100% inspection continues until i units in succession are found conforming and then sampling inspection begins. Step 2: During sampling inspection, units are inspected at rate f, sampling inspection continues until a non-conforming unit is found. When a non-conforming unit is found before finding k consecutive sampled units conforming, 100% inspection is invoked immediately as per (1) above. When a non-conforming unit is found after finding k consecutive sampled units conforming the following four units inspection begins. Step 3: During four units inspection, four consecutive units are inspected in the order of production. If all the four units inspected are found conforming, sampling inspection continues as per (2) above. If one the four units inspected is non-conforming, 100% inspection is invoked as per (1) above. Tightened CSP-3 (k = i) is a special case of tightened CSP-3 plan with k = i. 4. Operating characteristic function The OC function for Tightened CSP-3 (k =i) Plan is given by Pa (P) = q i /[ q i +f(1-q i +q k (q i -q 4 ))], k = i Construction of Tightened CSP-3 Plan indexed through AOQ cc

4 730 Radhakrishnan. R The general procedure for designing Tightened CSP-3 Plan indexed through a parameter AOQ cc which is a convex combination of AOQL and MAAOQ is given below: Step 1: Determine MAPD, MAAOQ and AOQL for Tightened CSP-3 for various values of n, i and find R 1 = MAAOQ/MAPD and R 2 = AOQL/MAPD. Step 2: Find AOQ cc = AOQL+ (1- ) MAAOQ for various values of and find R 3 =AOQ cc /MAPD. Step 3: The results of Step 1 and Step 2 for MAPD = , = 0.2 and = 0.7 using Excel Packages are presented in Table Selection of the plan Table 1 is used to construct the plan when the MAPD and AOQ cc are specified. One can find the ratio R 3 =AOQ cc /MAPD and locate the value in Table1 under the column R 3 (for fixed values of MAPD = , = 0.2 and = 0.7 and the corresponding values of n, k and i are noted. Example : For a specified MAAOQ = and MAPD= compute the ratio R 1 =MAPD/MAAOQ=1.6 which is associated with i = 624, n = 5 in Table 1 and f =1/n =0.2. Hence the Tightened CSP-3 Plan for specified MAAOQ = is i=624, k = 624 and f=0.2. For a specified AOQL = and MAPD= compute the ratio R 2 =MAPD/AOQL=1.85 which is associated with i = 1224, n = 23 in Table 1 and f =1/n =0.04. Hence the Tightened CSP-3 Plan for specified AOQL = is i= 1224, k = 1224 and f=0.04. For a specified value of MAAOQ = , AOQL= and =0.2,

5 Construction of Continuous Sampling 731 AOQcc= Compute the ratio R 3 =MAPD/AOQcc=1.64 which is associated with i = 733, n = 7 in Table 1 and f =1/n =0.14. Hence the Tightened CSP-3 Plan for specified AOQcc = and =0.2 is i=733, k = 733 and f=0.14. The OC curves for the Example is presented in Figure 1 Explanation: In a cell manufacturing company, if the producer fixes the quality level MAAOQ as (1563 defective cells out of ) and the consumer fixes the quality level AOQL as (1351 defective cells out of ) then a compromising quality level AOQ cc can be suggested as (152 defective cells out of ). Inspect cells consecutively in the order of production. This 100% inspection continues until 733 cells in succession are found conforming and then sampling inspection begins. During sampling inspection, cells are inspected at rate 0.14, sampling inspection continues until a non-conforming unit is found. When a non-conforming unit is found before finding 733 consecutive sampled cells conforming, 100% inspection is invoked immediately. When a defective cell is found after finding 733 consecutive sampled cells conforming the following four units inspection begins. During four units inspection, four consecutive cells are inspected and if all the four units inspected are found conforming, sampling inspection continues. If one the four cells inspected is non-conforming, 100% inspection is invoked. The AOQ curves for the Example are presented in Figure Conclusion In this paper a procedure for the construction and selection of Tightened CSP-3 Plan indexed through gain parameter, which is a convex combination of AOQL and MAAOQ is stated. A table also constructed for the easy selection of the plans when the indexing parameters and gain parameter are known. Readymade table is also provided in this paper for =0.2, =0.7

6 732 Radhakrishnan. R for MAPD of to take quick decisions on the nature of the sampling plan when the quality level of the producer and consumer are known. Similar tables can also be generated for various values of and MAPD based on the choice of the consumer and producer. References [1]. Fordice, J.J., 1972, A Tightened Multi-Level Continuous Sampling Plan CSP-T, Report No. QEM , Ammunition Procurement and Supply Agency, Joliety, Illinois. [2]. Kandasamy, C., 1993, Studies on Designing Certain Continuous Sampling Plans, Ph.D. Thesis, Bharathiar University, Tamilnadu, India. [3]. Liberman, G.J., and Solomon, H., 1955, Multilevel Continuous Sampling Plan, the Annals of Mathematical Statistics, Vol.26, No.4, PP [4]. Mandelson, J., 1962, The Statistician, The Engineer and sampling plans, Industrial Quality control, 19 5: [5]. Mayer, P. L., 1967, A note on sum of Poisson Probabilities and an application, Annals of Institute of Statistical Mathematics, 19: [6]. Radhakrishnan, R., 2002, Contribution to the study on selection of certain acceptance sampling plans, Ph.D thesis, Bharathiar University, Coimbatore, India. [7]. Radhakrishnan, R. and Esther Jenitha, (2011). Construction of Continuous Sampling Plan Indexed through AOQ cc. International Journal of Statistics & Systems. Volume 6, Number, PP [8]. Radhakrishnan, R. and Mallika, M., 2008, Designing of Sampling plans indexed through the convex combination of AOQL and MAAOQ, edited book of the National level Conference on IT and Business Intelligence organized by Institute of Management Technology, Nagpur, Maharashtra, India, Excel publishers Chapter 5: [9]. Radhakrishnan, R. and Mallika, M., 2009a, Construction of Sampling Plans indexed through AOQ cc, International Journal of Statistics and Systems, 4, 2, [10]. Radhakrishnan, R. and Mallika, M., 2009b, Construction of Chain Sampling Plan -1 indexed through Convex Combination of AOQL and MAAOQ, Global Journal of Pure and Applied Mathematics, 5, 1: [11]. Radhakrishnan, R. and Mallika, M., 2009c, Construction of Link Sampling Plan indexed through AOQ cc, Mathematics Applied in Sciences and Technology, 5,2. [12]. Radhakrishnan, R. and Mallika, M., 2010a, Construction of Double Sampling Plan through AOQ cc, International Journal of and Applied Mathematics and Statistics, 17, J10:

7 Construction of Continuous Sampling 733 [13]. Radhakrishnan, R. and Mallika, M., 2010b, Construction of Chain Sampling Plan-2 indexed through convex combination of AOQL and MAAOQ, Far East Journal of Theoretical Statistics, 31, 1: [14]. Sampathkumar, R., 2007, Construction and selection of mixed variables-attributes sampling plans, Ph.D thesis, Bharathiar University, Coimbatore, India. [15]. Soundararajan, V., 1975, Maximum allowable percent defective (MAPD) single sampling inspection by attribute plan, Journal of Quality Technology, 7, 4: [16]. Suresh, K. K. and Ramkumar, T.B., 1996, Selection of a sampling plan indexed with Maximum Allowable Average Outgoing Quality, Journal of Applied Statistics, 23, 6: Table 1: Characteristics of Tightened Continuous Sampling Plan-3 for MAPD = = 0.2 = 0.7 R 1 MAAOQ i n R 2 AOQL i n AOQ cc i n R 3 AOQ cc i n R Figure1: OC curves for i=1224, f=0.04 (AOQL).; i=624, f=0.2 (MAAOQ); i=733, f=0.14 (AOQ cc ) Figure2: AOQ curves for i=1224, f=0.04 (AOQL).; i=624, f=0.2 (MAAOQ); i=733, f=0.14 (AOQ cc )

8 734 Radhakrishnan. R List of Symbols The symbols used in this paper are as follows: N n i f p k - Lot size - Sample size - Clearance number - Sampling frequency - Gain parameter - Submitted lot quality of lot or process - The number of sample units to be found conforming in order that the inspection will continue to be in the sampling mode. p* - Maximum allowable percent defective P a (p) AOQ - Probability of acceptance for a given quality p - Average Outgoing Quality Biographical Sketch Dr. R. 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 32 years of teaching experience in teaching theoretical and applied Statistics, presented more than 250 papers in National and International seminars & conferences and published more than 115 articles in reputed National and International journals. He is also a quality auditor for ISO certifications and a certified Six Sigma Black Belt also. He gained 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.

9 Construction of Continuous Sampling 735 K. Esther Jenitha is having postgraduate and MPhil degrees in Statistics. She is an Assistant Professor in Statistics with Four years of teaching experience and doing research under the guidance of Dr. R. Radhakrishnan. Figure1: OC curves for i=1224, f=0.04 (AOQL).; i=624, f=0.2 (MAAOQ); i=733, f=0.14 (AOQ cc ) Figure2: AOQ curves for i=1224, f=0.04 (AOQL).; i=624, f=0.2 (MAAOQ); i=733, f=0.14 (AOQ cc )