SkSP-V Acceptance Sampling Plan based on Process Capability Index

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1 258 Chiang Mai J. Sci. 2015; 42(1) Chiang Mai J. Sci. 2015; 42(1) : Contributed Paper SkSP-V Acceptance Sampling Plan based on Process Capability Index Muhammad Aslam*[a], Nasrullah Khan [b] and Hina Khan [c] [a] Department of Statistics, Faculty of Sciences, King Abdulaziz University, Jeddah 21551, Saudi Arabia. [b] Department of Statistics, National College of Business Administration & Economics, Lahore, Pakistan. [c] Department of Statistics, GC University Lahore, Pakistan. *Author for correspondence; aslam_ravian@hotmail.com Received: 12 July 2012 Accepted: 8 September 2013 ABSTRACT In this paper, we present the SkSP-V acceptance sampling plan based on process capability index. The plan parameters of the proposed plan are determined by satisfying the producer s risk and the consumer s risk by minimizing the average sample number (ASN). The efficiency of the proposed plan is discussed over the existing acceptance sampling plan. Tables are discussed with the help of amplified pressure sensor data. Keywords: skip-lot sampling, process capability index, average sample number, operating characteristic curve 1. INTRODUCTION Acceptance sampling plans are designed to give protection to producer and consumer for a predefined level of product quality. The basic purpose of acceptance sampling plan is to reduce the cost involved in the inspection of lots by reducing the sample size. The decision about lot acceptance and rejection is based on the sample information; there is chance of rejecting a rejecting a good lot and accepting a bad lot. The probability that a good lot is rejected on the basis of sample information is called the produce s risk and the probability of accepting bad lot based on sample information is called the consumer s risk. The different acceptance sampling plans are available in the literature. These acceptance sampling plans have been widely used in the industries for the inspection of the submitted lot of the product. The acceptance sampling plans are classified in attribute acceptance plan and variable acceptance sampling. The first one is used, when the sampling plan product as good or bad. Later one is used when the purpose is measurement of the quality of the characteristic. The attribute acceptance sampling plans are easy to apply and cheap than the variable acceptance sampling plans. On the other hand, the variable acceptance sampling are more informative than the attribute acceptance sampling plan, for more details, reader may to See Pearn and Kotz [30], Pearn and Wu [31], Montgomery [25], Resnikoff and Liberman [35], Owen [29],Bender [16], Duncan [21],Moskowitz [26],Balamurali and Jun [14], Balakrishnan, et al. [12], Lio and Wu [24], Rao, et al. [34], Rao [33], Aslam, et al. [1],

2 Chiang Mai J. Sci. 2015; 42(1) 259 Fallahnezhad and Aslam [22], Aslam, et al. [7], Aslam, et al. [8] and Aslam, et al. [9]. The researchers are working to design acceptance sampling plans using the process capability index (PCI). The use of PCI in acceptance sampling may cause the better inspection of the submitted lot. The PCI uses both the process mean and the process dispersion from the center of specification limits, where C pk is defined as (2.1) where μ and σ are process mean and standard deviation, respectively, USL is the upper specification limit and LSL is the lower specification limit, d = USL LSL/ 2 is the half specification width and M = USL + LSL/ 2 is the midpoint of the two specification limits. According to Pearn and Kotz [30] the PCI is used in order to tell the capability of process, at level 1 means at most 2700 particle per million will lie outside the specification limits, and for at level 1.33 means that the imperfect particle are 66 ppm. In order to achieve ppm defect rate the level must be at 1.67, but to drop this defect rate only 2 particles per billion level must be at 2.0", for more detail, reader may see Pearn and Kotz [30] and Montgomery [25]. By exploring the literature, we note that various acceptance sampling schemes have been developed using PCI, for example, Pearn and Wu [31], Negrin, et al. [27], Negrin, et al. [28], Aslam, et al. [10] and Aslam, et al. [11]. The idea of SkSP-V sampling plan was originally developed by Balamurali and Jun [13]. Aslam, et al. [6] presented the procedure for the estimation of optimum parameters of SkSP-V sampling plan. By exploring the literature, we found no work on designing a SkSP-V acceptance sampling plan using PCI. In this paper, we will develop the SkSP-V plan using PCI. We assume that the quality characteristic follows the normal distribution. The details of Skip lot acceptance sampling is given in Section 2. The designing of the proposed plan is given in Section 3. The optimization solution is given in Section 4. An industrial example is given in next Section 5. Some concluding remarks are given in last section. 2. SKIP LOT SAMPLING In present era, industries are very serious for the betterment of their product in the market. As mentioned above, acceptance sampling schemes have been widely used to achieve the goal of in the industries. Among them, the skip lot acceptance sampling plans have been widely used in the industries like chemical industry. The skip lot acceptance sampling plans have been used in the industries when the inspection cost is very high. Dodge [19]introduced concept of continuous sampling is called CSP-1.Dodge [20] introduced skip lot sampling known as SkSP-1 and gave the applications for the inspection of material in bulk or product in successive batches. The details of skip lot acceptance sampling plan can be found in Aslam, et al. [2], Aslam, et al. [3], Aslam, et al. [4], Burnett [17], Balamurali and Jun [13], Perry [32], Aslam, et al. [6], Siraprapa and Sudasana and Sudasna-na Ayudthya [36], Aslam, et al. [5], Balamurali, et al. [15] and Vijayaraghavan [37]. 3. SkSP-V PLAN USING The operational procedure of SkSP-V sampling plan is given by Balamurali and Jun [13] and given as follows: (1) At the start we take a random sample by normal inspection, and specify the producer s risk α-risk and consumer s β- risk, and the acceptance quality level (AQL) and the limiting quality level(lql) based on index.

3 260 Chiang Mai J. Sci. 2015; 42(1) (2) Compute the value of based on random sample of size n where (3.1) and The sample means and standard deviation respectively. (3) The decision about lot sentencing is as follows (i) If k a, the lot is accepted. (ii) Otherwise lot is rejected. (4) If i consecutive lots are accepted on the normal inspection, then switch to skipping inspection by stopping the normal inspection. (5) In skipping inspection a random sample of size f that is only a fraction of the lot. Skipping inspection based on is continued upto the point a sample lot is rejected. (6) When a lot is rejected before k consecutive sample lot we stop the skipping inspection and shifted to the normal inspection in step 1. (7) If a lot is rejected after consecutive lots revert to normal inspection with a reduced clearance no x as in step 8 stated. (8) In normal inspection with reduced clearance no x. Lots are inspected individually one after another as they are submitted for inspection and continue till a lot is rejected or x lots are accepted which ever occur first. (9) If a lot is rejected during the reduced clearance no then immediately revert to normal inspection with clearance no. (10) If x lot are accepted then revert to skipping inspection by stopping the normal inspection as per 4. (11) Replace or correct all defective or nonconforming units in rejected lots 3.1 Designing of SKSP-V Using Balamurali and Jun [13] derived the operating characteristic (OC) function of SkSP-V sampling plan and is given as follows. = (3.2) For simplification it is assumed that k = x, where P is the lot acceptance probability for single acceptance sampling plan. or Here (3.3) (3.4) approximately follows the normal distribution as, see Duncan [21] From Aslam, et al. [11] (3.5) Let and are the probabilities that defective items lies outside the LSL and USL. By substituting By some simplification (3.6) (3.7) (3.8) (3.9)

4 Chiang Mai J. Sci. 2015; 42(1) DETERMINATION OF PARAMETERS OF PROPOSED PLAN 4.1 Symmetric Case For symmetry case, we will assume that = =. The OC function for this case is given as follows Aslam, et al. [11] acceptance probability should be smaller than β at limiting quality level (LQL), see Fertig and Mann [23]. Let AQL = p 1 and LQL = p 2. We will find the plan parameters of the proposed plan by satisfying following inequalities simultaneously. Minimize (4.1) (4.3) (4.4) (4.2) Let α be producer s risk, β be consumer s risk. The producer is interested that lot acceptance probability should be larger than 1 α, the producer s confidence level at acceptance quality level (AQL). On the other hand, the consumer s is also interested that the lot and (4.5) (4.6) We determined the plan parameters of the proposed plan for symmetric case for various values of AQL and LQL and placed in Table 1. Table 1. Plan parameters of SkSP-V based on process capability index ( ). p 1 p 2 n k a k i f ASN

5 262 Chiang Mai J. Sci. 2015; 42(1) From Table 1, as LQL increases for fixed value of AQL, we note the decreasing trend in sample size. 4.2 Asymmetric Case We will present the optimization solution for asymmetry case when,, and,. We will use the OC function given in Eq. (4.3). Table 2. Plan parameters of asymmetric case I. (4.7) (4.8) The plan parameters for asymmetry cases are presented in Table 2 and 3. We note the same behavior of plan parameters as in Table 1. Tables 2-3 are around here p 1 p 2 n k a k i f ASN Table 3. Plan parameters for asymmetric case II. p 1 p 2 n k a k i f ASN

6 Chiang Mai J. Sci. 2015; 42(1) 263 Table 3. (Continued). p 1 p 2 n k a k i f ASN ADVANTAGE OF SkSP-V BASED ON In this section, we will compare the efficiency of the proposed plan over the existing acceptance sampling. For the comparison purpose, we will select the same values of all the specified plan parameters. We will compare the performance of both acceptance sampling schemes in term of ASN. We present the plan parameters of the both sampling plans in Table 4. Table 4. Comparison between proposed and existing single plan. P 1 P 2 Proposed Single plan From Table 4, we can see that the proposed plan provides smaller values of ASN as compared to the existing plan. For example, when AQL=0.001 and LQL=0.002, the sample required from the proposed plan is 81 and from the existing sampling plan it is

7 264 Chiang Mai J. Sci. 2015; 42(1) 483. So, the proposed plan sufficiently reduce the sample size as compared to exiting sampling plan. 6. ILLUSTRATED EXAMPLE To see the application of purposed plan we applied the plan on real word industrial problem of amplified pressure sensor data, for detail of data of amplified pressure senor one can see study of Yen and Chang [18] and Aslam, et al. [11]. We considered the span of amplified pressure sensor as the variable of interest. The data of 32 values is reported in Table 1. Let the USL=2.1 V and LSL=1.9 V. Let p1 = 0.01, p2 = 0.02, α = 0.05 and β = 0.05 we find the optimal plan parameters from Table 1 as,, and. The proposed plan is implemented as follows: Step 1: Take a random sample of size 32 by normal inspection. Step 2: Compute the value of based on random sample of size n = 32 as follows: Calculate and Step 3: the decision about lot sentencing is as follows: As moved to Step-4, we accept the lot and Step 4: if consecutive lots are accepted on the normal inspection then switch to skipping inspection by stopping the normal inspection. Step 5 in skipping inspection a random sample of size of the lot is taken, that is only a fraction of the lot. Skipping inspection based on on is continued up to the point a sample lot is rejected. Step 6 when a lot is rejected before consecutive sample lot we stop the skipping inspection and shifted to the normal inspection in step 1. Step 7 if a lot is rejected after consecutive lots revert to normal inspection with a reduced clearance no x as in step 8 stated. Step 8 in normal inspection with reduced clearance no x, lots are inspected individually one after another as they are submitted for inspection and continue till up to point a lot is rejected are x lots are accepted which ever occur first. Step 9 If a lot is rejected during the reduced clearance no = 2 then immediately revert to normal inspection with clearance no. Step 10 if x lots are accepted then reverts to skipping inspection by stopping the normal inspection as per 4. Step 11 Replace or correct all defective or nonconforming units in rejected lot. 7. CONCLUSION The SkSP-V acceptance sampling plan using process capability index is proposed in this manuscript. The plan parameters of the proposed plan are determined for

8 Chiang Mai J. Sci. 2015; 42(1) 265 symmetry and asymmetry cases. A real industrial example is given. The proposed plan can be used in industries for the inspection of the product to save the cost and time of inspection. The proposed plan for some other sampling schemes can be considered as a future research. ACKNOWLEDGEMENTS The authors are deeply thankful to reviewer and the editor and their valuable suggestions to improve the quality of the manuscript. The author Muhammad Aslam is indebted to Deanship of Scientific Research, King Abdulaziz University, Jeddah, Saudi Arabia, for providing excellent research facilities. APPENDIX: Algorithm: Estimation of SkSP-V using index involves following steps. Input: Values, are taken as input Standardized normal varieties and are computed. and m=min (C1, C2) { ; (3: 500, nums, replace = TRUE) k < sample (1:8, nums, replace = TRUE), i < sample (1:8, nums, replace = TRUE), f < run if (nums, 0, 1) Random generation of natural parameters Compute LP1, LP2, ASN2, pa1 and pa2, Select those combination of LP1 and LP2, where (LP1 1 α and LP2 β ). Select the minimum value of ASN2 satisfying above constraint ASN [ i ] = ASN_min} Output: The optimized parameters value of proposed plan and x are estimated. Table 1: Plan parameters of SkSP-V based on Process Capability Index ( ) REFERENCES [1] Aslam M., Azam M., Lio Y.L. and Jun C.H., Two-stage group acceptance sampling plan for Burr type X percentiles, J. Testing Eval., 2013; 41(4): [2] Aslam M., Balamurali S. and Jun C.H., SkSP-V sampling plan with group sampling plan as reference based on truncated life test under weibull and generalized exponential distributions, Pak. J. Stat., 2013; 29(2): [3] Aslam M., Balamurali S., Jun C.H. and Ahmad M., Optimal design of skip lot group acceptance sampling plans for the Weibull distribution and the generalized exponential distribution, Qual. Eng., 2013; 25(3): DOI / [4] Aslam M., Balamurali S., Jun C.H. and Ahmad M., Optimal designing of a skip lot sampling plan by two point method, Pak. J. Stat., 2010; 26(4): [5] Aslam M., Balamurali S., Jun C.H. and Rasool M., Optimal designing of Skip lot sampling plan of type SkSP-2 with group acceptance sampling plan as reference plan under Burr type XII distribution, J. Stat. Comput. Sim., 2013; 83(1): DOI /

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