Design of six sigma quick switching variables sampling system of type indexed by six sigma AQL and six sigma AOQL

This article gives designing procedure of Six Sigma Quick Switching Variables Sampling System (SSQSVSS (nTσ, nNσ; kσ)) indexed by Six Sigma AQL and Six Sigma AOQL. The design procedure constructed tables for easy selection of system given indexed by six sigma quality levels by known and unknown σ respectively.

Introduction

The designing procedure for the SSQSVSS (n Tσ , n Nσ ; k σ ) indexed by SSAQL and SSAOQL is based on Govindaraju (1990) [3] procedures and tables, developing for the selection of single sampling plan for variables indexed by AQL and AOQL. Soundararajan (1981) [9] has developed procedures and tables for the selection of single sampling plans for attributes for given AQL and AOQL. Govindaraju (1990) [3] has developed procedures and tables for the selection of single sampling plans for variables indexed by AQL and AOQL. Later Soundarajan and Palanivel (2000) [8] have developed procedures and tables for the selection of quick switching single sampling variables systems indexed by AQL and AOQL. Based on above article Senthilkumar and Esha Raffie (2017) [6] have constructed SSQSVSS (n σ ; k Tσ , k Nσ ) indexed by Six Sigma AQL and Six Sigma AOQL. The extension work of SSQSVSS (n Nσ , n Tσ ; k σ ) indexed by Six Sigma AQL and Six Sigma AOQL are given in this article. This concept can be extended to variable quality characteristics of the study, the resulting plan would be designated as SSQSVSS and would be applied under the following conditions:  The production is steady, so that results on current and preceding lots are broadly indicative of a continuous process.  Lots are submitted substantially in the order of production.  Inspection is by variables, with the quality being defined as the fraction of nonconforming.  The sample units are selected from a large lot and production is continuous.  The production process should depend on automation and human handling is very negligible.  The industry may adopt system method with decision makers having an experience in adopting the six sigma quality initiatives.

Basic Assumptions

 The quality characteristic is represented by a random variable X measurable on a continuous scale.  Distribution of X is normal with mean and standard deviation.  An upper limit U, has been specified and a product is qualified as defective when X > U.

[when the lower limit L is Specified, the product is a defective one if X < L]  The Purpose of inspection is to control the fraction defective, p in the lot inspected.

The condition for application above basic assumptions of SSQSVSS (n Tσ , n Nσ ; k σ ) system are the same as that of the SSQSVSS (n σ ; k Tσ , k Nσ ) system. The fraction defective in a lot will be p=

Where z ~ N (0, 1). Under the σmethod plan, the lot would be accepted if X + k σ U, ≤

where U is the upper specification limit or X + k σ L, ≥

where L is the lower specification limit.

Procedure of SSQSVSS (nTσ, nNσ; kσ) with known σ variables plan as the reference plan

The Six Sigma Quick Switching Variables Sampling System with known σ variables plan as the reference plan has following operating procedure

Operating procedure

Operating procedure



Step 4: i) If X + k σ U or X k σ L    accept the lot and repeat Step 1, for the next lot.

ii) If X + k σ > U or X k σ < L,  reject the lot and repeat Step 3.

Where n Nσ and n Tσ are the sample sizes of normal and tightened single sampling variable plans respectively, and k is the acceptance constant under σ-method. Where X and σ are the average quality characteristic and standard deviation respectively. The Six Sigma Quick Switching Variables Sampling System as SSQSVSS (n T , n N ; k σ ). As the tightened plan sample size n T is greater than the normal plan sample size n N , for designing the SSQSVS system n T is fixed as a multiple of n N i.e. n T = m n N , where m >1. Thus the QSS can also be designated as SSQSVSS (mn N , n N ; k σ ).

Based on Romboski (1969), the OC function of SSQSVSS (n Tσ , n Nσ ; k σ ) can be written as

The fraction nonconforming in a given lot has been given in (1) with v= (U-µ)/σ and its probability of acceptance will be

If the quality of the accepted lot is p and all defective units found in the rejected lots are replaced by non-defective units in a rectifying inspection plan, the Six Sigma average outgoing quality (SSAOQ) can be approximated as SSAOQ = pP a (p) (5) where P a (p) is defined in equation (2). If p m is the proportion nonconforming items at which SSAOQ is maximum, one has

If SSAQL (p 1 ) is prescribed, then the corresponding value of v SSAQL or v 1 will be fixed and if P a (p) is fixed at 99.99966%, that is (1-α). Where, α = 0.0000034x10 - 1 should understand the limitations of plans indexed by SSAOQL. Sampling with rectifying of rejected lots on the one hand reduces the average percentage of nonconforming items in the lots, but on the other hand introduces nonhomogeneity in the series of lots finally accepted. That is, any particular lot will have a quality of p% or 0% nonconforming depending on whether the lot is accepted or rectified. Thus the assumption underlying the SSAOQL principle is that the homogeneity in the qualities of individual lots is unimportant and only the average quality matters. For plans listed in Table 1, if the individual lot quality happens to be the product quality p m at which SSAOQL occurs, then the associated probability of acceptance will be poor. Table 2 gives P a (p m ) values of plans given in Table 1. For example, for SSAQL is 0.000004 and SSAOQL is 0.00001, Table 2 gives P a (p m ) = 0.71. Then p m = SSAOQL/ P a (p m ) = 0.000014. In order to avoid such inconvenience, the producer should maintain the process quality more or less at the SSAQL. The higher rate of rejection of lots at p = p m will also indirectly put pressure on the producer to improve the submitted quality.

SSAOQL	SSAQL	—	—	—	—	—	—
	0.000001	0.000002	0.000003	0.000004	0.000005	0.00001	0.00005
0.000002	0.91
0.000003	0.89	0.91
0.000004	0.84	0.86	0.89
0.000005	0.81	0.83	0.86	0.89
0.000006	0.78	0.80	0.83	0.86	0.89
0.000007	0.76	0.78	0.81	0.84	0.87
0.000008	0.72	0.74	0.77	0.80	0.83
0.000009	0.71	0.73	0.76	0.79	0.82
0.00001	0.63	0.65	0.68	0.71	0.74
0.00002	0.61	0.63	0.66	0.69	0.72	0.75
0.00003	0.58	0.60	0.63	0.66	0.69	0.72
0.00004	0.54	0.56	0.59	0.62	0.65	0.68
0.00005	0.51	0.53	0.56	0.59	0.62	0.65
0.00006		0.41	0.44	0.47	0.50	0.53	0.58
0.00007			0.38	0.41	0.44	0.47	0.52
0.00008				0.35	0.38	0.41	0.46
0.00009					0.32	0.35	0.40
0.0001					0.29	0.32	0.37
0.0002					0.23	0.26	0.31
0.0003						0.22	0.27
0.0004							0.25
0.0005							0.21

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Pa (pm) Values of known σ plans

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SSAQL	SSAOQL	nT	nN	k	σ – level	nT	nN	k	σ – level
0.000004	0.000008	2101	1051	4.016	4.2	19046	9523	4.016	4.8
	0.000009	1541	771	4.001	4.1	13878	6939	4.001	4.8
	0.00001	959	480	3.987	3.9	8583	4291	3.987	4.6
	0.00002	316	158	3.975	3.5	2813	1406	3.976	4.5
0.000005	0.000006	7731	3865	3.941	4.6	67767	33884	3.941	5.2
	0.000007	6085	3043	3.920	4.5	52840	26420	3.920	5.1
	0.000008	5883	2942	3.904	4.5	50720	25360	3.904	5.1
	0.000009	4315	2158	3.889	4.4	36951	18476	3.889	5.0
	0.00001	2686	1343	3.875	4.3	22849	11424	3.875	4.9
	0.00002	885	442	3.863	3.9	7487	3744	3.863	4.6
	0.00003	714	357	3.781	3.8	5818	2909	3.781	4.5
	0.00004	605	303	3.732	3.7	4822	2411	3.732	4.5
	0.00005	481	241	3.695	3.7	3766	1883	3.696	4.5
0.00001	0.00002	973	487	3.751	3.9	7821	3910	3.751	4.6
	0.00003	785	393	3.669	3.9	6073	3036	3.669	4.5
	0.00004	666	333	3.620	3.8	5030	2515	3.620	4.5
	0.00005	529	265	3.583	3.7	3927	1964	3.584	4.5
0.00005	0.00006	1608	804	3.377	4.1	10779	5390	3.377	4.7
	0.00007	1272	636	3.355	4.1	8430	4215	3.355	4.7
	0.00008	1003	501	3.335	4.0	6580	3290	3.335	4.6
	0.00009	659	330	3.301	3.8	4252	2126	3.302	4.5
	0.0001	505	252	2.969	3.8	2729	1365	2.970	4.5

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0.000003	0.000004	2230	1115	4.224	4.2	22124	11062	4.224	4.9
	0.000005	1876	938	4.194	4.1	18373	9187	4.194	4.8
	0.000006	1534	767	4.165	4.0	14838	7419	4.165	4.8
	0.000007	1207	604	4.144	4.0	11575	5787	4.144	4.7
	0.000008	1167	584	4.128	4.0	11114	5557	4.128	4.7
	0.000009	856	428	4.113	3.8	8099	4049	4.113	4.6
	0.00001	533	266	4.099	3.7	5009	2505	4.100	4.5
0.000004	0.000005	4314	2157	4.082	4.4	40259	20129	4.082	5.0
	0.000006	2761	1380	4.053	4.3	25439	12719	4.053	4.9
	0.000007	2173	1087	4.032	4.2	19839	9920	4.032	4.8

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SSAQL	SSAOQL	nT	nN	k	σ – level	nT	nN	k	σ - level
0.000001	0.000002	2138	1069	4.491	4.1	23700	11850	4.491	4.9
	0.000003	1532	766	4.448	4.0	16688	8344	4.448	4.8
	0.000004	1448	724	4.418	4.0	15580	7790	4.418	4.8
	0.000005	1218	609	4.389	3.9	12950	6475	4.389	4.7
	0.000006	996	498	4.368	3.9	10498	5249	4.368	4.6
	0.000007	784	392	4.352	3.8	8209	4104	4.352	4.6
	0.000008	758	379	4.337	3.8	7887	3944	4.337	4.6
	0.000009	556	278	4.323	3.7	5752	2876	4.323	4.5
	0.00001	346	173	4.311	3.5	3561	1781	4.312	4.5
0.000002	0.000003	1685	843	4.336	4.1	17528	8764	4.336	4.8
	0.000004	1593	796	4.306	4.0	16360	8180	4.306	4.8
	0.000005	1340	670	4.277	4.0	13595	6797	4.277	4.7
	0.000006	1096	548	4.256	3.9	11019	5509	4.256	4.7
	0.000007	862	431	4.240	3.8	8615	4307	4.240	4.6
	0.000008	834	417	4.225	3.8	8276	4138	4.225	4.6
	0.000009	612	306	4.211	3.7	6034	3017	4.211	4.5
	0.00001	381	190	4.199	3.5	3736	1868	4.200	4.5

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SSQSVSS with known and unknown σ indexed by SSAQL and SSAOQL (nT = mnN, m=2)

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Procedure of unknown σ SSQSVSS for given SSAQL and SSLQL

If the population standard deviation σ is unknown, then it is estimated from the sample standard deviation S (n-1 as the divisor). If the sample size of the unknown sigma variables system (S-method) is n Ts , n Ns and the acceptance constant is k s , then the operating procedure is as follows

Step 1: Draw a sample of size n s from the lot, inspect and record the measurement of the quality characteristic for each unit of the sample. Compute the sample mean X and sample standard deviation S N .

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Selection of unknown σ SSQSVSS (nTs, nNs; ks) for Given SSAQL and SSAOQL

Table 1 also gives such matched s-method plan. For example, for given SSAQL is 0.00001 and SSAOQL is 0.00004, one obtains the parameters of the s-method plan from Table 1 to be n Ts = 5030, n Ns = 2515 and k s = 3.620, which is associated with 4.5 sigma level of SSQSVSS (n Ts, n Ns ; k s ). The sample size n Ts = m n Ns = (2) (2515) = 5030. Thus, for the requirem-ent, the SSQSVSS (n Ts, n Ns ; k s ) id specified by the parameters, n Ts = 5030, n Ns = 2515 and k s = 3.620.

Construction of Table 1

For constructing Table 1, a trial value of p m is assumed and the probability of acceptance at p m is found using (6) as P a (p m ) = SSAOQL / p m (7) The auxiliary variables v m , w Nm and w Tm corresponding to the values of p m and P a (p m ) respectively, are found using (1), ( 2), ( 3) and ( 4). For given values of (p 1 , α), determine the values of v 1 , w N1 and w T1 using the approximation (Abramwitz and Stegun (1972)) for the ordinate of the cumulative normal distribution. With the values of v m , w Nm and w Tm , the following equation is used for calculating n Nσ . Equation ( 8) is the formula for finding the sample size of a known σ SSQSVS system. With the values of n obtained from (14), it is then checked to see whether the assumed value of p m corresponds to the proportion non-conforming at which the SSAOQL occurs or not. That is, it is checked to see whether or not the trial value of p m satisfies the following condition.

Equation ( 8) was obtained from the following relation

If the assumed value of p m does not satisfy (10), then another trial value of p m is obtained from (10) by numerical methods. The methods of successive substation is often found to give good results and ( 10) is rewritten for this purpose as

After determining the next trial value of p m , again the values of v m , w Nm , w Tm and n σ are found and the condition (14) rechecked. This iterative procedure continues until the convergence of p m is achieved. Then the value of k σ is obtained from (3) and (4).

For obtaining the values of v 1 , w N and w T , the approximation for the ordinate of the cumulative normal distribution available in Abramowitz and Stegun (1972) was used. The S-method plans matching the σ-method plans were obtained using computer search routine through C++ programme. For selected combinations of SSAQL and SSAOQL, Table 1 was constructed following the above iterative procedure.

The iterative procedure given above may also be used to determine a Six Sigma Quick Switching Sampling System for given Six Sigma Indifference Quality Level (SSIQL) and SSAOQL as well as for given Six Sigma Limiting Quality Level(SSLQL) and SSAOQL with appropriate auxiliary variables v, w N and w T .

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3. Govindaraju K. Single sampling Plans for Variables Indexed by AQL and AOQL, Journal of Quality Technology. 1990; 22(4):310-313.
4. Romboski LD. An Investigation of Quick Switching Acceptance Sampling Systems, Doctoral Dissertation, Rutgers the State University, New Brunswick, New Jersey, 1969.
5. Senthilkumar D, Esha Raffie B. Six Sigma Quick Switching Variables Sampling System Indexed by Six Sigma Quality Levels, International Journal of Computer Science & Engineering Technology. 2012; 3(12):565-576.
6. Senthilkumar D, Esha Raffie B. Quick Switching Variables Sampling System Indexed by Six Sigma AQL and Six Sigma AOQL, International Journal of Current Research in Science and Technology. 2017; 3(10):21-28.
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