A web-based tool to design and analyze single- and double-stage acceptance sampling plans

Figures & data

Figure  1. OC-curves of several sampling plans where P_a denotes the acceptance probability: (a) attributes sampling plans and (b) variables sampling plans. The idealized OC-curves corresponding with complete inspection are shown in gray.

Figure 2. Algorithms for the design of two-point DSP plans. The parameters of the procedures determine the producer’s and consumer’s risk point. A constraint $n_2=r{n}_1$ on the sample sizes can be chosen by specifying $r\in {\mathbb{N}}_0.$

Figure 3. The ASN at $p_{\text{AQL}}$ as a function of the sample size n₁ = n₂ for several two-point DSP-$(n_1,n_2,k_1,k_2)$ plans that pass through two points $(p_{\text{AQL}},0.95)$ and $(p_{\text{RQL}},0.1).$ The dotted identity line correspond to an SSP-(n₁, k₁) where ASN $=n_1.$ The gray and black dots correspond to the plans obtained by Sommers (1981) and Algorithm 2, respectively.

Figure 4. The interface of our web tool that consists of a left panel to set the sampling plan parameters and a right panel to visualize the plan with an OC-curve and an ASN curve. The selection of the main sheet determines the type of inspection: attributes inspection or variables inspection. The left panel allows to implement single-stage and double-stage sampling plans.

Figure  5. Left panels of the interface: (a) the parameters of a DSP-$(n_1,n_2,c_1,c_2)$ plan for inspection by attributes, (b) the parameters of an SSP-(n, k) plan for inspection by variables, and (c) the parameters of a DSP-$(n_1,n_2,k_1,k_2)$ for inspection by variables. The panel in (a) is part of the main sheet entitled sampling plans for attributes inspection as shown in Figure 4. The panels in (b) and (c) are part of the main sheet entitled sampling plans for variables inspection.

Figure  6. (a) The ASN curve of a DSP-$(18,18,2.85,3.02)$ for variables inspection. (b) The risks associated to an individual ISO 3951-1 SSP-$(40,2.97)$ plan for inspection by variables.

Figure A1. Examples of curves $n_l(c_2)$ and $n_u(c_2)$ as defined in [11][11] $$n_l\left(c\right)=\inf \left\{n\right|{\varphi }_a^{\mathit{ssp}}(p_{\text{RQL}},n,c)\le \beta \left\}\hspace{1em} \text{and} \hspace{1em}{n}_u\right(c)=\sup \{n\left|{\varphi }_a^{\mathit{ssp}}\right(p_{\text{AQL}},n,c)\ge 1-\alpha \},$$[11] for an SSP-$(n,c_2)$ plan and $n_{1l}(n_2,c_1,c_2)$ and $n_u(n_2,c_1,c_2)$ as defined in [A.1][A.1] $$\begin{array}{l}{n}_{1l}(n_2,c_1,c_2)=\inf \left\{n_1\right|{\varphi }_a^{\mathit{dsp}}(p_{\text{RQL}},n_1,n_2,c_1,c_2)\le \beta \}\hspace{1em}\text{and} \\ \hspace{1em}{n}_{1u}(n_2,c_1,c_2)=\sup \left\{n_1\right|{\varphi }_a^{\mathit{dsp}}(p_{\text{AQL}},n_1,n_2,c_1,c_2)\ge 1-\alpha \}.\end{array}$$[A.1] for a DSP-$(n_1,n_2,c_1,c_2)$ plan with fixed choices of c₁ and n₂: (a) $c_1=0$ and $n_2=50,$ with $p_{\text{AQL}}=1\%$ and $p_{\text{RQL}}=10\%,$ (b) $c_1=10$ and $n_2=350,$ with $p_{\text{AQL}}=2\%$ and $p_{\text{RQL}}=4\%.$ Risks are set to $\alpha =\beta =5\%.$ The gray and black dotted lines correspond to the curves $n_l(c_2)-n_2$ and $n_u(c_2)-n_2$ respectively to which $n_{1l}(n_2,c_1,c_2)$ and $n_u(n_2,c_1,c_2)$ converge for large c₂.

Table B1. Table of two-point SSP and DSP plans indexed by $p_{\text{AQL}}$ and $p_{\text{RQL}}.$

		SSP-(n, c)		DSP-$(n_1,n_2,c_1,c_2)$				SSP-(n, k)		DSP-$(n_1,n_2,k_1,k_2)$
$p_{\text{AQL}}$	$p_{\text{RQL}}$	n	c	n1	c1	c2	ASN	n	k	n1	k1	k2	ASN
0.001	0.003	3922	7	2391	3	8	2913.4	74	2.90	54	2.86	2.96	60.8
	0.004	2317	5	1384	2	5	1604.8	45	2.85	33	2.80	2.93	37.1
	0.005	1335	3	692	0	3	1033.9	33	2.80	24	2.75	2.90	26.9
	0.006	1112	3	681	1	3	779	26	2.77	19	2.71	2.87	21.3
	0.007	759	2	416	0	2	554	22	2.74	16	2.67	2.84	17.8
0.002	0.006	1960	7	1195	3	8	1455.8	64	2.67	47	2.63	2.74	53.2
	0.008	1158	5	692	2	5	802.4	39	2.61	29	2.57	2.70	32.4
	0.01	667	3	346	0	3	517	29	2.57	21	2.51	2.66	23.4
	0.012	555	3	340	1	3	388.8	23	2.54	17	2.47	2.62	18.6
	0.014	379	2	208	0	2	277	19	2.50	14	2.43	2.6	15.4
0.005	0.015	783	7	477	3	8	580.5	53	2.35	39	2.31	2.42	43.3
	0.02	462	5	276	2	5	319.7	32	2.29	23	2.23	2.39	26.2
	0.025	266	3	138	0	3	206.2	23	2.23	17	2.17	2.33	18.8
	0.03	221	3	135	1	3	154.2	18	2.19	13	2.11	2.33	14.8
	0.035	151	2	83	0	2	110.5	15	2.15	11	2.07	2.27	12.2
0.01	0.03	390	7	238	3	8	289.3	44	2.08	32	2.03	2.16	35.9
	0.04	198	4	137	2	5	158.4	26	2.00	19	1.94	2.12	21.5
	0.05	132	3	69	0	3	103.2	19	1.95	14	1.88	2.05	15.4
	0.06	110	3	67	1	3	76.4	15	1.90	11	1.82	2.00	12.0
	0.07	75	2	41	0	2	54.5	12	1.85	9	1.77	1.98	9.90
0.02	0.04	616	18	331	7	19	445	94	1.88	69	1.85	1.94	77.6
	0.05	306	10	175	4	11	222.8	52	1.83	38	1.78	1.90	42.6
	0.06	194	7	118	3	8	142.9	35	1.78	26	1.73	1.85	28.7
	0.07	131	5	82	2	6	100.4	26	1.73	19	1.67	1.84	21.3
	0.08	98	4	68	2	5	78.4	21	1.69	15	1.62	1.81	16.9
0.03	0.05	807	32	415	10	33	706.2	154	1.75	114	1.72	1.79	128.1
	0.06	410	18	243	9	20	290.9	81	1.70	60	1.67	1.76	67.1
	0.07	252	12	130	4	12	175.7	53	1.65	39	1.61	1.72	43.5
	0.08	175	9	93	3	9	121.3	38	1.61	28	1.57	1.70	31.5
	0.09	129	7	68	2	7	90.7	30	1.58	22	1.52	1.67	24.5
0.05	0.07	1196	72	598	10	72	1196	300	1.55	222	1.53	1.58	249.5
	0.08	572	37	286	10	37	529.1	149	1.51	111	1.49	1.55	124.1
	0.09	348	24	185	10	25	244.5	93	1.47	69	1.44	1.53	77.1
	0.1	233	17	127	7	18	165.5	65	1.44	48	1.40	1.51	54.0
	0.11	170	13	94	5	14	125	49	1.41	36	1.37	1.49	40.8

Double-stage plans are chosen such that the ASN at $p_{\text{AQL}}$ is minimized and n₁ = n₂. Producer’s and consumer’s risk are set to $\alpha =5\%$ and $\beta =10\%,$ respectively.

Sommers, D. 1981. Two-point double variables sampling plans. Journal of Quality Technology 13 (1):25–30. doi: 10.1080/00224065.1981.11980982.

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