Linear combination of chi-squares for multinomial process monitoring

Talmoudi, Ramzi; Achouri, Ali; Taleb, Hassen

doi:10.1590/1806-9649-2021v28e41

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Sumário

THEMATIC SECTION: STATISTICAL PROCESS MONITORING AND CONTROL • Gest. Prod. 28 (3) • 2021 • https://doi.org/10.1590/1806-9649-2021v28e41 copy

Linear combination of chi-squares for multinomial process monitoring

Combinação linear de qui-quadrados para monitoramento de processo multinomial

Authorship SCIMAGO INSTITUTIONS RANKINGS

Abstract:

Marcucci (1985) proposed a chi square goodness of fit statistic based generalized p-chart for multinomial process monitoring. A chi square distribution quantile was considered as a control chart limit. A weighted chi square goodness of fit statistic-based control chart is proposed for multinomial process monitoring in this paper, where more important weights are advocated to poor quality categories. The statistic distribution is approximated by a well-known linear combination of chi squares distribution. The approximation is assessed through a simulation, an extreme percentile of the approximated distribution is used as an upper control chart limit and a comparison is carried out with a chi square goodness of fit statistic-based control chart. The average run length is used as a benchmark and the comparison is performed using simulations considering two process shifts scenarios. Under some restrictions, the weighted statistic-based control chart allows an earlier detection of process shift in case of deterioration and postpones out of control signals in case of improvement. This benefit is clearer when the process is improved by a decrease in the poor quality probability category and an increase in the best quality category probability.

Keywords:
Multinomial Process; Generalized p-Chart; Distribution Approximation; Simulation

w₁, …, w_q	á = 0.99	á = 0.975	á = 0.95	á = 0.9	á = 0.1	á = 0.05	á = 0.025	? = 0.01
0.10,0.40,0.50	0.0313	0.0593	0.0976	0.1643	2.1873	2.8175	3.4503	4.2908
1.00,1.00,1.00	0.1148	0.2157	0.3518	0.5843	6.2513	7.8147	9.3484	11.3448
1.00,4.00,5.00	0.3139	0.5936	0.9760	1.6438	21.8738	28.1754	34.5032	42.9080
0.40,0.50,5.00	0.1163	0.2212	0.3667	0.6273	14.1921	15.0605	15.2203	15.2930
0.20,0.30,10.0	0.1000	0.1936	0.3294	0.5922	7.4066	7.4389	7.4448	7.4516
0.10,0.20,0.40	0.0230	0.0434	0.0712	0.1191	1.5171	1.9716	2.4398	3.0766
0.20,0.30,0.50	0.0357	0.0672	0.1098	0.183	2.1214	2.7075	3.3021	4.1022
0.40,0.60,1.80	0.0871	0.1644	0.2696	0.4519	6.1697	8.182	10.2868	13.1681
0.05,0.10,0.40	0.0145	0.0276	0.0455	0.0769	1.262	1.7119	1.9708	2.0150
$e^{(j - \frac{q}{2} - 1)}$	0.0702	0.1329	0.2189	0.3698	5.5635	7.4118	8.8541	9.1435
0.05,0.10,0.20,0.30	0.0396	0.0651	0.0964	0.1465	1.3374	1.6958	2.0582	2.5438
0.05,0.15,0.30,0.40	0.0524	0.0866	0.1289	0.1972	1.8617	2.3563	2.8533	3.5223
0.05,0.10,0.20,0.40	0.0427	0.0703	0.1044	0.1593	1.5717	2.0260	2.4940	3.1316
0.05,0.10,0.20,0.50	0.0452	0.0747	0.1112	0.1703	1.8141	2.3745	2.9585	3.7887
0.05,0.10,0.25,0.30	0.0419	0.0690	0.1024	0.1559	1.4461	1.8302	2.2152	2.7256
0.05,0.10,0.15,0.30	0.0367	0.0604	0.0893	0.1355	1.2335	1.5744	1.9249	2.4015
1.00,2.00,3.00,5.00	0.7018	1.1510	1.7007	2.5729	22.3576	28.2192	34.1631	42.1611
0.50,1.00,3.00,5.00	0.5030	0.8321	1.2422	1.9096	20.2029	26.0361	32.0016	40.8856
1.00,3.00,5.00,10.0	1.0618	1.7539	2.6133	4.0032	39.7681	51.1285	62.8500	80.0873
1.00,1.00,1.00,1.00	0.2971	0.4844	0.7107	1.0636	7.7794	9.4877	11.1432	13.2767
2.00,5.00,10.0,20.0	2.0285	3.3504	4.9922	7.6501	78.0008	100.7196	124.1711	158.3079
$e^{(j - \frac{q}{2} - 1)}$	0.1869	0.3118	0.4702	0.7338	9.3129	12.3603	15.5398	19.9112
1.00,1.00,1.00,1.00,1.00	0.5542	0.8312	1.1454	1.6103	9.2363	11.0704	12.8325	15.0862
0.05,0.10,0.20,0.30,0.50	0.0981	0.1494	0.2093	0.3012	2.2890	2.8753	3.4764	4.4908
0.05,0.10,0.15,0.20,0.50	0.0849	0.1289	0.1801	0.2583	2.0067	2.5652	3.1473	3.9988
0.05,0.10,0.25,0.40,0.50	0.1094	0.1669	0.2346	0.3390	2.5866	3.2273	3.9065	4.7318
0.05,0.10,0.20,0.40,0.50	0.1044	0.1592	0.2235	0.3227	2.5004	3.1330	3.7664	4.6068
0.50,1.00,3.00,4.00,5.00	1.1371	1.7369	2.4427	3.5340	26.7899	33.2844	39.7262	48.2125
0.50,2.00,2.50,3.00,5.00	1.1809	1.7967	2.5147	3.6110	25.2738	31.2622	37.2654	45.2826
0.05,0.10,0.20,0.40,1.20	0.1267	0.1950	0.2769	0.4066	4.1783	5.5218	6.9284	9.6310

p₁, p₂, p₃	n=3000	n=1000	n=500	n=300	n=200
0.8000,0.1600,0.0400	0.0018	0.0018	0.0019	0.0021	0.0024
0.6600,0.3000,0.0400	0.0016	0.0017	0.0019	0.0020	0.0023
0.8500,0.1000,0.0500	0.0018	0.0019	0.0019	0.0021	0.0022
0.7400,0.2000,0.0600	0.0016	0.0018	0.0018	0.0019	0.0020
0.5500,0.3800,0.0700	0.0014	0.0015	0.0016	0.0017	0.0019
0.8000,0.1300,0.0700	0.0016	0.0017	0.0018	0.0019	0.0021
0.9000,0.0700,0.0300	0.0020	0.0022	0.0023	0.0028	0.0037
0.7000,0.2700,0.0300	0.0017	0.0019	0.0021	0.0025	0.0034
0.9425,0.0500,0.0075	0.0025	0.0029	0.0043	0.0047	0.0053
0.5500,0.4425,0.0075	0.0022	0.0024	0.0036	0.0040	0.0047
0.9600,0.0300,0.0100	0.0024	0.0027	0.0033	0.0045	0.0056
0.6500,0.3400,0.0100	0.0021	0.0023	0.0028	0.0040	0.0047
0.9300,0.0500,0.0200	0.0021	0.0023	0.0026	0.0039	0.0038
0.5000,0.4800,0.0200	0.0017	0.0019	0.0021	0.0035	0.0029
0.9725,0.0200,0.0075	0.0026	0.0030	0.0045	0.0046	0.0060
0.9835,0.0100,0.0065	0.0027	0.0038	0.0050	0.0052	0.0095
0.9750,0.0200,0.0050	0.0026	0.0033	0.0049	0.0057	0.0054
0.9775,0.0175,0.0050	0.0027	0.0033	0.0050	0.0056	0.0059
0.9700,0.0200,0.0100	0.0024	0.0027	0.0033	0.0047	0.0059
0.9675,0.0200,0.0125	0.0022	0.0030	0.0034	0.0043	0.0058

p₁, p₂, p₃, p₄	n=3000	n=1000	n=500	n=300	n=200
0.8000,0.1000,0.0600,0.0400	0.0019	0.0021	0.0023	0.0025	0.0027
0.6000,0.2500,0.1100,0.0400	0.0018	0.0019	0.0021	0.0023	0.0024
0.5500,0.2500,0.1500,0.0500	0.0017	0.0017	0.0018	0.0021	0.0021
0.8500,0.0500,0.0500,0.0500	0.0019	0.0021	0.0022	0.0025	0.0027
0.7000,0.1500,0.0900,0.0600	0.0017	0.0018	0.0019	0.0020	0.0022
0.5000,0.2900,0.1500,0.0600	0.0016	0.0017	0.0017	0.0018	0.0020
0.6000,0.3400,0.0525,0.0075	0.0023	0.0030	0.0036	0.0044	0.0055
0.5400,0.2500,0.2025,0.0075	0.0020	0.0027	0.0032	0.0039	0.0051
0.6000,0.3700,0.0200,0.0100	0.0022	0.0027	0.0035	0.0051	0.0058
0.8000,0.1400,0.0500,0.0100	0.0023	0.0028	0.0034	0.0048	0.0054
0.7000,0.1700,0.1100,0.0200	0.0020	0.0022	0.0025	0.0029	0.0035
0.5500,0.3500,0.0800,0.0200	0.0020	0.0022	0.0025	0.0029	0.0036
0.9450,0.0300,0.0150,0.0100	0.0025	0.0031	0.0039	0.0054	0.0064
0.9700,0.0125,0.0100,0.0075	0.0027	0.0036	0.0047	0.0059	0.0070
0.9650,0.0150,0.0100,0.0100	0.0026	0.0033	0.0042	0.0060	0.0069
0.9675,0.0150,0.0100,0.0075	0.0027	0.0036	0.0046	0.0057	0.0069
0.9430,0.0300,0.0150,0.0120	0.0024	0.0030	0.0037	0.0051	0.0055
0.9425,0.0300,0.0150,0.0125	0.0024	0.0029	0.0038	0.0045	0.0061
0.9350,0.0300,0.0200,0.0150	0.0024	0.0028	0.0035	0.0041	0.0054

p₁, p₂, p₃, p₄, p₅	n=3000	n=1000	n=500	n=300	n=200
0.7400,0.10,0.070,0.050,0.0400	0.0020	0.0021	0.0023	0.0026	0.0029
0.6500,0.13,0.100,0.080,0.0400	0.0019	0.0019	0.0021	0.0023	0.0026
0.6400,0.12,0.100,0.080,0.0600	0.0017	0.0018	0.0019	0.0021	0.0022
0.5500,0.20,0.100,0.080,0.0700	0.0017	0.0017	0.0018	0.0020	0.0021
0.5500,0.20,0.120,0.080,0.0500	0.0018	0.0018	0.0020	0.0021	0.0024
0.8000,0.05,0.050,0.050,0.0500	0.0019	0.0020	0.0022	0.0024	0.0028
0.6000,0.25,0.070,0.050,0.0300	0.0020	0.0022	0.0024	0.0027	0.0031
0.4000,0.27,0.200,0.100,0.0300	0.0018	0.0019	0.0021	0.0023	0.0026
0.7000,0.20,0.050,0.030,0.0200	0.0022	0.0025	0.0029	0.0033	0.0043
0.5000,0.23,0.150,0.100,0.0200	0.0019	0.0021	0.0025	0.0028	0.0035
0.8000,0.10,0.050,0.040,0.0100	0.0023	0.0029	0.0036	0.0046	0.0059
0.6000,0.15,0.140,0.100,0.0100	0.0021	0.0026	0.0032	0.0041	0.0053
0.9100,0.05,0.020,0.010,0.0100	0.0026	0.0034	0.0044	0.0058	0.0073
0.9105,0.05,0.020,0.010,0.0095	0.0027	0.0034	0.0044	0.0056	0.0067
0.9425,0.03,0.010,0.010,0.0075	0.0027	0.0038	0.0048	0.0060	0.0078
0.9250,0.04,0.020,0.010,0.0050	0.0029	0.0041	0.0059	0.0071	0.0073
0.9200,0.05,0.015,0.010,0.0050	0.0029	0.0041	0.0059	0.0071	0.0075
0.9000,0.05,0.020,0.015,0.0150	0.0024	0.0029	0.0037	0.0047	0.0058
0.9080,0.05,0.020,0.010,0.0120	0.0026	0.0032	0.0040	0.0053	0.0065
0.9060,0.05,0.020,0.012,0.0120	0.0025	0.0032	0.0039	0.0051	0.0062
0.9030,0.05,0.020,0.010,0.0170	0.0025	0.0030	0.0038	0.0048	0.0057

p₁, p₂, p₃	n=3000	n=1000	n=500	n=300	n=200
0.8000,0.1600,0.0400	0.0022	0.0023	0.0024	0.0029	0.0036
0.6600,0.3000,0.0400	0.0021	0.0021	0.0022	0.0028	0.0034
0.8500,0.1000,0.0500	0.0022	0.0023	0.0026	0.0026	0.0029
0.7400,0.2000,0.0600	0.0020	0.0020	0.0021	0.0022	0.0028
0.5500,0.3800,0.0700	0.0018	0.0018	0.0019	0.0019	0.0023
0.8000,0.1300,0.0700	0.0020	0.0020	0.0022	0.0022	0.0025
0.9000,0.0700,0.0300	0.0024	0.0025	0.0028	0.0032	0.0039
0.7000,0.2700,0.0300	0.0021	0.0022	0.0024	0.0027	0.0036
0.9425,0.0500,0.0075	0.0029	0.0034	0.0058	0.0058	0.0056
0.5500,0.4425,0.0075	0.0024	0.0028	0.0053	0.0056	0.0049
0.9600,0.0300,0.0100	0.0028	0.0032	0.0041	0.0050	0.0056
0.6500,0.3400,0.0100	0.0023	0.0028	0.0034	0.0041	0.0049
0.9300,0.0500,0.0200	0.0026	0.0027	0.0032	0.0043	0.0039
0.5000,0.4800,0.0200	0.0021	0.0022	0.0026	0.0036	0.0033
0.9725,0.0200,0.0075	0.0030	0.0035	0.0060	0.0064	0.0062
0.9835,0.0100,0.0065	0.0030	0.0042	0.0067	0.0065	0.0120
0.9750,0.0200,0.0050	0.0030	0.0042	0.0056	0.0059	0.0074
0.9775,0.0175,0.0050	0.0030	0.0042	0.0056	0.0060	0.0059
0.9700,0.0200,0.0100	0.0027	0.0034	0.0042	0.0049	0.0062
0.9675,0.0200,0.0125	0.0027	0.0034	0.0038	0.0060	0.0057

p₁, p₂, p₃, p₄	n=3000	n=1000	n=500	n=300	n=200
0.8000,0.1000,0.0600,0.0400	0.0024	0.0026	0.0028	0.0030	0.0033
0.6000,0.2500,0.1100,0.0400	0.0022	0.0024	0.0025	0.0026	0.0029
0.5500,0.2500,0.1500,0.0500	0.0024	0.0025	0.0027	0.0030	0.0032
0.8500,0.0500,0.0500,0.0500	0.0021	0.0022	0.0022	0.0024	0.0025
0.7000,0.1500,0.0900,0.0600	0.0022	0.0022	0.0023	0.0026	0.0026
0.5000,0.2900,0.1500,0.0600	0.0020	0.0020	0.0022	0.0023	0.0024
0.6000,0.3400,0.0525,0.0075	0.0028	0.0034	0.0044	0.0051	0.0060
0.5400,0.2500,0.2025,0.0075	0.0025	0.0031	0.0040	0.0047	0.0054
0.6000,0.3700,0.0200,0.0100	0.0028	0.0033	0.0040	0.0054	0.0065
0.8000,0.1400,0.0500,0.0100	0.0025	0.0027	0.0030	0.0036	0.0039
0.7000,0.1700,0.1100,0.0200	0.0024	0.0027	0.0030	0.0036	0.0040
0.5500,0.3500,0.0800,0.0200	0.0024	0.0027	0.0030	0.0036	0.0040
0.9450,0.0300,0.0150,0.0100	0.0030	0.0037	0.0046	0.0058	0.0068
0.9700,0.0125,0.0100,0.0075	0.0033	0.0042	0.0055	0.0067	0.0076
0.9650,0.0150,0.0100,0.0100	0.0032	0.0040	0.0050	0.0063	0.0076
0.9675,0.0150,0.0100,0.0075	0.0033	0.0042	0.0056	0.0065	0.0077
0.9430,0.0300,0.0150,0.0120	0.0031	0.0036	0.0046	0.0060	0.0063
0.9425,0.0300,0.0150,0.0125	0.0030	0.0036	0.0043	0.0053	0.0067
0.9350,0.0300,0.0200,0.0150	0.0029	0.0034	0.0040	0.0047	0.0059

p₁, p₂, p₃, p₄, p₅	n=3000	n=1000	n=500	n=300	n=200
0.7400,0.10,0.070,0.050,0.0400	0.0025	0.0026	0.0028	0.0031	0.0034
0.6500,0.13,0.100,0.080,0.0400	0.0024	0.0025	0.0026	0.0029	0.0031
0.6400,0.12,0.100,0.080,0.0600	0.0022	0.0023	0.0024	0.0025	0.0027
0.5500,0.20,0.100,0.080,0.0700	0.0021	0.0022	0.0023	0.0024	0.0026
0.5500,0.20,0.120,0.080,0.0500	0.0023	0.0023	0.0024	0.0026	0.0029
0.8000,0.05,0.050,0.050,0.0500	0.0025	0.0026	0.0028	0.0030	0.0033
0.6000,0.25,0.070,0.050,0.0300	0.0025	0.0027	0.0030	0.0033	0.0037
0.4000,0.27,0.200,0.100,0.0300	0.0022	0.0024	0.0026	0.0029	0.0032
0.7000,0.20,0.050,0.030,0.0200	0.0028	0.0030	0.0035	0.0040	0.0049
0.5000,0.23,0.150,0.100,0.0200	0.0024	0.0026	0.0030	0.0033	0.0040
0.8000,0.10,0.050,0.040,0.0100	0.0029	0.0035	0.0041	0.0055	0.0064
0.6000,0.15,0.140,0.100,0.0100	0.0026	0.0031	0.0037	0.0049	0.0057
0.9100,0.05,0.020,0.010,0.0100	0.0033	0.0041	0.0050	0.0068	0.0081
0.9105,0.05,0.020,0.010,0.0095	0.0033	0.0041	0.0052	0.0063	0.0075
0.9425,0.03,0.010,0.010,0.0075	0.0034	0.0045	0.0056	0.0070	0.0086
0.9250,0.04,0.020,0.010,0.0050	0.0035	0.0047	0.0068	0.0077	0.0082
0.9200,0.05,0.015,0.010,0.0050	0.0035	0.0047	0.0068	0.0078	0.0083
0.9000,0.05,0.020,0.015,0.0150	0.0031	0.0036	0.0045	0.0053	0.0067
0.9080,0.05,0.020,0.010,0.0120	0.0032	0.0039	0.0048	0.0062	0.0071
0.9060,0.05,0.020,0.012,0.0120	0.0031	0.0038	0.0047	0.0061	0.0069
0.9030,0.05,0.020,0.010,0.0170	0.0031	0.0037	0.0044	0.0056	0.0065

Brick Classification
Chipped
Time	Standard	Face	Cull	Total	$Y_{i}^{2}$	$Q_{i}$
1	242	8	4	254	0.25	0.24
2	199	5	3	207	0.58	0.48
3	228	10	5	243	1.05	0.69
4	193	5	3	201	1.78	0.38
5	214	15	3	232	10.05	6.83
6	132	4	2	138	0.22	0.21
7	206	7	5	218	0.13	0.12
8	146	5	4	155	0.30	0.28
9	207	7	7	221	1.57	1.53
10	174	24	8	206	57.44	38.71
11	223	12	10	245	8.66	7.40
12	204	12	5	221	4.59	3.03
13	196	8	8	212	3.76	3.66
14	225	10	10	245	6.52	6.03
15	225	7	5	237	0.16	0.01
16	141	2	5	148	2.75	2.30

In Control Process	n=3000		n=1000		n=500		n=300		n=200
In Control Process	X²	Q	X²	Q	X²	Q	X²	Q	X²	Q
0.9725,0.02,0.0075	316.87	370.13	246.70	332.84	210.47	222.92	188.27	216.04	127.96	167.82
Process Improvement
0.9835,0.0100,0.0065	1.23	1.68	14.13	61.17	172.15	405.29	366.79	350.37	302.48	302.79
0.9750,0.0200,0.0050	52.42	41.43	425.85	1179.14	407.01	1505.66	439.29	957.015	292.88	628.05
0.9775,0.0175,0.0050	23.30	26.26	485.46	1602.78	1297.15	3544.37	1166.94	1784.83	620.21	1050.71
Process Deterioration
0.9425,0.05,0.0075	1.00	1.00	1.01	1.03	1.19	1.39	1.69	2.15	2.34	3.24
0.9700,0.02,0.0100	12.68	10.44	30.79	31.11	48.10	38.92	56.80	53.74	46.44	51.44
0.9675,0.02,0.0125	2.16	1.96	6.88	6.79	14.15	11.52	21.15	19.30	20.64	21.50
0.6500,0.34,0.0100	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00

In Control Process	n=3000		n=1000		n=500		n=300		n=200
In Control Process	X²	Q	X²	Q	X²	Q	X²	Q	X²	Q
0.9450,0.030,0.015,0.0100	325.38	396.67	276.96	313.40	222.44	257.66	184.64	185.12	144.78	154.62
Process Improvement
0.9700,0.0125,0.01,0.0075	1.00	1.00	1.83	6.13	15.2	160.15	299.03	852.26	733.43	738.50
0.9650,0.0150,0.01,0.0100	1.01	1.11	3.44	15.84	29.86	116.25	161.11	176.65	220.72	184.93
0.9675,0.0150,0.01,0.0075	1.01	1.07	3.00	11.77	32.64	300.08	490.02	993.93	870.21	806.19
Process Deterioration
0.9430,0.030,0.015,0.0120	40.21	30.46	79.43	62.24	90.47	77.39	95.81	72.61	87.45	72.47
0.9125,0.030,0.015,0.0125	23.81	17.03	59.08	45.64	72.56	60.15	78.59	58.14	74.97	59.59
0.9350,0.030,0.020,0.0150	1.84	1.77	6.93	6.39	13.21	12.25	19.22	16.77	23.28	20.28

In Control Process	n=3000		n=1000		n=500		n=300		n=200
In Control Process	X²	Q	X²	Q	X²	Q	X²	Q	X²	Q
0.91,0.05,0.02,0.01,0.01	324.3	380.0	276.5	297.1	214.2	231.8	173.9	170.7	134.8	147.2
Process Improvement
0.9425,0.03,0.010,0.01,0.0075	1.0	1.0	1.8	6.7	10.6	92.7	59.8	227.9	180.9	243.9
0.9250,0.04,0.020,0.01,0.0050	1.8	2.3	25.6	62.3	123.9	301.7	224.1	375.2	232.6	345.4
0.9105,0.05,0.02,0.01,0.0095	350.6	478.9	309.5	403.5	247.33	297.7	192.6	211.3	164.6	161.6
Process Deterioration
0.900,0.05,0.02,0.015,0.015	1.6	1.5	5.9	5.1	11.9	10.0	17.2	14.0	22.1	17.6
0.908,0.05,0.02,0.010,0.010	49.6	32.9	91.9	65.1	98.9	75.3	95.8	73.0	94.3	67.9
0.906,0.05,0.02,0.012,0.012	19.7	16.3	48.2	40.1	60.1	52.1	62.1	53.0	64.1	51.6
0.903,0.05,0.02,0.010,0.017	1.8	1.5	6.7	4.7	13.2	9.4	19.9	13.4	24.9	15.7

Universidade Federal de São Carlos Departamento de Engenharia de Produção , Caixa Postal 676 , 13.565-905 São Carlos SP Brazil, Tel.: +55 16 3351 8471 - São Carlos - SP - Brazil
E-mail: gp@dep.ufscar.br

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