Monitoring multinomial processes based on a weighted chi-square control chart

Ali, Achouri; Khedhiri, Emira; Talmoudi, Ramzi; Taleb, Hassen

doi:10.1590/1806-9649-2021v28e43

Acessibilidade / Reportar erro

Brasil

Español English

sumário « anterior atual seguinte »

Sumário

THEMATIC SECTION: STATISTICAL PROCESS MONITORING AND CONTROL • Gest. Prod. 28 (3) • 2021 • https://doi.org/10.1590/1806-9649-2021v28e43 copy

Monitoring multinomial processes based on a weighted chi-square control chart

Monitoramento de processos multinomiais com base em um gráfico de controle qui-quadrado ponderado

Authorship SCIMAGO INSTITUTIONS RANKINGS

Abstract:

Interpreting an out-of-control signal is a crucial step in monitoring categorical processes. For the Chi-Square Control Chart (CSCC), an out-of control situation does not specify if it was a process deterioration or a process improvement. For this reason, a weighted chi-square statistical control chart WSCC is proposed with different weighting categories in order to enable an accelerated disclosure of a control situation after a shift due to a deterioration of quality and on the other hand, decelerate an out of control situation after a shift due to a quality improvement. Furthermore, in comparison with Marcucci’s method, the new procedure provides an accurate and easier way to interpret several signals. In other words, the WSCC allows a faster detection of an out-of control situation in the case of a quality deterioration, however, an out-of control situation is not quickly detected in the case of a quality improvement. Indeed, comparative studies have been performed to find the best control chart for each combination. Concluding remarks with comments and recommendations are given based on Average Run Length (ARL) and standard deviation run length (SDRL).

Keywords:
Multinomial processes; Categorical processes; Chi-square control chart; Weighted chi-square statistic; ARL and SDRL

Time	Conforming	Non-conforming Type A	Non-conforming Type B	Total	$Z i^{2}$	$Y i^{2}$
1	242	8	4	254	0.25	0.23
2	199	5	3	207	0.58	0.81
3	228	10	5	243	1.05	0.78
4	193	5	3	201	0.46	0.65
5	214	15	3	232	10.05	3.11
6	132	4	2	138	0.22	0.35
7	206	7	5	218	0.13	0.2
8	146	5	4	155	0.3	0.79
9	207	7	7	221	1.57	4.28
10	174	24	8	206	57.44	75.93
11	223	12	10	245	8.66	26.09
12	204	12	5	221	4.59	2.34
13	196	8	8	212	3.9	14.79
14	225	10	10	245	6.52	24.67
15	225	7	5	237	0.02	0.02
16	141	2	5	148	2.75	12.83

Time	Conforming	Non-conforming Type A	Non-conforming Type B	Total	$Z_{i}^{2}$
17	200	10	50	260	395.5
18	260	0	0	260	13.7

	$C_{1}$			…	$C_{q}$
	$j - \frac{q}{2} - 1$	$(\frac{X_{j} - n p_{j}}{\sqrt{n p_{j}}})$	$W_{j}$	…	$j - \frac{q}{2} - 1$	$(\frac{X_{j} - n p_{j}}{\sqrt{n p_{j}}})$	$W_{j}$	$Y_{i}^{2}$
$C_{1}$ ↗	-	+	$W_{j} \to 0$	…	+	-	$W_{j} \to 0$	↘	In control
$C_{q}$ ↘	-	+	$W_{j} \to 0$	…	+	-	$W_{j} \to 0$	↘	In control
$C_{1}$ ↘	-	-	$W_{j} \to +$ ∞	…	+	+	$W_{j} \to +$ ∞	↗	Out of control
$C_{q}$ ↗	-	-	$W_{j} \to +$ ∞	…	+	+	$W_{j} \to +$ ∞	↗	Out of control

Baseline proportions	n = 1000
Baseline proportions	q=5
P = [0.50 0.30 0.10 0.07 0.03]	UCL( $Z_{i}^{2}$ )	SDRL( $Z_{i}^{2}$ )	UCL( $Y_{i}^{2}$ )	SDRL( $Y_{i}^{2}$ )
	9.487729	19.1013	91.036737	19.453656
	ARL( $Z_{i}^{2}$ )		ARL( $Y_{i}^{2}$ )
	19.607843		19.96008
Improvement	ARL( $Z_{i}^{2}$ )	SDRL( $Z_{i}^{2}$ )	ARL( $Y_{i}^{2}$ )	SDRL( $Y_{i}^{2}$ )
P = [0.52 0.30 0.10 0.07 0.01]	1.0486577	0.2258877	136.9863	136.48538
P = [0.55 0.30 0.10 0.02 0.03]	1	0	35.460993	34.957417
P = [0.55 0.30 0.05 0.07 0.03]	1.0002	0.0141435	1.0316723	0.1807635
P = [0.58 0.22 0.10 0.07 0.03]	1.0007005	0.0264762	1.0001	0.0100005
P = [0.60 0.27 0.09 0.03 0.01]	1	0.0264762	2.7785496	2.2230133
Deterioration	ARL( $Z_{i}^{2}$ )	SDRL( $Z_{i}^{2}$ )	ARL( $Y_{i}^{2}$ )	SDRL( $Y_{i}^{2}$ )
P = [0.48 0.30 0.10 0.07 0.05]	1.2227929	0.5219479	1.0887316	0.3108133
P = [0.43 0.30 0.17 0.07 0.03]	1	0.0283062	1.0053282	0.0731887
P = [0.43 0.37 0.10 0.07 0.03]	1.0091836	0.0962701	1.0085729	0.0929860
P = [0.40 0.32 0.12 0.10 0.08]	1	0	1	0

Baseline proportions	n = 100
Baseline proportions	q=3
P = [0.7 0.2 0.1]	UCL( $Z_{i}^{2}$ )	SDRL( $Z_{i}^{2}$ )	UCL( $Y_{i}^{2}$ )	SDRL( $Y_{i}^{2}$ )
	5.9914646	21.13911	11.79648	19.334806
	ARL( $Z_{i}^{2}$ )		ARL( $Y_{i}^{2}$ )
	21.645022		19.84127
Improvement	ARL( $Z_{i}^{2}$ )	SDRL( $Z_{i}^{2}$ )	ARL( $Y_{i}^{2}$ )	SDRL( $Y_{i}^{2}$ )
P = [0.75 0.20 0.05]	4.0866367	3.5516141	81.967213	81.465679
P = [0.75 0.15 0.10]	6.0096154	5.4868809	6.4267352	5.9056067
P = [0.76 0.20 0.04]	2.5826446	2.021734	75.757576	75.255915
P = [0.78 0.16 0.06]	3.3590863	2.8150265	10.438413	9.9258276
P = [0.80 0.10 0.10]	1.512173	0.8800535	1.5304561	0.9010215
Deterioration	ARL( $Z_{i}^{2}$ )	SDRL( $Z_{i}^{2}$ )	ARL( $Y_{i}^{2}$ )	SDRL( $Y_{i}^{2}$ )
P = [0.68 0.20 0.12]	9.5785441	9.0647649	6.7204301	6.2003025
P = [0.65 0.23 0.12]	5.7208238	5.1968261	4.4622936	3.9306196
P = [0.60 0.20 0.20]	1.2669454	0.5815542	1.1500863	0.4154662
P = [0.55 0.25 0.20]	1.1082788	0.3464146	1.0625863	0.2578824

Baseline proportions	n = 1000
Baseline proportions	q=4
P = [0.60 0.25 0.10 0.05]	UCL( $Z_{i}^{2}$ )	SDRL( $Z_{i}^{2}$ )	UCL( $Y_{i}^{2}$ )	SDRL( $Y_{i}^{2}$ )
	7.8147279	20.327185	31.619325	19.453656
	ARL( $Z_{i}^{2}$ )		ARL( $Y_{i}^{2}$ )
	20.833333		19.96008
Improvement	ARL( $Z_{i}^{2}$ )	SDRL( $Z_{i}^{2}$ )	ARL( $Y_{i}^{2}$ )	SDRL( $Y_{i}^{2}$ )
P = [0.64 0.25 0.10 0.01]	1	0	98.039216	97.537934
P = [0.65 0.25 0.05 0.05]	1.0002	0.0141435	2.9664788	2.4152676
P = [0.69 0.25 0.05 0.01]	1	0	3.866976	3.3296437
P = [0.70 0.15 0.10 0.05]	1	0	1	0
P = [0.75 0.15 0.05 0.05]	1	0	1	0
Deterioration	ARL( $Z_{i}^{2}$ )	SDRL( $Z_{i}^{2}$ )	ARL( $Y_{i}^{2}$ )	SDRL( $Y_{i}^{2}$ )
P = [0.55 0.25 0.10 0.10]	1	0	1	0
P = [0.52 0.25 0.15 0.08]	1	0	1	0
P = [0.50 0.35 0.10 0.05]	1	0	1.0001	0.0100005
P = [0.50 0.30 0.15 0.05]	1	0	1	0

Baseline proportions	n = 1000
Baseline proportions	q=3
P = [0.7 0.2 0.1]	UCL( $Z_{i}^{2}$ )	SDRL( $Z_{i}^{2}$ )	UCL( $Y_{i}^{2}$ )	SDRL( $Y_{i}^{2}$ )
	5.9914646	19.374265	11.225403
	ARL( $Z_{i}^{2}$ )		ARL( $Y_{i}^{2}$ )
	19.880716		1.96008
Improvement	ARL( $Z_{i}^{2}$ )	SDRL( $Z_{i}^{2}$ )	ARL( $Y_{i}^{2}$ )	SDRL( $Y_{i}^{2}$ )
P = [0.75 0.20 0.05]	1	0	67.567568	67.065704
P = [0.75 0.15 0.10]	1.0306091	0.1776120	1.0291242	0.1731254
P = [0.76 0.20 0.04]	1	0	67.567568	67.065704
P = [0.78 0.16 0.06]	1.0001	0.0100005	1.1419436	0.4026060
P = [0.80 0.10 0.10]	1	0	1	0
Deterioration	ARL( $Z_{i}^{2}$ )	SDRL( $Z_{i}^{2}$ )	ARL( $Y_{i}^{2}$ )	SDRL( $Y_{i}^{2}$ )
P = [0.68 0.20 0.12]	2.1838829	1.6079371	1.6697278	1.0574796
P = [0.65 0.23 0.12]	1.1384335	0.3969853	1.0887316	0.3108133
P = [0.60 0.20 0.20]	1	0	1	0
P = [0.55 0.25 0.20]	1	0	1	0

Baseline proportions	n = 500
Baseline proportions	q=5
P = [0.50 0.30 0.10 0.07 0.03]	UCL( $Z_{i}^{2}$ )	SDRL( $Z_{i}^{2}$ )	UCL( $Y_{i}^{2}$ )	SDRL( $Y_{i}^{2}$ )
	9.487729	19.1013	95.113769
	ARL( $Z_{i}^{2}$ )		ARL( $Y_{i}^{2}$ )
	19.607843		19.96008
Improvement	ARL( $Z_{i}^{2}$ )	SDRL( $Z_{i}^{2}$ )	ARL( $Y_{i}^{2}$ )	SDRL( $Y_{i}^{2}$ )
P = [0.52 0.30 0.10 0.07 0.01]	1.7969452	1.19669	112.35955	111.85843
P=[0.55 0.30 0.10 0.02 0.03]	1.0031096	0.0558504	27.624309	27.1197
P = [0.55 0.30 0.05 0.07 0.03]	1.0654166	0.2639999	2.2727273	1.7007534
P = [0.58 0.22 0.10 0.07 0.03]	1.0659844	0.2652138	1.0539629	0.2384846
P = [0.60 0.27 0.09 0.03 0.01]	1.00040020	0.0200090	6.6269052	6.1064693
Deterioration	ARL( $Z_{i}^{2}$ )	SDRL( $Z_{i}^{2}$ )	ARL( $Y_{i}^{2}$ )	SDRL( $Y_{i}^{2}$ )
P = [0.48 0.30 0.10 0.07 0.05]	1.866368	1.271598	1.406272	0.7558630
P = [0.43 0.30 0.17 0.07 0.03]	1.0094892	0.0978736	1.1591515	0.4295122
P = [0.43 0.37 0.10 0.07 0.03]	1.1978917	0.4868807	1.1676787	0.4424871
P = [0.40 0.32 0.12 0.10 0.08]	1.0025063	0.0501257	1.0015023	0.4424871

Baseline proportions	n = 500
Baseline proportions	q=4
P = [0.60 0.25 0.10 0.05]	UCL( $Z_{i}^{2}$ )	SDRL( $Z_{i}^{2}$ )	UCL( $Y_{i}^{2}$ )	SDRL( $Y_{i}^{2}$ )
	7.8147279	19.334736	31.426705	19.453656
	ARL( $Z_{i}^{2}$ )		ARL( $Y_{i}^{2}$ )
	19.8412		19.96008
Improvement	ARL( $Z_{i}^{2}$ )	SDRL( $Z_{i}^{2}$ )	ARL( $Y_{i}^{2}$ )	SDRL( $Y_{i}^{2}$ )
P = [0.64 0.25 0.10 0.01]	.0017029	0.0413013	103.09278	102.59156
P = [0.65 0.25 0.05 0.05]	1.0446046	0.2158568	15.151515	14.642981
P = [0.69 0.25 0.05 0.01]	1	0	46.296296	45.793566
P = [0.70 0.15 0.10 0.05]	1.0007005	0.0264762	1.0003001	0.0173260
P = [0.75 0.15 0.05 0.05]	1	0	1.0001	0.0100005
Deterioration	ARL( $Z_{i}^{2}$ )	SDRL( $Z_{i}^{2}$ )	ARL( $Y_{i}^{2}$ )	SDRL( $Y_{i}^{2}$ )
P = [0.55 0.25 0.10 0.10]	1.021868	0.1494865	1.0060362	0.0779271
P = [0.52 0.25 0.15 0.08]	1.0092854	0.0968071	1.0131712	0.1155192
P = [0.50 0.35 0.10 0.05]	1.0034116	0.0585085	1.0083695	0.0918670
P = [0.50 0.30 0.15 0.05]	1.0928962	0.3186313	1.0084712	0.0918670

Baseline proportions	n = 500
Baseline proportions	q=3
P = [0.7 0.2 0.1]	UCL( $Z_{i}^{2}$ )	SDRL( $Z_{i}^{2}$ )	UCL( $Y_{i}^{2}$ )	SDRL( $Y_{i}^{2}$ )
	5.9914646	20.725393	11.48985	19.334806
	ARL( $Z_{i}^{2}$ )		ARL( $Y_{i}^{2}$ )
	21.231423		19.84127
Improvement	ARL( $Z_{i}^{2}$ )	SDRL( $Z_{i}^{2}$ )	ARL( $Y_{i}^{2}$ )	SDRL( $Y_{i}^{2}$ )
P = [0.75 0.20 0.05]	1.0276436	0.1685460	63.694268	63.19229
P = [0.75 0.15 0.10]	1.3504389	0.6879290	1.3417416	0.6771476
P = [0.76 0.20 0.04]	1.0015023	0.0387886	64.102564	63.600599
P = [0.78 0.16 0.06]	1.0239607	0.1566359	1.8261505	1.2282814
P = [0.80 0.10 0.10]	1	0	1	0
Deterioration	ARL( $Z_{i}^{2}$ )	SDRL( $Z_{i}^{2}$ )	ARL( $Y_{i}^{2}$ )	SDRL( $Y_{i}^{2}$ )
P = [0.68 0.20 0.12	3.8109756	3.2730046	2.6226069	2.0628766
P = [0.65 0.23 0.12]	1.69549	1.0859081	1.4976786	0.8633438
P = [0.60 0.20 0.20	1.0001	0.0100005	1	0
P = [0.55 0.25 0.20]	1	0	1	0

Baseline proportions	n = 300
Baseline proportions	q=5
P = [0.50 0.30 0.10 0.07 0.03]	UCL( $Z_{i}^{2}$ )	SDRL( $Z_{i}^{2}$ )	UCL( $Y_{i}^{2}$ )	SDRL( $Y_{i}^{2}$ )
	9.487729	19.453656	86.818157	19.453656
	ARL( $Z_{i}^{2}$ )		ARL( $Y_{i}^{2}$ )
	20.576132		19.96008
Improvement	ARL( $Z_{i}^{2}$ )	SDRL( $Z_{i}^{2}$ )	ARL( $Y_{i}^{2}$ )	SDRL( $Y_{i}^{2}$ )
P = [0.52 0.30 0.10 0.07 0.01}	3.7664783	19.453655	19.96008	19.453656
P = [0.55 0.30 0.10 0.02 0.03]	1.0954102	19.453655	34.965035	34.461408
P = [0.55 0.30 0.05 0.07 0.03}	1.4361626	19.453655	7.3855244	6.8673464
P = [0.58 0.22 0.10 0.07 0.03}	1.3640704	19.453655	1.3390466	0.6737946
P = [0.60 0.27 0.09 0.03 0.01]	1.0302905	19.453655	11.600928	11.089662
Deterioration	ARL( $Z_{i}^{2}$ )	SDRL( $Z_{i}^{2}$ )	ARL( $Y_{i}^{2}$ )	SDRL( $Y_{i}^{2}$ )
P = [0.48 0.30 0.10 0.07 0.05]	2.7847396	19.453655	1.9219681	1.3311624
P = [0.43 0.30 0.17 0.07 0.03]	1.1093854	19.453655	1.4755792	0.8377080
P = [0.43 0.37 0.10 0.07 0.03]	1.6943409	19.453655	1.5239256	0.8935455
P = [0.40 0.32 0.12 0.10 0.08]	1.0493179	19.453655	1.0171905	0.1322347

Baseline proportions	n = 300
Baseline proportions	q=4
P = [0.60 0.25 0.10 0.05]	UCL( $Z_{i}^{2}$ )	SDRL( $Z_{i}^{2}$ )	UCL( $Y_{i}^{2}$ )	SDRL( $Y_{i}^{2}$ )
	7.8147279	19.178522	34.163768	19.453656
	ARL( $Z_{i}^{2}$ )		ARL( $Y_{i}^{2}$ )
	19.685039		19.96008
Improvement	ARL( $Z_{i}^{2}$ )	SDRL( $Z_{i}^{2}$ )	ARL( $Y_{i}^{2}$ )	SDRL( $Y_{i}^{2}$ )
P = [0.64 0.25 0.10 0.01]	1.0980564	0.3281333	133.33333	132.83239
P = [0.65 0.25 0.05 0.05	1.3299641	0.6624503	22.421525	21.915822
P = [0.69 0.25 0.05 0.01]	1.0012014	0.0346820	80	79.498428
P = [0.70 0.15 0.10 0.05]	1.0303967	0.1769764	1.0234367	0.1548741
P = [0.75 0.15 0.05 0.05]	1.0002	0.0141435	1.0188487	0.1385784
Deterioration	ARL( $Z_{i}^{2}$ )	SDRL( $Z_{i}^{2}$ )	ARL( $Y_{i}^{2}$ )	SDRL( $Y_{i}^{2}$ )
P = [0.55 0.25 0.10 0.10]	1.1405109	0.4003176	1.0622477	0.2571429
P = [0.52 0.25 0.15 0.08	1.0976948	0.3274738	1.1201972	0.3669395
P = [0.50 0.35 0.10 0.05]	1.0678057	0.2690786	1.0992635	0.3303282
P = [0.50 0.30 0.15 0.05]	1.0928962	0.3186313	1.0949305	0.3224008

Baseline proportions	n = 300
Baseline proportions	q=4
P = [0.7 0.2 0.1]	UCL( $Z_{i}^{2}$ )	SDRL( $Z_{i}^{2}$ )	UCL( $Y_{i}^{2}$ )	SDRL( $Y_{i}^{2}$ )
	5.9914646	18.577242	12.064423
	ARL( $Z_{i}^{2}$ )		ARL( $Y_{i}^{2}$ )
	19.083969		19.96008
Improvement	ARL( $Z_{i}^{2}$ )	SDRL( $Z_{i}^{2}$ )	ARL( $Y_{i}^{2}$ )	SDRL( $Y_{i}^{2}$ )
P = [0.75 0.20 0.05]	1.2189176	0.5165680	70.921986	70.420211
P = [0.75 0.15 0.10]	2.0661157	1.4841558	2.0682523	1.4864102
P = [0.76 0.20 0.04]	1.039177	0.2017717	74.074074	73.572375
P = [0.78 0.16 0.06]	1.1944577	0.4819455	3.3557047	2.811592
P = [0.80 0.10 0.10]	1.0049241	0.0703445	1.0042177	0.0650806
Deterioration	ARL( $Z_{i}^{2}$ )	SDRL( $Z_{i}^{2}$ )	ARL( $Y_{i}^{2}$ )	SDRL( $Y_{i}^{2}$ )
P = [0.68 0.20 0.12]	5.4112554	4.8857374	3.8774719	3.3402569
P = [0.65 0.23 0.12]	2.463661	1.8989377	2.1710812	1.5945258
P = [0.60 0.20 0.20]	1.002004	0.0448109	1.0006004	0.0245104
P = [0.55 0.25 0.20]	1.0001	0.0100005	1	0

Baseline proportions	n = 100
Baseline proportions	q=5
P = [0.50 0.30 0.10 0.07 0.03]	UCL( $Z_{i}^{2}$ )	SDRL( $Z_{i}^{2}$ )	UCL( $Y_{i}^{2}$ )	SDRL( $Y_{i}^{2}$ )
	9.487729	19.062916	122.05381	19.453656
	ARL( $Z_{i}^{2}$ )		ARL( $Y_{i}^{2}$ )
	19.569472		19.96008
Improvement	ARL( $Z_{i}^{2}$ )	SDRL( $Z_{i}^{2}$ )	ARL( $Y_{i}^{2}$ )	SDRL( $Y_{i}^{2}$ )
P = [0.52 0.30 0.10 0.07 0.01]	16.528926	16.021126	99.009901	98.508632
P = [0.55 0.30 0.10 0.02 0.03]	4.199916	3.6659758	26.525199	26.020396
P = [0.55 0.30 0.05 0.07 0.03]	5.3763441	4.8506424	26.315789	25.810947
P = [0.58 0.22 0.10 0.07 0.03]	3.9184953	3.3817318	4.3994721	3.8672836
P = [0.60 0.27 0.09 0.03 0.01]	2.6226069	2.0628766	41.322314	40.819252
Deterioration	ARL( $Z_{i}^{2}$ )	SDRL( $Z_{i}^{2}$ )	ARL( $Y_{i}^{2}$ )	SDRL( $Y_{i}^{2}$ )
P = [0.48 0.30 0.10 0.07 0.05]	5.6338028	5.1093964	3.875969	3.3387373
P = [0.43 0.30 0.17 0.07 0.03]	2.18436	1.6084367	4.2753313	3.7420751
P = [0.43 0.37 0.10 0.07 0.03]	4.4802867	3.948757	4.456328	3.9246059
P = [0.40 0.32 0.12 0.10 0.08]	1.7476407	3.948757	1.5807777	0.9581651

Baseline proportions	n = 100
Baseline proportions	q=4
P = [0.60 0.25 0.10 0.05]	UCL( $Z_{i}^{2}$ )	SDRL( $Z_{i}^{2}$ )	UCL( $Y_{i}^{2}$ )	SDRL( $Y_{i}^{2}$ )
	7.8147279	21.56926	32.038648	19.453656
	ARL( $Z_{i}^{2}$ )		ARL( $Y_{i}^{2}$ )
	22.075055		19.96008
Improvement	ARL( $Z_{i}^{2}$ )	SDRL( $Z_{i}^{2}$ )	ARL( $Y_{i}^{2}$ )	SDRL( $Y_{i}^{2}$ )
P = [0.64 0.25 0.10 0.01]	6.0569352	5.5343951	151.51515	151.01432
P = [0.65 0.25 0.05 0.05]	4.7641734	4.2347579	26.385224	25.88039
P = [0.69 0.25 0.05 0.01]	1.699813	1.0906655	133.33333	132.83239
P = [0.70 0.15 0.10 0.05]	1.9912386	1.4049173	1.9853087	1.3986214
P = [0.75 0.15 0.05 0.05]	1.2674271	0.5821893	1.7979144	1.1977403
Deterioration	ARL( $Z_{i}^{2}$ )	SDRL( $Z_{i}^{2}$ )	ARL( $Y_{i}^{2}$ )	SDRL( $Y_{i}^{2}$ )
P = [0.55 0.25 0.10 0.10]	2.1181953	1.5390114	1.6594756	1.0461279
P = [0.52 0.25 0.15 0.08]	1.9868865	1.4002969	1.9508389	1.3619594
P = [0.50 0.35 0.10 0.05]	2.0395676	1.4561141	2.1168501	1.5375969
P = [0.50 0.30 0.15 0.05]	2.1168501	1.5375969	2.093364	1.512881

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

Acompanhe os números deste periódico no seu leitor de RSS