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

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

Brasil

Español English

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

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

Autoria SCIMAGO INSTITUTIONS RANKINGS

Resumo:

Interpretar um sinal fora de controle é uma etapa crucial no monitoramento de processos categóricos. Para o gráfico de controle do qui-quadrado (CSCC), uma situação fora de controle não especifica se foi uma deterioração do processo ou uma melhoria do processo. Por esta razão, um gráfico de controle estatístico qui-quadrado ponderado WSCC é proposto com diferentes categorias de ponderação, a fim de permitir uma divulgação acelerada de uma situação de controle após uma mudança devido a uma deterioração da qualidade e, por outro lado, desacelerar uma situação fora de controle após um turno devido a uma melhoria da qualidade. Além disso, em comparação com o método de Marcucci, o novo procedimento fornece uma maneira precisa e mais fácil de interpretar vários sinais. Em outras palavras, o WSCC permite uma detecção mais rápida de uma situação fora de controle no caso de uma deterioração da qualidade, entretanto, uma situação fora de controle não é detectada rapidamente no caso de uma melhoria da qualidade. Na verdade, estudos comparativos foram realizados para encontrar o melhor gráfico de controle para cada combinação. As observações finais com comentários e recomendações são fornecidas com base no comprimento médio de execução (ARL) e comprimento de execução de desvio padrão (SDRL).

Palavras-chave:
Processos multinomiais; Processos categóricos; Gráfico de controle qui-quadrado; Estatística qui-quadrado ponderada; ARL e 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

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