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The Trinomial ATTRIVAR control chart
Journal Pre-proof The Trinomial ATTRIVAR control chart
Felipe Domingues Simões, Antonio Fernando Branco Costa, Marcela Aparecida Guerreiro Machado PII:
S0925-5273(19)30393-7
DOI:
https://doi.org/10.1016/j.ijpe.2019.107559
Reference:
PROECO 107559
To appear in:
International Journal of Production Economics
Received Date:
23 July 2019
Accepted Date:
13 November 2019
Please cite this article as: Felipe Domingues Simões, Antonio Fernando Branco Costa, Marcela Aparecida Guerreiro Machado, The Trinomial ATTRIVAR control chart, International Journal of Production Economics (2019), https://doi.org/10.1016/j.ijpe.2019.107559
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Journal Pre-proof The Trinomial ATTRIVAR control chart Felipe Domingues Simões a,*, Antonio Fernando Branco Costa b, Marcela Aparecida Guerreiro Machado a.
a
Universidade Estadual Paulista “Júlio de Mesquita Filho”, Campus de Guaratinguetá, Departamento de Produção, Brazil. b
Universidade Federal de Itajubá, Brazil
* Corresponding author: Phone: +55 (11) 98624-3284 E-mail adresses: [email protected] (F.D. Simões), [email protected] (A.F.B. Costa), [email protected] (M.A.G. Machado)
Journal Pre-proof The Trinomial ATTRIVAR Control Chart
ABSTRACT In this article, we propose the Trinomial - ATTRIVAR (T-ATTRIVAR) control chart where attribute and variable sample data are used to control the process mean. Firstly, two discriminating limits sort the sample items into three excluding categories; that is, items in categories A, B, or AB, are, respectively, items with X dimensions smaller than the lower discriminating limit, larger than the upper discriminating limit, or neither smaller than the lower discriminating limit nor larger than the upper discriminating limit. Depending on the number of sample items in each category, one of three decisions is made: the process is declared in-control, the process is declared out-of-control, or all sample items are also measured. In this last case, the sample mean of X is used to decide the state of the process. Aslam et al. (2015) worked with the particular case where the sample items are classified as defective (items in category - A plus items in category - B) or not-defective (items in category AB). The strategy of splitting defectives into two excluding categories (A and B) enhances the performance of the ATTRIVAR chart. It is worth to emphasize that the previous attribute classification truncates the X distribution. Consequently, the mathematical development to obtain the ARLs is complex – the Average Run length (ARL) is the average number of samples the control chart requires to signal. With the density function of the sum of truncated X distributions, we obtained the exact ARLs. The exact minimum ARLs are lower than the minimum ARLs Ho & Aparisi (2016) obtained with the Genetic Algorithm.
Keywords: Shewhart control chart; ATTRIVAR control chart; Truncated normal distributions; Average run length; Monitoring process mean
Journal Pre-proof 1. Introduction The control chart for attributes was originally designed to control processes in which the monitored quality characteristics cannot be measured on a continuous numerical scale. The number of defectives bulbs from samples of size 50 is a typical example. However, the focus of many recent studies lies on the control of variable-type quality characteristics with attribute charts. Wu & Jiao (2008) were the first to propose an attribute chart to control the mean of a continuous variable X. Basically, when X is larger than a threshold the sample unit is classified as nonconforming. A sample of size n with more than m nonconforming units (referred to as MON) is a red sample. The MON chart produces an outof-control signal when the interval between two red samples is smaller than a threshold. Their article inspired many other studies where attribute charts are used to control the mean, the mean vector, the variance or the covariance matrix. Wu et al. (2009) discusses the monitoring of shafts with np charts and with 𝑋 charts. With the np chart, gauge rings are used to distinguish conforming and nonconforming shafts. With the 𝑋 chart, a micrometer is used to measure the diameters. On the basis of the monitoring costs, the attribute inspection usually outperforms the variable inspection. Quinino et al. (2015, 2017) and Aparisi et al. (2018) also designed attribute charts to control the mean of a continuous variable. Ho and Quinino (2016) and Bezerra et al. (2018) explored the idea of monitoring the process variance with attribute charts. Haridy et al. (2014) and Mosquera et al. (2018) investigated the efficiency of monitoring both, the mean and the variance of a continuous variable, with an attribute chart. Very recently, Zhou et al. (2019) proposed a chart similar to the MON chart, which was designed to detect mean shifts faster. Ho and Costa (2015) and Melo et al. (2017) built attribute charts to control the mean vector of bivariate charts. More recently, Machado et al. (2018) studied the efficiency of the attribute charts in signaling changes in the covariance matrix. The idea of working with attribute and variable sample data (ATTRIVAR data) was introduced by Sampaio et al. (2014). In their study, the samples are split in two subsamples, in such a way that, the items of the first subsample are submitted to a go/no-go gauge test, and the items of the second subsample are measured. It is simple to obtain the properties of the ATTRIVAR charts when the items submitted to the attribute-type inspection are not the same ones submitted to the variable-type inspection. This sampling strategy was also adopted in Ho and Quinino (2016), Ho and Aparisi (2016) and in Aparisi and Ho (2018). In Aslam et al. (2015) the sample items are submitted to the attributetype inspection and, depending on the results, they are also submitted to the variable-type inspection. In this case, the sample mean of the X observations decides the state of the process. It is worthwhile to stress that the attribute-type inspection always truncates the X distribution and, because of that, the
Journal Pre-proof mathematical development to obtain the 𝑋 distribution is not trivial, see Leoni and Costa (2019). Reminding that the Average Run Length (ARL) is the average number of samples the control chart requires to signal. Ho and Aparisi (2016) used a Genetic Algorithm (GA) to obtain the optimum parameters of the Aslam’s chart, that is, the parameters that lead to the lowest ARLs. After obtaining the exact ARLs, we could find ARLs that are lower than the minimum ARLs Ho & Aparisi (2016) obtained with the GA solution. The usual simulation was helpful to confirm that the solution presented by the GA doesn’t lead to the lowest ARLs. The Aslam et al (2015)'s idea of submitting the sample items to the attribute inspection before being measured is a milestone contribution to the Monitoring Process field. But, we cannot forget that the attribute inspection truncates the X distribution and, with truncated observations, it is not trivial to obtain the 𝑋 distribution. In this article, we also work with attribute and variable data collected from the same sample. However, in our model two discriminant limits sort the sample items into three excluding categories; that is, items in categories A, B, or AB, are, respectively, items with X dimensions smaller than the lower discriminating limit, larger than the upper discriminating limit, or neither smaller than the lower discriminating limit nor larger than the upper discriminating limit. Depending on the number of sample items in each category, one of three decisions is made: the process is declared in-control, the process is declared out-of-control, or all sample items are also measured. In this last case, the sample mean of X is used to decide the state of the process. Our model considers that the in-control mean and standarddeviation of the process are known. The term Trinomial (Trinomial ATTRIVAR chart) is used to emphasize that, during the attribute-type inspection, the sample items are grouped into three excluding categories. 2. The Binomial ATTRIVAR chart The Binomial ATTRIVAR chart (B-ATTRIVAR) is similar to the Mixed chart proposed by Aslam et al. (2015); except the Mixed chart was proposed to signal only increases in the process mean, because of that the lower discriminating limit LDL doesn’t exist. The Mixed and the B-ATTRIVAR charts are the type of charts where the sample items are always submitted to an attribute inspection and, depending on the results, their quality characteristic X may be measured too. During the attribute inspection, the number of disapproved items 𝑌𝐷 is determined - an item is not approved when its quality characteristic X is smaller than a lower discriminating limit LDL or it is greater than an upper discriminating limit UDL. If 𝑌𝐷 is smaller than a warning limit 𝑊𝑦 the process is assumed in-control, if 𝑌𝐷 is equal to, or greater than an upper control limit CLy the process is assumed out-of-control. However, if 𝑌𝐷 is equal or greater than 𝑊𝑦, but smaller than 𝐶𝐿𝑦, more information is necessary to
Journal Pre-proof decide the state of the process. In this case, the inspection moves to a more complex level, where the quality characteristic X of all sample items, approved and disapproved, are measured. The statistic 𝑋 is now used to decide the state of the process. One of the referees suggested to limit the variable-type inspection to the subsample of disapproved items, that is, the approved items are not submitted to this type of inspection. The size of the subsample of disapproved items is a discrete random variable, because of that, additional work is necessary to obtain the 𝑋 distribution; reminding that the X distribution of disapproved items is always truncated. The referee’s idea of measuring only the reproved items is worth of investigation. It is highly expected that by eliminating the measurement of the approved items, we are improving the relationship between sampling costs and the speed with which the ATTRIVAR chart signals. Fig. 1 illustrates the B-ATTRIVAR chart. The black points are the 𝑌𝐷 points; they were plotted using the left-hand side scale of the chart. Whenever a black point falls below the warning limit 𝑊𝑦 the process is declared in control, but whenever a black point falls above the control limit 𝐶𝐿𝑦 - the process is declared out of control. Otherwise, the X quality characteristic of the sample items is measured and the resulting 𝑋 is also plotted, but now as a white point and using the right-hand side scale of the chart (see samples 3, 5 and 6). The process is declared in control/out of control when the white point falls inside/outside the (LCL; UCL) limits; LCL is the lower control limit of the 𝑋 chart, and the UCL is its upper control limit.
𝑌𝐷
𝑋 𝑈𝐶𝐿
𝐶𝐿𝑦
𝑊𝑦 𝐿𝐶𝐿
1
2
3
4
5
6
⋯
Sampling Points
Fig. 1. The B-ATTRIVAR chart
Journal Pre-proof The B-ATTRIVAR chart and the ATTRIVAR-1 chart discussed in Ho and Aparisi (2016) are the same. It is always possible to find values for the B-ATTRIVAR chart’s parameters that leads to lower ARLs than the ones given by GA solution. It is worth to stress that the X distribution of an approved/disapproved item is truncated, because of that, the mathematical development to obtain the 𝑋 distribution is complex. Ho and Aparisi (2016) presented the general procedure to obtain the density function of Sn X when the X distributions are normal but left-tail, right-tail, or centrally truncated, X ~ N t ( ; ) . The way the expressions of the probability Pr[ Sn S0 | Xs ~ N t ( ; )] are built was presented by Ho and Aparisi (2016); their construction and computation are time-consuming, because of that, Ho and Aparisi (2016) opted to work with GA to obtain the properties of the B-ATTRIVAR chart. To obtain a deeper understanding of the B-ATTRIVAR chart’s properties, we wrote two programs to calculate the Average Run Length (ARL) of the B-ATTRIVAR chart. The first one obtains the ARLs by simulation and the second one calculates the ARLs using the exact distribution of X ( Sn / n) | YD ; the binomially distributed variable 𝑌𝐷 is the number of approved items by the attribute-type inspection. An important performing parameter of the B-ATTRIVAR chart is % X - the in-control rate with which the X quality characteristic of the sample items are measured. It is given by:
𝐶𝐿𝑦 ― 1
𝑛!
%𝑋𝐵 = ∑𝑖 = 𝑊𝑦 𝑖!(𝑛 ― 𝑖)! × (𝑝)𝑖 × (𝑞)(𝑛 ― 𝑖) In expression (1), n is the sample size, and 𝑞 = 1 ― 𝑝 = Φ
(
(1)
𝑈𝐷𝐿 ― 𝜇0
𝐿𝐷𝐿 ― 𝜇0
𝜎
𝜎
) ― Φ(
), where
0
is the in-
control process mean and 𝜎 is the process standard-deviation. The magnitude of the mean shift is expressed in units of the standard-deviation; if is the shifted mean, then +. The go/no-go gauges devices are built taking into account the in-control mean: LDL=-kg and UDL=+kg The width factor kg is a function of the in-control rate with which the sample items are measured (%𝑋). The 𝑋 chart’s control limits are: LCL=-k 𝑛 and UCL=+k 𝑛 The k factor works as a fine adjust to obtain the desired ARL0. In Table 1, the ARLs obtained by simulation (ARLsim) and the exact ARLs ( ARL X |YD ) calculated with the X | YD distributions are compared. The percentage difference, between exact and simulated ARLs is pretty low. All of the exact ARLs ( ARL X |YD ) calculated are inside the
Journal Pre-proof confidence intervals (CI), calculated with 20 simulation values for each scenario and with a confidence level of 95%. It means that the program that calculates the ARLs works properly with a confidence level of 95%. With large samples (n=10 or 15), the running time to obtain the exact ARLs is huge. Because of that, and based on the fact that the differences between simulated and exact ARLs are insignificant, the ARLs for large samples were obtained by simulation only.
Table 1: Exact and simulated ARLs of the B-ATTRIVAR chart ARL X |YD
n 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
%X̅B 25 25 25 25 25 25 15 15 15 15 15 15 5 5 5 5 5 5 25 25 25 25 25 25 15 15 15 15 15 15 5 5 5 5
δ 0.25 0.5 1 1.25 1.5 2 0.25 0.5 1 1.25 1.5 2 0.25 0.5 1 1.25 1.5 2 0.25 0.5 1 1.25 1.5 2 0.25 0.5 1 1.25 1.5 2 0.25 0.5 1 1.25
(I) 199.98 69.60 11.03 5.48 3.15 1.53 185.72 61.81 9.94 5.03 2.94 1.48 188.59 63.68 10.54 5.38 3.16 1.57 133.63 33.61 4.53 2.41 1.58 1.08 134.08 33.90 4.57 2.42 1.59 1.08 138.46 36.06 5.02 2.66
ARLsim
(II) 202.08 69.88 11.00 5.48 3.15 1.53 186.26 61.99 9.94 5.05 2.94 1.48 188.84 63.83 10.57 5.38 3.16 1.57 134.15 33.82 4.54 2.41 1.58 1.08 134.35 33.96 4.57 2.44 1.59 1.08 137.28 35.99 5.02 2.67
CI (95%) Δ 100 (I)-(II) / (I)
1.05% 0.40% 0.27% 0.00% 0.00% 0.00% 0.29% 0.29% 0.00% 0.40% 0.00% 0.00% 0.13% 0.24% 0.28% 0.00% 0.00% 0.00% 0.39% 0.64% 0.14% 0.10% 0.23% 0.12% 0.20% 0.19% 0.01% 0.70% 0.07% 0.10% 0.85% 0.20% 0.08% 0.23%
(199.429, 201.093) (68.835, 70.067) (10.967, 11.146) (5.474, 5.514) (3.135, 3.155) (1.528, 1.537) (184.774, 186.304) (61.386, 61.936) (9.873, 9.972) (5.007, 5.040) (2.935, 2.954) (1.479, 1.487) (187.598, 189.290) (63.524, 63.929) (10.535, 10.618) (5.361, 5.402) (3.144, 3.170) (1.568, 1.575) (133.543, 134.753) (33.563, 33.821) (4.512, 4.546) (2.403, 2.421) (1.575, 1.584) (1.078, 1.081) (133.400, 134.735) (33.756, 34.086) (4.566, 4.599) (2.413, 2.435) (1.581, 1.590) (1.079, 1.082) (137.720, 139.239) (35.711, 36.133) (5.010, 5.038) (2.655, 2.674)
Journal Pre-proof 5 5
5 5
1.5 2
1.72 1.12
1.73 1.12
0.39% 0.34%
(1.718, 1.726) (1.120, 1.123)
In Table 2, the ARLs of the B-ATTRIVAR chart designed by the GA (ARLGA) are compared with the ARLs obtained with the X | YD distribution ( ARL X |YD ) and by simulation (ARLsim). Confidence intervals (CI) were calculated with 20 simulation values for each scenario and with a confidence level of 95%. The ARLs of the B-ATTRIVAR chart designed by the GA are larger than the exact ARLs we obtained working with the X | YD distributions. Ho and Aparisi (2016) didn’t publish the values of the chart’s parameters that lead to their optimum ARLs, consequently, we couldn’t undertake further investigation, comparing the ARLs of the B-ATTRIVAR chart designed by GA algorithm with the ARLs X |YD for a set of pre-specified values of the chart’s parameters. It is important to stress that the
GA algorithm was built with the following constraint: %X̅ ≤ %X̅max, where %X̅ is the in-control rate with which the items of the samples are submitted to measurements and %X̅max is a pre-specified value. The optimum %X̅ (%X̅opt) is not necessarily equal to the %X̅max; depending on the input parameters, the %X̅opt <%X̅max.
Table 2: Comparing the ARLs obtained by different methods n 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 5
max %X̅B 25 25 25 25 25 25 15 15 15 15 15 15 5 5 5 5 5 5 25
δ ARLGA 0.25 0.5 1 1.25 1.5 2 0.25 0.5 1 1.25 1.5 2 0.25 0.5 1 1.25 1.5 2 0.25
189.84 61.29 10.19 5.16 3.00 1.51 189.84 61.29 10.19 5.20 3.03 1.61 192.85 64.78 11.65 5.83 3.36 1.73 139.68
Optimum %X̅B 10 10 10 10 10 10 10 10 10 10 10 10 5 5 5 5 5 5 25
ARLsim
CI (95%)
185.19 61.47 9.88 5.06 2.96 1.49 185.19 61.47 9.88 5.06 2.96 1.49 188.84 63.83 10.57 5.38 3.16 1.57 134.15
(184.841, 186.377) (61.015, 61.651) (9.893, 9.971) (5.017, 5.068) (2.942, 2.965) (1.487, 1.493) (184.841, 186.377) (61.015, 61.651) (9.893, 9.971) (5.017, 5.068) (2.942, 2.965) (1.487, 1.493) (188.208, 189.699) (63.286, 63.910) (10.489, 10.591) (5.367, 5.404) (3.135, 3.161) (1.565, 1.574) (132.780, 134.284)
ARL X |YD 185.20 61.43 9.93 5.04 2.95 1.49 185.20 61.43 9.93 5.04 2.95 1.49 188.59 63.68 10.54 5.38 3.16 1.57 133.63
Journal Pre-proof 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 15 15 15 15 15 15 15 15 15 15 15 15 15
25 25 25 25 25 15 15 15 15 15 15 5 5 5 5 5 5 25 25 25 25 25 25 15 15 15 15 15 15 5 5 5 5 5 5 25 25 25 25 25 25 15 15 15 15 15 15 5
0.5 1 1.25 1.5 2 0.25 0.5 1 1.25 1.5 2 0.25 0.5 1 1.25 1.5 2 0.25 0.5 1 1.25 1.5 2 0.25 0.5 1 1.25 1.5 2 0.25 0.5 1 1.25 1.5 2 0.25 0.5 1 1.25 1.5 2 0.25 0.5 1 1.25 1.5 2 0.25
34.64 4.61 2.51 1.59 1.08 140.04 35.64 4.78 2.61 1.71 1.11 145.48 40.98 5.29 2.76 2.25 1.15 79.55 14.18 1.99 1.36 1.06 1.02 77.82 14.31 2.01 1.36 1.08 1.00 89.55 19.27 2.66 1.76 1.30 1.04 49.91 7.62 1.85 1.15 1.07 1.00 53.56 9.52 1.51 1.31 1.07 1.00 69.82
25 25 25 25 25 15 15 15 15 15 15 5 5 5 5 5 5 25 25 25 25 25 25 15 15 15 15 15 15 5 5 5 5 5 5 25 25 25 25 25 25 15 15 15 15 15 15 5
33.61 4.53 2.41 1.58 1.08 134.08 33.90 4.60 2.44 1.60 1.09 138.46 36.06 5.02 2.66 1.72 1.13 -
33.82 4.54 2.41 1.58 1.08 134.35 33.96 4.59 2.45 1.60 1.09 137.28 35.99 5.02 2.67 1.73 1.13 73.93 13.17 1.84 1.23 1.05 1.00 76.57 13.96 1.91 1.27 1.07 1.00 85.94 16.92 2.40 1.49 1.16 1.01 49.34 7.47 1.30 1.05 1.01 1.00 51.62 8.12 1.39 1.09 1.01 1.00 63.32
(33.497, 33.784) (4.512, 4.550) (2.404, 2.416) (1.572, 1.582) (1.078, 1.081) (133.368, 134.605) (33.838, 34.065) (4.569, 4.612) (2.439, 2.454) (1.597, 1.606) (1.086, 1.091) (137.753, 139.395) (35.855, 36.205) (4.989, 5.030) (2.647, 2.668) (1.721, 1.730) (1.124, 1.131) (73.754, 74.322) (13.144, 13.259) (1.828, 1.841) (1.229, 1.237) (1.049, 1.055) (1.000, 1.001) (76.169, 77.046) (13.861, 13.965) (1.906, 1.920) (1.267, 1.274) (1.068, 1.070) (1.000, 1.002) (85.836, 86.526) (16.864, 17.009) (2.384, 2.401) (1.481, 1.491) (1.155, 1.160) (1.008, 1.010) (49.386, 49.849) (7.439, 7.503) (1.230, 1.305) (1.050, 1.057) (1.007, 1.010) (1.000, 1.001) (51.575, 52.098) (8.071, 8.142) (1.386, 1.393) (1.089, 1.096) (1.010, 1.014) (1.000, 1.001) (63.012, 63.637)
Journal Pre-proof 15 15 15 15 15
5 5 5 5 5
0.5 1 1.25 1.5 2
13.23 1.97 1.54 1.09 1.01
5 5 5 5 5
-
11.25 1.77 1.22 1.05 1.00
(11.242, 11.320) (1.770, 1.781) (1.220, 1.228) (1.050, 1.052) (1.000, 1.001)
A VBA program was written to obtain the simulated ARLs of the B-ATTRIVAR chart. Fig. 2 presents the screen of the VBA program (hosted in: http://www.fbranco.unifei.edu.br). The number of simulated Run Length (RL) with which the ARL is calculated is an input parameter (Sim. Runs). The desired % X can be reached by varying (kg, Wy, CLy), and the ARL0 by varying k.
Fig. 2. Screen of the VBA program with the % X and ARLs of the B-ATTRIVAR Chart
Journal Pre-proof 3. The Trinomial ATTRIVAR chart An upward mean shift increases the probability of classifying an item as defective - with an X value exceeding the Upper Discriminant Limit (𝑋 ≥ 𝑈𝐷𝐿), and decreases the probability of classifying an item as defective - with an X value lower than the Lower Discriminant Limit (𝑋 ≤ 𝐿𝐷𝐿). Similar comments can be made when the mean shift is a downward shift. With the usual samples of size five or close to five, it seems reasonable to admit the process is in-control whenever, among the sample items classified as defectives, two of them were declared defectives by different gauges: for instance, the first was reproved by the no-go gauge (𝑋 ≤ 𝐿𝐷𝐿), and the second one was reproved by the go gauge (𝑋 ≥ 𝑈𝐷𝐿). When the B-ATTRIVAR is in use, these two types of defectiveness are not distinguished and, independent of being “too small or too big”, the items are simply classified as defectives. In general, the assignable cause increases or decreases the process mean, consequently, it is pretty important to distinguish the defective item with 𝑋 ≥ 𝑈𝐷𝐿 from the defective item with 𝑋 ≤ 𝐿𝐷𝐿. The Trinomial version of the ATTRIVAR chart (T-ATTRIVAR) has been proposed with this purpose. When the T-ATTRIVAR chart is in use, a sample of size 𝑛 is taken and all of those 𝑛 items are classified in three excluding categories by a go/no-go gauge test; the items with 𝑋 ≤ 𝐿𝐷𝐿 are reproved by the no-go gauge test and goes to category A, the items with 𝑋 ≥ 𝑈𝐷𝐿 are reproved by the go gauge test and goes to category B, the remaining items with 𝐿𝐷𝐿 < 𝑋 < 𝑈𝐷𝐿 goes to category AB. The number of defectives in each of the three categories are 𝑌𝐴, 𝑌𝐵, and 𝑌𝐴𝐵. If YA > 0 and YB > 0; YA =0 and YB 0 and 𝑌𝐵 = 0; 𝑌𝐷 = 𝑌𝐵 whenever 𝑌𝐵 > 0 and 𝑌𝐴 = 0; and 𝑌𝐷 = 0 whenever 𝑌𝐴 > 0 and 𝑌𝐵 > 0.
Journal Pre-proof
Fig. 3. The flowchart of the T-ATTRIVAR chart
Journal Pre-proof
When the T-ATTRIVAR chart is in use, the in-control rate of samples submitted to the variable inspections is:
%𝑋𝑇 =
Where 𝑃𝐴 = Φ
(
∑
𝐶𝐿𝑦 ― 1
𝑛! 𝑛! ― 𝑖) ― 𝑖) × 𝑃𝑖𝐴 × 𝑃(𝑛 + × 𝑃𝑖𝐵 × 𝑃(𝑛 𝐴𝐵 𝐴𝐵 𝑖!(𝑛 ― 𝑖)! 𝑖!(𝑛 ― 𝑖)! 𝑖 = 𝑊𝑦
𝐿𝐷𝐿 ― 𝜇0 𝜎
) and 𝑃
𝐵
=1―Φ
(
𝑈𝐷𝐿 ― 𝜇0 𝜎
) and 𝑃
𝐴𝐵
= (1 ― 𝑃 𝐴 ― 𝑃 𝐵 ) .
In Fig. 4, the performance of the B- and the T- ATTRIVAR charts are compared. The parameters of the ATTRIVAR charts are in Table 3. All of them lead to an ARL0 of 370.4.
Table 3: Parameters of ATTRIVAR charts Chart n 𝑊𝑦 𝐶𝐿𝑦 B1
5
1
5
B2
5
2
5
T1
5
2
5
B3
3
1
3
T2
3
1
3
According to Fig. 4, the T-ATTRIVAR chart is highly recommended in two situations: a) when samples of size n=5 are taken from the process and it is not allowed to submit many of these samples to the variable inspection (the %𝑋𝑇 cannot exceed 15%). In reality, with samples of size 5, it is impossible to design a T-ATTRIVAR chart with the in-control properties of ARL0=370.4 and %𝑋𝑇> 15%. b) when samples of size n=3 are taken from the process and it is allowed to submit many of these samples to the variable inspection (the %𝑋𝑇 exceeds 15%).
Journal Pre-proof
Fig. 4. The B- and T-ATTRIVAR charts (𝑛 = 5 left and 𝑛 = 3 right)
A second VBA program was written to obtain the simulated ARLs of the T-ATTRIVAR chart. Fig. 5 presents the screen of the VBA program (hosted in: http://www.fbranco.unifei.edu.br).
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Fig. 5. Screen of the VBA program with the % X and ARLs of the T-ATTRIVAR Chart
4. Comparing the T-ATTRIVAR chart with other charts In this section, the T-ATTRIVAR chart is compared with the np chart, and also with the X rec and
np X X charts. In fact, if the sample items can be submitted to a device such as a go/no-go gauge test, then the np chart can be used to control the process mean. The sample item is approved when its X dimension is larger than a lower discriminating limit LDL and smaller than an upper discriminating limit UDL. The number D of approved items is binomially distributed; during the in-control period,
D ~ Bin(n, p0 ) with p0 Pr[ LDL X UDL] , and during the out-of-control period, D ~ Bin(n, p1 ) with p1 Pr[ LDL X UDL] . The np chart signals when D CLnp , where CLnp is the upper control limit of the np chart; the lower control limit of the np chart is zero. In Table 4, the TATTRIVAR chart is compared with the np chart. To compare the control charts in an economic perspective, the following cost function was adopted: 𝐶𝑇(𝑛) = 𝐶𝑎 × 𝑛 + 𝐶𝑣 × 𝑛 × %𝑋𝑇
(2)
Journal Pre-proof In expression (2), 𝐶𝑇(𝑛)
is the sampling cost of the T-ATTRIVAR chart, 𝐶𝑎 is the attribute
inspection cost per item, 𝑛 is the ATTRIVAR sample size , 𝐶𝑣 is the variable inspection cost per item and %𝑋𝑇 is the in-control rate of samples submitted to variable inspection. The T-ATTRIVAR chart with n=5 signals faster than the np chart with n=12 (𝑈𝐷𝐿 = ― 𝐿𝐷𝐿 = 1.696; CLnp 4). As only 5% of the in-control samples of the T-ATTRIVAR chart are submitted to the variable-type inspection, the ratio between the inspection cost by variable and by attribute might be as high as 28:1; if the attribute inspection cost is 1$, totalizing 12$ the sampling cost of the np chart, then the variable inspection cost might be as high as 28$, once 𝐶𝑇(5) = 1 × 5 + 28 × 5 × 0.05 = 12. Table 4 also compares the TATTRIVAR chart with n=3 with the np chart with n=6 (𝑈𝐷𝐿 = ―𝐿𝐷𝐿 = 1.931; CLnp 2). In this second case, the ratio between the inspection cost by variable and by attribute might be as high as 20:1. As the usual ratio between inspection costs by variable and by attribute is lower than 28:1 or 20:1, the T-ATTRIVAR is an interesting alternative to the np chart.
Table 4: Comparing the T-ATTRIVAR and the np charts T-ATTRIVAR
np
T-ATTRIVAR
np
δ\n
5
12
3
6
0
370.40
370.40
370.40
370.40
0.25
134.67
232.29
188.49
254.35
0.5
34.11
75.57
63.60
103.56
1
4.63
6.63
10.53
13.66
1.25
2.46
2.80
5.37
5.98
1.5
1.61
1.60
3.15
3.13
2
1.09
1.04
1.57
1.42
The X rec chart proposed by Quinino et al. (2015) requires the adoption of two discriminating limits to sort the sample items into three excluding categories; that is, items in categories A, B, or AB, are, respectively, items with X dimensions smaller than the lower discriminating limit, larger than the upper discriminating limit, or neither smaller than the lower discriminating limit nor larger than the upper discriminating limit. After that, N1, N2 and N3 observations are, respectively, randomly generate from the normal distributions N(-N1,1), N(N2,1), and N (0,1). The N1, N2 and N3 are, respectively, the number of sample items in A, B, and AB categories. The monitoring statistic X rec is the average of the N1, N2
Journal Pre-proof and N3 observations. The T-ATTRIVAR chart with n=5 and %𝑋 = 5% defeats the X rec chart with n=6, see Table 5. It is worth to stress that the X rec charts are pretty complicated to deal with, once they work with simulated observations. The idea of monitoring the process mean exploring attribute and variable sample data was introduced by Sampaio et al. (2014). With their np X X chart in use, the samples of size n are split in two subsamples of sizes n1 and n2, in such a way that, the n1 items of the first subsample are submitted to a go/no-go gauge test, and the n2 items of the second subsample are measured. The items of the first sample are classified as defective or non-defective, if the number of non-defectives exceeds a threshold, the second subsample is also inspected, but now the items of the second subsample are submitted to the variable-type inspection, ending up with an 𝑋 value used to decide the state of the process. The
np X X chart with n1=2, n2=4, and %𝑋 = 25%, that is, with in-control average sample size (ASS) of 3, is defeated by the T-ATTRIVAR chart with n=3 and %𝑋 = 20%, see Table 5. We built Table 5 with the ARLs of the X rec chart published by Quinino et al. (2015). In Sampaio et al. (2014), only upward shifts were considered to calculate the ARLs of the np X X control chart. So, for fair comparisons between its performance and the performance of the T-ATTRIVAR chart, Table 5 was built with the ARLs of the two-sided np X X chart, that is, the version of the np X X chart for upward and downward shifts detection.
Table 5: Comparing the T-ATTRIVAR, X rec and np X X charts T-ATT
T-ATT
T-ATT
3 371.54
np X X 3 (=ASS) 370.33
3 500.53
np X X 3 (=ASS) 500.16
δ\n 0
5 370.60
X rec 6 370.03
0.25
133.57
157.65
182.77
204.26
239.67
263.73
0.5
33.55
45.30
60.11
68.49
75.89
83.41
1
4.52
6.65
9.70
9.78
11.43
10.86
1.5
1.57
2.12
2.90
2.91
3.21
3.03
2
1.08
1.24
1.47
1.59
1.55
1.60
Journal Pre-proof 5. Illustrative example of the T-ATTRIVAR chart A numerical example was presented by Ho and Aparisi (2016) to illustrate the use of the BATTRIVAR chart. They adopted in-control observations normally distributed with 𝜇0 = 1000, 𝜎0 = 20, samples of size 𝑛 = 5, %𝑋𝑚𝑎𝑥 = 5%, and a process mean shift of one standard deviation (𝛿 = 1). The optimum solution presented by their GA was: 𝐿𝐷𝐿 = 971.18, 𝑈𝐷𝐿 = 1028.82, 𝑊𝑦 = 3, 𝐶𝐿𝑦 = 5 , 𝐿𝐶𝐿 = 976.0, 𝑈𝐶𝐿 = 1024.0 and %𝑋 = 2.66%, leading to an ARL=5.69. However, working with the maximum %𝑋 specified, that is, with %𝑋 = 5%, 𝐿𝐷𝐿 = 973.76, 𝑈𝐷𝐿 = 1026.24, 𝑊𝑦 = 3, 𝐶𝐿𝑦 = 5, 𝐿𝐶𝐿 = 976.0, 𝑈𝐶𝐿 = 1024.0, the ARL decreases around 10% (from 5.69 to 5.15); the GA algorithm was unable to discover that. According to Table 6, the B-ATTRIVAR chart signals faster when %𝑋 = 5%; additionally, Table 6 shows that the T-ATTRIVAR chart always defeats its competitor, the B-ATTRIVAR chart. With the input conditions (ARL0=370.4; %𝑋𝑚𝑎𝑥 = 5%; =1), the optimum parameters of the T-ATTRIVAR chart are 𝐿𝐷𝐿 = 968.62, 𝑈𝐷𝐿 = 1031.38, 𝑊𝑦 = 2, 𝐶𝐿𝑦 = 5, 𝐿𝐶𝐿 = 973.57 and 𝑈𝐶𝐿 = 1026.43. These optimum values are free of uncertainties, once the optimization worked with the exact ARLs, thanks to the construction and use of the X | YD distributions.
Table 6: The ARLs of the numerical example δ 0.25 0.5 1 1.25 1.5 2
(B/T) - ATTRIVAR Chart ( %𝑋) T (5%) B (2.66%) B (5%) 134.67 145.98 143.27 34.11 39.60 37.54 4.63 5.69 5.15 2.46 3.00 2.71 1.61 1.90 1.74 1.09 1.18 1.13
With the T-ATTRIVAR chart in use, samples of size 𝑛 are regularly taken from the production line and the n items are classified in three excluding categories by a go/no-go gauge test; the items with 𝑋 ≤ 𝐿𝐷𝐿 are reproved by the no-go gauge test and goes to category A, the items with 𝑋 ≥ 𝑈𝐷𝐿 are reproved by the go gauge test and goes to category B, the remaining items with 𝐿𝐷𝐿 < 𝑋 < 𝑈𝐷𝐿 goes to category AB. The number of sample items in each of the three categories are 𝑌𝐴, 𝑌𝐵, and 𝑌𝐴𝐵.
Journal Pre-proof Table 8 presents the 𝑌𝐴, 𝑌𝐵, and 𝑌𝐴𝐵 values of 24 samples of size five. The go no-go gauge test was built to attend the two discriminant limits: (𝐿𝐷𝐿 = 968.62; 𝑈𝐷𝐿 = 1031.38); it was assumed an in-control X~ N (1000; 20). The attribute control chart limits are 𝑊𝑦 = 2 and 𝐶𝐿𝑦 = 5, and the variable control chart limits are 𝐿𝐶𝐿 = 973.57 and 𝑈𝐶𝐿 = 1026.43. Fig. 6 presents the T-ATTRIVAR chart with the 𝑌𝐷 and 𝑋 values of the 24 samples. The monitoring statistic 𝑌𝐷 is a function of the 𝑌𝐴 and 𝑌𝐵, see Tables 7 and 8.
Table 7: The 𝑌𝐷 as a function of the 𝑌𝐴 and 𝑌𝐵 YA
YB
YD
See samples
Process decision
=0
=0
=0
>0
>0
=0
=0
< Wy
= YB
1, 4, 7, 8, 13, 15, 17, 18, 19, 22 2, 10, 16, 21, 23 6, 11, 20
< Wy
=0
= YA
3, 9, 12, 14
=0
≥ Wy < CLy
= YB
24
≥ Wy < CLy
=0
= YA
5
depends on the 𝑋
=0
≥ CLy
= YB
-
out of control
≥ CLy
=0
= YA
-
𝑋
Process decision
𝑋 ≥UCL or 𝑋 ≤ LCL LCL<𝑋 and 𝑋 < UCL
out of control
in control
in control
Table 8: The YD and 𝑋values # 1 2 3 4 5 6 7 8 9 10 11 12 13
YA 0 1 1 0 2 0 0 0 1 1 0 1 0
YB 0 1 0 0 0 1 0 0 0 1 1 0 0
YAB 5 3 4 5 3 4 5 5 4 3 4 4 5
YD 0 0 1 0 2 1 0 0 1 0 1 1 0
𝑋 985.61 -
# 14 15 16 17 18 19 20 21 22 23 24
YA 1 0 1 0 0 0 0 2 0 1 0
YB 0 0 1 0 0 0 1 1 0 1 4
YAB 4 5 3 5 5 5 4 2 5 3 1
YD 1 0 0 0 0 0 1 0 0 0 4
𝑋 1033.62
Journal Pre-proof Sample 24, where 𝑋 =1033.62 > 𝑈𝐶𝐿, leads to an alarm that an assignable cause is increasing the process mean; discarding the case of being a false alarm.
YD
6 5
1026.43
4 3 973.57
2 1 0 1
6
11
16 # Sample
21
26
Fig. 6. The T-ATTRIVAR chart with observations of twenty-four samples
6. Conclusions In this paper, we discussed the design of the B-ATTRIVAR chart, including the optimum designs presented by the GA developed by Ho and Aparisi (2016). We were able to find designs that lead to lower ARLs, that is, to lower delays with which the B-ATTRIVAR chart signals. For several scenarios, the GA optimization approach generated designs with corresponding ARLs that are more than 10% higher than the lowest ones we found in our studies with the computation of the exact ARLs. The trinomial version of the ATTRIVAR chart proved to be more efficient than the binomial version, especially when 𝑛 = 5 and less than 15% of the in-control samples are submitted to the variable inspection, or when 𝑛 = 3 and more than 15% of the in-control samples are submitted to the variable inspection. A suggestion of future work for this research is the investigation of the efficiency of a new ATTRIVAR control chart in which only the disapproved items are submitted to the variable-type inspection. This idea was proposed by one of the referees as mentioned in section 2.
Journal Pre-proof
Acknowledgements The authors would like to thank the referees for their valuable comments and suggestions. The research for this article was supported by FAPESP – Fundação de Amparo à Pesquisa do Estado de São Paulo, project 2018/07147-0 and by CNPq – Conselho Nacional de Desenvolvimento Científico e Tecnológico, projects 306671/2015-0 and 304599/2015-8.
References Aparisi, F., Epprecht, E.K., Mosquera, J., 2018. Statistical Process Control Based on Optimum Gages. Qual. Reliab. Eng. Int. 34, 2–14. doi:10.1002/qre.2135 Aparisi, F., Lee Ho, L., 2018. M-ATTRIVAR: An attribute-variable chart to monitor multivariate process means. Qual. Reliab. Eng. Int. 34, 214–228. Aslam, M., Azam, M., Khan, N., Jun, C.-H., 2015. A mixed control chart to monitor the process. Int. J. Prod. Res. 53, 4684–4693. Bezerra, E.L., Ho, L.L., da Costa Quinino, R., 2018. GS2 An optimized attribute control chart to monitor process variability. Int. J. Prod. Econ. 195, 287–295. Haridy, S., Wu, Z., Lee, K.M., Rahim, M.A., 2014. An attribute chart for monitoring the process mean and variance. Int. J. Prod. Res. 52, 3366–3380. Ho, L.L., Aparisi, F., 2016. ATTRIVAR: Optimized control charts to monitor process mean with lower operational cost. Int. J. Prod. Econ. 182, 472–483. Ho, L.L., Costa, A., 2015. Attribute Charts for Monitoring the Mean Vector of Bivariate Processes. Qual. Reliab. Eng. Int. 31, 683–693. Ho, L.L., da Costa Quinino, R., 2016. Combining attribute and variable data to monitor process variability: MIX S 2 control chart. Int. J. Adv. Manuf. Technol. 87, 3389–3396. Khoo, M.B.C., Wu, Z., Castagliola, P., Lee, H.C., 2013. A multivariate synthetic double sampling T2 control chart. Comput. Ind. Eng. 64, 179–189. Lee Ho, L., Quinino, R.C., 2013. An attribute control chart for monitoring the variability of a process, in: International Journal of Production Economics. pp. 263–267. Leoni, R.C. and, Costa, A.F.B., 2019. The performance of the truncated mixed control chart. Commun. Stat. - Theory Methods. 48, 4294–4301. Machado, M.A.G., Ho, L.L., Costa, A.F.B., 2018. Attribute control charts for monitoring the covariance matrix of bivariate processes. Qual. Reliab. Eng. Int. 34, 257–264. Melo, M.S., Ho, L.L., Medeiros, P.G., 2017. Max D: an attribute control chart to monitor a bivariate process mean. Int. J. Adv. Manuf. Technol. 90, 489-498.
Journal Pre-proof Mosquera, J., Aparisi, F., Epprecht, E.K., 2018. A global scheme for controlling the mean and standard deviation of a process based on the optimal design of gauges. Qual. Reliab. Eng. Int. 34, 718–730. Quinino, R.C., Lee Ho, L., Galindo Trindade, A.L., 2015. Monitoring the process mean based on attribute inspection when a small sample is available. J. Oper. Res. Soc. 66, 1860–1867. Quinino, R.C., Bessegato, L.F., Cruz, F.R.B., 2017. An attribute inspection control chart for process mean monitoring. Int. J. Adv. Manuf. Technol. 90, 2991–2999. Sampaio, E.S., Ho, L.L., de Medeiros, P.G., 2014. A Combined npx−X¯ Control Chart to Monitor the Process Mean in a Two-Stage Sampling. Qual. Reliab. Eng. Int. 30, 1003–1013. Wu, Z., Jiao, J., 2008. A control chart for monitoring process mean based on attribute inspection. Int. J. Prod. Res. 46, 4331–4347. Wu, Z., Khoo, M.B.C., Shu, L., Jiang, W., 2009. An np control chart for monitoring the mean of a variable based on an attribute inspection. Int. J. Prod. Econ. 121, 141–147. Zhou, W., Liu, N., Zheng, Z., 2019. A synthetic control chart for monitoring the small shifts in a process mean based on an attribute inspection. Commun. Stat. - Theory Methods 1–16.
Journal Pre-proof Authors’ Contributions
Felipe Domingues Simões: Software, Validation, Formal analysis, Investigation, Resources, Writing - Original Draft; Antonio Fernando Branco Costa: Conceptualization, Methodology, Supervision, Funding acquisition; Marcela Aparecida Guerreiro Machado: Project administration, Writing - Review & Editing, Visualization.
Journal Pre-proof Highlights
Scripts developed to calculate exact Average Run Lengths (ARLs) for ATTRIVAR charts Designs that lead to lower ARLs were found for the Binomial ATTRIVAR chart The Trinomial ATTRIVAR chart proposed is more efficient than the binomial version
Felipe Domingues Simões, Antonio Fernando Branco Costa, Marcela Aparecida Guerreiro Machado PII:
S0925-5273(19)30393-7
DOI:
https://doi.org/10.1016/j.ijpe.2019.107559
Reference:
PROECO 107559
To appear in:
International Journal of Production Economics
Received Date:
23 July 2019
Accepted Date:
13 November 2019
Please cite this article as: Felipe Domingues Simões, Antonio Fernando Branco Costa, Marcela Aparecida Guerreiro Machado, The Trinomial ATTRIVAR control chart, International Journal of Production Economics (2019), https://doi.org/10.1016/j.ijpe.2019.107559
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier.
Journal Pre-proof The Trinomial ATTRIVAR control chart Felipe Domingues Simões a,*, Antonio Fernando Branco Costa b, Marcela Aparecida Guerreiro Machado a.
a
Universidade Estadual Paulista “Júlio de Mesquita Filho”, Campus de Guaratinguetá, Departamento de Produção, Brazil. b
Universidade Federal de Itajubá, Brazil
* Corresponding author: Phone: +55 (11) 98624-3284 E-mail adresses: [email protected] (F.D. Simões), [email protected] (A.F.B. Costa), [email protected] (M.A.G. Machado)
Journal Pre-proof The Trinomial ATTRIVAR Control Chart
ABSTRACT In this article, we propose the Trinomial - ATTRIVAR (T-ATTRIVAR) control chart where attribute and variable sample data are used to control the process mean. Firstly, two discriminating limits sort the sample items into three excluding categories; that is, items in categories A, B, or AB, are, respectively, items with X dimensions smaller than the lower discriminating limit, larger than the upper discriminating limit, or neither smaller than the lower discriminating limit nor larger than the upper discriminating limit. Depending on the number of sample items in each category, one of three decisions is made: the process is declared in-control, the process is declared out-of-control, or all sample items are also measured. In this last case, the sample mean of X is used to decide the state of the process. Aslam et al. (2015) worked with the particular case where the sample items are classified as defective (items in category - A plus items in category - B) or not-defective (items in category AB). The strategy of splitting defectives into two excluding categories (A and B) enhances the performance of the ATTRIVAR chart. It is worth to emphasize that the previous attribute classification truncates the X distribution. Consequently, the mathematical development to obtain the ARLs is complex – the Average Run length (ARL) is the average number of samples the control chart requires to signal. With the density function of the sum of truncated X distributions, we obtained the exact ARLs. The exact minimum ARLs are lower than the minimum ARLs Ho & Aparisi (2016) obtained with the Genetic Algorithm.
Keywords: Shewhart control chart; ATTRIVAR control chart; Truncated normal distributions; Average run length; Monitoring process mean
Journal Pre-proof 1. Introduction The control chart for attributes was originally designed to control processes in which the monitored quality characteristics cannot be measured on a continuous numerical scale. The number of defectives bulbs from samples of size 50 is a typical example. However, the focus of many recent studies lies on the control of variable-type quality characteristics with attribute charts. Wu & Jiao (2008) were the first to propose an attribute chart to control the mean of a continuous variable X. Basically, when X is larger than a threshold the sample unit is classified as nonconforming. A sample of size n with more than m nonconforming units (referred to as MON) is a red sample. The MON chart produces an outof-control signal when the interval between two red samples is smaller than a threshold. Their article inspired many other studies where attribute charts are used to control the mean, the mean vector, the variance or the covariance matrix. Wu et al. (2009) discusses the monitoring of shafts with np charts and with 𝑋 charts. With the np chart, gauge rings are used to distinguish conforming and nonconforming shafts. With the 𝑋 chart, a micrometer is used to measure the diameters. On the basis of the monitoring costs, the attribute inspection usually outperforms the variable inspection. Quinino et al. (2015, 2017) and Aparisi et al. (2018) also designed attribute charts to control the mean of a continuous variable. Ho and Quinino (2016) and Bezerra et al. (2018) explored the idea of monitoring the process variance with attribute charts. Haridy et al. (2014) and Mosquera et al. (2018) investigated the efficiency of monitoring both, the mean and the variance of a continuous variable, with an attribute chart. Very recently, Zhou et al. (2019) proposed a chart similar to the MON chart, which was designed to detect mean shifts faster. Ho and Costa (2015) and Melo et al. (2017) built attribute charts to control the mean vector of bivariate charts. More recently, Machado et al. (2018) studied the efficiency of the attribute charts in signaling changes in the covariance matrix. The idea of working with attribute and variable sample data (ATTRIVAR data) was introduced by Sampaio et al. (2014). In their study, the samples are split in two subsamples, in such a way that, the items of the first subsample are submitted to a go/no-go gauge test, and the items of the second subsample are measured. It is simple to obtain the properties of the ATTRIVAR charts when the items submitted to the attribute-type inspection are not the same ones submitted to the variable-type inspection. This sampling strategy was also adopted in Ho and Quinino (2016), Ho and Aparisi (2016) and in Aparisi and Ho (2018). In Aslam et al. (2015) the sample items are submitted to the attributetype inspection and, depending on the results, they are also submitted to the variable-type inspection. In this case, the sample mean of the X observations decides the state of the process. It is worthwhile to stress that the attribute-type inspection always truncates the X distribution and, because of that, the
Journal Pre-proof mathematical development to obtain the 𝑋 distribution is not trivial, see Leoni and Costa (2019). Reminding that the Average Run Length (ARL) is the average number of samples the control chart requires to signal. Ho and Aparisi (2016) used a Genetic Algorithm (GA) to obtain the optimum parameters of the Aslam’s chart, that is, the parameters that lead to the lowest ARLs. After obtaining the exact ARLs, we could find ARLs that are lower than the minimum ARLs Ho & Aparisi (2016) obtained with the GA solution. The usual simulation was helpful to confirm that the solution presented by the GA doesn’t lead to the lowest ARLs. The Aslam et al (2015)'s idea of submitting the sample items to the attribute inspection before being measured is a milestone contribution to the Monitoring Process field. But, we cannot forget that the attribute inspection truncates the X distribution and, with truncated observations, it is not trivial to obtain the 𝑋 distribution. In this article, we also work with attribute and variable data collected from the same sample. However, in our model two discriminant limits sort the sample items into three excluding categories; that is, items in categories A, B, or AB, are, respectively, items with X dimensions smaller than the lower discriminating limit, larger than the upper discriminating limit, or neither smaller than the lower discriminating limit nor larger than the upper discriminating limit. Depending on the number of sample items in each category, one of three decisions is made: the process is declared in-control, the process is declared out-of-control, or all sample items are also measured. In this last case, the sample mean of X is used to decide the state of the process. Our model considers that the in-control mean and standarddeviation of the process are known. The term Trinomial (Trinomial ATTRIVAR chart) is used to emphasize that, during the attribute-type inspection, the sample items are grouped into three excluding categories. 2. The Binomial ATTRIVAR chart The Binomial ATTRIVAR chart (B-ATTRIVAR) is similar to the Mixed chart proposed by Aslam et al. (2015); except the Mixed chart was proposed to signal only increases in the process mean, because of that the lower discriminating limit LDL doesn’t exist. The Mixed and the B-ATTRIVAR charts are the type of charts where the sample items are always submitted to an attribute inspection and, depending on the results, their quality characteristic X may be measured too. During the attribute inspection, the number of disapproved items 𝑌𝐷 is determined - an item is not approved when its quality characteristic X is smaller than a lower discriminating limit LDL or it is greater than an upper discriminating limit UDL. If 𝑌𝐷 is smaller than a warning limit 𝑊𝑦 the process is assumed in-control, if 𝑌𝐷 is equal to, or greater than an upper control limit CLy the process is assumed out-of-control. However, if 𝑌𝐷 is equal or greater than 𝑊𝑦, but smaller than 𝐶𝐿𝑦, more information is necessary to
Journal Pre-proof decide the state of the process. In this case, the inspection moves to a more complex level, where the quality characteristic X of all sample items, approved and disapproved, are measured. The statistic 𝑋 is now used to decide the state of the process. One of the referees suggested to limit the variable-type inspection to the subsample of disapproved items, that is, the approved items are not submitted to this type of inspection. The size of the subsample of disapproved items is a discrete random variable, because of that, additional work is necessary to obtain the 𝑋 distribution; reminding that the X distribution of disapproved items is always truncated. The referee’s idea of measuring only the reproved items is worth of investigation. It is highly expected that by eliminating the measurement of the approved items, we are improving the relationship between sampling costs and the speed with which the ATTRIVAR chart signals. Fig. 1 illustrates the B-ATTRIVAR chart. The black points are the 𝑌𝐷 points; they were plotted using the left-hand side scale of the chart. Whenever a black point falls below the warning limit 𝑊𝑦 the process is declared in control, but whenever a black point falls above the control limit 𝐶𝐿𝑦 - the process is declared out of control. Otherwise, the X quality characteristic of the sample items is measured and the resulting 𝑋 is also plotted, but now as a white point and using the right-hand side scale of the chart (see samples 3, 5 and 6). The process is declared in control/out of control when the white point falls inside/outside the (LCL; UCL) limits; LCL is the lower control limit of the 𝑋 chart, and the UCL is its upper control limit.
𝑌𝐷
𝑋 𝑈𝐶𝐿
𝐶𝐿𝑦
𝑊𝑦 𝐿𝐶𝐿
1
2
3
4
5
6
⋯
Sampling Points
Fig. 1. The B-ATTRIVAR chart
Journal Pre-proof The B-ATTRIVAR chart and the ATTRIVAR-1 chart discussed in Ho and Aparisi (2016) are the same. It is always possible to find values for the B-ATTRIVAR chart’s parameters that leads to lower ARLs than the ones given by GA solution. It is worth to stress that the X distribution of an approved/disapproved item is truncated, because of that, the mathematical development to obtain the 𝑋 distribution is complex. Ho and Aparisi (2016) presented the general procedure to obtain the density function of Sn X when the X distributions are normal but left-tail, right-tail, or centrally truncated, X ~ N t ( ; ) . The way the expressions of the probability Pr[ Sn S0 | Xs ~ N t ( ; )] are built was presented by Ho and Aparisi (2016); their construction and computation are time-consuming, because of that, Ho and Aparisi (2016) opted to work with GA to obtain the properties of the B-ATTRIVAR chart. To obtain a deeper understanding of the B-ATTRIVAR chart’s properties, we wrote two programs to calculate the Average Run Length (ARL) of the B-ATTRIVAR chart. The first one obtains the ARLs by simulation and the second one calculates the ARLs using the exact distribution of X ( Sn / n) | YD ; the binomially distributed variable 𝑌𝐷 is the number of approved items by the attribute-type inspection. An important performing parameter of the B-ATTRIVAR chart is % X - the in-control rate with which the X quality characteristic of the sample items are measured. It is given by:
𝐶𝐿𝑦 ― 1
𝑛!
%𝑋𝐵 = ∑𝑖 = 𝑊𝑦 𝑖!(𝑛 ― 𝑖)! × (𝑝)𝑖 × (𝑞)(𝑛 ― 𝑖) In expression (1), n is the sample size, and 𝑞 = 1 ― 𝑝 = Φ
(
(1)
𝑈𝐷𝐿 ― 𝜇0
𝐿𝐷𝐿 ― 𝜇0
𝜎
𝜎
) ― Φ(
), where
0
is the in-
control process mean and 𝜎 is the process standard-deviation. The magnitude of the mean shift is expressed in units of the standard-deviation; if is the shifted mean, then +. The go/no-go gauges devices are built taking into account the in-control mean: LDL=-kg and UDL=+kg The width factor kg is a function of the in-control rate with which the sample items are measured (%𝑋). The 𝑋 chart’s control limits are: LCL=-k 𝑛 and UCL=+k 𝑛 The k factor works as a fine adjust to obtain the desired ARL0. In Table 1, the ARLs obtained by simulation (ARLsim) and the exact ARLs ( ARL X |YD ) calculated with the X | YD distributions are compared. The percentage difference, between exact and simulated ARLs is pretty low. All of the exact ARLs ( ARL X |YD ) calculated are inside the
Journal Pre-proof confidence intervals (CI), calculated with 20 simulation values for each scenario and with a confidence level of 95%. It means that the program that calculates the ARLs works properly with a confidence level of 95%. With large samples (n=10 or 15), the running time to obtain the exact ARLs is huge. Because of that, and based on the fact that the differences between simulated and exact ARLs are insignificant, the ARLs for large samples were obtained by simulation only.
Table 1: Exact and simulated ARLs of the B-ATTRIVAR chart ARL X |YD
n 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
%X̅B 25 25 25 25 25 25 15 15 15 15 15 15 5 5 5 5 5 5 25 25 25 25 25 25 15 15 15 15 15 15 5 5 5 5
δ 0.25 0.5 1 1.25 1.5 2 0.25 0.5 1 1.25 1.5 2 0.25 0.5 1 1.25 1.5 2 0.25 0.5 1 1.25 1.5 2 0.25 0.5 1 1.25 1.5 2 0.25 0.5 1 1.25
(I) 199.98 69.60 11.03 5.48 3.15 1.53 185.72 61.81 9.94 5.03 2.94 1.48 188.59 63.68 10.54 5.38 3.16 1.57 133.63 33.61 4.53 2.41 1.58 1.08 134.08 33.90 4.57 2.42 1.59 1.08 138.46 36.06 5.02 2.66
ARLsim
(II) 202.08 69.88 11.00 5.48 3.15 1.53 186.26 61.99 9.94 5.05 2.94 1.48 188.84 63.83 10.57 5.38 3.16 1.57 134.15 33.82 4.54 2.41 1.58 1.08 134.35 33.96 4.57 2.44 1.59 1.08 137.28 35.99 5.02 2.67
CI (95%) Δ 100 (I)-(II) / (I)
1.05% 0.40% 0.27% 0.00% 0.00% 0.00% 0.29% 0.29% 0.00% 0.40% 0.00% 0.00% 0.13% 0.24% 0.28% 0.00% 0.00% 0.00% 0.39% 0.64% 0.14% 0.10% 0.23% 0.12% 0.20% 0.19% 0.01% 0.70% 0.07% 0.10% 0.85% 0.20% 0.08% 0.23%
(199.429, 201.093) (68.835, 70.067) (10.967, 11.146) (5.474, 5.514) (3.135, 3.155) (1.528, 1.537) (184.774, 186.304) (61.386, 61.936) (9.873, 9.972) (5.007, 5.040) (2.935, 2.954) (1.479, 1.487) (187.598, 189.290) (63.524, 63.929) (10.535, 10.618) (5.361, 5.402) (3.144, 3.170) (1.568, 1.575) (133.543, 134.753) (33.563, 33.821) (4.512, 4.546) (2.403, 2.421) (1.575, 1.584) (1.078, 1.081) (133.400, 134.735) (33.756, 34.086) (4.566, 4.599) (2.413, 2.435) (1.581, 1.590) (1.079, 1.082) (137.720, 139.239) (35.711, 36.133) (5.010, 5.038) (2.655, 2.674)
Journal Pre-proof 5 5
5 5
1.5 2
1.72 1.12
1.73 1.12
0.39% 0.34%
(1.718, 1.726) (1.120, 1.123)
In Table 2, the ARLs of the B-ATTRIVAR chart designed by the GA (ARLGA) are compared with the ARLs obtained with the X | YD distribution ( ARL X |YD ) and by simulation (ARLsim). Confidence intervals (CI) were calculated with 20 simulation values for each scenario and with a confidence level of 95%. The ARLs of the B-ATTRIVAR chart designed by the GA are larger than the exact ARLs we obtained working with the X | YD distributions. Ho and Aparisi (2016) didn’t publish the values of the chart’s parameters that lead to their optimum ARLs, consequently, we couldn’t undertake further investigation, comparing the ARLs of the B-ATTRIVAR chart designed by GA algorithm with the ARLs X |YD for a set of pre-specified values of the chart’s parameters. It is important to stress that the
GA algorithm was built with the following constraint: %X̅ ≤ %X̅max, where %X̅ is the in-control rate with which the items of the samples are submitted to measurements and %X̅max is a pre-specified value. The optimum %X̅ (%X̅opt) is not necessarily equal to the %X̅max; depending on the input parameters, the %X̅opt <%X̅max.
Table 2: Comparing the ARLs obtained by different methods n 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 5
max %X̅B 25 25 25 25 25 25 15 15 15 15 15 15 5 5 5 5 5 5 25
δ ARLGA 0.25 0.5 1 1.25 1.5 2 0.25 0.5 1 1.25 1.5 2 0.25 0.5 1 1.25 1.5 2 0.25
189.84 61.29 10.19 5.16 3.00 1.51 189.84 61.29 10.19 5.20 3.03 1.61 192.85 64.78 11.65 5.83 3.36 1.73 139.68
Optimum %X̅B 10 10 10 10 10 10 10 10 10 10 10 10 5 5 5 5 5 5 25
ARLsim
CI (95%)
185.19 61.47 9.88 5.06 2.96 1.49 185.19 61.47 9.88 5.06 2.96 1.49 188.84 63.83 10.57 5.38 3.16 1.57 134.15
(184.841, 186.377) (61.015, 61.651) (9.893, 9.971) (5.017, 5.068) (2.942, 2.965) (1.487, 1.493) (184.841, 186.377) (61.015, 61.651) (9.893, 9.971) (5.017, 5.068) (2.942, 2.965) (1.487, 1.493) (188.208, 189.699) (63.286, 63.910) (10.489, 10.591) (5.367, 5.404) (3.135, 3.161) (1.565, 1.574) (132.780, 134.284)
ARL X |YD 185.20 61.43 9.93 5.04 2.95 1.49 185.20 61.43 9.93 5.04 2.95 1.49 188.59 63.68 10.54 5.38 3.16 1.57 133.63
Journal Pre-proof 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 15 15 15 15 15 15 15 15 15 15 15 15 15
25 25 25 25 25 15 15 15 15 15 15 5 5 5 5 5 5 25 25 25 25 25 25 15 15 15 15 15 15 5 5 5 5 5 5 25 25 25 25 25 25 15 15 15 15 15 15 5
0.5 1 1.25 1.5 2 0.25 0.5 1 1.25 1.5 2 0.25 0.5 1 1.25 1.5 2 0.25 0.5 1 1.25 1.5 2 0.25 0.5 1 1.25 1.5 2 0.25 0.5 1 1.25 1.5 2 0.25 0.5 1 1.25 1.5 2 0.25 0.5 1 1.25 1.5 2 0.25
34.64 4.61 2.51 1.59 1.08 140.04 35.64 4.78 2.61 1.71 1.11 145.48 40.98 5.29 2.76 2.25 1.15 79.55 14.18 1.99 1.36 1.06 1.02 77.82 14.31 2.01 1.36 1.08 1.00 89.55 19.27 2.66 1.76 1.30 1.04 49.91 7.62 1.85 1.15 1.07 1.00 53.56 9.52 1.51 1.31 1.07 1.00 69.82
25 25 25 25 25 15 15 15 15 15 15 5 5 5 5 5 5 25 25 25 25 25 25 15 15 15 15 15 15 5 5 5 5 5 5 25 25 25 25 25 25 15 15 15 15 15 15 5
33.61 4.53 2.41 1.58 1.08 134.08 33.90 4.60 2.44 1.60 1.09 138.46 36.06 5.02 2.66 1.72 1.13 -
33.82 4.54 2.41 1.58 1.08 134.35 33.96 4.59 2.45 1.60 1.09 137.28 35.99 5.02 2.67 1.73 1.13 73.93 13.17 1.84 1.23 1.05 1.00 76.57 13.96 1.91 1.27 1.07 1.00 85.94 16.92 2.40 1.49 1.16 1.01 49.34 7.47 1.30 1.05 1.01 1.00 51.62 8.12 1.39 1.09 1.01 1.00 63.32
(33.497, 33.784) (4.512, 4.550) (2.404, 2.416) (1.572, 1.582) (1.078, 1.081) (133.368, 134.605) (33.838, 34.065) (4.569, 4.612) (2.439, 2.454) (1.597, 1.606) (1.086, 1.091) (137.753, 139.395) (35.855, 36.205) (4.989, 5.030) (2.647, 2.668) (1.721, 1.730) (1.124, 1.131) (73.754, 74.322) (13.144, 13.259) (1.828, 1.841) (1.229, 1.237) (1.049, 1.055) (1.000, 1.001) (76.169, 77.046) (13.861, 13.965) (1.906, 1.920) (1.267, 1.274) (1.068, 1.070) (1.000, 1.002) (85.836, 86.526) (16.864, 17.009) (2.384, 2.401) (1.481, 1.491) (1.155, 1.160) (1.008, 1.010) (49.386, 49.849) (7.439, 7.503) (1.230, 1.305) (1.050, 1.057) (1.007, 1.010) (1.000, 1.001) (51.575, 52.098) (8.071, 8.142) (1.386, 1.393) (1.089, 1.096) (1.010, 1.014) (1.000, 1.001) (63.012, 63.637)
Journal Pre-proof 15 15 15 15 15
5 5 5 5 5
0.5 1 1.25 1.5 2
13.23 1.97 1.54 1.09 1.01
5 5 5 5 5
-
11.25 1.77 1.22 1.05 1.00
(11.242, 11.320) (1.770, 1.781) (1.220, 1.228) (1.050, 1.052) (1.000, 1.001)
A VBA program was written to obtain the simulated ARLs of the B-ATTRIVAR chart. Fig. 2 presents the screen of the VBA program (hosted in: http://www.fbranco.unifei.edu.br). The number of simulated Run Length (RL) with which the ARL is calculated is an input parameter (Sim. Runs). The desired % X can be reached by varying (kg, Wy, CLy), and the ARL0 by varying k.
Fig. 2. Screen of the VBA program with the % X and ARLs of the B-ATTRIVAR Chart
Journal Pre-proof 3. The Trinomial ATTRIVAR chart An upward mean shift increases the probability of classifying an item as defective - with an X value exceeding the Upper Discriminant Limit (𝑋 ≥ 𝑈𝐷𝐿), and decreases the probability of classifying an item as defective - with an X value lower than the Lower Discriminant Limit (𝑋 ≤ 𝐿𝐷𝐿). Similar comments can be made when the mean shift is a downward shift. With the usual samples of size five or close to five, it seems reasonable to admit the process is in-control whenever, among the sample items classified as defectives, two of them were declared defectives by different gauges: for instance, the first was reproved by the no-go gauge (𝑋 ≤ 𝐿𝐷𝐿), and the second one was reproved by the go gauge (𝑋 ≥ 𝑈𝐷𝐿). When the B-ATTRIVAR is in use, these two types of defectiveness are not distinguished and, independent of being “too small or too big”, the items are simply classified as defectives. In general, the assignable cause increases or decreases the process mean, consequently, it is pretty important to distinguish the defective item with 𝑋 ≥ 𝑈𝐷𝐿 from the defective item with 𝑋 ≤ 𝐿𝐷𝐿. The Trinomial version of the ATTRIVAR chart (T-ATTRIVAR) has been proposed with this purpose. When the T-ATTRIVAR chart is in use, a sample of size 𝑛 is taken and all of those 𝑛 items are classified in three excluding categories by a go/no-go gauge test; the items with 𝑋 ≤ 𝐿𝐷𝐿 are reproved by the no-go gauge test and goes to category A, the items with 𝑋 ≥ 𝑈𝐷𝐿 are reproved by the go gauge test and goes to category B, the remaining items with 𝐿𝐷𝐿 < 𝑋 < 𝑈𝐷𝐿 goes to category AB. The number of defectives in each of the three categories are 𝑌𝐴, 𝑌𝐵, and 𝑌𝐴𝐵. If YA > 0 and YB > 0; YA =0 and YB
Journal Pre-proof
Fig. 3. The flowchart of the T-ATTRIVAR chart
Journal Pre-proof
When the T-ATTRIVAR chart is in use, the in-control rate of samples submitted to the variable inspections is:
%𝑋𝑇 =
Where 𝑃𝐴 = Φ
(
∑
𝐶𝐿𝑦 ― 1
𝑛! 𝑛! ― 𝑖) ― 𝑖) × 𝑃𝑖𝐴 × 𝑃(𝑛 + × 𝑃𝑖𝐵 × 𝑃(𝑛 𝐴𝐵 𝐴𝐵 𝑖!(𝑛 ― 𝑖)! 𝑖!(𝑛 ― 𝑖)! 𝑖 = 𝑊𝑦
𝐿𝐷𝐿 ― 𝜇0 𝜎
) and 𝑃
𝐵
=1―Φ
(
𝑈𝐷𝐿 ― 𝜇0 𝜎
) and 𝑃
𝐴𝐵
= (1 ― 𝑃 𝐴 ― 𝑃 𝐵 ) .
In Fig. 4, the performance of the B- and the T- ATTRIVAR charts are compared. The parameters of the ATTRIVAR charts are in Table 3. All of them lead to an ARL0 of 370.4.
Table 3: Parameters of ATTRIVAR charts Chart n 𝑊𝑦 𝐶𝐿𝑦 B1
5
1
5
B2
5
2
5
T1
5
2
5
B3
3
1
3
T2
3
1
3
According to Fig. 4, the T-ATTRIVAR chart is highly recommended in two situations: a) when samples of size n=5 are taken from the process and it is not allowed to submit many of these samples to the variable inspection (the %𝑋𝑇 cannot exceed 15%). In reality, with samples of size 5, it is impossible to design a T-ATTRIVAR chart with the in-control properties of ARL0=370.4 and %𝑋𝑇> 15%. b) when samples of size n=3 are taken from the process and it is allowed to submit many of these samples to the variable inspection (the %𝑋𝑇 exceeds 15%).
Journal Pre-proof
Fig. 4. The B- and T-ATTRIVAR charts (𝑛 = 5 left and 𝑛 = 3 right)
A second VBA program was written to obtain the simulated ARLs of the T-ATTRIVAR chart. Fig. 5 presents the screen of the VBA program (hosted in: http://www.fbranco.unifei.edu.br).
Journal Pre-proof
Fig. 5. Screen of the VBA program with the % X and ARLs of the T-ATTRIVAR Chart
4. Comparing the T-ATTRIVAR chart with other charts In this section, the T-ATTRIVAR chart is compared with the np chart, and also with the X rec and
np X X charts. In fact, if the sample items can be submitted to a device such as a go/no-go gauge test, then the np chart can be used to control the process mean. The sample item is approved when its X dimension is larger than a lower discriminating limit LDL and smaller than an upper discriminating limit UDL. The number D of approved items is binomially distributed; during the in-control period,
D ~ Bin(n, p0 ) with p0 Pr[ LDL X UDL] , and during the out-of-control period, D ~ Bin(n, p1 ) with p1 Pr[ LDL X UDL] . The np chart signals when D CLnp , where CLnp is the upper control limit of the np chart; the lower control limit of the np chart is zero. In Table 4, the TATTRIVAR chart is compared with the np chart. To compare the control charts in an economic perspective, the following cost function was adopted: 𝐶𝑇(𝑛) = 𝐶𝑎 × 𝑛 + 𝐶𝑣 × 𝑛 × %𝑋𝑇
(2)
Journal Pre-proof In expression (2), 𝐶𝑇(𝑛)
is the sampling cost of the T-ATTRIVAR chart, 𝐶𝑎 is the attribute
inspection cost per item, 𝑛 is the ATTRIVAR sample size , 𝐶𝑣 is the variable inspection cost per item and %𝑋𝑇 is the in-control rate of samples submitted to variable inspection. The T-ATTRIVAR chart with n=5 signals faster than the np chart with n=12 (𝑈𝐷𝐿 = ― 𝐿𝐷𝐿 = 1.696; CLnp 4). As only 5% of the in-control samples of the T-ATTRIVAR chart are submitted to the variable-type inspection, the ratio between the inspection cost by variable and by attribute might be as high as 28:1; if the attribute inspection cost is 1$, totalizing 12$ the sampling cost of the np chart, then the variable inspection cost might be as high as 28$, once 𝐶𝑇(5) = 1 × 5 + 28 × 5 × 0.05 = 12. Table 4 also compares the TATTRIVAR chart with n=3 with the np chart with n=6 (𝑈𝐷𝐿 = ―𝐿𝐷𝐿 = 1.931; CLnp 2). In this second case, the ratio between the inspection cost by variable and by attribute might be as high as 20:1. As the usual ratio between inspection costs by variable and by attribute is lower than 28:1 or 20:1, the T-ATTRIVAR is an interesting alternative to the np chart.
Table 4: Comparing the T-ATTRIVAR and the np charts T-ATTRIVAR
np
T-ATTRIVAR
np
δ\n
5
12
3
6
0
370.40
370.40
370.40
370.40
0.25
134.67
232.29
188.49
254.35
0.5
34.11
75.57
63.60
103.56
1
4.63
6.63
10.53
13.66
1.25
2.46
2.80
5.37
5.98
1.5
1.61
1.60
3.15
3.13
2
1.09
1.04
1.57
1.42
The X rec chart proposed by Quinino et al. (2015) requires the adoption of two discriminating limits to sort the sample items into three excluding categories; that is, items in categories A, B, or AB, are, respectively, items with X dimensions smaller than the lower discriminating limit, larger than the upper discriminating limit, or neither smaller than the lower discriminating limit nor larger than the upper discriminating limit. After that, N1, N2 and N3 observations are, respectively, randomly generate from the normal distributions N(-N1,1), N(N2,1), and N (0,1). The N1, N2 and N3 are, respectively, the number of sample items in A, B, and AB categories. The monitoring statistic X rec is the average of the N1, N2
Journal Pre-proof and N3 observations. The T-ATTRIVAR chart with n=5 and %𝑋 = 5% defeats the X rec chart with n=6, see Table 5. It is worth to stress that the X rec charts are pretty complicated to deal with, once they work with simulated observations. The idea of monitoring the process mean exploring attribute and variable sample data was introduced by Sampaio et al. (2014). With their np X X chart in use, the samples of size n are split in two subsamples of sizes n1 and n2, in such a way that, the n1 items of the first subsample are submitted to a go/no-go gauge test, and the n2 items of the second subsample are measured. The items of the first sample are classified as defective or non-defective, if the number of non-defectives exceeds a threshold, the second subsample is also inspected, but now the items of the second subsample are submitted to the variable-type inspection, ending up with an 𝑋 value used to decide the state of the process. The
np X X chart with n1=2, n2=4, and %𝑋 = 25%, that is, with in-control average sample size (ASS) of 3, is defeated by the T-ATTRIVAR chart with n=3 and %𝑋 = 20%, see Table 5. We built Table 5 with the ARLs of the X rec chart published by Quinino et al. (2015). In Sampaio et al. (2014), only upward shifts were considered to calculate the ARLs of the np X X control chart. So, for fair comparisons between its performance and the performance of the T-ATTRIVAR chart, Table 5 was built with the ARLs of the two-sided np X X chart, that is, the version of the np X X chart for upward and downward shifts detection.
Table 5: Comparing the T-ATTRIVAR, X rec and np X X charts T-ATT
T-ATT
T-ATT
3 371.54
np X X 3 (=ASS) 370.33
3 500.53
np X X 3 (=ASS) 500.16
δ\n 0
5 370.60
X rec 6 370.03
0.25
133.57
157.65
182.77
204.26
239.67
263.73
0.5
33.55
45.30
60.11
68.49
75.89
83.41
1
4.52
6.65
9.70
9.78
11.43
10.86
1.5
1.57
2.12
2.90
2.91
3.21
3.03
2
1.08
1.24
1.47
1.59
1.55
1.60
Journal Pre-proof 5. Illustrative example of the T-ATTRIVAR chart A numerical example was presented by Ho and Aparisi (2016) to illustrate the use of the BATTRIVAR chart. They adopted in-control observations normally distributed with 𝜇0 = 1000, 𝜎0 = 20, samples of size 𝑛 = 5, %𝑋𝑚𝑎𝑥 = 5%, and a process mean shift of one standard deviation (𝛿 = 1). The optimum solution presented by their GA was: 𝐿𝐷𝐿 = 971.18, 𝑈𝐷𝐿 = 1028.82, 𝑊𝑦 = 3, 𝐶𝐿𝑦 = 5 , 𝐿𝐶𝐿 = 976.0, 𝑈𝐶𝐿 = 1024.0 and %𝑋 = 2.66%, leading to an ARL=5.69. However, working with the maximum %𝑋 specified, that is, with %𝑋 = 5%, 𝐿𝐷𝐿 = 973.76, 𝑈𝐷𝐿 = 1026.24, 𝑊𝑦 = 3, 𝐶𝐿𝑦 = 5, 𝐿𝐶𝐿 = 976.0, 𝑈𝐶𝐿 = 1024.0, the ARL decreases around 10% (from 5.69 to 5.15); the GA algorithm was unable to discover that. According to Table 6, the B-ATTRIVAR chart signals faster when %𝑋 = 5%; additionally, Table 6 shows that the T-ATTRIVAR chart always defeats its competitor, the B-ATTRIVAR chart. With the input conditions (ARL0=370.4; %𝑋𝑚𝑎𝑥 = 5%; =1), the optimum parameters of the T-ATTRIVAR chart are 𝐿𝐷𝐿 = 968.62, 𝑈𝐷𝐿 = 1031.38, 𝑊𝑦 = 2, 𝐶𝐿𝑦 = 5, 𝐿𝐶𝐿 = 973.57 and 𝑈𝐶𝐿 = 1026.43. These optimum values are free of uncertainties, once the optimization worked with the exact ARLs, thanks to the construction and use of the X | YD distributions.
Table 6: The ARLs of the numerical example δ 0.25 0.5 1 1.25 1.5 2
(B/T) - ATTRIVAR Chart ( %𝑋) T (5%) B (2.66%) B (5%) 134.67 145.98 143.27 34.11 39.60 37.54 4.63 5.69 5.15 2.46 3.00 2.71 1.61 1.90 1.74 1.09 1.18 1.13
With the T-ATTRIVAR chart in use, samples of size 𝑛 are regularly taken from the production line and the n items are classified in three excluding categories by a go/no-go gauge test; the items with 𝑋 ≤ 𝐿𝐷𝐿 are reproved by the no-go gauge test and goes to category A, the items with 𝑋 ≥ 𝑈𝐷𝐿 are reproved by the go gauge test and goes to category B, the remaining items with 𝐿𝐷𝐿 < 𝑋 < 𝑈𝐷𝐿 goes to category AB. The number of sample items in each of the three categories are 𝑌𝐴, 𝑌𝐵, and 𝑌𝐴𝐵.
Journal Pre-proof Table 8 presents the 𝑌𝐴, 𝑌𝐵, and 𝑌𝐴𝐵 values of 24 samples of size five. The go no-go gauge test was built to attend the two discriminant limits: (𝐿𝐷𝐿 = 968.62; 𝑈𝐷𝐿 = 1031.38); it was assumed an in-control X~ N (1000; 20). The attribute control chart limits are 𝑊𝑦 = 2 and 𝐶𝐿𝑦 = 5, and the variable control chart limits are 𝐿𝐶𝐿 = 973.57 and 𝑈𝐶𝐿 = 1026.43. Fig. 6 presents the T-ATTRIVAR chart with the 𝑌𝐷 and 𝑋 values of the 24 samples. The monitoring statistic 𝑌𝐷 is a function of the 𝑌𝐴 and 𝑌𝐵, see Tables 7 and 8.
Table 7: The 𝑌𝐷 as a function of the 𝑌𝐴 and 𝑌𝐵 YA
YB
YD
See samples
Process decision
=0
=0
=0
>0
>0
=0
=0
< Wy
= YB
1, 4, 7, 8, 13, 15, 17, 18, 19, 22 2, 10, 16, 21, 23 6, 11, 20
< Wy
=0
= YA
3, 9, 12, 14
=0
≥ Wy < CLy
= YB
24
≥ Wy < CLy
=0
= YA
5
depends on the 𝑋
=0
≥ CLy
= YB
-
out of control
≥ CLy
=0
= YA
-
𝑋
Process decision
𝑋 ≥UCL or 𝑋 ≤ LCL LCL<𝑋 and 𝑋 < UCL
out of control
in control
in control
Table 8: The YD and 𝑋values # 1 2 3 4 5 6 7 8 9 10 11 12 13
YA 0 1 1 0 2 0 0 0 1 1 0 1 0
YB 0 1 0 0 0 1 0 0 0 1 1 0 0
YAB 5 3 4 5 3 4 5 5 4 3 4 4 5
YD 0 0 1 0 2 1 0 0 1 0 1 1 0
𝑋 985.61 -
# 14 15 16 17 18 19 20 21 22 23 24
YA 1 0 1 0 0 0 0 2 0 1 0
YB 0 0 1 0 0 0 1 1 0 1 4
YAB 4 5 3 5 5 5 4 2 5 3 1
YD 1 0 0 0 0 0 1 0 0 0 4
𝑋 1033.62
Journal Pre-proof Sample 24, where 𝑋 =1033.62 > 𝑈𝐶𝐿, leads to an alarm that an assignable cause is increasing the process mean; discarding the case of being a false alarm.
YD
6 5
1026.43
4 3 973.57
2 1 0 1
6
11
16 # Sample
21
26
Fig. 6. The T-ATTRIVAR chart with observations of twenty-four samples
6. Conclusions In this paper, we discussed the design of the B-ATTRIVAR chart, including the optimum designs presented by the GA developed by Ho and Aparisi (2016). We were able to find designs that lead to lower ARLs, that is, to lower delays with which the B-ATTRIVAR chart signals. For several scenarios, the GA optimization approach generated designs with corresponding ARLs that are more than 10% higher than the lowest ones we found in our studies with the computation of the exact ARLs. The trinomial version of the ATTRIVAR chart proved to be more efficient than the binomial version, especially when 𝑛 = 5 and less than 15% of the in-control samples are submitted to the variable inspection, or when 𝑛 = 3 and more than 15% of the in-control samples are submitted to the variable inspection. A suggestion of future work for this research is the investigation of the efficiency of a new ATTRIVAR control chart in which only the disapproved items are submitted to the variable-type inspection. This idea was proposed by one of the referees as mentioned in section 2.
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Acknowledgements The authors would like to thank the referees for their valuable comments and suggestions. The research for this article was supported by FAPESP – Fundação de Amparo à Pesquisa do Estado de São Paulo, project 2018/07147-0 and by CNPq – Conselho Nacional de Desenvolvimento Científico e Tecnológico, projects 306671/2015-0 and 304599/2015-8.
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Journal Pre-proof Mosquera, J., Aparisi, F., Epprecht, E.K., 2018. A global scheme for controlling the mean and standard deviation of a process based on the optimal design of gauges. Qual. Reliab. Eng. Int. 34, 718–730. Quinino, R.C., Lee Ho, L., Galindo Trindade, A.L., 2015. Monitoring the process mean based on attribute inspection when a small sample is available. J. Oper. Res. Soc. 66, 1860–1867. Quinino, R.C., Bessegato, L.F., Cruz, F.R.B., 2017. An attribute inspection control chart for process mean monitoring. Int. J. Adv. Manuf. Technol. 90, 2991–2999. Sampaio, E.S., Ho, L.L., de Medeiros, P.G., 2014. A Combined npx−X¯ Control Chart to Monitor the Process Mean in a Two-Stage Sampling. Qual. Reliab. Eng. Int. 30, 1003–1013. Wu, Z., Jiao, J., 2008. A control chart for monitoring process mean based on attribute inspection. Int. J. Prod. Res. 46, 4331–4347. Wu, Z., Khoo, M.B.C., Shu, L., Jiang, W., 2009. An np control chart for monitoring the mean of a variable based on an attribute inspection. Int. J. Prod. Econ. 121, 141–147. Zhou, W., Liu, N., Zheng, Z., 2019. A synthetic control chart for monitoring the small shifts in a process mean based on an attribute inspection. Commun. Stat. - Theory Methods 1–16.
Journal Pre-proof Authors’ Contributions
Felipe Domingues Simões: Software, Validation, Formal analysis, Investigation, Resources, Writing - Original Draft; Antonio Fernando Branco Costa: Conceptualization, Methodology, Supervision, Funding acquisition; Marcela Aparecida Guerreiro Machado: Project administration, Writing - Review & Editing, Visualization.
Journal Pre-proof Highlights
Scripts developed to calculate exact Average Run Lengths (ARLs) for ATTRIVAR charts Designs that lead to lower ARLs were found for the Binomial ATTRIVAR chart The Trinomial ATTRIVAR chart proposed is more efficient than the binomial version