3.1. Data and variables
The data were taken from a privately-owned ceramic factory. The research includes 14 variables determined by the expert opinion. A data set including 1000 cases of these variables were used. The variables are various factors that affect the ceramic tile production or various materials that are used during the production of the ceramic tiles. These variables are size, glaze density, belt speed, press surface moisture, glaze weight, engobe weight, engobe density, clay type, printing color, drying temperature, firing time, firing temperature, moisture after drying, and shift variables. In the study, the effects of these factors on the 12 types of defects, which occur during the production of the ceramic tiles, were examined. These defect types are burr, crack, drip, deformation, pinhole, pitting, glaze blistering, color tone, black spot, plucking, tear, and surface defects. Glaze is a substance that is applied on ceramics to add shine, and engobe can be defined as a layer, which exists between clay and glaze and plays a role in hiding the color of the underlying clay. The variables used in this study are listed in Table 1 with their levels and explanations.
Table 1
Variables used in the study, their explanations, and levels
Variable | Definition | Levels |
Size | Indicates the two types of ceramic tile sizes | Wall tile Floor tile |
Glaze Density | Indicates the density of the glaze substance on the ceramic tiles (g/l) | 0-1780 1780–1800 1800 and over | Low Moderate High |
Band Speed | Indicates the number of ceramic tiles passing on the production band per minute (number/min) | 0–50 50 and over | Low High |
Press Surface Moisture | Indicates the amount of moisture at the surface of the press during the pressing process (%) | 0-0.05 0.05–0.06 0.06 and over | Low Moderate High |
Glaze Weight | Indicates the weight of the glaze substance on the ceramic tiles (g) | 0–64 64–80 80 and over | Low Moderate High |
Engobe Weight | Indicates the weight of the engobe substance on the ceramic tiles (g) | 0–65 65–70 70 and over | Low Moderate High |
Engobe Density | Indicates the density of the Engobe substance on the ceramic tiles (g/l) | 0-1760 1760–1810 1810 and over | Low Moderate High |
Clay Type | Indicates the two different clay types used in the production | Dry grinding Spray |
Printing Color | Indicates the two main printing color types used to color the ceramic tiles | Light colored Dark colored |
Drying Temperature | Indicates the temperature that the ceramic tiles are subjected to during the drying process (°C) | 0-255 255–267 267 and over | Low Moderate High |
Firing Time | Indicates the duration of time when the ceramic tiles are kept in the kiln during the firing process (min) | 0–40 40 and over | Low High |
Firing Temperature | Indicates the temperature that the ceramic tiles are subjected to in the kiln during the firing process (°C) | 0-1118 1118–1124 1124 and over | Low High High |
Moisture After Drying | Indicates the percentage of the moisture in the ceramic tiles after leaving the drying unit (%) | 0-0.40 0.40–0.55 0.55–0.80 | Low Moderate High |
Shift | Indicates the shift hours in the factory | 16:00–24:00 24:00–08:00 08:00–16:00 |
Defect Type | Indicates the 12 defect types observed on the ceramic tiles | Burr Crack Drip Deformation Pinhole Pitting Glaze blistering Color tone Black spot Plucking Tear Surface defect |
3.2. The constructed logistic regression model
The logistic regression model was built by using the 12 variables presented in Table 1. Weka (2021) software was employed to construct the logistic regression model of the defect type variable on the independent variables, which are size, glaze density, belt speed, press surface moisture, glaze weight, engobe weight, engobe density, clay type, printing color, drying temperature, firing time, firing temperature, moisture after drying, and shift.
Receiver operating characteristic (ROC) curves can be used to measure the estimation performances of the constructed models (Hand, 1997). Area under the curve (AUC) value, which can be defined as the area under the ROC curve, is a score ranging from zero to one. If the AUC value is one, the estimation is error-free. The estimation performance of the logistic regression model, for each type of defect, is presented in Table 2 with the corresponding AUC scores sorted in a descending order.
Table 2
The AUC scores of the constructed logistic regression model
Defect type | AUC Scores |
Deformation | 0.67 |
Glaze blistering | 0.63 |
Drip | 0.63 |
Pinhole | 0.62 |
Plucking | 0.62 |
Tear | 0.60 |
Black spot | 0.58 |
Crack | 0.53 |
Burr | 0.52 |
Pitting | 0.51 |
Color tone | 0.48 |
Surface defect | 0.46 |
When the AUC scores in Table 2 are examined, it is seen that the logistic regression model estimated the probability of the deformation defect occurrence with the highest rate of 67%. The probabilities of the glaze blistering and the drip defects were estimated with the second highest success level of 63% for each. The color tone defect and the surface defect, however, are the least successfully estimated defect types with the rates of 48% and 46% respectively.
3.3. The constructed Bayesian network model
To examine the relationships among the factors, the materials, the ceramic types, and the ceramic defect types, three different Bayesian network models were built in GeNIe (2019) software applying Bayesian Search, Peter-Clark, and Greedy Thick Thinning algorithms. Model selection can be made with the LL values obtained by Eq. 4. The algorithms experimented and the corresponding log-likelihood values are given in Table 3.
Table 3
Log-likelihood (LL) values
Algorithm | Log-likelihood (LL) |
Bayesian Search | -7340.11 |
PC | -7705.80 |
Greedy Thick Thinning | -8246.99 |
It is seen that the largest LL value in Table 3 belongs to Bayesian Search algorithm as -7340.11. Thus, the Bayesian network estimated by the Bayesian Search algorithm was selected as the model to be used in the analysis. The Bayesian network model, which was constructed with the help of GeNIe (2019) software, was re-arranged in Netica (2019) software and given in Fig. 2. The scores behind the bars are percentages, which show the probabilities of the corresponding levels.
To evaluate the estimation performance of the constructed Bayesian network for each type of defect, the calculated AUC scores are presented in Table 4 in a descending order.
Table 4
The AUC scores of the constructed Bayesian network model
Defect type | AUC Scores |
Crack | 0.86 |
Pinhole | 0.86 |
Color tone | 0.86 |
Glaze blistering | 0.83 |
Plucking | 0.82 |
Surface defect | 0.82 |
Burr | 0.81 |
Deformation | 0.81 |
Drip | 0.80 |
Black spot | 0.80 |
Tear | 0.79 |
Pitting | 0.77 |
Given the AUC scores in Table 4, the constructed Bayesian network model estimated the crack, the pinhole, and the color tone defects with the highest probability of 0.86 for each. The next most accurate estimate belongs to the glaze blistering defect with a success rate of 0.83. The third most successful estimate was made for the plucking defect and the surface defects with the rate of 0.82 for each. The pitting defect, however, was the least accurately estimated type of defect with the lowest probability of 0.77. In general, it is possible to conclude that the performance of the Bayesian network model to estimate the defect types is at a satisfactory level for all the defect types.
To make a comparison between the estimation performances of the constructed Bayesian network and the logistic regression models, the AUC scores of both models are presented in Table 5 side by side.
Table 5
AUC scores of the Bayesian network model and the logistic regression model
| AUC Scores |
Defect type | The Bayesian network model | The Logistic Regression model |
Crack | 0.86 | 0.53 |
Pinhole | 0.86 | 0.62 |
Color tone | 0.86 | 0.48 |
Glaze blistering | 0.83 | 0.63 |
Plucking | 0.82 | 0.62 |
Surface defect | 0.82 | 0.46 |
Burr | 0.81 | 0.52 |
Deformation | 0.81 | 0.67 |
Drip | 0.80 | 0.63 |
Black spot | 0.80 | 0.58 |
Tear | 0.79 | 0.60 |
Pitting | 0.77 | 0.51 |
When the AUC scores presented in Table 5 are examined, it is possible to see that the AUC scores of the Bayesian network model are considerably higher than the ones of the logistic regression model for every defect type. Thus, it is possible to conclude that the Bayesian network model performed quite better than the logistic regression model in estimating the probabilities of the defect types. Moreover, unlike the logistic regression model, the Bayesian network model also has the superiority of examining the bilateral or multilateral dependencies among the variables simultaneously, which allows a more complex analysis.
3.4. Sensitivity analysis
Entropy, which is commonly used in information theory, was introduced by Shannon (1948). In Bayesian networks, it is possible to perform a sensitivity analysis to determine to what extent the levels of any variable are affected by the change in the levels of other variables. Entropy score, which shows the irregularity of a system, can be used for sensitivity analysis. With the help of entropy score, it is possible to see how likely an unexpected situation will occur in a system.
Let \(X\) be a discrete random variable with possible values\({ x}_{1},\dots ,{x}_{n}\) and \(P\left(X\right)\) be its probability mass function. Then, entropy score \(H\left(X\right)\) can be calculated with the following equation.
\(H\left(x\right)=-\sum _{i=1}^{n}P\left({x}_{i}\right){log}_{b}P({x}_{i}\) ) (8)
where \(b\) is the base of the logarithm used. The entropy scores for the levels of the defect type variable with respect to the other variables are given in Table 6.
Table 6
The entropy scores of the factors causing the defects on the ceramic tiles
Variables | Entropy |
Firing Temperature | 0.03670 |
Drying Temperature | 0.03549 |
Press Surface Moisture | 0.03372 |
Moisture After Drying | 0.03303 |
Printing Color | 0.01940 |
Glaze Density | 0.01325 |
Firing Time | 0.01285 |
Size | 0.01059 |
Glaze Weight | 0.00869 |
Band Speed | 0.00862 |
Engobe Density | 0.00822 |
Clay Type | 0.00753 |
Engobe Weight | 0.00748 |
Shift | 0.00006 |
Table 6 presents the factors, which are the levels of the defect type variable, that cause various defects on the ceramic tiles and the corresponding entropy scores sorted in a descending order. According to the entropy values, the defects on the ceramic tiles are mostly affected by the firing temperature. The second most effective factor appears to be the drying temperature, while the third important factor is emerging as the press surface moisture. The finding that the firing temperature is the most effective factor causing the defects agrees with the finding by Jin et al. (2017) that the most important factor leading to shell deformation on industrial ceramics is firing temperature.
3.5. Analyses regarding the defect types
Detailed analyses of the results obtained from the constructed Bayesian network model are presented in the following lines for each type of defect.
3.5.1. Analysis results regarding the burr defect
The conditions where burr defect is observed at the highest rate appear to be the low levels of the moisture after drying, the high levels of the firing temperature, and the high levels of the press surface moisture. It is observed that the probability of the burr defect occurrence increases in the production conditions, where the engobe weight is at the medium levels and the engobe density is used at high levels. The burr defect is more commonly seen on the light-colored ceramics and the ceramics produced using spray type of clay. To reduce the occurrence probability of the burr defect, the optimal firing times can be suggested as the optimal firing temperatures can be suggested as 0-1118°C and the optimal drying temperatures as 255–267°C.
3.5.2. Analysis results regarding the crack defect
The crack defect is the third most common type of defect seen on the ceramic tiles, and it is more likely to be observed, when the high firing temperatures and the high levels of the moisture after drying variable are used. The high levels of firing time and the high levels of glaze weight are also determined as the factors increasing the occurrence of the crack defect. In contrast, the low levels of the press surface moisture and the moderate levels of the engobe weight increase the emergence of the crack defect. The crack defect was found to occur on the light-colored ceramics more frequently. Likewise, when using the spray type of clay, the risk of the crack defect occurrence increases. The most probable (69.8%) reason that played a role in the formation of the crack defect was found to be the firing time 40 min or over. To reduce the probability of the crack defect occurrence, the optimal firing temperatures can be suggested as 0-1118°C, and the optimal drying temperatures as 0-255°C.
3.5.3. Analysis results regarding the drip defect
It was determined that the high levels of the moisture after drying, the firing time, the glaze density, and the engobe density are the factors that increase the probability of the drip defect occurrence. However, the medium levels of the firing temperature, the drying temperature, the engobe weight, and the press surface moisture play a role in increasing the likelihood of the drip defect occurrence. The drip defect, unlike the other defect types, is more likely to occur at the low levels of the glaze weight. Unlike the burr and the crack defects, the drip defect is more likely to be observed in production conditions including dry grinding type of clay. Moreover, it appears that the drip defect is more likely to occur on the light-colored ceramic tiles, as well as the burr and the crack defects. It is understood that the most likely (56.6%) reason for the occurrence of the drip defect is the use of dry grinding clay type. To reduce the probability of occurrence of the drip defect, the optimal firing temperatures can be suggested as 0-1118°C and the optimal drying temperatures as 0-255°C.
3.5.4. Analysis results regarding the deformation defect
When the factors leading to the deformation defect were analyzed, it was found that the high levels of the firing temperature, the moisture after drying, the glaze density, the engobe weight, and the glaze weight variables had effects increasing the deformation defect. On the other hand, the low levels of the firing time, the engobe density, the press surface moisture, and the glaze density variables were found to increase the deformation defect occurrence. It is also observed that the deformation defect is more likely to occur on the light-colored tiles and on the tiles produced using spray type of clay. Additionally, it was revealed that the most probable cause of the deformation defect is to keep the firing temperatures at the levels of 1124°C and over. To reduce the probability of the deformation defect occurrence, the optimal firing temperatures can be suggested as 0-1118°C and the optimal drying temperatures as 255–267°C.
3.5.5. Analysis results regarding the pinhole defect
When the analysis results related to the pinhole defect are examined, it is possible to say that pinhole is the most common defect type observed on the ceramic tiles. The pinhole defect is more likely to occur at the high levels of the firing temperature, the firing time, the drying temperature, and the engobe density factors. However, the pinhole defect is observed more frequently at the low levels of the moisture after drying variable. Unlike the moderate levels of the other factors, the probability of the pinhole defect increases, when the engobe weight is used at its moderate levels 60–70 g. The pinhole defect tends to occur at approximately equal rates, when the tiles are produced using dry grinding or spray type of clay. It was found that the most likely (59.8%) cause of the occurrence of the pinhole defect is the high levels (267°C and over) of the drying temperature variable. Finally, to reduce the probability of the pinhole defect, the optimal levels of the firing temperature and the drying temperature can be suggested as 0-1118°C and 0-255°C respectively.
3.5.6. Analysis results regarding the pitting defect
It was observed that the most probable factors that increase the probability of the pitting defect are the high levels of the firing temperature and the firing time variables. Additionally, keeping the moisture after drying variable at the low levels slightly increases the probability of the pitting defect. On the other hand, the low levels of the moisture after drying variable, and the medium levels of the engobe weight variable are the only factors increasing the probability of the pitting defect. Like the pinhole defect, the pitting defect occurs with approximately equal probabilities, when dry grinding or spray type of clay is used during the production. Moreover, the most probable (64.9%) cause of the pitting defect is to keep the ceramic tiles in the kiln for durations 40 min and over. Finally, to reduce the probability of the pitting defect occurrence, the optimal firing temperatures can be suggested as 0-1118°C and the optimal drying temperatures as 0-255°C.
3.5.7. Analysis results regarding the glaze blistering defect
Unlike other defect types, the glaze blistering defect is the only defect type that is more likely to be observed on the dark colored ceramic tiles. The high levels of the firing temperature, the engobe density, the glaze density, and the drying temperature variables increase the probability of the glaze-blistering defect. The variables that increase the probability of the glaze blistering defect occurrence at their moderate levels were determined as the engobe weight and the press surface moisture. The Glaze blistering defect is also one of the defect types that occurs approximately equally in the use of dry grinding or spray type of clay. The most likely cause of the glaze blistering defect is the firing times of 40 min and over with a probability of 59.6%. To reduce the occurrence of the glaze blistering defects, the optimal firing temperatures can be suggested as 0-1118°C and the optimal drying temperatures as 0-255°C.
3.5.8. Analysis results regarding the color tone defect
According to the results of the analysis, the color tone defect is the second most common defect type observed on the ceramic tiles. The main reasons that increase the probability of the occurrence of the color tone defect are the high levels of the firing temperature, the glaze weight, and the firing time variables. The low levels of the moisture after drying, the engobe density, the engobe weight, and the drying temperature variables also increase the color tone defect. Additionally, the color tone defect is more likely to occur on the light-colored ceramic tiles. The most probable (56%) cause of the color tone defect is the use of spray type clay during the production. To reduce the probability of the color tone defect, the optimal firing temperatures can be suggested as 1118–1124°C and optimal drying temperatures as 0-255°C.
3.5.9. Analysis results regarding the black spot defect
Considering the variables leading to the black spot defect, it was found that the high levels of the moisture after drying, the firing temperature, the firing time, the press surface moisture, and the engobe density variables had an increasing effect on the probability of this defect type. The only factor whose moderate levels increase the probability of the black spot defect is the engobe weight variable. However, unlike the other defect types, it turned out that the low levels of any variables did not lead to a black spot defect. It is also seen the black spot defect is more likely to be observed on the light-colored ceramics. Firing times 40 min. and over are the most probably cause (57.8%) of the black spot defect with a probability of 57.8%. To reduce the probability of the black spot defect, the optimal firing temperatures can be suggested as 0-1118°C and the optimal drying temperatures as 0-255°C.
3.5.10. Analysis results regarding the plucking defect
Analysis of the factors leading to the plucking defect revealed that the high levels of the moisture after drying, the firing temperature, the firing time, the press surface moisture, the glaze weight, and the drying temperature variables had an increasing effect on the probability of observing this defect type. Just like the black spot defect, the moderate levels of the engobe weight variable increase the probability of the plucking defect. In addition, the plucking defect is also more likely to be observed on the light-colored ceramics. The probabilities of the two most likely factors causing the plucking defect are close to each other. These factors are the use of spray type of clay and the high levels of the press surface moisture, with probabilities 59.4% and 57.6% respectively. To reduce the occurrence rate of the plucking defect, the optimal firing temperatures can be suggested as 1118–1124°C and the optimal drying temperatures as 0-255°C.
3.5.11. Analysis results regarding the tear defect
When the factors leading to the tear defect are examined, it can be concluded that the high levels of the glaze density and the engobe density have an increasing effect on the probability of this defect type. Moreover, the moderate levels of the engobe weight and the moisture after drying variables also appear to be among the reasons increasing the tear defect probability. Unlike the other defect types, the tear defect is more likely to occur at the low levels of the firing temperature. Additionally, this defect type is also more likely to be observed on the light-colored ceramics. Use of dry grinding type of clay is another factor increasing the formation of the tear defect. The most likely (50.6%) cause of the tear defect is 40 min and over durations of the firing time. To reduce the probability of the tear defect, the optimal firing temperatures can be suggested as 1118–1124°C and the optimal drying temperatures as 255–267°C.
3.5.12. Analysis results regarding the surface defect
It is understood that the high levels of the variables the firing temperature and the firing time increase the probability of the surface defect. However, the moderate levels of the engobe weight and the drying temperature variables increase this kind of defect. In addition, the only factor whose low levels increase the probability of observing the surface defect appears to be the moisture after drying variable. The surface defect is also more likely to occur on the light-colored ceramics. Another result of the analysis is that the reason with the highest probability (67.6%) for the appearance of surface defect is again the firing times of 40 min. and over. To reduce the probability of the surface defect, the optimal firing temperatures can be suggested as 0-1118°C and the optimal drying temperatures as 0-255°C.
In addition to the analyses and the suggestions provided for each defect type above, some general comments and suggestions for all the defect types can be given as follows.
According to the findings obtained from the analyses, it was observed that the high levels of the band speed significantly increased the probability of every defect type except the deformation defect. Although, the clay type factor has partial effects on the defects in general, spray type of clay appears to cause more defects than dry grinding type of clay. Another result shows that light colored ceramics are more exposed to all kinds of the defects, except for the glaze blistering. This difference is especially evident in the black spot, the color tone, the deformation, the drip, and the burr defects. The effect of the shift hours in the factory, however, was found to be insignificant on the occurrence of the defects. Additionally, the high levels of the glaze density increase all the defect types but the deformation defect. As far as the firing temperature is concerned, it quite significantly increases all the defects except the tear defect. Firing time variable gives similar results too. The high levels of the engobe weight and the engobe density variables were found to be increasing the defects in general. In addition, the high levels of the glaze density also increase the defects. Speaking of the high levels of the firing temperature, the firing time, the engobe weight, and the engobe density variables, they are the factors that increase the occurrence most of the defects. Moreover, it was observed that the high levels of the drying temperature variable increase some of the defects, yet they have an adverse effect on some defect types. While the effects of the clay type on the defects were observed to be limited, it was seen that there is no significant influence of the shift variable on the defect types. In general, it is possible to comment that the light-colored ceramic tiles are more exposed to the defects than the dark colored tiles. Finally, based on the findings summarized above, it can be said that it would be a logical approach not to keep the belt speeds at the high levels to prevent the defects. In addition, it should not be preferred to keep the ceramic tiles in kilns at the high temperatures and for long times. Choosing the firing temperatures and the firing times optimally, will be effective in reducing the defects. The same interpretation can be made also for the glaze weight variable.