The Bayesian network has been used in various cases to represent the knowledge in the corresponding domain. For example Del Águila and Del Sagrado (2012) used BNs to create a decision support system in software engineering domain, Gupta and Pedro (2004) developed a Bayesian network to represent the human common sense in linguistics for humanoid robots to be able to respond to situations and Sedki and Beaufort (2012) developed a method that uses cognitive mapping to produce a Bayesian network for representing the knowledge in a domain.
In the field of Additive manufacturing, Wang et al. (2018) developed a knowledge man- agement system in which they developed a BN to model the knowledge in the domain of AM to help designers in early-stage design steps. Their model consists of an overview model and a detailed information layer, and they used the knowledge and data available in the literature to form the structure of the model and learn the parameters of the model. The model in this study contains and visualizes the experts’ knowledge in the domain trough probability distributions of the independent nodes and the structure of the network. The knowledge about the system under study is also augmented into the model through the causal graph and the constraints.
A general view of the marginal probability tables of each independent and performance node is depicted in Figure 37.
Figure 37. The probability of occurrence of each interval for each node for in- dependent and performance variables
Bayesialab software demonstrates the marginal probability table for each variable as a box, which is called a monitor (Bayesialab, 2018). Each box shows a set of information, including the name of the variable on the top of the box, the mean value and the standard deviation of the probability distribution of each node and a “value” which is here it is equal to the mean value of the distributions. The marginal distribution for each interval of the independent variables is displayed with the blue bars, having the probability values on the left side of them and the names of the intervals on the right side.
The first two boxes with red frames on the upper right corner of Figure 37 shows the marginal probability tables for performance variables. For example, given the current setting for the independent variables and the constraints, there is only 5.38% chance that the final part would have a curling defect with more than 10mm. This shows that the experts’ knowledge has been extracted for the variable configuration, at least in theory,
in most of the cases will lead to having a very small defect. Although the results are mostly in the safe region, i.e., less than 0.05mm, there is still a need for models to predict the cases which have a higher amount of defect.
The other element which is reflected in the model is that given the expert’s knowledge about their preference on intervals and the constraint of the system, the marginal prob- ability for each state of each value is recalculated. For example, for the variable “Width of the part” the probability for the intervals obtained through AHP is shown in Table 18.
Table 18. Probabilities for intervals of “Width of the part” variable obtained with AHP
Width of the part Probabilities
Low 23.85%
Medium 62.50%
High 13.65%
But after setting constraints, these probabilities have changed to the values shown in Table 19.
Table 19. Probabilities for intervals of “Width of the part” variable after setting constraints
Width of the part Probabilities
Low 1.28%
Medium 75.31%
High 23.41%
This is due to the removal of the values which create unwanted combinations through constraints set for the model.
The causal graph represents dependencies and independencies between variables of the system and Bayesian inference principles simulate the interactions between varia- bles based on these relations. A simple example is demonstrated as follows, to demonstrate some of the basic interactions between the variables of the model.
Based on the description of the DACM model in the section 4.1.1, an increase in the number of supports has a positive effect, which is reducing the curling defect, and a negative effect, which is increasing the total mass of the supports. So, using the monitors for the variables “Number of Supports”, “Total mass of supports” and “Curling defect”, this sort of interaction between the nodes can be demonstrated. The default values for marginal probabilities for each of these nodes shown in the top row of Figure 38.
Figure 38. Changes in Curling defect and Mass of the supports due to changes in the number of supports
Hard evidence can be set for any of the states of this node in the monitor to show the effect of changing the number of supports. Here, the hard evidence represents choosing one of the intervals of a variable, and by doing so, the software propagates the effect of this evidence through the network and demonstrates the posterior marginal distributions for the other nodes. As an example, if in a small number of supports are chosen in the design phase without any change in the other variables, after calculating the posterior distributions, according to the model we can expect that the value for the curling defect increases by 37.98% in average and the value for the mass of the supports decrease by 77.40% in average, as shown in the second row of Figure 38. This means that if the value for the number of supports is fixed within its first interval, all combinations of all other variables will lead to a probability distribution for the performance variables with the mean values and standard deviations described in the middle row of Figure 38. On the other hand, if the number of supports increases, this should lead to less curling defect in average and more mass for the supports in average, which is evident in the last row of Figure 38. By choosing the interval with the highest values for the variable “num- ber of supports” in the monitor window and after calculation of posterior distributions, the value of curling defect decrease by 34.22% in average and the value of the mass of supports increases by 69.04% in average for all possible combinations of the other var- iables.
Changes for marginal probabilities can also be seen in both variables in Figure 38. For example, having the highest values for the variable” number of supports” increases the chance for having a curling defect in the lowest interval, i.e., less than 0.02mm, from
16.23% to 21.95% for all combinations for the other variables. This change can be observed by comparing the value in the first row of Figure 38 with the third row.