In order to quantify the optimal model we need to design an objective function that combines and represents our criteria. We are interested in finding a model with high linearity (l), moderate state compactness (sc), low cross state similarity (css) and includes only important states (si). The idea of weighted parameters is widely used in multi-objective function optimisation [133] which is in the form of
maxPn
i=1wi∗ fi(x), where w is the weight and f (x) is the desired function
Adopting this principle can help us in designing our multi-objective function especially when the goal is not for finding a global optimum solution.
We have developed the following multi-objective functions to find the most desirable model for process abstraction. Two types for optimisation are developed which are soft optimisation and strict optimisation.
1- Soft optimisation uses multi-objective function that is presented in the Equation(5.5) for optimising candidate models space. This function represents only unconstrained criteria; linearity, state compactness and cross state similarity. It has a tolerance toward unimportant states. The purpose of this flexibility is to pick a model that represents the mainstream pro- cess regardless how many cases related in each state, hence model selected using this type of optimisation may hold a level of detail about the process.
f (1) = max{ 2 (li) | {z } Term2 −1 2(sci) − cssi | {z } Term1 : i = 1..n} (5.5)
2- Strict optimisation uses multi-objective function that is showed in the Equation(5.6) for optimising candidates model space. This function represents both constrained and uncon- strained criteria; linearity, state compactness, cross state similarity and state importance. As its name means, this type of optimisation takes state coverage into consideration and tends to penalize model with unimportant states. This function is developed to pick the best model that
99 5.6. Designing the multi-objective optimisation function
represents the mainstream process without detail about the process.
f (2) = max{1 − sii | {z } Term3 [2 (li) | {z } Term2 −1 2(sci) − cssi] | {z } Term1 : i = 1..n} (5.6)
state importance here is normalized using the following simple normalisation method:
normalized state importance (si) = original value - minimum valuemaximum value - minimum value
The proposed multi-objective functions have main terms:
Term1 consists of two functions which are state compactness and cross state similarity. State compactness decreases as we get high similarity inside states which leads to low state variance. knowing that state compactness keeps decreasing has derived us to reduce the weight of state compactness to half. This helps in penalising a model of having very low state variance. Cross state similarity decreases as we get less overlapping processes between states which is preferable. Term2 is linearity which increases as we get a more linear model and this rewards a model for providing better segmentation/partitioning. It is multiplied by 2 to ensure equal forces in the space. For more clarification, choosing this weight for linearity is a result of several empirical investigations as will be explained in the next section.
Term3 further penalises a model that has unimportant states.
This final version of the strict and soft multi-objective optimisation is a result of several attempts to understand each function individually and how all criteria may affect on each other.
5.6.1
Steps for designing the proposed multi-objective function
Before demonstrating our steps for designing the multi-objective function, it is important to take into consideration the property of each criteria and the relations between them. This has helped in criteria weightings and explaining how the criteria could contribute in a trade-off relation.
Criteria trade-off and weightings
Finding a solution in search space requires finding a balance between conflict criteria. This is because a model with high linearity is unlikely to have high state compactness or high dis- similarity between states. The weighting here focuses on unconstrained criteria since state importance will be treated as a penalty term after optimising the space using soft optimisation.
Figure 5.7: Process abstraction model and different forces
We have applied simple heuristic for criteria weighting tuning which has done using our synthetic logs and real data of colorectal cancer log. The following presented figures are generated from the colorectal cancer event log to avoid redundancy, however, other synthetic event logs that were presented in Chapter 4 have shown the same behaviour of criteria tuning. The calculation of the criteria using colorectal cancer data is reported in Table 5.1.
Table 5.1: Criteria calculation of colorectal cancer event log.
Model 1 2 3 4 5 6 7 8 9 10 11 12 13 14 States 2s 3s 4s 5s 6s 7s 8s 9s 10s 11s 12s 13s 14s 15s Linearity 0.99 0.75 0.78 0.79 0.78 0.72 0.71 0.71 0.65 0.74 0.73 0.66 0.67 0.65 compactness 0.76 0.42 0.44 0.26 0.28 0.23 0.18 0.16 0.15 0.15 0.35 0.32 0.33 0.28 cross sim. 0.89 0.84 0.76 0.50 0.47 0.44 0.45 0.51 0.43 0.47 0.47 0.47 0.43 0.42
During weights tuning process, we aimed to investigate the behaviour of state compactness and cross state similarity with linearity. Basically, combining state compactness and linearity in one function shows the dominance of the compactness over linearity where the model of the maximum value is the model of the best compactness score as shown in Figure 5.8.
f0= max{linearityi− state compactnessi: i = 1..n} (5.7)
Figure 5.8: State compactness and linearity in colorectal cancer log Hence, we have minimised the weight of state compactness to the half as below;
f1= max{linearityi−
state compactnessi
101 5.6. Designing the multi-objective optimisation function
Figure 5.9: State compactness and linearity after weighing
It can be clearly seen in Figure 5.9, that linearity has started to affect on function behaviour where model number 4, which has high linearity, seems to be equivalent to model with high compactness.
On the other hand, the relation between cross state similarity and linearity without weights has provided a balance between their forces where no one of these criteria has a dominance effect on the function behaviour.
f2= max{linearityi− cross states similarityi: i = 1..n} (5.9)
Figure 5.10: Cross state similarity and linearity criteria
The best model as presented in Figure 5.10, is model number 5, which is free of controlling by linearity or cross state similarity thus, there is no need to tuning their weights.
Investigating the behaviour of our multi-objective function after involving the three criteria has shown the dominance of the compactness again. This result seems to be plausible since the equation includes two forces which are compactness and cross state similarity and both pull the selection of the best model towards the right, high number of state as presented in Figure 5.11
f3= max{linearityi− 1
Figure 5.11: The three criteria in one function
Therefore, increasing the weight of linearity is suggested to provide a balance point in the space. Hence, the candidate space now has two forces (compactness and cross state similarity) that encourage the selection to go to high number of states and one force (linearity) with double weight to prefer the model of less number of states, see the following equation.
f4= max{2∗linearityi− 1
2∗(state compactnessi)−cross states similarityi : i = 1..n} (5.11)
Figure 5.12: Fitted weights for all criteria
Modifying the weight of linearity as suggested above has helped our multi-objective function to select a balanced model that is non-dominant by one of the criteria and represents resistance to all forces as illustrated in Figure 5.12.
This equation is the proposed multi-objective function which guarantees non of the criteria is dominant and assumes a balance of different forces is found.