Modelling the impact size of disruptive events using intelligent machine learning

To quantify the impact sizes of disruptive events on the cost and time of the construction activities, the ANFIS, as an intelligent machine learning technique, was employed. According to Asgari et al. (2016) quantification of these factors with classical methods, such as probability analysis and influence diagrams, are very difficult. Efficient applications and quantification techniques are difficult and complex, and furthermore, exact data are required (Chen and Zhang, 2014). Unfortunately, such data relevant to this study either do not exist at all or are hard to obtain.

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Moreover, most of the classical mathematical assessment methods, such as differential equations, are not able to examine the relationship between input variables and an output variable and they are not well suited for uncertainty problems (Zhang et al., 2009).

Also, these methods are based on statistical or computing techniques, and they cannot cover qualitative data which are used in the evaluation of uncertainties. The fuzzy inference system (FIS), on the other hand, is used in modelling the qualitative aspects without employing precise quantitative analyses; it provides standard practical methods for transformation into rulebased, as well as effective methods for turning membership functions (MF) for better performance index, and it is very useful in undertaking complex problems (Carr and Tah, 2001).

As discussed, earlier ANFIS combines the strengths of the artificial neural network with fuzzy inference systems and provides uncertainties capability to handle uncertainty, nonlinearity, and complex problems an efficient method for analysing the impact of uncertainties and risks

The magnitude of influence of disruptive events was assessed using two variables, probability of occurrence and severity of the event. Figure 4.35 illustrates the developed fuzzy system for assessing the disruptive events in the study.

Figure 4.35: Fuzzy system for disruptive events assessment

The value of probability of occurrence and severity of each event on the cost and time of activity were obtained from the 5-point Likert scale closed-ended questionnaire (Database, 5x5 rulebases). In the next step, the linguistic values of probability of occurrence, the severity of the event, and the impact size of the event were converted to fuzzy values. To convert a linguistic value to a fuzzy value, triangular fuzzy numbers were used, since the probability of occurrence and time variables underlie triangular distribution (Tang and Ang, 2007). The fuzzy values and fuzzy graphical diagram of each of these linguistic variables in terms of the probability of occurrence, severity, and impact are shown in Table 4.3.

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Table 4.3: Linguistic and fuzzy values of the probability, severity, and impact size

Linguistic value Fuzzy value Fuzzy graphical diagram

Probability of occurrence Rare (-0.1, 0.1, 0.3) Unlikely (0.1, 0.3, 0.5) Possible (0.3, 0.5, 0.7) Likely (0.5, 0.7, 0.9) Almost Certain (0.7, 0.9, 1.1) Severity of event Insignificant (-1, 1, 3) Minor (1, 3, 5) Moderate (3, 5, 7) Major (5, 7, 9) Catastrophic (7, 9, 11) Impact size Minimal (0, 1, 2) Low (1, 2,3) Moderate (2, 3, 4) High (3, 4, 5) Extreme (4, 5, 6)

After converting the linguistic values of variables into the fuzzy values, the center of area (COA) method was used to defuzzify the fuzzy values into a numerical value using best non-fuzzy performance (BNP) (Chen and Lu, 2001). The deterministic values of probability of occurrence, severity, and impact size of all 76 disruptive events for cost and time calculated by BNP using Equation 4.30 is outlined in Table 4.4.

𝑀𝐴 =

𝑎𝑙+ 2𝑎𝑚+ 𝑎𝑢

4 [4.30]

Where: 𝐴̃ = (𝑎𝑙+ 𝑎𝑚+ 𝑎𝑢)

Table 4.4: Conversion value of linguistic to numerical

Probability of Occurrence Severity of Event Impact Size Linguistic value Numerical

Value Linguistic Code

Numerical

Value Linguistic Code

Numerical Value

Rare 0.1 Insignificant 1 Minimal 1

Unlikely 0.3 Minor 3 Low 2

Possible 0.5 Moderate 5 Moderate 3

Likely 0.7 Major 7 High 4

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Table 4.3 shows the numerical values of probability of occurrence, the severity of the event, and impact size based on Table 4.4 fuzzy values.

4.9.6.2 Developing ANFIS structure

The Takagi and Sugeno ANFIS model was used for a systematic approach to generating fuzzy rules from a given input-output dataset. In this study, the first-order Sugeno fuzzy inference system proposed by Takagi and Sugeno (1983) was employed to assess the impact size of disruptive events in construction cost and duration of highway projects. In this inference system, the output of each rule is a linear combination of two input variables (occurrence and severity of event) added to by a linear term of “AND” logic. The final output is the weighted average of each rule’s output (Buragohain and Mahanta, 2008). Figure 4.36 illustrates the Takagi and Sugeno ANFIS structure in 5 layers which were developed for assessing the impact of disruptive events on the cost and time of highway structures. To model each ANFIS, the following 25 fuzzy rules, “If-Then”, were considered.

𝐼𝑓 ( 𝑃 ∣∣ 𝑝𝑖)𝐴𝑁𝐷 ( 𝑃𝑆 ∣∣ 𝑠𝑖) 𝑇ℎ𝑒𝑛 𝑓𝑖= 𝑎𝑝𝑖+ 𝑏𝑠𝑖+ 𝑟𝑖 _[4.31] Where P and S are numerical inputs while pi and si are numerical variables. Membership functions identify these variables. Also, ai, bi, and ri are parameters that determine the relationship between input and output.

Figure 4.36: Proposed Takagi and Sugeno ANFIS structure

1st layer – Input layer: Designates the numerical input values, probability of occurrence (P) and severity of event (S) to the different fuzzy set.

𝜇𝑝𝑖(𝑝) 𝑖 = 1,2,3,4,5 𝜇𝑠𝑖(𝑠) 𝑖 = 1,2,3,4,5

[4.32] Where, p and s are the membership functions for fuzzy sets of probability and severity. There are eight membership functions which are shown in Tables 4.10 and 4.11.

2nd layer – Inference layer: The “AND” operator was used for achieving the output (firing strength), which shows the degree of satisfaction of each 25 fuzzy rules in a different value of two input variables. P1 Pi P5 N N Π P S P S ∑ Probability of occurrence Severity of event Impact Size S1 Si S5 Π

Layer 1 Layer 2 Layer 3 Layer 4 Layer 5

ω1

ω2

𝜔ഥ1

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𝜔𝑖 = 𝜇𝑝𝑖(𝑝). 𝜇𝑠𝑖(𝑠) 𝑖 = 1,2,3,4,5 _[4.33] 3rd layer – Implication layer: The firing strengths were normalised in this layer.

𝜔ഥ𝑖 = 𝜔𝑖 ∑ 𝜔𝑖

𝑖 = 1,2,3,4,5 _[4.34]

4th layer – Aggregation layer: The normalised firing strengths were multiplied with the function of the Sugeno fuzzy rules which the consequent parameter set that was adjusted with the hybrid learning algorithm.

𝜔ഥ𝑖𝑓𝑖 = 𝜔ഥ𝑖(𝑎𝑝𝑖+ 𝑏𝑠𝑖+ 𝑟𝑖) 𝑖 = 1,2,3,4,5 _[4.35] Where ai, bi and ri are the consequent parameters set that adjusted with the learning algorithm.

5th layer – Defuzzification layer: The weighted averaged method was used to perform the process of defuzzification, which transforms the fuzzy result into a single numerical (crisp) output.

𝐹 = ∑ 𝜔ഥ𝑖𝑓𝑖 𝑖 = 1,2,3,4,5 _[4.36]

The designed ANFIS structure for this study was modelled in the MATLAB environment to forecast the impact size of disruptive events on the cost and time of highway construction projects. Eighty percent of the data collected from the census of South African highway project managers was used for training of the system, while the balance of 20% was used for checking the neural network which sets the system parameters. The model structure of the ANFIS that was designed for this study is illustrated in Figure 4.37

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Figure 4.37 illustrates the five membership functions of each input variable, 25 rules (AND rules), 25 membership functions for output variables, and a single output of the disruptive event in an impact assessment model in ANFIS.

Figure 4.38: 3D Surface Diagram of Rules in the FIS.

From the surface 3D diagram in Figure 4.38, it can be inferred that, with an increase in the probability of occurrence and the effect of the severity of the event, the impact size of the disruptive event on the cost and completion time of the highway construction projects was increased, which was aligned with the risk impact matrix shown in the Figure 4.38. The capability of neural networks in learning by train and adapting the network's parameters with training data is the main advantage of using a neural network in the design of fuzzy systems (Carr and Tah, 2001). To train FIS and determine the relationship between input and output variables of the current study, the hybrid learning algorithms were used.

To identify the best membership function, the designed ANFIS was modelled with the eight more common membership functions namely: Triangular (Trimf), Trapezoidal (Trapmf), Generalized bell-shaped (Gbellmf), Gaussian curve (Gaussmf), Gaussian combination (Gauss2mf), Π-shaped (Pimf), Difference between two sigmoidal (Dsigmf), and Product of two sigmoidal (Psigmf). The performance of each model for cost and time was evaluated by four types of error tests namely RMSE (Root Mean Square Error), NRMSE (Normalised Root Mean Square Error), MAPE (Mean Absolute Percentage Error), MSE (Mean Squared Error) and correlation coefficient value between the real data and predicted data as shown in Tables 4.5 and 4.6. The model with the least errors and R-Value closest to 1 is chosen as the best membership function.

As shown in Table 4.5, the Gaussian membership function (Gaussmf) had the least errors among all four tests and also the R-Value closest to 1 for the time variable. On the other hand, Table 4.6 revealed that Triangular membership function (Trimf) had the least errors and R-Value closest to 1 for the cost variable.

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Table 4.5: ANFIS membership functions for performance evaluation for cost

Model Membership

Function RMSE NRMSE MAPE MSE R-Value*

1 Trimf 3.12E-07 7.83E-08 4.35E-08 2.14E-14 0.999999999999997

2 Trapmf 2.41E-07 7.17E-08 3.74E-08 2.07E-14 0.999999999999996

3 Gbellmf 1.47E-07 7.36E-08 3.60E-08 2.17E-14 0.999999999999998

4 Gaussmf 1.14E-07 5.84E-08 2.23E-08 1.07E-14 0.999999999999999

5 Gauss2mf 1.28E-07 6.41E-08 2.98E-08 1.65E-14 0.999999999999999

6 Pimf 1.47E-07 7.36E-08 3.77E-08 2.17E-14 0.999999999999998

7 Dsigmf 1.34E-07 6.68E-08 3.17E-08 1.78E-14 0.999999999999998

8 Psigmf 1.85E-07 7.12E-08 3.53E-08 2.95E-14 0.999999999999996

* The R-Values are presented with 15 decimals to show the closeness to 1

Table 4.6: ANFIS membership functions for performance evaluation for time

Model Membership

Function RMSE NRMSE MAPE MSE R-Value*

1 Trimf 1.26E-07 6.28E-08 2.90E-08 1.58E-14 0.999999999999999

2 Trapmf 1.30E-07 6.49E-08 3.08E-08 1.68E-14 0.999999999999999

3 Gbellmf 1.47E-07 7.36E-08 3.60E-08 2.17E-14 0.999999999999998

4 Gaussmf 1.49E-07 7.45E-08 3.68E-08 2.22E-14 0.999999999999998

5 Gauss2mf 1.28E-07 6.41E-08 2.98E-08 1.65E-14 0.999999999999999

6 Pimf 1.47E-07 7.36E-08 3.77E-08 2.17E-14 0.999999999999998

7 Dsigmf 1.34E-07 6.68E-08 3.17E-08 1.78E-14 0.999999999999998

8 Psigmf 1.33E-07 6.64E-08 3.22E-08 1.77E-14 0.999999999999999

* The R-Values are presented with 15 decimals to show the closeness to 1

The low value of test error indicated the reliability of the model for impact assessment and the closeness of the R-value to 1 verified the fitness of the model for impact evaluation. Therefore, Trimf and Gaussmf were selected as the best membership functions to model and assess the impact size of disruptive events on construction cost and time of highway construction projects, respectively.

Since disruptive events affect construction cost and duration of infrastructure projects, the cost and time of the unit of the project, where the event occurred, were larger than the cost and time of a unit project, where the disruptive event did not occur. The cost and time of all unit lengths were summed up to obtain the total cost and time of a highway project for a simulation run, which was one point in the cost-time scatterplot. Since each simulation run was different, the occurrences of the disruptive event were changed: it may not occur, it may occur once, or it may occur more than once.

When modelling the disruptive event in addition to the variability and correlations, the cloud of points is expected to expand, and the range of possible total costs and total time increases dramatically. Figure 4.39 presents the flowchart of modelling disruptive events in infrastructure projects.

103 Start

Estimate the probability of occurrence of uncertainty event Estimate the severity of

uncertainty event on cost

Cost – Time scatterplot

End Impact Matrix

Estimate the severity of uncertainty event on time

Impact cost size ANFIS

Impact time size ANFIS

Total Cost _{Total Time}

Unit i Event occurrence model Markov No Occur Not Occur i=n Yes i=i+1

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