PREDICTION OF AVERAGE TOTAL PROJECT DURATION USING ARTIFICIAL NEURAL NETWORKS, FUZZY LOGIC, AND REGRESSION MODELS

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International Journal of Management (IJM)
Volume 13, Issue 7, July 2022, pp. 1-13, Article ID: IJM_13_07_001
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ISSN Print: 0976-6502 and ISSN Online: 0976-6510
© IAEME Publication Scopus Indexed
PREDICTION OF AVERAGE TOTAL PROJECT
DURATION USING ARTIFICIAL NEURAL
NETWORKS, FUZZY LOGIC, AND
REGRESSION MODELS
Bendada Larbi1, Brioua Mourad2, Djeffal Selman3, Morakchi Mohamed Razi4
1
Department of Mechanics, Laboratory of CMASMTF,
University of Larbi Ben M’hidi Oum-El-Bouaghi, Algeria
2
Department of Mechanics, Laboratory LRPI, University of Batna 2, Algeria
3
Department of Mechanics, Laboratory of CMASMTF,
University of Larbi Ben M’hidi Oum-El-Bouaghi, Algeria
4
Department of Electrical Engineering, Laboratory LGE University of M’sila,
M’sila, Algeria
ABSTRACT
The prediction of project‘s expectancy life is an important issue for entrepreneurs
since it helps them to avoid the expiration time of projects. To properly address this
issue, Neural Network-based approach, fuzzy logic and regression methods are used to
predict the necessary time that can be consumed to put an end to the targeted project.
Before applying the three aforementioned approaches, the modeling and simulation of
the activities network are introduced for calculating the total average time of project.
Then, comparatively speaking, the neural network, fuzzy logic and regression method
approach are compared in terms prediction’s accuracy. The generated error from the
three methods is compared, namely different types of errors are calculated. Basically,
the input variables consist of the probability of success (PS), the coefficient of
improvement (Coef_PS) and the coefficient of learning (CofA), while the output variable
is the average total duration of the project (DTTm). The Predicted mean square error
(MSE) values are purposefully used to compare the three models. Interestingly, the
results show that the optimum prediction model is the fuzzy logic model with accurate
results. It is noteworthy to say that the application in this paper can be applied on a real
case study.
Key words: projects duration prediction, artificial neural network, fuzzy logic, regression model,
uncertainty, prediction
Cite this Article: Bendada Larbi, Brioua Mourad, Djeffal Selman and Morakchi Mohamed Razi,
Prediction of Average Total Project Duration Using Artificial Neural Networks, Fuzzy Logic, and
Regression Models, International Journal of Management (IJM), 13(7), 2022, pp. 1-13.
https://iaeme.com/Home/issue/IJM?Volume=13&Issue=7

Regression Models
1. INTRODUCTION
The growth competition in the industrial market and, the complexity of the manufacturing
systems has created a very difficult environment to manage and plan in order to bring out a new
product. Companies undergoes remarkable pressure to create and maintain a competitive
advantage by reducing product development time taking into account the level of quality which
has to be always high [1]. A complex design or construction mega-project involves the
execution of a very large number of tasks with the participation of specialists from different
disciplines. With the increasing complexity of the process iterations become a reality, in order
to eliminate unnecessary iterations, project managers must take into account each possible
challenge and failure of each task during the course of the project. Therefore, a project must be
completed in a shorter time. By iterations, we refer to the possibility of redoing the actual task’
step or its precedent ones.
Gantt and PERT are commonly used in the planning of design processes yet it proves not
efficient, because it lacks the modeling of design activities with their iterations [2] [3] Basically,
several models have been developed to characterize the time and cost of repetitive (iterative)
design processes. These models include iterative sequential tasks, iterative parallel tasks, and
coupled models. Among the research works that have addressed these models,[4] have
developed a model based on the Markov chain using the DSM method with iteration
probabilities and task duration. In [5] have introduced iterations, which are the basic
characteristics of the product development process. Design iteration makes the product
development process more complex and difficult to analyze. In[6] have proposed a fuzzy
critical chain method for project planning under resource constraints and uncertainties. In [7]
have established a practical and appropriate algorithm that is provided for order production
planning under conditions of uncertainty. Producing the order production projects using
different methods faces uncertainty and therefore specific techniques are required to determine
their production planning program. Network models combining fuzzy logic to define sequence
of stages and production uncertainty are suitable tools for order production planning modeling.
In [8] have developed a convenient and appropriate algorithm for producing array estimation
scheduling software with the waterfall method under fuzzy conditions. The production of
software which deals with projects using different methodologies is highly likely to face
uncertainty and therefore specific techniques are required to determine their estimation
production scheduling maps. Network models combining fuzzy logic to define stage sequence
and production uncertainty are appropriate tools for software projects and their production
scheduling for graphical estimation modeling. In [9] where the average surface roughness
values obtained when setting AISI 4140 grade tempered steel with a hardness of 51 HRC, it is
modeled using fuzzy logic, artificial neural networks (ANN) and multi-regression equations.
[10] whose goal is to simultaneously optimize the time, cost and quality of a project by
analyzing their trade-offs. In [11] have used Classical back propagation neural network data
and general regression neural network data from 112 construction projects in Hong Kong are
used to examine the influence of rework on various project performance indicators, such as cost
overruns, time overruns and contract claims.
Within the same context, in this study, the objective is to predict the necessary time to finish
the targeted products’ development using three different models: fuzzy logic, artificial neural
networks, and regression models. In order to achieve this objective, this article is structured as
follows: in section 2, we present a brief overview on the construction of the model, the modeling
of activity networks, their parameters and the simulation approach. Section 3 is devoted to the
presentation of the three aforementioned models retained in our study, namely the modeling of
the average total duration of the project using artificial neural networks (ANN), fuzzy logic
(FLM) and regression model. Section 4 is dedicated to the comparison of the three predictive

Bendada Larbi, Brioua Mourad, Djeffal Selman and Morakchi Mohamed Razi
models developed in this study in terms of prediction’s accuracy. The results from these
comparisons and a discussion of these results are thoroughly presented. Finally, the last section
provides with a conclusion about the possibilities offered by the comparison of the three models
to calculate the predicted values of the mean square error (MSE) as well as some future works
regarding this work.
2. THE MODELING OF ACTIVITY NETWORKS
We are interested in the modeling of activity networks. For this reason, we have proposed a
process model that contains a set of sequential activities with feedback loops and iterative cycle.
The model represents various scenarios for moving from one activity to another and for
challenging previous activities.
Figure 1. Sequence of four activities and the possible reconsideration of each activity
2.1. Parameterization of the Model and the Process
In order to illustrate our approach, we considered the example of an activity network comprising
100 activities in series.
2.1.1. Duration of Activities
Some temporal indicators that are based on durations can be used, such as the sum of durations
of activities, or the average, or even the variance of these values. In the problem that concerns
us, any task ‘I’ is not defined by a fixed duration, but by an indicative duration between two
bounds [12] .
A variety of distributions or probability density functions (PDFs), notably the beta
distribution, have been used to represent the uncertainty of activity duration. These distributions
are generally positively skewed due to the tendency for expansion work to fill available time -
and human nature to relax when leading up - so activities are less likely to end early. Positive
skew can be represented by a triangular distribution, which is simple to understand and
construct, requiring only three data points per activity: optimistic, possibility, and pessimistic
[13]. Similar distributions based on a duration probability distribution according to a triangular
law. The duration of each activity = a + (b – a) * (rand(n1,1) + rand(n1,1)) / 2
With a = min =1; b = max = 20
2.1.2. Success of an Activity
Any disturbance may lead to infeasibility or to a result that does not comply with the
specifications of the activity. To integrate the effect of disturbances in the simulation, we assign
to each activity a probability of achievement or success, denoted (PS). Depending on the
behavior of each activity, this probability can be represented by different distribution functions
or simply by discrete values.

Regression Models
2.1.3. Failure of an Activity
We consider that any failure leads to a questioning. If an activity fails in its mission (probability
of success not reached), a second probability (PRC) makes it possible to choose the path to
rework. Table 1 presents the value of PS, COEF_PS and CofA for each level.
Table 1 The values of three coefficients (three inputs) for each level.
Level PS Coef_PS Cof A
1 0,795 0,00 0,6
2 0,825 0,05 0,8
3 0,950 0,10 1,0
2.2. Modeling the Average Total duration of the Project using Artificial Neural
Networks (ANN)
The ANN Technique has emerged as a powerful modeling tool that can be applied to many
scientific and/or engineering applications, such as: model reorganization, classification, data
processing, and process control. An artificial neural network simulates the computational
human brain mechanism to implement behavior [14].
The simulation results were used to develop an ANN model to predict the average total
project duration (DTTm) using MATLAB Neural Network Toolbox. There are three inputs and
one output in the ANN model. The input variables are PS, Coef_PS and CofA, and the output
variable is DTTm. Basically, 22 results from the total of 27 results obtained from the
simulations are purposefully used in the training of the network. The remaining data is used for
testing. Table 2 shows the simulation data chosen for the test. Many network architectures have
been tried. Then, the network structure of 3X3X3X1 (MSE with the lowest), which provides
the best results, are used. The chosen network structure is shown in Figure 2. The Feed-back
forward propagation algorithm, Levenberg-Marquardt (LM) training function and tan-sigmoid
activation function, are adopted to train the ANN. Neurons are arranged in the form of layers
in the forward feeding ANN, and the output of cells from one layer are fed to the next layer by
input weights. Two hidden layers were used and ANN formation was completed in 200 cycles.
Figure 2 Architecture of an artificial neural network.
Network performance is determined by (the predicted values of Mean Squared Errors) MSE
which should be minimized. MSE is calculated using equation (1). Table 3 shows the predicted

values of DTTm, which were obtained by ANN (Artificial Neural Networks) test data. The
MSE value, obtained from the ANN, is estimated to be 161673986.048591.
𝑀𝑆𝐸 =
1
𝑄
∑ 𝑒
𝑄
𝑘=1 (𝑘)2
=
1
𝑄
∑ [𝑡(𝑘) − 𝑦(𝑘)]2
𝑄
𝑘=1 (1)
Where, e (k) is the error between the target and the ANN output, t(k) is the target, y(k) is
value of the output per ANN, and Q is the total data count (Harun et al., 2011).
Table 2 Selected data for testing.
N° 2 5 6 23 25
Input
Probabilité de succès 0,795 0,795 0,795 0,950 0,950
Coefficient d’amélioration 0,000 0,050 0,050 0,050 0,100
Coefficient d’apprentissage 0,800 0,800 1,000 0,800 0,600
Output Durée totale moyenne du projet 3476,03 2278,57 2889,78 1098,16 1086,19
Table 3 The results of simulation execution and ANN_DTTm.
Test data Simulation DTTm ANN-DTTm
2 3476,03043652707 20891,4420337734
5 2278,56748869743 402,695521071452
6 2889,77748202287 -2784,42688690875
23 1098,15481620437 21977,7803292064
25 1086,18731810697 6865,38421031622
Figure 3. Validation Performance from the Neural Networks Model
2.2.1. Modeling the average total duration of the project using the fuzzy logic method
(FLM)
Although fuzzy control has great potential to solve complex control algorithms, design
procedure is very complex and specific. Moreover, fuzzy mathematics does not belong to the
domain of mathematics and it is not basic mathematical operator. For example, there is no
inverse adding in fuzzy math. It is very difficult to solve fuzzy equations. To solve various
equations, applications and practice of traditional control theory are used. So, in the absence of
mathematical tools, fuzzy control problem become solvable [15] .

Regression Models
A fuzzy expert system with multiple inputs and a single output are designed and developed
using fuzzy MATLAB Toolbox. For the fuzzy modeling mechanism of logical inference, we
adopted the Takagi–Sugeno approach. Figure 4 shows the structure of the developed fuzzy logic
model. The input variables are the probability of success, the improvement coefficient, and the
learning coefficient. These variables are defined using three membership degrees and triangle
membership functions. A total of 27 rules are established as it is shown in Figure 5. The results
of the fuzzy logic model (FLM) are given in Table 4. The MSE value are calculated as MSE =
4044021.83839137 at the end of the FLM.
Table 4 ANFIS parameters’
1 Number of nodes 78
2 Number of linear parameters 108
3 Number of nonlinear parameters 27
4 Total number of parameters 135
5 Number of training data pairs 22
6 Number of checking data pairs 0
7 Number of fuzzy rules 27
Figure 4. ANFIS system based on the Takagi–Sugeno model

Figure 5. Viewing membership rules
Figure 6. Possibility surface

Regression Models
Table 5 The results of simulation and FLM_DTTm.
Test data Simulation DTTm (Time unity) FLM-DTTm (Time unity)
2 3476,03043652707 943,058444833034
5 2278,56748869743 447,645136318505
6 2889,77748202287 5824,58708246683
23 1098,15481620437 286,381978561251
25 1086,18731810697 0
2.2.2. Modeling DTTM using the Regression Model
Multiple regression is a statistical technique that determines the correlation between variables.
It can be used with different types of data. Thus, it is well suited for predicting the average total
project duration where the aim (objective) is to find correlations between the average total
project duration and multiple parameters of the design process activities[16]
2.2.3. Regression Analysis
After obtaining the average total duration of the project for all the simulations, a table must be
filled in so as to obtain several values for the analysis. In order to obtain estimates of the
regression coefficients β0, β1, β2, β3, …..., β10, it is necessary to solve the system of linear
equations.
2.2.4. The Mathematical Model of the Average Total Project Duration
The correlations between the main activity parameters are obtained by regression. The
regression can be mathematically represented as follows:
DTTm = 861242,821436381* (PS)^2 + 440859,022666301 * (Coef_PS)^2 +
24427,5749095907 * (CofA)^2+ 628136,732401372* PS * Coef_PS + (-248767,504108431)
* Coef_PS * CofA + (-156776,439648829)* PS * CofA + ( -1437846,92721839)* PS -
412303,255634733 * Coef_PS + 114912,916994168 * CofA + 600288,720400987
Table 6 The results of simulation execution and Reg_DTTm.
Test data Simulation DTTm (Time unity) Reg-DTTm (Time unity)
2 3476,03043652707 9381,57340151735
5 2278,56748869743 4886,29312506365
6 2889,77748202287 9247,67454610195
23 1098,15481620437 392,452693875064
25 1086,18731810697 7910,09644879075
2.2.5. Model Validation Criteria
These criteria make it possible to judge the quality of the models, by giving an overall and
numerical estimate of the difference between the calculated results and the observed data.
The following formulas are commonly adopted to measure accuracy [5] . The sum of
squares error (SSE) is a measure of the difference between the data and predicted model. A
small SSE value indicates a good fit of the model to the data. SSE can be mathematically
represented by:
𝑆𝑆𝐸 = ∑(𝑦𝑖 − 𝑦
̂𝑖)2
𝑛
𝑖=1

Multiple Correlation Coefficient, R2
𝑅2
= 1 −
∑ (𝑦𝑖 − 𝑦
̂𝑖)2
𝑛
𝑖=1
∑ (𝑦𝑖 − 𝑦
̅)2
𝑛
𝑖=1
= 1 −
𝑀𝑆𝐸
𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒
Where yi is the observation value, y ̂i is the observation value estimate value, y ̅ is the
observation mean value, and MSE stands at the root mean square error.
𝑀𝑆𝐸 =
1
𝑛
∑(𝑦𝑖 − 𝑦
̂𝑖)2
𝑛
𝑖=1
, 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 =
1
𝑛
∑(𝑦𝑖 − 𝑦
̅𝑖)2
𝑛
𝑖=1
The greater R2
, the more accurate the model. The smaller the MSE, the more accurate the
model.
Squared Root of Mean Squared Error (RMSE)
RMSE is a good measure of prediction accuracy. It is frequently used to measure the differences
between values predicted by a model and the values actually observed from the thing being
modeled. These individual differences are also called residuals.
𝑅𝑀𝑆𝐸 = √
∑ (𝑦𝑖 − 𝑦
̂𝑖)2
𝑛
𝑖=1
𝑛
= √𝑀𝑆𝐸
Where 𝑦𝑖 are observed values, 𝑦
̂𝑖 are predicted values for rainfall and n is the number of
observation.
𝐴𝑐𝑐𝑢𝑟𝑎𝑐𝑦 = 100 − 𝑅𝑀𝑆𝐸
Mean Absolute Percentage Error (MAPE)
𝑀𝐴𝑃𝐸 =
1
𝑛
∑ |
𝑦𝑖 − 𝑦
̂𝑖
𝑦
̂𝑖
|
𝑛
𝑖=1
× 100
The Coefficient of Determination (R2)
R2
is a number between 0 and 1 that demonstrates the goodness of the model where 1.0 indicates
that a regression line fits the data well.
𝑅2
= 1 −
∑ (𝑦𝑖 − 𝑦
̂𝑖)2
𝑛
𝑖=1
∑ (𝑦𝑖 − 𝑦
̅)2
𝑛
𝑖=1
= 1 −
𝑀𝑆𝐸
𝑅2
= 1 −
𝑆𝑆𝐸
𝑆𝑆𝑇
= 1 −
𝑀𝑆𝐸
𝑆𝑆𝑇 = ∑(𝑦𝑖 − 𝑦
̅)2
𝑛
𝑖=1
𝑆𝑆𝐸 = ∑(𝑦𝑖 − 𝑦
̂𝑖)2
𝑛
𝑖=1
Relative Error (RE)
Indicates how large the error is relative to the correct value, y. It provides a comparison of the
error to the size of the measurement and can be computed using the following equation :

Regression Models
𝑅𝐸 = √
∑ (𝑦𝑖 − 𝑦
̂𝑖)2
𝑛
𝑖=1
∑ 𝑦𝑖
2
𝑛
𝑖=1
× 100
Mean Absolute Error (MAE)
The smaller the MAE, the better the model fit.
𝑀𝐴𝐸 =
|𝑦𝑖 − 𝑦
̂𝑖|
𝑛
Where 𝑦𝑖 are observed values, 𝑦
̂𝑖 are predicted values for rainfall and n is the number of
observation.
Nash Criterion (T)
Introduced by NASH and SUTCCLIFFE [17] it is defined by:
𝑇 = (1 −
∑ (𝑦𝑖 − 𝑦
̂𝑖)2
𝑛
𝑖=1
∑ (𝑦𝑖 − 𝑦
̅𝑖)2
𝑛
𝑖=1
) × 100
With :
𝑦𝑖 : The observed duration;
𝑦
̂𝑖 : The duration calculated by the model;
𝑦
̅𝑖 : Average of the durations observed.
The Nash criterion can be interpreted as being the proportion of the variance of the observed
discharge explained by the model. If T = 100%, the adjustment is perfect, on the other hand if
T < 0, the flow calculated by the model is a worse estimate than the simple average flow.
(XANTHOULIS, 1985).
Table 7 Statistical parameters of the ANN model, MLF and the regression model
Static parameters FLM 2nd
regression RNA
1 MSE 4044021,83839137 25832455,4185245 161673986,048591
2 RMSE 2010,97534504811 5082,5638627099 12715,1085739993
3 R2
15,1931991022687 124,027174568377 53,6635315076617
4 MAPE 164990,762371677 423924,343113793 1647858,46387904
5 MAE 3327,04489937909 20989,37267379 36524,1576659
6 PVAL 0,392618523622048 0,232516570334013 0,858040607119924
7 RHO 0,498534888398132 0,652647921199482 -0,111727533560749
8 T 15,1931991022687 124,027174568377 53,6635315076617
2.2.6. Comparison of different Predictive Models
The average MSE values of three different models are given in Table 7. According to the MSE
results, the fuzzy logic model produces the best prediction for MTTm, followed in second place
by the regression model, and finally the artificial neural networks. The comparison of the
simulation results and those obtained using the three methods are shown in Figure 7.

Figure 7. Comparison between the mean square error (MSE) values of the three models
Figure 8. Comparison between simulation and prediction results by regression model, neural networks
and fuzzy logic
3. RESULTS AND DISCUSSION AND INTERPRETATIONS
To better illustrate the simulation quality of the models, we present the simulation and
validation results, in order to better analyze the robustness of the three models as well as their
prediction’s accuracy. If we make a comparison between the models, we notice that the fuzzy
logic provides better results.
For the model based on neural networks, the results obtained are less efficient than the other
two models, As shown in Figure 7, FLM gives the best values for DTTm and MSE obtained as
MSE = 4044021.83839137. It is obviously clear that the values predicted by FLM are very
close to the values of the simulation. The results of the simulations demonstrate the ability of
the proposed process to predict effective performance of the average total project duration
commonly encountered in the design process.
0
20000000
40000000
60000000
80000000
100000000
120000000
140000000
160000000
180000000
MSEann MSEpoly MSEfis
simulation'
s
values
The used methods
Results by the three methods
Series1
-5000
0
5000
10000
15000
20000
25000
1 2 3 4 5
Difference
Between
Methods
Test Data
Comparison between simulation and prediction
results by Reg, ANN and ANFIS
Simulation DTTm
Reg
ANN
ANFIS

Regression Models
Interestingly, As a result of the ANOVA and regression, Coef_PS is the most important
factor that contributes to the average total project duration (see Pareto chart). Based on the
values obtained from MSE, the fuzzy logic model produces respectively the best prediction of
DTTm, followed by the second order regression model, then the artificial neural network model.
It is noteworthy to say that other artificial intelligence methods can also be used successfully
by increasing data number.
4. CONCLUSION
Project‘s expectancy life is of a great interest for managers in particular when it is necessary to
finish the given project’s tasks in the scheduled period. For this reason in this paper, three
models are used to successfully deal with this issue, namely Neural Network-based approach,
fuzzy logic and regression methods are adopted to predict the probable time that can be
consumed to finish the targeted project. Based on the obtained results, Fuzzy logic is a relatively
recent approximation technique which makes it possible to evaluate the inaccuracies of the
results of a simulation. The values of the average total project duration (ATDm) obtained by
simulation are successfully modeled using multiple approaches, including FLM, ANN and
multi-regression, through which, the predictive models developed demonstrated the ability to
accurately model the average total duration of the project. According to the MSE results, the
fuzzy logic model produces the best prediction for MTTm, followed in second place by the
regression model, and finally the artificial neural networks. The possible future works resides
in testing the newly emerging types of artificial intelligence-based models to predict the time
duration of projects.
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