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CASE STUDY IN MALAYSIA FOR LOGISTIC GROWTH MODELING IN
NEUTROSOPHIC FUZZY ENVIRONMENT: A COMPARATIVE STUDY WITH
CLASSICAL SOLUTIONS
Abstract
Classical model may not be effective in solving population
modeling involving parameter and initial data uncertainty. In
this study, a logistic differential equation is solved under both
classical and neutrosophic fuzzy environments to overcome these
limitations. For more effectively modeling uncertainty in this
study, both analytical and numerical solutions are contrasted
under different environments with closer consideration of
neutrosophic representation. Both of the solutions in classical
environment are computed using MATLAB meanwhile, the
solutions in neutrosophic environment is computed using
Microsoft Excel. In the neutrosophic fuzzy environment, the
growth rate, carrying capacity, and initial population are taken
as Triangular Neutrosophic Numbers, and solutions are derived
for the truth, indeterminacy, and falsity components
individually. The measures such as mean absolute percentage
error, mean squared error, and root mean squared error are
applied to compare the solutions from all the methods. The
results demonstrate that the solutions in neutrosophic
environment can provide a wider perspective of potential results
under uncertainty, whereas the classical solutions are in
sufficient when involving uncertainty due to their assumption on
precise input values. This article highlights the advantages of
applying neutrosophic logic to mathematical modeling,
especially to real-world systems with incomplete or imprecise
data. Future research is proposed to generalize analytical
multiple-parameter neutrosophic analysis and apply the
approach to more complex population or ecological models.
Methodology
Result
Name RAIHAN SYAFIQAH BINTI RAMLI (K242/36)
Supervisor DR ROLIZA BINTI MD YASIN
Problem Statement
Good population estimates have been helpful in informal national development and resources management, urban
planning and policy making. The popular mathematical model for population dynamics is the logistic growth, a
model containing such essential parameters as growth rate, r and carrying capacity, K. For the predicted value of r,
we get from the exponential model meanwhile for predicted value of K, the formula from the study of Islam and
Ahmed (2017) in establishing carrying capacity, K for Bangladesh will be used. Since this deterministic model takes
into consideration some limitations due to missing records or unstable migration rates, neutrosophic logic is a new
way to quantify uncertainty and indeterminacy of the date regarding populations.
Parikh and Sahni (2024)applied the neutrosophic logic to logistic growth models in projections of population
growth in India when facing with uncertainty. In doing so, the researchers established the capability of neutrosophic
logic in enhancing reliability for population models operating under conditions of uncertainty. However, to our best
knowledge, there is limited research on the neutrosophic logic to logistic growth models in projections of population
growth in Malaysia. Hence, it seems that this approach can be applied to the Malaysia demographic context in order
to understand how the theory of neutrosophic handles uncertainties.
There is a numerical method available to solve logistic models named the RK4 method. Its uniqueness is that this
method is known for its efficient in solving linear first order differential equations with respect to accuracy. The
latest research by Azis et al. (2024) shows the results of their studies which indicate that the RK4 method will be
applicable to predict population size and thus would show their strength in the solution of complex logistic
equations. By comparing the predicted population in analytical and numerical solutions with the actual populations,
the accuracy of the population prediction can be determined.
Objectives
1.To obtain the solution of logistic growth model in neutrosophic fuzzy environment.
2.To define the predicted Malaysian population using the solutions of logistic growth model in neutrosophic fuzzy
environment.
3.To define the predicted Malaysian population using the Runge-Kutta 4th Order method in the classical environment.
4.To compare the analytical solution of the classical and neutrosophic fuzzy environment, and numerical solution in
classical environment with the actual population data.
Implementation
1) Find the values of K and r 4) Determination of Parameters in Exponential Model
a) The Value of A
2) Implementation of Analytical Solution in Classical
b) The Value of Growth Rate by Exponential Model
Environment
5) Implementation of Analytical Solution in
Neutrosophic Fuzzy Environment
3) Construction of the Initial Value Problem in the
Form of Triangular Neutrosophic Number
th
6) Implementation of Runge-Kutta 4 order in
Classical Environment
Conclusion and Recommendation
Therefore, after comparing, all the three measures of evaluation which are MAPE, MSE, and RMSE are observed that all
of them gave the same relative accuracy ranking persistently. The analytical solution in neutrosophic fuzzy
environment shows the least error followed by numerical solution in classical environment and lastly analytical
solution in classical environment showed the highest error. Thus, the presence of uncertainty using neutrosophic fuzzy
environment proved that it can increases the precision in population prediction under imprecise or uncertain
circumstances. However, due to the use of an oversimplified logistic model of growth that approximates real
population behavior, the commonality of error among all the models must be noted. Thus, these results may not reflect
complex real-world behavior.
It is recommended to apply more advanced population growth models such as the Gompertz model (Welagedara et
al., 2019) and the Allee effect model (Petrovskii and Li, 2003) for additional research. In addition, increasing the
dataset such testing the model on multiple countries and applying neutrosophic modeling to more complex cases can
provide deeper insights into uncertainty systems and improve prediction.