Exploring the giant dipole resonance width in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mi>Pb</mml:mi> <mml:mprescripts/> <mml:none/> <mml:mn>208</mml:mn> </mml:mmultiscripts> </mml:math> and its constraints o

Rather than using the geometric method employed in the original Extended Quantum Molecular Dynamics (EQMD) model, this article employs a stochastic approach to analyze the collision term and examine the width of the isovector giant dipole resonance (GDR) in ${}^{208}$Pb. Based on the ``soft" EQMD model, which we recently developed, the response and strength functions are self-consistently determined for various symmetry energy coefficients and in-medium reduction factor values. The results confirm that the peak position and GDR width in ${}^{208}$Pb are highly sensitive to the symmetry energy and the in-medium nucleon-nucleon ({\it NN}) cross section. This provides an opportunity to study the nuclear equation of state (EoS) and the medium effect. A significant reduction in free {\it NN} elastic cross sections within the medium is necessary to accurately reproduce the GDR width, as demonstrated by a comparison with the evaluation data.