In this article, we present a novel methodology for inventory management in the pharmaceutical industry, considering the nature of its supply chain. Traditional inventory models often fail to capture the particularities of the pharmaceutical sector, characterized by limited storage space, product degradation, and trade credits. To address these particularities, using fuzzy logic, we propose models that are adaptable to real-world scenarios. The proposed models are designed to reduce total costs for both vendors and clients, a gap not explored in the existing literature. Our methodology employs pentagonal fuzzy number (PFN) arithmetic and Kuhn–Tucker optimization. Additionally, the integration of the naive Bayes (NB) classifier and the use of the Weka artificial intelligence suite increase the effectiveness of our model in complex decision-making environments. A key finding is the high classification accuracy of the model, with the NB classifier correctly categorizing approximately 95.9% of the scenarios, indicating an operational efficiency. This finding is complemented by the model capability to determine the optimal production quantity, considering cost factors related to manufacturing and transportation, which is essential in minimizing overall inventory costs. Our methodology, based on machine learning and fuzzy logic, enhances the inventory management in dynamic sectors like the pharmaceutical industry. While our focus is on a single-product scenario between suppliers and buyers, future research hopes to extend this focus to wider contexts, as epidemic conditions and other applications.

In this paper, we propose and derive a new system called pure hybrid fuzzy neutral delay differential equations. We apply the classical fourth-order Runge–Kutta method (RK-4) to solve the proposed system of ordinary differential equations. First, we define the RK-4 method for hybrid fuzzy neutral delay differential equations and then establish the efficiency of this method by utilizing it to solve a particular type of fuzzy neutral delay differential equation. We provide a numerical example to verify the theoretical results. In addition, we compare the RK-4 and Euler solutions with the exact solutions. An error analysis is conducted to assess how much deviation from exactness is found in the two numerical methods. We arrive at the same conclusion for our hybrid fuzzy neutral delay differential system since the RK-4 method outperforms the classical Euler method.

Decision-making in real-world scenarios faces uncertainty. Fuzzy theory has been a means to represent such uncertainty. In this study, we propose an approach that incorporates bipolarity into multi-criteria decision-making processes applied to digital marketing. The proposal considers both the positive and negative dimensions of data, leading to better-informed decisions. Our contribution integrates bipolarity into Pythagorean fuzzy matrices, a framework that broadens the utility of bipolar fuzzy theory. Through computational experimentation, we identify the most effective strategy for digital marketing platforms. When compared to existing techniques, our approach shows advantages, underlining its potential to improve decision-making in uncertain scenarios and offering insights for businesses that refine their digital marketing strategies.

Background: Mathematical models reflecting the epidemiological dynamics of dengue infection have been discovered dating back to 1970. The four serotypes (DENV-1 to DENV-4) that cause dengue fever are antigenically related but different viruses that are transmitted by mosquitoes. It is a significant global public health issue since 2.5 billion individuals are at risk of contracting the virus. Methods: The purpose of this study is to carefully examine the transmission of dengue with a time delay. A dengue transmission dynamic model with two delays, the standard incidence, loss of immunity, recovery from infectiousness, and partial protection of the human population was developed. Results: Both endemic equilibrium and illness-free equilibrium were examined in terms of the stability theory of delay differential equations. As long as the basic reproduction number (R0) is less than unity, the illness-free equilibrium is locally asymptotically stable; however, when R0 exceeds unity, the equilibrium becomes unstable. The existence of Hopf bifurcation with delay as a bifurcation parameter and the conditions for endemic equilibrium stability were examined. To validate the theoretical results, numerical simulations were done. Conclusions: The length of the time delay in the dengue transmission epidemic model has no effect on the stability of the illness-free equilibrium. Regardless, Hopf bifurcation may occur depending on how much the delay impacts the stability of the underlying equilibrium. This mathematical modelling is effective for providing qualitative evaluations for the recovery of a huge population of afflicted community members with a time delay.

Understanding disease dynamics is crucial for accurately predicting and effectively managing epidemic outbreaks. Mathematical modeling serves as an essential tool in such understanding. This study introduces an advanced susceptible, infected, recovered, and dead (SIRD) model that uniquely considers the evolution of the death parameter, alongside the susceptibility and infection states. This model accommodates the varying environmental factors influencing disease susceptibility. Moreover, our SIRD model introduces fractional changes in death cases, which adds a novel dimension to the traditional counts of susceptible and infected individuals. Given the model’s complexity, we employ the Laplace-Adomian decomposition method. The method allows us to explore various scenarios, including non-fuzzy non-fractional, non-fuzzy fractional, and fuzzy fractional cases. Our methodology enables us to determine the model’s equilibrium positions, compute the basic reproduction number, confirm stability, and provide computational simulations. Our study offers insightful understanding into the dynamics of pandemic diseases and underscores the critical role that mathematical modeling plays in devising effective public health strategies. The ultimate goal is to improve disease management through precise predictions of disease behavior and spread.

COVID-19, a new pandemic disease is becoming one of the major threats for surviving. Many new models are arrived to study the disease mathematically. Here we are introducing a new model in which instead of studying a day by day changes we are studying the average of 14 day transmission because its life or the patients incubation period is about an average of 14 days. Also, since this is pandemic, and being not aware of susceptible population among the world's population, we considered the model without S-susceptible population. i.e., IRD-Infectious, Recovered, Death-model. In this new model, we are also introducing a new method of calculating new number called N₀-average transmission number. This is used to study the average spread of infection instead of basic reproduction number R₀. The motto of this paper is not to predict the daily cases but to control the current spread of disease and deaths by identifying the threshold number, exceeding which will increase the spread of infection and number of deaths due to this pandemic. Also if the 14 day average IRD-populations are maintained under this threshold number, will definitely control this pandemic disease globally. Stability analysis and test for stiff system of differential equations are studied. Our main aim is to present the medical world, a threshold population of infected, recovered and death cases for every average of 14 days to quickly overcome this pandemic disease COVID-19.