Dynamic items delivery network: prediction and clustering.

Items delivery companies generally use a model to minimize delivery costs. From a mathematical perspective, the model is an objective function that involves constraints. Meanwhile, from a practical point of view, these constraints include aspects that affect item delivery, for example, delivery zones, number of delivery vehicles, vehicle capacity, trip routes, etc. However, the models built so far have not paid attention to changes in road density. This aspect can result in a nonoptimal delivery model, which results in not a minimum delivery cost. For this reason, this paper discusses how to divide zones using the clustering method and predict changes in the shipping zone of a dynamic network using predictive distribution. So, the model can work optimally if the delivery zones and delivery strategies are suitable.

Generating roots of cubic polynomials by Cardano's approach on correspondence analysis.

Cardano's formula is among the most popular cubic formula to solve any third-degree polynomial equation. In this paper, we propose the Cardano's approach as the alternative solution to generate the roots of the cubic characteristic polynomial analytically. In the context of correspondence analysis, these roots referred to eigenvalues, which play an important role in assessing the quality of the correspondence plot. Considering the correspondence analysis on the I × J contingency table for I = 4 and J = 4 , 5 , ⋯ , we obtained a cubic characteristic polynomial (since zero is one of its eigenvalues). Therefore, Cardano's formula allows us to obtain the eigenvalues directly without involving numerical processes, e.g., using singular value decomposition. We note several advantages of using Cardano's approach, such as (1) it produces the roots with the same result as singular value decomposition, as well more precise because without errors involving, (2) the algorithm is simpler and does not depend on initial guess, hence the computation time becomes shorter than numerical process, and (3) the manual calculation is easy because it uses a formula. The results show that the matrix operations on correspondence analysis can be replaced by a formula for determining eigenvalues and eigenvectors of the standard residual matrix directly. Some mathematical results are also presented.

Estimation of the Basic Reproductive Ratio for Dengue Fever at the Take-Off Period of Dengue Infection.

Estimating the basic reproductive ratio ℛ 0 of dengue fever has continued to be an ever-increasing challenge among epidemiologists. In this paper we propose two different constructions to estimate ℛ 0 which is derived from a dynamical system of host-vector dengue transmission model. The construction is based on the original assumption that in the early states of an epidemic the infected human compartment increases exponentially at the same rate as the infected mosquito compartment (previous work). In the first proposed construction, we modify previous works by assuming that the rates of infection for mosquito and human compartments might be different. In the second construction, we add an improvement by including more realistic conditions in which the dynamics of an infected human compartments are intervened by the dynamics of an infected mosquito compartment, and vice versa. We apply our construction to the real dengue epidemic data from SB Hospital, Bandung, Indonesia, during the period of outbreak Nov. 25, 2008-Dec. 2012. We also propose two scenarios to determine the take-off rate of infection at the beginning of a dengue epidemic for construction of the estimates of ℛ 0: scenario I from equation of new cases of dengue with respect to time (daily) and scenario II from equation of new cases of dengue with respect to cumulative number of new cases of dengue. The results show that our first construction of ℛ 0 accommodates the take-off rate differences between mosquitoes and humans. Our second construction of the ℛ 0 estimation takes into account the presence of infective mosquitoes in the early growth rate of infective humans and vice versa. We conclude that the second approach is more realistic, compared with our first approach and the previous work.