The completeness of the group classification of systems of two linear second-order ordinary differential equations with constant coefficients is delineated in the paper. The new cases extend what has been done in the literature. These cases correspond to the type of equations where the commutative property of the coefficient matrices with respect to the dependent variables and the first-order derivatives in the considered system does not hold. A discussion of the results as well as a note on the extension to linear systems of second-order ordinary differential equations with more than two equations are given.
Davao City, the largest city in the Philippines with 11 legislative districts, is among the top cities with high rabies incidence. To eliminate the rabies incidence, the City Veterinarian’s Office (CVO) has been implementing the Intensified Rabies Control Program (IRCP) that is composed of mass dog vaccination, dog neutering, dog impounding, and information and education campaigns (IEC). In this paper, we formulated a linear programming (LP) model that minimizes the IRCP operational cost while satisfying the different targets set by the CVO for each of the IRCP component as well as its allocated budget. From the analysis, the CVO should be lessening the number of dogs to be vaccinated in districts whose vaccine coverage is already above 70%. This is to ensure that the number of dogs to be vaccinated in other districts can be increased and achieve 70% vaccine coverage. However, the model suggests that it is optimal that 90% of the dogs in the Talomo district (the most populous district in the city) should be vaccinated. Furthermore, with the 24:1 ratio of the number of vaccinated dogs to the impounded dogs, it is optimal to neuter one dog for every 239 vaccinated dogs. Additionally, the model suggests impounding more dogs to highly populated districts. Lastly, the model suggests maintaining the average number of IEC participants. By doing these interventions firmly, dogs in Davao City can potentially achieve herd immunity, and the human’s risk to contract rabies will be lowered.
Problems concerning low-profit-generating production occur in many small producer enterprises, including banana flour production by small-scale Cavendish banana farmers. One usual difficulty is the inability of these small producers to adjust to market conditions due to limited resources. These farmers are unable to allocate adequate resources such as transport vehicles, equipment, and laborers to various activities in the production process. This paper formulated a mixed-integer linear programming model based on a supply chain network design for banana flour production. This model aimed to determine the optimal number of resources for a producer organization to maximize its total profit. The deterministic branch-and-bound technique and metaheuristic binary firefly algorithm were implemented to obtain the model solution. Based on a case study, both approaches consistently showed that to maximize the profit for small-scale banana flour production, they must operate one mill and one truck with a 4,000- kg maximum capacity and hire nine non-regular and 14 regular laborers. The methodology developed in this study can also be applied to other banana producer organizations with a similar supply chain network to explore alternative enterprises and improve profitability.
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