Articulo

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versión impresa ISSN 1409-2433

Rev. Mat vol.31 no.1 San José ene./jun. 2024

http://dx.doi.org/10.15517/rmta.v31i1.55440

Artículo

Vacating a parking lot under uncertainty

Desocupar un estacionamiento bajo incertidumbre

1Universidad Popular Autónoma del Estado de Puebla, Departamento de matemáticas, Puebla, México joseantonio.martinez03@upaep.edu.mx

2Universidad Popular Autónoma del Estado de Puebla, Departamento de matemáticas, Puebla, México damianemilio.gibaja@upaep.mx

3Universidad Popular Autónoma del Estado de Puebla, Departamento de matemáticas, Puebla, México juanjose.reyes@upaep.mx

4Universidad Popular Autónoma del Estado de Puebla, Departamento de matemáticas, Puebla, México joseluis.martinez01@upaep.mx

Abstract

First-mile problems have become a major problem for the automobile industry since moving cars from production plants to selling destinations is characterized by using vehicle carriers with limited space. Although supply chain processes have automatized last-mile operations to improve productivity and increase benefits, first-mile analysis has been widely ignored. For example, in the automotive industry, cars are stored in parking lots until they are demanded, which negatively impacts delivery times and increases transportation costs. The previous issues impact the first-mile logistics of the automobile industry to the detriment of copying with delivery times and increasing the operation cost. In this paper, we deal with the previous issues by modeling the movement of cars from the parking lot to the car Carrier as an optimal control problem. Considering that not all cars should leave the parking lot, we search for conditions that guarantee the existence of a unique optimal path when the cars’ requisition is uncertain. Theoretical results provide a closed-form solution that indicates the optimal path to fill the car carrier in a time window. Such solutions allow us to study the impact of exogenous parameters (such as the parking lot size, the starting point, andmarginal costs) on the behavior and features of the optimal path.

Keywords: calculus of variations; optimization; uncertainty; transshipment; optimal paths

Resumen

Los problemas de la primera milla se han convertido en un problema importante para la industria automotriz, ya que el traslado de los automóviles desde las plantas de producción a los destinos de venta se caracteriza por el uso de portavehículos con espacio limitado. Aunque los procesos de la cadena de suministro han automatizado las operaciones de última milla para mejorar la productividad y aumentar los beneficios, el análisis de la primera milla ha sido ampliamente ignorado. Por ejemplo, en la industria automotriz, los automóviles se almacenan en estacionamientos hasta que se demandan, lo que impacta negativamente en los tiempos de entrega y aumenta los costos de transporte. Lo anterior impacta la logística de primera milla de la industria automotriz en detrimento de copiar tiempos de entrega y aumentar el costo de operación. En este artículo, abordamos las cuestiones anteriores modelando el movimiento de los automóviles desde el estacionamiento hasta el portavehículos como un problema de control óptimo. Considerando que no todos los autos deben salir del estacionamiento, buscamos condiciones que garanticen la existencia de un único camino óptimo cuando la requisición de los autos es incierta. Los resultados teóricos proporcionan una solución de forma cerrada que indica la ruta óptima para llenar el portavehículos en una ventana de tiempo. Estas soluciones nos permiten estudiar el impacto de parámetros exógenos (como el tamaño del estacionamiento, el punto de partida y los costos marginales) sobre el comportamiento y las características de la ruta óptima.

Palabras clave: cálculo de variaciones; optimización; incertidumbre; transporte; caminos óptimos

Mathematics Subject Classification: Primary: 49Q22, 35Q49, 90B06.

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Acknowledgement

JAMD acknowledges support from Consejo Nacional de Humanidades, Ciencias y Tecnologías (CONAHCyT), Universidad Popular Autónoma del Estado de Puebla (UPAEP), and Asociación Sindical del Personal Académico de la BUAP (ASPABUAP).

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Received: June 20, 2023; Accepted: January 08, 2024

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