Optimal Layout Plan of Stands at the Macao Food Festival via Minimizing the Electrostatic Potential Energy with the Effective Charge as Popularity of Stands

Abstract

We proposed a mathematical model for designing the layout diagram of stand locations at the Macao Food Festival. The optimal layout diagram may be defined in such a way that, while requiring the distance between every pair of stands should not be too far away from each other, the crowd control is well managed so that people may patronize stands more effectively. More popular stands may have larger patronage, resulting in higher pedestrian flow nearby. Therefore, to avoid customers from packing shoulder to shoulder around more popular stands, we may treat every stand as a charged particle carrying an effective charge: the more popular a stand is, the higher the effective charge it carries. Under this assumption, the problem is then converted to the minimization problem of Coulomb electrostatic potential energy on a specific configuration of charge locations, with which the global minimum may be found by the Simulated Annealing and Metropolis Algorithm. Electrostatic energy density is interpreted as density of customers, while electric field the reversed crowd flow. Therefore, at a certain location we are able to predict the customer density by calculating the energy density and the net crowd flow with electric field lines. We also concluded that even though the required computation time to obtain a configuration of stand locations with the energy value close to the global minimum with a tolerable difference may be irrelevant to the randomly generated initial configuration of stand locations, setting up an appropriate initial configuration could be one of the key issues to find out the actual global minimum.