Probabilistic Location- Allocation-Routing Problem for Optimal Planning of Schools and Urban Transportation System

In all educational systems, selection of appropriate locations for schools in the city and allocation of students to these schools are part of the basic decisions. Also, finding the optimal route for the transportation of students is very necessary. In order to daily presence of students in schools, the traveling population in streets increases significantly. Thus, the required time for travelling a street increases. Also, stochastic events such as accidents and traffics can affect the travel time between two regions. It is obvious that with increase in the population flow in the street, probabilities of occurrence of these events increase. In the provided model, contrary to existing models in this field, the impact of population travelling and stochastic events on the location of schools, the allocation of students to the schools and routing are considered simultaneously. Generally, the aim is to determine appropriate locations as schools locations, allocate the existing students in each region to schools and find the movement path of each student to reach its corresponding school by considering direct impact of the stochastic factors and the population flow on the probabilistic travel times so that the total expected transportation time is minimized. Here, it is assumed that schools and streets have limited capacities for accepting the population. First, a function to compute the population-dependent travel times is defined and then, considering stochastic factors, a mixed-intiger nonlinear programming model is provided. To solve large problems, a hybrid algorithm incorporating genetic algorithm and simulated annealing algorithm is introduced. To validate the proposed model, a sample problem is considered and analyzed. Comparative numerical results demonstrate the potential effectiveness of the presented algorithms.