Assumed you are the planner $($i$.$e$.,$ the government$),$ the objective of the Transit Oriented Development $($TOD$)$ design is to maximize the social welfare of the urban system by determining the optimal location, number and size of the TOD zones along a rail line corridor and the train headway on that rail line.

Questions:

a$)$ Please formulate a travel cost function for a single passenger.

b$)$ Please establish the household residential location choice models.

c$)$ Please draft an elastic demand function based on travel cost.

d$)$ Please arrange the modeling to assess social welfare and its constraints. Please include the model objective function and necessary constraints.