System-Optimal Route Flows

1. Two routes connect an origin and destination with performance functions t1 = 5 + 3×1 and t2 = 7 + x2, with t’s in minutes and x’s in thousands of vehicles per hour. Total origin–destination demand is 7000 vehicles in the peak hour. What are user- equilibrium and system-optimal route flows and total travel times?


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2.Two routes connect an origin and a destination. Their performance functions are t1 = 3 + 1.5(x1/c1)2 and t2 = 5 + 4(x2/c2), with t’s in minutes and x’s and c’s being route flows and capacities, respectively. The origin–destination demand is 6000 vehicles per hour, and c1 and c2 are equal to 2000 and 1500 vehicles per hour, respectively. Proposed capacity improvements will increase c2 by 1000 vehicles per hour. It is known that the routes are currently in user equilibrium, and

it is estimated that each 1-minute reduction in route travel time will attract an additional 500 vehicles per hour (from latent travel demand and mode shifts). What will the user-equilibrium flows and total hourly origin–destination demand be after the capacity improvement?