1. Two routes connect an origin–destination pair with performance functions t1 = 5 + 4×1 and t2 = 7 + 2×2, with t’s in minutes and x’s in thousands of vehicles per hour. Assuming that both routes are used, can user-equilibrium and system-optimal solutions be equal at some feasible value of total origin-destination demand (q)? (Prove your answer.)
2. Three routes connect an origin–destination pair with performance functions t1 = 5 + 1.5×1, t2 = 12 + 3×2, and t3 = 2 + 0.2 x23 (with t’s in minutes and x’s in thousands of vehicles per hour). Determine user- equilibrium flows if q = 4300 veh/h.
3. Two routes connect an origin–destination pair with performance functions t1 = 6 + 4×1 and t2 = 2 + 0.5(x2)2 (with t’s in minutes and x’s in thousands of vehicles per hour). The origin–destination demand is 4000 veh/h at a travel time of 2 minutes, but for each additional minute beyond these 2 minutes, 100 fewer vehicles depart. Determine user-equilibrium route flows and total vehicle travel time.