assume that the cost function on each link is: Route 1 a, + bV Route 2...

  

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assume that the cost function on each link is: Route 1 a, + b₂V₁ Route 2 a₂ + b₂V/₂ Without loss of generality, assume (U₁,U2) be the user equilibrium solution and (S₁, S₂) be the system optimal solution. Consider the same graph as in Question 1. Now (1) Express the total system travel time under UE and SO using above notations. (10pts) (2) Show that the total system travel time under UE is at most 1/3 worse than the total system. travel time under SO. (30 Bonus Points) Hint #1: Hint #2: Hint #3: (a₁ + b₁U₁)U₁ + (a₂ + b₂ U₂)U₂ - (a₁ + b₁S₁)S₁ - (az + b₂S₂)Sz <= (a₁ + b₁U₂)S₁ + (a₂ + b₂U₂)Sz (a₁ + b₁S₁)S₁ - (a₂ + b₂S₂)Sz S₁b, (U₁-S₁)=b₁(S₁U₁-S2)=b₁| b₂ ( (+) ² - (S₁ - 4+) ³) ≤ b₂ (4) ² b,U SU,(a, + b₂ U₂) Question 1 (30 points): There are two routes between origin A and destination B, and their volume delay functions are shown in the figure below (T is the travel time in minutes, V is the traffic volume in vehicle trips). The total travel demand from A to B is 1500 trips. Use the user equilibrium and system optimal approaches to assign trips on two routes. (1) How many trips are assigned to Route #1 under UE and SO? (2) Why does the system optimal approach provide lower total travel cost than the user equilibrium approach? (3) How would you modify the cost function of the two routes such that the outcome of the UE is the same as the outcome of the SO? Route#1 T₁=10+ 0.05 V₁ B Route#2 T₂ = 15 +0.005 V₂ assume that the cost function on each link is: Route 1 a, + b₂V₁ Route 2 a₂ + b₂V/₂ Without loss of generality, assume (U₁,U2) be the user equilibrium solution and (S₁, S₂) be the system optimal solution. Consider the same graph as in Question 1. Now (1) Express the total system travel time under UE and SO using above notations. (10pts) (2) Show that the total system travel time under UE is at most 1/3 worse than the total system. travel time under SO. (30 Bonus Points) Hint #1: Hint #2: Hint #3: (a₁ + b₁U₁)U₁ + (a₂ + b₂ U₂)U₂ - (a₁ + b₁S₁)S₁ - (az + b₂S₂)Sz <= (a₁ + b₁U₂)S₁ + (a₂ + b₂U₂)Sz (a₁ + b₁S₁)S₁ - (a₂ + b₂S₂)Sz S₁b, (U₁-S₁)=b₁(S₁U₁-S2)=b₁| b₂ ( (+) ² - (S₁ - 4+) ³) ≤ b₂ (4) ² b,U SU,(a, + b₂ U₂) Question 1 (30 points): There are two routes between origin A and destination B, and their volume delay functions are shown in the figure below (T is the travel time in minutes, V is the traffic volume in vehicle trips). The total travel demand from A to B is 1500 trips. Use the user equilibrium and system optimal approaches to assign trips on two routes. (1) How many trips are assigned to Route #1 under UE and SO? (2) Why does the system optimal approach provide lower total travel cost than the user equilibrium approach? (3) How would you modify the cost function of the two routes such that the outcome of the UE is the same as the outcome of the SO? Route#1 T₁=10+ 0.05 V₁ B Route#2 T₂ = 15 +0.005 V₂