Problem 3. Traffic Assignment Problem Consider a simple network with 3 links and 3 nodes (see Figure below). Traffic is originated at node"" and is bounded for destinations j" and k, according to the O/D Table below. Note that drivers travelling from "i" to K have two possible routes: route s if they travel through links a and b, and route r if they travel through link c. Drivers travelling from "j" to "k" are stuck to route p" composed of link b. Given this scenario, and the performance functions for each link provided below (in some units of costs and as a function of the number of drivers travelling on the link, N), you are to: a) Derive analytically the traffic assignment under User Equilibrium (UE). b) Derive graphically the traffic assignment under UE. c) Derive the traffic assignment under User Equilibrium using an adaptive simulation. d) Derive the traffic assignment under System Optimal (SO) assignment. e) Compare the total time spent (TTS) collectively by all drivers under UE with the TTS under System Optimal (SO) assignment. Note: time here is equivalent to cost. Route "s" 0/D i j k (a) (b) i 0 0 30 Ca = 10+2N Co = 3+N, C-15+N j 0 0 10 Route" (c) k 0 0 0 Route" Note: In order to solve part c) you should fill the following table for each iteration: Iter Link Flows Link Costs Route Costs Rational Flow Adjustments # Na Nb Nc Cc C Cs Nrat Nsrat Nyrat Narat P. rat COC Nerat 1 2 Problem 3. Traffic Assignment Problem Consider a simple network with 3 links and 3 nodes (see Figure below). Traffic is originated at node"" and is bounded for destinations j" and k, according to the O/D Table below. Note that drivers travelling from "i" to K have two possible routes: route s if they travel through links a and b, and route r if they travel through link c. Drivers travelling from "j" to "k" are stuck to route p" composed of link b. Given this scenario, and the performance functions for each link provided below (in some units of costs and as a function of the number of drivers travelling on the link, N), you are to: a) Derive analytically the traffic assignment under User Equilibrium (UE). b) Derive graphically the traffic assignment under UE. c) Derive the traffic assignment under User Equilibrium using an adaptive simulation. d) Derive the traffic assignment under System Optimal (SO) assignment. e) Compare the total time spent (TTS) collectively by all drivers under UE with the TTS under System Optimal (SO) assignment. Note: time here is equivalent to cost. Route "s" 0/D i j k (a) (b) i 0 0 30 Ca = 10+2N Co = 3+N, C-15+N j 0 0 10 Route" (c) k 0 0 0 Route" Note: In order to solve part c) you should fill the following table for each iteration: Iter Link Flows Link Costs Route Costs Rational Flow Adjustments # Na Nb Nc Cc C Cs Nrat Nsrat Nyrat Narat P. rat COC Nerat 1 2