Problem 1 In modeling transportation network flows, different assumptions can be made to describe how individuals or shippers select the routes they or their commodities will use. In class, we described two of these assumptions: user equilibrium and system optimal. The network shown below will be analyzed using each of these two assumptions: 1 3 4 2 Demand: Q14 = 2000 vehicles/hour Q24 = 1000 vehicles/hour Supply: the link performance functions are given below (note that the subscripts refer to the starting and ending nodes of each link): 113 = 0.5 +0.001X13 t23 = 0.5 +0.001X23 t14 = 1.0 +0.0005X14 t24 = 1.0 +0.001X24 t34 = 1.0 +0.002X34 a. Write down the equations that determine the equilibrium solution under the System Optimal rule and solve for the volumes and travel times. The following relationships are necessary: the objective function, the flow conservation equations, and the link performance functions. What is the total system-wide travel time? b. Write down the User Equilibrium volumes and travel times. What is the total system-wide travel time in this case? c. Describe a transportation case (the actors, their locations on the network, and their objectives, the facilities making up the links, the nodes) for the above network, in which: (i) System Optimal would be the appropriate assumption; (ii) User Equilibrium would be the appropriate assumption