10.8. Problems

1. Calculate the system travel time and link flows by doing user equilibrium assignment for the network in the given figure. Verify that the flows are at user equilibrium.

This problem is asking you to perform a user equilibrium assignment for a specific transport network. User equilibrium is where no driver can reduce their travel costs by switching routes. To solve this, you need to calculate both the total travel time across the network and the flow of traffic on each road link based on the provided travel time equations and any constraints set by the problem. This involves using the equations from the previous sections to derive a function that minimizes the travel time, considering the trip distribution and the capacities of each route. Finally, you verify the results to ensure the flows align with the user equilibrium condition.

Think of it like a busy marketplace where each stall represents a route. If customers want to buy items, they will naturally choose the stalls that have the best prices (low travel time). However, if one stall gets too crowded, customers may switch to a less popular stall that offers similar products at comparable prices. The user equilibrium ensures that no customer can find a quicker route to their desired stall without causing congestion elsewhere.

Solution Solution Substituting the travel time in equation 10.4 to 10.8 yield to x1 x2 min Z(x)= 12+3x dx + 10+5x dx 1 1 2 2 ∫0 ∫0 3x2 5x2 =12x + 1 +10x + 2 1 2 2 2 subject to x +x =12 1 2 Substituting x =12 x , in the above formulation will yield the unconstrained formulation as below: 2 1 − 3x2 5(12 x )2 min Z(x)=12x + 1 +10(12 x)+ − 1 min Z(x)=4x2 58x +480.

In this chunk, the derivation of the travel time function and flow assignment is presented. You substitute the given travel time equations into the optimization problem Z(x), which represents the total travel time as a function of traffic flow. You then integrate these equations to establish a formula that calculates the overall travel time. After substituting specific constraints, you arrive at a minimization problem that helps to determine the flow on each route. The goal here is to find the optimal distribution of traffic that satisfies both the total demand and the user equilibrium conditions.

Imagine a delivery service needing to optimize routes for multiple packages. They can calculate the fastest delivery times if they adjust traffic based on current order volumes—like how you might reconfigure the routes drivers take based on current street congestions. By updating their routes according to new deliveries, they minimize overall time—which is exactly the process we’re following in the problem statement.

Differentiate the above equation with respect to x and equating to zero, leads to the solution for x1 and x2: x1 = 7.25, x2 = 4.75

To solve for the optimal values of x1 and x2, you differentiate the travel time function Z(x) with respect to x and set the derivative equal to zero. This mathematical technique finds the points at which the total travel time is minimized. After performing this differentiation and solving the resulting equation, you find the specific flow values for x1 (traffic flow on the first route) and x2 (traffic flow on the second route). These represent the optimal traffic distribution that minimizes total travel time while maintaining user equilibrium.

Consider a person trying to decide how long to spend on two different tasks to minimize their total time. By evaluating the time benefit of working on Task A versus Task B and adjusting their focus, they find a sweet spot, where changing their focus further would only lead to increased time—a balance that mirrors our search for optimal route usage in traffic distribution.

Travel times are t1 = 12 + 3(7.25) = 33.75, t2 = 10 + 5(4.75) = 33.75, i.e., t1 = t2

To confirm that the calculated flows are at user equilibrium, you need to check that the travel times on both routes are equal (t1 = t2). This is a key characteristic of user equilibrium—when flows are balanced, drivers are indifferent between routes, as neither presents a faster option at the current volume level. By substituting the values of x1 and x2 back into the travel time equations, you find that both travel times equate, validating your solution.

Think about two friends heading to the same destination from different starting points. If one friend assures the other that they will arrive simultaneously after considering traffic, it highlights user equilibrium—neither route offers a faster alternative given the current circumstances. By confirming both routes have the same travel time, you verify that drivers would be equally satisfied choosing either path.

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