QQuestionAnatomy and Physiology
QuestionAnatomy and Physiology
# Problem 4:
Bureau of Public Roads (BPR) function (shown below) is a travel time function used in urban transportation planning to calculate travel time on a link given volume on the link:
t=t_{0}\left(1 +\alpha\left(\frac{v}{C}\right)^{\beta}\right)
Where, $t=$ travel time on the link $t_{0}=$ free-flow travel time on the link $v=$ volume on the link $C=$ capacity of the link $\alpha, \beta=$ parameters of the model estimated for local condition Demand between a given origin-destination (OD) pair in a transportation network is 10,100 vehicles/hour. There are three independents links connecting the OD pair, whose travel times can be represented using the BPR function. Parameters of the BPR function for the three links are given below.
| BPR parameter | Link 1 | Link 2 | Link 3 |
| --- | --- | --- | --- |
| Free-flow travel time, $t_{0}$ (minutes) | 17.0 | 15.5 | 12.5 |
| Capacity, $C$ (vehicles/hour) | 3800 | 4200 | 6600 |
| $\alpha$ | 0.2 | 0.3 | 0.4 |
| $\beta$ | 2.8 | 3.7 | 4.3 |
Perform three iterations of the method of successive averages (MSA) to find user equilibrium assignment flows between the given OD pair. Show hard calculations for each iteration. At the end, summarize your calculations in a table by showing link flows and travel times at the end of each iteration. It's OK if flow on a link exceeds its capacity (this method allows for such scenarios, but penalizes such routes through rapidly increasing travel times).
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Step 1: Initialize the link flows and calculate the initial travel times.
Link 3: $$t_{3} = 12.5(1 + 0.4(3600/6600)^{4.3}) = 18.13 \text{ minutes}
We start by assuming an initial flow for each link. Let's assume the initial flow for Link 1 is 3000 vehicles/hour, Link 2 is 3500 vehicles/hour, and Link 3 is 3600 vehicles/hour. Using the BPR function, we can calculate the initial travel times:
Step 2: Calculate the average travel time and update the link flows.
v'_{3} = 3600 \frac{21.24}{18.13} = 4102.59 \approx 4103 \text{ vehicles/hour}
Using the initial travel times calculated in Step 1: Now, we update the link flows based on the average travel time. The updated flow for Link 1 is: Similarly, for Link 2 and Link 3:
Final Answer
The table above shows the link flows and travel times at the end of each iteration of the method of successive averages (MSA) for finding user equilibrium assignment flows between the given OD pair.
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