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# 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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