16-Civ-A6 Highway Design, Construction, and Maintenance · May 2014
Question 6 of 7: User-Equilibrium Assignment and the Braess Paradox
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
Paper format. National Examination, 98-Civ-A6 Transportation Planning & Engineering, May 2014 — 3 hours, closed book (one two-sided aid sheet). Seven questions; any five constitute a complete examination and all are of equal value (20 marks). All seven are solved below as a study resource.
Given. Two parallel routes with $t_1=14+V_1/600$, $t_2=15+V_2/1400$ (minutes); total demand $Q=5000$ veh/h. Part (b) adds Route 3 with $t_3=16+V_3/2800$.
Find. UE volumes and travel times with two routes (a) and three routes (b); and whether a new route always lowers travel time (c).
Non‑overlapping parallel routes between origin O and destination D. Route 3 (green, dashed) is added in part (b).
Approach. Apply Wardrop's first principle: at user equilibrium every used route has equal (and minimal) travel time. Invert each linear time function to $V_i(t)$, impose $\sum V_i=Q$, and solve for the common time $t$.
UE condition, two routes (a). Set $t_1=t_2$ with $V_1+V_2=5000$: $14+\dfrac{V_1}{600}=15+\dfrac{5000-V_1}{1400}$. Solving, $V_1=\boxed{1920}$ and $V_2=3080\ \text{veh/h}$.
Equilibrium time (a). $t_1=14+1920/600=17.2$ min and $t_2=15+3080/1400=17.2$ min — equal, confirming UE. The common travel time is $\boxed{17.2\ \text{min}}$.
UE with three routes (b). Write each volume as a function of the common time: $V_1=600(t-14)$, $V_2=1400(t-15)$, $V_3=2800(t-16)$. Summing to 5000: $600(t-14)+1400(t-15)+2800(t-16)=5000$, i.e. $4800\,t-74\,200=5000$, giving $t=\boxed{16.5\ \text{min}}$.
Volumes (b). Back‑substitute: $V_1=600(2.5)=1500$; $V_2=1400(1.5)=2100$; $V_3=2800(0.5)=1400$ veh/h (all positive, so all three routes are used and sum to 5000). Every route now takes $16.5$ min — a $0.7$ min improvement over part (a).
Does a new route always help? (c). No. Here Route 3 lowered travel time, but adding capacity does not guarantee improvement at UE — this is the Braess paradox. Because equilibrium is reached by individually optimising travellers (not a system optimum), a new link can attract flow in a way that raises congestion on shared segments and leaves everyone worse off. It only helps when, as here, the added route provides genuinely independent capacity between the same origin and destination.
