Solution: Part A
What is the flow and travel time on each link? Complete the table below for Network A:
Link Attributes
Link
|
Link Performance Function
|
Flow
|
Time
|
---|
o-p |
 |
|
|
p-r |
 |
|
|
o-q |
 |
|
|
q-r |
 |
|
|
These four links are really 2 links O-P-R and O-Q-R, because by conservation of flow Qop = Qpr and Qoq = Qqr.
Link Attributes
Link
|
Link Performance Function
|
Flow
|
Time
|
---|
o-p-r |
 |
|
|
o-q-r |
 |
|
|
By Wardrop's Equilibrium Principle, the travel time (cost) on each used route must be equal. Therefore
.
OR
By the conservation of flow principle
By substitution
Check
Check (within rounding error)
Link Attributes
Link
|
Link Performance Function
|
Flow
|
Time
|
---|
o-p-r |
 |
2.84 |
42.01
|
o-q-r |
 |
3.15 |
42.01
|
or expanding back to the original table:
Link Attributes
Link
|
Link Performance Function
|
Flow
|
Time
|
---|
o-p |
 |
2.84 |
14.2
|
p-r |
 |
2.84 |
27.84
|
o-q |
 |
3.15 |
26.3
|
q-r |
 |
3.15 |
15.75
|
User Equilibrium: Total Delay = 42.01 * 6 = 252.06
Part B
What is the system optimal assignment?
Conservation of Flow:
Analytic Solution requires minimizing total delay
And we can compute the SO travel times on each path
Note that unlike the UE solution,
Total Delay = 3.04(25+ 6*3.04) + 2.96(20+7*2.96) = 131.45+120.53= 251.98
Note: one could also use software such as a "Solver" algorithm to find this solution.
Part C
What is the Price of Anarchy?
User Equilibrium: Total Delay =252.06
System Optimal: Total Delay = 251.98
Price of Anarchy = 252.06/251.98 = 1.0003 < 4/3