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Amount of travel on road or transit line is
o a e
1.204 Lecture 18
Continuous constrained nonlinear
optimization:optimization:
Convex combinations 1:
Network equilibrium
Transportation network flows
- Amount of travel on any road or transit line isany
result of many individuals’ decisions
- These depend on price and quality of service
- Congestion in urban areas is a significant factor
- Analyzing passenger flows on networks relies on:
- Graph data structures
- Shortest path algorithms
- NetNetwork assignment algorithms that assign travelers to rk assignment lgorithms that assign tra lers to a particular set of streets or transit lines, based on travel time, cost and other service measures
- Demand models are also used
- Based on discrete choice theory (take 1.202!)
t t t t t t
Transportation network equilibrium
- Users make their own, ‘selfish’ decisions on the
best path through a network
- When congestion exists, traveler choices affect travel times, which in turn affect traveler choices, which…
- Users switch routes (and modes and time of day and trip frequency and location) in response to changes in service quality
- We model this as a market that reaches supply-demand equilibrium on every arc in a network
Figures from Sheffi
Definition of equilibrium
- Links (including intersections) have a supply function:
- Definition of equilibrium:
- For each origin-destination pair:
- Travel time for all used paths is equal, and is
- LLess thhan ((or equall to)) thhe travell tiime on any unusedd pathh
(Transit is messier, because it has a route structure as well as a network structure, but the same principles apply)
Q Quantity
Q*
P *
P Price
Supply function
Demand function
Figure by MIT OpenCourseWare.
1
3
2
5 4
1 2 3 4
5 6
7 8 9
10
Figure by MIT OpenCourseWare.
Link Travel Time [min]
Link Flow [veh/hr]
Free flow travel time
X3 Capacity (^) ω
t
Figure by MIT OpenCourseWare.
Formulation example
..
min ( ) ( 2 ) ( 1 2 ) 0 0
1 2 = + + + ∫ ∫
s t
zx d d
x x
ω ω ω ω
..
min ( ) ( 2 ) ( 1 2 )
1 5
0 , 0
5
5
0 0
2 1
1 2
1 2
1 1 = + + +
− = −
≥ ≥
∫ ∫
−
s t
zx d d
Convertto Dbysettingx x
x x
x x
x x ω ω ω ω
0 3
( )
( ) 1. 5 9 30
:
0 , 5 0
..
1 1
1
1
2 1
1 1
= ⇒ =
= − +
≥ − ≥
x dx
dzx
zx x x
Integrateanalytically
x x
st
z(x)= 2x 1 + x 12 /2 + (5-x 1 ) + (5-x 1 )^2
= 2x 1 + 0.5x 12 + 5 -x 1 + 25 - 10x 1 + x 12
Formulation
- The formulation has no economic or physical significancep y g
- It happens to produce the desired first-order conditions for an optimum
- They require that the time on all routes used between an origin and destination be equal
- And the time on routes not used must be greater
- We can view the objective function as a convergence criterion for the equilibrium solution
- Nonetheless, equilibrium is a key conceptq y p
- And it’s a nonlinear optimization problem, techniques for which we want to cover in this course
- This is a constrained continuous nonlinear optimization problem
Solution method: convex combinations
..
min ( )
i
hij xi b j j
st
z x
∑ ≥^ ∀
( )/|| ||
.
( , ,..., )
,
( , ,..., )
1 2
1 2
n n n
n n I
n n n
n I
n n n
i
Directionfromx toyisunitvector y x y x
maximumdecrease
solutiony y y y sodirectionfromx toygives
Tofinddescentdirectionwewishtofindauxiliaryfeasible
Assumecurrentsolutionisx x x x
− −
=
=
)
( ) ,...,
( ) ,
( ) ( ) ( || ||
( ) ( )
( ) ( )
(|| || )
( ) || ||
1 2 j
n n
n T n
n n
x
z x
x
z x
x
z x where z x y x
y x zx
Slopeofz x indirectionof y x
v means v v
f y y y
∂
∂
∂
∂
∂
∂ ∇ = −
− −∇ ⋅
− =
⋅
Solution method: convex combinations 2
min ( ) ( ) ( ) ( )
n n n n T L
hy b
st
z y zx zx y x
Rewriteoriginalproblemaslinearapproximation
min ( ) ( )
i i (^) i
n n n T L
n n
n n
i
ij i j
y x
zx z y zx y
Also zx isconstantatx x sowecandropitleaving
Atx x valueof objectivefunctionisconstant:wecandropzx
hy b
n n
i
ij i j
Itgivesadescentdirection y x
Thisisalinearprogramwhosesolutionisy
hy b
st
Next time
- We’ll apply the convex combinations method to the network
equilibrium problem
pp y
- Formulation
- Algorithms
- Direction finding
- Shortest path algorithm solves the linear program
- Compute y flow vector (auxiliary solution)
- Line search
- Bisection solves the line search problem
- – Must compute derivative of objective functionMust compute of objective function
- Move
- Update x flows on network as linear combination of x and y flows
- Update arc travel times; both of these steps are just algebra
- Convergence test
- Compute change in flows as simplest measure
- Java implementation
derivative
