IEDC-2010 Conference Proceedings (Download ... - NED University

3) If
k 1 k
2
, where g is a predefined tolerance, e.g.,
5
10 , then stop. Otherwise, go
to step (2).
In which, it focus on the case u
t
1, i.e., v ( x t
) ( xt
) , which implies that all drivers might reconsider
routes every day.
Where ( * i * i *
vt
( x )(1 vt
( x )) i j
x ) tij
( x*)
i * j *
vt
( x ) vt
( x ) i j
(12)
Dv x ) (1 u ) I u D ( F(
x )) DF(
x ) , D ( F(
x))
pij ( F(
x))
(13)
(
t 1 t 1 t 1
t 1
t 1
p ( F ( x
ij
(14)
t
t 1
))
i
( x
i
t 1
( x
)(1
t 1
)
j
i
( x
( x
t 1
t 1
)),
), i
x
p, t
v
)) ,
p
( xt
1)
p
( xt
1)
ui
( xt
1)
xi,
t 1
(1 u
p
( xt
1
x
p,
t 1
i P
i
j
j
(15)
Note: The relationship between the invariant covariance matrix of link flows and path flows
*
is = A A , where is the covariance matrix for the link flow vector and
* is the covariance matrix
for the path flow vector in the network.
3.3 Bi-Level Programming: Stochastic User Equilibrium
Owing to in the lower level it is difficult to get the analytical closed form both for the mean link flows
and link flow covariance matrix related to the capacity enhancement, it is tough to reformulate the bilevel
model to a single level nonlinear optimization. Therefore we combine the two levels together, and
aim at proposing an iterative algorithm to solve this SSO problem. In recent years, evolutionary
computation techniques have gained lots of attention regarding their potential as optimization techniques
for complex numerical functions. The appeal of Genetic Algorithms (GAs) comes from their simplicity
and elegance as robust heuristic search algorithms as well as from their power to discover solutions
rapidly for difficult high-dimensional problems. They are based on the concept of penalty functions,
which penalize unfeasible solutions. The fit solution in the final generation is the one that minimizes the
objective (fitness) function; this solution can be thought of as the GA has recommended choice. The bilevel
SSO problem is modeled as follows:
p,
t
exp(
j
exp(
F(
x
p,
t 1
F(
x
))
j,
t 1
))
min
Ca
a
2 2 4
0 aa 5
aa a
a a 4 a
F
0
Ca
Ca
F
t
2 1 0 2 1 0
5
(16)
s.t.
Where
* *
X ~ tMVN( , ) X 0; Ca
0, a A;
r laCa
B
C
a
is the independent variable and the design index.
C t ,
rs
rs
k a a k
a
a is obtained from: ,
a A
k
f q ,
rs
k
rs
P rs 1 1 exp( C rs C rs )) f rs q k, r,
s ,
k k j k rs
j
t t 1 C C
0 0
a a a a a
aa
is obtained from the convergent stationary matrix
* through the iterative algorithm
*
*
* T
( x ) Dv(
x ) Dv(
x )
k 1
k
4
Proceedings of IEDC 2010, 1-3 July, 2010
258
IEDC-2351-111