Coping with Variability in
Dynamic Routing
Problems
Tom Van Woensel (TU/e)
Christophe Lecluyse (UA), Herbert Peremans
(UA), Laoucine Kerbache (HEC) and Nico
Vandaele (UA)
Problem Definition
1200
96
94
1000
92
800
90
600
Flow
88
Speed
86
400
84
200
82
23
22
21
20
19
18
17
16
15
14
13
12
11
9
10
8
7
6
80
5
0
1200
96
94
1000
92
800
1200
90
96
600
94
88
Flow
Speed
1000
86
92
400
800
84
90
1200
96
200
94
1000
Speed
92
23
22
21
20
19
18
17
16
15
14
13
12
9
11
400
10
5
86
80
8
0
7
88
82
6
600
Flow
800
90
84
200
600
82
86
400
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
80
5
0
88
84
200
82
94
1000
1200
96
92
94
800
1000
90
92
600
88
Flow
Speed
800
90
86
400
600
88
84
86
200
82
400
84
200
82
23
22
21
20
19
18
17
16
15
14
13
12
11
9
10
8
7
80
6
0
5
23
22
21
20
19
18
17
16
15
14
13
12
11
9
10
8
7
6
80
5
0
Flow
Speed
23
22
21
20
19
18
17
16
15
14
13
12
11
9
8
10
80
7
96
6
1200
5
0
Flow
Speed
Previous work
 Deterministic Dynamic Routing Problems
 Inherent stochastic nature of the routing
problem due to travel times
 Average travel times modeled using queueing
models
 Heuristics used:



Ant Colony Optimization
Tabu Search
Significant gains in travel time observed
 Did not include variability of the travel times
A refresher on the queueing approach
to traffic flows
Speed
vf
Speed-density
diagram
Speed-flow diagram
v2
v1
Density
Traffic flow
kj
k2
k1
q
q
qmax
Flow-density diagram
Traffic flow
qmax
Queueing framework
Queue
Service Station (1/kj)
1
kj
v
W
Queueing
vf
v
1 T
T: Congestion parameter
Travel Time Distribution: Mean
 P periods of equal length Δp with a different
travel speed associated with each time period
p (1 < p < P)
 TT  k * p
 Decision variable is number of time zones k

Depends upon the speeds in each time zone
and the distance to be crossed
Travel Time Distribution: Variance I
 TT  k * p (Previous slide)
 Var(TT)  p2 Var(k)
 Variance of TT is dependent on the variance
of k, which depends on changes in speeds

i.e. Var(k) is a function of Var(v)
 Relationship between (changes in k) as a
result of (changes in v) needs to be
determined: k =  v
Travel Time Distribution: Variance III
Speed v
A
vavg
v
B
t0
Area A + Area B = 0  k =  v
Time zones k
Travel Time Distribution: Variance IV
 k   v (and  ~ f(v, kavg, p))
 Var(k)  2 Var(v)
 Var(v) ?
 1 
 kj 
Var (v)  Var

 W 


1
1
 Var (v)  2 Var 
kj
W 
Travel Time Distribution: Variance V
 What is Var(1/W)?


Not a physical meaning in queueing theory
Distribution is unknown but:


Assume that W follows a lognormal distribution
(with parameters  and )
Then it can be proven that: (1/W) also follows a
lognormal distribution with (parameters - and )
 See Papoulis (1991), Probability, Random Variables
and Stochastic Processes, McGraw-Hill for general
results.
Travel Time Distribution: Variance VI
With (1/W) following a Lognormal distribution,
the moments of its distribution can be related
to the moments of the distribution for W as
follows:
2
c
1
  W 1
E  
 W  E (W )
1
2   1 
Var    cW  E  
W 
  W 
2
Var (W )
c 
E (W ) 2
2
W
W ~ LN
Travel Time Distribution
 1/W ~ LN
 v ~ LN
 TT ~ LN
 Assumption is acceptable:
 If W ~ LN

Production management often W ~ LN


E.g. Vandaele (1996); Simulation + Empirics
Traffic Theory often TT ~ LN

Empirical research: e.g. Taniguchi et al. (2001) in
City Logistics
Travel Time Distribution: Overview
 TT ~ Lognormal distribution
E (TT )  kp
p  2  c  1 
cW 
Var (TT ) 

2
kj
 E (W ) 
2
2
2
W
2

 Var (W )
 1
2
2 2

p  Var (W ) E (W ) 


Var (TT ) 
2
2
k j E (W )   E (W ) 


2
E(W) and Var(W) see e.g. approximations Whitt for GI/G/K queues
Finding solutions for the Stochastic
Dynamic Routing Problem
1200
96
94
1000
92
800
90
600
88
86
400
Flow
Speed
Data generation:
Routing problem
Traffic generation
84
200
82
Tabu Search
23
22
21
20
19
18
17
16
15
14
13
12
11
9
10
8
7
6
80
5
0
Heuristics
Solutions
Ant Colony Optimization
Objective Functions I
 Results for F1(S):


Significant and consistent improvements in
travel times observed (>15% gains)
Different routes
Objective Functions II
 Objective Function F2(S)


No complete results available yet
Preliminary insights:




Not necessarily minimal in Total Travel Time
Variability in Travel Times is reduced
Recourse: Less re-planning is needed
Robust solutions
Conclusions
 Travel Time Variability in Routing Problems
 Travel Times


Lognormal distribution
Expected Travel Times and Variance of the
Travel Times via a Queueing approach
 Stochastic Routing Problems
 Time Windows !
Questions?