Dipartimento di Informatica e Sistemistica Dir. Prof. Bruno Fadini

1
MODELS AND METHODS FOR THE OPTIMAL LOCATION OF
TRAFFIC SENSORS AND VMSs A. Sforza DIS -
Università di Napoli Federico II Corso di
Ottimizzazione su Rete A.A. 2010/11
2
Outline of presentation

• Context
• Flow intercepting facility location problems
• Applications in Traffic Management and Control
• Optimization models proposed in literature
• Computational experience
• Proposals of new constraints
• A simple heuristic and some improving
  modifications
• Application to Traffic Network in Naples

3
Facility Location Problems

•  
• - Flow generating and/or attracting facilities
• vertex point path
• - Flow intercepting facilities
• in vertices on links

4
Flow generating and/or attracting
facilitiesvertex location

5
Flow generating facilitiesService reaches the
clients or vice-versa

6
Flow intercepting facilities

7
Flow intercepting facilities

8
Flow intercepting facilities O-D demand
flows
9
Flow intercepting facilities in the
vertices (two facilities)
10
Flow intercepting facilities on the links
(three facilities)
11

The flow intercepting facility location problem
is a problem of path covering
12
Applications in Traffic Management and Control

• Location of
• Traffic counting sensors (for o-d matrix
  estimation)
• To know a set of link flows or all the link flows
• Variable message systems
• Fixed
• Mobile
• Traffic checkpoints

13
Applications Service Facilities A
classification scheme

• Voluntary service facilities
• Car service stations, automatic teller machine
• Unconscious service facilities
• Traffic counting sensors
• Unvoluntary service facilities
• Variable message signs
• Compulsory service facilities
• Traffic check points
• Inspection Stations

14
Traffic Management and Control Applications

• Traffic counting sensors
• No need of double counting
• Variable message systems
• There could be the need of
• double (or more) intercepting

15
m 2 facilities

• No double counting Double counting for
  path p2
• p1 p2 p3 p1 p2
  p3

16
Available information

• - Information on path flows
• Information on link flows
• Assumption
• The flow pattern is not modified by facility
  location
• This is surely true for traffic sensors
• It could be not true for VMS

17
Information on path flows- Problem variables
G (N, A) N, set of vertices A, set of
links p path, P set of paths
18
 
Model P1 Maximization of the intercepted flow
with a fixed number of facilities
Max ? fp xp
p ?P
n
s.t. ? yj m
j1
? yj ? xp ? p?P
j ?p
yj 0, 1 xp 0, 1
 
19
 
Model P2 Minimization of the facility number to
intercept a fixed of the total demand
n
Min ? yj
j1
s.t. ? yj ? xp ? p?P (1)
j?p
? fp xp ? C?. (2)
p?P
yj 0, 1 ? j?N
xp 0, 1 ? p?P
 
20
Intercepting all the demanded flows

• ? p?P fpxp ? C?
• If we want to intercept all the demanded flows
• that is if C? ? p?P fp
• ? p?P fpxp ? ? p?P fp ? xp 1 ? p?P
• The second constraint disappears
• The first set of constraints becames
• ? j? p yj ? 1

21
 
Model P3 Minimization of the facility number
to intercept the total demand (i.e. to cover
all the paths)
n
Min ? yj
j1
s.t. ? yj ? 1 ? p?P
j? p
yj 0, 1 ? j?N
 
22
Model Output

• Solving the model P1 produces the location of the
  m facilities giving the maximization of the
  intecepted flows, but it does not always give the
  exact values of the yp variables
• Solving the model P2 produces the number and the
  location of the facilities needed to intercept a
  fixed percent of the total demand
• and the list of the covered paths (i.e. exact
  values of yp variables)
• Solving the model P3 produces the number and the
  location of the facilities needed to intercept
  the total demand (i.e. all the paths)

23
Location in vertices Location on links

• Location in vertices is powerful for sensor
  location
• to counting the flows of all the junction
  movement
• It is possible from the technological viewpoint
• using cameras and virtual sensors for each lane
• and so for each movement in the junction.
• Unfortunatly its result can be affected by
  errors,
• sometimes relevant as we will see after.
• For VMS location vertex location is not
  practicable,
• because users have to be informed in the middle
  of the link

24
Transform a vertex model in a link modelthrough
a dummy vertex

• In any case a vertex model is much more
  manageable, because the number of variables is
  more tractable with respect to the number of
  variables of a link model.
• Really it is possible to adopt a vertex model as
  a link model using a dummy vertex for each link
• For a single direction
• For both directions

25
Computational tests problem P1Nnodes_o/dpairs_pat
hsforodpairs_nodesforpaths
Network nodes plants Sol. value Gap Time (secs)
N100_5_3_5 100 5 91 0.00 0.06
N200_10_3_10 200 10 183 0.00 0.05
N300_30_4_15 300 15 777 0.00 321.95
N500_50_5_20 500 25 1620 1.73 1h
N1000_100_5_25 1000 50 3020 5.40 1h
26
Computational tests problem P2 (60)Nnodes_o/dpai
rs_pathsforodpairs_nodesforpaths
Network nodes plants Gap Time (secs)
N100_5_3_5 100 2 0.00 0.02
N200_10_3_10 200 3 0.00 0.03
N300_30_4_15 300 7 0.00 0.78
N500_50_5_20 500 9 0.00 4.45
N1000_100_5_25 1000 16 0.00 593.22
27
Computational tests problem P3Nnodes_o/dpairs_pat
hsforodpairs_nodesforpaths
Network nodes plants Gap Time (secs)
N100_5_3_5 100 3 0.00 0.02
N200_10_3_10 200 8 0.00 0.02
N300_30_4_15 300 17 0.00 134.56
N500_50_5_20 500 23 1.68 1h
N1000_100_5_25 1000 47 9.86 1h
28
Modification 1 of P2 model for traffic sensors
location

• The constraint (2) can be referred to a single
  o/d pair
• ? p?Pod fpxp ? C?
• for each o/d pair of a given set of o/d pair
• where Pod is the set of paths used to serve
  this o/d pair

29
Modification 2 of P2 model for traffic sensors
and VMS location

• To ensure that at least k paths of an od pair
  are intercepted the model can be integrated with
  the constraint
• ? p?Pod xp ? K
• for each o/d pair of a given set of o/d pair
• where Pod is the set of paths used to serve
  this o/d pair

30
Modification 3 of P2 or P3 modelsfor VMS Location

• To ensure that at least h plants intercept a
  path p
• the model can be integrated with the constraint
• ? j?p yj? h
• for each path p of a given set of relevant
  paths

31
Computational Times (sec)
32
CT Modification 1 of P2 Model
33
CT Modification 2 of P2 Model
34
Need of heuristic

• For real networks with medium-large size
• an heuristic approach seems unavoidable

35
A small network

1
4
3
2
7
6
5
Path 1 1- 2 - 5 Path 2 1 - 2 4 Path 3 1
3 4 Path 4 1 3 7 Path 5 2 - 5 Path 6 2
4 - 6 Path 7 3 4 - 6 Path 8 3 7
36
O/D paths

1
4
3
2
7
6
5
Path 1 1- 2 5 (1) Path 2 1 - 2 4 (2) Path
3 1 3 4 (2) Path 4 1 3 7(1) Path 5 2
5 (1) Path 6 2 4 - 6 (1) Path 7 3 4 6
(1) Path 8 3 7 (1)
37
A greedy heuristicBerman et al. (1992), Yang
and Zhou (1998)

• Coverage matrix B
• (path/link incidence matrix)
• The rows correspond to the paths p ?p ?P
• The columns correspond to the links a ?a ?A
• Each element bpa 1 if link a belongs to the
  path p
• 0 otherwise
• The coverage matrix can be obtained
• with an assignment model

38
The coverage matrix B
Link Path(flow) 1-2 1-3 2-4 3-4 2-5 4-6 3-7
p1(1) 1 0 0 0 1 0 0
p2(2) 1 0 1 0 0 0 0
p3(2) 0 1 0 1 0 0 0
p4(1) 0 1 0 0 0 0 1
p5(1) 0 0 0 0 1 0 0
p6(1) 0 0 1 0 0 1 0
p7(1) 0 0 0 1 0 1 0
p8(1) 0 0 0 0 0 0 1
39
A greedy heuristicBerman et al. (1992), Yang
and Zhou (1998)

• Scheme of the heuristic
• Step 0 set k0. Let B(k) be the coverage matrix
• Step 1 Compute fa(k) f ?a(k) , ?a ?A
• Step 2 Find aj fJ (k) max a ?A fa(k) and
  locate a facility in link aj
• (if more than one choose the link with lowest
  index,or better,
• choose the link belonging to the greatest number
  of paths)
• Step 3 Update the coverage matrix and generate
  B(k1)
• deleting the column corresponding to link aj
• (bpj(k1)0 ?p ?P)
• deleting the rows corresponding to the paths
  intercepted from aj)
• (bpa(k1)0 ?a ?A, for each p such that bpj(k)1
• Step 4 if bpa0 ?p ?P, ?a ?A , then STOP.
• otherwise, set kk1 and return to step 1

40
First step of the heuristic
Link Path(flow) 1-2 (3) 1-3 (3) 2-4 (3) 3-4 (3) 2-5 (2) 4-6 (2) 3-7 (2)
p1(1) 1 0 0 0 1 0 0
p2(2) 1 0 1 0 0 0 0
p3(2) 0 1 0 1 0 0 0
p4(1) 0 1 0 0 0 0 1
p5(1) 0 0 0 0 1 0 0
p6(1) 0 0 1 0 0 1 0
p7(1) 0 0 0 1 0 1 0
p8(1) 0 0 0 0 0 0 1
41
Second step
Link Path(flow) 1-2 (3) 1-3 (3) 2-4 (1) 3-4 (3) 2-5 (1) 4-6 (2) 3-7 (2)
p1(1) 1 0 0 0 1 0 0
p2(2) 1 0 1 0 0 0 0
p3(2) 0 1 0 1 0 0 0
p4(1) 0 1 0 0 0 0 1
p5(1) 0 0 0 0 1 0 0
p6(1) 0 0 1 0 0 1 0
p7(1) 0 0 0 1 0 1 0
p8(1) 0 0 0 0 0 0 1
42
Third step
Link Path(flow) 1-2 (3) 1-3 (3) 2-4 (1) 3-4 (1) 2-5 (1) 4-6 (2) 3-7 (1)
p1(1) 1 0 0 0 1 0 0
p2(2) 1 0 1 0 0 0 0
p3(2) 0 1 0 1 0 0 0
p4(1) 0 1 0 0 0 0 1
p5(1) 0 0 0 0 1 0 0
p6(1) 0 0 1 0 0 1 0
p7(1) 0 0 0 1 0 1 0
p8(1) 0 0 0 0 0 0 1
43
Forth step
Link Path(flow) 1-2 (3) 1-3 (3) 2-4 (0) 3-4 (0) 2-5 (1) 4-6 (2) 3-7 (1)
p1(1) 1 0 0 0 1 0 0
p2(2) 1 0 1 0 0 0 0
p3(2) 0 1 0 1 0 0 0
p4(1) 0 1 0 0 0 0 1
p5(1) 0 0 0 0 1 0 0
p6(1) 0 0 1 0 0 1 0
p7(1) 0 0 0 1 0 1 0
p8(1) 0 0 0 0 0 0 1
44
Fifth and last step
Link Path(flow) 1-2 (3) 1-3 (3) 2-4 (0) 3-4 (0) 2-5 (1) 4-6 (2) 3-7 (1)
p1(1) 1 0 0 0 1 0 0
p2(2) 1 0 1 0 0 0 0
p3(2) 0 1 0 1 0 0 0
p4(1) 0 1 0 0 0 0 1
p5(1) 0 0 0 0 1 0 0
p6(1) 0 0 1 0 0 1 0
p7(1) 0 0 0 1 0 1 0
p8(1) 0 0 0 0 0 0 1
45
Comparison between heuristic and exact approach

• This heuristic produces very fast solution,
• but the result can be much far from the exact
  solution

46
Model P3 exact solution

• 4 facilities on links 2-4, 3-4, 2-5, 3-7

1
4
3
2
7
6
5
Path 1 1- 2 5 (1) Path 2 1 - 2 4 (2) Path
3 1 3 4 (2) Path 4 1 3 7(1) Path 5 2
5 (1) Path 6 2 4 - 6 (1) Path 7 3 4 6
(1) Path 8 3 7 (1)
47
Greedy solution

• 5 facilities on links 1-2, 1-3, 2-5, 4-6, 3-7

1
4
3
2
7
6
5
Path 1 1- 2 5 (1) Path 2 1 - 2 4 (2) Path
3 1 3 4 (2) Path 4 1 3 7(1) Path 5 2
5 (1) Path 6 2 4 - 6 (1) Path 7 3 4 6
(1) Path 8 3 7 (1)
48
A simple improvement of the heuristic

• The heuristic can be improved in the step 2
• Step 2 Find aj fJ (k) max a ?A fa(k) and
  locate a facility in link a
• (if more than one choose the link with lowest
  index)
• Alternative
• 1. Choose the link belonging to the greatest
  number of paths
• 2. Modify the selection criterion of the links

49
A simple network

3
10
6
8
2
5
7
9
4
1
O/D pair 1 9 2 9 2
10 3 10 Path 1 1-4-7-9
Path 2 2-5-7-9 Path 3 2-5-8-10
Path 4 3-6-8-10
50
Possible solution 1 (sub-optimal)

3
10
6
8
2
5
7
9
4
1
O/D pair 1 9 2 9 2
10 3 10 Path 1 1-4-7-9
Path 2 2-5-7-9 Path 3 2-5-8-10
Path 4 3-6-8-10
51
Possible solution 2 (optimal)

3
10
6
8
2
5
7
9
4
1
O/D pair 1 9 2 9 2
10 3 10 Path 1 1-4-7-9
Path 2 2-5-7-9 Path 3 2-5-8-10
Path 4 3-6-8-10
52
The coverage matrix B
1-4 (1) 2-5 (2) 3-6 (1) 4-7 (1) 5-7 (1) 5-8 (1) 6-8 (1) 7-9 (2) 8-10 (2)
p1 1 1 1
p2 1 1 1
p3 1 1 1
p4 1 1
53
Step 1a
1-4 (1) 2-5 (2) 3-6 (1) 4-7 (1) 5-7 (1) 5-8 (1) 6-8 (1) 7-9 (2) 8-10 (2)
p1 1 1 1
p2 1 1 1
p3 1 1 1
p4 1 1
54
Step 1b
1-4 (1) 2-5 (2) 3-6 (1) 4-7 (1) 5-7 (1) 5-8 (1) 6-8 (1) 7-9 (2) 8-10 (2)
p1 1 1 1
p2 1 1 1
p3 1 1 1
p4 1 1 1
55
Step 1b
1-4 (1) 2-5 (2) 3-6 (1) 4-7 (1) 5-7 (1) 5-8 (1) 6-8 (1) 7-9 (2) 8-10 (2)
p1 1 1 1
p2 1 1 1
p3 1 1 1
p4 1 1 1
56
Step1c
1-4 (1) 2-5 (0) 3-6 (1) 4-7 (1) 5-7 (01) 5-8 (01) 6-8 (1) 7-9 (11) 8-10 (11)
p1 1 1 1
p2 1 1 1
p3 1 1 1
p4 1 1 1
57
Step 2a
1-4 (1) 2-5 (0) 3-6 (1) 4-7 (1) 5-7 (01) 5-8 (01) 6-8 (1) 7-9 (11) 8-10 (11)
p1 1 1 1
p2 1 1 1
p3 1 1 1
p4 1 1 1
58
Step 2b

1-4 (01) 2-5 (0) 3-6 (1) 4-7 (1) 5-7 (01) 5-8 (01) 6-8 (1) 7-9 (11) 8-10 (11)
p1 1 1 1
p2 1 1 1
p3 1 1 1
p4 1 1 1
59
Step 2b

1-4 (01) 2-5 (0) 3-6 (1) 4-7 (1) 5-7 (01) 5-8 (01) 6-8 (1) 7-9 (11) 8-10 (11)
p1 1 1 1
p2 1 1 1
p3 1 1 1
p4 1 1 1
60
Step 2c

1-4 (01) 2-5 (0) 3-6 (1) 4-7 (1) 5-7 (01) 5-8 (01) 6-8 (1) 7-9 (0) 8-10 (11)
p1 1 1 1
p2 1 1 1
p3 1 1 1
p4 1 1 1
61
Step 3a
1-4 (01) 2-5 (0) 3-6 (1) 4-7 (01) 5-7 (01) 5-8 (01) 6-8 (1) 7-9 (0) 8-10 (11)
p1 1 1 1
p2 1 1 1
p3 1 1 1
p4 1 1 1
62
Step 3b
1-4 (01) 2-5 (0) 3-6 (1) 4-7 (01) 5-7 (01) 5-8 (01) 6-8 (1) 7-9 (0) 8-10 (11)
p1 1 1 1
p2 1 1 1
p3 1 1 1
p4 1 1 1
63
Step 3b
1-4 (01) 2-5 (0) 3-6 (1) 4-7 (01) 5-7 (01) 5-8 (01) 6-8 (1) 7-9 (0) 8-10 (11)
p1 1 1 1
p2 1 1 1
p3 1 1 1
p4 1 1 1
64
Step 3c (all the flows are intercepted)
1-4 (01) 2-5 (0) 3-6 (01) 4-7 (01) 5-7 (01) 5-8 (01) 6-8 (01) 7-9 (0) 8-10 (0)
p1 1 1 1
p2 1 1 1
p3 1 1 1
p4 1 1 1
65
Check of the solution links
1-4 (01) 2-5 (0) 3-6 (01) 4-7 (01) 5-7 (01) 5-8 (01) 6-8 (01) 7-9 (0) 8-10 (0)
p1 1 1 1
p2 1 1 1
p3 1 1 1
p4 1 1 1
66
Final Solution
1-4 (01) 2-5 (0) 3-6 (01) 4-7 (01) 5-7 (01) 5-8 (01) 6-8 (01) 7-9 (0) 8-10 (0)
p1 1 1 1
p2 1 1 1
p3 1 1 1
p4 1 1 1
67
Revised greedy heuristic

• Step 0 set k0. Let B(k) be the coverage matrix
• Step 1 For each link a ?A, compute f1a(k),
  f2a(k)
• Where f1a(k) is the flow to intercept
• f2a(k) is the flow already intercepted
• Step 2 Sort the links in decreasing lexicografic
  order with respect to the couple f1a(k), f2a(k)
  and locate a facility in the first link aj
• Step 3 Update the coverage matrix and generate
  B(k1)
• - deleting the column corresponding to link aj
• (bpj(k1)0 ?p ?P)
• -deleting the rows corresponding to the paths
  intercepted with link aj
• (bpa(k1)0 ?a ?A, for each p such that
  bpj(k)1)
• Step 4 if bpa0 ?p ?P, ?a ?A , then GoTo the
  Step 5.
• otherwise, set kk1 and return to Step 1
• Step 5 Check the links inserted in the solution
• If a link intercept flows intercepted from other
  links,
• remove it from the solution.

68
A modification of the heuristic

• The heuristic can be adapted
• to the VMS location problem
• when it is necessary
• to intercept twice or more a specific path

69
I step of the modified heuristic
Link Path(flow) 1-2 (3) 1-3 (3) 2-4 (3) 3-4 (3) 2-5 (2) 4-6 (2) 3-7 (2)
p1(1) 1 0 0 0 1 0 0
p2(2) 1 0 1 0 0 0 0
p3(2) 0 1 0 1 0 0 0
p4(1) 0 1 0 0 0 0 1
p5(1) 0 0 0 0 1 0 0
p6(1) 0 0 1 0 0 1 0
p7(1) 0 0 0 1 0 1 0
p8(1) 0 0 0 0 0 0 1
? L
1 1
1 2
0 2
0 1
0 1
0 1
0 1
0 1
70
II step of the modified heuristic
Link Path(flow) 1-2 (0) 1-3 (3) 2-4 (3) 3-4 (3) 2-5 (1) 4-6 (2) 3-7 (2)
p1(1) 1 0 0 0 1 0 0
p2(2) 1 0 1 0 0 0 0
p3(2) 0 1 0 1 0 0 0
p4(1) 0 1 0 0 0 0 1
p5(1) 0 0 0 0 1 0 0
p6(1) 0 0 1 0 0 1 0
p7(1) 0 0 0 1 0 1 0
p8(1) 0 0 0 0 0 0 1
? L
1 1
1 2
1 2
1 1
0 1
0 1
0 1
0 1
71
III step of the modified heuristic
Link Path(flow) 1-2 (0) 1-3 (0) 2-4 (3) 3-4 (3) 2-5 (1) 4-6 (2) 3-7 (1)
p1(1) 1 0 0 0 1 0 0
p2(2) 1 0 1 0 0 0 0
p3(2) 0 1 0 1 0 0 0
p4(1) 0 1 0 0 0 0 1
p5(1) 0 0 0 0 1 0 0
p6(1) 0 0 1 0 0 1 0
p7(1) 0 0 0 1 0 1 0
p8(1) 0 0 0 0 0 0 1
? L
1 1
2 2
1 2
1 1
0 1
1 1
0 1
0 1
72
IV step of the modified heuristic
Link Path(flow) 1-2 (0) 1-3 (0) 2-4 (0) 3-4 (3) 2-5 (1) 4-6 (1) 3-7 (1)
p1(1) 1 0 0 0 1 0 0
p2(2) 1 0 1 0 0 0 0
p3(2) 0 1 0 1 0 0 0
p4(1) 0 1 0 0 0 0 1
p5(1) 0 0 0 0 1 0 0
p6(1) 0 0 1 0 0 1 0
p7(1) 0 0 0 1 0 1 0
p8(1) 0 0 0 0 0 0 1
? L
1 1
2 2
2 2
1 1
0 1
1 1
1 1
0 1
73
V step of the modified heuristic
Link Path(flow) 1-2 (0) 1-3 (0) 2-4 (0) 3-4 (0) 2-5 (1) 4-6 (0) 3-7 (1)
p1(1) 1 0 0 0 1 0 0
p2(2) 1 0 1 0 0 0 0
p3(2) 0 1 0 1 0 0 0
p4(1) 0 1 0 0 0 0 1
p5(1) 0 0 0 0 1 0 0
p6(1) 0 0 1 0 0 1 0
p7(1) 0 0 0 1 0 1 0
p8(1) 0 0 0 0 0 0 1
? L
1 1
2 2
2 2
1 1
1 1
1 1
1 1
0 1
74
VI step of the modified heuristic
Link Path(flow) 1-2 (0) 1-3 (0) 2-4 (0) 3-4 (0) 2-5 (0) 4-6 (0) 3-7 (1)
p1(1) 1 0 0 0 1 0 0
p2(2) 1 0 1 0 0 0 0
p3(2) 0 1 0 1 0 0 0
p4(1) 0 1 0 0 0 0 1
p5(1) 0 0 0 0 1 0 0
p6(1) 0 0 1 0 0 1 0
p7(1) 0 0 0 1 0 1 0
p8(1) 0 0 0 0 0 0 1
? L
1 1
2 2
2 2
1 1
1 1
1 1
1 1
1 1
75
VII and last step of the modified heuristic
Link Path(flow) 1-2 (0) 1-3 (0) 2-4 (0) 3-4 (0) 2-5 (0) 4-6 (0) 3-7 (0)
p1(1) 1 0 0 0 1 0 0
p2(2) 1 0 1 0 0 0 0
p3(2) 0 1 0 1 0 0 0
p4(1) 0 1 0 0 0 0 1
p5(1) 0 0 0 0 1 0 0
p6(1) 0 0 1 0 0 1 0
p7(1) 0 0 0 1 0 1 0
p8(1) 0 0 0 0 0 0 1
? L
1 1
2 2
2 2
1 1
1 1
1 1
1 1
1 1
76
Greedy solution

• 6 facilities on links 1-2, 1-3, 2-4, 3-4, 2-5,
  3-7

1
4
3
2
7
6
5
Path 1 1- 2 5 (1) Path 2 1 - 2 4 (2) Path
3 1 3 4 (2) Path 4 1 3 7(1) Path 5 2
5 (1) Path 6 2 4 - 6 (1) Path 7 3 4 6
(1) Path 8 3 7 (1)
77
Applications to TM in Naples

• ATENA Project (1999-2002)
• (MURST, City of Naples, FIAT, University of
  Naples)
• Low emission vehicle fleet experimentation
• Telematic system for traffic management
• Traffic monitoring and VMS
• Traffic Supervisor

78
Work perspectives

• Methodological scheme
• Sensor location
• UTM
• VMS location
• Process Scheme in ATIS scenario
• Flow monitoring
• Traffic Management
• Message to the users
• User behaviour and modification of the flow
  pattern
• Return to Flow monitoring and Iterate

79
Joint research perspectives

• Proposal of research project (Prin 2003)
• Infomobility and Transportation Network Design
• Roma La Sapienza (coordination)
• Camerino
• Genova
• Milano Politecnico
• Napoli Federico II

80
References

• 1 Apollonio N., Caccetta L., Simeone B.. On
  complexity of path covering problems in graphs.
  AIRO Winter 2003 Conference.
• 2 Berman O., Hodgson M.J., Krass D.. Flow
  Interception Problem, in Facility Location. A
  survey of application and methods. Ed. Drezner
  Z., Springler-Verlag. 389-426. (1995).
• 3 Berman O., Larson R.C., Fouska N.. Optimal
  Locating of Discretionary Facilities.
  Transportation Sciences. 26. 201-211. (1992).
• 4 Berman O., Krass D., Xu C.W.. Locating
  Discretionary Service Facilities Based on
  Probabilistic Customer Flows. Transportation
  Sciences. 29 (3). 276-290. (1995).
• 5 Berman O., Krass D., Xu C.W.. Locating
  Flow-InterceptingFacilities New Approaches and
  Results. Annals of Operations Research. 60.
  121-143. (1995).
• 6 Berman O., Krass D., Xu C.W.. A Generalized
  Discretionary Service Facility Location Models
  with Probabilistic Customer Flows. Stochastic
  Models. 13 (1). (1997).
• 7 Berman O., Krass D.. Flow intercepting
  spatial interaction model a new approach to
  optimal location of competitive facilities.
  Location Science. 6. 41-65. (1998).
• 8 Bianco L., Confessore G., Reverberi P.. A
  network based model for traffic sensor location
  with implications on O/D matrix estimates.
  Transportation Sciences. 35 (1). 50-60. (2001).
• 9 Cascetta E., Nguyen S.. A unified framework
  for estimating or updating Origin/Destination
  trip matrices from traffic counts.
  Transportation Research B. 22. 437-455. (1988).
• 10 Confessore G., DellOlmo P., Gentili M.. An
  Approximation Algorithm for the Dominating by
  Path Problem. AIRO Winter 2003 Conference.
• 11 Hodgson M.J.. A Flow Capturing Location
  Allocation Model. Geographical Analysis. 22.
  270-279. (1990).
• 12 Lam W.H.K., Lo H.P.. Accuracy of O/D
  estimates from traffic counts. Traffic
  Engineering and Control. 31. 358-370. (1990).
• 13 Yang H., Zhou J.. Optimal Traffic Counting
  location for Origin-Destination matrix
  estimation. Transportation Research. 32B (2).
  109-126. (1998).