A Game Theoretic Approach to Emergency Units Location Problem
A Game Theoretic Approach
to Emergency Units Location Problem
Vito FRAGNELLI
Università del Piemonte Orientale - Dipartimento di Scienze e Tecnologie Avanzate
[email protected]
joint work with:
Stefano GAGLIARDO
Università di Genova - Dipartimento di Matematica
e-mail: [email protected]
Fabio GASTALDI
Università del Piemonte Orientale - Dipartimento di Scienze e Tecnologie Avanzate
e-mail: [email protected]
Game Theory at Universities of Milano
Milano, 12-13 May 2011
1
A Game Theoretic Approach to Emergency Units Location Problem
Outline
1. Motivation
2. Preliminaries
3. Two games
4. Case Study 1
5. Real-world Example 1
6. Bankruptcy Approach
7. Case Study 2
8. Real-world Example 2
9. Concluding Remarks and Further Researches
2
A Game Theoretic Approach to Emergency Units Location Problem
1
Motivation
Emergency management −→ hard task
Various types of emergency:
• medical
• environmental
• urban
• ...
Different features ⇒ different ways, means and tools
Emergency ⇔ urgency ⇔ fast intervention
• medical −→ minutes
• environmental −→ hours
• earthquake −→ days
3
4
A Game Theoretic Approach to Emergency Units Location Problem
Utility of the intervention
utility 6
utility 6
-
distance
real
utility 6
-
distance
theoretical
-
distance
approximated
distance may be measured as ground distance, time distance, linear distance, etc.
Ambulances in Italy → red codes statistics account only intervention in 8 minutes
Don’t care! If the ambulance arrives later the intervention is anyhow completed
For fire planes and helicopters it is necessary to take into account the fuel tank capacity and
the distance from a place for charging water
5
A Game Theoretic Approach to Emergency Units Location Problem
Candidate locations may be viewed as interacting agents
⇓
The choice of deploying an emergency unit in a candidate location
takes into account its own characteristics as covered area and call frequency,
but should consider also the location of the other emergency units
2
1
2
1
1
The value in each box represents the weight of the zone
An ambulance located in a zone covers it and the two adjacent ones
If two ambulances are available, they should not be located in the maximal weight zones and
at least one should be located in a minimal weight zone, in order to cover the whole area
The second zones has the highest aggregate weight
A Game Theoretic Approach to Emergency Units Location Problem
6
Game theory allows to emphasize the interactions among the ambulances deployed on the area
under analysis
Location games were introduced for cost sharing problems (Tamir, 1992, Puerto-Garcia JuradoFernandez, 2001, Goemans-Skutella, 2004)
Here Game theory supports the location of the facilities
A Game Theoretic Approach to Emergency Units Location Problem
7
Complexity of emergency management allows and requires the synergic use of different methods
and approaches
• statistical analysis (Daskin, 1983, Fujiwara et al., 1987)
• simulation (Goldberg et al., 1990)
• topography and GIS (Geographical Information System) (Branas et al., 2000, Derekenaris
et al., 2001)
• graph theory
• mathematical programming (ReVelle-Hogan, 1989, Ball-Lin, 1993)
• stochastic optimization (Goldberg et al., 1990)
• ...
• Game Theory (of course!)
A Game Theoretic Approach to Emergency Units Location Problem
2
8
Preliminaries
A TU-game is a pair (N, v)
where N = {1, 2, . . . , n} finite set of players
v : 2N → R
characteristic function, with v(∅) = 0
A group of players S ⊆ N is a coalition and v(S) is the worth of the coalition; N is the grand
coalition
An allocation is a vector (xi )i∈N ∈ Rn, where xi represents the amount assigned to i ∈ N
P
An allocation (xi )i∈N is efficient if i∈N xi = v(N )
A solution is a function ψ that assigns an allocation ψ(v) to every TU-game in the class of
games with player set N
Here, Shapley value (1953):
1 X
φi(v) =
[v (P (π; i) ∪ {i}) − v (P (π; i))]
n!
π∈Π
where Π
set of the permutations of N
P (π; i) set of players preceding player i in permutation π
A Game Theoretic Approach to Emergency Units Location Problem
3
9
Two Games
Possible aims:
• efficiency : satisfying the maximum number of calls received from the area we attend to;
a priori analysis, based on historical data on the average frequency of emergency calls received
the results need to be validated with an a posteriori experimental study on the real situation under investigation
• equity : maximizing the area covered by all the ambulances, without considering the number
of calls received
• intermediate: maximizing the number of people living in the covered area
A Game Theoretic Approach to Emergency Units Location Problem
10
First idea: Take into account the area that each subset of candidate locations may cover
Hypothesis (minimal interference)
the calls originated in a given time interval from the area covered by an ambulance have a time
distribution that requires the minimum number of ambulances
Define a TU-game (N, v 1)
where N
set of candidate locations
v 1(S) sum of the weights of zones covered by at least one ambulance located in S
11
A Game Theoretic Approach to Emergency Units Location Problem
Example 1
- An area is described as grid of 18 zones, each with a weight of emergency calls
- 3 locations are candidated for an ambulance (bullets)
- The intervention time allows the ambulances to cover 9 zones
a
b
e
f
1.3
d
3.0
2.1
1.8
3.6
w
h
1.4
2.8
g
2.4
i
j
2.4
3.2
3.4
1.8
l
2.4
m
n
p
q
r
k
o
c
w
2.3
1.9
w
1.6
1.5
1.7
v 1(•) = 19.6
v 1(•) = 20.7 ← max
v 1(•) = 12.2
v 1(••) = 28.7
v 1(••) = 29.4 ← max
v 1(••) = 28.1
v 1(•••) = 36.1
The Shapley value, that is rooted in the concept of marginal contribution, represents very well
the interaction of the ”players” that depends on the weights of additional zones
It may be computed “dividing” the weight of each zone among the locations that cover it
The result may be proved decomposing the game w.r.t. the zones (and using the axioms)
12
A Game Theoretic Approach to Emergency Units Location Problem
For a simple computation, it is possible to refer to the coverage matrix C where cij = 1 if
location i covers zone j
•
a
b
c
1
1
1
d
e
f
g
h
i
1
1
1
1
1
1
1
1
1
•
•
j
1
1
and to the division matrix D where dij =
a
b
c
• 1.3 1.8 2.8
d
e
f
g
k
1
P
wj
h
i
k=1,...,n ckj
j
l
m
n
1
1
1
1
o
p
q
1
1
1
r
if cij = 1
k
l
m
n
o
p
q
r
2.1 1.8 1.2 0.8 1.6
•
1.8 1.2 0.8 1.6 3.4
•
0.8
1.2 1.6 1.7
1.8 1.2
1.4 2.3 1.9
The Shapley Value may be obtained summing up the values of each candidate location, so
φ = (13.4, 13.3, 9.4) and the ordering is •, •, •
The results of this game could be affected when two different locations overlap on several
zones, so that both may have a great relevance even if only one of them is actually important
A Game Theoretic Approach to Emergency Units Location Problem
13
Second idea: Take into account the area that each subset of candidate locations may cover
more than once
Define a TU-game (N, v 2)
where N
set of candidate locations
v 2(S) sum of the weights of zones covered by at least two ambulances located in S
14
A Game Theoretic Approach to Emergency Units Location Problem
Example 2 Referring to Example 1, the game is:
a
b
e
f
1.3
d
3.0
2.1
1.8
3.6
w
h
1.4
2.8
g
2.4
i
j
2.4
3.2
3.4
1.8
l
2.4
m
n
p
q
r
k
o
c
w
2.3
1.9
w
1.6
1.5
1.7
v 2(•) = 0
v 2(•) = 0
v 2(•) = 0
v 2(••) = 11.6
v 2(••) = 2.4
v 2(••) = 4.8
v 2(•••) = 14.0
The Shapley value may be used for the same motivation
It can be computed “dividing” the weight of each multicovered zone among the locations that
cover it
The result may be proved decomposing the game w.r.t. the multicovered zones (and using the
axioms)
15
A Game Theoretic Approach to Emergency Units Location Problem
It is possible to refer to the multicoverage matrix:
a
b
c
d
e
f
g
h
i
•
1
1
1
1
•
1
1
1
1
•
j
k
l
m
n
o
p
q
r
m
n
o
p
q
r
1
1
1
and to the division matrix:
a
b
c
d
e
f
g
h
i
•
1.8 1.2 0.8 1.6
•
1.8 1.2 0.8 1.6
•
j
k
0.8
l
1.2
1.2
The Shapley Value may be obtained summing up the values of each candidate location:
φ = (5.4, 6.6, 2.0)
In this case the highest the value, the highest the average multiple coverage of zones → the
ordering has to be reverted, so it is •, •, •
For this game, an isolated location, i.e. whose covered area has empty intersection with the area
of any other location, has always a null marginal contribution; so, its Shapley value is always
zero and this location has always the highest ranking
16
A Game Theoretic Approach to Emergency Units Location Problem
Third idea: Take into account both orderings, via a Borda count
Referring again to Example 1:
•
•
•
v1
1
2
3
v2
2
3
1
total
3
5
4
so, the ranking according to the Borda count is •, •, •
17
A Game Theoretic Approach to Emergency Units Location Problem
4
-
Case Study 1
An area is described as a set of 30 zones (circles) distributed on a unitary grid
All the zones have the same probability of generating an emergency call
5 zones are candidate locations for an ambulance (a bullet inside the corresponding circle)
It is possible to deploy 3 ambulances
The intervention time allows the ambulances to move two steps on the grid
The distance is measured according to the Manhattan metric
@
@
d
d
@
@
@
@
@
@
d
@
@
@
@
@
@
dr
@
@
d
dr
@
@
@
@
@
@ d
d
d
@
@
@
@
@
@
@ d @ @
@
@ @
@
@
d
d
@
1
d
@
@
4
d
@
@
d@
d
@
@
@
d@
2
dr
@
@
d
@
@
d
@
@
@
@
@ d
@
@
@
d
@
@
@
d @
@
dr@
d
@
@
@
d
d@
@
3
5
@ dr
@
d
@
@
@
@
@
@
d
@
@
@
A Game Theoretic Approach to Emergency Units Location Problem
18
v1
v2
Borda count
φ
(3.0, 4.0, 5.5, 4.5, 5.0) (2.0, 2.0, 2.5, 0.5, 3.0)
−
Ordering
3, 5, 4, 2, 1
4, 1–2, 3, 5
4, 3, 2–5, 1
The sign ”–” among two locations represents a draw in the ordering
• The two games produce large differences, justifying the use of both of them and of the
Borda count in order to balance the characteristics of the two approaches
• The same example was used by two researchers of the University of Milan that approached
the ambulance location problem using a mathematical programming method, with results
consistent with ours
• Our approach may have a dynamic interpretation: Given an ordering of the locations, we
have the solution also when the number of ambulances varies
The Borda count implies to deploy the three ambulances in locations 4, 3, 2 (or 5). If a
further ambulance becomes available it will be located at 5 (or 2); on the opposite, if one of
the three ambulances is temporarily out of order, it is possible to decide to use only locations
4 and 3
19
A Game Theoretic Approach to Emergency Units Location Problem
5
Real-world Examples 1
Extra-urban area of Milan; 117 municipalities, each coinciding with a zone
65 possible locations, those in which an emergency service is located
Covered area is computed according to the distances among the centers of the municipalities;
a town is covered if it may be reached in less than 18 minutes
The fast computation allows for both equity and efficiency solutions
Looking for efficiency, 12 scenarios has been identified
via the results of the research of University of Milan
as parts of the year in which the call trend is similar
and which may be treated in the same way; the weight
of each zone is the average number of calls per hour
originated from it in the corresponding scenario
W1
W2
W3
SAmf 1
SAmf 2
SAmf 3
SAss 1
SAss 2
SAss 3
S1
S2
S3
Winter
Winter
Winter
Spring-Autumn MO - FR
Spring-Autumn MO - FR
Spring-Autumn MO - FR
Spring-Autumn SA - SU
Spring-Autumn SA - SU
Spring-Autumn SA - SU
Summer
Summer
Summer
00
08
14
00
07
13
00
07
13
00
07
14
-
08
14
24
07
13
24
07
13
24
07
14
24
Looking for equity, the weight of each zone corresponds to the surface of the municipality
We suppose to locate 28 units
Each location may host at most one ambulance
37
20
61
15
53
11
31
48
3
25
54
35
6
62
59
32
18
28 56 41
2
8
22 26 19
9 57
39
43
23
21 16
64
55
58
34
12
17
24
27
60
46
49
51
50
4
7
14
63
44
10
1
29 5
40
45
42
52
33
36
65
38
30
47
13
A Game Theoretic Approach to Emergency Units Location Problem
21
In the following table, the second column refers to the equity approach, while the following 12
refer to the efficiency approach
Candidate locations which an ambulance is never assigned to are omitted: Bellinzago Lombardo (5), Bollate (8), Cassano d’Adda (13), Ceriano Laghetto (15), Cislago (20), Cuggiono
(25), Inzago (30), Lainate (31), Novate Milanese (39), Pozzuolo Martesana (47), Solaro (59),
Uboldo (VA - 61)
Towns red already host an ambulance
22
A Game Theoretic Approach to Emergency Units Location Problem
1
2
3
4
6
7
9
10
11
12
14
16
17
18
19
21
22
23
24
26
27
28
29
32
33
34
35
Town
Abbiategrasso
Arese
Arluno
Basiglio
Bernate Ticino
Binasco
Bresso
Buccinasco
Caronno Pertusella
Carugate
Cassina De’ Pecchi
Cernusco sul Naviglio
Cesano Boscone
Cesate
Cinisello Balsamo
Cologno Monzese
Cormano
Cornaredo
Corsico
Cusano Milanino
Gaggiano
Garbagnate Milanese
Gorgonzola
Limbiate
Locate Triulzi
Magenta
Marcallo con Casone
W
2 3
X X
X X
X X
SAmf
1 2 3
X X X
X X X
X X X
SAss
1 2 3
X X X
X X X
X X X
E 1
X X
X
X X
X
X
X X X X X X X
X X X X X X X X X
X
1
X
X
X
S
2
X
X
X
3
X
X
X
X X X
X X X
X
X
X X
X X X X X X X X X X X X
X X X
X X X X
X X X X
X X X X
X X X X
X X X X
X X X X X
X X X X
X
X
X X X X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X X X
X
X
X
X
X
X
X
X
X X X
X X X
X X X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
36
37
38
40
41
42
43
44
45
46
48
49
50
51
52
53
54
55
56
57
58
60
62
63
64
65
W SAmf SAss
S
Town
E 1 2 3 1 2 3 1 2 3 1 2
Melegnano X X X X X X X X X X X X
Melzo X
X
X
Misinto
Opera
Paderno Dugnano X X X X X X X X X X X
Paullo X
Pero X X X X X X X X X X X X
Peschiera Borromeo
Pieve Emanuele X
X
Pioltello X X X X X X X X X X X
Rho X
X
Rodano X
Rozzano X X X X X X X X X X X X
San Donato Milanese X X X X X X X X X X X X
San Giuliano Milanese X X X X X X X X X X X X
Saronno
Sedriano X X X X X X X X X X X
Segrate
X X X X X X X X
Senago X X X X X X X X X X X
Sesto San Giovanni X X X X X X X X X X X
Settimo Milanese X X X X X X X X X X X X
Trezzano sul Naviglio
X
X
Vanzago X X
X
X X
Vignate X
X
Vimodrone X X X X X X X X X X X
Vizzolo Predabissi X
3
X
X
X
X
X
X
X
X
X
X
X
X
X
23
A Game Theoretic Approach to Emergency Units Location Problem
6
Bankruptcy Approach
An estate E has to be divided among a set N = {1, 2, ..., n} of claimants, each of them with
a claim ci, i = 1, . . . , n on the estate, and the total claim exceeds the available estate
Formally, it is a triplet (N, E, c) where E ∈ R+, c ∈ Rn+ and E ≤ c1 + c2 + ... + cn
A solution is a vector x = (x1 , x2, ..., xn) ∈ Rn+ such that:
0 ≤ xi ≤ ci,
n
X
xi = E,
for each i ∈ N,
i=1
xi is the part of E assigned to i
A division rule is a map f that assigns a solution to each bankruptcy problem
24
A Game Theoretic Approach to Emergency Units Location Problem
Classical division rules:
(i) Proportional
P ROPi (N, E, c) = P
ci
j∈N cj
E, i ∈ N
(ii) Constrained Equal Award
CEAi (N, E, c) = min{ci, α}, i ∈ N,
P
where α is the unique real number such that i∈N CEAi (N, E, c) = E
(iii) Constrained Equal Loss
CELi (N, E, c) = max{ci − β, 0}, i ∈ N,
P
where β is the unique real number such that i∈N CELi (N, E, c) = E
Bankruptcy problems may be applied to allocate a scarce resource, in this case ambulances
A Game Theoretic Approach to Emergency Units Location Problem
25
A priori aggregation
N = {a1 , a2, . . . , al } is the set of claimants (candidate locations)
P
cj = k∈Sj wk , aj ∈ N are the claims, where Sj is the set of zones covered by a unit in aj
E is the estate (number of ambulances)
A posteriori aggregation
N = {1, 2, . . . , mn} is the set of claimants (zones)
wj , j ∈ N are the claims
E is the estate (number of ambulances)
P
Let b = (b1 , . . . , bnm) be a solution, then location aj receives k∈Sj bk ambulances
Each location has to be assigned an integer number of vehicles; for instance:
(i) let aJ be the location which received the largest number M of vehicles;
(ii) assign to aJ the minimum between an integer approximation of M and the number of
remaining ambulances and delete it;
(iii) if the number of remaining ambulances is larger than 0, go to step (i).
26
A Game Theoretic Approach to Emergency Units Location Problem
Using P ROP , CEA and CEL, both with the a priori (1) and the a posteriori (2) aggregation,
there are six solutions, but only four are significant:
(1)
(2)
• before rounding P ROPaj = P ROPaj (non-manipulability)
• CEA(1) is influenced by the ordering and by the rounding method (the ambulances may be
assigned to the first candidate locations or all to the last one)
A Game Theoretic Approach to Emergency Units Location Problem
7
Case Study 2
Consider a city represented by a 13 × 13 grid with 20 candidate locations
The color of each box indicates the average number of calls (the darker, the more)
A vehicle can cover a distance of two steps in the Manhattan metric
10 ambulances have to be allocated
27
A Game Theoretic Approach to Emergency Units Location Problem
Location PROP(1) PROP(2) CEA(1) CEA(2) CEL(1) CEL(2)
1
0
0
1
0
0
0
2
1
1
1
1
0
0
3
1
1
1
0
0
1
4
1
1
1
1
4
5
5
1
1
1
1
5
4
6
1
1
1
0
0
0
7
1
1
1
1
0
0
8
0
0
1
0
0
0
9
0
0
1
0
0
0
10
1
1
1
1
0
0
11
0
0
0
0
0
0
12
0
0
0
0
0
0
13
0
0
0
1
0
0
14
0
0
0
1
0
0
15
0
0
0
0
0
0
16
1
1
0
1
0
0
17
1
1
0
1
1
0
18
0
0
0
0
0
0
19
0
0
0
0
0
0
20
1
1
0
1
0
0
Only locations 4 and 5 obtain at least 1 ambulance under each type of significant solution
CEL solution is the only one that may assign more than one ambulance to a location
28
A Game Theoretic Approach to Emergency Units Location Problem
8
29
Real-world Examples 2
Using a priori aggregation, ambulances may be located iteratively, one by one in the location
with the highest claim
After each step, the claims are reduced in the covered area, with different methods, as some
calls are satisfied
For instance, looking for equity we may consider the area as fully covered
A vehicle covers six steps in the Manhattan metric
A Game Theoretic Approach to Emergency Units Location Problem
30
A Game Theoretic Approach to Emergency Units Location Problem
#
1
2
3
4
5
6
7
8
9
10
11
12
%
29.74
55.58
70.82
80.80
84.36
89.14
95.56
96.13
98.04
99.08
99.97
100.00
31
A Game Theoretic Approach to Emergency Units Location Problem
9
32
Concluding Remarks and Further Researches
• satisfactory and encouraging results
• the computation of the Shapley value does not require to determine the characteristic functions
• the hypothesis of minimal interference is very tight
• the resulting ranking may help to dramatically reduce the number of candidate locations
and then apply a mathematical programming approach to a smaller problem
A Game Theoretic Approach to Emergency Units Location Problem
33
• an emergency unit may serve only a limited number of calls per day, so more than one player
could be associated to each location
• assign different values to the areas covered by more than one location, as multiple coverage
corresponds to a better service
• the bankruptcy approach suggested possible theoretical extension in three directions:
– the estate is not continuously divisible
– the agents have non independent claims
– the claims and the estate are not homogeneous