Single and multi-group evacuation in medium and maxi health

Transcript

Single and multi-group evacuation in medium and
maxi health-care emergencies
Edoardo Amaldi
DEI, Politecnico di Milano
joint work with Stefano Coniglio and Claudio Iuliano
Game Theory Workshop @ Milano – May 13, 2011
Edoardo Amaldi (Politecnico di Milano)
Single and multi-group evacuation in medium and maxi health-care emergencies
1 / 17
Outline
1
The problem: evacuation in medium and maxi emergencies
2
Mixed integer linear programming formulation
3
Computational results
2 / 17
DECEMbRIA project
DECEMbRIA (DECisioni in EMergenza sanitaRIA) project funded by
Lombardy Region (2008-10)
involving
Operations Center of 118 Emergency Medical Service,
Ospedale Niguarda Ca’ Granda, Milano
Politecnico di Milano, Univ. di Milano, Univ. del Piemonte Orientale
Collaboration with Dr. Cristiano Cozzi, head of the EU of 118
3 / 17
Medium and maxi emergencies
Large accidents with many injured people (large intoxication, major
car accident, building collapse, airplane crash,...)
4 / 17
Medium and maxi emergencies
Large accidents with many injured people (large intoxication, major
car accident, building collapse, airplane crash,...)
Flow of events
1
establishment of an aid station on the emergency site
2
diagnosis: determination for each patient of severity color code
(red, yellow, green) and of needed medical specialties
3
assignment of each patient to a hospital
4 / 17
The problem
Given
for each patient: color code and needed medical specialties
(general reanimation, coronary, neuro-reanimation, operating room)
for each hospital: distance, capacities (number of curable
patients) and estimated competences w.r.t. color codes and
medical specialties
number of available ambulances
select for each patient an ambulance and a hospital so as to optimize
total travel distance
balance the residual capacities
quality of treatments
5 / 17
The problem
Peculiarities of the assignment problem
packing on ambulances
1 single red-code
1 yellow-code possibly with 1 green-code
up to 3 green-code
activation of internal emergency plan for massive injured people
inflow (PEIMAF) to increase the color-code capacities
6 / 17
Previous work
urban evacuation, e.g., [Zeng, Wang, and Kukreti, 2008]
building evacuation (flow over time), e.g., [Hoppe and Tardos, 1994]
military aeromedical evacuation, e.g., [Kott, Saks, and Mercer, 1999]
The problem is currently tackled on a case-by-case basis, not addressed
as an optimization problem
7 / 17
Scenarios
Efficiently support decisions in the critical setting of a medium and
maxi emergency
single-group evacuation: all patients simultaneously
Solving a mixed integer linear programming formulation
multi-group evacuation: patients in groups (e.g., when extracted
from the rubble after a building collapse)
Solving the single-group formulation iteratively, avoiding early
consumption of scarce resources
8 / 17
Parameters
C = {r , y , g }: color codes (red, yellow, green)
9 / 17
Parameters
S: medical specialties (general reanimation, coronary,
neuro-reanimation, operating room)
9 / 17
Parameters
P: patients, with Sp ⊆ S needed specialties for each p ∈ P
9 / 17
Parameters
Pc and Ps ⊆ P: patients with color code c and requiring specialty s
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Parameters
[
H=
Hi : set of hospitals partitioned into geographic zones i ∈ G
i∈G
For each h ∈ H:
dh ∈ R+ distance to emergency site
0
capacity khc and khs
per color code and specialty
0
competences bhc and bhs
in {0, . . . , 10} per color code and specialty
extra capacity with PEIMAF k̂hc per color
PEIMAF activation cost wh
9 / 17
Parameters
[
H=
i∈G
For each h ∈ H:
0
0
number of available ambulances a0
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Parameters
[
H=
i∈G
For each h ∈ H:
0
0
number of available ambulances a0
upper bound w0 on “cost” for activating PEIMAFs
9 / 17
Variables
Decision variables
xph ∈ {0, 1}: 1 if patient p is assigned to hospital h and 0 otherwise
yh ∈ {0, 1}: 1 if PEIMAF is activated in h and 0 otherwise
zh ∈ Z+ : number of ambulances sent to hospital h
uc ∈ Z+ : minimum residual capacity for color code c over all zones
us0 ∈ Z+ : minimum residual capacity for specialty s over all zones
10 / 17
Variables
Decision variables
xph ∈ {0, 1}: 1 if patient p is assigned to hospital h and 0 otherwise
yh ∈ {0, 1}: 1 if PEIMAF is activated in h and 0 otherwise
zh ∈ Z+ : number of ambulances sent to hospital h
uc ∈ Z+ : minimum residual capacity for color code c over all zones
us0 ∈ Z+ : minimum residual capacity for specialty s over all zones
Auxiliary variables
rhc ∈ [0, khc ]: residual capacity for color code c in hospital h
0
0
rhs
∈ [0, khs
]: residual capacity for specialty s in hospital h
zh0 ∈ Z+ : packing on ambulances for hospital h
10 / 17
Formulation
X
min
αc xph dh −
−
γc uc −
c∈C
c∈C ,p∈Pc ,h∈H
X
X
γs0 us0
s∈S
X
βc xph bhc −
X
0
βs0 xph bhs
s∈S,p∈Ps ,h∈H
s.t.
X
∀p ∈ P
xph = 1
(assignment)
h∈H
X
wh yh ≤ w0
(PEIMAF limit)
h∈H
X
xph + rhc ≤ khc + yh k̂hc
∀h ∈ H, c ∈ C
(color residual)
0
0
xph + rhs
= khs
∀h ∈ H, s ∈ S
(specialty residual)
p∈Pc
X
p∈Ps
...
11 / 17
Formulation
X
min
αc xph dh −
−
γc uc −
c∈C
X
X
γs0 us0
s∈S
X
βc xph bhc −
X
0
βs0 xph bhs
s∈S,p∈Ps ,h∈H
s.t.
X
∀p ∈ P
xph = 1
(assignment)
h∈H
X
wh yh ≤ w0
(PEIMAF limit)
h∈H
X
∀h ∈ H, c ∈ C
(color residual)
0
0
xph + rhs
= khs
∀h ∈ H, s ∈ S
p∈Pc
X
p∈Ps
...
11 / 17
Formulation
X
min
αc xph dh −
−
γc uc −
c∈C
X
X
γs0 us0
s∈S
X
βc xph bhc −
X
0
βs0 xph bhs
s∈S,p∈Ps ,h∈H
s.t.
X
∀p ∈ P
xph = 1
(assignment)
h∈H
X
wh yh ≤ w0
(PEIMAF limit)
h∈H
X
∀h ∈ H, c ∈ C
(color residual)
0
0
xph + rhs
= khs
∀h ∈ H, s ∈ S
p∈Pc
X
p∈Ps
...
11 / 17
Formulation
X
min
αc xph dh −
−
γc uc −
c∈C
X
X
γs0 us0
s∈S
X
βc xph bhc −
X
0
βs0 xph bhs
s∈S,p∈Ps ,h∈H
s.t.
X
∀p ∈ P
xph = 1
(assignment)
h∈H
X
wh yh ≤ w0
(PEIMAF limit)
h∈H
X
∀h ∈ H, c ∈ C
(color residual)
0
0
xph + rhs
= khs
∀h ∈ H, s ∈ S
p∈Pc
X
p∈Ps
...
11 / 17
Formulation
X
min
αc xph dh −
γc uc −
X
γs0 us0
X
βc xph bhc −
X
s∈S
c∈C
−
X
0
βs0 xph bhs
s∈S,p∈Ps ,h∈H
s.t.
uc ≤
X
rhc
∀c ∈ C , i ∈ G
(min res. color)
0
rhs
∀s ∈ S, i ∈ G
(min res. specialty)
h∈Hi
us0 ≤
X
h∈Hi
X
z h ≤ a0
(available ambulances)
h∈H
zh =
X
xph + zh0
∀h ∈ H
(amb. number)
∀h ∈ H
(amb. packing)
p∈(Pr ∪Py )
P
zh0
≥
p∈Pg
xph −
3
P
p∈Py
xph
12 / 17
Formulation
X
min
αc xph dh −
γc uc −
X
γs0 us0
X
βc xph bhc −
X
s∈S
c∈C
−
X
0
βs0 xph bhs
s∈S,p∈Ps ,h∈H
s.t.
uc ≤
X
rhc
∀c ∈ C , i ∈ G
(min res. color)
0
rhs
∀s ∈ S, i ∈ G
h∈Hi
us0 ≤
X
h∈Hi
X
z h ≤ a0
h∈H
zh =
X
xph + zh0
∀h ∈ H
(amb. number)
∀h ∈ H
(amb. packing)
p∈(Pr ∪Py )
P
zh0
≥
p∈Pg
xph −
3
P
p∈Py
xph
12 / 17
Formulation
X
min
αc xph dh −
γc uc −
X
γs0 us0
X
βc xph bhc −
X
s∈S
c∈C
−
X
0
βs0 xph bhs
s∈S,p∈Ps ,h∈H
s.t.
uc ≤
X
rhc
∀c ∈ C , i ∈ G
(min res. color)
0
rhs
∀s ∈ S, i ∈ G
h∈Hi
us0 ≤
X
h∈Hi
X
z h ≤ a0
h∈H
zh =
X
xph + zh0
∀h ∈ H
(amb. number)
∀h ∈ H
(amb. packing)
p∈(Pr ∪Py )
P
zh0
≥
p∈Pg
xph −
3
P
p∈Py
xph
12 / 17
Formulation
X
min
αc xph dh −
γc uc −
X
γs0 us0
X
βc xph bhc −
X
s∈S
c∈C
−
X
0
βs0 xph bhs
s∈S,p∈Ps ,h∈H
s.t.
uc ≤
X
rhc
∀c ∈ C , i ∈ G
(min res. color)
0
rhs
∀s ∈ S, i ∈ G
h∈Hi
us0 ≤
X
h∈Hi
X
z h ≤ a0
h∈H
zh =
X
xph + zh0
∀h ∈ H
(amb. number)
∀h ∈ H
(amb. packing)
p∈(Pr ∪Py )
P
zh0
≥
p∈Pg
xph −
3
P
p∈Py
xph
12 / 17
Characteristics
Optimal patient-to-ambulance assignment
implicitly accounted for by the last constraints
explicitly derived by a greedy procedure after solving the model
Proposition: Integrality of the solution guaranteed even with continuous
assignment variables xph
Extension to multiple simultaneous emergencies at different sites
13 / 17
Hard instances:
10 − 300 patients
random color-code in {r , y , g } (respectively 12%, 25% and 63%)
unique medical specialty
small groups in multi-group scenario (5 patients)
minimum number of ambulances in both scenarios
regular and PEIMAF color-code capacities, specialty capacities and
competences based on the experience of the EU operators
Milan and hinterland subdivided into 10 geographic zones
Averages on 30 instances, MIP formulation(s) solved via CPLEX 11
Intel Xeon 2.0 GHz, with 4 GB RAM (1 core and RAM usage < 1 GB)
14 / 17
|P|
10
15
20
25
30
35
40
45
50
60
70
80
90
100
120
150
180
200
250
300
single-group
|A|
Time [s]
6
0.02
9
0.03
11
0.03
13
0.03
16
0.04
19
0.05
21
0.08
24
0.05
26
0.06
31
0.07
36
0.14
41
0.13
46
0.23
51
0.23
60
0.29
75
0.54
91
0.98
101
1.18
127
1.86
150
4.15
multi-group
|A|
Time [s]
6
0.04
9
0.05
11
0.07
14
0.09
16
0.11
19
0.13
21
0.16
24
0.18
27
0.20
31
0.27
37
0.31
42
0.37
47
0.43
52
0.53
62
0.66
77
1.08
94
1.26
106
1.66
133
2.09
157
3.50
Gap [%]
0.00
0.00
0.00
1.93
0.20
0.47
0.81
0.96
0.97
4.83
3.15
3.10
3.36
2.87
2.88
3.10
2.49
2.62
2.82
1.75
15 / 17
Graphical user interface
16 / 17
Concluding remarks
We propose
a mixed integer linear formulation for the single-group evacuation
an iterative approach for the multi-group scenario
CPU time required to solve our formulations is sufficiently small to suit
the limited time available in critical emergency situations
Need for up-to-date and accurate data from the hospitals to use our
optimization tool
17 / 17
Concluding remarks
We propose
a mixed integer linear formulation for the single-group evacuation
an iterative approach for the multi-group scenario
CPU time required to solve our formulations is sufficiently small to suit
the limited time available in critical emergency situations
Need for up-to-date and accurate data from the hospitals to use our
optimization tool
17 / 17

Single and multi-group evacuation in medium and maxi health

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