1. Locations and opening hours of facilities which maximally cover flows in a network
Ken-ichi TANAKA
Department of Management Science,
Tokyo University of Science, Japan
Thank you Professor Sasaki.
I am Ken-ichi TANAKA from Tokyo University of Science.
Today, I will talk about
A time-dependent maximal flow-covering location problem on a continuous plane and a network space.

2. Introduction of temporal dimension in maximum flow-covering problem
locations and opening hours of facilities which maximally cover flows
facility
temporal axis
A potential customer:
A commuter who can stop over at one of the facilities for a given period of time and can go back home by a given time
arrival
time limit
potential customer
end
opening hours
Extended model of Berman’s
maximum flow-covering problem
start
O. Berman (1997): Deterministic Flow-demand Location Problems, Journal of Operational Research Society, 48,
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3. Dynamic Location Models
Introducing Temporal Dimension in Location Problems
Future uncertainty of demands
Mirchandani and Odoni (1979), Wearver and Church (1983), Drezner (1995),
Current, Ratick and ReVelle (1998), Owen and Daskin (1998), Snyder (2006)
Timing of location and relocation of a facility in the planning horizon
Drezner and Wesolowsky (1991), Farahani, et al. (2008)
Facility management model on a daily basis is also important
A new model determining opening hours of facilities
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4. Outline of the presentation
1. Single facility problem using a continuous formulation in a linear city
2. Single facility problem using the railway network data in Tokyo Metropolitan area
3. Integer programming formulation of multi-facility problem
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5. 1. Single facility problem using a
continuous formulation in a linear city
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6. Concert Problem
maximal flow-covering location problem + temporal dimension
Optimal concert plan for a organizer
potential customer
home
Concert Problem: finding
(1) the best location
(2) the best start time
which maximize the number of
potential customers.
11:00
p.m.
concert hall
temporal axis
10:30
3 hours
7:30
workplace
Simultaneous determination of
Where to find a concert hall ?
When to start the concert ?
6:30
p.m.
spatial axis
Space-time plane
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7. Definition of potential customers
Potential customer: Commuter who can attend the concert
from start to end after work and can go back home by a given time
= arrival time limit
subscript “h” means home
potential customer
home
If concert starts too early:
Few people can be in time for the start time
If concert starts too late:
Few people can get back home by
concert hall
temporal axis
3 hours
An example of other application area:
e.g.) Attending evening graduate school after work
What is the optimal timetable for graduate school
to attract lots of working people
workplace
spatial axis
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8. Assumptions and notations
home
concert information:
location of concert hall
start time of concert
length of the linear city
length of concert time
location of workplace
location of home
trip density from to
departure-time distribution
of commuters
length of concert time
movement speed of commuters
start time
workplace
concert hall
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9. Definition of departure-time distribution of commuters
cumulative distribution of
departure-time t of commuters
home =
the proportion of commuters that can leave their workplace by t
assumption:
length of concert time
70%
workplace
concert hall
8:00 p.m.
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10. Formulation of the concert problem
: the number of potential customers
arrival time limit
11:00
p.m.
hours
: location of the concert hall
: start time of the concert
5:00
p.m. 1
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11. Derivation of objective function
: the number of commuters that can fully attend the concert and can get back home by a given time
Movement of people can be represented by a straight line with a slope of
home
(1)
(2) be in time for the start time
To be a potential customer
and
workplace
home =
(2’) leave workplace by time
travel time from workplace to the concert hall
The number of potential customers
between two small black segments
workplace
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12. Derivation of objective function
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13. Numerical example 11:00 p.m. 9:00 p.m. hours 7:00 p.m. 5:00 p.m. 1
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14. 2. Single facility problem using the railway network data
in Tokyo Metropolitan area
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15. Railway network in Tokyo Metropolitan area
Tokyo Station
Tokyo Bay
Map of Japan
1796 stations
145km
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16. Concert Problem: network model
potential customer: Commuter who can fully attend the concert after work and can go back home by a given time
Input Data
(1) length of concert time
(2) arrival time limit
(3) railway network data
(4) OD data
(5) departure time distribution
temporal axis
concert hall
arrival
time limit
potential customer
Departure time distribution of commuters traveling from i to j
concert hours
end
5:00 p.m.
9:00 p.m.
start
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17. Railway network in Tokyo Metropolitan area (Origin：Tokyo Station)
Network and OD data km
Input Data
(1) length of concert time
(2) arrival time limit
(3) railway network data
(4) OD data
(5) departure time distribution km
assumption:
shortest travel time route between
each pair of stations
# of stations：
# of links：
# of railways：
# of commuters： 7.87 Million
Railway network in Tokyo Metropolitan area (Origin：Tokyo Station)
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18. Formulation of the Concert Problem
: number of commuters traveling from station to station
: set of stations (workplaces) from which travel time
to the concert hall is within
: set of stations (homes) to which travel time
from the concert hall is within
: traveling time from station to station
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19. The number of Potential Customers
[Million]
(1) Tokyo Sta.
(2) Musashisakai
Sta.
(2)
(1)
(3)
(3) Hachioji
Sta.
arrival time limit
1.0
5:00 p.m.
6:00
7:00
8:00
9:00
10:00
11:00 p.m.
start time
optimal start time
Tokyo　　 :58
Musashisakai　18:54
Hachioji　 :48
optimal service hours
time to go back home
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20. The number of Potential Customers
The animation is created by varying the start time from
5:00 p.m. to 8:00 p.m.
with a step of 5 minutes.
departure time distribution
5:00 p.m.
11:00 p.m.
9:00
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21. The number of Potential Customers
departure time distribution
5:00 p.m.
11:00 p.m.
9:00
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22. 3. Integer programming formulation of multi-facility problem
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23. Concert Problem Integer Programming Formulation of Concert Problem
Assumption: concert hours are the same for all facilities
facility
Objective function:
# of commuters who can stop over at one of the facilities for a given period of time and can go back home by a given time
temporal axis
arrival
time limit
potential customer
start time is discretized
end
opening hours
:whether a facility is located at station k opening at time t
Variables (Binary)
:whether commuters traveling from i to j departing
at time u can be covered or not
start
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24. Concert Problem Integer Programming Formulation of Concert Problem
Assumption: concert hours are the same for all facilities
max.
sub. to
(1)
(2)
(3)
(4)
(5)
Concert Problem
:whether a facility is located at station k opening at time t
Variables (Binary)
:whether commuters traveling from i to j departing
at time u can be covered or not
Parameters
:# of commuters traveling from i to j departing at time u
:whether a facility at station k opening at time t can cover commuters traveling from i to j departing at time u
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25. Some future work 1. Heuristic algorithm for the p-concert problem
2. Competitive location problem
3. Other formulation such as median and center
This figure shows the relations between the number of potential customers
for each start time corresponding to 3 locations as shown in this figure.
As shown in this figure, the concert organizer should find a concert hall at the city center.
The best start time is 7.
The most interesting point is that the optimal concert start time depends on concert locations.
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