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.

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, 75--81. 2008/9/17 EWGLA08

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 2008/9/17 EWGLA08

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 2008/9/17 EWGLA08

1. Single facility problem using a continuous formulation in a linear city 2008/9/17 EWGLA08

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 2008/9/17 EWGLA08

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 2008/9/17 EWGLA08

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 2008/9/17 EWGLA08

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. 2008/9/17 EWGLA08

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 2008/9/17 EWGLA08

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 2008/9/17 EWGLA08

Derivation of objective function 2008/9/17 EWGLA08

Numerical example 11:00 p.m. 9:00 p.m. hours 7:00 p.m. 5:00 p.m. 1 2008/9/17 EWGLA08

2. Single facility problem using the railway network data in Tokyo Metropolitan area 2008/9/17 EWGLA08

Railway network in Tokyo Metropolitan area Tokyo Station Tokyo Bay Map of Japan 1796 stations 145km 2008/9/17 EWGLA08

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 2008/9/17 EWGLA08

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： 1796 # of links： 2417 # of railways： 125 # of commuters： 7.87 Million Railway network in Tokyo Metropolitan area (Origin：Tokyo Station) 2008/9/17 EWGLA08

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 2008/9/17 EWGLA08

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　　 18:58 Musashisakai　18:54 Hachioji　 18:48 optimal service hours time to go back home 2008/9/17 EWGLA08

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 2008/9/17 EWGLA08

The number of Potential Customers departure time distribution 5:00 p.m. 11:00 p.m. 9:00 2008/9/17 EWGLA08

3. Integer programming formulation of multi-facility problem 2008/9/17 EWGLA08

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 2008/9/17 EWGLA08

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 2008/9/17 EWGLA08

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. 2008/9/17 EWGLA08