The Optimal Location Problem for Urban Facilities Considering the Time Axis
Cities contain various types of facilities. These range widely from familiar ones like restaurants, supermarkets, bank ATMs, and post offices to those that provide public services, such as schools, hospitals, fire stations, and city halls. How are the locations of these facilities determined? The location of a facility should significantly affect the convenience for service users and the sales of a store.
The problem of finding desirable locations for facilities within a given space is a fascinating topic that many researchers have long addressed. Here, I would like to introduce this problem by discussing the "facility location problem," which is one of the mathematical models studied in my field of expertise, operations research (OR).
The facility location problem is the problem of determining desirable locations for facilities, given a set of user residences and a set of candidate facility locations within a target space. Many variations exist, depending on the type of facility and the evaluation criteria for facility location. For example, a facility location problem called the maximal covering location problem is the problem of determining a facility arrangement that maximizes the total number of users residing within a certain distance of a facility (covered users).
The facility location problem is a spatial decision-making model, and in many cases, the flow of time is completely ignored. In reality, however, the demand for facility services changes significantly throughout the day. Furthermore, facilities are not open all day (with some exceptions, such as convenience stores). Therefore, pursuing the problem of determining the optimal method of service provision in a spatio-temporal domain—a higher dimension created by incorporating temporal elements into the facility location problem—seems to have great value, both academically and practically. In the latter half of this article, I will introduce some of the attempts I have made in this area (for details, please refer to reference [1] at the end).
Figure 1 shows a spatio-temporal domain and the flow of people within it. The horizontal cross-section of this diagram represents physical space, such as railway and road networks, while the vertical axis represents the time axis. Let's assume a situation where people stop by a facility at a station on their way home from work by train after their workday. This could be, for example, a scenario like going to a classical music concert. Of course, this can be replaced with activities like attending an English conversation school or a *Juku*, or watching a movie or a sporting event. The primary concern for service providers is likely to have as many people as possible use their services. In this context, while the selection of the station where the facility is located is certainly important, the timing of the service start is also crucial. If the service starts too early, fewer people will be able to make it in time after leaving work. Conversely, if it starts too late, fewer people will be able to get home after the service ends. In either case, it becomes impossible to secure a sufficient number of potential customers.
Figure 2 shows the ease of stopping by at railway stations in the Tokyo metropolitan area, calculated using actual railway flow data. The value for each station represents the number of people (as a percentage of the total number of travelers) who can stop by a facility for three hours after work and return home by 11:00 p.m., assuming a single facility is placed at that station and the optimal start time is chosen. The origin of the diagram is Tokyo Station, and the unit is 1 km. The station with the maximum number of potential visitors is Shinjuku Station, and stations near Shinjuku, such as Shibuya, as well as stations around Tokyo Station, are prominent. It is understandable that many facilities are located near these stations.
Next, let's consider a scenario where multiple facilities are located simultaneously. This is a situation where a single business entity places multiple stores in a chain. The problem here is to determine the location station and start time for each facility to maximize the total number of users who can stop by at least one of the located facilities. When there are multiple facilities, it is virtually impossible to find the best plan by enumerating all possible solutions, even with the fastest computers. Therefore, I implemented a solution method as a computer program that repeatedly applies a procedure to improve a given solution, and I will introduce the results obtained.
Figure 3 shows the result for the case of three facilities, where Shinjuku, Ochanomizu, and Yokohama were selected. A major feature is that two very close stations, Shinjuku and Ochanomizu, were chosen, and their start times differ by 30 minutes. If the time axis is ignored, there is no advantage to selecting two stations in close proximity. In reality, however, by simultaneously placing facilities in central urban areas that are easy for many people to visit and staggering their service start times, it is possible to efficiently capture flows of people with different spatio-temporal characteristics. Thus, by introducing a time axis into spatial decision-making problems, many new developments can be expected.
In recent years, large-scale and detailed geographic data have been made publicly available in various forms. Furthermore, with the advancement of information and communication technology, attempts are being made to compile large-scale flow data based on people's actual behavioral histories. Under these circumstances, I believe that the scope for the application of OR, which also studies methodologies for modeling reality, will continue to expand in the future.
References
[1] K. Tanaka: Maximum flow-covering location and service start time problem and its application to Tokyo metropolitan railway network. Journal of the Operations Research Society of Japan ,Vol. 54-4, 2011, pp. 237-258.