A common practice in spatial analysis is to represent the population of a spatial unit, such as a county or census tract, by a single point, and to use this point when measuring the distance between the population and other places such as service centers. In theoretical spatial systems, distance measurements obtained under this practice may differ from true distances by as much as eight percent, and the difference may be greater for real spatial systems. The presence and magnitude of these measurement errors have important implications for spatial analysis, and particularly for evaluating alternative facility location plans. x y dP --. ~i J FIGURE I. EXISTENCE OF SOURCE A ERROR.

The p-median problem is to select p facility sites from among n locations to minimize the average distance from the populations at the n locations to their nearest facility. A set of linear constraints and a linear objective function describe the problem. By varying the way that the objective function coefficients are derived, many other location problems can be defined as special cases of the same general mathematical form of the p-median model. These models include maximum distance-covering problems, problems with facility costs, and problems having multiple objectives. The diversity of these special cases suggests the use of the model as the core of a computer software system for location—allocation and spatial analyses.

This study examined whether multimodal trip planners can be developed using open-source software and open data sources. OpenStreetMap (OSM), maintained by the non-profit OpenStreetMap Foundation, is an open, freely available international repository of geographic data that individuals contribute about their communities. In the transit industry, Google's offer of a free online transit trip planner based on the General Transit Feed Specification (GTFS) has made GTFS a de facto standard for describing transit systems and a platform for many other Web and mobile applications. Over 125 public transportation agencies in the U.S. have put their data into GTFS format. Bus stop locations can link OSM and GTFS data. OpenTripPlanner is an open-source multimodal trip planning software system with an active developer community. The study team set up an instance of OpenTripPlanner for Tampa, Florida, using biking and walking data from OSM, and GTFS data from local transit agencies, to examine the tool's ability to route using multimodal data. The study team also recorded multimodal data for the Tampa region in OSM to examine the current OSM coding conventions and determine the coding system's ability to support functions required of a multimodal trip planner, such as providing information on access to transit, wheelchair accessibility, or conditions that could affect the safety of a trip (e.g., intersection crossings). This study also investigated the use of opensource software to quickly increase the amount of multimodal data available in OpenStreetMap. The research team created GTFS-OSM-Sync (GO_Sync), a framework and open-source software tool for synchronizing transit data between the transit agency's official GTFS dataset and OSM. GO_Sync connects the wealth of data from GTFS datasets to the ability of the OSM community to augment and improve the data. During a test deployment of GO_Sync in Tampa, OSM users corrected 173 bus stop locations. The project demonstrated that it is feasible to implement a multimodal trip planner using open-source software and open data sources. Based on existing practices regarding GTFS and OpenTripPlanner, transit schedule and route data are best obtained directly from transit agencies' GTFS files. Data on infrastructure for walking and cycling can be obtained from OSM or from other locally available public-domain data. This report suggests a few changes to the OSM coding conventions that would improve OSM's ability to meet the needs of a multimodal trip planner. The principal barrier to developing a multimodal trip planner remains the availability of data and, when using OSM as a source of data, the relatively low participation of U.S. residents in the project, compared to Europe. The OSM community recognizes this as a problem, but additional research is needed on how best to overcome it. Additional research also is needed on how best to communicate results from a trip planner to users who may have varying skill and comfort levels when it comes to bicycling and walking.

Khumawala's article [ 1 ] presents an efficient heuristic algorithm for solving the p-median problem with maximum distance constraints. This note compares the accuracy and efficiency of his heuristic with another recently published for this problem [3], which, in turn, is based on one published in 1968 [4]. Since large problems cannot be solved by the exact algorithms currently available, it is important t o know the performance capabilities of alternative heuristics. Khumawala's paper is exemplary in the detail it gives on the performance of his heuristic.The algorithm we use to recompute solutions t o the trial problems described by Khumawala is the Teitz and Bart heuristic [4] with two modifications. Large values are placed in the weighted-distance matrix whenever a possible source vertex is not within the given distance constraint of the corresponding demand node, and several errors of detail in their original paper were corrected. These are described elsewhereKhumawala's computations were made on the 30-community data of Toregas et al. [ 5 , p. 13671 with population data provided in an unpublished research paper [2, p. 121 .' Results of our own computations on this data (Table 1) are shown in a form identical with that of Khumawala [I, pp. 318-191. A single computer run, (which used 79 seconds of CPU time on an IBM 360/65) generated all but the seven asterisked entries in Table 1. A comparison of each entry from the initial run with the better entry from Khumawala's two tables shows that 43.8 percent of the 315 entries in the table are the same but that 48.3 percent are superior and 7.9 percent are inferior t o his results. Some local minima Edward L. Hillsman is a graduate student and Gerard Rushton is an associate professor in the department o f geography at the University of Iowa.[3, pp. 166-721.

Optimal p-median solutions were computed for six test problems on a network of forty-nine demand nodes and compared with solutions from two heuristic algorithms. Comparison of the optimal solutions with those from the Teitz and Bart heuristic indicates that this heuristic is very robust. Tests of the Maranzana heuristic, however, indicate that it is efficient only for small values of p (numbers of facilities) and that its robustness decreases rapidly as problem size increases.