(2) The probability that there will be a specific delay before Probability of Waiting Time Greater Than (T) Minutes; Third In each time interval, the chance that response to a call for service will be delayed when there are six engine companies is more than ten times as great as the chance when there are nine engine companies. (3) The fraction of time that the companies will be servicing calls The actual utilization factor of 2.6% for the Alexandria Fire Department's present seven companies agrees well with the numbers predicted by the model. 5.0 APPLICATION OF LOCATION MODELS If unlimited funds were available to supply resources, the Fire Department could theoretically maintain individual stations to protect each city block. However, since public service facilities are constrained by budget provisions, it is necessary to assign the available resources to the best strategic locations to ensure maximum effectiveness. Location analysis can be applied to this spatial aspect of fire service problems. Because response time is an integral factor in saving lives and reducing property loss, "the objective function" or goal of various location models is to reduce the time required to reach all demands for service by the nearest resource to each call. Since building-by-building or block-by-block identification would yield an excessive number of data items for the types of computation required, the area under study is divided into "zones", and response times are calculated from the nearest fire station to the zone. By comparing the various response times from actual and potential station locations, it is possible to evaluate the relative efficiency of different configurations of station locations. The function of a location model is to determine the "optimum" locational pattern, where the optimum is defined as the configuration yielding the minimum average response times. Two types of location analyses were used to generate and evaluate alternatives: an adaption of the Maranzana transportation algorithm [5] and a modification of the REDIST algorithm used to determine voting districts [6]. Both location methods use a heuristic (i.e., a decision-rule that doesn't guarantee a precisely optimal solution) to determine a "good" solution at reasonable computer cost. The locational patterns generated by the Maranzana and REDIST methods were used as two additional alternatives to be compared with alternatives suggested by the Alexandria Fire Department through the use of a simulation method to be described in the next chapter. 5.1 Assumptions Several assumptions (the first three of which reflect normal fire service operating procedures) were made in applying the Maranzana and REDIST models: 1. The important response time is that of the first due engine company, and it is that response time which should be minimized. 2. Each engine company is assigned to one station at a fixed location. The company always responds from the station, and is assumed to be in its station when not servicing calls. 3. 4. 5. A given fire demand zone is served exclusively by the closest All alarms occurring within a zone are treated as occurring at Travel times from zone centers to each potential station location are known. A simplified method was used to determine the zones and travel times in Alexandria. The steps were as follows: 1. Zones were equated with the 1500 x 1500 feet cell divisions mentioned in Section 3.4. 2. Calls for service within a cell were assumed to originate at the center of that cell. 3. An (x,y) coordinates system was developed with coordinates (0,0) at the northwest corner (see Figure 11). For example, the midpoint of the shaded cell has coordinates (0.5, 0.5). 4. The existing fire stations were placed on this coordinate system as the initial set of locations for the model to improve upon. 5. The distances between existing and proposed fire station locations and the centers of the cells were measured along theoretical streets paralleling the coordinate grid. 6. The travel time from each cell center to each existing and potential station site was calculated by dividing the travel distance by the assumed speed of 25 m.p.h. 5.2 Maranzana The Maranzana method consists of the following procedure: 1. Initialize Make an arbitrary selection from the possible station locations (in Alexandria, the current configuration). 2. District Assign each zone to the nearest fire station. A district is 3. Move For each district, determine if the current fire station location minimizes the objective function. If not, determine the location within that district which does so. This becomes the new fire station site for that district. 4. Terminate If the objective function in any district has been reduced, The locations selected in Alexandria by the Maranzana solution method are depicted in Figure 11. where W is the number of calls from the i-th cell and t. 1 t1 is the travel time to the i-th cell from the closest fire station, is used by the Maranzana method as the objective function to be minimized. This application used 46K of computer core and needed five passages through Steps 2-4 to terminate. Each such iteration took 40 seconds of computer time on a UNIVAC 1108 with Exec 2 system. Total running time was 3 minutes and 26 seconds. 5.3 REDIST REDIST uses almost the same objective function. In this case, the squares of the response times are used in order to make the districting more compact. However, the model also attempts to balance the work load (number of calls handled) among the various fire stations. Therefore, there is a major procedural change. Steps one and two are the same as in the Maranzana approach. However, step three is modified to include: Balance Redistribute zones to be served from each fire station to equalize the number of calls handled by each station as much as possible. The model then continues with step four of Maranzana. The locations and districts selected by the REDIST solution method in Alexandria are shown in Figure 12 (7). REDIST ran with 17K of computer core and needed five iterations at 41 seconds per iteration. Total running time was 3 minutes and 35 seconds. The application of these two methods of locating fire stations yielded two distinct locational patterns. These patterns are very much a function of the specific objective function used, the limitations imposed, and the assumptions made. It is impractical to experiment by physically relocating fire stations and observing the results. Therefore, the analytic methods of Location theory are a valuable managerial tool for discovering reasonable alternatives to be tested through simulation. The two alternative locational patterns and the present configuration were compared and evaluated using the simulation model discussed in the next chapter. |