Determining the positions of facilities, and allocating demands to them, is a vitally important problem. Location-allocation problems are optimization NP-hard procedures. This article evaluates the ordered capacitated multi-objective location-allocation problem for fire stations, using simulated annealing and a genetic algorithm, with goals such as minimizing the distance and time as well as maximizing the coverage. After tuning the parameters of the algorithms using sensitivity analysis, they were used separately to process data for Region 11, Tehran. The results showed that the genetic algorithm was more efficient than simulated annealing, and therefore, the genetic algorithm was used in later steps. Next, we increased the number of stations. Results showed that the model can successfully provide seven optimal locations and allocate high demands (280,000) to stations in a discrete space in a GIS, assuming that the stations' capacities are known. Following this, we used a weighting program so that in each repetition, we could allot weights to each target randomly. Finally, by repeating the model over 10 independent executions, a set of solutions with the least sum and the highest number of non-dominated solutions was selected from among many non-dominated solutions as the best set of optimal solutions.

The Vector Assignment Ordered Median Problem (VAOMP) is a new unified approach for location-allocation problems, which are one of the most important forms of applied analysis in GIS (Geospatial Information System). Solving location-allocation problems with exact methods is difficult and time-consuming, especially when the number of objectives and criteria increases. One of the most important criteria in location-allocation problems is the capacity of facilities. Firstly, this study develops a new VAOMP approach by including capacity as a criterion, resulting in a new model known as VAOCMP (Vector Assignment Ordered Capacitated Median Problem). Then secondly, the results of applying VAOMP, in scenario 1, and VAOCMP, in scenario 2, for the location-allocation of fire stations in Tehran, with the objective of minimizing the arrival time of fire engines to an incident site to no more than 5 min, are examined using both the Tabu Search and Simulated Annealing algorithms in GIS. The results of scenario 1 show that 52,840 demands were unable to be served with 10 existing stations. In scenario 2, given that each facility could not accept demand above its capacity, the number of demands without service increased to 59,080, revealing that the number of stations in the study area is insufficient. Adding 35 candidate stations and performing relocation-reallocation revealed that at least three other stations are needed for optimal service. Thirdly, and finally, the VAOMP and VAOCMP were implemented in a modest size problem. The implementation results for both algorithms showed that the Tabu Search algorithm performed more effectively.

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Abstract. Location-allocation analysis is one of the most GIS useful analysis, especially in allocating demands to facilities. One of these facilities is the fire stations, which the correct locations and optimal demand allocations to those have most importance. Each facility has a specific capacity that should be considered in locating the facilities and allocating the demand to those. In recent years, the use of unified models in solving allocation problems is too common because these models can solve a variety of problems, but in most of these models, the capacity criterion for facilities has been ignored. The present study tries to investigate the location-allocation problem of the fire stations with the aid of two Tabu and Genetic algorithms with the goal of maximizing the coverage using the (Vector Assignment Ordered Median Problem) VAOMP model, taking into account the capacity criterion and regardless of it. The results of using two algorithms in problem-solving show that the Genetic algorithm produces better quality solutions over a shorter time. Also, considering the capacity criterion that brings the problem closer to real-world space, in the study area, 59,640 demands will not be covered by any station within a 5-minute radius and will be highly vulnerable to potential hazards. Also, by adding 3 stations to the existing stations and re-allocating, the average of allocated demands with the help of Genetic was 93.39% and 92.74% for the Tabu algorithm. So it is necessary to consider the capacity of the facilities for optimal services.