Spatial Pattern Analysis for finding Weighted Candidate Set for p-median Problem inLocating Emergency Facilities

Facility location problems are well known problem in Industrial optimization. Locating emergency facility typically involves optimal location allocation of facilities with an objective to locate limited number of facilities such that the average response time is minimized. This paper describes an approach for finding a candidate set of emergency facility locations based on spat ial pattern analysis of incident data for input to the location-allocation problem. Keywords— p-median; Spatial Pattern Analysis; Location Allocation; Emergency Facilities; ArcGIS; I. INTRO DUCTION The location of facilit ies for handling incidents affects significantly on the safety and well-being of the community. The performance of the siting of these facilities depends on the response time of the emergency facilit ies. The core objective of emergency facility allocation problem is to locate the facility where the average response time (time between the receipt of a call and the arrival of emergency vehicle) is minimized. In past a lot of researches has been done for mathematically representing and solving this combitorial optimization problem based on various objectives. The pmedian model is one of the most popular and common model for defining facility location problem in the public and private sector. The p-median problem and its various variants have been used for deciding the location of facilit ies and allocate demand points toone or multip le facilities.The p-median problem on general g raphs with arbitrary p is computationally difficult to solve by exact methods as it is NP-hard [3].The difficulty of solving the p-median problem using exact methods has led researchers to consider sub optimal solutions generated by heuristic approaches. GIS as a tool stores, manipulate, analyse and visualize spatially referenced data. Earlier the facility location problem has been solved with more emphas is on graph solving strategies and less on the spatial nature of the facility location and demand.Here spatial pattern analysis techniques are discussed and appropriately used for identifying patterns in incident data (location and attribute data of incidents) which can help in determining the possible facility locations. Here basically spatial pattern analysis is used for determining the hot spots (high value clusters) in the incident data. II. P-MEDIAN MO DEL FOR FACILITY ALLOCATION The p-median model, one of the popular location allocation models was formulated by Hakimi in the mid-sixt ies. It is defined as: determine the location of p facilit ies to minimize the average (total) d istance between demands and their closest facility. Formally, the p-median problem is stated as: Given a graph or a network G = V , E , find Vp subset of V such that Vp = p, where p may either be variable or fixed, and that the sum ofthe shortest distances from the vert ices in V −Vp to their nearest vertex in Vp is minimized. In past a number of authors[6][7][8], used the p-median problem solution to locate emergency facilit ies, A brief review of p-median and other location model problem in emergency facility is d iscussed and used for solving large scale problems [4].A modified p-median is modelled for addressing the issue of dynamic behaviour of t raffic and uncertainty of demands [10]. The p-median problem in context to emergency facilit ies can be defined as determin ing the location of p emergency facilit ies from a set of possible candidates for emergency facilit ies with an objective to minimize the weighted distance between emergency points and their closest new emergency facility. The mapping of p-median problem in real world is done by partitioning the geographical region into a number of subregions based on certain criteria and constructing a corresponding graph. Each node of this graph represents the centre of the subregion and the edges represent the links in the Volume 2, Issue 5, May 2012 www.ijarcsse.com © 2012, IJARCSSE All Rights Reserved Page | 70 transportation network weighted by average travel time. Th is paper describes a method for finding the subregion based on spatial pattern analysis on the incident data. III. SPATIAL PATTERN ANALYSIS TECHNIQUES The spatial attribute/factor is the one that is common between location allocation problem and GIS. A review on location analysis and contribution of GIS functionalit ies for enhancement of this field is provided in [2].Spatial analysis functionalities can enhance the decision fo r site selection based on various spatial and non-spatial factors. Spatial pattern analysis refers to the use of quantitative methods for describ ing and analysing the distribution pattern of spatial features.At a general level, pattern analysis techniques describe whether the distribution pattern is random, dispersed or clustered. More specific techniques find high/low clustering with in the distribution pattern. Spatial autocorrelat ion concept helps in analysing the patterns, it measures the relationship among values of a variable according to the spatial arrangements of the values [11] and it finds wheater the data is clustered, random or unclustered based on the similarity of the values and their spatial closeness. Moran’s I is a popular technique that measures the spatial autocorrelation and thus detects the presence of the clustering of similar values[11].Moran's autocorrelation coefficient (often denoted as I) is an extension of Pearson productmoment correlation coefficient to a univariate series . More illustratively Pearson’s correlationbetween two variables x and yboth of length n measures whether, onaverage, x i and yi are associated.Moran's autocorrelation coefficientapplied to a single variable, say x , measures whether xiand xj, with i ≠ j, are associated. Mathematical Formula for Moran’s I. I = n S0 wij x i − x xj − x n j =1 n i=1 x i − x 2 n i=1 1 Where x i and xj are attribute values for features iand j, wi ,j is the spatial weight between feature i and j, x is the mean value of attribute and S0 is the sum of allwij ’s. Moran’s I however cannot tell whether the clustering is made of h igh values or low values .G-statistic can tell whetherthe spatial distribution of high values or low values in the dataset is more spatially clustered [12]. Various versions of G-statistic consider Local and Global d istribution of dataset. The General G-statistic is the global version o f G-statistic which assesses the overall pattern and trend of the data helps in finding High/Low clustering in the dataset.Itis most appropriate when there are unexpected spatial spikes of high values in an evenly distributed spatial dataset. Mathematical equation for General G-statistic The General GStatistic of overall spatial association is given as: G = wi ,jx ixj n j =1 n i=1 x ixj n j =1 n i=1 , ∀j ≠ i 2 Where x i and xj are attribure values for features iandj, and wi ,j is the spatial weight between feature iand j.