Optimal Deployment of Resources for the U.S. Coast Guard

Optimal Deployment of Resources for the U.S Coast Guard David Afshartous1, Yongtao Guan2, Anuj Mehrotra3, Joel Magnussen4 Abstract— Each year, the U.S Coast guard receives thousands of distress calls from both commercial and private vessels in the Caribbean Given limited resources, the Coast Guard must allocate resources across stations in order to best respond to these distress calls We present a mathematical framework that can be employed to develop optimal station locations and allocations of resources Additionally, we analyze the distribution of the actual distress calls for a five-month period with respect to locations of air stations within U.S Coast Guard District Initial analyses of base proximity distances and resource allocations identify several stations where resource allocation may be inappropriate Finally, we discuss a statistical model for the distress call process that may be utilized in the aforementioned mathematical models for resource optimization Key words: resource allocation, spatial statistics, Voronoi diagrams, stochastic optimization Background Each year, the U.S Coast guard receives thousands of distress calls from both commercial and private vessels in the Caribbean These distress calls range from simple overdue vessels, to cruise ship medical emergencies and large scale commercial shipping accidents Based on the nature of the information received, critical search and rescue (SAR) response decisions must be made The strategic question of where to position the limited resources given the fixed station locations is of keen interest to the Coast Guard The Coast Guard recently embarked on a major data collection effort for distress calls received in District (Caribbean) Such analysis has not been previously conducted and is considered very important to better deploy the resources and improve responsiveness Indeed, a FY04 operating budget of $538.95 million (search and rescue) combined with increased demands due to Homeland Security creates an urgent need to effectively deploy resources to meet the multifaceted missions This paper initiates a systematic approach to analyze the Coast Guard resource allocation by defining the underlying problems and describing a potentially suitable framework for further analysis The paper is organized as follow: In section two, we present mathematical models that quantify some strategic resource optimization problems In section 3, we describe and analyze the dataset of distress calls collected by the Coast Guard, respectively The statistical analysis consists of considering proximity distances, nearest neighbor distances, and the base deployments for the distress calls In section 4, we discuss a statistical model for the distress call process that may be utilized in the aforementioned mathematical models for resource optimization Section provides a brief summary of our initial conclusions and some direction of ongoing and future research Mathematical Models The ability of each station to respond to multiple distress calls depends on the type and number of aircraft available for deployment from that station Even when the distress calls not arrive Department of Management Science, School of Business, University of Miami, Coral Gables, FL 33124 afshar@miami.edu Department of Management Science, School of Business, University of Miami, Coral Gables, FL 33124 yguan@miami.edu Department of Management Science, School of Business, University of Miami, Coral Gables, FL 33124 anuj@miami.edu U.S Coast Guard, 1200 Brickell Avenue, Miami, FL 33129 FY 2003 Report, United States Coast Guard simultaneously, multiple resources may be needed at a station due to maintenance, training, and back-up considerations Hence, allocating the number of aircraft proportionate to the number of distress calls received by a station is a good proxy for actual needs The types of aircraft allocated to a station are largely driven by the speed, range, and capability of each aircraft These factors determine the appropriate aircraft for a distress call and the travel time One of the important considerations is to minimize travel time for the deployed aircraft to enhance the potential success of the mission For each distress call, we calculate the distance to each station We assume that only one base will respond (in reality, multiple bases may be involved, and ideally this base will be the closest base Thus, a naïve travel cost function is as follows: n C i 1 f (|| xi  b ji ||) , where x1 , , x n are the locations of the distress calls, b1 , ,b4 are the locations of the four air stations, ji is the index of the closest base to xi , and || || represents Euclidean distance in R The function f is conditional upon the aircraft at each base and provides the travel time for the most appropriate aircraft Regarding appropriateness, this is dichotomized according to whether the distress call requires a helicopter or fixed wing aircraft.6 This scheme employs the observed distress calls within the travel cost function Viewing the these distress calls as the outcome of a random process, we may generalize our cost function by employing a intensity function to model the distribution of distress calls, and integrate over the entire region S to obtain the average travel cost:  C  f (|| x  b ji ||) ( x )dx S We may formulate the problem of minimizing average travel cost via Voronoi diagrams For n distinct points x1 , , x n in N-dimensional Euclidean space R N , we define a partition of R N into n convex polyhedral regions Vi (i= 1,…,n) such that every x  Vi is closer to xi than any other x j ( j i ) Let V  S {V1  S , ,V4  S ) be the Voronoi diagram generated by P {b1 , , b4 } and bounded by the region S As in Okabe et al (1992), write V {V1 , ,V4 ) for V  S {V1  S , ,V4  S ) Any point in Vi is closer to bi than b j , j i by definition The travel cost from bi to a general point x may be modeled as a function of the squared Euclidean distance f (|| x  b j ||2 ) Employing the Voronoi diagrams, we directly write the total travel cost: F (b1 , , b4 )   f (|| x  b || )( x)dx i i 1 Vi (1) We no longer need the sub-indexed notation to indicate the closest base, since this follows directly from the Voronoi diagram definition (1) is non-linear and non-convex Normally, the objective is to minimize (1) with respect to the locations of the bases b1 , ,b4 However, as the bases are fixed, our goal is the minimization of (1) with respect to the allocation of resources To make this more explicit, we write equation (1) as follows: This assumption ignores the scenario where a fixed wing aircraft is initially used for the search, and a helicopter is subsequently used for the rescues (man in water) This assumption will be relaxed in the sequel F (b1 , , b4 ; y1, , y )   f (|| x  b i i 1 ||2 , y i ) ( x )dx Vi , (2) where y i represents the resources at base i, possibly a vector with counts for multiple types of resources Assuming a fixed pool of resources, see seek to distribute the resources as y1, , y to minimize (2) The practical considerations mentioned earlier (maintenance, training, back-up), combined with the fact that multiple resources may be deployed for a single distress call, require the incorporation of multiple resources in our travel cost function For instance, if the function f simply measures the travel time for the most appropriate aircraft to arrive on scene, we may alter f by dividing by the number of resources at the station , ni , thus imposing a penalty to stations with fewer resources Given the total travel cost in (2), the minimization will result in the “closer” stations receiving proportionally more and better resources Formally, the function f is as follows: f (|| x  bi ||, y i , ni )  || x  bi || 1 t xi vi ni ni , where vi represents the velocity of the most appropriate aircraft amongst y i ; thus t xi represents the time to arrive at distress call x from bi Regarding the determination of the most appropriate aircraft, let  equal the probability that a given distress call will require a helicopter and   equal the probability that a given distress call will require a fixed wing aircraft We assume that only one resource is deployed per distress call Assuming that there exists only one type of helicopter and one type of fixed wing aircraft, we have: f (|| x  bi ||, y i , ni )  || x  bi || || x  bi || 1  (1   )  (t x1 )  (1   )t x2 i i v1i n1i v 2i n 2i n1i n 2i (3) where v 1i is the velocity of the helicopter, v 12i is the velocity of the plane, n1i is the number of helicopters, n 2i is the number of planes, t x1i is the time to arrive at distress call x from bi for a helicopter, and t x2i is the time to arrive at distress call x from bi for a plane Since the objective function F is non-linear and non-convex, we must resort to computational methods, e.g., the steepest descent method See p.448 of Okabe et al (1992) for the derivation of the first derivatives of F in general Our problem will be slightly different due to the form of the function f Among the more strategic problems is the determination of optimal placement of the stations This may involve potentially re-locating the existing stations and/or opening new stations Similar problems of facility location and allocation have been considered in many applications (such as distribution and logistics) and widely studied Among some successful approaches for solving similar problem in absence of stochastic elements is the use of Mixed Integer Programming models The problem of determining suitable locations of the air stations and the allocation of appropriate resources should be solved in conjunction Such a problem in presence of uncertainty of distress call locations results in stochastic non-linear programming optimization models Such an analysis, which is underway, entails developing suitable models and solution techniques Here we focus on some preliminary analysis to point out the potential benefits of using a more rigorous framework to analyze these problems 3 The Distress Call Data The data consists of 450 distress calls, spanning a six month interval from September 2001 to February 2002 For each distress call, there exists information regarding location, the nature of the case, and the resources that were deployed In addition, the air and sea stations from which these resources were deployed can be determined The Coast Guard reports position locations in a geographic coordinate system7 known as Word Geodetic Society 1984 (WGS84) In order to display the data in two dimensions, the data from the spherical earth must be converted onto a flat plane We have chosen the Azmuthal Equidistant projection; the most significant characteristic of this projection is that both distance and direction are accurate from the central point Figure below displays these distress calls under this projection Only 317 cases are displayed due to missing data Figure 1: Distribution of Distress Calls Although many cases are near the Florida coast, several cases are widely distributed across the Caribbean Regarding distances, one observes that some distress calls are a few miles from Miami while others are over 500 miles away District includes four air stations and eighteen sea stations Table below displays the air base assets A geographic coordinate system uses a three-dimensional surface to define locations on the earth Air Base Name No of Assets Savannah, GA Borinquen, Puerto Rico Miami, FL Clearwater, FL 11 18 Table 1: Asset Distribution 3.1 Basic Distance Analysis The distribution of the distress calls may be analyzed from a variety of perspectives Figure below illustrates the distribution of the distances of these distress calls to the U.S coastline Over half the calls are over 175 miles (excluding few outliers of calls over 2000 miles from the coast.) Moreover, the skewed distribution illustrates a concentration of calls that are very close, and a long tail of outlying calls that are much further away Figure 2: Distribution of distance to the US coastline For each distress call, we calculate the distance to each of the air bases and thereby determine the base which is closest to a given distress call Base Name Miami Clearwater Borinquen Savannah No times closest (%) 187 (58.9%) 55 (17.4%) 44 (13.9%) 31 (9.8%) Table 2: Closest base ranking Air Station Miami is clearly in an advantageous position with respect to the locations of the actual distress calls in this time period Savannah, on the other hand, is rarely the closest base to a distress calls, somewhat expected given its location in Georgia Figure illustrates the proximity ordering versus the number of resources at each base Figure 3: Air Proximity Frequency versus Resource Allocation The x-axis measures the number of times a particular base was the closest, while the y-axis measures the number of aircraft at a base Thus, Miami is the closest 187 out of 317 calls and it has 11 planes Clearwater has 18 planes and was the closest base 55 times Thus, it appears that the resources allocated to Miami/Clearwater need to be adjusted Next, for the distress calls for which a particular base is the closest base, we examine the distribution of the distance from the base to the distress calls Figure below displays the distribution of these distances for each air base (outliers removed, > 2500 miles) Figure 4: Distribution of closest air base distance, outliers removed (N=312) For Borinquen, the distribution of closest distances is wide: when it is the closest base, half of the time it is almost 500 miles away (median = 461); 25% of the time it is over 1000 miles away Borinquen was the closest base 44 out of 317 times Other bases have a more narrow distribution, with Savannah having the smallest inter quartile range 3.2 Nearest Neighbor Distance Analysis The analysis above has not investigated whether the closest base is significantly closer than the other bases If a base is the closest base, but there exists another base that is only miles further away from the distress call, this is much different than the scenario where the next closest base is 200 miles further away Thus, we analyze the distribution of this differential distance Figure below illustrates the distribution of how much further away the next closest base is to the distress call for each closest base Figure 5: Next closest base distance for each air base, N=317 Thus, when Borinquen is the closest base, over half the time the next closest base is over 240 miles further from the call, and these distances exhibit a range up to 1000 miles This underscores the strategic importance of the base: a) Borinquen is often the closest base, and b) the next nearest base is often very far away Given the relatively few resources at Borinquen, the Coast Guard may want to put more resources in Borinquen Note: there exists less variance with respect to next closest distances than with respect to closest distances (Fig vs Fig 4) The frequencies for bases being the next closest base are provided below: Air Base Savannah Borinquen Miami Clearwater # Times Closest Base 31 44 187 55 # Times Next Closest Base 21 16 81 199 Table 3: Closest, Next Closest air bases, frequencies Counts for Miami and Clearwater are in opposite directions, i.e., Miami is the closest base more than twice as often as it is the next closest base, while Clearwater is the next closest base almost four times as often as it is the closest base We propose a summary measure to combine the information in the above tables in order to better assess the strategic importance of each base with respect to the distribution of distress calls A simple measure is a weighted sum of the closest and next closest frequencies for a given base, the weights depending on the total number of bases For example, for the air bases the weights could be and The tables above indicate that this measure would add value since a base that has a high frequency as the closest base can have a low frequency as the next closest base To be sure, one would expect a reciprocal relationship between the closest bases and the next closest bases For instance, for a given distress call, Base A may be the closest base and Base B may be the next closest base, but for a different distress call this relationship may be reversed Moreover, if the differential distance for such a base pairings is usually very small, then these two bases not provide much “leverage” with respect to response area In other words, responsiveness would not be greatly affected by relocating the resources from Base A to Base B or vice versa Figure 11 illustrates the breakdown of the next closest bases for the occasions when each air base is the closest air base For example, when Savannah is the closest base (N=31), Borinquen is the next closest base once, Miami is the next closest base twice, and Clearwater is the next closet base for the remaining 28 cases Note: there is always one base that is rarely the next closet base when a given base is the closest base, e.g., Clearwater is never the next closest base when Borinquen is the closest base (N=44) As a result, the next closest base is usually distributed among two of the possible three bases.8 Which of these two bases is more often the next closest base depends upon their relative locations and the distribution of the actual distress calls Figure 6: Closest Base, Next Closest Base Relationship (Air) 3.3 Base Involvement As mentioned earlier, we also have data regarding the involvement of each base for each distress call Initially, we focus on the first involved base For the 317 calls, we have at least one base involved 190 times, i.e., 128 of the 318 cases did not have any resources deployed Of these 190 cases, the first involvement is an air base 128 times and the remaining 61 involvements are sea bases Moreover, a second base is involved only 52 times, i.e., 138 of the 190 resource deployments involve resources from a single base Of the 52 second base deployments, 23 are from air bases and 29 are from sea bases The distribution of the cases vary according to the resources that were deployed, e.g., resources from air, sea, and no bases The no-response cases are widely Given that we have four bases, we may draw a triangle connecting the closest base and the two bases closest to it, with the final base lying outside this triangle; thus, the base outside the triangle will be the base that is rarely be the next closest base Two exceptions : Savannah-Borinqen, Miami-Savannah (XY) dispersed, the air cases are less dispersed with a Florida coast and Bahamas concentration, while the sea cases exhibit a tight Florida coastline pattern with some Caribbean deployments Involvement statistics may be with respect to proximity In Table below, base involvement statistics are calculated amongst both sea and air bases, while base proximity statistics are calculated within air bases.9 Air Base Savannah Borinquen Miami Clearwater # times deploye d 1st # times deploye d 1st and closest #times deploye d 1st and 2nd closest # times deploye d 2nd #times deploye d 2nd and closest #times deploye d 2nd and 2nd closest # times closest (closest and resources deployed) 16 71 40 14 57 0 20 14 0 31 (20) 44 (24) 187 (113) 55 (33) # times 2nd closest (2nd closest and resources deployed) 21 (10) 16 (5) 81 (48) 199 (127) 128 80 29 23 14 317 317 Table 4: Air Base involvement versus proximity Thus, we observe that of the 128 times the first base was an air base, the base was not the closest or next closest only 27 times (36=128 – 80 - 29); of the 23 times the second base was an air base, the base was not the closest or next closest only twice We highlight the following: When a plane is the first asset deployed, it is usually from the closest air base Borinquen was involved first only once, in spite of the fact that it was the closest base 44 times, 24 of which resulted in base involvement (sea=6, air=21; [air Miami = 8])) Thus, we highlight the following: Quite often when an air base is needed, and Borinquen is the most appropriate with respect to proximity, other air bases must be used instead Given the variability of how much further away the next closest base is, response time is clearly being impacted by not choosing to deploy from Borinquen Statistical Model for Distress Call Process To address the issues raised in Section 2, an estimate for the intensity function  ( x ) over the region S is needed For this we consider two approaches, a parametric approach and a nonparametric approach, respectively 4.1 A parametric approach Figure in Section 3.1 reveals that the likelihood for a distress call to occur at any location x, or equivalently the intensity function  ( x ) , is closely related to the distance of this location to the U.S coastline: the further the location is away from the coastline, the less likely for a distress call to occur at that location In addition, Figure suggests that this likelihood decays at an exponential In other words, we are interested in proximity with respect to bases of its type, and involvement with respect to all types of bases It does not matter if an air base is closer to the call if an air base is not necessary rate with respect to the distance to the coastline In light of these facts, we parameterize  ( x ) as follows  ( x )  exp{   || xc ||} , (4) where || xc || denotes the Euclidean distance of x from the U.S coastline We assume further that these locations of distress calls occurring independently, or equivalently, they are generated from an inhomogeneous Poisson process The distribution associated with the observed locations of distress calls, x1 , x2 ,L , xn , of this process on S can be factored as the product of a Poisson distribution with mean   S  ( x ) dx for the number of calls n, and a set of mutually independent variables with density  ( x ) /  for the call locations xi The parameters  ,  in (4) can then be obtained by maximizing the log-likelihood n n i 1 i 1 L ( ,  | x1 , x2 , L , xn )   log{ ( xi )}  S  ( x) dx   {   || xc ,i ||}  S  ( x ) dx (5) Note that to maximize (5), the integral S  ( x ) dx needs to be calculated Berman and Turner (1992) discuss different quadrature schemes for this term Their approach can be applied to our data set 4.2 A nonparametric approach A visual examination of Figure indicates that although the distance of a distress call to the U.S coastline plays a critical role in determining the value of the intensity function at that location, it by no means is the only factor that will affect this outcome For example, a location that is far away from the U.S coastline may be close to a major island in the Caribbean, where  ( x ) may be higher than what is given by (4) In addition, the intensity of distress calls in the Florida coastline seems much higher than in other states Thus, the aforementioned parametric approach may not provide a satisfactory fit to the data For this reason we also consider a nonparametric approach Let B(x,h) denote the disc with centre x and radius h, | S I B ( x, h ) | denote the intersected area of S and B(x,h), and N(x,h) denote the number of distress calls that fall within B(x,h) An edgecorrected estimator for  ( x ) is given by ˆ ( x )  N ( x, h) (6) | S I B ( x, h) | The above estimator assumes all the calls within B(x,h), independent of their respective distances to the U.S coastline, are equally important in estimating  ( x ) In light of Figure and the discussion in Section 4.1, this may not be very desirable Instead, we modify (6) as follows ˆ ( x )  N ( x, h, r ) , (7) | S I E ( x, h, r ) | where E ( x, h, r ) is an ellipse centered at x and N(x,h,r) now denotes the number of calls within E(x,h,r) The length of the major axes of the ellipse is equal to h and that of the minor axes is equal rh The orientations of the axes of the ellipse are the direction of the straight line connecting x with the closest point in the U.S coastline and the direction perpendicular to it, respectively The value 10 of r depends on how fast  ( x ) changes at x with respect to the distance to the coastline One possible solution is as follows: r c  '(|| xc ||)  c for some constant c < (8) (8) indicates that if  '() changes very quickly, which is the case if a location is near the coastline, r should be kept small; while if  '() changes fairly slowly, which is the case if a location is very far away the coastline, r should be chosen around one Summary We have proposed mathematical models to quantify the resource deployment problems faced by the Coast Guard for distress call response In addition, we have analyzed actual distress calls for a five-month period Initial analyses of basic distances and resource allocations have helped identify cases where resource allocation may be inappropriate Specifically, there may be too many resources at Clearwater and too few resources at Miami The next nearest-neighbor distances provide a risk assessment for each base, while next nearest-neighbor analyses quantify inter-base dependencies; the Coast Guard could use these relationships to guide resource allocation We have also examined actual resource deployments, e.g., whether resources were indeed deployed and from which bases they were deployed This information quantifies the operational decisions taken by the Coast Guard; air bases are overwhelmingly relied upon for the distress calls By dividing the distress calls according to the type of response (air/sea/none), we observe a clear distinction according to response type Finally we have proposed both a parametric and a nonparametric approach to estimate the intensity of the distress calls at any given location These are necessary to apply our proposed mathematical models To be sure, this analysis is specific to our data; it will be necessary to cross-validate across longer time spans.10 This analysis represents the initial step in a larger study where the goal is to optimize resource allocation to the bases In the sequel, we utilize the statistical model of the distress call process as an intensity function for the mathematical models for resource optimization References: Berman, M and Turner, T.R (1992) “Approximating point process likelihood with GLIM,” Applied Statistics, 41, 31-8 Cornuejols, G., Nemhauser, G.L., and Wolsey, L.A (1984) “The Uncapacitated Facility Location Problem,” Report No 605, Operations Research and Industrial Engineering, Cornell University Kennedy, Melita and Kopp, Steve (2000) “Understanding Map Projections,” Environmental Systems Research Institute (ESRI): Redlands, CA Nemhauser, G., Wolsey L.A (1988) Integer and Combinatorial Optimization New York: John Wiley & Sons Okabe, A., Boots, B., and Sugihara, K (1992) Spatial Tesselations: Concepts and Applications of Voronoi Diagrams New York: John Wiley & Sons United States Coast Guard, FY 2003 Report 10 We also analyzed the distress call data with respect to the eighteen sea bases, but have omitted these results due to space limitations 11 ... denotes the number of calls within E(x,h,r) The length of the major axes of the ellipse is equal to h and that of the minor axes is equal rh The orientations of the axes of the ellipse are the direction... is the velocity of the helicopter, v 12i is the velocity of the plane, n1i is the number of helicopters, n 2i is the number of planes, t x1i is the time to arrive at distress call x from bi for. .. 2002 For each distress call, there exists information regarding location, the nature of the case, and the resources that were deployed In addition, the air and sea stations from which these resources