Innovative Applications of O.R.

US Coast Guard air station location with respect to distress calls: A spatialstatistics and optimization based methodology

David Afshartous a,*, Yongtao Guan b, Anuj Mehrotra c

aDivision of Clinical Pharmacology, Miller School of Medicine, University of Miami, Miami, FL 33129, United StatesbDepartment of Management Science, University of Miami, School of Business, Coral Gables, FL 33124-8237, United StatescDepartment of Biostatistics, Yale School of Public Health, Yale University, New Haven, CT 06520-8034, United States

a r t i c l e i n f o

Article history:Received 8 March 2007Accepted 9 April 2008Available online 14 April 2008

Keywords:Spatial statisticsSimulationRobust optimizationFacility locationLocation uncertainty

a b s t r a c t

We study the problem of suitably locating US Coast Guard air stations to respond to emergency distresscalls. Our goal is to identify robust locations in the presence of uncertainty in distress call locations. Ouranalysis differs from the literature primarily in the way we model this uncertainty. In our optimizationand simulation based methodology, we develop a statistical model and demonstrate our procedure usinga real data set of distress calls. In addition to guiding strategic decisions of placement of various stations,our methodology is also able to provide guidance on how the resources should be allocated acrossstations.

! 2008 Elsevier B.V. All rights reserved.

1. Search and rescue mission

Each year, the US Coast Guard receives thousands of distresscalls from both commercial and private vessels. Given the emer-gency nature of many of these calls, a short response time is oftenthe difference between life and death. Green and Kolesar (2004)examined the context, content, and nature of MS/OR research inemergency responsiveness and note that there has been a dearthof recent research in this area and that management science tech-niques can be particularly useful. This observation is particularlyapplicable to the US Coast Guard operations.

As part of the department of Homeland Security, the US CoastGuard must balance the resources across their various missions:maritime mobility (ice operations, aids to navigation), national de-fense, maritime security, protection of natural resources, and mar-itime safety. Search and Rescue (SAR), a component of maritimesafety, concerns the actions that the US Coast Guard takes in orderto respond to emergency distress calls arising in the maritime envi-ronment. These distress calls range from medical emergencies oncruise ships to large scale shipping accidents to recreational boat-ers in peril. In 2003 alone, an estimated 5104 lives were saved bythe Coast Guard, 655 lives were lost, 481 lives were unaccountedfor, and 31,562 distress calls were received (USCG FY2003 Report).

Fig. 1 displays 4042 distress calls received during the year 2000which received a Coast Guard air response.

This paper concerns the strategic aspect of the emergency dis-tress call response problem, i.e., the determination of the actuallocations of Coast Guard stations such that they are adequately lo-cated with respect to distress calls that vary over time.

Discussions with Coast Guard officers have revealed that thisquestion has not been sufficiently investigated with respect tothe observed pattern of distress calls. Procedures at the tacticaland operational level are better developed. For instance, resourcedeployment decisions are made in conjunction with computer-as-sisted modeling of the probability of detection (POD) under variousdeployment scenarios. Operational decisions are guided via estab-lished survival tables to guide the search and rescue efforts andwell-defined procedures and advanced technology designed tomaximize the probability of detecting a target. While resource allo-cation and tactical/operational procedures are important, the focusof this paper is on the strategic question of locating the stations.

In the next section, we discuss the associated facilities locationproblem and describe why the methods in the literature are insuf-ficient for our problem. Our main technical contribution is thedevelopment of a spatial statistics model in Section 3 to providea simulation and optimization based methodology that enablesdetermination of robust locations for stations. Based on a real dataset of distress calls, our approach develops a statistical model withvariable distress call intensity in order to simulate distress calllocations. Integer programming models are then used to determine

0377-2217/$ - see front matter ! 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.ejor.2008.04.010

* Corresponding author.E-mail addresses: afshar@med.miami.edu (D. Afshartous), yongtao.guan@yale.

edu (Y. Guan), anuj@miami.edu (A. Mehrotra).

European Journal of Operational Research 196 (2009) 1086–1096

Contents lists available at ScienceDirect

European Journal of Operational Research

journal homepage: www.elsevier .com/locate /e jor

optimal locations for responding stations. The combination of sim-ulation and optimization provides information with respect to therobustness of initial solutions, which is otherwise difficult to ob-tain. Unlike traditional stochastic and robust optimization meth-ods, our approach does not require the specification of a set ofscenarios or a probability density function. We illustrate our pro-posed approach in Section 4 with respect to the Coast Guard Dis-trict 7 (headquarters Miami), and we conclude in Section 5.Although significant effort has been expended with respect tousing realistic information, this section is intended as a vehicle toillustrate the methodology rather than a full application study.

2. Optimally locating stations

Facilities location problems form a very important aspect ofstrategic planning for a wide range of firms in the public and pri-vate sector (Owen and Daskin, 1998). Some related examples in-clude the classic warehouse location problem (Weber, 1909), thesiting of equipment for oil spills (Psaraftis et al., 1986), and locatingbase stations or switching centers in a communications network(Amaldi et al., 2003; Mathar and Niessen, 2000; Hakimi, 1964).

The location literature may be broadly divided into medianproblems, center problems, and covering problems. The p-median(p-MP) problem is the problem of choosing p facilities that mini-mize the total weighted sum of distances of the clients (distresscalls) to the closest facility (station). The p-center (p-CP) problemrepresents the problem of choosing p facilities that minimizesthe maximum distance of any client to the closest facility. Theuncapacitated facility location problem (UFLP) may be viewed asa relaxation of p-MP where p is not fixed and a setup cost is in-cluded in the UFLP since the ‘‘p-facilities” constraint is omitted.Covering problems focus on ensuring that all or as many clientsas possible are within a pre-defined distance to a facility, and theseare very common in the emergency services field (Fitzsimmons,

1971; Gendreau et al., 1997). In all of the aforementioned prob-lems, although it may be assumed that each facility is able to serveall the potential clients a capacity constraint is usually implicit. Wecan formulate the basic framework as follows:

m: the finite number of clients, indexed by i, i 2 I = {1, . . . ,m},n: the finite number of sites for potential facilities, indexed by j,

j 2 J = {1, . . . ,n},p: the number of facilities to be opened, 1 6 p 6 n.

Often, the literature suggests that demand points be aggregatedinto sectors, each of which becomes a client in the model (Franciset al., 2000). However, potential errors introduced by aggregatingdemand points or by approximating a zone by a point can be se-vere depending on the type of objective function employed (Hills-man and Rhoda, 1978). Our methodology does not requireaggregation of points. Since generation of points is trivial oncethe intensity function is estimated, little would be gained by aggre-gation into sectors.

In general, the clients have some pre-determined demand for acommon good or product, in this case emergency rescue servicefrom the Coast Guard. For each existing and potential station,and each station-client pair, we define:

fj: the fixed cost of establishing facility j,cij: the total variable cost of serving client i’s demand from facil-

ity j.

Total cost is usually modeled as cij = wi(hj + tij), where

wi: the number of units demanded by client i,hj: the unit operating cost at facility j (including variable pro-

duction and administrative costs),tij: the unit cost of shipping to client i from facility j.

Fig. 1. Distress calls (year 2000).

D. Afshartous et al. / European Journal of Operational Research 196 (2009) 1086–1096 1087

We consider demand for each client to be equal, although thiscould be changed to model situations where different distress callsreceive more weight according to how many lives are at stake.Moreover, we assume that all of the demand is served by a singlefacility and not spread over facilities. Location problems that relaxthis assumption are known as vector assignment p-median(VAPMP) problems. Each demand node has a vector of utilizations:the first component is the fraction of demand that will be servicedby the closest facility; the second component is the fraction of de-mand that will be serviced by the second closest facility, etc. SeeWeaver and Church (1985) for details. Capacitated facility locationmodels without single-sourcing constraints may also split a client’sdemand in the optimal solution. We initially assume that each sta-tion has similar operating costs, such that hj may be ignored, andthat Euclidean distance dij is a good proxy for shipping costs.

Depending on the particular optimization goal, our locationproblem may be considered as a p-MP, p-CP, UFLP, or coveringproblem. For the purpose of illustration, we use a hybrid modelknown as the p-uncapacitated facility location problem (p-UFLP);this model is the same as the UFLP, but for the fact that the numberof facilities to be selected is pre-determined to be p. Although it iscommon to utilize coverage objectives in emergency services, weemploy a median objective for simplicity of illustration and to viewthe problem in terms of average distance per call; this informationis important to the Coast Guard. The p-UFLP may be solved viainteger linear programming where each integer variable is re-stricted to be either 0 or 1. Formally, we use the following binaryvariables:

yj ¼1; if facility j is established;0; otherwise;

!

xij ¼1; if client i is served from facility j;0; otherwise;

!

to state the linear integer program:

minX

j2Jfjyj þ

X

i2I

X

j2Jcijxij

X

j2Jxij ¼ 1; all i; i 2 I;

yj # xij P 0; all i; j; i 2 I; j 2 J;X

j2Jyj ¼ p;

yj 2 ð0;1Þ; all j 2 J; xij 2 ð0;1Þ; all i 2 I; j 2 J: ð1Þ

The first set of constraints ensures that all demands are satisfied,while the second set of constraints ensures that clients are onlyserved from an open facility.

Although facilities location models have originally been studiedfrom a deterministic perspective, there also exists work from a sto-chastic perspective (for a extensive review, see Snyder, 2006).These stochastic extensions may be divided into two groups: sce-nario-based and probabilistic-based. Scenario-based models con-sider multiple future scenarios, usually with respect to clientdemand and solve an optimization problem to satisfy various crite-ria (Daskin et al., 1997). Specifically, these approaches include thestochastic optimization approach of minimizing expected costacross scenarios (Louveaux, 1986; Mirchandani et al., 1985; Wea-ver and Church, 1983) and the robust optimization approach ofoptimizing worst-case cost or regret across scenarios (Serra andMarianov, 1998). In addition to client demand, the travel distanceon a network is often treated as uncertain, e.g., depending upon thetime of day for the ambulance location problem. In scenario-basedapproaches, the potential set of scenarios (not all of which need bechosen) is specified a priori and this set is fixed. Scenario-based ap-proaches have two main drawbacks: (1) the identification of thescenarios is itself often a difficult task, and (2) the range of scenar-

ios that can be specified is often limited due to computational rea-sons (Snyder, 2006). This latter drawback has recently beenmitigated by Ntaimo and Sen (2005).

Probabilistic approaches explicitly introduce random variablesto model the probability distributions of parameters of interest,once again mainly for the client demand (Carbone, 1974; Mirchan-dani and Odoni, 1979). Other research in stochastic location con-cerns the issue of facility availability, employing queueing theoryto model the demand for limited resources (Daskin, 1982, 1983).Similar research exists in the emergency response literature (Greenand Kolesar, 2004). However, there exists very little research withrespect to the stochastic nature of the existence and locations of theclients. Two exceptions are Belardo et al. (1984) and Psaraftis et al.(1986), both for the case of locating oil spill response resourceswhere the clients are the spills. However, for both models,although actual occurrence of clients (spills) is stochastic, the loca-tions are taken as the centroids of the different geographic regionsand are thus fixed. While this is plausible for the oil response prob-lem where oil shipping patterns are relatively well known, this isnot applicable to Coast Guard distress calls where there is signifi-cantly greater heterogeneity in boating traffic patterns and thusdistress call locations.

Other attempts to model the client locations stochastically in-clude Cooper (1974) and Ntaimo and Sen (2005). Although Coo-per’s (1974) approach is a step closer towards a stochastic modelof the clients, it is limited by the fact that it is a parametric ap-proach and thus one needs to select a single probability densityfunction and identify parameter values of the probability distribu-tion. Ntaimo and Sen (2005) study solution procedures for the sto-chastic server location problem (SSLP) and introduce uncertainty inthe network of client locations via a Bernoulli random variablewith p = 0.5. This is not appropriate for the Coast Guard problemsince we are not dealing with a stochastic network.

In addition to scenario-based approaches, there exist othermethods to investigate the stability or robustness of a solution.Hodgson (1991) used simulation to investigate the impact of dataerrors on the p-median model. His focus is on the impact of theseerrors, while our focus is the impact of the actual spatial processfor the client locations. His simulations are with respect to differ-ent error realizations; the spatial locations remain fixed after theyare initially generated. Moreover, Hodgson implicitly assumes thatthere exists some ‘‘correct” and static set of clients; stability onlyexists with respect to the deviation from the ‘‘correct” solution.While such an assumption is valid in many location problems, itdoes not apply to the case of Coast Guard distress calls. AlthoughCooper (1978) provides an attractive approach for accounting forinsufficient data, the approach also relies on the assumption thatwe have a fixed set of points for which there exists only measure-ment uncertainty. Other common measures of robustness includeminimax cost and minimax regret (see Snyder, 2006).

In summary, although there have been many stochastic exten-sions to facilities location problems, these approaches employmethods or assumptions which make them unsuitable for studyingour desired problem. We propose a spatial statistical model to ad-dress distress call location uncertainty and employ a simulationand optimization methodology to determine robust solutions forthese facilities locations problems.

3. Simulation and optimization based methodology

We first develop a statistical model to enable the simulation ofdistress call patterns. These patterns can then serve as input to theoptimization model and appropriate solutions that are robust overthe simulations can then be identified. In an indirect way, the ac-tual allocation of resources to each station can also be determined

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once the station locations are chosen. In this initial study, we onlyconsider the air stations.

Unlike other stochastic models for clients that focus either onclient demand or are restricted to a fixed set of candidate clientlocations, our simulation and optimization approach enables usto model the distress call process and allows for comparing thesolutions based on actual data with solutions that dominate acrosssimulations.

The strength of our approach lies in its simplicity: aggregationis not necessary, scenarios need not be specified a priori, and prob-ability density functions need not be specified; the uncertainty indistress call location is expressed via a non-parametric spatial sta-tistics model, through which the stability of solutions is nicely as-sessed. Although certain components of our method are somewhatcomplex, the overall simulation–optimization routine is simpleand easy to communicate. Furthermore, our method is a statisticalapproach: by examining realizations of distress calls derived fromour statistical model, we are simulating a population distribution.The number of simulations that are generated is arbitrary and onlylimited by computing time. In summary, our method should beconsidered as a data-driven statistical simulation approach, similarto commonly employed re-sampling methods in the statistics liter-ature, e.g., the bootstrap (Efron and Tibshirani, 1993), Jackknife,and Markov Chain Monte Carlo (MCMC) estimation methods (Gilkset al., 1996).

3.1. Statistical model

We focus on Coast Guard District 7, headquartered in Miami, FL,which essentially covers the southeast portion of Fig. 1. Fig. 2 dis-plays the relevant area and the calls during the year 2000 which re-ceived a Coast Guard air response.

The relevant area of interest S is defined to be the area runningeast–west from #59 to #87.5 degrees longitude, and south–northfrom 10 to 34.5 degrees latitude.

In order to simulate distress calls, we assume that the calls aregenerated by an inhomogeneous (also called non-homogeneous)Poisson process with an unknown intensity function /(x), where/(x) depends only on the location x (a two-dimensional vector that

contains both the longitude and latitude information). Thisassumption is equivalent to requiring that the distress calls withinthe study region occur independently from each other. In the con-text of our data, the Poisson assumption implies that the drivingforce determining the spatial distribution of distress calls is the(unobserved) traffic intensity of boats within the study region.Although there may be some concern of clustering possibly attrib-uted to severe weather (which conceptually may increase thechance of a distress call) within a given time period and over a spe-cific region, we believe that this is negligible due to the large regionand the long time span under consideration and consequently thevery sparse nature of our data. Moreover, we believe that the clus-tering is more likely to occur on the arriving time of the distresscalls but not as much on their actual locations, which are the mainfocus of our paper. In discussions with Coast Guard personnel, wehave further confirmed that the Poisson assumption is a practicallyreasonable assumption.

The intensity function /(x) determines the likelihood for a dis-tress call to occur at a given location x. To obtain an estimate for /(x) at any location x over the region S, we propose a non-parametricapproach, which will be described below. We do not consider aparametric approach (see Diggle, 2003) because this would requirethe specification of important factors affecting the distribution ofdistress calls as covariates. Such a specification, however, is notfeasible here due to the difficulty of identifying and further quan-tifying these factors. In a broader sense, the non-parametric esti-mation procedures being considered in this paper can beregarded as a special class of parametric estimations. Specially,the parameters involved are the smoothing parameters used whileforming the estimators, e.g., h in (2) and h and rx in (3).

Let B(x,h) denote a circular disc with center x and radius h,N(x,h) denote the number of distress calls that fall within the re-gion S \ B(x,h), and jS \ B(x,h)j denote the area of the correspond-ing region. A standard estimator for /(x) is given by

/̂ðxÞ ¼ Nðx;hÞj S \ Bðx;hÞ j

: ð2Þ

The parameter h in (2) is often referred to as the smoothingparameter (or bandwidth) in the statistics literature. By definition,the value of h controls the amount of smoothing which in turn af-fects the bias and variability of an estimated intensity function. Ingeneral, a small h yields an estimator with small bias but large var-iability while a large h leads to an estimator with small variabilitybut potentially large bias. An h that minimizes the mean square er-ror of /̂ðxÞ is often selected by a data-driven method (see Diggle,1985).

There exist two main issues with respect to standard intensityestimation methods for Eq. (2). One, the use of a circular regionB(x,h) precludes accounting for any directional features that mayexist in the data. Two, existing approaches determine a commonvalue of h for all x, regardless of their location. This is not desirablefor a spatial process that exhibits a rapid change in intensity acrossthe study region. For the Coast Guard data we observe a clear direc-tional feature in the distribution of distress calls. Specifically, theintensities near the coastline appear to be more similar in thedirections parallel to the main coastline directions than perpendic-ular to them. In addition, the spatial pattern exhibits rapid changesin intensity across the study region. To account for these two is-sues, we need to generalize Eq. (2).

To account for directional features in the data, we propose to re-place B(x,h) by an elliptical area where the major axis roughly par-allels the closest US coastline. Given that the US coastline isnonlinear, the requirement that the major axis of the elliptical areais parallel to the coastline is somewhat problematic. We solve thisproblem by approximating the coastline with a collection of linesegments. The number of line segments employed represents aFig. 2. District 7 distress calls – year 2000.

D. Afshartous et al. / European Journal of Operational Research 196 (2009) 1086–1096 1089

tradeoff between the accuracy of the coastline approximation andthe computational complexity involved in requiring the major axesof the elliptical areas to be parallel to the coastline. Indeed, if theelliptical area is relatively large and the direction of the coastlinechanges rapidly due using many short line segments, it will be-come unclear how to satisfy our requirement on the major axes.Based on these observations, we modify (2) as follows:

/̂eðxÞ ¼ Nðx; h; rxÞ= j S \ Eðx;h; rxÞ j; ð3Þ

where E(x,h, rx) is an ellipse centered at x and N(x,h, rx) now denotesthe number of calls within S \ E(x,h, rx). The lengths of the major andminor axis of E(x,h, rx) are equal to h and rxh, respectively, and theorientations of these axes are as in the foregoing discussion. Weset rx = 1 # kxck#a where a is a pre-defined positive constant and kxckrepresents the distance of x to the nearest coastline. Note that inten-sities for locations far from the coastline will be calculated via essen-tially circular ellipses since rx will be close to 1, while intensities forlocations closer to the coastline will be calculated via flatter ellipses.

To account for rapidly changing intensity across the study re-gion, we replace the bandwidth in (3) with a variable bandwidthhx that is location dependent. For each location, we choose hx asthe smallest bandwidth such that at least k distress calls are con-tained in E(x,hx,rx); this is in the spirit of the nearest neighbor ap-proach discussed in Silverman (1986, Section 5.2). The variablebandwidth is discrete and may vary between a pre-specified min-imum and maximum value, with the step granularity pre-specifiedas well. Observe that a larger hx will be needed to include k pointsat low intensity locations and a smaller hx will be needed to includek points at high intensity locations. The principal smoothingparameter now becomes k where a larger k corresponds to a great-er amount of smoothing; this parameter is often referred to as thenearest neighbor parameter.

3.2. Simulation of distress calls

A common practice to generate an inhomogeneous Poissonprocess involves two steps. First, a homogeneous Poisson processwith intensity equal to the largest value of the estimated inten-sities across the region is generated. Then, each simulated pointfrom the first step is retained with probability equal to the ratioof the estimated intensity at the simulated point to the largestestimated intensity in the region (Lewis and Shedler, 1979). Toobtain an estimate for the largest intensity in region, a fine gridis usually laid over the study region and the estimated largestintensity is set equal to the maximum of the estimated intensityfunction values at all grid nodes. The result of this procedureyields an inhomogeneous Poisson process with the estimatedintensity function. For the Coast Guard data, we have foundthe method above to be computationally infeasible. This ismainly due to the extreme magnitude of the global estimate ofmaximum intensity, resulting in the generation of too many callsin the first step of the method, and necessitating the retentioncheck on each of these calls in the second step of the method.

As an alternative method we propose to divide the study regioninto sub-blocks and perform the aforementioned method sepa-rately for these sub-blocks and then combine them together toyield the desired inhomogeneous Poisson process. For each sub-block the local maximum intensity within the sub-block is usedto simulate the homogeneous Poisson process from the first stepabove, instead of employing a global estimate of maximum inten-sity for the entire region. As a result, fewer points are simulated inlow intensity sub-blocks and this makes the procedure computa-tionally feasible. Note that the use of sub-blocks does not affectthe estimation of the underlying intensity. Regardless of the num-ber of sub-blocks being used, the simulated calls will have the dis-

tribution given by the estimated intensity function of the process.This is guaranteed by the property of a Poisson process that eventsobserved at disjoint regions are independent. To further reduce thecomputational cost we approximate the intensity function value atthe points generated in the first step of the procedure by the esti-mated intensity function values at the nearest node of the laid grid.The resolution of this laid grid should be fine enough such that thisapproximation maintains the integrity of the continuous intensitypattern. For the Coast Guard data, we settled on a grid resolution of1000 & 1000 since this resulted in grid points being within a dis-tance of 2 miles; any closer would not provide any more value gi-ven the nature distress call occurrence. In general, one may definevarious criteria to insure that the change in intensity for adjacentgrid points is not too large. For example, one may examine thechange in intensity at each grid point with respect to the averageor maximum intensity across the grid, or merely put an upperbound on the maximum percentage change in intensity for adja-cent grid points. These criteria were not useful for our data giventhe nature of distress calls occurring in isolated areas. For othertypes of data where isolated points are not as common, e.g., for-estry or crime data, such criteria may be more appropriate.

Defining values for the various parameters is an important com-ponent of the simulation process. If these parameters are not prop-erly selected, the simulation will not sufficiently emulate theactual point process. Indeed, parameter selection must be per-formed with respect to the actual application data and ideallythrough the use of some data-driven methods. Generally speaking,the intensity estimation for an inhomogeneous Poisson process isclosely related to the density estimation for identical and indepen-dently distributed observations, for which a large number of data-driven procedures have been proposed for the selection of thesmoothing parameters involved (see, e.g., Silverman, 1986). In par-ticular, in the case when the kernel smoothing given in (2) is used,these procedures can be directly used to select the bandwidth h. Inthe proposed case of selecting the nearest neighbor parameter k,however, Silverman (1986, p. 99) recommended against the useof such procedures. Instead he suggested by trying several differentvalues of k and choosing the one that gives the most satisfactoryresults. A similar recommendation was given by Zhuang et al.(2002) where they suggested using k values between 10 and 100.Following these recommendations, we experimented with severalk as well as a and the maximum bandwidth values and selected thecombination that yielded the number of calls per iteration beingclosest to the number of calls in the original data on average anda distributional pattern similar to the pattern of the original data.The final values for all the parameters are provided in Table 1.

The maximum bandwidth in Table 1 is used to prevent selectingan overly large value for the bandwidth in low intensity regions, asthis would lead to a biased intensity estimate at such a point. It isworth noting that if a point is very far away from the coastline, itcan be the case less than k or even no observations will be includeddue to the use of the maximum bandwidth. Even though that at

Table 1Simulation parameter values

Parameter Value

Grid resolution 1000No. of blocks to divide whole region into 192

Minimum no. of points in elliptical regions (k) 20Minimum bandwidth 2000 mMaximum bandwidth 20,000 mNo. of bandwidths to consider between minimum and maximum

bandwidth10

a = quantity that determines how the ratio of the two axes of an ellipsechange at different locations

0.1

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some locations (or even in some sub-blocks) the estimated intensi-ties are zero, we feel that this is still acceptable since even thoughthe true intensities were not zero, they should be extremely small.This basically says that it’s extremely unlikely for any distress callsto occur at such locations and thus the exclusion of such locationswhen developing any operation strategies seems reasonable. Notethat this typically was not a problem for locations close to thecoastline since the much denser observed distress calls. Finally,Fig. 3 illustrates our variable bandwidth approach for estimatingthe intensities at grid points via elliptical regions parallel to thenearest coastline. For the purpose of illustration, we have drawnonly a 25 & 25 grid and have not drawn the elliptical regions toscale. Not all of the grid is visible since we have restricted the area

to provide better illustration of the elliptical regions. The simula-tion routine is summarized in Appendix A via pseudocode.

4. An illustration: Distress calls from Coast Guard District 7

We apply our simulation–optimization methodology to the sta-tion location problem in Coast Guard District 7. Similar to theapplication of Psaraftis et al.’s (1986) model of oil spill responsein New England, the main goal is to demonstrate the versatilityof our methodology, perform sensitivity analysis, and provide in-sight into the various issues in strategic planning for distress callresponse in this particular geographic area. Although significant ef-fort has been expended with respect to using realistic information,this section is simply an illustration of the methodology and not anexhaustive analysis of this one data set and the correspondingsimulations.

Fig. 2 displays the calls from Coast Guard District 7 from theyear 2000, consisting of 1631 points in our pre-defined region ofinterest. For each distress call, there exists information regardinglocation, the nature of the case, and the resources that were de-ployed. In addition, the stations from which these resources weredeployed can be determined. The Coast Guard reports positionlocations in a geographic coordinate system known as Word Geo-detic Society 1984 (WGS84). In order to display the data in twodimensions, the data from the spherical earth must be convertedonto a flat plane. We have chosen the Azmuthal Equidistant projec-tion; the most significant characteristic of this projection is thatboth distance and direction are accurate from the central point.Fig. 4 displays the four existing air stations. In conjunction withCoast Guard personnel, eleven additional candidate locations wereselected for consideration as future air stations. To be sure, theopening or closing of a Coast Guard station is often a politicallycharged issue; thus, a thorough location analysis is important be-cause it may quantify value of existing stations, along with the im-pact of either opening a new station or closing an existing station.

In essence, we are interested in providing insight with respectto the following questions:Fig. 3. Grid points, variable bandwidth, changing elliptical regions.

Fig. 4. Existing (triangle) and candidate (circle) stations.

D. Afshartous et al. / European Journal of Operational Research 196 (2009) 1086–1096 1091

(1) Are the current stations located appropriately?(2) If the current stations could be closed and new stations

opened, where should stations be located?(3) What is the potential value of additional stations?(4) Are the answers to the above questions relatively stable as

the distress call distribution is varied?(5) Do the distress calls exhibit a significant seasonal pattern,

indicating that resources should be deployed accordingly?(6) Do we obtain similar results by analyzing a smaller sample

of data?

To investigate the above questions, we first solve two differentinteger programs and allow up to ten total stations to be selected(with minor appropriate modifications):

(1) Fixed-4 Problem: Choose a subset of p stations, with four ofthe p stations being the existing stations, such that total tra-vel distance is minimized. p = 4, . . . ,10.

(2) Free-4 Problem: Choose a subset of any p stations that min-imizes the total travel distance. p = 4, . . . ,10.

Indeed, we want to study these questions not just with re-gard to the original data but also with simulated data to inves-tigate the robustness of the solution. Thus, we first solve theoptimization problems utilizing only the original distress callsfor the year 2000. Next, we invoke our simulation–optimizationmethodology in order to assess the stability of this solution, anddetermine the solution that is best across the simulations. Wesimulate 100 iterations of distress calls; previous experimentsutilizing 1000 iterations provided similar results but requiredsignificantly greater computing time. We solve each of the aboveproblems via integer linear programming using CPLEX version6.5. Most Free-4 problems were solvable in half an hour ofCPU time on a DEC ALPHA 3000 (Model 900) platform. For thesimulations, the initial intensity estimation via sub-blocks overthe 1000 & 1000 grid required approximately 45 minutes ofcomputing time, and the generation of the sets of distress callsrequired only 4–5 seconds per iteration. The simulation codewas written in Matlab v6.5.1, and run on a server platform withWindows 2003 Server Enterprise Edition OS with Dual Xeon3.2 GHz processors and 4 GB or RAM.

4.1. Optimal locations of stations for actual data

Table 2 illustrates the solution for the actual data for the year2000 for the Fixed-4 problem of adding new stations, where theoriginal stations must be included. A cell entry of ‘&’ indicates thatthis station is part of the optimal solution. For example, the firstrow trivially contains the original four stations as the solution,and the second row solves the problem of adding an additional sta-tion (‘‘pick 5”).

We have normalized the optimized cost (total travel distance)by dividing by the number of calls (N = 1631), in essence yieldingthe distance per call provided by the given solution. This quantifiesthe value of adding a station(s). For instance, the cost/call for theoriginal four stations is approximately 199 km, while that for thesolution that adds Key West (S5) as the fifth station is approxi-mately 181 km, signifying that a distress call is almost 20 km closerto an air station, on average. This represents an improvement of8.9% in reducing travel distance. Although an improvement of20 km may seem small, this must be viewed with respect to thediscrete nature of distress calls response and the potential life-sav-ing impact. The corresponding results for the Free-4 problem areprovided in Table 3.

The information above allows us to examine the appropriate-ness of the original stations, i.e., whether the original stations arestill part of the optimal solution across the various scenarios. Theoverall indication is that some of the original stations are indeedappropriate, as Clearwater (S2), Savannah (S3), and Borinquen(S4) are included in all the solutions; Miami (S1), however, is onlyincluded in the extreme scenario of adding six new stations (pick10). This does not necessarily imply that Miami is poorly located,rather that the candidate stations we consider seem to provide po-tential alternatives that better optimize travel cost. Regarding theissue of where to put new stations if the current stations couldbe closed and new stations opened, Key West (S5) and Charleston(S12) appear most frequently across the scenarios, lending weightto their viability as a new station.

In addition, Table 4 displays the percentage cost savings foreach scenario with respect to the cost/call for the original four sta-tions. For example, for the Fixed-4 case, the biggest incrementalimprovement in reducing travel cost occurs when adding the firstadditional station (8.9%), while subsequent additional stations

Table 2Original year 2000 data solutions – Fixed-4 problem

S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 S14 S15 Cost/call (km)

Pick4 & & & & 1995 & & & & & 1816 & & & & & & 1707 & & & & & & & 1608 & & & & & & & & 1509 & & & & & & & & & 144

10 & & & & & & & & & & 140

Table 3Original year 2000 data solutions – Free-4 problem

S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 S14 S15 Cost/call (km)

Pick4 & & & & 1965 & & & & & 1786 & & & & & & 1647 & & & & & & & 1548 & & & & & & & & 1489 & & & & & & & & & 143

10 & & & & & & & & & & 140

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(pick 6 and higher) provide an incremental reduction between 2.3%and 5.3%. This is also the case for the Free-4 case. With respect tothe cost savings difference, the results indicate a relatively smallcost/call savings between the Fixed-4 and Free-4 solution for allscenarios. For instance, the Free-4 solution for having any four sta-tions has an optimized cost/call of 196 km, while the correspond-ing number for the original stations is approximately 199 km, adifference of only 3 km or 1.5%. This indicates that if the CoastGuard can only have four stations, the existing stations are almostas good as any set of four stations taken from the fifteen stations.Thus, although we may obtain different nominal solutions whenallowing any stations to be picked, the gains with respect to reduc-ing cost per call is relatively small.

4.2. Stability of solutions

In order to investigate the robustness of the solutions above, weinvoke our simulation–optimization methodology. Specifically,using the year 2000 data to estimate the intensity of distress calls,we employ our statistical model to generate 100 sets of simulateddistress calls. The number of distress calls per iteration will varyand has a Poisson distribution with mean and variance of 1631(the number of distress calls in the original data used for intensityestimation). And for each of the simulated sets of distress calls, weconsider the same optimization problems as for the original data.Table 5 shows how the results over the simulations compare withthe original solution for the Fixed-4 problem. We omit the corre-sponding table for the Free-4 problem due to space considerationsand the fact that the differences are minor. These results were sim-ilar (identical for all but one problem) when employing the mini-max regret criteria instead of selecting the solution that has themost frequent lowest cost per call over the simulations.

For instance, when adding only one station (pick 5), the originaldata solution is the optimal solution for each of the 100 simula-tions – Key West (S5) is selected every time. However, for the sce-nario of adding two new stations (pick 6), the original solutionoccurs only 19 times over the simulations, while another solutionoccurs 58 times; a total of 4 solutions exist over the 100 solutions.Thus, the original solution is not completely ‘‘stable” or robust overthe simulations, and the simulations have identified an alternative

solution. Specifically, the original data favors Apalachicola (S15)while the simulations indicate that Melbourne (S9) is a viable can-didate as well. Does this alternative solution provide significantgains with respect to reduced cost/call? The original solution withApalachicola (S15) has a median cost/call value of 168 km over the100 simulations, while that for the alternative solution with Mel-bourne is 167 km. Thus, the answer initially appears to be no.

The boxplots in Fig. 5 provide information via the relative per-formance of each solution compared to the original solution(S1–S5,15) at each of the 100 iterations. The dominant solution(S1–S5,9) over the simulations has a lower cost per call (positivedifference) in over 75% of the iterations, and has a 1.4 km (95% con-fidence interval = [1.15,1.65]) lower average distance per callwhen comparing directly. In terms of maximum regret, the maxi-mum regret of the alternative solution is 3.2 km (which is theminimax regret of all the solutions), while that of the original solu-tion is 5.5 km (which is the ‘‘maximax” regret if all the solutions).Thus, the alternative solution provides a better hedge in terms ofregret, which is not immediately apparent when examining theaverage distance per call. Although the simulations are primarilyintended to assess the robustness of the original solutions, both Ta-ble 5 and Fig. 5 may be analyzed from the perspective of selectingthe ‘best’ solution. For instance, in Table 5 consider the pick-6problem where there exist four solutions. We may transform theresults into a multinomial problem as in Bechhofer et al. (1959),where each solution has a fixed probability pi of producing the low-est cost per call over the simulations, and the problem is to insurethat the probability of correctly selecting the solution with thelargest pi is large. Bechhofer et al. (1959) presents tables that illus-trate the probability of correct selection under various design con-ditions where the selection method is to pick the solution with thehighest frequency across the simulations. To be sure, this will de-pend upon the selected parameters that drive the tables, especiallythe user defined constant h representing the ratio of the largest tosecond largest pi. For the case of 4 cells/solutions, a sample size of30 provides a probability of correct selection of 0.95 when h = 2.6and 0.72 when h = 1.7. All tables have n = 30 as the upper limit.Since the estimate of h for our data in Table 5 is approximately3.2, we may be confident that the probability of correction selec-tion is at least 0.95. In addition, note that the percentages of thesolutions were very stable over a large number of iterations, andsince this procedure merely selects solution with the largest ‘best’frequency over the iterations, we can be confident that asymptoticresults agree with our results with 100 iterations. Thus, whenframed as a multinomial problem, Table 5 can be used to statisti-cally select the solution with the highest probability of producingthe lowest cost/call.

With respect to Fig. 5 for the pick-6 problem, this may beviewed as a problem of correctly selecting the solution with the

Table 4Percent cost/call savings from the current stations

Pick Fixed-4 Free-4 Difference

4 0 1.5 1.55 8.9 10.5 1.66 14.2 17.7 3.57 19.5 23.2 3.78 24.6 26.0 1.49 27.3 28.3 1.0

10 29.6 29.6 0.0

Table 5Simulation results for the Fixed-4 problem: Frequency0: Frequency of the solution forthe actual data being an optimal solution in the simulated data

Pick Frequency0 FrequencyS SolutionS NumberS

4 100 – – 15 100 – – 16 19 58 S1–S5, S9 47 10 40 S1–S5, S9, S12 58 86 – – 29 55 – – 5

10 22 51 S1–S5, S8, S10, S12, S13, S15 8

FrequencyS: Frequency of the most frequent optimal solution for the simulateddata, if different from the original data solution. SolutionS: Identity of the mostfrequent optimal solution for the simulated data, if different from the solution tothe actual data. NumberS: Number of solutions across the simulations.

Cost

/Cal

l (ki

lom

eter

s)

S1 - 5 ,1 5 - S1 - 5 ,1 2S1 - 5 ,1 5 - S1 - 5 ,1 0S1 - 5 ,1 5 - S1 - 5 ,9

5.0

2.5

0.0

-2.5

-5.0

Fig. 5. Distribution of cost/call difference from original solution; Fixed-4 year 2000,pick-6.

D. Afshartous et al. / European Journal of Operational Research 196 (2009) 1086–1096 1093

lowest average cost/call. This may be viewed as a ranking andselection problem in the context of optimization via simulation(Kim and Nelson, 2002; Goldsman and Nelson, 2001; Pichitlamkenand Nelson, 2003). We employed a simple Bonferroni procedure formultiple hypothesis testing to address the statistical significance ofthe best solution. Given the four solutions, we conducted threehypothesis tests for a pairwise difference greater than 0, compar-ing the best solution to the other three. Technically, only one actualtest is needed, the best against the second best, but one must stilladjust the significance level accordingly as if doing all six pairwisecomparisons, since the best two are not a priori identifiable.Regardless, even with the Bonferroni correction to control overallType I error at the 0.01 level, the best solution (S1–S5,S9) was sta-tistically significant. Given that the Bonferroni procedure is conser-vative, other less stringent methods would give the same result.

For the pick-7 and pick-10 problems, we also obtain differentsolutions in the simulations than in the original data. This is thecase for both the Fixed-4 and Free-4 problems.

As with the static original data, Key West (S5) is always addedas a new station. Moreover, both the original data and simulationsindicate that when picking more than 8 stations (adding more than4 stations), Melbourne (S9) is no longer part of the solution. Sincemore stations are allowable, leverage is obtained by including Ft.Pierce (S8) and St. Augustine (S10), the stations adjacent to Mel-bourne (see Fig. 4).

4.3. Seasonality issues

Regarding the aforementioned issue of seasonality, we dividedthe data into two subsets and solved the corresponding optimiza-tion problems. The subsets of the data represented two contiguousblocks of months, April–August (N = 698) and September–March(N = 928). This specific partition was chosen due to the observedchanging volume of distress calls across these two periods. Theoverall trend in results was similar to the trend in the tables above.In other words, although the volume of distress calls may changebetween the seasons, the pattern does not change enough to alterthe solutions to our location problems. We also carried out a for-mal statistical test (nearest neighbor approach, Cuzick and Ed-wards, 1990) to investigate for a seasonal affect, and thestatistical test also indicated a change in volume without a statis-tically significant change in spatial pattern. We note, however, thatthere are some changes with respect to the cost/call measure in thecorresponding simulations. In Fig. 6, we present the distribution forcost/call for the original simulations and the simulations for thetwo seasons for the original solution to the Fixed-4 pick-6 problem.Thus, when broken down seasonally, there is a slight increase invariability for both seasons, and also an increase of about 12 km

in the median cost/call level for the September–March season.These results also hold when analyzing other solutions as well.This slight difference may be a manifestation of the increased var-iability of distress call locations during the hurricane season, indi-cating that the Coast Guard might require a slightly differentresource allocation in order to maintain similar responsiveness.

4.4. Sufficiency of data set

With respect to whether we obtain similar results by analyzing asmaller sample of data, we performed analogous analyses using thelast half of the year 2000 instead of the full year. This partition re-sulted in reducing the total number of distress calls to N = 823. TheFixed-4 and Free-4 optimization problems were solved for boththe static case and over simulations. Although therewere somemin-or differences, the overall results were aligned with those obtainedwhen employing the full year 2000. This indicates that (1) the prob-lemfor thestaticdata canbeobtained fromasmall sampleofdistresscalls, and (2) the robustness analyses of this static data solutionmayalso be obtained when estimating the intensity in the region with asmaller sample of data. Although we had the advantage of access tothe larger data sample, this will not necessarily be the case in otherapplied problems. Due to space considerations, we do not presentthe corresponding tables for the half-year data.

4.5. Tactical issues

The results of the simulation–optimization routine may also beused to answer the tactical question of how to distribute resourcesacross the air stations. For instance, along with the identities ofeach station for a given optimization scenario, we obtain the break-down of call assignments that led to the optimized cost. SinceEuclidean distance is the optimization criteria, this essentially pro-vides the frequencies for which each station (within the solutionset) was the closest station to a distress call. For the original staticdata and the simulations, we observe that Savannah (S3) usuallyreceives the largest assignment of calls for the various optimiza-tion problems. Fig. 7 illustrates the variability of call assignmentfor solution S1–S5, S9, i.e., the solution adding Key West and Mel-bourne for the pick-6 Fixed-4 problem based on the year 2000data. Given a fixed total of planes to be distributed across six sta-tions, the Coast Guard could distribute the planes in proportionto the median call assignment level, a proxy for the ‘‘busyness”of a station. Additionally, the variability of the call assignmentsacross the iterations provides sunny day and rainy day forecaststhat could be used for contingency resource planning. To be sure,this model assumes that the Coast Guard will operate under therule of sending the resource from the closest station for a given dis-tress call. This model may be generalized by considering differenttypes of distress calls that require different types of resources,e.g., helicopter versus fixed wing aircraft.

Given the rather wide design space we could continue analyz-ing different dimensions of the problems and solutions. This initialanalysis has been intended as an illustration of a newway to exam-ine an important problem. The simulations provide a check on thestability of the original solution, and examining the Fixed-4 vsFree-4 problem lends insight into the strategic importance andleverage of certain stations. Moreover, by considering many pick-p optimization problems, we observe the dynamic nature of thesolution space.

5. Discussion/summary

The proposed simulation–optimization methodology is usefulin investigating the problem of locating Coast Guard stations. By

Fig. 6. Distribution of cost/call; Fixed-4 problem: 2000, 2 seasons, pick-6; givensolution.

1094 D. Afshartous et al. / European Journal of Operational Research 196 (2009) 1086–1096

introducing variability into the distress calls process, we provide amore realistic approach to the problem. Based on a real data set ofdistress calls, we have developed a new statistical model with var-iable distress call intensity in order to simulate distress call loca-tions. A spatial simulation approach is taken in order to modelthe inherent uncertainty in distress call locations and their varia-tion over space and time. Integer programming models are thenused to determine optimal locations for responding stations. Ournon-parametric statistical model generalizes previous attemptsto model the uncertainty of client locations, and the combinationof simulation and optimization provides information with respectto the robustness of initial solutions. Unlike traditional stochasticand robust optimization methods, our approach does not requirethe specification of a set of scenarios or a probability densityfunction.

Our initial results indicate that the current stations are locatedwell. The results quantify the percentage improvement that ispossible to obtain as the number of stations is expanded to in-clude newer stations. This improvement is quantified with re-spect to the distance per call of a solution, showing thereduction in distance per call when adding an additional station.If the Coast Guard could only maintain four stations, the currentstations are almost as good as any combination of four stationsfrom the complete set of fifteen candidate stations. However,for some of the optimization problems, the simulations identifysolutions that are different than the solution obtained when onlyconsidering the original data. While any reduction in the distanceper call is welcome, whether this reduction will help save livesmust be ultimately decided by the Coast Guard. Regarding sea-sonal effects, the analysis indicates that partitioning by seasondoes not produce different solutions to the various optimizationproblems. There is, however, a slight increase in the variabilityof the cost per call for our seasonal partition, with the Septem-ber–March segment also showing a slight increase in the mediancost per call. Finally, the allocation of distress calls to variousopen stations in the simulations can be a good proxy for the re-sources to be allocated to each station.

The cost of opening a new station versus the operational savingsobtained are now quantifiable and provide guidance to plan expan-sion of US Coast Guard operations. On the practical front, our re-search demonstrates the possible operational improvements andprovides a potential framework for such an analysis. Other logisticsproblems of locating emergency response stations such as fire sta-tions, or strategically locating warehouses in presence of uncertaindistribution of demand can also benefit from similar analysis. Wehave illustrated that a spatial simulation approach is useful, andthe Coast Guard considers our approach to be a significant contri-bution towards expanding their capabilities to provide further ana-lytical frameworks for SAR analysis and other problems includingplanning, budgeting, and public resource management. Indeed,

the Coast Guard places a very high priority on SAR and is increas-ingly aware of the need to connect operations research and busi-ness modeling capability to policy-making.

Acknowledgements

The authors thank Dan Yadgar for assistance in managing thedistress call and simulated data, and the editor and two anony-mous referees for many helpful comments. The quality of this pa-per has also greatly benefited from many discussions with thefollowing Coast Guard personnel: Cmdr. Joel Magnussen, Lt. Cmdr.Chris Button, Lt. Kristey Bernstein, Mr. Richard Schaefer, and Mr.Frank Wood. This paper is dedicated to the memory of Cmdr. Mag-nussen deceased October 1, 2003. Cmdr. Magnussen had a keeninterest in this research as Director of Information Managementfor USCG District 7.

Appendix A. Pseudocode of the statistical simulation

1. Lay a grid of dimension 1000 & 1000 over the region(1000 = resolution defined in Section 3.2).

2. Estimate the intensity at each grid point.a. Obtain the direction of and the distance to the closest poly-

gon used to approximate the coastline.b. Increase the bandwidth (hx) from the minimum allowed

bandwidth. For each hx, obtain the ellipse centered at thegrid point as defined in Section 3.2 and count the numberof distress calls contained in the ellipse.

c. Obtain the smallest hx such that at least k points are includedin the resulting ellipse as the final bandwidth value; if suchan hx does not exists, use the maximum allowed hx as thefinal bandwidth value.

d. Obtain the intensity estimate using Eq. (3) and the finalbandwidth value determined in c.

3. Divide the region into smaller non-overlapping sub-blocks.4. Simulate distress calls in each sub-block region.

a. Obtain the maximum value of the intensity estimates fromstep 2 within the sub-block.

b. Determine the total number of points N to be simulated inthe sub-block: N is a Poisson random variable with meanequal to the product of the maximum intensity in and thearea of the sub-block.

c. Simulate N points from a uniform distribution within thesub-block.

d. Approximate the intensity at each simulated point by theintensity at the closest grid point.

e. Retain a simulated point with probability equal to the ratioof the (approximate) intensity at this location to the maxi-mum intensity within the sub-block.

5. Combine the simulated points across all the sub-blocks.6. Repeat 4 and 5 (100 times) to obtain multiple realizations of

distress calls.

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