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Development of a Model to Optimize Maritime Surveillance and Patrol Operations in Colombian Navy.

Colombian Navy carries out maritime operations against threats such as terrorism, drugs traffic, smuggling and illegal fishing. Maritime surveillance is the detection and classification process of surface targets within the interest area through the
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    1 Development of a Model to Optimize Maritime Surveillance and Patrol Operations inColombian Navy. Camilo Ernesto Segovia ForeroIndustrial Engineer Department. Universidad de los Andes. Bogotá , Colombia.Advisor : José Fidel Torres Delgado PhD.June 2008 Abstract Colombian Navy carries out maritime operations against threats such as terrorism, drugs traffic, smugglingand illegal fishing. Maritime surveillance is the detection and classification process of surface targets withinthe interest area through the use of air and surface platforms. It is then necessary to implement tacticsenabling to find efficient routes maximizing covering within the interest area. The scenario is made up by 2Naval Bases and 155 waypoints, which were obtained in accordance with illegal activities in 2006. Suchscenario is similar to a MDVRP (Multi-depot Vehicle Route Problem). The model enabling to work thisproblem as a CETSP (Close Enough Traveling Salesman Problem) through the use of a three stages method:(1) Assignment the waypoints to Naval Bases, (2) Grouping the waypoints in clusters, in accordance withradar range and (3) Routing platforms toward clusters centers obtained in stage 2. The model can be used as aquantitative tool in decision making process and in planning maritime operations. Palabras Claves : CETSP, GA, MDVRP, Maritime Patrol Operations, Maritime Surveillance Operations. 1. Introducción The Colombian Navy mission is to contribute tothe defense of the nation through the effective useof a flexible naval power within the maritime,fluvial and land spaces under its responsibility, inorder to fulfill the Constitutional function andparticipate in the development of sea power andprotection of Colombian interests. Based in suchmission, the Colombian Navy is responsible forsafeguarding the sovereignty and guaranteeingfree and safety navigation in Colombian maritimeareas who have an approximate extension of 928.660 km 2 (ARC-PEN, 2007).Colombian geo-strategic position and maritime jurisdiction are permanent objective of groupsoutside the law through the use   of maritime routesfor the development of unlawful   actions, such asdrug trafficking (ARC - Closing Spaces, 2007).In order to reach a wider covering in controllingmaritime space, due to its large extension and thelimited number of naval units, the ColombianNavy requires to have the necessary tools forplanning, decision process and execution of surveillance and patrol operations, optimizingavailable resources by maximizing the areasurveyed and minimizing operational costs.The problem of marine surveillance and itssolution methods in current literature shall bedealt with in section 2. The three stages method toreduce the complexity of the problem shall bediscussed in Section 3, where problemformulation, proposed scenario, mathematical andsome heuristic models are submitted as solutionmethods. Section 4 presents computational resultsobtained and conclusions shall be submitted insection 5. 2. Maritime Surveillance Problem The objective of maritime surveillance and patrolproblem is to survey an specific interest area ( IA )within a given marine region (Kilby, Tobin,Luscombe, Barry & Hickson, 2007), by usingoptimization methods maximizing the number of surface contacts detected and identified in theshortest time possible within the IA (Mercer,Barry, Marlow & Kilby, 2008). Such surveillanceis carried out with different types of naval unitssuch as frigates, patrol vessels, onboardhelicopters and patrol airplanes, called platforms.They obtain and maintain the information of surface targets found in the IA (Grob, 2006).In accordance with Grob (2006) the marinesurveillance problem has 6 phases: detection,    2localization, recognition, identification, pursuitand communicating the information, all of themdepending on the capacity of naval unit sensors, aswell as radar range determining surface targetsdetection. Targets identification not only dependson sensors capacity, but also from climaticconditions, target speed, direction and size sinceits classification is almost visually made (Merceret al., 2008).The objective of planning in patrol, surveillance,recognizance, search and rescue operations is todetermine one route visiting all points required,while the searching route total cost is minimized(Jacobson, McLay, Hall, Henderson & Vaughan,2006).Cost minimization to cover a larger IA is done byfinding the shortest route (Mercer et al., 2008).Cross, Marlow and Looker (2007) worked it as theTraveling Salesman Problem (TSP), where theclients were the points to be visited within anestablished search pattern (patrol route) for eachplatform taking part in patrolling within the IA .Grob (2006) interprets the optimization of thisproblem as a vehicle routing that works as anextension of the TSP problem, where the routeshall be updated in as far as new targets aredetected. This type of problem is named OLTSP(On-Line Traveling Salesman Problem). Cross etal. (2007) propose an application of the NTSP(Non-Stationary TSP) (Jiang, Sarker & Abbas,2005) for the problem of maritime surveillance,where targets move to constant speed and theyhave a weighing factor providing them somepreviously established priority according to thetype of craft and threats in the area.Kilby et al. (2007) posed the use of CETSP(Close-enough TSP) (Gulczynski, Heath & Price,2006), to perform the routing of vehicles, withouthaving to visit all required points exactly, butcoming closer enough to detect and identifysurface targets, using sensors from each platform.Jacobson et al. (2006) propose the use   of SGHC(Simultaneous Generalized Hill ClimbingAlgorithms) as optimal search strategy for thesolution and to determine patrolling patterns inrecognize, surveillance and/or search and rescueoperations using platforms with differentcapacities, whether individually orsimultaneously. 3. Formulation of the Problem and ProposedSolution through the Three Stages Method. The maritime surveillance problem as stated inSection 2 can be seen as a routing problem (Grob,2006), which is a generalization of TSP, where thesurveillance pattern shall be carried out by severalplatforms (Ehlers, 1998) leaving from the same orfrom different naval bases, with the purpose of visiting the points required in which case it isnamed MDVRP (Multi-depot Vehicle RouteProblem). 3.1. Proposed scenario Although Colombian maritime jurisdiction isapproximately 928.660 km 2 , the scenario shall bethe Colombian Caribbean that covers anapproximate area of 589.160 km 2 (ARC-ClosingSpaces, 2007).Due to the extension of jurisdictional waters andthe number of targets that can be detected duringthe operations of surveillance and recognition, it isnecessary to determine a route that visits all pointsrequired, minimizing the cost of the route(Jacobson et al., 2006).The selection of the points to be visited wascreated based in the routes used by crafts thatcarried out illegal activities in 2006. Obtainingthus 155 waypoints to create the surveillancepatterns and only 02 Naval Bases are used fromwhere platforms participating in the surveillanceand patrol operations shall leave. The proposedscenario can be seen in figure 1 on the COL. 005chart and the jurisdiction of the Colombian Navyin the Colombian Caribbean (ARC-PEN, 2007). BN1 BN4   BN1 BN4   Figure 1. Proposed scenario.    3The proposed scenario approaches a SMDVRP(Symetric Multi-depot Vehicle Route Problem)problem, where clients can be assisted from one orseveral deposits; each deposit has a group of navalunits available, which shall return to the samedeposit. One of the most common techniques tosolve this type of problems is using the two phasesmethod, where clients are assigned to the nearestdeposit and VRP sub-problems are solved for eachdeposit (Tasini, Urquahart & Viera, 2000). 3.2. Three Stages Method The VRP problem is NP-Complete and it is ageneralization of the TSP (Toth & Vigo 2002). Inorder to reduce the complexity through thereduction of waypoints, the proposed method hasthree stages: (1) Assignment to Naval Bases of waypoints, (2) Grouping and reduction of thewaypoints in accordance with radar range of available platforms and (3) Routing of theplatforms toward the waypoints obtained in stage2. 3.2.1. Stage 1: Assignment Keeping in mind that the proposed scenario has 02naval bases located in different geographicalpositions in the Colombian Caribbean and 155waypoints corresponding to the historical behaviorof crafts that carried out illegal activities in 2006,these points were assigned to the nearest navalbase (Tasini et al. 2000), transforming theMDVRP problem into two VRP problems. 3.2.2. Stage 2: Clustering Uses the principle practiced in the clusteringalgorithm by distance thresholds (A distancethreshold clustering algorithm) (Bow, 1984) thatenables to group the points to be visited assignedto each base in accordance with radar range ( RR )of platforms available in each of them. Thecenters of these groups will be now the newwaypoints to be visited for the routing problem.The clustering of points to be visited was carriedout with Euclidean metric. The algorithmdetermines the number of groups required since itis self-organized. The algorithm is an iterativeapplication of k-means algorithm (MacQueen,1967), starting from a determined τ distance,where τ = RR wanted and k the number of initialgroups, for our case k=2. Once the first twogroups have been obtained through k-means, theEuclidian distance of the points from each groupto their respective centers is measured, if this ishigher than τ , then a (k=k+1) group is iterativelyadded until all points are in groups whosedistances to their centers are less than or equal to τ . Figure 2 shows the pseudo-code used forgrouping by distance.The algorithm inputs are:1. The set of points X={x 1 ....... x Q }, where Q is thenumber of waypoints.2. The value of wanted radar range τ .3. Number of initial groups k=2.The outputs are:1. Number of k final groupings.2. Position of each group centers, which shall bethe new waypoints. Figure 2. Distance clustering seudo-code.  The algorithm was implements in Matlab 7 and itsimportance means that it enables to carry outStage 2 of the method, using the RR of platformsas input in nautical miles (NM), reducing thus thecomplexity of the problem. The values in theassignation are seen from the waypoints to thenearest base (Stage 1) on table 1 and the clusteringfor distance with an RR of 24 NM (Stage 2),    4which can be seen graphically in figure 3. (Anautical mile is equal to 1852 meters). Table 1. Results stage 1 and 2 with RR = 24 NM. 3.2.3. Stage 3: Routing This problem can be classified within a networkproblem and to be represented through anondirected graph, where G=(V,A,d), Vbeing theset of vertexes, A is the set of arches and d ij is thedistance associate between arches (Annaballi,2002). The cost matrix of the problem isrepresented in distances or monetary units. Theobjective of the problem is to find the best routethat connects all nodes with the followingrestrictions:1. Each of the waypoints shall be visited onlyonce.2. All routes shall begin and finish in the samenaval base.3. The route longitude shall not exceed thecapacity in distance (autonomy) of eachvehicle. Figure 3. Assignment to Naval Bases BN1 (red) , BN4(blue) and distance clustering with RR= 24 NM.   3.3. Mathematical formulation The two naval bases shall count at least withresources available as seen in table 2.Radar range ( RR ) is the distance through whichthe surface platforms are able to detect andidentify a target, for the air platforms the RR isthe detection distance and the identificationdistance shall be 24 nautical miles (NM). Naval Platforms BN1 BN4RadarRangeFrigate (FF) 1 1 24 NMOceanic Patrol (PO) 1 1 12 NMSea Patrol (PM) 1 1 12 NMCoastal Patrol (PO) 1 1 12 NMPatrol Boat (BP) 1 1 12 NMAir Patrol (MPA) 1 1 60 NM Table 2. Platforms available in the Naval Bases. 3.3.2. Mathematical model The proposed mathematical model corresponds toa VRP 4 (Toth & Vigo, 2002), using the subtouresbreak proposed by Miller, Tucker & Zemlin(1960) that enables these restrictions to havepolynomial cardinality. A restriction of maximumdistance that each platform can travel (autonomy)was added. Sets n = number of waypoints. V  = Vertex sets. V=0, 1, 2,3.. n   i = Start nodes. i Є  V, i=0,1,2,3..n   k  = Platforms available. k=1,2,3..K    u ik    = Load of the platform k  after visiting node i.   u  jk    = Load of the platform k  after visiting node  j.   Parameters CO k    = Operational Cost. CO k  =  fuelconsumtion k  *fuelcost  k  ($gln/h)  dis ij = Distance from i until  j (NM).   C  k  = Maximum capacity platform k  . d  i = Customer i demand.  A k    = Platform k  autonomy in NM .   l k    = Travelled NM cost (NM*$)   h MN eed economicsphlCost Operational k   / $   c ijk    = Cost to move platform k  from i until  j. ($)   ) / ($*)( MN l MN disc k ijijk    Variables otherwise juntilinode fromgok  platformif   x ijk    01   NavalBasesStage 0Stage 1AssignmentStage 2ClusteringInitialwaypointsWaypointsAssignmentWaypointsRR (24 NM)BN115599 31BN4 56 18TOTAL 155 155 49    5 otherwiseinodetheinstart k  platformif   y ik  01   Objetive Function K k nin jijk ijk  xc Z  1 0 0 *min  Constraints )8(...,3,2,1,,1,0 )7(...,3,2,1,,1,0 )6(,...,3,2,1 ...3,2,1 )5(,,,* )4(,...,3,2,1, )3(,..,2,1..,3,2,1,0 )2(, )1(,..3,2,1,01 0 000101 K k V  ji y K k V  ji x K k  A xdis K k C d d donde  jiV  jid C  xC uu K k V iC ud  K k ni y x x K  yni y ik ijk nin jk ijk ijk  ji jk ijk k  jk ik  k ik iik n j jik n jijk K k k K k ik   The objective function minimize financial costs,equation (1) imposes that each of the points i , bevisited by one platform k , equation (2) shows thatthe K platforms leaving shall return to the basewhere they have left, equation (3) is the balancerestriction indicating that every platform k ,arriving to a point i shall leave toward point  j .Equations (4) and (5) shall eliminate those sub-tours imposing capacity requirements andconnectivity (Miller et al., 1960). Restriction (6)prevents overcoming capacities from everyplatform k and restrictions (7) and (8) indicatewhich binary variables are.The mathematical model was implemented inXpress Modeling & Optimization from thecompany Dash Optimization (Ortiz, 2005), whichproduced optimal results with 5 surface platformsand with smaller instances to 34 points to bevisited.Given that the RR of all surface platforms was 24NM and using the points of Stage 2 to be visitedfound in the table 1, BN4 with 18 points and BN1with 31 points, the model was ran and the resultsof the problem were compared with objective of financial cost functions and with costs in distancein BN4 and BN1 as seen in tables 3 and 4. Number K of Platforms   Platforms(Types)   O.F /60(NM)   O.F(MILES $)   1   PM   16.3416   179.757   1   PO   16.3416   670.005   1   FF   16.3416   947.812   2   PM-PC   17.1495   195.706   2   PO-PM   17.1495   218.725   2   PM-BP   17.2686   188.996   3   PO-PM-PC   18.0894   239.578   4   PO-PM-PC-BP   20.0033   293.624   4   FF-PM-PC-BP   20.0033   327.128   Table 3. Problem solution in BN4, assuming that allsurface platforms have a RR=24 NM (18 waypoints),organized by distance cost between points. (PM: Seapatrol, BP: Patrol boat, PC: Coastal patrol, PO: Oceanicpatrol, FF: Frigate). Based in tables 3 and 4, it is then concluded thatthe financial minimization in the operation costsdoesn't guarantee the most efficient route. On theother hand, not all naval platforms have an RR of 24 NM, the complexity of the problem isincreased when using radar reach of surfaceplatforms from table 2, solutions found by themodel were not optimal anymore and theirrunning times were very high, therefore, it wasnecessary to implement a heuristic and a metaheuristic to obtain solutions close to the globaloptimal. Número K dePlataformasPlataformas(Tipos)F.O /60(NM)F.O(MILES $)1 PM 27.77991 305.5791 FF 27.77991 1706.772 PM-PC 29.0422 564.6132 FF-PM N.O. 29.0475 N.O. 626.4792 FF-PC 29.847 1521.33 FF-PO-PM 32.7091 440.0012 FF-PO N.O. 32.7941 1520.212 PO-PM N.O. 36.7532 399.6044 FF-PO-PM-PC N.O 36.3809 625.633 Table 4. Problem solution in BN1, assuming that allsurface platforms have a RR=24 NM (31 waypoints),organized by distance cost between points. (PM: Seapatrol, BP: Patrol boat, PC: Coastal patrol, PO: Oceanicpatrol, FF: Frigate). 3.4. Heuristic methods used to Solve theProblem Since the proposed mathematical model was notefficient in instances higher to 34 points andbearing in mind: (1) that the lower cost doesn'tguarantee the most efficient route and (2) that theRadar range ( RR ) from each platform is used in
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