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Comput Manag SciDOI 10.1007/s10287-013-0179-1

ORIGINAL PAPER

An orienteering model for the search and rescueproblem

Adel Guitouni Hatem Masri

Received: 6 February 2013 / Accepted: 3 July 2013 Springer-Verlag Berlin Heidelberg 2013

Abstract In this paper, we propose a new model for the search and rescue problem.We focus on the case of a single airborne search asset through a connected spaceand continuous time with a maximum travel time T . The intent is to maximize thedetection of a cooperative target (search and rescue). The proposed model is basedon the assumption of existing a priori information (e.g., result of information fusionprocess) to establish a spatial distribution of probability of containment in possiblegeographic locations. The possibility area is defined using a cut threshold on theprobability of containment and the search path as well as the allocation of the level ofeffort to each region in the search space is obtained based on an orienteering model.We illustrate the application of the proposed model on an empirical example.

Keywords Search and rescue problem Orienteering problem

1 Introduction

In this paper, we consider the search path problem for a single airborne search assetin a continuous space and time for a maximum mission time T , through a connectedregion. The objective is to maximize the detection of a cooperative target (search andrescue). A priori information (e.g., sensors data, experts opinion) is available to build

A. GuitouniCommand and Control Decision Support Systems Section, Defence R&D Canada,2459 Pie XI North, Quebec, QC G3J 1X5, Canadae-mail: adel.guitouni@drdc-rddc.gc.ca

H. Masri (B)College of Business Administration, University of Bahrain,P. O. Box 32038, Sakhir, Kingdom of Bahraine-mail: hmasri@uob.edu.bh

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a spatial distribution of the possible location of the target. Such distribution might bea probability, possibility, fuzzy or any other type of uncertainty modelling functions.A seeker or a searcher is an airborne platform equipped with sensors (e.g., scanningradar, imaging radar, electro-optic, infrared). The airborne platform has a maximumfly time that might be seen as time to refuel or any other system or human constraint.The problem is therefore to sequence a set of search activities in time and space inorder to maximize the chances to find the target. Search activities are in this case thetransit between two regions and active search in each region. The search plan shouldthen provide a path plan as well as the level of effort to be spent in each location.

The search-path problem has been closely related to search effort allocation prob-lem (Kierstead and Balzo 2003). Search theory supposes that targets and seekers aremodelled probabilistically (Stone 1975). Search theory was developed for the first timeby Koopman (1980) and the US Navys operations research group in order to developbetter strategies for anti-submarine warfare during the Second World War. Since then,search theory was mainly applied in application fields such as: surveillance, SAR mis-sions and exploration (e.g. finding the US Nuclear Submarine Scorpion lost in 1968and Clearing unexploded ordnance in the Suez Canal).

The search path planning problem has been subject to many ongoing researchactivities. In this paper, we propose a novel modelling for the search problem froma mathematical programming perspective. This paper is organized as follows. Wepresent an overview of the search and rescue problem and we review related work inSect. 2. In Sect. 3, we describe a variant of the search and rescue problem and theway that we are going to proceed to solve that problem. First, we start by buildingthe possibility area in Sect. 4 and then adapting in Sect. 5 an orienteering model tobuild the search path and to allocate the search effort to each region in the possibilityarea. We present an empirical validation of the model in Sect. 6. The conclusion ofthis paper is given in Sect. 7.

2 Search and rescue problem

Resource management for conducting search and rescue, surveillance and reconnais-sance mission is a very important problem for many organizations. This problem ischaracterized by the employment of mobile (e.g. maritime patrol aircraft, helicopters,UAVs, ships) and fixed surveillance assets (e.g. land radar) to a large geographic areain order to identify, assess and track the maximum number of moving, stopped or drift-ing objects. The observed objects are not necessarily aware of being observed and arecooperative or non-cooperative, and friendly or hostile. The scarce surveillance (e.g.electro-optical, infra-red, and synthetic aperture radar sensors), tracking capabilities(normal radar modes), scarce platform, time and space constraints makes it a verydifficult problem.

A search mission is usually planned for a given search area while minimizing costin term of time, distance and exposure to threats, and maximizing the chances ofsuccess subject to a set of constraints imposed by the amount of effort available forthe search. The search path planning problem is intractable and complex because it is

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Orienteering model for the search and rescue problem

multifaceted and NP-hard even for stationary targets (Trummel and Weisinger 1986).Many elements should be considered including:

the platform configurations and properties which generally related to the physicaland functional properties of the platform (aircraft),

the number of platforms and their configurations, the sensors, or the instruments used to carry out the seeker such as the eye, radar,

sonar, television, and cameras, the object or the target being searched which may be stationary or moving, coop-

erative or non cooperative, friendly or hostile, the geographic or physical search area that may be continuous (in Euclidean

n-space) or discrete (a set of cells), closed or open, and the available search effort may be continuous (i.e. measured by time or track length)

or discrete (i.e. measured by a finite number of scans or looks).Abi-Zeid and Frost (2005) developed a decision support system (SARPlan) based onsearch theory and optimization to enhance the SAR mission effectiveness. SARPlanaims at maximizing the efficient use of resources and increasing the chances of findingsurvivors in a shorter time. Based on search theory, SARPlan suggests an optimal over-land air plan for a SAR mission. In SARPlan, Constraint Satisfaction Programming(CSP) and traditional optimization techniques are used to derive the optimal plan.Note that an optimal plan is a plan that maximizes the probability of finding the searchobject and the efficient use of resources while minimizing search time, search area,and associated costs. Dor et al. (2009) proposed an extension of the search theoryfor search and rescue. They proposed the theory of belief functions for informationcombination and update for the optimal search planning context.

The discrete path problem is based on the assumption that the searcher or seekerand the target are roaming in a discrete time and space dimensions. Moreover, discrete-time moving-target search model usually assumes the Markovian property of thetargets movement (Hong et al. 2009). Dambreville and Cadre (2002) proposed analgorithm for optimal search for a target following a Markovian movement or a con-ditionally deterministic motion when search resources are renewed with generalizedlinear constraints.

An optimal dynamic programming method is proposed by Eagle (1984). Stewart(1979) has proposed a search efforts relocation heuristic where the effort is restrictedto paths. Eagle and Yee (1990) proposed an extension of the work of Stewart (1979).They proposed an optimal branch-and-bound algorithm limited 7 7 search grids.Other extensions of this work might be found in Dell et al. (1996) and Hohzaki andIida (1997).

Hong et al. (2009) proposed a pseudo-polynomial heuristic for the single-searcherpath-constrained discrete-time Markovian-target search. The heuristic is based on anapproximate of non-detection probability computed from the conditional probabil-ity that reflects the search history over a fixed time windows. Jacobson and McLay(2006) proposed simultaneous generalized hill climbing algorithms to determine opti-mal search strategies over multiple search platforms. Kierstead and Balzo (2003)proposed a genetic algorithm (GA) for planning search paths against a moving target.In his paper, Janez (2007) proposed to model the sensors planning problem as vehicle

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Destress CallOr

IntelligencePrioritization

Process

Fusion Process

Sensor 2

Sensor 3

Sensor 1

Spatial Density

(Possibility) Distribution

Filtering Process

Resource Allocation Path Planning

Tasking Orders

Platform Flying the mission

Action/EffectsAnd

Assessment

Fig. 1 Simplified process

routing problem (VRP). The formulation is based on the assumption that the coordi-nates of the targets to be visited is known a priori. Unlike previously mentioned works,targets are stationary and their location is well defined. The problem addressed is thenthe allocation of different surveillance assets to visiting each target.

3 The problem description

Let a seeker s be a combination of a given platform with appropriate sensors for agiven search situation. Multiple seekers might be assigned to different search areas.Search path in each area might follow a static or dynamic pattern. This paper addressesoptimal path planning given a particular seeker. We focus on airborne seekers.

We assume the process shown in Fig. 1. A distress call might be received by asearch and rescue centre, which will activate its prioritization process. Informationis received from different information sources (e.g., sensors). An information fusionprocess produces a spatial distribution of possible location of the target. Through afiltering process, one can generate a synthesized distribution function over the searcharea.

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Orienteering model for the search and rescue problem

We assume that a priori information is available through information fusionprocesses about a plausible distribution that represent the possibility that the target iswithin a particular containment area (Kao et al. 2001). The intensity of the distributionmight refer to a higher degree of confidence (e.g., probability) that the object mightbe in a particular more confined region of that containment area. At this point, thedistribution function is not necessarily a probability distribution function. Differentclustering techniques might be applied in order to create a more intelligible distributionmap.

A good search plan should maximize the probability of success (POS), which is theprobability of finding the search object. To compute the POS, two other probabilitiesare used. The probability of containment (POC), which is the probability that the searchobject is in a particular cell of the searched area, and the probability of detection (POD),which is the probability that the sensor detects the search object given that it is in thearea searched. Note that the probability POD is function of the effort provided by thesensor on a specific area. For simplicity of the model, we assume that the probabilityof detecting the target is function of the time spent searching in a particular geographicregion. We simplify the conditional probability to be expressed in term of the lateralrange function of the sensor, its sweep width W , and the total time spent in the region(Frost 1998).

We assume that the probability of detection is given by the following expression:

POD = 1 e(

W j v jA j

)t j (1)

where W j is the effective sweep width in region j, A j is the surface of region j, v j isthe platform speed in region j and t j is the level of effort allocated to region j . In thispaper, we propose a path planning algorithm to maximize the probability of findingthe cooperative target within a given time window. The effort may be measured bytime, area searched, track length or any other appropriate measure. The relationshipbetween these three probabilities is as follows:

POS = POC POD (2)

Now, the problem is to define the possibility area and to fly an airborne seeker withthe intent to find the target while optimizing a set of objectives under constraints.

4 The possibility area

The first step of this process consists in identifying the boundaries of the possibilityarea (PA) that may contain the search object. Subjective (e.g., rules, experts judg-ments) and objective (e.g., historical data, fused sensors data) knowledge are generallyused to define the limits of this area. For instance, the International Aeronautical andMaritime Search and Rescue (IAMSAR, 1999) Manual contains some guidelines toestablish the possibility area. Other approaches like possibility distribution functionor fuzzy functions might be used. Then, the possibility area is divided into J cells

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Fig. 2 2D view of a distribution of possible locations of the target

Fig. 3 Cut

( j = 1, . . ., J ) that represent the smallest regions over which search effort can beallocated. A search within each cell might follow a predefined or dynamic pattern.

Lets assume the search problem shown in Fig. 2. An airborne platform shouldfly over a geographic area following a given path to maximize the chances to findthe target. The aircraft has a limited flying time and the target might have a limitedsurvival time. The different contours represent distribution function of the probabilityof containment. The concentration of the contours represents higher probabilities.

We propose to apply a cut threshold and then we obtain the distribution of Fig. 3.The value of threshold may have an impact on the search area. A higher value of thethreshold can allow the airborne platform to fly over different regions and spend moretime where the contours are denser and less elsewhere. Such flying path might berepresented by Fig. 4.

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Orienteering model for the search and rescue problem

Fig. 4 Flight plan

1

2

i j(tij)

i j(tij)

3

4

5

7

68

(ti)

(t3)

(t4)

(t5)

(t6)

(t7)

(t8)

(t2)

Fig. 5 Example of flying path

Therefore, the search regions are the vertices of a graph G = (V, E) (see Fig. 5)where the search platform starts at the vertex i = 1 to visit the other regions (vertices).To travel between two vertices i and j , the search platform takes ti j time (tii = 0 forall i). Note that, we consider only vertices that verify {i | t1i + ti1 TM } where TMis the time window for the search and rescue mission.

5 A orienteering model

The search and rescue problem using only one search platform (SARP1) presentssome similarity compared to the orienteering problem with variable profits (OPVP).The OPVP is to find a set of vehicle tours that begins and ends at the start vertex,traverses a subset of the set of vertices and must have a total duration not exceeding

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a given limit (maximum mission time) where at every vertex the vehicle collects apercentage of the profit...