Following current international SAR procedures we consider three phases to the SAR operation - hasty, efficient and thorough. We define the three phases over the search region graph as incorporating:
• Phase One: edge searching of required edges, also incorporating required vertices.
• Phase Two: edge and triangular region searching of selected regions of the search
region graph utilizing efficient search patterns.
• Phase Three: edge and triangular region searching of selected regions of the graph utilizing thorough search patterns.
The main differences between search Phases Two and Three are the search patterns utilized. In the third phase these patterns are more thorough and result in higher POD levels. Generally this increase in thoroughness is offset by increases in manpower require ments, as searchers are spaced at closer intervals. These patterns have the lowest net benefit return per searcher hour of effort.
Within this chapter the fundamental and unique aspects of the SAR problem are explored in relation to existing problems in the literature.
6 . 1 Dynamism
The SAR problem is dynamic in that it evolves in real-time with the search revealing new clues, or lack of clues, to the subject's whereabouts, as terrain is searched. In addition, the problem is stochastic in that it is affected by uncontrollable elements, such
as weather, and changing factors such as the number of available searchers. In a dynamic problem the time dimension is essential to the crux of the problem and information regarding future events is unknown or tentative. The problem itself may be open-ended (having an indefinite time period) and the model of a dynamic routing problem may be categorized by paths, in preference to tours, due to this. Generally events happening in the 'near-term' are considered to be of greater importance. Psaraftis [141, page 226] considers it "unwise" to commit resources too far into the future in a dynamic scenario, as "other intermediate events may make such decisions suboptimal, and because such future information may change anyway."
The objective function and time constraints of a dynamic problem may also be of a different structure to that of static problems, and are defined for subproblems (de compositions in time or space) of the overall problem. Dror and Powell [44, page 12] note that dynamic and static problems do not possess a standard formulation and are "often characterized by the lack of a well-defined objective function." Often surrogate objectives will be utilized for tractability and these may be more representative of ob jective functions associated with static problems. Additionally, Psaraftis [141] considers that objective functions should meaningfully incorporate known future information to some extent. Often time constraints will be defined as 'soft' constraints.l The author considers that meaningful performance measures for dynamic problems are throughput or productivity related.
As there are no firm guidelines on how long a SAR operation will be conducted before suspension occurs and there is no means of establishing how long it will take to detect the subject, the SAR problem is indefinite in length. The problem is time critical and future information inputs are unknown. Re-scheduling and re-assigning search tasks to resources with respect to priorities derived from new information must be incorporated within the solution structure. Due to constraints of urgency all effort must be put into the immediate time frame; hence the problem displays all of the essential elements of a dynamic problem. The SAR problem is not only a dynamic problem with stochastic elements but, due to the urgency of response required, it also displays all the elements of a real-time decision problem.
6 . 2 Coverage
The classical definition of coverage from search theory is the ratio of search effort to the size of the search area and is defined in Section 2.7. We introduce a new definition of
IPsaraftis [141] defines a 'soft' constraint as one which can be violated at a cost, and one which is subject to update and revision.
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coverage, a visibility cover, which is analogous to the traditional definition of coverage within the Operations Research (OR) literature. This traditional definition applies to problems where locations are to be found for facilities which service portions of a network. If a portion of the network is within a certain distance of the facility it is deemed to be
"covered" by the facility.
We define terrain falling within a search resource's visibility measure (vrn), or alter
natively within their sound measure (srn), as falling within the resource's visibility cover
(the term 'visibility' is used interchangeably for both sound and visual searches) . Hence,
a search object positioned at a point on the TIN located at a distance d from the resource
is visibly covered by that resource if d < vrn for a visual search and if d < sm for a sound
search.
As the resource passing by such a search object has a positive, non-zero probability of detecting the object, the object can be viewed as being 'covered' by that resource. An object may however need to be 'covered' in such a way on more than one occasion for detection to actually result. In this sense the visibility cover definition differs from the OR coverage definition and, unlike most traditional measures, visibility coverage also differs in its dynamism. As a search resource's visibility measure alters with weather and light conditions so does the resource's visibility cover.
In light of this definition of coverage it is possible that some edges within the search area need not be explicitly traversed, i. e., visiting either or both of its end vertices may result in all terrain along an edge being visibly covered by a resource. Thus, in order for such an edge to be visibly covered it is not required that this edge should feature in one of the final search paths, only that one, or both, of its end vertices must. Accordingly, edges are partitioned into three disjoint classifications of visibility measure. In increasing level of restriction, the terrain represented by an edge falls within a resource's visibility cover if:
1. only one of the end vertices of the edge is visited;
2. both the end vertices of the edge are visited;
3. the edge is physically traversed completely.
Henceforth we refer to an edge as falling within Class 1, Class 2 or Class 3 as defined.
With changing visibility conditions over the search operation it is possible that an edge may fall within a different edge class at different time points.
The definition of explicit and implicit searching is also utilized. An edge which is physically traversed as a component of a resource's search path is described as being
explicitly searched, while an edge in Class 1 or Class 2 is implicitly searched if its required vertices are a component of a resource's search path and it is not explicitly searched.
Coverage-routing problems in the literature are now examined with respect to their applicability to the SAR problem.
6.2.1 Coverage - Routing Problems in the Literature
Initially those problems which seek a single tour or path under some measure of coverage are considered.
6.2.2 Vertex Coverage
6.2.2.1 Tour Routes
The Covering Salesman Problem (CSP), first introduced by Current in 1981 [36, 68], has the objective of finding the tour of minimal length over a subset of given vertices, such that every vertex which is not on the tour is within a predetermined covering distance of a vertex on the tour. The CSP is applied particularly to bilevel transportation networks where a service is provided at each vertex on the tour and those 'customers' off the tour must be close enough to access this service at at least one vertex on the tour. Particular applications include rural health delivery and postal services. Current and Schilling [36] also develop a bi-criterion formulation for the CSP that additionally considers the objective of minimizing the cost of stopping at each vertex on the tour.
Gendreau et. al. [69] further refine the CSP to define the Covering Tour Pro blem (CTP). Unlike the CSP, the CTP is defined over a graph consisting of a set of vertices which can be visited by the final tour - a subset of which must be visited - and an additional set of vertices which must be covered. Vertices lying off the final tour are covered if they lie within a specified distance from a vertex on the tour and the minimal length tour is sought.
Gendreau et. al. [68], and Current and Schilling [36], show that the CTP and CSP, respectively, reduce to the TSP when the vertices which must be covered by the tour are required to be within a distance of zero of the tour, i. e. the vertices must lie on the tour itself. Hence the problems are of NP-hard complexity.
The Geometric Covering Salesman Problem2 is defined by Arkin and Hassin
[5] as the problem where a salesman needs to determine a minimal length tour which intersects geometrically defined neighbourhoods of a set of buyers i.e., a specified set of regions of a plane. The neighbourhoods are defined by each buyer as the distance that
2The Geometric Covering Salesman Problem is also referenced as the Travelling Salesman with Neigh bourhoods problem [5).
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they are prepared to travel from their location to meet with the salesman. The authors note that the problem generalizes the Euclidean TSP where buyers are represented by single vertices; the GCSP is also NP-hard. The Geometric Covering Salesman Problem is described by Gendreau et.al. [68] as a continuous version of the CTP.
Two problems which are related to the CSP jCTP are the Median Tour Problem
(MTP) and the Maximal Covering Tour Problem (MCTP), introduced by Cur
rent and Schilling [37]. Both problems are also shown as NP-hard extensions of the TSP which visit only a subset, p, of the vertices in the graph. (This number could be allowed to vary as an additional criterion.) Both the MTP and the MCTP are formulated as bi-criterion problems where the first criterion is to minimize the total length of the tour. In the MTP each vertex has an associated demand and the second criterion is then to minimize the total demand weighted distance that must be travelled from vertices not on the tour, to the vertex in the tour closest to each. The second criterion of the MCTP is to minimize the demand at the vertices not covered (by some prespecified maximal travelling distance) by a vertex on the tour. The M CTP can be expressed as a special instance of the MTP by editing the distance matrix.
Essentially such problems consider two decisions; namely - which vertices to include on the tour and how best to route the selected vertices. Generally a conflict exists between the length of the tour and the accessibility objectives.
The CTP is applicable to a reconnaissance search phase where searching is restricted to specified vertices, such as huts, and the implicit searching of Class 1 and Class 2 edges. In this instance the set of vertices which must be visited are the identified huts and the end vertices of the edges required for each edge to be visibly covered. The vertex of a Class 1 edge which is not included in the final tour is covered by the tour in the manner described by the CTP, as it lies within a prespecified distance from a vertex in the tour. This distance is the visibility measure of that edge. While the predetermined covering distance is one fixed measure for all vertices in the problems described in the literature, this is not translated to the SAR model where visibility measures will differ depending on the terrain, and will also alter temporally when affected by changing weather and light conditions. The CTP restricts resources to beginning and terminating their search at the same vertex, e.g. the search base.
The objective of minimizing the length of the tour applies well to the reconnaissance phase where speed is considered an essential factor. When the coverage of a Class 1 or Class 2 edge is further modelled by delaying the search resource at the end vertex lying on the tour, this can be viewed as the cost of stopping at that vertex. In this case the bi criterion formulation of Current and Shilling is applicable. If an objective of maximizing POS were instead sought, then the MCTP of Current and Shilling could be adapted such
that the demand attached to a vertex not covered (and hence a Class 1 or Class 2 edge not covered) was equal to the predicted POS value of the uncovered edge incident to it.
As the problems are formulated for single tours, the final tour would need to be split for multiple resource deployment or the problems would need to be adapted to produce multiple tours. The tours also route only vertices, hence they are unsuitable for the inclusion of Class 3 edges in the reconnaissance phase.
6.2.2.2 Path Routes
The problem addressed by the Shortest Covering Path Problem (S CPP) is to find the path of least cost between two specified vertices such that all vertices in the network are covered. Each vertex is covered if it is within a prespecified distance of a vertex on the path, where this distance does not have to be identical for every vertex [33]. The SCPP was first formulated by Current et. al. [34]. The shortest path problem and the hamiltonian path problem are both special cases of the SCPP. The shortest path problem occurs when the covering distance is significantly large and the hamiltonian path problem occurs when the covering distance is significantly small. Current et. al.
[33] prove that the SCPP is NP-complete via a transformation to the hamiltonian path problem, indicating that the complexity of the problem can be reduced by utilizing a limited network where some arcs already exist.
The SCPP combines the theory of covering, in relation to facility location, with the shortest path problem to jointly address both network design and routing. As for the covering tour problems, the SCPP is especially applicable to bilevel routing.
Current et. al. [34] extend the SCPP to consider multiobjectives such as the trade-off between minimizing the path length and minimizing the maximum covering distance - effectively placing a budget constraint into the objective function. A trade-off curve can be generated for these objectives considering differing values for the covering distance.
Current et. al. [33] perform computational results on randomly generated symmetric networks ranging from 10 to 90 vertices, of differing densities, and conclude that the problems with an intermediate covering distance are the most difficult to solve.
Current et. al. [35] further extend the SCPP to the Maximum Covering Short est Path Problem (MCSPP). The objective of this problem is to find the minimum cost path between a predetermined starting vertex and a finishing vertex, such that the selected vertices on the path maximize the total demand covered by the path. In the MCSPP a demand exists at each vertex in the graph and this demand is met if the vertex is located on the path or if the vertex lies within a specified distance of a vertex on the path. The MCSPP is hence a multiobjective problem whose objectives conflict.
6.2. Coverage 1 5 1
Current et. al. [35] when the covering distance for all vertices in the graph is equal to zero, i. e., the vertices must be visited explicitly by the path for their demand to be met. This is a special case of the MCSPP and has the same objectives.
Both the MCSPP and MPSPP are defined over a graph consisting of non-directed arcs . . Current et. al. [35] emphasize the flexibility of the problem formulation which can also be extended to cater for: instances where specific arcs or intermediate vertices are required to be on the path; cases where multiple coverage of vertices is desirable; the inclusion of mandatory closeness constraints3 which could be varied for differing vertices; and situations where varying service demands may occur at vertices - each service being represented by an additional covering objective. The authors state that minimizing coverage distance can be expressed as a third objective in the MCSPP if it is not a well defined value.
The SCPP is applicable to planning search resource paths, as paths, not tours, are specifically detailed, although the problem addressed is once again a single, not multiple, path problem. Again the notion of coverage is identical to that of the various covering tour problems and fits well with the visibility cover in the search, as defined by searcher visibility. In the formulation of the SCPP, coverage distance is also able to be varied for differing vertices. However, once again, vertices are explicitly included in the path, not a specified set of edges. Additionally, symmetric distances are properties of the underlying network for the covering path problems which are not features of the search region graph.
The MPSPP by regarding vertices as covered, only if they are explicitly present on the path, could be seen as a special case of a reconnaissance search phase where searchers concentrate their searching on those edges and vertices physically traversed, and do not slow to investigate other areas in their proximity. If conditions were so severe as to restrict visibility to the terrain directly before, and underfoot, of the searcher, this constraint would be a true representation of the problem at hand.
6.2.2.3 Tree Networks
The Minimum Cost Covering Subtree Problem (MCCSP) is introduced by Aaronson
Hutson and ReVelle [1]. The problem is that of finding the minimum cost group of arcs which form a subtree, where all vertices are within a prespecified distance from some vertex in the subtree. Hence the problem is effectively a SCPP where the path must form a subtree and the underlying network forms a tree. Aaronson-Hutson and ReVelle define direct coverage as the coverage pertaining to vertices on the subtree and indirect
coverage pertaining to the cover of vertices not on the subtree. They further define two
classifications of indirect coverage. The first classification applies to vertices which are within a specified distance of a vertex on the subtree. In the second classification, vertices are covered if they are within a specified distance to an arc of the subtree.
Aaronson-Hutson and ReVelle [1] introduce the Maximal Indirect Covering Sub tree Problem