Finding a risk-constrained shortest path for an unmanned combat vehicle

Computers & Industrial Engineering 80 (2015) 245–253

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Finding a risk-constrained shortest path for an unmanned combat vehicle Kyung-Yeol Bae a, Yeong-Dae Kim b,⇑, Jun-Hee Han b a b

Korea Army Training and Doctrine Command, Yuseong-gu, Daejeon 308-878, South Korea Department of Industrial Engineering, Korea Advanced Institute of Science and Technology, Yuseong-gu, Daejeon 305-701, South Korea

a r t i c l e

i n f o

Article history: Received 27 March 2012 Received in revised form 11 December 2014 Accepted 13 December 2014 Available online 20 December 2014 Keywords: Resource-constrained shortest-path problem Unmanned combat vehicle Multiple-choice knapsack problem Dynamic programming

a b s t r a c t We consider a problem of finding a reconnaissance route of an unmanned combat vehicle (UCV) in a terrain, which is modeled as a grid. It is assumed that the traverse time to pass through each cell in the grid and risk level associated with each cell are given and that the cells where the reconnaissance points to be visited by the UCV are located and the visiting sequence of such cells are given in advance as in real situation of military operation. We focus on the problem with the objective of minimizing the total travel time of the UCV for a given limit on the sum of risk level values associated with the cells on the path of the UCV. We develop an optimal solution algorithm based on a dynamic programming algorithm for multiple-choice knapsack problems. We also present a heuristic algorithm, which can be used for large-size problems. For evaluation of the performance of the proposed algorithms, computational experiments are performed on a number of problem instances, and results show that the proposed algorithms give optimal or good solutions within an acceptable time for real military operations. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction In this paper, we consider a problem of finding a route of an unmanned combat vehicle (UCV) in the military battlefield. The UCV is a remote-controlled vehicle with no person on board that carries out various missions, such as surveillance, reconnaissance, and mine detection. It is usually equipped with a telescope, mine detection device, and/or guns. To carry out a mission assigned to a UCV, it should visit a number of locations. In general, the sequence of the locations the UCV should visit is determined by the commanding unit (before determining the route of UCVs) with consideration of the probabilities that there are enemy’s military units in certain regions, the aggressiveness and threats (on the friendly units) of those units, and the total length of the resulting route of the UCV by using the information send from satellites. The operator of the UCV has to determine the best route to minimize the time required to complete a given mission, considering the risk of being found and attacked by the enemy during the operation. In this study, we focus on a risk-constrained shortest path problem of minimizing the total travel time of a UCV for a given sequence of locations (reconnaissance points) to be visited while limiting the risk level associated with the path within a predetermined value. ⇑ Corresponding author. Tel.: +82 42 350 3120; fax: +82 42 350 3110. E-mail address: [email protected] (Y.-D. Kim). http://dx.doi.org/10.1016/j.cie.2014.12.016 0360-8352/Ó 2014 Elsevier Ltd. All rights reserved.

The problem considered in this paper is similar to the travelling salesman problem (TSP) in that both problems are concerned with finding a shortest path to visit a given set of locations and return to the initial location. However, a path between each pair of locations (to be visited) is given in the typical TSP and hence there is at most one arc between each pair of nodes (locations) in the associated network, while in the problem considered in this study, there may be multiple paths between each pair of locations (to be visited), from which one is to be taken, although the sequence of locations to be visited is given. In addition, if risk is regarded as resource, our problem is closely related to the resource-constrained shortest path problem (RCSPP), which is to find a shortest route from the source node to the destination node under the constraint that resource(s) required by the arcs on the path should not exceed a given limit(s). The difference is that there are multiple destinations (locations to be visited) in our problem, while there is a single destination in the typical RCSPP. There are a few studies on path planning of UCVs. Stentz (1994) suggests a two-step algorithm in which an initial path is determined based on known initial information and the path is modified by using updated information. Also, many metaheuristics have been developed for path planning of UCVs. Harder, Hill, and Moore (2004) and Shetty, Sudit, and Nagi (2008) suggest tabu search algorithm for unmanned combat aerial vehicles (UCAVs), and Edison and Shima (2011) develop a genetic algorithm based on graph theory. Moreover, Park, Kim, and Jeong (2012) suggest

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two-phase heuristic algorithms, and Han, Kim, and Lee (2014) suggest a heuristic algorithm based on the label-correcting algorithm. On the other hand, Cunningham and Roberts (2001) develop an adaptive algorithm to generate paths for multiple UCAVs, and Rathbun, Kragelund, Pongpunwattana, and Capozzi (2002) presents a path planning algorithm for a UCAV considering uncertain obstacles in a dynamically changing environment. Another problem that is similar to our problem is the path planning problem for robots. For this problem, Rhee and Rhee (1996) develop an algorithm for finding a collision-free path using learning techniques, and Choi and Kim (1996) develop an algorithm based on graph theory for generating a road-map and finding collision-free paths. Also, Chakraborty, Akella, and Wen (2010) present a two-step approach for a multiple-robot path planning problem, assigning destinations to the robots and finding paths for the robots. Meanwhile, Besada-Portas, de la Torre, and de la Cruz (2010) present an algorithm based on evolutionary algorithms, and Darrah et al. (2013) and Sahingoz (2014) suggests a genetic algorithm for path planning for multiple unmanned aerial vehicles. Although research on our problem is rare, if any, there are many research articles on TSPs and RCSPPs. For TSPs, Held and Karp (1970), Padberg and Rinaldi (1987), Fischetti and Toth (1997), and Bektas (2006), among many others, give optimal solution approaches, and many researchers develop various heuristic algorithms including those of Croes (1958), Lin and Kernighan (1973), Ozgur and Brown (1995), Russell (1977), Suh and Kim (2001), Samanlioglu, Ferrel, and Kurz (2008), Zhao, Li, Sun, and Mei (2009), and McWilliams (2010). The RCSPP, which is shown to be NP-complete (Handler & Zang, 1980), has been used as a model for various real-world problems in Mingozzi, Boschetti, Ricciardelli, and Bianco (1999), Holmberg and Yuan (2003), Wilhelm, Arambula, and Choudhry (2006), Wilhelm, Choudhry, and Damodaran (2007), and Shin, Yang, and Baek (2006) among others. There have been a variety of solution approaches to the RCSPP. For instance, a number of algorithms have been developed based on dynamic programming (Aneja, Aggarwal, & Nair, 1983; Desrochers & Soumis, 1988; Joksch, 1966), Lagrangean relaxation methods (Beasley & Christofides, 1989; Handler & Zang, 1980), and a column generation method (Chen & Shen, 2013). On the other hand, Wilhelm and Tarmy (2003) give a pseudo-polynomial-time two-phase algorithm for RCSPPs. Approximation algorithms have also been proposed for RCSPPs, including schemes by Henig (1986) and Warburton (1987) for multi-objective problems. Recently, Shi (2010) and Puglia Pugliese and Guerriero (2013) suggest approximation algorithms for shortest path problems with multiple constraints. In this study, we develop an optimal solution method for finding a shortest route for a given sequence of locations to be visited within a pre-specified total risk level. We also present a heuristic that can be used for large instances, for which optimal solutions cannot be found in a reasonable amount of time. In both algorithms, a number of (promising) paths are generated for each pair of consecutive locations in the sequence, and then one path is selected for each pair using a dynamic programming algorithm. The remainder of this paper is organized as follows. In Section 2, we describe the problem in detail, and we develop an optimal solution algorithm and a heuristic in Sections 3 and 4, respectively. These algorithms are evaluated through a series of computational experiments and results are reported in Section 5. Finally, Section 6 concludes the paper with a short summary and discussions on future research.

is represented with a grid map. The grid and each cell of the grid are rectangular. Two types of information are known for each cell, traverse time and risk level. These are estimated for each cell from the configuration of the ground, i.e., topography, of the cell (given in military operations maps) and the locations of the enemy’s units. (Such information is obtained by satellites and various intelligence assets of friendly units and updated frequently.) Note that a path from one point in the terrain to another can be represented by a sequence of cells through which the UCV passes, since there is no road (between the points in the terrain) on which the UCV can move, as is the case with the battlefield. Also, note that the probabilities that the UCV is detected and attacked by the enemy may differ in different cells. The problem is to find a path that requires the shortest traverse time for visiting a set of locations (reconnaissance points) according to a given sequence and returning to the base. The sum of risk levels associated with cells the UCV passes through should not exceed a given upper limit (UR) on the cumulative risk level. As stated earlier, the problem can be considered as a resource-constrained shortest path problem (RCSPP) if the risk level is considered as resource, and it is denoted as RCSPP-UCV in this paper. In this study, the terrain, which is represented on an M  N grid, is also represented by network G = (V, A), where V and A are the index sets of nodes and arcs, respectively. The nodes represent cells in the grid and each arc represents a connection or path between two adjacent cells. From each cell except for the cells on the border of the grid map, the UCV can move to eight adjacent cells as shown in Fig. 1. Fig. 2 shows a simple example of paths between two cells in the grid. Reconnaissance points are also represented by nodes. In addition, it is assumed that in order to reconnoiter a point, the UCV must pass through the cell in which the reconnaissance point is located. This is the way paths are determined and moving times are estimated in the battlefield (since there are no well-defined roads between the reconnaissance points). That is, the moving time between two points through a path is estimated as the sum of traverse times of cells included in the path. In this study, a path between two consecutive reconnaissance points in a given sequence is called a partial route. Each partial route can be defined by a sequence of nodes and/or arcs included in the route. A complete route, i.e., a path starting from the source (base), visiting all reconnaissance points according to a given sequence, and returning back to the source, is given as a concatenation of such partial routes. The sum of risk level values of the nodes (cells) included in a route is called the cumulative risk level of the route, and a route is called feasible if the cumulative risk of the route is not greater than a pre-specified limit (UR) on the cumulative risk of a complete route.

2. Problem description We consider a shortest path problem for an unmanned combat vehicle (UCV) which reconnoiters a focused area at night time. As in actual military operations, the terrain, or the area of interest,

Fig. 1. Possible directions of movement from a cell.

K.-Y. Bae et al. / Computers & Industrial Engineering 80 (2015) 245–253

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3. Optimal solution algorithm In the algorithm developed in this study, solutions are obtained in two steps. In the first step, we generate feasible partial routes, i.e., paths between two consecutive reconnaissance points in a given sequence of reconnaissance points to be visited. In the second step, we select the best set of partial routes that constitute a complete route from the base, to the reconnaissance points in the given sequence, and back to the base. The selected routes should satisfy the constraints on the cumulative risk level. Note that the problem of selecting partial routes from the set of candidates (for those partial routes) generated in the first step can be regarded as a multiplechoice knapsack problem. In the following, we give more detailed description of each step, or phase, of the algorithm. 3.1. Phase 1: enumerating feasible partial routes Fig. 2. Examples of paths between two nodes.

Assumptions made (based on the real battlefield situation) in this research are summarized below. (1) The traverse time and risk level associated with each cell on the grid map is given and does not change during a mission of the UCV or the planning horizon in the problem. (2) The risk level involved in passing though each cell in the grid map is given in six grades with integer values, 0–5, which have been estimated and specified already by a commanding unit. (3) The order (sequence) of reconnaissance points to be visited is given. (4) The time the UCV stays at a reconnaissance point for a mission is not considered (since it does not depend on the route). (5) Each cell can be visited multiple times. That is, a cell can be included in a path between a pair of consecutive reconnaissance points in the given sequence and another path between another pair of consecutive reconnaissance points. (6) The battery of the UCV is fully charged so that the UCV can visit all reconnaissance points and return to the base without recharging. (7) There is no failure (breakdown) of the UCV during the planning horizon. Before presenting algorithms for the problem in the following sections, we give notation used in the description of the algorithms. m n i, j p, q ap bp UR rik

tik r min i xik = 1

number of reconnaissance points that must be visited by the UCV. number of nodes (cells) in the grid map, n = M  N. indices for reconnaissance points (0 and m + 1 represent the base). indices for nodes. risk level associated with node p. traverse time for the UCV to pass through node p. pre-specified limit on the cumulative risk level of a complete route. risk level of the kth (candidate) partial route from the ith to the (i + 1)st reconnaissance points (in the sequence of reconnaissance points). traverse time of the kth partial route from the ith to the (i + 1)st reconnaissance points. risk level of the minimum-risk partial route from the ith to the (i + 1)st reconnaissance points. if the kth candidate partial route between the ith and (i + 1)st reconnaissance points is selected, and 0 otherwise.

In this phase, all feasible partial routes are enumerated for each pair of consecutive reconnaissance points in the given sequence. Among those enumerated partial routes, dominated ones are eliminated from consideration. Here, partial route A between two consecutive reconnaissance points is said to be dominated by partial route B between the same points, if the best feasible complete route in which partial route B is included is better (have shorter traverse time) than the best feasible complete route including partial route A. In the following, we present dominance properties that can be used for identification of dominated partial routes. Proposition 1. If the cumulative risk levels of two (or more) partial routes between a pair of reconnaissance points in the given sequence are the same, a partial route with a longer traverse time between the points is dominated by the one with a shorter traverse time. Proof. Let P and P0 denote, respectively, the partial routes with the longer traverse time and the shorter traverse time. Consider a complete route, S, in which P is included as a part and another complete route S0 , which is identical to S except that P is replaced with P0 . Then, the cumulative risk levels of S and S0 are the same, but the traverse time of S0 is shorter than that of S. Therefore, P is dominated by P0 . h Proposition 2. If the traverse times of two (or more) partial routes between a pair of reconnaissance points in the given sequence are the same, a partial route with a higher cumulative risk level between the points can be deleted from consideration in the problem to find a shortest-traverse-time route. Proof. It is obvious that the shortest route between a pair of destinations under a tighter constraint on the cumulative risk level for complete routes, i.e., with a lower UR value, is not shorter than the shortest route between the same pair of destinations under a looser constraint. Let P and P0 denote, respectively, the partial routes with the higher cumulative risk level and the lower cumulative risk level. Consider S, the best feasible complete route in which P is included as a part and S0 , the best feasible complete route including P0 . Since the limit on the cumulative risk level for the other parts (except for P) of S is smaller than that of S0 , S0 is at least as good as S. Therefore, P can be deleted from consideration. h Note that S and S0 may have the same traverse time, and hence S may not be dominated by S0 in an exact sense in this case. However, even though we delete P from consideration, we can always find a complete route with the shortest traverse time. From the above two propositions, we have the following dominance property.

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Corollary 1. A partial route with a higher cumulative risk level and a longer traverse time between a pair of reconnaissance points in the given sequence is dominated by a partial route with a lower cumulative risk level and a shorter traverse time between the two points.

Note that the dominance properties related to Propositions 1 and 2 are used in Steps 1 and 2 (a shortest-traverse-time path is generated for each risk level value), while the dominance property related to Proposition 3 is used in Step 3. 3.2. Phase 2: selecting partial routes for a complete route

In addition to the above dominance properties, we check feasibility of partial routes using an upper bound on the risk level value of a feasible partial route between two reconnaissance points. Using the following proposition, we identify infeasible partial routes and delete them from consideration in the algorithm for generating all non-dominated partial routes. Proposition 3. If the risk level of a partial route between the i-th and the (i + 1)st reconnaissance points in the given sequence exceeds nP o m min UR   Ui, that partial route is infeasible, that is,  r min j¼0 r j i the partial route results in infeasible solutions. P min Proof. Since m  r min is a lower bound on the sum of risk j¼0 r j i levels of all partial routes of a feasible complete route except the partial route between the ith and (i + 1)st reconnaissance points, Ui is an upper bound on the risk level of a partial route between the ith and (i + 1)st reconnaissance points. Then, the proposition follows. h In the algorithm suggested in this study, we generate all feasible partial routes between pairs of consecutive reconnaissance points but consider only non-dominated partial routes to find the best feasible complete route. All partial routes are generated with the algorithm developed by Hassin (1992) for the resource-constrained shortest path problem (RCSPP) based on Joksch’s algorithm (Joksch, 1966). Note that the problem of finding a route between two consecutive reconnaissance points can be viewed as an RCSPP. However, Hassin’s algorithm for RCSPPs gives only one shortest route, while we need all feasible routes that are not dominated by others. In this study, we modify Hassin’s algorithm to generate all feasible routes that are not dominated by others. The procedure for generating non-dominated partial routes between each pair of consecutive reconnaissance points, say between the ith point and the (i + 1)st point, can be summarized as follows. In the procedure, cpq(v) denotes the minimum cumulative traverse time of partial routes from node p to node q with cumulative risk of v or less, and Ui denotes the upper limit on the risk level of a partial route between the i-th and the (i + 1)st reconnaissance points in the given sequence, as defined in Proposition 3. In addition, nodes associated with the ith and the (i + 1)st reconnaissance points are denoted by p° and p⁄, respectively. Procedure 1. (Generating non-dominated partial routes between two nodes) Step 0. Set cpp(v) 0 for v = 0, . . . , UR, and cpq(v) 1, for v < 0 and all p and q. For all p and q (p – q): if nodes p and q are adjacent nodes, i.e., if the UCV can move from node p directly to node q, let cpq(aq) bq; otherwise, let cpq(v) 1 for all v. Set k = 1, and v = 0. Step 1. For l – p°: let min fcp k ðv  1Þ; cp l0 ðv  v 0 Þ þ cl0 l ðv 0 Þg. cp l ðv Þ 0 v 0 v ;l –l; p

cp°p⁄(v), rik v and Step 2. If cp°p⁄(v) < cp°p⁄(v  1), let tik k k + 1. Go to Step 1. Step 3. If v < Ui, let v v + 1 and go to Step 1; otherwise, terminate.

Once all feasible non-dominated partial routes are generated for each pair of consecutive reconnaissance points in the given sequence, we need to select partial routes, one for each pair of consecutive points, that minimize the total traverse time while satisfying the risk-level constraint. This selection problem can be modeled as the following multiple-choice knapsack problem (MCKP).

Minimize

XX t ik xik i

k

XX subject to rik xik  U R i

X xik ¼ 1 8i

ð1Þ

k

ð2Þ

k

xik 2 f0; 1g 8i; k

ð3Þ

The MCKP is NP-hard as a special case of the problem is the knapsack problem. There have been various algorithms developed for MCKPs (Dyer, Kayal, & Walker, 1984; Nauss, 1978; Shinha & Zoltners, 1979; Zemel, 1980). In this study, we solve the MCKP using a dynamic programming recursion presented in Dudzinski and Walukieiez (1987). Recursive equations for the dynamic programming algorithm used in this study are given below. Here, Tj(v) denotes the minimum cumulative traverse time of the feasible routes from the base to the jth reconnaissance point at cumulative risk of v or less, and Qi denotes the set of candidate partial routes from the ith to the (i + 1)st reconnaissance points. With Tj(v) = 1 for v < 0, the recursive equations are given as:

T 0 ðv Þ ¼ 0 for all

m P 0; and

T j ðv Þ ¼ min½tjk þ T j1 ðv  rik Þ; for k2Q j

m¼

j X i¼0

rmin ; . . . ; UR  i

m X

r min ; i

i¼jþ1

j ¼ 1; 2; . . . ; m þ 1: The solution is given as Tm+1(UR). 4. Heuristic algorithm The optimal solution algorithm presented in the previous section cannot solve large-size problems in a reasonable amount of time. Note that no longer than 10 min can be spent to solve the problem in an actual military situation (according to the current field manual). Hence, in this section, we suggest a heuristic for large instances of practical size. The basic idea of the heuristic is to solve the problem of generating partial routes between each pair of consecutive reconnaissance points in the given sequence by limiting the terrain to a (smaller) rectangular region that encompasses the two points. The heuristic follows the logic of the optimal solution algorithm, but it generates only feasible non-dominated partial routes within the rectangular region associated with each pair of consecutive reconnaissance points, and the multiple-choice knapsack problem is solved using those partial routes. The dynamic programming algorithm used in the optimal solution algorithm is used in this heuristic as well. We consider three variations of the heuristic, each of which uses a different way of defining the rectangular region. The first one defines the rectangular region as the minimum-size rectangle

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Fig. 3. Examples of rectangular regions to be considered in the heuristics.

Table 1 Results of the tests on 30  30 grid maps. NPa

b c

Solution qualityc

CPU time (s) OPT

H0

H1

H2

4

1.8 2.0 2.2

2.52 (2.20), 0 3.71 (3.22), 1 3.57 (2.98), 0

1.03 (0.95), 1 0.54 (0.94), 2 0.37 (0.63), 2

0.62(1.07), 2 0.54(0.94), 2 0.00(0.00), 3

87.34 100.21 123.28

0.82 0.74 0.82

1.61 1.33 1.45

3.04 2.25 2.38

6

1.8 2.0 2.2

4.42 (2.66), 0 6.41 (7.31), 0 2.91 (3.53), 1

2.42 (0.31), 0 4.65 (6.57), 1 1.49 (1.76), 2

1.12(1.45), 1 3.77(5.06), 1 0.42(0.44), 2

177.10 190.41 187.85

0.47 0.53 0.65

0.92 1.17 1.29

1.87 2.38 2.53

8

1.8 2.0 2.2

1.30 (2.26), 2 4.45 (2.45), 0 1.66 (1.84), 1

1.30 (2.26), 2 4.45 (2.45), 0 0.78 (0.73), 1

1.12(1.94), 2 2.22(2.57), 1 0.12(0.21), 2

246.12 271.04 282.81

1.61 1.76 1.94

3.37 3.56 3.96

5.85 6.34 6.96

10

1.8 2.0 2.2

0.08 (0.14), 2 2.13 (2.30), 1 2.34 (2.11), 1

0.08 (0.14), 2 0.61 (1.06), 2 1.44 (1.28), 1

0.00(0.00), 3 0.61(1.06), 2 0.49(0.85), 2

325.56 355.88 371.93

0.64 0.70 0.76

1.39 1.54 1.73

2.83 3.07 3.44

12

1.8 2.0 2.2

1.25 (2.17), 2 0.88 (1.52), 2 0.98 (0.47), 0

0.00 (0.00), 3 0.35 (0.61), 2 0.67 (0.43), 0

0.00(0.00), 3 0.00(0.00), 3 0.10(0.17), 2

405.94 428.66 526.19

1.49 1.63 1.89

2.99 3.25 3.82

5.43 5.90 7.39

1.75 (1.69), 13

0.85 (1.44), 12

0.36 (1.03), 31

272.02

1.10

2.22

4.11

Overall a

qb

H0

H1

H2

Number of reconnaissance points to be visited. Parameters used to control the tightness of the risk-level constraints. Average (and standard deviation in parenthesis) percentage error of the heuristic solutions from the optimal solutions and the number of instances (out of 3, or 45 in the last row) for which the heuristics found optimal solutions.

enclosing two consecutive reconnaissance points. The second and third define the rectangular regions by extending the first one in each direction by one more cell and two more cells, respectively (unless it is restricted by the boundary of the original terrain). Fig. 3 shows examples of the three types of rectangular regions to be considered in the sub-problems of finding partial routes between two consecutive locations. These three variations of the heuristic are to be denoted as H0, H1, and H2, for the minimumsize rectangular region, one-cell augmented rectangular region, and two-cell augmented rectangular region, respectively. In H0, if the two reconnaissance points in the sub-problem are on the same row or on the same column of the grid, there may be only one path between the two points. In particular, if this path contains cell(s) through which the UCV cannot pass, we cannot generate any partial route between the two points with H0. Therefore, to generate reasonably good partial routes in such cases, a one-cell augmented rectangle is used in H0. Similarly, in such cases, a two-cell augmented rectangle and a three-cell augmented rectangle are used in H1 and H2, respectively.

5. Computational experiments To evaluate performance of the algorithms suggested in this paper, computational experiments were performed on a number

of problem instances. These instances were generated randomly based on information of a real terrain, of size 3 km  3 km, in Korea. A total of 135 instances were generated, three instances for each of all combinations of five levels (4, 6, 8, 10, and 12) for m, the number of reconnaissance points to be visited by the unmanned combat vehicle (UCV), three levels (30 cells  30 cells, 40 cells  40 cells, and 50 cells  50 cells) for the size of the grid map, or the resolution of the map, and three levels (1.8, 2.0 and 2.2) for a parameter, q, used to control the tightness of the risk-level constraint. The following discussion details how this parameter was used. The location of the base is selected randomly from the bottom third part (relative to the vertical axis) and the middle half part (relative to the horizontal axis), i.e., the second and third quarter parts, of the terrain. Locations of the reconnaissance points were set as follows. First, the terrain was divided into m columns of an equal width, and then one reconnaissance point was selected randomly in each column. When the location of a point is too close to the base, say less than 0.5 km, the point is abandoned and another point is generated randomly in the same column. Note that the region within a reasonably short distance from the base can be checked from the base without using UCVs. The sequence of visiting the points was determined arbitrarily, but in a way that the length of the route would not be much longer than the shortestdistance route. Note that in practice the sequence is determined

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Table 2 Results of the tests on 40  40 grid maps. NPa

qb

Solution qualityc H0

H1

H2

Optimal

H0

H1

H2

4

1.8 2.0 2.2

2.91 (2.56), 1 2.37 (2.27), 1 2.18 (1.12), 0

1.81 (1.73), 1 1.20 (2.09), 2 0.56 (0.53), 1

0.88(1.53), 2 0.70(1.22), 2 0.21(0.37), 2

1369.87 1486.49 1595.80

4.98 5.81 6.03

9.05 9.88 10.68

13.90 15.06 16.31

6

1.8 2.0 2.2

3.18 (3.21), 1 1.82 (1.89), 1 1.55 (2.13), 1

2.19 (2.64), 1 0.72 (1.24), 2 0.60 (0.57), 1

0.68(0.97), 1 0.18(0.31), 2 0.22(0.39), 2

2293.87 2400.25 2392.18

3.66 3.98 4.40

7.15 7.78 8.36

12.76 13.82 14.82

8

1.8 2.0 2.2

1.44 (1.40), 1 1.53 (1.35), 1 1.07 (1.23), 1

1.03 (1.19), 1 0.97 (0.84), 2 0.34 (0.38), 1

0.31(0.54), 2 0.46(0.43), 1 0.09(0.15), 2

3211.99 3504.63 3771.48

11.07 11.68 11.96

18.60 19.33 20.36

37.74 38.90 40.42

10

1.8 2.0 2.2

1.27 (0.34), 0 1.01 (0.54), 0 1.03 (0.94), 1

0.84 (0.73), 1 0.68 (0.61), 1 0.57 (0.98), 2

0.62(0.55), 1 0.10(0.17), 2 0.26(0.45), 2

3469.13 3642.39 3867.57

12.84 14.75 17.44

20.48 23.53 26.99

31.63 36.12 40.85

12

1.8 2.0 2.2

1.19 (1.03), 1 1.15 (0.68), 0 1.18 (0.30), 0

1.03 (0.93), 1 0.86 (0.94), 2 0.32 (0.56), 2

0.44(0.38), 1 0.60(0.54), 1 0.19(0.33), 2

3354.56 3464.22 3651.78

3.98 4.26 4.65

7.89 8.19 8.53

12.90 13.31 14.52

1.66 (1.50), 10

0.91 (1.13), 21

0.40 (0.60), 25

2898.41

8.10

13.79

23.54

Overall a,b,c

CPU time (s)

See footnotes in Table 1.

Table 3 Results of the tests on 50  50 grid maps. NPa

qb

Solution qualityc H0

H1

H2

OPT

H0

H1

H2

4

1.8 2.0 2.2

3.00 (2.44), 0 0.72 (1.25), 2 0.21 (0.36), 2

2.73 (2.82), 1 0.72 (1.25), 2 0.00 (0.00), 3

2.73(2.82), 1 0.72(1.25), 2 0.00(0.00), 3

4708.96 4752.66 5364.04

29.01 19.96 21.91

57.27 39.82 42.60

110.85 69.37 81.38

6

1.8 2.0 2.2

2.74 (1.80), 0 2.21 (1.91), 1 1.57 (1.43), 1

0.72 (0.61), 1 0.62 (1.07), 2 0.25 (0.33), 2

0.12(0.18), 2 0.12(0.21), 2 0.00(0.00), 3

6102.46 6352.13 6641.59

17.06 19.15 20.35

30.80 33.90 38.47

59.37 62.47 68.59

8

1.8 2.0 2.2

4.17 (4.81), 0 1.05 (0.50), 0 1.02 (1.28), 0

1.54 (2.66), 2 0.33 (0.57), 2 0.17 (0.23), 2

0.32(0.56), 2 0.08(0.14), 2 0.17(0.23), 2

7406.05 7692.87 8763.84

13.27 14.42 15.59

26.76 28.59 28.30

54.48 56.66 57.69

10

1.8 2.0 2.2

0.33 (0.36), 1 1.06 (1.43), 1 0.61 (0.39), 1

0.18 (0.31), 2 0.76 (1.31), 2 0.34 (0.33), 1

0.06(0.10), 2 0.34(0.60), 2 0.15(0.23), 2

14404.11 15547.81 12862.79

10.70 11.56 15.93

17.56 19.07 29.79

29.59 31.76 56.75

12

1.8 2.0 2.2

1.74 (1.58), 1 1.26 (1.12), 1 1.09 (1.05), 1

1.02 (0.92), 1 0.53 (0.60), 1 0.33 (0.22), 1

0.07(0.12), 2 0.24(0.41), 2 0.20(0.28), 2

16767.22 18287.79 18646.91

19.69 20.57 11.28

31.48 33.28 19.10

62.73 67.57 36.70

2.05 (1.09), 12

1.53 (0.69), 25

1.47 (0.68), 31

9028.19

17.59

32.65

61.67

Overall a,b,c

CPU time (s)

See footnotes in Table 1.

with the consideration of the probability that there are enemy’s military units in a certain region and estimated aggressiveness and threats of those units as well as the (estimated) total length of the resulting route of the UCV. The risk level associated with a cell was generated from DU(0, 5), where DU(a, b) denotes the discrete uniform distribution with range [a, b], and the traverse time through each cell was selected from 1, 2, 3, 4, 5, and the infinity (1), with probabilities 0.1, 0.2, 0.2, 0.2, 0.2, and 0.1, respectively. Note that a cell with an infinite traverse time represents the region that the UCV cannot pass through. Also, note that the traverse time and risk level associated with each cell are given in six different (discrete) levels, not in a continuous scale, as is done in actual military operations. In each problem instance, the upper limit of the cumulative risk level of a complete route, UR, was set by multiplying q, a parameter used to control the tightness of the risk-level constraint, to the risk level of the minimum-risk route, which was obtained without considering traverse time. All the algorithms, the optimal solution algorithm and the three variations of the heuristic, were coded in

C++ and the tests were performed on a personal computer with a Pentium 4 processor running at 2.8 GHz clock speed. Results are given in Tables 1–3, which show the CPU time required for the algorithms and percentage error of the heuristic solutions from optimal solutions. Here, OPT represents the optimal solution algorithm. For all sizes of the grid maps considered in this series of tests, the heuristics gave solutions within very short time. However, OPT failed to give optimal solutions for 40  40 and 50  50 grid-map instances within 10 min, although 30  30 grid-map instances were solved to optimality within 10 min. It required longer CPU time when the risk-level constraint was looser (when q is larger, or UR is larger) since more values for v have to be considered to generate partial routes. The computation time required for OPT is more affected by the grid map size than by the number of reconnaissance points to be visited. This may be because the number of possible paths between a pair of reconnaissance points increases more quickly as the grid map size increases. In both optimal-solution and heuristic algorithms, almost all computation time was spent for Phase 1. It took less than

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3 4

risk level

2

1 2 1

3

5

4 5 ೂ B

Fig. 6. Grid map showing risk levels.

6

Fig. 4. The map of the terrain with locations of the base and reconnaissance points (the numbers denote the sequence of the reconnaissance points to be visited and ‘‘B’’ denotes the base).

time 1 2 3 4 5

We further evaluated performance of the algorithms with a test on a real map. The map is given in Fig. 4, and the 30  30 grid maps with information on the cells of the grid are given in Figs. 5 and 6. Note that 30  30 is the resolution currently used in practice. There are six reconnaissance points to be visited by the UCV, of which the locations are marked in the map of Fig. 4 along with the location of the base. The upper limit on the cumulative risk level of a complete route is 373 in this instance. The optimal solution algorithm and the three variations of the heuristic were applied to this real instance and results (routes) are given in Fig. 7. In this real-map instance, H2 gave the optimal route with the cumulative traverse time of 230. The cumulative traverse times were 239 and 233 in the solutions of H0 and H1, respectively and the percentage errors of the solutions of H0 and H1 from the optimal solution were 3.9% and 1.3%, respectively. The CPU times for H0, H1, H2, and the optimal solution algorithms were 0.61, 1.58, 3.31 and 228.25 s, respectively. That is, all the algorithms solved the problem within 10 min for this 30  30 grid-map instance. To see if larger problems (those with higher resolutions of the map) can be solved within 10 min, we tested the heuristic on

ೂ Fig. 5. Grid map showing traverse time.

0.1 s for Phase 2 in each of 30  30, 40  40, 50  50 grid-map instances. The average percentage error of the heuristic solutions of H0, H1 and H2 was reasonably small. As can be easily expected, H2 worked best (in terms of the solution quality) among the three variations of the heuristic. H0, H1 and H2 found optimal solutions in 13, 21, and 31 instances of 30  30 grid maps (out of 45), respectively. The numbers of instances of 40  40 and 50  50 grid maps for which optimal solutions were found by the heuristic were not much different from those of the instances of 30  30 grid maps. The performance of the heuristic, in terms of the percentage error, deteriorated slightly as the resolution or size of the grid map increased. This may be because the fraction of the terrain considered in the heuristic (out of the entire terrain) is smaller when the size of the map is larger, and hence it may be less likely that an optimal route can be found in the rectangular regions considered in the heuristic. Also, as the risk-level constraint becomes tighter (as q decreases), the percentage errors of the heuristic solutions increased. This may be because in cases where the constraint is tighter, one may have to explore wider ranges of cells to find good feasible routes, not just the cells within the rectangular regions as specified in the heuristic.

route obtained from OPT and H2 route obtained from H1 route obtained from H0 Fig. 7. The complete route obtained by the algorithms.

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Table 4 Results of the tests of H0, H1 and H2 on 60  60 and 75  75 grid maps. Map size

NPa

Solution qualityb

H0

H1

H2

60  60

4 6 8 10 12 Overall

1.55 1.27 1.05 0.56 0.38 0.96

(0.62) (0.13) (0.81) (0.28) (0.14) (0.58)

0.56 0.51 0.51 0.18 0.00 0.35

(0.79) (0.24) (0.72) (0.26) (0.00) (0.44)

50.86 33.28 18.02 13.77 16.53 26.49

105.58 50.05 31.74 25.53 29.78 48.54

161.34 68.32 53.21 47.33 51.28 76.30

70  70

4 6 8 10 12 Overall

0.98 0.58 0.72 0.49 0.48 0.91

(0.41) (0.82) (1.02) (0.09) (0.40) (0.71)

0.63 0.07 0.56 0.04 0.29 0.40

(0.90) (0.10) (0.79) (0.06) (0.13) (0.46)

486.77 254.25 451.34 238.73 222.5 330.73

707.07 392.75 659.96 385.62 370.37 503.15

1006.63 630.33 953.02 615.76 591.99 759.55

H0

a b

Average CPU time (s) H1

Number of reconnaissance points to be visited. Average (and standard deviation in parenthesis) percentage error of the solutions of H0 and H1 from those of H2.

another set of problem instances. The optimal solution algorithm is not included in this test because of excessive computation time required for these problem instances. Note that it took hours, much longer than 10 min, for the optimal solution algorithm to solve instances with 50  50 grid maps. Table 4 gives results of tests on 60  60 and 75  75 grid map instances, which were generated randomly with the method similar to the one used in the generation of smaller-size instances tested above. The table shows the percentage gaps of the solution values of H0 and H1 from those of H2, and CPU times required for an instance. As can be seen from the table, all 60  60 grid-map instances were solved by the three variations of the heuristic within 10 min. However, some of the 75  75 grid-map instances were not solved within the time. That is, H0, H1, and H2 failed to solve 1, 8, and 9 instances, respectively, out of 15 instances of 75  75 grid maps within 10 min. Differences in the solution qualities were not very large (less than 1% on average) among the solutions of the three variations of the heuristic. From the above test results (on problem instances of a real map and practical-size maps as well as large-size maps), it may be argued that the algorithms presented in this paper are viable tools for path planning of a UCV. They gave optimal or reasonably good paths within a reasonable amount of time. However, while the optimal solution algorithm (OPT) may be used for problems of 30  30 grid maps, which are currently used in practice, the heuristic may have to be used for grid maps with higher resolutions because of the computation time allowed for finding paths. 6. Concluding remarks In this paper, we considered a problem of finding reconnaissance routes of an unmanned combat vehicle (UCV) in a grid map, and presented an optimal solution algorithm and a heuristic that are composed of two phases. In the first phase of the algorithms, feasible partial routes between each pair of consecutive reconnaissance points are enumerated. In the second phase, a set of those partial routes is selected by solving a multiple-choice knapsack problem using a dynamic programming algorithm. Results of computational tests showed that the optimal solution algorithm solves problems of a practical size (with a reasonable resolution of the grid map for the terrain) within 10 min. For grid maps of a higher resolution, say 50 m  50 m or 75 m  75 m for each cell, the optimal solution algorithm could not solve the problem within 10 min, while the heuristic gave reasonably good solutions. Note that the military currently uses a resolution of 100 m  100 m for each grid in practice, but wants the resolution to be as high as 50 m  50 m in the future.

This research can be extended in several ways. For example, we could consider cases involving multiple UCVs, since some units (battalions) use two or three UCVs at the same time. In such cases, reconnaissance points need to be assigned to each UCV and the route of each UCV must be determined. In such cases, Also, one may need to consider cases in which information on the enemy’s military units and risk levels associated with the cells may change over time and the information can be updated in real time, and as a result, the location of reconnaissance points and the sequence of the visiting points may change. Also, after visiting of a reconnaissance point, risk levels of the cells near that point need to be changed. In such cases, one may have to develop a method to find a path on a real-time basis. In this paper, we minimize the length of the path while limiting the risk, which are the objective and constraint used in practice. However, in other circumstances, one may need to consider other objectives such as minimize the risk while limiting the length of the path and a bi-criterion objective of minimizing the risk and the length of the path.

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