[American Institute of Aeronautics and Astronautics AIAA Guidance, Navigation, and Control...

Constrained Optimization for UAV Task Assignment

Corey Schumacher* and Phillip Chandler† Air Force Research Laboratory (AFRL/VACA)

Wright-Patterson AFB, OH 45433-7531

Meir Pachter‡ Air Force Institute of Technology (AFIT/ENG)

Wright-Patterson AFB, OH 45433-7765

Lior Pachter § University of California at Berkeley

Berkeley, CA 94720-3840

The optimal timing of air-to-ground tasks is undertaken. Specifically, a scenario where multiple air vehicles are required to prosecute geographically dispersed targets is considered. Multiple tasks are to be successively performed on each target. The targets must be classified, attacked, and verified as destroyed. The optimal performance of these tasks requires cooperation amongst the vehicles such that critical timing constraints are satisfied. In this paper, the optimal task assignment and timing problem is posed as a mixed integer linear program (MILP). The solution of the MILP assigns all tasks to the vehicles in an optimal manner, including staged departure times, for groups of air vehicles. Coupled tasks involving timing and task order constraints are addressed. When the air vehicles have sufficient endurance, the existence of a solution is guaranteed. For the case of homogeneous air vehicles, a single vehicle typically performs both classify and attack consecutively on an individual target. Accordingly, in this work, classification and attack are grouped into a single task, substantially simplifying the solution of the MILP while resulting in little loss of solution flexibility.

Nomenclature

i = index for assignment start nodes

j = index for assignment completion nodes

J = cost function

k = index for tasks

n = number of targets

( )kvijt , = time required to complete a task

( )kjt = time task k is completed on target j

T = maximum endurance of any UAV

Ti,j = flight time between nodes

Tv = endurance of UAV v

v = index for vehicles

w = number of UAVs

* Research Aerospace Engineer, Control Sciences Division, Senior Member, AIAA † Aerospace Engineer, Control Sciences Division, Member, AIAA ‡ Professor, Electrical Engineering Department, Associate Fellow, AIAA § Professor, Department of Mathematics

AIAA Guidance, Navigation, and Control Conference and Exhibit16 - 19 August 2004, Providence, Rhode Island

AIAA 2004-5352

This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.

2 American Institute of Aeronautics and Astronautics

( )kvjix ,

, = binary task assignment variable

( )vwnix 1, ++ = binary assignment variable for null tasks

I. Introduction The optimization of air-to-ground operations is undertaken. A scenario where multiple Air Vehicles (AVs) are required to service geographically dispersed targets is considered. Multiple tasks must be successively performed on each target, viz., the targets must be classified, attacked, and the damage inflicted on the targets must be assessed. Multi-role AVs are considered s.t. each AV can perform all of the tasks. A case in point: Autonomous wide area search munitions (WASM) are small Unmanned Air Vehicles (UAV’s), each with a turbojet engine and sufficient fuel to fly for a short period of time. They are deployed in groups from aircraft flying at higher altitudes. They are typically deployed in groups of four, although larger formations are certainly possible. They are individually capable of autonomously searching for, recognizing, and attacking targets. The ability to communicate target information to one another, and consequently to cooperate, will greatly improve the effectiveness of future WASM. Thus, the problem is posed of planning the performance of the munitions’ tasks such that critical timing constraints are satisfied.

In [1-3], a time-phased network optimization model was used to perform task allocation for a team of WASM. The model is run simultaneously on all munitions at discrete points in time, and assigns each vehicle one or more tasks each time it is run. The network optimization model is run iteratively so that all of the known targets will be prosecuted by the resulting allocation. The model is solved each time new information is brought into the system, typically because a new target has been discovered or an already-known target’s status has been changed, thus achieving feedback action. Classification, attack, and battle damage assessment tasks can all be assigned to different vehicles when a target is found, resulting in the target being more quickly serviced. A single vehicle can also be given multiple task assignments to be performed in succession, if that is more efficient than having multiple vehicles perform the tasks individually. In [2], variable path lengths are included to guarantee that feasible trajectories will be calculated for all tasks. This method is computationally efficient and scales well, however the iterative procedure is heuristic and suboptimal.

This paper proposes an optimal formulation for solving the coupled multiple -assignment problem. Continuous timing variables are introduced. Formulating the optimization problem as a Mixed Integer Linear Program (MILP) allows the optimal solution to be found while satisfying all timing constraints. A MILP formulation for a similar task assignment problem was first presented in [4]. The MILP formulation given in [4] included the requirement for three tasks being completed for each target. Due to the complexity of the resulting MILP, only very limited problem sizes could be solved quickly. In this paper, classify and attack are grouped into a single task, allowing the MILP to be reformulated with only two tasks required per target. The resulting reduction in problem complexity allows real-time solution for problems of practical size. The assumption that classification and attack will be performed by the same vehicle is reasonable for homogeneous vehicles, as the same vehicle will almost always be assigned to both classify and attack the same target, even if this is not required. This behavior can be seen in [1-4]. In this paper, time is treated as a continuous variable and a rigorous optimal task assignment algorithm is developed. This requires the solution of a mixed integer linear program [5,6].

An alternative MILP formulation for UAV task assignment with coupled tasks and timing constraints, and where time is a continuous variable, is given in [5]. However, significant differences exist between the two formulations. The MILP formulation in [5] requires an enumeration of the shortest paths between all waypoints, and enumeration of all feasible permutations of these path combinations, subject to the timing constraints, which our method does not require. We rely on the equality and inequality constraints to enforce feasibility. A related MILP formulation is presented in [7]. The formulation in [7], however, requires discretization of time, substantially increasing the size of the resulting combinatorial optimization problem. The formulation presented here can be solved optimally for some realistic problem sizes, without requiring approximate solution. Loiter is also handled differently, with [5] requiring loiter, if it is needed, at target waypoints, while the method presented here allows staged departure times, before vehicles begin their tour of targets and tasks. Also, our MILP formulation is flexible enough to allow the consideration of many interesting cost functions, e.g., mission completion in minimum time, shortest total paths lengths traveled by the vehicles, or maximization of the number of air vehicles which survive the mission.

Tabu search can be used to solve difficult combinatorial optimization problems, e.g. the vehicle routing problem with (fixed) time windows [8-10]. The method presented here can also accommodate fixed time windows,


although arbitrary fixed time windows can make the problem, independent of solution methodology, infeasible. Moreover, the MILP presented here can also accommodate dynamic constraints on task performance, as is the case in scheduling problems. In this work, without time windows, feasibility is guaranteed, as long as the number of air vehicles exceeds the number of targets, even with three or more tasks per target.

II. Scenario Assume there is a number of wide area search munitions or air vehicles searching an area for unknown targets. Typically, four munitions will be deployed as a team, although multiple pods of four could be linked together to form a larger team. Vehicles travel in a pre-specified zamboni search pattern with a sensor that is capable of detecting and identifying potential targets. As the vehicles search, they come across enemy targets. When a potential target is discovered, a preliminary classification of the target is made. It is necessary to have a second vehicle examine the target and confirm the classification, to be confident that it is a valid target, and the target must be attacked and then verified as having been destroyed. We will group the reclassification and attack tasks into a single task, and the attack would be aborted if the reclassification determines that the potential target is not, in fact, a real target. This situation is addressed in a suboptimal manner in [1-3]. Here, we present an optimal mixed-integer linear programming solution for the n-target w-vehicle assignment problem that is applicable to realistic proble m sizes. Suppose we have n geographically dispersed targets with known position and w air vehicles (AV). We assume 1+≥ nw . We then have n+w+1 nodes: n target nodes, w source (or vehicle) nodes, and one sink node. An example State Transition Diagram for n=2 targets and w=3 vehicles is shown in Figure 1. Nodes 1,…,n are located at the n target positions. Nodes n+1,…n+w are located at the vehicle initial positions. Node n+w+1 is the “sink”. An air vehicle with no future target assignments is relegated to the sink, and will search for unknown targets. A vehicle located at the sink cannot be reassigned during the present assignment computation. The flight time of AV v

from node i to node j is ( ) 0, ≥kvijt . The indices i=1,…,n+w, j=1,…n, and v = 1,…,w . The index k designates the

task to be performed at node j. The time to travel from node i to node j depends on the particular AV’s airspeed and the assigned task k . Two tasks must be performed on each target: k=1 – Classification and Attack, and k=2 – Target Damage Assessment (Verification). Furthermore, once an AV attacks a target, the vehicle is lost and can no longer perform additional tasks. This is certainly the case for powered munitions, but if the AV is a reusable aircraft, one has to account for the depletion of its store of ammunition following each attack. The tasks must be performed on each target in the order listed. This results in critical timing constraints, which set this problem apart from the classical Vehicle Routing Problem (VRP) [11]. In the latter, rigid time windows for the arrival of the vehicles can be specified, however, the coupling brought about by the need to schedule the various tasks is absent. Evidently, our problem features some aspects of job shop scheduling [12].

In the operational scenario considered, the number of problem parameters ( )kvijt , is 2wn+2n(n-1)w = 2n2w.

When Euclidean distances are used, the dimension of the parameter space is reduced to 0.5n(n-1)+wn= 0.5n(n+2w-1). Finally, the endurance of AV v is .,...,1, wvTv = Figure 1 illustrates a scenario where 2 stationary ground targets are engaged by three AVs. The potential targets’ positions are known at the beginning of the optimization, but not the classification.

III. MILP Model

The MILP model uses a discrete approximation of the real world based on nodes that represent discrete start and end positions for segments of a UAVs path. Nodes representing target positions range from 1…n and nodes for UAV positions range from 1+n…w+n. There is also an additional logical node for the sink n+w+1. The sink node is used when a UAV is not assigned to attack a target; it goes to the sink when it is done with all of its tasks, or when it is not assigned another task. In practice, when a UAV enters the sink it is then used for searching the battlespace. The MILP model requires the minimum costs or times for a UAV to fly from one node to another node. We assume that any flight time larger than these minimum times is continuously achievable. These known flight times are constants

represented by ( )kvjit,

, , the time it takes UAV v to fly from node i to node j to perform task k . The flight times are

positive real numbers, ( ) 0,, ≥kv

jit .

A. Decision Variables


The binary decision variable ( ) 1,, =kv

jix if AV v is assigned to fly from node i to node j and perform task k at node j,

and 0 otherwise; i = 1,…n+w, j = 1,…,n, ,ji ≠ v = 1,…,w , and k = 1,2. Thus far, we have 2n2w binary decision variables. A vehicle does not have the option of starting and stopping a task at the same node, as no such achievable tasks exist in this scenario.

We also have the following additional binary decision variables. The decision variable ( ) 11, =++v

wnix if AV

v is assigned to fly from node i to the sink n+w+1, and is 0 otherwise; v=1,…,w and i = 1,…,n+w. This adds (n+1)w binary decision variables. Entering the sink can also be thought of as being assigned to the search task. Continuous decision variables:

The time of performance of task k on target j is ( ) 0>kjt ; k = 1,2 and j = 1,…,n. Thus, we have 2n

continuous decision variables. We also have w additional continous decision variables: the time AV v leaves node j = n + v is tv; v = 1,…,w . In total we have (2n2+n+1)w binary decision variables and 2n+w continuous non-negative decision variables.

B. Cost Functions A variety of cost functions are possible, depending on the exact application, and other variations in the problem formulation. Possible cost functions include:

1. Minimize the total flight time of the AVs

( ) ( )∑ ∑ ∑ ∑== =

+

= =

3

1 1 1 1

,,

,,

k

w

v

wn

i

n

j

kvji

kvji xtJ (1)

2. Alternatively, minimize the total engagement time. The target j is visited for the last time at time ( )2jt . Let

tf be the time at which all targets have been through Verification. Introduce an additional continuous

decision variable 1+ℜ∈ft . The cost function is then ftJ = and we minimize J subject to the constraints

( ) njtt fj ,,1,3 K=≤ (2)

We also add a small weight to the time of performance of each individual task, to encourage each individual task to be completed as quickly as possible. Then

( ) ( ) 3,2,1,,,1, ==+= knjtctJ kj

kjf K , (3)

where ( ) 0>kjc is a small weight on the completion time of each individual task. To weight the time of

performance of individual tasks more heavily, one could use ( ) ( ) .3,2,1,,,1, === knjtcJ kj

kj K

3. Other cost functions could also be formulated. For example, the problem could be formu lated to maximize a benefit function, similar to that used in [1-3]. This would allow direct incorporation of competing search tasks, if all target tasks were not required to be completed.

C. Non-Timing Constraints

Inclusion of all of the required constraints is critical to achieving the desired vehicle behavior.

1. Mission completion requires that each task is performed on each target exactly one time:

( )nj

kx

w

v

wn

jii

kvji ,...,1

2,1,1

1 ,1

,, =

=∑ =∑=

+

≠=, (4)

This yields 2n constraints.


2. Only two tasks can be performed on each target, and the first task (classify and attack) results in the loss of the attacking agent. Therefore, it should not be possible for any particular vehicle to be assigned to visit the same target twice. An AV v can visit target j at most once:

( )njwv

xk

wn

i

kvji ,...,1

,...,1,1

2

1 1

,, =

=∑ ≤∑=

+

= (5)

In addition, each AV v can only enter the sink once:

( ) wvxwn

i

vwni ,...,1,1

11, =≤∑

+

=++ (6)

This yields (n+1)w constraints. 3. A munition is perishable. An AV v can be assigned to attack at most one target. Thus,

( ) .,...,1,11 1

1,, wvx

n

j

wn

i

vji =∀∑ ≤∑

=

+

= (7)

This yields w constraint equations. 4. Due to Constraint 2, a vehicle can only enter a particular target node one time. Also, task 1 (classify and attack) is a “fatal” task, resulting in the loss of the vehicle. Thus, if AV v is assigned to fly to target (node) j to perform task k=1, then the AV v cannot, at any time, leave target (node) j. Thus

( ) ( ) ( )∑ ∑−≤+∑=

+

=++

≠=

2

1 1

1,,1,.

,1

,, ,1

k

wn

i

vji

vwnj

n

jii

kvij xxx (8)

where j=1,…,n; v=1,…,w. This yields nw constraints. 5. Due to continuity requirements, a vehicle cannot leave a node which it has not entered. Also, it cannot leave a node which it attacked. Therefore, an AV v leaves a target (node) j if and only if it entered that target (node) for the purpose of performing task 2 (verification):

∑ ∑=+∑=

+

≠=++

≠=

2

1 ,1

21,

,1, ,

k

wn

jii

vij

vwnj

n

jii

vkij xxx (9)

where j=1,…,n; v=1,…,w. This yields nw constraints. 6. All AVs leave the source nodes. An AV leaves the source node even if this entails a direct assignment to the sink.

( ) ( ) .,...,1,12

11,

1

,, wvxx

k

vwnvn

n

j

kvjvn =∀∑ =+∑

=+++

=+ (10)

This yields w constraints. Thus, there are a total of n(w+2)+w equality non-timing constraints, and 2w(n+1) inequality non-timing constraints.

D. Timing Constraints

Nonlinear equations which enforce the timing constraints are easily derived, and are given in [4]. We are however interested in an alternative formulation which uses linear inequalities so that a MILP formulation is achieved. Thus, let

{ } ,max 1wvvv TT =≡ (11)

where Tv , we recall, is the available endurance of vehicle v.

Then the linear timing constraints become:


( ) ( ) ( ) ( ) ( ) wTxxtttwn

ill

vil

kvji

kvjii

kj

∑−−++≤+

≠= ,1

2,,

,,

,,

2 2. (12)

( ) ( ) ( ) ( ) ( ) wTxxtttwn

ill

vil

kvji

kvjii

kj

∑−−−+≥+

≠= ,1

2,,

,,

,,

2 2. (13)

for i=1,…,n; j=1,…,n; ji ≠ ;v=1,…w; k=1,2 . This yields 4n(n-1)w constraints. These constraints affect the timing of tasks performed by vehicles coming from a target node. Also,

( ) ( ) ( ) wTxttt kvjvn

kvjvnv

kj

−++≤ ++

,,

,, 1. (14)

( ) ( ) ( ) wTxttt kvjvn

kvjvnv

kj

−−+≥ ++

,,

,, 1. (15)

for all j=1,…,n; k = 1,2; v = 1,…,w. This yields 4nw constraints. These constraints affect the timing of the first task performed by each vehicle, when it originates from its source node.

These timing constraints operate in pairs. They are loose inequalities, which do not come into play for

assignments ( )kvjix ,

, which do not occur, but effectively become hard equality constraints for assignments which do

occur, when the lower and upper bounds on the constraints are equal. Thus the time that a task k is performed on target j by AV v will be equal to the time that the preceding task was performed by AV v at node i, plus the time it will take AV v to fly from node i to node j. A similar constraint applies if AV v left its source node n+v to fly to node j. Furthermore,

( ) njtt jj ,...,1,21 =∀≤+ α , and (16)

( ) njtt fj ,...,1,2 =∀< , (17)

where α can be chosen to represent any desired minimum delay between tasks k=1 and k=2. This yields 2n constraints. The timing requirements thus add 4n2w+2n linear inequality constraints. E. Extensions Additional constraints can be included.

9. A vehicle’s assigned path cannot be longer than its remaining endurance Tv:

( ) ( ) ,2

1 1 ,1

,,

,, v

k

wn

i

n

ijj

kvji

kvji Txt ≤∑ ∑ ∑

=

+

= ≠= wv ,,1 K= (18)

This yields w constraints. 10. It is fairly easy to specify additional rigid time window constraints akin to the VRP, e.g., for time critical

targets one could demand that the attack on target j take place after time ( )2jt , and not before time ( )2

jt , i.e.

( ) ( ) ( )222jjj ttt ≤≤ , nj ,,1 K=∀ (19)

11. Numerous other constraints can also be included, such as: specific vehicles performing certain tasks, minimum time delays between tasks, simultaneous completion of attack tasks, and requiring the vehicle that classifies a target to also attack it. With some constraints, such as vehicle endurance (Constraint 9), the existence of a solution is no longer guaranteed.

12. Hetergeneous vehicles: For some applications, a set of heterogeneous vehicles would be used, with different capabilities. Some might be sensor platforms with no attack capability. Or some vehicles might simply have used all their ordinance, or not be carrying the proper ordinance to attack certain targets. In

such cases, we add the constraint ( ) 0,, =kv

jix for any combination where vehicle v cannot perform task k on

target j. 13. Partially prosecuted targets: If this algorithm was used for task assignment by a group of UAV’s, additional targets and tasks could be added to the overall task list while some previously-known targets were already partly prosecuted. In this case, fewer tasks would be required for some targets, when the assignments were recalculated. For already completed tasks, we modify Constraint 1 such that


( )nj

kx

w

v

wn

jii

kvji ,...,1

2,1,0

1 ,1

,, =

=∑ =∑=

+

≠=, (20)

for any target j and task k that have already been completed.

IV. Examples A. One Target and Two AVs To illustrate the method, we shall first consider the case of one target and two AVs, i.e. n=1 and w=2, as the problem is small and may be described in detail. In this case, we have 8 binary decision variables and 4 continuous decision variables. Minimizing the time the final task occurs will add an additional continuous decision variable tf, for a total of 13 decision variables. In this single-target case, we could exclude the additional variable and simply

minimize 21t , but the additional variable will be included to demonstrate the additional variable that would be

required for 2≥n . An example State Transition Diagram is given in Figure 1, for the more complex n=2, w=3 case. In the simpler case discussed here, there are only four total nodes: 1 target node, 2 source nodes, and 1 sink node. There are 8 binary decision variables:

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

=

=

,,,,,...,

,,,,,...,

24,3

14,2

24,1

14,185

2,21,3

1,21,3

2,11,2

1,11,241

xxxxxx

xxxxxx (21)

There are 5 continuous decision variables:

( ) ( ) ( ) ,,,,,,..., 21

1121139

= ftttttxx (22)

We wish to minimize

( )121113 1.0 xxxJ ++= (23) subject to the following constraints: From Constraint 1:

11

42

31=+=+

xxxx

(24)

From Constraint 2:

1111

86

75

43

21

≤+≤+≤+≤+

xxxxxxxx

(25)

From Constraint 3:

11

3

1≤≤

xx

(26)

In this simple case with n=1, Constraint 3 does not provide additional information and could be removed from the problem. However, for 2≥n , Constraint 3 is required. And Constraint 4 gives:

11

63

51≤+≤+

xxxx

(27)


From Constraint 5:

11

36

25=−=−

xxxx

(28)

From Constraint 6:

11

843

721=++=++

xxxxxx

(29)

Thus, we so far have six equality and eight inequality constraints. The timing constraints, still need to be addressed. With only 1 target node, the Constraints associated with Eq. (12-13) are not meaningful and are not part of the formulation. So we are left with the following timing constraints, from Eq. (14-15):

( ) ( )( ) ( )wTxtxx

wTxtxx

11,11,2911

11,11,2911

1

1

−−+≥

−++≤

( ) ( )( ) ( )wTxtxx

wTxtxx

31,2

1,31011

31,2

1,31011

1

1

−−+≥

−++≤ (30)

and

( ) ( )( ) ( )wTxtxx

wTxtxx

22,1

1,2912

22,1

1,2912

1

1

−−+≥

−++≤

( ) ( )( ) ( )wTxtxx

wTxtxx

42,2

1,31012

42,2

1,31012

1

1

−−+≥

−++≤ (31)

Also, from Eq. (16), we have:

α−≤ 1211 xx (32)

where α >0 is a small positive constant. We will set α=0.1. This enforces a small delay between each task being performed on a target. Finally, from Eq. (17):

1312 xx ≤ . (33) Thus the full set of constraints contains 6 equality constraints and 18 inequality constraints, for 24 total constraints. A few of them are redundant for this case, but would not be for a more complex problem with 2≥n . Let us make the simplifying assumption that the time to travel from node i to node j to perform task k is independent

of which task is required, and which vehicle is performing the task. Then ( )kv

jit,

, simply becomes ti,,j. For this

example, let

07.7

83.5

1,3

1,2

=

=

t

t

We will set T =Tv= 100 as the endurance of all of the AVs, so that endurance is not a constraint. Then the optimal assignment is:

xi = 1, i=1,4, 6


xi =0, i=2,3,5,7,8 xi = 0, i=9,10 x11=5.83 x12 = 7.07 x13 = 7.07

This corresponds with both vehicles immediately leaving their source nodes (x9-x10=0), and Vehicle 1 performing classify and attack on the target at t=5.83, with vehicle two performing verification at T=7.07. Vehicle 2 then proceeds to the sink (returns to search). Suppose that a longer period of time is required for a vehicle to perform task 1, e.g. let α=2.0. Then we achieve a similar solution, but with x10=0.76, and x12=7.83. In this case, Vehicle 2 is required to delay 0.76 before proceeding to the target to complete task 2. B. Larger Examples The size of the MILP expands rapidly as problem size increases. However, many practically-sized problems are amendable to optimal solution with this mixed-integer linear program formulation. For n targets, w vehicles, and 2 tasks per target, the problem size scales as follows: There are 2n2w+n(w+2)+2w+1 decision variables. Of these, 2n+w+1 are continuous timing variables, and the rest are binary decision variables. The number of constraints likewise grows rapidly. There are 4n2w+3nw+4n+3w constraints. Of these, n(w+2)+w are equality constraints. The rest are inequality constraints, including 2w(n+1) inequality non-timing constraints, and 4n2w+2n inequality timing constraints. For n=2, w=3, there are 33 binary decision variables, 8 continuous decision variables, 13 linear equality constraints, and 76 linear inequality constraints. For n=2, w=4, there are 44 binary decision variables, 9 continuous decision variables, 16 linear equality constraints, and 100 linear inequality constraints. For n=3, w=4, there are 88 binary decision variables, 11 continuous decision variables, 22 linear equality constraints, and 194 linear inequality constraints. The number of constraints and variables is linear in the number of vehicles, but quadratic in the number of targets. For n=3, w=8, there are 176 binary decision variables, 15 continuous decision variables, 38 linear equality constraints, and 382 linear inequality constraints. For n=4, w=5, there are 185 binary decision variables, 14 continuous decision variables, 33 linear equality constraints, and 398 linear inequality constraints. The initial condition of a problem is defined by the relative distances, or flight times, between the nodes. For the following examples, let Ti,j be defined as:

=

10.7703 12.2066 6.0828 6.7082 13.0000 14.4222 8.2462 8.6023 8.0623 8.9443 4.4721 3.1623 10.8167 12.7279 5.8310 8.2462 9.2195 10.2956 5.0990 4.4721 3.6056 3.1623 8.6023 10.0000

10.4403 10.0000 8.4853 3.1623 5.0990 8.0623 3.6056 9.2195

0 3.0000 5.0000 7.2801 3.0000 0 7.2111 7.0711 5.0000 7.2111 0 5.8310 7.2801 7.0711 5.8310 0

, jiT (34)

where the start node i is indexed down the rows, and the end node j is indexed over the columns. So the time for a vehicle to fly from node 4 to node 3, T4,3 = 3.0, and so on. The diagonal elements, i=j, correspond with a vehicle starting and ending a task at the same node, which is not allowed. The sink position only exists conceptually, so the


time to reach it would not be meaningful. A vehicle that is assigned to the sink continues to search for potential targets along a predefined search path. The given Tij has 12 ro ws and four columns, sufficient for a MILP with up to n=4 targets and up to w=8 vehicles. We will also require a delay of α=0.5 between the two tasks being performed on a specific target. For n=2, v=3, the optimal solution is for: Vehicle 1 to classify/attack Target 1 at t=7.07, for Vehicle 3 to classify/attack Target 2 at t=3.61, and for Vehicle 2 to verify Target 2 at t=5.0, and verify Target 1 at t=10.83. This solution has an optimal minimum cost of J=13.48. All tasks are completed by t=10.38. This optimization problem required 0.05 seconds to solve using 700MHz. All computation times given in this paper are for typical examples of the stated problem size using unoptimized MatLab code and GnuTools on a 700 MHz processor. For n=2, v=4, the optimal solution is for: Vehicle 4 to classify/attack Target 1 at t=3.16, Vehicle 3 to classify/attack Target 2 at t=3.61, Vehicle 1 to verify Target 1 at t=7.07, and Vehicle 2 to verify Target 2 at t=5.0 . This solution has an optimal minimum cost of J=9.67. All tasks are completed by t=7.1 . In this case, the availability of an additional vehicle which started out closer to Target 1 allowed the tasks to be completed much more quickly. This optimization problem took 5.17 seconds to solve. The n=2, v=5 solution is identical, with the addition that vehicle 5 is assigned directly to the sink. All of the target tasks are completed by the same Vehicles, at the same times, as in the n=2, v=4 case. The n=2, v=5 case is solved in 0.13 seconds. The MILP can be quickly solved for any reasonable number of vehicles, if only n=2 targets are included. The MILP requires 1.1 seconds to solve with 8 vehicles and 2 targets. If the vehicles are locating unknown targets via search techniques, and assigning tasks as targets are found, they would commonly only have to assign tasks for one or two targets at a time. If using the MILP to assign agents to a set of known targets beforehand, solution times would be longer, but more time would typically be available for computation. With n=3, v=4, the problem is becoming more complex. A node diagram of the optimal solution, with the given initial conditions, is shown in Figure 2. Transition arrows are shown for each binary assignment variable with a non-zero value. The arrows are color-coded for each UAV. UAV 1 is green, UAV 2 is red, UAV 3 is blue, and UAV 4 is black. Completion times for each task are also given. The optimal solution is for: Vehicle 2 to classify/attack Target 2 at t=3.61, Vehicle 1 to classify/attack Target 3 at t=3.00, Vehicle 4 to delay 0.34, then verify Target 3 at t=3.50, and then to classify/attack Target 1 at t=10.57, and for Vehicle 3 to verify Target 2 at t=8.49, and then verify Target 1 at t=14.32. This solution has an optimal minimum cost of J=18.66. All tasks are completed by t=14.32. In this case, Vehicle 4 exhibits a non-zero path length extension (delay) before beginning its set of tasks. If it had immediately proceeded to its first task without delay, it would have performed verification on Target 3 before the required time delay between tasks 1 and 2 had occurred. For this case, the optimization problem took 5.99 seconds to solve. Larger problems with 3 targets and w vehicles can also be solved in reasonable periods of time. An n=3, w=8 problem can be solved in 19.25 seconds. The problem complexity expands greatly for 4≥n targets. The n=4, v=5 example was solved in 18.6 minutes. A node diagram of the optimal solution is given in Figure 3. This solution illustrates interesting behavior, with Vehicle 5 (orange) performing three verify tasks, and Vehicle 4 performing a verify followed by a classify/attack task. Larger problems will be more computationally difficult to solve. However, the method can be used for many practical problem sizes. Much faster processors than 700 MHz could be available, speeding up solutions somewhat. Typical problems for many applications will be in the n=2, v=4 size range. Wide Area Search Munitions, e.g., are commonly dropped in pods of four, and four munitions can only fully prosecute a maximum of three targets, or four targets with the last target attacked but not verified destroyed. Four or fewer search vehicles finding many targets simultaneously is also unlikely. Many other UAV applications may also lend themselves to quickly-solveable MILP problem sizes. Efficient suboptimal solutions can be found for much larger problems by decomposing a larger problem into multiple smaller problems. A large problem with n=4, w=8 can be decomposed into smaller subproblems and solved very quickly. C. Three Tasks per Target

Reference [4] discusses the use of a similar MILP formulation for the case where the classify and attack tasks must be separated, and thus three tasks are required for each target. This results in more decision variables, and more constraints, for a given number of vehicles and targets. A complete discussion of the constraints required for the 3-task MILP formulation can be found in [4].


Here, we wish only to compare the MILP sizes and computational requirements for the two formulations. Inclusion of a third task requires more variables, and more constraints, not simply due to the existence of more tasks required to be assigned, but more constraints are also required to prevent looping and other not-physically-achievable assignments from being possible in the MILP solution. Table 1 summarizes the computational advantages achieved by reformulating the MILP for two tasks per target, instead of three. Of course, some applications could require three (or more!) separate tasks per target, and not allow this simplification. As can be seen from the table, required computation time scales up much more rapidly with three tasks per target, than it does with only two. Computation

times consistent with practical real-time implementations can be achieved for n=3 with two tasks per target, with solution times not being on the order of minutes until n=4. With three tasks per target, however, practical real-time solutions are only achievable for n=2, with the smallest possible 3-target case, n=3, w=4, requiring 27 minutes to solve. We have not attempted to solve the n=4, w=5 case for three tasks per target, as even the n=3, w=5 case requires excessive computation.

V. Conclusions This paper presents a Mixed Integer Linear Program (MILP) formulation for finding the optimal solution to a difficult multiple-task assignment/scheduling problem where the tasks are coupled by timing and task precedence constraints. Preserving linearity and casting this scheduling aspects - dominated combinatorial optimization problem into a MILP formulation, without having recourse to direct enumeration, is a major contribution of this paper. This formulation allows staged vehicle departure times to guarantee that timing constraints are satisfied, and directly incorporates the varying task completion times into the optimization. Optimally staged departures bring about a guarantee of feasibility, without a need to loiter. Moreover, the formulation is flexible enough to allow for various alternative performance functional, e.g., minimum time to mission completion, minimum path length traveled by the air vehicles, or minimum number of air vehicles required to accomplish the mission. This is a rigorous formulation, which allows a true optimal solution for a very challenging assignment and scheduling problem. Solution results were presented for practical problem sizes, with real-time implementable optimal solutions obtainable for many realistic problems.

References 1. Schumacher, C, Chandler, P. R, Rasmussen, S. J., “Task Allocation for Wide Area Search Munitions Via

Iterative Network Flow”, Proceedings of the 2002 AIAA Guidance, Navigation, and Control Conference. 2. Schumacher, C, Chandler, P. R, Rasmussen, S. J., “Task Allocation for Wide Area Search Munitions with

Variable Path Length”, Proceedings of the 2003 American Control Conference 3. Schumacher, C, Chandler, P. R, Rasmussen, S. J., “Path Elongation for Task Assignment”, Proceedings of

the 2003 AIAA Guidance, Navigation, and Control Conference. 4. Schumacher C., Pachter, M., Pachter, Chandler P., L.S., “UAV Task Assignment with Timing

Constraints,” Proceedings of the 2003 AIAA Guidance, Navigation, and Control Conference. 5. Alighanbardi, M., Kuwata, Y., How, J., “Coordination and Control of Multiple UAVs with Timing

Constraints and Loitering,” Proceedings of the 2003 American Control Conference. 6. Nemhauser, G. L., and Wolsey, L.A., “Integer and Combinatorial Optimization”, Wiley, 1988. 7. Richards, A., Bellingham, J., Tillerson, M., How, J.P., “Co -ordination and Control of Multiple UAVs,”

Proceedings of the 2002 AIAA Guidance, Navigation, and Control Conference. 8. Glover, F., Laguna, M., “Tabu Search,” Kluwer Academic Publishers, 1997.

Table 1. Comparison of MILP Size, Solution Time vs # Tasks per Target

# Vehicles w

# Targets n

# Tasks per Target

Number of Variables

Number of Constraints

Computation Time

2 41 89 0.05 seconds 3

2 3 61 183 1.06 seconds

2 65 143 0.50 seconds 5

2 3 97 297 8.4 seconds

2 99 216 6.07 seconds 4

3 3 150 498 27 minutes

2 122 267 20.9 seconds 5

3 3 185 618 193 minutes

2 199 431 18.6 minutes 5

4 3 303 1059 ???


9. O’Rourke, K.P., Bailey, T.G., Hill, R., Carlton, W.B., “Dynamic Routing of Unmanned Aerial Vehicles Using Reactive Tabu Search,” Military Operations Research Journal, Vol. 6, 2000.

10. Gendreau, M., Hertz, A. Laporte, G., “A Tabu Search Heuristic for the Vehicle Routing Problem,” Management Science, Vol. 40, pp. 1276-1289, 1994.

11. Toth, Paob, and Daniele Vipo, “The Vehicle Routing Problem,” SIAM 2002. 12. Pinedo, M., “Scheduling,” Prentice Hall, 2002. Figure 1. State Transition Diagram for n=2 Targets, w=3 Vehicles

4

2

1

3

5

( )16,3x

1v

( )kvkv xt ,1,2

,1,2 ,

( )kk xt ,11,3

,11,3 ,

( )kk xt ,21,4

,21,4 ,

6 Sink

( )kk xt ,12,3

,12,3 ,

( )36,5x

( )26,4x

( )vx 6,2

( )vx 6,1

( )kvkv xt ,2,1

,2,1 ,( )kk xt ,2

2,4,22,4 ,

2v

3v

Vehicles

Targets

( )kk xt ,31,5

,31,5 ,

( )kk xt ,32,5

,32,5 ,

i = 1,2,3,4,5 j = 1,2 v = 1,2,3 k = 1,2 1t

2t


Figure 2. Task Assignments for n=3 Targets, w=4 Vehicles

5

2

1

4

6

1v

( ) 12,31,2 =x

8 Sink

( ) 11,13,4 =x ( ) 13

6,1 =x

( ) 11,22,5 =x2v

3v

Vehicles

Targets

( ) 12,32,6 =x

i = 1,…,8 j = 1,2,3 v = 1,2,3,4 k = 1,2

1t

2t

3 7 4v 3t

( ) 12,43,7 =x

( ) 11,41,3 =x

32.14,23.10 21

11 == tt

48.8,60.3 22

12 == tt

50.3,00.3 23

13 == tt

Vehicle 1 Vehicle 2 Vehicle 3 Vehicle 4


Figure 3. Task Assignments for n=4 Targets, w=5 Vehicles

6

2

1

5

7

1v

0 Sink

( ) 11,12,5 =x

( ) 156,3 =x

( ) 11,21,6 =x

2v

3v

Vehicles

Targets

( ) 11,34,7 =x

i = 1,…,10 j = 1,2,3 ,4 v = 1,2,3,4 ,5 k = 1,2

1t

2t

3

8

4v

3t( ) 12,4

1,8 =x

( ) 11,43,1 =x

4 9

5v

( ) 12,52,9 =x

( ) 12,54,2 =x

( ) 12,53,4 =x

4t

47.4,16.3 21

11 == tt

83.5,61.3 22

12 == tt

83.13,54.11 23

13 == tt

83.10,61.3 24

14 == tt

Vehicle 1 Vehicle 2 Vehicle 3 Vehicle 4 Vehicle 5

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