Efficient fleet size optimization for task allocation of multiple unmanned vehicles

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ScienceDirect IFAC PapersOnLine 52-21 (2019) 297–301

Efficient fleet size optimization for task Efficient Efficient fleet fleet size size optimization optimization for for task task allocation of unmanned vehicles Efficient fleet size optimization task allocation of multiple multiple unmanned for vehicles allocation of multiple unmanned vehicles allocationSukmin of multiple Yoon and unmanned Jinwhan Kim ∗∗ vehicles

Sukmin Yoon and Jinwhan Kim ∗ Sukmin Sukmin Yoon Yoon and and Jinwhan Jinwhan Kim Kim ∗∗ Sukmin YoonEngineering, and Jinwhan KimDaejeon, Korea Department of Mechanical KAIST, Department of KAIST, Department of Mechanical Mechanical Engineering, Engineering, KAIST, Daejeon, Daejeon, Korea Korea (e-mail: [email protected], [email protected]). Department of Mechanical Engineering, KAIST, Daejeon, Korea (e-mail: [email protected], [email protected]). ∗ (e-mail: [email protected]). Department of Mechanical Engineering, KAIST, Daejeon, Korea (e-mail: [email protected], [email protected], [email protected]). (e-mail: [email protected], [email protected]). Abstract: An An efficient efficient task task allocation allocation algorithm algorithm is is crucial crucial for for multi-vehicle multi-vehicle collaboration. collaboration. To To Abstract: Abstract: An efficient task allocation algorithm is crucial for multi-vehicle collaboration. To achieve the high computational efficiency for task allocation of multiple vehicles, this study Abstract: An efficient task allocation algorithm is crucial for multi-vehicle collaboration. To achieve the high computational efficiency for task allocation of multiple vehicles, this study achieve high computational efficiency for task multiple vehicles, this Abstract: efficient task allocation algorithm crucial forof collaboration. To proposesthe anAn efficient fleet sizing algorithm based onallocation the concept concept of virtual virtual tasks and and new achieve the high computational efficiency for taskison allocation ofmulti-vehicle multiple vehicles, this aastudy study proposes an efficient fleet sizing algorithm based the of tasks new proposes an efficient fleet sizing algorithm based on the concept of virtual tasks and a new achieve the high computational efficiency for task allocation of multiple vehicles, this study strategy to reduce the associated computational complexity. The improved computational proposes an reduce efficientthe fleetassociated sizing algorithm based on the concept of improved virtual tasks and a new strategy to computational complexity. The computational strategy to reduce the associated computational complexity. The improved computational proposes an efficient fleetassociated sizing based algorithm on the concept of improved virtual tasks a new efficiency to andreduce the practical practical utilityalgorithm of computational the proposed proposed algorithm are The shown through the and simulation strategy the complexity. computational efficiency and the utility of the are shown through the simulation efficiency and the practical utility of the proposed algorithm are shown through the simulation strategy to reduce the associated computational complexity. The improved computational of a maritime autonomous search mission with multiple autonomous systems. efficiency and the practical utility of the proposed algorithm are shown through the simulation of a maritime autonomous search mission with multiple autonomous systems. of autonomous with autonomous systems. efficiency and the practical search utility mission of the proposed algorithm are shown through the simulation of a a maritime maritime autonomous search mission with multiple multiple autonomous systems. Copyright © 2019.autonomous The Authors. Published by Elsevier Ltd. All rights reserved. systems. of a maritime search mission with multiple autonomous Keywords: Multi-vehicle Multi-vehicle systems, systems, Task Task allocation-sharing allocation-sharing and and job job design, design, Autonomous Autonomous Keywords: Keywords: Multi-vehicle systems,Mission Task allocation-sharing allocation-sharing and job job design,Trajectory Autonomous vehicles, Fleet size optimization, planning and decision making, and path path Keywords: Multi-vehicle systems, Task and design, Autonomous vehicles, Fleet size optimization, Mission planning and decision making, Trajectory and vehicles, Fleet size optimization, Mission planning and decision making, Trajectory and path Keywords: Multi-vehicle systems, Task allocation-sharing and job design, Autonomous planning vehicles, planning Fleet size optimization, Mission planning and decision making, Trajectory and path planning Fleet size optimization, Mission planning and decision making, Trajectory and path vehicles, planning planning 1. INTRODUCTION INTRODUCTION plexity of of the the problem problem is is described. described. For For this, this, the the problem problem 1. plexity 1. INTRODUCTION INTRODUCTION plexity of the the problem problem is described. described. For the of task task allocation allocation for multiple multiple vehicles isthis, addressed in the the 1. plexity of is Foris this, the problem problem of for vehicles addressed in of task allocation allocation for multiple vehicles isin addressed in the the 1. INTRODUCTION plexity of the problem is described. For this, the problem directed graph to prevent the increase the size of of task for multiple vehicles is addressed in directed graph to prevent the increase in the size of the Recently, the the use use of of unmanned unmanned systems systems has has increased increased concon- of directed graph to for prevent the vehicles increase inaddressed the tasks. size of of the task allocation isin in the cost matrix caused bymultiple introducing the virtual virtual directed graph to prevent the increase the size Recently, cost matrix caused by introducing the tasks. Recently, theause use of range unmanned systems has has increased con- directed siderably in in wide of applications applications such as environenvironcost matrix matrix caused by introducing introducing the virtual virtual tasks. graph to prevent the increase in the tasks. size of the Recently, the of unmanned systems increased concost caused by the siderably a wide range of such as In the the following, the task allocation problem is defined defined siderably in wide range of al., applications such as environenvironRecently, theaause of (Zoss unmanned systems has increased con- cost mental monitoring monitoring (Zoss et 2018), intelligent intelligent surveilIn following, allocation is matrix causedthe by task introducing the problem virtual tasks. siderably in wide range of applications such as mental et al., 2018), surveilIn the following, following, the task allocation allocation problemcomplexity is defined defined in Section 2. The limitations and proposed In the the task problem is mental monitoring (Zoss et al., 2018), intelligent surveilsiderably in a wide range of applications such as environlance and reconnaissance (Meuth et al., 2009), and disaster mental monitoring (Zoss (Meuth et al., 2018), intelligent surveil- in Section 2. The limitations and proposed complexity lance and reconnaissance et al., 2009), and disaster in Section 2. Thethe limitations and proposed proposed complexity the following, taskdescribed allocation is3. mitigation strategy are described inproblem Sectioncomplexity 3.defined Next, in Section 2. The limitations and lance and reconnaissance (Meuth et al., al., 2009), andWoerner disaster mental monitoring (Zoss et al.,Beck 2018), intelligent surveil- In response (Murphy et al., (Meuth 2016; Beck et 2009), al., 2018; mitigation strategy are in Section Next, lance and reconnaissance et and disaster response (Murphy et al., 2016; et al., 2018; Woerner mitigation strategy are described in Section 3. Next, in Section 2. The limitations and proposed complexity 4, the validity of the proposed approach is strategy are described in Section 3. Next, response (Murphy et al., al., (Meuth 2016; Beck et 2009), al., 2018; 2018; Woerner lance reconnaissance et al., andWoerner disaster et al., al.,and 2017). In particular, particular, the Beck cooperative operations in- mitigation in Section 4, the validity of the proposed approach is response (Murphy et 2016; et al., et 2017). In the cooperative operations inin Section 4, the validity of the proposed approach is mitigation strategy are described in Section 3. Next, demonstrated through numerical simulations. in Section 4, the validity of the proposed approach is et al., 2017). In particular, the cooperative operations inresponse (Murphy et al., 2016; et al., 2018; Woerner volving multiple unmanned systems have been emphasized et al., 2017). In particular, the Beck cooperative operations in- demonstrated through numerical simulations. volving multiple unmanned systems have been emphasized demonstrated through numerical simulations. Section 4, through the validity of thesimulations. proposed approach is demonstrated numerical volving multiple unmanned of systems have systems, been emphasized et 2017). particular, theunmanned cooperative operations in- in to al., increase theIncapabilities capabilities unmanned and the the volving multiple unmanned systems have been emphasized to increase the of systems, and demonstrated through numerical simulations. to increase the capabilities of unmanned systems, and the volving multiple unmanned systems have been emphasized technologies necessary for task allocation in multi-vehicle to increase the capabilities of unmanned and the 2. CONSTRAINT CONSTRAINT PROGRAMMING-BASED PROGRAMMING-BASED TASK TASK technologies necessary for task allocationsystems, in multi-vehicle 2. technologies necessary for task task allocation in multi-vehicle multi-vehicle to increase the capabilities of unmanned systems, and the collaboration have drawn significant research attention 2. CONSTRAINT CONSTRAINT ALLOCATION PROGRAMMING-BASED TASK TASK technologies necessary for allocation in 2. PROGRAMMING-BASED collaboration have drawn significant research attention ALLOCATION collaboration have drawn significant research attention technologies necessary for2019). task allocation in multi-vehicle (Thompson and and Guihen, 2019). ALLOCATION 2. CONSTRAINT ALLOCATION PROGRAMMING-BASED TASK collaboration have drawn significant research attention (Thompson Guihen, (Thompson and Guihen, 2019). collaboration have drawn significant research attention ALLOCATION (Thompson and Guihen, 2019). The task allocation problem in this this paper is is defined defined on on Task allocation allocation isGuihen, problem that task allocation problem in Task aa problem that determines determines the the desirable desirable The (Thompson andis 2019). athisa paper a The task allocation problem in paper is defined on an undirected graph G = (V , E ), where V is the Task allocation is a problem that determines the desirable a a a task allocation in athis paper is defined on sequence of tasks tasks performed by determines an agent agent (or (or vehicle) The Task allocation is aperformed problem that theaadesirable an undirected graphproblem G = (V ,, E where V is sequence of by an vehicle) an undirected graph G = = set (V E aa ), ),paper where V aa and is the the task allocation in athis isset, defined onaa node composed of aa problem vehicle and aa where task E sequence of tasks performed by determines an agent agent (or vehicle) Task is aof problem thenumber an undirected graph G (V , E ), V is the when allocation the of number of tasks is isthat greater than the the number of The sequence tasks performed by an (or aadesirable vehicle) node composed of vehicle set and task set, and E when the number tasks greater than of a and a a task set,a and E a node composed of a vehicle set an undirected graph G = (V , E ), where V is the aa set of edges the nodes. The superscript when the number of taskstype is greater greater than isthe the number of is sequence tasks of performed by an agent (or a vehicle) node composed ofconnecting a vehicle set and a task set, and E aa agentsthe (orof vehicles). This of problem known as an when number tasks is than number of is set of edges connecting the nodes. The superscript agents (or vehicles). This type of problem is known as an is a set of edges connecting the nodes. The superscript node composed of a vehicle set and a task set, E a indicates actual nodes, and the superscript v virtual agents (orproblem, vehicles). This type of problem problem isthe known as an an when the number ofand tasks is greater than number of is a set of edges The superscript NP-hard and itstype degradation of is computational agents (or vehicles). This of known as a indicates actualconnecting nodes, andthethenodes. superscript v and virtual NP-hard problem, its degradation of computational a indicates actual nodes, and the superscript v virtual is a set of edges connecting the nodes. The superscript nodes and edges. The two-index edge is considered as a NP-hard problem, and its degradation of computational agents (or vehicles). This type of problem is known as an indicates actualThe nodes, and the superscript v virtual efficiency problem, and solution solution quality varies significantly significantly depend- a NP-hard andquality its degradation of computational nodes and edges. two-index edge is considered as a efficiency and varies dependnodes and edges. The two-index edge is considered as a a indicates actual nodes, and the superscript v virtual decision variable. Then, the integer programming based efficiency and solution quality varies significantly dependNP-hard problem, and its degradation of computational nodes and edges. The two-index edge is considered as a ing on the problem size defined by the number of vehicles efficiency solution significantly variable. Then, the integer programming based ing on theand problem sizequality definedvaries by the number of dependvehicles decision decision variable. Then, the integer programming based nodes and edges. The two-index edge is considered as a formulation (Bektas, 2006) is described as follow: ing on the problem size defined by the number of vehicles efficiency and solution quality varies significantly dependdecision variable. Then, the integer programming based and tasks (Baldacci et al., 2012; Whitbrook et al., 2018). ing the (Baldacci problem size defined the number (Bektas, 2006) is described as follow: and on tasks et al., 2012;by Whitbrook et of al.,vehicles 2018). formulation formulation (Bektas, 2006) is described as follow: decision variable. Then, the integer programming based and tasks (Baldacci et al., al., 2012;by Whitbrook et of al.,vehicles 2018). ing on the (Baldacci problem defined the number Furthermore, it is is size often computationally prohibitive to formulation (Bektas, 2006) is described as follow: and tasks et 2012; Whitbrook et al., 2018). Furthermore, it often computationally prohibitive to  formulation (Bektas, 2006) is described  as follow: Furthermore, it is issolutions often computationally prohibitive to and tasks (Baldacci et al., 2012; Whitbrook et al., 2018). compute optimal even for problems with mod  Furthermore, it often computationally prohibitive to compute optimal solutions even for problems with mod      compute optimal solutions even for problems with modFurthermore, it is often computationally prohibitive to erate dimension, which can lead to failure in completing   compute optimal which solutions problems with mod erate dimension, can even lead for to failure in completing min (1)   C(i,     ,,  min (1) C(i, j)e j)eij erate dimension, which can lead for to failure failure in completing completing compute optimal solutions even problems withaa modthe given given tasks. Thus, Thus, an algorithm for obtaining highij erate dimension, which can lead to in a C(i, j)eij  a  min (1) the tasks. an algorithm for obtaining high  ,,   j∈V i∈V min (1) C(i, j)e ij a a the given tasks. Thus, Thus, an algorithm for obtaining obtaining a is higherate dimension, which can lead to failure in completing i∈V a j∈V a quality solution with moderate computational cost rethe given tasks. an algorithm for a high min i∈V , (1) C(i, j)e quality solution with moderate computational cost is rej∈V a a ij i∈V j∈V quality solution with moderate computational cost is rethe given tasks. Thus, an algorithm for obtaining a highquired and and often with preferred in many many practical applications. applications. quality solution moderate computational cost is re- subject to quired often preferred in practical i∈V a j∈V a to  quired and and often preferred in practical applications. quality solution moderate computational cost is re- subject quired often with preferred in many many practical applications.  a subject to to Severaland allocation algorithms have been been considered in oror- subject ≤ N ∈ V (2)  a, a ,, ∀i Several allocation algorithms have considered in quired often preferred in many practical applications. A  eeij ≤ N ∀i ∈ V (2) a, subject to ij a A Several allocation algorithms have been considered in order to increase computational efficiency with no significant a eij ≤ Na , ∀i ∈ V a , (2) Several allocation algorithms have been considered in or j∈V A e ≤ N , ∀i ∈ V , (2) der to increase computational efficiency with no significant a a ij T A j∈V a der to increase increase computational efficiency with no significant significant Several allocation algorithms have beenwith considered in ora e T degradation of computational solution quality throughout the literature (2) der to efficiency no  j∈V a ij ≤ Na , ∀i ∈ VA , degradation of solution quality throughout the literature T j∈V Ta degradation of solution quality throughout the literature der to computational efficiency no significant (Kir´ lyincrease and of Abonyi, 2015; Vidal et al., al.,with 2013; Yoon and ≤ Na , ∀j ∈ V aa ,, (3)  degradation solution quality throughout the literature j∈V T eeij (Kir´ aaly and Abonyi, 2015; Vidal et 2013; Yoon and (3) ij ≤ Na , ∀j ∈ VA a a, a eij ≤ Na , ∀j ∈ VA (Kir´ aly ly and Nevertheless, Abonyi, 2015; Vidal et al., al., are 2013; Yoon and (3) degradation of solution quality throughout the literature Kim, 2017). these methods still not read i∈V A (Kir´ a and Abonyi, 2015; Vidal et 2013; Yoon and e ≤ N , ∀j ∈ V , (3) a ij a Kim, 2017). Nevertheless, these methods are still not readA i∈VT a a e T Kim, 2017). Nevertheless, these methods are still not readread(Kir´ aly and Nevertheless, Abonyi, 2015;these Vidal et involve al., are 2013; Yoon and (3) ily applicable applicable to the the problems problems that involve optimal fleet   i∈V a ij ≤ Na , ∀j ∈ VA , Kim, 2017). methods still not T i∈V ily to that optimal fleet   a a T a e = |V (r )| , ∀k ∈ V , (4) ily applicable to the problems that involve optimal fleet   Kim, 2017). Nevertheless, these methods are still not reada a sizing. ij k i∈V A ily applicable to the problems that involve optimal fleet   T eij = |V a (rk )| , ∀k ∈ VA (4) sizing. a, (4) a (r ) j∈V a (r ) e sizing. ij = |V a (rk )| , ∀k ∈ V a ,, ily applicable to the problems that involve optimal fleet   i∈V (4) k sizing. a k eij = |V (rk )| , ∀k ∈ VA A a a i∈V a (r ) j∈V (r ) In this study, we present an algorithm for optimal fleet k k a (r ) j∈V a (r ) eij = |V (rk )| , ∀k ∈ V , (4)  i∈V In this study, we present an algorithm for optimal fleet sizing. a (rk ) j∈V a (rk ) A i∈V k k  a In thisand study, weplanning present based an algorithm algorithm for optimal optimal fleet sizing route on the the concept concept of virtual virtual = 1, ∀j ∈ V , (5) In this study, we present an for fleet  a a (re) ij T sizing and route planning based on of i∈V a (rk ) j∈V  ek ij = 1, ∀j ∈ V , (5) a T sizing route planning based onthe the concept of virtual virtual a In thisand study, weplanning present an algorithm for optimal fleet tasks, and the strategy to computational com= 1, ∀j ∈ V (5) sizing route based on the concept of aa eeij i∈V T ,, = 1, ∀j ∈ V (5) tasks, the strategy to mitigate mitigate the computational comij T i∈V a a tasks, and the strategy to mitigate the computational comsizing route planning based on the concept of virtual e = 1, ∀j ∈ V , (5) i∈V tasks, and the strategy to mitigate the computational coma ij T i∈V tasks, andCopyright the strategy mitigate computational 2405-8963 © 2019.to The Authors.the Published by Elsevier comLtd. All rights reserved. i∈V a ∗ ∗ ∗ ∗

Peer review under responsibility of International Federation of Automatic Control. 10.1016/j.ifacol.2019.12.323

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 i∈S j∈S



j∈V a

eij = 1, ∀i ∈ VTa ,

(6)

eij ≤ |S| − 1 , ∀S ⊆ VTa , S = ∅, E a ∈ {0, 1}

N ×N

.

(7) (8)

In order to minimize the sum of the total travel distance of the vehicles, the objective function is set as Eq. (1). Equations (2) ∼ (4) indicate the multi-depot and closed path constraints. rk in the Eq. (4) denotes the route of vehicle k and |V a (rk )| is the number of actual nodes associated with each route. The number of the Hamilton cycles in graph G should be less than or equal to the total number of vehicles Na , some vehicles (referred to as redundant vehicles) may not carry out any tasks. The conflict-free constraints in Eqs. (5) and (6) ensure that each task is assigned to only one vehicle. The subtour elimination constraints in Eq. (7) prevent the task nodes from forming cycles between them. To solve this problem, the constraint programming-based task allocation algorithm (Yoon and Kim, 2019) can be used which is a two-phase solution approach consisting of bundle construction and path stretching. The procedure of the algorithm is illustrated in Fig. 1 and summarized in the following. Placing virtual tasks: At the beginning of the algorithm, a virtual task is defined and placed near each vehicle so as not to significantly affect the overall cost. The modified cost matrix including the edge costs corresponding to the virtual tasks can be expressed as follows:

C=



CAA CAT a CAT v CT a A CT a T a CT a T v CT v A CT v T a CT v T v



.

(9)

The cost matrix consists of a set of submatrices representing the edge costs between the different node sets. The CAA indicates the edge distance between the vehicle nodes. Likewise, the CAT a and CAT v are the edge costs between the vehicle nodes to actual task nodes and the vehicle node to virtual tasks nodes, respectively. Obtaining a bundle and optimal fleet: With the cost matrix mentioned above, the bundle which indicates the set of assigned tasks for each agent is obtained. For this, the modified cost matrix at each branch node is used as an input to the Hungarian algorithm (Kuhn, 1955; Munkres, 1957) to obtain a bundle. Initially, the edge costs between the task nodes in the upper triangular part of the cost matrix are set to be infinity. As a result, the initial modified-cost matrix contains many deactivated edges, and the initial solution is usually not optimal. By iteratively decreasing the number of deactivated edges, the solution quality can be improved. Then, the optimal fleet can be obtained by excluding the vehicles with no task assigned. Path stretching: The obtained bundle can be improved by resolving any detours (i.e., edge-crossing and z-shape path) caused by the remaining deactivated edge. For the crossing edge, the detour is resolved by flipping the

sequence of the nodes between the crossing point (Cr), {A1, T 1, Cr, T 2, T 3, Cr, A1} → {A1, T 1, T 3, T 2, A1}. A bundle with many crossings can be resolved by iteratively employing the edge flipping method until the revised bundle no longer has any crossing (van Leeuwen et al., 1980). 3. METHODOLOGY 3.1 Computational complexity The introduction of virtual tasks increases the computational complexity of task allocation. The number of deactivated edges and also increase the number of subtour elimination constraints, as shown below. N  +Ntv



k=1

(N  + Ntv )! , N  = Nta − Na , k! (N  − k + Ntv )!

(10)

where Ntv is the number of virtual tasks. Nta and Na are the number of actual task nodes and the number of vehicle nodes, respectively. The original problem in Fig. 1 has three deactivated edges. However, the number of deactivated edges increases to ten by adding the two virtual tasks, which increases the dimension of the problem. The virtual task also increases the computational complexity of the task assignment with each branch node during the graph-search procedure. The Hungarian algorithm (Munkres, 1957) used to solve the assignment problem at each branch node has a worst-case time complexity of O(N 3 ). Hence, the increase in the number of nodes (N ) at each branch node has a significant influence on the overall calculation efficiency. 3.2 Decision variable reduction We assume that each vehicle first performs a virtual task placed close to the vehicle. Based on this assumption, the decision variable matrix (E) can be defined as follows. 

0

0

E =  ET a A ET a T a E T v A ET v T a

 INa ×Ntv , 0 0

(11)

where I denotes the identity matrix, and it implies that each vehicle performs the virtual task placed in the vicinity. To satisfy the conflict-free constraints, the submatrices in the same row and column as the identity matrix are set to be zero. The remaining sub-matrices located at the lower left part (ET a A , ET a T a , ET v T a and ET v A ) are used to define the new cost matrix, and then, the task allocation problem is defined on a directed graph G = (V, A), where V is a node which includes virtual tasks, and A is a set of arcs connecting to nodes. Here, the size of the new cost matrix is equal to the size of the original cost matrix before without the virtual tasks. 4. SIMULATION RESULT The validity of the proposed method is demonstrated through numerical simulations. First, the computational



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Fig. 1. The procedure of the proposed task allocation algorithm with two virtual tasks (Vehicle = 2, Task = 3): The dark gray cells represent the deactivated edges. Vehicle 2 (A2) in this example only assign the virtual task and is considered a redundant vehicle. efficiency of the proposed algorithm is evaluated. Then, the algorithm is applied to a maritime autonomous search mission with multiple autonomous systems and its result is shown and discussed.

Table 1. Nodes position [x, y]: Each row corresponds to the index number of each node. Agent [20.06, [33.43, [ 1.47, [ 8.33, [15.31, [23.31, [36.17, [17.97, [27.61, [ 9.71,

4.1 Fleet size optimization and Route planning In order to evaluate the effect of performing the task with an optimal fleet, a simple example simulation was performed. A total of 10 vehicles and 10 tasks were distributed in the 40m × 40m square field. Their positions are shown in Table 1. To facilitate an intuitive interpretation of the results, the tasks are mainly placed in the lower part of the area and the vehicles are mainly placed in the upper part. Figure 2 and Table 2 summarize the result of the fleet size optimization and route planning. For comparison, the optimal reference solution is obtained by solving the task allocation problem by exhaustive search. The bar chart in Fig. 2-(c) compares the operational cost over different fleet combinations. In this example, the optimal fleet is the set {2, 3, 9}, and the corresponding planned routes are shown in Fig. 2-(a). The result confirms that performing the tasks with the optimal fleet is obviously more desirable and efficient than using all the vehicles. Also, Table 2 shows that the computational efficiency has been much improved without any degradation of the solution quality by using the proposed method. In order to assess the performance of the proposed method in more general settings, the problems with different node conditions in terms of node positions and the number of nodes are considered and their computation time and optimality values are averaged and compared. Figure 3 compares the performance by averaging 50 runs per case. In the case of full fleet operations (i.e., all the vehicles are used), the operational cost is significantly higher than the cases

nodes 35.00] 17.62] 4.31] 32.93] 33.83] 36.36] 19.84] 31.78] 6.58] 24.21]

Task nodes [23.61, 8.93] [26.61, 1.43] [32.47, 21.86] [ 9.42, 13.80] [ 0.58, 1.00] [ 0.92, 4.89] [27.86, 0.41] [14.21, 8.51] [30.13, 25.22] [25.27, 21.48]

Table 2. Comparison of the optimality of the solution with respect to the fleet size. The fleet size optimization without the decision variable reduction strategy (FSO w/o DVR) means that the solution is obtained by using the algorithm proposed in previous work (Yoon and Kim, 2019) Optimal fleet Total fleet FSO w/o DVR Proposed ∗

a

Fleet {2, 3, 9} {1 ∼ 10} {2, 3, 9} {2, 3, 9}

Obj. value 80.6 293.6 80.6 80.6

Computation 26.3 sec 0.06 sec 0.12 sec 0.03 sec

The results were generated using MATLAB on a PC with Core i7, 3.70 GHz processor with 16 GB of RAM.

of considering fleet size optimization (FSO). However, the na¨ıve application of FSO with virtual tasks induces a significant computational burden due to the increased number of decision variables. This computational burden can be avoided by incorporating the proposed method for reducing the number of a decision variable in section 3.2 into the FSO procedure. Table 2 shows that the proposed method achieve satisfactory performance improvement in

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Table 3. Vehicle specifications and parameters for simulations

USV 1 & 2 USV 3 & 4 UAV 1 ∼ 4 (a) Optimal fleet solution

(b) Full fleet solution

(c) Objective values

Fig. 2. An example of the fleet size optimization (Na = 10, Nt = 10): The green bar represents the optimal fleet solution, the red bar and the black bar indicate the total fleet solution and the worst fleet solution, respectively. The yellow star mark represents the solution provided by the proposed algorithm. terms of computation time without deteriorating the quality of the solution.

Sensing radius/range 120◦ /7m 120◦ /10m 180◦ /20m

Operating speed 2.5 m/s 2.5 m/s 5.0 m/s

Operating altitude 0m 0m 20 m

Operating time 30 min 240 min 30 min

multiple autonomous systems is considered. The scenario assumes that 4 unmanned aerial vehicles (UAVs) and 4 unmanned surface vessels (USVs) are used to perform 15 tasks randomly distributed in a 100m × 100m square field and as the simulation environment considered effect of the ocean current, the tasks moved with an average velocity of 0.1m/s and 45◦ to the x-axis. Both UAVs and USVs are assumed to be capable of detecting tasks, however it is assumed that the recognized tasks can be completed only by USVs. The specifications and parameters of the unmanned systems used in the simulation are shown in Table 3. At the beginning of the mission, the vehicles are commanded to follow the predetermined search path to detect the tasks, and the tasks are considered detected if they are within the sensing range of each vehicle. Figure 4 shows the simulation environment and the initial search path for each vehicle. In this simulation, task allocation for completing the detected tasks is performed every 60 seconds. Based on the result of the task allocation with FSO, some USVs are commanded to approach the tasks for completion and the remaining vehicles continue to search. The vehicles that have completed the given tasks are commanded to return to the search. The result of the FSO with the proposed decision variable reduction scheme is compared with those of the conventional FSO method and the without FSO (constraint programming-based algorithm). As shown in Fig. 5, the performance improvement by the employed FSO scheme in terms of the task completion ratio and the rate of convergence is evident. According to the result, it can be said that the computation efficiency of an allocation algorithm is significant in the time-varying environment because the environment may change between the time when the task allocation is started and the time when the calculation result is applied. Therefore, the proposed FSO scheme has an advantage in mission efficiency when using multiple autonomous systems in various missions. 5. CONCLUSION

Fig. 3. Comparison of the operational cost and computation time. Note that the computation time is written in log scale. 4.2 Application to an autonomous search mission In order to show the practical utility of the proposed algorithm, a maritime autonomous search mission with

This study described an efficient optimal fleet sizing and route planning algorithm with the concept of virtual tasks. In particular, to improve the computational efficiency of FSO, a new strategy to improve computational performance by reducing the number of decision variables was presented. This allowed converting the original task allocation problem defined on an undirected graph into the problem on a directed graph, and the cost matrix became more informative than the original cost matrix of the same size. The validity of the proposed strategy was evaluated through numerical simulations. The computational efficiency of the proposed algorithm was evaluated in various simulation settings. In addition, the effectiveness of the task allocation algorithm involving FSO and route



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REFERENCES

Fig. 4. The simulation environment and initial search path.

Fig. 5. The comparison of the task completion ratio: Task allocation performed every 60 seconds. planning was demonstrated through the simulation of a maritime autonomous search mission. ACKNOWLEDGEMENTS This research was supported by the projects titled “Development of a core technology and infra technology for the operation of USV with high reliability,” and “Development of Management Technology for HNS Accident,” funded by the Ministry of Oceans and Fisheries, Korea.

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