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Task allocation in robot systems with multi-modal capabilities
Proceedigs of the 15th IFAC Symposium on Proceedigs of the 15th IFAC Symposium on Information of Control Problems in Manufacturing Proceedigs the IFAC on Proceedigs the 15th 15th IFAC Symposium Symposium on Information of Control Problems in Manufacturing Available online at www.sciencedirect.com May 11-13, 2015. Ottawa, Canada Information Control Problems in Information Control Problems in Manufacturing Manufacturing May 11-13, 2015. Ottawa, Canada May May 11-13, 11-13, 2015. 2015. Ottawa, Ottawa, Canada Canada
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IFAC-PapersOnLine 48-3 (2015) 2109–2114
Task Task Task Task
allocation in robot systems allocation in robot systems allocation in robot systems allocation in robot systems multi-modal capabilities multi-modal capabilities multi-modal capabilities multi-modal capabilities
with with with with
Maciej Maciej Hojda Hojda Maciej Maciej Hojda Hojda Wroclaw University of Technology Wroclaw University of Technology Wroclaw, Wroclaw, Poland Poland Wroclaw University of [email protected]) Wroclaw(e-mail: University of Technology Technology Wroclaw, Wroclaw, Poland Poland (e-mail: [email protected]) (e-mail: (e-mail: [email protected]) [email protected]) Abstract: Abstract: This This work work focuses focuses on on a a problem problem of of joint joint task task allocation allocation and and routing routing in in aa multimultiAbstract: This work focuses on a problem of joint task allocation and routing in multirobot system. Executors, which are capable of movement are required to perform tasks Abstract: This work focuses problemofofmovement joint taskare allocation routingtasks in aa spread multirobot system. Executors, whichon area capable required and to perform spread robot system. Executors, which are capable of are to perform tasks spread over working space. are complex in they joint of executors robot which capable of movement movement are required required to perform tasks spread over a a system. workingExecutors, space. Tasks Tasks are are complex in that that they require require joint effort effort of multiple multiple executors over aa working space. Tasks are complex in that they require joint effort of multiple executors in order to be performed. Decision making is done under resource constraints and minimized over working space. Tasks are complex in that they require joint effort of multiple executors in order to be performed. Decision making is done under resource constraints and minimized in order to performed. Decision making is under resource constraints and is total resource used. Additionally, executors capable of in execution in order to be be performed. making is done doneare under resource constraints and minimized minimized is the the total resource used. Decision Additionally, executors are capable of working working in several several execution is the total resource used. Additionally, executors are capable of working in several execution modes depending on the chosen intensity of execution and the chosen set of tools (sensors and is the total resource Additionally, are and capable of working several execution modes depending on used. the chosen intensityexecutors of execution the chosen set ofintools (sensors and modes depending on the chosen intensity of execution and the chosen set of tools (sensors and actuators). Formulated decision making problem is proven to be NP-hard even in its feasibility modes depending on the chosen intensity of execution and the chosen set of tools (sensors and actuators). Formulated decision making problem is proven to be NP-hard even in its feasibility actuators). Formulated decision making problem proven to be even its version. solution algorithm execution bounded actuators). Formulated decision makingprovides problem aais is means proven of to efficient be NP-hard NP-hard even in inwith its feasibility feasibility version. Presented Presented solution algorithm provides means of efficient execution with bounded version. Presented solution algorithm provides a means of efficient execution with bounded constraint violations. For a special case of identical executors, an approximation ratio derived. version. Presented solution algorithm provides a executors, means of an efficient executionratio withis constraint violations. For a special case of identical approximation isbounded derived. constraint violations. For a special case of identical executors, an approximation ratio is derived. Empirical evaluations supplement the work with conclusions for other cases. constraint violations. For a special case of identical executors, for an approximation Empirical evaluations supplement the work with conclusions other cases. ratio is derived. Empirical evaluations supplement the work conclusions for cases. Empirical evaluations supplement work with withControl) conclusions forbyother other cases. © 2015, IFAC (International Federationthe of Automatic Hosting Elsevier Ltd. All rights reserved. Keywords: autonomous mobile robots, task allocation, MRTA, multi-modal Keywords: autonomous mobile robots, task allocation, MRTA, multi-modal execution, execution, robot robot Keywords: autonomous mobile robots, task allocation, MRTA, multi-modal execution, robot robot coordination Keywords: autonomous mobile robots, task allocation, MRTA, multi-modal execution, coordination coordination coordination 1. is 1. INTRODUCTION INTRODUCTION is also also relevant relevant for for multi-radio multi-radio platforms; platforms; see see Yong Yong (2012); (2012); 1. INTRODUCTION INTRODUCTION is also(2008). relevant for for multi-radio multi-radio platforms; platforms; see see Yong Yong (2012); (2012); Jung 1. is also relevant Jung (2008). Jung (2008). (2008). Jung This This work work focuses focuses on on aa problem problem of of multi-robot multi-robot task task alloalloIn recent years, expansion in the fields of robotics, agent In recent years, expansion in the fields of robotics, agent This This work focuses on a problem problem of multi-robot taskwhich allocation (MRTA) in such systems where executors, work focuses on a of multi-robot task allocation (MRTA) in such systems where executors, which In recent years, expansion in the fields of robotics, agent systems and artificial intelligence led to increased interest In recentand years, expansion in the led fields robotics, agent cation systems artificial intelligence to of increased interest (MRTA) in such such systems systems where tasks executors, which are of can in (MRTA) in where executors, which are capable capable of movement, movement, can perform perform tasks in multiple multiple systems and artificial artificial intelligence led to toinincreased increased interest cation towards of cooperating systems and intelligence led interest towards systems systems of robots robots cooperating in order order to to perform perform are capable of movement, can perform tasks in multiple modes and are required to cooperate in order to perform are capable of required movement, can perform taskstoinperform multiplea modes and are to cooperate in order a towards systems of robots robots cooperating in robots order to to(or perform a set In systems, more towards of cooperating in order perform a given given systems set of of tasks. tasks. In such such systems, robots (or more modes modes andofare are required to cooperate cooperate in order order to perform perform given set tasks. To solve the problem, decisions have and required to in to aa given set of tasks. To solve the problem, decisions have a given set of tasks. In such systems, robots (or more generally: executors) are expected to complete tasks which agenerally: given set of tasks.are Inexpected such systems, robots (or which more given executors) to complete tasks set of tasks. tasks. To solve solve the problem,are decisions have to made regarding which executors assigned to of To problem, decisions have to be be set made regarding whichthe executors are assigned to generally: executors) arediverse. expected to complete complete tasks which given are and The complexity demands generally: executors) are expected to tasks which are both both complex complex and diverse. The complexity demands to be made regarding which executors are assigned to which tasks and in which modes of execution. An approprito be made regarding which executors are assigned to which tasks and in which modes of execution. An appropriare both complex and diverse. The complexity demands cooperation of multiple robots while diversity requires are both complex and diverse. The complexity demands cooperation of multiple robots while diversity requires which which tasks and and in in which modes modes of execution. execution. An appropriapproprioptimization problem is This tasks which of An ate constrained constrained optimization problem is formulated. formulated. This cooperation of multiple multiple robotssets while diversity requires ate the of of (actuators). cooperation of robots while diversity requires the availability availability of distinctive distinctive sets of tools tools (actuators). ate constrained optimization problem is its formulated. This problem is proven to be NP-hard even in feasibility verate constrained optimization problem is formulated. This problem is proven to be NP-hard even in its feasibility verthe availability of distinctive sets of tools (actuators). Performing a task results in spending resources which are the availability distinctive sets ofresources tools (actuators). Performing a taskofresults in spending which are problem problem is proven proven to be be NP-hard even in in its its feasibility version, therefore effort is made to provide efficient methods is to NP-hard even feasibility version, therefore effort is made to provide efficient methods Performing a task task results results in spending spending resources which are subject to constraints. Those expenditures can vary for Performing a in resources which are subject to constraints. Those expenditures can vary for sion, sion, therefore effort is iswith made to provide provide efficientviolations. methods solutions bounded constraint therefore effort made to efficient methods of obtaining obtaining solutions with bounded constraint violations. subject to constraints. constraints. Those Those expenditures expenditures can vary vary for of different executors operate subject can for differenttoexecutors. executors. Furthermore, Furthermore, executors may may operate of obtaining solutions with bounded constraint violations. The algorithm proposed is assessed through empirical evalobtaining solutions with boundedthrough constraint violations. The algorithm proposed is assessed empirical evaldifferent executors. Furthermore, executorseither may different operate of in multiple which different Furthermore, executors may operate in one one of of executors. multiple modes modes which represent represent either different The algorithm proposed is assessed assessed through empirical evaluations and an approximation ratio is derived for a special The algorithm proposed is through empirical evaluations and an approximation ratio is derived for a special in one of multiple modes which represent either different power levels (intensity of execution) or execution with in one of multiple modesofwhich represent either different power levels (intensity execution) or execution with uations and an an approximation approximation ratio is is derived derived for for aa special special case executors. and ratio case of of identical identical executors. power levels (intensity of of execution) execution) or or execution execution with with uations different toolsets. power (intensity differentlevels toolsets. case of identical executors. case of identical executors.five sections, the first one being different toolsets. toolsets. This different This paper paper is is divided divided into into five sections, the first one being Common Common applications applications of of cooperating cooperating autonomous autonomous robots robots This This paper is divided into five five sections, the first first one being this introduction. Second section contains the formulapaper is divided into sections, the being this introduction. Second section contains the one formulaCommon applications of production, cooperating inspection autonomous robots include elastic systems and monCommon applications of cooperating autonomous robots include elastic systems production, inspection and mon- this this introduction. Second section contains the formulation of the task allocation and routing problem and Second section contains the formulaof the task allocation and routing problem and its its include elastic systems of production, production, inspection and mon- tion introduction. itoring; see (2009); Hojda Coltin (2010); include systems of inspection and monitoring; elastic see Correl Correl (2009); Hojda (2009); (2009); Coltin (2010); tion of the task allocation and routing problem and itsa complexity is proven therein. Third section provides tion of the is task allocation andThird routing problem and its complexity proven therein. section provides a itoring; see Correl (2009); Hojda (2009); Coltin (2010); Nagarajan (2014). Of executors, it is further assumed that itoring; see(2014). CorrelOf (2009); Hojda (2010); Nagarajan executors, it is(2009); furtherColtin assumed that complexity complexity is proven proven therein. Thirdproblem sectionwith provides solution algorithm for aatherein. substitutive weaker is Third section provides aa solution algorithm for substitutive problem with weaker Nagarajan (2014). Of executors, it is further assumed that they can selectively act on the task they were assigned Nagarajan (2014). Ofact executors, is further assumed that solution they can selectively on the ittask they were assigned algorithm for aa substitutive substitutive problem withthrough weaker constraints. This is evaluated solution algorithm for problem with weaker constraints. This algorithm algorithm is further further evaluated through they can selectively act on on the task task theytask were assigned to, they can portions of in an they selectively act the they were to, i.e. i.e.can they can perform perform portions of the the task in assigned an indeinde- constraints. constraints. This algorithm is further evaluated through simulations in the fourth section. Final section presents a This is further through simulations in the algorithm fourth section. Final evaluated section presents a to, i.e. they they can perform perform portions of the the task in an an indeindependent fashion. Examples of as in to, i.e. can portions in pendent fashion. Examples of such suchoftasks, tasks,task as presented presented in simulations simulations in the fourth section. Final section presents a short summary and lists possible venues of future works. in the fourth section. venues Final section presents a short summary and lists possible of future works. pendent fashion. Examples of such tasks, as presented in Akyildiz (2004), include monitoring and inspection, with pendent Examples of such tasks, as presented in short summary and lists possible venues of future works. Akyildiz fashion. (2004), include monitoring and inspection, with short summary and lists possible venues of future works. Akyildiz (2004), include include monitoring and andmonitoring inspection, with specific such or batAkyildiz (2004), monitoring inspection, specific problems problems such as as environmental environmental monitoring or with bat2. specific problems such as environmental monitoring or battlefield Melodia (2010); 2. PROBLEM PROBLEM FORMULATION FORMULATION specific problems suchFollowing as environmental or battlefield surveillance. surveillance. Following Melodia monitoring (2010); Abargoub Abargoub 2. PROBLEM PROBLEM FORMULATION FORMULATION tlefield surveillance. Following Melodia (2010); Abargoub 2. (2010); Zhenqiang (2014), we observe that the benefits tlefield surveillance. Following Melodia (2010); Abargoub (2010); Zhenqiang (2014), we observe that the benefits Given (2010); Zhenqiang (2014), workforce we observe observebecome that the the benefits from a evident in (2010); Zhenqiang (2014), we that benefits Given is is aa set set of of (indexes (indexes of) of) executors executors II {1, {1, 2, 2, .. .. .. ,, I}, I}, from deploying deploying a robotic robotic workforce become evident in Given is aa set set of2,(indexes (indexes of) executors executors I {1, {1, 2, . . . , I}, from deploying a robotic workforce become evident in tasks J {1, . . . , J} and modes L {1, L} wireless sensor networks, where the application of robots Given is of of) I 2, from deploying a roboticwhere workforce become evident in tasks J {1, 2, . . . , J} and modes L {1, 2, 2, ... ... ... ,,, I}, L} wireless sensor networks, the application of robots tasks J of {1, {1, 2,L.. .. elements . ,, J} J} and andrespectively. modes L L {1, 2, 2, ..has . .. ,, L} L} wireless sensorlatency networks, where the overall application of robots robots consisting I, J, Decision reduces cost, and enhances capabilities of tasks J 2, . modes {1, . wireless sensor networks, where the application of to reduces cost, latency and enhances overall capabilities of consisting of I, J, L elements respectively. Decision has to consisting of I, I, J, J, L L the elements respectively. Decision has to to reduces cost, In latency and enhances enhances overall capabilities of consisting be allocation of to in the significant portion those applications, of elements respectively. Decision has reduces cost, latency and overall of be made made regarding regarding the allocation of executors executors to tasks tasks in the network. network. In significant portion of of thosecapabilities applications, be made regarding the allocation of executors to tasks in the network. In significant portion of those applications, a chosen mode of execution. This decision is described ex. in Tekdas (2009), robots supplement a set of small made regarding allocation of decision executorsistodescribed tasks in the network. significant portion of thosea applications, a chosen mode of the execution. This ex. in TekdasIn(2009), robots supplement set of small be chosen mode xof of execution. execution. This decision decisionwith is described described ex. in Tekdas Tekdas (2009), robots supplement set of of small small aabychosen sensing devices (motes) with or ]i∈I,j∈J,k∈J,l∈L elements mode This is ex. in supplement aa set by the the variable variable x [x [xi,j,k,l sensing devices (2009), (motes) robots with calculation calculation or communication communication with elements i,j,k,l ]i∈I,j∈J,k∈J,l∈L byi,j,k,l the =variable variable x [x [x sensing devices (motes) with calculation calculation orexecution communication ]]i∈I,j∈J,k∈J,l∈L with elements capabilities. In this context, multi-modal refers x 1 if executor i performs task k in mode l directly i,j,k,l by the x sensing devices (motes) with or communication with i∈I,j∈J,k∈J,l∈L capabilities. In this context, multi-modal execution refers xi,j,k,l = 1 if executor i i,j,k,l performs task k in mode lelements directly capabilities. In this context, multi-modal execution refers x = 1 if executor i performs task k in mode l to transmission power control and sensor power control. It after executing task j (0 if otherwise). Let us define the i,j,k,l capabilities. In this context, execution refers = 1 if executor performs task k in l directly directly i,j,k,lexecuting to transmission power controlmulti-modal and sensor power control. It x after task j i (0 if otherwise). Letmode us define the to transmission power control and sensor power control. It after executing task j (0 if otherwise). Let us define to transmission power control and sensor power control. It after executing task j (0 if otherwise). Let us define the the Copyright 2015 IFAC 2183Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2015, 2015 IFAC 2183 Copyright 2015 IFAC 2183 Peer review© of International Federation of Automatic Copyright ©under 2015 responsibility IFAC 2183Control. 10.1016/j.ifacol.2015.06.400
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domain of x as X Ri∈I,j∈J,k∈J,l∈L . It is clear that xi,j,k,l are binary variables, therefore (1) ∀i ∈ I, j ∈ J, k ∈ J, l ∈ L xi,j,k,l ∈ {0, 1}. Execution modes differ in resources ei,k,l spent on execution and the portion of the task ηi,k,l that is completed throughout this execution. Costs of movement (or more generally, setup costs) are given by µi,j,k for ith executor performing task k after driving up from task j. It is clear that ei,k,l ≥ 0, ηi,k,l ≥ 0, µi,j,k ≥ 0. For simplicity, we define and use a compact form of those constants e [ei,k,l ]i∈I,k∈J,l∈L , η [ηi,k,l ]i∈I,k∈J,l∈L , µ [µi,j,k ]i∈I,j∈J,k∈J . At most one mode of execution is allowed for each executor-task pair ∀i ∈ I, j ∈ J, k ∈ J xi,j,k,l ≤ 1. (2) l∈L
Task is considered completed when its completion rate of E has been achieved ∀k ∈ J ηi,k,l xi,j,k,l ≥ E. (3) i∈I,j∈J,l∈L
On the other hand, the amount of resource for each executor is limited ∀i ∈ I (ei,k,l + µi,j,k )xi,j,k,l ≤ F. (4) j∈J,k∈J,l∈L
It is clear that E > 0 and F > 0.
The following constraints ensure an uniform starting location, connectivity of the routes, lack of loops and lack of subcycles. ∀i ∈ I xi,1,k,l = 1, (5) k∈J,l∈L
∀i ∈ I, k ∈ J
xi,j,k,l =
j∈J,l∈L
∀i ∈ I, k ∈ J
∀S ⊂ J ∧ S = ∅ ∧ S = J
xi,k,j,l ,
(6)
j∈J,l∈L
j∈J,l∈L
xi,j,k,l ≤ 1,
i∈I,j∈S ,k∈S,l∈L
(7)
xi,j,k,l ≥ 2. (8)
We consider a criterion based on the total amount of resources spent on execution Q(x) xi,j,k,l (ei,k,l + µi,j,k ). (9) i∈I,j∈J,k∈J,l∈L
Finally, we can formulate the main problem of this work which is as follows. Problem 1. (TAR – task allocation and routing). Given: I, J, L, e, η, µ, E, F Find: x∗ where (10) x∗ arg min Q(x),
Problem 2. (R||Cmax ). Given: I, J, e, F ≥ 0 ˜ where Find: x ˜ [˜ xi,k ]i∈I,k∈J ∈ D ˜ {˜ ˜ Xi∈I,k∈J R : (13) ∧ (14) ∧ (15)}, D x∈X ∀i ∈ I, k ∈ J x ˜i,k ∈ {0, 1}, x ˜i,k = 1, ∀k ∈ J ∀i ∈ I
k∈J
(12) (13) (14)
i∈I
x ˜i,k ei,k,1 ≤ F.
(15)
Proof. Consider a specific version of TARF where |L| = 1, ∀i ∈ I, j ∈ J, k ∈ J : ηi,k,1 = 1, µi,j,k = 0, E = 1. Let us call R||Cmax corresponding to an instance of TARF (and vice versa) iff both problems have the same parameters I, J, F, e. Notice that every instance of R||Cmax has a corresponding TARF formulation. We finish the proof in two steps. In the first stem we show that for every feasible solution to TARF there exists a solution to a corresponding R||Cmax . In the second step we show how to obtain a solution to TARF from its corresponding R||Cmax solution. First consider a modification of R||Cmax with the constraint (14) replaced by ∀k ∈ J x ˜i,k ≥ 1. (16) i∈I
˜ RCS . Denote this problem as RCS and its feasible set as D For every solution of RCS one can create a solution of R||Cmax by removing unnecessary allocations and this can be done in polynomial time. Now define a matrix x [xi,k ]i∈I,k∈J with elements obtained through the following transformation xi,k min{1, xi,j,k,1 } i ∈ I, k ∈ J (17) j∈J
where x ∈ D is a solution of TARF. Let us prove that x ∈ ˜ RCS . Constraint (13) is satisfied from (1)∧(7). Constraint D (16) is satisfied from (3). Finally, (15) is satisfied from (4). Next let us show that for every instance of R||Cmax where ˜ = ∅ there exists a solution of a corresponding TARF. D This can be proven by providing a method of constructing ˜ Let Mi {k ∈ J : x ˜i,k = 1} x ∈ D from x ˜ ∈ D. i i and let c [cm ]m∈{1,...,|Mi |} be a series of elements from Mi sorted in ascending order. Then define matrix x [xi,j,k,1 ]i∈I,j∈J,k∈J with elements given by i i 1 if ∃m ∈ Mi : cm = j ∧ cm+1 = k ∨ xi,j,k,1 . (18) m = |Mi | ∧ k = 1 0 if otherwise
Denote as TARF a problem of finding a feasible solution of TAR. Solving TARF is an already difficult problem as is stated in the following theorem. Theorem 1. (about the complexity of TARF). Problem R||Cmax can be reduced to an instance of TARF.
Elements from x define |I| routes with no subcycles or loops that execute task 1 ∈ J (go through the corresponding location) and as such they satisfy constraints (5), (6), (7), (8). Constraint (1) is satisfied from the definition of x. Then (2) is satisfied from (1) and from |L| = 1. Constraint (3) is satisfied from (14) and finally (4) is satisfied from (15). The fact that transformation (17) is executed in polynomial time concludes the proof.
Before proving the theorem about the complexity of TARF, let us recall the linear programming formulation of R||Cmax .
Theorem about the complexity of TARF guarantees that neither approximation algorithms nor any heuristics with polynomial execution time exist (unless P=NP) for the
x∈D
D {x ∈ X : (1) ∧ . . . ∧ (8)}.
(11)
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general case of TAR. Bearing such difficulty in mind, this work develops an approach based on solving a substitutive problem with weaker constraints and relates its solution to a potential solution of TAR.
∀i ∈ I, l ∈ L x˙ i,k,l ∈ {0, 1}, ∀i ∈ I x˙ i,k,l ≤ 1,
3. SOLUTION ALGORITHM
l∈L
∀k ∈ J and a resource limit ∀i ∈ I
x ˘i,k,l ηi,k,l ≥ E,
(21)
x ˘i,k,l ei,k,l ≤ F.
(22)
i∈I,l∈L
k∈J,l∈L
The decision making problem is as follows. Problem 3. (MTA – multi-task allocation). Given: I, J, L, η, e, E, F Find: x ˘∗ where ˘ x), Q(˘ ˘ x) x ˘i,k,l ei,k,l , (23) x ˘∗ arg min Q(˘ ˘ x ˘ ∈D
i∈I,k∈J,l∈L
˘ {˘ ˘ Xi∈I,k∈J,l∈L R : (19) ∧ . . . ∧ (22)}. D x∈X
(24)
∀i ∈ I
and let x˙ k [x˙ i,k,l ]i∈I,l∈L be a decision variable (for a fixed k ∈ J) with binary elements. The problem of a single task allocation is given as follows. Problem 4. (STA – single task allocation). Given: k ∈ J, E, F, I, η, e, x ˘ Find: x˙ ∗k where x˙ i,k,l ei,k,l (27) x˙ ∗k arg min ˙k x˙ k ∈D
i∈I,l∈L
˙ k {x˙ k ∈ ×i∈I,l∈L R : (29) ∧ (30) ∧ (31) ∧ (32)} (28) D
x˙ i,k,l ηi,k,l ≥ E,
l∈L
ˆk x ˆ k ∈D
x˙ i,k,l eˆi,k,l ≤ F.
(31) (32)
i∈I,l∈Li
ˆ k {ˆ D xk ∈ ×i∈I,l∈Li R : (35) ∧ (36) ∧ (37)} ∀i ∈ I, l ∈ Li x ˆi,k,l ≥ 0, ∀i ∈ I x ˆi,k,l ≤ 1,
i∈I,l∈Li
(34) (35) (36)
l∈Li
x ˆi,k,l ηi,k,l ≥ E.
(37)
RTA is a linear programming problem and can therefore be solved in polynomial time. This solution can then be rounded to become a feasible solution of STA. Rounding algorithm is given as follows. Algorithm 1. (RA – rounding algorithm). Given a solution of RTA x ˆ∗k do the following: (1) For each i ∈ I, l ∈ L\Li set xi,k,l := 0. (2) For each i ∈ I, l ∈ Li if x ˆ∗i,k,l ∈ {0, 1} then set ∗ xi,k,l := x ˆi,k,l . (3) If ∃!(i1 , l1 ) : i1 ∈ I, l1 ∈ Li1 for which x ˆ∗i1 ,l1 ∈ (0, 1) then set xi1 ,k,l1 = 1. (4) If ∃(i1 , l1 ), (i2 , l2 ) : i1 , i2 ∈ I, l1 ∈ Li1 , l2 ∈ Li2 for which x ˆ∗i1 ,k,l1 , x ˆ∗i2 ,l2 ∈ (0, 1) then: if ηi1 ,k,l1 ≥ ηi2 ,k,l2 , set xi1 ,k,l1 = 1, xi2 ,k,l2 = 0. Otherwise, set xi1 ,k,l1 = 0, xi2 ,k,l2 = 1. (5) Rounded solution of RTA is xk [xi,k,l ]i∈I,l∈L .
j∈J,l∈L
j∈J\k,l∈L
(30)
This problem can be solved approximately by solving and rounding its relaxed version which is given as follows. Let Li {k ∈ J : ei,k,l ≤ d ∧ eˆi,k,l ≤ F } for a given k ∈ J, d ≥ 0 and x ˘. Let x ˆk = [ˆ xi,k,l ]i∈I,l∈Li be a decision variable with binary elements. Problem 5. (RTA – relaxed task allocation). Given: k ∈ J, E, I, Li , η, e, x ˘ Find: x ˆ∗k where x ˆ∗k arg min x ˆi,k,l ei,k,l (33)
The MTA problem is also hard in the sense that its feasibility version can be reduced to R||Cmax . Proof is analogous to the proof of Theorem 1; see details in Hojda (2014). Nonetheless, a substitutive version can be solved approximately provided that the solution to MTA exists. Let SMTA be a version of MTA with the following constraint instead of (22) x ˘i,k,l ei,k,l ≤ 2IF. (25) ∀i ∈ I ˘ be the feasible set of SMTA and let x ˘∗ be its Let D solution. A feasible solution to SMTA can be obtained by solving a set of relaxed problems of single-task allocation. For a fixed x ˘, let eˆ [ˆ ei,k,l ]i∈I,k∈J,l∈L where x ˘i,j,l ei,j,l (26) eˆi,k,l ei,k,l +
(29)
l∈L
i∈I,l∈L
In this section we introduce and solve several intermediate problems before tackling TAR itself. Let us first consider the following problem of multi-task allocation where the decision variable x ˘ [˘ xi,k,l ]i∈I,k∈J,l∈L determines the task-executor allocation and mode of execution with partial decisions x ˘i,k,l = 1 if executor i is assigned to task k in mode l (0 if otherwise). The constraints given are binary requirement on decision variables (19) ∀i ∈ I, k ∈ J, l ∈ L x ˘i,k,l ∈ {0, 1}, single mode execution ∀i ∈ I, k ∈ J x ˘i,k,l ≤ 1, (20) task completion
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Rounding algorithm can then be incorporated into a 2approximate algorithm for solving MTA. This algorithm is as follows. Algorithm 2. (MTAsol). (1) For k := 1 to k = J do (a) Using binary search obtain the solution to RTA x ˆk for the smallest possible d. Use x = 0 as the parameter of RTA. (b) Using RA obtain xk from x ˆk . (2) Solution: x ˘r := [xi,k,l ]i∈I,k∈J,l∈L . ˘ ∧ Q(˘ ˘ xr ) ≤ 2Q(˘ ˘ x∗ ) Theorem 2. x ˘r ∈ D While Reader is referred to Hojda (2014) for a detailed proof and an elaboration on this theorem, let us nevertheless recall all the main points of that explanation.
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Proof. (1) Solution of RTA (any cornerpoint solution of the linear programming problem) has at most two nonintegral variables. Those variables are set to integrals in RA. (2) For a solution obtained with RA it holds that xi,k,l ei,k,l ≤ 2 x˙ ∗i,k,l ei,k,l . (38) i∈I,l∈L
i∈I,l∈L
∗
(3) Let x˙ xk be a solution of STA for x ˘=x ˘∗ . There exists ∗ x ˘ for which ∗ (39) ∀i ∈ I, l ∈ L x˙ xi,k,l = x∗i,k,l .
(4) Let x˙ 0k be a solution of STA for x ˘ = 0 it then holds that ∗ x˙ xi,j,k ei,k,l ≥ x˙ 0i,j,k ei,k,l . (40) i∈I,l∈L
i∈I,l∈L
(5) Combining (38), (39), (40) we show that xi,k,l ei,k,l ≤ 2Q(˘ x∗ ) ≤ 2IF.
(41)
i∈I,k∈J,l∈L
(6) Verification of (19), (20), (21) for x ˘r ends the proof.
Let us relate the solution of MTA to the solution of TAR. ˘ x∗ ) ≤ Q(x∗ ). Lemma 3. Q(˘ Proof. Consider w(x) [w(x)i,k,l ]i∈I,k∈J,l∈L with elements defined as follows xi,j,k,l . (42) w(x)i,k,l j∈J
˘ and It is sufficient to show that x ∈ D =⇒ w(x) ∈ D ˘ Q[w(x)] = Q(x). Constraints of MTA are satisfied for w(x) as follows: (19) from the definition of w(x), (20) from (2) and (7), (21) from (3) and (7), finally (22) from (4) and ˘ (7). Then we expand Q[w(x)] to obtain ˘ Q[w(x)] = w(x)i,k,l ei,k,l = (43) i∈I,k∈J,l∈L
=
xi,j,k,l ei,k,l = Q(x).
(44)
˜i ∀j ∈ J
yi,1,k = 1,
˜i k∈J
yi,j,k =
˜i j∈J
yi,k,j ,
(47) (48)
˜i j∈J
˜ i ∧ S = ∅ ∧ S = J ˜i ∀S ⊂ J
j∈S,k∈S
Also consider quality criterion Qiy (yi )
yi,j,k ≥ 2.
yi,j,k µi,j,k .
(49)
(50)
˜ i ,k∈J ˜i j∈J
The problem of mobile routing for a given executor and an instance of SMTA is as follows. Problem 6. (MR – mobile routing). ˜ i , µ Find: y ∗ where Given: i ∈ I, J i (51) yi∗ arg mini Qiy (yi ), y∈Dy
Diy
{yi ∈ Xj∈J˜ i ,k∈J˜ i R : (46) ∧ (47) ∧ (48) ∧ (49)}. (52)
MR is in fact an instance of the Travelling Salesman Problem. Since we assumed that µ is metric, so is this TSP. It can therefore be solved approximately (for example, by the means of the cheapest insertion algorithm). The approximation ratio is then given as follows (53) Qiy (y i ) ≤ αQiy (yi∗ )
under the assumption that y i ∈ Diy was derived using an α-approximate algorithm. Another important property of MR in relation to TAR is given as follows. Lemma 4. Qiy (y i ) ≤ α i∈I,j∈J,k∈J,l∈L x∗i,j,k,l µi,j,k . Proof. Let us notice that x∗ defines a route (over all executors) that visits all J tasks. Let C be the cost of the cheapest route (costs µ) over all J tasks. MR finds the ˜ i ⊂ J therefore cheapest route over J Qiy (yi∗ ) ≤ C ≤ x∗i,j,k,l µi,j,k , (54) i∈I,j∈J,k∈J,l∈L
From (53) and (54) we have Qiy (y i ) ≤ α
x∗i,j,k,l µi,j,k .
(55)
i∈I,j∈J,k∈J,l∈L
i∈I,k∈J,l∈L j∈J
In consequence, we have ∗ ˘ ˘ x∗ ) ≤ Q[w(x )] ≤ Q(x∗ ) Q(˘
˜i ∀j ∈ J
(45)
which ends the proof.
The following analysis is performed under two assumptions: that µ defines a metric (one for each executor) and that executors are identical. We derive an approximation algorithm for a substitutive version of TAR under those assumptions and empirically evaluate the algorithm when they do not hold. Let us now consider a set of routing problems, one for each executor. Those are formulated for a given instance and a ˜ i {1, . . . , J˜i } be a set feasible solution of SMTA x ˘r . Let J of tasks to which the ith executor was assigned. Then for a ˜ i {k ∈ J : ˘ri,k,l = 1}. Define given i ∈ I we have J l∈L x a decision variable yi [yi,j,k ]j∈J,k∈ ˜ ˜ i with elements J yi,j,k = 1 if task k is executed directly after task j (0 if otherwise). Consider the following constraints. ˜i, k ∈ J ˜ i yi,j,k ∈ {0, 1}, ∀j ∈ J (46)
˜ i (= 0 if otherwise) be Let y˘i,j,k = y i,j,k if j, k ∈ J the approximate solution for instances of MA solved for x ˘r obtainedy through MTAsol. Consider the following constraint xi,j,k,l (ei,k,l +µi,j,k ) ≤ (2+α)|I|F. (56) ∀i ∈ I j∈J,k∈J,l∈L
The substitutive TAR problem is given as follows. Problem 7. (STAR – substitutive TAR). Given: I, J, L, e, η, µ, F, E ∗ Find: x where ∗ x arg min Q(x),
(57)
x∈D
D {x ∈ X : (1) ∧ (2) ∧ (3) ∧ (56) ∧ (5) ∧ . . . ∧ (8)}. (58) Consider x ´ [´ xi,j,k,l ]i∈I,j∈J,k∈J,l∈L with elements defined as follows x ´i,j,k,l y˘i,j,k x ˘ri,k,l . (59) The following algorithm solves STAR in an approximate fashion.
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Algorithm 3. (STARsol). (1) Formulate and solve SMTA to obtain x ˘r . (2) Formulate and solve MR under x ˘r . Obtain y. (3) Use (59) to obtain x ´. This algorithm works in polynomial time since all its components work in polynomial time. It provides a solution with quality given by the following theorem. Theorem 5. x ´ ∈ D ∧ Q(x) ≤ (2 + α|I|)Q(x∗ ). ˘ and Proof. Property x ´ ∈ D comes directly from x ˘r ∈ D i y ∈ Dy . Then let us notice that x ´i,j,k,l µi,j,k = i∈I,j∈J,k∈J,l∈L
=
˜ i ,k∈J ˜i i∈I,j∈J
and
Qiy (y i )
(60)
˘ xr ). x ˘ri,k,l ei,k,l = Q(˘
(61)
y˘i,j,k µi,j,k =
Fig. 1. Experiment 1: Dependence of Q on J.
i∈I
x ´i,j,k,l ei,k,l =
i∈I,j∈J,k∈J,l∈L
=
˜ i ,l∈L i∈I,k∈J
Expanding Q(x) and from (60), (61), Lemma 3 and Lemma 4 we have Q(x) = y˘i,j,k x ˘ri,k,l ei,k,l + (62)
Fig. 2. Experiment 2: Dependence of Q on J.
i∈I,j∈J,k∈J,l∈L
+
y˘i,j,k x ˘ri,k,l µi,j,k =
(63)
i∈I,j∈J,k∈J,l∈L
˘ xr ) + = Q(˘
i∈I
Qiy (y i ) ≤ (2 + α|I|)Q(x∗ )
(64)
which ends the proof.
Algorithm for solving STAR can still be used as a heuristic algorithm in cases when executors are not identical or µ is not metric. Both cases, identical and unrelated executors are evaluated empirically in the following section. 4. ALGORITHM EVALUATION Let us first present selected preliminary empirical evaluations of the algorithms presented in the previous section. Those evaluations will be followed by comments regarding consequences of the results from the viewpoint of practical implementations. Experiment 1 illustrates the performance of the algorithm for a case of identical executors and tasks located on a line with even distances between neighbouring tasks. The case of unrelated executors is analysed in Experiment 2 and Experiment 3. Results are presented in respective figures. Experiment 1. Consider TAR with parameters I = 5, 5 ≤ J ≤ 25, L = 2, E ∈ {1, 3, 5}, F = 200, ei,k,l = 10 + k, ηi,k,l = l, µi,j,k = |j − k|. Experiment 2. Consider TAR with parameters I = 5, 5 ≤ J ≤ 25, L = 2, E ∈ {20, 25, 30}, F = 200, ei,k,l = 10 + 5i, ηi,k,l = l · i · k, µi,j,k = |j − k|. Experiment 3. Consider TAR with parameters I = 5, 5 ≤ J ≤ 25, L = 2, E ∈ {20, 25, 30}, F = 200, ei,k,l = 10 + 5i, ηi,k,l = l · i · k, µi,j,k = 10|j − k|.
Fig. 3. Experiment 3: Dependence of Q on J. In the first experiment, cost of execution grows with the number of tasks. This is reflected in the value of the objective function which increases at a growing rate. The rate of growth is dependent on the completion rate E in a proportional manner. It is quite unlike that in the second experiment when decrease from E = 30 to E = 25 causes a smaller impact than decrease from E = 25 to E = 20. This is caused by removal of less efficient executors from the execution process. First experiment also illustrates a clear violation of constraints of the TAR problem which is visible for the case of E = 5. Such violations are to be expected in the light of the properties proven in the previous sections. Finally, in the third experiment we can observe the influence of increased distances on the solution algorithm. Clearly visible nonlinearities are a result of greater influence of distance on the solution. As routes become more costly, so does the the value of the quality criterion. It the light of the results obtained through empirical evaluation it is essential to comment on possible applications of the evaluated algorithms. As already stated in Section 2, from Theorem 1 directly follows that no algorithm can
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obtain a feasible solution to TAR in polynomial time. It is therefore expected that solutions obtained through the use of STARsol will not, in general case, satisfy all the constraints imposed by TAR and this constraint violation is clearly visible in the empirical evaluations. The most straightforward application of such a result is to treat it as a cue to increase the available resources by the necessary amount ordained by the solution. However, for the case of identical executors and metric distances, another approach might be also possible, depending on actual numerical data of the problem. That approach is to solve STAR for a reduced amount of resources, i.e. for a new limit in constraint (56) calculated as F˜ = F/(2+α)|I|. If a solution to STAR with the new limit can be obtained through STARsol then the resulting x ´ is a feasible solution of TAR with the resource limit of F . There are, unfortunately, no guarantees that such a solution to STAR exists for every TAR formulation. 5. SUMMARY
Proceedings of the 27th IEEE Conference on Computer Communications, pages 1112–1120, 2008. T. Melodia, D. Pompili, I. Akyildiz. Handling mobility in wireless sensor and actor networks. IEEE Transactions on Mobile Computing, volume 10 (2), pages 160–173, 2010. T. Nagarajan, A. Thondiyath. An algorithm for cooperative task allocation in scalable, constrained multiple robot systems. Intelligent Service Robotics, volume 7 (4), 2014. O. Tekdas, V. Isler. Using Mobile Robots to Harvest Data from Sensor Fields. IEEE Wireless Communications, volume 16 (1), pages 22–28, 2009. F. Yong, S. Mo, G. Hackmann, L. Chenyang. Practical control of transmission power for wireless sensor networks. Proceedings of 20th IEEE International Conference on Network Protocols, pages 1–10, 2012. M. Zhenqiang, Y. Yang, M. Huan, W. Dandan. Connectivity preserving task allocation in mobile robotic sensor network. Proceedings of IEEE International Conference on Communications, pages 136–141, 2014.
Tackled in the paper was a problem of joint task allocation and routing for mobile executors with resource constraints. Proven difficulty of finding even a feasible solution in polynomial time led to the development of an approximation algorithm for a substitutive problem. An approximation ratio was derived for a special case of metric distances and uniform distributors. Nevertheless, it was shown that the algorithm is also applicable for cases where those assumptions do not hold. It is crucial for future research to further evaluate the possibility of obtaining a solution algorithm with better approximation bound for both, the criterion and the constraint violation as well as expanding analytical results on other cases, especially for the most practical case of unrelated executors. REFERENCES A. Abuarqoub, M. Hammoudeh, T. Alsboui. An overview of information extraction from mobile wireless sensor networks. Lecture Notes in Computer Science. Internet of Things, Smart Spaces, and Next Generation Networking Springer-Verlag, volume 7469, pages 95–106, 2012. I. Akyildiz, I. Kasimoglu. Wireless sensor and actor networks: Research challenges. Ad Hoc Networks, volume 2 (4), pages 351–367, 2004. B. Coltin, M. Veloso. Mobile Robot Task Allocation in Hybrid Wireless Sensor Networks. Proceedings of International Conference on Intelligent Robots and Systems, pages 2932–2937, 2010. N. Correll, A. Martinoli. Multirobot inspection of industrial machinery. IEEE Robotics and Automation Magazine, volume 16, pages 103–112, 2009. M. Hojda, J. Jozefczyk. Decision making algorithm for a class of two-level manufacturing systems. Kybernetes, volume 38, pages 1355–1372, 2009. M. Hojda. Multi-robot task allocation algorithms in systems with energy constraints. (in Polish) Automation of Discrete Processes: theory and applications (pl. Automatyzacja Procesw Dyskretnych: teoria i zastosowania), volume 2, pages 105–113, Gliwice, Poland, 2014. D. Jung, A. Savvides. An energy efficiency evaluation for sensor nodes with multiple processors, radios and sensor. 2188
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IFAC-PapersOnLine 48-3 (2015) 2109–2114
Task Task Task Task
allocation in robot systems allocation in robot systems allocation in robot systems allocation in robot systems multi-modal capabilities multi-modal capabilities multi-modal capabilities multi-modal capabilities
with with with with
Maciej Maciej Hojda Hojda Maciej Maciej Hojda Hojda Wroclaw University of Technology Wroclaw University of Technology Wroclaw, Wroclaw, Poland Poland Wroclaw University of [email protected]) Wroclaw(e-mail: University of Technology Technology Wroclaw, Wroclaw, Poland Poland (e-mail: [email protected]) (e-mail: (e-mail: [email protected]) [email protected]) Abstract: Abstract: This This work work focuses focuses on on a a problem problem of of joint joint task task allocation allocation and and routing routing in in aa multimultiAbstract: This work focuses on a problem of joint task allocation and routing in multirobot system. Executors, which are capable of movement are required to perform tasks Abstract: This work focuses problemofofmovement joint taskare allocation routingtasks in aa spread multirobot system. Executors, whichon area capable required and to perform spread robot system. Executors, which are capable of are to perform tasks spread over working space. are complex in they joint of executors robot which capable of movement movement are required required to perform tasks spread over a a system. workingExecutors, space. Tasks Tasks are are complex in that that they require require joint effort effort of multiple multiple executors over aa working space. Tasks are complex in that they require joint effort of multiple executors in order to be performed. Decision making is done under resource constraints and minimized over working space. Tasks are complex in that they require joint effort of multiple executors in order to be performed. Decision making is done under resource constraints and minimized in order to performed. Decision making is under resource constraints and is total resource used. Additionally, executors capable of in execution in order to be be performed. making is done doneare under resource constraints and minimized minimized is the the total resource used. Decision Additionally, executors are capable of working working in several several execution is the total resource used. Additionally, executors are capable of working in several execution modes depending on the chosen intensity of execution and the chosen set of tools (sensors and is the total resource Additionally, are and capable of working several execution modes depending on used. the chosen intensityexecutors of execution the chosen set ofintools (sensors and modes depending on the chosen intensity of execution and the chosen set of tools (sensors and actuators). Formulated decision making problem is proven to be NP-hard even in its feasibility modes depending on the chosen intensity of execution and the chosen set of tools (sensors and actuators). Formulated decision making problem is proven to be NP-hard even in its feasibility actuators). Formulated decision making problem proven to be even its version. solution algorithm execution bounded actuators). Formulated decision makingprovides problem aais is means proven of to efficient be NP-hard NP-hard even in inwith its feasibility feasibility version. Presented Presented solution algorithm provides means of efficient execution with bounded version. Presented solution algorithm provides a means of efficient execution with bounded constraint violations. For a special case of identical executors, an approximation ratio derived. version. Presented solution algorithm provides a executors, means of an efficient executionratio withis constraint violations. For a special case of identical approximation isbounded derived. constraint violations. For a special case of identical executors, an approximation ratio is derived. Empirical evaluations supplement the work with conclusions for other cases. constraint violations. For a special case of identical executors, for an approximation Empirical evaluations supplement the work with conclusions other cases. ratio is derived. Empirical evaluations supplement the work conclusions for cases. Empirical evaluations supplement work with withControl) conclusions forbyother other cases. © 2015, IFAC (International Federationthe of Automatic Hosting Elsevier Ltd. All rights reserved. Keywords: autonomous mobile robots, task allocation, MRTA, multi-modal Keywords: autonomous mobile robots, task allocation, MRTA, multi-modal execution, execution, robot robot Keywords: autonomous mobile robots, task allocation, MRTA, multi-modal execution, robot robot coordination Keywords: autonomous mobile robots, task allocation, MRTA, multi-modal execution, coordination coordination coordination 1. is 1. INTRODUCTION INTRODUCTION is also also relevant relevant for for multi-radio multi-radio platforms; platforms; see see Yong Yong (2012); (2012); 1. INTRODUCTION INTRODUCTION is also(2008). relevant for for multi-radio multi-radio platforms; platforms; see see Yong Yong (2012); (2012); Jung 1. is also relevant Jung (2008). Jung (2008). (2008). Jung This This work work focuses focuses on on aa problem problem of of multi-robot multi-robot task task alloalloIn recent years, expansion in the fields of robotics, agent In recent years, expansion in the fields of robotics, agent This This work focuses on a problem problem of multi-robot taskwhich allocation (MRTA) in such systems where executors, work focuses on a of multi-robot task allocation (MRTA) in such systems where executors, which In recent years, expansion in the fields of robotics, agent systems and artificial intelligence led to increased interest In recentand years, expansion in the led fields robotics, agent cation systems artificial intelligence to of increased interest (MRTA) in such such systems systems where tasks executors, which are of can in (MRTA) in where executors, which are capable capable of movement, movement, can perform perform tasks in multiple multiple systems and artificial artificial intelligence led to toinincreased increased interest cation towards of cooperating systems and intelligence led interest towards systems systems of robots robots cooperating in order order to to perform perform are capable of movement, can perform tasks in multiple modes and are required to cooperate in order to perform are capable of required movement, can perform taskstoinperform multiplea modes and are to cooperate in order a towards systems of robots robots cooperating in robots order to to(or perform a set In systems, more towards of cooperating in order perform a given given systems set of of tasks. tasks. In such such systems, robots (or more modes modes andofare are required to cooperate cooperate in order order to perform perform given set tasks. To solve the problem, decisions have and required to in to aa given set of tasks. To solve the problem, decisions have a given set of tasks. In such systems, robots (or more generally: executors) are expected to complete tasks which agenerally: given set of tasks.are Inexpected such systems, robots (or which more given executors) to complete tasks set of tasks. tasks. To solve solve the problem,are decisions have to made regarding which executors assigned to of To problem, decisions have to be be set made regarding whichthe executors are assigned to generally: executors) arediverse. expected to complete complete tasks which given are and The complexity demands generally: executors) are expected to tasks which are both both complex complex and diverse. The complexity demands to be made regarding which executors are assigned to which tasks and in which modes of execution. An approprito be made regarding which executors are assigned to which tasks and in which modes of execution. An appropriare both complex and diverse. The complexity demands cooperation of multiple robots while diversity requires are both complex and diverse. The complexity demands cooperation of multiple robots while diversity requires which which tasks and and in in which modes modes of execution. execution. An appropriapproprioptimization problem is This tasks which of An ate constrained constrained optimization problem is formulated. formulated. This cooperation of multiple multiple robotssets while diversity requires ate the of of (actuators). cooperation of robots while diversity requires the availability availability of distinctive distinctive sets of tools tools (actuators). ate constrained optimization problem is its formulated. This problem is proven to be NP-hard even in feasibility verate constrained optimization problem is formulated. This problem is proven to be NP-hard even in its feasibility verthe availability of distinctive sets of tools (actuators). Performing a task results in spending resources which are the availability distinctive sets ofresources tools (actuators). Performing a taskofresults in spending which are problem problem is proven proven to be be NP-hard even in in its its feasibility version, therefore effort is made to provide efficient methods is to NP-hard even feasibility version, therefore effort is made to provide efficient methods Performing a task task results results in spending spending resources which are subject to constraints. Those expenditures can vary for Performing a in resources which are subject to constraints. Those expenditures can vary for sion, sion, therefore effort is iswith made to provide provide efficientviolations. methods solutions bounded constraint therefore effort made to efficient methods of obtaining obtaining solutions with bounded constraint violations. subject to constraints. constraints. Those Those expenditures expenditures can vary vary for of different executors operate subject can for differenttoexecutors. executors. Furthermore, Furthermore, executors may may operate of obtaining solutions with bounded constraint violations. The algorithm proposed is assessed through empirical evalobtaining solutions with boundedthrough constraint violations. The algorithm proposed is assessed empirical evaldifferent executors. Furthermore, executorseither may different operate of in multiple which different Furthermore, executors may operate in one one of of executors. multiple modes modes which represent represent either different The algorithm proposed is assessed assessed through empirical evaluations and an approximation ratio is derived for a special The algorithm proposed is through empirical evaluations and an approximation ratio is derived for a special in one of multiple modes which represent either different power levels (intensity of execution) or execution with in one of multiple modesofwhich represent either different power levels (intensity execution) or execution with uations and an an approximation approximation ratio is is derived derived for for aa special special case executors. and ratio case of of identical identical executors. power levels (intensity of of execution) execution) or or execution execution with with uations different toolsets. power (intensity differentlevels toolsets. case of identical executors. case of identical executors.five sections, the first one being different toolsets. toolsets. This different This paper paper is is divided divided into into five sections, the first one being Common Common applications applications of of cooperating cooperating autonomous autonomous robots robots This This paper is divided into five five sections, the first first one being this introduction. Second section contains the formulapaper is divided into sections, the being this introduction. Second section contains the one formulaCommon applications of production, cooperating inspection autonomous robots include elastic systems and monCommon applications of cooperating autonomous robots include elastic systems production, inspection and mon- this this introduction. Second section contains the formulation of the task allocation and routing problem and Second section contains the formulaof the task allocation and routing problem and its its include elastic systems of production, production, inspection and mon- tion introduction. itoring; see (2009); Hojda Coltin (2010); include systems of inspection and monitoring; elastic see Correl Correl (2009); Hojda (2009); (2009); Coltin (2010); tion of the task allocation and routing problem and itsa complexity is proven therein. Third section provides tion of the is task allocation andThird routing problem and its complexity proven therein. section provides a itoring; see Correl (2009); Hojda (2009); Coltin (2010); Nagarajan (2014). Of executors, it is further assumed that itoring; see(2014). CorrelOf (2009); Hojda (2010); Nagarajan executors, it is(2009); furtherColtin assumed that complexity complexity is proven proven therein. Thirdproblem sectionwith provides solution algorithm for aatherein. substitutive weaker is Third section provides aa solution algorithm for substitutive problem with weaker Nagarajan (2014). Of executors, it is further assumed that they can selectively act on the task they were assigned Nagarajan (2014). Ofact executors, is further assumed that solution they can selectively on the ittask they were assigned algorithm for aa substitutive substitutive problem withthrough weaker constraints. This is evaluated solution algorithm for problem with weaker constraints. This algorithm algorithm is further further evaluated through they can selectively act on on the task task theytask were assigned to, they can portions of in an they selectively act the they were to, i.e. i.e.can they can perform perform portions of the the task in assigned an indeinde- constraints. constraints. This algorithm is further evaluated through simulations in the fourth section. Final section presents a This is further through simulations in the algorithm fourth section. Final evaluated section presents a to, i.e. they they can perform perform portions of the the task in an an indeindependent fashion. Examples of as in to, i.e. can portions in pendent fashion. Examples of such suchoftasks, tasks,task as presented presented in simulations simulations in the fourth section. Final section presents a short summary and lists possible venues of future works. in the fourth section. venues Final section presents a short summary and lists possible of future works. pendent fashion. Examples of such tasks, as presented in Akyildiz (2004), include monitoring and inspection, with pendent Examples of such tasks, as presented in short summary and lists possible venues of future works. Akyildiz fashion. (2004), include monitoring and inspection, with short summary and lists possible venues of future works. Akyildiz (2004), include include monitoring and andmonitoring inspection, with specific such or batAkyildiz (2004), monitoring inspection, specific problems problems such as as environmental environmental monitoring or with bat2. specific problems such as environmental monitoring or battlefield Melodia (2010); 2. PROBLEM PROBLEM FORMULATION FORMULATION specific problems suchFollowing as environmental or battlefield surveillance. surveillance. Following Melodia monitoring (2010); Abargoub Abargoub 2. PROBLEM PROBLEM FORMULATION FORMULATION tlefield surveillance. Following Melodia (2010); Abargoub 2. (2010); Zhenqiang (2014), we observe that the benefits tlefield surveillance. Following Melodia (2010); Abargoub (2010); Zhenqiang (2014), we observe that the benefits Given (2010); Zhenqiang (2014), workforce we observe observebecome that the the benefits from a evident in (2010); Zhenqiang (2014), we that benefits Given is is aa set set of of (indexes (indexes of) of) executors executors II {1, {1, 2, 2, .. .. .. ,, I}, I}, from deploying deploying a robotic robotic workforce become evident in Given is aa set set of2,(indexes (indexes of) executors executors I {1, {1, 2, . . . , I}, from deploying a robotic workforce become evident in tasks J {1, . . . , J} and modes L {1, L} wireless sensor networks, where the application of robots Given is of of) I 2, from deploying a roboticwhere workforce become evident in tasks J {1, 2, . . . , J} and modes L {1, 2, 2, ... ... ... ,,, I}, L} wireless sensor networks, the application of robots tasks J of {1, {1, 2,L.. .. elements . ,, J} J} and andrespectively. modes L L {1, 2, 2, ..has . .. ,, L} L} wireless sensorlatency networks, where the overall application of robots robots consisting I, J, Decision reduces cost, and enhances capabilities of tasks J 2, . modes {1, . wireless sensor networks, where the application of to reduces cost, latency and enhances overall capabilities of consisting of I, J, L elements respectively. Decision has to consisting of I, I, J, J, L L the elements respectively. Decision has to to reduces cost, In latency and enhances enhances overall capabilities of consisting be allocation of to in the significant portion those applications, of elements respectively. Decision has reduces cost, latency and overall of be made made regarding regarding the allocation of executors executors to tasks tasks in the network. network. In significant portion of of thosecapabilities applications, be made regarding the allocation of executors to tasks in the network. In significant portion of those applications, a chosen mode of execution. This decision is described ex. in Tekdas (2009), robots supplement a set of small made regarding allocation of decision executorsistodescribed tasks in the network. significant portion of thosea applications, a chosen mode of the execution. This ex. in TekdasIn(2009), robots supplement set of small be chosen mode xof of execution. execution. This decision decisionwith is described described ex. in Tekdas Tekdas (2009), robots supplement set of of small small aabychosen sensing devices (motes) with or ]i∈I,j∈J,k∈J,l∈L elements mode This is ex. in supplement aa set by the the variable variable x [x [xi,j,k,l sensing devices (2009), (motes) robots with calculation calculation or communication communication with elements i,j,k,l ]i∈I,j∈J,k∈J,l∈L byi,j,k,l the =variable variable x [x [x sensing devices (motes) with calculation calculation orexecution communication ]]i∈I,j∈J,k∈J,l∈L with elements capabilities. In this context, multi-modal refers x 1 if executor i performs task k in mode l directly i,j,k,l by the x sensing devices (motes) with or communication with i∈I,j∈J,k∈J,l∈L capabilities. In this context, multi-modal execution refers xi,j,k,l = 1 if executor i i,j,k,l performs task k in mode lelements directly capabilities. In this context, multi-modal execution refers x = 1 if executor i performs task k in mode l to transmission power control and sensor power control. It after executing task j (0 if otherwise). Let us define the i,j,k,l capabilities. In this context, execution refers = 1 if executor performs task k in l directly directly i,j,k,lexecuting to transmission power controlmulti-modal and sensor power control. It x after task j i (0 if otherwise). Letmode us define the to transmission power control and sensor power control. It after executing task j (0 if otherwise). Let us define to transmission power control and sensor power control. It after executing task j (0 if otherwise). Let us define the the Copyright 2015 IFAC 2183Hosting by Elsevier Ltd. 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domain of x as X Ri∈I,j∈J,k∈J,l∈L . It is clear that xi,j,k,l are binary variables, therefore (1) ∀i ∈ I, j ∈ J, k ∈ J, l ∈ L xi,j,k,l ∈ {0, 1}. Execution modes differ in resources ei,k,l spent on execution and the portion of the task ηi,k,l that is completed throughout this execution. Costs of movement (or more generally, setup costs) are given by µi,j,k for ith executor performing task k after driving up from task j. It is clear that ei,k,l ≥ 0, ηi,k,l ≥ 0, µi,j,k ≥ 0. For simplicity, we define and use a compact form of those constants e [ei,k,l ]i∈I,k∈J,l∈L , η [ηi,k,l ]i∈I,k∈J,l∈L , µ [µi,j,k ]i∈I,j∈J,k∈J . At most one mode of execution is allowed for each executor-task pair ∀i ∈ I, j ∈ J, k ∈ J xi,j,k,l ≤ 1. (2) l∈L
Task is considered completed when its completion rate of E has been achieved ∀k ∈ J ηi,k,l xi,j,k,l ≥ E. (3) i∈I,j∈J,l∈L
On the other hand, the amount of resource for each executor is limited ∀i ∈ I (ei,k,l + µi,j,k )xi,j,k,l ≤ F. (4) j∈J,k∈J,l∈L
It is clear that E > 0 and F > 0.
The following constraints ensure an uniform starting location, connectivity of the routes, lack of loops and lack of subcycles. ∀i ∈ I xi,1,k,l = 1, (5) k∈J,l∈L
∀i ∈ I, k ∈ J
xi,j,k,l =
j∈J,l∈L
∀i ∈ I, k ∈ J
∀S ⊂ J ∧ S = ∅ ∧ S = J
xi,k,j,l ,
(6)
j∈J,l∈L
j∈J,l∈L
xi,j,k,l ≤ 1,
i∈I,j∈S ,k∈S,l∈L
(7)
xi,j,k,l ≥ 2. (8)
We consider a criterion based on the total amount of resources spent on execution Q(x) xi,j,k,l (ei,k,l + µi,j,k ). (9) i∈I,j∈J,k∈J,l∈L
Finally, we can formulate the main problem of this work which is as follows. Problem 1. (TAR – task allocation and routing). Given: I, J, L, e, η, µ, E, F Find: x∗ where (10) x∗ arg min Q(x),
Problem 2. (R||Cmax ). Given: I, J, e, F ≥ 0 ˜ where Find: x ˜ [˜ xi,k ]i∈I,k∈J ∈ D ˜ {˜ ˜ Xi∈I,k∈J R : (13) ∧ (14) ∧ (15)}, D x∈X ∀i ∈ I, k ∈ J x ˜i,k ∈ {0, 1}, x ˜i,k = 1, ∀k ∈ J ∀i ∈ I
k∈J
(12) (13) (14)
i∈I
x ˜i,k ei,k,1 ≤ F.
(15)
Proof. Consider a specific version of TARF where |L| = 1, ∀i ∈ I, j ∈ J, k ∈ J : ηi,k,1 = 1, µi,j,k = 0, E = 1. Let us call R||Cmax corresponding to an instance of TARF (and vice versa) iff both problems have the same parameters I, J, F, e. Notice that every instance of R||Cmax has a corresponding TARF formulation. We finish the proof in two steps. In the first stem we show that for every feasible solution to TARF there exists a solution to a corresponding R||Cmax . In the second step we show how to obtain a solution to TARF from its corresponding R||Cmax solution. First consider a modification of R||Cmax with the constraint (14) replaced by ∀k ∈ J x ˜i,k ≥ 1. (16) i∈I
˜ RCS . Denote this problem as RCS and its feasible set as D For every solution of RCS one can create a solution of R||Cmax by removing unnecessary allocations and this can be done in polynomial time. Now define a matrix x [xi,k ]i∈I,k∈J with elements obtained through the following transformation xi,k min{1, xi,j,k,1 } i ∈ I, k ∈ J (17) j∈J
where x ∈ D is a solution of TARF. Let us prove that x ∈ ˜ RCS . Constraint (13) is satisfied from (1)∧(7). Constraint D (16) is satisfied from (3). Finally, (15) is satisfied from (4). Next let us show that for every instance of R||Cmax where ˜ = ∅ there exists a solution of a corresponding TARF. D This can be proven by providing a method of constructing ˜ Let Mi {k ∈ J : x ˜i,k = 1} x ∈ D from x ˜ ∈ D. i i and let c [cm ]m∈{1,...,|Mi |} be a series of elements from Mi sorted in ascending order. Then define matrix x [xi,j,k,1 ]i∈I,j∈J,k∈J with elements given by i i 1 if ∃m ∈ Mi : cm = j ∧ cm+1 = k ∨ xi,j,k,1 . (18) m = |Mi | ∧ k = 1 0 if otherwise
Denote as TARF a problem of finding a feasible solution of TAR. Solving TARF is an already difficult problem as is stated in the following theorem. Theorem 1. (about the complexity of TARF). Problem R||Cmax can be reduced to an instance of TARF.
Elements from x define |I| routes with no subcycles or loops that execute task 1 ∈ J (go through the corresponding location) and as such they satisfy constraints (5), (6), (7), (8). Constraint (1) is satisfied from the definition of x. Then (2) is satisfied from (1) and from |L| = 1. Constraint (3) is satisfied from (14) and finally (4) is satisfied from (15). The fact that transformation (17) is executed in polynomial time concludes the proof.
Before proving the theorem about the complexity of TARF, let us recall the linear programming formulation of R||Cmax .
Theorem about the complexity of TARF guarantees that neither approximation algorithms nor any heuristics with polynomial execution time exist (unless P=NP) for the
x∈D
D {x ∈ X : (1) ∧ . . . ∧ (8)}.
(11)
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general case of TAR. Bearing such difficulty in mind, this work develops an approach based on solving a substitutive problem with weaker constraints and relates its solution to a potential solution of TAR.
∀i ∈ I, l ∈ L x˙ i,k,l ∈ {0, 1}, ∀i ∈ I x˙ i,k,l ≤ 1,
3. SOLUTION ALGORITHM
l∈L
∀k ∈ J and a resource limit ∀i ∈ I
x ˘i,k,l ηi,k,l ≥ E,
(21)
x ˘i,k,l ei,k,l ≤ F.
(22)
i∈I,l∈L
k∈J,l∈L
The decision making problem is as follows. Problem 3. (MTA – multi-task allocation). Given: I, J, L, η, e, E, F Find: x ˘∗ where ˘ x), Q(˘ ˘ x) x ˘i,k,l ei,k,l , (23) x ˘∗ arg min Q(˘ ˘ x ˘ ∈D
i∈I,k∈J,l∈L
˘ {˘ ˘ Xi∈I,k∈J,l∈L R : (19) ∧ . . . ∧ (22)}. D x∈X
(24)
∀i ∈ I
and let x˙ k [x˙ i,k,l ]i∈I,l∈L be a decision variable (for a fixed k ∈ J) with binary elements. The problem of a single task allocation is given as follows. Problem 4. (STA – single task allocation). Given: k ∈ J, E, F, I, η, e, x ˘ Find: x˙ ∗k where x˙ i,k,l ei,k,l (27) x˙ ∗k arg min ˙k x˙ k ∈D
i∈I,l∈L
˙ k {x˙ k ∈ ×i∈I,l∈L R : (29) ∧ (30) ∧ (31) ∧ (32)} (28) D
x˙ i,k,l ηi,k,l ≥ E,
l∈L
ˆk x ˆ k ∈D
x˙ i,k,l eˆi,k,l ≤ F.
(31) (32)
i∈I,l∈Li
ˆ k {ˆ D xk ∈ ×i∈I,l∈Li R : (35) ∧ (36) ∧ (37)} ∀i ∈ I, l ∈ Li x ˆi,k,l ≥ 0, ∀i ∈ I x ˆi,k,l ≤ 1,
i∈I,l∈Li
(34) (35) (36)
l∈Li
x ˆi,k,l ηi,k,l ≥ E.
(37)
RTA is a linear programming problem and can therefore be solved in polynomial time. This solution can then be rounded to become a feasible solution of STA. Rounding algorithm is given as follows. Algorithm 1. (RA – rounding algorithm). Given a solution of RTA x ˆ∗k do the following: (1) For each i ∈ I, l ∈ L\Li set xi,k,l := 0. (2) For each i ∈ I, l ∈ Li if x ˆ∗i,k,l ∈ {0, 1} then set ∗ xi,k,l := x ˆi,k,l . (3) If ∃!(i1 , l1 ) : i1 ∈ I, l1 ∈ Li1 for which x ˆ∗i1 ,l1 ∈ (0, 1) then set xi1 ,k,l1 = 1. (4) If ∃(i1 , l1 ), (i2 , l2 ) : i1 , i2 ∈ I, l1 ∈ Li1 , l2 ∈ Li2 for which x ˆ∗i1 ,k,l1 , x ˆ∗i2 ,l2 ∈ (0, 1) then: if ηi1 ,k,l1 ≥ ηi2 ,k,l2 , set xi1 ,k,l1 = 1, xi2 ,k,l2 = 0. Otherwise, set xi1 ,k,l1 = 0, xi2 ,k,l2 = 1. (5) Rounded solution of RTA is xk [xi,k,l ]i∈I,l∈L .
j∈J,l∈L
j∈J\k,l∈L
(30)
This problem can be solved approximately by solving and rounding its relaxed version which is given as follows. Let Li {k ∈ J : ei,k,l ≤ d ∧ eˆi,k,l ≤ F } for a given k ∈ J, d ≥ 0 and x ˘. Let x ˆk = [ˆ xi,k,l ]i∈I,l∈Li be a decision variable with binary elements. Problem 5. (RTA – relaxed task allocation). Given: k ∈ J, E, I, Li , η, e, x ˘ Find: x ˆ∗k where x ˆ∗k arg min x ˆi,k,l ei,k,l (33)
The MTA problem is also hard in the sense that its feasibility version can be reduced to R||Cmax . Proof is analogous to the proof of Theorem 1; see details in Hojda (2014). Nonetheless, a substitutive version can be solved approximately provided that the solution to MTA exists. Let SMTA be a version of MTA with the following constraint instead of (22) x ˘i,k,l ei,k,l ≤ 2IF. (25) ∀i ∈ I ˘ be the feasible set of SMTA and let x ˘∗ be its Let D solution. A feasible solution to SMTA can be obtained by solving a set of relaxed problems of single-task allocation. For a fixed x ˘, let eˆ [ˆ ei,k,l ]i∈I,k∈J,l∈L where x ˘i,j,l ei,j,l (26) eˆi,k,l ei,k,l +
(29)
l∈L
i∈I,l∈L
In this section we introduce and solve several intermediate problems before tackling TAR itself. Let us first consider the following problem of multi-task allocation where the decision variable x ˘ [˘ xi,k,l ]i∈I,k∈J,l∈L determines the task-executor allocation and mode of execution with partial decisions x ˘i,k,l = 1 if executor i is assigned to task k in mode l (0 if otherwise). The constraints given are binary requirement on decision variables (19) ∀i ∈ I, k ∈ J, l ∈ L x ˘i,k,l ∈ {0, 1}, single mode execution ∀i ∈ I, k ∈ J x ˘i,k,l ≤ 1, (20) task completion
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Rounding algorithm can then be incorporated into a 2approximate algorithm for solving MTA. This algorithm is as follows. Algorithm 2. (MTAsol). (1) For k := 1 to k = J do (a) Using binary search obtain the solution to RTA x ˆk for the smallest possible d. Use x = 0 as the parameter of RTA. (b) Using RA obtain xk from x ˆk . (2) Solution: x ˘r := [xi,k,l ]i∈I,k∈J,l∈L . ˘ ∧ Q(˘ ˘ xr ) ≤ 2Q(˘ ˘ x∗ ) Theorem 2. x ˘r ∈ D While Reader is referred to Hojda (2014) for a detailed proof and an elaboration on this theorem, let us nevertheless recall all the main points of that explanation.
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Proof. (1) Solution of RTA (any cornerpoint solution of the linear programming problem) has at most two nonintegral variables. Those variables are set to integrals in RA. (2) For a solution obtained with RA it holds that xi,k,l ei,k,l ≤ 2 x˙ ∗i,k,l ei,k,l . (38) i∈I,l∈L
i∈I,l∈L
∗
(3) Let x˙ xk be a solution of STA for x ˘=x ˘∗ . There exists ∗ x ˘ for which ∗ (39) ∀i ∈ I, l ∈ L x˙ xi,k,l = x∗i,k,l .
(4) Let x˙ 0k be a solution of STA for x ˘ = 0 it then holds that ∗ x˙ xi,j,k ei,k,l ≥ x˙ 0i,j,k ei,k,l . (40) i∈I,l∈L
i∈I,l∈L
(5) Combining (38), (39), (40) we show that xi,k,l ei,k,l ≤ 2Q(˘ x∗ ) ≤ 2IF.
(41)
i∈I,k∈J,l∈L
(6) Verification of (19), (20), (21) for x ˘r ends the proof.
Let us relate the solution of MTA to the solution of TAR. ˘ x∗ ) ≤ Q(x∗ ). Lemma 3. Q(˘ Proof. Consider w(x) [w(x)i,k,l ]i∈I,k∈J,l∈L with elements defined as follows xi,j,k,l . (42) w(x)i,k,l j∈J
˘ and It is sufficient to show that x ∈ D =⇒ w(x) ∈ D ˘ Q[w(x)] = Q(x). Constraints of MTA are satisfied for w(x) as follows: (19) from the definition of w(x), (20) from (2) and (7), (21) from (3) and (7), finally (22) from (4) and ˘ (7). Then we expand Q[w(x)] to obtain ˘ Q[w(x)] = w(x)i,k,l ei,k,l = (43) i∈I,k∈J,l∈L
=
xi,j,k,l ei,k,l = Q(x).
(44)
˜i ∀j ∈ J
yi,1,k = 1,
˜i k∈J
yi,j,k =
˜i j∈J
yi,k,j ,
(47) (48)
˜i j∈J
˜ i ∧ S = ∅ ∧ S = J ˜i ∀S ⊂ J
j∈S,k∈S
Also consider quality criterion Qiy (yi )
yi,j,k ≥ 2.
yi,j,k µi,j,k .
(49)
(50)
˜ i ,k∈J ˜i j∈J
The problem of mobile routing for a given executor and an instance of SMTA is as follows. Problem 6. (MR – mobile routing). ˜ i , µ Find: y ∗ where Given: i ∈ I, J i (51) yi∗ arg mini Qiy (yi ), y∈Dy
Diy
{yi ∈ Xj∈J˜ i ,k∈J˜ i R : (46) ∧ (47) ∧ (48) ∧ (49)}. (52)
MR is in fact an instance of the Travelling Salesman Problem. Since we assumed that µ is metric, so is this TSP. It can therefore be solved approximately (for example, by the means of the cheapest insertion algorithm). The approximation ratio is then given as follows (53) Qiy (y i ) ≤ αQiy (yi∗ )
under the assumption that y i ∈ Diy was derived using an α-approximate algorithm. Another important property of MR in relation to TAR is given as follows. Lemma 4. Qiy (y i ) ≤ α i∈I,j∈J,k∈J,l∈L x∗i,j,k,l µi,j,k . Proof. Let us notice that x∗ defines a route (over all executors) that visits all J tasks. Let C be the cost of the cheapest route (costs µ) over all J tasks. MR finds the ˜ i ⊂ J therefore cheapest route over J Qiy (yi∗ ) ≤ C ≤ x∗i,j,k,l µi,j,k , (54) i∈I,j∈J,k∈J,l∈L
From (53) and (54) we have Qiy (y i ) ≤ α
x∗i,j,k,l µi,j,k .
(55)
i∈I,j∈J,k∈J,l∈L
i∈I,k∈J,l∈L j∈J
In consequence, we have ∗ ˘ ˘ x∗ ) ≤ Q[w(x )] ≤ Q(x∗ ) Q(˘
˜i ∀j ∈ J
(45)
which ends the proof.
The following analysis is performed under two assumptions: that µ defines a metric (one for each executor) and that executors are identical. We derive an approximation algorithm for a substitutive version of TAR under those assumptions and empirically evaluate the algorithm when they do not hold. Let us now consider a set of routing problems, one for each executor. Those are formulated for a given instance and a ˜ i {1, . . . , J˜i } be a set feasible solution of SMTA x ˘r . Let J of tasks to which the ith executor was assigned. Then for a ˜ i {k ∈ J : ˘ri,k,l = 1}. Define given i ∈ I we have J l∈L x a decision variable yi [yi,j,k ]j∈J,k∈ ˜ ˜ i with elements J yi,j,k = 1 if task k is executed directly after task j (0 if otherwise). Consider the following constraints. ˜i, k ∈ J ˜ i yi,j,k ∈ {0, 1}, ∀j ∈ J (46)
˜ i (= 0 if otherwise) be Let y˘i,j,k = y i,j,k if j, k ∈ J the approximate solution for instances of MA solved for x ˘r obtainedy through MTAsol. Consider the following constraint xi,j,k,l (ei,k,l +µi,j,k ) ≤ (2+α)|I|F. (56) ∀i ∈ I j∈J,k∈J,l∈L
The substitutive TAR problem is given as follows. Problem 7. (STAR – substitutive TAR). Given: I, J, L, e, η, µ, F, E ∗ Find: x where ∗ x arg min Q(x),
(57)
x∈D
D {x ∈ X : (1) ∧ (2) ∧ (3) ∧ (56) ∧ (5) ∧ . . . ∧ (8)}. (58) Consider x ´ [´ xi,j,k,l ]i∈I,j∈J,k∈J,l∈L with elements defined as follows x ´i,j,k,l y˘i,j,k x ˘ri,k,l . (59) The following algorithm solves STAR in an approximate fashion.
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Algorithm 3. (STARsol). (1) Formulate and solve SMTA to obtain x ˘r . (2) Formulate and solve MR under x ˘r . Obtain y. (3) Use (59) to obtain x ´. This algorithm works in polynomial time since all its components work in polynomial time. It provides a solution with quality given by the following theorem. Theorem 5. x ´ ∈ D ∧ Q(x) ≤ (2 + α|I|)Q(x∗ ). ˘ and Proof. Property x ´ ∈ D comes directly from x ˘r ∈ D i y ∈ Dy . Then let us notice that x ´i,j,k,l µi,j,k = i∈I,j∈J,k∈J,l∈L
=
˜ i ,k∈J ˜i i∈I,j∈J
and
Qiy (y i )
(60)
˘ xr ). x ˘ri,k,l ei,k,l = Q(˘
(61)
y˘i,j,k µi,j,k =
Fig. 1. Experiment 1: Dependence of Q on J.
i∈I
x ´i,j,k,l ei,k,l =
i∈I,j∈J,k∈J,l∈L
=
˜ i ,l∈L i∈I,k∈J
Expanding Q(x) and from (60), (61), Lemma 3 and Lemma 4 we have Q(x) = y˘i,j,k x ˘ri,k,l ei,k,l + (62)
Fig. 2. Experiment 2: Dependence of Q on J.
i∈I,j∈J,k∈J,l∈L
+
y˘i,j,k x ˘ri,k,l µi,j,k =
(63)
i∈I,j∈J,k∈J,l∈L
˘ xr ) + = Q(˘
i∈I
Qiy (y i ) ≤ (2 + α|I|)Q(x∗ )
(64)
which ends the proof.
Algorithm for solving STAR can still be used as a heuristic algorithm in cases when executors are not identical or µ is not metric. Both cases, identical and unrelated executors are evaluated empirically in the following section. 4. ALGORITHM EVALUATION Let us first present selected preliminary empirical evaluations of the algorithms presented in the previous section. Those evaluations will be followed by comments regarding consequences of the results from the viewpoint of practical implementations. Experiment 1 illustrates the performance of the algorithm for a case of identical executors and tasks located on a line with even distances between neighbouring tasks. The case of unrelated executors is analysed in Experiment 2 and Experiment 3. Results are presented in respective figures. Experiment 1. Consider TAR with parameters I = 5, 5 ≤ J ≤ 25, L = 2, E ∈ {1, 3, 5}, F = 200, ei,k,l = 10 + k, ηi,k,l = l, µi,j,k = |j − k|. Experiment 2. Consider TAR with parameters I = 5, 5 ≤ J ≤ 25, L = 2, E ∈ {20, 25, 30}, F = 200, ei,k,l = 10 + 5i, ηi,k,l = l · i · k, µi,j,k = |j − k|. Experiment 3. Consider TAR with parameters I = 5, 5 ≤ J ≤ 25, L = 2, E ∈ {20, 25, 30}, F = 200, ei,k,l = 10 + 5i, ηi,k,l = l · i · k, µi,j,k = 10|j − k|.
Fig. 3. Experiment 3: Dependence of Q on J. In the first experiment, cost of execution grows with the number of tasks. This is reflected in the value of the objective function which increases at a growing rate. The rate of growth is dependent on the completion rate E in a proportional manner. It is quite unlike that in the second experiment when decrease from E = 30 to E = 25 causes a smaller impact than decrease from E = 25 to E = 20. This is caused by removal of less efficient executors from the execution process. First experiment also illustrates a clear violation of constraints of the TAR problem which is visible for the case of E = 5. Such violations are to be expected in the light of the properties proven in the previous sections. Finally, in the third experiment we can observe the influence of increased distances on the solution algorithm. Clearly visible nonlinearities are a result of greater influence of distance on the solution. As routes become more costly, so does the the value of the quality criterion. It the light of the results obtained through empirical evaluation it is essential to comment on possible applications of the evaluated algorithms. As already stated in Section 2, from Theorem 1 directly follows that no algorithm can
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obtain a feasible solution to TAR in polynomial time. It is therefore expected that solutions obtained through the use of STARsol will not, in general case, satisfy all the constraints imposed by TAR and this constraint violation is clearly visible in the empirical evaluations. The most straightforward application of such a result is to treat it as a cue to increase the available resources by the necessary amount ordained by the solution. However, for the case of identical executors and metric distances, another approach might be also possible, depending on actual numerical data of the problem. That approach is to solve STAR for a reduced amount of resources, i.e. for a new limit in constraint (56) calculated as F˜ = F/(2+α)|I|. If a solution to STAR with the new limit can be obtained through STARsol then the resulting x ´ is a feasible solution of TAR with the resource limit of F . There are, unfortunately, no guarantees that such a solution to STAR exists for every TAR formulation. 5. SUMMARY
Proceedings of the 27th IEEE Conference on Computer Communications, pages 1112–1120, 2008. T. Melodia, D. Pompili, I. Akyildiz. Handling mobility in wireless sensor and actor networks. IEEE Transactions on Mobile Computing, volume 10 (2), pages 160–173, 2010. T. Nagarajan, A. Thondiyath. An algorithm for cooperative task allocation in scalable, constrained multiple robot systems. Intelligent Service Robotics, volume 7 (4), 2014. O. Tekdas, V. Isler. Using Mobile Robots to Harvest Data from Sensor Fields. IEEE Wireless Communications, volume 16 (1), pages 22–28, 2009. F. Yong, S. Mo, G. Hackmann, L. Chenyang. Practical control of transmission power for wireless sensor networks. Proceedings of 20th IEEE International Conference on Network Protocols, pages 1–10, 2012. M. Zhenqiang, Y. Yang, M. Huan, W. Dandan. Connectivity preserving task allocation in mobile robotic sensor network. Proceedings of IEEE International Conference on Communications, pages 136–141, 2014.
Tackled in the paper was a problem of joint task allocation and routing for mobile executors with resource constraints. Proven difficulty of finding even a feasible solution in polynomial time led to the development of an approximation algorithm for a substitutive problem. An approximation ratio was derived for a special case of metric distances and uniform distributors. Nevertheless, it was shown that the algorithm is also applicable for cases where those assumptions do not hold. It is crucial for future research to further evaluate the possibility of obtaining a solution algorithm with better approximation bound for both, the criterion and the constraint violation as well as expanding analytical results on other cases, especially for the most practical case of unrelated executors. REFERENCES A. Abuarqoub, M. Hammoudeh, T. Alsboui. An overview of information extraction from mobile wireless sensor networks. Lecture Notes in Computer Science. Internet of Things, Smart Spaces, and Next Generation Networking Springer-Verlag, volume 7469, pages 95–106, 2012. I. Akyildiz, I. Kasimoglu. Wireless sensor and actor networks: Research challenges. Ad Hoc Networks, volume 2 (4), pages 351–367, 2004. B. Coltin, M. Veloso. Mobile Robot Task Allocation in Hybrid Wireless Sensor Networks. Proceedings of International Conference on Intelligent Robots and Systems, pages 2932–2937, 2010. N. Correll, A. Martinoli. Multirobot inspection of industrial machinery. IEEE Robotics and Automation Magazine, volume 16, pages 103–112, 2009. M. Hojda, J. Jozefczyk. Decision making algorithm for a class of two-level manufacturing systems. Kybernetes, volume 38, pages 1355–1372, 2009. M. Hojda. Multi-robot task allocation algorithms in systems with energy constraints. (in Polish) Automation of Discrete Processes: theory and applications (pl. Automatyzacja Procesw Dyskretnych: teoria i zastosowania), volume 2, pages 105–113, Gliwice, Poland, 2014. D. Jung, A. Savvides. An energy efficiency evaluation for sensor nodes with multiple processors, radios and sensor. 2188