Research ArticleA Balanced Heuristic Mechanism for Multirobot TaskAllocation of Intelligent Warehouses

Luowei Zhou Yuanyuan Shi Jiangliu Wang and Pei Yang

Department of Control and System Engineering School of Management and Engineering Nanjing University Nanjing 210093 China

Correspondence should be addressed to Pei Yang yangpeinjueducn

Received 19 August 2014 Revised 21 October 2014 Accepted 21 October 2014 Published 11 November 2014

Academic Editor Hari M Srivastava

Copyright copy 2014 Luowei Zhou et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper presents a newmechanism for the multirobot task allocation problem in intelligent warehouses where a team of mobilerobots are expected to efficiently transport a number of given objectsWemodel the systemwith unknown task cost and the objectiveis twofold that is equally allocating the workload as well as minimizing the travel cost A balanced heuristic mechanism (BHM)is proposed to achieve this goal We raised two improved task allocation methods by applying this mechanism to the auction andclustering strategies respectivelyThe results of simulated experiments demonstrate the success of the proposed approach regardingincreasing the utilization of the robots as well as the efficiency of the whole warehouse system (by 5sim15) In addition the influenceof the coefficient 120572 in the BHM is well-studied Typically this coefficient is set between 07sim09 to achieve good system performance

1 Introduction

Autonomous guided vehicles (AGVs) have been used toperform tasks in warehouses for more than fifty years [12] In the past few years they have been mostly used totransport large or heavy objects such as rolls of uncut paper orengine blocks Recently as autonomous robots [3ndash5] becomecheaper smaller and more capable they are widely usedin the logistic industry for example being substituted forthe manual labor in the typical pick-pack-and-ship processin warehouses [6] Moreover in [7 8] it is pointed outthat the multirobot coordination if being really efficientand robust will put significant impact on improving thelogistical efficiencyThemultirobot coordination involves theallocation and execution of individual tasks with an efficientmechanism In this paper we mainly focus on the taskallocation process in which a team of autonomous robotsmust fulfill a set of orders in optimal routes meeting withcertain criteriaThis problem is also known asmultirobot taskallocation problem (MRTA) [9]

There are already some methods to deal with MRTA In1998 Yamauchi presented a greedy strategy [10] to coordinaterobots with which each robot chooses the closest task assoon as they finish their previous tasks Gerkey and Mataric

introduced a dynamic task allocation method [11] for groupsof autonomous robots with which the robots bid on allunallocated targets and the bids depend on the distancebetween their last targets and those unallocated targetsThe algorithm proposed by Sandholm [12] known as thecombinatorial auction method is considered more efficientbecause it divides tasks into different clusters according tocorrelations between tasks then robots bid on clusters of tasks[13] In 2004 Lagoudakis et al used Prim Allocation [14] togenerate the MSF (minimum spanning forest) of tasks forrobots and then transformed the forest in paths by the depth-first traversal Later some researchers [15 16] improved thedepth-first traversal process in the prim allocation methodand better results are obtained

As a matter of fact most of current attempts focuson minimizing the travel cost of the robots group withoutconcerning the travel time that is balancing the individualtravel cost of each robot There are few research worksputting emphasis on improving the utilization ofmultirobotsespecially in many time sensitive applications such as thelogistics industry rescue and exploration environmentsPuig et al [17] improved Yamauchirsquos greedy algorithm tosolve the balancing-target allocation problem in the planetaryexploration By this method after finishing its current task

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 380480 10 pageshttpdxdoiorg1011552014380480

2 Mathematical Problems in Engineering

every robot moves to the frontier cell until all the tasks havebeen explored However since every robot only acts for itselfwith no coordination with other robots this method maylead to a rather suboptimal solution in travel distance whichis not acceptable in our logistic application In 2011 Elangoet al [18] proposed the 119870-means clustering and auctionbasedmechanism (referred as119870CABmechanism) to balancethe task allocation in multirobot systems This approachconsiders the dual goal of bothminimizing the travel distanceand efficient sharing of the workload Nevertheless thecomplexity of this algorithm is relatively high and we willcompare this method with our proposed approaches later

Considering the specific characteristics of our problem(i) balancing the task allocation as well as minimizingthe travel distance (ii) the cost of tasks are unknownand (iii) large scale and dynamic environment a balancedheuristic mechanism (BHM) is proposed to coordinate theautonomous robots and is applied to several traditionalapproaches such as the auction allocationmethod and the119870-means clusteringmethod To evaluate the performance of dif-ferent strategies three criteria are provided as the travel time(TT) coefficient of variation (CV) between different robotsand total travel cost (TTC) Simulation results strongly provethat the BHM has a better balancing performance comparedwith traditional methods

The rest of this paper is organized as follows In Section 2we give the formal description and model of the order ful-fillment process in the warehouse In Section 3 we describethe overall control structure of the system and presentthe balanced heuristic mechanism (BHM) Then simulatedexperimental results are provided and compared with severaltraditional methods in Section 4

2 Problem Formulation

21 Problem Statement In this paper we focus on thedynamic task allocation problem for multirobots in anintelligent warehouse To characterize the system someassumptions are listed as follows

(i) the system is composed of large-scale physicalembedded robots

(ii) the robots and the tasks are homogenous(iii) the environment of the system is known to each robot(iv) each robot in the system is self-interested and fulfills

its own tasks independently(v) the robotsmay conflictwith each otherwhen fulfilling

their tasks and they communicate with each otherwhen being close to other vehicles (in purpose ofobstacle avoidance)

(vi) the time consuming at the station is the same for allrobots

Usually the multirobot task allocation is described asfollows given a sequence of tasks and a set of robots the goalis to assign tasks to robots in an efficient manner based onwhat the system is trying to optimize In our approach westrive to optimize the system in a dual objective function to

Picking station

Robot Storage shelves

Pod

Figure 1 Configuration of a warehouse system

keep the total travel cost as low as possible and keep the indi-vidual travel time as equal as possible respectivelyThese twoobjectives correspond to a balance between time expensesand robot expenses both of which deserve consideration in aconcrete physical system As shown in Figure 1 each storageshelf consists of several inventory pods and each pod consistsof several resources By order a robot lifts and carries a podat a time along a preplanned path deliver it to the specificstation which is appointed in the order and finally returns thepod backThemain consideration is how to allocate the tasksto the robots efficiently and equally while avoiding conflicts

Another point that must be taken into account is that tra-ditional task allocationmethods are based on the assumptionthat the cost of each task is already known However in thisproblem the input to the warehouse system is a sequenceof orders from which we can only know the location of thetask and the specific station to which the inventory pod issupposed to be delivered The cost of each task as well as thetravel cost from one location to another is totally unknown

22 Modelling The model formulated to solve the balancedmultirobot task allocation problem mentioned above in thewarehouse system is as follows (as shown in Figure 2)

Let a set of robots 119877 = 1199031 119903

2 119903

119898to complete a set of

tasks 119879 = 1199051 119905

2 119905

119899 The cost of task 119905

119894is 119908119894 which refers

to the travel cost of carrying the inventory pod to the specificstation and then delivering it back ignoring the time cost ofobstacle avoidance and 119862

119894119895(119894 119895 isin 119877 cup 119879) is the travel cost

between every two locations (usually from the robot to theinventory pod assigned in the order) Suppose 119903

119894have 119896 tasks

119879 = 119905

1198941 119905

1198942 119905

119894119896 then the total cost of all the assigned tasks

in 119879119894is expressed as119882(119903

119894) = 119908

1198941+119908

1198942+ sdot sdot sdot +119908

119894119896 More details

about the basic parameters and indexes using in the modelare shown in Figure 2

Rewrite 119879 as Γ = 1198791 119879

2 119879

119898 which means a partition

of the set of tasks where task set 119879119894are allocated to robot 119903

119894

Then define the individual travel cost as ITC(119903119894 119879

119894) which

represents the sum cost for 119903119894to fulfill its task set 119879

119894and can

be calculated by

ITC (119903119894 119879

119894) = 119862

119903119894 1199051198941+

119896minus1

sum

119895=1

119862

119905119894119895 119905119894(119895+1)+119882(119903

119894) (1)

Mathematical Problems in Engineering 3

T1

t11

t13

t12

t14

t15

t16t21

t22

t23t24

t25

T2

r1 r2

middot middot middot

middot middot middotCt16 t22

Ct11t14

Ct23 t25

Cr1t14Cr2t15

Tasks T1 assigned to r1 Tasks T2 assigned to r2

Figure 2 Basic parameters and indexes

where 119862119903119894 1199051198941

represents the travel cost for 119903119894to come to its first

task 1199051198941 and sum119896minus1

119895=1119862

119905119894119895 119905119894(119895+1)represents the total travel cost for

the rest 119896 minus 1 tasks Also119882(119903119894) represents the total task cost

of all the assigned tasks in 119879119894 Our dual goal can be expressed

as

119878 = minΓ

max119894

ITC (119903119894 119879

119894) (2)

which aims to minimize the travel cost as well as balances theindividual travel cost of each robot

Furthermore we provide three criteria to evaluate theperformance of different strategies travel time (TT) whichequals to the maximum of all the individual robotsrsquo travelcosts total travel cost (TTC) which reveals the total travelcost of all the robots evaluates the power consumption andmechanical loss of robots coefficient of variation (CV)whichis a statistical measurement commonly used for comparingthe diversity in work groups [19]

TT = max119894

ITC (119903119894 119879

119894)

TTC =119898

sum

119894=1

ITC (119903119894 119879

119894)

CV = standard deviation (120590)mean (120583)

(3)

where 120590 represents the standard deviation of ITC(119903119894 119879

119894) and

120583 is the mean of ITC(119903119894 119879

119894) 119894 = 1 2 119898

It is worth to point out that we use TTC to evaluate thepower consumption of the system On one hand comparedto the travel energy cost of the robots other energy costs suchas the communication cost and central controller cost arerelatively small so that they can be neglectedwithout affecting

the overall power consumption On the other handsum

119898

119894=1119882(119903

119894) (distance that robots running with pods) is

constant for a system which means the power consumptionof this part is unchanged for a certain system So we need tominimize sum119898

119894=1(119862

119903119894 1199051198941+ sum

119896minus1

119895=1119862

119905119894119895 119905119894(119895+1)) (distance that robots

running without pods) which can be fulfilled by minimizingTTCThus TTC can be used to evaluate the power consump-tion of the system

3 Method

In this section we develop the BHM based approach for thetask allocation of multirobots for the intelligent warehouseWe firstly provide an overview of the whole control systemThen the BHM principles are described in detail By intro-ducing thismechanism into the traditional auction allocationmethod and the 119870-means clustering method two improvedmethods based on BHM are presented and analyzed for thetask-allocation problem

31 Overall System The work space of the warehouse canbe divided into several grids with inventory storage zonesin the middle surrounding with the stations and workersAutonomous robots transport movable inventory pods fromstorage locations to stations where workers can pick itemsoff and pack them up then those packages are sent to thecustomers

In technical aspects the task allocation process is regu-lated by the central controller while the path planning andmotion planning for example the obstacle avoidance arefulfilled by robots In the task allocation process many cur-rent allocation mechanisms tend to use the dynamic method[20 21] however it may result in suboptimal performanceThus an approach is proposed combining both dynamic andstatic methods by means of the task pool The continuouslyarriving orders accumulate in the task pool The controllerrequires a certain number of tasks from the task pool after theprevious batch of tasks has been completed and uses a certainapproach to allocate those tasks to robotsThen the controllersends sets of tasks to corresponding robots by wirelesscommunication such as ZigBee and the robots completetheir tasks sequentially by a standard Alowast algorithm Usuallywe use the Alowast algorithm to find the shortest path betweenstorage locations and stations in a static environment whilein an uncertain environment the learning methods (egreinforcement learning [22 23]) may be necessary to find theshortest path which will be our future work and is beyondthe scope of this paper

In addition as for the obstacle avoidance process robotsfirstly use infrared detectors to detect other robots withincertain distance (usually several grids in front of them)Then they communicate with each other by wireless devicesso that they can verify their relative priorities Generallyspeaking the robot with more tasks left to be completedhas priority to occupy front grids Therefore the robot withfewer tasks left is likely to give way to the robot with moretasks to be completed So the actual balancing performance

4 Mathematical Problems in Engineering

Costevaluation

Path planning and motion(obstacle avoidance)

Decentralized controller

Central controller

(2) Task completion

Orderinput

Wirelesscommunication

R1

R2

R3

middot middot middotTask allocation(using BHM)

(1) Accurate Wij Cij

Figure 3 Control system structure

could be further enhanced in the real world comparedwith the simulated balancing performance of the simulatedexperiment This mechanism is referred as more-task-priormechanism in this paper The whole process is shown inFigure 3

32 Balanced Heuristic Mechanism (BHM) Considering thegoal of minimizing the travel cost as well as balancing theindividual travel cost of each robot a heuristic function isgiven in the following for guiding the task allocation process

119862 (119903

119894 119905

119895) = 120572 times dist (119903

119894 119905

119895) + (1 minus 120572) times 119862119882(119903

119894) (4)

where 119862(119903119894 119905

119895) describes the cost (considering both the time

and distance) for robot 119903119894to finish task 119905

119895 dist(119903

119894 119905

119895) is the

travel cost from robot 119903119894to task 119905

119895 and 119862119882(119903

119894) is the current

value of119882(119903119894) (refer to Section 2)

More specifically as both task cost and travel cost fromone location to another are unknown we evaluate 119908

119894(the

cost of each task) and119862119894119895(the travel cost from one location to

another) before the task allocation by using the standard Alowastalgorithm a simple but significantly effective path planningalgorithm When a robot fulfills a task or moves from onelocation to another in real environment more complexmotions should be taken into account such as the obstacleavoidance Then the more accurate value is shared with thecentral controller and the original estimated value stored inthe controller is replaced with the newest value which canhelp the controller to make a better task allocation Thus thefollowing allocation process is in the light of new informationand can be more accurate and predictable

BHM makes tasks to be assigned to the robot who hasthe lowest 119862(119903

119894 119905

119895) By introducing a parameter 120572 targets are

more inclined to be assigned to the nearby robot with fewerallocated tasks In practice an effective value of 120572 can be setbetween 07 and 09 according to the numbers of robots andtasks The detailed relationship between 120572 and the systemperformance will be given in Section 4

At the beginning stage as the value of119882(119903119894) is quite small

this mechanism has limited effects Thus the performance

of the system is not significantly superior to the one bytraditional methods that merely focus on minimizing thetotal travel distance However as the allocation proceeds therole of (1 minus 120572) times 119862119882(119903

119894) comes into effect by sharing the

workload with those who have fewer assigned tasks Aboveall the method can obtain an approximately optimal resultby considering the total travel cost and the workload balanceof robots A sample illustration of this heuristic mechanismis given in Figure 4

33 Application of BHM Recently the auction-based taskallocation method and the cluster-based task allocationmethod have been widely used to solve the MRTA problems[18 24] So we introduce our BHM approach into the classicsingle-item auction method and the 119870-means clusteringmethod respectively and proposed two BMH-based meth-ods to solve the balancing MRTA problem

331 An Improved Auction Based Task Allocation MethodUsing BHM A typical seal-bid single-round single-itemauction process generally consists of four steps

(i) task announcement(ii) matrix evaluation(iii) bid submission(iv) close of auction (to determine the winning bids and

notify the winning robot)

At the second stage thematrix is usually defined as a functionof the optimistic travel cost [11] as shown in the followingmatrix

(

(

119889

1199031 1199051119889

1199031 1199052sdot sdot sdot sdot sdot sdot 119889

1199031 119905119899

119889

1199032 1199051119889

1199032 1199052sdot sdot sdot sdot sdot sdot 119889

1199032 119905119899

d

d

119889

119903119898 1199051119889

119903119898 1199052sdot sdot sdot sdot sdot sdot 119889

119903119898 119905119899

)

)

(5)

Mathematical Problems in Engineering 5

tj

rmrm rnrn

Dist (rn tj) = 20 Dist (rn tj) = 20 Dist (rn tj) = 10Dist (rn tj) = 10

CW(rm) = 300 CW(rm) = 300CW(rn) = 340 CW(rn) = 320

C(rm tj) = 07 times 20 + 03 times 300 = 104radic

C(rn tj) = 07 times 10 + 03 times 340 = 109

C(rm tj) = 07 times 20 + 03 times 300 = 104

C(rn tj) = 07 times 10 + 03 times 320 = 103radic

tj

Figure 4 Example of the balanced heuristic mechanism

C(r1 t1) C(r1 t2)

C(r2 t1) C(r2 t2)

C(ri t1) C(ri t2)

C(rm t1) C(rm t2)

C(r1 tj)

C(r2 tj)

C(rm tj)

C(r1 tn)

C(r2 tn)

C(ri tn)

C(rm tn)

middot middot middot

middot middot middot

middot middot middot

middot middot middot middot middot middot middot middot middot

middot middot middot middot middot middot

middot middot middot

middot middot middot middot middot middot

middot middot middot

⋱

⋱

⋱⋱

⋱

Delete

RefreshC(ri tj)

Figure 5 Evaluation matrix refreshment

where 1199051 119905

2 119905

119899 are all the open tasks which have not

been carried out by robots In our method the BHM isintroduced into the matrix evaluation process In orderto strengthen the balancing performance of the auctionthe matrix evaluation is defined by 119862

119903119894 119905119895rather than the

optimistic travel distance between 119903119894and 119905119895

In each round after all of the robots submit their bidsthe single-round single-item auction is closedThe controllerwill decide thewinning bidderwith the smallest bid value andnotify the winner Then the bid matrix will be refreshed withfollowing steps (as shown in Figure 5)

Step 1 Delete the task 119905119896from the open-task set

Step 2 Recalculate the row (119862119903119894 1199051 119862

119903119894 1199051 119862

119903119894 119905119899)

As described in (4) the value of 119862(119903119894 119905

119895) is both related to

the distance between 119903119894and 119905119895as well as the already assigned

tasks of 119903119894 Firstly the current position of 119903

119894is replaced by

the location of 119905119895 since 119903

119894wins the bid and has to move

from its former position to 119905119895in order to carry out the

order Secondly the119862119882(119903119894) increases by the cost of the newly

allocated task which can be mathematically expressed as119862119882(119903

119894) = 119862119882(119903

119894) + 119908

119894119895 After the matrix is refreshed next

single-item auction process is proceeding until all the taskshave been allocated

332 A Balanced K-Means ClusteringMethod Using BHM Aclustering method can be generally divided into two stagesthat is firstly finding the cluster centers and then assigningthe sample data into the proper clusterThe119870-means cluster-ing is a commonly used partitioning method [25] the mainidea of which is to find119870 cluster centers (119888

1 119888

2 119888

119896) which

minimize119863 in the following by iteration

119863 =

119899

sum

119894minus1

min119903=12119896

119889 (119909

119894 119888

119903)

2

(6)

In (6) 119889(119909119894 119888

119903) represents the distance between the point 119909

119894

and 119888119903 known as the Euclidean distance When 119863 converges

to a stable value the appropriate (1198881 119888

2 119888

119896) are the cluster

centers for those119870 subgroupsAt the second stage the sample data 119909

119894is assigned into

the proper cluster with the smallest 119889(119909119894 119888

119903) 119903 = 1 2 119896

6 Mathematical Problems in Engineering

However this method only guarantees an optimal allocationscheme minimizing the travel cost while the size differencesbetween each cluster are ignored

In order to solve the balancing MRTA problem we makesome modifications on both of the two stages As for the firststage given that robots can only move along the grid lineswe substitute the Euclidean distance with the Manhattandistance in (6) In a rectangular coordinate system theManhattan distance between point 119875

1 (119909

1 119910

1) and point

119875

2 (119909

2 119910

2) can be expressed as

119889manhattan (1198751 1198752) =1003816

1003816

1003816

1003816

119909

1minus 119909

2

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119910

1minus 119910

2

1003816

1003816

1003816

1003816

(7)

Thus the main purpose is to find 119870 cluster centers(119888

1 119888

2 119888

119896) which minimize 119863manhattan in the following by

iteration

119863manhattan =119899

sum

119894minus1

min119903=12119896

119889manhattan (119909119894 119888119903) (8)

Then to reduce the value119863manhattan iteratively we find theManhattan center rather than the center of the collected tasksFinally a partial optimal solution is formed in a finite numberof iterations Generally speaking there are three steps of thefirst stage

(1) Initial step give an initial set of119870 cluster centers(2) Assignment step assign each point to the cluster center

which has the shortest Manhattan distance(3) Update step calculate the Manhattan Center which

can reduce the overall value of 119863manhattan The Man-hattan Center is defined as follows

It is supposed that there are 119898 points in a rectan-gular coordinate system as 119901

1 119901

2 119901

119898 coordinates of

which are denoted as (1199091 119910

1) (119909

2 119910

2) (119909

119898 119910

119898)Then the

descending order of the sequence 1199091 119909

2 119909

119898is denoted

as 11990910158401 119909

1015840

2 119909

1015840

119898and the descending order of the sequence

119910

1 119910

2 119910

119898is denoted as 1199101015840

1 119910

1015840

2 119910

1015840

119898 Our purpose is to

find the point 119875119898= (119909

119898119888 119910

119898119888) which minimizes (8)

min119909119898119888119910119898119888

119898

sum

119894=1

(

1003816

1003816

1003816

1003816

119909

119898119888minus 119909

119894

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119910

119898119888minus 119910

119894

1003816

1003816

1003816

1003816

) (9)

Since the values of 119909119898119888

and 119910119898119888

do not affect each otherthe value of sum119898

119894=1|119909

119898119888minus 119909

119894| and sum119898

119894=1|119910

119898119888minus 119910

119894| is minimized

separately It is easy to prove that when 119898 is odd 119909119898119888

is119909

1015840

(119898+1)2so that sum119898

119894=1|119909

119898119888minus 119909

119894| is minimum when 119898 is even

119909

119898119888is between 1199091015840

(119898+1)2and 1199091015840

((119898+1)2)+1so thatsum119898

119894=1|119909

119898119888minus119909

119894|

is minimum We can get 119910119898119888

in the same way Thereforethe Manhattan center 119901

119898119888can be calculated by following

equations

119909

119898119888=

119909

1015840

(119898+1)2119898 is odd

119909

1015840

(119898+1)2+ 119909

1015840

((119898+1)2)+1

2

119898 is even

119910

119898119888=

119910

1015840

(119898+1)2119898 is odd

119910

1015840

(119898+1)2+ 119910

1015840

((119898+1)2)+1

2

119898 is even

(10)

p4(36)

p1(14)

p2(22)

p3(43)

p5(88)

p6(96)

p7(91)

poc (56)

pmc (44)

Original center

Manhattan center

Figure 6 Example of the manhattan center

Figure 7 Example of the clustering result

As mentioned before the Manhattan center has theminimum overall Manhattan distance to these 119898 points bywhich the value of119863manhattan is reduced An example is shownin Figure 6 For the original center 119901

119900119888sum119898119894=1(|119909

119900119888minus119909

119894| + |119910

119900119888minus

119910

119894|) = 37 while for the Manhattan Center 119901

119898119888 sum119898119894=1(|119909

119898119888minus

119909

119894| + |119910

119898119888minus 119910

119894|) = 34

Now we concisely prove the convergence of the algo-rithm Given that the119863manhattan generated by the algorithm isstrictly decreasing and there only exist a finite number of suchpartitions the algorithm will reach a partial optimal solutionin a finite number of iterations

After obtaining the cluster centers the balanced heuristicmechanism is introduced into the second stage of cluster-ing In traditional methods when the 119898 cluster centers(119903

1 119903

2 119903

119898) are found in the first stage of the 119870-means

algorithm the remaining active tasks are allocated into theproper cluster with the smallest 119889(119909

119894 119888

119903) 119903 = 1 2 119896 How-

ever in our approach remaining active tasks are allocatedinto the proper cluster with the smallest 119862(119903

119894 119905

119895) defined in

(4)The main idea is that instead of merely considering the

travel cost the balance of the cluster size is taken into accountThe task tends to be allocated into the nearby cluster withfewer tasks When a new task is added into the cluster 119895the 119862119882(119903

119894) will be refreshed After all the active tasks have

been allocated the genetic-based TSP algorithm [26] is usedto plan the route for tasks in each cluster One example ofthe task allocation is demonstrated in Figure 7 (119898 = 4 119899 =50 120572 = 08) where different colors represent tasks allocatedto different robots Blocks with119883 are clustering centers

Mathematical Problems in Engineering 7

Table 1 Balance behavior of single-item auction versus BHM auc-tion

119877 = 8 Single-item auction BHM auction (120572 = 08)119898 120583 120590 CV = 120590120583 120583 120590 CV = 12059012058350 685 312 456 711 61 86100 1374 315 229 1469 53 36150 1969 553 281 2128 41 19200 2591 449 173 2827 50 18250 3234 439 136 3555 32 09300 3897 609 156 4221 21 05350 4418 701 159 4794 26 05400 5085 775 152 5499 19 04

4 Simulated Experiments

41 Simulation Description Several groups of simulatedexperiments are implemented with OPENGL and the exper-imental process is as follows

Step 1 Initialization

Step 2 Creating orders in the task pool while the arriving rateof orders meets the Poisson distribution

Step 3 Sending a specific number (eg 200) of orders fromthe task pool to the controller

Step 4 Using different methods to allocate the orders (tasks)More specifically there are four approaches namely 119870-means BHM 119870-means auction and BHMauction and theirperformances are compared

Step 5 Sending the allocated tasks by the controller to thecorresponding robots which begin to fulfil their tasks (usingthe Alowast algorithm to calculate the path and using the more-task-prior mechanism to avoid collision) Then go to Step 3

42 Simulation Results Analysis In this part three criteriamentioned in Section 2 are used to evaluate the performanceof the proposed balanced heuristic mechanism that iscoefficient of variation (CV) travel time (TT) and total travelcost (TTC)

First of all we compare the balance behavior of theimproved methods with the typical auction method and the119870-means method results of which are given in Tables 1and 2 where the data are collected by average during fiftytimes of simulation experiments Both Tables 1 and 2 showthat using the proposed BHM the diversity degree in workgroups is significantly reduced which indicates the moreequal workload distribution among robots

As shown in Figures 8 and 9 the travel time (TT) ofthe multirobot system is provided by both the traditionalmethod and the improved method We can conclude that byapplying the BHM based methods the time spent to finish

Table 2 Balance behavior of typical 119870-means versus BHM 119870-means

119877 = 8 119870-means BHM K-means (120572 = 08)119898 120583 120590 CV = 120590120583 120583 120590 CV = 12059012058350 714 310 434 749 50 67100 1390 513 369 1508 57 38150 2058 764 371 2204 49 22200 2751 1116 406 2949 63 21250 3441 1076 313 3684 69 19300 4095 1254 306 4440 65 15350 4776 1484 311 5214 83 16400 5465 1882 344 5895 66 11

50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

7000

8000

Task number

TT

AuctionBHM auction

Figure 8 Time spent by single-item auction versus BHM auction

the given task is much less than the one by using thetraditional method due to an efficient sharing of workload

Comparison results for traditional methods and BHMbasedmethods are demonstrated in Figure 10 bymeans of thetotal travel cost (TTC) from which we can conclude that inbalanced heuristic cases although robots pay more attentionto balancing the workload rather thanmerelyminimizing thetravel cost as traditional methods do the TTC only increasesby about 8 Concerning the obvious increasement of therobots utilization this slightly increased power consumptionis quite acceptable

With the traditional auction method and 119870-meansmethod the task allocation process only concentrates onminimizing the travel cost Once there is a robot nearest toan open task other robots do not attempt to compete for thetask even if they are idle Thus the utilization of the system isnot enough and the time cost to fulfill the given tasks is highWith BHM based methods proposed in this paper all robots

8 Mathematical Problems in Engineering

50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

7000

8000

9000

Task number

TT

K-meansBHM K-means

Figure 9 Time spent by 119870-means versus BHM 119870-means

50 100 150 200 250 300 350 400012345

TTC

Task number

times104

AuctionBHM auction

(a)

50 100 150 200 250 300 350 400012345

Task number

TTC

times104

K-meansBHM K-means

(b)

Figure 10 Total travel cost by traditional methods versus BHM adapted methods

can be utilized in a balanced and effective manner while thetravel cost is quite low

Then we study the influence of the variable 120572 on thesystem performance The value of 120572 describes the weight ofthe system energy saving demand and time saving demandFigure 11 demonstrates the relationship between 120572 and thetravel time (TT) It shows that in order to keep a good balancebetween minimizing the travel distance and efficient sharingof the workload the coefficient is usually set between 07 and09 to achieve the minimum of TT

Finally besides comparing BHM-based approaches withthe traditional auctionmethod and119870-meansmethod we usea more competing algorithm (119870CAB mechanism in [18]) totestify the scientific merit of the BHM scheme (as shownin Figures 12 and 13) We can conclude that by BHM basedmethods the time spent to finish the given task is 5sim15less than the119870CABmechanism meanwhile their total travelcosts are almost the same Thus the benefit of the balancedheuristic mechanism to warehouse systems is significant

5 Conclusion

In this paper a novel balanced heuristic mechanism for themultirobot task allocation problem is presented with respectto the task cost and travel cost To evaluate the proposedapproach we give three criteria namely travel time (TT)total travel cost (TTC) and coefficient of variation (CV)and compare some traditional methods with BHM-basedmethods under those criterions Simulation results show thatthe proposed approach greatly enhances the balance behaviorof the multirobot system reduces the total travel time whileachieves almost the same total travel cost as the traditionalmethods do In addition the influence of the variable 120572 onthe system performance is also studied in detail In our futurework BHM will be applied into more complicated envi-ronments which consist of heterogeneous tasks and robotsFurthermore we will dedicate to enhance the robustness ofthe proposed approach in dynamic environments Finally itis worth to point out that this new mechanism is of great

Mathematical Problems in Engineering 9

01 02 03 04 05 06 07 08 09 108

09

10

11

12

13

14

15

TT

120572

Task number = 50Task number = 100

Task number = 200Task number = 300

Normalized TT versus 120572 (m = 8)

Figure 11 Relationship between 120572 and travel time (TT)

50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

7000

Task number

TT

BHM auction

KCAB mechanismBHM K-means

Figure 12 Comparison between method provided in [15] versusBHM-based approaches (TT)

applicability to other task allocation problems such as theWSANs problem in [27]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Authorsrsquo Contribution

Luowei Zhou and Yuanyuan Shi contributed equally to thiswork

50 100 150 200 250 300 350 400

TTC

Task number

times104

BHM auction

KCAB mechanismBHM K-means

5

45

4

35

3

25

2

15

1

05

Figure 13 Comparison between method provided in [15] versusBHM-based approaches (TTC)

Acknowledgment

This work was supported by the Natural Science Foundationof China under Grants no 61273327 and no 61432008

References

[1] P R Wurman R DrsquoAndrea and M Mountz ldquoCoordinatinghundreds of cooperative autonomous vehicles in warehousesrdquoAI Magazine vol 29 no 1 pp 9ndash19 2008

[2] X Zhai J E Ward and L B Schwarz ldquoCoordinating aone-warehouse N-retailer distribution system under retailer-reportingrdquo International Journal of Production Economics vol134 no 1 pp 204ndash211 2011

[3] C Chen H-X Li and D Dong ldquoHybrid control forrobot navigationmdasha hierarchical Q-learning algorithmrdquo IEEERobotics amp AutomationMagazine vol 15 no 2 pp 37ndash47 2008

[4] S Chen and C Chen ldquoProbabilistic fuzzy system for uncertainlocalization and map building of mobile robotsrdquo IEEE Trans-actions on Instrumentation and Measurement vol 61 no 6 pp1546ndash1560 2012

[5] D Dong C Chen J Chu and T-J Tarn ldquoRobust quantum-inspired reinforcement learning for robot navigationrdquo IEEEASME Transactions on Mechatronics vol 17 no 1 pp 86ndash972012

[6] T Niemueller D Ewert S Reuter et al ldquoTowards benchmarking cyber-physical systems in factory automation scenariosrdquoinKI 2013 Advances in Artificial Intelligence vol 8077 of LectureNotes in Computer Science pp 296ndash299 2013

[7] M Brambilla E Ferrante M Birattari and M Dorigo ldquoSwarmrobotics a review from the swarm engineering perspectiverdquoSwarm Intelligence vol 7 no 1 pp 1ndash41 2013

[8] C Nieto-Granda J G Rogers and H I Christensen ldquoCoor-dination strategies for multi-robot exploration and mappingrdquoExperimental Robotics vol 33 no 4 pp 519ndash533 2013

10 Mathematical Problems in Engineering

[9] B-Y Shih C-Y Chen and W Chou ldquoAn enhanced obstacleavoidance and path correction mechanism for an autonomousintelligent robot withmultiple sensorsrdquo Journal of Vibration andControl vol 18 no 12 pp 1855ndash1864 2012

[10] B Yamauchi ldquoFrontier-based exploration using multiplerobotsrdquo in Proceedings of the 2nd International Conference onAutonomous Agents pp 47ndash53 ACM May 1998

[11] B P Gerkey and M J Mataric ldquoSold auction methods formultirobot coordinationrdquo IEEE Transactions on Robotics andAutomation vol 18 no 5 pp 758ndash768 2002

[12] T Sandholm ldquoAlgorithm for optimal winner determination incombinatorial auctionsrdquo Artificial Intelligence vol 135 no 1-2pp 1ndash54 2002

[13] M Berhault H Huang P Keskinocak et al ldquoRobot explorationwith combinatorial auctionsrdquo in Proceedings of the IEEERSJInternational Conference on Intelligent Robots and Systems vol1ndash4 pp 1957ndash1962 October 2003

[14] M G Lagoudakis M Berhault S Koenig P Keskinocak andA J Kleywegt ldquoSimple auctions with performance guaranteesfor multi-robot task allocationrdquo in Proceedings of the IEEERSJInternational Conference on Intelligent Robots and Systems(IROS rsquo04) pp 698ndash705 IEEE October 2004

[15] L Liu and D A Shell ldquoLarge-scale multi-robot task allocationvia dynamic partitioning and distributionrdquoAutonomous Robotsvol 33 no 3 pp 291ndash307 2012

[16] S Ozturk and A E Kuzucuoglu ldquoOptimal bid valuationusing path finding for multi-robot task allocationrdquo Journal ofIntelligent Manufacturing 2014

[17] D Puig M A Garcia and L Wu ldquoA new global optimizationstrategy for coordinated multi-robot exploration developmentand comparative evaluationrdquo Robotics and Autonomous Sys-tems vol 59 no 9 pp 635ndash653 2011

[18] M Elango S Nachiappan and M K Tiwari ldquoBalancing taskallocation inmulti-robot systems using K-means clustering andauction based mechanismsrdquo Expert Systems with Applicationsvol 38 no 6 pp 6486ndash6491 2011

[19] A G Bedeian and K W Mossholder ldquoOn the use of thecoefficient of variation as ameasure of diversityrdquoOrganizationalResearch Methods vol 3 no 3 pp 285ndash297 2000

[20] S S Chiddarwar and N R Babu ldquoConflict free coordinatedpath planning for multiple robots using a dynamic path mod-ification sequencerdquo Robotics and Autonomous Systems vol 59no 7-8 pp 508ndash518 2011

[21] J Capitan M T J Spaan L Merino and A Ollero ldquoDecen-tralizedmulti-robot cooperationwith auctioned POMDPsrdquoTheInternational Journal of Robotics Research vol 32 no 6 pp 650ndash671 2013

[22] D Dong C Chen H Li and T-J Tarn ldquoQuantum rein-forcement learningrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 38 no 5 pp 1207ndash12202008

[23] C Chen D Dong H-X Li J Chu and T-J Tarn ldquoFidelity-based probabilistic Q-learning for control of quantum systemsrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 25 no 5 pp 920ndash933 2014

[24] M Elango G Kanagaraj and S G Ponnambalam ldquoSandholmalgorithm with K-means clustering approach for multi-robottask allocationrdquo in Swarm Evolutionary and Memetic Com-puting vol 8298 pp 14ndash22 Springer International Publishing2013

[25] M E Celebi H A Kingravi and P A Vela ldquoA comparativestudy of efficient initialization methods for the k-means clus-tering algorithmrdquo Expert Systems with Applications vol 40 no1 pp 200ndash210 2013

[26] M Shao ldquoResearch on improved convergence co-cevolutiongenetic algorithm for TSPsrdquo International Journal of Conver-gence Computing vol 1 no 2 pp 118ndash126 2014

[27] I Mezei M Lukic V Malbasa and I Stojmenovic ldquoAuctionsand iMesh based task assignment in wireless sensor andactuator networksrdquo Computer Communications vol 36 no 9pp 979ndash987 2013

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

every robot moves to the frontier cell until all the tasks havebeen explored However since every robot only acts for itselfwith no coordination with other robots this method maylead to a rather suboptimal solution in travel distance whichis not acceptable in our logistic application In 2011 Elangoet al [18] proposed the 119870-means clustering and auctionbasedmechanism (referred as119870CABmechanism) to balancethe task allocation in multirobot systems This approachconsiders the dual goal of bothminimizing the travel distanceand efficient sharing of the workload Nevertheless thecomplexity of this algorithm is relatively high and we willcompare this method with our proposed approaches later

Considering the specific characteristics of our problem(i) balancing the task allocation as well as minimizingthe travel distance (ii) the cost of tasks are unknownand (iii) large scale and dynamic environment a balancedheuristic mechanism (BHM) is proposed to coordinate theautonomous robots and is applied to several traditionalapproaches such as the auction allocationmethod and the119870-means clusteringmethod To evaluate the performance of dif-ferent strategies three criteria are provided as the travel time(TT) coefficient of variation (CV) between different robotsand total travel cost (TTC) Simulation results strongly provethat the BHM has a better balancing performance comparedwith traditional methods

The rest of this paper is organized as follows In Section 2we give the formal description and model of the order ful-fillment process in the warehouse In Section 3 we describethe overall control structure of the system and presentthe balanced heuristic mechanism (BHM) Then simulatedexperimental results are provided and compared with severaltraditional methods in Section 4

2 Problem Formulation

21 Problem Statement In this paper we focus on thedynamic task allocation problem for multirobots in anintelligent warehouse To characterize the system someassumptions are listed as follows

(i) the system is composed of large-scale physicalembedded robots

(ii) the robots and the tasks are homogenous(iii) the environment of the system is known to each robot(iv) each robot in the system is self-interested and fulfills

its own tasks independently(v) the robotsmay conflictwith each otherwhen fulfilling

their tasks and they communicate with each otherwhen being close to other vehicles (in purpose ofobstacle avoidance)

(vi) the time consuming at the station is the same for allrobots

Usually the multirobot task allocation is described asfollows given a sequence of tasks and a set of robots the goalis to assign tasks to robots in an efficient manner based onwhat the system is trying to optimize In our approach westrive to optimize the system in a dual objective function to

Picking station

Robot Storage shelves

Pod

Figure 1 Configuration of a warehouse system

keep the total travel cost as low as possible and keep the indi-vidual travel time as equal as possible respectivelyThese twoobjectives correspond to a balance between time expensesand robot expenses both of which deserve consideration in aconcrete physical system As shown in Figure 1 each storageshelf consists of several inventory pods and each pod consistsof several resources By order a robot lifts and carries a podat a time along a preplanned path deliver it to the specificstation which is appointed in the order and finally returns thepod backThemain consideration is how to allocate the tasksto the robots efficiently and equally while avoiding conflicts

Another point that must be taken into account is that tra-ditional task allocationmethods are based on the assumptionthat the cost of each task is already known However in thisproblem the input to the warehouse system is a sequenceof orders from which we can only know the location of thetask and the specific station to which the inventory pod issupposed to be delivered The cost of each task as well as thetravel cost from one location to another is totally unknown

22 Modelling The model formulated to solve the balancedmultirobot task allocation problem mentioned above in thewarehouse system is as follows (as shown in Figure 2)

Let a set of robots 119877 = 1199031 119903

2 119903

119898to complete a set of

tasks 119879 = 1199051 119905

2 119905

119899 The cost of task 119905

119894is 119908119894 which refers

to the travel cost of carrying the inventory pod to the specificstation and then delivering it back ignoring the time cost ofobstacle avoidance and 119862

119894119895(119894 119895 isin 119877 cup 119879) is the travel cost

between every two locations (usually from the robot to theinventory pod assigned in the order) Suppose 119903

119894have 119896 tasks

119879 = 119905

1198941 119905

1198942 119905

119894119896 then the total cost of all the assigned tasks

in 119879119894is expressed as119882(119903

119894) = 119908

1198941+119908

1198942+ sdot sdot sdot +119908

119894119896 More details

about the basic parameters and indexes using in the modelare shown in Figure 2

Rewrite 119879 as Γ = 1198791 119879

2 119879

119898 which means a partition

of the set of tasks where task set 119879119894are allocated to robot 119903

119894

Then define the individual travel cost as ITC(119903119894 119879

119894) which

represents the sum cost for 119903119894to fulfill its task set 119879

119894and can

be calculated by

ITC (119903119894 119879

119894) = 119862

119903119894 1199051198941+

119896minus1

sum

119895=1

119862

119905119894119895 119905119894(119895+1)+119882(119903

119894) (1)

Mathematical Problems in Engineering 3

T1

t11

t13

t12

t14

t15

t16t21

t22

t23t24

t25

T2

r1 r2

middot middot middot

middot middot middotCt16 t22

Ct11t14

Ct23 t25

Cr1t14Cr2t15

Tasks T1 assigned to r1 Tasks T2 assigned to r2

Figure 2 Basic parameters and indexes

where 119862119903119894 1199051198941

represents the travel cost for 119903119894to come to its first

task 1199051198941 and sum119896minus1

119895=1119862

119905119894119895 119905119894(119895+1)represents the total travel cost for

the rest 119896 minus 1 tasks Also119882(119903119894) represents the total task cost

of all the assigned tasks in 119879119894 Our dual goal can be expressed

as

119878 = minΓ

max119894

ITC (119903119894 119879

119894) (2)

which aims to minimize the travel cost as well as balances theindividual travel cost of each robot

Furthermore we provide three criteria to evaluate theperformance of different strategies travel time (TT) whichequals to the maximum of all the individual robotsrsquo travelcosts total travel cost (TTC) which reveals the total travelcost of all the robots evaluates the power consumption andmechanical loss of robots coefficient of variation (CV)whichis a statistical measurement commonly used for comparingthe diversity in work groups [19]

TT = max119894

ITC (119903119894 119879

119894)

TTC =119898

sum

119894=1

ITC (119903119894 119879

119894)

CV = standard deviation (120590)mean (120583)

(3)

where 120590 represents the standard deviation of ITC(119903119894 119879

119894) and

120583 is the mean of ITC(119903119894 119879

119894) 119894 = 1 2 119898

It is worth to point out that we use TTC to evaluate thepower consumption of the system On one hand comparedto the travel energy cost of the robots other energy costs suchas the communication cost and central controller cost arerelatively small so that they can be neglectedwithout affecting

the overall power consumption On the other handsum

119898

119894=1119882(119903

119894) (distance that robots running with pods) is

constant for a system which means the power consumptionof this part is unchanged for a certain system So we need tominimize sum119898

119894=1(119862

119903119894 1199051198941+ sum

119896minus1

119895=1119862

119905119894119895 119905119894(119895+1)) (distance that robots

running without pods) which can be fulfilled by minimizingTTCThus TTC can be used to evaluate the power consump-tion of the system

3 Method

In this section we develop the BHM based approach for thetask allocation of multirobots for the intelligent warehouseWe firstly provide an overview of the whole control systemThen the BHM principles are described in detail By intro-ducing thismechanism into the traditional auction allocationmethod and the 119870-means clustering method two improvedmethods based on BHM are presented and analyzed for thetask-allocation problem

31 Overall System The work space of the warehouse canbe divided into several grids with inventory storage zonesin the middle surrounding with the stations and workersAutonomous robots transport movable inventory pods fromstorage locations to stations where workers can pick itemsoff and pack them up then those packages are sent to thecustomers

In technical aspects the task allocation process is regu-lated by the central controller while the path planning andmotion planning for example the obstacle avoidance arefulfilled by robots In the task allocation process many cur-rent allocation mechanisms tend to use the dynamic method[20 21] however it may result in suboptimal performanceThus an approach is proposed combining both dynamic andstatic methods by means of the task pool The continuouslyarriving orders accumulate in the task pool The controllerrequires a certain number of tasks from the task pool after theprevious batch of tasks has been completed and uses a certainapproach to allocate those tasks to robotsThen the controllersends sets of tasks to corresponding robots by wirelesscommunication such as ZigBee and the robots completetheir tasks sequentially by a standard Alowast algorithm Usuallywe use the Alowast algorithm to find the shortest path betweenstorage locations and stations in a static environment whilein an uncertain environment the learning methods (egreinforcement learning [22 23]) may be necessary to find theshortest path which will be our future work and is beyondthe scope of this paper

In addition as for the obstacle avoidance process robotsfirstly use infrared detectors to detect other robots withincertain distance (usually several grids in front of them)Then they communicate with each other by wireless devicesso that they can verify their relative priorities Generallyspeaking the robot with more tasks left to be completedhas priority to occupy front grids Therefore the robot withfewer tasks left is likely to give way to the robot with moretasks to be completed So the actual balancing performance

4 Mathematical Problems in Engineering

Costevaluation

Path planning and motion(obstacle avoidance)

Decentralized controller

Central controller

(2) Task completion

Orderinput

Wirelesscommunication

R1

R2

R3

middot middot middotTask allocation(using BHM)

(1) Accurate Wij Cij

Figure 3 Control system structure

could be further enhanced in the real world comparedwith the simulated balancing performance of the simulatedexperiment This mechanism is referred as more-task-priormechanism in this paper The whole process is shown inFigure 3

32 Balanced Heuristic Mechanism (BHM) Considering thegoal of minimizing the travel cost as well as balancing theindividual travel cost of each robot a heuristic function isgiven in the following for guiding the task allocation process

119862 (119903

119894 119905

119895) = 120572 times dist (119903

119894 119905

119895) + (1 minus 120572) times 119862119882(119903

119894) (4)

where 119862(119903119894 119905

119895) describes the cost (considering both the time

and distance) for robot 119903119894to finish task 119905

119895 dist(119903

119894 119905

119895) is the

travel cost from robot 119903119894to task 119905

119895 and 119862119882(119903

119894) is the current

value of119882(119903119894) (refer to Section 2)

More specifically as both task cost and travel cost fromone location to another are unknown we evaluate 119908

119894(the

cost of each task) and119862119894119895(the travel cost from one location to

another) before the task allocation by using the standard Alowastalgorithm a simple but significantly effective path planningalgorithm When a robot fulfills a task or moves from onelocation to another in real environment more complexmotions should be taken into account such as the obstacleavoidance Then the more accurate value is shared with thecentral controller and the original estimated value stored inthe controller is replaced with the newest value which canhelp the controller to make a better task allocation Thus thefollowing allocation process is in the light of new informationand can be more accurate and predictable

BHM makes tasks to be assigned to the robot who hasthe lowest 119862(119903

119894 119905

119895) By introducing a parameter 120572 targets are

more inclined to be assigned to the nearby robot with fewerallocated tasks In practice an effective value of 120572 can be setbetween 07 and 09 according to the numbers of robots andtasks The detailed relationship between 120572 and the systemperformance will be given in Section 4

At the beginning stage as the value of119882(119903119894) is quite small

this mechanism has limited effects Thus the performance

of the system is not significantly superior to the one bytraditional methods that merely focus on minimizing thetotal travel distance However as the allocation proceeds therole of (1 minus 120572) times 119862119882(119903

119894) comes into effect by sharing the

workload with those who have fewer assigned tasks Aboveall the method can obtain an approximately optimal resultby considering the total travel cost and the workload balanceof robots A sample illustration of this heuristic mechanismis given in Figure 4

33 Application of BHM Recently the auction-based taskallocation method and the cluster-based task allocationmethod have been widely used to solve the MRTA problems[18 24] So we introduce our BHM approach into the classicsingle-item auction method and the 119870-means clusteringmethod respectively and proposed two BMH-based meth-ods to solve the balancing MRTA problem

331 An Improved Auction Based Task Allocation MethodUsing BHM A typical seal-bid single-round single-itemauction process generally consists of four steps

(i) task announcement(ii) matrix evaluation(iii) bid submission(iv) close of auction (to determine the winning bids and

notify the winning robot)

At the second stage thematrix is usually defined as a functionof the optimistic travel cost [11] as shown in the followingmatrix

(

(

119889

1199031 1199051119889

1199031 1199052sdot sdot sdot sdot sdot sdot 119889

1199031 119905119899

119889

1199032 1199051119889

1199032 1199052sdot sdot sdot sdot sdot sdot 119889

1199032 119905119899

d

d

119889

119903119898 1199051119889

119903119898 1199052sdot sdot sdot sdot sdot sdot 119889

119903119898 119905119899

)

)

(5)

Mathematical Problems in Engineering 5

tj

rmrm rnrn

Dist (rn tj) = 20 Dist (rn tj) = 20 Dist (rn tj) = 10Dist (rn tj) = 10

CW(rm) = 300 CW(rm) = 300CW(rn) = 340 CW(rn) = 320

C(rm tj) = 07 times 20 + 03 times 300 = 104radic

C(rn tj) = 07 times 10 + 03 times 340 = 109

C(rm tj) = 07 times 20 + 03 times 300 = 104

C(rn tj) = 07 times 10 + 03 times 320 = 103radic

tj

Figure 4 Example of the balanced heuristic mechanism

C(r1 t1) C(r1 t2)

C(r2 t1) C(r2 t2)

C(ri t1) C(ri t2)

C(rm t1) C(rm t2)

C(r1 tj)

C(r2 tj)

C(rm tj)

C(r1 tn)

C(r2 tn)

C(ri tn)

C(rm tn)

middot middot middot

middot middot middot

middot middot middot

middot middot middot middot middot middot middot middot middot

middot middot middot middot middot middot

middot middot middot

middot middot middot middot middot middot

middot middot middot

⋱

⋱

⋱⋱

⋱

Delete

RefreshC(ri tj)

Figure 5 Evaluation matrix refreshment

where 1199051 119905

2 119905

119899 are all the open tasks which have not

been carried out by robots In our method the BHM isintroduced into the matrix evaluation process In orderto strengthen the balancing performance of the auctionthe matrix evaluation is defined by 119862

119903119894 119905119895rather than the

optimistic travel distance between 119903119894and 119905119895

In each round after all of the robots submit their bidsthe single-round single-item auction is closedThe controllerwill decide thewinning bidderwith the smallest bid value andnotify the winner Then the bid matrix will be refreshed withfollowing steps (as shown in Figure 5)

Step 1 Delete the task 119905119896from the open-task set

Step 2 Recalculate the row (119862119903119894 1199051 119862

119903119894 1199051 119862

119903119894 119905119899)

As described in (4) the value of 119862(119903119894 119905

119895) is both related to

the distance between 119903119894and 119905119895as well as the already assigned

tasks of 119903119894 Firstly the current position of 119903

119894is replaced by

the location of 119905119895 since 119903

119894wins the bid and has to move

from its former position to 119905119895in order to carry out the

order Secondly the119862119882(119903119894) increases by the cost of the newly

allocated task which can be mathematically expressed as119862119882(119903

119894) = 119862119882(119903

119894) + 119908

119894119895 After the matrix is refreshed next

single-item auction process is proceeding until all the taskshave been allocated

332 A Balanced K-Means ClusteringMethod Using BHM Aclustering method can be generally divided into two stagesthat is firstly finding the cluster centers and then assigningthe sample data into the proper clusterThe119870-means cluster-ing is a commonly used partitioning method [25] the mainidea of which is to find119870 cluster centers (119888

1 119888

2 119888

119896) which

minimize119863 in the following by iteration

119863 =

119899

sum

119894minus1

min119903=12119896

119889 (119909

119894 119888

119903)

2

(6)

In (6) 119889(119909119894 119888

119903) represents the distance between the point 119909

119894

and 119888119903 known as the Euclidean distance When 119863 converges

to a stable value the appropriate (1198881 119888

2 119888

119896) are the cluster

centers for those119870 subgroupsAt the second stage the sample data 119909

119894is assigned into

the proper cluster with the smallest 119889(119909119894 119888

119903) 119903 = 1 2 119896

6 Mathematical Problems in Engineering

However this method only guarantees an optimal allocationscheme minimizing the travel cost while the size differencesbetween each cluster are ignored

In order to solve the balancing MRTA problem we makesome modifications on both of the two stages As for the firststage given that robots can only move along the grid lineswe substitute the Euclidean distance with the Manhattandistance in (6) In a rectangular coordinate system theManhattan distance between point 119875

1 (119909

1 119910

1) and point

119875

2 (119909

2 119910

2) can be expressed as

119889manhattan (1198751 1198752) =1003816

1003816

1003816

1003816

119909

1minus 119909

2

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119910

1minus 119910

2

1003816

1003816

1003816

1003816

(7)

Thus the main purpose is to find 119870 cluster centers(119888

1 119888

2 119888

119896) which minimize 119863manhattan in the following by

iteration

119863manhattan =119899

sum

119894minus1

min119903=12119896

119889manhattan (119909119894 119888119903) (8)

Then to reduce the value119863manhattan iteratively we find theManhattan center rather than the center of the collected tasksFinally a partial optimal solution is formed in a finite numberof iterations Generally speaking there are three steps of thefirst stage

(1) Initial step give an initial set of119870 cluster centers(2) Assignment step assign each point to the cluster center

which has the shortest Manhattan distance(3) Update step calculate the Manhattan Center which

can reduce the overall value of 119863manhattan The Man-hattan Center is defined as follows

It is supposed that there are 119898 points in a rectan-gular coordinate system as 119901

1 119901

2 119901

119898 coordinates of

which are denoted as (1199091 119910

1) (119909

2 119910

2) (119909

119898 119910

119898)Then the

descending order of the sequence 1199091 119909

2 119909

119898is denoted

as 11990910158401 119909

1015840

2 119909

1015840

119898and the descending order of the sequence

119910

1 119910

2 119910

119898is denoted as 1199101015840

1 119910

1015840

2 119910

1015840

119898 Our purpose is to

find the point 119875119898= (119909

119898119888 119910

119898119888) which minimizes (8)

min119909119898119888119910119898119888

119898

sum

119894=1

(

1003816

1003816

1003816

1003816

119909

119898119888minus 119909

119894

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119910

119898119888minus 119910

119894

1003816

1003816

1003816

1003816

) (9)

Since the values of 119909119898119888

and 119910119898119888

do not affect each otherthe value of sum119898

119894=1|119909

119898119888minus 119909

119894| and sum119898

119894=1|119910

119898119888minus 119910

119894| is minimized

separately It is easy to prove that when 119898 is odd 119909119898119888

is119909

1015840

(119898+1)2so that sum119898

119894=1|119909

119898119888minus 119909

119894| is minimum when 119898 is even

119909

119898119888is between 1199091015840

(119898+1)2and 1199091015840

((119898+1)2)+1so thatsum119898

119894=1|119909

119898119888minus119909

119894|

is minimum We can get 119910119898119888

in the same way Thereforethe Manhattan center 119901

119898119888can be calculated by following

equations

119909

119898119888=

119909

1015840

(119898+1)2119898 is odd

119909

1015840

(119898+1)2+ 119909

1015840

((119898+1)2)+1

2

119898 is even

119910

119898119888=

119910

1015840

(119898+1)2119898 is odd

119910

1015840

(119898+1)2+ 119910

1015840

((119898+1)2)+1

2

119898 is even

(10)

p4(36)

p1(14)

p2(22)

p3(43)

p5(88)

p6(96)

p7(91)

poc (56)

pmc (44)

Original center

Manhattan center

Figure 6 Example of the manhattan center

Figure 7 Example of the clustering result

As mentioned before the Manhattan center has theminimum overall Manhattan distance to these 119898 points bywhich the value of119863manhattan is reduced An example is shownin Figure 6 For the original center 119901

119900119888sum119898119894=1(|119909

119900119888minus119909

119894| + |119910

119900119888minus

119910

119894|) = 37 while for the Manhattan Center 119901

119898119888 sum119898119894=1(|119909

119898119888minus

119909

119894| + |119910

119898119888minus 119910

119894|) = 34

Now we concisely prove the convergence of the algo-rithm Given that the119863manhattan generated by the algorithm isstrictly decreasing and there only exist a finite number of suchpartitions the algorithm will reach a partial optimal solutionin a finite number of iterations

After obtaining the cluster centers the balanced heuristicmechanism is introduced into the second stage of cluster-ing In traditional methods when the 119898 cluster centers(119903

1 119903

2 119903

119898) are found in the first stage of the 119870-means

algorithm the remaining active tasks are allocated into theproper cluster with the smallest 119889(119909

119894 119888

119903) 119903 = 1 2 119896 How-

ever in our approach remaining active tasks are allocatedinto the proper cluster with the smallest 119862(119903

119894 119905

119895) defined in

(4)The main idea is that instead of merely considering the

travel cost the balance of the cluster size is taken into accountThe task tends to be allocated into the nearby cluster withfewer tasks When a new task is added into the cluster 119895the 119862119882(119903

119894) will be refreshed After all the active tasks have

been allocated the genetic-based TSP algorithm [26] is usedto plan the route for tasks in each cluster One example ofthe task allocation is demonstrated in Figure 7 (119898 = 4 119899 =50 120572 = 08) where different colors represent tasks allocatedto different robots Blocks with119883 are clustering centers

Mathematical Problems in Engineering 7

Table 1 Balance behavior of single-item auction versus BHM auc-tion

119877 = 8 Single-item auction BHM auction (120572 = 08)119898 120583 120590 CV = 120590120583 120583 120590 CV = 12059012058350 685 312 456 711 61 86100 1374 315 229 1469 53 36150 1969 553 281 2128 41 19200 2591 449 173 2827 50 18250 3234 439 136 3555 32 09300 3897 609 156 4221 21 05350 4418 701 159 4794 26 05400 5085 775 152 5499 19 04

4 Simulated Experiments

41 Simulation Description Several groups of simulatedexperiments are implemented with OPENGL and the exper-imental process is as follows

Step 1 Initialization

Step 2 Creating orders in the task pool while the arriving rateof orders meets the Poisson distribution

Step 3 Sending a specific number (eg 200) of orders fromthe task pool to the controller

Step 4 Using different methods to allocate the orders (tasks)More specifically there are four approaches namely 119870-means BHM 119870-means auction and BHMauction and theirperformances are compared

Step 5 Sending the allocated tasks by the controller to thecorresponding robots which begin to fulfil their tasks (usingthe Alowast algorithm to calculate the path and using the more-task-prior mechanism to avoid collision) Then go to Step 3

42 Simulation Results Analysis In this part three criteriamentioned in Section 2 are used to evaluate the performanceof the proposed balanced heuristic mechanism that iscoefficient of variation (CV) travel time (TT) and total travelcost (TTC)

First of all we compare the balance behavior of theimproved methods with the typical auction method and the119870-means method results of which are given in Tables 1and 2 where the data are collected by average during fiftytimes of simulation experiments Both Tables 1 and 2 showthat using the proposed BHM the diversity degree in workgroups is significantly reduced which indicates the moreequal workload distribution among robots

As shown in Figures 8 and 9 the travel time (TT) ofthe multirobot system is provided by both the traditionalmethod and the improved method We can conclude that byapplying the BHM based methods the time spent to finish

Table 2 Balance behavior of typical 119870-means versus BHM 119870-means

119877 = 8 119870-means BHM K-means (120572 = 08)119898 120583 120590 CV = 120590120583 120583 120590 CV = 12059012058350 714 310 434 749 50 67100 1390 513 369 1508 57 38150 2058 764 371 2204 49 22200 2751 1116 406 2949 63 21250 3441 1076 313 3684 69 19300 4095 1254 306 4440 65 15350 4776 1484 311 5214 83 16400 5465 1882 344 5895 66 11

50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

7000

8000

Task number

TT

AuctionBHM auction

Figure 8 Time spent by single-item auction versus BHM auction

the given task is much less than the one by using thetraditional method due to an efficient sharing of workload

Comparison results for traditional methods and BHMbasedmethods are demonstrated in Figure 10 bymeans of thetotal travel cost (TTC) from which we can conclude that inbalanced heuristic cases although robots pay more attentionto balancing the workload rather thanmerelyminimizing thetravel cost as traditional methods do the TTC only increasesby about 8 Concerning the obvious increasement of therobots utilization this slightly increased power consumptionis quite acceptable

With the traditional auction method and 119870-meansmethod the task allocation process only concentrates onminimizing the travel cost Once there is a robot nearest toan open task other robots do not attempt to compete for thetask even if they are idle Thus the utilization of the system isnot enough and the time cost to fulfill the given tasks is highWith BHM based methods proposed in this paper all robots

8 Mathematical Problems in Engineering

50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

7000

8000

9000

Task number

TT

K-meansBHM K-means

Figure 9 Time spent by 119870-means versus BHM 119870-means

50 100 150 200 250 300 350 400012345

TTC

Task number

times104

AuctionBHM auction

(a)

50 100 150 200 250 300 350 400012345

Task number

TTC

times104

K-meansBHM K-means

(b)

Figure 10 Total travel cost by traditional methods versus BHM adapted methods

can be utilized in a balanced and effective manner while thetravel cost is quite low

Then we study the influence of the variable 120572 on thesystem performance The value of 120572 describes the weight ofthe system energy saving demand and time saving demandFigure 11 demonstrates the relationship between 120572 and thetravel time (TT) It shows that in order to keep a good balancebetween minimizing the travel distance and efficient sharingof the workload the coefficient is usually set between 07 and09 to achieve the minimum of TT

Finally besides comparing BHM-based approaches withthe traditional auctionmethod and119870-meansmethod we usea more competing algorithm (119870CAB mechanism in [18]) totestify the scientific merit of the BHM scheme (as shownin Figures 12 and 13) We can conclude that by BHM basedmethods the time spent to finish the given task is 5sim15less than the119870CABmechanism meanwhile their total travelcosts are almost the same Thus the benefit of the balancedheuristic mechanism to warehouse systems is significant

5 Conclusion

In this paper a novel balanced heuristic mechanism for themultirobot task allocation problem is presented with respectto the task cost and travel cost To evaluate the proposedapproach we give three criteria namely travel time (TT)total travel cost (TTC) and coefficient of variation (CV)and compare some traditional methods with BHM-basedmethods under those criterions Simulation results show thatthe proposed approach greatly enhances the balance behaviorof the multirobot system reduces the total travel time whileachieves almost the same total travel cost as the traditionalmethods do In addition the influence of the variable 120572 onthe system performance is also studied in detail In our futurework BHM will be applied into more complicated envi-ronments which consist of heterogeneous tasks and robotsFurthermore we will dedicate to enhance the robustness ofthe proposed approach in dynamic environments Finally itis worth to point out that this new mechanism is of great

Mathematical Problems in Engineering 9

01 02 03 04 05 06 07 08 09 108

09

10

11

12

13

14

15

TT

120572

Task number = 50Task number = 100

Task number = 200Task number = 300

Normalized TT versus 120572 (m = 8)

Figure 11 Relationship between 120572 and travel time (TT)

50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

7000

Task number

TT

BHM auction

KCAB mechanismBHM K-means

Figure 12 Comparison between method provided in [15] versusBHM-based approaches (TT)

applicability to other task allocation problems such as theWSANs problem in [27]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Authorsrsquo Contribution

Luowei Zhou and Yuanyuan Shi contributed equally to thiswork

50 100 150 200 250 300 350 400

TTC

Task number

times104

BHM auction

KCAB mechanismBHM K-means

5

45

4

35

3

25

2

15

1

05

Figure 13 Comparison between method provided in [15] versusBHM-based approaches (TTC)

Acknowledgment

This work was supported by the Natural Science Foundationof China under Grants no 61273327 and no 61432008

References

[1] P R Wurman R DrsquoAndrea and M Mountz ldquoCoordinatinghundreds of cooperative autonomous vehicles in warehousesrdquoAI Magazine vol 29 no 1 pp 9ndash19 2008

[2] X Zhai J E Ward and L B Schwarz ldquoCoordinating aone-warehouse N-retailer distribution system under retailer-reportingrdquo International Journal of Production Economics vol134 no 1 pp 204ndash211 2011

[3] C Chen H-X Li and D Dong ldquoHybrid control forrobot navigationmdasha hierarchical Q-learning algorithmrdquo IEEERobotics amp AutomationMagazine vol 15 no 2 pp 37ndash47 2008

[4] S Chen and C Chen ldquoProbabilistic fuzzy system for uncertainlocalization and map building of mobile robotsrdquo IEEE Trans-actions on Instrumentation and Measurement vol 61 no 6 pp1546ndash1560 2012

[5] D Dong C Chen J Chu and T-J Tarn ldquoRobust quantum-inspired reinforcement learning for robot navigationrdquo IEEEASME Transactions on Mechatronics vol 17 no 1 pp 86ndash972012

[6] T Niemueller D Ewert S Reuter et al ldquoTowards benchmarking cyber-physical systems in factory automation scenariosrdquoinKI 2013 Advances in Artificial Intelligence vol 8077 of LectureNotes in Computer Science pp 296ndash299 2013

[7] M Brambilla E Ferrante M Birattari and M Dorigo ldquoSwarmrobotics a review from the swarm engineering perspectiverdquoSwarm Intelligence vol 7 no 1 pp 1ndash41 2013

[8] C Nieto-Granda J G Rogers and H I Christensen ldquoCoor-dination strategies for multi-robot exploration and mappingrdquoExperimental Robotics vol 33 no 4 pp 519ndash533 2013

10 Mathematical Problems in Engineering

[9] B-Y Shih C-Y Chen and W Chou ldquoAn enhanced obstacleavoidance and path correction mechanism for an autonomousintelligent robot withmultiple sensorsrdquo Journal of Vibration andControl vol 18 no 12 pp 1855ndash1864 2012

[10] B Yamauchi ldquoFrontier-based exploration using multiplerobotsrdquo in Proceedings of the 2nd International Conference onAutonomous Agents pp 47ndash53 ACM May 1998

[11] B P Gerkey and M J Mataric ldquoSold auction methods formultirobot coordinationrdquo IEEE Transactions on Robotics andAutomation vol 18 no 5 pp 758ndash768 2002

[12] T Sandholm ldquoAlgorithm for optimal winner determination incombinatorial auctionsrdquo Artificial Intelligence vol 135 no 1-2pp 1ndash54 2002

[13] M Berhault H Huang P Keskinocak et al ldquoRobot explorationwith combinatorial auctionsrdquo in Proceedings of the IEEERSJInternational Conference on Intelligent Robots and Systems vol1ndash4 pp 1957ndash1962 October 2003

[14] M G Lagoudakis M Berhault S Koenig P Keskinocak andA J Kleywegt ldquoSimple auctions with performance guaranteesfor multi-robot task allocationrdquo in Proceedings of the IEEERSJInternational Conference on Intelligent Robots and Systems(IROS rsquo04) pp 698ndash705 IEEE October 2004

[15] L Liu and D A Shell ldquoLarge-scale multi-robot task allocationvia dynamic partitioning and distributionrdquoAutonomous Robotsvol 33 no 3 pp 291ndash307 2012

[16] S Ozturk and A E Kuzucuoglu ldquoOptimal bid valuationusing path finding for multi-robot task allocationrdquo Journal ofIntelligent Manufacturing 2014

[17] D Puig M A Garcia and L Wu ldquoA new global optimizationstrategy for coordinated multi-robot exploration developmentand comparative evaluationrdquo Robotics and Autonomous Sys-tems vol 59 no 9 pp 635ndash653 2011

[18] M Elango S Nachiappan and M K Tiwari ldquoBalancing taskallocation inmulti-robot systems using K-means clustering andauction based mechanismsrdquo Expert Systems with Applicationsvol 38 no 6 pp 6486ndash6491 2011

[19] A G Bedeian and K W Mossholder ldquoOn the use of thecoefficient of variation as ameasure of diversityrdquoOrganizationalResearch Methods vol 3 no 3 pp 285ndash297 2000

[20] S S Chiddarwar and N R Babu ldquoConflict free coordinatedpath planning for multiple robots using a dynamic path mod-ification sequencerdquo Robotics and Autonomous Systems vol 59no 7-8 pp 508ndash518 2011

[21] J Capitan M T J Spaan L Merino and A Ollero ldquoDecen-tralizedmulti-robot cooperationwith auctioned POMDPsrdquoTheInternational Journal of Robotics Research vol 32 no 6 pp 650ndash671 2013

[22] D Dong C Chen H Li and T-J Tarn ldquoQuantum rein-forcement learningrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 38 no 5 pp 1207ndash12202008

[23] C Chen D Dong H-X Li J Chu and T-J Tarn ldquoFidelity-based probabilistic Q-learning for control of quantum systemsrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 25 no 5 pp 920ndash933 2014

[24] M Elango G Kanagaraj and S G Ponnambalam ldquoSandholmalgorithm with K-means clustering approach for multi-robottask allocationrdquo in Swarm Evolutionary and Memetic Com-puting vol 8298 pp 14ndash22 Springer International Publishing2013

[25] M E Celebi H A Kingravi and P A Vela ldquoA comparativestudy of efficient initialization methods for the k-means clus-tering algorithmrdquo Expert Systems with Applications vol 40 no1 pp 200ndash210 2013

[26] M Shao ldquoResearch on improved convergence co-cevolutiongenetic algorithm for TSPsrdquo International Journal of Conver-gence Computing vol 1 no 2 pp 118ndash126 2014

[27] I Mezei M Lukic V Malbasa and I Stojmenovic ldquoAuctionsand iMesh based task assignment in wireless sensor andactuator networksrdquo Computer Communications vol 36 no 9pp 979ndash987 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

T1

t11

t13

t12

t14

t15

t16t21

t22

t23t24

t25

T2

r1 r2

middot middot middot

middot middot middotCt16 t22

Ct11t14

Ct23 t25

Cr1t14Cr2t15

Tasks T1 assigned to r1 Tasks T2 assigned to r2

Figure 2 Basic parameters and indexes

where 119862119903119894 1199051198941

represents the travel cost for 119903119894to come to its first

task 1199051198941 and sum119896minus1

119895=1119862

119905119894119895 119905119894(119895+1)represents the total travel cost for

the rest 119896 minus 1 tasks Also119882(119903119894) represents the total task cost

of all the assigned tasks in 119879119894 Our dual goal can be expressed

as

119878 = minΓ

max119894

ITC (119903119894 119879

119894) (2)

which aims to minimize the travel cost as well as balances theindividual travel cost of each robot

Furthermore we provide three criteria to evaluate theperformance of different strategies travel time (TT) whichequals to the maximum of all the individual robotsrsquo travelcosts total travel cost (TTC) which reveals the total travelcost of all the robots evaluates the power consumption andmechanical loss of robots coefficient of variation (CV)whichis a statistical measurement commonly used for comparingthe diversity in work groups [19]

TT = max119894

ITC (119903119894 119879

119894)

TTC =119898

sum

119894=1

ITC (119903119894 119879

119894)

CV = standard deviation (120590)mean (120583)

(3)

where 120590 represents the standard deviation of ITC(119903119894 119879

119894) and

120583 is the mean of ITC(119903119894 119879

119894) 119894 = 1 2 119898

It is worth to point out that we use TTC to evaluate thepower consumption of the system On one hand comparedto the travel energy cost of the robots other energy costs suchas the communication cost and central controller cost arerelatively small so that they can be neglectedwithout affecting

the overall power consumption On the other handsum

119898

119894=1119882(119903

119894) (distance that robots running with pods) is

constant for a system which means the power consumptionof this part is unchanged for a certain system So we need tominimize sum119898

119894=1(119862

119903119894 1199051198941+ sum

119896minus1

119895=1119862

119905119894119895 119905119894(119895+1)) (distance that robots

running without pods) which can be fulfilled by minimizingTTCThus TTC can be used to evaluate the power consump-tion of the system

3 Method

In this section we develop the BHM based approach for thetask allocation of multirobots for the intelligent warehouseWe firstly provide an overview of the whole control systemThen the BHM principles are described in detail By intro-ducing thismechanism into the traditional auction allocationmethod and the 119870-means clustering method two improvedmethods based on BHM are presented and analyzed for thetask-allocation problem

31 Overall System The work space of the warehouse canbe divided into several grids with inventory storage zonesin the middle surrounding with the stations and workersAutonomous robots transport movable inventory pods fromstorage locations to stations where workers can pick itemsoff and pack them up then those packages are sent to thecustomers

In technical aspects the task allocation process is regu-lated by the central controller while the path planning andmotion planning for example the obstacle avoidance arefulfilled by robots In the task allocation process many cur-rent allocation mechanisms tend to use the dynamic method[20 21] however it may result in suboptimal performanceThus an approach is proposed combining both dynamic andstatic methods by means of the task pool The continuouslyarriving orders accumulate in the task pool The controllerrequires a certain number of tasks from the task pool after theprevious batch of tasks has been completed and uses a certainapproach to allocate those tasks to robotsThen the controllersends sets of tasks to corresponding robots by wirelesscommunication such as ZigBee and the robots completetheir tasks sequentially by a standard Alowast algorithm Usuallywe use the Alowast algorithm to find the shortest path betweenstorage locations and stations in a static environment whilein an uncertain environment the learning methods (egreinforcement learning [22 23]) may be necessary to find theshortest path which will be our future work and is beyondthe scope of this paper

In addition as for the obstacle avoidance process robotsfirstly use infrared detectors to detect other robots withincertain distance (usually several grids in front of them)Then they communicate with each other by wireless devicesso that they can verify their relative priorities Generallyspeaking the robot with more tasks left to be completedhas priority to occupy front grids Therefore the robot withfewer tasks left is likely to give way to the robot with moretasks to be completed So the actual balancing performance

4 Mathematical Problems in Engineering

Costevaluation

Path planning and motion(obstacle avoidance)

Decentralized controller

Central controller

(2) Task completion

Orderinput

Wirelesscommunication

R1

R2

R3

middot middot middotTask allocation(using BHM)

(1) Accurate Wij Cij

Figure 3 Control system structure

could be further enhanced in the real world comparedwith the simulated balancing performance of the simulatedexperiment This mechanism is referred as more-task-priormechanism in this paper The whole process is shown inFigure 3

32 Balanced Heuristic Mechanism (BHM) Considering thegoal of minimizing the travel cost as well as balancing theindividual travel cost of each robot a heuristic function isgiven in the following for guiding the task allocation process

119862 (119903

119894 119905

119895) = 120572 times dist (119903

119894 119905

119895) + (1 minus 120572) times 119862119882(119903

119894) (4)

where 119862(119903119894 119905

119895) describes the cost (considering both the time

and distance) for robot 119903119894to finish task 119905

119895 dist(119903

119894 119905

119895) is the

travel cost from robot 119903119894to task 119905

119895 and 119862119882(119903

119894) is the current

value of119882(119903119894) (refer to Section 2)

More specifically as both task cost and travel cost fromone location to another are unknown we evaluate 119908

119894(the

cost of each task) and119862119894119895(the travel cost from one location to

another) before the task allocation by using the standard Alowastalgorithm a simple but significantly effective path planningalgorithm When a robot fulfills a task or moves from onelocation to another in real environment more complexmotions should be taken into account such as the obstacleavoidance Then the more accurate value is shared with thecentral controller and the original estimated value stored inthe controller is replaced with the newest value which canhelp the controller to make a better task allocation Thus thefollowing allocation process is in the light of new informationand can be more accurate and predictable

BHM makes tasks to be assigned to the robot who hasthe lowest 119862(119903

119894 119905

119895) By introducing a parameter 120572 targets are

more inclined to be assigned to the nearby robot with fewerallocated tasks In practice an effective value of 120572 can be setbetween 07 and 09 according to the numbers of robots andtasks The detailed relationship between 120572 and the systemperformance will be given in Section 4

At the beginning stage as the value of119882(119903119894) is quite small

this mechanism has limited effects Thus the performance

of the system is not significantly superior to the one bytraditional methods that merely focus on minimizing thetotal travel distance However as the allocation proceeds therole of (1 minus 120572) times 119862119882(119903

119894) comes into effect by sharing the

workload with those who have fewer assigned tasks Aboveall the method can obtain an approximately optimal resultby considering the total travel cost and the workload balanceof robots A sample illustration of this heuristic mechanismis given in Figure 4

33 Application of BHM Recently the auction-based taskallocation method and the cluster-based task allocationmethod have been widely used to solve the MRTA problems[18 24] So we introduce our BHM approach into the classicsingle-item auction method and the 119870-means clusteringmethod respectively and proposed two BMH-based meth-ods to solve the balancing MRTA problem

331 An Improved Auction Based Task Allocation MethodUsing BHM A typical seal-bid single-round single-itemauction process generally consists of four steps

(i) task announcement(ii) matrix evaluation(iii) bid submission(iv) close of auction (to determine the winning bids and

notify the winning robot)

At the second stage thematrix is usually defined as a functionof the optimistic travel cost [11] as shown in the followingmatrix

(

(

119889

1199031 1199051119889

1199031 1199052sdot sdot sdot sdot sdot sdot 119889

1199031 119905119899

119889

1199032 1199051119889

1199032 1199052sdot sdot sdot sdot sdot sdot 119889

1199032 119905119899

d

d

119889

119903119898 1199051119889

119903119898 1199052sdot sdot sdot sdot sdot sdot 119889

119903119898 119905119899

)

)

(5)

Mathematical Problems in Engineering 5

tj

rmrm rnrn

Dist (rn tj) = 20 Dist (rn tj) = 20 Dist (rn tj) = 10Dist (rn tj) = 10

CW(rm) = 300 CW(rm) = 300CW(rn) = 340 CW(rn) = 320

C(rm tj) = 07 times 20 + 03 times 300 = 104radic

C(rn tj) = 07 times 10 + 03 times 340 = 109

C(rm tj) = 07 times 20 + 03 times 300 = 104

C(rn tj) = 07 times 10 + 03 times 320 = 103radic

tj

Figure 4 Example of the balanced heuristic mechanism

C(r1 t1) C(r1 t2)

C(r2 t1) C(r2 t2)

C(ri t1) C(ri t2)

C(rm t1) C(rm t2)

C(r1 tj)

C(r2 tj)

C(rm tj)

C(r1 tn)

C(r2 tn)

C(ri tn)

C(rm tn)

middot middot middot

middot middot middot

middot middot middot

middot middot middot middot middot middot middot middot middot

middot middot middot middot middot middot

middot middot middot

middot middot middot middot middot middot

middot middot middot

⋱

⋱

⋱⋱

⋱

Delete

RefreshC(ri tj)

Figure 5 Evaluation matrix refreshment

where 1199051 119905

2 119905

119899 are all the open tasks which have not

been carried out by robots In our method the BHM isintroduced into the matrix evaluation process In orderto strengthen the balancing performance of the auctionthe matrix evaluation is defined by 119862

119903119894 119905119895rather than the

optimistic travel distance between 119903119894and 119905119895

In each round after all of the robots submit their bidsthe single-round single-item auction is closedThe controllerwill decide thewinning bidderwith the smallest bid value andnotify the winner Then the bid matrix will be refreshed withfollowing steps (as shown in Figure 5)

Step 1 Delete the task 119905119896from the open-task set

Step 2 Recalculate the row (119862119903119894 1199051 119862

119903119894 1199051 119862

119903119894 119905119899)

As described in (4) the value of 119862(119903119894 119905

119895) is both related to

the distance between 119903119894and 119905119895as well as the already assigned

tasks of 119903119894 Firstly the current position of 119903

119894is replaced by

the location of 119905119895 since 119903

119894wins the bid and has to move

from its former position to 119905119895in order to carry out the

order Secondly the119862119882(119903119894) increases by the cost of the newly

allocated task which can be mathematically expressed as119862119882(119903

119894) = 119862119882(119903

119894) + 119908

119894119895 After the matrix is refreshed next

single-item auction process is proceeding until all the taskshave been allocated

332 A Balanced K-Means ClusteringMethod Using BHM Aclustering method can be generally divided into two stagesthat is firstly finding the cluster centers and then assigningthe sample data into the proper clusterThe119870-means cluster-ing is a commonly used partitioning method [25] the mainidea of which is to find119870 cluster centers (119888

1 119888

2 119888

119896) which

minimize119863 in the following by iteration

119863 =

119899

sum

119894minus1

min119903=12119896

119889 (119909

119894 119888

119903)

2

(6)

In (6) 119889(119909119894 119888

119903) represents the distance between the point 119909

119894

and 119888119903 known as the Euclidean distance When 119863 converges

to a stable value the appropriate (1198881 119888

2 119888

119896) are the cluster

centers for those119870 subgroupsAt the second stage the sample data 119909

119894is assigned into

the proper cluster with the smallest 119889(119909119894 119888

119903) 119903 = 1 2 119896

6 Mathematical Problems in Engineering

However this method only guarantees an optimal allocationscheme minimizing the travel cost while the size differencesbetween each cluster are ignored

In order to solve the balancing MRTA problem we makesome modifications on both of the two stages As for the firststage given that robots can only move along the grid lineswe substitute the Euclidean distance with the Manhattandistance in (6) In a rectangular coordinate system theManhattan distance between point 119875

1 (119909

1 119910

1) and point

119875

2 (119909

2 119910

2) can be expressed as

119889manhattan (1198751 1198752) =1003816

1003816

1003816

1003816

119909

1minus 119909

2

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119910

1minus 119910

2

1003816

1003816

1003816

1003816

(7)

Thus the main purpose is to find 119870 cluster centers(119888

1 119888

2 119888

119896) which minimize 119863manhattan in the following by

iteration

119863manhattan =119899

sum

119894minus1

min119903=12119896

119889manhattan (119909119894 119888119903) (8)

Then to reduce the value119863manhattan iteratively we find theManhattan center rather than the center of the collected tasksFinally a partial optimal solution is formed in a finite numberof iterations Generally speaking there are three steps of thefirst stage

(1) Initial step give an initial set of119870 cluster centers(2) Assignment step assign each point to the cluster center

which has the shortest Manhattan distance(3) Update step calculate the Manhattan Center which

can reduce the overall value of 119863manhattan The Man-hattan Center is defined as follows

It is supposed that there are 119898 points in a rectan-gular coordinate system as 119901

1 119901

2 119901

119898 coordinates of

which are denoted as (1199091 119910

1) (119909

2 119910

2) (119909

119898 119910

119898)Then the

descending order of the sequence 1199091 119909

2 119909

119898is denoted

as 11990910158401 119909

1015840

2 119909

1015840

119898and the descending order of the sequence

119910

1 119910

2 119910

119898is denoted as 1199101015840

1 119910

1015840

2 119910

1015840

119898 Our purpose is to

find the point 119875119898= (119909

119898119888 119910

119898119888) which minimizes (8)

min119909119898119888119910119898119888

119898

sum

119894=1

(

1003816

1003816

1003816

1003816

119909

119898119888minus 119909

119894

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119910

119898119888minus 119910

119894

1003816

1003816

1003816

1003816

) (9)

Since the values of 119909119898119888

and 119910119898119888

do not affect each otherthe value of sum119898

119894=1|119909

119898119888minus 119909

119894| and sum119898

119894=1|119910

119898119888minus 119910

119894| is minimized

separately It is easy to prove that when 119898 is odd 119909119898119888

is119909

1015840

(119898+1)2so that sum119898

119894=1|119909

119898119888minus 119909

119894| is minimum when 119898 is even

119909

119898119888is between 1199091015840

(119898+1)2and 1199091015840

((119898+1)2)+1so thatsum119898

119894=1|119909

119898119888minus119909

119894|

is minimum We can get 119910119898119888

in the same way Thereforethe Manhattan center 119901

119898119888can be calculated by following

equations

119909

119898119888=

119909

1015840

(119898+1)2119898 is odd

119909

1015840

(119898+1)2+ 119909

1015840

((119898+1)2)+1

2

119898 is even

119910

119898119888=

119910

1015840

(119898+1)2119898 is odd

119910

1015840

(119898+1)2+ 119910

1015840

((119898+1)2)+1

2

119898 is even

(10)

p4(36)

p1(14)

p2(22)

p3(43)

p5(88)

p6(96)

p7(91)

poc (56)

pmc (44)

Original center

Manhattan center

Figure 6 Example of the manhattan center

Figure 7 Example of the clustering result

As mentioned before the Manhattan center has theminimum overall Manhattan distance to these 119898 points bywhich the value of119863manhattan is reduced An example is shownin Figure 6 For the original center 119901

119900119888sum119898119894=1(|119909

119900119888minus119909

119894| + |119910

119900119888minus

119910

119894|) = 37 while for the Manhattan Center 119901

119898119888 sum119898119894=1(|119909

119898119888minus

119909

119894| + |119910

119898119888minus 119910

119894|) = 34

Now we concisely prove the convergence of the algo-rithm Given that the119863manhattan generated by the algorithm isstrictly decreasing and there only exist a finite number of suchpartitions the algorithm will reach a partial optimal solutionin a finite number of iterations

After obtaining the cluster centers the balanced heuristicmechanism is introduced into the second stage of cluster-ing In traditional methods when the 119898 cluster centers(119903

1 119903

2 119903

119898) are found in the first stage of the 119870-means

algorithm the remaining active tasks are allocated into theproper cluster with the smallest 119889(119909

119894 119888

119903) 119903 = 1 2 119896 How-

ever in our approach remaining active tasks are allocatedinto the proper cluster with the smallest 119862(119903

119894 119905

119895) defined in

(4)The main idea is that instead of merely considering the

travel cost the balance of the cluster size is taken into accountThe task tends to be allocated into the nearby cluster withfewer tasks When a new task is added into the cluster 119895the 119862119882(119903

119894) will be refreshed After all the active tasks have

been allocated the genetic-based TSP algorithm [26] is usedto plan the route for tasks in each cluster One example ofthe task allocation is demonstrated in Figure 7 (119898 = 4 119899 =50 120572 = 08) where different colors represent tasks allocatedto different robots Blocks with119883 are clustering centers

Mathematical Problems in Engineering 7

Table 1 Balance behavior of single-item auction versus BHM auc-tion

119877 = 8 Single-item auction BHM auction (120572 = 08)119898 120583 120590 CV = 120590120583 120583 120590 CV = 12059012058350 685 312 456 711 61 86100 1374 315 229 1469 53 36150 1969 553 281 2128 41 19200 2591 449 173 2827 50 18250 3234 439 136 3555 32 09300 3897 609 156 4221 21 05350 4418 701 159 4794 26 05400 5085 775 152 5499 19 04

4 Simulated Experiments

41 Simulation Description Several groups of simulatedexperiments are implemented with OPENGL and the exper-imental process is as follows

Step 1 Initialization

Step 2 Creating orders in the task pool while the arriving rateof orders meets the Poisson distribution

Step 3 Sending a specific number (eg 200) of orders fromthe task pool to the controller

Step 4 Using different methods to allocate the orders (tasks)More specifically there are four approaches namely 119870-means BHM 119870-means auction and BHMauction and theirperformances are compared

Step 5 Sending the allocated tasks by the controller to thecorresponding robots which begin to fulfil their tasks (usingthe Alowast algorithm to calculate the path and using the more-task-prior mechanism to avoid collision) Then go to Step 3

42 Simulation Results Analysis In this part three criteriamentioned in Section 2 are used to evaluate the performanceof the proposed balanced heuristic mechanism that iscoefficient of variation (CV) travel time (TT) and total travelcost (TTC)

First of all we compare the balance behavior of theimproved methods with the typical auction method and the119870-means method results of which are given in Tables 1and 2 where the data are collected by average during fiftytimes of simulation experiments Both Tables 1 and 2 showthat using the proposed BHM the diversity degree in workgroups is significantly reduced which indicates the moreequal workload distribution among robots

As shown in Figures 8 and 9 the travel time (TT) ofthe multirobot system is provided by both the traditionalmethod and the improved method We can conclude that byapplying the BHM based methods the time spent to finish

Table 2 Balance behavior of typical 119870-means versus BHM 119870-means

119877 = 8 119870-means BHM K-means (120572 = 08)119898 120583 120590 CV = 120590120583 120583 120590 CV = 12059012058350 714 310 434 749 50 67100 1390 513 369 1508 57 38150 2058 764 371 2204 49 22200 2751 1116 406 2949 63 21250 3441 1076 313 3684 69 19300 4095 1254 306 4440 65 15350 4776 1484 311 5214 83 16400 5465 1882 344 5895 66 11

50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

7000

8000

Task number

TT

AuctionBHM auction

Figure 8 Time spent by single-item auction versus BHM auction

the given task is much less than the one by using thetraditional method due to an efficient sharing of workload

Comparison results for traditional methods and BHMbasedmethods are demonstrated in Figure 10 bymeans of thetotal travel cost (TTC) from which we can conclude that inbalanced heuristic cases although robots pay more attentionto balancing the workload rather thanmerelyminimizing thetravel cost as traditional methods do the TTC only increasesby about 8 Concerning the obvious increasement of therobots utilization this slightly increased power consumptionis quite acceptable

With the traditional auction method and 119870-meansmethod the task allocation process only concentrates onminimizing the travel cost Once there is a robot nearest toan open task other robots do not attempt to compete for thetask even if they are idle Thus the utilization of the system isnot enough and the time cost to fulfill the given tasks is highWith BHM based methods proposed in this paper all robots

8 Mathematical Problems in Engineering

50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

7000

8000

9000

Task number

TT

K-meansBHM K-means

Figure 9 Time spent by 119870-means versus BHM 119870-means

50 100 150 200 250 300 350 400012345

TTC

Task number

times104

AuctionBHM auction

(a)

50 100 150 200 250 300 350 400012345

Task number

TTC

times104

K-meansBHM K-means

(b)

Figure 10 Total travel cost by traditional methods versus BHM adapted methods

can be utilized in a balanced and effective manner while thetravel cost is quite low

Then we study the influence of the variable 120572 on thesystem performance The value of 120572 describes the weight ofthe system energy saving demand and time saving demandFigure 11 demonstrates the relationship between 120572 and thetravel time (TT) It shows that in order to keep a good balancebetween minimizing the travel distance and efficient sharingof the workload the coefficient is usually set between 07 and09 to achieve the minimum of TT

Finally besides comparing BHM-based approaches withthe traditional auctionmethod and119870-meansmethod we usea more competing algorithm (119870CAB mechanism in [18]) totestify the scientific merit of the BHM scheme (as shownin Figures 12 and 13) We can conclude that by BHM basedmethods the time spent to finish the given task is 5sim15less than the119870CABmechanism meanwhile their total travelcosts are almost the same Thus the benefit of the balancedheuristic mechanism to warehouse systems is significant

5 Conclusion

In this paper a novel balanced heuristic mechanism for themultirobot task allocation problem is presented with respectto the task cost and travel cost To evaluate the proposedapproach we give three criteria namely travel time (TT)total travel cost (TTC) and coefficient of variation (CV)and compare some traditional methods with BHM-basedmethods under those criterions Simulation results show thatthe proposed approach greatly enhances the balance behaviorof the multirobot system reduces the total travel time whileachieves almost the same total travel cost as the traditionalmethods do In addition the influence of the variable 120572 onthe system performance is also studied in detail In our futurework BHM will be applied into more complicated envi-ronments which consist of heterogeneous tasks and robotsFurthermore we will dedicate to enhance the robustness ofthe proposed approach in dynamic environments Finally itis worth to point out that this new mechanism is of great

Mathematical Problems in Engineering 9

01 02 03 04 05 06 07 08 09 108

09

10

11

12

13

14

15

TT

120572

Task number = 50Task number = 100

Task number = 200Task number = 300

Normalized TT versus 120572 (m = 8)

Figure 11 Relationship between 120572 and travel time (TT)

50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

7000

Task number

TT

BHM auction

KCAB mechanismBHM K-means

Figure 12 Comparison between method provided in [15] versusBHM-based approaches (TT)

applicability to other task allocation problems such as theWSANs problem in [27]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Authorsrsquo Contribution

Luowei Zhou and Yuanyuan Shi contributed equally to thiswork

50 100 150 200 250 300 350 400

TTC

Task number

times104

BHM auction

KCAB mechanismBHM K-means

5

45

4

35

3

25

2

15

1

05

Figure 13 Comparison between method provided in [15] versusBHM-based approaches (TTC)

Acknowledgment

This work was supported by the Natural Science Foundationof China under Grants no 61273327 and no 61432008

References

[1] P R Wurman R DrsquoAndrea and M Mountz ldquoCoordinatinghundreds of cooperative autonomous vehicles in warehousesrdquoAI Magazine vol 29 no 1 pp 9ndash19 2008

[2] X Zhai J E Ward and L B Schwarz ldquoCoordinating aone-warehouse N-retailer distribution system under retailer-reportingrdquo International Journal of Production Economics vol134 no 1 pp 204ndash211 2011

[3] C Chen H-X Li and D Dong ldquoHybrid control forrobot navigationmdasha hierarchical Q-learning algorithmrdquo IEEERobotics amp AutomationMagazine vol 15 no 2 pp 37ndash47 2008

[4] S Chen and C Chen ldquoProbabilistic fuzzy system for uncertainlocalization and map building of mobile robotsrdquo IEEE Trans-actions on Instrumentation and Measurement vol 61 no 6 pp1546ndash1560 2012

[5] D Dong C Chen J Chu and T-J Tarn ldquoRobust quantum-inspired reinforcement learning for robot navigationrdquo IEEEASME Transactions on Mechatronics vol 17 no 1 pp 86ndash972012

[6] T Niemueller D Ewert S Reuter et al ldquoTowards benchmarking cyber-physical systems in factory automation scenariosrdquoinKI 2013 Advances in Artificial Intelligence vol 8077 of LectureNotes in Computer Science pp 296ndash299 2013

[7] M Brambilla E Ferrante M Birattari and M Dorigo ldquoSwarmrobotics a review from the swarm engineering perspectiverdquoSwarm Intelligence vol 7 no 1 pp 1ndash41 2013

[8] C Nieto-Granda J G Rogers and H I Christensen ldquoCoor-dination strategies for multi-robot exploration and mappingrdquoExperimental Robotics vol 33 no 4 pp 519ndash533 2013

10 Mathematical Problems in Engineering

[9] B-Y Shih C-Y Chen and W Chou ldquoAn enhanced obstacleavoidance and path correction mechanism for an autonomousintelligent robot withmultiple sensorsrdquo Journal of Vibration andControl vol 18 no 12 pp 1855ndash1864 2012

[10] B Yamauchi ldquoFrontier-based exploration using multiplerobotsrdquo in Proceedings of the 2nd International Conference onAutonomous Agents pp 47ndash53 ACM May 1998

[11] B P Gerkey and M J Mataric ldquoSold auction methods formultirobot coordinationrdquo IEEE Transactions on Robotics andAutomation vol 18 no 5 pp 758ndash768 2002

[12] T Sandholm ldquoAlgorithm for optimal winner determination incombinatorial auctionsrdquo Artificial Intelligence vol 135 no 1-2pp 1ndash54 2002

[13] M Berhault H Huang P Keskinocak et al ldquoRobot explorationwith combinatorial auctionsrdquo in Proceedings of the IEEERSJInternational Conference on Intelligent Robots and Systems vol1ndash4 pp 1957ndash1962 October 2003

[14] M G Lagoudakis M Berhault S Koenig P Keskinocak andA J Kleywegt ldquoSimple auctions with performance guaranteesfor multi-robot task allocationrdquo in Proceedings of the IEEERSJInternational Conference on Intelligent Robots and Systems(IROS rsquo04) pp 698ndash705 IEEE October 2004

[15] L Liu and D A Shell ldquoLarge-scale multi-robot task allocationvia dynamic partitioning and distributionrdquoAutonomous Robotsvol 33 no 3 pp 291ndash307 2012

[16] S Ozturk and A E Kuzucuoglu ldquoOptimal bid valuationusing path finding for multi-robot task allocationrdquo Journal ofIntelligent Manufacturing 2014

[17] D Puig M A Garcia and L Wu ldquoA new global optimizationstrategy for coordinated multi-robot exploration developmentand comparative evaluationrdquo Robotics and Autonomous Sys-tems vol 59 no 9 pp 635ndash653 2011

[18] M Elango S Nachiappan and M K Tiwari ldquoBalancing taskallocation inmulti-robot systems using K-means clustering andauction based mechanismsrdquo Expert Systems with Applicationsvol 38 no 6 pp 6486ndash6491 2011

[19] A G Bedeian and K W Mossholder ldquoOn the use of thecoefficient of variation as ameasure of diversityrdquoOrganizationalResearch Methods vol 3 no 3 pp 285ndash297 2000

[20] S S Chiddarwar and N R Babu ldquoConflict free coordinatedpath planning for multiple robots using a dynamic path mod-ification sequencerdquo Robotics and Autonomous Systems vol 59no 7-8 pp 508ndash518 2011

[21] J Capitan M T J Spaan L Merino and A Ollero ldquoDecen-tralizedmulti-robot cooperationwith auctioned POMDPsrdquoTheInternational Journal of Robotics Research vol 32 no 6 pp 650ndash671 2013

[22] D Dong C Chen H Li and T-J Tarn ldquoQuantum rein-forcement learningrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 38 no 5 pp 1207ndash12202008

[23] C Chen D Dong H-X Li J Chu and T-J Tarn ldquoFidelity-based probabilistic Q-learning for control of quantum systemsrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 25 no 5 pp 920ndash933 2014

[24] M Elango G Kanagaraj and S G Ponnambalam ldquoSandholmalgorithm with K-means clustering approach for multi-robottask allocationrdquo in Swarm Evolutionary and Memetic Com-puting vol 8298 pp 14ndash22 Springer International Publishing2013

[25] M E Celebi H A Kingravi and P A Vela ldquoA comparativestudy of efficient initialization methods for the k-means clus-tering algorithmrdquo Expert Systems with Applications vol 40 no1 pp 200ndash210 2013

[26] M Shao ldquoResearch on improved convergence co-cevolutiongenetic algorithm for TSPsrdquo International Journal of Conver-gence Computing vol 1 no 2 pp 118ndash126 2014

[27] I Mezei M Lukic V Malbasa and I Stojmenovic ldquoAuctionsand iMesh based task assignment in wireless sensor andactuator networksrdquo Computer Communications vol 36 no 9pp 979ndash987 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

Costevaluation

Path planning and motion(obstacle avoidance)

Decentralized controller

Central controller

(2) Task completion

Orderinput

Wirelesscommunication

R1

R2

R3

middot middot middotTask allocation(using BHM)

(1) Accurate Wij Cij

Figure 3 Control system structure

could be further enhanced in the real world comparedwith the simulated balancing performance of the simulatedexperiment This mechanism is referred as more-task-priormechanism in this paper The whole process is shown inFigure 3

32 Balanced Heuristic Mechanism (BHM) Considering thegoal of minimizing the travel cost as well as balancing theindividual travel cost of each robot a heuristic function isgiven in the following for guiding the task allocation process

119862 (119903

119894 119905

119895) = 120572 times dist (119903

119894 119905

119895) + (1 minus 120572) times 119862119882(119903

119894) (4)

where 119862(119903119894 119905

119895) describes the cost (considering both the time

and distance) for robot 119903119894to finish task 119905

119895 dist(119903

119894 119905

119895) is the

travel cost from robot 119903119894to task 119905

119895 and 119862119882(119903

119894) is the current

value of119882(119903119894) (refer to Section 2)

More specifically as both task cost and travel cost fromone location to another are unknown we evaluate 119908

119894(the

cost of each task) and119862119894119895(the travel cost from one location to

another) before the task allocation by using the standard Alowastalgorithm a simple but significantly effective path planningalgorithm When a robot fulfills a task or moves from onelocation to another in real environment more complexmotions should be taken into account such as the obstacleavoidance Then the more accurate value is shared with thecentral controller and the original estimated value stored inthe controller is replaced with the newest value which canhelp the controller to make a better task allocation Thus thefollowing allocation process is in the light of new informationand can be more accurate and predictable

BHM makes tasks to be assigned to the robot who hasthe lowest 119862(119903

119894 119905

119895) By introducing a parameter 120572 targets are

more inclined to be assigned to the nearby robot with fewerallocated tasks In practice an effective value of 120572 can be setbetween 07 and 09 according to the numbers of robots andtasks The detailed relationship between 120572 and the systemperformance will be given in Section 4

At the beginning stage as the value of119882(119903119894) is quite small

this mechanism has limited effects Thus the performance

of the system is not significantly superior to the one bytraditional methods that merely focus on minimizing thetotal travel distance However as the allocation proceeds therole of (1 minus 120572) times 119862119882(119903

119894) comes into effect by sharing the

workload with those who have fewer assigned tasks Aboveall the method can obtain an approximately optimal resultby considering the total travel cost and the workload balanceof robots A sample illustration of this heuristic mechanismis given in Figure 4

33 Application of BHM Recently the auction-based taskallocation method and the cluster-based task allocationmethod have been widely used to solve the MRTA problems[18 24] So we introduce our BHM approach into the classicsingle-item auction method and the 119870-means clusteringmethod respectively and proposed two BMH-based meth-ods to solve the balancing MRTA problem

331 An Improved Auction Based Task Allocation MethodUsing BHM A typical seal-bid single-round single-itemauction process generally consists of four steps

(i) task announcement(ii) matrix evaluation(iii) bid submission(iv) close of auction (to determine the winning bids and

notify the winning robot)

At the second stage thematrix is usually defined as a functionof the optimistic travel cost [11] as shown in the followingmatrix

(

(

119889

1199031 1199051119889

1199031 1199052sdot sdot sdot sdot sdot sdot 119889

1199031 119905119899

119889

1199032 1199051119889

1199032 1199052sdot sdot sdot sdot sdot sdot 119889

1199032 119905119899

d

d

119889

119903119898 1199051119889

119903119898 1199052sdot sdot sdot sdot sdot sdot 119889

119903119898 119905119899

)

)

(5)

Mathematical Problems in Engineering 5

tj

rmrm rnrn

Dist (rn tj) = 20 Dist (rn tj) = 20 Dist (rn tj) = 10Dist (rn tj) = 10

CW(rm) = 300 CW(rm) = 300CW(rn) = 340 CW(rn) = 320

C(rm tj) = 07 times 20 + 03 times 300 = 104radic

C(rn tj) = 07 times 10 + 03 times 340 = 109

C(rm tj) = 07 times 20 + 03 times 300 = 104

C(rn tj) = 07 times 10 + 03 times 320 = 103radic

tj

Figure 4 Example of the balanced heuristic mechanism

C(r1 t1) C(r1 t2)

C(r2 t1) C(r2 t2)

C(ri t1) C(ri t2)

C(rm t1) C(rm t2)

C(r1 tj)

C(r2 tj)

C(rm tj)

C(r1 tn)

C(r2 tn)

C(ri tn)

C(rm tn)

middot middot middot

middot middot middot

middot middot middot

middot middot middot middot middot middot middot middot middot

middot middot middot middot middot middot

middot middot middot

middot middot middot middot middot middot

middot middot middot

⋱

⋱

⋱⋱

⋱

Delete

RefreshC(ri tj)

Figure 5 Evaluation matrix refreshment

where 1199051 119905

2 119905

119899 are all the open tasks which have not

been carried out by robots In our method the BHM isintroduced into the matrix evaluation process In orderto strengthen the balancing performance of the auctionthe matrix evaluation is defined by 119862

119903119894 119905119895rather than the

optimistic travel distance between 119903119894and 119905119895

In each round after all of the robots submit their bidsthe single-round single-item auction is closedThe controllerwill decide thewinning bidderwith the smallest bid value andnotify the winner Then the bid matrix will be refreshed withfollowing steps (as shown in Figure 5)

Step 1 Delete the task 119905119896from the open-task set

Step 2 Recalculate the row (119862119903119894 1199051 119862

119903119894 1199051 119862

119903119894 119905119899)

As described in (4) the value of 119862(119903119894 119905

119895) is both related to

the distance between 119903119894and 119905119895as well as the already assigned

tasks of 119903119894 Firstly the current position of 119903

119894is replaced by

the location of 119905119895 since 119903

119894wins the bid and has to move

from its former position to 119905119895in order to carry out the

order Secondly the119862119882(119903119894) increases by the cost of the newly

allocated task which can be mathematically expressed as119862119882(119903

119894) = 119862119882(119903

119894) + 119908

119894119895 After the matrix is refreshed next

single-item auction process is proceeding until all the taskshave been allocated

332 A Balanced K-Means ClusteringMethod Using BHM Aclustering method can be generally divided into two stagesthat is firstly finding the cluster centers and then assigningthe sample data into the proper clusterThe119870-means cluster-ing is a commonly used partitioning method [25] the mainidea of which is to find119870 cluster centers (119888

1 119888

2 119888

119896) which

minimize119863 in the following by iteration

119863 =

119899

sum

119894minus1

min119903=12119896

119889 (119909

119894 119888

119903)

2

(6)

In (6) 119889(119909119894 119888

119903) represents the distance between the point 119909

119894

and 119888119903 known as the Euclidean distance When 119863 converges

to a stable value the appropriate (1198881 119888

2 119888

119896) are the cluster

centers for those119870 subgroupsAt the second stage the sample data 119909

119894is assigned into

the proper cluster with the smallest 119889(119909119894 119888

119903) 119903 = 1 2 119896

6 Mathematical Problems in Engineering

However this method only guarantees an optimal allocationscheme minimizing the travel cost while the size differencesbetween each cluster are ignored

In order to solve the balancing MRTA problem we makesome modifications on both of the two stages As for the firststage given that robots can only move along the grid lineswe substitute the Euclidean distance with the Manhattandistance in (6) In a rectangular coordinate system theManhattan distance between point 119875

1 (119909

1 119910

1) and point

119875

2 (119909

2 119910

2) can be expressed as

119889manhattan (1198751 1198752) =1003816

1003816

1003816

1003816

119909

1minus 119909

2

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119910

1minus 119910

2

1003816

1003816

1003816

1003816

(7)

Thus the main purpose is to find 119870 cluster centers(119888

1 119888

2 119888

119896) which minimize 119863manhattan in the following by

iteration

119863manhattan =119899

sum

119894minus1

min119903=12119896

119889manhattan (119909119894 119888119903) (8)

Then to reduce the value119863manhattan iteratively we find theManhattan center rather than the center of the collected tasksFinally a partial optimal solution is formed in a finite numberof iterations Generally speaking there are three steps of thefirst stage

(1) Initial step give an initial set of119870 cluster centers(2) Assignment step assign each point to the cluster center

which has the shortest Manhattan distance(3) Update step calculate the Manhattan Center which

can reduce the overall value of 119863manhattan The Man-hattan Center is defined as follows

It is supposed that there are 119898 points in a rectan-gular coordinate system as 119901

1 119901

2 119901

119898 coordinates of

which are denoted as (1199091 119910

1) (119909

2 119910

2) (119909

119898 119910

119898)Then the

descending order of the sequence 1199091 119909

2 119909

119898is denoted

as 11990910158401 119909

1015840

2 119909

1015840

119898and the descending order of the sequence

119910

1 119910

2 119910

119898is denoted as 1199101015840

1 119910

1015840

2 119910

1015840

119898 Our purpose is to

find the point 119875119898= (119909

119898119888 119910

119898119888) which minimizes (8)

min119909119898119888119910119898119888

119898

sum

119894=1

(

1003816

1003816

1003816

1003816

119909

119898119888minus 119909

119894

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119910

119898119888minus 119910

119894

1003816

1003816

1003816

1003816

) (9)

Since the values of 119909119898119888

and 119910119898119888

do not affect each otherthe value of sum119898

119894=1|119909

119898119888minus 119909

119894| and sum119898

119894=1|119910

119898119888minus 119910

119894| is minimized

separately It is easy to prove that when 119898 is odd 119909119898119888

is119909

1015840

(119898+1)2so that sum119898

119894=1|119909

119898119888minus 119909

119894| is minimum when 119898 is even

119909

119898119888is between 1199091015840

(119898+1)2and 1199091015840

((119898+1)2)+1so thatsum119898

119894=1|119909

119898119888minus119909

119894|

is minimum We can get 119910119898119888

in the same way Thereforethe Manhattan center 119901

119898119888can be calculated by following

equations

119909

119898119888=

119909

1015840

(119898+1)2119898 is odd

119909

1015840

(119898+1)2+ 119909

1015840

((119898+1)2)+1

2

119898 is even

119910

119898119888=

119910

1015840

(119898+1)2119898 is odd

119910

1015840

(119898+1)2+ 119910

1015840

((119898+1)2)+1

2

119898 is even

(10)

p4(36)

p1(14)

p2(22)

p3(43)

p5(88)

p6(96)

p7(91)

poc (56)

pmc (44)

Original center

Manhattan center

Figure 6 Example of the manhattan center

Figure 7 Example of the clustering result

As mentioned before the Manhattan center has theminimum overall Manhattan distance to these 119898 points bywhich the value of119863manhattan is reduced An example is shownin Figure 6 For the original center 119901

119900119888sum119898119894=1(|119909

119900119888minus119909

119894| + |119910

119900119888minus

119910

119894|) = 37 while for the Manhattan Center 119901

119898119888 sum119898119894=1(|119909

119898119888minus

119909

119894| + |119910

119898119888minus 119910

119894|) = 34

Now we concisely prove the convergence of the algo-rithm Given that the119863manhattan generated by the algorithm isstrictly decreasing and there only exist a finite number of suchpartitions the algorithm will reach a partial optimal solutionin a finite number of iterations

After obtaining the cluster centers the balanced heuristicmechanism is introduced into the second stage of cluster-ing In traditional methods when the 119898 cluster centers(119903

1 119903

2 119903

119898) are found in the first stage of the 119870-means

algorithm the remaining active tasks are allocated into theproper cluster with the smallest 119889(119909

119894 119888

119903) 119903 = 1 2 119896 How-

ever in our approach remaining active tasks are allocatedinto the proper cluster with the smallest 119862(119903

119894 119905

119895) defined in

(4)The main idea is that instead of merely considering the

travel cost the balance of the cluster size is taken into accountThe task tends to be allocated into the nearby cluster withfewer tasks When a new task is added into the cluster 119895the 119862119882(119903

119894) will be refreshed After all the active tasks have

been allocated the genetic-based TSP algorithm [26] is usedto plan the route for tasks in each cluster One example ofthe task allocation is demonstrated in Figure 7 (119898 = 4 119899 =50 120572 = 08) where different colors represent tasks allocatedto different robots Blocks with119883 are clustering centers

Mathematical Problems in Engineering 7

Table 1 Balance behavior of single-item auction versus BHM auc-tion

119877 = 8 Single-item auction BHM auction (120572 = 08)119898 120583 120590 CV = 120590120583 120583 120590 CV = 12059012058350 685 312 456 711 61 86100 1374 315 229 1469 53 36150 1969 553 281 2128 41 19200 2591 449 173 2827 50 18250 3234 439 136 3555 32 09300 3897 609 156 4221 21 05350 4418 701 159 4794 26 05400 5085 775 152 5499 19 04

4 Simulated Experiments

41 Simulation Description Several groups of simulatedexperiments are implemented with OPENGL and the exper-imental process is as follows

Step 1 Initialization

Step 2 Creating orders in the task pool while the arriving rateof orders meets the Poisson distribution

Step 3 Sending a specific number (eg 200) of orders fromthe task pool to the controller

Step 4 Using different methods to allocate the orders (tasks)More specifically there are four approaches namely 119870-means BHM 119870-means auction and BHMauction and theirperformances are compared

Step 5 Sending the allocated tasks by the controller to thecorresponding robots which begin to fulfil their tasks (usingthe Alowast algorithm to calculate the path and using the more-task-prior mechanism to avoid collision) Then go to Step 3

42 Simulation Results Analysis In this part three criteriamentioned in Section 2 are used to evaluate the performanceof the proposed balanced heuristic mechanism that iscoefficient of variation (CV) travel time (TT) and total travelcost (TTC)

First of all we compare the balance behavior of theimproved methods with the typical auction method and the119870-means method results of which are given in Tables 1and 2 where the data are collected by average during fiftytimes of simulation experiments Both Tables 1 and 2 showthat using the proposed BHM the diversity degree in workgroups is significantly reduced which indicates the moreequal workload distribution among robots

As shown in Figures 8 and 9 the travel time (TT) ofthe multirobot system is provided by both the traditionalmethod and the improved method We can conclude that byapplying the BHM based methods the time spent to finish

Table 2 Balance behavior of typical 119870-means versus BHM 119870-means

119877 = 8 119870-means BHM K-means (120572 = 08)119898 120583 120590 CV = 120590120583 120583 120590 CV = 12059012058350 714 310 434 749 50 67100 1390 513 369 1508 57 38150 2058 764 371 2204 49 22200 2751 1116 406 2949 63 21250 3441 1076 313 3684 69 19300 4095 1254 306 4440 65 15350 4776 1484 311 5214 83 16400 5465 1882 344 5895 66 11

50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

7000

8000

Task number

TT

AuctionBHM auction

Figure 8 Time spent by single-item auction versus BHM auction

the given task is much less than the one by using thetraditional method due to an efficient sharing of workload

Comparison results for traditional methods and BHMbasedmethods are demonstrated in Figure 10 bymeans of thetotal travel cost (TTC) from which we can conclude that inbalanced heuristic cases although robots pay more attentionto balancing the workload rather thanmerelyminimizing thetravel cost as traditional methods do the TTC only increasesby about 8 Concerning the obvious increasement of therobots utilization this slightly increased power consumptionis quite acceptable

With the traditional auction method and 119870-meansmethod the task allocation process only concentrates onminimizing the travel cost Once there is a robot nearest toan open task other robots do not attempt to compete for thetask even if they are idle Thus the utilization of the system isnot enough and the time cost to fulfill the given tasks is highWith BHM based methods proposed in this paper all robots

8 Mathematical Problems in Engineering

50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

7000

8000

9000

Task number

TT

K-meansBHM K-means

Figure 9 Time spent by 119870-means versus BHM 119870-means

50 100 150 200 250 300 350 400012345

TTC

Task number

times104

AuctionBHM auction

(a)

50 100 150 200 250 300 350 400012345

Task number

TTC

times104

K-meansBHM K-means

(b)

Figure 10 Total travel cost by traditional methods versus BHM adapted methods

can be utilized in a balanced and effective manner while thetravel cost is quite low

Then we study the influence of the variable 120572 on thesystem performance The value of 120572 describes the weight ofthe system energy saving demand and time saving demandFigure 11 demonstrates the relationship between 120572 and thetravel time (TT) It shows that in order to keep a good balancebetween minimizing the travel distance and efficient sharingof the workload the coefficient is usually set between 07 and09 to achieve the minimum of TT

Finally besides comparing BHM-based approaches withthe traditional auctionmethod and119870-meansmethod we usea more competing algorithm (119870CAB mechanism in [18]) totestify the scientific merit of the BHM scheme (as shownin Figures 12 and 13) We can conclude that by BHM basedmethods the time spent to finish the given task is 5sim15less than the119870CABmechanism meanwhile their total travelcosts are almost the same Thus the benefit of the balancedheuristic mechanism to warehouse systems is significant

5 Conclusion

In this paper a novel balanced heuristic mechanism for themultirobot task allocation problem is presented with respectto the task cost and travel cost To evaluate the proposedapproach we give three criteria namely travel time (TT)total travel cost (TTC) and coefficient of variation (CV)and compare some traditional methods with BHM-basedmethods under those criterions Simulation results show thatthe proposed approach greatly enhances the balance behaviorof the multirobot system reduces the total travel time whileachieves almost the same total travel cost as the traditionalmethods do In addition the influence of the variable 120572 onthe system performance is also studied in detail In our futurework BHM will be applied into more complicated envi-ronments which consist of heterogeneous tasks and robotsFurthermore we will dedicate to enhance the robustness ofthe proposed approach in dynamic environments Finally itis worth to point out that this new mechanism is of great

Mathematical Problems in Engineering 9

01 02 03 04 05 06 07 08 09 108

09

10

11

12

13

14

15

TT

120572

Task number = 50Task number = 100

Task number = 200Task number = 300

Normalized TT versus 120572 (m = 8)

Figure 11 Relationship between 120572 and travel time (TT)

50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

7000

Task number

TT

BHM auction

KCAB mechanismBHM K-means

Figure 12 Comparison between method provided in [15] versusBHM-based approaches (TT)

applicability to other task allocation problems such as theWSANs problem in [27]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Authorsrsquo Contribution

Luowei Zhou and Yuanyuan Shi contributed equally to thiswork

50 100 150 200 250 300 350 400

TTC

Task number

times104

BHM auction

KCAB mechanismBHM K-means

5

45

4

35

3

25

2

15

1

05

Figure 13 Comparison between method provided in [15] versusBHM-based approaches (TTC)

Acknowledgment

This work was supported by the Natural Science Foundationof China under Grants no 61273327 and no 61432008

References

[1] P R Wurman R DrsquoAndrea and M Mountz ldquoCoordinatinghundreds of cooperative autonomous vehicles in warehousesrdquoAI Magazine vol 29 no 1 pp 9ndash19 2008

[2] X Zhai J E Ward and L B Schwarz ldquoCoordinating aone-warehouse N-retailer distribution system under retailer-reportingrdquo International Journal of Production Economics vol134 no 1 pp 204ndash211 2011

[3] C Chen H-X Li and D Dong ldquoHybrid control forrobot navigationmdasha hierarchical Q-learning algorithmrdquo IEEERobotics amp AutomationMagazine vol 15 no 2 pp 37ndash47 2008

[4] S Chen and C Chen ldquoProbabilistic fuzzy system for uncertainlocalization and map building of mobile robotsrdquo IEEE Trans-actions on Instrumentation and Measurement vol 61 no 6 pp1546ndash1560 2012

[5] D Dong C Chen J Chu and T-J Tarn ldquoRobust quantum-inspired reinforcement learning for robot navigationrdquo IEEEASME Transactions on Mechatronics vol 17 no 1 pp 86ndash972012

[6] T Niemueller D Ewert S Reuter et al ldquoTowards benchmarking cyber-physical systems in factory automation scenariosrdquoinKI 2013 Advances in Artificial Intelligence vol 8077 of LectureNotes in Computer Science pp 296ndash299 2013

[7] M Brambilla E Ferrante M Birattari and M Dorigo ldquoSwarmrobotics a review from the swarm engineering perspectiverdquoSwarm Intelligence vol 7 no 1 pp 1ndash41 2013

[8] C Nieto-Granda J G Rogers and H I Christensen ldquoCoor-dination strategies for multi-robot exploration and mappingrdquoExperimental Robotics vol 33 no 4 pp 519ndash533 2013

10 Mathematical Problems in Engineering

[9] B-Y Shih C-Y Chen and W Chou ldquoAn enhanced obstacleavoidance and path correction mechanism for an autonomousintelligent robot withmultiple sensorsrdquo Journal of Vibration andControl vol 18 no 12 pp 1855ndash1864 2012

[10] B Yamauchi ldquoFrontier-based exploration using multiplerobotsrdquo in Proceedings of the 2nd International Conference onAutonomous Agents pp 47ndash53 ACM May 1998

[11] B P Gerkey and M J Mataric ldquoSold auction methods formultirobot coordinationrdquo IEEE Transactions on Robotics andAutomation vol 18 no 5 pp 758ndash768 2002

[12] T Sandholm ldquoAlgorithm for optimal winner determination incombinatorial auctionsrdquo Artificial Intelligence vol 135 no 1-2pp 1ndash54 2002

[13] M Berhault H Huang P Keskinocak et al ldquoRobot explorationwith combinatorial auctionsrdquo in Proceedings of the IEEERSJInternational Conference on Intelligent Robots and Systems vol1ndash4 pp 1957ndash1962 October 2003

[14] M G Lagoudakis M Berhault S Koenig P Keskinocak andA J Kleywegt ldquoSimple auctions with performance guaranteesfor multi-robot task allocationrdquo in Proceedings of the IEEERSJInternational Conference on Intelligent Robots and Systems(IROS rsquo04) pp 698ndash705 IEEE October 2004

[15] L Liu and D A Shell ldquoLarge-scale multi-robot task allocationvia dynamic partitioning and distributionrdquoAutonomous Robotsvol 33 no 3 pp 291ndash307 2012

[16] S Ozturk and A E Kuzucuoglu ldquoOptimal bid valuationusing path finding for multi-robot task allocationrdquo Journal ofIntelligent Manufacturing 2014

[17] D Puig M A Garcia and L Wu ldquoA new global optimizationstrategy for coordinated multi-robot exploration developmentand comparative evaluationrdquo Robotics and Autonomous Sys-tems vol 59 no 9 pp 635ndash653 2011

[18] M Elango S Nachiappan and M K Tiwari ldquoBalancing taskallocation inmulti-robot systems using K-means clustering andauction based mechanismsrdquo Expert Systems with Applicationsvol 38 no 6 pp 6486ndash6491 2011

[19] A G Bedeian and K W Mossholder ldquoOn the use of thecoefficient of variation as ameasure of diversityrdquoOrganizationalResearch Methods vol 3 no 3 pp 285ndash297 2000

[20] S S Chiddarwar and N R Babu ldquoConflict free coordinatedpath planning for multiple robots using a dynamic path mod-ification sequencerdquo Robotics and Autonomous Systems vol 59no 7-8 pp 508ndash518 2011

[21] J Capitan M T J Spaan L Merino and A Ollero ldquoDecen-tralizedmulti-robot cooperationwith auctioned POMDPsrdquoTheInternational Journal of Robotics Research vol 32 no 6 pp 650ndash671 2013

[22] D Dong C Chen H Li and T-J Tarn ldquoQuantum rein-forcement learningrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 38 no 5 pp 1207ndash12202008

[23] C Chen D Dong H-X Li J Chu and T-J Tarn ldquoFidelity-based probabilistic Q-learning for control of quantum systemsrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 25 no 5 pp 920ndash933 2014

[24] M Elango G Kanagaraj and S G Ponnambalam ldquoSandholmalgorithm with K-means clustering approach for multi-robottask allocationrdquo in Swarm Evolutionary and Memetic Com-puting vol 8298 pp 14ndash22 Springer International Publishing2013

[25] M E Celebi H A Kingravi and P A Vela ldquoA comparativestudy of efficient initialization methods for the k-means clus-tering algorithmrdquo Expert Systems with Applications vol 40 no1 pp 200ndash210 2013

[26] M Shao ldquoResearch on improved convergence co-cevolutiongenetic algorithm for TSPsrdquo International Journal of Conver-gence Computing vol 1 no 2 pp 118ndash126 2014

[27] I Mezei M Lukic V Malbasa and I Stojmenovic ldquoAuctionsand iMesh based task assignment in wireless sensor andactuator networksrdquo Computer Communications vol 36 no 9pp 979ndash987 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

tj

rmrm rnrn

Dist (rn tj) = 20 Dist (rn tj) = 20 Dist (rn tj) = 10Dist (rn tj) = 10

CW(rm) = 300 CW(rm) = 300CW(rn) = 340 CW(rn) = 320

C(rm tj) = 07 times 20 + 03 times 300 = 104radic

C(rn tj) = 07 times 10 + 03 times 340 = 109

C(rm tj) = 07 times 20 + 03 times 300 = 104

C(rn tj) = 07 times 10 + 03 times 320 = 103radic

tj

Figure 4 Example of the balanced heuristic mechanism

C(r1 t1) C(r1 t2)

C(r2 t1) C(r2 t2)

C(ri t1) C(ri t2)

C(rm t1) C(rm t2)

C(r1 tj)

C(r2 tj)

C(rm tj)

C(r1 tn)

C(r2 tn)

C(ri tn)

C(rm tn)

middot middot middot

middot middot middot

middot middot middot

middot middot middot middot middot middot middot middot middot

middot middot middot middot middot middot

middot middot middot

middot middot middot middot middot middot

middot middot middot

⋱

⋱

⋱⋱

⋱

Delete

RefreshC(ri tj)

Figure 5 Evaluation matrix refreshment

where 1199051 119905

2 119905

119899 are all the open tasks which have not

been carried out by robots In our method the BHM isintroduced into the matrix evaluation process In orderto strengthen the balancing performance of the auctionthe matrix evaluation is defined by 119862

119903119894 119905119895rather than the

optimistic travel distance between 119903119894and 119905119895

In each round after all of the robots submit their bidsthe single-round single-item auction is closedThe controllerwill decide thewinning bidderwith the smallest bid value andnotify the winner Then the bid matrix will be refreshed withfollowing steps (as shown in Figure 5)

Step 1 Delete the task 119905119896from the open-task set

Step 2 Recalculate the row (119862119903119894 1199051 119862

119903119894 1199051 119862

119903119894 119905119899)

As described in (4) the value of 119862(119903119894 119905

119895) is both related to

the distance between 119903119894and 119905119895as well as the already assigned

tasks of 119903119894 Firstly the current position of 119903

119894is replaced by

the location of 119905119895 since 119903

119894wins the bid and has to move

from its former position to 119905119895in order to carry out the

order Secondly the119862119882(119903119894) increases by the cost of the newly

allocated task which can be mathematically expressed as119862119882(119903

119894) = 119862119882(119903

119894) + 119908

119894119895 After the matrix is refreshed next

single-item auction process is proceeding until all the taskshave been allocated

332 A Balanced K-Means ClusteringMethod Using BHM Aclustering method can be generally divided into two stagesthat is firstly finding the cluster centers and then assigningthe sample data into the proper clusterThe119870-means cluster-ing is a commonly used partitioning method [25] the mainidea of which is to find119870 cluster centers (119888

1 119888

2 119888

119896) which

minimize119863 in the following by iteration

119863 =

119899

sum

119894minus1

min119903=12119896

119889 (119909

119894 119888

119903)

2

(6)

In (6) 119889(119909119894 119888

119903) represents the distance between the point 119909

119894

and 119888119903 known as the Euclidean distance When 119863 converges

to a stable value the appropriate (1198881 119888

2 119888

119896) are the cluster

centers for those119870 subgroupsAt the second stage the sample data 119909

119894is assigned into

the proper cluster with the smallest 119889(119909119894 119888

119903) 119903 = 1 2 119896

6 Mathematical Problems in Engineering

However this method only guarantees an optimal allocationscheme minimizing the travel cost while the size differencesbetween each cluster are ignored

In order to solve the balancing MRTA problem we makesome modifications on both of the two stages As for the firststage given that robots can only move along the grid lineswe substitute the Euclidean distance with the Manhattandistance in (6) In a rectangular coordinate system theManhattan distance between point 119875

1 (119909

1 119910

1) and point

119875

2 (119909

2 119910

2) can be expressed as

119889manhattan (1198751 1198752) =1003816

1003816

1003816

1003816

119909

1minus 119909

2

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119910

1minus 119910

2

1003816

1003816

1003816

1003816

(7)

Thus the main purpose is to find 119870 cluster centers(119888

1 119888

2 119888

119896) which minimize 119863manhattan in the following by

iteration

119863manhattan =119899

sum

119894minus1

min119903=12119896

119889manhattan (119909119894 119888119903) (8)

Then to reduce the value119863manhattan iteratively we find theManhattan center rather than the center of the collected tasksFinally a partial optimal solution is formed in a finite numberof iterations Generally speaking there are three steps of thefirst stage

(1) Initial step give an initial set of119870 cluster centers(2) Assignment step assign each point to the cluster center

which has the shortest Manhattan distance(3) Update step calculate the Manhattan Center which

can reduce the overall value of 119863manhattan The Man-hattan Center is defined as follows

It is supposed that there are 119898 points in a rectan-gular coordinate system as 119901

1 119901

2 119901

119898 coordinates of

which are denoted as (1199091 119910

1) (119909

2 119910

2) (119909

119898 119910

119898)Then the

descending order of the sequence 1199091 119909

2 119909

119898is denoted

as 11990910158401 119909

1015840

2 119909

1015840

119898and the descending order of the sequence

119910

1 119910

2 119910

119898is denoted as 1199101015840

1 119910

1015840

2 119910

1015840

119898 Our purpose is to

find the point 119875119898= (119909

119898119888 119910

119898119888) which minimizes (8)

min119909119898119888119910119898119888

119898

sum

119894=1

(

1003816

1003816

1003816

1003816

119909

119898119888minus 119909

119894

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119910

119898119888minus 119910

119894

1003816

1003816

1003816

1003816

) (9)

Since the values of 119909119898119888

and 119910119898119888

do not affect each otherthe value of sum119898

119894=1|119909

119898119888minus 119909

119894| and sum119898

119894=1|119910

119898119888minus 119910

119894| is minimized

separately It is easy to prove that when 119898 is odd 119909119898119888

is119909

1015840

(119898+1)2so that sum119898

119894=1|119909

119898119888minus 119909

119894| is minimum when 119898 is even

119909

119898119888is between 1199091015840

(119898+1)2and 1199091015840

((119898+1)2)+1so thatsum119898

119894=1|119909

119898119888minus119909

119894|

is minimum We can get 119910119898119888

in the same way Thereforethe Manhattan center 119901

119898119888can be calculated by following

equations

119909

119898119888=

119909

1015840

(119898+1)2119898 is odd

119909

1015840

(119898+1)2+ 119909

1015840

((119898+1)2)+1

2

119898 is even

119910

119898119888=

119910

1015840

(119898+1)2119898 is odd

119910

1015840

(119898+1)2+ 119910

1015840

((119898+1)2)+1

2

119898 is even

(10)

p4(36)

p1(14)

p2(22)

p3(43)

p5(88)

p6(96)

p7(91)

poc (56)

pmc (44)

Original center

Manhattan center

Figure 6 Example of the manhattan center

Figure 7 Example of the clustering result

As mentioned before the Manhattan center has theminimum overall Manhattan distance to these 119898 points bywhich the value of119863manhattan is reduced An example is shownin Figure 6 For the original center 119901

119900119888sum119898119894=1(|119909

119900119888minus119909

119894| + |119910

119900119888minus

119910

119894|) = 37 while for the Manhattan Center 119901

119898119888 sum119898119894=1(|119909

119898119888minus

119909

119894| + |119910

119898119888minus 119910

119894|) = 34

Now we concisely prove the convergence of the algo-rithm Given that the119863manhattan generated by the algorithm isstrictly decreasing and there only exist a finite number of suchpartitions the algorithm will reach a partial optimal solutionin a finite number of iterations

After obtaining the cluster centers the balanced heuristicmechanism is introduced into the second stage of cluster-ing In traditional methods when the 119898 cluster centers(119903

1 119903

2 119903

119898) are found in the first stage of the 119870-means

algorithm the remaining active tasks are allocated into theproper cluster with the smallest 119889(119909

119894 119888

119903) 119903 = 1 2 119896 How-

ever in our approach remaining active tasks are allocatedinto the proper cluster with the smallest 119862(119903

119894 119905

119895) defined in

(4)The main idea is that instead of merely considering the

travel cost the balance of the cluster size is taken into accountThe task tends to be allocated into the nearby cluster withfewer tasks When a new task is added into the cluster 119895the 119862119882(119903

119894) will be refreshed After all the active tasks have

been allocated the genetic-based TSP algorithm [26] is usedto plan the route for tasks in each cluster One example ofthe task allocation is demonstrated in Figure 7 (119898 = 4 119899 =50 120572 = 08) where different colors represent tasks allocatedto different robots Blocks with119883 are clustering centers

Mathematical Problems in Engineering 7

Table 1 Balance behavior of single-item auction versus BHM auc-tion

119877 = 8 Single-item auction BHM auction (120572 = 08)119898 120583 120590 CV = 120590120583 120583 120590 CV = 12059012058350 685 312 456 711 61 86100 1374 315 229 1469 53 36150 1969 553 281 2128 41 19200 2591 449 173 2827 50 18250 3234 439 136 3555 32 09300 3897 609 156 4221 21 05350 4418 701 159 4794 26 05400 5085 775 152 5499 19 04

4 Simulated Experiments

41 Simulation Description Several groups of simulatedexperiments are implemented with OPENGL and the exper-imental process is as follows

Step 1 Initialization

Step 2 Creating orders in the task pool while the arriving rateof orders meets the Poisson distribution

Step 3 Sending a specific number (eg 200) of orders fromthe task pool to the controller

Step 4 Using different methods to allocate the orders (tasks)More specifically there are four approaches namely 119870-means BHM 119870-means auction and BHMauction and theirperformances are compared

Step 5 Sending the allocated tasks by the controller to thecorresponding robots which begin to fulfil their tasks (usingthe Alowast algorithm to calculate the path and using the more-task-prior mechanism to avoid collision) Then go to Step 3

42 Simulation Results Analysis In this part three criteriamentioned in Section 2 are used to evaluate the performanceof the proposed balanced heuristic mechanism that iscoefficient of variation (CV) travel time (TT) and total travelcost (TTC)

First of all we compare the balance behavior of theimproved methods with the typical auction method and the119870-means method results of which are given in Tables 1and 2 where the data are collected by average during fiftytimes of simulation experiments Both Tables 1 and 2 showthat using the proposed BHM the diversity degree in workgroups is significantly reduced which indicates the moreequal workload distribution among robots

As shown in Figures 8 and 9 the travel time (TT) ofthe multirobot system is provided by both the traditionalmethod and the improved method We can conclude that byapplying the BHM based methods the time spent to finish

Table 2 Balance behavior of typical 119870-means versus BHM 119870-means

119877 = 8 119870-means BHM K-means (120572 = 08)119898 120583 120590 CV = 120590120583 120583 120590 CV = 12059012058350 714 310 434 749 50 67100 1390 513 369 1508 57 38150 2058 764 371 2204 49 22200 2751 1116 406 2949 63 21250 3441 1076 313 3684 69 19300 4095 1254 306 4440 65 15350 4776 1484 311 5214 83 16400 5465 1882 344 5895 66 11

50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

7000

8000

Task number

TT

AuctionBHM auction

Figure 8 Time spent by single-item auction versus BHM auction

the given task is much less than the one by using thetraditional method due to an efficient sharing of workload

Comparison results for traditional methods and BHMbasedmethods are demonstrated in Figure 10 bymeans of thetotal travel cost (TTC) from which we can conclude that inbalanced heuristic cases although robots pay more attentionto balancing the workload rather thanmerelyminimizing thetravel cost as traditional methods do the TTC only increasesby about 8 Concerning the obvious increasement of therobots utilization this slightly increased power consumptionis quite acceptable

With the traditional auction method and 119870-meansmethod the task allocation process only concentrates onminimizing the travel cost Once there is a robot nearest toan open task other robots do not attempt to compete for thetask even if they are idle Thus the utilization of the system isnot enough and the time cost to fulfill the given tasks is highWith BHM based methods proposed in this paper all robots

8 Mathematical Problems in Engineering

50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

7000

8000

9000

Task number

TT

K-meansBHM K-means

Figure 9 Time spent by 119870-means versus BHM 119870-means

50 100 150 200 250 300 350 400012345

TTC

Task number

times104

AuctionBHM auction

(a)

50 100 150 200 250 300 350 400012345

Task number

TTC

times104

K-meansBHM K-means

(b)

Figure 10 Total travel cost by traditional methods versus BHM adapted methods

can be utilized in a balanced and effective manner while thetravel cost is quite low

Then we study the influence of the variable 120572 on thesystem performance The value of 120572 describes the weight ofthe system energy saving demand and time saving demandFigure 11 demonstrates the relationship between 120572 and thetravel time (TT) It shows that in order to keep a good balancebetween minimizing the travel distance and efficient sharingof the workload the coefficient is usually set between 07 and09 to achieve the minimum of TT

Finally besides comparing BHM-based approaches withthe traditional auctionmethod and119870-meansmethod we usea more competing algorithm (119870CAB mechanism in [18]) totestify the scientific merit of the BHM scheme (as shownin Figures 12 and 13) We can conclude that by BHM basedmethods the time spent to finish the given task is 5sim15less than the119870CABmechanism meanwhile their total travelcosts are almost the same Thus the benefit of the balancedheuristic mechanism to warehouse systems is significant

5 Conclusion

In this paper a novel balanced heuristic mechanism for themultirobot task allocation problem is presented with respectto the task cost and travel cost To evaluate the proposedapproach we give three criteria namely travel time (TT)total travel cost (TTC) and coefficient of variation (CV)and compare some traditional methods with BHM-basedmethods under those criterions Simulation results show thatthe proposed approach greatly enhances the balance behaviorof the multirobot system reduces the total travel time whileachieves almost the same total travel cost as the traditionalmethods do In addition the influence of the variable 120572 onthe system performance is also studied in detail In our futurework BHM will be applied into more complicated envi-ronments which consist of heterogeneous tasks and robotsFurthermore we will dedicate to enhance the robustness ofthe proposed approach in dynamic environments Finally itis worth to point out that this new mechanism is of great

Mathematical Problems in Engineering 9

01 02 03 04 05 06 07 08 09 108

09

10

11

12

13

14

15

TT

120572

Task number = 50Task number = 100

Task number = 200Task number = 300

Normalized TT versus 120572 (m = 8)

Figure 11 Relationship between 120572 and travel time (TT)

50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

7000

Task number

TT

BHM auction

KCAB mechanismBHM K-means

Figure 12 Comparison between method provided in [15] versusBHM-based approaches (TT)

applicability to other task allocation problems such as theWSANs problem in [27]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Authorsrsquo Contribution

Luowei Zhou and Yuanyuan Shi contributed equally to thiswork

50 100 150 200 250 300 350 400

TTC

Task number

times104

BHM auction

KCAB mechanismBHM K-means

5

45

4

35

3

25

2

15

1

05

Figure 13 Comparison between method provided in [15] versusBHM-based approaches (TTC)

Acknowledgment

This work was supported by the Natural Science Foundationof China under Grants no 61273327 and no 61432008

References

[1] P R Wurman R DrsquoAndrea and M Mountz ldquoCoordinatinghundreds of cooperative autonomous vehicles in warehousesrdquoAI Magazine vol 29 no 1 pp 9ndash19 2008

[2] X Zhai J E Ward and L B Schwarz ldquoCoordinating aone-warehouse N-retailer distribution system under retailer-reportingrdquo International Journal of Production Economics vol134 no 1 pp 204ndash211 2011

[3] C Chen H-X Li and D Dong ldquoHybrid control forrobot navigationmdasha hierarchical Q-learning algorithmrdquo IEEERobotics amp AutomationMagazine vol 15 no 2 pp 37ndash47 2008

[4] S Chen and C Chen ldquoProbabilistic fuzzy system for uncertainlocalization and map building of mobile robotsrdquo IEEE Trans-actions on Instrumentation and Measurement vol 61 no 6 pp1546ndash1560 2012

[5] D Dong C Chen J Chu and T-J Tarn ldquoRobust quantum-inspired reinforcement learning for robot navigationrdquo IEEEASME Transactions on Mechatronics vol 17 no 1 pp 86ndash972012

[6] T Niemueller D Ewert S Reuter et al ldquoTowards benchmarking cyber-physical systems in factory automation scenariosrdquoinKI 2013 Advances in Artificial Intelligence vol 8077 of LectureNotes in Computer Science pp 296ndash299 2013

[7] M Brambilla E Ferrante M Birattari and M Dorigo ldquoSwarmrobotics a review from the swarm engineering perspectiverdquoSwarm Intelligence vol 7 no 1 pp 1ndash41 2013

[8] C Nieto-Granda J G Rogers and H I Christensen ldquoCoor-dination strategies for multi-robot exploration and mappingrdquoExperimental Robotics vol 33 no 4 pp 519ndash533 2013

10 Mathematical Problems in Engineering

[9] B-Y Shih C-Y Chen and W Chou ldquoAn enhanced obstacleavoidance and path correction mechanism for an autonomousintelligent robot withmultiple sensorsrdquo Journal of Vibration andControl vol 18 no 12 pp 1855ndash1864 2012

[10] B Yamauchi ldquoFrontier-based exploration using multiplerobotsrdquo in Proceedings of the 2nd International Conference onAutonomous Agents pp 47ndash53 ACM May 1998

[11] B P Gerkey and M J Mataric ldquoSold auction methods formultirobot coordinationrdquo IEEE Transactions on Robotics andAutomation vol 18 no 5 pp 758ndash768 2002

[12] T Sandholm ldquoAlgorithm for optimal winner determination incombinatorial auctionsrdquo Artificial Intelligence vol 135 no 1-2pp 1ndash54 2002

[13] M Berhault H Huang P Keskinocak et al ldquoRobot explorationwith combinatorial auctionsrdquo in Proceedings of the IEEERSJInternational Conference on Intelligent Robots and Systems vol1ndash4 pp 1957ndash1962 October 2003

[14] M G Lagoudakis M Berhault S Koenig P Keskinocak andA J Kleywegt ldquoSimple auctions with performance guaranteesfor multi-robot task allocationrdquo in Proceedings of the IEEERSJInternational Conference on Intelligent Robots and Systems(IROS rsquo04) pp 698ndash705 IEEE October 2004

[15] L Liu and D A Shell ldquoLarge-scale multi-robot task allocationvia dynamic partitioning and distributionrdquoAutonomous Robotsvol 33 no 3 pp 291ndash307 2012

[16] S Ozturk and A E Kuzucuoglu ldquoOptimal bid valuationusing path finding for multi-robot task allocationrdquo Journal ofIntelligent Manufacturing 2014

[17] D Puig M A Garcia and L Wu ldquoA new global optimizationstrategy for coordinated multi-robot exploration developmentand comparative evaluationrdquo Robotics and Autonomous Sys-tems vol 59 no 9 pp 635ndash653 2011

[18] M Elango S Nachiappan and M K Tiwari ldquoBalancing taskallocation inmulti-robot systems using K-means clustering andauction based mechanismsrdquo Expert Systems with Applicationsvol 38 no 6 pp 6486ndash6491 2011

[19] A G Bedeian and K W Mossholder ldquoOn the use of thecoefficient of variation as ameasure of diversityrdquoOrganizationalResearch Methods vol 3 no 3 pp 285ndash297 2000

[20] S S Chiddarwar and N R Babu ldquoConflict free coordinatedpath planning for multiple robots using a dynamic path mod-ification sequencerdquo Robotics and Autonomous Systems vol 59no 7-8 pp 508ndash518 2011

[21] J Capitan M T J Spaan L Merino and A Ollero ldquoDecen-tralizedmulti-robot cooperationwith auctioned POMDPsrdquoTheInternational Journal of Robotics Research vol 32 no 6 pp 650ndash671 2013

[22] D Dong C Chen H Li and T-J Tarn ldquoQuantum rein-forcement learningrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 38 no 5 pp 1207ndash12202008

[23] C Chen D Dong H-X Li J Chu and T-J Tarn ldquoFidelity-based probabilistic Q-learning for control of quantum systemsrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 25 no 5 pp 920ndash933 2014

[24] M Elango G Kanagaraj and S G Ponnambalam ldquoSandholmalgorithm with K-means clustering approach for multi-robottask allocationrdquo in Swarm Evolutionary and Memetic Com-puting vol 8298 pp 14ndash22 Springer International Publishing2013

[25] M E Celebi H A Kingravi and P A Vela ldquoA comparativestudy of efficient initialization methods for the k-means clus-tering algorithmrdquo Expert Systems with Applications vol 40 no1 pp 200ndash210 2013

[26] M Shao ldquoResearch on improved convergence co-cevolutiongenetic algorithm for TSPsrdquo International Journal of Conver-gence Computing vol 1 no 2 pp 118ndash126 2014

[27] I Mezei M Lukic V Malbasa and I Stojmenovic ldquoAuctionsand iMesh based task assignment in wireless sensor andactuator networksrdquo Computer Communications vol 36 no 9pp 979ndash987 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

However this method only guarantees an optimal allocationscheme minimizing the travel cost while the size differencesbetween each cluster are ignored

In order to solve the balancing MRTA problem we makesome modifications on both of the two stages As for the firststage given that robots can only move along the grid lineswe substitute the Euclidean distance with the Manhattandistance in (6) In a rectangular coordinate system theManhattan distance between point 119875

1 (119909

1 119910

1) and point

119875

2 (119909

2 119910

2) can be expressed as

119889manhattan (1198751 1198752) =1003816

1003816

1003816

1003816

119909

1minus 119909

2

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119910

1minus 119910

2

1003816

1003816

1003816

1003816

(7)

Thus the main purpose is to find 119870 cluster centers(119888

1 119888

2 119888

119896) which minimize 119863manhattan in the following by

iteration

119863manhattan =119899

sum

119894minus1

min119903=12119896

119889manhattan (119909119894 119888119903) (8)

Then to reduce the value119863manhattan iteratively we find theManhattan center rather than the center of the collected tasksFinally a partial optimal solution is formed in a finite numberof iterations Generally speaking there are three steps of thefirst stage

(1) Initial step give an initial set of119870 cluster centers(2) Assignment step assign each point to the cluster center

which has the shortest Manhattan distance(3) Update step calculate the Manhattan Center which

can reduce the overall value of 119863manhattan The Man-hattan Center is defined as follows

It is supposed that there are 119898 points in a rectan-gular coordinate system as 119901

1 119901

2 119901

119898 coordinates of

which are denoted as (1199091 119910

1) (119909

2 119910

2) (119909

119898 119910

119898)Then the

descending order of the sequence 1199091 119909

2 119909

119898is denoted

as 11990910158401 119909

1015840

2 119909

1015840

119898and the descending order of the sequence

119910

1 119910

2 119910

119898is denoted as 1199101015840

1 119910

1015840

2 119910

1015840

119898 Our purpose is to

find the point 119875119898= (119909

119898119888 119910

119898119888) which minimizes (8)

min119909119898119888119910119898119888

119898

sum

119894=1

(

1003816

1003816

1003816

1003816

119909

119898119888minus 119909

119894

1003816

1003816

1003816

1003816

+

1003816

1003816

1003816

1003816

119910

119898119888minus 119910

119894

1003816

1003816

1003816

1003816

) (9)

Since the values of 119909119898119888

and 119910119898119888

do not affect each otherthe value of sum119898

119894=1|119909

119898119888minus 119909

119894| and sum119898

119894=1|119910

119898119888minus 119910

119894| is minimized

separately It is easy to prove that when 119898 is odd 119909119898119888

is119909

1015840

(119898+1)2so that sum119898

119894=1|119909

119898119888minus 119909

119894| is minimum when 119898 is even

119909

119898119888is between 1199091015840

(119898+1)2and 1199091015840

((119898+1)2)+1so thatsum119898

119894=1|119909

119898119888minus119909

119894|

is minimum We can get 119910119898119888

in the same way Thereforethe Manhattan center 119901

119898119888can be calculated by following

equations

119909

119898119888=

119909

1015840

(119898+1)2119898 is odd

119909

1015840

(119898+1)2+ 119909

1015840

((119898+1)2)+1

2

119898 is even

119910

119898119888=

119910

1015840

(119898+1)2119898 is odd

119910

1015840

(119898+1)2+ 119910

1015840

((119898+1)2)+1

2

119898 is even

(10)

p4(36)

p1(14)

p2(22)

p3(43)

p5(88)

p6(96)

p7(91)

poc (56)

pmc (44)

Original center

Manhattan center

Figure 6 Example of the manhattan center

Figure 7 Example of the clustering result

As mentioned before the Manhattan center has theminimum overall Manhattan distance to these 119898 points bywhich the value of119863manhattan is reduced An example is shownin Figure 6 For the original center 119901

119900119888sum119898119894=1(|119909

119900119888minus119909

119894| + |119910

119900119888minus

119910

119894|) = 37 while for the Manhattan Center 119901

119898119888 sum119898119894=1(|119909

119898119888minus

119909

119894| + |119910

119898119888minus 119910

119894|) = 34

Now we concisely prove the convergence of the algo-rithm Given that the119863manhattan generated by the algorithm isstrictly decreasing and there only exist a finite number of suchpartitions the algorithm will reach a partial optimal solutionin a finite number of iterations

After obtaining the cluster centers the balanced heuristicmechanism is introduced into the second stage of cluster-ing In traditional methods when the 119898 cluster centers(119903

1 119903

2 119903

119898) are found in the first stage of the 119870-means

algorithm the remaining active tasks are allocated into theproper cluster with the smallest 119889(119909

119894 119888

119903) 119903 = 1 2 119896 How-

ever in our approach remaining active tasks are allocatedinto the proper cluster with the smallest 119862(119903

119894 119905

119895) defined in

(4)The main idea is that instead of merely considering the

travel cost the balance of the cluster size is taken into accountThe task tends to be allocated into the nearby cluster withfewer tasks When a new task is added into the cluster 119895the 119862119882(119903

119894) will be refreshed After all the active tasks have

been allocated the genetic-based TSP algorithm [26] is usedto plan the route for tasks in each cluster One example ofthe task allocation is demonstrated in Figure 7 (119898 = 4 119899 =50 120572 = 08) where different colors represent tasks allocatedto different robots Blocks with119883 are clustering centers

Mathematical Problems in Engineering 7

Table 1 Balance behavior of single-item auction versus BHM auc-tion

119877 = 8 Single-item auction BHM auction (120572 = 08)119898 120583 120590 CV = 120590120583 120583 120590 CV = 12059012058350 685 312 456 711 61 86100 1374 315 229 1469 53 36150 1969 553 281 2128 41 19200 2591 449 173 2827 50 18250 3234 439 136 3555 32 09300 3897 609 156 4221 21 05350 4418 701 159 4794 26 05400 5085 775 152 5499 19 04

4 Simulated Experiments

41 Simulation Description Several groups of simulatedexperiments are implemented with OPENGL and the exper-imental process is as follows

Step 1 Initialization

Step 2 Creating orders in the task pool while the arriving rateof orders meets the Poisson distribution

Step 3 Sending a specific number (eg 200) of orders fromthe task pool to the controller

Step 4 Using different methods to allocate the orders (tasks)More specifically there are four approaches namely 119870-means BHM 119870-means auction and BHMauction and theirperformances are compared

Step 5 Sending the allocated tasks by the controller to thecorresponding robots which begin to fulfil their tasks (usingthe Alowast algorithm to calculate the path and using the more-task-prior mechanism to avoid collision) Then go to Step 3

42 Simulation Results Analysis In this part three criteriamentioned in Section 2 are used to evaluate the performanceof the proposed balanced heuristic mechanism that iscoefficient of variation (CV) travel time (TT) and total travelcost (TTC)

First of all we compare the balance behavior of theimproved methods with the typical auction method and the119870-means method results of which are given in Tables 1and 2 where the data are collected by average during fiftytimes of simulation experiments Both Tables 1 and 2 showthat using the proposed BHM the diversity degree in workgroups is significantly reduced which indicates the moreequal workload distribution among robots

As shown in Figures 8 and 9 the travel time (TT) ofthe multirobot system is provided by both the traditionalmethod and the improved method We can conclude that byapplying the BHM based methods the time spent to finish

Table 2 Balance behavior of typical 119870-means versus BHM 119870-means

119877 = 8 119870-means BHM K-means (120572 = 08)119898 120583 120590 CV = 120590120583 120583 120590 CV = 12059012058350 714 310 434 749 50 67100 1390 513 369 1508 57 38150 2058 764 371 2204 49 22200 2751 1116 406 2949 63 21250 3441 1076 313 3684 69 19300 4095 1254 306 4440 65 15350 4776 1484 311 5214 83 16400 5465 1882 344 5895 66 11

50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

7000

8000

Task number

TT

AuctionBHM auction

Figure 8 Time spent by single-item auction versus BHM auction

the given task is much less than the one by using thetraditional method due to an efficient sharing of workload

Comparison results for traditional methods and BHMbasedmethods are demonstrated in Figure 10 bymeans of thetotal travel cost (TTC) from which we can conclude that inbalanced heuristic cases although robots pay more attentionto balancing the workload rather thanmerelyminimizing thetravel cost as traditional methods do the TTC only increasesby about 8 Concerning the obvious increasement of therobots utilization this slightly increased power consumptionis quite acceptable

With the traditional auction method and 119870-meansmethod the task allocation process only concentrates onminimizing the travel cost Once there is a robot nearest toan open task other robots do not attempt to compete for thetask even if they are idle Thus the utilization of the system isnot enough and the time cost to fulfill the given tasks is highWith BHM based methods proposed in this paper all robots

8 Mathematical Problems in Engineering

50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

7000

8000

9000

Task number

TT

K-meansBHM K-means

Figure 9 Time spent by 119870-means versus BHM 119870-means

50 100 150 200 250 300 350 400012345

TTC

Task number

times104

AuctionBHM auction

(a)

50 100 150 200 250 300 350 400012345

Task number

TTC

times104

K-meansBHM K-means

(b)

Figure 10 Total travel cost by traditional methods versus BHM adapted methods

can be utilized in a balanced and effective manner while thetravel cost is quite low

Then we study the influence of the variable 120572 on thesystem performance The value of 120572 describes the weight ofthe system energy saving demand and time saving demandFigure 11 demonstrates the relationship between 120572 and thetravel time (TT) It shows that in order to keep a good balancebetween minimizing the travel distance and efficient sharingof the workload the coefficient is usually set between 07 and09 to achieve the minimum of TT

Finally besides comparing BHM-based approaches withthe traditional auctionmethod and119870-meansmethod we usea more competing algorithm (119870CAB mechanism in [18]) totestify the scientific merit of the BHM scheme (as shownin Figures 12 and 13) We can conclude that by BHM basedmethods the time spent to finish the given task is 5sim15less than the119870CABmechanism meanwhile their total travelcosts are almost the same Thus the benefit of the balancedheuristic mechanism to warehouse systems is significant

5 Conclusion

In this paper a novel balanced heuristic mechanism for themultirobot task allocation problem is presented with respectto the task cost and travel cost To evaluate the proposedapproach we give three criteria namely travel time (TT)total travel cost (TTC) and coefficient of variation (CV)and compare some traditional methods with BHM-basedmethods under those criterions Simulation results show thatthe proposed approach greatly enhances the balance behaviorof the multirobot system reduces the total travel time whileachieves almost the same total travel cost as the traditionalmethods do In addition the influence of the variable 120572 onthe system performance is also studied in detail In our futurework BHM will be applied into more complicated envi-ronments which consist of heterogeneous tasks and robotsFurthermore we will dedicate to enhance the robustness ofthe proposed approach in dynamic environments Finally itis worth to point out that this new mechanism is of great

Mathematical Problems in Engineering 9

01 02 03 04 05 06 07 08 09 108

09

10

11

12

13

14

15

TT

120572

Task number = 50Task number = 100

Task number = 200Task number = 300

Normalized TT versus 120572 (m = 8)

Figure 11 Relationship between 120572 and travel time (TT)

50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

7000

Task number

TT

BHM auction

KCAB mechanismBHM K-means

Figure 12 Comparison between method provided in [15] versusBHM-based approaches (TT)

applicability to other task allocation problems such as theWSANs problem in [27]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Authorsrsquo Contribution

Luowei Zhou and Yuanyuan Shi contributed equally to thiswork

50 100 150 200 250 300 350 400

TTC

Task number

times104

BHM auction

KCAB mechanismBHM K-means

5

45

4

35

3

25

2

15

1

05

Figure 13 Comparison between method provided in [15] versusBHM-based approaches (TTC)

Acknowledgment

This work was supported by the Natural Science Foundationof China under Grants no 61273327 and no 61432008

References

[1] P R Wurman R DrsquoAndrea and M Mountz ldquoCoordinatinghundreds of cooperative autonomous vehicles in warehousesrdquoAI Magazine vol 29 no 1 pp 9ndash19 2008

[2] X Zhai J E Ward and L B Schwarz ldquoCoordinating aone-warehouse N-retailer distribution system under retailer-reportingrdquo International Journal of Production Economics vol134 no 1 pp 204ndash211 2011

[3] C Chen H-X Li and D Dong ldquoHybrid control forrobot navigationmdasha hierarchical Q-learning algorithmrdquo IEEERobotics amp AutomationMagazine vol 15 no 2 pp 37ndash47 2008

[4] S Chen and C Chen ldquoProbabilistic fuzzy system for uncertainlocalization and map building of mobile robotsrdquo IEEE Trans-actions on Instrumentation and Measurement vol 61 no 6 pp1546ndash1560 2012

[5] D Dong C Chen J Chu and T-J Tarn ldquoRobust quantum-inspired reinforcement learning for robot navigationrdquo IEEEASME Transactions on Mechatronics vol 17 no 1 pp 86ndash972012

[6] T Niemueller D Ewert S Reuter et al ldquoTowards benchmarking cyber-physical systems in factory automation scenariosrdquoinKI 2013 Advances in Artificial Intelligence vol 8077 of LectureNotes in Computer Science pp 296ndash299 2013

[7] M Brambilla E Ferrante M Birattari and M Dorigo ldquoSwarmrobotics a review from the swarm engineering perspectiverdquoSwarm Intelligence vol 7 no 1 pp 1ndash41 2013

[8] C Nieto-Granda J G Rogers and H I Christensen ldquoCoor-dination strategies for multi-robot exploration and mappingrdquoExperimental Robotics vol 33 no 4 pp 519ndash533 2013

10 Mathematical Problems in Engineering

[9] B-Y Shih C-Y Chen and W Chou ldquoAn enhanced obstacleavoidance and path correction mechanism for an autonomousintelligent robot withmultiple sensorsrdquo Journal of Vibration andControl vol 18 no 12 pp 1855ndash1864 2012

[10] B Yamauchi ldquoFrontier-based exploration using multiplerobotsrdquo in Proceedings of the 2nd International Conference onAutonomous Agents pp 47ndash53 ACM May 1998

[11] B P Gerkey and M J Mataric ldquoSold auction methods formultirobot coordinationrdquo IEEE Transactions on Robotics andAutomation vol 18 no 5 pp 758ndash768 2002

[12] T Sandholm ldquoAlgorithm for optimal winner determination incombinatorial auctionsrdquo Artificial Intelligence vol 135 no 1-2pp 1ndash54 2002

[13] M Berhault H Huang P Keskinocak et al ldquoRobot explorationwith combinatorial auctionsrdquo in Proceedings of the IEEERSJInternational Conference on Intelligent Robots and Systems vol1ndash4 pp 1957ndash1962 October 2003

[14] M G Lagoudakis M Berhault S Koenig P Keskinocak andA J Kleywegt ldquoSimple auctions with performance guaranteesfor multi-robot task allocationrdquo in Proceedings of the IEEERSJInternational Conference on Intelligent Robots and Systems(IROS rsquo04) pp 698ndash705 IEEE October 2004

[15] L Liu and D A Shell ldquoLarge-scale multi-robot task allocationvia dynamic partitioning and distributionrdquoAutonomous Robotsvol 33 no 3 pp 291ndash307 2012

[16] S Ozturk and A E Kuzucuoglu ldquoOptimal bid valuationusing path finding for multi-robot task allocationrdquo Journal ofIntelligent Manufacturing 2014

[17] D Puig M A Garcia and L Wu ldquoA new global optimizationstrategy for coordinated multi-robot exploration developmentand comparative evaluationrdquo Robotics and Autonomous Sys-tems vol 59 no 9 pp 635ndash653 2011

[18] M Elango S Nachiappan and M K Tiwari ldquoBalancing taskallocation inmulti-robot systems using K-means clustering andauction based mechanismsrdquo Expert Systems with Applicationsvol 38 no 6 pp 6486ndash6491 2011

[19] A G Bedeian and K W Mossholder ldquoOn the use of thecoefficient of variation as ameasure of diversityrdquoOrganizationalResearch Methods vol 3 no 3 pp 285ndash297 2000

[20] S S Chiddarwar and N R Babu ldquoConflict free coordinatedpath planning for multiple robots using a dynamic path mod-ification sequencerdquo Robotics and Autonomous Systems vol 59no 7-8 pp 508ndash518 2011

[21] J Capitan M T J Spaan L Merino and A Ollero ldquoDecen-tralizedmulti-robot cooperationwith auctioned POMDPsrdquoTheInternational Journal of Robotics Research vol 32 no 6 pp 650ndash671 2013

[22] D Dong C Chen H Li and T-J Tarn ldquoQuantum rein-forcement learningrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 38 no 5 pp 1207ndash12202008

[23] C Chen D Dong H-X Li J Chu and T-J Tarn ldquoFidelity-based probabilistic Q-learning for control of quantum systemsrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 25 no 5 pp 920ndash933 2014

[24] M Elango G Kanagaraj and S G Ponnambalam ldquoSandholmalgorithm with K-means clustering approach for multi-robottask allocationrdquo in Swarm Evolutionary and Memetic Com-puting vol 8298 pp 14ndash22 Springer International Publishing2013

[25] M E Celebi H A Kingravi and P A Vela ldquoA comparativestudy of efficient initialization methods for the k-means clus-tering algorithmrdquo Expert Systems with Applications vol 40 no1 pp 200ndash210 2013

[26] M Shao ldquoResearch on improved convergence co-cevolutiongenetic algorithm for TSPsrdquo International Journal of Conver-gence Computing vol 1 no 2 pp 118ndash126 2014

[27] I Mezei M Lukic V Malbasa and I Stojmenovic ldquoAuctionsand iMesh based task assignment in wireless sensor andactuator networksrdquo Computer Communications vol 36 no 9pp 979ndash987 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Table 1 Balance behavior of single-item auction versus BHM auc-tion

119877 = 8 Single-item auction BHM auction (120572 = 08)119898 120583 120590 CV = 120590120583 120583 120590 CV = 12059012058350 685 312 456 711 61 86100 1374 315 229 1469 53 36150 1969 553 281 2128 41 19200 2591 449 173 2827 50 18250 3234 439 136 3555 32 09300 3897 609 156 4221 21 05350 4418 701 159 4794 26 05400 5085 775 152 5499 19 04

4 Simulated Experiments

41 Simulation Description Several groups of simulatedexperiments are implemented with OPENGL and the exper-imental process is as follows

Step 1 Initialization

Step 2 Creating orders in the task pool while the arriving rateof orders meets the Poisson distribution

Step 3 Sending a specific number (eg 200) of orders fromthe task pool to the controller

Step 4 Using different methods to allocate the orders (tasks)More specifically there are four approaches namely 119870-means BHM 119870-means auction and BHMauction and theirperformances are compared

Step 5 Sending the allocated tasks by the controller to thecorresponding robots which begin to fulfil their tasks (usingthe Alowast algorithm to calculate the path and using the more-task-prior mechanism to avoid collision) Then go to Step 3

42 Simulation Results Analysis In this part three criteriamentioned in Section 2 are used to evaluate the performanceof the proposed balanced heuristic mechanism that iscoefficient of variation (CV) travel time (TT) and total travelcost (TTC)

First of all we compare the balance behavior of theimproved methods with the typical auction method and the119870-means method results of which are given in Tables 1and 2 where the data are collected by average during fiftytimes of simulation experiments Both Tables 1 and 2 showthat using the proposed BHM the diversity degree in workgroups is significantly reduced which indicates the moreequal workload distribution among robots

As shown in Figures 8 and 9 the travel time (TT) ofthe multirobot system is provided by both the traditionalmethod and the improved method We can conclude that byapplying the BHM based methods the time spent to finish

Table 2 Balance behavior of typical 119870-means versus BHM 119870-means

119877 = 8 119870-means BHM K-means (120572 = 08)119898 120583 120590 CV = 120590120583 120583 120590 CV = 12059012058350 714 310 434 749 50 67100 1390 513 369 1508 57 38150 2058 764 371 2204 49 22200 2751 1116 406 2949 63 21250 3441 1076 313 3684 69 19300 4095 1254 306 4440 65 15350 4776 1484 311 5214 83 16400 5465 1882 344 5895 66 11

50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

7000

8000

Task number

TT

AuctionBHM auction

Figure 8 Time spent by single-item auction versus BHM auction

the given task is much less than the one by using thetraditional method due to an efficient sharing of workload

Comparison results for traditional methods and BHMbasedmethods are demonstrated in Figure 10 bymeans of thetotal travel cost (TTC) from which we can conclude that inbalanced heuristic cases although robots pay more attentionto balancing the workload rather thanmerelyminimizing thetravel cost as traditional methods do the TTC only increasesby about 8 Concerning the obvious increasement of therobots utilization this slightly increased power consumptionis quite acceptable

With the traditional auction method and 119870-meansmethod the task allocation process only concentrates onminimizing the travel cost Once there is a robot nearest toan open task other robots do not attempt to compete for thetask even if they are idle Thus the utilization of the system isnot enough and the time cost to fulfill the given tasks is highWith BHM based methods proposed in this paper all robots

8 Mathematical Problems in Engineering

50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

7000

8000

9000

Task number

TT

K-meansBHM K-means

Figure 9 Time spent by 119870-means versus BHM 119870-means

50 100 150 200 250 300 350 400012345

TTC

Task number

times104

AuctionBHM auction

(a)

50 100 150 200 250 300 350 400012345

Task number

TTC

times104

K-meansBHM K-means

(b)

Figure 10 Total travel cost by traditional methods versus BHM adapted methods

can be utilized in a balanced and effective manner while thetravel cost is quite low

Then we study the influence of the variable 120572 on thesystem performance The value of 120572 describes the weight ofthe system energy saving demand and time saving demandFigure 11 demonstrates the relationship between 120572 and thetravel time (TT) It shows that in order to keep a good balancebetween minimizing the travel distance and efficient sharingof the workload the coefficient is usually set between 07 and09 to achieve the minimum of TT

Finally besides comparing BHM-based approaches withthe traditional auctionmethod and119870-meansmethod we usea more competing algorithm (119870CAB mechanism in [18]) totestify the scientific merit of the BHM scheme (as shownin Figures 12 and 13) We can conclude that by BHM basedmethods the time spent to finish the given task is 5sim15less than the119870CABmechanism meanwhile their total travelcosts are almost the same Thus the benefit of the balancedheuristic mechanism to warehouse systems is significant

5 Conclusion

In this paper a novel balanced heuristic mechanism for themultirobot task allocation problem is presented with respectto the task cost and travel cost To evaluate the proposedapproach we give three criteria namely travel time (TT)total travel cost (TTC) and coefficient of variation (CV)and compare some traditional methods with BHM-basedmethods under those criterions Simulation results show thatthe proposed approach greatly enhances the balance behaviorof the multirobot system reduces the total travel time whileachieves almost the same total travel cost as the traditionalmethods do In addition the influence of the variable 120572 onthe system performance is also studied in detail In our futurework BHM will be applied into more complicated envi-ronments which consist of heterogeneous tasks and robotsFurthermore we will dedicate to enhance the robustness ofthe proposed approach in dynamic environments Finally itis worth to point out that this new mechanism is of great

Mathematical Problems in Engineering 9

01 02 03 04 05 06 07 08 09 108

09

10

11

12

13

14

15

TT

120572

Task number = 50Task number = 100

Task number = 200Task number = 300

Normalized TT versus 120572 (m = 8)

Figure 11 Relationship between 120572 and travel time (TT)

50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

7000

Task number

TT

BHM auction

KCAB mechanismBHM K-means

Figure 12 Comparison between method provided in [15] versusBHM-based approaches (TT)

applicability to other task allocation problems such as theWSANs problem in [27]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Authorsrsquo Contribution

Luowei Zhou and Yuanyuan Shi contributed equally to thiswork

50 100 150 200 250 300 350 400

TTC

Task number

times104

BHM auction

KCAB mechanismBHM K-means

5

45

4

35

3

25

2

15

1

05

Figure 13 Comparison between method provided in [15] versusBHM-based approaches (TTC)

Acknowledgment

This work was supported by the Natural Science Foundationof China under Grants no 61273327 and no 61432008

References

[1] P R Wurman R DrsquoAndrea and M Mountz ldquoCoordinatinghundreds of cooperative autonomous vehicles in warehousesrdquoAI Magazine vol 29 no 1 pp 9ndash19 2008

[2] X Zhai J E Ward and L B Schwarz ldquoCoordinating aone-warehouse N-retailer distribution system under retailer-reportingrdquo International Journal of Production Economics vol134 no 1 pp 204ndash211 2011

[3] C Chen H-X Li and D Dong ldquoHybrid control forrobot navigationmdasha hierarchical Q-learning algorithmrdquo IEEERobotics amp AutomationMagazine vol 15 no 2 pp 37ndash47 2008

[4] S Chen and C Chen ldquoProbabilistic fuzzy system for uncertainlocalization and map building of mobile robotsrdquo IEEE Trans-actions on Instrumentation and Measurement vol 61 no 6 pp1546ndash1560 2012

[5] D Dong C Chen J Chu and T-J Tarn ldquoRobust quantum-inspired reinforcement learning for robot navigationrdquo IEEEASME Transactions on Mechatronics vol 17 no 1 pp 86ndash972012

[6] T Niemueller D Ewert S Reuter et al ldquoTowards benchmarking cyber-physical systems in factory automation scenariosrdquoinKI 2013 Advances in Artificial Intelligence vol 8077 of LectureNotes in Computer Science pp 296ndash299 2013

[7] M Brambilla E Ferrante M Birattari and M Dorigo ldquoSwarmrobotics a review from the swarm engineering perspectiverdquoSwarm Intelligence vol 7 no 1 pp 1ndash41 2013

[8] C Nieto-Granda J G Rogers and H I Christensen ldquoCoor-dination strategies for multi-robot exploration and mappingrdquoExperimental Robotics vol 33 no 4 pp 519ndash533 2013

10 Mathematical Problems in Engineering

[9] B-Y Shih C-Y Chen and W Chou ldquoAn enhanced obstacleavoidance and path correction mechanism for an autonomousintelligent robot withmultiple sensorsrdquo Journal of Vibration andControl vol 18 no 12 pp 1855ndash1864 2012

[10] B Yamauchi ldquoFrontier-based exploration using multiplerobotsrdquo in Proceedings of the 2nd International Conference onAutonomous Agents pp 47ndash53 ACM May 1998

[11] B P Gerkey and M J Mataric ldquoSold auction methods formultirobot coordinationrdquo IEEE Transactions on Robotics andAutomation vol 18 no 5 pp 758ndash768 2002

[12] T Sandholm ldquoAlgorithm for optimal winner determination incombinatorial auctionsrdquo Artificial Intelligence vol 135 no 1-2pp 1ndash54 2002

[13] M Berhault H Huang P Keskinocak et al ldquoRobot explorationwith combinatorial auctionsrdquo in Proceedings of the IEEERSJInternational Conference on Intelligent Robots and Systems vol1ndash4 pp 1957ndash1962 October 2003

[14] M G Lagoudakis M Berhault S Koenig P Keskinocak andA J Kleywegt ldquoSimple auctions with performance guaranteesfor multi-robot task allocationrdquo in Proceedings of the IEEERSJInternational Conference on Intelligent Robots and Systems(IROS rsquo04) pp 698ndash705 IEEE October 2004

[15] L Liu and D A Shell ldquoLarge-scale multi-robot task allocationvia dynamic partitioning and distributionrdquoAutonomous Robotsvol 33 no 3 pp 291ndash307 2012

[16] S Ozturk and A E Kuzucuoglu ldquoOptimal bid valuationusing path finding for multi-robot task allocationrdquo Journal ofIntelligent Manufacturing 2014

[17] D Puig M A Garcia and L Wu ldquoA new global optimizationstrategy for coordinated multi-robot exploration developmentand comparative evaluationrdquo Robotics and Autonomous Sys-tems vol 59 no 9 pp 635ndash653 2011

[18] M Elango S Nachiappan and M K Tiwari ldquoBalancing taskallocation inmulti-robot systems using K-means clustering andauction based mechanismsrdquo Expert Systems with Applicationsvol 38 no 6 pp 6486ndash6491 2011

[19] A G Bedeian and K W Mossholder ldquoOn the use of thecoefficient of variation as ameasure of diversityrdquoOrganizationalResearch Methods vol 3 no 3 pp 285ndash297 2000

[20] S S Chiddarwar and N R Babu ldquoConflict free coordinatedpath planning for multiple robots using a dynamic path mod-ification sequencerdquo Robotics and Autonomous Systems vol 59no 7-8 pp 508ndash518 2011

[21] J Capitan M T J Spaan L Merino and A Ollero ldquoDecen-tralizedmulti-robot cooperationwith auctioned POMDPsrdquoTheInternational Journal of Robotics Research vol 32 no 6 pp 650ndash671 2013

[22] D Dong C Chen H Li and T-J Tarn ldquoQuantum rein-forcement learningrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 38 no 5 pp 1207ndash12202008

[23] C Chen D Dong H-X Li J Chu and T-J Tarn ldquoFidelity-based probabilistic Q-learning for control of quantum systemsrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 25 no 5 pp 920ndash933 2014

[24] M Elango G Kanagaraj and S G Ponnambalam ldquoSandholmalgorithm with K-means clustering approach for multi-robottask allocationrdquo in Swarm Evolutionary and Memetic Com-puting vol 8298 pp 14ndash22 Springer International Publishing2013

[25] M E Celebi H A Kingravi and P A Vela ldquoA comparativestudy of efficient initialization methods for the k-means clus-tering algorithmrdquo Expert Systems with Applications vol 40 no1 pp 200ndash210 2013

[26] M Shao ldquoResearch on improved convergence co-cevolutiongenetic algorithm for TSPsrdquo International Journal of Conver-gence Computing vol 1 no 2 pp 118ndash126 2014

[27] I Mezei M Lukic V Malbasa and I Stojmenovic ldquoAuctionsand iMesh based task assignment in wireless sensor andactuator networksrdquo Computer Communications vol 36 no 9pp 979ndash987 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

7000

8000

9000

Task number

TT

K-meansBHM K-means

Figure 9 Time spent by 119870-means versus BHM 119870-means

50 100 150 200 250 300 350 400012345

TTC

Task number

times104

AuctionBHM auction

(a)

50 100 150 200 250 300 350 400012345

Task number

TTC

times104

K-meansBHM K-means

(b)

Figure 10 Total travel cost by traditional methods versus BHM adapted methods

can be utilized in a balanced and effective manner while thetravel cost is quite low

Then we study the influence of the variable 120572 on thesystem performance The value of 120572 describes the weight ofthe system energy saving demand and time saving demandFigure 11 demonstrates the relationship between 120572 and thetravel time (TT) It shows that in order to keep a good balancebetween minimizing the travel distance and efficient sharingof the workload the coefficient is usually set between 07 and09 to achieve the minimum of TT

Finally besides comparing BHM-based approaches withthe traditional auctionmethod and119870-meansmethod we usea more competing algorithm (119870CAB mechanism in [18]) totestify the scientific merit of the BHM scheme (as shownin Figures 12 and 13) We can conclude that by BHM basedmethods the time spent to finish the given task is 5sim15less than the119870CABmechanism meanwhile their total travelcosts are almost the same Thus the benefit of the balancedheuristic mechanism to warehouse systems is significant

5 Conclusion

In this paper a novel balanced heuristic mechanism for themultirobot task allocation problem is presented with respectto the task cost and travel cost To evaluate the proposedapproach we give three criteria namely travel time (TT)total travel cost (TTC) and coefficient of variation (CV)and compare some traditional methods with BHM-basedmethods under those criterions Simulation results show thatthe proposed approach greatly enhances the balance behaviorof the multirobot system reduces the total travel time whileachieves almost the same total travel cost as the traditionalmethods do In addition the influence of the variable 120572 onthe system performance is also studied in detail In our futurework BHM will be applied into more complicated envi-ronments which consist of heterogeneous tasks and robotsFurthermore we will dedicate to enhance the robustness ofthe proposed approach in dynamic environments Finally itis worth to point out that this new mechanism is of great

Mathematical Problems in Engineering 9

01 02 03 04 05 06 07 08 09 108

09

10

11

12

13

14

15

TT

120572

Task number = 50Task number = 100

Task number = 200Task number = 300

Normalized TT versus 120572 (m = 8)

Figure 11 Relationship between 120572 and travel time (TT)

50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

7000

Task number

TT

BHM auction

KCAB mechanismBHM K-means

Figure 12 Comparison between method provided in [15] versusBHM-based approaches (TT)

applicability to other task allocation problems such as theWSANs problem in [27]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Authorsrsquo Contribution

Luowei Zhou and Yuanyuan Shi contributed equally to thiswork

50 100 150 200 250 300 350 400

TTC

Task number

times104

BHM auction

KCAB mechanismBHM K-means

5

45

4

35

3

25

2

15

1

05

Figure 13 Comparison between method provided in [15] versusBHM-based approaches (TTC)

Acknowledgment

This work was supported by the Natural Science Foundationof China under Grants no 61273327 and no 61432008

References

[1] P R Wurman R DrsquoAndrea and M Mountz ldquoCoordinatinghundreds of cooperative autonomous vehicles in warehousesrdquoAI Magazine vol 29 no 1 pp 9ndash19 2008

[2] X Zhai J E Ward and L B Schwarz ldquoCoordinating aone-warehouse N-retailer distribution system under retailer-reportingrdquo International Journal of Production Economics vol134 no 1 pp 204ndash211 2011

[3] C Chen H-X Li and D Dong ldquoHybrid control forrobot navigationmdasha hierarchical Q-learning algorithmrdquo IEEERobotics amp AutomationMagazine vol 15 no 2 pp 37ndash47 2008

[4] S Chen and C Chen ldquoProbabilistic fuzzy system for uncertainlocalization and map building of mobile robotsrdquo IEEE Trans-actions on Instrumentation and Measurement vol 61 no 6 pp1546ndash1560 2012

[5] D Dong C Chen J Chu and T-J Tarn ldquoRobust quantum-inspired reinforcement learning for robot navigationrdquo IEEEASME Transactions on Mechatronics vol 17 no 1 pp 86ndash972012

[6] T Niemueller D Ewert S Reuter et al ldquoTowards benchmarking cyber-physical systems in factory automation scenariosrdquoinKI 2013 Advances in Artificial Intelligence vol 8077 of LectureNotes in Computer Science pp 296ndash299 2013

[7] M Brambilla E Ferrante M Birattari and M Dorigo ldquoSwarmrobotics a review from the swarm engineering perspectiverdquoSwarm Intelligence vol 7 no 1 pp 1ndash41 2013

[8] C Nieto-Granda J G Rogers and H I Christensen ldquoCoor-dination strategies for multi-robot exploration and mappingrdquoExperimental Robotics vol 33 no 4 pp 519ndash533 2013

10 Mathematical Problems in Engineering

[9] B-Y Shih C-Y Chen and W Chou ldquoAn enhanced obstacleavoidance and path correction mechanism for an autonomousintelligent robot withmultiple sensorsrdquo Journal of Vibration andControl vol 18 no 12 pp 1855ndash1864 2012

[10] B Yamauchi ldquoFrontier-based exploration using multiplerobotsrdquo in Proceedings of the 2nd International Conference onAutonomous Agents pp 47ndash53 ACM May 1998

[11] B P Gerkey and M J Mataric ldquoSold auction methods formultirobot coordinationrdquo IEEE Transactions on Robotics andAutomation vol 18 no 5 pp 758ndash768 2002

[12] T Sandholm ldquoAlgorithm for optimal winner determination incombinatorial auctionsrdquo Artificial Intelligence vol 135 no 1-2pp 1ndash54 2002

[13] M Berhault H Huang P Keskinocak et al ldquoRobot explorationwith combinatorial auctionsrdquo in Proceedings of the IEEERSJInternational Conference on Intelligent Robots and Systems vol1ndash4 pp 1957ndash1962 October 2003

[14] M G Lagoudakis M Berhault S Koenig P Keskinocak andA J Kleywegt ldquoSimple auctions with performance guaranteesfor multi-robot task allocationrdquo in Proceedings of the IEEERSJInternational Conference on Intelligent Robots and Systems(IROS rsquo04) pp 698ndash705 IEEE October 2004

[15] L Liu and D A Shell ldquoLarge-scale multi-robot task allocationvia dynamic partitioning and distributionrdquoAutonomous Robotsvol 33 no 3 pp 291ndash307 2012

[16] S Ozturk and A E Kuzucuoglu ldquoOptimal bid valuationusing path finding for multi-robot task allocationrdquo Journal ofIntelligent Manufacturing 2014

[17] D Puig M A Garcia and L Wu ldquoA new global optimizationstrategy for coordinated multi-robot exploration developmentand comparative evaluationrdquo Robotics and Autonomous Sys-tems vol 59 no 9 pp 635ndash653 2011

[18] M Elango S Nachiappan and M K Tiwari ldquoBalancing taskallocation inmulti-robot systems using K-means clustering andauction based mechanismsrdquo Expert Systems with Applicationsvol 38 no 6 pp 6486ndash6491 2011

[19] A G Bedeian and K W Mossholder ldquoOn the use of thecoefficient of variation as ameasure of diversityrdquoOrganizationalResearch Methods vol 3 no 3 pp 285ndash297 2000

[20] S S Chiddarwar and N R Babu ldquoConflict free coordinatedpath planning for multiple robots using a dynamic path mod-ification sequencerdquo Robotics and Autonomous Systems vol 59no 7-8 pp 508ndash518 2011

[21] J Capitan M T J Spaan L Merino and A Ollero ldquoDecen-tralizedmulti-robot cooperationwith auctioned POMDPsrdquoTheInternational Journal of Robotics Research vol 32 no 6 pp 650ndash671 2013

[22] D Dong C Chen H Li and T-J Tarn ldquoQuantum rein-forcement learningrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 38 no 5 pp 1207ndash12202008

[23] C Chen D Dong H-X Li J Chu and T-J Tarn ldquoFidelity-based probabilistic Q-learning for control of quantum systemsrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 25 no 5 pp 920ndash933 2014

[24] M Elango G Kanagaraj and S G Ponnambalam ldquoSandholmalgorithm with K-means clustering approach for multi-robottask allocationrdquo in Swarm Evolutionary and Memetic Com-puting vol 8298 pp 14ndash22 Springer International Publishing2013

[25] M E Celebi H A Kingravi and P A Vela ldquoA comparativestudy of efficient initialization methods for the k-means clus-tering algorithmrdquo Expert Systems with Applications vol 40 no1 pp 200ndash210 2013

[26] M Shao ldquoResearch on improved convergence co-cevolutiongenetic algorithm for TSPsrdquo International Journal of Conver-gence Computing vol 1 no 2 pp 118ndash126 2014

[27] I Mezei M Lukic V Malbasa and I Stojmenovic ldquoAuctionsand iMesh based task assignment in wireless sensor andactuator networksrdquo Computer Communications vol 36 no 9pp 979ndash987 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

01 02 03 04 05 06 07 08 09 108

09

10

11

12

13

14

15

TT

120572

Task number = 50Task number = 100

Task number = 200Task number = 300

Normalized TT versus 120572 (m = 8)

Figure 11 Relationship between 120572 and travel time (TT)

50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

7000

Task number

TT

BHM auction

KCAB mechanismBHM K-means

Figure 12 Comparison between method provided in [15] versusBHM-based approaches (TT)

applicability to other task allocation problems such as theWSANs problem in [27]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Authorsrsquo Contribution

Luowei Zhou and Yuanyuan Shi contributed equally to thiswork

50 100 150 200 250 300 350 400

TTC

Task number

times104

BHM auction

KCAB mechanismBHM K-means

5

45

4

35

3

25

2

15

1

05

Figure 13 Comparison between method provided in [15] versusBHM-based approaches (TTC)

Acknowledgment

This work was supported by the Natural Science Foundationof China under Grants no 61273327 and no 61432008

References

[1] P R Wurman R DrsquoAndrea and M Mountz ldquoCoordinatinghundreds of cooperative autonomous vehicles in warehousesrdquoAI Magazine vol 29 no 1 pp 9ndash19 2008

[2] X Zhai J E Ward and L B Schwarz ldquoCoordinating aone-warehouse N-retailer distribution system under retailer-reportingrdquo International Journal of Production Economics vol134 no 1 pp 204ndash211 2011

[3] C Chen H-X Li and D Dong ldquoHybrid control forrobot navigationmdasha hierarchical Q-learning algorithmrdquo IEEERobotics amp AutomationMagazine vol 15 no 2 pp 37ndash47 2008

[4] S Chen and C Chen ldquoProbabilistic fuzzy system for uncertainlocalization and map building of mobile robotsrdquo IEEE Trans-actions on Instrumentation and Measurement vol 61 no 6 pp1546ndash1560 2012

[5] D Dong C Chen J Chu and T-J Tarn ldquoRobust quantum-inspired reinforcement learning for robot navigationrdquo IEEEASME Transactions on Mechatronics vol 17 no 1 pp 86ndash972012

[6] T Niemueller D Ewert S Reuter et al ldquoTowards benchmarking cyber-physical systems in factory automation scenariosrdquoinKI 2013 Advances in Artificial Intelligence vol 8077 of LectureNotes in Computer Science pp 296ndash299 2013

[7] M Brambilla E Ferrante M Birattari and M Dorigo ldquoSwarmrobotics a review from the swarm engineering perspectiverdquoSwarm Intelligence vol 7 no 1 pp 1ndash41 2013

[8] C Nieto-Granda J G Rogers and H I Christensen ldquoCoor-dination strategies for multi-robot exploration and mappingrdquoExperimental Robotics vol 33 no 4 pp 519ndash533 2013

10 Mathematical Problems in Engineering

[9] B-Y Shih C-Y Chen and W Chou ldquoAn enhanced obstacleavoidance and path correction mechanism for an autonomousintelligent robot withmultiple sensorsrdquo Journal of Vibration andControl vol 18 no 12 pp 1855ndash1864 2012

[10] B Yamauchi ldquoFrontier-based exploration using multiplerobotsrdquo in Proceedings of the 2nd International Conference onAutonomous Agents pp 47ndash53 ACM May 1998

[11] B P Gerkey and M J Mataric ldquoSold auction methods formultirobot coordinationrdquo IEEE Transactions on Robotics andAutomation vol 18 no 5 pp 758ndash768 2002

[12] T Sandholm ldquoAlgorithm for optimal winner determination incombinatorial auctionsrdquo Artificial Intelligence vol 135 no 1-2pp 1ndash54 2002

[13] M Berhault H Huang P Keskinocak et al ldquoRobot explorationwith combinatorial auctionsrdquo in Proceedings of the IEEERSJInternational Conference on Intelligent Robots and Systems vol1ndash4 pp 1957ndash1962 October 2003

[14] M G Lagoudakis M Berhault S Koenig P Keskinocak andA J Kleywegt ldquoSimple auctions with performance guaranteesfor multi-robot task allocationrdquo in Proceedings of the IEEERSJInternational Conference on Intelligent Robots and Systems(IROS rsquo04) pp 698ndash705 IEEE October 2004

[15] L Liu and D A Shell ldquoLarge-scale multi-robot task allocationvia dynamic partitioning and distributionrdquoAutonomous Robotsvol 33 no 3 pp 291ndash307 2012

[16] S Ozturk and A E Kuzucuoglu ldquoOptimal bid valuationusing path finding for multi-robot task allocationrdquo Journal ofIntelligent Manufacturing 2014

[17] D Puig M A Garcia and L Wu ldquoA new global optimizationstrategy for coordinated multi-robot exploration developmentand comparative evaluationrdquo Robotics and Autonomous Sys-tems vol 59 no 9 pp 635ndash653 2011

[18] M Elango S Nachiappan and M K Tiwari ldquoBalancing taskallocation inmulti-robot systems using K-means clustering andauction based mechanismsrdquo Expert Systems with Applicationsvol 38 no 6 pp 6486ndash6491 2011

[19] A G Bedeian and K W Mossholder ldquoOn the use of thecoefficient of variation as ameasure of diversityrdquoOrganizationalResearch Methods vol 3 no 3 pp 285ndash297 2000

[20] S S Chiddarwar and N R Babu ldquoConflict free coordinatedpath planning for multiple robots using a dynamic path mod-ification sequencerdquo Robotics and Autonomous Systems vol 59no 7-8 pp 508ndash518 2011

[21] J Capitan M T J Spaan L Merino and A Ollero ldquoDecen-tralizedmulti-robot cooperationwith auctioned POMDPsrdquoTheInternational Journal of Robotics Research vol 32 no 6 pp 650ndash671 2013

[22] D Dong C Chen H Li and T-J Tarn ldquoQuantum rein-forcement learningrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 38 no 5 pp 1207ndash12202008

[23] C Chen D Dong H-X Li J Chu and T-J Tarn ldquoFidelity-based probabilistic Q-learning for control of quantum systemsrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 25 no 5 pp 920ndash933 2014

[24] M Elango G Kanagaraj and S G Ponnambalam ldquoSandholmalgorithm with K-means clustering approach for multi-robottask allocationrdquo in Swarm Evolutionary and Memetic Com-puting vol 8298 pp 14ndash22 Springer International Publishing2013

[25] M E Celebi H A Kingravi and P A Vela ldquoA comparativestudy of efficient initialization methods for the k-means clus-tering algorithmrdquo Expert Systems with Applications vol 40 no1 pp 200ndash210 2013

[26] M Shao ldquoResearch on improved convergence co-cevolutiongenetic algorithm for TSPsrdquo International Journal of Conver-gence Computing vol 1 no 2 pp 118ndash126 2014

[27] I Mezei M Lukic V Malbasa and I Stojmenovic ldquoAuctionsand iMesh based task assignment in wireless sensor andactuator networksrdquo Computer Communications vol 36 no 9pp 979ndash987 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

[9] B-Y Shih C-Y Chen and W Chou ldquoAn enhanced obstacleavoidance and path correction mechanism for an autonomousintelligent robot withmultiple sensorsrdquo Journal of Vibration andControl vol 18 no 12 pp 1855ndash1864 2012

[10] B Yamauchi ldquoFrontier-based exploration using multiplerobotsrdquo in Proceedings of the 2nd International Conference onAutonomous Agents pp 47ndash53 ACM May 1998

[11] B P Gerkey and M J Mataric ldquoSold auction methods formultirobot coordinationrdquo IEEE Transactions on Robotics andAutomation vol 18 no 5 pp 758ndash768 2002

[12] T Sandholm ldquoAlgorithm for optimal winner determination incombinatorial auctionsrdquo Artificial Intelligence vol 135 no 1-2pp 1ndash54 2002

[13] M Berhault H Huang P Keskinocak et al ldquoRobot explorationwith combinatorial auctionsrdquo in Proceedings of the IEEERSJInternational Conference on Intelligent Robots and Systems vol1ndash4 pp 1957ndash1962 October 2003

[14] M G Lagoudakis M Berhault S Koenig P Keskinocak andA J Kleywegt ldquoSimple auctions with performance guaranteesfor multi-robot task allocationrdquo in Proceedings of the IEEERSJInternational Conference on Intelligent Robots and Systems(IROS rsquo04) pp 698ndash705 IEEE October 2004

[15] L Liu and D A Shell ldquoLarge-scale multi-robot task allocationvia dynamic partitioning and distributionrdquoAutonomous Robotsvol 33 no 3 pp 291ndash307 2012

[16] S Ozturk and A E Kuzucuoglu ldquoOptimal bid valuationusing path finding for multi-robot task allocationrdquo Journal ofIntelligent Manufacturing 2014

[17] D Puig M A Garcia and L Wu ldquoA new global optimizationstrategy for coordinated multi-robot exploration developmentand comparative evaluationrdquo Robotics and Autonomous Sys-tems vol 59 no 9 pp 635ndash653 2011

[18] M Elango S Nachiappan and M K Tiwari ldquoBalancing taskallocation inmulti-robot systems using K-means clustering andauction based mechanismsrdquo Expert Systems with Applicationsvol 38 no 6 pp 6486ndash6491 2011

[19] A G Bedeian and K W Mossholder ldquoOn the use of thecoefficient of variation as ameasure of diversityrdquoOrganizationalResearch Methods vol 3 no 3 pp 285ndash297 2000

[20] S S Chiddarwar and N R Babu ldquoConflict free coordinatedpath planning for multiple robots using a dynamic path mod-ification sequencerdquo Robotics and Autonomous Systems vol 59no 7-8 pp 508ndash518 2011

[21] J Capitan M T J Spaan L Merino and A Ollero ldquoDecen-tralizedmulti-robot cooperationwith auctioned POMDPsrdquoTheInternational Journal of Robotics Research vol 32 no 6 pp 650ndash671 2013

[22] D Dong C Chen H Li and T-J Tarn ldquoQuantum rein-forcement learningrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 38 no 5 pp 1207ndash12202008

[23] C Chen D Dong H-X Li J Chu and T-J Tarn ldquoFidelity-based probabilistic Q-learning for control of quantum systemsrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 25 no 5 pp 920ndash933 2014

[24] M Elango G Kanagaraj and S G Ponnambalam ldquoSandholmalgorithm with K-means clustering approach for multi-robottask allocationrdquo in Swarm Evolutionary and Memetic Com-puting vol 8298 pp 14ndash22 Springer International Publishing2013

[25] M E Celebi H A Kingravi and P A Vela ldquoA comparativestudy of efficient initialization methods for the k-means clus-tering algorithmrdquo Expert Systems with Applications vol 40 no1 pp 200ndash210 2013

[26] M Shao ldquoResearch on improved convergence co-cevolutiongenetic algorithm for TSPsrdquo International Journal of Conver-gence Computing vol 1 no 2 pp 118ndash126 2014

[27] I Mezei M Lukic V Malbasa and I Stojmenovic ldquoAuctionsand iMesh based task assignment in wireless sensor andactuator networksrdquo Computer Communications vol 36 no 9pp 979ndash987 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of