research article minimizing the number of mobile chargers to...

Research ArticleMinimizing the Number of Mobile Chargers to Keep Large-ScaleWRSNs Working Perpetually

Cheng Hu and Yun Wang

Key Lab of Computer Network and Information Integration MOE School of Computer Science and EngineeringSoutheast University Nanjing 210096 China

Correspondence should be addressed to Cheng Hu guyuecanhuigmailcom

Received 6 March 2015 Accepted 7 June 2015

Academic Editor Pierre Leone

Copyright copy 2015 C Hu and Y Wang 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

Wireless Rechargeable SensorNetworks in whichmobile chargers (MCs) are employed to recharge the sensor nodes have attractedwide attention in recent years Under proper charging schedules the MCs could keep all the sensor nodes working perpetuallySince MCs can be very expensive this paper tackles the problem of deciding the minimum number of MCs and their chargingschedules to keep every sensor nodeworking continuouslyThis problem isNP-hard we divide it into two subproblems and proposea GCHA (Greedily Construct Heuristically Assign) scheme to solve them First the GCHA greedily addresses a Tour ConstructionProblem to construct a set of tours to 1-cover theWRSN Energy of the sensor nodes in each of these tours can be timely replenishedby oneMC according to the decision condition derived from a Greedy Charging Scheme (GCS) Second the GCHA heuristicallysolves a Tour Assignment Problem to assign these tours to minimum number of MCs Then each of the MCs can apply the GCSto charge along its assigned tours Simulation results show that on average the number ofMCs obtained by the GCHA scheme isless than 11 over a derived lower bound and less than 05 over related work

1 Introduction

During the past decades Wireless Sensor Networks (WSNs)have been the focus of the research communityWSNs can bewidely used in scientific and military data gathering envi-ronmental surveillance targets monitoring and tracking andhealth care among others WSNs are composed of manysmall-sized sensor nodes Due to cost and size limitationshowever the battery volume of each sensor node is very smallTherefore energy problem poses a barrier for large-scaledeployment ofWSNs Toprolong the lifespan ofWSNsmeth-ods of energy conservation [1] incremental deployment [2]energy harvesting [3] and wireless charging are proposedAmong these methods wireless charging is the most promis-ing in removing the energy bottleneck ofWSNs [4] By apply-ing the technique of wireless charging which refers to mag-netic resonant coupling [5] in this paper a new paradigm ofWireless Rechargeable Sensor Networks (WRSNs) is proposedand immediately received wide attention In WRSNs mobilechargers (MCs) with high volume batteries are employed to

replenish the depleted energy of rechargeable sensor nodesThe batteries of the MCs can be quickly renewed at aservice station (S) In this way lifespan of WRSNs can besignificantly prolonged or theoretically speaking to infinityApplications include sustainable WRSNs in smart grids [6]and underground sensor networks powered by UAVs [7]

Most existing works on WRSNs only use one MC toconduct charging However an open issue in these works istheir poor scalability Even when the MC can simultane-ously charge all sensor nodes within its charging area themaximum network size is still limited by its charging powermoving speed and total energyTherefore multipleMCs areemployed to address the problem

Since an MC can be very expensive the first and fore-most issue in designing a WRSN with multiple MCs is todecide the minimum number of MCs required so thatno sensor node will die This problem named MinimumMobile Chargers Problem (MMCP) [8] is closely related tothe Distance-Constrained Vehicle Routing Problem (DVRP)[9] Given a network 119866 = (119864 119881) and a distance bound 119863

Hindawi Publishing CorporationInternational Journal of Distributed Sensor NetworksVolume 2015 Article ID 782952 15 pageshttpdxdoiorg1011552015782952

2 International Journal of Distributed Sensor Networks

N1

N2

N3

N4

N5

N6

N7

N8

N9

N10

N11

N12

S

R1

R2

R3

R4

30∘

r

Figure 1 A 12-node ring-type WRSN

the DVRP [9] is to find a minimum cardinality set of toursoriginating from a depot that covers all the vertices in119881 Eachtour is required to have length at most119863 SinceMCs are alsoenergy-constrained researchers reduce MMCP to DVRP sothat solutions for DVRP can be applied to address MMCP[8] We however argue that DVRP is only a subproblem ofMMCP This is because an MC can conduct charging onseveral tours

To demonstrate this viewpoint consider a demo WRSNshown in Figure 1 in which 12 sensor nodes are uniformlydeployed along a circle with radius 119903 Suppose that in thisWRSN due to energy limitation of theMCs anyMC is notable to charge all the sensor nodes in one tour Instead atmost3 sensor nodes can be charged within one tour Then at least4 tours are needed to cover the wholeWRSN as shown in thefigure Therefore in this scenario the answer to the variedDVRP is 4 Suppose the time of charging along each tour isat most 119905 while the lifetime of any sensor node is at least 4119905then only one MC is required the MC charges along tour119877119894during time [(4119896 + 119894 minus 1) sdot 119905 (4119896 + 119894) sdot 119905] (1 le 119894 le 4

119896 = 0 1 2 ) In this way energy of all the sensor nodes canbe timely replenished In summary in this case the answer tothe varied DVRP is 4 while the answer to the MMCP is 1

Therefore in this paper we divide the MMCP into twosubproblems that is a Tour Construction Problem (TCP) anda Tour Assignment Problem (TAP) The TCP is to constructminimumnumber of schedulable tours to cover aWRSNTheTAP is to assign the obtained tours to minimum number ofMCs so that all sensor nodes can be charged before theirbatteries are used upThese two problems are closely coupledWe propose a two-step solution namedGCHA (Greedily Con-struct Heuristically Assign) First we construct the shortestHamilton cycle through S and all the sensor nodes in aWRSN Based on a decision condition we greedily split thecycle into several schedulable subtours Second we prove thattheTAP isNP-hard andpropose a heuristic algorithm to solveit Following these two steps we finally solve the MMCP

The objectives of this paper are to decide the minimumnumber of MCs needed to keep all the sensor nodes in

a WRSN working continuously and to construct a validcharging schedule accordinglyThe contributions of the paperare the following

(i) We illustrate the essential difference between theMMCPand theDVRPanddivide theMMCP into twoNP-hard subproblems

(ii) We derive an efficient decision condition of whether asequence of sensor nodes can be charged by oneMCin one tour The decision condition can be used toconstruct a set of schedulable tours to cover aWRSNCharging schedule of charging along each tour canbe easily obtained by applying the Greedy ChargingScheme (GCS first proposed in [10])

(iii) We propose a heuristic algorithm to assign a set ofschedulable tours to minimum number ofMCs Thedeveloped heuristic rules can minimize the conflictratio of anMC so that it can charge along maximumnumber of tours

(iv) We develop a lower bound for MMCP The lowerbound is used to evaluate the performance of oursolution

The remainder of the paper is organized as followsSection 2 introduces latest related work Section 3 describesmodel assumptions and the problem under study The solu-tions of the TCP and the TAP are detailed in Sections 4 and5 Section 6 evaluates performance of the GCHA scheme andSection 7 concludes the paper

2 Related Work

Most existing works employ only one MC for chargingPeng et al [11] presented the design and implementation ofa wireless rechargeable sensor system and especially studiedthe charging planning problem Xie et al [10] constructed arenewable energy cycle of aWRSNby solving an optimizationproblem with the objective of minimizing the overall energyconsumption Hu et al [12] fully considered the influence ofimbalanced energy consumption in WRSNs and proposedan on-demand charging scheme and a method of deployingS He et al [13] laid a theoretical foundation for the on-demand mobile charging problem where individual sensornodes request charging from the mobile charger when theirenergy runs low Wang et al [14] provided stochastic analysisof wireless energy replenishment and proposed battery-awareenergy replenishment strategies with corresponding sensoractivation schemes In addition literatures [15ndash17] assumethat the MC can function as a mobile sink Zhao et al [15]designed an adaptive solution that jointly selects the sensorsto be charged and finds the optimal data gathering schemesuch that network utility can be maximized while main-taining perpetual operations of the network Guo et al [16]studied the problem of joint wireless energy replenishmentand mobile data gathering for WRSNs The target was tomaximize the overall network utility in terms of data gatheredby theMC Shi et al [17] aimed to jointly optimize a dynamicmultihop data routing a traveling path and a charging

International Journal of Distributed Sensor Networks 3

schedule such that the ratio of the MCrsquos vacation time overthe cycle time could be maximized

Nevertheless these works can only address the energyproblem in small-scale WRSNs To address the scalabilityproblem many researchers utilize the technique of simulta-neous power transfer [18] so that anMC can simultaneouslycharge all sensor nodes within its charging area Tong etal [19] considered monitoring a field with multiple postsof interest where at least one sensor node was deployedThey addressed problems of node deployment and routinggeneration with the target of minimizing the total chargingcost Xie et al extended the concept of renewal cycle tomultinode case in [20] Li et al [21] optimized the usage ofthe entire energy reserve on the MC by heuristic solutionsFu et al [22] planned the optimal movement strategy of theMC which wasmeanwhile a RFID reader such that the timeto charge all nodes in the network above their usable energythreshold was minimized

Although charging multiple nodes simultaneously canrelieve the scalability problem to some extent the size of suchWRSNs is still constrained by charging power moving speedand total energy of the MC Therefore multiple MCs arerequired to serve large-scaleWRSNs Dai et al [8] studied theproblem of minimizing the number of energy-constrainedMCs to cover a 2D WRSN By conducting appropriatetransformations they casted the problem into DVRP andsolved it by applying classical solutions for DVRP Zhang etal [23] used a number ofMCs with limited energy to chargea one-dimensional WRSN They assumed that energy canbe transferred among MCs with 100 efficiency and thenproposed collaborative approaches to maximize the energyefficiency of theMCs Liang et al [24] studied the use ofmin-imum number of mobile chargers to charge sensor nodes in aWRSN so that none of the nodes runs out of its energy subjectto the energy capacity imposed on mobile vehicles Theyformulated an optimization problem of scheduling mobilevehicles to charge lifetime-critical sensorswith an objective tominimize the number of mobile vehicles deployed subject tothe energy capacity constraint on each mobile vehicle Thenthey devised an approximation algorithm with a provableperformance guarantee

Different from most existing works this paper employsmultiple MCs to accomplish the charging mission in aWRSN so that scalability problem can be settled Literature[23] assumes 100 charging efficiency of MCs and theobjective is to maximize the ratio of the amount of payloadenergy to overhead energy while we aim to minimize thenumber ofMCs in a 2DWRSN whereMCs cannot transferenergy to each other Literatures [8 24] only solve the TCPwhile we further propose and solve the TAP to reduce thenumber of MCs used (A previous work of this paper ispresented in [25 26])

3 System Model and Problem Formulation

Consider a 2D WRSN with homogeneous rechargeable sen-sor nodesN = 1198731 1198732 119873119899 andMC119904 M = 11987211198722 119872119898 Each sensor node 119873

119894(1 le 119894 le 119899) has a battery with

capacity 119864119878and consumes energy at a constant rate 119901

119894

(applying the techniques in [26] our methods can be usedin situations where sensor nodes consume energy at randombut bounded rates)The sensor nodes are equippedwithwire-less energy receivers which can receive and store energytransferred from anMC Assume that charging and commu-nication use different frequencies such as the implementa-tion in [11] then a sensor node can communicate while beingcharged

Suppose that theMCs can move freely in the WRSN at aconstant speed V without any constraint of movement tracksThe energy consumed by each MC for moving per meter isa constant 119902

119898 The MCs record the positions of all sensor

nodes and can wirelessly charge the sensor nodes at a powerrate 119902

119888with efficiency 120578 Suppose that each MC can only

charge one sensor node at the same time Since 120578 dropsrapidly as the charging distance increases we let the MCsmove to each sensor node to conduct charging Consequently119902119888and 120578 are viewed as constants Charging and moving

activities of each MC are powered by the same battery withvolume 119864

119872 which can be renewed at a service station S in

an infinitely short period To simplify the presentation S isalso denoted by1198730

The charging mission in a WRSN is to schedule thecharging activities of theMCs to ensure that all sensor nodeswill never exhaust their energy AnMC at any time must bein one of the following three states [12] (a)moving state thatis moving from one place to another (b) charging state thatis charging a sensor node (c) resting state that is renewingits own battery and resting at S Note that it only consumesenergy inmoving and charging states To accomplish a charg-ingmission we let theMCs periodically charge sensor nodesalong a set of tours R = 1198771 1198772 119877119911 Define that119872119896 cancharge along a set of tours R

119896= 1198771 1198772 119877119911119896 if119872119896 can

ensure liveness of all the sensor nodes inR119896 then each of the

tours is called schedulable and these tours can be assigned to119872119896 Suppose that all tours are independently charged along

that is eachMCmust finish charging along one tour before itcharges along the next A charging schedule determines whendo theMCs chargewhich sensor nodes inwhat sequence andfor how long A charging schedule is called valid if it can beapplied to accomplish the charging mission

Tour 119877119894= ⟨1198730 1198731 119873119903119894⟩ determines an MCrsquos charg-

ing sequence of the 119903119894sensor nodes Obviously the union of

the sensor nodes in tours R should fully cover the WRSN119911

⋃

119894=1119877119894= NcupS (1)

An MC periodically charges along tour 119877119894with charging

period 119879119894(1 le 119894 le 119911) As will be proved in Section 4 119879

119894

of tour 119877119894ranges in a feasible region [119879119897

119894 119879119906

119894] Assume that

119879119894is rational Within each charging period 119879

119894 the total time

an MC spends in moving and charging states is called theworking time of 119877

119894 denoted by 119879119908

119894(119879119908119894le 119879119894) Consequently

we also use a tuple (119879119894 119879119908

119894) to substitute for 119877

119894 However if

the charging period of 119877119894has not been determined the tour

can be denoted by a triple (119879119897119894 119879119906

119894 120596119894) where 120596

119894is a working

time function dependent on 119879119894 The three representations of

119877119894will be used interchangeably in the rest of the paper where

no ambiguity occurs


Timeline

Tour R1Tour R2

Tw1 = 1 Tw

2 = 15

T2 = 5

T1 = 75

Figure 2 Assigning 1198771 an 1198772 to oneMC

Suppose 119911119896tours R

119896= 1198771 1198772 119877119911119896 are assigned to

119872119896(1 le 119911

119896le 119911) then we define the element period of119872

119896

denoted by 119866119896 as the greatest common divisor of charging

periods of these 119911119896tours Define the conflict ratio of119872

119896as

120575119896=sum119911119896

119894=1 119879119908

119894

119866119896

119866119896= GCD (1198791 1198792 119879119911119896) (2)

where function GCD(sdot) returns the greatest common divisorof the given numbers Informally speaking 120575

119896indicates the

maximum work load of 119872119896within any element period 119866

119896

during119866119896119872119896charges along119877

119894isin R119896for atmost onceWewill

prove that if 120575119894of119872119894is nomore than 1 a valid charging sched-

ule of charging along these 119896 tours can be easily constructedFor instance in Figure 2 1198771 = (75 1) and 1198772 = (5 15) canbe assigned to one MC Because the element period of theMC is 119866 = GCD(1198791 1198792) = GCD(75 5) = 25 the conflictratio of theMC is 120575 = (1198791199081 + 119879

119908

2 )119866 = (1 + 15)25 = 1Then the problem addressed in this paper can be formu-

lated as follows

MinimumMobile Chargers Problem (MMCP)Given 119899 sensornodes N = 119873

119894(119901119894 119864119878) | 1 le 119894 le 119899 and their

positions in a WRSN decide the minimum size 119898 of M =

119872119894(119902119888 119902119898 V 119864119872) | 1 le 119894 le 119898 to accomplish the charging

mission

When obtaining the minimum number of MCs a validcharging schedule should be constructed at the same time Tothis end the GCHA scheme constructs a set of schedulabletours that 1-covers the WRSN Then it assigns the tours tominimum number of MCs Each MC applies the GCS tocharge along its assigned tours The notations appearing inthis paper are listed in Notations for readersrsquo convenience

4 The Tour Construction Problem

Tours in R should fully cover the WRSN we consider ascenario that these tours form an exact 1-cover of the WRSNIn other words intersection of any two tours is empty andany sensor node can appear in any tour for at most once

119877119894cap119877119895= 119877

119894 119877119895isin R

119873119894= 119873119895 119873119894 119873119895isin 119877119896 119877119896isin R

(3)

Then we propose and address the following problem

Tour Construction Problem (TCP) Given 119899 sensor nodes N =119873119894(119901119894 119864119878) | 1 le 119894 le 119899 and their positions in a WRSN

construct minimum number of toursR schedulable toMCs(119902119888 119902119898 V 119864119872) to exactly 1-cover the WRSN

The decision form of the TCP is NP-hard since TCP con-tains DVRP as a subproblem [8] A simple transformation isto let 119899sdot119864

119878rarr 0 then theTCP can be reduced to aDVRPwith

distance constraint 119864119872119902119898 To address the TCP we firstly

propose a sufficient condition for anMC to be able to chargealong a tourThe condition is derived from aGreedy ChargingScheme (GCS) Then based on the decision condition wegreedily split the shortest Hamilton cycle through S and Ninto minimum number of schedulable subtours The proce-dure is elaborated below

41 The Greedy Charging Scheme When charging along atour 119877 = ⟨1198730 1198731 119873119903⟩ (1 le 119903 le 119899) an119872 isin M appliesthe GCS as follows In each period 119872 starts from 1198730sequentially fully charges each sensor node then returns to1198730 quickly renews its battery and rests until next periodDenote that119872 arrives at119873

119894at time 119905

119894and charges119873

119894for time

120591119894 then we have

119905119894=

119894minus1sum

119895=0120591119895+

119894

sum

119895=1

119889119895minus1119895

V 1 le 119894 le 119903 (4)

In (4) 119889119894119895

is the distance between 119873119894and 119873

119895and 1205910 ge 0 is

the duration that119872 rests at S and let 1199050 = 0 to simplify thepresentation Then 119872 can periodically charge along 119877 withcharging period

119879 =

119903

sum

119895=0120591119895+119863

V (5)

where 119863 = sum119903119895=1 119889119895minus1119895 + 1198891199030 is the total traveling distance of

the tour Then within each charging period the energy con-sumed by119873

119894can be fully replenished by119872 that is120591119894sdot 119902119888sdot 120578 = 119901

119894sdot 119879 1 le 119894 le 119903 (6)

To ensure the sensor nodes are working continuously theirenergy levels should never fall below 0 during 119879 that is

119864119878minus119901119894sdot (119879 minus 120591

119894) ge 0 1 le 119894 le 119903 (7)

Meanwhile the total energy of119872 should be enough to con-duct charging that is

119863 sdot 119902119898+

119903

sum

119895=1sdot 119902119888sdot 120591119895le 119864119872 (8)

Therefore119872 can charge along tour 119877 applying the GCSif and only if it satisfies (4)ndash(8)

42 Deriving the Decision Condition According to the for-mulation of the GCS we can derive an efficient decision con-dition to decide whether a tour is schedulable The decisioncondition named Tour Schedulable (TS) condition can beused to construct schedulable tours to exactly 1-cover theWRSN

Equation (6) can be rewritten as

120591119894=119901119894sdot 119879

119902119888sdot 120578 1 le 119894 le 119903 (9)

Thus the charging duration of each sensor node is dependenton its charging period 119879 In fact 119879 determines the charging


schedule of 119872 applying the GCS according to (4) and (9)Furthermore 119879 also determines the working time of 119877

119879119908=

119899

sum

119894=1120591119894+119863

V=119901sum119902119888sdot 120578sdot 119879 +

119863

V (10)

Obviously 119879119908 increases as 119879 increases however 119879119908119879decreases as 119879 increases Literature [12] presented that 119879119908119879indicates the total energy consumption of 119872 in chargingalong 119877 thus it should be minimized Plugging (9) into (5)we have

119879 = 1205910 +sum119903

119894=1 119901119894 sdot 119879

119902119888sdot 120578

+119863

Vgesum119903

119894=1 119901119894 sdot 119879

119902119888sdot 120578

+119863

V (11)

or equivalently

119879 ge119863 sdot 119902119888sdot 120578

V sdot (119902119888sdot 120578 minus 119901sum)

119901sum =119903

sum

119894=1119901119894 (12)

Let 119879119897 = (119863 sdot 119902119888sdot 120578)(V sdot (119902

119888sdot 120578 minus 119901sum)) then feasible charging

period of tour119877 cannot be less than119879119897 otherwise119872 does nothave enough time to charge all the sensor nodes in 119877 Notethat to make (12) work a necessary condition should be met119902119888sdot 120578 gt 119901sumPlugging (9) into (7) we have

119864119878ge119901119894sdot (119902119888sdot 120578 minus 119901

119894)

119902119888sdot 120578

sdot 119879 1 le 119894 le 119903 (13)

Since 119902119888sdot 120578 gt 119901sum gt 119901119894 the inequality is equal to

119879 le119864119904sdot 119902119888sdot 120578

119901mx 119901mx = max

1le119894le119903119901119894sdot (119902119888sdot 120578 minus 119901

119894) (14)

Let 119879119906 = (119864119904sdot 119902119888sdot 120578)119901mx then feasible charging period of 119877

cannot exceed119879119906 or some sensor node will run out of energybefore it is charged Combining (12) and (14) we have

119863 sdot 119902119888sdot 120578


le119864119904sdot 119902119888sdot 120578

119901mx(15)

or equivalently V ge (119863 sdot 119901mx)(119864119878 sdot (119902119888 sdot 120578 minus 119901sum)) whichis also a necessary condition for 119877 to have feasible chargingperiod Plugging (9) into (8) we have

119864119872ge 119863 sdot 119902

119898+119901sum120578sdot 119879 (16)

Formula (16) shows the minimum energy required for anMC to charge along 119877 with charging period 119879 Let 119879 = 119879119906we have 119864

119872ge (119864119878sdot 119902119888sdot 119901sum)119901mx + 119863 sdot 119902119898 By now we have

derived a sufficient condition for119872 to charge along 119877

TS Condition An MC applying the GCS can charge alongtour 119877 = ⟨119873

0 1198731 119873

119903⟩ if (a) 119902

119888sdot 120578 gt 119901sum (b) V ge (119863 sdot

119901mx)(119864119878 sdot (119902119888 sdot120578minus119901sum)) (c) 119864119872 ge 119863sdot119902119898+(119901sum sdot119864119878 sdot119902119888)119901mx

Note that in deriving the TS condition we let 119879 = 119879119906to decide the minimum energy requirement of anMC Onecan set other feasible119879 to derive a new sufficient condition orsay anchoring other values of 119879 for short In the next sectionhowever we will show that anchoring 119879 = 119879119906 can minimizethe total energy consumption of the MCs as well as theconflict ratio it generated

Although the TS condition is not necessary for a tour tobe schedulable [26] 119902

119888sdot 120578 gt 119901sum is a hard condition for any

MC to be able to charge along any tour Furthermore if thetotal charging power of M is not larger than the total energyconsumption rate of N in the WRSN the charging missioncannot be accomplishedTherefore we obtain a lower boundof the number ofMCs needed to solve the MMCP

Lower Bound119898lowastGiven 119899 sensor nodesN = 119873119894(119901119894 119864119878) | 1 le

119894 le 119899 and their positions in a WRSN the minimum numberof MCs (119902

119888 119902119898 V 119864119872) needed to accomplish the charging

mission is not less than119898lowast = lceilsum119899119894=1119901119894(119902119888sdot 120578)rceil

The lower bound 119898lowast is not tight since it only considersthe influence of charging Obviously an MC also spendstime and energy onmovingTherefore amore rigorous lowerbound should be not less than119898lowast However determining119898lowastis very simple and efficient thus we use 119898lowast for performanceevaluation in Section 4

Based on the TS condition we propose Algorithm 1 todecide whether a tour can be assigned to an MC (lines (1)ndash(7)) If the tour is schedulable then Algorithm 1 returns thefeasible region of its charging period (lines (8)-(9)) and thecorresponding working time function (line (10))

43 Obtaining Schedulable Subtours Since TCP is NP-hardbased on the TS condition we propose a greedy solution asshown in Algorithm 2 First we construct a Hamilton tour119867through S and N by applying LKH algorithm [27] (lines (2)-(3)) Then we replace edges 119890

119894119894+1 and 119890119895119895+1 by 1198900119894+1 and 1198901198950(lines (5) (7)) and decide whether an MC can charge alongtour ⟨1198730 119873119894+1 119873119895⟩ using Algorithm 1 (line (8)) A sub-tour is constructed if no more sensor nodes can be inserted(lines (9)ndash(13)) Repeat the procedure until all sensor nodesare covered (line (4)) In this way we greedily split 119867 into119911 schedulable subtours Figure 3 shows a demonstration ofAlgorithm 2 in which two schedulable tours are constructedto cover the 5-node WRSN Computation complexity ofverifying Proposition 2 is 119874(119903) If variables such as 119901sumand 119901mx can be saved in the algorithm the complexity ofupdating these information is119874(1)Then lines (3)ndash(15) can beexecuted with complexity 119874(119899) Therefore the computationcomplexity of Algorithm 2 is dependent on that of LKHwhich is 119874(11989922) [27] Note that 119867 is not necessarily theshortest Hamilton cycle through the WRSN In fact it onlyprovides a sequence of sensor nodes Therefore 119867 can beany permutation of the sensor nodes For instance119867 can beobtained by sorting the sensor nodes in ascending order oftheir energy consumption rates In this way sensor nodes ineach constructed subtour have similar energy consumptionrates and greatly relief the load imbalance phenomenon [12]


Step 1 Construct shortest Hamilton cycle Hthroughout the WRSN the TS condition is met

Step 2 Construct subtourR1 insert N1 into R1the TS condition is metStep 3 Construct subtourR1 insert N2 into R1

the TS condition is metStep 4 Construct subtourR1 insert N3 into R1

the TS condition is met All sensor nodes arecovered the tour construction algorithm ends

Step 6 Construct subtourR2 insert N5 into R2

condition is met

Step 5 Construct subtourR1 insert N4 into R1the TS condition is not met Pop N4 from R1construct subtour R2 Insert N4 into R2 the TS

N1

N2

N3

N4

N5

S(N0)

N1

N2

N3

N4

N5

S(N0)

N1

N2

N3

N4

N5

S(N0)

N1

N2

N3

N4

N5

S(N0)

N1

N2

N3

N4

N5

S(N0)

N1

N2

N3

N4

N5

S(N0)

Figure 3 Demonstration of the tour construction algorithm

Require119877 = ⟨1198730 1198731 119873119903⟩ 119864119878 119902119898 119902119888 V 119864119872 120578

EnsureSchedulability of 119877 (119879119897 119879119906 120596)

(1) calculate path length119863 of 119877(2) inquire energy consumption rates 119901

119894 of the sensor nodes in 119877

(3) 119901sum =119903

sum

119894=1

119901119894

(4) 119901mx = max1le119894le119903

119901119894sdot (119902119888sdot 120578 minus 119901

119894)

(5) if the conditions in Proposition 2 are not satisfied then(6) return unschedulable(7) end if(8) 119879119897 =

119863 sdot 119902119888sdot 120578


(9) 119879119906 =119864119904sdot 119902119888sdot 120578

119901mx

(10) 120596(119879) =119901sum

119902119888sdot 120578sdot 119879 +

119863

V

(11) return schedulable

Algorithm 1 Decide whether anMC can charge along 119877

which causes great waste in periodic charging schemes Fromthis viewpoint a periodic charging scheme using multipleMCs is similar to an on-demand charging scheme using oneMC Nevertheless total distance of each subtour should beminimized [10] Therefore it is a compromise that we madein designing Algorithm 2

5 The Tour Assignment Problem

After obtaining schedulable subtours that 1-cover all thesensor nodes we further assign these subtours to minimumnumber of MCs To this end we firstly formulate the TourAssignment Problem (TAP) and prove its NP-hardness Then


RequireN = 119873

119894| 1 le 119894 le 119899

EnsureR = 119877

119894| 1 le 119894 le 119911

(1) 119911 = 1 119894 = 1(2) apply LKH on S cup N to obtain119867(3) renumber sensor nodes in119867 as1198730 1198731 119873119899 (1198730 = S)(4) while 119894 le 119899 do(5) push1198730 into 119877119911(6) for 119895 = 119894 to 119899 do(7) push119873

119895into 119877

119911

(8) apply Algorithm 1 to decide the schedulability of 119877119911

(9) if 119877119911is unschedulable then

(10) pop119873119895from 119877

119911

(11) 119894 = 119895 119911 = 119911 + 1(12) break(13) end if(14) end for(15) end while

Algorithm 2 Construct a Hamilton tour119867 through S and N split119867 into 119911 schedulable subtours

we propose a heuristic algorithm to solve the problem Everytour discussed in this section is schedulable

51 The Tour Assignment Problem According to the above-mentioned analysis feasible charging period 119879

119894of tour 119877

119894

ranges from119879119897119894to119879119906119894 which also decides the working time119879119908

119894

of 119877119894in each period 119879119908

119894= 120596119894(119879119894) = (119901

119894

sum(119902119888 sdot 120578)) sdot 119879119894 +119863119894VThen the assignment problem can be formulated as follows

Tour Assignment Problem (TAP) Given 119911 tours R = 119877119894=

(119879119897

119894 119879119906

119894 120596119894) | 1 le 119894 le 119911 and enough MCs (119902

119888 119902119898 V 119864119872)

assign tours R to minimum number ofMCs

To prove the NP-hardness of the TAP we firstly consider119903 tours with equal charging period 119879 and investigate thesufficient and necessary condition of assigning these 119903 toursto oneMC

Proposition 1 Given 119903 tours 119877119894= (119879 119879

119908

119894) | 1 le 119894 le 119903 they

can be assigned to oneMC if and only if sum119903119894=1 119879119908

119894le 119879

Proof Suppose these 119896 tours can be assigned to the sameMC then sum119903

119894=1 119879119908

119894le 119879 Otherwise the MC cannot finish

charging along all tours within one charging period 119879causing conflicts in the next period

On the other hand if sum119903119894=1 119879119908

119894le 119879 one can construct a

valid schedule as followsTheMC charges sensor nodes in119877119894

during time [119896 sdot 119879 +sum119894minus1119895=1 119879119908

119895 119896 sdot 119879 +sum

119894

119895=1 119879119908

119895] (119896 = 0 1 2 )

Since sum119903119894=1 119879119908

119894le 119879 this schedule is valid

Therefore the proposition follows

Based on Proposition 1 we reduce the TAP to a set-partition problem and consequently prove that the decisionform of the problem is NP-hard

Proposition 2 Deciding whether 119911 tours can be assigned to119898MCs is NP-hard

Proof Consider 119911 tours 119877119894= (119879 119879

119908

119894) | 1 le 119894 le 119911 and

supposesum119911119894=1 119879119908

119894= 2119879 Let119898 = 2 then the decision problem

can be reformulated as whether 119877119894 can be assigned to 2

MCsWe now construct a set-partition problem as follows

Given 119911 numbers 1198861 1198862 119886119911 in which 119886119894= 119879119908

119894119879 (1 le

119894 le 119911) then according to Proposition 1 the above questionis equivalent to whether the 119911 numbers can be evenlypartitioned into two sets Since the set-partition problem isNP-hard the original problem is also NP-hard

52TheHeuristic Solution Wepropose a heuristicmethod todecide (a) the charging period of each tour and (b) whethera set of tours with fixed charging periods can be assigned tooneMC

We firstly generalize the assumption in Proposition 1 andassume that tours have different charging periods Then wederive a necessary condition for these tours to be assignedto the same MC Suppose that the periods of all tours arerational numbers and their greatest common divisor is 119866 (119866can be a decimal) then we have the following proposition

Proposition 3 Given 119896 set of tours 119877119895119894= (119886119895sdot119866 119879119895

119894) | 1 le 119894 le

119903119895 1 le 119895 le 119896 where 119877119895

119894is the 119894th tour of the 119895th set and 119886

119895

are coprime integers these tours can be assigned to one MC

only if sum119896119895=1(sum119903119895

119894=1 119879119895

119894119886119895) le 119866

Proof Suppose these tours can be assigned to oneMC whilesum119896

119895=1(sum119903119895

119894=1 119879119895

119894119886119895) gt 119866 Denote the least common multiple of

1198861 1198862 119886119896 by 119875 Multiplying both sides of the inequality by119875 we have

119875

1198861sdot

1199031

sum

119894=11198791119894+119875

1198862sdot

1199032

sum

119894=11198792119894+ sdot sdot sdot +

119875

119886119896

sdot

119903119896

sum

119894=1119879119896

119894gt 119866 sdot 119875 (17)

where 119875119886119895means how many times does the MC charge

along tours 119877119895119894 during 119866 sdot 119875 andsum119903119895

119894=1 119879119895

119894is the total working


time of the 119895th set of tours within each charging period 119886119895sdot 119866

It means that the total working time of all the tours during119866 sdot 119875 is larger than 119866 sdot 119875 which is a contradiction


Unfortunately the condition is not sufficient because itrequires that the 119895th set of tours can be evenly partitioned into119886119895parts such that the total working time of tours in each part

is the same Otherwise these tours are not surely schedulableFor example tours 2119879 21198793 2119879 1198793 and 119879 1198792 cannotbe charged by one MC even if they satisfy Proposition 3Besides deciding whether the 119895th set of tours can be evenlypartitioned into 119886

119895parts is NP-hard since it can be easily

reduced to a set-partition problemTherefore we propose thefollowing sufficient condition for efficiently deciding whether119903 tours can be assigned to oneMC

Proposition 4 Given 119903 tours 119877119894= (119886119894sdot 119866 119879

119908

119894) | 1 le 119894 le

119903 where 119886119894 are coprime integers they can be assigned to one

MC if sum119903119894=1 119879119908

119894le 119866


119908

119894) | 1 le 119894 le 119903

if they can be assigned to one MC obviously 119877119894 can be

assigned to the same MC Since sum119903119894=1 119879119908

119894le 119866 according to

Proposition 3 1198771015840119894 can be assigned to one MC Therefore

119877119894 can be assigned to oneMC

Note that Proposition 4 is not necessary For instanceconsider tour 119877119860 = (119879 1198792) and 119903 tours 119877119861

119894= (119903 sdot 119879 1198792) |

119903 gt 1 1 le 119894 le 119903 Since GCD(119879 119903 sdot 119879) = 119879 and the totalworking time of the 119903 + 1 tours is ((119903 + 1)2) sdot 119879 gt 119879 thecondition in Proposition 4 is not met However one can stillassign them to one MC let the MC charge along tour 119877119860

during time [119895 sdot 119879 (119895 + 12) sdot 119879] and charge along tour 119877119861119894

during time [(119895 sdot 119903 + 119894 minus 12) sdot 119879 (119895 sdot 119903 + 119894) sdot 119879] 119895 = 0 1 2 It can be readily proved that the charging schedule is valid

Proposition 4 inspires us to minimize the conflict ratioof each MC 120575 = sum119903

119894=1(119879119908

119894119866) If it is no more than 1 valid

charging schedule can be readily constructed Based on thisidea we propose the following heuristic rule to determine thecharging period of each tour

Proposition 5 Suppose tours 1198771 = (119879119897

1 119879119906

1 1205961) and 1198772 =(119879119897

2 119879119906

2 1205962) are assigned to119872 where (119896minus1)sdot1198791199061 lt 119879119897

2 le 119896sdot119879119906

1 le119879119906

2 (119896 ge 1) then the conflict ratio 120575 of119872 is minimized when1198791 = 119879

119906

1 and 1198792 = 119896 sdot 1198791199061

Proof For each tour 119877119894 120596(119879119894) = (119901

119894

sum(119902119888 sdot 120578)) sdot 119879119894 +119863119894V Tosimplify the presentation denote that 120596(119879

119894) = 119886119894sdot 119879119894+ 119887119894

Consider 120575when 1198792 varies within time interval [119894 sdot 1198791 (119894 +1) sdot 1198791] (119894 = 1 2 )

1205751 = 1198861 + 119894 sdot 1198862 +1198871 + 11988721198791

1198792 = 119894 sdot 1198791

1205752 = 1198861 + (119894 + 1) sdot 1198862 +1198871 + 11988721198791

1198792 = (119894 + 1) sdot 1198791

1205753 =1198861 sdot 1198791 + 1198862 sdot 1198792

119866+1198871 + 1198872119866

119894 sdot 1198791 lt 1198792 lt (119894 + 1) sdot 1198791

(18)

where 119866 = GCD(1198791 1198792) When 119894 sdot 1198791 lt 1198792 lt (119894 + 1) sdot 1198791119866 le 11987912 we hence have

1205753 =1198861 sdot 1198791 + 1198862 sdot 1198792

119866+1198871 + 1198872119866

ge 21198861 + 2119894 sdot 1198862 +2 (1198871 + 1198872)1198791

gt 1198861 + (119894 + 1) sdot 1198862 +1198871 + 11988721198791

= 1205752

(19)

In other words 1205751 lt 1205752 lt 1205753 Therefore in this scenario 120575 isminimized when 1198792 = 119894 sdot 1198791 Then 120575 = 1198861 + 119894 sdot 1198862 + (1198871 +1198872)1198791

On the other hand 120575 decreases as 1198791 increases Tominimize 120575 we have 1198791 = 119879

119906

1 Since (119896 minus 1) sdot 1198791199061 lt 119879119897

2 le119896 sdot 119879119906

1 le 119879119906

2 we have 119894 = 119896 that is 1198792 = 119896 sdot 119879119906

1 In summary120575 isminimizedwhen1198791 = 119879

119906

1 and1198792 = 119896sdot119879119906

1 then the proposition follows

In fact Proposition 5 requires the element period 119866 of119872 to be maximized This is an important reason that weanchor 119879 = 119879119906 when deriving the TS condition Note thatone can set 119879 = 119879

1015840 (119879119897 le 1198791015840lt 119879119906) in deriving the TS

condition In this way the number of subtours may bereduced in Algorithm 2 Nevertheless these tours are harderto be assigned to one MC according to Proposition 5 Moreimportantly anchoring smaller value of 119879 will increase thetotal working time ratio (denoted by Φ

119908) of theMCs

Φ119908=

119898

sum

119894=1

119911119894

sum

119895=1

119879119908

119895

119879119895

=

119911

sum

119894=1

119879119908

119894

119879119894

(20)

We have proved in [12] that the working time ratio of anMC indicates the total energy consumption of the MCin conducting charging Therefore Φ

119908indicates the total

energy consumption of all theMCs Based on the above tworeasons we anchor 119879 = 119879

119906 in deriving the TS conditionComparisons of anchoring different values of 119879 are madethrough simulations

Based on the heuristic rules formulated in Propositions 4and 5 we propose Algorithm 3 to greedily assign maximumnumber of tours to each MC First the tours obtained byAlgorithm 2 are sorted by 119879119906 in ascending order (line (2))Then for each MC sequentially consider if tour 119877

119894can be

assigned to it (lines (3)ndash(18)) According to Proposition 5 theelement period 119866

119896of119872119896is set to 119879119906

119894of its first assigned tour

119877119894(line (7)) For any subsequent tour 119877

119895 it is assigned to119872

119896

(lines (6) (13)) if

(a) 119877119895has not been assigned to anyMC (lines (4) (9))

(b) feasible period of 119877119895contains a multiple of 119866

119896(lines

(10)-(11))(c) after assigning the conflict ratio 120575

119896of 119872119896is still no

more than 1 (line (11))

Then charging period 119879119895of 119877119895is determined according to

Proposition 5 (lines (7) (12)) It can be readily proved that theworst case computation complexity of Algorithm 3 is 119874(1199112)where 119911 is the number of tours


RequireR = 119877

119894| 1 le 119894 le 119911

EnsureM = 119872

119894| 1 le 119894 le 119898 tour assignment of eachMC

(1) 119898 = 0(2) sort 119877

119894 by 119879119906

119894in ascending order

(3) for 119894 = 1 to 119911 do(4) if 119877

119894has not been assigned then

(5) 119898+ = 1(6) assign 119877

119894to119872119898

(7) 119866119898= 119879119894= 119879119906

119894 120575

119898=120596119894(119879119894)

119866119898

(8) for 119895 = 119894 + 1 to 119911 do(9) if 119877


(10) calculate the 119896 described in Proposition 5(11) if 119896 exists and 120575

119898+ (120596119895(119896 sdot 119879

119906

119894)119866119898) le 1 then

(12) 119879119895= 119896 sdot 119879

119906

119894 120575

119898+ =

120596119895(119879119895)

119866119898

(13) assign 119877119895to119872119898

(14) end if(15) end if(16) end for(17) end if(18) end for

Algorithm 3 Sort tours R by 119879119906 greedily assign the tours to minimumMCs based on Propositions 4 and 5

After tour assignment each MC applies the GCS toperiodically charge along its assigned tours Suppose 119911

119909tours

R119909= 119877119894= (119886119894sdot 119866119909 119879119908

119894) are assigned to119872

119909(1 le 119894 le 119911

119909le 119899

1 le 119909 le 119898) then a valid charging schedule is that119872119909charges

119877119894during time [119896 sdot 119886

119894sdot 119866119909+ sum119894minus1119895=1 119879119908

119895 119896 sdot 119886119894sdot 119866119909+ sum119894

119895=1 119879119908

119895]

(119896 = 0 1 2 ) and rests at S at other times The details ofthe charging schedule are shown in Algorithm 4 Howeverwhen119872

119909charges 119877

119894for the first time since the sensor nodes

are initially fully charged the actual working time of tour 119877119894

is less than 119879119908119894 This may influence subsequent periods To

address this issue one can make119872119909stay at sensor node 119873

119895

in 119877119894for time (119901

119895sdot 119879119894)(119902119888sdot 120578) and leave with119873

119895fully charged

In this way the actual working time of 119877119894is 119879119908119894afterwards

53 The Complete Procedure of the GCHA Solution To sum-marize the complete procedure of the GCHA solution ispresented as follows First apply Algorithm 2 to construct119911 schedulable tours to 1-cover the WRSN During this stepAlgorithm 1 is constantly called to decide whether a tour ofsensor nodes can be timely charged by one MC Secondapply Algorithm 3 to decide the charging period of each tourand assign these tours to 119898 MCs Then the MCs applyAlgorithm 4 to conduct charging along their assigned toursThe computation complexity of the solution is dependent onAlgorithm 2 which is 119874(11989922)

6 Numerical Results

In this section we evaluate the performance of the GCHAsolution To this end WRSNs are randomly generated overa 1000m times 1000m square area The sink node and service

station are located at the center of the area The sensornodes generate data at a random rate within [10 100] kbpsand their communication radius is 200m The energy con-sumption model is adopted from [10] in which only energyconsumption of sending and receiving data is consideredParticularly energy consumption of receiving a bit of data isa constant while energy consumption of sending a bit of datais additionally dependent on the transmitting distance Therouting scheme is the well known GPSR [28] which is wellscalable and also based on location information By defaultwe set 119864

119878= 10 kJ 119902

119898= 5 Jm V = 5ms 119864

119872= 40 kJ and 120578 =

1 which are adopted from [10] or with reasonable estimation

61 Demonstration of the GCHA Solution To better demon-strate the procedures of the GCHA solution we randomlygenerate a 100-node WRSN as shown in Figure 4 and set119902119888= 5W The S and sink node are located at the center

of the WRSN According to Algorithm 2 we firstly applyLKH algorithm to construct a Hamilton tour 119867 through Sand N as shown in Figure 5 Then based on the decisionresults returned by Algorithm 1 we greedily split 119867 into 10schedulable subtours as shown in Figure 6 It can be foundthat the tours nearer to the sink cover fewer sensor nodesbecause sensor nodes around the sink can be exhausted fasterdue to larger forwarding burden Then we call Algorithm 3to assign these tours to 4 MCs and each MC appliesAlgorithm 4 to conduct charging along its assigned tours

Figure 7 depicts the energy consumption of each MCwhen charging along each tour within one charging periodin which the lighter grey part of a bar indicates the energyconsumption of charging while the darker grey part of


RequireR119896= 119877119894| 1 le 119894 le 119911

119896

Ensure(119877119894 ⟨119905119894

119895⟩ ⟨120591119894

119895⟩) | 1 le 119894 le 119911

119896 1 le 119895 le 119903

119894

(1) 12059110 = 0(2) for 119894 = 2 to 119911

119896do

(3) 120591119894

0 = 120591119894minus10 + 119879

119908

119894minus1(4) end for(5) for 119894 = 1 to 119911

119896do

(6) 119905119894

0 = 0

(7) 120591119894

119895=119879119894sdot 119901119895

119902119888sdot 120578

1 le 119895 le 119903119894

(8) 119905119894

119895= 119905119894

119895minus1 + 120591119894

119895minus1 +119889119895minus1119895

V 0 le 119895 le 119903

119894

(9) end for(10) 119896 = 0(11) while ++119896 do(12) set current time to 0(13) for 119894 = 1 to 119911 do(14) if 119896 119886

119894= 0 then

(15) MC leaves S to charge along tour 119877119894at time 1205911198940

(16) for 119895 = 1 to 119903119894do

(17) MC arrives119873119894119895at 119905119894119895 and charges it for 120591119894

119894

(18) end for(19) MC returns to S end of charging along tour 119877

119894

(20) end if(21) end for(22) end while

Algorithm 4 Construct a valid charging schedule of119872119896along its assigned toursR

119896

0 500 10000

500

1000

X (m)

Y (m

)

S ampB

Figure 4 Routing of the 100-node WRSN

a bar indicates the energy consumption of moving Thefigure shows that the energy consumed in charging alongdifferent tours varies greatly This is because in the assigning

0 500 10000

500

1000

X (m)

Y (m

)

S ampB

Figure 5 Shortest Hamilton cycle of the 100-node WRSN

procedure the GCHA scheme selects 119879 of 1198771 sim 1198776 nearlythe same as their 119879119906 while it selects 119879 of 1198777 sim 11987710 muchsmaller than their 119879119906 In consequence when charging along


0 500 10000

500

1000

X (m)

Y (m

)

MC1 (3 tours)MC2 (3 tours)


S ampB

Figure 6 Schedulable subtours to cover the 100-node WRSN

1 2 3 4 5 6 7 8 9 100

05

1

15

2

25

3

35

4

Tour ID

Ener

gy co

nsum

ptio

n of

each

tour

(J)

times104

Figure 7 Energy consumption when charging along each tour

1198777 sim 11987710 the energy consumption of the MC is much lessthan 119864

119872 In turn these tours contribute less conflict ratio

of the MC as shown in Figure 8 so that the MC cancharge along more tours The ratio of the grey area in abar is the conflict ratio of an MC It can be seen that 1198724achieves a relatively low conflict ratio This is caused by thegreedy nature of our solution Balanced energy consumptionofMCs however is not a concern of this paper

0 1 2 3 40

02

04

06

08

1

12

14

16

18

2

MC ID

R (s

)

times104

120591rG

Tw

Figure 8 Conflict ratios of eachMC

62 Performance Evaluation We then compare the numberof MCs 119898 obtained by our solution with the lower bound119898lowast to evaluate the performance Specifically we evaluate the

approximation ratio of our results over the lower bound 120572 =119898119898lowast

621 Varying Charging Power of MCs The total energyconsumption rate of the WRSN shown in Figure 4 is 119901sum =1524W and the corresponding 119898lowast = lceil15245rceil = 4Therefore we actually give an optimum solution in thedemonstration regarding number ofMCs used We furtherrange charging power of theMCs from 3W to 9W and thecomparison between 119898 and 119898lowast is shown in Figure 9 where119911 is the number of subtours It is clear that in most cases ourmethod achieves optimum solutions of theMMCP Howeverwhen 119902

119888= 4W and 119902

119888= 8W our results are one more than

119898lowast The reason will be explained later

622 Varying Anchored Charging Period We have explainedthe reason to anchor 119879 = 119879119906 in deriving the TS condition Toverify its effectiveness we anchor119879 = 119879119897 ((119879119897+119879119906))4 ((119879119897+119879119906))2 (3(119879119897 + 119879119906))4 119879119906 and compare the performance of

these schemes We randomly generate 10 200-node WRSNsto test each anchored value of 119879 the results are shown inFigure 10 The blue stars show the averaged results achievedby the GCHA scheme the red dots show the mean valuesof total working time ratios of the MCs It can be seenthat 119898 increases first and then decreases as 119879 increases 119898is minimum when 119879 = 119879

119906 This is because when 119879 issmaller each obtained subtour can cover a larger numberof sensor nodes thus the number of tours is smaller when


2 3 4 5 6 7 8 9 10

Num

bers

0

2

4

6

8

10

12

qc middot 120578

z

m

mlowast

Figure 9 Results under different charging powers

Num

bers

12

13

14

15

16

T

m120601w

Tl (Tl + Tu)2 Tu

Figure 10 Results of anchoring different charging periods

119879 is larger each obtained subtour contributed less conflictratio of itsMC thus more tours can be assigned to the sameMC Consequently when anchoring intermediate values of119879 the number of obtained tours is relatively large and thesetours are hard to be assigned to the same MC leading tolarger results of 119898 As for Φ

119908 it monotonically increases as

119879 increases According to (10) 119879119908119879 of each tour decreasesas 119879 increases Furthermore as 119879 increases the influence ofthe decrease in119879119908119879 is larger than that of the increase in tour

n100 200 300 400 500

Num

bers

0

5

10

15

20

25

m

mlowast

m998400

Figure 11 Comparison between GCHA and NMV varying networksize

numbersTherefore anchoring a larger value of 119879 can reducethe total energy consumption ofMCs

In conclusion the results validate the effectiveness ofanchoring 119879 = 119879119906 in deriving the TS condition

623 Varying Network Size We compare the GCHA schemewith the NMV scheme proposed in [24] NMV is an on-demand charging scheme In [24] when the residual lifetimeof a sensor node falls in the predefined critical lifetime inter-val the sensor nodewill send an energy-recharging request toS for its energy replenishment OnceS receives the requestsfrom sensor nodes it performs the NMV scheme to dispatcha number of MCs to charge the requested sensor nodesTo use minimum number of MCs to meet the chargingdemand the NMV scheme firstly constructs a minimum-cost spanning tree of the network Then it decomposes thetree into minimum number of subtrees based on energyconstraints of eachMC each of the subtrees can be assignedto one MC Then it derives closed charging tours from thesubtrees and the number of the tours equals the number ofMCs required Since the NMV scheme is designed for low-power WRSNs we let the sensor nodes generate data at arandom rate within [5 30] kbps and range 119899 from 100 to 500With each size of 119899 we randomly generate 10 WRSNs forevaluation The results are shown in Figure 11 and Table 1 inwhich1198981015840 is the results achieved by the NMV scheme

Figure 11 shows that 119898 is very close to 119898lowast and muchsmaller than 1198981015840 On average the ratio of 1198981198981015840 is less than05 There are two main reasons that the NMV scheme usesmuch more MCs than the GCHA scheme (a) The NMVscheme is an on-demand charging scheme it happens thata large number of sensor nodes are running out of energy


Table 1 Approximation ratios of varying network size

119899 = 100

119898lowast 1 1 1 1 1 2 2 2 2 2

120572 = 1067119898 1 1 1 2 2 2 2 2 2 2119901sum 4445 4507 4617 4817 4888 5055 5233 5311 5340 5532

119902119888sdot 120578 sdot 119898

lowast 5 5 5 5 5 10 10 10 10 10

119899 = 200

119898lowast 2 2 2 3 3 3 3 3 3 3

120572 = 1074119898 2 3 3 3 3 3 3 3 3 3119901sum 9397 9605 9991 10122 10218 10362 10399 10555 11055 11073

119902119888sdot 120578 sdot 119898

lowast 10 10 10 15 15 15 15 15 15 15

119899 = 300

119898lowast 3 4 4 4 4 4 4 4 4 4

120572 = 1026119898 4 4 4 4 4 4 4 4 4 4119901sum 14996 15107 15158 15272 15508 15528 15600 15617 15750 15974

119902119888sdot 120578 sdot 119898

lowast 15 20 20 20 20 20 20 20 20 20

119899 = 400

119898lowast 4 5 5 5 5 5 5 5 5 5

120572 = 1020119898 5 5 5 5 5 5 5 5 5 5119901sum 19684 20223 20723 20853 21381 21461 21780 21789 22054 22153

119902119888sdot 120578 sdot 119898

lowast 20 25 25 25 25 25 25 25 25 25

119899 = 500

119898lowast 6 6 6 6 6 6 6 6 6 6

120572 = 1017119898 6 6 6 6 6 6 6 6 6 7119901sum 25765 26522 26810 26810 26880 27043 27044 27196 27432 29071

119902119888sdot 120578 sdot 119898

lowast 30 30 30 30 30 30 30 30 30 30

thus manyMCs are required for charging in these situationsWhile MCs applying the GCHA scheme charge the sensornodes periodically thus such situations never occur (b)Due to aforementioned reason many MCs applying theNMV scheme are idle in most cases causing great waste ofresources while in the GCHA scheme charging ability ofeach MC is fully utilized by assigning to it as many toursas possible

To analyze when does the GCHA scheme use moreMCsthan the lower bound we list the detailed results in Table 1The inconsistent situations are bold Although in most cases119898 = 119898

lowast in situations where 119902119888sdot 120578 sdot 119898

lowast is very close to119901sum 119898 minus 119898

lowast= 1 This is because the lower bound 119898lowast

only relates to the total energy consumption rate of a WRSNand the charging power of each MC In fact the MCs alsospend energy and time in moving Therefore if the totalcharging power of MCs is not significantly larger than thetotal energy consumption rate of the WRSN 119898lowast may not beachievable even though according to the results the averageapproximation ratio of our results is below 11 compared withthe lower bound119898lowast

7 Conclusion and Future Work

In this paper we exploit the Minimum Mobile ChargersProblem (MMCP) in Wireless Rechargeable Sensor NetworksWe divide the problem into two NP-hard subproblemsthe Tour Construction Problem (TCP) and the Tour Assign-ment Problem (TAP) We propose the Greedily ConstructHeuristically Assign (GCHA) scheme to solve the problemsThe GCHA scheme firstly constructs the shortest Hamiltoncycle through all nodes in a WRSN Then based on theTour Schedulable (TS) condition it greedily splits the cycle

into schedulable subtours Subsequently these subtours areheuristically assigned to minimum number of MCs so thatthe MCs can apply the Greedy Charging Scheme (GCS) tocharge along their assigned toursThe computation complex-ity of the GCHA scheme is 119874(11989922) which can be furtherreduced by applying a simpler TSP solution Simulationsare conducted to evaluate the performance of our solutionNumerical results show that the GCHA scheme only uses lessthan half of theMCs that are required by the NMV schemeIn most scenarios our method generates optimum solutionsforMMCP Comparedwith the Lower Bound119898lowast ourmethodachieved an approximation ratio less than 11

Future work includes the following aspects (a) We willdevelop a more rigorous lower bound of MMCP takingmovement of MCs into consideration Consequently theapproximation ratio of our method will be theoreticallyanalyzed (b) The energy efficiency ofMCs can be improvedby carefully merging traveling paths of some tours (c) Loadbalance betweenMCs will be studied

Notations

120575119894 Conflict ratio of119872

119894

119889119894119895 Distance between119873

119894and119873

119895

119863119894 Total distance of 119877

119894

119890119894119895 The path between119873

119894and119873

119895

119864119872 Battery capacity of eachMC

119864119878 Battery capacity of each sensor node

119866119896 Element period of119872

119896

M The set of mobile chargers in the WRSNM = 1198721 119872119898

119898lowast A lower bound of number ofMCs

required in a WRSN


N The set of sensor nodes in the WRSNN = 1198731 1198732 119873119899

119873119895

119894 The 119894th sensor node in tour 119877

119895

119901119894 Energy consumption rate of119873

119894

119901mx For tour 119877119896

119901mx = max1le119894le119903119896119901119894 sdot (119902119888 sdot 120578 minus 119901119894)119901sum For tour 119877119896 119901sum = sum

119903119896

119894=1 119901119894119902119888 Energy consumption rate of eachMC

in charging119902119898 Energy consumption rate of eachMC

in movingR Set of tours that fully cover the WRSN

R = 1198771 1198772 119877119911R119896 Tours that are assigned to119872

119896

R119896= 1198771 1198772 119877119911119896

119877119894 A charging tour denoted by119877119894= ⟨1198730 1198731 119873119903119894⟩ or

119877119894= (119879119897

119894 119879119906

119894 120596119894) or 119877

119894= (119879119894 119879119908

119894) under

different situationsS The service station of a WRSN also

denoted by1198730120591119895

119894 Charging time of anMC on119873119895

119894

119905119895

119894 Arrival time of anMC at119873119895

119894

119879119894 Charging period of 119877

119894 119879119897119894le 119879119894le 119879119906

119894

119879119908

119894 Working time of 119877

119894within a charging

period 119879119908119894= 120596119894(119879119894)

V Moving speed of eachMC

Conflict of Interests

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

Acknowledgments

The authors sincerely thank the anonymous reviewers fortheir valuable comments that led to the present improvedversion of the original paper This research work is partiallysupported by the NSF of China under Grant no 60973122the 973 Program in China and the Scientific ResearchFoundation of Graduate School of Southeast University

References

[1] G Anastasi M Conti M Di Francesco and A PassarellaldquoEnergy conservation in wireless sensor networks a surveyrdquoAdHoc Networks vol 7 no 3 pp 537ndash568 2009

[2] A Howard M J Matarıc and G S Sukhatme ldquoAn incrementalself-deployment algorithm for mobile sensor networksrdquo Auto-nomous Robots vol 13 no 2 pp 113ndash126 2002

[3] S Sudevalayam and P Kulkarni ldquoEnergy harvesting sensornodes survey and implicationsrdquo IEEE Communications Surveysand Tutorials vol 13 no 3 pp 443ndash461 2011

[4] L Xie Y Shi Y T Hou and A Lou ldquoWireless power transferand applications to sensor networksrdquo IEEE Wireless Communi-cations vol 20 no 4 pp 140ndash145 2013

[5] A Kurs A Karalis RMoffatt J D Joannopoulos P Fisher andM Soljacic ldquoWireless power transfer via strongly coupled mag-netic resonancesrdquo Science vol 317 no 5834 pp 83ndash86 2007

[6] M Erol-Kantarci and H T Mouftah ldquoSuresense sustainablewireless rechargeable sensor networks for the smart gridrdquo IEEEWireless Communications vol 19 no 3 pp 30ndash36 2012

[7] B Griffin and C Detweiler ldquoResonant wireless power transferto ground sensors from a UAVrdquo in Proceedings of the IEEEInternational Conference on Robotics and Automation (ICRArsquo12) pp 2660ndash2665 May 2012

[8] H P Dai X B Wu L J Xu G Chen and S Lin ldquoUsing mini-mum mobile chargers to keep large-scale wireless rechargeablesensor networks running foreverrdquo in Proceedings of the IEEE2013 22nd International Conference on Computer Communica-tion and Networks (ICCCN rsquo13) pp 1ndash7 August 2013

[9] V Nagarajan and R Ravi ldquoApproximation algorithms fordistance constrained vehicle routing problemsrdquo Networks vol59 no 2 pp 209ndash214 2012

[10] L Xie Y Shi Y T Hou and H D Sherali ldquoMaking sensornetworks immortal an energy-renewal approach with wirelesspower transferrdquo IEEEACMTransactions onNetworking vol 20no 6 pp 1748ndash1761 2012

[11] Y Peng Z Li and W Zhang ldquoProlonging sensor networklifetime through wireless chargingrdquo in Proceedings of the IEEE31st Real-Time Systems Symposium (RTSS 10) pp 129ndash139 SanDiego Calif USA November-December 2010

[12] C Hu Y Wang and L Zhou ldquoMake imbalance useful anenergy-efficient charging scheme in wireless sensor networksrdquoin Proceedings of the International Conference on PervasiveComputing and Application (ICPCA rsquo13) pp 160ndash171 December2013

[13] L He Y Gu J Pan and T Zhu ldquoOn-demand charging in wire-less sensor networks theories and applicationsrdquo in Proceedingsof the 10th IEEE International Conference onMobile Ad-Hoc andSensor Systems (MASS rsquo13 pp 28ndash36 IEEE Hangzhou ChinaOctober 2013

[14] CWang Y Y Yang and J Li ldquoStochastic mobile energy replen-ishment and adaptive sensor activation for perpetual wirelessrechargeable sensor networksrdquo in Proceedings of the IEEEWireless Communications and Networking Conference (WCNCrsquo13) pp 974ndash979 April 2013

[15] M Zhao J Li and Y Y Yang ldquoJoint mobile energy replen-ishment and data gathering in wireless rechargeable sensornetworksrdquo in Proceedings of the 23rd International TeletrafficCongress (ITC 11) pp 238ndash245 2011

[16] S T Guo C Wang and Y Y Yang ldquoMobile data gatheringwith wireless energy replenishment in rechargeable sensor net-worksrdquo in Proceedings of the 32nd IEEE Conference on ComputerCommunications (IEEE INFOCOM rsquo13) pp 1932ndash1940 April2013

[17] L Shi J H Han D Han X Ding and Z Wei ldquoThe dynamicrouting algorithm for renewable wireless sensor networks withwireless power transferrdquo Computer Networks vol 74 pp 34ndash522014

[18] A Kurs R Moffatt and M Soljacic ldquoSimultaneous mid-rangepower transfer to multiple devicesrdquo Applied Physics Letters vol96 no 4 Article ID 044102 2010

[19] B Tong Z Li G L Wang andW Zhang ldquoHow wireless powercharging technology affects sensor network deployment androutingrdquo in Proceedings of the 30th IEEE International Confer-ence on Distributed Computing Systems (ICDCS rsquo10) pp 438ndash447 IEEE Genoa Italy June 2010


[20] L Xie Y Shi Y T Hou W Lou H D Sherali and S F MidkiffldquoOn renewable sensor networks with wireless energy transferthe multi-node caserdquo in Proceedings of the 9th Annual IEEECommunications Society Conference on SensorMesh andAdHocCommunications and Networks (SECON rsquo12) pp 10ndash18 June2012

[21] K Li H Luan and C-C Shen ldquoQi-ferry energy-constrainedwireless charging in wireless sensor networksrdquo in Proceedings ofthe IEEE Wireless Communications and Networking Conference(WCNC rsquo12) pp 2515ndash2520 April 2012

[22] L K Fu P Cheng Y Gu J Chen and T He ldquoMinimizingcharging delay in wireless rechargeable sensor networksrdquo inProceedings of the 32nd IEEE Conference on Computer Commu-nications (IEEE INFOCOM rsquo13) pp 2922ndash2930 April 2013

[23] S Zhang JWu and S L Lu ldquoCollaborativemobile charging forsensor networksrdquo in Proceedings of the IEEE 9th InternationalConference onMobile Adhoc and Sensor Systems (MASS rsquo12) pp84ndash92 2012

[24] W F Liang W Z Xu X J Ren X Jia and X Lin ldquoMaintainingsensor networks perpetually via wireless recharging mobilevehiclesrdquo in Proceedings of the IEEE 39th Conference on LocalComputer Networks (LCN 14) pp 270ndash278 Edmonton CanadaSeptember 2014

[25] CHu andYWang ldquoMinimizing the number ofmobile chargersin a large-scale wireless rechargeable sensor networkrdquo in Pro-ceedings of the IEEE Wireless Communications and NetworkingConference (WCNC 15) New Orleans La USA March 2015

[26] C Hu and Y Wang ldquoSchedulability decision of charging mis-sions in wireless rechargeable sensor networksrdquo in Proceedingsof the 11th Annual IEEE International Conference on SensingCommunication and Networking (SECON rsquo14) pp 450ndash458Singapore June 2014

[27] K Helsgaun ldquoAn effective implementation of the Lin-Kerni-ghan traveling salesman heuristicrdquo European Journal of Oper-ational Research vol 126 no 1 pp 106ndash130 2000

[28] B Karp and H T Kung ldquoGPSR greedy perimeter statelessrouting for wireless networksrdquo in Proceedings of the 6th AnnualInternational Conference on Mobile Computing and Networking(MobiCom rsquo00) pp 243ndash254 2000

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



N1

N2

N3

N4

N5

N6

N7

N8

N9

N10

N11

N12

S

R1

R2

R3

R4

30∘

r

Figure 1 A 12-node ring-type WRSN

the DVRP [9] is to find a minimum cardinality set of toursoriginating from a depot that covers all the vertices in119881 Eachtour is required to have length at most119863 SinceMCs are alsoenergy-constrained researchers reduce MMCP to DVRP sothat solutions for DVRP can be applied to address MMCP[8] We however argue that DVRP is only a subproblem ofMMCP This is because an MC can conduct charging onseveral tours

To demonstrate this viewpoint consider a demo WRSNshown in Figure 1 in which 12 sensor nodes are uniformlydeployed along a circle with radius 119903 Suppose that in thisWRSN due to energy limitation of theMCs anyMC is notable to charge all the sensor nodes in one tour Instead atmost3 sensor nodes can be charged within one tour Then at least4 tours are needed to cover the wholeWRSN as shown in thefigure Therefore in this scenario the answer to the variedDVRP is 4 Suppose the time of charging along each tour isat most 119905 while the lifetime of any sensor node is at least 4119905then only one MC is required the MC charges along tour119877119894during time [(4119896 + 119894 minus 1) sdot 119905 (4119896 + 119894) sdot 119905] (1 le 119894 le 4

119896 = 0 1 2 ) In this way energy of all the sensor nodes canbe timely replenished In summary in this case the answer tothe varied DVRP is 4 while the answer to the MMCP is 1

Therefore in this paper we divide the MMCP into twosubproblems that is a Tour Construction Problem (TCP) anda Tour Assignment Problem (TAP) The TCP is to constructminimumnumber of schedulable tours to cover aWRSNTheTAP is to assign the obtained tours to minimum number ofMCs so that all sensor nodes can be charged before theirbatteries are used upThese two problems are closely coupledWe propose a two-step solution namedGCHA (Greedily Con-struct Heuristically Assign) First we construct the shortestHamilton cycle through S and all the sensor nodes in aWRSN Based on a decision condition we greedily split thecycle into several schedulable subtours Second we prove thattheTAP isNP-hard andpropose a heuristic algorithm to solveit Following these two steps we finally solve the MMCP

The objectives of this paper are to decide the minimumnumber of MCs needed to keep all the sensor nodes in

a WRSN working continuously and to construct a validcharging schedule accordinglyThe contributions of the paperare the following

(i) We illustrate the essential difference between theMMCPand theDVRPanddivide theMMCP into twoNP-hard subproblems

(ii) We derive an efficient decision condition of whether asequence of sensor nodes can be charged by oneMCin one tour The decision condition can be used toconstruct a set of schedulable tours to cover aWRSNCharging schedule of charging along each tour canbe easily obtained by applying the Greedy ChargingScheme (GCS first proposed in [10])

(iii) We propose a heuristic algorithm to assign a set ofschedulable tours to minimum number ofMCs Thedeveloped heuristic rules can minimize the conflictratio of anMC so that it can charge along maximumnumber of tours

(iv) We develop a lower bound for MMCP The lowerbound is used to evaluate the performance of oursolution

The remainder of the paper is organized as followsSection 2 introduces latest related work Section 3 describesmodel assumptions and the problem under study The solu-tions of the TCP and the TAP are detailed in Sections 4 and5 Section 6 evaluates performance of the GCHA scheme andSection 7 concludes the paper

2 Related Work

Most existing works employ only one MC for chargingPeng et al [11] presented the design and implementation ofa wireless rechargeable sensor system and especially studiedthe charging planning problem Xie et al [10] constructed arenewable energy cycle of aWRSNby solving an optimizationproblem with the objective of minimizing the overall energyconsumption Hu et al [12] fully considered the influence ofimbalanced energy consumption in WRSNs and proposedan on-demand charging scheme and a method of deployingS He et al [13] laid a theoretical foundation for the on-demand mobile charging problem where individual sensornodes request charging from the mobile charger when theirenergy runs low Wang et al [14] provided stochastic analysisof wireless energy replenishment and proposed battery-awareenergy replenishment strategies with corresponding sensoractivation schemes In addition literatures [15ndash17] assumethat the MC can function as a mobile sink Zhao et al [15]designed an adaptive solution that jointly selects the sensorsto be charged and finds the optimal data gathering schemesuch that network utility can be maximized while main-taining perpetual operations of the network Guo et al [16]studied the problem of joint wireless energy replenishmentand mobile data gathering for WRSNs The target was tomaximize the overall network utility in terms of data gatheredby theMC Shi et al [17] aimed to jointly optimize a dynamicmultihop data routing a traveling path and a charging










119894











119896= 1198771 1198772 119877119911119896 if119872119896 can







⋃

119894=1119877119894= NcupS (1)



119894


119894 119879119906

119894] Assume that




119894 denoted by 119879119908

119894(119879119908119894le 119879119894) Consequently



119894 However if



119894 120596119894) where 120596

119894is a working



no ambiguity occurs


Timeline

Tour R1Tour R2

Tw1 = 1 Tw

2 = 15

T2 = 5

T1 = 75



119896= 1198771 1198772 119877119911119896 are assigned to

119872119896(1 le 119911


119896



119896as

120575119896=sum119911119896

119894=1 119879119908

119894

119866119896

119866119896= GCD (1198791 1198792 119879119911119896) (2)


119896indicates the


119896





119908


lated as follows


119894(119901119894 119864119878) | 1 le 119894 le 119899 and their



mission




119877119894cap119877119895= 119877

119894 119877119895isin R

119873119894= 119873119895 119873119894 119873119895isin 119877119896 119877119896isin R

(3)









119894at time 119905


119894for time


119905119894=

119894minus1sum

119895=0120591119895+

119894

sum

119895=1

119889119895minus1119895

V 1 le 119894 le 119903 (4)

In (4) 119889119894119895


119895and 1205910 ge 0 is


119879 =

119903

sum

119895=0120591119895+119863

V (5)




119894sdot 119879 1 le 119894 le 119903 (6)


119864119878minus119901119894sdot (119879 minus 120591

119894) ge 0 1 le 119894 le 119903 (7)


119863 sdot 119902119898+

119903

sum

119895=1sdot 119902119888sdot 120591119895le 119864119872 (8)




120591119894=119901119894sdot 119879

119902119888sdot 120578 1 le 119894 le 119903 (9)




119879119908=

119899

sum

119894=1120591119894+119863

V=119901sum119902119888sdot 120578sdot 119879 +

119863

V (10)


119879 = 1205910 +sum119903

119894=1 119901119894 sdot 119879

119902119888sdot 120578

+119863

Vgesum119903

119894=1 119901119894 sdot 119879

119902119888sdot 120578

+119863

V (11)

or equivalently

119879 ge119863 sdot 119902119888sdot 120578


119901sum =119903

sum

119894=1119901119894 (12)





119894)

119902119888sdot 120578

sdot 119879 1 le 119894 le 119903 (13)


119879 le119864119904sdot 119902119888sdot 120578

119901mx 119901mx = max


119894) (14)



119863 sdot 119902119888sdot 120578


le119864119904sdot 119902119888sdot 120578

119901mx(15)


119864119872ge 119863 sdot 119902

119898+119901sum120578sdot 119879 (16)





0 1198731 119873

119903⟩ if (a) 119902





















condition is met


N1

N2

N3

N4

N5

S(N0)

N1

N2

N3

N4

N5

S(N0)

N1

N2

N3

N4

N5

S(N0)

N1

N2

N3

N4

N5

S(N0)

N1

N2

N3

N4

N5

S(N0)

N1

N2

N3

N4

N5

S(N0)


Require119877 = ⟨1198730 1198731 119873119903⟩ 119864119878 119902119898 119902119888 V 119864119872 120578




(3) 119901sum =119903

sum

119894=1

119901119894

(4) 119901mx = max1le119894le119903

119901119894sdot (119902119888sdot 120578 minus 119901

119894)


119863 sdot 119902119888sdot 120578


(9) 119879119906 =119864119904sdot 119902119888sdot 120578

119901mx

(10) 120596(119879) =119901sum

119902119888sdot 120578sdot 119879 +

119863

V







RequireN = 119873

119894| 1 le 119894 le 119899

EnsureR = 119877

119894| 1 le 119894 le 119911


119895into 119877

119911



(10) pop119873119895from 119877

119911





119894of tour 119877

119894


119894

of 119877119894in each period 119879119908

119894= 120596119894(119879119894) = (119901

119894



(119879119897

119894 119879119906


119888 119902119898 V 119864119872)




119908

119894) | 1 le 119894 le 119903 they


119894le 119879


119894=1 119879119908







119895 119896 sdot 119879 +sum

119894

119895=1 119879119908

119895] (119896 = 0 1 2 )

Since sum119903119894=1 119879119908






119908

119894) | 1 le 119894 le 119911 and

supposesum119911119894=1 119879119908





119894119879 (1 le





119894) | 1 le 119894 le

119903119895 1 le 119895 le 119896 where 119877119895


119895


only if sum119896119895=1(sum119903119895

119894=1 119879119895

119894119886119895) le 119866


119895=1(sum119903119895

119894=1 119879119895



119875

1198861sdot

1199031

sum

119894=11198791119894+119875

1198862sdot

1199032

sum

119894=11198792119894+ sdot sdot sdot +

119875

119886119896

sdot

119903119896

sum

119894=1119879119896

119894gt 119866 sdot 119875 (17)



119894=1 119879119895











119908

119894) | 1 le 119894 le


MC if sum119903119894=1 119879119908

119894le 119866


119908

119894) | 1 le 119894 le 119903







119894= (119903 sdot 119879 1198792) |





119894=1(119879119908




1 119879119906

1 1205961) and 1198772 =(119879119897

2 119879119906


2 le 119896sdot119879119906

1 le119879119906


119906

1 and 1198792 = 119896 sdot 1198791199061


119894


119894) = 119886119894sdot 119879119894+ 119887119894


1205751 = 1198861 + 119894 sdot 1198862 +1198871 + 11988721198791

1198792 = 119894 sdot 1198791

1205752 = 1198861 + (119894 + 1) sdot 1198862 +1198871 + 11988721198791

1198792 = (119894 + 1) sdot 1198791

1205753 =1198861 sdot 1198791 + 1198862 sdot 1198792

119866+1198871 + 1198872119866

119894 sdot 1198791 lt 1198792 lt (119894 + 1) sdot 1198791

(18)


1205753 =1198861 sdot 1198791 + 1198862 sdot 1198792

119866+1198871 + 1198872119866

ge 21198861 + 2119894 sdot 1198862 +2 (1198871 + 1198872)1198791

gt 1198861 + (119894 + 1) sdot 1198862 +1198871 + 11988721198791

= 1205752

(19)



119906


2 le119896 sdot 119879119906

1 le 119879119906



119906

1 and1198792 = 119896sdot119879119906





119908) of theMCs

Φ119908=

119898

sum

119894=1

119911119894

sum

119895=1

119879119908

119895

119879119895

=

119911

sum

119894=1

119879119908

119894

119879119894

(20)






119894can be


119896of119872119896is set to 119879119906




119896

(lines (6) (13)) if



119896(lines


119896of 119872119896is still no





RequireR = 119877

119894| 1 le 119894 le 119911

EnsureM = 119872


(1) 119898 = 0(2) sort 119877

119894 by 119879119906


(3) for 119894 = 1 to 119911 do(4) if 119877


(5) 119898+ = 1(6) assign 119877

119894to119872119898

(7) 119866119898= 119879119894= 119879119906

119894 120575

119898=120596119894(119879119894)

119866119898

(8) for 119895 = 119894 + 1 to 119911 do(9) if 119877



119898+ (120596119895(119896 sdot 119879

119906

119894)119866119898) le 1 then

(12) 119879119895= 119896 sdot 119879

119906

119894 120575

119898+ =

120596119895(119879119895)

119866119898

(13) assign 119877119895to119872119898




119909tours

R119909= 119877119894= (119886119894sdot 119866119909 119879119908


119909(1 le 119894 le 119911

119909le 119899


119877119894during time [119896 sdot 119886

119894sdot 119866119909+ sum119894minus1119895=1 119879119908

119895 119896 sdot 119886119894sdot 119866119909+ sum119894

119895=1 119879119908

119895]


119909charges 119877





119895

in 119877119894for time (119901


119895fully charged



6 Numerical Results



119878= 10 kJ 119902

119898= 5 Jm V = 5ms 119864

119872= 40 kJ and 120578 =






RequireR119896= 119877119894| 1 le 119894 le 119911

119896

Ensure(119877119894 ⟨119905119894

119895⟩ ⟨120591119894

119895⟩) | 1 le 119894 le 119911

119896 1 le 119895 le 119903

119894

(1) 12059110 = 0(2) for 119894 = 2 to 119911

119896do

(3) 120591119894

0 = 120591119894minus10 + 119879

119908


119896do

(6) 119905119894

0 = 0

(7) 120591119894

119895=119879119894sdot 119901119895

119902119888sdot 120578

1 le 119895 le 119903119894

(8) 119905119894

119895= 119905119894

119895minus1 + 120591119894

119895minus1 +119889119895minus1119895

V 0 le 119895 le 119903

119894


119894= 0 then


(16) for 119895 = 1 to 119903119894do


119894


119894



119896

0 500 10000

500

1000

X (m)

Y (m

)

S ampB



0 500 10000

500

1000

X (m)

Y (m

)

S ampB




0 500 10000

500

1000

X (m)

Y (m

)



S ampB


1 2 3 4 5 6 7 8 9 100

05

1

15

2

25

3

35

4

Tour ID

Ener

gy co

nsum

ptio

n of

each

tour

(J)

times104





0 1 2 3 40

02

04

06

08

1

12

14

16

18

2

MC ID

R (s

)

times104

120591rG

Tw





119888= 4W and 119902







2 3 4 5 6 7 8 9 10

Num

bers

0

2

4

6

8

10

12

qc middot 120578

z

m

mlowast


Num

bers

12

13

14

15

16

T

m120601w

Tl (Tl + Tu)2 Tu





n100 200 300 400 500

Num

bers

0

5

10

15

20

25

m

mlowast

m998400








119899 = 100

119898lowast 1 1 1 1 1 2 2 2 2 2

120572 = 1067119898 1 1 1 2 2 2 2 2 2 2119901sum 4445 4507 4617 4817 4888 5055 5233 5311 5340 5532

119902119888sdot 120578 sdot 119898

lowast 5 5 5 5 5 10 10 10 10 10

119899 = 200

119898lowast 2 2 2 3 3 3 3 3 3 3

120572 = 1074119898 2 3 3 3 3 3 3 3 3 3119901sum 9397 9605 9991 10122 10218 10362 10399 10555 11055 11073

119902119888sdot 120578 sdot 119898

lowast 10 10 10 15 15 15 15 15 15 15

119899 = 300

119898lowast 3 4 4 4 4 4 4 4 4 4

120572 = 1026119898 4 4 4 4 4 4 4 4 4 4119901sum 14996 15107 15158 15272 15508 15528 15600 15617 15750 15974

119902119888sdot 120578 sdot 119898

lowast 15 20 20 20 20 20 20 20 20 20

119899 = 400

119898lowast 4 5 5 5 5 5 5 5 5 5

120572 = 1020119898 5 5 5 5 5 5 5 5 5 5119901sum 19684 20223 20723 20853 21381 21461 21780 21789 22054 22153

119902119888sdot 120578 sdot 119898

lowast 20 25 25 25 25 25 25 25 25 25

119899 = 500

119898lowast 6 6 6 6 6 6 6 6 6 6

120572 = 1017119898 6 6 6 6 6 6 6 6 6 7119901sum 25765 26522 26810 26810 26880 27043 27044 27196 27432 29071

119902119888sdot 120578 sdot 119898

lowast 30 30 30 30 30 30 30 30 30 30











Notations


119894


119894and119873

119895


119894

119890119894119895 The path between119873

119894and119873

119895




119896



required in a WRSN



119873119895


119895


119894

119901mx For tour 119877119896


119903119896





119896

R119896= 1198771 1198772 119877119911119896


119877119894= (119879119897

119894 119879119906

119894 120596119894) or 119877

119894= (119879119894 119879119908

119894) under


denoted by1198730120591119895


119894

119905119895


119894


119894 119879119897119894le 119879119894le 119879119906

119894

119879119908



period 119879119908119894= 120596119894(119879119894)




Acknowledgments


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


















119894











119896= 1198771 1198772 119877119911119896 if119872119896 can







⋃

119894=1119877119894= NcupS (1)



119894


119894 119879119906

119894] Assume that




119894 denoted by 119879119908

119894(119879119908119894le 119879119894) Consequently



119894 However if



119894 120596119894) where 120596

119894is a working



no ambiguity occurs


Timeline

Tour R1Tour R2

Tw1 = 1 Tw

2 = 15

T2 = 5

T1 = 75



119896= 1198771 1198772 119877119911119896 are assigned to

119872119896(1 le 119911


119896



119896as

120575119896=sum119911119896

119894=1 119879119908

119894

119866119896

119866119896= GCD (1198791 1198792 119879119911119896) (2)


119896indicates the


119896





119908


lated as follows


119894(119901119894 119864119878) | 1 le 119894 le 119899 and their



mission




119877119894cap119877119895= 119877

119894 119877119895isin R

119873119894= 119873119895 119873119894 119873119895isin 119877119896 119877119896isin R

(3)









119894at time 119905


119894for time


119905119894=

119894minus1sum

119895=0120591119895+

119894

sum

119895=1

119889119895minus1119895

V 1 le 119894 le 119903 (4)

In (4) 119889119894119895


119895and 1205910 ge 0 is


119879 =

119903

sum

119895=0120591119895+119863

V (5)




119894sdot 119879 1 le 119894 le 119903 (6)


119864119878minus119901119894sdot (119879 minus 120591

119894) ge 0 1 le 119894 le 119903 (7)


119863 sdot 119902119898+

119903

sum

119895=1sdot 119902119888sdot 120591119895le 119864119872 (8)




120591119894=119901119894sdot 119879

119902119888sdot 120578 1 le 119894 le 119903 (9)




119879119908=

119899

sum

119894=1120591119894+119863

V=119901sum119902119888sdot 120578sdot 119879 +

119863

V (10)


119879 = 1205910 +sum119903

119894=1 119901119894 sdot 119879

119902119888sdot 120578

+119863

Vgesum119903

119894=1 119901119894 sdot 119879

119902119888sdot 120578

+119863

V (11)

or equivalently

119879 ge119863 sdot 119902119888sdot 120578


119901sum =119903

sum

119894=1119901119894 (12)





119894)

119902119888sdot 120578

sdot 119879 1 le 119894 le 119903 (13)


119879 le119864119904sdot 119902119888sdot 120578

119901mx 119901mx = max


119894) (14)



119863 sdot 119902119888sdot 120578


le119864119904sdot 119902119888sdot 120578

119901mx(15)


119864119872ge 119863 sdot 119902

119898+119901sum120578sdot 119879 (16)





0 1198731 119873

119903⟩ if (a) 119902





















condition is met


N1

N2

N3

N4

N5

S(N0)

N1

N2

N3

N4

N5

S(N0)

N1

N2

N3

N4

N5

S(N0)

N1

N2

N3

N4

N5

S(N0)

N1

N2

N3

N4

N5

S(N0)

N1

N2

N3

N4

N5

S(N0)


Require119877 = ⟨1198730 1198731 119873119903⟩ 119864119878 119902119898 119902119888 V 119864119872 120578




(3) 119901sum =119903

sum

119894=1

119901119894

(4) 119901mx = max1le119894le119903

119901119894sdot (119902119888sdot 120578 minus 119901

119894)


119863 sdot 119902119888sdot 120578


(9) 119879119906 =119864119904sdot 119902119888sdot 120578

119901mx

(10) 120596(119879) =119901sum

119902119888sdot 120578sdot 119879 +

119863

V







RequireN = 119873

119894| 1 le 119894 le 119899

EnsureR = 119877

119894| 1 le 119894 le 119911


119895into 119877

119911



(10) pop119873119895from 119877

119911





119894of tour 119877

119894


119894

of 119877119894in each period 119879119908

119894= 120596119894(119879119894) = (119901

119894



(119879119897

119894 119879119906


119888 119902119898 V 119864119872)




119908

119894) | 1 le 119894 le 119903 they


119894le 119879


119894=1 119879119908







119895 119896 sdot 119879 +sum

119894

119895=1 119879119908

119895] (119896 = 0 1 2 )

Since sum119903119894=1 119879119908






119908

119894) | 1 le 119894 le 119911 and

supposesum119911119894=1 119879119908





119894119879 (1 le





119894) | 1 le 119894 le

119903119895 1 le 119895 le 119896 where 119877119895


119895


only if sum119896119895=1(sum119903119895

119894=1 119879119895

119894119886119895) le 119866


119895=1(sum119903119895

119894=1 119879119895



119875

1198861sdot

1199031

sum

119894=11198791119894+119875

1198862sdot

1199032

sum

119894=11198792119894+ sdot sdot sdot +

119875

119886119896

sdot

119903119896

sum

119894=1119879119896

119894gt 119866 sdot 119875 (17)



119894=1 119879119895











119908

119894) | 1 le 119894 le


MC if sum119903119894=1 119879119908

119894le 119866


119908

119894) | 1 le 119894 le 119903







119894= (119903 sdot 119879 1198792) |





119894=1(119879119908




1 119879119906

1 1205961) and 1198772 =(119879119897

2 119879119906


2 le 119896sdot119879119906

1 le119879119906


119906

1 and 1198792 = 119896 sdot 1198791199061


119894


119894) = 119886119894sdot 119879119894+ 119887119894


1205751 = 1198861 + 119894 sdot 1198862 +1198871 + 11988721198791

1198792 = 119894 sdot 1198791

1205752 = 1198861 + (119894 + 1) sdot 1198862 +1198871 + 11988721198791

1198792 = (119894 + 1) sdot 1198791

1205753 =1198861 sdot 1198791 + 1198862 sdot 1198792

119866+1198871 + 1198872119866

119894 sdot 1198791 lt 1198792 lt (119894 + 1) sdot 1198791

(18)


1205753 =1198861 sdot 1198791 + 1198862 sdot 1198792

119866+1198871 + 1198872119866

ge 21198861 + 2119894 sdot 1198862 +2 (1198871 + 1198872)1198791

gt 1198861 + (119894 + 1) sdot 1198862 +1198871 + 11988721198791

= 1205752

(19)



119906


2 le119896 sdot 119879119906

1 le 119879119906



119906

1 and1198792 = 119896sdot119879119906





119908) of theMCs

Φ119908=

119898

sum

119894=1

119911119894

sum

119895=1

119879119908

119895

119879119895

=

119911

sum

119894=1

119879119908

119894

119879119894

(20)






119894can be


119896of119872119896is set to 119879119906




119896

(lines (6) (13)) if



119896(lines


119896of 119872119896is still no





RequireR = 119877

119894| 1 le 119894 le 119911

EnsureM = 119872


(1) 119898 = 0(2) sort 119877

119894 by 119879119906


(3) for 119894 = 1 to 119911 do(4) if 119877


(5) 119898+ = 1(6) assign 119877

119894to119872119898

(7) 119866119898= 119879119894= 119879119906

119894 120575

119898=120596119894(119879119894)

119866119898

(8) for 119895 = 119894 + 1 to 119911 do(9) if 119877



119898+ (120596119895(119896 sdot 119879

119906

119894)119866119898) le 1 then

(12) 119879119895= 119896 sdot 119879

119906

119894 120575

119898+ =

120596119895(119879119895)

119866119898

(13) assign 119877119895to119872119898




119909tours

R119909= 119877119894= (119886119894sdot 119866119909 119879119908


119909(1 le 119894 le 119911

119909le 119899


119877119894during time [119896 sdot 119886

119894sdot 119866119909+ sum119894minus1119895=1 119879119908

119895 119896 sdot 119886119894sdot 119866119909+ sum119894

119895=1 119879119908

119895]


119909charges 119877





119895

in 119877119894for time (119901


119895fully charged



6 Numerical Results



119878= 10 kJ 119902

119898= 5 Jm V = 5ms 119864

119872= 40 kJ and 120578 =






RequireR119896= 119877119894| 1 le 119894 le 119911

119896

Ensure(119877119894 ⟨119905119894

119895⟩ ⟨120591119894

119895⟩) | 1 le 119894 le 119911

119896 1 le 119895 le 119903

119894

(1) 12059110 = 0(2) for 119894 = 2 to 119911

119896do

(3) 120591119894

0 = 120591119894minus10 + 119879

119908


119896do

(6) 119905119894

0 = 0

(7) 120591119894

119895=119879119894sdot 119901119895

119902119888sdot 120578

1 le 119895 le 119903119894

(8) 119905119894

119895= 119905119894

119895minus1 + 120591119894

119895minus1 +119889119895minus1119895

V 0 le 119895 le 119903

119894


119894= 0 then


(16) for 119895 = 1 to 119903119894do


119894


119894



119896

0 500 10000

500

1000

X (m)

Y (m

)

S ampB



0 500 10000

500

1000

X (m)

Y (m

)

S ampB




0 500 10000

500

1000

X (m)

Y (m

)



S ampB


1 2 3 4 5 6 7 8 9 100

05

1

15

2

25

3

35

4

Tour ID

Ener

gy co

nsum

ptio

n of

each

tour

(J)

times104





0 1 2 3 40

02

04

06

08

1

12

14

16

18

2

MC ID

R (s

)

times104

120591rG

Tw





119888= 4W and 119902







2 3 4 5 6 7 8 9 10

Num

bers

0

2

4

6

8

10

12

qc middot 120578

z

m

mlowast


Num

bers

12

13

14

15

16

T

m120601w

Tl (Tl + Tu)2 Tu





n100 200 300 400 500

Num

bers

0

5

10

15

20

25

m

mlowast

m998400








119899 = 100

119898lowast 1 1 1 1 1 2 2 2 2 2

120572 = 1067119898 1 1 1 2 2 2 2 2 2 2119901sum 4445 4507 4617 4817 4888 5055 5233 5311 5340 5532

119902119888sdot 120578 sdot 119898

lowast 5 5 5 5 5 10 10 10 10 10

119899 = 200

119898lowast 2 2 2 3 3 3 3 3 3 3

120572 = 1074119898 2 3 3 3 3 3 3 3 3 3119901sum 9397 9605 9991 10122 10218 10362 10399 10555 11055 11073

119902119888sdot 120578 sdot 119898

lowast 10 10 10 15 15 15 15 15 15 15

119899 = 300

119898lowast 3 4 4 4 4 4 4 4 4 4

120572 = 1026119898 4 4 4 4 4 4 4 4 4 4119901sum 14996 15107 15158 15272 15508 15528 15600 15617 15750 15974

119902119888sdot 120578 sdot 119898

lowast 15 20 20 20 20 20 20 20 20 20

119899 = 400

119898lowast 4 5 5 5 5 5 5 5 5 5

120572 = 1020119898 5 5 5 5 5 5 5 5 5 5119901sum 19684 20223 20723 20853 21381 21461 21780 21789 22054 22153

119902119888sdot 120578 sdot 119898

lowast 20 25 25 25 25 25 25 25 25 25

119899 = 500

119898lowast 6 6 6 6 6 6 6 6 6 6

120572 = 1017119898 6 6 6 6 6 6 6 6 6 7119901sum 25765 26522 26810 26810 26880 27043 27044 27196 27432 29071

119902119888sdot 120578 sdot 119898

lowast 30 30 30 30 30 30 30 30 30 30











Notations


119894


119894and119873

119895


119894

119890119894119895 The path between119873

119894and119873

119895




119896



required in a WRSN



119873119895


119895


119894

119901mx For tour 119877119896


119903119896





119896

R119896= 1198771 1198772 119877119911119896


119877119894= (119879119897

119894 119879119906

119894 120596119894) or 119877

119894= (119879119894 119879119908

119894) under


denoted by1198730120591119895


119894

119905119895


119894


119894 119879119897119894le 119879119894le 119879119906

119894

119879119908



period 119879119908119894= 120596119894(119879119894)




Acknowledgments


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Timeline

Tour R1Tour R2

Tw1 = 1 Tw

2 = 15

T2 = 5

T1 = 75



119896= 1198771 1198772 119877119911119896 are assigned to

119872119896(1 le 119911


119896



119896as

120575119896=sum119911119896

119894=1 119879119908

119894

119866119896

119866119896= GCD (1198791 1198792 119879119911119896) (2)


119896indicates the


119896





119908


lated as follows


119894(119901119894 119864119878) | 1 le 119894 le 119899 and their



mission




119877119894cap119877119895= 119877

119894 119877119895isin R

119873119894= 119873119895 119873119894 119873119895isin 119877119896 119877119896isin R

(3)









119894at time 119905


119894for time


119905119894=

119894minus1sum

119895=0120591119895+

119894

sum

119895=1

119889119895minus1119895

V 1 le 119894 le 119903 (4)

In (4) 119889119894119895


119895and 1205910 ge 0 is


119879 =

119903

sum

119895=0120591119895+119863

V (5)




119894sdot 119879 1 le 119894 le 119903 (6)


119864119878minus119901119894sdot (119879 minus 120591

119894) ge 0 1 le 119894 le 119903 (7)


119863 sdot 119902119898+

119903

sum

119895=1sdot 119902119888sdot 120591119895le 119864119872 (8)




120591119894=119901119894sdot 119879

119902119888sdot 120578 1 le 119894 le 119903 (9)




119879119908=

119899

sum

119894=1120591119894+119863

V=119901sum119902119888sdot 120578sdot 119879 +

119863

V (10)


119879 = 1205910 +sum119903

119894=1 119901119894 sdot 119879

119902119888sdot 120578

+119863

Vgesum119903

119894=1 119901119894 sdot 119879

119902119888sdot 120578

+119863

V (11)

or equivalently

119879 ge119863 sdot 119902119888sdot 120578


119901sum =119903

sum

119894=1119901119894 (12)





119894)

119902119888sdot 120578

sdot 119879 1 le 119894 le 119903 (13)


119879 le119864119904sdot 119902119888sdot 120578

119901mx 119901mx = max


119894) (14)



119863 sdot 119902119888sdot 120578


le119864119904sdot 119902119888sdot 120578

119901mx(15)


119864119872ge 119863 sdot 119902

119898+119901sum120578sdot 119879 (16)





0 1198731 119873

119903⟩ if (a) 119902





















condition is met


N1

N2

N3

N4

N5

S(N0)

N1

N2

N3

N4

N5

S(N0)

N1

N2

N3

N4

N5

S(N0)

N1

N2

N3

N4

N5

S(N0)

N1

N2

N3

N4

N5

S(N0)

N1

N2

N3

N4

N5

S(N0)


Require119877 = ⟨1198730 1198731 119873119903⟩ 119864119878 119902119898 119902119888 V 119864119872 120578




(3) 119901sum =119903

sum

119894=1

119901119894

(4) 119901mx = max1le119894le119903

119901119894sdot (119902119888sdot 120578 minus 119901

119894)


119863 sdot 119902119888sdot 120578


(9) 119879119906 =119864119904sdot 119902119888sdot 120578

119901mx

(10) 120596(119879) =119901sum

119902119888sdot 120578sdot 119879 +

119863

V







RequireN = 119873

119894| 1 le 119894 le 119899

EnsureR = 119877

119894| 1 le 119894 le 119911


119895into 119877

119911



(10) pop119873119895from 119877

119911





119894of tour 119877

119894


119894

of 119877119894in each period 119879119908

119894= 120596119894(119879119894) = (119901

119894



(119879119897

119894 119879119906


119888 119902119898 V 119864119872)




119908

119894) | 1 le 119894 le 119903 they


119894le 119879


119894=1 119879119908







119895 119896 sdot 119879 +sum

119894

119895=1 119879119908

119895] (119896 = 0 1 2 )

Since sum119903119894=1 119879119908






119908

119894) | 1 le 119894 le 119911 and

supposesum119911119894=1 119879119908





119894119879 (1 le





119894) | 1 le 119894 le

119903119895 1 le 119895 le 119896 where 119877119895


119895


only if sum119896119895=1(sum119903119895

119894=1 119879119895

119894119886119895) le 119866


119895=1(sum119903119895

119894=1 119879119895



119875

1198861sdot

1199031

sum

119894=11198791119894+119875

1198862sdot

1199032

sum

119894=11198792119894+ sdot sdot sdot +

119875

119886119896

sdot

119903119896

sum

119894=1119879119896

119894gt 119866 sdot 119875 (17)



119894=1 119879119895











119908

119894) | 1 le 119894 le


MC if sum119903119894=1 119879119908

119894le 119866


119908

119894) | 1 le 119894 le 119903







119894= (119903 sdot 119879 1198792) |





119894=1(119879119908




1 119879119906

1 1205961) and 1198772 =(119879119897

2 119879119906


2 le 119896sdot119879119906

1 le119879119906


119906

1 and 1198792 = 119896 sdot 1198791199061


119894


119894) = 119886119894sdot 119879119894+ 119887119894


1205751 = 1198861 + 119894 sdot 1198862 +1198871 + 11988721198791

1198792 = 119894 sdot 1198791

1205752 = 1198861 + (119894 + 1) sdot 1198862 +1198871 + 11988721198791

1198792 = (119894 + 1) sdot 1198791

1205753 =1198861 sdot 1198791 + 1198862 sdot 1198792

119866+1198871 + 1198872119866

119894 sdot 1198791 lt 1198792 lt (119894 + 1) sdot 1198791

(18)


1205753 =1198861 sdot 1198791 + 1198862 sdot 1198792

119866+1198871 + 1198872119866

ge 21198861 + 2119894 sdot 1198862 +2 (1198871 + 1198872)1198791

gt 1198861 + (119894 + 1) sdot 1198862 +1198871 + 11988721198791

= 1205752

(19)



119906


2 le119896 sdot 119879119906

1 le 119879119906



119906

1 and1198792 = 119896sdot119879119906





119908) of theMCs

Φ119908=

119898

sum

119894=1

119911119894

sum

119895=1

119879119908

119895

119879119895

=

119911

sum

119894=1

119879119908

119894

119879119894

(20)






119894can be


119896of119872119896is set to 119879119906




119896

(lines (6) (13)) if



119896(lines


119896of 119872119896is still no





RequireR = 119877

119894| 1 le 119894 le 119911

EnsureM = 119872


(1) 119898 = 0(2) sort 119877

119894 by 119879119906


(3) for 119894 = 1 to 119911 do(4) if 119877


(5) 119898+ = 1(6) assign 119877

119894to119872119898

(7) 119866119898= 119879119894= 119879119906

119894 120575

119898=120596119894(119879119894)

119866119898

(8) for 119895 = 119894 + 1 to 119911 do(9) if 119877



119898+ (120596119895(119896 sdot 119879

119906

119894)119866119898) le 1 then

(12) 119879119895= 119896 sdot 119879

119906

119894 120575

119898+ =

120596119895(119879119895)

119866119898

(13) assign 119877119895to119872119898




119909tours

R119909= 119877119894= (119886119894sdot 119866119909 119879119908


119909(1 le 119894 le 119911

119909le 119899


119877119894during time [119896 sdot 119886

119894sdot 119866119909+ sum119894minus1119895=1 119879119908

119895 119896 sdot 119886119894sdot 119866119909+ sum119894

119895=1 119879119908

119895]


119909charges 119877





119895

in 119877119894for time (119901


119895fully charged



6 Numerical Results



119878= 10 kJ 119902

119898= 5 Jm V = 5ms 119864

119872= 40 kJ and 120578 =






RequireR119896= 119877119894| 1 le 119894 le 119911

119896

Ensure(119877119894 ⟨119905119894

119895⟩ ⟨120591119894

119895⟩) | 1 le 119894 le 119911

119896 1 le 119895 le 119903

119894

(1) 12059110 = 0(2) for 119894 = 2 to 119911

119896do

(3) 120591119894

0 = 120591119894minus10 + 119879

119908


119896do

(6) 119905119894

0 = 0

(7) 120591119894

119895=119879119894sdot 119901119895

119902119888sdot 120578

1 le 119895 le 119903119894

(8) 119905119894

119895= 119905119894

119895minus1 + 120591119894

119895minus1 +119889119895minus1119895

V 0 le 119895 le 119903

119894


119894= 0 then


(16) for 119895 = 1 to 119903119894do


119894


119894



119896

0 500 10000

500

1000

X (m)

Y (m

)

S ampB



0 500 10000

500

1000

X (m)

Y (m

)

S ampB




0 500 10000

500

1000

X (m)

Y (m

)



S ampB


1 2 3 4 5 6 7 8 9 100

05

1

15

2

25

3

35

4

Tour ID

Ener

gy co

nsum

ptio

n of

each

tour

(J)

times104





0 1 2 3 40

02

04

06

08

1

12

14

16

18

2

MC ID

R (s

)

times104

120591rG

Tw





119888= 4W and 119902







2 3 4 5 6 7 8 9 10

Num

bers

0

2

4

6

8

10

12

qc middot 120578

z

m

mlowast


Num

bers

12

13

14

15

16

T

m120601w

Tl (Tl + Tu)2 Tu





n100 200 300 400 500

Num

bers

0

5

10

15

20

25

m

mlowast

m998400








119899 = 100

119898lowast 1 1 1 1 1 2 2 2 2 2

120572 = 1067119898 1 1 1 2 2 2 2 2 2 2119901sum 4445 4507 4617 4817 4888 5055 5233 5311 5340 5532

119902119888sdot 120578 sdot 119898

lowast 5 5 5 5 5 10 10 10 10 10

119899 = 200

119898lowast 2 2 2 3 3 3 3 3 3 3

120572 = 1074119898 2 3 3 3 3 3 3 3 3 3119901sum 9397 9605 9991 10122 10218 10362 10399 10555 11055 11073

119902119888sdot 120578 sdot 119898

lowast 10 10 10 15 15 15 15 15 15 15

119899 = 300

119898lowast 3 4 4 4 4 4 4 4 4 4

120572 = 1026119898 4 4 4 4 4 4 4 4 4 4119901sum 14996 15107 15158 15272 15508 15528 15600 15617 15750 15974

119902119888sdot 120578 sdot 119898

lowast 15 20 20 20 20 20 20 20 20 20

119899 = 400

119898lowast 4 5 5 5 5 5 5 5 5 5

120572 = 1020119898 5 5 5 5 5 5 5 5 5 5119901sum 19684 20223 20723 20853 21381 21461 21780 21789 22054 22153

119902119888sdot 120578 sdot 119898

lowast 20 25 25 25 25 25 25 25 25 25

119899 = 500

119898lowast 6 6 6 6 6 6 6 6 6 6

120572 = 1017119898 6 6 6 6 6 6 6 6 6 7119901sum 25765 26522 26810 26810 26880 27043 27044 27196 27432 29071

119902119888sdot 120578 sdot 119898

lowast 30 30 30 30 30 30 30 30 30 30











Notations


119894


119894and119873

119895


119894

119890119894119895 The path between119873

119894and119873

119895




119896



required in a WRSN



119873119895


119895


119894

119901mx For tour 119877119896


119903119896





119896

R119896= 1198771 1198772 119877119911119896


119877119894= (119879119897

119894 119879119906

119894 120596119894) or 119877

119894= (119879119894 119879119908

119894) under


denoted by1198730120591119895


119894

119905119895


119894


119894 119879119897119894le 119879119894le 119879119906

119894

119879119908



period 119879119908119894= 120596119894(119879119894)




Acknowledgments


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119879119908=

119899

sum

119894=1120591119894+119863

V=119901sum119902119888sdot 120578sdot 119879 +

119863

V (10)


119879 = 1205910 +sum119903

119894=1 119901119894 sdot 119879

119902119888sdot 120578

+119863

Vgesum119903

119894=1 119901119894 sdot 119879

119902119888sdot 120578

+119863

V (11)

or equivalently

119879 ge119863 sdot 119902119888sdot 120578


119901sum =119903

sum

119894=1119901119894 (12)





119894)

119902119888sdot 120578

sdot 119879 1 le 119894 le 119903 (13)


119879 le119864119904sdot 119902119888sdot 120578

119901mx 119901mx = max


119894) (14)



119863 sdot 119902119888sdot 120578


le119864119904sdot 119902119888sdot 120578

119901mx(15)


119864119872ge 119863 sdot 119902

119898+119901sum120578sdot 119879 (16)





0 1198731 119873

119903⟩ if (a) 119902





















condition is met


N1

N2

N3

N4

N5

S(N0)

N1

N2

N3

N4

N5

S(N0)

N1

N2

N3

N4

N5

S(N0)

N1

N2

N3

N4

N5

S(N0)

N1

N2

N3

N4

N5

S(N0)

N1

N2

N3

N4

N5

S(N0)


Require119877 = ⟨1198730 1198731 119873119903⟩ 119864119878 119902119898 119902119888 V 119864119872 120578




(3) 119901sum =119903

sum

119894=1

119901119894

(4) 119901mx = max1le119894le119903

119901119894sdot (119902119888sdot 120578 minus 119901

119894)


119863 sdot 119902119888sdot 120578


(9) 119879119906 =119864119904sdot 119902119888sdot 120578

119901mx

(10) 120596(119879) =119901sum

119902119888sdot 120578sdot 119879 +

119863

V







RequireN = 119873

119894| 1 le 119894 le 119899

EnsureR = 119877

119894| 1 le 119894 le 119911


119895into 119877

119911



(10) pop119873119895from 119877

119911





119894of tour 119877

119894


119894

of 119877119894in each period 119879119908

119894= 120596119894(119879119894) = (119901

119894



(119879119897

119894 119879119906


119888 119902119898 V 119864119872)




119908

119894) | 1 le 119894 le 119903 they


119894le 119879


119894=1 119879119908







119895 119896 sdot 119879 +sum

119894

119895=1 119879119908

119895] (119896 = 0 1 2 )

Since sum119903119894=1 119879119908






119908

119894) | 1 le 119894 le 119911 and

supposesum119911119894=1 119879119908





119894119879 (1 le





119894) | 1 le 119894 le

119903119895 1 le 119895 le 119896 where 119877119895


119895


only if sum119896119895=1(sum119903119895

119894=1 119879119895

119894119886119895) le 119866


119895=1(sum119903119895

119894=1 119879119895



119875

1198861sdot

1199031

sum

119894=11198791119894+119875

1198862sdot

1199032

sum

119894=11198792119894+ sdot sdot sdot +

119875

119886119896

sdot

119903119896

sum

119894=1119879119896

119894gt 119866 sdot 119875 (17)



119894=1 119879119895











119908

119894) | 1 le 119894 le


MC if sum119903119894=1 119879119908

119894le 119866


119908

119894) | 1 le 119894 le 119903







119894= (119903 sdot 119879 1198792) |





119894=1(119879119908




1 119879119906

1 1205961) and 1198772 =(119879119897

2 119879119906


2 le 119896sdot119879119906

1 le119879119906


119906

1 and 1198792 = 119896 sdot 1198791199061


119894


119894) = 119886119894sdot 119879119894+ 119887119894


1205751 = 1198861 + 119894 sdot 1198862 +1198871 + 11988721198791

1198792 = 119894 sdot 1198791

1205752 = 1198861 + (119894 + 1) sdot 1198862 +1198871 + 11988721198791

1198792 = (119894 + 1) sdot 1198791

1205753 =1198861 sdot 1198791 + 1198862 sdot 1198792

119866+1198871 + 1198872119866

119894 sdot 1198791 lt 1198792 lt (119894 + 1) sdot 1198791

(18)


1205753 =1198861 sdot 1198791 + 1198862 sdot 1198792

119866+1198871 + 1198872119866

ge 21198861 + 2119894 sdot 1198862 +2 (1198871 + 1198872)1198791

gt 1198861 + (119894 + 1) sdot 1198862 +1198871 + 11988721198791

= 1205752

(19)



119906


2 le119896 sdot 119879119906

1 le 119879119906



119906

1 and1198792 = 119896sdot119879119906





119908) of theMCs

Φ119908=

119898

sum

119894=1

119911119894

sum

119895=1

119879119908

119895

119879119895

=

119911

sum

119894=1

119879119908

119894

119879119894

(20)






119894can be


119896of119872119896is set to 119879119906




119896

(lines (6) (13)) if



119896(lines


119896of 119872119896is still no





RequireR = 119877

119894| 1 le 119894 le 119911

EnsureM = 119872


(1) 119898 = 0(2) sort 119877

119894 by 119879119906


(3) for 119894 = 1 to 119911 do(4) if 119877


(5) 119898+ = 1(6) assign 119877

119894to119872119898

(7) 119866119898= 119879119894= 119879119906

119894 120575

119898=120596119894(119879119894)

119866119898

(8) for 119895 = 119894 + 1 to 119911 do(9) if 119877



119898+ (120596119895(119896 sdot 119879

119906

119894)119866119898) le 1 then

(12) 119879119895= 119896 sdot 119879

119906

119894 120575

119898+ =

120596119895(119879119895)

119866119898

(13) assign 119877119895to119872119898




119909tours

R119909= 119877119894= (119886119894sdot 119866119909 119879119908


119909(1 le 119894 le 119911

119909le 119899


119877119894during time [119896 sdot 119886

119894sdot 119866119909+ sum119894minus1119895=1 119879119908

119895 119896 sdot 119886119894sdot 119866119909+ sum119894

119895=1 119879119908

119895]


119909charges 119877





119895

in 119877119894for time (119901


119895fully charged



6 Numerical Results



119878= 10 kJ 119902

119898= 5 Jm V = 5ms 119864

119872= 40 kJ and 120578 =






RequireR119896= 119877119894| 1 le 119894 le 119911

119896

Ensure(119877119894 ⟨119905119894

119895⟩ ⟨120591119894

119895⟩) | 1 le 119894 le 119911

119896 1 le 119895 le 119903

119894

(1) 12059110 = 0(2) for 119894 = 2 to 119911

119896do

(3) 120591119894

0 = 120591119894minus10 + 119879

119908


119896do

(6) 119905119894

0 = 0

(7) 120591119894

119895=119879119894sdot 119901119895

119902119888sdot 120578

1 le 119895 le 119903119894

(8) 119905119894

119895= 119905119894

119895minus1 + 120591119894

119895minus1 +119889119895minus1119895

V 0 le 119895 le 119903

119894


119894= 0 then


(16) for 119895 = 1 to 119903119894do


119894


119894



119896

0 500 10000

500

1000

X (m)

Y (m

)

S ampB



0 500 10000

500

1000

X (m)

Y (m

)

S ampB




0 500 10000

500

1000

X (m)

Y (m

)



S ampB


1 2 3 4 5 6 7 8 9 100

05

1

15

2

25

3

35

4

Tour ID

Ener

gy co

nsum

ptio

n of

each

tour

(J)

times104





0 1 2 3 40

02

04

06

08

1

12

14

16

18

2

MC ID

R (s

)

times104

120591rG

Tw





119888= 4W and 119902







2 3 4 5 6 7 8 9 10

Num

bers

0

2

4

6

8

10

12

qc middot 120578

z

m

mlowast


Num

bers

12

13

14

15

16

T

m120601w

Tl (Tl + Tu)2 Tu





n100 200 300 400 500

Num

bers

0

5

10

15

20

25

m

mlowast

m998400








119899 = 100

119898lowast 1 1 1 1 1 2 2 2 2 2

120572 = 1067119898 1 1 1 2 2 2 2 2 2 2119901sum 4445 4507 4617 4817 4888 5055 5233 5311 5340 5532

119902119888sdot 120578 sdot 119898

lowast 5 5 5 5 5 10 10 10 10 10

119899 = 200

119898lowast 2 2 2 3 3 3 3 3 3 3

120572 = 1074119898 2 3 3 3 3 3 3 3 3 3119901sum 9397 9605 9991 10122 10218 10362 10399 10555 11055 11073

119902119888sdot 120578 sdot 119898

lowast 10 10 10 15 15 15 15 15 15 15

119899 = 300

119898lowast 3 4 4 4 4 4 4 4 4 4

120572 = 1026119898 4 4 4 4 4 4 4 4 4 4119901sum 14996 15107 15158 15272 15508 15528 15600 15617 15750 15974

119902119888sdot 120578 sdot 119898

lowast 15 20 20 20 20 20 20 20 20 20

119899 = 400

119898lowast 4 5 5 5 5 5 5 5 5 5

120572 = 1020119898 5 5 5 5 5 5 5 5 5 5119901sum 19684 20223 20723 20853 21381 21461 21780 21789 22054 22153

119902119888sdot 120578 sdot 119898

lowast 20 25 25 25 25 25 25 25 25 25

119899 = 500

119898lowast 6 6 6 6 6 6 6 6 6 6

120572 = 1017119898 6 6 6 6 6 6 6 6 6 7119901sum 25765 26522 26810 26810 26880 27043 27044 27196 27432 29071

119902119888sdot 120578 sdot 119898

lowast 30 30 30 30 30 30 30 30 30 30











Notations


119894


119894and119873

119895


119894

119890119894119895 The path between119873

119894and119873

119895




119896



required in a WRSN



119873119895


119895


119894

119901mx For tour 119877119896


119903119896





119896

R119896= 1198771 1198772 119877119911119896


119877119894= (119879119897

119894 119879119906

119894 120596119894) or 119877

119894= (119879119894 119879119908

119894) under


denoted by1198730120591119895


119894

119905119895


119894


119894 119879119897119894le 119879119894le 119879119906

119894

119879119908



period 119879119908119894= 120596119894(119879119894)




Acknowledgments


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















condition is met


N1

N2

N3

N4

N5

S(N0)

N1

N2

N3

N4

N5

S(N0)

N1

N2

N3

N4

N5

S(N0)

N1

N2

N3

N4

N5

S(N0)

N1

N2

N3

N4

N5

S(N0)

N1

N2

N3

N4

N5

S(N0)


Require119877 = ⟨1198730 1198731 119873119903⟩ 119864119878 119902119898 119902119888 V 119864119872 120578




(3) 119901sum =119903

sum

119894=1

119901119894

(4) 119901mx = max1le119894le119903

119901119894sdot (119902119888sdot 120578 minus 119901

119894)


119863 sdot 119902119888sdot 120578


(9) 119879119906 =119864119904sdot 119902119888sdot 120578

119901mx

(10) 120596(119879) =119901sum

119902119888sdot 120578sdot 119879 +

119863

V







RequireN = 119873

119894| 1 le 119894 le 119899

EnsureR = 119877

119894| 1 le 119894 le 119911


119895into 119877

119911



(10) pop119873119895from 119877

119911





119894of tour 119877

119894


119894

of 119877119894in each period 119879119908

119894= 120596119894(119879119894) = (119901

119894



(119879119897

119894 119879119906


119888 119902119898 V 119864119872)




119908

119894) | 1 le 119894 le 119903 they


119894le 119879


119894=1 119879119908







119895 119896 sdot 119879 +sum

119894

119895=1 119879119908

119895] (119896 = 0 1 2 )

Since sum119903119894=1 119879119908






119908

119894) | 1 le 119894 le 119911 and

supposesum119911119894=1 119879119908





119894119879 (1 le





119894) | 1 le 119894 le

119903119895 1 le 119895 le 119896 where 119877119895


119895


only if sum119896119895=1(sum119903119895

119894=1 119879119895

119894119886119895) le 119866


119895=1(sum119903119895

119894=1 119879119895



119875

1198861sdot

1199031

sum

119894=11198791119894+119875

1198862sdot

1199032

sum

119894=11198792119894+ sdot sdot sdot +

119875

119886119896

sdot

119903119896

sum

119894=1119879119896

119894gt 119866 sdot 119875 (17)



119894=1 119879119895











119908

119894) | 1 le 119894 le


MC if sum119903119894=1 119879119908

119894le 119866


119908

119894) | 1 le 119894 le 119903







119894= (119903 sdot 119879 1198792) |





119894=1(119879119908




1 119879119906

1 1205961) and 1198772 =(119879119897

2 119879119906


2 le 119896sdot119879119906

1 le119879119906


119906

1 and 1198792 = 119896 sdot 1198791199061


119894


119894) = 119886119894sdot 119879119894+ 119887119894


1205751 = 1198861 + 119894 sdot 1198862 +1198871 + 11988721198791

1198792 = 119894 sdot 1198791

1205752 = 1198861 + (119894 + 1) sdot 1198862 +1198871 + 11988721198791

1198792 = (119894 + 1) sdot 1198791

1205753 =1198861 sdot 1198791 + 1198862 sdot 1198792

119866+1198871 + 1198872119866

119894 sdot 1198791 lt 1198792 lt (119894 + 1) sdot 1198791

(18)


1205753 =1198861 sdot 1198791 + 1198862 sdot 1198792

119866+1198871 + 1198872119866

ge 21198861 + 2119894 sdot 1198862 +2 (1198871 + 1198872)1198791

gt 1198861 + (119894 + 1) sdot 1198862 +1198871 + 11988721198791

= 1205752

(19)



119906


2 le119896 sdot 119879119906

1 le 119879119906



119906

1 and1198792 = 119896sdot119879119906





119908) of theMCs

Φ119908=

119898

sum

119894=1

119911119894

sum

119895=1

119879119908

119895

119879119895

=

119911

sum

119894=1

119879119908

119894

119879119894

(20)






119894can be


119896of119872119896is set to 119879119906




119896

(lines (6) (13)) if



119896(lines


119896of 119872119896is still no





RequireR = 119877

119894| 1 le 119894 le 119911

EnsureM = 119872


(1) 119898 = 0(2) sort 119877

119894 by 119879119906


(3) for 119894 = 1 to 119911 do(4) if 119877


(5) 119898+ = 1(6) assign 119877

119894to119872119898

(7) 119866119898= 119879119894= 119879119906

119894 120575

119898=120596119894(119879119894)

119866119898

(8) for 119895 = 119894 + 1 to 119911 do(9) if 119877



119898+ (120596119895(119896 sdot 119879

119906

119894)119866119898) le 1 then

(12) 119879119895= 119896 sdot 119879

119906

119894 120575

119898+ =

120596119895(119879119895)

119866119898

(13) assign 119877119895to119872119898




119909tours

R119909= 119877119894= (119886119894sdot 119866119909 119879119908


119909(1 le 119894 le 119911

119909le 119899


119877119894during time [119896 sdot 119886

119894sdot 119866119909+ sum119894minus1119895=1 119879119908

119895 119896 sdot 119886119894sdot 119866119909+ sum119894

119895=1 119879119908

119895]


119909charges 119877





119895

in 119877119894for time (119901


119895fully charged



6 Numerical Results



119878= 10 kJ 119902

119898= 5 Jm V = 5ms 119864

119872= 40 kJ and 120578 =






RequireR119896= 119877119894| 1 le 119894 le 119911

119896

Ensure(119877119894 ⟨119905119894

119895⟩ ⟨120591119894

119895⟩) | 1 le 119894 le 119911

119896 1 le 119895 le 119903

119894

(1) 12059110 = 0(2) for 119894 = 2 to 119911

119896do

(3) 120591119894

0 = 120591119894minus10 + 119879

119908


119896do

(6) 119905119894

0 = 0

(7) 120591119894

119895=119879119894sdot 119901119895

119902119888sdot 120578

1 le 119895 le 119903119894

(8) 119905119894

119895= 119905119894

119895minus1 + 120591119894

119895minus1 +119889119895minus1119895

V 0 le 119895 le 119903

119894


119894= 0 then


(16) for 119895 = 1 to 119903119894do


119894


119894



119896

0 500 10000

500

1000

X (m)

Y (m

)

S ampB



0 500 10000

500

1000

X (m)

Y (m

)

S ampB




0 500 10000

500

1000

X (m)

Y (m

)



S ampB


1 2 3 4 5 6 7 8 9 100

05

1

15

2

25

3

35

4

Tour ID

Ener

gy co

nsum

ptio

n of

each

tour

(J)

times104





0 1 2 3 40

02

04

06

08

1

12

14

16

18

2

MC ID

R (s

)

times104

120591rG

Tw





119888= 4W and 119902







2 3 4 5 6 7 8 9 10

Num

bers

0

2

4

6

8

10

12

qc middot 120578

z

m

mlowast


Num

bers

12

13

14

15

16

T

m120601w

Tl (Tl + Tu)2 Tu





n100 200 300 400 500

Num

bers

0

5

10

15

20

25

m

mlowast

m998400








119899 = 100

119898lowast 1 1 1 1 1 2 2 2 2 2

120572 = 1067119898 1 1 1 2 2 2 2 2 2 2119901sum 4445 4507 4617 4817 4888 5055 5233 5311 5340 5532

119902119888sdot 120578 sdot 119898

lowast 5 5 5 5 5 10 10 10 10 10

119899 = 200

119898lowast 2 2 2 3 3 3 3 3 3 3

120572 = 1074119898 2 3 3 3 3 3 3 3 3 3119901sum 9397 9605 9991 10122 10218 10362 10399 10555 11055 11073

119902119888sdot 120578 sdot 119898

lowast 10 10 10 15 15 15 15 15 15 15

119899 = 300

119898lowast 3 4 4 4 4 4 4 4 4 4

120572 = 1026119898 4 4 4 4 4 4 4 4 4 4119901sum 14996 15107 15158 15272 15508 15528 15600 15617 15750 15974

119902119888sdot 120578 sdot 119898

lowast 15 20 20 20 20 20 20 20 20 20

119899 = 400

119898lowast 4 5 5 5 5 5 5 5 5 5

120572 = 1020119898 5 5 5 5 5 5 5 5 5 5119901sum 19684 20223 20723 20853 21381 21461 21780 21789 22054 22153

119902119888sdot 120578 sdot 119898

lowast 20 25 25 25 25 25 25 25 25 25

119899 = 500

119898lowast 6 6 6 6 6 6 6 6 6 6

120572 = 1017119898 6 6 6 6 6 6 6 6 6 7119901sum 25765 26522 26810 26810 26880 27043 27044 27196 27432 29071

119902119888sdot 120578 sdot 119898

lowast 30 30 30 30 30 30 30 30 30 30











Notations


119894


119894and119873

119895


119894

119890119894119895 The path between119873

119894and119873

119895




119896



required in a WRSN



119873119895


119895


119894

119901mx For tour 119877119896


119903119896





119896

R119896= 1198771 1198772 119877119911119896


119877119894= (119879119897

119894 119879119906

119894 120596119894) or 119877

119894= (119879119894 119879119908

119894) under


denoted by1198730120591119895


119894

119905119895


119894


119894 119879119897119894le 119879119894le 119879119906

119894

119879119908



period 119879119908119894= 120596119894(119879119894)




Acknowledgments


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










RequireN = 119873

119894| 1 le 119894 le 119899

EnsureR = 119877

119894| 1 le 119894 le 119911


119895into 119877

119911



(10) pop119873119895from 119877

119911





119894of tour 119877

119894


119894

of 119877119894in each period 119879119908

119894= 120596119894(119879119894) = (119901

119894



(119879119897

119894 119879119906


119888 119902119898 V 119864119872)




119908

119894) | 1 le 119894 le 119903 they


119894le 119879


119894=1 119879119908







119895 119896 sdot 119879 +sum

119894

119895=1 119879119908

119895] (119896 = 0 1 2 )

Since sum119903119894=1 119879119908






119908

119894) | 1 le 119894 le 119911 and

supposesum119911119894=1 119879119908





119894119879 (1 le





119894) | 1 le 119894 le

119903119895 1 le 119895 le 119896 where 119877119895


119895


only if sum119896119895=1(sum119903119895

119894=1 119879119895

119894119886119895) le 119866


119895=1(sum119903119895

119894=1 119879119895



119875

1198861sdot

1199031

sum

119894=11198791119894+119875

1198862sdot

1199032

sum

119894=11198792119894+ sdot sdot sdot +

119875

119886119896

sdot

119903119896

sum

119894=1119879119896

119894gt 119866 sdot 119875 (17)



119894=1 119879119895











119908

119894) | 1 le 119894 le


MC if sum119903119894=1 119879119908

119894le 119866


119908

119894) | 1 le 119894 le 119903







119894= (119903 sdot 119879 1198792) |





119894=1(119879119908




1 119879119906

1 1205961) and 1198772 =(119879119897

2 119879119906


2 le 119896sdot119879119906

1 le119879119906


119906

1 and 1198792 = 119896 sdot 1198791199061


119894


119894) = 119886119894sdot 119879119894+ 119887119894


1205751 = 1198861 + 119894 sdot 1198862 +1198871 + 11988721198791

1198792 = 119894 sdot 1198791

1205752 = 1198861 + (119894 + 1) sdot 1198862 +1198871 + 11988721198791

1198792 = (119894 + 1) sdot 1198791

1205753 =1198861 sdot 1198791 + 1198862 sdot 1198792

119866+1198871 + 1198872119866

119894 sdot 1198791 lt 1198792 lt (119894 + 1) sdot 1198791

(18)


1205753 =1198861 sdot 1198791 + 1198862 sdot 1198792

119866+1198871 + 1198872119866

ge 21198861 + 2119894 sdot 1198862 +2 (1198871 + 1198872)1198791

gt 1198861 + (119894 + 1) sdot 1198862 +1198871 + 11988721198791

= 1205752

(19)



119906


2 le119896 sdot 119879119906

1 le 119879119906



119906

1 and1198792 = 119896sdot119879119906





119908) of theMCs

Φ119908=

119898

sum

119894=1

119911119894

sum

119895=1

119879119908

119895

119879119895

=

119911

sum

119894=1

119879119908

119894

119879119894

(20)






119894can be


119896of119872119896is set to 119879119906




119896

(lines (6) (13)) if



119896(lines


119896of 119872119896is still no





RequireR = 119877

119894| 1 le 119894 le 119911

EnsureM = 119872


(1) 119898 = 0(2) sort 119877

119894 by 119879119906


(3) for 119894 = 1 to 119911 do(4) if 119877


(5) 119898+ = 1(6) assign 119877

119894to119872119898

(7) 119866119898= 119879119894= 119879119906

119894 120575

119898=120596119894(119879119894)

119866119898

(8) for 119895 = 119894 + 1 to 119911 do(9) if 119877



119898+ (120596119895(119896 sdot 119879

119906

119894)119866119898) le 1 then

(12) 119879119895= 119896 sdot 119879

119906

119894 120575

119898+ =

120596119895(119879119895)

119866119898

(13) assign 119877119895to119872119898




119909tours

R119909= 119877119894= (119886119894sdot 119866119909 119879119908


119909(1 le 119894 le 119911

119909le 119899


119877119894during time [119896 sdot 119886

119894sdot 119866119909+ sum119894minus1119895=1 119879119908

119895 119896 sdot 119886119894sdot 119866119909+ sum119894

119895=1 119879119908

119895]


119909charges 119877





119895

in 119877119894for time (119901


119895fully charged



6 Numerical Results



119878= 10 kJ 119902

119898= 5 Jm V = 5ms 119864

119872= 40 kJ and 120578 =






RequireR119896= 119877119894| 1 le 119894 le 119911

119896

Ensure(119877119894 ⟨119905119894

119895⟩ ⟨120591119894

119895⟩) | 1 le 119894 le 119911

119896 1 le 119895 le 119903

119894

(1) 12059110 = 0(2) for 119894 = 2 to 119911

119896do

(3) 120591119894

0 = 120591119894minus10 + 119879

119908


119896do

(6) 119905119894

0 = 0

(7) 120591119894

119895=119879119894sdot 119901119895

119902119888sdot 120578

1 le 119895 le 119903119894

(8) 119905119894

119895= 119905119894

119895minus1 + 120591119894

119895minus1 +119889119895minus1119895

V 0 le 119895 le 119903

119894


119894= 0 then


(16) for 119895 = 1 to 119903119894do


119894


119894



119896

0 500 10000

500

1000

X (m)

Y (m

)

S ampB



0 500 10000

500

1000

X (m)

Y (m

)

S ampB




0 500 10000

500

1000

X (m)

Y (m

)



S ampB


1 2 3 4 5 6 7 8 9 100

05

1

15

2

25

3

35

4

Tour ID

Ener

gy co

nsum

ptio

n of

each

tour

(J)

times104





0 1 2 3 40

02

04

06

08

1

12

14

16

18

2

MC ID

R (s

)

times104

120591rG

Tw





119888= 4W and 119902







2 3 4 5 6 7 8 9 10

Num

bers

0

2

4

6

8

10

12

qc middot 120578

z

m

mlowast


Num

bers

12

13

14

15

16

T

m120601w

Tl (Tl + Tu)2 Tu





n100 200 300 400 500

Num

bers

0

5

10

15

20

25

m

mlowast

m998400








119899 = 100

119898lowast 1 1 1 1 1 2 2 2 2 2

120572 = 1067119898 1 1 1 2 2 2 2 2 2 2119901sum 4445 4507 4617 4817 4888 5055 5233 5311 5340 5532

119902119888sdot 120578 sdot 119898

lowast 5 5 5 5 5 10 10 10 10 10

119899 = 200

119898lowast 2 2 2 3 3 3 3 3 3 3

120572 = 1074119898 2 3 3 3 3 3 3 3 3 3119901sum 9397 9605 9991 10122 10218 10362 10399 10555 11055 11073

119902119888sdot 120578 sdot 119898

lowast 10 10 10 15 15 15 15 15 15 15

119899 = 300

119898lowast 3 4 4 4 4 4 4 4 4 4

120572 = 1026119898 4 4 4 4 4 4 4 4 4 4119901sum 14996 15107 15158 15272 15508 15528 15600 15617 15750 15974

119902119888sdot 120578 sdot 119898

lowast 15 20 20 20 20 20 20 20 20 20

119899 = 400

119898lowast 4 5 5 5 5 5 5 5 5 5

120572 = 1020119898 5 5 5 5 5 5 5 5 5 5119901sum 19684 20223 20723 20853 21381 21461 21780 21789 22054 22153

119902119888sdot 120578 sdot 119898

lowast 20 25 25 25 25 25 25 25 25 25

119899 = 500

119898lowast 6 6 6 6 6 6 6 6 6 6

120572 = 1017119898 6 6 6 6 6 6 6 6 6 7119901sum 25765 26522 26810 26810 26880 27043 27044 27196 27432 29071

119902119888sdot 120578 sdot 119898

lowast 30 30 30 30 30 30 30 30 30 30











Notations


119894


119894and119873

119895


119894

119890119894119895 The path between119873

119894and119873

119895




119896



required in a WRSN



119873119895


119895


119894

119901mx For tour 119877119896


119903119896





119896

R119896= 1198771 1198772 119877119911119896


119877119894= (119879119897

119894 119879119906

119894 120596119894) or 119877

119894= (119879119894 119879119908

119894) under


denoted by1198730120591119895


119894

119905119895


119894


119894 119879119897119894le 119879119894le 119879119906

119894

119879119908



period 119879119908119894= 120596119894(119879119894)




Acknowledgments


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


















119908

119894) | 1 le 119894 le


MC if sum119903119894=1 119879119908

119894le 119866


119908

119894) | 1 le 119894 le 119903







119894= (119903 sdot 119879 1198792) |





119894=1(119879119908




1 119879119906

1 1205961) and 1198772 =(119879119897

2 119879119906


2 le 119896sdot119879119906

1 le119879119906


119906

1 and 1198792 = 119896 sdot 1198791199061


119894


119894) = 119886119894sdot 119879119894+ 119887119894


1205751 = 1198861 + 119894 sdot 1198862 +1198871 + 11988721198791

1198792 = 119894 sdot 1198791

1205752 = 1198861 + (119894 + 1) sdot 1198862 +1198871 + 11988721198791

1198792 = (119894 + 1) sdot 1198791

1205753 =1198861 sdot 1198791 + 1198862 sdot 1198792

119866+1198871 + 1198872119866

119894 sdot 1198791 lt 1198792 lt (119894 + 1) sdot 1198791

(18)


1205753 =1198861 sdot 1198791 + 1198862 sdot 1198792

119866+1198871 + 1198872119866

ge 21198861 + 2119894 sdot 1198862 +2 (1198871 + 1198872)1198791

gt 1198861 + (119894 + 1) sdot 1198862 +1198871 + 11988721198791

= 1205752

(19)



119906


2 le119896 sdot 119879119906

1 le 119879119906



119906

1 and1198792 = 119896sdot119879119906





119908) of theMCs

Φ119908=

119898

sum

119894=1

119911119894

sum

119895=1

119879119908

119895

119879119895

=

119911

sum

119894=1

119879119908

119894

119879119894

(20)






119894can be


119896of119872119896is set to 119879119906




119896

(lines (6) (13)) if



119896(lines


119896of 119872119896is still no





RequireR = 119877

119894| 1 le 119894 le 119911

EnsureM = 119872


(1) 119898 = 0(2) sort 119877

119894 by 119879119906


(3) for 119894 = 1 to 119911 do(4) if 119877


(5) 119898+ = 1(6) assign 119877

119894to119872119898

(7) 119866119898= 119879119894= 119879119906

119894 120575

119898=120596119894(119879119894)

119866119898

(8) for 119895 = 119894 + 1 to 119911 do(9) if 119877



119898+ (120596119895(119896 sdot 119879

119906

119894)119866119898) le 1 then

(12) 119879119895= 119896 sdot 119879

119906

119894 120575

119898+ =

120596119895(119879119895)

119866119898

(13) assign 119877119895to119872119898




119909tours

R119909= 119877119894= (119886119894sdot 119866119909 119879119908


119909(1 le 119894 le 119911

119909le 119899


119877119894during time [119896 sdot 119886

119894sdot 119866119909+ sum119894minus1119895=1 119879119908

119895 119896 sdot 119886119894sdot 119866119909+ sum119894

119895=1 119879119908

119895]


119909charges 119877





119895

in 119877119894for time (119901


119895fully charged



6 Numerical Results



119878= 10 kJ 119902

119898= 5 Jm V = 5ms 119864

119872= 40 kJ and 120578 =






RequireR119896= 119877119894| 1 le 119894 le 119911

119896

Ensure(119877119894 ⟨119905119894

119895⟩ ⟨120591119894

119895⟩) | 1 le 119894 le 119911

119896 1 le 119895 le 119903

119894

(1) 12059110 = 0(2) for 119894 = 2 to 119911

119896do

(3) 120591119894

0 = 120591119894minus10 + 119879

119908


119896do

(6) 119905119894

0 = 0

(7) 120591119894

119895=119879119894sdot 119901119895

119902119888sdot 120578

1 le 119895 le 119903119894

(8) 119905119894

119895= 119905119894

119895minus1 + 120591119894

119895minus1 +119889119895minus1119895

V 0 le 119895 le 119903

119894


119894= 0 then


(16) for 119895 = 1 to 119903119894do


119894


119894



119896

0 500 10000

500

1000

X (m)

Y (m

)

S ampB



0 500 10000

500

1000

X (m)

Y (m

)

S ampB




0 500 10000

500

1000

X (m)

Y (m

)



S ampB


1 2 3 4 5 6 7 8 9 100

05

1

15

2

25

3

35

4

Tour ID

Ener

gy co

nsum

ptio

n of

each

tour

(J)

times104





0 1 2 3 40

02

04

06

08

1

12

14

16

18

2

MC ID

R (s

)

times104

120591rG

Tw





119888= 4W and 119902







2 3 4 5 6 7 8 9 10

Num

bers

0

2

4

6

8

10

12

qc middot 120578

z

m

mlowast


Num

bers

12

13

14

15

16

T

m120601w

Tl (Tl + Tu)2 Tu





n100 200 300 400 500

Num

bers

0

5

10

15

20

25

m

mlowast

m998400








119899 = 100

119898lowast 1 1 1 1 1 2 2 2 2 2

120572 = 1067119898 1 1 1 2 2 2 2 2 2 2119901sum 4445 4507 4617 4817 4888 5055 5233 5311 5340 5532

119902119888sdot 120578 sdot 119898

lowast 5 5 5 5 5 10 10 10 10 10

119899 = 200

119898lowast 2 2 2 3 3 3 3 3 3 3

120572 = 1074119898 2 3 3 3 3 3 3 3 3 3119901sum 9397 9605 9991 10122 10218 10362 10399 10555 11055 11073

119902119888sdot 120578 sdot 119898

lowast 10 10 10 15 15 15 15 15 15 15

119899 = 300

119898lowast 3 4 4 4 4 4 4 4 4 4

120572 = 1026119898 4 4 4 4 4 4 4 4 4 4119901sum 14996 15107 15158 15272 15508 15528 15600 15617 15750 15974

119902119888sdot 120578 sdot 119898

lowast 15 20 20 20 20 20 20 20 20 20

119899 = 400

119898lowast 4 5 5 5 5 5 5 5 5 5

120572 = 1020119898 5 5 5 5 5 5 5 5 5 5119901sum 19684 20223 20723 20853 21381 21461 21780 21789 22054 22153

119902119888sdot 120578 sdot 119898

lowast 20 25 25 25 25 25 25 25 25 25

119899 = 500

119898lowast 6 6 6 6 6 6 6 6 6 6

120572 = 1017119898 6 6 6 6 6 6 6 6 6 7119901sum 25765 26522 26810 26810 26880 27043 27044 27196 27432 29071

119902119888sdot 120578 sdot 119898

lowast 30 30 30 30 30 30 30 30 30 30











Notations


119894


119894and119873

119895


119894

119890119894119895 The path between119873

119894and119873

119895




119896



required in a WRSN



119873119895


119895


119894

119901mx For tour 119877119896


119903119896





119896

R119896= 1198771 1198772 119877119911119896


119877119894= (119879119897

119894 119879119906

119894 120596119894) or 119877

119894= (119879119894 119879119908

119894) under


denoted by1198730120591119895


119894

119905119895


119894


119894 119879119897119894le 119879119894le 119879119906

119894

119879119908



period 119879119908119894= 120596119894(119879119894)




Acknowledgments


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










RequireR = 119877

119894| 1 le 119894 le 119911

EnsureM = 119872


(1) 119898 = 0(2) sort 119877

119894 by 119879119906


(3) for 119894 = 1 to 119911 do(4) if 119877


(5) 119898+ = 1(6) assign 119877

119894to119872119898

(7) 119866119898= 119879119894= 119879119906

119894 120575

119898=120596119894(119879119894)

119866119898

(8) for 119895 = 119894 + 1 to 119911 do(9) if 119877



119898+ (120596119895(119896 sdot 119879

119906

119894)119866119898) le 1 then

(12) 119879119895= 119896 sdot 119879

119906

119894 120575

119898+ =

120596119895(119879119895)

119866119898

(13) assign 119877119895to119872119898




119909tours

R119909= 119877119894= (119886119894sdot 119866119909 119879119908


119909(1 le 119894 le 119911

119909le 119899


119877119894during time [119896 sdot 119886

119894sdot 119866119909+ sum119894minus1119895=1 119879119908

119895 119896 sdot 119886119894sdot 119866119909+ sum119894

119895=1 119879119908

119895]


119909charges 119877





119895

in 119877119894for time (119901


119895fully charged



6 Numerical Results



119878= 10 kJ 119902

119898= 5 Jm V = 5ms 119864

119872= 40 kJ and 120578 =






RequireR119896= 119877119894| 1 le 119894 le 119911

119896

Ensure(119877119894 ⟨119905119894

119895⟩ ⟨120591119894

119895⟩) | 1 le 119894 le 119911

119896 1 le 119895 le 119903

119894

(1) 12059110 = 0(2) for 119894 = 2 to 119911

119896do

(3) 120591119894

0 = 120591119894minus10 + 119879

119908


119896do

(6) 119905119894

0 = 0

(7) 120591119894

119895=119879119894sdot 119901119895

119902119888sdot 120578

1 le 119895 le 119903119894

(8) 119905119894

119895= 119905119894

119895minus1 + 120591119894

119895minus1 +119889119895minus1119895

V 0 le 119895 le 119903

119894


119894= 0 then


(16) for 119895 = 1 to 119903119894do


119894


119894



119896

0 500 10000

500

1000

X (m)

Y (m

)

S ampB



0 500 10000

500

1000

X (m)

Y (m

)

S ampB




0 500 10000

500

1000

X (m)

Y (m

)



S ampB


1 2 3 4 5 6 7 8 9 100

05

1

15

2

25

3

35

4

Tour ID

Ener

gy co

nsum

ptio

n of

each

tour

(J)

times104





0 1 2 3 40

02

04

06

08

1

12

14

16

18

2

MC ID

R (s

)

times104

120591rG

Tw





119888= 4W and 119902







2 3 4 5 6 7 8 9 10

Num

bers

0

2

4

6

8

10

12

qc middot 120578

z

m

mlowast


Num

bers

12

13

14

15

16

T

m120601w

Tl (Tl + Tu)2 Tu





n100 200 300 400 500

Num

bers

0

5

10

15

20

25

m

mlowast

m998400








119899 = 100

119898lowast 1 1 1 1 1 2 2 2 2 2

120572 = 1067119898 1 1 1 2 2 2 2 2 2 2119901sum 4445 4507 4617 4817 4888 5055 5233 5311 5340 5532

119902119888sdot 120578 sdot 119898

lowast 5 5 5 5 5 10 10 10 10 10

119899 = 200

119898lowast 2 2 2 3 3 3 3 3 3 3

120572 = 1074119898 2 3 3 3 3 3 3 3 3 3119901sum 9397 9605 9991 10122 10218 10362 10399 10555 11055 11073

119902119888sdot 120578 sdot 119898

lowast 10 10 10 15 15 15 15 15 15 15

119899 = 300

119898lowast 3 4 4 4 4 4 4 4 4 4

120572 = 1026119898 4 4 4 4 4 4 4 4 4 4119901sum 14996 15107 15158 15272 15508 15528 15600 15617 15750 15974

119902119888sdot 120578 sdot 119898

lowast 15 20 20 20 20 20 20 20 20 20

119899 = 400

119898lowast 4 5 5 5 5 5 5 5 5 5

120572 = 1020119898 5 5 5 5 5 5 5 5 5 5119901sum 19684 20223 20723 20853 21381 21461 21780 21789 22054 22153

119902119888sdot 120578 sdot 119898

lowast 20 25 25 25 25 25 25 25 25 25

119899 = 500

119898lowast 6 6 6 6 6 6 6 6 6 6

120572 = 1017119898 6 6 6 6 6 6 6 6 6 7119901sum 25765 26522 26810 26810 26880 27043 27044 27196 27432 29071

119902119888sdot 120578 sdot 119898

lowast 30 30 30 30 30 30 30 30 30 30











Notations


119894


119894and119873

119895


119894

119890119894119895 The path between119873

119894and119873

119895




119896



required in a WRSN



119873119895


119895


119894

119901mx For tour 119877119896


119903119896





119896

R119896= 1198771 1198772 119877119911119896


119877119894= (119879119897

119894 119879119906

119894 120596119894) or 119877

119894= (119879119894 119879119908

119894) under


denoted by1198730120591119895


119894

119905119895


119894


119894 119879119897119894le 119879119894le 119879119906

119894

119879119908



period 119879119908119894= 120596119894(119879119894)




Acknowledgments


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










RequireR119896= 119877119894| 1 le 119894 le 119911

119896

Ensure(119877119894 ⟨119905119894

119895⟩ ⟨120591119894

119895⟩) | 1 le 119894 le 119911

119896 1 le 119895 le 119903

119894

(1) 12059110 = 0(2) for 119894 = 2 to 119911

119896do

(3) 120591119894

0 = 120591119894minus10 + 119879

119908


119896do

(6) 119905119894

0 = 0

(7) 120591119894

119895=119879119894sdot 119901119895

119902119888sdot 120578

1 le 119895 le 119903119894

(8) 119905119894

119895= 119905119894

119895minus1 + 120591119894

119895minus1 +119889119895minus1119895

V 0 le 119895 le 119903

119894


119894= 0 then


(16) for 119895 = 1 to 119903119894do


119894


119894



119896

0 500 10000

500

1000

X (m)

Y (m

)

S ampB



0 500 10000

500

1000

X (m)

Y (m

)

S ampB




0 500 10000

500

1000

X (m)

Y (m

)



S ampB


1 2 3 4 5 6 7 8 9 100

05

1

15

2

25

3

35

4

Tour ID

Ener

gy co

nsum

ptio

n of

each

tour

(J)

times104





0 1 2 3 40

02

04

06

08

1

12

14

16

18

2

MC ID

R (s

)

times104

120591rG

Tw





119888= 4W and 119902







2 3 4 5 6 7 8 9 10

Num

bers

0

2

4

6

8

10

12

qc middot 120578

z

m

mlowast


Num

bers

12

13

14

15

16

T

m120601w

Tl (Tl + Tu)2 Tu





n100 200 300 400 500

Num

bers

0

5

10

15

20

25

m

mlowast

m998400








119899 = 100

119898lowast 1 1 1 1 1 2 2 2 2 2

120572 = 1067119898 1 1 1 2 2 2 2 2 2 2119901sum 4445 4507 4617 4817 4888 5055 5233 5311 5340 5532

119902119888sdot 120578 sdot 119898

lowast 5 5 5 5 5 10 10 10 10 10

119899 = 200

119898lowast 2 2 2 3 3 3 3 3 3 3

120572 = 1074119898 2 3 3 3 3 3 3 3 3 3119901sum 9397 9605 9991 10122 10218 10362 10399 10555 11055 11073

119902119888sdot 120578 sdot 119898

lowast 10 10 10 15 15 15 15 15 15 15

119899 = 300

119898lowast 3 4 4 4 4 4 4 4 4 4

120572 = 1026119898 4 4 4 4 4 4 4 4 4 4119901sum 14996 15107 15158 15272 15508 15528 15600 15617 15750 15974

119902119888sdot 120578 sdot 119898

lowast 15 20 20 20 20 20 20 20 20 20

119899 = 400

119898lowast 4 5 5 5 5 5 5 5 5 5

120572 = 1020119898 5 5 5 5 5 5 5 5 5 5119901sum 19684 20223 20723 20853 21381 21461 21780 21789 22054 22153

119902119888sdot 120578 sdot 119898

lowast 20 25 25 25 25 25 25 25 25 25

119899 = 500

119898lowast 6 6 6 6 6 6 6 6 6 6

120572 = 1017119898 6 6 6 6 6 6 6 6 6 7119901sum 25765 26522 26810 26810 26880 27043 27044 27196 27432 29071

119902119888sdot 120578 sdot 119898

lowast 30 30 30 30 30 30 30 30 30 30











Notations


119894


119894and119873

119895


119894

119890119894119895 The path between119873

119894and119873

119895




119896



required in a WRSN



119873119895


119895


119894

119901mx For tour 119877119896


119903119896





119896

R119896= 1198771 1198772 119877119911119896


119877119894= (119879119897

119894 119879119906

119894 120596119894) or 119877

119894= (119879119894 119879119908

119894) under


denoted by1198730120591119895


119894

119905119895


119894


119894 119879119897119894le 119879119894le 119879119906

119894

119879119908



period 119879119908119894= 120596119894(119879119894)




Acknowledgments


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 500 10000

500

1000

X (m)

Y (m

)



S ampB


1 2 3 4 5 6 7 8 9 100

05

1

15

2

25

3

35

4

Tour ID

Ener

gy co

nsum

ptio

n of

each

tour

(J)

times104





0 1 2 3 40

02

04

06

08

1

12

14

16

18

2

MC ID

R (s

)

times104

120591rG

Tw





119888= 4W and 119902







2 3 4 5 6 7 8 9 10

Num

bers

0

2

4

6

8

10

12

qc middot 120578

z

m

mlowast


Num

bers

12

13

14

15

16

T

m120601w

Tl (Tl + Tu)2 Tu





n100 200 300 400 500

Num

bers

0

5

10

15

20

25

m

mlowast

m998400








119899 = 100

119898lowast 1 1 1 1 1 2 2 2 2 2

120572 = 1067119898 1 1 1 2 2 2 2 2 2 2119901sum 4445 4507 4617 4817 4888 5055 5233 5311 5340 5532

119902119888sdot 120578 sdot 119898

lowast 5 5 5 5 5 10 10 10 10 10

119899 = 200

119898lowast 2 2 2 3 3 3 3 3 3 3

120572 = 1074119898 2 3 3 3 3 3 3 3 3 3119901sum 9397 9605 9991 10122 10218 10362 10399 10555 11055 11073

119902119888sdot 120578 sdot 119898

lowast 10 10 10 15 15 15 15 15 15 15

119899 = 300

119898lowast 3 4 4 4 4 4 4 4 4 4

120572 = 1026119898 4 4 4 4 4 4 4 4 4 4119901sum 14996 15107 15158 15272 15508 15528 15600 15617 15750 15974

119902119888sdot 120578 sdot 119898

lowast 15 20 20 20 20 20 20 20 20 20

119899 = 400

119898lowast 4 5 5 5 5 5 5 5 5 5

120572 = 1020119898 5 5 5 5 5 5 5 5 5 5119901sum 19684 20223 20723 20853 21381 21461 21780 21789 22054 22153

119902119888sdot 120578 sdot 119898

lowast 20 25 25 25 25 25 25 25 25 25

119899 = 500

119898lowast 6 6 6 6 6 6 6 6 6 6

120572 = 1017119898 6 6 6 6 6 6 6 6 6 7119901sum 25765 26522 26810 26810 26880 27043 27044 27196 27432 29071

119902119888sdot 120578 sdot 119898

lowast 30 30 30 30 30 30 30 30 30 30











Notations


119894


119894and119873

119895


119894

119890119894119895 The path between119873

119894and119873

119895




119896



required in a WRSN



119873119895


119895


119894

119901mx For tour 119877119896


119903119896





119896

R119896= 1198771 1198772 119877119911119896


119877119894= (119879119897

119894 119879119906

119894 120596119894) or 119877

119894= (119879119894 119879119908

119894) under


denoted by1198730120591119895


119894

119905119895


119894


119894 119879119897119894le 119879119894le 119879119906

119894

119879119908



period 119879119908119894= 120596119894(119879119894)




Acknowledgments


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










2 3 4 5 6 7 8 9 10

Num

bers

0

2

4

6

8

10

12

qc middot 120578

z

m

mlowast


Num

bers

12

13

14

15

16

T

m120601w

Tl (Tl + Tu)2 Tu





n100 200 300 400 500

Num

bers

0

5

10

15

20

25

m

mlowast

m998400








119899 = 100

119898lowast 1 1 1 1 1 2 2 2 2 2

120572 = 1067119898 1 1 1 2 2 2 2 2 2 2119901sum 4445 4507 4617 4817 4888 5055 5233 5311 5340 5532

119902119888sdot 120578 sdot 119898

lowast 5 5 5 5 5 10 10 10 10 10

119899 = 200

119898lowast 2 2 2 3 3 3 3 3 3 3

120572 = 1074119898 2 3 3 3 3 3 3 3 3 3119901sum 9397 9605 9991 10122 10218 10362 10399 10555 11055 11073

119902119888sdot 120578 sdot 119898

lowast 10 10 10 15 15 15 15 15 15 15

119899 = 300

119898lowast 3 4 4 4 4 4 4 4 4 4

120572 = 1026119898 4 4 4 4 4 4 4 4 4 4119901sum 14996 15107 15158 15272 15508 15528 15600 15617 15750 15974

119902119888sdot 120578 sdot 119898

lowast 15 20 20 20 20 20 20 20 20 20

119899 = 400

119898lowast 4 5 5 5 5 5 5 5 5 5

120572 = 1020119898 5 5 5 5 5 5 5 5 5 5119901sum 19684 20223 20723 20853 21381 21461 21780 21789 22054 22153

119902119888sdot 120578 sdot 119898

lowast 20 25 25 25 25 25 25 25 25 25

119899 = 500

119898lowast 6 6 6 6 6 6 6 6 6 6

120572 = 1017119898 6 6 6 6 6 6 6 6 6 7119901sum 25765 26522 26810 26810 26880 27043 27044 27196 27432 29071

119902119888sdot 120578 sdot 119898

lowast 30 30 30 30 30 30 30 30 30 30











Notations


119894


119894and119873

119895


119894

119890119894119895 The path between119873

119894and119873

119895




119896



required in a WRSN



119873119895


119895


119894

119901mx For tour 119877119896


119903119896





119896

R119896= 1198771 1198772 119877119911119896


119877119894= (119879119897

119894 119879119906

119894 120596119894) or 119877

119894= (119879119894 119879119908

119894) under


denoted by1198730120591119895


119894

119905119895


119894


119894 119879119897119894le 119879119894le 119879119906

119894

119879119908



period 119879119908119894= 120596119894(119879119894)




Acknowledgments


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119899 = 100

119898lowast 1 1 1 1 1 2 2 2 2 2

120572 = 1067119898 1 1 1 2 2 2 2 2 2 2119901sum 4445 4507 4617 4817 4888 5055 5233 5311 5340 5532

119902119888sdot 120578 sdot 119898

lowast 5 5 5 5 5 10 10 10 10 10

119899 = 200

119898lowast 2 2 2 3 3 3 3 3 3 3

120572 = 1074119898 2 3 3 3 3 3 3 3 3 3119901sum 9397 9605 9991 10122 10218 10362 10399 10555 11055 11073

119902119888sdot 120578 sdot 119898

lowast 10 10 10 15 15 15 15 15 15 15

119899 = 300

119898lowast 3 4 4 4 4 4 4 4 4 4

120572 = 1026119898 4 4 4 4 4 4 4 4 4 4119901sum 14996 15107 15158 15272 15508 15528 15600 15617 15750 15974

119902119888sdot 120578 sdot 119898

lowast 15 20 20 20 20 20 20 20 20 20

119899 = 400

119898lowast 4 5 5 5 5 5 5 5 5 5

120572 = 1020119898 5 5 5 5 5 5 5 5 5 5119901sum 19684 20223 20723 20853 21381 21461 21780 21789 22054 22153

119902119888sdot 120578 sdot 119898

lowast 20 25 25 25 25 25 25 25 25 25

119899 = 500

119898lowast 6 6 6 6 6 6 6 6 6 6

120572 = 1017119898 6 6 6 6 6 6 6 6 6 7119901sum 25765 26522 26810 26810 26880 27043 27044 27196 27432 29071

119902119888sdot 120578 sdot 119898

lowast 30 30 30 30 30 30 30 30 30 30











Notations


119894


119894and119873

119895


119894

119890119894119895 The path between119873

119894and119873

119895




119896



required in a WRSN



119873119895


119895


119894

119901mx For tour 119877119896


119903119896





119896

R119896= 1198771 1198772 119877119911119896


119877119894= (119879119897

119894 119879119906

119894 120596119894) or 119877

119894= (119879119894 119879119908

119894) under


denoted by1198730120591119895


119894

119905119895


119894


119894 119879119897119894le 119879119894le 119879119906

119894

119879119908



period 119879119908119894= 120596119894(119879119894)




Acknowledgments


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119873119895


119895


119894

119901mx For tour 119877119896


119903119896





119896

R119896= 1198771 1198772 119877119911119896


119877119894= (119879119897

119894 119879119906

119894 120596119894) or 119877

119894= (119879119894 119879119908

119894) under


denoted by1198730120591119895


119894

119905119895


119894


119894 119879119897119894le 119879119894le 119879119906

119894

119879119908



period 119879119908119894= 120596119894(119879119894)




Acknowledgments


References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article minimizing the number of mobile chargers to...

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