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tinllege SAccepted 17 September 2010Available online 29 September 2010Keywords:Data gatheringWireless sensor networksenergy resources. We employ a hierarchical topology and routing structure with multiplecontrol trafc; and in industrial settings to track invento-ries and the state of other resources [37]. Recently, data-gathering WSNs nd increasingly widespread applicationsin ecological and environmental monitoring [13,14,29].performing computations on-board in a sensor [36].In general, the lifetime of a sensor network can bedened as the time frame between two successive sensordeployments, i.e., a deployment cycle. A deployment cycleconsists of successive periods of xed time length for whichtopology and/or routing decisions are made. Thus, prolong-ing the network lifetime corresponds to obtaining themaximum number of successive periods that the data1570-8705/$ - see front matter 2010 Elsevier B.V. All rights reserved. Corresponding author. Tel.: +1 979 845 9573.E-mail address: uster@tamu.edu (H. ster).Ad Hoc Networks 9 (2011) 835851Contents lists available at ScienceDirectAd Hoc Ne.e lsdoi:10.1016/j.adhoc.2010.09.010Wireless sensors are used to sense a wide range ofnatural or articial phenomena including temperature,pressure, humidity, light, motion, weight, noise, etc. Wire-less sensor networks (WSNs) can be employed for datagathering purposes in inhospitable environments and dif-cult-to-reach terrains, such as forests, urban or rural battle-elds, and borderlines; in wild habitats and oceans tomonitor and observe natural phenomena; in disaster pre-vention and relief; in urban environments to monitor andefcient design and operation of WSNs are topology control,which refers to the determination of an underlying networktopology and routing, which refers to the determinationdata transfer paths over this network. The relationship be-tween these problems is emphasized byWSN-specic attri-butes energy efciency and computation-communicationtrade-off. Energy efciency is important because each sen-sor is equipped with an on-board nonrenewable powerunit. The communication-computation trade-off refers tothe fact that communication consumes more energy thanNetwork design modelsAlgorithms1. Introductionsinks and devise a topology control scheme via usable energy fraction at the sensors.We develop and examine three different mathematical models whose solutions prescribeclusterhead and sink locations and data routing from sensors to sinks in a period of adeployment cycle. We develop a heuristic solution algorithm which provides very smalloptimality gaps for the models. The approach utilizes two types of solution representa-tions, a combination of multiple neighborhoods, and objective value-based cut inequalitiesfor improving the evaluation of candidate solutions. We present extensive numerical testresults and analysis of the models and the solution approach. We determine that our pro-posed model, which minimizes average energy usage and the range of remaining energydistribution at the sensors, captures important characteristics of topology control and rout-ing integration in WSN design and exhibits signicantly better performance than ourbenchmark models and a well-known protocol HEED in extending network lifetime. 2010 Elsevier B.V. All rights reserved.Two fundamental and related problems for effective andArticle history:Received 30 January 2010Received in revised form 30 July 2010This study considers an integrated topology control and routing problem in wireless sensornetworks (WSNs), which are employed to gather data via use of sensors with limitedIntegrated topology control and roufor prolonged network lifetimeHalit ster , Hui LinDepartment of Industrial and Systems Engineering, Texas A&M University, Coa r t i c l e i n f o a b s t r a c tjournal homepage: wwwg in wireless sensor networkstation, TX 77843-3131, United Statestworksevier .com/locate /adhocredundancy mentioned above. We assume that the836 H. ster, H. Lin / Ad Hoc Networks 9 (2011) 835851generated at the sensors can reach the user. In our case, theend of a deployment cycle is reached when it is not possi-ble to obtain a feasible solution to the problem of transmit-ting data generated at the sensors to the user.In this paper, we develop power-aware mathematicalmodels and solution approaches for the integrated topologycontrol and data routing problems to prolong the lifetime ofa WSN. To this end, we consider three models that differmainly in terms of their objective functions. The objectivesinclude minimization of (1) total or average energy usage inthe system, (2) maximum energy used at a sensor node, and(3) a weighted sum of the range of end-of-period remainingenergy distribution at the sensors and the average energyused in the system.We adopt a hierarchical data ow structure in whichdata generated at the sensors are rst routed to the sensorsdesignated as clusterheads (CHs). Each sensor is assignedto at least one CH which reduces the total data size thatit receives from sensors via aggregation. Each CH routesdata to a sink either through other CHs, which act onlyas relays without aggregation, or directly. Such a structureis benecial in terms of energy efciency in three ways:(1) Since the sensors in close proximity of each other arelikely to be in the same cluster and may generate very sim-ilar data, data aggregation at CHs helps to reduce redun-dancy and energy consumption in communication. (2)Hierarchical structure distributes the energy usage to mul-tiple sensors on multi-hop paths, thus eliminating thequick expiration of the sensors away from the sinks. (3)Since energy dissipation in communication is proportionalto the square of the distance, compared to direct commu-nication, the total energy dissipation due to communica-tion is less on a multi-hop route [31].Our contributions in this paper can be summarized asfollows.1. We devise three mathematical models for integratedtopology and routing decisions for data-gatheringWSNs. The objectives, minimization of total (or aver-age) energy usage in the network and minimization ofthe maximum energy usage at a sensor, are commonlyconsidered in devising communication protocols (e.g.,[15,16,25,27,28,38]). However, this has not been donefrom an integrated mathematical modelling perspectiveas in our case. We consider these two models as bench-mark models for our third proposed model, which min-imizes the total energy and the range of remainingenergy distribution in the network.2. In devising our models, we consider the use of multiplesinks. This is helpful for energy efciency since multiplesinks create an opportunity for better proximity to sen-sors, thus saving energy in communication. It is possi-ble to route the data so that the energy drainage inthe network is more evenly distributed to the sensorsby changing the locations of the sinks and the CHs ineach period. Xue et al. [39] also consider multiple sinks,however, with known locations as opposed to our casewhere the locations are also determined.3. We suggest a new approach to achieve topology controlvia limiting the usable energy at a sensor as a fraction ofits total available energy. We show that how this usableaggregation ratio increases with increasing sensordeployment density. In previous studies with dataaggregation (e.g., [15,16,19,42]), we observe thataggregation of data into a single signal at each CH,i.e., regardless of the amount of data received, is com-mon which is applicable in such cases as monitoringmaximum temperature in the sensor eld.5. Since the models dictate large discrete optimization for-mulations, obtaining exact solutions are highly imprac-tical using exact optimization methods such as abranch-and-cut algorithm. Thus, we develop a heuristicsolution algorithm which provides very small optimal-ity gaps for the models. The approach utilizes two typesof solution representations, a combination of multipleneighborhoods, and objective value-based cut inequali-ties for improving the evaluation of candidate solutions.Computational evidence demonstrates that our pro-posed third model performs signicantly better thanboth benchmark models. Furthermore, we comparethe performance of the proposed model to a well-known protocol HEED [42] and show that our proposedmodel performs signicantly better in terms of networklifetime.The rest of this paper is organized as follows. After aliterature review in Section 2, we introduce the notation,problem denition and the optimization models in Sec-tion 3. In Section 4, we develop algorithmic approachesfor solving our mathematical models. In Section 5, weprovide computational results illustrating the perfor-mance of our algorithms, impact of usable energy fractionon network lifetime, and a comparison of the models insingle-and multi-period settings as well as a detailedanalysis based on energy characteristics and network life-times. Finally, in Section 6, we present a summary andconclusions.2. Related literatureIn the WSN literature, extensive effort has been in-vested in developing energy efcient protocols and routingparadigms to maintain a requested WSN topology. In thissection, we provide an overview of the topology controland routing studies, and point out most relevant studiesto our work.Topology control is mainly achieved by the adjustmentof sensors transmission ranges, which is related to thepower level settings at the sensors [30,31]. Topologyenergy determined is important and difcult in our twobenchmark models. On the other hand, the solution inthe proposed third model is insensitive to this charac-teristic as the control of energy distribution is implicitlyaccounted in the objective.4. We consider cases where an overall view of a measure,such as spatial and temporal temperature/humidiy/pressure gradients in a large sensor eld deployedfor environmental monitoring, is of interest. To thisend, we employ a general data aggregation approachat the CHs that represents the elimination of dataH. ster, H. Lin / Ad Hoc Networks 9 (2011) 835851 837control is not an activity that directly causes energy usageas in the case of routing data. An energy related objectivein topology control is essentially a proxy measureadopted for instilling energy efciency at this stage toachieve real benets in the routing stage. There are sev-eral studies, e.g., [12,22,25,27,35,38], that consider thetopology control from this perspective and aim to set upa strongly connected network by considering an energyusage related objective. Minimization of total energy usedand the maximum energy used by a sensor are two com-mon metrics. As opposed to a hierarchical structure and aconvergecast pattern, these studies take the perspectiveof a at topology and a unicast or broadcast among thesensors. A topology control approach for a convergecastis given by Rodoplu and Meng [28]. The authors describea distributed algorithm which generates a minimum totalenergy, a tree rooted at the master-site to which all sen-sors can reach. Recently, there are also some researchersaddressing the coverage problem for the desired topology,while maintaining the connectivity. Zhang and Hou [43]present a distributed algorithm for maintaining sensingcoverage and connectivity by keeping a minimum num-ber of active sensors. Aleri et al. [7] present a mixedinteger linear programming model to exploit data redun-dancy by dening subset of sensors active in differenttime periods. Column generation approach and a heuristicalgorithm are suggested for model solution. Soro andHeinzelman [34] consider the coverage preservation prob-lem in the context of cluster-based sensor networks.Several coverage-aware cost metrics are explored forselecting the set of CH and active sensors that provide fullnetwork coverage, as well as the set of routers thatforward data to the sink node.The majority of the routing studies focus on develop-ing communications protocols while others adopt span-ning tree, shortest path or multi-commodity ow basedapproaches. Heinzelman et al. [15] develop a data aggre-gating cluster-based routing protocol Low Energy Adap-tive Clustering Hierarchy (LEACH) for a convergecast toa single sink at a xed known location. Each CH actsboth as an aggregator and a relay to the sink. LEACHaims to maximize lifetime via a localized approach usingrandomized CH selections and a minimum energy basedassignment of sensors to CHs. LEACH is one of the rsthierarchical routing approaches in WSNs and later, muchwork (e.g., [8,16,21,24]) focuses on extending their workwith the same general setting. Although LEACH performsbetter than direct-from-sensor-to-sink routing, the mini-mum transmission energy (MTE) based routing [33], anda static clustering based routing approach (instead of CHrotations), certain issues about its underlying assump-tions remain. LEACH assumes that all nodes can reachthe sink with enough power, which is not always truefor WSNs, due to the limited energy at the sensors.LEACH adopts the randomized rotation of CHs to ensurea balanced energy consumption, but this may causesome nodes not to have a CH in their vicinity. To accountfor the shortcomings of LEACH, among others, the HybridEnergy-Efcient Distributed clustering (HEED) protocolby Younis and Fahmy [42] is well-known for its perfor-mance. In HEED, the primary goal is to identify CHs andorganize sensors into clusters so as to better utilize thelimited energy resources. The CHs are probabilisticallyselected based on their remaining energy and the sensorsjoin clusters so that the communication cost is minimized.In comparison to LEACH, HEED assumes a multi-hopconnection between CHs or to the sink node and, at eachperiod, it selects CHs with high remaining energy. HEEDprovides an effective method for prolonging the networklifetime. However, it does not necessarily ensure abalanced energy consumption because of the followingreasons: (1) In HEED, the sensors chosen as CHs closer tothe sink may have quick drainage due to their heavy loadin forwarding data packets. (2) At each period, HEEDattempts to chooses the highest energy sensors asCHs, which may be biased from the long-term networklifetime perspective.As opposed to the localized algorithms, someresearchers address the routing scheme based on a theo-retical formation of the problem. Chang and Tass [11]consider the maximum lifetime routing problem withat topologies. They present a linear programming modelwhere the objective is lifetime (single timeframe) maxi-mization (equivalently maximizing data ow). Xueet al. [39] extend the framework of a multi-commodityow problem and suggest the use of multiple data sinksto increase the network lifetime. However, they do notconsider data aggregation to eliminate redundancy. Huaand Yum [17] aim to maximize the network lifetime byjointly optimizing data aggregation and routing. Themain drawback in these studies is that they assumethe static network topology, which may not be optimalfor balancing the energy consumption over the periods.Kalpakis et al. [19] study the maximum lifetime dataaggregation (MLDA) problem and formulate the lifetimeas a maximization linear program. The solution givesthe overall arc ows and it is later decomposed to deter-mine spanning (routing) trees, one for each period,rooted at the sink. Al-Karaki et al. [6] propose an energyefcient routing scheme Grid-based Routing and Aggre-gator Selection Scheme (GRASS). GRASS aims to maxi-mize the network lifetime by jointly addressing thecluster-based routing problem with application specicdata aggregation. Mathematical formulation and heuristicsolution approaches are suggested. Reviews on routingare given in [1,2,5] and excellent reviews of WSNs in-clude [3,4,20,41].Recently, cross-layer design and optimization in WSNshave received signicant attention. In the cross-layer ap-proach, different layers of the protocol stack can integrateand share information among each other to enhance net-work performance and maximize the lifetime. Burri et al.[10] propose a data-gathering protocol Dozer, that jointlyconsiders medium access control (MAC) layer, topologycontrol and routing to save energy. Dozer employs atree-based network structure to route the data, coordi-nates the nodes sleep schedules and achieves low radioduty cycles. Madan et al. [26] consider the link, MACand routing layers with optimization of transmissionschemes to maximize the lifetime. A simple networktopology is given to compute energy consumption andnetwork lifetime.highly dependent on the specic application and networkw elo d -su eac llo h lsi nw e fca oth -tia y eCH s yef m tse n gus838 H. ster, H. Lin / Ad Hoc Networks 9 (2011) 835851parameters such as deployment density. We also assumea xed number of CHs being active in a period which canbe controlled by the user. In our computational studies,similar to previous studies [16,34,42], we consider varyingnumber of CHs, approximately 820% of the number ofsensors.For energy dissipation, we employ a widely adoptedrst order radio model (e.g. see [15]) in which energy dis-sipation is w (J/bit) to run the circuitry and v (J/bit/m2) forthe transmit amplier. Then, transmitting xab (bits) of datafrom node a to node b dissipates w vD2abxab and to re-ceive the same amount of data dissipates wxab, where Dabis the distance (in meters) between a and b. In addition,to account for the power dissipation due to data aggrega-tion/processing efforts at a CH, we employ a dissipationrate of c (J/bit). We employ the values of the energy dissi-pation related parameters as in [15]; specically, v = 100pJ/bit/m2, w = 50 nJ/bit, and we use a c value of 5 nJ/bit(note that pJ = pico-joules and nJ = nano-joules).As mentioned above, we limit the usable energy that asensor or a CH can use. We represent the usable amountas a fraction p of the total available energy at a sensorand refer to it also as topology control parameter. ThisWe note that these studies adopt a predeterminedtopology structure such as tree-based and linear topologyfor data routing. Research on the relation between topol-ogy control and routing is very limited. Specically, if theissues of routing are not taken into consideration in thetopology control problem, then the underlying topologymight not be suited for supporting an efcient routingscheme. Considering this deciency in the current litera-ture, we investigate the integrated mathematical modelsand their solution algorithms for topology control androuting, along with consideration of multiple sinks, basedon optimization techniques for the design of WSNs. In thiscontext, we devise new models and propose a new objec-tive, minimizing the average energy usage and the rangeof remaining energy distribution in the network. We showthat the new approach integrates topology and routingdecisions very effectively without a need for adjustingusable energy levels which we suggest for alternative mod-els for prolonging network lifetime. Furthermore, we em-ploy a generalized aggregation approach suitable fordata-gathering related applications such as in environmen-tal monitoring.3. Mathematical modelsBefore we describe our models, we summarize theassumptions and dene the notation.We assume that once the sensors are randomly de-ployed in the sensor eld, they approximate their positionsvia triangulation [32,9]. This information along with pre-congured sensor IDs are communicated to the user via aminimum cost forwarding protocol [40].In a given period, a sensor collects data at an averagerate given by bits/unit time and forwards it to its CH. Weuse an average aggregation ratio at a CH, mainly to reectthe elimination of data overlap, and note that this ratio isxujk data ow per unit time from aCH j to a sink node kzcj 1 if a node j is set up as a CH, 0o.w.zuk 1 if a node k is set up as a sink,0 o.w.ei energy consumed by a sensornode iecm energy consumed by a CH mECmax maximum energy consumed ata sensorERmax maximum remaining energy ata sensorERmin minimum remaining energy ata sensorGiven common characteristics, such as underlying net-work topology and ow structure and energy consumptioncalculations, we state the following set of common con-straints for our models.xccij data ow per unit time from aCH i to a CH jtime from a sensor i to a CH jxcij fraction of data ow per unitDecision variablesTusable fraction of Ei at a sensorthe length of a periodpiEi available energy (J) at a sensoran average data aggregationratioUsnumber of CHsnumber of sinksHnodes a and bDab distance (m) between any twotime) at a sensor iRi data generation rate (bits/unitKset of candidate CHs, j 2 Jset of candidate sinks, k 2 KModel parameters IJset of sensors, i 2 Ied as CHs.g the same sensors from beinciency because it pronsors by circumventian essential property for energotes uniform power drainage al CHs. Clearly, in evers can change, which iwe consider the sensors as potenperiod the active sinks and thndidate sink locationse sensor eld. For CHs,ployed), we assume that a set oare also known with respect tnks. Since the sensorhich the sensors are deld is known a priori (withications from which tconsider a nite set of potentiae models choose as operationame that the sinks arecessible by a user. Wethe CHs in each period. We asnot energy-constrained and arIn addition to data ocations of the sinks an.s, our models also determine thsors whole energy to be usable can easily make the sensorvulnerable to quick energy depletion since it can be se-provides the ability to manage topology control implicitly.This is an important characteristic because allowing a sen-lected as a CH repeatedlyComXk2KXj2JXk2KERmin 6 Ei ei 8i 2 I 22network. Specically, the objective in M3 directly ad-dresseremaiimizinenerg4. SolOumingthe uCPLEX) is not helpful due to high memory and runtimeH. ster, H. Lin / Ad Hoc Networks 9 (2011) 835851 839Xi2IRixcim 0 8m 2 J 3Xj2Jxcij 1 8i 2 I 4xcij 6 zcj 8i 2 I ;8j 2 J 5xccmj 6Xi2IRizcj 8m; j 2 J 6xujk 6Xi2IRizuk 8j 2 J ;8k 2 K 7xcjk 6Xi2IRizcj 8j 2 J ;8k 2 K 8ei 6 pEi 8i 2 I 9ecj 6 pEj 8j 2 J 10Xj2Jzcj H 11Xk2Kzuk U 12zcj ; zuk 2 f0;1g 8i 2 I ; j 2 J 13xcij; xccij ; xujk; ei; ecj P 0 8i 2 I ; j 2 J ; k 2 K 14Constraints (1) and (2) determine energy consumed by aCH and a sensor, respectively. The energy dissipation dueto aggregation is embedded in the last term of theleft-hand side in (1) in such a way that more aggregation(a higher s value) results in higher energy dissipation.Constraints (3) state the ow balance at each CH and alsoensure that the data collected at a sensor is aggregatedwith data from other sensors only once after it is trans-ferred to a CH. Constraints (4) guarantee that the data gen-erated at each sensor reaches a CH. Constraints (5)(8)assign the values of binary variables related to CH and sinkselections. Constraint (9) and (10) ensure that the total en-ergy consumed at a sensor cannot exceed its total availableenergy. Constraints (11) and (12) establish the requirednumber of CHs and sinks, respectively, and (13) and (14)are for variable integrality and non-negativity.Alternative modelsM1 Minimize the total energy used:MinXm2Jecm Xi2Iei 15subject to 114M2 Minimize the maximum energy consumed by a sensor node:Min ECmax 16subject to 114ei 6 ECmax 8i 2 I 17ecj 6 ECmax 8j 2 J 18ECmax P 0 19mon constraintsw vD2mkTxumk Xj2Jnfmgw vD2mj TxccmjXj2JnfmgwTxccjm Xi2Iw csRiTxcim ecm 8m 2 J 1w vD2ijRiTxcij ei 8i 2 I 2xumk Xj2Jnfmgxccmj Xj2Jnfmgxccjm 1 s !requirements. Therefore, we study the development of ef-cient heuristic solution approaches which take advantageof the availability of explicit model formulations andunderlying model and solution characteristics.s the energy usage and also the variation in thening energy distribution at the sensor nodes by min-g its range. That is, M3 aims to uniformly distributey usage across the network.ution approachesr models of interest dictate mixed integer program-problems and, for relatively large-scale instances,se of branch-and-cut solution algorithms (as inzcj Ej ecj 6 ERmax 8j 2 J 231 zci Ei ei 6 ERmax 8i 2 I 24ERmax; ERmin P 0 25Constraint set pairs (21) and (22) and (23) and (24) expressthe minimum and the maximum remaining energy levelsfor each sensor node, respectively.As discussed in the previous section, the objectives inM1 andM2 are commonly considered in devising protocolsrather than in a mathematical modelling context as in thisstudy. We observe that, when a lifetime measure is consid-ered, these objectives do not exactly capture the energydepletion pattern that we would like to see in the network.More specically,M1minimizes overall energy dissipa-tion, which may lead to an energy drainage at certainnodes due to their successive usage in several periods ifthey happen to be on more favorable paths requiring lesstransmission energy. On the other hand, since M2 aimsto minimize the highest energy usage at a sensor node, itcan do so by creating long multi-hop routes from sensorsto the sinks so that each sensor dissipates only a smallamount of energy. In both cases, we face the issue of quickenergy drainage which occurs at certain nodes in the M1case and in the whole network in the M2 case. Limitingthe usable energy reserve at sensors can alleviate thisproblem by making only a fraction of the energy availableat the sensor nodes in each period.We develop the third model in order to address thedrawbacks ofM1 andM2 via the objective function forma-tion that promotes, from the perspective of lifetime maxi-mization, a more favorable energy depletion pattern in aConstraint (17) and (18) impose a variable upper bound onthe energy consumed at a sensor.M3 Minimize the weighted (where t is the weight) sum of averageenergy consumption and range of remaining energy levels:Min tXm2Jecm Xi2Iei !,jIj ERmax ERmin 20subject to 112 13and14ERmin 6 Ej ecj 8j 2 J 21t ecm ei jI j ERmax ERmin 6 bZ ; 28representations.for M2 and the other for M1 and M3.all that in Men consumed eis ast one sens tin the sensor eva Since the en atr ission, it m tar ead all over -ua sor or a sma -tio y high energ dof over the se -ri given in Alg t,w te that Cons ,i.e core algori ;an e best of al en lution.h start rstfr he candidat t840 H. ster, H. Lin / Ad Hoc Networks 9 (2011) 835851In the rst one, we represent a solution by a set of xedCHs given by set C fj 2 J : zcj 1g. To evaluate the good-ness (objective value) of such a solution, we solve a sub-problem, denoted by SubC, by simply xing thecorresponding binary variables in C at 1 in the model ofinterest. Thus, we obtain the sink locations and data rout-ing along with the objective value ZC. In the second case,we represent a solution by the xed sink locations, i.e.,D fk 2 K : zuk 1g. To evaluate the goodness of a givensolution, we solve a subproblem, SubU, which is derivedfrom the model of interest by using D, and obtain theCHs and data routing along with the objective value ZD.m2J i2Irespectively. Furthermore, forM3, using the following twological inequalitiesERmax ERmin P 0 andXi2IEi Xm2Jecm Xi2Iei ! !,jIj 6 ERmaxin conjunction with Eq. (28), we obtain an additional cutinequality given bytXi2IEi=jIj ERmax !6 bZ 29Thus, we are able to benchmark the quality of our heuristicsolutions by employing a wider set of problem instances.We develop an efcient overall solution procedure that ad-dresses all three models,M1,M2 andM3, which differ onlyin the initial solution construction component. Before wepresent this complete procedure, we discuss its compo-nents in detail.4.1. Solution representations and subproblemsAlthough the models considered here differ structurally,they embody the same sets of binary variables. Thus, weutilize the binary variables to obtain two solutionWhile developing the solution approaches and conduct-ing the computational studies using both the heuristicmethods and the exact branch-and-cut approach, we alsoincorporate objective function value-based cuts into themodels (in exact approach) or subproblems (in heuristicapproach) in order to signicantly improve their solutiontimes. In particular, supposing that an upper bound bZ forthe model of interest is available, then for M1, M2, andM3, we utilize the cut inequalitiesXm2Jecm Xi2Iei 6 bZ ; 26ECmax 6 bZ ; 27andX X !,of the CHs (H 1 of them) is picked based on distance insuch a way that a good spread of CHs in the sensor eldis promoted. Distance dj; C between a node j and the setof already selected CHs (C) is measured as mini2Cdj; i.We pick the next CH as being the farthest away from thealready selected CHs (lines 79). Once a solution Sc is ob-tained after all of the CHs are picked this way, its goodnessis evaluated by solving the problem SubC described above.If Sc is better than the best solution so far (Sb), then it be-comes the new Sb (lines 1114). In our computationalstudies, we use a Maxiter value of jJ j=2 if jJ j < 100 andjJ j=5 otherwise.Algorithm 1. ConstructM2()1: initialize ZSb 1, Maxiter;2: while Maxiter > 0 do3: C ;, h = 1;4: Randomly pick a node i from J ;5: C CSfig, C J n fig;6: while h < H do7: j argmaxfdj; C : j 2 Cg;8: C CSfjg, C C n fjg;9: h = h + 1;10: end while11: Sc C, solve SubC to obtain ZSc;12: if ZSc < ZSb then13: Sb Sc , ZSb ZSc;14: end if15: Maxiter =Maxiter 1;16: end while Sb and ZSb4.2.2. Construction Heuristic for M1 and M3Both M1 and M3 mainly aim to minimize energyconsumption with an additional requirement for M3,om trandomly picks a single node as a CHe set of CHs (lines 45). Then the resal soEacd th l the Maxiter solutions obtained is th., its thm (lines 315) is run Maxiter timese no tructM2() is a multi-start approachstic, orithm 1, we proceed as follows. FirsCHs nsor eld, in the ConstructM2() heunall y consumption. To ensure this spreal sen ll set of sensors are subjected to excepe spr the sensor eld. This way, no individansm akes sense to employ a set of CHs thalue. ergy consumption is largely due to datall s) uniquely determines the objectivat le or whose energy consumption (highesergy by a sensor. Then, in a solution, therRec 2, we aim to minimize the maximum4.2.1. Construction heuristic for M24.2. Construction heuristicsWe devise two heuristics to nd initial solutions, onew y nis m sto eCH e -in s oth ne dto s nin or ewhole sensor eld in two nested parts, a box centeredwe later utilize in our complete procedure for improvingthe solutions obtained via construction heuristics. In theby a construction heuristic. Inspired by the variabledepth search approach presented by Lin and Kernighamte hite n ga b gth m tso na dfu n -chse ), ngo s -pl at fth ist )in r -tio , -se ex eva e tse es enu r nin h yoc Networks 9 (2011) 835851 841in the sensor eld and the band around it. In particular,the candidate CHs in the rst part, denoted by F 1, arethe ones that are in a square of size b centered in thesensor eld. Then, the set of candidate CHs in the secondpart (close to periphery of the eld), called F 2, is simplygiven by J n F1.In this process, we also attempt to avoid coincidentallywell-positioned sensors (from an energy dissipation mini-mization point-of-view) being selected as CHs repeatedlyin successive periods and to protect low-energy sensorsfrom being selected as CHs. For this purpose, we consideronly a subset of sensors with higher-energy as the set ofcandidate CHs J . We denote this subset as IR since it isa subset of sensor set I . Specically, to determine the IRset, we use a threshold value THW calculated as W%of the average initial energy level at the sensors, i.e.,TH W W=100 Pi2IEi=jI j) and IR fi 2 I : Ei P TH Wg.The ConstructM1M3() heuristic is also a multi-startapproach as ConstructM2() and we use the same maxi-mum number of iterations, Maxiter, as in ConstructM2().In each start, we proceed as follows (lines 523 in Algo-rithm 2). We rst pick about a fraction a of total requiredCHs in F 2 (periphery band), specically we pick a total ofba Hc CHs from F 2. For this, we start by randomly pickinga CH and then choose the rest of the CHs one at a time atthe median distance from the currently selected ones (lines512). Then, the rest of the CHs (i.e., H ba Hc) arepicked from set F 1 similarly (lines 1318). We determinethe goodness of the obtained solution Sc by solving theassociated SubC. We update the best solution Sb by Sc ifnecessary (lines 1923).In our implementations, before we choose the CHs, wecheck the number of candidates in F 1 and F 2 starting withan a value of 0.80 and a b (size of the center box) valuedetermined as bN/2c 2 where N is the size of a squaresensor eld. If there are not enough candidates in sets F 1or F 2, we increase or decrease b by one unit. Note that,by changing the value of a, the ratio of CHs located aroundthe periphery to those in the inner part of the sensor eldcan be changed. Also, we use aW value of 60 in our numer-ical studies.Algorithm 2. ConstructM1M3()1: initializeF1 = F2 = ;, ZSb 1, Maxiter;2: Form F 1#J using the center square of size b;3: F 2 J n F 1;4: while Maxiter >0 do5: C ;, h = 1;6: Randomly pick a node i from F 2;7: C CSfig, C F 2 n fig;8: while h < bH ac doAlg ithm 2, aims to achieve this by dividing thward sinks. The ConstructM1M3() heuristic, giveat e rgy efcient natural paths can be establisheg CH are chosen from the center of the sensor eld ss ar selected close to the periphery while the remainward the sinks, it is preferable to ensure that somalso inimized. Since the data ow in the network ihereb the range of the remaining energy distributioH. ster, H. Lin / Ad Hmbeeacof neighboring solutions of a current solutioiteration. We enhance its efciency further barch a large solution space as it generates a largriabl depth search approach is a technique that of changeable CHs. Furthermore, we note that thn Sc we use a derived set, Scfree, to represent this subgene ating neighborhood solutions. For a current solue ex ing CHs can be exchanged (assigned as non-CHain l er in Section 4.4, we assume that only a subset oodne s is monotonously improved. However, as we exnsor h is increased up to hmax as long as the solutioange (a single pair-exchange of a CH and a non-CHnctiorticular, we employ an h-exchange neighborhoowhere, in general terms, starting with a 1-exlutioIn p.e sa e number of CHs as non-CHs in the currennum er of non-CHs serve as CHs while unassigninratio of the algorithm, CH set is modied by makinnded form of exchange neighborhood on C. In eac[23], it searches the solution space by applying an ex-rst method, we employ the CH set C as the solution repre-sentation and accordingly utilize subproblem SubC to eval-uate the goodness of the solution. In the second one, werepresent a solution by the set of selected sinks D and em-ploy the subproblem SubU for evaluating the goodness of asolution. This approach also includes nding a randomizedinitial solution, similar to nding CHs in ConstructM2()heuristic.4.3.1. The Search SubC() Improvement ProcedureThis procedure starts with a feasible solution, de-noted by Sbinit , which includes the CHs as determined9: Pick j* which has the median dj; C;8j 2 C;10: C CSfjg, C C n fjg;11: h = h + 1;12: end while13: C F 1;14: while h < H do15: Pick j* which has the median dj; C;8j 2 C;16: C CSfjg, C C n fjg;17: h = h + 1;18: end while19: Sc C, solve SubC to obtain ZSc;20: if ZSc < ZSb then21: Sb Sc , ZSb ZSc;22: end if23: Maxiter=Maxiter 1;24: end while25: return Sb and ZSb4.3. Solution improvement proceduresIn this section, we devise two search procedures that20: if gh > G then21: G* = g ;23: *4. he Seahis me don k selec(s oblem -in cess. A sty ly not -da s J , a -ch s, sim ytim onsum eso n spac yem ding a oan rithm eso a Subre spec nin rithm -ite ich we j,e. j=2 as ytw mpone ea ion by dth ers by .Th proach -tio at sink hot will pr t842 H. ster, H. Lin / Ad Hoc Networks 9 (2011) 835851utilizing both the proximity and the energy level infor-mation at the sensor nodes leading to a considerationof restricted and dynamic neighborhoods for CHs. In par-ticular, we use distance and energy information as fol-lows: for a CH to be exchanged, we consider only thenon-CH sensors in J which are within its q radius andhave at least a THW energy level. Then, the neighborhoodfor each node is clearly restricted to a subset of J . Also,it is very likely that the possible non-CHs that can be ex-changed with a CH vary in each period due to changingenergy levels. This neighborhood generation approachhelps to limit the neighborhood size with only a smallimpact on the network lifetime since the low-energysensors are prevented from being repeatedly employedas CHs.The SearchSubC() heuristic proceeds as detailed inAlgorithm 3. The outside while loop, lines 228, controlsthe search procedure by conditioning on the maximumallowable exchange neighborhood size and the continuityof improvements over the best solution Sb. Since, as hincreases, the generation of a neighborhood and the eval-uation of the solutions require longer runtimes, we limitthe exchange neighborhood size by hmax. For a given hvalue, we generate the neighborhoods of j-exchange,j = 1, . . . ,h, in order (lines 318). For each j, we performat most Maxiter j-exchanges (lines 513). For each suchexchange i, we randomly pick j nodes from the Scfree setof the current solution and generate a neighborhood solu-tion set Xi. The generation of Xi for j = 1 is straightfor-ward. For jP 2, we proceed by generating individualneighborhoods of j CHs (as in j = 1 case) and then, weconsider all feasible combinations for a j-exchange. Toevaluate the goodness of a solution generated, we employthe subproblem SubC and assign the most improvingsolution after Maxiter iterations as the current solutionSc. If an improvement gj over the best solution Sb (line14) is obtained, then we update the best solution (lines1517). After this for loop is completed, considering allpossible j values (lines 318), in line 19, we calculatethe cumulative improvement gh for h-exchange neighbor-hood search, and if it improves the highest so far improve-ment quantity G*, G* is also updated (lines 2022). Theoperations on lines 2325 aim mainly at improving thesolution time in the neighborhood search. Specically, ifthe highest objective improvement amount is greaterthan a maximum preset value Gmax and the neighborhoodradius q is greater than a preset minimum value, wedecrease q by one unit so that, in the next maximumneighborhood size h, the generation and evaluation ofneighboring solutions in each iteration i (lines 69) areless time consuming.We note that, while evaluating a neighboring solutionvia solving SubC on line 8, we also employ the cut inequal-ities (26) for M1, (27) for M2, and (28) and (29) for M3. Inall cases, we replace bZ with ZSb and observe signicantimprovement in solution times. In our numerical studies,given in Section 5, we set the values of hmax, initial q, qmin,and Gmax as 3, 5, 2 and 0:1 ZSbinit. In addition, we use aMaxiter value of H if jJ j < 100, H/5 if jJ j > 200, and H/2otherwise. locations; thus, utilizing a spread set of sinks leads to moren thhers that are signicantly separated from eacomote a divergent data ow to well-apare ap in this rst part is motivated by the observae oth ensuring a good separation between themsolut picking the rst sink location randomly ano co nts. In the rst part (lines 39), we generatg. jK in Section 5. The procedure involves mainlr, wh recommend to be set much less than the jKAlgo 4, includes a xed number of iterations, MaxMo ically, the procedure SearchSubU(), givelving U whenever needed.algo and by employing the cut inequalities whilbed randomized construction component intlutio e over a set K can efciently be searched be c ing. On the other hand, we observe that thange ilar to the one described above, is still verte CHvery large when compared to the set of candineighborhood search procedure based on exg propicallthough the candidate set of sink nodes K iubpr for xed sinks D) is generally a time-consumsin tion. Finding the optimum solution of SubUIn t thod, we use a solution representation base3.2. T rch SubU() Improvement Procedure27: end while S and ZS 26: h = h + 1;b b25: end if24: q = q 1;if G > Gmax and qP qmin thenh22: end ifAlgorithm 3. SearchSubC()1: initialize h = 1, hmax, G* = 0, g0 =1,Sbinit (from a construction heuristic),Maxiter, Gmax, q, qmin, J ;2: while h 6 hmax and G* = gh1 do3: for j = 1 to h do4: Sc Sbinit5: for i = 1 to Maxiter do6: Randomly pick j nodes from Scfree7: Generate a neighborhood solutions set Xi;8: For each C 2 Xi, solve SubC to obtain ZC;9: Si argminfZC : C 2 Xig10: if ZSi < ZSc then11: Sc Si;12: end if13: end for14: gj ZSb ZSc;15: if gj > 0 then16: Sb Sc , ZSb ZSc;17: end if18: end for19: gh Phj1gj;*solution so far, Sb, is updated if improved by the currenttion ag tcu q yas Xva yef -po f -pe e nit eneAl8: u = u + 1;tial solution Sb and record the objective value ZSb (line 7).Al i5:H. ster, H. Lin / Ad Hoc Networks 9 (2011) 835851 8439: end while10: Sc D, solve SubU with EpGap, TiLim, andcut with bZ ZSb to obtain ZSc;11: if ZSc < ZSb then12: Sb Sc , ZSb ZSc;13: end if14: Maxiter =Maxiter 1;15: end while16: Solve SubU with TiLim and cut with bZ ZSb;Sb and ZSb.4.4. The complete procedureOur complete procedure brings together the abovecomponents including construction and improvementheuristics. We apply the complete procedure to solve M1,M2, and M3 by varying only the construction heuristiccomponent, denoted by ConstructM (), in Algorithm 5.In particular, to obtain diversication in searching thesolution space, we initially generate Q solutions whereeach solution is represented by a set of CHs, C(lines 23). One of these Q solutions is found by using7: D DSfkg, D D n fkg;6: k argmaxfdk;D : k 2 Dg;5: while u < U do4: D DSfkg, D K n fkg;3: Randomly pick a node k from K;2: while Maxiter > 0 doMaxiter, D ;, u = 1;1: initialize Sb (the best available solution),gorithm 4. SearchSubU()xt.is emcomputational times. This is especially true whebedded in the overall procedure that we describse onsivexploring the solution space efciently with inexfective in terms of solution quality and serves the purlue of 10 . Although this procedure is simple, it is verthe stopping criterion, i.e., EpGap is the default CPLE6t ineain by solving the subproblem SubU with the besuality; however, this time we consider TiLim onlsolution Sc , and a new iteration is started. Once the itera-tions are completed, we attempt to improve the nal solu-uniform energy depletion in the network. In the second part(lines 1014), we assign the solution obtained from therst part as the current solution, and solve the associatedsubproblem SubU by also incorporating the cut inequalitycorresponding to the particular model being solved, specif-ically, (26) for M1, (27) for M2, and (28) and (29) for M3,with a bZ value equivalent to ZSb. While solving the sub-problem, again to alleviate the problem of excessive run-times, we employ a stopping criterion given by a EpGapoptimality gap or TiLim time limit, whichever is reachedrst (EpGap and TiLim are CPLEX parameters). The best8: Apply SearchSubU() using Sb as the inputsolution. Sb and ZSb5. Computational studyThe objective of our computational studies in this sec-tion is threefold. First, we examine the impact of usable en-ergy fraction p on network lifetime. Second, we consider asingle-period setting and evaluate the performance of ouralgorithms on the basis of solution quality and time via uti-lizing exact solutions for benchmarking. Third, utilizingour algorithms, we evaluate the effectiveness of the threemodels in prolonging the network lifetime in a multi-per-iod setting. To this end, we compare M3 with both ourbenchmark models M1 and M2 as well as with HEED [42].Unless stated otherwise, all of the input and algorithmicparameter values are set as mentioned previously. Further-more, we generate jI j sensor coordinates randomly using auniform distribution in a square of size N meters(m). Thecandidate sites for sinks, K, are also generated randomlyon the periphery of the sensor eld. We set the periodlength as T = 4000 time-units and the aggregation ratio as7:Determine the Sfree associated with S ;Apply SearchSubC() to Sb and record ZSb;6:Assign the best of Q solutions as the Sb;b b4:randomly;Improve each of Q solutions using SearchSubC();3:ConstructM ()Generate Q 1 CH based initial solutions2: Generate an initial solution usingConstructM (), SearchSubC(), andSearchSubU();initialize Algorithmic parameters for1:gor thm 5. Complete approachIn the last stage (line 8), we utilize the procedure SearchSubU() with Sb used in the initialization step.the appropriate construction heuristic and the others aregenerated randomly. Note that both ConstructM2() andConstructM1M3() already employ the C set to representa solution. We then apply the neighborhood search proce-dure SearchSubC() to each of these solutions indepen-dently, assuming Sbfree Sb (line 4). That is, we perform anintensied search in the exchange neighborhood of each ofQ solutions. Upon completion, we identify the common, ifany, CHs that appear in all of theQ solutions obtained; theseCHs are clearly favored in each solution, thus, we choose tokeep them in the nal solution. We also identify the bestsolution among the Q solutions, assign it as the Sb (line 5),and form the set Sbfree as the CHs in Sb that are not commonin all of the Q solutions (line 6). As mentioned in Section4.3.1, subsetSbfree represents the exchangeable CHs in a solu-tion Sb. We treat the CHs that are common to all of the Qsolutions as a preferable (non-exchangeable) CHs, and, thus,we do not engage them in the improvement procedureSearchSubC().We then apply the SearchSubC()with ini-tings with number of sensors, jI j, less than 100 and 3 forent instance sets forM1, 18 forM2, and 12 forM3 obtainedby varying the values of jI j and the number of requiredCHs H, we solve 30 randomly generated instances.The values in columns 47 of Table 1 are averaged overthe instances. The fourth (TOave) and the fth (TOCave ) columnsreport the exact solution times (in seconds). In the former,we report the average runtimes (ns represents the in-stances not solved in 20 h), whereas in the latter, we reportthe same measure; however, this time we also employ cutinequalities in solving the models exactly as described inSection 4. Clearly, for an instance, the optimum objectivefunction values, ZO, for both of these cases are the same.However, the solution times are signicantly improvedwhen cut inequalities are employed, as observed in columnfor TOCave . The sixth column (THave) includes the average run-times obtained using our complete approach given in Algo-rithm 5. We observe that they are quite signicantly lowerthan the runtimes for exact solutions. In our tests, we con-sider only one additional random initial solution in Algo-rithm 5 which provides enough diversication and844 H. ster, H. Lin / Ad Hoc Networks 9 (2011) 835851s = 0.3. In addition, we set the weight for the average en-ergy dissipation component of the objectives M3 as t = 5which we empirically determined as a reasonably goodvalue.The computational studies are performed on a machinewith Pentium D 3.2 GHz CPU and 2.0GB RAM and the algo-rithms are implemented in C++utilizing STL (StandardTemplate Library) and Concert Technology when CPLEXwas used.5.1. The Impact of Usable Energy Fraction p on NetworkLifetimeTo gain insight into the models M1, M2, and M3, andverify their effectiveness, we empirically examine the im-pact of topology control parameter p, which is the fractionof total energy available at a sensor made usable in a per-iod, on network lifetime. For this, we employ a data set of10 sample instances and optimally solve the models foreach instance with varying p values using exact branch-and-cut implementation in CPLEX 9.0. The test instancesare generated randomly with xed N, jKj, jI j, U, H, and Rivalues of 50, 8, 15, 2, 3, and 10 bits/time-unit, respectively.We run each instance for a full deployment cycle (life-time) 10 times, where each of the 10 runs corresponds toinitial p values (denoted by pI) ranging from 0.1 to 1.0.We start the rst period of an instance with a pI value asthe current p value and, whenever the solution to the mod-el is infeasible for a period, we increase the current p valueby an increment (inc) value for which we assume a value of0.1. The most recent value of p in a period is carried as thecurrent p value for the next period. The end of the deploy-ment cycle is marked when p is 1.0 (whole available energyis available to use at the sensors) and the models solutionis infeasible. In general, a topology control scheme speci-ed as such can be represented as (pI, inc) and it shouldbe calibrated, i.e., the values of pI and inc should be deter-mined for a specic application via experimentation.We summarize the results in Fig. 1 where the averagelifetimes over 10 instances for each pI value are plotted.We observe that the performance of M1 and M2 is highlysensitive to the initial value of p. If we do not restrict themaximum energy usage for each node, i.e. start the deploy-ment cycle with a large p value, the M1 and M2 performvery poorly. It seems that determining a good p value, pos-sibly via a conservative use of energy from the beginning ofa deployment cycle and a calibration of its increments, pro-vides a good lifetime measure. On the other hand, it isinteresting to observe that M3 is inuenced very little, ifat all, by the choice of pI value, i.e., it is highly robust tothe changes in the amount of energy reserved for usageat the sensors. We can explain these differences in the per-formance of the models with respect to the p value as fol-lows. If the energy usage is not constrained in M1 and M2,then the same underlying network, given by the optimummodel solution, is repeatedly obtained. This leads to quickenergy depletion at certain sensors, thus shortening thenetwork lifetime. M3 is a multi-objective approach; itincorporates remaining energy measures into the optimi-zation and attempt to address energy dissipation both interms of minimizing its average in the network and itsthe ones with jI j values greater than or equal to 100. Wealso assume that the size of the sensor eld, N, is 50 forall the settings, except the ones with jI j equal to 150 forwhich N is 75. Furthermore, we set the data generation rateRi to 10 for settings with jI j less than 75, and to 9, 8, and 7for the ones with jI j values of 75, 100, and 150, respec-tively. For each problem setting, where we have 15 differ-variation among sensors. Thus, it promotes a favorabletransition in terms of energy status in the network fromperiod-to-period. The choice of usable energy fraction pis insignicant in M3 because its role is already effectivelyembedded in the model, leaving no question about howthe value of p should be calibrated from period-to-period.5.2. Performance of heuristicsTable 1 summarizes the results for all three modelswhere we assume that the number of candidate sink loca-tions, jKj, is 8 and required sink locations, U, is 2 for set-10203040506070800.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Initial pAverageNumberofPeriods(M1) (M2) (M3)Fig. 1. Average network lifetimes under varying initial p values.352H. ster, H. Lin / Ad Hoc Networks 9 (2011) 835851 845Table 1Performance results for solving single period problems.Model jI j H TOaveM1 25 2 0.634 2.165 2.5350 4 22.428 19.0910 17.7875 6 112.5212 83.9015 63.53100 8 963.0716 567.5220 377.12150 12 6646.524 3600.330 2620.8M2 15 2 5.10notable improvements in solution quality. In terms of solu-tion quality, we calculate the optimality gap asD = 100 (ZH ZO)/ZO, where ZH represents the heuristicobjective value and ZO is determined using cut inequalitieswhenever necessary. The average percentage optimalitygaps Dave, reported in the last column, illustrate that ourapproach is highly effective.5.3. Network lifetime comparison of modelsTo examine the performance of the models in terms ofnetwork lifetime measure, we use our heuristic procedurein a multi-period setting under two topology controlschemes (pI, inc) as (1.0,0.0) and (0.1,0.1). In Table 2, wepresent our results for a number of instance sets, each gi-ven in a row, and 10 randomly generated instances foreach set. The rst part of the table corresponds to the case3 25.224 58.9820 2 19.233 123.474 424.3725 2 87.144 1845.175 7698.0930 2 220.044 8339.406 ns40 4 ns6 ns8 ns50 4 ns8 ns10 nsM3 15 2 2.793 7.534 17.0920 2 8.363 38.264 125.1525 2 23.504 755.865 3105.9630 2 56.024 2243.286 nsTOCave THaveDave0.98 2.05 0.872.32 2.21 0.662.75 2.41 0.5824.69 13.09 0.7924.48 13.05 0.1119.00 13.96 0.21127.15 82.81 0.1382.63 67.67 0.3079.31 63.34 0.19902.92 220.51 0.50649.54 132.20 0.12472.15 122.78 0.125962.81 427.96 1.163227.57 245.97 0.273483.89 274.84 0.412.80 1.98 1.09where no calibration for usable energy fraction is done; p isset to 1.0 so that the complete energy at each sensor isavailable for usage. Once again, M3 performs signicantlybetter in terms of network lifetime thanM1 andM2, whichperform particularly poorly.Thus, we further consider the (pI, inc) = (0.1,0.1) schemeas reported in the second part of Table 2. In this case, life-time performance with bothM1 andM2 improves dramat-ically, illustrating the effectiveness of employing a usableenergy fraction p. Observe that while M2 performs betterthan M1 under p = 1.0, this is reversed under the (0.1,0.1)scheme. On the other hand, it is clear that even M1 stillcannot perform as well as M3. Although we try only onescheme out of numerous possible ones, it is clear that agood strategy for adjusting the value of p, i.e., initial usableenergy fraction pI and its increments inc, in a deploymentcycle is needed for extending network lifetimes. However,2.30 2.08 1.392.54 1.99 1.333.28 4.35 1.045.87 4.63 1.136.68 4.57 1.1015.75 9.09 0.7022.31 9.27 0.8233.88 9.14 0.8920.72 16.66 0.5377.84 17.64 0.70179.87 16.57 0.79300.36 52.10 0.52944.07 49.71 0.571241.67 49.07 0.671186.85 126.37 0.61848.82 108.19 0.844938.61 106.98 1.112.68 3.06 0.007.15 11.77 0.1416.12 29.30 0.166.72 8.81 0.2129.27 23.69 1.09145.50 72.65 0.2623.38 13.22 0.1282.20 70.22 0.101081.02 99.12 1.2650.87 52.16 1.40212.30 60.01 0.228233.12 279.70 0.66thod.)M122122122123123123123134846 H. ster, H. Lin / Ad Hoc Networks 9 (2011) 835851there are two issues associated with such an approach.Table 2Comparison of different objective in the multi-period based on heuristic me(pI, inc) = (1.0,0.0Problem setting H M1Set1 2 9jI j 25, jKj = 8, 4 11U = 2, N = 50, Ri = 10 5 12Set2 4 7jI j 50, jKj = 8, 8 9U = 2, N = 50, Ri = 10 10 11Set3 6 5jI j 75, jKj = 8, 12 12U = 2, N = 50, Ri = 9 15 14Set4 8 20jI j 100, jKj = 16, 16 15U = 3, N = 50, Ri = 8 20 13Set5 12 10jI j 150, jKj = 16, 24 12U = 3, N = 75, Ri = 7 30 18Set6 16 13jI j 200, jKj = 16, 32 10U = 3, N = 75, Ri = 6 40 22Set7 20 7jI j 250, jKj = 16, 40 9U = 3, N = 100, Ri = 5 50 12Set8 24 11jI j 300, jKj = 16, 48 24U = 3, N = 100, Ri = 4 60 20First, when changing problem size and environmentdepending on the application, calibrating for a good valuep for M1 and an inc for a (pI,inc) scheme during a deploy-ment cycle is very difcult and impractical. Second, evenif a topology control scheme, such as (pI, inc) = (0.1,0.1)),is employed, nding the feasible p in each period impliessolving the same problem multiple times with varying pvalues. On the other hand, we observe that none of thesedifculties is associated with M3 since it is very robust tothe changes in p, and, thus, it truly integrates the topologycontrol and the routing problems effectively.We next examine the remaining energy distributions ina relatively large problem setting both from the modelsperformance perspective and their impact on the redeploy-ment strategies. For this purpose, we consider the Set6. InFig. 2ac, we plot the minimum, the average, and the max-imum remaining energy levels (ERmin, ERave, and ERmax, respec-tively) at the sensor nodes after each period for eachmodel. Furthermore, in Fig. 3ac, we give the remainingenergy levels of the sensor nodes at the end of the deploy-ment cycle.In Fig. 2ac, we observe that the variations in remainingenergy levels forM1 andM2 are very large when comparedto M3. More specically, for M1, the ERmin and ERmax valuesare apart from each other, thus implying a large variationin remaining energy levels. This is also illustrated by theend-of-deployment-cycle energy levels, which are quitescattered for the sensors, as observed in Fig. 3a. Since M1concentrates on minimizing total energy usage, some sen-sors are more frequently chosen as CHs and deplete theirenergy more. On the other hand, sinceM2 does not address(pI, inc) = (0.1,0.1)2 M3 M1 M23 61 60 492 66 59 576 69 63 512 62 50 530 67 58 555 69 51 543 71 59 511 76 62 596 78 54 555 87 77 585 93 81 631 93 67 654 78 63 445 85 70 520 87 56 586 92 80 538 100 78 584 102 78 656 92 62 438 100 80 576 103 74 649 115 95 546 126 91 684 130 87 61any network-wide energy usage measures, the objective ofminimizing the maximum energy usage implies one ormore critical sensors whose energy usage is determinantof the overall performance. The usage level determinedfor the critical sensor(s) also dictates the energy usageallowance (upper bound) for other sensors, and the relativecloseness of the end-of-period ERave and ERmin values in Fig. 2aand b show that the sensors dissipate their energy at sim-ilar rates. This is also observed in Fig. 3b where most of thesensors remaining energy levels, although somewhat scat-tered, are more clustered close to zero. In bothM1 andM2,the energy depletion rate at the sensors is also implicitlydetermined by the (pI, inc) = (0.1,0.1) scheme. The use ofmore stringent schemes generally causes infeasibilities,and, thus, the fraction p is incremented early in the deploy-ment cycle, and an exact calibration of p is very difcult.Coupled with the results given in Table 2, it is clear thatthe energy reserve scheme is the main reason we obtainrelatively better lifetime measures with M1 and M2. InFig. 2a and c, we clearly observe that M3 always has thehighest ERmin and the lowest ERmax. Furthermore, these valuesare very close to each other, presenting a narrow range (asmall variation) in the remaining energy levels, which im-plies that most of the sensors deplete their energy more orless at the same rate during the progression of the deploy-ment cycle. This is also clearly reected in Fig. 3c in whichthe remaining energy levels form a very narrow band.Since the minimization of total energy usage is also consid-ered in M3, the depletion rate is slow, providing a goodnetwork lifetime measure without relying on an explicitH. ster, H. Lin / Ad Hoc Networks 9 (2011) 835851 847control of the usable energy reserve scheme. This is notunexpected, since we formulate the objective function ofM3 to incorporate this characteristic as well.Fig. 2. Remaining energy characteristics (progression).Finally, from a redeployment strategy perspective, M3is appealing since it contains primarily low-energy sensorsFig. 3. Remaining energy distributions (end-of-deployment-cycle).848 H. ster, H. Lin / Ad Hoc Networks 9 (2011) 835851at the end of the deployment cycle. This can facilitate arandom deployment in the beginning of each cycle, so asto start with similar initial settings. On the other hand,M1 and M2 require special deployment and/or furtherattention to topology control in successive deploymentcycles due to a relatively high overall variation in theend-of-deployment-cycle energy levels.5.4. Network lifetime comparison of M3 and HEEDTo further evaluate the performance of our proposedmodel, we compare model M3 with Hybrid Energy-Ef-cient Distributed clustering (HEED) [42] which is a well-known method for its performance in terms of networklifetime. As mentioned in Section 2, HEEDs main goal isto identify CHs and assign sensors to clusters for better en-ergy efciency. The CHs are probabilistically selected basedon their remaining energy and the sensors join clusters soas to minimize the communication cost. HEED does notspecify a particular scheme for routing CHs to the sink afterthe clusters and CHs are determined; however, the authorsspecically mention routing to achieve minimum powerusage across the network as a possible approach amongothers. For detailed description on HEED protocol, we referthe reader to Younis and Fahmy [42].We choose HEED for comparison because it has the fol-lowing features similar to the setting we consider: (1) It isa cluster-based routing protocol for data gathering pur-pose; HEED assumes a multi-hop connection betweenCHs and to the sink and, at each period, it selects CHs withhigh remaining energy. (2) Data collection is performedperiodically which is suitable for continuous monitoring.(3) Data aggregation is performed at each clusterhead forenergy efciency.On the other hand, there are also some differences inthe model assumptions of HEED andM3 : (1) In HEED, onlyone sink node is considered and each sensor can onlychoose one CH to transmit its data; (2) The cluster radiusis explicitly specied, i.e., each sensor can only transmitits data to a CH within the specied range. (3) The data isaggregated into a single representation signal at each CH,i.e., regardless of the amount of data received, a CH trans-mits a xed amount of data out towards the sink, thus thesink eventually receives this xed amount of data from thenetwork (e.g., maximum temperature in the sensor eld).(4) HEED adopts the radio model where both the free space(typical D2 power loss) and the multipath fading (D4 powerloss) models are used. To transmit xij (bits) of data fromnode i to node j dissipates f(Dij)xij, where f(Dij) is dened asf Dij w v1D2ij if Dij < d0w v2D4ij if Dij P d0(30with parameters set as d0 = 75m, v1 = 10 pJ/bit/m2,v2 = 0.0013 pJ/bit/m4, and w = 50 nJ/bit.In theM3model, we consider multiple sinks and a gen-eral aggregation scheme in which, instead of a single rep-resentative data for the sensor eld, a more general viewof a measure is of interest, e.g., temperature/humidiy/pres-sure gradients in a large sensor eld employed for environ-mental monitoring. More importantly, M3 also integratesselection of CHs and sinks with routing decisions. There-fore, for comparison purposes, we modify model M3 tohandle the several characteristics of the setting in HEEDby introducing cluster radius, single sink, and the new dis-tance representation as follows.First, we introduce two new parameters into M3: r asthe cluster radius and k as the total amount of data per per-iod generated at each sensor. Furthermore, we redene thevariable xcij as a binary variable with a value of 1 if a sensor iis assigned to a CH j, and 0 otherwise. We also note thatsince there is a single sink we use the variable xum0, whereindex 0 represents the sink, for ow from CH j to the sink.Then, the modied formulation, called eM3, is(i 2 I ; j;m 2 J unless stated otherwise)Min t1=jIjXm2Jecm Xi2Iei ! ERmax ERmin 31s.t.f Dm0kxum0 Xj2Jnfmgf Dmjkxccmj Xj2Jnfmgwkxccjm w cskXi2Ixcim ecm 8m 32Xj2Jf Dijkxcij ei 8i 33xum0 Xj2J nfmgxccmj Xj2Jnfmgxccjm 1 sXi2Ikxcim ! 0 8m34Xj2Jxcij 1 8i 35xcij 6 br=Dijczcj 8i;8j 36xccmj 6 kjI jzcj 8m; j 37xcj0 6 kjI jzcj 8j 38ei 6 pEi 8i 39ecj 6 pEj 8j 40Xj2Jzcj H 41zcj Ej ecj 6 ERmax 8j 421 zci Ei ei 6 ERmax 8i 43ERmin 6 Ei ei 8i 44ERmin 6 Ej ecj 8j 45xcij; zcj ;2 f0;1g 8i; j 46xccij ; xuj0; ei; ecj ; ERmax; ERmin P 0 8i; j 47(32) and (33) assign the values of the total energy con-sumed by a CH and a sensor node, respectively. Constraints(34) state the data ow balance at each CH node and con-straint (35) guarantees that each sensor is assigned to oneCH. Constraints (36)(38) assign the values of binary vari-ables related to CH location selections. Note that con-straints (36) also ensure that only the sensors within thecluster radius r can transmit the data to the associatedCH. Constraint sets (39) and (40) ensure that the total en-ergy consumed at a node cannot exceed the total availableenergy at the corresponding sensors. Constraints (41)H. ster, H. Lin / Ad Hoc Networks 9 (2011) 835851 849establish the required number of CHs. Constraint sets4245 give the maximum and minimum remaining energyat a sensor node. Finally, (46) and (47) include the integral-ity and non-negativity of the decision variables.Our solution approach in Section 4 is directly applicableto solve the modied model as well. However, since onlyone sink is employed in eM3, steps 7 and 8 in the Algo-rithm 5 are excluded.As mentioned above, the main component of the HEEDis the selection of CHs and the assignment of sensors toCHs i.e., forming the clusters); no specic routing protocolto compute inter-cluster paths between CHs or to the sinkis given in HEED. Thus, as suggested in the context ofHEED, we minimize energy usage by employing a subprob-lem of our M1 model (which minimizes average energyusage in the network) while determining data routing fromsensors to the sink via CHs. In doing so, we readily incorpo-rate the same general aggregation in M3 in the routingscheme sinceM1 employs the same aggregation approach.In each period, we obtain the formation of clusters, i.e., thevalues of xcij and zcj using HEED algorithm, then we solve themodel M1 by xing the binary variables xcij and zcj to thesevalues to obtain data routing given by variables xccij and xuj0.For our studies, similar to the setting in HEED, we as-sume the nodes are randomly distributed in square of size100m sensor eld (with its lower left one corner at the ori-gin) and the sink node is located at coordinates (50, 175).The cluster radius r is set 25m and the amount of sensordata generated per period, k, is taken as 2000 bits. Theaggregation ratio, s, values of 0.15, 0.20, 0.25, and 0.30;and the number of CHs, H, of 9, 10, 13, and 15 are usedfor sensor elds with the number of sensors, I , of 150,200, 250, and 300, respectively. In doing so, we ensure thatthe number of CHs is about 5% of the number of sensors,similar to HEED, and the aggregation ratio increases (i.e.,amount of data eliminated increases) as the sensor densityin the eld increases. Finally, as before, the initial energy atthe sensors are randomly drawn from U[0.1,0.5].We summarize the average lifetime results over our in-stances in Table 3. It can easily be observed that the inte-grated approach, which we devise via integratingtopology and routing decisions, signicantly performs bet-ter than HEED; the network lifetime is about sixfold longeron average with the integrated approach eM3.Table 3Network lifetime via HEED and M3 approaches.Number of nodes, jI jApproach 150 200 250 300HEED 42 50 55 50eM3 264 283 294 3026. Summary and conclusionsIn this paper, we introduce three alternative mathemat-ical models for integrated topology and routing decisionsfor data-gathering WSNs so as to prolong their lifetimesubject to limited energy at the sensors. In doing so, wealso consider a hierarchical network structure withmulti-hop routing, multiple sinks, and a general dataaggregation approach and devise the models to determineclusterhead and active sink locations as well as data owroutes from sensors to sinks in each period. We also de-scribe a topology control scheme on how the usable energyfraction (of the available energy) at a sensor changes fromperiod-to-period in a deployment cycle.The rst two approaches, M1 and M2, which have theobjectives of average energy usage minimization and min-imization of the maximum energy usage at a sensor, areconsidered previously in the literature while prescribingcommunication protocols as opposed to an integratedmathematical modelling perspective as in this study. Thus,these models can be considered as benchmark models forthe performance of the proposed third model, M3, whichhas an objective of the minimizing the total energy andthe range of remaining energy distribution at the sensors.Since the models dictate large discrete optimization for-mulations, employing exact optimization approaches ishighly impractical, thus, we develop a general heuristicalgorithm, applicable for each model, that performs verywell in our computational tests. Our procedure utilizes ef-cient construction heuristics, two types of solution repre-sentations, a combination of multiple neighborhoods, andan objective value based on cut inequalities for better ef-ciency in evaluating candidate solutions. Heuristic solu-tions are also employed via cut inequalities to improvethe time performance and alleviate the memory difcultiesassociated with an exact branch-and-cut implementation.Using our heuristic approach, we conduct numericaltests and analyses of the models in a multi-period setting.We observe that M2 and especially M1 perform verypoorly when usable energy fraction p is set to 1.0 duringa deployment cycle. When the usable energy scheme ischanged to a conservative one given by (pI, inc) = (0.1,0.1),i.e. initial p is 0.1 and increased by 0.1 whenever energylevels are insufcient in a period, bothM1 (which now per-forms better than M2) and M2 exhibit better network life-time performance; however, they still perform inferior toM3 and appear to be highly sensitive to the topology con-trol scheme.Our proposed modelM3 is highly insensitive to the set-ting of usable energy fraction p and performs very well interms of network lifetime. It incorporates both energyusage and variation in the end-of-period remaining energylevels in its objective, and, thus, truly integrates topologycontrol and routing decisions without the need to exoge-nously set and calibrate a p value. In our numerical tests,we also observe that, even when the rst period energylevels in a deployment cycle are varied, M3 performs verywell with a full available energy (i.e. p = 1.0). This propertyalso contributes to efciencies in terms of redeploymentsince the reconguration via topology and routing deci-sions in each period leads to a self-adjustment in the net-work. As a result, the energy levels at the end of adeployment cycle are conned in a narrow band which im-plies that uniform deployment strategies can be effectivelyemployed.Finally, we also compare the performance of the pro-posed M3 model to a well-known protocol HEED devisedspecically to determine CH locations and sensor-to-CH850 H. ster, H. Lin / Ad Hoc Networks 9 (2011) 835851assignments in a WSN. For data routing in HEED, we em-ploy a mathematical optimization model based on ourmodels; and we also slightly modifyM3 to obtain a specialcase, which is still solvable by our algorithm, so that thetwo approaches can be compared. In our computationalstudies, we show that our proposed model performs signif-icantly better in this comparison as well.In summary, our modelling approaches, solution algo-rithms, and extensive analyses illustrate that M3 hasattractive properties capturing important characteristicsof integrated topology and routing decisions to improveenergy efciency and prolong lifetime of data-gatheringWSNs.AcknowledgmentsThe authors thank the anonymous referees and associ-ate editor for their helpful comments and suggestionson an earlier version of this paper. This research wassupported by the National Science Foundation GrantCMMI-0428831.References[1] A. Abbasi, M. 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Hou, Maintaining sensing coverage and connectivity inlarge sensor networks, Ad Hoc & Sensor Wireless Networks 1 (12)(2005) 89124.Halit ster is an Associate Professor in theDepartment of Industrial and Systems Engi-neering at Texas A&M University. He holds aPh.D. in Management Science/Systems fromMcMaster University, Ontario, Canada. Hisresearch interests are in the design of net-worked systems in logistics and communica-tions, and applied optimization. Hispublications appeared in Computers andOperations Research, European Journal ofOperational Research, Interfaces, IIE Transac-tions, Naval Research Logistics, and Trans-portation Science among others. He is an Associate Editor for IIETransactions and a member of IIE and INFORMS.Hui Lin is a Ph.D. candidate in the Departmentof Industrial and Systems Engineering atTexas A&M University. She holds two B.S.degrees in Automation and Engineering Eco-nomics, and an M.S. degree in ElectricalEngineering from Tianjin University, China.Her research interests include wireless sensornetwork design, applied optimization andhigh performance computing. She is a studentmember of INFORMS.H. ster, H. Lin / Ad Hoc Networks 9 (2011) 835851 851Integrated topology control and routing in wireless sensor networks for prolonged network lifetimeIntroductionRelated literatureMathematical modelsSolution approachesSolution representations and subproblemsConstruction heuristicsConstruction heuristic for M2Construction Heuristic for M1 and M3Solution improvement proceduresThe Search SubC() Improvement ProcedureThe Search SubU() Improvement ProcedureThe complete procedureComputational studyThe Impact of Usable Energy Fraction p on Network LifetimePerformance of heuristicsNetwork lifetime comparison of modelsNetwork lifetime comparison of M3 and HEEDSummary and conclusionsAcknowledgmentsReferences