Optimization Problems, Models, and Heuristicsin Wireless Sensor Networks

Vinicius Morais, Fernanda S. H. Souza, and Geraldo R. Mateus

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Problems Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Optimization Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Coverage and Density Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Routing or Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Clustering and Mobile Sink . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Heuristic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Sensor Localization, Placement, and Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Density Control, Clustering, Routing, and Sink mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Integrated Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Abstract

This chapter provides an overview and a comprehensive discussion of problems,models, algorithms, and applications in a vast and growing literature of wirelesssensor networks. Being a particular kind of ad hoc network, many power man-agement and communication protocols may be designed specifically for thosenetworks. The critical issues considered in these protocols are the objectives, the

V. Morais (�) • G.R. MateusDepartamento de Ciência da Computação, Universidade Federal de Minas Gerais, BeloHorizonte, Brazile-mail: vwcmorais@dcc.ufmg.br; mateus@dcc.ufmg.br

F.S.H. SouzaDepartamento de Ciência da Computação, Universidade Federal de São João del-Rei, São Joãodel-Reil, Brazile-mail: fsumika@ufsj.edu.br

© Springer International Publishing AG 2016R. Martí et al. (eds.), Handbook of Heuristics,DOI 10.1007/978-3-319-07153-4_53-1

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quality of communication, the energy consumption, and the network lifetime.Moreover, due to the large-scale aspect inherent in some applications, traditionalexact solution approaches are not practical, so heuristics may be adopted instead.The chapter starts by introducing the main concepts in the design of WSN and awide range of problems and applications. Basic formulations and algorithms arealso discussed, together with their benefits and drawbacks.

KeywordsOptimization problems • Heuristics wireless sensor networks

Introduction

The wireless sensor networks (WSN) are a special kind of ad-hoc networks usedto monitor events and phenomena in a given place [1, 2]. They have becomepopular due to their wide applicability in many different areas such as environmental(wildlife and environment), industrial (mines, production, manufacture), and health,in order to monitor temperature, humidity, pressure, and movement [3–5]. The studyof such networks has been a fruitful area of research over the years and continues toenable the permanent evolution of technologies.

WSN consist of autonomous, small, and compact devices capable of performingactivities such as sensing, processing, and communication. These devices aresensor nodes, and among them, may have one or more special nodes called sink,which is responsible for collecting the sensed data and managing the network.The sensors are made up of sensing boards, processor, communication radio, andbattery. Because of their purposes, size, and low cost, they have serious restrictionsof energy, low performance, and small radius of communication. For networksinstalled in areas with difficult access, it becomes unviable exchange or rechargethe battery of a sensor. These characteristics imply a limited lifetime of the sensorsand of the network itself.

The sensors are used to monitor a working area within a certain radius ofcoverage, besides being able to exchange information with other sensors and sinks.Two major communication patterns are often adopted: data collection and datadissemination. Data collection consists of sensors sending the collected data to asink, while data dissemination concerns information from a sink being sent to thesensors. Those communication processes can be performed through one (single) ormore (multi) hops or links. In this chapter whenever referring to the transmission ofinformation, both senses are being considered. A sink aggregates the informationreceived from sensors and usually has unlimited or renewable energy. It can bestatic or mobile, visiting the monitored region and collecting information from thesensors.

The sensor nodes are generally static but may also have mobility [6–8]. MobileWSN consist of a number of sensors that can move on their own and interactwith the physical environment. A sensor in a mobile network can also sense,process information, and communicate like a static node. Mobility may be applied

Optimization Problems, Models, and Heuristics in Wireless Sensor Networks 3

to the whole network or only to a subset of sensors. The degree of mobility canvary, including constant movement of the sensors or alternating periods of mobil-ity/immobility. Those factors impact directly the network dynamics. Alongside withmobile networks, there are four more types of networks, which are distinguishedmainly with respect to the place where the sensors are deployed. The sensors canbe deployed on land, underground, and underwater [2]. A terrestrial network, forinstance, consists of several sensors deployed on land, either in an ad-hoc or in apreplanned manner. In an underground network, the sensors are placed in caves,mines, or underground, while on an underwater network, the devices are deployedinto rivers or oceans. Among these networks one can define a multimedia network,where the sensors are used to handle multimedia data. While a dense deploymentof sensors is often employed in a terrestrial WSN, in an underwater WSN, a sparsedeployment is used. The maintenance cost and difficulty of implementation of eachnetwork are directly dependent on the application constraints.

The classification of WSN depends primarily on their application. They arecommonly classified into monitoring and tracking categories. Monitoring appli-cations include patient monitoring in medical centers, security detection, habitatmonitoring, power, inventory location, factory and automation processes, seismicand structural monitoring in the industrial or public context, and home-officeenvironmental monitoring to provide basic services to smart environments, amongothers. Tracking applications include military missions, wildlife tracking, and trafficcontrol. Consequently, the technologies used for each WSN are directly dependenton the network type. Refer to Yick et al. [2] for a survey of sensor technologies inpractical situations.

WSN may also be classified into homogeneous or heterogeneous. WSN are saidto be homogeneous, when composed by devices with the same hardware capabilities(processor, memory, battery, and communication device features). Conversely, WSNare heterogeneous when composed by devices with different capabilities. Anotherpossibility to classify WSN is based on their organization. The networks can behierarchical or flat. A sensor network is hierarchical when nodes are grouped for thepurpose of communication, otherwise it is flat.

The lifetime of a WSN can be defined as the network operation time until thefirst sensor fails due to lack of energy. This chapter does not treat cases of failurefor other reasons, such as mechanical failure or if a sensor is destroyed. Anotherdefinition of lifetime is given by the monitoring time until the coverage of the entireworking area and network connectivity are not guaranteed.

Many factors influence the design of WSN. Fault tolerance, scalability, energyconsumption, production cost, environment, network topology, hardware con-straints, and transmission rate are some points to be considered. Due to the lowbattery capacity of the sensors, the energy consumption is one of the most importantconstraints on WSN [9]. In this context several problems are addressed to ensurecoverage, monitoring, and connectivity, aiming to maximize the network lifetimewithin a minimum quality standard or quality of service (QoS) (QoS usuallyrefers to quality as perceived by the user and/or application). In spite of energy

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consumption being the most critical factor in WSN design, other parameters suchas number of sensors, coverage, and network connectivity must also be considered.

The problems that appear in the context of WSN are studied independently or inan integrated manner. In general, to define them it is supposed that the working areato be covered is divided into small squares which represent points or demand nodesthat require coverage by at least one sensor. This is a well-known approach appliedin wireless network design. Each point concentrates the demand of its square. Apoint is considered covered by a sensor when the referred sensor is able to monitorit and if the point is within the radius of coverage of that sensor. The smaller arethe square dimensions, the closer they are to describing continuous area. In theliterature there are several works that do not make use of this discretization process,opting to explore the continuous space. Another assumption usually found in theliterature is the use of unit disk graph to model the communication among sensors.A homogeneous transmission range for the sensors is considered and taken as thedisk radius. Many works employ this model, although transmission ranges can beheterogeneous and also not defined as perfect disks. However, due to its theoreticalsimplicity, it is commonly applied. An evolution of this model, named quasi unitdisk graph is proposed in [10].

Many algorithms can be devised to deal with WSN problems. Regardingoptimality guarantees, they can be divided into exact approaches and heuristics.The former comprise algorithms developed to solve mathematical models based oninteger linear programming (ILP), while the latter include proposals which relieveoptimality conditions in order to provide scalability. Also, algorithms can performin a centralized or distributed fashion. Centralized algorithms concern decision-making is done by a single entity in the network owning global information of theentire network. This entity is usually a sink node which is able to disseminate thesolution for all sensors in the network. On the other hand, distributed algorithmsindicate that many entities in the network take a joint decision based only on localinformation. Global information can improve the optimization process but is notalways available and may not be applied in practice due to the overhead not allowedin real-time applications. Local information seems more reasonable in practice, butsuffers from lower-quality solutions compared to the ones with global information.An alternative to take advantage of both methods is to use clustering, which canoptimize locally inside each cluster. Thus, both exact methods and heuristics can beaccomplished in such hybrid solutions. Distributed algorithms concern a wide topicof research [11–14] and are out of scope of this chapter.

The remainder of the chapter is organized as the following. A brief definition ofclassical problems appearing in WSN is presented in section “Problems Definition”.Mixed integer linear programming formulations for these problems are outlined insection “Optimization Models”. The aim is to formally describe some problemsand give some intuition about exact solution approaches. Heuristic methods appliedon the design of WSN are addressed in section “Heuristic Methods”. Finally, thechapter is closed in section “Conclusion”.

Optimization Problems, Models, and Heuristics in Wireless Sensor Networks 5

Problems Definition

To monitor inhospitable and inaccessible environments, the sensors are deployedin an ad-hoc manner, launched from an aircraft or an unmanned vehicle. Thus,it is not possible to know beforehand their exact locations. However, knowingthe location of these sensors is generally important to route data and to ensureconnectivity. The problem of defining the positions of the sensors is known aslocalization problem. Some localization methods include Global Positioning System(GPS), beacon (anchor) nodes, and proximity-based localization that make use ofthe coordinates of neighbor nodes to determine their own localization. A review oflocation problems in WSN is given in [15].

It is straightforward to note that a fundamental requirement in the design of WSNis to ensure coverage of the working area. This problem is named as the coverageproblem. The coverage level can be total or partial. In general, full coverage is oftenpreferred. Therefore, the coverage problem chooses a set of active sensors such thateach point is within a range of at least one sensor. The loss of coverage in at leastone point can be characterized as end of the network lifetime.

To monitor accessible environments, the sensors may be optimally deployed ina preplanned manner. The placement (deployment) problem seeks to minimize thenumber of sensors deployed in order to guarantee connectivity and coverage of theentire monitoring area. The placement problem is a variant of the well-known artgallery problem [16], the problem of placing the fewest number of guards in a givenarea such that the surveillance is guaranteed. For the following problems, a usualassumption is to consider that the location of the sensors is known.

Another fundamental requirement in the design of WSN is the connectivity. Thedata collected by the sensors must be delivered to the sinks. Thus, routes must beestablished connecting each active sensor to a sink. A feasible route consists oflinks between pairs of sensors/sinks that respect the communication radius of eachnode. The fewer is the number of links, the smaller is the energy consumption.The routing problem aims to find feasible routes from each active sensor to thesinks, or vice versa, while optimizing a given resource. A solution for the routingproblem consists of rooted trees, centered at the sinks, spanning a subset of activesensors. Automatically, the sensors nearest to the sinks may spend too much energygiven the number of routes traversing those nodes. Such a drawback is knownas the sink neighborhood problem [17]. There are two kinds of routing methods:reactive routing methods and proactive routing methods. In reactive methods theroutes are calculated just in time. In proactive methods the routing tables are createdand stored regardless of the moment the routes are used. However, in networkswith large number of sensors, storing the routing tables can lead to excessiveusage of memory. To overcome this problem, the sensors can be clustered andcommunication hierarchies be defined.

The sensors consume energy to perform communication, process, and aggregatedata, besides maintenance, activation, or deactivation processes. There are severalways of dealing with the energy consumption. One possibility to extend the network

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lifetime is to use a large number of sensors to ensure the desired coverage. However,to avoid spending too much energy at a redundant coverage, the sensors do notperform at the same time. In this case, each sensor can be active, working on thenetwork, or stay on standby, sleep mode, since energy consumption is extremely lowin this mode. When active sensors fail due to depletion of their batteries, new sensorsmust be activated. The optimization problem arising in this context is the densitycontrol problem (DCP). The purpose of DCP is to schedule a sequence of activationand deactivation of sensor nodes in order to minimize the energy consumption andmaximize the network lifetime. The network lifetime comes to an end when thenetwork requirements are no longer ensured.

All the problems above have been considered for flat WSN. Whereas thecommunication among sensors and sinks is one of the most expensive operationsin terms of energy consumption, an alternative to flat networks is to set hierarchies.In hierarchical networks, the sensors are clustered, and a single sensor is electedas the cluster-head in each cluster. It is up to the cluster-head aggregating andtransmitting the information directly to a sink or through other cluster-heads. Insuch a clustering problem, from time to time, the cluster-head role can be swappedamong the sensors within the same cluster to avoid premature failure due to lack ofenergy. Some classical combinatorial optimization problems such as p-median andp-center may apply in a clusterization scheme.

In the routing problem for hierarchical WSN, two kinds of routes are considered:intra-cluster route and intercluster route. In intra-cluster routes the communicationoccurs among sensors and cluster-head within the same cluster through single ormulti-hop strategies. In intercluster routes the connections are among cluster-headsand sinks. In both levels a tree topology must be defined to transmit the informationfrom the sensors to their cluster-heads and from the cluster-heads to the sinks.

In some applications the mobility of the sinks is explored to gather sensedinformation through the network. This approach prevents the sensors to spend theirlimited energy in relaying of messages through a long path until the sinks. However,the message delivery latency can increase significantly, thanks to the long routetraversed by a sink to visit the whole working area. In order to go around of suchproblem, clustering schemes must be used so that each sink only visits a reducedset of sensors (for instance, the cluster-heads). In this context, the communicationbetween sensors and sink is only possible through a cluster-head using multi-hopor single-hop approaches to define the intra-cluster routes. Classical optimizationproblems such as shortest path and traveling salesman problem (TSP) are used tomodel the trails traversed by the mobile sinks.

Figure 1 illustrates optimization problems in WSN. Different tasks and objectivesmay be desired requiring specific approaches to deal with the problems described. InFig. 2, for instance, two different approaches are presented exploring the mobilityof the sink. In the first approach, a mobile sink visits each sensor in the workingarea to collect sensed data. This problem can be modeled with the TSP. Anotheralternative is to define a reduced TSP route, defining a subset of sensors to be visitedby the mobile sink such that all the other sensors are close enough to establish thecommunication. This approach can be seen as a TSP by neighborhood.

Optimization Problems, Models, and Heuristics in Wireless Sensor Networks 7

sensor

demandnode

coverage

clustering

cluster head

monitoring area

routing

density control

sinksink

Fig. 1 Illustration of classical optimization problems in WSN

mobile sink modeled as a TSP mobile sink modeled as a reduced TSP (neighborhood)

monitoring area

Fig. 2 Illustration of mobile sink approaches modeled with the TSP

Optimization Models

As presented earlier, many optimization problems arise in the design of WSN. Manyof these problems are variants of classical combinatorial optimization problems thatcan be formulated as linear mixed-integer programming (MIP) models [18]. In thefollowing, mathematical models for some of these problems are presented.

Coverage and Density Control

Mathematical formulations for coverage and density control problems can be foundin [19–21]. The models consider the following notation: S D f1; : : : ; ng is the setof sensors; P D f1; : : : ; pg is the set of demand points to be covered by the sensors;matrix A.n � p/ indicates whether a sensor i 2 S is able to cover (aij D 1) or not(aij D 0) the point j 2 P regarding the sensing range Ris . To each sensor i 2 S , anactivation energy Ei

a is also assigned.

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A formulation to the coverage and density control problems uses binary variablesfxi 2 B W i 2 Sg and fyij 2 B W i 2 S; j 2 P g to define a subset of sensors selectedto be active and the coverage relationship between S and P . If xi D 1 holds, sensori 2 S is active. If yi;j D 1, demand point j 2 P is covered by sensor i 2 S . Then,the coverage and density control formulation is given by:

min

(Xi2S

Eiaxi W .x; y/ 2 (2) � (5)

)(1)

Xi2S

aij yij � 1 8j 2 P I (2)

aij yij � xi 8i 2 S; 8j 2 P I (3)

yij 2 f0; 1g 8i 2 S; 8j 2 P I (4)

xi 2 f0; 1g 8i 2 S: (5)

Constraints (2), (3), (4), and (5) assure that active sensors cover the set of demandpoints. Constraints (2) guarantee that every demand point j 2 P is covered by atleast one sensor i 2 S . Constraints (3) ensure that only active sensors can providethe coverage of demand points. Constraints (4)–(5) impose variables to be binary.Finally, objective function (1) minimizes the energy consumed by the activation ofsensor nodes.

Routing or Connectivity

Mathematical models exploring connectivity in the context of WSN are describedin [19, 21, 22]. The models consider a set of static sinks M D f1; : : : ; mg and acommunication range (Ric) associated with each sensor i 2 S . The matrix B.n �.nCm// indicates whether a sensor or sink i 2 S and j 2 S [M relies (bij D 1)or not (bij D 0) within the communication range Ric of each other. To each sensori 2 S is also assigned the routing energy (Ei

r ) required to communicate with othersensors or sinks.

Let fzijk 2 B W i 2 S; j 2 S; k 2 S [ M g denote decision variables used

to indicate whether the sensors j 2 S and k 2 S [ M belong (zijk = 1) or not

(zijk = 0) to the communication path from sensor i 2 S to a sink node k. Alongsidewith the decision variables xi defined in the first model, the routing problem,modeled as routing trees, can be formulated as:

min

8<:Xi2S

Eia xi C

Xi2S

Xj2S

Xk2S[M

Eir zijk W .x; z/ 2 (7) � (12)

9=; (6)

Optimization Problems, Models, and Heuristics in Wireless Sensor Networks 9

Xk2S[M

bjk zijk � xj 8i 2 Snfkg; 8j 2 SnfkgI (7)

Xj2S

bjk zijk � xk 8i 2 Snfkg; 8k 2 Snfj g [M I (8)

Xj2S

bjk zijk �X

l2S[M

bkl zikl D 0 8i 2 S; 8k 2 SnfigI (9)

�X

l2S[M

bkl zikl D �xi 8i 2 S; i D kI (10)

zijk 2 f0; 1g 8i 2 S; j 2 S; k 2 S [M (11)

xi 2 f0; 1g 8i 2 S: (12)

Here, constraints (7), (8), (9), (10), (11), and (12) ensure the connectivityamong sensors and sinks. Constraints (7)–(8) assure that the communication isonly allowed among active sensors. Constraints (9) guarantee the flow balance fortransshipment nodes. Constraints (10) indicate that the flow originates at activesensors. Constraints (11)–(12) impose the domain of the variables. Finally, theobjective function (1) minimizes the routing and activation energies consumed.

Clustering and Mobile Sink

Valle et al. [23] proposed an integrated model for clustering and routing problems.The model considers a set of mobile sinks M D f1; : : : ; mg and a set of arcs A Df.i; j /; .j; i/ W 8i; j 2 S; i ¤ j g. Each arc .i; j / 2 A denotes a possible point-to-point movement for a sink. Additionally, dij is the Euclidean distance between iand j 2 S . It is assumed that all sinks start from an initial sensor i D 1 and returnto an artificial node 0, such that NA D A [ f.i; 0/ W 8i 2 Snf1gg and f.i; 0/ W di0 Dd1i ;8i 2 Snf1gg. It is important to point out that each cluster-head is visited onlyonce and by exactly one sink; a sensor i 2 S must be a cluster-head or be coveredby a cluster-head; at least one cluster-head should belong to the set of neighbors ofsensor i : N.i/ D fj 2 S W dij � Ricg.

In order to model the integrated clustering and routing problem, assume thefollowing decision variables: frmi 2 B W m 2 M; i 2 Sg, taking value rmi = 1 ifsensor i 2 S is a cluster-head in the route of the sink m 2 M , otherwise rmi = 0;fwmij 2 B W m 2 M; .i; j / 2 NAg that takes value wmij = 1 if arc .i; j / 2 NA belongs

to route of the sink m 2 M , otherwise wmij = 0; fvmij 2 RC W m 2 M; .i; j / 2 NAg,

indicating the flow, originated atm, traversing arc .i; j / 2 NA; and, finally, fh 2 RCg

that denotes the longest length among all routes. The integrated clustering androuting problems can be formulated by the following MIP model:

min fh W .r;w; v; h/ 2 (14) � (27)g (13)

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Xi2Snf1g

vm1;i DX

i2f0g[Snf1g

rmi 8m 2M I (14)

Xj2f0g[Snf1g

vmij �Xj2S

vmji D �rmi 8i 2 Snf1g; 8m 2M I (15)

Xi2S

vmi;0 D 1 8m 2M I (16)

vmij � nwmij 8.i; j / 2 NA; 8m 2M I (17)

wmij � rmi 8.i; j / 2 NA; 8m 2M I (18)

wmij � rmj 8.i; j / 2 NA; 8m 2M I (19)X

m2M

rmi � 1 8i 2 Snf1gI (20)

Xm2M

Xj2N.i/

rmj CXm2M

rmi � 1 8i 2 Snf1gI (21)

Xj2S[f0g

wmij � 1 8i 2 S; 8m 2M I (22)

h �X.i;j /2 NA

dijwmij 8m 2M I (23)

rm1 D rm0 D 1 8m 2M I (24)

rmi 2 f0; 1g 8i 2 S [ f0g; 8m 2M I (25)

vmij � 0 8.i; j / 2 NA; 8m 2M I (26)

wmij 2 f0; 1g 8.i; j / 2 NA; 8m 2M: (27)

Constraints (14), (15), (16), (17), (18), (19), (20), (21), (22), (23), (24), (25), (26),and (27) ensure the integration of clustering and routing problems using mobilesinks. Constraints (14), (15), and (16) guarantee the flow conservation for nodesf1g; Snf1g and f0g, respectively. Note that constraints (14) assure that the flow ofcommoditym starting at node 1 is equal to the number of cluster-heads to be visitedin the route plus the artificial node f0g. On the other hand, constraints (15) imposethat each node i visited by m 2 M must consume a unit of flow. Inequalities (17),(18), and (19) are coupling constraints, ensuring that commodities only traversearcs selected for the route and arcs of the route have endpoints consuming a unitof flow. Constraints (20) guarantee that a cluster-head is visited once by a singleroute. Constraints (21) assure that each sensor is a cluster-head or has a cluster-head as neighbor. Taken together, constraints (14), (15), (16) and (22) guarantee thetopology of selected arcs induce M elementary routes connecting vertices 1 and 0.Constraints (23) define the longest length among all routes to be minimized by the

Optimization Problems, Models, and Heuristics in Wireless Sensor Networks 11

objective function (13). Constraints (24) impose that the starting node 1 and the finalartificial node 0 are part of all routes. Finally, constraints (25), (26), and (27) definethe domain of variables.

Models (1), (6), and (13) can be implemented in optimization packages andsolved with a branch-and-bound (BB) algorithm. However, the main drawbackof this approach is that only instances of limited size are solvable in feasiblecomputational time. Since WSN are designed as large-scale networks, it is notexpected that a traditional BB is able to solve the problem to optimality. Therefore,one must resort to heuristic-based methods to tackle the problems.

Heuristic Methods

This section draws attention to heuristic-based methods used in the design of WSN.In general, heuristics are designed to face challenging and complex optimizationproblems in which exact optimization methods fail, especially when dealing withtime-constrained online applications that are combinatorial in nature. Due to thepractical and technological relevance of WSN, heuristics are preferentially themethods to be used. However, each heuristic is designed to tackle a specificproblem. So, a given approach may not be generalized to solve different problems.Motivated by the need of general solution strategies, optimization researches havebeen focusing on the study of metaheuristics. A metaheuristic combines intensi-fication and diversification procedures to perform guided searches through one ormore neighborhood structures. Some metaheuristics used in the design of WSNare greedy randomized adaptive search procedure (GRASP) [24], path relinking(PR) [25], simulated annealing (SA) [26], iterated local search (ILS) [27], tabusearch (TS) [28], variable neighborhood search (VNS) [29, 30], genetic algorithms(GA) [31], memetic algorithms (MA) [32], ant colony optimization (ACO) [33],and particle swarm optimization (PSO) [34]. For supplementary reading, Gloverand Kochenberger [35] are referred to provide a good overview of the most popularmetaheuristics. In addition to [36–40], here, the reader can find some hints ofhow to use heuristics to solve practical WSN optimization problems when studiedindependently or in an integrated manner.

Sensor Localization, Placement, and Coverage

As indicated above, the design of WSN has to concern with several classical prob-lems. These include localization, placement, coverage, density control, clustering,routing, and sink mobility. In general, the purpose of solving these problems isto maximize the network lifetime, keeping connectivity, ensuring coverage, andreducing cost. Many existing applications require information about the geographicposition of each node in the network. So, the localization problem must besolved first. The heuristics proposed in the literature for localization are generallybased on GPS system, beacon (anchor) nodes, and proximity-based localization.

12 V. Morais et al.

These methods make use of the position of neighbor sensors, obtained duringthe installation of the network, to determine the position of a sensor. Molina andAlba [41], for instance, proposed heuristics based on SA, GA, and PSO to solve thelocalization problem. Their heuristics determine the sensor nodes locations usingthe trilateration technique. The global objective of their heuristic is to minimize themeasured distance error comparing the node-to-node distances with the distancemeasured when using GPS. Shahrokhzadeh et al. [42] proposed a centralized(solved by a sink) SA-based heuristic for solving sensor nodes localization problem.Their heuristic is based on Euclidean distance and mathematical techniques, such astrilateration and triangulation.

The placement of the sensors is a key factor in the design of WSN. Thedeployment may be an activity performed during the installation of a networkor may also be a continuous process to replace failed sensors or to improve thecoverage over a certain area. As stated above, the placement problem seeks tominimize the number of sensors deployed in order to guarantee connectivity andcoverage of the entire monitoring area. A number of deterministic greedy heuristicscan be described to determine the minimum number of sensor needed. In general,those methods try to anticipate the sensor deployment in all candidates’ points andkeep the best position selected. Brazil et al. [43] modeled the placement problemas a version of the Steiner problem and solved it by means of a greedy-basedheuristic. Sasikumar et al. [44] proposed two-phase heuristics for placing sensors ina heterogeneous network, called nearest to base station and max residual capacity.In this work, they separate the nodes on: sensors, relay nodes (RN), and basestation (sink node). The RN node performs functions equivalent to a cluster-head,but may have unlimited energy. In the first phase, their heuristics place the RNnodes, wherein the placement problem is modeled as the minimum set coveringproblem. Then the sensors are placed on the second phase with a greedy criterionbased either on the distance to the RN nodes or on residual energy remaining onthe node. To solve the sink placement problem, Laszka et al. [45] provided a GA-based heuristic. SA-based heuristics for placement and coverage problems were alsodeveloped in [46].

Deschinkel [47] concentrates on a centralized scheme to solve the coverageproblem, where a large number of sensors are randomly deployed in the monitoringarea. The problem was modeled as the non-disjoint cover sets problem and solvedby means of a column generation (CG) heuristic. In this procedure the subproblemis solved with a heuristic for the classical set covering problem that uses the dualmultipliers from the master problem. They compare the results obtained by theheuristic with that provided by a MIP solver. Cardei et al. [48] also deal withcoverage problem. The interesting aspect of their work is given when the authorsconsider the property that each sensor has adjustable sensing ranges. To solve theproblem, a greedy-based heuristic was used.

Optimization Problems, Models, and Heuristics in Wireless Sensor Networks 13

Density Control, Clustering, Routing, and Sink mobility

The solution algorithms for the placement problem may define a dense network.This characteristic increases the robustness against failures, however, leads to alarge redundancy that can result in congestion and waste of energy. To overcomethis problem, the density control problem is used to select a subset of sensors to beactive while keeping the others inactive (sleep mode), which allows the reduction ofthe initial topology. Delicato et al. [49] formulated the DCP problem as a knapsackproblem. In this approach, the expected time horizon is divided in small timeperiods. For each period, a subset of sensors must be activated. By formulatingthe DCP as a knapsack problem, each period is equivalent to a knapsack, so theproblem is to find a set of nodes (items) to keep active (to be inserted in a knapsack)per period. In their model, coverage and connectivity are also considered. To solvethe problem, a greedy-based heuristic was proposed. In this heuristic the sensors aresorted by their residual energy and then, following the given preference, the sensorsare set to be active (inserted in a knapsack).

An interesting and effective multi-start (MS)-based heuristic to solve the densitycontrol problem is given by Karasabun et al. [50]. Their heuristic is composed bytwo phases, a constructive routine to define a set of initial active sensors and aniterative improving local search procedure. To ensure the network connectivity, theSteiner problem is solved over the active sensor nodes.

Clustering is another optimization problem considered when defining the net-work topology. The clustering problem directly contributes for the scalability ofnetworks. Many existing protocols for defining clusters and selecting cluster-headsassociated with them are based on random algorithms or heuristics to classicalcombinatorial optimization problems, such as p-median, dominating set, and setcovering problems. In such approaches, every sensor node is either in a set ofcluster-heads or is assigned to a cluster-head. In general, among the cluster-heads,a multi-hop communication scheme is used, while inside a cluster multi- and/orsingle-hop communication could be used. Santos et al. [51] modeled the clusteringproblem as the independent dominating set problem. Two greedy heuristics aredescribed in that reference.

Matos et al. [52] call attention to the importance of rotating the sensor that actsas cluster-head within a single cluster. Cluster-heads consume much more energythan a sensor that does not perform this feature, since they need to aggregate, send,and receive sensing data. Thus, cluster-head rotation helps to avoid a prematuredeath of a sensor, disconnecting the network. The rotation can be done from timeto time or after some amount of data has been transferred by the network. Inaddition, Matos et al. [52] also proposed a centralized GRASP-based heuristicwith path relinking as an intensification phase to solve the clustering problem. Theproblem was modeled as the p-median problem, where the cluster-heads are chosenamong the alive sensors. The intra-clusters communication follows the single-hopstrategy. Their heuristic is composed of three phases. In the constructive phase, afeasible solution is generated by the randomized greedy algorithm described in [53].

14 V. Morais et al.

Then, a swap local search is applied. Finally, a path relinking routine is applied asintensification phase. To attest the effectiveness of their approach, they comparedtheir heuristic with methods described in [54, 55]. Albath et al. [56] consider thata sensor to be selected as cluster-head must have enough residual energy. In theirwork, the problem of properly choosing cluster-heads was modeled as the minimumdominating set problem.

Looking for metaheuristics methods, Ferentinos and Tsiligiridis [57] described aGA-based heuristic used in an agriculture application. Their algorithm is designedto solve clustering and density control problem while respecting connectivityconstraints. Another GA-based heuristic for clustering problem is addressed byKuila et al. [58]. Ting and Liao [59] described the clustering problem as the k-coverproblem and solved it with a MA-based heuristic.

The transmission of sensing data and the dissemination of information in WSNare referred as the routing problem. This problem is defined mainly when the nodesposition and the cluster-heads selected are already known. The main objective whensolving the routing problem is to guarantee the network connectivity. Routing inWSN design is tackled by many authors [19,21,22,60] through a multi-hop strategyand fixed sink nodes as well as through mobile sinks [23, 61, 62].

Minimum spanning tree, shortest path, and TSP are preferentially used to modelthe routing problem in WSN design. In general the edge weight (length or cost)represents the energy needed to transmit data. Thus, routing algorithm based onKruskal, Prim, Dijkstra, Bellman-Ford, or TSP methods can be applied. As statedin section“Problems Definition”, the sink neighborhood problem [17] arises whendealing with fixed sink nodes. To overcome that problem, some alternatives takeplace. Some works consider multiple sinks or, in some contexts, explore the networkby mobile sinks. More sinks result in shorter routes from sensors to their closestsink, while mobility may lead to an efficient scheme for the energy control.

Centralized and distributed heuristics to control and coordinate the currentmovement of multiple sinks seeking for the lifetime maximization in WSN aredescribed by Basagni et al. [63]. In this reference, a sink is considered active whenit is stopped, waiting for the sensor data, and inactive when it is moving through thenetwork. In their approach at least one sink must move at a time, so it is necessaryto define a schedule for the movements of sinks. The routing scheme was modeledwith TSP problem and solved by a Christofides’ heuristic [64].

To control the sink movements, Basagni et al. [17] introduced a distributedheuristic, named greedy maximum residual energy. In their heuristic, a sink definesits own route by moving from its current location to a new position by givingpreference to areas with the highest residual energy. More precisely, the greedycriterion of the proposed heuristic is the amount of energy left in the sensors arounda mobile sink that moves in direction of the sensors with higher energy to collecttheir data.

Optimization Problems, Models, and Heuristics in Wireless Sensor Networks 15

Integrated Problems

Most the works found in the WSN literature describe hierarchical approaches to thedesign of networks. Those works are drawn for solving WSN problems in multiplesteps. Although less, there are works in the literature that address a number ofclassical WSN problems in an integrated manner [21, 65–72]. A natural clusteringand routing problem is the minimum connected dominating set problem [73–75],the problem of finding a connected dominating set of cluster-heads with smallestcardinality. Aioffi et al. [66], for instance, have provided a constructive heuristic toan integrated problem, named integrated clustering and routing problem (ICRP).Their proposed heuristic is based on the cheapest insertion algorithm for theEuclidean TSP. In such method, a set of routes is iteratively defined, one for each

Table 1 Description of the cited references on heuristic-based methods for WSN problems.Consider the following acronyms list: coverage problem (CP), density control (DCP), localizationproblem (LP), routing problem (RP), clustering problem (CL), and placement problem (PP). Multi-hop communication (MhC), fixed sink (FS), mobile sink (MoS), and multiple sink (MuS)

Reference Problems Methods

Aioffi et al. [82] CP DCP RP CL – – MhC – MoS – Greedy

Aioffi et al. [66] – – RP CL – – – – MoS MuS GRASP, ILS

Alba and Molina [46] CP – – – PP – – – – SA

Albath et al. [56] – – – CL – – – FS – – Greedy

Basagni et al. [17] – – – – – – – – MoS – Greedy

Basagni et al. [63] – – RP – PP – – FS MoS MuS Greedy

Brazil et al. [43] – – – – PP – – – – – Greedy

Cardei et al. [48] CP DCP – – – – – – – – Greedy

Delicato et al. [49] – DCP – – – – – – – – Greedy

Deschinkel [47] CP – – – – – – – – – CG heuristic

Ferentinos and Tsiligiridis [57] – DCP RP CL – – – FS – – GA

Guney et al. [68] CP – RP – – LP – – – – TS

Karasabun et al. [50] – DCP – – – – – – – – MS

Kuila et al. [58] – – – CL – – – FS – – GA

Laszka et al. [45] – – – – PP LP – – – – GA

Lin and Uster [83] – – – – PP – – – – – MS

Matos et al. [52] CP – RP – – – MhC – – – GRASP, PR

Molina and Alba [41] – – – – – LP – – – – SA, GA, PSO

Santos et al. [51] – – – CL – – – FS – – Greedy

Sasikumar et al. [44] – – – – PP – – – – – Greedy

Shahrokhzadeh et al. [42] – – RP – PP – – – – – SA

Ting and Liao [59] – – – CL – – – – – – MA

Turkogullari et al. [76] – – RP – PP – – – – – MIP-heu.

Uster and Lin [67] – DCP – CL – – – – – MuS Constructiveheu.

Xing et al. [65] – – RP CL – – – – – – Greedy

16 V. Morais et al.

mobile sink, at the same time as the sensors are grouped into clusters. Each sinknode visits some cluster-heads, in such a way that every sensor node in the networkis either a cluster-head visited by a route or lies within a cluster associated witha cluster-head in one of the possible routes. The greedy criterion of the proposedheuristic is based on the distance among the last cluster-head assigned to a route andthe other cluster-heads not routed yet. Hybrid heuristics based on GRASP and ILSmetaheuristics were also applied to solve the problem. Another integrated approachis due to Xing et al. [65] that also studied clustering and routing problems. Tosolve the problem, a heuristic was used in the following schemes. First, it solvesthe Steiner problem to select the cluster-heads and then uses a TSP local searchheuristic to find a route for each mobile sink.

A TS-based heuristic for an integrated coverage, sink location, and routingproblems can be found in Guney et al. [68]. The heuristic starts by finding near-optimal sensor locations satisfying the coverage requirements, and then it solvesthe sink location and the data routing problem modeled as the p-median problem.Türkogullari et al. [76] proposed a hybrid heuristic to solve large instance ofan integrated problem involving sensor placement, density control, sink location,and routing problem with low conservation, energy consumption, and budgetconstraints. In such algorithm, the optimal sensor-to-sink routes are solved as alinear program (LP), while the sink location problem is modeled as a set coveringproblem and solved by a disjoint sets heuristic. Another heuristic based on exactapproach, used to solve large and real-life sink location and routing problems, couldbe found in [77].

Table 1 summarizes the previous cited works, highlighting the problems solvedand the heuristic methods used. For an overview of the mathematical aspects ofnetwork optimization problems, we also refer to [78–81].

Conclusion

In this chapter, an overview of classical optimization problems and solution methodsin the design of wireless sensor networks are presented. WSN have been widelystudied due to their importance in many practical applications. Unlike a generalnetwork, WSN are designed for specific applications and take into an account theapplication goals, the associated costs, the hardware capabilities, and other systemconstraints. Therefore, optimization techniques are shown to be an essential partin the design of new protocols. The integration of two or more problems, suchas coverage, routing, density control, clustering, and so on, requires even moresophisticated approaches to achieve high-quality solutions. It’s straightforward toconclude that optimization tools play a key role in the design of WSN. The referredmethods in this chapter have the capacity to improve significantly metrics such asnetwork lifetime, coverage rate, and communication delay. Looking to the future,WSN represent an attractive research area with several new possibilities where themajor challenge is to approximate the theoretical tools for practical applications.

Optimization Problems, Models, and Heuristics in Wireless Sensor Networks 17

Acknowledgements This work was partially supported by the Brazilian National Council forScientific and Technological Development (CNPq), the Foundation for Support of Research ofthe State of Minas Gerais, Brazil (FAPEMIG), and Coordination for the Improvement of HigherEducation Personnel, Brazil (CAPES). Vinicius Morais is funded by CAPES BEX 7461/14-3.

Cross-References

�Genetic Algorithms�GRASP� Iterated Local Search�Memetic Algorithms�Network Optimization� Particle Swarm Methods� Path Relinking� Simulated Annealing�Tabu Search�Variable Neighborhood Search

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