Optimal and Approximate Approaches for Deployment of Heterogonous Sensing Devices

In this paper, a modeling framework for the problem of deploying a set of heterogeneous sensors in a field with time-varying differential surveillance requirements is presented. The problem is formulated as a mixed integer mathematical program with the objective is to maximize coverage of a given field. Two meta-heuristics are also used to solve this problem. The first heuristic adopts a Genetic Algorithm (GA) approach while the second heuristic implements a Simulated Annealing (SA) algorithm. A set of experiments are used to illustrate capabilities of the developed models and to compare their performance. The experiments investigate the effect of parameters related to the size of the sensor deployment problem including number of deployed sensors, size of the monitored field and length of the monitoring horizon. They also examine several endogenous parameters of the developed GA and SA algorithms.


Introduction
Latest advances in wireless sensing technologies have considerably expanded their applications including military, homeland and border security, roadway safety and traffic surveillance, habitat monitoring, and wildlife and environment protection [3,16,17].In most of these applications, a network of individual wireless sensors is used to collect state-describing data from a given field.This data is then transmitted through the network to one or more predefined sink nodes for processing.Clearly, the performance of a Wireless Sensor Network (WSN) would largely depend on characteristics and deployment scheme of individual sensors used to construct this network.Sensors could be characterized by their lifespan, power saving capabilities, mobile capabilities, reliability, coverage range and communication range.Using sensors with superior sensing capabilities together with accurate placement of these sensors in the field's hot spots would result in more effective surveillance.In this context, WSNs could generally be classified into a) homogenous vs. heterogeneous and b) ad hoc vs. organized [8,15].Homogenous WSNs use identical set of sensors, while heterogeneous WSNs consider sensors that differ in one or more of the above characteristics.In ad hoc WSNs, sensors are randomly placed mainly due to limited access to the monitored field.However, in organized WSNs, full access to the monitored field is granted and hence the exact location of each used sensor over the monitoring horizon is predefined.This paper studies organized heterogeneous WSNs.A modeling framework for the problem of deploying a set of heterogeneous sensors in a field with time-varying differential surveillance requirements is presented.In this framework, the problem is formulated as a mixed integer mathematical program with the objective is to maximize coverage of a given field.A set of constraints are defined for this mathematical program to guarantee that each zone in the monitored field achieves its required surveillance requirements.Constraints are also defined to ensure that no sensor is used beyond its capacity.The solution of this mathematical program yields the deployment scheme for each used sensor.Two meta-heuristics are also used to solve this problem.The first heuristic adopts a Genetic Algorithm (GA) approach while the second heuristic implements a Simulated Annealing (SA) algorithm.A set of experiments are used to illustrate capabilities of the developed models and to compare their performance.The experiments investigate the effect of parameters related to the size of the sensor deployment problem including number of deployed sensors, size of the monitored field and length of the monitoring horizon.They also examine endogenous parameters of the developed GA and SA algorithms.
The contribution of this research work is fourfold.First, the modeling framework considers the deployed of heterogeneous set of sensors.Most existing sensor deployment algorithms assume the deployment of identical sensors (see for example, [3,4,6] ).Thus, a more general framework is needed for large-scale surveillance operations in which multiple sets of heterogeneous sensors are integrated in one deployment plan.Second, main characteristics of the sensors such as lifespan, power saving and mobile capabilities and reliability are explicitly considered.Third, the framework considers monitoring fields with time-varying differential surveillance requirements.In other words, the framework develops a deployment scheme that is responsive to the temporal-spatial variation in the criticality of the different zones of the monitored field.Finally, developed algorithms in this framework generate near-optimal solution for large-size deployment problems in reasonable running time, which enable the use of these algorithms in applications that requires real-time sensor deployment.
Early contribution to the problem of surveillance device deployment returns to Chvatal, 1975 [21], who introduced the Art Gallery problem.In this problem, the goal is to determine the minimum number of observers required to secure an art gallery with a non-uniform geometry.Different versions of this problem have been studied to include mobile guard and guards with limited visibility (e.g., [11]).Nonetheless, research in the area of sensing devices deployment has rapidly advanced with the emergence of wireless sensors networks.Most of the research work in this area has concentrated on studying the optimal formation of a WSN that can be used to collect data from a given field and to transmit this data to one or more sink points (e.g., [12,14]).
For example, Chakrabarty in [12] proposed a mathematical programming approach for sensor and target location in distributed sensor networks.The formulation assumes homogenous sensing devices with perfect accuracy.Isler in [22] proposed concurrent and incremental deployment methods using sampling theory in which new nodes are deployed based on samples taken from some other randomly deployed nodes.Liu and Mohapatra [25] proposed an integer linear program to maximize the overall lifetime of WSNs.A heuristic is proposed where nodes are forced to send collected data as far as they could and bypass some intermediate nodes to save their energy.Hu et al. [23] proposed the deployment of superior set of sensors, called microservers in a hybrid deployment framework.These microservers are used to filter and route the data in order to reduce the load on other devices.Lee et al. [9] shows mathematically and using simulation that using sensors with different lifetime might prolong the overall wireless sensor network's lifetime.Deployment of mobile nodes has been described in several contexts.One common approach assumes availability of a superior leader that guides several mobile sensor nodes to their deployment positions.Wang et al. [7] proposed an algorithm that uses mobile nodes to cover the blind spots in the monitoring area based on static nodes biding data.In addition, Poduri et al. [20] introduced an algorithm based on artificial potential fields for self-deployment of mobile sensor nodes.The algorithm aims to achieve maximum coverage of the monitoring fields.Furthermore, Howard et al. [2] presented an incremental deployment approach that uses the information gathered from previously deployed nodes in unknown environment to guide deploying the rest of the nodes.Issues related to sensors reliability are presented in [18,27,28] considering the deployment of stationary sensing devices with imprecise detection capabilities.
The objective is to maximize the monitoring field coverage for target detection using a set of devices with probabilistic precessions.In addition, the effect of sensors aging on coverage performance is studied in [10].Furthermore, sensor self-scheduling (state-switching) capability for energy saving purposes is considered in [24,26].The goal is to prolong the network life time through scheduling sensor nodes to be inactive during periods with slow or no activities (e.g., off-peak periods).
The paper is organized as follows.The sensor deployment problem is formally defined in section 2 which also describes the mathematical program developed for the problem.Section 3 presents the GA and SA algorithms used to solve this problem.Experiments that illustrate the performance of these algorithms are presented in Section 4. Finally, the paper concludes in section 5.

A field
) (A G is given.This field is divided into A zones that monitored for a horizon of length T using a set of surveillance devices

Off
otherwise.
In addition, a surveillance device could be stationary or mobile.If a stationary device is deployed in a zone, this device is assumed to remain in this zone for the entire lifespan of the device.On the contrary, a mobile device can cover multiple zones over the surveillance period T .
All mobile devices are assumed to have no restriction on the start or the end locations of their deployment tours.We define the binary variable

 (1)
Subject to: Assignment Constraints Mobility Constraints As shown in (1), the objective function maximizes the field coverage which is described as the sum over all time intervals, zones and surveillance devices of the products of the observation weight t i w , the decision variable  (20) ensure that each surveillance device is not utilized beyond its lifespan.The consumption of a device's lifespan is computed as the sum over all intervals in which the device is active plus the loss in the device lifespan associated with its moves along the different zones.Finally, the integrality of all binary variables is preserved in constraints (21).

Meta Heuristic Approximate Approaches
Due to the intractability of this problem, seeking optimal deployment schemes for largescale problems might be impractical.In this section, we present two approximate meta-heuristics to solve large-scale sensor deployment problems.These heuristics adopt Genetic Algorithm and Simulated Annealing approaches, respectively.Details on modeling and implementation issues of these two algorithms are provided in the following subsections.

The Genetic Algorithm Approach
GA is a machine-learning model, which adopts its behavior from the processes of evolution in nature.This is done by the creation of a population of individuals represented by chromosomes.Individuals in the population continuously go through a process of evolution to increase their fitness and adaptiveness to their environments.The evolution is made up through exchanging characteristics with other elements of the population (crossover) or through selfchanges in the element (mutation).New generations appear from cloning of the current population, in proportion to their fitness.The fitness is a single objective (evaluation) function of the chromosome that returns a numerical value to differentiate between good and bad ones.
Figure 1 gives a flow chart of a typical GA.The details of the algorithm and its advanced features can be found in [1,5,13].A typical GA starts by generating a set of initial chromosomes (initial population).This is usually done following a random procedure.Then, each of these chromosomes is evaluated through objective (evaluation) function to obtain their fitness.While a satisfactory chromosome is not obtained, the algorithm selects from the previous population a set where n is a predefined parameter.The initial n chromosomes can be generated randomly or based on a guided algorithm.In case a guided algorithm is used, the first chromosome is generated randomly; then the second chromosome is generated through applying one of the two crossover operators described above.These two chromosomes are then used to generate the rest of the population as described hereafter.For n new chromosomes to be generated, crossover and mutation operators are applied on chromosomes generated in the previous iteration.Two different crossover operators are adopted, which are Time-Exchange (TE) and Best-Chromosome (BC).The TE crossover is similar to the case where a single chromosome per iteration is generated.However, in the case where multiple chromosomes per iteration are generated, the sensor deployment patterns, in the same time interval, in two different chromosomes are exchanged.Figure 5 presents an example on the application of the TE crossover operator.Two chromosomes exchanges their sensor deployment patterns in time interval t1 resulting in two new constraints.The BC crossover operator is similar to the TE operator with the exception that only the two fittest chromosomes are used as parents for all new chromosomes.Following the crossover operations, the mutation operation is used to prevent the search from getting trapped in the local minima and also to prevent chromosomes repetition.In the mutation step, some of the generated chromosomes genes are randomly altered (from zero to one or vise versa).Thus, the search is directed to a new area in the search space.The fitness of each generated chromosomes is measured using the fitness function given below.One should notice the similarity between this fitness function and the objective function of the mathematical program in §2.
, where x is the chromosome identifier.
Given the fitness value for each new chromosome, chromosomes are added as part of the current population only if they outperform the current available solution.The algorithm stopping criteria could be based on a fixed number of iterations or a given number of iterations in which the solution does not improve.

The Simulated Annealing Approach
Simulated Annealing (SA) is a randomized search technique for highly nonlinear problems [19].In its search process, the algorithm is similar to using a bouncing ball that can bounce over mountain from valley to valley based on the ball's temperature until the highest tip is found.The algorithm starts by generating an initial feasible solution and computing its performance.This solution is stored as the best solution obtained so far.Neighborhood of this solution is searched and a new solution is generated.If the new solution's performance is greater than the highest gain found so far (uphill move), the new solution is accepted and saved.If the gain of the new solution is less than the upper bound performance found so far, still accept this new inferior solution but with some probability (downhill move).The probability of accepting inferior solutions is reduced after each iteration (increase of the ball temperature).The process continues until no better solution is found indicating that the maximum possible temperature is achieved.A formal description of the SA algorithm can be described using the following main steps: Define, X 0 = initial solution, and the best solution so far,


= heating rate for temperature schedule, P(X k+1 , X k ) = probability of acceptance a new solution (X k+1 ) given that the solution is (X k ).This probability is calculated as follows: Step 1: Set k=1 and select the initial temperature 1  and the final temperature f  .
Select an initial solution X 1 and set X 0 =X 1 .
Step 2: Select a new solution X k+1 from N(X k ).
Applying the SA algorithm for the sensors deployment problem, a solution is represented by a string of integers.The length of this string is |S|*|T|.In this string, each sensor-time interval is assigned a zone such that no two sensors are allowed to be deployed on the same zone in the same time interval.If a sensor is not used in one time interval, the corresponding cell in this solution string is assigned zero.Figure 7 illustrates an example for representing a solution generated by SA algorithm.Similar to the GA algorithm, generated solutions are subjected to feasibility check to ensure the satisfaction of all constrained described in §2.In addition, the algorithm can be extended to generate multiple solutions per iteration.In such case, different neighborhoods are explored at the same time.The incombent value X ˆmaintains the best solution from all of generated solutions per iteration.A comparison of the SA algorithm performance considering single solution and multiple solution implementations is presented hereafter.t1 t2 t3 z1 0 z4 z2 z4 z3 0 z4 z2 s1 s2 s3 s1 s2 s3 s1 s2 s3 Figure 7: example on the simulated annealing solution in which 4 zones are monitored by 3 sensors for 3 time unites

GA and SA Benchmarking and Comparison
The mathematical program described in §2 is used to provide an optimal solution for the sensor deployment problem.The commercial optimization package CPLEX 8.0 running on a 2.4 GHz machine with 2 GB memory is used to generate the optimal solution for different problem settings.This optimal solution is used to benchmark the performance of solutions obtained by the GA and SA algorithms.Three different sets of experiments are conducted.These experiments study the effect of increasing number of zones, number of sensors and time horizon on the running time required to generate the optimal solution, respectively.In all experiments, the time-varying observations on the different zones were generated randomly following a uniform distribution U(0,200).In addition, a heterogeneous set of sensors is assumed.The sensors' operational characteristics s L , s M and s P are generated randomly as function of the length of monitored horizon.For example, if the monitoring horizon is T intervals, the sensor lifespan, maximum allowed number of moves, and maximum allowed number of state switching are generated randomly using the uniform distribution U(1,T ).All sensors are assumed to be 100% reliable.
As illustrated in Table 1, the running time required to generate the optimal solution increases exponentially with the increase in the size of the problem.For instance, a running time of 1542.4 seconds is recorded for a problem of 10 zones, five sensors and a horizon of 12 intervals.This running time jumps to 59031.1 seconds when number of zones is increased to 30.
Problem settings with dimensions beyond the ones presented in the table could not be generated using the machine mentioned above.The results indicate that both GA and SA algorithms provide high quality solutions.In the experiment with lowest performance (experiment 5 using the SA), 85% of the optimal objective function value is obtained.Furthermore, up to 99% of the corresponding optimal performance is recorded when the GA algorithm is used in experiments 1 and 9.The running time of both algorithms is noticeably small compared to that of the optimal solution.For example, in experiment 1, the running times of the GA and SA algorithms are observed to be 0.05% and 0.06% of the optimal solution's running time, respectively.In most of these experiments, the GA outperforms the SA in terms of objective function.
On the other hand, the SA seems to converge faster than GA algorithm.These results are confirmed in Figure 8 which illustrates the comparison results of the GA and SA algorithms when different numbers of chromosomes/solutions per iteration are considered.sensors capabilities are generated randomly as mentioned above.As shown in Figure 8a, the genetic algorithms outperform the simulated annealing algorithm in terms of the objective function.On average, the recorded objective performance for the SA algorithm is almost 96% of the genetic algorithm performance.However, the SA running time is less than that of the GA running time by 9%.This set of experiments also illustrates the impact of the number of chromosomes/solutions per iteration on the solution performance.In general, increasing the number of solutions per iteration resulted in convergence at a better objective function for both algorithms.This is achieved on the expense of the running time, however.For example, the objective performance of a single chromosome/solution is almost 70% of that when 50 chromosome/solutions per iteration are generated.The required time for a single chromosome/solution is approximately 0.02% of the 50 chromosomes/solutions per iteration case.

GA Related Results
In this section, we present results related to the GA algorithm.First, the algorithm convergence pattern is presented for the cases of single and multiple chromosomes.Then, the effect of crossover and mutation strategies on the solution quality is illustrated.Figure 9 shows the objective performance and corresponding running time for a deployment problem with 100 zones, 50 sensors and 12 time intervals.The value of the objective performance and the corresponding cumulative running time are recorded after every iteration.SE crossover operator and 100% mutation are used in these experiments.As shown in Figure 9a, generating 10 chromosomes per iteration results in convergence at higher objective using less number of iterations.For instance, in the multiple chromosomes case, an objective of 24841 units is recorded at iteration 64.This value is achieved at iteration 156 in the single chromosome case.
On the other hand, the running time of multiple chromosomes are higher than the time recorded for the single chromosome case.As shown in the Figure 9b, the running time in the single chromosome implementation is almost 60% of that recorded in the multiple chromosomes implementation.in greater running time.Thus, for higher coverage performance, one might recommend using SE crossover operator as long as number of sensors is less than the number of zones.Otherwise, TE is recommended from the running time point of view.
As the horizon length increases, the TE crossover strategy becomes more superior in terms of coverage performance.This is also associated with an increase in the running time.As illustrated in Figure 11, for horizons beyond 20 time intervals, the SE strategy yields higher coverage performance.A corresponding pattern is recorded for the algorithm running time.The time difference shown in the Figures 10b and 11b, returns to the additional processing time required for the SE operator to ensure feasibility of generated chromosomes.As mentioned above, mutation in GA plays an important role in directing the solution towards different search spaces.Figure 12 illustrates the effect of using different mutation percentage on the average objective performance.Mutation percentage is measured as the ratio between number of altered genes and total chromosome length.A field of 100 zones is monitored for 12 units of time using 50 sensors.Ten chromosomes per iteration and TE crossover operator are used throughout these experiments.The mutation percentage ranges from 0% to 100%.The results show that as the mutation percentage increases, the objective performance increases.For instance, at 20% mutation rate, an objective of 9376 units is recorded.The objective increased to 10389 units at 100% mutation rate.

SA Related Results
The SA objective function convergence pattern is presented in this section.For this purpose, the SA algorithm is applied for a problem of 100 zones, 12 time units and 50 sensors.
The heating rate is selected to be 0.99 and a sample of 300 iterations is recorded.The starting temperature is assumed to be 2 temperature units and the final temperature is set to 50.As shown in Figure 13, the algorithm starts by pivoting at solutions with low objective values since the acceptance probability of a new solution is initially high.This may lead the algorithm to fall in a local minimum such as the fall occurred at iteration 154.Also, the incumbent value X maintains the highest objective value which is 7939 reached at iteration 114.As the temperature increases, the acceptance probability is decreased, and the chance of pivoting at low performance solutions decreases.This pattern is clearly showed starting at iteration 200.
Two implementations are considered for the SA algorithm: single solution per iteration and multiple solutions per iteration.Multiple solutions per iteration algorithm direct the search into multiple search spaces.To compare these two implementations, problem settings of 50 sensors 12 time units and number of zones that range from 100 to 1000 zones.For the multiple solutions case, 10 solutions per iteration are generated.The starting and final temperatures are set to 2 and 1000, respectively.The temperature rate is assumed to be 0.99.As shown in Figure 14, based on the conducted experiments, the multiple solutions implementation seems to outperform the single solution.For example, for a problem with 1000 zones, the objective function of single solution implementation is 10% less than that of the multiple solutions implementation.This 10% improvement in the coverage performance is associated with about 40% increase in the running time.

Conclusions
A modeling framework for the problem of deploying a set of heterogeneous sensors in a field with time-varying differential surveillance requirements is presented.In this framework, the problem is formulated as a mixed integer mathematical program with the objective is to maximize coverage of a given field.A set of constraints are defined to ensure that sensors are not used beyond their capacity.Two meta-heuristics are also used to solve this problem.The first heuristic adopts a Genetic Algorithm (GA) approach while the second heuristic implements a Simulated Annealing (SA) algorithm.A set of experiments are used to illustrate capabilities of the developed models and to compare their performance.The results indicate that both GA and SA algorithms provide high quality solutions in terms of objective function and running time.In addition, GA seems to yield better coverage performance than the SA.However, its running time was always higher in all considered experimental settings.

Figure 5 :Figure 6 :
Figure 5: Time-exchange crossover operation between two chromosomes; (a) before crossover (b) after crossover.t1 in both chromosomes are exchanged to generate two new chromosomes of the current solution, G(X k ) = performance of the current solution, (highest temperature value),

Figure 8 :
Figure 8: (a) A comparison between genetic and simulated annealing algorithms with different chromosomes/solutions per iteration, (b) The running time of genetic and simulated annealing algorithms

Figure 9 :
Figure 9: GA performance progress with number of iterations (a) Objective performance (b) Running time

Figure 10 :Figure 11 :
Figure 10: A comparison between Time-exchange and Sensor-exchange crossover operators performance with increasing number of sensors (a) Objective performance (b) Running time

Table 1 :
Performance of the GA and SA algorithms compared to the optimal solution.