Multi-Objective Simulated Annealing Approach for Optimal Routing in Time-Driven Sensor Networks

María Luisa Santamaría and Sebastià Galmés Dept. of Mathematics and Computer Science

Universitat de les Illes Balears Palma de Mallorca, Spain

maria-luisa.santamaria, sebastia.galmes@uib.es

Abstract—In this work we propose multi-objective simulated annealing as a heuristic technique for optimal routing in time-driven sensor networks. Unlike previous algorithms and methods, this technique is intended to tackle multiple performance-related and other design objectives in a computationally feasible way. Since these objectives are usually in conflict, the general solution is formulated as the so-called Pareto set, which is the set of non-dominated design vectors representing different tradeoffs.

Keywords-time-driven wireless sensor network; time-division multiple access; spanning tree; simulated annealing.

I. INTRODUCTION It is widely recognized that one of the main design issues in

sensor networks is lifetime, which is directly related to how fast sensor nodes drain their energy. The fact is that every node in a Time-Driven Sensor Network (TD-WSN) wastes a certain amount of energy per communication round, which can be basically attributed to the transmission and reception of packets. Given the importance of this issue, usually the problem of constructing an optimal data-gathering tree has consisted of determining a tree that maximizes the time until first node death.

One solution to this problem is provided in [1], which demonstrates its NP-hard complexity. Thus it proposes a heuristic approach on the basis of several simplifying but too strict assumptions: absence of power control, which eliminates the dependence on transmission distance, and use of a strong packet aggregation scheme that reduces the number of packets to be processed by a node to the number of its direct descendents (children). In contrast, other works account for the transmission distance by applying the well-known Minimum Spanning Tree (MST) algorithm, but again the total traffic load supported by every node is ignored. Other algorithms, like the well-known Collection Tree Protocol (CTP) [2], do not even focus on maximizing network lifetime, but on performance, and relegate the issue of energy consumption to the MAC layer (ZigBee, for instance). More specifically, CTP is aimed at finding high-throughput paths from nodes to sink on the basis of link quality estimates - use of the ETX (Expected Transmissions) metric [3].

The common denominator to these routing schemes (see [4] for a complete survey) is that they only focus on a single optimization objective (be it lifetime, throughput or any other), and in addition they address such objective by relying on a single factor associated with the node local environment (be it distance, link quality or number of direct descendants). In contrast, the work proposed in this paper is intended to face up the construction of data-gathering trees by considering multiple optimization objectives as well as global-scale factors in the evaluation of such objectives. Accordingly, the rest of the paper is organized as follows. In Section II, we formulate the problem and propose a general solution method based on the technique of simulated annealing – the so-called Multi-Objective Simulated Annealing (MOSA). In Section III, we formulate a MOSA algorithm adapted to TD-WSN. In Section IV we provide some preliminary results and, in Section V, we describe subsequent research tasks.

II. PROBLEM CHARACTERIZATION Although lifetime is of primary importance in the design of

sensor networks, other objectives related with network performance and planning may be relevant too. Without being exhaustive, Table I highlights the main design objectives in the construction of a data-gathering tree and the optimization criteria that best fit these objectives (the degree of fitting is indicated as a small or large dot). In particular, note that lifetime can be enhanced via different optimization criteria. One of them focuses exclusively on link distance, given its relevance in the evaluation of the energy wasted in the transmission of packets. In this case, the Minimum Spanning Tree (MST) criterion, with link cost function equal to Euclidean distance, becomes appropriate. Other criteria are intended to enhance network lifetime by homogenizing the distribution of energy consumption over the network. Examples like “choose that route with smallest reluctance”, where reluctance is defined as the reciprocal of residual node battery, fall within this category. In this case, the Shortest Path Routing (SPR) with link cost equal to reluctance becomes the appropriate choice. The advantage of these optimization criteria, which are all based on assigning simple costs to links, is that they can be easily implemented by means

19th Annual IEEE International Symposium on Modelling, Analysis, and Simulation of Computer and Telecommunication Systems

1526-7539/11 $26.00 © 2011 IEEE

DOI 10.1109/MASCOTS.2011.55

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19th Annual IEEE International Symposium on Modelling, Analysis, and Simulation of Computer and Telecommunication Systems

1526-7539/11 $26.00 © 2011 IEEE

DOI 10.1109/MASCOTS.2011.55

458

19th Annual IEEE International Symposium on Modelling, Analysis, and Simulation of Computer and Telecommunication Systems

1526-7539/11 $26.00 © 2011 IEEE

DOI 10.1109/MASCOTS.2011.55

458

of well-known efficient algorithms (see [5] for more details). However, these link cost-based algorithms can only yield limited results in terms of lifetime, either because they do not tackle all relevant factors (transmission distance, traffic load and others) or because they do not directly focus on maximizing the time until first node death (lifetime definition).

TABLE I. DESIGN OBJECTIVES AND OPTIMIZATION CRITERIA

Design Objectives Optimization Criteriaa

MHC MST SPR STML

Lifetime

PER, PLRb

Path delay

Throughput

Number of relay nodes

a. MHC: Minimum Hop Count; MST: Minimum Spanning Tree (with link cost equal to Euclidean distance); SPR: Shortest Path Routing (with generic link cost); STML: Spanning Tree with Maximum Lifetime.

b. PER: Packet Error Rate; PLR: Packet Loss Rate.

Altogether this means that maximizing network lifetime can only be based on a full spanning tree exploration, which we refer to in Table I as finding the Spanning Tree with Maximum Lifetime (STML). Unfortunately, the problem of exploring all possible spanning trees is known to be NP-hard (if the network contains N nodes and 1 base station, the number of different spanning trees is given by 1)1( −+ NN - Cayley’s formula [6]) and thus it can only be solved via heuristic techniques unless the network size is very small.

Other design objectives in Table I deserve some attention. For instance, path delay and packet error/loss rate can be optimized via some form of SPR, although for the majority of time-driven sensor networks, the Minimum Hop Count (MHC) criterion suffices (MHC is a particular case of SPR). Regarding the throughput, selecting the route with maximum sum of ETX link metrics has been proved to generate high-throughput paths between the source nodes and the base station [3]. This relies again on SPR via an appropriate choice of ETX-based link costs. Finally, the MST criterion, based again on Euclidean distance, or other related criteria such as the spanning tree with minimum number of Steiner points, entail the minimization of the number of relay nodes that must be inserted in order to create a connected network from sparsely deployed nodes.

Unfortunately, the design objectives described so far are usually in conflict. For instance, MHC would be better accomplished by connecting every source node directly to the base station, but this could lead to poor lifetimes or require too many additional relay nodes. Also, if nodes had power control capabilities, their lifetime could be increased by reducing as much as possible their transmission distances, but again this could require excessive relay nodes. A third case is highlighted in

[3], where it is shown that high-throughput paths are not necessarily those with minimum number of hops.

The most complete formulation of the solution to a multi-objective optimization problem with competing objectives is the so-called Pareto set, which is the set of non-dominated solutions. In order to formalize this concept for our graph-related problem, let us first adopt the following definitions:

• V: set of nodes, including the base station, whose geographic location is assumed to be fixed.

• N: size of the sensor network, that is, number of sensor nodes. Obviously, 1−= VN .

• G = (V, A): complete graph on the N+1 vertices that represent the nodes and the base station. Note that this graph has an arc (link) between every pair of vertices (nodes). Thus, A represents the set of all possible arcs.

• SG: Set of all possible spanning trees in G.

With no loss of generality, let us assume that all objectives are of minimization type (a maximization objective can be easily adapted by simply taking its reciprocal). Then, a multi-objective variable is formulated as o = [o1, Ωom], where oj, j = 1Ωm denotes an arbitrary optimization (minimization) objective and m is the total number of objectives. Note that in fact o = o(t) and oj = oj(t), where t is a feasible solution, that is, an arbitrary spanning tree (t œ SG). Then, it is said that solution t œ SG dominates solution u œ SG (denoted as t u) if and only if o(t) < o(u), which means that

)()(:1)()(,1 uotomkuotomj kkjj <∈∃∧≤∈∀ …… .(1)

This is the concept of Pareto dominance. In case that two feasible solutions t and u are such that both t u and u t are false, it is said that t and u constitute a pair of non-dominated solutions (denoted as t ~ u).

A standard technique for generating the Pareto set in multi-objective optimization problems is to minimize weighted sums of the different objectives for different settings of the weights. However, aside from the difficulties associated with the a priori selection of weights, in [7] it is demonstrated that such technique can leave significant parts of the Pareto set inaccessible. Instead, current techniques exploit the concept of dominance outlined above in order to determine a rich set of Pareto solutions (then, any criterion can be applied to select a particular solution within such a representative set, but this is out of the scope of this paper). One of these powerful techniques is MOSA, which is considered in the next section.

III. MOSA PROPOSAL FOR OPTIMAL ROUTING IN TD-WSN The feasibility of a multi-objective approach for optimal

routing in TD-WSN is highly dependent on its computational

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complexity, since the number of nodes can be very large - recall that, if at least one of the criteria is STML, the theoretical complexity is given by 1)1( −+ NN . In this sense, we made a first step in [8], where we considered a single optimization objective (lifetime) by using a more complete node energy consumption model where both, transmission distance and traffic load, were taken into account. More specifically, we proposed simulated annealing (SA) as a heuristic solving-method and we showed that this approach allows for obtaining the optimal or a close-to-optimal solution in linear complexity with the number of nodes. Then, it is because of these satisfactory results that we are currently working on the multi-objective extension of this methodology to optimal routing in TD-WSN. To the best of our knowledge, this is the first time that such a multi-objective framework (and particularly MOSA) is proposed.

1 Set PS = t, T = T0; /t is any initial spanning tree/ 2 while (T > Tf) 3 for i = 1ΩLE 4 t’ = Perturb(t) 5 if (t’ t ¤ t’ ~ t) 6 t = t’ 7 PS = UpdatePS (t, PS) 8 else if (exp(-Do/T) > Random(0,1) 9 t = t’ 10 end if 11 end for 12 T = aÿT 13 end while

Figure 1. MOSA Core algorithm.

Our proposal for the pseudo-code of the MOSA algorithm is shown in Fig. 1 (MOSA Core). In this figure, PS represents the current Pareto set, T is the so-called temperature or control parameter, T0 is an initial value for this temperature, Tf is the final temperature (frozen condition), a is a real number less than 1, LE is the length of each iteration, and Perturb and UpdatePS are two associated modules described respectively in Figs. 2 and 3. Line 8 in MOSA Core corresponds to the application of the so-called Metropolis criterion, which essentially allows for avoiding high-climbing. Do is some positive function that quantifies the multi-objective variation, that is, the discrepancy between o(t’) and o(t). This function may be taken as the maximum (minimum) among all single-objective variations, although the most recommended implementation is the average over all such single-objective variations.

Regarding the pseudo-code for Perturb, the formulation suggested in [9] has initially been adapted to the problem under consideration, although it may be subject to variations. In this module, P(i, t) denotes the parent node of node i in a given tree t. Note that line 4 are intended to avoid cycles when transforming t into t’. Finally, in UpdatePS (Fig. 3), Size(PS) represents a simple module that returns the current number of elements in PS, PS(i) is the i:th element in PS (it is assumed that some ordination has been previously defined in PS, being the particular order irrelevant), Remove(PS(i)) is a procedure that eliminates the i:th

element from PS and Return(PS) is a terminating module that returns PS in its current state.

1 Select one node i = 1ΩN arbitrarily 2 Select another node j ∫ i arbitrarily 3 P(i, t’) = j 4 if cycle = true P(j, t’) = P(i, t)

Figure 2. Perturb (perturbation) module.

IV. PRELIMINARY RESULTS In order to preliminarily assess the correctness and viability

of the proposed MOSA approach to optimal routing, we performed several tests over small-size sensor networks in which full exploration was feasible. We considered three optimization criteria, namely maximizing network lifetime (STML), minimizing average link distance (MST), which is directly related with the sum of all link distances, and minimizing average hop count (MHC). For the evaluation of lifetime, we adopted the following model for the energy consumed by node i, E(i):

iidmEiig

mEiigiEf

w

elec

∀⋅⋅⋅++

⋅⋅+=

,)())()((

))(2)(()(

σ

σ (2)

1 for i = 1ΩSize(PS) 2 if (t PS(i)) 3 Remove(PS(i)) 4 else if (t PS(i)) 5 Return(PS) 6 end if 7 end for 8 PS = PS ( t 9 Return(PS)

Figure 3. UpdatePS module.

Here, g(i) and σ(i) stand respectively for the number of packets generated and forwarded by node i during a communication round (traffic load), d(i) is the transmission distance of node i (link distance between node i and its parent node in a given tree), Eelec is the energy dissipated by the transceiver circuitry to transmit or receive a single bit, )(idE f

w ⋅ is the energy radiated to the wireless medium to transmit a single bit over a link of distance )(id , f is the path-loss exponent and m is the packet size in bits. In turn, the distance-dependent component depends on the distance itself: if d(i) § d0 (reference distance), then Ew = Efs and f = 2 (free-space propagation); else if d ¥ d0, then Ew = Emp and f > 2 (multi-path propagation). Specifically, the values adopted for these parameters were as follows: Eelec = 50 nJ/bit, Efs = 10 pJ/bit/m2, Emp = 0.0013 pJ/bit/m4, f = 4 and d0 = 75 m. The battery capacity was set to 15kJ.

As a first step, we implemented a MOSA version with hierarchical objectives, which means that the algorithm was driven by the maximization of one of them (lifetime), but it was

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also entrusted to check the rest of objectives for each explored spanning tree, even is this tree was to be discarded by the Metropolis criterion. Moreover, we used a slightly inefficient implementation in order to widen the range of explored (feasible) solutions. Specifically, the average complexity in number of iterations, as a function of the network size, was approximately given by 40000 + 2495ÿN (see [8]). Table II shows the results for network sizes between 3 and 8 sensor nodes. For each size, the results shown in the table correspond to a single random deployment of nodes in a region of 250 m x 250 m, with the base station located in the middle of one side. Although a more detailed analysis is required in order to get more precise estimates (by generating multiple random deployments for each network size), the table reveals encouraging results in such a preliminary assessment of correctness and practical viability. Two results are particularly significant. One is the fact that the increasing trend of the size of the exact Pareto set is much smoother than that of the theoretical complexity. The other refers to the capture percentage, which is defined as the percentage of solutions within the exact Pareto set that are present in the detected Pareto set. Note that this is a conservative metric because it only reflects the degree of perfect matching between the two sets. However, this does not mean that the rest of solutions in the detected Pareto are poor; in fact, the non-intersecting subsets in both exact and detected Pareto sets are prone to be close, as it can be expected from any heuristic methodology that in general produces a suboptimal solution.

TABLE II. EXECUTION MTERICS FOR DIFFERENT NETWORK SIZES

Execution Metrics

Number of Sensor Nodes 3 4 5 6 7 8

Theoretical complexity

16 125 1296 16807 262144 4782969

Size of exact Pareto set 6 8 17 15 39 77

Size of detected Pareto set 6 8 17 15 33 48

Capture percentage 100 100 100 100 66.67 37.66

Number of iterations (x1000) 51 51 52 52 86 85

For illustration purposes, Fig. 4 shows the bi-dimensional distribution of feasible solutions for a random deployment of 3 nodes that resulted in the following coordinates with respect to the base station: (113, -16), (103, 92) and (167, 115) (in meters). The x-axis corresponds to the network lifetime (L) and the y-axis to the average link distance (ALD), which is proportional to the sum of all link distances. For simplicity, the third magnitude - the average hop count (AHC), has been indicated as labels only for the elements of the Pareto front (Pareto set in the decision space). Note also that only a subset of the Pareto set are the solutions that would be obtained by applying the MHC, MST and STML algorithms in isolation.

V. CONCLUSIONS AND FURTHER WORK In this work, we have proposed MOSA as a general method

for optimal routing in TD-WSN. Specifically, we have developed a pseudo-code for this method and obtained preliminary satisfactory results. Further research is devoted to analyze the case of non-hierarchical objectives, which is expected to provide better tradeoffs between complexity and capture percentage.

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ACKNOWLEDGMENT This work is supported by the Spanish Ministry of Science

and Innovation under contract TIN10-16345.

REFERENCES [1] Y. Wu, S. Fahmy, and N. B. Shroff, “On the construction of a

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[4] K. Akkaya, and M. Younis, “A survey on routing protocols for wireless sensor networks,” Ad Hoc Networks, vol. 3, pp. 325-349, 2005.

[5] D. Bertsekas, and R. Gallager, Data Networks. Prentice-Hall, 1992.

[6] M. Aigner, and G. M. Ziegler, Proofs from The Book. Springer, 1998.

[7] I. Das, and J. E. Dennis, “A closer look at drawbacks of minimizing weighted sums of objectives for Pareto set generation in multicriteria optimization problems," Structural and Multidisciplinary Optimization, vol. 14, pp. 63-69, 1997.

[8] M. L. Santamaría, S. Galmés, and R. Puigjaner, “Simulated annealing approach to optimizing the lifetime of sparse time-driven sensor networks,” Proc. of IEEE/ACM MASCOTS 2009 (London, UK, September 21-23, 2009).

[9] B. Suman, and P. Kumar, “A survey of simulated annealing as a tool for single and multiobjective optimization,” Journal of the Operational Research Society, vol. 57, pp. 1143-1160, 2006.

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