On Scheduling the Dwells of a Multifunction Radar

Emilie WINTER, Ludovic LuPINSKIRadar Technical DirectorateTHALES AIRBORNE SYSTEMS78851 Elancourt, France

{emilie.winter, ludovic.lupinski}@,fr.thalesgroup.com

Abstract Among several other tasks, the radar of a fighter hasto search, track and identify potential targets. The waveformsused by the radar for each of these tasks are most oftenincompatible and hence, cannot be processed simultaneously.Moreover, these tasks are repeated several times in a cyclicfashion. Altogether, this deflnes a complex scheduling problemthat impacts a lot on the quality of the radar's output. In thispaper, we define a formal framework for this real timescheduling problem and we introduce a local search method tocompute efficient schedules for the radar. We also describe aset of lower bounds for the scheduling problem. Simulatedresults are provided.

Keywords Scheduling, Local Search, Linear Programming,Airborne Radar.

I. INTRODUCTIONThis paper is focused on recent radars with an

Electronically Steered Antenna (ESA). An ESA is a planararray antenna made of many individual radiating elements.Unlike a mechanically steered antenna, which is pivoting soas to direct the beam, an ESA lies in a fix position on theaircraft. The phase of the radiowaves is controlledelectronically so that the radar beam lights up the desireddirection. See [1], [2] and [3] for a more detailed descriptionof airborne radars.

One of the key advantages of ESAs is that the beam isextremely agile. As it is not subjected to the mechanicalinertia of the antenna, it can be moved instantaneouslyanywhere, anytime, even outside the search domain (seeFigure 1). Moreover, the radar can switch instantaneously toan appropriate waveform.

The following tasks have to be achieved by themultifunction radar:

* Research. The radar is sweeping across a domain todetect the potential presence of targets inside.

* Tracking. The radar is closely monitoring thebehavior of targets (detected at the Research stage).

Figure 1. Functions and Advantages of ESA Radars.

* Data Link. The radar is used as a communicationtool with other platforms.

* Calibration. The radar is performing cycliccalibration to ensure a high reliability level.

Research, Tracking, Data Link and Calibration requireincompatible waveforms. So, all corresponding tasks have tobe scheduled to ensure they do not overlap in time. Moreover,Research, Tracking, Data Link and Calibration tasks have tobe repeated in a more or less regular fashion. Depending onthe nature of the task, the periodicity constraint can beextremely important or not. For instance, as it is absolutelyforbidden to lose a close target, tracking dwells have to berepeated with high regularity. Likewise, data link dwells areplayed at regular intervals, but there, the regularity constraintis much higher. Finally, to ensure a surveillance of greatquality, in any circumstances, the radar has to play at least aminimum amount of research dwells.

The aim of the study is to make the radar "moreefficient" while meeting all constraints informally describedabove. We also want to take into account situations wherethe radar is overloaded and where some tasks have to berejected by the system. Barbaresco [4] describes a stronglyrelated framework: Tasks are scheduled on a frame durationand are executed while the next schedule is computed. Toschedule the tasks, Barbaresco associate deadlines totasks and use a heuristic called EDF (Earliest DeadlineFirst). If one of the most important tasks is completedafter its deadline then some tasks are removed from thesystem to reduce the load and the scheduling procedure isrun again. The empirical study led in [4] shows that simple

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scheduling heuristics often improve the behavior of theradar.

We use a model of the problem introduced in [5] to beable to evaluate the quality of solutions (Section 2). Then,section 3 shows that when the sequence order of the tasks isknown, the problem can be solved in polynomial time byLinear Programming. Finally, in section 4, we introduce alocal search method and give some experimental results.

The problem can also be seen as a timing problem inwhich the tasks are scheduled to minimize an Earliness-Tardiness criterion [6].

Another scheduling problem for radars is the problem ofinterleaving tasks corresponding to receiving and sendingdata [7, 8]. We study a situation where we do not haveinterleaving and both the sending and the receiving tasks aremodeled as a unique task.

II. MODEL

We define a 'job" as a single radar function. A jobcould be either the track of a given target or a data link orthe "research". To perform a job, several dwells are repeatedwith more or less regularity. Therefore, we consider that ajob consists in a set of non-preemptive "operations" whichmust obey a kind of periodicity rule.

As shown on figure 2, jobs are not necessarily executedwith strict periodicity. The temporal gap between the twofirst operations of job 1 is greater than the gap between theoperations 2 and 3. This schedule is feasible if performanceconstraints are ensured. The radar is seen as a singlemachine on which jobs have to be processed.

To model the problem, we associate cost functions 8i tothe distance between starting times of consecutiveoperations of the same job i. They are V-Shaped, i.e., 81(x)

max(a(li-x), A/(x-l,)) where li, ai-0 and A6»0 arerespectively the ideal distance between two starting timesand the penalty weights associated to a smaller (resp. larger)inter-distance.

For each kind of job i, we have defined, in collaborationwith radar engineers, cost functions features that fit real lifescenari. An example of what cost functions can be is givenfigure 3.

SIJS21 1 SI, r

_I 031 I

lilyackinig

Research

(11

I / At0

Figure 3. Cost Functions

We are now ready to formally define the problem. Given njobs 1, 2, ..., n. Each job i is made of n(i) consecutive andidentical operations .il,*--, Oin(i) with processing time pi. Foreach job i, we have a cost function 8i associated to thedistance between starting times of consecutive operations.We are also given an integerH > Zi n(i) pi that represents thehorizon of the schedule.

A set of starting times Si defines a feasible schedule if andonly if (1) operations start after or at 0 and are notcompleted later than H and (2) operations do not overlap,for any pair of operations Ou, OiyY Si +pi < Sij or Sij +pi <Su. The cost associated to the schedule is exactly the sumover all jobs i of y 1 'i (Si,+, - S)

Note that, in practice, the problem is solvedcontinuously and the schedule followed in the past interactswith the schedule under construction. Stated another way,many jobs have been started a long time ago. This featuredoes not change the combinatorial structure of the problemand to keep things simple, we omit all details about it.

It is proven in [5] that the problem is NP-Hard in thestrong sense [9].

III. TIMING PROBLEM

We study a variant of the problem, in which the orderbetween all operations is known, that can be solved in

S1,3 S2,3 1,4

_II 2M.S I [01 02

400 timet

Figure 2. Regularity Constraints

polynomial time by linear programming. The timingproblem is used later on to search for an optimal schedule.

Consider the variant of the problem in which the orderof all operations is known. Now the question is to computestarting times (meeting the order mentioned above) thatminimize the total cost. This problem can be solved in

too 2000

IE:Tm

I

polynomial time because cost functions are V-shaped (ingeneral this problem is strongly NP-Hard). We show thatthe problem can be modeled as a Linear Program [10].

From now on, assume that for any job i and anyoperation j, prd(i,j) (respectively succ(i,j)) is the couple ofindices (ij) that corresponds to the operation Oij, preceding(resp. following) immediately OQ in the sequence. Assumethat prd(i,j) = (0,0) (resp. succ(i,j) = (0,0)) if Oi is the first(resp. last) operation in the sequence.

In [5], we show that the following Linear Program (LP)computes the optimal starting times. In LP, Su denotes thestarting time of OQ and Wi is the cost associated to thedistance between Ou andOi1.

min

St

Figure 4). Neighbors are obtained by either swapping tworandomly chosen operations (SWR) or by swappingconsecutive operations (SWC), or by inserting a randomlychosen operation at some other place in the sequence (IR).The "moves" SWR and IR can lead to sequence ofoperations that do not meet the natural order of operationswithin one job. As all jobs' operations have the sameprocessing times and the same cost functions, the sequenceis "repaired" by renumbering operations. We have alsodesigned two other variants of the moves SWRL and IRL(where "L" stands for Limited) in which sequences that donot meet the natural order of operations are not considered.

A move is performed if the schedule corresponding tothe new sequence improves the best known total cost. Thisstep requires the computation of a linear program and, asshown by initial experiments, this turns to be very costly interms of CPU time. To improve the behavior of the

5i,j, + pi, < Sii I < i < n, < j < n(i)(i' j') = prd(i, j) X (0,0)

Wij a,(1, -Si +Sij_) 1< i<n,2< j <n(i)Wij > Pi(Sij -Sijl -li)I <i<n,2< j <n(i)

0<Sii <H-p, 1<i<n,1<j<n(i)

Wij >0 1<i<n,2< j<n(i)LP

IV. HEURISTICS

The schedule of the radar has to be built within somefractions of a second. We then have to derive fast real timeheuristics to solve the problem. In the following we describea local search method.

A. A Local Search MethodLocal search is a well-known methodology used for

obtaining good solutions of NP-hard combinatorialoptimization problems. A local search algorithm starts witha feasible solution x and tries to find a better solution y inthe neighborhood of x. If such a solution y is found then thecurrent solution becomes y and the algorithm iterates.otherwise, the algorithm ends with the locally optimalsolution x. We refer to [11] for an overview of local searchand to [12,13] for a detailed study of local search methodsapplied to scheduling problems. A key feature of efficientLocal Search techniques lies in the definition of theneighborhood.

Recall that when the sequence of all operations is known,the problem can be solved in polynomial time by linearprogramming (Section3). Hence, a solution can be definedby the sequence of operations. Our local search obviouslyrelies on this encoding of solutions that allows us to rely onwell-known neighborhood structures. We start with somerandom sequence that we iteratively try to improve (see

Figure 4. An Overview of Local Search.

algorithm, we have designed a set of dominance rules thatallows us to discard some of the moves that do not improvethe solution. Due to space limitation, this important featureof the algorithm is not described in the paper.

B. ExperimentsWe have generated 27 random instances that represent

real life situations. The total number of operations variesfrom 44 to 96, the load of the radar varies from 45% to 100% and the slopes of the cost functions a and 6 have beenchosen to model different kind of situations. All

experiments have made run on DELL Latitude D600 laptoprunning Linux. We have used GNU Linear ProgrammingKit [14] as a LP solver.

We first report computational results of the 5 local searchmethods SWR, SWC, IR, SWRL, IRL over the 27 instances.For each instance, we denote by ube the value of theobjective function after t seconds of CPU time for the localsearch X (either SWR, SWC, IR, SWRL or IRL). We define*uboo as the best solution found over all methods. To reporton the average behavior of the algorithms, we normalize allobjective values. Figure 5 reports the average normalizedobjective function (ubt - ubo)1(uebXO- ubo) of the 5 methodsover time. This clearly shows that SWRL, IRL and SWRoutperform all other methods. Moreover, the local searchalgorithms converge fast.

swc

&WR

------SWRL

IRE3 ge \0X, .. ~~~~~~~~IRL

E _

03~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

CPU time(secondes}

Figure 5. Local Search Results.

V. CONCLUSION

We have introduced an efficient local search method,based on a compact and efficient neighborhood, to schedulean airborne radar. Although it is extremely fast, it stillrequires too much time to be used in practice. We are

currently trying to improve the method by reducing thesearch space. We also wish to develop simple heuristics thatcompute competitive schedules.

ACKNOWLEDGMENT

The authors would like to thank Ph. Baptiste (ECOLEPOLYTECHNIQUE, LIX, CNRS) for his active participation inthis work and his expertise on combinatorial optimization.

REFERENCES

[1] J.-P Hardange, Ph. Lacomme and J.-C. Marchais, Airborne andSpaceborne Radars, Masson, 1995.

[2] F. Le Chevalier, Principles of Radar and Sonar Signal Processing,Artech House Publishers, 2002.

[3] G. W. Stimson. Introduction to Airborne Radar, 2nd Edition.SciTech Publishing, Inc., 1998.

[4] F. Barbaresco, "Approche Cognitive de la Gestion Radar",proceedings of Cogis 2003, 2003

[5] Ph. Baptiste, E. Winter, "On Scheduling a Multifunction Radar",proceedings of Cogis 2006, Paris, march 2006.

[6] Y. Hendel, "Contributions a l'ordonnancement juste-a-temps", thesisreport, Pierre et Marie Curie University, LIP6 laboratory, 2005.

[7] M. Elshafei, H. D. Sherali, J. C. Smith. "Radar pulse interleaving formulti-target tracking", Naval Research Logistics, Volume 51, Issue 1,Pages 72--94, 2003.

[8] A. J. Orman, C. N. Potts. "On the Complexity of Coupled-taskScheduling", Discrete Applied Mathematics, 72(1-2): 141-154, 1997.

[9] M. R. Garey and D. S. Johnson. "Computers and Intractability. AGuide to the Theory of NP-Completeness", W. H. Freeman andCompany, 1979.

[10] M. Minoux, Programmation Mathematique, Theorie et algorithmes,Collection Technique et Scientifique des Telecommunications. Bordas& CNET-ENST, Paris, France, 1983.

[11] E. Aarts and J.K. Lenstra, "Local Search in CombinatorialOptimization", John Wiley & Sons, New York, 1997.

[12] P. Brucker, J. Hurink, and F.Werner, "Improving local searchheuristics for some scheduling problems - Part I.", Discrete AppliedMathematics, 65, 97-122, 1996.

[13] P. Brucker, J. Hurink, and F.Werner, "Improving local searchheuristics for some scheduling problems - Part II.", Discrete AppliedMathematics, 72, 47-69, 1997.

[14] GNU Linear Programming Kit, home page.http://www.gnu.org/software/glpk/glpk.html