minimizing the cost of availability of coverage from a constellation of satellites: evaluation of...

Minimizing the Cost ofAvailability of Coveragefrom a Constellation ofSatellites: Evaluation ofOptimization MethodsClifford Kelley and Maged Dessouky*

Daniel J. Epstein Department of Industrial and Systems Engineering, University of Southern California, Los Angeles, CA90089-0193

MINIMIZING THE COST OF AVAILABILITY OF COVERAGE FROM SATELLITE CONSTELLATIONS

Received 27 August 2003; Accepted 21 October 2003, after one or more revisionsDOI 10.1002/sys.10059

ABSTRACT

Satellite systems to provide terrestrial surface coverage are currently optimized by inde-pendently optimizing parts of the system or by locally starting at a “baseline” configurationand making small changes. Even when using a “baseline,” the various components such asconstellation configuration, satellite design, and acquisition have generally been optimizedseparately. A model was developed to minimize the life-cycle cost of the system as a wholewith the appropriate linkages between the major parts of the system in both cost andperformance. The model consists of a cost model, and a performance model. The cost modelincludes the cost of purchasing and maintaining an inventory of satellites and launch vehicles,launch operations, and orbit operations. The performance model evaluates the availabilityover geographic grid averaging areas over a given period of time. The performance of localoptimization techniques (Simplex and Complex) was evaluated against global optimizationtechniques (Simulated Annealing and Genetic Algorithms) for providing availability of cover-age. The global optimization techniques performed much better with Genetic Algorithmsperforming slightly better than Simulated Annealing. The quality of the solutions found wasbenchmarked against a trade study and local optimization methods, and our proposed global

Regular Paper

*Author to whom all correspondence should be addressed([email protected]).

Systems Engineering, Vol. 7, No. 2, 2004© 2004 Wiley Periodicals, Inc.

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systems approach provided significant cost savings over the existing techniques. © 2004 WileyPeriodicals, Inc. Syst Eng 7: 113–122, 2004

Key words: satellites, optimization, coverage

1. INTRODUCTION

In recent years satellite constellations (a group of satel-lites working together to provide a service) have beenproposed and deployed. They provide communication,navigation, surveillance, and other services. As withany system, the designers are challenged to providethese services at the lowest possible cost. As noted inBlanchard and Fabrycky [1998], many systems havebeen planned, designed, produced, and operated withlittle concern for economic issues. Often, when cost isconsidered, it is concentrated on acquisition costs,which are only the tip of the iceberg. Blanchard andFabrycky [1998] also noted that the largest impact onthe cost of a system is in the planning and conceptualstage. Because of this, it is critical that a cost andperformance model be used during this part of the lifeof a program. In this paper, we develop a completelife-cycle model for minimizing the cost of availabilityof coverage from a constellation of satellites.

Satellite systems to provide coverage are currentlyoptimized by independently optimizing parts of thesystem or by locally starting at a “baseline” configura-tion and making small changes. Even when using a“baseline,” the various components such as constella-tion configuration, satellite design, and acquisition havegenerally been optimized separately. As illustrated inWertz and Larson [1996], the satellite design life hasbeen optimized to provide the lowest cost per year forsatellite and launch vehicle production. Constellationshave been optimized (see also Ballard [1980], Draim[1985, 1986], Lang [1987], Walker [1970, 1971, 1982,1984], and Wertz [2001]) by choosing the smallestconstellation at the lowest altitude that can provide therequired coverage (without failures). Satellite andlaunch vehicle acquisition are optimized separatelybased on an estimate of launch demand (based ondesign life and constellation size).

Although it is recognized that when satellites failcoverage suffers, this service availability has not beentreated explicitly in the literature. It is usually statedthat, to maintain coverage with outages, spares will beused. The goals of this research were to develop a modelto minimize the life-cycle cost of a satellite system in aglobal sense with the acquisition, replenishment, andoperations costs closely linked to a performance model,which determines availability of service. Further objec-

tives were to determine the best optimization method touse, and to determine how much cost savings could beachieved over current practices.

2. PROBLEM DESCRIPTION

The service examined in this research was the availabil-ity of coverage. Coverage is the existence of a clear lineof sight from a satellite to a user of the service. In orderto determine service availability, it must be clearlydefined. A service area or volume must be defined,which specifies where the service will be provided (onthe surface of the earth, up to a specified altitude, etc.)along with a criterion. The criterion may be that theavailability is an average over the entire service volume,or for every location in the service volume, or there maybe a number of defined areas within the service area.For many satellite-based systems, grid points acrosscommon latitudes are averaged. When the ground trackdoes not repeat for a large number of orbits, the cover-age is averaged over the latitude. Since life-cycle costis used to define the cost of the system, the availabilityof service is also defined over the life of the system. Aset of time points is chosen to determine the serviceavailability for a set of grid averaging areas. Generallythese are chosen to span one orbit (if all of the satelliteorbits have a common period) or the least commonmultiple of all of the satellite orbital periods. The timestep is chosen to assure that service to a grid point is notskipped which is directly related to the relative angularmovement of the satellite and grid point spacing.

The state of the constellation is modeled as a 2-di-mensional Markov chain. The number of satellites inthe nominal constellation that have failed and the num-ber of spare satellites that have been used defines eachstate. The rate at which satellites fail moves the chaininto higher failure states with lower service availability.The launch rate (which is assumed to be constant for allof the states) moves the chain back into the lower stateswith higher service availability.

The cost model computes the life-cycle cost of thesystem and must take into account all of the factors thatdetermine performance. To avoid solutions that makeno sense, it is critical that every part of the model thataffects availability also has a cost associated with it. Agraphical representation of the problem is shown inFigure 1, which includes the supply chain of obtaining

114 KELLEY AND DESSOUKY

the satellites and launch vehicles, storing them, launch-ing them, operating them to provide service, and dis-posing of the satellites when they fail. Since coverageis so fundamental, it is also the most studied. Availabil-ity of coverage though has not been formally treated inthe literature. For many systems, although coverage isnecessary, it is not a sufficient condition for service.

Figure 2 illustrates the visibility of the satellites tothe surface of the earth. This case is the lowest altitudeand smallest number of satellites (in circular orbits)required to achieve single coverage above the horizonworldwide. At this instant in time, everywhere on theearth has at least one satellite in view. A large area ofthe earth has two satellites in view and some small areashave three satellites in view. Figure 3 shows the sametime and same constellation but with one satellite failed.This illustrates how drastically coverage may be af-fected when even a single satellite fails.

The rest of the paper is organized as follows. Thecomplete life-cycle cost model and the availability ofcoverage is given in Section 3. Section 4 describes the

solution techniques along with the results of the analy-sis of the techniques. We benchmark our global systemapproach versus the current practice of locally solvingthe problem in piecemeal in Section 5. Conclusions anddirections for further research are provided in Section6.

3. MODEL DEVELOPMENT

The life-cycle model for minimizing the cost of avail-ability of coverage from a constellation of satellites iscomplex and requires nonlinear optimization. The com-plete mathematical description of the model is providedin Kelley [2003]. In this section, we provide an over-view of the components of the model.

Independent variables: The independent variablesassociated with the satellites are the reliability beta ofthe Weibull reliability distribution, the ratio of designlife variance to design life, and the satellite order leadtime which includes the development cycle. The inde-pendent booster variables are the order cost and leadtime. Since there is assumed to be no development cyclefor the booster, this is only the production time. Theonly independent variable associated with the launch isthe probability of launch success. The independentvariables associated with the service are the geographicgrid points, time points, and rephasing delta velocity.The independent variables for the total system are theinterest rate, life-cycle time, and the required serviceavailability.

Decision variables: Most of these variables are as-sociated with the satellite constellation such as the orbitaltitude, inclination, number of satellites, number ofspares, number of planes, and phasing of the satellitesin each plane. The manufacturing decision variables aresatellite design life and reliability. The purchasing vari-

Figure 1. Schematic of providing service from a constellationof satellites. [Color figure can be viewed in the online issue,which is available at www.interscience.wiley.com.]

Figure 2. Coverage plot for 5/5/1 with no satellites failed.

Figure 3. Coverage plot for 5/5/1 with 1 satellites failed.

MINIMIZING THE COST OF AVAILABILITY OF COVERAGE FROM SATELLITE CONSTELLATIONS 115

ables are the satellite and booster inventory availability.The launch portion determines the mean launch timebetween the decision to launch, and the time the satelliteis placed in its assigned slot and starts to provideservice.

Computed parameters: The satellite failure rate andvariance are computed based on the combination of anormal distribution for the satellite design life and aWeibull distribution for random failures. The optimalreorder point and order quantity are computed based onthe input parameters of order cost, interest rate, and thedecision variable satellite availability using an inven-tory model based on Silver [1998]. The booster inven-tory computations are similar and use the launch vehicleavailability from inventory to compute the optimal re-order point and order quantity. The mean rephase timeis computed assuming a fixed delta orbital velocity andis a ratio to the orbit radius. The last variables computedare the total life-cycle cost and service availability.

Computed functions: These are the probability ofoccurrence of each constellation state along with theservice availability for each of these states. A Markovchain is used to model the failure states of the constel-lation where the transitions represent the launch ofsatellites, the failure of satellites, and the rephasing ofsatellites. The procurement model affects the launchrate because of the probability that both satellites andlaunch vehicles are available to support a launch. Foreach of these failure states, the availability of servicemust be computed for each point in the user servicevolume. In many cases, the computation time can begreatly reduced due to the limited number of satellitesin the constellation that are able to provide service to auser at any given time.

3.1. Life-Cycle Cost

The total life-cycle cost Ctotal consists of replenishment,launch, satellite, launch vehicle, and operations costsover the life of the system TL.

Ctotal = (Replenishment + Launch + Satellite +Launch Vehicle + Operations)TL.

The replenishment cost is the cost/month of placing theorders and purchasing the satellites and launch vehicles.Replenishment is linked to the performance model bythe fact that if the satellite or launch vehicle is notavailable from stock, the launch rate will be decreased.Since the satellite and launch vehicle costs are a func-tion of the service and the orbit, only the cost of capitaland ordering costs are considered as replenishmentcosts.

The launch cost is the cost/month of the personneland facilities required to prepare and launch the satel-

lites and launch vehicles. It links to the performancemodel by increasing or decreasing the launch rate. Atthe expense of running more shifts and overtime, thelaunch rate can be increased to increase the probabilityof being in a state with fewer satellite failures, whichwill in turn increase the availability of the service.

The satellite cost is the cost/month for the satellites.The cost of the satellites are a function of the relation-ship of the number of satellites, the service being pro-vided, the altitude, and their failure rate. The failure ratedirectly affects the state of the constellation and drivesthe cost of the satellite by requiring more subsystemredundancy, larger solar arrays, etc. Since the satellitesare assumed to not be an off-the-shelf design wheneveran order is placed, the development cost is included.

The operations cost is the cost/month of the infra-structure required to maintain the constellation of sat-ellites. There is a fixed cost for facilities such as officespace, antenna sites to communicate with the satellites,and a variable cost of people on hand to monitor andtroubleshoot the satellites. Even with a large degree ofautomation it has been estimated that it takes from 2 to5 persons per satellite to maintain a constellation.

The launch vehicle cost assumes that unlike whenthe satellite is ordered the launch vehicle is an off-the-shelf design so that the development cost is not in-cluded. The cost is primarily determined by the mass ofthe satellites carried into orbit, the orbit altitude, andinclination of the orbit.

The satellite development, production, and launchvehicle production costs were based on NAFCOM(NASA Air Force COst Model). The rest of the costparameters were based on cost trends and other datafrom Wertz and Larson [1999] and Koelle [2000].While the general trends of the model are based oncommon practices in government and industry, themodel was written to be general enough that an analystcould enter his own CERs (Cost Estimating Relation-ship) and MERs (Mass Estimating Relationship forestimating the mass of a satellite to determine launchvehicle costs) parameters.

3.2. Availability Model

The availability model consists of two parts. The firstpart determines the availability as a function of thenumber of failed satellites. The second part determinesthe probability the constellation will have a given num-ber of failed satellites. For coverage, a histogram ofnumber of satellites in view is computed for every gridaveraging area. A statistical model is then used tocompute the availability for every combination of failedsatellites.


To determine the probability that a given number ofsatellites are failed, a 2-dimensional Markov chainmodel was embedded into the overall optimizationmodel. Figure 4 shows an example of a 4 satelliteconstellation in 2 planes with 2 spares. A simpleMarkov chain for constellation states was described byDurand and Caseau [1990] and Phlong and Elrod[1994]. Material on queuing theory and its applicationsis available from Kleinrock [1975, 1976] and Kao[1997].

The state probabilities of the “nominal” constella-tion are determined by the position along the horizontalaxis. These satellites are used in the computation ofservice availability. If the required service availabilitycan be achieved with satellites in this “nominal” con-stellation, the number of satellites that can fail and stillachieve the required availability are considered “ex-cess.” The vertical axis is the state probabilities forspare satellites which are not counted for availability.This implies that the “spares” are not positioned toprovide service and thus do not contribute to the per-formance of the constellation. Spares contribute byincreasing the probability that the constellation hasfewer satellites failed because they can be quicklymoved to replace a failed satellite. By using a Markovchain, we imply that when the constellation reaches thedesigned size the launches stop. When spares are avail-able, the preference is to use them. That is, the constel-lation design, launch, and replenishment strategyallows the constellation to be larger on occasion thanneeded for basic (no satellites failed) service in order tohave satellites available to handle outages. An addi-tional assumption made in this model is that the satel-lites are not moved once they are placed into a particularorbital “slot” and spares are moved only once duringtheir lifetime.

As seen in Figure 4, although many transitions areconceivable, some are not desirable, for example, when

in states 11–14, a satellite could be launched. But, sincenone of the spares have been used, it is much faster torephase a spare. This same philosophy is taken for states6–9. If a spare is available in that plane, then it will berephased. Otherwise, a launch will occur. In this exam-ple, the probability is 50% for each of these transitionsbecause we have 2 planes and 2 spares. The otherstrategy can be seen in states 5 and 10. Launches toplace spares only occur when the nominal constellationis full.

Another way to reduce cost is to launch more thanone satellite per launch vehicle. The cost of launchingmultiple satellites on a larger launch vehicle is less thanthe cost of single launches on a smaller launch vehicle.The model assumes that multiple-launched satellitescannot be placed into different planes. If this is not thecase, it could be accounted for by increasing the launchrate. As shown in Figure 5, this complicates the Markovchain and changes some of the launch strategies. Figure5 uses the same constellation as in Figure 4 which is a4-satellite constellation in 2 planes with 2 spares. In thiscase when we launch nominal satellites, we skip a statesince 2 satellites are on board. This dual launch alsoallows us to transition from state 9 to 15 which launchesa spare and a nominal satellite into the same plane.Again, similar to Figure 4, there are no launch transi-tions from states 11 through 14. Because a dual launchwould place 2 spares into the same plane from state 5 itis excluded. For this particular constellation, this strat-egy activates the spares only after failures have oc-curred.

4. SOLUTION TECHNIQUES

Because both the cost and availability functions arenonlinear with continuous and integer variables, robustsolution techniques are required to solve the model. As

Figure 4. Single launch Markov chain. [Color figure can beviewed in the online issue, which is available at www.interscience.wiley.com.]

Figure 5. Dual launch Markov chain. [Color figure can beviewed in the online issue, which is available at www.interscience.wiley.com.]


shown in Kelley [2003], the problem can have multiplelocal minimums, and therefore global optimizationtechniques are expected to perform better than localtechniques. Two general global optimization tech-niques were used: Simulated Annealing (Aarts andKorts [1989]) and Genetic Algorithms (Chambers[2001], Mitchell [1999], Goldberg [1989]). Eight vari-ations of simulated annealing and eight variations ofgenetic algorithms were evaluated. As with any heuris-tic method, there is no guarantee that they will find thebest global optimum.

The simulated annealing method had three user-specified settings. Each setting had two options. Thefirst setting determined the choice of using either theclassical or wormhole method. The classical simulatedannealing method takes a “seed” solution and makessmall changes to it. If the new solution meets therequired availability and is less expensive, it is accepted.If it does not meet the required availability or is moreexpensive, the solution may be accepted depending onthe quality of the solution and the current “temperature”level. At the start of the simulation, the temperature isset to a high value, and virtually any solution is ac-cepted. As the simulation progresses, the temperature isreduced and increasingly only better solutions are ac-cepted. If the number of solutions tried is large enoughand the cooling is slow enough, the model will find agood solution. Alternatively, the wormhole methodonly accepts better solutions but creates solutions in alarge area of the solution space around the “seed”solution. Thus it can take a “wormhole” path to a newsolution without moving through the points in between.As the simulation progresses, the area of the solutionspace of possible moves is contracted. If the number ofsolutions tried is large enough and the contraction rateof the wormhole area is slow enough, the model willfind a good solution.

The second setting determines the availability com-putation. It has two options. It can be computed eitheras a deterministic or stochastic value. Although thestochastic availability can be computed much faster, thecooling rate or contraction rate must be slower due tothe uncertainty in the result.

The third setting sets the solution space to be eitherpartitioned or nonpartitioned. A partitioned solutionspace is used when the number of variables in thesolution is allowed to change. Constellations can bedesigned to have uniform or non-uniform spacing. Uni-formly spaced constellations have 5 degrees of free-dom, orbital altitude, inclination, number of satellites,number of planes (each plane equal), and plane-to-plane phasing (satellite–satellite spacing equal). Theorbital eccentricity and argument of perigee are set tozero. When any of these constraints is relaxed, the

number of degrees of freedom can be as large as sixtimes the number of satellites. Since this analysis useduniform constellation, the partitioned option was notused.

The genetic algorithm method had two settings. Onesetting had four options while the other had two. Ge-netic algorithms optimize by mimicking biological or-ganism evolution. An initial population of solutions iscreated from a “seed” solution. Each member is as-signed a “fitness,” which is based on the cost of thesolution when it meets the availability constraint, and apenalty cost is added when it does not meet the con-straint. The solutions can “mate” and produce an off-spring that is similar to its “parents.” In addition,random “mutations” are introduced to avoid the popu-lation from being captured by local optima. With a largeenough population and many generations, the solutionswill tend to cluster around a good solution. The firstsetting selects the method used to decide which solu-tions can “mate” and produce an offspring. The choicesare Roulette, Elitist, Tournament, and Uniform. TheRoulette method randomly chooses the two parents inproportion to their fitness. The elitist method allowsonly a subset of the population (the elite) to mate. Theuniform method is the opposite of elitist in that allmembers have an equal chance to “mate.”

Similar to simulated annealing, the second settingfor the genetic algorithm determines the availabilitycomputation. Although the stochastic availability canbe computed much faster, the number of generationsused must be larger due to the uncertainty in the result.

Two local optimization techniques were also used:the Nelder-Mead simplex method described in Press[1997] and Bazaraa [1993], and the Complex methoddescribed by Gupta [1974]. In the simplex method anumber of closely spaced separate solutions (one morethan the dimension of the problem) are used to find thebest direction for the “simplex” to move in search of abetter solution. The complex method uses at least twomore solutions than the dimension of the problem. Thesolutions initially span the solution space and graduallyconverge on the solution centroid by successively mov-ing the worst solution toward the current centroid.

To determine the best solution procedure, the cost ofproviding coverage to every location on the earth of 1satellite greater than or equal to 99% of the time with agrazing angle of 0.0 degrees above the local horizonwas minimized. The problem has a total of 12 dimen-sions. The constellation was constrained to a uniformconfiguration, and the satellites were constrained to allbe at the same altitude and inclination. While satellitesat different altitudes and inclinations could be used, theearth’s gravitational anomalies would cause the constel-


lation configuration to be unstable, and a large amountof fuel would be required to maintain it.

The dimensions are:

1. Number of satellites (integer) 2. Number of planes (integer, common divisor of

number of satellites) 3. Phasing between planes (integer, 0 to number of

planes 1) 4. Orbit semimajor axis (km) 5. Inclination (deg) 6. Satellite design life (months) 7. Satellite reliability (probability) 8. Probability of satellite available for launch from

inventory 9. Probability of launch vehicle available for

launch from inventory.10. Number of spare satellites on orbit (integer)11. Number of satellites launched on each launch

vehicle (integer)12. Mean time to launch a satellite(s) (months).

A total of 30 runs with different random number seedswere run for each method.

4.1. Results

Tables I and II summarize the results. Since the constel-lations were constrained to uniform configurations, theproblem did not require partitioning so it was not oneof the chosen simulated annealing methods. Table Icompares the performance of each method. Since thecomputation of service availability and cost dominatethe time to run the optimization, the number of functionevaluations indicates the relative amount of computertime to arrive at an answer. As expected, for a problemwith multiple local minima, the Simplex method, whilevery fast, performed poorly, with the best solutionalmost 30% higher than the global methods. The Com-plex method did better than the Simplex method (andfaster than Simulated Annealing and Genetic Algo-rithms). While searching over more of the trade space,many of the 30 runs would get “stuck” in local solutionsand did not show a consistent trend of decreasing costwith the number of evaluations. Although the stochasticperformance computations appeared interesting, the ex-tra overhead of more evaluations to compensate for theuncertainly of the performance appeared to outweighany computation time advantages it had. Both Simu-lated Annealing and Genetic Algorithms performed

Table I. Computational Results of the Different Solution Techniques


well. All of the Simulated Annealing techniquesshowed a consistent trend of improving the solutionwith a larger number of function evaluations whilesome of the Genetic Algorithm techniques got “stuck”in higher cost solutions. In general, when the GeneticAlgorithm converged, it consistently appeared to con-verge somewhat faster than the Simulated Annealingtechniques and also was able to find better solutions.While one of the features of the simulation was to makea number of optimization techniques accessible to theanalyst, overall the Genetic Algorithms are recom-mended.

Table II lists the resulting decision variables of thebest solutions found. The best every grid point is takenfrom the optimization comparison and was generatedby the Genetic Algorithm Elitist deterministic availabil-ity method. Using this same method the problem wasalso optimized for service provided at every latitude.The solutions have a number of similarities and a num-ber of differences. As expected, service provided atevery latitude is less expensive than at every grid pointsince service is averaged across common latitudes. Thetwo solutions are similar in their inventory policy, sat-ellite design parameters, and inclination. The majordifferences are in their replenishment policies and acorresponding difference in their constellations. Theevery grid point solution used spares and reduced costwith multiple satellite launches and a small number ofplanes (2). The every latitude solution was at the otherextreme with no spares, a single satellite launch, and thesame number of planes as satellites (5). In addition,every latitude solution used a higher orbit.

5. BENCHMARKING

In order to determine what advantages this methodprovides, it was benchmarked against current designpractices. In current practice, two general methods areused, the separate trade study method and the localoptimization method.

In the separate trade study method, it is assumed thatthe parts of the system are orthogonal and thus may beoptimized separately. In the coverage problem, it isassumed that the satellite costs, constellation costs, andreplenishment costs may be optimized independentlyand when combined produce the lowest cost solution.In the local optimization method, solutions from thetrade study method or expert opinions of possible com-binations of options are combined and then used as the“seed” for a local optimization method.

The separate trade study method used constellationsfrom 5 to 9 satellites ranging from 2 to 6 planes. Thisrange was chosen based on providing combinations thatwould range from using excess satellites, spares, orcombinations thereof. The mean launch time, reliabil-ity, design life, and inventory policies were treated asgiven based on other analysis.

The benchmark solution results are presented inTable III. The resulting constellations spanned the rangefrom 5 satellites in 5 planes to 6 satellites in 2 planesand 7 satellites in 7 planes. In each case, these constel-lations are very similar to the Walker starting points.Even though the constellations have similar configura-tions to the global optimization results, their inclina-tions and orbital altitudes are significantly different.This is likely due to the inclusion of availability ofservice. Since these constellations were designed for100% coverage with no satellite failures, one would notexpect to get the same answer. The fact that the localoptimization methods didn’t find answers as good asthe global methods illustrates that many local optimaexist.

The benchmarking methods might provide startingpoints close enough to the global optimum if the prob-lem were intuitive and the number of dimensions issmall. The resulting global solutions, even though theywere uniform configurations, were different in bothinclination and altitude from what a simple optimiza-tion for altitude and single satellite coverage wouldsuggest. This indicates that when outages are consid-ered, a different inclination and altitude has a signifi-cant, unanticipated effect on the availability ofcoverage, and the constellation design is not orthogonalto the rest of the system. As shown in Tables II and IIIthe globally optimized solutions provided life-cyclecost reductions of 64–235 million dollars or approxi-mately 5–17% over the benchmark methods.

Table II. Coverage Global Optimization Results


6. CONCLUSIONS AND DIRECTIONS FORFURTHER RESEARCH

A model that tightly integrates the computation of costand availability of coverage was developed for optimiz-ing a system consisting of a constellation of satellites.The problem is nonlinear and can have multiple localminimums. A number of local and global optimizationmethods were tried and compared. A number of vari-ations of both simulated annealing and genetic algo-rithms as well as simplex and complex methods wereevaluated. Both Simulated Annealing and Genetic Al-gorithms consistently converged to good answers withGenetic Algorithms appearing to converge slightlyfaster and producing slightly better answers.

Further areas for research include more sophisti-cated Markov chain models for the constellation state,adding the cost of the user equipment, and expandingthe optimization to a variable number of dimensions.When nonuniform constellations are considered thenumber of dimensions of the problem varies. The struc-ture of the Genetic Algorithms would make the optimi-zation very easy to implement where SimulatedAnnealing requires complex partitioning methods. Thisis equivalent to a number of species competing for thebest solution. In addition, Genetic Algorithms wouldallow proposed solutions to “seed” the initial popula-tion and compete directly with other potential solutions.

This model was developed in response to a govern-ment initiative to use cost as an independent variable orCAIV. It would also be useful to provide the option torun the optimization backwards (i.e., to constrain the

life-cycle cost and maximize availability). Often whendeveloping a new system, the budgeting process forcesa design to cost (DTC) methodology. Funding systemson a year-to-year basis forces the designers to minimizethe development costs (generally creating a system withhigher life cycle cost). Additionally, studies to deter-mine under what conditions excess satellite are pre-ferred over spares could help analysts reduce thenumber of dimensions of the problem.

ACKNOWLEDGMENTS

This paper is derived from portions of the Ph.D. disser-tation of Kelley [2003]. We would like to thank Dr.Elliot Axelband, Dr. George Freidman, Dr. F. StanSettles, and Dr. Satish Bukkapatnam of the Universityof Southern California for their helpful comments inguiding the research.

REFERENCES

E. Aarts and J. Korts, Simulated annealing and Boltzmanmachines: A stochastic approach to combinatorial optimi-zation and neural computing, Wiley, New York, 1989.

A.H. Ballard, Rosette constellations of earth satellites, IEEETrans Aerospace Electron Syst 16 (1980), 656–673.

M.S. Bazaraa, H.D. Sherali, and C.M. Shetty, Nonlinearprogramming: Theory and algorithms, Wiley, New York,1993.

D.C. Beste, Design of satellite constellations for optimalcontinuous coverage, IEEE Trans Aerospace ElectronSyst 14 (1978), 466–473.

Table III. Coverage Benchmark Results


B.S. Blanchard and W.J. Fabrycky, Systems engineering andanalysis, Prentice Hall, Englewood Cliffs, NJ, 1998.

L. Chambers, The practical handbook of genetic algorithmsapplications, Chapman & Hall/CRC, Boca Raton, FL,2001.

J.E. Draim, Three and four-satellite continuous-coverageconstellations, J Guidance 8 (1985), 725–730.

J.E. Draim, A common period four-satellite continuous globalcoverage constellation, 1986 AIAA/AAS AstrodynamicsConference Proceedings, 1986, pp. 112–120.

J.-M. Durand and A. Caseau, GPS availability, part II evalu-ation of state probabilities for 21 satellite and 24 satelliteconstellations, Navigation J Inst Navigation 37 (1990),285–296.

D.E. Goldberg, Genetic algorithms in search, optimization &machine learning, Addison-Wesley, New York, 1989.

S.K. Gupta, Optimal stabilization policy under price controls:An application of optimal control techniques to economicpolicy analysis, Ph.D. Dissertation, Washington Univer-sity, Sever Institute of Technology, St. Louis, MO, 1974.

E. Kao, An introduction to stochastic processes, DuxburyPress, Belmont, CA, 1997.

C.W. Kelley, Minimizing the cost of availability of servicesfrom a constellation of satellites, Ph.D. Dissertation, Uni-versity of Southern California, Los Angles, CA, 2003.

L. Kleinrock, Queueing systems Volume I: Theory, Wiley,New York, 1975.

L. Kleinrock, Queueing systems Volume II: Computer appli-cations, Wiley, New York, 1976.

D.E. Koelle, Handbook of cost engineering for space trans-portation systems with TRANSCOST 7.0 statistical-ana-lytical model for cost estimation and economicaloptimization of launch vehicles, Report No. TCS-TR-168/00, TCS—TransCostSystems, Liebigweg, Germany,2000.

T.J. Lang, Symmetric circular orbit satellite constellations forcontinuous global coverage, AAS 87 (1987), 1111–1132.

M. Mitchell, An introduction to genetic algorithms, MITPress, Boston, MA, 1999.

W.S. Phlong and B.D. Elrod, Availability characteristics ofGPS and augmentation alternatives, Navigation J InstNavigation 40 (1994), 409–428.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Nu-merical recipes in C, Cambridge University Press, Mel-bourne, 1997.

E.A. Silver, D.F. Pyke, and R. Peterson, Inventory manage-ment and production planning and scheduling,3rd edition,Wiley, New York, 1998.

J.G. Walker, Circular orbit patterns providing continuouswhole earth coverage, RAE Technical Report 70211,Royal Aircraft Establishment, 1970.

J.G. Walker, Some circular orbit patterns providing continu-ous whole earth coverage, J Brit Interplanetary Soc 24(1971), 369–384.

J.G. Walker, Coverage predictions and selection criteria forsatellite constellations, RAE Technical Report 82116,Royal Aircraft Establishment, 1982.

J.G. Walker, Satellite constellations, J Brit Interplanetary Soc37 (1984), 559–572.

J. R. Wertz, Mission geometry; orbit and constellation designand management, Microcosm Press, El Segundo, CA,Kluwer Academic Publishers, Dordrecht, The Nether-lands, 2001.

J.R. Wertz and W.J. Larson, Space mission analysis anddesign, 2nd edition, Microcosm Press, Torrance, CA, Klu-wer Academic Publishers, Dordrecht, The Netherlands,1996.

J.R. Wertz and W.J. Larson, Reducing space mission cost,Microcosm Press, El Segundo, CA, Kluwer AcademicPublishers, Dordrecht, The Netherlands, 1999.

Clifford Kelley holds a B.S. in Physics, a B.S. in Electrical Engineering, and an M.S in SystemsEngineering and recently completed his Ph. D. in Industrial and Systems Engineering at the Universityof Southern California. He started his career at Boeing working on computer modeling of missile guidancesystems, high energy lasers, and synthetic aperture radar. Later he worked for Rockwell International onthe B-1B program in the Infrared and Radar countermeasures area, the communications and navigationsystem, and the computer system. He currently works for Boeing as a Senior Systems Engineer on theGPS IIA, GPS IIF and GPS III programs.

Maged M. Dessouky is an Associate Professor of Industrial and Systems Engineering, University ofSouthern California. Prior to joining USC, Dr. Dessouky was employed at Hewlett-Packard (SystemsAnalyst), Bellcore (Member of Technical Staff), and Pritsker Corporation (Senior Systems Analyst). Dr.Dessouky has an in-depth theoretical and practical understanding of models and heuristic methods forhigh technology system optimization. He is currently an investigator on several transit-related projects,including the NSF-sponsored Real-time Scheduling of Demand Responsive Systems Project. He receivedhis Ph.D. in Industrial Engineering from University of California, Berkeley, and M.S. and B.S. degreesfrom Purdue University. He is the Area Editor of Planning and Scheduling for the International Journalof Computers & Industrial Engineering.


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