RUNWAY SCHEDULE DETERMINATION BY SIMULATION OPTIMIZATION. Thomas Curtis Holden Frederick Wieland

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1 Proceedngs of the 2003 Wnter Smulaton Conference S. Chck P. J. Sánchez D. Ferrn and D. J. Morrce eds RUNWAY SCHEDULE DETERMINATION BY SIMULATION OPTIMIZATION Thomas Curts Holden Frederck Weland The MITRE Corporaton 755 Colshre Drve McLean VA 2202 U.S.A. ABSTRACT Many arport runway expanson projects are restrcted by space lmtatons mposed by development n the vcnty of the arport. Ths often causes planners to choose confguratons for new runways that lmt the use of these runways n tme and/or space. Studes that model arports wth new runways that are not yet operatonal need to develop plausble operatonal models for these new runways snce hstorcal data s not avalable. We look at a runway schedule problem encountered durng the confguraton and valdaton step of an earler study. We develop a method usng smulaton optmzaton to approach the runway schedule problem and compare t to a manual approach developed n the earler study. We use the Total Arspace and Arport Modeler to model the arport and arspace operatons and Fast Smulated Annealng for the optmzaton. INTRODUCTION Economc and avaton ndustry analyss suggests that passenger ar travel demand n the Unted States wll grow n the medum and long term despte the recent slowdown (FAA 2002a). The plans to deal wth ths growth nvolve the addton of new runways to arports n congested areas (FAA 2002b). Some arports are severely constraned by geography and local development causng planners to choose confguratons for new runways that lmt the use of the new runways n tme and/or space. Integratng these constraned runways nto the exstng avaton nfrastructure s a challengng task that has been the focus of a sgnfcant amount of research. We look at a runway schedule problem encountered durng the confguraton and valdaton step of an earler study. The prevous study models an arport wth a new runway that s restrcted to ether departures or arrvals but not both at any gven tme. Snce no hstorcal data exsted for the new runway at the tme of the prevous study the analysts needed to develop a plausble schedule for the new runway. We use smulaton optmzaton to fnd such a schedule. The method developed here s offered as an alternatve to a manual optmzaton approach used n the prevous study. The arport model used n ths study s taken from the prevous study. Ths model s bult for the Total Arspace and Arport Modeler (TAAM). TAAM s a determnstc fast-tme tme-stepped smulaton that models arports arspace and flghts. Arports can be modeled as smple pont sources or n great detal as a network of taxways connectng runways to gates. Flghts are modeled as ndvdual physcal arplanes that have state (alttude locaton velocty mass etc.) and propertes (average clmb rate fuel capacty maxmum speed etc.) that depend on the type of the arplane. The state of each flght evolves n smulaton tme and s determned by the flght s schedule ts nteracton wth other flghts and the arport and arspace rules. Smplfed laws of knetcs that depend on the propertes of each arplane govern the ncremental movement of flghts. Arport and arspace rules can be added to make smulated flghts conform to standard departure enroute and approach paths. The rest of ths paper s organzed as follows. In secton 2 the prevous study and the runway schedule optmzaton problem are dscussed n more detal. In secton 3 the smulaton optmzaton approach to the runway schedule optmzaton problem s developed. Secton 4 dscusses the smulaton optmzaton results and contrasts them wth the prevous study approach. In secton 5 we summarze wth a concluson. 2 THE RUNWAY SCHEDULE PROBLEM SETTING 2. Runway Operatons The runway consdered n ths study s restrcted to operatons at one end. We refer to each end of the runway by drectonal orentaton φ 0 and φ. The three possble

2 states of the runway are llustrated n Fgure. These states are the followng:. the φ drecton s open for departure the φ drecton s open for arrval. (c) 3. the runway s closed. When the runway s open for departure n the φ 0 drecton t obvously cannot accept arrvals n the φ drecton. To prevent operatonal conflct between departures and arrvals the runway s not allowed to be open for arrval or departure smultaneously. Lmtng the runway to arrvals or departures at any one tme forces controllers to decde when φ 0 should be open for departures and when φ should be open for arrvals. A good schedule for the runway that optmzes some metrc lke runway utlzaton or delay wll nvolve tradeoffs between satsfyng arrval and departure demands. The relatonshp between theses tradeoffs s qute complex snce the arrval and departure demand as a functon of tme depends on many factors. 2.2 The Manual Optmzaton Approach The manual approach nvolves runnng the smulaton wth two baselne schedules one wth the runway n state all-day and the other wth the runway n state φ φ (c) φ 0 φ 0 all-day. The arrval delay assocated wth the all day schedule and the departure delay assocated wth the all day schedule s plotted vs. tme of day on one graph. Ths graph for the model used n ths study s shown n Fgure 2. The estmate of the best schedule s developed by pckng the tme perods when delay for the all day schedule s hgh and makng φ open for arrval durng those tmes. Durng the rest of the day φ 0 s set open for departure. Delay per 5-mnute perod (mnutes) Departure Delay for sgma All Day Schedule Arrval Delay for sgma 0 All Day Schedule 4:00 6:00 8:00 0:00 2:00 4:00 6:00 8:00 20:00 22:00 0:00 Tme of day (hour) Fgure 2: Arrval and Departure Delay 3 THE SIMULATION OPTIMIZATION APPROACH The relatonshp between the runway schedule and delay s probably non-lnear. Because of ths the manual approach to mnmze delay s probably lmted n terms of how much delay can be reduced. Here s a lst of some addtonal factors that may lmt the manual approach:. Certan flghts n the model cannot use the runway so delay contrbutons by these flghts should not be consdered n the same way as flghts that can use the runway. 2. It s not clear whch runway should be open when delay peaks for the all-day schedule and the all-day schedule occur durng the same perod and are of comparable magntude. A smulaton optmzaton technque may be able to account for all of these factors. 3. Overvew of Smulaton Optmzaton φ φ - Runway s open for arrval - Runway s open for departure Fgure : Runway States 0 Smulaton optmzaton s the use of search methods to fnd nput parameter settngs that mprove selected output measures of a smulated system (Boesel 200). The motvaton for dong smulaton optmzaton s to support analytcal studes that use smulaton to study real world systems. Applcatons of ths technque nclude transportaton systems manufacturng systems supply chans call centers and fnance (Fu 200).

3 Most smulaton optmzaton approaches nclude the followng components: an optmzaton algorthm an objectve functon a set of constrants and a smulaton engne. The optmzaton algorthm attempts to fnd a mnmum or maxmum value for the objectve functon. The objectve functon s a wrapper for the smulaton that translates parameters from the optmzaton algorthm to a confguraton object that the smulaton uses. The objectve functon also gathers values from the smulaton output to generate a sngle result. The constrants defne vald solutons based on the objectve functon nput parameters and/or results. 3.2 Runway Schedule Objectve Functon The runway schedule objectve functon creates a new TAAM runway preferences confguraton fle (a.prf fle) each tme t s called. The TAAM runway preferences fle defnes a set of consecutve tme wndows and correspondng states for each runway at the arport. Each tme entry n the preferences fle defnes the end tme of one wndow and the startng tme of the next wndow. The runway schedule objectve functon needs to translate the values passed to t by the optmzaton algorthm nto a vald runway preferences fle. These values are represented by a vector we wll call θ. Each element n θ s represented by θ where s the ndex of the value. The θ are ndexed n the followng manner: θ0 θ θ 2 θ n where n s the number of elements n θ. It s clear that two classes of thngs need to be represented n θ : tme wndows and the correspondng runway states. To address these requrements each θ value s taken to be the end tme for the th tme wndow and the state for the th tme wndow s taken to be a state. The start tme of each tme wndow s the end tme of the prevous tme wndow; therefore the perod spanned by tme wndow + s θ θ. A fxed actve tme perod + durng one day s defned: ( s) s the start tme for the frst tme wndow whle s the end tme of the last tme wndow. Each represents a number of mnutes past ( s) θ. The tme resoluton of the preferences fle s to the mnute so the objectve functon essentally operates on a dscrete set of parameters. In the prevous study t was determned that the runway needs to be closed for a short perod of tme when operatons transton from arrvals to departures or departures to arrvals. Ths closure s requred to flush out any flghts from the old state so that nterference between flghts s avoded. It was determned that the runway needs to be closed for 5 mnutes when transtonng from departures on φ 0 to arrvals on φ and 0 mnutes when transtonng from arrvals on φ to departures on φ 0. Snce we are nterested n a soluton that takes the form of a set of tme wndows that each represent one of two states { } the smplest form of any soluton wll be a sequence of consecutve tme wndows that alternate state between 0 ) and wth the approprate ( flush perods ncluded n each tme wndow. We mmc ths form by alternatng the states n the followng way: = when s odd when s even. Each tme wndow ncludes the approprate flush perod whch we wll call δ at the end of the tme wndow. δ s determned by the states and +. The state before s and s set to the state of the frst tme (s) wndow 0 whle the state after s n and s set to the state of the last tme wndow n. Fgure 3 s a dagram of the nformaton represented by three hypothetcal elements of θ. The values are: θ = 430 θ + = 460 θ + 2 = 500. The axs at the bottom of Fgure 3 represents the tme of day n mnutes past mdnght whle the shadng represents the state of the runway at a partcular tme. The tme spanned by each of the three θ elements and the flush perods are marked on the dagram. ( 0) ( θ θ ) ( θ + + ( ) θ ) + 2 ( c) ( θ θ ) Tme of day (mnutes past mdnght) Fgure 3: Example of θ Informaton It s possble that the dfference between a set of consecutve θ values wll be less than the ntervenng δ perods causng the δ to overlap. When overlap clusters lke ths form the tme wndows represented by θ are replaced n the TAAM preferences fle by a sngle tme wndow wth a new state and flush perod δ that depend on the and δ values. When these overlap clusters form the states represented by θ are squeezed out and the resultng preferences fle wll have less than n tme wndows. Snce the absolute optmal schedule may have fewer than n tme wndows allowng tme wndows to be

4 squeezed out wll allow the optmal soluton to exst n the soluton space of θ. The overlap clusters mentoned n the paragraph above are defned by any contguous set { θ 0 m m < n} where the followng hold: θ m θ m > δ m when m 0 θ θ δ when m n m + m > m + + θ δ + θ n m < m. The frst two nequaltes defne condtons on the bounds m and m whch span all the θ n a overlap cluster whle the last nequalty ndcates that only the θ that qualfy for the overlap cluster are permtted n the set. The wndow used n the runway preferences fle n place of a overlap cluster s a wndow wth state = m and flush perod δ whch s set to: δ = δ where δ m s defned by the recursve relaton and δ + + δ δ + = ( θ + θ ) n m < m δ + m when = + or 0 when = δ = 5 when = m when +. The total perod of the wndow used n the preferences fle s θ m θ m. Ths method of squeezng out tme wndows provdes for smooth varaton n the sze of the overlappng flush perods δ wth respect to the tme dfference between the squeezed tme wndows n the cluster. It s possble that the optmal schedule may have more than n tme wndows. In ths case the absolute optmal schedule wll not be n the soluton space of θ. The choce for the sze of θ wll be lmted by the number of tmes a runway can practcally change state n the perod of (s) tme defned by { }. m δ m = δ m and and m + m + = = The constrants on θ are: ( s) ( e). θ n 0 < n 2. θ θ + n 0 < n. The frst constrant s a smple projecton that lmts the range of the values n θ whle the second constrant requres the elements to be n ascendng order whch s requred by the representaton of the { } states. The result of the runway schedule objectve functon s a sngle number ρ that represents average total delay. The average total delay s calculated by averagng the sequencng and departure delay values for all flghts arrvng and departng from the arport. The flght delay values are taken from the seq/dep delay feld of the TAAM report (.rep) fle. 3.3 Optmzaton Strategy The optmzaton strategy used n ths study utlzes a varant of Smulated Annealng (SAN) called Fast Smulated Annealng (FSAN) (Szu and Hartley 987). SAN s a technque for random search optmzaton based on an analogy to the condensed matter process of annealng. SAN was formalzed for combnatoral optmzaton by Krkpatrck et al. (98) and was later extended to apply to general contnuous and dscrete optmzaton problems. Annealng s the process of heatng a sold to a hgh temperature and then coolng t at a slow rate so that the fnal state s at or near the lowest energy state. The ground state or lowest energy state for many solds has a specal form such as a crystallne structure. Physcal annealng occurs naturally n magma ntrusons n the crust of Earth and s also used n the laboratory and n ndustral producton to create sold materals wth very specfc propertes such as specalzed metals and slcon wafers. The basc SAN algorthm has three man functonal components: the acceptance functon the generaton functon and the coolng schedule. SAN s an teratve algorthm that successvely calls the objectve functon wth a pont n the soluton space returned by the generaton functon. The generaton functon pcks a random pont from a unmodal dstrbuton whch samples the entre soluton space wth non-zero probablty. Ths dstrbuton s centered on a reference pont θ ˆ whch s taken as the current estmate of the optmum. The unmodal dstrbuton bases the search to a neghborhood around θ ˆ. At each teraton new ponts are accepted as θˆ wth probablty: ( ) ( ˆ) L θ L θ ( ˆ ) mn{ t A θ θ t = e A } A where L(θ) s the value of the objectve functon at θ and t s the acceptance temperature. L(θ) s often referred A

5 to as the energy n the context of SAN. The value of s controlled by the coolng schedule and s a decreasng functon of the teraton k. Ths acceptance functon allows SAN to accept new solutons wth hgher energy than the reference soluton wth non-zero probablty. The non-zero probablty of acceptng ponts wth hgher energy as the new reference pont prevents SAN from searchng only one neghborhood of the soluton space. Ths can be benefcal when searchng objectve functons that have many undesrable local mnma snce SAN wll tend to wander from the vcnty of one local mnmum to another. Ths wanderng behavor allows SAN to converge to a global mnmum as k when certan condtons are met (Locatell 2002). FSAN uses a generaton functon that pcks ponts from the Cauchy dstrbuton whle the classc form of SAN sometmes known as Boltzmann annealng pcks ponts from the Normal dstrbuton. The mplementaton used n ths study scales the Cauchy dstrbuton wth a generaton temperature t whch decays as a functon of G k the teraton k. Ths reduces the sze of the neghborhood that s searched as the number of teratons ncreases allowng the algorthm to eventually focus ts search on one neghborhood (presumably a neghborhood that contans the global mnmum). Szu and Hartley (987) showed that when the Cauchy dstrbuton s used to generate canddate ponts the temperature schedules and t can decay as fast as / k and stll guarantee convergence. Ths s much faster than the / log( k) decay schedule requred for classc Boltzmann annealng. We also ran the objectve functon aganst a smple blnd random search optmzaton algorthm. The blnd random search pcks ponts from a unform dstrbuton that samples the entre soluton space wth equal probablty. The blnd random search does not focus ts search on a partcular neghborhood of the soluton space. If the neghborhood search behavor of FSAN s any beneft when FSAN s appled to the runway schedule objectve functon we wll expect FSAN to perform better than the blnd random search. Snce both FSAN and blnd random search evaluate the objectve functon once for each teraton the number of objectve functon evaluatons s equvalent between FSAN and blnd random search when the number of teratons s the same. Ths pont s mportant because the computatonal tme requred to evaluate the objectve functon s orders of magntude longer than the tme taken by any other part of the optmzaton algorthm. 3.4 TAAM Smulaton The verson of TAAM used n ths study s TAAM Plus V.2. Release compled on SunOS 5.8 for 86pc. The t Ak G k t Ak results from the runway schedule optmzaton used n the prevous study cannot be compared drectly to the results obtaned usng the smulaton optmzaton technque developed n ths study because ths study used a dfferent verson of TAAM. Therefore to facltate a comparson wth the smulaton optmzaton method developed n ths study we use the method developed n the prevous study to create a schedule. 3.5 Model Setup The TAAM model was set up based on the model defned n the prevous study. The model defnes one full day of traffc whch ncludes 784 departures and 783 arrvals. All TAAM runs were done wth the arport ground model turned off. Ths s consstent wth the prevous study and elmnates any delay assocated wth taxng and gate usage. The runtme of the model was about 200 seconds on a 700 MHz Xeon machne. 4 RESULTS We obtan a soluton usng FSAN that result n lower average total delay than the blnd random search and the manual method (see Fgure 4). We ran 5 replcatons of FSAN blnd random search and the manual method usng the runway schedule objectve functon. The parameters for the FSAN runs are lsted n Table whle the blnd random search parameters are lsted n Table 2. The start (s) tme for the actve perod s set to 360 because there s very lttle traffc before 360. Average total delay (seconds) Iteraton Smulated Annealng Blnd Random Search Manual Approach Fgure 4: FSAN and Blnd Random Search Results Table : Parameters Used for FSAN Runs FSAN Objectve functon Iteratons t t Sze of θ (s) G 0 A

6 Table 2: Parameters Used for Blnd Random Search Runs Blnd random search Objectve functon Iteratons Sze of θ (s) The manual data was generated by 5 runs of TAAM wth 5 dfferent varatons of the TAAM preferences fle. The preferences fles were generated usng the procedure descrbed n secton 2.2. The number of tme wndows used n the preferences fle vared from 9 to 3. The sample mean of the average total delay for the reference soluton of the FSAN runs and the sample mean of the average total delay for the best soluton of the blnd random search runs are plotted as a functon of teraton n Fgure 4. The sample mean value of the manual runs also appears n Fgure 4. The 500 th teraton sample mean value s sgnfcantly lower for the FSAN runs than the blnd random search runs. The sample mean delay for the FSAN reference soluton matches the manual soluton after about 75 teratons whle the blnd random search matches t after about 50 teratons. The FSAN curve appear to be decreasng at teraton 500 whch suggests that FSAN was stll not close to convergence at teraton 500. Ths s supported by the fact that FSAN runs that were allowed to run out to 800 teratons acheved solutons wth average total delay as low as seconds. Table 3 summarzes statstcs on the runs. ρ m s the value of the mean average total delay of all the manual runs and s the mean average total delay of the reference soluton at teraton 500 for the FSAN and blnd random search runs. The confdence nterval was generated usng 95% confdence nterval t-statstcs. Notce that the FSAN 95% confdence nterval does not overlap wth ether the blnd random search or the manual 95% confdence nterval. Ths confdence nterval data provdes further statstcal evdence that the mprovement of the FSAN approach over the blnd random search and manual approaches s sgnfcant. The confdence nterval for the manual runs s much larger than the FSAN and blnd random search runs whch suggests that the optmzaton algorthms generate more consstent results than the manual method. Although ths s true for these partcular runs t s mportant to note that the manual confdence nterval and sample mean may vary dependng on who pcks the schedule snce the results wll depend sgnfcantly on the judgment of the person pckng the canddate schedules. The mean total runtme s Table 3: Summary of Results Optmzaton ρ m (s) Confdence Mean Total Approach Interval Runtme (s) FSAN ρ m ± Blnd search ρ m ± Manual ρ m ± approach the mean total tme requred to do one run of an optmzaton method. The FSAN approach s somewhat faster than the blnd search. Ths s because TAAM runs faster wth schedules that produce lower average total delay. The manual soluton requred only one run so t obvously ran much faster than the FSAN and blnd random search runs. The mean total runtme data for the manual runs do not take nto account the tme requred to do the two runs needed to make Fgure 2. The 500 th teraton reference solutons for the 5 FSAN runs appear n Fgure 5 and the ntal (or 0 th teraton) reference soluton for all the FSAN and blnd random search runs appear n Fgure 6. The states for each tme wndow follow the shadng scheme defned n Fgure 3 and the flush perods are not ncluded. As we would expect 500 th teraton solutons have a bas towards the state ( 0) after around 6:00 when the arrval delay for the all day schedule s hgh (see Fgure 2). The manual schedules developed based on the delay graph shown n Fgure 2 also have ths bas but the FSAN mean average total delay s lower. The dfferences between the schedules developed usng the manual method and the FSAN runs may be due to the lmtatons dscussed n Secton th teraton FSAN Reference Soluton :00 7:00 8:00 9:00 0:00 :00 2:00 3:00 4:00 5:00 6:00 7:00 8:00 9:00 20:00 2:00 22:00 23:00 0:00 Tme of day (hour) Fgure 5: 500 th Iteraton FSAN Solutons 6:00 7:00 8:00 9:00 0:00 :00 2:00 3:00 4:00 5:00 6:00 7:00 8:00 9:00 20:00 2:00 22:00 23:00 0:00 5 CONCLUSIONS Tme of day (hour) Fgure 6: Intal FSAN Soluton In ths study we developed a method to do runway schedule optmzaton usng smulaton optmzaton. Ths method used the TAAM smulaton n conjuncton wth Fast Smulated Annealng (Szu and Hartley 987). The method apples when there are two possble states for the runway. The method results compare favorably to a man-

7 ual method developed by a prevous study and results from a blnd random search algorthm. The tme and resources requred to do smulaton optmzaton usng TAAM are not trval. The sample mean runtme of teraton runs of FSAN usng the TAAM-based objectve functon s about 26 hours on a 700 MHz Xeon machne. If smulaton optmzaton s to be used to calbrate and valdate TAAM models the tme and effort must be carefully weghed aganst an alternatve approach (such as a manual approach). It would be nterestng to explore the possblty of usng a runway schedule smulaton optmzaton technque n a real-world settng. Ths could be used to help ground controllers decde what runway schedule to use. Ths applcaton would requre a fast hgh fdelty smulaton and a smulaton optmzaton procedure that can be updated as the real tme state evolves. It would also requre that actual arlne schedules ncludng provsons for general avaton traffc that have fled flght plans be used by the model durng the optmzaton phase. As external changes to the schedule take place (cancellatons substtutons addtons) then the optmzaton procedure may need to be repeated wth the changed traffc. ACKNOWLEDGMENTS We would lke to thank Preston Avaton for provdng the TAAM software for ths project. We would also lke to thank Mke Yablonsk for provdng the model used n the study. Mke also provded crucal tps and nsght on the basc and not so basc aspects of usng TAAM. John Kuchenbrod provded us wth addtonal vtal tps and trcks about usng TAAM. The contents of ths materal reflect the vews of the authors. Nether the Federal Avaton Admnstraton nor the Department of Transportaton makes any warranty or guarantee or promsed expressed or mpled concernng the content or accuracy of the vews expressed heren. R.R. Barton K. Kang and P.A. Fshwck eds. pp Pscataway New Jersey: Insttute of Electrcal and Electroncs Engneers. Krkpatrck S. C.D. Gellatt and M.P. Vwecch 98. Optmzaton by smulated annealng. Scence 220: Locatell M Smulated annealng algorthms for contnuous global optmzaton Handbook of Global Optmzaton volume 2 P.M. Pardalos H.E. Romejn eds. Dordrecht The Netherlands: Kluwer Academc Publshers. Szu H. and R. Hartley 987. Fast Smulated Annealng Physcs Letters A 22: No AUTHOR BIOGRAPHYS THOMAS CURTIS HOLDEN s a Senor Smulaton and Modelng Engneer at The MITRE Corporaton s Center for Advanced Avaton Systems Desgn (CAASD). Hs e-mal address s <tholden@mtre.org>. FREDERICK WIELAND holds a PhD n nformaton technology/appled probablty theory from George Mason Unversty. He s the developer of numerous smulatons for the U.S. Department of Defense as well as the Federal Avaton Admnstraton ncludng CTLS DPAT Matrx and others and has done extensve research n the parallelzaton of large-scale smulatons such as TAAM. He has been workng n the smulaton feld for 20 years. Hs e-mal address s <fweland@mtre.org>. REFERENCES Boesel J. R.O. Bowden Jr. F. Glover J.P. Kelly and E. Westwg 200. Future of Smulaton Optmzaton. In Proceedng of the 200 Wnter Smulaton Conference J.A. Jones R.R. Barton K. Kang and P.A. Fshwck eds. pp Pscataway New Jersey: Insttute of Electrcal and Electroncs Engneers. Federal Avaton Admnstraton (FAA) 2002a. FAA Aerospace Forecasts: Fscal Year Washngton DC: US Department of Transportaton. Federal Avaton Admnstraton (FAA) 2002b. Natonal Arspace System Operatonal Evoluton Plan Washngton DC: US Department of Transportaton. Fu M.C Smulaton Optmzaton. In Proceedng of the 200 Wnter Smulaton Conference J.A. Jones