Location of Rescue Helicopters in South Tyrol

Transcription

1 Locaton of Rescue Helcopters n South Tyrol Monca Talwar Department of Engneerng Scence Unversty of Auckland New Zealand talwar_monca@yahoo.co.nz Abstract South Tyrol s a popular destnaton n Northern Italy for toursts from the north and south of the Alpne mountan ranges. The growng demand for tourst actvtes such as skng, hkng and clmbng has led to a large number of accdents n the regon over the years. Ths has resulted n an ncreased demand for rescue helcopters, as most of the accdent stes are not accessble by vehcles on-ground. In ths proect we focus on optmsng the locatons of three rescue helcopters coordnated by a South Tyrolean rescue organsaton called Weßes Kreuz n order mprove the emergency response tmes. Two models were formulated to solve ths problem. The frst model mnmses the average response tmes and the second mnmses the worst response tmes. We solve the two models usng some effectve heurstcs. These heurstcs do not guarantee optmalty but do provde local optmal solutons that can mprove the response tmes consderably. The methods mplemented have shown that the model that mnmses the worst response tme results n more evenly dstrbuted accdent stes to the new helcopter base locatons. 1 Introducton Ths paper presents a study on the locaton of rescue helcopters n South Tyrol, Italy. The study focuses on a South Tyrolean rescue organsaton called Weßes Kreuz that provdes rescue facltes for all the accdents that occur wthn the regon. The followng sectons descrbe the locaton problem n detal. 1.1 Background on The Weßes Kreuz and South Tyrol Weßes Kreuz s a non-proft rescue organsaton assocated wth Samartan Internatonal and s operatng n South Tyrol whch s stuated n the northern part of Italy. It coordnates three rescue helcopters. Two of whch, named Pelkan 1 and Pelkan 2, are owned by the Weßes Kreuz. The thrd one s called Aut Alpne Dolomtes (AAD n remanng text) and s operated by the mountaneerng rescue organsaton n the Dolomtes regon of South Tyrol. A sngle call centre answers the emergency calls and also dspatches the three helcopters [1]. South Tyrol s a popular destnaton for toursts from the north and south of the Alpne Mountan ranges. The growng demand for tourst actvtes such as hkng and clmbng n the summer and skng n wnter has led to an ncreased number of accdents n recent years. Due to the mountanous terran, many of the accdent stes are not

2 accessble by cars or vans whch has therefore led to an ncreased demand for rescue va helcopters [1]. 1.2 Problem Descrpton Currently, the frst rescue helcopter, Pelkan 1, s statoned n Bozen, the captal cty of South Tyrol, Pelkan 2 s statoned n a town called Brxen and AAD s statoned n Kastelruth as depcted n Fgure 1. The problem wth the current stuaton s that the three helcopters are statoned too close together n the central part of South Tyrol. Pelkan 1 s statoned too far n the east of the regon and Pelkan 2 s statoned too far n the west of the regon. Ths worsens the emergency response tmes. 1.3 Focus of Proect Fgure 1: Map of South Tyrol [2] The am of ths proect s to fnd ways to mprove or optmse the current base locatons of the helcopters coordnated by Weßes Kreuz, n order to shorten the emergency response tmes. For ths proect, we take nto consderaton that the bases can be located anywhere n the regon as long as they are accessble and can be staffed permanently. We have used varous heurstcs, algorthms and mathematcal technques to fnd solutons to the base locaton problem. Although these methods may not guarantee optmalty, they can provde close-to-optmal solutons that reduce the emergency response tmes sgnfcantly. 1.4 Data The data provded by Weßes Kreuz conssted of a lst of 1928 mssons flown over the years 1996 to 1997 [1]. We were gven the coordnates of 109 accdent stes and 3

3 helcopter bases n the South Tyrol regon where the coordnates had been measured usng an offcal map of the South Tyrolean Statstcal Insttute. The locaton of the admnstratve centres of the towns n the area of whch the accdents took place were taken to be the pont representng each of these accdent stes because the coordnates for the exact locaton of the accdent ste were not avalable [1]. We were also provded wth the number of mssons to the area of each town (see Fgure 1). Ths defnes the weghtng on a partcular accdent ste. 2 Mathematcal Models The crteron for fndng a good base locaton for the helcopters requres the mprovement of the response tmes to the emergency calls. The response tmes depend on the dstances between the helcopter bases and the accdent stes. Ths gves rse to two basc models, one that wll mprove the average response tmes by mnmsng the total weghted dstance between the helcopter bases and the accdent stes. And the second that wll mprove the worst response tmes by mnmsng the maxmum un-weghted dstance between a helcopter base and the accdent ste. 2.1 Model 1: Mnmsng the Total Weghted Dstance to Improve the Average Response Tmes Ths model fnds p ponts X 1,, X p for the locaton of the new emergency facltes where X = (x, y ) such that the total weghted dstance between the accdent stes,.e. the demand ponts at locatons P 1,, P n, and ther closest new emergency faclty s mnmsed. Ths model wll mnmse the average response tmes n the regon snce t takes the weghtng nto account and therefore s termed the Weghted p-medan Model. mn w = n mn X,..., X 1,..., p = 1,.., 1 d ( X p Snce we are dealng wth helcopters, the Eucldean dstance was consdered as beng the realstc dstance between a helcopter base and an accdent ste. The Eucldean dstance s a measure of the straght lne between two ponts; n ths case the two ponts are the accdent ste and the base locaton. Therefore, n the mathematcal model above d(x, P ) s the Eucldean dstance between X (base locaton ) and P (accdent ste ) where: d 2 ( X, P ) = ( x Px ) + ( y Py ) 2, Fndng the mnmum of d X, P ) s dffcult because ths s a non-convex ( optmsaton problem. Ths s because we know that the Eucldean dstance s a quadratc functon and the mnmum of many quadratc functons forms a non-convex functon. Therefore the Weghted p-medan Model s hard [3]. 2.2 Model 2: Mnmsng the Maxmum Un-weghted Dstance to Improve the Worst Case Response Tmes Ths model fnds p ponts X 1,, X p for the locaton of the new emergency facltes such that the maxmum un-weghted dstance between the accdent stes,.e. the demand P )

4 ponts at locatons P 1,, P n, and ther closest new emergency faclty s mnmsed. Ths model wll only mnmse the worst response tme n the regon snce t does not take the weghtng nto account and therefore s termed the Un-weghted p-centre Model. mn X 1... X max p = 1.. n mn = 1.. p d ( X, P ) Once agan, ths model results n a non-convex functon because the nteror part of the model stll requres fndng the mnmum d X, P ), as was the case for Model 1, ( and so the Un-weghted p-centre Model s hard, too [4]. For ths reason, we have developed a heurstc to solve Model 1 and 2. 3 Heurstc Soluton Approach The appled heurstc allows us to allocate each of the accdent stes to ts nearest helcopter base. Ths helps avod multple allocatons. We then form a subdvson by groupng all the accdent stes that are allocated to a partcular helcopter base together. Model 1 and 2 can then be solved by treatng each of the subdvded regons as a separate problem. Once the regon s subdvded, Model 1 for each regon becomes: mn w d ( X, P ) (1) = n X 1,..., and s termed the Weghted Medan Model because t reduces the average response tme by mnmsng the total weghted dstance travelled by the three helcopters. Model 2 for each regon becomes: mn max d ( X, P ) X = 1.. n (2) and s termed the Un-weghted Centre Model because t reduces the worst response tme by mnmsng the maxmum un-weghted or the longest dstance travelled by a helcopter. Here ranges from regon 1 to regon 3 for the three helcopter bases. By solvng Models (1) and (2) we obtan a new base locaton n each of the subdvded regons. We can then reallocate each of the accdent stes to ts nearest helcopter base to form a new subdvson and contnue ths process untl there are no more changes n the base locatons. We show how to solve Models (1) and (2) n Secton 4 and 5. 4 Soluton Approach for Model Generatng Startng Solutons Usng the Squared Eucldean Dstance Usng the Squared-Eucldean dstance was the frst step to fndng a soluton for the locaton of the three helcopter bases. Our model for ths approach s: mn X = 1,..., n w d ( X, P ) 2

5 We then dfferentate the expanded form of the squared Eucldean dstance and set t to zero to obtan a formula for x and y whch gves us the coordnates of the helcopter base: x n = 1 = n = 1 w Px w y n = 1 = n = 1 w Py w Where Px and Py are the coordnates of the accdent stes. 4.2 Incorporatng True Dstances Usng the Weszfeld Algorthm In order to mnmse the true dstance we need to dfferentate the expresson for the total weghted dstance. The dervatve for the true dstance s not defned at (x, y ) = (Px, Py ) and there s no explct formula for computng the optmal locatons for the helcopter bases. Therefore we have to use an teratve procedure to solve ths problem. In ths proect we have used a gradent steepest descent teratve method called the Weszfeld algorthm n order to compute the new helcopter base locatons [5]. 4.3 Soluton Procedure for Model 1 Fgure 2 llustrates a schematc of the procedure for computng the optmal base locatons for Pelkan 1, Pelkan 2 and AAD usng Model 1 (Weghted Medan). To fnd the optmal locaton for each base, we frst use the formula derved by mnmsng the Squared- Eucldean dstance to obtan a startng soluton (Refer Secton 4.1). Ths s then fed nto the Weszfeld Algorthm whch fnds the optmal base and returns t to the man loop where we update the subdvded regons by reallocatng the accdent stes to ther nearest new bases. If there are any changes n the base locatons, we resolve the problem usng the new subdvded regons; otherwse the man loop stops executon. Choose Bases Subdvde Regon Use Squared- Eucldean dstance formula Yes Fnd Optmal Bases Are there any more changes? Startng soluton Update x, y for bases usng Weszfeld Algorthm No STOP Is Stoppng Crteron satsfed? No STOP Yes Fgure 2: Procedure for fndng optmal base locatons usng Model 1.

6 5 Soluton Approach for Model 2 In order to solve Model 2, we appled some basc geometrc propertes of trangles. One of the propertes states that f you move from sde AB to the nsde of the trangle ABC provded that the dstance to the vertex A s the same as that to B, then you are movng along the perpendcular bsector of AB. The same apples for sdes AC and BC. Therefore the pont of ntersecton of any two bsectors of the trangle s equdstant from all three vertces of the trangle and the perpendcular bsector of the thrd sde should also ntersect at that pont. The pont of ntersecton s therefore the centre of the crcumscrbed crcle or the crcle that passes through the three vertces of the trangle centred at the ntersecton of any two bsectors [6]. For our problem, ths centre pont defnes the optmal locaton of the helcopter base for three accdent stes A, B, C. 5.1 Fndng the Optmal Radus The optmal radus s the smallest radus that encrcles all other accdent stes wthn a subdvded regon and s centred at the ntersecton pont of any two of the perpendcular bsectors of a trangle. However, f the trangle happens to be obtuse, then the pont of ntersecton les outsde the crcle. In ths case we can take the mdpont of the largest sde of the trangle (sde opposte the obtuse angle) as beng the centre of the crcle. Ths gves rse to what we call the two-pont search because two of the three vertces of the trangle le on the crcle and the thrd one les nsde the crcle. If the trangle n consderaton happens to be acute, then the ntersecton of any two of the perpendcular bsectors of the trangle gves us the centre of the crcle. Ths gves rse to the three-pont search because, n ths case, all the three ponts.e. the three vertces of the trangle le on the crcle. In order to calculate the rad, frst the centre of the crcle has to be found. The followng sectons descrbe the methods for fndng the centre for the two-pont and three-pont search technques. 5.2 Two-Pont Search If we apply the property of perpendcular bsectors to obtuse trangles that have two vertces lyng on the crcle, then the crcle wll be centred at the mdpont of the straght lne through the two ponts as stated above. We can then examne how the radus of ths crcle can be used to search for possble optmal locatons for the helcopter bases. The radus s only consdered a possble optmal one f t encrcles all the other accdent stes n the regon. If the dstance between the centre of the crcle and another accdent ste n the regon s larger than the radus, t s smply gnored and a new set of ponts are chosen through whch another crcle s formed. 5.3 Three-Pont Search By reapplyng the perpendcular bsector property (Secton 5) for three ponts, where all three ponts le on the crcle, the ntersecton of the bsectors of the sdes of the trangle gves us the centre of the crcumscrbed crcle. Ths pont s equdstant from the three vertces of the trangle. To ensure that the pont of ntersecton falls nsde the trangle defned by the three ponts, the angles of the trangle must be less than 90. We then search for a possble optmal radus that encrcles all the other accdent stes n the regon as we dd for the two-pont search technque.

7 5.4 Soluton Procedure for Model 2 Fgure 3 follows the procedure for computng the optmal base locatons for Pelkan 1, Pelkan 2 and AAD usng Model 2 (Un-weghted Centre). We use the two-pont and three-pont search heurstcs (See Secton 5.2 and 5.3) to fnd the smallest rad that can encrcle all allocated stes n a subdvson. The crcle formed by ths radus s an optmal crcle and the centre of ths crcle s the optmal locaton of the helcopter. These locatons are returned to the man loop where we employ the same heurstc framework as we dd for Model 1. Yes Choose Bases Subdvde Regon Fnd Optmal Bases Are there any more changes? Use Two-Pont Search Use Three-Pont Search Choose the smaller radus Return radus And Optmal Bases STOP 6 Results Fgure 3: Procedure for fndng optmal base locatons usng Model 2. The optmal locaton of each helcopter was subsequently found by mplementng the heurstcs descrbed n prevous sectons. The resultng allocaton of the accdent stes to each of the newfound helcopter bases s shown n Fgure 4 and Soluton for Mnmsaton of the Total Weghted Dstance Allocatons and optmal bases found by Model 1 x Mühlbach y Tsens St. Chrstna Locatons served by P1 Locatons served by P2 Locatons served by AAD Pelkan1 Pelkan 2 AAD Fgure 4: New Base locatons found by Model 1.

8 The uneven allocatons for Model 1, as depcted n Fgure 4, are most lkely to be the result of a locally optmal soluton found by the appled heurstcs. From Table 1 we can see that the number of stes (59) allocated to Pelkan 1 has not changed and s far greater than the number of accdent stes allocated to Pelkan 2 and AAD whch only have 34 and 16 allocated stes respectvely. Helcopter Locatons x y Total Weghted Dstance (km) Maxmum Un-weghted Dstance (km) of Mssons of Allocated Stes Pelkan Pelkan AAD Table 1: Results for Model 1. Compare wth Orgnal Values: Helcopter Locatons x y Total Weghted Dstance (km) Maxmum Un-weghted Dstance (km) of Mssons of Allocated Stes Pelkan Pelkan AAD Table 2: Orgnal total weghted dstance and maxmum un-weghted dstance. Although Model 1 has not resulted n evenly allocated accdent stes, t has mproved the total weghted dstance by 7% and the maxmum un-weghted dstance by 14%. 6.2 Soluton for Mnmsaton of the Maxmum Un-weghted Dstance Allocatons and optmal bases found by Model 2 x Schnals St. Lorenzen y 60 Bozen allocated to P1 P1 Base allocated to P2 P2 Base allocated to AAD AAD Base Fgure 5: New Base locatons found by Model 2. Fgure 5 shows that the allocated regons obtaned by solvng the Un-weghted Centre model are far better than those obtaned by the Weghted Medan Model because of the even dstrbuton of accdent stes to ther respectve helcopter bases (See Table 3).

9 Helcopter Locatons x y Total Weghted Dstance (km) Maxmum Un-weghted Dstance (km) of Mssons of Allocated Stes Pelkan Pelkan AAD Table 3: Results for Model 2. Results show an mprovement n the maxmum covered dstance of about 46% whereas the total weghted dstance has only mproved by 4%. These results verfy that Model 2 would be more sutable to use for fndng the optmal base locatons for the rescue helcopters. 6.3 Relatng Model 1 wth Model 2 We can explore the possblty of fndng better solutons by studyng the effect of movng the bases found by the Weghted Medan Model n a straght lne to the bases found by the Un-weghted Centre Model because the optmal locatons found by the latter resulted n more evenly allocated regons. To do ths, we moved the three bases smultaneously n a straght lne through 100 steps and evaluated the maxmum unweghted dstance and total weghted dstance at each pont. The resultng relaton s shown n Fgure 6. Mnmsng the maxmum unweghted soluton Vs Mnmsng total weghted soluton 65 Soluton from Model 1 60 max unweghted dstance Soluton from Model 2 Soluton found by mergng bases of Model 1 to Model total weghted dstance Fgure 6: Obectve of Model 1 versus Model 2 Solutons of Model 1 and 2 From Fgure 6 we can nfer that a better soluton does exst but has not been found by the appled heurstcs. It also confrms that the soluton obtaned by mnmsng the total weghted dstance s not a globally optmal soluton because the pont ( , ) n the fgure (Fgure 6) gves a lower total weghted dstance than the one found by mplementng the Weszfeld Algorthm n the heurstc for Model 1. Ths s certanly the best soluton found so far. If the coordnates of the helcopter bases at ths pont ( , ) are evaluated, the optmal allocatons can be obtaned. Results n Table 4 show that the accdent stes are more evenly allocated to the three helcopters. Table 4 also shows an mprovement n the total weghted dstance of 9% and n the maxmum un-weghted dstance of 43%. The total weghted dstance s lower than that

10 found by solvng Model 1 and Model 2 and the maxmum un-weghted dstance (40.36km) s not much worse than the one found by solvng Model 2 (38.50km). Helcopter Locatons x y Total Weghted Dstance (km) Maxmum Un-weghted Dstance (km) of Mssons of Allocated Stes Pelkan Pelkan AAD Table 4: Optmal locatons found by shftng bases to locatons found n Model 2 In order to overcome these setbacks of solvng the models separately we could n the future consder a b-crtera approach wth both the Weghted p-medan and the Unweghted p-centre obectve functons. Ths wll help fnd all possble effcent solutons for a combnaton of the two obectves from whch we can establsh a trade-off between the two models n order to decde whch of the effcent solutons s most sutable for the Weßes Kreuz. 7 Conclusons Our heurstc soluton approach ncorporated two basc models n order to solve the helcopter base locaton problem n South Tyrol. The frst model was the Weghted Medan Model whch was used to mprove the average response tmes and the second was the Un-weghted Centre Model to mprove the worst response tmes. The frst model gave us a local optmal soluton for the helcopter locatons whch had uneven allocatons of accdent stes to the bases. The Un-weghted Centre Model, however, has shown an mprovement of 46% by reducng the largest unweghted dstance from 7l km to 38 km along wth more evenly dstrbuted accdent stes to the helcopters as opposed to Model 1. By movng the optmal bases of Model 1 to those of Model 2 we were able to mprove the average response tmes consderably to present a good soluton to the Weßes Kreuz. Acknowledgements I would lke to thank my supervsor, Dr. Matthas Ehrgott for hs contnuous gudance, support and encouragement through the course of ths proect. Thanks also to the Weßes Kreuz Rescue Organsaton n South Tyrol, Italy for provdng us wth vtal nformaton that was requred to mplement methods and algorthms n the proect. 8 References [1] Ehrgott M. (2001). Locaton of Rescue Helcopters n South Tyrol. Internatonal Journal of Industral Engneerng, 9(1): pp [2] Autonome Provnz, Bozen- Südtrol. HILFE UrbanBrowser 3.0. [Onlne]. Avalable: [2002, September 4] [3] Arora S., Raghavan P., Rao S. (1998). Approxmaton schemes for Eucldean and k-medans and related problems. Proceedngs of Symposum on Theory of Computng (Stoc 98): pp [4] Megddo N. and Supowt K.J. (1984). On the Complexty of Some Geometrc Locaton Problems. SIAM Journal on Computng, 13: pp [5] Weszfeld E. Sur le Pont pour Lequel la Somme des Dstances de n Ponts Donnés est Mnmum TShoku Mathematcs Journal 43, 1937, pp [6] Anatomy of Trangles, Part of the Geometry Prmer for Mathematcs 337 at the Unversty of Brtsh Columba [Onlne]. Avalable: