ON A PROBLEM OF SPATIAL ARRANGEMENT OF SERVICE STATIONS. Alexander Andronov, Andrey Kashurin

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1 Pat I Pobobabystc Modes Comute Moden and New Technooes 007 Vo No 3-37 Tansot and Teecommuncaton Insttute Lomonosova Ra LV-09 Latva ON A PROBLEM OF SPATIAL ARRANGEMENT OF SERVICE STATIONS Aeande Andonov Andey Kashun Ra Technca Unvesty Kau St Ra LV-658 Latva E-ma: AesandsAndonovs@tuv andesasuns@sovv A obem o sevce staton aanement n sata sace s consdeed A densty uncton o sevced obect ocaton and a uncton that descbes the coesondn oss ae nown As ctea o aanement s an aveae tota oss Fo the otmsaton the adent method s used Numeca eames ustate the suested aoach to settn the obem souton Keywods: sata aanement sevce statons adent method Intoducton Let us consde a ea sace X o that concete ont w be maed by х o ane t s twodmensona vecto t s avaabe to consde anothe dmenson too A dstance * s detemned o onts and * that satses usua condtona o dstance aoms: 0 * 0 * ' ' * Some obects ae aaned n the sace o eame men anmas statones Let us name as х obect the obect that s at the ont х The densty o obect aanement s descbed by nown densty uncton 0 so Χ Some sevce statons must be aaned n the sace the numbe s It s necessay to detemne those coodnates I a х obect s sevced by th staton then coesondn oss s eua to o eame Let us ca o as oss uncton and suose that t s a symmety accodn to eo and conve down A amount o sevce o the х obect s devated between vaous sevce statons accodn to nvese ooton o the dstances om the х obect and the staton Most ecsey a at o х obect sevce that beons to the -th staton s δ Now a obem can be omuated as oows: to nd coodnates aanement that mnmes the tota oss: d o staton The atce s oaned n the oown way At st one-dmensona case o sace X and the coesondn eame s consdeed Then we consde two-dmensona case The atce ends by some concuson emas The Aend contans the anaytca nvestaton o the smest one-dmensona case when and densty and oss uncton o ae symmetc unctons 3

2 Pat I Pobobabystc Modes 3 One-mensona Case At st we consde a case when sace X s ea as R - We w use a adent method o the mnmaton o cteon Fo that am et us cacuate a coesondn adent Fo a ata devatve o we have the oown eesson: d d d 3 Now we ae abe to ewte the adent o as T 4 To acceeate a conveence o the adent method we use a two-stae ocedue At the st stae we use comonent-wse coodnate-wse modcaton o the adent method It means that a seuence o cyces s eomed Each cyce contans teatons un the -th teaton uncton s mnmed wth esect to coodnate at the same tme othe coodnates do not chane Fo that mnmaton the adent method wth adent 3 s used The cyces end when the chane o uncton s ma In the second stae we cabate the obtaned esut by usn the usua adent method wth adent 4 3 Eame o One-mensona Case Let densty uncton be a mtue o noma dstbutons wth means and vaances and wehted coecents : < < e π 5 Futhe et us use the oown dstance uncton and oss uncton: 6 7 Then we have the oown devatves: < othewse 8 9 Now we ae abe to use omua 3 o otmaton

3 Pat I Pobobabystc Modes Let us consde the oown numeca data: 4 9 and T T T The Fue contans an accodn ahc o densty uncton Fue Pot o uncton o one-dmensona case We ben the st stae o the otmaton ocedue wth the vaues o coodnates 3 4 T 4 7 T It coesonds to 9766 vaue o cteon Tabe contans the esuts o seuenta cyces TABLE Resuts o seuenta cyces o one-dmensona case Iteaton numbe We see that mnma vaue o cteon s eua to 7468 that s cacuated by T Futhe we eom the second stae o the otmaton ocedue and nay et mnma vaue 7445 that coesonds to coodnates T 4 Two-mensona Case Now we consde a case when sace X s ea ane R Then the coodnates o an obect ae T scaa devatve 3 we have two-dmensona vecto coodnates o the -st staton ae T Now nstead o 33

4 Pat I Pobobabystc Modes 34 T Anaoousy to 3 we have o ata devatve : 0 d d d Now nstead o 4 we have the -mat o the ata devatves Fo the otmaton we aan use the two-stae ocedue At the st stae the comonent-wse coodnate-wse modcaton s used as oows un the -th teaton uncton s mnmed wth esect to both coodnates o the -st staton at the same tme othe coodnates do not chane Accodn to the adent method we move aon the adent wth esect to ecacuatn the one contnuay At the second stae we wo wth the u adent 5 Eame o Two-mensona Case As beoe et densty uncton be a mtue o two-dmensona noma dstbutons wth means T T and vaances T T T and wehted coecents : e T ρ ρ ρ π Futhe et us use the oown dstance uncton and oss uncton: 3 4 Then we have the oown devatves : 5

5 Pat I Pobobabystc Modes < othewse Now we ae abe to use omua 3 o otmaton Let us consde the oown numeca data: 4 9 and ρ The Fue contans an accodn ahc o densty uncton 6 Fue Pot o uncton o two-dmensona case Tabe contans the esuts o seuenta cyces o the otmaton ocedue TABLE Resuts o seuenta cyces o two-dmensona case Iteaton numbe Fom the Tabe we can see how the adent method moves the cteon o vaue contnuay 35

6 Pat I Pobobabystc Modes Concuson A obem o sevce staton aanement n sata sace s consdeed The eaboated aothm o the obem souton s based on the adent method The consdeed numeca eames show ts ecency The authos ntend to ay the suested aoach to sovn the actca aanement obems APPENIX Now we w consde the smest one-dmensona case when one staton ony and densty and oss uncton o ae symmetc unctons Let have a mama vaue at the symmety ont m and have mnma vaue * at the symmety ont 0 Now nstead o we have the oown ctea: d m m d 7 As s a symmetc uncton esectvey m m m then o the sum o two onts m and m we have the oown sum o the ntea eesson n 7: m m m m m m m The convety o uncton ves us m 0 m m m m The owe mt s obtaned m whch s obvousy cea d m m Theeoe m m Tan devatve wth esect to m d m d 0 0 m and euate one to eo we et m d 0 m m d m d As uncton has the mnmum at the ont 0 and then the devatve om s neatve o < 0 and s ostve o > 0 Theeoe m m m d m m m d 36

7 Obvousy the unue souton s eesson: m m d m d Pat I Pobobabystc Modes m Theeoe o otma vaue we have the oown Reeences Svastava M S Methods o Mutvaate Statstcs Wey Sees n Pobabty and Statstcs New Yo: John Wey & Sons Inc 00 Tunton A Mat Cacuaton and Zeo-One Matces: Statstca and Econometc Acatons Cambde: Cambde Unvesty Pess 00 37