A Linear Programming Approach to the Train Timetabling Problem

Transcription

1 A Lnear Programmng Aroach to the Tran Tmetablng Problem V. Cacchan, A. Carara, P. Toth DEIS, Unversty of Bologna (Italy) e-mal (vcacchan, acarara,

2 The Tran Tmetablng Problem (on a sngle one-ay track) ams at determnng an otmal tmetable for a set of trans hch does not volate track caacty constrants: a mnmum tme nterval (d) beteen 2 deartures s requred, a mnmum tme nterval (a) beteen 2 arrvals s requred, no overtakng s alloed beteen to consecutve statons.

3 Trans must be run every day of a gven tme horzon (e.g. 6-2 months). We are gven a so-called deal tmetable on nut hch s tycally nfeasble. To obtan a feasble tmetable to knds of modfcatons are alloed: change the dearture tme of some trans from ther frst staton (shft) and/or ncrease the mnmum stong tme n some of the ntermedate statons (stretch). Travel tme beteen to consecutve statons s assumed to be fxed for each tran.

4 Tmes are dscretzed and exressed as ntegers from to 440 (mnutes n a day). S ={,..., s} set of statons T ={,..., t} S = { f,..., l } set of trans set of statons vsted by tran The tmetable ndcates the deal dearture tme from f, the deal arrval tme n l and the deal arrval and dearture tmes for each ntermedate staton. f,..., l +

5 Each tran s also assgned an deal roft π deendng on the tye of the tran (eurostar, freght, etc). If the tran s shfted and/or stretched the roft s decreased. If the roft becomes null or negatve the tran s cancelled. π φ ( ) ν γ μ ν μ shft stretch (.e. sum of the stretches n all statons) The obectve s to maxmze the overall roft of the trans.

6 A grah formulaton G=(V,A) drected acyclc multgrah V set of nodes: { σ, τ} U ( U U... UU ) U ( W U... UW 2 s s ) U W Arrval nodes n staton Dearture nodes from staton A set of arcs: A t U... A U... U A To tyes of arcs: Segment arcs Staton arcs, T

7 tme staton σ de_node_ Tran f segment_arc_ arr_node_2 staton_arc_2 de_node_2 segment_arc_2 arr_node_3 staton_arc_3 de_node_3 segment_arc_3 arr_node_4 l τ

8 Dearture Constrants: de_ de_2 stat stat 2 Arrval Constrants: stat stat 2 arr_ arr_2 Overtakng Constrants: stat stat 2

9 Ideal ath Ne ath Stat deal_de_nst Shft Stat 2 sto Stat 3 sto Stretch Stat 4

10 Tmetable for tran Feasble ath from σ to τ usng arcs n A Overall tmetable Set of feasble aths, at most one for each tran. P Set of feasble aths for tran T P = P U... U P t Set of feasble aths for all the trans We ntroduce a bnary varable for each feasble ath for a tran. x = { 0, } P x x = = 0 ath n the soluton otherse

11 Proft of a ath P : π = π a a π a = roft of arc a: π φ (ν ) for the frst arc leavng σ corresondng to shftν 0 γ μ for the segment arcs for staton arcs corresondng to stretch μ

12 V notaton Set of nodes belongng to tran W Set of dearture nodes from staton U Set of arrval nodes to staton P Set of feasble aths for tran T P = P U... U P t Set of feasble aths for all the trans π Actual roft for ath P u P u P Set of aths that vst node u, u U, Set of aths belongng to tran, that vst node u S T, u U, S

13 P π x max P x, T + d W P x :, {} W s S, \ + a u u u U u P u x :, { } U u S, \ T V W P x 2 :, I { },, \ W s S { }, 0, x P T g t = ),,..., ( 2

14 Soluton methods: branch-and-cut-andrce constructve heurstcs local search

15 All the soluton methods use the LP-relaxaton for: obtanng an uer bound on the otmum soluton value suggestng good choces n the constructon of the soluton.

16 LP-relaxaton s solved usng column generaton and constrant searaton as the number of varables and constrants s huge for realorld nstances.

17 Column generaton Add varables th ostve reduced rofts. Reduced roft of ath n a π α a v v P αv = Sum of dual varables of constrants nvolvng node v V For each tran, fnd maxmum roft ath n an acyclc grah.

18 Constrant searaton Add constrants volated by the current soluton. dearture/arrval constrants overtakng constrants

19 Dearture/arrval constrant searaton For each staton, sum the values of the varables corresondng to aths that vst nodes n each ndo smaller than mnmum tme nterval beteen 2 deartures (analogously for arrvals). mnmum tme nterval beteen 2 deartures

20 Overtakng constrant searaton (overtakng beteen 2 trans,k) For each S, and 2 k 2 for each k k W IV let :,..., 2,..., 2 W = g + IV k k k k ( ) = max{ + d, + t + a analogous t sum the values of the varables of tran n the ndo: sum the values of the varables of tran k k n the ndo: k } k 2 k 2 +

21 Overtakng constrant searaton (overtakng among 2 or more trans) Exact method (dynamc rogrammng). Order aths by decreasng seed. Clque: set of ncomatble aths. Status: b,c. u b c u d + b c z + a + z

22 UB Lagr UB LP UB LP (2 trans) Instances Value Tme (sec) Value Tme (sec) Value Tme (sec) PC-BO- (22,7) PC-BO-2 (93,7) PC-BO-3 (60,7) PC-BO-4 (40,7) MU-VR (54,48) BZ-VR (28,27) CH-RM (4,02) BN-BO (68,48) CH-MI (94,6) MO-MI- (6,7)

23 HS lagr Best Heur BB_arcs Instances Value Tme (sec) Value Tme (sec) Value Tme (sec) PC-BO- (22,7) PC-BO-2 (93,7) PC-BO-3 (60,7) PC-BO-4 (40,7) MU-VR (54,48) BZ-VR (28,27) CH-RM (4,02) BN-BO (68,48) CH-MI (94,6) MO-MI- (6,7)