COLUMN GENERATION HEURISTICS FOR SPLIT PICKUP AND DELIVERY VEHICLE ROUTING PROBLEM FOR INTERNATIONAL CRUDE OIL TRANSPORTATION

Transcription

1 12/03/2013 CAPD Annual Meetng Carnege Melln Unversty U.S.A. COLUMN GENERATION HEURISTICS FOR SPLIT PICKUP AND DELIVERY VEHICLE ROUTING PROBLEM FOR INTERNATIONAL CRUDE OIL TRANSPORTATION Mathematcal Scence fr Scal Systems Graduate Schl f Engneerng Scence Osaa Unversty JAPAN Tatsush Nsh March Slde 1

2 12/03/2013 CAPD Annual Meetng Carnege Melln Unversty U.S.A. Outlne 1. Intrductn 2. Sh schedulng rblem fr nternatnal crude l transrtatn 3. Prblem mdelng and frmulatn 4. Slutn arach: Clumn generatn heurstcs 5. Cmutatnal exerments 6. Cnclusn and future wrs Slde 2

3 12/03/2013 CAPD Annual Meetng Carnege Melln Unversty U.S.A. Intrductn Ol s ne f the mst cnsumed energy resurce n Jaan. Jaan has t mrt crude l frm ther cuntres. Table: Value f mrts f crude l n 2011 Ran Cuntry The value f mrts (US $ bllns) 1 Unted States Chna Jaan Inda (Reference: htt://ecdb.net/ranng/mf_tmg.html) Pcu and delvery transrtatn rblem s sgnfcant fr glbal lgstcs. Slde 3

4 12/03/2013 CAPD Annual Meetng Carnege Melln Unversty U.S.A. Intrductn Pcu and delvery crude l transrtatn schedulng rblem. The bjectve s t mnmze the ttal cst durng cu and delvery. Slde 4

5 12/03/2013 CAPD Annual Meetng Carnege Melln Unversty U.S.A. Intrductn Befre human s decsn (calculatn by hand r exerence) When the scale s t large decsn-mang becmes dffcult. Our urse s t decde faster and lan better schedules. Glbal lgstcs rblem The mrtance wll ncrease Slde 5

6 12/03/2013 CAPD Annual Meetng Carnege Melln Unversty U.S.A. Prevus wrs n VRP wth slt delveres Exact algrthms taes much cmutng tme Heurstc algrthm (Savng methd Passen et al 2011) Otmalty cannt be ensured. Clumn generatn tmal fr cntnuus relaxatn f Dantzg-Wlfe refrmulatn Clumn generatn fr slt delvery VRP (Jn et al 2007) Branch and rce and cut (Brønm et al 2010) (Henng et al 2012) Hwever ractcal cnstrants fr crude l transrtatn s nt cnsdered. Objectve f ths wr We rse a clumn generatn heurstc fr real case study f nternatnal crude l transrtatn rblems. Slde 6

7 12/03/2013 CAPD Annual Meetng Carnege Melln Unversty U.S.A. Prblem descrtn Lad lannng (Slt cu) Delvery lannng (Slt delvery) Ttal Demand Demand Ladng laces Ladng lannng Start(mdnt) Base Unladng Demand laces Unladng lannng Objectve functn: T mnmze the ttal dstances the cst msed by vstng ladng laces Slde 7

8 12/03/2013 CAPD Annual Meetng Carnege Melln Unversty U.S.A. Prblem descrtn The number f avalable taners s fxed. (Cannt ncrease) Caacty f taners are dfferent fr each ne. The lmts f ladng vlume and unladng vlume n each ladng lace and unladng lace are dfferent fr each lace resectvely. The number f vstng tme s lmted. The assgnments f ladng laces and unladng laces fr each l are dfferent. (Ol t unladng laces s ne-t-many) Etc. Slde 8

9 06/20/2012 ISFA2012 Inut data and decsn varable Gven Dstances between ladng laces Demand vlume f ls Lmts f ladng vlume and unladng vlume at each lace Caacty f taners Decsn varable x j {01} {01} vstng sequence assgnment Objectve functn: t mnmze the ttal dstances and rt charge a b ladng vlume unladng vlume

10 Prblem frmulatn 12/03/2013 CAPD Annual Meetng Carnege Melln Unversty U.S.A. Slde 9 L U q b ' ' U j j w ' ' ' ' U j j w ' ' ' ' T O ul D b ' ' T L D a Demand cnstrant } { s L j j x } { s L j j x Assgnment cnstrant m L Lmtatn f vstng tme 0 ) 1 ( j j j x M t T t Subtur elmnatn O a a a max mn Mnmum and maxmum Ladng vlume T H Requred number f taner frm demands The rblem s nwn t be NP-cmlete

11 12/03/2013 CAPD Annual Meetng Carnege Melln Unversty U.S.A. Dantzg-Wlfe refrmulatn taes a value f 1 f lan s adted fr taner mn ( w d x w c 1 j j 2 T j T ) T a D Set arttnng cnstrants 1 {01} 0 1

12 12/03/2013 CAPD Annual Meetng Carnege Melln Unversty U.S.A. Clumn generatn and Lagrangan relaxatn Cntnuus Relaxatn f DW refrmulatn mn ( w 1 T j w T 2 T q d j c x j D ) Dual Lagrangan Dual f Orgnal Prblem max w w K T 1 T j c D j j 2 T d x q Smlex algrthm Lwer bund Clumn generatn heurstcs Uer bund Lwer bund Uer bund Subgradent algrthm Whch s better bund? Lagrangan relaxatn heurstcs Lwer bunds are theretcally equal. but smlex algrthm Is better than subgradent methd

13 12/03/2013 CAPD Annual Meetng Carnege Melln Unversty U.S.A. Clumn generatn arach Orgnal rblem Dual varable Dantzg-Wlfe decmstn & refrmulatn Master rblem Prcng rblem The tmal cmbnatn f taners schedules s decded. New clumn A clumn whch mrves the bjectve f master rblem s generated. Lwer bund Heurstcs methd But t s nt feasble slutn. A feasble slutn s derved! Slde 10

14 12/03/2013 CAPD Annual Meetng Carnege Melln Unversty U.S.A. Restrcted master rblem Ttal dstance Ttal rt charge Weghted ceffcent Cnstrants Ttal demand cnstrants One rutng lan can be adted fr each taner Relaxatn f bnary cnstrants n Decsn varable Whether lan fr s adted r nt. Ceffcents deendent rutng lans Whether taner f lan vsts frm t j r nt. Whether taner f lan vsts r nt Lnear Prblem(LP) Ladng vlume Slde 11

15 12/03/2013 CAPD Annual Meetng Carnege Melln Unversty U.S.A. Prcng rblem The bjectve s t mnmze the reduced cst. Plan f taner s cnsdered. Dual varable f RMP There are cnstrants all fr each sngle taner. Ladng lannng Start(mdnt) Unladng lannng T determne a sequence ladng vlume and unladng vlume. Ladng lace Base Unladng lace Slde 12

16 12/03/2013 CAPD Annual Meetng Carnege Melln Unversty U.S.A. Cnstructn f an ntal slutn When usng clumn generatn an ntal feasble slutn s needed. In ths rblem real cases are sused. S we cannt ncrease the number f avalable taners. Challenge T derve an ntal slutn n fxed number f taners s dffcult because f set arttnng cnstrants (demand cnstrants) and a taner cannt vst mre than tw ladng laces. We rse a methd cnstructng an ntal slutn by usng nly exstng taners. Slde 13

17 12/03/2013 CAPD Annual Meetng Carnege Melln Unversty U.S.A. Cnstructn f an ntal slutn Place 2 Q2 = 1235 Place 3 Q3 = 900 Place 4 Q4 = 1193 Place 5 Q5 = 1070 Place 1 Q1 = 810 Place 6 Q6 = 961 Start(md) nt Taner Caacty Ttal ladng vlume

18 12/03/2013 CAPD Annual Meetng Carnege Melln Unversty U.S.A. Cnstructn f an ntal slutn Place 2 Q2 = 0 Place 3 Q3 = 0 Place 4 Q4 = 1193 Place 5 Q5 = 0 Place 1 Q1 = 0 Sequence f Taner 3 Place 6 Q6 = 961 Start(md) nt Taner Caacty Ttal ladng vlume All the ls are laded. An added cnstrant: Each taner has t leave at least a threshld unts f l r n l n each lace. Ths threshld s determned by gradually ncreasng the value.

19 12/03/2013 CAPD Annual Meetng Carnege Melln Unversty U.S.A. Cnstructn f a feasble slutn In the master rblem the tmal cmbnatn f lans s decded. The decsn varable taes 1 f lan s adted. 0 f lan s nt adted. Hwever the master rblem has LP relaxatn s fractnal value. We cannt decde whether the lan s adted r nt. taes We rse an algrthm t derve a feasble slutn frm fractnal slutn n the master rblem. Slde 15

20 12/03/2013 CAPD Annual Meetng Carnege Melln Unversty U.S.A. Cnstructn f a feasble slutn The lnear relaxed master rblem Plan (Clumn) a b c d e Value f Feasble slutn (Uer Bund) Slde 16

21 12/03/2013 CAPD Annual Meetng Carnege Melln Unversty U.S.A. Clumn Generatn Heurstc Basc Idea: secfy threshld value t elmnate nn-rmsng ndes () Slve an ntal rblem () Delete all the clumns nt fxed And execute clumn generatn agan () If the slutn s nfeasble and the lwer bund s larger than uer bund Bactrac and slve anther rblem (v) Outut the slutn whch has the best bjectve n all f cmbnatns Slde 17

22 12/03/2013 CAPD Annual Meetng Carnege Melln Unversty U.S.A. Branch and rce Branchng eratn 1. Number f taners vstng at a ladng lace 2. Number f taners vstng tw ladng laces sequentally 3. A taner vsts a ladng lace r nt 4. A taner vsts tw ladng laces sequentally r nt number f vstng tme fr taner t ladng lace s calculated If the number f vstng t a ladng lace s 2.5 Cnstrant s added such that the number f vstng s less than 2 Cnstrant s added such that the number f vstng s greater than 3 Slde 18

23 12/03/2013 CAPD Annual Meetng Carnege Melln Unversty U.S.A. Reductn f cmutatnal effrt Lwer bund f clumn generatn taes a lt f cmutng tme. We utlze cntnuus relaxatn f the rgnal rblem fr bundng rcedure befre clumn generatn. It can reduce cmutng tme. If the slutn f restrcted master rblem s 1.0 the slutn after the fxng f the varable s the same. It can elmnate clumn generatn rcedure.

24 12/03/2013 CAPD Annual Meetng Carnege Melln Unversty U.S.A. Cmutatnal exerments Case study wth ractcal data Ladng lannng (SDVRP) Small scale: 4 taners 22 ladng laces 26ls 8 unladng laces Medum scale: 13 taners 22 ladng laces 26 ls 8 unladng laces Ladng and unladng lannng (SPDVRP) Large scale: 18 taners 22 ladng laces 26 ls 8 unladng laces Cmutatnal envrnment Intel Cre(TM)2 Du 3.06GHz wth 2GB memry s used fr the cmutatns. RMP and SP are slved by IBM ILOG CPLEX12.1. Slde 19

25 12/03/2013 CAPD Annual Meetng Carnege Melln Unversty U.S.A. Cmutatnal results (small scale nstance) 4 Taners nstances Branch and bund Clumn generatn +heurstcs Branch and rce Number f taners Uer bund Lwer bund DGa(%) Cmutatn tme[s] :DGa = 100 (UB - LB)/LB Slde 20

26 12/03/2013 CAPD Annual Meetng Carnege Melln Unversty U.S.A. Cmutatnal results(medum scale nstance) 13 Taners nstances Methd UB LB DGAP clumns tme[s] Intal CG-BB PM(0.5) * PM(0.6) * PM(0.7) PM(0.8) PM(0.9) BB * Oeratr ( ): Threshld value arameter fr selectng ndes 20% cst reductn by the rsed methd Slde 21

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28 12/03/2013 CAPD Annual Meetng Carnege Melln Unversty U.S.A. Practcal cnstrants In rder t create a lan wth full caacty the rrty f clumn generatn heurstc s set t (Ladng vlume) n the lan. Sme secfc ls shuld be delvered n a secfed rat. We ncluded the cnstrants n the rcng rblem. The demanded tems f ls are gven as rrty frm the database gven frm exert eratr. Slde 23

29 12/03/2013 CAPD Annual Meetng Carnege Melln Unversty U.S.A. Cmutatnal results (large scale nstance) 18 Taners nstance Methd UB LB DGAP Number f clumns tme[s] Intal CG-BB PM(0.5) * PM(0.6) * PM(0.7) * PM(0.8) * PM(0.9) BB * B&B methd cannt derve a feasble slutn ( ): Threshld value arameter fr selectng ndes Slde 24

30 12/03/2013 CAPD Annual Meetng Carnege Melln Unversty U.S.A. Cnclusn and future wrs Cnclusn A clumn generatn arach has been rsed t slve the slt cu and delvery vehcle rutng rblem fr crude l transrtatn. We rsed a ractcal algrthm t generate a feasble slutn wth clumn generatn. The case study has demnstrated that the effectveness f the rsed methd cmared wth human eratr s result. Future wrs We shuld cnsder mre detaled cnstrants n unladng lannng. We wll try t aly branch-and-rce algrthm fr ths rblem. Integratn f rductn lannng and sh schedulng wll be requred. Slde 25