Fixed-Charge Network Flow Problems with a Hybrid Optimization and. Constraint Programming Approach. Solving. John Hooker, Hak-Jin Kim

Transcription

1 Solvng Fed-Charge Network Flow Problems wth a Hbrd Optmaton and Constrant Programmng Approach John Hooker, Hak-Jn Km Carnege ellon Unverst ISP, August 2

2 An Approach that Combnes Optmaton and Constrant Programmng Fed charge network flow problems llustrate a general approach to modelng and soluton that combnes elements of optmaton and constrant programmng. Logcal epressons capture the dscrete element (whether to nstall an arc) wthout nteger varables.

3 A Combned Approach A relaaton s formed usng the contnuous varables onl ( proected relaaton). The relaaton solves rapdl because t s a mncost network flow problem. The constrants are gathered up nto a global constrant. The global constrant trggers generaton of the relaaton as well as constrant propagaton.

4 odelng Advantage In general the modelng advantage s a more natural and succnct model. The structure of the model ndcates how procedures should nteract to solve t. In ths case, the model s not much smpler than an IP model. But the global constrant dentfes structure that speeds soluton.

5 Algorthmc Advantage In ths case, constrant propagaton at ever node tends to shrnk the search tree. The absence of flow cuts n the relaaton tends to make the search tree larger. Flow cuts would destro the network structure of the proected relaaton. But fast soluton of network flow problems speeds the processng of each node. The method appears to work well for fed-charge transportaton problems but not warehouse locaton problems, both specal cases.

6 Fed-Charge Network Flow: -1 Formulaton s.t. Flow on arc mn Total cost c {,1} Varable arc cost Fed arc cost f,, s, all Arc capact Net suppl at node all all,, (cost) (flow balance) (bg )

7 Fed-Charge Network Flow: Logc-Based Formulaton Total cost mn s.t. c ( ) s f,, all, all all,, (arc ncluded) (arc omtted) (flow balance) Propostonal varable (true or false)

8 Proected Relaaton for Arc (,) c f f Proected relaaton f c

9 mn s.t. Contnuous Relaatons Lfted Proected (mn-cost network flow) c 1 f s mn s.t. c f s The relaatons are equvalent

10 Branch-and-bound Soluton Branch on propostonal varables. Appl constrant propagaton to f s when possble. Solve proected relaaton wth fed to ero whenever false. Soluton at current node s feasble f c f whenever > (analogous to ntegral soluton n -1 programmng).

11 Constrant Propagaton Use arc capactes to f some s to 1. Use reduced costs to f some s to. Use nogoods n the form of Benders cuts to f some s on whch the search has branched. Ccle through these steps at each node untl no further varables can be fed.

12 Constrant Propagaton on Arc Capactes For each node use the vald nequalt not fed to s to f some s to 1 (true) f possble (.e., make sure there s enough capact n outgong arcs to carr net suppl), and smlarl for ncomng arcs.

13 Constrant Propagaton Usng Reduced Costs Can use reduced costs to f logcal varables. F to false f the reduced cost of n the -1 model mples that t can be fed to. The reduced cost of n the -1 model s easl recoverable, even though the -1 model s not used. It s tmes the reduced cost of n the proected relaaton.

14 Constrant Propagaton Usng Benders Cuts as Nogoods When some dscrete varables have been fed, nogoods dentf wh ths partal soluton s unsatsfactor and rule out all solutons that are unsatsfactor for the same reason. Nogoods contan varables currentl fed b branchng, whereas cuttng planes contan varables not et fed. Benders cuts can serve as nogoods, where the master problem contans the varables currentl fed. Nogoods are used here for propagaton, not as part of the relaaton.

15 Suppose s currentl fed to for F. The subproblem s: mn s.t. F F c s f F 1, Generate Benders cut: F v F ( f v ) u s,, F u F F F w s f w ( u ) ( v ) ( v ) ( w ) Dual varables when feasble when nfeasble

16 Although -1 relaaton s not solved, Benders cuts can be recovered from the subproblem for the proected relaaton. mn s.t. F F, c,, f F s The Benders cut s one of the followng: F F 1 1 ( c f ) ( ) F t F t f u s F u s F t fed to true to false t ( u ) ( t ) ( t ) ( t )

17 Global Constrant Formulaton Formulate usng global constrant, to trgger specaled relaaton and constrant propagaton. mn s.t. FedCostNetworkFlow(,, s, )

18 Specal Case: Warehouse Locaton K warehouse capact s total demand arc capact s s s d demand at f warehouse fed cost f Warehouse Demand pont

19 Lfted relaaton of mn cost flow problem Proected relaaton mn s.t. c d 1 K f mn s.t. c These are equvalent. d K f K

20 Lfted relaaton of mn cost flow problem Stronger lfted relaaton used for -1 model mn s.t. c d 1 K f mn s.t. ust add ths to obtan stronger relaaton. Its proecton loses network structure. c d K 1 f

21 The proected relaaton of the warehouse locaton problem s weaker than the usual relaaton of -1 model. The dfference ma be neglgble f the arc capactes are large. If the arc capactes are tght, tradtonal methods ma be better. Also, the warehouse locaton problem has relatvel few -1 varables relatve to contnuous varables.

22 Specal Case: Fed Charge Transportaton Problem The fed charge transportaton problem s a more promsng applcaton of the proected relaaton. The proected relaaton s equvalent to the relaaton of the standard -1 model. There are man -1 varables.

23 Fed Charge Transportaton Problem s suppl arc capact s d demand f fed arc cost s s Orgn Destnaton

24 Generaton of Benders Cuts Benders cuts at nfeasble nodes correspond to the soluton of ma flow/mn cut problems on the transportaton network.

25 Suppose has been fed to false for all orange arcs. There s no feasble flow. s 1 2 d , all, 1 s Orgn Destnaton

26 Convert to a ma flow problem (orange arcs have capact ero). 1 3 Suppl demand 75 n capact cut a flow 5 Benders cut s