A Modeling System to Combine Optimization and Constraint. Programming. INFORMS, November Carnegie Mellon University.

Transcription

1 A Modelng Sstem to Combne Optmzaton and Constrant Programmng John Hooker Carnege Mellon Unverst INFORMS November 000

2 Based on ont work wth Ignaco Grossmann Hak-Jn Km Mara Axlo Osoro Greger Ottosson Erlendr Thorstensson

3 Goal Desgn a modelng langage whose sntax ndcates how optmzaton and constrant programmng can combne to solve the problem.

4 Wh Combne Optmzaton and Constrant Programmng? Some constrants n a gven problem propagate well and others relax well. Constrant programmng ses constrant propagaton to redce domans sets of possble vales of varables and so redce branchng. Optmzaton ses easl-solved relaxatons to obtan bonds on the optmal vale and so redce branchng. Constrant programmng ses global constrants to nform the solver abot specal strctre n the model.

5 General Form of Model mn s.t. q x f x X D h x I obectve fncton condtonal constrants solton varables search varables Checkable constrant belongs to NP Us contnos Set of solble constrants belong to NP co-np Us dscrete

6 Addtonal Tpes of Constrants p d g x x I I I 4 checkable constrants solble constrants global constrants Defnable n terms of a set of condtonal constrants Degenerate cases of a condtonal constrant

7 The Basc Idea Branch on search varables. At each node of the search tree: Appl constrant propagaton to checkable and global constrants to redce domans. Create a relaxaton. If antecedent of a condtonal s tre fre the condtonal b addng conseqent to relaxaton. Add relaxatons of global constrants to relaxaton. Use optmal vale of relaxaton to prne tree f possble. Backtrack or tr to fnd feasble vales of search varables consstent wth solton varables

8 Relaxaton at a Gven Node of the Search Tree mn s.t. f x g x I h x relaxaton of x X d Solble constrants I Conseqents of fred condtonal constrants for whch x q I 4 s tre Specalzed relaxatons of global constrants

9 Processng Network Desgn 4 Unt x x Unt Unt x 4 x 5 x 6 x 4 x 5 x 6 Unt 4 5 Unt 5 6 Unt 6

10 Processng Network Desgn The model ses search varables to ndcate the presence or absence of a nt. It ses condtonal constrants to reqre that the fxed cost be ncrred or the nt sht down.

11 max s.t. 0.6 = x 4 + x = x = x 4 0. = x 5 + x 6 x c / 0 r z = b = Ax Bx = tre z = d = false = 0 flow thr nts flow balance nt s open nt s closed nt capactes

12 Processng Network Desgn Add don t-be-stpd constrants to ensre that a nt s not opened nless downstream nts are opened.

13 be - stpd - don't 0 nt capactes nt s closed 0 false nt s open tre flow balance flow thr nts s.t. max / = = = = = = x c d z Bx b Ax z r

14 Processng Network Desgn Use an neqalt-or global constrant to obtan good relaxaton of dsnctve constrants. Use cnf global constrant to nvoke resolton algorthm for don t-be-stpd constrants.

15 global constrant cnf 0 nt capactes global constrant 0 neqalt - or flow balance flow thr nts s.t. max / = = = x c d z Bx b Ax z r Part of relaxaton Inclde relaxaton of global constrant Process smbolc

16 Knapsack Problem wth All-dfferent Orgnal problem mn s.t dfferent {4} 0 As modeled here solved b branchng and doman redcton onl.

17 Knapsack Problem wth All-dfferent The contnos predcate adds a contnos relaxaton and an desred cttng planes. mn s.t. Replace obectve fncton wth 5x + 8x + 4x contnos5 contnos - dfferent {4} Add x + 5x + x 0 and lnk x and possbl knapsack cts

18 Knapsack Problem wth All-dfferent The ct predcate generates cts n the search varables so that doman redcton s appled to cts. Contnos adds contnos relaxaton of problem and cts. mn s.t. z contnos ct - dfferent {4} z

19 Cmlatve Global Constrant Ensres that total resorces consmed b obs at an one tme do not exceed C. cmlatvet t n d d n r r n C Job start tmes Job dratons Job resorce reqrements

20 Manfactrng Unt Prodcton Schedlng Storage Tanks Capact C Capact C Capact C Packng Unts

21 Fllng of StorageTank Level Need to enforce capact constrant here onl t t + b/r + b/s Fllng starts Packng starts Fllng ends Batch sze Manfactrng rate Packng ends Packng rate

22 mn s.t. T T t cmlatve b v = + s b cmlatve R t s r + b s + 0 t v m t s b s C b s Makespan n n Job release tme e p m storage tanks Job draton Tank capact p packng nts

23 mn s.t. T T t cmlatve b v = + s b cmlatve R t + s r b s + 0 t v m t s b s C b s n n Part of relaxaton Appl doman redcton relaxaton et to be developed e p