Optimization Methods for Engineering Design. Logic-Based. John Hooker. Turkish Operational Research Society. Carnegie Mellon University

Transcription

1 Logc-Based Optmzaton Methods for Engneerng Desgn John Hooker Carnege Mellon Unerst Turksh Operatonal Research Socet Ankara June 1999

2 Jont work wth: Srnas Bollapragada General Electrc R&D Omar Ghattas Cl and Enronmental Engneerng CMU. Ignaco Grossmann Chemcal Engneerng CMU

3 Idea of logc-based methods An alternate to nteger and mxed nteger/lnear programmng. Represent dscrete choces wth logcal propostons rather than nteger arables. Sole wth branch-and-bound methods as n nteger programmng but wth a dfference: use logcal nference and doman reducton methods from constrant programmng as well as lnear programmng. use new relaxatons for logcal constrants rather than the tradtonal lnear programmng relaxaton

4 Adantages of logc-based approach to desgn Desgn tpc noles a combnaton of dscrete choces and contnuous parameters. Logc framework prodes more natural modelng of dscrete choces. Soluton approach harnesses power of logcal nference and remoes unnecessar nteger arables from the lnear relaxatons. B dstngushng specal cases logc sngulartes can be aoded.

5 Outlne Logc-based optmzaton for chemcal processng network desgn (process snthess). Logc-based optmzaton for truss structure desgn. Computatonal results Integraton of optmzaton and constrant satsfacton.

6 Processng network desgn Desgn a network of processng unts such as reactors or dstllaton unts Meet demand whle mnmzng fxed and arable costs. Dscrete choce s whch unts to nst. Problem presented here s lnear but heat exchange and other processes ge rse to nonlnear models.

7 4-component separaton network A BCD 1 B CD 3 BC D 4 C D 8 ABCD ABC D 2 AB CD 5 A BC 6 AB C 7 B C 9 A B 10

8 Model for separaton problem Unt s nsted Unt flow cost mn s.t. Unt s not nsted c x + x 0 x x α ( z k k k z x x f k x z k ) 0 0 Fxed cost Flow olume } Flow balance } Unt output } Capact

9 Integer programmng model Replace dscrete constrants wth 0 x k 0-1 arable A contnuous (lnear programmng) relaxaton s mportant for solng the problem. It can be obtaned b replacng {01} wth 0 1

10 Relaxaton for logc constrants The logc constrants Can be relaxed: z 1 f M ( z x z f ) 0 x 0 Where M upper bound on output of unt So nteger arables add needless oerhead to soluton of lnear programmng relaxaton at each node.

11 Addtonal logc constrants Constrants such as can speed processng at each node b rulng out solutons that cannot be optmal.

12 Logc-based branch & bound x 1 true x 1 false Sole as an LP: lnear part of problem plus relaxaton of condtonal constrants plus: 1 f 1 z x 2 true x 2 false Sole same problem except replace z 1 f 1 wth: z 1 0 x 1 0 Because x 3 x 1 and x 4 x 1 fx x 3 x 4 false and add constrants smlar to aboe

13 Truss structure desgn Fnd truss structure that supports a gen load whle mnmzng total weght of bars. An gen bar ma be present of absent from the truss. Its cross-sectonal area must be one of seeral dscrete alues. The model s nonlnear. Logc-based modelng aods the sngulartes that occur n tradtonal models when the bar sze goes to zero.

14 Planar 10-bar cantleer truss 0 deg. freedom 2 deg. freedom Total 8 degrees of freedom Load

15 Notaton elongaton of bar s force along bar h length of bar d node dsplacement A cross-sectonal area of bar p load along d.f.

16 nonlnear mn s. t. E h L L h cos cos A A U d d d \ / k ( A A k θ θ } Mnmze total weght s d s U ) p } Equlbrum } Compatblt } Hooke s law } Elongaton bounds } Dsplacement bounds } Logcal dsuncton Area must be one of seeral dscrete alues A k Constrants can be mposed for multple loadng condtons

17 Logc-based branch & bound Rather than branch on arables branch b splttng the range of areas A At each node add the constrants L A A A where the bounds are equal to one of the dscrete areas bar can hae U

18 Mxed nteger model ntroduces man addtonal arables mn s. t. h cos θ d E h d k k L L k k A d A k s k U d U p k cos θ 1 k k s k 0-1 arables ndcatng sze of bar Elongaton arable dsaggregated b bar sze Hooke s law becomes lnear A useful relaxaton can be obtaned wthout the extra arables...

19 Lnear relaxaton Use the change of arables: Current bounds on areas A A 0 L A U (1 ) The s are not 0-1 arables but are contnuous arables that ar n the nteral [01] ntroduced onl to form a relaxaton. The resultng relaxaton s not a true relaxaton but prodes a ald bound on the optmal alue...

20 d d d s A A h E d p s s t A A h U L U L U L U L U L 1 0 ) (1 ) (1 ) ( cos cos.. )] (1 [ mn θ θ Hooke s law s lnearzed Elongaton bounds splt nto 2 sets of bounds

21 Parel wth branch & bound In branch & bound soluton of the lnear relaxaton s often ntegral whch reduces branchng. In logc-based branchng soluton of relaxaton often puts areas at endponts of ther ranges n whch case the hae one of the permssble dscrete alues.

22 Computatonal testng Logc-based: Branch on upper half of nteral frst. MILP: Sole wth CPLEX 4.0 wth automatc SOS detecton turned on. Problems: 10-bar truss 25-bar transmsson tower rectangular buldng ( bars) Bars are lnked when the must hae the same sze.

23 Computatonal results Problem Lnkng groups Logc-based seconds CPLEX MILP seconds 10-bar truss bar truss bar truss bar truss bar truss + L bar truss + D bar truss + D L 2 loadng condtons D dsplacement bounds

24 Computatonal results Problem Lnkng groups Logcbased seconds CPLEX MILP seconds 25-bar tower + L bar buldng + L bar buldng + L > bar buldng + L > bar structure 96 >36000 >72000 L 2 loadng condtons

25 An alternate model Remoe nonlneartes b wrtng Hooke s law drectl as a dsuncton \/ E A k s h k that can be relaxed wth a sngle nequalt for each. The dsuncte constrant ( A A ) can now be dropped. \/ k k

26 Possble enhancements Lmt total number of bar szes n structure. Enforce smmetres n topolog een when bar szes need not match. Enforce connectedness and other propertes logc. Fnd logcal characterzaton of stablt.

27 Constrant programmng Constrant programmng s based prmarl on logc-based methods known as constrant satsfacton methods. Dscrete arables tend to be multalued (here the were 2-alued). Global constrants explot problem structure and speed soluton; e.g. -dfferent( 1 n ) cumulate. Inference takes the form of doman reducton algorthms partcularl when appled to global constrants.

28 Optmzaton + constrant programmng Combnng optmzaton and constrant programmng s a er acte area of research; e.g. OPL modelng language. The logc-based method descrbed here s an example of one approach.