A Tour of Modeling Techniques

Transcription

1 A Tour of Modelng Technques John Hooker Carnege Mellon Unversy EWO Semnar February 8 Slde

2 Oulne Med neger lnear (MILP) modelng Dsuncve modelng Knapsack modelng Consran programmng models Inegraed Models Slde

3 Med Ineger/Lnear Modelng MILP Models Dsuncve Modelng Knapsack Modelng Slde 3

4 MILP models An med neger lnear programmng (MILP) model has he form mn c + dy A + by b, y y neger Slde 4

5 A prncpled approach o MILP modelng MILP modelng combnes wo dsnc knds of modelng. Modelng of subses of connuous space, usng - aulary varables. Knapsack modelng, usng general neger varables. MILP can model subses of connuous space ha are unons of polyhedra. ha s, represened by dsuncons of lnear sysems. So a prncpled approach s o analyze he problem as dsuncons neger of lnear + knapsack sysems nequales Slde 5

6 Dsuncve Modelng Theorem. A subse of connuous space can be represened by an MILP model f and only f s he unon of fnely many polyhedra havng he same recesson cone. Recesson cone of polyhedron Polyhedron Unon of polyhedra wh he same recesson cone (n hs case, he orgn) Slde 6

7 Modelng a unon of polyhedra Sar wh a dsuncon of lnear sysems o represen he unon of polyhedra. The kh polyhedron s { A k b} mn c ( k k A b ) k Inroduce a - varable y k ha s when s n polyhedron k. Dsaggregae o creae an k for each k. mn c k k k A b y, all k k k y = y k k = k {,} k Slde 7

8 Tgh Relaaons Basc fac: The connuous relaaon of he dsuncve MILP model provdes a conve hull relaaon of he dsuncon. Ths s he ghes possble lnear model for he dsuncon. Unon of polyhedra Conve hull relaaon (ghes lnear relaaon) Slde 8

9 Eample: Fed charge funcon Mnmze a fed charge funcon: mn f = f + c f > Slde 9

10 Fed charge problem Mnmze a fed charge funcon: mn f = f + c f > Feasble se Slde

11 Fed charge problem Mnmze a fed charge funcon: mn f = f + c f > Unon of wo polyhedra P, P P Slde

12 Fed charge problem Mnmze a fed charge funcon: mn f = f + c f > Unon of wo polyhedra P, P P P Slde

13 Fed charge problem Mnmze a fed charge funcon: mn f = f + c f > The polyhedra have dfferen recesson cones. P P Slde 3 P recesson cone P recesson cone

14 Fed charge problem Mnmze a fed charge funcon: Add an upper bound on mn f = f + c f > M The polyhedra have he same recesson cone. P P Slde 4 M P recesson cone P recesson cone

15 Fed charge problem Sar wh a dsuncon of lnear sysems o represen he unon of polyhedra mn = M f + c P P Slde 5 M

16 Fed charge problem Sar wh a dsuncon of lnear sysems o represen he unon of polyhedra Inroduce a - varable y k ha s when s n polyhedron k. Dsaggregae o creae an k for each k. mn = M f + c mn = My + c fy { } y + y =, y, k = +, = + Slde 6

17 To smplfy, replace wh snce = mn = My + c fy { } y + y =, y, k = +, = + Slde 7

18 To smplfy, replace wh snce = mn + = + { } y + y =, y, k My c fy Slde 8

19 Replace wh because plays no role n he model mn + = + { } y + y =, y, k My c fy Slde 9

20 Replace wh Because plays no role n he model mn { } y + y =, y, k My c + fy Slde

21 Replace y wh y Because y plays no role n he model mn { } y + y =, y, k My c + fy Slde

22 Replace y wh y Because y plays no role n he model mn My y c + fy {, } or mn c + fy y My {,} Bg M Slde

23 Eample: Uncapacaed facly locaon m possble facory locaons n markes Locae facores o serve markes so as o mnmze oal fed cos and ranspor cos. No lm on producon capacy of each facory. Fed cos f c Transpor cos Slde 3

24 Uncapacaed facly locaon m possble facory locaons n markes Dsuncve model: mn z + c, all =, all, all z f z = =, all Fracon of marke s demand sasfed from locaon Fed cos f c Transpor cos Facory a locaon No facory a locaon Slde 4

25 Uncapacaed facly locaon Dsuncve model: mn z + c MILP formulaon: Slde 5 mn, all, =, all, z f y, all z =, all = + z = z + z y, all =, all, all z f z = =, all z + c y =, all { },,,

26 Uncapacaed facly locaon = Le snce z = z Le snce = z = MILP formulaon: Slde 6 mn, all, =, all, z f y, all z =, all = + z = z + z y z + c y =, all { },,,

27 Uncapacaed facly locaon Le snce z = = z Le snce = z = MILP formulaon: Slde 7 mn y, all, z y, all {,} z + c f y =, all

28 Uncapacaed facly locaon Le snce z = = z Le snce = z = MILP formulaon: Slde 8 mn y, all, z y, all {,} z + c f y =, all or mn {,} f y + c y, all, y =, all

29 Uncapacaed facly locaon Mamum oupu from locaon MILP formulaon: mn y, all, y {,} f y + c =, all Begnner s model: mn y ny, all {,} f y + c =, all Based on capacaed locaon model. I has a weaker connuous relaaon Slde 9 Ths begnner s msake can be avoded by sarng wh dsuncve formulaon.

30 Eample: Lo szng wh seup coss Ma producon level Seup cos ncurred = Demand = D D D D 3 D 4 D 5 D 6 Deermne lo sze n each perod o mnmze oal producon, nvenory, and seup coss. Slde 3

31 Fed-cos varable Fed cos Producon capacy Producon level v f v v C C = () Sar producon (ncurs seup cos) Logcal condons: () Connue producon (no seup cos) (3) Produce nohng (no producon cos) () In perod () or () n perod () In perod neher () nor () n perod Slde 3

32 () Sar producon () Connue producon (3) Produce nohng v f v v C C = Conve hull MILP model of dsuncon: v f y Cy v Cy v 3 3 = k k =, =, = k k = k = k = v v y y y k {,}, k =,,3 Slde 3

33 To smplfy, defne z = y y = y Conve hull MILP model of dsuncon: v f y Cy v Cy v 3 3 = k k =, =, = k k = k = k = v v y y y k {,}, k =,,3 Slde 33

34 To smplfy, defne z = y y = y Conve hull MILP model of dsuncon: v f z C z v Cy v 3 3 = 3 3 k k =, =, z + y k = k = v v z, y {,}, k =,,3 Slde 34 = for sarup = for connued producon

35 Snce = se 3 = + Conve hull MILP model of dsuncon: v f z Cz v Cy v 3 3 = 3 3 k k k = k = v = v, =, z + y z, y {,}, k =,,3 Slde 35

36 Snce = se 3 = + Conve hull MILP model of dsuncon: C ( z + y ) v f z v v 3 3 k k = v = v, z + y z, y {,}, k =,,3 Slde 36

37 Snce v occurs posvely n he obecve funcon, 3 and v, v do no play a role, le v = v Conve hull MILP model of dsuncon: v f z C ( z + y ) v v 3 3 k k = v = v, z + y z, y {,}, k =,,3 Slde 37

38 Snce v occurs posvely n he obecve funcon, 3 and v, v do no play a role, le v = v Conve hull MILP model of dsuncon: v f z C ( z + y ) + y z, y {,}, k =,,3 z Slde 38

39 Formulae logcal condons: () In perod () or () n perod () In perod neher () nor () n perod v f z C ( z + y ) z y z, y {,}, k =,,3 z + y z + y z y Slde 39

40 Add obecve funcon Un producon cos Un holdng cos n mn ( p + h s + v ) = v f z C ( z + y ) + y z, y {,}, k =,,3 z y z + y z z y Slde 4

41 Knapsack Models Ineger varables can also be used o epress counng deas. Ths s oally dfferen from he use of - varables o epress unons of polyhedra. Slde 4

42 Eample: Fregh Transfer Transpor 4 ons of fregh usng 8 rucks, whch come n 4 szes Truck sze Number avalable Capacy (ons) Cos per ruck Slde

43 Number of rucks of ype Knapsack packng consran mn {,,,3} 8 Knapsack coverng consran Truck ype Number avalable Capacy (ons) Cos per ruck Slde

44 Eample: Fregh Packng and Transfer Transpor packages usng n rucks Each package has sze a. Each ruck has capacy Q. Slde 44

45 Knapsack componen The rucks seleced mus have enough capacy o carry he load. n = Q y a = f ruck s seleced Slde 45

46 Dsuncve componen (wh embedded knapsack consran) Cos varable Slde 46 Truck seleced Truck no seleced z c z a Q =, all Use connuous relaaon because we wan a dsuncon of lnear sysems Cos of operang ruck = f package s loaded on ruck

47 Dsuncve componen (wh embedded knapsack consran) Truck seleced Truck no seleced z c z a Q =, all Conve hull MILP formulaon z a c y Q y y Slde 47

48 The resulng model = = mn n = a Q y, all y, all, n n Q y =, all, y {,} c y a Dsuncve componen Logcal condon (each package mus be shpped) Knapsack componen Slde 48

49 The resulng model = = mn n = a Q y, all y, all, n n Q y =, all, y {,} c y a The y s redundan bu makes he connuous relaaon gher. Ths s a modelng rck, par of he folklore of modelng. Slde 49

50 The resulng model = Slde 5 = mn n = a Q y, all y, all, n n Q y =, all, y {,} c y a The y s redundan bu makes he connuous relaaon gher. Ths s a modelng rck, par of he folklore of modelng. Convenonal modelng wsdom would no use hs consran, because s he sum of he frs consran over. Bu radcally reduces soluon me, because generaes knapsack cus. Ths argues for a prncpled approach o modelng.

51 Consran Programmng Models Global Consrans Employee Schedulng Slde 5

52 Global consrans A global consran represens a se of consrans wh specal srucure. The srucure s eploed by flerng algorhms n he CP solver. Slde 5

53 Some general-purpose global consrans Alldff - Requres ha all he lsed varables ake dfferen values. Among - Bounds he number of lsed varables ha ake one of he values n a ls. Cardnaly - Bounds he number of lsed varables ha ake each of he values n a ls. Elemen - Requres ha a gven varable ake he yh value n a ls, where y s an neger varable. Pah - Requres ha a gven graph conan a pah of a mos a gven lengh. Slde 53

54 Some global consrans for schedulng Dsuncve - Requres ha no wo obs overlap n me. Cumulave - Lms he resources consumed by obs runnng a any one me. In parcular, can lm he number of obs runnng a any one me. Srech - Bounds he lengh of a srech of conguous perods assgned he same ob. Sequence A se of overlappng among consrans. Regular Generalzes srech and sequence. Dffn - Requres ha no wo boes n a se of muldmensonal boes overlap. Used for space or space-me packng. Slde 54

55 Eample: Employee Schedulng Schedule four nurses n 8-hour shfs. A nurse works a mos one shf a day, a leas 5 days a week. Same schedule every week. No shf saffed by more han wo dfferen nurses n a week. A nurse canno work dfferen shfs on wo consecuve days. A nurse who works shf or 3 mus do so a leas wo days n a row. Slde 55

56 Two ways o vew he problem Assgn nurses o shfs Sun Mon Tue Wed Thu Fr Sa Shf A B A A A A A Shf C C C B B B B Shf 3 D D D D C C D Assgn shfs o nurses Sun Mon Tue Wed Thu Fr Sa Nurse A Nurse B Nurse C 3 3 Nurse D Slde 56 = day off

57 Use boh formulaons n he same model! Frs, assgn nurses o shfs. Le w sd = nurse assgned o shf s on day d alldff( w, w, w ), all d The varables w d, w d, d d 3d w 3d ake dfferen values Tha s, schedule 3 dfferen nurses on each day Slde 57

58 Use boh formulaons n he same model! Frs, assgn nurses o shfs. Le w sd = nurse assgned o shf s on day d alldff( w, w, w ), all d d 3d ( w A B C D ) cardnaly (,,, ),(5,5,5,5),(6,6,6,6) d A occurs a leas 5 and a mos 6 mes n he array w, and smlarly for B, C, D. Tha s, each nurse works a leas 5 and a mos 6 days a week Slde 58

59 Use boh formulaons n he same model! Frs, assgn nurses o shfs. Le w sd = nurse assgned o shf s on day d ( w d wd w3d ) d ( w A B C D ) alldff,,, all cardnaly (,,, ),(5,5,5,5),(6,6,6,6) ( ) nvalues w,..., w,, all s s,sun s,sa The varables w s,sun,, w s,sa ake a leas and a mos dfferen values. Slde 59 Tha s, a leas and a mos nurses work any gven shf.

60 Remanng consrans are no easly epressed n hs noaon. So, assgn shfs o nurses. Le y d = nurse assgned o shf s on day d alldff (,, ) y y y, all d d d 3d Assgn a dfferen nurse o each shf on each day. Slde 6 Ths consran s redundan of prevous consrans, bu redundan consrans speed soluon.

61 Remanng consrans are no easly epressed n hs noaon. So, assgn shfs o nurses. Le y d = nurse assgned o shf s on day d alldff ( y, y, y ) d d 3d ( y y P ),Sun, all srech,, (,3),(,),(6,6),, all,sa d Every srech of s has lengh beween and 6. Every srech of 3 s has lengh beween and 6. So a nurse who works shf or 3 mus do so a leas wo days n a row. Slde 6

62 Remanng consrans are no easly epressed n hs noaon. So, assgn shfs o nurses. Le y d = nurse assgned o shf s on day d alldff ( y, y, y ) d d 3d ( y y P ),Sun, all srech,, (,3),(,),(6,6),, all,sa d Here P = {(s,),(,s) s =,,3} Whenever a srech of a s mmedaely precedes a srech of b s, (a,b) mus be one of he pars n P. So a nurse canno swch shfs whou akng a leas one day off. Slde 6

63 Now we mus connec he w sd varables o he y d varables. Use channelng consrans: w =, all y d d y = s, all s w sd d Channelng consrans ncrease propagaon and make he problem easer o solve. Slde 63

64 The complee model s: ( w d wd w3d ) d ( w A B C D ) alldff,,, all cardnaly (,,, ),(5,5,5,5),(6,6,6,6) ( ) nvalues w,..., w,, all s alldff s,sun ( y, y, y ) d d 3d s,sa ( y y P ),Sun, all srech,, (,3),(,),(6,6),, all,sa d w =, all y d d y = s, all s w sd d Slde 64

65 Inegraed Models Produc Confguraon Machne Assgnmen and Schedulng Slde 65

66 Eample: Produc Confguraon Ths eample combnes MILP modelng wh varable ndces, used n consran programmng. I can be solved by combnng MILP and CP echnques. Slde 66

67 The problem Choose wha ype of each componen, and how many Memory Memory Memory Memory Memory Memory Personal compuer Power supply Dsk drve Dsk drve Dsk drve Dsk drve Dsk drve Power supply Power supply Power supply Slde 67

68 Inegraed model Un cos of producng arbue Amoun of arbue produced (< f consumed): memory, hea, power, wegh, ec. mn c v v = q A, all k L v U, all Amoun of arbue produced by ype of componen Quany of componen nsalled Slde 68

69 Inegraed model Ths s reformulaed mn c v v = q A, all k L v U, all s a varable nde v z, all = ( n ) elemen,( q, A,, q A ), z, all, Slde 69

70 Inegraed model Ths s reformulaed mn c v v = q A, all k L v U, all s a varable nde v z, all = (,, n z ) elemen,( q A q A ),, all, Slde 7 Se z equal o he h em n he red ls.

71 Machne Assgnmen and Schedulng Assgn obs o machnes and schedule he machnes assgned o each machne whn me wndows. The obecve s o mnmze makespan. Tme lapse beween sar of frs ob and end of las ob. Combne MILP and CP modelng Slde 7

72 Machne Schedulng The model s Sar me varable for ob mn M M s + p, all r s d p, all Processng me of ob on machne Machne assgned o ob ( = = ) dsuncve ( s ),( p ), all Makespan Slde 7

73 Machne Schedulng The model s Release me for ob mn M M s + p, all r s d p, all ( = = ) dsuncve ( s ),( p ), all Tme wndows Deadlne for ob Slde 73

74 Machne Schedulng The model s mn M M s + p, all r s d p, all ( = = ) dsuncve ( s ),( p ), all Sar mes of obs assgned o machne Dsuncve global consran requres ha Jobs do no overlap Slde 74

75 Machne Schedulng The problem can be solved by logc-based Benders decomposon. mn M M s + p, all r s d p, all ( = = ) dsuncve ( s ),( p ), all Maser problem s hs plus Benders cus, solved as an MILP Slde 75

76 Machne Schedulng The problem can be solved by logc-based Benders decomposon. mn M M s + p, all r s d p, all ( = = ) dsuncve ( s ),( p ), all Maser problem s hs plus Benders cus, solved as an MILP Subproblem s hs, solved by CP Slde 76