A Principled Approach to MILP Modeling

Transcription

1 A Prncpled Approach o MILP Modelng John Hooer Carnege Mellon Unvers Augus 008 Slde

2 Proposal MILP modelng s an ar, bu need no be unprncpled. Slde

3 Proposal MILP modelng s an ar, bu need no be unprncpled. I has wo basc componens: Dsuncve modelng of subses of connuous space. Knapsac modelng of counng deas. Slde 3

4 Proposal MILP modelng s an ar, bu need no be unprncpled. I has wo basc componens: Dsuncve modelng of subses of connuous space. Knapsac modelng of counng deas. MILPs can model subses of connuous space ha are unons of polhedra. ha s, represened b dsuncons of lnear ssems. Slde 4

5 Proposal MILP modelng s an ar, bu need no be unprncpled. I has wo basc componens: Dsuncve modelng of subses of connuous space. Knapsac modelng of counng deas. MILPs can model subses of connuous space ha are unons of polhedra. ha s, represened b dsuncons of lnear ssems. So a prncpled approach s o analze he problem as Slde 5 dsuncons neger of lnear + napsac ssems nequales

6 Proposal Jeroslow s Represenabl Theorem provdes heorecal bass for dsuncve modelng. Bounded MILP represenabl assumes bounded neger varables. Ths s nadequae for napsac modelng. Slde 6

7 Proposal Jeroslow s Represenabl Theorem provdes heorecal bass for dsuncve modelng. Bounded MILP represenabl assumes bounded neger varables. Ths s nadequae for napsac modelng. We wll generalze Jeroslow s heorem. Knapsac modelng accommodaed. Ineger varables can be unbounded. Slde 7

8 Oulne Bounded med neger represenabl Bounded represenabl heorem. Conve hull formulaon Eample: Fed charge problem Wh he dsuncve model wors Mulple dsuncons Eample: Facl locaon Eample: Lo szng wh seup coss Bg-M dsuncve formulaon Slde 8 Eample: Healh care benefs

9 Oulne General med neger represenabl Knapsac models General represenabl heorem. Conve hull formulaon Eample: Facl locaon Wh a sngle recesson cone Eample: Fregh pacng and ransfer Research ssues Slde 9

10 Slde 0 Bounded MILP Represenabl

11 Bounded represenabl heorem Defnon of R. Jeroslow: R A subse S of s bounded MILP represenable f S s he proecon ono of he feasble se of some MILP consran se of he form n A + Bz + D b n R, z R {0,} p m Bounded general neger varables can be encoded as 0- varables Aular connuous varables can be used Slde

12 Bounded represenabl heorem Theorem (Jeroslow). A subse of connuous space s bounded MILP represenable f and onl f s he unon of fnel man polhedra havng he same recesson cone. Recesson cone of polhedron Polhedron Unon of polhedra wh he same recesson cone (n hs case, he orgn) Slde

13 Conve hull formulaon Sar wh a dsuncon of lnear ssems o represen he unon of polhedra. ( A b ) The h polhedron s { A b} Inroduce a 0- varable ha s when s n polhedron. Dsaggregae o creae an for each. A b, all = = { 0,} Slde 3

14 Conve hull formulaon Sar wh a dsuncon of lnear ssems o represen he unon of polhedra. ( A b ) The h polhedron s { A b} Inroduce a 0- varable ha s when s n polhedron. Dsaggregae o creae an for each. Ever bounded MILP represenable se has a model of hs form. A b, all = = { 0,} Slde 4

15 Conve Hull Formulaon The connuous relaaon of hs dsuncve MILP provdes a conve hull relaaon of he dsuncon. Srcl, descrbes he closure of he conve hull. Unon of polhedra Conve hull relaaon (ghes lnear relaaon) Slde 5

16 Idea behnd he conve hull formulaon Sar b formulang a conve hull formulaon of he relaaon of he dsuncon Wre each soluon as a conve combnaon of pons n he polhedron A b, all = = [0,] Conve hull relaaon Slde 6

17 Idea behnd he conve hull formulaon Now appl a change of varable Wre each soluon as a conve combnaon of pons n he polhedron A b, all = = [0,] Change of varable = A b, all = = [0,] Conve hull relaaon Slde 7

18 Idea behnd he conve hull formulaon Now mae s 0- varables o ge an MILP represenaon A b, all = = { 0,} Mae s 0- A b, all = = [0,] Conve hull formulaon Slde 8

19 Idea behnd he conve hull formulaon When s hs a vald formulaon? Le s loo a an eample frs A b, all = = { 0,} Conve hull formulaon Slde 9

20 Eample: Fed charge funcon Mnmze a fed charge funcon: mn 0 f = 0 f + c f > 0 0 Slde 0

21 Fed charge problem Mnmze a fed charge funcon: mn 0 f = 0 f + c f > 0 0 Feasble se (epgraph) Slde

22 Fed charge problem Mnmze a fed charge funcon: mn 0 f = 0 f + c f > 0 0 Unon of wo polhedra P, P P Slde

23 Fed charge problem Mnmze a fed charge funcon: mn 0 f = 0 f + c f > 0 0 Unon of wo polhedra P, P P P Slde 3

24 Fed charge problem Mnmze a fed charge funcon: mn 0 f = 0 f + c f > 0 0 The polhedra have dfferen recesson cones. P P Slde 4 P recesson cone P recesson cone

25 Fed charge problem Dsuncve model descrbes conve hull relaaon bu no he feasble se. mn = f + c P P Slde 5

26 Fed charge problem Sar wh a dsuncon of lnear ssems o represen he unon of polhedra mn = f + c Inroduce a 0- varable ha s when s n polhedron. Dsaggregae o creae an for each. mn = c f + =, [0,] = +, = + Slde 6

27 To smplf, replace wh snce = 0 mn = c f + =, [0,] = +, = + Slde 7

28 To smplf, replace wh snce = 0 mn 0 + = + 0 c f + =, [0,] Slde 8

29 Replace wh because plas no role n he model mn 0 + = =, [0,] c f Slde 9

30 Replace wh because plas no role n he model mn 0 c + f + =, [0,] Slde 30

31 Replace wh because plas no role n he model mn 0 c + f + =, [0,] Slde 3

32 Replace wh because plas no role n he model mn [0,] 0 c + f Slde 3

33 The conve hull s hs. mn [0,] 0 c + f P P Slde 33

34 mn [0,] 0 c + f Relaaon correcl descrbes closure of conve hull P P Slde 34

35 mn 0 c + f {0,} Bu MILP model does no descrbe feasble se P P Slde 35

36 To f he problem Add an upper bound on mn 0 f = 0 f + c f > 0 0 M The polhedra have he same recesson cone. P P Slde 36 M P recesson cone P recesson cone

37 Fed charge problem mn The dsuncon s now = 0 0 M 0 f + c P P Slde 37 M

38 Fed charge problem mn = 0 0 M 0 f + c The dsuncve model s mn = 0 0 M 0 + c f { } + =, 0, = +, = + Slde 38

39 Ths smplfes as before mn 0 M c + f { 0, } Slde 39

40 Ths smplfes as before mn 0 M c + f { 0, } Prevous model mn 0 c + f { 0, } Slde 40

41 Ths smplfes as before mn 0 M c + f { 0, } or mn c + f 0 M { 0,} Bg M Prevous model mn 0 c + f { 0, } Slde 4

42 The model now correcl mn c + f descrbes he feasble se. { 0,} 0 M Bg M P P Slde 4 M

43 Wh he dsuncve model wors P l Recesson cone of polhedra mn c A b, all = = { 0,} Le S be feasble se. P Slde 43

44 Wh he dsuncve model wors P l mn c A b, all = = { 0,} P Le S be feasble se. S some P Slde 44

45 Wh he dsuncve model wors P l mn c A b, all = = { 0,} P Le S be feasble se. S some P sasfes he model for =, oher s = 0 l =, oher s = 0 l Slde 45

46 Wh he dsuncve model wors P l mn c A b, all = = { 0,} P Conversel, suppose,, s sasf he model some = P Slde 46

47 Wh he dsuncve model wors P l mn c A b, all = = { 0,} P Conversel, suppose,, s sasf he model some = P A l l 0 for oher ls Slde 47

48 Wh he dsuncve model wors P l mn c A b, all = l { 0,} = P Conversel, suppose,, s sasf he model some = P A l l l s are recesson drecons for oher 0 for oher ls P s l Slde 48

49 Wh he dsuncve model wors P l mn c A b, all = l { 0,} = P Conversel, suppose,, s sasf he model some = P A l l l s are recesson drecons for P 0 for oher ls Slde 49

50 Wh he dsuncve model wors P l l = + l mn c A b, all = = { 0,} Slde 50 P Conversel, suppose,, s sasf he model some = P A l l l s are recesson drecons for P l l A 0 A = A + b l 0 for oher ls

51 Wh he dsuncve model wors P l l = + l mn c A b, all = = { 0,} Slde 5 P Conversel, suppose,, s sasf he model some = P A l l l s are recesson drecons for P l l A 0 A = A + b l P S 0 for oher ls

52 Mulple dsuncons Combnng ndvdual conve hull formulaons for wo dsuncons ( A a ) ( B b ) does no necessarl produce a conve hull formulaon for he par Theorem. unless he dsuncons have no common varables. Slde 5

53 Eample: Facl locaon Capac m possble facor locaons n mares Locae facores o serve mares so as o mnmze oal fed cos and ranspor cos. C D Demand Fed cos f c Transpor cos Slde 53

54 Fed cos Facl locaon m possble facor locaons C f Slde 54 c Transpor cos n mares D Dsuncve model: mn C = 0, all z f, all z 0 = 0, all = D, all z + c Facor a locaon Amoun shpped from facor o mare No facor a locaon

55 Facl locaon Slde 55 Dsuncve model: MILP formulaon: mn C = 0, all z f, all z 0 = 0, all = D, all mn C, all = D, all {0,}, 0, all, z + c f + c

56 Uncapacaed facl locaon Begnner s msae: Model as specal case of capacaed problem mn n, all { 0,} f + c =, all Facor has ma oupu n Fracon of demand sasfed b facor Ths s no he bes model. We can oban a gher model b sarng wh dsuncve formulaon. Slde 56

57 Uncapacaed facl locaon m possble facor locaons n mares Dsuncve model: Fracon of demand sasfed b facor mn z + c Fed cos f c Transpor cos 0, all = 0, all, all z f z 0 = =, all Facor a locaon No facor a locaon Slde 57

58 Uncapacaed facl locaon Slde 58 MILP formulaon: Begnner s model: mn 0, all, mn f + c =, all {0,}, all n, all =, all {0,}, all f + c Ths s he eboo model. More consrans, bu gher relaaon.

59 Eample: Lo szng wh seup coss Ma producon level Seup cos ncurred = Demand = D 0 D D D 3 D 4 D 5 D 6 Deermne lo sze n each perod o mnmze oal producon, nvenor, and seup coss. Slde 59

60 Fed-cos varable Fed cos Producon capac Producon level v f v 0 v 0 0 C 0 C = 0 () 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 60

61 () Sar producon () Connue producon (3) Produce nohng v f v 0 v 0 0 C 0 C = 0 Conve hull MILP model of dsuncon: 0 v f C 0 v 0 C v 3 3 = =, =, = = = = v v {0,}, =,,3 Slde 6

62 To smplf, defne z = = Conve hull MILP model of dsuncon: 0 v f C 0 v 0 C v 3 3 = =, =, = = = = v v {0,}, =,,3 Slde 6

63 To smplf, defne z = = Conve hull MILP model of dsuncon: 0 v f z C z v 0 0 C v 3 3 = =, =, z + = = v v z, {0,}, =,,3 Slde 63 = for sarup = for connued producon

64 Snce = 0 se 3 = + Conve hull MILP model of dsuncon: 0 v f z Cz 0 v 0 C v 3 3 = = = v = v, =, z + z, {0,}, =,,3 Slde 64

65 Snce = 0 se 3 = + Conve hull MILP model of dsuncon: 0 C ( z + ) v f z v 0 v = v = v, z + z, {0,}, =,,3 Slde 65

66 Snce v occurs posvel n he obecve funcon, 3 and v, v do no pla a role, le v = v Conve hull MILP model of dsuncon: v f z 0 C ( z + ) v 0 v = v = v, z + z, {0,}, =,,3 Slde 66

67 Snce v occurs posvel n he obecve funcon, 3 and v, v do no pla a role, le v = v Conve hull MILP model of dsuncon: v f z 0 C ( z + ) + z, {0,}, =,,3 z Slde 67

68 Formulae logcal condons: () In perod () or () n perod () In perod neher () nor () n perod v f z 0 C ( z + ) z z, {0,}, =,,3 z + z + z Slde 68

69 Add obecve funcon Un producon cos Un holdng cos n mn ( p + h s + v ) = v f z 0 C ( z + ) + z, {0,}, =,,3 z z + z z Slde 69

70 Logcal varables To ghen an MILP formulaon of A B C D E F G ( ) A B E Pu logcal consran n CNF: A B E Replace negave wh posve varables: And add conve hull formulaon of hs clause. C D E Conecure: hs does no ghen he formulaon when he dsuncons have no varables n common. Slde 70

71 Bg-M Dsuncve Formulaon Agan sar wh a dsuncon of lnear ssems. ( A b ) Bg M s when s n polhedron. M s a vecor of bounds ha maes ssem nonbndng when = 0. { } ( ) A b M, all = 0,, all M = b mn A A b l l ( l ) Slde 7

72 Bg-M Dsuncve Formulaon Agan sar wh a dsuncon of lnear ssems. ( A b ) Bg M s when s n polhedron. M s a vecor of bounds ha maes ssem nonbndng when = 0. { } ( ) A b M, all = 0,, all M = b mn A A b l l ( l ) Ever bounded MILP-represenable se has a model of hs form (as well as a conve hull dsuncve model). Slde 7

73 Eample: Healh Care Benefs Dsrbue lmed healh benefs o wo persons. Person receves ul u. u Two crera: If u u, Rawlsan: ma mn{u,u } If u u >, ularan: ma u + u u Mamze welfare of person who s more serousl ll, unless hs requres oo much sacrfce from he oher person. Slde 73

74 Eample: Healh Care Benefs Dsrbue lmed healh benefs o wo persons. Person receves ul u. u Two crera: If u u, Rawlsan: ma mn{u,u } If u u >, ularan: ma u + u Opmzaon problem: Slde 74 S u ma z { } mn u, u + f u u z u + u oherwse u, u S

75 Eample: Healh Care Benefs Dsrbue lmed healh benefs o wo persons. Person receves ul u. u Two crera: If u u, Rawlsan: ma mn{u,u } If u u >, ularan: ma u + u Opmzaon problem: Slde 75 S u ma z { } mn u, u + f u u z u + u oherwse u, u S Ensures connu

76 Eample: Healh Care Benefs Dsrbue lmed healh benefs o wo persons. Person receves ul u. u Ignorng S, we would le a conve hull MILP model of he epgraph. Can we do? No! Opmzaon problem: Slde 76 u ma z { } mn u, u + f u u z u + u oherwse u, u S

77 Eample: Healh Care Benefs Dsrbue lmed healh benefs o wo persons. Person receves ul u. u Epgraph s unon of wo polhedra: P has recesson cone {( α, β, z) z α + β, α, β 0} u Slde 77

78 Eample: Healh Care Benefs Dsrbue lmed healh benefs o wo persons. Person receves ul u. u Epgraph s unon of wo polhedra: P has recesson cone {( α, β, z) z α + β, α, β 0} P has recesson cone {(,, z) 0 z } { (,0,0),(0,,0) } Slde 78

79 Eample: Healh Care Benefs Dsrbue lmed healh benefs o wo persons. Person receves ul u. u Soluon: Add consran u u M M No need o bound u, u ndvduall M u Slde 79

80 Eample: Healh Care Benefs Dsrbue lmed healh benefs o wo persons. Person receves ul u. u Soluon: Add consran u u M M No need o bound u, u ndvduall P has recesson cone (,, z) 0 z { } M u Slde 80

81 Eample: Healh Care Benefs Dsrbue lmed healh benefs o wo persons. Person receves ul u. u Soluon: Add consran u u M M No need o bound u, u ndvduall P has recesson cone (,, z) 0 z { } So does P M u Slde 8

82 Eample: Healh Care Benefs Dsrbue lmed healh benefs o wo persons. Person receves ul u. M u Bg-M model: z u + + ( M ) z u + + ( M ) z u + u + ( ) u u M, u u M { } u, u 0, 0, M u Slde 8

83 Eample: Healh Care Benefs Dsrbue lmed healh benefs o wo persons. Person receves ul u. M u Bg-M model: z u + + ( M ) z u + + ( M ) z u + u + ( ) u u M, u u M { } u, u 0, 0, Theorem: Ths s a conve hull formulaon. M u Slde 83

84 Eample: Healh Care Benefs Dsrbue lmed healh benefs o wo persons. Person receves ul u. M u Bg-M model: z u + + ( M ) z u + + ( M ) z u + u + ( ) u u M, u u M { } u, u 0, 0, Theorem: Ths s a conve hull formulaon. M u Model s no gher f we use u, u M Slde 84

85 Eample: Healh Care Benefs Opmzaon problem for he n-person case: ma z { } n { } { } z ( n ) + nmn u + ma 0, u mn u u u M, all, u 0, u S = Slde 85

86 Eample: Healh Care Benefs Opmzaon problem for he n-person case: ma z { } n { } { } z ( n ) + nmn u + ma 0, u mn u u u M, all, u 0, u S = Bg-M dsuncve model: Slde 86 ma z z + w, all { } = w + u + ( M ), all, w u + ( ), all, = 0, all u u M, all, n u S 0,, all,

87 Eample: Healh Care Benefs Opmzaon problem for he n-person case: ma z { } n { } { } z ( n ) + nmn u + ma 0, u mn u u u M, all, u 0, u S = Bg-M dsuncve model: Slde 87 ma z z + w, all { } = w + u + ( M ), all, w u + ( ), all, = 0, all u u M, all, n u S 0,, all, Theorem: Ths s a conve hull formulaon.

88 Slde 88 General MILP Represenabl

89 Knapsac Models Ineger varables can also be used o epress counng deas. Ths s oall dfferen from he use of 0- varables o epress unons of polhedra. Eamples: Knapsac nequales Pacng and coverng Logcal clauses Cos bounds Slde 89

90 Knapsac Models Dsuncve represenabl does no accommodae napsac consrans n a naural wa. Knapsac consrans are bounded MILP represenable onl f neger varables are bounded. and onl n a echncal sense. B regardng each neger lace pon as a polhedron. Slde 90

91 General represenabl heorem Ineger varables can now be unbounded: n p R Z A subse S of s MILP represenable f S s he proecon ono of he feasble se of some MILP consran se of he form A + Bz + D b R Z, z R {0,} n p m q Some modelng varables are connuous, some neger Aular connuous varables can be used Slde 9

92 General represenabl heorem Ineger varables can be unbounded: n p R Z A subse S of s MILP represenable f S s he proecon ono of he feasble se of some MILP consran se of he form A + Bz + D b R Z, z R {0,} n p m q Assume ha A, B, D, b conss of raonal daa Slde 9

93 General represenabl heorem Ineger varables can be unbounded: n p R Z A subse S of s MILP represenable f S s he proecon ono of he feasble se of some MILP consran se of he form A + Bz + D b R Z, z R {0,} n p m q Assume ha A, B, D, b conss of raonal daa A med neger polhedron s an se of he form Slde 93 { n p R Z A b}

94 General represenabl heorem Raonal vecor d s a recesson drecon of a med n neger polhedron P R Z f s a recesson drecon of n+ p some polhedron Q R for whch P ( n p R Z ) = Q p Med neger polhedron P Slde 94

95 General represenabl heorem Raonal vecor d s a recesson drecon of a med n neger polhedron P R Z f s a recesson drecon of n+ p some polhedron Q R for whch P ( n p R Z ) = Q p Med neger polhedron P Polhedron Q Slde 95

96 General represenabl heorem Raonal vecor d s a recesson drecon of a med n neger polhedron P R Z f s a recesson drecon of n+ p some polhedron Q R for whch P ( n p R Z ) = Q p Recesson cone of Q = recesson cone of P Med neger polhedron P Polhedron Q Slde 96

97 General represenabl heorem n+ p Lemma. All polhedra nr havng he same nonemp n p nersecon whr Z have he same recesson cone. Recesson cone of Q = recesson cone of P Med neger polhedron P Polhedron Q Slde 97

98 General represenabl heorem n p Theorem. A nonemp subse of R Z s MILP represenable f and onl f s he unon of fnel man med neger polhedra n p n R Z havng he same recesson cone. Recesson cone of Q = recesson cone of P Med neger polhedron P Unon of med neger polhedra wh he same recesson cone (n hs case, he orgn) Polhedron Q Slde 98

99 Conve Hull Formulaon Sar wh a dsuncon of lnear ssems o represen he unon of med neger polhedra. The h polhedron s { R n Z p A b } ( A b ) Asde from doman of, he dsuncve model s he same as before. A b, all = = { } n p R Z, 0, Slde 99

100 Conve Hull Formulaon Sar wh a dsuncon of lnear ssems o represen he unon of med neger polhedra. The h polhedron s { R n Z p A b } ( A b ) Asde from doman of, he dsuncve model s he same as before. Ever MILP represenable se has a model of hs form. A b, all = = { } n p R Z, 0, Slde 00

101 Conve Hull Formulaon Sar wh a dsuncon of lnear ssems o represen he unon of med neger polhedra. The h polhedron s { R n Z p A b } ( A b ) Asde from doman of, he dsuncve model s he same as before. Ever MILP represenable se has a model of hs form. also a model n dsuncve bg-m form. A b, all = = { } n p R Z, 0, Slde 0

102 Conve Hull Formulaon Theorem. If each med neger polhedron has a conve hull formulaon A b, he dsuncve model s a conve hull formulaon of he dsuncon. Unon of med neger polhedra wh conve hull descrpons Conve hull relaaon Slde 0

103 Eample: Facl locaon m possble facor locaons n mares Locae facores o serve mares so as o mnmze oal facor cos and ranspor cos. C D Fed cos ncurred for each vehcle used. f Fed cos Slde 03 c K Transpor cos per vehcle

104 f Fed cos Facl locaon m possble facor locaons C Slde 04 c K n mares Transpor cos per vehcle Dsuncve model: mn C 0, all 0, all Kw =, all z z 0 f = w, all Z = D, all z + c w Facor a locaon Number of vehcles from facor o mare No facor a locaon

105 f Fed cos Facl locaon m possble facor locaons C Slde 05 c K n mares Transpor cos per vehcle Dsuncve model: mn C 0, all 0, all Kw =, all z z 0 f = w, all Z = D, all z + c w Descrbes med neger polhedron Facor a locaon Number of vehcles from facor o mare No facor a locaon

106 Facl locaon Slde 06 Dsuncve model: MILP formulaon: mn C 0, all 0, all Kw =, all z z 0 f = w, all Z = D, all mn C, all = D, all 0 K w, all, {0,}, w Z, all, z + c w f + c w

107 Wh a Sngle Recesson Cone Suppose S s represened b A + Bz + D b R Z, z R {0,} n p m q For each bnar, hs descrbes a med neger polhedron P( ). So S s a unon of med neger polhedra. Slde 07

108 Wh a sngle recesson cone Suppose S s represened b A + Bz + D b R Z, z R {0,} n p m q For each bnar, hs descrbes a med neger polhedron P( ). So S s a unon of med neger polhedra. Now s a recesson drecon of nonemp P( ) ff (,u, ) s a recesson drecon of A B D 0 n p m q u + : 0 0 u 0 R Z R 0 0 Slde 08

109 Wh a sngle recesson cone Suppose S s represened b A + Bz + D b R Z, z R {0,} n p m q For each bnar, hs descrbes a med neger polhedron P(). So S s a unon of med neger polhedra. Now s a recesson drecon of nonemp P() ff (,u, ) s a recesson drecon of A B D 0 n p m q u + : 0 0 u 0 R Z R 0 0 Tha s, ff A B D u Slde 09

110 Wh a sngle recesson cone Suppose S s represened b A + Bz + D b R Z, z R {0,} n p m q For each bnar, hs descrbes a med neger polhedron P(). So S s a unon of med neger polhedra. Now s a recesson drecon of nonemp P() ff (,u, ) s a recesson drecon of A B D 0 n p m q u + : 0 0 u 0 R Z R 0 0 Tha s, ff A B D u Bu hs s ndependen of. Slde 0

111 Eample: Fregh Pacng and Transfer Transpor pacages usng n rucs Each pacage has sze a. Each ruc has capac Q. Slde

112 Knapsac componen The rucs seleced mus have enough capac o carr he load. n = Q a = f ruc s seleced Slde

113 Dsuncve componen Cos varable Truc seleced Truc no seleced z c z 0 a Q = 0 { 0, }, all Cos of operang ruc = f pacage s loaded on ruc Slde 3

114 Dsuncve componen Cos varable Truc seleced Truc no seleced z c z 0 a Q = 0 { 0, }, all Descrbes med neger polhedron Cos of operang ruc = f pacage s loaded on ruc Slde 4

115 Dsuncve componen Truc seleced Truc no seleced z c z 0 a Q = 0 { 0, }, all 0 Conve hull MILP formulaon z a c Q Slde 5

116 The resulng model = = mn n = a Q, all 0, all, n n Q =, all, {0,} c a Dsuncve componen Logcal condon (each pacage mus be shpped) Knapsac componen Slde 6

117 The resulng model = = mn n = a Q, all 0, all, n n Q =, all, {0,} c a The s redundan bu maes he connuous relaaon gher. Ths s a modelng rc, par of he follore of modelng. Slde 7

118 The resulng model = = mn n = a Q, all 0, all, n n Q =, all, {0,} c a The s redundan bu maes he connuous relaaon gher. Ths s a modelng rc, par of he follore of modelng. Convenonal modelng wsdom would no use hs consran, because s he sum of he frs consran over. Bu radcall reduces soluon me, because generaes lfed napsac cus. Slde 8

119 Research ssues Can he smplfcaon of a conve hull MILP formulaon be auomaed? Wha are some condons under whch a bg-m dsuncve model s a conve hull formulaon? When does conve hull formulaon of logcal consrans ghen he model? How can a modelng ssem faclae and encourage prncpled modelng? Slde 9