A Principled Approach to. MILP Modeling. Columbia University, August Carnegie Mellon University. Workshop on MIP. John Hooker.

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1 Slide A Principled Approach o MILP Modeling John Hooer Carnegie Mellon Universiy Worshop on MIP Columbia Universiy, Augus 008

2 Proposal MILP modeling is an ar, bu i need no be unprincipled. Slide

3 Proposal MILP modeling is an ar, bu i need no be unprincipled. I has wo basic componens: Disjuncive modeling of subses of coninuous space. Knapsac modeling of couning ideas. Slide 3

4 Proposal MILP modeling is an ar, bu i need no be unprincipled. I has wo basic componens: Disjuncive modeling of subses of coninuous space. Knapsac modeling of couning ideas. MILPs can model subses of coninuous space ha are unions of polyhedra. ha is, represened by disjuncions of linear sysems. Slide 4

5 Proposal MILP modeling is an ar, bu i need no be unprincipled. I has wo basic componens: Disjuncive modeling of subses of coninuous space. Knapsac modeling of couning ideas. MILPs can model subses of coninuous space ha are unions of polyhedra. ha is, represened by disjuncions of linear sysems. So a principled approach is o analyze he problem as Slide 5 disjuncions ineger of linear + napsac sysems inequaliies

6 Proposal Jeroslow s Represenabiliy Theorem provides heoreical basis for disjuncive modeling. Bounded MILP represenabiliy assumes bounded ineger variables. This is inadequae for napsac modeling. Slide 6

7 Proposal Jeroslow s Represenabiliy Theorem provides heoreical basis for disjuncive modeling. Bounded MILP represenabiliy assumes bounded ineger variables. This is inadequae for napsac modeling. We will generalize Jeroslow s heorem. Knapsac modeling accommodaed. Ineger variables can be unbounded. Slide 7

8 Slide 8 Ouline Bounded mied ineger represenabiliy Bounded represenabiliy heorem. Conve hull formulaion Eample: Fied charge problem Why he disjuncive model wors Muliple disjuncions Eample: Faciliy locaion Eample: Lo sizing wih seup coss Big-M disjuncive formulaion Eample: Healh care benefis

9 Slide 9 Ouline General mied ineger represenabiliy Knapsac models General represenabiliy heorem. Conve hull formulaion Eample: Faciliy locaion Why a single recession cone Eample: Freigh pacing and ransfer Research issues

10 Slide 0 Bounded MILP Represenabiliy

11 Bounded represenabiliy heorem Definiion of R. Jeroslow: n A subse S of R is bounded MILP represenable if S is he projecion ono of he feasible se of some MILP consrain se of he form A + Bz + Dy b y R, z R n m {0,} p Bounded general ineger variables can be encoded as 0- variables Auiliary coninuous variables can be used Slide

12 Bounded represenabiliy heorem Theorem (Jeroslow). A subse of coninuous space is bounded MILP represenable if and only if i is he union of finiely many polyhedra having he same recession cone. Recession cone of polyhedron Polyhedron Union of polyhedra wih he same recession cone (in his case, he origin) Slide

13 Conve hull formulaion Sar wih a disjuncion of linear sysems o represen he union of polyhedra. The h polyhedron is { A b} Inroduce a 0- variable y ha is when is in polyhedron. Disaggregae o creae an for each. Slide 3 ( A b ) A b y, all y = = y { 0, }

14 Conve hull formulaion Sar wih a disjuncion of linear sysems o represen he union of polyhedra. The h polyhedron is { A b} Inroduce a 0- variable y ha is when is in polyhedron. Disaggregae o creae an for each. Every bounded MILP represenable se has a model of his form. Slide 4 ( A b ) A b y, all y = = y { 0, }

15 Conve Hull Formulaion The coninuous relaaion of his disjuncive MILP provides a conve hull relaaion of he disjuncion. Sricly, i describes he closure of he conve hull. Union of polyhedra Conve hull relaaion (ighes linear relaaion) Slide 5

16 Idea behind he conve hull formulaion Sar by formulaing a conve hull formulaion of he relaaion of he disjuncion Wrie each soluion as a conve combinaion of poins in he polyhedron A b, all y = y y = [0,] Conve hull relaaion Slide 6

17 Idea behind he conve hull formulaion Now apply a change of variable Wrie each soluion as a conve combinaion of poins in he polyhedron A b, all y = y y = [0,] Change of variable = y A b y, all y = y = [0,] Conve hull relaaion Slide 7

18 Idea behind he conve hull formulaion Now mae y s 0- variables o ge an MILP represenaion A b, all y = y y = { 0, } Mae y s 0- A b y, all y = y = [0,] Conve hull formulaion Slide 8

19 Idea behind he conve hull formulaion When is his a valid formulaion? Le s loo a an eample firs A b, all y = = y y { 0, } Conve hull formulaion Slide 9

20 Eample: Fied charge funcion Minimize a fied charge funcion: min 0 if = 0 f + c if > 0 0 Slide 0

21 Fied charge problem Minimize a fied charge funcion: Feasible se (epigraph) Slide min 0 if = 0 f + c if > 0 0

22 Fied charge problem Minimize a fied charge funcion: Union of wo polyhedra P, P P Slide min 0 if = 0 f + c if > 0 0

23 Fied charge problem Minimize a fied charge funcion: Union of wo polyhedra P, P P P Slide 3 min 0 if = 0 f + c if > 0 0

24 Fied charge problem Minimize a fied charge funcion: The polyhedra have differen recession cones. P P Slide 4 min 0 if = 0 f + c if > 0 0 P recession cone P recession cone

25 Fied charge problem Disjuncive model describes conve hull relaaion bu no he feasible se. P P Slide 5 min = f + c

26 Fied charge problem Sar wih a disjuncion of linear sysems o represen he union of polyhedra Inroduce a 0- variable y ha is when is in polyhedron. Disaggregae o creae an for each. Slide 6 min = f + c min = c + fy y + y =, y [0,] = +, = +

27 Slide 7 To simplify, replace wih since 0 = min = c + fy y + y =, y [0,] = +, = +

28 Slide 8 To simplify, replace wih since 0 = min c fy y + y = y = +, [0,]

29 Slide 9 Replace wih because plays no role in he model min 0 0 c + fy y + y = y = +, [0,]

30 Slide 30 Replace wih because plays no role in he model min 0 c + fy y + y =, y [0,]

31 Slide 3 Replace y wih y because y plays no role in he model min 0 c + fy y + y =, y [0,]

32 Slide 3 Replace y wih y because y plays no role in he model min y [0,] 0 c + fy

33 Slide 33 min y [0,] 0 c + fy P The conve hull is his. P

34 Slide 34 min y [0,] 0 c + fy P Relaaion correcly describes closure of conve hull P

35 Slide 35 min y {0,} 0 c + fy P Bu MILP model does no describe feasible se P

36 To fi he problem Add an upper bound on The polyhedra have he same recession cone. P P Slide 36 min 0 f + c if > 0 0 if = 0 M M P recession cone P recession cone

37 Fied charge problem min The disjuncion is now = 0 0 M 0 f + c P P Slide 37 M

38 Fied charge problem The disjuncive model is Slide 38 min = 0 0 M 0 f + c min = 0 0 My 0 c + fy { } y + y =, y 0, = +, = +

39 Slide 39 This simplifies as before min 0 M c + y { 0, } y fy

40 Slide 40 This simplifies as before min 0 My c + fy y { 0, } Previous model min 0 c + fy y { 0, }

41 Slide 4 This simplifies as before min 0 My c + fy y { 0, } or 0 min c + fy My y { 0, } Previous model min 0 c + fy y { 0, } Big M

42 Slide 4 The model now correcly min c + fy describes he feasible se. y { 0, } 0 My P P M Big M

43 Why he disjuncive model wors P l Recession cone of polyhedra min A b y, all y c = y = { 0, } Le S be feasible se. P Slide 43

44 Why he disjuncive model wors P l min c A b y, all y = = y { 0, } P Le S be feasible se. S some P Slide 44

45 Why he disjuncive model wors P l min c A b y, all y = = y { 0, } P Le S be feasible se. S some P saisfies he model for y =, oher y s = 0 l = l =, oher s 0 Slide 45

46 Why he disjuncive model wors P l min c A b y, all y = = y { 0, } P Conversely, suppose, y, s saisfy he model some y = P Slide 46

47 Why he disjuncive model wors P l min c A b y, all y = = y { 0, } P Conversely, suppose, y, s saisfy he model some y = P l l A 0 for oher l s Slide 47

48 Why he disjuncive model wors P l l min c A b y, all y = = y { 0, } P Conversely, suppose, y, s saisfy he model some y = P l l A 0 for oher l s s are recession direcions for oher s P l l Slide 48

49 Why he disjuncive model wors P l l min c A b y, all y = = y { 0, } P Conversely, suppose, y, s saisfy he model some y = P l l A 0 for oher l s s are recession direcions for P l Slide 49

50 Why he disjuncive model wors P l l = + l min c A b y, all y = = y { 0, } Slide 50 P Conversely, suppose, y, s saisfy he model some y = P A l l 0 for oher l s l s are recession direcions for P A l 0 A = A + l b l

51 Why he disjuncive model wors P l l = + l min c A b y, all y = = y { 0, } Slide 5 P Conversely, suppose, y, s saisfy he model some y = P A l l 0 for oher l s l s are recession direcions for P A l 0 A = A + l b l P S

52 Muliple disjuncions Combining individual conve hull formulaions for wo disjuncions ( A a ) ( B b ) does no necessarily produce a conve hull formulaion for he pair Theorem. unless he disjuncions have no common variables. Slide 5

53 Eample: Faciliy locaion Capaciy m possible facory locaions n mares Locae facories o serve mares so as o minimize oal fied cos and ranspor cos. C j f i c ij i j Fied cos Transpor cos D j Demand Slide 53

54 Faciliy locaion m possible facory locaions n mares C j D j f i c ij i j Fied cos Transpor cos Slide 54 Amoun shipped from facory i o mare j Disjuncive model: min i ij ij i ij ij C i j ij = 0, all j z i f i, all i z i 0 = ij 0, all j = D, all j i ij j z + c Facory a locaion i No facory a locaion i

55 Faciliy locaion Disjuncive model: MILP formulaion: Slide 55 min i ij ij i ij ij C i j ij = 0, all j z i f i, all i z i 0 = ij 0, all j = D, all j i ij j z + c min j i i i ij ij i ij C y, all i ij i i = D, all j ij j y {0,}, 0, all i, j i ij f y + c

56 Uncapaciaed faciliy locaion Beginner s misae: Model i as special case of capaciaed problem min y j i i i i ij ij i ij ny, all i ij i { 0, } =, all j ij f y + c Facory i has ma oupu n Fracion of demand j saisfied by facory i This is no he bes model. We can obain a igher model by saring wih disjuncive formulaion. Slide 56

57 Uncapaciaed faciliy locaion m possible facory locaions n mares Disjuncive model: Fracion of demand j saisfied by facory i f i c ij i j Fied cos Transpor cos min i ij ij i ij 0 ij, all j ij = 0, all j z i f i z i 0 = =, all j i ij z + c Facory a locaion i No facory a locaion i, all i Slide 57

58 Uncapaciaed faciliy locaion MILP formulaion: min i i ij ij i ij 0 y, all i, j i ij i =, all j ij y {0,}, all i i f y + c Slide 58 Beginner s model: min j i i i ij ij i ij ny, all i ij i =, all j ij y {0,}, all i i f y + c This is he eboo model. More consrains, bu igher relaaion.

59 Eample: Lo sizing wih seup coss Seup cos incurred Ma producion level = Demand = D 0 D D D 3 D 4 D 5 D 6 Deermine lo size in each period o minimize oal producion, invenory, and seup coss. Slide 59

60 Slide 60 Fied-cos variable Fied cos Producion capaciy Producion level v f v 0 v 0 0 C 0 C = 0 () Sar producion (incurs seup cos) () Coninue producion (no seup cos) (3) Produce nohing (no producion cos) Logical condiions: () In period () or () in period () In period neiher () nor () in period

61 () Sar producion () Coninue producion (3) Produce nohing v f v 0 v 0 0 C 0 C = 0 Conve hull MILP model of disjuncion: 0 v f y C y 0 v 0 v C y 3 3 = =, =, = = = = v v y y y {0,}, =,,3 Slide 6

62 To simplify, define z = y y = y Conve hull MILP model of disjuncion: 0 v f y C y 0 v 0 v C y 3 3 = =, =, = = = = v v y y y {0,}, =,,3 Slide 6 0 0

63 To simplify, define z = y y = y Conve hull MILP model of disjuncion: 0 v f z C z v 0 v 0 C y 3 3 = = = v v,, = = z, y {0,}, =, z + y,3 Slide 63 = for sarup = for coninued producion

64 Since se 3 = 0 = + Conve hull MILP model of disjuncion: 0 v f z C z 0 v 0 v C y 3 3 = 3 3 = = v = v, =, z + y z, y {0,}, =,,3 Slide

65 Since se 3 = 0 = + Conve hull MILP model of disjuncion: 0 C ( z + y ) v f z v 0 3 = v = v, z + y z, y {0,}, =,,3 Slide 65 3 v 0

66 Since v occurs posiively in he objecive funcion, v v 3 and, do no play a role, le v = v Conve hull MILP model of disjuncion: v f z 0 C ( z + y ) v 0 3 v 0 3 = v = v, z + y z, y {0,}, =,,3 Slide 66

67 Since v occurs posiively in he objecive funcion, v v 3 and, do no play a role, le v = v Conve hull MILP model of disjuncion: v f z 0 C ( z + y ) z, y {0,}, =,,3 z + y Slide 67

68 Formulae logical condiions: () In period () or () in period () In period neiher () nor () in period z v f z 0 C ( z + y ), y {0,}, =,,3 z z z + y y z + y y Slide 68

69 Add objecive funcion Uni producion cos Uni holding cos m n in ( p + h s + v ) = v f z 0 C ( z + y ) z, y {0,}, =,,3 z + y y z + y z z y Slide 69

70 Logical variables To ighen an MILP formulaion of A B C D E F G ( y y ) y A B E Pu logical consrain in CNF: y A y B y E Replace negaive wih posiive variables: And add conve hull formulaion of his clause. C D E Conjecure: his does no ighen he formulaion when he disjuncions have no variables in common. Slide 70

71 Big-M Disjuncive Formulaion Again sar wih a disjuncion of linear sysems. ( A b ) Big M y is when is in polyhedron. M is a vecor of bounds ha maes sysem nonbinding when y = 0. { } ( ) A b y M, all y 0,, all y = ( l l ) l M = b min A A b Slide 7

72 Big-M Disjuncive Formulaion Again sar wih a disjuncion of linear sysems. ( A b ) Big M y is when is in polyhedron. M is a vecor of bounds ha maes sysem nonbinding when y = 0. { } ( ) A b y M, all y 0,, all y = ( l l ) l M = b min A A b Every bounded MILP-represenable se has a model of his form (as well as a conve hull disjuncive model). Slide 7

73 Eample: Healh Care Benefis Disribue limied healh benefis o wo persons. Person i receives uiliy u i. u Two crieria: If u u, Rawlsian: ma min{u,u } If u u >, uiliarian: ma u + u u Maimize welfare of person who is more seriously ill, unless his requires oo much sacrifice from he oher person. Slide 73

74 Eample: Healh Care Benefis Disribue limied healh benefis o wo persons. Person i receives uiliy u i. u Two crieria: If u u, Rawlsian: ma min{u,u } If u u >, uiliarian: ma u + u Opimizaion problem: Slide 74 S u ma z z { } u + u oherwise min u, u + if u u u, u S

75 Eample: Healh Care Benefis Disribue limied healh benefis o wo persons. Person i receives uiliy u i. u Two crieria: If u u, Rawlsian: ma min{u,u } If u u >, uiliarian: ma u + u Opimizaion problem: Slide 75 S u ma z z { } u + u oherwise min u, u + if u u u, u S Ensures coninuiy

76 Eample: Healh Care Benefis Disribue limied healh benefis o wo persons. Person i receives uiliy u i. u Ignoring S, we would lie a conve hull MILP model of he epigraph. Can we do i? No! Opimizaion problem: Slide 76 u ma z z { } u + u oherwise min u, u + if u u u, u S

77 Eample: Healh Care Benefis Disribue limied healh benefis o wo persons. Person i receives uiliy u i. u Epigraph is union of wo polyhedra: P has recession cone { ( α, β, z ) z α + β, α, β 0 } u Slide 77

78 Eample: Healh Care Benefis Disribue limied healh benefis o wo persons. Person i receives uiliy u i. u Epigraph is union of wo polyhedra: P has recession cone { ( α, β, z ) z α + β, α, β 0 } P has recession cone { (,, z ) 0 z } { (,0,0),(0,,0) } Slide 78

79 Eample: Healh Care Benefis Disribue limied healh benefis o wo persons. Person i receives uiliy u i. u Soluion: Add consrain u u M M No need o bound u, u individually M u Slide 79

80 Eample: Healh Care Benefis Disribue limied healh benefis o wo persons. Person i receives uiliy u i. u Soluion: Add consrain u u M M No need o bound u, u individually P has recession cone (,, z ) 0 z { } M u Slide 80

81 Eample: Healh Care Benefis Disribue limied healh benefis o wo persons. Person i receives uiliy u i. u Soluion: Add consrain u u M M No need o bound u, u individually P has recession cone (,, z ) 0 z { } So does P M u Slide 8

82 Eample: Healh Care Benefis M Disribue limied healh benefis o wo persons. Person i receives uiliy u i. u Big-M model: z u + + ( M ) y z u + + ( M ) y z u + u + ( y ) u u M, u u M { } u, u 0, y 0, M u Slide 8

83 Eample: Healh Care Benefis M Disribue limied healh benefis o wo persons. Person i receives uiliy u i. u Big-M model: z u + + ( M ) y z u + + ( M ) y z u + u + ( y ) u u M, u u M { } u, u 0, y 0, Theorem: This is a conve hull formulaion. M u Slide 83

84 Eample: Healh Care Benefis M Disribue limied healh benefis o wo persons. Person i receives uiliy u i. u Big-M model: z u + + ( M ) y z u + + ( M ) y z u + u + ( y ) u u M, u u M { } u, u 0, y 0, Theorem: This is a conve hull formulaion. M u Model is no igher if we use u, u M Slide 84

85 Eample: Healh Care Benefis Opimizaion problem for he n-person case: ma z { } n { } { } z ( n ) + n min u + ma 0, u min u u u M, all i, j i j u 0, u S j j j j j j = Slide 85

86 Eample: Healh Care Benefis Opimizaion problem for he n-person case: ma z { } n { } { } z ( n ) + n min u + ma 0, u min u u u M, all i, j i j u 0, u S j j j j j j = Big-M disjuncive model: Slide 86 ma z + w, all i { } w + u + y ( M ), all i, j w u + ( y ), all i, j y = 0, all i ii u u M, all i, j y 0,, all i, j j n ij i ij ij j ij i j u S ij z = ij

87 Eample: Healh Care Benefis Opimizaion problem for he n-person case: ma z { } n { } { } z ( n ) + n min u + ma 0, u min u u u M, all i, j i j u 0, u S j j j j j j = Big-M disjuncive model: Slide 87 ma z + w, all i { } w + u + y ( M ), all i, j w u + ( y ), all i, j y = 0, all i ii u u M, all i, j y 0,, all i, j j n ij i ij ij j ij i j u S ij z = ij Theorem: This is a conve hull formulaion.

88 Slide 88 General MILP Represenabiliy

89 Knapsac Models Ineger variables can also be used o epress couning ideas. This is oally differen from he use of 0- variables o epress unions of polyhedra. Eamples: Knapsac inequaliies Pacing and covering Logical clauses Cos bounds Slide 89

90 Knapsac Models Disjuncive represenabiliy does no accommodae napsac consrains in a naural way. Knapsac consrains are bounded MILP represenable only if ineger variables are bounded. and only in a echnical sense. By regarding each ineger laice poin as a polyhedron. Slide 90

91 General represenabiliy heorem Ineger variables can now be unbounded: n p A subse S of R Z is MILP represenable if S is he projecion ono of he feasible se of some MILP consrain se of he form A + Bz + Dy b y R Z, z R n p m {0,} q Some modeling variables are coninuous, some ineger Auiliary coninuous variables can be used Slide 9

92 General represenabiliy heorem Ineger variables can be unbounded: n p A subse S of R Z is MILP represenable if S is he projecion ono of he feasible se of some MILP consrain se of he form A + Bz + Dy b y R Z, z R n p m {0,} q Assume ha A, B, D, b consis of raional daa Slide 9

93 General represenabiliy heorem Ineger variables can be unbounded: n p A subse S of R Z is MILP represenable if S is he projecion ono of he feasible se of some MILP consrain se of he form A + Bz + Dy b y R Z, z R n p m {0,} q Assume ha A, B, D, b consis of raional daa A mied ineger polyhedron is any se of he form Slide 93 { n p R Z A b }

94 General represenabiliy heorem Raional vecor d is a recession direcion of a mied n p ineger polyhedron P R Z if i is a recession direcion of some polyhedron for which Q R ( n p ) P = Q R Z n + p Slide 94 Mied ineger polyhedron P

95 General represenabiliy heorem Raional vecor d is a recession direcion of a mied n p ineger polyhedron P R Z if i is a recession direcion of some polyhedron for which Q ( n p ) P = Q R Z R n + p Slide 95 Mied ineger polyhedron P Polyhedron Q

96 General represenabiliy heorem Raional vecor d is a recession direcion of a mied n p ineger polyhedron P R Z if i is a recession direcion of some polyhedron for which Q R ( n p ) P = Q R Z n + p Slide 96 Recession cone of Q = recession cone of P Mied ineger polyhedron P Polyhedron Q

97 General represenabiliy heorem Lemma. All polyhedra in R having he same nonempy n p inersecion wih R Z n + p have he same recession cone. Slide 97 Recession cone of Q = recession cone of P Mied ineger polyhedron P Polyhedron Q

98 General represenabiliy heorem n p Theorem. A nonempy subse of R Z is MILP represenable if and only if i is he union of finiely many mied ineger polyhedra n p in R Z having he same recession cone. Recession cone of Q = recession cone of P Mied ineger polyhedron P Union of mied ineger polyhedra wih he same recession cone (in his case, he origin) Polyhedron Q Slide 98

99 Conve Hull Formulaion Sar wih a disjuncion of linear sysems o represen he union of mied ineger polyhedra. The h polyhedron is { R n Z p A b } ( A b ) Aside from domain of, he disjuncive model is he same as before. A b y, all y = = { } n p R Z, y 0, Slide 99

100 Conve Hull Formulaion Sar wih a disjuncion of linear sysems o represen he union of mied ineger polyhedra. The h polyhedron is { R n Z p A b } ( A b ) Aside from domain of, he disjuncive model is he same as before. Every MILP represenable se has a model of his form. A b y, all y = = { } n p R Z, y 0, Slide 00

101 Conve Hull Formulaion Sar wih a disjuncion of linear sysems o represen he union of mied ineger polyhedra. The h polyhedron is { R n Z p A b } ( A b ) Aside from domain of, he disjuncive model is he same as before. Every MILP represenable se has a model of his form. also a model in disjuncive big-m form. A b y, all y = = { } n p R Z, y 0, Slide 0

102 Conve Hull Formulaion Theorem. If each mied ineger polyhedron has a conve hull formulaion A b, he disjuncive model is a conve hull formulaion of he disjuncion. Union of mied ineger polyhedra wih conve hull descripions Conve hull relaaion Slide 0

103 Eample: Faciliy locaion m possible facory locaions n mares C j D j Fied cos Slide 03 f i i j c ij K ij Transpor cos per vehicle Locae facories o serve mares so as o minimize oal facory cos and ranspor cos. Fied cos incurred for each vehicle used.

104 Faciliy locaion m possible facory locaions n mares C j f i Fied cos Slide 04 i j c ij K ij Transpor cos per vehicle Number of vehicles from facory i o mare j Disjuncive model: min i ij ij i ij ij C i j 0, all ij 0, all ij K ij w j ij j =, all i z z i 0 i f = w ij, all j Z = D, all j i ij j z + c w No facory a locaion i Facory a locaion i

105 f i Fied cos Faciliy locaion m possible facory locaions n mares C j Slide 05 i j Disjuncive model: min i ij ij i ij ij C i j 0, all ij 0, all ij K ij w j ij j =, all i z z i 0 i f = w ij, all j Z = D, all j i ij j c ij Transpor Describes K ij cos per vehicle mied ineger polyhedron z + c w Facory a locaion i Number of vehicles from facory i o mare j No facory a locaion i

106 Faciliy locaion Disjuncive model: MILP formulaion: Slide 06 min i ij ij i ij ij C i j 0, all ij 0, all ij K ij w j ij j =, all i z z i 0 i f = w ij, all j Z = D, all j i ij j z + c w min j i i i ij ij i ij C y, all i ij i i = D, all j ij j 0 K w, all i, j ij ij ij y {0,}, w Z, all i, j i ij f y + c w

107 Why a Single Recession Cone Suppose S is represened by A + Bz + Dy b y R Z, z R n p m {0,} q For each binary y, his describes a mied ineger polyhedron P ( y ). So S is a union of mied ineger polyhedra. Slide 07

108 Why a single recession cone Suppose S is A + Bz + Dy b represened by R Z, z R y n p m {0,} q For each binary y, his describes a mied ineger polyhedron P ( y ). So S is a union of mied ineger polyhedra. Now is a recession direcion of nonempy P ( y ) iff (,u,y ) is a recession direcion of y A B D 0 n p m q u + : 0 0 u 0 R Z R y 0 0 y y Slide 08

109 Why a single recession cone Suppose S is represened by A + Bz + Dy b y R Z, z R n p m {0,} q For each binary y, his describes a mied ineger polyhedron P(y). So S is a union of mied ineger polyhedra. Now is a recession direcion of nonempy P(y) iff (,u,y ) is a recession direcion of A B D 0 n p m q u + : 0 0 u 0 R Z R y 0 0 y y Tha is, iff A B D u y 0 Slide 09

110 Why a single recession cone Suppose S is represened by A + Bz + Dy b y R Z, z R n p m {0,} q For each binary y, his describes a mied ineger polyhedron P(y). So S is a union of mied ineger polyhedra. Now is a recession direcion of nonempy P(y) iff (,u,y ) is a recession direcion of A B D 0 n p m q u + : 0 0 u 0 R Z R y 0 0 y y Tha is, iff A B D u y 0 Bu his is independen of y. Slide 0

111 Eample: Freigh Pacing and Transfer Transpor pacages using n rucs Each pacage j has size a j. Each ruc i has capaciy Q i. Slide

112 Knapsac componen The rucs seleced mus have enough capaciy o carry he load. n Q y a i i j i = j = if ruc i is seleced Slide

113 Disjuncive componen Cos variable Truc i seleced Truc i no seleced z i c i z i 0 a j ij Q i j ij = 0 ij { 0, }, all j Cos of operaing ruc i = if pacage j is loaded on ruc i Slide 3

114 Disjuncive componen Cos variable Truc i seleced Truc i no seleced z i c i z i 0 a j ij Q i j ij = 0 ij { 0, }, all j Describes mied ineger polyhedron Cos of operaing ruc i = if pacage j is loaded on ruc i Slide 4

115 Disjuncive componen Truc i seleced Truc i no seleced z i c i z i 0 a j ij Q i j ij = 0 ij { 0, }, all j Slide 5 Conve hull MILP formulaion j 0 z c y i i i a Q y j ij i i y ij i

116 The resuling model j min i n i i a Q y, all i j ij i i 0 y, all i, j i n = n ij i =, all j ij i i j i = j, y {0,} ij i = c y Q y a Slide 6 Disjuncive componen Logical condiion (each pacage mus be shipped) Knapsac componen

117 The resuling model j min i n i i a Q y, all i j ij i i 0 y, all i, j i n = n ij i =, all j ij i i j i = j, y {0,} ij i = c y Q y a Slide 7 The y i is redundan bu maes he coninuous relaaion igher. This is a modeling ric, par of he follore of modeling.

118 The resuling model j min i n i i a Q y, all i j ij i i 0 y, all i, j i n = n ij i =, all j ij i i j i = j, y {0,} ij i = c y Q y a Slide 8 The y i is redundan bu maes he coninuous relaaion igher. This is a modeling ric, par of he follore of modeling. Convenional modeling wisdom would no use his consrain, because i is he sum of he firs consrain over i. Bu i radically reduces soluion ime, because i generaes lifed napsac cus.

119 Research issues Can he simplificaion of a conve hull MILP formulaion be auomaed? Wha are some condiions under which a big-m disjuncive model is a conve hull formulaion? When does conve hull formulaion of logical consrains ighen he model? How can a modeling sysem faciliae and encourage principled modeling? Slide 9