Modeling with Metaconstraints and Semantic Typing of Variables

Transcription

1 Modelng wth Metaconstrants and Semantc Typng of Varables André Cré Unversty of Toronto John Hooker Carnege Mellon Unversty Tallys Yunes Unversty of Mam INFORMS Journal on Computng (2016) 1-13

2 Basc Problem Metaconstrants (n partcular, global constrants) n a model help convey problem structure to the solver. But they pose a fundamental problem of varable management. How to solve t? Slde 2

3 Basc Problem Metaconstrants (n partcular, global constrants) n a model help convey problem structure to the solver. But they pose a fundamental problem of varable management. How to solve t? Treat varable declaratons as database queres. In a system of semantc typng. Slde 3

4 Explotng Problem Structure You can t solve hard problems wthout explotng specal structure (No Free Lunch Theorem). Slde 4

5 Explotng Problem Structure You can t solve hard problems wthout explotng specal structure (No Free Lunch Theorem). For SAT solvers: Effcent encodng of problem n SAT form Slde 5

6 Explotng Problem Structure You can t solve hard problems wthout explotng specal structure (No Free Lunch Theorem). For SAT solvers: Effcent encodng of problem n SAT form For CP solvers: Careful choce of global constrants Redundant constrants, search strategy, etc. Slde 6

7 Explotng Problem Structure You can t solve hard problems wthout explotng specal structure (No Free Lunch Theorem). For SAT solvers: Effcent encodng of problem n SAT form For CP solvers: Careful choce of global constrants Redundant constrants, search strategy, etc. For MIP solvers: Careful choce of varables for tght formulaton Addton of vald nequaltes Slde 7

8 Conveyng structure to the solver(s) Formulate problem wth global constrants or metaconstrants to reveal structure Automatcally convert these to optmal formulaton for the solvers(s) Best choce of varables. Reformulaton of constrants. For effectve propagaton or tght relaxaton Best choce of doman flters. Generaton of vald nequaltes Slde 8

9 Conveyng structure to the solver(s) Formulate problem wth global constrants or metaconstrants to reveal structure Automatcally convert these to optmal formulaton for the solvers(s) Best choce of varables. Reformulaton of constrants. For effectve propagaton or tght relaxaton Best choce of doman flters. Generaton of vald nequaltes However, metaconstrants pose a fundamental problem of varable management Slde 9

10 Varable management problem Reformulaton typcally ntroduces new varables Dfferent metaconstrants may ntroduce varables that are functonally the same varable or related n some other way. Recognzng these relatonshps s essental to obtanng a good model (e.g., a tght contnuous relaxaton) How can the solver understand what s gong on n the model? Slde 10

11 Varable management problem Reformulaton typcally ntroduces new varables Dfferent metaconstrants may ntroduce varables that are functonally the same varable or related n some other way. Recognzng these relatonshps s essental to obtanng a good model (e.g., a tght contnuous relaxaton) How can the solver understand what s gong on n the model? Proposal: Model wth semantc typng of varables. Slde 11

12 Varable management problem Example: Let x j = worker assgned to job j c j = cost of assgnng worker to job j Fnd mn-cost assgnment: mn j c alldff ( x,, x ) 1 jx j n Slde 12

13 Varable management problem Example: Let x j = worker assgned to job j c j = cost of assgnng worker to job j Fnd mn-cost assgnment: mn Objectve functon s reformulated wth 0-1 varables: j c alldff ( x,, x ) 1 jx j n where x j yj mn cy j j j Slde 13

14 Varable management problem Example: Let x j = worker assgned to job j c j = cost of assgnng worker to job j Fnd mn-cost assgnment: mn Objectve functon s reformulated wth 0-1 varables: j c alldff ( x,, x ) 1 jx j n where x j yj mn cy j j j Alldff constrant s reformulated wth 0-1 varables: y 1, all j; y 1, all j j j Slde 14

15 Varable management problem Slde 15 Example: Let x j = worker assgned to job j c j = cost of assgnng worker to job j Fnd mn-cost assgnment: mn Objectve functon s reformulated wth 0-1 varables: Alldff constrant s reformulated wth 0-1 varables: j c alldff ( x,, x ) 1 jx j n where x j yj mn cy j j j y 1, all j; y 1, all How does the solver know that we want y j = y j, allowng the problem to be solved rapdly as a classcal assgnment problem? j j j

16 Semantc typng We assume that varables are declared. Slde 16

17 Semantc typng We assume that varables are declared. Semantc typng assgns a dfferent meanng to each varable By assocatng the varable wth a mult-place predcate and keyword. The keyword queres the relaton denoted by the predcate, as one queres a relatonal database. Slde 17

18 Semantc typng We assume that varables are declared. Semantc typng assgns a dfferent meanng to each varable By assocatng the varable wth a mult-place predcate and keyword. The keyword queres the relaton denoted by the predcate, as one queres a relatonal database. Advantage: Ths allows the solver to deduce relatonshps between varables, both orgnal or ntroduced. Can automatcally add channelng constrants. Slde 18 It s also good modelng practce.

19 How varables are ntroduced A model may nclude two formulatons of the problem that use related varables. Common n CP, because t strengthens propagaton. Slde 19

20 How varables are ntroduced A model may nclude two formulatons of the problem that use related varables. Common n CP, because t strengthens propagaton. For example, x job assgned to worker y j worker assgned to job j Solver should generate channelng constrants to relate the varables to each other: j x, y y j x Slde 20

21 How varables are ntroduced The solver may reformulate a dsjuncton of lnear systems k A x k b k usng a convex hull (or bg-m ) formulaton: k k A x b y, all k k k x x, y 1 y k k k k 0,1, all k k Other constrants may be based on same set of alternatves, and correspondng auxlary varables (y k etc.) should be equated. Slde 21

22 How varables are ntroduced A nonlnear or global solver may use McCormck factorzaton to replace nonlnear subexpressons wth auxlary varables to obtan a lnear relaxaton. Slde 22

23 How varables are ntroduced A nonlnear or global solver may use McCormck factorzaton to replace nonlnear subexpressons wth auxlary varables to obtan a lnear relaxaton. For example, blnear term xy can be lnearzed by replacng t wth new varable z and constrants L x L y L L z L x U y L U y x x y y x x y U x U y U U z U x L y U L y x x y y x x y where x Lx, Ux, y Ly, U y Factorzaton of dfferent constrants may create varables for dentcal subexpressons. They should be dentfed to get a tght relaxaton. Slde 23

24 How varables are ntroduced The solver may reformulate dfferent global constrants from CP by ntroducng varables that have the same meanng. Slde 24

25 How varables are ntroduced The solver may reformulate dfferent global constrants from CP by ntroducng varables that have the same meanng. For example, sequence constrant lmts how many jobs of a gven type can occur n gven tme nterval: and cardnalty constrant lmts how many tmes a gven job appears cardnalty x, x job n poston j Both may ntroduce varables that should be dentfed. sequence x, x job n poston j y 1 when job j occurs n poston j Slde 25

26 How varables are ntroduced The solver may ntroduce equvalent varables whle nterpretng metaconstrants desgned for classcal MIP modelng stuatons: Fxed-charge network flow Faclty locaton Lot szng Job shop schedulng Assgnment (3-dm, quadratc, etc.) Pecewse lnear Slde 26

27 Motvatng example Allocate 10 advertsng spots to 5 products x how many spots allocated to product yj 1 f j spots allocated to product A B C D E Slde 27

28 Motvatng example Allocate 10 advertsng spots to 5 products 4 spots per product x how many spots allocated to product yj 1 f j spots allocated to product A B C D E Slde 28

29 Motvatng example Allocate 10 advertsng spots to 5 products 4 spots per product Advertse 3 products x how many spots allocated to product yj 1 f j spots allocated to product A B C D E Slde 29

30 Motvatng example Allocate 10 advertsng spots to 5 products x how many spots allocated to product yj 1 f j spots allocated to product 4 spots per product Advertse 3 products 4 spots for at least one product A B C D E Slde 30

31 Motvatng example Allocate 10 advertsng spots to 5 products 4 spots per product Advertse 3 products x how many spots allocated to product yj 1 f j spots allocated to product A B C D E 4 spots for at least one product P j = proft from allocatng j spots to product Objectve: maxmze proft Slde 31

32 Motvatng example spots n {0..4} product n {A,B,C,D,E}) Index sets Slde 32

33 Motvatng example spots n {0..4} product n {A,B,C,D,E} data P{product,spots} Data nput Slde 33

34 Motvatng example spots n {0..4} product n {A,B,C,D,E} data P{product,spots} Declaraton of varable x x[] s howmany spots allocate(product ) Slde 34

35 Motvatng example spots n {0..4} product n {A,B,C,D,E} data P{product,spots} Declaraton of varable x x[] s howmany spots allocate(product ) Ths makes t a varable declaraton Slde 35

36 Motvatng example spots n {0..4} product n {A,B,C,D,E} data P{product,spots} Declaraton of varable x x[] s howmany spots allocate(product ) Ths s the semantc type Slde 36

37 Motvatng example spots n {0..4} product n {A,B,C,D,E} data P{product,spots} Declaraton of varable x x[] s howmany spots allocate(product ) Slde 37 Indcates an nteger quantty Other keywords: howmuch whether

38 Motvatng example spots n {0..4} product n {A,B,C,D,E} data P{product,spots} Declaraton of varable x x[] s howmany spots allocate(product ) How many of what? Slde 38

39 Motvatng example spots n {0..4} product n {A,B,C,D,E} data P{product,spots} Declaraton of varable x x[] s howmany spots allocate(product ) 2-place predcate assocated wth varable x Slde 39 Every varable s assocated wth a predcate that gves t meanng

40 Motvatng example spots n {0..4} product n {A,B,C,D,E} data P{product,spots} Declaraton of varable x x[] s howmany spots allocate(product ) Other term of the predcate Slde 40

41 Motvatng example spots n {0..4} product n {A,B,C,D,E} data P{product,spots} Declaraton of varable x x[] s howmany spots allocate(product ) Assocates ndex of x[] wth ndex set product Slde 41

42 Motvatng example max x 10, y 2, y y 1, x jy, all spots n {0..4} j j product n {A,B,C,D,E} j j data P{product,spots} x[] s howmany spots allocate(product ) maxmze sum{product } P[,x[]] Objectve functon P x Slde 42

43 Motvatng example max x 10, y 2, y y 1, x jy, all spots n {0..4} product n {A,B,C,D,E} data P{product,spots} j j x[] s howmany spots allocate(product ) maxmze sum{product } P[,x[]] sum{product } x[] <= spots avalable P x j j Slde 43

44 Motvatng example max x 10, y 2, y y 1, x jy, all spots n {0..4} j j product n {A,B,C,D,E} j j data P{product,spots} x[] s howmany spots allocate(product ) maxmze sum{product } P[I,x[]] sum{product } x[] <= 10 y[,j] s whether allocate(product, spots j) Declare y j P x Indcates 0-1 varable Slde 44

45 Motvatng example max x 10, y 2, y y 1, x jy, all spots n {0..4} j j product n {A,B,C,D,E} j j data P{product,spots} x[] s howmany spots allocate(product ) maxmze sum{product } P[,x[]] sum{product } x[] <= 10 y[,j] s whether allocate(product, spots j) Declare y j P x Assocated wth same predcate as x[] Slde 45

46 Motvatng example spots n {0..4} product n {A,B,C,D,E} data P{product,spots} max x 10, y 2, y y 1, x jy, all x[] s howmany spots allocate(product ) maxmze sum{product } P[,x[]] sum{product } x[] <= 10 y[,j] s whether allocate(product, spots j) sum{product } y[,0] >= 2 At least 2 products not advertsed j P x j j j Slde 46

47 Motvatng example spots n {0..4} product n {A,B,C,D,E} data P{product,spots} x[] s howmany spots allocate(product ) maxmze sum{product } P[,x[]] sum{product } x[] <= 10 y[,j] s whether allocate(product, spots j) sum{product } y[,0] >= 2 sum{product } y[,4] >= 1 max x x 10, y 2, y j P y 1, x jy, all j j j At least 1 product gets 4 spots Slde 47

48 Motvatng example max x 10, y 2, y y 1, x jy, all spots n {0..4} j j product n {A,B,C,D,E} j j data P{product,spots} x[] s howmany spots allocate(product ) maxmze sum{product } P[,x[]] sum{product } x[] <= 10 y[,j] s whether allocate(product, spots j) sum{product } y[,0] >= 2 sum{product } y[,4] >= 1 {product } sum{spots j} y[,j] = 1 {product } x[] = sum{spots j} j*y[,j] Solver generates lnkng constrants because x[] and y[,j] are assocated wth the same predcate. P x Slde 48

49 Motvatng example max spots n {0..4} product n {A,B,C,D,E} data P{product,spots} j j x[] s howmany spots allocate (product ) maxmze sum{product } P[,x[]] Ths constrant must be lnearzed. Solver generates x 10, y 2, y 1 y 1, x jy, all y [,j] s whether allocate(product, spots j) z j j z P y, y 1, x jy, all j j j j j 0 j 0 j 0 Slde 49

50 Motvatng example max spots n {0..4} product n {A,B,C,D,E} data P{product,spots} j j x[] s howmany spots allocate (product ) maxmze sum{product } P[,x[]] Ths constrant must be lnearzed. Solver generates x 10, y 2, y 1 y 1, x jy, all y [,j] s whether allocate(product, spots j) y and y are dentfed because they have the same type: y[,j] s whether allocate(product, spots j) z j j z P y, y 1, x jy, all j j j j j 0 j 0 j 0 Slde 50

51 Predcates and relatons Predcate allocate denotes 2-place relaton (set of tuples). Schematcally ndcated by: 1 2 product spots x Slde 51

52 Predcates and relatons Predcate allocate denotes 2-place relaton (set of tuples). Schematcally ndcated by: 1 2 product spots x Column correspondng to a varable must be a functon of other columns. Slde 52

53 Predcates and relatons Predcate allocate denotes 2-place relaton (set of tuples). Schematcally ndcated by: 1 2 product spots x Declaraton of x[] as howmany spots allocate (product ) and y[,j] as whether allocate (product, spots j) query the relaton for how many and whether. Slde 53

54 Predcates and relatons Predcate allocate denotes 2-place relaton (set of tuples). Schematcally ndcated by: 1 2 product spots x Declaraton of x[] as howmany spots allocate (product ) and y[,j] as whether allocate (product, spots j) query the relaton for how many and whether. In general, keywords are queres (analogous to relatonal database) Slde 54

55 Predcates and relatons Relaton table reveals channelng constrants. For example, x[] s whch job assgn(worker ) y[j] s whch worker assgn(job ) 1 2 job worker j, x, y j Slde 55 We can read off the channelng constrants j x x y y j y x

56 Predcates and relatons If several jobs can be assgned to a worker, we declare z[] s whchset job assgn(worker ) The channelng constrants are j z y Slde 56

57 Prevous work Model management uses semantc typng to help combne models and use nhertance. Orgnally nspred by object-orented programmng Bradley & Clemence (1988) Quddty: a rgorous attempt to analyze condtons for varable dentfcaton Bhargava, Kmbrough & Krshnan (1991) SML uses typng n a structured modelng framework Geoffron (1992) Ascend uses strongly-typed, object-orented modelng Bhargava, Krshnan & Pela (1998) Slde 57

58 Prevous work Our semantc typng dffers: Less ambtous because t doesn t attempt model management. There s only one model. More ambtous because we recognze relatonshps other than equvalence. We manage varables ntroduced by solver. Slde 58

59 Prevous work Modelng systems that convey some structure to solver: CP modelng systems use global constrants. AIMMS uses typed ndex sets. MnZnc reformulates metaconstrants for specfc solvers. Savle Row uses common subexpresson elmnaton. OPL, Xpress-Kals, Comet, etc., use nterval varables. SAT solver SymChaff uses hgh-level AI plannng language PDDL. SIMPL has full metaconstrant capablty. Slde 59

60 Prevous work However, none of these systems deals systematcally wth the varable management problem. We address t wth semantc typng of varables. Slde 60

61 Assgnment problem mn worker n {1..m} job n {1..n} data C{worker,job} x[] s whch job assgn(worker ) mnmze sum{worker } C[,x[]] alldff{x[*]} alldff x,, c x 1 x n Slde 61

62 Assgnment problem mn worker n {1..m} job n {1..n} data C{worker,job} x[] s whch job assgn(worker ) mnmze sum{worker } C[,x[]] alldff{x[*]} alldff x,, c x 1 x n Objectve functon s formulated max c y, x y, all j j j j y[,j] s whether assgn(worker, job j) Slde 62

63 Assgnment problem mn worker n {1..m} job n {1..n} data C{worker,job} x[] s whch job assgn(worker ) mnmze sum{worker } C[,x[]] alldff{x[*]} alldff x,, c x 1 x n Objectve functon s formulated max c y, x y, all j j j j y[,j] s whether assgn(worker, job j) Alldff s formulated Slde 63 y 1, all, y 1, all j, x jy, all j j j j j y [,j] s whether assgn(worker, job j) Solver dentfes y and y to create classcal AP.

64 Latn squares j Numbers n every row and column are dstnct. We wll use three formulatons to mprove propagaton. Slde 64

65 Latn squares j Numbers n every row and column are dstnct. We wll use three formulatons to mprove propagaton. alldff x, x, all 1 alldff x, x, all j 1 j 1 1k n nj alldff y, y, all alldff n y y, all k j1 1k nk alldff z, x, all j alldff z, x, all k jn nk row, col, num n {1..n} x[,j] s whch num assgn(row, col j) y[,k] s whch col assgn(row, num k) z[j,k] s whch row assgn(col j, num k) Slde 65

66 Latn squares j Numbers n every row and column are dstnct. We wll use three formulatons to mprove propagaton. alldff x, x, all 1 alldff x, x, all j 1 j 1 1k n nj alldff y, y, all alldff n y y, all k j1 1k nk alldff z, x, all j alldff z, x, all k jn nk row, col, num n {1..n} x[,j] s whch num assgn(row, col j) y[,k] s whch col assgn(row, num k) z[j,k] s whch row assgn(col j, num k) {row } alldff{x[,*]); {col j} alldff{x[*,j]) {row } alldff{y[,*]); {num k} alldff{y[*,j]) {col j} alldff{z[j,*]); {num k} alldff{z[*,k]) Slde 66

67 Latn squares The predcate assgn denotes the 3-place relaton num col row k, x j j, y k, z jk row, col, num n {1..n} x[,j] s whch num assgn(row, col j) y[,k] s whch col assgn(row, num k) z[j,k] s whch row assgn(col j, num k) {row } alldff{x[,*]); {col j} alldff{x[*,j]) {row } alldff{y[,*]); {num k} alldff{y[*,j]) {col j} alldff{z[j,*]); {num k} alldff{z[*,k]) Slde 67

68 Latn squares The predcate assgn denotes the 3-place relaton num col row k, x j j, y k, z jk We can read off the channelng constrants: k x, j y, z, all, j, k z y z x y x jk k jk j k kj whch can be propagated. Slde 68

69 Latn squares {row } alldff{x[,*]); {col j} alldff{x[*,j]) {row } alldff{y[,*]); {num k} alldff{y[*,j]) {col j} alldff{z[j,*]); {num k} alldff{z[*,k]) The 3 formulatons generate 3 dentcal MIP models: x x x x x k ; 1, all, j; 1, all, k; 1, all j, k j jk jk jk jk k k j y y y y y j, 1, all, k; 1, all, j; 1, all j, k k jk jk jk jk j j k z z z z z jk jk, jk 1, all j, k; jk 1 j jk, all, ; 1, all, k j k Slde 69

70 Latn squares {row } alldff{x[,*]); {col j} alldff{x[*,j]) {row } alldff{y[,*]); {num k} alldff{y[*,j]) {col j} alldff{z[j,*]); {num k} alldff{z[*,k]) The 3 formulatons generate 3 dentcal MIP models: x x x x x k ; 1, all, j; 1, all, k; 1, all j, k y y y y y j, 1, all, k; 1, all, j; 1, all j, k z z z z z jk jk, jk 1, all j, k; jk 1 j jk Slde 70 j jk jk jk jk k k j k jk jk jk jk j j k The solver declares, all, ; 1, all, x y z,, jk jk jk k whether assgn(row, col j, num k) So t treats them as the same varable and generates only 1 MIP model. j k

71 Multple whch varables In general, an n-place predcate that denotes the relaton 1 k k + 1 n term 1 term k term k+1 term n 1 1, x (1) x k 1, k k ( k ) n for whch varables, where j ( ) 1 j1 j1 n generates the channelng constrants j x, all,,, j 1,, k j 1 j1 j1 k (1) ( j1) ( j1) ( k ) k 1 n x x x x 1 n Slde 71

72 Multple whether varables whether keywords serve as projecton operators on the relaton. y[,j,d] s whether assgn(worker, job j, day d) Project out d : y1[,j] s whether assgn(worker, job j) Project out j and d : y2[] s whether assgn(worker ) Slde 72

73 Short forms Declare x to be cost of actvty : x[] s howmuch cost(actvty ) whch s short for the formal declaraton x[] s howmuch cost cost(actvty ) n whch a new term cost s generated Declare x to be cost: x s howmuch cost whch s short for x s howmuch cost cost() Slde 73

74 Pecewse lnear Pecewse lnear functon z = f(x) Breakponts n A, ordnates n C f(x) x s howmuch output ndex n {1..n} data A,C{ndex} A z s howmuch cost pecewse(x,z,a,c) ths metaconstrant defnes z = f(x) C x Slde 74

75 Pecewse lnear Pecewse lnear functon z = f(x) Breakponts n A, ordnates n C x s howmuch output ndex n {1..n} data A,C{ndex} z s howmuch cost pecewse(x,z,a,c) Solver generates the locally deal model n1 n1 c 1 c x a1 x, z c1 x a a C f(x) ( a a ) x ( a a ), 0,1, 1,, n 1 A x We need to declare auxlary varables, x Slde 75

76 Pecewse lnear Pecewse lnear functon z = f(x) Breakponts n A, ordnates n C x s howmuch output ndex n {1..n} data A,C{ndex} z s howmuch cost pecewse(x,z,a,c) pecewse constrant nduces solver to declare a new ndex set that assocates ndex wth A, and use t to declare, x xbar[] s howmuch output.a(ndex ) delta[] s whether lastpostve output.a(ndex ) C f(x) A x Both declaratons create predcates nherted from output and A Slde 76

77 Pecewse lnear f(x) Suppose there s another pecewse functon on the same break ponts f(x) x s howmuch output ndex n {1..n} x data A,C{ndex} A z s howmuch cost pecewse(x,z,a,c) data C {ndex} z s howmuch proft pecewse(x,z,a,c ) x [] s howmuch cost output.a(ndex ) delta [] s whether lastpostve output.a(ndex) Slde 77

78 Pecewse lnear f(x) Suppose there s another pecewse functon on the same break ponts x s howmuch output ndex n {1..n} data A,C{ndex} z s howmuch cost pecewse(x,z,a,c) data C {ndex} z s howmuch proft pecewse(x,z,a,c ) x [] s howmuch cost output.a(ndex ) delta [] s whether lastpostve output.a(ndex) A f(x) Because new pecewse constrant s assocated wth the same x and A, solver agan creates output.a. x The solver creates varables and x wth same types as and x and so dentfes them. Slde 78

79 Interval varables cumulatve x, D, R, L Each job j runs for a tme nterval x j. We wsh to schedule jobs so that total resource consumpton never exceeds L. job n {1..n} tme n {t..t} data W,D,R{job} wndow, duraton, resource runnng n [tme,tme] makes runnng an nterval varable x[j] s when runnng sched(job j) subset W[j] cumulatve(x,d,r,l) L x W, all j j j Slde 79

80 Interval varables Each job j runs for a tme nterval x j. We wsh to schedule jobs so that total resource consumpton never exceeds L. job n {1..n} tme n {t..t} data W,D,R{job} wndow, duraton, resource Solver generates the model delta[j,t] s whether runnng.start sched(job j, tme t) ph[j,t] s whether runnng sched(job j, tme t) Slde 80 t 1, all j; R L, all t jt j jt j, all t, t wth 0 t t D, all j jt jt j cumulatve x, D, R, L x W, all j runnng n [tme,tme] makes runnng an nterval varable x[j] s when runnng sched(job j) subset W[j] cumulatve(x,d,r,l) j j

81 Interval varables Suppose we want fnsh tmes to be separated by at least T 0 cumulatve x, D, R, L job n {1..n} tme n {t..t} data W,D,R{job} runnng n [tme,tme] x[j] s when runnng sched(job j) subset W[j] cumulatve(x,d,r,l) {job j, job k j<>k} x[j].end x[k].end >= T0 end j end k x W, all j j j x x T, all j, k, j k delta[j,t] s whether runnng.start sched(job j, tme t) ph[j,t] s whether runnng sched(job j, tme t) 0 Slde 81

82 Interval varables Suppose we want fnsh tmes j j to be separated by at least T 0 end end x j xk T0, all j, k, j k job n {1..n} tme n {t..t} data W,D,R{job} runnng n [tme,tme] x[j] s when runnng sched(job j) subset W[j] cumulatve(x,d,r,l) {job j, job k j<>k} x[j].end x[k].end >= T0 delta[j,t] s whether runnng.start sched(job j, tme t) ph[j,t] s whether runnng sched(job j, tme t) Solver generates cumulatve x, D, R, L jt kt 1, all t, t wth 0 t t T0, all j, t wth j k epslon[j,t] s whether runnng.end sched(job j, tme t) x W, all j Slde 82

83 Interval varables Varables jt and jt are related by an offset. Solver assocates runnng.end n declaraton of jt wth runnng.start n declaraton of jt and deduces e, jt, all j, t j t D j delta[j,t] s whether runnng.start sched(job j, tme t) ph[j,t] s whether runnng sched(job j, tme t) epslon[j,t] s whether runnng.end sched(job j, tme t) Slde 83

84 Interval varables Varables jt and jt are related by an offset. Solver assocates runnng.end n declaraton of jt wth runnng.start n declaraton of jt and deduces e, jt, all j, t j t D j Solver also assocates runnng.end n declaraton of jt wth runnng n declaraton of jt and deduces the redundant constrants, all t, twth 0 t t D, all j jt jt j delta[j,t] s whether runnng.start sched(job j, tme t) ph[j,t] s whether runnng sched(job j, tme t) epslon[j,t] s whether runnng.end sched(job j, tme t) Slde 84

85 TSP wth Sde Constrants Travelng salesman problem wth mssng arcs and precedence constrants. mn alldff x, crcut s x x, all, j wth prec 1 Succ cty, poston n {1..n} data D{cty, cty} Dstances data Prec{cty, cty} Prec[,j]=1 f must precede j data Succ{cty} Succ[j] = set of successors of cty j s D s j j Slde 85

86 TSP wth Sde Constrants Travelng salesman problem wth mssng arcs and precedence constrants. mn alldff x, crcut cty, poston n {1..n} data D{cty, cty} Dstances data Prec{cty, cty} Prec[,j]=1 f must precede j data Succ{cty} Succ[j] = set of successors of cty j Two varable systems: x[] s whch poston orderng(cty ) s[] s successor cty orderng(cty ) subset Succ[] D s x x, all, j wth prec 1 s Succ s j j Slde 86

87 TSP wth Sde Constrants Travelng salesman problem wth mssng arcs and precedence constrants. mn alldff x, crcut cty, poston n {1..n} data D{cty, cty} Dstances data Prec{cty, cty} Prec[,j]=1 f must precede j data Succ{cty} Succ[j] = set of successors of cty j Two varable systems: x[] s whch poston orderng(cty ) s[] s successor cty orderng(cty ) subset Succ[] s x x, all, j wth prec 1 Succ Precedence constrants requre x varables prec{cty, cty j Prec[,j] = 1}: x[] < x[j] Mssng arc constrants (mplct n data Succ) requre s varables s D s j j Slde 87

88 TSP wth Sde Constrants Travelng salesman problem wth mssng arcs and precedence constrants. cty, poston n {1..n} data D{cty, cty} Dstances data Prec{cty, cty} Prec[,j]=1 f must precede j data Succ{cty} Succ[j] = set of successors of cty j Two varable systems: x[] s whch poston orderng(cty ) s[] s successor cty orderng(cty ) subset Succ[] Precedence constrants requre x varables prec{cty, cty j Prec[,j] = 1}: x[] < x[j] Mssng arc constrants (mplct n data Succ) requre s varables mn sum {cty } D[,s[]] Objectve functon Slde 88 mn alldff D x, crcut s x x, all, j wth prec 1 s Succ s j j

89 TSP wth Sde Constrants The solver can gve alldff(x) a conventonal assgnment model usng z k = whether cty s n poston k. z[,k] s whether orderng(cty, poston k) Slde 89

90 TSP wth Sde Constrants The solver can gve alldff(x) a conventonal assgnment model usng z k = whether cty s n poston k. z[,k] s whether orderng(cty, poston k) For crcut(s), the solver can ntroduce w j = whether cty mmedately precedes cty j. w[,j] s whether successor orderng(cty, cty j) Slde 90

91 TSP wth Sde Constrants The solver can gve alldff(x) a conventonal assgnment model usng z k = whether cty s n poston k. z[,k] s whether orderng(cty, poston k) For crcut(s), the solver can ntroduce w j = whether cty mmedately precedes cty j. w[,j] s whether successor orderng(cty, cty j) Declaraton of z tells solver that predcate s orderng(cty,poston), not orderng(cty,cty). Slde 91

92 TSP wth Sde Constrants The solver can gve alldff(x) a conventonal assgnment model usng z k = whether cty s n poston k. z[,k] s whether orderng(cty, poston k) For crcut(s), the solver can ntroduce w j = whether cty mmedately precedes cty j. w[,j] s whether successor orderng(cty, cty j) Declaraton of z tells solver that predcate s orderng(cty,poston), not orderng(cty,cty). Solver generates cuttng planes n w-space and s-space. Slde 92

93 TSP wth Sde Constrants The solver can gve alldff(x) a conventonal assgnment model usng z k = whether cty s n poston k. z[,k] s whether orderng(cty, poston k) For crcut(s), the solver can ntroduce w j = whether cty mmedately precedes cty j. w[,j] s whether successor orderng(cty, cty j) Declaraton of z tells solver that predcate s orderng(cty,poston), not orderng(cty,cty). Solver generates cuttng planes n w-space and s-space. The successor keyword tells solver how z and w relate. Slde 93, all t, twth 0 t t D, all j jt jt j

94 TSP wth Sde Constrants Suppose we also have constrants on whch cty s n poston k. Smply declare y[k] = whch cty orderng(poston k) The solver generates the channelng constrants between y[k] and x[] = whch poston s cty Slde 94

95 TSP wth Sde Constrants Suppose we also have constrants on whch cty s n poston k. Smply declare y[k] = whch cty orderng(poston k) The solver generates the channelng constrants between y[k] and x[] = whch poston s cty The solver can also ntroduce a second (equvalent) objectve functon mn sum{poston k} D[y[k],y[k+1]] whch may mprove boundng. MIP Slde 95

96 Pros and Cons of Semantc Typng Pros Conveys problem structure to the solver(s) by allowng use of metaconstants Incorporates state of the art n formulaton, vald nequaltes Allows solver to expand repertory of technques Doman flterng, propagaton, cuttng plane algorthms Good modelng practce Self-documentng Bug detecton Slde 96

97 Pros and Cons of Semantc Typng Cons Modeler must be famlar wth a large collecton of metaconstrants Rather than few prmtve constrants Slde 97

98 Pros and Cons of Semantc Typng Cons Modeler must be famlar wth a large collecton of metaconstrants Rather than few prmtve constrants Response Modeler must be famlar wth the underlyng concepts anyway Modelng system can offer sophstcated help, mprove modelng Slde 98

99 Pros and Cons of Semantc Typng Cons Modeler must be famlar wth a large collecton of metaconstrants Rather than few prmtve constrants Response Modeler must be famlar wth the underlyng concepts anyway Modelng system can offer sophstcated help, mprove modelng OR, SAT communty s not accustomed to hgh-level modelng Typed languages lke Ascend never really caught on, resstance to CP. Slde 99

100 Pros and Cons of Semantc Typng Cons Modeler must be famlar wth a large collecton of metaconstrants Rather than few prmtve constrants Slde 100 Response Modeler must be famlar wth the underlyng concepts anyway Modelng system can offer sophstcated help, mprove modelng OR, SAT communty s not accustomed to hgh-level modelng Typed languages lke Ascend never really caught on, resstance to CP. Response Tran the next generaton!