An Introduction to Constraint Based Scheduling

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1 Lesson 1: An Inroducion o Consrain Based Scheduling Michele Lombardi <michele.lombardi2@unibo.i> DEIS, Universiy of Bologna

2 Ouline Course ouline Lecure 1: An Inroducion o Consrain Based Scheduling Lecure 2: Advanced scheduling opics Lecure 3: Resource allocaion and scheduling Hands-on Session: ogeher wih Prof. Van Hoeve

Ouline Some Disclaimers This is a course on CP for scheduling This choice leaves ou scheduling heurisics such as Geneic Algorihms, Paricle Swarm

3 Ouline Some Disclaimers This is a course on CP for scheduling This choice leaves ou scheduling heurisics such as Geneic Algorihms, Paricle Swarm Opimizaion, Simulaed Annealing... I will ry o survey (a leas briefly) sae of he ar mehods This choice leaves ou some hisorically imporan approach.

4 Ouline Ouline of his lecure Par 1: An inroducion o scheduling problems Par 2: Consrain Based Scheduling Par 2.1: Filering for he cumulaive consrains Par 2.2: Search Sraegies for scheduling problems

5 Abou he Slides Sorry for he slides, bu......we ve jus go married!

6 Abou he Slides I s all available a hp://ai.unibo.i (Jus look for my web page in he people secion)

7 Lesson 1, Par 1: An Inroducion o Consrain Based Scheduling So, wha s his scheduling abou? DEIS, Universiy of Bologna

8 Scheduling: a Definiion Scheduling: allocaing scarce resources o aciviies over ime

9 Scheduling: a Definiion Scheduling: allocaing scarce resources o aciviies over ime Classical Inerpreaion: Scheduling: ordering resource-requiring aciviies over ime A very imporan (and ough) problem class Many pracical applicaions! Some examples follow...

10 Applicaion Examples Parallel Compuing, boh a server-room and single chip level

11 Aircraf Landing Applicaion Examples

12 Operaing Indusrial Robos Applicaion Examples

13 Consrucion Projecs Applicaion Examples

14 Transporaion and courier operaion Applicaion Examples

15 Timeabling for Spor Evens Applicaion Examples

16 The Resource Consrained Projec Scheduling We will focus on a paricular ype of scheduling problem Resource Consrained Projec Scheduling Problem Given a se of aciviies, conneced by precedence consrains and requiring a se of finie capaciy resources, find a feasible schedule, opimal w.r.. some performance meric Finding a feasible soluion is easy Finding an opimal soluion is NP-hard (and very ough!)

More on precedence consrains: A precedence consrain beween and a j forces aciviy a i o

17 The Resource Consrained Projec Scheduling More on aciviies: An aciviy performed a i is some operaion o be A0 Aciviies are non-inerrupible An aciviy a i has fixed duraion d i More on precedence consrains: A precedence consrain beween and a j forces aciviy a i o end before sars a j a i A5 A6 Aciviies and precedence consrains form a DAG, called Projec Graph A8

18 The Resource Consrained Projec Scheduling More on resources: A resource r k is an energy provider over ime Each resource r k has finie capaciy c k, renewed a each ime insan Each aciviy requires some amoun for he duraion of is execuion rq ik 0 of each resource r k c k Ai rq ik d i

The Resource Consrained Projec Scheduling More on schedules: A schedule is an assignmen of sar imes s i ( ) o each aciviy A schedule is feasible is no precedence consrain is violaed and

19 The Resource Consrained Projec Scheduling More on schedules: A schedule is an assignmen of sar imes s i ( ) o each aciviy A schedule is feasible is no precedence consrain is violaed and no resource capaciy is exceeded More on performance merics: A performance meric is a schedule dependen cos funcion A common performance meric is he makespan mkspan = max a i s i (a i )+d i

A Sample Problem More on schedules: A Projec Graph A is he se of aciviies is a se of aciviy pairs, represening he precedence consrains Aciviy duraions ha, Ei d i

20 A Sample Problem More on schedules: A Projec Graph A is he se of aciviies is a se of aciviy pairs, represening he precedence consrains Aciviy duraions ha, Ei d i a i E (a i,a j ) Se R of resources r k, wih capaciy c k All resource requiremens r ik duraion req for r 0 ( c 0 =2) A5 2/1 1/2 A0 0/0 1/1 1/2 1/2 A6 3/2 2/1 A8 0/0

21 A Sample Problem The source and sink aciviies are fake and can herefore be disregarded This is a (makespan) opimal schedule 2 A5 2/1 1/2 1/1 1/2 1/2 A6 3/2 2/1 makespan A5 A6

22 Exensions and Generalizaions There are a lo of problem varians! Times lags & ime windows Time lags are bounds on he difference beween he end ime and he sar ime of wo aciviies: [ i,j, i,j] Ai Aj i,j apple s j ( ) (s i ( )+d i ) apple i,j Minimal ime lags are easy do deal wih Wih max. ime lags, finding a feasible soluion becomes NP-hard! Time windows are ime lags from an aciviy wih s 0 ( )=d 0 =0

23 Exensions and Generalizaions (Sequence Dependen) Seup Times Defined for unary resources If ai and aj are scheduled in sequence, hen hey mus obey a separaion consrain 1 A B

24 Exensions and Generalizaions (Sequence Dependen) Seup Times Defined for unary resources If ai and aj are scheduled in sequence, hen hey mus obey a separaion consrain 1 A seup ime B

25 Exensions and Generalizaions (Sequence Dependen) Seup Times Defined for unary resources If ai and aj are scheduled in sequence, hen hey mus obey a separaion consrain 1 B A

26 Exensions and Generalizaions (Sequence Dependen) Seup Times Defined for unary resources If ai and aj are scheduled in sequence, hen hey mus obey a separaion consrain Oher ypes of resources: Reservoirs 3 Ai Aj

27 Exensions and Generalizaions (Sequence Dependen) Seup Times Defined for unary resources If ai and aj are scheduled in sequence, hen hey mus obey a separaion consrain Oher ypes of resources: Reservoirs Sae Resources Ai Aj requires requires oven a 1500 C oven a 750 C

28 Exensions and Generalizaions (Sequence Dependen) Seup Times Defined for unary resources If ai and aj are scheduled in sequence, hen hey mus obey a separaion consrain Oher ypes of resources: Reservoirs Sae Resources Oher varians (see [1,3])

29 Exensions and Generalizaions Oher performance measures (Weighed) Tardiness coss: X w i max(0,s i ( )+d i dl i ) a i 2A where dl i is a deadline (Weighed) Earliness coss: X w i max(0,dl i s i ( ) d i ) a i 2A

30 Special Cases Job Shop Scheduling Aciviies are organized in sequences (jobs) Only unary capaciy resources (machines) Aciviies in a job require disinc machines A0 a job A5 A6 A8 requires machine 0 requires machine 1 requires machine 2

Special Cases Job Shop Scheduling Aciviies are organized in sequences (jobs) Only unary capaciy resources (machines) Aciviies in a job require disinc machines Unary duraions, unary

31 Special Cases Job Shop Scheduling Aciviies are organized in sequences (jobs) Only unary capaciy resources (machines) Aciviies in a job require disinc machines Unary duraions, unary resources More relaed o imeabling or spor scheduling Closer o racabiliy Typically deal wih using differen mehods (e.g. GCC consrain) A nice uorial by Claude-Guy Quimper a CPAIOR 2012

32 Special Cases Cyclic Varians The projec graph is indefiniely repeaed over ime Obj: maximize hroughpu Flow-shop scheduling Cyclic RCPSP Specific mehods available (periodic schedules)

33 Lesson 1, Par 2: An Inroducion o Consrain Based Scheduling Consrain Based Scheduling DEIS, Universiy of Bologna

34 Consrain Based Scheduling Consrain Based Scheduling = solving scheduling problems wih CP The erm originaes from he [1] Scheduling conceps are mapped o CP variables and consrains Dedicaed search sraegies

.eoh] Duraion consrain: s i + d i = e i Precedence Consrains They are sraighforward o model!

35 CBS: Aciviies & Precedence Consrsains Aciviies Aciviies are modeled wih 2 decision variables + 1 consrain: Sar of a i! s i 2 [0..eoh] End of a i! e i 2 [0..eoh] Duraion consrain: s i + d i = e i Precedence Consrains They are sraighforward o model! End-o-sar: End-o-sar wih ime-lags: e i apple s j i,j apple s j e i apple i,j Time windows modeled by resricing he and domains s i e i

36 CBS: Resources (Renewable) Resources Modeled via he cumulaive consrain (inroduced in [2]): For a resource r k 2 R cumulaive([s i ], [d i ], [rq i,k ],c k ) The consrain is saisfied iff: X s i apple<s i +d i rq i,k apple c k 8 =0..eoh I.e., if here is no resource over-usage unil he End of Horizon

37 RCPSP Model A full model for a RCPSP insance looks like his: min z = max a i 2A e i e i = s i + d i 8a i 2 A e i apple s j 8(a i,a j ) 2 E cumulaive([s i ], [d i ], [rq i,k ],c k ) 8r k 2 R The model is simple and compac ( 2 A variables) Two key issues: Enforcing Consisency of he cumulaive consrain Very large variable domains

38 Lesson 1, Par 2.1: An Inroducion o Consrain Based Scheduling Filering for he Cumulaive Consrain DEIS, Universiy of Bologna

Issue 1: Enforcing Cumulaive Consisency cumulaive([s i ], [d i ], [rq i,k ],c k ) The cumulaive consrain encodes a single-resource scheduling problem wih no precedence consrain This is NP-hard in

39 Issue 1: Enforcing Cumulaive Consisency cumulaive([s i ], [d i ], [rq i,k ],c k ) The cumulaive consrain encodes a single-resource scheduling problem wih no precedence consrain This is NP-hard in he general case Hence, incomplee filering Which kind of filering? There are a many available algorihms We will see he main ones, rying o highligh he key ideas Some of he algorihm deails will be omied

40 Timeable Filering Compulsory pars s i Bound of he and variables have convenional names: e i min(s i ) = Earlies Sar Time, max(s i ) = Laes Sar Time min(e i ) = Earlies End Time, max(e i ) = Laes End Time c k Ai EST LET

41 Timeable Filering Compulsory pars s i Bound of he and variables have convenional names: e i min(s i ) = Earlies Sar Time, max(s i ) = Laes Sar Time min(e i ) = Earlies End Time, max(e i ) = Laes End Time c k Ai EST EET LET

42 Timeable Filering Compulsory pars s i Bound of he and variables have convenional names: e i min(s i ) = Earlies Sar Time, max(s i ) = Laes Sar Time min(e i ) = Earlies End Time, max(e i ) = Laes End Time c k Ai EST LST EET LET

43 Timeable Filering If, hen is a compulsory par EET i >LST i EET i LST i c k EST LST EET LET

44 Timeable Filering By aggregaing he resource usage over he compulsory pars of all aciviies, we obain an opimisic usage profile Then we can perform sweep-filering c k A0

45 Timeable Filering Then we can perform sweep-filering (see [3]) The algorihm mainains A lis of evens (for he sar/end of he compulsory pars) A sweep line An efficien daa srucure (ime able) is used o sore he res. usage c k A0

46 Timeable Filering In order o prune EST i We sar from he curren for an aciviy EST i We probe all evens, searching for he firs one when here is enough room o schedule a i a i c k A0

47 Timeable Filering enough capaciy here c k A0

48 Timeable Filering bu no here => no enough room c k A0

49 Timeable Filering c k A0

50 Timeable Filering c k A0

51 Timeable Filering Ok! c k A0

52 Timeable Filering This is he new EST 4 c k A0

53 Timeable Filering Timeable filering runs in O(n cos of imeable acc.) 80% of he imes, his is all you need! Bu someimes i does no work (guess when?) Luckily, we have somehing more... c k A0

54 Edge Finding Edge finding = deecing (emporary) precedence relaions Given a subse of aciviies EST =min ai 2 EST i LET = max ai 2 LET i E = P a i 2 r i,k d i Required energy: Available energy: A C = c k (LET EST ) Rule 1 (overload checking): if E >C, hen fail

55 Edge Finding EST =0 LET =5 C = 15 E = 10 3 C A B

56 Edge Finding Assume we wan o schedule an New available energy: New resource consumpion: a i /2 so ha c k (LET EST [ai ) E + rq i,k d i e i apple LET D 3 C EST =0 LET =5 A E = 10 B C = 15

57 Edge Finding Assume we wan o schedule an New available energy: New resource consumpion: a i /2 so ha c k (LET EST [ai ) E + rq i,k d i e i apple LET Rule 2 (edge finder): D E + rq i,k d i >c k (LET EST,) => e i >LET [ai 3 C A B

58 Edge Finding Rule 2 (edge finder): E + rq i,k d i >c k (LET EST,) => e i >LET [ai D E +2 3 = 16 3 (LET 0) = 15 3 C EST =0 LET =5 A E = 10 B C = 15

59 Edge Finding Rule 2 (edge finder): E + rq i,k d i >c k (LET EST,) => e i >LET [ai 3 D C E +2 3 = 16 3 (LET 0) = 15 EET i > 5 ) EST i > 2 EST =0 LET =5 A E = 10 B C = 15

60 Edge Finding Rule 2 (edge finder): E + rq i,k d i >c k (LET EST,) => e i >LET [ai 3 D C E +2 3 = 16 3 (LET 0) = 15 EET i > 5 ) EST i > 2 EST =0 LET =5 A E = 10 B C = 15

61 Edge Finding Rule 2 (edge finder): E + rq i,k d i >c k (LET EST,) => e i >LET [ai However, if we schedule a i so ha compees wih aciviies in, we may delay heir end By aking his ino accoun, we can compue a beer EST i bound

62 Edge Finding reserved for : conflic (c k rq i,k ) (LET EST ) Bu we now ha a i mus end before all aciviies in Hence all mus be exhaused before sars E a i 3 A D C B

63 Edge Finding reserved for : (c k rq i,k ) (LET EST ) conflic Bu we now ha a i mus end before all aciviies in Hence all mus be exhaused before sars E a i 3 A D C B

64 Edge Finding E (c k rq i,k )(LET EST ) EST i EST + rq i,k remaining capaciy is he same as rq i,k Since canno be scheduled a, his holds for each a i EST i 0 The highes bound gives he updae (here, B and C) 3 C A D B

65 Edge Finding E (c k rq i,k )(LET EST ) EST i EST + rq i,k remaining capaciy is he same as rq i,k Since canno be scheduled a, his holds for each a i EST i 0 The highes bound gives he updae (here, B and C) 3 C A B EST i =4

66 Edge Finding Here s he full rule! Rule 2 (edge finder): E + rq i,k d i >c k (LET EST [ai ), => 8 0 res( 0,a i ) > 0 : EST i EST 0 + res( 0,a i ) rq i,k where: res( 0,a i )=E 0 (c k rq i,k ) (LET 0 EST 0)

) O(k n 2 ) for exended edge-finding (differen rigger rule) Oher edge-finder relaed algorihms: No-firs/no-las rule [4]

67 Edge Finding Sae-of-he-ar edge finder algorihms use powerful daa srucures (such as rees) o do he enumeraion in poly-ime O(k n log n) for sandard edge-finding (k = #disinc requiremens - check he references!) O(k n 2 ) for exended edge-finding (differen rigger rule) Oher edge-finder relaed algorihms: No-firs/no-las rule [4] Timeable edge-finding ([5], runs in O(n 2 )) Used in CP-Opimizer Takes ino accoun compulsory pars Energeic reasoning ([1], runs in O(n 3 ) Precedence Graph based (wo algorihms, in [6])

68 Some Noable Dominance Relaions Edge Finder Timeabling Timeable Edge Finder No-firs/No-las Energeic Reasoning Precedence Graph Rules of humb Firs ry wih imeabling Add edge-finding (very good when he ime windows are small) Energeic reasoning seldom helps

69 Lesson 1, Par 2.2: An Inroducion o Consrain Based Scheduling Search Sraegies for Scheduling Problems DEIS, Universiy of Bologna

4, 12) Trivial horizon: 158 Hence: s i, e i 2 [0.

70 Issue 2: Dealing wih Large Domains Figures of a ypical RCPSP Insance j301_1 from he PSPLIB [6bis] 30 aciviies, 48 arcs 4 resources (12, 13, 4, 12) Trivial horizon: 158 Hence: s i, e i 2 [0..158] How do we search for a soluion? Forge abou min size domain... We need a specialized search sraegy

71 Back o Our Sample Problem [0,5]/[2,7] [0,7]/[1,8] A5 [3,10]/[4,11] [2,7]/[3,8] A6 [3,8]/[6,11] [1,8]/[2,9] [2,9]/[4,11] A5 A6 2

72 Back o Our Sample Problem [0,5]/[2,7] [0,7]/[1,8] A5 [3,10]/[4,11] [2,7]/[3,8] Which aciviy o selec? Naural choice: A6 [3,8]/[6,11] focus on ready ones [1,8]/[2,9] [2,9]/[4,11] A5 A6 2

73 Back o Our Sample Problem [0,5]/[2,7] Ready aciviies have min EST [0,7]/[1,8] A5 [2,7]/[3,8] How o break ies? We need a prioriy rule A6 [3,8]/[6,11] (e.g. ighes deadline) [1,8]/[2,9] [3,10]/[4,11] [2,9]/[4,11] A5 A6 2

74 Back o Our Sample Problem [0,5]/[2,7] [0,7]/[1,8] A5 [3,10]/[4,11] [2,7]/[3,8] A6 [3,8]/[6,11] [1,8]/[2,9] [2,9]/[4,11] A5 A6 2

75 Back o Our Sample Problem Schedule a he EST [0,7]/[1,8] A5 [3,10]/[4,11] [2,7]/[3,8] A6 [3,8]/[6,11] [1,8]/[2,9] [2,9]/[4,11] A5 A6 2

76 Back o Our Sample Problem [0,7]/[1,8] A5 [3,10]/[4,11] [2,7]/[3,8] A6 [3,8]/[6,11] [2,8]/[3,9] [3,9]/[5,11] A5 A6 2

77 Back o Our Sample Problem A5 [3,10]/[4,11] [2,7]/[3,8] A6 [3,8]/[6,11] [2,8]/[3,9] [3,9]/[5,11] A5 A6 2

78 Back o Our Sample Problem A5 [3,10]/[4,11] A6 [3,8]/[6,11] [3,8]/[4,9] [4,9]/[6,11] A5 A6 2

79 Back o Our Sample Problem A5 [4,10]/[5,11] A6 [4,8]/[7,11] A5 A6 [4,9]/[6,11] 2

80 Back o Our Sample Problem Break furher ies based on aciviy index A5 [4,10]/[5,11] A6 [4,8]/[7,11] A5 A6 [4,9]/[6,11] 2

81 Back o Our Sample Problem A6 [5,8]/[8,11] [5,9]/[7,11] A5 A6 2 A5

82 Back o Our Sample Problem [8,9]/[10,11] A5 A6 2 A5 A6

83 Back o Our Sample Problem This is called Prioriy Rule based Scheduling [7] One of he oldes heurisic approaches for scheduling! Very fas and ypically effecive We used he parallel Schedule Generaion Scheme (no necessarily he bes one) A5 A6 2 A5 A6

84 Back o Our Sample Problem Bu we missed opimaliy! 2 A5 A6 A5 A6 2 A5 A6

85 Backracking Sraegy #1 Schedule anoher aciviy wih min EST A5 A6 2 A5 A6

86 Backracking Sraegy #1 [0,5]/[2,7] [0,7]/[1,8] A5 [3,10]/[4,11] [2,7]/[3,8] A6 [3,8]/[6,11] [1,8]/[2,9] [2,9]/[4,11] A5 A6 2

87 Backracking Sraegy #1 [0,5]/[2,7] A5 [3,10]/[4,11] [2,7]/[3,8] A6 [3,8]/[6,11] [1,8]/[2,9] [2,9]/[4,11] A5 A6 2

88 Backracking Sraegy #1 We have no chance o ge his! A5 [3,10]/[4,11] 2 [2,7]/[3,8] A6 A5 A6 [1,8]/[2,9] [3,8]/[6,11] [2,9]/[4,11] A5 A6 2

89 Backracking Sraegy #2 Schedule anoher aciviy wih min EST Schedule anoher aciviy Very (...) inefficien A5 A6 2 A5 A6

90 Backracking Sraegy #3 Schedule anoher aciviy wih min EST Schedule anoher aciviy On backracking, pospone he previously scheduled aciviy When propagaion updaes is EST, de-pospone i A5 A6 2 A5 A6

91 Backracking Sraegy #3 [0,5]/[2,7] [0,7]/[1,8] A5 [3,10]/[4,11] [2,7]/[3,8] A6 [3,8]/[6,11] [1,8]/[2,9] [2,9]/[4,11] A5 A6 2

92 Backracking Sraegy #3 [0,5]/[2,7] A5 [3,10]/[4,11] [2,7]/[3,8] A6 [3,8]/[6,11] [1,8]/[2,9] [2,9]/[4,11] A5 A6 2

93 Backracking Sraegy #3 [2,5]/[4,7] A5 [3,10]/[4,11] [2,7]/[3,8] A6 [3,8]/[6,11] A5 A6 [2,9]/[4,11] 2

94 Backracking Sraegy #3 [2,5]/[4,7] A5 [3,10]/[4,11] [2,7]/[3,8] A6 [3,8]/[6,11] A5 A6 [2,9]/[4,11] 2

95 Backracking Sraegy #3 A5 [3,10]/[4,11] [2,7]/[3,8] A6 [3,8]/[6,11] A5 A6 [2,9]/[4,11] 2

96 Backracking Sraegy #3 And here we go! 2 [2,7]/[3,8] A5 [3,10]/[4,11] A6 A5 A6 [3,8]/[6,11] [2,9]/[4,11] A5 A6 2

apple EST i LST j apple EST i EST i EST j a i wih minimum, depospone i a i EST i Before scheduling a i if here exiss an

97 Backracking Sraegy #3 This sraegy is known as Schedule or Pospone [1]: On he firs branch schedule an aciviy Break ies according o any rule On backracking, pospone When propagaion updaes a i EST i An enhancemen ( a Dominance Rule ): EET j apple EST i LST j apple EST i EST i EST j a i wih minimum, depospone i a i EST i Before scheduling a i if here exiss an aciviy a j ha should or mus be scheduled before, i.e. if: hen abor he branch, since propagaion will never be modified by

The sraegy may be incomplee When a scheduling decision may affec aciviies saring earlier When here are maximal ime

98 Backracking Sraegy #3 Schedule or Pospone works very nicely In paricular for makespan minimizaion Bu here are drawbacks! The sraegy may be incomplee When a scheduling decision may affec aciviies saring earlier When here are maximal ime lags (or negaive minimal ime lags, since hey are equivalen) When he cos funcion is non-regular A cos funcion is regular if is value can only decrease when he sar ime of any aciviy is decreased Wach ou for more complex cases!

99 Lesson 1: An Inroducion o Consrain Based Scheduling References DEIS, Universiy of Bologna

100 References [1] Bapise, P., Le Pape, C., & Nuijen, W. (2001). Consrain-based scheduling. Kluwer Academic Publishers. [2] Aggoun, A., Beldiceanu, N.: Exending CHIP in order o solve complex scheduling and placemen problems. Mahemaical and Compuer Modelling 17(7), (1993) [3] Beldiceanu, N., Carlsson, M.: Sweep as a Generic Pruning Technique Applied o he Non-Overlapping Recangles Consrain. In Proc. of he 7h CP, , Paphos, (2001). [4] Schu, A.,Wolf, A., Schrader, G.: No-firs and no-las deecion for cumulaive scheduling in O(n3 logn). In: Declaraive Programming for Knowledge Managemen, pp Springer-Verlag (2006). [5] Vilím, P. (2011). Timeable Edge Finding Filering Algorihm for Discree Cumulaive Resources. Proc. of CPAIOR, Rerieved from hp://

101 References Some imporan filering algorihms based on he so-called Precedence Graph are repored in: [6] Laborie, P. (2003). Algorihms for propagaing resource consrains in AI planning and scheduling: Exising approaches and new resuls. Arificial Inelligence, 143(2), doi: / S (02) For he PSPLIB [6bis] Kolisch, R. (1997). PSPLIB - A projec scheduling problem library. European Journal of Operaional Research, 96(1),

Comments on Window-Constrained Scheduling

Comments on Window-Constrained Scheduling Commens on Window-Consrained Scheduling Richard Wes Member, IEEE and Yuing Zhang Absrac This shor repor clarifies he behavior of DWCS wih respec o Theorem 3 in our previously published paper [1], and describes