Multicriteria Scheduling: Theory and Models

1. Multicriteria Scheduling: Theory and Models Vincent T’KINDT Laboratoire d’Informatique (EA 2101) Dépt. Informatique - Polytech’Tours Université François-Rabelais de Tours – France tkindt@univ-tours.fr

2. Structure • Theory of Multicriteria Scheduling, • Optimality definition, • How to solve a multicriteria scheduling problem, • Application to a bicriteria scheduling problem, • Considerations about the enumeration of optimal solutions. • Some models and algorithms, • Scheduling with intefering job sets, • Scheduling with rejection cost. • Solution of bicriteria single machine problem by mathematical programming

3. What is Multicriteria Scheduling? • Multicriteria Optimization: How to optimize several conflicting criteria? • Scheduling: How to determine the « optimal » allocation of tasks (jobs) to resources (machines) over time? • Multicriteria Optimization: How to optimize several conflicting criteria? • Scheduling: How to determine the « optimal » allocation of tasks (jobs) to resources (machines) over time? • Multicriteria Scheduling = • Scheduling + Multicriteria Optimization.

4. Theory of Multicriteria Scheduling • What about multicriteria optimization? • K criteria Zi to minimize, • The notion of optimality is defined • by means of Pareto optimality, • We distinguish between: • Strict Pareto optimality, • Weak Pareto optimality. Z2 Z1 A solution x is a strict Pareto optimum iff there does not exist another solution y such that Zi(y) ≤Zi(x), i=1,…,K, with at least one strict inequality. A solution x is a weak Pareto optimum iff there does not exist another solution y such that Zi(y) < Zi(x), i=1,…,K. E  WE

5. Theory of Multicriteria Scheduling • Multicriteria scheduling (straigth extension), • Determine one or more Pareto optimal (preferrably strict) allocations of tasks (jobs) to resources (machines) over time. • General fundamental considerations, • How to calculate a strict Pareto optimum ? • How to calculate the “best” strict Pareto optimum ?  This depends on Decision Maker’s preferences.

6. Theory of Multicriteria Scheduling • How can be expressed decision maker’s preferences? • By means of weights (wi for criterion Zi), • By means of goals (fex: Zi [LB;UB]), • By means of bounds (Zi ei), • By means of an absolute order. • Numerous studies can be found in the literature, • Convex combination of criteria (Geoffrion’s theorem), • e-constraint approach, • Lexicographic approach, • Parametric approach, • …

7. Theory of Multicriteria Scheduling • How to calculate the “best” strict Pareto optimum ?

8. Theory of Multicriteria Scheduling • Convex combination of criteria, • Min SiaiZi(x) • st • x  S • ai  [0;1], Siai = 1 • Strong convex hypothesis (Geoffrion’s theorem). • Discrete case: supported vs non supported Pareto optima. • e-constraint approach, • Min Z1(x) • st • x  S • Zi  ei, i=2,…,K • Weak Pareto optima, • Often used in a posteriori algorithms. • Lexicographic approach: Z1 Z2  …  ZK

9. Theory of Multicriteria Scheduling • Illustration on an example problem: 1|di|Lmax, C • A single machine is available, • n jobs have to be processed, • pi : processing time, • di : due date, • Minimize Lmax=maxi(Ci-di) and C=Si Ci, p1 C1 C2 1 C3 2 3 1 2 3 Machine time d2 d1 d3

10. Theory of Multicriteria Scheduling • Illustration on an example problem: 1|di|Lmax, C • Design of an a posteriori algorithm1 • A strict Pareto optimum is calculated by means of the e-contraint approach • Known results : • The 1||C problem is solved to optimality by Shortest Processing Times first rule (SPT), • The 1|di|Lmax problem is solved to optimality by Earliest Due Date first rule (EDD), 1 L. van Wassenhove and L.F. Gelders (1980). Solving a bicriterion scheduling problem, EJOR, 4:42-48.

11. Theory of Multicriteria Scheduling • To calculate a Pareto optimum, solve the 1|di|e(C/Lmax) problem: • Lmax e • maxi(Ci-di)  e • Ci-di  e, i=1,…,n • Ci  Di =di + e, i=1,…,n

12. Theory of Multicriteria Scheduling • Decision Aid module, • Solve the 1|di|Lmax problem => Lmax* value. • Solve the 1||C problem => s0, C(s0), Lmax(s0). • E={s0}, e=Lmax(s0)-1. • While e> Lmax* Do • Solve the 1|Di=di+ e | C problem => s, • E=E//{s}, e=Lmax(s)-1. • End While. • Return E; Lmax Lmax(s0) e s e e Lmax* C(s0) C

13. Theory of Multicriteria Scheduling • Scheduling module (how to solve the 1|Di|C problem), 1 2 3 2 1 3 Machine 7 0 time D1 D2 D3

14. Theory of Multicriteria Scheduling • Candidate list based algorithm, • This a posteriori algorithm is optimal, • The scheduling module works in O(nlog(n)), • There are at most n(n+1)/2 non dominated criteria vectors, • This enumeration problem is easy, • A polynomial time algorithm for calculating a strict Pareto optimum, • A polynomial number of non dominated criteria vectors.

15. Theory of Multicriteria Scheduling • The enumeration of Pareto optima is a challenging issue, • How hard is it to perform the enumeration? •  Complexity theory. • How conflicting are the criteria? •  A priori evaluation, •  Algorithmic evaluation, •  A posteriori evaluation (experimental evaluation).

16. Theory of Multicriteria Scheduling • From a theoretical viewpoint… complexity theory, • Originally dedicated to decision problems, • Scheduling problems are often optimisation problems,

17. Theory of Multicriteria Scheduling • But now what happen for multicriteria optimisation? • We minimise K criteria Zi, • Enumeration of strict Pareto optima, • Counting problem C • Input data, or instance, denoted by I (set DO). • Question: how many optimal solutions are there regarding the objective of problem O? • Enumeration problem E • Input data, or instance, denoted by I (set DO). • Goal: find the set SI the optimal solutions regarding the objective of problem O.

18. Theory of Multicriteria Scheduling Problems which can be solved in polynomial time in the input size and number of solutions Spatial complexity vs Temporal complexity, V. T’kindt, K. Bouibede-Hocine, C. Esswein (2007). Counting and Enumeration Complexity with application to Multicriteria Scheduling, Annals of Operations Research, 153:215-234.

19. Theory of Multicriteria Scheduling • There are some links between classes, • If E  P then O  PO and C  FP, • If O  NPOC and C  #PC then E  ENPC. • .. in practice… • …if O  NPOC then E  ENPC

20. Theory of Multicriteria Scheduling • A priori conflicting measure: analysis on the potential number of strict Pareto optima, • Cone dominance, • Consider the following bicriteria / bivariable MIP problem: • Min Si ci1xi • Min Si ci2 xi • st • Ax  b • x  N2 x2 c1 c2 c1 and c2 are the generators of cone C x1

21. Theory of Multicriteria Scheduling x2 x1

22. Theory of Multicriteria Scheduling • Consider the following problem: 1||Si uiCi, Si viCi • The criteria can be formulated as: • Si uiCi = SiSk ui pk xki • and • Si viCi = SiSk vi pk xki • with xki = 1 if Jk precedes Ji

23. Theory of Multicriteria Scheduling • The generators are: • c1 = [u1p1,…,u1pn,u2p1,…,u2pn,…,unpn] • and • c2 = [v1p1,…,v1pn,v2p1,…,v2pn,…,vnpn] • The cone C is defined by: • C={y  Rn2 / c1.y≥ 0 and c2.y ≥ 0} • If C is tight, then the number of Pareto optima is possibly high. c1 C c2

24. Theory of Multicriteria Scheduling • The maximum angle between c1 and c2 is obtained for : • ui=0, i=1,…,l, and ui≥0, i=l+1,…,n • and • vi ≥0, i=1,…,l, and vi=0, i=l+1,…,n • as the weights are non negative. • This can be helpful to identify/generate instances with a potentially high number of strict Pareto optima.

25. Theory of Multicriteria Scheduling • Drawback: the number of strict Pareto optima also depends on the spreading of solutions (constraints), • Drawback: not easy to generalize to max criteria. • Generally, the number of strict Pareto optima is evaluated by means of an algorithmic analysis, • See for instance the 1|di|Lmax, wCsum problem, • But we have a bound on the number of non dominated criteria vectors.

26. Structure • Theory of Multicriteria Scheduling, • Optimality definition, • How to solve a multicriteria scheduling problem, • Application to a bicriteria scheduling problem, • Considerations about the enumeration of optimal solutions. • Some models and algorithms, • Scheduling with interfering job sets, • Scheduling with rejection cost. • Solution of bicriteria single machine problem by mathematical programming

27. Some models and algorithms • A classification based on model features and not simply on machine configurations, • Scheduling with controllable data, • Scheduling with setup times, • Just-in-Time scheduling, • Robust and flexible scheduling, • Scheduling with interfering job sets, • Scheduling with rejection costs, • Scheduling with completion times, • Scheduling with only due date based criteria, • ….

28. Scheduling with interfering job sets • 2 sets of jobs to schedule, • Set A: nA, evaluated by criterion ZA, • Set B: nB, evaluated by criterion ZB, • Potentially large number of Pareto optima (remember the cone dominance approach).

29. Scheduling with interfering job sets • Consider the 1||Fl(Cmax, wCsum) problem, • Fl(Cmax, wCsum) = CAmax + awCBsum wi‘=awi p1‘=p1+p2+p3 / w1’=1 4 5 6 1’ 1 2 3 wCBsum CAmax 4 1’ 5 6 Machine 0 time WSPT on the fictitious A job and B jobs with weights wi’

30. Scheduling with interfering job sets

31. Scheduling with interfering job sets • Multiple machines problems,

32. Scheduling with rejection costs • A set of n jobs to be scheduled, • A job can be scheduled or rejected, • Minimize a « classic » criterion Z, • Minimize the rejection cost RC=Si rci, •  Often Fl(Z,RC)=Z+RC is minimized.

33. Scheduling with rejection cost • Consider the 1||Fl(Csum, RC) problem, • Fl(Csum, RC) = Csum + RC Job i: pi: processing time, rci: rejection cost. 1 2 3 4 1 2 3 4 Machine 0 time Fl=23 Compute the variations in the objective function Di: Di =[ -2;-3;-10;-7] SPT to get the initial sequencing

34. Scheduling with rejection cost • Consider the 1||Fl(Csum, RC) problem, • Fl(Csum, RC) = Csum + RC 3 1 2 4 Machine 0 time Fl=13 Compute the variations in the objective function Di: Di =[ -1;-1;--;-3]

35. Scheduling with rejection cost • Consider the 1||Fl(Csum, RC) problem, • Fl(Csum, RC) = Csum + RC 3 4 1 2 Machine 0 time Fl=10 Compute the variations in the objective function Di: Di =[ 0;1;--;--]

36. Scheduling with rejection costs • Single machine problems,

37. Scheduling with rejection costs • Multiple machines problems,

38. Structure • Theory of Multicriteria Scheduling, • Optimality definition, • How to solve a multicriteria scheduling problem, • Application to a bicriteria scheduling problem, • Considerations about the enumeration of optimal solutions. • Some models and algorithms, • Scheduling with interfering job sets, • Scheduling with rejection cost. • Solution of bicriteria single machine problem by mathematical programming

39. Bicriteria scheduling and Math. Prog. • Nous considérons le problème d’ordonnancement suivant, • Le problème est noté 1|di| Lmax, Uw, • n travaux, • pi : durée de traitement, • di : date de fin souhaitée, • wi : un poids associé au retard. • On souhaite calculer un optimum de Pareto pour les critères Lmax et Uw. • Lmax=maxi(Ci-di), le plus grand retard algébrique, • Uw =SiwiUi, avec Ui=1 si Ci>di et 0 sinon, nombre pondéré de travaux en retard. • NP-difficile (quel sens ?) • Baptiste, Della Croce, Grosso, T’kindt (2007). Sequencing a single machine with due dates and deadlines: an ILP-BasedApproach to SolveVery Large Instances, à paraître dans Journal of Scheduling.

40. Bicriteria scheduling and Math. Prog. • Utilisation de l’approche e-contrainte, • Minimiser Uw • sc • Lmax  e (A) • La contrainte (A) est équivalente à : • Ci  Di=di+ e, i=1,…,n • Pour calculer un optimum de Pareto on résout le problème noté 1|di , Di| Uw

41. Bicriteria scheduling and Math. Prog. • Qu’avons-nous fait pour résoudre le problème 1|di , Di| Uw ? • Partant d’un modèle mathématique… • … proposition d’une heuristique (borne inférieure) • …mise en place de techniques de réduction de problème • Tous ces éléments ont été intégrés dans une PSE.

42. Bicriteria scheduling and Math. Prog. • Modélisation linéaire en variables bivalentes, • xi = 1 si Ji est en avance, • Bt = {i/Dit} et At = {i/di>t}, • Formulation indexée sur le temps (|T|2n), • T={di,Di}i

43. Bicriteria scheduling and Math. Prog. • Calcule d’une borne inférieure (heuristique), • Propriété : Soit pipj, dj di, Di Dj, wj wi, avec au moins une inégalité stricte. On a (i >> j) : • 1. Si i est en retard, j l’est aussi, • 2. Si j est en avance, i l’est aussi. • Algorithme basé sur le LP et la notion de « core problem », • Mettre dans le « core problem » les variables fractionnaires, • Mettre les variables entières non dominées,

44. Bicriteria scheduling and Math. Prog. • Résoudre le « core problem » à l’aide du MIP (5% des var), • La solution du MIP donne la LB, • Recherche locale en O(n3) par swap de travaux en avance et en retard.

45. Bicriteria scheduling and Math. Prog. • Preprocessing : traitement visant à réduire l’espace de recherche (parfois en réduisant la taille du problème), • Différents types de preprocessing, • Contraintes : • - ajout de contraintes redondantes, • - élimination de contraintes redondantes, • - … • Variables : • - réduction des bornes, • - fixation de variables, • - … • On s’est intéressé à des techniques de preprocessing sur les variables.

46. Bicriteria scheduling and Math. Prog. • Une technique générale de fixation de variables, • Basée sur la résolution de la relaxation linéaire, • Soit LB une borne inférieure et UBlp la borne relachée, • On sait que pour toute solution x du problème mixte : • cx=UBlp+ SjHB rj xj • avec HB l’ensemble des variables hors base dans une solution donnant UBlp. • avec rj le coût réduit (négatif ou nul) associé à xj • UBlp+ SjHB rj xj ≤ LB • SjHB rj xj ≤ LB-UBlp

47. Bicriteria scheduling and Math. Prog. • On en déduit la condition de fixation suivante : • Si rj≥ LB-UBlp alors xj=0 • De même on peut fixer des variables à 1 en introduisant des variables d’écart sj : • xj+sj=1 • … et en tenant le même raisonnement si sj est fixé à 0 alors xj doit être fixé à 1.

48. Bicriteria scheduling and Math. Prog. • On utilise également une technique de fixation basée sur les pseudocostsuj et lj • Soit xjune variable réelle de base du LP et on pose : • lj: une binf sur la diminution unitaire du coût si xj=0 • uj : une binf sur la diminution unitaire du coût si xj=1 • Si (1-xj)*uj≤ UBlp-LBalors xj=0 • Si xj*lj≤ UBlp-LBalors xj=1 • Pour calculer lj et uj on peut utiliser les pénalités de Dantzig1 • 1 Dantzig (1963). LinearProgramming and Extensions, Princeton UniversityPress, Princeton.

49. Bicriteria scheduling and Math. Prog. • Algorithme de preprocessing, • Résoudre le LP, • Fixer des variables par les coûts réduits, • Fixer des variables par les pseudocosts, • Si l’étape 3 a permis de fixer des variables, aller en (1). •  Permet de fixer environ 95% des variables.

50. Bicriteria scheduling and Math. Prog. • Algorithme de la PSE proposée : • Preprocessing, • Branchement sur une variable binaire, • Choix de la variable : • La variable avec le max des pseudo-costs. • Profondeur d’abord, • UB: LP + procédure de réduction, • Si à un nœud il y a moins de 1.4 107 coefficients non nuls on résout le sous problème directement par le MIP.