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Multicriteria Scheduling: Theory and Models Vincent TKINDT Laboratoire dInformatique (EA 2101) Dépt. Informatique - PolytechTours Université François-Rabelais.

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Présentation au sujet: "Multicriteria Scheduling: Theory and Models Vincent TKINDT Laboratoire dInformatique (EA 2101) Dépt. Informatique - PolytechTours Université François-Rabelais."— Transcription de la présentation:

1 Multicriteria Scheduling: Theory and Models Vincent TKINDT Laboratoire dInformatique (EA 2101) Dépt. Informatique - PolytechTours Université François-Rabelais de Tours – France

2 Multicriteria Scheduling: Theory and Models Vincent Tkindt2 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 Multicriteria Scheduling: Theory and Models Vincent Tkindt3 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 Scheduling = Scheduling + Multicriteria Optimization. Multicriteria Optimization: How to optimize several conflicting criteria? Scheduling: How to determine the « optimal » allocation of tasks (jobs) to resources (machines) over time?

4 Multicriteria Scheduling: Theory and Models Vincent Tkindt4 Theory of Multicriteria Scheduling What about multicriteria optimization? –K criteria Z i to minimize, –The notion of optimality is defined by means of Pareto optimality, –We distinguish between: Strict Pareto optimality, Weak Pareto optimality. A solution x is a strict Pareto optimum iff there does not exist another solution y such that Z i (y) Z i (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 Z i (y) < Z i (x), i=1, …,K. EWE Z1Z1 Z2Z2

5 Multicriteria Scheduling: Theory and Models Vincent Tkindt5 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 Makers preferences.

6 Multicriteria Scheduling: Theory and Models Vincent Tkindt6 Theory of Multicriteria Scheduling How can be expressed decision makers preferences? –By means of weights (w i for criterion Z i ), –By means of goals (fex: Z i [LB;UB]), –By means of bounds (Z i i ), –By means of an absolute order. Numerous studies can be found in the literature, – Convex combination of criteria (Geoffrions theorem), – -constraint approach, – Lexicographic approach, – Parametric approach, – …

7 Multicriteria Scheduling: Theory and Models Vincent Tkindt7 Theory of Multicriteria Scheduling How to calculate the best strict Pareto optimum ?

8 Multicriteria Scheduling: Theory and Models Vincent Tkindt8 Theory of Multicriteria Scheduling Convex combination of criteria, Min i i Z i (x) st x S i [0;1], i i = 1 Strong convex hypothesis (Geoffrions theorem). Discrete case: supported vs non supported Pareto optima. -constraint approach, Min Z 1 (x) st x S Z i i, i=2,…,K Weak Pareto optima, Often used in a posteriori algorithms. Lexicographic approach: Z 1 Z 2 … Z K

9 Multicriteria Scheduling: Theory and Models 2 Vincent Tkindt9 Theory of Multicriteria Scheduling Illustration on an example problem: 1|d i |L max, C A single machine is available, Machine n jobs have to be processed, p i : processing time, d i : due date, time d1d1 d2d2 d3d3 321 p1p1 Minimize Lmax=max i (C i -d i ) and C= i C i, 1 3 C1C1 C2C2 C3C3

10 Multicriteria Scheduling: Theory and Models Vincent Tkindt10 Theory of Multicriteria Scheduling Illustration on an example problem: 1|d i |L max, C Design of an a posteriori algorithm 1 1 L. van Wassenhove and L.F. Gelders (1980). Solving a bicriterion scheduling problem, EJOR, 4: A strict Pareto optimum is calculated by means of the -contraint approach Known results : –The 1||C problem is solved to optimality by Shortest Processing Times first rule (SPT), –The 1|d i |L max problem is solved to optimality by Earliest Due Date first rule (EDD),

11 Multicriteria Scheduling: Theory and Models Vincent Tkindt11 Theory of Multicriteria Scheduling To calculate a Pareto optimum, solve the 1|d i | (C/L max ) problem: L max max i (C i -d i ) C i -d i, i=1,…,n C i D i =d i +, i=1,…,n

12 Multicriteria Scheduling: Theory and Models Vincent Tkindt12 s Theory of Multicriteria Scheduling Decision Aid module, i.Solve the 1|d i |L max problem => L max * value. ii.Solve the 1||C problem => s 0, C(s 0 ), L max (s 0 ). iii.E={s 0 }, =L max (s 0 )-1. iv.While > L max * Do i.Solve the 1|D i =d i + | C problem => s, ii.E=E//{s}, =L max (s)-1. v.End While. vi.Return E; C L max L max * C(s 0 ) L max (s 0 )

13 Multicriteria Scheduling: Theory and Models Vincent Tkindt13 Theory of Multicriteria Scheduling Scheduling module (how to solve the 1|D i |C problem), time 0 Machine 123 D1D1 D2D2 D3D

14 Multicriteria Scheduling: Theory and Models Vincent Tkindt14 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 Multicriteria Scheduling: Theory and Models Vincent Tkindt15 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 Multicriteria Scheduling: Theory and Models Vincent Tkindt16 Theory of Multicriteria Scheduling From a theoretical viewpoint… complexity theory, – Originally dedicated to decision problems, Scheduling problems are often optimisation problems,

17 Multicriteria Scheduling: Theory and Models Vincent Tkindt17 Theory of Multicriteria Scheduling But now what happen for multicriteria optimisation? –We minimise K criteria Z i, –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 Multicriteria Scheduling: Theory and Models Vincent Tkindt18 Theory of Multicriteria Scheduling Spatial complexity vs Temporal complexity, Problems which can be solved in polynomial time in the input size and number of solutions V. Tkindt, K. Bouibede-Hocine, C. Esswein (2007). Counting and Enumeration Complexity with application to Multicriteria Scheduling, Annals of Operations Research, 153:

19 Multicriteria Scheduling: Theory and Models Vincent Tkindt19 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 Multicriteria Scheduling: Theory and Models Vincent Tkindt20 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 i c i 1 x i Min i c i 2 x i st Ax b x N 2 x2x2 x1x1 c1c1 c2c2 c 1 and c 2 are the generators of cone C

21 Multicriteria Scheduling: Theory and Models Vincent Tkindt21 Theory of Multicriteria Scheduling x2x2 x1x1

22 Multicriteria Scheduling: Theory and Models Vincent Tkindt22 Theory of Multicriteria Scheduling Consider the following problem: 1|| i u i C i, i v i C i The criteria can be formulated as: i u i C i = i k u i p k x ki and i v i C i = i k v i p k x ki with x ki = 1 if J k precedes J i

23 Multicriteria Scheduling: Theory and Models Vincent Tkindt23 Theory of Multicriteria Scheduling The generators are: c 1 = [u 1 p 1,…,u 1 p n,u 2 p 1,…,u 2 p n,…,u n p n ] and c 2 = [v 1 p 1,…,v 1 p n,v 2 p 1,…,v 2 p n,…,v n p n ] The cone C is defined by: C={y R n2 / c 1.y 0 and c 2.y 0} If C is tight, then the number of Pareto optima is possibly high. c1c1 c2c2 C

24 Multicriteria Scheduling: Theory and Models Vincent Tkindt24 Theory of Multicriteria Scheduling The maximum angle between c 1 and c 2 is obtained for : u i =0, i=1,…,l, and u i 0, i=l+1,…,n and v i 0, i=1,…,l, and v i =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 Multicriteria Scheduling: Theory and Models Vincent Tkindt25 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|d i |L max, wCsum problem, But we have a bound on the number of non dominated criteria vectors.

26 Multicriteria Scheduling: Theory and Models Vincent Tkindt26 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 Multicriteria Scheduling: Theory and Models Vincent Tkindt27 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 Multicriteria Scheduling: Theory and Models Vincent Tkindt28 Scheduling with interfering job sets 2 sets of jobs to schedule, Set A: n A, evaluated by criterion Z A, Set B: n B, evaluated by criterion Z B, Potentially large number of Pareto optima (remember the cone dominance approach).

29 Multicriteria Scheduling: Theory and Models Vincent Tkindt29 Scheduling with interfering job sets Consider the 1||F l (C max, wC sum ) problem, F l (C max, wC sum ) = C A max + wC B sum time 0 Machine C A max wC B sum 1 p 1 =p 1 +p 2 +p 3 / w 1 =1 w i = w i WSPT on the fictitious A job and B jobs with weights w i 1456

30 Multicriteria Scheduling: Theory and Models Vincent Tkindt30 Scheduling with interfering job sets ProblemReferenceNote 1|d i |F l (C max,L max )Baker and Smith (2003) Yuan et al. (2005) Polynomial in O(n B log(n B )). 1|d i |F l (C max,wCsum)Baker and Smith (2003)Polynomial in O(n B log(n B )). 1|d i |F l (L max,wCsum)Baker and Smith (2003) Yuan et al. (2005) NP-hard. Polynomial for w i =1. 1|| (f A max /f B max ) Agnetis et al. (2004)O(n 2 A +n B log(n B )). At most n A n B Pareto. 1|| (wCsum A /f B max ) Agnetis et al. (2004)NP-hard. Polynomial for w i =1 (at most n A n B Pareto). 1|d i | (U A /f B max ) Agnetis et al. (2004)O(n A log(n A )+n B log(n B )). 1|d i | (U A /U B ) Agnetis et al. (2004)O(n 2 A n B +n 2 B n A ). 1|d i | j wU j Cheng and Juan (2006)m job sets. Strongly NP-hard. 1|d i | (wCsum A /U B ) Agnetis et al. (2004)NP-hard. 1|| (Csum A /Csum B ) Agnetis et al. (2004)NP-hard (at most 2 n Pareto).

31 Multicriteria Scheduling: Theory and Models Vincent Tkindt31 Scheduling with interfering job sets ProblemReferenceNote J|d i |Z A,Z B Agnetis et al. (2000)Z A and Z B are quasi-convexe functions of the due dates. Enumerate the Pareto. F2|| (C max A /C max B ) Agnetis et al. (2004)NP-hard. O2|| (C max A /C max B ) Agnetis et al. (2004)NP-hard. Multiple machines problems,

32 Multicriteria Scheduling: Theory and Models Vincent Tkindt32 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= i rc i, Often F l (Z,RC)=Z+RC is minimized.

33 Multicriteria Scheduling: Theory and Models Vincent Tkindt33 Scheduling with rejection cost Consider the 1||F l (C sum, RC) problem, F l (C sum, RC) = C sum + RC time 0 Machine 3214 Job i: p i : processing time, rc i : rejection cost. SPT to get the initial sequencing 3214 ipipi rc i Compute the variations in the objective function i : i =[ -2;-3;-10;-7] F l =23

34 Multicriteria Scheduling: Theory and Models Vincent Tkindt34 Scheduling with rejection cost Consider the 1||F l (C sum, RC) problem, F l (C sum, RC) = C sum + RC time 0 Machine ipipi rc i Compute the variations in the objective function i : i =[ -1;-1;--;-3] F l =13

35 Multicriteria Scheduling: Theory and Models Vincent Tkindt35 Scheduling with rejection cost Consider the 1||F l (C sum, RC) problem, F l (C sum, RC) = C sum + RC time 0 Machine ipipi rc i Compute the variations in the objective function i : i =[ 0;1;--;--] F l =10

36 Multicriteria Scheduling: Theory and Models Vincent Tkindt36 Scheduling with rejection costs ProblemReferenceNote 1||F l (wC sum,RC)Engels et al. (1998)Weakly NP-hard. Dyn. Prog and approx. scheme. Polynomial if w i =w or p i =p. 1|r i,prec|F l (wC sum,RC)Engels et al. (1998)Approximation scheme. 1|d i |F l (L max,RC)Sengupta (1999)Weakly NP-hard. Dyn. Prog and approx. scheme. 1|r i,d i,p i contr|F l ( i (R i -w i T i -c i x i ),RC) Yang and Geunes (2007)R i : profit, x i : compression amount, c i : compression cost. NP-hard. Heuristic. Single machine problems,

37 Multicriteria Scheduling: Theory and Models Vincent Tkindt37 Scheduling with rejection costs ProblemReferenceNote P||F l (C max, RC)Bartal et al. (2000)Approximation algo for the off-line case and competitive algo for the on- line case. P|pmtn|F l (C max, RC)Seiden (2001)Competitive algorithm for the on-line case. P,Q|pmtn|F l (C max,RC)Hoogeveen et al. (2000)Weakly NP-hard. Approx. scheme. R|pmtn|F l (C max,RC)Hoogeveen et al. (2000)Strongly NP-hard. Approx. scheme. O|pmtn|F l (C max,RC)Hoogeveen et al. (2000)Strongly NP-hard. Approx. scheme. Multiple machines problems,

38 Multicriteria Scheduling: Theory and Models Vincent Tkindt38 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 Multicriteria Scheduling: Theory and Models Vincent Tkindt39 Bicriteria scheduling and Math. Prog. Nous considérons le problème dordonnancement suivant, Le problème est noté 1|d i | L max, U w, n travaux, p i : durée de traitement, d i : date de fin souhaitée, w i : un poids associé au retard. On souhaite calculer un optimum de Pareto pour les critères L max et U w. L max =max i (C i -d i ), le plus grand retard algébrique, U w = i w i U i, avec U i =1 si C i >d i et 0 sinon, nombre pondéré de travaux en retard. NP-difficile (quel sens ?) Baptiste, Della Croce, Grosso, Tkindt (2007). Sequencing a single machine with due dates and deadlines: an ILP-Based Approach to Solve Very Large Instances, à paraître dans Journal of Scheduling.

40 Multicriteria Scheduling: Theory and Models Vincent Tkindt40 Bicriteria scheduling and Math. Prog. Utilisation de lapproche -contrainte, Minimiser U w sc L max (A) La contrainte (A) est équivalente à : C i D i =d i +, i=1,…,n Pour calculer un optimum de Pareto on résout le problème noté 1|d i, D i | U w

41 Multicriteria Scheduling: Theory and Models Vincent Tkindt41 Bicriteria scheduling and Math. Prog. Quavons-nous fait pour résoudre le problème 1|d i, D i | U w ? Partant dun modèle mathématique… … proposition dune heuristique (borne inférieure) …mise en place de techniques de réduction de problème Tous ces éléments ont été intégrés dans une PSE.

42 Multicriteria Scheduling: Theory and Models Vincent Tkindt42 Bicriteria scheduling and Math. Prog. Modélisation linéaire en variables bivalentes, x i = 1 si J i est en avance, B t = {i/D i t} et A t = {i/d i >t}, Formulation indexée sur le temps (|T| 2n), T={d i,D i } i

43 Multicriteria Scheduling: Theory and Models Vincent Tkindt43 Bicriteria scheduling and Math. Prog. Calcule dune borne inférieure (heuristique), Propriété : Soit p i p j, d j d i, D i D j, w j w i, avec au moins une inégalité stricte. On a (i >> j) : 1. Si i est en retard, j lest aussi, 2. Si j est en avance, i lest 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 Multicriteria Scheduling: Theory and Models Vincent Tkindt44 Bicriteria scheduling and Math. Prog. Résoudre le « core problem » à laide du MIP (5% des var), La solution du MIP donne la LB, Recherche locale en O(n 3 ) par swap de travaux en avance et en retard.

45 Multicriteria Scheduling: Theory and Models Vincent Tkindt45 Bicriteria scheduling and Math. Prog. Preprocessing : traitement visant à réduire lespace 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 sest intéressé à des techniques de preprocessing sur les variables.

46 Multicriteria Scheduling: Theory and Models Vincent Tkindt46 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 UB lp la borne relachée, On sait que pour toute solution x du problème mixte : cx=UB lp + j HB r j x j avec HB lensemble des variables hors base dans une solution donnant UB lp. avec r j le coût réduit (négatif ou nul) associé à x j UB lp + j HB r j x j LB j HB r j x j LB-UB lp

47 Multicriteria Scheduling: Theory and Models Vincent Tkindt47 Bicriteria scheduling and Math. Prog. On en déduit la condition de fixation suivante : Si r j LB-UB lp alors x j =0 De même on peut fixer des variables à 1 en introduisant des variables décart s j : x j +s j =1 … et en tenant le même raisonnement si s j est fixé à 0 alors x j doit être fixé à 1.

48 Multicriteria Scheduling: Theory and Models Vincent Tkindt48 Bicriteria scheduling and Math. Prog. On utilise également une technique de fixation basée sur les pseudocosts u j et l j Soit x j une variable réelle de base du LP et on pose : l j : une binf sur la diminution unitaire du coût si x j =0 u j : une binf sur la diminution unitaire du coût si x j =1 Si (1-x j )*u j UB lp -LB alors x j =0 Si x j *l j UB lp -LB alors x j =1 Pour calculer l j et u j on peut utiliser les pénalités de Dantzig 1 1 Dantzig (1963). Linear Programming and Extensions, Princeton University Press, Princeton.

49 Multicriteria Scheduling: Theory and Models Vincent Tkindt49 Bicriteria scheduling and Math. Prog. Algorithme de preprocessing, (1)Résoudre le LP, (2)Fixer des variables par les coûts réduits, (3)Fixer des variables par les pseudocosts, (4)Si létape 3 a permis de fixer des variables, aller en (1). Permet de fixer environ 95% des variables.

50 Multicriteria Scheduling: Theory and Models Vincent Tkindt50 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 dabord, UB: LP + procédure de réduction, Si à un nœud il y a moins de coefficients non nuls on résout le sous problème directement par le MIP.

51 Multicriteria Scheduling: Theory and Models Vincent Tkindt51 Bicriteria scheduling and Math. Prog. Quelques résultats, Cplex seul résout jusquà n=4000 en moins de 290s en moyenne, G=100*(UB-Opt)/Opt G=100*(LB-Opt)/Opt

52 Multicriteria Scheduling: Theory and Models Vincent Tkindt52 Bicriteria scheduling and Math. Prog. Pas de résultat sur lénumération des optima de Pareto, Approche testée sur un autre problème dordonnancement 1, Le problème F2|d i =d, d unknown | d, U, Le calcul dun optimum de Pareto se fait jusquà n=3000 (Cplex limité à n=2000 et la litérature à n=900), On fixe environ 85% des variables. Lénumération des (n+1) optima de Pareto strict se fait jusquà n=500 en moins de 800s. 1 Tkindt, Della Croce, Bouquard (2007). Enumeration of Pareto Optima for a Flowshop Scheduling Problem with Two Criteria, Informs JOC, 19(1):64-72.

53 Multicriteria Scheduling: Theory and Models Vincent Tkindt53 Now whats going on? Investigation of structural properties of the Pareto set for scheduling problems, How to quickly calculate a Pareto optimum starting with a known one? Generalized dominance conditions, Measuring the conflictness of criteria: from cone dominance to the complexity of counting problems, Complexity of exponential algorithms, …

54 Multicriteria Scheduling: Theory and Models Vincent Tkindt54 Now whats going on? Investigation of emerging models, Scheduling with interfering job sets, Scheduling with rejection costs, Scheduling for new orders, Combined models: scheduling with rejection costs and new orders, …

55 Multicriteria Scheduling: Theory and Models Vincent Tkindt55 Now whats going on? Industrial applications, Are often multicriteria by nature, Practical application of theoretical models.

56 Multicriteria Scheduling: Theory and Models Vincent Tkindt56 You want to know more? V. Tkindt, JC. Billaut (2006). Multicriteria Scheduling: Theory, Models and Algorithms. Springer.


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