Multicriteria Scheduling: Theory and Models Vincent TKINDT Laboratoire dInformatique (EA 2101)...

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Multicriteria Scheduling: Theory and Models

Vincent T’KINDTLaboratoire d’Informatique (EA 2101)Dépt. Informatique - Polytech’Tours

Université François-Rabelais de Tours – Francetkindt@univ-tours.fr

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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

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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?

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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.

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.

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• 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.

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• 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 i),– By means of an absolute order.

• Numerous studies can be found in the literature,– Convex combination of criteria (Geoffrion’s theorem),– -constraint approach,– Lexicographic approach,– Parametric approach,– …

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Module which takes into account the criteria

Algorithms based on an a priori method

Module which solves the scheduling problem

value of the parameters

a Pareto optimum

Algorithms based on an interactive method

a Pareto optimum

• How to calculate the “best” strict Pareto optimum ?

Algorithms based on an a posteriori method

a Pareto optimum

a set of Pareto optima

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• Convex combination of criteria,Min i iZi(x)

stx Si [0;1], i i = 1

Strong convex hypothesis (Geoffrion’s theorem). Discrete case: supported vs non supported Pareto optima.

• -constraint approach,Min Z1(x)

stx SZi i, i=2,…,K

Weak Pareto optima, Often used in a posteriori algorithms.

• Lexicographic approach: Z1 Z2 … ZK

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• Illustration on an example problem: 1|di|Lmax, C

• A single machine is available,

Machine

• n jobs have to be processed,• pi : processing time,• di : due date,

timed1 d2 d3

332211

• Minimize Lmax=maxi(Ci-di) and C=i Ci,

C1 C2 C3

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• Illustration on an example problem: 1|di|Lmax, C

Design of an a posteriori algorithm1

1 L. van Wassenhove and L.F. Gelders (1980). Solving a bicriterion scheduling problem, EJOR, 4:42-48.

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|di|Lmax problem is solved to optimality by Earliest Due

Date first rule (EDD),

• To calculate a Pareto optimum, solve the 1|di|(C/Lmax) problem:

Lmax maxi(Ci-di) Ci-di , i=1,…,n Ci Di =di + , i=1,

• Decision Aid module,

i. Solve the 1|di|Lmax problem => Lmax* value.

ii. Solve the 1||C problem => s0, C(s0), Lmax(s0).

iii. E={s0}, =Lmax(s0)-1.

iv. While > Lmax* Do

i. Solve the 1|Di=di+ | C problem => s,

ii. E=E//{s}, =Lmax(s)-1.

v. End While.vi. Return E;

Lmax(s0)

• Scheduling module (how to solve the 1|Di|C problem),

Machine

11 22 33

331122

• 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.

• 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).

• From a theoretical viewpoint… complexity theory,– Originally dedicated to decision problems,

• Scheduling problems are often optimisation problems,

• But now what happen for multicriteria optimisation?– We minimise K criteria Zi,

– Enumeration of strict Pareto optima,

Counting problem CInput data, or instance, denoted by I (set DO).Question: how many optimal solutions are there regarding the objective of problem O?

Enumeration problem EInput data, or instance, denoted by I (set DO).Goal: find the set SI the optimal solutions regarding the objective of problem O.

Spatial complexity vs Temporal

complexity,

Problems which can be solved in polynomial time in the input size and number of solutions

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.

• 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

• A priori conflicting measure: analysis on the potential number of strict Pareto optima,

• Cone dominance,Consider the following bicriteria / bivariable MIP

problem:Min i ci

Min i ci2 xi

st Ax b x N2

c2c1 and c2 are the generators of cone

• Consider the following problem: 1||i uiCi, i viCi

• The criteria can be formulated as:i uiCi = i k ui pk xki

and i viCi = i k vi pk xki

with xki = 1 if Jk precedes Ji

• The generators are:c1 = [u1p1,…,u1pn,u2p1,…,u2pn,…,unpn]andc2 = [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

• The maximum angle between c1 and c2 is obtained for :ui=0, i=1,…,l, and ui≥0, i=l+1,…,nandvi ≥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.

• 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.

Structure

• Some models and algorithms,– Scheduling with interfering job sets,– Scheduling with rejection cost.

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,• ….

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).

Scheduling with interfering job sets• Consider the 1||Fl(Cmax, wCsum) problem,

Fl(Cmax, wCsum) = CAmax + wCB

Machine

44 55 66332211

wCBsum

1’1’

p1‘=p1+p2+p3 / w1’=1

wi‘=wi

WSPT on the fictitious A job and B jobs with weights wi’

1’1’44 55 66

Scheduling with interfering job sets

Problem Reference Note

1|di|Fl(Cmax,Lmax) Baker and Smith (2003)Yuan et al. (2005)

Polynomial in O(nB log(nB)).

1|di|Fl(Cmax,wCsum)

Baker and Smith (2003)

Polynomial in O(nB log(nB)).

1|di|Fl(Lmax,wCsum)

Baker and Smith (2003)Yuan et al. (2005)

NP-hard. Polynomial for wi=1.

1||(fAmax/fB

max) Agnetis et al. (2004) O(n2A+nB log(nB)). At most nAnB

Pareto.

1||(wCsumA/fBmax) Agnetis et al. (2004) NP-hard. Polynomial for wi=1 (at most

nAnB Pareto).

1|di|(UA/fBmax) Agnetis et al. (2004) O(nA log(nA)+nB log(nB)).

1|di|(UA/UB) Agnetis et al. (2004) O(n2A nB+n2

B nA).

1|di|jwUj Cheng and Juan (2006)

m job sets. Strongly NP-hard.

1|di|(wCsumA/UB) Agnetis et al. (2004) NP-hard.

1||(CsumA/CsumB) Agnetis et al. (2004) NP-hard (at most 2n Pareto).

Scheduling with interfering job sets

J|di|ZA,ZB Agnetis et al. (2000) ZA and ZB are quasi-convexe functions of the due dates. Enumerate the Pareto.

F2||(CmaxA/Cmax

B) Agnetis et al. (2004) NP-hard.

O2||(CmaxA/Cmax

B) Agnetis et al. (2004) NP-hard.

• Multiple machines problems,

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 rci,

Often Fl(Z,RC)=Z+RC is minimized.

Scheduling with rejection cost

• Consider the 1||Fl(Csum, RC) problem,

Fl(Csum, RC) = Csum + RC

Machine

332211 44

Job i: pi: processing time, rci: rejection cost.

SPT to get the initial sequencing

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i pi rci

Compute the variations in the objective function i:i =[ -2;-3;-10;-7]

Machine

2211 44

i pi rci

Compute the variations in the objective function i:i =[ -1;-1;--;-3]

Machine

i pi rci

Compute the variations in the objective function i:i =[ 0;1;--;--]

1||Fl(wCsum,RC) Engels et al. (1998) Weakly NP-hard. Dyn. Prog and approx. scheme. Polynomial if wi=w or pi=p.

1|ri,prec|Fl(wCsum,RC) Engels et al. (1998) Approximation scheme.

1|di|Fl(Lmax,RC) Sengupta (1999) Weakly NP-hard. Dyn. Prog and approx. scheme.

1|ri,di,pi contr|Fl(i (Ri-wiTi-cixi),RC)

Yang and Geunes (2007)

Ri: profit, xi: compression amount, ci: compression cost. NP-hard. Heuristic.

• Single machine problems,

P||Fl(Cmax, RC) Bartal et al. (2000) Approximation algo for the off-line case and competitive algo for the on-line case.

P|pmtn|Fl(Cmax, RC) Seiden (2001) Competitive algorithm for the on-line case.

P,Q|pmtn|Fl(Cmax,RC) Hoogeveen et al. (2000)

Weakly NP-hard. Approx. scheme.

R|pmtn|Fl(Cmax,RC) Hoogeveen et al. (2000)

Strongly NP-hard. Approx. scheme.

O|pmtn|Fl(Cmax,RC) Hoogeveen et al. (2000)

Strongly NP-hard. Approx. scheme.

• Multiple machines problems,

Structure

• Some models and algorithms,– Scheduling with interfering job sets,– Scheduling with rejection cost.

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 =i wiUi, 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-Based Approach to Solve Very Large Instances, à paraître dans Journal of Scheduling.

• Utilisation de l’approche -contrainte,Minimiser Uw

scLmax (A)

• La contrainte (A) est équivalente à :

Ci Di=di+, i=1,…,n

• Pour calculer un optimum de Pareto on résout le problème noté 1|di , Di| Uw

Bicriteria scheduling and Math. Prog.• Qu’avons-nous fait pour résoudre le problème 1|di , Di|

• 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.

• Modélisation linéaire en variables bivalentes,

• xi = 1 si Ji est en avance,• Bt = {i/Dit} et At = {i/di>t},• Formulation indexée sur le temps (|T|2n),

T={di,Di}i

• Calcule d’une borne inférieure (heuristique),Propriété : Soit pipj, 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,

• 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.

• 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.

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+ jHB 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+ jHB rj xj ≤ LB jHB rj xj ≤ LB-UBlp

• 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.

• On utilise également une technique de fixation basée sur les pseudocosts uj et ljSoit xj une 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-LB alors xj=0

Si xj*lj ≤ UBlp-LB alors xj=1

• Pour calculer lj et uj on peut utiliser les pénalités de Dantzig1

1 Dantzig (1963). Linear Programming and Extensions, Princeton University Press, Princeton.

• Algorithme de preprocessing,(1)Résoudre le LP,(2)Fixer des variables par les coûts réduits,(3)Fixer des variables par les pseudocosts,(4)Si l’étape 3 a permis de fixer des variables, aller

en (1).

Permet de fixer environ 95% des variables.

• 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.

• Quelques résultats,• Cplex seul résout jusqu’à n=4000 en moins de

290s en moyenne,

G=100*(UB-Opt)/Opt

G=100*(LB-Opt)/Opt

Bicriteria scheduling and Math. Prog.• Pas de résultat sur l’énumération des optima de

Pareto,

• Approche testée sur un autre problème d’ordonnancement1,• Le problème F2|di=d, d unknown | d, U,• Le calcul d’un optimum de Pareto se fait jusqu’à

n=3000 (Cplex limité à n=2000 et la litérature à n=900),• On fixe environ 85% des variables.

• L’énumération des (n+1) optima de Pareto strict se fait jusqu’à n=500 en moins de 800s.

1 T’kindt, Della Croce, Bouquard (2007). Enumeration of Pareto Optima for a Flowshop Scheduling Problem with Two Criteria, Informs JOC, 19(1):64-72.

Now what’s 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,• …

• 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, …

• Industrial applications,• Are often multicriteria by nature,• Practical application of theoretical models.

You want to know more?

V. T’kindt, JC. Billaut (2006). Multicriteria Scheduling: Theory, Models and Algorithms. Springer.

Multicriteria Scheduling: Theory and Models Vincent TKINDT Laboratoire dInformatique (EA 2101)...

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