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Multicriteria Scheduling: Theory and Models Vincent TKINDT Laboratoire dInformatique (EA 2101)...
Transcript of Multicriteria Scheduling: Theory and Models Vincent TKINDT Laboratoire dInformatique (EA 2101)...
Multicriteria Scheduling: Theory and Models
Vincent T’KINDTLaboratoire d’Informatique (EA 2101)Dépt. Informatique - Polytech’Tours
Université François-Rabelais de Tours – [email protected]
Multicriteria Scheduling: Theory and Models
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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.
E WE
Z1
Z2
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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.
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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 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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Theory of Multicriteria Scheduling
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
Module which takes into account the criteria
Module which solves the scheduling problem
value of the parameters
a Pareto optimum
a Pareto optimum
• How to calculate the “best” strict Pareto optimum ?
Algorithms based on an a posteriori method
Module which takes into account the criteria
Module which solves the scheduling problem
value of the parameters
a Pareto optimum
a set of Pareto optima
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Theory of Multicriteria Scheduling
• 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
Multicriteria Scheduling: Theory and Models
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Theory of Multicriteria Scheduling
• 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
p1
• Minimize Lmax=maxi(Ci-di) and C=i Ci,
11 33
C1 C2 C3
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Theory of Multicriteria Scheduling
• 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),
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Theory of Multicriteria Scheduling
• 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,
…,n
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s
Theory of Multicriteria Scheduling
• 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;
C
Lmax
Lmax*
C(s0)
Lmax(s0)
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Theory of Multicriteria Scheduling
• Scheduling module (how to solve the 1|Di|C problem),
time0
Machine
11 22 33
D1
D2
D3
331122
7
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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.
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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).
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Theory of Multicriteria Scheduling
• From a theoretical viewpoint… complexity theory,– Originally dedicated to decision problems,
• Scheduling problems are often optimisation problems,
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Theory of Multicriteria Scheduling
• 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.
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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. T’kindt, K. Bouibede-Hocine, C. Esswein (2007). Counting and Enumeration Complexity with application to Multicriteria Scheduling, Annals of Operations Research, 153:215-234.
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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
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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 ci
1xi
Min i ci2 xi
st Ax b x N2
x2
x1
c1
c2c1 and c2 are the generators of cone
C
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Theory of Multicriteria Scheduling
x2
x1
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Theory of Multicriteria Scheduling
• 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
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Theory of Multicriteria Scheduling
• 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
c2
C
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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,…,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.
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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.
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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 interfering job sets,– Scheduling with rejection cost.
• Solution of bicriteria single machine problem by mathematical programming
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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,• ….
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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).
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Scheduling with interfering job sets• Consider the 1||Fl(Cmax, wCsum) problem,
Fl(Cmax, wCsum) = CAmax + wCB
sum
time0
Machine
44 55 66332211
CAmax
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
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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).
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Scheduling with interfering job sets
Problem Reference Note
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,
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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.
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Scheduling with rejection cost
• Consider the 1||Fl(Csum, RC) problem,
Fl(Csum, RC) = Csum + RC
time0
Machine
332211 44
Job i: pi: processing time, rci: rejection cost.
SPT to get the initial sequencing
332211 44
i pi rci
1 1 2
2 2 4
3 4 1
4 5 5
Compute the variations in the objective function i:i =[ -2;-3;-10;-7]
Fl=23
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Scheduling with rejection cost
• Consider the 1||Fl(Csum, RC) problem,
Fl(Csum, RC) = Csum + RC
time0
Machine
33
2211 44
i pi rci
1 1 2
2 2 4
3 4 1
4 5 5
Compute the variations in the objective function i:i =[ -1;-1;--;-3]
Fl=13
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Scheduling with rejection cost
• Consider the 1||Fl(Csum, RC) problem,
Fl(Csum, RC) = Csum + RC
time0
Machine
33
2211
44
i pi rci
1 1 2
2 2 4
3 4 1
4 5 5
Compute the variations in the objective function i:i =[ 0;1;--;--]
Fl=10
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Scheduling with rejection costs
Problem Reference Note
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,
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Scheduling with rejection costs
Problem Reference Note
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,
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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 interfering job sets,– Scheduling with rejection cost.
• Solution of bicriteria single machine problem by mathematical programming
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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.
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Bicriteria scheduling and Math. Prog.
• 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
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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.
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Bicriteria scheduling and Math. Prog.
• 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
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Bicriteria scheduling and Math. Prog.
• 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,
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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.
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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.
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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
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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.
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Bicriteria scheduling and Math. Prog.
• 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.
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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 l’étape 3 a permis de fixer des variables, aller
en (1).
Permet de fixer environ 95% des variables.
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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.
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Bicriteria scheduling and Math. Prog.
• 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
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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.
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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,• …
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Now what’s 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, …
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Now what’s going on?
• Industrial applications,• Are often multicriteria by nature,• Practical application of theoretical models.
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You want to know more?
V. T’kindt, JC. Billaut (2006). Multicriteria Scheduling: Theory, Models and Algorithms. Springer.