Multicriteria Scheduling: Theory and Models

1
Multicriteria Scheduling Theory and Models

• Vincent TKINDT
• Laboratoire dInformatique (EA 2101)
• Dépt. Informatique - PolytechTours
• Université François-Rabelais de Tours France
• tkindt_at_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), ?i1,,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)
lt Zi(x), ?i1,,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 Makers preferences.
6
Theory of Multicriteria Scheduling

• How can be expressed decision makers preferences?

• By means of weights (wi for criterion Zi),
• By means of goals (fex Zi ? LBUB),
• By means of bounds (Zi ?? ei),
• By means of an absolute order.

• Numerous studies can be found in the literature,
• Convex combination of criteria (Geoffrions
  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 Si aiZi(x) st x ? S ai ? 01, Si ai 1

• Strong convex hypothesis (Geoffrions theorem).
• Discrete case supported vs non supported Pareto
  optima.

• e-constraint approach,

Min Z1(x) st x ? S Zi ? ei, ? i2,,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 1diLmax, C

• A single machine is available,

• n jobs have to be processed,

• pi processing time,

• di due date,

• Minimize Lmaxmaxi(Ci-di) and CSi Ci,

p1
3
2
1
C1
C2
C3
2
1
3
Machine
time
d2
d1
d3
10
Theory of Multicriteria Scheduling

• Illustration on an example problem 1diLmax, C

Design of an a posteriori algorithm1
A strict Pareto optimum is calculated by means of
the e-contraint approach

• Known results
• The 1C problem is solved to optimality by
  Shortest Processing Times first rule (SPT),
• The 1diLmax 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,
442-48.
11
Theory of Multicriteria Scheduling

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

• Lmax ? e
• maxi(Ci-di) ? e
• Ci-di ? e, ? i1,,n
• Ci ? Di di e, ? i1,,n

12
Theory of Multicriteria Scheduling

• Decision Aid module,

1. Solve the 1diLmax problem gt Lmax value.
2. Solve the 1C problem gt s0, C(s0), Lmax(s0).
3. Es0, eLmax(s0)-1.
4. While e gt Lmax Do
5. Solve the 1Didi e C problem gt s,
6. EE//s, eLmax(s)-1.
7. End While.
8. Return E

Lmax
Lmax(s0)
e
s
e
e
Lmax
C
C(s0)
13
Theory of Multicriteria Scheduling

• Scheduling module (how to solve the 1DiC
  problem),

1
2
3
3
1
2
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(n1)/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. Tkindt, K. Bouibede-Hocine, C. Esswein
(2007). Counting and Enumeration Complexity with
application to Multicriteria Scheduling, Annals
of Operations Research, 153215-234.
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 1Si uiCi, Si
  viCi
• The criteria can be formulated as
• Si uiCi Si Sk ui pk xki
• and
• Si viCi Si Sk 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
• Cy ? 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
• ui0, ?i1,,l, and ui0, ?il1,,n
• and
• vi 0, ?i1,,l, and vi0, ?il1,,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 1diLmax, 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 1Fl(Cmax, wCsum) problem,
• Fl(Cmax, wCsum) CAmax awCBsum

wiawi
p1p1p2p3 / w11
4
5
6
3
2
1
1
wCBsum
CAmax
1
4
5
6
Machine
0
time
WSPT on the fictitious A job and B jobs with
weights wi
30
Scheduling with interfering job sets
Problem Reference Note
1diFl(Cmax,Lmax) Baker and Smith (2003) Yuan et al. (2005) Polynomial in O(nB log(nB)).
1diFl(Cmax,wCsum) Baker and Smith (2003) Polynomial in O(nB log(nB)).
1diFl(Lmax,wCsum) Baker and Smith (2003) Yuan et al. (2005) NP-hard. Polynomial for wi1.
1e(fAmax/fBmax) Agnetis et al. (2004) O(n2AnB log(nB)). At most nAnB Pareto.
1e(wCsumA/fBmax) Agnetis et al. (2004) NP-hard. Polynomial for wi1 (at most nAnB Pareto).
1die(UA/fBmax) Agnetis et al. (2004) O(nA log(nA)nB log(nB)).
1die(UA/UB) Agnetis et al. (2004) O(n2A nBn2B nA).
1diSjwUj Cheng and Juan (2006) m job sets. Strongly NP-hard.
1die(wCsumA/UB) Agnetis et al. (2004) NP-hard.
1e(CsumA/CsumB) Agnetis et al. (2004) NP-hard (at most 2n Pareto).
31
Scheduling with interfering job sets

• Multiple machines problems,

Problem Reference Note
JdiZA,ZB Agnetis et al. (2000) ZA and ZB are quasi-convexe functions of the due dates. Enumerate the Pareto.
F2e(CmaxA/CmaxB) Agnetis et al. (2004) NP-hard.
O2e(CmaxA/CmaxB) Agnetis et al. (2004) NP-hard.
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 RCSi rci,
• ? Often Fl(Z,RC)ZRC is minimized.

33
Scheduling with rejection cost

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

i pi rci
1 1 2
2 2 4
3 4 1
4 5 5
Job i pi processing time, rci rejection
cost.
3
2
1
4
3
2
1
4
Machine
0
time
Fl23
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 1Fl(Csum, RC) problem,
• Fl(Csum, RC) Csum RC

i pi rci
1 1 2
2 2 4
3 4 1
4 5 5
3
2
1
4
Machine
0
time
Fl13
Compute the variations in the objective function
Di Di -1-1---3
35
Scheduling with rejection cost

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

i pi rci
1 1 2
2 2 4
3 4 1
4 5 5
3
4
2
1
Machine
0
time
Fl10
Compute the variations in the objective function
Di Di 01----
36
Scheduling with rejection costs

• Single machine problems,

Problem Reference Note
1Fl(wCsum,RC) Engels et al. (1998) Weakly NP-hard. Dyn. Prog and approx. scheme. Polynomial if wiw or pip.
1ri,precFl(wCsum,RC) Engels et al. (1998) Approximation scheme.
1diFl(Lmax,RC) Sengupta (1999) Weakly NP-hard. Dyn. Prog and approx. scheme.
1ri,di,pi contrFl(Si (Ri-wiTi-cixi),RC) Yang and Geunes (2007) Ri profit, xi compression amount, ci compression cost. NP-hard. Heuristic.
37
Scheduling with rejection costs

• Multiple machines problems,

Problem Reference Note
PFl(Cmax, RC) Bartal et al. (2000) Approximation algo for the off-line case and competitive algo for the on-line case.
PpmtnFl(Cmax, RC) Seiden (2001) Competitive algorithm for the on-line case.
P,QpmtnFl(Cmax,RC) Hoogeveen et al. (2000) Weakly NP-hard. Approx. scheme.
RpmtnFl(Cmax,RC) Hoogeveen et al. (2000) Strongly NP-hard. Approx. scheme.
OpmtnFl(Cmax,RC) Hoogeveen et al. (2000) Strongly NP-hard. Approx. scheme.
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 dordonnancement
  suivant,
• Le problème est noté 1di 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.
• Lmaxmaxi(Ci-di), le plus grand retard
  algébrique,
• Uw Si wiUi, avec Ui1 si Cigtdi 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
Bicriteria scheduling and Math. Prog.

• Utilisation de lapproche e-contrainte,
• Minimiser Uw
• sc
• Lmax ? e (A)
• La contrainte (A) est équivalente à
• Ci ? Didi e, ?i1,,n
• Pour calculer un optimum de Pareto on résout le
  problème noté 1di , Di Uw

41
Bicriteria scheduling and Math. Prog.

• Quavons-nous fait pour résoudre le problème 1di
  , Di Uw ?
• 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
Bicriteria scheduling and Math. Prog.

• Modélisation linéaire en variables bivalentes,

• xi 1 si Ji est en avance,
• Bt i/Di?t et At i/digtt,
• Formulation indexée sur le temps (T?2n),
• Tdi,Dii

43
Bicriteria scheduling and Math. Prog.

• Calcule dune borne inférieure (heuristique),
• Propriété Soit pi?pj, dj ?di, Di ?Dj, wj ?wi,
  avec au moins une inégalité stricte. On a (i gtgt
  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
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(n3) par swap de travaux en
  avance et en retard.

45
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
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
• cxUBlp Sj?HB rj xj
• avec HB lensemble 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 xj0
• De même on peut fixer des variables à 1 en
  introduisant des variables décart sj
• xjsj1
• 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 pseudocosts uj et lj
• Soit xj une variable réelle de base du LP et on
  pose
• lj une binf sur la diminution unitaire du coût
  si xj0
• uj une binf sur la diminution unitaire du coût
  si xj1
• Si (1-xj)uj UBlp-LB alors xj0
• Si xjlj UBlp-LB alors xj1
• Pour calculer lj et uj on peut utiliser les
  pénalités de Dantzig1
• 1 Dantzig (1963). Linear Programming and
  Extensions, Princeton University Press,
  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 létape 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 dabord,
• 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.

51
Bicriteria scheduling and Math. Prog.

• Quelques résultats,
• Cplex seul résout jusquà n4000 en moins de 290s
  en moyenne,

G100(UB-Opt)/Opt
G100(LB-Opt)/Opt
52
Bicriteria scheduling and Math. Prog.

• Pas de résultat sur lénumération des optima de
  Pareto,
• Approche testée sur un autre problème
  dordonnancement1,
• Le problème F2did, d unknown d, U,
• Le calcul dun optimum de Pareto se fait jusquà
  n3000 (Cplex limité à n2000 et la litérature à
  n900),
• On fixe environ 85 des variables.
• Lénumération des (n1) optima de Pareto strict
  se fait jusquà n500 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
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
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
Now whats going on?

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

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