Optimisation combinatoire multiobjectif (slide #2)
GT “Contraintes et RO” – June 7, 2005
Optimisation combinatoire multiobjectif (slide #3)
GT “Contraintes et RO” – June 7, 2005
Content
Optimisation combinatoire multiobjectif
• Some definitions and characteristics of MOCO
Xavier GANDIBLEUX
• Some problems and some applications
LINA - Laboratoire d’Informatique de Nantes Atlantique Universite de Nantes 2 rue de la Houssiniere BP92208, F-44322 Nantes cedex 03 – FRANCE
[email protected]
• Exact Solution Methods and MOCO Journ´ ees du groupe de travail “Contraintes et RO” du GdR CNRS ALP “PPC et optimisation multi-objectifs ou d´ ecision multi-crit` eres” R´ eunion du 7 juin 2005 — Lens, France
• MultiObjective MetaHeuristics and MOCO
with Matthias EHRGOTT University of Auckland – NEW ZEALAND
• Epilogue
Xavier Gandibleux
Optimisation combinatoire multiobjectif (slide #4)
GT “Contraintes et RO” – June 7, 2005
Optimisation combinatoire multiobjectif (slide #5)
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Fundamental definitions
Xavier Gandibleux
Optimisation combinatoire multiobjectif (slide #6)
Objective functions • s ∈ X, wj : A → Z ,
Decisions - Decision space
j = 1, . . . , p
• Finite set A = {a1 , . . . , an }
Some definitions and characteristics
weight functions
• X ⊆ 2A
◦ S-type: z j (s) =
• s∈X
◦ M-type: z j (s) = maxa∈s wj (a)
a∈s
wj (a)
Example
• Fundamental definitions
MultiObjective Combinatorial Optimization problem
◦ A = edges of graph
• Definition of optimal solution
◦ X = paths
• Approximate solutions
“min”(z1 (s), . . . , zp (s))
Xavier Gandibleux
GT “Contraintes et RO” – June 7, 2005
Xavier Gandibleux
Optimisation combinatoire multiobjectif (slide #8)
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Example 1: (MOAP) Formulation in terms of binary variables :
⎡
• a variable xi for each element ai ∈ A
j ⎢ min z (x) = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
• representation of s ∈ X as binary vector x ∈ {0, 1}n ⎧ ⎨ 1 ai ∈ s xi = ⎩ 0 else
n n i=1 l=1
n i=1 n
cjil xil
xil = 1 xil = 1
j = 1, . . . , p l = 1, . . . , n i = 1, . . . , n
l=1
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
• k constraints of specific structure defining X
Xavier Gandibleux
Optimisation combinatoire multiobjectif (slide #9)
GT “Contraintes et RO” – June 7, 2005
⎡ j ⎢ max z (x) = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
n i=1
n
⎤
cji xi
j = 1, . . . , p
wil xi ≤ ωl
l = 1, . . . , k
i=1
wil ≥ 0
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
xi ∈ {0, 1}
Multiobjective Assignment Problem
k = 1 → Multiobjective Knapsack Problem k > 1 → Multiobjective Multidimensionnal Knapsack Prob.
INP -hard (Serafini 1986) (Reason: partition structure)
INP -hard (Reason: knapsack structure)
• n variables xi , i = 1, . . . , n, • p objectives z j , j = 1, . . . , p
Xavier Gandibleux
Example 2: (MO[M]KP) ⎤
xil ∈ {0, 1}
(MOCO) is a discrete optimization problem, with
(MOCO)
s∈X
◦ s = a path
Optimisation combinatoire multiobjectif (slide #7)
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Xavier Gandibleux
Xavier Gandibleux
Optimisation combinatoire multiobjectif (slide #10)
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Optimisation combinatoire multiobjectif (slide #11)
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Definition of optimal solution
Example 3: (MOSPP) ⎡
n
j ⎢ max z (x) = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
i=1
n
⎤
cji xi
j = 1, . . . , p
til xi ≤ 1
l = 1, . . . , k
• Decision space X ⊂ (0, 1)n
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
i=1
til ∈ {0, 1}
A solutions x ∈ X
Optimisation combinatoire multiobjectif (slide #12)
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Definition of optimal solution
• Objective space Z = z(X) ⊂ Zp A vector (point) z(x) ∈ Z
• Two solutions x, x ∈ X
z2
• Vectors (points) : z(x), z(x ) ∈ Z z(x’)
Dominance relation
z2
x2
xi ∈ {0, 1}
• z(x) dominates z(x ) iff z(x) ≤ z(x ) and z(x) = z(x )
z ∈Z
x ∈X
Multiobjective Set Packing Problem INP -hard (Reason: knapsack structure)
z1
x1
z(x)
• z(x) strongly dominates z(x ) iff z(x) < z(x )
z1
∴ Xavier Gandibleux
Optimisation combinatoire multiobjectif (slide #13)
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Definitions - Optimal solution (2) 1 2 • x1 = lexmin x∈X (z (x), z (x))
min
x∈X z 1 (x)≤z 1 (x1 )
Optimisation combinatoire multiobjectif (slide #14)
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Definitions - Optimal solution (2) z2
min (z 1 (x)) 1) x1 = x∈X 2) x1 =
Xavier Gandibleux
Weak efficiency z(x1')
(z 2 (x))
• Weakly efficient solutions: XW z(x2)
Lexicographic optimality
z1
• x is an efficient solution iff x ∈ X and x ∈ X such that z(x) dominates z(x )
A
B
A
B
• Efficient solutions: XE ⊆ XW
• When x ∈ XW then z(x ) is a weakly non-dominated point
C
z1
Xavier Gandibleux
GT “Contraintes et RO” – June 7, 2005
z1
Xavier Gandibleux
Optimisation combinatoire multiobjectif (slide #17)
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Number of efficient solutions – theoretical results
Definitions - Optimal solution (4)
• When x ∈ XE then z(x ) is a nondominated point • Non-dominated points: ZN
• Weakly non-dominated pts: ZW N
• with respect to 1 or all permutations of the objective functions z j
Optimisation combinatoire multiobjectif (slide #16)
z2
C
z(x2')
• z(x1) 100
• Telecommunications and computer network (2-3 objectives) → ‘Multicriteria shortest path problem’ (Randriamasy et al. 2002) • Trip organization (2-5 objectives) → ‘Preference-based Multicriteria TSP ALS : TPP’ (Godart 2001) • Vehicle routing problem (2-7 objectives) → ‘Vehicle routing problems’ (El-Sherbeny 2001, Jozefowiez 2004) • Airline crew scheduling (2 objectives) → ‘Bicriteria set partitioning problems’ (Ehrgott and Ryan 2001) • Railway network infrastructure capacity (2 objectives) → ‘Bicriteria set packing problems’ (Delorme et al. 2001)
∴ Xavier Gandibleux
Optimisation combinatoire multiobjectif (slide #31)
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Xavier Gandibleux
Optimisation combinatoire multiobjectif (slide #32)
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Xavier Gandibleux
Optimisation combinatoire multiobjectif (slide #33)
Weighted Sums Method
MOCO in practice: characteristic
Solve In general (and it is not a surprize)... min
Exact Solution Methods & MOCO
large size combinatorial optimisation problems
0 ≤ λj ≤ 1
• Weighted Sums Method
with multiple objectives !!!
÷
j=1
⎩
λj z j (x) : x ∈ X
j=1
⎫ ⎬ (Pλ )
⎭
λj = 1
• All supported efficient solutions found
• Dynamic Programming
• Early papers: nonsupported efficient solutions: not known, neglected
• Ranking
• Usually p = 2, what if p ≥ 3?
• Specific Methods: Labeling
• Assignment (Dathe 1978), transportation (Srinivasan/Thompson 1976), knapsack (Rosenblatt/Sinuany-Stern, 1989), etc.
• Two Phases Method
Xavier Gandibleux
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p
⎧ p ⎨
• Compromise Programming • ε-Constraint ...to solve.
Optimisation combinatoire multiobjectif (slide #34)
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Xavier Gandibleux
Optimisation combinatoire multiobjectif (slide #35)
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Xavier Gandibleux
Optimisation combinatoire multiobjectif (slide #36)
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Example (1/2):
Example (2/2):
Compromise Programming
z(x) = λ1 z 1 (x) + λ2 z 2 (x) where λ1 ≥ 0, λ2 ≥ 0 and λ1 + λ2 = 1.
b : optimal solution of the weighted sum ? Never !
Minimize distance to ideal point z¯j := min z j (x) x∈X
Use Tchebycheff norm as distance measure p min max{λj |z j (x) − z¯j } : x ∈ X j=1
• All efficient solutions found • Usually INP -hard (shortest path, Murthy/Her, 1992)
The weighted sum for each vectors : z(a) = −12000λ1 + 420644λ2 = 420644 − 432644λ1 z(b) = −17250λ1 + 429359λ2 = 429359 − 446609λ1 z(c) = −21250λ1 + 443005λ2 = 443005 − 464255λ1
for λ1 ∈ [0, 22361 31611 ] c is optimal
for λ1 ∈ [ 22361 31611 , 1] a is optimal.
• With l1 norm same as weighted sum • With lp norm, 1 < p < ∞, nonlinear objective • Impact on interactive methods
When b is the optimal solution of the weighted sum ? Xavier Gandibleux
Xavier Gandibleux
Xavier Gandibleux
Optimisation combinatoire multiobjectif (slide #37)
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Optimisation combinatoire multiobjectif (slide #38)
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Optimisation combinatoire multiobjectif (slide #39)
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Example : z˜ = z¯ − Scalarizing function s(z(x), λ, z˜) : s(z(x), λ, z˜) = max
1≤j≤p
p j z j (x) , ρ > 0 λj z (x) − z˜j − ρ j=1
A suggested point : min s(z(x), λ, z˜)
x∈X
Xavier Gandibleux
Optimisation combinatoire multiobjectif (slide #40)
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Xavier Gandibleux
Optimisation combinatoire multiobjectif (slide #41)
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Specific Methods: Labeling
Specific Methods: Labeling
Shortest path problem: multiobjective label setting (Martins 1984)
Shortest path problem: multiobjective label setting (Martins 1984)
2
1
(10,1,1,1) (4,0,0,3)
6
(5,1,3,7)
(6,1,18,10)
3
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13,5,26,17
5
Xavier Gandibleux
Optimisation combinatoire multiobjectif (slide #43)
20,15,15,12
(6,0,0,6)
(1,4,8,1)
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Phase 1: Find supported efficient solutions Phase 2: Use information of phase 1 solutions to generate nonsupported efficient solutions: reduced costs, bounds, etc.
(10,4,2,10) (6,2,10,10)
Optimisation combinatoire multiobjectif (slide #42)
Two Phases Method
4
(0,10,12,1)
Xavier Gandibleux
• Appropriate if single objective problem is in P • Used for 2 objectives problems, how to generalize? • Applications: network flow (Lee/Pulat 1993), assignment(Vis´ee et al. 1998), spanning tree (Ramos et al. 1998), knapsack (Ulungu/Teghem 1994)
Xavier Gandibleux
Optimisation combinatoire multiobjectif (slide #44)
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Xavier Gandibleux
Optimisation combinatoire multiobjectif (slide #45)
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Multiobjective metaheuristics Z2
Metaheuristic: definition (Osman/Laporte 96, Glover/Laguna 97)
λAB ZA
Iterative master strategy that guides and modifies the operations of subordinate heuristics by combining intelligently different concepts for exploring and exploiting the search space
MOMH & MOCO • Multiobjective metaheuristics
ZB
• Evolutionary vs Neighborhood Search Algorithms • Hybrid and Problem-dependent Algorithms
• Applicable generally to a large number of problems • Metaheuristic, a solution concept • instantiated to specific problem: heuristic, a solution method
• Steps in ‘Hybridation’ Schemes
◦ Hybrid metaheuristic ? 0
Z1
Xavier Gandibleux
Xavier Gandibleux
Xavier Gandibleux
Optimisation combinatoire multiobjectif (slide #46)
GT “Contraintes et RO” – June 7, 2005
• constraint programming, genetic algorithms, evolutionary methods, neural networks, simulated annealing, tabu search, greedy randomized adaptive search, ant colony systems, particle swarm optimization, noising methods, variable neighborhood search, scatter search, etc. • MOMH: Metaheuristic for multiobjective optimization
Optimisation combinatoire multiobjectif (slide #47)
ZN ZPN
Aims: good tradeoff between the – Quality of XP E
0
Main principles
• 1984 (PhD thesis): Schaffer, VEGA
Optimisation combinatoire multiobjectif (slide #49)
2. Self adaptation, i.e. independent evolution
ZPN
3. Cooperation, i.e. exchange of information between individuals
cooperation
z(P)
Z1
• starting in 1995: MOGA+LS (95), PSA (96), TAMOCO (97)... • around 2000, the ‘MOMH boouum’ + growing interest in MOCO ! • MOMH =f (EA, NSA, Opt, specific)
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subpop 1 shuffle entire population
Optimisation combinatoire multiobjectif (slide #50)
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Z2
apply genetic operators
Q subpopulation are created
individual n
individual n
individuals are now mixed
start all over again
generation (t)
contributes to the evolution process to generate XP E
Xavier Gandibleux
Optimisation combinatoire multiobjectif (slide #51)
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1. Uniform convergence How to accomplish both fitness assignment and selection, in order to guide the search toward the efficient frontier?
individual 1
2. Uniform distribution How to maintain a diversified population in order to avoid premature convergence and find a uniform distribution of solutions along the efficient frontier?
(crossover, mutation)
subpop Q
Z1
Parallel process where the whole population
MO-EA: two central questions
• Extension of GENESIS to Vector Evaluated GA (VEGA) individual 1
0
⇒
Xavier Gandibleux
Vector Evaluated Genetic Algorithm by Schaffer (1984)
initial population size : N
self adaptation
1992 (10th MCDM): Serafini, acceptation rules for SA 1992 (EURO), 1993 (PhD thesis): Teghem/Ulungu, MOSA 1996 (MOPGP), 1997 (LNEMS): G./Mezdaoui/Fr´eville, MOTS 1997 (DSI): Sun, TS for MOCO 1998 (14th MCDM): G./Vancoppenolle/Tuyttens, GRASP-biSCP
Xavier Gandibleux
individual n
1. Initial population P
Z2
Neighborhood Search Algorithms (NSA) wave
∴
create subpop.
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Hybrid and Problem-dependent Algorithms wave
– Time & memory requirements
individual 1
Optimisation combinatoire multiobjectif (slide #48)
Evolutionary Algorithms
MOMH (and MOCO): timeline Evolutionary Algorithms (EA) wave
• • • • •
Z2
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generation (t+1)
• Non Pareto based method
0
ranking / niching / sharing
Z1
• Generation process (parallel selection)
Xavier Gandibleux
Optimisation combinatoire multiobjectif (slide #52)
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MO-EA: usual components
Xavier Gandibleux
Optimisation combinatoire multiobjectif (slide #53)
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Significant MO-EA
• evolutionary operators (crossover, mutation) • an archive of elite solutions • a ranking method • a guiding method • a clustering method
no archive
bounded archive
unbounded archive
only from population
from population and archive
only from archive
dominance level
dominance grade
others (e.g. strength)
density based (e.g. fitness sharing)
population structure
population + archive
population
archiving
selection elitism
primary fitness
secondary fitness
none
(Laumanns et al. 2001)
• a fitness measure
• Nondominated Sorting Genetic Algorithm (NSGA) by Srinivas and Deb, 1994.
C.A. Coello. A comprehensive survey of evolutionary-based multiobjective optimization techniques. Knowledge and Information Systems, 1(3):269-308, 1999.
• Niched Pareto Genetic Algorithm (NPGA) by Horn, Nafpliotis and Goldberg, 1994.
C.A. Coello. An updated survey of GA-based multiobjective optimization techniques. ACM Computing Surveys, 32(2):109–143, 2000.
• Multiple Objective Genetic Algorithm (MOGA95) by Murata and Ishibuchi, 1995. • Morita-G.-Katoh Algorithm (MGK) by Morita, G. and Katoh, 1998, 2001.
C.A. Coello. EMO repository. http://www.lania.mx/~ccoello/EMOO/
• Strength Pareto Evolutionary Algorithm (SPEA) by Zitzler and Thiele, 1998.
D. Jones, S.K. Mirrazavi, and M. Tamiz. Multi-objective meta-heuristics: An overview of the current state-of-the-art. EJOR 137(1):1-9, 2002.
• Multiple Objective Genetic Local Search (MO-GLS) by Jaszkiewicz, 2001. • Multiple Objective Genetic Tabu Search (MOGTS) by Barichard and Hao, 2002.
Xavier Gandibleux
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C. M. Fonseca and P. J. Fleming. An Overview of Evolutionary Algorithms in Multiobjective Optimization. Evolutionary Computation, 3(1):1–16, Spring 1995.
• Multiple Objective Genetic Algorithm (MOGA93) by Fonseca and Fleming, 1993.
• Pareto Archived Evolution Strategy (PAES) by Knowles and Corne, 1999.
• a penalty strategy for infeasible solutions, etc.
Optimisation combinatoire multiobjectif (slide #54)
Several Surveys...
• Vector Evaluated Genetic Algorithm (VEGA) by Schaffer, 1984.
• a population of solutions
Xavier Gandibleux
...and books
Xavier Gandibleux
Xavier Gandibleux
Optimisation combinatoire multiobjectif (slide #55)
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Neighborhood Search Algorithms
MO-EA and MOCO Problems In 2002: Transportation Problem Spanning Tree Problem Travelling Salesperson Problem Knapsack Problem Multi-constraint Knapsack Problem Set Covering Problem Containership Loading Design Single Machine Scheduling Problem
Optimisation combinatoire multiobjectif (slide #56)
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MO-NSA: usual components - a neighborhood structure N (moves: swap, switch, heuristic...)
MO-NSA: usual components 2 1 1 3 1 1 1 3
Optimisation combinatoire multiobjectif (slide #57)
- an exploration strategy (partial, exhaustive)
Z2
• Initial solution x0 ZPN
(starter)
• Neighbors {x} ∈ N (xt )
(visit)
• Search direction(s) λ
(cover)
λ
or reference point(s) z˙
(attract)
• Selection mechanism
z(x0 )
• Iterative principle
0
(choose) (go-ahead)
Z1
- an acceptation rule (SA principle, TS principle) - a list of candidates - a scalarizing function (local aggregation) S(z(x), z, ˙ λ) - an oscillation strategy - a greedy (randomized) strategy - a path-relinking strategy, etc.
⇒ Sequential process in the objective space Z ∴ Xavier Gandibleux
Optimisation combinatoire multiobjectif (slide #58)
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MOSA by Ulungu/Teghem, 1992
Xavier Gandibleux
Optimisation combinatoire multiobjectif (slide #59)
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• Initial solution x0
• Use of wide diversified set of weights λ ∈ Λ
λ
zU
• Use a domination acceptance principle (isBetter) • Generate sets of potential efficient solutions XP E|λ
z(x) z(xn)
λ
N( z(xn) ) z1
• Use a scalarizing function which aggregates the multi objective information in a single objective one S(z(x), λ) Experiments on knapsack, assignment, scheduling, and VRP problems
• Neigh. structure N (z(x0 )) • Search directions λ • Tabu process / features
z2
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The period 84-95: EA and NSA for MOCO EA parallel convergence towards the non-dominated frontier, simple, slow, ‘ignore’ the combinatorial structure NSA ‘aggressive’ sequential convergence towards a subpart of the nondominated frontier, fast, sophisticated, must be guided
• Reference point z U • Local aggregation mechanism s(z(x), z U , λ) • Tabu memory to browse Z
TS for MOCO by Sun, 1997, same ideas but in an interactive procedure (using the preferences of the decision-maker)
Xavier Gandibleux
Optimisation combinatoire multiobjectif (slide #61)
Optimisation combinatoire multiobjectif (slide #60)
Hybrid and Problem-dependent Algorithms
MOTS by G./Mezdaoui/Fr´ eville, 1996
Based on the simulated annealing metaheuristic, plus :
Xavier Gandibleux
The period 95-04: MOMH for MOCO → fusion/collaborartion of ideas/principles from EA and NSA: ‘hybrid’ algorithms or simply a level of maturity of MOMH! → use information derived from the combinatorial structure: an useful source of knowledge → existing techniques in combinatorial optimization (e.g. cuts)
Xavier Gandibleux
Optimisation combinatoire multiobjectif (slide #62)
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Xavier Gandibleux
Optimisation combinatoire multiobjectif (slide #63)
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Steps in ‘Hybridation’ Schemes
Problem dependent components
2) EA as master strategy, NSA as secondary strategy:
1) EA components integrated in NSA: P drive local search processes
- specific crossover operators (e.g. Lacomme et al. 03 / biCARP) - specific neighborhood structures
Pareto Simulated Annealing (PSA) by Czyzak/Jaszkiewicz, 1996
- bound sets on the non-dominated frontier • Initial solutions: use sample S ⊂ X of solutions
- handling constraints in a relaxation strategy - numerical properties of the objective functions
Genetic plus local search: The memetic version of MOGA by Murata/Ishubushi, 1995
• Exploration principle: interaction between solutions guides the generation process through the values of λ
- properties of subsets of exact solutions (easily computed) - etc.
to make the EA aggressive in improving as far as possible good solutions with a NSA: a depth first search method, a basic (or truncated) tabu search, etc.
MGK by Morita/G./Katoh, 1998, 2001 MOG-LS by Jaszkiewicz, 2001 Genetic plus tabu search:
∴ Xavier Gandibleux
Tabu search for multiobjective combinatorial optimization (TAMOCO) by M.P. Hansen, 1997, 2000
Xavier Gandibleux
GTSMOKP by Barichard/Hao, 2002
Xavier Gandibleux
Optimisation combinatoire multiobjectif (slide #64)
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Example (1/4)
Optimisation combinatoire multiobjectif (slide #65)
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Example (2/4)
Optimisation combinatoire multiobjectif (slide #66)
Example (3/4)
MOMKP n=250, p=2, k=2
MOMKP n=250, p=2, k=2
MOMKP n=250, p=2, k=2
10500
10500
10500 E VEGA SPEA NSGA
E VEGA 10000
9500
9500
9500
9000
9000
9000
8500
8500
8500
8000
8000
8000
7500
8000
8500
9000
9500
7500 7000
10000
7500
8000
z1
8500
9000
9500
E VEGA SPEA NSGA MOGLS MOGTS
10000
10000
7500 7000
7500 7000
10000
7500
8000
z1
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8500
9000
9500
10000
z1
Xavier Gandibleux
Optimisation combinatoire multiobjectif (slide #67)
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Xavier Gandibleux
Optimisation combinatoire multiobjectif (slide #68)
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Xavier Gandibleux
Optimisation combinatoire multiobjectif (slide #69)
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Example (4/4) 3) Alternating schemes based on EA and NSA as a blackbox
MOMKP n=250, p=2, k=2
10500
E VEGA SPEA NSGA MOGLS MOGTS MGK
10000
A hybrid algorithm using both EA and NSA independently by Ben Abdelaziz/Chaouachi/Krichen, 1997, 1999. Step 1: EA (a GA) : to produce a first diversified approximation
9500
9000
8000
7500 7000
Experiment : the multiobjective knapsack problem. 7500
8000
8500
9000
9500
10000
z1
Experiment : the biobjective set packing problem.
Xavier Gandibleux
Optimisation combinatoire multiobjectif (slide #70)
Step 1: λ-GRASP to compute an initial set of very good solutions P0 Step 2: from P ⊇ P0 , to consolidate the approximation of the non-dominated frontier with A-SPEA (SPEA1 + all XP E + aggressive LS)
Step 2: NSA (a TS) : to improve the output of the GA
8500
A scheme based on an NSA interfaced with an EA by Delorme/G./Degoutin, 2003, 2005.
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Xavier Gandibleux
Optimisation combinatoire multiobjectif (slide #71)
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Xavier Gandibleux
Optimisation combinatoire multiobjectif (slide #72)
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Path-relinking in MOP:
4) EA + NSA + problem dependent components
(Path-relinking by Glover/Laguna, 1997: a component for TS)
Starting point : Similarities between efficient solutions EA : population, evolutionary operators NSA : LS, guiding strategy Problem dependent components : bound sets, XSEm z2
IB
A crossover using a ‘genetic map’ of the population by G./Morita/Katoh, 2003 (EMO) A guiding strategy based on a path-relinking using XP E by G./Morita/Katoh, 2003 (MIC), 2004 (JMMA)
the average percentage of exact solutions found (120 instances; 16 runs; same computational effort)
IA 0
N(IA)
0 z1
Xavier Gandibleux
Xavier Gandibleux
Xavier Gandibleux
Optimisation combinatoire multiobjectif (slide #73)
GT “Contraintes et RO” – June 7, 2005
Optimisation combinatoire multiobjectif (slide #74)
GT “Contraintes et RO” – June 7, 2005
Optimisation combinatoire multiobjectif (slide #75)
GT “Contraintes et RO” – June 7, 2005
5) Approximation method and exact procedure together Path-relinking operator: comparison with MOSA
Inside an approximation method : cuts which eliminate parts of the decision space where (provably) no exact efficient solution exists by G./Fr´eville, 2000 — Experiment : BiKP
M1 (improved MOSA) vs M1 (min/avg/max) & CPUt (avg) for rule 1+3 with PEinit 100 %
500 s 450 s
80 %
400 s
UB=105
UB-=104
: proposition 1
350 s
Inside an exact method: seek and cut method: “seek” computes a local ZP N used for “cutting” the search space of an implicit enumeration scheme by Przybylski/G./E., 2004 — Experiment : BiAP Results with the bound 2
: proposition 2
16000
2ph (PR1)(N1) (PR1)(N2) (PR2)(N1) (PR2)(N2) (PR3)(N1)
100%
40 %
90% 80%
200 s
10000 60% layers (1...n) 50% in % 40%
100 s M1 (min/avg/max) CPUt (avg) improved MOSA (M1)
50 s
0% 60
70
80
90
layers to visit
8000
6000
30%
0s 05 10 15 20 25 30 35 40 45 50
12000
layers eliminated
70%
150 s 20 %
14000
CPU_t (s)
250 s
CPUt(sec)
300 s
M1 (%)
60 %
4000
100 20%
Instance size (n x n) 10%
PowerPC G4 400 Mhz / RAM : 256 Mo / OS : Mac OS X 10.2.6 / Language : C / Compiler : gcc-3.1 / Optimizer option : -O3 Resolution repeated 5 times using different seeds / avg values / generations: 250 000 iterations / refreshment: each 100 000 iterations
0%
0
KP50
KP100
KP150
LB=44
KP200
KP250
LB+=78
KP300
KP350
KP400
KP450
GT “Contraintes et RO” – June 7, 2005
50
KP500
Optimisation combinatoire multiobjectif (slide #77)
GT “Contraintes et RO” – June 7, 2005
• missing: the decision-aid for the decision-maker (problem dependant), coming with real-life applications • no need of a new ‘sexy’ metaphor, but more and more efficient algorithms, “recycling” the 50 years of knowledge of (single objective) optimisation with new optimization technologies.
Xavier Gandibleux
GT “Contraintes et RO” – June 7, 2005
75 80 instance size
85
95
100
÷ Xavier Gandibleux
Optimisation combinatoire multiobjectif (slide #78)
GT “Contraintes et RO” – June 7, 2005
• M. Ehrgott, X. Gandibleux eds (2002) Multicriteria Optimization: A State-of-the-Art Annotated Bibliographic Survey; International Series in Operations Research and Management Science, Kluwer. • X. Gandibleux, M. Sevaux, K. S¨ orensen, V. T’kindt eds (2004) Metaheuristics for Multiobjective Optimisation. Lecture Notes in Economics and Mathematical Systems 535, Springer, Berlin.
Xavier Gandibleux
Optimisation combinatoire multiobjectif (slide #80)
90
• M. Ehrgott (2000) Multicriteria Optimization; Lecture Notes in Economics and Mathematical Systems 491, Springer, Berlin.
– the quality measures – the scalability problem – theoretical aspects of optimal solutions, etc.
∴
Optimisation combinatoire multiobjectif (slide #79)
70
Ressources: Books
• many hot topics:
• Some information and links
65
Some information and links
• MOMH =f (EA, NSA, Opt, specific)
• Conclusion and the future
60
Xavier Gandibleux
Conclusion and the future
Epilogue
55
instance
Xavier Gandibleux
Optimisation combinatoire multiobjectif (slide #76)
2000
layers eliminated
GT “Contraintes et RO” – June 7, 2005
Xavier Gandibleux
Optimisation combinatoire multiobjectif (slide #81)
GT “Contraintes et RO” – June 7, 2005
Ressources: Journal issues • X. Gandibleux, A. Jaszkiewicz, A. Fr´eville, R. Slowinski (2000), Special issue ‘Multiple Objective MetaHeuristics’. Journal of Heuristics, 6(3), August 2000. • M. Ehrgott, J. Figueira, X. Gandibleux (2005). Special issue ‘Multiple Objective Discrete and Combinatorial Optimization’. Annals of Operations Research, To appear in 2005.
www.terry.uga.edu/mcdm/
Xavier Gandibleux
Xavier Gandibleux
Xavier Gandibleux
Optimisation combinatoire multiobjectif (slide #82)
GT “Contraintes et RO” – June 7, 2005
Xavier Gandibleux
