Bounds and Bound Sets for Biobjective Combinatorial Optimization Problems :
set covering, set packing and set partitionning problems with two objectives
Xavier GANDIBLEUX1 and Matthias EHRGOTT2
(1) LAMIH - Recherche Op´ erationnelle et Informatique Universit´ e de Valenciennes et du Hainaut-Cambr´ esis Le Mont Houy, F-59313 Valenciennes cedex 9 – FRANCE
[email protected]
(2) Department of Engineering Science University of Auckland Private Bag 92019, Auckland – NEW ZEALAND
[email protected]
MCDM 2002 Winter Conference February 18-23, 2002 — Semmering, Austria
MCDM 2002, February 18-23, 2002
Bounds and Bound Sets
First study, 15th MCDM Int. Conf.
• Bounds : ideal point, nadir point. • Bound sets :
linear relaxation, greedy bounds, specific bounds (MartelloToth bounds, Christofides’bound, . . . ). • Biobjective combinatorial problems :
Assigment problem, Travelling salesman problem, Knapsack problem.
X. Gandibleux, M. Ehrgott
MCDM 2002, February 18-23, 2002
Bounds and Bound Sets
Today, 16th MCDM Int. Conf. :
• Bound sets : linear relaxation, greedy bounds.
• Feedback following the numerical experiments
. limit of a general IP solver (LPSOLVE, CPLEX),
. link between functions and difficulties to solve, . characteristics of solutions observed,
. quality of approximations compared to exact solutions. • Biobjective combinatorial problems :
Set Covering Problem, Set Packing Problem, Set Partitioning Problem.
X. Gandibleux, M. Ehrgott
MCDM 2002, February 18-23, 2002
Bounds and Bound Sets
n X
ci1 xi
∈
{0, 1}
≥ 1 j = 1, . . . , m
The Set Covering Problem (SCP)
min
i=1
n X
ci2 xi aji xi
i=1
n X
min subject to
i=1
xi
where a aji ∈ {0, 1} and aji = 1 means variable xi covers constraint j.
X. Gandibleux, M. Ehrgott
MCDM 2002, February 18-23, 2002
Bounds and Bound Sets
n X
ci1 xi
∈
{0, 1}
≤ 1 j = 1, . . . , m
The Set Packing Problem (SPP)
max i=1
n X
ci2 xi aji xi
i=1
n X
max subject to
i=1
xi
where a aji ∈ {0, 1} and aji = aji0 = 1 means variable xi and xi0 are in conflict for ressource j.
X. Gandibleux, M. Ehrgott
MCDM 2002, February 18-23, 2002
Bounds and Bound Sets
n X
ci1 xi
∈
{0, 1}
= 1 j = 1, . . . , m
The Set Partitioning Problem (SPA)
min i=1
n X
ci2 xi aji xi
i=1
n X
min subject to
i=1
xi
where a aji ∈ {0, 1} and aji = 1 means constraint j can be satisfied by variable xi .
X. Gandibleux, M. Ehrgott
MCDM 2002, February 18-23, 2002
Bounds and Bound Sets
SCP/SPP/SPA : Important in practice
Airline crew scheduling
◦ Minimize cost
◦ Maximize robustness of solution → Biobjective SPA
Railway network infrastructure capacity
◦ Maximize number of trains
◦ Maximize robustness of solution → Biobjective SPP
X. Gandibleux, M. Ehrgott
Bounds and Bound Sets
MCDM 2002, February 18-23, 2002
Introduction Notations and definitions Numerical instances Algorithms Results Conclusion
X. Gandibleux, M. Ehrgott
Bounds and Bound Sets
MCDM 2002, February 18-23, 2002
Notations and definitions
Multiobjective combinatorial optimization problem (MOCO) “min” (z 1 (x), . . . , z Q (x)) x∈X
(MOCO) is a discrete optimization problem, with . X the decision space, . x a binary vector of variables x ∈ {0, 1}n , . n variables xi , i = 1, . . . , n, . Q objectives z j , j = 1, . . . , Q . m constraints of specific structure defining X
X. Gandibleux, M. Ehrgott
Bounds and Bound Sets
• Pareto optimality:
MCDM 2002, February 18-23, 2002
x ∈ X Pareto optimal if there does not exist x0 ∈ X such that z q (x0 ) ≤ z q (x) for all q = 1, . . . , Q and z p (x0 ) < z p (x) for some p • Efficiency:
x Pareto optimal then z(x) = (z 1 (x), . . . , z Q (x)) is efficient/nondominated • set of Pareto optimal solutions: XP ar • set of efficient values: E
X. Gandibleux, M. Ehrgott
MCDM 2002, February 18-23, 2002
Bounds and Bound Sets
min{Cx : Ax = b, x ≥ 0}
λj cj x : Ax = b, x ≥ 0
λj = 1
j=1,...,Q
PQ J=1
Supported and Nonsupported Efficient Solutions
Linear programming
min
E is set of solutions of X with 0 < λ < 1
(MOCO) → supported efficient solutions SE, nonsupported efficient solutions N E exist
X. Gandibleux, M. Ehrgott
MCDM 2002, February 18-23, 2002
Bounds and Bound Sets
Lower and upper bound sets (min)
Q A lower bound set for Z is a subset L ⊆ IR+ such that
1. for each z ∈ Z∃l ∈ L such that lq ≤ z q (x), q = 1, . . . , Q
2. there is no pair z ∈ Z, l ∈ L such that z dominates l
Q An upper bound set for Z is a subset U ⊆ IR+ such that
1. for each z ∈ Z∃u ∈ U such that z q ≤ uq , q = 1, . . . , Q
2. there is no pair z ∈ Z, u ∈ U such that u dominates z
X. Gandibleux, M. Ehrgott
Bounds and Bound Sets
MCDM 2002, February 18-23, 2002
Introduction Notations and definitions Numerical instances Algorithms Results Conclusion
X. Gandibleux, M. Ehrgott
MCDM 2002, February 18-23, 2002
Bounds and Bound Sets
n = #variables 100 . . . 1000 100 . . . 1000
m = #constraints 10. . . 200 10 . . . 25
Characteristics of numerical instances
The sizes : SCP/SPP SPA
11 100 10 high
41 200 40 low
42 400 40 low
43 200 40 high
61 600 60 low
62 600 60 high
81 800 80 low
82 800 80 high
101 1000 100 low
102 1000 100 high
201 1000 200 low
The constraints : reduced (SCP) SCP/SPP density = 2% . . . 34% max#1 = 10. . . 200 SPA max#1 = 0.15*n . . . 0.40*n Series : n m density
X. Gandibleux, M. Ehrgott
Bounds and Bound Sets
The objectives : four families A: random ci1 , ci2 randomly generated i = 1, . . . , n;
MCDM 2002, February 18-23, 2002
B: conflictual 2 1 ci1 randomly generated i = 1, . . . , n; cn−i+1 = c i i = 1, . . . , n;
C: patterns l1 =rnd(), l2 =rnd(),. . . ; v1 =rnd(), v2 =rnd(),. . . ; c11 = c21 = . . . = cl11 = v1 ; cl11 +1 = cl11 +2 = . . . = cl11 +l2 = v2 ; . . . D: conflictual patterns B and C combined;
11A, 11B, 11C, 11D, 41A, . . . , 201D : 44 instances available on the MCDM society WWW site
X. Gandibleux, M. Ehrgott
Bounds and Bound Sets
MCDM 2002, February 18-23, 2002
Introduction Notations and definitions Numerical instances Algorithms Results Conclusion
X. Gandibleux, M. Ehrgott
MCDM 2002, February 18-23, 2002
Bounds and Bound Sets
Exact LP & 01 solutions: a two phases algorithm
firstPhase : procedure () is
- -| Compute x(1) and x(2) , the lexicographically optimal solutions for - -| permutations (z 1 , z 2 ) and (z 2 , z 1 ) of the objectives. x(1) ← SolveLexicography (z1 ↓ , z2 ↓) x(2) ← SolveLexicography (z2 ↓ , z1 ↓) S ← {x(1) , x(2) } - -| Compute all solutions between x(1) and x(2) . - -| Update S with all new solutions generated. solveRecursion(x(1) ↓ , x(2) ↓ , S l) end firstPhase
X. Gandibleux, M. Ehrgott
Bounds and Bound Sets
solveRecursion : procedure ( x(A) ↓ , x(B) ↓ , S l) is
MCDM 2002, February 18-23, 2002
- -| Compute the optimal solutions x(C) of (Pλ ) : min{λ1 z 1 (x) + λ2 z 2 (x) | x ∈ X} - -| where λ1 = z 2 (x(A) ) − z 2 (x(B) ), and λ2 = z 1 (x(B) ) − z 1 (x(A) ). x(C) ← SolvePλ (λ ↓, z 1 (x(B) ) ↓, z 2 (x(A) ) ↓ ) if exist(x(C) ) then S ← S ∪ {x(C) } solveRecursion(x(A) ↓ , x(C) ↓ , S l) solveRecursion(x(C) ↓ , x(B) ↓ , S l) end if end solveRecursion
SolveLexicography, SolvePλ : CPLEX 6.6.1 library is called.
X. Gandibleux, M. Ehrgott
MCDM 2002, February 18-23, 2002
Bounds and Bound Sets
Approximated solutions: heuristics
• SCP : constructive greedy algorithm setting variables from 0 to 1
. choose smallest ci (λ) such that xi covers an additional constraint . stop when all constraints are covered (satisfied)
• SPP : constructive greedy algorithm setting variables from 0 to 1
. choose biggest ci (λ) such that xi satisfies an additional constraint
. stop when it is impossible to saturate again one constraint
• SPA : simulated annealing coupled with a local search
X. Gandibleux, M. Ehrgott
Bounds and Bound Sets
MCDM 2002, February 18-23, 2002
SPA : simulated annealing coupled with a local search 1. compute a SCP feasible solution 2. change the solution to have a SPP compatible solution 3. if this is not a SPA feasible solution then - -| start a simulated annealing using this solution
a. the objective function : min the # of unsatisfied constraints b. the move : flip01 and flip10
c. apply a local search ((1-1 Exchange) when SA accepts a neighbor
d. restart (change the solution to have a SPP compatible solution) when SA does not produce a feasible solution after a given condition endIf
4. improve the feasible solution with a local search (1-1 Exchange) using the convex combination of the objective
X. Gandibleux, M. Ehrgott
Bounds and Bound Sets
MCDM 2002, February 18-23, 2002
Introduction Notations and definitions Numerical instances Algorithms Results Conclusion
X. Gandibleux, M. Ehrgott
Bounds and Bound Sets
• Bounds sets (min)
The context
MCDM 2002, February 18-23, 2002
– Lower bound Computed as continuous relaxation
– Upper bound Obtained after application of a heuristic • Implementation – C language – Cplex : a mainframe / Unix
– Heuristics : PowerPC G4 450Mhz / 128Mb / MacOS X Remarks – performances of both machines are close – CPUt / heuristics / SCP-SPP not significant
X. Gandibleux, M. Ehrgott
18000 16000 14000 12000 10000 8000 6000 4000 2000 0 11A 43A
61A instance
62A
81A
Bi-objective set covering problem : number of solutions
42A
LP 01
82A
number of solutions
500 450 400 350 300 250 200 150 100 50
11
42
instance
61
62
81
82
01A 01B 01C 01D
MCDM 2002, February 18-23, 2002
43
Bi-objective set covering problem : number of solutions (series)
41
X. Gandibleux, M. Ehrgott
⇒ in any case, #solutions for instances “with patterns” is low.
⇒ #solutions grows significantly for instances “low density”, “without pattern”;
41A
SCP : Solutions
Bounds and Bound Sets
number of solutions
1400
Bi-objective set covering problem : CPU time
62A
81A
800
600
400
200
0 11A
LP 01
1400
800
600
400
200
0 11
42
instance
61
62
81
82
01A 01B 01C 01D
MCDM 2002, February 18-23, 2002
43
Bi-objective set covering problem : CPU time (series)
41
X. Gandibleux, M. Ehrgott
. 101/102/201 : difficult instances (specially “with patterns”)
. 61/62/81/82 : interesting instances
. 11/41/42/43 : easy instances
⇒ Until serie 82, CPLEX can solve;
82A
1200
61A instance
1200
43A
1000
42A
cpuTime
1000
41A
SCP : CPU time
Bounds and Bound Sets
cpuTime
700
600
500
400
300
200
100
0 11C instance
11D
Bi-objective set covering problem : mean CPU time per solution
11B
01
cpuTime
10
8
6
4
2
0 11A
11D
01
MCDM 2002, February 18-23, 2002
11C instance
Bi-objective set covering problem : mean CPU time per solution (zoom)
11B
X. Gandibleux, M. Ehrgott
⇒ Families C&D (“high density, with patterns”) seems the most difficult to solve for any instances.
⇒ Mean time to generate one solution could be important specially for instances 62/82 (“medium size, high density”).
11A
SCP : mean time
Bounds and Bound Sets
cpuTime
6000
5000
4000
3000
2000
1000
0 0
3000 z1
4000
Bi-objective set covering problem : 2scp42B
2000
LP 01 GR01
5000
6000
z2
6000
5000
4000
3000
2000
1000
0 0
• bad : - “high density, with patterns”
1000
4000
LP 01 GR01
5000
6000
MCDM 2002, February 18-23, 2002
3000 z1
Bi-objective set covering problem : 2scp82D
2000
X. Gandibleux, M. Ehrgott
- LP overlaps E (“low density without pattern”)
• good : - lower bound and E are very close (“low density”)
1000
SCP : Lower bounds (LP)
Bounds and Bound Sets
z2
6000
5000
4000
3000
2000
1000
0 0
3000 z1
4000
Bi-objective set covering problem : 2scp81D
2000
LP 01 GR01
5000
6000
z2
1400
1200
1000
800
600
400
200
0 0
200
z1
800
1000
LP 01 GR01
1200
1400
MCDM 2002, February 18-23, 2002
600
Bi-objective set covering problem : 2scp62D
400
- unusual frontier, greedy not well spread (“low density with patterns”)
• bad : - not well spread, clusters (“high density”)
X. Gandibleux, M. Ehrgott
• good : - greedy well distributed (“low density without pattern”)
1000
SCP : Upper bounds (GR01)
Bounds and Bound Sets
z2
Bounds and Bound Sets
1800
1600
1400
1200
600
1000
2scp41D
800
1200
1400
1600
MCDM 2002, February 18-23, 2002
1800
Gr LP SE NE Ê
GRASP metaheuristic to solve biSCP (14thMCDM)
z2
1000
800
600 z1
X. Gandibleux, M. Ehrgott
Bounds and Bound Sets
SPP : General remarks (11-82 / A-D)
MCDM 2002, February 18-23, 2002
• Cplex is able to solve biggest instances (efficient preprocessing before to solve the instance)
⇒ instances SCP not pertinent to draw the the limits and difficulty of a solver;
• Mean time to generate a solution is often greater than for the SCP;
• Upper bound (LP) : same comments than for the SCP;
• Lower bound (GR01) : in general, not famous, well distributed, some clusters (instances “with patterns”); • ⇒ all instances are easy except. . .
X. Gandibleux, M. Ehrgott
3000
2500
2000
1500
1000
500
0 0
500
1500 z1
2000
Bi-objective set packing problem : 2spp41D
1000
LP 01 GR01
2500
3000
cpuTime
8000
7000
6000
5000
4000
3000
2000
1000
0 11A
41A
42A
61A instance
62A
LP 01
82A
X. Gandibleux, M. Ehrgott
81A
MCDM 2002, February 18-23, 2002
43A
Bi-objective set packing problem : CPU time
SPP : two very hard instances : 41D and 42C !
Bounds and Bound Sets
z2
SPA : (10-25 / A) Bi-objective set partitioning problem : CPU time
20A
25A
0
10
20
30
LP 01
600
0
100
400
LP 01 SA_LS
500
600
MCDM 2002, February 18-23, 2002
300 z1
Bi-objective set partitioning problem : 2spa20A
200
X. Gandibleux, M. Ehrgott
No enough results to draw any robust conclusion, just some comments. . .
0
100
200
300
500
instance
50
15A
400
10A
z2
40
60
Bounds and Bound Sets
cpuTime
Bounds and Bound Sets
SPA : some comments • Lower bound (LP) is not good.
MCDM 2002, February 18-23, 2002
• Feasible solutions are interesting even if they are constructed without a strong optimization toward the efficient frontier. • Testing : – penalty function – reject rule based on performances
– Strength Pareto Evolutionary Algorithm (SPEA by Zitzler and Thiele, 1998)
X. Gandibleux, M. Ehrgott
Bounds and Bound Sets
MCDM 2002, February 18-23, 2002
Introduction Notations and definitions Numerical instances Algorithms Results Conclusion
X. Gandibleux, M. Ehrgott
Bounds and Bound Sets
Conclusion
MCDM 2002, February 18-23, 2002
• Not all results have been presented (supported and non-supported solutions, distance measures, etc.)
• SCP : highlighted a class of not friendly problems : SCP with high density and patterns : – difficult to solve – Bad lower bound (LP) : great distance – Bad upper bound (GR01) : clusters and holes • SPP : to build other instances • SPA : to continue. . .
X. Gandibleux, M. Ehrgott
