Yet another two-phase method for the biobjective
assignment problem
Charles Delort, Olivier Spanjaard
LIP6
6/16/2011–MCDM, Jyväskylä
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Charles Delort, Olivier Spanjaard (LIP6)
Biobjective assignment problem
6/16/2011–MCDM, Jyväskylä
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Biobjective Assignment Problem


(6, 1) (2, 4) (5, 1)
(2, 2) (1, 4) (6, 1)
(3, 4) (4, 1) (2, 1)
(6, 1) + (1, 4) + (2, 1) = (9, 6)
o2
14
Non-dominated points
12
Dominated points
10
8
Aim : find a minimal complete
set (Hansen, 1980) = exactly
one solution for each
non-dominated point.
NP-hard (Serafini, 1986).
Charles Delort, Olivier Spanjaard (LIP6)
Convex hull
6
4
2
2
4
6
8
10
12
14
16
o1
Figure: Objective space.
Biobjective assignment problem
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State of the art
A. Przybylski, X. Gandibleux, and M. Ehrgott. Two phase algorithms
for the bi-objective assignment problem. European Journal of
Operational Research, 185(2):509–533, March 2008.
C.R. Pedersen, L.R. Nielsen, and K.A. Andersen. The bicriterion multi
modal assignment problem: Introduction, analysis, and experimental
results. Informs Journal on Computing, 20(3):400411, 2008.
Both algorithms are two-phase methods, using a ranking procedure in the
second phase.
Charles Delort, Olivier Spanjaard (LIP6)
Biobjective assignment problem
6/16/2011–MCDM, Jyväskylä
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Two-phase method
Framework for solving
biobjective problems introduced
by Ulungu and Teghem (1993).
First phase: find all supported
solutions.
Second phase: find
non-supported solutions.
Non−dominated
convex hull
Search zone
supported point
non−supported
non−dominated point
Best used when the underlying
single objective problem is
“easy“ to solve (assignment
problem, minimum spanning
tree, knapsack ...).
Charles Delort, Olivier Spanjaard (LIP6)
Biobjective assignment problem
6/16/2011–MCDM, Jyväskylä
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Two-phase method: first phase
Agregate both objectives into
one using weighted sums.
Aneja and Nair’s method (1979)
p1
Charles Delort, Olivier Spanjaard (LIP6)
Biobjective assignment problem
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Two-phase method: first phase
p2
Agregate both objectives into
one using weighted sums.
Aneja and Nair’s method (1979)
p1
Charles Delort, Olivier Spanjaard (LIP6)
Biobjective assignment problem
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Two-phase method: first phase
p2
Agregate both objectives into
one using weighted sums.
Aneja and Nair’s method (1979)
p3
p1
Charles Delort, Olivier Spanjaard (LIP6)
Biobjective assignment problem
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Two-phase method: first phase
p2
Agregate both objectives into
one using weighted sums.
Aneja and Nair’s method (1979)
p3
p4
p1
Charles Delort, Olivier Spanjaard (LIP6)
Biobjective assignment problem
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Two-phase method: first phase
p2
Agregate both objectives into
one using weighted sums.
Aneja and Nair’s method (1979)
p3
p4
p1
Charles Delort, Olivier Spanjaard (LIP6)
Biobjective assignment problem
6/16/2011–MCDM, Jyväskylä
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Two-phase method: second phase
Explore search zones (triangles),
one at a time.
Non−dominated
convex hull
Search zone
supported point
Use enumerative methods:
branch and bound, dynamic
programming, ranking ...
non−supported
non−dominated point
The most time consuming
phase.
Charles Delort, Olivier Spanjaard (LIP6)
Biobjective assignment problem
6/16/2011–MCDM, Jyväskylä
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Limits of the ranking method

Large equivalence classes.
n! solutions, only one point: (n, n).
Instance
A-20-100
A-20-200
Number of supported solutions
minimum average maximum
233
374
1487
732
47271
888863
(1, 1) · · ·
 ..
..
 .
.
(1, 1) · · ·

(1, 1)
.. 
. 
(1, 1)
Number of supported points
minimum average maximum
67
89
129
107
136
159
Table: number of supported points and supported solutions.
Charles Delort, Olivier Spanjaard (LIP6)
Biobjective assignment problem
6/16/2011–MCDM, Jyväskylä
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Our algorithm: two-phase method, using Multi-Objective
Branch and Bound (MOBB)
Aneja and Nair’s method for the first phase.
Multi-objective branch and bound in the second phase.
Branching part: divide a
problem into several
subproblems.


a11 a12 a13
a21 a22 a23 
a31 a32 a33
Charles Delort, Olivier Spanjaard (LIP6)
a11
a22
Biobjective assignment problem
a23
a12
a21
a13
a23 a21
6/16/2011–MCDM, Jyväskylä
a22
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MOBB: defining bound sets
Idea : Using sets of points as
bounds, instead of a single point.
First proposed by Villareal and
Karwan (1981).
Independently implemented by
Ehrgott and Gandibleux (2007)
and Sourd and Spanjaard
(2008).
Charles Delort, Olivier Spanjaard (LIP6)
Biobjective assignment problem
6/16/2011–MCDM, Jyväskylä
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MOBB: computing UB4
We want to find a zone of the
objective space containing all
non-dominated points of the initial
problem: UB4
upper bound UB
UB: non-dominated points found so
far.
upper relaxation UB4
one defines the local nadir
points from UB;
UB4 : the area dominating these
points.
Charles Delort, Olivier Spanjaard (LIP6)
Biobjective assignment problem
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MOBB: computing UB4
We want to find a zone of the
objective space containing all
non-dominated points of the initial
problem: UB4
upper bound UB
UB: non-dominated points found so
far.
upper relaxation UB4
one defines the local nadir
points from UB;
UB4 : the area dominating these
points.
Charles Delort, Olivier Spanjaard (LIP6)
Biobjective assignment problem
6/16/2011–MCDM, Jyväskylä
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MOBB: computing UB4
We want to find a zone of the
objective space containing all
non-dominated points of the initial
problem: UB4
upper bound UB
UB: non-dominated points found so
far.
upper relaxation UB4
one defines the local nadir
points from UB;
UB4 : the area dominating these
points.
Charles Delort, Olivier Spanjaard (LIP6)
Biobjective assignment problem
UB4
6/16/2011–MCDM, Jyväskylä
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MOBB: computing LB<
We want to compute a zone of the
objective space containing all points
of the considered subproblem: LB<
lower bound LB
LB: non-dominated boundary of the
convex hull
lower relaxation LB<
LB< : the area dominated by LB.
Charles Delort, Olivier Spanjaard (LIP6)
Biobjective assignment problem
6/16/2011–MCDM, Jyväskylä
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MOBB: computing LB<
We want to compute a zone of the
objective space containing all points
of the considered subproblem: LB<
lower bound LB
LB: non-dominated boundary of the
convex hull
lower relaxation LB<
LB< : the area dominated by LB.
Charles Delort, Olivier Spanjaard (LIP6)
Biobjective assignment problem
6/16/2011–MCDM, Jyväskylä
15 / 28
MOBB: computing LB<
We want to compute a zone of the
objective space containing all points
of the considered subproblem: LB<
lower bound LB
LB: non-dominated boundary of the
convex hull
lower relaxation LB<
LB<
LB< : the area dominated by LB.
Charles Delort, Olivier Spanjaard (LIP6)
Biobjective assignment problem
6/16/2011–MCDM, Jyväskylä
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MOBB
At each node of the MOBB:
LB<
Compute LB< , from convex hull
using Aneja and Nair’s method.
Compare to UB4 , obtained
from local nadir points of
already found solutions.
If LB< ∩ UB4 = ∅, then discard
the node.
Charles Delort, Olivier Spanjaard (LIP6)
UB4
Biobjective assignment problem
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MOBB in the second phase
Less vertices computed
in LB< .
Faster computation of
the intersection.
LB<(n)
More nodes discarded.
T
Further improvements: reuse of the parent node’s LB< .
Charles Delort, Olivier Spanjaard (LIP6)
Biobjective assignment problem
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MOBB in the second phase
Less vertices computed
in LB< .
Faster computation of
the intersection.
T
LB<[T ]
UB4[T ]
More nodes discarded.
Further improvements: reuse of the parent node’s LB< .
Charles Delort, Olivier Spanjaard (LIP6)
Biobjective assignment problem
6/16/2011–MCDM, Jyväskylä
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Our algorithm
Principal steps of the algorithm:
First phase: find extreme solutions.
Local search: find a good upper bound UB.
Second phase: for each search zone T :
Local shaving: reduce the subproblem size.
MOBB: find non-supported non-dominated solutions in T .
Charles Delort, Olivier Spanjaard (LIP6)
Biobjective assignment problem
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Our algorithm
Local search

Start from given solutions and then iteratively move
(6, 1)
(2, 2)
to neighbor solutions.
Initialize with solutions found in the first phase. (3, 4)
(6, 1)
Neighborhood : solutions with two assignments
(2, 2)
switched.
(3, 4)
Goal: find a good upper bound for the
non-dominated points: UB.
(2, 4)
(1, 4)
(4, 1)
(2, 4)
(1, 4)
(4, 1)

(5, 1)
(6, 1)
(2, 1)

(5, 1)
(6, 1)
(2, 1)
Shaving
Forbid or make mandatory some assignments before solving the problem.
Forbid aij and compute LB< . If the intersection with UB4 is empty,
then this assignment is made mandatory.
Goal: reduce problem size.
Charles Delort, Olivier Spanjaard (LIP6)
Biobjective assignment problem
6/16/2011–MCDM, Jyväskylä
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Numerical results: non-correlated instances
Three different types of instances:
type A: c1ij ∈ {1, . . . , R}, c2ij ∈ {1, . . . , R};


(13, 10) (8, 18) (17, 6) (13, 20)
 (6, 4)
(13, 7)
(4, 3)
(10, 4) 


 (20, 3) (17, 16) (18, 6) (15, 6) 
(1, 13) (12, 12) (16, 20) (14, 12)
Charles Delort, Olivier Spanjaard (LIP6)
Biobjective assignment problem
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Numerical results: positively correlated instances
Three different types of instances:
type B: c1ij ∈ {1, . . . , R},
c2ij ∈ {max{c1ij − R/10, 1}, . . .,min{c1ij + R/10, R}};

(11, 11)
 (9, 10)

(18, 20)
(4, 3)
Charles Delort, Olivier Spanjaard (LIP6)

(18, 20) (18, 17) (4, 4)
(17, 16) (12, 12) (4, 3) 

(13, 11) (18, 18) (7, 7) 
(18, 18) (1, 3) (10, 11)
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Numerical results: negatively correlated instances
Three different types of instances:
type C: c1ij ∈ {1, . . . , R},
c2ij ∈ {max{R − R/10 − c1ij , 1}, . . .,min{R + R/10 − c1ij , 20}}

(11, 9)
(14, 6)

(5, 16)
(13, 5)
Charles Delort, Olivier Spanjaard (LIP6)
(15, 3)
(17, 1)
(8, 11)
(8, 12)

(18, 2) (10, 12)
(19, 2) (17, 2) 

(8, 14) (7, 11) 
(10, 9) (1, 17)
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Numerical results
Type
A-20-20
A-20-40
A-20-60
A-20-80
A-20-100
A-20-200
B-20-100
B-20-200
C-20-50
C-20-100
our algorithm1
avg.
max.
0.17
0.29
1.12
1.78
2.88
3.86
5.72
7.19
10.77 15.68
73.44 84.25
1.36
2.18
6.15
16.13
4.08
5.49
38.82
44.1
method
avg.
0.16
3.40
21.68
54.19
137.18
n.c.
n.c.
n.c.
n.c.
n.c.
PGE2
max.
0.25
5.73
32.71
85.11
194.08
n.c.
n.c.
n.c.
n.c.
n.c.
ε-constraint method1
avg.
max.
1.17
1.59
7.30
9.08
24.56
33.05
66.42
127.52
216
762
3010
10573
16.54
21.49
155.7
188.2
73.45
78.13
652.5
679.1
Table: Comparison of the resolution times of our method, the method PGE and
the ε-constraint method
1
2
processor used : Core 2 Duo E8400, 3 GHz
processor used : P4 EE 3, 4 GHz
Charles Delort, Olivier Spanjaard (LIP6)
Biobjective assignment problem
6/16/2011–MCDM, Jyväskylä
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Numerical results
type
A-20-100
A-50-100
A-100-100
B-20-100
B-50-100
B-100-100
type
C-20-20
C-50-20
C-100-20
exact
avg.
max.
10.77 15.68
50.85 72.82
159.4 216.7
1.34
2.21
4.19
7.7
9.77
15.8
exact
1.23
1.68
2.61
3.34
7.27
9.19
ε = 0.1%
avg. max.
1.69 2.64
3.43 5.94
4.18 5.8
0.09 0.34
0.03 0.2
0.08 0.19
ε = 1%
0.08 0.1
0.1 0.12
0.12 0.14
ε = 0.5%
avg. max.
0.07 0.15
0.11 0.18
0.12 0.29
0.02 0.04
0.03 0.05
0.04 0.07
ε = 5%
0.01 0.02
0.01 0.02
0.01 0.02
Table: Resolution times, in seconds, of our method, searching for a minimal
complete set, or an ε-cover.
Charles Delort, Olivier Spanjaard (LIP6)
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6/16/2011–MCDM, Jyväskylä
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Numerical results: ε-cover
ε-dominance
x 4ε y ⇐⇒ ∀i : xi ≤
(1 + ε)yi .
ε-cover of X
subset Xε ⊆ X s.t. any
point of X is
ε-dominated by a point
in Xε .
Charles Delort, Olivier Spanjaard (LIP6)
Biobjective assignment problem
6/16/2011–MCDM, Jyväskylä
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Numerical results
type
A-20-100
A-50-100
A-100-100
B-20-100
B-50-100
B-100-100
type
C-20-20
C-50-20
C-100-20
exact
avg.
max.
10.77 15.68
50.85 72.82
159.4 216.7
1.34
2.21
4.19
7.7
9.77
15.8
exact
1.23
1.68
2.61
3.34
7.27
9.19
ε = 0.1%
avg. max.
1.69 2.64
3.43 5.94
4.18 5.8
0.09 0.34
0.03 0.2
0.08 0.19
ε = 1%
0.08 0.1
0.1 0.12
0.12 0.14
ε = 0.5%
avg. max.
0.07 0.15
0.11 0.18
0.12 0.29
0.02 0.04
0.03 0.05
0.04 0.07
ε = 5%
0.01 0.02
0.01 0.02
0.01 0.02
Table: Resolution times, in seconds, of our method, searching for a minimal
complete set, or an ε-cover.
Charles Delort, Olivier Spanjaard (LIP6)
Biobjective assignment problem
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Conclusion
Using MOBB is promising in a two-phase framework.
Very efficient on instances with small range.
Doesn’t scale well with the range.
ε-cover gives good results, for higher range.
To do: comparison with Pedersen et al. (2008) on instances with
small range.
Further work: specify the method for other biobjective problems.
Charles Delort, Olivier Spanjaard (LIP6)
Biobjective assignment problem
6/16/2011–MCDM, Jyväskylä
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