multi-objective integer programming: an improved recursive ... · •let p be a clmoip problem, and...
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Multi-objective Integer Programming: An Improved Recursive Algorithm
Melih Ozlen, School of Mathematical and Geospatial Sciences
RMIT University, Australia
Benjamin A. Burton, School of Mathematics and Physics,
The University of Queensland, Australia
Outline
• Motivation
• Multi-Objective Optimization
• Bi-Objective Integer Programming
• Multi-Objective Integer Programming
• Results
• Conclusion
• Future Research
RMIT
University©2010
School of Mathematical
and Geospatial Sciences
Motivation
RMIT
University©2010
School of Mathematical
and Geospatial Sciences
Travelling Salesperson Problem (TSP)
Motivation
• Travelling Salesperson Problem
• n nodes to cover
• cost matrix from node i to node j
• time matrix from node i to node j
• find the minimum cost of covering all nodes
• minimum cost is $50k
• find the minimum duration of covering all nodes
• minimum duration is 100 minutes
RMIT
University©2010
School of Mathematical
and Geospatial Sciences
Motivation
Objective Constraint Cost ($k) Time (mins)
Minimise Cost 50* 100
Minimise Time Minimum Cost 50* 95**
f(Cost, Time) Time or Cost 57 90
f(Cost, Time) Time or Cost 60 86
f(Cost, Time) Time or Cost 62 83
Minimise Cost Minimum Time 65** 80*
Minimise Time 70 80*
RMIT
University©2010
School of Mathematical
and Geospatial Sciences
Integer Programming
• Logistics/Production planning
• Scheduling
• Network design
• Workforce management
• Pure Maths
– Computational Geometry
– Topology
Multiple Objectives
• Cost/Profit
• Environment impact
• Risk/Safety
• Sustainability
• Waste
Multi-Objective Integer Programming
RMIT
University©2010
School of Mathematical
and Geospatial Sciences
k objectives
Integer objective coefficients
Problem constraints
1 1 2 2
1 1 1
min ( ) , ( ) ,..., ( )n n n
j j j j k jk j
j j j
f x c x f x c x f x c x
1 2, ,...,j j jkc c c
, 0,x X x x
Multi-Objective Optimization
RMIT
University©2010
School of Mathematical
and Geospatial Sciences
A solution ' is if and only if
there is no such that ( ) ( ') for all {1,..., }
and ( ) ( ') for at least one .
The resulting objective vector ( ') is
said to be
i i
i i
x X
x X f x f x i k
f x f x i
f x
efficient
nondominated.
1 2
An efficient solution is if it
minimizes some convex combination of , ,... ( ),
( ), where 0, 1.
k
i i i i
x X
f x f x f x
w f x w w
supported efficient
A solution ' is if and only if
there is no such that ( ) ( ') for all {1,..., }
and ( ) ( ') for at least one .
The resulting objective vector ( ') is
said to be
i i
i i
x X
x X f x f x i k
f x f x i
f x
efficient
nondominated.
1 2
An efficient solution is if it
minimizes some convex combination of , ,... ( ),
( ), where 0, 1.
k
i i i i
x X
f x f x f x
w f x w w
supported efficient
A solution ' is if and only if
there is no such that ( ) ( ') for all {1,..., }
and ( ) ( ') for at least one .
The resulting objective vector ( ') is
said to be
i i
i i
x X
x X f x f x i k
f x f x i
f x
efficient
nondominated.
1 2
For a multi-objective optimization problem,
aiming to minimize all objectives, defined as,
Min ( ), ( ),... ( ), st. ,
a solution ' is if and only if
there is no such that ( )
k
i
f x f x f x x X
x X
x X f x
efficient
( ') for all {1,..., }
and ( ) ( ') for at least one . The resulting
objective vector ( ') is said to be
i
i i
f x i k
f x f x i
f x nondominated.
Multi-Objective Optimization
1 2
An efficient solution is ,
if it minimizes some convex combination of
, ,... ( ), ( ), where 0, 1.
If there exists no such convex combination for an
efficient so
k i i i i
x X
f x f x f x w f x w w
supported efficient
lution, , then it is .x X unsupported efficient
RMIT
University©2010
School of Mathematical
and Geospatial Sciences
A solution ' is if and only if
there is no such that ( ) ( ') for all {1,..., }
and ( ) ( ') for at least one .
The resulting objective vector ( ') is
said to be
i i
i i
x X
x X f x f x i k
f x f x i
f x
efficient
nondominated.
1 2
An efficient solution is if it
minimizes some convex combination of , ,... ( ),
( ), where 0, 1.
k
i i i i
x X
f x f x f x
w f x w w
supported efficient
Decision Space
Objective Space
f1(x)
f2(x)
1
3
4
5
6
2
1, 2, 3, 4, 6 nondominated
1,2,4,6 supported
3 unsupported
5 dominated
Multi-Objective Integer Programming
• Hierarchical optimization
• Simultaneous optimization
– Known utility function
• Linear – find the best supported
• Nonlinear – find the best nondominated
– Unknown utility function
• Linear – generate supported nondominated solutions
• Nonlinear – generate all nondominated solutions
Multi-Objective Integer Programming
RMIT
University©2010
School of Mathematical
and Geospatial Sciences
• Polynomially solvable
– Assignment
– Transportation/Transhipment
– Network Flow
• NP-Hard
– Bi-Objective Assignment
– Bi-Objective Transportation
– Bi-Objective Network Flow
Literature
– Known utility function
• Linear
• Abbas and Chaabane [2006], EJOR
• Jorge [2009], EJOR
• Nonlinear
• Ozlen et al. [2010]
– Unknown utility function
• Linear
• Przybylski et al. [2010], IJOC
• Özpeynirci and Köksalan [2010], MS
Literature
– Generating nondominated solutions
– Klein and Hannah (1982), EJOR – Sequentially solving single objective problems
– Sylva and Crema (2004), EJOR– Weighted objective to ensure generating all
– Laumanns et al. (2006), EJOR – Adaptive ε-constraint method
– Ozlen and Azizoglu (2009), EJOR– Recursive algorithm
– Przybylski et al. (2010), DO– Two phase method
– Supported
– Unsupported
Bi-Objective Integer Programming
RMIT
University©2010
School of Mathematical
and Geospatial Sciences
Two objectives
Integer objective coefficients
Problem constraints
1 1 2 2
1 1
min ( ) , ( )n n
j j j j
j j
f x c x f x c x
1 2,j jc c
, 0,x X x x
Bi-Objective Integer Programming
RMIT
University©2010
School of Mathematical
and Geospatial Sciences
Lexicographic optimization
Constraint on the secondary objective
1min ( ),s.t. f x x X
2 ( )f x l
*
2Initialize , update as ( ) -1, and stop when infeasiblel l f x
*
2 1 1min ( ),s.t. ( ) ( ),f x f x f x x X
RMIT
University©2010
School of Mathematical
and Geospatial Sciences
Bi-Objective Integer Programming
f1(x)
f2(x)
f2(x) ≤ f2GUB
1
2
3
f2GUBl1l2l3
f2(x) ≤ l1-1f2(x) ≤ l2-1f2(x) ≤ l3-1
infeasible
Ozlen and Azizoglu (2009)
For any MOIP problem, a nondominated solution ' providing
an function value, ( ), of all
nondominated solutions, should also be
upper bound on the objective
no with respndo ectminate to
all
d
other
t
k
h
x
fk x
1 2 1,1 obj ( ), ( ), , ( )ective .s kf x fk f x x
upper bound on the obje
For any constrained MOIP problem, a nondominated solution '
providing an function value, ( ), of all
nondominated solutions satisfying the constraints, sh
ct
ou
iv
ld also b
e k
thk
x
f x
1 2 1
e
with respect to all other , nondominat
( ),
ed 1 objectives
( ), , ( ).kf x f f x
k
x
Ozlen and Azizoglu (2009)
Constrained Lexicographic Multi Objective Integer Programming *
1 2 1Lexicographic objective 1: Min ( ), ( ),..., ( )
Lexicographic objective 1: Min ( )
s.t.
( )
k
k
k k
f x f x f x
f x
f x l
x X
f1(x) f2(x) f3(x)
10 20 9
14 18 12
19 12 6
25 9 13
Ozlen and Azizoglu (2009)
Improved Recursive Algorithm
Intermediate problem
1 2
1
1 1 2 2
1 2
Lexicographic objective 1: Min ( ), ( ),..., ( )
Lexicographic objective 1: Min ( )
s.t.
( ) , ( ) ,...., ( )
We denote such a problem using the notation ( , , ,..., )
q
q
q q q q k k
q q k
f x f x f x
f x
f x l f x l f x l
x X
q l l l .
Improved Recursive Algorithm
• Let P be a CLMOIP problem, and let R be a relaxation of P. If R is infeasible, then P is also infeasible.
• Let P be a CLMOIP problem, and let R be a relaxation of P. If every nondominated objective vector for R is also feasible for P, then the set of all nondominatedobjective vectors for P is precisely the set of all nondominated objective vectors of R.
• If R has even a single nondominated objective vector that is not feasible for P, then the solution to R cannot be used to avoid solving P.
Improved Recursive Algorithm
Since Algorithm 1 iterates by incrementally lowering the bounds as the algorithm progresses it becomes highly likely that we can find a relaxation of the current CLMOIP amongst our set of already solved problems.
There could be many relaxations to a given CLMOIP, each with different nondominated sets, all of the relaxations should be examined until one is found that allows us to avoid solving the current CLMOIP.
' ' '
2
'
1
1 2
'
The CLMOIP problem ( , , ,..., ) is a relaxation of
( , if for all, and if for so,..., ) . me
q q k
q q ik i i i
q l l l
q l l il l il l l
Improved Recursive Algorithm
Improved Recursive Algorithm
• Coding complexities
– Recursion or Stack Structure (B&B)
– Data structures
• Nondominated solutions
• Solved problem information
– Efficient Query methods
• Locating useful relaxations
Example
Example (contd)
Experiment
• # objectives – 3 or 4
• Objective coefficient – U[1,10] or U[1,20]
• Problem size - 25, 100, 225 decisions
• 20 random instances under each setting
• # IPs solved
• Not solver dependent
• CPU time is not relevant
Results
Results
Conclusion
• Improved recursive algorithm solves significantly less IPs compared to the original.
• The average number of IPs solved in order to identify each nondominated solution is not sensitive to problem size.
• Average number of IPs solved increases with increasing number of objectives.
RMIT
University©2010
School of Mathematical
and Geospatial Sciences
Current Research
• Parallel Implementation
– Dividing over the objective value ranges
• Works efficiently for large objective ranges
• Almost perfect parallelisation
– Starting with different objective orderings
• Works well for tight objective ranges
• Parallelisation efficiency drops with number of CPUs
• Open source and commercial solver integration
RMIT
University©2010
School of Mathematical
and Geospatial Sciences
Future Research
• Lexicographic Integer Programming
– Specialised efficient algorithms
• B&B
• B&C
• Multi-Objective Mixed Integer Programming
– Hybrid with Multi-Objective Linear Programming
• Developing problem specific algorithms
RMIT
University©2010
School of Mathematical
and Geospatial Sciences