Multi-objective Integer Programming: An Improved Recursive ... · •Let P be a CLMOIP problem, and let R be a relaxation of P. If R is infeasible, then P is also infeasible. •Let

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

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School of Mathematical

and Geospatial Sciences

Motivation

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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

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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



Minimise Cost Minimum Time 65** 80*

Minimise Time 70 80*

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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

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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

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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





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



( ), where 0, 1.

k

i i i i

x X

f x f x f x

w f x w w

supported efficient





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

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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



( ), 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


• 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


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• 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

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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


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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

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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


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


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 .


• 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.


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



• 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.

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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

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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

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Documents

Multi-objective Integer Programming: An Improved Recursive ... · •Let P be a CLMOIP problem, and let R be a relaxation of P. If R is infeasible, then P is also infeasible. •Let