MOEA/D: A Multiobjective Evolutionary Algorithm

Based on Decomposition

Reporter: Steven

Date: 2011/5/4

Why decompose the MOP ? Most MOPs may have many or even infinite Pareto

optimal vectors. It is very time-consuming to obtain the complete PF.

Decision maker may not be interested in having an unduly large number of Pareto optimal vectors to deal with due to overflow of information.

Many MO algorithms are to find a manageable number of Pareto optimal vectors which are evenly distributed along the PF, and thus good representatives of the entire PF.

DECOMPOSITION OF MULTIOBJECTIVE OPTIMIZATION A. Weighted Sum Approach

: be a weight vector

: Object solution

m

i i

Tm

1

1

1

),...,(

Tm xfxfxF ))(),...,(()( 1

m

iii

ws xfxgMaximize1

)()|(

DECOMPOSITION OF MULTIOBJECTIVE OPTIMIZATION B. Tchebycheff Approach

: is the reference point

|})(|{max*),|( *

1iii

mi

te zxfzxgMinimize

Tmzzz *)*,...,(* 1

1f

2f

0

*Z

|)(| *ii zxf

DECOMPOSITION OF MULTIOBJECTIVE OPTIMIZATION C. Boundary Intersection (BI) Approach

xdxFztosubject

dzxgMinimize bi

)(*

*),|(

DECOMPOSITION OF MULTIOBJECTIVE OPTIMIZATION C. Penalty-based Boundary Intersection (BI)

Approach

||)*()(||||||

||)(*||

,*),|(

121

21

dzxFdandxFz

dwhere

xtosubjectddzxgMinimizeT

bi

THE FRAMEWORK OF MOEA/D

At each generation , MOEA/D with the Tchebycheff approach maintains:

THE FRAMEWORK OF MOEA/D

The algorithm works as follows:

Step 1) Initialization:

Step 2) Update:

1f

2f

0

*Z

)(

)(,...)( 111

lB

xfxf T

)(

)(,...)( 212

kB

xfxf T

y

y‘

Step 3) Stopping Criteria: If stopping criteria is satisfied,then stop and

output EP. Otherwise, go to Step 2.

Discussions of MOEA/D 1) Why a Finite Number of Subproblems are

Considered in MOEA/D:

MOEA/D spends about the same amount of effort on each of the N aggregation functions, while MOGLS randomly generates a weight vector at each iteration, aiming at optimizing all the possible aggregation functions.

Since the computational resource is always limited, optimizing all the possible aggregation functions would not be very practical, and thus may waste some computational effort.

Discussions of MOEA/D 2) How Diversity is Maintained in MOEA/D:

NSGA-II and SPEA-II → crowding distancesMOEA/D → The “diversity” among these subproblems will naturally lead to diversity in the population.

3) Mating Restriction and the Role of in MOEA/D:T is too small :the solution could be very close to their parents, the algorithm lacks the ability to explore new areas in the search space.T is too large :the exploitation ability of the algorithm is weakened.

Multiobjective 0–1 Knapsack Problem Given a set of n items and a set of m

knapsacks, the multiobjective 0–1 knapsack problem (MOKP) can be stated as:

is the profit of item j in knapsack i

is the weight of item j in knapsack i

is the capacity of knapsack i

item i is selected and put in all the knapsacks.ic

1ix

0ijp

0ijw

Experimental Results- CPU time

Experimental Results- C metric C(A,B) is defined as the percentage of the

solutions in B that are dominated by at least one solution in A

Experimental Results- D metric Distance from Representatives in the PF ( D-

metric):

Fig. 4. Plots of the non-dominated solutions with the lowest D-metric in 30 runs of MOEA/D and MOGLS with the weighted sum approach for all the 2-objective MOKP test instances.

Fig. 5. Plots of the nondominated solutions with the lowest D-metric in 30 runsof MOEA/D and MOGLS with the Tchebycheff approach for all the 2-objectiveMOKP test instances.

We use five widely used bi-objective ZDT test instances and two 3-objective instances in comparing MOEA/D with NSGA-II

A Bit More Effort on MOEA/D: Can MOEA/D with other advanced decomposition methods such as the PBI approach find more evenly distributed solutions for 3-objective test instances like DTLZ1 and DTLZ 2

PBINSGA-11MOEA/D Te

A Bit More Effort on MOEA/D: Can MOEA/D with objective normalization perform better in the case of disparately scaled objectives as in ZDT3

normalization

f2→10f2

Without normalization

Sensitivity of in MOEA/D

MOEA/D is not very sensitive to the setting of , at least for MOPs that are somehow similar to these test instances

MOEA/D Using Small Population

Scalability