EMAPS II : AN EVOLUTIONARY ALGORITHM FOR MULTIOBJECTIVE OPTIMIZATION

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EMAPS II AN EVOLUTIONARY ALGORITHM FOR
MULTIOBJECTIVE OPTIMIZATION

• Banu Soylu Murat Köksalan
• Industrial Engineering Department
• Middle East Technical University
• Ankara-TURKEY
• XI ELAVIO 2005

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Content

• Aim of the Study
• Literature Review
• An Evolutionary Algorithm
• -Fitness Function
• -Favorable Weights
• -Insertion and Replacement Rules
• -Ranking and Fitness Updates
• -Generating the Initial Set of Solutions
• Steps of the Algorithm EMAPS II
• Computational Results
• Conclusion

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Aim of the Study

• Develop an EA to approximate efficient frontier
  in Multiobjective Problems
• Goals
• (1) evolve towards the efficient frontier
• (2) evenly distribute over frontier obtain a
  well-spread frontier

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Goals of the algorithm
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How to achieve the goals of the algorithm ?

• (1) Favor yourself over other members
• (2) Favor large distance from closest contender
• and Seed with good extreme solutions in each
  objective

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Bad Representation Example 1
(1)st goal is satisfied but not (2)nd
goal (Bad-spread representation)
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Bad Representation Example 2
(1)st goal is satisfied but not
(2)nd goal (Non-uniform distribution)
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Bad Representation Example 3
(2)nd goal is satisfied but not
(1)st goal (Unconvergence to the eff. frontier)
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Literature Review

• According to fitness assignment scheme

• Pareto Based MOEAs
• Goldberg (1989)
• MOGA Fonseca Fleming (1993)
• NPGA Horn et al. (1994)
• NSGA Srinivas Deb (1994)
• NSGA II Deb et al. (2000)
• SPEA Zitzler Thiele (1999)
• SPEA II Zitzler et al. (2001)
• PAES Knowles Corne (2000)

• Non-Pareto Based MOEAs
• VEGA Schaffer (1984)
• BGGA Allenson (1992)
• WBGA Hajela Lin (1992)
• VEA Hajela Lin (1993)

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Non-dominated Sorting (Determination of Frontiers)
Illustration of
frontiers
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Fitness Function

• Strength (?i), i.e. distance between
  and
• where is
• for a
  linear distance function
• for a
  Tchebycheff distance function

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Fitness Function
average of the strengths,
worst-case measure,
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Fitness Function
Adjust raw fitness with non-dominated sorting
idea Step 1. Rank the population (Determine the
frontiers) Step 2. Adjust raw fitness values
Let be the set of solutions with
rank j. For
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Favorable Weights
Compute favorable weights either by (1)
Considering a linear distance function or
by (2) Considering a Tchebycheff distance function
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Favorable Weights
Tchebycheff Distance Function
Definition (Steuer, 1986). Let z z and w ?
W. Then z is a definition point of the z-z
w8 contour if and only if
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Favorable Weights
Tchebycheff Distance Function
Illustration of contours,
diagonal direction and ?xi
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Insertion and Replacement Rules
An offspring is stillborn if it is dominated by
any elements of the worst frontier
If, Then xi ? X is a candidate for removal from
the population and xi ? X with the most
negative Di is removed from the population
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Insertion and Replacement Rules
Crowdence Measure
Step 1. For each objective function (k1,...,m),
determine the nearest neighbors (neigh.1,
neigh.2) of a solution by sorting. Step 2.
Assign high value to the members on the extremes
of each objective and compute the crowdence
measure for all other members
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Ranking and Fitness Updates
Update all fitness values with each xoff
Step 1. Find rank of xoff. Move all dominated
members by xoff to next frontier. Step 2. Update
raw fitness scores. Step 3. Adjust raw fitness
scores.
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Generating Initial Set of Solutions
Find close to best solutions in each objective to
use as seeds
(1) Run EMAPS II as a single objective EA (2)
Repair solutions obtained from heuristics (eg. LP
relaxation)
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EMAPS II Algorithm
Step 0. Initialization 0.1. Generate an
initial population and seed it 0.2. Estimate
the ideal point Step 1. Evaluation of the
initial population 1.1. Compute the favorable
weights, raw fitness scores 1.2. Determine the
ranks, adjust the raw fitness scores Step 2.
Selection (tournament selection with
replacement) Step 3. Crossover (one-point
crossover) Step 4. Mutation (bit wise
mutation) Step 5. Repair Step 6. Duplication
and stillborn check Step 7. Evaluate the
offspring Step 8. Insertion and Replacement Step
9. Update ranks and raw fitnesses. Step 10.
Termination
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Computational Results (Cont. Test Problems)
ZDT 1 (Convex)
, xi ? 0,1 for i 1,...,30
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Computational Results (Cont. Test Problems)
ZDT 1 (Convex)
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Computational Results (Cont. Test Problems)
ZDT 2 (Non-convex)
, xi ? 0,1 for i 1,...,30
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Computational Results (Cont. Test Problems)
ZDT 2 (Non-convex)
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Computational Results (Cont. Test Problems)
ZDT 3 (Convex, Disconnected)
, xi ? 0,1 for i 1,...,30
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Computational Results (Cont. Test Problems)
ZDT 3 (Convex, Disconnected)
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Computational Results (Cont. Test Problems)
ZDT 4 (Convex, Multimodal)
, x1 ? 0,1 and xi ? -5,5 for i 2,...,10
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Computational Results (Cont. Test Problems)
ZDT 4 (Convex, Multimodal)
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Computational Results (Cont. Test Problems)
ZDT 5 (Convex, Deceptive)
, xi ? 0,1 for i 1,...,11
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Computational Results (Cont. Test Problems)
ZDT 5 (Convex, Deceptive)
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Computational Results (Cont. Test Problems)
ZDT 6 (Non-convex, Non-uniformly spaced)
, xi ? 0,1 for i 1,...,10
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Computational Results (Cont. Test Problems)
ZDT 6 (Non-convex, Non-uniformly spaced)
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Computational Results (MOKP Problem)
pj,k profit of placing j in k wj,k weight
of j in k xj 1, if j is selected xj
0, otherwise
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Computational Results (MOKP Problem)
MOKP (m2, J250)
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Computational Results (MOKP Problem)
MOKP (m2, J500)
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Computational Results (MOKP Problem)
MOKP (m2, J750)
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Conclusions
An EA is developed for Multiobjective
Optimization. Tchebycheff distance function is
considered to compute fitness scores. Tested on
six continuous and three multiobjective
2 knapsacks problems. EMAPS II is
promising. Some difficulties exist with
multimodal functions.