An ordering Genetic Algorithm for Assembly Planning

1
An ordering Genetic Algorithm for Assembly
Planning

• P. De Lit, P. Latinne, B. Rekiek, A. Delchambre
• Department of Applied Mechanics
• Université Libre de Bruxelles

2
What will we talk about ?

• Presentation of the problem
• Ordering Genetic Algorithms
• Precedence constraints
• Sequence evaluation
• Conclusions

3
What will we talk about ?

• Presentation of the problem
• Ordering Genetic Algorithms
• Precedence constraints
• Sequence evaluation
• Conclusions

4
We aim to determine assembly plans

• From the links between the componentsLijcouple
  of connected components, possible assembly
  directions
• Interference matrices

DATA
5
Our illustrative example

• n components (7)
• m links (10)
• Precedence constraints between the links

6
Some links are implicitly defined

• Implicit links arise when assembling two
  subassemblies or a component to a subassembly

gen. support (2)
cover (1)
Smouse 6 5 8 7 3 2 4 9 0 1
6
5
circuit/ cable (3)
3
buttons (0)

• By realizing link 7, link 3 is implicitly defined

ball support (5)
ball (4)
7
8
7
Identification and management of the subassemblies

• A group vector keeps trace of the pertaining from
  a part to the several subassemblies, indexed from
  n (number of components) to 2n-1

Smouse 6 5 8 7 3 2 4 9 0 1
8
What will we talk about ?

• Presentation of the problem
• Ordering Genetic Algorithms
• Precedence constraints
• Sequence evaluation
• Conclusions

9
What are Genetic Algorithms ?

• Inspired from evolution of species in Nature
  (objective function acts as environment)
• Maintain a population of solutions
• Work with a representation of the solutions
  (chromosomes)
• New solutions created by combining the best
  members of the population (heredity)
• New solutions replace the worst members of the
  population (survival of the fittest)

10
The whole thing in a nutshell
11
Ordering problems
The order between the objects is important
Objects to order
An ordering
Two other solutions
12
The OGA encoding

• Each gene in the chromosome codes a link
• Sequence is read from the left to the right

Smouse 6 5 8 7 3 2 4 9 0 1
13
PMX Crossover
P1
P2
7
4
0
6
2
1
5
3
14
What will we talk about ?

• Presentation of the problem
• Ordering Genetic Algorithms
• Precedence constraints
• Sequence evaluation
• Conclusions

15
Chromosomes need to be repaired after the
crossover

• Univocal validation function V repairs the
  chromosome

Possible chromosome encoding
GA population
V
V
Valid solutions
16
Sequence is validated thanks to the precedence
constraints
gen. support (2)
cover (1)
9
2
6
circuit/ cable (3)
17
Each link in a chromosome receives a precedence
value ?ij
where Lxy is the set of links that must
precede Lij ?xy is the set of precedence values
associated to Lxy M is a chosen number greater
than m

• Links with a higher ?ij are performed first

18
Computation of the ?
L12 (9) must be preceded by L13 (2) and L23 (6)
The ? are computed recursively

• If ?13 is not known, ?12 cannot be computed

19
Links are ordered according to their precedence
values ?

• If two values are equal, the first link in the
  sequence to repair is placed first
• Example from

?13 ?23 9 and ?12 5 yields
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Seemingly correct sequences may be incorrect
because of implicit links

• Suppose 3 must precede 9 (and sequence is correct
  according to the ?)

gen. support (2)
cover (1)
9
2
6
5
circuit/ cable (3)

• Because of the implicit linksthe sequence is not
  valid

buttons (0)
4
21
Repair mechanism for implicit links

• The liaison implying invalid links is delayed
  (placed in a FIFO with the links on which it has
  a precedence constraint)
• Next link is analyzed
• As soon as a link of the original sequence is
  validated, we try to put a link in the FIFO in
  the sequence under construction
• If this is not possible, we go to the next link
  of the original sequence

22
Example of implicit link treatment
Smouse3 6 5 2 8 7 3 4 9 0 1

• 2 is put on the stack 6 5 ...
• 8 does not implies invalid links 6 5 8 ... and
  2 cannot be placed in the sequence
• 7 does not implies invalid links 6 5 8 7... and
  2 cannot be placed in the sequence
• 3 does not implies invalid links 6 5 8 7 3 ...
• 2 is placed in the sequence 6 5 8 7 3 2 ...

Finally, Smouse 6 5 8 7 3 2 4 9 0 1
23
Sequence decoding
Smouse 6 5 7 8 3 2 4 9 0 1
DATA
24
What will we talk about ?

• Presentation of the problem
• Ordering Genetic Algorithms
• Precedence constraints
• Sequence evaluation
• Conclusions

25
Individuals are evaluated thanks to several
criteria

• Number of turnings
• Stability
• Parallelism
• Early or late components
• ...

We need a method to compare the plans
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Number of turnings

• When connecting a component to a subassembly, we
  check check if the direction of insertion has
  changed
• When connecting two new components, a handability
  index is used
• When connecting two subassemblies, the first is
  considered to go on the second

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Stability and parallelism

• We compute the stability of the subassemblies
  along the axes thanks to contact matrices
• Stability is checked each time a link is
  performed, and instabilities are summed
• Parallelism index is incremented when two new
  parts or subassemblies are assembled

28
How to make the comparison ?
Classical aggregation
PrometheeII method

• w1 x c1 w2 x c2 ...
• Lack of coherence in the aggregation of several
  not comparable criteria
• Yields an absolute fitness for the individuals

• Multi-criteria decision aid method
• Coherent weights are given to each criterion
• Yields a relative fitness of the individuals in
  the population

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What will we talk about ?

• Presentation of the problem
• Ordering Genetic Algorithms
• Precedence constraints
• Sequence evaluation
• Conclusions

30
Let us conclude

• An Ordering Genetic algorithm is used to
  construct the assembly plans
• Precedence constraints are taken into account to
  generate valid solutions
• The plans are evaluated thanks to a multicriteria
  decision-aid method (PrometheeII)

31
screws 1 2 (6)
1
0
5
3
(7)
ball support (5)
ball (4)
7
8
(9)
32
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