Scatter Search
Laguna, Scatter search in Pardalos and
Resende (eds.), Handbook of Applied
Optimization, Oxford Academic Press, 2002
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Basic notions of SS
Population based metaheuristic
Differs in the way
It combines solutions
It updates the “reference solutions”
Claimed to be superior to “probabilistic learning
approaches”
Based on decision rules and (surrogate) constraints
in IP
Combining solutions is the primal counterpart to
the dual strategy of combining constraints
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Basic notions of SS (cont.)
SS operates on a set of solutions called the
reference set
A new solution is created from the linear (convex
or non-convex) combination of at least two
reference (or recently created) solutions
Reference set evolves by deleting old solutions
and adding new solutions
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Combining solutions in SS
A, B, C are reference
solutions to begin with
Solution 1 is a nonconvex combination of
A and B, generated
after a number of trials
Solution 3 is generated
from solutions C and
1, 4 from 1 and 2
C
2
1
4
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A
B
3
4
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Reference set
Unlike GA population, reference set in SS tends to
be small (20 solutions or less)
Instead of sampling from a population to choose
parents for recombination, SS considers at least all
pairs of reference solutions
All three-way, four-way (and so on) combinations
can also be tried
Solutions in reference set resemble elite solutions
of TS
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SS algorithm
Step 1:
Generate an initial set of solutions to guarantee a
critical level of diversity
Apply a heuristic process to improve these
solutions
Designate a subset of “best” solutions as the
reference set (best in terms of objective function
value or diversification)
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SS algorithm (cont.)
Step 2:
Create new solutions consisting of structured
combinations of subsets of reference solutions
Structured combinations are:
Inside or outside the convex region spanned by
reference solutions
Modified to yield acceptable solutions, e.g. if linear
combination is non-integer, apply rounding
Not necessarily feasible with respect to real constraints
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SS algorithm (cont.)
Step 3: Apply the heuristic improvement process in
Step 1 to the solutions created in Step 2
Step 4: Select a collection of best improved solutions
from Step 3 and form the new reference set
* Repeat Steps 2-4 until reference set does not
change any more
* Diversify reference set and restart from Step 1
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SS decisions
Diversification generation method (Step 1)
Heuristic improvement method (Steps 1 and 3)
Reference set update method (Steps 1 and 4)
Subset generation method (Step 2, to select
subsets of reference set to combine)
Solution combination method (Step 2)
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Example: 0-1 knapsack
Maximize 11x1 + 10 x2 + 9 x3 + 12 x4 + 10 x5
+6 x6 + 7 x7 + 5 x8 + 3 x9 + 8 x10
Subject to 33 x1 + 27 x2 + 16 x3 + 14 x4 + 29 x5 + 30 x6
+31x7 + 33 x8 + 14 x9 + 18 x10 ≤ 100
xi ∈{0,1} for i = 1,K,10
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Diversification generation
For n = 10 variables, max h = 5 and initial seed x = (0,0,...,0)
Type 1 solutions are created as follows:
Initialize xi′ = 0, i = 1,K, n
To find a solution for h = 1, K ,5, modify
x1′ = 1 − x1
x1′+ hk = 1 − x1+ hk , k = 1, K, n / h
Type 2 solutions are found as the complement of type 1, i.e.
xi′′ = 1 − xi′, i = 1, K, n
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Diversification generation (cont.)
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Heuristic improvement
If a trial solution is infeasible, change variables
from 1 to 0 starting with the one having the
smallest profit-to-weight ratio, until constraint is
no longer violated
If a trial solution is feasible, change variables from
0 to 1 starting with the one having the largest
profit-to-weight ratio, as long as the constraint is
satisfied
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Heuristic improvement (cont.)
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Heuristic improvement (cont.)
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Reference set update
Reference set of b=5 solutions including
b1=3 high quality solutions and
b2=2 diverse solutions
Improved solutions 1=9=10, 2, 8 are high quality
A diversity (distance) measure
(0, 1, 1, 1, 0, 0, 0, 0, 1, 1)
(0, 1, 1, 1, 1, 0, 0, 0, 1, 0)
(0+0+0+0+1+0+0+0+0+1) = 2
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Reference set update (cont.)
Choose diversified solutions as those whose
minimum distance to previously selected reference
solutions is maximum (solutions 3 and 7)
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Subset generation
Types of subsets considered are:
1) All 2-solution subsets
2) 3-solution subsets = 2-solution subsets + best (in
objective function value) not in the subset
3) 4-solution subsets = 3-solution subsets + best not
in the subset
4) Subsets consisting of the best i solutions, i=5,...,b
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Subset generation (cont.)
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Solution combination
Score for variable i with respect to solutions in
subset S is found as:
∑ OV ( j ) * xij
score(i ) =
j∈S
∑ OV ( j )
j∈S
Construct trial solution by rounding the score to
nearest integer, i.e.
1 if score(i) > 0.5
xi′ = 
0 if score(i ) ≤ 0.5
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Solution combination (cont.)
New solution is (0, 1, 0, 1, 0, 0, 0, 0, 0, 1)
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SS algorithm revisited
Generate 20 new solutions by combining solutions
in each subset (Step 2)
Improve new solutions (Step 3)
Update the reference set (Step 4)
If reference set has changed return to Step 2,
otherwise restart (e.g. with a different seed
solution)
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