Preliminary Background
Tabu Search
Genetic Algorithm
Faculty of Information Technology
University of Science
Vietnam National University of Ho Chi Minh City
March 2010
Problem used to illustrate
• General problem
min f(x)
xєX
• Assignment type problem: Assignment of resources j to activities i
min f(x)
Subject to ∑1≤ j≤ m xij = 1
xij = 0 or 1
1≤ i ≤ n
1≤ i ≤ n, 1≤ j ≤ m
Neighborhood (Local) Search Techniques
(NST)
• A Neighborhood (Local) Search Technique (NST) is an iterative procedure
starting with an initial feasible solution x0 .
• At each iteration:
- we move from the current solution x є X to a new one x' є X in its
neighborhood N(x)
- x' becomes the current solution for the next iteration
- we update the best solution x* found so far.
• The procedure continues until some stopping criterion is satisfied
Neighborhood
Neighborhood N(x) :
The neighborhood N(x) varies with the problem, but its elements are always
generated by slightly modifying x.
If we denote M the set of modifications (or moves) to generate neighboring
solutions, then
N(x) = {x' : x' = x  m , m  M }
Neighborhood for assigment type problem
• For the assignment type problem:
Let x be as follows: for each 1≤ i ≤ n,
xij(i) = 1
xij = 0 for all other j
The elements of the neighborhood
N(x) are generated by slightly
modifying x:
N(x) = {x' : x' = x  m , m  M }
Each solution x' є N(x) is obtained by selecting an activity i
and modifying its resource from j(i) to some other p
(i. e., the modification can be denoted m = [i, p] ):
x' ij(i) = 0
x' ip = 1
x' ij = xij for all other i, j
Descent Method (D)
• At each iteration, a best solution x' є N(x) is selected as the
current solution for the next iteration.
• Stopping criterion:
f(x') ≥ f(x)
i.e., the current solution cannot be improved or
a first local minimum is reached.
Tabu Search
• Tabu Search is an iterative neighborhood or
local search technique
• At each iteration we move from a current
solution x to a new solution x' in a neigborhood
of x denoted N(x),
• until we reach some solution x* acceptable
according to some criterion
Selecting x'
Selecting x'
Selecting x'
Selecting x'
Selecting x'
Selecting x'
Selecting x'
Tabu Search (TS)
Tabu Search (TS)
•
Initialize
Select an initial solution x0 є X
Let x:= x0
Tabu Search (TS)
•
Initialize
Select an initial solution x0 є X
Let x:= x0 and stop:= false
•
While not stop
Determine a subset NC(x) ⊆ N(x)
At each iteration, a best solution x' є NC(x) is
selected
Determine x' є NC(x) such that
x' := argmin z є NC(x) { f(z) }
Tabu Search (TS)
•
Initialize
Select an initial solution x0 є X
Let x:= x0 and stop:= false
•
While not stop
Determine a subset NC(x) ⊆ N(x)
At each iteration, a best solution x' є NC(x) is
selected
Determine x' є NC(x) such that
x' := argmin z є NC(x) { f(z) }
x:= x'
x' є NC(x) is the current solution for the next
iteration
Tabu Search (TS)
•
Initialize
Select an initial solution x0 є X
Let x:= x0 and stop:= false
•
While not stop
Determine a subset NC(x) ⊆ N(x)
Determine x' є NC(x) such that
x' := argmin z є NC(x) { f(z) }
x:= x'
•
As long as x' is better than x, the behavior of the
procedure is similar to that of the descent
method.
•
Otherwise, moving to x' as the next current
solution induces no improvement or a
deterioration of the objective function, but it
allows to move out of a local minimum
Tabu Search (TS)
•
Initialize
Select an initial solution x0 є X
Let TLk = Φ, k = 1, 2, …,p
Let x:= x0 and stop:= false
•
While not stop
Determine a subset NC(x) ⊆ N(x) of solutions
z = x  m such that
tk(m) is not in TLk , k = 1, 2, …, p
Determine x' є NC(x) such that
x' := argmin z є NC(x) { f(z) }
x:= x'
Update Tabu Lists TLk , k = 1, 2,…, p
•
As long as x' is better than x, the behavior of the
procedure is similar to that of the descent
method.
•
Otherwise, moving to x' as the next current
solution induces no improvement or a
deterioration of the objective function, but it
allows to move out of a local minimum
To prevent cycling, recently visited solutions are
eliminated from NC(x) using Tabu lists
Tabu Lists (TL)
• Short term Tabu lists TLk are used to remember attributes or characteristics
of the modification used to generate the new current solution
• A Tabu List often used for the assignment type problem is the following:
If the new current solution x' is generated from x by modifying the
resource of i from j(i) to p, then the pair (i, j(i)) is introduced in the Tabu
list TL
If the pair (i, j) is in TL, then any solution where resource j is to be
assigned to i is declared Tabu
• The Tabu lists are cyclic in order for an attribute to remain Tabu for a fixed
number nk of iterations
Tabu Search (TS)
•
Initialize
Select an initial solution x0 є X
Let TLk = Φ, k = 1, 2, …,p
Let x:= x0 and stop:= false
•
While not stop
Determine a subset NC(x) ⊆ N(x) of solutions
z = x  m such that
tk(m) is not in TLk , k = 1, 2, …, p
Determine x' є NC(x) such that
x' := argmin z є NC(x) { f(z) }
x:= x'
Update Tabu Lists TLk , k = 1, 2,…, p
•
As long as x' is better than x, the behavior of the
procedure is similar to that of the descent
method.
•
Otherwise, moving to x' as the next current
solution induces no improvement or a
deterioration of the objective function, but it
allows to move out of a local minimum
To prevent cycling, recently visited solutions are
eliminated from NC(x) using Tabu lists
Tabu Search (TS)
•
Initialize
Select an initial solution x0 є X
Let TLk = Φ, k = 1, 2, …,p
Let x* := x:= x0 and stop:= false
•
•
While not stop
Determine a subset NC(x) ⊆ N(x) of solutions
z = x  m such that
tk(m) is not in TLk , k = 1, 2, …, p
or
f(z) < f(x*)
Determine x' є NC(x) such that
x' := argmin z є NC(x) { f(z) }
x:= x'
Update Tabu Lists TLk , k = 1, 2,…, p
Since Tabu lists are specified in terms of
attributes of the modifications used, we required
an Aspiration criterion to bypass the tabu status
of good solutions declared Tabu without having
been visited recently
may include z in NC(x) even if z is Tabu
whenever
f(z) < f(x*)
where x* is the best solution found so far
Tabu Search (TS)
•
Initialize
Select an initial solution x0 є X
Let TLk = Φ, k = 1, 2, …,p
Let x* := x:= x0 and stop:= false
•
While not stop
Determine a subset NC(x) ⊆ N(x) of solutions
z = x  m such that
tk(m) is not in TLk , k = 1, 2, …, p
or
f(z) < f(x*)
Determine x' є NC(x) such that
x' := argmin z є NC(x) { f(z) }
x:= x'
If f(x) < f(x*) then x* := x ,
Update Tabu Lists TLk , k = 1, 2,…, p
Update x* the best solution found so far
Tabu Search (TS)
•
•
Initialize
Select an initial solution x0 є X
Let TLk = Φ, k = 1, 2, …,p
•
No monotonicity of the objective function!!!
Let x* := x:= x0 and stop:= false
•
Stopping criterion ???
While not stop
Determine a subset NC(x) ⊆ N(x) of solutions
z = x  m such that
tk(m) is not in TLk , k = 1, 2, …, p
or
f(z) < f(x*)
Determine x' є NC(x) such that
x' := argmin z є NC(x) { f(z) }
x:= x'
If f(x) < f(x*) then x* := x ,
Update Tabu Lists TLk , k = 1, 2,…, p
Tabu Search (TS)
•
Initialize
Select an initial solution x0 є X
Let TLk = Φ, k = 1, 2, …,p
Let iter := niter := 0
Let x* := x:= x0 and stop:= false
•
While not stop
iter := iter + 1 ; niter := niter + 1
Determine a subset NC(x) ⊆ N(x) of solutions
z = x  m such that
tk(m) is not in TLk , k = 1, 2, …, p
or
f(z) < f(x*)
Determine x' є NC(x) such that
x' := argmin z є NC(x) { f(z) }
x:= x'
If f(x) < f(x*) then x* := x , and niter := 0
If iter = itermax or niter = nitermax
then stop := true
Update Tabu Lists TLk , k = 1, 2,…, p
x* is the best solution generated
•
Stopping criteria:
- maximum number of iterations
- maximum number of successive
iterations where f(x*) does not improve
Improving Strategies
• Intensification
• Multistart diversification strategies:
- Random Diversification (RD)
- First Order Diversification (FOD)
• Variable Neighborhood Search (VNS)
• Exchange Procedure
Intensification
• Intensification strategy used to search more extensively a
promissing region
Diversification
• The diversification principle is complementary to the
intensification. Its objective is to search more extensively the
feasible domain by leading the NST to unexplored regions of
the feasible domain.
• Numerical experiences indicate that it seems better to apply a
short NST (of shorter duration) several times using different
initial solutions rather than a long NST (of longer duration).
Genetic Algorithm (GA)
• Population based algorithm
• At each iteration (generation) three different operators are first applied to
generate a set of new (offspring) solutions using the N solutions of the
current population:
selection operator: selecting from the current population parent-solutions
that reproduce themselves
crossover (reproduction) operator: producing offspring-solutions from each
pair of parent-solutions
mutation operator: modifying (improving) individual offspring-solution
• A fourth operator (culling operator) is applied to determine a new
population of size N by selecting among the solutions of the current
population and the offspring-solutions according to some strategy
Encoding the solution
• The phenotype form of the solution x є ℝn is encoded (represented) as a
genotype form vector z є ℝm (or chromozome) where m may be different
from n.
• For example in the assignment type problem:
let x be the following solution: for each 1≤ i ≤ n,
xij(i) = 1
xij = 0 for all other j
x є ℝnxm can be encoded as z є ℝn where
zi = j(i)
i = 1, 2, …, n
i.e., zi is the index of the resource j(i) assigned to activity i
Selection operator
• This operator is used to select an even number (2, or 4, or …, or N) of
parent-solutions.
• Each parent-solution is selected from the current population according to
some strategy or selection operator.
• Note that the same solution can be selected more than once.
• The parent-solutions are paired two by two to reproduce themselves.
• Selection operators:
Random selection operator
Proportional (or roulette whell) selection operator
Tournament selection operator
Diversity preserving selection operator
Crossover (recombination) operators
• Crossover operator is used to generate new solutions including
interesting components contained in different solutions of the
current population.
• The objective is to guide the search toward promissing regions
of the feasible domain X while maintaining some level of
diversity in the population.
• Pairs of parent-solutions are combined to generate offspringsolutions according to different crossover (recombination)
operators.
One point crossover
• The one point crossover generates two offspring-solutions from the two
parent-solutions
z1 = [ z11, z21, …, zm1]
z2 = [ z12, z22, …, zm2]
as follows:
i) Select randomly a position (index) ρ, 0 ≤ ρ ≤ m.
ii) Then the offspring-solutions are specified as follows:
oz1 = [ z11, z21, …, zρ1, zρ+12, …, zm2]
oz2 = [ z12, z22, …, zρ2, zρ+11, …, zm1]
Hence the first ρ components of offspring oz1 (offspring oz2) are the
corresponding ones of parent 1 (parent 2), and the rest of the components
are the corresponding ones of parent 2 (parent 1)
Two points crossover
• The two points crossover generates two offspring-solutions from the two
parent-solutions
z1 = [ z11, z21, …, zm1]
z2 = [ z12, z22, …, zm2]
as follows:
i) Select randomly two positions (indices) μ,ν, 1 ≤ μ ≤ ν ≤ m.
ii) Then the offspring-soltions are specified as follows:
oz1 = [ z11, …, zμ-11, zμ2, …, zν2, zν+11, …, zm1]
oz2 = [ z12, …, zμ-12, zμ1, …, zν1, zν+12, …, zm2]
Hence the offspring oz1 (offspring oz2) has components μ, μ+1, …, ν of
parent 2 (parent 1), and the rest of the components are the corresponding
ones of parent 1 (parent 2)
Uniform crossover
• The uniform crossover requires a vector of bits (0 or 1) of dimension m to
generate two offspring-solutions from the two parent-solutions
z1 = [ z11, z21, …, zm1] , z2 = [ z12, z22, …, zm2] :
i) Generate randomly a vector of bits, for example [0, 1, 1, 0, …, 1, 0]
ii) Then the offspring-solutions are specified as follows:
parent 1: [ z11, z21, z31, z41,…, zm-11, zm1]
parent 2: [ z12, z22, z32, z42,…, zm-12, zm2]
Vector of bits: [ 0 , 1 , 1 , 0 , …, 1 , 0 ]
Offspring oz1 : [ z11, z22, z32, z41,…, zm-12, zm1]
Offspring oz2: [ z12, z21, z31, z42,…, zm-11, zm2]
Uniform crossover
• The uniform crossover requires a vector of bits (0 or 1) of dimension m to
generate two offspring-solutions from the two parent-solutions
z1 = [ z11, z21, …, zm1] , z2 = [ z12, z22, …, zm2] :
i) Generate randomly a vector of bits, for example [0, 1, 1, 0, …, 1, 0]
ii) Then the offspring-solutions are specified as follows:
parent 1: [ z11, z21, z31, z41,…, zm-11, zm1]
th component of oz1
Hence
the
i
parent 2: [ z12, z22, z32, z42,…, zm-12, zm2]
(oz2) is the ith component of
parent 1 (parent 2) if the ith
Vector of bits: [ 0 , 1 , 1 , 0 , …, 1 , 0 ] component of the vector of bits
is 0, otherwise, it is equal to the
Offspring oz1 : [ z11, z22, z32, z41,…, zm-12, zm1] ith component of parent 2
Offspring oz2: [ z12, z21, z31, z42,…, zm-11, zm2] (parent 1)
Mutation operator
• Mutation operator is an individual process to modify offspring-solutions
• In traditional variants of Genetic Algorithm the mutation operator is used to
modify arbitrarely each componenet zi with a small probability:
For i = 1 to m
Generate a random number β є [0, 1]
If β < βmax then select randomly a new value for zi
where βmax is small enough in order to modify zi with a small probability
• Mutation operator simulates random events perturbating the natural
evolution process
• Mutation operator not essential, but the randomness that it introduces in the
process, promotes diversity in the current population and may prevent
premature convergence to a bad local minimum