Metaheuristics
• The idea: search the solution space directly. No math models, only a
set of algorithmic steps, iterative method. Find a feasible solution
and improve it. A greedy solution may be a good starting point.
• Goal: Find a near optimal solution in a given bounded time.
• Applied to combinatorial and constraint optimization problems
• Diversification and intensification of the search are the two
strategies for search in Metaheuristics. One must strike a balance
between them. Too much of either one will yield poor solutions.
Remember that you have only a limited amount of time to search
and you are also looking for a good quality solution. Quality vs Time
tradeoff.
– For applications such as design decisions focus on high quality solutions
(take more time) Ex. high cost of investment, and for control/operational
decisions where quick and frequent decisions are taken look for good
enough solutions (in very limited time) Ex: scheduling
– Trajectory based methods are heavy on intensification, while population
based methods are heavy on diversification.
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Metaheuristics
• The idea: search the solution space directly. No math models, only a
set of algorithmic steps, iterative method.
• Trajectory based methods (single solution based methods)
– search process is characterized by a trajectory in the search space
– It can be viewed as an evolution of a solution in (discrete) time of a
dynamical system
• Tabu Search, Simulated Annealing, Iterated Local search, variable
neighborhood search, guided local search
• Population based methods
– Every step of the search process has a population – a set of- solutions
– It can be viewed as an evolution of a set of solutions in (discrete) time of a
dynamical system
• Genetic algorithms, swarm intelligence - ant colony optimization, bee colony
optimization, scatter search
• Hybrid methods
• Parallel metaheuristics: parallel and distributed computingindependent and cooperative search
You will learn these techniques through several examples
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S-Metaheuristics – single solution based
• Idea: Improve a single solution
• These are viewed as walks through neighborhoods in the search
space (solution space)
• The walks are performed via iterative procedures that move from
the current solution to the next one
• Iterative procedure consist of generation and replacement from a
single current solution.
– Generation phase: A set of candidate solutions C(s) are generated from
the current solution s by local transformation.
– Replacement phase: Selection of s’ is performed from C(s) such that the
obj function f(s’) is better than f(s).
• The iterative process continues until a stopping criteria is reached
• The generation and replacement phases may be memoryless.
Otherwise some history of the search is stored for further
generation of candidate solutions.
• Key elements: define the neighborhood structure and the initial
solution.
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Neighborhood
• Representation of solutions
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–
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Vector of Binary values – 0/1 Knapsack, 0/1 IP problems
Vector of discrete values- Location , and assignment problems
Vector of continuous values on a real line – continuous, parameter optimization
Permutation – sequencing, scheduling, TSP
• k-distance
– For discrete values distance d(s,s’)<e, for continuous values: sphere of radius d
around s
– For binary vector of size n, 1-distance neighborhood of s will have n neighbors (flip
one bit at a time) Ex: hypercube, neighbors of 000 are 100, 010, 001.
• For permutation based representations
– k-exchange (swapping) or k-opt operator (for TSP)
• Ex: 2-opt: for permutations of size n, the size of the neighborhood is
neighbors of 231 are 321, 213 and 132
n(n-1)/2. The
– for scheduling problems
• Insertion operator 12345 14235
• Exchange operator 12345 14325
• Inversion operator 123456 154326
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Initial solution and objective function
• Random or greedy
• Or hybrid
• In most cases starting with greedy will reduce computational
time and yield better solutions, but not always
• Sometimes random solutions may be infeasible
• Sometimes expertise is used to generate initial solutions
• For population-based metaheuristics a combination of greedy
and random solutions is a good strategy.
• Complete vs incremental evaluation of the obj. function
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Distance and Landscape
• For binary and flip move operators
– For a problem of size n, the search space size is 2n and the maximum
distance of the search space is n (the maximum distance is by flipping
all n values).
• For permutation and exchange move operators
– For a problem of size n, the search space size is n! and the maximum
distance between 2 permutations is n-1
• Landscape
– Flat, plain; basin, valley;
rugged, plain;
rugged, valley
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Local search
• Hill climbing (descent), iterative improvement
• Select an initial solution
• Selection of the neighbor that improves the solution (obj
func)
– Best improvement (steepest ascent/descent). Exhaustive exploration
of the neighborhood (all possible moves). Pick the best one with the
largest improvement.
– First improvement (partial exploration of the neighborhood)
– Random selection – evaluate a few randomly selected neighbors and
select the best among them.
• Great method if there are not too many local optimas.
• Issues: search time depends on initial solution and not good if
there are many local optimas.
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Local search
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•
•
•
Maximize f(x)= x3-60x2+900x
Use binary search
Starting solution 10001 = 24+ 20 = f (17) = 2774
Find the neighbors for 10001
• Solution : local optima is 10000 = f(16) = 3136
• But global optima is 01010 = f(10)= 4000
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Escaping local optimas
• Accept nonimproving neighbors
– Tabu search and simulated annealing
• Iterating with different initial solutions
– Multistart local search, greedy randomized adaptive search procedure
(GRASP), iterative local search
• Changing the neighborhood
– Variable neighborhood search
• Changing the objective function or the input to the problem in
a effort to solve the original problem more effectively.
– Guided local search
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Tabu search – Job-shop Scheduling problems
• Single machine, n jobs, minimize total weighted tardiness, a job
when started must be completed, N-P hard problem, n! solutions
• Completion time Cj
• Due date dj
• Processing time pj
• Weight wj
• Release date rj
• Tardiness Tj = max (Cj-dj, 0)
• Total weighted tardiness = ∑ wj . Tj
• The value of the best schedule is also called aspiration criterion
• Tabu list = list the swaps for a fixed number of previous moves
(usually between 5 and 9 swaps for large problems), too few will
result in cycling and too many may be unduly constrained.
• Tabu tenure of a move= number of iterations for which a move is
forbidden.
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Tabu Search - Job shop scheduling example
• Single machine, 4 jobs, minimize total weighted tardiness, a
job when started must be completed, N-P hard problem, 4!
Solutions
• Use adjacent pair-wise flip operator
• Tabu list – last two moves cannot be swapped again.
Jobs j
pj
dj
wj
1
10
4
14
2
10
2
12
3
13
1
1
4
4
12
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Initial solution 2143 weighted tardiness = 500
• Stopping criterion: Stop after a fixed number of moves or
after a fixed number of moves show no improvements over
the current best solution.
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Tabu search
• Static and dynamic size of tabu list
• Short term memory – stores recent history of solutions to
prevent cycling. Not very popular because of high data
storage requirements.
• Medium term memory – Intensification of search around the
best found solutions. Only as effectives a the landscape
structure. Ex: intensification around a basin is useless.
• Long term memory – Encourages diversification. Keeps an
account of the frequency of solutions (often from the start of
the search) and encourages search around the most frequent
ones.
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