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Population-based metaheuristics
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Nature-inspired
Initialize a population
A new population of solutions is generated
Integrate the new population into the current one using one these
methods – by replacement which is a selection process from the new and
current solutions
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Evolutionary Algorithms – genetic algorithm
Estimation of distribution algorithm (EDA)
Scatter search
Evolutionary programming- genetic programming
Swarm Intelligence
• Ant colony
• Particle swarm optimization (PSO)
• Bee colony
– Artificial Immune system AIS
• Continue until a stopping criteria is reached
• The generation and replacement process could be memoryless or some
search memory is used
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Bee Colony Optimization
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TSP
Lowering makespan is equivalent to the profitability of a food source in terms of
distance and sweetness of nectar
5 steps: recruitment, local search, neighborhood shrinking, site abandonment, and
global search.
The mating process maintains healthy numbers of the reproductive queen, male
drones, (to generate new bees- solutions), non reproductive female workers (foragers
and scouts), and the new born broods.
Maintain a colony of bees (N solutions) where a bee will traverse the entire solution.
The scouts (ns ≤N) must visit every node once from the start to the sink
Once a feasible solution is found they will return to perform the waggle dance to
advertise their finds (evaluate the objective function and rank the best nb ≤ns to the
worst ) – create a neighborhood
The elite nb solutions are kept which recruit new foragers nbf, and the rest nb-ns also
recruit foragers but less than nbf. The numbers of recruits depends on fitness function
and the size of nb.
The new recruits (foragers) will search locally within the scouts neighborhood and
foragers better than scouts will now become new scouts. Otherwise the neighborhood
is shrunk.
If the neighborhood is gone then a new scout is picked from N
Replace nb-ns with new randomly generated solutions and add to N
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Artificial Immune system
• Mimics biological immune system
• Immune system is adaptive, parallel, self-organized
• Representations: components antigens, antibodies, cells, and
molecules
• Affinity: interactions between system components and with
the environment. Represented as a similarity or dissimilarity
index by using the distance measure
• Adaptation: Procedures that govern the system
• Population based AIS – clonal and natural selection theory
– Immune system stimulated by antigens
• Network based – immune network and danger theory
– Immune system stimulated by antigens and by other parts of the
network
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AIS – Clonal theory
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Antibody
Solution
Affinity
Obj fnc
Antigen
Optimization problem
Cloning
Reproduction of solutions
Somatic mutation multiple mutation of the solution
Affinity maturation Mutation and selection of best solution
Receptor editing Diversification
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Clonal Selection theory
• For Immune system response to infection
• Based on the concept of cloning and affinity maturation
• B and T lymphocytes are selected to destroy the antigens
invading the body
• When an antigen enters the body, the B cells that best bind
with the antigens proliferate by cloning. The B cells clone a
specific type of antibody
• The strength of binding is dependent on how well (closely)
the paratope on the antibody bind with the epitope of the
antigen. This property is called affinity
• A high rate of somatic mutation is applied to the cloned cell to
promote genetic diversity
• The selection pressure ensures that only cells with higher
affinity will survive
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Clonal Selection theory in Optimization
• Antibody and antigens are represented by a string of attributes and
are binary, integer or real valued
• Their matching is performed based on distance metric Euclidean,
Manhattan or Hamming
• Generate a population of N antibodies randomly. Select n from N
based on a selection criteria and clone and mutate to construct new
candidate population of antibodies
• The number of clones Nc generated is proportional to their affinities
• The mutation rate is inversely proportional to their affinities
• Receptor editing leads to extreme mutation, which results in
diversification
• Evaluate the cloned antibodies
• Replace the worst members of the n with the best from the above
process
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Timetabling Summary
• Timetabling with resource constraints
– Each job uses only one type of tool
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No precedence constraints
R identical units of the tool are available
Rj units needed by each job j
pj = 1 for all j
• Timetabling with tooling constraints
– Different types of tools but only 1 unit (quantity) of each type of tool
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pj = 1 for all j
Problem equivalent to graph coloring heuristic
No cost involved
No precedence relationship
– With minimizing timetabling costs
• pj = 1 for all j
• No precedence relationship
– Everything same as above but pj is different – more in workforce scheduling
• Timetabling with both tooling and resource constraints
– Different types of tools, multiple quantities, pj is different, and there is a
precedence constraint
• Project scheduling
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Project-Scheduling with Resource Constraints
• Chapter 4
• Each job requires a given amount of resource
• If jobs overlap then at any time the demand for a given
resource should not exceed the total amount available
• Obj fnc: Minimize makespan
• Let N be the number of different type of resources
• Wlj denote the amount job j needs of resource l
• Let Wl be the total amount of resource available
• Upper bound on makespan H = 𝑛𝑗=1 𝑝𝑗
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Project-Scheduling with Resource Constraints
• The integer program is NP Hard when the number of jobs, and
resources grow in size
• Hence, a combination of heuristics are needed for different
processing times
• Both FF and FFD can be adapted to solve this problem if pj = 1
for all j
• Also considering preemptions can improve the solution
– Job interruptions
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Project-Scheduling with Resource Constraints
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Precedence constraints
Machine sequences for each job
Tooling constraints
Resource constraints – workforce
Obj fnc: minimize makespan
One possible solution
– Start with shifting bottleneck and use only those sequences for lateness
calculation in which the precedence constraint is satisfied on each
machine.
– For each interval [t-1, t] check feasibility using the tooling constraint and
resource constraint.
– Resequence and check feasibility again.
• If pj = 1 for all j then use FFD to create an initial set of bins [t-1,t]
and identify the jobs in those bins
– Check for tooling conflicts
– Shift jobs between bins to avoid tool conflicts.
– Assign the jobs to the machines in each bin while checking for precedence
constraints
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Timetabling in Sports
• Chapter 10
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