Population-based metaheuristics 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 – 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 1

Bee Colony Optimization Job shop scheduling Lowering makespan is equivalent to the profitability of a food source in terms of distance and sweetness of nectar Maintain a colony of bees (N solutions) where a bee will traverse a potential disjunctive arc (machine schedule) The foragers (n out of 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 to the worst) The elite solutions are kept which are then followed by other bees The mating process maintains healthy numbers of the reproductive queen, male drones, (to generate new bees- solutions), non reproductive female workers (unemployed foragers -scouts and onlookers, and the employed foragers) and, the new born broods. 2

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 3

AIS – Clonal theory Antibody Solution AffinityObj fnc AntigenOptimization problem CloningReproduction of solutions Somatic mutationmultiple mutation of the solution Affinity maturationMutation and selection of best solution Receptor editingDiversification 4

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 5

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 N c 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 6

Timetabling Summary Timetabling with resource constraints – Each job uses only one type of tool No precedence constraints R identical units of the tool are available R j units needed by each job j p j = 1 for all j Timetabling with tooling constraints – Different types of tools but only 1 unit (quantity) of each type of tool p j = 1 for all j Problem equivalent to graph coloring heuristic No cost involved No precedence relationship – With minimizing timetabling costs p j = 1 for all j No precedence relationship – Everything same as above but p j is different – more in workforce scheduling Timetabling with both tooling and resource constraints – Different types of tools, multiple quantities, p j is different, and there is a precedence constraint Project scheduling 7

Project-Scheduling with Resource Constraints 8

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 p j = 1 for all j Also considering preemptions can improve the solution – Job interruptions 9

Project-Scheduling with Resource Constraints 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 p j = 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 10

Timetabling in Sports Chapter 10 11