Preliminary Background Tabu Search Genetic Algorithm

Problem used to illustrate General problem min f(x) x є X Assignment type problem: Assignment of resources j to activities i min f(x) Subject to ∑ 1≤ j≤ m x ij = 1 1≤ i ≤ n x ij = 0 or 1 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 x 0. 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, x ij(i) = 1 x ij = 0 for all other j 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 = x ij for all other i, j The elements of the neighborhood N(x) are generated by slightly modifying x: N(x) = {x' : x' = x  m, m  M }

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'

Tabu Search (TS)

Initialize Select an initial solution x 0 є X Let x:= x 0

Tabu Search (TS) Initialize Select an initial solution x 0 є X Let x:= x 0 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) } At each iteration, a best solution x' є NC(x) is selected

Tabu Search (TS) Initialize Select an initial solution x 0 є X Let x:= x 0 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' At each iteration, a best solution x' є NC(x) is selected x' є NC(x) is the current solution for the next iteration

Tabu Search (TS) Initialize Select an initial solution x 0 є X Let x:= x 0 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 x 0 є X Let TL k = Φ, k = 1, 2, …,p Let x:= x 0 and stop:= false While not stop Determine a subset NC(x) ⊆ N(x) of solutions z = x  m such that t k (m) is not in TL k, k = 1, 2, …, p Determine x' є NC(x) such that x' := argmin z є NC(x) { f(z) } x:= x' Update Tabu Lists TL k, 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 TL k 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 n k of iterations

Tabu Search (TS) Initialize Select an initial solution x 0 є X Let TL k = Φ, k = 1, 2, …,p Let x:= x 0 and stop:= false While not stop Determine a subset NC(x) ⊆ N(x) of solutions z = x  m such that t k (m) is not in TL k, k = 1, 2, …, p Determine x' є NC(x) such that x' := argmin z є NC(x) { f(z) } x:= x' Update Tabu Lists TL k, 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 x 0 є X Let TL k = Φ, k = 1, 2, …,p Let x* := x:= x 0 and stop:= false While not stop Determine a subset NC(x) ⊆ N(x) of solutions z = x  m such that t k (m) is not in TL k, 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 TL k, 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 x 0 є X Let TL k = Φ, k = 1, 2, …,p Let x* := x:= x 0 and stop:= false While not stop Determine a subset NC(x) ⊆ N(x) of solutions z = x  m such that t k (m) is not in TL k, 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 TL k, k = 1, 2,…, p Update x* the best solution found so far

Tabu Search (TS) Initialize Select an initial solution x 0 є X Let TL k = Φ, k = 1, 2, …,p Let x* := x:= x 0 and stop:= false While not stop Determine a subset NC(x) ⊆ N(x) of solutions z = x  m such that t k (m) is not in TL k, 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 TL k, k = 1, 2,…, p No monotonicity of the objective function!!! Stopping criterion ???

Tabu Search (TS) Initialize Select an initial solution x 0 є X Let TL k = Φ, k = 1, 2, …,p Let iter := niter := 0 Let x* := x:= x 0 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 t k (m) is not in TL k, 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 TL k, 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 ℝ n is encoded (represented) as a genotype form vector z є ℝ m (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, x ij(i) = 1 x ij = 0 for all other j x є ℝ nxm can be encoded as z є ℝ n where z i = j(i) i = 1, 2, …, n i.e., z i 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 offspring- solutions according to different crossover (recombination) operators.

One point crossover The one point crossover generates two offspring-solutions from the two parent-solutions z 1 = [ z 1 1, z 2 1, …, z m 1 ] z 2 = [ z 1 2, z 2 2, …, z m 2 ] as follows: i) Select randomly a position (index) ρ, 0 ≤ ρ ≤ m. ii) Then the offspring-solutions are specified as follows: oz 1 = [ z 1 1, z 2 1, …, z ρ 1, z ρ+1 2, …, z m 2 ] oz 2 = [ z 1 2, z 2 2, …, z ρ 2, z ρ+1 1, …, z m 1 ] Hence the first ρ components of offspring oz 1 (offspring oz 2 ) 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 z 1 = [ z 1 1, z 2 1, …, z m 1 ] z 2 = [ z 1 2, z 2 2, …, z m 2 ] as follows: i) Select randomly two positions (indices) μ,ν, 1 ≤ μ ≤ ν ≤ m. ii) Then the offspring-soltions are specified as follows: oz 1 = [ z 1 1, …, z μ-1 1, z μ 2, …, z ν 2, z ν+1 1, …, z m 1 ] oz 2 = [ z 1 2, …, z μ-1 2, z μ 1, …, z ν 1, z ν+1 2, …, z m 2 ] Hence the offspring oz 1 (offspring oz 2 ) 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 z 1 = [ z 1 1, z 2 1, …, z m 1 ], z 2 = [ z 1 2, z 2 2, …, z m 2 ] : 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: [ z 1 1, z 2 1, z 3 1, z 4 1,…, z m-1 1, z m 1 ] parent 2: [ z 1 2, z 2 2, z 3 2, z 4 2,…, z m-1 2, z m 2 ] Vector of bits: [ 0, 1, 1, 0, …, 1, 0 ] Offspring oz 1 : [ z 1 1, z 2 2, z 3 2, z 4 1,…, z m-1 2, z m 1 ] Offspring oz 2 : [ z 1 2, z 2 1, z 3 1, z 4 2,…, z m-1 1, z m 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 z 1 = [ z 1 1, z 2 1, …, z m 1 ], z 2 = [ z 1 2, z 2 2, …, z m 2 ] : 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: [ z 1 1, z 2 1, z 3 1, z 4 1,…, z m-1 1, z m 1 ] parent 2: [ z 1 2, z 2 2, z 3 2, z 4 2,…, z m-1 2, z m 2 ] Vector of bits: [ 0, 1, 1, 0, …, 1, 0 ] Offspring oz 1 : [ z 1 1, z 2 2, z 3 2, z 4 1,…, z m-1 2, z m 1 ] Offspring oz 2 : [ z 1 2, z 2 1, z 3 1, z 4 2,…, z m-1 1, z m 2 ] Hence the i th component of oz 1 (oz 2 ) is the i th component of parent 1 (parent 2) if the i th component of the vector of bits is 0, otherwise, it is equal to the i th component of parent 2 (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 z i 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 z i where βmax is small enough in order to modify z i 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