Preliminary Background Tabu Search Genetic Algorithm

1. Preliminary BackgroundTabu SearchGenetic Algorithm

2. 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≤ mxij = 1 1≤ i ≤ n xij = 0 or 1 1≤ i ≤ n, 1≤ j ≤ m

3. 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

4. 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 }

5. Neighborhood for assigment type problem For theassignment type problem: Let x be as follows: for each 1≤ i ≤ n, xij(i) = 1 xij = 0 for all other j Each solutionx'єN(x)is obtained byselecting an activity i and modifying its resource from j(i)tosome 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 The elements of the neighborhood N(x)are generated by slightly modifying x: N(x) = {x' : x' = x  m , m  M }

6. 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.

7. 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

8. Selecting x'

9. Selecting x'

10. Selecting x'

11. Selecting x'

12. Selecting x'

13. Selecting x'

14. Selecting x'

15. Tabu Search (TS)

16. Tabu Search (TS) Initialize Select an initial solution x0є X Let x:= x0

17. Tabu Search (TS) Initialize Select an initial solution x0є X Let x:= x0and stop:= false While not stop Determine a subsetNC(x) ⊆ N(x) Determine x' є NC(x)such that x' := argmin zє NC(x){ f(z) } At each iteration, a best solution x' є NC(x)is selected

18. Tabu Search (TS) Initialize Select an initial solution x0є X Let x:= x0and stop:= false While not stop Determine a subsetNC(x) ⊆ N(x) Determine x' є NC(x)such that x' := argmin zє 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

19. Tabu Search (TS) Initialize Select an initial solution x0є X Let x:= x0and stop:= false While not stop Determine a subsetNC(x) ⊆ N(x) Determine x' є NC(x)such that x' := argmin zє 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

20. Tabu Search (TS) Initialize Select an initial solution x0є X Let TLk = Φ, k = 1, 2, …,p Let x:= x0and stop:= false While not stop Determine a subsetNC(x) ⊆ N(x) of solutions z = x  m such that tk(m) is not inTLk , k = 1, 2, …, p Determine x' є NC(x)such that x' := argmin zє 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

21. Tabu Lists (TL) Short termTabu lists TLkare used to remember attributes or characteristics of the modification used to generate the new current solution A Tabu Listoften 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 theTabu 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

22. Tabu Search (TS) Initialize Select an initial solution x0є X Let TLk = Φ, k = 1, 2, …,p Let x:= x0and stop:= false While not stop Determine a subsetNC(x) ⊆ N(x) of solutions z = x  m such that tk(m) is not inTLk , k = 1, 2, …, p Determine x' є NC(x)such that x' := argmin zє 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

23. Tabu Search (TS) Initialize Select an initial solution x0є X Let TLk = Φ, k = 1, 2, …,p Let x* :=x:= x0and stop:= false While not stop Determine a subsetNC(x) ⊆ N(x) of solutions z = x  m such that tk(m) is not inTLk , k = 1, 2, …, p or f(z) < f(x*) Determine x' є NC(x)such that x' := argmin zє 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

24. Tabu Search (TS) Initialize Select an initial solution x0є X Let TLk = Φ, k = 1, 2, …,p Let x* := x:= x0and stop:= false While not stop Determine a subsetNC(x) ⊆ N(x) of solutions z = x  m such that tk(m) is not inTLk , k = 1, 2, …, p or f(z) < f(x*) Determine x' є NC(x)such that x' := argmin zє 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

25. Tabu Search (TS) Initialize Select an initial solution x0є X Let TLk = Φ, k = 1, 2, …,p Let x* := x:= x0and stop:= false While not stop Determine a subsetNC(x) ⊆ N(x) of solutions z = x  m such that tk(m) is not inTLk , k = 1, 2, …, p or f(z) < f(x*) Determine x' є NC(x)such that x' := argmin zє NC(x){ f(z) } x:= x' If f(x) < f(x*) then x* := x , Update Tabu Lists TLk , k = 1, 2,…, p No monotonicity of the objective function!!! Stopping criterion ???

26. Tabu Search (TS) Initialize Select an initial solution x0є X Let TLk = Φ, k = 1, 2, …,p Let iter := niter := 0 Let x* := x:= x0and stop:= false While not stop iter := iter + 1 ; niter := niter + 1 Determine a subsetNC(x) ⊆ N(x) of solutions z = x  m such that tk(m) is not inTLk , k = 1, 2, …, p or f(z) < f(x*) Determine x' є NC(x)such that x' := argmin zє NC(x){ f(z) } x:= x' If f(x) < f(x*) then x* := x ,and niter := 0 If iter= itermaxor 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

27. Improving Strategies Intensification Multistart diversification strategies: - Random Diversification (RD) - First Order Diversification (FOD) Variable Neighborhood Search (VNS) Exchange Procedure

28. Intensification • Intensification strategy used to search more extensively a promissing region

29. 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 shortNST (of shorter duration) several times using different initial solutions rather than a longNST (of longer duration).

30. 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

31. Encoding the solution • The phenotype form of the solution x єℝnis encoded(represented)as a genotype form vector z єℝm(or chromozome) where m may be different from n. • For example in the assignment type problem: let xbe the following solution: for each 1≤ i ≤ n, xij(i) = 1 xij = 0 for all other j x єℝnxmcan be encoded as z єℝnwhere zi = j(i) i = 1, 2, …, n i.e., zi is the index of the resource j(i)assigned to activity i

32. 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

33. 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.

34. 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 ρcomponentsof 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)

35. 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)

36. 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]

37. 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] Hence the ithcomponent of oz1 (oz2) is the ithcomponent of parent 1 (parent 2) if the ith component of the vector of bits is 0, otherwise, it is equal to the ithcomponent of parent 2 (parent 1)

38. 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 ziwith a small probability: For i = 1to m Generate a random numberβє[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 operatornot 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