A combinatorial particle swarm optimisation for solving permutation flowshop problems

Simulated Annealing as an Intensification Component in Hybrid Population-based Metaheuristics

“…This choice follows the recommendation in (Taillard, 2005) suggesting the use of absolute computational burden measures (i.e., independent of the kind of computer) in order to obtain results easier to be compared in the scientific community. As regards the values of the parameters characterizing the SA procedure included in both the hybrid algorithms here considered, we fixed θ=0.95, the initial temperature T 0 = -(0.2⋅Z 0 )/log(0.5) (such value is chosen to impose that at the initial iteration the probability of accepting a solution with a 20% deviation from objective value of the starting solution is 0.5), and imposing 10⋅n 2 non improving iterations, where n is the number of jobs of the considered scheduling problem, as SA stopping criterion (note that similar settings are used in (Jarboui et al, 2007)). …”

“…This problem is NP-hard in the strong sense (Garey et al, 1976) for m≥3 and only instances of limited size can be solved by exact solution methods in an acceptable computation time. Therefore numerous heuristics approaches have been proposed in the literature, among which constructive heuristics (e.g., (Palmer, 1965), (Campbell et al, 1970), (Taillard, 1990)) improvement heuristics (e.g, (Ho & Chang, 1991), (Woo & Yim, 1998), (Suliman, 2000)) and metaheuristics as SA ( (Osman & Potts, 1989), (Ishibuchi et al, 1995)), TS ( (Nowicki & Smutnicki, 1996), (Grabowski and Wodecki, 2004)), GA ( (Reeves, 1995), (Ruiz et al, 2006)), ACO ( (Rajendran & Ziegler, 2004)) and PSO algorithms ( , (Lian et al, 2006a), (Tasgetiren et al, 2007), (Jarboui et al, 2007)), some of which are taken as reference for the performance evaluation of the PSO-SA proposed in the following.…”

“…The SA intensification step included in the overall structures of both the ACO-SA and PSO-SA algorithms respectively in Figure 1 and 2 is performed using a SA procedure similar to the one adopted for the H-CPSO algorithm presented in (Jarboui et al, 2007). The SA algorithm, which originally took its inspiration from the simulation of the physical annealing of melted metals (Kirkpatric et al, 1983), iterates exploring the neighbourhood N(x) of the current solution x accepting a stochastically generated candidate x'∈N(x) with the following probabilistic rule: if ΔZ=Z(x)-Z(x')≤0 then x' becomes the new current solution, otherwise x' is randomly accepted according to the probability distribution P(ΔZ, T)=…”

“…In both the proposed hybrid algorithms to update T we adopt the exponential rule T k =T 0 ⋅θ k , where θ is a constant positive parameter. Similarly to (Jarboui et al, 2007) we use a stochastic definition of the neighbourhood N(x) of the current solution x based on the random selection of insert and swap moves. In particular, we apply either an insert or a swap move on the permutation associated with x to generate the solution candidates at each SA iteration: first the algorithm randomly chooses with probability 0.5 between the two kinds of move, then it randomly selects the job to be moved and its insertion point or the pair of jobs to be swapped.…”

Evolutionary Hybrid Particle Swarm Optimization Algorithm for Solving NP-Hard No-Wait Flow Shop Scheduling Problems

“…In metallurgy, the annealing process is the process where metals are cooled slowly to reach a state of low energy where they are very strong [37]. At high temperatures, the movements are random, whereas at low temperatures, little randomness is observed.…”