The two-machine permutation flow shop scheduling problem with buffer is studied for the special case that all processing times on one of the two machines are equal to a constant c . This case is interesting because it occurs in various applications, for example, when one machine is a packing machine or when materials have to be transported. Different types of buffers and buffer usage are considered. It is shown that all considered buffer flow shop problems remain NP -hard for the makespan criterion even with the restriction to equal processing times on one machine. However, the special case where the constant c is larger or smaller than all processing times on the other machine is shown to be polynomially solvable by presenting an algorithm (2BF-OPT) that calculates optimal schedules in O ( n log n ) steps. Two heuristics for solving the NP -hard flow shop problems are proposed: (i) a modification of the commonly used NEH heuristic (mNEH) and (ii) an Iterated Local Search heuristic (2BF-ILS) that uses the mNEH heuristic for computing its initial solution. It is shown experimentally that the proposed 2BF-ILS heuristic obtains better results than two state-of-the-art algorithms for buffered flow shop problems from the literature and an Ant Colony Optimization algorithm. In addition, it is shown experimentally that 2BF-ILS obtains the same solution quality as the standard NEH heuristic, however, with a smaller number of function evaluations.

The dynamics of Ant Colony Optimization (ACO) algorithms is studied using a deterministic model that assumes an average expected behavior of the algorithms. The ACO optimization metaheuristic is an iterative approach, where in every iteration, artificial ants construct solutions randomly but guided by pheromone information stemming from former ants that found good solutions. The behavior of ACO algorithms and the ACO model are analyzed for certain types of permutation problems. It is shown analytically that the decisions of an ant are influenced in an intriguing way by the use of the pheromone information and the properties of the pheromone matrix. This explains why ACO algorithms can show a complex dynamic behavior even when there is only one ant per iteration and no competition occurs. The ACO model is used to describe the algorithm behavior as a combination of situations with different degrees of competition between the ants. This helps to better understand the dynamics of the algorithm when there are several ants per iteration as is always the case when using ACO algorithms for optimization. Simulations are done to compare the behavior of the ACO model with the ACO algorithm. Results show that the deterministic model describes essential features of the dynamics of ACO algorithms quite accurately, while other aspects of the algorithms behavior cannot be found in the model.