Local search and trajectory metaheuristics for the flexible job shop scheduling problem with sequencing flexibility and position-based learning effect

TL;DR

The flexible job shop scheduling problem with sequencing flexibility and position-based learning effect is considered and a local search method and four trajectory metaheuristics are considered, showing that tabu search, built on the reduced neighborhood, stands out in relation to other the other three metaheuristics, namely, iterated local search, greedy randomized adaptive search procedure, and simulating annealing.

Abstract

The flexible job shop scheduling problem with sequencing flexibility and position-based learning effect is considered in the present work. In [K. A. G. Araujo, E. G. Birgin, and D. P. Ronconi, Technical Report MCDO02022024, 2024], models, constructive heuristics, and benchmark instances for the same problem were introduced. In the present work, we are concerned with the development of effective and efficient methods for its resolution. For this purpose, a local search method and four trajectory metaheuristics are considered. In the local search, we show that the classical strategy of only reallocating operations that are part of the critical path can miss better quality neighbors, as opposed to what happens in the case where there is no learning effect. Consequently, we analyze an alternative type of neighborhood reduction that eliminates only neighbors that are not better than the current solution. In addition, we also suggest a neighborhood cut and experimentally verify that this significantly reduces the neighborhood size, bringing efficiency, with minimal loss in effectiveness. Extensive numerical experiments with the local search and the metaheuristics are carried on. The experiments show that tabu search, built on the reduced neighborhood, when applied to large-sized instances, stands out in relation to other the other three metaheuristics, namely, iterated local search, greedy randomized adaptive search procedure, and simulating annealing. Experiments with classical instances without sequencing flexibility show that the introduced methods also stand out in relation to methods from the literature. All the methods introduced, as well as the instances and solutions found, are freely available. As a whole, we build a test suite that can be used in future work.

Figures (4)

• Figure 1: On the left, representation of the operations' precedence constraints by a DAG $D=(\mathcal{O},\widehat{A})$, where $\mathcal{O} = \{ 1, 2, \dots, 5 \}$ represents the set of operations and $\widehat{A} = \{(1,2), (2,3), (4,5)\}$ is the set of arcs that represents the precedence constraints. In this simple example, precedence constraints are given by a linear order, i.e. there is no sequencing flexibility. This instance has two machines and each of the five operations can be processed in any of the two machines, i.e. $\mathcal{F}=\{1,2\}$ and $\mathcal{F}_i=\mathcal{F}$ for all $i \in \mathcal{O}$. This means that there is full routing flexibility. The table on the right shows the standard processing times $p_{ik}$ of the five operations on each of the two machines.
• Figure 2: In this figure we consider the instance in Figure \ref{['fig1']} with learning rate $\alpha=1$. The digraph on the left (Figure \ref{['fig2']}a) represents a feasible solution in which machine 1 (associated with the cyan color) processes operation 2 only, while machine 2 (associated with the orange color) processes operations 1, 4, 5, and 3 in that order. The colored numbers represent the actual processing time of the operations, with the influence of the learning effect. The critical path, which length corresponds to the makespan, is given by the path $s, 1, 4, 5, 3, t$ (highlighted in yellow in the picture). The digraph on the right (Figure \ref{['fig2']}b) represents the feasible solution obtained by reallocating operation 2, that was not in the critical path, from machine 1 to machine 2 between operations 1 and 4. The constructed feasible solution, with critical path given by $s, 1, 2, 4, 5, 3, t$, has a makespan smaller than the original one (528 versus 658).
• Figure 3: This figure shows the value of $\bar{C}_{\max}$ as a function of the CPU time for each of the seven methods applied to the large-sized instances. The average makespan shown in the graphic is the average of the different values shown in Tables \ref{['tab5']}, \ref{['tab6']}, and \ref{['tab7']}, that separate the instances by DA type, Y type and learning rates $\alpha \in \{0.1, 0.2, 0.3\}$.
• Figure 4: This figure is similar to Figure \ref{['fig3']}. The difference is that, for methods that solve each instance five times, instead of considering the best of the five makespan, we consider the average of the five makespan. This measure is more fair when in the comparison there are methods that do not posses a random ingredient and, therefore, are run only one time per instance.