Flow Shop with Job–Dependent Buffer Requirements—a Polynomial–Time Algorithm and Efficient Heuristics

Iterated Local Search and Other Algorithms for Buffered Two-Machine Permutation Flow Shops with Constant Processing Times on One Machine

“…The flow shop problem with spanning buffer is also NP-complete (Lin et al, 2009). Examples for methods from the literature for this flow shop type are a Variable Neighborhood Search by Kononova and Kochetov (2013), where Integer Linear Programming was also used to solve small instances, a Branch-and-Bound algorithm used to calculate lower bounds and optimally solve small instances with up to 18 jobs (Lin et al, 2009) as well as a heuristic based on Lagrangian relaxation and bin packing (Kononov et al, 2019). This buffer type is also analyzed by Gu et al (2018) Figure 1: Overview of works containing comparisons between algorithms for buffered flow shops with intermediate buffers.…”

“…As shown in Section 4, the buffered flow shops with b i = c are NP-complete for all buffer models and buffer usage values considered here. However, there exist special cases that are solvable in polynomial time, such as F 2|spanningBuffer , s i = a i |C max with the additional condition that both max a i and max b i do not exceed Ω/5 (Kononov et al, 2019) and F 2|spanningBuffer , s i = a i |C max with the additional constraint max a i ≤ min b i (Min et al, 2019).…”

“…The studied problem is also relevant from a theoretical perspective as it explores the computational complexity of buffer flow shops, especially the boundary between N P -hard and efficiently solvable problems, similar to the analyses of flow shops with buffers by Kononov et al (2019) and Ernst et al (2019). For the studied buffered twomachine flow shop problems it is shown that they remain NP-hard for the makespan criterion even under the restriction to equal processing times on one machine.…”

On a borderline between the NP-hard and polynomial-time solvable cases of the flow shop with job-dependent storage requirements

“…Each job j seizes ω(j) units of storage space (buffer) at the beginning of its processing on M 1 and releases this portion of storage space only at the completion of the job's processing on M 2 . Similar to [14,15,16,20,21] it is assumed that, for each j ∈ N , ω(j) = a j . At any point in time, the total consumption of the storage space cannot exceed Ω -the storage capacity.…”

“…The computational complexity of the PP-problem motivated interest in branch-and-bound algorithms [14,19,20] as well as in integer programming-based and metaheuristic optimisation methods [16,18]. Another direction of research, triggered by the computational complexity of the PP-probem, was the study of its various particular cases [2,15,17,22,26]. Thus, [15] presents a polynomial-time algorithm and a proof that this algorithm constructs an optimal schedule for any instance of the PP-problem where the storage capacity is not less than five times the maximal processing time.…”