A branch and price approach for the container relocation problem

An exact algorithm for the unrestricted block relocation problem

“…In Table 5, our previous algorithm in [23] and an improved one TR using the dominance rules are compared. We also present the results of four existing exact approaches: the branch-and-price algorithm [16], the A * algorithm [17], and the IP approaches [21,28]. It should be noted that the average computation time in [21,28] is over non-trivial instances solved to optimality: Lower and upper bounds are computed for each instance in pre-processing, and it is considered as a non-trivial instance when the LB/UB gap is not zero.…”

“…It should be noted that the average computation time in [21,28] is over non-trivial instances solved to optimality: Lower and upper bounds are computed for each instance in pre-processing, and it is considered as a non-trivial instance when the LB/UB gap is not zero. With regard to [16,17], only the number of instances solved to optimality is shown in the table. It is because average computation time in [16] is provided separately for the root node and inner nodes of the search tree, which makes it difficult to evaluate overall computation time.…”

“…Jovanovic and Voß [15] proposed a chain heuristic for the restricted BRP with distinct priorities, which improves the look-ahead heuristic by introducing a heuristic function that takes into account future relocations. Zehendner and Feillet [16] proposed a branch-and-price algorithm for the restricted BRP with distinct priorities. Expósito-Izquierdo et al [17] applied a best-first A * algorithm for the restricted and unrestricted BRPs with distinct priorities, and proposed a domain-specific knowledge based heuristic for the unrestricted BRP with distinct priorities.…”

“…In addition to this difference in blocks, relocations are treated in different ways: Relocated slabs are moved back to the original stack in the same order immediately after the target slab is retrieved, or slabs are relocated at most once (a sufficient number of empty stacks are assumed). The pre-marshalling or [5] b&b Wan [6] ILP Ü nlüyurt and Aydın [11] b&b Caserta et al [3] ILP Zhu et al [13] IDA * Petering and Hussein [14] MILP Zehendner and Feillet [16] b&p Expósito-Izquierdo et al [17] A * , ILP Expósito-Izquierdo et al [18] b&b Zehendner et al [21] ILP Eskandari and Azari [19] ILP Tanaka and Takii [23] b&b Ku and Arthanari [24] b&b Tricoire et al [26] b&b de Melo da Silva et al [27] b&b †…”

Solving the pre-marshalling problem to optimality with A* and IDA*

“…In pre-marshalling, containers cannot be removed, thus we must define a much wider class of symmetries and branching rules to avoid repeated work. A branch-and-price algorithm is introduced in [53]; however, its reliance on container removals makes it impractical for pre-marshalling. The RLIDA* approach in [1] achieves state-of-the-art performance for the BRP by intelligently switching between different lower bound heuristics with differing levels of CPU time requirements and bound tightness.…”