The Three-Dimensional Bin Packing Problem and Its Practical Algorithm.

Three-dimensional cutting and packing problems have a range of important applications and are of particular relevance to the transportation of cargo in the form of container loading problems. Recent years have seen a marked increase in the number of papers examining a variant of the container loading problem ranging from largely theoretical to implementations that focus on meeting the many critical real-world constraints. In this paper, we review the literature focusing on the solution methodologies employed by researchers, with the aim of providing insight into some of the critical algorithmic design issues. In addition, we provide an extensive comparison of algorithm performance across the benchmark literature.

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“…This is the approach of Jin et al. (), who design several neighborhood moves with the aim of eliminating a target bin. Crainic et al.…”

Section: Improvement Heuristicsmentioning

confidence: 99%

A comparative review of 3D container loading algorithms

Zhao

Bennell

Bektaş

et al. 2014

Int Trans Operational Res

121

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“…(c)). Formally, the aforementioned conditions can be rewritten as follows (for more details, the reader can refer to Jin et al., ):

l_{italicij} = 1 \Rightarrow x_{i} + w_{i} \leq x_{j},

u_{italicij} = 1 \Rightarrow y_{i} + h_{i} \leq y_{j},

b_{italicij} = 1 \Rightarrow z_{i} + d_{i} \leq z_{j} .

…”

Section: Solving 3d‐sbsbpp With Hybrid Greedy Heuristicsmentioning

confidence: 99%

“…Unlike the model of Jin et al. (), in order to check whether two items are positioned in the same bin, the variable

c_{ij}

i \neq j = 1, ..., n

(see Fig. ), is applied such that

c_{italicij} = 1 \Leftrightarrow γ_{i} < γ_{j},

c_{italicij} = 0 \Leftrightarrow γ_{i} \geq γ_{j} .

In this case, two items

(i, j)

are placed in the same bin if and only if the following condition holds:

c_{italicij} + c_{j i} = 0 .

As the result, both C1 and C2 can be rewritten as follows:…”

Section: Solving 3d‐sbsbpp With Hybrid Greedy Heuristicsmentioning

confidence: 99%

“…1(c)). Formally, the aforementioned conditions can be rewritten as follows (for more details, the reader can refer to Jin et al, 2003): Unlike the model of Jin et al (2003), in order to check whether two items are positioned in the same bin, the variable c ij , i = j = 1, . .…”

Section: Notations For the 3d-sbsbppmentioning

confidence: 99%

“…We also note that several ILP models have been designed for the family of C&P problems (cf. Bekrar and Kacem, 2009;Egeblad and Pisinger, 1999;Gilmore and Gomory, 1961;Jin et al, 2003;Junqueira et al, 2012aJunqueira et al, , 2012bMack et al, 2004;Morabito and Arenales, 1994;Puchinger and Raidl, 2007;Valério de Carvalho, 2002). Finding an optimal solution for these models is extremely difficult and in some cases, searching for a feasible solution remains hard.…”

Section: Introductionmentioning

confidence: 99%

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Hybrid greedy heuristics based on linear programming for the three‐dimensional single bin‐size bin packing problem

Hifi

Negre

2013

Int Trans Operational Res

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In this paper, we propose to solve the three‐dimensional single bin‐size bin packing problem (3D‐SBSBPP) using a simple strategy based on integer linear programming (ILP) heuristics, without using any improvement based on metaheuristics. We first propose an ILP that is converted into a series of three‐dimensional single knapsack problems (3D‐SKP). Then, the first tailored heuristic can be viewed as a hybrid approach in which both “selection” and “positioning” phases are combined. The first phase serves to select a subset of items where each of these items is susceptible to belonging to an active container. The positioning phase serves to pack a subset of items already preselected by the selection phase. Then, both phases cooperate till packing all items into their corresponding containers. The second heuristic can be viewed as an extended version of the first one. Indeed, before deciding whether the current container is closed or a new container is activated, “a local reoptimization phase” is considered. Finally, both proposed heuristics are evaluated on a set of random instances obtained by using the standard generator scheme of the literature. The provided results show that both proposed heuristics remain competitive when compared to the results obtained by one of the best methods of the literature.

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