Heuristics and Approximation Algorithms for MD-KP

Moon and Moser [1967], among other results, present upper bounds on the area S of a two-dimensional bin necessary to pack a list I of items. This upper bound is based only on the total area of the items, _i* _i

i I

A l w

∈

=

∑

i I

D l w

∈

= . If all items in the list are squares, then the list can be packed in any square bin with side _{B B D, ≥ + A D−} 2_{. The list can also be packed in any rectangular bin with}

2 ,

S≥ A and shorter side B, if D ≤ B. If items and bin are rectangular, it is shown that the items can be packed in a bin with shorter side B, with D ≤ B, if _{S ≥2A B+} 2 _2.

The proof, in all cases, is constructive, and is based on a packing algorithm. The algorithm initially sorts items from largest to smallest, and then packs the items in layers, as in shelves, with the height of each layer defined by the largest item in it. Each item is always placed in the lowest, and leftmost, position that accepts the item, without violating the borders of the bin. Therefore, the algorithm fixes the order in which items are packed, defines where to pack each item, fixes the orientation, and generates only guillotine cut patterns. Figure V.1 presents a typical layout obtained with this approach.

Figure V.1 Typical packing pattern obtained with the packing algorithm described by Moon and Moser [1967].

Meir and Moser [1968], using a variation of the same algorithm, improve the upper bound on the area of the bin in the general case, with rectangular items and bins, and show that if _{S≥2A B+} 2 _{8, then the set of items can be packed.}

George and Robinson [1980] also explore the idea of using layers of items when packing three-dimensional boxes in a container. As in the work of Meir and Moser [1967], items are sorted by size and packed in layers on the length of container. The length of each layer is the length of the first item packed in it. If an empty space is encountered between layers, then a filling scheme is employed in this space. Whenever possible, boxes of same dimensions are packed together. The authors report good practical results, although the worst-case performance is not analyzed.

The algorithm proposed by Coffman et al [1980], First-Fit Decreasing-Height (FFDH), for 2D-SPP generates patterns similar to those produced by Moon and Moser [1967] and George and Robinson [1980]. In this case, the dimensions considered by the authors for the bin are width and height. The width of the bin is fixed and the objective is to minimize the height of packing all items. This algorithm obtains heights that are at most 1.7 times larger than the height observed in an optimal solution.

Wang [1983] presents a method for generating solutions for 2D-KP based on the idea of combining items together into larger rectangles and combining these again into larger rectangles, and solves instances with up to 20 items. Further work with this approach includes Oliveira and Ferreira [1990] and Daza et al [1995].

Murata et al [1995] introduce the idea of Sequence Pairs, and show that it can be used to represent and solve 2D-KP problems. A Sequence Pair representation of the

packing pattern of a set I ={ , , , ,...}a b c d of items consists of two strings, each with length

I , determined on an oblique grid as shown in Figure V.2.

Figure V.2 Using Sequence Pairs to represent a packing pattern for 2D-KP.

The packing pattern in (a), with items {a, b, c, d, e, f} is represented by the sequence pair (a b d e c f, c b f a d e). The order of the symbols in the string determines the position of the item in the oblique grid, and defines the packing pattern (after Kang and Dai [1998]).

Considering 3D-KP, Li and Cheng [1990] demonstrate that packing strategies as the FFDH have unbounded worst-case performance bounds in 3D-SPP and propose a new approximation algorithm which generates packing with height at most 3.25 times larger than in an optimal solution. Li and Cheng [1992] subsequently study other approximation algorithms.

Chen et al [1995] propose an exact algorithm for container loading based on a MIP model, and show it can be used to solve an instance of 3D-KP with six items.

Scheithauer and Sommerweiss [1998] propose another approach for 2D-KP, not based on layers. Their procedure, the Four-Block heuristic based on the G4-heuristic for PLP [Scheithauer and Terno 1996], packs items with identical dimensions in blocks, and uses up to four blocks placed in the corners of the bin. Each block is composed with different items, as in Figure V.3.

Figure V.3 Packing pattern generated with the 4-Block heuristic, proposed by Scheithauer and Sommerweiss [1998].

Scheithauer and Sommerweiss [1998] also discuss some practical conditions influencing the feasibility of packing patterns. These conditions, usually observed in real world applications, are:

Weight: The total weight of the items packed in a pallet has to be less than a structural limit.

Placement: Some items, because of their density, weight, or contents may not be placed on top of other items.

Splitting: If the demand of a given item is large enough to occupy a single pallet, then a pallet packed only with this item has to be used. This restriction has the object of reducing loading costs. This may also apply to a full layer of the same product on a pallet.

Connectivity: All items of a given type are to be packed as a block in a pallet. This reduces the number of trips the loader must perform to the storage area.

Stability: Packing patterns have to be stable for transport.

Bischoff and Ratcliff [1995] and Adams [1996] address these and other conditions related to container loading.