According to the typology summarized by Bortfeldt and W¨ascher (2012), the John Deere tractor shipping case (with only 40’ containers) is Multiple Bin-Size Bin Packing Problem (MBSBPP), packing a strongly heterogeneous set of cargo into a weakly heterogeneous assortment of containers such that the value of the used containers is minimized. If this shipping optimization case includes multiple sizes of containers, the case can be categorized under Residual Bin Packing Problem
(RBPP), packing a strongly heterogeneous set of cargo into a strongly heterogeneous assortment of containers such that the value of the used containers is minimized.
Along the road of searching for methods to tackle the problems that haven’t been tackled, the uniqueness of this case is that most SKD modules (shipments) are limited variation of orientation, compared to other cases researched by other. For example, no SKD modules over 100 kg have their orientation significantly altered from the original orientation on the tractor. Those SKD modules have the bottom surface towards the bottom all the time, only rotations around the vertical axis.
Front and rear axles have the shape at the boundary between regular and irregular, shown in Figure 7.2. The concept of principle axis is also introduced to represent the poses in a more visualized way, also prepare for placing with limited variation of orientations.
Figure 7.2. The axle SKD module from John Deere 9000R tractor with principle axes highlighted (permission obtained)
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Obtaining bounding box of the object is similar to identification of the principle axes. There are two types of bounding box, oriented bounding box (OBB) and axis-aligned bounding box (AABB). Oriented bounding box means that the construction of the bounding box follows predefined principle axes, while
axis-aligned bounding box stands for bounding box without predefined axes. Identifying the principle axes for regular shapes is similar to OBB, while for
irregular shapes is closer to the AABB approach. The general method of identifying the principle axes for irregular shape is shown in the Algorithm 7.1.
Algorithm 7.1 Logical process of identifying the principle axes for irregular shape Obtain the three-dimensional shape;
Place the shape on a reference platform as the original designed way;
Plane x − y is set to be parallel to the reference platform and through the CoG of the object;
Identify the longest straight line L connecting two arbitrary surface point of the object;
such that L is through the CoG of the object and L is parallel to the reference platform;
Line L is the x-axis of the object’s principle axis; Identify line M through the CoG of the object;
such that M perpendicular to L and M parallel to reference platform; Identify line N through the CoG of the object;
such that N perpendicular to reference platform;
Line L, M , N are x-, y-, z-axis of the object, as the principle axes;
Follow Cartesian right-hand-rule to identify positive directions of the principle axes;
For regular shapes, the x − y plane is assumed to be the reference platform that the object should sit on. The rest of the steps are same as the Algorithm 7.1 shown for irregular shape. Figure 7.3 demonstrate the principle axes of a shipping container.
Figure 7.3. Principle axes for sea container as a regular shape
The orientations discussed in the earlier chapters are from the perspective of the object itself. When from the container point of view, the orientation involves more constraints, such as the stability of the container and the possible fixture device in container. Figure 7.4 shows several possible orientations of the axle SKD module loading in the container.
Notice there are several orientation patterns that not from the limited variation of orientation as discussed before. The limited variation of orientation requires only right angle turns along the principle axis that perpendicular to the reference platform. However, from the perspective of the whole container, as a constrained space, the limited variation of orientation includes all possible orientations shown in Figure 7.4. With the enhanced flexibility, the limited
variation of orientation can actually provide more possible optimal configurations. If the approach of the elevated flexibility applies to the limited variation of orientation under the perspective of single object, there may also be more possibility of finding the optimal configuration. Illustrated in Figure 7.5, the axle SKD module
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Figure 7.4. Possible orientations for placing axle SKD modules in the container with more options than limited variation of orientations
from the John Deere 9000R tractor is viewed from the x-axis and placed together with a non-right-angle turn along the x-axis. This particular configuration expends range of the potential optimal configuration with more flexible limited variation of orientation and from higher level of view when placing objects.
Therefore, by removing the right-angle turn restriction from the definition of limited variation of orientation and apply free rotation to one or two principle axes, a new orientation type forms with enhanced flexibility compared with limited variation of orientation. Implementing this new orientation type requires more programming and constraint design for the engineers, also more resources consumed by calculations.
Figure 7.5. Possible group placing of axle SKD modules with more flexibility of limited variation of orientation