In this chapter we examine the construction of the dimensioned orthogonal geo metric dual, or layout, for any specified MPG with area requirements. We discuss in detail previous methods from the literature, providing enhancements to these methods where appropriate to facilitate implementation. Previously, little had been done to implement the more theoretical methods that have been presented. The modifications, either in the implementation, or in rounding out the theoret ical background, are in some instances to the point where the original algorithm is no longer recognisable, with only the motivational argument remaining. This effectively leads to some new methods for designing a layout, and indeed we will also see the motivation for, and implementation of, some rather different methods. Finally we present a characterisation of MPGs with special structural properties which enable a dimensionalisable layout to be easily constructed by exploiting the MPG structure.
4.1 D eltahedron
The first method that we will discuss is the Deltahedron layout method. As could probably be surmised, this �ethod uses the special structure of the Deltahedron MPG from Section 3.2 . 1 to create the layout. The principle behind this approach was first shown by G iffin, Foulds and Cameron (41] for Deltahedron MPGs with at most two distance classes, plus the exterior facility, and was fully developed by Giffin, Watson and Foulds [42] for arbitrary Deltahedron MPGs. The description of the Deltahedron Layout Algorithm which follows is based on that of the later.
44 Chapter 4. the Block Plan
4. 1 . 1 The D e ltahedron Layout Algorithm
In Section 3.2.1 we showed that the Deltahedron heuristic proceeds by inserting one vertex and three edges at a time into a face of the current MPG, starting with the Tetrahedron as the initial triangulation. The layout is constructed in essentially the same manner by considering an initial template representing the Tetrahedron, and then placing facilities in the layout in the same order in which they were inserted into the MPG. Throughout this description, we assume that the exterior facility, labelled 1 , is in the initial Tetrahedron, which is a requirement for the creation of the layout by this method. We will see in Section 4.1.3 that this requirement can always be satisfied by a rearranging of the insertion order used to generate the MPG, which forces the exterior to be in the initial Tetrahedron without changing the final MPG.
The initial tetrahedron may be constructed as shown in Figure 4. 1 . The wall intersections of the layout labelled J are the 3-joints, whereas those labelled j de note 2-joints. This labelling can be easily updated to reflect subsequent placements of facilities into the layout.
1(1,3,4)
j
1
j
J2
1(2,3,4)
1(1,2,4)
3
4
1(1,3,4)
JFigure 4. 1 : The Initial Deltahedron Layout
If a face (!1, f2 ,
h)
is deleted from the MPG by inserting a vertex f in this face, then in the layout f must be placed adjacent to the facilities represented in the MPG by f1, h and !J. In order to maximise the resultant regularity of these four facilities, we require a suitable placement mechanism, which maintains as much ofthe current structure of the-layout as possible. In order to attain this, we place f inside one of fh h , or f3 bordering the other two facilities. The dimensioning of the layout must also be addressed, and the placement of facilities in the layout must ensure that we can dimension the layout once the layout is completed, by ensuring that no faultlines are formed during the placement process. Figure 4.2 shows the four possibilities for the insertion of a facility 5 into the initial layout, corresponding to vertex 5 being placed in each of the four faces of the initial MPG. We describe such placements of a facility to be at a particular 3-joint.
In Figure 4 .2(a) facility 5 is being placed at 3-joint ( 1 , 2, 3), i. e. adjacent to facilities 1, 2, and 3. In order to maintain the rectangular layout perimeter the placement host must be either facility 2 or 3. If facility 2 were to be chosen, then the layout would take the form of Figure 4.3, with all facilities maintaining their rectangularity, and the adjacencies ( 1 , 5), (2, 5) and (3, 5) would all be assured, regardless of the facility areas. However if a4a5 > a2a3, then adjacency (2, 3) would not be satisfied. Therefore we do not allow placements of the form of Figure 4.3.
Some 3-joint labels require updating to reflect the placement of facility 5. In the MPG, face ( 1 , 2, 3) is removed, being replaced by faces ( 1 , 2, 5), ( 1 , 3, 5), and (2, 3, 5), which is parallelled i n the layout, with the 3-joint ( 1 , 2, 5) replacing ( 1 , 2, 3), and two new 3-joints ( 1 , 3, 5), and (2, 3, 5) being created. The placement of facility 5 in Figures 4.2(b) and 4 .2(c) are similar to that of Figure 4.2(a) , however the placement of facility 5 at the 3-joint (2, 3, 4) is different, since facility 5 cannot be adjacent to the exterior. This cannot be accommodated, while maintaining rectangularity of each facility, and either facility 3 or 4 must become 1-shaped as shown in Figure 4.2( d).
Two basic placement operations are easily identifiable, as those of Figures 4.2( a) and 4.2( d); each having an obvious variation if the placement host is 1-shaped rather than rectangular, and implicit variants determined by reflections and rota
tions of these operations. Describing fully these two operations will enable us to see the choice of placement hosts, and the use of placement directions. We call the two operations P 0 1 , the general form of which is given in Figure 4.4, and is akin to the operation of Figure 4 .2(a) , while the second operation of Figure 4.5 is called P02, and is akin to Figure 4.2( d) . Figure 4.4 shows facility h being placed in ft at 3-joint J. For simplicity, the fact that f1 is chosen as the placement host at J is indicated by an arrow emanating from J into f1 , called a placement direction.
46
5
3
5
1
2
3
(a)
5 at ( 1 ,2,3)1
2
(
c)
5 at (1 ,3,4)4
4
4. the Block Plan
3
3
1
2
4
(b) 5 at ( 1 ,2,4)1
2
5
(d) 5 at ( 2,3,4)5
4
1
5
2
3
4
Figure 4.3: A forbidden placement of facility 5 at ( 1 , 2, 3)
J
3-joint
2-joint
3-joint
48 Chapter 4. the Block Plan