In this chapter we identify two fundamental procedures which enable the quantifi able improvement of a layout with respect to at least one of the regularity measures described in Section 2.5. These procedures will work on both dimensionalisable and undimensionalisable layouts. Obtaining procedures which can work on all layouts are difficult, due to the possibility ·of foregoing adj acencies, or area requirements, because of the presence of faultlines in undimensionalisable layouts.
5 . 1 Rectilinear Segment Reduction
The first improvement procedure which we introduce here takes two adjacent facili ties which share at least 3 adjacent rectilinear segments, and attempts to eliminate two corners common to both facilities. This procedure is called Rectilinear Seg ment Reduction (RSR). A simple example of which is shown in Figure 5. 1 . There
1 30 Chapter 5. Improvement Procedures
01
�2 �1
02 �
Figure 5.2: Rectilinear Segment Reduction - Conditions of Use
are two things which we must be aware of when performing RSR. The first is re taining the correct areas for each of the facilities, hence eliminating the need for re-dimensioning, and thus retaining the improvement at a local level. The second is that in changing the segments no adjacencies should be violated.
Let us assume that there exist two facilities, i and j, which share three adjacent rectilinear segments as shown in Figure 5.2, where each corner k is defined by its coordinate (xk, Yk)· We wish to find new a and f3 coordinates. With reference to Figure 5.2, consider two lengths, Y1 = IYa - Ya(new) I and Y2 = IY� - Y�(new) I · Also let
x1 = lxcr - X51 1 , x2 = jx52 - x� l· Regardless of how much we change the segments, we are constrained by Equation 5.1 .
(5. 1 )
Now either the 3-joint a1 or the 3-joint a2 will constrain the RSR on the left, and the 3-joint (31 or the 3-joint (32 will constrain the RSR on the right, or we will be able to successfully perform the reduction. The 3-joints which constrain the RSR are called a*, and (3*, respectively. Let c be a minimum specified adjacency wall length , providing Equations 5.2 and 5.3.
YI (5.2)
Y2 = (5.3)
Choosing y1 and y2 via Equations 5.2 and 5.3 ensures that Equation 5. 1 is satisfied. Furthermore, if y1 satisfies the first minimisation term, y2 will satisfy its
second, and vice-versa. In the first instance we are constrained by the 3-joint a:* , and in the second by /3*. If both satisfy their third term, then the reduction will be successful.
If the reduction is successful, then the number of corners of each facility will be reduced by two. This will naturally lead to an increase in regularity with respect to the number of corners; further, reducing the perimeter of the facility, will also enhance the perimeter ratio values. We cannot guarantee an increase in the bounding polygon measures, but they will not decrease. The usable space polygon ratios may either increase or decrease.
RSR is useful mainly for methods which employ the ODA or a variant of it, encompassing SIMPLE, the VSA and the Contraction Algorithm. RSR may be incorporated into the VSA by deleting three rectilinear segments formed during the recoupling of the facilities in the layout, while in SIMPLE, and the Contraction Algorithm, RSR essentially provides the a: and f3 values used in the ODA, which is preferable to attempting to input them by hand or fixing them at a set value for all facilities.
Note that we can reduce quite complicated structures by using this one simple operation, as two facilities sharing k adjacent rectilinear segments can have their shape improved using repeated RSRs. Also note that the RSR may not be success ful in attaining a single straight line segment, but it may be beneficial in making y1
and y2 as long as possible i n order to reduce the perimeter ratio values, by reducing
l l81 - 82 1 ! (the length of the middle line segment) .
Figure 5 . 3 shows an example of a layout constructed using the Contraction Algorithm, before and after the use of RSR, complete with the respective regularity values in Table 5. 1 .
If we are constrained by /32 or o:2, as opposed to o:1 and /31 i n Figure 5.2, we may have sets of nested facilities of the form of Figure 5.4. By iteratively changing these segments as much as possible, we can eliminate such nestings, providing the constraining 3-joints allow this, even though to begin with there may seem initially to be no possible reductions.
1 32 5 10 6 I 7 9
(
a)
Chapter 5. Procedures 5 10 6 I l 7 2 3 9(
b)
Figure 5.3: The Layouts of a 1 0 Facility Problem
(
a)
Before and(
b)
After perform ing RSRMeasure Average Values % Increase
Before RSR After RSR
Enclosed Rectangle 0.8255 0.8568 3.8%
Enclosed Golden Rectangle 0.3385 0.3631 7.3%
Enclosed Square 0.2151 0 .2244 4.3%
Bounding Rectangle 0.7143 0 . 7367 3.1%
Bounding Golden Rectangle 0.3643 0 .3755 3.1%
Bounding Square 0.2252 0.2321 3.1%
Perimeter 0.6049 0 .6629 9.6%
Number of Corners 8.2222 4 .6667 -43.2% Table 5. 1 : Changes in Selected Regularity Values for Layouts in Figure 5.3
Figure 5.4: An example where there are no immediate RSR reductions, but iterative reduction may be successful
5 . 2 Linear Transformation
This improvement procedure is a global improvement, yet still requires minimal effort. We consider a Linear Transformation (LT) of the layout. Fundamental components of this linear transformation are that area
(
as the determinant of the transformation matrix is 1 ) and shape are preserved, allowing us to never need be concerned about redimensioning, or about losing adjacencies.Consider every 2- and 3-joint within a layout with coordinates (xi, Yi)· Let
(xo, Yo) denote the top left coordinate of any layout i. e. xo $ Xi, and Yo $ Yi, 'Vi. Then we can perform a linear transformation of the layout as shown in Equation 5.4, where a is a scale parameter.
[xi'w, yf"w] = [(x; - xo), (y; - Yo)]
[ � � ]
+ [xo, Yo] , a > 0 (5.4)The value of a corresponds to the factor by which we stretch the layout. The determination of the scale parameter is the key to the success of this improvement. Firstly note that the enclosed rectangle, bounding rectangle and number of corners regularity values are unchanged by this transformation. We can optimize the value of a with respect to one of the other regularity measures only. It is difficult to pre dict the behaviour of the other enclosing and bounding polygon measures, but the perimeter ratio follows a smooth curve, leading up to a point where the perimeter is minimised for a particular shape. Therefore we have chosen the perimeter ratio with which to optimize a .
1 34 (a) 0.1 0.6 0.4 0.2 fo .. (c)
Chapter 5. Improvement Procedures
(b) o.a 0.6 0.4 0.2 101 0 10 .. 10_, 10' (d)
Figure 5.5: An 1-shaped facility (a) and graphs of the regularity measures as
a varies; (b) perimeter ratio; (c) bounding polygon ratios; (d) enclosed polygon
ratios
the perimeter ratio is a smooth robust curve; the golden rectangle polygon ratios
bound the square polygon ratio. Note that the enclosed polygon measures have two local optima corresponding to the two values of a where each of the parts of the L become squares, or golden rectangles, respectively.
Note that we do not consider maximising the average perimeter ratio, as this may lead to very elongated layouts, which are impractical. For example, consider the layout i n Figure 5.6, where there are n + 1 facilities all of area 1 , and the top facility has degree n.
Figure 5.6: An example where maximising the average perimeter ratio will result in an impractical layout
If we maximize the average perimeter ratio, the optimal value of a, will be
where all facilities except the top facility will be squares, having perimeter ratios
of 1 . This means that the average perimeter ratio for the layout of Figure 5.6 is
given by E quation 5.5.
1 n+l 4yfai 1 4
n + 1 t=l
?::
-p· 1 = n + 1 (n + 2n +.l.. )
2n(5.5)
Now as n -+ oo, in Equation 5.5 the average perimeter ratio -+ 1, yet the perimeter ratio of the top facility -+ 0, leading to an impractical layout where the top facility cannot be used. Therefore when using the linear transformation, we seek to maximise the minimum perimeter ratio. This ensures that the layouts are attempting to maximize usability. This example also highlights the necessity for considering minimum and maximum statistics, in tandem with average statistics, in general.
Consider the example we saw in Figure 5.3. When we perform the linear trans formation, the changes in the regularity values given in Table 5.2.
1 36 Chapter 5. Improvement Procedures
Measure Average Values % Increase
Before LT After LT
Enclosed Golden Rectangle 0.3631 0.3837 5.7%
Enclosed Square 0.2244 0.2371 5.7%
Bounding Golden Rectangle 0.3755 0.4098 9. 1 %
Bounding Square 0.2321 0.2537 9.3%
Perimeter 0.6629 0.6689 1 .0%
Measure Minimum Values % Increase
Before LT After LT
Enclosed Golden Rectangle 0.1043 0.1 185 13.6%
Enclosed Square 0.0645 0.0733 13.6%
Bounding Golden Rectangle 0. 1553 0. 1 753 12.9%
Bounding Square 0.0960 0.1091 13.7%
Perimeter 0.4013 0.4079 1 .6%
Table 5.2: Changes in Selected Regularity Values for the Layout in Figure 5.3
(
b)
after application of the LTC o da:
In this chapter, we have introduced two methods for improving the regularity val ues of a layout. We have been able to implement these improvement procedures locally, and hence retain the duality with the MPG and area specifications. These improvement procedures can prove very effective, as we shall see in Section 6.4. The next chapter, in fact, more fully examines the methods described in Chapter 4, via an extensive computational experiment which considers a variety of different attributes of the layouts.