5.12 Design Methodologies
5.12.6 Interactive Optimization–Based Design
In the early 1980s it became clear that trying to use mathematical programming for solving large realistic cases was out of reach in location and layout design involving interaction between facilities. Researchers started to look for sub-problems which could be solved optimally or near-optimally using heuristics. A design methodology emerged from this trend, termed interactive optimization-based design (Montreuil 1982). The concept is to let the engineer in the driver seat like in interactive design, while giving him access to a variety of focused optimizers supporting the various design tasks.
The earliest such methodologies used optimization to generate more advanced design skeletons than simple flow graphs and relationship graphs, from which the engineer had to interactively generate a design. The three best-known layout design skeleton-based methodologies, respectively, rely on the maximum-weighted planar adjacency graph (Foulds et al. 1985, Leung 1992), the maximum-weighted matching adjacency graph (Montreuil et al. 1987), and the cut tree (Montreuil and Ratliff 1988b).
The adjacency graph methods exploit three properties of any 2D layout. The adjacency graph property is that for any layout, one can draw an adjacency graph where each node is an entity in the layout (center, aisle segment, the outside, etc.) and each link corresponds to a pair of entities being adjacent to each other.
The planar adjacency graph property states that the adjacency graph of a 2D layout is planar, meaning that it can be drawn without link crossings. Figure 5.18 illustrates these first two properties for the block layout of Figure 5.2e.
The matching adjacency graph property states that when assigning a value to each link equal to the boundary length shared by both entities defining the link, then the sum of all link values associated with a given entity is equal to the perimeter of that entity, defining the degree of the node representing the entity. For example, as shown in Figure 5.19, center A is adjacent to centers B and C and to the outside.
The adjacent boundaries between A and these three entities are respectively 11.9, 10.3, and 8.8 m long. The sum of these adjacencies is 31, which is the perimeter of center A.
Every layout has a planar adjacency graph. If one could find the adjacency graph of the optimal layout, then the engineer could generate the optimal layout itself. For example, given the building and center space requirements, one can use the adjacency graph of Figure 5.18 as a design skeleton from which can
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FIGuRE 5�18 Illustrating the adjacency graph property and planar adjacency graph property using the block layout of Figure 5.2e.
be drawn the layout of Figure 5.2e with much ease. Given a weight for each potential link, the weighted maximum planar graph problem (Osman et al. 2003) aims to find the planar graph whose sum of link weights is maximal. In layout design, the weight for each link corresponds either to the flow between the centers or their qualitative proximity relationship importance expressed through the weight of their desired proximity type (e.g., adjacent: 100, very near: 50, not far: 2, very far: −50). A heuristic can be used to generate rapidly a near-optimal maximum-weighted planar graph. The engineer interactively draws the planar graph. Then he generates layouts respecting as much as possible the relative positioning of centers in the drawn graph and the adjacencies suggested by its links. This may be easy or rather difficult since not all planar graphs can be transformed in feasible layouts respecting the spatial requirements of each center and the building.
A similar approach is used when exploiting the matching adjacency graph property. The maximum-weighted b-matching problem (Edmonds 1965) can be solved optimally in polynomial time. This problem finds the graph, respecting the degree of each node while embedding links into the graph and stating a usage for each link respecting its imposed lower and upper usage bounds, which maximizes the sum over all links of the product of their usage and their value. In layout design, each node corresponds to a center;
the value of each link is set as done earlier for the planar graph approach, yet here is divided by the upper usage bound; the degree of each node is bounded by desired lower and upper limits imposed on the center perimeter; a positive lower bound on a link forces the centers to be adjacent to a given extent; and, finally, the upper bound on a link indicates the maximum allowed adjacent boundary length between two cen-ters. For example, the maximum adjacency between a 12 × 20 rectangular center and a 15 × 30 rectangu-lar center is at most 20 m. The b-matching algorithm finds its optimal graph which is used by the engineer as a design skeleton representing the targeted adjacency graph. The engineer interactively generates a satisfying layout by iteratively drawing and adjusting a layout respecting the matching graph as much as possible, or resolving the b-matching model with adjusted link bounds to forbid or enforce specific adjacencies.
Cut trees are another type of design skeleton used in layout design. Cut trees can be computed from a flow graph in polynomial time (Gomory and Hu 1961). Figure 5.20 depicts the cut tree for the inter-center undirected loaded flow graph extracted from Table 5.6. Montreuil and Ratliff (1988b) prove that (i) the cut tree is the optimal inter-center travel network when the network links are all set to a unitary-length link and the travel network is restricted to have a noncyclic tree structure and (ii) if the centers have to be placed in two distinct facilities with a specific pair of centers forced to be separated from each other, then the cut tree will always indicate optimally which centers should be in each of the two facilities, assuming no restraining space constraints. For example, in Figure 5.20, if centers C and I have to be in distinct
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FIGuRE 5�19 Illustrating the matching adjacency graph property using the block layout of Figure 5.2e.
ties, then one has simply to find the single path between C and I in the cut tree, here C-E-PF-I, and then find the link with lowest value and cut it to find the optimal separation of centers. Here the lowest value link is C-E with a value of 490. Therefore, centers A, C, and D are best located in a facility and the remain-ing centers in the other facility. The 490 value indicates how much flow is to circulate between the facili-ties. In layout design, one seeks to decide what to put near each other and what to put far from each other.
As a design skeleton, the cut tree can guide an engineer into generating a layout. The cut tree can be molded at will to fit specific building constraints. The main rules are to systematically aim to locate centers so that higher value links and paths in the cut tree are as small as possible, and to avoid unnecessary link crossings. The cut tree can also be used for layout analysis, as shown in Figure 5.21, where the cut tree is overlaid on the current and alternative layouts of Figures 5.6 and 5.8, respectively. It is easy to see that the current design respects poorly the guidance of the cut tree, while the alternative layout, which has a significantly better travel score, does better even though it does not do it as best as could be.
Using design skeletons was the first stage of interactive optimization–based design. Montreuil et al. (1993b) later introduced a linear programming model for swiftly finding the optimal block layout, with located I/O stations, minimizing rectilinear flow travel given a set of flows and the relative posi-tions of centers as inferred by the drawing of a design skeleton. This allows the engineer to manipulate the design skeleton, then to request a layout optimization based on the drawing of the design skeleton, and to examine a few seconds later the resulting layouts, iterating until he is satisfied with the design.
Also developed were models and approaches for designing the travel network given a block layout with located I/O stations (e.g., Chhajed et al. 1992). Complementarily, a linear programming based model was introduced for optimizing the design of a net layout given a block layout and travel net-work (Montreuil and Venkatadri 1988, Montreuil et al. 1993a). The model optimally shrinks cell sizes from gross to net shapes, locates aisle segments appropriately and locates I/O stations so as to minimize aisle-based travel.
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FIGuRE 5�20 Inter-center cut tree based on the loaded flows of Table 5.6.
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FIGuRE 5�21 Cut tree over imposed on the current and alternative layouts of Figures 5.6 and 5.8.
The combination of these optimization models, used interactively by the engineer, allows him to gener-ate designs that, even though they may not be globally optimal, benefit from optimized components and from the human capability to integrate them creatively. The main advantage of interactive optimization is that it enables the engineer while leaving him in the driver seat. The focused and integrated usage of opti-mization lets him address large cases efficiently. The main disadvantage lies in the current lack of wide and open accessibility of design software capable of sustaining such rich interactive optimization.