1 INTRODUCTION
Spatial configuration is concerned with finding feasible locations and dimensions for a set of
interrelated objects that meet all design requirements and maximize design quality in terms
of design preferences. Spatial configuration is relevant to all physical design problems, so it
is an important area of inquiry. Research on automation of spatial configuration includes
component packing [11–13], route path planning [18], process and facilities layout, VLSI
design ½16; 17 , and architectural layout [3–10]. Architectural layout is particularly interesting
because in addition to common engineering objectives such as cost and performance, architectural
design is especially concerned with aesthetic and usability qualities of a layout,
which are generally more difficult to describe formally. Also, the components in a building
layout (rooms or walls) often do not have pre-defined dimensions, so every component of the
layout is resizable.
Reported attempts to automate the process of layout design started over 35 years ago [3].
Researchers have used several problem representations and solution search techniques to
describe and solve the problem.
One approach to spatial allocation is to define the available space as a set of grid
squares and use an algorithm to allocate each square to a particular room or activity [4–7]
(see Fig. 1). This problem is inherently discrete and multi-modal. Because of the combinatorial complexity, it cannot be solved exhaustively for reasonably-sized layout problems.
Several heuristic strategies have been developed to find solutions without searching
the design space exhaustively. Liggett and Mitchell [4] use a constructive placement strategy
followed by an iterative improvement strategy. In this method, space is allocated for rooms
one at a time based on the best probable design move at each step. Other researchers have
used stochastic algorithms for search [5–7].
Another approach to representing the building layout design space is to decompose the problem into two parts: topology and geometry. Topology refers to logical relationships between layout components. Geometry refers to the position and size of each component in the layout. Topological decisions define constraints for the geometric design space. For example, a topological decision that ‘‘room i is adjacent to the north wall of room j’’ restricts the geometric coordinates of room i relative to room j. Researchers have developed decisiontree-based combinatorial representations and used constraint satisfaction programming techniques to enumerate solutions without exhaustive search. Baykan and Fox [8] and Schwarz, Berry, and Saviv [9] developed variations of this model and have been able to enumerate solutions for a studio apartment and for a nine-room building respectively. Medjeoub and Yannou [10] developed a similar model, but they use a technique of first enumerating all topologies that can produce at least one feasible geometry. The designer is then able to review the feasible topological possibilities and select those which s=he wants to explore geometrically. This technique reduces computation dramatically, and they have shown success for up to twenty rooms.
Successful generation of global quality solutions has been achieved for medium-sized
problems; however, there is still a need for a strategy that can handle larger problems
computationally. It would be useful to take advantage of the speed of gradient-based
algorithms on the geometric aspects of the layout, because they involve continuous variables.
This article develops a mathematical model for the geometric decisions in the layout problem
that allows efficient solution with gradient-based and hybrid local–global methods. This
model is then embedded into another model used for topology decisions that is solved with
heuristic global methods. The geometric optimization process allows fast solution of large
complex problems that also enables a true interactive design process described in a sequel
article [2]. The topology optimization component has had limited success due to the
combinatorial nature of the topology decisions. The interactive optimization tool can be
downloaded from http:==ode.engin.umich.edu.
2 OPTIMIZATION OF GEOMETRY
The geometric optimization problem is posed as a process of finding the best location and
size of a group of interrelated rectangular units. A new decision model is formulated
where all objectives and constraints are continuous functions, and all design variables
have continuous domains.
A Unit is defined as a rectangular, orthogonal space allocated for a specific architectural
function. Examples of architectural functions include living spaces, storage spaces, facilities,
and accessibility spaces. For simplicity, this representation assumes that all Units can be
represented as rectangles or combinations of orthogonal rectangles. This simple representation
can model a large array of architectural layouts, and more complex shapes could be added to
the model to expand this array. Figure 2 shows a Unit represented as a point in space ðx; yÞ, and
the perpendicular distance from that point to each of the four walls: fN; S; E; and Wg. This
model has more variables than necessary to describe the shape; however, it allows an optimization
algorithm to change the position of a Unit independently without affecting its size (by
changing x or y), and it can change any of the four wall positions independently (by changing
N; S; E, or W). Although this model increases the problem dimensionality, it offers a lot of
flexibility to make the best design moves at each step of the optimization.
Units are grouped into several categories based on their function: Rooms, Boundaries,
Hallways, and Accessways. Rooms are Units used for sustained living activity as determined
by the designer. The differentiation between living space vs. non-living space is important
only in optimization objectives that maximize the amount of space used for living relative
to all other space. A Boundary is a Unit that has other Units constrained inside of it, and
it is not considered living space. A Hallway is a Unit with no physical walls that is not a
living space. Hallways function as pathways. An Accessway is a Hallway that is constrained
to geometrically intersect two Units. Accessways are generally restricted to be small,
and they are forced to intersect two other Units. They function to keep the two Units adjacent
and connected, and to ensure that there is room for a door or opening between the rooms.
In Figure 3, the external rectangle represents the building Boundary, the living room, bedroom,
and bathroom are Rooms, the hall is a Hallway, and the three Units labeled ‘‘A’’ are
Accessways that define space for a doorway between Units.
Units that are along external walls may also have windows for natural lighting. Window
height can be fixed for each Unit, and window width is a variable. oN ; oS ; oE; oW represent
the width of the north, south, east and west windows, respectively.
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