On Optimization in Computer Aided Architectural Design

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Building and Environment, Vol. 15, pp. 73-80 © Pergamon Press Ltd. 1980. Printed iff Great Britain

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On Optimization in Computer Aided Architectural Design ANTONY D. RADFORD* JOHN S. G E R O t This paper examines the role of computers in the provision of information for architectural design decision making and compares the potential contributions of simulation, generation and optimization techniques. It argues that optimization models are particularly well suited to the provision of design information because they produce results which are prescriptive, express design options and address the problems of the stability and sensitivity of solutions to changes over time. The difficulties posed by the multiple objectives which characterize architectural design problems are discussed and some solution approaches are described. The paper concludes that optimization concepts offer a powerful approach to design decision making and warrant much more research activity in the development of techniques and models for application in architecture.

allow new methods to become practicable that were previously impossible because of the quantity of computation involved. The new generation of mathematical models for the calculation of heating and cooling loads and energy use are prime examples [1, 2]. These models follow reality much more closely than was possible in manual methods and allow the potential for better designs more closely related to performance requirements. Moreover, with the increasing importance of energy and cost-effectiveness clients will come to expect this kind of analysis as part of the architect's design service. But as mathematical models these techniques could (theoretically) be operated manually by one man at a desk with a calculator and plenty of time. The third contribution of computers is in wholly new approaches to design. The concept of databases, of storing building descriptions mathematically and interactively modifying and operating on that description, depends wholly on computers. Databases lie behind the field of computer graphics which is perhaps the most visible impact of computers in practice.

INTRODUCTION THE USE OF computers as a design aid is being taken seriously by the architectural profession. From being the concern of a few research institutions and a small number of pioneering practices, computer methods have developed to the point where general practices in the U.K., U.S. and elsewhere are examining the choice of machines and systems and introducing them into practice. There is a growing belief that computers are going to have a major impact on design in the future and that we are on the threshold of computers in practices (or at least the larger practices) becoming the norm instead of the exotic exception. Impact o f computers

This increasing acceptance is due to sound economic reasoning. It is a result of practitioners' forecasts of what the profession will be like and clients will expect in the architecture of the future. Computers are being used because they allow a practice to offer a better service more efficiently. There are three quite distinct ways in which they can do this. Firstly, computers can do some of the work presently done manually in a faster and more accurate manner. Examples are the simple calculations of environmental performance, the organization of specifications and feasibility calculations. We can call this aspect of their role the mimicking of manual methods. Secondly, computers

Expectations

What can an architect expect from this new tool? It is easier to state what he cannot expect: the computer will not design for him, will never take over his design responsibilities. What it will offer is assistance, by providing information on which to make decisions, by taking over the drudgery of producing working drawings, by making it possible to alter designs through the modification of a single common database in place of rewriting a specification and altering a host of drawings at different scales and for different purposes. If the buildings produced by computer aided design look woeful (and some of the published examples do) it is firmly and absolutely the responsibility of the architect, not of the computer. For until such time as an

*Department of Architecture and Building Science University of Strathclyde Glasgow, U.K. Previously in the Department of Architectural Science, University of Sydney, Sydney, Australia. Now with Faulkner-Brown, Hendy, Watkinson, Stonor, at Dobson House, Northumbrian Way, Killingworth, Newcastle upon Tyne, U.K. tDepartment of Architecture and Building Science, University of Strathclyde, Glasgow, U.K. On leave from the Department of Architectural Science, University of Sydney, Sydney, Australia. 73

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architect finds a computer terminal as sympathetic a medium for the development of aesthetically pleasing sketch designs as the traditional pencil and drawing board, much of his design activity should remain centred on the drawing board. The emphasis in computer aided architectural design should always be on the architectural design and not on the computer. No one, after all, talks of pencil-aided architectural design: the pencil is simply a tool, a medium, a means towards the goal of good design. The computer will be no more (and possibly no less) important in the future. This paper, though, is not intended to be a treatise on computers. Its context is just one of the roles we have outlined above: the role of the computer as a provider of information for decision making. Its purpose is to look at information needs, the ways in which it can be obtained, and to advocate optimization as an insufficiently exploited and potentially powerful approach to the production of that information.

INFORMATION FOR DECISION MAKING

Information sources Design in architecture is a goal-directed activity in which decisions are taken about the physical form of buildings and their components in order to ensure their fitness for intended purposes. In order to take those decisions the architect needs information on the relationship between his goals and the means at his disposal for achieving them. At the early stages of design most information comes from previous experience in practice or education and from manuals and reference books which suggest a qualitative approach to the problem at hand. As design progresses this general guidance must be supplemented by specific information about the particular problem and proposals. The only practicable means of obtaining this information is through manipulatable models of the situation under study. A model is a representation of reality, where representation means the expression of chosen relevant characteristics of the observed reality [3]. Hence the architect sketches, erases and redraws in the manipulation of a two-dimensional model of the form of a building or component. Drawings model form, appearance, materials and through them the aesthetic qualities of space and movement through space. Mathematical models are concerned with those aspects drawings are poor at modelling: the performance of materials in terms of heat or light transmittance, the performance of spaces in terms of the movement and queuing of groups of people, the monetary and energy costs of a building proposal. Information from both sources is necessary for good design. If in the past the visual aspects of architecture have predominated and the performance aspects taken second place this is partly because the visual, physical, models have been far easier to set up and manipulate than the available mathematical models. Computers offer the potential for these performance qualities to take their rightful place alongside (but in no way replacing) the visual qualities of architecture.

Information needs What kind of information, then, would the architect like to have for decision making? In essence he wants to know what is the 'best' thing to do, what the 'best' solution is in design terms for his particular set of goals. Difficult enough to provide, but because in architectural design little is static or permanent he wants even more than this. It may not be possible to completely state his goals; other factors may come into play; some of his assumptions (such as the cost of energy) may change in the course of the life of the building. So he wants to know what the options are, by how much 'best' is 'best', what is second-best, what happens if his assumptions of relative cost or other matters should change. Yet no-one wants to be confronted by a mass of detail, by tables of printed data. The kind of information sought should be concise, relevant and easily accessible. As in most parts of life there is a gap between what is wanted and what can be achieved. We can, though, make some general statements which can be used as a guide for information provision. We want to know what is the 'best' thing to do: this means we should prefer prescriptive information, indicating the best solution approach, to evaluative information testing only our preconceived ideas. We want to know the design options: this means the information should concern fields and ranges of solutions rather than single solutions. We want to know what happens if our assumptions change: this means we need information on how much a solution's performance will alter given changes in design parameters (known as sensitivity analysis) and how stable is the choice of best solution given changes in those parameters (known as stability analysis). We do not want unnecessary detail: this means the degree of modelling must be appropriate to the scale of the problem. A thermal model useful in deciding the volumetric form of a building is very different in degree of modelling to one useful in making decisions about the size of heating plant. These are the kinds of measures we should use in looking at the form of mathematical models used in computer aided design. DESIGN MODELS With very few exceptions all computer aided design models fall into one of three categories: simulation, generation and optimization. In simulation the computer is used to predict the consequences of a set of design decisions by manipulating a mathematical model which describes the design. All decision making is external to the model. In generation the computer is used to explore the consequences of the recursive application of an ordered set of decision rules. Thus some decision making is internal to the model but is not purposeful: all decisions which conform to the rules are equally acceptable. In optimization the computer is used to prescribe a set of decisions in order to achieve a specified goal as closely as possible. Some decision making is internal to the model and is purposeful: decisions are chosen according to their ranking on an explicit measure of effectiveness.

On Optimization in Computer Aided Architectural Design Simulation Simulation-based evaluative models have been and remain very powerful tools. The great strength of simulation is that by fixing all the design variables one can examine their consequences in great detail, and on as many different aspects of the problem as there are prediction methods available. Examples in architectural design abound: thermal simulations [1, 2], acoustic simulations [4], daylight simulations [5], the movement of people and baggage [6] and structural simulations [7] to name a few. The disadvantage is that simulation requires the user to operate by trial and error. To get any quantitative information he must first have a solution and the design process therefore involves a cyclical procedure of postulationevaluation-modification where different possibilities are examined. Moreover, design options and sensitivity to changing assumptions can only be investigated by repeating the simulation many times with different sets of decisions.

Generation Generative models have received much less attention than simulation in the history of computer aids to design. If suitable decision rules can be formulated, generative models will produce an unranked catalogue of conforming solutions from which the designer can choose. Two examples will serve to illustrate the concept. The first is an intriguing investigation of style in architecture, the study of the rules and priorities that make a chosen architect's work distinctive. By analysing and then applying the rules used by Palladio to design the villas presented in his I Quattro libri dell'Architettura [8] it proved possible to generate a catalogue of the floor plans that Palladio might have used [9, 10]. The second example is a method for the design of fixed external sun shades [11]. The rule is to project at each intersection on a notional grid superimposed over a given window opening a perpendicular of length sufficient to cast a shadow to the edge of the window at all times during specified days and hours. The ends of these perpendiculars mark the edge of a shade necessary to prevent the penetration of direct solar radiation throughout the given period. The result is a graphical solution which expresses the full range of feasible solutions to the given problem, and makes possible the design of many variations of efficient nonstandard shades. By their nature, generative models provide a field solution which demonstrates the design options but the lack of purpose behind the decisions means that the set of solutions generated may be large and partly superfluous. The sun shade example is an exception because the rule itself ensures fitness for purpose in the solution.

Optimization Optimization models search the whole field of feasible solutions to identify those best suited to stated goals. Thus optimization directly approaches an answer to the designer's fundamental question of what is

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the 'best' solution. We write 'approaches an answer' rather than 'answers' because in most design problems there are many disparate and conflicting goals and the definition of a best solution depends on the designer's judgement about the relative importance of different objectives, a subject to which we shall return later in this paper. There are now a number of examples in architectural design, notably in the field of layout planning which is usually treated in this way [12] but extending to environmental design [13, 14], site development [15], building services [16] and other aspects of architecture. Design options can be presented by identifying near-optimal as well as optimal solutions and formal methods are available for examining the sensitivity and stability of solutions. The disadvantage of optimization is the difficulty of formulating meaningful quantifiable objectives in a discipline characterized by multiple and ill-defined objectives.

Comparison of approaches Simulation, generation and optimization therefore produce very different kinds of information. Simulation can produce a great deal of information about many aspects of building performance but for only one predefined solution at a time. It tells the designer nothing about that solution compared with other feasible solutions unless he repeats the analysis with different designs in a process of informal optimization. Generation produces a subset of unranked feasible solutions by following a predefined sequence of rules. It tells the designer nothing about the performance or relative merit of solutions in the generated set other than satisfaction of performance objectives ensured by the rules themselves. Optimization encompasses the whole field of feasible solutions and produces an ordered subset of those solutions which best satisfy a subset of specified performance objectives. It tells the designer nothing about performance in objectives outside this subset unless he further investigates the solutions produced in a process of simulation. Which approach is appropriate depends on the kind of information being sought. We have argued above that the designer's principal need is for information that is prescriptive, that expresses the design options and that addresses the problems of the sensitivity and stability of solutions given changes in the assumptions on which they are selected. Whilst it is not a panacea, optimization satisfies this need far more closely than either simulation or generation. Although we have stressed the differences between them, simulation, generation and optimization can be made to emulate and complement each other through the manner of their application. Thus simulation models offer the potential for providing a field solution through repetition, including exhaustive enumeration as an extreme, and the identification of a 'best' solution by inspection of a set of simulated performances. Conversely, optimization will emulate simulation if the feasible solution space is constrained to a single solution. Optimization usually subsumes generation and simulation: within an optimization model generation is necessary to create the solution space which is sear-

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ched and simulation is necessary to predict the performance which is to be optimized. The only exception is an analytical optimization approach which utilizes a non-numerical solution methodology. Analytical optimization is founded on the notions of differential calculus and does not need generation or simulation to operate on. The solution is produced in terms of symbols and is therefore a field solution for all pertinent values of the variables. Optimization models are inherently more complex than simulation or generation at the same degree of modelling. OPTIMIZATION

Problem characteristics The requirements for an optimization technique for use in this context relate to the characteristics of the information required, the types of problems and the objectives encountered. Historically, optimization has been treated as a mathematical problem in which the interest lay only in the extreme value of the objective and some means of obtaining that value. What these means were, the actual set of decisions required, were not particularly significant. Neither was any suboptimal solution. In design both the decisions and suboptimal solutions are very important. The decisions represent the physical form of a building or other proposal, the construction methods used and the placement of insulation, windows and other design elements. It is with the arrangement of these physical elements that the architect has traditionally been most concerned. The suboptimal solutions represent the design options, telling the designer about solution forms that are close to the optimal in terms of the stated objective but which may have more acceptable sets of decisions for their realization. Any technique adopted must therefore be able to produce field or ranges of solutions which can be examined both in terms of their relative merit (the values of the objective function) and in terms of the sets of decisions they represent. But this is only a part of the information required. We also require good information on the sensitivity of performance and stability of solution choice given possible changes in design parameters. In optimization such information is usually obtained by methods of postoptimality analysis after optimal and suboptimal solutions have been identified. Given this rather demanding specification for the information required it is unfortunate that many of the problems found in architecture have characteristics which are particularly unhelpful for easy application of optimization techniques. Architectural problems typically involve variables which are discrete and discontinuous and relationships which are nonlinear. Discrete variables are those which can only take specific values, for example pipe and brick sizes (to correspond with available materials) or numbers of rooms and floors (which must be integers to be meaningful). Discontinuous variables are those which exhibit discontinuity in their range of values, for example the required rate of provision of water closets in a building may change at threshold values imposed by building

regulations. Nonlinear relationships are those that do not vary uniformly with the variables on which they depend, for example area and the whole range of performance measures such as heat loss, daylighting and cost which are functions of area. Moreover, some of the relationships may be better represented as stochastic rather than deterministic, requiring probability distributions for their modelling. Problems involving the movement and queuing of people are frequently of this type since mankind rarely behaves in a deterministic manner.

Solution techniques If we examine available optimization techniques we find that although the choice is wide few will satisfy all these requirements. Classical calculus [17] provides an analytical solution very quickly to a range of design problems which can be formulated as continuous and differentiable equations and are subject to few constraints. In architecture it has been used to find a geometry for multi-storey built forms which will minimize conduction heat losses [18, 19], to find the thickness of insulation required to minimize capital plus amortized running costs [20], and for a small number of other relatively simple problems. Linear programming [21] is an efficient and well-developed numerical method which has been used to determine distributions of apartment types and land use in order to minimize development costs [15], to decide on the distribution of services within a building [22] and for other problems which can be modelled by linear relationships between the variables. For nonlinear relationships an algorithm by Graves and Whinston [23] has been used to dimension a given floor plan topology in order to minimize cost [24] and geometric programming [25] has been used to find the combination of boiler capacity, preheat time and thermal insulation which will minimize capital plus amortized running costs [13]. Dynamic programming [26] has been applied to a number of architectural problems including drainage design [27], the design of an elevator system [28] and the design of a floor/ceiling sandwich for multi-storey buildings [29]. Dynamic programming has properties which make it particularly appropriate for the kinds of problems with which design is concerned since it will handle discrete, discontinuous or stochastic variables and nonlinear relationships and is efficient with constraints. It does, however, lack a standard methodology and requires more thought about the structure of each separate problem than is sometimes necessary with other methods. OBJECTIVES Our examples so far have concerned design problems which can be expressed as having a single objective measurable by a quantitative objective function. In single objective problems the aim is simply to identify the solution that gives the optimal value of this objective function: the least cost, the least energy use, the least floor area are typical examples. Since all solutions are measured on a common scale there is no

On Optimization in Computer Aided Architectural Design difficulty in establishing which is best. But buildings and their component systems are rarely designed with a single aim in mind. Usually design problems have a number of quite distinct and disparate objectives which are important to a greater or lesser degree. Low capital cost and low energy use may both be important in a design situation, but the solution which minimizes capital cost is unlikely to also minimize energy use. Indeed, the chosen solution may well be neither the minimum cost nor the minimum energy solution but some compromise between the two which offers the best balance of performance in both the objectives. In this section we want to introduce the notions of multi-attribute and multi-criterion problems. In this paper we shall use 'criterion' to denote a noncommensurable measure of performance, so that the choice of best solution might depend on performance in a single criterion (for example cost) or in several different criteria (for example cost, energy use and expected life). We shall use 'attribute' to denote a commensurable component of a criterion, for example the cost of separate elements of a building where the objective is to minimize total cost. This allows us to classify objectives as single criterion, multi-attribute single criterion, multi-criterion or multi-attribute multicriterion. We can examine the meaning of this classification by introducing the notions of a decision space and a criteria space and resorting to some mathematical notation. If solutions can be described by the values of N different decision variables the set of all possible solutions can be graphically represented in an N dimensional decision space where each axis spans the feasible values of one decision variable. Similarly, if the performance of solutions can be described by the values of P different criteria the set of all possible performances can be represented in a P dimensional criteria space. Each feasible point in the decision space will have associated with it a corresponding point in the criteria space, and vice-versa. The design problem can be described quite simply as finding the best feasible point in the criteria space and the corresponding point in the decision space. For clarity of presentation in a two dimensional medium we shall restrict ourselves in the following discussion to a problem with only two decision variables, dl and d 2. To give them meaning we shall assume they represent wall insulation thickness and roof insulation thickness for a building, their type and all other variables being considered fixed.

Single criterion objectives We shall begin with the single criterion objective of minimum total amortized cost over the expected life of the building. In a single criterion problem the criterion space reduces to a one dimensional vector and each point in the decision space has associated with it a corresponding point in this vector (Fig. 1). We shall adopt the convention that increasing distance from the origin of the criterion vector or criteria space corresponds with increasing desirability, that is decreasing

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cost in our illustrative example. The whole set of feasible performances will occupy a segment of this vector and a solution is clearly dominated (bettered by) any solution with a better performance in the criterion. This is historically the most common form of optimization and we have cited other examples from architectural design earlier in this paper. d~

Decision space

~e

d2 i_

"I Criterion vector

Fig. 1. Decision space and single criterion vector.

Multi-attribute objectives We shall now examine the separate contributing effects of the cost of providing the insulation (the capital cost) and the cost of fuel (the running cost). These can be regarded as two attributes of solutions which contribute additively to the value of the objective. Thus each point in the decision space will have associated with it a point in a two dimensional attribute space as well as a point in the criterion vector (Fig. 2). We can then examine the stability of the optimal solution given changes in either of these attributes by applying methods of postoptimality analysis to the optimization results [30, 31]. In architectural design such studies of multi-attribute problems have been carried out for the constrained dimensioning of a small apartment floor plan, where the attributes were the areas of each room and the objective was to minimize total cost expressed as a function of the floor areas of different types of rooms [321 and for the design of a floor-ceiling sandwich for a multi-storey building, where the attributes were the costs of each component system and the additional costs of combining types of system and the objective was also to minimize total cost [31]. ~C

Decision space

Attribute space

Criterion vector

Fig. 2. Decision space, two-dimensional attribute space and single criterion vector.

Multi-criterion objectives Let us now assume that we have no knowledge of the future cost of fuel and instead of measuring the operating performance of the design in terms of amor-

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tized fuel cost we wish to measure it directly as annual energy consumption. We now have two noncommensurable criteria: initial cost in monetary units and energy consumption in megajoules or other energy units. For a two criteria problem the criteria space is twqt~xl~imensional and the whole set of feasible performances will occupy an area of this space known as the feasible region. Consider two solutions A and B (Fig. 3). The criteria space gives no clear indication of which solution is to be prefegred: A is better than B in criterion 1 and B is better than A in criterion 2. Consider, though, the third solution C in Fig. 3. Solution A offers more of both criteria 1 and 2 than does C and we can therefore state that C is dominated by A. If we identify the set of all solutions that are not dominated by any other solutions in both criteria we find that their performances lie along the boundary of the feasible region (Fig. 4). They are known formally as the set of non-dominated, non-inferior or Pareto optimal solutions. We shall adopt the term Pareto optimal to distinguish this specific meaning of dominance from other uses of the word. dI

SOLUTION

CI

C2

Decision space

domination, in the same way as an optimal solution in single criterion optimization is identified by a mechanistic comparison of objective values. But whereas single criterion optimization provides the decision maker with a clearly 'best' solution, Pareto optimization with multiple criteria provides a field or subset of solutions which the decision maker should investigate. Any of these Pareto solutions may be the preferred choice. Which is 'best' depends on the decision maker's perception of the relative importance of performance in the different criteria. The choice is essentially a matter of tradeoffs. Given two criteria C 1 and C> the decision maker has to decide how much worsening of C1 he is prepared to accept in return for an improvement in C 2. In terms of our illustrative example, the architect must decide bow much increase in capital cost he is prepared to accept in return for lower energy consumption. The problem, of course, is that the increase in capital cost one is prepared to accept in order to improve energy consumption depends on what the actual figures are.

Criteria space

Fig. 3. Decision space and criterion space for two criteria.

Pareto optimal values

C2

Criteria space

Fig. 4. The Pareto set in two-dimensional criteria spaces.

Extending this concept to the general case of P criteria, a feasible solution to a multi-criterion optimization problem is Pareto optimal if there exists no other feasible solution that will yield an improvement in performance in one criterion without causing a decrease in performance in at least one other criterion. For two criteria the Pareto set traces a curve along the edge of the criteria space. For three criteria the set traces a surface in three-dimensional space for which each point satisfies the definition above. Higher dimensions are difficult to visualize but the set of Pareto optimal performances for a problem with P criteria will form a surface in P-dimensional space. The Pareto set, then, is the closest equivalent in multi-criterion optimization to the concept of an optimal solution in single criterion optimization. It is defined by t h e mechanistic application of tests of

CHOICE WITH CRITERIA

MULTIPLE

How, then, can a best compromise solution be decided? The first approach is simply to consider a graphical representation of the Pareto optimal set or an approximation to it and use the tradeoff information it portrays to arrive at a personal choice of solution. We shall call this a non-preference method because it requires no explicit definition of the relative importance of criteria by the decision maker. This is a good approach; it uses the concepts of optimization to identify the performance options open to the designer and leaves him to make the decisions based on the information provided and any other information he has available. It may, though, mean the generation of rather a lot of alternatives and requires some ingenuity in the presentation of Pareto sets for more than three criteria [33, 34]. The second approach is to make some prior assumption of the relative importance of criteria and use this information in the generation of alternatives to identify a single choice or at most a small subset of the Pareto set. We shall call methods based on this approach preference methods. They provide a much smaller number of alternatives by attempting to determine in advance the needs of the decision maker.

Non-preference (Pareto ) methods We can compare the implications of using each of these two approaches in terms of the amount of computation they require to operate, the information which is necessary for them to work and the information they provide. We have to accept that Pareto optimization can be computationally expensive, especially with large or complex problems and where there are large numbers of criteria (the numbers of solutions in the Pareto set increases exponentially with the number of criteria). On the other hand the only

On Optimization in Computer Aided Architectural Design information required is that necessary to operate the mathematical models behind the optimization, the same information as is required for simulation. In return for the computational burden a great deal of information is provided for the decision maker to explore the implications of his decisions. They also provide a possible framework for the examination of non-quantifiable subjective criteria in relation to quantative criteria, one of the major problems of choice amongst alternatives with multiple objectives. The designer can define a class or characteristic of solutions which qualify them to be excluded from the domination procedure and therefore presented alongside the Pareto set. What should then become apparent is how much departure from the performance combinations that could be achieved in the quantifiable criteria is necessary to achieve the designer's notion of good performance in the non-quantifiable or subjective criterion. In architecture non-preference (Pareto) methods have been used in the development of environmental tradeoff diagrams which relate the performance achievable in different components of the internal physical environment within a room both to the building design and to each other [35]. By exploring the relationships between criteria such as predicted daylight factor, sound intrusion and peak summer and minimum winter internal temperatures the designer can make design decisions which are directed towards the balance of environmental performance he seeks. It is also possible to trace the whole range of feasible combinations of values for significant pairs or groups of the criteria and to examine the effects on these feasible combinations of allowing different forms of design and materials.

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nation level, noise level and insolation [36]. Because of the assumptions that are necessary, extensive sensitivity and stability analysis should be carried out whenever preference methods are adopted.

Comparison of approaches Non-preference methods place the major burden on computation and allow the designer to make decisions in full knowledge of their tradeoff implications. Preference methods reduce computation by placing additional burdens on the decision maker in requiring him to articulate his preferences in advance, without knowledge of their tradeoff implications. We believe that non-preference (Pareto) methods are of practical use in design despite the computational disadvantages. Pareto optimization produces a field of nondominated solutions which are all potentially 'best', depending on the designer's perception of the relative importance of the objectives. He can operate on this field to extract subsets of Pareto solutions with specific design characteristics or having performances in nominated criteria within specified ranges. Where there is some basis for integrating aspects of different criteria under a common denominator this can be expressed in the formulation of an additional hyperobjective. For example, given two criteria of daylighting and internal temperature one could, given the necessary information, superimpose contours of equal cost implications or equal energy implications on a graphical representation of the Pareto set. If the Pareto set, or an approximation to it, has been established then any choice, operation or the formulation of any hyperobjective is carried out with knowledge of the actual performance values and relationships.

Preference methods Preference methods involve much less computation because they generate only one or a small subset of the Pareto set. However, this saving is offset by the additional information required, the necessity of specifying in advance the tradeoff information between objectives. Obtaining this information can be time consuming and expensive. Moreover, the information provided is restricted to a point solution (the 'best compromise') or a small number of 'equally good' solutions from which the decision maker can choose. Preference methods do not provide the context information on the whole range of feasible solutions which is provided by the Pareto set. There are several solution techniques that incorporate preferences and make different assumptions about dominance and the means by which a best compromise solution should be identified [34]. The most common approach is to reduce the multicriterion problem to a multi-attribute single criterion problem by interpreting the non-commensurable criteria in terms of a common denominator, either an abstract measure of utility or a real measure such as cost or energy. In architecture cost-benefit analysis has been used to establish an optimal window design for a room taking criteria of internal temperature, illumi-

CONCLUSION In this paper we have examined the use of optimization concepts in the computer's role as a provider of information for design decision making. We have argued that the designer needs information that is prescriptive, expresses his design options, tells him about the sensitivity and stability of solutions given changes in the parameters on which they are selected, and is presented at a level of detail appropriate to the stage in the design process. We have outlined three categories of mathematical models for computer aided design, simulation, generation and optimization, and argued that optimization models are best suited to the provision of this kind of information. We have described the requirements of an optimization technique for use in this context and enumerated some possible approaches. Finally we have examined the relationship between single criterion problems, multiple attribute problems and multiple criteria problems and discussed some approaches to the typical architectural problem with multiple criteria. We have argued that a nonpreference approach based on Pareto optimality which allows the designer to make tradeoff decisions can provide practical and useful information for design.

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Antony D. RadJbrd and John S. Gero

We stated early in this paper t h a t optimization is a potentially powerful but insufficiently exploited technique for c o m p u t e r aided architectural design. Most of the examples of its application that we have quoted have been research studies, part of the development of ideas a n d methodologies. There remains a great need for reliable a n d widely applicable optimization models for architectural practice which incorporate the con-

cepts we have put forward. P e r h a p s the most important conclusion we can m a k e is t h a t further research interest a n d activity in this area is likely to prove particularly fruitful.

Acknowledgements This work is directly supported by the Australian Research Grants Committee, the Science Research Council and the Association of Commonwealth Universities through a Commonwealth Visiting Professorship.

REFERENCES 1. J.A. Clarke, Energy implications in building design: a thermal simulation design model, Proc. 3rd Int. Symp. on The Use of Computers for Environmental Engineering Related to Buildings, National Research Council of Canada, Ottowa, pp. 3---20 (1979). 2. T. Kusada, NBSLD, Computer Program for Heating and Cooling loads in Buildings, Center for Building Technology, Institute for Applied Technology, National Bureau of Standards, Washington, D.C. (1974). 3. M. Echenique, Models: A Discussion, Working Paper No. 6, Center for Land Use and Built Form Studies, University of Cambridge, Cambridge (1968). 4. M.R. Schroeder, Computer models for concert hall acoustics, Amer. J. Phys. 41,461 471 (1973). 5. S. Bensasson and K. Burgess, Computer Programs for Daylighting in Buildings, Evaluation Report No. 3, Design Office Consortium, Cambridge (1978). 6. L. W. W. Laing, A computer simulation model for the design of airport terminal buildings, Computer Aided Design 7, 37-42 (1975). 7. S.J. Fenves (ed.) STRESS: A Reference Manual, MIT Press, Cambridge, Mass. (1965). 8. A. Palladio, The Four Books of Architecture, reprinted from the 1738 translation by Isaac Ware of I Quattro Libri dell'Architettura, Dover, New York (1965). 9. G. Stiny and W. J. Mitchell, The Palladian grammar. Envir. Planning B 5, 5 18 (1978). 10. G. Stiny and W. J. Mitchell, Counting Palladian plans. Envir. Planning B 5, 189 198 (1978). 11. E. Shaviv, The determination of the form of windows and sunshades in a hot climate. Proc. CAD78, IPC Science and Technology Press, Guildford, pp. 11~120 (1978). 12. C. M. Eastman (Ed.), Spatial Synthesis in Computer-Aided Building Design, Wiley, New York (1975). 13. A. J. Wilson and A. B. Templeman, An approach to the optimal thermal design of office buildings. Bldg Envir. 11, 39 50 (1976). 14. S, A. Jurovics, Optimization applied to the design of an energy efficient building. IBM J. Res. Dev, 22, 378-385 (1978). 15. R.J. Aguilar and J. R. Hand, A generalized linear model for optimization of architectural planning. Proc. Spring Joint Computer Con[erence, Thompson Books, Washington, D.C., pp. 81 88 (1968). 16. M. Kovarik, Constrained optimization in the design of ducts. Proc. 5th Australian Computer Con]erence, ACS, Sydney, pp. 184- 188 (1972). 17. B.S. Gottfried and J. Weisman, Introduction to Optimization Theory, Prentice Hall, Englewood Cliffs, N.J. (1973). 18. J.K. Page, The optimization of building shape to conserve energy. J. Archit. Res. 3, 2(~28 (1974). 19. A. H. Bridges, The inter-relationship of design and performance variables. Proc. CAD76, IPC Science and Technology Press, Guildford, pp. 22 30 (1976). 20. R.L. Hennessy, Optimum wall insulation for cold climates. Proc. Am. Soc. cir. Eng. 103, 529-535 (1977). 21. D.E. Grawoig, Decision Mathematics. McGraw Hill, New York (1967). 22. E. E. Dudnik, Uncertainty and the design of building subsystems--a linear programming approach. Bldg Envir. 12, 111 116 (1977). 23. G.W. Graves and A. B. Whinston, An algorithm for the quadratic assignment problem. Mgmt Sci. 16, 453 471 (1970). 24. W.J. Mitchell, J. P. Steadman and R. S. Liggett, Synthesis and optimization of small rectangular floor plans. Envir. Planning B, 3, 37 70, (1976). 25. R.J. Duffin, E. L. Peterson and C. M. Zener, Geometric Programmin~Theory and Applications, Wiley, New York (1967). 26. G.L. Nemhauser, Introduction to Dynamic Programming, Wiley, New York (1967). 27. A.J. Wilson, A. L. Britch and A. B. Templeman, The optimal design of drainage systems. Engng Optimization 1, 111 123, (1974). 28. B.A. Powell, Optimal elevator banking under heavy up-traffic Transpn Sci. 5, 109 121 (1971). 29. J. S. Gero, P. J. Sheehan and J. M. Becker, Building design using feedforward nonserial dynamic programming. Engng Optimization 3, 183 192, (1978). 30. J.S. Gero, A. D. Radford and J. Cameron, Postoptimality analysis for multi-attributive objective functions in dynamic programming. Engng Optimization 4, 65 72, (1979). 31. M. A. Rosenman, A. D. Radford and J. S. Gero, Postoptimality analysis in dynamic programming. Engng. Optimization (in press). 32. J.S. Gero, Design by optimization and beyond, Proc. PArC 79, AMK, Berlin, pp. 393--402 (1979). 33. W.S. Meisel, Tradeoff decisions in multiple criteria decision making. Multiple Criteria Decision Making, J..L. Cochrane and M. Zeleny (Eds.) University of South Carolina Press, South Carolina, pp. 461476 (1973). 34. J.L. Cohon, Multiobjective Programming and Planning, Academic Press, New York (1978). 35. A. D. Radford and J. S. Gero, Tradeoff diagrams for the integrated design of the physical environment in buildings. Bldg Envir. 15, 3 15 (1980). 36. T.A. Markus, The real cost of a window. Trans. Bartlett Soc. 8, 35-58 (1969 70).