Architectural Design for Space Layout by Genetic Algorithms

Transcription

1 Architectural Design for Space Layout by Genetic Algorithms Özer Ciftcioglu, Sanja Durmisevic and I. Sevil Sariyildiz Delft University of Technology, Faculty of Architecture Building Technology, 2628 CR Delft, The Netherlands Tel: ; Fax: Abstract-A novel method to produce space layout topology for architectural design is described. A required space-layout for an architectural design is identified by the method of genetic algorithms according to a given norm and metric function. The design solution is based on graph representation of the layout so that the desired relations between the pairs of nodes are, in general, considered to be independent variables of appropriate series of multivariable functions representing the requirements. Referring to this graph structure, the design solution is considered as an optimization problem with multi-objective criteria. Having obtained such design solution by genetic algorithm, the architectural layout partitions based on this structure is carried out afterwards as architectural design exercise searching for optimal decisions among various alternatives fulfilling some subtle preferences. 1. INTRODUCTION One of the important architectural design tasks is related to space layout and the problem is addressed and documented in literature (Steadman 1983; Wong and Liu 1986; Lai and Leinwarnd 1988; Koakutsu et al 199; Sutanthavibul et al. 1991; Koakutsu et al 1992; Damski 1997). Referring to these works, the task can be tackled by different approaches and these extend from dimensionless form of rectangular units on a plan to area, width and length of each space and the optimal dimensions according to some criteria. However in most cases, the criteria used are not satisfactory enough for a design since the formulation of the solution leads to single objective function, generally is referred to as "cost function" while architectural designs usually concern multi-objective considerations to be fulfilled individually. Solution to such tasks is generally is in the form of approximation to some design objectives and consequently such solutions are not often conclusive but indicative. Therefore, the systems, which deliver such indicative solutions, are referred to as design decision support systems. In essence, the problem stems from its ill-conditioned nature. That is, architectural problems are rather soft relative to engineering problems and therefore engineering approach to such soft problems may not be always conclusive. Particularly, in the second half of the last decade, evolutionary algorithms are developed to tackle ill-posed problems. One can encounter such problems in architectural design activities. For architectural design, several researchers used genetic algorithm (GA) approach (Damski 1997, Jagielski 1997). Genetic algorithms have been conventionally applied almost exclusively to singleattribute problems. However, multiple attributes can be treated in a similar way with appropriate conditionings in the algorithm. In this respect, GAs are rather appealing for architectural design tasks as architectural design problems are multi-attribute. In addition to multi-objective assignments where even the objectives might be conflicting among themselves, there is another important issue worth to mention. In architectural design, the actual locations of the nodes represented in the corresponding graph are most relevant to the layout rather than the boundaries determined by the method being used. In other words, for an architect it may be most desirable to identify the central locations of the units on the layout, so that he can further elaborate on the architectural design determining the unit boundaries in most convenient way directed by the requirements of the actual utilization space. This means, s/he can consider several subtle design variations among preferential alternatives. Such flexibility provides the architect with an additional dimension in his professional domain. The optimization algorithms, so far endeavor to establish the unit boundaries in the solution space so that from the design viewpoint, the case can be seen as a computer enhanced design, rather than an architectural design. The present approach intends to introduce robust multiobjective design solutions by GAs for architectural design problems providing architect with the additional architectural dimension described above. For this purpose, design requirements are computed according to given norms and metric functions. The system is based on graph representation of the layout so that the desired relations (attributes) on the graph are considered to be independent variables related to the design constraints. Since such functional relations are often discrete and soft, they pose ill-defined problems and for this case the method of GA can provide effective solution procedures. Referring to preceding considerations, the organization of the paper is as follows. Section 2 briefly describes the method of GA. Section 3 describes the application of GA to a two- dimensional simulated space layout problem and reports the analysis results, which is followed by conclusions 2. BASICS OF GENETIC ALGORITHMS The original GA invented by Holland (1975) and its many variants, collectively known as genetic algorithms, are

2 computational procedures that mimic the natural process of evolution as they are inspired by the evolution of populations. They work by evolving a population of individuals over a number of generations. In the terminology of GAs, population is a collection of several alternative solutions to a given problem. Each individual in the population is expressed in the form of a sequence of number called string which is mostly a sequence of binary numbers (Goldberg 1989) although in general, it may be sequence of real numbers as well (Michalewicz 1998; Haupt and Haupt 1999). The string is named a genotype. This is coded information. For example, a binary string represents a binary number bearing several binary coded parameter values. The decoded information is called phenotype. Every individual solution in the form of a string is named chromosome, in analogy to chromosomes in natural systems. Often these individuals are coded as binary strings, and the individual characters or symbols in the strings are referred to as genes and their varying values as alleles. In the algorithm, as result of each iteration process, a new generation is evolved from the existing population in an attempt to obtain better solutions. The population size determines the amount of information stored by the GA and the population is evolved over a number of generations. A fitness value is assigned to each individual in the population, where the fitness computation depends on the application. For each generation, individuals are selected from the population for reproduction. These individuals are chromosomes and are crossed to generate new chromosomes, which are mutated with some low mutation probability. The aim of genetic algorithms is to use simple representations to encode complex structures and simple operations to improve these structures. Their representations and operators therefore characterize genetic algorithms. Basic genetic operators of GA are as follows: Coding - An essential characteristic of a genetic algorithm is the coding of the variables that describe the problem where the variables are transformed to a binary string of specific length called chromosome. Reproduction Production of the next generation of population as result of a selection process based on a problem specific criterion known as fitness function. Crossover-Mutation By crossover, two members of the population are selected randomly and exchange part of their chromosomal information with a specified probability. By mutation, certain digits of the chromosome are altered with a specified probability. Decoding By decoding, the solution for each specific instance is determined and the value of the objective function that corresponds to the individuals is evaluated. As result of this evaluation, if necessary, the same steps are repeated from the reproduction phase onwards until the desired convergence is reached. The basic parameters of a simple genetic algorithm are the population size of the generation, the probability of crossover and the probability of mutation. By varying these parameters, the convergence of the problem is controlled. GAs are often used to solve complex optimization problems in diverse applications (Gen and Cheng 1997). They are suitable for design optimization problems in Architecture and as algorithm, they are different from traditional optimization methods in the following respects. they work with a coding of the variables set and not with the variables themselves they search from a population of points rather than by improving a single point they use objective function information without any gradient information their transition scheme is probabilistic they can deal with a multi-objective optimization tasks GAs are especially capable of handling optimization problems in which the objective function are discontinuous or non-differentiable, non-convex, multi-modal or noisy. Since the algorithm operate on a population instead of a single point in the search space, they can climb many peaks in parallel and therefore reduce the probability of finding local minima. 3. SPACE LAYOUT DESIGN EXERCISE BY GA For solution to a space layout design problem by GA, a simulated design task is devised with a graph representation. In the graph, adjacency between two nodes in the layout topology is arbitrarily defined with some hypothetical design requirements. In general, the adjacency can be assigned through elaboration of the following aspects (Ciftcioglu et al 2): 1. Connectivity pattern defined by type of the nodes Depending on the functionality of the spaces (e.g., offices), different adjacency values can be attached to the spaces that are directly connected with each other and to those that are indirectly connected for example via a corridor. This can be shown through adequate adjacency value 2. Optimal distance in relation with the requirements This is connected to various factors like functional and structural factors as examples. For example, for a certain level of sound attenuation, less soundproof walls are used, as the distance between two spaces (represented as nodes) can be larger, and vice-versa. 3. Cost function derived from the above-mentioned aspects, etc. By combining all of the above-mentioned aspects, a final model can be accomplished, which would represent the optimization of all aspects considered together as a whole. The graph representation would give to the architect enough space for the final design, since in a way, the results provided would serve him as the guidelines for design approach, indicating spatial layout, but still not determining the final shape of the units. The architect can see immediately, what the consequences are for the design by

3 appointing higher importance to certain aspects. S/he can also choose a simple design decision-making, which would mean that all aspects are of the same importance. The soft computing approach being presented in fact gives to Architect additional freedom of design flexibility in contrast with the traditional design optimization outcomes. S/he also sees directly the consequences of each design solution with respect to marginal client s needs next to the design requirements. This is especially of importance, since each design and circumstances surrounding it, would require considerations of certain design factors at the cost of some others. By the traditional methods dealing with such different design objectives is not an easy task. For example privacy and spatial closeness between units are two different design qualities which can be separately considered by soft computing and the design outcomes can be easily be integrated with final architectural judgement. This is in contrast with the traditional design optimization methods, which often lead to inconvenient solid design solutions because of a single objective function to be optimized and also difficulties due to dealing with such vague concepts. In the latter case, therefore the results are hardly of practical use if they are not inconclusive at all. Below, a design example is provided, showing the possible layout matching requirements based on multi-objective criteria. In the simulated design task a simple architectural floor plan design with five vertices is considered. The simulation is devised in two parts. In the first part, from initially known actual vertex locations, characteristic design properties are identified by computation. In the second part, the computed design properties are considered as design requirements and the actual vertex locations are requested, as design task. In this way, the outcomes from the design task would be easily verified by comparing them with the initially computed design properties. As characteristic design properties, the relationships between the nodes are determined by means of computation from the simulation model with five vertices. The adjacency can represent similarity, closeness, privacy etc. relative to each pair of locations denoted as nodes. In A this exercise, the relationships are characteristic values describing the relative position of each pair of nodes within the graph assessed as degree of congestion and it can be determined by a given relationship involving the Euclidean distance between the nodes. Some possible relationships will be mentioned later in the text. The graph representation of the layout of present example is shown in figure 1. The adjacency matrix for this graph is given by where A = [a ij ] is a 5 5 matrix and each row and each column of A corresponds to a distinct vertex of V. Then, a ij =1 if vertex v i is adjacent to vertex v j and a ij = otherwise. Note that a ii = for each i=1,2,.,5. The adjacency matrix is a symmetric (,1)-matrix, with zeros down the main diagonal. The adjacency matrix contains all the structural information about the plan. Considering the graph-theoretic definition of adjacency, the graded architectural relationships between the nodes are referred to as proximity, in place of adjacency Figure 1. Graph representation of a design layout. Following the clockwise direction, circles indicate nodes 1 and 2 and triangles nodes 5 and 6 The designer states the design attributes. These attributes are normally the objectives in the genetic algorithm. In space layout topology, design is connected to norms defined between the units in that topology. The distance between any pair of units can be computed from a metric function with a selected norm. Let the distance between any two point x and y be denoted by d(p,q). This is the minimum length of a p-q path in the graph. The following properties hold for the distance function d: d(p,q) and d(p,q)= if, and only if p=q d(p,q)= d(q,p) d(p,q)+d(q,z) d(p,z) These three properties define what is normally called a metric function on the vertex set of a graph. In general, if the nodes in the graph represent units of interest, the difference between the units can be expressed by the distance function. If we define a proximity function representing the relative status of p and q, as prox(p,q), we write prox(p,q) 1 and prox(q,q) = 1 where the proximity would diminish with the increasing distance. The way of diminishing is dependent on the design problem at hand and might take various forms, like prox(p,q)= 1/[1+d(p,q)] or prox(p,q) = e -d(p,q)

4 as examples. In the space layout problem, the nodes (vertices in the graph) represent the locations and the proximity measure can be used to represent any desired (complex) relationship between the nodes. After the locations having been properly determined in accordance with the design requirements the architect determines final space layout as appropriate partitions. For instance, the relative difference can be calculated as the attenuation of noise level between any two locations and it is dependent on the Euclidean distance between the same locations used in the relevant P c distance function To demonstrate the potential of genetic algorithm for the P space layout problem, the proximity value, for a particular Euclidean distance r=d(p,q) between any two locations p and q, is given by = exp[-c 1 (r) 2 ]+ C 2 where C 1, C 2 are some constants. In particular C 1 =.5 and C 2 varies between.1 and.4 in the GA implementation. With this definition all proximity values between the pair of nodes are calculated. Since the proximity matrix is symmetrical, for the calculations only the upper triangle excluding the main diagonal is considered. In this particular example, with five vertices there are only ten pairs to consider. The computed proximity matrix (P c ) bearing the congestion values in a unit square layout area is given by columns which corresponds to the general form In the simulated design task, the information given by the congestion matrix above is considered as space layout design requirement and the appropriate locations of the nodes matching the requirement are sought. The solution by GA is given in figure 2(a) and the GA solution together with the set of locations used as a base to compute the congestion matrix P c is shown in figure 2(b). 3 (a) 5 (b) Figure 2: Locations 1 identified by GA 2 for given congestion factors (a) and the same locations together with the locations initially used for designing the exercise (b) The estimated congestions corresponding to figure 2(b) is found to be 4 column numbers Referring to figure 2, the graph representation of P e is given in figure 3 together with that of computed congestion matrix P c for easy comparison. It is interesting to note that, the two graphs are rather similar indicating the accuracy of the GA solution. Since only adjacency between the pair of nodes is considered, the graphs in the solution space need not necessarily to overlap and this is the case in figure 3. Referring to this exercise, in general, the graphs might be rotated as well as translated, relative to one another. Therefore a better comparison would be through the congestion values of the matrices. 5 P e

5 Figure 3. Graph of design solution estimated by GA and its initial counterpart used for devising the exercise (with dashed lines) The comparison of matrices P c (computed proximity/congestion) and P e (estimated proximity /congestion) is illustrated in figure 4 where two sets of data are virtually the same marking the accuracy of the GA solution. The nodes above are the representatives of each space. Figure 5 provides a possible layout interpretation of the required congestion/adjacency values for the 5 units. In this example, all the values that are above 1.1 (P e >1.1) mean that there is a high congestion and below 1.1 value (P e <1.1) have less congestion, thus those spaces would require more privacy. This requirement may be satisfied by choice of material applied to separation walls. If the value is very high it may be decided that no physical partition between two spaces is required at all. So, in this respect choice of materials or amount of openings between two spaces can influence the design. For example, for spaces 1-4; 1-5; 3-4 and 3-5 condition P e >1.1 is applicable. If we consider 1-4 and 1-5 it means that wall separating space 1 and 4 and the one separating 1 and 5 can be made transparent in order to have better control of space 1. Opposite to this, space 2 requires less congestion, meaning that space 2 should be Figure 4. Required congestion/proximity values indicated by circles and the identified counterparts by genetic algorithm. Symbols in figure 2 and the corresponding nodes in the congestion matrix P e are given below. 1 o 2 * more isolated from other spaces since all relations of space 2 to other spaces are less than 1.1 value. This means that relations and openings with other spaces should be minimum and the walls in this case for space 2 should be solid rather than transparent. Such decisions, as explained through this example can influence design costs and prevent unnecessary investments. One of possible layout designs is given in figure 5. With such method, an architect can still determine the shape and the size of the spaces and influence the layout, while he has in mind the optimal position of these spaces in relation to each other. Some details of the algorithm are as follows. Population size = 1. Number of iterations = 25 Crossover and mutation probability =.8 and.2 respectively Bit length of the chromosome per variable = 12 Number of variables (proximity/congestion) =

6 Figure 5. One possible layout organization as a subtle preference based on the graph from figure 2(a). Partitions 1-4, 1-5, 3-4 and 3-5 are transparent (see text) 4. Conclusions Various optimization procedures as architectural problem solutions are generally based on uni-objective optimization. However such problems are generally complex and need multi-objective considerations and heuristic search algorithms are rather suitable tools to deal with the related space layout problems. As one of the heuristic search algorithm, the method of genetic algorithm is used for architectural space layout design by means of a simulated design exercise. Such a seemingly simple, however rather complex design optimization task is conveniently accomplished by genetic algorithm. The design layout established by the present approach indicates the central locations of the architectural design units represented by nodes in the relevant graph rather than the imperative partitions as this is the case for traditional design optimization based outcomes. Therefore desirably, architect can exercise own professional considerations, preferences among subtle alternatives and creativity to obtain optimal layout design with appropriate partitions based on final human-intelligence based decisions. Jagielski R. and J.S. Gero (1997). A Genetic Programming Approach to the Space Layout Planning Problem in Proc. 7 th International Conference on Computer Aided Architectural Design Futures, R. Junge (Ed.),, 4-6 August Munich, Germany Koakutsu S. Sugai Y. and Hirata H. (199). Block Placement by Improved Simulated Annealing Based On genetic Algorithm, Trans. of the Institute of Electronics, Information and Communication Engineers of Japan, Vol.J73A, No.1, pp Koakutsu S. Sugai Y. and Hirata H. (1992). Floorplanning by Improved Simulated Annealing Based on Genetic Algorithm, Trans. of the Institute of Electronics, Information and Communication Engineers of Japan, Vol.112-C, No.7, pp Lai Y.T. and Leinwarnd (1988). Algorithms for Floorplan Design via Rectangular Dualization, IEEE Trans. Computer- Aided Design, Vol. CAD-7, No.12, pp Michalewicz Z. (1999). Genetic Algorithms + Data Structures = Evolution Programs, Springer, Berlin Steadman, J.P.(1983). Architectural Morphology - An Introduction to the Geometry of Building Plans, Pion, London Sutanthavibul S., Shragowitz E. and Rosen J.B. (1991). An Analytical Approach to Floorplan Design and Optimization, IEEE Trans, Computer-Aided Design, Vol. CAD-1, No.6, pp Wong D. and C. Liu (1986). A New Algorithm for Floorplan Design, in Proc. 23 rd ACM-IEEE Design Automation Conference, Boston, Wizard V. and Yannakakis M. (Eds.), pp References Ciftcioglu Ö., Durmisevic S. and Sariyildiz S. (2). Design for Space Layout Topology by Neural Network in Proc. 5 th International Conference on Design and Decision Support Systems in Architecture and Urban Planning, Nijkerk, August 22-25, the Netherlands Damski Jose C. and John S. Gero (1997). An Evolutionary Approach to Generating Constrained Based Space Layout Topologies in Proc. of the 7 th International Conference on Computer Aided Architectural Design Futures, R. Junge (Ed.), 4-6 August, Munich, Germany, Gen M. and Cheng R. (1997). Genetic Algorithms and Engineering Design, John Wiley & Sons Inc., New York Goldberg, D.E. (1989). Genetic Algorithms in Search, Optimization, and Machine Learning, Reading, MA Addison Wesley Haupt R.L. and Haupt S.E. (1998). Practical Genetic Algorithms, John Wiley & Sons, Inc, New York, Toronto Holland J. (1975). Adaptation in Natural and Artificial Systems, University of Michigan Press, Ann Arbor Horn J. and N. Nafpliotis (1994). Multiobjective Optimization Using the Niched Pareto Genetic Algorithm in, IEEE World Congress on Computational Intelligence (ICEC'94), Vol.1