ARCHITECTURAL LAYOUT DESIGN THROUGH SIMULATED ANNEALING ALGORITHM HAO ZHENG1 and YUE REN2 1University of Pennsylvania, Philadelphia, USA [email protected] 2University College London, London, UK [email protected] Abstract. Simulated Annealing is an artificial intelligence algorithm for finding the optimal solution of a proposition in an ample search space, which is based on the similarity between the physical annealing process of solid materials and the combinatorial optimization problem. In architectural layout design, although architects usually rely on their subjective design concepts to arrange buildings in a site, the judging criteria hidden in their design concepts are understandable. They can be summarized and parameterized as a combination of penalty and reward functions. By defining the functions to evaluate a design plan, then using the simulated annealing algorithm to search the optimal solution, the plan can be optimized and generated automatically. Six penalty and reward functions are proposed with different parameter weights in this article, which become a guideline for architectural layout design, especially for residential area planning. Then the results of several tests are shown, in which the parameter weights are adjusted, and the importance of each function is integrated. Lastly, a recommended weight and “temperature” setting are proposed, and a system of generating architectural layout is invented, which releases architects from building arranging work in an early stage. Keywords. Architectural Layout; Simulated Annealing; Artificial Intelligence; Computational Design. 1. Introduction to the Simulated Annealing Algorithm 1.1. METROPOLIS ALGORITHM Metropolis et al. (1953) firstly developed a method, especially running in computers, for finding the minimum or maximum value of a function, which may be complicated and hard to describe. Rather than recursively comparing values in the function and finding the direction of optimizing to reach the objective value, in the metropolis algorithm, there is a distinct possibility that the program will accept a worse result and its optimizing direction so that the optimizing process will not get trapped in a local optimum. Figure 1 shows a minimum optimizing process. In the optimizing process, the current state xn results in its value in the function f(xn). When the process moves RE: Anthropocene, Proceedings of the 25th International Conference of the Association for Computer-Aided Architectural Design Research in Asia (CAADRIA) 2020, Volume 1, 275-284. © 2020 and published by the Association for Computer-Aided Architectural Design Research in Asia (CAADRIA), Hong Kong. 276 H. ZHENG AND Y. REN forward, the next state xn+1results in its value in the function f(xn+1), which is compared with the current value. If f(xn+1) is smaller than f(xn), such as f(x2) compared with f(x1), the current state will move to xn+1 and waiting for the next loop. Figure 1. Optimizing Process of Metropolis Algorithm. However, if f(xn+1) is greater than f(xn), such as f(x3) compared with f(x2), in the metropolis algorithm, the program will calculate a probability of acceptance, which indicates whether the state will ignore the result and still move forward, as following:( ) f(x ) − f(x ) p = exp − n+1 n f(x )≥f(x ) (1) T n+1 n Therefore ideally, with a proper setting of variable T, which is an inputted parameter controlling the updating possibility and frequency, the process will jump out of the local optimum f(x2) and eventually reach the global optimum f(x5). 1.2. SIMULATED ANNEALING AND ITS APPLICATIONS In the metropolis algorithm, the setting of variable T will primarily affect the accuracy and efficiency of the optimizing process. To find the best setup, Kirkpatrick, Gelatt, and Vecchi (1983) developed the Simulated Annealing Algorithm, changing the variable T into a process-related variable, which will decrease gradually so that the probability of accepting a worse result will become smaller and smaller, and finally reach a stable status without any acceptances of the worse outcomes. This process is similar to the physical annealing, where variable T acts as a temperature value to control the annealing rate, so it’s called Simulated Annealing. The detailed procedure can be expressed as the following four steps: (1) Set the starting temperature value as ts, the ending temperature value as te, the function to update the T value as Tt, and the initial x value as x0. Update the optimized x value xopt by x0 and calculate f(xopt). (2) Apply a random change to xopt and get xn, then calculate f(xn). (3) If f(xn) is smaller than f(xopt), update xopt by xn. If f(xn) is greater than f(xopt), calculate the probability of acceptance p(xn; xopt), then generate a random number pn between 0 and 1. If pn is greater than p(xn; xopt), update xopt by xn. (4) Update the current T value tn by T (tn). If tn is greater than te, execute (2) and loop the procedure. ARCHITECTURAL LAYOUT DESIGN THROUGH SIMULATED 277 ANNEALING ALGORITHM Thus, for any given objective function f(x), the Simulated Annealing Algorithm can find the best way to approach the global optimum. Sen and Stoffa (1991) applied the Simulated Annealing Algorithm in engineering, minimizing the error energy between the data and theories in seismic waveform inversion. Abramson (1991) built a system to find the best solution for the school timetabling problem. Tracey, Clark, and Mander (1998) constructed an optimized data generation framework to find program flaws. Ekren and Ekren (2010) used the simulated annealing algorithm to optimize the size of the wind hybrid energy conversion system. In urban, structure, and architecture fields, the Simulated Annealing Algorithm also has a wide range of applications. Hong (2011) built a model to achieve accurate forecasting of inter-urban traffic flow. Sonmez (2008) developed a method to optimize a structure to be lightweight, high-performance, and low-cost. AlHalawani et al. (2013) built a framework to produce a factored facade representation and a facade analysis algorithm with the help of Simulated Annealing. Yeh (2006) solved the facility layout problem in the interior design of a hospital by generating floor plans, while Michalek, Choudhary, and Papalambros (2002) developed a similar method to arrange rooms in a house. 1.3. PROJECT GOAL Different from the previous research, this paper focuses on the Simulated Annealing Algorithm in arranging buildings in the site, especially for residential area planning, developing unique functions to describe the main aspects when designing the architectural layout, optimizing and making a balance between each design focus. Through this research, we want to provide a useful tool for architects that the layout of the residential buildings can be generated by controlling the parameters of the design tendency, thus showing an example of translating design ideas and strategies into parameters with a more mathematical method. 2. Penalty and Reward Functions in Architectural Layout Design 2.1. PARAMETERS AND MEANINGS To clarify the question and the working limit, parameters to represent the condition and the requirements of the design were proposed firstly (figure 2). For the site, when designing, all parameters are already set and cannot be changed. Parameters include Pent for the position of entrance, Gbou for the boundary geometry of the design area, Asun for the local sunshine angle in the winter solstice, and Griv for the geometry of the river or any landscape attractions. These four parameters stay constant and will be used to define penalty and reward functions. 278 H. ZHENG AND Y. REN Figure 2. Parameters in an Architectural Layout Design. Then for the buildings, which need to be placed into the site, parameters include Gn for the boundary geometry of each building, Pn for the entrance position of each building, Hn for the height, An for the rotating angle, Xn and Yn for the relative position of each building. Among them, Gn Pn and Hn are pre-designed and should keep constant, while Xn Yn and An are the main parameters, which will be fed into the Simulated Annealing Algorithm as x value for the objective function f(x). 2.2. PENALTY FUNCTIONS After defining all parameters, the objective function f(x) was developed as the weighted sum of 6 sub-functions. First, to avoid any glaring geometric errors for architectural design, two penalty functions were proposed (figure 3). Figure 3. Penalty Functions. One is called Out-of-Boundary Penalty, which means, when the geometries of buildings are partly or wholly outside the site boundary, a minus point will be ARCHITECTURAL LAYOUT DESIGN THROUGH SIMULATED 279 ANNEALING ALGORITHM added into the objective function as the following formula: Xn f1 = − k1 · Area(RegionDefference(Gi;Gbou)) (2) i=1 Also, when two buildings have an overlapping area, it doesn’t make sense. Thus another penalty function is called Overlapping Penalty, to avoid the collision of buildings, as follows: nX−1Xn f2 = − k2 · Area(RegionIntersection(Gi;Gj)) (3) i=1 j=i In the two formulas, k1 and k2 are factors that adjust the order of magnitude, since the unit in these two formulas is m2, which is different from the following formulas. And the proper settings of k1 and k2 will be discussed in the following pages. 2.3. REWARD FUNCTIONS In addition to the penalty functions, 4 reward functions were also constructed, to guide the system to find the best solution (figure 4). Figure 4. Reward Functions. First, if the entrance of a building is close to the access of the site, the system will give a reward point to the function, because it will be convenient for the building users. It’s called Convenience Reward, based on the following formula. 280 H. ZHENG AND Y. REN The shorter the distance between two entrances, the higher score it will gain.