2.1. Building performance simulation

With the growing demand for energy-efficient buildings, numerous building energy simulation tools have been developed. Green building standards such as LEED (USGBC, 2009) have been issued and implemented in many countries. A more sustainable and energy-saving type of design has been advocated by designers, engineers, and developers. The applications for building simulations include building heating and cooling load calculation, daylighting calculation and reflective roof analysis, building energy management and control system design, building regulations, code checking, cost analysis, and so forth (Hong et al., Reference Hong, Chou and Bong2000). Although many simulation tools exist today, the optimization process, which involves iterative simulations, is inefficient and time consuming.

2.2. Design by optimization and problem statement

Architectural design problems are generally complex and often involve multiple design objectives. Besides energy efficiency, other design objectives architects might consider throughout the comprehensive architectural design process include aesthetic, structural, functional, social, historical, behavioral, and so on (Lobos & Donath, Reference Lobos and Donath2010). Thus, architectural design can be seen as a multiple-objective optimization process, aiming to find optimal solutions for multiple objectives that often conflict with one another (Radford & Gero, Reference Radford and Gero1987). In general, there are two ways to solve multiple-objective optimization problems: convert a multiple-objective optimization into a single objective optimization by using a weighted sum of the objective functions with often arbitrary weights; or search for the Pareto optimal set, which is more widely accepted for practical multiple-objective optimization, because normally there exists a set of optimal solutions (design options) instead of a single solution for multiple-objective optimization. In this study we use the first method of design optimization. While the present methods can work for both of the two ways described above, in this study, we start testing our methods using the simpler way of optimization, converting a multiple-objective optimization into a single objective optimization. We will discuss the Pareto optimal optimization in a future study.

Because architectural design problems often having discrete, nonlinear, and stochastic decision variables, optimization methods such as classical calculus, linear programming, and dynamic programming are not applicable to most of the complex optimization problems in architecture (Radford & Gero, Reference Radford and Gero1987). GAs model natural selection and the process of evolution (Holland, Reference Holland1975). Using the mechanisms of natural selection in evolving individuals who adapt to an environment over time, GAs provide a robust search process and have been used in optimizing complex and poorly understood scientific and engineering problems (Gero & Louis, Reference Gero and Louis1995) such as automotive design (Vidal et al., Reference Vidal, Crainic, Gendreau, Lahrichi and Rei2012; Mahmoodabadi et al., Reference Mahmoodabadi, Safaie, Bagheri and Nariman-Zadeh2013), engineering design (Gen & Cheng, Reference Gen and Cheng2000; Deb, Reference Deb2012), medicine (Chen & Chen, Reference Chen and Chen2011), and so on. In the field of architecture, the applications for GAs include structural design (Kociecki & Adeli, Reference Kociecki and Adeli2013), green building design (Wang et al., Reference Wang, Zmeureanu and Rivard2005; Attia et al., Reference Attia, Gratia, De Herde and Hensen2012), and space planning (Jo & Gero, Reference Jo and Gero2006; Rawat et al., Reference Rawat, Shahani, Natu, Badami and Hingorani2012). Many tools (Gerber et al., Reference Gerber, Lin, Pan and Solmaz2012) are under development for the purpose of generation of design alternatives.

In addition to the improvement of multiple-objective GAs from the general computing point of view, there is a significant need to research a more efficient optimization tool that will couple with design simulations (e.g., energy simulations) more efficient during the optimization process. In this paper we present two proposed techniques for improving optimization as employed in a case study for space layout: an offline simulation technique, and a divide and conquer (D&C) method, customized for architectural design problems. The development of both methods is based on architects’ (as opposed to computer scientists’) domain of knowledge.

Time and computational complexity analyses are applied to evaluate an algorithm's usage of computing and time resources. A well-known limitation of GAs is that it often can take a tremendous amount of search and evaluation iterations before reaching optimal solutions, especially when the task involves multiple objectives. A GA optimization process could take days to complete using today's conventional computers. In an optimization study of insulation usage and space conditioning load, it took a GA 50 min to run 25 different designs and 2 h and 18 min to run 100 different designs on a standard laptop (1.7-GHz CPU, 512 MB RAM; Shi, Reference Shi2011). In a study of green building design optimization using multiple-objective GAs, a computer with Windows XP (3.06-GHz Pentium IV processor, 512 MB RAM) took 30 h to run 29 solutions (Wang et al., Reference Wang, Zmeureanu and Rivard2005). This time-consuming aspect of GAs could be a serious impediment to design automation, effectively discouraging designers from conducting more simulation studies. The number of generations and solutions in the above GA examples are too few to be of any practical use. When GA execution time can be saved by using a smaller search space, global optimality is unlikely to be achieved. In contrast, increasing GA search space to allow for a greater number of optimal solutions will result in a longer execution time, which in turn will prevent GAs from being useful in the design process. Thus, reducing the run time of GAs is essential to its use in design optimization.

Previous research studies have shown that building simulation is a very time-consuming and labor-intensive process. For instance, in a high-rise office building energy simulation study, the computer will need more than 30 h to perform a 1-year hourly simulation, and the time required for monthly simulations will vary from 36 min to around 6 h, using four desktop computers running at the same time (Ahn et al., Reference Ahn, Young-Jin, Deu-Woo, Sung-Hwan and Cheol-Soo2013). Researchers point out that the time and resources needed for simulation are one of the reasons that simulation tools have not been fully embraced by architects (Shi, Reference Shi2011). Architectural design firms have been using GAs as a way to search for the optimal design options to assist with the design decision-making process (Besserud et al., Reference Besserud, Skidmore and Merrill2008; Claussnitzer et al., Reference Claussnitzer, Katz, Shaxted, Park and Yori2014). However, due to the tight project timeline, architects do not have the time and resources. Most optimization projects have a small population. Therefore, the optimal solutions are not guaranteed to be found.

When a standard GA is used with simulation for optimization, a significant computer effort is expected (Renner & Ekárt, Reference Renner and Ekárt2003). As a result, various methods have been researched for increasing the speed of GAs. The literature on this topic can be divided into four groups. One involves improving the infrastructure, running GAs on a faster machine, or using multiple machines working together. The parallel GA uses a group of cooperating computers to solve complex problems in less time (Muhlenbein, Reference Muhlenbein and Rawlins1991). Cloud computing (Armbrust et al., 2000) has also been introduced as a way to reduce the computing time of optimization problems. Cloud services are now being used in many design firms to speed up the optimization process (Claussnitzer et al., Reference Claussnitzer, Katz, Shaxted, Park and Yori2014).

The second group of research in the literature investigates the construction of approximate simulation models: surrogate models. Surrogate models mimic the actual simulation model but use less computing power (Forrester et al., Reference Forrester, Bressloff and Keane2006). The computational cost of GAs can be reduced by replacing computationally expensive fitness evaluations with cheap approximation models such as surrogate models. (Ong et al., Reference Ong, Nair and Keane2003) A review paper (Jin, Reference Jin2005) points out that approximation models are beneficial when computation of the fitness function evaluation in GAs is time consuming; no analytical fitness function model exists (the fitness function needed to be evaluates by humans, for example, in music composition and art design); the results of GAs is not always consistent; and the fitness landscape has multiple local optimals. Surrogate models are used to replace the accurate and time-commuting fitness evaluation. It is recommended to use a surrogate model with the real/original fitness function to avoid having a false optimum in the surrogate model (Jin et al., Reference Jin, Olhofer and Sendhoff2000). The trade-off between the fidelity (approximation accuracy) and computational cost is shown in Figure 1 (Jin, Reference Jin2011). Generally speaking, fitness evaluations with high fidelity are more time consuming, and fitness evaluations with low fidelity are usually less time consuming. The user should balance between the needs for accuracy and available computational cost. Surrogate models are applicable to many operators in GAs. For example, population generation, crossover (Anderson & Hsu, Reference Anderson and Hsu1999), mutation (Abboud & Schoenauer, Reference Abboud, Schoenauer, Collet, Fonlupt, Hao, Lutton and Schoenauer2002), local search (Ong et al., Reference Ong, Nair and Lum2006), global search (Simpson et al., Reference Simpson, Mauery, Korte and Mistree2001), and fitness evaluations (Buche et al., Reference Buche, Schraudolph and Koumoutsakos2005; Jin, Reference Jin2011). Multiple surrogate models can be applied together to a GA problem. For example, researchers (Zhou et al., Reference Zhou, Ong, Nair, Keane and Lum2007) combine global and local surrogate models to improve the speed of an evolutionary optimization. However, surrogate models require simulation knowledge, that is, how to construct approximate simulation models, which architects usually do not possess. This inspired the present project in the reuse of offline or presimulation that architects already know how to conduct in the optimization process. The proposed methods are expected to be simpler for implementation than surrogate models because the original simulation models can be used in the optimization process.

Fig. 1.

The trade-off between fidelity and computational cost (adapted from Jin, Reference Jin2011).

The third group of research suggests to use simplified building simulation tools or reduce the number of simulation needs. Sentient building simulation systems construct a result database of a portion of the search space and interpolate the results based on designer's need (Negendahl et al., Reference Negendahl, Perkov and Heller2014).

The fourth group in the literature is focused on making the GAs work more efficiently. Associative parametric models can be used to describe a large and complex geometric system with fewer variables, which would make the optimization process faster (von Buelow et al., Reference von Buelow, Falk and Turrin2010). The evolutionary D&C algorithm reduces the complexity of the problem by dividing a large problem into simple subproblems (Valenzuela & Jones, Reference Valenzuela and Jones1993).

Based on the literature review, in order to improve a GA's application in architectural design, we first try to reduce the run time or the number of simulation runs in the optimization process; and second, we try to make GAs work more efficiently. In this paper we present two proposed techniques for improving optimization as employed in a case study for space layout: an offline simulation technique, which is a method to reduce the simulation needs, and D&C, which is a method to accelerate GAs, customized for architectural design problems. The development of both methods is based on architects’ (as opposed to computer scientists’) domain knowledge. The improvements can be additive to existing improvements, for example, the computing infrastructure improvements. Overall, this paper analyzes the existing problems and limitations of design optimization when associated with building simulation; presents the two proposed methods: offline simulation and D&C; and demonstrates a case study that could be used to validate these methods. This research also describes experiments performed on the use of parametric design, design optimization, daylight simulation, and design automation in the early conceptual design process.

4.1. Design objectives

Validations of the method is presented within the context of a case study for parametric form-finding in a nursing unit design with two design objectives: to minimize nurses’ travel distance from the nurses’ station to each patient room; and to maximize the daylight illuminance in all patient rooms, use LEED standard (healthcare supplement; USGBC, 2009) as a reference.

In any study related to the behavior and working efficiency of the nursing staff, one of the most important variables is the distance that a nurse is obligated to walk in a hospital. Walking has been identified as a major time-consuming activity for nurses, and evidence from previous studies suggests that the time saved by walking can be turned into more time spent on patient care activities (Zimring et al., Reference Zimring, Joseph and Choudhary2004). Individual nurses across all study units travel between 1 and 5 miles per 10-h daytime shift. Average travel distance ranges between 2.4 and 3.4 miles with a median of 3.0 miles per 10 h (Hendrich et al., Reference Hendrich, Chow, Skierczynski and Lu2008). Unnecessary walking may lead to time waste and add to fatigue and stress. In the case study, the objective is to minimize the total nurses’ walking distance.

Daylight is an essential element for patients’ well-being. Research shows that patients in sunny rooms feel less pain and stress, and take less medication as compared to patients in rooms with less sunlight (Walch et al., Reference Walch, Rabin, Day, Williams, Choi and Kang2005). This case study uses the LEED standard as a reference to construct the fitness function for the daylight illuminance level calculation (Rahmani Asl et al., Reference Rahmani Asl, Zarrinmehr and Yan2013). The objective is to achieve maximum daylighting under LEED requirement. LEED requires 75% or more of the perimeter area to achieve a daylight illuminance level between 110 and 5400 lux. Therefore, in this case study, the daylight illuminance in every inpatient unit layout design is expected to meet this requirement. The current tools and techniques used are introduced in Section 4.2, and the measurable baseline model is discussed in Section 4.3.

4.2. Design platform and tools

Parametric models can generate a very complex building geometry with a number of variables, rules, and constraints that are defined by the designers. Several design software tools offer parametric modeling features. Of these software options, the integrated Rhino/Grasshopper program has widely been used because of its powerful modeling capability, intuitive interface, and abundance of plugins that greatly expand its functionality. It also provides a ready-to-use GA plugin, Galapagos, which can be used for optimization. Hence, this case study uses Rhino/Grasshopper as the design platform. The following is a complete list of current tools and techniques used in this case study:

1. 1. Rhinoceros (Rhino), a NURBS-based three-dimensional modeling program;

2. 2. Grasshopper, a visual programming plugin for parametrically editing models in Rhino;

3. 3. Galapagos, a GA tool in Grasshopper; and

4. 4. DIVA, a thermal and daylight simulation plugin for Grasshopper. DIVA conducts daylight analyses for Rhino models through Radiance and DAYSIM daylight simulation engines.

The Galapagos-DIVA has been used in previous work for daylight analysis and optimization in the early architectural design process (e.g., Portugal & Guedes, Reference Portugal and Guedes2012; Gallas & Halin, Reference Gallas and Halin2013).

4.3. Baseline model

Here we present the simplified yet representative case study to validate the two methods we propose: offline simulation and D&C. The design problem is to find the optimal nursing unit layout that would allow for the least travel distance for nurses and a daylight illuminance that meets the LEED standard. The layout of the nursing unit is defined by a set of parameters, restrictions, and objectives. The parameters include constants and variables. The constants include a 15 × 15 grid with a total number of 225 cells, each measuring 15 × 15 ft. as possible room spaces; a central nurses’ station represented by the blue square, sized 29 × 29 ft.; an 8-ft. corridor outside of the central nurses’ station; the city of Boston as the location of the building in DIVA; and a window size of 6 × 6 ft. in every patient room. Figure 2 shows the constant parameters in a possible layout solution; the small red (dark) squares represent the patient rooms.

Fig. 2.

The constant parameters in a possible layout solution.

The variables are the locations of 12 patient rooms. A patient room can be located in any cell, with the following restrictions: a patient room cannot overlap with any other patient room, nurses’ station, or the corridor; and in order to introduce natural light into each room, a patient room cannot be surrounded by other rooms in all four directions. In other words, at least one of the four cells surrounding each room should be vacant for window opening.

In this case study, we convert two objectives into a single objective by using a weighted sum of the objective functions with predefined (architects’ subjective) weights. For both travel distance and daylight illuminace, 100 points are given as the highest fitness score. The total nurses’ travel distance is calculated as the sum of the distances from the center of the central nurses’ station to the centers of all the patient rooms. The calculation of the nurses’ station to patient room distance and the total distance are simplified in the case study. In a practical situation, nurses’ walking distances are usually affected by the distance from the nurses’ station to the patient bed, following each nurse's travel path and affected by the order of that nurse's activities. Neither daylight nor travel distance subfitness functions should be linear in the overall fitness function. For daylight illuminance, losing the same amount of daylight affects more a darker room than a brighter room (Rutten, Reference Rutten2011; see Fig. 3 left). The same additional increment of travel distance makes a nurse feel much more fatigued, if that nurse has traveled a longer distance than someone who has just started a shift (see Fig. 3 right).

Fig. 3.

(Left) The daylight illuminance subfitness function is nonlinear. (Right) The travel distance subfitness function is nonlinear.

A fitness score of 100 is given to the nursing unit layout with the shortest total travel distance possible, 388 ft. (see Fig. 4, left), and 0 is given to the layout with the furthest total travel distance possible, 1,708 ft. (see Fig. 4, right). For any nursing unit layout solution, the fitness score for the travel distance is defined as follows. (We chose the power of 2 in the case study for simplicity of calculation.)

Fig. 4.

Unit layouts with the (left) shortest travel distance and (right) furthest travel distance.

$$\eqalign{Fitness_{travel distance} & = 100 \left(1 - \left(\displaystyle{Distance - 388 \over 1708 - 388} \right)^{\!\!2} \right)\comma \cr & \qquad \le Distance \le 1708.}$$

Daylight illuminance is evaluated by the percentage of sensor grid points in a room achieving a daylight illuminance level between 110 and 5400 lux. In this case study, the above percentage is 89% if all rooms face south and 70% if all rooms face north (and in between for east and west). Fitness values of 100 and 0 are assigned to rooms with south-facing and north-facing windows, respectively. For any nursing unit layout solution, the fitness score for daylight illuminance is defined as follows:

$$\eqalign{Fitnes{s_{daylighting}} & = 100 \left(Sqrt \left(1 - \displaystyle{{89\% - DaylightPercentage} \over {89\% - 70\%}} \right)\right)\comma \cr &70\% \le DaylightPercentage \le 89\%.} $$

According to LEED IEQ Credit 8.1 “Daylight and Views-Daylight,” daylight illuminance simulations should be conducted under clear sky conditions at two different times. In this research, those times are 9 a.m. and 3 p.m. on September 21. Thus, in DIVA, “clear sky with sun” and “illuminance” are selected for the sky conditions. However, DIVA can only conduct simulations at a specified time point in each run. In order to satisfy the LEED requirement, the optimization process must run two separate times, one on September 21 at 9 a.m. and the other at 3 p.m. Although the setup of both optimization processes have the same objectives and parameters (except the solar time) in DIVA, the results may be different because daylighting conditions differ. We perform both a 9 a.m. and 3 p.m. calculation in our case study.

The DIVA daylight simulation accuracy increases if more sensors are used in the patient rooms. However, the more sensors it uses, the more computing time it needs. Consequently, designers should balance the need for accuracy with the associated computing time, based on the project's requirements. In the baseline model used in this research, 100 sensors are evenly distributed in each room (1,200 sensors in 12 rooms total) at the level of desk height. All simulations are run on a standard laptop (ThinkPad series, Windows 7 64-bit, Intel^R Core^™ 2.50-GHz i5-2520M CPU with 8 GB of memory). In Galapagos, the population size per generation is 50, initial boost is 2, inbreeding factor is 75%, and maintain factor is 5%. For this particular study, a single run of the daylight simulation in DIVA requires approximately 1 to 2 min to complete. However, when DIVA is associated with Galapagos in the optimization process, it can be very time consuming; each solution requires a simulation. In our experiment, 78 h of DIVA run time + Galapagos could only calculate 13 generations of the GA, and the results were not nearly optimal in our examination. The complete optimization could take days, which is not practical even for this simplified case study.

Here, we summarize the problems we encountered during design optimization when using GAs and energy simulation as the platform. Due to the limitations of the software (Grasshopper + Galapagos + DIVA), only one specific time of a day (e.g., 9 a.m.) can be calculated per simulation. If design optimization involves a building simulation at a different time (e.g., 3 p.m.), the entire process must be reperformed. Thus, the information from both building simulations cannot be combined into one design optimization to find the optimal solution. Design optimization is very time consuming. Most architectural design projects have tight schedules. Therefore, it is unrealistic for architects to spend days on one optimization problem.

4.4. Offline simulation

Because the current design optimization process has the above-mentioned problems, offline simulation is introduced to integrate multiple building simulations at different times (e.g., 9 a.m. and 3 p.m.) into one optimization problem; and to reduce computing time when a building simulation is coupled with design optimization by separating the simulation from the optimization process. We separate the building simulation from the GA optimization process and conduct all required simulations in advance. This way, the simulation results can be reused in similar situations (e.g., all the rooms with windows facing the same direction will have the same daylight illuminance results in any GA generation and across the generations), and time-consuming simulations will not need to be run for each solution.

Because the locations of patient rooms are variables, one question is how to define the window opening directions. In terms of daylight, a very large window facing north and a relatively small window facing south might both satisfy the LEED daylight requirement. However, window-opening directions not only affect the interior daylight level but also have a significant impact on building energy consumption. In LEED, a project can earn up to 24 points in the “energy performance” category. The points that a building can earn depend upon the percentage of improvement in building energy consumption as compared to its baseline performance. This is calculated using a computer simulation model for the entire project based on Appendix G of the ANSI/ASHRAE/IESNA Standard 90.1-2007 (ASHRAE, 2007). The baseline model performance is defined as an average of the results of four simulations from four orientations: the original orientation of the building, and the orientation rotated by 90°, 180°, and 270°. The results of the thermal energy simulation using DIVA shows that the building consumes the least amount of heating and cooling energy when the window is facing south. The second best orientation is east, the third is west, and the worst is north. Based on this analysis, we conclude that the window-opening priority order should be south–east–west–north. Using this predefined priority list saves time spent computing the GA; if the priority list is not used, all possible directions must be included in the search space and evaluated by the simulation and fitness functions.

We divide the floor plan into four sections (see Fig. 5). A VB Script node written in Grasshopper is used to implement the following rules, the goal of which is to determine automatically each window's opening direction for the patient rooms during the GA process: if the center of a room is in Section 1, the priority of window-opening direction is south–east–west–north. VB will check the availability of adjacent rooms in the above sequence; if the center of the room is in Section 2, the priority is west–south–east–north; if the center of the room is in Section 3, the priority is north–south–east–west; if the center of the room is in Section 4, the priority is east–south–west–north. The center of a room is possible to be on a line that separates two sections. The priority rules for window-opening directions still hold in this situation: if the center of the room is on the line that separates Sections 1 and 2, or 1 and 4, the room has a south window; if the center is on the line that separates Sections 3 and 4, the room has an east window; and if the center is on the line that separates Sections 2 and 3, the room has a west window. Patient rooms are not required to be located next to the corridor. Any of the four directions of every room is possible for window opening, following the priority rules.

Fig. 5.

The floor plan is divided into four sections in order to determine the priority of window-opening directions. The windows’ locations are indicated in the figure as dark line segments on the edges of the rooms.

In more realistic simulations than this present, simplified study, further design modifications are required to make the design more practical in real projects: if the patient rooms are away from the corridor (not connected), additional corridors need to be added into the design layout; if the door of a room is blocked by other patient rooms, further modification to the design layout is needed. For example, in Figure 5, one room in the upper left corner needs access to the corridor.

One of the research questions can be stated as follows: because the parameters are consistently changing during the optimization process, how do we categorize the results of each simulation so that each category of simulation only needs a single simulation prior to the optimization process? For daylighting simulations, all patient rooms in this study share the same parameters (room dimensions, window size and location, building materials, number and location of sensors in DIVA), except for room locations and window-opening directions. However, as long as the patient rooms have the same window directions, the daylighting values of the rooms can be regarded as also being the same (in a simplified experiment, when shading is ignored). Based on this, the daylight simulations of all patient rooms in this project can be simplified as simulating four patient rooms with windows facing south, east, west, and north, respectively. Therefore, before the start of GA optimization, a DIVA daylight simulation is conducted to calculate the daylight level in every patient room. The location of rooms does not matter for daylight evaluation, only for walking distance calculation. The offline, presimulated daylight illuminance levels of all directions (south, east, west, and north) are used in the GA process. During the optimization process, although the room location and the window-opening direction may change in any design solution, we can obtain the daylight fitness score of the entire design solution by counting the total percentage of sensors that meet the LEED illuminance value.

Comparing the workflows of existing design optimizations (see Fig. 6) and offline simulation-based optimizations (see Fig. 7), the difference is that in offline simulation-based optimizations, building simulations are performed prior to the optimization process. This change can reduce a significant amount of computing time. In this simplified case study, it took the computer approximately 40 min to calculate 13 generations of the GA when using the offline simulation method (plus about 8 min for the simulations of the four rooms with windows facing the four different directions), while the existing method uses 13 h for 13 generations of the GA. However, due to the large population size, although the obtained best nursing unit layout is close to the (perceived) optimal solution (an elliptical layout of patient rooms surrounding the nurses’ station), the improvement in solutions gets very slow after the 13th generation. Again, improvements to the GA process still need to be made.

Fig. 6.

The traditional genetic algorithm optimization workflow.

Fig. 7.

The improved genetic algorithm optimization workflow with offline simulation.

4.5. D&C

The method of D&C is evaluated here in an effort to further improve GA efficiency and save computing time. This method suggests the use of a GA to search in subproblems with fewer populations instead of the entire population. As mentioned in Section 3.2, one requirement for the applicability of D&C is that the decision variables are independent of one another. This requirement can be satisfied in the present case study. In this study, when all 12 genomes (room locations) are used in a GA, the total population size is 225 × 224 × · · · × 214 = 1.249 × 10²⁸, where 225 is the number of possible locations for the first genome, 224 is the number of possible locations for the second genome after the first genome has selected a spot, and so on. However, if only one genome is used in one GA run, the population size can be reduced to 225 for the first room, 224 for the second room, and so on. The idea here is that instead of directly solving an optimization problem with 12 genomes, we solve another equivalent problem: optimizing one genome at a time for 12 times. Figure 8 shows the working process of the nursing unit layout optimization when applying the D&C method. Each time, one room is added to the previous result. This way, the nursing unit layout optimization is completed in 20 min.

Fig. 8.

The working process of the nursing unit layout optimization with the divide and conquer method.

When the D&C method is applicable, we can further improve the efficiency of the GA and simplify the GA coding by linking only one genome (the decision variable) to the fitness function. In this study, for the GA with the D&C method, the fitness function for the travel distance is as follows (the fitness function for daylighting remains the same in this study):

$$\eqalign{Fitnes{s_{travel\,\,distance}} & = 100 \left(1 - {\left(\displaystyle{{Distance - 30} \over {143 - 30}}\right)^{\!\!2}}\right)\hbox{,} \cr &30\,ft \le Distance \le 143\,ft.}\hfill$$