# Space-Constrained Scheduling Optimization Method for Minimizing the Effects of Stacking of Trades
## 1 Introduction  
Construction projects are carried out by various resources (e.g., workers, equipment, and materials), which are allocated according to the schedule plan. The resources have physical volumes and occupy a set amount of space when executing work. Existing studies named this \"workspace\" [1]. Construction projects are performed within a limited space and are resource-intensive; therefore, problems with workspace interference can frequently occur if the project does not have a proper schedule. Workspace interference causes a variety of problems. Generally, workspace interferences are known to affect declines in work productivity and work safety [1]. The work efficiency of labor decreases if they are not provided with sufficient workspace, and the risk of accidents increases when the workspace of construction equipment and the workspace of laborers overlap. Furthermore, if the problem goes beyond workspace interference and conflicts occur, that is, if workspaces for executing work are not provided, it becomes impossible to execute the work. Workspace conflicts inevitably lead to delayed project completion time and additional costs. Therefore, in academia, the problem of developing schedules that minimize workspace interference is named space-constrained scheduling (SCS), and many studies were conducted to address this problem [2, 3, 4, 5, 6, 7]. Existing studies on SCS have had the goal of minimizing workspace interference. The general process for doing so is to (1) identify workspaces required for executing activities, (2) determine whether any workspaces are occupied simultaneously according to the sequence of activity execution, and (3) establish a plan for minimizing workspace interference. Previous SCS studies determined whether workspace interference occurs based on the workspaces required to execute activities. This study defines such a method as an activity-oriented workspace generation and interference identification method. The activity-oriented method assumes that productivity and safety issues will not occur if activity workspaces do not overlap, even when several tasks are executed within a limited workarea. However, studies on construction productivity [8; 9; 10; 11; 12] have asserted that an increase in resources (e.g., workers, equipment, and materials) in a limited work area leads to declines in productivity. Previous studies called this phenomenon \"stacking of trades.\" [8; 9; 10; 11; 12]. The trades are a group of skilled workers such as pipefitters, electricians, etc. [PERSON] [8] used statistical methods to predict declines in work productivity due to stacking of trades. The independent variable that was used in these predictions was work area density (m\({}^{2}\)/worker), whereas the dependent variable was labor productivity efficiency (percent productivity). Figure 1 shows a productivity-efficiency function for work area density in electrical work proposed by [PERSON] [8]. In short, productivity decreases as work area density increases. Similarly, work area density is closely related to safety issues. As mentioned previously, the goal of performing SCS is to ensure work productivity and safety. Therefore, the appropriate approach for achieving the goals of SCS is an area-oriented method rather than an activity-oriented method. That is, it is necessary to have an SCS method that determines whether work areas with a certain predefined size are occupied as activities are executed, estimates the density of work areas, forecasts to what extent productivity declines when the work area is occupied simultaneously by multiple activities, and identifies a solution for minimizing workspace interference. The area-oriented SCS reduces the stacking of trades by minimizing the simultaneous occupation of the work areas that has a certain area size by multiple activities. In addition, because SCS has a quite large solution space, it is difficult to solve by intuition, and it is easy to produce poorly made plans.