# Space-Constrained Scheduling Optimization Method for Minimizing the Effects of Stacking of Trades
## 3 Space-Constrained Scheduling Optimization Method
### Defining the Occupation Density Function  
The user with expertise in construction planning and management divides the construction site into specific work areas considering the characteristics of the workspace (Step 3). For example, rooms that are closed on all sides, transporting passages, and material loading spaces, etc., can be classified into specific work areas. Certainly, it is simply possible to set up the specific work areas by dividing the site into grid unit planes. However, it should be noted that the specific work areas (_h_) that are defined in this step are the standard for defining the occupation density function in Step 5. If the construction site is divided into too many specific work areas, it may affect the computation speed of finding the optimal solution. A defined specific work area (_h_) can be converted into a polygonal plane that is made up of several vertices. SSO stores the vertex coordinates set {P\({}_{1}\)(x\({}_{1}\), y\({}_{1}\)), P\({}_{2}\)(x\({}_{2}\), y\({}_{2}\)),...,P\({}_{n}\)(x\({}_{n}\), y\({}_{n}\))} and z value of the specific work area (_h_) in the matrix \(S_{D}\) (Equation (2)). \[S_{D}=\left[(x_{h,n},y_{h,n}),z_{h}\right],\ n=1:N,\ h=1:H \tag{2}\]  
Next, it is required to determine a function for the density at which a certain work area is occupied due to the workspace needed to execute an activity (_i_) in Step 5. In this study, this is named the occupation density function (_f_s). The occupation density of the work area may change according to the progress rate of the activity. The occupation density may be low during the planning stage and high during the end stage. The opposite scenario may also be true. In addition, the work areas that are occupied may vary according to the rate of progress. For example, in the case of masonry work, transport routes are necessary for the initial process, but the routes are no longer required for stacking bricks in the subsequent process. In this way, occupied work areas and occupation density vary with characteristics of activity and rate of progress. SSO identifies the occupation density function (_f_s,_h_), where the _x_-axis is the rate of progress (_p_, \(p\) = 0:100) and the y axis is the occupation density (_w_, \(w\) = 0:1) for the specific work area (_h_) of each activity (_i_), as shown in Equation (3). An occupation density of 1 means the maximum allowable value at which work execution is possible in the work area. \[f_{s_{i,h}}(p)=\frac{r_{i,h}(p)+e_{i,h}(p)+m_{i,h}(p)+f_{i,h}(p)}{s_{h}} \tag{3}\]  
where \(h\) is the index of specific work areas defined in step 3; \(r_{i,h}(p)\) is the area occupied by the workers in the work area \(h\) when the rate of progress for activity \(i\) is \(p\); \(e_{i,h}(p)\) is the area occupied by equipment in work area \(h\) when the rate of progress for activity \(i\) is \(p\); \(m_{i,h}(p)\) is the area occupied by material in work area \(h\) when the rate of progress for activity \(i\) is \(p\); \(f_{i,h}(p)\) is the area occupied by temporary facilities in work area \(h\) when the rate of progress for activity \(i\) is \(p\); and \(s_{h}\) is the area size of work area \(h\). \(r_{i,h}(p)\), \(e_{i,h}(p)\) and \(m_{i,h}(p)\) are calculated as unit size \(\times\) number of inputs. The unit size for each variable is obtained from the material DB, and the number of inputs is obtained from the matrix (\(N_{D}\)) defined in Step 1. \(s_{h}\) is calculated using the polygon-area function (Equation (4)). The vertex coordinates of work area (_h_), which are variables used in the \(s_{h}\) calculation, are obtained from the matrix \(S_{D}\) stored in Step 4. \[s_{h}=\mathit{polygonarea}(x_{n},y_{n},n)=\frac{1}{2}\left|\sum_{n=1}^{N}(x_{ n}+x_{n+1})(y_{n}-y_{n+1})\right| \tag{4}\]Figure 3 is a simple example of explaining the concept of the occupation density function. It is assumed that the work area (_h_) was divided into four areas (A, B, C, and D) in Step 3. Activity \(i\) is executed by occupying the A work area until the rate of progress is 40%, and the area occupation density is 0.4.