## 5 Implementation
### Modified Problem-Solution Procedure  
The modified problem-solution procedure leads to the formulation of a modified linear programming problem to resolve issues that arise when adopting a linear programming method. In using the optimization problem in Formulation 2, isolated lines can still occur in some cases, and such lines are undesirable for axial line analysis. The zero-one integer linear programming method is modified to accommodate this situation. While the original constraints of the optimal problem do not prevent isolated lines that are undesirable for axial line analysis, the isolated lines can be suppressed by adding more constraints to the optimization problem. To avoid isolated axial lines, a group of constraints are added to the original formulation. If this modified formulation of zero-one integer linear programming can provide a solution to the optimization problem, then the solution will be guaranteed to be an optimum. Therefore, an optimization problem that includes constraints for suppressing isolated lines is formulated. The new formulation of the problem (Formulation 3) is stated as follows: minimize \(\sum_{i=1}^{n}x_{i}\), subject to  
where  
\[\mathbf{A}=\left[\begin{array}{ccccc}a_{1,1}&a_{1,2}&\cdots&a_{1,n-1}&a_{1,n} \\ a_{2,1}&a_{2,2}&&a_{2,n}\\ \vdots&&\ddots&&\vdots\\ a_{m-1,1}&&&a_{m-1,n-1}&a_{m-1,n}\\ a_{m,1}&a_{m,2}&\cdots&a_{m,n-1}&a_{m,n}\\ \end{array}\right]\mathbf{X}=\left[\begin{array}{c}x_{1}\\ x_{2}\\ \vdots\\ x_{n-1}\\ x_{n}\end{array}\right]\mathbf{B}=\left[\begin{array}{c}b_{1}\\ b_{2}\\ \vdots\\ b_{m-1}\\ b_{m}\end{array}\right]\]  
\[\mathbf{C}=\left[\begin{array}{ccccc}c_{1,1}&c_{1,2}&\cdots&c_{1,n-1}&c_{1,n}\\ c_{2,1}&c_{2,2}&&c_{2,n}\\ \vdots&&\ddots&&\vdots\\ c_{n-1,1}&&&c_{n-1,n-1}&c_{n-1,n}\\ c_{n,1}&c_{n,2}&\cdots&c_{n,n-1}&c_{n,n}\\ \end{array}\right],\mathbf{D}=\left[\begin{array}{c}d_{1}\\ d_{2}\\ \vdots\\ d_{n-1}\\ d_{n}\end{array}\right],\]  
\(a_{ij}=1\) if the \(i\)th s-line has an intersection with the \(j\)th axial line, and 0 otherwise,  
\(b_{i}=1\),  
\(c_{ij}=1\) if the \(i\)th axial line has an intersection with the \(j\)th axial line, and 0 otherwise,  
\(c_{ii}=-1\),  
\(d_{i}=0\),  
\(x_{i}=1\) if the \(i\)th axial line is selected, and 0 otherwise,  
\(m=\) the number of s-lines,  
\(n=\) the number of axial lines.  
Figure 2: A minimal set of axial lines that “get everywhere” (\(n=11\)).  
**Formulation 3.** Modified problem for finding a minimal set of axial lines  
This modified linear programming problem is examined in a hypothetical plan (Figure 1a) and it prevents isolated axial lines in the axial line reduction like Figure 3. In the circle appearing in Figure 3, all of the lines are connected to other lines, but in Figure 2, the isolated line is not connected to other lines. Adding more constraints into the linear programming problem enables Formulation 3 to find a minimal set of axial lines without an isolated line. In addition, the number of axial lines according to the modified formulation is expected to equal to or greater than that calculated by the original formulation (which could include isolated lines). For example, as shown in Figure 3, the modified optimization problem prevents isolated axial lines and the number of lines is expected to remain the same as previously (\(n=11\)). This output is tested for several cases discussed in other literature, e.g., [1, 6]. Three cases of buildings and two cases of urban complexes are examined with this modified linear programming problem, as illustrated in Figures 4 and 5. These cases have been used to verify whether various methods in previous studies generate appropriate results in built environments [1, 2].  
Figure 4: Computation procedure of the modified problem.  
Figure 3: A minimal set of axial lines that “get everywhere” (\(n=11\), isolated lines suppressed).