## 7 Conclusions  
The modified linear programming problem discussed in Section 5.2 shows a valid algorithm for finding a minimal set of axial lines that represent a given geometry. This paper devises an algorithm for finding a minimal set of axial lines that represent a given building plan. A linear programming problem was formulated based on the algorithm, then a linear programming software package was used to solve this problem. In this paper, the algorithm that uses a linear programming solver provides promising results for developing an axial line set for the spatial analysis of built environments. In addition, according to the examination of the linear programming problem in this paper, it is not necessary to use s-lines to find a minimal set of axial lines. This paper's method can use user-supplied lines or any other type of lines (believed to be) representing the entire space of interest. Given a set of lines of any type that represents the space, we can find a minimal subset of the given set representing the same space by using the linear programming method. The most significant contribution is the provision of a novel method for automatically generating axial maps of space syntax to enable the spatial analysis and evaluation of building spaces. The developed method will have the advantage of consistent results, fast speed, large capacity and error immunity in analyzing large sets of data. Thus, this method will allow people to effectively generate axial line maps for analyzing built environments. Integrating the algorithm presented in this study into other software tools for spatial analysis is highly feasible; many linear programming solvers, free or commercial, are available. The second contribution of this paper is that it contributes to improving the quality of built environments by enabling the evaluation and analysis of spatial quality in existing or proposed construction projects. Scholars and professionals will be able to use the developed method to appropriately examine various spatial aspects in the given circumstances of built environments. With such an assessment of spatial qualities, appropriate alternatives can be proposed. However, it is desired that every inner wall be enclosed by axial lines in order to represent the topology of the spatial configuration. While [PERSON] et al. [2] demonstrated that a circulation loop connection occurred in spatial configuration, the formulations in this paper leave this circulation loop connection requirement unconsidered. Therefore, some inner walls can be unenclosed by a minimal set of axial lines. Finding a set of axial lines enclosing every inner wall can follow the second step of [PERSON] et al. [2] once a minimal set is found by using the proposed methods in this paper. This proposed method guarantees a minimal set of axial lines representing an entire geometry of building and urban layout. For the same reason, preventing isolated groups of axial lines is unconsidered. In future work, dynamic linear programming will be sought to keep axial lines connected and incorporate \"inner wall enclosure\" constraints. This paper describes the initial attempt to find a minimal set of axial lines using the optimization problem. Unlike a greedy backward elimination and a sub-set elimination, the optimization problem simultaneously examines the entire space of a given plan. This paper achieves the first step in finding a minimal set of axial lines, each of which has at least one connection to the others. Developing the approach using the linear programming method, the authors are planning to formulate a more devised problem for solving the challenges in this paper. In future research, while non-dynamic linear programming solvers are incapable of solving the circulation loop problem, it is possible that a branch-and-bound algorithm, which is one of the methods for solving an integer linear programming problem, will be devised to solve the problem.