# Linear Project-Scheduling Optimization Considering a Reverse Construction Scenario
## 4 Design of a Solving Algorithm for the LSM-CBPS Model
### Improved Grey Wolf Optimization Algorithm Design
2) When \(ED_{j,l}-ED_{i,k}>ED_{i,k}-SD_{j,l}\), \(i_{k}\) selects forward direction and \(j_{l}\) selects reverse direction, \(j_{l}\) achieves an earlier start time compared with when both constructions are forward. 3) When \((SD_{j,l}-ED_{i,k})/pr_{i,k}>(ED_{j,k}-ED_{j,l})/pr_{j,l}\), \(i_{k}\) selects forward construction and \(j_{l}\) selects reverse construction, \(j_{l}\) achieves an earlier start time compared with when both constructions are forward. We may not be able to discuss all the scenarios here, but these interesting findings are meaningful for on-site scheduling. Moreover, by eliminating the ideas provided in this section, more situations and conclusions can be further analyzed and obtained. ## 6 Conclusions

In summary, this study focused on scheduling optimization in construction projects under the LSM framework and made the following contributions. 1. For linear scheduling optimization, scenarios with reverse construction activities were considered. A systematic spatiotemporal logical constraint system consisting of 48 scenarios was proposed to fully describe the temporal and spatial logical relations faced by various reverse construction scenarios in linear project construction. To the best of our knowledge, this is the first study on the spatiotemporal logical relationship description method and constraint portrayal system for reverse construction activities under LSM. The proposed constraint system may serve as a general foundation for future research in this field;
2. An LSM-based scheduling optimization model named LSM-CBPS was proposed. This model is compatible with the 48 types of reverse construction spatiotemporal logic constraints mentioned above, as well as other classic constraints such as crew and mode constraints, and achieves the flexible matching ability of multisection-multicrew, multicrew-multimode, and multicrew-multiconstruction directions. These practical scenario-oriented designs enhance the practicality of the model;
3. The application of the GWO algorithm in scheduling optimization based on the LSM framework was explored. An improved GWO algorithm with four-layer coding rules, discrete position update operators, dynamic constraint processing, cross-mutation operators, and heuristic rules was designed. This algorithm demonstrated superior performance compared with other algorithms, i.e., original GWO, GA, and PSO, in terms of solution quality, stability, and efficiency. To the best of our knowledge, this is the first application of GWO-based algorithm in linear scheduling optimization. The effectiveness of the proposed model and algorithm was verified through a practical railway construction project, thereby highlighting its advantages in handling large-scale optimization problems and generating construction schedules with optimized durations. However, this study still has some limitations. It considers only the optimization of construction duration but does not address other objectives such as resource leveling. Multi objective optimization has received widespread attention in various fields [59], however, considering the characteristics of the GWO algorithm, this study focuses on single objective optimization and does not address the impact of reverse construction activities scenarios on multiple objectives synchronous optimization. Furthermore, the construction process of the construction crew is simplified, thereby neglecting various uncertainties that occur in real construction scenarios. Future research should address these limitations and explore additional factors to further improve construction scheduling optimization. **Author Contributions:** Conceptualization, Y.T. and Z.Y. ; methodology, Y.T. and Z.Y. ; validation, Z.Y., C.W. and Z.H. ; formal analysis, Z.Y., Y.Z. and Z.H. ; data curation, C.W. and Y.Z. ; writing--original draft preparation, Z.Y. ; writing--review and editing, Z.Y. and C.W. ; visualization, Z.Y. and Y.Z. ; supervision, Y.T. ; project administration, Y.T. ; funding acquisition, Y.T.