# Linear Project-Scheduling Optimization Considering a Reverse Construction Scenario
## 3 Development of the LSM-CBPS Model
### Constraints
#### 3.5.2 Spatiotemporal Constraints in Reverse Construction Scenarios  
To address the practical scenarios of reverse construction in linear project construction and explore its impact for scheduling optimization in terms of project duration, this section describes the 48 types of spatiotemporal logic constraints involved in reverse construction scenarios.  
(1) Time constraints  
1) If a minimum time constraint exists between subactivity \(j_{l}\) and subactivity \(i_{k}\), scenarios are categorized and discussed based on activity type, construction direction, construction rate, and activity mileage. The minimum time constraint between \(j_{l}\) and \(i_{k}\) is subdivided into the following 12 cases, which are shown in Figure 3:  
\[\text{Case 1}:\;st_{i,k}+T_{i,k,SD_{j,l},ED_{i,k}}+Tmin_{i,j}\leq st_{j,l}; \tag{5}\]  
\[\text{Case 2}:\;st_{i,k}+T_{i,k,SD_{i,k},ED_{i,k}}+Tmin_{i,j}\leq st_{j,l}+T_{ j,l,SD_{j,l},SD_{i,k}}; \tag{6}\]  
\[\text{Case 3}:\;st_{i,k}+T_{i,k,SD_{i,k},ED_{i,k}}+Tmin_{i,j}\leq st_{j,l}+T_{ j,l,ED_{i,k},ED_{j,l}}; \tag{7}\]  
Figure 2: Illustration of the problem description.  
Figure 3: Minimum time buffer constraints between the activities under reverse direction.  
Figure 4: Maximum time buffer constraints between activities under reverse direction.  
2. Distance constraints 1) If a minimum distance constraint exists between subactivity \(j_{l}\) and subactivity \(i_{k}\), scenarios are categorized and discussed based on activity type, construction direction, construction rate, and activity mileage. The minimum distance constraint between \(j_{l}\) and \(i_{k}\) is subdivided into the following 12 cases, which are shown in Figure 5. \[\text{Case 1}:\ SD_{i,k}+D_{i,k,st_{i,j},e_{i,k}}+Dmin_{i,j}\leq SD_{j,l};\] (29) \[\text{Case 2}:\ SD_{i,k}+D_{i,k,st_{i,k},e_{i,k}}+Dmin_{i,j}\leq SD_{j,l}+D_{j, l,st_{i,j},st_{i,k}};\] (30) \[\text{Case 3}:\ SD_{i,k}+D_{i,k,st_{i,k},e_{i,k}}+Dmin_{i,j}\leq SD_{j,l}+D_{j, l,d_{i,k},e_{i,j}};\] (31) \[\text{Case 4}:\ SD_{i,k}+D_{i,k,st_{i,k},e_{i,l}}+Dmin_{i,j}\leq SD_{j,l};\] (32) \[\text{Case 5}:\ SD_{i,k}+Dmin_{i,j}\leq SD_{j,l}+D_{j,l,e_{i,k},e_{j,j}};\] (33) \[\text{Case 6}:\ SD_{i,k}+D_{i,k,e_{i,j},e_{i,k}}+Dmin_{i,j}\leq SD_{j,l};\] (34) \[\text{Case 7}:\ SD_{i,k}+D_{i,k,st_{i,j},e_{i,k}}+Dmin_{i,j}\leq SD_{j,l}+D_{j, l,st_{i,j},e_{i,j}};\] (35) \[\text{Case 8}:\ SD_{i,k}+D_{i,k,st_{i,k},e_{i,k}}+Dmin_{i,j}\leq SD_{j,l}+D_{j,l,st_{i,k},e_{i,j}};\] (36) \[\text{Case 9}:\ SD_{i,k}+D_{i,k,st_{i,j},e_{i,k}}+Dmin_{i,j}\leq SD_{j,l};\] (37) \[\text{Case 10}:\ SD_{i,k}+D_{i,k,st_{i,k},e_{i,k}}+Dmin_{i,j}\leq SD_{j,l};\] (38) \[\text{Case 11}:\ ED_{i,1}+Dmin_{i,j}\leq SD_{j,l}+D_{j,l,d_{i,1},e_{i,j}};\] (39) \[\text{Case 12}:\ ED_{i,1}+Dmin_{i,j}\leq SD_{j,l}.\] (40)  
2) Similarly, the maximum distance constraint between \(j_{l}\) and \(i_{k}\) is subdivided into the following 12 cases, which are shown in Figure 6. \[\text{Case 1}:\ SD_{i,k}+Dmax_{i,j}\geq SD_{j,l}+D_{j,l,st_{i,j},e_{i,k}};\] (41) \[\text{Case 2}:\ SD_{i,k}+D_{i,k,e_{i,j},e_{i,k}}+Dmax_{i,j}\geq SD_{j,l}+D_{j, l,st_{i,j},e_{i,j}};\] (42) \[\text{Case 3}:\ SD_{i,k}+D_{i,k,st_{i,k},e_{i,l}}+Dmax_{i,j}\geq SD_{j,l}+D_{j,l,st_{i,j},e_{i,j}};\] (43) \[\text{Case 4}:\ SD_{i,k}+Dmax_{i,j}\geq SD_{j,l}+D_{j,l,st_{i,k},e_{ i,j}};\] (44) \[\text{Case 5}:\ SD_{i,k}+D_{i,k,st_{i,j},e_{i,k}}+Dmax_{i,j}\geq SD_{j,l}+D_{j,l,st_{i,j},e_{i,j}};\] (45) \[\text{Case 6}:\ SD_{i,k}+D_{i,k,st_{i,k},e_{i,k}}+Dmax_{i,j}\geq SD_{j,l}+D_{j,l,st_{i,k},e_{i,j}};\] (46) \[\text{Case 7}:\ SD_{i,k}+Dmax_{i,j}\geq SD_{j,l}+D_{j,l,e_{i,k},e_{ i,j}};\] (47)\[\text{Case 8}:\ SD_{i,k}+D_{i,k,et_{j,l},et_{i,k}}+Dmax_{i,j}\geq SD_{j,l}; \tag{48}\]  
\[\text{Case 9}:\ SD_{i,k}+Dmax_{i,j}\geq SD_{j,1}; \tag{49}\]  
\[\text{Case 10}:\ SD_{i,k}+D_{i,k,et_{j,l},et_{i,k}}+Dmax_{i,j}\geq SD_{j,1}; \tag{50}\]  
\[\text{Case 11}:\ ED_{i,1}+Dmax_{i,j}\geq SD_{j,l}+D_{j,l,st_{j,l},et_{j,l}}; \tag{51}\]  
\[\text{Case 12}:\ ED_{i,1}+Dmax_{i,j}\geq SD_{j,l}+D_{j,l,st_{i,k},et_{j,l}}. \tag{52}\]