## 4 Results
### Tasks Redundancy for Fault Tolerance  
Here we explore the effects of the objective-function term C (2) on the scheduling results. For this, we solved the problem instance consisting of constellations with two satellites, while adding term C in the objective function with a weight of 10. The resulting schedule is presented in Figure 7, along with the energy analysis.  
Notice that Figure 7a,c show each instant \(\beta\) assumed value 1 meaning a specific constellation task started running on more than one satellite simultaneously. Comparing this result with the previous result reported in Figure 3, without C, we can see that now fifteen tasks start running simultaneously, while none were invoked before to run simultaneously. While C is an effective approach to introduce redundancy and fault tolerance in the constellation, its impact on energy dynamics is minimal, as seen in Figure 7b,d, showing a consistent power usage that adheres to constraints (10), very similar to the behavior depicted in Figure 3c,d.  
It is relevant to assess the impact that different values of the C weight can have on the values of the objective function terms and the solving time for a given problem. For this, the problem instance of size 2 is solved using 10 different weight values, ranging from 0 to 18. The results are shown in Figure 8.  
Figure 6: Mission planning for a two-satellite constellation considering the two Pareto-optimal extreme points: maximizing A (tasks 1 to 5) and maximizing B (tasks 6 to 10). **(a)** Scheduling for satellite 1, maximizing A. **(b)** Scheduling for satellite 2, maximizing A. **(c)** Scheduling for satellite 1, maximizing B. **(d)** Scheduling for satellite 2, maximizing B.  
Figure 7: _Cont._It can be observed that as the weight increases, the value of the objective-function term C also increases, but not always proportionally, which would indicate no gains in terms of redundancy--as the term C is not directly proportional to the weight, then the \(\beta\) variable must be increasing. If more weight is given to term C, the solving time, term A, and term B values decrease modestly, remaining practically stable. This indicates that the term C only encourages redundancy, without causing a significant loss in the values that can be extracted from the mission.  
Finally, we study how the introduction of term C impacts on term A, term B and the solve time as the constellation size increases. The same experiment depicted in Figure 4 is repeated here, although now with C and a weight of 10. The results are depicted in Figure 9. We can conclude that A and B change only marginally, while the solution time increases considerably for size 5. Furthermore, as more satellites are admitted, more redundancy is achieved as term C increases although its weight remains constant.  
Figure 8: Collective (A), individual (B) and redundancy (C) objective values, and the respective solving time, for the constellation with two satellites, solved with different weight values for the objective term C.  
Figure 7: Mission planning for the two-satellite constellation, accounting for term C in the objective function. (**a**) Scheduling in satellite 1. (**b**) Energy in satellite 1. (**c**) Scheduling in satellite 2. (**d**) Energy in satellite 2.