## 6 Observability of the graph
### Binary-Integer Programming Model Considering an Unlimited Number of Channel Capacities
The focus is to adopt the BBA, which is utilized to solve relaxed programming problems to achieve bounds on the objective function value of the original MILP model within a binary domain [13; 14; 15; 16; 17; 18]. The BBA solves LP-relaxed problems where the binary restriction on the decision variables is substituted by the relaxed restriction \(\overrightarrow{0}\leq\overrightarrow{x}\leq\overrightarrow{1}\)[13; 14; 15; 16; 17; 18]. The next steps of the algorithms are to implement and solve the linear relaxed problems [37; 38; 41; 51]. If the algorithm's output is integer, then the solution is achieved [13; 14; 15; 16; 17; 18]. In a different outcome, two sub-problems are constructed by utilizing a branch methodology on a fractional variable [37; 38; 41; 51]. The gap between the upper and lower bounds is inside a pre-given tolerance [13; 14; 15; 16; 37; 38; 41; 51]. The upper bound is the incumbent cost value, and the current lower bound is the minimum quantity of the lower bounds produced by finding a solution to the candidate sub-problems [13; 14; 15; 16; 37; 38; 41; 51]. To run this optimization process, MILP solvers can be called to achieve an accurate solution point within a range given by the users. **Definition 4**. : _The optimality gap is stated as follows:_  
\[Gap=\mid upper-lower\mid<\in,\ \in>0 \tag{10}\]  
**Definition 5**. : _Definition of the percentage relative gap:_  
\[\mbox{percentage relative gap}=100\times\mid\mbox{Upper}-\mbox{Lower}\mid/ \left(1+\mid\mbox{Upper}\mid\right) \tag{11}\]The termination criterion equal to the optimality gap \(Gap=0\) means that a global solution has been reached [13; 14; 15; 37; 38; 41; 51]. The incumbent solution is the best solution delivered by the repetitive strategy in the BBA search tree [13; 14; 15; 16]. The tolerance gap is a need for a successful termination of the BBA [13; 14; 15; 16; 37; 38; 41; 51]. The optimization procedure leads to a \(\epsilon\)-optimality criterion [13; 14; 15; 16]. The relative absolute gap is a criterion to measure how close the solution is to being global [13; 14; 15; 16]. The steps of the BBA implementation are outlined in Algorithm 1. ```
1. Calculate the lower bound: A procedure is used to achieve the lower bound on the objective function to solve the given sub-problem and deliver an estimated point. 2. Calculate the upper Bound: An upper bound is evaluated to give the solution. 3. Branching method: The sub-problem is portioned to deliver more than two children by a procedure strategy
4. Methodology search: A search order is programmed to construct the binary tree. 5. If the stopping criterion is satisfied, then end up the iterative scheme. 6. The BBA ends delivering a solution as an output. ```  
**Algorithm 1**Steps of BBA methodology  
The \(0-1\) integer program can be solved utilizing genetic algorithms and binary particle swarm optimization [18; 51]. They follow a probabilistic search of the feasible region due to a random selection of the initial population, searching and finding an optimum solution [25; 26; 51]. Strategies such as special creation, crossover, and mutation functions are utilized to enforce the variables to be declared in the integer problem solving [72]. GA tries to minimize a penalty function, not necessarily the fitness one, and a penalty term is added to consider any infeasibility while the feasible region is explored [25; 26; 27; 50; 51]. The penalty function is performed with a binary tournament choice to find individuals for subsequent generations [25; 26; 50; 51]. If the member is feasible, the penalty function value of a member included in the population is the fitness function [51]. Otherwise, the objective function counts the maximum fitness function among feasible members of the population, plus a sum of the constraint violations in case of an infeasible point if the member is found to be infeasible [72]. The steps of the GA code are outlined in Algorithm 2.