## 2 Background  
The facility location problem is a combinatorial optimization problem pertaining to operations, distribution, and logistics. The problem involves finding the optimal placement of facilities in a set of demand centers to minimize the total cost. The demand centers represent locations with varying degrees of demand for a service, and facilities must be placed at these locations to service the demand. The cost of operating a facility depends on the distance between the facility and its corresponding demand center, as well as the fixed cost of operating the facility. A solution to the FLP is a set of facility locations that minimize the total cost, where each demand center can have at most one facility within it.  
Examples of the FLP can be found in various real-world scenarios. For example, in the context of city management, the FLP can arise in the placement of fire department facilities, where they must consider the costs of maintaining equipment, firefighters, and the response time. In this case, the demand centers can be the neighborhoods within the city, each with a different demand value corresponding to the neighborhood's population. Another example is the arrangement of networked sensors in a space to minimize the number of sensors required while maximizing the coverage area [12; 13].  
Formally, the FLP can be defined as three distinct multi-objective optimization problems [7], where:  
\(D\): Set of all demand centers;  
\(F\): Set of all candidate facilities;  
\(G\): Set of open facilities;  
\(W\): Set of open facilities that did not fail;_P(W)_: Power set of W;  
\(f_{i}\): Cost of opening a facility at location \(i\);  
\(c_{ij}\): Cost of assigning demand center \(j\) to facility \(i\);  
\(y_{i}\): Binary decision variable that indicates whether facility \(i\) is open;  
\(x_{ij}\): Binary decision variable that indicates whether location \(j\) is assigned to facility \(i\);  
\(u_{ij}\): Binary decision variable that indicates whether location \(j\) is assigned to facility \(i\) after failures have occurred;  
\(v_{i}\): Binary decision variable that indicates whether open facility \(i\) has failed.  
1. First subproblem: 1. \(\underset{i\in F}{minimize_{y}}\sum\limits_{i\in F}f_{i}y_{i}\) 2. \(\underset{i\in F}{minimize_{x}}\sum\limits_{i\in F}\sum\limits_{j\in D}d_{j}c_ {ij}x_{ij}\). The first objective function minimizes the cost of opening facilities, and the second objective function minimizes the total distance without failures. These two objectives allow decision makers to understand the impact of one objective on another (i.e., how opening more facilities reduces the total distance).
2. Second subproblem: 1. \(\underset{i\in F}{minimize_{v}}\sum\limits_{m\in G}f_{m}v_{m}\); 2. \(\underset{i\in W}{maximize_{u}}\sum\limits_{k\in W}\sum\limits_{j\in D}d_{j}c_ {kj}u_{kj}\); 3. subject to: 1. \(\underset{j\in W}{y_{m}}-v_{m}\leq k\); 2. \(\underset{v_{m}\in\{0,1\}}\forall m\in G\); 3. \(u_{kj}\leq x_{kj}\forall k\in W,j\in D\); 4. \(\underset{k\in W}{\sum}u_{kj}=1,\forall j\in D\). The first objective function minimizes the number of facilities that fail. The second objective function maximizes the distance after failures. Constraint \(i\) ensures that only open facilities can fail. Constraint \(ii\) ensures that \(v\) is a binary decision variable. Constraint \(iii\) ensures that demand centers are reassigned to open facilities, and constraint \(iv\) ensures that demand centers are reassigned.
3. Third subproblem: 1. \(\underset{i\in F}{minimize_{x}}\sum\limits_{i\in F}\sum\limits_{j\in D}d_{j}c_ {ij}x_{ij}\); 2. \(\underset{i\in W}{minimize_{u}}\sum\limits_{k\in W}\sum\limits_{j\in D}d_{j}c_ {kj}u_{kj}\). The first objective function minimizes the distance before failure and the second objective function minimizes the distance after failure.