## 2 Problem Description
### Mathematical Model  
In this proposed model, it is assumed that there are several hubs at different levels in the studied area, and the level of demand coverage of these hubs needs to be examined. If there are changes in the capacity or location of the hubs that are economically justified, these changes in the location and capacity of the existing hubs are made. Ultimately, a new structure of hubs is presented, where the location and capacity of some hubs have changed, but the demand coverage has increased. In this section, the mathematical model of the research is presented. We seek to examine the efficiency of existing hubs with the goal of achieving a higher level of demand coverage. To this end, in our analysis, we consider the economic viability of the following actions: establishment of hubs, removal of hubs, increase, and decrease in facility capacity. Sets, indices, parameters, and decision variables in this section are introduced. Sets and indices. Objective functions and constraints. Given the sets, indices, parameters, and decision variables presented in the previous section, the objective functions and constraints of the proposed model are provided as follows:  
\[\text{Max}\sum_{i}hiz_{i}^{0} \tag{1}\]  
\[\text{Min}\sum_{i,j,t}F_{j,t}^{l}x_{j,t}^{l}+\sum_{i,j}r_{j}^{l}(\varphi) \tag{2}\]  
\[\text{Max}\sum_{i,j,t}G_{j,t}^{l}x_{j,t}^{l} \tag{3}\]  
\[\text{s.t.}\]  
\[\sum_{j\in N_{b}}a_{i,j}^{l,s}\geq s_{i}^{s}\quad\forall i\in N_{b},b\in B,l\in \{w,h\},s \tag{4}\]  
\[\sum_{j\in F_{r}}\left(a_{i,j}^{c,s}+a_{i,j}^{h,s}\right)\geq s_{i}^{s}\ \ \forall i\in F_{r},r\in R,s \tag{5}\]  
\[\sum_{j\in N_{b}}y_{j,t}^{m}\geq 1\quad\forall b\in B \tag{6}\]  
Figure 1: Different scenarios of demand point coverage. \[\sum_{j\in N_{b},t}y^{c}_{j,t}+\sum_{j\in N_{b},t}y^{h}_{j,t}\leq 1\ \ \ \forall b\in B \tag{7}\] \[\sum_{j\in F_{s},t}y^{h}_{j,t}\leq 1\ \ \forall r\in R\] (8) \[\text{Min}\sum_{i,j,t}F_{j,t}^{l}x_{j,t}^{l}+\sum_{i,j}r_{j}^{l}(\varphi) \tag{2}\]0 (9) \[\text{Min}\sum_{i,j,t}F_{j,t}^{l}x_{j,t}^{l}+\sum_{i,j}r_{j}^{l}(\varphi) \tag{2}\]1 (10) \[\text{Min}\sum_{i,j,t}F_{j,t}^{l}x_{j,t}^{l}+\sum_{i,j}r_{j}^{l}(\varphi) \tag{2}\]2 (11) \[\text{Min}\sum_{i,j,t}F_{j,t}^{l}x_{j,t}^{l}+\sum_{i,j}r_{j}^{l}(\varphi) \tag{2}\]3 (12) \[\text{Min}\sum_{i,j,t}F_{j,t}^{l}x_{j,t}^{l}+\sum_{i,j}r_{j}^{l}(\varphi) \tag{2}\]4 (13) \[\text{Min}\sum_{i,j,t}F_{j,t}^{l}x_{j,t}^{l}+\sum_{i,j}r_{j}^{l}(\varphi) \tag{2}\]5 (14) \[\text{Min}\sum_{i,j,t}F_{j,t}^{l}x_{j,t}^{l}+\sum_{i,j}r_{j}^{l}(\varphi) \tag{2}\]6 (15) \[\text{Min}\sum_{i,j,t}F_{j,t}^{l}x_{j,t}^{l}+\sum_{i,j}r_{j}^{l}(\varphi) \tag{2}\]7  
Objective Function (1) represents the amount of demand covered. Objective Function (2) represents the cost incurred due to changes in the structure of hubs, where a binary variable \(x^{l}_{j,t}\) takes a value of one if a new hub is located or if there is a change in the capacity of a hub or its relocation, and the cost of this change is calculated. In the second part of the equation, the cost of the coverage radius of each hub is calculated as a function of the radius. Objective Function (3) maximizes the level of greenness in the hub location. Each demand point is covered only if it is allocated to at least one hub at level w and one hub at level h in its own region; this condition is considered in Equation (4). According to the model assumptions, a hub at level h can make it unnecessary for a demand point to be allocated to a hub at level c; therefore, in Equation (4), this assumption is applied that allocating a demand point to a hub at level h in one region does not necessarily require assigning that point to a hub at level c.  
Equation (5) expresses the requirement to establish at least one level-w hub in each region. The assumption of the prohibition of simultaneously establishing two hubs at levels h and c in one region is considered in Equation (7). Equation (8) limits the establishment of more than one level-h hub in one region. Equation (9) restricts the establishment of more than one hub at a single point.