## 4 Solution Procedure
### Sub-Gradiant Method
If the number of located facilities is less, then the number of the covered demands and the assigned modules would be less. These values would increase by augmenting the number of located facilities as shown in Table 3. On the other hand, for the modules assigned to the facilities, by having more located facilities the model assigns more modules to the facilities and as a result, more demand points would be covered. Figure 3 illustrates the values for the objective function, the number of the assigned modules and the covered demand points. #### 5.1.4 Sensitivity Analysis for the Cost of Module Assignment

Another important parameter that influences the results is the cost of the modules assignment to the facilities. Table 4 shows the effects of changing the module assignment cost. According to the results, as the cost of module assignment is increasing, while other parameters are fixed, fewer modules would be decided to allocate to the facilities and relatively the objective function values decrease. Having a fewer number of modules, the number of covered demand points also decreases. Figure 4, depicts the results of Table 4. These results validate the correctness of the proposed model. Figure 3: Results of the sensitivity analysis for the number of the facilities. \begin{table}
\begin{tabular}{c c c c c c c} \hline \# & \(\beta_{i}\) & \(h_{iik,t}\) & P & Obj & Y & X \\ \hline \(8\) & (60,80) & 20,40 & 2 & 52.9 & 7 & 88.9 \\ \(9\) & (60,80) & 20,40 & 3 & 76.2 & 10 & 124.2 \\ \(10\) & (60,80) & 20,40 & 4 & 94.8 & 11 & 133.3 \\ \hline \end{tabular}
\end{table} 
Table 3: Computational results to observe the effect of the increasing number of located facilities. Figure 2: Results of the sensitivity analysis for the capacity of the facilities. ### Comparison with the Conventional Methods

In this section, some numerical examples are generated and tested to analyze the model performance as well as the solution approaches. The coordination of the points in the test problems is randomly generated using a uniform distribution between \([0,1000]\). Demands of points are randomly generated, using a uniform distribution between \([1,5]\). It is assumed that modules 1 and 4 have three sizes while there are two sizes for modules 2 and 3. Costs of establishing modules are randomly generated using a uniform distribution between \([40,60]\). The parameter \(\alpha_{j}\) is fixed to 1 for all points. The parameter \(p_{ij}\) is considered as

\[g_{ij}=\left\{ \begin{array}{rl}1\;if\;d_{ij}\leq 300\\ 0.75\;if\;300<d_{ij}\leq 400\\ 0.5\;if\;400<d_{ij}\leq 500\\ 0.25\;if\;500<d_{ij}\leq 600\\ 0\;otherwise\end{array} \right.\]

The problems are solved using GAMS (CPLEX solver) software (25.1) on a PC with a 1.6-GHz Core i5-8250U CPU and 8 GB of RAM running Windows 10 (64 bit). Table 5 contains the information for the rest of the parameters. Furthermore, parameter \(\rho=0.4\) showed better performances for the test problems. Problem instances 1-8 are categorized as small size problems, based on the capability of CPLEX in solving these problems. Problem instances 9-20 with demand points more than 300 and candidate location of facilities more than 50, belong to the large-scale problems category in this study, for which CPLEX could not solve these problems and only bounds from different approaches are available for these problems. The maximum iteration to stop all algorithms is set for 40 iterations for large scale problem. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \# & \(\beta_{i}\) & \(h_{i,k,t}\) & **P** & **Obj** & **Y** & **X** \\ \hline
11 & (60,80) & 10,30 & 3 & 191.3 & 12 & 146.2 \\
12 & (60,80) & 20,40 & 3 & 85.81 & 11 & 136.2 \\
13 & (60,80) & 30,50 & 3 & 13.5 & 3 & 32.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Computational results for the effect of the increasing module assignment cost. Figure 4: Results of the sensitivity analysis for the module assignment cost.