# Optimization of the Weighted Multi-Facility Location Problem Using MS Excel
## 3 Performance Evaluation
### Benchmark Instances  
The MFLP-W task calculations will be performed on three benchmarks. The topology of points for the artificial benchmark A (Figure 1) are designed and weighted in a special way, so that it is possible to calculate the optimal solution as the reference value. Benchmark B is a combination of twenty randomly generated groups of five points, their position in the area is designed so that there is no mutual influence (Figure 2). The optimal center positions cannot be directly calculated. For benchmark B, the reference value was obtained in three ways:  
* Each group is solved separately by an evolutionary method in Excel as a Single Facility Location Problem (SFLP) task. The sum of the resulting values is then the solution of the MFLP problem with this topology (best result 360,186.787);
* Computed in MATLAB--the function _fmincon_ from the \"optimization toolbox\" is used to find a minimum of nonlinear multivariable functions, for more details see example [23] or [24] (best result 360,186.79);
* The lowest value of 300 calculations by the metaheuristic method on three hardware configurations (the best result 360,186.7863)--as this method provided the best solution, it serves as a reference value. \begin{table}
\begin{tabular}{c c} \hline \hline
**Name of Term** & **Explanation** \\ \hline Reference & Optimal or reference solution for an instance. \\ Minimal & Minimal value from set of 100 results \\ Average & Average value from set of 100 results \\ Dif. ref-min & Difference between reference and minimal value (\%) \\ Dif. ref-avg & Difference between reference and average value (\%) \\ Standard dev. & Standard deviation for set of 100 results \\ Coefficient of variation & Relative standard deviation for set of 100 results \\ Avg. time & Average time of computing one result in set of 100 results \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Definitions of terms used. \begin{table}
\begin{tabular}{c c} \hline \hline
**Name of Acronym** & **Explanation** \\ \hline EHW1 & Results of operations with evolutionary method on HW1 \\ EHW2 & Results of operations with evolutionary method on HW2 \\ EHW3 & Results of operations with evolutionary method on HW3 \\ MHW1 & Results of operations with metaheuristic algorithm on HW1 \\ MHW2 & Results of operations with metaheuristic algorithm on HW2 \\ MHW3 & Results of operations with metaheuristic algorithm on HW3 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Definitions of abbreviations used. For benchmark C, values of positions of points are generated by the Excel _RANDBE-TWEEN_ function in the range 0 to 45,000 for \(x\) coordinate, range 0 to 60,000 for \(y\) coordinate, for weight in range 1 to 25 (Figure 3). For benchmark C, the reference solution is calculated using the metaheuristic method: the lowest value of 300 calculations on three HW configurations is used. Table 4 shows the reference values for the benchmark instances. Figure 1: Benchmark instance A.  
Figure 2: Benchmark instance B. For each benchmark task, 100 optimizations were performed both in the Excel-Solver software (see Section 2.1 for details) as well as using the metaheuristic algorithm (see Section 2.2 for details). Each task is solved in six ways (thus, in total, 1800 optimizations trials are performed):  
* by Excel-Solver evolutionary method on HW1;
* by Excel-Solver evolutionary method on HW2;
* by Excel-Solver evolutionary method on HW3;
* by metaheuristic algorithm on HW1;
* by metaheuristic algorithm on HW2;  
For each task, method and hardware configuration, the following parameters and outputs calculated from all individual solutions are recorded:  
* minimal solution;
* average solution;
* standard deviation;
* coefficient of variation;
* difference between the reference and the minimal solutions in percent;
* difference between the reference and the average solutions in percent;
* histogram, classes are from 1 percent to 10 percent deviation from reference;
* graph with coordinates for result for the reference and the minimal solution.