# Facility Layout Problem with Alternative Facility Variants
## 6 Results
### Experiment 3  
Figure 5 clearly demonstrates the ability of the MOEA to efficiently evolve a population of candidate solutions and to converge to a diverse set of high-quality solutions even when the optimization criteria are a mixture of explicit (minCost) and implicit (maxFreeSpace) ones. It shows all non-dominated solutions from the initial and final populations of all independent runs. One can see that, starting from very poor solutions with the initial population, the MOEA converges to much better solutions in the end. In particular, the final non-dominated fronts have shifted both to the right and downwards, thus significantly improving both objectives. However, the median maxFreeSpace value of the final non-dominated solutions still remains well below the maxFreeSpace value obtained with a single-objective EA from the data set \(D_{A}\) (i.e., the value of EA_F_T in Table 1). We hypothesize this can be attributed to the MOEA having to deal with a mixture of explicit and implicit objectives, where the explicit ones are much easier to improve than the implicit ones. Thus, the optimization algorithms need to be further enhanced to cope with this situation efficiently. On the contrary, the vast majority of final solutions lie below the \(cost_{static}\) value of minCost. Many of them are also very good in terms of the maxFreeSpace value.  
We also experimented with different genetic operators, namely the 1-point and 2-point crossovers. Figure 6 shows the final sets of non-dominated solutions obtained with MOEA_T_T using the 1-point and MOEA_T_T using the 2-point crossover. As for the  
Figure 5: Evolution of non-dominated solutions using MOEA_T_T. Initial (red) and final (blue) sets of non-dominated solutions generated in 100 independent runs are shown. The horizontal dashed line indicates the minCost = \(cost_{static}\) value of solutions produced for the data set \(D_{A}\) (i.e., without alternative workstations). The vertical solid line indicates the median maxFreeSpace value obtained with single-objective EA on data set \(D_{A}\).  
median minCost value, the results are almost equal. A bigger difference, in favor of the 2-point crossover, can be observed with respect to the median maxFreeSpace value. The 2-point crossover also outperforms the 1-point one in terms of the hypervolume dominated by the final set of solutions. The median hypervolume dominated by the 2-point crossover solutions is \(2.83\times 10^{9}\), while the median hypervolume of the 1-point crossover solutions is \(2.72\times 10^{9}\). This is a statistically significant difference, as indicated by the \(p\)-value of \(4.3\times 10^{-4}\). This observation clearly shows the importance of using appropriate sampling operators.  
Figure 7 shows illustrative examples of two floor plans evolved in a single run of MOEA_T_T. Several workstations are realized differently in these two floor plans; examples are the three workstations marked 'A', 'B', and 'C'.  
Figure 6: Comparison of MOEA_T_T using 1-point and MOEA_T_T using 2-point crossover. The sets of non-dominated solutions generated in 100 independent runs are shown.  
Figure 7: Examples of two floor plans using different workstation variants: (**a**) a floor plan with maxFreeSpace = 10.010 m\({}^{2}\) and minCost = 261.500. (**b**) a floor plan with maxFreeSpace = 10.620 m\({}^{2}\) and minCost = 243.900. Workstations marked with ‘A’, ‘B’, and ‘C’ are examples of workstations realized differently in the two floor plans.