# Multi-Criteria Optimization in Operations Scheduling Applying Selected Priority Rules
## 5 Experimental Results  
The performed optimization experiments were of two categories. Firstly, the single-objective optimizations exploring 4096 scenarios for each of the three priority rules were conducted to find the reference points involved in the robust transformation of production objectives (with the aims given in Table 5). All results were obtained under the same conditions, which are described in previous section with respect to design space, objective functions, constraints, and conditions of simulation runs. The column graph in Figure 2 shows a comparison of ideal point values when applying different priority rules. Analogically, Figure 3 depicts a comparison of obtained values of the worst point. All data were presented in Table 6 above. Secondly, the experiments focused on multi-criteria optimization carried out to acquire the optimal daily production results according to given a priori preferences. When applying the WPM method, the minimization of objective function regarding the best and worst values of production performance, introduced in Box 2 and subjected to constraints, was performed. On the contrary, the objective function built via WSM scalarization method presented in Box 1 was maximized. In contrast to the approach in the works of [PERSON] and [PERSON] (2014) or [PERSON] et al. (2017), where the distance of the solution to the ideal point was minimized, we modified the structure of scalar objective function so that the maximization of the distance between solution and the worst point was needed. Figure 3: The comparison of the worst point values with respect to different priority rules. Figure 2: The comparison of the ideal point values with respect to different priority rules. The individual values of five production objectives for obtained optimal solutions in objective space applying WSM and WPM methods linked to three selected scheduling principles are shown in Table 7.  
The graphical comparison of production goals obtained employing WSM and WPM methods for three different priority rules is presented in Figure 4a-e. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Scalarization Method**} & \multirow{2}{*}{**Priority Rule**} & **Optimum of the Objective** & **Number of Products** & **Flow Time** & **Work in** & **Machine** & **Costs Per Part** \\  & & **the Objective** & **Products** & **(min.)** & **Progress** & **Utilization** & **Unit** \\  & & **Function** & **(pcs)** & & **(pcs)** & **(\%)** & **(\%)** \\ \hline \multirow{3}{*}{WSM} & FIFO & 0.826 & 460 & 69.173 & 5.544 & 86.569 & 31.940 \\  & EDD & 0.879 & 472 & 60.377 & 4.453 & 87.923 & 31.577 \\  & STR & 0.883 & 475 & 60.649 & 5.000 & 88.039 & 31.719 \\ \hline \multirow{3}{*}{WPM} & FIFO & 1.042 & 447 & 47.946 & 3.826 & 82.756 & 31.766 \\  & EDD & 1.050 & 452 & 45.360 & 3.004 & 83.590 & 32.043 \\ \cline{1-1}  & STR & 1.014 & 455 & 46.399 & 2.778 & 84.315 & 31.971 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Results of multi-criteria optimization based on scalarization for three priority rules. Figure 4: _Cont._  
Figure 4: The comparison of production goals when applying two multi-criteria methods with different priority rules. (**a**) Number of products; (**b**) machine utilization; (**c**) flow time; (**d**) work in progress; (**e**) costs per part unit. On the basis of obtained production goals as nondominated solutions in objective space presented in Table 7, the four design variables--the times between arrivals for entry of each of four parts P1, P2, P3, and P4 (the inter-arrival times (IATP1--IATP4)) were identified as effective solutions in the design space. They are presented for both methods and three priority rules in Table 8.  
In this work, we address not only solving multi-objective optimization problem in scheduling itself, but also the way, how to evaluate the impact of the applied priority rule to the production results via obtaining the trade-off solution, and comparing it to the best values of many production goals determined by limits of the studied system.