# Multi-Criteria Optimization in Operations Scheduling Applying Selected Priority Rules
## 3 Multi-Criteria Optimization in Scheduling
### Applied Multi-Criteria Optimization Methods  
In this study, the utility approach was employed to compare production system performance under different priority rules via WSM and Weighted Product Method (WPM).  
WSM belongs to the most common scalarization procedure in the literature. It is a special type of global criterion method, expressed by the weighted sum of \(k\) individual objectives \(F_{i}\) (3):  
\[U=\sum_{i=1}^{k}w_{i}.F_{i};w_{i}>0\,;\,\sum_{i=1}^{k}w_{i}=1 \tag{3}\]  
where weights \(w_{i}\) as method parameters are used to model preferences. Assuming the vector \(\mathbf{w}\) is fixed, then minimizing \(U\) function (3) is a necessary and sufficient condition for Pareto optimality under convex constraint conditions ([PERSON]'s theorem [39]). This method suffers from a well-known disadvantage related to the geometric point of view. Since the scalar MOO function is generated as a linear combination of individual objective functions, the ability to yield the Pareto optimal solutions is restricted to the convex part of Pareto front only. Despite it, this method is rather popular in the application in a job shop scheduling area. To mention works that solved multicriteria problems applying WSM or via sum approach, e.g., [PERSON] et al. [14] published an extensive study of model types of production systems with seven objective functions studied alone or in different mutual combinations, and compared to the results in the literature. To perform multicriteria optimization, they used the multi-objective simulated annealing procedure for solving a given combinatorial optimization problem. [PERSON] et al. [9] studied dynamic job shop scheduling problems considering makespan and tardiness simultaneously when employing a variable neighborhood search method and ANN for finding optimal rescheduling. The work of [PERSON] and [PERSON] [41] addressed a multi-objective job-shop scheduling problem with equal-size lot splitting. It was solved via hybrid ant colony optimization-based algorithms. In the paper, the authors minimized scalar objective function generated in the form of the weighted sum of makespan, tardiness, and lot-splitting cost with an accent to issue related to lot-splitting of jobs and tradeoff between lot-splitting costs and makespan. To compare the effect of the first employed multi-objective method, the Weighted Product Method (WPM) was selected. The utility function \(U\) based on the multiplication of \(k\) objectives \(F_{i}\) was expressed in the form (4).  
\[U=\prod_{i=1}^{k}F_{i}^{w_{i}};w_{i}>0\,;\,\sum_{i=1}^{k}w_{i}=1 \tag{4}\]  
Weights \(w_{i}\) represent the relative importance of the individual objectives [24]. WPM is known mostly as a component of decision making methods in the Multiple Criteria Decision Making (MCDM) domain [42; 43; 44]. It supposed that the potential nonlinearities in the utility function and resulting computational issues can be the reason for lack of extensive application of this method [24]. According to [PERSON] [45], a major weakness of the method is that it overestimates the extremes leading to undesirable results, as it significantly favors/disfavors the final evaluation of any alternative that is far from the average in relation to one criterion.