# EPO Extension of Dispatching Rules to Minimize Effects of Time Uncertainty in Production Scheduling
## 4 Case Study
### Methodology  
To verify the effectiveness of the EPO extension in the considered company, we first need to determine \(PTU_{*,JTD}\) for all the jobs to be planned. Eventually, based on the specificity of the company and the interviews with the employees, we have selected the following formulas:  
1. \(PTU_{E,JTD}\): mean,
2. \(PTU_{P,JTD}\): 1 quartile,
3. \(PTU_{O,JTD}\): 3 quartile.  
The Similarity function, used to compute the \(PTU_{*,I_{TD}}\) values was implemented using decision rules. The use of decision rules enabled us to consider different times horizons (_TH_). Based on analysis of historical data and interviews with employees, the maximum value of _TH_ was set to one year. The analyzes performed were based on the production data that covers jobs starting a year before the start date of the earliest scheduled task. The data was manually verified to detect and correct any potential errors or inconsistencies.  
To analyze the performance of the proposed approach, we decided to use the forward method because it is the closest to the scheduling method currently used in the company considered. To evaluate the quality of the schedules, we use two objective functions: makespan (\(C_{\text{max}}\)) and average completion time (_AC_). Moreover, each operation carried out as part of the production process of a given order will be referred to as a single job. The next job (\(J_{next}\)) to be scheduled at the time \(t\) will be selected using the LPT dispatching rule, which is based on the longest processing time (_LPT_):  
\[LPT_{M}^{t}\gets J_{next}:PT_{next}=\max_{J\in JF_{M}}PT_{J}, \tag{14}\]  
where \(JF_{M}\) is the set jobs that are feasible (admissible) on machine \(M\), and \(PT_{J}\) is the processing time of the job \(J\) (analogously, the set of jobs that are infeasible on machine \(M\) will be denoted as \(JI_{M}\)).  
The LPT rule was chosen to illustrate the proposed approach because it is the best approximation of the actual rule used in the company, which is confidential and cannot be quoted. However, it should be emphasized that the EPO extension is universal and can be implemented with other dispatching rules, as well as the LPT rule. The analysis was performed on jobs with production start dates ranging between January 2022 and August 2022. We have selected nine random non-overlapping samples of jobs to be scheduled. To ensure greater representativeness of the results, the samples covered periods of different lengths: 5 days, 10 days, and 20 days. The analysis of the efficiency of the proposed EPO extension was performed in the following steps:  
1. Schedule jobs from a given sample using both the LPT dispatching rule and the EPO extension that integrates the LPT dispatching rule (14), called EPO.
2. Compute the actual objective functions for the schedules obtained from LPT and EPO by replacing the theoretical processing times with the actual processing times.
3. Compare the objective function values from the first step and the second step. The pseudo-code of the EPO-forward algorithm is presented in Algorithm 1.  
The above-described three-step procedure was used to perform computations on real data coming from the considered company. The obtained results are presented and discussed in the next section.