*4.5. Experimental Results*

The first of the verification objectives is to compare the nominal and robust production schedules in terms of evaluation criteria. The values of the evaluation indicators of the schedules have been determined and are summarized below (Table 7).


**Table 7.** Evaluation criteria in the nominal and robust schedules.

From the presented data, it can be seen that the implementation of service time buffers increases the completion times of all jobs. As a consequence, in each of the analyzed cases, one additional shift is required to complete the production process. This effect is not at all unexpected, given that incorporating service time buffers is inseparably connected with elongation. It should be noted, however, that in the robust schedule the time spent in the production system is not extended owing to the fact that the mean flow time is subject to slight elongation.

Figure 2 presents the visual interpretation of the nominal and robust SPT schedules. Service time buffers are represented by crossed white blocks.

**Figure 2.** Schedules obtained using the SPT (shortest processing time) rule: (**a**) nominal, (**b**) robust.

A further indicator of robustness that is of great importance in scheduling is the number of critical operations. Scheduling should minimize its value because the stability of the executed process is compromised with the rising number of critical operations. In the analyzed example, the number of critical operations is considered in relation to individual jobs (*YKJ*) and machines (*YKM*). The robust scheduling results with respect to the number of critical operations are presented in Table 8.


The incorporation of service time buffers is shown to have a positive effect on the considered parameters. The number of critical operations is reduced by up to 20%. This confirms the legitimacy of implementing service time buffers, which generate additional space in the schedule and thus can prove to be beneficial in the event of machinery failure or other process disruptions.

Simulation tests are conducted to indicate which of the schedules features a makespan closer to the simulated completion time of all jobs. The tests follow the procedure presented in the preceding section and their results are given below, in Tables 9 and 10.


**Table 9.** Results from simulation: nominal and robust schedules (LPT—longest processing time).

The schedules generated with the LPT and SPT dispatching rules are shown to outperform the nominal schedule. Their accuracy of predictions is closer to the production data established in simulations. At a closer investigation, the LPT schedules (Table 9) exhibit good compliance of robust and simulated makespans. The schedule is robust for an average of 1.56 h longer than the simulated process; however, considering the nominal schedule, the completion time of all jobs is on average 4.65 h shorter. The comparable makespan length of the robust schedule and the executed production schedule is further confirmed by the mean value of indicator *ECmax* , which amounts to 1.03 for the robust schedule, and 0.91 for the nominal schedule.

A similarity of a comparable magnitude is also shown to occur in the production process simulations conducted according to the SPT schedules (Table 10). The mean makespan of the nominal schedule is −5.75 h, while of the robust schedule 2.67 h. In the same case, the mean relative increase is 0.89 for the nominal schedule and 1.05 for the robust schedule.

Nevertheless, it should be noted that in several simulations (for both the LPT and SPT rules), the nominal schedules display a closer resemblance to the executed production process; still, the robust and the executed schedules also show a good fit (e.g., simulation 2 for the LPT rule, or simulation 18 for the SPT rule).

To summarize, the data obtained in the study clearly indicate that the schedule with service time buffers achieves a closer resemblance to the simulated makespan.


**Table 10.** Results from simulation: nominal and robust schedules (SPT).

Figures 3 and 4 display the results for makespan increase indicators, which provide further evidence confirming the legitimacy of our solutions. The proximity of the robust schedules to the simulated schedules is again highlighted by their being situated close to the dashed line.

**Figure 3.** Increase of makespan in LPT and SPT schedules.

**Figure 4.** Relative increase of makespan in LPT and SPT schedules.

Robust production schedules generated with the application of our solutions determine makespans closer to the simulated completion times of all jobs in the simulated production conditions under technological machinery failure uncertainty. This is evidenced by several indications, e.g., the fact that for the robust schedules, the values of Δ*C*max tend to be close to 0 (Figure 3), whereas in the case of *ECmax* around 1 (Figure 4). Values 0 and 1 of the considered indicators denote igh compliance of the robust schedule with the production execution (simulation).

#### **5. Summary and Conclusions**

The execution of production processes is associated with the occurrence of various uncertainty factors. Disruptions generate problems that may have a marked effect on production schedules. Therefore, more effort is required in developing techniques and methods that affirm the relevance of uncertainty factors in manufacturing and propose viable solutions. Robust scheduling exhibits the required potential to cope with disruptions and, thus, should be studied further.

In this investigation, the aim was to design a robust production scheduling method with the implementation of Markov chain theory and ARIMA models that will provide for the negative effects of technological machinery failure. The analyses reveal that the inclusion of machine failure in the production schedule results in the extension of the performance indicators, mean flow time, mean job completion time, as well as the central criterion describing the performance of the production system—the completion time of all jobs (makespan). However, the elongation remains within the reasonable limits given that the production is carried out according to failure-inclusive schedules. The simulations evidence that the robust schedules bear a closer similarity to the simulated production process than their nominal equivalents. In other words, the proposed model generates high-accuracy makespan while increasing the robustness and stability of the schedule.

To extend our research in the future, we intend to develop improved models that will: provide for the management of other uncertainty factors in production scheduling (e.g., disruptions related to transport, availability of materials or employee absence), enable reactive scheduling of production jobs or extend the versatility of the proposed solutions over other manufacturing systems. Our current findings and methodologies should make a noteworthy contribution to the theory of production scheduling, as well as appeal to practitioners representing various manufacturing industries and different-sized enterprises.

**Author Contributions:** A.G. gave the theoretical and substantive background for the developed solution and conceived and designed the experiments, E.K. prepared and provided mathematical description of the method, Ł.S. prepared conception of proposed method and conducted experimental verification of the solution. All authors have read and agreed to the published version of the manuscript.

**Funding:** The project/research was financed from the Lublin University of Technology Project—Regional Initiative of Excellence from the funds of the Ministry of Science and Higher Education on the basis of a contract No. 030/RID/2018/19.

**Conflicts of Interest:** The authors declare no conflict of interest.
