4. Statistical analysis of scheduling results

Figure 7 shows the makespan *Cmax* of schemes 5–8 running 10 times separately. For scheme 7, the average value of makespan *Cmax* in 10 tests was 298.1, which was the worst among the four schemes. Although scheme 5 could sometimes obtain better results, the results obtained in 10 tests were not stable enough and were with large fluctuations, easily falling into the local extremum sometimes. Its average value of 10 tests was 295.2. Compared with the other three schemes, the results obtained by the scheme 6 were relatively stable. However, its average result was poor, at 296.3. The average value of the test results of scheme 8 was 292.3, which was the best of all the schemes. The fluctuations of the curve indicate that the magnitude of the change in the results of scheme 8 in 10 tests is small. This shows that when solving multi-queue limited buffers scheduling problems in a flexible flow shop with setup times, compared with the CGA, the ICGA has not only greatly improved the optimization performance, but also improved the stability of the optimization results.

**Figure 7.** Results of 10 instance tests.

#### **6. Conclusions**

This study explored the multi-queue limited buffers scheduling problems in a flexible flow shop with setup times in a bus manufacturer. The study proposed an ICGA for global optimization to better improve the global optimization results, aiming at the shortcomings of the CGA that it is easy to fall into the local extremum and rapidly stops evolving. This algorithm employed the probability density function of the Gaussian distribution to map the original probabilistic model to a new probabilistic model so as to enhance the evolutionary vigor of the CGA. The job was processed on the specified online sequence in accordance with the individual decoding. Considering the impact of multi-queue limited buffers, the problems of the job into and out of the buffer during the subsequent scheduling were emphasized. When the job enters the buffer, the remaining capacity of buffers should be taken into account to reduce machining blocking and stagnation. When the job leaves the buffer and is assigned to the machine at the next stage, it is influenced by the setup times. In this study, the setup time of the machine was calculated based on the change in the properties of successively processed jobs. The SST rule was used to reduce the setup times. Finally, the findings of simulation experiments proved that combining the ICGA with local dispatching rules could better solve the multi-queue limited buffers scheduling problems in a flexible flow shop with setup times.

**Author Contributions:** Z.H. conceived and designed the research; Q.Z. performed the experiment and wrote the manuscript; H.S. and J.Z. checked the results of the whole manuscript.

**Funding:** This research was funded by the Liaoning Provincial Science Foundation, China (grant number: 2018106008), the Natural Science Foundation of China (grant number: 61873174), Project of Liaoning Province Education Department, China (grant number: LJZ2017015) and Shenyang Municipal Science and Technology Project, China (grant number: Z18-5-102).

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