*Article* **Modified Multi-Crossover Operator NSGA-III for Solving Low Carbon Flexible Job Shop Scheduling Problem**

**Xingping Sun, Ye Wang, Hongwei Kang \*, Yong Shen \*, Qingyi Chen and Da Wang**

School of Software, Yunnan University, Kunming 650000, China; sunxp@ynu.edu.cn (X.S.); wangye@mail.ynu.edu.cn (Y.W.); devas9@ynu.edu.cn (Q.C.); wangda@mail.ynu.edu.cn (D.W.) **\*** Correspondence: hwkang@ynu.edu.cn (H.K.); sheny@ynu.edu.cn (Y.S.)

**Abstract:** Low carbon manufacturing has received increasingly more attention in the context of global warming. The flexible job shop scheduling problem (FJSP) widely exists in various manufacturing processes. Researchers have always emphasized manufacturing efficiency and economic benefits while ignoring environmental impacts. In this paper, considering carbon emissions, a multi-objective flexible job shop scheduling problem (MO-FJSP) mathematical model with minimum completion time, carbon emission, and machine load is established. To solve this problem, we study six variants of the non-dominated sorting genetic algorithm-III (NSGA-III). We find that some variants have better search capability in the MO-FJSP decision space. When the solution set is close to the Pareto frontier, the development ability of the NSGA-III variant in the decision space shows a difference. According to the research, we combine Pareto dominance with indicator-based thought. By utilizing three existing crossover operators, a modified NSGA-III (co-evolutionary NSGA-III (NSGA-III-COE) incorporated with the multi-group co-evolution and the natural selection is proposed. By comparing with three NSGA-III variants and five multi-objective evolutionary algorithms (MOEAs) on 27 wellknown FJSP benchmark instances, it is found that the NSGA-III-COE greatly improves the speed of convergence and the ability to jump out of local optimum while maintaining the diversity of the population. From the experimental results, it can be concluded that the NSGA-III-COE has significant advantages in solving the low carbon MO-FJSP.

**Keywords:** multi-objective optimization; flexible job shop scheduling problem; low carbon; genetic algorithm; multi-crossover operator; co-evolution
