# Research on Low-Carbon-Emission Scheduling of Workshop under Uncertainty
## 1 Introduction
The remainder of the paper is organized as follows: the second section constructs the low-carbon-emission scheduling model, the third section proposes the NSGA-III-ST algorithm, the fourth section solves the optimization model, the fifth section discusses the algorithm's advantages and disadvantages, and the sixth section presents the conclusions. The method flow chart is shown in Figure 1.  
**2. Low-Carbon-Emission Scheduling Optimization Modeling**  
_2.1. Description and Handling of Uncertainty_  
There are many uncertainties in the manufacturing process, and the emergence of uncertainties will strongly influence the accuracy of workshop scheduling [25]. A workshop's  
Figure 1: The flow of the method. uncertainties can be divided into four categories based on different sources: the system's inherent uncertainty, uncertainty of equipment, uncertainty of operation personnel, and external factors caused by sale behavior or purchasing behavior. Because the system's intrinsic uncertainty is generally not the focus of research, and the uncertainties caused by personnel or management methods are challenging to measure and calculate, uncertainty research focuses on the equipment, which consists mainly of machine breakdown, order insertion, order cancellation, workpiece processing time, unknown workpiece arrival time, order delivery time and raw material delay, and so on. After the weights of uncertain factors and the objective function are determined by the expert scoring method, and sensitivity analysis is performed by an analytic hierarchy process, the importance of the workshop uncertainty is determined as the workpiece processing time, order delivery time, machine breakdown, raw material delay, unknown workpiece arrival time, order insertion, and order cancellation. Meanwhile, the uncertainty of the processing time leads to uncertainty of the order delivery time, which affects the satisfaction degree (\(SD\)), and the uncertainty of processing time affects the makespan and carbon emissions. Therefore, the processing and order delivery times are selected as the uncertainties of the low-carbon-emission workshop scheduling problem. The processing time cannot be precisely measured in manufacturing workshops; fuzzy sets are frequently used to describe it [26; 27]. The triangular fuzzy number expresses the fuzzy processing time. Furthermore, because trapezoidal fuzzy numbers are commonly used when considering advance and delay, a trapezoidal fuzzy number expresses the fuzzy delivery time. (a) Fuzzy processing time  
The triangular fuzzy number \(\tilde{A}=(a_{1},a_{2},a_{3})\) is used to denote the fuzzy processing time of the workpiece, where \(a_{1}\) denotes the shortest processing time of the process, \(a_{2}\) is the most probable processing time, and \(a_{3}\) denotes the longest processing time. The probability distribution function of the affiliation function \(\mu_{\tilde{A}}\) is illustrated in Equation (1). When \(a_{2}=(a_{1}+a_{3})/2\), the affiliation degree value of the triangular fuzzy number is 1. \[\mu_{\tilde{A}}= \begin{cases}\frac{x-a_{1}}{a_{2}-a_{1}},a_{1}\leq x\leq a_{2} \\ \frac{x-a_{2}}{a_{2}-a_{3}},a_{2}\leq x\leq a_{3}\\ 0,others\end{cases}  \tag{1}\]  
Set the triangular fuzzy number \(\tilde{B}=(b_{1},b_{2},b_{3})\). The addition of triangular fuzzy numbers can be expressed as \(\tilde{A}+\tilde{B}=(a_{1},a_{2},a_{3})+(b_{1},b_{2},b_{3})=(a_{1}+b_{1},a_{2} +b_{2},a_{3}+b_{3})\), where \(b_{1}\) denotes the shortest processing time of the process, \(b_{2}\) is the most probable processing time, and \(b_{3}\) denotes the longest processing time. The symbol \(\vee\) denotes taking the more significant number of fuzzy numbers, and the operation of taking the larger one is \(\tilde{A}\vee\tilde{B}=(a_{1},a_{2},a_{3})\vee(b_{1},b_{2},b_{3})\simeq(a_{1} \lor b_{1},a_{2}\lor b_{2},a_{3}\lor b_{3})\). The chosen large operation is chosen to approximate the fuzzy number, because the fuzzy number obtained by the more extensive operation is not always the triangular fuzzy number.