## 2 Problem Description and Modeling
### Mathematical Model
\begin{array}{ll}\left|\frac{MWL-WL^{\prime}{}_{1,i}}{MCL- CL^{\prime}{}_{1,i}}-\frac{MWL}{MCL}\right|&CL^{\prime}{}_{i}\
eq MCL\\ \\ \left|\left(MWL-WL^{\prime}{}_{1,i}\right)-\frac{MWL}{MCL}\right|&CL^{\prime}{}_ {i}=MCL\end{array} \right.\] This is continued for other units to acquire \(F_{2}=\frac{\sum\limits_{i=1}^{K}Z_{i}}{c}\). \(H_{1,i}\) is the absolute value of the difference between the updated used working life and the used calendar life ratio and the ideal ratio of each equipment. When \(CL^{\prime}_{1,i}=MCL\), the used working life of the equipment is calculated as a ratio. 3. The average utilization function, \(F_{3}\). It is defined that the standard for equipment to enter into consumption is that the working life must be less than 3 h or the calendar life must be less than 1 month. When the equipment reaches the consumption standard, it can be considered that the utilization rate is 100%. The process is as follows: For \(u_{1}\), define \(k_{1,1}:=0\), \(k_{1,2}:=0\), \(RWL_{1}:=0\), \(RCL_{1}:=0\). Traverse the updated data. When \(CL^{\prime}{}_{1,i}:=0\), \(k_{1,1}:=k_{1,1}+1\); When \(WL^{\prime}{}_{1,i}\geq w\), \(RWL_{1}:=RWL_{1}+WL^{\prime}{}_{1,i}\); When \(WL^{\prime}{}_{1,i}<w\), \(RWL_{1}:=RWL_{1}+0\); When \(WL^{\prime}{}_{1,i}<w\), \(k_{1,2}:=k_{1,2}+1\), \(RCL_{1}:=RCL_{1}+CL^{\prime}{}_{1,i}\). The utilization function of \(u_{1}\) is shown in Formula (3). \[R_{1}=\left\{ \begin{array}{ll}\frac{k_{1,1}+MWL-RWL_{1}}{k_{1,1}+MWL}\frac{ k_{1,2}+MCL-RCL_{1}}{k_{1,2}+MCL}\\ \\ \frac{k_{1,1}+MWL-RWL_{1}}{k_{1,3}+MWL_{1}}*100\%&k_{1,1}>0\&k_{1,2}>0\\ \\ \frac{k_{1,2}+MCL-RCL_{1}}{k_{1,2}+MCL}*100\%&k_{1,2}>0\&k_{1,1}=0\end{array} \right.\] (3)  
\(k_{1,1}\) is the quantity of equipment with an exhausted calendar life of \(u_{1}\), \(k_{1,2}\) is quantity of equipment with an exhausted calendar life of \(u_{1}\), \(RWL_{1}\) is the sum of the remaining working life of the equipment with an exhausted calendar life after updating \(u_{1}\), and \(RCL_{1}\) is the sum of the remaining calendar life of the equipment with an exhausted working life after updating \(u_{1}\). This is continued for other units to acquire \(F_{3}=\frac{\sum\limits_{i=1}^{K}R_{i}}{c}\). 4. The rotation-quantity waste function, \(F_{4}\). The transportation cost and risk level caused by rotation can be replaced by the equipment rotation quantity. In order to reduce the equipment transportation cost and risk caused by rotation, the number of rotating pieces of equipment should be as small as possible. Let us assume that the number of pieces of equipment rotated from a certain unit at each time cannot exceed \(g\), the rotation phase can be divided into from \(u_{1}\) to \(u_{2}\), from \(u_{1}\) to \(u_{3}\),..., from \(u_{1}\) to \(u_{c}\),..., and from \(u_{c}\) to \(u_{c-1}\), where the total time of rotations is \(c*(c-1)\), and the maximum quantity of rotations is \(c*(c-1)*g\). Let us assume that the rotation quantity represented by an individual chromosome is \(P\), the rotation-quantity waste rate function of the individual \(F_{4}\) is shown in Formula (4). \[F_{4}=\frac{c*(c-1)*g-P}{c*(c-1)*g} \tag{4}\]  
Aiming at the multi-objective assignment problem with an uncertain rotation quantity, with the task quantity completed by each equipment under the conditions of rotation, this paper constructed a decision-making model for the rotation and echelon usage of dual-life equipment and proposed an improved NSGA-III method with a variable length chromosome (I-NSGA-III-VLC) to solve it. This method can obtain a variety of optimization schemes in a relatively short time. Through scheme screening, the problem of poor working-life echelon status, low matching between calendar life and working life and the low utilization of the working life of equipment in different units can be solved to a certain extent with as few rotations as possible.