## 3 Hybrid Algorithms Design
### Local Search Algorithm  
This LS algorithm searches the neighborhood from a solution of production quantity under the condition that the number of workers remains unchanged to improve \(Z_{1}\). The problem of optimization of production quantity is a special case of the MCF problem, according to [20]. Thus, the LS algorithm for production quantity can be designed based on augmenting cycle.  
First, production planning can be dealt with as a network model. Any two unequal periods \(t_{1},t_{2}\in[1,T]\) can form a cycle. The maximum adjustment production quantity of \(i\) is limited by the minimum value of the production quantity of period \(t_{1}\), the inventory capacity from period \(t_{1}\) to \(t_{2}\), and the production quantity of period \(t_{2}\). The maximum adjustment amount can be determined according to the limit quantity analysis of three aspects for the formed anticlockwise circle and clockwise circle, respectively.  
(1) The formed anticlockwise cycle  
The production quantity of period \(t_{1}\): The maximum production quantity of the period \(t_{1}\) will limit the increased flow, which can be expressed as \(\text{MaxPP}_{it_{1}}-PP_{t_{1}}\).  
Inventory remaining capacity: The minimum remaining inventory capacity from period \(t_{1}\) to \(t_{2}\) is \(\min\{CN_{i}-CI_{it}|t_{1}<t\leq t_{2}\}\).  
The production quantity of period \(t_{2}\): If there is no stockout from period \(t_{1}\) to \(t_{2}\), the adjustment amount of the period \(t_{2}\) limit is \(PP_{t_{2}}-\text{MinPP}_{t_{2}}\) (as shown in Figure 3a). Otherwise, the adjustment amount can be increased to the total stockout quantity from period \(t_{1}\) to \(t_{2}\), \(\sum\limits_{t^{\prime\prime}=t_{1}+1}^{t_{2}}\sum\limits_{t^{\prime\prime}=t ^{\prime\prime}}^{T}B_{it^{\prime\prime}-1}(t^{\prime}-t^{\prime\prime}+1)\) (as shown in Figure 3b).  
Figure 2: Running results of GA for 10 times.  
The maximum adjustment amount of anticlockwise cycle for product \(i\) can be obtained as follows:  
\[FP_{i}=\min\{\text{MaxPP}_{it_{1}}-PP_{it_{1}},CN_{i}-CI_{it},PP_{it_{2}}-\text{ MinPP}_{it_{2}}+\sum_{t^{\prime\prime}=t_{1}+1}^{t_{2}}\sum_{t^{\prime\prime}=t^{ \prime\prime}}^{\mathcal{T}}B_{it^{\prime\prime}-1}(t^{\prime}-t^{\prime\prime }+1)\,|t_{1}<t\leq t_{2}\,\}. \tag{28}\]  
The formed clockwise cycle  
The production quantity of period \(t_{1}\): If there is no stockout from period \(t_{1}\) to \(t_{2}\), the adjustment amount of the period \(t_{1}\) limit is \(PP_{it_{1}}-\text{MinPP}_{it_{1}}\) (as shown in Figure 4a). Otherwise, the adjustment amount can be increased to the minimum stockout quantity from period \(t_{1}\) to \(t_{2}\): \(\min\{B_{it^{\prime\prime}-1}(t^{\prime}-t^{\prime\prime}+1)|t_{1}\leq t^{ \prime\prime}<t_{2},t^{\prime\prime}\leq t^{\prime}\leq T\}\) (as shown in Figure 4b).  
Inventory remaining capacity: The minimum remaining inventory capacity from period \(t_{1}\) to \(t_{2}\) is \(\min\{CI_{it}|CI_{it}>0,t_{1}<t\leq t_{2}\,\}\).  
The production quantity of period \(t_{2}\): The maximum production quantity of the period \(t_{2}\) will limit the increased flow, which can be expressed as \(\text{MaxPP}_{it_{2}}-PP_{it_{2}}\).  
The maximum adjustment amount of clockwise cycle for product \(i\) can be obtained by  
\[\begin{array}{l}FP_{i}=\min\{\text{MaxPP}_{it_{2}}-PP_{it_{2}},CI_{it},B_{it ^{\prime\prime}-1}(t^{\prime}-t^{\prime\prime}+1),\\ PP_{it_{1}}-\text{MinPP}_{it_{1}}+\min\{B_{it^{\prime\prime}-1}(t^{\prime}-t^{ \prime\prime}+1)\}\big{|}CI_{it}>0,t_{1}<t\leq t_{2}\}.\end{array} \tag{29}\]  
LS algorithm can significantly improve the quality of solution, especially in large-scale experiments, as shown in Figure 5. However, the running time will be greatly increased for each experiment.  
Figure 4: Illustration of augmenting flow in clockwise cycle.  
Figure 3: Illustration of augmenting flow in anticlockwise cycle.