## 3 Solution Approach
### Local Neighborhood Search Algorithm  
To find more high-quality solutions, we proposed a local neighborhood search algorithm for the current feasible solution. The neighborhood structures are defined as follows.  
(1) The neighborhood of the lot move  
The first neighborhood structure is for the shifting of lots between two periods for a single item. Assume that there exists an item \(i\), and, for any two periods \(t_{1}\) and \(t_{2}\), there are two lots \(x_{it_{1}}>0,x_{it_{2}}>0\), in the corresponding period. Again, assume that there are no positive lots among \(t_{1}\) and \(t_{2}\), i.e., the two periods \(t_{1}\) and \(t_{2}\) can be discontinuous, but there must be no production occurring between them. The neighborhood of the lot move is then defined for all possible moves between \(t_{1}\) and \(t_{2}\) as stated above. As an illustrative example, let \(\Delta>0\) be the lot from period \(t_{1}\) to period \(t_{2}\); then, we have  
\[\begin{cases}x_{it_{1}}=x_{it_{1}}-\Delta\\ x_{it_{2}}=x_{it_{2}}+\Delta^{\prime}\end{cases} \tag{19}\]  
Note that the lot move may be bidirectional. A forward move reduces the inventories, while a backward move may reduce the setups. The cost change can be computed from Formulas (20) and (21).  
\[\text{cost}=-\lambda h_{i}\cdot\Delta-s_{i},\text{ forforwardmoves} \tag{20}\]  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline \(100^{9}\) & & & & \(80^{8}\) & \(40^{11}\) & & & \(150^{7}\) & & & \(80^{14}\) & 450 & 1 \\  & \(150^{6}\) & \(10^{15}\) & \(50^{12}\) & & & & & \(180^{5}\) & & & \(10^{13}\) & \(\mathbb{X}\) & 400 & 2 \\  & & \(190^{4}\) & & & & \(40^{3}\) & \(40^{10}\) & & & \(180^{2}\) & & & \(450\) & 3 \\  & & & & & & & & & & & & & \(170^{14}\) & \(130^{14}\) & 300 & 4 \\
100 & 150 & 200 & 50 & 80 & 40 & 40 & 40 & 150 & 180 & 180 & 180 & 210 & & \\ \hline \hline \end{tabular}
\end{table}
Table 7: The fourth basic feasible solution and closed loop.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline \(100^{9}\) & & & & \(80^{8}\) & \(40^{11}\) & & & \(150^{7}\) & & & \(80^{14}\) & 450 & 1 \\  & \(150^{6}\) & \(10^{15}\) & \(50^{12}\) & & & & & \(180^{5}\) & & & \(10^{13}\) & 40\[\text{cost}= \lambda h_{i}\cdot\Delta-s_{i},\text{ forbackwardmoves} \tag{21}\]  
When the inventory cost \(h_{i}\) changes from period to period, the formula needs to be adjusted correspondingly. The cardinal of the lot-move neighborhood is \(O(n^{3})\).  
The neighborhood of the 2-opt lot exchanges, which change the lots for two items in two continuous periods. Assume that there are two items \(i_{1}\), \(i_{2}\) and two continuous periods, \(t_{1}\) and \(t_{2}\), and the related lots are \(x_{i_{1}t_{1}},x_{i_{1}t_{2}},x_{i_{2}t_{1}},x_{i_{2}t_{2}}\), respectively. Now, we swap two lots for two items \(i_{1}\) and \(i_{2}\) in two periods \(t_{1}\) and \(t_{2}\), and assume that \(\Delta_{1}\leq\text{ min}(x_{i_{1}t_{2}},x_{i_{2}t_{1}})\), \(\Delta_{2}\leq\text{ min}(x_{i_{1}t_{1}},x_{i_{2}t_{2}})\). Then, we have  
\[\begin{cases}x_{i_{1}t_{1}}=x_{i_{1}t_{1}}+\Delta_{1}\\ x_{i_{1}t_{2}}=x_{i_{1}t_{2}}-\Delta_{1}\\ x_{i_{1}t_{1}}=x_{i_{1}t_{1}}-\Delta_{1}\\ x_{i_{2}t_{2}}=x_{i_{2}t_{2}}+\Delta_{1}\end{cases}\text{ for forward exchange, }\begin{cases}x_{i_{1}t_{1}}=x_{i_{1}t_{1}}-\Delta_{2}\\ x_{i_{1}t_{2}}=x_{i_{1}t_{2}}+\Delta_{2}\\ x_{i_{1}t_{1}}=x_{i_{1}t_{1}}+\Delta_{2}\\ x_{i_{2}t_{2}}=x_{i_{2}t_{2}}-\Delta_{2}\end{cases}\]  
The cardinal of the neighborhood is \(O(n^{4})\).  
As an illustrative example, below, a 3-item \(\times\) 4-period lot-sizing problem is considered [15]. The data are referred to in the literature. The 2-opt lot exchange local neighborhood search results are shown in Figure 1.  
An initial solution is obtained from the DS heuristic, and the objective function value is 1415.7. Then, the two 2-opt lot exchanges are performed and two improved solutions are obtained with values of 1414 and 1412.6, which are optimal.