## 3 Solution Approach
### Subgradient Algorithm
It can be seen that the results of the LRFN proposed in this paper are better compared to the others. For problems with looser capacity constraints, like TVW3 and TVW4, their capacity utilization ratios are below 73%, and the LRFN obtains the optimal solution. For problems with tighter capacity constraints, such as TVW2, the capacity utilization ratio is 91%. The LRFN obtains the optimal solution, and the others obtain near-optimal solutions. LV has a larger error, about 11.3%. For the tightest capacity problem, TVW1, its capacity utilization ratio reaches 93%, the LRFN obtains the best near solution, and the error is below 1%, while the errors of DS and LV are 3.3% and 6.4%, respectively. The computational results show that the LRFN has the best performance. The performance comparison is shown in Figure 2. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Problem** & **LP** & **Optimal** & **LR** & **LRFN** & **DS** & **LV** \\
**ID** & **Relaxation** & **Solution** & **LR** & **LRFN** & **DS** & **LV** \\ \hline TVW1 & 7996.67 & 8430 & 8710 & **8520** & 8710 & 8970 \\ TVW2 & 7722.27 & 7910 & 7930 & **7910** & 7930 & 8800 \\ TVW3 & 7534.17 & 7610 & 7610 & **7610** & 7970 & 7970 \\ TVW4 & 7446.17 & 7520 & 7520 & **7520** & 8000 & 8000 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Computational results for benchmarks. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Item**} & \multirow{2}{*}{\(h_{i}\)} & \multirow{2}{*}{\(s_{i}\)} & \multicolumn{8}{c}{**Period**} & \multirow{2}{*}{**TBO**\({}^{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{ \text{ \text{  }}}}}}}} }{}}{}}}}}}}}\) \\ \cline{5-10}  & & & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & \\ \hline
1 & 1 & 100 & - & 70 & 50 & 100 & 20 & 80 & - & 100 & 2 \\
2 & 1 & 200 & 20 & 40 & 50 & 10 & 30 & - & 40 & 50 & 3.65 \\
3 & 1 & 200 & 40 & 50 & - & 100 & 40 & 80 & 90 & 160 & 2.33 \\
4 & 1 & 300 & - & 100 & 100 & 150 & 160 & 90 & 100 & 100 & 2.5 \\
5 & 1 & 400 & 50 & - & 20 & 40 & 10 & 10 & 20 & 10 & 6.32 \\
6 & 1 & 250 & 70 & 40 & 40 & 40 & 100 & 20 & 40 & 50 & 3.16 \\
7 & 1 & 500 & - & 20 & 50 & 10 & 20 & 60 & 40 & 40 & 5.77 \\
8 & 1 & 300 & 10 & 20 & - & - & 10 & 10 & 20 & 30 & 6.93 \\ \hline \hline \end{tabular}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{Available demand} & \multirow{2}{*}{190} & \multirow{2}{*}{340} & \multirow{2}{*}{310} & \multirow{2}{*}{450} & \multirow{2}{*}{390} & \multirow{2}{*}{350} & \multirow{2}{*}{540} & \multirow{2}{*}{2920} \\ TVW1 & & 350 & 350 & 350 & 400 & 400 & 400 & 400 & 500 & 3150 (93\%) \\ TVW2 & & 400 & 400 & 400 & 400 & 400 & 400 & 400 & 3200 (91\%) \\ TVW3 & & 500 & 500 & 500 & 500 & 500 & 500 & 500 & 4000 (73\%) \\ TVW4 & & 600 & 600 & 600 & 600 & 600 & 600 & 600 & 4800 (61\%) \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Data for problems TVW1-4. Figure 2: Computational results comparison. ### More Computational Examples

The benchmarks that we previously computed have been regarded as traditional problems in lot-sizing publications. However, the difficulty of lot-sizing problems can be evaluated with the following criteria. 1. The number of time periods \(\mid\!T\!\mid\) and the number of items \(\mid\!I\!\mid\). 2. The tightness of the capacity utilization: \(\frac{\sum\limits_{i=1}^{T}\frac{I}{A_{ii}}}{\sum\limits_{i=1}^{T}C_{i}}\). Thus, we compute larger problems at different difficulty levels. The data of these problems are randomly generated and adhere to the following limitations:

1. the maximal number of items is set to 512;
2. the number of periods is set to 48;
3. the maximal utilization ratio is set to 93%. The other problem-specific data are randomly generated--for example, \(C_{t}\): rand (800, 2000), \(d_{ii}\): rand (0,100), \(s_{i}\): rand (100, 2000), \(p_{i}\): (1,8), \(h_{i}\): (1,4) and \(a_{i}=1\), and so on.