## 3 Solution Approach
### Subgradient Algorithm
The computational results for these problems are listed in Table 11.

In Table 11, the column \"Iteration Found\" indicates that the best solution is found in a certain iteration. The meanings of the other columns are clear. As shown in Table 11, the average dual gap is within 2% (exactly 1.76), with the exceptions of some extreme examples, like problem No. 15, where the dual gap is 9.54. The reason for this result may be the excessively tight capacity utilization. Usually, the looser the capacity utilization ratio, the smaller the dual gap is, and vice versa. ## 5 Conclusions

In this paper, we propose a Lagrange relaxation approach to solve the capacitated multi-item lot-sizing problem, the CLSP. The CLSP is often encountered in industry production settings and the study of its solution approaches is significant. Our approach represents a deliberate attempt to solve such industrial problems. The main work includes (1) developing a stepping-stone algorithm to solve the resulting transportation problem to obtain the upper bounds; (2) developing a fix-up procedure to obtain feasible solutions; (3) developing a local search heuristic to obtain high-quality solutions; and (4) providing a complete Lagrange relaxation algorithm framework. The computational experiments on the benchmarks and randomly generated lot-sizing problems verify the effectiveness and efficiency of the proposed solution approach. There are some limitations to our method: \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Problem** & **Problem Size** & **LB** & **Optimal** & **Iteration** & **Gap** & **CPU** \\
**No.** & **(Items \(\times\) Periods)** & & **Solution** & **Found** & **(\%)** & **Time (s)** \\ \hline
1 & 5 \(\times\) 36 & 82,131.54 & 85,374 & 201 & 3.95 & 68.73 \\
2 & 5 \(\times\) 48 & 111,368.25 & 115,136 & 215 & 3.38 & 215 \\
3 & 10 \(\times\) 10 & 84,904.29 & 88,094 & 2 & 3.76 & 5.33 \\
4 & 10 \(\times\) 15 & 67,167.7 & 67,906 & 49 & 1.1 & 1.5 \\
5 & 10 \(\times\) 15 & 71,694.02 & 71,705 & 8 & 0.02 & 0.03 \\
6 & 10 \(\times\) 20 & 151,859.54 & 157,470 & 87 & 3.69 & 40.75 \\
7 & 10 \(\times\) 20 & 85,426.11 & 85,619 & 1 & 0.23 & 2.47 \\
8 & 15 \(\times\) 20 & 160,088.16 & 161,128 & 14 & 0.65 & 7.4 \\
9 & 10 \(\times\) 24 & 103,383.29 & 103,866 & 10 & 0.47 & 82.97 \\
10 & 15 \(\times\) 36 & 57,341.8 & 58,184 & 40 & 1.47 & 66.08 \\
11 & 10 \(\times\) 36 & 220,486.05 & 224,047 & 214 & 1.62 & 380.94 \\
12 & 10 \(\times\) 30 & 224,231.93 & 226,571 & 7 & 1.04 & 161.01 \\
13 & 20 \(\times\) 20 & 261,226.71 & 267,422 & 2 & 2.37 & 16.65 \\
14 & 20 \(\times\) 30 & 379,853.7 & 383,533 & 3 & 0.97 & 50.51 \\
15 & 10 \(\times\) 48 & 358,551 & 392,750 & 1 & 9.54 & 1147.91 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Computational results of large-sized problems. * Parameter setting in the subgradient algorithm;
* A feasible solution fix-up policy;
* Subgradient algorithm substitution, such as multiplier adjustment methods. These also constitute the directions of our ongoing research. Conceptualization, Z.G. and D.L. ; data curation, D.L. ; formal analysis, D.W.; methodology, Z.G. ; experimental design and testing, D.L. ; software, D.W. and Z.Y. ; supervision, Z.G. ; validation, Z.Y. ; writing--original draft, Z.G. ; writing--review and editing, D.W. All authors have read and agreed to the published version of the manuscript. This study was partly supported by the Major Program of the National Natural Science Foundation of China (72192830, 72192831) and the 111 Project (B16009). Not applicable. The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author. The authors declare no conflicts of interest. ## References

* (1) [PERSON]; [PERSON] Lot sizing and scheduling-survey and extensions. _Eur. J. Oper. Res._**1997**, _99_, 228-249. [CrossRef]
* (2) [PERSON]; [PERSON] Multi-item single-level capacitated dynamic lot-sizing heuristics: A general review. _J. Oper. Res. Soc._**1988**, _39_, 991-1004.