## 2 Literature Review  
The study of lot-sizing problems can be traced back to the EOQ problem [18], i.e., the economic ordering quantity model. The EOQ is essentially a trade-off among the ordering/setup cost and inventory holding cost, seeking to achieve the optimal economic lot sizes. The EOQ supposes that the problem is for a single stationary product, an invariable demand and an unlimited production capacity. The EOQ can be solved with differential methods. While some complex characteristics are combined in the \"lot sizes\" problem, this problem is very difficult to solve. For example, when the product demand changes over time, more products are produced. However, one of the most essential changes for problem-solving is the existence of production capacity constraints. Consequently, the complexity of a lot-sizing problem can be characterized in terms of the length of the horizon, number of items and capacity constraints. Based on these characteristics, lot-sizing problems can be classified into the uncapacitated single-item lot-sizing problem (SILSP), the capacitated single-item lot-sizing problem (CSILSP), the uncapacitated multi-item lot-sizing problem (ULSP) and the capacitated single-level multi-item lot-sizing problem (CLSP). Among them, the CLSP is the most common and difficult problem and has become the core of lot-sizing problem research. Meanwhile, the SILSP is the simplest lot sizing problem, specific to a single product, with a finite planning horizon of \(T\) periods and no capacity constraints. SILSPs are solvable in \(O\) (\(T^{2}\)), except for some variants with some essential extensions, such as the setup time and (or) cost, backlogs, time windows, etc., which are NP-hard. ULSPs are an extension of SILSPs with multiple items and thus are also solvable in \(O\) (\(T^{2}\)). CSILSPs are an extension of the SILSPs with capacity constraints and are NP-hard [4]. The CLSP is the extension of the CSILSP with multiple items and is thus NP-hard. Because of the difficulties of the CLSP, most of the existing solution approaches are heuristic and the exact solution approaches are only suitable for small-sized problems. In general, the solution approaches to the CLSP can be divided into three classes: common heuristics, mathematic programming-based and meta-heuristics approaches. Heuristics is sometimes the only available choice when the capacity constraints are too tight in a CLSP. The well-known common heuristics include lot-for-lot, least unit cost (LUC) [19], part-period balancing (PPB) [20; 21], Silver-Meal (SM) [22], Lambrecht-Vanderveken (LV) [23] and Dixon-Silver (DS) [13]. The lot-for-lot heuristic directly converts the demand matrix into a lot-size matrix to obtain an initial solution. This approach maximizes the setup costs and minimizes the inventory holding costs. It is suitable for problems with looser capacity constraints and can quickly obtain an initial feasible solution. The LUC heuristic tracks the lowest cost per unit for production, and it can be expressed mathematically as  
\[\underset{t}{\text{argmin}}\ k(t)=\frac{s+h\sum\limits_{\tau=1}^{t}(\tau-1)d_{ \tau}}{\sum\limits_{\tau=1}^{t}d_{\tau}},\ t=1,\ldots,T \tag{8}\]  
Hereafter, \(s\) indicates the setup cost and \(h\) is the unit inventory holding cost. The PPB heuristic aims at seeking the maximum production over the demand of several periods when the total holding cost does not exceed a single setup cost. It is expressed by the formula  
\[\underset{t}{\text{argmax}}\ h\sum\limits_{\tau=1}^{t}(\tau-1)d_{\tau}-s\leq 0,\ t=1,\ldots,T \tag{9}\]The SM heuristic attempts to minimize the average cost per unit time for each lot. Mathematically, it can be expressed as  
\[\underset{t}{\text{argmin}}\ k(t)=\frac{s+h\sum\limits_{\tau=1}^{t}(\tau-1)d_{ \tau}}{t},t=1,\ldots,T \tag{10}\]  
The LV heuristic is an extended Eisenhut heuristic [24]. LV's idea is analogous to that of [PERSON]'s heuristic, a period-by-period heuristic.