## 2 Literature Review
Because we always suppose that the current period is 1, the LV then computes the maximum order up to the period  
\[\min_{t}\ B_{t}=\sum\limits_{\tau=2}^{t}\ (\sum\limits_{i=1}^{l}d_{i\tau}-C_{ \tau})>0,\ t=2,\ldots,T \tag{11}\]  
Suppose that \(B_{t^{*}}>0\); this means that, from period 2 to period \(t^{*}\), the cumulative demands exceed the cumulative capacity by \(B_{t^{*}}\) units, which must be backward-shifted to period 1. We repeat the above process until the current period is \(T\) and a feasible solution is obtained. The DS heuristic can be regarded as an extension of the SM heuristic for multiple items, because the single-item SM heuristic for the multiple-item lot-sizing problems is not necessarily optimal or even feasible if capacity limitations exist. Consequently, for a multiple-item problem, the DS would seek to increase the items, which results in the largest decrease in the average costs per unit time per unit capacity. \[\underset{u_{i}}{\text{argmax}}\ \ u_{i}=\frac{AC_{i}(T_{i})-AC_{i}(T_{i}+1)}{d_{ i,T_{i}+1}}>0,\ i=1,\ldots,I \tag{12}\]  
where, \(AC_{i}(T_{i})=(s_{i}+h_{i}\sum\limits_{\tau=1}^{T_{i}}(\tau-1)d_{i\tau})/\,T_{ i},T_{i}=1,\ldots,T\) denotes the average cost per unit time of a lot of item \(i\) that will satisfy \(T_{i}\) periods' requirements. Therefore, \(u_{i}\) indicates the marginal decrease in the average costs per unit of capacity absorbed when increasing the production of item \(i\) from \(T_{i}\) to \(T_{i}+1\). Mathematic programming-based approaches are a broad class of methods, suited for lot-sizing problems with complex constraints and multiple items. The well-known W-W [25] dynamic programming for the uncapacitated single-item lot-sizing problem is representative. It solves the USLP effectively in time \(O\) (\(T^{2}\)). Moreover, [PERSON] et al. [26] proposed a linear programming (LP)-based approach for the CLSP. This approach embeds valid inequalities into a branch-and-bound framework to solve the linear programming relaxation of the CLSP. Up to 20-item \(\times\) 13-period CLSPs are solved optimally with this approach. [PERSON] and [PERSON] [27] used column generation and a generalized upper bound to solve the CLSP. Up to 393-item \(\times\) 6-period CLSPs are solved with good approximation. In addition, [PERSON] [28] proposed a column generation heuristic for the CLSP, where the time-consuming W-W dynamic programming algorithm is replaced by the PPB heuristic in computing shadow prices, thus reducing the CPU time, with a slight loss in optimality. [PERSON] [29] proposed a variable redefinition approach for the CLSP. He uses computationally efficient approaches, such as the shortest-path and minimum-cost network flow, to obtain lower bounds and upper bounds in a branch-and-bound framework. [PERSON] and [PERSON] [30] proposed a Lagrangian relaxation approach for the CLSP. The Lagrangian relaxation problem consists of \(\left|\left|\left|\left|\left|\left|\left|\left|\left|\left|\left|\left| \left|\left|\left|\left|\left|\left|\left|\left|\left|\left|\left|\left|\left| \left|\left|\left|\left|\left|\left|\left|\left|\left|\left|\left|\left|\left| \left|\left|\left|\left|\left|\left|\left|\left|\left|\left|\left|\left|\left| \left|\leftheuristic, (2) linear programming-based relaxation and (3) an extension of the solution procedure of [PERSON] and [PERSON] [30] with setup times and multiple resources. Meta-heuristics have also been used to solve the CLSP. [PERSON] and [PERSON] [34] proposed a heuristic genetic algorithm (GA) for the CLSP. They designed a domain-specific encoding scheme for lot sizes and provided a heuristic shifting procedure for decoding for feasibility. The computational results show that the algorithm converges a solution within 200 generations for most of the small-scale examples (\(N\leq 10\)) and within 500 generations for most of the modest examples (\(N\leq 20\)). [PERSON] [35] applied a genetic algorithm (GA) to solve a lot-sizing problem and considered batch ordering and backorders.