## 1 Introduction  
The capacitated multi-item lot-sizing problem, referred to as the CLSP, is often encountered in industry production settings, where multiple products are produced and a resource bottleneck exists [1; 2; 3]. The problem is to determine the production lot size of each product and its timing over a finite number of periods, so that the total cost, consisting of the production costs, setup costs and inventory costs, is minimal and the production demand of each product in each period is met. Mathematically, the CLSP can be formulated as follows. (CLSP)  
\[\text{minimize}\ z=\sum_{l=1}^{T}\sum_{i=1}^{I}\left(p_{it}x_{it}+h_{it}I_{it}+s _{it}y_{it}\right) \tag{1}\]  
\[I_{it-1}+x_{it}-I_{it}=d_{it},i\in I,t\in T \tag{2}\]  
\[\sum_{i\in I}a_{it}x_{it}\leq C_{t},t\in T \tag{3}\]  
\[x_{it}\leq My_{it},i\in I,t\in T \tag{4}\]  
\[x_{it},I_{it}\geq 0,i\in I,t\in T \tag{5}\]  
\[I_{it}=0,i\in I \tag{6}\]  
\[y_{it}\in\{0,1\},i\in I,t\in T \tag{7}\]where the meanings of the notations are as follows. Indices and sets:  
\(i\)--product;  
\(t\)--period;  
\(I\)--set of products;  
\(T\)--set of periods. Parameters:  
\(p_{it}\)--unit linear production cost for product \(i\) in period \(t\);  
\(h_{it}\)--unit inventory holding cost for one unit of product \(i\) between periods \(t\) and \(t+1\);  
\(s_{it}\)--setup cost for product \(i\) in period \(t\);  
\(d_{it}\)--demand for product \(i\) in period \(t\);  
\(a_{it}\)--unit capacity requirement for product \(i\);  
\(C_{\Gamma}\)--available capacity in period \(t\);  
\(M\)--a large positive number. Variables:  
\(x_{it}\)--amount of product \(i\) produced in period \(t\);  
\(I_{it}\)--inventory of product \(i\) at the end of period \(t\);  
\(y_{it}\)--binary variable that = 1 if \(x_{it}>0\) and = 0 if \(x_{it}=0\). In the CLSP model, the objective function (1) minimizes the total production, inventory holding and setup costs. Constraint (2) represents the inventory balance; for any product \(i\), the production in period \(t\) plus the remaining inventory at the end of period \(t-1\) minus the demand in period \(t\) is equal to the inventory at the end of period \(t\). Constraint (3) is the capacity constraint for the resource bottleneck, which means that the capacity consumption of all products in each period \(t\) cannot exceed the available amount in that period. Constraint (4) is the associated relation of variables \(x\) and \(y\), where a product \(i\) is produced in a period \(t\) if and only if it has been set up in that period. Constraints (5)-(7) define the decision variables. Note that inventories are non-negative, meaning that no backlogs are allowed. Note also that when the production costs do not change over time, the first cost in the objective function is constant and can be removed. CLSPs are the core of production planning and scheduling. In the enterprise production environment, the production capacity of the key equipment is often limited, and many products must share the key equipment resource, such as a key machine tool or a processing center--for example, in the manufacturing industry, these include tires, bearings, chemical products and pharmaceuticals. In these industries, the organization of production is often in the form of an MRP; in this process, the main production schedule (MPS) is devised according to the bill of materials (BOM), used to calculate the material demand. Often, the product demand is beyond the key equipment capacity in many periods; in addition, the key equipment capacity is not fully utilized in many periods, resulting in stock shortages and cost increases. Therefore, it is important to consider the production capacity limit in the process of production planning. The CLSP is useful to solve this kind of problem and its main aim is to determine the lot sizes. However, the CLSP problem is NP-hard [4]; thus, the search for an effective solution to the CLSP has been of interest in both industry [5,6] and academic communities [7, 8, 9].