## 1 Introduction  
Aggregate production planning (APP) is an operational activity that determines the minimum cost, workforce, and production plans required to meet customer demands. APP simultaneously establishes optimal production, inventory, and employment levels over a given finite planning horizon to meet the total demand for all products that share the same limited resources [1]. As the implementation of JIT (Just in Time) practice becomes increasingly popular, each echelon in a supply chain tends to carry smaller inventories, and thus the whole supply chain is made more vulnerable to lost sales and/or backorders [2]. Furthermore, with the development of the manufacturing industry, more dynamic characteristics of production need to be considered in practical application. Therefore, the APP model should include multiple factors related to production and inventory, such as production cost, labor cost, inventory cost, etc. and should also consider some extension conditions, such as backorder cost, overtime cost, and lost sales cost. APP is one of the most critical areas of planning performed in the design of production systems, and it has attracted considerable interest from both practitioners and academics [1]. Many researchers have considered the backordering decisions in the APP model. We summarized the most important and related studies that consider backordering for APP in Table 1.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Article** & **Model Category** & **Objective Function** & **Considerations** & **Solving Approaches** \\ \hline [PERSON] and [PERSON] [3] & Multiple objective linear programming model & Total production costs; Carrying and backordering costs; Costs of changes in labor levels & Inventory levels; labor levels; machine capacity; warehouse space; the time value of money & Solution algorithm based on linear programming problem \\  & & The market demand; production cost; & \\ [PERSON] et al. [4] & A fuzzy random APP & The chance of obtaining the profit more than the predetermined profit & subcontracting cost; inventory & A hybrid optimization \\  & & The chance of obtaining the profit more than the & carrying cost; backorder cost; & \\ [PERSON] et al. [4] & A fuzzy random APP & The chance of obtaining the profit more than the & \\ [PERSON] et al. Throughout the review, multi-objective programming has been widely used in this area, and most of the existing research on APP mainly consider the constraints on the balance equation for production, inventory capacity, inventory and demand, production capacity, and labor capacity. Moreover, the production cost, inventory and backorder level are the other main objectives that have been taken into consideration. The main methods applied to management science techniques for APP problems are: linear programming, piecewise linear programming, nonlinear programming, etc. Table 2 shows some of the characteristics of the existing models. Furthermore, [PERSON] et al. [17] analyzed the computational complexity of a class of deterministic production planning problems with various types of cost functions and proved that the single material problem under the constraints of linear cost function and time-varying capacity and some special cases are NP-hard. [PERSON] and [PERSON] [18] also proved that the multi-item capacitated production planning problem is strongly NP-hard. In addition, according to [PERSON] et al. [19] and [PERSON] et al. [13], APP problems with multi-phase production are among strongly NP-hard issues. Moreover, metaheuristics have proved to be efficient techniques for solving APP problems. Among the metaheuristics, genetic algorithms (GAs) ([PERSON] et al. [6], [PERSON] et al. [11], [PERSON] et al. [13], [PERSON] et al. [19], [PERSON] et al. [20]), and particle swarm optimization (PSO) ([PERSON] et al. [16], [PERSON] et al. [21], [PERSON] et al. [22]) have been used to deal with APP models.