## 1 Introduction  
The Aggregate Production Plan (APP) aims to meet customer requirements at the lowest investment and operation cost [1], considering production, inventory and workforce in a planning horizon [2]. In a highly competitive environment for companies, it is necessary to find effective solutions in the shortest possible time [3], increasing customer service [4] and managing uncertainty in different variables [5]. Considering the uncertainty in demand and service levels makes the problem highly complex [6]; also, real-world problems are multi-product and multi-period [7], causing insufficiency of traditional methods [3, 8]. For this reason, additional methods to deterministic and stochastic optimization have been proposed to deal with the complexity, adding to the literature methods, such as extended for order performance [2], hierarchical production planning [4], fuzzy logic [5], hybrid algorithm based in simulated annealing (SA) and simplex downhill (SD) [9] particle swarm optimization algorithm (PSO) which considered labor characteristics, modeled as integer linear programming [10], among others. The multi-product APP is the model that considers more than one family product [11], minimizing total production costs and improving efficiency of equipment. [PERSON], [PERSON] and [PERSON] [12] proposed a multi-product APP as a bi-objective problem, the first objective maximizing the profit through the improvement of the learning curve in workers and the second objective minimizing the repairs and deterioration of the machines. APP includes families of products or products that have the same production processes [13]. [PERSON] et al. [7] consider multi-site APP problems in multi-period, multi-product and multiple transportation modes under uncertainty, which means, restrictions that deal with real situations. Additionally, it has been necessary to adapt the problem to specific cases, such as the agrifood supply chain [5], power systems [1], pharmaceutical industry [14] and apparel production [7], because it is related to its objectives and specific characteristics. There is no single production plan; each company works its APP differently according to the purposes of its organization. The formulation of an APP through mathematical programming seeks to minimize costs or maximize profits. One of the most important aspects to consider is the types of variables. The variables included in the model are important because they allow the APP to be adequate or not for an enterprise. Similarly, they impact the costs (the value of the objective function). Most of the variables used are demand, number of workers, production capacity, quantity to be produced and inventory, as is usual in the literature [1,2]. Some models may include more or fewer variables, depending on the needs of the organization; in the same way, they may have one or more objectives, as well as various types of constraints [11,12]. Ideally, an APP should be aligned with the production and contracting capabilities of the organization, forecasted demand (if it is forecasted), inventory levels, availability of raw materials, resources for processing and external factors that may affect the organization such as the local economy or competitors. APP is used in medium-term horizons with periods from 3 to a maximum of 18 months [15]. Similarly, it is usual that one or more of the parameters in the APP are considered random (the most used in the literature is demand), but not limited to the percentage of defective products (product quality), production costs, machine failure, quality of inputs, among others, allowing planning to be more realistic. The motivation of this work was extending a real developed project of [PERSON] et al.