## 2 Theoretical Background  
Traditional PAP problem models can be classified into six categories: (i) linear programming (LP); (ii) linear decision rule (RDL); (iii) method of transport; (iv) management coefficient approach; (v) search decision rule (RDB); and (vi) simulation [17]. [PERSON] et al. [18] complemented with three more methods: (i) dynamic programming; (ii) transport tables, and (iii) fuzzy logic mathematical programming. When using any of the planning models, the objectives and inputs of the model are generally considered to be deterministic, and problems can only be solved if they have the sole objective of minimizing the cost during the planning period [15]. According to [PERSON] [19], studies have been using models to support aggregate planning in natural systems since the 1980s. For example, [PERSON] et al. [20] developed a bi-objective optimization model for an aggregate planning problem with work learning effect and machine deterioration. The first objective maximizes profit, and the second minimizes costs associated with machine repairs and deterioration. The objective was to obtain the appropriate production rates at regular and overtime levels, inventory, and shortages, hiring and firing of workers, and the quantities of subcontractors. To validate the mathematical formulation of the model, it was converted into a single-objective model using the fuzzy objective programming method, based on which computational experiments are performed on a set of small-sized random instances solved by the software LINGO. [PERSON] and [PERSON] [21] performed a linear programming model with three objective functions, which minimize operational cost, energy cost, and carbon emission. It is assumed that several types of energy can be used. They differ in the amount of pollution, calorific value, and cost. The model was applied to the foundry factory, which produced a wide range of products such as aluminum, copper, and lead, and used different machines and manufacturing equipment capable of consuming different types of fossil fuels with different carbon emissions. The objective was to determine the most effective means of satisfying anticipated aggregate demand by adjusting production rates, inventory levels, backlogs, and other controllable variables. The model was solved as a single-objective model, applying a goal-setting technique, and to overcome the effects of uncertain input data, a robust optimization approach was applied to the model. These objective functions sought to minimize operating costs, energy, and carbon footprints. The authors concluded that by increasing the budget level of uncertainty, the amount of increase for electricity cost is much greater than other objective function costs. [PERSON] et al. [22] presented an optimization model for aggregate production planning under the assumption that electricity supply is subject to uncertain interruptions caused by participation in uninterruptible load contracts. The objective was to minimize the cost of electricity used in production by providing a robust production plan to guarantee demand satisfaction in all possible outage scenarios. The model considered the operational level problem, aggregate production, and inventory planning with the electricity supply uncertainty and deterministic demand. Production decisions were separated from interruption decisions, as the former belonged to the industrial company, while the latter related to the electricity retailer. A heuristic was developed to find a viable solution to the model, in which it found an optimal solution for each instance in 7 periods. The outage uncertainty framework allows different contract and operational rules to be incorporated into the production planning problem simultaneously, such as limiting production in post-outage recovery or prohibiting production level increases in specific periods. [PERSON] et al.