## 2 Materials and Methods
### Mathematical Approach to Aggregate Planning  
The following mathematical techniques are the best known and can be used for a more complex solution of the aggregate planning area:  
Linear programming--is an optimisation technique that offers to find the maximal profit, receipts or minimal costs on the basis of limited resources. The special type of linear programming known as the transport model can be used for creating an aggregate plan that will help balance the production capacity and demand at minimal costs. However, only a few decisions of aggregate planning are compatible with the assumptions of linear programming [32]. Mixed integer linear programming (MILP)--the MILP can be very useful for those aggregate plans that are prepared for a whole wide range of products where the plan is a summarisation of the plans for individual product lines. This approach offers a method for determining the number of units that are to be manufactured for each group of products [33]. Linear decision rule (LDR)--an optimisation technique that looks for minimising the production costs (work, overtime, hiring/unpaid leave, storage costs for overproduction) utilises a range of functions for an approximate cost estimation (three of them are quadratic) to achieve a single quadratic equation. After that (using the calculation) two linear equations can be derived from the quadratic equation, one of them for planning production for each time period and the second one for planning the workforce for each time period [33]. Management coefficients model--formulated by [34] is based on an assumption that the size of production is adjusted by this general decision-making rule:  
\[P_{t}=a*W_{t-1}-b*I_{t-1}+c*F_{t+1}+K \tag{1}\]  
where \(P_{t}\) is the production amount per \(t\), \(W_{t-1}\) is the workforce for the previous time period, \(I_{t-1}\) are the inventories at the end of the previous time-period, \(F_{t+1}\) is the demand forecast for the next time-period, \(a\), \(b\), \(c\), \(K\) are the constants. Then the regression analysis for estimating the values \(a\), \(b\), \(c\), and \(K\) is used. The final result is the decision-making rule based on the previous management behaviour without any further functions of the explicit costs. There is an assumption that the managers know what is important although it is not easy to identify these costs. This method completes the usage of the experience assessment [33]. Search decision rule (SDR)--is a methodology that does not have several limitations of the linear assumption of the costs in the linear programming framework. The SDR allows the user to determine the values of the costs only on a very general level. That requires the computer programme to be created in such a way it will assess the costs for each plan. Then we look for that plan among all available ones which has the lowest production costs. Compared with linear programming there is no certainty of finding any optimum [35]. Simulation--we can use a lot of simulation models for aggregate planning. The development of the aggregate plan (in the simulation model environment) can be tested under a lot of conditions to find a satisfactory plan. These models can be also included in the decision support system that can help plan and analyse the alternative check rules. These models can have several integrated conflict objectives in the manufacturing strategy by using various quantitative measurements of productivity, customer services and flexibility [36]. Approach of functional objective search (FOS)--is a computer aggregate planning system that contains a wide spectrum of real planning conditions. It can create realistic, low-cost operating plans that provide possibilities to achieve different planning objectives. The systems work on the basis of comparing the basic planned load and the available capacity.