## 4 Multidimensional, Epistemic Fuzzy Arithmetic of Type-2 Fuzzy Numbers
Scope I concerns a very rare situation where \(SB\) is grown on a poor quality soil and the fertilization dose is very low. Scope III is the range which is usually most frequent in the fertilizing trials. The range IV is the range of the excessive amount of the nutrient which causes mineral imbalance in the soil. Looking at this curve it is easy to understand how a major role plays appropriate amount of N for achieving high yield. However, it should be remembered that the grower operates under uncertainty and the fertilization curve for one and the same field changes gradually every year. Mathematically, the nitrogen fertilization curve can be approximated by the 2 nd-order Equation (55). It is assumed that the range I, as it is very rare, does not apply to the field under consideration. \[p=a_{2}d^{2}+a_{1}d+a_{0}=(a_{2}d+a_{1})d+a_{0} \tag{55}\]  
In the Equation (55), \(p\) denotes the \(SB\) yield [q/ha] and \(d\) denotes the \(N\)-fertilize dose [kg/ha]. The coefficients \(a_{0}\), \(a_{1}\), \(a_{2}\) have their practical sense. The \(a_{0}\) [q/ha] coefficient means yield from 1 ha with complete lack of fertilization. The \(a_{1}\) factor represents (approximately) the increment of the yield due to the fertilization increase from \(d=0\) to \(d=1\) [kg/ha]. The coefficient \(a_{2}\) can be called the \"braking\" coefficient. It is negative and it represents a reduction in the impact of each next kilogram of fertilizer due to soil saturation. Note that, in order to identify very, very roughly the values of all 3 coefficients \(a_{0}\), \(a_{1}\), \(a_{2}\), it is necessary to perform at least a 3-year field fertilization experiment, and still only average values of these coefficients for this 3-year period will be obtained. In the following years, slightly different values of these coefficients will be obtained due to the variability of environmental influences. The research on the values of the coefficients is tedious and requires many years. Usually, for the field of interest to us, we have a small number of measurements. Let us now assume that, on the basis of the conducted experiments for a certain field, it was found that the coefficient \(a_{0}\) of zero-fertilization ranges from \([250,450]\), but usually it is within the range \([300,350]\). The yield increment coefficient \(a_{1}\) is generally within the range \([3.0,3.6]\), but usually within the range \([3.0,3.3]\). The inhibition factor \(a_{2}\) is generally in the range \([-0.012,-0.010]\), but most of its values are in the range \([-0.011,-0.010]\). There are global ranges of the values of the coefficients and the ranges of the highest reliability of these coefficients. Knowledge of the fertilization rates can be represented in the form of T2 FNs shown in Figures 32-34. Figure 31: The most common qualitative response curve for nitrogen (\(N\)) fertilization. Knowledge about the values of the fertilization rates does not necessarily have to be provided in the form of 3D drawings. We can more easily present it in the form of FOUs of \(\tilde{A}_{0}\), \(\tilde{A}_{1}\), \(\tilde{A}_{2}\) in 2D space, as shown in Figure 35.  
The formula for the optimal dose value of \(N\)-fertil \(d_{opt}\) can be derived on the basis of the 1 st derivative of the yield \(p\) from Formula (55). If the fertilization rates are known exactly, the result is given by Formula (56). \[d_{opt}=-\frac{a_{1}}{2a_{2}} \tag{56}\]  
Figure 34: Knowledge of the value of the \(a_{2}\left[\mathrm{kg/ha}\right]^{2}\) coefficient of inhibition of yield increase caused by fertilization, expressed in the form of T2 FN. Figure 29:0 Footprints of uncertainty FOUs of T2 FNs \(\tilde{A}_{0}\), \(\tilde{A}_{1}\), \(\tilde{A}_{2}\) representing knowledge about fertilization coefficients. Figure 29:1 Knowledge about the value of the zero fertilization coefficient \(a_{0}\left[\mathrm{q/ha}\right]\) presented in the form of T2 FN. The optimal \(d_{opt}\) fertilization value corresponds to the maximum \(SB\)-yield \(p_{\max}\), Formula (57).