## 1 Introduction
Figure 1: Formation of the transverse distribution of the sprayed liquid obtained from the boom of a field sprayer utilising flat fan sprayers: (**a**) streams viewed from the side; (**b**) streams viewed from above; (**c**) combining the streams of individual nozzles. Every agricultural sprayer should undergo obligatory technical condition inspections once every three years, except for new machines that are inspected before a five-year period expiry date. So far, in practice it is usually assumed that when some nozzles of a field sprayer are worn out, it is necessary to replace the set of flat fan sprayers in the sprayer concerned [8; 10; 11]. This inevitably results in the purchase of an expensive set of new flat fan sprayers. However, already in the work [6] it is indicated that the homogeneity of the sprayed liquid distribution and the application of pesticides is conditioned not only by the improvement of the manufacturers of sprayer components in meeting the operating standards of these entire devices but by maintaining the parameter uniformity of the newly produced flat fan nozzles. [PERSON] and [PERSON] note that the uniformity of the distribution of chemical plant protection products, as required by relevant regulations, can also be obtained by permutating the worn-out flat fan sprayers on the beam of field sprayers [12]. The methods of the monitoring search for the extremum of the objective function are characterised by great simplicity and naturalness, because they consist in finding the value of the function at each point of the set of feasible solutions. However, indicating such a permutation of 24 flat fan nozzles, which is close to the optimal permutation regarding the variation coefficient out of all 24! = 6.20448 \(\times\) 10\({}^{+23}\) permutations of flat fan nozzles with a varying wear degree, turns out to be an \(NP\)-difficult computational problem. (The problem becomes even more computationally complicated when we consider the possibility of rotating each of the nozzles on the boom by 180\({}^{\circ}\). Then, as many as 2\({}^{(24-1)}\).24! = 5.2047 \(\times\) 10\({}^{+30}\) different nozzle-positioning results may be obtained.) Illustratively, this means that in a reasonable amount of time this problem cannot be ultimately and optimally solved even with the use of good equipment. Therefore, the authors of the cited work suggested a search for the best satisfactory solution out of only 10,000 randomly drawn permutations related to the ordering of a group of only 20 flat fan nozzles. Unfortunately, this means drawing conclusions about the distribution of the coefficient of variation based on a random sample constituting only a 1.61174 \(\times\) 10\({}^{-20}\) part of the population of all permutations (respectively, a 1.92134 \(\times\) 10\({}^{-27}\) part of the population of all various possible sprayer settings) and searching for the best global minimum of the examined parameter only in such an insignificant part of all permutations (respectively, all possible nozzle settings). The solution obtained in this way is optimal only for the group of 10,000 permutations taken into consideration, but it is impossible to apply this result to the actual optimal solution among all permutations (respectively, all possible settings of the nozzles). Moreover, this method guarantees neither approximation to the requirements of the standard nor approximation to the global optimum. Thus, even with its multiple application to selected flat fan sprayers, one cannot achieve any resolution as to whether it is worth looking even further by selecting blindly another group of random, e.g., 10,000, permutations. The purpose of the present work is to propose a way out of this often endless sequence of computations performed for a blind sampling of 10,000 permutations.