# 6.9.6. A genuinely optimum fire bucket - часть 5

**Fig. 6.44. the "Rotated" optimum fire bucket**

Step 10 in fig. 6.44 is the nucleus of the solution of our: in this, a two-parameter membership function is generated by the merging (akin to multiplication) of two single-parameter membership functions.

In FST there are no traditional concepts of addition, subtraction, multiplication etc.(those represented by the mathematical operators "+", "-", "×" etc. in the Mathcad environment). In FST, multiplication (crossing of sets – logical AND) is replaced with an operation of searching for a minimum; and addition (merging of sets – logical OR) by searching for a maximum. The mathematics of precise sets is a special case (a subset of the mathematics of fuzzy sets) where these operators/functions are genuinely equivalent. This means that in the Mathcad environment, where there are no built- in AND and OR operators, we can create equivalents using the **min** and **max** functions we described in Etude 3.

In our task the membership function mu_rh is generated by the fuzzy addition (**min**) of the functions mu_r, mu_h and mu_v. That is, the fuzzy set "the convenient bucket " is the intersection of three other fuzzy sets: "convenient radius of bucket" (step 1), "convenient height of bucket" (step 6) and "a not heavy bucket" (step 9).

Step 11. The top of the 'mountain' – the surface plot of function mu_rh – is the point where the parameters of most convenient fire bucket lie.

**Fig. 6.45. Designing the optimum fire bucket**

Step 12 (fig. 6.45). Searching for the maximum of the function mu_rh in the Mathcad environment can be done in various ways (see Etudes 2 and 3). We'll proceed this way: we'll imagine metal disks of various radius R (from 10mm-500 mm with a step of 1 mm) and cut each size into 2 to 10 identical buckets. This gives a large variety of bucket sizes and geometries, and we'll consider the optimum (the most convenient) bucket to be the one for which the membership function mu_rh is maximised.