Using this method, one looks for the optimum in the sense of the function extreme by its differentiation. This method leads to the exact determination of the global extreme and all local extremes, which can be treated as the solution of the task if it is located in the area of permissible decisions and if there is no need to examine the edge of the area.
Fig. 6.92 Scheme of construction with a reinforcing rib
Example
The dimensions of the transverse cross section of the socle rib shown in Fig. 6.92 should be chosen so that the weight of the length of its section is the smallest. The construction before the introduction of rib has the stiffness of 20,000 N/m, and after the improvement, it should have the rigidity of 30,000 N/m. The rib is made of pinewood, for which the elasticity modulus amounts to E = 12,000 MPa.
We will begin solving these tasks from building an optimisation model. For this purpose, we determine decision variables. In the discussed case, the cross section of the rib must be described by the dimensions q x p. Restrictions in this task are the conditions for the stiffness of the entire construction k = 20,000 N/m and stiffness of the rib itself kz > 30,000 – 20,000 = 10,000 N/m, whereby
3 3
kz = – E • > 10,000 N/m,
4 (a2 + b2)
and also dimensions of the socle:
0<y = p < 80 mm 0<x = q < 80 mm.
The function of purpose should correspond to the desired minimisation of the weight of the rib. With constant cross-sectional dimensions, along the length of the rib, the condition of the minimum weight corresponds to the condition of the minimum area of cross section. Function of purpose can be therefore formulated as follows:
When using the above relations, the minimum value of the function of purpose is obtained, with the accuracy of the variable y, that is,
= 40000^Щ
3 • E • p
Because the function of purpose does not have an extreme, the searched optimum must be placed on the edge of the area of permissible decisions (Fig. 6.93). This task is best solved using the graph of the area of permissible decisions, with the values of purpose f(x, y) = const for:
P • q > 3E • ^/(a2 + b2)3 = 795,046.4 mm4. (6.266)
In Fig. 6.93, it can be seen that the sought optimum lies at the intersection of the edge determined by the inequalities. Assuming, therefore, that q = 80 mm, we obtain the optimal result p = 1.55 andq = 80 mm, for which the function of purpose has the value f(p, q) = 124 mm2. Of course, a question must be asked, whether such a construction can be made? Probably very hard, due to technological problems related to mounting of the rib. Therefore, if we want to include technology in the mathematical model, additional technological conditions must be taken into account, in this case:
20 <p < 80. (6.267)
The obtained solution q = 80 and p = 20 is then the optimum for both in terms of construction and in terms of technology.
