702 Modern Methods of Optimization
5.Carry out the mutation operation using the mutation probabilitypmto find the
new generation ofmstrings.
6.Evaluate the fitness valuesFi, i = 1 , 2 ,... , m, of themstrings of the new
population. Find the standard deviation of themfitness values.
7.Test for the convergence of the algorithm or process. Ifsf≤ (sf)max, the con-
vergence criterion is satisfied and hence the process may be stopped. Otherwise,
go to step 8.
8.Test for the generation number. Ifi≥imax, the computations have been per-
formed for the maximum permissible number of generations and hence the
process may be stopped. Otherwise, set the generation number asi=i+1 and
go to step 3.13.2.6 Numerical Results
The welded beam problem described in Section 7.22.3 (Fig. 7.23) was considered by
Deb [13.20] with the following data: population size=100, total string length=40,
substring length for each design variable=10, probability of crossover=0.9, and prob-
ability of mutation=0.01. Different penalty parameters were considered for different
constraints in order to have the contribution of each constraint violation to the objec-
tive function be approximately the same. Nearly optimal solutions were obtained after
only about 15 generations with approximately 0. 9 × 100 × 15 =1350 function evalua-
tions. The optimum solution was found to bex 1 ∗= 0. 2489 , x∗ 2 = 6. 1730 , x 3 ∗= 8. 1789 ,
x∗ 4 = 0. 2 533, andf∗= 2 .43,which can be compared with the solution obtained from
geometric programming,x 1 ∗= 0. 2455 , x 2 ∗= 6. 1960 , x 3 ∗= 8. 2730 , x 4 ∗= 0. 2 455, and
f∗= 2. 3 9 [13.21]. Although the optimum solution given by the GAs corresponds to
a slightly larger value off∗, it satisfies all the constraints (the solution obtained from
geometric programming violates three constraints slightly).13.3 Simulated Annealing
13.3.1 Introduction
The simulated annealing method is based on the simulation of thermal annealing of
critically heated solids. When a solid (metal) is brought into a molten state by heating
it to a high temperature, the atoms in the molten metal move freely with respect to each
other. However, the movements of atoms get restricted as the temperature is reduced.
As the temperature reduces, the atoms tend to get ordered and finally form crystals
having the minimum possible internal energy. The process of formation of crystals
essentially depends on the cooling rate. When the temperature of the molten metal is
reduced at a very fast rate, it may not be able to achieve the crystalline state; instead,
it may attain a polycrystalline state having a higher energy state compared to that of
the crystalline state. In engineering applications, rapid cooling may introduce defects
inside the material. Thus the temperature of the heated solid (molten metal) needs to
be reduced at a slow and controlled rate to ensure proper solidification with a highly
ordered crystalline state that corresponds to the lowest energy state (internal energy).
This process of cooling at a slow rate is known as annealing.