13.3 Simulated Annealing 705
computational effort. A smaller value ofn, on the other hand, might result either in a
premature convergence or convergence to a local minimum (due to inadequate explo-
ration of the design space for the global minimum). Unfortunately, no unique set of
values are available forT,n, andcthat will work well for every problem. However,
certain guidelines can be given for selecting these values. The initial temperatureT
can be chosen as the average value of the objective function computed at a number
of randomly selected points in the design space. The number of iterationsncan be
chosen between 50 and 100 based on the computing resources and the desired accu-
racy of solution. The temperature reduction factorccan be chosen between 0.4 and
0.6 for a reasonable temperature reduction strategy (also termed the cooling schedule).
More complex cooling schedules, based on the expected mathematical convergence
rates, have been used in the literature for the solution of complex practical optimiza-
tion problems [13.19]. In spite of all the research being done on SA algorithms, the
choice of the initial temperatureT, the number of iterationsnat any specific tem-
perature, and the temperature reduction factor (or cooling rate)cstill remain an art
and generally require a trial-and-error process to find suitable values for solving any
particular type of optimization problems. The SA procedure is shown as a flowchart
in Fig. 13.2.
13.3.4 Features of the Method
Some of the features of simulated annealing are as follows:
1.The quality of the final solution is not affected by the initial guesses, except
that the computational effort may increase with worse starting designs.
2.Because of the discrete nature of the function and constraint evaluations, the
convergence or transition characteristics are not affected by the continuity or
differentiability of the functions.
3.The convergence is also not influenced by the convexity status of the feasible
space.
4.The design variables need not be positive.
5.The method can be used to solve mixed-integer, discrete, or continuous
problems.
6.For problems involving behavior constraints (in addition to lower and upper
bounds on the design variables), an equivalent unconstrained function is to be
formulated as in the case of genetic algorithms.
13.3.5 Numerical Results
The welded beam problem of Section 7.22.3 (Fig. 7.23) is solved using simu-
lated annealing. The solution is given byx 1 ∗= 0. 2471 , x∗ 2 = 6. 1451 , x∗ 3 = 8. 2721 ,
x 4 ∗= 0. 2 495, andf∗= 2. 4 148. This solution can be compared with the solutions
obtained by genetic algorithms (x 1 ∗= 0. 2489 , x 2 = 6. 1730 , x 3 ∗= 8. 1789 , x∗ 4 = 0. 2 533,
and f∗= 2. 4 331) and geometric programming (x 1 ∗= 0. 2536 , x∗ 2 = 7. 1410 , x 3 ∗=
7. 1044 , x 4 ∗= 0. 2 536, andf∗= 2. 3 398). Notice that the solution given by geometric
programming [13.21] violated three constraints slightly, while the solutions given by
the genetic algorithms [13.20] and simulated annealing satisfied all the constraints.
