124 A.F. Shapiro Insurance: Mathematics and Economics 26 2000 119–132
Fig. 9. Reproduction, crossover and mutation.
p
j
= fy
j
P
j
f y
j
, and the solutions with the high- est p
j
’s would be bred with one another. A flowchart of the process for generating new pop-
ulations of solutions is depicted in Fig. 9. As indicated, there are three ways to develop a new
generation of solutions: reproduction, crossover, and mutation. Reproduction adds a copy of a fit individ-
ual to the next generation. In the previous example, reproduction would take place by randomly choosing
a solution from the population, where the probability a given solution would be chosen would depend on
its p
j
value. Crossover emulates the process of cre- ating children, and involves the creation of new in-
dividuals children from the two fit parents by a re- combination of their genes parameters. In the ex-
ample, crossover would take place in two steps: first, the fit parents would be randomly chosen on the basis
of their p
j
values; second, there would be a recom- bination of their genes. If, e.g., the fit parents were
11000 and 01101, crossover might result in the two children 11001 and 01100. Under mutation, a small
number of gene values in the population are replaced with randomly generated values. This has the poten-
tial effect of introducing good gene values that may not have occurred in the initial population or which
were eliminated during the iterations. In this illustra- tion, the process is repeated until the new generation
has the same number of individuals as the current one.
Fig. 10. Hill-climbing Metropolis criterion.
6. Simulated annealing
Annealing is the physical process of heating a solid in a heat bath until it melts and then slowly cooling it
down until it crystallizes into a state of perfect struc- ture. The free energy stress of the solid is minimized
during this process.
Simulated annealing is a heuristic that mimics the annealing process in as much as it allows solutions
that increase the value of the objective function to be accepted with a certain probability. Thus, unlike
decent methods, in which only sequences that de- crease the value of the objective function are accepted
for further consideration, simulated annealing is a randomized improvement method that sometimes ac-
cepts a new sequence even though its objective value exceeds that of the old sequence. This procedure,
which is known as hill-climbing, is represented in Fig. 10.
13
The figure shows the interaction of four parame- ters: i, the current solution configuration; j, an alter-
nate configuration, randomly chosen from the neigh- borhood of i; 1C
ij
, the change in the cost function as a result of using configuration j rather than i; and
c, a control parameter.
14
As indicated, j is the next configuration in the series if 1C
ij
≤ 0; otherwise, the
probability that j is the next configuration in the series is exp−1C
ij
c. For a given c, this process continues until an equilibrium is reached.
13
This figure is based on the discussion in Aarts and Van Laarhoven 1987, Chapter 2.
14
In the annealing process, 1C
ij
and c would represent the change in energy and temperature, respectively.
A.F. Shapiro Insurance: Mathematics and Economics 26 2000 119–132 125
Fig. 11. The price of cotton.
The control parameter c is high in the initial stages of the search so that many increases in the objective
function are accepted. It is systematically lowered until it reaches a small value for which virtually no further
deteriorations take place. The last configuration in the process is taken as the solution to the problem.
The Traveling Salesman Problem Lawler et al., 1985 is a well-known problem in combinatorial opti-
mization
15
which can be solved using simulated an- nealing. The problem involves the pair i, C, where i
represent the potential tours of the salesman and C is a cost function which assigns a length of time to each
tour. The problem is to find the tour for which C is a minimum, i.e., the tour with the shortest length.
7. Fractal analysis
