Procedure [22] 

The simulated annealing method simulates the process of slow cooling of molten metal to achieve the minimum function value in a minimization prob- lem。 The cooling phenomenon of the molten metal is simulated by introducing a temperature-like parame- ter and controlling it using the concept of Boltzmann’s probability distribution。 The Boltzmann’s probability distribution implies that the energy (E) of a system   in 

thermal equilibrium at temperature T is distributed probabilistically according to the relation: 

 E

P (E ) e kT                                                                                                          (46) 

where P(E) denotes the probability of achieving the  energy level E, and k is called the Boltzmann’s cons- tant。 Equation (46) shows that at high   temperatures the system has nearly a uniform probability of    being  at any energy state。 However, at low temperatures, the system has a small probability of being at a high- energy state。 This indicates that when the search  process is assumed to follow Boltzmann’s probability  distribution, the convergence of the simulated anneal- ing algorithm can be controlled by controlling the tem- perature  T。  The  method  of  implementing  the Boltz-

mann’s probability distribution in simulated  thermody-

Note that the probability of accepting the point Xi+1 is not same in all situations。 As can be seen from Eq。 (50), this probability depends on the values of ∆E and T。 If the temperature T is large, the probability will be high for design points Xi+1 with larger function values  (with larger values of ∆E = ∆f)。 Thus at high tempe- ratures, even worse design points Xi+1 are likely to be accepted because of larger probabilities。 However, if the temperature T is small, the probability of  accept- ing worse design points Xi+1  (with larger values of ∆E  = 

= ∆f) will be small。 Thus as the temperature values 

get smaller (that is, as the process gets closer to the optimum solution), the design points Xi+1 with larger  function values compared to the one at Xi are less likely to be accepted。 

ΔE

namic  systems,  suggested  by  Metropolis  et al。 [18] 

P (E i 1) e

kT                                                                                                       (50) 

can also be used in the context of minimization of functions。 In the case of function minimization, let the current design point (state) be Xi, with the corres- ponding value of the objective function given by fi  = 

= f(Xi)。 Similar to the energy state of a thermodynamic 

system, the energy Ei at state Xi is given by: 

Ei  fi   f (X i  )                                                         (47) 

Then, according to the Metropolis criterion, the pro- bability of the next design point (state) Xi+1 depends on the difference in the energy state or function va- lues at the two design points (states) given by: