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Appendix B
Simulated Annealing Method
In this Appendix, we summarize the simulated annealing method which is reported by Kirk-patrick et al. [14] and Laarhoven and Aarts [24]. Simulated annealing is an effective method for finding a good optimal solution to an optimization problem. Usually, finding an optimal solu-tion for some optimizasolu-tion can be a difficult task. This is not only because when the problem becomes sufficiently large we need to search enormous number of feasible solutions to find the optimal, but also because even using modern computing methods there are still too many feasible solutions to consider. In our model, we need to search five parameters, which areM, pM, pL, h, andm. The vector xk denotes the five parameters at state k in the searching precess, and f(x) is the objective function of our optimization model. It has over 90 million feasible solutions to be considered. Therefore, we can extremely decrease computing time via simulated annealing.
Firstly, let us figure out how simulated annealing works. The main idea of simulated anneal-ing algorithm is inspired from the procedure of annealanneal-ing in metallurgy. Annealanneal-ing involves heating and cooling a metal to change its internal structure and physical properties. When the structure becomes fixed, metal consequently can retain its new obtained properties. So, in simu-lated annealing, we will set up a parameter called temperature to simulate the heating or cooling process. As the algorithm runs, the temperature will be allowed to decrease slowly. While the temperature parameter at high value, the simulated annealing algorithm will be allowed to accept feasible solutions that are worse than the current solution more frequency. This useful property
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gives the algorithm the ability to jump out of the local optimums. Therefore, simulated anneal-ing algorithm is remarkably effective for findanneal-ing a close to optimum solution when dealanneal-ing with searching numerous local optimums.
Secondly, before shows the algorithm, we should introduce the acceptance probability func-tion. Since simulated annealing will occasionally accept worse solutions, it depends on Boltz-mann’s function to decide which solutions to accept. That is
PR(f(xk)) =min{1, e
(−∆ f Tk
)
},
wherePR(f(xk))is the probability that the algorithm will accept the current statexk,∆ f is the difference of the objective function value between current statexkand the previous state xk−1, andTkis the control temperature in statek. After having the PR(f(xk)), we compute a random variableV between 0 and 1 to compare with it. If V >PR(f(xk)), then we abandon the statexk, otherwise accept it. Eventually, in our algorithm, we check if the neighbor solution is better than our current solution. If it is, we accept it unconditionally. If however, the neighbor solution is worse than the current solution, we use Boltzmann’s function to decide whether accept it or not.
From the Boltzmann’s function we know that the algorithm is more likely accept solutions at high temperatures. Actually, the temperature parameterT is not constant, we set Tk+1=0.95Tk.
Finally, we give the pseudo-code of the simulated annealing in Algorithm 1.
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Algorithm 1 Simulated Annealing Algorithm
Require: f(x): objective function;x0: initial solution;T0: initial temperature;K: step size;
Ensure: optimal best valuexb
1: Initialk = 0, current best state xb = x0, current best value f(xb) = f(x0), current tem-peratureTC = T0;
2: whilek <K do
3: Compute the neighbor ofxk, which isxk+1;
4: if f(xk+1) ≤ f(xk)then
5: xb =xk+1, f(xb) = f(xk+1);
6: else
7: Compute the PR(f(xk))and the random variableV;
8: if V ≤ PR(f(xk))then
9: xb = xk+1, f(xb) = f(xk+1);
10: else
11: xb = xk, f(xb) = f(xk);
12: end if
13: end if
14: Tk+1 =0.95Tkandk=k+1;
15: end while
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