## 4 Optimal Solution Applied to an Elevator Group Control System
### Simulated Annealing Algorithm  
Simulated annealing is a famous iterative metaheuristic [21]. To obtain an approximate solution (close to the optimum) for computational optimization problems, the SA method based on only the point search technique has been commonly favored by experts as well as researchers. SA begins with the initial solution and tries to improve it in time by producing modifications to that solution at every step. The principal characteristic of SA is that obtaining a worse solution can be accepted with some likelihood that changes in time depending on the temperature and the adjustment of the solution. Consequently, while looking for the best solution to a given problem, the value of every candidate solution should be assessed, and this is accomplished by utilizing an objective function. The algorithm of SA is presented in the following Algorithm 1. ```
begin
1 Input: Solinitial; Tinitial; Trimin; reduction factor \(\alpha\)
2 Output: Solbest
3 Current \(\leftarrow\) Solinitial; Optimal \(\leftarrow\) Current; T \(\leftarrow\) Tinitial
4 while (T > Trim) do
5 for i = 0 to N (maxIter)
6 Next \(\leftarrow\) a randomly selected successor of Current
7 if fitness (Next) < fitness (Current) then
8 Current \(\leftarrow\) Next
9 if fitness (Next) < fitness (Optimal) then
10 Optimal \(\leftarrow\) Next
11 Top \(\leftarrow\) T
12 end if
13 else
14 Afitness = fitness (Next) \(-\) fitness (Current)
15 r \(\leftarrow\) random (0, 1)
16 if r < exp (\(\Lambda\)fitness/kT) then
17 Current \(\leftarrow\) Next // Current = Next only with probability exp(\(-\Delta\)E/KT)
18 end if
19 end if
20 end for
21 T \(\leftarrow\)\(\alpha\) x T
22 end while end. ```  
**Algorithm 1** Simulated Annealing  
The local search is executed N times (maximum of iterations); after that, the temperature T is decreased as in T \(\leftarrow\)\(\alpha\)T, where the parameter \(\alpha\) is considered as the reduction factor (\(0\leq\alpha\leq 1\)). In the SA algorithm, it is essential to obtain precise values for three parameters: the initial temperature (T), the reduction factor (\(\alpha\)), and the maximum number of iterations permitted for each temperature value (N). These parameters are obtained as a product of calibration tests. The calibration process of simulated annealing is as follows:  
1. The initial temperature T is considered in the range between 10 and 180;
2. The reduction factor is tested at the range between 0.1 and 1;
3. The maximum number of iterations N is verified in the range between 1 and 18. Significant research efforts have been made to identify optimal parameters for the SA algorithm [22]. However, it was proven that an optimal and universally applicable configuration does not exist. In fact, various problems require particular parameters specific to each. If each algorithm parameter is conceptualized as a variable and SA performance is conceptualized as an objective value, the search for the best parameter settings is equivalent to navigating this landscape to find the global optimum, a task that varies from problem to problem and involves identifying the most favorable configuration in this broad and diverse parameter space. In establishing parameter values for SA, a common approach involves parameter tuning, relying on systematic experimentation of different parameter values. This method involves conducting multiple tests to determine the values that produce optimal results and then adopting these selected values for the final execution of the algorithm. Importantly, this parameterization technique complies with a fixed value paradigm during algorithm execution, where the selected values remain constant, without dynamic change during runtime. Figure 3 demonstrates that choosing the reduction factor (\(\alpha\)) is of supreme significance. A reduction factor that is too small (from 0.1 to 0.6) will lead to a very bad solution (the average waiting time is very high), whereas a larger value will cause the average waiting time to continue oscillating around the global minimum.