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If the granularity for the exit gauge at each stand is 0.01, the total number of binary bits representing h1 should be 7, since 26≤(2.5-1.54)/0.01≤27; 6 bits for representing h2 since 25≤(1.54-1.03)/0.01≤26; 5 bits for representing h3 since 24≤（1.03-0.74）/0.01≤25; and 5 bits for representing h4 since 24≤（0.74-0.56）/0.01≤25: The binary bits representing each exit gauge are finally linked together as a binary chain. So the total bit number of the final chain is 23.
The initial population is usually chosen at random, or by a heuristic approach. In order to get the best solution, as many points in the search space as possible should be searched. Thus, some variations are introduced into the new population by genetic operations, which include reproduction, crossover and mutation.
3.3.1.2. Operation of the genetic algorithms.
To maximize the possibility of an improvement being achieved by a new population, reproduction is employed to select some parent binary strings (also called chromosomes) with high fitness values. This means that if a binary structure represents a point in a high-performance area of the search space, it may lead to further exploration in this area of the search space. The selected populations are then duplicated, and go on to the next generation. The most important genetic operation is crossover. Under the crossover operator, two binary structures in the new population exchange portions of their binary bits. This is implemented by randomly choosing a crossover point, and exchanging the segments to the right of the crossover point. Also, multiple crossover points could be randomly assigned to get higher evolutionary efficiency. To avoid convergence to a local minimum area during searching, mutation is employed to make the search jump outside the local minimum area. This is implemented by changing the binary value 1 to 0 or 0 to 1 at a randomly selected bit of a binary representation. The whole genetic operation is a systematic binary structure recombination process.
3.3.1.3. Evaluation.
A fitness function is used to measure the performance of each binary string. How close a particular binary string is to the final optimization destination is judged by its corresponding fitness value, and whether the rolling parameters resulting from it violate the constraints described in Section 3.2. In this paper, the total cost function (Eq. (33)) is employed as the fitness function. All the binary strings are ranked on the basis of how small their fitness values are. The higher a string is ranked, the smaller its fitness value is. The top 20 binary strings will be selected as the candidates for the next generation. The whole evolutionary process ceases once the termination criterion is met. The termination criterion in this research is the total number of trials, 5000. Numerical experiments show that this termination criterion is suitable for the research addressed here.
The general genetic model has been constructed in C/C++. Numerical calculations were carried out on Pentium 200, Pentium II 266 and 400 PCs. The whole optimization process took 33 s on the Pentium II 400 MHz PC, 43 s on the Pentium II 266 MHz PC and 99s on a Pentium 200 MHz PC. The final optimization results are reported in the following section.
3.3.2. Optimization parameters and results
3.3.2.1. Optimization parameters.
The total number of trials is 5000 for this optimization procedure. The initial population size is assigned to 20, which means the number of initial randomly generated binary strings is 20. The binary structure length is 23. The crossover and mutation rates are set to 0.65 and 0.005, respectively. The numerical calculation is done under the following three conditions:
1. The coefficients of the cost function are assigned to 0.4 for K1, 0.2 for K2, and 0.4 for K3:
2. The coefficients of the cost function are assigned to 0.35 for K1, 0.3 for K2, and 0.35 for K3: