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3.2. Constraints for validity checks
3.2.1. Roll force and roll torque
The total roll force and roll torque are limited to the corresponding maximum values due to the mechanical design limits imposed by manufacturers of the rolling mill and electrical drive motors. This constraint can be described as follows:
3.2.2. Strip exit thickness
The exit strip thickness is kept within its upper and lower limits as follows:
The bounds h min and h max are determined on the basis of the validity check requirements of the rolling mill. Usually, they are physically determined by the mechanical design limits of the mill. The validity checks are used to ensure that operator adjustments do not result in an infeasible set of screw positions on adjacent stands.
3.2.3. Tension force
The tension stress at each stand should not exceed its upper and lower limits.
The t i f min is based on a lower limit on the tension, set by measurement noise, so that the strip does not loop between two neighboring stands. The t i f max is set by strip tearing and skidding considerations. The i f max is usually assigned a value of about one-third of the yield stress of the strip.
After the construction of the cost functions and constraints, the schedule-optimization problem may be described as: starting from an initial searching point h1, h2, h3, h4, h5}, find a satisfactory combination of the strip exit gauges at minimum cost without violating the constraint limitations.
3.3. Genetic-algorithm-based optimization
3.3.1. Genetic algorithms
Genetic algorithms were first proposed by Holland as heuristic searching mechanisms for intelligent systems. Though their use has been growing since the early 1970s (Holland, 1969, 1975), only recently has their commercial potential been demonstrated (Gold-berg, 1989; Davis, 1991; Fogel, 1995). The main operations in a GA include parent selection, reproduction, crossover, and mutation. The goal of these operations is to generate meaningful offspring. A typical step-by-step procedure for a GA is described as follows:
1. Define a fitness function for the optimization problem, e.g. Eq. (33) in this paper.
2. Encode the variables into binary codes, e.g. h1, h2, h3, h4, and then combine the individual binary code for each variable together into a binary chain, as a combined single variable.
3. Initialize a population as the population for the first generation.
4. Evaluate each individual in the population.
5. Implement the genetic operations: reproduction, crossover, and mutation.
6. Rank the population.
7. Delete the lowest-ranked genomes in the population, and keep high-ranked individuals.
8. Repeat Steps 5-7 until the evolutionary termination criterion is satisfied.
3.3.1.1. Genetic encoding of rolling scheduling problem.
In the optimization of the rolling schedule, as the exit thickness at the last stand is fixed, for a five-stand tandem rolling mill, only the exit gauges at the first four stands h1, h2, h3, h4 need to be encoded. For such a multiple variable encoding problem, each variable should be encoded first, and then linked together as a chain. For example, suppose that a steel strip with an initial gauge of 2.5 mm is to be rolled to 0.5 mm, with the following exit gauges based on the empirical reduction patterns:
h1=1.874, h2=1.214, h3=0.853, h4=0.623, h5=0.50
If the average of two adjacent exit gauges is taken as the boundary between the possible value fields of these two exit gauges, the possible value fields for the exit gauges are as follows:
1.54≤h1≤2.5, 1.03≤h2≤1.54, 0.74≤h3≤1.03, 0.56≤h4≤0.74, h5=0.50