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Abstract：Scheduling for tandem cold mills refers to the determination of inter-stand gauges, tensions and speeds of a specified product. Optimal schedules should result in maximized throughput and minimized operating cost. This paper presents a genetic algorithm based optimization procedure for the scheduling of tandem cold rolling mills. The optimization procedure initiates searching from a logical staring point Ð an empirical rolling schedule Ð and ends with an optimum cost. Cost functions are constructed to heuristically direct the genetic algorithm's searching, based on the consideration of power distribution, tension, strip flatness4508
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and rolling constraints. Numerical experiments have shown that the proposed method is more promising than those based on semi-empirical formulae. The results generated from a case study show that the proposed approach could significantly improve empirically derived settings for the tandem cold rolling mills. © 2000 Elsevier Science Ltd. All rights reserved.
Keywords: Rolling schedules; Evolutionary algorithms; Tandem cold rolling; Process optimization
1. Introduction
Automation systems for tandem cold rolling mills are continuously being improved due to today's stringent high throughput, quality and low scrap loss requirements for products. To consolidate competitive strengths in the global market, many steel companies are engaged in maximizing the reduction and consequently minimizing the cost of manufacture (Bryant, 1973; Yuen and Nguyen, 1996; Ozsoy et al., 1992). Rolling scheduling is an important aspect in the operation of tandem cold rolling mills. It defines stand reductions, tensions, rolling forces, roll torque, mill maximum speeds, and threading adjustments. The optimized scheduling should lead to improved thickness, surface finish and shape performance of the products.
In the last two decades, only a few papers have addressed the rolling scheduling problem, especially for tandem cold rolling (Wang et al., 1998). An early work has led to the development of mill scheduling systems achieving correct output and satisfactory shape in a tandem cold rolling mill. The scheduling is described as a constrained two-point boundary-value problem that is solved using conjugate gradient and projection techniques. The cost functions defined for the optimization problem include the strip shape cost, tension cost and thermal-crown cost. Although the results generated from the optimized schedules are better than those from the original empirical schedules, power cost is not considered, while a uniform power distribution is desirable for the tandem cold rolling mills. Moreover, the calculation of the costs relies mostly on some linear equations, with linear coefficients given to the rolling parameters. Although the conjugate gradient methods are frequently used in practice, even when the cost function for the optimization problem is not convex, there are reasons to believe that such use leads to the computation of a local minimum (Polak, 1971; Luenberger, 1984; Nash and Ariela, 1996). The computing equipment available at that time also limits the calculation capacity. Ozsoy et al. employed a nonlinear programming method called the hill-climbing algorithm to optimize rolling schedules for a hot rolling process. The results show
that although the optimization problem cannot be solved in a closed form because of the non-linearity of the defining equations and the amplitude constraints on the system variables, it can be solved numerically on a digital computer by nonlinear programming. However, the convergent behavior of the nonlinear programming method employed in (Ozsoy et al., 1992) is directly affected by the initial searching point used. Some other nonlinear programming methods, such as sequential quadratic programming, also have a number of disadvantages besides their added complexity (derivative calculations, etc.), such as the local minimum problem, nonguaranteed convergence, and expensive calculation cost. As an intelligent searching mechanism, genetic algorithms (GA) have potential, and seem to be flexible enough to overcome the abovementoined disadvantages.