The Charge Batching Planning Problem in Steelmaking Process Using Lagrangian Relaxation Algorithm

Abstract

In this paper, we investigate the charge batching planning problem (CBP) arising from practical steelmaking production. The CBP transforms the primary order requirements into various production batches (charges) subject to the steelmaking processing constraints and composite batch conditions according to the similarity in steelgrade, dimension, physical property, and due-date of orders. On the basis of a practical steelmaking process, a novel mixed-integer programming model for the CBP is presented by considering above constraints and features here, and two kinds of Lagrangian relaxation (LR) methods are proposed to solve the CBP by using different relaxation methods. In the first LR method, the relaxed problem is separated into subproblems by relaxing assignment constraints which are solved optimally by dynamic programming. In the second method, variable splitting is presented by introducing identical copies of some subsets of the original variables. To guarantee the equivalence to the primal problem, a number of equality coupling constraints are added into the model which are relaxed during the course of the second Lagangian relaxation. The multipliers in all above LR methods are then iteratively updated along subgradient directions. Computational experiments have been carried out and the results show that both LR methods can produce satisfactory average duality gaps and the second LR method is little better than the first method. The iron and steel industry has played an important role in the global market economy during the past decade. Along with rapid development, the iron and steel industry also faces fierce competition. To enhance their competitive power, many iron and steel corporations have changed production mode by transforming large lot production into small lot, with multiple varieties for satisfying their customer’s diverse requirements. Since most of production equipment in the iron and steel industry is very large, it often operates in batch mode to save resources and energy consumption, but there is significant conflict between large batch mode and the customer’s diverse requirements. Batching planning groups customer requirements into batches in order to resolve this contradiction and to improve the utilization of large production equipment. This paper investigates a charge batching planning problem arising from practical steelmaking production operation management. A simplified steelmaking production process is illustrated in Figure 1. The steel production flow begins with iron making in the blast furnace. Iron ore is converted to pig iron which is transported by torpedo car to the steelmaking mill. Then pig iron is transformed to liquid steel in the converter. At last the melted steel is solidified into slab in continuous casters. In the steelmaking stage, a charge that is a basic production unit for steelmaking production refers to concurrent smelting in the same converter (or electric arc furnace). In the continuous casting stage, the charges are grouped into different casts. Each cast consists of several charges with similar steelgrades that are processed consecutively on the same continuous caster using the same crystallizer. Reasonable design of charges and casts can improve productivity and cut down resource and energy consumption. The design of charges or casts is also regarded as charge batching planning or cast batching planning, respectively. The charge batching planning (CBP) is a key element of the production operation management in the process industry. It is converting the primary order requirements into various production batches (charges) subject to the steelmaking processing constraints. Figure 2 illustrates the process of making a charge batching plan. The rectangles in the left column in different colors denote steel orders with different steelgrade and specification. The orders with similar steelgrade and specification are denoted by close colors. As shown in Figure 2, orders in similar gray color are put into the first charge and orders in similar black color are put into the second charge. The batching decision problem has recently received more attention from researchers. Trautmann and Schwindt 2 proposed a resource-constrained project scheduling method to deal with