A GRASP heuristic for slab scheduling at continuous casters

Abstract

In this paper, we develop an approach for scheduling slabs at continuous casters in the steel industry. The scheduling approach incorporates specific constraints such as flexible production orders, material supply in batches and different setup types. We further introduce a continually adjustable casting width, which corresponds to a technological control parameter. We present a new MILP model formulation, which integrates slab design and scheduling. Solutions for the model are obtained by a greedy randomized adaptive search procedure. We analyze the applicability and performance of the approach in a numerical case study which is based on real world data. High-valued feasible production plans can be obtained in reasonable computing time. The approach is able to solve industry size problem instances in reasonable time. As compared to the status-quo, on average savings in the number of charges of 10.6 % are obtained.

Appendices

Appendix A

There are two types of nonlinearities in the problem. On the one hand, there are products of binary decision variables in the constraints (3), (4), (6), (7) and (13) and on the other hand products of a binary and a continuous decision variable in the constraints (10). As a result, two different types of linearization are necessary.

To linearize the products of binary decision variables, for each pair of multiplied binary decision variables $a$ and $b$, a new continuous decision variable $c$ is introduced, which has to fulfill the following constraints.

$$\begin{aligned}&c \ge a + b - 1 \end{aligned}$$

(20)

$$\begin{aligned}&c \le a \end{aligned}$$

(21)

$$\begin{aligned}&c \le b \end{aligned}$$

(22)

$$\begin{aligned}&c \ge 0 \end{aligned}$$

(23)

Thus, if both $a$ and $b$ are $1$, $c$ has to be $1$ as well, due to constraint (20). Otherwise, $c$ only can be $0$, due to either constraint (21) or (22). Constraint (23) is only necessary to prevent the continuous decision variable $c$ from being negative, if both $a$ and $b$ are $0$. It is dispensible if $c$ is treated as binary decision variable.

To linearize constraints (10), another transformation is used. Since the formulation is an exclusive-or composition depending on $\Delta _j$, the transformation is done as follows.

$$\begin{aligned}&\displaystyle \left( mat_{j-1} + z_{j-1}\cdot Ch - \sum _{i=1}^{n} m_{i} \cdot x_{ij-1}\right) - \Delta _j \cdot M \le mat_j \quad \forall j = 2\ldots n \qquad \quad \end{aligned}$$

(24)

$$\begin{aligned}&\displaystyle mat_j \le \left( mat_{j-1} + z_{j-1}\cdot Ch - \sum _{i=1}^{n} m_{i} \cdot x_{ij-1}\right) + \Delta _j \cdot M \quad \forall j = 2\ldots n \qquad \quad \end{aligned}$$

(25)

$$\begin{aligned}&\displaystyle - M \cdot \left( 1-\Delta _{j} \right) \le mat_j \quad \forall j = 2\ldots n \end{aligned}$$

(26)

$$\begin{aligned}&\displaystyle mat_j \le M \cdot \left( 1-\Delta _{j} \right) \quad \forall j = 2\ldots n \end{aligned}$$

(27)

Thus, if $\Delta _j$ is 0, constraints (24) and (25) bound $mat_j$ as required by the equality sign in constraints (10). Besides, constraints (26) and (27) have no bounding effect. Otherwise, if $\Delta _j$ is 1, constraints (26) and (27) force $mat_j$ to 0, while constraints (24) and (25) have no bounding effect.

Appendix B

The introduced bias function $f(c_i) = c_{i}^{\mathrm{log}_{10}(m\cdot m)}$ is used to determine selection probabilities for each job $p_{i}=f(c_i)/\sum _{k=1}^{m}{f(c_i)}$. It dynamically adopts to the amount of high-evaluated jobs. Besides it assigns selection probabilities to each job, which represent their relative evaluation.

In Table 10, exemplary job portfolios, arranged by job evaluations, are given. Each job portfolio consists of $m=51$ jobs. In portfolio (a), only one job has a high evaluation of 10,000, while all 50 other jobs have a low evaluation of 1. Using the introduced bias function, the high-evaluated job is selected greedily with a selection probability $p_i$ of 100.0 %, while, the low-evaluated jobs are neglected. In portfolio (b), the first job has an evaluation of 10,000, while the evaluation of the following jobs decrease by 200. Note that the evaluation of the last job is set to 1, due to the transformation function introduced in Section 3.1.1. Using the introduced bias function, the selection probabilities decrease from 8.5 % for the first job down to 0.0 % for the last job. The cumulated selection probability for the best 8 jobs is 53.3 %. Thus, in more than the half of selections, one of the best 8 jobs is selected. The cumulated selection probability increases to 81.4 % (95.5 %) until job 16 (25). In reverse, the worst 50 % of jobs have a cumulated selection probability of less than 5 %. Doing so, it is very likely that a good-evaluated job is selected randomly. In portfolio (c), two very good, two medium evaluated and a large number of the weak jobs are within the job portfolio. Using the introduced bias function, the selection probabilities for the very good-evaluated jobs are set to 52.9 % (37.0 %), while even the medium-evaluated jobs have a selection probability of 5.0 % each. The large number of weak evaluated jobs has a cumulated selection probability of 0.0 %. Thus, only good- or medium-evaluated jobs can be selected, while good-evaluated jobs are much more likely to be selected. As a result, the bias function reproduces greediness, if there is only one very good evaluated job, and randomness with regard to the evaluation (and, therefore, to the fitting of charges). Besides, weak-evaluated jobs are neglected. Thus, setting the size of a RCL and selecting a proper bias function on the RCL are unnecessary.

Table 10 Selection probabilities for different job portfolio evaluations