A GRASP heuristic for slab scheduling at continuous casters

Matthias Gerhard Wichmann; Thomas Volling; Thomas Stefan Spengler

doi:10.1007/s00291-013-0330-y

OR Spectrum

July 2014, Volume 36, Issue 3, pp 693–722 | Cite as

A GRASP heuristic for slab scheduling at continuous casters

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Authors and affiliations

Matthias Gerhard WichmannEmail author
Thomas Volling
Thomas Stefan Spengler

Regular Article

First Online: 07 June 2013

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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.

Keywords

Continuous casting Flexible orders Scheduling GRASP

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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

$i$	(a)		(b)		(c)
	$c_i$	$p_{i}\ (\%)$	$c_i$	$p_{i}\ (\%)$	$c_i$	$p_{i}\ (\%)$
1	10,000	100.0	10,000	8.5	10,000	52.9
2	1	0.0	9,800	7.9	9,000	37.0
3	1	0.0	9,600	7.4	5,000	5.0
4	1	0.0	9,400	6.8	5,000	5.0
5	1	0.0	9,200	6.4	200	0.0
$\ldots $	$\ldots $	$\ldots $	$\ldots $	$\ldots $	$\ldots $	$\ldots $
49	1	0.0	400	0.0	200	0.0
50	1	0.0	200	0.0	200	0.0
51	1	0.0	1	0.0	1	0.0

References

Bellabdaoui A, Teghem J (2006) A mixed-integer linear programming model for the continuous casting planning. Int J Prod Econ 104(2):260–270CrossRefGoogle Scholar
Binato S, Hery WJ, Loewenstern D, Resende MGC (2001) A greedy randomized adaptive search procedure for job shop scheduling. In: Hansen P, Ribeiro CC (eds) Essays and surveys on metaheuristics. Kluwer Academic Publishers, DordrechtGoogle Scholar
Bresina JL (1996) Heuristic-biased stochastic sampling. In: Proceedings of the AAAIGoogle Scholar
Chang S, Chang MR, Hong Y (2000) A lot-grouping algorithm for a continuous slab caster in an integrated steel mill. Prod Plan Control 11(4):363–368CrossRefGoogle Scholar
Cicirello VA, Smith SF (2005) Enhancing stochastic search performance by value-biased randomization of heuristics. J Heuristics 11(1):5–34CrossRefGoogle Scholar
Dawande M, Kalagnanam J, Lee HS, Reddy C, Siegel S, Trumbo M (2004) The slab design problem in the steel industry. Interfaces 34(3):215–225CrossRefGoogle Scholar
Fabian T (1958) A linear programming model of integrated iron and steel production. Manag Sci 4:415–449CrossRefGoogle Scholar
Feo TA, Resende MG (1989) A probabilistic heuristic for a computationally difficult set covering problem. Oper Res Lett 8(2):67–71CrossRefGoogle Scholar
Festa P, Resende MGC (2009) An annotated bibliography of grasp—Part I: Algorithms. Int Trans Oper Res 16(1):1–24CrossRefGoogle Scholar
Frisch AM, Miguel I, Walsh T (2001) Modelling a steel mill slab design problem. In: Proceedings of the IJCAI 01 workshop on modelling and solving problems with constraints, pp 39–45Google Scholar
Kruskal JB (1956) On the shortest spanning subtree of a graph and the travelling salesman problem. Proc Am Math Soc 7:48–50CrossRefGoogle Scholar
Lee HS, Murthy SS, Haider SW, Morse DV (1996) Primary production scheduling at steelmaking industries. IBM J Res Dev 40(2):231–252CrossRefGoogle Scholar
Li T, Xi Y (2007) An optimal two-stage algorithm for slab design problem with flexible demands. In: Proceedings of the IEEE, pp 1789–1793Google Scholar
Prais M, Ribeiro C (2000) Reactive grasp: an application to a matrix decomposition problem in tdma traffic assignment. INFORMS J Computm 164–176Google Scholar
Redwine CN, Wismer DA (1974) A mixed integer programming model for scheduling orders in a steel mill. J Optim Theory Appl 14(3):305–318CrossRefGoogle Scholar
Resende M, Ribeiro C (2003) Greedy randomized adaptive search procedures. In: Glover F, Kochenberger G (eds) Handbook of metaheuristics. International Series in Operations Research and Management Science, vol. 57. Springer, New York, pp 219–249Google Scholar
Ross GT, Soland RM (1975) A branch and bound algorithm for the generalized assignment problem. Math Program 8:91–103CrossRefGoogle Scholar
Salzgitter AG (2011) Geschäftsbericht 2010. http://www.salzgitter-ag.de/MDB/Investor_Relations/Downloads/Finanzberichte/2010/szag_gb_2010.pdf
Spengler TS, Seefried O (2003) Optimale Belegungsplanung von Stranggießanlagen mittels 2-dimensionaler Bin-Packing-Modelle. Oper Res Proc 2002:53–58Google Scholar
Stahleisen (ed) (2003) Steel manual. Verlag Stahleisen GmbH, DüsseldorfGoogle Scholar
Stahleisen (ed) (2010) STAHL-Statistik. Verlag Stahleisen GmbH, Düsseldorf. http://www3.stahleisen.de/LinkClick.aspx?fileticket=EQfilXMNNGw%3d
Tang L, Liu G (2007) A mathematical programming model and solution for scheduling production orders in shanghai baoshan iron and steel complex. Eur J Oper Res 182:1453–1468CrossRefGoogle Scholar
Tang L, Liu J, Rong A, Yang Z (2001) A review of planning and scheduling systems and methods for integrated steel production. Eur J Oper Res 133(1):1–20Google Scholar
Tang L, Wang G (2008) Decision support system for the batching problems of steelmaking and continuous-casting production. Omega 36(6): 976–991. A Special Issue Dedicated to the 2008 Beijing Olympic GamesGoogle Scholar

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Authors and Affiliations

Matthias Gerhard Wichmann
- 1
Email author
Thomas Volling
- 1
Thomas Stefan Spengler
- 1

1.Institute of Autmotive Management and Industrial ProductionTechnische Universität Braunschweig BraunschweigGermany

A GRASP heuristic for slab scheduling at continuous casters

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\(i\)	(a)		(b)		(c)
	\(c_i\)	\(p_{i}\ (\%)\)	\(c_i\)	\(p_{i}\ (\%)\)	\(c_i\)	\(p_{i}\ (\%)\)
1	10,000	100.0	10,000	8.5	10,000	52.9
2	1	0.0	9,800	7.9	9,000	37.0
3	1	0.0	9,600	7.4	5,000	5.0
4	1	0.0	9,400	6.8	5,000	5.0
5	1	0.0	9,200	6.4	200	0.0
\(\ldots \)	\(\ldots \)	\(\ldots \)	\(\ldots \)	\(\ldots \)	\(\ldots \)	\(\ldots \)
49	1	0.0	400	0.0	200	0.0
50	1	0.0	200	0.0	200	0.0
51	1	0.0	1	0.0	1	0.0