Elsevier

Computers & Chemical Engineering

Volume 106, 2 November 2017, Pages 133-146
Computers & Chemical Engineering

Review
An improved Lagrangian relaxation approach to scheduling steelmaking-continuous casting process

https://doi.org/10.1016/j.compchemeng.2017.05.026Get rights and content

Highlights

  • We study a mixed integer mathematical model for the scheduling in the steelmaking-continuous casting process.

  • The concave–convex procedure is introduced to deal with the subproblems of the relaxed problem.

  • The convergence of the concave–convex procedure is established under some appropriate assumptions.

  • An improved conditional surrogate subgradient algorithm is proposed to solve the Lagrangian dual problem.

  • A simple heuristic algorithm is designed to construct a feasible schedule by adjusting the solutions of the relaxed problem.

Abstract

In the steelmaking continuous-casting (SCC) process, scheduling problem is a key issue for the iron and steel production. To improve the productivity and reduce material consumption, optimal models and approaches are the most useful tools for production scheduling problems. In this paper, we firstly develop a mixed integer nonlinear mathematical model for the SCC scheduling problem. Due to its combinatorial nature and complex practical constraints, it is extremely difficult to cope with this problem. In order to obtain a near-optimal schedule in a reasonable computational time, Lagrangian relaxation approach is developed to solve this SCC scheduling problem by relaxing some complicated constraints. Owing to the existence of the nonseparability coming from the product of two binary variables, it is still hard to deal with this relaxed problem. By making use of their characteristics, the subproblems of the relaxed problem can be converted into different difference of convex (DC) programming problems, which can be solved effectively by using the concave–convex procedure. Under some reasonable assumptions, the convergence of the concave–convex procedure can be established. Furthermore, we introduce an improved conditional surrogate subgradient algorithm to solve the Lagrangian dual (LD) problem and analyze its convergence under some appropriate assumptions. In addition, we present a simple heuristic algorithm to construct a feasible schedule by adjusting the solutions of the relaxed problem. Lastly, some numerical results are reported to illustrate the efficiency and effectiveness of the proposed method.

Introduction

The iron and steel industry, one of the cornerstone industries, makes a material contribution to the world economy by providing raw materials for a number of other important industries, such as, machinery manufacturing, shipbuilding, petro-chemical and construction industry. Unlike other industries, the process of iron and steel production runs at high-temperature and high-weight material flow with complicated technological processes and extensive energy consumption (Li et al., 2012). In the iron and steel industry, SCC process plays a significant role, since it is one of the largest bottlenecks in the manufacturing process. Production scheduling in the iron and steel industry has been recognized as one of the most difficult industrial scheduling problems (Harjunkoski and Grossmann, 2001). In the SCC process, two main tasks of scheduling are to determine which orders are allocated to each machine, and assign the sequence of orders allocated at each stage (Li et al., 2011, Tang et al., 2000). Due to its combinatorial nature, strict requirements on material continuity and complex practical constraints, it is extremely challenging to solve the scheduling problems of the SCC process. Optimal scheduling of the SCC process can effectively improve machine productivity, reduce material and energy consumption and minimize production cost (Li et al., 2016b, Ye et al., 2014). Therefore, it is critical to develop an effective and efficient optimization model and approach to cope with the complicated scheduling problem of the SCC process.

In recent years, most published works on optimization models and approaches for scheduling problems of the SCC process can be roughly classified into three categories: mathematical programming methods, artificial intelligence methods and heuristic methods. Using mathematical programming methods, Bellabdaoui and Teghem (2006) presented a mixed integer mathematical model for the scheduling of steelmaking continuous casting production, which can be solved by using some standard software packages. Harjunkoski and Grossmann (2001) presented a decomposition algorithm to split the large scheduling problem of steel industry into smaller subproblems that can often be solved optimally by using mathematical programming methods. Mao et al. (2014) modeled the SCC scheduling problem as a mixed-integer linear programming problem and proposed a novel Lagrangian relaxation approach to solve this problem. Mao et al. (2015) presented a time-index formulation for the SCC scheduling problem and developed an effective subgradient method and dynamic programming approach to deal with this scheduling problem. Tang et al. (2002) formulated a novel integer programming formulation with a separable structure for SCC scheduling problem and developed an improved solution method by combining Lagrangian relaxation, dynamic programming and heuristics to solve this problem. Ye et al. (2014) introduced robust optimization and stochastic programming approaches for addressing a medium-term production scheduling of the large-scale steelmaking continuous casting process under demand uncertainty. With respect to artificial intelligence methods, Atighehchian et al. (2009) investigated a novel iterative algorithm by combining ant colony optimization and non-linear optimization methods for scheduling of the SCC production. Jiang et al. (2015) investigated a mathematic programming model for the SCC scheduling problem with controllable processing times and proposed a meta-heuristic algorithm by comparing differential evolution algorithm with a variable neighborhood decomposition search to address this problem. Li et al. (2014) formulated a realistic hybrid flowshop scheduling problem model for steelmaking casting process and developed an effective fruit fly optimization algorithm to solve the steelmaking casting problem. Li et al. (2016) proposed a hybrid fruit fly optimization algorithm and successfully applied to solve the hybrid flowshop rescheduling problem with flexible processing time in steelmaking casting systems. Long et al. (2016) studied a dynamic scheduling model with NP-hard feature for the SCC scheduling problem under the continuous caster breakdown and developed a hybrid algorithm featuring a genetic algorithm combined with a general variable neighbourhood search to solve this model. Pan (2016) addressed a new SCC scheduling problem arising from iron and steel production process, modeled this problem as a combination of two coupled sub-problems and presented a novel cooperative co-evolutionary artificial bee colony algorithm with two sub-swarms to address the sub-problems of this scheduling problem, respectively. Tang and Wang (2010) designed an improved particle swarm optimization algorithm for the hybrid flowshop scheduling problem in the integrated production process of steelmaking continuous-casting. Tang et al. (2014) studied an improved differential evolution algorithm to solve a challenging problem of dynamic scheduling in the SCC production. Zhao et al. (2011) formulated a mathematical programming model for the SCC scheduling problem and proposed a tabu search algorithm to deal with the allocation and sequencing decisions. As for heuristic methods, Missbauer et al. (2009) proposed a mixed integer linear programming model for the SCC scheduling problem and presented a three-stage heuristic solution procedure to improve the schedule by means of a linear programming model. Pacciarelli and Pranzo (2004) modeled the SCC scheduling problem by means of the alternative graph and described a beam search procedure to tackle with this problem. Yu and Pan (2012) proposed a three-stage rescheduling method including the batches splitting, forward scheduling method and backward scheduling method for solving a novel multi-objective nonlinear programming model of the SCC production process. Yu et al. (2016) considered a job start-time delay issue for the SCC rescheduling problem and carried out an effective heuristic rescheduling algorithm for the SCC production system to quickly respond to any disruption with a proper rescheduling plan.

Inspired by the above existing literatures, the motivation and main contribution of this paper are in following directions. Firstly, the optimization models for the SCC scheduling problems are usually described by adopting a big-M strategy (Harjunkoski and Grossmann, 2001, Jiang et al., 2015, Li et al., 2016a, Long et al., 2016, Missbauer et al., 2009, Mao et al., 2014, Pan, 2016, Tang et al., 2002, Tang et al., 2014, Tang and Wang, 2008, Ye et al., 2014), which play a significant role in improving the productivity and reducing the cost of the entire production process. In the big-M strategy, the main drawbacks are that the computation time will increase owing to the existence of redundant constraints (Tang et al., 2013, Vallada and Ruiz, 2011) and the big-M formulation usually produces much looser lower bound (Mao et al., 2015). As a result, we address a new mixed integer nonlinear mathematical model for the SCC scheduling problem without using the big-M strategy to avoid above weaknesses. Secondly, in most cases, scheduling problems of the iron and steel industry are NP-hard, which implies that no algorithm can optimally solve these problems within a reasonable computation time (Chen and Luh, 2003). In 1988, Gupta (1988) has proved that the two-stage flowshop problem with identical multiple machines at each stage is NP-hard and two-stage flowshop problem is also NP-hard even if the number of machines at one of the two stage is one. Due to the complexity, the SCC scheduling problem addressed in this paper is much more complicated than the two stages flowshop scheduling problem (Gupta, 1988), which means that the SCC scheduling problem is also NP-hard. Therefore, the SCC scheduling problem cannot be solved optimally within the reasonable computation time. Thus, Lagrangian relaxation approach is introduced to deal with the SCC scheduling problem, because this approach can provide a lower bound to evaluate the optimality of solutions and yield a near-optimal schedule in a reasonable computational time (Nishi and Hiranaka, 2013). Up to now, published works on the Lagrangian relaxation approaches have mainly focused on relaxing the complicated constraints to decompose relaxed problems into some simple subproblems, whose optimal solutions can be obtained easily (Buil et al., 2012, Fu and Diabat, 2015, Mao et al., 2014, Mao et al., 2015, Nishi et al., 2010, Nishi and Hiranaka, 2013, Sun and Yu, 2015, Tang et al., 2002, Tang and Liu, 2007). Nevertheless, for some complicated scheduling problems, it is extremely difficult to decompose the relaxed problem into some simple subproblems, due to the nonseparability coming from the product of two binary variables. To the best of our knowledge, few studies have been reported on how to cope with the nonseparable issue of the relaxed problem for the SCC scheduling problems. By relaxing the complex constraints, the relaxed problem can be decomposed into two separable subproblems. However, it is extremely hard to obtain the optimal solutions of these two subproblems, because the Lagrange function is nonseparable. By making use of the characteristics of the relaxed problem, the subproblems are converted into different DC programming problems that can be solved by using the concave–convex procedure. Thirdly, the multipliers are usually updated along the subgradient directions that can be obtained by fully optimizing the relaxed problem, which means that the convergence of the concave–convex procedure plays a crucial role in updating the Lagrangian multipliers. Under some reasonable assumptions, its convergence is analyzed to guarantee that the sequence generated by the concave–convex procedure can converge to a stationary point of the relaxed problem. Fourthly, another challenge task in the Lagrangian relaxation approach is to maximize the dual function effectively and efficiently (Bertsekas, 1999). Due to its low memory requirements, subgradient method is frequently used to optimize the dual function. However, zigzagging phenomena will occur for the standard subgradient method and the subgradient directions are obtained by fully optimizing the relaxed problem, which can influence the convergence speed. To improve the efficiency, we design an improved conditional surrogate subgradient algorithm to solve the LD problem and prove its convergence under some appropriate assumptions. Lastly, the solution obtained by solving the LD problem may not satisfy machine capacity constraint or operation precedence relationship constraint of the SCC scheduling problem, which means that it is not a feasible schedule. Hence, a heuristic algorithm is constructed to obtain a feasible schedule by adjusting appropriately the solutions of the relaxed problem.

The rest of the paper is organized as follows. In the next section, we give a brief description of the production process in steelmaking-continuous casting. In Section 3, we formulate a mixed integer nonlinear mathematical model for the SCC scheduling problem. In Section 4, an improved solution methodology is designed by combining Lagrangian relaxation, the concave–convex procedure, conditional surrogate subgradient algorithm and a simple heuristic algorithm to solve the SCC scheduling problem. Some numerical results are reported to show the efficiency and effectiveness of the conditional surrogate subgradient method in Section 5. In the last section, we make some conclusions about this study.

Section snippets

Process description of steelmaking-continuous casting

The SCC production is regarded as a bottleneck in the steel production because its production capacity is in general lower than that of hot rolling and cold rolling production (Tang and Wang, 2008). The main production process of SCC can be broadly classified into three stages: steelmaking, refining and continuous casting as illustrated in Fig. 1.

Steelmaking process is one of the most energy-intensive processes for producing molten steel from the main raw materials including iron ore and scrap

Mathematical formulation of the SCC scheduling problem

The following notations are adopted to establish a mathematical model for the SCC scheduling problem.

Sets and number of elements
j:index of stage;
i, r:charge index;
n:cast index;
Mj:number of the identical parallel machines in stage j;
Ω:the set of all charges, |Ω| is the total number of charges;
Ωn:the set of all charges in the nth cast, n = {1, 2, …, N}, where N is the total number of casts, Ωn1Ωn2=,Ωn1Ωn2Ωnn=Ω, for all n1  n2  {1, 2, …, N};
Bk:the set of indices of all casts on the kth machine

Lagrangian relaxation

In the model formulated above, there exist different variables in the constraints (9) and (10), which make this model difficult to deal with. By relaxing the constraints (9) and (10), we can receive the relaxed problem, given as followsL(μ,λ)=min(F1+F2+F3+F4),withF1=i=1|Ω|j=1S1Cj(ti,j+1ti,j),F2=i=1N(D1ts(n1)+1,Sl+D2ts(n1)+1,Su),F3=i=1|Ω|r=1,ri|Ω|j=1S1k=1Mjμirjk(yrik+yirkxijkxrjk),F4=i=1|Ω|r=1,ri|Ω|j=1S1k=1Mjλirjkyrik(ti,jtr,jPr,j).Subject to (5)–(8), (11)–(13), (15) and

Numerical experiments

In this section, some numerical results are provided to test the performance of the proposed method. In our experiment, all algorithms are implemented in C# language and run on PC with Intel Core i7-4770 3.4 GHz CPU, and windows 10 operation system (64 bit). In all algorithms, upper bounds, lower bounds, duality gaps, iteration numbers and running time are used to evaluate the performance of the conditional surrogate subgradient method by comparing with the improved surrogate subgradient method

Conclusions

In this paper, we study a new mixed integer nonlinear mathematical model for a real-world scheduling problem arising from the SCC process. Due to its complexity, Lagrange relaxation approach is adopted to solve this problem, which can yield near-optimal schedules within a reasonable computational time. By relaxing the complicated constraints, the relaxed problem can be decomposed into three subproblems by using the concave–convex procedure. To improve the computational efficiency, we devise an

Acknowledgements

The authors would like to thank the editor and anonymous referees for their valuable comments and suggestions, which improve this paper greatly. This work was partly supported by the National Natural Science Foundation of China (61333006, 61473074, 51634002).

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