A Hybrid Approach for the Integrated Scheduling of Steel Plants

Abstract

The integrated scheduling for the continuous casting (CC) and hot rolling (HR) process remains challenging in the iron and steel production. Taking into account the technological and practical constraints, we establish a mathematical model with the objective of maximizing the number of slabs processed in hot charge rolling (HCR) or direct hot charge rolling (DHCR) mode. Using the decomposition strategy, a hybrid algorithm is proposed based on the combination of the mathematical programming and constraint programming (CP) methods. Computational results demonstrate that the hybrid algorithm is efficient and effective for solving the integrated scheduling problem (ISP) of steel plants.

1. Introduction

Since the development of the automation and integration techniques in the iron and steel manufacture, the appropriate integrated production planning and scheduling for the continuous casting (CC) and hot rolling (HR) process has been of great significance to reduce the energy cost and improve the production efficiency. The integrated process requires that the production management should guarantee the time consistency, as well as the balance of material flow between two production stages. As the bottleneck process, CC can cast the high-temperature molten steel into slabs, which is a primary process to the upstream and downstream steps of the production line. In addition, the HR process substantially affects the quality of finished products, the due date of contract and the satisfaction degree of customers. Thus, the integrated scheduling for CC and HR process plays an important role in reducing the energy consumption, shortening the cycle life of products, enhancing the product quality, and controlling the inventory of intermediate products (slabs) and finished products (coils).

Currently, planning and scheduling decisions are made independently at different production stages. At each stage, the scheduling only considers the production constraints within the stage in question. Generally, the HR production plan is made firstly according to customer orders. Meanwhile, the required slabs are provided to the preceding production process. Thereafter, the steelmaking scheduling plan is made based on the HR production plan. However, this organization tends to induce substantial workload, thus delaying the production in the HR process. Therefore, it is essential and necessary to make an effective integrated scheduling plan.

In the iron and steel manufacture, production planning and scheduling were received considerable attention to study. In view of the importance to the theoretical researches and the practical applications, a large number of approaches^1,2,3,4,5) were proposed to solve the scheduling problems. A comprehensive review of previous studies regarding the integrated steel production planning and scheduling could be found in Tang et al.⁶⁾ Meanwhile, numerous approaches such as tabu search,⁷⁾ genetic algorithm,⁸⁾ and particle swarm optimization algorithm⁹⁾ were developed to solve the HR batch scheduling problem over the past decade. For the integrated production plan, Tang et al.¹⁰⁾ investigated two batching problems for the steelmaking and CC production, and proposed two heuristic algorithms for the corresponding problems. With respect to the HR scheduling problem, Zhao et al.¹¹⁾ formulated the batch planning of staple material as a vehicle routing problem with time windows (VRPTW), and optimized the batches of established units by adjusting the rolling sequences to reach a higher hot charge ratio.

In this study, we summarize the novelties of the integrated scheduling problem (ISP) as follows: (1) Unlike the problem for an independent stage, the integrated scheduling simultaneously considers the technological constraints within the CC and HR production process to guarantee the balance of material flow and the synchronization of scheduling time; (2) Our optimal objective is to maximize the total number of slabs processed in modes of hot charge rolling (HCR) or direct hot charge rolling (DHCR), further enhance the HCR and DHCR ratios to ensure reducing the energy consumption and shortening the preparation time before rolling, where HCR (DHCR) ratio is the percent of the amount for slabs processed in HCR (DHCR) mode with the total number of slabs. In the modern integrated production process, the HCR ratio and DHCR ratio are important indices for evaluating the performance of the iron and steel production.

In addition, a hybrid solution approach based on the decomposition strategy is developed to solve the complex ISP, which is motivated from the work of Jain and Grossmann.¹²⁾ An integration algorithm for mixed-integer linear programming (MILP)/constraint programming (CP) was initially developed by Jain and Grossmann based on the Benders decomposition to solve a class of scheduling problems. Thereafter, various integration schemes^13,14,15,16) with taking advantage of the complementary strengths of MILP and CP were proposed for solving the complicated combinatorial optimization problems.

2. Problem Description and Mathematical Model

2.1. Problem Description

The iron and steel production flow consists of three key stages, i.e., steelmaking, CC and HR. At the steelmaking stage, the mixture of molten iron and raw materials is reduced to desirable molten steel with the main alloy elements. At the immediate stage, CC casts the molten steel into slabs and provides slabs for the following production process. At the HR stage, the slabs are rolled into required steel plates or coils according to customer orders. The slab yard between the CC and HR stage includes a heat preservation pit and a cold slab yard, which are regarded as buffers. In the general production practice, the CC slabs may be transported to the customers as intermediate products, delivered to the slab yard to wait for rolling, transported to the heat furnace for reheating, or even directly rolled.

According to the temperature of slabs and the way through which the CC provides slabs for the HR, the linkage mode between the CC and HR process defines the following production processes, cold charge rolling (CCR), HCR, DHCR, and hot direct rolling (HDR), as illustrated in Fig. 1.

Fig. 1.

Linkage modes between CC and HR process.

In the traditional CCR process, CC slabs are transported firstly by transfer rails to the slab yard. Thereafter, the slabs are loaded into the heat furnace according to the rolling plan. The charge temperature is generally below 400°C.

In the HCR process, tepid slabs are sent to the heat preservation pit after the CC process. Then, the slabs are lifted from the pit and charged into the heat furnace for reheating according to the rolling requirements. The temperature of initial slabs ranges from 400 to 700°C.

In the DHCR process, hot and immaculate slabs are directly charged into the heat furnace through transfer rails, and then the HR process is undertaken. The charge temperature varies from 700 to 1000°C.

The main features of the HDR process are as follows: without being heated in the heat furnace, hot and immaculate slabs are directly sent to the HR machine after the slab edges have been heated by the heater. In general, the slabs temperature is kept over 1100°C before rolling.

As the process objects (casts, rolling units) are cast or rolled continuously at a high temperature, more direct linkages between the CC and HR stage indicate less energy consumption and shorter production lead time. It is obvious that the HDR technique can achieve the highest energy savings, followed by DHCR, HCR and CCR. Similar trends exist in the preparation time before rolling. Due to the necessity for a special configuration of the production line and the high organization level, the HDR technique is seldom implemented in the modern iron and steel manufacture.

The steel strips in which rolling unit and the sequence of strips are defined at the planning level with considering multiple factors, such as the orders and products delivery date. In the steelmaking plan, the slabs included in which casts and processed sequence in the corresponding cast are also determined with considering the HR plan and the limitations of the CC stage.

As specifically defined here, there is only one machine at the HR stage. In addition, the following assumptions are made for the ISP.

(1) ACC (HR) machine can process at most one cast (rolling unit) at one time. A cast (rolling unit) can only be processed on one CC (HR) machine at any time.

(2) The set-up time is required between two continuous casts on the same CC machine, which is independent with the casting sequence and casts’ properties.

(3) For two consecutive rolling units, only when the former has been finished and after a relative set-up time of the rolling machine, can the immediate next one be started.

(4) Slabs included in the same cast (rolling unit) are continuously processed on the CC (HR) machine.

(5) For the required rolling slabs from the CC machines, only when the casting process has been finished and transported to the rolling machine, can the rolling operation be started.

(6) The cold slab yard can infinitely provide slabs for the HR production process.

(7) The required slabs in the rolling process can be completed punctually by the heat furnace.

Our integrated scheduling aims to provide an effective scheduling plan within a given time horizon and the objective is to maximize the number of slabs processed in the HCR or DHCR mode. The tasks of the ISP are as follows: (1) Determine the processing equipment for each cast, as well as the processing sequence of casts and rolling units; (2) Decide the start time for casts and rolling units, from which the start time of slabs in casts and rolling units can be derived. Furthermore, the linkage mode of slabs can be fixed based on the interval time, which eventually help to determine the destination (cold slab yard, heat preservation pit, heat furnace, and HR machine) of the CC slabs.

2.2. Mathematical Programming Formulation

The following notations are employed to describe the ISP:

Indices and sets:

n^C: total number of casts;

n^R: total number of rolling units;

M^C: total number of CC machines;

i, j: index for casts, i, j = 0, …, n^C+1, where 0 and n^C+1 denote the fictitious cast;

q, g: index for rolling units, q, g = 0,…, n^R+1, where 0 and n^R+1 denote the virtual rolling unit;

m: index for CC machines, m = 1, …, M^C;

$N_i^C$ : set of slabs in the ith cast;

N^C: set of slabs in all casts, $N^C=N_1^C\cup N_2^C\dots \cup N_{n^C}^C$ ;

$N_q^R$ : set of slabs in the qth rolling unit;

l: index for slabs, l ∈ N^C, l ∈ N^R;

$k_{il}^C$ : position of slab l in cast i;

$k_{ql}^R$ : position of slab l in rolling unit q;

D^C: set-up time of CC machines between two consecutive casts;

D^R: set-up time of the HR machine between two adjacent rolling units;

t^C: processing time for each slab on CC machines;

t^R: processing time for each slab on the HR machine;

$p_i^C$ : processing time of cast i on CC machines;

$p_q^R$ : processing time of rolling unit q on the HR machine;

Tr: transportation time between the CC and HR stage;

$U_{il}^C$ : binary constant, 1 if slab l is included in cast i, 0 otherwise;

v: identifier of linkage mode between the CC and HR stage, v = 1denotes the DHCR process, v = 2 expresses the HCR process, and v = 3 expresses the CCR process;

$T_v^{min}$ : minimum interval time for slabs for v process;

$T_v^{max}$ : maximum interval time for v process, $T_1^{min}<T_1^{max}<T_2^{min}<T_2^{max}<T_3^{min}<T_3^{max}$ ;

H: time horizon for the production plan;

${\delta }_{ql}$ : interval time of slab l in rolling unit q between the CC and HR stage;

$s_{il}^C$ : start time of slab l in cast i;

$s_{ql}^R$ : start time of slab l in rolling unit q;

M: a sufficient large number;

Decision variables:

x_ijm: binary variable, 1 if cast i and j are assigned to the same CC machine m, and cast j is processed directly after i, 0 otherwise;

y_im: binary variable, 1 if cast i is processed by the CC machine m, 0 otherwise;

z_qg: binary variable, 1 if rolling unit q is the direct predecessor for g, 0 otherwise;

$S_i^C$ : continuous variable, start time for cast i;

$S_q^R$ : continuous variable, start time for rolling unit q;

Accordingly, the mathematical model ISP_M1for ISP is presented as follows:   

$$max\cdot \sum _{q=1}^{n^R}\sum _{l\in N_q^R\cap N^C}min\left(1,max\left(0,T_2^{max}-{\delta }_{ql}\right)\right)$$(1)

  

$$s.t.\sum _{j=1,j\ne i}^{n^C+1}{x}_{ijm}=\sum _{j=0,j\ne i}^{n^C}{x}_{jim}=y_{im},\mathrm{\hspace{0.17em} }i=1,\dots ,n^C,\mathrm{\hspace{0.17em} }m=1,\dots ,M^C$$(2)

  

$$\sum _{m=1}^{M^C}{y}_{im}=1,\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }i=1,\mathrm{\hspace{0.17em} }\dots ,\mathrm{\hspace{0.17em} }{n}^C$$(3)

  

$$s_{il}^C=S_i^C+\left(k_{il}^C-1\right)\cdot t^C,\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }i=1,\dots ,n^C,\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }l\in N_i^C$$(4)

  

$$S_i^C+p_i^C+D^C-M\left(3-y_{im}-y_{jm}-x_{ijm}\right)\le S_j^C,\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }i,j=1,\dots ,n^C$$(5)

  

$$\sum _{g=1,g\ne q}^{n^R+1}{z}_{qg}=\sum _{g=0,g\ne q}^{n^R}{z}_{gq}=1,\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }q=1,\dots ,n^R$$(6)

  

$$s_{ql}^R=S_q^R+\left(k_{ql}^R-1\right)\cdot t^R,\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }q=1,\dots ,n^R,\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }l\in N_q^R$$(7)

  

$$S_q^R+p_q^R+D^R-M\left(1-z_{qg}\right)\le S_g^R,\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }q,g=1,\dots ,n^R$$(8)

  

$${\delta }_{ql}=s_{ql}^R-\sum _{i=1}^{n^C}{U}_{il}^C\left(s_{il}^C+t^C\right)\ge tr,\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }q=1,\dots ,n^R,\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }l\in N_q^R\cap N^C$$(9)

  

$$\begin{array}{r}{x}_{ijm},y_{im},z_{qg}\in \left\{0,1\right\},\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }i,j=0,\dots ,n^C+1,\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }m=1,\dots ,M^C,\\ \hfill q,g=0,\dots ,n^R+1\end{array}$$(10)

  

$$\begin{array}{r}{S}_i^C,S_q^R\ge 0,\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }{S}_i^C<H-p_i^C,\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }{S}_q^R\le H-p_q^R\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }i=1,\dots ,n^C,\\ \hfill q=1,\dots ,n^R\end{array}$$(11)

The objective function (1) maximizes the number of slabs processed in the HCR or DHCR mode. Constraint (2) ensures that there is only one predecessor and one successor for each cast. Constraint (3) guarantees that each cast is assigned to only one CC machine. Equation (4) calculates the casting start time for slabs. Constraint (5) guarantees the set-up time between two consecutive casts processed by the same CC machine. Constraint (6) ensures that there is only one predecessor and one successor for each rolling unit. Equation (7) computes the rolling start time for slabs. Constraint (8) defines that the set-up time between two consecutive rolling units. Constraint (9) ensures that the interval time for the slab between CC and HR stage is larger than the transportation time. Constraints (10) and (11) define the value range of decision variables.

3. Proposed Hybrid Algorithm

Comparing to the traditional scheduling problems, the technological constraints related to the CC and HR production process are considered in the ISP, as well as the consistency between two stages. The main decisions involved in the ISP include the assignment of casts to CC machines, the processing sequence and the start time for casts and rolling units. Thus, the difficulties led us to develop a hybrid algorithm based on the decomposition strategy for solving the ISP.

In the present work, the original ISP is split into a master problem and a CP subproblem in a natural way using the special features of the integrated production. To obtain potentially good solutions, the master problem is used to find the partial solutions, and the CP subproblem is to check the feasibility of the obtained solutions and generate a complete schedule plan. Specially, we propose an iterative scheme where iterate between a master problem and a CP subproblem, in a similar fashion as in Jain and Grossmann.¹²⁾ The start time of the casts and rolling units, the sequence of the rolling units are determined in the master problem, while the CP subproblem is used to derive a feasible complete solution. It means that the CP subproblem should identify the processing machine for casts and the corresponding casting sequence for casts processed by the same CC machine with considering the obtained start time. If a feasible solution is obtained, it becomes the final solution. Otherwise, the causes of infeasibilities infer cuts, which are then added to the master problem to perform the next iteration. A simplified flow diagram of the proposed algorithm for the ISP is shown in Fig. 2.

Fig. 2.

Schematic diagram of the proposed hybrid algorithm.

3.1. Master Problem

For the master problem, the continuous variables (S_i^C, S_q^R) are considered, which are important to specify the processing mode for slabs. Since the production practice that there is only one machine at the HR stage, the sequence of the rolling units is also considered in the master problem to optimize the high level solutions. Especially, the sequence of the rolling units is important to derive the feasible casting sequence for casts. Otherwise, more infeasible solutions would be generated to satisfy the technology constraints of the DHCR or HCR process excluding the resource (machines) constraint. Thus, it is necessary to consider the sequence of the rolling units in the master problem. Then, the model ISP_M2 for the master problem can be written as follows:

obj: (1)

s.t. (4), (6) — (11)

Since the slabs are processed continuously at the CC and HR stage, the master problem should determine the start time of the batches (casts, rolling units), and then the start time of the slabs can be derived. The interval time of the slabs between two production stages simultaneously exists in constraint (9) and the objective function (1), where constraint (9) gives a lower bound of the interval time, whereas each linkage mode gives the domain of this time. This implies that the interval time should be short enough to ensure a large number of slabs charged in the DHCR or HCR mode.

3.2. CP Subproblem

After the master problem is solved, the CP subproblem decides if this partial solution can be extended to a complete solution by fixing the assignment variables (y_im) and the sequencing variables (x_ijm). In this paper, we model the CP subproblem using the modeling language of ILOG’s OPL, which has a number of global constraints and special constructs for the scheduling problems. In the CP subproblem, the parallel CC machines are assumed to be unary resources in OPL. The defining attribute of a unary resource is that the CC machine can process only one cast at a time. Using this basic framework of OPL the CP model ISP_M3 can be written as follows.   

$$s.t.\hspace{1em}{c}_i.start=S_{c_i}^C,\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }{c}_i=c_1,\dots ,c_{n^C}$$(12)

  

$$c_i.duration=p_{c_i}^C,\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }{c}_i=c_1,\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\dots ,\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }{c}_{n^C}$$(13)

  

$$alternative\text{ (}{c}_i,assign[c_i,m\text{ in 1},\dots M^C]\text{)}$$(14)

  

$$\begin{array}{c}noOverlap\text{(}se{q}^C,D^C\text{)},se{q}^C\text{(}assign[c_i\text{ in }{c}_1,\dots ,c_{nC},m]\text{)},\\ m=\text{ 1},\dots ,M^C\end{array}$$(15)

The CP subproblem is used to determine the processing machine and the feasible casting sequence for casts; therefore, no objective function is included in the ISP_M3. Constraint (12) establishes the relationship between the master problem and the CP subproblem according to the start time of the casts. Constraint (13) expresses that the duration of a cast is independent with the CC machine. Constraint (14) plays the same role as constraint (3), where assign [c_i, m] is used to denote the binary assignment variable. Similarly, constraint (15) is equivalent to constraint (5), and seqC denotes the binary sequencing variables.

3.3. Generating Cuts

The cut is used to forbid the infeasible values and tighten the domain of variables, which is important for the master problem. Jain and Grossmann’s cut-generating scheme relies largely on the fact that the resource units (machines) are independent. Therefore, the cuts can be separately generated for each individual machine. In the present study, it is challenging to design the effective cuts generation method due to the interaction between the rolling sequence and the casting sequence.

The following notations are adopted to introduce the cuts generation method:

$P{S}_m^t$ : the set of the casts started processing before time t (t < H) by the CC machine m.

i_m: the index for the cast with the maximal starting time in $P{S}_m^t$ , and $i_m=arg\underset{i\in P{S}_m^t}{\max }{S}_i^C$ .

Then, the following observation concerning the feasibility of the schedule plan is introduced:

Observation 1. The solution is infeasible if, for some t, (1) the duration of cast i overlap with that of some cast j, or (2) the interval time between cast i and cast j is less than the set-up time, where $i=arg\underset{m=1,\mathrm{..},M^C}{\min }\left(S_{i_m}^C+p_{i_m}^C\right)$ .

In an infeasible solution, it is assumed that at least one of the conditions in Observation 1 is satisfied for some cast j and $t=S_j^C$ . In this case, the temporary precedence relationship i→j is determined, and the cut $S_j^C-\left(S_i^C+p_i^C\right)\ge D^C$ is generated, where $i=arg\underset{m=1,\dots ,M^C}{\min }\left(S_{i_m}^C+p_{i_m}^C\right)$ .

4. Computational Experiments and Results

4.1. Experimental Design

Since there are no benchmarks test sets available for the ISP, several instances are generated according to the collective production data from an iron and steel plant in China. The solution methodology is implemented in C++, and performed on a PC with Intel Core(TM) 2, 2.66 GHz using the Windows XP operating system. The MIP optimizer ILOG CPLEX 11.2 and CP 2.1 are used for obtaining the optimal solutions.

The parameter settings used in our experiments are as follows: M^C = 3, Tr = 8 min, D^C = 20 min, D^R = 10 min, t^C = 5 min, t^R = 2 min, [ $T_1^{min},\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }{T}_1^{max}$ ] = [0, 60] min, [ $T_2^{min},\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }{T}_2^{max}$ ] = [61, 120] min.

The sizes of the experimental instances are illustrated in Table 1. Since the 24 h operation period of the steel plant is divided into three shifts, 3 groups of experimental instances are generated accordingly. In Table 1, N^RC denotes the number of required slabs at the rolling stage from the CC machines.

Table 1.Sizes of experimental instances.

inst	H(m)	nC	|NC|	nR	|NR|	NRC
11	480	5	143	3	104	87
12		8	199	5	168	147
13		9	217	6	185	158
21	960	9	267	5	216	193
22		10	308	6	258	236
23		11	348	7	281	252
24		12	387	8	343	303
25		13	423	9	389	335
31	1440	11	457	5	349	324
32		12	497	6	406	382
33		13	546	7	457	419
34		14	594	8	514	466

4.2. Computational Results

In order to evaluate the performance of the proposed algorithm, the optimizer CPLEX is used to solve the model ISP_M1. The computational results for solving the 3 group sizes of instances by CPLEX and the decomposition algorithm are respectively shown in Tables 2 and 3. Of note, the model sizes and the search process of the master problem, and the CP subproblem in the last iteration are shown in Table 3. “r_DHCR” and “r_HCR ” respectively present the DHCR ratio and HCR ratio. “cuts” represents the number of cuts inferred by the CP subproblems. “obj” represents the objective value.

Table 2.Computational results for the model ISP_M1.

inst	nodes	iteration	variables	constraints	time (s)	rDHCR (%)	rHCR (%)	obj
11	107	1672	1 51	315	3.656	20.69	74.71	83
12	3826	125080	320	593	5.812	24.49	72.11	142
13	49624	2048867	395	686	64.875	57.59	36.71	149
21	5712	69286	381	743	6.735	34.2	62.69	187
22	413591	6885765	462	906	308.657	36.02	57.63	221
23	1603828	71136199	551	1023	2809.45	38.89	59.52	248
24	4103061	176432119	648	1218	6901.11b	31.68	54.46	261a
25	1968637	74570569	753	1383	3701.3b	46.57	34.93	273a
31	1475784	667999044	521	1139	3071	35.19	54.32	290
32	1689931	74279046	614	1344	4628.44	34.29	53.93	337
33	3873950	88383317	715	1515	6864.69	34.12	54.89	373
34	1641603	42469713	824	1714	3601.47b	27.47	54.29	381a

^a  Suboptimal solution.

^b  computational time when the suboptimal solution obtained.

Table 3.Computational results for the model ISP_M2-ISP_M3.

inst	ISP_M2					ISP_M3					cuts	rDHCR (%)	rHCR (%)	obj
	nodes	iteration	variables	constraints	time (s)	branch	Choice point	variable	constraints	time (s)
11	38	741	25	194	0.891	3	5	23	8	0.047	1	40.22	55.17	83
12	66	1518	50	335	3.344	4	6	35	8	0.047	2	36.05	60.54	142
13	327	5908	65	373	9.813	3	5	39	8	0.188	5	53.16	39.24	146
21	194	2996	51	430	5.766	3	5	39	8	0.109	3	36.79	60.11	187
22	488	7242	66	531	32.828	4	6	43	8	0.203	7	44.92	48.73	221
23	206	4305	83	577	25.438	5	7	47	8	0.234	6	48.8	49.6	248
24	345152	5807110	102	691	495.39	5	7	51	8	0.047	1	32.01	60.4	280
25	18545	183989	123	777	354.718	5	7	55	8	0.141	4	45.07	48.96	315
31	638	19306	53	690	22.719	4	6	47	8	0.063	3	43.52	45.99	290
32	36459	383073	68	818	153.828	5	7	51	8	0.109	2	39.53	48.69	337
33	13610	293395	85	911	281.86	5	7	55	8	0.188	6	36.28	52.74	373
34	7214	83635	104	1019	950.484	5	7	59	8	0.156	3	33.05	58.58	427

It can be clearly seen that all instances are successfully solved by the decomposition algorithm, while CPLEX fail to obtain optimal solutions for 3 instances as the memory space is occasionally exceeded. Thus, the suboptimal solutions are obtained within the maximum allowable time for these instances. In all problem sizes, the first data set are relative easy to solve for both the hybrid algorithm and CPLEX. The sizes of the model ISP_M1 and ISP_M2 increase much rapidly with the size of instances. To a certain extent, the optimal solutions are obtained by the master problem with the cost of long computational time for the big M constraint (8). Therefore, the computational time of the model ISP_M1 and ISP_M2 increase with the size of instances and the time horizon. For all the experimental instances, the CP time is less than the model ISP_M2 in terms of the determined start time for casts. Furthermore, comparing to the model ISP_M2, the CP time varies slightly with the size of instances.

The HCR ratio obtained by the hybrid algorithm is higher than the DHCR ratio for most instances. Although the effective integrated scheduling plan relies largely on the production plan, the constraints of the HCR process are relatively easy to meet in comparison to the DHCR technology. For some instances with the optimal solutions obtained by CPLEX, it can be seen that the DHCR ratio obtained by the hybrid algorithm is relatively higher than the model ISP_M1. It means more energy saving, as well as the shorter lead time before the HR process. According to the computational results, it is concluded that the hybrid algorithm is more effective and shows a great superiority in efficiency for solving the ISP.

5. Conclusions

This study focused on the ISP for the CC and HR production process in the iron and steel production. Our aim is to generate an effective scheduling plan within the given time horizon in order to ensure more slabs charged in the HCR or DHCR mode. For the established mathematical model, a hybrid algorithm was proposed based on the decomposition strategy. Cuts generation scheme based on the temporal precedence relationship was devised for the master problem. Computational results indicate that the hybrid algorithm is capable of solving the ISP efficiently. Compared to CPLEX, the proposed algorithm is more effective.

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