Research on hot rolling scheduling problem based on Two-phase Pareto algorithm

Under the background of excess capacity and energy saving in iron and steel enterprises, the hot rolling batch scheduling problem based on energy saving is a multi-objective and multi constraint optimization problem. In this paper, a hybrid multi-objective prize-collecting vehicle routing problem (Hybrid Price Collect Vehicle Routing Problem, HPCVRP) model is established to ensure minimum energy consumption, meet process rules, and maximize resource utilization. A two-phase Pareto search algorithm (2PPLS) is designed to solve this model. The improved MOEA/D with a penalty based boundary intersection distance (PBI) algorithm (MOEA/D-PBI) is introduced to decompose the HPCVRP in the first phase. In the second phase, the multi-objective ant colony system (MOACS) and Pareto local search (PLS) algorithm is used to generate approximate Pareto-optimal solutions. The final solution is then selected according to the actual demand and preference. In the simulation experiment, the 2PPLS is compared with five other algorithms, which shows the superiority of 2PPLS. Finally, the experiment was carried out on actual slab data from a steel plant in Shanghai. The results show that the model and algorithm can effectively reduce the energy consumption in the process of hot rolling batch scheduling.


Introduction
Hot rolling is one of the three processes required for iron and steel production. It needs to meet the requirements of multiple objectives and multiple constraints. The hot rolling batch scheduling (HRBS) [1] problem has a significant influence on the quality, cost, and energy consumption of steel production. HRBS is an energy intensive production process in the iron and steel industry. It converts steel slabs at high temperatures in hot rolling process into steel strips. This process consumes a lot of power. Recently, as China has proposed a national policy on energy conservation and emission reduction, and because of rising energy prices, the steel industry must reduce energy consumption and reduce production costs to cope with the above problems during the hot rolling process. The planning task addressed in the HRBS is to generate a production schedule, i.e. to determine the production sequence of the slabs. Hot rolling steel is a customer individual product; the final dimensions of the steel strip after hot rolling are determined by the customer's requirement. This process requires a different heating temperature; a reasonable planning task of HRBS can effectively solve the problem of energy consumption [2][3][4].

PLOS ONE
PLOS ONE | https://doi.org/10.1371/journal.pone.0241077 December 28, 2020 1 / 14 a1111111111 a1111111111 a1111111111 a1111111111 a1111111111 In the HRBS, the order contracts for the original data set and the pre-production contract revenue pool will be grouped into slabs of hot rolling units. The process sequence for a number of rolling units will be then determined according to the constraint conditions of the hot rolling batch scheduling, in order to meet the standards of the continuous casting billet collecting pool (see Fig 1 for the slab structure). The development of the hot rolling batch scheduling program has a significant effect on the product quality, production efficiency, and energy consumption.
As hot-rolling process is the final procedure among the three steps of steel making, the slab rolling order directly affects the quality of products and energy consumption; therefore, the slab preparation staff should develop a reasonable schedule based on the process, resources, and energy constraints. Since the general delivery time for an order, in an actual enterprise situation is 6-10 days, the plan will not affect the delivery time. Therefore, in this paper, the daily data of the slabs is taken as the object of study, and the slab is divided into several rolling units. In the rolling batch schedule, each rolling unit has strict requirements of slab size, rolling order, steel, convergence of the unit, and rolling volume; and it aims at improving productivity and reducing energy losses, while simultaneously ensuring quality. A reasonable scheduling scheme needs to meet the following objectives: 1. Process conditions to be met: To ensure that the slab meets the surface and physical requirements for the front and rear slab width, thickness, and hardness, for the minimum penalty jump.

Resource utilization:
To ensure machining quality of the rolling element, maximum roll utilization, reduction in the utilization rate of roll changing, and reduction in the cost of slab production, for the rolling elements of the slab.
3. Saving energy and reducing consumption: In the hot rolling production process, due to the different grades of steel, the hardness and material properties of different slabs, the roll energy-consumption load fluctuation, and the power of the rolling load as the main source of power consumption, the slab rolling sequence will lead to different types of steel rolling at different values power consumption. Therefore, the difference in power consumption between slabs will be the minimum penalty jump. The transition between the slabs for power consumption should be smooth, and should not have any repeated jumping. The process for hot rolling batch scheduling preparation is shown in Fig 2. The HRBS will greatly affect the hot rolling energy consumption issues. Also, due to the relatively few studies on the energy consumption optimization of the HRBS, motivation of this research prompted us to further study the energy consumption of HRBS based on the new optimization model. The model contains the optimization objectives and constraints of process, resources and energy consumption, multi-objective optimization algorithm was designed and solves the model by this paper.
The main contributions of this paper are as follows: 1. Considering the previous research, the minimum number of rolling plans for minimum power loss and the minimum processing per ton of steel plate, HRBS is summed into a hybrid prize collecting model.

2.
We used the 2PPLS algorithm to solve the constructed HPCVRP model. The effectiveness of the proposed algorithm was verified by comparing it with the other MOEAs. In this study, we used the MOEA/D-PBI algorithm [12] to decompose the HPCVRP, and to enhance the efficiency of finding the local optimal solution, we then used the MOACS and Pareto local search (PLS) method to obtain the final solution. to the roll slab j immediately after slab i, where d ij = P w ij +P g ij +P h ij , P w ij 、P g ij 、P h ij represent the penalties caused by the jumps in width, gauge, and hardness when moving from slab i to slab j; P e ij , The penalties caused by the power consumption (per ton of steel) when moving from slab i to slab j; P i , The prize (penalty) obtained if slab i is (is not) scheduled in the rolling batch. P i represents the desirability of the processing slab i, and can be set as a product of the priority and the rolling length of slab i. Therefore, a slab with a higher priority and a longer rolling length obtains a higher price (penalty) than that with a lower priority and a shorter rolling length.; Q i , The rolling length of slab i, where i2 N; Q L , The lower bound of the rolling length of the slabs scheduled in each rolling unit; Q S , The upper bound of the rolling length of the slabs scheduled in each rolling unit.
Finally, the simulation data was compared with the manual scheduling data (actual data of an iron and steel enterprise in Shanghai) to ensure further optimization of the hot rolling batch scheduling problem.
The remainder of this paper is organized as follows. The literature review and related work is presented in Section II. The mathematical model for the hot rolling batch scheduling is presented in Section III. Section IV presents the 2PPLS algorithm. Section V includes the experimental studies for demonstrating the efficiency of the proposed method in addition to some discussion. Finally, the conclusions are presented in Section VI.

Literature review and related work
This paper reviews the literature from two aspects of hot rolling scheduling model and solving algorithm.
In hot rolling scheduling modeling, HRBS is a typical non-deterministic polynomial (NPhard) combinatorial optimization problem with a large number of multi-objectives and constraints (Lopez, Carter & Gendreau, 1998), it is almost impossible to produce optimal scheduling solutions by manual scheduling or intelligent optimization methods [5]. Atilla Ö zgüra et al reviewed 90 hot rolling mill scheduling articles from 1989-2020.Diversity in optimization methods, constraints incorporated in the analysis and level of abstraction and data availability are analyzed [6]. Quan Ke Pan, Liang Gao and Wang Ling addressed a hot-rolling scheduling problem from compact strip production pro-cesses. A mathematical model that consists of two coupled sub-problems is presented, a multi-objective evolutionary algorithm (MOEA/D) is developed to find Pareto optimal or near-optimal solutions of the sheet-strip sequencing problem [7]. A new mixed-integer problem formulation model for hot rolling mill that focuses on energy consumption was established by Karen . They used the greedy random adaptive search algorithm to solve the model, and achieved good results with respect to energy consumption [8]. From the above analysis, it can be seen that the HRBS is more and more closely related to energy consumption. HRBS is mostly approximate mathematical model.
In terms of solving algorithm, Liu et al (2015) used a combination of particle swarm optimization (PSO) and simulated annealing algorithm (SA) to solve the HRBS. The Metropolis criterion was introduced to search for Pbest (Pbestis the local optimization solution) and Gbest (Gbestis the global optimization solution) twice. The algorithm obtained good results, when compared to the results of manual scheduling. However, the solution process uses a simple weighting method, in which the weight coefficient is given by experience, to convert the multiobjective problem into a single objective problem without considering the consistency between multiple objectives, to obtain the scheduling results; therefore, it needs to be optimized further [9]. Zangari M, Pozo A, Santana R, et al (2017) have presented a preliminary study on an algorithm based on MOEA/D framework and the bio-inspired metaheuristic called binary ant colony optimization (BACO) [10], The results have shown that the proposed MOEA/D-BACO has outperformed MOEA/D, which uses genetic operators, in most of the test instances. As the HRBS is an NP-hard combinatorial optimization problem, and has multi-objectives and multi-constraints; the ACO is one of the best algorithms to solve the classic TSP problem. Rui Zhang et al (2020) proposed a hybrid metaheuristic algorithm, which combines ant colony system and enhanced local search to provide an efficient solution to the HRBS [11]. Tan et al (2019) have presented a hybrid algorithm that combines mixed integer programming and constraint programming is designed to solve each subprolem [12]. The hybrid algorithm and enhanced local search strategy can be used to solve the HRBS.
As discussed above, the traditional hot rolling process consumes a large amount of energy and cannot meet the requirements of energy saving and consumption reduction. We must analyze and study these characteristics to build a new mathematical model. In addition, we require a new algorithm to solve HRBS when the number of objectives is more than 3 and contains many constraints. In this paper, the problem of hot rolling batch optimization is formulated as a hybrid multi-objective prize collecting vehicle routing problem (HPCVRP). We propose a two-phase Pareto local search (2PPLS) algorithm for the real-world multi-objectives optimization problem. In the first phase, the algorithm generates a set of high-quality solutions by MOEA/D-PBI, and in the second phase, we employ the Pareto local search (PLS) method to generate approximate Pareto-optimal solutions

Problem modeling
At present, there are many models for VRP research, but the considerations are often rather single and far from wide practical application. However, there are many factors to be considered in HRBS.
1. To fully ensure the capacity of rolling unit, it is necessary to set a vehicle path model with unknown number of vehicles; 2. To fully reflect the preferential processing temperature of the slab, bonuses collection mode, that bonus to collect vehicle routing model; 3. To meet the wear life constraints of the roll, a vehicle path model that can constrain the vehicle path, that is, the ability to restrain needs to be adopted.
In this paper, a new hot rolling optimization model is established according to the characteristics of hot rolling batch scheduling. The HPCVRP model is combined with the characteristics of VRP and PCTSP. Each slab corresponds to a store, each rolling unit corresponds to a car, and each store requires a corresponding slab rolling length, which is the distance between the penalty values of the corresponding store slabs. As shown in Fig 3, the distribution centre of the distribution vehicles must pass and eventually be returned to the distribution centre in HPCVRP. Thus, it is a closed loop process; however, since the hot rolling process is an open loop process, the "virtual slab" concept is introduced at the beginning and end of the slab rolling process. Each slab must be and can only be placed in a rolling unit. In order to describe the mathematical model, the parameters are defined as follows: Decision variables: x ijk -in the rolling units k, the slab i is followed by j; y ik -the rolling units k contains the slab i; HRBS can be formulated as the following multi-objective HPCVRP model: x ijk � n k À 1 ð6Þ 8 > > > < > > > : ( Objective (1) is to minimize the total penalties caused by the jumps in width, gauge, and hardness between adjacent slabs. Objective (2) is to minimize the penalties due to unscheduled slabs.
Objective (3) is to minimize the total penalties caused by the jumps in the power consumption (per ton of steel) slabs. Constraints (4)-(5) specify the sequence of a rolling unit. Constraints (6)-(7) ensure that the slabs are scheduled only once, while the virtual slab 0 must be scheduled in each rolling unit. Constraint (8) indicates the capacity constraint of each rolling unit. Constraint (9) ensures that the decision variables x ijk and y ik take the integer values of only 0 or 1.

Solution algorithm
In the HPCVRP, the continuous slab and rolling unit are considered as the service customers and delivery vehicles, respectively. During the rolling process, the total value of the width, thickness, and hardness of the adjacent slab represents the transportation cost. To ensure minimization of the energy cost of production, the hot rolling slab not only needs to meet the process and resource conditions, but also needs to ensure maximum energy saving. It can be seen from equation (1)-(9) that the HPCVRP model contains three objectives and multiple constraints, and the optimization of this model is many objective optimization problems (MaOPs). As the common intelligent algorithms, such as the ant colony algorithm, genetic algorithm, etc., have poor performance in dealing with MaOPs, we need to use a new algorithm to improve the performance of combination optimization problems, and to deal with MaOPs.

The proposed Two-phase Pareto search algorithm
Two-phase Pareto search algorithm. HPCVRP is a combinatorial optimization problem, when the number of slabs is large, it will bring "combinatorial explosion", and there are more optimization objectives problem, In addition, for the MaOPs, the weighting method is generally used, but the dimensions of the multi-objective in the rolling plan model are not the same, and the weight coefficient of the objective is difficult to be determined reasonably. So it is very difficult to solve it accurately. In this paper, we present 2PPLS to find a suitable approximation solution to the above problem. The flow chart for the algorithm is shown in   [11] for MOPs was proposed. Recently, An MOEA/D-ACO with PBI for MaOPs was proposed by (Ling and Wang.2018). As the number of objectives in HPCVRP is more than three, the MOEA/D-ACO uses the weighted sum and Tchebycheff approaches to solve the problem, but the correlation between the algebraic and the geometrical interpretation of the weight sets is not valid. In this paper, we introduce the PBI approach to MOEA/D. Therefore, we decompose HPCVRP into multiple sub-problems through MOEA/D-PBI algorithm. The purpose of PBI is to find the intersection of the top boundary and a group of lines. When these lines are uniformly distributed, the interval of the result provides an approximate solution for the Pareto optimal boundary of the whole population. This is helpful to deal with the non-convex problem of Pareto optimal bound. The optimization problem of the PBI approach is defined as: Subject to x 2 O Where d1 ¼k ðFðxÞ À z�Þ T w k = k w k d2 ¼k FðxÞ À ðz � þ d1w= k w kk Fig 5 illustrate d 1 and d 2 of a solution x i . G pbi (x|w,z � ) is a composite measure of x for both convergence and diversity, d 1 is used to measure the convergence of x i towards the EF, d 2 is a kind of measure for the population diversity. The distance between d 1 and d 2 is controlled by the parameter θ = 5.0 (θ is a penalty parameter).
MOEA/D-PBI was shown in algorithm1. Second phase. MOACS is used to solve the HPCVRP. Each ant is responsible for solving one sub-problem. During the search, each ant also records the best solution found so far for its sub-problem. An ant updates its current solution if it has found a better one in terms of its own objective. The path (rolling plan) constructed by ant i. The PLS method is a local search algorithm developed by various authors [14]. In PLS, the neighbourhood of every solution is explored. If the neighbour is not dominated by a Pareto-optimal solution, then the neighbour is added to the population and the set of approximate Pareto-optimal solutions. The algorithm stops when it is no longer possible to find any more dominated neighbours from the population. The PLS does not prescribe any numerical parameters based on the notion of Pareto local optimum set [15][16][17]. In this paper, the set of Pareto-optimal solution denoted by X E contains all the approximate Pareto-optimal solutions. The main purpose for the second phase is the generation of the approximate Pareto-optimal solutions X E . The pseudo-code of the PLS is shown in algorithm 2.

Algorithm 2 Pareto Local search algorithm
Input: initial population X S Output: an approximate Pareto-optimal solution X E (1) X E P ini //initialization of X E and population P 0 with the initial population P ini (2)P 0 P ini (3)P a ;//initialization of an auxiliary population P n (4)added true //added is a Boolean variable (5)while P 0 6 ¼;do// generation of all the neighbours p n of each solution p 0 2P 0 (6)for all p 0 2P 0 do (7)for all p n 2N(p 0 ) do (8) if z(p 0 )>z(p n ) then (9) for all x2X E do // add solution step 9-step16 (10) if z(x)�z(p 0 ) then// a solution x, its evolution z(x) (11)added false (12) Break (13) 18-24 (19) added false (20) break (21) if z(p 0 ) �z© then (22) X E X E \{x} (23)if added = true then (24) X E X E [ {p 0 } (25) P a ;// re-initialization of P a Solving process. According to the 2ppls algorithm mentioned above, it is used to solve the HPCVRP model. The specific process is shown in algorithm 3.

Algorithm 3 2PPLS for HPCVRP
Input: initial parameters (1) Decomposing HPCVRP into several sub-problems by MOEA/D-PBI (2)Select ant i from X E and starts from virtual slab 0 and selects the slab with the largest width from the candidate slab set as the first slab of the k-th rolling unit main material (3) Ant i select the next slab from the feasible slab set NF according to the state transfer strategy If N F = 0, then it returns to virtual slab 0 and moves to the next step. Otherwise, continue to select the next slab  (4) Let t = t + 1, if t � m, turn to step (2) to start the construction of the next rolling unit. After all ants complete the path construction, if t > m, according to the Pareto local search strategy, the path (rolling plan) constructed by ant i is searched locally to improve the quality of the solution (5) Update the pheromone of the non dominated solution successfully updated to the external file in each iteration. If the algorithm meets the termination conditions, output the non dominated solution and the corresponding approximate Pareto front end in the external file and stop the algorithm; otherwise, transfer to step 2

Simulation results
In this section, we compare the 2PPLS algorithm with five state-of-the-art algorithms  [18][19][20]. The test problems are introduced in Sect. 5.1. Sect 5.2 describes the quality indicators. The five state-of-the-arts algorithms used for comparison and the corresponding parameter settings are briefly introduced in Sect. 5.3.

Parameter settings
The parameters for the six MOEAs considered in this study are listed below. 7. Number of points in Monte Carlo sampling: It was set to 1,000,000 to ensure precision [21][22][23].

Results and discussion.
To prove the advantages of the 2PPLS algorithm in solving HRBS, three groups of practical production data were collected from Shanghai Baoshan Iron & Steel Co., Ltd. The data consisted of different number of slabs of 151, 271, and 331. They were divided into three problems. The problems were named based on the number of slabs. E. g., Problem 331 contains 331 slabs. The approximate front returned by each algorithm corresponding to each batch-scheduling problem is shown in Fig 6, in which the horizontal axis denotes the total penalties caused by the jumps between adjacent slabs, and the vertical axis denotes the penalties for unscheduled slabs, the conclusions are given below.
1. 2PPLS has the best convergence performance, as its approximate fronts cover most of those returned by the other algorithms, especially for the larger scheduling problems. Furthermore, 2PPLS can generate some non-dominated solutions with the collection of higher prizes, which is important for HPCVRB, because collection of higher prizes imply that more prior slabs can be scheduled in the rolling batch.
2. MOEA/D-ACO W can also generate good solutions for the smaller scheduling problems such as 151; however, it has a worse convergence performance for the larger scheduling problems such as 331.
3. MOEA/D-ACO T can obtain well-spread approximate fronts; however, some of its solutions were dominated by 2PPLS.
4. IABC, HVNS, and HPSA are good if and only if the number of slabs is small, and only a few solutions are better than that of 2PPLS.
5. It can be seen from Fig 6 that 2PPLS dominates in most solutions and that the advantage of 2PPLS is more pronounced as the number of slabs increases.
Actual data validation. Table 1 shows the results between, 2PPL, MOEA/D-ACO w , HPSA and manual scheduling method. Table 2 shows the results of a comparison between the 2PPLS, the manual algorithm for HRBS, the results of the manual and 2PPLS scheduling methods. From these results, it can be seen that the solutions obtained from the 2PPLS are better than those from the other algorithms. Compared with manual scheduling algorithm, it can be noticed that the values of objective1, objective2, and objective3 were reduced by 15.43%, 15.98% and 10.01%, respectively for the unit 151, by 19.75%, 21.2%, and 15.89%, respectively for the unit 271, and by 23.98%, 23.33%, and 17.28%, respectively for the unit 331. The energy saving rates was 10.01%, 15.89%, and 17.28%. These results and comparison show that 2PPLS is definitely better than the manual algorithm, in terms of convergence precision, when searching for optimal results of HPCVRP.

Conclusions
The major contribution of this paper is proposal of an HPCVRP module for HRBS. With regard to saving energy, reducing consumption, and improving production efficiency, the HPCVRP model can reduce the energy consumption in the process for hot rolling batch scheduling, while satisfying the process requirements and resource utilization rate. A set of high-quality solutions is generated by MOEA/D-PBI in the first phase of the 2PPLS. Due to the number of model optimization objectives, the PBI aggregate function was selected to decompose the objectives. In the second phase, we employ the MOACS and PLS method to generate approximate Pareto-optimal solutions. Based on the actual slab data for scheduling, it could be seen that the results of the 2PPLS scheduling method are significantly better than those of the manual scheduling method. Therefore, the planner can select a satisfactory solution based on their experience or preference. The results show that the 2PPLS is better than the manual method with regard to scheduling performance. Our future work aims at analysing  further, how to deal with HPCVRP and dynamic constraint processing, and to optimize the number of objectives more effectively. Manual  25  20  24  5  4  1  2  3  32  33  34  31  35  11  13   2PPLS  1  2  3  32  33  34  31  35  18  19  11  12  13  15  16   Manual  15  12  29  28  30   2PPLS  17  27  28  29  30 https://doi.org/10.1371/journal.pone.0241077.t002