# Research on Energy-Saving Production Scheduling Based on a Clustering Algorithm for a Forging Enterprise

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^{†}

## Abstract

**:**

## 1. Introduction

_{2}emissions in Taiwan is presented as well as the government policies and regulatory infrastructure for shifting to a low-carbon society, centering on feasible solutions with renewable energy, energy efficiency, and nuclear power [12]. As for low carbon manufacturing or energy-efficient production, a decision making framework for implementing Environmentally Benign Manufacturing (EBM) based on the Genetic Simulated Annealing Algorithm and low-carbon production based on an IPO (Input-Process-Output) model to improve environmental performance was provided. Another new mathematical programming model of flow shop scheduling problems, considering peak power load, energy consumption, and associated carbon footprint in addition to cycle time, was proposed as well as demonstrated by a simple case study of a flow shop with two machines to produce a variety of parts [13]. An energy-efficient model for flexible flow shop scheduling (FFS) based on an energy-efficient mechanism was proposed to solve multi-objective optimization with an improved, genetic-simulated annealing algorithm to make a significant trade-off between the makespan and the total energy consumption to implement feasible scheduling [14]. Holistic metrics to evaluate the energy efficiency in manufacturing companies, considering the different organization levels of production, such as machine or equipment level, production line level, and factory level, were presented [15]. Traditional production scheduling always considers performance indicators, such as processing time, cost, and quality, as optimization objectives in manufacturing systems, and little consideration is given to environment or energy related objectives. Besides, our literature review also suggests that few researches have considered scheduling for forging enterprises.

## 2. Production Investigation and Analysis of DVR Enterprise

Department | Component | Functions |
---|---|---|

Production department | Forging workshop | Forging production, post-forging heat treatment, mold manufacturing and management… |

Production department | Machining workshop | Rough machining, secondary machining, mold modification, forgings correction… |

Production department | Heat treatment workshop | Normalizing, Quenching, tempering hardness test… |

Production department | Comprehensive workshop | Cutting, product packaging, transferring of semi-finished products, outsourcing, raw materials and finished products management. |

Service department | equipment maintenance workshop | Equipment management & maintenance |

Service department | Process department | Prepare process plan |

Service department | Quality assurance department | Inspection and document preparation |

- (1)
- Cutting. Cutting is the start of production and carried out by the sawing machine. Since steel products are widely used in the enterprise, cutting can be primarily used for assessing cost which is normally expressed by material utilization.
- (2)
- Forging. It is a critical procedure for production and also the focus of workshop management. The whole production process is mainly organized according to lead time and quality of forgings. The most important step in forgings’ production is the charging, which means forgings shall be reasonably arranged in a heating furnace and heated up to the initial temperature.
- (3)
- Machining. The machining of forgings is similar to common machining. To ensure individual production and small lot production, universal machine tools are arranged in workshops. Since the machining is not the critical business for a foundry works, the machining is normally outsourced.
- (4)
- Heat treatment. It is usually the final but a critical procedure in production. The heat treatment could comprise normalizing, annealing, solution treatment, quenching and tempering, and the commonly used treatment is thermal refining (quenching + tempering). The key point of heat treatment is similar to forging (how to arrange the heating furnace). However, the principles behind the arrangement vary greatly.

## 3. Forging Production Scheduling Based on Improved Genetic Algorithm

#### 3.1. Scheduling Problem Description

_{j}denote the aggregate of sawing machines available for cutting of workpiece j, and t

_{ij}and d

_{j}denote the processing time and delivery time of workpiece j on sawing machine i from M

_{j}, respectively. So, the following matrix T regarding process time and delivery time can be formed as follows:

_{j}(${W}_{j}={\displaystyle \sum _{i=1}^{m}{t}_{ij}}$, m denotes the number of processes/machines) and T

_{j}(${T}_{j}=\mathrm{max}\{{W}_{j}-{d}_{j},0\}$) denote completion time and delay of workpiece j, respectively, and μ

_{j}denotes delay penalty coefficient of workpiece j. N = {1, 2, …, n} and M = {1, 2, …, m} denotes the aggregate of steels and aggregate of sawing machines, respectively.

_{j}.

_{min}:

#### 3.2. Coding

#### 3.3. Population Initialization

_{1}, x

_{2}, …, x

_{n}) (x

_{i}, i ∈ N). For each x

_{i}, randomly select a machine from M

_{j}as the machine for processing it and code this machine as y

_{i}to generate the initial machine gene (y

_{1}, y

_{2}, …, y

_{n}). Then, Z = [x

_{1}y

_{1}, x

_{2}y

_{2}, ..., x

_{n}y

_{n}] is generated. This combination method for initialization provides for population diversity and process constraints between sawing machines and workpieces.

#### 3.4. Genetic Operator

- (1)
- Selection

- (2)
- Crossover
- a.
- Extended order crossover operator is adopted as follows:
- b.
- Provided two parent individuals for crossover are A and B. Randomly generate k numbers from steels set N = {1, 2, …, n} to form Γ = (Γ
_{1}, Γ_{2}, …, Γ_{k}). Amplify Γ_{i}100 times to form Ψ = (Ψ_{1}, Ψ_{2}, …,Ψ_{k}) and then select a gene cluster from A to form Z (Z_{1}, Z_{2}, …, Z_{x}) within the interval [Ψ_{1},Ψ_{1}+ 100]. - c.
- For each gene Z
_{i}, select a gene from B within the interval [Ψ_{1},Ψ_{1}+ 100]. for exchange. - d.
- As the above, exchange all Z
_{i}so as to generate two filial generations.

- a.
- Randomly generate four numbers 2, 3, 4 and 5 from the aggregate of steels and amplify those numbers 100 times to form Ψ = (200, 300, 400, 500). Then, select genes from parent A within intervals [200,300], [300,400], [400,500] and [500,600], respectively, to form a gene cluster Z = (202, 301, 403, 502) (see the hatched section in Figure 2).
- b.
- For each gene Z
_{i}in the cluster (take the gene Z_{i}= {202} as an example), select a gene {203} from B within the interval [200,300] and exchange them. - c.
- Follow the above method to replace the gene cluster Z = (202, 301, 403, 502) to Z’ = (203, 302, 404, 501) so as to generate two filial generations A’ and B’.

- (3)
- Mutation

_{j}(but excluding the gene for mutation) to replace the original to generate a plurality of individuals. Select the best one as filial generation.

_{i}= {202} is M

_{j}= {1, 2, 3}. So, the gene clusters after variation are A1 and A2 (see Figure 3).

#### 3.5. Case Study

No. | Size | No. | Size |
---|---|---|---|

1 | 12” ingot and 14” ingot | 6 | Φ200 less, □200 less |

2 | 17” ingot | 7 | Φ200~Φ300, □200–□300 |

3 | 22” ingot | 8 | Φ300~Φ500, □300–□500 |

4 | 24” ingot | 9 | Φ500~Φ650, □500–□650 |

5 | 26” and bigger ingots | 10 | Φ650~Φ800, □650–□800 |

No. | 1 | 2 | 3 | 4 | 5 | Delivery Time |
---|---|---|---|---|---|---|

1 | 8.5 | 8.2 | 8.5 | 0 | 0 | 15.5 |

2 | 5 | 0 | 0 | 0 | 0 | 12.5 |

3 | 7.4 | 7.2 | 0 | 0 | 0 | 10 |

4 | 0 | 0 | 0 | 3.4 | 3.6 | 12 |

5 | 6 | 5.7 | 0 | 0 | 0 | 4 |

6 | 10.7 | 10.2 | 10.3 | 0 | 0 | 18.5 |

7 | 0 | 0 | 0 | 4.2 | 4.5 | 20 |

8 | 0 | 0 | 5.5 | 5.6 | 5.9 | 8 |

9 | 0 | 0 | 4.5 | 4.8 | 4.9 | 14.5 |

10 | 5.2 | 5.2 | 0 | 0 | 0 | 17 |

11 | 0 | 0 | 5.7 | 5.4 | 5.6 | 5.5 |

12 | 0 | 9.7 | 9.4 | 0 | 0 | 13.6 |

13 | 4.5 | 4.8 | 4.7 | 0 | 0 | 17 |

14 | 5.6 | 5.9 | 0 | 0 | 0 | 7.5 |

15 | 0 | 0 | 0 | 3.3 | 3.7 | 20.5 |

No. (Machine) | No. (Workpiece) | No. (Machine) | No. (Workpiece) |
---|---|---|---|

1 | 14, 2, 10, 13 | 4 | 11, 9, 15 |

2 | 5, 3, 1 | 5 | 4, 7, 8 |

3 | 12, 6 |

## 4. Stacking Combination Optimization for Free Forging

#### 4.1. Stacking Problems Description

- (1)
- The height of piled up material shall not exceed maximum height of furnace. Workpieces shall be stacked within effective heating area.
- (2)
- Workpieces stacked in one furnace should be easily distinguished by weights and specifications. Workpieces with similar specifications shall not be stacked into one furnace. If they have to be stacked into a furnace, label them with marks.
- (3)
- Workpieces with the same material and production batch number shall be stacked into one furnace as far as possible.
- (4)
- Workpieces with different kinds of material can be stacked into one furnace if heating process permits. However, stacking workpieces with the same specification but different kinds of materials shall be strictly prohibited.
- (5)
- Ingot shall be placed around 200 mm away from bottom and wall and at least 200 mm away from other ingots.
- (6)
- Mixing cold ingot with hot ingot is strictly prohibited. Unless otherwise specified, use minimum values for stacking temperature, heating temperature and heating rate (which means to follow the heating regulations for cold ingot).

- (1)
- Maximum stacking coefficient η

- (2)
- Deviation coefficient ε

- (3)
- Deviation ratio ζ

- Classify the material to be forged according to process and specifications. The material classified into a same type can be combined together for heating. A dynamic clustering method is proposed in this paper to realize automatic material classification based on stacking rules.
- For materials which can be combined, optimize the piling up according to the heating process curve, which is fundamental to increasing yield, saving energy for low-carbon production and increasing profit.

#### 4.2. Dynamic Clustering for Forgings

_{1}, w

_{2}, …,w

_{n}}, representing all forgings to be classified (in W, indicating the number of forgings to be classified). w

_{i}= (w

_{i1}, …, w

_{ik}, …, w

_{im}) (i = 1, 2, …, n) represents the parameters set used to describe the forgings for classification (wherein m represents the number of characters’ parameters).

- (1)
- Normalization

- a.
- For indexes where the smaller, the better,$${w}_{ik}^{\prime}=\frac{{w}_{k\mathrm{max}}-{w}_{ik}}{{w}_{k\mathrm{max}}-{w}_{k\mathrm{min}}}$$
- b.
- For indexes where the bigger, the better,$${w}_{ik}^{\prime}=\frac{{w}_{ik}-{w}_{k\mathrm{min}}}{{w}_{k\mathrm{max}}-{w}_{k\mathrm{min}}}$$
_{kmax}, x_{kmin}represent maximum and minimum of $\left\{{w}_{1k},{w}_{2k},\mathrm{...},{w}_{nk}\right\}$, respectively.

_{ik}′ after normalization is still denoted by w

_{ik}.

- (2)
- Determination of similarity coefficient

_{ij}∈ [0,1] be used to represent the similarity coefficient between elements w

_{i}and w

_{j}. If r

_{ij}= 0, w

_{i}is entirely different from w

_{j}without similarity. If r

_{ij}= 1, they are completely similar to or same with each other. When i = j, r

_{ij}is identically equal to 1. The normalized w

_{ik}is used to determine r

_{ij}.

_{ij}can be determined by Euclidean distance dot product, max-min and geometrical average minimum. The Euclidean distance method not only can compare the differences in overall respect, but can indicate the area with great differences, and has a high accuracy and maneuverability. The most important is the index below used to calculate the similarity of forgings that can be considered with the same weight. So, the Euclidean distance here is adopted.

_{ij}).

- a.
- Reflexivity. That is $\forall i,{r}_{ii}=1$.
- b.
- Symmetry. That is $\forall i,\forall j,{r}_{ij}={r}_{ji}\in [0,1]$.
- c.
- For clustering, the abovementioned matrix R has to be transformed to an equivalent matrix in order to enable transitivity within the matrix [17].

^{+}on set X such that R

^{+}contains R and R

^{+}is minimal. ${R}^{+}=\underset{i=\{1,2,\mathrm{3...}\}}{{\displaystyle \mathrm{U}}}{\mathrm{R}}^{\mathrm{i}}$ where ${\mathrm{R}}^{\mathrm{i}}$ is the i-th power of R, defined inductively by ${\mathrm{R}}^{1}=\mathrm{R}$ and for i > 0, ${\mathrm{R}}^{\mathrm{i}+1}=\mathrm{R}\xb0{\mathrm{R}}^{\mathrm{i}}$ ($\xb0$ denotes composition of relations) [18]. R* can be constructed by square self-synthesis method [19]. The square self-synthesis method is logical operation. The elements of ${\mathrm{R}}^{2}$ can be calculated as follows:

- (3)
- Determination of λ

_{ij}≥ λ, then r

_{ij}is replaced with 1, and otherwise, replaced with 0. Since R is a symmetrical matrix, corresponding Boolean matrix R

_{λ}can be obtained. In R

_{λ}, if r

_{ij}(λ) (i ≠ j) is equal to 1, then w

_{i}and w

_{j}can be clustered into one type. Therefore, various classifications can be obtained with different λ.

#### 4.3. Stacking Optimization

#### 4.3.1. Problem Analysis of Stacking Optimization

Material | Weight or Ingot Size | |
---|---|---|

2400kg or More Octagon Ingot | 2400–1200kg 22” Ingot | |

20, 35, 45, Q235 (A_{3}), A105, Q34516Mn, 16MnD, 30Mn |

#### 4.3.2. Mathematic Model for Stacking Optimization

_{i}refers to the unit weight of forgings α

_{i}). The number of forgings α

_{i}is θ

_{j}($\theta =[{\theta}_{1},{\theta}_{2},\mathrm{...},{\theta}_{n}]$ is a number set). So, the total stacked gross weight of forgings is:

- (1)
- Selection of heating furnace

- (2)
- Objective function

_{i}refers to the gross weight of forgings in each stacking; and ${\overline{x}}_{i}$ refers to theoretical weight set by heating curve corresponding to x

_{i}.

- (3)
- Constraints

- (4)
- Optimization

_{i}, randomly assign a value within interval from 0 to θ

_{i}and record as τ

_{i}.

_{i}indicates the number of workpieces actual scheduled).

_{i}with the theoretical weight set using the heating curve to obtain the absolute difference, namely the average deviation coefficient f. Then, update $\theta =[{\theta}_{1}-{\tau}_{1},{\theta}_{2}-{\tau}_{2},\mathrm{...},{\theta}_{n}-{\tau}_{n}]$.

_{n × k}matrix can be obtained.

_{ij}(i = 1, 2, …, n, j = 1, 2, …, k) refers to the number of workpieces i piled up in furnace j, n is the total number of forgings, k is the number of heating furnaces, and Matrix T refers to the stacking combination matrix.

#### 4.4. Case Study

No. | Forging Tonnage (T)↑ | Initial Temperature (°C)↓ | Weight (kg)↑ | Number of Heat (Times)↓ | Stacking Temperature (°C)↓ |
---|---|---|---|---|---|

1 | 5 | 1100 | 300 | 1 | 1200 |

2 | 3 | 1200 | 500 | 1 | 900 |

3 | 3 | 1100 | 400 | 1 | 1200 |

4 | 5 | 1100 | 500 | 2 | 1200 |

5 | 5 | 1100 | 400 | 2 | 900 |

6 | 5 | 1200 | 500 | 2 | 1200 |

7 | 5 | 1200 | 300 | 2 | 900 |

8 | 5 | 1200 | 500 | 1 | 1200 |

max | 5 | 1200 | 500 | 2 | 1200 |

min | 3 | 1100 | 300 | 1 | 900 |

- (1)
- After being normalized, we can get$${({w}_{ik})}_{8\times 5}=\left|\begin{array}{ccccc}1& 1& 0& 1& 0\\ 0& 0& 1& 1& 1\\ 0& 1& 0.5& 1& 0\\ 1& 1& 1& 0& 0\\ 1& 1& 0.5& 0& 1\\ 1& 0& 1& 0& 0\\ \begin{array}{l}1\\ 1\end{array}& \begin{array}{l}0\\ 0\end{array}& \begin{array}{l}0\\ 1\end{array}& \begin{array}{l}0\\ 1\end{array}& \begin{array}{l}1\\ 0\end{array}\end{array}\right|$$
- (2)
- Obtain similar matrix

- (3)
- Forgings classification (clustering)$${R}^{8}={R}^{4}=\left|\begin{array}{cccccccc}1& 0.500& 0.605& 0.500& 0.500& 0.500& 0.500& 0.500\\ 0.500& 1& 0.500& 0.500& 0.500& 0.500& 0.500& 0.500\\ 0.605& 0.500& 1& 0.500& 0.500& 0.500& 0.500& 0.500\\ 0.500& 0.500& 0.500& 1& 0.650& 0.646& 0.605& 0.500\\ 0.500& 0.500& 0.500& 0.650& 1& 0.646& 0.605& 0.500\\ 0.500& 0.500& 0.500& 0.646& 0.646& 1& 0.605& 0.500\\ 0.500& 0.500& 0.500& 0.605& 0.605& 0.605& 1& 0.500\\ 0.500& 0.500& 0.500& 0.5& 0.500& 0.500& 0.500& 1\end{array}\right|$$

- (4)
- Stacking optimization

No. | Unit Ingot Weight (kg) | Quantity |
---|---|---|

1 | 310 | 25 |

3 | 518 | 10 |

4 | 415 | 15 |

5 | 520 | 5 |

6 | 309 | 20 |

7 | 516 | 26 |

Furnace No. | Forging No. | Quantity | Gross Weight (T) | Deviation from Theoretical Value (T) | Average Difference f (%) |
---|---|---|---|---|---|

1 | 1 | 25 | 7.75 | 0.25 | 3.23 |

2 | 03, 04 | 10, 6 | 7.67 | 0.33 | 4.3 |

3 | 04, 05, 06 | 9, 5, 5 | 7.8 | 0.2 | 2.56 |

4 | 06, 07 | 15, 2 | 5.667 | 0.333 | 4.26 |

5 | 7 | 11 | 5.632 | 0.368 | 6.53 |

6 | 7 | 13 | 6.708 | 0.282 | 4.2 |

## 5. Conclusions

- (1)
- The presented work focuses on two types of scheduling for a forging enterprise. One is for cutting and machining scheduling, which is similar to traditional machining scheduling, and the other is for forging and heat treatment scheduling, characterized by stacking and heat treatment.
- (2)
- Dynamic clustering is proposed for forging combination before stacking optimization. The forgings to be optimized are clustered according to certain rules, which can greatly reduce the computations required in stacking optimization in order to more easily obtain a solution.
- (3)
- In reality, the production manager judges different forgings based on empirical criteria. This empirical standard is generally difficult to portray. In the fuzzy clustering analysis, λ (0 < λ < 1) is utilized to indicate the empirical criteria. The clustering results vary with λ, resulting in various heating plans with different production efficiency, energy consumption and carbon emissions.
- (4)
- The proposed stacking optimization involves ensuring the gross weight of forgings is as close to the maximum batch capacity as possible.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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## Share and Cite

**MDPI and ACS Style**

Tong, Y.; Li, J.; Li, S.; Li, D.
Research on Energy-Saving Production Scheduling Based on a Clustering Algorithm for a Forging Enterprise. *Sustainability* **2016**, *8*, 136.
https://doi.org/10.3390/su8020136

**AMA Style**

Tong Y, Li J, Li S, Li D.
Research on Energy-Saving Production Scheduling Based on a Clustering Algorithm for a Forging Enterprise. *Sustainability*. 2016; 8(2):136.
https://doi.org/10.3390/su8020136

**Chicago/Turabian Style**

Tong, Yifei, Jingwei Li, Shai Li, and Dongbo Li.
2016. "Research on Energy-Saving Production Scheduling Based on a Clustering Algorithm for a Forging Enterprise" *Sustainability* 8, no. 2: 136.
https://doi.org/10.3390/su8020136