# Multi-Objective Bi-Level Programming for the Energy-Aware Integration of Flexible Job Shop Scheduling and Multi-Row Layout

^{*}

## Abstract

**:**

## 1. Introduction

- In much research, scheduling is done based on the assumption that the transportation time between machines is either neglected or determined. However, in the actual workshop, the positions of machines will significantly affect the transportation time of jobs. This may make the enterprises’ production cycle longer or generate production stagnation, which leads to more idle energy consumption. Therefore, the generated scheduling schemes are somehow unrealistic and cannot be readily executed in the workshop, resulting in the optimum scheduling scheme often becoming infeasible;
- After the layout is set, the performance of scheduling schemes is highly dependent on the determined layout scheme. Moreover, if the type and requirement of the product change greatly, the scheduling scheme will change accordingly. As a result, the material handling information between machines will be greatly affected, which may cause the original layout scheme to become inefficient;
- Separate optimization of scheduling and layout planning does not guarantee optimality of the whole manufacturing system since scheduling or layout planning has more than one criterion to be considered, in which many criterions are often conflicting. For example, in the real manufacturing process, each operation could be implemented on a set of machines, including dedicated machines and universal machines. Generally, when an operation is processed on the dedicated machine, the corresponding processing time and energy consumption are minimal. In this manner, the scheduling scheme displays a short completion time and low processing energy consumption, while the corresponding layout scheme may lead to high transportation energy consumption and material handling quantity since the jobs are frequently transported between machines. On the contrary, the scheduling scheme may lead to a long completion time and high processing energy consumption, while the corresponding layout scheme may lead to low transportation energy consumption and material handling quantity. Neither of these manners can obtain a high production efficiency and low energy consumption solution.

- (1)
- These studies cannot provide effective guidance for enterprises. Since most of these studies focus on the job shop scheduling problem (JSSP) and discrete workshop layout problem (DWLP), the flexible processing route of jobs and size of machines are neglected, which means that these studies cannot solve more realistic problems, such as the flexible job shop scheduling problem (FJSSP), single-row workshop layout problem (SRWLP), multi-row workshop layout problem (MRWLP), and so on;
- (2)
- The optimality of the layout scheme cannot be ensured. Because most studies of ISLP only consider scheduling objectives, the layout problem is simply appended to the scheduling problem as a constraint, which ignores the interaction between scheduling and layout planning. For the integrated model that only considers scheduling objectives, it is difficult to ensure the feasibility of the scheduling and layout schemes simultaneously. For example, if only the makespan is optimized, a scheduling scheme with lower makespan may be accompanied by an unreasonable layout scheme. The unreasonable layout scheme may result in frequent job delays and processing interruptions, which greatly offsets the economic advantage;
- (3)
- Only a few studies of ISLP consider the energy consumption indicator. If only optimizing the efficiency objectives, a solution with a higher production efficiency may also be a solution with higher energy consumption. The higher energy cost will have an adverse impact on the final profit of enterprises. Admittedly, we should seek a solution that balances energy consumption and production efficiency in solving the ISLP problem.

- (1)
- An MEIFM problem is proposed for balancing the production efficiency and energy consumption;
- (2)
- Based on the interaction of FJSSP and MRWLP, an MOBLP model is formulated to depict the integrated problem;
- (3)
- An IMHGA is proposed to solve the bi-level programming model for optimizing the FJSSP and MRWLP simultaneously.

## 2. Literature Review on Solution Strategies

## 3. Bi-Level Programming Model for MEIFM

#### 3.1. Problem Description

- (1)
- All jobs and machines are available at zero time, and machines can only be shut down if all jobs on them have been completed;
- (2)
- The job processing cannot be interrupted after starting processing, and each machine can only process one job at a time;
- (3)
- The loading and unloading time should be neglected in the process of jobs transportation;
- (4)
- The centers of machines located on the same row are in the same horizontal line;
- (5)
- The transportation time and energy consumption of jobs are only related to the distance between machines.

#### 3.2. Model Formulation

#### 3.2.1. Notations

#### 3.2.2. Multi-Objective Bi-Level Programming Model

_{lk}and d

_{lk}. First, the lower-level model regards the f

_{lk}of the upper-level model as constraints to optimize the layout scheme. Then, the d

_{lk}of the lower-level model feeds back to the upper-level model to affect the evaluation indexes of the upper-level model.

_{ij}on machine k. Figure 5 shows two cases of calculating dt

_{ijk}, in which Figure 5a expresses that when ct

_{i’j’k}is less than or equal to the sum of ct

_{i}

_{(j−1)l}and tt

_{lk}, dt

_{ijk}= 0 and Figure 5b shows that when ct

_{i’j’k}is greater than the sum of ct

_{i}

_{(j−1)l}and tt

_{lk}, dt

_{ijk}= ct

_{i’j’k}− tt

_{lk}− ct

_{i}

_{(j−1)l}. Equation (6) calculates the transportation time of jobs between machines. Equation (7) calculates the start time of O

_{ij}on machine k. Equation (8) computes the material handling frequency between machines. Equation (9) calculates the idle time of each machine. Equations (10)–(12) are used to get the total processing energy consumption, total idle energy consumption, and total transportation energy consumption, respectively.

## 4. Model Solution

#### 4.1. Algorithm Construction

_{max}and TEC, and the lower-level GA is employed to obtain the optimal layout scheme by minimizing the MHQ.

_{lk}. Thus, to improve the convergence speed of the lower-level GA and find the optimal layout scheme, we adopt the tournament selection operator with the parent-offspring competition strategy as the selection operation. This selection operation can reserve elite individuals and conduct a centralized search of their neighborhood. Besides this, it can improve the convergence speed remarkably [35]. Specifically, the flowchart of IMHGA is given in Figure 6.

#### 4.2. The Upper-Level Algorithm for FJSSP

#### 4.2.1. Encoding and Initialization Population

_{21}. Similarly, the number 2 written in the fifth gene of the OS chromosome is the second appearance of operation 2, referring to operation 2 of job 2, i.e., O

_{22}. In Figure 7b, O

_{11}written outside the first gene of the MA chromosome indicates that operation 1 of job 1 will be processed on machine 1. Similarly, O

_{12}and O

_{13}illustrate that operation 2 of job 1 will be processed on machine 4 and operation 3 of job 1 will be processed on machine 5.

#### 4.2.2. Fitness Evaluation

#### 4.2.3. Selection Operator

#### 4.2.4. Multi-Parent Crossover Operator

Algorithm 1 The procedure of MIPOX |

Input: Three parent operation sequence chromosomes |

Output: Two offspring operation sequence chromosomes |

1: Randomly divide the set of job numbers $\{1,2,\dots ,n\}$ into two nonempty exclusive subsets J1 and J2; |

2: Copy those numbers in J2 from parent 1 to offspring 1 and from parent 3 to offspring 2, preserving their order; |

3: Copy those numbers in J1 from parent 2 to offspring 2 and from parent 3 to offspring 1, preserving their order. |

Algorithm 2 The procedure of MMPX |

Input: Three parent machine assignment chromosomes |

Output: Two offspring machine assignment chromosomes |

1: Generate a random set Rand0_1, which consists of integer 0 and 1, and has the same length as machine assignment chromosomes; |

2: If Rand0_1 = 0, machine assignment number copies directly from Parent 1 to Offspring 1 and from Parent 2 to Offspring 2; |

3: If Rand0_1 = 1, machine assignment number copies randomly from Parent 2 and Parent 3 to Offspring 1 and from Parent 1 and Parent 3 to Offspring 2. |

#### 4.2.5. Mutation Evaluation

#### 4.3. The Lower-Level Algorithm for MRWLP

#### 4.3.1. Encoding and Decoding

#### 4.3.2. Fitness Evaluation

#### 4.3.3. Tournament Selection Operator with Parent-Offspring Competition Strategy

_{l}) and corresponding offspring population (O

_{l}) constitute a temporary population (T

_{l}). Then, the T

_{l}is sorted according to the fitness value. Finally, the first N

_{l}(the size of P

_{l}) chromosomes are selected to form the next generation parent population (P

_{l}

_{+1}), so as to ensure that the population size remains unchanged. The parent-offspring competition strategy not only preserves elite individuals and avoids loss of the best solution, but also improves the fitness of the overall population. The specific process is shown in Figure 11.

#### 4.3.4. Crossover Operator

#### 4.3.5. Mutation Operator

## 5. Computation Experiments

#### 5.1. Description of Test Data and Parameter Setting

#### 5.2. Experimental Analyses

_{max}and PE of solution 13 are less than solution 1, the TEC of solution 13 is still greater than that of solution 1. This is because each operation in solution 13 chooses the machine with a minimum processing time, which causes jobs to move frequently between machines, resulting in excessive TE and MHQ. Therefore, the PE and TE curves generate the peak-trough correspondence phenomenon. The phenomenon further illustrates that the coordinated optimization of scheduling and layout planning can not only quick respond to the changes of market demand, but also balance the production efficiency and energy consumption of enterprises.

#### 5.3. Algorithm Comparison

_{1}(1,7), a

_{2}(2, 4), a

_{3}(3, 3), a

_{4}(4, 2), a

_{4}(7, 1)] and B = [b

_{1}(2, 6), b

_{2}(3, 5), b

_{3}(4, 4), b

_{4}(5, 3), b

_{5}(6, 2)]. In Figure 19, it is obvious that the convergence of A is better than B and the distribution of B is better than A. From the perspective of the convergence metric, C(A,B) = 5/5 = 1 and C(B,A) = 0/5 = 0. Besides, from the perspective of the spacing metric, d

_{a}

_{1}= min{|1 − 2| + |7 − 4|, |1 − 3| + |7 − 3|, |1 − 4| + |7 − 2|, |1 − 7| + |7 − 1|} = min{4, 6, 8, 12} = 4; d

_{a}

_{2}= 2; d

_{a}

_{3}= 2; d

_{a}

_{4}= 2; d

_{a}

_{5}= 4; d

_{b}

_{1}= 2; d

_{b}

_{2}= 2; d

_{b}

_{3}= 2; d

_{b}

_{4}= 2; d

_{b}

_{5}= 2; $\overline{{d}_{A}}$ = 2.8 and $\overline{{d}_{B}}$ = 2. Accordingly, S(A) = $\sqrt{\frac{1}{1-5}{{\displaystyle \sum}}_{i=1}^{5}(2.8-{d}_{\mathit{ai}})}$ = 1.0954 and S(B) = 0. It is clear that the results of the two metrics consist of the actual characterization of A and B, and illustrate that the larger the convergence value, the better the convergence, and the smaller spacing value, the better the distribution.

## 6. Conclusions and Future Work

- (1)
- Separate optimization of scheduling and layout planning can limit the performance of the manufacturing system because the interaction between them is ignored. Therefore, the coordination optimization of scheduling and layout planning is necessary and can greatly improve the compatibility of the manufacturing system;
- (2)
- The solutions of the MEIFM problem proposed by this paper not only improve the responsiveness of enterprises facing rapid changes of market demand, but also provide energy-saving methods from a systematic optimization perspective for manufacturing enterprises;
- (3)
- The methodology developed in this paper will provide efficient guidance and reference for solving complex bilevel optimization problems.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Symbol | Meaning | |
---|---|---|

Optimization objectives | C_{max} | Makespan |

TEC | Total energy consumption | |

MHQ | Material handling quantity | |

Index | i | Index of job, i = 1, 2, $\cdots $, n |

j | Index of operation for job i, j = 1, 2, $\cdots $, o_{i} | |

k, l | Index of machine, k, l = 1, 2, $\cdots $, m | |

r | Index of machine row number, r = 1, 2, $\cdots $, g | |

Intermediate variables | O_{ij} | j-th operation of the job i |

ct_{ijk} | Completion time of O_{ij} on machine k | |

ct_{i’j’k} | Completion time of immediate operation of O_{ij} on machine k | |

st_{ijk} | Start time of O_{ij} on machine k | |

it_{k} | Idle time of machine k | |

tt_{lk} | Transportation time from machine l to machine k | |

dt_{ijk} | Delay time of O_{ij} on machine k due to machine resource constraints | |

d_{lk} | Distance between machine l to machine k | |

f_{lk} | Material handling frequency from machine l to machine k | |

PE | Processing energy consumption of all machines | |

IE | Idle energy consumption of all machines | |

TE | Transportation energy consumption of all jobs in workshop | |

x_{l} | Horizontal coordinate of machine l in workshop | |

y_{l} | Vertical coordinate of machine l in workshop | |

Input variables | pt_{ijk} | Processing time of O_{ij} on machine k |

v | Transportation speed of transporter in workshop | |

pe_{k} | Processing energy consumption per unit time of machine k | |

ie_{k} | Idle energy consumption per unit time of machine k | |

te | Transportation energy consumption per unit time of transporter | |

e_{lk} | Minimal distance between machine l and machine k that must be maintained in horizontal direction | |

Δ_{lk} | Net distance between machine l and machine k in horizontal direction | |

e_{l} | Length of machine l in horizontal direction | |

w_{l} | Width of machine l in vertical direction | |

s | Center distance of two adjacent rows | |

E | Length of workshop in horizontal direction | |

W | Width of workshop in vertical direction | |

Decision variables | x_{ijk} | Binary variable, if O_{ij} is processed on machine k, then x_{ijk} = 1; otherwise, x_{ijk} = 0 |

x_{ilk} | Binary variable, if job i is transported from machine l to machine k, then x_{ilk} = 1; otherwise, x_{ilk} = 0 | |

z_{lr} | Binary variable, if machine l is located on r-th row in the workshop, then z_{lr} = 1; otherwise z_{lr} = 0 |

Job | Operation | Processing Time (min) | |||||||
---|---|---|---|---|---|---|---|---|---|

M1 | M2 | M3 | M4 | M5 | M6 | M7 | M8 | ||

Job 1 | O_{1,1} | 5 | 3 | 5 | 3 | 3 | — | 10 | 9 |

O_{1,2} | 10 | — | 5 | 8 | 3 | 9 | 9 | 6 | |

O_{1,3} | — | 10 | — | 5 | 6 | 2 | 4 | 5 | |

Job 2 | O_{2,1} | 5 | 7 | 3 | 9 | 8 | — | 9 | — |

O_{2,2} | — | 8 | 5 | 2 | 6 | 7 | 10 | 9 | |

O_{2,3} | — | 10 | — | 5 | 6 | 4 | 1 | 7 | |

O_{2,4} | 10 | 8 | 9 | 6 | 4 | 7 | — | — | |

Job 3 | O_{3,1} | 10 | — | — | 7 | 6 | 5 | 2 | 4 |

O_{3,2} | — | 10 | 6 | 4 | 8 | 9 | 10 | — | |

O_{3,3} | 1 | 4 | 5 | 6 | — | 10 | — | 7 | |

Job 4 | O_{4,1} | 3 | 1 | 6 | 5 | 9 | 7 | 8 | 4 |

O_{4,2} | 12 | 11 | 7 | 8 | 10 | 5 | 6 | 9 | |

O_{4,3} | 4 | 6 | 2 | 10 | 3 | 9 | 5 | 7 | |

Job 5 | O_{5,1} | 3 | 6 | 7 | 8 | 9 | — | 10 | — |

O_{5,2} | 10 | — | 7 | 4 | 9 | 8 | 6 | — | |

O_{5,3} | — | 9 | 8 | 7 | 4 | 2 | 7 | — | |

O_{5,4} | 11 | 9 | — | 6 | 7 | 5 | 3 | 6 | |

Job 6 | O_{6,1} | 6 | 7 | 1 | 4 | 6 | 9 | — | 10 |

O_{6,2} | 11 | — | 9 | 9 | 9 | 7 | 6 | 4 | |

O_{6,3} | 10 | 5 | 9 | 10 | 11 | — | 10 | — | |

Job 7 | O_{7,1} | 5 | 4 | 2 | 6 | 7 | — | 10 | — |

O_{7,2} | — | 9 | — | 9 | 11 | 9 | 10 | 5 | |

O_{7,3} | — | 8 | 9 | 3 | 8 | 6 | — | 10 | |

Job 8 | O_{8,1} | 2 | 8 | 5 | 9 | — | 4 | — | 10 |

O_{8,2} | 7 | 4 | 7 | 8 | 9 | — | 10 | — | |

O_{8,3} | 9 | 9 | — | 8 | 5 | 6 | 7 | 1 | |

O_{8,4} | 9 | — | 3 | 7 | 1 | 5 | 8 | — |

Machine Number | M1 | M2 | M3 | M4 | M5 | M6 | M7 | M8 | |
---|---|---|---|---|---|---|---|---|---|

Energy Consumption | |||||||||

pe_{k} (kw) | 4.0 | 7.0 | 9.0 | 14.0 | 6.0 | 5.0 | 8.0 | 4.0 | |

ie_{k} (kw) | 1.0 | 1.2 | 0.9 | 0.8 | 0.6 | 0.9 | 0.8 | 0.8 |

Machine Number | M1 | M2 | M3 | M4 | M5 | M6 | M7 | M8 | |
---|---|---|---|---|---|---|---|---|---|

Size | |||||||||

e_{k} (m) | 1.9 | 3.0 | 2.0 | 2.0 | 2.5 | 3.0 | 3.0 | 5.0 | |

w_{k} (m) | 1.8 | 2.0 | 1.0 | 1.8 | 1.5 | 3.0 | 2.8 | 3.0 |

Solution Number | C_{max} (min) | TEC (kw/h) | MHQ (kg) | PE (kw/h) | IE (kw/h) | TE (kw/h) |
---|---|---|---|---|---|---|

1 | 23.4752 | 853.0703 | 82.8975 | 562 | 83.8266 | 207.2437 |

2 | 23.4752 | 853.2411 | 82.9660 | 562 | 83.8260 | 207.4151 |

3 | 23.5124 | 857.1738 | 73.6347 | 592 | 81.0870 | 184.0868 |

4 | 23.4751 | 866.7355 | 88.3640 | 562 | 83.8254 | 220.9101 |

5 | 23.4794 | 871.7218 | 79.5463 | 592 | 80.8560 | 198.8658 |

$\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ |

46 | 33.3751 | 1133.8233 | 24.9989 | 952 | 119.3260 | 62.4973 |

47 | 33.3750 | 1148.3944 | 27.4677 | 954 | 125.7250 | 68.6693 |

48 | 33.3838 | 1156.5963 | 23.0329 | 978 | 120.9869 | 57.5823 |

49 | 33.3769 | 1158.3316 | 24.8773 | 978 | 118.1384 | 62.1932 |

50 | 33.3750 | 1188.4205 | 26.4382 | 1004 | 118.3250 | 66.0955 |

Solution Number | C_{max} (min) | TEC (kw/h) | MHQ (kg) | PE (kw/h) | IE (kw/h) | TE (kw/h) |
---|---|---|---|---|---|---|

13 | 22.0077 | 904.4978 | 107.4975 | 550 | 85.7541 | 268.7437 |

14 | 21.5220 | 906.5748 | 106.9683 | 550 | 89.1541 | 267.4207 |

15 | 22.7642 | 909.2554 | 105.9224 | 550 | 94.4495 | 264.8059 |

16 | 22.6255 | 910.7705 | 106.9167 | 550 | 93.4788 | 267.2918 |

Algorithm | Convergence Metric | Spacing Metric | |
---|---|---|---|

C(IMHGA, MHGA) | C(MHGA, IMHGA) | ||

IMHGA | 0.9338 | — | 3.2292 |

MHGA | — | 0.2000 | 5.9856 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhang, H.; Ge, H.; Pan, R.; Wu, Y.
Multi-Objective Bi-Level Programming for the Energy-Aware Integration of Flexible Job Shop Scheduling and Multi-Row Layout. *Algorithms* **2018**, *11*, 210.
https://doi.org/10.3390/a11120210

**AMA Style**

Zhang H, Ge H, Pan R, Wu Y.
Multi-Objective Bi-Level Programming for the Energy-Aware Integration of Flexible Job Shop Scheduling and Multi-Row Layout. *Algorithms*. 2018; 11(12):210.
https://doi.org/10.3390/a11120210

**Chicago/Turabian Style**

Zhang, Hongliang, Haijiang Ge, Ruilin Pan, and Yujuan Wu.
2018. "Multi-Objective Bi-Level Programming for the Energy-Aware Integration of Flexible Job Shop Scheduling and Multi-Row Layout" *Algorithms* 11, no. 12: 210.
https://doi.org/10.3390/a11120210