Multi-Objective Bi-Level Programming for the Energy-Aware Integration of Flexible Job Shop Scheduling and Multi-Row Layout
<p>Relations between FJSSP and MRWLP.</p> "> Figure 2
<p>Structures of three methods for solving the ISLP problem.</p> "> Figure 3
<p>Leader-follower coordinated structure of the MEIFM problem.</p> "> Figure 4
<p>Structure of the proposed bi-level model.</p> "> Figure 5
<p>Two cases of calculating <span class="html-italic">dt<sub>ijk</sub></span>.</p> "> Figure 6
<p>The flowchart of IMHGA.</p> "> Figure 7
<p>A chromosome example for the scheduling scheme.</p> "> Figure 8
<p>An example for MIPOX.</p> "> Figure 9
<p>An example for MMPX.</p> "> Figure 10
<p>Two examples of calculating <span class="html-italic">fit</span>.</p> "> Figure 11
<p>Tournament selection operator with the parent-offspring competition strategy.</p> "> Figure 12
<p>The convergence curve of three objective functions.</p> "> Figure 13
<p>The Pareto solutions’ distribution for three objective functions.</p> "> Figure 14
<p>The Gantt chart of solution 1.</p> "> Figure 15
<p>The layout scheme of solution 1.</p> "> Figure 16
<p>The distribution curves for three energy consumption components.</p> "> Figure 17
<p>The Gantt chart of solution 13.</p> "> Figure 18
<p>The layout scheme of solution 13.</p> "> Figure 19
<p>An example of calculating the convergence and spacing metric.</p> ">
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
4. Model Solution
4.1. Algorithm Construction
4.2. The Upper-Level Algorithm for FJSSP
4.2.1. Encoding and Initialization Population
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 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
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
5.3. Algorithm Comparison
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 | Cmax | Makespan |
TEC | Total energy consumption | |
MHQ | Material handling quantity | |
Index | i | Index of job, i = 1, 2, , n |
j | Index of operation for job i, j = 1, 2, , oi | |
k, l | Index of machine, k, l = 1, 2, , m | |
r | Index of machine row number, r = 1, 2, , g | |
Intermediate variables | Oij | j-th operation of the job i |
ctijk | Completion time of Oij on machine k | |
cti’j’k | Completion time of immediate operation of Oij on machine k | |
stijk | Start time of Oij on machine k | |
itk | Idle time of machine k | |
ttlk | Transportation time from machine l to machine k | |
dtijk | Delay time of Oij on machine k due to machine resource constraints | |
dlk | Distance between machine l to machine k | |
flk | 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 | |
xl | Horizontal coordinate of machine l in workshop | |
yl | Vertical coordinate of machine l in workshop | |
Input variables | ptijk | Processing time of Oij on machine k |
v | Transportation speed of transporter in workshop | |
pek | Processing energy consumption per unit time of machine k | |
iek | Idle energy consumption per unit time of machine k | |
te | Transportation energy consumption per unit time of transporter | |
elk | Minimal distance between machine l and machine k that must be maintained in horizontal direction | |
Δlk | Net distance between machine l and machine k in horizontal direction | |
el | Length of machine l in horizontal direction | |
wl | 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 | xijk | Binary variable, if Oij is processed on machine k, then xijk = 1; otherwise, xijk = 0 |
xilk | Binary variable, if job i is transported from machine l to machine k, then xilk = 1; otherwise, xilk = 0 | |
zlr | Binary variable, if machine l is located on r-th row in the workshop, then zlr = 1; otherwise zlr = 0 |
Job | Operation | Processing Time (min) | |||||||
---|---|---|---|---|---|---|---|---|---|
M1 | M2 | M3 | M4 | M5 | M6 | M7 | M8 | ||
Job 1 | O1,1 | 5 | 3 | 5 | 3 | 3 | — | 10 | 9 |
O1,2 | 10 | — | 5 | 8 | 3 | 9 | 9 | 6 | |
O1,3 | — | 10 | — | 5 | 6 | 2 | 4 | 5 | |
Job 2 | O2,1 | 5 | 7 | 3 | 9 | 8 | — | 9 | — |
O2,2 | — | 8 | 5 | 2 | 6 | 7 | 10 | 9 | |
O2,3 | — | 10 | — | 5 | 6 | 4 | 1 | 7 | |
O2,4 | 10 | 8 | 9 | 6 | 4 | 7 | — | — | |
Job 3 | O3,1 | 10 | — | — | 7 | 6 | 5 | 2 | 4 |
O3,2 | — | 10 | 6 | 4 | 8 | 9 | 10 | — | |
O3,3 | 1 | 4 | 5 | 6 | — | 10 | — | 7 | |
Job 4 | O4,1 | 3 | 1 | 6 | 5 | 9 | 7 | 8 | 4 |
O4,2 | 12 | 11 | 7 | 8 | 10 | 5 | 6 | 9 | |
O4,3 | 4 | 6 | 2 | 10 | 3 | 9 | 5 | 7 | |
Job 5 | O5,1 | 3 | 6 | 7 | 8 | 9 | — | 10 | — |
O5,2 | 10 | — | 7 | 4 | 9 | 8 | 6 | — | |
O5,3 | — | 9 | 8 | 7 | 4 | 2 | 7 | — | |
O5,4 | 11 | 9 | — | 6 | 7 | 5 | 3 | 6 | |
Job 6 | O6,1 | 6 | 7 | 1 | 4 | 6 | 9 | — | 10 |
O6,2 | 11 | — | 9 | 9 | 9 | 7 | 6 | 4 | |
O6,3 | 10 | 5 | 9 | 10 | 11 | — | 10 | — | |
Job 7 | O7,1 | 5 | 4 | 2 | 6 | 7 | — | 10 | — |
O7,2 | — | 9 | — | 9 | 11 | 9 | 10 | 5 | |
O7,3 | — | 8 | 9 | 3 | 8 | 6 | — | 10 | |
Job 8 | O8,1 | 2 | 8 | 5 | 9 | — | 4 | — | 10 |
O8,2 | 7 | 4 | 7 | 8 | 9 | — | 10 | — | |
O8,3 | 9 | 9 | — | 8 | 5 | 6 | 7 | 1 | |
O8,4 | 9 | — | 3 | 7 | 1 | 5 | 8 | — |
Machine Number | M1 | M2 | M3 | M4 | M5 | M6 | M7 | M8 | |
---|---|---|---|---|---|---|---|---|---|
Energy Consumption | |||||||||
pek (kw) | 4.0 | 7.0 | 9.0 | 14.0 | 6.0 | 5.0 | 8.0 | 4.0 | |
iek (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 | |||||||||
ek (m) | 1.9 | 3.0 | 2.0 | 2.0 | 2.5 | 3.0 | 3.0 | 5.0 | |
wk (m) | 1.8 | 2.0 | 1.0 | 1.8 | 1.5 | 3.0 | 2.8 | 3.0 |
Solution Number | Cmax (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 |
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 | Cmax (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 |
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Share and Cite
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
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 StyleZhang, 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