A Bi-Objective Approach to Minimize Makespan and Energy Consumption in Flow Shops with Peak Demand Constraint
Abstract
:1. Introduction
2. Literature Review
3. Problem Statement
3.1. Problem Description
3.2. Mathematical Programming Model
- If job is processed at the kth position on machine with speed , it is 1; else, 0.
- If there is a PM immediately before , ; else, 0.
- If there is a setup immediately before , ; else, 0.
- If is started in the interval and there is a setup before , it is 1; else, 0.
- Actual processing time of .
- Actual energy consumption per unit time when processing .
- Start time of .
- Finish time of .
- Machine’s age immediately before .
- Machine’s age immediately after .
- Energy consumption of machine during the interval.
- Energy consumption of caused by during the interval.
- Energy consumption of caused by idle time before during the interval.
4. Algorithm Designing
4.1. Framework
4.2. Decoding Method
- Step 1.
- Set . Set .
- Step 2.
- Set .
- Step 3.
- Calculate the energy consumption of in the interval, which is . Then, we have . If , then change the speed level of and update . Calculate the finish time of for each machine, which is .
- Step 4.
- Set .
- Step 5.
- If , then set , go to Step 2. Else, if , set , go to Step 6. Else, go to Step 6.
- Step 6.
- If , then set , , go to Step 5; else, go to Step 7.
- Step 7.
- Let be the successor of . Let be the basic processing time of . If , then set and perform a PM to set .
- Step 8.
- Find the earliest allowed start time of , which is . If , then go to Step 9; else, go to Step 11.
- Step 9.
- If , then set , go to Step 5; else, go to Step 10.
- Step 10.
- Calculate the energy consumption of the remaining idle time of this interval, which is . If , then set , , go to Step 5. Else, set , , go to Step 5.
- Step 11.
- If , then try to start the job as early as possible, update , go to Step 12; else, go to Step 14.
- Step 12.
- If the idle consumption between and is smaller than , then set . If the length of this idle time is longer than and , then set . Go to Step 13.
- Step 13.
- If the finish time of is smaller than , then set , consider the next operation in this machine, go to Step 7. Else, this job will be one job started in the interval and finished in the following intervals, set , go to Step 5.
- Step 14.
- If , then set , , go to Step 11. Else, set , go to Step 5.
5. Numerical Results
5.1. Validation of Algorithm
5.2. Impact of Constraints
5.3. Comparison between Different Solutions
5.4. Discussions
6. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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Reference | Production Scheduling | Maintenance Planning | Energy Controlling | |||||
---|---|---|---|---|---|---|---|---|
Production | Jobs Sequencing | Fixed | Flexible | Objective | On/Off | Speed Selection | Peak Demand | |
[12,13,14] | √ | √ | ✕ | ✕ | ✕ | ✕ | ✕ | ✕ |
[15,16,17,18,19,20,21,22,23,24] | √ | √ | √ | ✕ | ✕ | ✕ | ✕ | ✕ |
[25,26,27] | √ | √ | ✕ | √ | ✕ | ✕ | ✕ | ✕ |
[28,29,30,31,32] | √ | √ | ✕ | ✕ | √ | ✕ | ✕ | ✕ |
[33,34,35,36,37,38,39] | √ | √ | ✕ | ✕ | √ | √ | ✕ | ✕ |
[40,41,42,43,44,45,46,47] | √ | √ | ✕ | ✕ | √ | ✕ | √ | ✕ |
[48] | √ | √ | ✕ | ✕ | √ | √ | √ | ✕ |
[49] | √ | √ | ✕ | ✕ | √ | ✕ | ✕ | √ |
[50] | √ | ✕ | ✕ | ✕ | √ | ✕ | √ | √ |
[51,52] | √ | ✕ | ✕ | ✕ | √ | √ | ✕ | √ |
[53,54] | √ | √ | ✕ | ✕ | √ | ✕ | √ | √ |
This paper | √ | √ | ✕ | √ | √ | √ | √ | √ |
pt | n | Peak Limit | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
120 | 135 | 150 | 165 | 180 | |||||||
60 | 50 | 1 | 0 | 1 | 0 | 0.96 | 0 | 1 | 0 | 1 | 0 |
100 | 1 | 0 | 1 | 0 | 0.94 | 0 | 0.87 | 0 | 1 | 0 | |
200 | 0.98 | 0 | 0.98 | 0 | 0.89 | 0 | 0.77 | 0 | 0.94 | 0 | |
180 | 50 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |
100 | 1 | 0 | 1 | 0 | 0.98 | 0 | 1 | 0 | 1 | 0 | |
200 | 1 | 0 | 1 | 0 | 0.97 | 0 | 1 | 0 | 1 | 0 | |
300 | 50 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 0.99 | 0 |
100 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | |
200 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |
pt | n | Peak Limit | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
120 | 135 | 150 | 165 | 180 | |||||||
NS-II | CH | NS-II | CH | NS-II | CH | NS-II | CH | NS-II | CH | ||
60 | 50 | 200 | 18 | 200 | 25 | 200 | 30 | 200 | 35 | 200 | 37 |
100 | 200 | 16 | 200 | 22 | 200 | 27 | 200 | 31 | 200 | 33 | |
200 | 200 | 16 | 200 | 21 | 200 | 25 | 200 | 29 | 200 | 32 | |
180 | 50 | 200 | 17 | 200 | 23 | 200 | 28 | 200 | 31 | 200 | 33 |
100 | 200 | 18 | 200 | 24 | 200 | 28 | 200 | 32 | 200 | 35 | |
200 | 200 | 18 | 200 | 24 | 200 | 26 | 200 | 30 | 200 | 33 | |
300 | 50 | 200 | 19 | 200 | 23 | 200 | 29 | 200 | 33 | 200 | 34 |
100 | 200 | 17 | 200 | 22 | 200 | 26 | 200 | 28 | 200 | 30 | |
200 | 200 | 15 | 200 | 21 | 200 | 24 | 200 | 26 | 200 | 29 |
pt | n | “Left Solution” | “Right Solution” | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Peak = 120 | Peak = 150 | Peak = 180 | Peak = 120 | Peak = 150 | Peak = 180 | ||||||||
Cmax | TEC | Cmax | TEC | Cmax | TEC | Cmax | TEC | Cmax | TEC | Cmax | TEC | ||
60 | 50 | 1438 | 10,117 | 1335 | 10,428 | 1127 | 11,355 | 1896 | 8543 | 1903 | 8539 | 1927 | 8531 |
100 | 2894 | 20,093 | 2653 | 20,878 | 2208 | 23,036 | 3715 | 17,209 | 3744 | 17,233 | 3708 | 17,230 | |
200 | 5958 | 41,418 | 5451 | 42,558 | 4674 | 45,587 | 7387 | 35,092 | 7395 | 35081 | 7396 | 35,072 | |
180 | 50 | 1560 | 10,125 | 1333 | 11,102 | 1211 | 11,489 | 2023 | 8545 | 2023 | 8533 | 2023 | 8551 |
100 | 3101 | 20,271 | 2896 | 21,022 | 2456 | 22,565 | 4021 | 17,112 | 4030 | 17,146 | 4045 | 17,160 | |
200 | 6417 | 40,994 | 5965 | 42,450 | 4871 | 47,640 | 7880 | 35,035 | 7846 | 35,049 | 7852 | 35,060 | |
300 | 50 | 1685 | 10,115 | 1422 | 11,246 | 1337 | 11,421 | 2146 | 8536 | 2132 | 8554 | 2139 | 8557 |
100 | 3364 | 20,011 | 3032 | 21,367 | 2691 | 22,836 | 4315 | 17,101 | 4411 | 17,075 | 4294 | 17,121 | |
200 | 6924 | 40,962 | 5813 | 46,168 | 5288 | 48,373 | 8382 | 35,013 | 8378 | 35,053 | 8448 | 35,035 |
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Cui, W.; Lu, B. A Bi-Objective Approach to Minimize Makespan and Energy Consumption in Flow Shops with Peak Demand Constraint. Sustainability 2020, 12, 4110. https://doi.org/10.3390/su12104110
Cui W, Lu B. A Bi-Objective Approach to Minimize Makespan and Energy Consumption in Flow Shops with Peak Demand Constraint. Sustainability. 2020; 12(10):4110. https://doi.org/10.3390/su12104110
Chicago/Turabian StyleCui, Weiwei, and Biao Lu. 2020. "A Bi-Objective Approach to Minimize Makespan and Energy Consumption in Flow Shops with Peak Demand Constraint" Sustainability 12, no. 10: 4110. https://doi.org/10.3390/su12104110