Uncertain Production Scheduling Based on Fuzzy Theory Considering Utility and Production Rate
Abstract
:1. Introduction
2. Problem State
3. Deterministic Scheduling Model Based on USEBCT Modeling Method
3.1. Basic Concepts of USEBCT Model
- (1)
- Event: time measurement, each event has one start time and finish time;
- (2)
- State: raw materials, intermediates and final products;
- (3)
- Task: action of producing or consuming one state;
- (4)
- Unit: machines in which the task can be performed.
3.2. Deterministic Scheduling Model
4. Fuzzy Scheduling Model and Defuzzification
4.1. Uncertainty Description
4.2. Defuzzification of Fuzzy Scheduling Model
5. Case Study
6. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
Nomenclature
Upper layer: | |
Indices | |
task | |
unit | |
event | |
state | |
utility | |
Sets | |
tasks | |
tasks which can be performed in unit j | |
tasks which produce state | |
tasks which consume state | |
tasks which consume utility | |
units | |
units which are suitable for performing task | |
total events postulated in the scheduling horizon | |
states | |
states that are raw materials | |
states that are final products | |
states that are intermediates | |
utilities | |
utilities that are required by task | |
intermediate states with dedicated finite intermediate storage | |
Parameters | |
minimum batch size of task | |
maximum batch size of task | |
initial amount available for state | |
maximum storage capacity for state | |
proportion of state consumed by task | |
proportion of state produced by task | |
coefficient between production rate and consumption of utility consumed by task | |
minimum production rate of task | |
price of state | |
large positive number in big-M constraints | |
short-term scheduling horizon | |
coefficient between consumption of utility and amount of material consumed/processed by task at event | |
maximum availability of utility for task | |
limit on the maximum number of events over which a task is allowed to continue | |
Binary variables | |
binary variable for task active at event | |
Positive variables | |
amount of material processed by task at event | |
production rate of task at event | |
excess amount of state that needs to be stored at event | |
start time of a task at event | |
end time of a task at event | |
consumption of utility by task at event | |
start time at which there is a change in the consumption of utility at event |
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Task 1 (R1) | Task 2 (R2) | Task 3 (R3) | Task 4 (R4) | |||
---|---|---|---|---|---|---|
RI1 | 40 | 80 | 0.25 | 0.3 | - | - |
RI2 | 25 | 50 | 0.25 | 0.3 | - | - |
RII | 40 | 80 | - | - | 0.2 | 0.5 |
F1 | F2 | I1 | I2 | I3 | P1 | P2 | |
---|---|---|---|---|---|---|---|
STsmax | 1000 | 1000 | 200 | 100 | 500 | 1000 | 1000 |
STs0 | 400 | 400 | 0 | 0 | 0 | 0 | 0 |
Prices | 0 | 0 | 0 | 0 | 0 | 30 | 40 |
Task 1 | Task 2 | Task 3 | Task 4 | ||||
---|---|---|---|---|---|---|---|
RI1 | RI2 | RI1 | RI2 | RII | RII | ||
three prominent values | (0.8 1 1.3) | (0.4 0.6 0.7) | (0.55 0.6 0.62) | (0.4 0.5 0.6) | (1.9 2 2.2) | (0.9 1 1.2) | |
actual data | event 1 | 0.92 | 0.65 | 0.60 | 0.45 | 1.95 | 0.95 |
event 2 | 0.90 | 0.60 | 0.61 | 0.50 | 2.10 | 1.00 | |
event 3 | 0.98 | 0.62 | 0.60 | 0.50 | 2.00 | 0.98 | |
event 4 | 1.00 | 0.58 | 0.59 | 0.51 | 1.98 | 1.10 | |
event 5 | 1.15 | 0.50 | 0.58 | 0.55 | 2.00 | 1.00 |
Task 1 | Task 2 | Task 3 | Task 4 | ||||
---|---|---|---|---|---|---|---|
RI1 | RI2 | RI1 | RI2 | RII | RII | ||
three prominent values | (23 24 25) | (20 24 27) | (27 30 35) | (28 30 31) | (22 24 27) | (29 30 33) | |
actual data | event 1 | 23.5 | 25.5 | 32.5 | 30.0 | 25.4 | 29.7 |
event 2 | 24.0 | 23.0 | 31.2 | 30.5 | 23.6 | 31.5 | |
event 3 | 23.8 | 24.0 | 28.5 | 30.0 | 24.0 | 29.6 | |
event 4 | 24.5 | 22.0 | 30.0 | 29.4 | 24.0 | 30.0 | |
event 5 | 24.0 | 24.7 | 29.6 | 29.0 | 23.0 | 30.8 |
Results Based on Actual Data | Deterministic Model | Fuzzy Model | ||||
---|---|---|---|---|---|---|
Objective Function ($) | Completion Time (h) | Objective Function ($) | Completion Time (h) | Objective Function ($) | Completion Time (h) | |
5448 | 11.20 | 5400 | 12.30 | 0.5 | 5426.25 | 11.42 |
0.6 | 5421.00 | 12.10 | ||||
0.7 | 5415.75 | 12.19 | ||||
0.8 | 5410.50 | 11.37 |
Deterministic Model | Fuzzy Model | |||
---|---|---|---|---|
Objective Function ($) | Completion Time (h) | Objective Function ($) | Completion Time (h) | |
0.88% | 9.82% | 0.5 | 0.40% | 1.96% |
0.6 | 0.50% | 8.04% | ||
0.7 | 0.59% | 8.84% | ||
0.8 | 0.69% | 1.52% | ||
average value | 0.545% | 5.09% |
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Wang, Y.; Jin, X.; Xie, L.; Zhang, Y.; Lu, S. Uncertain Production Scheduling Based on Fuzzy Theory Considering Utility and Production Rate. Information 2017, 8, 158. https://doi.org/10.3390/info8040158
Wang Y, Jin X, Xie L, Zhang Y, Lu S. Uncertain Production Scheduling Based on Fuzzy Theory Considering Utility and Production Rate. Information. 2017; 8(4):158. https://doi.org/10.3390/info8040158
Chicago/Turabian StyleWang, Yue, Xin Jin, Lei Xie, Yanhui Zhang, and Shan Lu. 2017. "Uncertain Production Scheduling Based on Fuzzy Theory Considering Utility and Production Rate" Information 8, no. 4: 158. https://doi.org/10.3390/info8040158
APA StyleWang, Y., Jin, X., Xie, L., Zhang, Y., & Lu, S. (2017). Uncertain Production Scheduling Based on Fuzzy Theory Considering Utility and Production Rate. Information, 8(4), 158. https://doi.org/10.3390/info8040158