Single-Machine Maintenance Activity Scheduling with Convex Resource Constraints and Learning Effects
Abstract
:1. Introduction
- Single-machine maintenance scheduling with convex resource constraint and learning effect is modeled and studied;
- Four algorithms are provided for the following two objective functions: (1) minimize the sum of scheduling cost (including the weighted sum of earliness, tardiness and common due date (flow allowance), where the weight is the position-dependent weight) and resource consumption cost; and (2) the resource consumption cost has an upper bound, minimizing the dispatch cost.
- It is shown that these problems can be solved in polynomial time, and the effectiveness of the algorithms is presented by numerical study.
2. Problem Description
3. Main Properties
4. Optimal Analysis
4.1. Results of
Algorithm 1: Solution for problem . |
Initialization: Let , , , and . Step 1. For Step 2. If , then obtain the minimum value and the schedule by using (14)–(19); If , then let , , , and ; If , then obtain the minimum value and the schedule by using (14)–(19); If , then let , , , and . Step 3. Choose the minimum value , and obtain the corresponding schedule , , and . |
4.2. Results of
Algorithm 2: Solution for problem . |
Initialization: Let , , , and . Step 1. For Step 2. If , then obtain the minimum value and the schedule by using (26)–(30); If , then let , , , and ; If , then obtain the minimum value and the schedule by using (26)–(30); If , then let , , , and . Step 3. Choose the minimum value , and obtain the corresponding schedule , , and . |
4.3. Results of
Algorithm 3: Solution for problem . |
Initialization: Let , , , and . Step 1. For Step 2. If , then obtain the minimum value and the schedule by using (43)–(46); If , then let , , , and ; If , then obtain the minimum value and the schedule by using (43)–(46); If , then let , , , and . Step 3. Choose the minimum value , and obtain the corresponding schedule , , and . |
4.4. Results of
Algorithm 4: Solution for problem . |
Initialization: Let , , , and . Step 1. For Step 2. If , then obtain the minimum value and the schedule by using (52)–(56); If , then let , , , and ; If , then obtain the minimum value and the schedule by using (52)–(56); If , then let , , , and . Step 3. Choose the minimum value , and obtain the corresponding schedule , , and . |
5. An Example and Numerical Study
5.1. An Example
5.2. Numerical Study
- (1)
- , and ;
- (2)
- , and ;
- (3)
- () is drawn from a discrete uniform distribution in [1, 100] (i.e., );
- (4)
- () ∼ [0.5, 1];
- (5)
- () ∼ [1, 40];
- (6)
- () ∼ [1, 50].
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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References | Scheduling Problem | Time Complexity |
---|---|---|
Wang et al. [29] | ||
Zhu et al. [30] | ||
Wang and Wang [15] | ||
Bai et al. [17] | ||
Ji et al. [22] | ||
Zhao et al. [25] | ||
This article | ||
Notation | Meaning |
---|---|
n | the number of jobs |
the j-th job | |
the job scheduled in the j-th position | |
(resp. ) | the normal processing time of job (resp. ) |
() | the actual processing time of job (resp. ) |
the modifying rate of job | |
the actual processing time of job in position r | |
the learning factor | |
t | the maintenance duration |
l | the location of the maintenance activity |
(resp. ) | the resource allocated to job (resp. ) |
(resp. ) | the completion time of job (resp. ) |
(resp. ) | the start time of job (resp. ) |
(resp. ) | the earliness of job (resp. ) |
(resp. ) | the tardiness of job (resp. ) |
(resp. ) | the cost when allocating unit resource to job (resp. ) |
the position-dependent (but job-independent) weight (cost) of the j-th job | |
( ) | the given constant |
9 | 11 | 4 | 13 | 22 | 6 | |
7 | 2 | 8 | 5 | 4 | 10 | |
5 | 4 | 2 | 12 | 6 | 9 | |
22 | 16 | 15 | 20 | 7 | 9 |
j∖r | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|
1 | 67.89213 | |||||
2 | 18.9772 | |||||
3 | ||||||
4 | ||||||
5 | ||||||
6 | 36 |
l | u | ([l]) | ||
---|---|---|---|---|
1 | 6.23569 | |||
2 | ||||
3 | ||||
4 | ||||
5 | ||||
6 |
j∖r | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|
1 | ||||||
2 | ||||||
3 | ||||||
4 | ||||||
5 | ||||||
6 |
l | u | |||
---|---|---|---|---|
1 | ||||
2 | ||||
3 | ||||
4 | ||||
5 | ||||
6 |
Algorithm 1 | Algorithm 2 | Algorithm 3 | Algorithm 4 | |||||
---|---|---|---|---|---|---|---|---|
n | mean | max | mean | max | mean | max | mean | max |
35 | 367.09 | 390.12 | 408.67 | 442.16 | 302.86 | 352.29 | 322.18 | 349.82 |
45 | 896.46 | 925.06 | 1250.46 | 1346.82 | 759.28 | 829.51 | 762.24 | 837.24 |
55 | 1864.52 | 1876.59 | 2386.35 | 2587.35 | 1562.37 | 1582.65 | 2118.32 | 2297.14 |
65 | 3624.21 | 3703.29 | 4103.54 | 4346.54 | 2993.52 | 3121.57 | 3314.17 | 3504.02 |
75 | 6735.58 | 6898.34 | 7827.50 | 8786.35 | 5615.58 | 5754.39 | 6849.91 | 6928.42 |
85 | 13,683.45 | 13,968.32 | 15,072.53 | 16,855.50 | 11,394.35 | 11,528.39 | 13,864.47 | 14,941.57 |
95 | 23,877.03 | 24,120.57 | 26,147.35 | 27,845.36 | 21,572.70 | 23,489.86 | 24,478.50 | 26,116.25 |
105 | 34,453.85 | 34,635.58 | 40,527.32 | 42,965.50 | 28,712.15 | 28,892.27 | 34,785.23 | 37,123.85 |
115 | 53,679.54 | 54,008.94 | 61,296.86 | 63,085.45 | 44,731.62 | 45,021.23 | 52,246.21 | 54,004.32 |
125 | 77,623.53 | 77,985.72 | 89,614.36 | 95,238.62 | 64,685.83 | 65,023.21 | 76,372.23 | 81,823.32 |
135 | 114,304.61 | 114,892.56 | 136,835.45 | 139,834.75 | 95,153.30 | 95,802.52 | 117,325.23 | 120,115.84 |
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Wei, Z.-J.; Wang, L.-Y.; Zhang, L.; Wang, J.-B.; Wang, E. Single-Machine Maintenance Activity Scheduling with Convex Resource Constraints and Learning Effects. Mathematics 2023, 11, 3536. https://doi.org/10.3390/math11163536
Wei Z-J, Wang L-Y, Zhang L, Wang J-B, Wang E. Single-Machine Maintenance Activity Scheduling with Convex Resource Constraints and Learning Effects. Mathematics. 2023; 11(16):3536. https://doi.org/10.3390/math11163536
Chicago/Turabian StyleWei, Zong-Jun, Li-Yan Wang, Lei Zhang, Ji-Bo Wang, and Ershen Wang. 2023. "Single-Machine Maintenance Activity Scheduling with Convex Resource Constraints and Learning Effects" Mathematics 11, no. 16: 3536. https://doi.org/10.3390/math11163536