Abstract
This work is inspired by very challenging issues arising in space logistics. The problem of scheduling a number of activities, in a given time elapse, optimizing the resource exploitation is discussed. The available resources are not constant, as well as the request, relative to each job. The mathematical aspects are illustrated, providing a time-indexed MILP model. The case of a single resource is analysed first. Extensions, including the multi-resource case and the presence of additional conditions are considered. Possible applications are suggested and an in-depth experimental analysis is reported.
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Acknowledgements
The author is very grateful to the two referees whose suggestions contributed to the improvement of the original version of this chapter, significantly. Thanks are also due to Jane Evans for her very valuable support in revising the whole manuscript.
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Appendix
Appendix
1.1 Test Set F Power Consumption
Cycle type | Power consumption per sub-interval (units) | Max. No. of cycles | Cycle type | Power consumption per sub-interval (units) | Max. No. of cycles |
|---|---|---|---|---|---|
1 | 0.5,1.4,0.9 | 700 | 26 | 1.9 × 23 | 100 |
2 | 1.3,0.4,0.3 | 700 | 27 | 1.7 × 10, 2.3 × 13 | 50 |
3 | 0.7, 4.6, 2.9 | 300 | 28 | 0.3, 1.6 × 9, 6.1 × 3, 0.5 × 12 | 50 |
4 | 0.3, 0.2, 4.4, 6.9, 0.8 | 500 | 29 | 0.3 × 9, 0.7 × 18 | 100 |
5 | 0.7, 1.7, 4.7, 4.2, 1.3 | 200 | 30 | 0.3 × 9, 0.9 × 16, 4.7 × 3, 0.3 × 2 | 70 |
6 | 0.5, 2.5, 3.7, 6.9, 2.2 | 150 | 31 | 1.1 × 30 | 50 |
7 | 1.7, 2.6, 2.8, 2.9, 2.2, 3.3, 3.1 | 150 | 32 | 0.4 × 9, 0.9 × 21 | 100 |
8 | 1.2, 2.2 × 2, 3.2 × 2, 5.2, 5, 4.2 | 100 | 33 | 1.9 × 10, 0.2 × 21 | 70 |
9 | 0.3, 2.7, 4.5, 4.4, 6.3, 6.6, 4.9 | 100 | 34 | 1.9 × 11, 0.9 × 19 | 70 |
10 | 0.9, 0.8, 1.0, 2.1, 6.2, 8.4, 8.6, 0.5, 0.6, 0.5 | 100 | 35 | 0.7 × 9, 0.9 × 21, 1.8 × 2 | 100 |
11 | 1.3 × 3, 2.6 × 8 | 100 | 36 | 2.8 × 3, 0.9 × 27, 0.2 × 3 | 70 |
12 | 2.7, 2.5, 2.7, 3.9, 3.8, 3.7, 3.0, 3.1, 3.3, 3.5, 3.7 | 100 | 37 | 0.7 × 9, 0.9 × 21, 2.8 × 3 | 70 |
13 | 4.1 × 5, 10.7 × 6 | 30 | 38 | 0.9 × 25, 1.9 × 5, 2.8 × 3 | 70 |
14 | 2.7 × 5, 10.2 × 6, 1.9 × 4 | 30 | 39 | 0.8 × 9, 0.7 × 3, 0.9 × 18, 6.9, 0.2 × 3 | 70 |
15 | 4.9 × 5, 13.7 × 6, 2.1 × 4 | 30 | 40 | 0.3 × 31, 4.1 × 4 | 50 |
16 | 1.3 × 5, 15.5 × 3, 2.7, 2.9 × 8 | 50 | 41 | 1.9, 0.8 × 8, 0.7 × 3, 0.9 × 23 | 70 |
17 | 1.9 × 2, 0.4 × 3, 14.7 × 5, 1.1 × 7 | 30 | 42 | 4.5, 0.9, 0.8 × 7, 0.9 × 26 | 70 |
18 | 0.3 × 5, 16.7 × 12 | 30 | 43 | 0.3 × 3, 0.2 × 5, 0.9 × 19, 0.6 × 6, 4.9 × 2 | 70 |
19 | 0.5 × 10, 4.9 × 5, 07 × 4 | 70 | 44 | 1.7 × 9, 1.8 × 3, 1.9 × 7, 1.7 × 11, 1.3 × 5, 2.9 | 50 |
20 | 0.1 × 7, 4.5 × 7, 0.7 × 5 | 70 | 45 | 2.8 × 9, 2.9 × 6, 1.9 × 8, 1.7 × 7, 0.6, 0.5, 0.4, 0.3, 0.2, 0.1 | 30 |
21 | 0.4 × 9, 4.3 × 7, 0.9 × 3 | 70 | 46 | 1.3, 1.7, 1.9, 1.1, 1.3, 1.2, 1.5, 1.7, 1.9, 1.3 × 8, 1.9 × 9, 1.7 × 4, 0.8 × 7 | 50 |
22 | 0.7 × 9, 12.9 × 7, 0.7 × 3 | 30 | 47 | 1.7, 1.8, 1.7, 1.8, 1.7, 1.8, 1.7, 1.8, 1.7, 1.3 × 5, 1.9 × 5, 1.7 × 2, 1.8, 1.9, 1.8, 1.9, 1.8, 1.9, 1.8, 1.9, 1.7, 2.1, 2.3, 2.5, 2.7, 2.9 × 3 | 30 |
23 | 0.3 × 3, 20.5 × 3, 0.3 × 15 | 50 | 48 | 1.5 × 4, 1.7 × 3, 1.2 × 2, 1.4, 1.5, 1.4, 1.5, 1.4, 1.5, 1.4, 1.5 × 2, 1.7 × 5, 1.9 × 7, 14.9, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6 | 30 |
24 | 0.7 × 7, 20.3 × 3, 0.6 × 11 | 50 | 49 | 0.9 × 5, 0.5 × 4, 0.8 × 10, 0.7, 2.7 × 8, 2.3 × 6, 2.9 × 4 | 50 |
25 | 0.3 × 13, 24.7 × 3, 0.9 × 5 | 30 | 50 | 2.5 × 6, 2.9 × 3, 0.9 × 5, 0.5 × 3, 0.7 × 9, 0.9 × 5, 0.4 × 5, 0.9 × 3 | 50 |
1.2 Test set B Power Function
25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 45, 45, 45, 45, 45, 25, 25, 25, 25, 25, 35, 35, 35, 35, 35, 35, 35, 35, 35, 35, 50, 50, 50, 30, 30, 30, 30, 30, 30, 30, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 35, 35, 35, 35, 35, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 35, 35, 35, 35, 35, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50
1.3 Test Set C Power Function
35, 35, 35, 35, 35, 35, 35, 35, 35, 35, 35, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 50, 50, 50, 50, 25, 25, 25, 35, 35, 35, 35, 35, 35, 35, 35, 35, 35, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 30, 30, 30, 30, 30, 50, 50, 50, 50, 50, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 35, 35, 35, 35, 35, 35, 35, 35, 35, 35, 30, 30, 30, 30, 30, 30, 30, 50, 50, 50, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 35, 35, 35, 35, 35, 25, 25, 25, 25, 25, 35, 35, 35, 35, 35, 35, 35, 35, 35, 35, 50, 50, 50, 30, 30, 30, 30, 30, 30, 30, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 35, 35, 35, 35, 25, 25, 25, 25, 35, 35, 35, 35, 35, 35, 45, 45, 45, 45, 45, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 35, 35, 35, 35, 35, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50
1.4 Test Set D Power Function
35, 35, 35, 35, 35, 35, 35, 35, 35, 35, 35, 40, 40, 40, 40, 40, 40, 40, 40, 40, 40, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 50, 50, 50, 50, 45, 45, 45, 40, 40, 40, 35, 35, 35, 35, 35, 35, 35, 40, 40, 40, 45, 45, 45, 50, 50, 50, 50, 45, 45, 45, 40, 40, 40, 45, 45, 45, 40, 40, 40, 40, 40, 40, 35, 35, 35, 30, 30, 27, 27, 27, 27, 33, 33, 33, 40, 40, 40, 35, 35, 35, 35, 35, 35, 35, 35, 35, 35, 37, 39, 41, 43, 45, 47, 49, 50, 50, 50, 49, 47, 45, 43, 41, 39, 37, 35, 33, 31, 35, 35, 35, 35, 35, 29, 27, 25, 25, 25, 30, 30, 30, 35, 35, 37, 37, 37, 35, 35, 39, 43, 50, 45, 40, 35, 30, 40, 43, 47, 50, 50, 50, 50, 50, 47, 45, 43, 41, 39, 37, 35, 31, 29, 27, 25, 30, 33, 35, 35, 37, 37, 37, 39, 41, 43, 45, 47, 49, 50, 50, 50, 50, 50, 47, 45, 43, 41, 39, 37, 35, 33, 31, 30, 29, 28, 27, 26, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 32, 33, 34, 35, 35, 37, 39, 41, 43, 45, 47, 49, 41, 37, 29, 27
1.5 Test Set E Power Function
25, 27, 29, 31, 33, 35, 35, 37, 39, 41, 43, 75, 75, 75, 75, 75, 25, 25, 25, 25, 25, 25, 25, 25, 75, 75, 75, 75, 75, 75, 75, 75, 75, 50, 50, 50, 50, 50, 45, 43, 41, 39, 37, 35, 25, 25, 25, 25, 25, 25, 25, 40, 40, 40, 40, 40, 70, 70, 70, 70, 70, 50, 50, 50, 50, 50, 50, 50, 45, 45, 45, 40, 40, 40, 37, 37, 37, 35, 35, 30, 30, 27, 27, 27, 27, 33, 33, 33, 40, 40, 40, 35, 35, 35, 35, 35, 35, 35, 35, 35, 35, 37, 39, 41, 43, 45, 47, 49, 50, 53, 57, 59, 63, 67, 71, 75, 35, 35, 35, 31, 31, 35, 35, 35, 35, 35, 29, 27, 25, 25, 25, 30, 30, 30, 35, 35, 37, 37, 37, 35, 35, 39, 43, 70, 67, 65, 63, 61, 59, 57, 53, 50, 50, 50, 50, 50, 47, 45, 43, 41, 39, 37, 35, 31, 29, 27, 25, 30, 33, 35, 35, 37, 37, 37, 39, 41, 43, 45, 47, 49, 75, 75, 75, 75, 75, 74, 74, 63, 63, 63, 63, 63, 59, 59, 59, 59, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 32, 33, 34, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 37, 39, 41, 43, 47, 49, 51, 53, 55, 61, 63, 67, 69, 71, 73, 75, 75, 75, 75
1.6 Test set F Power Function
25.2, 25.3, 25.5, 25.9, 25.8, 25.7, 25.4, 25.3, 25.5, 25.6, 25.8, 50.9, 50.1, 50.1, 50.2, 50.5, 50.7, 50.8, 50.9, 50.6, 50.5, 25.7, 25.8, 25.6, 25.5, 25.4, 25.7, 25.0, 25.1, 25.4, 25.7, 50.8, 50.9, 50.2, 50.3, 50.4, 50.6, 50.9, 50.6, 50.7, 50.3, 25.5, 25.2, 25.1, 25.4, 25.5, 25.9, 25.6, 25.7, 25.4, 25.7, 50.5, 50.6, 50.7, 50.6, 50.8, 50.9, 50.0, 50.1, 50.2, 50.4, 25.2, 25.5, 25.6, 25.8, 25.4, 25.6, 25.9, 25.0, 25.9, 25.9, 50.4, 50.5, 50.6, 50.7, 50.8, 50.2, 50.4, 50.9, 50.7, 50.8, 25.4, 25.2, 25.4, 25.5, 25.1, 25.3, 25.2, 25.3, 25.5, 25.7, 25.8, 25.9, 25.0, 25.2, 25.5, 25.7, 25.6, 25.3, 25.3, 25.4, 50.7, 50.9, 50.8, 50.0, 50.4, 50.2, 50.1, 50.3, 50.2, 50.5, 40.3, 40.2, 40.7, 40.8, 40.9, 40.2, 40.3, 40.7, 40.6, 40.9, 45.1, 45.3, 45.2, 45.6, 45.5, 25.8, 25.9, 25.9, 25.6, 25.7, 35.5, 35.6, 35.4, 35.3, 35.3, 35.1, 35.9, 35.1, 35.5, 35.3, 50.5, 50.6, 50.7, 30.6, 30.8, 30.9, 30.0, 30.3, 30.2, 30.5, 25.4, 25.6, 25.5, 25.8, 25.7, 25.1, 25.1, 25.2, 25.3, 25.4, 35.8, 35.7, 35.9, 35.4, 35.4, 25.5, 25.6, 25.9, 25.5, 25.9, 25.0, 25.7, 25.8, 25.9, 25.5, 25.6, 25.1, 25.3, 25.4, 25.8, 50.9, 50.4, 50.7, 50.2, 50.7, 50.9, 50.1, 50.3, 50.5, 50.8, 40.6, 40.9, 40.2, 40.6, 40.8, 40.4, 40.6, 40.1, 40.3, 40.2, 30.6, 30.7, 30.4, 30.5, 30.9, 30.0, 30.7, 30.8, 30.1, 30.3, 45.2, 45.4, 45.3, 45.7, 45.6, 45.9, 45.8, 45.1, 45.6, 45.3, 30.9, 30.8, 30.4, 30.1, 30.2, 30.5, 30.4, 30.5, 30.5, 30.7, 30.7, 30.7, 30.9, 30.0, 35.2, 35.4, 35.3, 35.5, 35.1, 50.6, 50.8, 50.5, 50.9, 50.0, 50.2, 50.4, 50.5, 50.6, 50.1, 50.8
1.7 Test Set G: Resource 1 Function
35, 35, 35, 35, 35, 35, 35, 35, 35, 35, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 35, 35, 35, 35, 35, 35, 35, 35, 35, 35, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 25, 25, 25, 40, 40, 40, 40, 40, 40, 40, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 35, 35, 35, 35, 35, 35, 35, 27, 27, 27, 27, 27, 27, 39, 39, 39, 25, 25, 25, 25, 25, 43, 43, 43, 43, 43, 43, 25, 25, 25, 25, 25, 25, 25, 25, 25, 35, 35, 35, 35, 35, 35, 35, 35, 35, 35, 35, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 29, 29, 29, 29, 29, 29, 29
1.8 Test Set G: Resource 2 Function
20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 23, 23, 23, 23, 23, 23, 23, 23, 29, 29, 29, 29, 29, 29, 29, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 11, 11, 11, 11, 11, 11, 11, 11, 27, 27, 27, 27, 27, 27, 27, 27, 25, 25, 25, 29, 29, 29, 29, 29, 29, 29, 29, 29, 21, 21, 21, 21, 21, 21, 21, 21, 21, 23, 23, 23, 23, 23, 23, 23, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11
1.9 Test Set G: Resource 3 Function
17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 23, 23, 23, 23, 23, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 22, 22, 22, 22, 22, 22, 22, 22, 23, 23, 23, 25, 25, 25, 25, 25, 19, 19, 19, 19, 19, 19, 19, 19, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 21, 21, 21, 21, 21, 21, 21
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Fasano, G. (2016). Resource-Constrained Scheduling with Non-constant Capacity and Non-regular Activities. In: Fasano, G., Pintér, J.D. (eds) Space Engineering. Springer Optimization and Its Applications, vol 114. Springer, Cham. https://doi.org/10.1007/978-3-319-41508-6_4
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