Abstract
Scheduling and managing projects are very important and hot topics in project management science. Multi-mode resource-constrained project scheduling problem (MM-RCPSP) is a RCPSP with special features in which each activity may be executed in more than one mode. Each mode has different options of cost, execution time, resources availabilities, and resources requirements. There are many well known measuring criteria related to the complexity and performance measures for scheduling projects, especially for the single mode projects. In this paper, two selection criteria for dealing with multi-mode resource constrained projects were suggested. According to these selection criteria, some well-known complexity and performance measures were modified for dealing with multi-mode projects. Five single-mode projects and five multi-mode projects are considered as test problems for applying the modified complexity and performance measures based on the suggested selection criteria. The obtained results rendered by the suggested selection criteria for test problems are compared by the existing criteria measures and the results are in the same trend and very promising. Also, we proposed new complexity measures and performance measures for MMRCP. The proposed complexity and performance measures also applied to the test problems. The obtained results rendered by the proposed complexity and performance measures are tested against the results obtained by existing complexity and performance measures. The new results are also promising and having the same trends.
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Appendices
Appendix 1
1.1 Definitions
A | Number of activities in project |
CP | Critical path length |
CPmax | Calculation of project activities considering the modes providing max free float for those have more than one mode |
CPmin | Calculation of project activities considering the modes having minimum times for those having more than one mode |
DOCM | Degree of complexity measure |
Mi | Number of modes of activity I |
Mij | A parameters equal 1 for the single mode activities or M number of available modes for the activities |
AMM | Number of multi-mode activities in the project |
NPC | Number of parallel paths in network > 1 |
Ni | Number of modes for each multi-mode activity |
PCM | Proposed complexity measure |
PPM | Proposed performance measure |
Ri | Is the maximum resource of type i available |
Rt | Required resource for scheduling at time t |
ti | The expected duration for activity i |
Wi | The weight of resource type i |
WN | The weight of resource type N |
Appendix 2
2.1 Test problems
Five Single-Mode test projects will be converted into multi-mode projects based on some assumption these assumptions are
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1.
At least 27% of network activities are of multi-mode activities.
-
2.
27% multi-mode is selected randomly; at least one of 27% of multi-mode is critical activity and selected randomly.
-
3.
The modes of 27% of Multi-mode are ranges in their mode from 2 to 3 modes.
-
4.
The activities of multi-mode have no relations among each other in their modes of execution.
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5.
The times and resources added to the activity as it represents the second or third mode for the activity may be less than or greater than the original times and resources of the activities, this means that there are no relationships between time multiplied by its resources for the different modes for the same activity.
2.2 Five single-mode projects are
Project # 1
Number of activities = 10
Available resources = 2
Resource 1 = 4
Resource 2 = 3
Act | Di | Res1 | Res 2 | Precedence |
|---|---|---|---|---|
1 | 2 | 2 | 2 | |
2 | 3 | 1 | 2 | |
3 | 2 | 3 | 2 | |
4 | 3 | 1 | 2 | 1 |
5 | 1 | 1 | 3 | 2 |
6 | 2 | 4 | 2 | 3 |
7 | 4 | 2 | 2 | 3 |
8 | 2 | 4 | 3 | 4 |
9 | 3 | 2 | 1 | 4 |
10 | 2 | 1 | 2 | 5, 6, 9 |
Project # 2
Number of activities = 11
Available resources = 2
Resource 1 = 4
Resource 2 = 3
Act | Di | Res1 | Res2 | Precedence |
|---|---|---|---|---|
1 | 1 | 2 | 1 | |
2 | 5 | 2 | 3 | 1 |
3 | 2 | 2 | 2 | 1 |
4 | 2 | 2 | 1 | 2 |
5 | 2 | 3 | 4 | 2 |
6 | 2 | 2 | 3 | 3 |
7 | 3 | 3 | 3 | 3 |
8 | 2 | 1 | 2 | 5, 6 |
9 | 3 | 2 | 2 | 8 |
10 | 3 | 2 | 1 | 4, 9 |
11 | 2 | 2 | 1 | 7.10 |
Project # 3
Number of activities = 12
Available resources = 2
Resource 1 = 9
Resource 2 = 3
Act | Di | Res1 | Res2 | Precedence |
|---|---|---|---|---|
1 | 12 | 4 | 2 | |
2 | 8 | 2 | 3 | |
3 | 7 | 7 | 2 | |
4 | 9 | 9 | 3 | 1 |
5 | 0 | 0 | 0 | 1 |
6 | 3 | 2 | 1 | 2 |
7 | 12 | 2 | 3 | 2 |
8 | 5 | 5 | 3 | 2 |
9 | 2 | 8 | 2 | 3; 8 |
10 | 7 | 1 | 2 | 5, 6 |
11 | 6 | 3 | 2 | 7, 10 |
12 | 10 | 2 | 3 | 9, 11 |
Project # 4
Number of activities = 13
Available resources = 2
Resource 1 = 5
Resource 2 = 3
Act | Di | Res1 | Res2 | Precedence |
|---|---|---|---|---|
1 | 2 | 2 | 1 | |
2 | 3 | 3 | 1 | |
3 | 1 | 1 | 1 | |
4 | 4 | 5 | 2 | 2 |
5 | 5 | 5 | 3 | 1.4 |
6 | 2 | 3 | 1 | 1.4 |
7 | 6 | 4 | 3 | 2 |
8 | 4 | 2 | 2 | 2 |
9 | 3 | 4 | 2 | 6.7 |
10 | 3 | 2 | 2 | 6.7 |
11 | 7 | 1 | 2 | 3.8.10 |
12 | 3 | 5 | 2 | 9.11 |
13 | 2 | 4 | 1 | 5.12 |
Project # 5
Number of activities = 11
Available resources = 2
Resource 1 = 9
Resource 2 = 3
Act | Di | Res1 | Res2 | Precedence |
|---|---|---|---|---|
1 | 4 | 4 | 2 | |
2 | 4 | 1 | 1 | |
3 | 2 | 6 | 2 | |
4 | 4 | 7 | 3 | 1 |
5 | 3 | 3 | 3 | 3 |
6 | 3 | 9 | 2 | 2 |
7 | 3 | 5 | 3 | 2 |
8 | 8 | 3 | 2 | 5.7 |
9 | 1 | 6 | 3 | 4 |
10 | 4 | 2 | 2 | 6 |
11 | 3 | 4 | 2 | 9.10 |
2.3 Five multi-mode projects based on the previous assumption
Project # 1
Number of activities = 10
Available resources = 2
Resource 1 = 4
Resource 2 = 3
Multi-mode activities 2, 7, and 9
Act | Di | Res1 | Res2 | Precedence |
|---|---|---|---|---|
1 | 2 | 2 | 2 | |
2 | 3 | 1 | 2 | |
2 | 3 | 1 | ||
3 | 2 | 3 | 2 | |
4 | 3 | 1 | 2 | 1 |
5 | 1 | 1 | 3 | 2 |
6 | 2 | 4 | 2 | 3 |
7 | 4 | 2 | 2 | 3 |
2 | 4 | 3 | ||
6 | 2 | 1 | ||
8 | 2 | 4 | 3 | 4 |
9 | 3 | 2 | 1 | 4 |
1 | 3 | 3 | ||
10 | 2 | 1 | 2 | 5, 6, 9 |
Project # 2
Number of activities = 11
Available resources = 2
Resource 1 = 4
Resource 2 = 3
Multi-mode activities 2, 7, and 10
Act | Di | Res1 | Res2 | Precedence |
|---|---|---|---|---|
1 | 1 | 2 | 1 | |
2 | 5 | 2 | 3 | 1 |
2 | 4 | 2 | ||
3 | 2 | 2 | 2 | 1 |
4 | 2 | 2 | 1 | 2 |
5 | 2 | 3 | 4 | 2 |
6 | 2 | 2 | 3 | 3 |
7 | 3 | 3 | 3 | 3 |
4 | 2 | 4 | ||
8 | 2 | 1 | 2 | 5, 6 |
9 | 3 | 2 | 2 | 8 |
10 | 3 | 2 | 1 | 4, 9 |
5 | 3 | 4 | ||
1 | 1 | 2 | ||
11 | 2 | 2 | 1 | 7.10 |
Project # 3
Number of activities = 12
Available resources = 2
Resource 1 = 9 and Resource 2 = 3
Multi-mode activities 1, 4, 7 and 12
Act | Di | Res1 | Res2 | Precedence |
|---|---|---|---|---|
1 | 12 | 4 | 2 | |
8 | 5 | 3 | ||
14 | 4 | 1 | ||
2 | 8 | 2 | 3 | |
3 | 7 | 7 | 2 | |
4 | 9 | 9 | 3 | 1 |
5 | 6 | 3 | ||
5 | 0 | 0 | 0 | 1 |
6 | 3 | 2 | 1 | 2 |
7 | 12 | 2 | 3 | 2 |
8 | 4 | 3 | ||
8 | 5 | 5 | 3 | 2 |
9 | 2 | 8 | 2 | 3;8 |
10 | 7 | 1 | 2 | 5,6 |
11 | 6 | 3 | 2 | 7,10 |
12 | 10 | 2 | 3 | 9,11 |
4 | 7 | 2 |
Project # 4
Number of activities = 13
Available resources = 2
Resource 1 = 5 and Resource 2 = 3
Multi-mode activities 5, 7, 10 and 11
Act | Di | Res1 | Res2 | Precedence |
|---|---|---|---|---|
1 | 2 | 2 | 1 | |
2 | 3 | 3 | 1 | |
3 | 1 | 1 | 1 | |
4 | 4 | 5 | 2 | 2 |
5 | 5 | 5 | 3 | 1.4 |
2 | 4 | 2 | ||
6 | 2 | 3 | 1 | 1.4 |
7 | 6 | 4 | 3 | 2 |
8 | 2 | 1 | ||
8 | 4 | 2 | 2 | 2 |
9 | 3 | 4 | 2 | 6.7 |
10 | 3 | 2 | 2 | 6.7 |
1 | 4 | 3 | ||
11 | 7 | 1 | 2 | 3.8.10 |
3 | 5 | 2 | ||
9 | 4 | 3 | ||
12 | 3 | 5 | 2 | 9.11 |
13 | 2 | 4 | 1 | 5.12 |
Project # 5
Number of activities = 11
Available resources = 2
Resource 1 = 9 and Resource 2 = 3
Multi-mode activities 2, 7, and 8
Act | Di | Res1 | Res2 | Precedence |
|---|---|---|---|---|
1 | 4 | 4 | 2 | |
2 | 4 | 1 | 1 | |
1 | 6 | 3 | ||
3 | 2 | 6 | 2 | |
4 | 4 | 7 | 3 | 1 |
5 | 3 | 3 | 3 | 3 |
6 | 3 | 9 | 2 | 2 |
7 | 3 | 5 | 3 | 2 |
1 | 7 | 2 | ||
8 | 8 | 3 | 2 | 5.7 |
11 | 2 | 3 | ||
9 | 1 | 6 | 3 | 4 |
10 | 4 | 2 | 2 | 6 |
11 | 3 | 4 | 2 | 9.10 |
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Abdel-Basset, M., Atef, A. & Hussein, AN. Some appraisal criteria for multi-mode scheduling problem. J Ambient Intell Human Comput 10, 1641–1654 (2019). https://doi.org/10.1007/s12652-018-0771-x
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DOI: https://doi.org/10.1007/s12652-018-0771-x