Application of Three Metaheuristic Algorithms to Time-Cost-Quality Trade-Off Project Scheduling Problem for Construction Projects Considering Time Value of Money
Abstract
:1. Introduction
- What is the significance of considering time value money in time-cost-quality (TCQ) project scheduling problems?
- Which methods can be used to optimize the proposed multi-objective mathematical programming model for scheduling construction projects?
- Is it possible to implement the proposed model in a real case construction project?
- Which method can find the best solution among the Pareto frontier?
2. Literature Review
- Considering time value of money in the TCQ trade-off problem for construction projects;
- Applying three metaheuristic algorithms of MOGWO, NSGA-II, and MOPSO for solving the proposed problem;
- Implementing the proposed model on a real case of bridge construction project with 88 activities;
- Using Shannon’s entropy technique and WASPAS method for finding the best Pareto solution.
3. Materials and Methods
3.1. The Assumptions, Indices, Parameters and Decision Variables
- Activity-on-node (AON) is used for project network representation.
- Finish-to-start precedence relationship is considered.
- Both renewable and non-renewable resources are considered.
- The parameters are deterministic.
- The quality of each activity should be higher than a predefined minimum level.
- The available renewable resources are limited in each time period.
Sets : Set of activities where : Set of modes (executive mode for activity where ) : Set of renewable resources where : Set of nonrenewable resources where t: The period of time |
Parameters : Earliest start time of activity : Latest start time of activity : Precedence relationship between activity and activity . : Minimum delay for activities and with precedence relationship finish to start : Duration of activity in execution mode : Amount of renewable resource for executing activity in mode in each time. : Amount of nonrenewable resource for executing activity in mode . : The availability of the nonrenewable resource . : The availability of the nonrenewable resource . : The quality level of activity in mode . : The cost of one unit of renewable resource in each time. : The cost of one unit of non-renewable resource . : Minimum level quality of activity . : Horizon time of project : Worth of activity in mode over the duration of activity per time. It is the actual cost of activity including renewable and non-renewable resources. : Worth of completed activity representing the holding cost, % per time. : Single-payment present worth factor : return rate |
Variables : Lateness of activity : Non-renewable and renewable resource costs in each time of executing activities : Project worth obtained by completing each activity |
Binary variables 1 if activity starts in mode at time t, 0 otherwise 1 if activity execute in mode at time t, 0 otherwise 1 if activity finishes in mode at time t, 0 otherwise |
3.2. Mathematical Programming Model
3.3. Solution Methodology
3.3.1. Augmented ε-Constraint (AEC)
3.3.2. Solution Representation
3.3.3. The NSGA-II Metaheuristic Algorithm
Crossover Operator
Mutation Operator
3.3.4. The MOPSO Metaheuristic Algorithm
3.3.5. The MOGWO Metaheuristic Algorithm
3.4. Searching, Siege and Hunting Prey
3.5. The Evaluation Metrics
3.5.1. CPU Computational Time
3.5.2. Mean Ideal Distance (MID)
3.5.3. Spread of Non-Dominance Solutions (SNS)
3.5.4. The Rate of Achievement to Two Objectives Simultaneously (RAS)
3.5.5. Spacing (S)
3.5.6. Diversification Matrix (DM)
3.6. Parameter Tuning
Determining the Normalized Weight Vector
3.7. The WASPAS Method
4. Results and Discussion
4.1. The Validation of the Proposed Model
4.2. Performance Analysis of the Algorithms
4.3. Case Study
4.3.1. 3-Dimensional Objective Space
4.3.2. Finding the Best Solution through MCDM Methods
4.4. Discussion
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Authors | Year | Multi-Objective | MCDM | Time Value of Money | Holding Cost | Case Study |
---|---|---|---|---|---|---|
Mahmoudi and Javed [17] | 2020 | ✓ | ||||
Sharma and Trivedi [18] | 2020 | ✓ | ✓ | |||
Moghadam et al. [19] | 2020 | ✓ | ✓ | |||
Jeunet and Bou Orm [20] | 2020 | ✓ | ✓ | |||
Keshavarz and Shoul [21] | 2020 | ✓ | ||||
Banihashemi and Khalilzadeh [22] | 2021 | ✓ | ✓ | |||
Mao et al. [23] | 2020 | ✓ | ✓ | |||
Panwar and Neeraj Jha [24] | 2021 | ✓ | ||||
Hosseinzadeh et al. [25] | 2021 | ✓ | ✓ | |||
Luong et al. [26] | 2021 | ✓ | ✓ | |||
Hamta et al. [27] | 2021 | ✓ | ✓ | |||
Nguyen et al. [28] | 2021 | ✓ | ✓ | |||
Huynh et al. [29] | 2021 | ✓ | ✓ | |||
Banihashemi et al. [30] | 2021 | ✓ | ✓ | |||
Liu et al. [31] | 2021 | ✓ | ✓ | |||
This paper | 2021 | ✓ | ✓ | ✓ | ✓ | ✓ |
Parameter Levels of the NSGA-II Algorithm | ||||
---|---|---|---|---|
Factor | Level | Levels | ||
Low | Medium | High | ||
Crossover Percentage (PC) | 3 | 60% | 70% | 80% |
Mutation Percentage (PM) | 3 | 25% | 35% | 40% |
Mutation Rate (MU) | 3 | 2% | 3% | 4% |
Parameter Levels of the MOPSO Algorithm | ||||
Factor | Level | Levels | ||
Low | Medium | High | ||
Inertia weight (A) | 3 | 0.5 | 1 | 1.5 |
Inertia weight damping rate (B) | 3 | 0.85 | 0.9 | 0.95 |
Personal experience weight (C) | 3 | 0.02 | 0.03 | 0.04 |
Leader weight (D) | 3 | 1 | 2 | 3 |
Number of grids (E) | 3 | 2 | 3 | 4 |
Inflation rate for grids (F) | 3 | 9 | 10 | 11 |
Leader selection pressure (G) | 3 | 1 | 0.15 | 0.2 |
Deletion selection pressure (H) | 3 | 2 | 4 | 6 |
Mutation rate (J) | 3 | 0.02 | 0.1 | 0.8 |
Parameter Levels of the MOGWO Algorithm | ||||
Factor | Level | Levels | ||
Low | Medium | High | ||
alpha | 3 | 0.1 | 0.15 | 0.2 |
beta | 3 | 3 | 4 | 5 |
nGrid | 3 | 9 | 10 | 11 |
A | 3 | 1 | 0 | 2 |
Parameters | Random Distribution Functions |
---|---|
PSPLIB | |
PSPLIB | |
PSPLIB | |
PSPLIB | |
PSPLIB | |
PSPLIB | |
PSPLIB | |
PSPLIB | |
[0, 8] | |
[0.7, 0.92], Ref. [64] | |
[100, 150] | |
[10, 15] | |
0.1, Ref. [35] |
1 | 2 | 3 | 1 | 2 | 3 | H | |||
---|---|---|---|---|---|---|---|---|---|
1 | 0.91 | 0.87 | 0.73 | 1 | 2 | 3 | 4 | 0.1 | 19 |
2 | 0.89 | 0.79 | 0.68 | 2 | 3 | 3 | 4 | 0.2 | |
3 | 0.92 | 0.80 | 0.71 | 3 | 3 | 4 | 4 | 0.2 | |
4 | 0.89 | 0.81 | 0.73 | 4 | 3 | 4 | 4 | 0.1 | |
5 | 0.97 | 0.82 | 0.77 | 5 | 2 | 3 | 4 | 0.1 | |
6 | 0.91 | 0.81 | 0.69 | 6 | 2 | 3 | 3 | 0.2 | |
7 | 0.92 | 0.83 | 0.72 | 7 | 2 | 2 | 3 | 0.1 |
Mode 1 | Mode 2 | Mode 3 | |||||||
---|---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 | |
1 | 4 | 3 | 3 | 3 | 2 | 2 | 2 | 1 | 1 |
2 | 5 | 4 | 4 | 4 | 3 | 3 | 4 | 2 | 2 |
3 | 3 | 3 | 3 | 2 | 2 | 2 | 2 | 1 | 2 |
4 | 3 | 3 | 3 | 2 | 2 | 2 | 2 | 1 | 1 |
5 | 4 | 4 | 4 | 3 | 3 | 3 | 2 | 2 | 2 |
6 | 5 | 5 | 5 | 4 | 4 | 4 | 3 | 3 | 3 |
7 | 4 | 4 | 4 | 4 | 4 | 3 | 3 | 3 | 3 |
Availability of resources | 9 | 7 | 7 | 9 | 7 | 7 | 9 | 7 | 7 |
Cost of using one unit of renewable resource r | 1 | 2 | 3 |
400 | 500 | 200 | |
Holding cost factor (S) | 0.1 |
Time | Time | ||
---|---|---|---|
1 | 14,566 | 9 | 0 |
2 | 14,566 | 10 | 7900 |
3 | 6866 | 11 | 7900 |
4 | 3637 | 12 | 0 |
5 | 7637 | 13 | 11,750 |
6 | 7637 | 14 | 11,750 |
7 | 7637 | 15 | 7075 |
8 | 4000 | 16 | 7075 |
Parameters | Random Distribution Functions |
---|---|
+ | |
+ min{} | |
[1, 6] | |
[0, 8] | |
[0, 8] | |
[0, 2] | |
[9, 11] | |
[0.55, 0.95] | |
[100, 150] | |
[10, 15] | |
0.6 | |
0.1 |
Small Size | Medium Size | Large Size | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Example Number | Number of Activities | Number of Renewable Resources | Number of Modes | Example Number | Number of Activities | Number of Renewable Resources | Number of Modes | Example Number | Number of Activities | Number of Renewable Resources | Number of Modes |
1 | 8 | 2 | 2 | 11 | 30 | 3 | 3 | 21 | 55 | 4 | 4 |
2 | 10 | 2 | 2 | 12 | 32 | 3 | 3 | 22 | 60 | 4 | 4 |
3 | 12 | 2 | 2 | 13 | 34 | 3 | 3 | 23 | 65 | 4 | 4 |
4 | 14 | 2 | 2 | 14 | 36 | 3 | 3 | 24 | 70 | 4 | 4 |
5 | 16 | 2 | 2 | 15 | 38 | 3 | 3 | 25 | 75 | 4 | 4 |
6 | 18 | 2 | 2 | 16 | 40 | 3 | 3 | 26 | 80 | 4 | 4 |
7 | 20 | 2 | 2 | 17 | 42 | 3 | 3 | 27 | 85 | 4 | 4 |
8 | 22 | 2 | 2 | 18 | 44 | 3 | 3 | 28 | 90 | 4 | 4 |
9 | 24 | 2 | 2 | 19 | 46 | 3 | 3 | 29 | 95 | 4 | 4 |
10 | 26 | 2 | 2 | 20 | 48 | 3 | 3 | 30 | 100 | 4 | 4 |
Example Size | Solver | Time | Diversity | Spacing | MID | SNS | RAS |
---|---|---|---|---|---|---|---|
Small size | NSGA-II | 27 | 9127 | 289 | 50266 | 2234 | 0.1194 |
MOPSO | 18 | 10432 | 315 | 49621 | 2882 | 0.1198 | |
MOGWO | 26 | 13785 | 329 | 49248 | 4438 | 0.0443 | |
Medium size | NSGA-II | 85 | 190507 | 5330 | 654818 | 46415 | 0.0413 |
MOPSO | 68 | 197849 | 6822 | 608938 | 51107 | 0.0233 | |
MOGWO | 82 | 293482 | 7310 | 625251 | 88450 | 0.015 | |
Large size | NSGA-II | 315 | 322508 | 4331 | 1147337 | 71679 | 0.0566 |
MOPSO | 242 | 311015 | 6220 | 1087265 | 76031 | 0.0521 | |
MOGWO | 291 | 488381 | 8814 | 1081765 | 142794 | 0.0217 |
Criteria | Cost | Time | Quality |
Weight | 0.333517668 | 0.333343 | 0.333139 |
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Kebriyaii, O.; Heidari, A.; Khalilzadeh, M.; Antucheviciene, J.; Pavlovskis, M. Application of Three Metaheuristic Algorithms to Time-Cost-Quality Trade-Off Project Scheduling Problem for Construction Projects Considering Time Value of Money. Symmetry 2021, 13, 2402. https://doi.org/10.3390/sym13122402
Kebriyaii O, Heidari A, Khalilzadeh M, Antucheviciene J, Pavlovskis M. Application of Three Metaheuristic Algorithms to Time-Cost-Quality Trade-Off Project Scheduling Problem for Construction Projects Considering Time Value of Money. Symmetry. 2021; 13(12):2402. https://doi.org/10.3390/sym13122402
Chicago/Turabian StyleKebriyaii, Omid, Ali Heidari, Mohammad Khalilzadeh, Jurgita Antucheviciene, and Miroslavas Pavlovskis. 2021. "Application of Three Metaheuristic Algorithms to Time-Cost-Quality Trade-Off Project Scheduling Problem for Construction Projects Considering Time Value of Money" Symmetry 13, no. 12: 2402. https://doi.org/10.3390/sym13122402
APA StyleKebriyaii, O., Heidari, A., Khalilzadeh, M., Antucheviciene, J., & Pavlovskis, M. (2021). Application of Three Metaheuristic Algorithms to Time-Cost-Quality Trade-Off Project Scheduling Problem for Construction Projects Considering Time Value of Money. Symmetry, 13(12), 2402. https://doi.org/10.3390/sym13122402