# Application of Three Metaheuristic Algorithms to Time-Cost-Quality Trade-Off Project Scheduling Problem for Construction Projects Considering Time Value of Money

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## Abstract

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## 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 $i$: Set of activities where $i=\left\{0,1,\dots ,N,N+1\right\}$ ${m}_{i}$: Set of modes (executive mode for activity $i$ where ${m}_{i}=\left\{1,2,\dots ,{M}_{i}\right\}$) $r$: Set of renewable resources where $r=\left\{1,2,\dots ,R\right\}$ $l$: Set of nonrenewable resources where $l=\left\{1,2,\dots ,L\right\}$ t: The period of time |

Parameters $e{s}_{i}$: Earliest start time of activity $i$ $l{s}_{i}$: Latest start time of activity $i$ $Ef{s}_{ij}$: Precedence relationship between activity $i$ and activity $j$. $f{s}_{ij}$: Minimum delay for activities $i$ and $j$ with precedence relationship finish to start $D{u}_{i{m}_{i}}$: Duration of activity $i$ in execution mode ${m}_{i}$ $D{r}_{i{m}_{i}r}$: Amount of renewable resource $r$ for executing activity $i$ in mode ${m}_{i}$ in each time. ${l}_{i{m}_{i}l}$: Amount of nonrenewable resource $l$ for executing activity $i$ in mode ${m}_{i}$. ${N}_{l}$: The availability of the nonrenewable resource $l$. ${A}_{r}$: The availability of the nonrenewable resource $r$. ${q}_{i{m}_{i}}$: The quality level of activity $i$ in mode ${m}_{i}$. ${C}_{r}$: The cost of one unit of renewable resource $r$ in each time. ${C}_{l}$: The cost of one unit of non-renewable resource $l$. $Qmi{n}_{i}$: Minimum level quality of activity $i$. $H$: Horizon time of project ${w}_{i{m}_{i}}$: Worth of activity $i$ in mode ${m}_{i}$ over the duration of activity per time. It is the actual cost of activity $i$ including renewable and non-renewable resources. $S$: Worth of completed activity representing the holding cost, % per time. $S{S}_{t}$: Single-payment present worth factor $\alpha $: return rate $S{S}_{t}=\frac{1}{{\left(1+\alpha \right)}^{t}}$ |

Variables $Ltn{s}_{i}$: Lateness of activity $i$ ${W}_{t}$: Non-renewable and renewable resource costs in each time of executing activities $W{P}_{t}$: Project worth obtained by completing each activity |

Binary variables ${x}_{i{m}_{i}t}:$ 1 if activity $i$ starts in mode ${m}_{i}$ at time t, 0 otherwise ${u}_{i{m}_{i}t}:$ 1 if activity $i$ execute in mode ${m}_{i}$ at time t, 0 otherwise ${F}_{i{m}_{i}t}:$ 1 if activity $i$ finishes in mode ${m}_{i}$ 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

_{1}and r

_{2}randomly change between 0 and 1 [51].

#### 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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**Figure 12.**Pareto sets of solutions for the three-criteria and two-criteria multi-objective optimization problem.

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 |
---|---|

$e{s}_{i}$ | PSPLIB |

$l{s}_{i}$ | PSPLIB |

$H$ | PSPLIB |

$D{D}_{i}$ | PSPLIB |

$D{u}_{im}$ | PSPLIB |

$D{r}_{imr}$ | PSPLIB |

$Ef{s}_{ij}$ | PSPLIB |

${A}_{r}$ | PSPLIB |

${l}_{iml}$ | $\sim \mathrm{U}$ [0, 8] |

${q}_{im}$ | $\sim \mathrm{U}$ [0.7, 0.92], Ref. [64] |

${C}_{r}$ | $\sim \mathrm{U}$ [100, 150] |

${C}_{l}$ | $\sim \mathrm{U}$ [10, 15] |

$S$ | 0.1, Ref. [35] |

${\mathit{q}}_{\mathit{i}\mathit{m}}$ | 1 | 2 | 3 | $\mathit{D}{\mathit{u}}_{\mathit{i}\mathit{m}}$ | 1 | 2 | 3 | ${\mathit{V}}_{\mathit{i}}$ | 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 |

${\mathrm{x}}_{\mathrm{1.3.2}}=1\text{\hspace{1em}\hspace{1em}}{\mathrm{du}}_{1.1}=4$ | |||

${\mathrm{u}}_{\mathrm{1.3.2}}=1$ | ${\mathrm{u}}_{\mathrm{1.3.3}}=1$ | ${\mathrm{u}}_{\mathrm{1.3.4}}=1$ | ${\mathrm{u}}_{\mathrm{1.3.5}}=1$ |

${\mathrm{x}}_{\mathrm{2.2.1}}=1\text{\hspace{1em}\hspace{1em}}{\mathrm{du}}_{2.2}=3$ | |||

${\mathrm{u}}_{\mathrm{2.2.1}}=1$ | ${\mathrm{u}}_{\mathrm{2.2.2}}=1$ | ${\mathrm{u}}_{\mathrm{2.2.3}}=1$ | |

${\mathrm{x}}_{\mathrm{3.2.7}}=1\text{\hspace{1em}\hspace{1em}}{\mathrm{du}}_{3.2}=4$ | |||

${\mathrm{u}}_{\mathrm{3.2.7}}=1$ | ${\mathrm{u}}_{\mathrm{3.2.8}}=1$ | ${\mathrm{u}}_{\mathrm{3.2.9}}=1$ | ${\mathrm{u}}_{\mathrm{3.2.10}}=1$ |

${\mathrm{x}}_{\mathrm{4.3.5}}=1\text{\hspace{1em}\hspace{1em}}{\mathrm{du}}_{4.3}=4$ | |||

${\mathrm{u}}_{\mathrm{4.3.5}}=1$ | ${\mathrm{u}}_{\mathrm{4.3.6}}=1$ | ${\mathrm{u}}_{\mathrm{4.3.7}}=1$ | ${\mathrm{u}}_{\mathrm{4.3.8}}=1$ |

${\mathrm{x}}_{\mathrm{5.1.10}}=1\text{\hspace{1em}\hspace{1em}}{\mathrm{du}}_{5.1}=2$ | |||

${\mathrm{u}}_{\mathrm{5.1.10}}=1$ | ${\mathrm{u}}_{\mathrm{5.1.11}}=1$ | ||

${\mathrm{x}}_{\mathrm{6.1.13}}=1\text{\hspace{1em}\hspace{1em}}{\mathrm{du}}_{6.1}=2$ | |||

${\mathrm{u}}_{\mathrm{6.4.13}}=1$ | ${\mathrm{u}}_{\mathrm{6.4.14}}=1$ | ||

${\mathrm{x}}_{\mathrm{7.2.15}}=1\text{\hspace{1em}\hspace{1em}}{\mathrm{du}}_{7.2}=2$ | |||

${\mathrm{u}}_{\mathrm{7.2.15}}=1$ | ${\mathrm{u}}_{\mathrm{7.2.16}}=1$ |

Time | ${\mathit{W}}_{\mathit{t}}\left(\mathbf{Dollar}\right)$ | Time | ${\mathit{W}}_{\mathit{t}}\left(\mathbf{Dollar}\right)$ |
---|---|---|---|

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 |
---|---|

$H$ | ${\mathrm{es}}_{\mathrm{N}}$ + ${\mathrm{Du}}_{\mathrm{Nm}}$ |

$D{D}_{i}$ | ${\mathrm{es}}_{\mathrm{i}}$ + min{${\mathrm{Du}}_{\mathrm{im}}$} |

$D{u}_{im}$ | [1, 6] |

$D{r}_{imr}$ | [0, 8] |

${l}_{iml}$ | [0, 8] |

$f{s}_{ij}$ | [0, 2] |

${A}_{r}$ | [9, 11] |

${q}_{im}$ | [0.55, 0.95] |

${C}_{r}$ | [100, 150] |

${C}_{l}$ | [10, 15] |

$Qmi{n}_{i}$ | 0.6 |

$S$ | 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 |

**Table 11.**The average of six metrics for three proposed algorithms in small, medium, and large sizes.

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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## Share and Cite

**MDPI and ACS Style**

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

**AMA Style**

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 Style**

Kebriyaii, 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