# Application of a Genetic Algorithm for Proactive Resilient Scheduling in Construction Projects

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Resilient Scheduling Framework

#### 2.2. Optimization in Resilient Scheduling

#### 2.3. Optimization Problem

- Minimize the duration of a project;
- Maximize the final profit (i.e., minimize the overall expenses);
- Maximize the surrogate measure for resilience.

_{i}, as shown in Equation (1).

_{i}are obtained by summing $A{S}_{i}$, $D{P}_{i}$, $A{C}_{i}$ and $R{U}_{i}$, which are expressed as shown in Equations (2)–(6), respectively.

- Equation (2) calculates the relative number of successors per activity, $A{S}_{i}$. The number of both direct and indirect successors ${N}_{succ}$ for the activity $i$ is divided by the number of all activities in a project $n$ (both dummy start and dummy end are included);
- Equation (3) measures the relative duration of the activity, $D{P}_{i}$;
- Equation (4) states the relative cost of the activity, $A{C}_{i}$;
- Equation (5) determines relative resource usage as required by the activity, $R{U}_{i}$.
- Equation (6) calculates the weight of the activity, ${w}_{i}$.

- Each activity, including the dummy start and end, can be started only once;
- Precedence relations between activities must be respected;
- At the end of each month, the cumulative cash gap (before receiving the payment from the investor) must not exceed the permitted credit limit;
- Resource constraints must be respected at all times;
- No pre-emption of activities is allowed.

#### 2.4. Research Design

#### 2.5. Customized Genetic Algorithm

#### 2.5.1. Initialization for the Genetic Algorithm

#### 2.5.2. Evaluation

#### 2.5.3. Survival and Selection

#### 2.5.4. Genetic Operators

## 3. Application of Customized Algorithm on a Test Case

#### 3.1. Project Description

#### 3.2. Steps for Resilience Analysis

#### 3.3. Realized State Simulation

#### 3.4. Resilience Analysis on Project Data

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

Symbol | Description |
---|---|

T | Length of the planning horizon (t = 1, 2, …, m) |

${x}_{\mathit{it}}$ | Binary decision variable which equals 1 if activity i starts at the time t, 0 otherwise |

A | Set of project activities (i = 1, 2, …, n), including dummy start 0 and dummy end n + 1 |

E | Set of precedence relations |

R | Set of project resources (r = 1, 2, …, k) |

${w}_{i}$ | Weight of activity i |

${d}_{i}$ | Expected duration for activity i |

${c}_{i}$ | Deterministic cost of activity i |

${u}_{ir}$ | Consumption of resource r as required by activity i |

${a}_{r}$ | Availability of resource r during project time T |

q | Start of the time period for which the resource constraint is checked |

${\mathit{FF}}_{i}$ | Resource-technology free float for activity i |

$P$ | Final profit at the end of a project |

$\mathit{EOM}$ | End of the month considering project timeline (time step used when calculating Cash Flow), (eom = 1, 2, …, l) |

${\mathit{CG}}_{\mathit{eom}}$ | Cumulative cash flow value at the end of the month eom |

$P{T}_{l}$ | Payment at the end of the project timeline |

${I}_{\mathit{eom}}$ | Total interest charges at the end of the month eom |

ir | Interest rate per period |

$W$ | Credit limit for the project |

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**Figure 2.**General workflow considering the application of GA on the resilient scheduling problem for construction projects.

**Figure 4.**Another common approach to building a baseline schedule is to utilize the parallel schedule generation scheme (PSGS), which iterates over the decision points in time, represented by completion times of previously scheduled activities [28]. However, in this research, serial schedule generation scheme (SSGS) is selected over PSGS because the latter method searches in a smaller solution space than the former, so the optimal schedule may not be found when considering a regular performance measure, such as project minimization [29].

**Figure 7.**Results of the resilience analysis: comparing the most resilient solution with an original baseline.

**Figure 8.**Cumulative resilience analysis results: (

**a**) SM value distribution in the obtained Pareto front; (

**b**) absolute start time deviations for all Pareto solutions; (

**c**) makespan and profit deviation distributed by SM value when all scenarios are included.

Reference | Optimization Model | Solution Procedure | Surrogate Measure | Cash Flow Calculation |
---|---|---|---|---|

[5] | N/A | Heuristic algorithm | Mean-variance model | No |

[4] | Multi-objective RCPSP | NSGA-II | Time buffers and activity floats | No |

[6] | Bi-objective RCPSP | N/A | Resource-technology free float | No |

[7] | Multi-objective RCPSP | Hierarchical approach with exact algorithm | Weighted sum of resource-technology free floats | Yes |

Current | Multi-objective RCPSP | Customized NSGA-II | Weighted sum of resource-technology free floats | Yes |

Symbol | Data | Value | Units |
---|---|---|---|

OP | Overhead percentage | 0.15 | % of Total Direct Costs (TDC) |

MP | Mobilization percentage | 0.05 | % of (TDC + Overheads) |

TP | Tax percentage | 0.02 | % of (TDC + Overheads + Mobilization) |

MP | Markup percentage | 0.20 | % of (TDC + Overheads + Mobilization + Tax) |

BP | Bond percentage | 0.01 | % of (TDC + Overheads + Mobilization + Tax + Markup) |

ADV | Advance | 0.10 | % of (TDC + Overheads + Mobilization + Tax + Markup + Bond) |

D | Penalty (per day of prolongation) | 0.0001 | % of (TDC + Overheads + Mobilization + Tax + Markup + Bond) |

RET | Retainage | 0.05 | % of monthly payment from investor to contractor |

ir | Interest | 0.008 | % of cumulative interest charges per month |

h | Surplus | 0.005 | % of cumulative monthly cash flow after payment |

k | Interest on unused credit | 0.002 | % of unused portion of credit |

W | Credit limit | 700 | thousands of financial units (EUR) |

Scenario 2 | Scenario 3 | Scenario 4 | |
---|---|---|---|

Modified activities ID | 6, 25, 26 | 2, 28, 35 | 3, 18, 36 |

Hypothetical durations | 35, 50, 22 | 95, 15, 25 | 40, 40, 40 |

Hypothetical costs | 44.258, 420.212,170.815 | 75, 157.54, 102.06 | 210, 135.55, 56 |

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Milat, M.; Knezić, S.; Sedlar, J. Application of a Genetic Algorithm for Proactive Resilient Scheduling in Construction Projects. *Designs* **2022**, *6*, 16.
https://doi.org/10.3390/designs6010016

**AMA Style**

Milat M, Knezić S, Sedlar J. Application of a Genetic Algorithm for Proactive Resilient Scheduling in Construction Projects. *Designs*. 2022; 6(1):16.
https://doi.org/10.3390/designs6010016

**Chicago/Turabian Style**

Milat, Martina, Snježana Knezić, and Jelena Sedlar. 2022. "Application of a Genetic Algorithm for Proactive Resilient Scheduling in Construction Projects" *Designs* 6, no. 1: 16.
https://doi.org/10.3390/designs6010016