# Resilient Scheduling as a Response to Uncertainty in Construction Projects

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Resilience Approach to Manage Uncertainty in Construction Projects

#### 2.1. Understanding Uncertainty

#### 2.2. Taxonomy

- Member facet tends to describe all aspects of human attributes together with their mutual relationships which could lead to project perturbations. Members in a construction project could be individuals as well as groups of people, i.e., organizations.
- The project is described as features of a construction project which represent substantial means of how a project affects uncertainty existence.
- Impact means all the direct effects of uncertainty sources on traditionally determined project objectives (schedule-budget-quality goals), with additional sustainability aspects [59].
- Causes are states, conditions, and events in which sources of uncertainty are triggered.
- The life cycle presents stages of a construction project in which uncertainties may arise. This facet is the time dimension of the universe of discourse.

#### 2.3. Resilience as a Response to Uncertainty in Construction Projects

#### 2.4. Optimization Model for Resilient Scheduling Problem

#### 2.4.1. Surrogate Measure and Activity Weights

_{i}), it is more important to provide bigger FF

_{i}, so the baseline schedule is as resilient as possible. Activity weight is calculated based on four indices:

- AS
_{i}—Equation (1) states the relative number of successors per activity. The number of 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 included); - DP
_{i}— Equation (2) calculates relative duration of the activity, i; - AC
_{i}— Equation (3) determines relative cost of the activity, i; - RU
_{i}— Equation (4) counts relative resource usage as required by the activity, i.

#### 2.4.2. Optimization Problem

## 3. Application of Resilient Scheduling on a Test Problem Instance

#### 3.1. Solving the Multi-Objective Optimization Problem

- Unfavorable weather conditions might interfere with the execution of Activity 2, which is performed on the construction site;
- Architectural innovation involves an element of uncertainty while implementing the Activity 4;
- Change orders in Activity 6 might cause technical difficulty for accomplishing the execution in the estimated time frame.

#### 3.2. Validating Resilient Scheduling Process

_{a}) at a simulated duration (T

_{a}) consists mainly of material or equipment and labour. For example, if the cost of activity consists mainly of material, it will have a lower ratio value, R, while higher dependence on equipment and labour will be represented with a higher R-value [70]. In the simulation analysis, R is set to 0.6 for all activities. To calculate activities’ cost C

_{a}at a simulated duration T

_{a}, Equation (14) is employed from the Reference [70]:

_{m}represents the original cost of activity at the deterministic duration, T

_{m}.

_{i}) probability of reaching baseline due date, (Eq

_{ii}) probability of reaching baseline due date multiplied with coefficient 1.1, (Eq

_{iii}) average tardiness amount for the proposed baseline, (Eq

_{iv}) average tardiness amount for activities’ start times, (Eq

_{v}) the percentage of simulations where credit limit was broken, and (Eq

_{vi}) probability of reaching or exceeding the baseline profit. The last equilibrium dimension (Eq

_{vi}) is calculated by considering only those simulations for which credit limit, W, was not broken. From the simulation results, as shown in Table 8, it is evident that the first case (single optimal solution from the previous section) generates better results in all equilibrium calculations than the second case, for which profit maximization was considered as a more important objective than maximizing SM value. With an increase in SM value, there is a constant rise in the probability of reaching both strict (Eq

_{i}) and relaxed (Eq

_{ii}) baseline duration, as well as a constant decrease of average project tardiness (Eq

_{iii}), and a decrease of average tardiness for activities’ start times (Eq

_{iv}). Moreover, the percentage of simulations for which the credit limit is broken (Eq

_{v}) is slightly lower in the first case, where the SM value was higher. Finally, the probability of reaching baseline profit or attaining even higher profit values (Eq

_{vi}) is significantly better for the test case with a higher SM value.

_{i}– Eq

_{vi}) when comparing the baseline schedule with higher SM value to another solution in which profit maximization dominated SM improvement. The optimal trade-off between project duration, stability, and profit is obtained for a baseline schedule calculated as a result of the resilient scheduling approach presented in this study. According to validation results, it can be stated that the contractor will benefit from the enhanced resilience of the baseline schedule, since its schedule is improved for the case with the higher SM value. Therefore, this research contributes to the project-management body of knowledge by exploring an approach to develop resilient baseline schedules which will maximize the probability of reaching project goals. With the introduction of the financing aspects in the underlying resilience scheduling problem, the essential issues in construction-management reality are considered. The final output of the MOO problem is given in a form of a baseline schedule which simultaneously minimizes duration of a project, maximizes its resilience, and maximizes final profit from for a contractor. This way, project performances are improved, since the baseline calculations can be accepted with improved confidence levels.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Derbe, G.; Li, Y.; Wu, D.; Zhao, Q. Scientometric review of construction project schedule studies: Trends, gaps and potential research areas. J. Civ. Eng. Manag.
**2020**, 26, 343–363. [Google Scholar] [CrossRef][Green Version] - Faghihi, V.; Nejat, A.; Reinschmidt, K.F.; Kang, J.H. Automation in construction scheduling: A review of the literature. Int. J. Adv. Manuf. Technol.
**2015**, 81, 1845–1856. [Google Scholar] [CrossRef] - Cajzek, R.; Klanšek, U. Cost optimization of project schedules under constrained resources and alternative production processes by mixed-integer nonlinear programming. Eng. Constr. Archit. Manag.
**2019**, 26, 2474–2508. [Google Scholar] [CrossRef] - García-Nieves, J.; Ponz-Tienda, J.; Ospina-Alvarado, A.; Bonilla-Palacios, M. Multipurpose linear programming optimization model for repetitive activities scheduling in construction projects. Autom. Constr.
**2019**, 105, 102799. [Google Scholar] [CrossRef] - Zou, X.; Fang, S.; Huang, Y.; Zhang, L. Mixed-Integer Linear Programming Approach for Scheduling Repetitive Projects with Time-Cost Trade-Off Consideration. J. Comput. Civ. Eng.
**2017**, 31, 06016003. [Google Scholar] [CrossRef] - Klanšek, U. Mixed-Integer Nonlinear Programming Model for Nonlinear Discrete Optimization of Project Schedules under Restricted Costs. J. Constr. Eng. Manag.
**2016**, 142, 04015088. [Google Scholar] [CrossRef] - Liu, Z.; Zhang, Y.; Yu, M.; Zhou, X. Heuristic algorithm for ready-mixed concrete plant scheduling with multiple mixers. Autom. Constr.
**2017**, 84, 1–13. [Google Scholar] [CrossRef] - Sonmez, R.; Iranagh, M.; Uysal, F. Critical Sequence Crashing Heuristic for Resource-Constrained Discrete Time–Cost Trade-Off Problem. J. Constr. Eng. Manag.
**2016**, 142, 04015090. [Google Scholar] [CrossRef] - Li, H.; Xu, Z.; Demeulemeester, E. Scheduling Policies for the Stochastic Resource Leveling Problem. J. Constr. Eng. Manag.
**2015**, 141, 04014072. [Google Scholar] [CrossRef] - Tran, D.; Chou, J.; Luong, D. Multi-objective symbiotic organisms optimization for making time-cost tradeoffs in repetitive project scheduling problem. J. Civ. Eng. Manag.
**2019**, 25, 322–339. [Google Scholar] [CrossRef][Green Version] - Agdas, D.; Warne, D.; Osio-Norgaard, J.; Masters, F. Utility of Genetic Algorithms for Solving Large-Scale Construction Time-Cost Trade-Off Problems. J. Comput. Civ. Eng.
**2018**, 32, 04017072. [Google Scholar] [CrossRef][Green Version] - Aminbakhsh, S.; Sonmez, R. Pareto Front Particle Swarm Optimizer for Discrete Time-Cost Trade-Off Problem. J. Comput. Civ. Eng.
**2017**, 31, 04016040. [Google Scholar] [CrossRef] - Sroka, B.; Rosłon, J.; Podolski, M.; Bożejko, W.; Burduk, A.; Wodecki, M. Profit optimization for multi-mode repetitive construction project with cash flows using metaheuristics. Arch. Civ. Mech. Eng.
**2021**, 21, 1–17. [Google Scholar] [CrossRef] - Tao, S.; Wu, C.; Hu, S.; Xu, F. Construction project scheduling under workspace interference. Comput.-Aided Civ. Infrastruct. Eng.
**2020**, 35, 923–946. [Google Scholar] [CrossRef] - Amer, F.; Koh, H.; Golparvar-Fard, M. Automated Methods and Systems for Construction Planning and Scheduling: Critical Review of Three Decades of Research. J. Constr. Eng. Manag.
**2021**, 147, 03121002. [Google Scholar] [CrossRef] - ElMenshawy, M.; Marzouk, M. Automated BIM schedule generation approach for solving time–cost trade-off problems. Eng. Constr. Archit. Manag.
**2021**. Epub ahead of printing. [Google Scholar] - Wang, Z.; Azar, E.R. BIM-based draft schedule generation in reinforced concrete-framed buildings. Constr. Innov.
**2019**, 19, 280–294. [Google Scholar] [CrossRef] - Abbasi, S.; Taghizade, K.; Noorzai, E. BIM-Based Combination of Takt Time and Discrete Event Simulation for Implementing Just in Time in Construction Scheduling under Constraints. J. Constr. Eng. Manag.
**2020**, 146, 04020143. [Google Scholar] [CrossRef] - Dasović, B.; Galić, M.; Klanšek, U. A Survey on Integration of Optimization and Project Management Tools for Sustainable Construction Scheduling. Sustainability
**2020**, 12, 3405. [Google Scholar] [CrossRef][Green Version] - Nusen, P.; Boonyung, W.; Nusen, S.; Panuwatwanich, K.; Champrasert, P.; Kaewmoracharoen, M. Construction Planning and Scheduling of a Renovation Project Using BIM-Based Multi-Objective Genetic Algorithm. Appl. Sci.
**2021**, 11, 4716. [Google Scholar] [CrossRef] - Xie, L.; Chen, Y.; Chang, R. Scheduling Optimization of Prefabricated Construction Projects by Genetic Algorithm. Appl. Sci.
**2021**, 11, 5531. [Google Scholar] [CrossRef] - Wang, H.; Lin, J.; Zhang, J. Work package-based information modeling for resource-constrained scheduling of construction projects. Autom. Constr.
**2020**, 109, 102958. [Google Scholar] [CrossRef] - Sbiti, M.; Beddiar, K.; Beladjine, D.; Perrault, R.; Mazari, B. Toward BIM and LPS Data Integration for Lean Site Project Management: A State-of-the-Art Review and Recommendations. Buildings
**2021**, 11, 196. [Google Scholar] [CrossRef] - Perminova, O.; Gustafsson, M.; Wikström, K. Defining uncertainty in projects—A new perspective. Int. J. Proj. Manag.
**2008**, 26, 73–79. [Google Scholar] [CrossRef] - Project Management Institute. Construction Extension to the PMBOK Guide; Project Management Institute, Inc.: Newtown Square, PA, USA, 2016; pp. 29–30. [Google Scholar]
- Ahmad, Z.; Thaheem, M.; Maqsoom, A. Building information modeling as a risk transformer: An evolutionary insight into the project uncertainty. Autom. Constr.
**2018**, 92, 103–119. [Google Scholar] [CrossRef] - Badran, D.; AlZubaidi, R.; Venkatachalam, S. BIM based risk management for design bid build (DBB) design process in the United Arab Emirates: A conceptual framework. Int. J. Syst. Assur. Eng. Manag.
**2020**, 11, 1339–1361. [Google Scholar] [CrossRef] - Xiong, J.; Chen, Y.; Zhou, Z. Resilience analysis for project scheduling with renewable resource constraint and uncertain activity durations. J. Ind. Manag. Optim.
**2016**, 12, 719–737. [Google Scholar] - Yeganeh, F.T.; Zegordi, S.H. A multi-objective optimization approach to project scheduling with resiliency criteria under uncertain activity duration. Ann. Oper. Res.
**2020**, 285, 161–196. [Google Scholar] [CrossRef] - Milat, M.; Knezic, S.; Sedlar, J. A new surrogate measure for resilient approach to construction scheduling. Proc. Comp. Sci.
**2021**, 181, 468–476. [Google Scholar] [CrossRef] - Zhao, M.; Wang, X.; Yu, J.; Xue, L.; Yang, S. A construction schedule robustness measure based on improved prospect theory and the Copula-CRITIC method. Appl. Sci.
**2020**, 10, 2013. [Google Scholar] [CrossRef][Green Version] - Zhao, M.; Wang, X.; Yu, J.; Bi, L.; Xiao, Y.; Zhang, J. Optimization of Construction Duration and Schedule Robustness Based on Hybrid Grey Wolf Optimizer with Sine Cosine Algorithm. Energies
**2020**, 13, 2015. [Google Scholar] [CrossRef][Green Version] - Chapman, C.; Ward, S. Project Risk Management: Processes, Techniques and Insights, 2nd ed.; John Wiley & Sons Ltd.: Chichester, UK, 2003; pp. 1–15. [Google Scholar]
- Zhang, J.; El-Diraby, T.E. Social semantic approach to support communication in AEC. Int. J. Proj. Manag.
**2012**, 26, 90–104. [Google Scholar] [CrossRef] - Elghamrawy, T.; Boukamp, F.; Kim, H.S. Ontology-based, semi-automatic framework for storing and retrieving on-site construction problem information—An RFID-based case study. In Proceedings of the Construction Research Congress 2009: Building a Sustainable Future, Seattle, WA, USA, 5–7 April 2009. [Google Scholar]
- Tah, J.H.M.; Carr, V. Knowledge-based approach to construction project risk management. J. Comput. Civ. Eng.
**2001**, 15, 170–177. [Google Scholar] [CrossRef] - Tserng, H.P.; Yin, Y.L.S.; Dzeng, R.J.; Wou, B.; Tsai, M.D.; Chen, W.Y. A study of ontology-based risk management framework of construction projects through project life cycle. Autom. Constr.
**2009**, 18, 994–1008. [Google Scholar] [CrossRef] - Ding, L.Y.; Zhong, B.T.; Wu, S.; Luo, H.B. Construction risk knowledge management in BIM using ontology and semantic web technology. Saf. Sci.
**2016**, 87, 202–213. [Google Scholar] [CrossRef][Green Version] - El-Diraby, T.A.; Lima, C.; Feis, B. Domain taxonomy for construction concepts: Toward a formal ontology for construction knowledge. J. Comput. Civ. Eng.
**2005**, 19, 394–406. [Google Scholar] [CrossRef] - Costa, R.; Lima, C.; Sarraipa, J. Facilitating knowledge sharing and reuse in building and construction domain: An ontology-based approach. J. Intell. Manuf.
**2016**, 27, 263–282. [Google Scholar] [CrossRef] - Niu, J.; Issa, R.R.A. Developing taxonomy for the domain ontology of construction contractual semantics: A case study on the AIA A201 document. Adv. Eng. Inform.
**2015**, 29, 472–482. [Google Scholar] [CrossRef] - Fidan, G.; Dikmen, I.; Tanyer, M.A.; Birgonul, T.M. Ontology for relating risk and vulnerability to cost overrun in international projects. J. Comput. Civ. Eng.
**2011**, 25, 302–315. [Google Scholar] [CrossRef] - Zhang, L.; Issa, R.R.A. Ontology-based partial building information model extraction. J. Comput. Civ. Eng.
**2013**, 27, 576–584. [Google Scholar] [CrossRef][Green Version] - Baudrit, C.; Taillandier, F.; Tran, T.T.P.; Breysse, D. Uncertainty processing and risk monitoring in construction projects using hierarchical probabilistic relational models. Comp. Aid. Civ. Inf. Eng.
**2019**, 34, 97–115. [Google Scholar] [CrossRef] - Jiang, S.; Wang, N.; Wu, J. Combining BIM and ontology to facilitate intelligent green building evaluation. J. Comput. Civ. Eng.
**2018**, 32. [Google Scholar] [CrossRef] - Xing, X.; Zhong, B.; Luo, H.; Lic, H.; Wu, H. Ontology for safety risk identification in metro construction. Comp. Ind.
**2019**, 109, 14–30. [Google Scholar] [CrossRef] - Zhong, B.; Li, H.; Luo, H.; Zhou, J.; Fang, W.; Xing, X. Ontology-based semantic modeling of knowledge in construction: Classification and identification of hazards implied in images. J. Constr. Eng. Manag.
**2020**, 146, 04020013. [Google Scholar] [CrossRef] - Zhong, B.; Gan, C.; Luo, H.; Xing, X. Ontology-based framework for building environmental monitoring and compliance checking under BIM environment. Build. Environ.
**2018**, 141, 127–142. [Google Scholar] [CrossRef] - Elazouni, A.M.; Metwally, F.G. Finance-Based Scheduling: Tool to Maximize Project Profit Using Improved Genetic Algorithms. J. Constr. Eng. Manag.
**2005**, 131, 400–412. [Google Scholar] [CrossRef] - Fathi, H.; Afshar, A. GA-based multi-objective optimization of finance-based construction project scheduling. KSCE J. Civ. Eng.
**2010**, 14, 627–638. [Google Scholar] [CrossRef] - El-Abbasy, M.; Elazouni, A.; Zayed, T. Finance-based scheduling multi-objective optimization: Benchmarking of evolutionary algorithms. Autom. Constr.
**2020**, 120, 103392. [Google Scholar] [CrossRef] - Damian, D.; Knauss, A.; Zavala, E.; Marco, J.; Franch, X. SACRE: Supporting contextual requirements’ adaptation in modern self-adaptive systems in the presence of uncertainty at runtime. Exp. Syst. Appl.
**2018**, 98, 166–188. [Google Scholar] - Taillandier, F.; Taillandier, P.; Tepeli, E.; Breysse, D.; Mehdizadeh, R.; Khartabil, F. A multi-agent model to manage risks in construction project (SMACC). Autom. Constr.
**2015**, 58, 1–18. [Google Scholar] [CrossRef] - Gündüz, M.; Nielsen, Y.; Özdemir, M. Quantification of Delay Factors Using the Relative Importance Index Method for Construction Projects in Turkey. J. Manag. Eng.
**2012**, 29, 133–139. [Google Scholar] [CrossRef] - Ustinovičius, L. Uncertainty analysis in construction project’s appraisal phase. In Proceedings of the 9th International Conference Modern Building Materials, Structures and Techniques, Vilnius, Lithuania, 16–18 May 2007. [Google Scholar]
- Ali, Z.; Zhu, F.; Hussain, S. Identification and assessment of uncertainty factors that influence the transaction cost in public sector construction projects in Pakistan. Buildings
**2018**, 8, 157. [Google Scholar] [CrossRef][Green Version] - Sacco, G.; Tzitzikas, Y. Dynamic Taxonomies and Faceted Search; Springer: Berlin, Germany, 2013. [Google Scholar]
- Ranganathan, S.R. The Colon Classification; Rutgers University Press: New Brunswick, ME, Canada, 1965. [Google Scholar]
- Rafindadi, A.D.; Mikić, M.; Kovačić, I.; Cekić, Z. Global Perception of Sustainable Construction Project Risks. Procedia–Soc. Behav. Sci.
**2014**, 119, 456–465. [Google Scholar] [CrossRef][Green Version] - Warmink, J.J.; Janssen, J.A.E.B.; Booij, M.J.; Krol, M.S. Identification and classification of uncertainties in the application of environmental models. Environ. Model. Soft.
**2010**, 25, 1518–1527. [Google Scholar] [CrossRef] - Hazir, O.; Haouari, M.; Erel, E. Robust scheduling and robustness measures for the discrete time/cost trade-off problem. Eur. J. Oper. Res.
**2010**, 207, 633–643. [Google Scholar] [CrossRef] - Zahid, T.; Agha, M.H.; Schmidt, T. Investigation of surrogate measures of robustness for project scheduling problems. Comput. Ind. Eng.
**2019**, 129, 220–227. [Google Scholar] [CrossRef] - Au, T.; Hendrickson, C. Profit Measures for Construction Projects. J. Constr. Eng. Manag.
**1986**, 112, 273–286. [Google Scholar] [CrossRef] - Ahuja, H. Construction Performance Control by Networks; Wiley: New York, NY, USA, 1976. [Google Scholar]
- Al-Shihabi, S.; AlDurgam, M. A max-min ant system for the finance-based scheduling problem. Comput. Ind. Eng.
**2017**, 110, 264–276. [Google Scholar] [CrossRef] - Hendrickson, C. Project Management for Construction: Fundamental Concepts for Owners, Engineers, Architects and Builders. Available online: https://www.cmu.edu/cee/projects/PMbook/ (accessed on 18 May 2021).
- Demeulemeester, E.; Herroelen, W. Project Scheduling; Kluwer Academic Publishers: New York, NY, USA, 2002; p. 67. [Google Scholar]
- Hapke, M.; Jaszkiewicz, A.; Słowiński, R. Interactive analysis of multiple-criteria project scheduling problems. Eur. J. Oper. Res.
**1998**, 107, 315–324. [Google Scholar] [CrossRef] - Chaturvedi, S.; Rajasekar, E.; Natarajan, S. Multi-objective Building Design Optimization under Operational Uncertainties Using the NSGA II Algorithm. Buildings
**2020**, 10, 88. [Google Scholar] [CrossRef] - Al-Sadek, O.; Carmichael, D. On simulation in planning networks. Civ. Eng. Syst.
**1992**, 9, 59–68. [Google Scholar] [CrossRef]

**Figure 2.**Layout of the proposed resilience procedure for uncertainty management in construction projects.

**Figure 4.**Baseline schedules with additional resource constraints for (

**a**) Case 1 with the highest SM value and lower profit than in Case 2, and (

**b**) Case 2 with the highest profit and lower SM value than in Case 1.

**Figure 5.**Simulation results: (

**a**) total project duration (months); (

**b**) total project profit for all iterations in simulation, CL may be broken (thousand of financing units).

Abstraction Category | Facet Question | Facet | Description |
---|---|---|---|

Personality | Who | Member | Uncertainty related to the member in a construction project. |

Matter | What | Project | Uncertainty related to a construction project itself. |

Energy | How | Impact | Uncertainty impact on the project objective. |

Space | Where | Cause | Cause of uncertainty source in a construction project. |

Time | When | Life cycle | The time dimension of a construction project presenting a stage when uncertainty may arise. |

**Table 2.**The second level of faceted hierarchy for uncertainty sources in a construction project, where the primary node is “Uncertainty sources in a construction project”.

Member | Project | Impact | Cause | Life Cycle |
---|---|---|---|---|

Behavior | Activities | Cost | Location | Conceptualization |

Capacity | Complexity | Quality | Threats | Planning |

Decision making | Novelty | Schedule | Type | Execution |

Interactions | Type | Sustainability | - | Termination |

Nature | - | - | - | - |

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

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

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) |

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

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

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

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

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

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

$\mathrm{P}$ | Final profit at the end of a project |

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

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

$\mathrm{W}$ | Credit limit for the project |

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

${\mathrm{d}}_{\mathrm{i}}$, $\mathrm{i}\in \mathrm{A}$ | Duration of activities |

${\mathrm{s}}_{\mathrm{i}}$, $\mathrm{i}\in \mathrm{A}$ | Start times for activities from baseline schedule |

${\mathrm{c}}_{\mathrm{i}}$, $\mathrm{i}\in \mathrm{A}$ | Total direct cost per each activity in thousand of financial units |

OP | Overhead percentage in decimal form |

MP | Mobilization percentage in decimal form; |

TP | Tax percentage in decimal form |

MP | Markup percentage in decimal form |

BP | Bond percentage in decimal form |

N | Negotiated duration of the project in months |

S | Realized duration of the project in months |

ADV | Advance payment (percentage of TBP in decimal form) |

D | Late completion penalty in thousand of cost units |

RET | Percentage of retainage for investors’ payments (decimal) |

ir | Interest percentage in decimal form |

h | Surplus percentage in decimal form (h < ir) |

k | Percentage for interest on the unused portion of a credit |

W | Specified credit limit of the overdraft |

Activity Index | Activity Description |
---|---|

${\mathrm{A}}_{0}$ | Dummy start |

${\mathrm{A}}_{1}$ | Procurement |

${\mathrm{A}}_{2}$ | Field mobilization and site work |

${\mathrm{A}}_{3}$ | Landscape equipment mobilization |

${\mathrm{A}}_{4}$ | Systems work |

${\mathrm{A}}_{5}$ | Structure work |

${\mathrm{A}}_{6}$ | Construction finishing operations |

${\mathrm{A}}_{7}$ | Landscape earthwork |

${\mathrm{A}}_{8}$ | Landscape surfaces |

${\mathrm{A}}_{9}$ | Dummy end |

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

OP | Overhead percentage | 0.15 |

MP | Mobilization percentage | 0.05 |

TP | Tax percentage | 0.02 |

MP | Markup percentage | 0.20 |

BP | Bond percentage | 0.01 |

ADV | Advance | 0.10 |

D | Penalty | 2 |

RET | Retainage | 0.05 |

ir | Interest | 0.008 |

h | Surplus | 0.005 |

k | Interest on unused credit | 0.002 |

Instance | Baseline | Duration | SM | Profit |
---|---|---|---|---|

Case 1 | 0, 5, 0, 0, 8, 5, 14, 10, 13, 18 | 18 | 0.487 | 904.12 |

Case 2 | 0, 0, 3, 0, 5, 5, 14, 7, 10, 18 | 18 | 0.353 | 911.81 |

Instance | SM Value | Profit | W | Eq _{i} | Eq _{ii} | Eq _{iii} | Eq _{iv} | Eq _{v} | Eq _{vi} |
---|---|---|---|---|---|---|---|---|---|

Case 1 | 0.487 | 904.12 | 280 | 0.3 | 0.9135 | 0.66 | 0.12 | 1.65% | 0.3769 |

Case 2 | 0.353 | 911.81 | 280 | 0.2046 | 0.8249 | 0.93 | 0.42 | 1.73% | 0.0978 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Milat, M.; Knezić, S.; Sedlar, J. Resilient Scheduling as a Response to Uncertainty in Construction Projects. *Appl. Sci.* **2021**, *11*, 6493.
https://doi.org/10.3390/app11146493

**AMA Style**

Milat M, Knezić S, Sedlar J. Resilient Scheduling as a Response to Uncertainty in Construction Projects. *Applied Sciences*. 2021; 11(14):6493.
https://doi.org/10.3390/app11146493

**Chicago/Turabian Style**

Milat, Martina, Snježana Knezić, and Jelena Sedlar. 2021. "Resilient Scheduling as a Response to Uncertainty in Construction Projects" *Applied Sciences* 11, no. 14: 6493.
https://doi.org/10.3390/app11146493