Designing a Robust Multi-objective Model for Project Scheduling with Limited Resources and Time-cost Balance

Document Type : Research Paper


1 MSc., Department of Construction Project Management, Faculty of Architecture, Khatam University, Tehran, Iran.

2 Associate Prof., Department of Management, Faculty of Management and Finance, Khatam University, Tehran, Iran.

3 Assistant Prof., Department of Management, Faculty of Management and Finance, Khatam University, Tehran, Iran.



Objective: One of the important issues in project management is project scheduling. Since projects are faced with high uncertainty, project scheduling is of high importance under uncertain conditions. The purpose of this research is to present a robust multi-objective model to optimize project scheduling with limited resources by considering uncertainty, in which activities have several execution modes with uncertain duration, costs, and resources.
Methods: After reviewing the extant theoretical literature on project scheduling, the assumptions, parameters, and variables of the mathematical model were determined. Then, considering the goals and limitations of the project scheduling problem in deterministic conditions, a mathematical model was developed. This model was transformed into a single-objective model by the epsilon constraint method. To consider the uncertainty in the parameters of the problem, robust optimization, and Bertsimas and Sim's approach were used. Also, Robust optimization of the single-objective model was developed to consider the uncertainty.
Results: According to the obtained results, while the duration of the project increases with the increase in the tolerance of non-deterministic parameters; the percentage of project duration changes decreases for higher values ​​of robust parameters.
Conclusion: The robust parameters are with negative coefficient in the objective function, so their increase leads to a decrease in the objective function value. The highest reduction of the objective function is when the robust parameter is changed from two to one. This coefficient decreases with the increasing value of the corresponding parameter i.e., as the value of stability parameters increases, its effect on changing the value of the objective function decreases. The cost parameter was changed between -40% to +40% for the value of gamma 10 (the state where 10 activities of the project are non-deterministic) for different variation values of parameters of non-deterministic activities. Its effect on the values of the objective function indicated that the variation of the cost parameter in the range of 0 to +40% turns it into an unnecessary constraint and that its change has no effect on the value of the objective function (project duration). Also, in the range of 0 to -40%, the cost reduction caused a decrease in the value of the objective function (increasing the project duration) and the maximum impact of the reduction of the project budget related to the situation where the uncertain parameters of time and cost change by 40% and 50%.


Main Subjects

Artigues, C., Leus, R., & Talla Nobibon, F. (2013). Robust optimization for resource-constrained project scheduling with uncertain activity durations. Flexible Services and Manufacturing Journal, 25, 175-205.
Azar, A., Rezaei Pandari, A. (2017). Advanced Operation Research, Tehran, Negah Danesh Publication. (in Persian)
Ballestin, F. & Leus, R. (2009). Resource‐constrained project scheduling for timely project completion with stochastic activity durations. Production and operations management, 18(4), 459-474.
Balouka, N. & Cohen, I. (2021). A robust optimization approach for the multi-mode resource-constrained project scheduling problem. European journal of operational research, 291(2), 457-470.
Banihashemi, S. A., & Khalilzadeh, M. (2022). A Robust Bi-objective Optimization Model for Resource Levelling Project Scheduling Problem with Discounted Cash Flows. KSCE Journal of Civil Engineering, 26(6), 2539-2554.
Bold, M., & Goerigk, M. (2022). A faster exact method for solving the robust multi-mode resource-constrained project scheduling problem. Operations Research Letters, 50(5), 581-587.
Bruni, M. E., Pugliese, L. D. P., Beraldi, P., & Guerriero, F. (2017). An adjustable robust optimization model for the resource-constrained project scheduling problem with uncertain activity durations. Omega, 71, 66-84.
Cao, Q., Zou, X., & Zhang, L. (2022). Multi-objective Robust Optimization Model for Generating Stable and Makespan-Protective Repetitive Schedules. Journal of construction Engineering and Management, 148(9).
Chakrabortty, R. K., & Ryan, M. J. (2020). Robust optimization based heuristic approach for solving stochastic multi-mode resource constrained project scheduling problem. 2020 IEEE International Conference on Industrial Engineering and Engineering Management (IEEM).
Cohen, I., & Balouka, N. (2018). Robust Project Planning under Resource Constraints. 2018 IEEE Technology and Engineering Management Conference (TEMSCON).
Guldemond, T. A., Hurink, J. L., Paulus, J. J., & Schutten, J. M. (2008). Time-constrained project scheduling. Journal of Scheduling, 11, 137-148.
Hazır, Ö., Haouari, M., & Erel, E. (2015). Robust optimization for the discrete time-cost tradeoff problem with cost uncertainty. Handbook on Project Management and Scheduling, 2, 865-874.
Lock, D. (2007). Project management (9th edition). Burlington, VT: Gower.
Ma, G., Gu, L., & Li, N. (2015). Scenario-based proactive robust optimization for critical-chain project scheduling. Journal of construction Engineering and Management, 141(10), 04015030.
Moradi, M., Hafezalkotob, A., & Ghezavati, V. (2019). Robust resource-constrained project scheduling problem of the project’s subcontractors in a cooperative environment under uncertainty: Social complex construction case study. Computers & industrial engineering, 133, 19-28.
Nabipoor Afruzi, E., Aghaie, A., & Najafi, A. A. (2020). Robust optimization for the resource-constrained multi-project scheduling problem with uncertain activity durations. Scientia Iranica, 27(1), 361-376.
Tabrizi, B. H., & Ghaderi, S. F. (2016). A robust bi-objective model for concurrent planning of project scheduling and material procurement. Computers & industrial engineering, 98, 11-29.
Tavana, M., Abtahi, A.-R., & Khalili-Damghani, K. (2014). A new multi-objective multi-mode model for solving preemptive time–cost–quality trade-off project scheduling problems. Expert Systems with Applications, 41(4), 1830-1846. doi:10.1016/j.eswa.2013.08.081
Tirkolaee, E. B., Goli, A., Hematian, M., Sangaiah, A. K., & Han, T. (2019). Multi-objective multi-mode resource constrained project scheduling problem using Pareto-based algorithms. Computing, 101, 547-570.
Van Peteghem, V., & Vanhoucke, M. (2010). A genetic algorithm for the preemptive and non-preemptive multi-mode resource-constrained project scheduling problem. European Journal of Operational Research, 201(2), 409-418.
Zhang, Z., & Zhong, X. (2018). Time/resource trade-off in the robust optimization of resource-constraint project scheduling problem under uncertainty. Journal of Industrial and Production Engineering, 35(4), 243-254.