Abstract
Nowadays the advent of new types of projects such as startups, maintenance, and education make a revolution in project management, so that, classical project scheduling methods are incapable in analyzing of these stochastic projects. This study considers a time–cost trade-off project scheduling problem, where the structure of the project is uncertain. To deal with the uncertainties, we implemented Graphical Evaluation and Review Technique (GERT). The main aim of the study is to balance time and the amount of a non-renewable resource allocated to each activity considering the finite-time horizon and resource limitations. To preserve the generality of the model, we considered both discrete and continuous distribution functions for the activity’s duration. From a methodological standpoint, we proposed an analytical approach based on the Markov Decision Process (MDP) and Semi-Markov Decision Process (SMDP) to find the probability distribution of project makespan. These models are solved using the value iteration and a finite-horizon Linear Programming (LP) model. Two randomly generated examples explain the value iteration for models in detail. Furthermore, seven example groups each with five instances are adopted from a well-known data set, PSPLIB, to validate the efficiency of the proposed models in contrast to the two extensively-studied methods, Genetic algorithm (GA) and Monte-Carlo simulation. The convergence of the GA and simulation results to those of MDP and SMDP represent the efficiency of the proposed models. Besides, conducting a sensitivity analysis on the project completion probability with respect to the available resource, gives a good insight to managers to plan their resources.
Similar content being viewed by others
References
Aouam, T., & Vanhoucke, M. (2019). An agency perspective for multi-mode project scheduling with time/cost trade-offs. Computers and Operations Research, 105, 167–186. https://doi.org/10.1016/j.cor.2019.01.012
Arisawa, S., & Elmaghraby, S. E. (1972). Optimal time–cost trade-offs in GERT networks. Management Science, 18(11), 589–599. https://doi.org/10.1287/mnsc.18.11.589
Aytulun, S. K., & Ermis, M. (2009). Space launch vehicle project reliability analysis by using GERT. in RAST 2009 - Proceedings of 4th international conference on recent advances space technologies, (pp. 241–248). https://doi.org/10.1109/RAST.2009.5158205
Aytulun, S. K., & Guneri, A. F. (2008). Business process modelling with stochastic networks. International Journal of Production Research, 46(10), 2743–2764. https://doi.org/10.1080/00207540701543601
Azaron, A., & Tavakkoli-Moghaddam, R. (2007). Multi-objective time-cost trade-off in dynamic PERT networks using an interactive approach. European Journal of Operational Research, 180(3), 1186–1200. https://doi.org/10.1016/j.ejor.2006.05.014
Bellas, C. (1971). The use of GERT in the planning and control of marketing research. Journal of Marketing Research, 8(3), 335–339.
Cheng, J., Fowler, J., Kempf, K., & Mason, S. (2015). Multi-mode resource-constrained project scheduling problems with non-preemptive activity splitting. Computers and Operations Research, 53, 275–287. https://doi.org/10.1016/j.cor.2014.04.018
Eiben, A. E., Rau, P., & Ruttkay, Z. (1994). Genetic algorithms with multi-parent recombination Genetic algorithms with multi-parent recombination. In Proceedings of the international conference on evolutionary computation. The Third conference on parallel problem solving from nature, October, (pp. 78–87). https://doi.org/10.1007/3-540-58484-6
Eshtehardian, E., Afshar, A., & Abbasnia, R. (2009). Fuzzy-based MOGA approach to stochastic time-cost trade-off problem. Automation in Construction, 18(5), 692–701. https://doi.org/10.1016/j.autcon.2009.02.001
Feng, C., Liu, L., & Burns, S. A. (1997). Using genetic algorithms to solve construction time-cost trade-off problems. Journal of Computing in Civil Engineering, 11, 184–189.
Feng, C., Liu, L., & Burns, S. A. (2000). Stochastic construction time-cost trade-off analysis. Journal of Computing in Civil Engineering, 14(April), 117–126.
Fix, W., & Neumann, K. (1979). Project scheduling by special GERT networks. Computing, 23(3), 299–308. https://doi.org/10.1007/BF02252134
Golenko-Ginzburg, D., Gonik, A., & Laslo, Z. (2003). Resource constrained scheduling simulation model for alternative stochastic network projects. Mathematics and Computers in Simulation, 63(2), 105–117. https://doi.org/10.1016/S0378-4754(03)00050-8
Hazir, Ö., Erel, E., & Gunalay, Y. (2011). Robust optimization models for the discrete time/cost trade-off problem. International Journal of Production Economics, 130(1), 87–95. https://doi.org/10.1016/j.ijpe.2010.11.018
Hogg, G. L., Phillips, D. T., Maggard, M. J., & Lesso, W. G. (1975a). Gerts qr: A model for multi-resource constrained queueing systems part I: Concepts, notation, and examples. AIIE Transactions, 7(2), 89–99. https://doi.org/10.1080/05695557508974991
Hogg, G. L., Phillips, D. T., Maggard, M. J., & Lesso, W. G. (1975b). Gerts qr: A model for multi-resource constrained queueing systems part II: An analysis of parallel-channel, dual-resource constrained queueing systems with homogeneous resources. AIIE Transactions, 7(2), 100–109. https://doi.org/10.1080/05695557508974992
Ke, H., Ma, W., & Chen, X. (2012). Modeling stochastic project time-cost trade-offs with time-dependent activity durations. Applied Mathematics and Computation, 218(18), 9462–9469. https://doi.org/10.1016/j.amc.2012.03.035
Ke, H., Ma, W., & Ni, Y. (2009). Optimization models and a GA-based algorithm for stochastic time-cost trade-off problem. Applied Mathematics and Computation, 215(1), 308–313. https://doi.org/10.1016/j.amc.2009.05.004
Klerides, E., & Hadjiconstantinou, E. (2010). A decomposition-based stochastic programming approach for the project scheduling problem under time/cost trade-off settings and uncertain durations. Computers and Operations Research, 37(12), 2131–2140. https://doi.org/10.1016/j.cor.2010.03.002
Kosugi, T., Hayashi, A., Matsumoto, T., Akimoto, K., Tokimatsu, K., Yoshida, H., Tomoda, T., & Kaya, Y. (2004). Time to realization: Evaluation of CO2 capture technology R&Ds by GERT (Graphical Evaluation and Review Technique) analyses. Energy, 29(9–10), 1297–1308. https://doi.org/10.1016/j.energy.2004.03.088
Littman, M. L., Dean, T. L., & Kaelbling, L. P. (2013). On the complexity of solving markov decision problems. In Proceedings of the 11th Onference on Uncertainty in Artificial Intelligence, (pp. 394–402).
Liu, D., Li, H., Wang, H., Qi, C., & Rose, T. (2020). Discrete symbiotic organisms search method for solving large-scale time-cost trade-off problem in construction scheduling. Expert Systems with Applications, 148, 113230. https://doi.org/10.1016/j.eswa.2020.113230
Machado-Domínguez, L. F., Paternina-Arboleda, C. D., Vélez, J. I., & Barrios-Sarmiento, A. (2021). A memetic algorithm to address the multi-node resource-constrained project scheduling problem. Journal of Scheduling, 24(4), 413–429. https://doi.org/10.1007/s10951-021-00696-5
Puterman, M. L. (1994). Markov Decision Processes. John Wiley and Sons. Inc.
Mirjalili, S. (2019). Genetic algorithm. Studies in Computational Intelligence. https://doi.org/10.1007/978-3-662-43631-8_3
Nelson, R. G., Azaron, A., & Aref, S. (2016). The use of a GERT based method to model concurrent product development processes. European Journal of Operational Research, 250(2), 566–578. https://doi.org/10.1016/j.ejor.2015.09.040
Neumann, K. (1985). EOR project networks. Computing, 34, 1–15.
Norouzi, G., Heydari, M., Noori, S., & Bagherpour, M. (2015). Developing a mathematical model for scheduling and determining success probability of research projects considering complex-fuzzy networks. Journal of Applied Mathematics, 2015, 1–15. https://doi.org/10.1155/2015/809216
Phillips, D. T., & Hogg, G. L. (1976). Stochastic network analysis with resource constraints, cost parameters, and queueing capabilities using gerts methodologies. Computers and Industrial Engineering, 1(1), 13–25. https://doi.org/10.1016/0360-8352(76)90004-8
Pritsker, A. A. B. (1966). GERT: Graphical Evaluation and Review Technique (p. 152). The RAND Corporation. https://doi.org/10.4018/978-1-4666-4940-8.ch009
Raju, G. V. S. (1971). Sensitivity Analysis of GERT Networks. AIIE Transactions, 3(2), 133–141. https://doi.org/10.1080/05695557108974796
Ren, S., & Yuan, Z. (2012). Simulation analysis of criticality indexes of activities in GERT networks. In Proceedings of the 2012 5th international conference on business intelligence and financial engineering, BIFE 2012, 567–571. Doi: https://doi.org/10.1109/BIFE.2012.124
Rostami, S., Creemers, S., & Leus, R. (2018). New strategies for stochastic resource-constrained project scheduling. Journal of Scheduling, 21(3), 349–365. https://doi.org/10.1007/s10951-016-0505-x
Rubinstein, R. Y., & Kroese, D. P. (2007). Simulation and the Monte Carlo method (Wiley series in probability and statistics). John Wiley & Sons.
Said, S. S., & Haouari, M. (2015). A hybrid simulation-optimization approach for the robust Discrete Time/Cost Trade-off Problem. Applied Mathematics and Computation, 259, 628–636. https://doi.org/10.1016/j.amc.2015.02.092
Sonmez, R., & Bettemir, Ö. H. (2012). A hybrid genetic algorithm for the discrete time-cost trade-off problem. Expert Systems with Applications, 39(13), 11428–11434. https://doi.org/10.1016/j.eswa.2012.04.019
Tao, L., Su, X., & Javed, S. A. (2022). Time-cost trade-off model in GERT-type network with characteristic function for project management. Computers and Industrial Engineering, 169(May), 108222. https://doi.org/10.1016/j.cie.2022.108222
Tao, L., Wu, D., Liu, S., & Lambert, J. H. (2017). Schedule risk analysis for new-product development: The GERT method extended by a characteristic function. Reliability Engineering and System Safety, 167(2016), 464–473. https://doi.org/10.1016/j.ress.2017.06.010
Tao, S., & Dong, Z. S. (2018). Multi-mode resource-constrained project scheduling problem with alternative project structures. Computers and Industrial Engineering, 125(August), 333–347. https://doi.org/10.1016/j.cie.2018.08.027
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(42), 1830–1846. https://doi.org/10.1016/j.eswa.2013.08.081
Taylor, B. W., Keown, A. J., & Barrett, R. T. (1982). Analyzing court system congestion with Q-Gert network modeling and simulation. Computers and Operations Research, 9(3), 163–171. https://doi.org/10.1016/0305-0548(82)90016-8
Vanhoucke, M., & Debels, D. (2007). The discrete time/cost trade-off problem: Extensions and heuristic procedures. Journal of Scheduling. https://doi.org/10.1007/s10951-007-0031-y
Whitehouse, G. E., & Pritsker, A. A. B. (1969). Gert: Part iii—further statistical results; counters, renewal times, and correlations. AIIE Transactions, 1(1), 45–50. https://doi.org/10.1080/05695556908974412
Yu, L., & Zuo, M. (2008). An estimating method for IT project expected duration oriented to GERT. IFIP International Federation for Information Processing, 255, 1557–1566. https://doi.org/10.1007/978-0-387-76312-5_89
Project Scheduling Problem Library (PSPLIB). Data set of Multi-Mode Resource Constrained Project Scheduling Problem (MRCPSP). Retrieved from http://www.omdb.wi.tum.de/psplib/getdata.cgi?mode=mm.
Funding
The authors of this paper declare that there is no conflict of interest associated with this publication and this study is not funded by a specific project grant.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Ethical approval
This article does not contain any studies with human participants or animals performed by any of the authors.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Sadri, S., Ghomi, S.M.T.F. & Dehghanian, A. Analysis of a time–cost trade-off in a resource-constrained GERT project scheduling problem using the Markov decision process. Ann Oper Res (2024). https://doi.org/10.1007/s10479-024-05896-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10479-024-05896-8