1Assistant Professor, Faculty of Industrial and Mechanical Engineering, Islamic Azad University (Qazvin Branch)
2M.Sc., Faculty of Industrial and Mechanical Engineering, Islamic Azad University (Qazvin Branch)
3Ph.D., Department of Mechanical and Industrial Engineering, Northeastern University, Boston, USA
Project planning is part of project management, which is relates to the use of schedules such as Gantt charts to plan and subsequently report progress within the project environment. Initially, the project scope is defined and the appropriate methods for completing the project are determined. In this paper a new approach for the critical path analyzing a project network with random fuzzy activity times has been proposed. The activity times of a project are assumed to be random fuzzy. Linear programming formulation has been applied to determine the critical path. The critical path method (CPM) problem has been solved using the expected duration optimization model and the mean-variance model, along with Liu’s definition for random fuzzy variables. Furthermore, a numerical example problem is solved for illustrating the proposed method.
 Chanas, S.; Dubois, D. and Zielin´-Ski, P.; “On the Sure Criticality of Tasks in Activity Networks with Imprecise Durations,” IEEE Transactions on Systems, Man and Cybernetics–Part B: Cybernetics, Vol. 4, No. 32, pp. 393–407, 2002.
 Chanas, S. and Zielin´-Ski, P.; “Critical Path Analysis in the Network with Fuzzy Activity Times,” Fuzzy Sets and Systems, Vol. 122, No. 2, pp. 195–204, 2001.
 Chanas, S. and Zielin´-Ski, P.; “The Computational Complexity of the Criticality Problems in a Network with Interval Activity Times,” European Journal of Operational Research, Vol. 136, No. 2, pp. 541–550, 2002.
 Chanas, S. and Zielin´-Ski, P.; “On the Hardness of Evaluating Criticality of Activities in a Planar Network with Duration Intervals,” Operations Research Letters, Vol. 31, No. 1, pp. 53–59, 2003.
 Chanas, S. and Kamburowski, J.; “The Use of Fuzzy Variables in PERT,” Fuzzy Set Systems, Vol. 5, No. 1, pp. 11–9, 1981.
 Chen, S. P.; “Analysis of Critical Paths in a Project Network with Fuzzy Activity Times,” European Journal of Operational Research, Vol. 183, No. 1, pp. 442–459, 2007.
 Tseng, C. and KO, P.; “Measuring Schedule Uncertainty for a Stochastic Resource-Constrained Project Using Scenario-Based Approach with Utility- Entropy Decision Model,” Journal of Industrial and Production Engineering, pp. 1–10, 2016.
 Ding, C. and Zhu, Y.; “Two Empirical Uncertain Models for Project Scheduling Problem,” Journal of the Operational Research Society, Vol. 66, No. 9 , pp. 1471–1480, 2015.
 Elmaghraby, S.; “On Criticality and Sensitivity in Activity Networks,” International Journal of Production Research, Vol. 127 No. 2, pp. 220–38, 2000.
 Hassanzadeh, R.; Mahdavi, I.; Mahdavi-Amiri, N. and Tajdin, A.; “A Genetic Algorithm for Solving Fuzzy Shortest Path Problems with Mixed Fuzzy Arc Lengths,” Mathematical and Computer Modelling, Vol. 57, No. 1, pp. 84–99, 2013.
 Hasuike, T.; Katagiri, H. and Ishii, H.; “Portfolio Selection Problems with Random Fuzzy Variable Returns,” Fuzzy Sets and Systems, Vol. 160, pp. 2579–2596, 2009.
 Hillier, F.S. and Lieberman, G. J.; “Introduction to Operations Research,” McGraw-Hill, Singapore, 7th ed., 2001.
 Kaur, P. and Kumar, A.; “Linear Programming Approach for Solving Fuzzy Critical Path Problems with Fuzzy Parameters,” Applied Soft Computing, Vol. 21, pp. 309–319, 2014.
 Ke, H. and Liu, B.; “Project Scheduling Problem with Mixed Uncertainty of Randomness and Fuzziness,” European Journal of Operational Research, Vol. 183, No. 9, pp. 135–147, 2007.
 Kelley, J. E.; “Critical Path Planning and Scheduling–Mathematical Basis,” Operational Research, Vol. 9, No. 3, pp. 296–320, 1961.
 Li, X. and Liu, B.; “New Independence Definition of Fuzzy Random Variable and Random Fuzzy Variable,” World Journal of Modelling and Simulation, Vol. 2, No. 5, pp. 338–342, 2006.
 Li, X.; Qin, Z. and Kar, S.; “Mean-Variance- Skewness Model for Portfolio Selection with Fuzzy Returns,” European Journal of Operational Research, Vol. 202, No. 1, pp. 239–247, 2010.
 Lin, L.; Lou, T. and Zhan, N. “Project Scheduling Problem with Uncertain Variables,” Applied Mathematics, Vol. 5, pp. 685–690, 2014.
 Liu, B.; “Theory and Practice of Uncertain Programming,” Physica-Verlag, Heidelberg, 2002.
 Liu, B.; “Uncertainty Theory: An Introduction to its Axiomatic Foundations,” Springer-Verlag, Berlin, 2004.
 Madhuri, K. U.; Saradhi, B. P. and Shankar, N. R.; “Fuzzy Linear Programming Model for Critical Path Analysis,” Int. J. Contemp. Math. Sciences, Vol. 8, No. 2, pp. 93–116, 2013.
 Malcolm, D. G.; Roseboom, J. H.; Clark, C. E. and Fazar, W.; “Application of a Technique for Research and Development Project Evaluation,” Operational Research, Vol. 7, pp. 646–69, 1959.
 Guide, A.; “Project Management Body of Knowledge (PMBOK® GUIDE),” Project Management Institute, 2001.
 Sadjadi, S. J.; Pourmoayed, R. and Aryanezhad, M. B.; “A Robust Critical Path in an Environment with Hybrid Uncertainty,” Applied Soft Computing, Vol. 12, No. 3, pp. 1087–1100, 2012.
 Van-Slyke, R. M.; “Monte-Carlo Method and the PERT Problem,” Operational Research, Vol. 11, No. 5, pp. 839–60, 1963.
 Yakhchali, S. H. and Ghodsypour, S. H.; “On the Latest Starting Times and Criticality of Activities in a Network with Imprecise Durations,” Appllied Mathemathical Modelling, Vol. 34, No. 8, pp. 2044– 2058, 2010.
 Zadeh, L. A.; “The Concept of a Linguistic Variable and its Application to Approximate Reasoning,” Information Sciences, Vol. 8, No. 3, pp. 199–249, 1975.
 Zammori, F. A.; Braglia, M. and Frosolini, M.; “A Fuzzy Multi-Criteria Approach for Critical Path Definition,” International Journal of Project Management, Vol. 27, No. 3, pp. 278–291, 2009.
 Zareei, A.; Zaerpour, F.; Bagherpour, M.; Noora, A. and Vencheh, A.; “A New Approach for Solving Fuzzy Critical Path Problem Using Analysis of Events,” Expert Systems with Applications, Vol. 38, No. 3, pp. 87–93, 2011.
 Zielin´-Ski, P.; “On Computing the Latest Starting Times and Floats of Activities in a Network with Imprecise Durations,” Fuzzy Sets and Systems, Vol. 150, No. 3, pp. 53–76, 2005.