Fuzzy Optimization and Decision Making

, Volume 9, Issue 2, pp 219–231 | Cite as

New fuzzy models for time-cost trade-off problem

  • Hua Ke
  • Weimin Ma
  • Xin Gao
  • Weihua Xu


The time-cost trade-off problem is a specific type of the project scheduling problem which studies how to modify project activities so as to achieve the trade-off between the completion time and the project cost. In real projects, the trade-off between the project cost and the completion time, and the uncertainty of the environment are both considerable aspects for managers. In this paper, three new fuzzy time-cost trade-off models are proposed, in which credibility theory is applied to describe the uncertainty of activity duration times. A searching method by integrating fuzzy simulation and genetic algorithm is produced to search the quasi-optimal schedules under some decision-making criteria. The purpose of the paper is to reveal how to obtain the optimal balance of the completion time and the project cost in fuzzy environments.


Time-cost trade-off Credibility theory Fuzzy simulation Genetic algorithm 


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  1. Azaron A., Perkgoz C., Sakawa M. (2005) A genetic algorithm approach for the time-cost trade-off in PERT networks. Applied Mathematics and Computation 168(2): 1317–1339MATHCrossRefMathSciNetGoogle Scholar
  2. Butcher W. S. (1967) Dynamic programming for project cost-time curve. Journal of Construction Division 93(C01): 59–73Google Scholar
  3. Chanas S., Kamburowski J. (1981) The use of fuzzy variables in PERT. Fuzzy Sets and Systems 5(1): 11–19MATHCrossRefMathSciNetGoogle Scholar
  4. Charnes A., Cooper W. W. (1959) Chance-constrained programming. Management Science 6: 73–79MATHCrossRefMathSciNetGoogle Scholar
  5. Charnes A., Cooper W. W. (1962) A network interpretation and a direct sub-dual algorithm for critical path scheduling. Journal of Industrial Engineering 13: 213–219Google Scholar
  6. Chua D. K. H., Chan W. T., Govindan K. (1997) A time-cost trade-off model with resource consideration using genetic algorithm. Civil Engineering Systems 14: 291–311CrossRefGoogle Scholar
  7. Eshtehardian E., Afshar A., Abbasnia R. (2008) Time-cost optimization: Using GA and fuzzy sets theory for uncertainties in cost. Construction Management and Economics 26(7): 679–691CrossRefGoogle Scholar
  8. Feng C. W., Liu L., Burns S. A. (1997) Using genetic algorithms to solve construction time-cost trade-off problems. Journal of Construction Engineering and Management 11(3): 184–189Google Scholar
  9. Freeman R. J. (1960) A generalized network approach to project activity sequencing. IRE Transactions on Engineering Management 7(3): 103–107CrossRefGoogle Scholar
  10. Ghazanfari M., Shahanaghi K., Yousefli A. (2008) An application of possibility goal programming to the time-cost trade off problem. Journal of Uncertain Systems 2(1): 22–28Google Scholar
  11. Ghazanfari M., Yousefli A., Ameli M. S. J., Bozorgi-Amiri A. (2009) A new approach to solve time-cost trade-off problem with fuzzy decision variables. The International Journal of Advanced Manufacturing Technology 42: 408–414CrossRefGoogle Scholar
  12. Goldratt E. (1997) Critical chain. The North River Press, Great Barrington, MAGoogle Scholar
  13. Golenko-Ginzburg D., Gonik A. (1997) Stochastic network project scheduling with non-consumable limited resources. International Journal of Production Economics 48(1): 29–37CrossRefGoogle Scholar
  14. Gutjahr W. J., Strauss C., Wagner E. (2000) A stochastic branch-and-bound approach to activity crashing in project management. INFORMS Journal on Computing 12(2): 125–135MATHCrossRefGoogle Scholar
  15. Hapke M., Jaszkiewicz A., Slowinski R. (1994) Fuzzy project scheduling system for software development. Fuzzy Sets and Systems 67(11): 101–117CrossRefMathSciNetGoogle Scholar
  16. Hapke M., Slowinski R. (1993) A DSS for resource-constrained project scheduling under uncertainty. Journal of Decision Systems 2(2): 111–128Google Scholar
  17. Hapke M., Slowinski R. (1996) Fuzzy priority heuristics for project scheduling. Fuzzy Sets and Systems 83(3): 291–299CrossRefGoogle Scholar
  18. Hapke M., Slowinski R. (2000) Fuzzy set approach to multi-objective and multi-mode project scheduling under uncertainty. In: Hapke M., Slowinski R. (eds) Scheduling under fuzziness. Physica-Verlag, Heidelberg, pp 197–221Google Scholar
  19. Jin C., Ji Z., Lin Y., Zhao Y., Huang Z. (2005) Research on the fully fuzzy time-cost trade-off based on genetic algorithms. Journal of Marine Science and Application 4(3): 18–23CrossRefGoogle Scholar
  20. Kaufmann A., Gupta M. M. (1988) Fuzzy mathematical models in engineering and management science. Amsterdam, North-HollandMATHGoogle Scholar
  21. Ke H., Liu B. (2005) Project scheduling problem with stochastic activity duration times. Applied Mathematics and Computation 168(1): 342–353MATHCrossRefMathSciNetGoogle Scholar
  22. Ke H., Liu B. (2010) Fuzzy project scheduling problem and its hybrid intelligent algorithm. Applied Mathematical Modelling 34(2): 301–308MATHCrossRefMathSciNetGoogle Scholar
  23. 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: 308–313MATHCrossRefMathSciNetGoogle Scholar
  24. Kelley J. E. Jr. (1961) Critical path planning and scheduling mathematical basis. Operations Research 9(3): 296–320MATHCrossRefMathSciNetGoogle Scholar
  25. Laslo Z. (2003) Activity time-cost tradeoffs under time and cost chance constraints. Computers & Industrial Engineering 44: 365–384CrossRefGoogle Scholar
  26. Leu S. S., Chen A. T., Yang C. H. (2001) A GA-based fuzzy optimal model for construction time-cost trade-off. International Journal of Project Management 19: 47–58CrossRefGoogle Scholar
  27. Liu B. (1997) Dependent-chance programming: A class of stochastic programming. Computers & Mathematics with Applications 34(12): 89–104MATHCrossRefMathSciNetGoogle Scholar
  28. Liu B., Iwamura K. (1998a) Chance constrained programming with fuzzy parameters. Fuzzy Sets and Systems 94: 227–237MATHCrossRefMathSciNetGoogle Scholar
  29. Liu B., Iwamura K. (1998b) A note on chance constrained programming with fuzzy coefficients. Fuzzy Sets and Systems 100: 229–233MATHCrossRefMathSciNetGoogle Scholar
  30. Liu B. (1999) Dependent-chance programming with fuzzy decisions. IEEE Transactions on Fuzzy Systems 7: 354–360CrossRefGoogle Scholar
  31. Liu B. (2002) Theory and practice of uncertain programming. Physica-Verlag, HeidelbergMATHGoogle Scholar
  32. Liu B., Liu Y. K. (2002) Expected value of fuzzy variable and fuzzy expected value models. IEEE Transactions on Fuzzy Systems 10: 445–450CrossRefGoogle Scholar
  33. Liu B. (2004) Uncertainty theory: An introduction to its axiomatic foundations. Springer-Verlag, BerlinMATHGoogle Scholar
  34. Liu B. (2006a) A survey of credibility theory. Fuzzy Optimization and Decision Making 5: 387–408CrossRefMathSciNetGoogle Scholar
  35. Liu Y. K. (2006b) Convergent results about the use of fuzzy simulation in fuzzy optimization problems. IEEE Transactions on Fuzzy Systems 14: 295–304CrossRefGoogle Scholar
  36. Long L. D., Ohsato A. (2008) Fuzzy critical chain method for project scheduling under resource constraints and uncertainty. International Journal of Project Management 26(6): 688–698CrossRefGoogle Scholar
  37. Özdamar L., Alanya E. (2001) Uncertainty modelling in software development projects (with case study). Annals of Operations Research 102(1–4): 157–178MATHCrossRefMathSciNetGoogle Scholar
  38. Phillips S. Jr., Dessouky M. I. (1997) Solving the project time/cost tradeoff problem using minimal cut concept. Management Science 24(4): 393–400CrossRefGoogle Scholar
  39. Prade H. (1979) Using fuzzy set theory in a scheduling problem: A case study. Fuzzy Sets and Systems 2(2): 153–165MATHCrossRefGoogle Scholar
  40. Siemens N. (1971) A simple CPM time-cost trade-off algorithm. Management Science 17(6): 354–363CrossRefMathSciNetGoogle Scholar
  41. Talbot F. B. (1982) Resource-constrained project scheduling with time-resource tradeoffs: The non-preemptive case. Management Science 28: 1197–1210MATHCrossRefGoogle Scholar
  42. Wang J. R. (1999) A fuzzy set approach to activity scheduling for product development. Journal of the Operational Research Society 50: 1217–1228MATHGoogle Scholar
  43. Wang J. R. (2002) A fuzzy project scheduling approach to minimize schedule risk for product development. Fuzzy Sets and Systems 127(2): 99–116MATHCrossRefMathSciNetGoogle Scholar
  44. Wollmer R. D. (1985) Critical path planning under uncertainty. Mathematical Programming Study 25: 164–171MATHMathSciNetGoogle Scholar
  45. Zadeh L. A. (1965) Fuzzy sets. Information and Control 8: 338–353MATHCrossRefMathSciNetGoogle Scholar
  46. Zadeh L. A. (1978) Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems 1: 3–28MATHCrossRefMathSciNetGoogle Scholar
  47. Zahraie B., Tavakolan M. (2009) Stochastic time-cost-resource utilization optimization using nondominated sorting genetic algorithm and discrete fuzzy sets. Journal of Construction Engineering and Management 135: 1162–1171CrossRefGoogle Scholar
  48. Zheng D. X. M., Ng S. T. (2005) Stochastic time-cost optimization model incorporating fuzzy sets theory and nonreplaceable front. Journal of Construction Engineering and Management 131(2): 176–186CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.School of Economics and ManagementTongji UniversityShanghaiChina
  2. 2.School of Mathematics and PhysicsNorth China Electric Power UniversityBeijingChina

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