Neural Computing and Applications

, Volume 28, Issue 8, pp 2193–2206 | Cite as

Applying fuzzy multi-objective linear programming to a project management decision with nonlinear fuzzy membership functions

  • Ehsan EhsaniEmail author
  • Nima KazemiEmail author
  • Ezutah Udoncy Olugu
  • Eric H. Grosse
  • Kurt Schwindl
Original Article


This paper investigates a multi-objective project management problem where the goals of the decision maker are fuzzy. Prior research on this topic has considered linear membership functions to model uncertain project goals. Linear membership functions, however, are not much flexible to model uncertain information of projects in many situations, and therefore, fuzzy models with linear membership functions are not suitable to be applied in many practical situations. Hence, the purpose of this paper is to apply nonlinear membership functions in order to develop a better representation of fuzzy project planning in practice. This approach supports managers in examining different solution strategies and in planning projects more realistically. In doing so, a fuzzy mathematical project planning model with exponential fuzzy goals is developed first which takes account of (a) the time between events, (b) the crashing time for activities, and (c) the available budget. Following, a weighted max–min model is applied for solving the multi-objective project management problem. The performance of the developed solution procedure is compared with the literature that applied linear membership functions to this problem, and it is shown that the model developed in this paper outperforms the existing solution.


Project management Nonlinear fuzzy membership function Exponential membership function Fuzzy multi-objective programming Weighted max–min approach 



The second and third authors gratefully acknowledge financial support from University of Malaya under the grant RP018a-13aet.


  1. 1.
    Amid A, Ghodsypour S, O’Brien C (2011) A weighted max–min model for fuzzy multi-objective supplier selection in a supply chain. Int J Prod Econ 131:139–145CrossRefGoogle Scholar
  2. 2.
    Arıkan F, Güngör Z (2001) An application of fuzzy goal programming to a multiobjective project network problem. Fuzzy Sets Syst 119:49–58MathSciNetCrossRefGoogle Scholar
  3. 3.
    Baker BN, Murphy DC, Fisher D (2008) Factors affecting project success. Project Management Handbook, 2nd edn, pp 902–919Google Scholar
  4. 4.
    Belassi W, Tukel OI (1996) A new framework for determining critical success/failure factors in projects. Int J Project Manage 14:141–151CrossRefGoogle Scholar
  5. 5.
    Bellman RE, Zadeh LA (1970) Decision-making in a fuzzy environment. Manag Sci 17:B-141–B-164Google Scholar
  6. 6.
    Bells S (1999) Flexible membership functions.
  7. 7.
    Belout A, Gauvreau C (2004) Factors influencing project success: the impact of human resource management. Int J Project Manage 22:1–11CrossRefGoogle Scholar
  8. 8.
    Bhaskar T, Pal MN, Pal AK (2011) A heuristic method for RCPSP with fuzzy activity times. Eur J Oper Res 208:57–66MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Buller A (2002) Fuzzy sets with dynamic memberships. FSKD 2:18–22Google Scholar
  10. 10.
    Çebi F, İrem O (2015) A fuzzy multi-objective model for solving project network problem with bonus and incremental penalty cost. Comput Ind Eng 82:143–150CrossRefGoogle Scholar
  11. 11.
    Chanas S, Zieliński P (2001) Critical path analysis in the network with fuzzy activity times. Fuzzy Sets Syst 122:195–204MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Chanas S, Zieliński P (2002) The computational complexity of the criticality problems in a network with interval activity times. Eur J Oper Res 136:541–550MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Chen C-T, Huang S-F (2007) Applying fuzzy method for measuring criticality in project network. Inf Sci 177:2448–2458CrossRefzbMATHGoogle Scholar
  14. 14.
    Cooke-Davies T (2002) The “real” success factors on projects. Int J Project Manage 20:185–190CrossRefGoogle Scholar
  15. 15.
    Díaz-Madroñero M, Peidro D, Vasant P (2010) Vendor selection problem by using an interactive fuzzy multi-objective approach with modified S-curve membership functions. Comput Math Appl 60:1038–1048MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Fallaphour A, Olugu EU, Musa SN (in press) An integrated model for supplier performance evaluation under fuzzy environment: application of analytical hierarchy process and multi-expression programming, Neural Comput Appl. doi: 10.1007/s00521-015-2078-6
  17. 17.
    Fallahpour A, Olugu EU, Musa SN, Wong KY, Khezrimotlagh D (2015) An integrated model for green supplier selection under fuzzy environment: application of data envelopment analysis and genetic programming approach. Neural Comput Appl. doi: 10.1007/s00521-015-1890-3 Google Scholar
  18. 18.
    Göçken T (2013) Solution of fuzzy multi-objective project crashing problem. Neural Comput Appl 23:2167–2175CrossRefGoogle Scholar
  19. 19.
    Gupta P, Mehlawat MK (2009) Bector-Chandra type duality in fuzzy linear programming with exponential membership functions. Fuzzy Sets Syst 160:3290–3308MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Hartmann S, Briskorn D (2010) A survey of variants and extensions of the resource-constrained project scheduling problem. Eur J Oper Res 207:1–14MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Herroelen W, Leus R (2005) Project scheduling under uncertainty: survey and research potentials. Eur J Oper Res 165:289–306CrossRefzbMATHGoogle Scholar
  22. 22.
    Hersh HM, Caramazza A (1976) A fuzzy set approach to modifiers and vagueness in natural language. J Exp Psychol Gen 105:254CrossRefGoogle Scholar
  23. 23.
    Huang M, Yuan J, Xiao J (2015) An adapted firefly algorithm for product development project scheduling with fuzzy activity duration. Math Probl Eng. doi: 10.1155/2015/973291 Google Scholar
  24. 24.
    Jadidi O, Zolfaghari S, Cavalieri S (2014) A new normalized goal programming model for multi-objective problems: A case of supplier selection and order allocation. Int J Prod Econ 148:158-165 CrossRefGoogle Scholar
  25. 25.
    Kazemi N, Ehsani E, Glock CH (2014) Multi–objective supplier selection and order allocation under quantity discounts with fuzzy goals and fuzzy constraints. Int J Appl Decis Sci 7(1):66–96Google Scholar
  26. 26.
    Kazemi N, Ehsani E, Glock CH, Schwindl K (2015) A mathematical programming model for a multi-objective supplier selection and order allocation problem with fuzzy objectives. Int J Serv Oper Manag 21(4):435–465Google Scholar
  27. 27.
    Kazemi N, Ehsani E, Jaber M (2010) An inventory model with backorders with fuzzy parameters and decision variables. Int J Approx Reason 51(8):964–972MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Kazemi N, Olugu EU, Abdul-Rashid SH, Bin Raja Ghazilla RA (2015) Development of a fuzzy economic order quantity model for imperfect quality items using the learning effect on fuzzy parameters. J Intell Fuzzy Syst 28:2377–2389MathSciNetCrossRefGoogle Scholar
  29. 29.
    Kazemi N, Shekarian E, Cárdenas-Barrón LE, Olugu EU (2015) Incorporating human learning into a fuzzy EOQ inventory model with backorders. Comput Ind Eng 87:540–542CrossRefGoogle Scholar
  30. 30.
    Shekarian E, Glock CH, Pourmousavi Amiri SM, Schwindl K (2014) Optimal manufacturing lot size for a single stage production system with rework in a fuzzy environment. J Intell Fuzzy Syst 27:3067–3080MathSciNetzbMATHGoogle Scholar
  31. 31.
    Shekarian E, Jaber M, Kazemi N, Ehsani E (2014) A fuzzified version of the economic production quantity (EPQ) model with backorders and rework for a single–stage system. Euro J Ind Eng 8(3):291–324CrossRefGoogle Scholar
  32. 32.
    Shekarian S, Udoncy Olugu E, Hanim Abdul-Rashid S, Bottani E (In Press) A fuzzy reverse logistic inventory system integrating economic order/production quantity models. Int J Fuzzy Syst. doi: 10.1007/s40815-015-0129-x
  33. 33.
    Ke H, Liu B (2010) Fuzzy project scheduling problem and its hybrid intelligent algorithm. Appl Math Model 34:301–308MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Leberling H (1981) On finding compromise solutions in multicriteria problems using the fuzzy min-operator. Fuzzy Sets Syst 6:105–118MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Lewis JP (2010) Project planning, scheduling, and control: the ultimate hands-on guide to bringing projects in on time and on budget, 5th edn. McGraw-Hill Osborne Media, New YorkGoogle Scholar
  36. 36.
    Liang T-F (2009) Application of fuzzy sets to multi-objective project management decisions. Int J Gen Syst 38:311–330MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Liang T-F (2009) Fuzzy multi-objective project management decisions using two-phase fuzzy goal programming approach. Comput Ind Eng 57:1407–1416CrossRefGoogle Scholar
  38. 38.
    Liang T-F (2010) Applying fuzzy goal programming to project management decisions with multiple goals in uncertain environments. Expert Syst Appl 37:8499–8507CrossRefGoogle Scholar
  39. 39.
    Lin C-C (2004) A weighted max–min model for fuzzy goal programming. Fuzzy Sets Syst 142:407–420MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Mahmoodzadeh S, Shahrabi J, Pariazar M, Zaeri M (2007) Project selection by using fuzzy AHP and TOPSIS technique. Int J Hum soc sci 1(3):135–140 Google Scholar
  41. 41.
    Masmoudi M, Haït A (2013) Project scheduling under uncertainty using fuzzy modelling and solving techniques. Eng Appl Artif Intell 26:135–149CrossRefGoogle Scholar
  42. 42.
    Nieto-Morote A, Ruz-Vila F (2011) A fuzzy approach to construction project risk assessment. Int J Project Manage 29:220–231CrossRefzbMATHGoogle Scholar
  43. 43.
    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. J Appl Math. doi: 10.1155/2015/809216 MathSciNetGoogle Scholar
  44. 44.
    Ozmehmet Tasan S, Gen M (2013) An integrated selection and scheduling for disjunctive network problems. Comput Ind Eng 65:65–76CrossRefGoogle Scholar
  45. 45.
    Sadjadi SJ, Pourmoayed R, Aryanezhad M-B (2012) A robust critical path in an environment with hybrid uncertainty. Appl Soft Comput 12:1087–1100CrossRefGoogle Scholar
  46. 46.
    Shi Q, Blomquist T (2012) A new approach for project scheduling using fuzzy dependency structure matrix. Int J Project Manage 30:503–510CrossRefGoogle Scholar
  47. 47.
    Singh S, Olugu EU, Musa SN (2015) Strategy selection for sustainable manufacturing with integrated AHP -VIKOR method under interval-valued fuzzy environment. Int J Adv Manuf Technol. doi: 10.1007/s00170-015-7553-9 Google Scholar
  48. 48.
    Singh S, Olugu EU, Musa SN, Mahat AB (2015) Fuzzy-based sustainability evaluation method for manufacturing SMEs using balanced scorecard framework. J Intell Manuf. doi: 10.1007/s10845-015-1081-1 Google Scholar
  49. 49.
    Slyeptsov AI, Tyshchuk TA (2003) Fuzzy temporal characteristics of operations for project management on the network models basis. Eur J Oper Res 147:253–265MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Söderholm A (2008) Project management of unexpected events. Int J Project Manage 26:80–86CrossRefGoogle Scholar
  51. 51.
    Verma R, Biswal M, Biswas A (1997) Fuzzy programming technique to solve multi-objective transportation problems with some non-linear membership functions. Fuzzy Sets Syst 91:37–43MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Wang RC, Liang TF (2006) Application of multiple fuzzy goals programming to project management decisions. Int J Ind EngTheory Appl Pract 13:219–228Google Scholar
  53. 53.
    Watada J (1997) Fuzzy portfolio selection and its applications to decision making. Tatra Mt Math Publ 13:219–248MathSciNetzbMATHGoogle Scholar
  54. 54.
    Węglarz J, Józefowska J, Mika M, Waligóra G (2011) Project scheduling with finite or infinite number of activity processing modes—a survey. Eur J Oper Res 208:177–205MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Yakhchali SH (2012) A path enumeration approach for the analysis of critical activities in fuzzy networks. Inf Sci 204:23–35CrossRefzbMATHGoogle Scholar
  56. 56.
    Yakhchali SH, Ghodsypour SH (2010) Computing latest starting times of activities in interval-valued networks with minimal time lags. Eur J Oper Res 200:874–880CrossRefzbMATHGoogle Scholar
  57. 57.
    Yang M-F, Lin Y (2013) Applying fuzzy multi-objective linear programming to project management decisions with the interactive two-phase method. Comput Ind Eng 66:1061–1069CrossRefGoogle Scholar
  58. 58.
    Yazenin A (1987) Fuzzy and stochastic programming. Fuzzy Sets Syst 22:171–180MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353CrossRefzbMATHGoogle Scholar
  60. 60.
    Zammori FA, Braglia M, Frosolini M (2009) A fuzzy multi-criteria approach for critical path definition. Int J Project Manage 27:278–291CrossRefGoogle Scholar

Copyright information

© The Natural Computing Applications Forum 2016

Authors and Affiliations

  1. 1.Young Researchers and Elite Club, Sari BranchIslamic Azad UniversitySariIran
  2. 2.Center for Product Design and Manufacturing, Department of Mechanical Engineering, Faculty of EngineeringUniversity of MalayaLembah PantaiMalaysia
  3. 3.Institute of Production and Supply Chain ManagementTechnische Universität DarmstadtDarmstadtGermany
  4. 4.Department of Business and Industrial EngineeringUniversity of Applied Sciences Würzburg-SchweinfurtSchweinfurtGermany

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