Expected shortfall for the makespan in activity networks under imperfect information

  • Carlo MeloniEmail author
  • Marco Pranzo


This paper deals with the evaluation of the expected shortfall or the conditional value-at-risk for the makespan in scheduling problems represented as temporal networks under incomplete and uncertain information. We consider temporal activity network representations of scheduling problems affected by uncertainties related to the activity durations and we assume that for these uncertainties only incomplete or imperfect information is available. More precisely, for each activity only the interval for its integer valued duration is known to the scheduler. We address the evaluation of the expected shortfall associated to a feasible schedule discussing its importance in scheduling applications. We propose lower and upper bounds, heuristics to determine a fast computational estimation of the expected shortfall, and an exact method for a class of activity networks. The experimental results show that the proposed method can enable to use the expected shortfall as optimization criterion for wide classes of scheduling approaches considering risk-aversion in different practical contexts.


Expected shortfall CVaR Project scheduling Makespan Uncertainty Activity networks 



  1. Artigues C, Leus R, Talla Nobibon F (2013) Robust optimization for resource-constrained project scheduling with uncertain activity durations. Flex Serv Manuf J 25:175–205CrossRefGoogle Scholar
  2. Atakan S, Bülbül K, Noyan N (2017) Minimizing value-at-risk in single-machine scheduling. Ann Oper Res 248(1–2):25–73MathSciNetzbMATHCrossRefGoogle Scholar
  3. Baker KR (2014) Setting optimal due dates in a basic safe-scheduling model. Comput Oper Res 41:109–114MathSciNetzbMATHCrossRefGoogle Scholar
  4. Baker K, Trietsch D (2009) Principles of sequencing and scheduling. Wiley, New YorkzbMATHCrossRefGoogle Scholar
  5. Bang-Jensen J, Gutin G (2008) Digraphs: theory, algorithms and applications, 2nd edn. Springer, LondonzbMATHGoogle Scholar
  6. Bein WM, Kamburowski J, Stallmann MFM (1992) Optimal reduction of two-terminal directed acyclic graphs. SIAM J Comput 21(6):1112–1129MathSciNetzbMATHCrossRefGoogle Scholar
  7. Bertsimas D, Lauprete GJ, Samarov A (2004) Shortfall as a risk measure: properties, optimization and applications. J Econ Dyn Control 28(7):1353–382MathSciNetzbMATHCrossRefGoogle Scholar
  8. Canon LC, Jeannot E (2010) Evaluation and optimization of the robustness of DAG schedules in heterogeneous environments. IEEE Trans Parallel Distrib Syst 21(4):532–546CrossRefGoogle Scholar
  9. Catalão JPS, Pousinho HMI, Contreras J (2012) Optimal hydro scheduling and offering strategies considering price uncertainty and risk management. Energy 37:237–244CrossRefGoogle Scholar
  10. Chanas S, Zieliński P (2002) The computational complexity of the critical problems in a network with interval activity times. Eur J Oper Res 136:541–550zbMATHCrossRefGoogle Scholar
  11. Chanas S, Dubois D, Zieliński P (2002) On the sure criticality of tasks in activity networks with imprecise durations. IEEE Trans Syst Man Cybern B Cybern 32(4):393–407CrossRefGoogle Scholar
  12. Chang Z, Song S, Zhang Y, Ding J-Y, Zhang R, Chiong R (2017) Distributionally robust single machine scheduling with risk aversion. Eur J Oper Res 256:261–274MathSciNetzbMATHCrossRefGoogle Scholar
  13. Crabill T, Maxwell W (1969) Single machine sequencing with random processing times and random due-dates. Nav Res Logist Q 16:549–555MathSciNetzbMATHCrossRefGoogle Scholar
  14. Damelin S, Miller W (2011) The mathematics of signal processing. Cambridge University Press, CambridgezbMATHCrossRefGoogle Scholar
  15. Daniels RL, Kouvelis P (1995) Robust scheduling to hedge against processing time uncertainty in single-stage production. Manag Sci 41(2):363–376zbMATHCrossRefGoogle Scholar
  16. De P, Ghosh JB, Wells CE (1992) Expectation-variance analyss of job sequences under processing time uncertainty. Int J Prod Econ 28(3):289–297CrossRefGoogle Scholar
  17. Demeulemeester EL, Herroelen WS (2002) Project scheduling a research handbook. Kluwer Academic, DordrechtzbMATHGoogle Scholar
  18. De Reyck B, Demeulemeester E, Herroelen W (1999) Algorithms for scheduling projects with generalized precedence relations. In: Wȩglarz J (ed) Project scheduling, vol 14. International series in operations research and management science. Springer, BostonzbMATHCrossRefGoogle Scholar
  19. Elmaghraby SE (1977) Activity networks: project planning and control by network models. Wiley, New YorkzbMATHGoogle Scholar
  20. Elmaghraby SE (1990) Project bidding under deterministic and probabilistic activity durations. Eur J Oper Res 49:14–34zbMATHCrossRefGoogle Scholar
  21. Elmaghraby SE (2005) On the fallacy of averages in project risk management. Eur J Oper Res 165(2):307–313MathSciNetzbMATHCrossRefGoogle Scholar
  22. Fang C, Kolisch R, Wang L, Mu C (2015) An estimation of distribution algorithm and new computational results for the stochastic resource-constrained project scheduling problem. Flex Serv Manuf J 27(4):585–605CrossRefGoogle Scholar
  23. Framinan JM, Leisten R, Ruiz García R (2014) Manufacturing scheduling systems. An integrated view on models, methods and tools. Springer, LondonzbMATHCrossRefGoogle Scholar
  24. Fulkerson DR (1962) Expected critical path lengths in PERT networks. Oper Res 10:808–817zbMATHCrossRefGoogle Scholar
  25. García-González J, Parrilla E, Mateo A (2007) Risk-averse profit-based optimal scheduling of a hydro-chain in the day-ahead electricity market. Eur J Oper Res 181:1354–1369zbMATHCrossRefGoogle Scholar
  26. Hagstrom JN (1988) Computational complexity of PERT problems. Networks 18(2):139–147MathSciNetCrossRefGoogle Scholar
  27. Hall NG (2016) Research and teaching opportunities in project management. INFORMS Tutor Oper Res.
  28. Hartmann S, Kolisch R (2000) Experimental evaluation of state-of-the-art heuristics for resource constrained project scheduling. Eur J Oper Res 127(2):394–407zbMATHCrossRefGoogle Scholar
  29. Herroelen WS, Leus R (2005) Project scheduling under uncertainty: survey and research potentials. Eur J Oper Res 165(2):289–306zbMATHCrossRefGoogle Scholar
  30. Ivanescu CV, Fransoo JC, Bertrand JWM (2002) Makespan estimation and order acceptance in batch process industries when processing times are uncertain. OR Spectrum 24:467–495zbMATHCrossRefGoogle Scholar
  31. Kalinchenko K, Veremyev A, Boginski V, Jeffcoat DE, Uryasev S (2011) Robust connectivity issues in dynamic sensor networks for area surveillance under uncertainty. Pac J Optim 7:235–248MathSciNetzbMATHGoogle Scholar
  32. Kelley JE Jr (1961) Critical-path planning and scheduling: mathematical basis. Oper Res 9(3):296–320MathSciNetzbMATHCrossRefGoogle Scholar
  33. Kolisch R, Hartmann S (2006) Experimental investigation of heuristics for resource-constrained project scheduling: an update. Eur J Oper Res 174(1):23–37zbMATHCrossRefGoogle Scholar
  34. Kolisch R, Sprecher A (1996) PSPLIB—a project scheduling library. Eur J Oper Res 96:205–216zbMATHCrossRefGoogle Scholar
  35. Lai T-C, Sotskov YN (1999) Sequencing with uncertain numerical data for makespan minimisation. J Oper Res Soc 50:230–243zbMATHCrossRefGoogle Scholar
  36. Larsen R, Pranzo M (2019) A framework for dynamic rescheduling problems. Int J Prod Res 57(1):16–33CrossRefGoogle Scholar
  37. Lawrence SR, Sewell EC (1997) Heuristic, optimal, static, and dynamic schedules when processing times are uncertain. J Oper Manag 15:71–82CrossRefGoogle Scholar
  38. Li Z, Ierapetritou M (2008) Process scheduling under uncertainty: review and challenges. Comput Chem Eng 32:715–727CrossRefGoogle Scholar
  39. Luh PB, Chen D, Thakur LS (1999) An effective approach for job-shop scheduling with uncertain processing requirements. IEEE Trans Robot Autom 15(2):715–727CrossRefGoogle Scholar
  40. Meloni C, Pacciarelli D, Pranzo M (2004) A rollout metaheuristic for job shop scheduling problems. Ann Oper Res 131(1–4):215–235MathSciNetzbMATHCrossRefGoogle Scholar
  41. Pinedo M (2001) Scheduling: theory, algorithms, and systems, 2nd edn. Prentice Hall, Upper Saddle, NJGoogle Scholar
  42. Pranzo M, Meloni C, Pacciarelli D (2003) A new class of greedy heuristics for job shop scheduling problems. Lect Notes Comput Sci 2647:223–236MathSciNetzbMATHCrossRefGoogle Scholar
  43. Pranzo M, Pacciarelli D (2016) An iterated greedy metaheuristic for the blocking job shop scheduling problem. J Heurist 22(4):587–611CrossRefGoogle Scholar
  44. Ramponi FA, Campi MC (2017) Expected shortfall: heuristics and certificates. Eur J Oper Res 267(3):1003–1013MathSciNetzbMATHCrossRefGoogle Scholar
  45. Rockafellar RT, Uryasev S (2000) Optimization of conditional value-at-risk. J Risk 2(3):21–41CrossRefGoogle Scholar
  46. Rockafeller RT (2007) Coherent approaches to risk in optimization under uncertainty. INFORMS Tutor Oper Res.
  47. Rothkopf MH (1966) Scheduling with random service times. Manag Sci 12(9):707–713MathSciNetzbMATHCrossRefGoogle Scholar
  48. Sabuncuoglu I, Bayiz M (2000) Analysis of reactive scheduling problems in a job shop environment. Eur J Oper Res 126:567–586zbMATHCrossRefGoogle Scholar
  49. Sarin SC, Nagarajan B, Liao L (2010) Stochastic scheduling: expectation-variance analysis of a schedule. Cambridge University Press, New YorkzbMATHCrossRefGoogle Scholar
  50. Sarin SC, Sherali HD, Liao L (2014) Minimizing conditional-value-at-risk for stochastic scheduling problems. J Sched 17:5–15MathSciNetzbMATHCrossRefGoogle Scholar
  51. Słowiński R, Hapke M (eds) (2010) Scheduling under fuzziness. Physica-Verlag, HeidelbergzbMATHGoogle Scholar
  52. Szelke E, Kerr RM (1994) Knowledge-based reactive scheduling. Prod Plan Control 5(2):124–145CrossRefGoogle Scholar
  53. Tao L, Wu DD, Liu S, Dolgui A (2018) Optimal due date quoting for a risk-averse decision-maker under CVaR. Int J Prod Res 56(5):1934–1959CrossRefGoogle Scholar
  54. Urgo M, Váncza J (2018) A branch-and-bound approach for the single machine maximum lateness stochastic scheduling problem to minimize the value-at-risk. Flex Serv Manuf J. CrossRefGoogle Scholar
  55. Vieira GE, Herrmann JW, Lin E (2003) Rescheduling manufacturing systems: a framework of strategies, policies and methods. J Sched 6:39–62MathSciNetzbMATHCrossRefGoogle Scholar
  56. Wiesemann W (2012) Optimization of temporal networks under uncertainty. Springer, HeidelbergCrossRefGoogle Scholar
  57. Wu CW, Brown KN, Beck JC (2009) Scheduling with uncertain durations: modeling \(\beta\)-robust scheduling with constraints. Comput Oper Res 36:2348–2356MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Dipartimento di Ingegneria Elettrica e dell’InformazionePolitecnico di BariBariItaly
  2. 2.Dipartimento di Ingegneria dell’Informazione e Scienze MatematicheUniversità di SienaSienaItaly

Personalised recommendations