Mathematical Programming

, Volume 135, Issue 1–2, pp 437–471 | Cite as

Robust resource allocations in temporal networks

  • Wolfram WiesemannEmail author
  • Daniel Kuhn
  • Berç Rustem
Full Length Paper Series A


Temporal networks describe workflows of time-consuming tasks whose processing order is constrained by precedence relations. In many cases, the durations of the network tasks can be influenced by the assignment of resources. This leads to the problem of selecting an ‘optimal’ resource allocation, where optimality is measured by network characteristics such as the makespan (i.e., the time required to complete all tasks). In this paper we study a robust resource allocation problem where the task durations are uncertain, and the goal is to minimise the worst-case makespan. We show that this problem is generically \({\mathcal{NP}}\) -hard. We then develop convergent bounds on the optimal objective value, as well as feasible allocations whose objective values are bracketed by these bounds. Numerical results provide empirical support for the proposed method.


Robust optimisation Temporal networks Resource allocation problem 

Mathematics Subject Classification (2010)

90-02 90B15 90C25 


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  1. 1.
    Adlakha V.G., Kulkarni V.G.: A classified bibliography of research on stochastic pert networks: 1966–1987. INFOR 27(3), 272–296 (1989)zbMATHGoogle Scholar
  2. 2.
    Atamtürk A., Zhang M.: Two-stage robust network flow and design under demand uncertainty. Oper. Res. 55(4), 662–673 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Averbakh I.: On the complexity of a class of combinatorial optimization problems with uncertainty. Math. Program. 90(2), 263–272 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Ben-Tal A., El Ghaoui L., Nemirovski A.: Robust Optimization. Princeton University Press, Princeton (2009)zbMATHGoogle Scholar
  5. 5.
    Bertsimas, D., Natarajan, K., Teo, C.-P.: Applications of semidefinite optimization in stochastic project scheduling. Technical report, High Performance Computation for Engineered Systems, Singapore–MIT Alliance (2002)Google Scholar
  6. 6.
    Bertsimas D., Sim M.: Robust discrete optimization and network flows. Math. Program. 98(1–3), 49–71 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Bertsimas D., Sim M.: Tractable approximations to robust conic optimization problem. Math. Program. 107(1–2), 5–36 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Birge J.R., Maddox M.J.: Bounds on expected project tardiness. Oper. Res. 43(5), 838–850 (1995)zbMATHCrossRefGoogle Scholar
  9. 9.
    Boyd S.P., Kim S.-J., Patil D.D., Horowitz M.A.: Digital circuit optimization via geometric programming. Oper. Res. 53(6), 899–932 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Brucker P., Drexl A., Möhring R., Neumann K., Pesch E.: Resource-constrained project scheduling: notation, classification, models, and methods. Eur. J. Oper. Res. 112(1), 3–41 (1999)zbMATHCrossRefGoogle Scholar
  11. 11.
    Calafiore G., Campi M.C.: Uncertain convex programs: randomized solutions and confidence levels. Math. Program. 102(1), 25–46 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Chen X., Sim M., Sun P.: A robust optimization perspective on stochastic programming. Oper. Res. 55(6), 1058–1071 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Chen X., Sim M., Sun P., Zhang J.: A linear-decision based approximation approach to stochastic programming. Oper. Res. 56(2), 344–357 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Cohen I., Golany B., Shtub A.: The stochastic time-cost tradeoff problem: a robust optimization approach. Networks 49(2), 175–188 (2006)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Demeulemeester E., Vanhoucke M., Herroelen W.: RanGen: a random network generator for activity-on-the-node networks. J. Schedul. 6(1), 17–38 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Demeulemeester E.L., Herroelen W.S.: Project Scheduling—A Research Handbook. Kluwer, Dordrecht (2002)zbMATHGoogle Scholar
  17. 17.
    Elmaghraby S.E.: On criticality and sensitivity in activity networks. Eur. J. Oper. Res. 127(2), 220–238 (2000)zbMATHCrossRefGoogle Scholar
  18. 18.
    Eppstein, D.: Finding the k shortest paths. In: IEEE Symposium on Foundations of Computer Science, pp. 154–165 (1994)Google Scholar
  19. 19.
    Erera A.L., Morales J.C., Savelsbergh M.: Robust optimization for empty repositioning problems. Oper. Res. 57(2), 468–483 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Feige, U., Jain, K., Mahdian, M., Mirrokni, V.: Robust combinatorial optimization with exponential scenarios. In: Fischetti, M., Williamson, D.P. (eds.) Integer Programming and Combinatorial Optimization, Lecture Notes in Computer Science, pp. 439–453. Springer: Berlin (2007)Google Scholar
  21. 21.
    Fulkerson D.R.: A network flow computation for project cost curves. Manage. Sci. 7(2), 167–178 (1961)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Garey M.R., Johnson D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co, San Francisco (1979)zbMATHGoogle Scholar
  23. 23.
    Goel V., Grossmann I.E.: A class of stochastic programs with decision dependent uncertainty. Math. Program. 108(2–3), 355–394 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Hagstrom J.N.: Computational complexity of PERT problems. Networks 18(2), 139–147 (1988)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Hettich R., Kortanek K.O.: Semi-infinite programming: theory, methods, and applications. SIAM Rev. 35(3), 380–429 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Horst R., Pardalos P.M., Thoai N.V.: Introduction to Global Optimization, 2nd edn. Kluwer, Dordrecht (2000)zbMATHGoogle Scholar
  27. 27.
    Ibaraki T.: Approximate algorithms for the multiple-choice continuous knapsack problems. J. Oper. Res. Soc. Japan 23(1), 28–63 (1980)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Janak S.L., Lin X., Floudas C.A.: A new robust optimization approach for scheduling under uncertainty: II. uncertainty with known probability distribution. Comput. Chem. Eng. 31(3), 171–195 (2007)CrossRefGoogle Scholar
  29. 29.
    Kelley J.E. Jr: Critial-path planning and scheduling: mathematical basis. Oper. Res. 9(3), 296–320 (1961)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Khandekar, R., Kortsarz, G., Mirrokni, V., Salavatipour, M.R.: Two-stage robust network design with exponential scenarios. In: Halperin, D., Mehlhorn, K. (eds.) Algorithms—ESA 2008, Lecture Notes in Computer Science, pp. 589–600. Springer, Berlin (2008)Google Scholar
  31. 31.
    Kouvelis P., Yu G.: Robust Discrete Optimization and its Applications. Kluwer, Dordrecht (1997)zbMATHGoogle Scholar
  32. 32.
    Kwow Y.-K., Ahmad I.: Static scheduling algorithms for allocating directed task graphs to multiprocessors. ACM Comput. Surveys 31(4), 406–471 (1999)CrossRefGoogle Scholar
  33. 33.
    Li Z., Ierapetritou M.: Process scheduling under uncertainty: review and challenges. Comput. Chem. Eng. 32(4–5), 715–727 (2008)CrossRefGoogle Scholar
  34. 34.
    Liebchen C., Lübbecke M.E., Möhring R.H., Stiller S.: The concept of recoverable robustness, linear programming recovery, and railway applications. In: Ahuja, R.K., Möhring, R.H., Zaroliagis, C.D. (eds) Robust and Online Large-Scale Optimization, Lecture Notes in Computer Science, pp. 1–27. Springer, Berlin (2009)CrossRefGoogle Scholar
  35. 35.
    Limongelli, C., Pirastu, R.: Exact solution of linear equation systems over rational numbers by parallel p-adic arithmetic. RISC Report Series 94–25, University of Linz, Austria (1994)Google Scholar
  36. 36.
    Lin X., Janak S.L., Floudas C.A.: A new robust optimization approach for scheduling under uncertainty: I. Bounded uncertainty. Comput. Chem. Eng. 28(6–7), 1069–1085 (2004)CrossRefGoogle Scholar
  37. 37.
    Ludwig A., Möhring R.H., Stork F.: A computational study on bounding the makespan distribution in stochastic project networks. Ann. Oper. Res. 102, 49–64 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Malcolm D.G., Roseboom J.H., Clark C.E., Fazar W.: Application of a technique for research and development program evaluation. Oper. Res. 7(4), 646–669 (1959)CrossRefGoogle Scholar
  39. 39.
    Meilijson I., Nádas A.: Convex majorization with an application to the length of critical paths. J. Appl. Probab. 16(3), 671–677 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Möhring R.H.: Scheduling under uncertainty: bounding the makespan distribution. In: Alt, H. (ed) Conputational Discrete Mathematics, Lecture Notes in Computer Science., pp. 79–97. Springer, Berlin (2001)Google Scholar
  41. 41.
    Mulvey J.M., Vanderbei R.J., Zenios S.A.: Robust optimization of large-scale systems. Oper Res. 43(2), 264–281 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Neumann K.: Scheduling of projects with stochastic evolution structure. In: Weglarz, J. (eds) Project Scheduling: Recent Models, Algorithms, and Applications, pp. 309–332. Kluwer, Dordrecht (1999)Google Scholar
  43. 43.
    Ordóñez F., Zhao J.: Robust capacity expansion of network flows. Networks 50(2), 136– (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Ruszczyński, A., Shapiro, A. (eds): Stochastic Programming. Elsevier, Amsterdam (2003)zbMATHGoogle Scholar
  45. 45.
    Sahinidis N.V.: Optimization under uncertainty: State-of-the-art and opportunities. Comput. Chem. Eng. 28(6–7), 971–983 (2004)CrossRefGoogle Scholar
  46. 46.
    Schwindt C.: Resource Allocation in Project Management. Springer, Berlin (2005)Google Scholar
  47. 47.
    Shabtay D., Steiner G.: A survey of scheduling with controllable processing times. Discr. Appl. Math. 155(13), 1643–1666 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Stiller, S.: Extending Concepts of Reliability. Network Creation Games, Real-time Scheduling, and Robust Optimization. PhD thesis, Institut für Mathematik, Technische Universität Berlin (2009)Google Scholar
  49. 49.
    Wiesemann, W., Kuhn, D., Rustem, B.: Multi-resource allocation in stochastic project scheduling. Ann. Oper. Res. (2008, in press)Google Scholar
  50. 50.
    Zeng L., Benatallah B., Ngu A.H.H., Dumas M., Kalagnanam J., Chang H.: QoS-Aware middleware for web services composition. IEEE Tran. Softw. Eng. 30(5), 311–327 (2004)CrossRefGoogle Scholar

Copyright information

© Springer and Mathematical Optimization Society 2011

Authors and Affiliations

  1. 1.Imperial College of Science, Technology and MedicineLondonUK

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