Advertisement

Mathematical Methods of Operations Research

, Volume 76, Issue 3, pp 289–319 | Cite as

The opportunistic replacement problem: theoretical analyses and numerical tests

  • Torgny Almgren
  • Niclas Andréasson
  • Michael Patriksson
  • Ann-Brith Strömberg
  • Adam WojciechowskiEmail author
  • Magnus Önnheim
Original Article

Abstract

We consider a model for determining optimal opportunistic maintenance schedules w.r.t. a maximum replacement interval. This problem generalizes that of Dickman et al. (J Oper Res Soc India 28:165–175, 1991) and is a natural starting point for modelling replacement schedules of more complex systems. We show that this basic opportunistic replacement problem is NP-hard, that the convex hull of the set of feasible replacement schedules is full-dimensional, that all the inequalities of the model are facet-inducing, and present a new class of facets obtained through a \({\{0, \frac{1}{2}\}}\) -Chvátal–Gomory rounding. For costs monotone with time, a class of elimination constraints is introduced to reduce the computation time; it allows maintenance only when the replacement of at least one component is necessary. For costs decreasing with time, these constraints eliminate non-optimal solutions. When maintenance occasions are fixed, the remaining problem is stated as a linear program and solved by a greedy procedure. Results from a case study on aircraft engine maintenance illustrate the advantage of the optimization model over simpler policies. We include the new class of facets in a branch-and-cut framework and note a decrease in the number of branch-and-bound nodes and simplex iterations for most instance classes with time dependent costs. For instance classes with time independent costs and few components the elimination constraints are used favorably. For fixed maintenance occasions the greedy procedure reduces the computation time as compared with linear programming techniques for all instances tested.

Keywords

Maintenance optimization Mixed integer programming Complexity analysis Polyhedral analysis 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Andréasson N (2004) Optimization of opportunistic replacement activities in deterministic and stochastic multi-component systems. Licentiate thesis. Department of Mathematics, Chalmers University of Technology and Göteborg University Göteborg, SwedenGoogle Scholar
  2. Barlow R, Proschan F (1965) Mathematical theory of reliability. Wiley, New YorkzbMATHGoogle Scholar
  3. Besnard F, Patriksson M, Strömberg A-B, Wojciechowski A, Bertling L (2009) An optimization framework for opportunistic maintenance of offshore wind power systems. In: Proceedings of IEEE PowerTech2009 conference, pp 2970–2976Google Scholar
  4. Bohlin M, Doganay K, Kreuger P, Steinert R, Wärja M (2010) Searching for gas turbine maintenance schedules.. AI Mag 31(1): 21–36Google Scholar
  5. Caprara A, Fischetti M (1996) Chvátal–Gomory cuts. Math Program 74: 221–235MathSciNetzbMATHGoogle Scholar
  6. Cigolini R, Fedele L, Garetti M, Macchi M (2008) Recent advances in maintenance and facility management. Prod Plan Control 19: 279–286CrossRefGoogle Scholar
  7. Corio MR, Costantini LP (1989) Frequency and severity of forced outages immediately following planned or maintenance outages. In: Generating availability trends summary report. North American Electric Reliability CouncilGoogle Scholar
  8. Day JA, George LL (1982) Opportunistic replacement of fusion power system parts. Conference presentation, the reliability and maintainability symposium, Los Angeles, CA, USA, January 26–28Google Scholar
  9. Dekker R, Wildeman RE, van der Duyn Schouten FA (1997) A review of multi-component maintenance models with economic dependence. Math Methods Oper Res 45: 411–435MathSciNetzbMATHCrossRefGoogle Scholar
  10. Dickman B, Epstein S, Wilamowsky Y (1988) A 0-1 mathematical programming formulation for multi-component deterministic opportunistic replacement. In: Naumes W, Pavan R (eds) Northeast decision sciences institute proceedings, pp 116–118. Newport, RI, USAGoogle Scholar
  11. Dickman B, Wilamowsky Y, Epstein S (1990) Modeling deterministic opportunistic replacement as an integer programming problem. Am J Math Manag Sci 10(3–4): 323–339zbMATHGoogle Scholar
  12. Dickman B, Epstein S, Wilamowsky Y (1991) A mixed integer linear programming formulation for multi-component deterministic opportunistic replacement. J Oper Res Soc India 28: 165–175zbMATHGoogle Scholar
  13. Epstein S, Wilamowsky Y (1980) A disk replacement policy for jet engines. Ann Soc Logist Eng 5: 35–36Google Scholar
  14. Epstein S, Wilamowsky Y (1982) A replacement schedule for multicomponent life-limited parts. Naval Res Logist Q 29: 685–692zbMATHCrossRefGoogle Scholar
  15. Epstein S, Wilamowsky Y (1985) Opportunistic replacement in a deterministic environment. Comput Oper Res 12(3): 311–322zbMATHCrossRefGoogle Scholar
  16. Epstein S, Wilamowsky Y (1986) An optimal replacement policy for life limited parts. J Oper Res Soc India 23: 151–163MathSciNetzbMATHGoogle Scholar
  17. Garey MR, Johnson D (1979) Computers and intractability: a guide to the theory of NP-completeness. W.H. Freeman and Company, New YorkzbMATHGoogle Scholar
  18. George LL, Lo YH (1980) An opportunistic look-ahead replacement policy. Ann Soc Logist Eng 14(4): 51–55Google Scholar
  19. Gurobi Optimization (2011) Gurobi Optimizer (2011) www.gurobi.com; visited December 19th 2011
  20. Jorgenson DW, Radner R (1960) Optimal replacement and inspection of stochastically failing equipment. Paper P-2074, Rand Corporation, Santa Monica, CA, USAGoogle Scholar
  21. Junger M, Liebling T, Naddef D, Nemhauser G, Pulleyblank W (2009) 50 years of integer programming 1958–2008: from the early years to the state-of-the-art. Springer, BerlinGoogle Scholar
  22. Mobley RK (2004) Maintenance fundamentals, 2nd edn. Elsevier Amsterdam, The NetherlandsGoogle Scholar
  23. Nemhauser GL, Wolsey LA (1988) Integer and combinatorial optimization. Wiley, New YorkzbMATHGoogle Scholar
  24. Nicolai RP, Dekker R (2008) Optimal maintenance of multi-component systems: a review. In: Kobbacy KAH, Murthy DNP (eds) Complex system maintenance handbook, Springer series in reliability engineering. Springer, Berlin, pp 263–286Google Scholar
  25. Nilsson J, Patriksson M, Strömberg A-B, Wojciechowski A, Bertling L (2009) An opportunistic maintenance optimization model for shaft seals in feed-water pump systems in nuclear power plants. In: Proceedings IEEE PowerTech2009 conference, pp 2962–2969Google Scholar
  26. Patriksson M, Strömberg, A-B, Wojciechowski A (2012a) The stochastic opportunistic replacement problem, part I: models incorporating individual component lives. Ann Oper Res. doi: 10.1007/s10479-012-1131-4
  27. Patriksson M, Strömberg A-B, Wojciechowski A (2012b) The stochastic opportunistic replacement problem, part II: a two stage solution approach. Ann Oper Res. doi: 10.1007/s10479-012-1134-1
  28. Pintelon LM, Gelders LF (1992) Maintenance management decision making. Eur J Oper Res 58: 301–317CrossRefGoogle Scholar
  29. Robertson R, Jones A (2004) Pay day. Plant Eng Maint 28: 18–25Google Scholar
  30. Svensson J (2007) Survival estimation for opportunistic maintenance. Doctoral thesis, Department of Mathematics, Chalmers University of Technology and Göteborg University, Göteborg, SwedenGoogle Scholar
  31. Xia L, Zhao Q, Jia QS (2008) A structure property of optimal policies for maintenance problems with safety-critical components. IEEE Trans Autom Sci Eng 5: 519–531CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Torgny Almgren
    • 1
  • Niclas Andréasson
    • 2
  • Michael Patriksson
    • 3
    • 4
  • Ann-Brith Strömberg
    • 3
    • 4
  • Adam Wojciechowski
    • 3
    • 4
    Email author
  • Magnus Önnheim
    • 3
    • 4
  1. 1.Volvo Aero CorporationTrollhättanSweden
  2. 2.SmålandsstenarSweden
  3. 3.Department of Mathematical SciencesChalmers University of TechnologyGothenburgSweden
  4. 4.Department of Mathematical SciencesUniversity of GothenburgGothenburgSweden

Personalised recommendations