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A Graphical Approach to Solve an Investment Optimization Problem

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Journal of Mathematical Modelling and Algorithms in Operations Research

Abstract

We consider a project investment problem, where a set of projects and an overall budget are given. For each project, a piecewise linear profit function is known which describes the profit obtained if a specific amount is invested into this project. The objective is to determine the amount invested into each project such that the overall budget is not exceeded and the total profit is maximized. For this problem, a graphical algorithm (GrA) is presented which is based on the same Bellman equations as the best known dynamic programming algorithm (DPA) but the GrA has several advantages in comparison with the DPA. Based on this GrA, a fully-polynomial time approximation scheme is proposed having the best known running time. The idea of the GrA presented can also be used to solve some similar scheduling or lot-sizing problems in a more effective way, e.g., the related problem of finding lot-sizes and sequencing several products on a single imperfect machine.

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References

  1. Lazarev, A.A., Werner, F.: A graphical realization of the dynamic programming method for solving NP-hard problems. Comput. Math. Appl. 58(4), 619–631 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Gafarov, E.R., Dolgui, A., Werner, F.: Dynamic programming approach to design FPTAS for single machine scheduling problems. Research Report LIMOS UMR CNRS 6158 (2012)

  3. Kellerer, H., Pferschy, U., Pisinger, D.: Knapsack Problems. Springer-Verlag, Berlin (2004)

    Book  MATH  Google Scholar 

  4. Shaw, D.X., Wagelmans, A.P.M.: An algorithm for single-item capacitated economic lot sizing with piecewise linear production costs and general holding costs. Manag. Sci. 44(6), 831–838 (1998)

    Article  MATH  Google Scholar 

  5. Kameshwaran, S., Narahari, Y.: Nonconvex piecewise linear knapsack problems. Eur. J. Oper. Res. 192, 56–68 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chubanov, S., Kovalyov, M.Y., Pesch, E.: An FPTAS for a single-item capacitated economic lot-sizing problem with monotone cost structure. Math. Program. Ser. A 106, 453–466 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dolgui, A., Kovalyov, M.Y., Shchamialiova, K.: Multi-product lot-sizing and sequencing on a single imperfect machine. Comput. Optim. Appl. 50, 465–482 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Schmeleva, K., Delorme, X., Dolgui, A., Grimaud, F., Kovalyov, M.Y.: Lot-sizing on a single machine, ILP models. Comput. Ind. Eng. 65, 561–569 (2013)

    Article  Google Scholar 

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Authors and Affiliations

  1. Ecole Nationale Superieure des Mines, FAYOL-EMSE, CNRS:UMR6158, LIMOS, 42023, Saint-Etienne, France

    Evgeny R. Gafarov & Alexandre Dolgui

  2. Institute of Control Sciences of the Russian Academy of Sciences, Profsoyuznaya st. 65, 117997, Moscow, Russia

    Evgeny R. Gafarov & Alexander A. Lazarev

  3. Higher School of Economics (National Research University), Moscow, Russia

    Alexander A. Lazarev

  4. Lomonosov Moscow State University, Moscow, Russia

    Alexander A. Lazarev

  5. Moscow Institute of Physics and Technology (State University), Moscow, Russia

    Alexander A. Lazarev

  6. Fakultät fu̇r Mathematik, Otto-von-Guericke-Universität Magdeburg, PSF 4120, 39016, Magdeburg, Germany

    Frank Werner

Authors
  1. Evgeny R. Gafarov
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  2. Alexandre Dolgui
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  3. Alexander A. Lazarev
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  4. Frank Werner
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Corresponding author

Correspondence to Alexandre Dolgui.

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Gafarov, E.R., Dolgui, A., Lazarev, A.A. et al. A Graphical Approach to Solve an Investment Optimization Problem. J Math Model Algor 13, 597–614 (2014). https://doi.org/10.1007/s10852-013-9248-2

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  • DOI: https://doi.org/10.1007/s10852-013-9248-2

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