Abstract
Capital rationing is a major problem in managerial decision making. The classical mathematical formulation of the problem relies on a multi-dimensional knapsack model with known input parameters. Since capital rationing is carried out in conditions where uncertainty is the rule rather than the exception, the hypothesis of deterministic data limits the applicability of deterministic formulations in real settings. This paper proposes a stochastic version of the capital rationing problem which explicitly accounts for uncertainty. In particular, a mathematical formulation is provided in the framework of stochastic programming with joint probabilistic constraints and a novel solution approach is proposed. The basic model is also extended to include specific risk measures. Preliminary computational results are presented and discussed.
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Ahmed, S.: Convexity and decomposition of mean-risk stochastic programs. Math. Program. 106, 433–446 (2006)
Artzner, P., Delbaen, F., Eber, J.-M., Heath, D.: Coherent measures of risk. Math. Finance 9(3), 203–228 (1999)
Baumol, W., Quandt, R.: Investment and discount rates under capital rationing—a programming approach. Econom. J. 75, 317–329 (1965)
Beraldi, P., Bruni, M.E.: An exact approach for solving integer problems under probabilistic constraints with random technology matrix. Ann. Oper. Res. doi:10.1007/s10479-009-0670-9
Beraldi, P., Bruni, M.E.: Capital rationing problem under uncertainty. Technical Report ParCoLab, DEIS, University of Calabria (2009)
Beraldi, P., Ruszczyński, A.: The probabilistic set covering problem. Oper. Res. 50, 956–967 (2002)
Beraldi, P., Ruszczyński, A.: A branch and bound method for stochastic integer problems under probabilistic constraints. Optim. Methods Softw. 17, 359–382 (2002)
Bisschop, J., Roelofs, M.: AIMMS. 3.7 User’s guide. Paragon Decision Technology B.V., The Netherlands (2006)
Bonami, P., Lejeune, M.: An exact solution approach for portfolio optimization problems under stochastic and integer constraints. Oper. Res. 57(3), 650–670 (2009)
Byrne, R., Charnes, A., Cooper, W.W., Kortanek, K.: Some new approaches to risk. Account. Rev. 43, 18–37 (1968)
Chandra, A., Menon, N.M., Mishra, B.K.: Budgeting for information technology. Int. J. of Account. Inf. Syst. 8, 264–282 (2007)
Costa, A., Paixao, J.M.: An approximate solution approach for a scenario based capital budgeting problem. Comput. Manag. Sci. doi:10.1007/s10287-009-0117-4
CPLEX. ILOG, C.P.L.E.X.: 6.5: Users manual. CPLEX Optimization, Inc., Incline Village, NV (1999)
Eldenburg, L., Krishnan, R.: Management accounting and control in health care: an economics perspective. In: Handb. of Manag. Account. Res, vol. 2, pp. 859–883 (2006)
Kim, D.: Capital budgeting for new projects: on the role of auditing in information acquisition. J. Account. Econ. 41, 257–270 (2006)
Kira, D., Kusy, M., Ratta, I.: The effect of project risk on capital rationing under uncertainty. Eng. Econ. 45, 37–55 (2000)
Levy, H.: Stochastic dominance and expected utility: survey and analysis. Manag. Sci. 38, 555–593 (1992)
Lorie, J.H., Savage, L.J.: Three problems in capital rationing. J. Bus. 28, 229–239 (1955)
Luedtke, J.: An integer programming decomposition approach for optimization with probabilistic constraints. In: The 20th International Symposium of Mathematical Programming, Chicago, IL (2009)
Luedtke, J., Ahmed, S.: A sample approximation approach for optimization with probabilistic constraints. Math. Program. 100, 589–611 (2008)
Luedtke, J., Ahmed, S., Nemhauser, G.: An integer programming approach for linear programs with probabilistic constraints. Math. Program. 122(2), 247–272 (2010)
Markovitz, H.M.: Mean-Variance Analysis in Portfolio Choice and Capital Markets. Blackwell, Oxford (1987)
Meier, H., Christofides, N., Salkin, G.: Capital budgeting under uncertainty - an integrated approach using contingent claims analysis and integer programming. Oper. Res. 49, 196–206 (2001)
Myers, S.C.: A note on linear programming and capital budgeting. J. Finance 27, 89–92 (1972)
Ogryczak, W., Ruszczynski, A.: On consistency of stochastic dominance and mean-semideviation models. Math. Program., Ser. B 89, 217–232 (2001)
Ogryczak, W., Ruszczynski, A.: Dual stochastic dominance and related mean–risk models. SIAM J. Optim. 13, 60–78 (2002)
Padberg, M., Wilczak, M.: Optimal project selection when borrowing and lending rates differ. Math. Comput. Model. 29, 63–78 (1999)
Pagnoncelli, B.K. Ahmed, Shapiro, A.: Sample average approximation method for chance constrained programming: theory and applications. J. Optim. Theory Appl. 142, 399–416 (2009) doi:10.1007/s10957-009-9523-6
Pinter, J.: Deterministic approximations of probability inequalities. ZOR, Z. Oper.-Res. 33, 219–239 (1989)
Porter, R.B.: Semivariance and stochastic dominance: a comparison. Am. Econ. Rev. 64, 200–204 (1974)
Prékopa, A.: Dual method for a one-stage stochastic programming with random rhs obeying a discrete probability distribution. ZOR, Z. Oper.-Res. 34, 441–461 (1990)
Prékopa, A.: Stochastic Programming. Kluwer, Boston (1995)
Ruszczyski, A.: Probabilistic programming with discrete distributions and precedence constrained knapsack polyhedra. Math. Program. 93, 195–215 (2002)
Ruszczynski, A., Shapiro, A.: Stochastic Programming. Handbook in Operations Research and Management Science. Elsevier, Amsterdam (2000)
Sarper, H.: Capital rationing under risk: a chance constrained approach using uniformly distributed cash flows and budgets. Eng. Econ. 39, 49–76 (1993)
Seda, M.: Solving resource-constrained project scheduling problem as a sequence of multi-knapsack problems. WSEAS Trans. Inf. Sci. Appl. 3, 1785–1791 (2006)
Weingartner, H.M.: Mathematical Programming and the Analysis of Capital Budgeting Problems. Prentice-Hall, Englewood Cliffs (1963)
Weingartner, H.M.: Capital budgeting of interrelated project: survey and synthesis. Manag. Sci. 12, 485–516 (1966)
Weingartner, H.M.: Criteria for programming investment project selection. J. Ind. Econ. 11, 65–76 (1966)
Zenios, S.A., Ziemba, W.: Handbook on Asset and Liability Management, vol. B: Applications and Case Studies. North-Holland, Amsterdam (2006)
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Beraldi, P., Bruni, M.E. & Violi, A. Capital rationing problems under uncertainty and risk. Comput Optim Appl 51, 1375–1396 (2012). https://doi.org/10.1007/s10589-010-9390-y
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DOI: https://doi.org/10.1007/s10589-010-9390-y