Capital rationing problems under uncertainty and risk

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.

Threshold Boolean form for joint probabilistic constraints with random technology matrix

“…To generate the problem instances, we use the approach described by Beraldi et al [4] and decompose a planning horizon into a finite number of periods. The budget available is known for the first period, while it is stochastic for the next ones.…”

“…Such numbers of projects are typical for the capital rationing problem, as the project selection is typically preceded by a screening phase in which homogeneous groups of projects are defined and are later on subjected to a joint evaluation [7]. The project outflows ξ i j and project values V j have been randomly generated from a uniform distribution (see [4,31]) defined on [300, 600] and [10,1000], respectively. We also generate the available budgets from a uniform distribution defined on…”

Chance-Constrained Binary Packing Problems

“…An approach based on studying the corresponding mixed-integer programming formulation has been effective for this case , Küçükyavuz 2012. Methods for more general finite scenario CCSPs, where the constraint matrixà may also be random, have been studied in Ruszczyński (2002), Beraldi and Bruni (2010), Tanner and Ntaimo (2010), Luedtke (2010Luedtke ( , 2014, and Beraldi et al (2012). None of these methods exploit the integrality of the discrete decision variables if they exist, in contrast to our method for lifting probabilistic cover inequalities.…”

“…Section 9 of the online supplement provides details on the time spent generating the different types of cuts in the probabilistic cover-based formulation. We also tried a specialized branch-and-bound algorithm proposed in Beraldi et al (2012), but found that although it worked well for an instance with n = 20 items, it was unable to solve any of these larger instances. We see from Table 3 that when solving the probabilistic cover-based formulation of the individual knapsack instances, the combination of local cuts and projection cuts yields improved performance over using either set of cuts individually (shown in Table 2), now allowing all instances to be solved within the time limit.…”

A multistage stochastic programming approach for capital budgeting problems under uncertainty

“…The first one relies on the use of chance constraints and it has in Byrne et al (1968) and Sarper (1993) some of its main representatives. We also mention the recent contribution of Beraldi et al (2011) where uncertainty and risk are dealt by probabilistic constraints jointly imposed on the limited budget restrictions and a mean-risk objective function. The approach based on probabilistic constraints suggests a 'proactive' strategy in that the investment plan is defined in order to hedge, with a given confidence level, the potential losses due to future adverse events.…”