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In this paper, we present bounds for multi-horizon stochastic optimization problems, a class of problems introduced in Kaut et al. (Comput Manag Sci 11:179–193, 2014) relevant in many industry-life applications typically involving strategic and operational decisions on two different time scales. After providing three general mathematical formulations of a multi-horizon stochastic program, we extend the definition of the traditional Expected Value problem and Wait-and-See problem from stochastic programming in a multi-horizon framework. New measures are introduced allowing to quantify the importance of the uncertainty at both strategic and operational levels. Relations among the solution approaches are then determined and chain of inequalities provided. Numerical experiments based on an energy planning application are finally presented.

This erratum is published due to proofing error as author corrections were overlooked.

In this paper, we consider the problem of tax evasion, which occurs whenever an individual or business ignores tax laws. Fighting tax evasion is the main task of the Economic and Financial Military Police, which annually performs fiscal controls to track down and prosecute evaders at national level. Due to limited financial resources, the tax inspector is unable to audit the population entirely. In this article, we propose a model to assist the Italian tax inspector (Guardia di Finanza, G.d.F.) in allocating its budget among different business clusters, via a controller-controlled Stackelberg game. The G.d.F. is seen as the leader, while potential evaders are segmented into classes according to their business sizes, as set by the Italian regulatory framework. Numerical results on the real Italian case for fiscal year 2015 are provided. Insights on the optimal number of controls the inspector will have to perform among different business clusters are discussed and compared to the strategy implemented by the G.d.F.

In this paper, we consider multi-stage robust optimization problems of the minimax type. We assume that the total uncertainty set is the cartesian product of stagewise compact uncertainty sets and approximate the given problem by a sampled subproblem. Instead of looking for the worst case among the infinite and typically uncountable set of uncertain parameters, we consider only the worst case among a randomly selected subset of parameters. By adopting such a strategy, two main questions arise: (1) Can we quantify the error committed by the random approximation, especially as a function of the sample size? (2) If the sample size tends to infinity, does the optimal value converge to the “true” optimal value? Both questions will be answered in this paper. An explicit bound on the probability of violation is given and chain of lower bounds on the original multi-stage robust optimization problem provided. Numerical results dealing with a multi-stage inventory management problem show that the proposed approach works well for problems with two or three time periods while for larger ones the number of required samples is prohibitively large for computational tractability. Despite this, we believe that our results can be useful for problems with such small number of time periods, and it sheds some light on the challenge for problems with more time periods.

The explicit consideration of uncertainty is essential in addressing most planning and operation issues encountered in the management of complex systems. Unfortunately, the resulting stochastic programming formulations, integer ones in particular, are generally hard to solve when applied to realistically-sized instances. A common approach is to consider the simpler deterministic version of the formulation, even if it is well known that the solution quality could be arbitrarily bad. In this paper, we aim to identify meaningful information, which can be extracted from the solution of the deterministic problem, in order to reduce the size of the stochastic one. Focusing on two-stage formulations, we show how and under which conditions the reduced costs associated to the variables in the deterministic formulation can be used as an indicator for excluding/retaining decision variables in the stochastic model. We introduce a new measure, the Loss of Reduced Costs-based Variable Fixing (LRCVF), computed as the difference between the optimal values of the stochastic problem and its reduced version obtained by fixing a certain number of variables. We relate the LRCVF with existing measures and show how to select the set of variables to fix. We then illustrate the interest of the proposed LRCVF and related heuristic procedure, in terms of computational time reduction and accuracy in finding the optimal solution, by applying them to a wide range of problems from the literature.

We apply a dynamic stochastic control (DSC) approach based on an open-loop linear feedback policy to a classical asset-liability management problem as the one faced by a defined-benefit pension fund (PF) manager. We assume a PF manager seeking an optimal investment policy under random market returns and liability costs as well as stochastic PF members’ survival rates. The objective function is formulated as a risk-reward trade-off function resulting in a quadratic programming problem. The proposed methodology combines a stochastic control approach, due to Primbs and Sung (IEEE Trans Autom Control 54(2):221–230, 2009), with a chance constraint on the PF funding ratio (FR) and it is applied for the first time to this class of long-term financial planning problems characterized by stochastic asset and liabilities. Thanks to the DSC formulation, we preserve the underlying risk factors continuous distributions and avoid any state space discretization as is typically the case in multistage stochastic programs (MSP). By distinguishing between a long-term PF liability projection horizon and a shorter investment horizon for the FR control, we avoid the curse-of-dimensionality, in-sample instability and approximation errors that typically characterize MSP formulations. Through an extended computational study, we present in- and out-of-sample results which allows us to validate the proposed methodology. The collected evidences confirm the potential of this approach when applied to a stylized but sufficiently realistic long-term PF problem.

A facility has to be located in a given area to serve a given number of customers. The position of the customers is not known. The service to the customers is carried out by several traveling salesmen. Each of them has a capacity in terms of the maximum number of customers that can be served in any tour. The aim was to determine the service zone (in a shape of a circle) that minimizes the expected cost of the traveled routes. The center of the circle is the location of the facility. Once the position of the customers is revealed, the customers located outside the service zone are served with a recourse action at a greater unit cost. For this problem, we compare the performance of two different solution approaches. The first is based on a heuristic proposed for the Capacitated Traveling Salesman Problem and the second on the optimal solution of a stochastic second-order cone formulation with an approximate objective function.

Multistage stochastic programs, which involve sequences of decisions over time, are usually hard to solve in realistically sized problems. Providing bounds for optimal solution may help in evaluating whether it is worth the additional computations for the stochastic program vs. simplified approaches. In this paper we generalize measures from the two-stage case, based on different levels of available information, to the multistage stochastic programming problems. A set of theorems providing chains of inequalities among the new quantities are proved. Numerical results on a case study related to a simple transportation problem illustrate the described relationships.

Stochastic programs are usually hard to solve when applied to real-world problems; a common approach is to consider the simpler deterministic program in which random parameters are replaced by their expected values, with a loss in terms of quality of the solution. The Value of the Stochastic Solution— VSS —is normally used to measure the importance of using a stochastic model. But what if VSS is large, or expected to be large, but we cannot solve the relevant stochastic program? Shall we just give up? In this paper we investigate very simple methods for studying structural similarities and differences between the stochastic solution and its deterministic counterpart. The aim of the methods is to find out, even when VSS is large, if the deterministic solution carries useful information for the stochastic case. It turns out that a large VSS does not necessarily imply that the deterministic solution is useless for the stochastic setting. Measures of the structure and upgradeability of the deterministic solution such as the loss using the skeleton solution and the loss of upgrading the deterministic solution will be introduced and basic inequalities in relation to the standard VSS are presented and tested on different cases.