Ward Romeijnders

A two-step method for forecasting spare parts demand using information on component repairs

Forecasting spare parts demand is notoriously difficult, as demand is typically intermittent and lumpy. Specialized methods such as that by Croston are available, but these are not based on the repair operations that cause the intermittency and lumpiness of demand. In this paper, we do propose a method that, in addition to the demand for spare parts, considers the type of component repaired. This two-step forecasting method separately updates the average number of parts needed per repair and the number of repairs for each type of component. The method is tested in an empirical, comparative study for a service provider in the aviation industry. Our results show that the two-step method is one of the most accurate methods, and that it performs considerably better than Croston’s method. Moreover, contrary to other methods, the two-step method can use information on planned maintenance and repair operations to reduce forecasts errors by up to 20%. We derive further analytical and simulation results that help explain the empirical findings.

Cost benefits of postponing time-based maintenance under lifetime distribution uncertainty

We consider the problem of scheduling time-based preventive maintenance under uncertainty in the lifetime distribution of a unit, with the understanding that every time a maintenance action is carried out, additional information on the lifetime distribution becomes available. Under such circumstances, typically either point estimates for the unknown parameters are used, or expected costs are minimized taking the uncertainty in the parameters into account. Both approaches, however, ignore that the uncertainty is reduced much faster if preventive maintenance actions are postponed. Although this initially leads to higher costs due to a higher risk of breakdowns, the obtained additional information can be exploited thereafter as it enables better maintenance decisions going forward. We assess the long-term benefits of initially postponing preventive maintenance, and perform a numerical study to identify under what circumstances these benefits are largest. This study is the first to recognize that the choice of a maintenance strategy influences the information that becomes available, and aims to initiate follow-up research in the area of maintenance planning.

Total variation bounds on the expectation of periodic functions with applications to recourse approximations

We derive a lower and upper bound for the expectation of periodic functions, depending on the total variation of the probability density function of the underlying random variable. Using worst-case analysis we derive tighter bounds for functions that are periodically monotone. These bounds can be used to evaluate the performance of approximations for both continuous and integer recourse models. In this paper, we introduce a new convex approximation for totally unimodular recourse models, and we show that this convex approximation has the best worst-case error bound possible, improving previous bounds with a factor 2. Moreover, we use similar analysis to derive error bounds for two types of discrete approximations of continuous recourse models with continuous random variables. Furthermore, we derive a tractable Lipschitz constant for pure integer recourse models.

Convex Approximations for Totally Unimodular Integer Recourse Models: A Uniform Error Bound

We consider a class of convex approximations for totally unimodular (TU) integer recourse models and derive a uniform error bound by exploiting properties of the total variation of the probability density functions involved. For simple integer recourse models this error bound is tight and improves the existing one by a factor 2, whereas for TU integer recourse models this is the first nontrivial error bound available. The bound ensures that the performance of the approximations is good as long as the total variations of the densities of all random variables in the model are small enough.

Assessing the quality of convex approximations for two-stage totally unimodular integer recourse models

We consider two types of convex approximations of two-stage totally unimodular integer recourse models. Although worst-case error bounds are available for these approximations, their actual performance has not yet been investigated, mainly because this requires solving the original recourse model. In this paper we assess the quality of the approximating solutions using Monte Carlo sampling, or more specifically, using the so-called multiple replications procedure. Based on numerical experiments for an integer newsvendor problem, a fleet allocation and routing problem, and a stochastic activity network investment problem, we conclude that the error bounds are reasonably sharp if the variability of the random parameters in the model is either small or large; otherwise, the actual error of using the convex approximations is much smaller than the error bounds suggest. Moreover, we conclude that the solutions obtained using the convex approximations are good only if the variability of the random parameters is ...

Convex hull approximation of TU integer recourse models:Counterexamples, sufficient conditions, and special cases

We consider a convex approximation for integer recourse models. In particular, we show that the claim of Van der Vlerk (2004) that this approximation yields the convex hull of totally unimodular (TU) integer recourse models is incorrect. We discuss counterexamples, indicate which step of its proof does not hold in general, and identify a class of random variables for which the claim in Van der Vlerk (2004) is not true. At the same time, we derive additional assumptions under which the claim does hold. In particular, if the random variables in the model are independently and uniformly distributed, then these assumptions are satisfied.

A Convex Approximation for Two-Stage Mixed-Integer Recourse Models with a Uniform Error Bound

We develop a convex approximation for two-stage mixed-integer recourse models, and we derive an error bound for this approximation that depends on the total variations of the probability density functions of the random variables in the model. We show that the error bound converges to zero if all these total variations converge to zero. Our convex approximation is a generalization of the one in Romeijnders, van der Vlerk, and Klein Haneveld [Math. Program., to appear] restricted to totally unimodular integer recourse models. For this special case it has the best worst-case error bound possible. The error bound in this paper is the first in the general setting of mixed-integer recourse models. As main building blocks in its derivation we generalize the asymptotic periodicity results of Gomory [Linear Algebra Appl., 2 (1969), pp. 451--558] for pure integer programs to the mixed-integer case, and we use the total variation error bounds on the expectation of periodic functions derived in Romeijnders, van der V...

An approximation framework for two-stage ambiguous stochastic integer programs under mean-MAD information

Abstract We consider two-stage recourse models in which only limited information is available on the probability distributions of the random parameters in the model. If all decision variables are continuous, then we are able to derive the worst-case and best-case probability distributions under the assumption that only the means and mean absolute deviations of the random parameters are known. Contrary to most existing results in the literature, these probability distributions are the same for every first-stage decision. The ambiguity set that we use in this paper also turns out to be particularly suitable for ambiguous recourse models involving integer decisions variables. For such problems, we develop a general approximation framework and derive error bounds for using these approximatons. We apply this approximation framework to mixed-ambiguous mixed-integer recourse models in which some of the probability distributions of the random parameters are known and others are ambiguous. To illustrate these results we carry out numerical experiments on a surgery block allocation problem.

Higher-order total variation bounds for expectations of periodic functions and simple integer recourse approximations

We derive bounds on the expectation of a class of periodic functions using the total variations of higher-order derivatives of the underlying probability density function. These bounds are a strict improvement over those of Romeijnders et al. (Math Program 157:3–46, 2016b), and we use them to derive error bounds for convex approximations of simple integer recourse models. In fact, we obtain a hierarchy of error bounds that become tighter if the total variations of additional higher-order derivatives are taken into account. Moreover, each error bound decreases if these total variations become smaller. The improved bounds may be used to derive tighter error bounds for convex approximations of more general recourse models involving integer decision variables.

The stochastic programming heritage of Maarten van der Vlerk

Maarten van der Vlerk was born in the year 1961 in Assen, The Netherlands, where he spent his childhood and attended school up to the A-level, confirming the requisite maturity for university studies. At the University of Groningen he became a student in the Department of Econometrics and commenced his doctoral studies there under the supervision of Wim Klein Haneveld and Leen Stougie in 1990. Invited and encouraged by his supervisors, Maarten placed a research focus on the integer programming side of stochastic optimization. Soon, he discovered his first original results culminating in a 1993 paper in Mathematical Programming with François Louveaux of the Facultés Universitaires Notre Dame de la Paix in Namur, on “Stochastic programming with simple integer recourse” (Louveaux and van der Vlerk 1993). With his thesis “Stochastic programming with integer recourse”, Maarten was awarded his Ph.D. degree in Economics from the University of Groningen in 1995 (van der Vlerk 1995). After a year as postdoc at the Center for Operations Research and Econometrics (CORE) at the Université Catholique de Louvain at Louvain-la-Neuve, Maarten returned to Groningen. In 1999 he received a prestigious research fellowship from the Royal Netherlands Academy of Sciences. In 2008, Maarten became Professor of Stochastic Optimization at the University of Groningen, Faculty of Economics and Business, Department of Operations. From the mid 1990s on, stochastic integer programming—with its structures to be detected, its algorithms to be constructed, and, last but not least, its real-life applications to be developed—became the field where Maarten van der Vlerk’s research earned highest recognition in the stochastic programming community and beyond. A permanently recurring research target of Maarten, and his various coauthors, has been convex approximation of the notoriously non-convex, discontinuous, mixed-integer