Closed-Form Solutions for Robust Inventory Management

Abstract

We propose and analyze robust optimization models of an inventory management problem, where cumulative purchase, inventory, and shortage costs over n periods are minimized for correlated nonidentically distributed demand. We assume that the means and covariance matrix of stochastic demand are known; the distributions are not needed. We derive closed-form ordering quantities for both symmetric and asymmetric uncertainty sets, under capacitated inventory constraints, in both static and dynamic settings. The behaviors of our robust strategies differ qualitatively depending on the symmetry of the uncertainty set. For instance, considering our simplest static problem, (1) if the uncertainty set is symmetric, then there is positive ordering in all periods, whereas (2) if the set is asymmetric, then there is a set of periods in the middle of the planning horizon with zero orders. We also link the symmetry of the uncertainty set to the symmetry of the demand distribution. Finally, we present encouraging computational results where our solution compares favorably to previously studied, more complex robust solutions.

Robust Inventory Management: An Optimal Control Approach

“…Iancu et al (2013) continue the study of affine policies, more fully characterizing the problem structures where affine policies are optimal in dynamic robust optimization. Mamani et al (2016) study a rolling-horizon variant of their basic static model, and show that it exhibits encouraging computational performance with respect to Bertsimas and Thiele (2006) and Bertsimas et al (2010). Solyalı et al (2016) also propose a rollinghorizon variant of their static model whose computational performance also compares well with Bertsimas and Thiele (2006), Ben-Tal et al (2004), See and Sim (2010), and others.…”

“…Bienstock and Özbay (2008) generalize (Bertsimas and Thiele 2006) in multiple directions and also analyze data-driven robust models, focusing on algorithmic issues. Mamani et al (2016) study a similar inventory problem to that in Bertsimas and Thiele (2006) and Bienstock and Özbay (2008), except that the uncertainty sets are motivated by the CLT, and closed-form solutions, in both static and dynamic contexts, are derived. Other researchers have studied robust inventory management from different perspectives.…”

“…Next, Bandi and Bertsimas (2012) provide an in-depth study of the use of CLT-style uncertainty sets; these sets have been applied to information theory by Bandi and Bertsimas (2011), option pricing by Bandi and Bertsimas (2014b), auction design by Bandi and Bertsimas (2014a), and queueing theory by Bandi et al (2015Bandi et al ( , 2016. More relevant to our paper is Mamani et al (2016), who studied robust inventory management and whose design of uncertainty sets was also motivated by the CLT. Our paper differs from this literature stream in that our uncertainty sets are motivated by strong laws of large numbers for stochastic processes, rather than the finite-dimensional CLT or the law of the iterated logarithm.…”

“…While many inventory applications in practice are dynamic, in the sense that decisions can be adjusted with new information, there is merit in studying static models. First, contractual obligations can require that all orders be specified in advanced; Ardestani-Jaafari and Delage (2016) (see Remark 7 in their section 6) and Mamani et al (2016) (see the introduction to their section 3) discuss such cases. Furthermore, since robust dynamic models are difficult in general (Ben-Tal et al 2004 prove that a generic adjustable dynamic robust model is NP-hard), static models are frequently used as subroutines in the design of rolling-horizon dynamic optimization models, as in Mamani et al (2016) (see their section 4) and Solyalı et al (2016) (see their section 4); we adopt a similar approach in later sections.…”

A Multiproduct Single‐Period Inventory Management Problem under Variable Possibility Distributions

“…Under interval demand uncertainty, Solyali et al [19] proposed a new robust formulation which could solve the intractability issue for large problem instances. As for recent development in stochastic inventory management problems, the interested reader may further refer to [20][21][22][23][24].…”