Stochastic programming for decentralized newsvendor with transshipment Nichalin S. Summerfield n , Moshe Dror MIS Department, The University of Arizona, Tucson, AZ 85721, United States article info Article history: Received 5 August 2011 Accepted 14 February 2012 Available online 23 February 2012 Keywords: Stochastic programming Inventory management Cooperative and non-cooperative game abstract This paper discusses a family of two-stage decentralized inventory problems using a unifying framework (taxonomy) depicted as a multilevel graph. This framework allows us to model and link different problems of competing retailers who independently procure inventory in response to uncertain demand and anticipated inventory decisions of other retailers. In this family of problems, in the ex-post stage, the retailers exercise recourse actions in response to the realized demand and competitors’ chosen procure- ment levels. For example, retailers could coordinate inventory transshipment to satisfy shortage with overage based on profit sharing agreements. Our framework provides a unifying parsimonious view using a single methodological prism for a variety of problems. Equally importantly, as recourse options are laid out, our framework clarifies and contributes a modeling connection between problems in a clear taxonomy of models. This unifying perspective explicitly links work that appeared in isolation and offers future research directions. & 2012 Elsevier B.V. All rights reserved. 1. Introduction The query in this paper pertains to a group of independent retailers, like independent car dealerships, aviation parts merchants, and others, who decide, each one independently, about their individual procurement strategy for a single product of uncertain demand. The retailers’ initial procurement decisions account for all the available options beyond the point of learning the actual demands. For instance, if a car dealership does not have a car requested by a customer, it might consider acquiring the car from a competing dealer. All parties have to be appropriately compensated. That is, the car dealership can anticipate ex post cooperation options with other dealerships by means of a trans- shipment mechanism, etc. We propose to view this setting and all its problems through the prism of the stochastic programming methodology. More to the point, we view the inventory procurement options of a retailer competing with similar retailers as a strategic decision of how much to order at the outset (ex ante) without knowing the exact nature of the demand. Nevertheless, it is presumed that the retailer has different options of communicating and collaborating with the competitor(s) at a later stage and responding ex post in a variety of options once the retailer learns the true demand. Without inferring any gender bias, we refer to an individual retailer in this setting by the generic ‘she.’ Stochastic programming is a mathematical methodology for solving optimization problems with time-dependent stochastic variables representing uncertainty of future values. The metho- dology was introduced by Dantzig (1955), Beale (1955), Walkup and Wets (1967), and among others. For basic exposition, definitions, concepts, and additional references, see Birge and Louveaux (1997). Stochastic programming model envisions two (or more) deci- sion stages: given a probability space (O, X, P ) and certain known initial information, e.g., cost parameters and demand uncertainty parameters, the first-stage decisions are taken at some cost. Subsequently, the true value of demand is revealed. At that point, another set of decisions is taken at some cost in response to the realized demand and the first stage decision. The common objective is to minimize the expected cost or to maximize the expected profit. Stochastic programming methodology has also been used to analyze centralized inventory system with demand uncertainty (Chen and Zhang, 2009; Ozen et al., 2009). As many stochastic programming professionals know well, for a majority of real-life problems, after the realization of the demand, the second-stage decision domain can be very large and computa- tionally ‘unmanageable’. To mitigate this problem, one has to consider restrictions in the recourse options by limiting the recourse policies in the second stage. We have to examine carefully what restrictions lead to computational resolution of the strategy domain and potential real-life implementation of effective solutions. In this exposition, we present a systematic exploration process of the different recourse restrictions that are applicable in the case of decentralized inventory with transshipment. The presumed aim of imposing recourse restrictions in the second stage of our stochastic Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/ijpe Int. J. Production Economics 0925-5273/$ - see front matter & 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2012.02.014 n Corresponding author. E-mail addresses: nichalin@email.arizona.edu (N.S. Summerfield), mdror@eller.arizona.edu (M. Dror). Int. J. Production Economics 137 (2012) 292–303