Production, Manufacturing and Logistics Stability and monotonicity in newsvendor situations Ulas ß Özen a,⇑ , Nesim Erkip b , Marco Slikker c a Alcatel-Lucent Bell Laboratories, Blanchardstown Industrial Park, Blanchardstown, Dublin 15, Ireland b Department of Industrial Engineering, Bilkent University, Ankara, Turkey c School of Industrial Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands article info Article history: Received 3 February 2011 Accepted 8 November 2011 Available online 17 November 2011 Keywords: Inventory centralization Newsvendor Game theory Core abstract This study considers a supply chain that consists of n retailers, each of them facing a newsvendor prob- lem, and a supplier. Groups of retailers might increase their expected joint profit by joint ordering and inventory centralization. However, we assume that the retailers impose some level of stock that should be dedicated to them. In this situation, we show that the associated cooperative game has a non-empty core. Afterwards, we concentrate on a dynamic situation, where several model cost parameters and the retailers’ dedicated stock levels can change. We investigate how the profit division might be affected by these changes. We focus on four monotonicity properties. We identify several classes of games with retailers, where some of the monotonicity properties hold. Moreover, we show that pairs of cooperative games associated with newsvendor situations do not necessarily satisfy these properties in general, when changes in dedicated stock levels are in concern. Ó 2011 Published by Elsevier B.V. 1. Introduction In this paper, we consider a distribution system that consists of a supplier and n independent retailers, each facing a stochastic de- mand. Each retailer solves a single period problem (newsvendor problem), i.e., at the start of the period, every retailer determines his order quantity that maximizes his expected profit anticipating that, after the products are delivered to the retailers, demands are realized and satisfied from the stock as much as possible. In this network, we study the inventory pooling coalitions in which the retailers can jointly invest in a common pool of inventory to be allocated after demand realization. In a specific cooperation sce- nario, we study the stability of these coalitions in static and dy- namic settings. Benefits of inventory pooling, i.e., cost savings and profit in- crease, have been studied in different inventory settings (Eppen, 1979; Eppen and Schrage, 1981; Chen and Lin, 1989; Chang and Lin, 1991; Cherikh, 2000). These early studies assume single own- ership of the system. Individual firms, however, are especially interested in what they can get for themselves from inventory cen- tralization. Several other papers have investigated the allocation of benefits (reduced cost or increased profit) problem and proposed several mechanisms. For instance, Gerchak and Gupta (1991) com- pared four simple allocation mechanisms and showed that only one of them guarantees lower cost for every store than its stand-alone cost. Robinson (1993) extended their analysis to other allocation mechanisms, i.e., the Shapley value (cf. Shapley, 1953) and the Lounderback allocation (Lounderback, 1976). Hartman and Dror (1996) examined allocation mechanisms for this setting using three criteria. These are core non-emptiness, computational ease and justifiability. The core concept, a measure of stability, has also received special interest by several other papers and the core non-emptiness has been shown for different newsvendor set- tings: newsvendors with a common pool of inventory (Hartman et al., 2000; Müller et al., 2002; Slikker et al., 2001), and newsvendors with lateral transshipment or multiple channels of supply (Slikker et al., 2005; Özen et al., 2008; Chen and Zhang, 2009). All of these studies assume complete pooling of inventory, i.e., inventory can be diverted to satisfy demand that creates the highest profit from any stock point. However, the benefits of pooling of stock can also be seen in restrictive settings. Anupindi et al. (2001) considered a distribution system where the retailers keep local inventory. After satisfying their local demand, the retailers cooperate by transship- ping excess inventory in one location to satisfy excess demand in another location. They derived a profit sharing mechanism based on dual prices of the optimal shipping problem after demand real- ization, which is a core element and leads to joint optimal orders being an equilibrium. The model of Anupindi et al. (2001) is ex- tended in several directions by Granot and Sošic ´ (2003) and Sošic ´ (2006). In this paper, we do not consider a complete consolidation of inventories when the retailers cooperate. Instead, we assume that the retailers invest in a common pool of inventory but each retailer asks a minimum amount of inventory to be dedicated for him, 0377-2217/$ - see front matter Ó 2011 Published by Elsevier B.V. doi:10.1016/j.ejor.2011.11.021 ⇑ Corresponding author. E-mail addresses: ulas.ozen@alcatel-lucent.com (U. Özen), nesim@bilkent.edu.tr (N. Erkip), m.slikker@tue.nl (M. Slikker). European Journal of Operational Research 218 (2012) 416–425 Contents lists available at SciVerse ScienceDirect European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor