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18
A stochastic programming duality approach to inventory centralization games," Operations Research, minor revision submitted
, 2007
"... In this paper, we present a unified approach to study a class of cooperative games arising from inventory centralization. The optimization problems corresponding to the inventory games are formulated as stochastic programs. We observe that the strong duality of stochastic linear programming not only ..."
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Cited by 16 (5 self)
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In this paper, we present a unified approach to study a class of cooperative games arising from inventory centralization. The optimization problems corresponding to the inventory games are formulated as stochastic programs. We observe that the strong duality of stochastic linear programming not only directly leads to a series of recent results concerning the nonemptiness of the cores of such games, but also suggests a way to find an element in the core. The proposed approach is also applied to inventory games with concave ordering cost. In particular, we show that the newsvendor game with concave ordering cost has a nonempty core. Finally, we prove that it is NPhard to determine whether a given allocation is in the core for the inventory games even in a very simple setting.
Matching Models for Preferencesensitive Group Purchasing TYLER LU, University of Toronto
"... Matching buyers and sellers is one of the most fundamental problems in economics and market design. An interesting variant of the matching problem arises when selfinterested buyers come together in order to induce sellers to offer quantity or volume discounts, as is common in buying consortia, and ..."
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Cited by 9 (2 self)
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Matching buyers and sellers is one of the most fundamental problems in economics and market design. An interesting variant of the matching problem arises when selfinterested buyers come together in order to induce sellers to offer quantity or volume discounts, as is common in buying consortia, and more recently in the consumer group couponing space (e.g., Groupon). We consider a general model of this problem in which a group or buying consortium is faced with volume discount offers from multiple vendors, but group members have distinct preferences for different vendor offerings. Unlike some recent formulations of matching games that involve quantity discounts, the combination of varying preferences and discounts can render the core of the matching game empty, in both the transferable and nontransferable utility sense. Thus, instead of coalitional stability, we propose several forms of Nash stability under various epistemic and transfer/payment assumptions. We investigate the computation of buyerwelfare maximizing matchings and show the existence of transfers (subsidized prices) of a particularly desirable form that support stable matchings. We also study a nontransferable utility model, showing that stable matchings exist; and we develop a variant of the problem in which buyers provide a simple preference ordering over “deals ” rather than specific valuations—a model that is especially attractive in the consumer space—which also admits stable matchings. Computational experiments demonstrate the efficacy and value of our approach.
Duality Approaches to Economic LotSizing Games
, 2007
"... We consider the economic lotsizing (ELS) game with general concave ordering cost. In this cooperative game, multiple retailers form a coalition by placing joint orders to a single supplier in order to reduce ordering cost. When both the inventory holding cost and backlogging cost are linear functio ..."
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Cited by 7 (3 self)
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We consider the economic lotsizing (ELS) game with general concave ordering cost. In this cooperative game, multiple retailers form a coalition by placing joint orders to a single supplier in order to reduce ordering cost. When both the inventory holding cost and backlogging cost are linear functions, it can be shown that the core of this game is nonempty. The main contribution of this paper is to show that a core allocation can be computed in polynomial time. Our approach is based on linear programming (LP) duality and is motivated by the work of Owen [19]. We suggest an integer programming formulation for the ELS problem and show that its LP relaxation admits zero integrality gap, which makes it possible to analyze the ELS game by using LP duality. We show that, there exists an optimal dual solution that defines an allocation in the core. An interesting feature of our approach is that it is not necessarily true that every optimal dual solution defines a core allocation. This is in contrast to the duality approach for other known cooperative games in the literature.
Cost Allocation for Joint Replenishment Models
, 2006
"... We consider the onewarehouse multiple retailer inventory model with a submodular joint setup cost function. The objective of this model is to determine an inventory replenishment policy that minimizes the longrun average system cost over an infinite time horizon. Although the optimal policy for t ..."
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Cited by 5 (2 self)
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We consider the onewarehouse multiple retailer inventory model with a submodular joint setup cost function. The objective of this model is to determine an inventory replenishment policy that minimizes the longrun average system cost over an infinite time horizon. Although the optimal policy for this problem is still unknown, a class of easytoimplement poweroftwo policies are 98 % effective. This paper focuses on how the cost, under an optimal poweroftwo policy, should be allocated to the retailers. This question generates an interesting cooperative game. We prove that this cooperative game has a nonempty core. The key to our result is a strong duality theorem for the onewarehouse multiple retailer problem under poweroftwo policies.
Cooperation in Service Systems
"... We consider a number of servers that may improve the efficiency of the system by pooling their service capacities to serve the union of the individual streams of customers. This economies of scope phenomenon is due to the reduction in the steadystate mean total number of customers in system. The qu ..."
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Cited by 4 (0 self)
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We consider a number of servers that may improve the efficiency of the system by pooling their service capacities to serve the union of the individual streams of customers. This economies of scope phenomenon is due to the reduction in the steadystate mean total number of customers in system. The question we pose is how the servers should split among themselves the cost of the pooled system. When the individual incoming streams of customers form Poisson processes and individual service times are exponential, we define a transferable utility cooperative game in which the cost of a coalition is the mean number of customers (or jobs) in the pooled system. We show that albeit the fact the characteristic function is neither monotone nor concave, the game (and its subgames) possess nonempty cores. In other words, for any subset of servers there exist costsharing allocations under which no partial subset can take advantage by breaking away and forming a separate coalition. We give an explicit expression for all (infinitely many) nonnegative core cost allocations of this game. Finally, we show that except for the case where all individual servers have the same cost, there exist infinitely many core allocations with negative entries, and we show how to construct a convex subset of the core where at least one server is being paid in order to join the grand coalition. Subject classification:
Approximating the Least Core Value and Least Core of Cooperative Games with Supermodular Costs
, 2010
"... We study the approximation of the least core value and the least core of supermodular cost cooperative games. We provide a framework for approximation based on oracles that approximately determine maximally violated constraints. This framework yields a.3 C "/approximation algorithm for computi ..."
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Cited by 4 (0 self)
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We study the approximation of the least core value and the least core of supermodular cost cooperative games. We provide a framework for approximation based on oracles that approximately determine maximally violated constraints. This framework yields a.3 C "/approximation algorithm for computing the least core value of supermodular cost cooperative games, and a polynomialtime algorithm for computing a cost allocation in the 2approximate least core of these games. This approximation framework extends naturally to submodular profit cooperative games. For scheduling games, a special class of supermodular cost cooperative games, we give a fully polynomialtime approximation scheme for computing the least core value. For matroid profit games, a special class of submodular profit cooperative games, we give exact polynomialtime algorithms for computing the least core value as well as a least core cost allocation.
Stochastic linear programming games with concave preferences
 European Journal of Operational Research
, 2015
"... Abstract We study stochastic linear programming games: a class of stochastic cooperative games whose payoffs under any realization of uncertainty are determined by a specially structured linear program. These games can model a variety of settings, including inventory centralization and cooperative ..."
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Cited by 1 (1 self)
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Abstract We study stochastic linear programming games: a class of stochastic cooperative games whose payoffs under any realization of uncertainty are determined by a specially structured linear program. These games can model a variety of settings, including inventory centralization and cooperative network fortification. We focus on the core of these games under an allocation scheme that determines how payoffs are distributed before the uncertainty is realized, and allows for arbitrarily different distributions for each realization of the uncertainty. Assuming that each player's preferences over random payoffs are represented by a concave monetary utility functional, we prove that these games have a nonempty core. Furthermore, by establishing a connection between stochastic linear programming games, linear programming games and linear semiinfinite programming games, we show that an allocation in the core can be computed efficiently under some circumstances.
Dynamic Cost Allocation for Economic Lot Sizing Games
, 2013
"... We consider a cooperative game defined by an economic lot sizing problem with concave ordering costs over a finite time horizon, in which each player faces demand for a single product in each period and coalitions can pool orders. We show how to compute a dynamic cost allocation in the strong sequen ..."
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Cited by 1 (1 self)
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We consider a cooperative game defined by an economic lot sizing problem with concave ordering costs over a finite time horizon, in which each player faces demand for a single product in each period and coalitions can pool orders. We show how to compute a dynamic cost allocation in the strong sequential core of this game, i.e. an allocation over time that exactly distributes costs and is stable against coalitional defections at every period of the time horizon. 1
Math. Program., Ser. A DOI 10.1007/s1010701104417 FULL LENGTH PAPER
, 2010
"... Integrated market selection and production planning: complexity and solution approaches ..."
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Integrated market selection and production planning: complexity and solution approaches