Duality Approaches to Economic Lot‐Sizing Games

Abstract

We consider the economic lot-sizing (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 non-empty. 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.

Dynamic cost allocation for economic lot sizing games

“…for some τ ≥ r, we can model the economic lot sizing problem for coalition R ⊆ N with the following linear program (Chen and Zhang 2006):…”

“…, k, for all j, k such that τ ≤ j ≤ k ≤ T . Although the decision variables are continuous, this interpretation is well-defined: Chen and Zhang (2006) showed that there always exists an optimal solution to (1) such that x jk ∈ {0, 1} for all j, k. The first term of the objective (1a) is the cost of greedily using the initial inventoryŝ r−1 R to satisfy demand in periods r, . .…”

“…Gopaladesikan et al (2012) proposed a polynomial-time combinatorial algorithm that solves f r (R,ŝ r−1 ) and its dual (actually, an affine transformation of its dual), and used the primal and dual optimal solutions in conjunction with some structural properties established by Chen and Zhang (2006) to compute an allocation in the core of the economic lot sizing game (N, f r (•,ŝ r−1 )). We summarize these results in the lemma below.…”

“…In a cooperative setting, this model would capture the situation described above, with each retailer facing its own demand over the planning horizon that must be satisfied with product orders or inventory, but with coalitions of retailers being able to order product together. The cooperative game derived from this model was proposed in van den Heuvel et al (2007) and subsequently studied in Chen and Zhang (2006), Gopaladesikan et al (2012). However, all of these papers approach the problem from a static perspective; that is, the authors assume an optimal solution is implemented over the entire horizon, and seek an up-front static cost allocation in the core of this cooperative game.…”

Simultaneous Penalization and Subsidization for Stabilizing Grand Cooperation

“…This is one of the central themes in cooperative game theory, which has wide applications. See, for example, facility location games (Goemans andSkutella 2000, Mallozzi 2011), inventory games (Anily and Haviv 2007, Chen 2009, Zhang 2009, Chen and Zhang 2016, and outsourcing games (Aydinliyim andVairaktarakis 2010, Cai andVairaktarakis 2012), to name a few. The set of such cost allocations is known as the core of a cooperative game (Shapley and Shubik 1969).…”