A primal–dual algorithm for computing a cost allocation in the core of economic lot-sizing games

Abstract

5We consider the economic lot-sizing game with general concave ordering cost functions. It is 6 well-known that the core of this game is nonempty when the inventory holding costs are linear. 7The main contribution of this work is a combinatorial, primal-dual algorithm that computes a cost 8 allocation in the core of these games in polynomial time. We also show that this algorithm can be 9 used to compute a cost allocation in the core of economic lot-sizing games with remanufacturing 10 under certain assumptions.

Dynamic cost allocation for economic lot sizing games

“…Lemma 1 (Chen andZhang 2006, Gopaladesikan et al 2012). Let x * and α * respectively be optimal solutions to f r (R, s r−1 ) and its dual, obtained using the algorithm by Gopaladesikan et al (2012). Then, x * and α * satisfy the following properties:…”

“…. , (a m , b m )} with a 1 = 1 and b m = T , obtained by the algorithm of Gopaladesikan et al (2012).…”

“…In previous approaches to economic lot sizing games (Chen and Zhang 2006, Gopaladesikan et al 2012, van den Heuvel et al 2007, the authors assume the entire cost of the optimal solution is allocated up front at the start of the horizon. Instead, we make the more realistic assumption that ordering costs must be allocated in the same period an order takes place, and holding costs for each period must be allocated in that same period.…”

“…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.…”

Cost allocation of capacity investment games

“…If the production is made at one period, then the capacity is 3 + η and the total cost is κ(3 + η) + 1; if the production is made at two periods, then the capacity is (3 + η)/2 and the total cost is κ(3 + η)/2 + 2; if the production is made at three periods, then the capacity is (3 + η)/3 and the total cost is κ(3 + η)/3 + 3. For κ = 1/2, it is optimal to set the capacity level at 3 + η when η ∈ [0, 1], the capacity level at (3 + η)/2 when η ∈ [1,9] and the capacity level at (3 + η)/3 when η ≥ 9. Thus, the optimal capacity level may decrease with demand.…”