The multilocation transshipment problem

The transshipment fund mechanism: Coordinating the decentralized multilocation transshipment problem

“…Thus, π t is a function of only the order quantities vector Q. One can show, using arguments similar to Herer et al [10], that π t is a concave function of Q.…”

“…The functionsĈ i (ā) satisfy (10). To see this note that for these payments, the derivatives in (10) …”

“…By the argument regarding retailer i's profit at the beginning of the proof, the marginal profit gain from announcing an additional unit which is actually not transshipped is zero, and when announcing an additional unit which is transshipped the marginal profit gain equals ∂ā i (ā) whenā i < 0 (thus it has a plus sign). Because (10) implies that this marginal profit gain is nonnegative forā i < a i and nonpositive for a i ≥ a i , announcingā i = a i is optimal. Similarly, announcingā i = a i is optimal when considering an actual excess supply/demand vector a with a i < 0.…”

“…adapted to a maximization problem from Herer et al [10]. Fix an optimal solution to this LP and denote it by T cp (a), standing for complete pooling transshipment quantities given the vector a, an extended interpretation of the complete pooling term for two retailers coined by Tagaras [18].…”

“…This includes for example the work of Krishnan and Rao [13], Tagaras [18], Robinson [14], and, more recently, Correspondence to: E. Hanany (hananye@post.tau.ac.il) Herer et al [10]. Other works have addressed decentralized inventory systems, either with transshipments, see for example, Rudi et al [15], or structured with more general (mostly vertical) supply chains, see for example, the survey by Cachon [3].…”