Uncapacitated lot-sizing: The convex hull of solutions

Packing and covering a tree by subtrees

“…The problem P2 with a(v)= 1 for all vE V was the major motivation for this work, as it generalises the tree packing problems considered in [1].…”

“…The connection between the recursive algorithm and the dual greedy algorithm is readily seen by observing, that (1) y…”

“…where a(p(r))=O by definition, and y be definedby (1), then xERI{I and yERl+Vl are optimal solutions to P1 and D~ respectively. …”

A Polyhedral Study of Multiechelon Lot Sizing with Intermediate Demands

“…For k = 3 and l = 4, we have x 1 1 + 4y 1 2 + 3y 1 3 + x 2 2 + x 2 4 ≥ 7 where T 1 = {2, 3}, T 2 = {2, 4}, T 3 = ∅, and x 1 1 + 4y 1 2 + 3y 1 3 + y 2 2 + x 2 4 ≥ 7 where (16) is equivalent to the ( , S) inequality of Barany et al (1984) for the second echelon only, where = l and T 3 = S. For example, x 2 1 + x 2 2 + y 2 3 ≥ 3 (17) is the ( , S) inequality for the second echelon only, with = 3 and S = {3}. In addition, for l = n, T 2 = [1, n], T 3 = ∅, inequality (16) is equivalent to the ( , S) inequality of Barany et al (1984) for the first echelon only, where = k and T 1 = S. For example,…”

“…Krarup and Bilde (1977) give an uncapacitated facility location extended formulation for ULS and show that the linear programming (LP) relaxation of this formulation always has an optimal solution with integer setup variables. Barany et al (1984) give a complete linear description of the ULS polyhedron using the so-called ( , S) inequalities. Since then, several extensions of the single-echelon ULS polyhedron have been considered, to incorporate backlogging Wolsey, 1988, Küçükyavuz andPochet, 2009), uncertainty in demands (Guan et al, 2006a,b), production or inventory capacities (Pochet and Wolsey, 1993, Atamtürk and Muñoz, 2004, Atamtürk and Küçükyavuz, 2005, among others (see Pochet and Wolsey (2006) for a review).…”