Definitions and notations. Let E be a set of quadruplets that can be written in REG. Given X ∈ E, its four fields are denoted X.rd, X.ℓvℓ, X.xx ℓ and X.xxℓ, respectively. Relations > and ≥ refer to the lexicographical ordering over X. Xxxxxxxx, where appropriate, we consider the array view[1..n] as the set {view[1], · · · , view[n]}. Definition 1. Let X, Y ∈ E . X ⊐ Y d=ef (X > Y ) ∧ [(X.rd > Y.rd) ∨ (X.xx ℓ)]. X ⊐ Y (X “dominates” Y ) if (a) X is lexicographically greater than Y and (b) the round of X is greater than the round of Y or X indicates a conflict (in this case, as X > Y , we have X.rd = Y.rd). At the operational level the algorithm ensures that the quadruplets it generates are totally ordered by the relation >. Differently, the relation ⊐ (which is a partial order on these quadruplets, see Section 4.2) captures the relevant part of this total order, and is consequently the key cornerstone on which the proof of the algorithm relies. When X ⊐ Y holds, we say “X strictly dominates Y ”. Similarly, “X dominates Y ”, denoted X ⊒ Y , means that (X ⊐ Y ) or (X = Y ) holds. Relations ⊏ and ⊑ are defined in the natural way.
Definitions and notations e The shipper’s decision problem involves selecting carriers, from a set E of candidates, over a multi- period planning horizon T = {1, 2, ..., N}. Every carrier e ∈ E operates its own set of geographically dispersed facilities, Ie, each of which is equipped for cross-docking operations. From the point of view of the shipper, the collection I = ∪e∈EIe forms the complete set of available cross-docking facilities. To use the facilities in Ie, the shipper must have previously signed a contract with carrier e. The conditions applying to a contract with e include its duration (He periods) and the minimum payment commitment (Mt) for every period t ∈ T . If a contract with a carrier e is in effect during period t, then the total payment due to e for period t, denoted ct , is at least equal to Mt or if it exceeds Mt,
