  We propose a method to generate cutting-planes from multiple covers of
knapsack constraints. The covers may come from different knapsack inequalities
if the weights in the inequalities form a totally-ordered set. Thus, we
introduce and study the structure of a totally-ordered multiple knapsack set.
The valid multi-cover inequalities we derive for its convex hull have a number
of interesting properties. First, they generalize the well-known (1,
k)-configuration inequalities. Second, they are not aggregation cuts. Third,
they cannot be generated as a rank-1 Chvatal-Gomory cut from the inequality
system consisting of the knapsack constraints and all their minimal cover
inequalities. We also provide conditions under which the inequalities are
facets for the convex hull of the totally-ordered knapsack set, as well as
conditions for those inequalities to fully characterize its convex hull. We
give an integer program to solve the separation and provide numerical
experiments that showcase the strength of these new inequalities.
