  In this paper we study lattice rules which are cubature formulae to
approximate integrands over the unit cube $[0,1]^s$ from a weighted reproducing
kernel Hilbert space. We assume that the weights are independent random
variables with a given mean and variance for two reasons stemming from
practical applications: (i) It is usually not known in practice how to choose
the weights. Thus by assuming that the weights are random variables, we obtain
robust constructions (with respect to the weights) of lattice rules. This, to
some extend, removes the necessity to carefully choose the weights. (ii) In
practice it is convenient to use the same lattice rule for many different
integrands. The best choice of weights for each integrand may vary to some
degree, hence considering the weights random variables does justice to how
lattice rules are used in applications.
  We also study a generalized version which uses $r$ constraints which we call
the cbc$r$c (component-by-component with $r$ constraints) algorithm. We show
that lattice rules generated by the cbc$r$c algorithm simultaneously work well
for all weights in a subspace spanned by the chosen weights
$\boldsymbol{\gamma}^{(1)},..., \boldsymbol{\gamma}^{(r)}$. Thus, in
applications, instead of finding one set of weights, it is enough to find an
$r$ dimensional convex polytope in which the optimal weights lie. The price for
this method is a factor $r$ in the upper bound on the error and in the
construction cost of the lattice rule. Thus the burden of determining one set
of weights very precisely can be shifted to the construction of good lattice
rules.

Numerical Analysis