  Andrews, Lewis and Lovejoy introduced the partition function $PD(n)$ as the
number of partitions of $n$ with designated summands. A bipartition of $n$ is
an ordered pair of partitions $(\pi_1, \pi_2)$ with the sum of all of the parts
being $n$. In this paper, we introduce a generalized crank named the $pd$-crank
for bipartitions with designated summands and give some inequalities for the
$pd$-crank of bipartitions with designated summands modulo 2 and 3. We also
define the $pd$-crank moments weighted by the parity of $pd$-cranks
$\mu_{2k,bd}(-1,n)$ and show the positivity of $(-1)^n\mu_{2k,bd}(-1,n)$. Let
$M_{bd}(m,n)$ denote the number of bipartitions of $n$ with designated summands
with $pd$-crank $m$. We prove a monotonicity property of $pd$-cranks of
bipartitions with designated summands and find that the sequence
$\{M_{bd}(m,n)\}_{|m|\leq n}$ is unimodal for $n\not= 1,5,7$.
