  We study the Min-Weighted Sum Bin Packing problem, a variant of the classical
Bin Packing problem in which items have a weight, and each item induces a cost
equal to its weight multiplied by the index of the bin in which it is packed.
This is in fact equivalent to a batch scheduling problem that arises in many
fields of applications such as appointment scheduling or warehouse logistics.
We give improved lower and upper bounds on the approximation ratio of two
simple algorithms for this problem. In particular, we show that the
knapsack-batching algorithm, which iteratively solves knapsack problems over
the set of remaining items to pack the maximal weight in the current bin, has
an approximation ratio of at most 17/10.
