  Multihead weighing machines (MWMs) are ubiquitous in industry for fast and
accurate packaging of a wide variety of foods and vegetables, small hardware
items and office supplies. A MWM consists of a system of multiple hoppers that
are filled with product which when discharged through a funnel fills a package
to a desired weight. Operating this machine requires first to specify the
product weight targets or setpoints that each hopper should contain on average
in each cycle, which do not need to be identical. The selection of these
setpoints has a major impact on the performance of a MWM. Each cycle, the
machine fills a package running a built-in knapsack algorithm that opens --or
leaves shut-- different combinations of hoppers releasing their content such
that the total weight of each package is near to its target, minimizing the
amount of product ``given away". In this paper, we address the practical open
problem for industry of how to determine the setpoint weights for each of the
hoppers given a desired total package weight for a widely used type of MWM. An
order statistic formulation based on a characterization of near-optimal
solutions is presented. This is shown to be computationally intractable, and a
faster heuristic that utilizes a lower bound approximation of the expected
smallest order statistic is proposed instead. The setup solutions obtained with
the proposed methods can result in substantial savings for MWM users.
Alternatively, the analysis presented could be used by management to justify
the acquisition of new MWM machines.
