Analysis and Optimal Targets Setup of a Multihead Weighing Machine

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.