Optimal setup of a multihead weighing machine

Multihead weighing machines are ubiquitous in industry for fast and accurate packaging of a wide variety of foods and vegetables, small hardware items and office supplies. These machines consist 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 the 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 setpoints selection has a major impact on the performance of a multihead weighing machine. 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 package weight is near to its target, minimizing the amount of product “given away”. In this paper, we address the open problem for industry of how to determine the setpoint weights for each of the hoppers before starting up the machine, given a desired total package weight. 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 solutions obtained with the proposed methods can result in substantial savings for users of multihead weighing machines. Alternatively, the analysis presented could be used by management to justify the acquisition of new machines of this type.