A printed circuit board (PCB) assembly task requires that a set of components be picked from their respective pickup locations and then be placed at their respective placement locations on the card being assembled. A pick-and-place robot is used for automated assembly of PCB's. The overall assembly time depends on two different decision variables: (i) the pickup locations of the components (in general there are several alternative pickup locations available, whereas the placement location of components is fixed and is determined by the card being assembled), and (ii) the sequence in which the pickup and placement of components is performed. In this paper, we develop a technique based on integer programming to determine both an optimal assignment of pickup locations as well as an optimal sequence of pickup and placements of the components. We demonstrate that the overall optimization problem is an instance of linear integer programming, and hence it is computationally intractable. We obtain near optimal solutions-that are computationally tractable-using the techniques of (i) minimum weight matching for determining an optimal assignment of pickup locations, and (ii) traveling salesman problem for determining an optimum sequence of pickups and placements. Near optimal solutions provide an upper bound for the optimal assembly time; we consider a linear programming relaxation of the problem to obtain a lower bound for the optimal assembly time. The gap between the upper bound and the lower bound provides a measure of closeness of near optimal solutions to an optimal one. Finally, we use simulations to compare the saving in overall assembly time using the techniques developed here and some of the techniques that are currently in use in industrial settings.
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