This paper presents a workable analytical method for finding the optimum set of dimension tolerances for a mechanical device that will minimize manufacturing costs and meet the imposed restraint conditions. The restraint conditions represent physical requirements that critical parts must fit relative to each other with a given precision. Using the mathematical description of the restraints and information on the cost of manufacturing each dimension as a function of the tolerance, the method utilizes Lagrange multipliers to minimize nonlinear cost functions subject to nonlinear restraint conditions. For all but the simplest problems the use of the method requires computer programming. The method is presented in sufficient detail to allow the reader to develop a computer code. A Fortran computer program that will handle essentially any mechanism and calculate tolerances on a sure fit or statistical-fit basis has proven that the method will give useful results.

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