TERWORTH Computer-AidedDesign,Vol. 27, No. 10, pp, 759-768, 1995 EINEMANN Copyright Q 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 00104495/95 $10.00 + 0.00 Deterministic tolerance synthesis: a comparative study Andrew Kusiak and Chang-Xue Feng The problem of the synthesis of discrete tolerances is mod- elled as a combinatorial optimization problem, which is then solved by an integer programming approach, a design of experiments method, and the Taguchi method. The paper is perhaps the first attempt to apply the DOE and TM ap- proaches to deterministic tolerance synthesis. Although the design of experiments approach and the Taguchi method are statistical tools for the design and analysis of experiments, they are useful in deterministic tolerance synthesis. A com- parative study has been conducted to investigate the advan- tages and disadvantages of the three approaches for solving the problem of the synthesis of discrete tolerances, with the objective of minimizing the manufacturing cost. The research shows that, while the integer programming approach is suit- able for solving linear deterministic problem, the design of experiments approach is more appropriate for nonlinear problems, and it can be used to solve probabilistic problep. Keywords: quality engineering, tolerance synthesis, statistical tools A careful analysis and assignment of tolerances can significantly reduce manufacturing costs. A design en- gineer usually selects tolerances on the basis of his/her experience and the functionality of the part (product). Then, a safety factor (i.e. a tightening up of the toler- ances by some factor) is used to compensate for the lack of manufacturing knowledge. There are two basic processes in tolerance design: analysis and synthesis. In zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFED tolerance analysis, the component tolerances are speci- fied, and the resulting assembly variation and yield are calculated. Tolerance synthesis involves the allocation of the specified assembly tolerances among the compo- nent dimensions of an assembly to ensure a specified yield. The literature on tolerance synthesis has been reviewed by Voelcker ‘, Juster 2 and Chase and Parkin- son3. A careful study of the previous research indicates that two categories of objectives have been used in the design of tolerances: (a) minimization of the direct manufacturing cost, i.e. the design of tolerances for cost (DTFC), and (b) minimization of the sensitivity of Intelligent Systems Laboratory, Department of Industrial Engineer- ing, University of Iowa, Iowa City, IA 52242-1527, USA Paper receiwd: 15 July 1994. Retied: 3 Nowmber 1994 tolerances to variations in manufacturing processes and the service environment, i.e. design of tolerances for quality (DTFQ) (see, for example, Reference 4), and design of tolerances for reliability (DTFR) (see, for example, Reference 5). From the certainty point of view, methods in the design of tolerances are classified as (a> deterministic, or (b) probabilistic (statistical, stochastic). Among the deterministic approaches, linear programming, nonlinear programming, and heuristic approaches have frequently been used. This paper presents two methods of deterministic tolerance synthesis, the design experiments (DOE) ap- proach and the Taguchi method (TM), along with the integer programming approach. The three methods are illustrated by numerical examples. A comparative study shows that, while the integer programming approach is suitable for linear deterministic problems, the DOE approach is a good alternative for deterministic toler- ance synthesis, regardless of the problem’s linearity and nonlinearity. Furthermore, the latter approach can be applied to solve probabilistic problems. Review of deterministic methods in tolerance synthesis Two models for calculating a tolerance stackup (accu- mulation) have been widely used? (a) the worst case (WC) model (sometimes called a sure-fit model, or the arithmetic law model), and (b) the root sum square (RSS) model (sometimes called a statistical-fit model, or the variance law model). The WC model assumes that all the component dimensional tolerances occur at their worst limit at the same time. It is used by product designers to ensure that all machining tolerances or assemblies meet the specified tolerance stackup or assembly limit. When the number of component toler- ances in machining or the number of parts in an assembly increases, the component tolerances must be reduced in order to meet the tolerance stackup or assembly limit, which results in a high manufacturing cost. In this paper, tolerance synthesis with the WC model is referred to as deterministic tolerance synthe- sis. Cotiputer-Aided Design Volume 27 Number 10 October 1995 759