Minimum Exponential Cost Allocation of Sure-Fit Tolerances

Abstract

The least cost allocation of sure-fit machine tolerances for Speckhart’s exponential cost model is solved in closed form, without numerical iteration, as a geometric program with zero degrees of difficulty. The results show the importance of an exponential cost sensitivity parameter defined as the “characteristic tolerance”. The theoretical minimum cost can be determined without specifying the corresponding tolerances. Specific minimum cost tolerances can be computed later in closed form if potential cost savings are significant.

A survey of research in the application of tolerance analysis to the design of mechanical assemblies

“…Numerous researchers have proposed different search algorithms and different forms of empirical cost functions, as summarized in Table 2. Reciprocal A + B/T Lagrange mult Nonlin prog Reciprocal Squared A + B/T 2 Lagrange mult Spotts [1973] Reciprocal Power A + B/T k Lagrange mult Sutherland & Roth [1975] Multi/Recip Powers B/T k i Nonlin prog Lee & Woo [1990] Lagrange mult Bennett & Gupta [1969] Lagrange mult Chase et al [1990] Nonlin prog Andersen [1990] Exponential B e -mT Lagrange mult Speckhart [1972] Geom prog Wilde & Prentice [1975] Graphical Peters [1970] Expon/Recip Power B e -mT /T k Nonlin prog Michael & Siddall [1981 Piecewise Linear A i -B i T i Linear prog Bjork [1989], Patel [1980] Empirical Data Discrete points Zero-one prog Ostwald & Huang [1977] Combinatorial Monte & Datseris [1982] Branch & Bound Lee & Woo [1989] The constant coefficient A represents the fixed costs, such as tooling, setup, prior operations, etc. The B term represents the cost of producing a single component dimension to a specified tolerance T. All costs are calculated on a per part basis.…”

A pseudo-boolean approach to determining least cost tolerances

“…Rule 3: If g;=hl(x)+h3(X)~d"g~=hl(X)+h2(X)~dkand h2(X)~dk-d, where hlx), h 2 (x ) and h 3 (x ) are non-negative functions, then, any x satisfying g;~d, also satisfies g~~dk' Proof: This was proven by Wilde and Prentice (1975). Since h 3(x)~O , h 1(x)~d ,.…”

“…Rule 5: If T p, + T p, > d, for some component p and every pair (s, I) of manufacturing process for p, then the logical constraint is redundant. Proof: This was proven by Wilde and Prentice (1975) as follows; if x satisfies g;';;;d" then for every pair (s,t) Tp,xp,+ I;"xp,';;;d, which implies that no two variables x p, and x pr can be unity simultaneously.…”

“…Another approach by Wilde and Prentice (1975) showed that the least cost tolerance for the exponential cost model is solved in a closed form without numerical iteration as a geometric program with zero degree of difficulty. Smather and Ostwald (1972) recognizing the relationship existing between the process, the processing cost and the tolerance, proposed that the solution to the least cost tolerance problem could be solved by a dynamic programming approach.…”

A study into the use of tolerance analysis in the mechanical design of medical devices

“…Although several researchers have studied how to balance tolerances in order to minimise manufacturing cost (Peters, 1970), (Spotts, 1973) and (Wilde and Prentice, 1975), little research is present in relation to the medical industry. Research concluding the effects of tolerance on mechanical performance is limited; evidence of the employment of tolerance analysis methods in relation to specific drug delivery devices is scarce (El-Haik and Mekki, 2008).…”