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MIL-HDBK-338 15 OCTOBER 1984 10.10.3.1 THE GEOMETRY OF SYSTEM R&M TRADEOFFS Let us examine the effect on availability of different amounts of reliability and maintainability and determine the least-cost combination of reliability R (obtained through equipment design) and maintainability M (obtained through test equipment, repair system, and sparing doctrine) which will produce a specified (mission-required) availability A. Since Ai = MTBF/(MTBF + MTTR), a three-dimensional graph or "response surface" will show the relationship among R, M, and A. Note that MTBF is a measure of R, but MTTR is a measure of 1/M. We can show such a response surface, as in Figure 10.10.3.1-1. Al through A4 are increasing levels of availability; any particular availability, for example, Al, may be obtained from many different combinations of R and M. Figure 10.10.3.1-2 shows a two-dimensional projection of such a surface; for the rest of this section we will use such projections as representatives of the response surface. Such projections may be thought of as contour lines on a map, just as we represent topographical features on a two-dimensional map. They are called "iso-availability" (constant availability) contours or "isoquants" for short. The isoquants shown represent availability, increasing from Al to A4. They say nothing about cost. Along any isoquant, many different costs are represented. The surface and its isoquants are convex to the original, showing decreasing marginal (incremental) returns (benefits). As reliability and maintainability are successively increased by a fixed increment, availability is increased by a decreasing increment. We can quickly see this from the formula Ai = MTBF/(MTBF + MTTR). (Try assuming values for either MTBF or MTTR, holding the other constant. Also, calculate the partial derivative of A with respect to either). The individual isoquants approach asymptotes to either axis because larger and larger amounts of R or M are required to produce a fixed A as M or R becomes small. R and M are "competitive substitutes" for each other, and the rate of such competitive substitution diminishes as we move along any isoquant. Again, reference to the formula shows the substitution effect. For a target availability A and the resultant constant K, if we take partial derivatives in Eq. (10.143), remembering that MTTR is proportional to 1/M, we obtain: MTTR _/l - A\ MTBF \ A / MR \ A / / A \/l\ (10.144) 10-119

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