Optimally balancing manufacturing cost and yield. The vector x∈Rn denotes the nominal parameters in a manufacturing process. The yield of the process, i.e., the fraction of manufactured goods that is acceptable, is given by Y(x). We assume that Y is log-concave (which is often the case; see example 3.43). The cost per unit to manufacture the product is given by cTx, where c∈Rn. The cost per acceptable unit is cTx/Y(x). We want to minimize cTx/Y(x), subject to some convex constraints on x such as a linear inequalities Ax⪯b. (You can assume that over the feasible set we have cTx>0 and Y(x)>0.)
This problem is not a convex or quasiconvex optimization problem, but it can be solved using convex optimization and a one-dimensional search. The basic ideas are given below; you must supply all details and justification.
(a) Show that the function f:R→R given by
f(a)=sup{Y(x)∣Ax⪯b,cTx=a},
which gives the maximum yield versus cost, is log-concave. This means that by solving a convex optimization problem (in x) we can evaluate the function f.
(b) Suppose that we evaluate the function f for enough values of a to give a good approximation over the range of interest. Explain how to use these data to (approximately) solve the problem of minimizing cost per good product.