The above problem is from Powell - Approximate Dynamic Programming

An airline has to decide when to bring an aircraft in for a major engine overhaul. Let s_t, represent the state of the engine in terms of engine wear, and let d_t be a nonnegative amount by which the engine deteriorates during period t. At the beginning of period t the airline may decide to continue operating the aircraft (z_t, = 0) or to repair the aircraft (z_t = 1) at a cost of c^R, which has the effect of returning the aircraft to s_t + 1 = 0. If the airline does not repair the aircraft, the cost of operation is c^o(s_t), which is a nondecreasing, convex function in s_t. Define what is meant by a control limit policy in dynamic programming, and show that this is an instance of a monotone policy. Formulate the one-period reward function C_t(s_t, z_t), and show that it is submodular. Show that the decision rule is monotone in s_t. (Outline the steps in the proof, and then fill in the details.) Assume that a control limit policy exists for this problem, and let y be the control limit. Now write C_t(s_t, z_t) as a function of one variable: the state s. Using the control limit structure of Section 8.2.2, we can write the decision z_t as the decision rule z^pi(s_t). Illustrate the shape of C_t (S_t z^pi(s)) by plotting it over the range 0 lessthanorequalto s lessthanorequalto 3 y(in theory, we may be given an aircraft with s > y initially)