Bang-bang optimal maintenance and replacement model. 10 Consider a machine whose resale value gradually declines over time, and its output is assumed to be proportional to its resale value. It is possible, somehow, to slow down the rate of decline of the resale value, applying preventive maintenance. The control problem consists of simultaneously determining the optimal rate of preventive maintenance and the optimal sale date T* of machine. This is a control problem where both the terminal point of the state variable and the terminal time are not fixed, that is, a terminal curve problem. The model can be formalized as follows: yðtÞ ¼ state variable; resale value of the machine: uðtÞ ¼ control variable; preventive maintenance at time t; such that uðtÞ˛U. gðtÞ ¼ maintenance effectiveness function; i:e: money added to the resale value per dollar spent on preventive maintenance: dðtÞ ¼ obsolescence function measured in terms of dollars subracted from yðtÞ at time t: r ¼ discount rate ðif applicableÞ. p ¼ constant production rate in dollars per unit time and per unit resale value: In terms of Maximum Principle, the problem can be then stated as: max fuðtÞg JðuðtÞÞ ¼ Z T 0 ½pyðtÞ  uðtÞe rt dt þ yðTÞe rT s:t: y_ðtÞ¼  dðtÞ þ gðtÞuðtÞ and yð0Þ ¼ y0 and where we denote the function outside the integral as hðyðTÞ; TÞ ¼ yðTÞerT. This problem turns out to be a bang-bang control problem with a linear Hamiltonian function. Build and solve for the optimal control policy u*(t), optimal path y*(t), and optimal switch time t* the following three scenarios of the problem.

a. The first model is without the discount rate and with terminal time of machine resale given, as T ¼ 36 months, while y*(T) is instead free.

b. The second model includes the discount rate r with terminal time T ¼ 36 months, while y*(T) is instead free.

c. The third model includes the discount rate r , and it is specified as terminal curve problem, so that neither y*(T) nor the optimal time of resale T* are preset and therefore are free, and they need to be found by the problem itself, setting up the required transversality conditions for y*(T) ¼ free as: vh vy ðy ðTÞÞ  l ðTÞ ¼ 0 and T ¼ free as: Hðy ðTÞ; u ðTÞ; l ðTÞ; TÞ¼  vh vt ðy ðTÞ; TÞ ¼ r,yðTÞe rT: In all the three scenarios, use the following information: u(t)˛[0, 1], y(0) ¼ 100, d(t) ¼ 2, p ¼ 0:1, r ¼ 0:05, and g(t) ¼ 2/(1 þ t) 1/2 and adopt the unit of time of 1 month.

where T is fixed, H and Q are real symmetric positive semidefinite matrices and R is a real symmetric positive definite matrix. It is assumed that state controls are not bounded and the vector y(T) is free. Solving for the Hamiltonian Maximum Principle conditions, the solution is given by the following 2n equations: 2 6 4 y_ðtÞ  p_ðtÞ 3 7 5 ¼ 2 6 4 AðtÞjBðtÞR1 ðtÞBTðtÞ  j   QðtÞj ATðtÞ 3 7 5 2 6 4 y ðtÞ  p ðtÞ 3 7 5 where this time we have indicated the Lagrange multipliers with p(t).