# Firm Dynamic Optimization Problem

A firm has received an order at time $0$ for $M$ units of product to be delivered by time $T$. It seeks a production schedule for filling this order at minimum cost. Let $x(t)$ denote inventory accumulated by time $t$, hence $x’(t)$ denotes the firm’s production at time $t$. There are two sources of cost: cost of production is $[x’(t)]^2$, the cost of holding $x(t)$ is $2x(t)$. Assume the firm discounts future costs with a discount rate $r > 0$.

I am trying to formulate the Hamiltonian for this firm problem. However, I want to make sure my objective and constraints are correct:

Is the problem of the form: $$min \int_{0}^{T} e^{-rt} [2x(t)+[x’(t)]^2]dt$$ i.e. $$-max \int_{0}^{T} e^{-rt} [2x(t)+[x’(t)]^2]dt$$ Is this the proper objective or am I completely off tanget?

• $2x(t)$ is the instantaneous cost of holding inventory $x(t)$? – Alecos Papadopoulos May 1 '18 at 1:25
• Except if there are aspects of the problem that are not included in the post, it appears that the law of motion of the state variable is, at the same time, the decision/control variable (how much to produce per instant of time). Without any constraints on the production capabilities per instant of time, the solution is trivial: produce $M$ units at time $T$, thus incurring zero inventory costs, and discounting the most the production costs. – Alecos Papadopoulos May 1 '18 at 1:44
• See, that's what I figured, and yes these are the only aspects to the question. Thank you for reinforcing my initial thoughts. – nl003 May 1 '18 at 1:47