WOLFRAM|DEMONSTRATIONS PROJECT

Maximizing Profit in Ore Mining

​
total extraction duration
300
initial ore in the ground
2.
ore price
1.
rate extraction
ore stock
costate
A mine owner has the right to extract ore for a time duration
T
. Initially there is
x(t=0)=
x
0
ore in the ground. As ore is extracted, the instantaneous stock of ore
x(t)
declines, and
dx
dt
=-u(t)
is the extraction rate. Assume that the ore sells at a constant price
p
and the extraction cost is
2
(u(t))
x(t)
.
This Demonstration finds the optimal extraction rate
u(t)
that maximizes the profit (or performance measure), defined as
J=
T
∫
0
px(t)-
2
(u(t))
x(t)
dt
.
When you set the value of the parameters
T
,
p
, and
x
0
, the Demonstration plots the rate of extraction
u(t)
, the instantaneous stock of ore
x(t)
, and the costate (also called shadow price)
λ(t)
. Assume that the ore remaining in the ground at time
T
has no value (i.e. there is no "scrap value", or
λ(T)=0
).
Two solution methods are adopted: (1) the discrete-time method (solutions indicated by the blue triangles), and (2) the continuous-time method (the red curves).
You can readily check that when the duration
T
is decreased, the extraction rate is larger. Also for a fixed value of
T
, if you increase the initial stock of ore in the mine, then both the profit
J
and the extraction rate are larger. The values of
J
for the discrete-time and continuous-time versions are indicated in blue and red, respectively. The two values show very close agreement. Finally, you can lower the ore price, which results in a lower profit.