SME
Human Geography · Problem type
Least-cost site: mine or market
Use this for “Smelting 1 t of copper takes 5 t of ore. The mine is 300 km from the market and freight is $0.10 a tonne-km. Build at the mine or the market?”
Chart
C(100) = 70
This is an example. Type your own values to start a new problem.
r
Freight per tonne-km
example · $/(t·km)
m
Raw material
example · t
q
Product
example · t
D
Mine to market
example · km
x
Site from mine
example · km
C
Transport cost
calculated · $
Cₘᵢₙₑ
Cost at the mine
worked out (not typed) · $
Cₘₖₜ
Cost at the market
worked out (not typed) · $
MI
Material index
worked out (not typed)
site
Cheaper site
worked out (not typed)
Formulas
site: the end with the lower of Cₘᵢₙₑ and Cₘₖₜ
30 < 150, so the mine
C
=
r
×
(
m
×
x
+
q
×
(
D
−
x
))
70 = 0.1 × (5 × 100 + 1 × (300 − 100))
Cₘᵢₙₑ
=
r
×
q
×
D
30 = 0.1 × 1 × 300
Cₘₖₜ
=
r
×
m
×
D
150 = 0.1 × 5 × 300
MI
=
m
q
5
=
5
1
Assumptions
•
Weber’s model: only transport cost changes with the site; labor and power cost the same everywhere.
•
The plant sits on the road from mine to market, s km from the mine; ore and product pay one freight rate.
•
Material index above 1: the process loses weight, so build at the mine; below 1, at the market.
Step-by-step
WE KNOW
x = 100 km, r = 0.1 $/(t·km), m = 5 t, q = 1 t, D = 300 km
FIND
transport cost (C), cost at the mine (Cₘᵢₙₑ), cost at the market (Cₘₖₜ), material index (MI), cheaper site (site)
Step 1 · Find transport cost (C)
Use C = r × (m × x + q × (D − x))
The
raw
material
rides
x
km
from
the
mine,
the
product
the
other
D
−
x
km
to
market.
C
=
0.1
×
(5
×
100
+
1
×
(300
−
100))
C
=
0.1
×
(5
×
100
+
200)
C
=
0.1
×
(500
+
200)
C
=
0.1
×
700
C
=
$70
Step 2 · Find cost at the mine (Cₘᵢₙₑ)
Use Cₘᵢₙₑ = r × q × D
At
the
mine
only
the
product
travels,
the
whole
D
km.
Cₘᵢₙₑ = 0.1 × 1 × 300
Cₘᵢₙₑ = $30
Step 3 · Find cost at the market (Cₘₖₜ)
Use Cₘₖₜ = r × m × D
At
the
market
only
the
raw
material
travels,
the
whole
D
km.
Cₘₖₜ = 0.1 × 5 × 300
Cₘₖₜ = 0.5 × 300
Cₘₖₜ = $150
Step 4 · Find material index (MI)
Use MI = m ÷ q
Tonnes of raw material per tonne of product: above 1, the plant sheds weight.
MI
=
5
1
MI = 5
Step 5 · Find cheaper site (site)
Use site: the end with the lower of Cₘᵢₙₑ and Cₘₖₜ
Cost changes steadily along the road, so the cheapest site is one end: build where the cost is lower.
30 < 150: the mine
site = the end with the lower of 30 and 150
site = the mine
Check
Put the numbers back in. Both sides should match.
30 < 150, so the mine ✓
70 = 0.1 × (5 × 100 + 1 × (300 − 100)) ✓
30 = 0.1 × 1 × 300 ✓
150 = 0.1 × 5 × 300 ✓
5
=
5
1
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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