SME
Engineering Economics · Problem type
Economic service life
Use this for “A $40,000 machine loses 20% of its market value a year and costs $5,000 to run, rising $1,500 a year. At 10%, which life costs least?”
Table
Life (n)
(yr)
Equivalent uniform annual cost (EUAC)
($)
EUAC against n, worked out for every n from 1 to 10 yr
The least EUAC ≈ 16,120 $ comes at n ≈ 4.959 yr
This is an example. Type your own values to start a new problem.
$
Sₙ
=
$
C
(1
−
%)
$
EUAC
=
$
C
(A/P,
i
%,
n
)
−
$
Sₙ
(A/F,
i
%,
n
)
+
$
A₁
+
$
G
(A/G,
i
%,
n
)
Formulas
Sₙ
=
C
×
(1 −
d
100
)
n
13,107
=
40,000
×
(1 −
20
100
)
5
EUAC
=
C
×
(
i
100
)
÷
(1
−
(1 +
i
100
)
−
n
)
−
Sₙ
×
(
i
100
)
÷
(
(1 +
i
100
)
n
−
1)
+
A₁
+
G
×
(
100
i
−
n
÷
(
(1 +
i
100
)
n
−
1))
16,120
=
40,000
×
(
10
100
)
÷
(1
−
(1 +
10
100
)
-5
)
−
13,107
×
(
10
100
)
÷
(
(1 +
10
100
)
5
−
1)
+
5,000
+
1,500
×
(
100
10
−
5
÷
(
(1 +
10
100
)
5
−
1))
Assumptions
•
The asset loses d of its market value each year: Sₙ = C(1 − d)ⁿ after n years.
•
Operating cost is A₁ the first year and rises by G each year after.
•
Costs are positive and lower is better (FE’s signed convention writes them negative).
•
The economic service life is the n of least EUAC: the ringed row of the table.
Step-by-step
WE KNOW
C = $40,000, d = 20%, A₁ = $5,000, G = $1,500, i = 10%, n = 5 yr
FIND
market value after n years (Sₙ), equivalent uniform annual cost (EUAC)
Step 1 · Find market value after n years (Sₙ)
Use Sₙ = C × (1 − d ÷ 100)ⁿ
Each
year
the
market
value
keeps
(1
−
d
)
of
what
it
was.
Sₙ
=
40,000
×
(1 −
20
100
)
5
Sₙ
=
40,000
×
(1 − 0.2)
5
Sₙ
=
40,000
×
0.8
5
Sₙ = 40,000 × 0.3277
Sₙ = about $13,107
Step 2 · Find equivalent uniform annual cost (EUAC)
Use EUAC = C × (i ÷ 100) ÷ (1 − (1 + i ÷ 100)⁻ⁿ) − Sₙ × (i ÷ 100) ÷ ((1 + i ÷ 100)ⁿ − 1) + A₁ + G × (100 ÷ i − n ÷ ((1 + i ÷ 100)ⁿ − 1))
Recover
the
first
cost,
take
back
the
market
value
at
year
n
,
and
add
the
operating
cost
as
an
even
amount.
Try
each
life:
the
least
EUAC
is
the
economic
service
life.
EUAC
=
C
×
0.2638
−
Sₙ
×
0.1638
+
A₁
+
G
×
1.8101
EUAC = 40,000 × 0.2638 − 13,107 × 0.1638 + 5,000 + 1,500 × 1.8101
(A/P)
=
0.1
1 −
1.1
-5
=
0.2638
(A/F)
=
0.1
1.1
5
− 1
=
0.1638
(A/
G
)
=
1
0.1
−
5
1.1
5
− 1
=
1.8101
EUAC = about $16,120
Check
Put the numbers back in. Both sides should match.
13,107
=
40,000
×
(1 −
20
100
)
5
✓
16,120
=
40,000
×
(
10
100
)
÷
(1
−
(1 +
10
100
)
-5
)
−
13,107
×
(
10
100
)
÷
(
(1 +
10
100
)
5
−
1)
+
5,000
+
1,500
×
(
100
10
−
5
÷
(
(1 +
10
100
)
5
−
1))
✓
Topic check
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