Solves the
economic lot-size model
or economic order quantity, EOQ model,
a classical case in inventory theory, characterized by:
(a) constant demand rate of d units per period;
(b) the order quantity, Q, arriving all at once when desired
(as when the inventory level drops to 0); and
(c) planned shortages not allowed.
A graph is made of z ($/yr) vs. Q (kg).
• Ordering cost ($): K + c Q
• Holding cost per period ($/kg-yr),
'average inv. level', 'holding cost', 'period length':
{(Q + 0) ⁄ 2} × {h} ×
{Q ⁄ d} = h Q² ⁄ (2d)
• Total cost in period ($):
{K + c Q} + {h Q² ⁄ (2d)}
• Total cost per unit time ($/yr):
z = [K + c Q + h Q² ⁄ (2d)]
⁄ (Q ⁄ d) =
{d c} + {h ⁄ 2} Q + {d K} ⁄ Q
• Setting the derivative with respect to Q equal to 0:
dz ⁄ dQ =
h ⁄ 2 − d K ⁄ Q² = 0
(Derivation.pdf,
.xls
.xls2)
The well-known EOQ (kg) formula (sometimes even
referred to just as the square root formula†) is:
Q* = √(2 d K / h)
. Hence:
min z = z* = d c + √(2Kdh);
corresponding period (yr),
θ* = Q* ⁄ d = √[2K⁄(dh)]
Examples
[Tavares et al., 1996, 163]:
d = 1200 m³, K = 15 $,
c = 20 $/m³, h = 5 $/m³-yr;
Q* ~= 85 m³, θ* = 0.071 yr,
z*inv ~= 424 $/yr.
[Wikipedia, EOQ.pdf]:
d (AR) = 10 000 units, K (CO) = 2 $,
c (CU) = 8 $/unit, rate = 2 % → h (CC) = 0.16 $/unit-yr;
Q* (EOQ) = 500 units, θ* = 0.05 yr,
z* = 80 080 $/yr.
† Remark that, for the sum of a straight line
(A + Bx) and a hyperbola (E ⁄ x),
it is: x* = √(E⁄B),
y* = A + 2 √(B E). |
