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Unformatted text preview: f MN ISyE Lecture 10: Inventory Models November 7, 2013 7 / 55 The basic EOQ model We will be ordering Q units in “cycles” of length Q/a, and our objective is to
determine Q
The purchasing and setup cost per cycle is
K + cQ
The holding cost per cycle is the area under one of the triangles:
ˆ Q/a
hQ2
h · (Q − at) dt =
2a
0
so the total cost per cycle is
K + cQ +
J. G. Carlsson, U of MN ISyE hQ2
2a Lecture 10: Inventory Models November 7, 2013 8 / 55 The basic EOQ model
The total cost per cycle is
hQ2
2a
and therefore, the total cost per unit time is
K + cQ + 2 K + cQ + hQ /2a
aK
hQ
=
+ ac +
Q/a
Q
2
We differentiate this with respect to Q to find the optimal Q∗ :
d
dQ ( aK
hQ
+ ac +
Q
2 ) aK h
= − 2 + = 0 =⇒ Q∗ =
2
Q √ 2aK
,
h the EOQ formula J. G. Carlsson, U of MN ISyE Lecture 10: Inventory Models November 7, 2013 9 / 55 Observations
The optimal order quantity Q∗ is √
∗ Q=
which costs us aK
hQ
+ ac +
Q
2 √ Q= 2aK
h
= √
2aKh + ac 2aK/h per unit time
Note that Q∗ becomes larger as a and K increase and smaller as h increases
Also note that the optimal cost doesn’t change if we change K → β K and
h → h/β because
√
√
2a(β K)(h/β ) + ac = 2aKh + ac
Also, the optimal cost doesn’t change if we change a → β a and h → h/β and
c → c/β because
√
√
2(β a)K(h/β ) + (β a)(c/β ) = 2aKh + ac
J. G. Carlsson, U of MN ISyE Lecture 10: Inventory Models November 7, 2013 10 / 55 Planned shortages
A shortage or stockout occurs when we cannot meet demand currently
because the inventory is depleted
Under certain circumstances, it may be desirable to permit a limited planned
shortage – this requires that customers be willing to accept a delay in fulfilling
their orders
The EOQ model with planned shortages addresses this kind of situation as
follows [1] :
When a shortage occurs, the affected customers will wait for the product to
become available again. Their backorders are filled immediately when the order
quantity arrives to replenish inventory.
We incorporate a shortage cost p per unit short per unit of time short:
$
unit × hour
We are also interested in the inventory level just after a batch of Q units is
added to inventory, which we’ll call S
p: [1] F.S. Hillier and G.J. Lieberman. Introduction to Operations Research.
Introduction to Operations Research. McGrawHill Higher Education, 2010.
J. G. Carlsson, U of MN ISyE Lecture 10: Inventory Models November 7, 2013 11 / 55 Planned shortages The costs per cycle (of length Q) are given as follows:
The purchasing and setup cost per cycle, as before, is
K + cQ
Inventory level is positive for time S/a, so the holding cost is the area under one
´ S/a
2
of the “positive triangles”: 0 h · (S − at) dt = hSa
2
The shortage cost is the area above one of the “negative triangles”:
´ Q/a
−2
− S/a p · (S − at) dt = p(Q2a S)
J. G. Carlsson, U of MN ISyE Lecture 10: Inventory Models November 7, 2013 12 / 55 Planned shortages The total cost per cycle is then
K + cQ + hS2
p(Q − S)2
+
2a
2a The total cost per unit time is
aK
hS2
p(Q − S)2
K + cQ + hS /2a + p(Q−S) /2a
=
+ ac +
+
Q/a
Q
2Q
2Q
2 2 This has two variables, and we can differentiat...
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