Inventory models

engineering management

1. Engineering Management 19/12/2013
M M HASAN, Lecturer, AIE, HSTU, DINAJPUR 1
MD MOUDUD HASAN
LECTURER
AIE, HSTU
DINAJPUR
An inventory problem exists when it is necessary to
stock physical goods or commodities for the purpose of
satisfying demand over a specified time horizon(finite
or infinite).

Almost every business must stock goods to ensure
smooth and efficient running of its operation.

Decisions regarding how much and when to order are
typical of every inventory problem.

2. Engineering Management 19/12/2013
M M HASAN, Lecturer, AIE, HSTU, DINAJPUR 2
1.Economic factors:
a)Setup cost:
•Fixed charge associated with the placement of an order
•Independent of the quantity ordered
b)Purchase price:
•Parameter is of special interest when quantity discounts
or price breaks can be secured
c)Selling price:
•Unit selling price may be constant or variable depending
on whether quantity discount is allowed.
d)Holding cost:
•The cost of carrying inventory in storage
•It includes the interest on invested capital, storage,
handling cost, depreciation cost, etc.
e.Shortage cost:
•The penalty cost incurred as a result of running out of
stock when the commodity is needed.
•Include costs due to loss in customers goodwill and
potential loss in income.
2.Demand:
a)Deterministic demand :
•It is assumed that the quantities needed over
subsequent periods of time are known with certainty.
b)Probabilistic demand:
•The requirements over a certain period of time are not
known with certainty
•Requirements pattern can be described by a known
probability distribution

3. Engineering Management 19/12/2013
M M HASAN, Lecturer, AIE, HSTU, DINAJPUR 3
3.Ordering Cycle:
a)Continuous review
•A record of the inventory level is updated continuously
until a certain lower limit is reached at which point a
new order is placed.
b)Periodic review
•Orders are placed usually at equally spaced intervals of
time.
4.Delivery Lags or Lead Times:
•Time between the placement of an order and its
receipt is called delivery lag or lead time.
5.Stock Replenishment:
•The actual replenishment of stock may occur
instantaneously or uniformly.
6.Time Horizon:
•The time horizon defines the period over which the
inventory level will be controlled.

7.Number of supply Echelons:
•An inventory system may consist of several stocking
points.

8.Number of items:
•An inventory system may involve more than one
item.

4. Engineering Management 19/12/2013
M M HASAN, Lecturer, AIE, HSTU, DINAJPUR 4

The simplest type of inventory model occurs when demand
is constant over time with instantaneous replenishment
and no shortages.

Typical situations to which this model may apply are-
1.Use of light bulbs in a building.
2.Use of clerical supplies, such as paper, pads, and pencils, in an
office.
3.Use of certain industrial supplies such as bolts and nuts.
etc.

5. Engineering Management 19/12/2013
M M HASAN, Lecturer, AIE, HSTU, DINAJPUR 5
Let,
K= Setup cost per unit time
h= Holding cost per unit inventory per unit time
β= Demand per unit time
Y= Inventory level
TCU= Total cost per unit time
t
o=Y/β

TCU as a function of Y can be written as
TCU (Y) = (Setup cost/ unit time)+(holding cost/unit time)
=
E
n
g
+ℎ×

M

TCU(Y)=
Ea

+

M

For minimum cost,
t
t1
/203=0
so that, −
Ea

N
+
m
M
=0
⟹
m
M
=


N

⟹3
∗
=
MEa
m

This is called as wilson’s economic lot size for single item
static model.

6. Engineering Management 19/12/2013
M M HASAN, Lecturer, AIE, HSTU, DINAJPUR 6
Total inventory cost,
/203
∗
= cuIe
•Assumptions in single item static model:
I.Single item
II.Deterministic demand
III.Instantaneous replenishment
IV.Zero delivery lag
V.Uniform demand per unit time.
The purchasing price per unit depends on the quantity
purchased.
This usually occurs in the form of discrete price breaks
or quantity discounts.
In such cases, the purchasing price should be
considered in the inventory model.
Consider the inventory model with instantaneous stock
replenishment and no shortage.
Let,
C
1= cost per unit for y<q
C
2= cost per unit for y≥q
The total cost per unit time for y<q is
/20
T3=
1+

3
+
ℎ
2
3

7. Engineering Management 19/12/2013
M M HASAN, Lecturer, AIE, HSTU, DINAJPUR 7
•The total cost per unit time for y≥q is
/20
M3= IU
2+

3
+
ℎ
2
3

•Let Y
m be the quantity at which the minimum
values of TCU
1 and TCU
2 occur.
3
J=
cuI
ℎ

Define y=q
1(>y
m) such that
TCU
1 (y
m) = TCU
2(q
1)

8. Engineering Management 19/12/2013
M M HASAN, Lecturer, AIE, HSTU, DINAJPUR 8
Total cost per unit

/20P
⋇
=
/20
2P
J, C≤
/20
MC, P
o<C<C
T

/20
1

P
J,C≥C
T>P
o

lxsw
⋇
=lxs
hw
k
,
d≤
lxsw
⋇
=lxs
hd, w
k<d<d
f
lxsw
⋇
=lxs
fw
k,d≥d
f>w
k
q
1 q
1
q
1

9. Engineering Management 19/12/2013
M M HASAN, Lecturer, AIE, HSTU, DINAJPUR 9
The daily demand for a commodity is approximately 100 unit.
Every time an order is placed, a fixed cost Tk. 100 is incurred.
The daily holding cost per unit inventory is TK. 0.02. if the lead
time is 12 days . Determine the willson’s economic lot size and
the reorder point.
Solution:
Given,
Demand, β= 100 unit/day
Setup cost, K= 100 Tk/order
Holding cost, h= 0.02 Tk/unit/day
Lead time= 12 days

We know,
Economic lot size, 3
∗
=
MEa
m
=
M×Tzz×Tzz
z.
=1000
Required,
i.Economic lot size =?
ii.Reorder point=?
Ordering time,
F
∗
=
3
∗
I
=
1000
100
=10
Effective Lead time= 12-10=2 days
Demand for effective lead time= 2β = 2 x 100 =200 unit
Reorder point= 200 unit
10 days

1000
unit

Reorder point

Lead
time 12
days

200 unit

10. Engineering Management 19/12/2013
M M HASAN, Lecturer, AIE, HSTU, DINAJPUR 10
Consider the inventory model with the following information.
Demand, β= 5 unit/day
Setup cost, K= 10 Tk/order
Holding cost, h= 1 Tk/unit/day
C
1=2, 0<q<15
C
2= 1, q≥15

We know,
First compute, 3
o=
MEa
m
=
M× ×6
T
=10
Since y
m< q, it is necessary to check whether q is less than q
1.
The value of q
1 is computed from
TCU
1 (y
m) = TCU
2(q
1)
IU
1+

P
o
+
ℎ
2
P
o=
2+

C
T
+
ℎ
2
C
T

⟹5×2+
10×5
10
+
1
2
×10=5×1+
10×5
C
T
+
1
2
C
T

⟹C
T
2
−30C
T+100=0

⟹C
T=
30±900−400
2


⟹C
T=26.18 3.82
By definition, q
1 is selected as the larger value.
Since q
1 > q, it follows that y*=q=15
Total cost per unit, sw
⋇
=lxs
hd, w
k<d<d
f
=×f+
×

+
f
h
×
= 15.83