2.
INVENTORY SYSTEM
InventoryInventory is the stock of any item or resource
used in an organization and can include: raw
materials, finished products, component parts,
supplies, and work-in-process
An inventory systeminventory system is the set of policies and
controls that monitor levels of inventory and
determines what levels should be maintained,
when stock should be replenished, and how
large orders should be
3.
INVENTORY COSTS
Holding (or carrying) costs
Costs for storage, handling, insurance, etc
Setup (or production change) costs
Costs for arranging specific equipment
setups, etc
Ordering costs
Costs of someone placing an order, etc
Shortage costs
Costs of canceling an order, etc
4.
E(1
)
Independent vs. Dependent Demand
Independent Demand (Demand for the final end-
product or demand not related to other items)
Dependent
Demand
(Derived demand
items for
component
parts,
subassemblies,
raw materials,
etc)
Finished
product
Component parts
5.
Inventory Systems
Single-Period Inventory Model
One time purchasing decision (Example:
vendor selling t-shirts at a football game)
Seeks to balance the costs of inventory
overstock and under stock
Multi-Period Inventory Models
Fixed-Order Quantity Models
Event Triggered (Example: running out of
stock)
Fixed-Time Period Models
Time triggered (Example: Monthly sales call
by sales representative)
6.
Fixed-order quantity model
also called economic order quantity
(EOQ)
an inventory control model where the
amount requisitioned is fixed and the
actual ordering is triggered by
inventory dropping to a specified level
of inventory
7.
Fixed-Order Quantity Model
Assumptions
Demand for the product is constant and uniform
throughout the period
Lead time (time from ordering to receipt) is
constant
Price per unit of product is constant
8.
Fixed-Order Quantity Model
Assumptions
Inventory holding cost is based on average
inventory
Ordering or setup costs are constant
All demands for the product will be satisfied (No
back orders are allowed)
9.
Basic Fixed-Order Quantity Model and
Reorder Point Behavior
R = Reorder point
Q = Economic order quantity
L = Lead time
L L
Q QQ
R
Time
Number
of units
on hand
1. You receive an order quantity Q.
2. Your start using
them up over time. 3. When you reach down to
a level of inventory of R,
you place your next Q
sized order.
4. The cycle then repeats.
10.
Cost Minimization Goal
Ordering Costs
Holding
Costs
Order Quantity (Q)
C
O
S
T
Annual Cost of
Items (DC)
Total Cost
QOPT
By adding the item, holding, and ordering costs
together, we determine the total cost curve, which in
turn is used to find the Qopt inventory order point that
minimizes total costs
By adding the item, holding, and ordering costs
together, we determine the total cost curve, which in
turn is used to find the Qopt inventory order point that
minimizes total costs
11.
Basic Fixed-Order Quantity
(EOQ) Model Formula
H
2
Q
+S
Q
D
+DC=TC
Total
Annual =
Cost
Annual
Purchase
Cost
Annual
Ordering
Cost
Annual
Holding
Cost
+ +
TC=Total annual
cost
D =Demand
C =Cost per unit
Q =Order quantity
S =Cost of placing
an order or setup
cost
R =Reorder point
L =Lead time
H=Annual holding
and storage cost
per unit of inventory
TC=Total annual
cost
D =Demand
C =Cost per unit
Q =Order quantity
S =Cost of placing
an order or setup
cost
R =Reorder point
L =Lead time
H=Annual holding
and storage cost
per unit of inventory
12.
Deriving the EOQ
Using calculus, we take the first derivative of
the total cost function with respect to Q,
and set the derivative (slope) equal to zero,
solving for the optimized (cost minimized)
value of Qopt
Using calculus, we take the first derivative of
the total cost function with respect to Q,
and set the derivative (slope) equal to zero,
solving for the optimized (cost minimized)
value of Qopt
Q =
2DS
H
=
2(Annual Demand)(Order or Setup Cost)
Annual Holding CostOPT
Reorder point, R = d L
_
(constant)timeLead=L
(constant)demanddailyaverage=d
_
We also need a
reorder point to
tell us when to
place an order
We also need a
reorder point to
tell us when to
place an order
13.
EOQ Example (1) Problem Data
Annual Demand = 1,000 units
Days per year considered in average
daily demand = 365
Cost to place an order = $10
Holding cost per unit per year = $2.50
Lead time = 7 days
Cost per unit = $15
Given the information below, what are the EOQ and
reorder point?
Given the information below, what are the EOQ and
reorder point?
14.
EOQ Example (1) Solution
Q =
2DS
H
=
2(1,000 )(10)
2.50
= 89.443 units orOPT 90 units
d =
1,000 units / year
365 days / year
= 2.74 units / day
Reorder point, R = d L = 2.74units / day (7days) = 19.18 or
_
20 units
In summary, you place an optimal order of 90 units. In
the course of using the units to meet demand, when
you only have 20 units left, place the next order of 90
units.
In summary, you place an optimal order of 90 units. In
the course of using the units to meet demand, when
you only have 20 units left, place the next order of 90
units.
15.
EOQ Example (2) Problem Data
Annual Demand = 10,000 units
Days per year considered in average daily
demand = 365
Cost to place an order = $10
Holding cost per unit per year = 10% of cost per
unit
Lead time = 10 days
Cost per unit = $15
Determine the economic order quantity
and the reorder point given the following…
Determine the economic order quantity
and the reorder point given the following…
16.
EOQ Example (2) Solution
Q =
2DS
H
=
2(10,000 )(10)
1.50
= 365.148 units, orOPT 366 units
d =
10,000 units / year
365 days / year
= 27.397 units / day
R = d L = 27.397 units / day (10 days) = 273.97 or
_
274 units
Place an order for 366 units. When in the course of
using the inventory you are left with only 274 units,
place the next order of 366 units.
Place an order for 366 units. When in the course of
using the inventory you are left with only 274 units,
place the next order of 366 units.
17.
Economic Production Quantity (EPQ)
Production done in batches or lots
Capacity to produce a part exceeds the
part’s usage or demand rate
Assumptions of EPQ are similar to EOQ
except orders are received
incrementally during production
18.
Assumptions:
Only one item involved
Annual demand is known
The usage rate is constant
Usage occurs continually, but
production occurs periodically
The production rate is constant
Lead time does not vary
There are no quantity discounts
20.
Formulas:
No. of runs per year = D/Q
Annual setup cost = (D/Q)S
TCmin = Carrying Cost + Setup Cost = ( )H+(D/Qo)S
Qo = ; where p = production or delivery
rate
u = usage rate
Cycle Time =
Run Time =
Imax = (p-u) and
Iaverage =
21.
Example 1
A toy manufacturer uses 48,000 rubber wheels
per year for its popular dump truck series. The
firm makes its own wheels, which it can
produce at a rate of 800 per day. The toy trucks
are assembled uniformly over the entire year.
Carrying cost is $1 per wheel a year. Setup cost
for a production run of wheels is $45. The firm
operates 240 days per year.
22.
Determine the
a.Optimal run size.
b.Minimum total annual cost for carrying
and setup.
c.Cycle time for the optimal run size.
d.Run time.
23.
Given:
D = 48,000 wheels per year
S = $45
H = $1 per wheel per year
p = 800 wheels per day
u = 48,000 wheels per 240 days, or 200
wheels per day
24.
a. Qo=
Qo= = 2,400 wheels
b. TCmin = ( )H+(D/Qo)S
Thus, you must first compute Imax:
Imax = (p-u) = (800-200) = 1,800 wheels
TC = ( )($1) + ( )($45) = $1,800
Solution:
25.
c. Cycle Time = = = 12 days
Thus, a run of wheels will be made every 12 days.
d. Run Time = = = 3 days
Thus, each run will require 3 days to complete.
26.
Example 2
The Dine Corporation is both a producer & a user of brass
couplings. The firm operates 220 days a year & uses the
couplings at a steady rate of 50 per day. Couplings can be
produced at a rate of 200 per day. Annual storage cost is $2 per
coupling, & machine setup cost is $70 per run.
a.Determine the economic run quantity.
b.Approximately how many runs per year will there be?
c.Compute the maximum inventory level.
28.
b. No. of runs = D = 11,000 = 10.86 ≈ 11
per year Q0 1,013
c. Imax = Qp (p - u) = 1,013 (200 - 50)
p
Imax = 759.75 or 760 units
29.
Quantity Discounts
• Quantity Discounts, also called Price-Break
Models, are price reductions for large orders
offered to costumers to induce them to buy in
large quantities.
• The buyer must weigh the potential benefits
of reduced purchase price and fewer orders
that will result from buying in large quantities
against the increase in carry costs caused by
higher average inventories.
30.
Quantity Discounts
• The buyer’s goal with quantity discounts is to select
the order quantity that will minimize total cost.
• Total cost is the sum of carrying cost, ordering cost
and purchasing cost.
31.
Price-Break Model Formula
CostHoldingAnnual
Cost)SetuporderDemand)(Or2(Annual
=
iC
2DS
=QOPT
Based on the same assumptions as the EOQ model,
the price-break model has a similar Qopt formula:
i = percentage of unit cost attributed to carrying inventory
C = cost per unit
Since “C” changes for each price-break, the formula
above will have to be used with each price-break cost
value
32.
Total Cost = Carrying Cost + Ordering Cost + Purchasing Cost
TC = (Q/2)H + (D/Q)S + PD
Where:
Q = Quantity P = Unit Price
D = Demand
S = Ordering Cost
33.
Example1
A company has a chance to reduce their inventory
ordering costs by placing larger quantity orders using
the price-break order quantity schedule below. What
should their optimal order quantity be if this company
purchases this single inventory item with an e-mail
ordering cost of $4, a carrying cost rate of 2% of the
inventory cost of the item, and an annual demand of
10,000 units?
A company has a chance to reduce their inventory
ordering costs by placing larger quantity orders using
the price-break order quantity schedule below. What
should their optimal order quantity be if this company
purchases this single inventory item with an e-mail
ordering cost of $4, a carrying cost rate of 2% of the
inventory cost of the item, and an annual demand of
10,000 units?
Order Quantity(units) Price/unit($)
0 to 2,499 $1.20
2,500 to 3,999 1.00
4,000 or more .98
34.
Solution
units1,826=
0.02(1.20)
4)2(10,000)(
=
iC
2DS
=QOPT
Annual Demand (D)= 10,000 units
Cost to place an order (S)= $4
First, plug data into formula for each price-break value of “C”
units2,000=
0.02(1.00)
4)2(10,000)(
=
iC
2DS
=QOPT
units2,020=
0.02(0.98)
4)2(10,000)(
=
iC
2DS
=QOPT
Carrying cost % of total cost (i)= 2%
Cost per unit (C) = $1.20, $1.00, $0.98
Interval from 0 to 2499, the
Qopt value is feasible
Interval from 2500-3999, the
Qopt value is not feasible
Interval from 4000 & more, the
Qopt value is not feasible
Next, determine if the computed Qopt values are feasible or not
35.
Solution
Since the feasible solution occurred in the first price-
break, it means that all the other true Qopt values occur
at the beginnings of each price-break interval. Why?
Since the feasible solution occurred in the first price-
break, it means that all the other true Qopt values occur
at the beginnings of each price-break interval. Why?
0 1826 2500 4000 Order Quantity
Total
annual
costs
So the candidates
for the price-
breaks are 1826,
2500, and 4000
units
So the candidates
for the price-
breaks are 1826,
2500, and 4000
units
Because the total annual cost function is
a “u” shaped function
Because the total annual cost function is
a “u” shaped function
36.
Solution
iC
2
Q
+S
Q
D
+DC=TC
Next, we plug the true Qopt values into the total cost
annual cost function to determine the total cost under
each price-break
Next, we plug the true Qopt values into the total cost
annual cost function to determine the total cost under
each price-break
TC(0-2499)=(10000*1.20)+(10000/1826)*4+(1826/2)(0.02*1.20)
= $12,043.82
TC(2500-3999)= $10,041
TC(4000&more)= $9,949.20
TC(0-2499)=(10000*1.20)+(10000/1826)*4+(1826/2)(0.02*1.20)
= $12,043.82
TC(2500-3999)= $10,041
TC(4000&more)= $9,949.20
Finally, we select the least costly Qopt, which is this
problem occurs in the 4000 & more interval. In
summary, our optimal order quantity is 4000 units
Finally, we select the least costly Qopt, which is this
problem occurs in the 4000 & more interval. In
summary, our optimal order quantity is 4000 units
37.
Example2
Consider the following case, where
D = 10,000 units (annual demand)
i = 20% of cost (annual carrying cost, storage, interest,
obsolescence, etc.)
S = $20 to place an order
C = Cost per unit (according to order size; 0-499 units,
$5 per unit; 500-999, $4.5 per unit; 1,000 and up, $3.9
per unit)
Consider the following case, where
D = 10,000 units (annual demand)
i = 20% of cost (annual carrying cost, storage, interest,
obsolescence, etc.)
S = $20 to place an order
C = Cost per unit (according to order size; 0-499 units,
$5 per unit; 500-999, $4.5 per unit; 1,000 and up, $3.9
per unit)
What quantity should be ordered?What quantity should be ordered?
38.
Solution
units632.46=
0.2(5)
20)2(10,000)(
=
iC
2DS
=QOPT
Annual Demand (D)= 10,000 units
Cost to place an order (S)= $20
First, plug data into formula for each price-break value of “C”
units666.67=
0.2(4.50)
20)2(10,000)(
=
iC
2DS
=QOPT
units716.11=
0.2(3.90)
20)2(10,000)(
=
iC
2DS
=QOPT
Carrying cost % of total cost (i)= 20%
Cost per unit (C) = $5, $4.5, $3.9
Interval from 0 to 499, the Qopt
value is feasible
Interval from 500-999, the Qopt
value is not feasible
Interval from 1000 & more, the
Qopt value is not feasible
Next, determine if the computed Qopt values are feasible or not
39.
Solution
iC
2
Q
+S
Q
D
+DC=TC
Next, we plug the true Qopt values into the total cost
annual cost function to determine the total cost under
each price-break
Next, we plug the true Qopt values into the total cost
annual cost function to determine the total cost under
each price-break
TC(0-499)=(10000*5)+(10000/634)*20+(634/2)(0.2*5)
= $50,632.46
TC(500-999)= $45,600
TC(1000&more)= $39,558.57
TC(0-499)=(10000*5)+(10000/634)*20+(634/2)(0.2*5)
= $50,632.46
TC(500-999)= $45,600
TC(1000&more)= $39,558.57
Finally, we select the least costly Qopt, which in this
problem occurs in the 1000 & more interval. In
summary, our optimal order quantity is 1000 units
Finally, we select the least costly Qopt, which in this
problem occurs in the 1000 & more interval. In
summary, our optimal order quantity is 1000 units
40.
A-B-C Classification
Classifies Inventory items according to
some measure of importance, Usually
annual dollar value, and then allocates
control efforts accordingly
41.
Three Classes of Items
A (Very Important Items)
Account for 10%-20% of the number of items in inventory
60%-70% of the dollar inventory of an inventory
B (Moderately Important)
Account for 20%-30% of the number of items in inventory
20%-50% of the dollar inventory of an inventory
C (Least Important)
Account for 50%-60% of the number of items in inventory
10%-15% of the dollar inventory of an inventory
42.
A typical A-B-C breakdown in relative annual dollar
value of items and number of items by category.
43.
Example
The annual dollar value of 12 items
has been calculated according to
annual demand and unit cost. The
annual dollar values were then arrayed
from highest to lowest to simplify
classification of items.
44.
Item
Number
Annual
Demand
Unit Cost
Annual Dollar
Value
Classification
8 $1,000 $4,000 $4,000,000 A
5 $3,900 $700 $2,730,000 A
3 $1,900 $500 $950,000 B
6 $1,000 $915 $915,000 B
1 $2,500 $330 $825,000 B
4 $1,500 $100 $150,000 C
12 $400 $300 $120,000 C
11 $500 $200 $100,000 C
9 $8,000 $10 $80,000 C
2 $1,000 $70 $70,000 C
7 $200 $210 $42,000 C
10 $9,000 $2 $18,000 C
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