Transcript ppt
Order Quantities when
Demand is Approximately
Level
Chapter 5
Inventory Management
Dr. Ron Tibben-Lembke
Inventory Costs
Costs associated with inventory:
Cost of the products
Cost of ordering
Cost of hanging onto it
Cost of having too much / disposal
Cost of not having enough (shortage)
Shrinkage Costs
How much is stolen?
2% for discount, dept. stores, hardware,
convenience, sporting goods
3% for toys & hobbies
1.5% for all else
Where does the missing stuff go?
Employees: 44.5%
Shoplifters: 32.7%
Administrative / paperwork error: 17.5%
Vendor fraud: 5.1%
Inventory Holding Costs
Category
% of Value
Housing (building) cost
6%
Material handling
3%
Labor cost
3%
Opportunity/investment
Pilferage/scrap/obsolescence
Total Holding Cost
11%
3%
26%
ABC Analysis
Divides on-hand inventory into 3 classes
Basis is usually annual $ volume
A class, B class, C class
$ volume = Annual demand x Unit cost
Policies based on ABC analysis
Develop class A suppliers more
Give tighter physical control of A items
Forecast A items more carefully
Classifying Items
as ABC
% Annual $ Usage
100
80
60
A
40
B
20
C
0
0
50
100
% of Inventory Items
150
ABC Classification Solution
Stock #
Vol.
206
105
019
144
207
26,000
200
2,000
20,000
7,000
Total
Cost
$ Vol.
$ 36 $936,000
600 120,000
55 110,000
4
80,000
10
70,000
1,316,000
%
ABC
ABC Classification Solution
Stock #
Vol.
206
105
019
144
207
26,000
200
2,000
20,000
7,000
Total
Cost
$ Vol.
$ 36 $936,000
600 120,000
55 110,000
4
80,000
10
70,000
%
71.1
9.1
8.4
6.1
5.3
1,316,000 100.0
ABC
A
A
B
B
C
Economic Order Quantity
Assumptions
Demand rate is known and constant
No order lead time
Shortages are not allowed
Costs:
A - setup cost per order
v - unit cost
r - holding cost per unit time
EOQ
Inventory
Level
Q*
Optimal
Order
Quantity
Decrease Due to
Constant Demand
Time
EOQ
Inventory
Level
Q*
Optimal
Order
Quantity
Instantaneous
Receipt of Optimal
Order Quantity
Time
EOQ
Inventory
Level
Q*
Reorder
Point
(ROP)
Time
Lead Time
EOQ
Inventory
Level
Q*
Average
Inventory Q/2
Reorder
Point
(ROP)
Time
Lead Time
Total Costs
Average Inventory = Q/2
Annual Holding costs = rv * Q/2
# Orders per year = D / Q
Annual Ordering Costs = A * D/Q
Annual Total Costs = Holding + Ordering
Q
D
TC (Q) vr * A *
2
Q
How Much to Order?
Annual Cost
Holding Cost
= H * Q/2
Order Quantity
How Much to Order?
Annual Cost
Ordering Cost
= A * D/Q
Holding Cost
= H * Q/2
Order Quantity
How Much to Order?
Annual Cost
Total Cost
= Holding + Ordering
Order Quantity
How Much to Order?
Total Cost
= Holding + Ordering
Annual Cost
Optimal Q
Order Quantity
Optimal Quantity
Total Costs =
Q
D
vr * A *
2
Q
Optimal Quantity
Total Costs =
Q
D
vr * A *
2
Q
Take derivative
with respect to Q =
vr
D
A* 2
2
Q
Optimal Quantity
Total Costs =
Q
D
vr * A *
2
Q
Take derivative
with respect to Q =
vr
D
A* 2 0
2
Q
Set equal
to zero
Optimal Quantity
Total Costs =
Q
D
vr * A *
2
Q
Take derivative
with respect to Q =
vr
D
A* 2 0
2
Q
Solve for Q:
vr DA
2
2 Q
Set equal
to zero
Optimal Quantity
Total Costs =
Q
D
vr * A *
2
Q
Take derivative
with respect to Q =
vr
D
A* 2 0
2
Q
Solve for Q:
vr DA
2
2 Q
2 AS
Q
vr
2
Set equal
to zero
Optimal Quantity
Total Costs =
Q
D
vr * A *
2
Q
Take derivative
with respect to Q =
vr
D
A* 2 0
2
Q
Set equal
to zero
Solve for Q:
vr DA
2
2 Q
2 AS
Q
vr
2
2 AS
Q
vr
Sensitivity
Suppose we do not order optimal EOQ, but
order Q instead, and Q is p percent larger
Q = (1+p) * EOQ
Percentage Cost Penalty given by:
p2
PCP 50
1 p
EOQ = 100, Q = 150, so p = 0.5
50*(0.25/1.5) = 8.33 a 8.33% cost increase
Figure 5.3 Sensitivity
Percentage Cost Penalty using Q different from the EOQ
30
25
20
PCP
15
10
5
0
-0.6
-0.4
-0.2
0
-5
p
0.2
0.4
0.6
A Question:
If the EOQ is based on so many
horrible assumptions that are never
really true, why is it the most
commonly used ordering policy?
Benefits of EOQ
Profit function is very shallow
Even if conditions don’t hold
perfectly, profits are close to optimal
Estimated parameters will not throw
you off very far
Tabular Aid 5.1
For A = $3.20 and r = 0.24%
Calculate Dv =total $ usage (or sales)
Find where Dv fits in the table
Use that number of months of supply
D = 200, v = $16, Dv=$3,200
From table, buy 1 month’s worth
Q = D/12 = 200/12 = 16.7 = 17
How do you get a table?
Decide which T values you want to
consider: 1 month, etc.
Use same v and r values for whole table
For each neighboring set of T’s, put them
into
288 A
Dv
T1T2 r
How do you get a table?
For example, A = $3.20, r = 0.24
To find the breakpoint between 0.25 and 0.5
Dv = 288 * 3.2 / (0.25 * 0.5 * 0.24)
= 921.6 / 0.03 = 30,720
So if Dv is less than this, use 0.25, more
than that, use 0.5
Find 0.5 and 0.75 breakpoint:
Dv = 288 * 3.2/(0.5 * 0.75 * 0.24) = 10,2240
Why care about a table?
Some simple calculations to get set up
No thinking to figure out lot sizes
Every product with the same ordering cost
and holding cost rate can use it
Real benefit - simplified ordering
Every product ordered every 1 or 2 weeks, or
every 1, 2, 3, 4, 6, 12 months
Order multiple products on same schedule:
Get volume discounts from suppliers
Save on shipping costs
Savings outweigh small increase from non-EOQ orders
Uncoordinated Orders
Time
Simultaneous Orders
Time
Same T = number months supply allows firm to order at
same time, saving freight and ordering expenses
Adjusted some T’s, changed order times
Offset Orders
Same T = number months supply allows firm to control
maximum inventory level by coordinating replenishments
With different T, no consistency
Quantity Discounts
How does this all change if price
changes depending on order size?
Explicitly consider price:
2 AD
Q
vr
Discount Example
D = 10,000
A = $20
Price
v = 5.00
4.50
3.90
Quantity
Q < 500
501-999
Q >= 1000
r = 20%
EOQ
633
666
716
Discount Pricing
Total Cost
Price 1
Price 2
Price 3
X 633
X 666
X 716
500
1,000
Order Size
Discount Pricing
Total Cost
Price 1
Price 2
Price 3
X 633
X 666
X 716
500
1,000
Order Size
Discount Example
Order 666 at a time:
Hold 666/2 * 4.50 * 0.2= $299.70
Order 10,000/666 * 20 = $300.00
Mat’l 10,000*4.50 = $45,000.00 45,599.70
Order 1,000 at a time:
Hold 1,000/2 * 3.90 * 0.2=$390.00
Order 10,000/1,000 * 20 = $200.00
Mat’l 10,000*3.90 = $39,000.00 39,590.00
Discount Model
1.Compute EOQ for each price
2.Is EOQ ‘realizeable’? (is Q in range?)
If EOQ is too large, use lowest
possible value. If too small, ignore.
3.Compute total cost for this quantity
4.Select quantity/price with lowest total
cost.
Adding Lead Time
Use same order size
Order before inventory depleted
R = DL where:
2 DA
Q
vr
D = annual demand rate
L = lead time in years