Order Quantities when Demand is Approximately Level Chapter 5 Inventory Management Dr. Ron...
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Order Quantities when Demand is Approximately Level Chapter 5 Inventory Management Dr. Ron Tibben-Lembke
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Order Quantities when Demand is
Approximately Level
Order Quantities when Demand is
Approximately LevelChapter 5
Inventory Management
Dr. Ron Tibben-Lembke
Inventory CostsInventory 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 CostsShrinkage 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
Inventory Holding Costs
Category % of Value
Housing (building) cost 6%
Material handling 3%
Labor cost 3%
Opportunity/investment 11%
Pilferage/scrap/obsolescence 3%
Total Holding Cost 26%
ABC Analysis
Divides on-hand inventory into 3 classes A class, B class, C class
Basis is usually annual $ volume $ 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
0
20
40
60
80
100
0 50 100 150
0
20
40
60
80
100
0 50 100 150
% of Inventory Items% of Inventory Items
% Annual $ Usage% Annual $ Usage
AA
BB CC
ABC Classification Solution
Stock # Vol. Cost $ Vol. % ABC
206 26,000 $ 36 $936,000
105 200 600 120,000
019 2,000 55 110,000
144 20,000 4 80,000
207 7,000 10 70,000
Total 1,316,000
ABC Classification Solution
Stock # Vol. Cost $ Vol. % ABC
206 26,000 $ 36 $936,000 71.1 A
105 200 600 120,000 9.1 A
019 2,000 55 110,000 8.4 B
144 20,000 4 80,000 6.1 B
207 7,000 10 70,000 5.3 C
Total 1,316,000 100.0
Economic Order Quantity
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
EOQEOQ
Time
Inventory Level
Q*OptimalOrderQuantity
Decrease Due toConstant Demand
EOQEOQ
Time
Inventory Level
Q*OptimalOrderQuantity
InstantaneousReceipt of OptimalOrder Quantity
EOQEOQ
Time
Inventory Level
Q*
Lead Time
ReorderPoint(ROP)
EOQEOQ
Time
Inventory Level
Q*
Lead Time
ReorderPoint(ROP)
Average Inventory Q/2
Total CostsTotal 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
DA
QvrQTC *
2*)(
How Much to Order?How Much to Order?
Annual Cost
Order Quantity
Holding Cost= H * Q/2
How Much to Order?How Much to Order?
Annual Cost
Order Quantity
Holding Cost= H * Q/2
Ordering Cost= A * D/Q
How Much to Order?How Much to Order?
Annual Cost
Order Quantity
Total Cost= Holding + Ordering
How Much to Order?How Much to Order?
Annual Cost
Order Quantity
Total Cost= Holding + Ordering
Optimal Q
Optimal QuantityOptimal Quantity
Q
DA
Qvr *
2* Total Costs =
Optimal QuantityOptimal Quantity
Q
DA
Qvr *
2* Total Costs =
2*
2 Q
DA
vr
Take derivative with respect to Q =
Optimal QuantityOptimal Quantity
Q
DA
Qvr *
2* Total Costs =
2*
2 Q
DA
vr
Take derivative with respect to Q =
Set equal to zero0
Optimal QuantityOptimal Quantity
Q
DA
Qvr *
2* Total Costs =
2*
2 Q
DA
vr
Take derivative with respect to Q =
Solve for Q:
22 Q
DAvr
Set equal to zero0
Optimal QuantityOptimal Quantity
Q
DA
Qvr *
2* Total Costs =
2*
2 Q
DA
vr
Take derivative with respect to Q =
Solve for Q:
22 Q
DAvr
Set equal to zero0
vr
ASQ
22
Optimal QuantityOptimal Quantity
Q
DA
Qvr *
2* Total Costs =
2*
2 Q
DA
vr
Take derivative with respect to Q =
Solve for Q:
22 Q
DAvr
Set equal to zero0
vr
ASQ
22 vr
ASQ
2
SensitivitySensitivity
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:
EOQ = 100, Q = 150, so p = 0.5
50*(0.25/1.5) = 8.33 a 8.33% cost increase
p
p
150PCP
2
Figure 5.3 SensitivityFigure 5.3 Sensitivity
Percentage Cost Penalty using Q different from the EOQ
-5
0
5
10
15
20
25
30
-0.6 -0.4 -0.2 0 0.2 0.4 0.6
p
PC
P
A Question: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 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 5.1Tabular 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?
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
rTT
ADv
21
288
How do you get a table?
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?
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 OrdersUncoordinated Orders
Time
Simultaneous OrdersSimultaneous Orders
TimeSame T = number months supply allows firm to order atsame time, saving freight and ordering expensesAdjusted some T’s, changed order times
Offset OrdersOffset Orders
Same T = number months supply allows firm to control maximum inventory level by coordinating replenishmentsWith different T, no consistency
Quantity DiscountsQuantity Discounts
How does this all change if price changes depending on order size?
Explicitly consider price:
vr
ADQ
2
Discount ExampleDiscount Example
D = 10,000 A = $20 r = 20%
Price Quantity EOQv = 5.00 Q < 500 633
4.50 501-999 6663.90 Q >= 1000 716
Discount PricingDiscount Pricing
Total Cost
Order Size500 1,000
Price 1 Price 2 Price 3
X 633
X 666
X 716
Discount PricingDiscount Pricing
Total Cost
Order Size500 1,000
Price 1 Price 2 Price 3
X 633
X 666
X 716
Discount ExampleDiscount Example
Order 666 at a time:Hold 666/2 * 4.50 * 0.2= $299.70Order 10,000/666 * 20 = $300.00Mat’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=$390.00Order 10,000/1,000 * 20 = $200.00Mat’l 10,000*3.90 = $39,000.00 39,590.00
Discount Model 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 TimeAdding Lead Time
Use same order size
Order before inventory depleted R = DL where:
D = annual demand rate L = lead time in years
vr
DAQ
2
