INVENTORY MANAGEMENT Operations Management Dr. Ron Tibben-Lembke
Mar 29, 2015
INVENTORY MANAGEMENT
Operations Management
Dr. Ron Tibben-Lembke
Purposes of Inventory
Meet anticipated demand Demand variability Supply variability
Decouple production & distribution permits constant production quantities
Take advantage of quantity discounts Hedge against price increases Protect against shortages
2006 13.81 1857 24.0% 446 801 58 1305 9.92007
US Inventory, GDP ($B)
-
2,000
4,000
6,000
8,000
10,000
12,000
14,000
1984
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
Business Inventories US GDP
US Inventories as % of GDP
0.0%
5.0%
10.0%
15.0%
20.0%
25.0%
1984
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
Year
% o
f G
DP
Source: CSCMP, Bureau of Economic Analysis
Two Questions
Two main Inventory Questions: How much to buy? When is it time to buy?Also:Which products to buy?From whom?
Types of Inventory
Raw Materials Subcomponents Work in progress (WIP) Finished products Defectives Returns
Inventory Costs
What costs do we experience because we carry inventory?
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 ValueHousing (building) cost 4%Material handling 3%Labor cost 3%Opportunity/investment 9%Pilferage/scrap/obsolescence 2%
Total Holding Cost 21%
Inventory Models
Fixed order quantity models How much always same, when changes Economic order quantity Production order quantity Quantity discount
Fixed order period models How much changes, when always same
Economic Order Quantity
Assumptions Demand rate is known and constant No order lead time Shortages are not allowed Costs:
S - setup cost per order H - holding cost per unit time
EOQ
Time
Inventory Level
Q*OptimalOrderQuantity
Decrease Due toConstant Demand
EOQ
Time
Inventory Level
Q*OptimalOrderQuantity
InstantaneousReceipt of OptimalOrder Quantity
EOQ
Time
Inventory Level
Q*
Lead Time
ReorderPoint(ROP)
EOQ
Time
Inventory Level
Q*
Lead Time
ReorderPoint(ROP)
Average Inventory Q/2
Total Costs
Average Inventory = Q/2 Annual Holding costs = H * Q/2 # Orders per year = D / Q Annual Ordering Costs = S * D/Q Cost of Goods = D * C Annual Total Costs = Holding + Ordering +
CoG
DCQ
DS
QHQTC **
2*)(
How Much to Order?
Annual Cost
Order Quantity
Holding Cost= H * Q/2
How Much to Order?
Annual Cost
Order Quantity
Holding Cost= H * Q/2
Ordering Cost= S * D/Q
How Much to Order?
Annual Cost
Order Quantity
Total Cost= Holding + Ordering
How Much to Order?
Annual Cost
Order Quantity
Total Cost= Holding + Ordering
Optimal Q
Optimal Quantity
Total Costs =
2*
2 Q
DS
H
Take derivative with respect to Q =
Solve for Q:
22 Q
DSH
Set equal to zero0
H
DSQ
22 H
DSQ
2
DCQ
DS
QH **
2*
Adding Lead Time
Use same order size
Order before inventory depleted R = * L where:
= average demand rate (per day) L = lead time (in days) both in same time period (wks, months,
etc.)
H
DSQ
2
dd
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? 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
Quantity Discounts
How does this all change if price changes depending on order size?
Holding cost as function of cost: H = I * C
Explicitly consider price:
Q2DS
I C
Discount Example
D = 10,000 S = $20 I = 20%
Price Quantity EOQc = 5.00 Q < 500
6334.50 501-999
6663.90 Q >= 1000 716
Discount Pricing
Total Cost
Order Size500 1,000
Price 1 Price 2 Price 3
X 633
X 666
X 716
Discount Pricing
Total Cost
Order Size500 1,000
Price 1 Price 2 Price 3
X 633
X 666
X 716
Discount 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
1. Compute EOQ for next cheapest price2. Is EOQ feasible? (is EOQ in range?)
If EOQ is too small, use lowest possible Q to get price.
3. Compute total cost for this quantity4. Repeat until EOQ is feasible or too big.5. Select quantity/price with lowest total
cost.
INVENTORY MANAGEMENT-- RANDOM DEMAND
Random Demand
Don’t know how many we will sell Sales will differ by period Average always remains the same Standard deviation remains constant
Impact of Random Demand
How would our policies change? How would our order quantity change? How would our reorder point change?
Mac’s Decision
How many papers to buy? Average = 90, st dev = 10 Cost = 0.20, Sales Price = 0.50 Salvage = 0.00 Cost of overestimating Demand, CO
CO = 0.20 - 0.00 = 0.20 Cost of Underestimating Demand, CU
CU = 0.50 - 0.20 = 0.30
Optimal Policy
G(x) = Probability demand <= xOptimal quantity:
Mac: G(x) = 0.3 / (0.2 + 0.3) = 0.6From standard normal table, z = 0.253=Normsinv(0.6) = 0.253Q* = avg + zs = 90+ 2.53*10 = 90 +2.53 =
93
uo
u
CC
C
Q)Pr(D
Optimal Policy
If units are discrete, when in doubt, round up
If u units are on hand, order Q - u units Model is called “newsboy problem,”
newspaper purchasing decision By time realize sales are good, no time to
order more By time realize sales are bad, too late,
you’re stuck Similar to the problem of # of Earth Day
shirts to make, lbs. of Valentine’s candy to buy, green beer, Christmas trees, toys for Christmas, etc., etc.
Random Demand – Fixed Order Quantity
If we want to satisfy all of the demand 95% of the time, how many standard deviations above the mean should the inventory level be?
Probabilistic Models
Safety stock = xm
From statistics,
From normal table z.95 = 1.65
Safety stock = zsL= 1.65*10 = 16.5
R = m + Safety Stock
Therefore, z = Safety stock & Safety stock = zsLsL
=350+16.5 = 366.5 ≈ 367
L
xz
Random Example
What should our reorder point be? demand over the lead time is 50 units, with standard deviation of 20 want to satisfy all demand 90% of the time (i.e., 90% chance we do not run out)
To satisfy 90% of the demand, z = 1.28 Safety stock = zσL= 1.28 * 20 = 25.6 R = 50 + 25.6 = 75.6
St Dev Over Lead Time
What if we only know the average daily demand, and the standard deviation of daily demand? Lead time = 4 days, daily demand = 10, standard deviation = 5,
What should our reorder point be, if z = 3?
St Dev Over LT
If the average each day is 10, and the lead time is 4 days, then the average demand over the lead time must be 40.
What is the standard deviation of demand over the lead time?
Std. Dev. ≠ 5 * 4
404*10* Ld
St Dev Over Lead Time
Standard deviation of demand =
R = 40 + 3 * 10 = 70
daydaysL L
1054
daydaysL LzLdzLdR **
Service Level Criteria
Type I: specify probability that you do not run out during the lead time Probability that 100% of customers go
home happy Type II: proportion of demands met from
stock Percentage that go home happy, on
average Fill Rate: easier to observe, is commonly
used G(z)= expected value of shortage, given z.
Not frequently listed in tables
RateFillQ
zGL
1)(
Two Types of Service
Cycle DemandStock-Outs
1 1800
2 750
3 23545
4 1400
5 1800
6 20010
7 1500
8 900
9 160010 400
Sum 1,45055
Type I: 8 of 10 periods80% service
Type II:1,395 / 1,450 =
96%
FIXED-TIME PERIOD MODELS
Fixed-Time Period Model
Every T periods, we look at inventory on hand and place an order
Lead time still is L. Order quantity will be different,
depending on demand
Fixed-Time Period Model: When to Order?
Time
Inventory Level Target maximum
Period
Fixed-Time Period Model: : When to Order?
Time
Inventory Level Target maximum
PeriodPeriod
Fixed-Time Period Model:When to Order?
Time
Inventory Level Target maximum
PeriodPeriod
Fixed-Time Period Model: When to Order?
Time
Inventory Level Target maximum
Period PeriodPeriod
Fixed-Time Period Model: When to Order?
Time
Inventory Level Target maximum
Period PeriodPeriod
Fixed-Time Period Model: When to Order?
Time
Inventory Level Target maximum
Period PeriodPeriod
Fixed Order Period
Standard deviation of demand over T+L =
T = Review period length (in days) σ = std dev per day Order quantity (12.11) =
IzLTdq LT )(
LTLT
Inventory Recordkeeping
Two ways to order inventory: Keep track of how many delivered, sold Go out and count it every so oftenIf keeping records, still need to double-check Annual physical inventory, or Cycle Counting
Cycle Counting
Physically counting a sample of total inventory on a regular basis
Used often with ABC classification A items counted most often (e.g., daily)
Advantages Eliminates annual shut-down for physical
inventory count Improves inventory accuracy Allows causes of errors to be identified
Fixed-Period Model
Answers how much to order Orders placed at fixed intervals
Inventory brought up to target amount Amount ordered varies
No continuous inventory count Possibility of stockout between intervals
Useful when vendors visit routinely Example: P&G rep. calls every 2 weeks
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
% of Inventory Items
% Annual $ Volume
A
B C
Items %$Vol %ItemsA 80 15B 15 30C 5 55
ABC Classification Solution
Stock # Vol. Cost $ Vol. % ABC
206 26,000 $ 36 $936,000 71.1
105 200 600 120,000 9.1
019 2,000 55 110,000 8.4
144 20,000 4 80,000 6.1
207 7,000 10 70,000 5.3
Total 1,316,000 100.0
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