Inventory Management

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Inventory Management

• Operations Management
• Dr. Ron Tibben-Lembke

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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

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2006 13.81 1857 24.0 446 801 58 1305 9.9
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(No Transcript)
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Source CSCMP, Bureau of Economic Analysis
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Two Questions

• Two main Inventory Questions
• How much to buy?
• When is it time to buy?
• Also
• Which products to buy?
• From whom?

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Types of Inventory

• Raw Materials
• Subcomponents
• Work in progress (WIP)
• Finished products
• Defectives
• Returns

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Inventory Costs

• What costs do we experience because we carry
  inventory?

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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)

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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

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Inventory Holding Costs

• Category of Value
• Housing (building) cost 4
• Material handling 3
• Labor cost 3
• Opportunity/investment 9
• Pilferage/scrap/obsolescence 2
• Total Holding Cost 21

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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

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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

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EOQ
Inventory Level
Q Optimal Order Quantity
Decrease Due to Constant Demand
Time
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EOQ
Inventory Level
Instantaneous Receipt of Optimal Order Quantity
Q Optimal Order Quantity
Time
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EOQ
Inventory Level
Q Optimal Order Quantity
Time
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EOQ w Lead Time
Inventory Level
Q Optimal Order Quantity
Time
Lead Time
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EOQ
Inventory Level
Q
Reorder Point (ROP)
Time
Lead Time
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EOQ
Inventory Level
Q
Average Inventory Q/2
Reorder Point (ROP)
Time
Lead Time
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Total Costs

• Average Inventory Q/2
• Annual Holding costs H Q/2
• Orders per year D / Q
• Annual Ordering Costs S D/Q
• Annual Total Costs Holding Ordering

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How Much to Order?
Annual Cost
Holding Cost H Q/2
Order Quantity
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How Much to Order?
Annual Cost
Ordering Cost S D/Q
Holding Cost H Q/2
Order Quantity
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How Much to Order?
Total Cost Holding Ordering
Annual Cost
Order Quantity
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How Much to Order?
Total Cost Holding Ordering
Annual Cost
Optimal Q
Order Quantity
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Optimal Quantity
Total Costs
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Optimal Quantity
Total Costs
Take derivative with respect to Q
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Optimal Quantity
Total Costs
Take derivative with respect to Q
Set equal to zero
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Optimal Quantity
Total Costs
Take derivative with respect to Q
Set equal to zero
Solve for Q
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Optimal Quantity
Total Costs
Take derivative with respect to Q
Set equal to zero
Solve for Q
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Optimal Quantity
Total Costs
Take derivative with respect to Q
Set equal to zero
Solve for Q
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Adding Lead Time

• Use same order size
• Order before inventory depleted
• R L where
• demand rate (per day)
• L lead time (in days)
• both in same time period (wks, months, etc.)

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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?

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Benefits of EOQ

• Profit function is very shallow
• Even if conditions dont hold perfectly, profits
  are close to optimal
• Estimated parameters will not throw you off very
  far

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Sensitivity

• Suppose we do not order optimal Q, but order Q
  instead.
• Percentage profit loss given by
• Should order 100, order 150 (50 over)
• 0.5(1.5 0.66) 1.08 an 8cost increase

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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

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Discount Example

• D 10,000 S 20 I 20
• Price Quantity EOQ
• c 5.00 Q lt 500 633
• 4.50 501-999 666
• 3.90 Q gt 1000 716

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Discount Pricing
Total Cost
Price 1
Price 2
Price 3
X 633
X 666
X 716
Order Size
500 1,000
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Discount Pricing
Total Cost
Price 1
Price 2
Price 3
X 633
X 666
X 716
Order Size
500 1,000
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Discount Example

• Order 666 at a time
• Hold 666/2 4.50 0.2 299.70
• Order 10,000/666 20 300.00
• Matl 10,0004.50 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
• Matl 10,0003.90 39,000.00 39,590.00

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Discount Model

• 1. Compute EOQ for next cheapest price
• 2. 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 quantity
• Repeat until EOQ is feasible or too big.
• Select quantity/price with lowest total cost.

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Inventory Management-- Random Demand
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Master of the Obvious?

• If you focus on the things the customers are
  buying its a little easier to stay in stock
• James Adamson CEO, Kmart Corp.
• 3/12/02
• Fired Jan, 2003

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Random Demand

• Dont know how many we will sell
• Sales will differ by period
• Average always remains the same
• Standard deviation remains constant

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Impact of Random Demand

• How would our policies change?
• How would our order quantity change?
• How would our reorder point change?

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Macs Decision

• How many papers to buy?
• Average 90, st dev 10
• Cost 0.20, Sales Price 0.50
• Salvage 0.00
• Overage CO 0.20 - 0.00 0.20
• Underage CU 0.50 - 0.20 0.30

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Optimal Policy

• F(x) Probability demand lt x
• Optimal quantity
• Mac F(Q) 0.3 / (0.2 0.3) 0.6
• From standard normal table, z 0.253
• Normsinv(0.6) 0.253
• Q avg zs 90 2.5310 90 2.53 93

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Optimal Policy

• Model is called newsboy problem, newspaper
  purchasing decision
• If units are discrete, when in doubt, round up
• If u units are on hand, order Q - u units

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Example Macs Newsstand
Probability Demand lt 9 10003 12246
19 / 52 0.3654
Macs sales are roughly normally distributed
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Mac Continued

• Calculate average sales 11.73
• Standard Deviation 4.74
• In the future, update exponentially

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Multiple Periods

• For multiple periods,
• salvage cost - holding cost
• Solve like a regular newsboy

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Random Demand

• 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?

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Probabilistic Models
Safety stock x ?m
From statistics,
From normal table z.95 1.65
Safety stock zs? 1.6510 16.5
ROP m Safety Stock
35016.5 366.5 367
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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
• To satisfy 90 of the demand, z 1.28
• R 50 25.6 75.6

Safety stock
z
s
1.28 20 25.6
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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?

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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

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St Dev Over Lead Time

• Standard deviation of demand
• R 40 3 10 70

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Service Level Criteria

• Type I specify probability that you do not run
  out during the lead time
• Chance that 100 of customers go home happy
• Type II proportion of demands met from stock
• 100 chance that this many go home happy, on
  average
• Service levels easier to estimate

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Two Types of Service

• Cycle Demand Stock-Outs
• 1 180 0
• 2 75 0
• 3 235 45
• 4 140 0
• 5 180 0
• 6 200 10
• 7 150 0
• 8 90 0
• 9 160 0
• 10 40 0
• Sum 1,450 55

Type I 8 of 10 periods 80 service Type
II 1,395 / 1,450 96
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Type I Service

• a desired service level
• We want F(R) a
• R m s z
• Example a 0.98, so z 2.05
• if m 100, and s 25, then
• R 100 2.05 25 151

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Type II Service

• b desired service level
• Number of mad cust. (1- b) EOQ
• L(z) EOQ (1- b) / s
• Example EOQ 100, b 0.98
• L(z) 100 0.2 / 25 0.8
• P. 835 z 1.02
• R 126 -- A very different answer

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Inventory Recordkeeping

• Two ways to order inventory
• Keep track of how many delivered, sold
• Go out and count it every so often
• If keeping records, still need to double-check
• Annual physical inventory, or
• Cycle Counting

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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

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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 PG rep. calls every 2 weeks

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Fixed-Period Model When to Order?
Inventory Level
Target maximum
Time
Period
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Fixed-Period Model When to Order?
Inventory Level
Target maximum
Time
Period
Period
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Fixed-Period Model When to Order?
Inventory Level
Target maximum
Time
Period
Period
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Fixed-Period Model When to Order?
Inventory Level
Target maximum
Time
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Fixed-Period Model When to Order?
Inventory Level
Target maximum
Time
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Fixed-Period Model When to Order?
Inventory Level
Target maximum
Time
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Fixed Order Period

• Standard deviation of demand over TL
• T Review period length (in days)
• s std dev per day
• Order quantity (12.11)

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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

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Classifying Items as ABC
Annual Usage
A
B
C
of Inventory Items
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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
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ABC Classification Solution