Inventory Management

INVENTORY MANAGEMENT

Operations Management

Dr. Ron 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


PeriodPeriod

Fixed-Time Period Model:When to Order?

Time


PeriodPeriod


Time


Period PeriodPeriod


Time


Period PeriodPeriod


Time


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

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Inventory Management