Inventory Management

Agenda

• Independent Demand Inventory– Dependent vs. independent demand– Basic Economic Order Quantity (EOQ) model.

Also known as Economic Lot Size Model– Models with Demand and Supply Uncertainty

• Fixed ordering costs: the base-stock model (s,S)

• No fixed ordering costs: the base-stock model (S)

– Risk pooling

Why do companies hold inventory?Why might they avoid doing so?

• WHY?– To meet anticipated customer demand

– To account for differences in production timing (smoothing)

– To protect against uncertainty (demand surge, price increase, lead time slippage)

– To maintain independence of operations (buffering)

– To take advantage of economic purchase order size

• WHY NOT?– Requires additional space

– Opportunity cost of capital

– Spoilage / obsolescence

E(1)

Independent vs. Dependent Demand

Independent Demand (Demand not related to other items or the final end-product)

Independent Demand (Demand not related to other items or the final end-product)

Dependent Demand

(Derived demand items for

component parts, subassemblies, raw materials,

etc.)

Dependent Demand

(Derived demand items for

component parts, subassemblies, raw materials,

etc.)

Ford Taurus

Body Assy.

Wheel Assy. (4)

Wheel (1) Tire (1)

Two Decisions in Inventory Management

• When is it time to reorder?

• If it is time to reorder, how much?

Economic Order Quantity Model:Where it all started….

On-

hand

Inv

ento

ry

Time

QDemand Rate, D

Time Between Orders

(Cycle Time) T = Q/D

Average CycleInventory, Q/2

Q/2

Basic EOQ Assumptions

• Constant Demand Rate

• Instantaneous replenishment

• Orders received in full after lead-time.

• Constant Unit Price (no discounts)

Economic Order Quantity Cost Model:Constant Demand, No Shortages

TC = total annual inventory costD = annual demand (units / year)Q = order quantity (units)K = cost of placing an order or setup cost ($)h = annual inventory carrying cost ($ / unit /year)

Total Annual Inventory Cost

=AnnualOrderingCost

TC = (D / Q) K + (Q / 2) h

AnnualHoldingCost

+

Trade-off in EOQ Model: Inventory Level vs. Number of Orders

Time

Q

On-

hand

Inv

ento

ry

Time

Q

On-

hand

Inv

ento

ry

Many orders, low inventory level

Few orders, high inventory level

Cost Relationships for Basic EOQ(Constant Demand, No Shortages)

TC

– A

nn

ual

Co

st

Total Cost

CarryingCost

OrderingCost

EOQ balances carryingcosts and ordering costs in this model.

Q* Order Quantity (how much)

EOQ Results (How Much to Order) (Constant Demand, No Shortages)

Economic Order Quantity

Number of Orders per year

Length of order cycle

Total cost = TC = (D / Q*) K + (Q* / 2) h

T = Q* / D

= D / Q*

= Q* = 2 D K

h

EOQ Example (How Much)

D = 1,000 units per year

K = $20 per order

h = $8.33 per unit per month

2 1000 20* 20

100Q EOQ:

= $100 per unit per year

BE CAREFUL!

Number of orders per year = 1000/20 = 50 orders

Length of order cycle = T = 365 days/50 orders = 7.3 days

Total cost = 20(1000/20)+100*20/2 = $2,000

EOQ Example (cont.) (D = 1,000, K = $20, h = $100)

Question: What if the company can only order in multiples of 12? (That is, order either 0 or 12 or 24 or 36, etc…)?

Answer: Q* = 20. Closest matches (above and below) are 12 and 24. Need to compute TC for both, and decide:

Q = 12 TC(12) = 20(1000/12)+100*12/2 = $2,266.67

Q = 24 TC(24) = 20(1000/24)+100*24/2 = $2,033.33

So, the company should order in lots of Q = 24

TC(Q) = (D / Q) K + (Q / 2) h

Robustness of EOQ model

Order Quantity

Annual Cost

Total Cost

Q*Q*-Q Q*+Q

TC

Would have to mis-specify Q* by quite a bit before total annual inventory costs would change significantly.

Very Flat Curve - Good!!

Example

Example: EOQ Robustness

• Suppose that in the last problem, you have mis-specified the order costs by 100% and the holding costs by 50%. That is, – K used in the computation = $40/order (actual cost = $20 / order)– h used in computation = $150 / unit / year (actual = $ 100)– Then, using these wrong costs, you would have gotten

2(1,000)40' 23.1

150Q

Your actual TC (computed substituting Q’ into TC, using correct costs of K = $20, and h = $100:

1,000 2320 100 $2,019

23 2TC Only 1% above minimum TC!

Variations of EOQ

Some assumptions so far• Instantaneous replenishment (zero lead time)

• Certain and constant demand rate

• Constant price

Some variations of EOQ• Positive lead times and uncertain lead times

• Uncertain demand

• EOQ with quantity discounts

EOQ with Positive Lead TimeO

n-ha

nd I

nven

tory

Time

QDemand Rate, D

Time Between Orders

(Cycle Time) T = Q/D

Average CycleInventory, Q/2

ReorderPoint, s

PlaceOrder

Receiveorder

LeadTime, L

Q/2

Example: Daily demand (d) = 1,000/365 = 2.74 /dayDelivery lead time (L) of 2 days

Determining When to Reorder

• Quantity to order (how much…) determined by EOQ • Reorder point (when…)determined by finding the

inventory level that is adequate to protect the company from running out during delivery lead time

• With constant demand and constant lead time, (EOQ assumptions), the reorder point is exactly the amount that will be sold during the lead time.

s = d*L = (2.74) (2) = 5.5 6 units

Effects of Demand / Lead Time Variability on Reorder Point (When)

Safety Stock level

s

Expected demand ataverage demand rate d

Placeorder

Receiveorder

L

Variable demand

QUESTION: How much inventory is

needed during lead time L?

KEY POINT: s is larger when there is uncertainty

about demand or L

Calculation of Appropriate Safety Stock Level

• Safety stock: stock carried to provide a level of protection against stockouts due to uncertainty of demand during lead time

• Stockout Criterion: Find s such that the probability of stockout (during the lead time) is

• Demand during lead time is a random variable– Estimate distribution from historical data (build

histogram of demand + frequencies)– Normal is frequently used if distribution is unknown

Computing s …

Probability {Demand during lead-time < s} = 1-

Service LevelService Level

1-

Assumption: Demand during lead-time is normally distributedAssumption: Demand during lead-time is normally distributed

s

Probability distribution of demand during lead time

Probability distribution of demand during lead time

Computing s: Taking Advantage of the Normal Distribution

1-

s

Probability distribution of demand during lead time:

Mean = ; Std Dev =

Probability distribution of demand during lead time:

Mean = ; Std Dev =

sz s z

sz s z

1-

z0

1 -

.90 .95 .98 .99 .999

z 1.28 1.64 2.05 2.33 3.09

From normal table or, in Excel, use: =normsinv (0.90)

Issue…

• The parameters and refer to mean and standard deviation of demand during lead time

• Normally, companies have statistics on demand and lead time per unit of time (say, days, weeks, months)– AVG = average demand per unit of time – STD = standard deviation of demand per unit of time – AVGL = average lead time – STDL = standard deviation of lead time – Just be consistent: if demand is given on a certain time unit, say, days,

then use lead time in the same time unit (in this case, days)

• How to we compute and from AVG, STD, AVGL, and STDL?

Safety stock SS

More specifically….

2 2 2s AVG AVGL z STDL AVG STD AVGL

Standard deviation of

demand during lead time ()

Safety factor (std

normal table)

Mean demand during lead

time ()

Note: This is a very good approximation even when demand is not normally distributed.

•If lead time is constant, 0STDL s AVG AVGL z STD AVGL

0STD •If demand is constant, s AVG AVGL z STDL AVG

The (s,S) Policy: When There Are Fixed Ordering Costs

s should be set to cover the lead time demand and together with a safety stock that insures the stock out probability is (When)

S depends on the fixed order cost – EOQ (How much)

Time

Inve

ntor

y

L

R

Orderplaced

Orderarrives

Average demandduring lead time

Safety Stock

sS

The (s,S) Policy: Fixed Ordering Costs

• Compute Q using the EOQ formula, using mean demand D = AVG (be careful about units…):

2 K AVGQ

h

2 K AVG

Qh• Set S = s + Q

• Order when: inventory position (IP) drops below s

2 2 2s AVG AVGL z STDL AVG STD AVGL 2 2 2s AVG AVGL z STDL AVG STD AVGL

• Order how much: bring IP to S (“big S”)

• Need to define inventory position (IP)

IP = On-Hand + On-Order– Backorder

Example: (s,S) Model

• Consider inventory management for a certain SKU at Home Depot. Supply lead time is variable (since it depends on order consolidation with other stores) and has a mean of 5 days and standard deviation of 2 days.

• Daily demand for the item is variable with a mean of 30 units and a coefficient of variation of 0.20.

• Assume a 95% service level.

• There are fixed ordering costs that are estimated at $50. Assume that holding costs are 15% of the product cost ($80) per year. Also, assume that the store is open 360 days a year. Propose an inventory policy for this SKU.

Solution

30; 0.2(30) 6AVG STD 5; 2AVGL STDL

Assume 95% service level z = 1.64

2 2 2

2 2 2 30 5 1.64 2 30 6 5 150 100.8 251

s AVG AVGL z STDL AVG STD AVGL

• Variable definitions and preliminary calculations:

• Compute s

(.15)80 / 360 0.0333; 50h K

2 2(30)50300

0.0333AVG K

Qh

251 300 551S s Q

• Compute Q

Example: (s,S) Model

• Consider inventory management for a certain SKU at WalMart. Supply lead time is variable and has a mean of 1 week and standard deviation of 2 weeks.

• Weekly demand for the item is variable with a mean of 125 units and a standard deviation of 50.

• Assume a 90% service level.

• There are fixed ordering costs that are estimated at $30. Assume that holding costs are 20% of the product cost ($40) per year. Also, assume that the store is open 52 weeks a year. Propose an inventory policy for this SKU.

Solution

125; 50AVG STD 1; 2AVGL STDL

Assume 90% service level z = 1.28

2 2 2

2 2 2 125 1 1.28 2 125 50 1 125 326.3 452

s AVG AVGL z STDL AVG STD AVGL

• Variable definitions and preliminary calculations:

• Compute s

(.20)40 / 52 0.1538; 30h K

2 2(125)30221

0.1538AVG K

Qh

452 221 673S s Q

• Compute Q

• Inventory policy: keep IP constant at s units (s is called the base stock level).– When: IP drops below s

• So, s is also the reorder point for this model

– How much: order to bring IP back to s

• Example: suppose inventory level on-hand is 10, s = 20, and there are 2 units already in order. Then,

IP = 10 + 2 = 12 units.

The firm should order 20 – 12 = 8 units.

The Base-Stock Policy s: No Fixed Ordering Costs

Example: Base-Stock Model s

Consider inventory management for a certain SKU at King Soopers Supply lead time is variable and has a mean of 2 days and standard deviation of 4 days. Daily demand for the item is variable with a mean of 24 units and a coefficient of variation of 0.30. Propose an inventory policy for this SKU. Assume a 98% service level.

24; 0.3(24) 8AVG STD 2; 4AVGL STDL

Assume 98% service level z = 2.05

2 2 2

2 2 2 24 2 2.05 4 24 8 2 48 198.2 247

s AVG AVGL z STDL AVG STD AVGL

Summary of Inventory Models

Is demand rate and lead time constant?

Use EOQ•How much: Q (EOQ formula)•When: d*L (reorder point)

no

yesAre there fixed ordering costs?

no

yesUse (s, S) policy

•How much: necessary to bring IP back to S, where S = s + Q (Q is from EOQ formula)•When: IP drops below s (base-stock policy formula)

Use base stock (s) policy•When: IP drops below s•How much: necessary to bring IP back to s

Risk Pooling

• Means and variances are additive• Stock is based on std. Deviations

– Square root law: stock for combined demands is less than the combined stocks

2 2 2

2 2

X Y X Y

X Y X Y

2 2 2

2 2

X Y X Y

X Y X Y

HP Example: Benefits of a Universal Product

N. America

N(200,60)

Europe

N(150,50)

Consider z = 2 (~ 98% of service level)

320 250 570NA Es s

Because of a different power supplies, HP had two laser printers, one for Europe and one for N. America. A universal product (with a universal power supply) has been proposed, but costs $30 extra. Is it worthwhile? Below is monthly demand for HP for the two markets (in thousands). Assume a one-month constant lead-time (STDL = 0) for both markets.

200(1) 2(60) 1 320

150(1) 2(50) 1 250

NA NA NA NA NA

E E E E E

s AVG AVGL z STD AVGL

s AVG AVGL z STD AVGL

HP Example (cont.): Benefits of a Universal Product

Demand seen by HP (NA and Europe)2 2(200 150, 50 60 ) (350,78.1)N N

350(1) 2(78.1) 1 506.2s AVG AVGL z STD AVGL