Supply Chain Inventory Management

Multi-period Problems

Read: Chap 10.1-10.2.

Chap 11.1-11.2 (except materials for fill rate)

Defining concepts – put aside

• Assemble-to-order (build-to-order)

– You start assembly operations only after you have received the order

– Often, you have the materials/subassemblies ready

• Dell has all the materials made ready by its suppliers, then “batch” assembles to the orders

• Make-to-order (build-to-order)

– You start processing the order after you receive it

– Typically, you do not have materials ready until the order arrives

• Most apparel subcontract manufacturers run in this mode

• Which mode takes a longer lead time to finish the order?

•

Inventory Decisions/Rules

• Imagine you’re a manager for a store/DC

– Park’n Shop: for shampoo, you can buy from Procter

& Gamble (寶潔), Johnson & Johnson (强生)

– P&G, J&J, may have their own DRPs for stock mgmt

– But Park’n Shop orders/replenishes by its own way

• Assume after you order, sometime later you will get it delivered

• Then, the decisions are …

Decisions

• When to order?

• In what quantity?

• How to get the order delivered?

• From whom?

• Prices?

• …

SKU: Stock keeping unit – the decision unit

• How much to order / produce? – Order quantity -- e.g., OUT, EOQ, …

• When to order / produce? – Reorder points, time periods

Use Safety Stock to cope with uncertainty

Two Inventory Decisions/Replenishment Rules

Multiple Period Problems

• Inventory may be sold/used and replenished in multiple periods - multiple period inventory problem What is the major difference between this and the

newsboy problem ?

You can carry inventory from one period to the next

Backlogging your demand is possible (but at some intangible cost)

• Two popular types of systems: periodic review/ and continuous review/replenishment

Periodic v.s. Continuous Review

• Periodic review model

– Common for B & C items

– “Economy of scales”

• Continuous review model

– Slow moving items or

– Important items

• When periodic/continuous review?

• Inventory policy: when and how much to order - tactical level

The ABC Classification

• Close examination of multi-SKU inventory systems reveals a statistical regularity in the usage rate of diff items ($)

– A class, B class, C class: 80/20 rule

• Policies based on ABC analysis

– Develop class A suppliers more ...

– Give tighter physical control of A items

– Forecast A items more carefully

Two Inventory Decisions

• How much to order / produce? – Order quantity -- e.g., OUT, EOQ, …

• When to order / produce? – Reorder points, time periods

Use Safety Stock to cope with uncertainty

Inventory Costs

• Order/Set-up costs – Trucking, receiving, inspection, calls, faxes,…

– GE estimated the cost of processing a typical replenishment order is about $50

– Trucking cost – a major part

Diminishing

over time

• Holding costs

• Stock-out / shortage /underage cost

• Salvage value

Inventory Holding Costs

Housing (building) cost 6%

Material handling costs 3%

Labor cost 3%

Inventory investment costs 11%

Pilferage, scrap, & obsolescence 3%

Total holding cost 26%

% of Category Inventory Value

EOQ Assumptions

• Known & constant demand

• Known & constant lead time

– Instantaneous receipt of material (or constant leadtime)

• No quantity discounts

• Only order (setup) cost & holding cost

• No stockouts

• Price is pre-fixed

EOQ Model: How Much to Order?

Order Quantity

Annual Cost

Holding Cost

Total Cost Curve

Order (Setup) Cost

Optimal

Order Quantity (Q*)

Reorder

Point

(ROP)

EOQ Model: When to Order?

Time

Inventory Level

Optimal

Order

Quantity

(Q*)

Average

Inventory (Q*/2)

Lead Time

Cycle

Inventory!

The EOQ Formula

• Economic

order quantity

• Total cost = order cost + carrying cost

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

• Optimal fixed order quantity

hc2DKQ*EOQ

Where:

D = annual demand (units)

K = setup cost/order

h = carrying charge ($/unit/year)

c = unit cost ($/unit)

The Key Insight of EOQ

1. There is a tradeoff between lot size and inventory

2. Holding and setup costs are fairly insensitive to lot size Q

TC(Q*) = cD +(D/Q)K+ (Q/2 ) h c

=> Sqrt (2 D*K*h*c)

Q’ -> TC(Q’)/TC(Q*) = *Q’/Q* + Q*/Q’+/2

e.g., Q’ = 2Q*, TC(Q’)/TC(Q*) = 1.25

Textbook

• Economic order quantity – lot size

??2??EOQhc

2DKEOQ

- The model is insensitive to

parameter values and robust!

Example: Broadway • Consider inventory management for a certain SKU at Broadway.

Supply lead time is 4 days. Daily demand for the item is variable

with a mean of 30 units and a coefficient of variation of 20%.

Assume that fixed ordering cost is estimated at $50 per order,

and inventory 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.

L = 4, AVG D’d = 30 per day

K = 50, hC = (0.15)(80)/360 = 0.0333 (converted to daily cost)

Use EOQ to calculate order quantity:

Q = sqrt[2(K)(AVG)/(hc)] = sqrt[2(50)(30)/0.0333] = sqrt(90000) = 300

Is that all?

Multiple Products/Items

• The setup cost can be shared among different products/items

• Replenish all jointly

• Replenish a subset each time

Time

Order

qu’ty All jointly

together

Subset jointly

together

Problems with EOQ?

• Deterministic demand?

• Moving to random/uncertain demand

– What will happen to the above EOQ charts?

• First consider a single product/item

Shortage/Underage Costs

• Emergency ordering, loss of goodwill, lost sales, hurting return buz.

• In multiple periods, they are notoriously difficulty to estimate

• Shortage/stockout occurs, what’ll happen?

– Lost-sales or backlogging/backordering

• Throughout, we assume backordering

Performance Evaluation of Material/Dist Mgt

• Imagine you are a distribution manager, you’re responsible for delivery of products to sales outlets - mkting gets customer & you deliver. How should you be evaluated?

Costs are clearly most relevant. But how to estimate

“shortage costs?”

What will happen if we ignore shortage costs?

Service Levels

• Inventory related service performance measurements

– Fill rate: fraction of demand being filled right away e.g., over past 52 wks, customer orders: 120,500 units,

filled upon arriving: 110,050 units => “customer backorders” = 10,450. Fill rate = 0.91.

– Ready rate: % of periods (or times) that there is no stock out - filling all the demand arriving in the periods (or upon arriving); e.g.,

among the past 52 wks (or orders), in 47 wks (or orders), no stock-out, ready rate = 90%. In your textbook, it is called “Cycle Service Level (CSL)” , also known as “In-Stock Rate”.

Item/Line-based service levels

WalMart: 95%-98%; Dell’s DC: 98.80%

Cycle-Service-Level = In-Stock Rate

Line and Order Based Service Levels

• Line/item based or item based service level: for only one SKU or similar SKUs.

• Order based: a customer order may request a number of different SKUs - lines - with varying #.

• Throughout this course, we mean item/line service level.

• Several Cases mention both types of service levels.

Multi-Product Availability

• Service level = fill rate <= suitable?

• Item-based = Line-based <=> order based

• Order “fulfillment rate”: % times of filling orders completely (prob. being fully fulfilled)

More about Fill/Ready Rate

• Off-shelf v.s. Time-window

– Off-shelf : instant availability

– Time-window : available or delivered within a certain time - ready rate within TW

• We confine ourselves within off-shelf fill /ready rates , item-based

Service Level

Line/Item Based

Fill rate In-stock %, Cycle Service Level

Off-shelf Window Based

Order Based

Using Service Level Specification to Determine Order-

up-to Level

• What would you do, as a dist. Manager, if you are evaluated only by service performances, say, fill rate: the higher fill rate, the higher your bonus?

– Or Supply chain planner

• Trade-off between service and costs - holding & ordering

How to Set a Service Level?

E.g., Cycle-Service-Level = In-Stock Rate:

Item Gross Margin x 52 weeks x (1 – In-Stock Rate) =

The Cost of carrying an item for a yr

) margin] gross item x [52

yr/ onefor iteman carrying ofcost -(1 RateStock -In

If review/replenishment is made at a frequency other than

weekly, we simply replace 52 in the formula with the number of

replenishment cycles in a yr. Say, monthly, 12.

An example: cost of holding an item for a year = 30% its value (at cost)

1) Item gross margin = 10 % its value (at cost)

Then, in-stock rate = 1-30%/(52*10%) =94%

2) If gross margin = 20% (50%), then it is 97% (99%)

If monthly (instead of weekly): 1) at margin 10%, it is 75%

2) at margin 20% (50%): the rate = 87.5% (95%)

Now, are we lowing the service and letting more stock out?

APPROPRIATE LEVEL

CSL

$

TC

SALES

Demands are uncertain!

• Inventory position (effective inventory level =

on hand + on order – backorders (or allocated)

• Rule: Whenever the inventory position goes below the reorder point R then order a fixed quantity Q.

Order point system (R,Q)

Best known as “Reorder Point &

Batch Size Policy”

• Two bin system ...

• Rule: when the big bin is empty, open the small bin and place an order.

A B

Order point system (R,Q)

X

Time

Inventory Level

Max.

Lead Time

Place

order Receive

order

Freq

Safety Stock (SS)

X

Time

Inventory Level

Max.

Lead Time

Place

order Receive

order

SS

Freq

Aver. DDLT

DDLT=demand during leadtime

TIME

CYCLE

SAFETY

Q

Ave

. In

ven

tory

How to determine (ROP, Q)?

• Lot size Q by EOQ

• ROP by a service level specification

• Key assumption here is “the same demand pattern over many periods or long time”

Safety Stock (SS)

X

Time

Inventory Level

Max.

Lead Time

Place

order Receive

order

SS

Freq

Aver. DDLT

ROP

An Example

You’re a buyer for General

Hospital. The demand for

hospital ER kits is normally

distributed. The mean demand

during the reorder period is 350

kits, with = 10 kits. The

hospital wants stockouts no more

than 5% of the time. What are

the safety stock & ROP?

Solution

X

= 350

Svc Level = .95

P(Stockout) = .05

Frequency

x = ?

= 10

Safety Stock = x -

Solution

.95

Safety stock

From statistics,

Safety stockTherefore, & Safety stock

From normal table, 1.65

Safety stock 1.65 10 16.5

ROP Safety stock

350 16.5 366.5 367

x

xz

z z

z

z

If sigma is bigger or leadtime longer, then SS ?

Evaluating In-stock Rate (Cycle Service Level) Given an (R, Q) Replenishment Policy (Ex. 11-2)

• Weekly demand for Palms: N(2,500, 500^2)

• LT = 2 wks. What the In-stock rate if R=6,000, Q=10,000?

R = Aver DDLT+ S.S. ,

Aver DDLT = 2x2500=5,000

S.S. = 6,000 -5,000 = 1,000,

Sigma_L = Sqr (2)*500 =707,

S.S. = z*sigma_L

z = 1000/707 = 1.4144: the rate or CSL = 92 %

z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359

0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753

0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141

0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517

0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879

0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224

0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549

0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852

0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133

0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389

1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621

1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830

1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015

1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177

1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319

1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441

1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545

1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633

1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706

1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767

2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817

2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857

2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890

2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916

2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936

2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952

2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964

2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974

2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981

2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986

3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990

3.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993

3.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995

3.3 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997

F(z)

0 z

Normal Dist. Table

R = Aver DDLT+ S.S. , Aver DDLT = 2x2500=5,000

S.S. = 6,000 -5,000 = 1,000, Sigma_L = Sqr (2)*500 =707,

S.S. = z*sigma_L -> z = 1000/707 = 1.4144: the rate or CSL = 92 %

Other Continuous Systems (relatively short “review” intervals)

• “Sell one then replenish one” – ( )

• Extended reorder-point, lot-size system

– (r, nQ): r=ROP

• Min – Max policy (system )

賣一補一

Time

Eff. Inventory Level

Max.

=ROP+Q

Lead Time

Place

order Receive

order

ROP

Periodic Review Systems Order up-to model definitions

• On-order inventory / pipeline inventory = the number of units that have been ordered but have not been received.

• On-hand inventory = the number of units physically in inventory ready to serve demand.

• Backorder = the total amount of demand that has has not been satisfied: – All backordered demand is eventually filled, i.e., there are no lost sales.

• Inventory level = On-hand inventory - Backorder. • Inventory position = On-order inventory + Inventory level.

• Order up-to level, S

– the maximum inventory position we allow. – sometimes called the base stock level.

13-47

Time

Inventory Level Target maximum

Period Period Period

When and how much is ordered?

Economy of scale?

SS

Aver. Demand

Periodic Review

• Ordering every period

SS = Reorder point (level) – Aver. DDLT

• Can (r, Q) policy work properly?

• Min – Max policy (system)

For your reference: Review interval is reduced from monthly

to weekly. Will inventory level to be lowered?

Multi-Item System Control

• Coordination is required for multiple product ordering. Why?

• One policy is (Min, c, Max) : can-order

ROP -

MIN

c – Can-order

Max

Item A

ROP -

MIN

c – Can-order

Max

Item B

Realism

• All models (stylized systems) assume stable markets

• When they are not, these are just approximations!

What learnt here?

• Two review/auditing systems

• Several ordering rules/policies

• How are the question of when and how much to order in each of this systems-policies answered?

• Reorder point/level can be determined by a service level requirement/specification

• The concept of safety stock

Managerial Insights

Insight 1: the higher variability of demand, the higher SS.

Insight 2: Shorter leadtime will result in lower SS

Insight 3: When supply and/or leadtime is uncertain, SS?

Homework 2 – Part 1 (Due date: Nov. 9, 7:pm)

1. What are some strategic, planning, and operational decisions that must be made by an apparel retailer like Giordano 佐丹奴? (Chapter 1)

2. Why does ZARA source products with uncertain demand (in EU) from local manufacturers and products with predictable demand from Asian contractors?

3. Offer an explain why Sasa’s inventory turns (3 times/yr) is so much slower than Circle-K (~20/yr)?

4. “Speculate” how different in their supply chains between Watsons (health and beauty products part) and Fortress, both belonging to

Each answer is at most 2 pages long! Try to

apply the framework/theory learnt.

(continued) 5. The following table shows financial data (yr 2007) for

two major retailers

Retailer A Retailer B

($mil) ($mil)

Inventories $3, 643 $29,447

Sales $48,106 $286,103

Cost of Goods Sold (COGS) $41,651 $215,493

Assume both retailers have an average annual holding cost rate of 30% (i.e., costs

both $3 hold an item that they procured for $10 for one entire yr).

a.How many days, on average, does a product stay in Retailer A before it is sold?

(Hint: inventory turns.) Assume that stores are operated 365 days a yr.

b.How much lower is, on average, the inventory cost for Retailer A compared to

Retailer B of a household cleaner valued at $100 COGS? Assume the unit costs

of the cleaner is the same for both.

6. Chap. 12, Discussion Question 1

7. Chap. 12, Discussion Question 2

8 Chap 12, Ex. 8. [Note, for Part a, you just calculate the underage cost Cu as the foregone profit.]

9. Chap 12, Exercise 9. [Note: salvage value = 75-15=60)

10. Chap 12, Ex. 10. parts a & d only. (Note that in a, the salvage value = $70- $10; in d, the salvage value $75 is the net after deducting the holding-transport costs already)

All questions are in the handout pages.

HW 2 – Part 2 (due date: ., Nov. 23, 7: 00 pm)

Exercises (Ch. 11) -- see your handout. You need to hand-in this part

after the Midterm exam.

1. Exercise (Ch. 11) 1. (Best Buy/Motorola)

2. Exercise (Ch. 11) 2. (Best Buy/Motorola)

3. Exercise (Ch. 11) 3

4. Exercise (Ch. 11) 4 (HP Printers at Sam’s Club): Skip the last

question regarding “fill rate”

5-6. Discussion Q. (Ch. 11), 7 (Amazon.com) & 9 (Bookstore Borders)

Remember: HP Case questions – hand in Nov. 2nd (in class, hard copy,

better before the class stars and in any case no later than 7:30 )