Inventory Management

Chapter 13

Learning Objectives• You should be able to:

1. Define the term inventory, list the major reasons for holding inventories, and list the main requirements for effective inventory management

2. Explain periodic and perpetual review systems3. Explain the objectives of inventory management4. Describe the A-B-C approach and explain how it is useful5. Describe the basic EOQ model and its assumptions and

solve typical problems6. Describe the economic production quantity model and

solve typical problems7. Describe the quantity discount model 8. Describe reorder point models9. Describe situations in which the single-period model

would be appropriate, and solve typical problems

2Instructor Slides

Inventory

• Inventory– A stock or store of goods

• Independent demand items– Items that are ready to be sold or used

• A “typical” firm (on “typical” times) has roughly 30% of its current assets and as much as 90% of its working capital invested in inventory

13-3

Types of Inventory

• Raw materials and purchased parts• Work-in-process (WIP) = partially

completed goods• Finished goods inventories

(manufacturing) or merchandise (retail)• Maintenance and repairs (MRO) inventory• Goods-in-transit to warehouses or

customers (pipeline inventory)

13-4

Inventory

• Inventories are a vital part of business:

– necessary for operations

– contribute to customer satisfaction

• But– Costly– May have limited shelf-life– Carrier may have limited space

13-5

Functions of Inventory• To meet anticipated demand

• “anticipation stock”

• To smooth production requirements• seasonal demand

• To permit operations• Work-In-Process

• To decouple operations• buffer between successive operations in case of a breakdown

• To protect against stock-outs• Delayed deliveries or increase in demand

• To take advantage of order cycles• minimize purchasing and holding costs or economies of

producing in large quantities

• To help hedge against price increases • To take advantage of quantity discounts 12-6

Inventory CostsPurchase cost

The amount paid to buy the inventoryHolding (carrying) costs

Cost to carry an item in inventory for a length of time, usually a year (rent, equipment, materials, labor, insurance, security,

interest and other direct expenses).Ordering costs

Costs of ordering and receiving inventorySetup costs (Analogous to ordering costs)

The costs involved in preparing equipment for a jobShortage costs

Costs resulting when demand exceeds the supply of inventory; often unrealized profit per unit

• Lead time• Time interval between ordering and receiving the order

13-7

Objectives of Inventory Control

The overall objective of inventory management is to achieve:

1. Satisfactory levels of customer service • Having the right goods available in the right quantity in

the right place at the right time• Focus on size and timing

2. While keeping inventory costs (ordering and carrying) within reasonable bounds (minimize)

13-8

Objectives of Inventory Control

• Measures of performance:1. Customer satisfaction

– Number and quantity of backorders– Customer complaints

2. Inventory turnover = (average) cost of goods sold (average) inventory investment

during a period

13-9

Inventory Management

• Management has two basic functions concerning inventory:1. Establish a system for tracking items in

inventory2. Make decisions about

• When to order• How much to order

13-10

ABC Classification System

• A-B-C approach– Classifying inventory according to some measure

of importance, and allocating control efforts accordingly

– A items (very important)• 10 to 20 percent of the number of items in inventory• about 60 to 70 percent of the annual dollar value

– B items (moderately important)– C items (least important)

• 50 to 60 percent of the number of items in inventory but • only about 10 to 15 percent of the annual dollar value

13-11

ABC Classification System

• How to classify?1. For each item,

multiply annual volume by unit price to get the annual dollar value.

2. Arrange annual values in descending order.

3. A items: the few with the highest annual dollar valueC items: the most with the lowest dollar value.

B items: those in

between13-12

# Annual demand

Unit price

Annual value

Clss % of items

% of value

8 1,000 4,000 4,000,000 A 5.3 52.7

3 2,400 500 1,200,000 B

6 1,000 1,000 1,000,000 B 31.4 40.8

1 2,500 360 900,000 B

4 1,500 100 150,000 C

10 500 200 100,000 C

9 8,000 10 80,000 C

2 1,000 70 70,000 C 63.3 6.5

5 700 70 49,000 C

7 200 210 42,000 C

18,800 7,591,000 100 100

Inventory Counting Systems• Periodic System (e.g. small retailer)

• Physical count of items in inventory made at periodic intervals– Many items ordered at the same time. Savings in processing and

shipping of orders.vs.– Lack of control between reviews. Having to keep extra stock to protects

against shortages.

• Perpetual Inventory System (e.g. bank transactions)• System that keeps track of removals from inventory continuously,

thus monitoring current levels of each item• An order is placed when inventory drops to a predetermined minimum

level (can optimize Q)Vs.• Added cost of record keeping. Usually has to be accompanied by a

periodic physical count.

• Note: RFID technology is used by inventory counting systems

13-13

Inventory Ordering Models

• Economic Order Quantity models:– identify the optimal order quantity

• by minimizing total annual costs that vary with order size and frequency

1. The basic Economic Order Quantity model (EOQ)2. The Economic Production Quantity model (EPQ)3. The Quantity Discount model*

• Other Models1. Reorder Point Ordering* (uncertainty, when to order)2. Fixed-Order-Interval Model*3. Single Period model (perishable items)

13-14

Basic Economic Order Quantity Model

• The basic EOQ model:– used to find a fixed order quantity that will minimize

total annual inventory costs– Purchase price is not included (cost is unaffected by it)

• Assumptions:1. Only one product is involved

2. Annual demand requirements are known

3. Demand is even throughout the year

4. Lead time does not vary

5. Each order is received in a single delivery

6. There are no quantity discounts

13-15

The Inventory Cycle (EOQ)

Profile of Inventory Level Over Time

Quantityon hand

Q

Receive order

Placeorder

Receive order

Placeorder

Receive order

Lead time

Reorderpoint

Usagerate

Time

13-16

Total Annual Cost

orderper cost Ordering

yearper unitsin usually Demand,

yearper usually unit,per cost (carrying) Holding

unitsin quantity Order

where

2

Cost Ordering AnnualCost Holding AnnualCost Total

S

D

H

Q

SQ

DH

Q

13-17

Number of orders

Average number of units in inventory

Goal: Total Cost Minimization

Order Quantity (Q)

The Total-Cost Curve is U-Shaped

Ordering Costs

QO

An

nu

al C

ost

(optimal order quantity)

Holding Costs

SQ

DH

QTC

2

13-18

Deriving EOQ

• Using calculus, we take the derivative of the total cost function and set the derivative (slope) equal to zero and solve for Q.

cost holdingunit per annual

cost)der demand)(or annual(22O

H

DSQ

13-19

2

0)1(

2'

2

2

H

Q

DS

Q

DSHTC

2

HQ

Q

DS

The total cost curve reaches its minimum where the carrying and ordering costs are equal.

Length of order cycle = Q/DNumber of orders = D/Q

Example

• Tire distributer• D (Demand)=9,600 tires per year• H (Holding cost)=$16 per unit per year• S (Ordering cost) = $75 per order

12-20

tires30016

75*600,9*220

H

DSQ

800,4400,2400,275300

600,916

2

300

2 S

Q

DH

QTC

Example • Tire distributer• D (Demand)=9,600 tires per year• H (Holding cost)=$16 per unit per year• S (Ordering cost) = $75 per order • Q0=300 tires

• TCmin = 4,800

• TC curve relatively flat at optimum12-21

880,4880,2000,275250

600,916

2

250

2250 S

Q

DH

QTC

000,5800,1200,375400

600,916

2

400

2400 S

Q

DH

QTC

Economic Production Quantity (EPQ)

• The batch mode is widely used in production. In certain instances, the capacity to produce a part exceeds its usage (demand rate)– Assumptions

1. Only one item is involved

2. Annual demand requirements are known

3. Usage rate is constant

4. Usage occurs continually, but production occurs periodically

5. The production rate is constant

6. Lead time does not vary

7. There are no quantity discounts

22Instructor Slides

EPQ: Inventory Profile

23Instructor Slides

Q

Qp

Imax

Productionand usage

Productionand usage

Productionand usage

Usageonly

Usageonly

Cumulativeproduction

Amounton hand

Time

EPQ – Total Cost

24Instructor Slides

rate Usage

ratedelivery or Production

inventory Maximum

where

2

Cost SetupCost CarryingTC

max

max

min

u

p

upp

Q

I

SQ

DH

I

p

up

p

H

DSQp

2

Solution:

Example• Toy manufacturer makes rubber wheels for dump truck

toys.• D=48,000 wheels per year• S=$45• H=$1 per wheel per year• P=800 wheels per day• u=200 wheels per day

25Instructor Slides

wheelsQ

up

p

H

DSQ

p

p

400,2200800

800

1

45)48000(2

2

Quantity Discount Model

• Take into account quantity discount offered by supplier (add purchase cost to model)

• Quantity discount– Price reduction for larger orders offered to customers to

induce them to buy in large quantities

26Instructor Slides

priceUnit

where

2

Cost PurchasingCost OrderingCost CarryingCost Total

P

PDSQ

DH

Q

Quantity Discounts*

27Instructor Slides

Adding PD does not change EOQ

The total-cost curve with quantity discounts is composed of a portion of the total-cost curve for each price

Reorder Point Ordering

• Reorder-Point– When the quantity on hand of an item drops to this

amount (quantity-trigger), the item is reordered.– Determinants of the Reorder-Point

1. The rate of demand2. The lead time3. The extent of demand and/or lead time variability4. The degree of stockout risk acceptable to

management

13-28

Safety Stock*

29Instructor Slides

Fixed-order-interval (FOI) model

• Fixed-order-interval (FOI) model– Orders are placed at fixed time intervals

• Reasons for using the FOI model– Supplier’s policy may encourage its use– Grouping orders from the same supplier can produce

savings in shipping costs– Some circumstances do not lend themselves to

continuously monitoring inventory position

30Instructor Slides

Fixed-Quantity (ROP) vs. Fixed-Interval Ordering*

31Instructor Slides

Fixed Interval

Fixed Quantity

Single-Period Model• Single-period model

– Model for ordering of perishables and other items with limited useful lives

• Period = life of the good.• Items are not carried over to the next period

• The goal of the single-period model is to identify the order quantity that will minimize the long-run excess and shortage costs

• Two categories of problems:1. Demand can be characterized by a continuous

distribution2. Demand can be characterized by a discrete distribution

13-32

Single-Period Model

• Shortage cost– Generally, the unrealized profit per unit– Cshortage = Cs = Revenue per unit – Cost per unit

• Excess cost– Different between purchase cost and salvage

value of items left over at the end of the period– Cexcess = Ce = Cost per unit – Salvage value per unit

13-33

Continuous Stocking LevelsUniform Demand

• Service Level = probability that demand will not exceed the stocking level (S).

• Shortage: Demand > S0

• Excess: Demand < S0

Service level

So Balance Point

Quantity

Ce Cs

So =OptimumStocking Quantity

unitper cost excess

unitper cost shortage

where

level Service

e

s

es

s

C

C

CC

C

13-34

Like a Seesaw, leverage

If demand is not uniformly distributed, consider cumulative probability of demand

So=min+SL*(max-min)

Continuous Stocking LevelsUniform Demand

• Sweet cider delivered to bar• Demand uniformly distributed

between 300 and 500 liters/week.• Cost=20 cent/liter• Revenue = 80 cent/liter.• Ce=• cost-salvage =.2 $/liter• Cs=• revenue-cost =.8-.2=.6 $/liter• SL=• Cs/(Cs+Ce) =.6/(.6+.2)=.75• So=• min+SL*(max-min) =300+.75*(500-300)=450 liter• Stockout risk = • 1-SL = 1- .75=.25

Service level=75%

So =450 Balance Point

Ce=.2 $/liter Cs=.6 $/liter

unitper cost excess

unitper cost shortage

where

level Service

e

s

es

s

C

C

CC

C

13-35

300 500

Discrete Stocking Levels

• Cs/(Cs+Ce) may not coincide with a feasible stocking level.

• Solution: Stock at the next higher level so that the desired service level is equaled or exceeded.

unitper cost excess

unitper cost shortage

where

level Service

e

s

es

s

C

C

CC

C

13-36

Discrete Stocking Levels

37

Q P(R≤Q) Expected profit

1 0.05 3.5

2 0.1 6.725

3 0.15 9.675

4 0.2 12.35

5 0.25 14.75

6 0.3 16.875

7 0.35 18.725

8 0.4 20.3

9 0.45 21.6

10 0.5 22.625

11 0.55 23.375

12 0.6 23.85

13 0.65 24.05

14 0.7 23.975

15 0.75 23.625

16 0.8 23

17 0.85 22.1

18 0.9 20.925

19 0.95 19.475

20 1 17.75

• Discrete• p = $5.5• v= $0• c= $2• SL= Cs/(Cs+Ce)= (p-c)/[(p-c)+(c-v)] = (5.5-2)/[(5.5-2)+(2-0)] = 3.5/(3.5+2)= .636• Q=13

Discrete Stocking Levels

• Shortage (downtime) cost = $4,200

• Excess (spare part) cost = $200

• SL = = 4200/(4200+200) = .84

• Next level - > 2 spare parts90% will not run out of spare parts

13-38

Spares used

Relative frequency

Cumulative frequency

0 .2 .2

1 .4 .6

2 .3 .9

3 .1 1

4 or more

0 1

Operations Strategy*

• Too much inventory:– Costly to maintain– Tends to hide problems (easier to live with

problems than to eliminate them)• Wise strategy = find ways to

– Improve demand forecasts = reduce safety stock

– Improve inventory management– Reduce Ordering costs– Reduce inventory Holding costs– Reduce variation (e.g., lead-time)– Lean Operations– Supply-Chain Management 12-39