inventory management and control

1

Inventory Management and

Control

2

AMAZON.com

• Jeff Bezos, in 1995, started AMAZON.com as a “virtual” retailer – no inventory, no warehouses, no overhead; just a bunch of computers.

• Growth forced AMAZON.com to excel in inventory management!

• AMAZON is now a worldwide leader in warehouse management and automation.

3

Order Fulfillment at AMAZON (1 of 2)

1. You order items; computer assigns your order to distribution center [closest facility that has the product(s)]

2. Lights indicate products ordered to workers who retrieve product and reset light.

3. Items placed in crate with items from other orders, and crate is placed on conveyor. Bar code on item is scanned 15 times – virtually eliminating error.

4

Order Fulfillment at AMAZON (2 of 2)

4. Crates arrive at a central point where items are boxed and labeled with new bar code.

5. Gift wrapping done by hand (30 packages per hour)

6. Box is packed, taped, weighed and labeled before leaving warehouse in a truck.

7. Order appears on your doorstep within a week

5

Inventory Defined

• Inventory is the stock of any item or resource held to meet future demand and can include: raw materials, finished products, component parts, supplies, and work-in-process

6

Inventory

Process stage

Demand Type

Number & Value Other

Raw Material WIP

Finished Goods

Independent Dependent

A Items B Items C Items

Maintenance Operating

Inventory Classifications

7

E(1)

Independent vs. Dependent Demand

B(4)

E(2)D(1)

C(2)

E(3)B(1)

A

Independent Demand (Demand for the final end-product or demand not related to other items; demand created by

external customers)

Dependent Demand

(Derived demand for component

parts, subassemblies,

raw materials, etc- used to produce final products)

Finishedproduct

Component parts

Independent demand is uncertain Dependent demand is certain

8

Inventory Models

• Independent demand – finished goods, items that are ready to be sold– E.g. a computer

• Dependent demand – components of finished products– E.g. parts that make up the computer

9

Types of Inventories (1 of 2)

• Raw materials & purchased parts

• Partially completed goods called work in progress

• Finished-goods inventories (manufacturing firms) or merchandise (retail stores)

10

Types of Inventories (2 of 2)

• Replacement parts, tools, & supplies

• Goods-in-transit to warehouses or customers

11

The Material Flow Cycle (1 of 2)

12

Run time: Job is at machine and being worked onSetup time: Job is at the work station, and the work station is

being "setup."Queue time: Job is where it should be, but is not being

processed because other work precedes it.Move time: The time a job spends in transitWait time: When one process is finished, but the job is waiting

to be moved to the next work area.Other: "Just-in-case" inventory.

The Material Flow Cycle (2 of 2)

WaitTime

MoveTime

QueueTime

SetupTime

RunTimeInput

Cycle Time

Output

13

Performance Measures

• Inventory turnover (the ratio of annual cost of goods sold to average inventory investment)

• Days of inventory on hand (expected number of days of sales that can be supplied from existing inventory)

14

Functions of Inventory (1 of 2)

1. To “decouple” or separate various parts of the production process, ie. to maintain independence of operations

2. To meet unexpected demand & to provide high levels of customer service

3. To smooth production requirements by meeting seasonal or cyclical variations in demand

4. To protect against stock-outs

15

Functions of Inventory (2 of 2)

5. To provide a safeguard for variation in raw material delivery time

6. To provide a stock of goods that will provide a “selection” for customers

7. To take advantage of economic purchase-order size

8. To take advantage of quantity discounts

9. To hedge against price increases

16

• Higher costs– Item cost (if purchased)– Ordering (or setup) cost– Holding (or carrying) cost

• Difficult to control

• Hides production problems

• May decrease flexibility

Disadvantages of Inventory

17

Inventory CostsHolding (or carrying) costs

Costs for storage, handling, insurance, etc

Setup (or production change) costs Costs to prepare a machine or process for

manufacturing an order, eg. arranging specific equipment setups, etc

Ordering costs (costs of replenishing inventory) Costs of placing an order and receiving goods

Shortage costs Costs incurred when demand exceeds supply

18

Holding (Carrying) Costs

• Obsolescence• Insurance• Extra staffing• Interest• Pilferage• Damage• Warehousing• Etc.

19

Inventory Holding Costs(Approximate Ranges)

Category

Housing costs (building rent, depreciation, operating cost, taxes, insurance)

Material handling costs (equipment, lease or depreciation, power, operating cost)

Labor cost from extra handlingInvestment costs (borrowing costs, taxes,

and insurance on inventory)

Pilferage, scrap, and obsolescence

Overall carrying cost

Cost as a % of Inventory Value

6%(3 - 10%)

3%(1 - 3.5%)

3%(3 - 5%)

11%(6 - 24%)

3% (2 - 5%)

26%

20

Ordering Costs

• Supplies

• Forms

• Order processing

• Clerical support

• etc.

21

Setup Costs

• Clean-up costs

• Re-tooling costs

• Adjustment costs

• etc.

22

Shortage Costs

• Backordering cost

• Cost of lost sales

23

Inventory Control System Defined

An inventory system is the set of policies and controls that monitor levels of inventory and determine what levels should be maintained, when stock should be replenished and how large orders should be

Answers questions as: When to order? How much to order?

24

Objective of Inventory Control

To achieve satisfactory levels of customer service while keeping inventory costs within reasonable bounds

Level of customer service

Costs of ordering and carrying inventory

25

A system to keep track of inventory

A reliable forecast of demand

Knowledge of lead times

Reasonable estimates of Holding costs

Ordering costs

Shortage costs

A classification system

Requirements of an Effective Inventory Management

26

Inventory Counting (Control) Systems• Periodic System

Physical count of items made at periodic intervals; order is placed for a variable amount after fixed passage of time

• Perpetual (Continuous) Inventory System System that keeps track of removals from inventory continuously, thus monitoring current levels of each item (constant amount is ordered when inventory declines to a predetermined level)

27

Inventory ModelsSingle-Period Inventory Model

One time purchasing decision (Example: vendor selling t-shirts at a football game)

Seeks to balance the costs of inventory overstock and under stock

Multi-Period Inventory Models Fixed-Order Quantity Models

• Event triggered (Example: running out of stock) Fixed-Time Period Models

• Time triggered (Example: Monthly sales call by sales representative)

28

Single-Period Inventory Model

29

• Single period model: model for ordering of perishables and other items with limited useful lives

• Shortage cost: generally the unrealized profits per unit

• Excess cost: difference between purchase cost and salvage value of items left over at the end of a period

Single Period Model

30

• Continuous stocking levels

– Identifies optimal stocking levels

– Optimal stocking level balances unit shortage and excess cost

• Discrete stocking levels

– Service levels are discrete rather than continuous

– Desired service level is equaled or exceeded

Single Period Model

31

Single-Period Model

uo

u

CC

CP

uo

u

CC

CP

sold be unit will y that theProbabilit

estimatedunder demand ofunit per Cost C

estimatedover demand ofunit per Cost C

:Where

u

o

P

This model states that we should continue to increase the size of the inventory so long as the probability of selling the last unit added is equal to or greater than the ratio of: Cu/Co+Cu

This model states that we should continue to increase the size of the inventory so long as the probability of selling the last unit added is equal to or greater than the ratio of: Cu/Co+Cu

32

Optimal Stocking Level

Service Level

So

Quantity

Ce Cs

Balance point

Service level =Cs

Cs + CeCs = Shortage cost per unitCe = Excess cost per unit

33

Single Period Example 1• Ce = $0.20 per unit

• Cs = $0.60 per unit

• Service level = Cs/(Cs+Ce) = .6/(.6+.2)

• Service level = .75

Service Level = 75%

Quantity

Ce Cs

Stockout risk = 1.00 – 0.75 = 0.25

34

Single Period Model Example 2

Our college basketball team is playing in a tournament game this weekend. Based on our past experience we sell on average 2,400 shirts with a standard deviation of 350. We make $10 on every shirt we sell at the game, but lose $5 on every shirt not sold. How many shirts should we make for the game?Cu = $10 and Co = $5; P ≤ $10 / ($10 + $5) = .667

Z.667 = .432 therefore we need 2,400 + .432(350) = 2,551 shirts

35

Multi-Period Inventory Models

Fixed-Order Quantity Models (Types of)Economic Order Quantity ModelEconomic Production Order Quantity (Economic Lot Size) ModelEconomic Order Quantity Model with Quantity Discounts

Fixed Time Period (Fixed Order Interval) Models

36

Fixed Order Quantity Models:Economic Order Quantity Model

37

Economic Order Quantity Model Assumptions (1 of 2):

• Demand for the product is known with certainty, is constant and uniform throughout the period

• Lead time (time from ordering to receipt) is known and constant

• Price per unit of product is constant (no quantity discounts)

• Inventory holding cost is based on average inventory

38

Economic Order Quantity Model Assumptions (2 of 2):

• Ordering or setup costs are constant

• All demands for the product will be satisfied (no back orders are allowed)

• No stockouts (shortages) are allowed

• The order quantity is received all at once. (Instantaneous receipt of material in a single lot)

The goal is to calculate the order quantitiy that minimizes total cost

39

Basic Fixed-Order Quantity Model and Reorder Point Behavior

R = Reorder pointQ = Economic order quantityL = Lead time

L L

Q QQ

R

Time

Numberof unitson hand(Inv. Level)

1. You receive an order quantity Q.

2. You start using them up over time. 3. When you reach down to

a level of inventory of R, you place your next Q sized order.

4. The cycle then repeats.

40

EOQ Model

Reorder Point

(ROP)

Time

Inventory LevelAverageInventory

(Q/2)

Lead Time

Order Quantity

(Q)

Demand rate

Order placed Order received

41

EOQ Cost Model: How Much to Order?

By adding the holding and ordering costs together, we determine the total cost curve, which in turn is used to find the optimal order quantity that minimizes total costs

Slope = 0Slope = 0

Total CostTotal Cost

Order Quantity, Order Quantity, QQ

Annual Annual cost ($)cost ($)

Minimum Minimum total costtotal cost

Optimal orderOptimal order QQoptopt

Carrying Cost =Carrying Cost =HHQQ

22

Ordering Cost =Ordering Cost =SSDD

QQ

42

• More units must be stored if more are ordered

Purchase OrderDescription Qty.Microwave 1

Order quantity


Order quantity

Why Holding Costs Increase?

43

Cost is spread over more units

Example: You need 1000 microwave ovens





1 Order (Postage $ 0.33) 1000 Orders (Postage $330)

Order quantity


Why Ordering Costs Decrease ?

44

Basic Fixed-Order Quantity (EOQ) Model Formula

H 2

Q + S

Q

D + DC = TC H

2

Q + S

Q

D + DC = TC

Total Annual =Cost

AnnualPurchase

Cost

AnnualOrdering

Cost

AnnualHolding

Cost+ +

TC=Total annual costD =Annual demandC =Cost per unitQ =Order quantityS =Cost of placing an order or setup costR =Reorder pointL =Lead timeH=Annual holding and storage cost per unit of inventory

TC=Total annual costD =Annual demandC =Cost per unitQ =Order quantityS =Cost of placing an order or setup costR =Reorder pointL =Lead timeH=Annual holding and storage cost per unit of inventory

45

EOQ Cost Model

Annual ordering cost =Annual ordering cost =S S DD

QQ

AnnualAnnual carrying costcarrying cost = =HHQQ

22

Total cost = +Total cost = +S S DD

QQH H QQ

22

TC = +S D

Q

H Q

2

= +S D

Q2

H

2TC

Q

0 = +S D

Q2

H

2

Qopt =2SD

H

Deriving Qopt Proving equality of costs at optimal point

=S D

Q

H Q

2

Q2 =2S D

H

Qopt =2 S D

H

Using calculus, we take the first derivative of the total cost function with respect to Q, and set the derivative (slope) equal to zero, solving for the optimized (cost minimized) value of Qopt

46

Deriving the EOQ

Q = 2DS

H =

2(Annual D em and)(Order or Setup Cost)

Annual Holding CostOPTQ =

2DS

H =

2(Annual D em and)(Order or Setup Cost)

Annual Holding CostOPT

Reorder point, R = d L_

Reorder point, R = d L_

d = average daily demand (constant)

L = Lead time (constant)

_We also need a reorder point to tell us when to place an order

We also need a reorder point to tell us when to place an order

How much to order?:

When to order?

47

Optimal Order Quantity

Expected Number of Orders

Expected Time Between Orders Working Days / Year

Working Days / Year

= =× ×

= =

= =

=

= ×

Q*D SH

ND

Q*

TN

dD

ROP d L

2

EOQ Model Equations

48

EOQ Example 1 (1 of 3)

Annual Demand = 1,000 unitsDays per year considered in average daily demand = 365Cost to place an order = $10Holding cost per unit per year = $2.50Lead time = 7 daysCost per unit = $15

Given the information below, what are the EOQ and reorder point?

Given the information below, what are the EOQ and reorder point?

49

EOQ Example 1(2 of 3)

Q = 2DS

H =

2(1,000 )(10)

2.50 = 89.443 units or OPT 90 unitsQ =

2DS

H =

2(1,000 )(10)

2.50 = 89.443 units or OPT 90 units

d = 1,000 units / year

365 days / year = 2.74 units / dayd =

1,000 units / year

365 days / year = 2.74 units / day

Reorder point, R = d L = 2.74units / day (7days) = 19.18 or _

20 units Reorder point, R = d L = 2.74units / day (7days) = 19.18 or _

20 units

In summary, you place an optimal order of 90 units. In the course of using the units to meet demand, when you only have 20 units left, place the next order of 90 units.

In summary, you place an optimal order of 90 units. In the course of using the units to meet demand, when you only have 20 units left, place the next order of 90 units.

50

EOQ Example I(3 of 3)

TCTCminmin = =SSDD

QQHHQQ

22

TCTCminmin = = (10)(1,000)(10)(1,000)

9090((2,52,5)()(990)0)

22

TCTCminmin = $ = $ 111 111 + $ + $111111 = = 22 22 $$

Orders per year =Orders per year = DD//QQoptopt

== 1000/1000/9900

== 1111 orders/year orders/year

Order cycle timeOrder cycle time== 365/(365/(DD//QQoptopt))

== 336565//1111 == 33.133.1daysdays

++

+

51


Annual Demand = 10,000 unitsDays per year considered in average daily demand = 365Cost to place an order = $10Holding cost per unit per year = 10% of cost per unitLead time = 10 daysCost per unit = $15

Determine the economic order quantity and the reorder point given the following…

Determine the economic order quantity and the reorder point given the following…

52


Q =2DS

H=

2(10,000 )(10)

1.50= 365.148 units, or OPT 366 unitsQ =

2DS

H=

2(10,000 )(10)

1.50= 365.148 units, or OPT 366 units

d =10,000 units / year

365 days / year= 27.397 units / dayd =

10,000 units / year

365 days / year= 27.397 units / day

R = d L = 27.397 units / day (10 days) = 273.97 or _

274 unitsR = d L = 27.397 units / day (10 days) = 273.97 or _

274 units

Place an order for 366 units. When in the course of using the inventory you are left with only 274 units, place the next order of 366 units.

Place an order for 366 units. When in the course of using the inventory you are left with only 274 units, place the next order of 366 units.

53

EOQ Example 3

HH = $0.75 per yard = $0.75 per yard SS = $150 = $150 DD = 10,000 yards = 10,000 yards

QQoptopt = =22 S S DD

HH

QQoptopt = =2(150)(10,000)2(150)(10,000)

(0.75)(0.75)

QQoptopt = 2,000 yards = 2,000 yards

TCTCminmin = + = +S S DD

QQH H QQ

22

TCTCminmin = + = +((150)(10,000)150)(10,000)

2,0002,000(0.75)(2,000)(0.75)(2,000)

22

TCTCminmin = $750 + $750 = $1,500 = $750 + $750 = $1,500

Orders per year = D/Qopt

= 10,000/2,000

= 5 orders/year

Order cycle time =311 days/(Order cycle time =311 days/(DD//QQoptopt))

== 311/5311/5

== 62.2 store days62.2 store days

54

When to Reorder with EOQ Ordering ?• Reorder Point – is the level of inventory at which a

new order is placed

ROP = d . L

• Safety Stock - Stock that is held in excess of expected demand due to variable demand rate and/or lead time.

• Service Level - Probability that demand will not exceed supply during lead time (probability that inventory available during lead time will meet demand) 1 - Probability of stockout

55

Reorder Point Example

Demand = 10,000 yards/year

Store open 311 days/year

Daily demand = 10,000 / 311 = 32.154 yards/day

Lead time = L = 10 days

R = dL = (32.154)(10) = 321.54 yards

56

Determinants of the Reorder Point

• The rate of demand

• The lead time

• Demand and/or lead time variability

• Stockout risk (safety stock)

57

Answer how much & when to order Allow demand to vary

Follows normal distribution Other EOQ assumptions apply

Consider service level & safety stock Service level = 1 - Probability of stockout Higher service level means more safety stock More safety stock means higher ROP

Probabilistic Models

58

Safety Stock

LT Time

Expected demandduring lead time

Maximum probable demandduring lead time

ROP

Qu

an

tity

Safety stock

Safety stock reduces risk ofstockout during lead time

59

Variable Demand with a Reorder Point

Reorderpoint, R

Q

LTLT

TimeTimeLTLT

Inve

nto

ry le

vel

0

60

Reorder Point with a Safety Stock

Reorderpoint, R

QQ

LT

Time

LT

Inve

nto

ry le

vel

0

Safety Stock

61

Reorder Point With Variable Demand

R = dL + zd Lwhere

d = average daily demandL = lead time

d = the standard deviation of daily demand

z = number of standard deviationscorresponding to the service levelprobability

zd L = safety stock

62

Reorder Point for Service Level

Probability of meeting demand during lead time = service level

Probability of a stockout

R

Safety stock

dLExpected Demand

zd L

The reorder point based on a normal distribution of LT demand

63

Reorder Point for Variable Demand (Example)

The carpet store wants a reorder point with a 95% service level and a 5% stockout probability

d = 30 yards per dayL = 10 daysd = 5 yards per day

For a 95% service level, z = 1.65

R = dL + z d L

= 30(10) + (1.65)(5)( 10)

= 326.1 yards

Safety stock = z d L

= (1.65)(5)( 10)

= 26.1 yards

64

Fixed Order Quantity Models:-Noninstantaneous Receipt-Production Order Quantity

(Economic Lot Size) Model

65

Production done in batches or lotsCapacity to produce a part exceeds that part’s usage or

demand rateAllows partial receipt of material

Other EOQ assumptions apply

Suited for production environment Material produced, used immediately Provides production lot size

Lower holding cost than EOQ modelAnswers how much to order and when to order

Production Order Quantity Model

66

EOQ POQ ModelWhen To Order

Time

Inve

ntor

y Le

vel

Both production and usage take

place Usage only takes placeMaximum

inventory level

67

EOQ POQ ModelWhen To Order

Reorder Point (ROP)

Time

Inventory Level

AverageInventory

Lead Time

Optimal Order Quantity(Q*)

68

POQ Model Inventory Levels (1 of 2)Inventory Level

TimeSupply Begins

Supply Ends

Production portion of cycle

Demand portion of cycle with no supply

Maximum inventory level

69

POQ Model Inventory Levels (2 of 2)

Time

Inventory Level

Production Portion of

Cycle

Max. Inventory Q·(1- u/p)Q*Q*

Supply Begins

Supply Ends

Inventory level with no demand

Demand portion of cycle with no supply

Average inventory Q/2(1- u/p)

70

D = Demand per year

S = Setup cost

H = Holding cost

d = Demand per day

p = Production per day

POQ Model Equations

Production Order Quantity

Setup Cost

Holding Cost

= =

-

= *

= *

=

Q

H* up

Q

D

QS

p*

1

(

1/2 * H * Q -u

p1

)-u

p1

( )

2*D*S

( )Maximum inventory level

QDS

H

p

p u0

2

71

Production Order Quantity Example (1 of 2)

H = $0.75 per yard S = $150 D = 10,000 yards

u = 10,000/311 = 32.2 yards per day p = 150 yards per day

POQopt = = = 2,256.8 yards

2 S D

H 1 - up

2(150)(10,000)

0.75 1 - - 32.2150

TC = + 1 - = $1,329up

S DQ

H Q2

Production run = = = 15.05 days per orderQp

2,256.8150

72

Production Quantity Example (2 of 2)

H = $0.75 per yard S = $150 D = 10,000 yards

u= 10,000/311 = 32.2 yards per day p = 150 yards per day

QQoptopt = = = 2,256.8 yards = = = 2,256.8 yards

22CCooDD

CCcc 1 - 1 - ddpp

2(150)(10,000)2(150)(10,000)

0.75 1 - 0.75 1 - 32.232.2150150

TCTC = + 1 - = $1,329 = + 1 - = $1,329ddpp

CCooDD

QQ

CCccQQ

22

Production run = = = 15.05 days per orderQp

2,256.8150

Number of production runs = = = 4.43 runs/yearDQ

10,0002,256.8

Maximum inventory level = Q 1 - = 2,256.8 1 -

= 1,772 yards

up

32.2150

73

Fixed-Order Quantity Models:Economic Order Quantity Model

with Quantity Discounts

74

• Answers how much to order & when to order

• Allows quantity discounts

– Price per unit decreases as order quantity increases

– Other EOQ assumptions apply

• Trade-off is between lower price & increased holding cost

Quantity Discount Model

TC = + + PDS D

Q

iC QQ

22Where P: Unit Price

Total cost with purchasing cost

75

Price-Break Model Formula

Cost Holding Annual

Cost) Setupor der Demand)(Or 2(Annual =

iC

2DS = QOPT

Based on the same assumptions as the EOQ model, the price-break model has a similar Qopt formula:

i = percentage of unit cost attributed to carrying inventoryC = cost per unit

Since “C” changes for each price-break, the formula above will have to be used with each price-break cost value

76

Total Costs with PDC

ost

EOQ

TC with PD

TC without PD

PD

0 Quantity

Adding Purchasing costdoesn’t change EOQ

77

Total Cost with Constant Carrying Costs

OC

EOQ Quantity

Tot

al C

ost TCa

TCc

TCbDecreasing Price

CC a,b,c

78

Quantity Discount – How Much to Order?

79

Price-Break Example 1 (1 of 3)

ORDER SIZE PRICE

0 - 99 $10

100 - 199 8 (d1)

200+ 6 (d2)

For this problem holding cost is given as a constant value, not as a percentage of price, so the optimal order quantity is the same for each of the price ranges. (see the figure)

80

Price Break Example 1 (2 of 3)

QQoptopt

Carrying cost Carrying cost

Ordering cost Ordering cost

Inve

nto

ry c

ost

($)

Inve

nto

ry c

ost

($)

QQ((dd1 1 ) = 100) = 100 QQ((dd2 2 ) = 200) = 200

TC TC ((dd2 2 = $6 ) = $6 )

TCTC ( (dd1 1 = $8 )= $8 )

TC TC = ($10 )= ($10 )

81

Price Break Example 1 (3 of 3)

QQoptopt

Carrying cost Carrying cost

Ordering cost Ordering cost

Inve

nto

ry c

ost

($)

Inve

nto

ry c

ost

($)

QQ((dd1 1 ) = 100) = 100 QQ((dd2 2 ) = 200) = 200

TC TC ((dd2 2 = $6 ) = $6 )

TCTC ( (dd1 1 = $8 )= $8 )

TC TC = ($10 )= ($10 )

The lowest total cost is at the second price break

82

Price Break Example 2

QUANTITYQUANTITY PRICEPRICE

1 - 491 - 49 $1,400$1,400

50 - 8950 - 89 1,1001,100

90+90+ 900900

SS = = $2,500 $2,500

HH = = $190 per computer $190 per computer

DD = = 200200

QQoptopt = = = 72.5 PCs = = = 72.5 PCs22SSDD

HH2(2500)(200)2(2500)(200)

190190

TCTC = + + = + + PD PD = $233,784 = $233,784 SSDD

QQoptopt

H H QQoptopt

22

For For QQ = 72.5 = 72.5

TCTC = + + = + + PD PD = $194,105= $194,105SSDD

QQ

H H QQ

22

For For QQ = 90 = 90

83


A company has a chance to reduce their inventory ordering costs by placing larger quantity orders using the price-break order quantity schedule below. What should their optimal order quantity be if this company purchases this single inventory item with an e-mail ordering cost of $4, a carrying cost rate of 2% of the inventory cost of the item, and an annual demand of 10,000 units?

A company has a chance to reduce their inventory ordering costs by placing larger quantity orders using the price-break order quantity schedule below. What should their optimal order quantity be if this company purchases this single inventory item with an e-mail ordering cost of $4, a carrying cost rate of 2% of the inventory cost of the item, and an annual demand of 10,000 units?

Order Quantity(units) Price/unit($)0 to 2,499 $1.202,500 to 3,999 1.004,000 or more .98

84

Price-Break Example (2 of 4)

units 1,826 = 0.02(1.20)

4)2(10,000)( =

iC

2DS = QOPT

Annual Demand (D)= 10,000 unitsCost to place an order (S)= $4

First, plug data into formula for each price-break value of “C”

units 2,000 = 0.02(1.00)

4)2(10,000)( =

iC

2DS = QOPT

units 2,020 = 0.02(0.98)

4)2(10,000)( =

iC

2DS = QOPT

Carrying cost % of total cost (i)= 2%Cost per unit (C) = $1.20, $1.00, $0.98

Interval from 0 to 2499, the Qopt value is feasible

Interval from 2500-3999, the Qopt value is not feasible

Interval from 4000 & more, the Qopt value is not feasible

Next, determine if the computed Qopt values are feasible or not

85


Since the feasible solution occurred in the first price-break, it means that all the other true Qopt values occur at the beginnings of each price-break interval. Why?

Since the feasible solution occurred in the first price-break, it means that all the other true Qopt values occur at the beginnings of each price-break interval. Why?

0 1826 2500 4000 Order Quantity

Total annual costs

So the candidates for the price-breaks are 1826, 2500, and 4000 units

So the candidates for the price-breaks are 1826, 2500, and 4000 units

Because the total annual cost function is a “u” shaped function

Because the total annual cost function is a “u” shaped function

86


iC 2

Q + S

Q

D + DC = TC iC

2

Q + S

Q

D + DC = TC

Next, we plug the true Qopt values into the total cost annual cost function to determine the total cost under each price-break

Next, we plug the true Qopt values into the total cost annual cost function to determine the total cost under each price-break

TC(0-2499)=(10000*1.20)+(10000/1826)*4+(1826/2)(0.02*1.20) = $12,043.82TC(2500-3999)= $10,041TC(4000&more)= $9,949.20

TC(0-2499)=(10000*1.20)+(10000/1826)*4+(1826/2)(0.02*1.20) = $12,043.82TC(2500-3999)= $10,041TC(4000&more)= $9,949.20

Finally, we select the least costly Qopt, which in this problem occurs in the 4000 & more interval. In summary, our optimal order quantity is 4000 units

Finally, we select the least costly Qopt, which in this problem occurs in the 4000 & more interval. In summary, our optimal order quantity is 4000 units

87

Multi-period Inventory Models:Fixed Time Period

(Fixed-Order- Interval) Models

88

Orders are placed at fixed time intervals

Order quantity for next interval? (inventory is brought up to target amount, amount ordered varies)

Suppliers might encourage fixed intervals

Requires only periodic checks of inventory levels (no continous monitoring is required)

Risk of stockout between intervals

Fixed-Order-Interval Model

89

Inventory Level in a Fixed Period System

Various amounts (Qi) are ordered at regular time intervals (p) based on the quantity necessary to bring inventory up to

target maximum

pp pp pp

QQ11 QQ22

QQ33

QQ44

Target maximum

TimeTime

d In

vent

ory

d In

vent

ory

90

Tight control of inventory items

Items from same supplier may yield savings in:

Ordering

Packing

Shipping costs

May be practical when inventories cannot be closely monitored

Fixed-Interval Benefits

91

Requires a larger safety stock Increases carrying cost Costs of periodic reviews

Fixed-Interval Disadvantages

92

Fixed-Time Period Model with Safety Stock Formula

order)on items (includes levelinventory current = I

timelead and review over the demand ofdeviation standard =

yprobabilit service specified afor deviations standard ofnumber the= z

demanddaily averageforecast = d

daysin timelead = L

reviewsbetween days ofnumber the= T

ordered be toquantitiy = q

:Where

I - Z+ L)+(Td = q

L+T

L+T

order)on items (includes levelinventory current = I

timelead and review over the demand ofdeviation standard =

yprobabilit service specified afor deviations standard ofnumber the= z

demanddaily averageforecast = d

daysin timelead = L

reviewsbetween days ofnumber the= T

ordered be toquantitiy = q

:Where

I - Z+ L)+(Td = q

L+T

L+T

q = Average demand + Safety stock – Inventory currently on handq = Average demand + Safety stock – Inventory currently on hand

93

Fixed-Time Period Model: Determining the Value of T+L

T+L di 1

T+L

d

T+L d2

=

Since each day is independent and is constant,

= (T + L)

i

2

T+L di 1

T+L

d

T+L d2

=

Since each day is independent and is constant,

= (T + L)

i

2

The standard deviation of a sequence of random events equals the square root of the sum of the variances

94

Order Quantity for a Periodic Inventory System

Q = d(tb + L) + zd T + L - Iwhere

d = average demand rateT = the fixed time between ordersL = lead time

d = standard deviation of demand

zd tb + L = safety stockI = inventory level

z = the number of standard deviations for a specified service level

95

Fixed-Period Model with Variable Demand (Example 1)

d = 6 bottles per dayd = 1.2 bottlestb = 60 daysL = 5 daysI = 8 bottlesz = 1.65 (for a 95% service level)

Q = d(tb + L) + zd tb + L - I

= (6)(60 + 5) + (1.65)(1.2) 60 + 5 - 8

= 397.96 bottles

96

Fixed-Time Period Model withVariable Demand (Example 2)(1 of 3)

Average daily demand for a product is 20 units. The review period is 30 days, and lead time is 10 days. Management has set a policy of satisfying 96 percent of demand from items in stock. At the beginning of the review period there are 200 units in inventory. The standard deviation of daily demand is 4 units.

Given the information below, how many units should be ordered?

Given the information below, how many units should be ordered?

97

Fixed-Time Period Model with Variable Demand (Example 2)(2 of 3)

T+ L d2 2 = (T + L) = 30 + 10 4 = 25.298 T+ L d

2 2 = (T + L) = 30 + 10 4 = 25.298

So, by looking at the value from the Table, we have a probability of 0.9599, which is given by a z = 1.75

So, by looking at the value from the Table, we have a probability of 0.9599, which is given by a z = 1.75

98

Fixed-Time Period Model with Variable Demand (Example 2) (3 of 3)

or 644.272, = 200 - 44.272 800 = q

200- 298)(1.75)(25. + 10)+20(30 = q

I - Z+ L)+(Td = q L+T

units 645

or 644.272, = 200 - 44.272 800 = q

200- 298)(1.75)(25. + 10)+20(30 = q

I - Z+ L)+(Td = q L+T

units 645

So, to satisfy 96 percent of the demand, you should place an order of 645 units at this review period

So, to satisfy 96 percent of the demand, you should place an order of 645 units at this review period

99

Miscellaneous Systems:Optional Replenishment System

Maximum Inventory Level, M

MActual Inventory Level, I

q = M - I

I

Q = minimum acceptable order quantity

If q > Q, order q, otherwise do not order any.

100

ABC Classification System

• Demand volume and value of items vary

• Items kept in inventory are not of equal importance in terms of:

– dollars invested

– profit potential

– sales or usage volume

– stock-out penalties

101

ABC Classification System

Classifying inventory according to some measure of importance and allocating control efforts accordingly.

AA - very important

BB - mod. important

CC - least important Annual $ value of items

AA

BB

CC

High

Low

Low HighPercentage of Items

102

Classify inventory into 3 categories typically on the basis of the dollar value to the firm $ volume = Annual demand x Unit cost

A class, B class, C class Policies based on ABC analysis– Develop class A suppliers more carefully – Give tighter physical control of A items– Forecast A items more carefully

ABC Analysis

103

% of Inventory Items

Classifying Items as ABC

0

20

40

60

80

100

0 50 100

% Annual $ Usage

AABB

CC

Class % $ Vol % ItemsA 70-80 5-15B 15 30C 5-10 50-60

104

ABC Classification

11 $ 60$ 60 909022 350350 404033 3030 13013044 8080 606055 3030 10010066 2020 18018077 1010 17017088 320320 505099 510510 6060

1010 2020 120120

PARTPART UNIT COSTUNIT COST ANNUAL USAGEANNUAL USAGE

105

ABC Classification

11 $ 60$ 60 909022 350350 404033 3030 13013044 8080 606055 3030 10010066 2020 18018077 1010 17017088 320320 505099 510510 6060

1010 2020 120120

PARTPART UNIT COSTUNIT COST ANNUAL USAGEANNUAL USAGETOTAL % OF TOTAL % OF TOTALPART VALUE VALUE QUANTITY % CUMMULATIVE

9 $30,600 35.9 6.0 6.08 16,000 18.7 5.0 11.02 14,000 16.4 4.0 15.01 5,400 6.3 9.0 24.04 4,800 5.6 6.0 30.03 3,900 4.6 10.0 40.06 3,600 4.2 18.0 58.05 3,000 3.5 13.0 71.0

10 2,400 2.8 12.0 83.07 1,700 2.0 17.0 100.0

$85,400

106

ABC Classification

11 $ 60$ 60 909022 350350 404033 3030 13013044 8080 606055 3030 10010066 2020 18018077 1010 17017088 320320 505099 510510 6060

1010 2020 120120


9 $30,600 35.9 6.0 6.08 16,000 18.7 5.0 11.02 14,000 16.4 4.0 15.01 5,400 6.3 9.0 24.04 4,800 5.6 6.0 30.03 3,900 4.6 10.0 40.06 3,600 4.2 18.0 58.05 3,000 3.5 13.0 71.0

10 2,400 2.8 12.0 83.07 1,700 2.0 17.0 100.0

$85,400

AA

BB

CC

107

ABC Classification

11 $ 60$ 60 909022 350350 404033 3030 13013044 8080 606055 3030 10010066 2020 18018077 1010 17017088 320320 505099 510510 6060

1010 2020 120120


9 $30,600 35.9 6.0 6.08 16,000 18.7 5.0 11.02 14,000 16.4 4.0 15.01 5,400 6.3 9.0 24.04 4,800 5.6 6.0 30.03 3,900 4.6 10.0 40.06 3,600 4.2 18.0 58.05 3,000 3.5 13.0 71.0

10 2,400 2.8 12.0 83.07 1,700 2.0 17.0 100.0

$85,400

AA

BB

CC

% OF TOTAL % OF TOTALCLASS ITEMS VALUE QUANTITY

A 9, 8, 2 71.0 15.0B 1, 4, 3 16.5 25.0C 6, 5, 10, 7 12.5 60.0

108

ABC Classification

100 100 –

80 80 –

60 60 –

40 40 –

20 20 –

0 0 –| | | | | |00 2020 4040 6060 8080 100100

% of Quantity% of Quantity

% o

f V

alu

e%

of

Val

ue

AA

BBCC

109

• Inventory accuracy refers to how well the inventory records agree with physical count.

• Physically counting a sample of total inventory on a regular basis

• Used often with ABC classification– A items counted most often (e.g., daily)

Inventory Accuracy and Cycle Counting

110

Advantages of Cycle Counting

• Eliminates shutdown and interruption of production necessary for annual physical inventories

• Eliminates annual inventory adjustments• Provides trained personnel to audit the accuracy of inventory• Allows the cause of errors to be identified and remedial

action to be taken• Maintains accurate inventory records

111

Last Words

Inventories have certain functions.

But too much inventory

- Tends to hide problems

- Costly to maintain

So it is desired

• Reduce lot sizes

• Reduce safety stocks

inventory management and control

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