inventory management (session 3,4) - ximb.ac. · pdf fileinventory management (session 3,4) 1....

Inventory ManagementInventory Management(Session 3,4)(Session 3,4)

1

OutlineOutline

Elements of Inventory ManagementInventory Control SystemsEconomic Order Quantity ModelsThe Basic EOQ ModelThe EOQ Model with Non-Instantaneous ReceiptThe EOQ Model with ShortagesQuantity DiscountsReorder PointDetermining Safety Stocks Using Service LevelsOrder Quantity for a Periodic Inventory System

2

Elements of Inventory ManagementRole of Inventory

Inventory is a stock of items kept on hand used to meet Inventory is a stock of items kept on hand used to meet customer demand.customer demand.A level of inventory is maintained that will meet anticipated A level of inventory is maintained that will meet anticipated demand.demand.If demand not known with certainty, safety (buffer) stocks If demand not known with certainty, safety (buffer) stocks are kept on hand.are kept on hand.Additional stocks are sometimes built up to meet seasonal Additional stocks are sometimes built up to meet seasonal or cyclical demand.or cyclical demand.

3

Elements of Inventory ManagementRole of Inventory

Large amounts of inventory sometimes purchased to take Large amounts of inventory sometimes purchased to take advantage of discounts.advantage of discounts.InIn--process inventories maintained to provide independence process inventories maintained to provide independence between operations.between operations.Raw materials inventory kept to avoid delays in case of Raw materials inventory kept to avoid delays in case of supplier problems.supplier problems.Stock of finished parts kept to meet customer demand in Stock of finished parts kept to meet customer demand in event of work stoppage.event of work stoppage.

4

Elements of Inventory ManagementDemand

Inventory exists to meet the demand of customers.Inventory exists to meet the demand of customers.Customers can be external (purchasers of products) or Customers can be external (purchasers of products) or internal (workers using material).internal (workers using material).Management needs accurate forecast of demand.Management needs accurate forecast of demand.Items that are used internally to produce a final product are Items that are used internally to produce a final product are referred to as referred to as dependent demand items.dependent demand items.

Items that are final products demanded by an external Items that are final products demanded by an external customer are customer are independent demand independent demand itemsitems..

5

Elements of Inventory ManagementInventory Costs

Carrying costs Carrying costs -- Costs of holding items in storageCosts of holding items in storage..

Vary with level of inventory and sometimes with length Vary with level of inventory and sometimes with length of time held.of time held.Include facility operating costs, record keeping, Include facility operating costs, record keeping, interest, etc.interest, etc.Assigned on a per unit basis per time period, or as Assigned on a per unit basis per time period, or as percentage of average inventory value (usually percentage of average inventory value (usually estimated as 10% to 40%).estimated as 10% to 40%).

6


Ordering costsOrdering costs -- costs of replenishing stock of inventory.costs of replenishing stock of inventory.Expressed as dollar amount per order, independent of Expressed as dollar amount per order, independent of order size.order size.Vary with the number of orders made.Vary with the number of orders made.Include purchase orders, shipping, handling, Include purchase orders, shipping, handling, inspection, etc.inspection, etc.

7


Shortage, or stockoutShortage, or stockout costs costs -- Costs associated with Costs associated with insufficient inventory.insufficient inventory.

Result in permanent loss of sales and profits for items Result in permanent loss of sales and profits for items not on hand.not on hand.Sometimes penalties involved; if customer is internal, Sometimes penalties involved; if customer is internal, work delays could result.work delays could result.

8

Inventory Control Systems

An inventory control system controls the level of inventory by determining how much (replenishment level) and whento order.Two basic types of systems -continuous (fixed-order quantity) and periodic (fixed-time).In a continuous system, an order is placed for the same constant amount when inventory decreases to a specified level.In a periodic system, an order is placed for a variable amount after a specified period of time.

9

Inventory Control SystemsContinuous Inventory System

A continual record of inventory level is maintained.Whenever inventory decreases to a predetermined level, the reorder point, an order is placed for a fixed amount to replenish the stock.

The fixed amount is termed the economic order quantity, whose magnitude is set at a level that minimizes the total inventory carrying, ordering, and shortage costs.

Because of continual monitoring, management is always aware of status of inventory level and critical parts, but system is relatively expensive to maintain.

10

Inventory Control SystemsPeriodic Inventory System

Inventory on hand is counted at specific time intervals and an order placed that brings inventory up to a specified level.Inventory not monitored between counts and system is therefore less costly to track and keep account of.Results in less direct control by management and thus generally higher levels of inventory to guard against stockouts.System requires a new order quantity each time an order is placed.Used in smaller retail stores, drugstores, grocery stores and offices.

11

Economic Order Quantity Models

Economic order quantity, or economic lot size, is the quantity ordered when inventory decreases to the reorder point.Amount is determined using the economic order quantity (EOQ) model.Purpose of the EOQ model is to determine the optimal order size that will minimize total inventory costs.Three model versions to be discussed:

Basic EOQ modelEOQ model without instantaneous receiptEOQ model with shortages

12

Economic Order Quantity ModelsBasic EOQ Model

A formula for determining the optimal order size that minimizes the sum of carrying costs and ordering costs.Simplifying assumptions and restrictions:

Demand is known with certainty and is relatively constant over time.No shortages are allowed.Lead time for the receipt of orders is constant.The order quantity is received all at once and instantaneously.

13

Economic Order Quantity ModelsBasic EOQ Model

Figure The Inventory Order Cycle 14

Basic EOQ ModelCarrying Cost

Carrying cost usually expressed on a per unit basis of time, traditionally one year.Annual carrying cost equals carrying cost per unit per year times average inventory level:

Carrying cost per unit per year = Cc

Average inventory = Q/2Annual carrying cost = CcQ/2.

15

Basic EOQ ModelCarrying Cost

Figure: Average Inventory 16

Basic EOQ ModelOrdering Cost

Total annual ordering cost equals cost per order (Co) times number of orders per year.Number of orders per year, with known and constant demand, D, is D/Q, where Q is the order size:

Annual ordering cost = CoD/QOnly variable is Q, Co and D are constant parameters.

Relative magnitude of the ordering cost is dependent on order size.

17

Basic EOQ ModelTotal Inventory Cost

2Q

cCQD

oCTC +=

Total annual inventory cost is sum of ordering and carrying cost:

TC = Total Inventory CostCo = Ordering CostCc = Inventory Carrying CostQ = Order QuantityD = Demand

18

Basic EOQ ModelTotal Inventory Cost

Figure: The EOQ Cost Model 19

Basic EOQ ModelEOQ and Min. Total Cost

EOQ occurs where total cost curve is at minimum value and carrying cost equals ordering cost:

The EOQ model is robust because Q is a square root and errors in the estimation of D, Cc and Co are dampened.

cCDoC

optQ

optQcC

optQDoC

TC

2

2min

=

+=

20

Example 1

I-75 Carpet Discount Store, Super Shag carpet sales.Given following data, determine number of orders to be made annually and time between orders given store is open every day except Sunday, Thanksgiving Day, and Christmas Day.

yd 000,2)75.0()000,10)(150(22

:sizeorder Optimal

10,000yd D $150, oC $0.75, cC

:parameters Model

===

===

cCDoC

optQ

21

days store 62.2 5311

/days 311 timecycleOrder

5000,2000,10

:yearper orders ofNumber

500,1$2)000,2()75.0(000,2

000,10)150(2min

:costinventory annual Total

===

==

=+=+=

optQD

optQD

optQcC

optQD

oCTC

22

Basic EOQ ModelEOQ Analysis over time

For any time period unit of analysis, EOQ is the same.Shag Carpet example on monthly basis:

yd 000,2)0625.0()3.833)(150(22

:sizeorder Optimal

monthper yd 833.3 Dorderper $150 oC

monthper ydper $0.0625 cC

:parameters Model

===

=

=

=

cCDoC

optQ23

$1,500 ($125)(12) cost inventory annual Total

monthper 125$

2)000,2()0625.0(000,2

)3.833()150(2min

:costinventory monthly Total

==

=

+=+= optQcC

optQD

oCTC

Basic EOQ ModelEOQ Analysis over time

24

EOQ ModelNon-Instantaneous Receipt Description

In the non-instantaneous receipt model the assumption that orders are received all at once is relaxed. (Also known as gradual usage or production lot size model.)The order quantity is received gradually over time and inventory is drawn on at the same time it is being replenished.

25


Figure: The EOQ Model with Non-Instantaneous Order Receipt 26


⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

−+=

−=

−=

−=

==

pdQ

cCQD

oC

pdQ

cC

pdQ

pdQ

12costinventory annual Total

12 cost carrying Total

12 levelinventory Average

1 levelinventory Maximum

demanded isinventory at which ratedaily dover timereceivedisorder heat which tratedaily p

ModelFormulation

27


)/1(2

:sizeorder Optimal

curvecost totalofpoint lowest at 12

pdcCDoC

optQ

QD

oCpdQ

cC

−=

=− ⎟⎟⎠

⎞⎜⎜⎝

⎛

Equation shown above is the optimal order size under non-Instantaneous receipt

28

Example

Super Shag carpet manufacturing facility:

yd 8.256,2

1502.32175.0

)000,10)(150(21

2 :sizeorder Optimal

dayper yd 150 pdayper yd 32.2 10,000/311 year per yd 10,000 D

unitper $0.75 cC

$150 oC

=

−=

−=

====

=

=

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛pd

cC

DoCoptQ

29

Example

yd 772,11502.3218.256,21 levelinventory Maximum

runs 43.48.256,2000,10

runs)n (productioyear per orders ofNumber

days 05.151508.256,2 length run Production

329,1$1502.3212

)8.256,2()075(.)8.256,2()000,10()150(

12costinventory annual minimum Total

=−=−=

==

=

===

=−+=

−+=

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

pdQ

QD

pQ

pdQ

cCQD

oC

30

EOQ ModelWith Shortages

In the EOQ model with shortages, the assumption that shortages cannot exist is relaxed.

Assumed that unmet demand can be backordered with all demand eventually satisfied.

31


Figure: The EOQ Model with Shortages 32


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+==

+==

+−+=

=

−==

sCcCcC

optQoptS

sCcCsC

cCDoC

optQ

QD

oCQSQ

cCQS

sC

QD

oC

QSQ

cCQS

sC

level Shortage

2quantityorder Optimal

22)(

22

costinventory Total

cost ordering Total

22)( costs carrying Total 2

2 costs shortage Total

33


Figure: Cost Model with Shortages 34

Example

I-75 Carpet Discount Store allows shortages; shortage cost Cs, is $2/yard per year.

yd 2.345,2275.02

75.0)000,10)(150(22

:quantityorder Optimal

yd 10,000 Dydper $2 sC

ydper $0.75 cC

$150 oC

=+=+

=

=

=

=

=

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

sCcCsC

cCDoC

optQ

35

Example

$1,279.20 639.60 465.16 $174.44

2.345,2)000,10)(150(

)2.345,2(22)6.705,1)(75.0(

)2.345,2(22)6.639)(2(

22)(

22

:costinventory Total

yd 6.63975.0275.02.345,2

:level Shortage

=++=

++=

+−+=

=+=+= ⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

QD

oCQSQ

cCQS

sCTC

sCcCcC

optQoptS

36

Example

days 19.9or year 0.064 10,000639.6

D 2 t

shortage a is re which theduring Time

days 53.2or 0.17110,000639.6-2,345.2 1 t

handon isinventory which during Time

days 0.7326.4311

orders ofnumber yearper days t ordersbetween Time

yd 6.705,16.6392.345,2 levelinventory Maximum

yearper orders 26.42.345,2000,10 orders ofNumber

====

==−==

====

=−=−=

===

S

DSQ

SQ

QD

37

Quantity Discounts

Price discounts are often offered if a predetermined number of units is ordered or when ordering materials in high volume.Basic EOQ model used with purchase price added:

where: P = per unit price of the itemD = annual demand

Quantity discounts are evaluated under two different scenarios:

With constant carrying costsWith carrying costs as a percentage of purchase price

PDQcCQ

DoCTC ++= 2

38

Quantity Discounts with Constant Cc

Optimal order size is the same regardless of the discount price.The total cost with the optimal order size must be compared with any lower total cost with a discount price to determine which is the lesser.

39

University bookstore: For following discount schedule offered by Comptek, should bookstore buy at the discount terms or order the basic EOQ order size?

Determine optimal order size and total cost:

5.72190)200)(500,2(2

cCDo2C

optQ

200D unit per $190cC $2,500 oC

===

===

Example

Quantity Quantity PricePrice1 1 –– 4949 $ 1400$ 14005050--8989 1100110090 + 90 + 900900

40

Example

Compute total cost at eligible discount price ($1,100):

Compare with total cost of with order size of 90 and price of $900:

Because $194,105 < $233,784, maximum discount price should be taken and 90 units ordered.

784,233$)200)(100,1(2)5.72()190()5.72(

)200)(500,2(

2min

=++=

++= PDoptQcC

optQDoC

TC

105,194$)200)(900(2)90)(190(

)90()200)(500,2(

2

=++=

++= PDQcCQ

DoCTC

41

Quantity Discounts with Cc

% of Price

University Bookstore example, but a different optimal order size for each price discount.Optimal order size and total cost determined using basic EOQ model with no quantity discount.This cost then compared with various discount quantity order sizes to determine minimum cost order.This must be compared with EOQ-determined order size for specific discount price.Data:

Co = $2,500D = 200 computers per year

42

Compute optimum order size for purchase price without discount and Cc = $210:

Compute new order size:

Quantity Price Carrying Cost0 - 49 $1,400 1,400(.15) = $21050 - 89 1,100 1,100(.15) = 16590 + 900 900(.15) = 135

69210)200)(500,2(22===

cCDoC

optQ

8.77165)200)(500,2(2 ==optQ


% of Price

43

Compute minimum total cost:

Compare with cost, discount price of $900, order quantity of 90:

Optimal order size computed as follows:

Since this order size is less than 90 units , it is not feasible,thus optimal order size is 90 units.

845,232$

)200)(100,1(2)8.77(1658.77

)200)(500,2(2

=

++=++= PDQcCQ

DoCTC

6301912009002

9013590

2005002 ,$))(())(())(,( =++=TC

1.86135)200)(500,2(2 ==optQ


% of Price

44

Reorder Point

The reorder point is the inventory level at which a new order is placed.Order must be made while there is enough stock in place to cover demand during lead time.Formulation:

R = dLwhere d = demand rate per time period

L = lead timeFor Carpet Discount store problem:

R = dL = (10,000/311)(10) = 321.54

45

Reorder Point

Figure: Reorder Point and Lead Time 46

Reorder Point

Figure: Inventory Model with Uncertain Demand

Inventory level might be depleted at slower or faster rate during lead time.When demand is uncertain, safety stock is added as a hedge against stockout.

47

Reorder Point (4 of 4)Reorder Point

Figure: Inventory model with safety stock 48

Determining Safety Stock by Service Level

Service level is probability that amount of inventory on hand is sufficient to meet demand during lead time (probability stockout will not occur).

The higher the probability inventory will be on hand, the more likely customer demand will be met.

Service level of 90% means there is a .90 probability that demand will be met during lead time and .10 probability of a stockout.

49

Reorder Point with Variable Demand

stocksafety

yprobabilit level service toingcorrespond deviations standard ofnumber

demanddaily ofdeviation standard the

timelead

demanddaily average

pointreorder

:where

=

=

=

=

=

=

+=

LdZ

Z

d

L

d

R

LdZLdR

σ

σ

σ

50

Reorder Point with Variable Demand

Figure: Reorder Point for a Service Level51

Example

I-75 Carpet Discount Store Super Shag carpet.For following data, determine reorder point and safety stock for service level of 95%.

26.1. : formulapoint reorder in termsecond isstock Safety

yd 1.326

1.26300)10)(5)(65.1()10(30

A)appendix 1,-A (Table 1.65 Zlevel, service 95%For

dayper yd 5 days 10 L

dayper yd 30

=

+=+=+=

=

===

Ld

ZLdR

d

d

σ

σ

52

Reorder Point with Variable Lead Time

stocksafety timelead during demand ofdeviation standard

timelead ofdeviation standard timelead average

demanddaily constant

:where

===

==

+=

LZdLd

LLd

LZdLdR

σσ

σ

σ

For constant demand and variable lead time:

53

Example

yd 5.4485.148300)3)(30)(65.1()10)(30(

level service 95% afor 1.65

days 3

days 10

dayper yd 30

=+=+=+=

=

=

=

=

LZdLdR

Z

L

L

d

σ

σ

Carpet Discount Store:

54

Reorder Point with Variable Demand and Lead Time

When both demand and lead time are variable:

stocksafety 22)(

2)(Z

timelead during demand ofdeviation standard 22)(

2)(

timelead average demanddaily average

:where

22)(

2)(

=+

=+

==

++=

dLLd

dLLd

Ld

dLLdZLdR

σσ

σσ

σσ

55

Example

Carpet Discount Store:

yds 450.8 150.8 300

(30)(3)(3)(30) (5)(5)(10)(1.65) (30)(10)

22)(

2)(

level service 95%for 1.65 days 3days 10

dayper yd 5 dayper yd 30

=+=

++=

++=

==

=

==

dLLdZLdR

ZL

Ld

d

σσ

σ

σ

56

Order Quantity for a Periodic Inventory System

A periodic, or fixed-time period inventory system is one in which time between orders is constant and the order size varies.Vendors make periodic visits, and stock of inventory is counted.An order is placed, if necessary, to bring inventory level back up to some desired level.Inventory not monitored between visits.At times, inventory can be exhausted prior to the visit, resulting in a stockout.Larger safety stocks are generally required for the periodic inventory system. 57

Order Quantity for Variable Demand

For normally distributed variable daily demand:

stockin inventory stocksafety

demand ofdeviation standard timelead

ordersbetween timefixed therate demand average

:where

)(

==+

====

−+++=

ILbtdZ

dLbtd

ILbtdZLbtdQ

σσ

σ

58

Order Quantity for Variable Demand

Corner Drug Store with periodic inventory system.Order size to maintain 95% service level:

bottles398 85 60)(1.65)(1.2 5) (6)(60

)(

level service 95%for 1.65 bottles 8 days 5

days 60 bottles 1.2

dayper bottles 6

=−+++=

−+++=

=====

=

ILbtdZLbtdQ

ZILbtd

d

σ

σ

59

Example 1 (Electronic Village Store)Electronic Village store stocks and sells a particular brand of PCs. It costs the store $450 each time it places an order with the manufacturer for the PCs. The annual cost of carrying the PCs in inventory is $ 170. the store manager estimates the annual demand for the PCs will be 1200 units.

a) Determine the optimal order quantity and the total min. inventory cost.

b) Assume that shortages are allowed and shortage cost is $600 per unit per year. Compute the optimal order quantity and the total minimum inventory cost.

60

Example 1

For data below determine:Optimal order quantity and total minimum inventory cost.Assume shortage cost of $600 per unit per year, compute optimal order quantity and minimum inventory cost.

Step 1 (part a): Determine the Optimal Order Quantity.

computers personal 7.79170)200,1)(450(22

450$

$170 computerspersonal 1,200

===

=

==

cCDoC

Q

oCcC

D

61

Example 1

Step 2 (part b): Compute the EOQ with Shortages.

$13,549.91

7.79200,14502

7.791702cost Total

=

==+= ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛QD

oCQcC

computers personal 3.90

600170600

170)1200)(450(22

600$

=

+=+

=

=

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

sCcCsC

cCDoC

Q

Cs

62

Example 1

$11,960.98

3.90200,1450)3.90(2

2)9.193.90(170)3.90(22)9.19)(600(

22)(

2

2 cost Total

computers personal 19.96001701703.90

=

+−+=

+−+=

=+=+=

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

QDoC

QSQ

cCQSsC

sCcCcC

QS

63

Example 2 (Computer Product Store)

A computer products store stocks color graphics monitors, and the daily demand is normally distributed with a mean of 1.6 monitors and a standard deviation of 0.4 monitors. The lead time to receive an order from the manufacturer is 15 days.

Determine the reorder point that will achieve a 98% service level.

64

Example 2

level) service 98% a(for 05.2

dayper monitors 0.4

days 15

dayper monitors 1.6

=

=

=

=

Z

d

L

d

σ

Sells monitors with daily demand normally distributed with a mean of 1.6 monitors and standard deviation of 0.4 monitors. Lead time for delivery from supplier is 15 days.Determine the reorder point to achieve a 98% service level.Step 1: Identify parameters.

65

Example 2

Step 2: Solve for R.

monitors18.2718.324

15)04)(.05.2()15)(6.1(

=+=

+=+= LdZLdR σ

66

ABC Classification System

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

dollars investeddollars invested

profit potentialprofit potential

sales or usage volumesales or usage volume

stockstock--out penalties out penalties

0

3060

30

60

AB

C

% of $ Value

% of Use

So, identify inventory items based on percentage of total dollar value, where “A” items are roughly top 15 %, “B”items as next 35 %, and the lower 65% are the “C” items

67

Inventory Accuracy and Cycle CountingInventory Accuracy and Cycle Counting

DefinedDefined

Inventory accuracy refers to how well the Inventory accuracy refers to how well the inventory records agree with physical inventory records agree with physical countcount

Cycle Counting is a physical inventoryCycle Counting is a physical inventory--taking technique in which inventory is taking technique in which inventory is counted on a frequent basis rather than counted on a frequent basis rather than once or twice a yearonce or twice a year

68

End of Session

69

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