inventory management - jeryfarel | ahmad jaeri berbagi ilmu · pdf fileintroduction to...

Introduction to Management Science

(8th Edition, Bernard W. Taylor III)

Chapter 16

Chapter 16 - Inventory Management 1

Inventory Management

Elements of Inventory Management

Inventory Control Systems

Economic Order Quantity Models

Chapter Topics


The Basic EOQ Model

The EOQ Model with Non-Instantaneous Receipt

The EOQ Model with Shortages

EOQ Analysis with QM for Windows

EOQ Analysis with Excel and Excel QM


Quantity Discounts

Reorder Point

Determining Safety Stocks Using Service Levels

Order Quantity for a Periodic Inventory System

Inventory is a stock of items kept on hand used to meet customer demand..

A level of inventory is maintained that will meet anticipated

Elements of Inventory ManagementRole of Inventory (1 of 2)

A level of inventory is maintained that will meet anticipated demand.

If demand not known with certainty, safety (buffer) stocks are kept on hand.

Additional stocks are sometimes built up to meet seasonal or cyclical demand.


or cyclical demand.

Large amounts of inventory sometimes purchased to take advantage of discounts.

In-process inventories maintained to provide independence between operations.

Raw materials inventory kept to avoid delays in case of

Elements of Inventory ManagementRole of Inventory (2 of 2)

Raw materials inventory kept to avoid delays in case of supplier problems.

Stock of finished parts kept to meet customer demand in event of work stoppage.


Inventory exists to meet the demand of customers.

Customers can be external (purchasers of products) or internal (workers using material).

Elements of Inventory ManagementDemand

internal (workers using material).

Management needs accurate forecast of demand.

Items that are used internally to produce a final product are referred to as dependent demand items.

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


customer are independent demand items.

Carrying costs - Costs of holding items in storage.

Vary with level of inventory and sometimes with length of time held.

Elements of Inventory ManagementInventory Costs (1 of 3)

of time held.

Include facility operating costs, record keeping, interest, etc.

Assigned on a per unit basis per time period, or as percentage of average inventory value (usually estimated as 10% to 40%).


estimated as 10% to 40%).

Ordering costs - costs of replenishing stock of inventory.

Expressed as dollar amount per order, independent of order size.


order size.

Vary with the number of orders made.

Include purchase orders, shipping, handling, inspection, etc.


Shortage, or stockout costs - Costs associated with 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.

Sometimes penalties involved; if customer is internal, work delays could result.


An inventory control system controls the level of inventory by determining how much (replenishment level) and whento order.

Inventory Control Systems

to 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


In a periodic system, an order is placed for a variable amount after a specified period of time.

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

Inventory Control SystemsContinuous Inventory Systems

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


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

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

Inventory Control SystemsPeriodic Inventory Systems

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


System requires a new order quantity each time an order is placed.

Used in smaller retail stores, drugstores, grocery stores and offices.

Economic order quantity, or economic lot size, is the quantity ordered when inventory decreases to the reorder point.


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 model

EOQ model without instantaneous receipt

EOQ model with shortages

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

Simplifying assumptions and restrictions:

Economic Order Quantity ModelsBasic EOQ Model (1 of 2)

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.

Economic Order Quantity ModelsBasic EOQ Model (2 of 2)

Figure 16.1The Inventory Order Cycle


Carrying cost usually expressed on a per unit basis of time, traditionally one year.

Annual carrying cost equals carrying cost per unit per year

Basic EOQ ModelCarrying Cost (1 of 2)

Annual carrying cost equals carrying cost per unit per year times average inventory level:

Carrying cost per unit per year = Cc

Average inventory = Q/2

Annual carrying cost = CcQ/2.


Basic EOQ ModelCarrying Cost (2 of 2)

Figure 16.4Average Inventory


Total annual ordering cost equals cost per order (Co) times number of orders per year.

Number of orders per year, with known and constant

Basic EOQ ModelOrdering Cost

Number of orders per year, with known and constant demand, D, is D/Q, where Q is the order size:

Annual ordering cost = CoD/Q

Only variable is Q, Co and D are constant parameters.

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


order size.

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

Basic EOQ ModelTotal Inventory Cost (1 of 2)

2QCc

QDCoTC


Basic EOQ ModelTotal Inventory Cost (2 of 2)


Figure 16.5The EOQ Cost Model

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

Basic EOQ ModelEOQ and Minimum Total Cost

The EOQ model is robust because Q is a square root and

CcCoDQopt

QoptCcQoptCoDTC

2

2

min


errors in the estimation of D, Cc and Co are dampened.

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

Basic EOQ ModelExample (1 of 2)

10,000ydD$150,Co$0.75,Cc

:parametersModel

made annually and time between orders given store is open every day except Sunday, Thanksgiving Day, and Christmas Day.


yd0002750

0001015022

:sizeorderOptimal

,).(

),)(( CcCoDQopt

5001000275000010150

:costinventory annualTotal

QoptCcDCoTC ,$),().(,)(min

Basic EOQ ModelExample (2 of 2)

311days311

5000200010

: yearper ordersofNumber

500120002750

000200010150

2

QoptD

QoptCcQopt

DCoTC

,,

,$),().(,,)(min


daysstore62.25

311days311timecycleOrder QoptD/

For any time period unit of analysis, EOQ is the same.

Shag Carpet example on monthly basis:

Basic EOQ ModelEOQ Analysis Over Time (1 of 2)

:sizeorderOptimal

monthper yd833.3Dorderper$150Co

monthper ydper$0.0625Cc

:parametersModel


yd000206250

383315022

:sizeorderOptimal

,).(

).)(( CcCoDQopt

0002062503833150

:costinventory monthly Total

),().().()(min QoptCcDCoTC

Basic EOQ ModelEOQ Analysis Over Time (2 of 2)

$1,500($125)(12)costinventory annualTotal

monthper125

2000206250

00023833150

2

$

),().(,

).()(min QoptCcQopt

DCoTC


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.)

EOQ ModelNon-Instantaneous Receipt Description (1 of 2)

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.


EOQ ModelNon-Instantaneous Receipt Description (2 of 2)


Figure 16.6 The EOQ Model with Non-Instantaneous Order Receipt

demandedisinventory whichatratedaily dtimeoverreceivedisorderthe whichatratedaily p

Non-Instantaneous Receipt ModelModel Formulation (1 of 2)

pdQCc

pdQ

pdQ

12

costcarryingTotal

12

levelinventory Average

1levelinventory Maximum


pdQCc

QDCo

pCc

12

costinventory annualTotal

12

costcarryingTotal

QDCop

dQCc

curvecosttotalofpointlowestat1

2

Non-Instantaneous Receipt ModelModel Formulation (2 of 2)

)/( pdCcCoDQopt

1

2:sizeorderOptimal


$150Co

Non-Instantaneous Receipt ModelExample (1 of 2)

Super Shag carpet manufacturing facility:

1502321750

000101502

12:sizeorderOptimal

dayper yd150pdayper yd32.210,000/311 yearper yd10,000D

unitper$0.75Cc $150Co

..

),)((

pdCc

CoDQopt


yd82562

150

.,

p

8256200010

12

costinventory annualminimumTotal

).,(),(

pdQCc

QDCo

Non-Instantaneous Receipt ModelExample (2 of 2)

runs)n(productio yearperordersofNumber

days0515150

82562lengthrunProduction

3291150

23212

8256207582562

00010150

..,

,$.).,()(.).,(),()(

QD

pQ


yd7721150

2321825621levelinventory Maximum

runs43482562

00010

,..,

..,

,

pdQ

Q

EOQ Model with ShortagesDescription (1 of 2)

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

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


EOQ Model with ShortagesDescription (2 of 2)


Figure 16.7The EOQ Model with Shortages

QSQCc

QSCs

22

costscarryingTotal 2

2costsshortageTotal )(

EOQ Model with ShortagesModel Formulation (1 of 2)

CcCsCoDQopt

QDCo

QSQCc

QSCs

QDC

2quantityorderOptimal

22

22

costinventory Total

0costorderingTotal

)(


CsCcCcQoptSopt

CsCcCs

CcCoDQopt

levelShortage

2quantityorderOptimal



Figure 16.8Cost Model with Shortages


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

:quantityorderOptimal

yd10,000D ydper$2Cs

ydper$0.75Cc $150Co


yd234522

7502750

0001015022 .,..

),)((

CsCcCs

CcCoDQopt

yd663975023452

level:Shortage

...,CcQoptSopt


22

22

22

:costinventory Total

yd66397502

75023452

)(

..

..,

QDCo

QSQCc

QSCsTC

CsCcCcQoptSopt


$1,279.20639.60465.16$174.44

2345200010150

234522267051750

234522266392

.,

),)(().,(

).,)(.().,(

).)((

yearper orders26.42.345,2

000,10ordersofNumber

QD


days53.2or 0.171639.6-2,345.2 t

handon isinventory which duringTime

days0.7326.4

311ordersofnumber yearper days tordersbetween Time

yd6.705,16.6392.345,2levelinventory Maximum

SQ

SQ


days19.9or year 0.06410,000639.6

D2 t

shortageaisre which theduringTime

days53.2or 0.17110,000

639.6-2,345.21 t

S

DSQ

EOQ Analysis with QM for Windows


Exhibit 16.1

EOQ Analysis with Excel and Excel QM (1 of 2)


Exhibit 16.2

EOQ Analysis with Excel and Excel QM (2 of 2)


Exhibit 16.3

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:

Quantity Discounts

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

PDQCcQDCoTC

2


Quantity discounts are evaluated under two different scenarios:

With constant carrying costs

With carrying costs as a percentage of purchase price

Quantity Discounts with Constant Carrying CostsAnalysis Approach

Optimal order size is the same regardless of the discount price.

The total cost with the optimal order size must be compared 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.


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

Quantity Discounts with Constant Carrying CostsExample (1 of 2)

terms or order the basic EOQ order size?

Determine optimal order size and total cost:

Quantity Price1- 4950 – 8990 +

$1,400 1,100 900


572190

20050022Cc

2CoDQopt

200D unitper$190Cc $2,500Co

.))(,(

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

2min PDQoptCc

QoptCoDTC

Quantity Discounts with Constant Carrying CostsExample (2 of 2)

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

78423320010012

5721905722005002

2

,$))(,().()().(

))(,(

Qopt

2 PDQCc

QCoDTC


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

1051942009002

9019090

2005002

2

,$))(())(()(

))(,(

Q

University Bookstore example, but a different optimal order size for each price discount.

Optimal order size and total cost determined using basic

Quantity Discounts with Carrying CostsPercentage of Price Example (1 of 3)

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.


specific discount price.

Data:

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

Quantity Price Carrying Cost

0 - 49 $1,400 1,400(.15) = $21050 - 89 1,100 1,100(.15) = 165


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

50 - 89 1,100 1,100(.15) = 16590 + 900 900(.15) = 135

69210

200500222 ))(,(CcCoDQopt


Compute new order size:

210Cc

877165

20050022 .))(,( Qopt

Compute minimum total cost:

20010012

8771658772005002

2))(,().(

.))(,( PDQCc

QCoDTC


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

Optimal order size computed as follows:

845232

28772

,$

.

Q

6301912009002

9013590

2005002 ,$))(())(())(,( TC


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.

186135

20050022 .))(,( Qopt

Quantity Discount ModelSolution with QM for Windows


Exhibit 16.4

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

Reorder Point (1 of 4)

Order must be made while there is enough stock in place to cover demand during lead time.

Formulation:

R = dL

where d = demand rate per time period

L = lead time


L = lead time

For Carpet Discount store problem:

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



Figure 16.9Reorder Point and Lead Time


Inventory level might be depleted at slower or faster rate during lead time.

When demand is uncertain, safety stock is added as a When demand is uncertain, safety stock is added as a hedge against stockout.


Figure 16.10Inventory Model with Uncertain Demand



Figure 16.11Inventory model with safety stock

Determining Safety Stocks Using Service Levels

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


:where

LdZLdR

Reorder Point with Variable Demand (1 of 2)

demanddaily ofdeviation standard the

timelead

demanddaily average

pointreorder

:where

d

L

d

R


stocksafety

yprobabilitlevelservice toingcorresponddeviationsstandardofnumber

demanddaily ofdeviation standard the

LdZ

Z

d

Reorder Point with Variable Demand (2 of 2)


Figure 16.12Reorder Point for a Service Level

I-75 Carpet Discount Store Super Shag carpet.

For following data, determine reorder point and safety stock for service level of 95%.

Reorder Point with Variable Demand Example

)appendix A1,- A(Table1.65Zlevel,service95%For

dayper yd5ddays10L

dayper yd30

d

for service level of 95%.


26.1. :formulapointreorderintermsecondisstockSafety

yd13261263001056511030

..))()(.()(

LdZLdR

Determining the Reorder Point with Excel

Determining Reorder Point with Excel

Determining the Reorder Point with Excel


Exhibit 16.5

LZdLdR

Reorder Point with Variable Lead Time

For constant demand and variable lead time:

timeleadduringdemandofdeviation standard timeleadofdeviation standard

timeleadaveragedemanddaily constant

:where

LdL

Ld


stocksafety LZdL

dayper yd30d

Reorder Point with Variable Lead Time Example

Carpet Discount Store:

yd5.4485.148300)3)(30)(65.1()10)(30(

levelservice95%afor 1.65

days3

days10

LZdLdR

Z

L

L


yd5.4485.148300)3)(30)(65.1()10)(30( LZdLdR

22)(

2)( dLLdZLdR

When both demand and lead time are variable:

Reorder PointVariable Demand and Lead Time

222

timeleadaveragedemanddaily average

:where

)()(

Ld

dLLdZLdR


stocksafety 22

)(2

)(Z

timeleadduringdemandofdeviation standard22

)(2

)(

dLLd

dLLd

dayper yd5dayper yd30

d

Carpet Discount Store:

Reorder PointVariable Demand and Lead Time Example

22)(

2)(

levelservice95%for 1.65days3days10

dayper yd5

dLLdZLdR

ZL

Ld


yds450.8 150.8300

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

)()(

dLLdZLdR

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.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.

For normally distributed variable daily demand:)( ILbtdZLbtdQ

Order Quantity for Variable Demand

demandofdeviation standard timelead

ordersbetween timefixed theratedemandaverage

:where

d

Lbtd


stockin inventory

stocksafety

I

LbtdZd

dayper bottles6d

Corner Drug Store with periodic inventory system.

Order size to maintain 95% service level:

Order Quantity for Variable Demand Example

levelservice95%for 1.65bottles8days5

days60

bottles1.2dayper bottles6

ZILbtd

d


bottles398 8560)(1.65)(1.25)(6)(60

)(

ILbtdZLbtdQ

Order Quantity for the Fixed-Period ModelSolution with Excel


Exhibit 16.6

For data below determine:

Optimal order quantity and total minimum inventory cost.

Example Problem SolutionElectronic Village Store (1 of 3)

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.

$170computerspersonal1,200

CcD


computerspersonal779170

200145022

450$170

.),)((

$

CcCoDQ

CoCc

7792001450

2779170

2costTotal

.,.

QDCoQCc


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

$13,549.91

600170600

170120045022

600

))((

$

CsCcCs

CcCoDQ

Cs


computerspersonal390

600170

.

CsCc


computerspersonal19.9600170

170390

.

CsCcCcQS

$11,960.98

3902001450

39022919390170

39022919600

22

22costTotal

.,

).()..(

).().)((

)(Q

CoDQSQCc

QCsS


$11,960.98

Example Problem SolutionComputer Products Store (1 of 2)

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.

days15

dayper monitors1.6

L

d

monitors. Lead time for delivery from supplier is 15 days.

Determine the reorder point to achieve a 98% service level.

Step 1: Identify parameters.


level)service98%a(for 05.2

dayper monitors0.4

Z

d

Example Problem SolutionComputer Products Store (2 of 2)

Step 2: Solve for R.

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

monitors18.2718.324

d


inventory management - jeryfarel | ahmad jaeri berbagi ilmu · pdf fileintroduction to...

Documents