Inventry Model

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Inventory Models


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Inventory System DefinedInventory System Defined

Inventory CostsInventory Costs

Economic Order QuantityEconomic Order Quantity

SingleSingle--Period Inventory ModelPeriod Inventory Model

MultiMulti--Period Inventory Models: Basic FixedPeriod Inventory Models: Basic Fixed--Order Quantity ModelsOrder Quantity Models

MultiMulti--Period Inventory Models: Basic FixedPeriod Inventory Models: Basic Fixed--Time Period ModelTime Period Model

Price Break ModelsPrice Break Models

Content


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Inventory SystemInventory System

Inventory is the stock of any item or resource used in anInventory is the stock of any item or resource used in anorganization and can include: raw materials, finished products,organization and can include: raw materials, finished products,

component parts, supplies, and workcomponent parts, supplies, and work--inin--processprocess

An inventory system is the set of policies and controls thatAn inventory system is the set of policies and controls that

monitor levels of inventory and determines what levels should bemonitor levels of inventory and determines what levels should be

maintained, when stock should be replenished, and how largemaintained, when stock should be replenished, and how large

orders should beorders should be


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Purposes of InventoryPurposes of Inventory

1. To maintain independence of operations1. To maintain independence of operations

2. To meet variation in product demand2. To meet variation in product demand

3. To allow flexibility in production scheduling3. To allow flexibility in production scheduling

4. To provide a safeguard for variation in raw material delivery time4. To provide a safeguard for variation in raw material delivery time

5. To take advantage of economic purchase5. To take advantage of economic purchase--order sizeorder size


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Inventory CostsInventory Costs

Holding (or carrying) costsHolding (or carrying) costs

Costs for storage, handling, insurance, etcCosts for storage, handling, insurance, etc

Setup (or production change) costsSetup (or production change) costs

Costs for arranging specific equipment setups, etcCosts for arranging specific equipment setups, etc Ordering costsOrdering costs

Costs of someone placing an order, etcCosts of someone placing an order, etc Shortage costsShortage costs

Costs of canceling an order, etcCosts of canceling an order, etc


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Cost Minimization GoalCost Minimization Goal

Ordering Costs

Holding

Costs

Order Quantity (Q)

COST

Annual Cost of

Items (DC)

Total Cost

QOPT

By adding the item, holding, and ordering costs

together, we determine the total cost curve, which inturn is used to find the Qopt inventory order point that

minimizes total costs


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Basic FixedBasic Fixed--Order Quantity (EOQ) Model FormulaOrder Quantity (EOQ) Model Formula

Total

Annual =

Cost

Annual

Purchase

Cost

Annual

Ordering

Cost

Annual

Holding

Cost+ +

TC=Total annual

cost

D =Demand

C =Cost per unit

Q =Order quantity

Co =Cost of placing

an order or setup

cost

R =Reorder point

L =Lead time

Ch=Annual holding

and storage cost

per unit of inventoryCh2Q+Co

QD+DC=TC


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Economic Ordering Quantity (EOQ)Economic Ordering Quantity (EOQ)

Using calculus, we take the first derivative of the total cost function withUsing calculus, we take the first derivative of the total cost function with

respect to Q, and set the derivative (slope) equal to zero, solving forrespect to Q, and set the derivative (slope) equal to zero, solving forthe optimized (cost minimized) value of Qthe optimized (cost minimized) value of Qoptopt

R eo rder p oint, R = d L_

d = average dailydemand (constant)

L = Leadtime (constant)

_W

e also need areorder point to

tell us when to

place an order

Q =2DCo

Ch =2(AnnualDemand)(OrderorSetup Cost)

Annual Holding CostOPT


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Inventory ModelsInventory Models


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

Seeks to balance the costs of inventory overstock and underSeeks to balance the costs of inventory overstock and understockstock

MultiMulti--Period Inventory ModelsPeriod Inventory Models FixedFixed--Order Quantity ModelsOrder Quantity Models Event triggered (Example: running out of stock)Event triggered (Example: running out of stock)

FixedFixed--Time Period ModelsTime Period Models Time triggered (Example: Monthly sales call by salesTime triggered (Example: Monthly sales call by sales

representative)representative)


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uo

u

CC

CP

e

This model states that we shouldcontinue to increase the size ofthe inventory so long as theprobability of selling the lastunit added is equal to or greater

than the ratio of: Cu/Co+Cu

soldbeunit willythattheProbabilit

estimatedunderdemandofunitperCostC

estimatedoverdemandofunitperCostC

:Where

u

o

!

!!

P


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Single Period Model ExampleSingle Period Model Example

Our college Cricketteamis playinginatournamentgamethisOur college Cricketteamis playinginatournamentgamethis

weekend. Based on our pastexperience wesell onaverage 2,500weekend. Based on our pastexperience wesell onaverage 2,500T shirts withastandarddeviation of 350. Wemake Rs100 onT shirts withastandarddeviation of 350. Wemake Rs100 onevery T shirt wesellatthegame, butlose Rs50 onevery T shirtevery T shirt wesellatthegame, butlose Rs50 onevery T shirtnotsold. How manyshirtsshould wemake for thegame?notsold. How manyshirtsshould wemake for thegame?

CCuu == Rs100 andRs100 and CCoo = Rs50;= Rs50; PP 100 / (100 + 50) = .667100 / (100 + 50) = .667

ZZ.667.667 = .432 (use NORMSDIST(.667) or Appendix E)= .432 (use NORMSDIST(.667) or Appendix E)

therefore we need 2,500 + .432(350) = 2,651 shirtstherefore we need 2,500 + .432(350) = 2,651 shirts


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MultiMulti--Period Models:Period Models:FixedFixed--Order Quantity Model Model AssumptionsOrder Quantity Model Model Assumptions

Demand for the product is constant and uniformDemand for the product is constant and uniformthroughout the periodthroughout the period

Lead time (time from ordering to receipt) isLead time (time from ordering to receipt) isconstantconstant

Price per unit of product is constantPrice per unit of product is constant


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MultiMulti--Period Models:Period Models:FixedFixed--Order Quantity Model Model AssumptionsOrder Quantity Model Model Assumptions

Inventory holding cost is based on average inventoryInventory holding cost is based on average inventory

Ordering or setup costs are constantOrdering or setup costs are constant

All demands for the product will be satisfied (No backAll demands for the product will be satisfied (No back

orders are allowed)orders are allowed)


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Fixed Order ModelFixed Order Model

QuantQuantity,ity,QQ

TTiimmeeLL



RR

TTiimm

eeLL

TTiimm

eeLL


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Basic FixedBasic Fixed--Order Quantity Model and Reorder Point BehaviorOrder Quantity Model and Reorder Point Behavior

R = Reorder point

Q = Economic order quantity

L = Lead time

L L

Q QQ

R

Time

Number

of unitson hand

1. Youreceive anorderquantityQ.

2. Yourstartusingthem up overtime. 3.Whenyoureach downto

a levelofinventoryof ,

you place yournextQ

sized order.

4.The cycle thenrepeats.


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EOQ Example (1) Problem DataEOQ Example (1) Problem Data

Annual Demand = 1,000 unitsDays per year considered in average

daily demand = 365Cost to place an order = Rs10

Holding cost per unit per year = Rs2.50Lead time = 7 daysCost per unit = Rs15

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


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EOQ Example (1) SolutionEOQ Example (1) Solution

d =1,000units / year

365days / year= 2.74units / day

eorder point, = d L = 2 .74units / day (7days) = 19.18 or_

20 units

In summary, you place an optimal order of 90 units. Inthe course of using the units to meet demand, when

you only have 20 units left, place the next order of 90

units.

Q =2DCo

Ch=

2(1,000)( 10)

2.50= 89.443un itsorOPT 90 units


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Quantity,Quantity,Q1Q1

Fixed TimeFixed Time

Quantity,Quantity,Q 2Q 2

Quantity,Quantity,

Q 3Q 3

Fixed TimeFixed TimeFixed TimeFixed Time

Fixed Period ModelFixed Period Model


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EOQ Example (2) Problem DataEOQ Example (2) Problem Data

Annual Demand = 10,000 unitsDays per year considered in average daily

demand = 365

Cost to place an order = Rs10

Holding cost per unit per year = 10% of costper unit

Lead time = 10 days

Cost per unit = Rs15

Determine the economic order quantity

and the reorder point given the following


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EOQ Example (2) SolutionEOQ Example (2) Solution

d =

10,000units / year

365 days / year = 27.397units / day

R = d L = 27.397 un its / day (10 days) = 273.97 o r_

274 its

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.

Q =2DCo

Ch=

2(10,000) (10)

1.50= 365.148un its, orOPT 366 units


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FixedFixed--Time Period Model with Safety Stock FormulaTime Period Model with Safety Stock Formula

order)onitems(includeslevelinventorycurrent=I

timeleadandrevieoverthedemandodeviationstandard=

yprobabilitservicespeci iedafordeviationsstandardofnumberthe=zdemanddailyaverageforecast=d

daysintimelead=L

revie sbetweendaysofnumberthe=T

orderedbetoquantitiy=q

:Where

I-Z+L)+(Td=q

L+T

L+T

W

W

q = Average demand + Safety stock Inventory currently on hand


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MultiMulti--Period Models: FixedPeriod Models: Fixed--Time Period Model:Time Period Model:Determining the Value ofDetermining the Value of WWT+LT+L

W W

W

W W

T+ L di 1

T+ L

d

T+ L d

2

=

Since each dayisindependent and is constant,

= (T + L)

i

2

!

The standard deviation of a sequence of randomThe standard deviation of a sequence of random

events equals the square root of the sum of theevents equals the square root of the sum of thevariancesvariances


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Example of the FixedExample of the Fixed--Time Period ModelTime Period Model

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

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


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Example of the FixedExample of the Fixed--Time Period Model: Solution (Part 1)Time Period Model: Solution (Part 1)

W W

T+ L d

2 2

= (T + L) =3

0 +1

04

= 25.

298

The value for z is found by using the Excel NORMSINVfunction. By adding 0.5 to all the values in Z table and finding the

value in the table that comes closest to the service probability, thez value can be read by adding the column heading label to therow label.

So, by adding 0.5 to the value from Z Table of 0.4599, we have aprobability of 0.9599, which is given by a z = 1.75


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Example of the FixedExample of the Fixed--Time Period Model: Solution (Part 2)Time Period Model: Solution (Part 2)

or644.272,=200-44.272800=q

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

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

units645

W

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


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PricePrice--Break Model FormulaBreak Model Formula

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 willhave to be used with each price-break cost value

CostHoldingAnnual

Cost)SetuporderDemand)(Or2(Annual

=iC

2DCo

=QOPT


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PricePrice--Break Example Problem DataBreak Example Problem Data

A company has a chance to reduce their inventory ordering costs by

placing larger quantity orders using the price-break order quantityschedule below. What should their optimal order quantity be if thiscompany purchases this single inventory item with an e-mail orderingcost of Rs4, 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 quantityschedule below. What should their optimal order quantity be if thiscompany purchases this single inventory item with an e-mail orderingcost of Rs4, 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(Rs)0to 2,499 Rs1.202,500to 3,999 1.004,000 or more .98


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PricePrice--Break Example SolutionBreak Example Solution

Annual Demand(D)=10,000 unitsCostto placean order (Co)= Rs4

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

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

Interval from0to 2499, theQopt valueis feasible

Interval from 2500-3999,theQopt valueisnotfeasibleInterval from 4000 & more,theQopt valueisnot

feasible

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

units1,826=0.02(1.20)

4)2(10,000)(=

iC

2DCo=QOPT

units2,000=0.02(1.00)

4)2(10,000)(=

iC

2DCo=QOPT

units2,020=0.02(0.98)

4)2(10,000)(=

iC

2DCo=QOPT


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

0 1826 2500 4000 Order Quantity

Totalannualcosts So the candidates

for the price-

breaks are 1826,2500, and 4000units

Because the total annual cost function is au shaped function


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

TC(0-2499)=(10000*1.20)+(10000/1826)*4+(1826/2)(0.02*1.20)

= Rs12,043.82TC(2500-3999)= Rs10,041TC(4000&more)= Rs9,949.20

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

4000 units

iC2

Q+Co

Q

D+DC=TC


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Inventry Model

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