1 1 Slide © 2008 Thomson South-Western. All Rights Reserved Slides by JOHN LOUCKS St. Edwards...

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© 2008 Thomson South-Western. All Rights Reserved© 2008 Thomson South-Western. All Rights Reserved

Slides by

JOHNLOUCKSSt. Edward’sUniversity

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Chapter 10, Part AChapter 10, Part AInventory Models: Deterministic DemandInventory Models: Deterministic Demand

Economic Order Quantity (EOQ) ModelEconomic Order Quantity (EOQ) Model Economic Production Lot Size ModelEconomic Production Lot Size Model Inventory Model with Planned ShortagesInventory Model with Planned Shortages Quantity Discounts for the EOQ ModelQuantity Discounts for the EOQ Model

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Inventory ModelsInventory Models

The study of The study of inventory modelsinventory models is concerned is concerned with two basic questions:with two basic questions:

• How muchHow much should be ordered each time should be ordered each time

• WhenWhen should the reordering occur should the reordering occur The objective is to The objective is to minimize total variable minimize total variable

costcost over a specified time period (assumed over a specified time period (assumed to be annual in the following review).to be annual in the following review).

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Inventory CostsInventory Costs

Ordering costOrdering cost -- salaries and expenses of -- salaries and expenses of processing an order, regardless of the order processing an order, regardless of the order quantityquantity

Holding costHolding cost -- usually a percentage of the value -- usually a percentage of the value of the item assessed for keeping an item in of the item assessed for keeping an item in inventory (including finance costs, insurance, inventory (including finance costs, insurance, security costs, taxes, warehouse overhead, and security costs, taxes, warehouse overhead, and other related variable expenses)other related variable expenses)

Backorder costBackorder cost -- costs associated with being out -- costs associated with being out of stock when an item is demanded (including of stock when an item is demanded (including lost goodwill)lost goodwill)

Purchase costPurchase cost -- the actual price of the items -- the actual price of the items Other costsOther costs

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Deterministic ModelsDeterministic Models

The simplest inventory models assume The simplest inventory models assume demand and the other parameters of the demand and the other parameters of the problem to be problem to be deterministicdeterministic and constant. and constant.

The deterministic models covered in this The deterministic models covered in this chapter are:chapter are:

• Economic order quantity (EOQ)Economic order quantity (EOQ)

• Economic production lot sizeEconomic production lot size

• EOQ with planned shortagesEOQ with planned shortages

• EOQ with quantity discountsEOQ with quantity discounts

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Economic Order Quantity (EOQ)Economic Order Quantity (EOQ)

The most basic of the deterministic inventory The most basic of the deterministic inventory models is the models is the economic order quantity (EOQ)economic order quantity (EOQ). .

The variable costs in this model are annual The variable costs in this model are annual holding cost and annual ordering cost. holding cost and annual ordering cost.

For the EOQ, annual holding and ordering For the EOQ, annual holding and ordering costs are equal.costs are equal.

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Economic Order QuantityEconomic Order Quantity

AssumptionsAssumptions

• Demand is constant throughout the year at Demand is constant throughout the year at DD items per year.items per year.

• Ordering cost is $Ordering cost is $CCoo per order. per order.

• Holding cost is $Holding cost is $CChh per item in inventory per per item in inventory per year.year.

• Purchase cost per unit is constant (no quantity Purchase cost per unit is constant (no quantity discount).discount).

• Delivery time (lead time) is constant.Delivery time (lead time) is constant.

• Planned shortages are not permitted.Planned shortages are not permitted.

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Economic Order QuantityEconomic Order Quantity

FormulasFormulas

• Optimal order quantity: Optimal order quantity: Q Q * = 2* = 2DCDCoo//CChh

• Number of orders per year: Number of orders per year: DD//Q Q * *

• Time between orders (cycle time): Time between orders (cycle time): Q Q */*/DD years years

• Total annual cost: [(1/2)Total annual cost: [(1/2)Q Q **CChh] + [] + [DCDCoo//Q Q *]*]

(holding + ordering)(holding + ordering)

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Example: Bart’s Barometer BusinessExample: Bart’s Barometer Business

Economic Order Quantity ModelEconomic Order Quantity Model

Bart's Barometer Business is a retail outlet Bart's Barometer Business is a retail outlet thatthat

deals exclusively with weather equipment. deals exclusively with weather equipment.

Bart is trying to decide on an inventoryBart is trying to decide on an inventory

and reorder policy for home barometers. and reorder policy for home barometers.

Barometers cost Bart $50 each andBarometers cost Bart $50 each and

demand is about 500 per year distributeddemand is about 500 per year distributed

fairly evenly throughout the year. fairly evenly throughout the year.

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Economic Order Quantity ModelEconomic Order Quantity Model

Reordering costs are $80 per order and Reordering costs are $80 per order and holdingholding

costs are figured at 20% of the cost of the costs are figured at 20% of the cost of the item. BBB isitem. BBB is

open 300 days a year (6 days a week and open 300 days a year (6 days a week and closed twoclosed two

weeks in August). Lead time is 60 working weeks in August). Lead time is 60 working days.days.

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Total Variable Cost ModelTotal Variable Cost Model

Total Costs Total Costs = (Holding Cost) + (Ordering Cost) = (Holding Cost) + (Ordering Cost)

TCTC = [= [CChh((QQ/2)] + [/2)] + [CCoo((DD//QQ)] )]

= [.2(50)(= [.2(50)(QQ/2)] + [80(500//2)] + [80(500/QQ)] )]

= 5= 5QQ + (40,000/ + (40,000/QQ))

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Optimal Reorder QuantityOptimal Reorder Quantity

Q Q * = 2* = 2DCDCo o //CChh = 2(500)(80)/10 = = 2(500)(80)/10 = 89.44 89.44 90 90

Optimal Reorder PointOptimal Reorder Point

Lead time is Lead time is m m = 60 days and daily demand is = 60 days and daily demand is dd = 500/300 or 1.667. = 500/300 or 1.667.

Thus the reorder point Thus the reorder point r r = (1.667)(60) = 100. = (1.667)(60) = 100. Bart should reorder 90 barometers when his Bart should reorder 90 barometers when his inventory position reaches 100 (that is 10 on inventory position reaches 100 (that is 10 on hand and one outstanding order).hand and one outstanding order).

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Number of Orders Per YearNumber of Orders Per Year

Number of reorder times per year = (500/90) = Number of reorder times per year = (500/90) = 5.56 or once every (300/5.56) = 54 working 5.56 or once every (300/5.56) = 54 working days (about every 9 weeks).days (about every 9 weeks).

Total Annual Variable CostTotal Annual Variable Cost

TCTC = 5(90) + (40,000/90) = 450 + 444 = = 5(90) + (40,000/90) = 450 + 444 = $894$894

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We’ll now use a spreadsheet to implementWe’ll now use a spreadsheet to implement

the Economic Order Quantity model. We’ll confirmthe Economic Order Quantity model. We’ll confirm

our earlier calculations for Bart’s problem andour earlier calculations for Bart’s problem and

perform some sensitivity analysis.perform some sensitivity analysis.

This spreadsheet can be modified to This spreadsheet can be modified to accommodateaccommodate

other inventory models presented in this chapter.other inventory models presented in this chapter.

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Partial Spreadsheet with Input DataPartial Spreadsheet with Input Data

A B1 BART'S ECONOMIC ORDER QUANTITY23 Annual Demand 5004 Ordering Cost $80.005 Annual Holding Rate % 206 Cost Per Unit $50.007 Working Days Per Year 3008 Lead Time (Days) 60

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Partial Spreadsheet Showing Formulas for OutputPartial Spreadsheet Showing Formulas for OutputA B C

10 Econ. Order Qnty. =SQRT(2*B3*B4/(B5*B6/100))11 Request. Order Qnty12 % Change from EOQ =(C11/B10-1)*1001314 Annual Holding Cost =B5/100*B6*B10/2 =B5/100*B6*C11/215 Annual Order. Cost =B4*B3/B10 =B4*B3/C1116 Tot. Ann. Cost (TAC) =B14+B15 =C14+C1517 % Over Minimum TAC =(C16/B16-1)*1001819 Max. Inventory Level =B10 =C1120 Avg. Inventory Level =B10/2 =C11/221 Reorder Point =B3/B7*B8 =B3/B7*B82223 No. of Orders/Year =B3/B10 =B3/C1124 Cycle Time (Days) =B10/B3*B7 =C11/B3*B7

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Partial Spreadsheet Showing OutputPartial Spreadsheet Showing OutputA B C

10 Econ. Order Qnty. 89.4411 Request. Order Qnty. 75.0012 % Change from EOQ -16.151314 Annual Holding Cost $447.21 $375.0015 Annual Order. Cost $447.21 $533.3316 Tot. Ann. Cost (TAC) $894.43 $908.3317 % Over Minimum TAC 1.551819 Max. Inventory Level 89.44 7520 Avg. Inventory Level 44.72 37.521 Reorder Point 100 1002223 No. of Orders/Year 5.59 6.6724 Cycle Time (Days) 53.67 45.00

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Summary of Spreadsheet ResultsSummary of Spreadsheet Results

• A 16.15% negative deviation from the EOQ A 16.15% negative deviation from the EOQ resulted in only a 1.55% increase in the resulted in only a 1.55% increase in the Total Annual Cost.Total Annual Cost.

• Annual Holding Cost and Annual Ordering Annual Holding Cost and Annual Ordering Cost are no longer equal.Cost are no longer equal.

• The Reorder Point is not affected, in this The Reorder Point is not affected, in this model, by a change in the Order Quantity.model, by a change in the Order Quantity.

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Economic Production Lot SizeEconomic Production Lot Size

The The economic production lot size modeleconomic production lot size model is a is a variation of the basic EOQ model. variation of the basic EOQ model.

A A replenishment orderreplenishment order is not received in one is not received in one lump sum as it is in the basic EOQ model. lump sum as it is in the basic EOQ model.

Inventory is replenished gradually as the order Inventory is replenished gradually as the order is produced (which requires the production rate is produced (which requires the production rate to be greater than the demand rate). to be greater than the demand rate).

This model's variable costs are annual holding This model's variable costs are annual holding cost and annual set-up cost (equivalent to cost and annual set-up cost (equivalent to ordering cost). ordering cost).

For the optimal lot size, annual holding and set-For the optimal lot size, annual holding and set-up costs are equal.up costs are equal.

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• Demand occurs at a constant rate of Demand occurs at a constant rate of DD items per items per year.year.

• Production rate is Production rate is PP items per year (and items per year (and PP > > D D ).).

• Set-up cost: $Set-up cost: $CCoo per run. per run.

• Holding cost: $Holding cost: $CChh per item in inventory per year. per item in inventory per year.

• Purchase cost per unit is constant (no quantity Purchase cost per unit is constant (no quantity discount).discount).

• Set-up time (lead time) is constant.Set-up time (lead time) is constant.


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FormulasFormulas

• Optimal production lot-size: Optimal production lot-size:

Q Q * = 2* = 2DCDCo o /[(1-/[(1-DD//P P ))CChh]]

• Number of production runs per year: Number of production runs per year: DD//Q Q **

• Time between set-ups (cycle time): Time between set-ups (cycle time): Q Q */*/DD years years

• Total annual cost: [(1/2)(1-Total annual cost: [(1/2)(1-DD//P P ))Q Q **CChh] + [] + [DCDCoo//Q Q *]*]

(holding + ordering)(holding + ordering)

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Example: Non-Slip Tile Co.Example: Non-Slip Tile Co.

Economic Production Lot Size ModelEconomic Production Lot Size Model

Non-Slip Tile Company (NST) has been usingNon-Slip Tile Company (NST) has been using

production runs of 100,000 tiles, 10 times per yearproduction runs of 100,000 tiles, 10 times per year

to meet the demand of 1,000,000 tilesto meet the demand of 1,000,000 tiles

annually. The set-up cost is $5,000 perannually. The set-up cost is $5,000 per

run and holding cost is estimated atrun and holding cost is estimated at

10% of the manufacturing cost of $110% of the manufacturing cost of $1

per tile. The production capacity of per tile. The production capacity of

the machine is 500,000 tiles per month. The the machine is 500,000 tiles per month. The factoryfactory

is open 365 days per year.is open 365 days per year.

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Total Annual Variable Cost ModelTotal Annual Variable Cost Model

This is an economic production lot size problem with This is an economic production lot size problem with

DD = 1,000,000, = 1,000,000, PP = 6,000,000, = 6,000,000, CChh = .10, = .10, CCoo = = 5,0005,000

TCTC = (Holding Costs) + (Set-Up Costs) = (Holding Costs) + (Set-Up Costs)

= [= [CChh((QQ/2)(1 - /2)(1 - DD//P P )] + [)] + [DCDCoo//QQ] ]

= .04167= .04167QQ + 5,000,000,000/ + 5,000,000,000/QQ

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Optimal Production Lot SizeOptimal Production Lot Size

Q Q * = 2* = 2DCDCoo/[(1 -/[(1 -DD//P P ))CChh]]

= 2(1,000,000)(5,000) /[(.1)(1 - 1/6)] = 2(1,000,000)(5,000) /[(.1)(1 - 1/6)]

= 346,410 = 346,410

Number of Production Runs Per YearNumber of Production Runs Per Year

DD//Q Q * = 2.89 times per year* = 2.89 times per year


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Total Annual Variable CostTotal Annual Variable Cost

How much is NST losing annually by using their How much is NST losing annually by using their present production schedule?present production schedule?

Optimal Optimal TC TC = .04167(346,410) + = .04167(346,410) + 5,000,000,000/346,4105,000,000,000/346,410

= $28,868= $28,868

Current Current TC TC = .04167(100,000) + = .04167(100,000) + 5,000,000,000/100,000 5,000,000,000/100,000

= $54,167= $54,167

Difference Difference = 54,167 - 28,868 = $25,299= 54,167 - 28,868 = $25,299

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Idle Time Between Production RunsIdle Time Between Production Runs

There are 2.89 cycles per year. Thus, each There are 2.89 cycles per year. Thus, each cycle lasts (365/2.89) = 126.3 days. The time to cycle lasts (365/2.89) = 126.3 days. The time to produce 346,410 per run = (346,410/6,000,000)365 produce 346,410 per run = (346,410/6,000,000)365 = 21.1 days. = 21.1 days. Thus, the machine is idle for:Thus, the machine is idle for:

126.3 - 21.1 = 105.2 days between runs.126.3 - 21.1 = 105.2 days between runs.

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Maximum InventoryMaximum Inventory

Current Policy:Current Policy:

Maximum inventory = (1-Maximum inventory = (1-DD//P P ))Q Q **

= (1-= (1-1/61/6)100,000 )100,000 83,333 83,333

Optimal Policy:Optimal Policy:

Maximum inventory = (1-Maximum inventory = (1-1/61/6)346,410 = )346,410 = 288,675288,675

Machine UtilizationMachine Utilization

Machine is producing Machine is producing DD//PP = 1/6 of the time. = 1/6 of the time.


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EOQ with Planned ShortagesEOQ with Planned Shortages

With the With the EOQ with planned shortages modelEOQ with planned shortages model, a , a replenishment order does not arrive at or before replenishment order does not arrive at or before the inventory position drops to zero. the inventory position drops to zero.

ShortagesShortages occur until a predetermined occur until a predetermined backorder quantity is reached, at which time the backorder quantity is reached, at which time the replenishment order arrives. replenishment order arrives.

The variable costs in this model are annual The variable costs in this model are annual holding, backorder, and ordering. holding, backorder, and ordering.

For the optimal order and backorder quantity For the optimal order and backorder quantity combination, the sum of the annual holding and combination, the sum of the annual holding and backordering costs equals the annual ordering backordering costs equals the annual ordering cost.cost.

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• Demand occurs at a constant rate of Demand occurs at a constant rate of DD items/year. items/year.

• Ordering cost: $Ordering cost: $CCoo per order. per order.

• Holding cost: $Holding cost: $CChh per item in inventory per year. per item in inventory per year.

• Backorder cost: $Backorder cost: $CCbb per item backordered per year. per item backordered per year.

• Purchase cost per unit is constant (no qnty. Purchase cost per unit is constant (no qnty. discount).discount).

• Set-up time (lead time) is constant.Set-up time (lead time) is constant.

• Planned shortages are permitted (backordered Planned shortages are permitted (backordered demand units are withdrawn from a replenishment demand units are withdrawn from a replenishment order when it is delivered).order when it is delivered).

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FormulasFormulas

• Optimal order quantity: Optimal order quantity:

Q Q * = 2* = 2DCDCoo//CChh ( (CChh++CCb b )/)/CCbb

• Maximum number of backorders: Maximum number of backorders:

S S * = * = Q Q *(*(CChh/(/(CChh++CCbb))))

• Number of orders per year: Number of orders per year: DD//Q Q **


• Total annual cost: Total annual cost:

[[CChh((Q Q *-*-S S *)*)22/2/2Q Q *] + [*] + [DCDCoo//Q Q *] + [*] + [S S *2*2CCbb/2/2Q Q *]*]

(holding + ordering + backordering)(holding + ordering + backordering)

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Example: Hervis Rent-a-CarExample: Hervis Rent-a-Car

EOQ with Planned Shortages ModelEOQ with Planned Shortages Model

Hervis Rent-a-Car has a fleet of 2,500 Hervis Rent-a-Car has a fleet of 2,500 RocketsRockets

serving the Los Angeles area. All Rockets areserving the Los Angeles area. All Rockets are

maintained at a central garage. Onmaintained at a central garage. On

the average, eight Rockets per the average, eight Rockets per

month require a new engine.month require a new engine.

Engines cost $850 each. ThereEngines cost $850 each. There

is also a $120 order costis also a $120 order cost

(independent of the number of (independent of the number of

engines ordered). engines ordered).

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EOQ with Planned Shortages ModelEOQ with Planned Shortages Model

Hervis has an annual holding cost rate of Hervis has an annual holding cost rate of 30% on30% on

engines. It takes two weeks to obtain the engines. It takes two weeks to obtain the engines afterengines after

they are ordered. For each week a car is out of they are ordered. For each week a car is out of service,service,

Hervis loses $40 profit. Hervis loses $40 profit.

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Optimal Order PolicyOptimal Order Policy

DD = 8 x 12 = 96; = 8 x 12 = 96; CCoo = $120; = $120; CChh = .30(850) = $255; = .30(850) = $255;

CCbb = 40 x 52 = $2080 = 40 x 52 = $2080

Q Q * = 2* = 2DCDCoo//CChh ( (CChh + + CCbb)/)/CCbb

= 2(96)(120)/255 x = 2(96)(120)/255 x (255+2080)/2080(255+2080)/2080

= 10.07 = 10.07 10 10

S S * = * = Q Q *(*(CChh/(/(CChh++CCbb)) ))

= 10(255/(255+2080)) = 1.09 = 10(255/(255+2080)) = 1.09 1 1


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Optimal Order Policy (continued)Optimal Order Policy (continued)

Demand is 8 per month or 2 per week. SinceDemand is 8 per month or 2 per week. Since

lead time is 2 weeks, lead time demand is 4. lead time is 2 weeks, lead time demand is 4.

Thus, since the optimal policy is to order 10 toThus, since the optimal policy is to order 10 to

arrive when there is one backorder, the order shouldarrive when there is one backorder, the order should

be placed when there are 3 engines remaining inbe placed when there are 3 engines remaining in

inventory.inventory.

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Stockout: When and How LongStockout: When and How Long

Question:Question:

How many days after receiving an order How many days after receiving an order doesdoes

Hervis run out of engines? How long is HervisHervis run out of engines? How long is Hervis

without any engines per cycle?without any engines per cycle?

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Stockout: When and How LongStockout: When and How Long

AnswerAnswer

Inventory exists for Inventory exists for CCbb/(/(CCbb++CChh) = ) = 2080/(255+2080)2080/(255+2080)

= .8908 of the order cycle. (Note, (= .8908 of the order cycle. (Note, (Q Q *-*-S S *)/*)/Q Q * = * = .8908.8908

also, before also, before Q Q * and * and S S * are rounded.) * are rounded.)

An order cycle is An order cycle is Q Q */*/DD = .1049 years = = .1049 years = 38.3 days.38.3 days.

Thus, Hervis runs out of engines .8908(38.3) = Thus, Hervis runs out of engines .8908(38.3) = 34 days34 days

after receiving an order. after receiving an order.

Hervis is out of stock for approximately 38 Hervis is out of stock for approximately 38 - 34 = 4- 34 = 4

days.days.

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EOQ with Quantity DiscountsEOQ with Quantity Discounts

The The EOQ with quantity discounts modelEOQ with quantity discounts model is is applicable where a supplier offers a lower purchase applicable where a supplier offers a lower purchase cost when an item is ordered in larger quantities. cost when an item is ordered in larger quantities.

This model's variable costs are annual holding, This model's variable costs are annual holding, ordering and purchase costs.ordering and purchase costs.

For the optimal order quantity, the annual holding For the optimal order quantity, the annual holding and ordering costs are and ordering costs are notnot necessarily equal. necessarily equal.

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• Demand occurs at a constant rate of Demand occurs at a constant rate of DD items/year.items/year.

• Ordering Cost is $Ordering Cost is $CCoo per order. per order.

• Holding Cost is $Holding Cost is $CChh = $ = $CCiiII per item in inventory per item in inventory per year (note holding cost is based on the cost per year (note holding cost is based on the cost of the item, of the item, CCii).).

• Purchase Cost is $Purchase Cost is $CC11 per item if the quantity per item if the quantity ordered is between 0 and ordered is between 0 and xx11, $, $CC22 if the order if the order quantity is between quantity is between xx11 and and xx2 2 , etc., etc.

• Delivery time (lead time) is constant.Delivery time (lead time) is constant.


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FormulasFormulas

• Optimal order quantity: the procedure for Optimal order quantity: the procedure for determining determining Q Q * will be demonstrated* will be demonstrated

• Number of orders per year: Number of orders per year: DD//Q Q * *


• Total annual cost: [(1/2)Total annual cost: [(1/2)Q Q **CChh] + [] + [DCDCoo//Q Q *] + *] + DCDC

(holding + ordering + (holding + ordering + purchase)purchase)

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Example: Nick's Camera ShopExample: Nick's Camera Shop

EOQ with Quantity Discounts ModelEOQ with Quantity Discounts Model

Nick's Camera Shop carries Zodiac instant printNick's Camera Shop carries Zodiac instant print

film. The film normally costs Nick $3.20film. The film normally costs Nick $3.20

per roll, and he sells it for $5.25. Zodiacper roll, and he sells it for $5.25. Zodiac

film has a shelf life of 18 months.film has a shelf life of 18 months.

Nick's average sales are 21 rolls perNick's average sales are 21 rolls per

week. His annual inventory holdingweek. His annual inventory holding

cost rate is 25% and it costs Nick $20 to place an cost rate is 25% and it costs Nick $20 to place an order order

with Zodiac.with Zodiac.

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EOQ with Quantity Discounts ModelEOQ with Quantity Discounts Model

If Zodiac offers a 7% discount on orders of If Zodiac offers a 7% discount on orders of 400400

rolls or more, a 10% discount for 900 rolls or rolls or more, a 10% discount for 900 rolls or more,more,

and a 15% discount for 2000 rolls or more, and a 15% discount for 2000 rolls or more, determinedetermine

Nick's optimal order quantity.Nick's optimal order quantity.

----------------------------------------

DD = 21(52) = 1092; = 21(52) = 1092; CChh = .25( = .25(CCii); ); CCoo = 20 = 20

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Unit-Prices’ Economical Order QuantitiesUnit-Prices’ Economical Order Quantities

• For For CC44 = .85(3.20) = $2.72 = .85(3.20) = $2.72

To receive a 15% discount Nick must orderTo receive a 15% discount Nick must order

at least 2,000 rolls. Unfortunately, the film's shelfat least 2,000 rolls. Unfortunately, the film's shelf

life is 18 months. The demand in 18 months (78life is 18 months. The demand in 18 months (78

weeks) is 78 x 21 = 1638 rolls of film. weeks) is 78 x 21 = 1638 rolls of film.

If he ordered 2,000 rolls he would have toIf he ordered 2,000 rolls he would have to

scrap 372 of them. This would cost more than thescrap 372 of them. This would cost more than the

15% discount would save.15% discount would save.

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• For For CC33 = .90(3.20) = $2.88 = .90(3.20) = $2.88

QQ33* = 2* = 2DCDCoo//CChh = 2(1092)(20)/[.25(2.88)] = 246.31 = 2(1092)(20)/[.25(2.88)] = 246.31 (not feasible) (not feasible)

The most economical, feasible quantity for The most economical, feasible quantity for CC33 is 900. is 900.

• For For CC22 = .93(3.20) = $2.976 = .93(3.20) = $2.976

QQ22* = 2* = 2DCDCoo//CCh h == 2(1092)(20)/[.25(2.976)] = 242.30 2(1092)(20)/[.25(2.976)] = 242.30

(not feasible)(not feasible)

The most economical, feasible quantity for The most economical, feasible quantity for CC22 is 400. is 400.

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• For For CC11 = 1.00(3.20) = $3.20 = 1.00(3.20) = $3.20

QQ11* = 2* = 2DCDCoo//CChh = 2(1092)(20)/.25(3.20) = = 2(1092)(20)/.25(3.20) = 233.67 233.67 (feasible)(feasible)

When we reach a When we reach a computedcomputed QQ that is feasible that is feasible we stop computing we stop computing Q'sQ's. (In this problem we have no . (In this problem we have no more to compute anyway.)more to compute anyway.)

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Total Cost ComparisonTotal Cost Comparison

Compute the total cost for the most economical, feasible Compute the total cost for the most economical, feasible order quantity in each price category for which a order quantity in each price category for which a Q Q * was * was computed.computed.

TCTCii = (1/2)( = (1/2)(QQii**CChh) + () + (DCDCoo//QQii*) + *) + DCDCii

TCTC3 3 = (1/2)(900)(.72) +((1092)(20)/900)+(1092)(2.88) = 3493= (1/2)(900)(.72) +((1092)(20)/900)+(1092)(2.88) = 3493

TCTC22 = (1/2)(400)(.744)+((1092)(20)/400)+(1092)(2.976) = 3453 = (1/2)(400)(.744)+((1092)(20)/400)+(1092)(2.976) = 3453

TCTC11 = (1/2)(234)(.80) +((1092)(20)/234)+(1092)(3.20) = 3681 = (1/2)(234)(.80) +((1092)(20)/234)+(1092)(3.20) = 3681

Comparing the total costs for 234, 400 and 900, the Comparing the total costs for 234, 400 and 900, the lowest total annual cost is $3453. Nick should order 400 rolls lowest total annual cost is $3453. Nick should order 400 rolls at a time. at a time.

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End of Chapter 10, Part AEnd of Chapter 10, Part A

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