1 1 Slide © 2001 South-Western College Publishing/Thomson Learning Anderson Sweeney Williams...

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© 2001 South-Western College Publishing/Thomson Learning© 2001 South-Western College Publishing/Thomson Learning

Anderson Sweeney Anderson Sweeney WilliamsWilliams

Anderson Sweeney Anderson Sweeney WilliamsWilliams

Slides Prepared by JOHN LOUCKSSlides Prepared by JOHN LOUCKS

QUANTITATIVE QUANTITATIVE METHODS FORMETHODS FORBUSINESS 8eBUSINESS 8e

QUANTITATIVE QUANTITATIVE METHODS FORMETHODS FORBUSINESS 8eBUSINESS 8e

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Chapter 13Chapter 13Inventory Management: Independent-Inventory Management: Independent-

DemandDemand Economic Order Quantity (EOQ) ModelEconomic Order Quantity (EOQ) Model Economic Production Lot Size ModelEconomic Production Lot Size Model An Inventory Model with Planned ShortagesAn Inventory Model with Planned Shortages Quantity Discounts for the EOQ ModelQuantity Discounts for the EOQ Model A Single-Period Inventory Model with A Single-Period Inventory Model with

Probabilistic DemandProbabilistic Demand An Order-Quantity, Reorder-Point Model with An Order-Quantity, Reorder-Point Model with

Probabilistic DemandProbabilistic Demand A Periodic-Review Model with Probabilistic A Periodic-Review Model with Probabilistic

DemandDemand

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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 costminimize total variable cost over a specified time period (assumed to be over a specified time period (assumed to be annual in the following review).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: QQ* = 2* = 2DCDCoo//CChh

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

• Time between orders (cycle time): Time between orders (cycle time): QQ*/*/DD yearsyears

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

(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 (BBB) is a retail Bart's Barometer Business (BBB) is a retail outlet which deals exclusively with weather outlet which deals exclusively with weather equipment. Currently BBB is trying to decide on an equipment. Currently BBB is trying to decide on an inventory and reorder policy for home barometers. inventory and reorder policy for home barometers.

Barometers cost BBB $50 each and demand Barometers cost BBB $50 each and demand is about 500 per year distributed fairly evenly is about 500 per year distributed fairly evenly throughout the year. Reordering costs are $80 per throughout the year. Reordering costs are $80 per order and holding costs are figured at 20% of the order and holding costs are figured at 20% of the cost of the item. cost of the item.

BBB is open 300 days a year (6 days a week BBB is open 300 days a year (6 days a week and closed two weeks in August). Lead time is 60 and closed two weeks in August). Lead time is 60 working days.working days.

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Total Variable Cost ModelTotal Variable Cost Model

Total Costs Total Costs = (Holding Cost) + (Ordering = (Holding Cost) + (Ordering Cost) 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 = 60 days and daily demand is is dd = 500/300 or1.667. = 500/300 or1.667.

Thus the reorder point Thus the reorder point rr = (1.667)(60) = = (1.667)(60) = 100. Bart should reorder 90 barometers when 100. Bart should reorder 90 barometers when his inventory position reaches 100 (that is 10 his inventory position reaches 100 (that is 10 on hand and one outstanding order).on 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 implement We’ll now use a spreadsheet to implement the Economic Order Quantity model. We’ll the Economic Order Quantity model. We’ll confirm our earlier calculations for Bart’s confirm our earlier calculations for Bart’s problem and perform some sensitivity analysis.problem and perform some sensitivity analysis.

This spreadsheet can be modified to This spreadsheet can be modified to accommodate other inventory models presented accommodate other inventory models presented in this chapter.in this chapter.

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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


Partial Spreadsheet with Input DataPartial Spreadsheet with Input Data

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

10 Optimal Order Quantity =SQRT(2*B3*B4/(B5*B6/100))11 Requested Order Quantity12 % Change from EOQ =(C11/B10-1)*1001314 Annual Holding Cost =B5/100*B6*B10/2 =B5/100*B6*C11/215 Annual Ordering Cost =B4*B3/B10 =B4*B3/C1116 Total Annual Cost =B14+B15 =C14+C1517 % Over Minimum TAC =(C16/B16-1)*1001819 Maximum Inventory Level =B10 =C1120 Average Inventory Level =B10/2 =C11/221 Reorder Point =B3/B7*B8 =B3/B7*B82223 Number of Orders Per 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 Optimal Order Quantity 89.4411 Requested Order Quantity 75.0012 % Change from EOQ -16.151314 Annual Holding Cost $447.21 $375.0015 Annual Ordering Cost $447.21 $533.3316 Total Annual Cost $894.43 $908.3317 % Over Minimum TAC 1.551819 Maximum Inventory Level 89.44 7520 Average Inventory Level 44.72 37.521 Reorder Point 100 1002223 Number of Orders Per 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 Total resulted in only a 1.55% increase in the Total Annual Cost.Annual Cost.

• Annual Holding Cost and Annual Ordering Cost Annual Holding Cost and Annual Ordering Cost are no longer equal.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 is Inventory is replenished gradually as the order is produced (which requires the production rate to produced (which requires the production rate to be greater than the demand rate). 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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AssumptionsAssumptions• Demand occurs at a constant rate of Demand occurs at a constant rate of DD items items

per year.per year.• Production rate is Production rate is PP items per year (and items per year (and P P

>>DD).).

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

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


• Set-up time (lead time) is constant.Set-up time (lead time) is constant.• Planned shortages are not permitted.Planned shortages are not permitted.

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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 yearsyears

• 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 Non-Slip Tile Company (NST) has been using production runs of 100,000 tiles, 10 times using production runs of 100,000 tiles, 10 times per year to meet the demand of 1,000,000 tiles per year to meet the demand of 1,000,000 tiles annually. The set-up cost is $5,000 per run and annually. The set-up cost is $5,000 per run and holding cost is estimated at 10% of the holding cost is estimated at 10% of the manufacturing cost of $1 per tile. The manufacturing cost of $1 per tile. The production capacity of the machine is 500,000 production capacity of the machine is 500,000 tiles per month. The factory is open 365 days tiles per month. The factory is open 365 days per year.per year.

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Total Annual Variable Cost ModelTotal Annual Variable Cost Model

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

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

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

The number of runs per year = The number of 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 How much is NST losing annually by using their present production schedule?using their 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 = produce 346,410 per run = (346,410/6,000,000)365 = 21.1 days. (346,410/6,000,000)365 = 21.1 days. Thus, Thus, the machine is idle for 126.3 - 21.1 = 105.2 days the machine is idle for 126.3 - 21.1 = 105.2 days between runs.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 83,33383,333

Optimal Policy:Optimal Policy:

maximum inventory = (1-maximum inventory = (1-1/61/6)346,410 = )346,410 = 288,675.288,675.

Machine UtilizationMachine Utilization

The machine is producing tiles The machine is producing tiles DD//PP = 1/6 of the = 1/6 of the time. 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 backorder occur until a predetermined backorder quantity is reached, at which time the 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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AssumptionsAssumptions• Demand occurs at a constant rate of Demand occurs at a constant rate of DD items per items per

year.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 per item backordered per year.year.


• 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 demand units are withdrawn from a replenishment order when it is delivered).replenishment 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 **• Time between orders (cycle time): Time between orders (cycle time): Q Q */*/DD years years• 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 Rockets serving the Los Angeles area. All Rockets serving the Los Angeles area. All Rockets are maintained at a central garage. On Rockets are maintained at a central garage. On the average, eight Rockets per month require a the average, eight Rockets per month require a new engine. Engines cost $850 each. There is new engine. Engines cost $850 each. There is also a $120 order cost (independent of the also a $120 order cost (independent of the number of engines ordered). number of engines ordered).

Hervis has an annual holding cost rate of Hervis has an annual holding cost rate of 30% on engines. It takes two weeks to obtain 30% on engines. It takes two weeks to obtain the engines after they are ordered. For each the engines after they are ordered. For each week a car is out of service, Hervis loses $40 week a car is out of service, Hervis loses $40 profit. 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) = = .30(850) = $255; $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. Demand is 8 per month or 2 per week. Since lead time is 2 weeks, lead time demand is Since lead time is 2 weeks, lead time demand is 4. 4.

Thus, since the optimal policy is to order Thus, since the optimal policy is to order 10 to arrive when there is one backorder, the 10 to arrive when there is one backorder, the order should be placed when there are 3 engines order should be placed when there are 3 engines remaining in inventory.remaining in inventory.

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Stockout: When and How LongStockout: When and How Long

How many days after receiving an order does How many days after receiving an order does Hervis run out of engines? How long is Hervis without Hervis run out of engines? How long is Hervis without any engines per cycle?any engines per cycle?

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Inventory exists for Inventory exists for CCbb/(/(CCbb++CChh) = ) = 2080/(255+2080) = .8908 of the order cycle. (Note, 2080/(255+2080) = .8908 of the order cycle. (Note, ((QQ*-*-SS*)/*)/QQ* = .8908 also, before * = .8908 also, before Q Q * and * and S S * are * are rounded.) rounded.)

An order cycle is An order cycle is Q Q */*/DD = .1049 years = 38.3 = .1049 years = 38.3 days. Thus, Hervis runs out of engines .8908(38.3) = days. Thus, Hervis runs out of engines .8908(38.3) = 34 days after receiving an order. 34 days after receiving an order.

Hervis is out of stock for approximately 38 - 34 Hervis is out of stock for approximately 38 - 34 = 4 days.= 4 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 applicable where a supplier offers a lower purchase cost when an item is ordered in larger purchase cost when an item is ordered in larger quantities. 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 For the optimal order quantity, the annual holding and ordering costs are holding and ordering costs are notnot necessarily necessarily equal.equal.

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AssumptionsAssumptions• Demand occurs at a constant rate of Demand occurs at a constant rate of DD items items

per year.per 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 per year (note holding cost is based on the cost of the item, cost 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.• Planned shortages are not permitted.Planned shortages are not permitted.

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FormulasFormulas

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

demonstrateddemonstrated• 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

yearsyears

• 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 print Nick's Camera Shop carries Zodiac instant print film. The film normally costs Nick $3.20 per roll, and film. The film normally costs Nick $3.20 per roll, and he sells it for $5.25. Zodiac film has a shelf life of 18 he sells it for $5.25. Zodiac film has a shelf life of 18 months. Nick's average sales are 21 rolls per week. months. Nick's average sales are 21 rolls per week. His annual inventory holding cost rate is 25% and it His annual inventory holding cost rate is 25% and it costs Nick $20 to place an order with Zodiac.costs Nick $20 to place an order with Zodiac.

If Zodiac offers a 7% discount on orders of 400 If Zodiac offers a 7% discount on orders of 400 rolls or more, a 10% discount for 900 rolls or more, rolls or more, a 10% discount for 900 rolls or more, and a 15% discount for 2000 rolls or more, determine and a 15% discount for 2000 rolls or more, determine Nick's optimal order quantity.Nick's optimal order quantity.

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

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Unit-Prices’ Economical, Feasible Order Unit-Prices’ Economical, Feasible Order QuantitiesQuantities

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

To receive a 15% discount Nick must To receive a 15% discount Nick must order order at least 2,000 rolls. Unfortunately, at least 2,000 rolls. Unfortunately, the film's shelf the film's shelf life is 18 months. The demand life is 18 months. The demand in 18 months (78 in 18 months (78 weeks) is 78 X 21 = 1638 rolls weeks) is 78 X 21 = 1638 rolls of film. of film.

If he ordered 2,000 rolls he would If he ordered 2,000 rolls he would have to have to scrap 372 of them. This would cost scrap 372 of them. This would cost more than the more than the 15% discount would save.15% discount would save.

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

• 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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Unit-Prices’ Economical, Feasible Order Unit-Prices’ Economical, Feasible Order QuantitiesQuantities

• 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 that is feasible we stop computing feasible we stop computing Q'sQ's. (In this problem . (In this problem we have no more to compute anyway.)we have no 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 lowest Comparing the total costs for 234, 400 and 900, the lowest total annual cost is $3453. Nick should order 400 rolls at a total annual cost is $3453. Nick should order 400 rolls at a time. time.

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Probabilistic ModelsProbabilistic Models

In many cases demand (or some other factor) is In many cases demand (or some other factor) is not known with a high degree of certainty and a not known with a high degree of certainty and a probabilistic inventory modelprobabilistic inventory model should actually be should actually be used. used.

These models tend to be more complex than These models tend to be more complex than deterministic models. deterministic models.

The probabilistic models covered in this chapter The probabilistic models covered in this chapter are: are: • single-period order quantitysingle-period order quantity• reorder-point quantityreorder-point quantity• periodic-review order quantityperiodic-review order quantity

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Single-Period Order QuantitySingle-Period Order Quantity

A A single-period order quantity modelsingle-period order quantity model (sometimes (sometimes called the newsboy problem) deals with a called the newsboy problem) deals with a situation in which only one order is placed for situation in which only one order is placed for the item and the demand is probabilistic. the item and the demand is probabilistic.

If the period's demand exceeds the order If the period's demand exceeds the order quantity, the demand is not backordered and quantity, the demand is not backordered and revenue (profit) will be lost. revenue (profit) will be lost.

If demand is less than the order quantity, the If demand is less than the order quantity, the surplus stock is sold at the end of the period surplus stock is sold at the end of the period (usually for less than the original purchase (usually for less than the original purchase price).price).

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AssumptionsAssumptions

• Period demand follows a known probability Period demand follows a known probability distribution:distribution:

normal: mean is normal: mean is µµ, standard deviation is , standard deviation is uniform: minimum is uniform: minimum is aa, maximum is , maximum is bb

• Cost of overestimating demand: $Cost of overestimating demand: $ccoo

• Cost of underestimating demand: $Cost of underestimating demand: $ccuu

• Shortages are not backordered.Shortages are not backordered.• Period-end stock is sold for salvage (not held Period-end stock is sold for salvage (not held

in inventory).in inventory).

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FormulasFormulas

• Optimal probability of no shortage: Optimal probability of no shortage:

P(demand P(demand << Q Q *) = *) = ccuu/(/(ccuu++ccoo))

• Optimal probability of shortage:Optimal probability of shortage:

P(demand > P(demand > Q Q *) = 1 - *) = 1 - ccuu/(/(ccuu++ccoo))

• Optimal order quantity, based on demand Optimal order quantity, based on demand distribution:distribution:

normal: normal: Q Q * = * = µµ + + zz uniform: uniform: Q Q * = * = aa + P(demand + P(demand << Q Q *)(*)(bb--aa))

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Example: McHardee PressExample: McHardee Press

Single-Period Order Quantity ModelSingle-Period Order Quantity Model

McHardee Press publishes the Fast Food McHardee Press publishes the Fast Food Restaurant Menu Book and wishes to determine Restaurant Menu Book and wishes to determine how many copies to print. There is a fixed cost how many copies to print. There is a fixed cost of $5,000 to produce the book and the of $5,000 to produce the book and the incremental profit per copy is $.45. Any unsold incremental profit per copy is $.45. Any unsold copies of the book can be sold at salvage at a copies of the book can be sold at salvage at a $.55 loss. $.55 loss.

Sales for this edition are estimated to be Sales for this edition are estimated to be normally distributed. The most likely sales normally distributed. The most likely sales volume is 12,000 copies and they believe there volume is 12,000 copies and they believe there is a 5% chance that sales will exceed 20,000. is a 5% chance that sales will exceed 20,000.

How many copies should be printed?How many copies should be printed?

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mm = 12,000. To find = 12,000. To find note that note that zz = 1.65 = 1.65 corresponds to a 5% tail probability. Therefore, corresponds to a 5% tail probability. Therefore,

(20,000 - 12,000) = 1.65(20,000 - 12,000) = 1.65 or or = 4848 = 4848

Using incremental analysis with Using incremental analysis with CCoo = .55 and = .55 and CCuu = = .45, .45, ((CCuu/(/(CCuu++CCoo)) = .45/(.45+.55) = .45)) = .45/(.45+.55) = .45

Find Find Q Q * such that P(* such that P(DD < < Q Q *) = .45. The *) = .45. The probability of 0.45 corresponds to probability of 0.45 corresponds to zz = -.12. Thus, = -.12. Thus,

Q Q * = 12,000 - .12(4848) = 11,418 * = 12,000 - .12(4848) = 11,418 booksbooks

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Single-Period Order Quantity (revised)Single-Period Order Quantity (revised)

If any unsold copies of the book can be sold at If any unsold copies of the book can be sold at salvage at a $.65 loss, how many copies should be salvage at a $.65 loss, how many copies should be printed? printed?

CCoo = .65, ( = .65, (CCuu/(/(CCuu + + CCoo)) = .45/(.45 + .65) = .4091)) = .45/(.45 + .65) = .4091

Find Find Q Q * such that P(* such that P(DD < < Q Q *) = .4091. *) = .4091. zz = -.23 = -.23 gives this probability. Thus, gives this probability. Thus,

Q Q * = 12,000 - .23(4848) = 10,885 books* = 12,000 - .23(4848) = 10,885 books

However, since this is less than the breakeven However, since this is less than the breakeven volume of 11,111 books (= 5000/.45), volume of 11,111 books (= 5000/.45), no copies no copies should be printedshould be printed because if the company produced because if the company produced only 10,885 copies it will not recoup its $5,000 fixed only 10,885 copies it will not recoup its $5,000 fixed cost of producing the book.cost of producing the book.

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Reorder Point QuantityReorder Point Quantity

A firm's A firm's inventory positioninventory position consists of the on- consists of the on-hand inventory plus on-order inventory (all hand inventory plus on-order inventory (all amounts previously ordered but not yet amounts previously ordered but not yet received). received).

An inventory item is reordered when the item's An inventory item is reordered when the item's inventory position reaches a predetermined inventory position reaches a predetermined value, referred to as the value, referred to as the reorder pointreorder point. .

The reorder point represents the quantity The reorder point represents the quantity available to meet demand during lead time. available to meet demand during lead time.

Lead timeLead time is the time span starting when the is the time span starting when the replenishment order is placed and ending when replenishment order is placed and ending when the order arrives. the order arrives.

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Reorder Point QuantityReorder Point Quantity

Under deterministic conditions, when both Under deterministic conditions, when both demand and lead time are constant, the reorder demand and lead time are constant, the reorder point associated with EOQ-based models is set point associated with EOQ-based models is set equal to lead time demand. equal to lead time demand.

Under probabilistic conditions, when demand Under probabilistic conditions, when demand and/or lead time varies, the reorder point often and/or lead time varies, the reorder point often includes safety stock. includes safety stock.

Safety stockSafety stock is the amount by which the reorder is the amount by which the reorder point exceeds the expected (average) lead time point exceeds the expected (average) lead time demand. demand.

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Safety Stock and Service LevelSafety Stock and Service Level

The amount of safety stock in a reorder point The amount of safety stock in a reorder point determines the chance of a stockout during lead determines the chance of a stockout during lead time. time.

The complement of this chance is called the The complement of this chance is called the service level. service level.

Service levelService level, in this context, is defined as the , in this context, is defined as the probability of not incurring a stockout during any probability of not incurring a stockout during any one lead time. one lead time.

Service level, in this context, also is the long-run Service level, in this context, also is the long-run proportion of lead times in which no stockouts proportion of lead times in which no stockouts occur.occur.

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Reorder PointReorder Point

AssumptionsAssumptions• Lead-time demand is normally distributed Lead-time demand is normally distributed

with mean with mean µµ and standard deviation and standard deviation ..• Approximate optimal order quantity: EOQApproximate optimal order quantity: EOQ• Service level is defined in terms of the Service level is defined in terms of the

probability ofprobability of

no stockouts during lead time and is no stockouts during lead time and is reflected in reflected in zz..• Shortages are not backordered.Shortages are not backordered.• Inventory position is reviewed continuously.Inventory position is reviewed continuously.

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Reorder PointReorder Point

FormulasFormulas

• Reorder point: Reorder point: rr = = µµ + + zz• Safety stock: Safety stock: zz• Average inventory: 1/2(Average inventory: 1/2(QQ) + ) + zz• Total annual cost: [(1/2)Total annual cost: [(1/2)Q Q **CChh] + [] + [zz CChh] + ] +

[[DCDCoo//Q Q *] *]

(holding(normal) + holding(safety) + (holding(normal) + holding(safety) + ordering)ordering)

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Example: Robert’s DrugExample: Robert’s Drug

Reorder Point ModelReorder Point Model

Robert's Drugs is a drug wholesaler supplying 55 Robert's Drugs is a drug wholesaler supplying 55 independent drug stores. Roberts wishes to determine independent drug stores. Roberts wishes to determine an optimal inventory policy for an optimal inventory policy for ComfortComfort brand brand headache remedy. Sales of headache remedy. Sales of ComfortComfort are relatively are relatively constant as the past 10 weeks of data indicate:constant as the past 10 weeks of data indicate:

Week Sales (cases) Week Sales Week Sales (cases) Week Sales (cases)(cases) 1 1 110 110 6 120 6 120

2 2 115 115 7 7 130130

3 3 125 125 8 115 8 115 4 4 120 120 9 110 9 110 5 5 125 125 10 130 10 130

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Each case of Each case of ComfortComfort costs Roberts $10 costs Roberts $10 and Roberts uses a 14% annual holding cost rate and Roberts uses a 14% annual holding cost rate for its inventory. The cost to prepare a purchase for its inventory. The cost to prepare a purchase order for order for ComfortComfort is $12. What is Roberts’ is $12. What is Roberts’ optimal order quantity?optimal order quantity?

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Optimal Order QuantityOptimal Order Quantity

The average weekly sales over the 10 The average weekly sales over the 10 week period is 120 cases. Hence week period is 120 cases. Hence DD = 120 X 52 = 120 X 52 = 6,240 cases per year; = 6,240 cases per year;

CChh = (.14)(10) = 1.40; = (.14)(10) = 1.40; CCoo = 12. = 12.

Q Q * = 2* = 2DCDCoo//CChh = (2)(6240)(12)/1.40 = = (2)(6240)(12)/1.40 = 327327

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The lead time for a delivery of The lead time for a delivery of Comfort Comfort has averaged four working days. Lead time has has averaged four working days. Lead time has therefore been estimated as having a normal therefore been estimated as having a normal distribution with a mean of 80 cases and a distribution with a mean of 80 cases and a standard deviation of 10 cases. Roberts wants standard deviation of 10 cases. Roberts wants at most a 2% probability of selling out of at most a 2% probability of selling out of ComfortComfort during this lead time. What reorder point during this lead time. What reorder point should Roberts use?should Roberts use?

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Optimal Reorder PointOptimal Reorder Point

Lead time demand is normally distributed Lead time demand is normally distributed with with mm = 80, = 80, = 10. = 10.

Since Roberts wants at most a 2% Since Roberts wants at most a 2% probability of selling out of probability of selling out of ComfortComfort, the , the corresponding corresponding zz value is 2.06. That is, value is 2.06. That is, PP((zz > > 2.06) = .0197 (about .02). 2.06) = .0197 (about .02).

Hence Roberts should reorder Hence Roberts should reorder ComfortComfort when supply reaches when supply reaches mm + + zz = 80 + 2.06(10) = = 80 + 2.06(10) = 101 cases. 101 cases.

The safety stock is 21 cases.The safety stock is 21 cases.

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Total Annual Inventory CostTotal Annual Inventory Cost

Ordering: (Ordering: (DCDCoo//Q Q *) = ((6240)(12)/327) *) = ((6240)(12)/327) = $229 Holding-Normal: (1/2) = $229 Holding-Normal: (1/2)Q Q **CCoo = = (1/2)(327)(1.40) = 229(1/2)(327)(1.40) = 229

Holding-Safety Stock: Holding-Safety Stock: CChh(21) = (1.40)(21) (21) = (1.40)(21) = 29 = 29

TOTAL TOTAL = $487 = $487

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Periodic Review SystemPeriodic Review System

A A periodic review systemperiodic review system is one in which the is one in which the inventory level is checked and reordering is done only inventory level is checked and reordering is done only at specified points in time (at fixed intervals usually). at specified points in time (at fixed intervals usually).

Assuming the demand rate varies, the order quantity Assuming the demand rate varies, the order quantity will vary from one review period to another. (This is will vary from one review period to another. (This is in contrast to the continuous review system in which in contrast to the continuous review system in which inventory is monitored continuously and a fixed-inventory is monitored continuously and a fixed-quantity order can be placed whenever the reorder quantity order can be placed whenever the reorder point is reached.)point is reached.)

At the time a periodic-review order quantity is being At the time a periodic-review order quantity is being decided, the concern is that the on-hand inventory decided, the concern is that the on-hand inventory and the quantity being ordered is enough to satisfy and the quantity being ordered is enough to satisfy demand from the time this order is placed until the demand from the time this order is placed until the next order is next order is received received (not placed).(not placed).

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Periodic Review Order QuantityPeriodic Review Order Quantity

AssumptionsAssumptions• Inventory position is reviewed at constant intervals Inventory position is reviewed at constant intervals

(periods).(periods).• Demand during review period plus lead time periodDemand during review period plus lead time period

is normally distributed with mean is normally distributed with mean µµ and standard and standard deviation deviation ..• Service level is defined in terms of the probability of Service level is defined in terms of the probability of

no stockouts during a review period plus lead time no stockouts during a review period plus lead time period and is reflected in period and is reflected in zz. . • On-hand inventory at ordering time: On-hand inventory at ordering time: II• Shortages are not backordered.Shortages are not backordered.• Lead time is less than the length of the review Lead time is less than the length of the review

period.period.

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Periodic Review Order QuantityPeriodic Review Order Quantity

FormulasFormulas

• Replenishment level: Replenishment level: MM = = µµ + + zz• Order quantity: Order quantity: QQ = = MM - - II

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Example: Ace BrushExample: Ace Brush

Periodic Review Order Quantity ModelPeriodic Review Order Quantity Model

Joe Walsh is a salesman for the Ace Brush Joe Walsh is a salesman for the Ace Brush Company. Every three weeks he contacts Dollar Company. Every three weeks he contacts Dollar Department Store so that they may place an order Department Store so that they may place an order to replenish their stock. Weekly demand for Ace to replenish their stock. Weekly demand for Ace brushes at Dollar approximately follows a normal brushes at Dollar approximately follows a normal distribution with a mean of 60 brushes and a distribution with a mean of 60 brushes and a standard deviation of 9 brushes. standard deviation of 9 brushes.

Once Joe submits an order, the lead time Once Joe submits an order, the lead time until Dollar receives the brushes is one week. until Dollar receives the brushes is one week. Dollar would like at most a 2% chance of running Dollar would like at most a 2% chance of running out of stock during any replenishment period. If out of stock during any replenishment period. If Dollar has 75 brushes in stock when Joe contacts Dollar has 75 brushes in stock when Joe contacts them, how many should they order?them, how many should they order?

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Demand During Uncertainty PeriodDemand During Uncertainty Period

The review period plus the following lead time totals 4 The review period plus the following lead time totals 4 weeks. This is the amount of time that will elapse before weeks. This is the amount of time that will elapse before the next shipment of brushes will arrive.the next shipment of brushes will arrive.

Weekly demand is normally distributed with:Weekly demand is normally distributed with:

Mean weekly demand, Mean weekly demand, µµ = 60 = 60

Weekly standard deviation, Weekly standard deviation, = 9 = 9

Weekly variance, Weekly variance, 22 = 81 = 81

Demand for 4 weeks is normally distributed with:Demand for 4 weeks is normally distributed with:

Mean demand over 4 weeks, Mean demand over 4 weeks, µµ = 4x60 = 240 = 4x60 = 240

Variance of demand over 4 weeks, Variance of demand over 4 weeks, 22 = 4x81 = = 4x81 = 324324

Standard deviation over 4 weeks, Standard deviation over 4 weeks, = (324) = (324)1/21/2 = 18 = 18

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Replenishment Level Replenishment Level

MM = = µµ + + zz where where zz is determined by the is determined by the desired stockout probability. For a 2% stockout desired stockout probability. For a 2% stockout probability (2% tail area), probability (2% tail area), zz = 2.05. Thus, = 2.05. Thus,

MM = 240 + 2.05(18) = 277 brushes = 240 + 2.05(18) = 277 brushes

As the store currently has 75 brushes in As the store currently has 75 brushes in stock, Dollar should order: stock, Dollar should order:

277 - 75 = 202 brushes277 - 75 = 202 brushes

The safety stock is:The safety stock is:

zz = (2.05)(18) = 37 brushes = (2.05)(18) = 37 brushes

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The End of Chapter 13The End of Chapter 13

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