Chapter 11, Part A Inventory Models: Deterministic Demand

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Chapter 11, Part AChapter 11, 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 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: 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 yearsyears

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

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

BBB is open 300 days a year (6 days a BBB is open 300 days a year (6 days a week and closed two weeks in August). Lead week and closed two weeks in August). Lead time is 60 working days.time is 60 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 or 1.667. = 500/300 or 1.667.

Thus the reorder point Thus the reorder point r r = (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 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 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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•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 > > D D ).).

•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.

•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 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

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:the machine is idle for:

126.3 - 21.1 = 105.2 days between 126.3 - 21.1 = 105.2 days between runs.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 = 288,675288,675

Machine UtilizationMachine Utilization

Machine is producing Machine is producing 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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•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 per item backordered per year.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 **

•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) = $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. 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 LongHow many days after receiving an order How many days after receiving an order

does Hervis run out of engines? How long is Hervis does Hervis run out of engines? How long is Hervis without any engines per cycle?without 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. 2080/(255+2080) = .8908 of the order cycle. (Note, ((Note, (Q Q *-*-S S *)/*)/Q Q * = .8908 also, before * = .8908 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 = 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 - Hervis is out of stock for approximately 38 - 34 = 4 days.34 = 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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•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 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.


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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 ModelNick's Camera Shop carries Zodiac instant Nick's Camera Shop carries Zodiac instant

print film. The film normally costs Nick $3.20 per print film. The film normally costs Nick $3.20 per roll, and he sells it for $5.25. Zodiac film has a roll, and he sells it for $5.25. Zodiac film has a shelf life of 18 months. Nick's average sales are shelf life of 18 months. Nick's average sales are 21 rolls per week. His annual inventory holding 21 rolls per week. 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 with Zodiac.order with Zodiac.

If Zodiac offers a 7% discount on orders of If Zodiac offers a 7% discount on orders of 400 rolls or more, a 10% discount for 900 rolls 400 rolls or more, a 10% discount for 900 rolls or more, and a 15% discount for 2000 rolls or or more, and a 15% discount for 2000 rolls or more, determine Nick's optimal order quantity.more, determine 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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Chapter 11, Part A Inventory Models: Deterministic Demand