1 1 slide © 2008 thomson south-western. all rights reserved slides by john loucks st. edward’s...

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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 BChapter 10, Part BInventory Models: Probabilistic DemandInventory Models: Probabilistic Demand

Single-Period Inventory Model with Probabilistic Single-Period Inventory Model with Probabilistic DemandDemand

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

Periodic-Review Model with Probabilistic DemandPeriodic-Review Model with Probabilistic Demand

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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 situation called the newsboy problem) deals with a situation in which only one order is placed for the item and in which only one order is placed for the item and the demand is probabilistic. the demand is probabilistic.

If the period's demand exceeds the order quantity, If the period's demand exceeds the order quantity, the demand is not backordered and revenue the demand is not backordered and revenue (profit) will be lost. (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 price).(usually for less than the original purchase 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


McHardee Press publishes the Fast Food McHardee Press publishes the Fast Food MenuMenu

Book and wishes to determine how manyBook and wishes to determine how many

copies to print. There is a fixed cost ofcopies to print. There is a fixed cost of

$5,000 to produce the book and the$5,000 to produce the book and the

incremental profit per copy isincremental profit per copy is

$0.45. Any unsold copies of the$0.45. Any unsold copies of the

the book can be sold at salvage at athe book can be sold at salvage at a

$.55 loss. $.55 loss.

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Sales for this edition are estimated to be Sales for this edition are estimated to be normallynormally

distributed. The most likely sales volume is distributed. The most likely sales volume is 12,00012,000

copies and they believe there is a 5% chance copies and they believe there is a 5% chance that salesthat sales

will exceed 20,000. 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 can be sold at salvage If any unsold copies can be sold at salvage

at a $.65 loss, how many copies should be at a $.65 loss, how many copies should be printed? printed?

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

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

Q Q * = 12,000 - .23(4848) = 10,885 books* = 12,000 - .23(4848) = 10,885 booksHowever, since this is less than the However, since this is less than the

breakeven volume of 11,111 books (= breakeven volume of 11,111 books (= 5000/.45), 5000/.45), no copies should be printedno copies should be printed because if because if the company produced only 10,885 copies it will the company produced only 10,885 copies it will not recoup its $5,000 fixed cost.not recoup its $5,000 fixed cost.

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Reorder Point QuantityReorder Point Quantity

A firm's A firm's inventory positioninventory position consists of the on-hand consists of the on-hand inventory plus on-order inventory (all amounts inventory plus on-order inventory (all amounts previously ordered but not yet received). previously ordered but not yet 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 service The complement of this chance is called the service level. 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 occur.proportion of lead times in which no stockouts occur.

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Reorder PointReorder Point


• 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 probability Service level is defined in terms of the probability of no stockouts during lead time and is reflected of no stockouts during lead time and is reflected in in zz..


• Inventory position is reviewed continuously.Inventory position is reviewed continuously.

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FormulasFormulas

Reorder point: Reorder point: rr = = µµ + + zz

Safety stock: Safety stock: zz

Average inventory: ½Average inventory: ½((Q Q ) + ) + zz

Total annual cost: [( ½ )Total annual cost: [( ½ )Q Q **CChh]] ++ [[zz CChh]] ++ [[DCDCoo//Q Q *] *]

(hold.(normal) + hold.(safety) (hold.(normal) + hold.(safety)

+ ordering)+ ordering)

Reorder PointReorder Point

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Reorder Point ModelReorder Point Model

Robert's Drugs is a drug wholesaler supplyingRobert's Drugs is a drug wholesaler supplying

55 independent drug stores. 55 independent drug stores.

Roberts wishes to determine anRoberts wishes to determine an

optimal inventory policy for optimal inventory policy for

ComfortComfort brand headache remedy. brand headache remedy.

Sales of Sales of ComfortComfort are relatively constant are relatively constant

as the past 10 weeks of data (on next slide) as the past 10 weeks of data (on next slide) indicate.indicate.

Example: Robert’s DrugExample: Robert’s Drug

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Reorder Point ModelReorder Point Model

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

2 2 115 115 7 7 130 130

3 3 125 125 8 8 115 115

4 4 120 120 9 9 110 110

5 5 125 125 10 10 130 130


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Each case of Each case of ComfortComfort costs Roberts $10 and costs Roberts $10 and

Roberts uses a 14% annual holding cost rate for itsRoberts uses a 14% annual holding cost rate for its

inventory. The cost to prepare a purchase order forinventory. The cost to prepare a purchase order for

ComfortComfort is $12. What is Roberts’ optimal order is $12. What is Roberts’ optimal order

quantity?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 = 120 X 52 = 6,240 cases per year; 52 = 6,240 cases per year;

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


*o h2 / (2(6240)(12))/ 1.40 327Q DC C *o h2 / (2(6240)(12))/ 1.40 327Q DC C

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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 hashas averaged four working days. Lead time has

therefore been estimated as having a normaltherefore been estimated as having a normal

distribution with a mean of 80 cases and a standarddistribution with a mean of 80 cases and a standard

deviation of 10 cases. Roberts wants at most a 2%deviation of 10 cases. Roberts wants at most a 2%

probability of selling out of probability of selling out of ComfortComfort during this lead during this lead

time. What reorder point should Roberts use?time. What reorder point should Roberts use?

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Optimal Reorder PointOptimal Reorder Point

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

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

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

The safety stock is The safety stock is zz = 21 cases. = 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= $229

Holding-Normal: (1/2)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 inventory level is checked and reordering is done only at specified points in time (at fixed intervals only at specified points in time (at fixed intervals usually). usually).

Assuming the demand rate varies, the order Assuming the demand rate varies, the order quantity will vary from one review period to quantity will vary from one review period to another. another.

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

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Periodic Review Order QuantityPeriodic Review Order Quantity


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

• Demand during review period plus lead time Demand during review period plus lead time period is normally distributed with mean period is normally distributed with mean µµ and and standard deviation standard deviation ..

• Service level is defined in terms of the Service level is defined in terms of the probability of no stockouts during a review probability of no stockouts during a review period plus lead time period and is reflected in period plus lead time period and is reflected in zz. .

• On-hand inventory at ordering time: On-hand inventory at ordering time: HH


• Lead time is less than the review period length.Lead time is less than the review period length.

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FormulasFormulas

Replenishment level: Replenishment level: MM = = µµ + + zz

Order quantity: Order quantity: QQ = = MM - - HH

Periodic Review Order QuantityPeriodic Review Order Quantity

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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 BrushJoe Walsh is a salesman for the Ace Brush

Company. Every three weeks he contacts DollarCompany. Every three weeks he contacts Dollar

Department Store so that theyDepartment Store so that they

may place an order to replenishmay place an order to replenish

their stock. Weekly demandtheir stock. Weekly demand

for Ace brushes at Dollarfor Ace brushes at Dollar

approximately follows a normalapproximately follows a normal

distribution with a mean of 60 brushes and adistribution with a mean of 60 brushes and a

standard deviation of 9 brushes. standard deviation of 9 brushes.

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Periodic Review Order Quantity ModelPeriodic Review Order Quantity ModelOnce Joe submits an order, the lead time Once Joe submits an order, the lead time

untiluntilDollar receives the brushes is one week. Dollar receives the brushes is one week. DollarDollarwould like at most a 2% chance of running out would like at most a 2% chance of running out ofofstock during any replenishment period. If stock during any replenishment period. If DollarDollarhas 75 brushes in stock when Joe contacts has 75 brushes in stock when Joe contacts them,them,how many should they order?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 weeks. This is the amount of time that will elapse before the next shipment of brushes will arrive.before 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 = 4 x 60 = 240 = 4 x 60 = 240

Variance of demand over 4 weeks, Variance of demand over 4 weeks, 22 = 4 x 81 = = 4 x 81 = 324324

Standard deviation over 4 weeks, Standard deviation over 4 weeks, = (324) = (324)1/21/2 = = 1818

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Replenishment LevelReplenishment 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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End of Chapter 10, Part BEnd of Chapter 10, Part B

1 1 slide © 2008 thomson south-western. all rights reserved slides by john loucks st. edward’s...

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