Special Inventory Mgmt Models

7/27/2019 Special Inventory Mgmt Models

1/42

Special InventoryManagement & ControlModels

Non-instantaneous replacement

Quantity discount (Price break model)

One period decision (Single period model)


2/42

NoninstantaneousReplenishment

Maximum cycle inventory to be decided!Item used or sold as it is completedUsually production rate, p, exceeds the demandrate, d , so there is a buildup of ( p d ) units per time periodBoth p and d expressed in same time intervalBuildup continues for Q / p days to reach themaxm cycle inventory


3/42


Product ion quant i ty Q

Maximum inventory

I max

Productionand demand

Demand or Consumption

onlyTBO

p d

Demand duringproduction interval

O n - h a n d

i n v e n

t o r y

Time

Lot Sizing with Noninstantaneous Replenishment

p

d


4/42

NoninstantaneousReplenishmentMaximum cycle inventory is:

Cycle inventory is no longer Q / 2, it is I max / 2

wherep = production rated = demand rate

Q = lot size

p

d p Q d p

p

Q I max


5/42


Total annual cost = Annual holding cost+ Annual ordering or setup cost

D is annual demand and Q is lot sized is daily demand; p is daily production rate

S Q

D H

p

d p Q S

Q

D H

I C

22max


6/42


Economic Production Lot Size (ELS): optimal lot size Derived by calculus Because the second term is greater than 1,

the ELS results in a larger lot size than theEOQ

d p

p

H

DS EL S

2


7/42

Finding the Economic Production LotSizeEXAMPLE D.1

A plant manager of a chemical plant must determine the lotsize for a particular chemical that has a steady demand of 30 barrels per day. The production rate is 190 barrels per day, annual demand is 10,500 barrels, setup cost is $200,annual holding cost is $0.21 per barrel, and the plantoperates 350 days per year.

a. Determine the economic production lot size (ELS)b. Determine the total annual setup and inventory

holding cost for this item

c. Determine the time between orders (TBO), or cyclelength, for the ELSd. Determine the production time per lot

What are the advantages of reducing the setup time by 10percent?


8/42

Finding the Economic ProductionLot Size

SOLUTIONa. Solving first for the ELS, we get

d p

p

H

DS 2ELS barrels4,873.430190

190210

200500102.$

$,

b. The total annual cost with the ELS is

S Q

D H

p

d p Q C

2

20048734

50010210190

301902

48734$

.,,

.$.,

828619143091430 .$.$.$


9/42

Finding the Economic ProductionLot Sizec. Applying the TBO formula to the ELS, we get

days/year 350ELSTBO ELS D

days162 or 162.4

d. The production time during each cycle is the lot size dividedby the production rate:

p

ELS

35050010

48734,

.,

days26 or 25.6190

48734 .,


10/42

Application D.1 A domestic automobile manufacturer schedules 12 two-person teams to assemble 4.6 liter DOHC V-8 engines per work day. Each team can assemble 5 engines per day. Theautomobile final assembly line creates an annual demand for the DOHC engine at 10,080 units per year. The engine andautomobile assembly plants operate 6 days per week, 48weeks per year. The engine assembly line also producesSOHC V-8 engines. The cost to switch the production linefrom one type of engine to the other is $100,000. It costs$2,000 to store one DOHC V-8 for one year.

a. What is the economic lot size?b. How long is the production run?c. What is the average quantity in inventory?d. What is the total annual cost?


11/42

Application D.1SOLUTIONa. Demand per day = d = 10,080/[(48)(6)] = 35

d p

p

H

DS 2ELS 3855513560

600002

000100080102.,

,,,

or 1,555 engines

b. The production run

p

Q days production 26 or 25.91

605551,


12/42

Application D.1

c. Average inventory

d. Total annual cost

engines32460

35602

5551,

S Q

D H

p

d p Q S

Q

D H

I C

22max

0001005551080100002

603560

25551 ,$

,,,$,

1482961231648917647

,,$

,$,$

p

d p Q I

22max


13/42

Quantity Discounts

Price incentives to purchase large quantities createpressure to maintain a large inventory

Items price is no longer fixed If the order quantity is increased enough, then the priceper unit is discounted

A new approach is needed to find the best lot size thatbalances:

Adv antages o f low er p r i ces fo r pu rchased m ate ri al s and fewer orders Disadvan tages o f th e inc reased c os t o f h o ld ing m ore inven to ry


14/42

Quantity Discounts

Total annual cost = Annual holding cost+ Annual ordering or setup cost+ Annual cost of materials

where P = price per unit, and

D = annual demand (consumption)

PDS Q D H QC 2


15/42

Quantity Discounts

Unit holding cost ( H ) is usual ly expressed as apercentage of unit priceThe lower the unit price ( P ) is, the lower the unitholding cost ( H ) is

The total cost equation yields U-shape total costcurves

There are cost curves for each price levelThe feasible total cost begins with the top curve, thendrops down, curve by curve, at the price breaksEOQs do not necessarily produce the best lot size

The EOQ at a particular price level may not be feasibleThe EOQ at a particular price level may be feasible but

may not be the best lot size


16/42

Two-Step Solution Procedure

Step 1. Beginning with lowest price, calculate the EOQfor each price level until a feasible EOQ isfound. It is feasible if it lies in the rangecorresponding to its price. Each subsequentEOQ is smaller than the previous one,because P , and thus H , gets larger andbecause the larger H is in the denominator of the EOQ formula.Step 2. If the first feasible EOQ found is for thelowest price level, this quantity is the bestlot size. Otherwise, calculate the total costfor the first feasible EOQ and for the larger price break quantity at each lower pricelevel. The quantity with the lowest total costis optimal.


17/42

Quantity Discounts

(a) Total cost curves with purchasedmaterials added

(b) EOQs and price break quantities

PD for P = $4.00 PD for

P = $3.50 PD for P = $3.00

EOQ 4.00

EOQ 3.50 EOQ 3.00

T o

t a l c o s

t ( d o

l l a r s

)

Purchase quantity ( Q )0 100 200 300

Firstprice

break

Secondprice

break

T o

t a l c o s

t ( d o

l l a r s

)

Purchase quantity ( Q )0 100 200 300

Firstpricebreak

Secondpricebreak

C for P = $4.00

C for P = $3.50C for P = $3.00

Total Cost Curves with Quantity Discounts


18/42

Find Q with QuantityDiscounts

EXAMPLE D.2 A supplier for St. LeRoy Hospital has introduced quantitydiscounts to encourage larger order quantities of aspecial catheter. The price schedule is

Order Quantity Price per Unit

0 to 299 $60.00

300 to 499 $58.80

500 or more $57.00

The hospital estimates that its annual demand for this item is936 units, its ordering cost is $45.00 per order, and its annualholding cost is 25 percent of the catheters unit price. Whatquantity of this catheter should the hospital order to minimizetotal costs? Suppose the price for quantities between 300 and499 is reduced to $58.00. Should the order quantity change?


19/42


SOLUTIONStep 1: Find the first feasible EOQ, starting with the lowest price

level:

H

DS 2EOQ 0057 . units77005725000459362

.$..$

A 77-unit order actually costs $60.00 per unit, instead of the$57.00 per unit used in the EOQ calculation, so this EOQ isinfeasible. Now try the $58.80 level:

H DS 2EOQ 8058 . units768058250 00459362 .$.

.$

This quantity also is infeasible because a 76-unit order is toosmall to qualify for the $58.80 price. Try the highest price level:


20/42


This quantity is feasible because it lies in the rangecorresponding to its price, P = $60.00

H

DS 2EOQ 0060 . units75006025000459362

.$..$

Step 2: The first feasible EOQ of 75 does not correspond tothe lowest price level. Hence, we must compare itstotal cost with the price break quantities (300 and 500units) at the lower price levels ($58.80 and $57.00):


21/42

Find Q with Quantity Discounts

PD S Q D H Q C

2

284579360060004575

93600602502

7575 ,$.$.$.$.C

38257936805800453009368058250

2300

300 ,$.$.$.$.C

9995693600570045500

93600572502

500500 ,$.$.$.$.C

The best purchase quantity is 500 units, which qualifies for thedeepest discount


22/42

Application D.2

A suppliers price schedule is: Order Quantity Price per Unit

0 99 $50

100 or more $45

If ordering cost is $16 per order, annual holding cost is 20percent of the purchase price, and annual demand is 1,800items, what is the best order quantity?


23/42

Application D.2

SOLUTIONStep 1:

H

DS 2EOQ 0045 . e)(infeasiblunits8020451680012

.,

H DS 2EOQ 0050 . (feasible)units762050

1680012.

,

Step 2:

76C 759908001501676

800120502

76,$,

,.

100C 738818001451610080012045

2100

,$,,

.

The best order quantity is 100 units


24/42

One-Period Decisions

Seasonal goods are a dilemma facing manyretailers.Newsboy problem

Step 1: List different demand levels and

probabilities.Step 2: Develop a payoff table that shows the profit

for each purchase quantity, Q , at eachassumed demand level, D .Each row represents a different order quantity and each column represents adifferent demand.The payoff depends on whether all units aresold at the regular profit margin whichresults in two possible cases.


25/42


If demand is high enough ( Q D), then all of the casesare sold at the full profit margin, p, during the regular season

If the purchase quantity exceeds the eventual demand(Q > D ), only D units are sold at the full profit margin,and the remaining units purchased must be disposedof at a loss, l , after the season

Payoff = (Profit per unit)(Purchase quantity) = pQ

Payoff = (Demand)Lossper unit

Profit per unit sold

duringseason

Amountdisposedof after season

= p X D l X (Q D)


26/42


Step 3: Calculate the expected payoff of each Q byusing the expected value decision rule. For a specific Q, first multiply each payoff by itsdemand probability, and then add theproducts.

Step 4: Choose the order quantity Q with thehighest expected payoff.


27/42

Finding Q for One-PeriodDecisions

EXAMPLE D.3One of many items sold at a museum of natural history is aChristmas ornament carved from wood. The gift shop makes a$10 profit per unit sold during the season, but it takes a $5loss per unit after the season is over. The following discreteprobability distribution for the seasons demand has beenidentified:

Demand 10 20 30 40 50

Demand Probability 0.2 0.3 0.3 0.1 0.1

How many ornaments should the museums buyer order?

Fi di f O P i d


28/42


SOLUTIONEach demand level is a candidate for best order quantity,so the payoff table should have five rows. For the firstrow, where Q = 10, demand is at least as great as thepurchase quantity. Thus, all five payoffs in this row are

This formula can be used in other rows but only for thosequantity demand combinations where all units are soldduring the season. These combinations lie in the upper-right portion of the payoff table, where Q D . For example,the payoff when Q = 40 and D = 50 is

Payoff = pQ = ($10)(10) = $100

Payoff = pQ = ($10)(40) = $400

d f d


29/42


The payoffs in the lower-left portion of the table representquantity demand combinations where some units must bedisposed of after the season ( Q > D). For this case, the payoff must be calculated with the second formula. For example,when Q = 40 and D = 30,

Using OM Explorer, we obtain the payoff table in Figure D.5

Payoff = pD l (Q D ) = ($10)(30) ($5)(40 30) = $250

Now we calculate the expected payoff for each Q by multiplying

the payoff for each demand quantity by the probability of thatdemand and then adding the results. For example, for Q = 30,

Payoff = 0.2($0) + 0.3($150) + 0.3($300) + 0.1($300) + 0.1($300)= $195


30/42

Application D.3

For one item, p = $10 and l = $5. The probability distribution for theseasons demand is:

Demand Demand(D) Probability10 0.220 0.330 0.340 0.150 0.1

Complete the following payoff matrix, as well as the column onthe right showing expected payoff. What is the best choice for Q ?


31/42

Application D.3D

ExpectedPayoff Q 10 20 30 40 5010 $100 $100 $100 $100 $100 $10020 50 200 200 200 200 17030 0 300 30040 50 100 250 400 400 175

50 100 50 200 350 500 140


32/42

Application D.3

Payoff if Q = 30 and D = 20:

pD l (Q D ) = 10(20) 5(30 20) = $150

Payoff if Q = 30 and D = 40:

Expected payoff if Q = 30:

pD = 10(30) = $300

0(0.2) + 150(0.3) + 300(0.3 + 0.1 + 0.1) = $195

Q = 30 has the highest payoff at $195.00

D

ExpectedPayoff Q 10 20 30 40 5010 $100 $100 $100 $100 $100 $10020 50 200 200 200 200 17030 0 300 30040 50 100 250 400 400 175

50 100 50 200 350 500 140

150 300 195


33/42

Solved Problem 1

Peachy Keen, Inc., makes mohair sweaters, blouses withPeter Pan collars, pedal pushers, poodle skirts, and other popular clothing styles of the 1950s. The average demandfor mohair sweaters is 100 per week. Peachys productionfacility has the capacity to sew 400 sweaters per week.

Setup cost is $351. The value of finished goods inventoryis $40 per sweater. The annual per-unit inventory holdingcost is 20 percent of the items value.

a. What is the economic production lot size (ELS)?b. What is the average time between orders (TBO)?c. What is the total of the annual holding cost and setup cost?


34/42

Solved Problem 1

SOLUTIONa. The production lot size that minimizes total cost is

d p

p

H

DS 2ELS100400

40040200

351521002$.

$

sweaters78034300456 ,

b. The average time between orders is

D ELSOTB ELS year 0.152005780,

Converting to weeks, we get

weeks7.8r weeks/yea52year 0.15TBO ELS


35/42

Solved Problem 1

c. The minimum total of setup and holding costs is

S Q

D H

p

d p Q C

2

3517802005402004001004002780 $,$.

r $4,680/year $2,340/year $2,340/yea


36/42

Solved Problem 2

A hospital buys disposable surgical packages from Pfisher, Inc.Pfishers price schedule is $50.25 per package on orders of 1 to 199packages and $49.00 per package on orders of 200 or morepackages. Ordering cost is $64 per order, and annual holding cost is20 percent of the per unit purchase price. Annual demand is 490packages. What is the best purchase quantity?

SOLUTIONWe first calculate the EOQ at the lowest price:

H

DS 2EOQ0049 .

packages8040060049200

00644902,

.$.

.$


37/42

Solved Problem 2

This solution is infeasible because, according to the priceschedule, we cannot purchase 80 packages at a price of $49.00 each. Therefore, we calculate the EOQ at the nextlowest price ($50.25):

H

DS 2EOQ 2550 . packages7924162550200

00644902,.$.

.$

This EOQ is feasible, but $50.25 per package is not the lowestprice. Hence, we have to determine whether total costs can be

reduced by purchasing 200 units and thereby obtaining aquantity discount.


38/42

Solved Problem 2

PD S Q D H Q C

2

4902550006479

49025502002

7979 .$.$.$.C

490004900642004900049200

2200

200 .$.$.$.C

/year $25,416.44$24,622.50ar $396.68/year $396.98/ye

/year $25,146.80$24,010.00ar $156.80/year $980.00/ye

Purchasing 200 units per order will save $269.64/year,compared to buying 79 units at a time.


39/42

Solved Problem 3

Swell Productions is sponsoring an outdoor conclave for owners of collectible and classic Fords. The concession standin the T-Bird area will sell clothing such as T-shirts and officialThunderbird racing jerseys. Jerseys are purchased fromColumbia Products for $40 each and are sold during the eventfor $75 each. If any jerseys are left over, they can be returnedto Columbia for a refund of $30 each. Jersey sales depend onthe weather, attendance, and other variables. The followingtable shows the probability of various sales quantities. Howmany jerseys should Swell Productions order from Columbiafor this one-time event?Sales Quantity Probability Quantity Sales Probability

100 0.05 400 0.34

200 0.11 500 0.11

300 0.34 600 0.05


40/42

Solved Problem 3

SOLUTIONTable next slide is the payoff table that describes this one-period inventory decision. The upper right portion of the tableshows the payoffs when the demand, D , is greater than or equal to the order quantity, Q. The payoff is equal to the per-unit profit (the difference between price and cost) multiplied bythe order quantity. For example, when the order quantity is100 and the demand is 200,

Payoff = ( p c )Q = ($75 - $40)100 = $3,500


41/42

Solved Problem 3

PAYOFFS table

Demand, D Expected

Payoff Q 100 200 300 400 500 600100 $3,500 $3,500 $3,500 $3,500 $3,500 $3,500 $3,500

200 $2,500 $7,000 $7,000 $7,000 $7,000 $7,000 $6,775

300 $1,500 $6,000 $10,500 $10,500 $10,500 $10,500 $9,555

400 $500 $5,000 $9,500 $14,000 $14,000 $14,000 $10,805

500 ($500) $4,000 $8,500 $13,000 $17,500 $17,500 $10,525

600 ($1,500) $3,000 $7,000 $12,000 $16,500 $21,000 $9,750


42/42

Solved Problem 3The lower-left portion of the payoff table shows the payoffswhen the order quantity exceeds the demand. Here thepayoff is the profit from sales, pD , minus the lossassociated with returning overstock, l (Q D), where l is thedifference between the cost and the amount refunded for each jersey returned and Q D is the number of jerseys

returned. For example, when the order quantity is 500 andthe demand is 200,Payoff = pD l (Q D ) = ($75 - $40)200 ($40 $30)(500 200)

= $4,000

The highest expected payoff occurs when 400 jerseys are

ordered:Expected payoff 400 = ($500 0.05) + ($5,000 0.11)

+ ($9,500 0.34) + ($14,000 0.34)+ ($14,000 0.11) + ($14,000 0.05)

= $10,805

Documents

Special Inventory Mgmt Models