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06-Jul-10
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Special Inventory ModelsSpecial Inventory ModelsD
D – 1Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
For For Operations Management, 9eOperations Management, 9e by by Krajewski/Ritzman/Malhotra Krajewski/Ritzman/Malhotra © 2010 Pearson Education© 2010 Pearson Education
PowerPoint Slides PowerPoint Slides by Jeff Heylby Jeff Heyl
Noninstantaneous Replenishment
Maximum cycle inventory
Item used or sold as it is completed Item used or sold as it is completed
Usually production rate, p, exceeds the demand rate, d, so there is a buildup of (p – d) units per time period
Both p and d expressed in same time interval
Buildup continues for Q/p days
D – 2Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
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Production quantityQ
Demand duringry
Noninstantaneous ReplenishmentNoninstantaneous Replenishment
Maximum inventoryImax
p – d
Demand during production interval
On
-han
d in
ven
tor
D – 3Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Production and demand
Demand only
TBO
Time
Figure D.1 – Lot Sizing with Noninstantaneous Replenishment
Noninstantaneous Replenishment
Maximum cycle inventory is:
dpQ
where
p = production rate
d = demand rate
pdp
QdppQ
Imax
D – 4Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Cycle inventory is no longer Q/2, it is Imax /2
Q = lot size
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Noninstantaneous ReplenishmentNoninstantaneous Replenishment
Total annual cost = Annual holding cost + Annual ordering or setup cost
D is annual demand and Q is lot size
d is daily demand; p is daily production rate
g p
SQD
Hp
dpQS
QD
HI
C
22max
D – 5Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
QpQ 22
Noninstantaneous ReplenishmentNoninstantaneous Replenishment
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 the EOQ
pDSELS
2
D – 6Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
dpHELS
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Finding the Economic Production Finding the Economic Production Lot SizeLot Size
EXAMPLE D.1
A plant manager of a chemical plant must determine the lot size for a particular chemical that has a steady demand of 30 barrelsfor 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 $0.21 per barrel, and the plant operates 350 days per year.
a. Determine the economic production lot size (ELS)
b. Determine the total annual setup and inventory holding cost for this item
D – 7Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
c. Determine the time between orders (TBO), or cycle length, for the ELS
d. Determine the production time per lot
What are the advantages of reducing the setup time by 10 percent?
Finding the Economic Production Finding the Economic Production Lot SizeLot Size
SOLUTION
a. Solving first for the ELS, we get
dpp
HDS
2ELS
barrels 4,873.4
30190190
210200500102
.$
$,
b. The total annual cost with the ELS is
SQD
Hp
dpQC
2
D – 8Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Qp 2
20048734
50010210
19030190
248734
$.,
,.$
.,
828619143091430 .$.$.$
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Finding the Economic Production Finding the Economic Production Lot SizeLot Size
c. Applying the TBO formula to the ELS, we get
days/year 350ELS
TBOELS D
days162 or 162.4
d. The production time during each cycle is the lot size divided by the production rate:
35050010
48734,
.,
D – 9Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
p
ELSdays 26 or 25.6
19048734
.,
Finding the Economic Production Finding the Economic Production Lot SizeLot Size
D – 10Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Figure D.2 – OM Explorer Solver for the Economic Production Lot Size Showing the Effect of a 10 Percent Reduction in Setup Cost
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Application D.1Application 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. The automobile fi l bl li t l d d f th DOHCfinal assembly line creates an annual demand for the DOHC engine at 10,080 units per year. The engine and automobile assembly plants operate 6 days per week, 48 weeks per year. The engine assembly line also produces SOHC V-8 engines. The cost to switch the production line from one type of engine to the other is $100,000. It costs $2,000 to store one DOHC V-8 for one year.
Wh t i th i l t i ?
D – 11Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
a. What is the economic lot size?
b. How long is the production run?
c. What is the average quantity in inventory?
d. What is the total annual cost?
Application D.1Application D.1
SOLUTION
a. Demand per day = d = 10,080/[(48)(6)] = 35
dpp
HDS
2ELS
385551
356060
0002000100080102
.,,
,,
or 1,555 engines
b. The production run
D – 12Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
b. The production run
pQ
days production 26 or 25.91605551
,
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Application D.1Application D.1
c. Average inventory
engines32435605551
,
dpQImax
d. Total annual cost
engines324602
,
SQ
DH
p
dpQS
Q
DH
IC
22max
p
pQ22
max
D – 13Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
000100555108010
000260
356025551
,$,,
,$,
1482961
231648917647
,,$
,$,$
Quantity Discounts Quantity Discounts
Price incentives to purchase large quantities create pressure to maintain a l i tlarge inventory
Item’s price is no longer fixed If the order quantity is increased enough, then
the price per unit is discounted
A new approach is needed to find the best lot size that balances:
D – 14Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
size that balances: Advantages of lower prices for purchased materials
and fewer orders
Disadvantages of the increased cost of holding more inventory
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Quantity Discounts Quantity Discounts
Total annual cost = Annual holding cost + Annual ordering or setup cost
where P = price per unit
g p+ Annual cost of materials
PDSQD
HQ
C 2
D – 15Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Quantity Discounts Quantity Discounts
Unit holding cost (H) is usually expressed as a percentage of unit price
Th l th it i (P) i th l th it The lower the unit price (P) is, the lower the unit holding cost (H) is
The total cost equation yields U-shape total cost curves There are cost curves for each price level
The feasible total cost begins with the top curve, then drops down curve by curve at the price breaks
D – 16Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
drops down, curve by curve, at the price breaks
EOQs do not necessarily produce the best lot size
The EOQ at a particular price level may not be feasible
The EOQ at a particular price level may be feasible but may not be the best lot size
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TwoTwo--Step Solution ProcedureStep Solution Procedure
Step 1. Beginning with lowest price, calculate the EOQ for each price level until a feasible EOQ is found. It is feasible if it lies in the range gcorresponding to its price. Each subsequent EOQ is smaller than the previous one, because P, and thus H, gets larger and because the larger H is in the denominator of the EOQ formula.
Step 2. If the first feasible EOQ found is for the lowest price level this quantity is the best lot
D – 17Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
lowest price level, this quantity is the best lot size. Otherwise, calculate the total cost for the first feasible EOQ and for the larger price break quantity at each lower price level. The quantity with the lowest total cost is optimal.
Quantity Discounts Quantity Discounts
EOQ 4.00
EOQ 3.50
EOQ 3.00
C for P = $4.00
C for P = $3.50
C for P = $3.00
PD forP = $4.00 PD for
P = $3.50 PD forP = $3.00
Tota
l co
st (
do
llar
s)
First price
Second price
Tota
l co
st (
do
llar
s)
First price
Second price
D – 18Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
(a) Total cost curves with purchased materials added
(b) EOQs and price break quantities
Purchase quantity (Q)0 100 200 300
price break
price break
Purchase quantity (Q)0 100 200 300
price break
price break
Figure D.3 – Total Cost Curves with Quantity Discounts
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Find Find QQ with Quantity Discountswith Quantity Discounts
EXAMPLE D.2
A supplier for St. LeRoy Hospital has introduced quantity discounts to encourage larger order quantities of a specialdiscounts to encourage larger order quantities of a special 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
D – 19Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
The hospital estimates that its annual demand for this item is 936 units, its ordering cost is $45.00 per order, and its annual holding cost is 25 percent of the catheter’s unit price. What quantity of this catheter should the hospital order to minimize total costs? Suppose the price for quantities between 300 and 499 is reduced to $58.00. Should the order quantity change?
Find Find QQ with Quantity Discountswith Quantity Discounts
SOLUTION
Step 1: Find the first feasible EOQ, starting with the lowest price level:level:
HDS2
EOQ 0057.
units 77
005725000459362
.$..$
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 is infeasible. Now try the $58.80 level:
D – 20Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
HDS2
EOQ 8058.
units 76
805825000459362
.$..$
This quantity also is infeasible because a 76-unit order is too small to qualify for the $58.80 price. Try the highest price level:
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Find Find QQ with Quantity Discountswith Quantity Discounts
HDS2
EOQ 0060.
units 75
006025000459362
.$..$
This quantity is feasible because it lies in the range corresponding to its price, P = $60.00
H 006050 .$.
Step 2: The first feasible EOQ of 75 does not correspond to the lowest price level. Hence, we must compare its
D – 21Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
total cost with the price break quantities (300 and 500 units) at the lower price levels ($58.80 and $57.00):
Find Find QQ with Quantity Discountswith Quantity Discounts
PDSQD
HQ
C 2
284579360060004575936
00602502
7575 ,$.$.$.$. C
3825793680580045300936
80582502
300300 ,$.$.$.$. C
9995693600570045936
0057250500
$$$$
D – 22Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
9995693600570045500936
00572502
500500 ,$.$.$.$. C
The best purchase quantity is 500 units, which qualifies for the deepest discount
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Find Find QQ with Quantity Discountswith Quantity Discounts
D – 23Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Figure D.4 – OM Explorer Solver for Quantity Discounts Showing the Best Order Quantity
Application D.2Application D.2
A supplier’s price schedule is:
Order Quantity Price per UnitOrder Quantity Price per Unit
0–99 $50
100 or more $45
If ordering cost is $16 per order, annual holding cost is 20 percent of the purchase price, and annual demand is 1,800 items, what is the best order quantity?
D – 24Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
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Application D.2Application D.2
SOLUTION
Step 1:
2 1680012
HDS2
EOQ 0045.
e)(infeasibl units 80
20451680012
.
,
HDS2
EOQ 0050.
(feasible) units 76
20501680012
.
,
Step 2:
C 75990800150168001
205076
$,
D – 25Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
76C 7599080015016768001
20502
76,$,
,.
100C 73881800145161008001
20452
100,$,
,.
The best order quantity is 100 units
OneOne--Period Decisions Period Decisions
Seasonal goods are a dilemma facing many retailers.
N b bl 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 each assumed demand level, D.Each row represents a different order
D – 26Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Each row represents a different order quantity and each column represents a different demand.The payoff depends on whether all units are sold at the regular profit margin which results in two possible cases.
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OneOne--Period Decisions Period Decisions
If demand is high enough (Q ≤ D), then all of the cases are sold at the full profit margin, p, during the regular seasonregular 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 disposed of at a loss, l, after the season
Payoff = (Profit per unit)(Purchase quantity) = pQ
D – 27Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
disposed of at a loss, l, after the season
Payoff = –(Demand)Lossperunit
Profit perunit soldduringseason
Amountdisposedof afterseason
OneOne--Period Decisions Period Decisions
Step 3: Calculate the expected payoff of each Q by using the expected value decision rule. For a g pspecific Q, first multiply each payoff by its demand probability, and then add the products.
Step 4: Choose the order quantity Q with the highest expected payoff.
D – 28Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
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Finding Finding QQ for Onefor One--Period DecisionsPeriod Decisions
EXAMPLE D.3
One of many items sold at a museum of natural history is a Christmas ornament carved from wood. The gift shop makes aChristmas ornament carved from wood. The gift shop makes a $10 profit per unit sold during the season, but it takes a $5 loss per unit after the season is over. The following discrete probability distribution for the season’s demand has been identified:
Demand 10 20 30 40 50
Demand Probability 0.2 0.3 0.3 0.1 0.1
D – 29Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
How many ornaments should the museum’s buyer order?
Finding Finding QQ for Onefor One--Period DecisionsPeriod Decisions
SOLUTION
Each demand level is a candidate for best order quantity, so the payoff table should have five rows. For the first row,the payoff table should have five rows. For the first row, where Q = 10, demand is at least as great as the purchase quantity. Thus, all five payoffs in this row are
This formula can be used in other rows but only for those quantity–demand combinations where all units are sold during th Th bi ti li i th i ht ti
Payoff = pQ = ($10)(10) = $100
D – 30Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
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)(40) = $400
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Finding Finding QQ for Onefor One--Period DecisionsPeriod Decisions
The payoffs in the lower-left portion of the table represent quantity–demand combinations where some units must be disposed of after the season (Q > D). For this case, the payoff
t b l l t d ith th d f l F lmust 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
D – 31Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Now we calculate the expected payoff for each Q by multiplying the payoff for each demand quantity by the probability of that demand 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
Finding Finding QQ for Onefor One--Period DecisionsPeriod Decisions
D – 32Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Figure D.5 – OM Explorer Solver for One-Period Inventory Decisions Showing the Payoff Table
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Finding Finding QQ for Onefor One--Period DecisionsPeriod Decisions
Using OM Explorer, Figure D.6 shows the expected payoffs
D – 33Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Figure D.6 – OM Explorer Solver Showing the Expected Payoffs
Application D.3Application D.3
For one item, p = $10 and l = $5. The probability distribution for the season’s demand is:
Demand Demand
(D) Probability
10 0.2
20 0.3
30 0.3
40 0.1
50 0.1
D – 34Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Complete the following payoff matrix, as well as the column on the right showing expected payoff. (Students complete highlighted cells) What is the best choice for Q?
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Application D.3Application D.3
D Expected PayoffQ 10 20 30 40 50
10 $100 $100 $100 $100 $100 $100
20 50 200 200 200 200 170
30 0 300 300
40 –50 100 250 400 400 175
50 –100 50 200 350 500 140
D – 35Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Application D.3Application D.3
D Expected PayoffQ 10 20 30 40 50
10 $100 $100 $100 $100 $100 $100
Payoff if Q = 30 and D = 20:
pD – l(Q – D) = 10(20) – 5(30 – 20) = $150
20 50 200 200 200 200 170
30 0 300 300
40 –50 100 250 400 400 175
50 –100 50 200 350 500 140
150 300 195
D – 36Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
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
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Solved Problem 1Solved Problem 1
Peachy Keen, Inc., makes mohair sweaters, blouses with Peter Pan collars, pedal pushers, poodle skirts, and other popular clothing styles of the 1950s. The average demand for mohair
t i 100 k P h ’ d ti f ilit h thsweaters is 100 per week. Peachy’s production facility has the capacity to sew 400 sweaters per week. Setup cost is $351. The value of finished goods inventory is $40 per sweater. The annual per-unit inventory holding cost is 20 percent of the item’s value.
a. What is the economic production lot size (ELS)?
b. What is the average time between orders (TBO)?
Wh t i th t t l f th l h ldi t d t t?
D – 37Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
c. What is the total of the annual holding cost and setup cost?
Solved Problem 1Solved Problem 1
SOLUTION
a. The production lot size that minimizes total cost is
dpp
HDS
2ELS
100400
40040200
351521002
$.$
sweaters 78034
300456 ,
b. The average time between orders is
D – 38Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
D
ELSOTB ELS year0.15
2005780
,
Converting to weeks, we get
weeks7.8rweeks/yea52year0.15TBOELS
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Solved Problem 1Solved Problem 1
c. The minimum total of setup and holding costs is
DdpQ
S
QD
Hp
dpQC
2
3517802005
40200400
1004002
780$
,$.
r$4,680/year$2,340/year$2,340/yea
D – 39Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solved Problem 2Solved Problem 2
A hospital buys disposable surgical packages from Pfisher, Inc. Pfisher’s price schedule is $50.25 per package on orders of 1 to 199 packages and $49.00 per package on orders of 200 or more
k O d i t i $64 d d l h ldipackages. Ordering cost is $64 per order, and annual holding cost is 20 percent of the per unit purchase price. Annual demand is 490 packages. What is the best purchase quantity?
SOLUTION
We first calculate the EOQ at the lowest price:
DS2 00644902 $
D – 40Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
HDS2
EOQ 0049.
packages 804006
004920000644902
,.$.
.$
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Solved Problem 2Solved Problem 2
This solution is infeasible because, according to the price schedule, we cannot purchase 80 packages at a price of $49.00 each. Therefore, we calculate the EOQ at the next lowest price ($50 25)($50.25):
HDS2
EOQ 2550.
packages 792416
255020000644902
,.$.
.$
This EOQ is feasible, but $50.25 per package is not the lowest price. Hence, we have to determine whether total costs can be
D – 41Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
price. Hence, we have to determine whether total costs can be reduced by purchasing 200 units and thereby obtaining a quantity discount.
Solved Problem 2Solved Problem 2
PDSQD
HQ
C 2
4902550006479490
25502002
7979 .$.$.$. C
49000490064200490
00492002
200200 .$.$.$. C
/year$25,416.44$24,622.50ar$396.68/year$396.98/ye
D – 42Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
2002
/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.
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Solved Problem 3Solved Problem 3
Swell Productions is sponsoring an outdoor conclave for owners of collectible and classic Fords. The concession stand in the T-Bird area will sell clothing such as T-shirts and official Th d bi d i j J h d fThunderbird racing jerseys. Jerseys are purchased from Columbia Products for $40 each and are sold during the event for $75 each. If any jerseys are left over, they can be returned to Columbia for a refund of $30 each. Jersey sales depend on the weather, attendance, and other variables. The following table shows the probability of various sales quantities. How many jerseys should Swell Productions order from Columbia for this one-time event?
D – 43Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Sales Quantity Probability Quantity Sales Probability
100 0.05 400 0.34
200 0.11 500 0.11
300 0.34 600 0.05
Solved Problem 3Solved Problem 3
SOLUTION
Table D.1 is the payoff table that describes this one-period inventory decision. The upper right portion of the table showsinventory decision. The upper right portion of the table shows 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 by the order quantity. For example, when the order quantity is 100 and the demand is 200,
Payoff = (p – c)Q = ($75 - $40)100 = $3,500
D – 44Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
06-Jul-10
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Solved Problem 3Solved Problem 3
TABLE D 1 | PAYOFFSTABLE D.1 | PAYOFFS
Demand, D Expected PayoffQ 100 200 300 400 500 600
100 $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
D – 45Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
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
Solved Problem 3Solved Problem 3
The lower-left portion of the payoff table shows the payoffs when the order quantity exceeds the demand. Here the payoff is the profit from sales, pD, minus the loss associated with
t i t k l(Q ) h l i th diff b treturning overstock, l(Q – D), where l is the difference 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 and the 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
D – 46Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
The highest expected payoff occurs when 400 jerseys are ordered:
Expected payoff400 = ($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
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D – 47Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.