Suppose SnowValley stocks 30,000 pairs of Beast 98’s.

BUS221 Practice for Exam 2 ? Answers
Spring 2021
Note: The real exam will be about half the length of this practice exam!
Question 1 [35 points]
Snow Valley operates a chain of ski stores. Because of long lead times, it must decide how many skis to stock
in April, well before the start of the ski season in November. We will focus on a particular type of ski, the
Beast 98. Demand is forecasted to be normally distributed with a mean of 23,000 pairs and a standard deviation
6,500 pairs. SnowValley purchases Beast 98’s direct from the manufacturer for all its stores at a price of
$320/pair. Beast 98’s sell for $650/pair during the ski season and leftover pairs are sold at end of season
or summer sales for $215/pair.
(a) Suppose SnowValley stocks 30,000 pairs of Beast 98’s.
Answer Use a Newsvendor (perishable demand) model (see e.g. the model User Specied Order Quantity Q in Excel workbook Perishable Demand Excel Models.xlsx with mean demand = 23,000 pairs,
std. dev. of demand = 6,500 pairs and Q = 30,000 pairs).
i. [5 points] How many pairs can SnowValley expect to sell during the ski season?
Answer E[Sales] = 22,533 pairs.
ii. [5 points] How many pairs should SnowValley expect to be left over to sell at a discount?
Answer E[Inventory] = 7,466.8 pairs.
iii. [5 points] What will be the probability that SnowValley stocks out of Beast 98’s?
Answer PrfStockoutg = 0.14076.
(b) [7 points] Find the cost of underage and the cost of overage for Beast 98 skis.
Answer
Cu = Selling Price ? Unit Cost = $650/pair ? $320/pair = $330/pair;
Co = Unit Cost ? Salvage Value = $320/pair ? $215/pair = $105/pair:
(c) [6 points] What is the optimal order quantity for Beast 98 skis?
Answer Using the model OPTIMAL ORDER QUANTITY Q in Excel workbook Perishable Demand
Excel Models.xlsx with mean demand = 23,000 pairs, std. dev. of demand = 6,500 pairs and Cu and Co
as in part b, we nd Q = 27,562 pairs pairs.
(d) [7 points] Many times, customers who want to buy skis will settle for a dierent type if ShowValley is
out of Beast 98’s. How will that change your answer to part c?
Answer Since SnowValley has a chance to make another sale if it has run out of Beast 98’s to sell, the
cost of underage is less than Cu as calculated in part b. Hence the critical ratio Cu=(Cu + Co) is reduced,
and the optimal order quantity is less than in part c
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Practice for Exam 2 ? Answers Page 2
Question 2 [35 points]
fast furniture is a supplier to IKEA. It uses 90,000 sqft/week of laminated wood in its production process.
FastFurniture purchases laminated wood from a nearby supplier with minimal lead time at a cost of $1.90/sqft
per order. In addition, the supplier charges a xed cost of $500 for every order placed. FastFurniture’s holding
cost is 25%/yr (1 year = 52 weeks).
Answer Use the EOQ (steady demand) model in Excel workbook Steady Demand Model.xlsx with
D = Demand rate = (90,000 sqft/week)  (52 wks/yr) = 4,680,000 sqft/yr;
F = Fixed cost of ordering = $500; and
H = Holding cost rate = ($1.90/sqft)  (25%/yr) = $0.475/sqft/yr:
(a) Suppose FastFurniture uses an order quantity of 200,000 sqft for laminated wood.
Answer Let Q = 200,000 sqft.
i. [5 points] What will be the average on-hand inventory?
Answer Avg Inventory = Q/2 = (200,000 sqft)/2 = 100,000 sqft.
ii. [5 points] How many orders for FastFurniture place per year?
Answer Order Frequency = D=Q = (4,680,000 sqft/yr)/(200,000 sqft) = 23.4/yr.
iii. [5 points] What will the total of ordering and holding cost be per year?
Answer We have
Annual Holding Cost = Avg Inventory  H = (100,000 sqft)  ($0.475/sqft/yr)
= $47,500/yr;
Annual Fixed Ordering Cost = Order Frequency  F = (23.4/yr)  $500
= $11,700/yr;
so
Annual Total Cost = Annual Holding Cost + Annual Fixed Ordering Cost
= $47,500/yr + $11,700/yr = $59,200/yr:
(b) [6 points] What order quantity minimizes the total annual cost of ordering and holding inventory?
Answer
EOQ =
r
2  F  D
H
=
s
2  ($500)  (4,680,000 sqft/yr)
$0.475/sqft/yr
= 99,260 sqft:
(c) [6 points] What will be FastFurniture’s total cost of holding and ordering per year with the order
quantity from part b?
Answer When using the EOQ, the average holding cost per time unit equals the average xed ordering
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Practice for Exam 2 ? Answers Page 3
cost per time unit, so
Annual Fixed Ordering Cost = Annual Holding Cost = 1
2  EOQ  H
= 1
2  (99,260 sqft)  ($0.475/sqft/yr)) = $23,574/yr:
The annual purchasing cost does not depend on the order quantity, so we now have
Annual Total Cost = Annual Holding Cost + Annual Fixed Ordering Cost
= $23,574/yr + $23,574/yr = $47,149/yr:
(d) [8 points] The supplier oers to reduce the xed cost for orders in full truck loads of 40,000 sqft square
feet. At what values for the xed cost would FastFurniture take advantage of this oer?
Answer From the answer to part c we know that annual total cost equals $47,149/yr. When using an
order quantity of QT = 40,000 sqft, the annual holding cost will be
1
2  QT  H = 1
2  (40,000 sqft)  ($0.475/sqft/yr) = $9,500/yr;
and the annual xed ordering cost will be
D
QT
 F =
4,680,000 sqft/yr
40,000 sqft
 F = (117.00 /yr)  F: (1)
So the full truckload order quantity will be cheaper than the EOQ order quantity when
$9,500/yr + (117.00 /yr)  F < $47,149/yr
, F <
$47,149/yr ? $9,500/yr
117.00 /yr
= $321.78:
Question 3 [35 points]
Maria and Luigi Vespucci operate a grocery store in Naples, NY. The store is known for its authentic Italian
cooking supplies. In keeping with the authentic theme, the Vespucci’s purchase olive oil for their store
from a distant cousin in the old country (i.e., in Naples, Italy). Weekly sales of olive oil follow a normal
distribution with a mean of 90 bottles bottles and a standard deviation of 50 bottles bottles. The olive oil
costs $$6.95/bottle per bottle and shipping costs $$0.95/bottle per bottle and takes 5 weeks weeks. The
store orders olive oil once every 4 weeks weeks. Holding cost is 25%/yr
(a) Suppose the store uses an order-up-to level of 800 bottles bottles.
Answer The order-up-to level is S = 800 bottles bottles.
i. [5 points] Assume current on-order inventory is 594 bottles and on-hand inventory is 85 bottles
bottles of olive oil. How many bottles should be ordered?
Answer The inventory position is
IP = On-hand Inventory + On-order Inventory ? Backorders
= (85 bottles) + (594 bottles) ? 0 = 679 bottles;
so the quantity to be ordered is
S ? IP = (800 bottles) ? (679 bottles) = 121 bottles bottles
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ii. [5 points] Assume current on-order inventory is 594 bottles bottles of olive oil and there are 37 bottles
bottles backordered. How many bottles should be ordered?
Answer The inventory position is
IP = On-hand Inventory + On-order Inventory ? Backorders
= 0 + (594 bottles) ? (37 bottles) = 557 bottles;
so the quantity to be ordered is
S ? IP = (800 bottles) ? (557 bottles) = 243 bottles bottles
iii. [5 points] What will be the long run average on-hand inventory?
Answer Use the USER SPECIFIED ORDER-UP-TO LEVEL model on sheet Normal Demand in Excel
workbook Frequent Orders Excel Models.xlsx with
 = Mean demand per week = 90 bottles;
 = Standard dev of demand per week = 50 bottles;
R = Review period = 4 weeks;
L = Lead time = 5 weeks;
S = Order-up-to level = 800 bottles:
From the model we nd Average On-hand Inventory = 54.974 bottles bottles.
iv. [5 points] What will be the average annual cost of holding on-hand inventory?
Answer The average annual cost of holding on-hand inventory can be found as
(Holding cost percentage per year)  (Price per bottle + Shipping per bottle)  (Average Inventory)
= (25%/yr)  ($6.95/bottle + $0.95/bottle)  (54.974 bottles) = $108.57/yr:
(b) [5 points] Suppose the store wants to maintain an in-stock probability of 0.75. What order-up-to level
should it use?
Answer Using the TARGET IN-STOCK PROBABILITY model on the same sheet and using the same
values for , , R and L, and with the Target In-Stock Probability set to 0.75, we nd S = 911.17 bottles.
(c) [10 points] Suppose the store wants to limit the number of stock-outs to 2/yr per year (= 52 weeks).
What in-stock target should it use?
Answer The number of orders placed by the store will be (52 weeks per year) / (4 weeks) = 13/yr. So the
expected number of stockouts is 13/yrPrfStockoutg. The store wants this not to exceed 2/yr, so it should
aim for 1?PrfIn-stockg = PrfStockoutg  (2/yr)=(13/yr), so PrfIn-stockg  1?(2/yr)=(13/yr) = 0.84615.
So the store should use a Target In-stock Probability of 0.84615.
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Practice for Exam 2 ? Answers Page 5
Question 4 [15 points]
Below are 10 plots labeled (A) thru (J). For each of the functions described in the table below, select the
plot that most closely resembles the actual shape of the function.
(A) (B) (C) (D) (E)
x
y
x
y
x
y
x
y
x
y
(F) (G) (H) (I) (J)
x
y
x
y
x
y
x
y
x
y
Function Description Plot
Order frequency as function of order quantity in EOQ model D
Stockout probability as function of order quantity in Newsvendor model J
Expected inventory as function of order quantity in Newsvendor model E
Average on-order inventory as function of lead time in frequent order model G
Optimal order quantity as function of demand rate in EOQ model B
Question 5 [15 points]
(a) [5 points] Brie y describe the dierence between make-to-stock and make-to-order.
Answer No answer provided.
(b) [5 points] Brie y explain the critical ratio.
Answer No answer provided.
(c) [5 points] Brie y describe the dierence between make-to-stock and make-to-order.
Answer No answer provided.
Question 6 [30 points]
Fairport Chocolate Factory (FCF) produces patented Fairport Luxury Easter Eggs (FLEEs) for sale during
the Easter selling season (about six weeks long, ending on Easter Sunday). FLEEs sells for $21.70/lb, and
costs $7.00/lb to produce. The production lead time is well over three months, mostly due to the long lead
time for some of the ingredients that have to come from South America. FLEEs that remain unsold until
after Easter are disposed of at a signicant discount: on average FCF sells them for $3.00/lb. FCF estimates
that demand during the regular pre-Easter selling season is normally distributed with mean 800 lbs and
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Practice for Exam 2 ? Answers Page 6
standard deviation 258 lbs.
Answer This problem is a Newsvendor (Perishable Inventory) problem with  = Mean demand = 800 lbs
and  = Std. dev. of demand = 258 lbs.
(a) [5 points] What is the cost of underage per pound for FLEEs?
Answer For every pound under-produced, FCF loses out on the selling price $21.70/lb but saves $7.00/lb
in acquisition cost. So the cost of underage is
Cu = $21.70/lb ? $7.00/lb = $14.70/lb:
(b) [5 points] What is the cost of overage per pound for FLEEs?
Answer For every pound not sold by Easter, FCF paid $7.00/lb while it will only get back $3.00/lb, so
the cost of overage is
Co = $7.00/lb ? $3.00/lb = $4.00/lb:
(c) Assume that FCF produces 860 lbs before the start of the season.
i. [3 points] What is the probability that there are no FLEEs left unsold by Easter?
Answer
PrfDemand  Qg = 1 ? NORM.DIST(Q; ; ; 1)
= 1 ? NORM.DIST(860 lbs; 800 lbs; 258 lbs; 1) = 0.40805:
ii. [3 points] How many FLEEs are expected to remain unsold by Easter?
Answer Let z = (Q ? )= = (860 lbs ? 800 lbs)=(258 lbs) = 0.23256, then
I(z) = NORM.S.DIST(z; 0) + z  NORM.S.DIST(z; 1) = 0.52596;
Expected Inventory = I(z) = (258 lbs)  I(0.23256) = (258 lbs)  0.52596 = 135.698 lbs:
iii. [3 points] Find the expected lost sales.
Answer
Expected Lost Sales = Mean Demand ? Expected Sales
= Mean Demand ? (Q ? Expected Inventory)
= (800 lbs) ? (860 lbs) + (135.698 lbs) = 75.698 lbs:
iv. [6 points] How much prot can FCF expect to make on FLEEs this year?
Answer
Expected Prot = Cu  Expected Sales ? Co  Expected Inventory
($14.70/lb)  (724.30 lbs) ? ($4.00/lb)  (135.698 lbs) = $10,104:
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(d) [5 points] How many FLEEs should be produced to maximize FCF’s expected prot?
Answer The critical ratio is Cu=(Cu+Co) = ($14.70/lb)=($14.70/lb+$4.00/lb) = 0.78610, so the optimal
order quantity is
Q = NORM.INV(0.78610; ; ) = 1004.58 lbs:
Question 7 [35 points]
Garden Equipment Corporation (GEC) supplies pool and garden equipment to construction companies and
private customers in Southern Florida. One of the products it sells is the Weber Summit S-750 6 burner
propane gas grill. GEC charges its customers $2,539.95 for one of these grills, and buys them from the Weber
Corporation for $1,919.00 each. Demand for these grills is steady over the entire year, and on average, GEC
sells 1.44 grills per week. GEC uses a periodic review order-up-to policy with a review period of 4 weeks
and the lead time is 4 weeks. Inventory holding costs are 23%/year.
Calculate the following quantities for the Weber Summit S-750.
(a) Suppose GEC uses an order-up-to level of 8 grills.
Answer To answer this part, use the Poisson sheet from Excel workbook Frequent Orders Excel Mod-
els.xlsx. For part a, the USER-SPECIFIED ORDER-UP-TO LEVEL (top) part of the sheet should be used.
The expected demand duriing lead time plus review period equals (4 weeks+4 weeks)(1.44 grills/week) =
11.52 grills.
i. [5 points] What is the average on-hand inventory?
Answer Using the spreadsheet model, the average on-hand inventory is IOH = 0.21451 grills.
ii. [5 points] What is the average on-order inventory?
Answer The average on-order inventory is
IOO = (Lead Time)  (Demand Rate)
= (4 weeks)  (1.44 grills/wk) = 5.76000 grills:
iii. [5 points] What is the stockout probability?
Answer
Pr(Stockout) = Pr(DL+R > Order-up-to Level) = 1 ? POISSON.DIST(8 grills,11.52 grills,1) = 0.81094:
iv. [5 points] What is the average annual inventory holding cost for Weber Summit S-750 grills?
Answer The average total inventory is
TI = IOH + IOO = 0.21451 grills + 5.76000 grills = 5.97451 grills:
The average holding cost per year is
AHC = (Holding Cost Percent per Year)  (Average Total Inventory)  (Cost per Unit)
= (23%/year)  (5.97451 grills)  ($1,919.00) = $2,636.97/year:
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(b) Now suppose GEC wants to achieve an in-stock probability of 0.66.
Answer For part b, the TARGET IN-STOCK PROBABILITY (bottom) part of the sheet should be
used. The expected demand duriing lead time plus review period still equals (4 weeks + 4 weeks) 
(1.44 grills/week) = 11.52 grills.
i. [5 points] What is the order-up-to level GEC should use?
Answer Using the spreadsheet model, the order-up-to level to use is 13 grills.
ii. [5 points] Now what is the average on-hand inventory?
Answer And the average on-hand inventory level is IOH = 2.23800 grills.
iii. [5 points] What are the annual inventory turns for Weber Summit S-750 grills. [Assume 1 year =
52 weeks.]
Answer The average total inventory is
TI = IOH + IOO = 2.23800 grills + 5.76000 grills = 7.99800 grills:
So
Turns =
Annual Demand
Average Total Inventory
=
(52 weeks/yr)  (1.44 grills/week)
7.99800 grills
= 9.3623/year:
Question 8 [20 points]
(a) [4 points] An order-up-to policy with a review period of 2 weeks is used to manage item X. Demands
during the previous ve weeks were as follows:
Week 11 12 13 14 15
Demand (in units) 35 31 46 41 33
It is now the start of week 16 and time to order. How many units should be ordered?
Answer The periodic review order-up-to policy always orders the demand since the previous order, which
here is the total demand over the last 2 weeks, i.e., 74 units.
(b) The EOQ policy is used to set the order quantity for item Y. Currently, the optimal order quantity is
1,680 units and the total annual cost of holding inventory and ordering is $36,900/year. Now suppose
demand increases by 27%.
i. [4 points] What is the new optimal order quantity?
Answer
EOQcurrent =
r
2  S  D
H
= 1,680 units;
so
EOQnew =
r
2  S  (D  (1 + 27%)
H
=
r
2  S  D
H

p
1.27 =
= (1,680 units)  1.1269 = 1,893.3 units:
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ii. [4 points] What is the new minimum total annual cost of holding inventory and ordering?
Answer
TCcurrent =
p
2  S  D  H = $36,900/year;
so
TCnew =
p
2  S  (D  (1 + 27%)  H =
p
2  S  D  H 
p
1.27 =
= ($36,900/year)  1.1269 = $41,584/year:
(c) Using a Newsvendor model with Normally distributed demand to choose the order quantity for item Z,
the optimal safety factor is 1.55. Mean demand is 8,660 units and standard deviation 5,237 units.
Answer Let D denote the random demand,  = E[D] = 8,660 units,  = st.dev.(D) = 5,237 units, and
z = 1.55 the optimal safety factor.
i. [4 points] What is the optimal order quantity?
Answer
Q =  + z   = 8,660 units + 1.55  5,237 units = 16,777 units:
ii. [4 points] What is the in-stock probability?
Answer
Pr(In-stock) = Pr(D  Q) = Pr

D ? 


Q ? 


= Pr(Z  z) = NORM.S.DIST(1.55,1) = 0.93943:
Question 9 [20 points]
(a) [5 points] Who will have lower supply-demand mismatch costs, a messenger bag manufacturer selling
through Amazon or Timbuk2. Explain why.
Answer Most likely Timbuk2 will have lower mismatch costs: it assembles to order, so the mismatch costs
at the nished product level are zero, while this is certainly not true for the manufacturer selling through
Amazon.
(b) [5 points] List three reasons for holding inventory.
Answer Reasons for carrying inventory are:
 Flow time: it takes time to transport physical products, and some products require signicant time
in production (e.g., cheese, wine, etc.).
 Seasonality: when demand is seasonal and production capacity is expensive, it is often cheaper
to produce at a constant rate and build up inventory before the saesonal peak arrives, or when
demand is level and raw material availability is seasonal (agricultural products such as fruits and
vegetables), it is often advantageous to process the harvest quickly because the processed product
(e.g. tomato paste) is much cheaper to store than the raw material (e.g., tomatoes) themselves.
 Batching: due to setup times, or due batch processing eciencies (ovens, trucks, containers, etc.).
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 Buers: allows adjacent processing steps to run independently to some degree so that a hickup in
one step does not immediately aect the entire process.
 Demand uncertainty: extra inventory is carried to avoid stockouts which could lead to costly lost
sales or production interruptions.
 Speculative inventory: carried when a price increase is anticipated.
(c) [5 points] If a rm loses sales, it does not have left-over inventory, and vice versa. Then how is it possible
that in the Newsvendor model both expected unsold inventory and expected lost sales are positive?
Answer For some demand realizations, the rm loses sales, while for others, the rm has left-over inven-
tory. Both occur with positive probability, so both the expected lost sales (= E[maxf0;D ? Qg]) and the
expected unsold inventory (= E[maxf0;Q ? Dg])are positive.
(d) [5 points] Explain how Amazon benets from risk pooling.
Answer Amazon uses location pooling: demand from a wide geographic area can be satised from a single
inventory stocking location. Since total demand over a wide region is much easier to predict than demand
in small parts (cities, towns, neighborhoods) of that reason, Amazon needs less inventory to achieve the
same service level or can oer a better service level with the same investment in inventory than smaller
competitors that serve a limited geographic region. Hence Amazon has a signicant advantage over these
smaller competitors.
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