Solutions to Exam #1
John H. Vande Vate
Fall, 2002
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Scores
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10
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0
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< 40
< 50
<60
<70
<80
<90
<100
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Stats
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Range: 37-89
Mean: 61
Std. Dev: 11
Curve: 11
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Concerns?
• Submit concerns about grading in writing
• Identify what you think is mis-graded and
why.
• Will return regraded exams by the end of
the semester.
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Question #1
• Our plant makes two products and ships them to customers
across the country. Demand for the products is relatively
constant at 10 units of each product each day at each
customer. We have 1,000 customers across the country.
• Product
Selling Price
Weight
• Product 1:
$100
99 lbs
• Product 2:
$100
1 lbs
The capacity of our trucks is 45,000 lbs. The average
distance to customers is 1,000 miles and we pay our carrier
$1.50/mile (one way, regardless of the load) plus an
additional $0.10/lb (regardless of the distance). Our
inventory carrying cost is 15%/year. Treat a year as 250
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days.
Question 1.A
What is the optimal frequency for direct
shipments to our customers?
First, we can ignore the $0.10/lb as the total amount
we pay under this heading will be the same
regardless of our shipping frequency.
Short Answer: Every unit incurs $0.10
Long Answer: $0.10*Q is how much we pay per
shipment
There will be D/Q shipments per year
We will pay $0.10D per year regardless of the
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frequency
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Question 1.A
• Use the version of the EOQ model for several
destinations:
• Q* = sqrt(2AD/hC)sqrt(P/(D+P))
• A is the cost per trip = $1,500
• D is the annual demand for a single customer or
2,500 packages consisting of 1 unit of product 1
and one unit of product 2
• h is 15% per year
• C is the value of a package or $200
• P is the production rate which we take to be 7
1,000D
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Question 1.A
• So, Q* =
sqrt(2*1500D/hC)sqrt(1,000D/1,001D) ≈
sqrt(3,000D/30) = sqrt(100D) = 10sqrt(2,500)
= 500.
• Total weight of 500 packages is 50,000 lbs
which exceeds the capacity of the truck.
• So, we should ship in full truckloads of 450
units each. That means the answer is to
ship every 45 working days.
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Question 1.B
You calculated the optimal frequency f* in Part A
using the average distance to customers. Some of
our customers are closer than average and others
are further away. We have decided that to keep our
operations simple, we will ship to each customer
either
i.
The optimal frequency f*
ii.
Twice the optimal frequency 2f*:
iii.
Half the optimal frequency f*/2
Which customers should be served with each
frequency? Be specific and be sure to show how
you obtained your answer.
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Question 1.B
Convention: If the optimal frequency is f* =
once every 45 days then 2f* is twice every
45 days and f*/2 is once every 90 days.
Since shipping every 45 days requires full
truckload shipments, reducing the shipping
frequency will never be advantageous
regardless of where the customer is unless
we change mode.
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Question 1.B
The decision of whether to ship at the frequency f* or
at the frequency 2f* will depend on the distance to
the customer. We can find the breakpoint by setting
the two cost formulas equal and finding the
appropriate distance:
AD/450 + 450hC/2 = AD/225 + 225hC/2
AD+ 450*225hC = 2AD + 225*225hC
225*225hC = AD
2252hC/D = A = $1.5*breakpoint distance
So, customers closer than about 400 miles we can
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ship to twice as frequently.
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Question 2
Our carrier’s trucks average about 50 miles per
hour and, because of Federal Transportation
Regulations, about 10 hours per 24 hour day.
Calculate the pipeline inventory costs inherent in
the distribution system you designed in answer
to Question 1, Part A.
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Question 2.A
It takes two days to reach the average customer.
So, every item we make is on the road for an
average of two days. Consequently, there is
approximately two days production in the
pipeline. That’s 2days * 10 packages per
day/customer * 1000 customers = 20,000
packages each worth $200. So the value of the
pipeline inventory is $4,000,000 and the inventory
costs associated with that are 15% of $4 million =
$600,000/year
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Question 2.B
Factor the pipeline inventory costs into your
model for calculating the optimal frequency.
How does this change your answer from
Question 1, Part A, Part B?
It doesn’t. Goods still take just as long to get
there. If we increase the shipping frequency, there
will just be more smaller shipments, but they will
still add up to the same amount of goods on the
road at any given time – 2 days worth of
production
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Question 3
1 We want to develop a flows-on-paths formulation of a
multi-modal logistics problem. You have been asked to
provide examples to help the engineers in building the
model. You have the following list of variables and are
asked to use them in formulating each of the following
constraints.
The variable Flow[origin, Path, destination] is the annual
quantity of goods shipped from origin to destination via the
path Path, where Path is a set of Edges from the origin to
the destination via a number of intermediate stops. Each
edge also indicates the mode used. So, for example, the
edge (Detroit, Kansas City, Unit Train) indicates a Unit
Train move from Detroit to Kansas City with no
intermediate stops.
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Question 3
The paths from Detroit to Los Angeles are:
Path1 := {(Detroit, Kansas City, Unit Train),
(Kansas City, Denver, Loose Car),
(Denver, Los Angeles, Loose Car)}
Path2 := {(Detroit, Los Angeles, Unit Train)};
Path3 := {(Detroit, Los Angeles, Loose Car)};
Path4 := {(Detroit, Kansas City, Loose Car),
(Kansas City, Los Angeles, Loose Car)};
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Question 3
The paths from Detroit to Seattle are:
Path5 := {(Detroit, Kansas City, Unit Train),
(Kansas City, Denver, Loose Car),
(Denver, Seattle, Loose Car)}
Path6 := {(Detroit, Seattle, Unit Train)};
Path7 := {(Detroit, Seattle, Loose Car)};
Path8 := {(Detroit, Kansas City, Loose Car),
(Kansas City, Seattle, Loose Car)};
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Project Topics
Provide a linear description in terms of the Flow
variables of:
The total quantity of goods shipped from Detroit
annually:
Sum{path in paths that start at Detroit, dest in
destinations} Flow[Detroit, path, dest];
Flow[Detroit, Path1, Los Angeles] + Flow[Detroit, Path2, Los Angeles] +
Flow[Detroit, Path3, Los Angeles] + Flow[Detroit, Path4, Los Angeles] +
Flow[Detroit, Path5, Seattle] + Flow[Detroit, Path6, Seattle] +
Flow[Detroit, Path7, Seattle] + Flow[Detroit, Path8, Seattle]
B
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AMPL Version
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Set CITIES;
Set ORIGINS;
Set DESTINS;
Set PATHS;
Set MODES;
Set EDGES within CITIES cross CITIES cross
MODES;
• Set PATHEDGES{PATHS} within EDGES;
• Param PathOrigin{PATHS};
• Param PathDestin{PATHS};
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AMPL Version
• Var Flow{orig in ORIGINS, path in
PATHS, dest in DESTINS:
PathOrigin[path] = orig and
PathDestin[path] = dest} >= 0;
• Sum{path in PATHS, dest in DESTINS:
PathOrigin[path] = ‘Detroit’}
Flow[‘Detroit’, path, destin]
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Question 3
The total quantity of goods loaded onto Unit Trains
in Detroit annually
Sum{path in the set of paths that start in Detroit with
unit train shipments, dest in destinations}
Flow[Detroit, path, dest];
The total quantity of goods passing through
Kansas City annually
Sum{orig in Origins, path in paths that include
Kansas City, dest in Destinations} Flow[orig, path,
dest};
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More Options
The total quantity of goods that passed
through Denver arriving in Seattle annually
Sum{orig in Origins, path in paths that end in
Seattle and include Kansas City}Flow[orig,
path, Seattle]
The total quantity of goods that pass through
both Kansas City and Denver each year.
Sum{orig in Origins, path in paths that include
both Kansas City and Denver, dest in
Destinations} Flow[orig, path, dest];
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Question 4
• In class we discussed two models for the
inventory at a production plant serving many
identical customers with staggered
shipments via a single loading dock.
Model 1: In this analysis, we calculated the
inventory impact of each customer and
showed that, when the number of customers
is large (say 100), any single customer’s
contribution to inventory at the plant is small
– much smaller than Q/2, where Q is the
quantity delivered in each shipment.
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Question 4
Model 2: In this analysis, we argued that,
whatever the contribution of a single customer,
the operations at a production facility serving
several customers with shipments of size Q
will be indistinguishable from those of a facility
serving a single customer with shipments of
shipments of size Q: The inventory at the plant
will grow from 0 to Q, at which point the
vehicle will depart and the inventory will return
to 0. This process repeats and so the average
inventory at the plant should be Q/2 in both
cases.
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Question 4
What will the average inventory at the
plant be? Assume the plant serves 100
identical customers each with annual
demand D. It sends each customer
shipments of size Q. To answer this
question completely, you must
The two models are identical!
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Resolution
Inventory per shipment: Q*Q/2P
Shipments per year: D/Q to each customer
Total Inventory contribution per customer: QD/2P.
With n identical customers P = nD
So each customer contributes Q/2n in annual
inventory at the plant.
But there are n customers, so average annual
inventory at the plant is … Q/2.
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