MD235, Sections 3 and 4
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There are 4 questions, each with several parts. The points for each part are given in parentheses. Do all
work on the test paper. You must show your work for full credit. That means showing your
calculations on problems requiring calculations and explaining your logic on reasoning questions.
There should be enough room on the front. Use the spaces provided to report your answers.
1. Speedy Bus Lines runs busses from Green Bay a nd Appleton to Chicago, and also from Chicago to
Peoria and Springfield, as shown below:
Green Bay Peoria
Speedy runs one bus on each route (shown by one a rrow) each day. Each bus has a capacity of 50
passengers. Passengers can buy a ticket for any one route, or for a trip from Green Bay or Appleton to
Peoria or Springfield with a stopover in ChicagoFor each possible ticket Speedy offers two types of
ticket, Economy and Premium. The price of each type of ticket, and the projected demand for each type
of ticket, is given in the table below:
Price (after $ sign) and demand (in parentheses) for each type of ticket
From: Chicago Peoria Springfield
Green Bay Economy: $30 (30) Economy: $42 (11) Economy: $40 (14)
Premium: $50 (15) Premium: $58 (9) Premium: $55 (17)
Appleton Economy: $20 (28) Economy: $35 (32) Economy: $38 (16)
Premium: $35 (12) Premium: $44 (12) Premium: $50 (12)
Chicago Economy: $32 (24) Economy: $29 (23)
Premium: $56 (20) Premium: $51 (19)
Page 1 of 12 Speedy has asked you to create a linear program to determine how many tickets of each type to offer so
as to maximize its total revenue. You have defined your decision variables as follows:
GCE = the number of Economy tickets to offer for a Green Bay-Chicago trip.
GCP = the number of Premium tickets to offer for a Green Bay-Chicago trip.
GPE = the number of Economy tickets to offer for a Green Bay-Peoria trip.
GPP = the number of Premium tickets to offer for a Green Bay-Peoria trip.
. . . and so on.
1a. (2 points) What is the total number of decision variables needed for your linear program?Show your
calculations or reasoning.
16. The decision variables are the number of tickets of each type to offer. You can find them by counting up the
total ticket types from the table.
1b. (3 points) Write the beginning of the objective function for the linear program, including the first five terms
Max Z = 30GCE + 50GCP + 42GPE + 58GPP + 40GSE + etc.
1c. (5 points) Write the constraint that applies to the capacity of the route from Chicago to Peoria.
GPE + GPP + APE + APP + CPE + CPP <= 50
(If you only add up GPE and GPP, you are leaving out tickets with connections that also go through the Green
1d. (3 points) Write the constraint that applies to the number of Premium class tickets from Appleton to Chicago.
ACP <= 12
Page 2 of 12 1e. (3 points) Write what the constraint requested in problem 1d would be if the fare for a Premium ticket from
Appleton to Chicago were increased by $5.
ACP <= 12
1f. (3 points) Write what the constraint requested in problem 1d would be if the demand for Premium tickets
from Appleton to Chicago increased by 5.
ACP <= 17
Page 3 of 12 2. Plevvy Investments is constructing a portfolio for a client with 20 million dollars to invest. The
investment classes it may use, their projected annual rates of return, and their risk levels, are as follows:
Investment class Annual rate of return (%) Risk level (on a scale of 0-10)
Grade A Bonds 2 2
Grade B Bonds 3.5 3
Blue Chip Stocks 6 5
Growth Industry Stocks 9 6.4
Early Stage Private Equity 17 9.2
All the money must be invested. The client wants to maximize annual return on the portfolio while
limiting the average risk to no more than 5.5. At the same time, no one class of investment can be more
than one-quarter of the total portfolio. Plevvy intends to decide the amount to be invested in each
investment class with a linear program.
2a. (4 points) In the table below, enter a symbol and a definition for each decision variable required for
the linear program. There may be more spaces in the table than needed.
X1 The amount to invest in Grade A bonds
X2 The amount to invest in Grade B bonds
X3 The amount to invest in blue chip stocks
X4 The amount to invest in growth industry stocks
X5 The amount to invest in early stage private equity
2b. (3 points) Write the correct objective function in the space below.
Max Z = .02X1 + .035X2 + .06X3 + .09X4 + .17X5
2c. (9 points) Write all the constraints in the table below, with no more than one constraint on each line.
There may be more spaces than needed.
X1 + X2 + X3 + X4 + X5 = 20
0.1X1 + 0.15X2 + 0.25X3 + 0.32X4 + 0.276X5 <= 5.5 (Alternatives: subtract RHS terms from
both sides, use 20 million instead of sum of Xs, etc.)
X1 <= 5
X2 <= 5
X3 <= 5
X4 <= 5
X5 <= 5
X1, X2, X3, X4, X5 ≥ 0
Page 4 of 12 2d. (5 points) The client has informed Plevvy that the amount invested in a lower-risk investment class
must be at least as much as the amount invested in any higher-risk investment class. In other words, the
amount invested in the lowest-risk investment class must be as much or more than for any other
investment class, the amount invested in the second lowest-risk investment class must be more than for
any other investment class except the lowest-risk investment class, etc. In the table below, write the
smallest set of additional constraints to the model that will produce this result. There may be more
spaces than necessary. Show your calculations or reasoning in the blank space below the table.
X1 >= X2
X2 >= X3
X3 >= X4
X4 >= X5
X1 must be >= X2, X3, X4, and X5 to make sure you don’t invest more in the lower-risk investments.
X2 must be > X3, X4, and X5.
But if X1 is >=X2 and X2 >= X3, then of course X1 >= X3.
That goes for all the other pairs, so you only need these four constraints.
Page 5 of 12 2e. (8 points) The client has changed his mind, and now wants to minimize the average risk of the
portfolio while getting an average annual return of at least 7%. In the table below in the spaces indicated,
write the new objective function (if any), the new constraints (if any), and any old constraints that must be
eliminated (if any) to change the linear program to do what the client wants. There may be more spaces in
the table than needed. Show your work or reasoning in the blank space below the table.
Write new objective function (if any) in the space below:
Min Z = 0.1X1 + 0.15X2 + 0.25X3 + 0.32X4 + 0.276X5
Write new constraints (if any) in the spaces below:
.02X1 + .035X2 + .06X3 + .09X4 + .17X5 >= 0.07 (X1 + X2 + X3 + X4 + X5) (This can
be written several ways)
Write any constraints that need to be eliminated (if any) in the spaces below:
0.1X1 + 0.15X2 + 0.25X3 + 0.32X4 + 0.276X5 <= 5.5
Since we’re now minimizing average risk it’s now the new objective function and the old constraint specifying a maximum risk
The average risk is:
(2X1 + 3X2 + 5X3 + 6.4X4 + 9.2X5)/(X1+X2+X3+X4+X5)
But since the o.f. must be linear and we have specified that X1+X2+X3+X4+X5=20 we can write:
2X1 + 3X2 + 5X3 + 6.4X4 + 9.2X5/20
Or 0.1X1 + 0.15X2 + 0.25X3 + 0.32X4 + 0.46X5
Since we have a minimum on average return, it is now a new constraint.
Page 6 of 12 3. Phelps Petroleum blends four different ingredients (A, B, C, and D) to produce three grades of
gasoline (Regular, Premium, and Super). It has orders currently for some of each grade. It has a limited
amount of each ingredient available to make them. Each ingredient contains some octane and some
oxides. Each grade of gasoline must contain at least a certain amount of octane and must not exceed a
certain amount of oxide. The data for all of these requirements are as follows:
Ingredient Cost ($/gallon) Gallons available Octane content Oxide content
A 5 42 87 20
B 9 49 98 27
C 6 53 88 18
D 3.5 35 84 16
Gasoline grade Min. required Octane Max. required Oxides Current orders (gal.)
Regular 87 27 30
Premium 89 25 40
Super 93 23 30
Phelps decided how to produce its products with a linear program. The results of the sensitivity analysis
are below on this page and the next:
Final Reduced Objective Allowable Allowable
Cell Name Value Cost Coefficient Increase Decrease
$E$5 A Regular 0.00 9.44E‐16 5 1.00E+30 0.00
$F$5 A Premium 30.11 0.00 5 0.01 0.32
$G$5 A Super 0.00 0.91 5 1.00E+30 0.91
$E$6 B Regular 6.43 0.00 9 0.02 5.09
$F$6 B Premium 7.83 0.00 9 1.50 0.00
$G$6 B S