Question

. A manufacturing company produces diesel engines in four factories located in Tucson, Seattle, Baltimore, and Detroit. Three trucking firms purchase these engines for their plants located in Nashville, Miami, and Charleston. The supplies and demands, along with the per engine transportation costs in dollars are given below:

Plant

Nashville Miami Charleston Supply

__________________________________________________________________

Tucson 800 1100 400 35

Factory Seattle 550 950 600 35

Baltimore 450 715 372 20

Detroit 520 805 414 35

__________________________________________________________________

Demand 50 30 35

(a) Formulate a linear programming problem to minimize total cost for this transportation problem by defining the decision variables, objective function, and all the constraints.

(b) Solve the linear programming formulation from part (a) by using either Excel or QM for Windows. Find and interpret the optimal solution and optimal value. Please also include the computer output with your submission.

The following questions are mathematical modeling questions.
Please answer by defining the decision variables, objective
function, and all the constraints. Write all details of the
formulation. Please do **NOT** solve the problems
after formulating.

2. A carpet company sells and installs floor covering for commercial buildings. The company was awarded a contract for four jobs. The company needs to assign an installation crew to each of the four jobs. Currently, four installation crews are available for assignment. Each crew is identified by a color code. The following table shows the time required (in hours) for each crew to complete each of the five jobs:

Time Required in Hours

Job 1 Job 2 Job 3 Job 4

_____________________________________________

Red 34 32 30 48

White 38 34 36 47

Crew Blue 28 42 35 37

Green 28 35 35 40

The company wants to assign Red Crew or Blue Crew to Job 2 because of customer preference. It does not want to assign Green Crew to Job 4 because of quality considerations.

The objective is to minimize the total cost of all assignments.

Formulate a linear programming model for the above situation by determining

(a) The decision variables

(b) Determine the objective function. What does it represent?

(c) Determine all the constraints. Briefly describe what each constraint represents.

**Note: Do NOT solve the problem after
formulating.**

3. Rachel wants to set up a trust fund for her two children using $1,600,000. The trust fund has three investment options: a bond fund, a stock fund, and treasury bills fund. The projected returns over the life of the investments are 4.2% for the bond fund, 6.2% for the stock fund, and 4.5% for the treasury bills fund. She wants to invest at least 25% of the total amount in the bond fund, at least 30% in the stock fund, and at least 20% in the treasury bills fund. She also wants the amount invested in the stock fund to be more than or equal to the amount invested in the bond fund. She wants to know how much money should be invested in each of the three alternatives to maximize the total projected returns.

Formulate a linear programming model for the above situation by determining

(a) The decision variables

(b) Determine the objective function. What does it represent?

(c) Determine all the constraints. Briefly describe what each constraint represents.

**Note: Do NOT solve the problem after
formulating.**

4. An ad campaign for a trip to Norway will be conducted in a limited geographical area and can use TV ads, radio ads, magazines ads, and newspaper ads. Information about each medium is shown below.

Medium |
Cost Per Ad |
Number of People Reached |

TV |
12,500 |
42000 |

Radio |
2000 |
8300 |

Magazine |
950 |
3400 |

Newspaper |
2000 |
8500 |

The number of TV ads cannot be more than 3. The total number of Radio and Newspaper ads must be more than the total number of TV ads. There must be at least a total of 10 ads in all four media. The advertising budget is $50,000. The objective is to maximize the total number of people reached.

Formulate a linear programming model for the above situation by determining

(a) The decision variables

(b) Determine the objective function. What does it represent?

(c) Determine all the constraints. Briefly describe what each constraint represents.

**Note: Do NOT solve the problem after
formulating.**

Answer #1

**Solution:**

Decision variables: Xij = Quantity to transport from each of the factories indicated by i to each of the plants indicated by j. Where i=1,2,3,4 for each of the factories located in Tucson, Seattle, Baltimore, Detroit and j=1,2,3 for each of the plants located in Nashville, Miami and Charleston

Objective: Minimize Z = 900X11 + 1200X12 + 500X13 + 650X21 + 1050X22 + 700X23 + 550X31 + 815X32 + 472X33 + 620X41 + 905X42 + 514X43

s.t.

X11 + X12 + X13 <= 80

X21 + X22 + X23 <= 100

X31 + X32 + X33 <= 40

X41 + X42 + X43 <= 100

X11 + X21 + X31 + X41 = 160

X12 + X22 + X32 + X42 = 60

X13 + X23 + X33 + X43 = 100

Xij >= 0

An ad campaign for a trip to Norway will be conducted in a
limited geographical area and can use TV ads, radio ads, magazines
ads, and newspaper ads. Information about each medium is shown
below.
Medium
Cost Per Ad
Number of People Reached
TV
12,500
42000
Radio
2000
8300
Magazine
950
3400
Newspaper
2000
8500
The number of TV ads cannot be more than 3. The total number of
Radio and Newspaper ads must be more than the total...

Let us move to linear programming
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Each 1000 dollars spent on radio ads, results in 70 new customers.
Similarly, each 1000 dollars spent on TV ads yield 105 new
customers. Each radio ad costs $250, and each TV ad costs $1500.
The combined number of ads used in the campaign cannot exceed 75.
The amount spent on radio ads, and the amount spent on TV ads
cannot differ from one another...

The manager of a department store in Seattle is attempting to
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television station, and newspaper to make presentations in which
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The television station representative indicates that a TV
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Constraints affecting the production quantities are the production
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Professional Printers, Inc.make two models of commercial
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for each printer is given below.
SHOW ALL WORK
FlowPrint Model
TypeIt Model
Hours Available
Plastic
2
4
38
Ink Assembly
2
5
44
Molding Time
5
3
34
â€‹The profit for the FlowPrint model is $1000 per printer and the
profit for the TypeIt model is $2000 per printer.
Formulate the linear programming problem only. Be sure to
define decision variables, provide objective function, and...

A brokerage firm has just been instructed by one of its clients
to invest $600,000 of her money. The analysts at the brokerage firm
are considering the following options for investment:
Projected
Rate
Investment
Option
of Return (%)
Municipal bonds
2.4
Company A
stocks
8.0
Company B
stocks
10.2
Company C stocks
9.5
The client has specified the following guidelines:
- Municipal bonds should constitute at least 30% of the money
invested.
- At least 50% of...

A company manufactures two products: large fans and medium fans.
Each large fan requires 3 hours of wiring and 2 hours of drilling.
Each medium fan requires 2 hours of wiring and 1 hour of drilling.
There are 480 hours of wiring time available and 280 hours of
drilling time available. Each large fan yields a profit of $25.
Each medium fan yields a profit of $18. The company wants to
manufacture at least 20 large fans. The objective is...

A company manufactures two products: large fans and medium fans.
Each large fan requires 3 hours of wiring and 2 hours of drilling.
Each medium fan requires 2 hours of wiring and 1 hour of drilling.
There are 480 hours of wiring time available and 280 hours of
drilling time available. Each large fan yields a profit of $25.
Each medium fan yields a profit of $18. The company wants to
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Stockco is considering four investments. Investment 1 will yield
a net present value (NPV) of $16,000; investment 2, an NPV of
$22,000; investment 3, an NPV of $12,000; and investment 4, an NPV
of $8,000. Each investment requires a certain cash outflow at the
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programming model clearly defining the decision variables,
objective function and constraints...

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