Problem 1 (10 points)
A cellular phone company wants to locate two new
communications towers to cover 4 regions. The company wants to minimize the
cost of installing the two towers. The regions that can be covered by each
tower site are indicated by a 1 in the following table:
Tower Sites
Region
1
2
3
4
A
1
1
B
1
1
1
C
1
1
1
D
1
1
COST ($000s)
200
150
190
250
a)
Word-process
the linear programming model below.
Variables
Xi
= 1, if the tower site 1 is selected and 0 otherwise.
where i = 1, 2, 3 and 4
Objective Function
Constraints
b)
Set up the
spreadsheet for Excel Solver. Copy and paste the spreadsheet below.
c)
Copy and
paste the Answer report below.
d)
Write the
Optimal Solution below. Note: The Optimal Solutions must include the optimal
values for the non-zero variables as well as the objective function.
Problem 2 (10 points)
An investor has $500,000 to invest and wants to
maximize the money they will receive at the end of one year. They can invest in
condos, apartments and houses. The profit after one year, the cost and the
number of units available are shown below.
Variable
Investment
Profit
($1,000)
Cost
($1,000)
Number Available
X1
Condos
6
50
10
X2
Apartments
12
90
5
X3
Houses
9
100
7
a)
Word-process
the linear programming model below.
Variables
X1 = Number of condos purchased
X2 = Number of apartments purchased
X3= Nummber of houses purchased
Objective Function
Constraints
b)
Set up the
spreadsheet for Excel Solver. Copy and paste the spreadsheet below.
c)
Copy and
paste the Answer report below.
d)
Write the
Optimal Solution below. Note: The Optimal Solutions should include the optimal
values for the non-zero variables as well as the objective function.
Problem 3 (10 points)
Finnish Furniture manufactures tables in
facilities located in three citiesâReno, Denver, and Pittsburgh. The tables are
then shipped to three retail stores located in Phoenix, Cleveland, and Chicago.
Management wishes to develop a distribution schedule that will meet the demands
at the lowest possible cost. The shipping cost per unit from each of the
sources to each of the destinations is shown in the following table:
FROM
PHOENIX
CLEVELAND
CHICAGO
RENO
10
16
19
DENVER
12
14
13
PITTSBURGH
18
12
12
The
available supplies are 120 units from Reno, 200 from Denver, and 160 from
Pittsburgh. Phoenix has a demand of 140 units, Cleveland has a demand ‘of 160
units, and Chicago has a demand of 180 units. How many units should be shipped
from each manufacturing facility to each of the retail stores if cost is to be
minimized? What is the total cost?
a)
Word-process
the linear programming model below.
Variables
Xij
= Number of tables shipped from Plant in City i to Retail Store in City j,
where i = 1, 2, 3 and j = 1, 2, 3
Objective Function
Constraints
b)
Set up the
spreadsheet for Excel Solver. Copy and paste the spreadsheet below.
c)
Copy and
paste the Answer report below.
e)
Write the
Optimal Solution below. Note: The Optimal Solutions should include the optimal
values for the non-zero variables as well as the objective function.
Problem 4 (10 points)
A company needs to ship 100 units from Seattle to Denver at
the lowest possible cost. The costs associated with shipping between the cities
are:
To
From
Portland
Spokane
Salt Lake City
Denver
Seattle
100
500
600
–
Portland
–
350
300
–
Spokane
–
–
250
200
Salt Lake City
–
–
–
200
a)
Use the following nodes to complete the diagram for this
problem: Node 1: Seattle, Node 2: Portland, Node 3: Spokane, Node 4: Salt Lake
City, and Node 5: Denver.
1
2
b)
Word-process the linear programming model below. Note: This
is a shortest route problem. It can be formulated assuming one unit or 100
units.
Variables
Xij
= Flow from City i to City j,
where i = 1, 2, 3, 4 and j = 2, 3, 4, 5
Objective Function
Constraints
c)
Set up the spreadsheet for Excel Solver. Copy and paste the
spreadsheet below.
d)
Copy and paste the Answer report below.
e)
Write the Optimal Solution below. Include the optimal
shortest route and the value of the objective function.
Problem 5 (10 points)
A railroad needs to move the maximum amount of material
through its rail network. The numbers given on the arcs are capacities (the
maximum amount that can flow along those arcs). Formulate the LP model to
determine this maximum possible material that can be moved through the network
based on the following network diagram.
a)
Word-process
the linear programming model below.
Variables
Xij
= Flow from Node i to Node j,
Objective Function
Constraints
b)
Set up the
spreadsheet for Excel Solver. Copy and paste the spreadsheet below.
c)
Copy and
paste the Answer report below.
d)
Write the
Optimal Solution below. Note: The Optimal Solutions should include the optimal
values for the non-zero variables as well as the objective function.
