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:

a) Word-process the linear programming model below.

Variables

X_i = 1, if the tower site 1 is selected and 0 otherwise.

wherei = 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.

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 cities—Reno, 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:

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

X_ij = Number of tables shipped from Plant in City i to Retail Store in City j,

wherei = 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:

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.

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

X_ij = Flow from City i to City j,

wherei = 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

X_ij = 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.

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