Assignment Linear Programming (LP)

Problem 1:

Personal Mini Warehouses is planning to open a new branch in Manitoba. In doing so, the company must determine how many storage rooms of each size to build. The company builds large and small rental spaces. After careful analysis of the expected monthly earnings of both types of rental space, as well as constraints (budget, area required for each size space, and projected rental limitations), Personal has formulated the following Linear Programming (LP) model:

x₁ = number of large space storage rooms to develop x₂ = number of small space storage rooms to develop

Maximize monthly earnings Z = 50x₁ + 20x₂ (profit) subject to 2x₁ + 4x₂ ≤ 400 (advertising budget in $) 100 x₁ + 50x₂ ≤ 8000 (space area in square metres) x₁ ≤ 60 (rental limit expected) x₁, x₂ ≥ 0 (non negativity)

Briefly explain or define each of these parts of the model:
a. The 50 in the objective function.
b. The product of the 20 and x₂ in the objective function (e.g. 20x₂)
c. The 8000 square meters in the space area constraint.
d. The 100 in the space area constraint.

e. The product of 2 and x₁ in the advertising budget constraint (e.g. 2x₁) Solve this LP problem using the graphical method:
f. Graph the constraints and identify the feasible region.

g. Determine the optimal solution and the maximum profit (show your calculations).

Include "managerial statements" that communicate the results of the analyses (i.e. describe verbally the results).

h. Determine the amount of slack for each of the constraint.

i. There is a single optimal solution to this problem. However, if the objective function had been parallel to one of the constraints, there would have been two equally optimal solutions. If the profit of x₁ remains at $50, what profit of x₂ would cause the objective function to be parallel to the space area constraint? Explain how you determined this.

Problem 2:

Solve the following linear programming problem graphically:

Minimize Z = 3x₁ + 6x₂

subject to

3x₁ + 2x₂ ≤ 18

x₁ + x₂ ≥ 5

x₁ ≤ 4

x₂ ≤ 7

x₂/x₁ ≤ 7/8 x₁, x₂ ≥ 0

a. Graph the constraints and identify the feasible region.

b. Determine the optimal solution and the minimum value of the objective function. (Show your calculation).

c. What would be the effect on the solution if the constraint x2 ≤ 7 is changed to x2 ≥ 7?

Problem 3

Solve the following linear programming problem graphically:

Maximize Z = 110x₁ + 75x₂ subject to

2x₁ + x₂ ≥ 40

- 6x₁ + 8x₂ ≤ 120

70x₁ + 105x₂ ≥ 2,100

x₁, x₂ ≥ 0

a. Graph the constraints and identify the feasible region.

b. Explain the solution result. (Show your work).

Problem 4:

Universal Claims Processors processes insurance claims for large national insurance companies. Most claim processing is done by a large pool of computer operators, some of whom are permanent and some temporary. A permanent operator can process 16 claims per day, whereas a temporary operator can process 12 claims per day, and on average the company processes at least 450 claims each day. The company has 40 computer workstations. A permanent operator will generate about claims with errors each day, whereas a temporary operator averages about 1.4 defective claims per day. The company wants to limit claims with errors to 25 per day. A permanent operator is paid $64 per day and a temporary operator is paid $42 per day. The company wants to determine the number of permanent and temporary operators to hire in order to minimize costs.

a. Formulate algebraically a linear programming model for the above problem. Define the decision variables, objective function, and constraints.

b. Draw the feasible region for the linear programming model.

c. Find the optimal solution(s) and optimal value of the objective function for the linear programing model. Justify why the solution is optimal. Describe also verbally how many permanent and temporary operators to hire (i.e. include a managerial statement).

d. Explain the effect on the optimal solution and optimal value of the objective function of changing the daily pay for a permanent operators from $64 to $54.

e. Explain the effect on the optimal solution and optimal value of the objective function of changing the daily pay for a temporary operators from $42 to $36.

f. Suppose that Universal Claims Processors decided not to try to limit the number of defective claims each day. What would be the effect on the optimal solution? Formulate a linear programming model on a spreadsheet to find out. SOLVE using Excel solver (Provide a printout of the corresponding "Excel Spreadsheet" and the "Answer Report"). Describe verbally how many permanent and temporary operators to hire.