Question description

 

rOPRE 315: Homework 2

 

Please answer the following questions giving all details:

 

1. The Charm City Clothiers Inc. makes coats and slacks. The two resources required are wool cloth and labor. The company has 200 square yards of wool and 300 hours of labor available. Each coat requires 5 square yards of wool and 10 hours of labor, whereas each pair of slacks requires 6 square yards of wool and 4 hours of labor. The profit for a coat is $16, and the profit for a pair of slacks is $10. The company wants to determine the number of coats and pairs of slacks to make so that profit will be maximized.

 

a. Formulate a linear programming model for this problem.

b. Solve this model by hand using the corner points graphical method.

 

2. Solve the following linear programming model graphically. In addition, write the problem in standard form and do a constraint analysis for the optimal solution.

 

Maximize Z = 10x + 15y

Subject to 

x + 2y < 20                  

x + y < 15                                      

x > 8 

x, y ≥ 0

 

3. The Charm City Mining Company owns two mines, each of which produces three grades of ore— high, medium, and low. The company has a contract to supply a smelting company with at least 100 tons of high-grade ore, 110 tons of medium-grade ore, and 120 tons of low-grade ore. Each mine produces a certain amount of each type of ore during each hour that it operates. Mine 1 produces 6.2 tons of high-grade ore, 3.2 tons of medium-grade ore, and 4.5 tons of low grade ore per hour. Mine 2 produces 3.5, 4.1, and 8 tons, respectively, of high-, medium-, and low-grade ore per hour. It costs the company $350 per hour to operate mine 1, and it costs $250 per hour to operate mine 2. The company wants to determine the number of hours it needs to operate each mine so that its contractual obligations can be met at the lowest cost.

 

(a) Define 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. Determine whether the following linear programming problem is infeasible, unbounded, or has multiple optimal solutions. Draw a graph and explain your conclusion.

 

Maximize Z = 60x₁ +480x₂

Subject to 

-3x₁ + 4x₂ < 60                                      

2x₁ + 3x₂ > 60   

4x₁ + 2x₂ > 80                  

x₁, x₂ ≥ 0

 

Notes:

 - Homework 2 is due by midnight Sunday, September 25^th.

 

- Each student must solve Homework 2 problems individually by himself/herself or within the team he/she is a member of. Please do not ask for help from anyone else to answer any questions in this homework.

 

- Please include all details and steps performed to find your answers. Just writing the final answers will not get you full credit.

 

- Some of the questions above require drawing graphs. Here is a list of options for you to include a graph in your answer.

 

Draw the graph using MS Paint or CorelDraw or some other software. Copy the graph in a MS Word file and post the file on the course website in appropriate category.

                                                 OR

Draw the graph by hand on a paper, scan it or take a photo and post it with your answer.

                                                OR

Describe the graph in words in your answer.

 

- Please post your answers in the assignments section of the course website.

 

- A team should post their answers in the assignments section of the course website by any one of the team members. The submission must contain names of all team members. A member whose name is missing from the assignment will not receive any credit.

 

A team submission will be graded based on the answers submitted by the team. If answer to one or more questions is missing because a member of the team did not do his/her part, then the team will not receive any credit for those questions.