**OPRE 315: Homework 2 **

Please answer the following questions giving all details:

1. A clothier makes coats and slacks. The two resources required are wool cloth and labor. The clothier has 150 square yards of wool and 200 hours of labor available. Each coat requires 3 square yards of wool and 10 hours of labor, whereas each pair of slacks requires 5 square yards of wool and 4 hours of labor. The profit for a coat is $45, and the profit for slacks is $35. The clothier 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 using graphical analysis.

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 = 20x_{1} + 32x_{2}

Subject to

2x_{1} + 4x_{2} < 40

x_{1} + x_{2} < 15

x_{1} > 8

x_{1}, x_{2} ≥ 0

3. The Baltimore Mining Company owns two mines, each of which produces two grades of ore— high and low. The company has a contract to supply a smelting company with at least 50 tons of high-grade ore and 40 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 5 tons of high-grade ore and 4 tons of low-grade ore per hour. Mine 2 produces 2 and 4 tons, respectively, of high-grade and low-grade ore per hour. It costs the company $350 per hour to mine each ton of ore from mine 1, and it costs $200 per hour to mine each ton of ore from mine 2. The company wants to operate Mine 1 for at least 4 hours. Determine the number of hours the company needs to operate each mine so that its contractual obligations can be met at the lowest cost.

Formulate a linear programming model for this problem.

**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 = 100x + 80y

Subject to

-6x + 8y < 120

70x + 105y > 2100

x, y ≥ 0

**Notes:**

- Homework 2 is due by **midnight Sunday, September 22 ^{nd}.**

- 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 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.

Subject | Mathematics |

Due By (Pacific Time) | 09/21/2013 08:00 pm |

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