1. LINEAR PROGRAMMING

1. Find the complete (including values for slack variables) optimal solution to this linear programming problem using. graphical method
   Min 5X + 6Y
   s.t. 3X + Y > 15
   X + 2Y > 12
   3X + 2Y > 24
   X, Y > 0
2. Find the complete (including values for slack variables) optimal solution to this linear programming problem using Excel Solver or QM for Windows .
   Max 5X + 3Y
   s.t. 2X + 3Y < 30
   2X + 5Y < 40
   6X - 5Y < 0
   X, Y > 0
3. Formulate and solve the following problem. Use method of your choice.

The Marketing Club at your college has decided to raise funds by selling three types of T-shirts: one with single-color "ordinary" design, one with a two-color "fancy" design, and one with a three-color "very fancy" design. The club feels that it can sell up to 300 T-shirts. "Ordinary" T-shirts will cost the club $6 each, "fancy" T-shirts $8 each, and "very fancy" T-shirts $10 each, and the club has a total purchasing budget of $3,000. It will sell "ordinary" T-shirts at profit of $4 each, "fancy" T-shirts at profit of $5 each, and "very fancy" T-shirts at a profit of $4 each. How many of each kind of T-shirt should the club order to maximize profit?

1. Formulate and solve the following problem. Use method of your choice.
   The production manager for Beer etc. produces 2 kinds of beer: light (L) and dark (D). Two resources used to produce beer are malt and wheat. He can get at most 4800 oz of malt per week and 3200 oz of wheat per week. Each bottle of light beer requires 12 oz of malt and 4 oz of wheat, while a bottle of dark beer uses 8 oz of malt and 8 oz of wheat. Profits for light beer are $2 per bottle, and profits for dark beer are $1 per bottle. What is the linear programming model for this problem?
   II. INTEGER PROGRAMMING
   1. Solve the following problem:
   Max X + 2Y
   s.t. 6X + 8Y < 48
   7X + 5Y > 35
   X, Y > 0
   Y integer
   1. Find the optimal solution to the LP Relaxation. Round down to find a feasible integer solution. Is this solution optimal?
   2. Find the optimal solution.

2. The Westfall Company has a contract to produce 10,000 garden hoses for a large discount chain. Westfall has four different machines that can produce this kind of hose. Because these machines are from different manufacturers and use differing technologies, their specifications are not the same.

1. The company wants to minimize total cost. Give the objective function.
2. Give the constraints for the problem.
3. Write a constraint to ensure that if machine 4 is used, machine 1 cannot be.

III. TRANSPORTATION, ASSIGNMENT, AND TRANSSHIPMENT PROBLEMS

1. Write the linear programming problem for this network.

2. Canning Transport is to move goods from three factories to three distribution centers. Information about the move is given below. Give the network model and the linear programming model for this problem.

Shipping costs are:

IV. interpretation of computer solution

1. The linear programming problem whose output follows is used to determine how many bottles of fire red nail polish (x ₁ ), bright red nail polish (x ₂ ), basil green nail polish(x ₃ ), and basic pink nail polish(x ₄ ) a beauty salon should stock. The objective function measures profit; it is assumed that every piece stocked will be sold. Constraint 1 measures display space in units, constraint 2 measures time to set up the display in minutes. Note that green nail polish does not require any time to prepare its display. Constraints 3 and 4 are marketing restrictions. Constraint 3 indicates that the maximum demand for fire red and green polish is 25 bottles, while constraint 4 specifies that the minimum demand combined for bright red, green and pink nail polish bottles is at least 50 bottles.

MAX 100x ₁ + 120x ₂ + 150x ₃ + 125x ₄

Subject to 1. x ₁ + 2x ₂ + 2x ₃ + 2x ₄ \(\le \) 108

2. 3x ₁ + 5x ₂ + x _{4 \(\le \)} 120

3. x ₁ + x ₃ \(\le \) 25

4. x ₂ + x ₃ + x _{4 \(\ge \)} 50

x ₁ , x ₂ , x ₃ , x _{4 \(\ge \)} 0

Optimal Solution:

Objective Function Value = 7475.000

Variable Value Reduced Costs

X1 8 0

X2 0 5

X3 17 0

X4 33 0

Constraint Slack / Surplus Dual Prices

1 0 75

2 63 0

3 0 25

4 0 -25

Objective Coefficient Ranges

Variable Lower Limit Current Value Upper Limit

X1 87.5 100 none

X2 none 120 125

X3 125 150 162

X4 120 125 150

Right Hand Side Ranges

Constraint Lower Limit Current Value Upper Limit

1 100 108 123.75

2 57 120 none

3 8 25 58

4 41.5 50 54

a) How many bottles of fire red nail polish, bright red nail polish, basil green polish and pink nail polish should be stocked?

b) What is the maximum profit?

c) How much space will be left unused?

d) How many minutes of idle time remaining for setting up the display?

e) To what value can the per bottle profit on fire red nail polish drop before the solution (product mix) would change?

f) By how much can the per bottle profit on green basil nail polish increase before the solution (product mix) would change?

Price: $26.69

Solution: The downloadable solution consists of 14 pages, 1269 words and 8 charts.
Deliverable: Word Document