1. Suppose P = 3x + 4y is the objective function (Profit function) in a linear programming (maximization) problem, where x is the number of units of product A and y is the number of units of product B.

1. What does the coefficient of x represent? How about the coefficient of y? (1 marks)
2. Given the linear programming problem:

X + y 4 Resource 1

and

2x + y 5 Resource 2

1. What is the linear inequality that represents h unit increase in Resource 1? (1 marks)
2. Write the inequality that represents k units of increase in Resource 2. (1 marks)

2. Explain the meaning of shadow pricing and binding constraints in your own words. (2 marks)

3. Use the method of corners to minimize the following objective function:

C = 2x + 7y subject to 3x + 5y 45 and 3x + 10y 60 knowing that x and y cannot be negative. Instructions: Show the graph and identify the corner points. Then plug points one by one into the objective function to find the optimizing solution (5 marks)

4. Consider the following linear programing problem:

P = 2x + 4y subject to

2x + 5y 19 Resource 1

3x + 2y 12 Resource 2

as well as x 0 and y 0.

1. Use the method of corners (showing your graph and corner points) to solve this problem.

(i.e. Find the number of x and y so that P is optimized)

Hint: If you are dealing with a bounded area, then it is a maximizing problem. (10 marks)

2. Will the optimal solution you found in part a remain optimal if coefficient of x changes to 4.5? (5 marks)
3. Will the solution remain optimal is amount of Resource 1 changes to 25? (5 marks)

5. Deluxe River cruises operates a fleet of river vessels. They have two type of vessels; type A with 60 deluxe cabins and 160 standard cabins, and type B with 80 deluxe and 120 standard cabins. Based on their contract, they have to provide at least 360 deluxe and 680 standard cabins for their next tour in July. Costs are $44,000 per type A, and $54,000 per type B vessel.

1. Write down the objective function and all of the binding restraints. (4 marks)
2. Graph the system of inequalities and determine the corner points. (10 marks)
3. How many of each type of vessel should be used to minimize the cost? (4 marks)