MUST SHOW ALL WORK. CANNOT USE COMPUTER SOFTWARE FOR GRAPHS

 

 

1. (a) Define the slack and surplus variables. What do they represent? What is (are) the difference(s) between a slack and a surplus variable?

 

 

 

(b) Briefly describe the important parts of each step needed to make a decision using decision sciences models.

 

 

 

(c) What are the different types of special situations that may occur while solving a linear programming problem? Briefly describe each of these special situations.

 

 

 

(d) What are the important properties of a straight line? Briefly describe each property. What are the different types of slopes possible for a straight line? Briefly describe each type of slope and give one example for each type.

 

 

 

 

 

2. Given the following linear programming problem

 

 

 

Maximize 20x + 15y

 

Subject to

 

  2x + 2y < 100

 

  2x + 3y < 120

 

            x > 10

 

        x, y > 0

 

 

 

(a)  Graph the constraints.

 

(b)  Find the coordinates of each corner point of the feasible region

 

(c)  Determine the optimal solution.

 

 

 

 

 

3. Given that the optimal solution of the following linear programming problem is x = 10 and y = 10, state the problem in standard form and do a constraint analysis for the optimal solution.

 

 

 

Maximize   100x + 80y

 

Subject to

 

      5x + 2y ≤ 80

 

      3x + 5y ≥ 60

 

         x + y ≤ 20

 

               x ≤ 10

 

            x, y > 0

 

 

 

 

 

4. A company manufactures two kinds of pinball machines, each requiring a different manufacturing technique. Each Super Ball machine requires 25 hours of labor, 8 hours of testing, and yields a profit of $350. Each Silver Ball machine requires 12.5 hours of labor, 8.5 hours of testing, and yields a profit of $200. There are 2500 hours of labor and 1200 hours of testing available.

 

                                      

 

The company has made contracts with the retailers to provide at least 60 Super Ball machines and at least 58 Silver Ball machines. The company wants to manufacture at most a combined total of 135 of Super Ball and Silver Ball machines.

 

 

 

How many of each kind of pinball machines the company should manufacture to maximize the total profit?

 

 

 

Formulate a linear programming model for the above situation by determining

 

(a) The decision variables.

 

(b) The objective function.

 

(c) All the constraints.

 

 

 

Note: Do NOT solve the problem after formulating.

 

 

 

 

 

5. Charm City Foods manufactures a snack bar by blending two ingredients:  a nut mix and a granola mix.  Information about the two ingredients (per ounce) is shown below.

 

 

 

Ingredient	Cost In Dollars	Fat Grams	Protein Grams	Calories
Nut Mix	0.90	20	6.5	650
Granola Mix	0.45	3	2.2	55

 

 

 

The company needs to develop a linear programming model whose solution would tell them how many ounces of each mix to put into the snack bar. The blend should contain no more than 1200 calories, at least 12 grams of protein, and no more than 32 grams of fat. In addition, at least one ounce of nut mix must be included in the blend. The objective is to minimize the total cost.

 

 

 

Formulate a linear programming model for the above situation by determining

 

(a) The decision variables.

 

(b) The objective function.

 

(c) All the constraints.

 

 

 

Note: Do NOT solve the problem after formulating.

 

 

 

 

 

6. Determine whether the following linear programming problem is infeasible, unbounded, or has multiple optimal solutions. Draw a graph to find the feasible region (if it exists) and explain your conclusion.

 

 

 

Maximize 22x_l + 32x₂

 

Subject to: 

 

          2x_l + x₂ < 15

 

                   x_l  > 10

 

                   x₂ < 10

 

              x_l, x₂ > 0