1. Find the complete optimal solution to this linear programming problem.

Min 3X + 3Ys.t. 12X + 4Y > 48

10X + 5Y > 50 4X + 8Y > 32

X , Y > 0

TOPIC: Graphical solution

2. The Sanders Garden Shop mixes two types of grass seed into a blend. Each type of grass has been rated (per pound) according to its shade tolerance, ability to stand up to traffic, and drought resistance, as shown in the table. Type A seed costs $1 and Type B seed costs $2. If the blend needs to score at least 300 points for shade tolerance, 400 points for traffic resistance, and 750 points for drought resistance, how many pounds of each seed should be in the blend? Which targets will be exceeded? How much will the blend cost?

Type A Type BShade Tolerance 1 1Traffic Resistance 2 1Drought Resistance 2 5

TOPIC: Modeling and graphical solution

3. The following linear programming problem has been solved by The Management Scientist. Use the output to answer the questions.

LINEAR PROGRAMMING PROBLEM

MAX 25X1+30X2+15X3

S.T.1) 4X1+5X2+8X3<1200

2) 9X1+15X2+3X3<1500

OPTIMAL SOLUTION

Objective Function Value = 4700.000

Variable Value Reduced Costs ---------- -------- ------------------ X1 140.000 0.000 X2 0.000 10.000 X3 80.000 0.000

Constraint Slack/Surplus Dual Prices ------------- ---------------- -------------- 1 0.000 1.000 2 0.000 2.333

OBJECTIVE COEFFICIENT RANGES

Variable Lower Limit Current Value Upper Limit ---------- --------------- ---------------- --------------- X1 19.286 25.000 45.000 X2 No Lower Limit 30.000 40.000 X3 8.333 15.000 50.000

RIGHT HAND SIDE RANGES

Constraint Lower Limit Current Value Upper Limit ------------ --------------- ----------------- --------------- 1 666.667 1200.000 4000.000 2 450.000 1500.000 2700.000

a. Give the complete optimal solution.b. Which constraints are binding?c. What is the dual price for the second constraint? What interpretation does this have?d. Over what range can the objective function coefficient of x2 vary before a new solution

point becomes optimal?e. By how much can the amount of resource 2 decrease before the dual price will change?f. What would happen if the first constraint's right-hand side increased by 700 and the

second's decreased by 350?

TOPIC: Interpretation of Management Scientist output

4. FarmFresh Foods manufactures a snack mix called TrailTime by blending three ingredients: a dried fruit mixture, a nut mixture, and a cereal mixture. Information about the three ingredients (per ounce) is shown below.

Ingredient Cost Volume Fat Grams CaloriesDried Fruit .35 1/4 cup 0 150Nut Mix .50 3/8 cup 10 400Cereal Mix .20 1 cup 1 50

The company needs to develop a linear programming model whose solution would tell them how many ounces of each mix to put into the TrailTime blend. TrailTime is packaged in boxes that will hold between three and four cups. The blend should contain no more than 1000 calories and no more than 25 grams of fat. Dried fruit must be at least 20% of the volume of the mixture, and nuts must be no more than 15% of the weight of the mixture. Develop a model that meets these restrictions and minimizes the cost of the blend.

5. Solve the following problem by the simplex method.

Max 14x1 + 14.5x2 + 18x3

s.t. x1 + 2x2 + 2.5x3 50x1 + x2 + 1.5x3 30

x1 , x2 , x3 0