91Ó°ÊÓ

Sketch the region that corresponds to the given inequalities, say whether the region is bounded or unbounded, and find the coordinates of all corner points (if any). $$ \begin{aligned} 2 x+y & \leq 4 \\ x-2 y &>2 \end{aligned} $$

Solve the LP problems. If no optimal solution exists, indicate whether the feasible region is empty or the objective function is unbounded. \(\vee\) Minimize \(c=-x+2 y\) subject to \(\begin{aligned} y & \leq \frac{2 x}{3} \\\ x & \leq 3 y \\ y & \geq 4 \\ x & \geq 6 \\ x+y & \leq 16 . \end{aligned}\)

We suggest the use of technology. Round all answers to two decimal places. \(\begin{array}{ll}\text { Minimize } & c=50.3 x+10.5 y+50.3 z \\ \text { subject to } & 3.1 x \quad+1.1 z \geq 28 \\ & 3.1 x+y-1.1 z \geq 23 \\ & 4.2 x+y-1.1 z \geq 28 \\ & x \geq 0, y \geq 0, z \geq 0\end{array}\)

Ruff, Inc. makes dog food out of chicken and grain. Chicken has 10 grams of protein and 5 grams of fat per ounce, and grain has 2 grams of protein and 2 grams of fat per ounce. A bag of dog food must contain at least 200 grams of protein and at least 150 grams of fat. If chicken costs 10 e per ounce and grain costs \(1 \phi\) per ounce, how many ounces of each should Ruff use in each bag of dog food in order to minimize cost? HINT [See Example 4.]

Each serving of Gerber Mixed Cereal for Baby contains 60 calories and 11 grams of carbohydrates. Each serving of Gerber Mango Tropical Fruit Dessert contains 80 calories and 21 grams of carbohydrates. \({ }^{11}\) If the cereal costs \(30 \phi\) per serving and the dessert costs 50 per serving, and you want to provide your child with at least 140 calories and at least 32 grams of carbohydrates, how can you do so at the least cost? (Fractions of servings are permitted.)

Purchasing Bingo's Copy Center needs to buy white paper and yellow paper. Bingo's can buy from three suppliers. Harvard Paper sells a package of 20 reams of white and 10 reams of yellow for \$60, Yale Paper sells a package of 10 reams of white and 10 reams of yellow for \(\$ 40\), and Dartmouth Paper sells a package of 10 reams of white and 20 reams of yellow for \(\$ 50\). If Bingo's needs 350 reams of white and 400 reams of yellow, how many packages should it buy from each supplier to minimize the cost? What is the least possible cost?

Resource Allocation Meow makes cat food out of fish and cornmeal. Fish has 8 grams of protein and 4 grams of fat per ounce, and cornmeal has 4 grams of protein and 8 grams of fat. A jumbo can of cat food must contain at least 48 grams of protein and 48 grams of fat. If fish and cornmeal both cost 5elounce, how many ounces of each should Meow use in each can of cat food to minimize costs? What are the shadow costs of protein and of fat? HINT [See Example 2.]

Your salami manufacturing plant can order up to 1,000 pounds of pork and 2,400 pounds of beef per day for use in manufacturing its two specialties: "Count Dracula Salami" and "Frankenstein Sausage." Production of the Count Dracula variety requires 1 pound of pork and 3 pounds of beef for each salami, while the Frankenstein variety requires 2 pounds of pork and 2 pounds of beef for every sausage. In view of your heavy investment in advertising Count Dracula Salami, you have decided that at least onethird of the total production should be Count Dracula. On the other hand, due to the health-conscious consumer climate. your Frankenstein Sausage (sold as having less beef) is earning your company a profit of \(\$ 3\) per sausage, while sales of the Count Dracula variety are down and it is earning your company only \(\$ 1\) per salami. Given these restrictions, how many of each kind of sausage should you produce to maximize profits, and what is the maximum possible profit? HINT [See Example 3.]

The Scottsville Textile Mill produces several different fabrics on eight dobby looms which operate 24 hours per day and are scheduled for 30 days in the coming month. The Scottsville Textile Mill will produce only Fabric 1 and Fabric 2 during the coming month. Each dobby loom can turn out \(4.63\) yards of either fabric per hour. Assume that there is a monthly demand of 16,000 yards of Fabric 1 and 12,000 yards of Fabric 2. Profits are calculated as 33 d per yard for each fabric produced on the dobby looms. a. Will it be possible to satisfy total demand? b. In the event that total demand is not satisfied, the Scottsville Textile Mill will need to purchase the fabrics from another mill to make up the shortfall. Its profits on resold fabrics ordered from another mill amount to \(20 \mathrm{~d}\) per yard for Fabric 1 and \(16 \mathrm{e}\) per yard for Fabric \(2 .\) How many yards of each fabric should it produce to maximize profits?

If a linear programming problem has a bounded, nonempty feasible region, then optimal solutions (A) must exist (B) may or may not exist (C) cannot exist