Deck 6: Linear Programming

Question 1

Write down (without solving) the dual linear programming problem. Minimize $C = 13 s + 35 t$ subject to the following. $$\begin{array} { l } 7 s - 3 t \geq 15 \\s + 4 t \geq 25 \\s \geq 0 , t \geq 0\end{array}$$

A)Maximize $P = 15 x + 25 y$ subject to the following. $$\begin{array} { l } 7 x + y \leq 13 \\- 3 x + 4 y \leq 35 \\x \leq 0 , y \leq 0\end{array}$$

B) Maximize $P = 15 x + 25 y$ subject to the following.
$$\begin{array} { l } 7 x + y \leq 13 \\3 x + 4 y \leq - 35 \\x \geq 0 , y \geq 0\end{array}$$

C) Maximize $P = 15 x + 25 y$ subject to the following.

$$\begin{array} { l } 7 x + y \geq 13 \\- 3 x + 4 y \geq 35 \\x \geq 0 , y \geq 0\end{array}$$

D) Maximize $P = 15 x + 25 y$ subject to the following.
$$\begin{array} { l } 7 x + y \leq 13 \\3 x + 4 y \leq 35 \\x \geq 0 , y \geq 0\end{array}$$

E) Maximize $P = 15 x + 25 y$ subject to the following.
$$\begin{array} { l } 7 x + y \leq 13 \\- 3 x + 4 y \leq 35 \\x \geq 0 , y \geq 0\end{array}$$

Answer 1

Maximize $P = 15 x + 25 y$ subject to the following.
$$\begin{array} { l } 7 x + y \leq 13 \\- 3 x + 4 y \leq 35 \\x \geq 0 , y \geq 0\end{array}$$

Question 2

Solve the standard minimization problem using duality.
Minimize $c = 5 s + 2 t$
Subject to $$\begin{array} { l } 3 s + t \geq 30 \\s + t \geq 20 \\s + 3 t \geq 30 \\s \geq 0 , t \geq 0\end{array}$$

A) $c = 55$ , $s = 5$ , $t = 15$
B) $c = 85$ , $s = 15$ , $t = 5$
C) $c = 25$ , $s = 5$ , $t = 0$
D) $c = 30$ , $s = 0$ , $t = 15$
E) $c = 10$ , $s = 0$ , $t = 5$

Answer 2

$c = 55$ , $s = 5$ , $t = 15$

Question 3

Solve the standard minimization problem using duality. $$\begin{array} { l } \text { Minimize } ^ { c = 3 s + 3 t + 3 u } \\3 s + 2 t + u \geq 150 \\2 s + t + 3 u \geq 150 \\\text { subject to } s + 3 t + 2 u \geq 150 \\s \geq 0 , t \geq 0 , u \geq 0 \\\end{array}$$
If you have several solutions, select one of them.

A) $c = 90$ , $s = 37.5$ , $t = 0$ , $u = 37.5$
B) $c = 225$ , $s = 25$ , $t = 25$ , $u = 25$
C) $c = 135$ , $s = 15$ , $t = 15$ , $u = 15$
D) $c = 180$ , $s = 20$ , $t = 20$ , $u = 20$
E) $c = 225$ , $s = 10$ , $t = 10$ , $u = 10$

Answer 3

$c = 225$ , $s = 25$ , $t = 25$ , $u = 25$

Question 4

Solve the standard minimization problem using duality.
Minimize $c = 0.4 s + 0.6 t$
Subject to $$\begin{array} { l } 0.2 s + 0.1 t \geq 1 \\0.15 s + 0.3 t \geq 1.5 \\10 s + 10 t \geq 80 \\s \geq 0 , t \geq 0\end{array}$$

A) $c = 3.6$ , $s = 6$ , $t = 2$
B) $c = 2.8$ , $s = 6$ , $t = 2$
C) $c = 1.2$ , $s = 6$ , $t = 2$
D) $c = 4.4$ , $s = 2$ , $t = 6$
E) $c = 2.8$ , $s = 2$ , $t = 6$

Question 5

Solve the games with the given payoff matrix.
$$P = \left[ \begin{array} { l l l } - 4 & - 6 & - 3 \\- 4 & - 3 & - 5\end{array} \right]$$

A) $C = \left[ \begin{array} { c } 0 \\\frac { 10 } { 81 } \\\frac { 5 } { 9 }\end{array} \right]$ , $R = \left[ \begin{array} { l l } \frac { 10 } { 81 } & \frac { 5 } { 9 }\end{array} \right]$ , $e = - \frac { 21 } { 5 }$
B) $C = \left[ \begin{array} { l } \frac { 2 } { 5 } \\\frac { 3 } { 5 } \\0\end{array} \right]$ , $R = \left[ \begin{array} { l l } \frac { 2 } { 5 } & \frac { 3 } { 5 }\end{array} \right]$ , $e = - \frac { 21 } { 5 }$
C) $C = \left[ \begin{array} { l } 0 \\\frac { 2 } { 5 } \\\frac { 3 } { 5 }\end{array} \right]$ , $R = \left[ \begin{array} { l l } \frac { 2 } { 5 } & \frac { 3 } { 5 }\end{array} \right]$ , $e = \frac { 21 } { 5 }$
D) $C = \left[ \begin{array} { c } 0 \\\frac { 10 } { 81 } \\\frac { 5 } { 9 }\end{array} \right]$ , $R = \left[ \begin{array} { l l } \frac { 10 } { 81 } & \frac { 5 } { 9 }\end{array} \right]$ , $e = \frac { 21 } { 5 }$
E) $C = \left[ \begin{array} { l } 0 \\\frac { 2 } { 5 } \\\frac { 3 } { 5 }\end{array} \right]$ , $R = \left[ \begin{array} { l l } \frac { 2 } { 5 } & \frac { 3 } { 5 }\end{array} \right]$ , $e = - \frac { 21 } { 5 }$

Question 6

Solve the standard minimization problem using duality.
Minimize $c = 4 s + 7 t + 9 u$
Subject to $\begin{array} { l } s + 2 t + 2 u \geq 60 \\2 s + t + 3 u \geq 60 \\s \geq 0 , t \geq 0 , u \geq 0\end{array}$

A) $c = 140$ , $s = 0$ , $t = 20$ , $u = 0$
B) $c = 80$ , $s = 20$ , $t = 0$ , $u = 0$
C) $c = 220$ , $s = 20$ , $t = 20$ , $u = 0$
D) $c = 260$ , $s = 20$ , $t = 0$ , $u = 20$
E) $c = 320$ , $s = 0$ , $t = 20$ , $u = 20$

Question 7

Solve the games with the given payoff matrix. $$P = \left[ \begin{array} { c c c } - 1 & 1 & 2 \\2 & - 1 & - 2 \\1 & 2 & - 1\end{array} \right]$$

A) $C = \left[ \begin{array} { l } \frac { 5 } { 9 } \\\frac { 2 } { 9 } \\\frac { 2 } { 9 }\end{array} \right]$ , $R = \left[ \begin{array} { l l l } \frac { 7 } { 18 } & \frac { 1 } { 18 } & \frac { 5 } { 9 }\end{array} \right]$ , $e = - \frac { 85 } { 18 }$
B) $C = \left[ \begin{array} { l } \frac { 5 } { 9 } \\\frac { 2 } { 9 } \\\frac { 2 } { 9 }\end{array} \right]$ , $R = \left[ \begin{array} { l l l } \frac { 7 } { 18 } & \frac { 1 } { 18 } & \frac { 5 } { 9 }\end{array} \right]$ , $e = - \frac { 31 } { 18 }$
C) $C = \left[ \begin{array} { c } \frac { 5 } { 9 } \\\frac { 7 } { 18 } \\\frac { 1 } { 18 }\end{array} \right]$ , $R = \left[ \begin{array} { l l l } \frac { 5 } { 9 } & \frac { 1 } { 18 } & \frac { 7 } { 18 }\end{array} \right]$ , $e = \frac { 5 } { 18 }$
D) $C = \left[ \begin{array} { c } \frac { 1 } { 18 } \\\frac { 7 } { 18 } \\\frac { 5 } { 9 }\end{array} \right]$ , $R = \left[ \begin{array} { l l l } \frac { 5 } { 9 } & \frac { 1 } { 18 } & \frac { 7 } { 18 }\end{array} \right]$ , $e = - \frac { 85 } { 18 }$
E) $C = \left[ \begin{array} { c } \frac { 5 } { 18 } \\\frac { 5 } { 18 } \\\frac { 4 } { 9 }\end{array} \right]$ , $R = \left[ \begin{array} { l l l } \frac { 5 } { 9 } & \frac { 1 } { 18 } & \frac { 7 } { 18 }\end{array} \right]$ , $e = - \frac { 31 } { 18 }$

Question 8

Minimize $p = 6 a + 4 b$ subject to the following constraints.
$\begin{array} { l } a + b \geq 10 \\6 a + b \geq 30 \\a \geq 0 , b \geq 0\end{array}$

A) $a = 4$ , $b = 6$ , $p = 48$
B) $a = 10$ , $b = 0$ , $p = 60$
C) $a = 0$ , $b = 30$ , $p = 180$
D) $a = 0$ , $b = 10$ , $p = 40$
E) $a = 6$ , $b = 6$ , $p = 60$

Question 9

Oz makes lion food out of giraffe and gazelle meat. Giraffe meat has 18 grams of protein and 36 grams of fat per pound, while gazelle meat has 36 grams of protein and 18 grams of fat per pound. A batch of lion food must contain at least 36,000 grams of protein and 54,000 grams of fat. Giraffe meat costs $4/pound and gazelle meat costs $8/pound. How many pounds of each should go into each batch of lion food in order to minimize costs What are the shadow costs of protein and fat Use the rounded numbers of pounds in your calculations. ?

A)1,333 pounds of giraffe meat and 333 pounds of gazelle, 22 cents per gram of protein and 15 cents per gram of fat
B) 333 pounds of giraffe meat and 333 pounds of gazelle, 15 cents per gram of protein and 15 cents per gram of fat
C) 1,333 pounds of giraffe meat and 333 pounds of gazelle, 15 cents per gram of protein and 22 cents per gram of fat
D) 1,333 pounds of giraffe meat and 333 pounds of gazelle, 22 cents per gram of protein and 22 cents per gram of fat
E) 333 pounds of giraffe meat and 1,333 pounds of gazelle, 22 cents per gram of protein and 15 cents per gram of fat

Question 10

Solve the standard minimization problem using duality. $$\begin{array} { l } \text { Minimize } & { c = 8 s + 7 t + 8 u } \\&s + t + u \geq 100 \\&2 s + t \geq 50 \\\text { subject to } &t + u \geq 50 \\&s \geq 0 , t \geq 0 , u \geq 0 \\\end{array}$$

A) $c = 800$ , $s = 100$ , $t = 0$ , $u = 0$
B) $c = 800$ , $s = 0$ , $t = 0$ , $u = 100$
C) $c = 700$ , $s = 0$ , $t = 100$ , $u = 0$
D) $c = 350$ , $s = 0$ , $t = 50$ , $u = 0$
E) $c = 400$ , $s = 50$ , $t = 0$ , $u = 0$

Question 11

Write down (without solving) the dual LP problem.
Minimize $c = s + 6 t + u$ subject to
$$\begin{array} { l } 9 s - t + v \geq 3,000 \\u - v \geq 1,000 \\s + t \geq 500 \\s \geq 0 , t \geq 0 , u \geq 0 , v \geq 0\end{array}$$

A)maximize $$p = x + 6 y + z$$ subject to $$\begin{array} { l } 9 x - y + w \geq 3,000 \\z - w \geq 1,000 \\x + y \geq 500 \\x \geq 0 , y \geq 0 , z \geq 0 , w \geq 0\end{array}$$

B) maximize $p = 3,000 x + 1,000 y + 500 z$ subject to
$$\begin{array} { l } 9 x + z \geq 1 \\- x + z \geq 6 \\y \geq 1 \\x - y \geq 0 \\x \geq 0 , y \geq 0 , z \geq 0\end{array}$$

C) maximize $p = 3,000 x + 1,000 y + 500 z$ subject to

$$\begin{array} { l } 9 x + z \leq 1 \\- x + z \leq 6 \\y \leq 1 \\x - y \leq 0 \\x \geq 0 , y \geq 0 , z \geq 0\end{array}$$

D) maximize $p = x + 6 y + z$ subject to
$$\begin{array} { l } 9 x - y + w \leq 3,000 \\z - w \leq 1,000 \\x + y \geq 500 \\x \geq 0 , y \geq 0 , z \geq 0 , w \geq 0\end{array}$$

E) maximize $p = x + 6 y + z$ subject to

$$\begin{array} { l } 9 x + y + w \geq 3,000 \\z + w \geq 1,000 \\x + y \geq 500 \\x \geq 0 , y \geq 0 , z \geq 0 , w \geq 0\end{array}$$

Question 12

Solve the standard minimization problem using duality.
Minimize $C = 18 s + 29 t$
Subject to $$\begin{array} { l } 3 s + 2 t \geq 60 \\6 s + 3 t \geq 60 \\s \geq 0 , t \geq 0\end{array}$$

A) $C = 360$ , $s = 20$ , $t = 0$
B) $C = 580$ , $s = 0$ , $t = 20$
C) $C = 290$ , $s = 0$ , $t = 10$
D) $C = 180$ , $s = 10$ , $t = 0$
E) $C = 650$ , $s = 20$ , $t = 10$

Question 13

The Enormous State University's Business School is buying computers. The school has two models to choose from. Each first model comes with 400 MB of memory and 80 GB of disk space, while each second model has 300 MB of memory and 100 GB of disk space. For reasons related to its accreditation, the school would like to be able to say that it has a total of at least 48,000 MB of memory and at least 12,800 GB of disk space. If both models cost $4,400 each, how many of each should the school buy to keep the cost as low as possible What are the shadow costs of memory and disk space

A)80 first models and 60 second models, $ $\frac { 77 } { 6 }$ per MB and $ $\frac { 385 } { 8 }$ per GB
B) 60 first models and 80 second models, $ $\frac { 385 } { 8 }$ per MB and $ $\frac { 385 } { 8 }$ per GB
C) 60 first models and 80 second models, $ $\frac { 77 } { 6 }$ per MB and $ $\frac { 385 } { 8 }$ per GB
D) 60 first models and 80 second models, $ $\frac { 77 } { 6 }$ per MB and $ $\frac { 77 } { 6 }$ per GB
E) 80 first models and 60 second models, $ $\frac { 385 } { 8 }$ per MB and $ $\frac { 77 } { 6 }$ per GB

Question 14

We are to minimize $p = 3 a + 2 b$ subject to the following constraints. $$\begin{array} { l } a + b \geq 4 \\3 a + b \geq 6 \\a , b \geq 0\end{array}$$
After using the simplex method for dual problem to arrive at the tableau below, what is the maximum value for $p$
X y s t p
$$\left[ \begin{array} { c c c c c } 0 & 1 & \frac { 1 } { 2 } & - \frac { 1 } { 2 } & 0 \\\frac { 1 } { 2 } & & & \\1 & 0 & - \frac { 1 } { 2 } & \frac { 3 } { 2 } & 0 \\\frac { 3 } { 2 } & & & & \\0 & 0 & 1 & 3 & 1 \\9 & & & &\end{array} \right]$$

A)9
B) 3
C) 5.5
D) $\frac { 1 } { 2 }$
E) $\frac { 3 } { 2 }$

Question 15

Solve the games with the given payoff matrix. $$P = \left[ \begin{array} { l l l } - 3 & - 5 & - 2 \\- 3 & - 2 & - 4 \\- 4 & - 3 & - 3\end{array} \right]$$

A) $C = \left[ \begin{array} { l } \frac { 1 } { 2 } \\\frac { 1 } { 6 } \\\frac { 1 } { 3 }\end{array} \right]$ , $R = \left[ \begin{array} { l l l } \frac { 1 } { 6 } & \frac { 1 } { 2 } & \frac { 1 } { 3 }\end{array} \right]$ , $e = - \frac { 19 } { 6 }$
B) $C = \left[ \begin{array} { l } \frac { 1 } { 6 } \\\frac { 1 } { 3 } \\\frac { 1 } { 2 }\end{array} \right]$ , $R = \left[ \begin{array} { l l l } \frac { 1 } { 3 } & \frac { 1 } { 2 } & \frac { 1 } { 6 }\end{array} \right]$ , $e = \frac { 19 } { 6 }$
C) $C = \left[ \begin{array} { l } \frac { 1 } { 6 } \\\frac { 1 } { 3 } \\\frac { 1 } { 2 }\end{array} \right]$ , $R = \left[ \begin{array} { l l l } \frac { 1 } { 3 } & \frac { 1 } { 2 } & \frac { 1 } { 6 }\end{array} \right]$ , $e = - \frac { 19 } { 6 }$
D) $C = \left[ \begin{array} { l } \frac { 1 } { 2 } \\\frac { 1 } { 6 } \\\frac { 1 } { 3 }\end{array} \right]$ , $R = \left[ \begin{array} { l l l } \frac { 1 } { 6 } & \frac { 1 } { 2 } & \frac { 1 } { 3 }\end{array} \right]$ , $e = \frac { 19 } { 6 }$
E) $C = \left[ \begin{array} { l } \frac { 1 } { 3 } \\\frac { 1 } { 2 } \\\frac { 1 } { 6 }\end{array} \right]$ , $R = \left[ \begin{array} { l l l } \frac { 1 } { 3 } & \frac { 1 } { 6 } & \frac { 1 } { 2 }\end{array} \right]$ , $e = \frac { 19 } { 6 }$

Question 16

Ruff makes dog food out of chicken and grain. Chicken has 10 grams of protein and 5 grams of fat/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 50 grams of fat. If chicken costs 15 cents per ounce and grain costs 2 cents per ounce, how many ounces of each should Ruff use in each bag of dog food in order to minimize cost What are the shadow costs of protein and fat ?

A)100 oz of chicken and no grain, 1 cents per gram of protein and 1 cents per gram of fat
B) 200 oz of chicken and no grain, 0 cents per gram of protein and 1 cents per gram of fat
C) 100 oz of chicken and no grain, 0.5 cents per gram of protein and 0.5 cents per gram of fat
D) 200 oz of grain and no chicken, 0.5 cents per gram of protein and 0.5 cents per gram of fat
E) 100 oz of grain and no chicken, 1 cents per gram of protein and 0 cents per gram of fat

Question 17

Solve the standard minimization problem using duality.
Minimize $c = 0.7 s + 0.3 t$
Subject to $$\begin{array} { l } 30 s + 20 t \geq 600 \\0.1 s + 0.4 t \geq 4 \\0.2 s + 0.3 t \geq 4.5 \\s \geq 0 . t \geq 0\end{array}$$

A) $c = 9 , s = 30 , t = 0$
B) $c = 21 , s = 0 , t = 30$
C) $c = 30 , s = 30 , t = 30$
D) $c = 9 , s = 0 , t = 30$
E) $c = 21 , s = 30 , t = 0$

Question 18

Solve the games with the given payoff matrix. $$\left[ \begin{array} { c c c c } 1 & - 1 & 2 & 0 \\1 & 2 & 0 & 1 \\0 & 1 & 1 & 0 \\2 & 0 & - 2 & 2\end{array} \right]$$

A) $C = \left[ \begin{array} { c } \frac { 1 } { 3 } \\\frac { 2 } { 3 } \\0 \\0\end{array} \right]$ , $R = \left[ \begin{array} { l l l l } 0 & \frac { 1 } { 3 } & 0 & \frac { 2 } { 3 }\end{array} \right]$ , $e = - \frac { 2 } { 3 }$
B) $C = \left[ \begin{array} { c } 0 \\0 \\\frac { 2 } { 3 } \\\frac { 1 } { 3 }\end{array} \right]$ , $R = \left[ \begin{array} { l l l l } \frac { 1 } { 3 } & \frac { 2 } { 3 } & 0 & 0\end{array} \right]$ , $e = \frac { 2 } { 3 }$
C) $C = \left[ \begin{array} { c } \frac { 1 } { 3 } \\\frac { 2 } { 3 } \\0 \\0\end{array} \right]$ , $R = \left[ \begin{array} { l l l l } 0 & 0 & \frac { 1 } { 3 } & \frac { 2 } { 3 }\end{array} \right]$ , $e = \frac { 2 } { 3 }$
D) $C = \left[ \begin{array} { c } 0 \\0 \\\frac { 2 } { 3 } \\\frac { 1 } { 3 }\end{array} \right]$ , $R = \left[ \begin{array} { l l l l } 0 & 0 & \frac { 1 } { 3 } & \frac { 2 } { 3 }\end{array} \right]$ , $e = - \frac { 2 } { 3 }$
E) $C = \left[ \begin{array} { c } 0 \\0 \\\frac { 2 } { 3 } \\\frac { 1 } { 3 }\end{array} \right]$ , $R = \left[ \begin{array} { l l l l } 0 & 0 & \frac { 2 } { 3 } & \frac { 1 } { 3 }\end{array} \right]$ , $e = - \frac { 2 } { 3 }$

Question 19

We are to minimize $p = 3 a + 2 b$ subject to the following constraints.
$$\left\{ \begin{array} { c } a + b \geq 4 \\3 a + b \geq 6 \\a , b \geq 0\end{array} \right\}$$
What is the dual problem in matrix form

A) $\left[ \begin{array} { l l l } 1 & 1 & 4 \\3 & 1 & 6 \\3 & 2 & 0\end{array} \right]$
B) $\left[ \begin{array} { l l l } 1 & 1 & 4 \\3 & 1 & 6 \\1 & 1 & 0\end{array} \right]$
C) $\left[ \begin{array} { c c c } 1 & 1 & 4 \\3 & 1 & 6 \\- 3 & - 2 & 0\end{array} \right]$
D) $\left[ \begin{array} { c c c } 1 & 3 & - 3 \\1 & 1 & - 2 \\4 & 6 & 0\end{array} \right]$
E) $\left[ \begin{array} { l l l } 1 & 3 & 3 \\1 & 1 & 2 \\4 & 6 & 0\end{array} \right]$

Question 20

Write down (without solving) the dual LP problem.
Maximize $p = x + y + z + w$ subject to
$$\begin{array} { l } x + y + z \leq 9 \\y + z + w \leq 7 \\x + z + w \leq 2 \\x + y + w \leq 4 \\x \geq 0 , y \geq 0 , z \geq 0 , w \geq 0\end{array}$$

A)minimize $c = 9 s + 7 t + 2 u + 4 v$ subject to $$\begin{array} { l } s + u + v \leq 1 \\s + t + v \leq 1 \\s + t + u \leq 1 \\t + u + v \leq 1 \\s \geq 0 , t \geq 0 , u \geq 0 , v \geq 0\end{array}$$

B) minimize $c = s + t + u + v$ subject to
$$\begin{array} { l } s + t + u \leq 7 \\t + u + v \leq 6 \\s + u + v \leq 3 \\s + t + v \leq 4 \\s \geq 0 , t \geq 0 , u \geq 0 , v \geq 0\end{array}$$

C) minimize $c = s + t + u + v$ subject to $$\begin{array} { l } s + t + u \geq 9 \\t + u + v \geq 7 \\s + u + v \geq 2 \\s + t + v \geq 4 \\s \geq 0 , t \geq 0 , u \geq 0 , v \geq 0\end{array}$$

D) minimize $c = s + t + u + v$ subject to

$$\begin{array} { l } s + t + u \geq 9 \\t + u + v \geq 7 \\s + u + v \geq 2 \\s + t + v \geq 4 \\s \geq 0 , t \geq 0 , u \geq 0 , v \geq 0\end{array}$$

E) minimize $c = 9 s + 7 t + 2 u + 4 v$ subject to
$$\begin{array} { l } s + u + v \leq 1 \\s + t + v \leq 1 \\s + t + u \leq 1 \\t + u + v \leq 1 \\s \geq 0 , t \geq 0 , u \geq 0 , v \geq 0\end{array}$$

Question 21

Federal Rent-a-Car is putting together a new fleet. It is considering package offers from three car manufacturers. Fred Motors is offering 5 small cars, 5 medium cars, and 10 large cars for $500,000. Admiral Motors is offering 5 small, 10 medium, and 5 large cars for $400,000. Chrysalis is offering 10 small, 5 medium, and 5 large cars for $300,000. Federal would like to buy at least 600 small cars, at least 500 medium cars, and at least 600 large cars. How many packages should it buy from each car maker to keep the total cost as small as possible What will be the total cost ?

A)15 packages of Fred Motors, 35 packages of Admiral Motors, and 15 packages of Chrysalis, the total cost is $26,000,000
B) 35 packages of Fred Motors, 15 packages of Admiral Motors, and 35 packages of Chrysalis, the total cost is $34,000,000
C) 35 packages of Fred Motors, 35 packages of Admiral Motors, and 15 packages of Chrysalis, the total cost is $36,000,000
D) 15 packages of Fred Motors, 35 packages of Admiral Motors, and 35 packages of Chrysalis, the total cost is $32,000,000
E) 15 packages of Fred Motors, 15 packages of Admiral Motors, and 35 packages of Chrysalis, the total cost is $24,000,000

Question 22

Write down (without solving) the dual LP problem.
​
Maximize subject to
​ ​
Use the letters s, t​, u, and v in the indicated order.
​
Minimize __________ subject to the constraints.
​
Please enter the inequalities, separated by commas.

Question 23

The Enormous State University's Business School is buying computers. The school has two models to choose from. Each first model comes with 400 MB of memory and 80 GB of disk space, while each second model has 300 MB of memory and 100 GB of disk space. For reasons related to its accreditation, the school would like to be able to say that it has a total of at least 48,000 MB of memory and at least 12,800 GB of disk space. If both models cost $800 each, how many of each should the school buy to keep the cost as low as possible

__________ first model(s) and __________ second model(s).

What are the shadow costs of memory and disk space

Please, enter first the shadow cost of MB and then the shadow cost of GB as improper fractions of integers without units. Separate your answers by a comma.

Question 24

Solve the standard minimization problem using duality.
​
Minimize ​
subject to ​ __________
​ __________
​ ___________

Question 25

Solve the standard minimization problem using duality.

Minimize $c = 6 s + 2 t$
subject to $$\begin{array} { l } s + t \geq 9 \\6 s + t \geq 24 \\s \geq 0 , t \geq 0\end{array}$$
After using the simplex method for dual problem to arrive at the tableau below, what is the maximum value for c
$\quad \quad x\quad y\quad z \quad w \quad c$
$\left[ \begin{array} { c c c c c } 0&1&\frac{1}{5}&\frac{1}{-5}&0\\\frac{4}{5}\\1&0&\frac{1}{-5}&\frac{6}{5}&0\\\frac{6}{5}\\0&0&3&6&1\\30\end{array} \right]$

Question 26

Ruff 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 100 grams of fat. If chicken costs 25 cents/ounce and grain costs 1 cents/ounce, how many ounces of each should Ruff use in each bag of dog food in order to minimize cost

__________ oz of chicken and __________ oz of grain.

What are the shadow costs of protein and fat

__________ cent(s) per gram of protein and __________ cent(s) per gram of fat.

Question 27

Solve the standard minimization problem using duality.
​
Minimize ​
subject to ​ _________
​ _________
​ __________
​ __________

Question 28

Oz makes lion food out of giraffe and gazelle meat. Giraffe meat has 18 grams of protein and 36 grams of fat per pound, while gazelle meat has 36 grams of protein and 18 grams of fat per pound. A batch of lion food must contain at least 36,000 grams of protein and 54,000 grams of fat. Giraffe meat costs $2/pound and gazelle meat costs $4/pound. How many pounds of each should go into each batch of lion food in order to minimize costs Please round the answers to the nearest pound.

__________ pound(s) of giraffe meat and __________ pound(s) of gazelle.

What are the shadow costs of protein and fat Please round the answers to the nearest cent.

__________ cents per gram of protein and __________ cents per gram of fat.

Question 29

Solve the standard minimization problem using duality.
​
Minimize ​
subject to ​ __________
​ __________
​ __________
​ __________

Question 30

Each serving of Gerber Mixed Cereal for Baby contains 60 calories and no Vitamin C. Each serving of Gerber Mango Tropical Fruit Dessert contains 80 calories and 45 percent of the U.S. Recommended Daily Allowance (RDA) of Vitamin C for infants. Each serving of Gerber Apple Banana Juice contains 60 calories and 120 percent of the RDA of Vitamin C for infants. The cereal costs 10 cents per serving, the dessert costs 53 cents per serving, and the juice costs 27 cents per serving. If you want to provide your child with at least 120 calories and at least 120 percent of the RDA of Vitamin C, how can you do so at the least cost ?

A)1 serving of cereal, 1 serving of juice, and no dessert
B) 1 serving of dessert, 1 serving of juice, and no cereal
C) 1 serving of cereal, 3 servings of dessert, and no juice
D) 3 servings of dessert, 1 serving of juice, and no cereal
E) 1 serving of cereal, 1 serving of dessert, and no juice

Question 31

Write down (without solving) the dual LP problem.
​
Minimize subject to
​ ​
Use the letters x, y, z, and w in the indicated order. Please, use the numbers in the expression for ​p in order they come in the given LP problem.
​
Maximize __________ subject to the constraints.
​
Please enter the inequalities, separated with commas.

Question 32

Solve the standard minimization problem using duality.
​
Minimize ​
subject to ​ __________
​ __________
​ __________

Question 33

Solve the standard minimization problem using duality.
​
Minimize ​
subject to ​ __________
​ __________
​ __________

Question 34

Solve the standard minimization problem using duality.
​
Minimize ​
subject to ​

Question 35

Solve the standard minimization problem using duality. Minimize subject to the following.
​ ​ __________
​ __________
​ __________

Question 36

The Miami Beach City Council has offered to subsidize hotel development in Miami Beach, and it is hoping for at least two hotels with a total capacity of at least 2,600. Suppose that you are a developer interested in taking advantage of this offer by building a small group of hotels in Miami Beach. You are thinking of three prototypes: a convention-style hotel with 500 rooms costing $100 million, a vacation-style hotel with 200 rooms costing $20 million, and a small motel with 50 rooms costing $4 million. The City Council will approve your plans, provided you build at least one convention-style hotel and no more than two small motels. How many of each type of hotel should you build to satisfy the city council's wishes and stipulations while minimizing your total cost ?

A)1 convention-style hotel, 13 vacation-style hotels, and 2 small motels
B) 1 convention-style hotel, 1 vacation-style hotel, and 2 small motels
C) 2 convention-style hotels, 10 vacation-style hotels, and 1 small motel
D) 2 convention-style hotels, 1 vacation-style hotel, and 1 small motel
E) 1 convention-style hotel, 10 vacation-style hotels, and 2 small motels

Question 37

Write down ( without solving ) the dual LP problem.

Minimize $$p = 5 a + 4 b$$ subject to
$$\left\{ \begin{array} { c } a + b \geq 8 \\5 a + b \geq 20 \\a \geq 0 , b \geq 0\end{array} \right\}$$
What is the dual problem in matrix form

Question 38

Solve the standard minimization problem using duality.
Minimize subject to ​ __________
​ __________
​ __________

Question 39

We return to your exploits coordinating distribution for the Tubular Ride Boogie Board Company. You will recall that the company has manufacturing plants in Tucson, Arizona and Toronto, Ontario, and you have been given the job of coordinating distribution of their latest model, the Gladiator, to their outlets in Honolulu and Venice Beach. The Tucson plant can manufacture up to 620 boards per week, while the Toronto plant, beset by labor disputes, can produce no more than 410 Gladiator boards per week. The outlet in Honolulu orders 480 Gladiator boards per week, while Venice Beach orders 510 boards per week. Transportation costs are as follows: Tucson to Honolulu: $10 per board; Tucson to Venice Beach: $5 per board; Toronto to Honolulu: $20 per board; Toronto to Venice Beach: $10 per board. Your manager has said that you are to be sure to fill all orders and ship the boogie boards at a minimum total transportation cost. But you have just been notified that workers at the Toronto boogie board plant have gone on strike, resulting in a total work stoppage. You are to come up with a revised delivery schedule by tomorrow with the understanding that the Tucson plant can push production to a maximum of 880 boards per week. Can you devise a delivery schedule with these accommodations in mind ?

A)no
B) yes

Question 40

Solve the standard minimization problem using duality.
​
Minimize ​
subject to ​ __________
​ __________
​ __________

Question 41

Minimize subject to the following (using the simplex method).
​ ​ __________
​ __________
​ __________

Question 42

Minimize $c = 5.45 x + y + 1.5 z + w$
Subject to
$$\begin{aligned}5.12 x - y + w & \geq 1,500 \\z + w & \geq 1,500 \\1.12 x + y & \leq 1,000 \\x \geq 0 , y \geq 0 , z & \geq 0\end{aligned}$$

A) $c = 1,500$ , $x = 0$ , $y = 1,500$ , $z = 0$ , $w = 0$
B) $c = 1,500$ , $x = 0$ , $y = 0$ , $z = 0$ , $w = 1,500$
C) $c = 1,500$ , $x = 0$ , $y = 0$ , $z = 1,500$ , $w = 0$
D) $c = 8,175$ , $x = 1,500$ , $y = 0$ , $z = 0$ , $w = 0$
E) $c = 2,250$ , $x = 0$ , $y = 0$ , $z = 1,500$ , $w = 0$

Question 43

Minimize $c = 4 x + 7 y$
Subject to
$\begin{array} { l } 8 x + 3 y \geq 30 \\2 x + 3 y \geq 12 \\x \geq 0 , y \geq 0\end{array}$

A) $c = 24$ , $x = 6$ , $y = 0$
B) $c = 21$ , $x = 0$ , $y = 3$
C) $c = 6$ , $x = 3$ , $y = 0$
D) $c = 42$ , $x = 0$ , $y = 6$
E) no solution

Question 44

Minimize $c = 2 x + y + 3 z$
Subject to
$$\begin{aligned}x + y + z & \geq 150 \\2 x + y & \geq 30 \\y + z & \geq 40 \\x \geq 0 , y \geq 0 , z & \geq 0\end{aligned}$$

A) $c = 450$ , $x = 0$ , $y = 0$ , $z = 150$
B) $c = 300$ , $x = 150$ , $y = 0$ , $z = 0$
C) $c = 150$ , $x = 0$ , $y = 150$ , $z = 0$
D) $c = 30$ , $x = 0$ , $y = 30$ , $z = 0$
E) no solution

Question 45

Maximize
​ ​
subject to
​ ​
​
If there is no solution, do not fill the fields.
​ __________
​ __________
​ __________
​ __________
​ __________

Question 46

Maximize $p = 2 x + 3 y + z + 4 w$
Subject to
$\begin{aligned}x + y + z + w & \leq 40 \\2 x + y - z - w & \geq 10 \\x + y + z + w & \geq 10 \\x \geq 0 , y \geq 0 , z \geq 0 , w & \geq 0\end{aligned}$

A) $p = 65$ , $x = 25$ , $y = 0$ , $z = 15$ , $w = 0$
B) $p = 135$ , $x = 0$ , $y = 25$ , $z = 0$ , $w = 15$
C) $p = 130$ , $x = 0$ , $y = 15$ , $z = 0$ , $w = 15$
D) $p = 55$ , $x = 15$ , $y = 0$ , $z = 25$ , $w = 0$
E) no solution

Question 47

Your small farm encompasses 100 acres, and you are planning to grow tomatoes, lettuce, and carrots in the coming planting season. Fertilizer costs per acre are: $5 for tomatoes, $4 for lettuce, and $2 for carrots. Based on past experience, you estimate that each acre of tomatoes will require an average of 4 hours of labor per week, while tending to lettuce and carrots will each require an average of 2 hours per week. You estimate a profit of $2,000 for each acre of tomatoes, $1,500 for each acre of lettuce and $500 for each acre of carrots. You would like to spend at least $300 on fertilizer (your niece owns the company that manufactures it) and your farm laborers can supply up to 500 hours per week. How many acres of each crop should you plant to maximize total profits In this event, will you be using all 100 acres of your farm ?

A)plant 100 acres of tomatoes and no other crops, you will be using all 100 acres of your farm
B) plant 60 acres of tomatoes and no other crops, you will not be using all 100 acres of your farm
C) plant 100 acres of lettuce and no other crops, you will be using all 100 acres of your farm
D) plant 40 acres of carrots and no other crops, you will be using all 100 acres of your farm
E) plant 20 acres of lettuce and no other crops, you will be using all 100 acres of your farm

Question 48

Minimize $c = 2.2 x + y + 3.3 z$
Subject to
$$\begin{aligned}x + 1.5 y + 1.2 z & \geq 150 \\2 x + 1.5 y & \geq 40 \\1.5 y + 1.1 z & \geq 40 \\x \geq 0 , y \geq 0 , z & \geq 0\end{aligned}$$
Round all answers to two decimal places.

A) $c = 320$ , $x = 100$ , $y = 100$ , $z = 0$
B) $c = 330$ , $x = 0$ , $y = 0$ , $z = 100$
C) $c = 100$ , $x = 0$ , $y = 100$ , $z = 0$
D) $c = 550$ , $x = 100$ , $y = 0$ , $z = 100$
E) $c = 220$ , $x = 100$ , $y = 0$ , $z = 0$

Question 49

Maximize $p = x + 2 y$
Subject to
$$\begin{aligned}x + y & \leq 20 \\y & \geq 8 \\2 x - y & \geq 0 \\x \geq 0 , y & \geq 0\end{aligned}$$

A) $p = 80$ , $x = 40$ , $y = 20$
B) $p = 100$ , $x = 20$ , $y = 40$
C) $p = \frac { 80 } { 3 }$ , $x = \frac { 40 } { 3 }$ , $y = \frac { 20 } { 3 }$
D) $p = \frac { 100 } { 3 }$ , $x = \frac { 20 } { 3 }$ , $y = \frac { 40 } { 3 }$
E) no solution

Question 50

Minimize $c = 50 x + 50 y + 11 z$
Subject to
$$\begin{aligned}2 x + z & \geq 8 \\2 x + y - z & \geq 4 \\3 x + y - z & \leq 6 \\x \geq 0 , y \geq 0 , z & \geq 0\end{aligned}$$

A) $c = 399$ , $x = 5$ , $y = 1$ , $z = 9$
B) $c = 494$ , $x = 5$ , $y = 4$ , $z = 4$
C) $c = 161$ , $x = 2$ , $y = 1$ , $z = 1$
D) $c = 344$ , $x = 2$ , $y = 4$ , $z = 4$
E) no solution

Question 51

Maximize $p = 12 x + 10 y$
Subject to
$$\begin{aligned}x + y & \leq 40 \\x & \geq 1 \\- x + 2 y & \geq 0 \\x \geq 0 , y & \geq 0\end{aligned}$$

A) $p = 1,360$ , $x = 80$ , $y = 40$
B) $p = \frac { 560 } { 3 }$ , $x = \frac { 40 } { 3 }$ , $y = \frac { 80 } { 3 }$
C) $p = \frac { 1,360 } { 3 } , x = \frac { 80 } { 3 } , y = \frac { 40 } { 3 }$
D) $p = 1,280$ , $x = 40$ , $y = 80$
E) no solution

Question 52

Minimize $c = 5 x + y + z + 2 w$
Subject to
$$\begin{aligned}5 x - y + w & \geq 1,000 \\z + w & \leq 2,000 \\x + y & \leq 500 \\x \geq 0 , y \geq 0 , z \geq 0 , w & \geq 0\end{aligned}$$

A) $c = 600$ , $x = 0$ , $y = 200$ , $z = 200$ , $w = 200$
B) $c = 1,600$ , $x = 200$ , $y = 200$ , $z = 200$ , $w = 200$
C) $c = 1,000$ , $x = 200$ , $y = 0$ , $z = 0$ , $w = 0$
D) $c = 200$ , $x = 0$ , $y = 200$ , $z = 0$ , $w = 0$
E) no solution

Question 53

Fancy Pineapple produces pineapple juice and canned pineapple rings. This year the company anticipates a demand of at least 10,000 pints of pineapple juice and 500 cans of pineapple rings. Each pint of pineapple juice requires 2 pineapples, and each can of pineapple rings requires 1 pineapple. The company anticipates using at least 5,000 pineapples for these products. Each pint of pineapple juice costs the company 20 cents to produce, and each can of pineapple rings costs 50 cents to produce. How many pints of pineapple juice and cans of pineapple rings should Fancy Pineapple produce to meet the demand and minimize total costs ?

A)25,000 pints of pineapple juice and 500 cans of pineapple rings
B) 500 pints of pineapple juice and 10,000 cans of pineapple rings
C) 1,000 pints of pineapple juice and 25,000 cans of pineapple rings
D) 10,000 pints of pineapple juice and 500 cans of pineapple rings
E) 25,000 pints of pineapple juice and 1,000 cans of pineapple rings

Question 54

Maximize $p = 7 x + 9 y$ subject to the following (using the simplex method).
$$\begin{aligned}9 x + 8 y & \leq 144 \\x + y & \geq 8 \\8 x + 9 y & \leq 144 \\x \geq 0 , y & \geq 0\end{aligned}$$

A) $p = 112$ , $x = 16$ , $y = 0$
B) $p = 144$ , $x = 0$ , $y = 16$
C) $p = 56$ , $x = 8$ , $y = 0$
D) $p = 72$ , $x = 0$ , $y = 8$
E) no solution

Question 55

Maximize subject to the following (using the simplex method).
​ ​ __________
​ __________
​ __________

Question 56

The political pollster Canter is preparing for a national election. It would like to poll at least 2,400 Democrats and 2,400 Republicans. Each mailing to the East Coast gets responses from 100 Democrats and 50 Republicans. Each mailing to the Midwest gets responses from 100 Democrats and 100 Republicans. And each mailing to the West Coast gets responses from 50 Democrats and 100 Republicans. Mailings to the East Coast cost $40 each to produce and mail, mailings to the Midwest cost $60 each, and mailings to the West Coast cost $50 each. How many mailings should Canter send to each area of the country to get the responses it needs at the least possible cost What will it cost ?

A)16 mailings to the East Coast, none to the Midwest, 16 to the West Coast, the cost is $1,440
B) No mailings to the East Coast, 24 to the Midwest, none to the West Coast, the cost is $1,440
C) 24 mailings to the East Coast, none to the Midwest, 24 to the West Coast, another solution resulting in the same cost is no mailings to the East Coast, 16 to the Midwest, none to the West Coast, the cost is $1,440
D) 16 mailings to the East Coast, none to the Midwest, 16 to the West Coast, another solution resulting in the same cost is no mailings to the East Coast, 24 to the Midwest, none to the West Coast, the cost is $1,440
E) No mailings to the East Coast, 16 to the Midwest, none to the West Coast, the cost is $1,440

Question 57

Minimize $c = 50 x + 11 y + 50 z$
Subject to
$$\begin{array} { r } 3 x + z \geq 6 \\3 x + y - z \geq 6 \\4 x + y - z \leq 8 \\x \geq 0 , y \geq 0 , z \geq 0\end{array}$$

A) $c = 400$ , $x = 8$ , $y = 0$ , $z = 0$
B) $c = 716$ , $x = 8$ , $y = 6$ , $z = 5$
C) $c = 88$ , $x = 0$ , $y = 0$ , $z = 8$
D) $c = 100$ , $x = 2$ , $y = 0$ , $z = 0$
E) no solution

Question 58

Maximize $p = 2 x + 3 y + 1.1 z + 4 w$
Subject to
$$\begin{aligned}1.2 x + y + z + w & \leq 42 \\2.2 x + y - z - w & \geq 15 \\1.2 x + y + z + 1.2 w & \geq 11.5 \\x \geq 0 , y \geq 0 , z \geq 0 , w & \geq 0\end{aligned}$$
Round all answers to two decimal places.

A) $p = 111$ , $x = 28.5$ , $y = 0$ , $z = 0$ , $w = 13.5$
B) $p = 139.5$ , $x = 0$ , $y = 28.5$ , $z = 0$ , $w = 13.5$
C) $p = 154.5$ , $x = 0$ , $y = 13.5$ , $z = 0$ , $w = 28.5$
D) $p = 71.85$ , $x = 28.5$ , $y = 0$ , $z = 13.5$ , $w = 0$
E) $p = 100.35$ , $x = 0$ , $y = 28.5$ , $z = 13.5$ , $w = 0$

Question 59

Minimize
​ ​
subject to
​ ​
If there is no solution, do not fill the fields.
​ __________
​ __________
​ __________
​ __________

Question 60

Maximize $p = 3 x + 5 y + z$
Subject to
$$\begin{aligned}2 x - 6 y + 2 z & \leq 20 \\x + 3 y - 4 z & \leq 11 \\x \geq 0 , y \geq 0 , z & \geq 0\end{aligned}$$

A) $p = 116$ , $x = 9$ , $y = 14$ , $z = 13$
B) $p = 116$ , $x = 13$ , $y = 9$ , $z = 14$
C) $p = 116$ , $x = 14$ , $y = 9$ , $z = 13$
D) $p = 116$ , $x = 14$ , $y = 13$ , $z = 9$
E) no solution

Question 61

The political pollster Canter is preparing for a national election. It would like to poll at least 900 Democrats and 900 Republicans. Each mailing to the East Coast gets responses from 100 Democrats and 50 Republicans. Each mailing to the Midwest gets responses from 100 Democrats and 100 Republicans. And each mailing to the West Coast gets responses from 50 Democrats and 100 Republicans. Mailings to the East Coast cost $40 each to produce and mail, mailings to the Midwest cost $60 each, and mailings to the West Coast cost $50 each. How many mailings should Canter send to each area of the country to get the responses it needs at the least possible cost

__________ mailings to the East Coast, none to the Midwest, and __________ to the West Coast.

Another solution resulting in the same cost is no mailings to the East Coast, __________ to the Midwest, and none to the West Coast.

What will it cost

$__________

Question 62

You have $600,000 and are considering investing in three municipal bond funds, Pimco, Fidelity Spartan, and Columbia Oregon. You have the following data: $$\begin{array} { | c | c | } \hline \text { Company } & \text { Yield } \\\hline \text { PNF (Pimco NY) } & 7 \% \\\hline \text { FDMMX (Fidelity Spartan Mass) } & 5 \% \\\hline \text { CMBFX (Columbia Oregon) } & 4 \% \\\hline\end{array}$$ Your broker has made the following suggestions:
At least 50 percent of your total investment should be in CMBFX.
No more than 10 percent of your total investment should be in PNF.
How much should you invest in each fund to maximize your anticipated returns while following your broker's advice

A)$60,000 in PNF, $240,000 in FDMMX, and $300,000 in CMBFX
B) $240,000 in PNF, $300,000 in FDMMX, and $60,000 in CMBFX
C) $300,000 in PNF, $240,000 in FDMMX, and $60,000 in CMBFX
D) $300,000 in PNF, $60,000 in FDMMX, and $240,000 in CMBFX
E) $60,000 in PNF, $300,000 in FDMMX, and $240,000 in CMBFX

Question 63

The Miami Beach City Council has offered to subsidize hotel development in Miami Beach, and it is hoping for at least two hotels with a total capacity of at least 1,800. Suppose that you are a developer interested in taking advantage of this offer by building a small group of hotels in Miami Beach. You are thinking of three prototypes: a convention-style hotel with 500 rooms costing $100 million, a vacation-style hotel with 200 rooms costing $20 million, and a small motel with 50 rooms costing $4 million. The City Council will approve your plans, provided you build at least one convention-style hotel and no more than two small motels. How many of each type of hotel should you build to satisfy the city council's wishes and stipulations while minimizing your total cost

__________ convention-style hotel(s), __________ vacation-style hotel(s), and __________ small motel(s).

Now assume that the city council will give developers 20 percent of the cost of building new hotels in Miami Beach, up to $70 million. Will the city's $70 million subsidy be sufficient to cover 20 percent of your total costs

Question 64

Each serving of Gerber Mixed Cereal for Baby contains 60 calories and no Vitamin C. Each serving of Gerber Mango Tropical Fruit Dessert contains 80 calories and 45 percent of the U.S. Recommended Daily Allowance (RDA) of Vitamin C for infants. Each serving of Gerber Apple Banana Juice contains 60 calories and 120 percent of the RDA of Vitamin C for infants. The cereal costs 10 cents per serving, the dessert costs 53 cents per serving, and the juice costs 27 cents per serving. If you want to provide your child with at least 360 calories and at least 240 percent of the RDA of Vitamin C, how can you do so at the least cost
?
__________ serving(s) of cereal, __________ serving(s) of dessert, and __________ serving(s) of juice.

Question 65

Minimize
​ ​
subject to
​ ​
If there is no solution, do not fill the fields.
​ __________
​ __________
​ __________
​ __________
​ __________

Question 66

Minimize
​ ​
subject to
​ ​
If there is no solution, do not fill the fields.
​ __________
​ __________
​ __________
​ __________

Question 67

Maximize
​ ​
subject to
​ ​
Please enter your answer as three numbers, separated by commas in the order p, x, y. If there is no solution, enter no solution.

Question 68

Minimize
​ ​
subject to
​ ​
If there is no solution, do not fill the fields.
​ __________
​ __________
​ __________
​ __________

Question 69

We return to your exploits coordinating distribution for the Tubular Ride Boogie Board Company. You will recall that the company has manufacturing plants in Tucson, Arizona and Toronto, Ontario, and you have been given the job of coordinating distribution of their latest model, the Gladiator, to their outlets in Honolulu and Venice Beach. The Tucson plant can manufacture up to 620 boards per week, while the Toronto plant, beset by labor disputes, can produce no more than 410 Gladiator boards per week. The outlet in Honolulu orders 470 Gladiator boards per week, while Venice Beach orders 500 boards per week. Transportation costs are as follows: Tucson to Honolulu: $10 per board; Tucson to Venice Beach: $5 per board; Toronto to Honolulu: $20 per board; Toronto to Venice Beach: $10 per board. Your manager has said that you are to be sure to fill all orders and ship the boogie boards at a minimum total transportation cost. But you have just been notified that workers at the Toronto boogie board plant have gone on strike, resulting in a total work stoppage. You are to come up with a revised delivery schedule by tomorrow with the understanding that the Tucson plant can push production to a maximum of 810 boards per week. What should you do

Question 70

Federal Rent-a-Car is putting together a new fleet. It is considering package offers from three car manufacturers. Fred Motors is offering 5 small cars, 5 medium cars, and 10 large cars for $500,000. Admiral Motors is offering 5 small, 10 medium, and 5 large cars for $400,000. Chrysalis is offering 10 small, 5 medium, and 5 large cars for $300,000. Federal would like to buy at least 750 small cars, at least 600 medium cars, and at least 750 large cars. How many packages should it buy from each car maker to keep the total cost as small as possible

__________ package(s) of Fred Motors, __________ package(s) of Admiral Motors, and __________ package(s) of Chrysalis.

What will be the total cost

$__________

Question 71

Maximize
​ ​
subject to
​ ​
Round answers to two decimal places, if necessary.
​
If there is no solution, do not fill the fields.
​ __________
​ __________
​ __________
​ __________
​ __________

Question 72

Minimize
​ ​
subject to
​ ​
If there is no solution, do not fill the fields.
​ __________
​ __________
​ __________
​ __________
​ __________

Question 73

Use the simplex method to maximize $2 x + 9 y = p$ , subject to the following. $$\left\{ \begin{array} { c } 2 x + y \leq 16 \\x + 2 y \leq 8 \\x \geq 0 , y \geq 0\end{array} \right\}$$

A) $p = 110$
B) $p = 144$
C) $p = 36$
D) $p = 120$
E) $p = 72$

Question 74

Fancy Pineapple produces pineapple juice and canned pineapple rings. This year the company anticipates a demand of at least 15,000 pints of pineapple juice and 2,500 cans of pineapple rings. Each pint of pineapple juice requires 2 pineapples, and each can of pineapple rings requires 1 pineapple. The company anticipates using at least 10,000 pineapples for these products. Each pint of pineapple juice costs the company 20 cents to produce, and each can of pineapple rings costs 50 cents to produce. How many pints of pineapple juice and cans of pineapple rings should Fancy Pineapple produce to meet the demand and minimize total costs

__________ pints of pineapple juice and __________ cans of pineapple rings.

Question 75

Minimize
​ ​
subject to
​ ​
Round answers to two decimal places, if necessary.
​
If there is no solution, do not fill the fields.
​ __________
​ __________
​ __________
​ __________

Question 76

Your publishing company is about to start a promotional blitz for its new book, Physics for the Liberal Arts. You have 35 salespeople stationed in Chicago and 10 in Denver. You would like to fly at most 20 into Los Angeles and at most 20 into New York. A round-trip plane flight from Chicago to LA costs $195; from Chicago to NY costs $182; from Denver to LA costs $395; and from Denver to NY costs $166. You want to spend at most $7,380 on plane flights. How many salespeople should you fly from each of Chicago and Denver to each of LA and NY to have the most salespeople on the road ?

A)20 people from Chicago to LA, 10 people from Chicago to NY, no people from Denver to LA, and 10 people from Denver to NY
B) 10.0000000000 people from Chicago to LA, 10.0000000000 people from Chicago to NY, 20.0000000000 people from Denver to LA, and no people from Denver to NY
C) 10.0000000000 people from Chicago to LA, no people from Chicago to NY, 10.0000000000 people from Denver to LA, and 20.0000000000 people from Denver to NY
D) 10 people from Chicago to LA, 20 people from Chicago to NY, no people from Denver to LA, and 10 people from Denver to NY
E) 10 people from Chicago to LA, no people from Chicago to NY, 10 people from Denver to LA, and 10 people from Denver to NY

Question 77

Maximize
​ ​
subject to
​ ​
Please enter your answer as three numbers, separated by commas in the order p, x, y. If there is no solution, enter no solution.

Question 78

The University's employee credit union has $4 million available for loans in the coming year. As VP in charge of finances, you must decide how much capital to allocate to each of four different kinds of loans, as shown in the table. $$\begin{array} { | c | c | } \hline \text { Type of Loan } & \begin{array} { c } \text { Annual Rate of } \\\text { Return }\end{array} \\\hline \text { Automobile } & 8 \% \\\hline \text { Furniture } & 10 \% \\\hline \text { Signature } & 12 \% \\\hline \text { Other secured } & 10 \% \\\hline\end{array}$$ State laws and credit union policies impose the following restrictions:
Signature loans may not exceed 10 percent of the total investment of funds.
Furniture loans plus other secured loans may not exceed automobile loans.
How much should you allocate to each type of loan to maximize the annual return

A)allocate $1,800,000 to automobile loans, $400,000 to signature loans, $1,800,000 to furniture loans and no allocation to other secured loans
B) allocate $1,800,000 to automobile loans, $400,000 to signature loans, no allocation to furniture loans and $1,800,000 other secured loans
C) allocate $400,000 to automobile loans, $1,800,000 to signature loans, and $1,800,000 to any combination of furniture loans and other secured loans
D) allocate $1,800,000 to automobile loans, $400,000 to signature loans, and $1,800,000 to any combination of furniture loans and other secured loans
E) allocate $1,800,000 to automobile loans, $400,000 to signature loans, and $400,000 to any combination of furniture loans and other secured loans

Question 79

Your small farm encompasses 150 acres, and you are planning to grow tomatoes, lettuce, and carrots in the coming planting season. Fertilizer costs per acre are: $5 for tomatoes, $4 for lettuce, and $2 for carrots. Based on past experience, you estimate that each acre of tomatoes will require an average of 4 hours of labor per week, while tending to lettuce and carrots will each require an average of 2 hours per week. You estimate a profit of $2,000 for each acre of tomatoes, $1,500 for each acre of lettuce and $500 for each acre of carrots. You would like to spend at least $600 on fertilizer (your niece owns the company that manufactures it) and your farm laborers can supply up to 900 hours per week. How many acres of each crop should you plant to maximize total profits

Plant __________ acres of tomatoes, __________ acres of lettuce, and __________ acres of carrots.

In this event, will you be using all 150 acres of your farm

Question 80

If x dollars is invested in a company that controls, say, 30 percent of the market with 5 brand-names, then $0.3 x$ is a measure of market exposure and $5 x$ is a measure of brand-name exposure. Now suppose you are a broker at a large securities firm, and one of your clients would like to invest up to $100,000 in recording industry stocks. You decide to recommend a combination of stocks in four of the world's largest companies: Warner Music, Universal Music, Sony, and EMI. $$\begin{array} { | c | c | c | c | c | } \hline & \text { Warner Music } & \text { Universal Music } & \text { Sony } & \text { EMI } \\\hline \text { Market Share } & 12 \% & 20 \% & 20 \% & 15 \% \\\hline \begin{array} { c } \text { Number of } \\\text { brand-names }\end{array} & 8 & 20 & 10 & 15 \\\hline\end{array}$$
You wanted your client to maximize his total market exposure but limit his brand-name exposure to 2 million or less (representing an average of 20 labels or fewer per company), and still invest at least 20 percent of the total in Universal. How much should you advise your client to invest in each company

A)$20,000 in Warner Music, $80,000 in Sony, and no investment in Universal Music and EMI
B) $80,000 in Warner Music, $20,000 in EMI, and no investment in Universal Music and Sony
C) $20,000 in Universal Music, $80,000 in Sony, and no investment in Warner Music and EMI
D) $80,000 in Warner Music, $20,000 in Sony, and no investment in Universal Music and EMI
E) $80,000 in Universal Music, $20,000 in Sony, and no investment in Warner Music and EMI