# Mathematics

**Question 1**

1.

Fractional relationships between variables are permitted in the standard form of a linear program.

Answer

True

False

**2 points **

**Question 2**

1.

In formulating a typical diet problem using a linear programming model, we would expect most of the constraints to be related to calories.

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Answer

True

False

**2 points **

**Question 3**

1.

A constraint for a linear programming problem can never have a zero as its right-hand-side value.

** **

Answer

True

False

**2 points **

**Question 4**

1.

In a media selection problem, instead of having an objective of maximizing profit or minimizing cost, generally the objective is to maximize the audience exposure.

** **

Answer

True

False

**2 points **

**Question 5**

1.

Product mix problems cannot have “greater than or equal to” (≥) constraints.

** **

Answer

True

False

**2 points **

**Question 6**

1.

When using a linear programming model to solve the “diet” problem, the objective is generally to maximize profit.

Answer

True

False

**2 points **

**Question 7**

1.

The owner of Black Angus Ranch is trying to determine the correct mix of two types of beef feed, A and B which cost 50 cents and 75 cents per pound, respectively. Five essential ingredients are contained in the feed, shown in the table below. The table also shows the minimum daily requirements of each ingredient.

Ingredient | Percent per pound in Feed A | Percent per pound in Feed B | Minimum daily requirement (pounds) |

1 | 20 | 24 | 30 |

2 | 30 | 10 | 50 |

3 | 0 | 30 | 20 |

4 | 24 | 15 | 60 |

5 | 10 | 20 | 40 |

The constraint for ingredient 3 is:

Answer

.5A + .75B = 20

.3B = 20

.3 B≤ 20

.3B ≥ 20

**2 points **

**Question 8**

1.

Let xij = gallons of component i used in gasoline j. Assume that we have two components and two types of gasoline. There are 8,000 gallons of component 1 available, and the demand gasoline types 1 and 2 are 11,000 and 14,000 gallons respectively. Write the supply constraint for component 1.

Answer

x21 + x22 ≤ 8000

x12 + x22 ≥ 8000

x11 + x12 ≤ 8000

x21 + x22 ≥ 8000

**2 points **

**Question 9**

1.

If Xij = the production of product i in period j, write an expression to indicate that the limit on production of the company’s 3 products in period 2 is equal to 400.

Answer

X21 + X22 + X23 ≥ 400

X21 + X22 + X23 ≤ 400

X12 + X22 + X32 ≥ 400

X12 + X22 + X32 ≤ 400

**2 points **

**Question 10**

1.

The production manager for the Softy soft drink company is considering the production of 2 kinds of soft drinks: regular and diet. Two of her resources are production time (8 hours = 480 minutes per day) and syrup (1 of the ingredients) limited to 675 gallons per day. To produce a regular case requires 2 minutes and 5 gallons of syrup, while a diet case needs 4 minutes and 3 gallons of syrup. Profits for regular soft drink are $3.00 per case and profits for diet soft drink are $2.00 per case. What is the time constraint?

Answer

2R + 4D ≤ 480

2D + 4R ≤ 480

2R + 3D ≤ 480

3R + 2D ≤ 480

**2 points **

**Question 11**

1.

Assume that x2, x7 and x8 are the dollars invested in three different common stocks from New York stock exchange. In order to diversify the investments, the investing company requires that no more than 60% of the dollars invested can be in “stock two”. The constraint for this requirement can be written as:

Answer

.4×2 – .6×7 – .6×8 ≤ 0

x2 ≥ .60 (x2 + x7 + x8)

.4×2 – .6×7 – .6×8 ≥ 0

-.4×2 + .6×7 + .6×8 ≤ 0

**2 points **

**Question 12**

1.

In a portfolio problem, X1, X2, and X3 represent the number of shares purchased of stocks 1, 2, an 3 which have selling prices of $15, $47.25, and $110, respectively. The investor has up to $50,000 to invest. The stockbroker suggests limiting the investments so that no more than $10,000 is invested in stock 2 or the total number of shares of stocks 2 and 3 does not exceed 350, whichever is more restrictive. How would this be formulated as a linear programming constraint?

Answer

X2 ≤ 10000

X2 + X3 ≤350

10,000 X2 ≤ 350X2 + 350X3

47.25X2 ≤10,000

X2 + X3 ≤ 350

47.25X2 ≤10,000

47.25 X2 + 110X3 ≤ 350

**2 points **

**Question 13**

1.

A systematic approach to model formulation is to first

Answer

construct the objective function

develop each constraint separately

define decision variables

all of the above

** ****2 points **

**Question 14**

1. The production manager for the Softy soft drink company is considering the production of 2 kinds of soft drinks: regular and diet. Two of her resources are constraint production time (8 hours = 480 minutes per day) and syrup (1 of her ingredient) limited to 675 gallons per day. To produce a regular case requires 2 minutes and 5 gallons of syrup, while a diet case needs 4 minutes and 3 gallons of syrup. Profits for regular soft drink are $3.00 per case and profits for diet soft drink are $2.00 per case. What is the optimal daily profit?

Answer

$220

$420

$320

$280

** ****2 points **

**Question 15**

1.

A croissant shop produces 2 products: bear claws (B) and almond filled croissants (C). Each bear claw requires 6 ounces of flour, 1 ounce of yeast, and 2 TS of almond paste. An almond filled croissant requires 3 ounces of flour, 1 ounce of yeast, and 4 TS of almond paste. The company has 6600 ounces of flour, 1400 ounces of yeast, and 4800 TS of almond paste available for today’s production run. Bear claw profits are 20 cents each, and almond filled croissant profits are 30 cents each. What is the optimal daily profit?

Answer

$380

$400

$420

$440

**2 points **

**Question 16**

1.

The following types of constraints are ones that might be found in linear programming formulations:

1. | ≤ |

2. | = |

3. | > |

Answer

1 and 2

2 and 3

1 and 3

all of the above

**2 points **

**Question 17**

1.

Small motors for garden equipment is produced at 4 manufacturing facilities and needs to be shipped to 3 plants that produce different garden items (lawn mowers, rototillers, leaf blowers). The company wants to minimize the cost of transporting items between the facilities, taking into account the demand at the 3 different plants, and the supply at each manufacturing site. The table below shows the cost to ship one unit between each manufacturing facility and each plant, as well as the demand at each plant and the supply at each manufacturing facility.

What is the demand constraint for plant B?

Answer

x 1B + x 2B +x 3B = 600

x B1 + x B2 +x B3 = 150

x 1B + x 2B +x 3B = 150

none of the above

**2 points **

Question 18

1.

Compared to blending and product mix problems, transportation problems are unique because

Answer

They maximize profit.

The constraints are all equality constraints with no “≤” or “≥” constraints.

They contain fewer variables.

The solution values are always integers.

**2 points **

Question 19

1.

Kitty Kennels provides overnight lodging for a variety of pets. An attractive feature is the quality of care the pets receive, including well balanced nutrition. The kennel’s cat food is made by mixing two types of cat food to obtain the “nutritionally balanced cat diet.” The data for the two cat foods are as follows:

Kitty Kennels wants to be sure that the cats receive at least 5 ounces of protein and at least 3 ounces of fat per day. What is the cost of this plan? *Express your answer with two places to the right of the decimal point.** **For instance, $9.32** **(nine dollars and thirty-two cents) would be written as 9.32*

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**2 points **

Question 20

1.

Quickbrush Paint Company makes a profit of $2 per gallon on its oil-base paint and $3 per gallon on its water-base paint. Both paints contain two ingredients, A and B. The oil-base paint contains 90 percent A and 10 percent B, whereas the water-base paint contains 30 percent A and 70 percent B. Quickbrush currently has 10,000 gallons of ingredient A and 5,000 gallons of ingredient B in inventory and cannot obtain more at this time. The company wishes to use linear programming to determine the appropriate mix of oil-base and water-base paint to produce to maximize its total profit. How many gallons of water based paint should the Quickbrush make? *Note: Please express your answer as a whole number, rounding the nearest whole number, if appropriate.*

Answer

** **

**2 points **