# Mathematics

**Question 1**

Management science techniques focus primarily on observation, model construction and implementation to find an appropriate solution to a problem.

**Question 2**

In linear programming problems, multiple optimal solutions occur when constraints are parallel to each other.

**Question 3**

A change in the value of an objective function coefficient will always change the value of the optimal solution.

**Question 4**

Fractional relationships between variables are not permitted in the standard form of a linear program.

**Question 5**

In a total integer model, all decision variables have integer solution values.

**Question 6**

In a transshipment problem, items may be transported from destination to destination and from source to source.

**Question 7**

The events in an experiment are mutually exclusive if only one can occur at a time.

**Question 8**

The minimax criterion minimizes the maximum payoff.

**Question 9**

Excel can be used to simulate systems that can be represented by both discrete and continuous random variables.

**Question 10**

Adjusted exponential smoothing is an exponential smoothing forecast adjusted for seasonality.

**Question 11**

Which of the following is an equation or an inequality that expresses a resource restriction in a mathematical model?

**Question 12**

If the price decreases but fixed and variable costs do not change, the break even point:

**Question 13**

A slack variable:

**Question 14**

Cully furniture buys 2 products for resale: big shelves (B) and medium shelves (M). Each big shelf costs $500 and requires 100 cubic feet of storage space, and each medium shelf costs $300 and requires 90 cubic feet of storage space. The company has $75000 to invest in shelves this week, and the warehouse has 18000 cubic feet available for storage. Profit for each big shelf is $300 and for each medium shelf is $150. What is the storage space constraint?

**Question 15**

The following is an Excel “Answer” and “Sensitivity” reports of a linear programming problem:

**The Answer Report:**

**The Sensitivity Report: **

Which additional resources would you recommend to be increased?

**Question 16**

Given the following linear programming problem that minimizes cost.

Min Z = 2x + 8y

Subject to 8x + 4y ≥ 64

2x + 4y ≥ 32

y ≥ 2

What is the sensitivity range for the third constraint, y ≥ 2?

**Question 17**

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

**Question 18**

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:

**Question 19**

The Wiethoff Company has a contract to produce garden hoses for a customer. Wiethoff has 4 different machines that can produce this kind of hose. Write a constraint to ensure that if machine 4 is used, machine 1 will not be used.

**Question 20**

If we are solving a 0-1 integer programming problem, the constraint x1 = x2 is a __________ constraint.

**Question 21**

The following table represents the cost to ship from Distribution Center 1, 2, or 3 to Customer A, B, or C.

The constraint that represents the quantity demanded by Customer B is:

**Question 22**

A professor needs help from 3 student helpers to complete 4 tasks. The first task is grading; the second is scanning; the third is copying, and the fourth is organizing student portfolios. The estimated time for each student to do each task is given in the matrix below.

Which of the following constraints represents the assignment for student A?

**Question 23**

Mutually exclusive events are:

**Question 24**

Jim is considering pursuing an MS in Information Systems degree. He has applied to two different universities. The acceptance rate for applicants with similar qualifications is 20% for University X and 45% for University Y. What is the probability that Jim will not be accepted at either university?

**Question 25**

Determining the worst payoff for each alternative and choosing the alternative with the best worst is called :

**Question 26**

A business owner is trying to decide whether to buy, rent, or lease office space and has constructed the following payoff table based on whether business is brisk or slow.

**Question 27**

For the following frequency distribution of demand, the random number 0.8177 would be interpreted as a demand of:

**Question 28**

In the Monte Carlo process, values for a random variable are generated by __________ a probability distribution.

**Question 29**

Given an actual demand of 59, a previous forecast of 64, and an alpha of .3, what would the forecast for the next period be using simple exponential smoothing?

**Question 30**

__________ moving averages react more slowly to recent demand changes than do __________ moving averages.

**Question 31**

A production process requires a fixed cost of $50,000. The variable cost per unit is $25 and the revenue per unit is projected to be $45. Find the break-even point.

**Question 32**

Tracksaws, Inc. makes tractors and lawn mowers. The firm makes a profit of $30 on each tractor and $30 on each lawn mower, and they sell all they can produce. The time requirements in the machine shop, fabrication, and tractor assembly are given in the table.

Formulation:

Let x = number of tractors produced per period

y = number of lawn mowers produced per period

MAX 30x + 30y

subject to 2 x + y ≤ 60

2 x + 3y ≤ 120

x ≤ 45

x, y ≥ 0

The graphical solution is shown below.

What is the shadow price for fabrication? Write your answers with two significant places after the decimal and do not include the dollar “$” sign.

**Question 33**

Consider the following linear program, which maximizes profit for two products, regular (R), and super (S):

MAX

50R + 75S

s.t.

1.2R + 1.6 S ≤ 600 assembly (hours)

0.8R + 0.5 S ≤ 300 paint (hours)

.16R + 0.4 S ≤ 100 inspection (hours)

**Sensitivity Report:**

**Final**

**Reduced**

**Objective**

**Allowable**

**Allowable**

**Cell**

**Name**

**Value**

**Cost**

**Coefficient**

**Increase**

**Decrease**

$B$7

Regular =

291.67

0.00

50

70

20

$C$7

Super =

133.33

0.00

75

50

43.75

**Final**

**Shadow**

**Constraint**

**Allowable**

**Allowable**

**Cell**

**Name**

**Value**

**Price**

**R.H. Side**

**Increase**

**Decrease**

$E$3

Assembly (hr/unit)

563.33

0.00

600

1E+30

36.67

$E$4

Paint (hr/unit)

300.00

33.33

300

39.29

175

$E$5

Inspect (hr/unit)

100.00

145.83

100

12.94

40

A change in the market has increased the profit on the super product by $5. Total profit will increase by __________. Write your answers with two significant places after the decimal and do not include the dollar “$” sign.

**Question 34**

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 optimal cost of this plan? Write your answers with two significant places after the decimal and do not include the dollar “$” sign.

**Question 35**

Find the optimal Z value for the following problem. Do not include the dollar “$” sign with your answer.

MAX Z = 5×1 + 8×2

s.t. x1 + x2 ≤ 6

5×1 + 9×2 ≤ 45

x1, x2 ≥ 0 and integer

**Question 36**

A company markets educational software products, and is ready to place three new products on the market. Past experience has shown that for this particular software, the chance of “success” is 80%. Assume that the probability of success is independent for each product. Find the probability that at least two of the three is successful. Round your result to four significant places after the decimal.

**Question 37**

The local operations manager for the IRS must decide whether to hire 1, 2, or 3 temporary workers. He estimates that net revenues will vary with how well taxpayers comply with the new tax code. The probabilities of low, medium, and high compliance are 0.20, 0.30, and 0.50 respectively. What are the expected net revenues for the number of workers he will decide to hire? Do not include the dollar “$” sign with your answer.

**Question 38**

An investor is consider 4 different opportunities, A, B, C, or D. The payoff for each opportunity will depend on the economic conditions, represented in the payoff table below.

.

. Economic Condition

. Poor Average Good Excellent

. __Investment (S1) (S2) (S3) (S4)__

. A 50 75 20 30

. B 80 15 40 50

. C -100 300 -50 10

. D 25 25 25 25

.

Suppose all states of the world are equally likely (each state has a probability of 0.25). What is the expected value of perfect information? Round your answer to the nearest integer.

**Question 39**

Consider the following decision tree. The objective is to choose the best decision among the two available decisions A and B. Find the expected value of the best decision. Do not include the dollar “$” sign with your answer.

**Question 40**

Recent past demand for product ZXT is given in the following table.