# Quantitative Methods: Management Science and Probability and Stats

Please refer to the attached ms word file.

#1: The Willow Furniture Company produces tables. The fixed monthly cost of production is $8,000, and the variable cost per table is $65. The tables sell for $180 a piece.

a. For a monthly volume of 300 tables, determine the total cost, total revenue, and profit.

b. Determine the monthly break-even volume for the Willow Furniture Company.

#3: The Retread Tire Company recaps tires. The fixed annual cost of the recapping operation is $60,000. The variable cost of recapping a tire is $9. The company charges $25 to recap a tire.

a. For a monthly volume of $18,000 yards of denim, determine the total cost, total revenue, and profit.

b. Determine the annual break-even volume for the Rolling Creek Textile Mill.

#7: Andy Mendoza makes handcrafted dolls, which he sells at craft fairs. He is considering mass producing the dolls to sell in stores. He estimates that the initial investment for plant and equipment will be $25,000, whereas labor, material, packaging, and shipping will be about $10 per doll. If the dolls are sold for $30 each, what sales volume is necessary for Andy to break even?

#9: If the maximum operating capacity of the Rolling Creek Textile Mill described in problem 3 is 25,000 yards of denim per month, determine the break-even volume as a percentage of capacity.

#17: Andy Mendoza in problem 7 is concerned that the demand for his dolls will not exceed the break-even point. He believes he can reduce his initial investment by purchasing used sewing machines and fewer machines. This will reduce his initial investment from $25,000 to $17,000. However, it will also require his employees to work more slowly and perform more operations by hand, thus increasing variable cost from $10 to $14 per doll. Will these changes reduce his breakeven point?

#23. Consider a model in which two products, x and y, are produced. There are 100 pounds of material and 80 hours of labor available. It requires 2 pounds of material and 4 hours of labor to produce a unit of x, and 1 pound of material and 5 hours of labor to produce a unit of y. The profit for x is $30 per unit and the profit for y is $50 per unit. If we want to know how many units of x and y to produce to maximize profit, the model is

maximize Z = 30x + 50y

subject to

2x + 4y = 100

x + 5y = 80

Determine the solution to this problem and explain your answer.

#17. The Ramshead Pub sells a large quantity of beer every Saturday. From past sales records the pub has determined the following probabilities for sales. Compute the expected number of barrels that will be sold on Saturday.

Barrels Probability

6 .10

7 .20

8 .40

9 .25

10 .05

1.00

#18. The following probabilities for grades in management science have been determined based on past records. The grades are assigned on a 4.0 scale, where an A is a 4.0, a B a 3.0, and so on. Determine the expected grade and variance for the course.

Grade Probability

A .10

B .30

C .40

D .10

F .10

29. A warehouse distributor of carpet keeps 6,000 yards of deluxe shag carpet in stock during a month. The average demand for carpet from the stores that purchase from the distributor is 4,500 yards per month with a deviation of 900 yards. What is the probability that a customer's order will not be met during a month? (This situation is referred to as a stockout.)

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#### Solution Summary

This solution answers problems related to total cost, revenue, profit and sales problems.

Quantitative Methods

Management Science

1. There is a fixed cost of $50,000 to start a production process. Once the process has begun, the variable cost per unit is $25. The revenue per unit is projected to be $45. Find the break-even point.

2. Administrators at a university are planning to offer a summer seminar. It costs $3000 to reserve a room, hire an instructor, and bring in the equipment. Assume it costs $25 per student for the administrators to provide the course materials. If we know that 20 people will attend, what price should be charged per person to break even?

Probability and Statistics

3. A new county hospital is attempting to determine whether it needs to add a particular specialist to its staff. Five percent of the general hospital population in the county contracts the illness the specialist would treat. If 12 patients check into the hospital in a day, what is the probability that 4 or more will have the illness?

4. A loaf of bread is normally distributed with a mean of 22 ounces and a standard deviation of 0.5 ounces. What is the probability that a loaf is larger than 21 ounces?

Decision Analysis

5. A payoff table (profits) is shown below.

States of Nature

Decisions S1 S2 S3

D1 10 8 6

D2 14 15 2

D3 7 8 9

a. Using the maximax criterion, what decision should be made by the decision maker?

b. Using the maximin criterion, what decision should be made by the decision maker?

c. Using an equal likelihood criterion, what decision should be made by the decision maker?

d. Using minimax regret criterion, what decision should be made by the decision maker?

e. If the probabilities of s1, s2, and s3 are 0.2, 0.4, and 0.4, respectively, what decision should be made by the decision maker?

6. 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 0.3, 0.4, and 0.3, respectively, and the payoff table is shown below. Using expected values, determine how many workers the company should hire.

# of workers Low compliance Medium compliance High compliance

1 50 50 50

2 20 60 100

3 -10 70 150

Simulation

7. The number of cars arriving at Joe Kelly's oil change and tune-up place during the last 200 hours of operation is observed to be the following.

Number of cars arriving Frequency

4 10

5 30

6 70

7 50

8 40

a. Determine the probability distribution, and the cumulative probability distribution of car arrivals.

b. Simulate 20 hours of car arrivals at Joe Kelly's oil change and tune-up place.

c. For the simulation in (b), what is the average number of cars arriving per hour?

Forecasting

8. Given the following data on hotel check-ins for a 6-month period:

Month Number of rooms

July 70

August 105

September 90

October 120

November 110

December 115

a. What is the 3-month moving average for January?

b. What is the 5-month moving average for January?

9. Recent actual and forecasted data for product XYZ is given in the following table.

Month Actual Demand 3-month Forecasted Demand

May 35 25

June 31 30

July 48 33

August 41 38

Determine the MSE, MAD, cumulative error and average error.

10. The following table summarizes data between money spent on gambling and winnings for Robert.

Money Spent Money Won

x y

12 62

10 54

16 86

18 100

15 80

10 57

5 26

12 60

22 105

25 140

Develop a linear regression equation for these data and forecast how much money Robert will win if he spends $30.

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