Attach computer work with the homework (add all printouts after the seventh problem).

1. Ace Advertising has won the contract to promote the Macaw Heads upcoming concert in Aldrich. They have identified four relevant advertising outlets and using Nielson and Arbitron data gathered the information in Table 1 about each advertising outlet. The advertising budget is limited to $50,000. At least 500 thousand of the effective audience contact should be individuals over the age of 40. For TV ads, the effective audience contact (and thus the percent of audience over 40) drops by 20% after the 20th ad (include a decision variable for both maximum and reduced effectiveness TV ad). The newspaper has agreed to cut the cost of an ad by 50% for every additional ad purchased after the fifth ad. Thus the first five ads cost the regular price. After the 6th regular price ad is purchased, the first reduced ad can be purchased. After the seventh regular price ad is purchased, the second reduced ad can be purchased (include a decision variable for the number of regular ads purchased and the number of reduced cost ads purchase and be sure to require that the number of regular cost ads is always five more than the reduced cost ads). At most, only 30 newspaper ads are possible. For every 106.7 radio ad, there must be at least two 103.1 Radio ads. Formulate the LP model to determine the number of ads in each outlet and maximize the effective audience contact.

.

Advertising Effective Audience Percent of Effective Cost per ad

Outlets Contact (000) Audience over 40

Radio station 103.1 4 80% $200

Radio station 106.7 6 20% $300

Newspaper 8 60% $400

TV 30 50% $1000

1a). How many individuals over the age of 40 will be exposed to the ads? _________

1b). What would be the effective audience contact if the advertising budget was $60,000?

1c). More ads would be purchased for Radio station 106.7 if the effective audience contact was equal to what value?

_____________

1d). How many newspaper ads should be purchased? _______________

2. Island Water Sports is a business that provides rental equipment and instruction for a variety of water sports in a resort town. On one particular morning, a decision must be made of how many Wildlife Raft Trips and how many Group Sailing Lessons should be scheduled. Each Wildlife Raft Trip requires one captain and one crew person, and can accommodate six passengers. The revenue per raft trip is $120. There are ten rafts available, and at least 30 people are on the list for reservations this morning. Each Group Sailing Lesson requires one captain and two crew people for instruction. Two boats are needed for each group. Four students will form each group. There are 12 sailboats available, and at least 20 people are on the list for sailing instruction this morning. The revenue per group sailing lesson is $160. The company has 12 captains and 18 crew available this morning. The company would like to maximize the number of customers served while generating at least $1500 in revenue and honoring all reservations. Formulate the LP model to determine how many Raft trips and how many Sailing lessons should be scheduled.

2a, How many customers will be served? _________

2b. How may rafting trips should be scheduled? ____________

2c. What will be the total revenue? _________

2d. How many customers would be served if only 11 captains were available? _______

2e. It would only be beneficial to offer more sailing lessons if the number of customers per sailing lesson was equal to what value?

_________

3. A director for a summer camp recreation is trying to choose activities for a rainy day. Information about possible choices is given in the table below.

Category

Activity Time

(minutes) Popularity

with Campers Popularity with

Counselors

Art 1. Painting 30 2 2

2. Drawing 20 1 2

3. Nature Craft 30 3 1

Music 4. Rhythm band 20 1 5

Sports 5. Relay races 45 4 1

6. Basketball 60 5 3

Computer 7. Internet 45 5 1

8. Creative writing 30 2 3

9. Games 40 5 2

The popularity ratings are defined so that 5 is the most popular.

The objective is to keep the campers as happy as possible.

Formulate the integer LP model with the following conditions also apply:

a. At most one art activity can be done.

b. No more than two computer activities can be done.

c. If basketball is chosen, then the music must be chosen.

d. At least 120, but not more than 165 minutes of activities may be selected

e. To keep the staff happy, the counselor rating should sum of the chosen activities should sum to at least 10.

3a. Which activities will be selected?

3b. What is the total popularity score with the campers for the chosen activities? _________

3c. What is the total popularity score with the counselors for the chosen activities? ________

d. How much time is required to complete all the chosen activities? _________

e. How many constraints are binding? ______

4. Phil’s Tennis Emporium Club has six indoor tennis courts. Phil is perplexed about how much time to set aside during the club’s prime time hours this winter for each of the six revenue producing activities listed in the table below. Prime time is from 5:00 PM to 10 PM Sunday through Friday (i.e, each of the six courts is available for five hours per night, six nights per week). To satisfy club members, Phil has decided that at least one-half of the available prime time hours each week must be set aside for individual members play. A session for individual members takes up one hour of court time and produces $10 revenue as shown in the table below. League play has been an effective means of attracting new members to the club. Therefore, Phil has scheduled four leagues for the winter term. Each league session requires four hours of court time and generates $50 per session. Individual lessons by the head tennis pro, M. Hall, requires one hour and generates $25 revenue, while individual lessons by the assistant pro, D. Martin, requires one hour of court time and generates $20 revenue. M. Hall and D. Martin work together during each group sessions (i.e., both are present during an adult group lesson and both are present during a youth group lesson). Head pro M. Hall is only willing to work 30 hours per week, while assistant pro D. Martin is willing to work up to 50 hours per week. Phil has decided that youth group lessons are important for sustaining high membership in the future. Therefore, Phil has decided that there must be at least five youth group sessions scheduled each week. Formulate the integer LP model to determine how many sessions of each of the six activities should be scheduled per week in order to maximize Phil’s weekly revenue.

Activity Court hours per session Revenue per session

Individual member play 1 hour $10

League Play 4 hours $50

Individual Lesson (head pro) 1 hour $25

Individual Lesson (assistant pro) 1 hour $20

Group Lesson – Adults 2 hours $75

Group Lesson – Youth 2 hours $40

4a. How many of each type of activity should be scheduled per week?

5. Acme Inc must send a representative to each of eight different satellite locations. Eight Acme representatives have been identified for the assignments. One representative will be sent to each of the eight satellite locations. In order to make the assignments each representative has been asked to rank their preferences from 1 – most favorable assignment to 8 – their least favorable assignment. The rankings appear in the table below. Determine which representative should be sent to each location in order to minimize the rankings and avoid sending representatives to the least favorable locations. Turn in computer printout along the answers

Locations

NYC CHI BOS N.O LA LV MIA MIN

Representatives

Professor Plum 6 4 1 3 7 8 5 2

Colonel Mustard 7 3 2 4 5 1 6 8

Ms Scarlett 8 2 6 5 1 3 4 7

Mrs. White 7 5 6 3 4 1 2 8

Miss Peacock 6 3 7 2 5 1 8 4

Mr. Green 7 6 5 3 8 1 2 4

The Butler 6 2 5 8 1 7 3 4

French Maid 5 3 1 4 2 8 6 7

Who should be assigned to each of the cities?

Representative

New York ¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬_________________

Chicago _________________

Boston _________________

New Orleans _________________

Los Angeles _________________

Las Vegas _________________

Miami _________________

Minnesota _________________

Total objective function value? __________________

6 Frugal Rent-A-Car has eight store lots in the Greater St. Louis Metropolitan area. At the beginning of each day, they would like to have a predetermined number of cars available at each lot. However, since customers renting a car may return the car to any of the eight lots, the number of cars available at the end of the day does not always equal the designated number of cars needed at the beginning of the day. Frugal would like to redistribute the cars in the lots to meet the minimum demand and minimize the time needed to move the cars.

Table I below, summarizes the results at the end of one particular day.

Table II below summarizes the time required to travel between the lots.

Solve the problem in order to determine how many cars should be transported from one lot to the next. Turn in computer printout along the answers

Table I Lot

Cars 1 2 3 4 5 6 7 8

Available 39 20 14 26 42 28 38 52

Desired 30 25 22 40 30 20 32 60

Table II To (in minutes)

From 1 2 3 4 5 6 7 8

1 -- 12 17 18 10 20 40 35

2 14 -- 10 19 16 15 28 32

3 14 10 -- 12 8 9 34 26

4 8 16 14 -- 12 15 44 38

5 11 21 16 18 -- 10 24 28

6 24 12 9 17 15 -- 36 35

7 38 30 32 40 25 38 -- 29

8 30 34 28 35 31 33 31 --

How many cars will be sent from and to each destination?

From To Number of Cars

________ ________ _______

________ ________ _______

________ ________ _______

________ ________ _______

________ ________ _______

________ ________ _______

7. Assume Acme Inc is considering introducing a new product - super Deluxe Widgets. The Widgets will sell for $149 per unit. It is anticipated that the first year administrative costs will be $50,000 and the first year advertising budget is projected to be $30,000. The direct labor costs are uncertain, but it is believed they can be accurately can be represented by a normal distribution with a mean of $75 and a standard deviation of $15. Experts in the area have subjectively estimated the parts cost can be simulated by the discrete probability distribution listed below. The forecasted demand for the first year may be represented by a uniform distribution with limits of 20,000 and 40,000. Obtain summary statistics for 50 simulated trials to answer the four questions listed below. (Turn in the computer printout along with this answer sheet).

Cost per unit Probability

$20 .20

$25 .20

$30 .35

$35 .15

$40 .10

7A). What is the mean profit on the 50 simulated trials?

7B) What is the standard deviation of the 50 simulated trials?

7C). How many simulated trials resulted in a loss?

7D). Of the 50 simulated trials, what was the maximum profit?

## This question was asked on Dec 09, 2010.

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