The Relax-and-EnjoyLake Development Corporation is developing a lakeside community at a privately owned lake and is in the business of selling property for vacation and/or retreat cottages. The primary market for these lakeside lots includes all middle and upper income families within approximately 100 miles of the development. Relax-and-Enjoy has employed the advertising firm of Boone, Phillips and Jackson to design the promotional campaign for the project.
After considering possible advertising media and the market to be covered, Boone has made the preliminary recommendation to restrict the first month's advertising to five sources. At the end of this month, Boone will then reevaluate its strategy based upon the month's results. Boone has collected data on a number of potential purchase families reached, the cost per advertisement, the maximum number of times each medium is available, the expected exposure for each of the five media. The expected exposure is measured in terms of an exposure unit, a management judgment measure of the relative value of one advertisement in each of the media. The measures based on Boone's experience in the advertising business take into account such factors as audience profile (age, income, and education of the audience reached), image presented, and quality of the advertisement. The information collected to date is presented below.
Advertising Media Alternatives for the
Relax-and-Enjoy Lake Development Corporation
Number of 
Potential Maximum
Purchase Cost per Times Expected
Families Advertise- Available Exposure
Advertising Media Reached ment per Month* Units
1. Daytime TV (1 min) 1000 $1500 15 65
Station WKLA
2. Evening TV (30s) 2000 $3000 10 90
Station WKLA
3. Daily newspaper 1500 $400 25 40
(full page), The
Morning Journal
4. Sunday newspaper 2500 $1000 4 60
Magazine (1/2 pg.
Color). The Sunday
Press.
5. Radio, 8:00 a.m. or 300 $100 30 20
5:00 p.m. news (30's)
__Station KNOP_________________________________________________________

*The maximum number of times the medium is available is either the maximum number of times the advertising medium occurs (e.g. four Sundays for medium 4) or the maximum number of times Boone will allow the medium to be used. 

Relax-and-Enjoy has provided Boone with an advertising budget of $30,000 for the first month's campaign. In addition Relax-and-Enjoy has the following goals and priorities regarding how Boone allocates these funds, as follows:

Goal 1: To utilize at least 10 television commercials (Priority 1).

Goal 2: To reach at least 50,000 potential purchases during the month 
(Priority 2).

Goal 3: To spend no more than $18,000 on television advertisements
(Priority 3).

Goal 4: To come as close as possible to achieving 2400 exposure units
(Priority 4).

Goal 5: To minimize the advertising budget (Priority 5).

(a) Formulate a goal programming model of this problem:

Economic constraints:
Goal constraints:
Objective function:
(b) Suppose that Boone is working on a similar but smaller problem for another client, and they are considering only two media alternatives, evening T.V. and the daily newspaper. This second problem can be modified as follows:

3000 X1 + 400 X2 = $24,000 Budget

X1 - d1+ + d1- = 7

X2 - d2++ d2- = 15

2000 X1 + 1500 X2 - d3+ + d3- = 30,000

Min Z= P1 d1- + P2 d2+ + P3 d3- + P4 (-d1+)