LINEAR PROGRAMMING FORMULATION PROBLEMS

PRODUCTION MODEL

1. A company produces framed photographs which are sold to retail print outlets. The company produces three sizes of a particular print. The large print is 2’ x 3’. The medium print is 1.5’ x 2’. The small print is 1’ x 1.5’. The company has an inventory of 1000 linear feet of the wood framing material used for framing all three sizes of this print. The company also has in inventory 600 square feet of the glass used with that framing material. The profit earned on a large print is $8.00. The profit earned on a medium print is $5.00 and $2.50 on a small print. At least 100 of the large prints must be manufactured for use in the stores as demonstration models. Determine how many of each picture should be produced in order to maximize total profit.

PORTFOLIO MIX SELECTION

2. A stock broker has a client who wants to invest $1,000,000 in the stock market only. The client wants a diversified stock portfolio. Currently the brokerage firm is suggesting that the broker choose from the following list of stocks:

The client wants diversity in the portfolio where no more than 15% of the portfolio is invested in any one stock. Although the investor wants a high return, e would like to control his level of risk. He wants no more than 30% of his stock acquisitions to be at a risk level of 7 or greater. He also would like at least 10% of his portfolio to have a risk rating of 3 or less. The investor would prefer that manufacturing, computer and electronic stocks make up 40% or less of the portfolio. The investor feels that the textile industry will become favorable soon, and would like to have at least ten textile shares in his portfolio. How many shares of each stock should be purchased to maximize the total annual return on the portfolio?

MEDIA SELECTION MODEL

3. A vacation resort is planning to increase its bookings by doing some advertising. However, the decision must be made as to where to spend the allocated advertising dollars in order to reach the largest number of potential visitors. The budget for this advertising campaign is set at $50,000. This budget is to be spent on a combination of magazine, radio station and television advertising. The magazine charges $1,500 for a 1/4 page ad, which reaches 185,500 people. For a 1 minute radio spot, which reaches 290,500 people, the cost is $3,000. A one minute television spot reaches 460,000 people at a cost of $5,500. (The assumption is made that all print and broadcasting advertisements have already been prepared and that the concern for cost now is only with running these ads in the respective mediums). It has been decided that at least one ad will run in each medium, with no more than fifteen magazine ads being run. At least 40% of the budget will be spent on TV advertising.

MARKET RESEARCH MODEL

4. A market research company has been hired by a firm to perform market research on the growing interest in the sport of indoor rock climbing. The research is to be done using a mail survey. To do this, the market research firm will buy mailing lists from various magazines. The company wants to reach both males and females already involved in some athletic or outdoor activity. Due to these requirements, the following magazines were chosen. The cost for a mailing list from the leading rock climbing magazine is $200 and will give a mailing list of 110,00 readers. Of these readers 75% are male and 25% are female. The leading men’s fitness magazine reaches 200,000 readers, 10% of which are women. The cost for this mailing list is $500. The leading women’s fitness magazine charges $400 for their mailing list and reaches 175,000 readers. Of these readers, 85% are women. The leading general outdoor recreation magazine has a readership of 250,000 of which 50% are women and 50% are men. The mailing list for this magazine will cost $350. The leading mail order outdoor outfitter firm, sells the mailing list of its catalog members for $100 and reaches 50,000 people, of which 40% are women and 60% men. The marketing firm has allocated $1000 for purchase of these mailing lists. The company feels that the larger part of those who are becoming indoor rock climbers are men and therefore would like to reach at least twice as many men as women. Determine which the most appropriate mailing lists for the marketing firm to purchase.

5. GRAIN BLENDING MODEL

Superior Equine Products is planning on producing a reconditioning feed for horses. The scientific department has determined that to serve as a reconditioning food, the final feed mix must contain 10% or less Fat, 30% Protein, at least 17% Fiber and 20% Carbohydrates. The feed will be a combination of Corn, Wheat, Alfalfa and Sorghum. For its first production of this feed, 2000 lb. will be produced. The following table contains both the nutritional values and associated costs for each of the four components of this feed:

Determine the mix that will meet all nutritional requirements at the lowest possible cost per pound.