1)

Acme Widgets produces four products: A, B, C and D. Each unit of product A requires 2 hours of milling, one hour of assembly and $2 worth of in-process inventory. Each unit of product B requires one hour of milling, 3 hours of assembly and $5 worth of in-process inventory. Each unit of product C requires 2 1/2 hours of milling, 3 1/2 hours of assembly and $4 worth of in-process inventory. Finally, each unit of product D requires 5 hours of milling, no assembly time and $16 worth of in-process inventory. The firm has 1,200 hours of milling time and 1,300 hours of assembly time available. Each unit of product A returns a profit of $40; each unit of B has a profit of $36; each unit of product C has a profit of $24; and each unit of product D has a profit of $48. Not more than 120 units of product A can be sold and not more than 96 units of product C can be sold. Any number of units of products B and D may be sold. However, at least 100 units of product D must be produced and sold to satisfy a contract requirement. It is otherwise assumed that whatever is produced can be sold. Formulate the above as a linear programming problem to maximize profits to the firm.

2)

The Thrifty Loan Company is planning its operations for the next year. The company makes five types of loans. The loans are listed below along with the annual return on the loans:
 

Legal requirements and company policy place the following limits upon the various types of loans:

Signature loans cannot exceed 10% of the total amount of loans. The amount of signature and furniture loans together cannot exceed 20% of the total amount of loans. First mortgages must be at least 40% of the total mortgages and at least 20% of the total amount of loans. Second mortgages may not exceed 25% of the total amount of loans. The firm can lend a maximum of $1.5 million.

Formulate the above as a linear programming problem to maximize the revenues from loans.

3)

Roscoe owns a used furniture store. He has 500 square feet of floor space available for new purchases. The following pieces of furniture are available to him:
 

 Roscoe does not want to stock more sofas than beds. For each patio set stocked he wants to have at least one of everything else. He has $450 allocated for these purchases. Formulate the above as a linear programming problem to maximize Roscoe's profit from his purchases.

4)

The marketing department for Omni World Enterprises would like to allocate next year's advertising budget among the various media to maximize the return to the firm. The year's expenditures for advertising are not to exceed $2 million, with not more than $1.1 million spent during the first six months. The media used are newspapers, magazines, radio and television. Spending on the different media is restricted by the following company policies:

a. At least $200,000 is to be spent on newspapers and magazines combined in each half of the year.
b. At most, 80% of the advertising expenditures are to be spent on television in each six-month period.
c. At least $50,000 is to be spent on radio for the year.
d. At least 25% of the advertising expenditures on television are to be spent in the second six-month period.

Returns from a dollar spent on advertising in each medium are as follows:
 

Formulate a linear programming problem for Omni's advertising budget.

5)

Rott Irony manufactures four types of light fixtures.  A fancy lamp yields a profit of $100, takes 10 hours of labor and 2 hours of machine time, and requires 10 square feet of sheet metal.  An ornate lamp yields a profit of $150, takes 8 hours of labor and 3 hours of machine time, and requires 20 square feet of sheet metal.  The plain and rococo lamps each yield a profit of $200, and involve one hour of machine time.  However, the rococo lamp requires 20 hours of labor and 30 square feet of metal, and the plain lamp requires 10 hours of labor and 15 square feet of metal.  Rott must produce at least twice as many plain as rococo lamps.  Only 1,000 hours of labor and 200 hours of machine time are available, and 5,000 square feet of sheet metal are in inventory.

Rott wants to determine how many of each type of lamp to make to maximize total profits.  Formulate the above as a linear programming problem.

6)

Licores Andinas, S.A. is engaged in the production and sale of two kinds of liquor.  Unlike a fully integrated distillery, the company is a rectifier only.  That is, it purchases intermediate-stage products in bulk, purifies them through repeated distillation, mixes them, bottles products under its own brand names and then sells them to commercial channels.  One product is a brandy branded as “El Presidente;” the other is a liqueur, “Canario,” made from sugar cane, sometimes referred to locally as rompope.  Sales of each product are independent of each other.  In the firm’s experience market limits on sales have never come into play short of the firm’s producing capacity, so that it can be assumed that everything produced is sold.

Labor is not a significant constraint on the firm.  Machine capacity, however, is inadequate to produce all that the firm might sell.  El Presidente requires three machine hours per bottle, but because of additional blending requirements Canario absorbs a total of four hours of machine time per bottle.  A total of 20,000 machine hours are available in the current production period.

Higher quality makes the direct operating costs (principally labor and materials) of El Presidente $3 per bottle, in contrast with the Canario, which costs $2 per bottle.  Excluding collections of receivables from sales made during the current production period, funds available to finance labor and cost of materials are planned at $4,000.  Collection experience on the El Presidente and the Canario sales varies from time to time.  However, it is anticipated that 44% of El Presidente sales and 31% of the Canario sales made from current production will be collected during the same production period, and that the cash proceeds will be available to finance operations.  All direct costs will be paid during the production period, and none accrued.

Prices tolerated by the market differ for the two products.  El Presidente is sold to the distributive channels at $5 per bottle and Canario at $4.50 per bottle.

Planning for company activities during the approaching production period has led to some disagreement among the members of management.  The Production and Marketing managers on the one hand, and the Treasurer-Controller on the other, are unable to agree on the most desirable product mix and production volume to schedule.  The Managing Director, however, has decreed that the principal objective of the firm should be to maximize profits.

Formulate the above as a linear programming problem and solve using the graphical solution method.  What is the best profit the firm can achieve?
 

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