The Foggy Bottom Development Corporation has just purchased a small hotel for conversion to condominium apartments. The building, in a popular area of Washington, D.C., near the U.S. State Department, will be highly marketable, and each condominium sale is expected to yield a good profit. The conversion process, however, includes several options. Basically, four types of condominiums can be designed out of the former hotel rooms. They are deluxe one-bedroom apartments, regular one-bedroom apartments, deluxe studios, and efficiency apartments. Each will yield a different profit, but each type also requires a different level of investment in carpeting, painting, appliances, and carpentry work. Bank loans dictate a limited budget that may be allocated to each of these needs. Profit and cost data, and cost of conversion requirements, for each apartment are shown in the accompanying table. Thus, we see that the cost of carpeting a deluxe one bedroom unit is $1,100, the cost of carpeting a regular one-bedroom unit is $1,000, and so on. A total of $35,000 is budgeted for all new carpet in the building.

Zoning regulations dictate that the building contains no more than 50 condominiums when the conversion is completed—and no fewer than 25 units. The development company also decides that to have a good blend of owners, at least 40% but no more than 70% of the units should be one-bedroom apartments. Not all money budgeted in each category need be spent, although profit is not affected by cost savings. But since the money represents a bank loan, under no circumstances may it be exceeded or even shifted from one area, such as carpeting, to another, such as painting.
(a) Formulate Foggy Bottom Development Corporation’s decision as a linear program to maximize profits.
(b) Convert your objective function and constraints to a form containing the appropriate slack, surplus, and artificialvariables.

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TYPE OF APARTMENT TOTAL DELUXE ONE-BEDROOM ($) REGULAR ONE-BEDROOM DELUXE STUDIO RENOVATION REQUIREMENT New carpet Painting New appliances Carpentry work Profit per unit EFFICIENCY BUDGETED 35.000 28,000 45.000 19,000 1,100 1,000 700 1,600 .200 1,000 6,000 5,000 3,500