The HHH building company has built five 20story apartment buildings in Saint Louis area, planning to sell units as topmarket condominiums. They had developed many properties before and had a reputation for high quality construction and furnishings, and therefore they were concerned that the proper materials were to be used in construction. Moreover, they knew that perspective buyers were very interested in the type of finishmaterials used. This fivebuilding project would include 750 condos. All apartments had two bathrooms and two bedrooms, a large entry hallway, living room and a large kitchen, and were approximately 1500 squarefeet each in area. HHH believed that the materials that were utilized in the three major areas of the units the bathrooms, kitchens and entryways (called the BKE area, a total of 450 squarefeet per apartment), would have the greatest impact on the pricing of the condos. Other than appliances, the most important features were the type of flooring and surface materials that were used in these rooms. These materials would affect the designation of the apartment unit as a luxury, highend, or averagemarket unit and determine the selling price of the condo. The materials to be chosen were marble, ceramic tile or wood. Any one of these could be used for the entire 450 squarefeet of BKE flooring or they could be combined in any squarefootage proportion. HHH had developed a materialrating system that calculated a weighted average of the quality level of the materials used. Marble surfaces were rated 9.0 (the highest being 10.0), ceramic tiles were rated 6.0, and the wood surfaces had a 4.0 rating. The overall quality rating for an apartment was the weighted average of the materialratings multiplied by the number of squarefeet of each material used in the BKE area divided by the total BKE area. For example, a unit that had an equal amount of marble, ceramic and wood (150 squarefeet of each), would have a qualityrating of:

(9*150+6*150+4*150)/450 = 6.33

HHH could buy all the wood they needed. Ceramic tile was more expensive and in moderate supplies on the market. For the time period relevant to this particular project, they felt that they could buy up to 100,000 squarefeet of ceramic tile. Marble was the most expensive material and in short supplies. Their importer could promise at most 60,000 sq.ft. of marble.

The luxury apartments, that could bring top prices of $900 per squarefoot, required quality ratings of at least 8.0. Highend units required a qualityrating of 6.0 or more, and varied in selling price, but it was believed that the average price would be $700 per sq.ft. They felt that the averagemarket units would sell for $500 per sq.ft. Since HHH had certain minimum requirements for any of their apartments, they would not allow any unit to be below 4.0 in quality rating. The markups on the condos were proportionately equal at varying cost levels, and therefore HHH thought that they could concentrate on maximizing their sales revenue, to maximize profit margins.

The real estate market at this time was booming. Buyers were everywhere and everyone wanted to purchase their own condominium rather than rent. Employees in the financial industries were flush with bonus money, and there seemed to be an unending demand for quality apartments. The real estate brokers who were marketing the properties for HHH informed them that they could sell a maximum of 150 of the units as luxury condos if they were constructed accordingly. The demand for highend condos was also strong, and the brokers estimated that they could easily get highend condo buyers for at most 200 units, and a maximum of 500 of the averagemarket units could also be sold.

Formulate the problem as a linear program. Specifically, perform all the steps below.

Note: You can use the Blackboard MathEditor function to type in the formula (click on the f_x symbol on the editing toolboard) Alternatively, you can use the subscript function to type symbols like x₁ and use ">=" for "greater or equal" sign.

1. List and explain carefully what the decision variables are

2. Write out the objective function. Explain carefully.

3. Write out the constraints. Explain carefully.

4. Write the linear program.

Remark: your LP does not need to be presented in its standard form but you need to make sure that the objective function and the constraints are all linear! Also, you are not asked to solve the problem, only to formulate it.

9x150 + 6 x 150 +4 x 150 =6.33 450 9x150 + 6 x 150 +4 x 150 =6.33 450