Ilbert Moss and Angela Pasaic spent several summers during their college years working at archaeological sites in the Southwest. While at these digs they learned from local artisans how to make ceramic tiles. After college they started a tile manufacturing firm called Mossaic Tiles, Ltd. They opened their plant in New Mexico, where they would have convenient access to a special clay to make a clay derivative for their tiles. Their manufacturing operation consists of a few simple but precarious steps, including molding the tiles, baking, and glazing.
Gilbert and Angela plan to produce two basic types of tile for use in home bathrooms, kitchens, sunrooms, and laundry rooms: a larger single-colored tile; and a smaller patterned tile. In the manufacturing process the color or pattern is added before a tile is glazed. Either a single color is sprayed over the top of a newly baked set of tiles or a stenciled pattern is sprayed on the top of a baked set of tiles.
The tiles are produced in batches of 100. The first step is to pour the clay derivative into specially constructed molds. It takes 18 minutes to mold a batch of 100 larger tiles and 15 minutes to prepare a mold for a batch of 100 smaller tiles. The company has 60 hours available each week for molding. After the tiles are molded they are baked in a kiln: 0.27 hour for a batch of 100 larger tiles and 0.58 hour for a batch of 100 smaller tiles. The company has 105 hours available each week for baking. After baking, the tiles are either colored or patterned and glazed. This process takes 0.16 hour for a batch of 100 or larger tiles and 0.20 hour for a batch of 100 smaller tiles. Forty hours are available each week for the "glazing" process. Each batch of 100 large tiles requires 32.8 pounds of the clay derivative to produce, while each batch of smaller tiles requires 20 pounds. The company has 6,000 pounds of the clay derivative available each week.
Mossaic Tile earns a profit of $190 for each batch of 100 of the larger tiles and $240 for each batch of 100 smaller patterned tiles. Angela and Gilbert want to know how many batches of each type of tile to produce each week in order to maximize profit. They also have some questions about resource usage they would like to answer.
A. Formulate a linear programming model for the Mossaic Tile Company to determine the mix of the tiles it should manufacture each week.
B. Transform the model into standard form.
C. Solve the linear programming model graphically.
D. Determine the resources left over and not used at the optimal solution point.
E. For artistic reasons Gilbert and Angela like to produce the smaller patterned tiles best. They also believe in the long run the smaller tiles will be a more successful product. What must be the profit for the smaller tiles in order for the company to produce only the smaller tiles?
F. Solve the linear programming model using the computer.
G. Mossaic believes it may be able to reduce the time required for molding to 16 minutes for a batch of larger tiles and 12 minutes for a batch of the smaller tiles. How will this affect the solution?
H. The company that provides Mossaic with clay has indicated that it can deliver an additional 100 pounds of clay each week. Should Mossaic agree to this offer?
I. Mossaic is considering adding capacity to one of its kilns to provide 20 additional glazing hours per week at a cost of $90,000. Should it make the investment?
J. The kiln for glazing had to be shut down for three hours, reducing the available kiln hours from 40 to 37. What effect will this have on the solution?