Gilbert Hua and Angela Pasaic spent several summers during their university years working at archaeological sites in the American Southwest. While at these digs they learned from local artisans how to make ceramic tiles. After university they started a tile manufacturing firm called Mosaic 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 moulding 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-coloured tile and a smaller patterned tile. In the manufacturing process, the colour or pattern is added before a tile is glazed. Either a single colour 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 moulds. It takes 18 minutes to mould a batch of 100 larger tiles and 15 minutes to prepare a mould for a batch of 100 smaller tiles. The company has 60 hours available each week for moulding. After the tiles are moulded, 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 coloured or patterned and glazed. This process takes 0.16 hour for a batch of 100 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 kilograms of the clay derivative to produce, while each batch of smaller tiles requires 20 kilograms. The company has 6000 kilograms of the clay derivative available each week. 

Mosaic 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 to maximize profit. They also have some questions about resource usage they would like to answer.
a. Formulate a linear programming model for the Mosaic 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 prefer to produce the smaller patterned tiles. 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 for the company to produce only the smaller tiles?
f.  Solve the linear programming model using Excel.
g. Mosaic believes that it may be able to reduce the time required for moulding 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 Mosaic with clay has indicated that it can deliver an additional 100 kilograms of clay each week. Should Mosaic agree to this offer?
i.  Mosaic 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?