# Question

Gilbert Moss and Angela Pasaic spent several summers during their college years working at archaeological sites in the Southwest. While at those digs, they learned how to make ceramic tiles from local artisans. After college they made use of their college experiences to start 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 they intend to use to make a clay derivative for their tiles. Their manufacturing operation consists of a few relatively 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. The two types of tile are a larger, singlecolored 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 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 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, whereas each batch of smaller tiles requires 20 pounds. The company has 6,000 pounds of the clay derivative available each week.

Mossaic Tiles 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. In addition, they have some questions about resource usage they would like answered.

A. Formulate a linear programming model for Mossaic Tiles, Ltd., and determine the mix of 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. Determine the sensitivity ranges for the objective function coefficients and constraint quantity values by using the graphical solution of the model.

F. For artistic reasons, Gilbert and Angela prefer to produce the smaller, patterned tiles. They also believe that in the long run, the smaller tiles will be a more successful product. What must the profit be for the smaller tiles in order for the company to produce only the smaller tiles?

G. Solve the linear programming model by using the computer and verify the sensitivity ranges computed in (E).

H. 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 smaller tiles. How will this affect the solution?

I. The company that provides Mossaic with clay has indicated that it can deliver an additional 100 pounds each week. Should Mossaic agree to this offer?

J. 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?

K. The kiln for glazing had to be shut down for 3 hours, reducing the available kiln hours from 40 to 37. What effect will this have on the solution?

Gilbert and Angela plan to produce two basic types of tile for use in home bathrooms, kitchens, sunrooms, and laundry rooms. The two types of tile are a larger, singlecolored 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 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 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, whereas each batch of smaller tiles requires 20 pounds. The company has 6,000 pounds of the clay derivative available each week.

Mossaic Tiles 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. In addition, they have some questions about resource usage they would like answered.

A. Formulate a linear programming model for Mossaic Tiles, Ltd., and determine the mix of 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. Determine the sensitivity ranges for the objective function coefficients and constraint quantity values by using the graphical solution of the model.

F. For artistic reasons, Gilbert and Angela prefer to produce the smaller, patterned tiles. They also believe that in the long run, the smaller tiles will be a more successful product. What must the profit be for the smaller tiles in order for the company to produce only the smaller tiles?

G. Solve the linear programming model by using the computer and verify the sensitivity ranges computed in (E).

H. 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 smaller tiles. How will this affect the solution?

I. The company that provides Mossaic with clay has indicated that it can deliver an additional 100 pounds each week. Should Mossaic agree to this offer?

J. 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?

K. The kiln for glazing had to be shut down for 3 hours, reducing the available kiln hours from 40 to 37. What effect will this have on the solution?

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