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, 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 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.
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?
l. What are the reduced costs for larger and smaller tiles? Explain.

Answer 1

per batch	Large Tile	Small Tile	Availability
Mold time	0.3	0.25	60
Baking	0.27	0.58	105
Coloring	0.16	0.2	40
Clay	32.8	20	6000
Profit	190	240

The unit of time is hour. Hence 18 min and 15 min have been converted to 0.3 hours and 0.25 hours respectively.

Formulating linear programming:

1. Decision Variables:

Let L and S be the batches of the tiles to be produced each week.

2. Objective Function:

Maximize profit

Z max = 190L+240S

3. Constraints:

Constraints
Mold time	0.3L+0.25S	<=	60
Baking	0.27L+0.58S	<=	105
Coloring	0.16L+0.2S	<=	40
Clay	32.8L+20S	<=	6000
	L,S	>=	0

4. Spreadsheet Model:

Formulae Used:

5. Solving by solver:

Parameters

Sensitiyvity Report: After clicking on solve, in the solver result, select sensitivity and select ok to get the solution along with the sensityvity report.

Solution:

j. The capacity of Klin would be increased by 20 hours with a cost of 90000.

The new capacity = 105+20 = 125

New Objective = Z max = 190L+240S-90000

Re run the solver:

The objective value is negative i.e it will lead to a loss. Hence, it should not make this investment.

k. The updated capacity of glazing by klin = 37 hours

in the sensitivity report, the shadow price for this constraint is 1170.1. This means any unit change in the capacity would change the objective value by 1170.1. Here the change is -3, So the updated profit would be = 47886.59-1170.1*3 = 44376.29

also check the same by updating the capacity in the spreadsheet model:

l. Reduced Cost for both Large and Small Tiles is 0. This means the final values of Large and Small tiles would NOT change if the value of the corresponding objective coefficient is changed. The range within which this is true is represented by the allowable increase and decrease value.

For example: If the value of the objective coefficient of Large Tile changes to 192, the final value would not change and the same is true for any value n between 190-78.27 = 111.73 to 190+2 = 192.