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 build 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.

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.

I don’t know how to do part l

Answer 1

Reduced cost is a part of the sensitivity analysis of Linear Programming. As it is mentioned specifically about part l, I am jumping to part l.

Let us assume that Mossaic Tiles has to produce X1 and X2 number of batches of Large and Small tiles respectively.

Z max = 190X1 + 240X2

Subject to constraints:

(18/60)X1+(15/60)X2 <= 60 or 0.3X1+0.25X2 <= 60

0.27X1+0.58X2 <= 105

0.16X1+0.2X2 <=40

32.8X1+20X2 <= 6000

X1 and X2 >= 0

Solution:

While getting the solution, generate the sensitivity report as well.

The solution and optimal batch quantities are as mentioned in the above screenshot.

Now for reduced cost, we need to look at the sensitivity report generated.

The reduced cost is 0 for both the variables.

This means any change in the objective coefficient results in no change in the final value of the variables. The optimal quantity (value of X1 and X2) would remain the same (objective value might change though) for the below range of objective coefficient values

Large Tile ,X1 = 111.7241 <= X1 <= 192

Small TIle, X2 = 237.5 <= X2 <= 408.1481

Lower limit = Objective Coefficient - Allowable Decrease

Upper Limit = Objective Coefficient+Allowable Increase