Note: Write the formula for every question depending upon the requirement.
Calculations: Answer all the questions Case Problems:
Case Problem:
Saudi Tiles, Ltd. Hafiz and Nahla 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 Saudi 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. Hafiz and Nahla 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. Saudi Tiles earns a profit of $180 for each batch of 100 of the larger tiles and $210 for each batch of 100 smaller patterned tiles. Hafiz and Nahla 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.
1-Formulate a linear programming model for Saudi Tiles, Ltd.
2-Transform the model into standard form.
3-Solve the linear programming model graphically.
4-Identify the feasible region and optimal solution.
5-Conduct a sensitivity analysis to determine the range of optimality for the objective function coefficient.
6-If the right-hand-side of molding constraint is marginally increased by one hour, what will be the dual price?
7-Suppose there is a simultaneous change in the objective function coefficient in larger and smaller tiles by $225 and $175. What will be new optimal solution?
8-Based on h, what is the percentage change in allowable increase and allowable decrease in the objective function coefficient?
9-Justify whether the optimal solution will change.