Discuss the below in detail:
Case Problems:
Sparkling Clothes Laundry
Question 1:
When Moni purchased the Sparkling Clothes Laundry, she thought that because it was in a good location near several high-income neighborhoods, she would automatically generate good business if she improved the laundry's physical appearance. Thus, she initially invested a lot of her cash reserves in remodeling the exterior and interior of the laundry. However, she just thought about break even in the year following her acquisition of the laundry, which she didn't feel was a sufficient return, given how hard she had worked. Moni didn't realize that the dry-cleaning business is very competitive and that success is based more on price and quality service, including quickness of service, than on the laundry's appearance.
In order to improve her service, Moni is considering purchasing new dry-cleaning equipment, including a pressing machine that could substantially increase the speed at which she can dry-clean clothes and improve their appearance. The new machinery fixed costs is $2,150 per month. Moni estimates her variable costs to be $0.25 per item dry-cleaned, which will not change if she purchases the new equipment. Her current fixed costs are $1,700 per month. She charges customers $1.10 per clothing item.
a. Write the mathematical expression for the total cost and total revenue for the current machine.
b. What is Moni's current monthly volume to reach break-even?
c. If Moni purchases the new equipment with a fixed cost of $2150 per month, how many additional items will she have to dry-clean each month to reach break-even?
d. Moni estimates that with the new equipment she can increase her volume to 4,300 items per month. What monthly profit would she realize with that level of business during the next 3 years?
e. Moni believes that if she doesn't buy the new equipment but lowers her price to $0.99 per item, she will increase her business volume. If she lowers her price, what will her new break-even volume be? If her price reduction results in a monthly volume of 3,800 items, what will her monthly profit be?
f. Moni estimates that if she purchases the new equipment and lowers her price to $0.99 per item, her volume will increase to about 4,700 units per month. Based on the local market, that is the largest volume she can realistically expect. What will be the profit and suggest what Moni should do?
g. Refer to questions b and c, and draw the graphs for breakeven.
Question 2:
The Saudia Juice Company produces two kinds of Juice-Saudia Nectar and Saudia Red. The juice is produced from 64 tons of grapes the company has acquired this season. A batch (1,000-gallon) of Nectar requires 4 tons of grapes, and a batch of Red requires 8 tons. However, production is limited by the availability of only 50 cubic yards of storage space and 120 hours of processing time. A batch of each type of juice requires 5 cubic yards of storage space. The processing time for a batch of Nectar is 15 hours, and the processing time for a batch of Red is 8 hours. Demand for each type of juice is limited to seven batches. The profit for a batch of Nectar is $9,000, and the profit for a batch of Red is $12,000. The company wants to determine the number of batches of Nectar (x1) and Red (x2) to produce in order to maximize profit. (Note: Solve using 2 decimal places)
a. Formulate a linear programming model for this problem.
b. Solve this model by using graphical analysis.
c. Identify the feasible region.
d. Find the extreme points and optimal solution
e. What is the standard form of the linear programming model?
f. How much of slack resource will be left unused at the optimal solution?
g. What are the binding, non-binding and redundant constraints?
h. What would be the effect on the optimal solution, if the profit for a batch of Nectar and Red is $10850 and $10750 respectively?
Question 3:
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.
a. Formulate a linear programming model for Saudi Tiles, Ltd.
b. Transform the model into standard form.
c. Solve the linear programming model graphically.
d. Identify the feasible region and optimal solution.
e. Conduct a sensitivity analysis to determine the range of optimality for the objective function coefficient.
f. If the right-hand-side of molding constraint is marginally increased by one hour, what will be the dual price?
g. 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?
h. Based on h, what is the percentage change in allowable increase and allowable decrease in the objective function coefficient? (1.5 marks)
i. Justify whether the optimal solution will change.