Assignment - Problems
Problem 1 - Sam is considering starting a healthy life style this year and is looking to join the local gym for the coming 10 months. He can choose between two plans A or B. In Plan A, the gym charges a sign-up fee of $120 and $30 per month, while in Plan B, there is no sign-up fee but the monthly fee is $45. Answer the following and show your analyses
a) Help Sam make a decision on which plan to choose? Show your work.
b) After how many months would both plans become indifferent? Show your work.
c) Suppose Sam saw a coupon that costs $10 and will give him a discount of $5/month for the first 6 months of being enrolled in Plan B. Should he purchase the coupon? Justify your answer and show your work.
Problem 2 - A company is considering launching two new products that have large sales potential. Product 1 requires some of the production capacity in Plants 1 and 3, but none in Plant 2. Product 2 needs only Plants 2 and 3. The marketing division has concluded that the company could sell as much of either product as could be produced by these plants. However, because both products would be competing for the same production capacity in Plant 3, it is not clear which mix of the two products would be most profitable. Each product will be produced in batches of 20, so the production rate is defined as the number of batches produced per week. Any combination of production rates that satisfies these restrictions is permitted, including producing none of one product and as much as possible of the other. The number of hours of production time available per week in each plant for both products as well as the number of hours of production time required for each batch produced of each new product in each plant is given in the table below. The profit per batch of product 1 is $3000 and of product 2 is $2000. Answer the following
|
Plant
|
Production Time per Batch, Hours
|
Production Time Available per Week, Hours
|
|
Product
|
|
1
|
2
|
|
1
|
1
|
0
|
4
|
|
2
|
0
|
2
|
12
|
|
3
|
3
|
2
|
18
|
a) Formulate a linear mathematical program to determine what the production rates should be for the two products in order to maximize their total profit, subject to the restrictions imposed by the limited production capacities available in the three plants. Clearly define your decision variables, constraints and objective function.
b) Use the graphical method to solve the model and find the optimal amounts of batches of each product and the optimal amount of profit.
c) Using manual calculations, determine by how much the profit per batch of each product can increase or decrease before the optimal solution changes.
d) By inspecting your graphical solution, how much would an additional hour of available hour in Plant 2 should be worth. Justify your answer based on your graph.
e) By inspecting your graphical solution, by how many hours, if any, can you reduce the time available in Plant 1 without affecting the optimal amount of profit?
f) If the new products had been required to return a net profit of at least $50,000 per week to justify discontinuing part of the current product line. What would be the additional constraint and how does this change the solution. Demonstrate this change on your graphical solution.
Problem 3 - Engineering Division has been asked to select at least 3 out of 7 possible sites for oil exploration at a cost of $50,000 per site evaluation. Evaluating Sites 1 and 3 will prevent you from exploring Site 7, and evaluating Sites 2 or 4 will prevent you from evaluating Site 5. Site 3 cannot be evaluated unless Site 4 is evaluated.
a) Formulate a mathematical linear program to minimize the total cost. Clearly define your decision variables, objective and constraints.
b) Implement the formulation in Excel and solve with Solver to find the optimal solution. Include a snapshot of your solution in the report.
Problem 4 - An agricultural office is planning the production for the coming year. The agricultural output of each region is limited by both the amount of available irrigable land and the quantity of water allocated for irrigation as shown in Table 1. The crops suited for these region include crops 1, 2, and 3, and these are being considered for the upcoming season. These crops differ primarily in their expected net return per acre and their consumption of water. In addition, the state has set a maximum quota for the total acreage that can be devoted to each of these crops as shown in Table 2. Because of the limited water available for irrigation, the state will not be able to use all its irrigable land for planting crops in the upcoming season. To ensure equity between the three regions, it has been agreed that every region will plant the same proportion of its available irrigable land. For example, if region 1 plants 200 of its available 400 acres, then region 2 must plant 300 of its 600 acres, while region 3 plants 150 acres of its 300 acres. Any combination of the crops, however, may be grown at any of the regions. The job facing this office is to plan how many acres to devote to each crop at the respective regions while satisfying the given restrictions. The objective is to maximize the total net return as a whole for all regions.
|
Region
|
Usable Land (Acres)
|
Water Allocation (Acre Feet)
|
|
1
|
400
|
600
|
|
2
|
600
|
800
|
|
3
|
300
|
375
|
|
Table 1
|
|
Crop
|
Maximum Quota (Acres)
|
Water consumption (Acres feet/acre)
|
Net Return ($/Acre)
|
|
Crop 1
|
600
|
3
|
1,000
|
|
Crop 2
|
500
|
2
|
750
|
|
Crop 3
|
325
|
1
|
250
|
|
Table 2
|
a) Formulate a linear mathematical model to solve this problem. Clearly define your decision variables, objective function and constraints.
b) Implement your formulation in Excel and use Solver to find the optimal solution. Provide answers for the optimal amounts and the corresponding amounts of return. Provide an organized screen shot of your solution and a snapshot of the sensitivity report.
c) Based on the sensitivity report to your solution in Part (b), if you are to eliminate one of the crops, which one would you eliminate and why? By how much does the net return per acer for that crop needs to be before it becomes part of the optimal solution? Justify your answer.
d) Based on your solution and sensitivity report in Part (b), if you have a choice to increase the usable land versus increasing the water allocation amounts, which one would you pick and why? By how much would you increase it and why?
e) Suppose that because of competition, it may be necessary to reduce the selling price and therefore the net profit per acer for Crop 1 across all three regions. From the sensitivity report in Part (b), by how much can you reduce that net profit per acre without affecting the optimal solution? Justify your answer.
Problem 5 - A coffee bean company blends four types of beans into three blends of coffee. One blend is sold to hotels, another to restaurants and a third to supermarkets. The company has four bean suppliers who independently supply the four different bean types. Below is a summary of the blend recipes, the cost and availability information for the different bean types. The percentages indicate the fraction of each bean type used in each blend.
The processor's plant can handle no more than 100,000 pounds per week and the company would like to operate at capacity. There is no problem selling the final blends although the marketing department requires minimum production levels of 10,000, 25,000, and 30,000 pounds for the hotel, restaurant and market blends respectively.
|
Bean Type
|
Composition
|
Cost/lb
|
Weekly availability
|
|
Hotel
|
Restaurant
|
Supermarket
|
|
1
|
20%
|
35%
|
10%
|
$0.60
|
40,000
|
|
2
|
40%
|
15%
|
35%
|
0.80
|
25,000
|
|
3
|
15%
|
20%
|
40%
|
0.55
|
20,000
|
|
4
|
25%
|
30%
|
15%
|
0.70
|
45,000
|
|
Whole sale price/lb
|
1.25
|
1.50
|
1.40
|
|
|
a) Formulate a Linear Programming model for the problem to maximize the weekly profit. Define clearly all of your decision variables, objective and constraints.
b) Solve the model using Excel and report the optimal amounts of each bean type to be purchased.
c) If there is a chance to increase the bean availability for any of the four types by 1000 pound, which one would you select? Explain your answer based on the solution you obtained above without resolving the model.
d) If the marketing department would like to increase the minimum production level by 500 pounds for one of the blends, which one would you recommend? Explain your answer based on the solution you obtained above without resolving the model.
Attachment:- Assignment Files.rar