Week 2: Assignment 1
A farmer's family is making plans for the year's planting. Its members are considering planting corn, tomatoes, potatoes, and strawberries. They have 50 acres to plant on. The aim is to determine what it costs to plant an acre of each crop, computing the yield in bushels, forecasting the revenue for a bushel of each crop, and choosing the combination of crops that will yield the most profit. The data they have collected, along with the availability of resources, is shown in the table.
|
Crop
|
Yield/Acre (bushels)
|
Revenue/ Bushel
|
Planting
Time Required (hours)
|
Tending Time Required
per Week
|
Required Feet of Water per Acre /Season
|
Fertilizer Required per Acre
(pounds)
|
Harvest Time Required
(hours)
|
|
Corn
|
50
|
$90
|
10
|
2
|
2.5
|
50
|
6
|
|
Tomatoes
|
38
|
$210
|
15
|
8
|
3
|
60
|
20
|
|
Potatoes
|
45
|
$50
|
12
|
2
|
2
|
45
|
9
|
|
Strawberries
|
56
|
$56
|
18
|
12
|
3
|
35
|
20
|
|
|
|
Total Available
|
770
|
550
|
300
|
3000
|
770
|
a) Determine the best mix of crops to maximize their revenue.
b) The farmers also have the opportunity to buy a 78-acre farm adjacent to their land. They would like to narrow their selection of crops to corn, strawberries, or a combination of the two. If they acquire the land, they will be able to increase the time available to 1,800 hours for planting, 825 hours for tending, and 1,400 hours for harvesting. All these increases are from the available limits in part (a). Between the two farms, there is 510 acre-feet of water available for the season. The farmers can obtain up to 7,000 pounds of fertilizer. The new farm has not been cultivated in a while, so the farmers believe that each acre of the new farm will take an extra 4 hours of labor to plant, and an extra 2 hours per acre to tend. Because of the condition of the new farm, they expect the yields to be down from 50 to 45 bushels per acre for corn and from 56 to 50 bushels per acre for strawberries. They want to know the best combination of crops to plant on each farm, with the goal to maximize revenue from at least 75% of each farm's acreage.
Week 2: Assignment 2
A couple has agreed to attend a "Casino Night" as part of a fundraiser for the local hospital. They do not like to gamble because they believe that gambling is generally a losing proposition. However, for the sake of the charity, they have decided to attend and spend $300 on the games. There will be four games, each involving standard decks of cards.
The first game, Jack in 52, is won if you select the Jack of a particular suit from the deck. The probability of actually doing this is 4 in 52 (4/52 or .0769). Gamblers may place bets of $1, $2, or $4 on this game. If they win, the payouts are $12.00 for a $1 bet, $24.55 for a $2 bet, and $49 for a $4 bet.
The second game, Red Face in 52, is won if you select a red face card (including the Jack, Queen, or King) from the deck. The probability of winning is 6 in 52 (.1154). Again, bets may be placed in denominations of $1, $2, and $4. Payouts are $8.10, $16.35, and $32.50, respectively.
The third game, Face in 52, is won if you select 1 of the 12 face cards from the deck. The probability of winning is 12 in 52 (.2308). Payouts are $4, $8.15, and $16 for $1, $2, and $4 bets.
The last game, Red in 52, is won if you select a red card from the deck. The probability of winning is 26 in 52 (.5). Payouts are $1.80, $3.80, and $7.50 for $1, $2, and $4 bets.
Given that they can calculate the expected return or loss for each type of game and level of wager, they have decided to see if they can minimize their expected loss by planning their evening using LP. For example, a $1 bet in Jack in 52 has a return of $12.00, but there is only a 1 in 13 chance of winning. Therefore, the expected value of the dollar bet is ($12.00*(1/13)) or $.9231. This computes to an expected loss of $1-.9231, or $.0769.
The couple wants to appear generous. Therefore, they will place at least 20 bets (of any value) on each of the four games. Further, they will spend at least $26 on 1-dollar bets, at least $50 on 2-dollar bets, and at least $72 on 4-dollar bets. They will bet exactly the agreed-upon $300. What should be their gambling plan, and what is their expected loss for the evening?
Week 3: Assignment 1
Minfly Golf Company manufactures golf balls at Philadelphia, Chicago, and Denver. Cases of golf balls are shipped to warehouses in Orlando, Dallas, and Los Angeles. The following table provides the weekly capacities of the factories and the minimum weekly requirements of the warehouses along with the cost per case to ship between them. Which shipping plan will minimize total costs?
|
Factories
|
Orlando
|
Dallas
|
Los Angeles
|
Capacities
|
|
Philadelphia
|
6
|
9
|
16
|
2500
|
|
Chicago
|
7
|
5
|
10
|
1500
|
|
Denver
|
13
|
8
|
5
|
2000
|
|
Requirements
|
3000
|
1200
|
1800
|
|
After the #1 player on the tour started using Minfly balls, demand from the three warehouses has increased by 1,000 cases per week. To meet this increased demand, an additional factory needs to be built with a capacity of 3,000 cases per week. The two potential locations being considered are Memphis and San Francisco. The cost per case to ship between these locations and the warehouses are as follows:
|
Factories
|
Orlando
|
Dallas
|
Los Angeles
|
Capacities
|
|
Memphis
|
6
|
4
|
12
|
3000
|
|
San Francisco
|
13
|
8
|
5
|
3000
|
|
Requirements
|
4000
|
2200
|
2800
|
|
Where should the new factory be located?
Week 3: Assignment 2
Telephone calls between San Francisco (SF) and Boston (BO) travel through the network shown in the diagram. Calls between these two cities can be routed through Denver (DN), Phoenix (PH), Houston (HO), Chicago (CH), or Atlanta (AT). The number of telephone lines between each location is indicated by the numbers on the arcs connecting each pair of cities. For example, there are 900 lines between San Francisco and Denver. What is the maximum number of calls that can take place between San Francisco and Boston at any one time?
Week 4: Assignment 1
Peters Financial needs to develop an investment portfolio for Mrs. Charles from the following list of possible investments.
|
Investment
|
Cash Required ($)
|
Expected Annual Return ($)
|
|
A
|
5,000
|
500
|
|
B
|
8,000
|
640
|
|
C
|
3,500
|
390
|
|
D
|
10,000
|
700
|
|
E
|
8,500
|
750
|
|
F
|
12,000
|
1,000
|
|
G
|
4,000
|
300
|
Ms. Charles has a total of $30,000 to invest. The following conditions must be met:
· If Investment F is chosen, then Investment G must also be part of the portfolio.
· Only one of investments, A or B, can be included.
Which stocks should be included in Ms. Charles’ portfolio?
Week 4: Assignment 2
Kent County plans to develop several new recreational facilities that must be completed within the $3.5-million budget. A survey of county residents has given information about the type of facilities that county residents would like to see built. The information is described in the following table. Specifically, this table provides the cost to construct and maintain each facility, the acres each facility will require, and the average monthly use of each facility. The county has decided that at least 15 facilities will be built and has set aside 55 acres for construction.
|
Facility
|
Cost per
Facility
|
Acres per Facility
|
Use in People per Month
|
Annual Maintenance
|
|
Basketball Courts
|
$300,000
|
3
|
700
|
$3,000
|
|
Baseball Fields
|
$250,000
|
5
|
1,000
|
$6,000
|
|
Playgrounds
|
$75,000
|
2
|
800
|
$3,000
|
|
Soccer Fields
|
$175,000
|
3
|
1,200
|
$7,000
|
The county has also established the following list of prioritized goals:
P1: The county would like to spend the entire budget.
P2: The county would like to build enough facilities so that 15,000 people or more each month can use them.
P3: The county does not want to use more than the 55 acres set aside for the project.
P4: The county does not want to spend more than $80,000 per year on maintenance costs for the new facilities.
How many of each type of facility should be constructed?
Question 1 of 2:
(A) Sal's International is a popular haircutting and styling salon near the campus of the University of New Orleans. Four barbers work full-time and spend an average of 15 minutes per customer. Customers arrive throughout the day at an average rate of 12 each hour. All arriving customers are assigned a waiting number. Arrivals tend to follow the Poisson distribution, while service time is exponentially distributed. Assuming an infinite population source, determine the following:
What is the average number of customers in the salon?
What is the average time that a customer spends in the salon?
What is the average time a customer spends waiting to be attended?
What is the average number of customers waiting to be attended?
(B) Sal is now considering changing the queuing characteristics of his salon. Upon arrival, instead of being assigned waiting numbers, customers will be able to choose the barbers they prefer. Assuming this selection does not change while the customers are waiting for their barbers to become available and the requests for each of the four barbers are evenly distributed, answer the following:
What is the average number of customers in the salon?
What is the average time that a customer spends in the salon?
What is the average time a customer spends waiting to be attended?
What is the average number of customers waiting to be attended?
(C) Explain why the results from parts A and B are different
Rob Johnson is a product manager at Diamond Chemicals, which is considering whether to launch a new product line that will require it to build a new facility. The technology required to produce the new product is yet untested. If Rob decides to build the new facility and the process is successful, Diamond Chemicals will realize a profit of $650,000. If the process does not succeed, the company will lose $800,000. Rob estimates that there is 60% probability that the process will succeed.
Rob can also decide to build a pilot plant for $50,000 to test the new process before deciding to build the full-scale facility. If the pilot plant succeeds, Rob feels there is 85% chance of the success of the full-scale facility. If the pilot plant fails, Rob feels there is only 20% chance of the success of the full-scale facility. The probability that the pilot plant will succeed is approximately 60%. Structure this problem using a decision tree and advise Rob what to do
The references below will help you create a decision tree by using the Excel add-in, Treeplan. Treesamp.xls is a sample case with examples of decision trees, treeplan.pdf contains detailed instructions for using the Treeplan add-in, and treeplan.zip is the Excel add-in. When opening treeplan.zip, be sure to select Save As in order to save the file to your computer. After opening the file, select Enable Macros, click the Add-Ins tab, and then select Decision Tree in the Menu Commands group. To get started, you must have an active Excel worksheet open. Contact your
Attachment:- requirement.zip