operation research

AREC 525

Spring 2020

Final Exam

FOUR required problems (100 points) + ONE bonus problem (10 points)

Instructions:

• Inform Dr. Yu ([email protected]) immediately when you receive the exam. If you do this correctly, it is worth 2 extra points.

• You have exactly 10 hours (12:30 pm – 10:30 pm on April 30) to complete and turn in the exam to Dr. Yu via email.

• You need to turn in your solutions in an Excel file. Name your solution file as “525_FinalExam_last name”.

• This is an open-book exam: you can use books, notes, computer, or internet sources to answer any of those problems. However, you CANNOT use other person for help.

• Carefully check the setup in Solver. The grade will be based on what you submit. A substantial portion of points will be deducted if Solver shows error in your model or

gives the solutions that are different from your results summary.

Problem 1. (15 points)

Tropicsun is a leading grower and distributor of fresh citrus products with three large citrus

groves scattered around central Florida in the cities of Mt. Dora, Eustis, and Clermont. Tropicsun

currently has 275,000 pounds (lbs) of citrus at the grove in Mt. Dora, 400,000 lbs at the grove in

Eustis, and 300,000 lbs at the grove in Clermont. Tropicsun has citrus processing plants in Ocala,

Orlando, and Leesburg with processing capacities to handle 200,000, 600,000, and 225,000 lbs,

respectively. Tropicsun contracts with a local trucking company to transport its fruit from the

groves to the processing plants. The trucking company charges a flat rate for every mile

regardless of how many pounds of fruit are shipped. The following table summarizes the

distances (in miles) between the groves and processing plants:

Table 1. Distance (in miles) between the groves and processing plants

Processing Plant

Grove Ocala Orlando Leesburg

Mt. Dora 21 50 40

Eustis 35 30 22

Clermont 55 20 25

Tropicsun wants to determine how many pounds to ship from each grove to each processing

plant to minimize the total transportation cost. Please formulate an integer linear programming

model for this problem. You need to define variables, notations, and subscripts clearly if you use

the general form.

Problem 2. (25 points)

Blue Ridge Hot Tubs manufactures and sells three models of hot tubs: Aqua-Spa, Hydro-Lux,

and Typhoon-Lagoons. Each Aqua-Spa requires 1 unit of pump, 9 hours of labor, and 12 feet of

tubing. Each Hydro-Lux requires 1 unit of pump, 6 hours of labor, and 16 feet of tubing. Each

Typhoon-Lagoon requires 1 pump, 8 hours of labor, and 13 feet of tubing. The cost of acquiring

a pump and a foot of tubing is $585 and $4.5, respectively, and the hourly labor cost is $22.5.

The company has 200 pumps, 1566 production labor hours, and 2880 feet of tubing available

during the next production cycle. Also, the company’s marketing research team estimates the

demand function for each hot tube model as below:

Aqua-Spa: Q Aqua

= 300 − 0.175×P Aqua

Hydro-Lux: Q Hydro

= 260 − 0.15×P Hydro

Typhoon-Lagoons: Q Typhoon

= 245 − 0.16×P Typhoon

The marketing team now reports that the prices for the similar products from the competing

brands range between $1,000 and $1,500. The company intends to meet the demand and does not

want to carry inventory. The company now consults you to determine the selling price of those

three hot tub models (i.e., P Aqua, PHydro, PTyphoon) to maximize its profits.

a) Formulate the mathematical programming model for this problem. b) Solve the model in a spreadsheet. c) Interpret and summarize the results. d) Which of the resource constraints are binding at the optimal solution? What is the shadow

price of that binding constraint?

e) Due to the COVID-19 issue, the available labor hours for the next production cycle drop by 20%, what are the new optimal solution?

Table 3. Hours of tractor use by critical month

Land

Quality

Cotton-

Conventional

Corn -

Conventional

Soybeans -

Conventional

Hours/acre

April 2 2 1

May 7 5 3

October 5 2 2

Problem 3. (25 points)

Farmer Eric has a total of 2500 acres of land in which 1000 acres are in a very good quality of

land (L1). The remaining 1500 acres of land are highly erodible with 1300 acres are classified as

a medium quality (L2) and 200 acres as a poor quality (L3). The Natural Resources Conservation

Service (NRCS) has indicated that the highly erodible lands cannot be used for low residue crops

such as cotton or soybeans. However, the NRCS allows Eric to plant cotton and soybeans on the

highly erodible land if he adopts no-till practices. Eric’s experience shows that the production

costs of no-till soybean and cotton will increase by 10% per acre. Also, no-till practices will

reduce cotton yield by 15% but have no impact on soybean yield. The yield records of corn,

cotton and soybeans for the three soils at Eric’s farm can be found in Table 1. The production

costs summary of those three crops is presented in Table 2. Eric also learns that one tractor will

provide a maximum of 400 hours of time per month and costs $15/acre-hour to run. The months

of April, May, and October will be critical to the farming operation. The estimated time for

conventional tillage is summarized in Table 3. Also, he expects a 40% reduction in the tractor

use time for cotton and soybeans in April and May if no-till practices are applied. The expected

selling price at harvest are $0.85/pound for cotton, $5.80/bushel for corn, and $9.35/bushel for

soybeans. Eric asks for your advice to determine what crops to plant, where to plant the crops,

and the acreage of the planted crops to maximize his farming profit.

a) Formulate an LP model for this problem. b) Solve the model using the spreadsheet modeling. c) Interpret and summarize the results. d) If now corn prices drops to $4.05/bushel, what will be the optimal solution?

Table 1. Yield records on Eric’s farm

Land

Quality

Cotton Corn Soybeans

Pounds Bushels

L1 1.6 145 42

L2 1.0 110 30

L3 0.9 95 28

Table 2. Cost records on Eric’s farm

Land

Quality

Cotton -

Conventional

Corn -

Conventional

Soybeans -

Conventional

Dollars/acre

L1 320 220 150

L2 335 190 130

L3 285 175 135

Problem 4. (35 points)

Mrs. Carol Ford approaches you for your advice to make an investment portfolio. She has paid

attention to the four stocks, namely Stock C, Stock P, Stock M, and Stock V, and obtained the

quarterly returns over the past five years (see the “Data.xlsx” file). She asks you to determine the

percentage of her money to be allocated to each of the stocks.

a) Estimate the average annual returns and associated variance/covariance of those four stocks. b) Suppose Mrs. Ford is completely risk averse. Determine the percentage allocated to each

stock in her portfolio, and the resulting expected risk and return using Solver.

c) Now suppose Mrs. Ford is completely insensitive to risk and intends to maximize possible return. Determine the percentage allocated to each stock in her portfolio, and the resulting

expected risk and return using Solver.

d) If Mrs. Ford wants to maximize the average returns on the stocks while minimizing the risk of her investment portfolio. Determine the solution that minimizes the maximum percentage

deviation from the two optimal objectives using the MINIMAX method with Solver.

e) Derive an efficient frontier for the portfolio by assigning the respective weight to returns and risk, i.e. (Wreturns, Wrisk), as (10,1), (5,1), (2,1), (1,1), (1,2), (1,5) and (1,10) in part d.

f) Mrs. Ford is now particularly interested in the stock P and would like to get the forecasts of the returns in the next four quarters (i.e., 2019 Q1 – 2019 Q4) for this stock. Please use

Holt’s method and Solver to minimize the MSE between the actual and predicted stock

returns. Determine the forecasted returns for the next four quarters using this technique.

g) Now use Holt-Winter’s method for additive seasonal effects and Solver to minimize the MSE between the actual and predicted stock returns. Determine the forecasted returns for the next

four quarters using this technique.

h) Which forecasting method will you recommend based on the results in parts f and g? Why?

Bonus Problem (10 points)

You are consulted by an owner of a local delivery service. He has three trucks to run the

business. Now he is contracted to deliver 15 shipping boxes of produces to customers in 15

locations. The weight of each box is summarized in Table 1 and the load capacities of each truck

is presented in Table 2.

Table 1. Weight of produces in each box

Box # Box 1 Box 2 Box 3 Box 4 Box 5 Box 6 Box 7 Box 8

Weight (lbs) 80 95 125 210 160 320 90 110

Box # Box 9 Box 10 Box 11 Box 12 Box 13 Box 14 Box 15

Weight (lbs) 70 210 260 170 240 80 180

Table 2. Load capacities of truck

Truck Weight Capacity Box Capacity Cost per pound

1 800 pounds 5 $0.34 2 900 pounds 6 $0.42 3 850 pounds 5 $0.25

The owner would like to load each truck with five boxes and 800 pounds while minimizing the

total shipping costs. There is an additional charge of $50 per box if trucks carrying extra boxes.

Similarly, a charge of $0.10 per pound will be imposed for trucks carrying less weight.

Formulate the integer goal programming model for this problem. You need to define variables,

notations, and subscripts clearly if you use the general form.