LGT 5431: Homework Assignment
Linear Programming
Due 11:59pm Wed, Feb 5
Instructions:
You must submit your solutions via the LEARN@PolyU Assignment tab by 11:59 pm on Wednesday, February 5. Any submission after this will be marked “ late” in LEARN@PolyU and will be penalized severely. Please make sure you submit on time.
What to submit to LEARN@PolyU:
Your homework write-up -- answers to the questions below. This can be a Word file or pdf. Note that you may hand-write your solutions. If you do, you should scan your answers and upload the files as one pdf document. Any handwritten solutions should be neat and easily legible. We cannot grade them if we cannot read them!
If the homework involves work on Excel, please remember to submit the corresponding Excel files as well.
What to bring to class on Feb 6:
You must turn in a hard copy of your homework; they should be identical to the files you uploaded to LEARN@PolyU. Any variation will be treated as a violation of the honor code. Do not bring hard copies of your Excel spreadsheets!
PARTIAL CREDIT IS AVAILABLE for the following questions, provided that your work is shown and clearly interpreted.
1. Agri-Pro is a company that sells agricultural products to farmers in several states. One service it
provides to its customers is custom feed mixing, whereby a farmer can order a specific amount of
livestock feed and specify the amount of corn, grain, and minerals the feed should contain. Agri-Pro stocks bulk amounts of four types of feeds that it can mix to meet a given customer’s specifications. The following table summarizes the four feeds, their composition of corn, grain, and minerals, and the cost per pound for each type.
Percent of Nutrient in
|
Nutrient
|
Feed 1
|
Feed 2
|
Feed 3
|
Feed 4
|
|
Corn
|
30%
|
5%
|
20%
|
10%
|
|
Grain
|
10%
|
30%
|
15%
|
10%
|
|
Minerals
|
20%
|
20%
|
20%
|
30%
|
|
Cost per pound
|
$0.25
|
$0.30
|
$0.32
|
$0.15
|
Agri-Pro has just received an order from a local chicken farmer for 8000 pounds of feed. The farmer wants this feed to contain at least 20% corn, 15% grain, and 15% mineral. What should Agri-Pro do to fill this order at minimum cost?
(a) [3 points] Define the following decision variables:
X1 = pounds of Feed 1 to use in the mix
X2 = pounds of Feed 2 to use in the mix
X3 = pounds of Feed 3 to use in the mix
X4 = pounds of Feed 4 to use in the mix
Write down in the space below the objective function and constraints, for the optimization problem to help Agri-Pro decide the amount of each feed to fill the order at minimum cost.
Objective function:
Constraints:
(b) [3 points] Set up an optimization model in Excel to solve this problem. Solve it using Solver. In the space below, report the objective function value found by Solver, as well as the recommended amount for each feed. For full credit, you must submit your complete spreadsheet in your completed Excel file on LEARN@PolyU.
|
Objective Function Value
|
|
|
Feed 1
|
|
|
Feed 2
|
|
|
Feed 3
|
|
|
Feed 4
|
|
Please perform. the analysis based on the sensitivity reports provided below, which may be different from what you obtained in Part (b).
(c) [3 points] Using the sensitivity table above, what will happen if corn requirement is
increased from at least 20% to at least 21%? Explain with reference to the quantities from the sensitivity report.
a. Cost will increase by $5
b. Cost will increase by $40
c. Cost will increase by $50
d. It is outside the allowable increase and we need to resolve the LP
2. Hong Kong Farm, a land and farming company, plans to grow two types of vegetables: tomato and green pepper. The company wants to maximize its net profit (i.e., gross revenue minus operating cost) subject to budget, time (in man days) and land constraints. The expected gross revenue, operating cost, time and land requirements for each unit of tomato and pepper are:
|
Vegetables
|
Gross Revenue
|
Operating Cost
|
Time (man days)
|
Land
Requirement
|
|
Tomato
|
$800
|
$500
|
12
|
1
|
|
Pepper
|
$600
|
$250
|
24
|
1
|
The company has set aside a $15,000 budget, a total of 720 man days of time, 45 acres of land. From past experience, the company decides that the amount of tomato should be at least half the amount of green pepper and at most twice the amount of green pepper.
(a) [4 points] Write down in the space below the decision variables, objective function, and constraints, for the optimization problem to help Hong Kong Farm decide the amount of vegetables to grow. Note that fraction of a unit is allowed.
Decision variables:
Objective function:
Constraints:
(b) [3 points] Set up an optimization model in Excel to solve this problem. Solve it using Solver. In the space below, report the objective function value found by Solver, as well as the recommended amount for each type of vegetable. For full credit, you must submit your complete spreadsheet in your completed Excel file on LEARN@PolyU.
|
Objective Function Value
|
|
|
Tomato
|
|
|
Green Pepper
|
|
Parts (c) and (d) are separate questions, which are unrelated to each other. Please perform the analysis based on the sensitivity reports provided below, which may be different from what you obtained in Part (b).
(c) [3 points] A new Consumer Reports predicts that the revenue of tomato will be $670 per unit. Should the company change its plan? Explain with reference to the quantities from the sensitivity report.
(d) [3 points] After surveying the land, the company finds that 5 acres of land are actually not suitable for growing vegetables (i.e., only 40 acres are available for growing vegetables). How does the company’s net profit change? Explain with reference to the appropriate quantities from the sensitivity report.