CS 731 – Mathematical Foundations of Computer Networking
Spring 2022
Home Assignment 4
(Due: April 7, 2022)
Submission Instructions: Online submissions on Blackboard are required. Feel free to edit this Word
document to insert your answers, and upload the resulting Word document on Blackboard. It is also possible
to write your answers on separate sheets, and upload on Blackboard a scan of your sheets. Multiple online
submissions on Blackboard are allowed, but only the last online submission made by April 7 will be graded.
Class Policy Reminder: Each student is expected to work independently on ALL home assignments.
Collaboration among students is not permitted. If you need help with any question, please seek it only from
the course instructor.
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1. A Kansas farmer owns 100 acres of land on which she plans to grow wheat and corn. Each acre of wheat
requires 5 hours of labor and $20 of capital, and each acre of corn requires 4 hours of labor and $40 of
capital. The farmer has at most 800 hours of labor and $2,400 of capital available. The profit from an acre
of wheat is $80 and that from an acre of corn is $100. The farmer would like to determine how many acres
of each crop she should plant to maximize her profit. To this end, she formulates this as a linear
programming problem.
a) [2 points] Introduce names for the necessary number of control parameters for this problem.
b) [2 points] Give the objective function the farmer needs to maximize.
c) [2 points] What constraint is imposed due to the labor limitation?
d) [2 points] What constraint is imposed due to the capital limitation?
e) [6 points] Write all other constraints present in this problem.
f) [2 points] Would this problem require any surplus or slack variables? (Yes or No)
g) [2 points] Is this an integer linear programming problem? (Yes or No)
2. Suppose we employ ILP to find the highest weight matching of the following bipartite graph:
𝟓. 𝟑
𝑢1
𝟒. 𝟔
𝑣1
𝟑. 𝟓
𝑢2
𝟐. 𝟎
𝑣2
𝟏. 𝟔
𝑣3
a) [5 points] Give the name of each control parameter.
b) [7 points] Write all constraints in this exercise, and the objective function.