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can be purchased in multiple quantities. This problem is known as the Knapsack
Problem due to its general structure—selecting a number of items (projects in
this case) for a limited budget (knapsack).
Available contract
types
# of contract
investments available Cost per contract
Value of contract
Project 1
4
1
2
Project 2
3
3
8
Project 3
2
4
11
Project 4
2
7
20
Budget <
=
20; X j =
# of contracts of project j with possible values 0, 1, 2,
...
.
a. Formulate the LP for this problem.
b. Solve the problem with Solver.
c. What is the marginal effect on the objective function of adding one unit of
each project to # of contract investments available ? Recall this sensitivity
analysis is done by looking at each decision variable, one at a time.
d. How does the solution change if the budget constraint is reduced by 5 units
to 15?
8. Nutrition Problem - A farmer raises hogs for proﬁt. They are organic pigs of the
highest caliber. The farmer is a former nutritionist and has devised the following
table of possible nutrients and minimum daily requirements for 3 nutritional
categories. For example, a Kilogram of corn provides 90 units of carbohydrates,
30 of protein, and 10 of vitamins. Also, there are 200 units of carbohydrates
required daily, and corn costs \$35 per kilogram. The farmer is interested in
knowing the kilograms of Corn, Tankage, and Alfalfa that will minimize his
cost.
Daily nutritional requirements
Min. daily
requirements
Kg. of Corn
Kg. of Tankage Kg. of Alfalfa
Carbohydrates
90
20
40
200
Protein
30
80
60
180
Vitamins
10
20
60
150
Cost \$
\$35
\$30
\$25
a. Formulate the LP for this problem.
b. Solve the problem with Solver.
c. What is the marginal effect on the objective function of adding one unit of
each ingredient (Min. daily requirements)? Recall this sensitivity analysis is
done by looking at each decision variable, one at a time.

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