1. Consider the following LP problem:
Maximize X1 + 2X2, Subject to:
3X1 + 4X2  24,
2X1 + 5X2  10, X1, X2 are non-negative.
a)
Graph the feasible region and find the optimal solution
1. Max X1 + 2X2
ST
3X1 + 4X2 <= 24
2X1 + 5X2 <= 10
X1, X2 >= 0
(20%).
Solve for extreme points
(8, 0) (0, 6)
(5, 0) (0, 2) (0,0) Feasible corners
a) Graph the feasible region
X2
C1: 3X1 + 4X2 =24
C2: 2X1 + 5X2 = 10
6
5
4
3
2
1
X1
1
2
This is the feasible
region.
3
4
5
6
7
8
The optimal solution is X1 = 5, X2 = 0.
This maximizes the solution at 5.
2. True/False (why?) Any number divided by itself is always equal to 1.
(5%)
3. Name a former president of the United States who is not buried in the USA. (5%)
4. Pick any three of the following questions, and explain each one in a short paragraph:
(15%)
a) Management Science:
b) Mathematical Model:
c) What is the structure of a Linear Program?
d) Binding Constraint:
e) Slack variable:
5. What are steps in making decisions process? (5%)
6. How does management science improve the abilities of a decision-maker? (5%)
7. What is the difference between a parameter and a decision variable in mathematical
terms?
(5%)
8. Palmgard International makes two products: inner glove and STS batting glove. Profit
is $5 for the inner glove and $7 for the STS batting glove. The inner glove requires 1.5
sq. ft. of leather and 6 inches of foam while the STS batting glove requires 1 sq. ft of
leather and 4 inches of foam. Palmgard receives 500 sq. ft of leather and 175 ft. of foam
each day for production. It takes 6 minutes to construct an inner glove and 4 minutes to
construct an STS batting glove and there are 80 hours available each day. Palmgard
requires that at least 275 inner gloves are made daily. Set up the Linear Program model
(do not solve it).
(15%)
Decision Variables are:
X1 = Number of inner glove made daily
X2 = Number of STS batting gloves made daily
Maximize daily profit 5X1 + 7X2
Subject to
1.5X1 + X2  500 (leather)
6X1 + 4X2  2100 (foam)
6X1 + 4X2  4800 (labor)
X1  275 (production)
X1, X2  0.
9. Find the optimal solution for the following production problem with n=3 products and
m=1 (resource) constraint:
Maximize 3X1 + 2X2 + X3
Subject to: 4X1 + 2X2 + 3X3  12
(15%)
all variables Xi's  0
Since the feasible region is bounded, following the Algebraic Method by setting all the
constraints at the binding position, we have the following system of equations:
4X1 + 2X2 + 3X3 = 12
X1 = 0
X2 = 0
X3 = 0
The (corner points) solutions obtained by solving all systems of equations with three
equations, the solutions from this system of equations are summarized in the following
table.
X1 X2 X3 Total Net Profit
0 0 4
4
0 6 0
12*
3 0 0
9
0 0 0
0
Thus, the optimal strategy is X1 = 0, X2 = 6, X3 = 0, with the maximum net profit of
$12.
10. Give a numerical example for LP with unbounded solution.
(10%)