Sample Questions for Stevens’ Exam I, Quantitative Methods
(Answers to 1-13 appear after those questions.)
Unless otherwise specified, all linear programs referred to are assumed to
have nonnegativity constraints on all decision (and, of course, slack)
variables.
1. Suppose that each question on an assignment requires 5 minutes of
thinking and 30 seconds of recording your answer. Let QUEST = the number
of questions, THINK = total # seconds spent thinking, and REC = total
# of second spent recording. Then the relationships among these variables
are given by the constraint(s)
a)
b)
c)
d)
e)
QUEST = THINK + REC
QUEST = 5 THINK + 30 REC
QUEST = 300 THINK + 30 REC
600 QUEST = THINK + 10 REC
300 QUEST = THINK, 30 QUEST = REC
2. Assuming that the objective function is not simply a constant (number),
the optimal OFL in a MAXIMIZE graphical problem will never
a)
b)
c)
d)
e)
touch the feasible region at more than one point
cut through the interior of the feasible region
lie so that the entire feasible region is on or above the line
correspond exactly to the original OFL
The events listed in a) through d) are all possible.
3. Assuming the x1 axis is horizontal and the x2 axis is vertical, the graph
of the constraint x1 >= 4x2 is
a) the single point (0,0)
b) the entire first quadrant.
c) all points on or above a certain oblique (slanted) line passing through
the origin.
d) all points on or below a certain oblique (slanted) line passing through
the origin.
e) all points on or to one side of the 45 degree line bisecting the first and
third quadrants.
4. A certain linear program has exactly two decision variables and exactly three
constraints. One of these constraints is x - 2y = 10. The program has no nonnegativity constraints.
Which of the following is impossible?
a)
b)
c)
d)
e)
The feasible region is nonexistent.
The feasible region is a single point.
The feasible region is a line segment.
The feasible region is a line.
The feasible region is a triangle.
5. Which of the following statements is NOT true in general? You may assume
that the program in question has two decision variables, A and B.
a) If the program's feasible region is unbounded, the program is unbounded.
b) If both nonnegativity constraints are binding on the optimal solution,
the origin is an optimal point.
c) If a constraint is added to an infeasible program, it remains infeasible.
d) If a constraint is removed from an unbounded program, it remains
unbounded.
e) If a program has an optimal point, the program is neither infeasible nor
unbounded.
6. Suppose we have a linear program involving exactly two decision
variables. We focus on a particular inequality constraint in this
program. Which of the following statements is NOT equivalent to the
others? Note that all of these statements would be made in reference to
a particular (not necessarily optimal) solution to a problem.
a)
b)
c)
d)
e)
The inequality constraint is binding.
The inequality constraint has zero slack.
The computed value of the two sides of the inequality are, in fact, equal.
The inequality constraint is satisfied.
When solved graphically, the inequality's constraint line passes through
the solution point.
7. A linear program has the objective MAX P = 3x + 5y + 2z. The optimal
value of P is 400. Then which of the following could NOT possibly occur?
a)
b)
c)
d)
e)
We find a feasible solution to the MAX which gives a P value of 405.
We find an infeasible solution to the MAX gives a P value of 300.
We find that in the optimal solution, x, = y = z.
We find that the optimal point is x = 100, y = 20, z = 0.
All of these represent possible occurrences.
8. In our study of linear programs, we have forbidden
a)
b)
c)
d)
e)
fractional values for variables.
negative values for the optimal objective function value in a MAX program.
strict inequalities in constraints (like "less than" or "greater than")
feasible regions of infinite extent (in some direction).
equality constraints.
9. Removing a constraint from a linear program (and otherwise leaving that
program unchanged) will
a)
b)
c)
d)
e)
always expand the feasible region of the program.
always change the optimal value of the objective function.
sometimes worsen the optimal objective function value.
sometimes cause a (previously feasible) problem to become infeasible.
sometimes cause a (previously bounded) problem to become unbounded.
10. Suppose I wish to maximize my profit from the sales of my product--wooden
bowls. Each bowl costs $2 to make, so if I sell it for S dollars, I
make (S - 2) dollars profit on it. The number that I can sell depends on
the selling price; if the selling price is S dollars, then I can sell
(100 - 10S) bowls. (Clearly, this formula only applies for selling
prices up through $10.) There are some further constraints on my
production and sales which do not concern us now. This problem cannot
properly be cast as a LINEAR program. Which of the following are valid
reasons for reaching this conclusion?
I. if I sell bowls for less than $2, profit would be negative.
II. the objective function depends on only one variable, S.
III. the objective function would not be linear.
a)
b)
c)
d)
e)
I only
II only
III only
I and II
II and III
11. The pressure inside an aerosol can increases LINEARLY with the
temperature of the can. When the can is at 60 degrees Fahrenheit, the
pressure is 12 units. When the temperature is 75 degrees Fahrenheit,
the pressure is 12.3 units. What would the pressure in the can be when
the temperature reaches 105 degrees Fahrenheit?
a)
b)
c)
d)
e)
12.6 units
12.7 units
12.8 units
12.9 units
over 13.0 units
Answers for 1-13:
1. e
2. b
3. d
4. e
5. a
6. d
7. a
8. c
9. e
10. c
11. d