DSCI 3870.002: Management Science
Exam # 1
10/02/2012
Time allotted: 75 minutes
Name:
SID#:
Please read this carefully
The questions, which you attempt today, consist of True/False and multiple-choice questions worth 200
points total. Some of these questions are based on descriptive cases. There are a total of 28 questions.
Please answer all the questions on the scantron sheet provided. After you are done, please turn in the
scantron and this question booklet.
Please note that you have to enter your name and Student ID Number (SID#) in the above area and on the
scantron. Failure to do so will result in a grade of zero .
This is an open book exam. As mentioned in the syllabus, you are not allowed to use a programmable
calculator and/or laptop computer. Any other type of calculator is allowed. Besides the textbook, you are
allowed to use only those notes, which have been handed out in class, and notes, which you have recorded
during, class sessions. Be sure to allocate you time wisely between the multiple choice and T/F questions.
Acronyms such as LP, LHS (left-hand side), RHS (right-hand side) etc. may have been used. If you are
unsure about an acronym, please ask me/the invigilator immediately.
All the best!!!
1. Consider a linear program (LP):
Min.
2X + Y
St.
3X + Y = 5
2X + Y = 10
X, Y > 0
a.
b.
c.
d.
e.
the feasible region for this LP does not exist*
the feasible region for this LP is a line segment
the feasible region for this LP is a bounded “polygon”
the feasible region for this LP is unbounded
the feasible region for this LP is a point
2. A Linear Program with a polygonal feasible region can have the following types of optimal
solutions:
I) Unbounded objective function value
II) Unique solution
III) Alternative solutions
a. II only.
b. II and III only. *
c. I and II only.
d. III only.
e. I, II and III.
3. The constraint 3x1 - 4x2 < 0 includes the point (20, 50).
a. True *
b. False
4. An Operations Research analyst at the Smith-Lawton Chemical Company formulated the
following constraints. Which is/are invalid for a linear programming problem?
a.
b.
c.
d.
e.
X1≠ 4X2+X3
1.5X1+6X2+3X3 > 21
19 X2- 0.25 X3=17
X1+ 4X2 + X3 < 1
both a & d *
5. If a Linear Program has a line segment as its feasible region, this means that the number of
solutions that satisfies all constraints is
a.
0
b.
1
c.
an infinite number. *
d.
2.
The following five questions relate to the GRAPHICAL LP PROBLEM given below :
The feasible region has been graphed below to help you answer the questions. Each question is to
be answered independently of the other questions.
Constraint 1
Constraint 2
Constraint 3
Constraint 4
Constraint 5
x+y>3
x – 3y < –3
x–y<3
y<5
3x – 2y > –4
x, y > 0
y
7
6
A
5
4
B
C
3
2
D
E
1
0
6.
a.
b.
c.
d.
e.
0
1
2
3
4
Which constraint is represented by letter D?
Constraint 1*
Constraint 2
Constraint 3
Constraint 4
Constraint 5
5
6
7
x
7.
a.
b.
c.
d.
e.
Which constraint is represented by letter B?
Constraint 1
Constraint 2
Constraint 3*
Constraint 4
Constraint 5
8.
a.
b.
c.
d.
e.
Which constraint is represented by letter C?
Constraint 5*
Constraint 4
Constraint 3
Constraint 2
Constraint 1
9.
a.
b.
c.
d.
The addition of which constraint below would NOT change the feasible region?
x>6
y>6
x+y>4
x < 10 *
10.
a.
b.
c.
d.
e.
If the objective function is to Maximize 2x + y, which of the points is optimal?
Intersection of A and C
Intersection of A and B*
Intersection of B and E
Intersection of C and D
Intersection of D and E
11. Which of the following would be an acceptable objective function for a linear model?
a. Max. -10X1 + 5X2
b. Min. 2 X1 + X2 / X1
c. Min. 12 X1 - 17 X2
d. Max. 7 X1 + Ln[X2]
e. Both a. and c. are acceptable*
The next three questions are based on the following case:
The relationship d= 4008p describes what happens to demand(d) as price (p) varies. Here,
price can vary between $20 and $40. The cost is $10 per unit.
12.
How many units can be sold at the $20 price and at the $35 price respectively?
a. 120, 240
b. 240, 120 *
c. 160, 280
d. 280, 160
e. None of the above.
13. Consider prices of $20, $30, and $40. Among these, which price alternative(s)
will maximize total revenue?
a. $20
b. $30
c. $40
d. $20 and $30 both *
e. $20 and $40 both
14. The expression for total profit is given as
a. 400 p8p2
b. 390 p8p2
c. 480 p8p2-4,000 *
d. 320 p8p2-4,000
e. None of the above.
15. Identification and definition of a problem
a.
cannot be done until alternatives are proposed.
b.
is the first step of decision making.*
c.
is the final step of problem solving.
d.
requires consideration of multiple criteria.
The next two questions are based on the following case
To establish a driver education school, organizers must decide how many cars, instructors, and
students to have. Costs are estimated as follows. Annual fixed costs to operate the school are
$120,000. The annual leasing cost per car is $7,000. The cost per instructor is $20,000 and two
instructors, working at different times, would share one car. Tuition for each student is $800. Let
x be the number of cars and y be the number of students.
16. The expression for total profit is given as,
a. 800y – (120,000 + 27,000x)
b. 800y – (120,000 + 34,000x)
c. 800y – (120,000 + 47,000x)*
d. 800x – (120,000 + 47,000y)
e. None of the above.
17. If they think, realistically, that 500 students will attend, then at most how many cars can be
leased without making a loss?
a. 6
b. 7
c. 8
d. 4
e. 5*
The next three questions are based on the following case:
A manufacturer of hot tubs makes two kinds of hot tubs – Wishy and Washy. Wishy requires 8
hours of labor and Washy requires 6 hours. Wishy needs 13 feet of tubing and Washy needs 15
feet. The manufacturer expects to have 1200 labor hours and 2500 feet of tubing available. A
single hot tub of type Wishy requires 1 pump whereas type Washy requires 2 pumps. He has
260 pumps at his disposal. He expects to make a profit of $600 on each type of hot tub Wishy
he sells and $400 on Washy.
Let X= the number of Wishy bathtubs manufactured and Y= the number of Washy bathtubs
manufactured.
18.
The Objective Function for the problem is:
a. Max 1200 X+2500Y
b. Max 600X+400Y*
c. Min 600X+400Y
d. Max X+2Y
e. Min 8X+6Y
19.
Which of the following constraints represents the restriction on tubing?
a. X+2Y ≤ 200
b. 8X+6Y < 1300
c. 8X+6Y ≥ 1300
d. 13X+15Y < 2500*
e. 13X+15Y > 2500
20.
The solution of 100 Wishy and 80 Washy tubs could be the optimal solution.
a. True.
b. False.*
c. Cannot be determined. Need more information.
The next three questions are based upon the following case
A company manufacturers a computer processing unit (CPU) consisting of a board and two chips.
That means, for example, if 50 CPUs are produced, 50 boards and 100 chips must be made. Each
board must be processed on a forging machine, a drilling machine, a grinder and a milling machine.
Each chip must be processed on a forging machine, a grinder, and a lathe. The table below shows
the hours required to process each shaft and frame on different machines.
Hours required to manufacture a single part
Machine
Board
Forging
0.3
Drilling
0.4
Grinding
0.6
Lathe
Milling
0.8
Chip
0.4
0.3
0.7
-
The company has 10 forging machines, 5 drilling machines, 12 grinding machines, 15 lathes, and
10 milling machines. Assume that each machine operates a maximum of 2000 hours per year. The
boards and chips have no demand as individual items. Formulate the problem of finding the
maximum number of CPUs that can be produced in a year, as a linear program.
Let, B = Number of boards produced per year
C = Number of chips produced per year
21. The constraint that captures the restriction on forging hours is given as:
a. 3B + 4C < 200,000
b. 3B + 2C < 200,000
c. 3B + 8C < 200,000
d. 0.3B + 0.4C < 20,000
e. both a. and d. *
22. In order to capture the restriction that boards and chips have no demand as individual items and
also that we need one board and two chips for each CPU, we should:
a. add the constraint B – C = 0
b. add the constraint 0.5B – C = 0
c. add the constraint B – 2C = 0
d. add the constraint 2B – C = 0 *
e. do nothing
23. The objective function is given as:
a. Max B - C
b. Max B*C
c. Min C
d. Max B *
e. b and d are both acceptable.
The next four questions based on this case and on MS Excel outputs that follow
Orange is a Denton, TX based start-up company, which plans to go into the lucrative business of
tablet computers. Orange plans to introduce four different products:
uRad (A powerful tablet PC with WiFi.)
uRad3G (Similar to uRad but also includes 3G connectivity.)
uRad4G (Similar to uRad but includes 4G connectivity, too.)
uRadX (A smaller, flash-drive version of uRad.)
The following table provides information regarding the profit margin and labor and assembly time
required for each product:
Profit Margin ($)
Labor (hours)
Assembly (hours)
uRad
70
1
0.8
uRad3G
85
1.5
1
uRad4G
95
1.5
1.4
uRadX
55
1
0.6
Orange needs to decide the monthly production of each type of product. They have the additional
restrictions/constraints/requirements:
* They have a monthly supply of 17,000 hard disks, which are used by all uRad, uRad3G, and
uRad4G tablets they manufacture.
* Each of the 4 products uses the same battery. They have a monthly supply of 20,000 batteries.
* They have 25,000 labor and 22,000 assembly hours available each month.
* They signed a contract with an electronics store to provide 5,000 uRadX’s monthly.
The following Linear Programming model is used to maximize monthly profit, assuming they can
sell all they produce:
Let x1 – Number of uRad tablets produced each month.
x2 – Number of uRad3G tablets produced each month.
x3 – Number of uRad4G tablets produced each month.
x4 – Number of uRadX tablets produced each month.
Max 70 x1 + 85 x2 + 95 x3 + 55 x4
s.t. x1 + x2 + x3 < 17,000
x1 + x2 + x3 + x4 < 20,000
1 x1 + 1.5 x2 + 1.5 x3 + 1 x4 < 25,000
0.8 x1 + 1 x2 + 1.4 x3 + 0.6 x4 < 22,000
x4 > 5,000
x1, x2, x3, x4 > 0
[Available Hard-disk Constraint]
[Available Battery Constraint]
[Labor Time Constraint]
[Assembly Time Constraint]
[uRadX Contract Constraint]
[Non-negativity Constraint]
Adjustable Cells
Cell
$A$9
$B$9
$C$9
$D$9
Name
x1- uRad
x2- uRad3G
x3- uRad4G
x4- uRadX
Final
Value
5000
0
10000
5000
Reduced Objective
Cost
Coefficient
0
70
-10
85
0
95
0
55
Allowable
Allowable
Increase
Decrease
25 6.666666667
10
1E+30
10
10
15
1E+30
Final
Value
Shadow
Price
Allowable
Increase
Constraints
Cell
$B$12
$B$13
$B$14
$B$15
$B$16
Name
Available Harddisk
Available Battery
Labor Time
Assembly Time
uRadX Contract
15000
20000
25000
21000
5000
0
20
50
0
-15
Constraint
R.H. Side
Allowable
Decrease
17000
1E+30
2000
20000
2000 1666.666667
25000 833.3333333
5000
22000
1E+30
1000
5000
5000
2000
24. What is the optimal profit (value of the objective function)?
a. $1,300,000
b. $1,225,000
c. $1,675,000
d. $20,000
e. None of the above*
25. Current optimal solution suggests not producing any uRad3G. Choose all of the profit margins
at which Orange would consider producing uRad3G?
I. $70.
II. $90.
III. $100.
IV. $110.
a.
b.
c.
d.
e.
I only.
IV only.
II and III.
III and IV. *
II, III and IV.
26. Which constraints have zero surplus or slack variables?
a. Constraints 2, 3 and 5 *
b. Constraints 1 and 4
c. Constraints 2 and 3
d. Constraint 5 only
e. None of the above.
27. If the available assembly time is reduced to 20,500, the optimal solution would change.
a. True*
b. False
c. Need more information.
28. The field of management science
a.
concentrates on the use of qualitative methods to assist in decision making.
b.
is another name for information systems.
c.
approaches decision making rationally, with techniques based on the scientific
method.*
d.
each of the above is true.