DSCI 3870.001: Management Science
Exam # 3
Version: Blue
December 13, 2012
Time allotted: 120 minutes
Name:
SID#:
Please read this carefully
The questions, which you attempt today, consist of True/False and multiple-choice
questions worth 133 points total. Some of these questions are based on descriptive
cases. There are a total of 40 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. You are allowed to use a programmable calculator and/or laptop
computer. 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, IP, NLP 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.
Best of luck!!
1. Consider a maximal flow problem in which vehicular traffic entering a city is routed among
several routes before eventually leaving the city. When represented with a network,
a. the nodes represent stoplights.
b. the arcs represent one way streets.
c. the nodes represent locations where speed limits change.
2.
a.
b.
c.
A sensitivity analysis report for integer linear programming
is available through MS Excel
has precisely the same interpretation as that from linear programming.
is not available through MS Excel.
3. Let x1 and x2 be 0 - 1 variables whose values indicate whether projects 1 (x1) and 2 (x2) are
not done (0) or are done (1) respectively. Which choice below indicates that project 2 can be
done only if project 1 is done?
a. x1 + x2 = 1
b. x1 – x2 < 0
c. x1 – x2 > 0
4. In case of nonlinear programming, a feasible solution is a global optimum if there are no other
feasible points with a better objective function value in the feasible region.
a. True
b. False
5. In class I discussed a puzzle that involved opening a safe with 6 dials and showed that the
problem could be modeled as a _______________.
a. spanning tree problem
b. non-linear programming problem
c. binary integer programming problem
6. Setting of the “Tolerance” option to anything other than zero in an integer programming model
runs the risk of:
a. not getting a feasible solution
b. getting a sub-optimal integer solution
c. getting a solution outside of the integer feasible region
7. Consider the network and the optimal solution in MS Excel for the Max Flow model
Max Flow Model
Capacity
From\To
Node 1
Node 1
Node 2
Node 3
6
5
Node 2
Node 3
Node 4
Node 5
8
4
3
4
Node 4
6
Node 5
3
Capacity
From\To
Node 1
Node 2
Node 3
Node 4
Node 5
Node 6
Total
From
Node 1
0
6
3
0
0
0
9
Node 2
0
0
0
6
3
0
9
Node 3
0
3
0
0
2
0
3
Node 4
0
0
0
0
0
6
6
Node 5
0
0
0
0
0
3
3
Total To
0
9
3
6
3
9
Question: Based on the output, what is the maximum flow?
a. 6
b. 9
c. 11
8.
Node 6
8. Consider the network below for a shortest path model. Bidirectional arrows () indicate that
travel is possible both ways. If we were to formulate an LP model to find the shortest path from
Node 1 to Node 7, then the constraint for Node 6 would be written as:
a. X16 + X56 X64− X67 = 0
b. −X16−X56 +X64+X65 +X67 = 0
c. −X16−X56 +X64+X65 +X67 = 1
The next five questions refer to the following case and the table that follows:
A company has six projects, each of which must be completed in-house or fully subcontracted. Labor
hours are limited to 6000.
Data on project costs and labor requirements is given below.
Project
Profit (in-house)
Profit (subcontract)
Labor requirements
1
$10,000
$5,000
1,200
2
$20,000
$4,000
1,000
3
$22,000
$3,000
1,500
4
$15,000
$8,000
1,100
Let, Xi = 1 if project i is done in-house, 0 if not
Yi = 1 if project i is subcontracted, 0 if not
where i = 1,2,3,4,5,6
9. The most appropriate labor hours constraint is given as:
a. 1200 X1 + 1000 X2 + 1500 X3 + 1100 X4 +1300 X5 +1600X6 = 6000
b. 1200 X1 + 1000 X2 + 1500 X3 + 1100 X4 +1300 X5 +1600X6 ≤ 6000
c. 1200 Y1 + 1000 Y2 + 1500 Y3 + 1100 Y4 +1300 Y5 +1600Y6 ≤ 6000
d. 1200 Y1 + 1000 Y2 + 1500 Y3 + 1100 Y4 +1300 Y5 +1600Y6 = 6000
5
$11,000
$6,200
1,300
6
$24,000
$4,500
1,600
10. If part of the optimal solution were, X1=0, X2=1, X3=0, X4=1,X5=1,X6 =0 then the optimal
profit would be,
a. $58,500
b. $50,000
c. $102,000
d. $18,200
.
11. If no more than 4 projects can be done in-house. This could be modeled as the constraint:
a. X4 ≥ 1
b. X4 ≤ 1
c. X1+X2+X3+X4+X5+X6 ≤ 4
d. X1+X2+X3+X4+X5+X6 ≥ 4
12. The director of the company adds the requirement that if projects 4 or 6 (or both) are done inhouse then project 2 has to be done in-house. This could be modeled as the constraint(s):
a. X2 ≤ X4 and X2 ≤ X6
b. X2 ≤ X4 and X2 ≤ X6 and X2 ≥ X4 + X6 -1
c. X2 ≥ X4 + X6
d. X2 ≥ X4 and X2 ≥ X6
13. If the director of the company adds the requirement that exactly one of projects 1 and 5 must be
done in-house. This could be modeled as the constraint:
a. X1 + X5 ≤ 1
b. X1 + X5 ≤ 0
c. X1 + X5 ≥ 1
d. X1 + X5 = 1
The next three questions refer to the following case and the picture that follows:
GoodDrinks produces two kinds of drinks: Thai Tea and Regular Coffee. The company is adding a
new special drink, which it hopes will help attract new customers. Management is planning to
promote this new drink in two media: radio and direct-mail advertising. A media budget of $5000 is
available for this promotional campaign. Based on past experience in promoting its other drinks,
GoodDrink obtained the following estimate of the relationship between sales and the amount spent
on promotion in these two media.
S =  3R2  12 M2  6 R M + 15 R + 42M
where,
S = total sales in thousands of dollars
R = thousands of dollars spent on radio advertising
M = thousands of dollars spent on direct-mail advertising
GoodDrinks would like to develop a promotional strategy that will lead to maximum sales subject to
the restriction provided by the media budget.
14. What would be the sales if $3000 is spent on radio advertising and $2000 is spent on direct-mail
advertising?
a. - $ 110.874 Million
b. $ 3000
c. $ 16000
d. $ 18000
15. The optimization model to solve GoodDrinks’ problem may be best classified as a:
a. Linear Programming Model
b. Mixed Integer Programming Model
c. Non-linear Integer Programming Model
d. Constrained Non-Linear Programming Model
16. Consider the feasible region (shaded) and the iso-contours (dotted) of the objective function
shown in the figure below with objective value displayed for two of the iso-contours. Approximately
how much should GoodDrinks spend on radio and direct-mail advertising respectively in order to
achieve their stated objective?
a. Nothing on radio, $3000 on direct-mail
b. $ 3500 on radio, $1500 on direct-mail
c. $ 1000 on radio, $1500 on direct-mail
d. $ 2500 on radio, Nothing on direct-mail
The next four questions are based on the following case:
Consider the Excel implementation and subsequent Sensitivity Analysis for a typical portfolio
optimization model below. Let x, y and z represent the fraction invested in stocks AT&T, GM and
USS respectively. The “Average Return” and the information in the “Covariance Matrix” represent
basic statistical analysis performed on returns data provided to us for AT&T, GM and USS. The
optimization model embedded in MS Excel appears below the covariance matrix.
Average Return
Covariance
Matrix
AT&T
GM
USS
Decision:
Stock %
Requirements
Expected Return
15.29%
11.86%
11.61%
AT&T
0.0348
0.0131
0.0280
GM
0.0131
0.0137
0.0069
USS
0.0280
0.0069
0.0659
Total
62.38%
<=75%
9.54%
37.62%
<=75%
4.46%
0.00%
<=75%
0.00%
100%
=100%
14%
Portfolio
Variance
0.0216
>=14.0%
Sensitivity Analysis
Variable Cells
Cell
$C$19
$D$19
$E$19
Name
Decision: Stock % AT&T
Decision: Stock % GM
Decision: Stock % USS
Final
Value
0.623786408
0.376213592
0
Reduced
Gradient
0
0
0.015404637
Name
Decision: Stock % Total
Expected Return Total
Final
Value
1
0.14
Lagrange
Multiplier
0.065264295
0.775240433
Constraints
Cell
$F$19
$F$21
17. The appropriate objective function is given as:
a.
Max. 0.0954x + 0.0446y + 0.000 z
b.
Max. 0.0348x2 +0.0137 y2 + 0.0659 z2 + 0.0131 x y + 0.028x z +0.0069y z
c.
Min. 0.0348x +0.0137 y + 0.0659 z + 0.0262 x y + 0.056 x z +0.0138 y z
d.
Min. 0.0348x2 +0.0137 y2 + 0.0659 z2 + 0.0262 x y + 0.056x z +0.0138 y z
18. One of the constraints in the problem would be:
a. 0.6238x + 0.3762y + 0.0 z ≤ 0.75
b. 0.0954 x + 0.0446 y + 0.0z = 0.14
c. 0.1529 x + 0.1186 y + 0.1161 z ≥ 0.14
d. 0.0954 x + 0.0446 y + 0.0z ≥ 0.14
19.
a.
b.
c.
d.
20.
If the investor demands one percent more return, then the additional risk that he/she will have
to bear would be approximately:
1.0216%
0.00775
0.065
0.0154
If the investor demands that at least 1% of the portfolio be USS stock, then the optimal
portfolio variance would _____ as compared to its current value.
a. decrease
b. increase
c. stay the same
21.
Consider the shaded feasible regions shown below:
a.
b.
c.
d.
both regions are convex
Region A is non-convex and Region B is convex
Region A is convex and Region B is non-convex
both regions are non-convex
The next three questions are based on the following case:
Consider the following integer programming problem and partial graphical solution which follows.
Answer the next two questions based on this information.
Objective function: 3 x + 5 y
Subject to:
2 x + 3 y < 16
x + 4y > 6
2x-y>3
x > 0, y > 0
x – Integer, y – Integer
y
4
: 1.0 x + 4.0 y = 6.0
: 2.0 x + 3.0 y = 16.0
3
: 2.0 x - 1.0 y = 3.0
2
1
0
0
1
2
3
4
5
6
7
8
x
: 2.0x + 3.0y <= 16.0
: 1.0x + 4.0y >= 6.0
: 2.0x - 1.0y >= 3.0
22. If the objective function were to be minimized, then the linear programming
relaxation of the problem would yield an objective function value:
a. slightly more than 11
b. slightly less than 40
c. slightly less than 11
d. exactly the same as the integer program
23. If the objective function were to be maximized, then the optimal objective function value for
the integer program would be.
a. 25.6
b. 25
c. 24
d. 40
24. In order to solve the problem by complete enumeration, we would have to calculate the
objective function value,
a. 2 times
b. 4 times
c. 12 times
d. ∞ times
Read the following case and answer the four questions that follow:
The new Keller Museum of Contemporary Art (KMCA) is considering installing a video camera
security system to reduce its insurance premiums. A diagram of the nine display rooms that
KMCA uses for exhibits is shown in the figure below; the openings between the rooms are
numbered 1-13. A security firm proposed that two-way cameras be installed at some room
openings. Each camera has the ability to monitor the two rooms between which the camera is
located. For example, if a camera were located at opening number 4, rooms 2 and 4 would be
covered; if a camera were located at opening 9, rooms 7 and 8 would be covered; and so on.
KMCA management decided not to locate a camera system at the “Main Entrance” to the display
rooms. However, they would like to provide surveillance for rooms and of course spend as little
as possible on the cameras installed, all of which are identical and cost the same.
 1 if a camera is located at opening i
xi  
Let
0 if not
Room 1
1
3
2
5
Room 4
Room 2
4
6
7
8
Room 6
Room 7
9
11
Room 8
Room 3
10
Room 5
13
12
Room 9
25. The minimum number of constraints (excluding non-negativity) for the most concise IP
formulation of the above problem would be:
a. 9
b. 10
c. 11
d. 13
26. The appropriate objective function for the problem is:
a. Max. x1  x13
b. Min. x1+ 2 x2 + 3 x3 +4 x4 + 5 x5 + 6 x6 + 7 x7 + 8 x8 +9 x9
c. Min. x1+ x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9
d. Min. x1+ x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11+x12 + x13
27. The inequality x5 + x6 + x9 ≥ 1 represents the constraint for:
a. Room 4
b. Room 6
c. Room 7
d. Rooms 4, 6 and 7
28. The constraint x1 + x2+ x4 + x6+ x7 3 would indicate that:
a. Room 4 needs surveillance with at least 3 cameras
b. Room openings 1,2,4,6 and 7 need three cameras
c. Rooms 1,2,4,6 and 7 need three cameras
d. Room 1 cannot have surveillance from openings 1,2,4,6,7
Read the following case and answer the four questions that follow:
Starbucks coffee company makes two types of Coffee: Arabian Mocha Java and Decaf Espresso
Roast. These two types of coffees are made by blending three varieties of coffee beans,
Plantation X, Plantation Y and Plantation Z. Further, the coffee beans are flavored by an
ingredient called chicory and the coffees must meet restrictions on chicory content. Too much
chicory content spoils the taste of coffee. The three varieties of coffee beans are shipped, mixed
together in two delivery trucks to the retail center. Plantation X, is shipped in truck 1, Plantation
Z is shipped in truck 2, and Plantation Y is shipped in truck1 and/or truck 2. No more than 1500
pounds of Arabian Mocha Java and 2400 pounds of Decaf Espresso Roast may be sold. Using
the data in the table below, we want to formulate a profit-maximizing non-linear program.
COFFEE
Arabian Mocha Java
Decaf Espresso Roast
SALES PRICE per POUND ($) CHICORY CONTENT (%)
37
NO MORE THAN 3
41
NO MORE THAN 8
COST PER POUND ($)
Plantation X
Plantation Y
Plantation Z
30
6
26
4
19
7
Assume the following:
X = pounds of Plantation X purchased
Y1, Y2 = pounds of Plantation Y purchased and shipped in trucks 1 and 2 respectively
Z = pounds of Plantation Z purchased
TS1, TT1 = pounds of beans from truck 1 blended into Arabian Mocha Java and
Decaf Espresso Roast respectively
TS2, TT2 = pounds of beans from truck 2 blended into Arabian Mocha Java and
Decaf Espresso Roast respectively
TC1, TC2 = chicory content percentage of beans in trucks 1 and 2, respectively
29. The objective function can be represented as:
a. Max. 37 (TS1+TS2) + 41(TT1+TT2)
b. Max. 37(TS1+TS2)+ 41(TT1+TT2)– 3TC1 – 8TC2
c. Max. 37(TS1+TS2)+ 41(TT1+TT2)– 6X – 4(Y1 + Y2) – 7Z
d. Max. 37(TS1+TS2)+ 41(TT1+TT2)– 30X – 26(Y1 + Y2) – 19Z
30. The chicory percentage of beans in Truck 1 is given as:
a. TC1 = 0.06X + 0.04Y1
b. TC1 = (0.06X + 0.04Y1) / (X + Y1)
c. TC1 = (0.06X + 0.04Y1) / (Y1 + Y2)
d. TC1 = (X + Y1) / (0.06 X + 0.04Y1)
31. The coffee shipped in truck 2 satisfies which of the following constraints:
a. Y2 + Z = TS2 + TT2
b. Y + Z = TS2 + TT2
c. Y2 = (TS2 + TT2)/Z
d. Both a and c are acceptable
32. The restriction on chicory content for Arabian Mocha Java is captured by the constraint:
a. TS1 TC1 + TS2 TC2 < 0.03
b. TS1 TC1 + TS2 TC2 <= 0.03
c. TS1 (0.06X + 0.04Y1) / (X + Y1) + TS2 (0.07Z + 0.04Y2) / (Z + Y2) <= 0.03
d. TS1 (0.06X + 0.04Y1) / (X + Y1) + TS2 (0.07Z + 0.04Y2) / (Z + Y2) <= 0.03 (TS1+TS2)
The next three questions are based on the following case:
Peaches are to be transported from three orchard regions to two canneries. Intermediate stops at a
consolidation station are possible.
Orchard
Supply
Station
Cannery
Capacity
Riverside (R)
1200
Waterford (W)
Sanderson (S)
2500
Sunny Slope (SS)
1500
Northside (N)
Millville (M)
3000
Old Farm (OF)
2000
Shipment costs (cij) of the flow (Xij) on arc (i,j) are shown in the table below. Where no
cost is given, shipments are not possible. The network flow diagram is given below.
R
R
SS
1
OF
W
N
5
4
5
OF
6
3
N
2
2
4
5
9
S
M
M
3
SS
W
S
2
33. Which of the following represents the objective function?
a. Max. 3XRS+ 5XRW +1XRSS +4XSSW +5XSSN +6XOFW +3XOFN +2XWN +2XNW +2XWS +4XWM
+5XNS +9XNM +2XMS
b. Min. 3XRS+ 5XRW +1XRSS +4XSSW +5XSSN +6XOFW +3XOFN +2XWN +2XNW +2XWS +4XWM
+5XNS +9XNM +2XMS
c. Min. 3XSR+ 5XWR +1XSSR +4XWSS +5XNSS +6XWOF +3XNOF +2XNW +2XWN +2XSW +4XMW
+5XSN +9XMN +2XSM
d. Min. XRS+ XRW +XRSS +XSSW +XSSN +XOFW +XOFN +XWN +XNW +XWS +XWM +XNS +XNM
+XMS
34. The overall flow constraint at Waterford is given as:
a. +XRW+ XSSW +XOFW +XNW +XWN +XWS +XWM ≤ 0
b. +XRW+ XSSW +XOFW =XNW +XWN +XWS +XWM = 1
c. +XRW+XSSW +XOFW +XNW –XWN –XWS –XWM ≥ 0
d. –XRW–XSSW –XOFW –XNW +XWN +XWS +XWM = 0
35. Which nodes are “pure” transshipment nodes i.e. with no net demand or supply?
a. R, S and OF
b. W and N
c. S and M
d. SS,W, N and S
Read the following case and answer the three questions that follow:
Mary has 5 textbooks: Applied Science (A), Biology (B), Chemistry (C), Decision Science (D),
and Economics (E). She wants to change them into eBooks, so she hired four people to help her
with different typing speed. The expected finishing time (in hours) for each person of each book
are shown in the table. Mary wants to assign the people to books so that she can get the eBooks
as soon as possible. Assume that the variables are Xij where i=1,2,3,4 denote each person and
j=A,B,C,D,E denote textbooks, and making sure that each book is assigned to a single person
and each person works on at most one textbook.
Textbooks
A
B
C
D
E
1
28
22
27
28
21
2
32
19
22
24
25
3
25
39
30
32
20
4
22
24
26
24
29
Person#
36.
a.
b.
c.
d.
The constraint for Biology is given as:
22X1B + 19X2B + 39X3B + 24X4B =1
X1B + X2B + X3B + X4B > 1
X1B + X2B + X3B + X4B < 1
X1B + X2B + X3B + X4B =1
37.
a.
b.
c.
d.
The constraint for person 3 is given as:
X3A + X3B + X3C + X3D =1
X3A + X3B + X3C + X3D + X3E =1
X3A + X3B + X3C + X3D < 1
25X3A + 39X3B + 30X3C + 32X3D + 20X3E > 1
38. Using the Greedy Heuristic discussed in class, what would be the assignment of cars to
workers?
a. 1-Chemisty , 2-Biology, 3-Economics,4-Applied Science
b. 1-Decision Science, 2-Applied Science, 3-Biology,4-Economics
c. 1-Chemisty , 2-Economics, 3-Decision Science,4-Applied Science
d. 1-Economics , 2-Biology, 3-Applied Science,4-Decision Science
39. The Lone Star Bank recently budgeted the opening up of new zonal office in Texas and solved a Max Cover problem for the same
(as we have seen in class). In the picture of the optimal solution shown below, how many zonal offices were budgeted to be opened?
a. 5
b. 30
c. 35
d. 254
40. Consider the optimal solution to the “Set Cover” version of problem 7.15 posted online and discussed in class. Based on this
optimal solution, which counties receive double coverage?
a. 9,10,11,12
b. 1,3,4,5
c. 5,6,7,8
d. None
----------------- END of EXAM ------------------------