ENTS 635
Decision Support Methods for Telecommunications Managers
Midterm Examination
Fall 2002
Name:
SOLUTIONS
Student ID:
Score
[Do not write here]
Section I. _______/10
Section IV _______/80
Section II. _______/20
Section IV _______/20
Section III_______/96
Total________/126
I. Terms and Definitions (2 pts. each, 10 total)
1. Define the term reduced cost in the context of linear programming.
The reduced cost of a variable is the net effect on the objective function of a unit increase in that
variable (marginal value).
2. Define the term shadow price in the context of linear programming.
A shadow price of a resource is the amount of gain in the objective function for each unit increase a
constrained resource.
3. Define the term lagrangian multiplier in the context of linear programming.
A Lagrangian multiplier is the non-linear programming equivalent of a shadow price, that is, the
amount of gain in the objective function for each unit increase of a constrained resource.
4. What is meant by a basic feasible solution to a linear programming problem?
A feasible solution in which the number of non-negative variables is exactly equal to the number of
equations.
5. What is meant by a mixed-integer programming problem? An integer program in which some of the
variables are constrained to be integer and some are not.
1
II. True or False (2 pts. each, 20 total)
(Circle the appropriate term)
1. True or False: Every network flow problem with all integer values in the right-hand side will yield an
integer solution.
2. True or False: A spanning tree problem is a special type of network flow problem.
3. True or False: An assignment problem is another name for a transportation problem.
4. True or False: Every integer programming problem has a linear programming relaxation.
5. True or False: Every spanning tree problem can be solved by a greedy algorithm.
6. True or False: As a manager, your utility curve should reflect your personal values and aversion to risk.
7. True or False: Every linear programming problem can be solved by the graphical method.
8. True or False: Every integer linear programming problem can be solved by the branch-and-bound
algorithm.
9. True or False: Every linear programming problem has a unique solution point.
10. True or False: Except for nonnegative bounds on variables, all constraints in a network flow problem
are based on conservation of flow.
III. Short Questions and Computations (6 pts. each, 96 total)
1. Consider a game in which you roll a six-sided die with the numbers 2, 2, 2, 2, 32, and 62 on it. You win
the amount showing on the die, unless it is a "2", in which case you lose. You pay $20 for each roll of the
die, no matter whether you win or lose. In the following questions, you can leave your computations
unsimplified,.
(a) Calculate the expected winnings for the game.
$X = 4/6 (-$20) + 1/6 (52)
(b) How much would you expect to win (or lose) on a single roll?
We would expect to lose $20 on a single role (this is the most likely outcome!
(c) After 1,000 plays of this game, how much would you expect to win (lose)?
After 1,000 plays, we would expect to win 1000 ($X), where X is given in part (a).
2
2. Consider the following linear combination of the two points (3, 1) and (2, 7): w1 (3, 1) + w2 (2, 7),
where w1 = 0.4 and w2 = 0.6.
(a) Is this a convex combination of the points? Why or why not?
Yes, it is a weighted sum of the points, where the weights are non-negative and add to 1.0.
(b) Consider the set of all points that are generated by allowing w1 to take on all values from 0 to 1, and
setting w2 = 1  w1. Geometrically, what does this set look like?
This is a line segment between the two points.
3. In a linear programming problem, the feasible region contains infinitely many potential solutions,
whenever it has more than one. What is the key fact about linear programming that allows the Simplex
Algorithm to avoid checking all the points in the feasible region?
The Simplex algorithm moves strategically around points on the boundary (the vertices of the simplex).
There are a finite number of them.
4. Suppose that you are about to make a managerial decision based on an expected value computation. A
fellow manager advises that bbefore you act on the expected value, you should compute the expected
opportunity loss of the problem. How would you respond?
There is no point to this, since each of these decision-based rules will yield the same option.
5. State how the maxi-min decision rule works, and give circumstances in which it might fail. Back this up
with an actual payoff matrix, if you can.
Choose the option with the highest payoff in its best-case scenario.
This can fail when one does not consider the worst-case scenario. For example, among the two
options below, option A is preferred by the maxi-min rule, but the worst-case scenario for B is only
slightly less than that of A.
A
B
1000
29
28
29
6.
(a) Give an example of a situation in which an expected value computation is meaningful.
Any repeatable experiment, such as rolling a die with associated payoffs.
3
(b) Give an example of a situation in which expected value might be misleading.
Any experiment or decision, such as buying a car, which is a one-time thing, that is, not repeatable.
In this case, one should also consider the most likely outcome.
7. Explain why it is true that if we solve the linear programming relaxation (LPR) of an integer program
(IP) and find that all integer constraints are satisfied then we have obtained a solution to the IP.
Any integer solution to the LPR is the best solution of all points in the feasible region of the LPR.
Since this includes the integer points, it is the best integer solution.
8. A company wishes to manufacture s switches and r routers each month. Their net profit on routers is the
same as it is for switches. They wish to make as much profit as possible. What would you advise them, if
they are faced with the following constraints on s and r?
3s + r >= 8
s + 3r >= 8
Under only these constraints, there is no limit to how much profit can be made (the problem is
unbounded).
9. Suppose we have a linear programming problem with 5 variables (including slack variables) and three
equations. For each of the following feasible solution vectors, state whether it is basic, nonbasic, or can't
tell (circle the appropriate response).
(a) (2, 2, 2, 0, 0)
Basic
Non-basic
Can't Tell
(b) (3, 4, 6, 1, 0)
Basic
Non-basic
Can't Tell
(c) (0, 0, 0, 1, 0)
Basic
Non-basic
Can't Tell
10. The graphical method is being used to solve a linear programming problem. All of the constraints, and
the feasible region, have been graphed in Figure 1. What will the optimal solution be, if the objective is to
maximize 3x  2y?
4
35
y
30
3
x
0
0
45
24
25
(0, 20) 20
(24, 14)
15
(0, 0)
10
Feasbile
5
Region
-2
y
0
20
0
14
0
-40
135
44
0
0
10
20
30
40
50
60
70
(45, 0)
80
90
x
Figure 1.
11. What makes integer programming so much harder than linear programming?
Unlike linear programming, we have lost the guarantee that an optimal solution will occur on the
boundary of the simplex. Hence, a great many more solutions need to be considered.
12. When looking at the matrix of constraint coefficients (i.e.,. the A in Ax <= b), what is the tell-tale sign
of a network flow problem?
In each column, there will be exactly one "1" and exactly one "-1", with all other entries zero,
13. Suppose that variables X1, …, X5 are binary decision variables for projects 1 through 5, respectively.
Write a mathematical (inequality) constraint to meet the written constraint.
(a) Choose exactly one of projects 1, 2, or 3.
X1 + X 2 + X 3 = 1
(b) If project 2 is chosen, then project 3 must be chosen as well.
X1 <= X2
(c) If project 1 or 2 is chosen (or both), then project 3 must be chosen.
X1 + X2 <= 2X3
14. What is the relationship (in terms of inequalities) between the following numbers?
IPmax = maximum objective function value of an integer program (IP)
LPmax = maximum objective function value of the linear programming relaxation of the above IP
Explain how you know this.
5
LPmax >= IPmax because the LP feasible region is a superset of the IP feasible region, and the max
value can only increase when more points are added to the IP feasible region.
15. Give the linear program relaxation of the following integer program.
Max 2x1 - 3x2 + x3
Subject to
2x1 + 2x2 <= 60
2x1 + 2x2 <= 80
x1, x2 >= 0
x1, x2 integer
LPR is the above math program with the bottom (integer) constraint removed.
16. Consider the feasible region for the LP shown in Figure 2. Draw the feasible region for the integer
program that is formed by restricting both of the variables to integer values.
The collection of black dots is the feasible region to the IP.
4
3
2
1
1
2
3
4
5
Figure 2
6
IV. Problems (10 points each, 80 total)
1. You are faced with a decision problem in which you must choose a subset of 50 different business
opportunities. You are pretty sure that you can formulate it as an integer program, but it is too large to be
solved by Excel Solver. So, you propose that it be solved using a commercial solver software package. A
fellow manager, Sandy, has pointed out that since each decision variable will be binary, there is a finite
number of potential solutions. Sandy suggests that rather than invest in new software, you should find the
optimal solution by writing some computer code to enumerate all the potential solutions and select the one
with the best objective function value. How would you respond to Sandy?
Dear Sandy,
Since each variable is binary, there are 250 possible solutions. This is an astronomical number of
solutions to check!
Love and Kisses,
Your fellow manager.
2. Technical analysts working for you are coding their own version of the branch-and-bound algorithm to
solve a particular type of mixed-integer programming problem. They are confronted with the following
situation when testing the code on an instance of the problem. The problem is to maximize the objective
function. So far, the algorithm has found an integer solution, I, with an objective function value of 32,100.
An LP relaxation has been solved for a current branch, and the resulting objective function value is 29,888.
The algorithm has decided that it does not need to pursue this branch any more. Having heard that you took
ENTS635 at the University of Maryland, they have come to you for advise. Specifically, has their
algorithm taken the right step? Why or why not?
Yes, the algorithm is OK. There is no point in pursuing this branch because the best-case scenario is
to find an integer point with optimal function value of 29,888 (the LP max is an upper bound for the
IP max in this case). But we already have an integer solution with a function value of 32,100.
3. Consider the flow diagram in Figure 3.
(a) What type of math programming problem is this?
An assignment problem, also known as a
tranportation problem.
-800
200
1
500
2
(d) What is the meaning of the figure 200?
This is the limit on the demand of
This is the per-unit cost of shipping goods from
node 1 to node 3.
+500
5
+500
300
700
(e) Write a conservation of flow constraint for node 1.
800 = X13 + X14 + X15 (inflow = outflow)
Figure 3.
7
4
400
-700
(d) What is the meaning of the figure -800?
This is the available flow from node 1.
+500
600
(b) What type of node is node 1?
A supply node.
(c) What type of node is node 3?
A demand node.
3
4. Figure 4 shows the feasible region to a linear program, which requires that profit = 24x + 24y be
maximized.
(a) How many optimal solutions are there to this problem? (Think carefully!)
There are infinitely many (but only two optimal vertices). All points on the edge between the points
shown are optimal solutions.
(b) Suppose that the objective function were changed to profit = 24x + 25y. Now how many optimal
solutions are there?
This tips the slope of the objective function a bit counter-clockwise, which will make the point (a, b)
the unique solution. Note: the new objective function can be written as y =  24/25x + P/25, and the
optimal function value will be P = 24a + 25b.
P = 24x + 24y
(c) Has the maximal profit increased or
decreased?
Maximal profit has increased, because
before, the optimal value was 24a + 25b.
Now it is 24a + 25b, so the function value
has increased by one unit of b. (Note: this
assumes that b >= 0.)
(a, b)
(c, d)
y
Feasible Region
(d) By how much has the maximal profit
changed?
By part (c), the profit has increased by
one unit of b.
x
Figure 4.
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5. One of your fellow-managers, Chris Marr, is on an extended vacation (family emergency). You have
found the decision tree in Figure 5 that Chris was working on.
(a) Describe the problem that you think Chris was trying to solve.
It looks like Chris had a choice between bidding on two projects (or something like that).
Project A has two possible outcomes, win or lose, with probabilities of 0.6 and 0.4, respectively.
Project B had three possible outcomes, win, re-bid, or lose with probabilities of 0.6, 0.1, and 0.3,
respectively. The payoffs are given below each branch.
(b) What is the meaning of the number 48 on the upper-right branch?
This is the net payoff, if Chris bids on project A and wins.
(c) What is this decision tree advising you to do, and why?
Based on expected value, the decision tree is advising that we bid on project B, which has an expected
payoff of 34.8
0.6
Win
48
Bid on A
-18
66
21.6
48
0.4
Lose
-18
0
2
-18
0.6
34.8
Win
68
80
68
0.1
Bid on B
Re-bid
-24
-12
34.8
-12
-24
0.3
Lose
-12
0
9
Figure 5.
-12
6. Bosnet, a high-tech electronics manufacturer for the Internet age, manufactures internet enable TV sets
(T), internet enable stereo systems (S), and high fidelity speakers (P). They have a limited supply of
common parts - Ethernet card (450 in inventory), LCD screen (250 in inventory), speaker cone (800 in
inventory), power supply (450 in inventory), electronics (600 in inventory) - that these products use. An
Internet enabled TV set requires an Ethernet card, a LCD screen, 2 speaker cones, a power supply, and 2
electronic units. An internet enable stereo requires an Ethernet card, 2 speaker cones, a power supply, and
an electronic unit. A high fidelity speaker requires a speaker cone and an electronic unit. The profit on TV
sets is $75, the profit on stereos is $50, and the profit on speakers is $35. We establish the following
notation.
T  Number of TV sets produced
S  Number of stereos produced
P  Number of speakers produced
We may write a model for this problem as follows.
Maximize 75T  50 S  35 P
subject to
T  S  450 (ethernet cards)
T  250 (LCD screens)
2T  2 S  P  800 (speaker cones)
T  S  450 (power supplies)
2T  S  P  600 (electronic units)
T , S, P  0
Use the Excel Solver output provided to answer the following questions. Please explain each of your
answers.
(a) Does the solution change if only 425 Ethernet cards are available?
No; this is not a binding constraint.
(b) Is it profitable to produce Speakers?
No; there are none produced in the optimal solution.
(c) Because of a change in production technology, the profit margin on stereos has increased to $60. Should
the production plan of Bosnet change? If so, what is their new profit?
No; this is within the range of the allowable increase.
(d) 50 speaker cones were found to be defective, reducing the number of available speaker cones to 750.
What will the profit be in this situation?
The shadow price on speaker cones is 12.5, so, they will lose 12.5 on each speaker cone, for a loss of
$12.5 times 50 = $625. Note: this computation is valid only because the corresponding constraint is
binding and the reduction by 50 units is within the allowable decrease.
10
(e) Another supplier is willing to sell electronic units to Bosnet. However, their prices for an electronic unit
is $20 higher than what Bosnet pays its regular supplier. Should Bosnet purchase these electronic units? If
so, how many units should they purchase at most?
They will make $25 more profit (which is net) on each electronic unit (see shadow price). If they pay
$20 more in cost per unit, then they will profit $5 on each unit. So, this is worth their while. But, the
allowable increas is good only up to 50 units.
Microsoft Excel 8.0e Sensitivity Report
Worksheet: [Bosnet.xls]LP
Report Created: 10/22/2002 10:50:45 PM
Adjustable Cells
Cell
Name
R10C2 Variables T
R10C3 Variables S
R10C4 Variables P
Final
Reduced
Objective
Allowable
Allowable
Value
Cost
Coefficient
Increase
Decrease
200
0
75
25 4.999999998
200
0
50
25
12.5
0 -2.499999999
35 2.499999999
1E+30
Constraints
Cell
R14C5
R15C5
R16C5
R17C5
R18C5
Name
Ethernet Cards Used
LCD Screens Used
Speaker Cones Used
Power Supplies Used
Electronic Units Used
Final
Value
400
200
800
400
600
Shadow
Price
0
0
12.5
0
25
Constraint
R.H. Side
450
250
800
450
600
Allowable
Increase
1E+30
1E+30
100
1E+30
50
Allowable
Decrease
50
50
100
50
200
Figure 6.
7. Once again, you have been asked to fill in for your fellow manager, Chris Marr. You have discovered
the spreadsheet shown in Figure 7 in Chris' computer files, under a folder marked "Prospective Products".
Apparently, Excel Solver has already been run to optimality.
(a) What was Chris trying to figure out?
It appears that Chris was solving a linear integer program, which must decide how many of each
type of patio furniture to sell (or maybe manufacture - its' hard to tell).
What's up with Chris, anyway? 
11
(b) In words, translate all of the information in the row marked "Assembly Hours" (as best you can; the
units might not be clear).
Each unit of the standard product requires 2 hours of assembly, each unit of the deluxe product
requires 1 hour of assembly, and each unit of the premium product requires 1 hour of assembly.
In all there are only 600 hours of assembly available. Moreover, in the optimal solution, all of these
600 hours are used.
(c) What information would you need from Excel to know how much one should be willing to pay for more
labor hours for painting?
We would need the shadow price of the "paint hours" constraint.
Deluxe
Premium
Variables
Profit
Standard
Patio Furniture
Problem
200
75
200
50
0
35
25000
<--Max
1
0
2
1
1
0
0
1
0
1
Used
400
200
800
400
600
Available Leftover
450
50
250
50
800
0
450
50
600
0
Constraints
Aluminum
1
Steel
1
Paint Hours
2
Design Hours
1
Assembly Hours
2
Figure 7.
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8. Senior management at Optel, an optical switch manufacturer, is faced with the problem of determining
whether or not to develop a new "terra-power" optical switch. The research and development (R&D) costs
of developing such a switch is estimated to be $25 million. If the company goes ahead with the R&D and
develops the switch, a crucial issue is whether or not the switch will be viewed by customers as superior to
existing optical switches available in the market (and thus worth paying more for). The company assesses a
probability of 0.6 that the switch will be viewed as superior, and 0.4 that the switch will be viewed as
inferior.
After the initial R&D decision is made, and dependent upon the product's image (i.e., after they know
whether it is viewed as superior or inferior), the company needs to decide whether to produce and market
the switch. If they decide to produce and market the switch, its success depends on whether Optel's main
competitor, Lucent Technologies, reacts and develops a competing product. Optel assesses a probability of
0.8 that there will be a competing optical switch by Lucent, if Optel's switch is viewed as a superior
product. They also assess a probability of 0.3 that there will be a competitive optical switch by Lucent, if
Optel's switch is viewed as an inferior product.
If Optel's switch is viewed as a superior product, then Optel estimates that it will gross $150 million in
profits (if produced and marketed) if there is a competitive product, and $350 million in profits (if produced
and marketed), if there is no competitive product. If Optel's switch is viewed as an inferior product, then
Optel estimates that it will lose $150 million (if produced and marketed) if there is a competitive product
and will make $30 million in profits (if produced and marketed) if there is no competitive product. All of
these figures do not include the R&D costs.
(a) Structure Optel's senior management's problem as a decision tree.
See tree below.
(b) Determine what course of action Optel should take under the expected monetary value (EMV) decision
rule to maximize profit.
Optel should develop the switch (follow the upper branch); the expected value is $115.4 million.
0.8
Lucent reacts
125
Produce & market
0
150
165
0.6
Viewed Superior
0.2
Lucent doesn't react
325
1
0
125
350
325
165
Don't produce & market
-25
0
-25
Develop Switch
-25
0.3
Lucent reacts
89
-175
Produce & market
0
-150
-49
0.4
Viewed Inferior
2
0
0.7
Lucent doesn't react
5
1
89
-175
30
5
-25
Don't produce & market
-25
0
-25
Don't develop Switch
0
0
0
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V. Advanced Problem (20 points)
This problem addresses utility curves, as
shown in Figure 8. The horizontal axis is
payoff in units of (thousands of) dollars,
while the vertical axis is relative preference
for a given payoff . Another way to view the
utility curve is to say that the vertical axis
(the "p" values) represent the probability
value at which a person would be indifferent
to a risky situation. For instance, consider
the two alternatives below.
Alt 1: Receive a cash amount of $X .
Alt 2: Win $10,000 with probability p and
lose $3000 with a probability (1  p).
Utility
risk averse
1.00
risk neutral
0.75
risk seeking
0.50
0.25
0.00
A point on the utility curve, (p, X ), would
tell you that p is the probability value that
would make a person indifferent to the two
alternatives when $X is the cash amount in
alternative 1. Roughly speaking, this is how
much the person values $X when weighed against the risky offer.
5k
10k
Figure 8
(a) Explain why the upper curve in Figure 8 is risk averse, while the lower curve is risk seeking.
Suppose that person U has the upper curve and person L has the lower curve. Fix a value of p. Draw
a horizontal line to the right. You will intersect the upper curve before the lower curve. This means
person U requires less cash to walk away from the same risk confronted by person L. Therefore, it
fears risk more than L. (A similar argument can be made by considering a fixed cash value (person
U will place more emphasis on the cash reward rather than the risky deal.) Note: The fact that the
straight curve is risk neutral is an arbitrary choice on our part, but reasonable, since it is in line with
expected value.
(b) What is the interpretation of the intersection of the vertical line ($5k) with the risk averse curve?
This is the value of p at which a person with the upper utility curve will be indifferent to taking the
cash reward or the risky offer.
(c) In light of the two-alternative interpretation of the utility curves, why is it reasonable to assume that all
of the curves intersect at the origin and at the point ($10k, 1.0)?
This says that , no matter who you are, it would seem reasonable to accept a guaranteed $10,000
rather than accept the risky offer (in which you could make at most $10,000, but might lose).
(d) In theory, a utility curve can take on any form. We usually assume that the person behind the utility
curve is "rational". Note that each of the curves in Figure 8 is monotonic, meaning that it is either nondecreasing or non-increasing. Give a careful argument why a non-monotonic curve would represent
irrational or inconsistent behavior on the part of the person to which the curve corresponds. Feel free to
refute this claim, if you have a counter-argument.
Consider a non-monotonic utility curve that rises to a probability level 0.5 at cash value 7k, then falls
to a probability value of 0.4 at cash value 8k. The point (0.5, 7k) says that this person would be on the
border of accepting the cash value 7k to avoid alternative 2. This implies that if the cash value were
increased slightly, say to 8k, and/or if the probability of winning in alternative 2 were decreased, to
say 0.4, then they would accept the 8k in cash to avoid alternative 2. But this contradicts the
interpretation of the point (0.4, 8k), which implies that they would be indifferent in this case.
(e) Write an essay that relays any thoughts, insights, ideas, suggestions, or criticisms you have on the topic
of utility curves in practical decision making. Open ended question. Student dependent. No right or
wrong answer.
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Payoff