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CHAPTER 2
Elements of Decision Problems
Notes
This chapter is intended to start the reader thinking about decision problems in decision-analysis terms.
Thus, we talk about decisions to make, uncertain events, and valuing consequences. To make sure that the
richness of the terrain is understood, we introduce the concepts of dynamic decision making, a planning
horizon, and trade-offs.
In our definition of terms, we refer to a decision makerโs objectives where the term values is used to refer to
the decision makerโs set of objectives and their structure. The terms decision and alternative are adopted,
and are used throughout the book rather than similar terms such as โchoiceโ and โoption.โ Likewise, we
have adopted the term uncertain event (and sometimes chance event), which then has outcomes. Finally,
and perhaps most significant, we have adopted Savageโs term consequence to refer to what the decision
maker experiences as a result of a combination of alternative(s) chosen and chance outcome(s). Another
term that we use that comes from Keeneyโs value-focused thinking is the notion of decision context. This
term is discussed in the text and still more thoroughly in Keeneyโs book. Briefly, it refers to the specific
identification of the problem (from which we might suspect that when one solves the wrong problem, one
has used the wrong decision context). It also can be used as a way to identify the class of alternatives that
one is willing to consider; a broader context (safety in auto travel as compared to specific traffic laws, for
example) leads a decision maker to consider a broader class of alternatives.
The time value of money appears in Chapter 2 and may seem out of place in some ways. It is here because
it is a fundamental way that streams of cash flows are valued, and because it provides a nice example of a
basic trade-off. Also, we have found that since most students have already been exposed to discounting, we
have been able to incorporate NPV calculations into problems and case studies throughout the book. For
the few students who have not encountered the topic, the early introduction to discounting in Chapter 2
provides enough information for them to proceed. Of course, the section on NPV may be skipped and used
as a reference later for problems that require discounting or for the discussion of trade-offs in Chapter 15.
Topical cross-reference for problems
Requisite models
โSecretaryโ problem
Sequential decisions
Time value of money
2.13
2.6
2.2, 2.6, Early Bird, Inc.
2.9-2.12, The Value of Patience
Solutions
2.1. a. Some objectives might be to minimize cost, maximize safety, maximize comfort, maximize
reliability, maximize cargo capacity (for shopping or vacationing), maximize maneuverability (in city
traffic). Students will undoubtedly come up with others as well.
b. In this new context, appropriate objectives might be minimize travel time, maximize exercise, minimize
total transportation cost, minimize use of fossil fuels, maximize ease (suitably defined) of visiting friends
and shopping. New alternatives to consider include using a bicycle or public transportation, walking,
rollerblading, skateboarding, motorcycle or scooter, renting a car, such as Zipcar. One might even consider
moving in order to live in a more convenient location.
2.2. Future alternatives can affect the eventual value of the consequence. For example, a university faculty
member, when accepting a position at a different institution, may not immediately resign his or her position
at the first university. Instead, a leave of absence may be taken. The leave of absence provides the
opportunity to decide in the future whether to stay at the new institution or return to the old one. A faculty
member would most likely think about the two different situations โ resigning the current position
immediately versus taking a leave and postponing a permanent decision โ in very different ways.
8
Another good example is purchasing a house. For many people in our mobile society, it is important to
think about the potential for selling the house in the future. Many purchasers might buy an unusual house
that suits them fine. However, if the house is too unusual, would-be purchasers might be afraid that, if they
decide to sell the house in the near future, it may be difficult to find a buyer and the sales price might be
lower than it would be for a more conventional house.
Finally, the current choice might eliminate a future valuable option. For example, our policy of powering
cars with fossil fuels reduces our options for using oil for potentially more valuable and less destructive
future activities.
2.3. In the first case, the planning horizon may be tied directly to the solution of the specific problem at
hand. If the problem is an isolated one not expected to repeat, this is a reasonable horizon. If more similar
problems are anticipated, the planning horizon might change to look forward in time far enough to
anticipate future such situations. If the firm is considering hiring a permanent employee or training existing
employees, then a planning horizon should be long enough to accommodate employee-related issues
(training, reviews, career advancement, and so on). In this broader context, the firm must consider
objectives related to hiring a new person (or training), which might include maximizing the welfare of
current employees, minimizing long-term costs of dealing with the class of problems, satisfying
affirmative-action requirements, or equity in treatment of employees.
2.4. In making any decision, it is important to 1) use all currently available information and 2) think
carefully about future uncertainty. Thus it is necessary to keep track of exactly what information is
available at each point in time. If information is lost or forgotten, then it will either be treated as an
uncertainty or simply not used when deciding. Clearly, the farmer would want to keep up to date on the
weather and incorporate any change to the forecast.
2.5. Some possibilities: insurance, hire another firm to manage the protection operation, press for
regulatory decisions and evaluations (i.e., get public policy makers to do the necessary analysis), do
nothing, develop a โcleanup cooperativeโ with other firms, or design and develop equipment that can serve
a day-to-day purpose but be converted easily to cleanup equipment. Students may come up with a wide
variety of ideas.
2.6. The employer should think about qualifications of the applicants. The qualifications that he seeks
should be intimately related to what the employer wants to accomplish (objectives โ e.g., increase market
share) and hence to the way the successful applicant will be evaluated (attributes โ e.g., sales). The
planning horizon may be critical. Is the employer interested in long-term or short-term performance? The
uncertainty that the employer faces, of course, is the uncertainty regarding the applicantโs future
performance on the specified attributes.
If the decision maker must decide whether to make a job offer at the end of each interview, then the
problem becomes a dynamic one. That is, after each interview the decision maker must decide whether to
make the offer (and end the search) or to continue the search for at least one more interview, at which time
the same decision arises. In this version of the problem, the decision maker faces an added uncertainty: the
qualifications of the applicants still to come. (This dynamic problem is sometimes known as the โSecretary
Problem,โ and has been analyzed extensively and in many different forms in the operations-research
literature. For example, see DeGroot (2004) Optimal Statistical Decisions, Hoboken, NJ: Wiley & Sons. P.
325.)
2.7. Decisions to make: How to invest current funds. Possible alternatives include do nothing, purchase
specific properties, purchase options, etc. Other decisions might include how to finance the purchase, when
to resell, how much rent to charge, and so on. Note that the situation is a dynamic one if we consider future
investment opportunities that may be limited by current investments.
Uncertain events: Future market conditions (for resale or renting), occupancy rates, costs (management,
maintenance, insurance), and rental income.
9
Possible outcomes: Most likely such an investor will be interested in future cash flows. Important trade-offs
include time value of money and current versus future investment opportunities.
2.8. Answers depend on personal experience and will vary widely. Be sure to consider current and future
decisions and uncertain events, the planning horizon, and important trade-offs.
2.9. NPV
=
-2500 1500
1700
+
+
0
1
1.13
1.13
1.132
= -2500 + 1327.43 + 1331.35
= $158.78.
Or use Excelโs function NPV:
=-2500+NPV(0.13,1500,1700) = $158.78
The Excel file, โProblem 2.9.xlsโ has the equation set-up as a reference to cells that contain the cash flows.
2.10. NPV
-12000 5000
5000 -2000
6000
6000
+
+
+
+
= 1.12 +
2
3
4
5
1.12
1.12
1.12
1.12
1.126
= -10,714.29 + 3985.97 + 3558.90 – 1271.04 + 3404.56 + 3039.79
= $2003.90
Using Excelโs NPV function:
=NPV(0.12,-12000,5000, 5000,-2000,6000,6000)
= $2,003.90
The internal rate of return (IRR) for this cash flow is approximately 19.2%.
The Excel file, โProblem 2.10.xlsโ has the equation set-up as a reference to cells that contain the cash
flows.
2.11.
If the annual rate = 10%, then the monthly (periodic) rate r = 10% / 12 = 0.83%.
NPV(0.83%)
90
= -1000 + 1.0083 +
90
90
+ … +
2
12
1.0083
1.0083
= $23.71.
Or use Excelโs NPV function, assume the 12 payments of $90 appear in cells B13:B24:
=-1000+NPV(0.1/12,B13:B24)= $23.71
(As shown in the Excel file โProblem 2.11.xlsโ)
If the annual rate = 20%, then the monthly (periodic) rate r = 20% / 12 = 1.67%.
NPV(1.67%)
90
= -1000 + 1.0167 +
90
90
+ … +
= $-28.44.
2
1.0167
1.016712
10
Or use Excelโs NPV function, assume the 12 payments of $90 appear in cells B13:B24:
=-1000+NPV(0.2/12,B13:B24)= $-28.44
(As shown in the Excel file โProblem 2.11.xlsโ)
The annual interest rate (IRR) that gives NPV=0 is approximately 14.45%. You can verify this result by
substituting 14.45% / 12 = 1.20% for r in the calculations above.
Or with Excelโs IRR function, IRR(Values, Guess), assume the series of payments (the initial $1000
payment and the series of 12 payments of $90) are in cells B12:B24:
=IRR(B12:B24,0) = 1.20%
(As shown in the Excel file โProblem 2.11.xlsโ)
2.12. a. If the annual rate = 10%, then the monthly rate r = 10%/12 = 0.83%. Always match the periodicity
of the rate to that of the payments or cash flows.
-55
NPV(Terry) = 600 + 1.0083 +
-55
-55
+ … +
2
1.0083
1.008312
= $-25.60.
Be sure to get the orientation correct. For Terry, the loan is a positive cash flow, and the payments are
negative cash flows (outflows). Thus, the NPV is negative. Because of the negative NPV, Terry should
know that this deal is not in his favor and that the actual interest rate being charged is not 10% annually. If
it were, then NPV should equal zero. The actual annual interest being charged must be greater than 10% as
NPV is less than zero.
Or with Excelโs NPV function, assume the series of 12 payments of $55 are in cells B12:B23.
=NPV(0.1/12,B12:B23)+600
= -$25.60
These calculations and those associated with the remaining parts of the question are shown in the Excel file
โProblem 2.12.xlsโ.
b. For the manager, the $600 loan is a negative cash flow, and the payments are positive cash flows. Hence,
NPV(Mgr)
55
= -600 + 1.0083 +
55
55
+ … +
1.00832
1.008312
= $25.60.
Or with Excelโs NPV function, assume the series of 12 receipts of $55 are in cells B12:B23.
=NPV(0.1/12,B12:B23)-600
= $25.60
c. If the annual rate is 18%, then NPV is about $-0.08. In other words, the actual rate on this loan (the
internal rate of return or IRR) is just under 18%.
Using Excelโs IRR function, and assuming the cash flows are in cells B11:B23:
11
=IRR(B11:B23,0)*12
= 17.97% annually
2.13. Should future decisions ever be treated as uncertain events? Under some circumstances, this may not
be unreasonable.
If the node for selling the car is included at all, then possible consequences must be considered. For
example, the consequence would be the price obtained if he decides to sell, whereas if he keeps the car, the
consequence would be the length of the carโs life and cost to maintain and repair it.
If the node is a decision node, the requisite model would have to identify the essential events and
information prior to the decision. If the node is a chance event, this amounts to collapsing the model, and
hence may be useful in a first-cut analysis of a complicated problem. It would be necessary to think about
scenarios that would lead to selling the car or not, and to evaluate the uncertainty surrounding each
scenario.
2.14. Vijayโs objectives include maximizing profit, minimizing unsavory behavior, minimizing legal costs,
and maximizing Rising Moonโs appeal. Students will think of other objectives. Vijayโs decision is to apply
for a liquor license, and if granted, then he could decide on how to manage drinking at Rising Moon. For
example, he might be able to create a separate area of his place, such as a beer garden, where drinking
alcohol is allowed. Vijay could also decide to broaden his menu in other ways than serving alcohol. The
uncertainties include future sales and profit for Rising Moon, market reaction to offering alcohol, amount
of disruption occurring from serving alcohol, and legal liabilities. Consequence measures for sales, profit,
and legal costs are clear. He could simply count the number of disruptions to the business due to alcohol or
he could try to associate a cost figure to the unsavory behavior. Rising Moonโs appeal could be measured
by the change in sales volume due to introducing alcohol.
Vijay will certainly, as law requires, hedge by carrying insurance, and he will want to think carefully about
the level of insurance. As mentioned, he might be able to have a designated area for drinking alcohol. He
could gather information now via surveys or speaking to other local merchants. And he can always change
his mind later and stop serving alcohol.
Case Study: The Value of Patience
The Excel solution for this case is provided in the file โValue of Patience case.xlsxโ.
1.
NPV
= -385,000 +
100,000 100,000
100,000
+
+ … +
= $-3847.
2
1.18
1.18
1.18 7
Thus, Union should not accept the project because the NPV is negative.
Using Excelโs NPV function and assuming the series of 7 payments of $100,000 are in cells B12:B18:
=-385000+NPV(0.18,B12:B18)
= -$3847
2.
NPV
= -231,000 +
50,000 50,000
50,000
+
+ … +
= $12,421.
2
1.10
1.10
1.10 7
This portion of the project is acceptable to Briggs because it has a positive NPV.
Using Excelโs NPV function and assuming the series of 7 payments of $50,000 are in cells E12:E18:
= -231,000+NPV(0.1,E12:E18)
= $12,421
12
3.
NPV
= -154,000 +
50,000 50,000
50,000
+
+ … +
= $36,576.
2
1.18
1.18
1.18 7
Thus, this portion of the project is profitable to Union.
Using Excelโs NPV function and assuming the series of 7 payments of $50,000 are in cells H12:H18:
= -154,000+NPV(0.18,H12:H18)
= $36,576
Some students will want to consider the other $231,000 that Union was considering investing as part of the
entire project. Note, however, that if Union invests this money at their 18% rate, the NPV for that particular
investment would be zero. Thus the NPV for the entire $385,000 would be the sum of the two NPVs, or
$36,576.
4. Patience usually refers to a willingness to wait. Briggs, with the lower interest rate, is willing to wait
longer than Union to be paid back. The higher interest rate for Union can be thought of as an indication of
impatience; Union needs to be paid back sooner than Briggs.
The uneven split they have engineered exploits this difference between the two parties. For Briggs, a
payment of $50,000 per year is adequate for the initial investment of $231,000. On the other hand, the less
patient Union invests less ($154,000) and so the $50,000 per year is satisfactory.
As an alternative arrangement, suppose that the two parties arrange to split the annual payments in such a
way that Union gets more money early, and Briggs gets more later. For example, suppose each invests half,
or $192,500. Union gets $100,000 per year for years 1-3, and Briggs gets $100,000 per year for years 4-7.
This arrangement provides a positive NPV for each side: NPV(Union) = $24,927, NPV(Briggs) = $45,657.
Briggs really is more patient than Union!
Case Study: Early Bird, Inc.
1. The stated objective is to gain market share by the end of this time. Other objectives might be to
maximize profit (perhaps appropriate in a broader strategic context) or to enhance its public image.
2. Early Birdโs planning horizon must be at least through the end of the current promotion. In a first-cut
analysis, the planning horizon might be set at the end of the promotion plus two months (to evaluate how
sales, profits, and market share stabilize after the promotion is over). If another promotion is being planned,
it may be appropriate to consider how the outcome of the current situation could affect the next promotion
decision.
3, 4.
Customer
response
New Morningโs
reaction
Move up promotion start date?
Reaction to
New Morning
Now
13
Market
Share or
Profits

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