Operations Research 128 Dr. Emad Elbeltagi CHAPTER 5 DECISION ANALYSIS Decision theory provides a framework and methodology for rational decision making. It treats decisions against nature, where the result (return) from a decision depends on action of another player (nature). For example, if the decision is to carry an umbrella or not, the return (get wet or not) depends on what action nature takes. It is important to note that, in this model, the returns accrue only to the decision maker. Nature does not care what the outcome is. This condition distinguishes decision theory from game theory. In game theory, both players have an interest in the outcome. In many cases, solving your problem involves choosing among alternatives. Your objective is to choose the alternative that is best, where “best” depends on what your goals are. Indeed, the first rule of decision making is to know what your goals are. For example, if your decision problem is which movie to see, then “best” means “most entertaining” (assuming being entertained is your goal). Having identified your goals, you next have to identify your alternatives. For some decision-making problems, your alternatives are obvious. For instance, if you are deciding which movie to see, then your alternatives are the movies playing plus, possibly, not seeing any movie at all. For other problems, however, identifying your alternatives is more difficult. For instance, if you are deciding which personal computer to buy, then it can be quite difficult to identify all your alternatives (e.g., you may not know all the companies that make computers or all the optional configurations available). Your choice of alternative will lead to some consequence. Depending on the decision- making problem you face, the consequence of choosing a given alternative will be either
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Operations Research 128 Dr. Emad Elbeltagi
CHAPTER 5
DECISION ANALYSIS
Decision theory provides a framework and methodology for rational decision making. It
treats decisions against nature, where the result (return) from a decision depends on
action of another player (nature). For example, if the decision is to carry an umbrella or
not, the return (get wet or not) depends on what action nature takes. It is important to note
that, in this model, the returns accrue only to the decision maker. Nature does not care
what the outcome is. This condition distinguishes decision theory from game theory. In
game theory, both players have an interest in the outcome.
In many cases, solving your problem involves choosing among alternatives. Your
objective is to choose the alternative that is best, where “best” depends on what your
goals are. Indeed, the first rule of decision making is to know what your goals are. For
example, if your decision problem is which movie to see, then “best” means “most
entertaining” (assuming being entertained is your goal).
Having identified your goals, you next have to identify your alternatives. For some
decision-making problems, your alternatives are obvious. For instance, if you are
deciding which movie to see, then your alternatives are the movies playing plus, possibly,
not seeing any movie at all. For other problems, however, identifying your alternatives is
more difficult. For instance, if you are deciding which personal computer to buy, then it
can be quite difficult to identify all your alternatives (e.g., you may not know all the
companies that make computers or all the optional configurations available).
Your choice of alternative will lead to some consequence. Depending on the decision-
making problem you face, the consequence of choosing a given alternative will be either
Operations Research 129 Dr. Emad Elbeltagi
known or uncertain. If you are driving in your neighborhood, then you know where you
will end up if you turn left at a given intersection. If you are investing in the stock
market, then you are uncertain about what returns you will earn. Typically, we will
suppose that even if you are uncertain about which particular consequence will occur,
you know the set of possible consequences. For instance, although you don’t know what
your stock price will be a year from now; you do know that it will be some non-negative
number. Moreover, you likely know something about which stock prices are more or less
likely. For example, you may believe that it is more likely that your stock’s price will
change by 20% or less than it will change by 21% or more.
5.1 Prototype Example
The X Company owns a land that may contain oil. A geologist has reported to
management that he/she believes there is a chance of ¼ of oil. Because of this an oil
company has offered to purchase the land for LE90,000. However, X is considering
holding the land in order to drill for oil itself. If oil is found, the company’s expected
profit will be approximately LE700,000. A loss of LE100,000 will be incurred if the land
is dry (no oil).
In this example, the decision maker must choose an action d from a set of possible
actions. The set contains all the feasible alternatives under consideration for how to
proceed. Table 5.1 summarizes the example data.
Table 5.1 Prospective profit for the X Company
decision
State of nature
Oil Dry
Drill for oil
Sell the land
LE700,000
LE90,000
-LE100,000
LE90,000
Chance of state ¼ ¾
Operations Research 130 Dr. Emad Elbeltagi
This choice of an action must be made in the face of uncertainty, because the outcome
will be affected by random factors that are outside the control of the decision maker.
Each of these possible situations is referred to as a state of nature. For each combination
of a decision di and a state of nature j, the decision maker knows what the resulting
payoff would be. The payoff is a quantitative measure of the value to the decision maker
of the consequences of the outcome. Frequently, payoff is represented by the monetary
gain (profit)
5.2 General Representation
In general, the decision analysis problem can be represented as given in Table 5.2. This
table shows that the fundamental piece of data for decision theory problems is the payoff.
This table is also called the payoff table. Table 5.2 provides the payoff for each
combination of a decision d and state of nature j.
Table 5.2 Payoff table for the decision analysis
decision
State of nature
1 2 …. m
d1
d2
….
dn
r11 r12 …. r1m
r21 r22 …. r2m
..... ..... .... ....
rn1 rn2 …. rnm
The entries rij are the payoffs for each possible combination of decision and state of
nature. The decision process can be summarized as follow:
- The decision maker selects one of the possible actions d1, d2, …, dn. Say di.
- After this decision is made, a state of nature occurs. Say state j.
- The return received by the decision maker is rij.
- The payoff table then should be used to find an optimal decision according to the
appropriate criterion.
Operations Research 131 Dr. Emad Elbeltagi
The question faced by the decision maker is: which decision to select? The decision will
depend on the decision maker's belief concerning what nature will do, that is, which state
of nature will occur. If we believe state j will occur, we select the decision di associated
with the largest number rij in column j of the payoff table. Different assumptions about
nature's behavior lead to different procedures for selecting “the best" decision.
If we know which state of nature will occur, we simply select the decision that yields the
largest return for the known state of nature. In practice, there may be infinitely many
possible decisions. If these possible decisions are represented by a vector d and the return
by the real-valued function r(d), the decision problem can then be formulated as:
max r(d) subject to feasibility constraints on d
As indicated in Table 5.1, the X Company has two possible actions (decisions; d1 and d2):
drill for oil or sell the land. With each decision, there is two possible states of natures: the
land contains oil or not. The prior probabilities of the two states of nature are 0.25 and
0.75 respectively.
5.3 Decisions Under Uncertainty
Choosing a specific decision depends upon the decision maker either if he/she is a risk
averse or a risk taker. Thus, the following three methods will be considered: Maximin
payoff criterion; the maximum likelihood criterion, and the expected value.
5.3.1 The Max-min Payoff Criterion
In this method, for each possible decision, the minimum payoff over all states of nature is
determined. Then, select the maximum of these minimum payoffs. Finally, choose the
decision whose minimum payoff gives this maximum. Going back to the prototype
example of section 5.1, the minimum payoff for the first decision "Drill for oil" is -
LE100,000 and that for the second decision "Sell the land" is LE90,000. Then the
selected decision is "Sell the land" as it has the maximum of the minimum payoffs.
Operations Research 132 Dr. Emad Elbeltagi
Such method provides the best guarantee of the payoff that will be obtained. This
criterion is not often used in decision analysis as it is extremely conservative criterion.
This criterion normally is of interest only to a very cautions decision maker.
5.3.2 The Maximum Likelihood Criterion
This method focuses on the most likely state of nature (i.e., the state of nature with
highest probability of occurrence). Using this method, one should identify the most likely
state of nature and then select the decision with the highest payoff in this state of nature.
In the prototype example of section 5.1, the "Dry" state of nature has the maximum
probability of occurrence "0.75". At this state of nature, the highest payoff is LE90,000
corresponding to second decision. Thus, the selected decision is "Sell the land".
This method assumes that the most important state of nature is the most likely one, and
the chosen decision is the best one for the most important state of nature. However, this
method does not rely on the probabilities of the other states of nature. This represents the
major drawback of this method as it ignores all other information.
5.3.3 The Expected Return
This method is the most commonly one. It uses the best available estimates of the
probabilities of the respective states of nature. Then, it calculates the expected value of
the payoff for each of the possible decisions, and then chooses the action with maximum
expected payoff. Expected value is a criterion for making a decision that takes into
account both the possible outcomes for each decision alternative and the probability that
each outcome will occur.
Illustrative Example
Consider that two persons (a and b) are playing with rolling a die. Player "a" will
pay to the other LE9, and then a fair die will be rolled. If the die comes up a 3, 4, 5,
Operations Research 133 Dr. Emad Elbeltagi
or 6, then player "b" will pay "a" LE15. If the die comes up 1 or 2, player "a" looses
the LE9. Furthermore, player "b" agrees to repeat this game as many times as
player "a" wishes to play. Should player "a" agree to play this game?
If a six-sided die is fair, there is a 1/6 probability that any specified side will come
up on a roll. Therefore there is a 4/6 (2/3) probability that a 3, 4, 5, or 6 will come
up and you will win. At first glance, this may not look like a good bet since "a"
may lose LE9, while he/she can only win LE6. However, the probability of winning
the LE6 is 2/3, while the probability of losing the LE9 is only 1/3. Perhaps this isn't
such a bad bet after all since the probability of winning is greater than the
probability of losing. The payoff table is shown below.
Table 5.2: Payoff Table of the Illustrative Example
decision
State of nature
Win Loose
Play
Do not play
LE6
LE0
-LE9
LE0
Chance of state 2/3 1/3
The key to analyze this decision is that "b" allows "a" plays this game as many
times as he/she wants. For example, how often would you expect to win if you play
the game 1,500 times? Based on probability theory, you know that the proportion of
games in which you will win over the long run is approximately equal to the
probability of winning a single game. Thus, out of the 1,500 games, you would
expect to win approximately (2/3) x 1500 = 1000 times. Therefore, over the 1,500
games, you would expect to win a total of approximately 1000 x LE6 + 500 x (-
LE9) = LE1500.
Based on this logic, what is each play of the game worth? If 1500 plays of the game
are worth LE1500, then one play of the game should be worth LE1500 / 1500 =
LE1. Accordingly, you will make an average of LE1 each time you play the game.
Operations Research 134 Dr. Emad Elbeltagi
A little thought about the logic of these calculations shows that you can directly
determine the average payoff from one play of the game by multiplying each
possible payoff from the game by the probability of that payoff, and then adding up
the results. For the die tossing game, this calculation is (2/3) x LE6 + (1/3) x LE(-9)
= LE1.
The quantity calculated is called the expected value for an alternative; this value is
a good measure of the value of an alternative since over the long run this is the
average amount that you expect to make from selecting the alternative. Expected
Value for an uncertain alternative is calculated by multiplying each possible
outcome of the uncertain alternative by its probability, and summing the results.
The expected value decision criterion selects the alternative that has the best
expected value. In situations involving profits where "more is better," the
alternative with the highest expected value is best, and in situations involving costs,
where "less is better," the alternative with the lowest expected value is best.
In calculating the expected return, we make the assumption that there is more than
one state of nature and that the decision maker knows the probability with which
each state of nature will occur. Let pj be the probability that state j will occur. If the
decision maker makes decision di, then the expected return ERi is:
ERi = ri1p1 + ri2p2 + ….. + rimpm.
The decision di* that maximizes ERi will be chosen namely
ERi* = maximum over all i of ERi.
Example 5.1
Let us consider the example of the newsboy problem: a newsboy buys papers from the
delivery truck at the beginning of the day. During the day, he sells papers. Leftover
papers at the end of the day are worthless. Assume that each paper costs 15 cents and
sells for 50 cents and that the following probability distribution is known.
Operations Research 135 Dr. Emad Elbeltagi
p0 = Prob { demand = 0 } = 2/10
p1 = Prob { demand = 1 } = 4/10
p2 = Prob { demand = 2 } = 3/10
p3 = Prob { demand = 3 } = 1/10
How many papers should the newsboy buy from the delivery truck?
To solve this exercise, we first construct the payoff table. Here rij is the reward achieved
when i papers are bought and a demand j occurs.
Decision
State of nature
0 1 2 3
0
1
2
3
0
-15
-30
-45
0
35
20
5
0
35
70
50
0
35
70
105
Next, we compute the expected returns for each possible decision.