THE COMPONENTS OF A DECISION PROBLEM .................................................. 1
Decision Making under Uncertainty ............................................................................... 3
Maximax Procedure ........................................................................................................ 3
Maximin procedure ......................................................................................................... 4
Equal Likelihood Procedure ........................................................................................... 5
Minimax Regret Procedure ............................................................................................. 5
Decision Making under Risk ............................................................................................ 6
Maximizing Expected Gain ............................................................................................ 6
Minimizing Expected Regret .......................................................................................... 7
The Expected Value of Perfect Information ................................................................... 8
Decision Trees: An Alternative to Payoff Tables ......................................................... 10
Nodes and Branches ...................................................................................................... 10
Paths and Payoffs .......................................................................................................... 11
Analyzing a Decision Tree .............................................................................................. 11
Multistage Decision Trees .............................................................................................. 13
Other Approaches to Analyzing a Decision Tree ......................................................... 14
Risk analysis ................................................................................................................. 16
Extreme Case Analysis ................................................................................................. 16
UTILITY: AN ALTERNATIVE TO DOLLAR VALUE ........................................... 17
SUMMARY ..................................................................................................................... 17
DISCUSSION QUESTIONS .......................................................................................... 18
SOLVED PROBLEMS ................................................................................................... 18
PROBLEMS .................................................................................................................... 24
CASE: TRANSFORMER REPLACEMENT AT MOUNTAIN STATES
ELECTRIC SERVICE ................................................................................................... 31
KEY TERMS ................................................................................................................... 32
The Components of a Decision Problem
The three basic components of a decision problem are the decision alternatives; the possible
environments, conditions, or states of nature within which a decision is to be implemented; and the
Excerpted from Integrated Operations Management, Mark Hanna and W. Rocky Newman, Prentice Hall,
2003. Supplementary lecture notes for educational purposes only.
outcomes or payoffs that will result from each possible combination of alternatives and states of
The decision alternatives are those actions from among which the decision maker must choose. That
is, they are the aspects of the decision situation over which the decision maker has direct control.
The states of nature are those aspects of the decision situation over which the decision maker does not
have direct control. They certainly include things over which there is no control, such as the weather,
the general state of the economy, or actions taken by the national government. They may also include
things which can be influenced to some extent but not completely. The market response to a new
product (which is partly determined by the price charged and advertising) or a competitor's actions
(which may be at least partly determined by choices made by the decision maker) are conditions the
decision maker cannot determine completely.
Making a decision and implementing it within a state of nature will result in some outcome, which
may be either a gain or a loss. The payoff function specifies what that outcome is for every possible
alternative/state-of-nature combination. The payoff function may be given either as an equation or,
when the possible decision alternatives and states of nature are relatively limited, as a payoff table or
matrix. For example, Table B.1 shows a payoff table for a proposed cardiac catheterization service at
Cheryl has been approached by the head of the hospital's cardiac care unit about the possibility of
starting a cardiac catheterization service at the hospital. Doing so would require either the
conversion of existing space in the hospital (which would allow the development of a small unit) or
the construction of an addition (which would make a larger unit possible). Additional equipment
would also have to be purchased. Before taking this idea to the capital expenditure review
committee for the chain that owns the hospital, Cheryl gets information on the costs of the two
possibilities from the hospital architect and asks the hospital's marketing department for a
preliminary assessment of the possible levels of demand for this type of service. Putting this
information together, Cheryl creates the payoff table in Table B.1. The hospital chain uses a five-
year planning horizon for major capital expenditures. The five-year returns shown in the table are
stated in millions of dollars of net present value over that time horizon.
Table B.1 Payoff Table for Cardiac Catheterization Service
Alternative Low Medium High
Build addition –3 1 7
Convert space –1 2 5
Do nothing 0 0 0
There are three basic types of decision problems, each with its own mode of analysis:
1. Decision making under certainty: In this type of problem, the state of nature is known, so the
payoff table has only one column. In principle, the analysis is simple: Choose the alternative
with the best payoff. In practice, it is not always that easy, since an "alternative" may actually
represent the values for many decision variables. However, solution procedures have been
developed for many problems of this type. For example, Supplement C presents an
introduction to linear programming, a popular technique for decision making under
2. Decision making under uncertainty: In this type of problem, decision makers have the payoff
table, but no information on the relative likelihood of the states of nature. A number of
approaches to this type of problem are possible, several of which will be discussed in this
3. Decision making under risk: In this type of problem, in addition to the payoff table, decision
makers have the probabilities for the different states of nature. The standard approach is
expected value analysis, which will be discussed in this supplement.
DECISION MAKING UNDER UNCERTAINTY
As noted above, in decision making under uncertainty we have a payoff table but no information about
the relative likelihood of the different states of nature. A number of procedures have been developed
for choosing an alternative under these conditions, each based on a different philosophy about what
constitutes a good choice. We shall consider four: maximax, maximin, equal likelihood, and minimax
The maximax procedure (called the minimin procedure if the payoffs are costs or losses rather than
profits or gains) is an optimistic approach. It assumes that no matter what alternative we choose,
"nature" will smile favorably on us and choose the state of nature that will benefit us the most (or hurt
us the least). Thus, we choose the alternative with the largest maximum gain (or the smallest
Find the maximax solution to Cheryl's cardiac catheterization service problem described in Example
From Exhibit B.1, the maximum payoffs for the decision alternatives are:
Build addition: 7
Convert space: 5
Do nothing: 0
The largest maximum payoff is 7, so the "maximax" decision is to build an addition.
The maximin procedure (called the minimax procedure if the payoffs are costs or losses) is a
pessimistic approach. It assumes that, no matter what alternative we choose, "nature" will choose the
state of nature that benefits us the least (or hurts us the most). Thus, we choose the alternative with the
largest minimum gain (or the smallest maximum loss). In fact, if "nature" is a competitor who gets to
choose his alternative after we have chosen ours, this could be a very good strategy. (This gets into an
area called game theory, which we will not discuss.)
Find the maximin solution to Cheryl's cardiac catheterization service problem described in Example
From Exhibit B.1, the minimum payoffs for the alternatives are:
Build addition: 3
Convert space: 1
Do nothing: 0
The largest minimum payoff is 0, so the "maximin" decision is to do nothing.
Equal Likelihood Procedure
One criticism of the maximax and maximin strategies is that they focus on only one payoff for each
alternative, ignoring all other states of nature and their payoffs. The equal likelihood procedure is
based on the assumption that if we cannot determine the relative likelihood of the states of nature, then
it is rational to presume that they are equally likely. If this is the case, then we should choose the
alternative with the largest average payoff (or the smallest average payoff for losses).
Find the equal likelihood solution to Cheryl's cardiac catheterization service problem described in
From Exhibit B.1, the averages of the rows of the payoff table are:
Build addition: 1.667
Convert space: 2.000
Do nothing: 0
The largest average payoff is 2.0, so the "equal likelihood" decision is to convert space.
Minimax Regret Procedure
We have all had the experience of making a decision and then, after observing the result, wishing that
we had made another choice instead, so that our payoff would have been better (or less bad). That is
the concept behind the minimax regret procedure: We choose the alternative that will yield the
smallest possible maximum regret. The procedure has two steps:
1. Construct a regret table by subtracting each payoff from the maximum payoff in its column.
(For a loss table, subtract the smallest loss in each column from each entry in its column.)
2. Apply the minimax procedure to the regret table: Find the maximum regret in each row; then
choose the alternative with the smallest maximum regret.
Find the minimax regret solution to Cheryl's cardiac catheterization service problem described in
Step 1: Construct the regret table from the payoff table in Table B.1. As shown in Exhibit B.2, this
is done in two steps. First, as shown in Exhibit B.2(a), find the maximum entry in each column of
the payoff table. Second, as shown in Exhibit B.2(b), each payoff table entry is subtracted from the
maximum in its column to get the regret table.
Step 2: Find the maximum regret for each alternative, as shown to the right of the regret table in
Figure B.2(b). The alternative with the smallest maximum regret is to convert space, with a
maximum regret of 2.
DECISION MAKING UNDER RISK
There are two basic approaches to decision making under risk: (1) maximizing expected gain or
minimizing expected loss, and (2) minimizing expected regret.
Maximizing Expected Gain
Under some reasonable assumptions about the characteristics of a rational decision maker, it can be
shown that when probabilities for the states of nature are known, the alternative that maximizes the
expected gain (or minimizes the expected loss) is the appropriate decision. To determine the expected
value of a decision alternative, multiply that alternative's payoff under each different state of nature by
that state of nature's probability and sum the resulting products.
Minimizing Expected Regret
As an alternative to using the payoff table to maximize the expected gain (or minimize the expected
loss), we can choose the alternative that minimizes the expected regret. As shown in Examples B.6
and B.7, the two approaches lead to the same decision.
(Refer to Example B.1.) Cheryl has gone back to the hospital's marketing department to get some
additional information about the relative likelihood of the different levels of demand for the
proposed cardiac catheterization service. After review of their marketing research, the department
gives Cheryl the following probability estimates: P(low) = .2, P(medium) = .7, P(high) = .1. Using
these probabilities, determine which alternative maximizes the expected value return to the hospital.
The projected returns are given by the rows of Table B.1. The expected returns for the three
Build addition: .2(–3) + .7(1) + .1(7)
= –.6 + .7 + .7 = .8
Convert space: .2(–1) + .7(2) + .1(5)
= –.2 + 1.4 + .5 = 1.7
Do nothing: .2(0) + .7(0) + .1(0) = 0
The alternative with the highest expected payoff is to convert space. Exhibit B.3 shows the Excel
computation of the expected values.
(Refer to Example B.1.) The regret matrix for the cardiac catheterization lab decision problem was
developed in Example B.5. Determine which decision alternative minimizes the hospital
management's expected regret.
Using the regret matrix from Exhibit B.2(b) and the probabilities given in Example B.6, the
expected regrets for the three decision alternatives are:
Build addition: .2(–3) + .7(1) + .1(0)
= .6 + .7 + 0 = 1.3
Convert space: .2(–1) + .7(0) + .1(2)
= –.2 + 0 + .2 = .4
Do nothing: .2(0) + .7(2) + .1(7)
= 0 + 1.4 + .7 = 2.1
The alternative with the lowest expected regret is to convert space.
Comparing the expected regrets found in Example B.7 with the expected payoffs in Example B.6, we
see that, for each pair of alternatives, the difference between the values of the expected gains and the
difference between the values of the expected regrets are identical. For example, the difference
between the expected gains for "build addition" and "convert space" is 1.7 – .8 = .9. The difference
between the expected regrets for the same pair is 1.3 – .4 = .9. This finding is not specific to this
particular example, but is a general result.
The Expected Value of Perfect Information
If, before having to make a final decision, we could find out exactly what the state of nature was going
to be, we could improve the quality of the decision. The expected value of perfect information (EVPI)
is a measure of the expected current worth to the decision maker of being able to find out, just before
having to make the decision, what the state of nature will be. It combines the current assessments of
the probabilities of the states of nature with the improved value of knowing the state of nature. Thus,
the EVPI is computed as the difference between the expected value of making the decision with
perfect information (EWPI), given by the weighted combination of the best payoffs for each state of
nature, and the expected value of the best decision with current information.
(Refer to Example B.1.) Assuming it would be possible to find out the demand level for the cardiac
catheterization service before making a final decision about whether to introduce the service and, if
so, how large a unit to build, determine the current expected value of having this perfect
The best decisions for each state of nature (demand level) and their values, as given in Exhibit B.1,
Demand Best Decision Value
Low Do nothing 0
Medium Convert space 2
High Build addition 7
Using the probabilities of the demand levels given in Example B.6, the expected value with perfect
information (EWPI) is:
EWPI = .2(0) +.7(2) + .1(7) = 0 + 1.4 + .7 = 2.1
The best decision without having perfect information, as found in Example B.6, is to convert space,
which has an expected value of 1.7. Thus, the expected value of perfect information is:
EVPI = EWPI – E(best decision) = 2.1 – 1.7 = .4
That is, the hospital could increase its expected net present value by $.4 million if it were able to
find out what the demand for this new service would be before committing to whether and how to
Note that the expected value of perfect information in Example B.8 is exactly the same as the
expected regret from Example B.7. Given that EVPI and expected regret are both computed by
comparing, for each state of nature, the value of a specific decision with the value of the best decision
for that state of nature, this result is not an accident. It will always be the case.
Since it is never possible (at least legally) to get perfect information before having to make a decision,
the expected value of perfect information is useful mainly as an upper bound on the expected value of
sample or imperfect information (such as market surveys or economic forecasts), which is often
available before a final decision must be made.
DECISION TREES: AN ALTERNATIVE TO PAYOFF TABLES
An alternative to using a payoff to compute the expected values of decision alternatives is to use a
decision tree. As shown in Figure B.1, a decision tree consists of two or more stages or levels, shown
in time order from left to right.
Figure B.1 Decision Tree for Cardiac Catheterization Service Proposal
Nodes and Branches
Each level of a decision tree consists of nodes (the squares and circles) and branches, which represent
alternatives. The square nodes represent decision points; each branch from a decision node represents
one of the decision alternatives available at that point. The circles, or chance nodes, represent problem
features that are determined by chance or probability, such as the states of nature.
Figure B.1 is actually the decision tree for Cheryl's problem of whether to propose starting a cardiac
catheterization service and, if so, how large a unit to build. The nodes have been numbered to
facilitate the description of the tree.
Node 1 is the decision node. There are three alternatives, represented by the three branches:
(1) build an addition, (2) convert existing space, and (3) do nothing.
Node 2 is a chance node representing the possible states of nature that might follow a decision
to build an addition. There are three possibilities: (1) low demand, which has a probability of
.2, (2) medium demand, which has a probability of .7, and (3) high demand, which has a
probability of .1. Notice that each state of nature's probability has been written next to its
Node 3 is another chance node, representing the possible states of nature that might follow a
decision to convert existing space. The branching is identical to the one for node 2.
Node 4 is the chance node that represents what might happen after a decision to do nothing.
While it could have branches identical to those for nodes 2 and 3, it is simpler to represent
this way. That is, with probability 1, nothing is going to happen since no action is being
Paths and Payoffs
A connected series of branches that starts at the extreme left side of the tree (node 1) and goes through
all levels of the tree is called a path. Each path represents one of the possible sequences of decisions
and chance results for the problem represented by the tree. For example, "build addition" from node 1
and "medium demand" from node 2 is one possible path or decision alternative and demand level
At the extreme right edge of the tree is the net payoff for each of these possible paths, as given
originally in the payoff table in Table B.1. For example, building an addition and experiencing
medium demand for the service will result in a net present value of $1 million.
ANALYZING A DECISION TREE
The standard approach to analyzing a decision tree is to choose on the basis of expected value. The
procedure for doing this is called averaging out and folding back.
Averaging out means replacing each branching by a single number. For a chance branching, the
number used is the expected value, which is found by multiplying the probability of each branch by
the value at its right end and summing. For a decision branching, the number used is the value of the
best alternative in the branching.
Folding back means that this process starts at the right-hand edge of the tree and proceeds back to the
start of the tree, working from right to left.
Use the process of averaging out and folding back to analyze the decision tree shown in Figure B.1.
Determine which decision alternative maximizes the expected return for the cardiac catheterization
decision problem described in Example B.1.
The first round of averaging out consists of replacing each chance branching in the right-most level
of the tree by its expected value:
The chance branching from node 2 is replaced by its expected value:
.2(–3) + .7(1) + .1(7) = –.6 + .7 + .7 = .8
The chance branching from node 3 is replaced by its expected value:
.2(–1) + .7(2) + .1(5) = –.2 + 1.4 + .5 = 1.7
The chance branching from node 4 is replaced by its expected value, which is 0.
The result of this first round of averaging out is the reduced tree shown in Figure B.2(a), in which
each second-level chance branching has been replaced by its expected value.
Figure B.2 Analysis of Cardiac Catheterization Service Proposal Decision Tree
We now back up one level in the tree (folding back). In the second round of "averaging out," node
1, which is in the new right-most level of the tree of the reduced tree shown in Figure B.2(a), is
replaced by the value of the best alternative, which is 1.7 for "convert space." To show this, the
value of the alternative chosen has been written above the decision node and the branches not
chosen have been marked out with slashes as shown in Figure B.2(b).
Since the tree now has only one decision node, we can readily see that the decision strategy that
maximizes the expected return to the hospital over the planning horizon is: Convert space for an
expected net present value of $1.7 million.
In practice, we would consolidate the entire analysis process into a single decision tree rather than
redrawing successively smaller trees after each round of averaging out and folding back. To do this,
the result of each averaging out is written above the node to which it applies, with, as suggested in
Example B.9, the decision alternatives not chosen being marked out with slashes. The result of
applying this summarization process to the analysis in Example B.9 is shown in Figure B.3, with the
averaging out results being shown in red.
Figure B.3 Analyzed Decision Tree for Cardiac Catheterization Service Proposal
MULTISTAGE DECISION TREES
While we can certainly use a decision tree to analyze a one-decision problem, as just illustrated for
Cheryl's cardiac catheterization service proposal, we really will not gain anything we could not get
from a payoff table analysis. The real benefit of using a decision tree comes in more complicated
problems with multiple levels of decisions or states of nature, particularly if the decision alternatives,
the states of nature, or their probabilities depend on what precedes them in the tree. While we could
still use payoff tables to analyze these types of problems, structuring the tables would be difficult. A
tree shows the structure and relationships in such a problem much better than a table does.
Fred has just returned from a weeklong trip to China, where he was part of a team sent by his
company to explore a joint venture with a Chinese company to manufacture and market consumer
electronics in China and Southeast Asia. Part of the decision about whether and how to enter into
the arrangement is the decision about the size of the facility to build. The two alternatives discussed
were (1) to build a large plant initially or (2) to build a small plant and then, if warranted, expand
later or, if business is not good enough, sell out to the Chinese partner.
The possible decisions—along with preliminary estimates of the sales levels, their probabilities, and
the resulting net present values of the different combinations—are shown in the decision tree in
Figure B.4. Notice two particular features of this tree:
It is possible to have two or more successive chance nodes, as in the "high" result of node 2
(sales level during the first two years) being followed by node 4 rather than a decision.
(Technically, there is a decision to "stay" or "sell out," but the decision is obvious and is not
included in the analysis.)
The probabilities of the states of nature for nodes 4, 8, 9, 10, and 11, all of which deal with
the sales level during years 3–5, depend on the sales level during the first two years. That
is, the later years' sales probabilities are conditional on the earlier years' results.
Determine what Fred's strategy should be. That is, determine whether they should initially build a
large or small plant and, subsequently, what to do after the end of the first two years.
The averaging out and folding back analysis of this decision is shown in red on the tree in Figure
B.4. The best strategy is to build a large plant (node 1 choice) and, whether the first two years' sales
are high or low (node 4), to stay with the project. The overall expected net present value from
following this strategy is $20.0 million.
Figure B.4 Decision Tree for Chinese Joint Venture Proposal
OTHER APPROACHES TO ANALYZING A DECISION TREE
While expected value analysis is the standard approach to analyzing a decision tree, it is not the only
method possible. Two alternative approaches are risk analysis, which recognizes the various possible
outcomes and their probabilities, and extreme case analysis, which, like the maximax and maximin
strategies, takes either an optimistic or pessimistic view rather than working with probabilities.
Explanations of both approaches follow.
Refer back to the decision tree in Figure B.1. Construct cumulative probability distributions for the
payoffs for the three decision alternatives.
"Build addition" has possible payoffs of –3, with probability .2, 1 with probability .7, and 7, with
probability 1. The cumulative probability distribution for "build addition" is, therefore:
Cumulative Value Probability
Similarly, the cumulative probability distributions for the alternatives "convert space" and "do
Convert space Do nothing
–1 .2 0 1.0
Figure B.5 shows graphs of these three cumulative probability distributions. The decision maker can
choose among the three alternatives by comparing the cumulative distributions on whatever basis he
or she prefers.
Figure B.5 Risk Analysis Distributions
This approach to analyzing a decision tree recognizes each possible decision combination and the set
of possible payoffs and their probabilities. These are then converted into a cumulative probability
distribution and graphed for comparison purposes.
Extreme Case Analysis
This approach combines the averaging out and folding back approach described for expected value
analysis with the procedures used earlier for decision making under certainty. The folding back part is
the same, but in averaging out, rather than replacing a chance branching by its expected value, use the
highest branch value (for the "best case" analysis) or use the lowest branch value (for the "worst case"
Refer back to Figure B.1. Perform a "best case" analysis on the cardiac catheterization service
problem. (The "worst case" analysis will be left for the problems.)
The results of averaging out and folding back for a best case analysis are shown in red in the tree
diagram in Figure B.6. The probability values from Figure B.1 have been eliminated since they are
not relevant for a best case analysis.
In the first round of averaging out:
The branching from node 2 is replaced by 7, the value for high demand.
The branching from node 3 is replaced by 5, the value for high demand.
The branching from node 4 is replaced by 0, which is the only possible payoff.
The second round of averaging out uses the results from the first round. Since node 1 is a decision
node, its branching is replaced with the value of the alternative chosen. Given that we are following
a maximax or "best case" approach, that alternative is "build addition," with value 7.
Decision Tree for Cardiac Catheterization Proposal, Using "Best Case"
Utility: An Alternative to Dollar Value
In some decisions, money is not an appropriate measure of the quality of the various outcomes. For
example, how could you put a dollar value on losing your health or winning the top international
award in your field? In other cases, the dollar value of an outcome may have implications beyond its
monetary significance. A big loss, for example, may result in the bankruptcy of the organization,
which is more significant than the dollar value of the loss.
When dollar value alone is not an adequate measure of the value of an outcome, an alternative
measure, utility, may be used. How to determine values for a utility function for a given situation is
outside the scope of this supplement. However, a fairly complete discussion may be found in any text
on decision theory, such as Raiffa, Decision Analysis: Introductory Lectures on Choices Under
Uncertainty (Addison-Wesley, 1970). Making decisions based on utility, however, is no different
from making decisions based on money, as described in this supplement.
All decision problems have three basic components: decision alternatives, implementation
environments or states of nature, and a payoff function that gives the gain or loss for each
In decision making under uncertainty, we assume that these three components are all that are known.
A number of procedures based on different philosophies of what constitutes a good approach—
maximax, maximin, equal likelihood, and minimax regret—were presented and illustrated.
Maximization of expected value or minimization of expected cost is the standard procedure for
decision making under risk, in which probabilities for the states of nature are known in addition to the
three basic components described.
An alternative to using payoff tables for decision making under risk is the decision tree, in which
branchings represent either the decision alternatives or the states of nature relevant at a given point in
the process. Decision trees are more flexible than payoff tables and can be more easily used to
represent multistage decision problems and situations in which the probabilities of the states of nature
depend on what has happened up to that point in the tree. The standard analysis approach for a
decision tree with probabilities is averaging out and folding back, in which, starting at the right-most
side of the tree, each branching is replaced by a single number—the expected value of a chance
branching or the value of the alternative chosen in a decision branching. Two other possible
approaches to analyzing a decision tree were also presented: risk analysis and best case or worst case
1. Identify the basic components of a decision-making problem.
2. Describe the differences among decision making under certainty, uncertainty, and risk.
3. Identify four different procedures for decision making under uncertainty and what the basic
concept or philosophy is for each.
4. Describe the basic idea of the expected value of perfect information.
5. Why might using a decision tree be preferable to using a decision table?
6. Describe the process of averaging out and folding back.
1. A high school band's parents organization operates a Christmas tree lot every year to raise
funds. Trees are bought in batches of 100 for $1,000 and sold for $20 each. Leftover trees are
taken away by a landscaping company (at no cost or revenue to the band) to be shredded for
mulch. Based on past experience, the organization estimates that they can sell either 3, 4, 5, or
6 batches of trees.
a. Find the maximax, maximin, equally likely, and minimax regret decisions based on
b. Based on prior years' experience, the organization estimates that there is a 10%
chance of selling 3 batches, a 30% chance of selling 4 batches, a 40% chance of
selling 5 batches, and a 20% chance of selling 6 batches. Based on these probability
estimates, determine the number of batches to buy to maximize expected profits.
a. First, develop a profit payoff table as shown in Table B.2. The table entries are
determined as follows:
Profit = Revenue – Cost = 2,000(sold) – 1,000(bought), so: If demand < bought,
profit = 2,000(demand) – 1,000(bought). If demand > bought, profit = (2,000 –
The maximum, minimum, and average profits for the alternatives are shown in
Table B.2 Payoff Table for Solved Problem 1
Bought (batches) 3 4 5 6
3 3,000 3,000 3,000 3,000
4 2,000 4,000 4,000 4,000
5 1,000 3,000 5,000 5,000
6 0 2,000 4,000 6,000
Maximax: The largest maximum profit is $6,000 from stocking 6 batches.
Maximin: The largest minimum profit is $3,000 from stocking 3 batches.
Equally likely: The largest average profit is $3,500 from stocking either 4 or 5 batches.
Minimax regret: The regret matrix and the maximum regret for each possible stocking level
are shown in Exhibit B.4b. The smallest maximum regret is $2,000 from stocking either 4 or
c. The expected values of the different possible stocking levels are:
Stock 3: .1(3,000) + .3(3,000) + .4(3,000) + .2(3,000) = 3,000
Stock 4: .1(2,000) + .3(4,000) + .4(4,000) + .2(4,000) = 3,800
Stock 5: .1(1,000) + .3(3,000) + .4(5,000) + .2(5,000) = 4,000
Stock 6: .1(0) + .3(2,000) + .4(4,000) + .2(6,000) = 3,400
The stocking level that maximizes expected profits is 5 batches for an expected profit
2. A resort development company has the opportunity to buy all or a portion of the acreage
surrounding a lake for development. They must decide the size of the development for which
they should buy land and put in roads and utilities. The company's owners are considering
either a small or large development. Similarly, they have identified two general levels of
market acceptance of their project: low or high.
Estimates of the cost of land, road development, and utility installation for the different
development sizes and of how many lots would be sold for each market acceptance level
result in the profit estimates (in millions of dollars) for each combination of size and
acceptance level shown in Table B.3.
Table B.3 Payoff Table for Lakeside Development Problem
Size of Development Low High
Small 5 7.5
Large –5 25
a. Assuming that the company cannot make probability estimates for the different
market acceptance levels, determine the appropriate decisions with the maximax,
maximin, equally likely, and minimax regret procedures.
b. Using some basic market research, the owners estimate that the probabilities of the
different levels of acceptance are P(low) = .6 and P(high) = .4. Using these
probabilities, find the alternative that maximizes expected profits and the expected
value of perfect information.
c. Construct and analyze a decision tree using the expected profit criterion.
d. The developers' chief financial officer (CFO) has proposed that the company
consider adopting a two-stage development approach. Under this approach, the
company will initially buy enough property for a small development and also
purchase a three-year option on the rest of the property. At the end of the three years,
the company can, if it seems warranted, then buy the rest of the property and increase
the size of the development. Due to the cost of the option, all final payoffs in Table
B.3 will be reduced by 1 ($1 million), but the opportunity to take advantage of the
initial acceptance results to change the size decision may make up for this reduction.
The CFO believes that the probabilities of the two market acceptance levels during
the first three years will remain as currently estimated: P(low) = .6 and P(high) = .4.
However, he also believes that the probabilities of the final levels of market
acceptance can be re-estimated on the basis of the experience during the first three
years. Specifically, he estimates these probabilities as follows:
P(low final acceptance|low initial acceptance) = .8
P(high final acceptance|low initial acceptance) = .2
P(low final acceptance|high initial acceptance) = .3
P(high final acceptance|high initial acceptance) = .7
Develop a decision tree to model the CFO's alternative proposal. Analyze this tree to
determine whether this two-stage development strategy is preferable to the decision
reached in part c.
a. The maximum, minimum, and average rows are shown in Exhibit B.5(a). From them we
can determine that the decisions for the different procedures are:
Maximax: The larger row maximum is 25, so the decision is a large development.
Maximin: The larger row minimum is 5, so the decision is a small development.
Equally likely: The larger row average is 10, which comes from a small development.
The regret matrix and its maximum rows are shown in Exhibit B.5(b). The alternative with the
smallest maximum regret is a large development, with a maximum regret of 10.
b. The expected profits of the alternatives are:
Alternative Expected Profit
Small .6(5) + .4(7.5) = 6
Large .6(–5)+ .4(25) = 7
c. The alternative with the higher expected profit is a large development.
Using the highest payoffs for the different states of nature, the expected value with
perfect information (EWPI) is:
EWPI = .6(5) + .4(25) = 13
The expected value of perfect information is:
EVPI = EWPI – E(Large development) = 13 – 7 = 6 or $6 million.
d. The decision tree and the analysis (in red) are shown in Figure B.7. The averaging out
and folding back goes as follows:
Node 2: The expected profits are .6(5) + .4(7.5) = 6.
Node 3: The expected profits are .6(–5) + .4(25) = 7.
Moving back to the first level:
Node 1: The alternative with the highest expected value is a large development,
so mark out (with slashes) the other alternative.
The alternative with the highest expected profit is a large development with expected
profit = 7 or $7,000,000.
Figure B.7 Decision Tree for Basic Resort Development Problem
d. The decision tree (with analysis in red) is shown in Figure B.8. The expected payoffs
for the two original decision possibilities (small development without an option to
expand and large development) are taken from the decision tree in Figure B.7. Based
on the analysis shown in Figure B.8, the company should initially build a small
development with an option to expand after three years. If the initial level of
acceptance is low, the company should not expand. If, however, the initial level of
acceptance is high, the company should expand. Following this two-stage
development strategy increases the expected profit from $7 million for initially
adopting a large development, to $8.7 million.
Figure B.8 Decision Tree for Expanded Resort Development Problem
B.1. A toy company has developed a new toy for the upcoming Christmas season. Since this toy is
considerably different from the ones it has manufactured previously, the company will need to
develop a new production facility for it. Three facility sizes—small, medium, and large—are under
consideration. Given the nature of the toy market, the company is unsure as to what demand level it
will encounter. The preliminary analysis is to be based on the demand being low, average, or high. A
small amount of subcontracting will be available, so that if the production facility is undersized it will
be possible to meet some of the excess demand. The accompanying table shows the estimated profits,
in $1,000s, of the various facility-size–demand-level combinations.
Facility Size Low Average High
Small 750 900 900
Medium 350 1,100 1,300
Large –250 600 2,000
a. Determine the best production facility size using maximax, maximin, equally likely, and
b. The company's initial assessment of the probabilities of the different market sizes is: P(low) =
.5, P(average) = .3, P(high) = .2. Determine the production facility size that maximizes
expected profits and find the expected value of perfect information.
B.2. A publisher has received an unsolicited manuscript of a first novel. The decision is whether to
offer the author a contract. Based on an initial reading of the manuscript, the publisher estimates the
following profits if a contract is offered: If the sales level is high, profits will be $100,000; if
moderate, profits will be $20,000; if low, they will lose $30,000. The publisher estimates the
probabilities for the sales level to be: P(high) = .1, P(moderate) = .4, P(low) = .5. Determine, based on
expected profits, whether the author should be offered a contract or not. Determine the expected value
of perfect information.
B.3. A plumbing contractor has the opportunity to bid on a contract to do the plumbing work for a new
office building. After reviewing the blueprints and specifications, the contractor estimates that the job
will cost $300,000. The possible bids the contractor might make and his estimates of the probability of
winning the contract at each bid level are:
Bid Probability Win
What should the contractor bid if he wishes to maximize his expected profits?
B.4. A developer is planning a new office complex, which may include some retail space. The
possible percentages of retail space that the developer is considering are: none, 20%, or 40%. The
desirability of the various percentages of retail space depends on the demand for office space. The
estimated yearly profits (in $1,000s) for the different retail percentages and office space demand levels
are given in the accompanying table.
Office Space Demand
Retail Percentage Low Medium High
None –100 100 250
20 percent 150 200 200
40 percent 300 150 100
a. Determine what the percentage allocation of retail space should be using the maximax,
maximin, equally likely, and minimax regret procedures.
b. The developer's assessments of the probabilities of the different office space demand levels
are: P(low) = .3, P(moderate) = .4, P(high) = .3. Determine the percentage allocation of retail
space that maximizes expected yearly profits. Find the expected value of perfect information.
B.5. A recent business school finance graduate has just received an inheritance of $10,000. Trying to
decide how she should invest the money, she has identified three possible alternatives: stocks,
commodities, and T-bills. The success of any alternative will depend on the performance of the
economy over the next year. The accompanying table shows the gain (in $100s) from each investment
alternative for each performance level of the economy.
Performance of Economy
Investment Alternatives Recession Stagnant Growth
Stocks –10 0 20
Commodities –50 5 50
T-Bills 7 7 7
a. Determine the best way to invest the money using the maximax, maximin, equally likely, and
minimax regret approaches.
b. Her estimates of the probabilities of the different possible performance levels of the economy
are: P(recession) = .1, P(stagnant) = .6, P(growth) = .3. Determine how to invest the money to
maximize her expected gain. Find the expected value of perfect information.
B.6. Conduct a "worst case" analysis of the decision tree for the cardiac catheterization service
proposal in Figure B.1 and Example B.1.
B.7. A hardware store orders snow blowers during the summer for delivery in the fall. Each snow
blower costs the store $400 and, if sold prior to or during the winter, sells for $550. The store's
manager doesn't want to carry any unsold snow blowers in inventory from one year to the next, so he
reduces the price to $350 in the spring in order to get rid of any leftovers. Based on past experience,
the store's manager expects that the demand for snow blowers at full price will be between 6 and 10.
a. Identify the decision alternatives and states of nature and construct a payoff table for the store
manager's snow blower stocking problem.
b. Determine the number of snow blowers to stock if the maximax criterion is used. Repeat for
maximin, equally likely, and minimax regret.
c. Based on the long-range forecasts for the upcoming winter's weather, the store manager
estimates the probabilities of selling different numbers of snow blowers at full price to be:
P(6) = .35, P(7) = .30, P(8) = .20, P(9) = .10, and P(10) = .05. Determine the number of snow
blowers to stock to maximize expected profits.
d. Using the same probabilities, find the number of snow blowers to stock to minimize expected
regret. Verify that the decision is the same as in part c.
e. Using the probabilities in part c, find the expected value of perfect information.
B.8. Dot'z Bakery bakes fresh apple pies each morning for sale that day. A pie costs $2 to make and
sells for $4. Any pies left at the end of the day are sold the following day at a discounted price of
$1.50. Based on her past experience, the bakery's manager expects to sell between 8 and 12 pies per
a. Identify the decision alternatives and states of nature and construct a payoff table for the
bakery manager's apple pie stocking problem.
b. Determine the number of apple pies to bake if the maximax criterion is used. Repeat for
maximin, equally likely, and minimax regret.
c. Based on historical sales records, the bakery manager estimates the probabilities of the
different apple pie demand levels as: P(8) = .1, P(9) = .2, P(10) = .4, P(11) = .2, P(12) = .1.
Determine the number of apple pies to bake to maximize expected profits.
d. Using the same probabilities, find the number of apple pies to bake to minimize expected
regret. Verify that the decision is the same as in part (c).
e. Using the probabilities in part c, find the expected value of perfect information.
B.9. Considerable research and a great deal of practical experience show that the production of goods
or services is generally subject to learning or experience curve effects. That is, records kept on the
operation of many manufacturing and service delivery systems show that the time and cost required to
produce a unit of output decrease at a fairly predictable rate as experience with that production
increases. Given this predictable improvement, some companies follow a strategy of initially pricing a
new product below its production cost as a way of building demand for the product, recognizing that,
due to the learning curve effects, the cost will eventually be lower than the sale price, generating
profits that will more than compensate for the initial losses.
A company is considering using this type of pricing strategy for a new product it is about to introduce
but is uncertain as to what learning rate to expect and, therefore, what pricing strategy to use. If a low
initial price is set based on the assumption of a high rate of improvement and the improvement is
slower, then the long-term profits will be lower than anticipated or nonexistent. If, however, a higher
initial price is set, then demand would probably be lower and, even if the rate of learning is high,
profits will not meet the hoped-for levels. Initial assessments of the profit levels under various
combinations of pricing and learning rates are summarized in the accompanying table.
Rate of Learning
Pricing Strategy Low Moderate High
Price Low –10 –3 12
Price Medium –4 3 10
Price High 1 6 7
a. Determine the pricing strategy to follow if the maximax criterion is used. Repeat for
maximin, equally likely, and minimax regret.
b. Based on the company's experience with other new products, the operations manager
estimates the probabilities of the different rates of learning to be: P(low) = .2, P(moderate) =
.6, P(high) = .2. Determine the pricing strategy to follow to maximize expected profits.
c. Using the same probabilities, find the pricing strategy to follow to minimize expected regret.
Verify that the decision is the same as in part c.
d. Using the probabilities in part c, find the expected value of perfect information.
B.10. Tom's airline is considering a coupon promotion to increase business. Under this program, the
airline will give anyone purchasing a full-fare ticket a coupon worth half off on any future flight. The
airline is considering doing this for two weeks, a month, or not all. The possible results of undertaking
this program are that it will be very successful, moderately successful, or not successful at all in
developing new business over the next year after the program is over.
Using other airlines' experiences with this type of program, the airline marketing manager estimates
the net profits (in millions of dollars) for the various combinations of program times and market
responses shown in the accompanying table.
Level of Success
Program Length Not Moderate Very
Month –5 –1 10
Two Weeks –2.5 2 7
Do nothing 0 0 0
a. Determine the amount of time for which to run the promotion if the maximax criterion is
used. Repeat for maximin, equally likely, and minimax regret.
b. Again based on other airlines' experiences with this type of coupon program, the airline's
marketing manager estimates the probabilities of the different levels of program success to
be: P(very successful) = .20, P(moderately successful) = .25, P(not successful) = .55. Which
alternative should the airline adopt if management wishes to maximize expected net profits
from the program? Determine the expected value of perfect information.
B.11. A distributor of greeting cards and related products has the opportunity to participate in the
merchandising activities associated with a forthcoming children's movie. Due to the production lead-
time and the relatively short expected product life, the distributor must make a decision now about
how much of this special party package to order. The distributor's marketing manager estimates that
demand for this product will be between 400 and 800 units, in increments of 100 units. A unit, which
consists of 1,000 party packages, will cost $3,000 and sell for $5,000 at full price. If demand is too
low, the selling price will be cut to $2,000 to clear out the excess inventory.
a. Determine the number of units of product to order if the maximax criterion is used. Repeat for
maximin, equally likely, and minimax regret.
b. Based on her past experience with similar types of movie-based special products, the
marketing manager estimates that the probabilities of the different possible demand levels are:
P(400) = .2, P(500) = .4, P(600) = .2, P(700) = .1, and P(800) = .1. Determine the number of
units to order to maximize expected profit. Also determine the expected value of perfect
c. The manufacturer of the party packages will offer the distributor a discounted cost of $2,500
per unit if a minimum of 600 units are ordered. Does the availability of this quantity discount
make any difference in the decision about how many units to order if the expected profit
criterion is used?
B.12. A company is being sued for damages as a result of injuries incurred due to product failure
under normal use. The management is consulting with lawyers to determine how much to offer the
plaintiff as a settlement to avoid having to go to trial. The lawyers have suggested taking a
low/medium/high offer approach.
In an attempt to get off relatively cheaply, the company could initially offer the plaintiff $100,000, an
amount that the lawyers believe would have only a 20% chance of being accepted. Alternatively, they
could offer $150,000, which the lawyers believe would have a 50-50 chance of acceptance, or offer
$200,000, which the lawyers believe would definitely be accepted.
If the plaintiff does not accept the initial offer, the company would then make a second, higher offer,
which the lawyers believe would have to be higher than a comparable offer made initially to get the
same probability of acceptance. Specifically, the lawyers believe that the company would have to
offer $175,000 to have a 50-50 chance of acceptance in the second stage and offer $225,000 to be
guaranteed of acceptance.
Finally, if the plaintiff accepts neither the first nor the second offer, the lawyers believe that a final
offer of $250,000 would definitely be accepted.
a. Determine what strategy the company should follow to minimize the expected cost of the
settlement. That is, determine what the initial offer should be and, if it is not accepted, what
the second offer should be.
b. Suppose that there is a 30% chance of the plaintiff accepting an initial offer of $100,000
(nothing else in the description given changes). Would this affect the strategy that minimizes
the expected cost?
c. Determine the minimum probability of acceptance of an initial offer of $100,000 that would
make the strategy starting with that amount optimal.
B.13. The city manager of Bridgeport is developing a recommendation to the city council on when to
undertake repairs needed on one of the main bridges crossing the river that runs through the middle of
town. The city engineer has informed her that the bridge is deteriorating at a rate that will make major
repairs necessary sometime within the next three years.
If the repairs are done this year, the cost is estimated to be $1.2 million. If the repairs are delayed until
next year and the condition of the bridge does not get significantly worse, then inflation and increased
minor deterioration are estimated to raise the cost by 10%. If, however, there is significant additional
deterioration, which is a function of the severity of the winter, the repairs will have to be done and the
cost will increase to $1.8 million. At this point, the weather service to which the city subscribes
estimates only a 20% chance that the winter will be severe.
If the upcoming winter is not severe, then the city will again have the option of delaying the repairs
for another year. Again, if the following winter is not severe, inflation and normal increased
deterioration will raise the cost by an additional 10% over the cost of repair in the second year. If the
winter between the second and third years is severe, the added deterioration to the bridge will raise the
cost to $2.4 million. Normal weather patterns suggest that there is a 50-50 chance of a severe winter
two years from now.
If this were all there were to the decision, the city manager would not have a problem. Since the cost
keeps going up, regardless of how severe the winter is, the repairs should be done this year. However,
the city manager has been informed by the local state representative that there is a possibility that the
city could get a $600,000 grant from the state that would help pay for the repairs, thus reducing the
cost to the city. While the grant will not be available for the current year, the representative estimates
a 60% chance that the grant will be available in the second year. Furthermore, if the city does not get a
grant in year 2, there is a 75% chance they will get one in year 3.
a. Using this information, develop a recommendation for the city manager that will minimize
the expected cost to the city of repairing the bridge.
b. Changing nothing else, what is the minimum probability of getting a grant in year 3 (given
that one had not been received in year 2) that would make delaying the repair to year 3 (if
possible) the optimal strategy?
B.14. The Pittsburgh Steelers have just won the AFC title, and the owner of a sports specialty store in
Pittsburgh has to decide how many of the special "AFC Champions" T-shirts he should order. The T-
shirt manufacturer will only sell the shirts in cases of 100 at $1,000 per case. The shirts will be priced
for sale at $15 each, with any left over by the time of the Super Bowl (which will be in two weeks)
being sold on the discount table at $6 each. To simplify the analysis, assume that demand for the T-
shirts is in whole cases. The sports store owner believes that his store will sell between one and three
cases of shirts.
a. Determine the number of cases of shirts the store owner should order if the maximax criterion
is used. Repeat for maximin, equally likely, and minimax regret.
b. Based on T-shirt sales during previous Steeler appearances in the Super Bowl, the store
owner estimates the probabilities of the different demand levels as: P(1) = .12, P(2) = .48,
P(3) = .40. Determine the number of cases of T-shirts to order to maximize expected profits.
c. Using the same probabilities, find the number of cases of T-shirts to order to minimize
expected regret. Verify that the decision is the same as in part b.
d. Using the probabilities in part b, find the expected value of perfect information.
B.15. (Continuation of Problem B.14.) To provide some protection against getting stuck with too
many unsold T-shirts, the store owner has asked the T-shirt producer about the possibility of buying
some shirts now and, if desired, more a week later. The producer says he is willing to do this, but,
because he will have to set his equipment up to produce a second batch later, the reorder will cost
$1,200 per case rather than $1,000.
The store owner believes that during the first week there is a 60% chance that there will be demand for
one case of shirts and a 40% chance of demand for two cases. He also believes that the first week's
demand will be a good indication of what the demand will be in the second week. Specifically, he
estimates the second week's demand (in cases) to be:
If the first week's demand is for one case
P(2nd week's demand = 0) = .2
P(2nd week's demand = 1) = .6
P(2nd week's demand = 2) = .2
If the first week's demand is for two cases
P(2nd week's demand = 0) = .3
P(2nd week's demand = 1) = .7
a. Using a decision tree, determine the ordering strategy that will maximize the store owner's
b. Compare the expected profit from following the optimal two-stage ordering strategy
developed in part a with the expected profit if all shirts have to be ordered initially, as
determined in part b of Problem B.14.
CASE: Transformer Replacement at Mountain States Electric Service
Mountain States Electric Service is an electrical utility company serving several states in the Rocky
Mountain region. It is considering replacing some of its equipment at a generating substation and is
attempting to decide whether it should replace an older, existing PCB transformer. (PCB is a toxic
chemical known formally as polychlorinated biphenyl.) Even though the PCB generator meets all
current regulations, if an incident occurred, such as a fire, and PCB contamination caused harm either
to neighboring businesses or farms or to the environment, the company would be liable for damages.
Recent court cases have shown that simply meeting utility regulations does not relieve a utility of
liability if an incident causes harm to others. Also, courts have been awarding large damages to
individuals and businesses harmed by hazardous incidents.
If the utility replaces the PCB transformer, no PCB incidents will occur, and the only cost will be that
of the transformer, $85,000. Alternatively, if the company decides to keep the existing PCB
transformer, then management estimates there is a 50-50 chance of there being a high likelihood of an
incident or a low likelihood of an incident. For the case in which there is a high likelihood that an
incident will occur, there is a .004 probability that a fire will occur sometime during the remaining life
of the transformer and a .996 probability that no fire will occur. If a fire occurs, there is a .20
probability that it will be bad and the utility will incur a very high cost of approximately $90 million
for the cleanup, whereas there is a .80 probability that the fire will be minor and a cleanup can be
accomplished at a low cost of approximately $8 million. If no fire occurs, then no cleanup costs will
occur. For the case in which there is a low likelihood of an incident occurring, there is a .001
probability that a fire will occur during the life of the existing transformer and a .999 probability that a
fire will not occur. If a fire does occur, then the same probabilities exist for the incidence of high and
low cleanup costs, as well as the same cleanup costs, as indicated for the previous case. Similarly, if
no fire occurs, there is no cleanup cost.
Perform a decision tree analysis of this problem for Mountain States Electric Service and indicate the
recommended solution. Is this the decision you believe the company should make? Explain your
This case was adapted from W. Balson, J. Welsh, and D. Wilson, "Using Decision Analysis and Risk
Analysis to Manage Utility Environmental Risk," Interfaces 22, no. 6 (November-December 1992): 126-
The possible environments, conditions, or states of nature within which a decision is to be
The process of replacing each branching by a single number. For a chance branching, the
number used is the expected value, which is found by multiplying the probability of each
branch by the value at its right end and summing. For a decision branching, the number used
is the value of the best alternative in the branching.
Problem features that are determined by chance or probability, such as the states of nature.
decision making under certainty
A basic type of decision problem where decision makers have the payoff table, but no
information on the relative likelihood of the states of nature. In principle, the analysis is
simple: Choose the alternative with the best payoff. In practice, it is not always that easy,
since an "alternative" may actually represent the values for many decision variables.
decision making under risk
A basic type of decision problem where the state of nature is known, so the payoff table has
only one column. In principle, the analysis is simple: Choose the alternative with the best
decision making under uncertainty
A basic type of decision problem where decision makers have the payoff table, but no
information on the relative likelihood of the states of nature.
A tool that facilitates the description of the decision tree. There are three alternatives,
represented by the three branches: (1) build an addition, (2) convert existing space, and (3) do
An alternative to using a payoff to compute the expected values of decision alternatives,
which consists of two or more stages or levels, shown in time order from left to right.
The conditions, or states of nature, within which a decision is to be implemented.
A procedure based on the assumption that if we cannot determine the relative likelihood of
the states of nature, then it is rational to presume that they are equally likely.
expected value of perfect information (EVPI)
A measure of the expected current worth to the decision maker of being able to find out, just
before having to make the decision, what the state of nature will be.
The process which starts at the right-hand edge of the tree and proceeds back to the start of
the tree, working from right to left.
An optimistic approach that assumes that no matter what alternative we choose, "nature" will
smile favorably on us and choose the state that will benefit us the most. Thus, we choose the
alternative with the largest maximum gain.
A pessimistic approach that assumes that no matter what alternative we choose, "nature" will
choose the state that benefits us the least. Thus, we choose the alternative with the largest
A procedure taken if the payoffs are costs or losses. A pessimistic approach that assumes that
no matter what alternative we choose, "nature" will choose the state that benefits us the least.
Thus, we choose the alternative with the largest minimum gain (or the smallest maximum
The approach in which we choose the alternative that will yield the smallest possible
An approach that assumes that no matter what alternative we choose, "nature" will smile
favorably on us and choose the state of nature that will hurt us the least. Thus, we choose the
alternative with the smallest minimum loss.
Payoffs that will result from each possible combination of alternatives and states of nature.
Outcomes that will result from each possible combination of alternatives and states of nature.
A tool which specifies what that outcome is for every possible alternative/state-of-nature
combination. It may be given either as an equation or, when the possible decision alternatives
and states of nature are relatively limited, as a payoff table or matrix.
A matrix that specifies the outcome for every possible alternative/state-of-nature when the
states-of-nature are relatively limited.
An evaluation of the various possible outcomes and their probabilities, and extreme case
analysis, which, like the maximax and maximin strategies, takes either an optimistic or
pessimistic view rather than working with probabilities.
states of nature
Aspects of the decision situation over which the decision maker does not have direct control.
An alternative measure when dollar value alone is not an adequate measure of the value of an