A payoff table is limited to a single decision situation. If a decision requires a series of decisions, a payoff table cannot be created, and a sequential decision treemust be used. We demonstrate the use of a decision tree in the following example.
• Excel File
Expected value of perfect information (EVPI):
the maximum value of perfect information to the decision maker.
Sequential decision tree:
a graphical method for analyzing decision situations that require a sequence of decisions over time.
The Southern Textile Company is considering two alternatives: to expand its existing produc- tion operation to manufacture a new line of lightweight material; or to purchase land on which to construct a new facility in the future. Each of these decisions has outcomes based on product market growth in the future that result in another set of decisions (during a 10-year planning horizon), as shown in the following figure of a sequential decision tree. In this figure the square nodes represent decisions, and the circle nodes reflect different states of nature and their probabilities.
The first decision facing the company is whether to expand or buy land. If the company expands, two states of nature are possible. Either the market will grow (with a probability of
Example S1.3
A Sequential Decision Tree
(Continued)
0.60) or it will not grow (with a probability of 0.40). Either state of nature will result in a pay- off. On the other hand, if the company chooses to purchase land, three years in the future another decision will have to be made regarding the development of the land.
At decision node 1, the decision choices are to expand or to purchase land. Notice that the costs of the ventures ($800,000 and $200,000, respectively) are shown in parentheses. If the plant is expanded, two states of nature are possible at probability node 2: The market will grow, with a probability of 0.60, or it will not grow or will decline, with a probability of 0.40.
If the market grows, the company will achieve a payoff of $2,000,000 over a 10-year period.
However, if no growth occurs, a payoff of only $225,000 will result.
If the decision is to purchase land, two states of nature are possible at probability node 3.
These two states of nature and their probabilities are identical to those at node 2; however, the payoffs are different. If market growth occurs for a three-year period, no payoff will occur, but the company will make another decision at node 4 regarding development of the land. At that point, either the plant will be expanded at a cost of $800,000 or the land will be sold, with a payoff of $450,000. The decision situation at node 4 can occur only if market growth occurs first. If no market growth occurs at node 3, there is no payoff, and another decision situation becomes necessary at node 5: A warehouse can be constructed at a cost of $600,000 or the land can be sold for $210,000. (Notice that the sale of the land results in less profit if there is no market growth than if there is growth.)
If the decision at decision node 4 is to expand, two states of nature are possible: The mar- ket may grow, with a probability of 0.80, or it may not grow, with a probability of 0.20. The probability of market growth is higher (and the probability of no growth is lower) than before because there has already been growth for the first three years, as shown by the branch from node 3 to node 4. The payoffs for these two states of nature at the end of the 10-year period are $3,000,000 and $700,000, respectively.
If the company decides to build a warehouse at node 5, two states of nature can occur:
Market growth can occur, with a probability of 0.30 and an eventual payoff of $2,300,000, or no growth can occur, with a probability of 0.70 and a payoff of $1,000,000. The probability of market growth is low (i.e., 0.30) because there has already been no market growth, as shown by the branch from node 3 to node 5.
Solution
We start the decision analysis process at the end of the decision tree and work backward to- ward a decision at node 1.
First, we must compute the expected values at nodes 6 and 7:
EV(node 6) 0.80($3,000,000) 0.20($700,000) $2,540,000 EV(node 7) 0.30($2,300,000) 0.70($1,000,000) $1,390,000 Expand
(–$800,000) 2
1
3
4
5
6
7 Purchase
land (–$200,000)
Market growth
No market growth Market growth (3 years,
$0 payoff)
$3,000,000
$700,000
$2,300,000
$1,000,000 Expand
(–$800,000)
Sell land
Market growth No market
growth Market growth No market
growth Sell land
Warehouse (–$600,000)
$210,000
$450,000
$225,000
$2,000,000
0.40 0.60 0.40 0.60
No market growth (3 years,
$0 payoff)
0.70 0.30 0.20 0.80
Decision trees allow the decision maker to see the logic of decision making by providing a picture of the decision process. Decision trees can be used for problems more complex than this example without too much difficulty.
These expected values (as well as all other nodal values) are shown in boxes in the figure.
At decision nodes 4 and 5, a decision must be made. As with a normal payoff table, the de- cision is made that results in the greatest expected value. At node 4 the choice is between two values: $1,740,000, the value derived by subtracting the cost of expanding ($800,000) from the expected payoff of $2,540,000, and $450,000, the expected value of selling the land computed with a probability of 1.0. The decision is to expand, and the value at node 4 is $1,740,000.
The same process is repeated at node 5. The decisions at node 5 result in payoffs of
$790,000 (i.e., $1,390,000 600,000 $790,000) and $210,000. Since the value $790,000 is higher, the decision is to build a warehouse.
Next, the expected values at nodes 2 and 3 are computed:
EV(node 2) 0.60($2,000,000) 0.40($225,000) $1,290,000 EV(node 3) 0.60($1,740,000) 0.40($790,000) $1,360,000
(Note that the expected value for node 3 is computed from the decision values previously determined at nodes 4 and 5.)
Now the final decision at node 1 must be made. As before, we select the decision with the greatest expected value after the cost of each decision is subtracted.
Expand: $1,290,000 800,000 $490,000 Land: $1,360,000 200,000 $1,160,000
Since the highest net expected value is $1,160,000, the decision is to purchase land, and the payoff of the decision is $1,160,000.
Expand (–$800,000)
2
1
3
4
5
6
7 Purchase
land (–$200,000)
Market growth No market
growth
Market growth (3 years,
$0 payoff)
$3,000,000
$700,000
$2,300,000
$1,000,000 Expand
(–$800,000)
Sell land
Market growth No market
growth Market growth No market
growth Sell land
Warehouse (–$600,000)
$210,000
$450,000
$225,000
$2,000,000
0.40 0.60 0.40 0.60
No market growth (3 years,
$0 payoff)
0.70 0.30 0.20
$1,160,000 0.80
$1,290,000
$1,360,000
$1,740,000
$790,000
$2,540,000
$1,390,000
S U M M A R Y
In this supplement we have provided a general overview of decision analysis. To a limited extent, we have also shown that
the logic of such operational decisions throughout the organi- zation are interrelated to achieve strategic goals.
S U M M A R Y O F K E Y FO R M U L A S
Expected Value
EV(x) = a
n i=1
p(xi)xi
Expected Value of Perfect Information
EVPI expected value given perfect information expected value without perfect information
S U M M A R Y O F K E Y T E R M S
coefficient of optimism (␣)a measure of a decision maker’s opti- mism, from 0 (completely pessimistic) to 1 (completely optimistic), used in the Hurwicz decision criterion.
decision analysis a set of quantitative decision-making tech- niques to aid the decision maker in dealing with decision situations in which uncertainty exists.
equal likelihood (La Place) criteriona decision criterion in which each state of nature is weighted equally.
expected value a weighted average of decision outcomes in which each future state of nature is assigned a probability of occurence.
expected value of perfect information (EVP) the maximum value that a decision maker would be willing to pay for perfect information about future states of nature.
Hurwicz criteriona decision criterion in which the decision pay- offs are weighted by a coefficient of optimism,.
maximax criteriona decision criterion that results in the maxi- mum of the maximum payoffs.
maximin criteriona decision criterion that results in the maxi- mum of the minimum payoffs.
minimax regret criterion a decision criterion that results in the minimum of the maximum regrets for each alternative.
payoffthe outcome of a decision.
payoff tablea means of organizing and illustrating the payoffs from different decisions given various states of nature.
sequential decision tree a graphical method for analyzing decision situtations that require a sequence of decisions over time.
S O LV E D P R O B L E M S
Consider the following payoff table for three product decisions (A, B, and C) and three future market conditions (payoffs $ millions).
• Animated Demo Problem
SOLUTION
Step 1. Maximax criterion
Decision: Product B Decision: Product A
Determine the best decision using the following decision criteria.
1. Maximax 2. Maximin
Step 2. Maximin criteria MARKET CONDITIONS
DECISION 1 2 3
A $1.0 $2.0 $0.5
B 0.8 1.2 0.9
C 0.7 0.9 1.7
MAXIMUM PAYOFFS
A $2.0 ← Maximum
B 1.2
C 1.7
MINIMUM PAYOFFS
A 0.5
B 0.8 ← Maximum
C 0.7
Economic/Political Climate
Country Decline Same Improve
South Korea 21.7 19.1 15.2
China 19.0 18.5 17.6
Taiwan 19.2 17.1 14.9
Poland 22.5 16.8 13.8
Mexico 25.0 21.2 12.5
Interest Rates
Investments 5% 6% 7% 8% 9%
Money market fund 1.7 2.8 3.0 3.6 4.5
Stock growth fund 5 3 3.5 5 7.5
Bond fund 5 4 3.5 3 2
Government fund 4 3.6 3.2 2.8 2.1
Risk fund 12 7 4.2 9.3 16.7
Savings bonds 3 3 3.2 3.4 3.5
Interest Rates
Project Decline Stable Increase
Office building 0.5 1.7 4.5
Parking lot 1.5 1.9 2.4
Warehouse 1.7 1.4 1.0
Shopping mall 0.7 2.4 3.6
Condominiums 3.2 1.5 0.6
S1-1. Telecomp is a U.S.-based manufacturer of cellular tele- phones. It is planning to build a new manufacturing and distribution facility in either South Korea, China, Taiwan, Poland, or Mexico. The cost of the facility will differ between countries and will even vary within countries depending on the economic and political climate, including monetary exchange rates. The company has estimated the facility cost (in $ millions) in each country under three dif- ferent future economic/political climates as follows.
PROBLEMS
• GO Tutorial
S1-4. In Problem S1-3 the Landloc real estate development firm has hired an economist to assign a probability to each direc- tion interest rates may take over the next five years. The economist has determined that there is a 0.50 probability that interest rates will decline, a 0.40 probability that rates will re- main stable, and a 0.10 probability that rates will increase.
a. Using expected value, determine the best project.
b. Determine the expected value of perfect information.
S1-5. Nicole Nelson has come into an inheritance from her grandparents. She is attempting to decide among several investment alternatives. The return after one year is depen- dent primarily on the interest rate during the next year. The rate is currently 7%, and she anticipates it will stay the same or go up or down by at most 2 points. The various investment alternatives plus their returns ($10,000s) given the interest rate changes are shown in the following table.
Determine the best decision using the following decision criteria. (Note that since the payoff is cost, the maximax cri- teria becomes minimin and maximin becomes minimax.) a. Minimin
b. Minimax
c. Hurwicz ( 0.40) d. Equal likelihood
S1-2. A global economist hired by Telecomp, the U.S.-based computer manufacturer in Problem S1-1, estimates that the probability that the economic and political climate over- seas and in Mexico will decline during the next five years is 0.30, the probability that it will remain approximately the same is 0.40, and the probability that it will improve is 0.30. Determine the best country to construct the new facil- ity in and the expected value of perfect information.
S1-3. Landloc, a real estate development firm, is considering sev- eral alternative development projects. These include build- ing and leasing an office building, purchasing a parcel of land and building a parking lot, buying and leasing a ware- house, building a shopping mall, and building and selling condominiums. The financial success of these projects depends on interest rate movement in the next five years.
The various development projects and their five-year finan- cial return ($ millions) given that interest rates will decline, remain stable, or increase are shown in the following pay- off table.
Determine the best investment using the following deci- sion criteria.
a. Maximax b. Maximin c. Equal likelihood d. Hurwicz ( 0.3)
Determine the best investment using the following decision criteria.
a. Maximax b. Maximin c. Equal likelihood
d. Assume that Nicole, with the help of a financial newsletter and some library research, has been able to assign probabilities to each of the possible interest rates during the next year as follows:
Interest Rate 5% 6% 7% 8% 9%
Probability 0.1 0.2 0.4 0.2 0.1
Using expected value, determine her best investment decision.
S1-6. The Tech football coaching staff has six basic plays it runs every game. Tech has an upcoming game against State on Saturday, and the coaches know State employs five differ- ent defenses. The coaches have estimated the number of yards Tech will gain with each play against each defense, as shown in the following payoff table.
a. Compute the expected value for each decision and select the best one.
b. Determine how much the firm would be willing to pay to a market research firm to gain better information about future market conditions.
c. Assume that probabilities cannot be assigned to future market conditions, and determine the best decision using the maximax, maximin, minimax regret, and equal likeli- hood criteria.
S1-8. John Wiley Publishing Company publishes an operations management textbook that is scheduled for a revision. The book has been moderately successful, but each year more new books enter the market, some existing books are drop- ped by publishers, and various innovative pedagogical approaches are introduced by authors and publishers, such that the competitive market is always highly uncertain. In addition, the role the Internet will play in future textbook publishing is an unknown. As a result, Wiley is trying to decide whether to publish the next edition of the OM book as a smaller paperback, publish a new edition very similar in size and content to the current edition, significantly re- vise the book with an emphasis on services and processes, or make a major revision with significant physical changes including adding color and more graphics. The following payoff table summarizes the possible revision decisions with profits (or losses) for the three-year lifecycle of the new edition, and the future states of nature relative to the competitive market.
Defense
Play 54 63 Wide Tackle Nickel Blitz
Off tackle 2 2 6 8 2
Option 3 7 1 10 16
Toss Sweep 5 14 5 6 12
Draw 2 3 2 8 7
Pass 10 20 12 7 6
Screen 5 3 8 7 16
a. If the coaches employ an offensive game plan, they will use the maximax criterion. What will their best play be?
b. If the coaches employ a defensive plan, they will use the maximin criterion. What will their best play be?
c. What will their best play be if State is equally likely to use any of its defenses?
d. The Tech coaches have reviewed game films and have determined the following probabilities that State will use each of its defenses.
Defense 54 63 Wide Tackle Nickel Blitz
Probability 0.40 0.10 0.20 0.20 0.10
Using expected value, rank Tech’s plays from best to worst. During the actual game, Tech has a third down and 10 yards to go and the coaches are 60% certain State will blitz, with a 10% chance of any of the other four defenses.
What play should Tech run, and is it likely Tech will make the first down?
S1-7. The Dynamax Company is going to introduce one of three new products: a widget, a hummer, or a nimnot. The mar- ket conditions (favorable, stable, or unfavorable) will de- termine the profit or loss the company realizes, as shown in the following payoff table.
Market Conditions
Favorable Stable Unfavorable
Product 0.2 0.5 0.3
Widget $160,000 $90,000 $50,000
Hummer 70,000 40.000 20,000
Nimnot 45,000 35,000 30.000
Competitive Market Publication Decision Unfavorable Same Favorable
Paperback $68,000 $170,000 $395,000
Similar revision 24,000 375,000 672,000 Major content revision 31,000 515,000 725,000 Major physical revision —105,000 280,000 972,000
Determine the best decision for the publisher using the fol- lowing criteria.
a. Maximax b. Minimax c. Equal likelihood d. Hurwicz ( .35)
S1-9. In Problem S1-8, if Wiley Publishing is able to assign prob- abilities of occurrence of 0.23 to unfavorable market condi- tions, 0.46 for the same market conditions, and 0.31 for favorable market conditions, what is the best decision using expected value? Based on the results in Problem S1-8 and the expected value result in this problem, does there appear to be an overall “best” decision? Compute the expected value of perfect information, and explain its meaning.
S1-10.Amtrex International is a major U.S.-based electronics firm that manufactures a number of electronic components for domestic and global consumer electronics companies. It imports most of its materials and the components used in its products to the United States from overseas suppliers.
Amtrex is in the process of trying to improve its global
States of Nature
Declining Same Growth
Decision Conditions Conditions Conditions
Hong Kong $31 $28 $67
Singapore 24 33 71
Shanghai 28 35 55
Busan 17 25 49
Kaohsiung 15 41 59
States of Nature
Declining Same Growth
Decision Conditions Conditions Conditions
Hong Kong .27 .45 .28
Singapore .18 .51 .31
Shanghai .22 .61 .17
Busan .15 .45 .40
Kaohsiung .25 .38 .37
Energy Prices
Decrease Same Increase
Lease Decision .17 .34 .49
1-year $156,000 $93,000 $16,000
2-year 427,000 150,000 42,000
3-year 642,000 319,000 171,000
4-year 933,000 473,000 337,000
5-year 1,228,000 516,000 551,000
supply chain operations, and as part of this process the company wants to determine a single supplier located at one of the major ports around the world to contract with for the majority of its business. The company is consider- ing six suppliers, each located at one of the following ports: Hong Kong, Singapore, Shanghai, Busan, and Kaoh- siung. The company has estimated the possible profit (or loss) it might achieve with each of the potential suppliers depending on a variety or possible future company and port conditions, including IT capability, port growth and expansion, ship and container availability, security, re- gional market and political environment, and transport to the port from the supplier’s suppliers. Depending on these various factors, further supplier and port conditions could decline, grow and expand, or remain the same. The follow- ing payoff table summarizes the increased outcomes (in $ millions) for the potential suppliers and the possible future sates of nature for a specific time frame.
S1-12.The Willow Café is located in an open-air mall. Its lease expires this year and the restaurant owner has the option of signing a 1-, 2-, 3-, 4-, or 5-year lease. However, the owner is concerned about recent energy price increases (including the price of gasoline), which affect virtually every aspect of the restaurant operation including the price of food items and materials, delivery costs, and its own utilities.
The restaurant was very profitable when energy prices were lower, and the owner believes if prices remain at ap- proximately their current level profits will still be satisfac- tory; however, if prices continue to rise he believes that he might be forced to close. In these latter circumstances a longer term lease could be a financial disaster, but with a shorter term lease the mall landlord could always rent the restaurant’s space out from under it when the lease expires.
As such, the restaurant owner’s estimates of future profits must also reflect the possibility that the lease will not be re- newable. The following payoff table summarizes the owner’s profit (and loss) estimates for each future state of nature of energy prices (over a five-year period).
Determine the best decision using each of the following criteria.
a. Maximax b. Maximin c. Equal likelihood d. Hurwicz ( 0.65) e. Minimax regret
S1-11. In Problem S1-10, suppose Amtrex is able to assign probabil- ities to each of the states of nature for each of the suppliers/
ports as follows:
a. Using expected value, determine the port/supplier Amtrex should use.
b. Based on the results from Problem S1-10, and the re- sult from part a, is there a best overall decision?
Determine the best decision using expected value.
S1-13.Compute the expected value of perfect information for the Willow Café in Problem S1-12. Explain what this value means and how such information might be obtained.
S1-14.The Weight Club (see Case Problem 1.4) is considering adding a new service facility among several possible alter- natives including a child care center, a swimming pool, new locker rooms and showers, a health-oriented food court, and a spa. The success of each alternative depends on their demand (i.e., new members who would join be- cause of the new facility), which is uncertain. The follow- ing payoff table summarizes the returns (based on costs and increased enrollments) for each alternative service fa- cility given three future levels of demand.
Demand
Service Facility Poor Moderate High Child care center $17,000 $27,000 $41,000
Swimming pool —75,000 26,000 71,000
New lockers and showers 12,000 37,000 57,000
Food court —31,000 19,000 87,000
Spa 6,000 25,000 32,000