Chapter Decision Analysis Problem Formulation Decision Making without Probabilities Decision Making with Probabilities Risk Analysis and Sensitivity Analysis Decision Analysis with Sample Information Computing Branch Probabilities Problem Formulation A decision problem is characterized by decision alternatives, states of nature, and resulting payoffs. The decision alternatives are the different possible strategies the decision maker can employ. The states of nature refer to future events, not under the control of the decision maker, which may occur. States of nature should be defined so that they are mutually exclusive and collectively exhaustive. Influence Diagrams An influence diagram is a graphical device showing the relationships among the decisions, the chance events, and the consequences. Squares or rectangles depict decision nodes. Circles or ovals depict chance nodes. Diamonds depict consequence nodes. Lines or arcs connecting the nodes show the direction of influence.
Payoff Tables The consequence resulting from a specific combination of a decision alternative and a state of nature is a payoff. A table showing payoffs for all combinations of decision alternatives and states of nature is a payoff table. Payoffs can be expressed in terms of profit, cost, time, distance or any other appropriate measure. Decision Trees A decision tree is a chronological representation of the decision problem. Each decision tree has two types of nodes; round nodes correspond to the states of nature while square nodes correspond to the decision alternatives. The branches leaving each round node represent the different states of nature while the branches leaving each square node represent the different decision alternatives. At the end of each limb of a tree are the payoffs attained from the series of branches making up that limb. Decision Making without Probabilities Three commonly used criteria for decision making when probability information regarding the likelihood of the states of nature is unavailable are: the optimistic approach the conservative approach the minimax regret approach.
Optimistic Approach The optimistic approach would be used by an optimistic decision maker. The decision with the largest possible payoff is chosen. If the payoff table was in terms of costs, the decision with the lowest cost would be chosen. Conservative Approach The conservative approach would be used by a conservative decision maker. For each decision the minimum payoff is listed and then the decision corresponding to the maximum of these minimum payoffs is selected. (Hence, the minimum possible payoff is maximized.) If the payoff was in terms of costs, the maximum costs would be determined for each decision and then the decision corresponding to the minimum of these maximum costs is selected. (Hence, the maximum possible cost is minimized.) Minimax Regret Approach The minimax regret approach requires the construction of a regret table or an opportunity loss table. This is done by calculating for each state of nature the difference between each payoff and the largest payoff for that state of nature. Then, using this regret table, the maximum regret for each possible decision is listed. The decision chosen is the one corresponding to the minimum of the maximum regrets. 3 3
Example Consider the following problem with three decision alternatives and three states of nature with the following payoff table representing profits: States of Nature s s s 3 d 4 4 - Decisions d 0 3 - d 3 5-3 Example: Optimistic Approach An optimistic decision maker would use the optimistic (maximax) approach. We choose the decision that has the largest single value in the payoff table. Maximax decision Decision Maximum Payoff d 4 d 3 d 3 5 Maximax payoff Example: Optimistic Approach Formula Spreadsheet A B C D E F PAYOFF TABLE 3 Decision State of Nature Maximum Recommended 4 Alternative s s s3 Payoff Decision 5 d 4 4 - =MAX(B5:D5) =IF(E5=$E$9,A5,"") 6 d 0 3 - =MAX(B6:D6) =IF(E6=$E$9,A6,"") 7 d3 5-3 =MAX(B7:D7) =IF(E7=$E$9,A7,"") 9 Best Payoff =MAX(E5:E7) 4 4
Example: Optimistic Approach Solution Spreadsheet A B C D E F PAYOFF TABLE 3 Decision State of Nature Maximum Recommended 4 Alternative s s s3 Payoff Decision 5 d 4 4-4 6 d 0 3-3 7 d3 5-3 5 d3 9 Best Payoff 5 Example: Conservative Approach A conservative decision maker would use the conservative (maximin) approach. List the minimum payoff for each decision. Choose the decision with the maximum of these minimum payoffs. Maximin decision Minimum Decision Payoff d - d - d 3-3 Maximin payoff Example: Conservative Approach Formula Spreadsheet A B C D E F PAYOFF TABLE 3 Decision State of Nature Minimum Recommended 4 Alternative s s s3 Payoff Decision 5 d 4 4 - =MIN(B5:D5) =IF(E5=$E$9,A5,"") 6 d 0 3 - =MIN(B6:D6) =IF(E6=$E$9,A6,"") 7 d3 5-3 =MIN(B7:D7) =IF(E7=$E$9,A7,"") 9 Best Payoff =MAX(E5:E7) 5 5
Example: Conservative Approach Solution Spreadsheet A B C D E F PAYOFF TABLE 3 Decision State of Nature Minimum Recommended 4 Alternative s s s3 Payoff Decision 5 d 4 4 - - 6 d 0 3 - - d 7 d3 5-3 -3 9 Best Payoff - Example: Minimax Regret Approach For the minimax regret approach, first compute a regret table by subtracting each payoff in a column from the largest payoff in that column. In this example, in the first column subtract 4, 0, and from 4; etc. The resulting regret table is: s s s 3 d 0 d 4 0 d 3 3 0 Example: Minimax Regret Approach For each decision list the maximum regret. Choose the decision with the minimum of these values. Minimax decision Decision Maximum Regret d d 4 d 3 3 Minimax regret 6 6
Example: Minimax Regret Approach Formula Spreadsheet A B C D E F PAYOFF TABLE Decision State of Nature 3 Altern. s s s3 4 d 4 4-5 d 0 3-6 d3 5-3 7 OPPORTUNITY LOSS TABLE 9 Decision State of Nature Maximum Recommended 0 Altern. s s s3 Regret Decision d =MAX($B$4:$B$6)-B4 =MAX($C$4:$C$6)-C4 =MAX($D$4:$D$6)-D4 =MAX(B:D) =IF(E=$E$4,A,"") d =MAX($B$4:$B$6)-B5 =MAX($C$4:$C$6)-C5 =MAX($D$4:$D$6)-D5 =MAX(B:D) =IF(E=$E$4,A,"") 3 d3 =MAX($B$4:$B$6)-B6 =MAX($C$4:$C$6)-C6 =MAX($D$4:$D$6)-D6 =MAX(B3:D3) =IF(E3=$E$4,A3,"") 4 Minimax Regret Value =MIN(E:E3) Example: Minimax Regret Approach Solution Spreadsheet A B C D E F PAYOFF TABLE Decision State of Nature 3 Alternative s s s3 4 d 4 4-5 d 0 3-6 d3 5-3 7 OPPORTUNITY LOSS TABLE 9 Decision State of Nature Maximum Recommended 0 Alternative s s s3 Regret Decision d 0 d d 4 0 4 3 d3 3 0 3 4 Minimax Regret Value Decision Making with Probabilities Expected Value Approach If probabilistic information regarding the states of nature is available, one may use the expected value (EV) approach. Here the expected return for each decision is calculated by summing the products of the payoff under each state of nature and the probability of the respective state of nature occurring. The decision yielding the best expected return is chosen. 7 7
Expected Value of a Decision Alternative The expected value of a decision alternative is the sum of weighted payoffs for the decision alternative. The expected value (EV) of decision alternative d i is defined as: N EV( d ) P ( s ) V i j ij j where: N = the number of states of nature P(s j ) = the probability of state of nature s j V ij = the payoff corresponding to decision alternative d i and state of nature s j Example: Burger Prince Burger Prince Restaurant is considering opening a new restaurant on Main Street. It has three different models, each with a different seating capacity. Burger Prince estimates that the average number of customers per hour will be 0, 00, or 0. The payoff table for the three models is on the next slide. Payoff Table Average Number of Customers Per Hour s = 0 s = 00 s 3 = 0 Model A $0,000 $5,000 $4,000 Model B $,000 $,000 $,000 Model C $ 6,000 $6,000 $,000
Expected Value Approach Calculate the expected value for each decision. The decision tree on the next slide can assist in this calculation. Here d, d, d 3 represent the decision alternatives of models A, B, C, and s, s, s 3 represent the states of nature of 0, 00, and 0. Decision Tree d d d 3 3 4 s s s 3 s s s 3 s s s 3.4..4.4..4.4..4 Payoffs 0,000 5,000 4,000,000,000,000 6,000 6,000,000 Expected Value for Each Decision Model A Model B d d EMV =.4(0,000) +.(5,000) +.4(4,000) = $,600 EMV =.4(,000) +.(,000) +.4(,000) = $,600 3 Model C d 3 EMV =.4(6,000) +.(6,000) +.4(,000) = $4,000 4 Choose the model with largest EV, Model C. 9 9
Expected Value Approach Formula Spreadsheet A B C D E F PAYOFF TABLE 3 Decision State of Nature Expected Recommended 4 Alternative s = 0 s = 00 s3 = 0 Value Decision 5 d = Model A 0,000 5,000 4,000 =$B$*B5+$C$*C5+$D$*D5 =IF(E5=$E$9,A5,"") 6 d = Model B,000,000,000 =$B$*B6+$C$*C6+$D$*D6 =IF(E6=$E$9,A6,"") 7 d3 = Model C 6,000 6,000,000 =$B$*B7+$C$*C7+$D$*D7 =IF(E7=$E$9,A7,"") Probability 0.4 0. 0.4 9 Maximum Expected Value =MAX(E5:E7) Expected Value Approach Solution Spreadsheet A B C D E F PAYOFF TABLE 3 Decision State of Nature Expected Recommended 4 Alternative s = 0 s = 00 s3 = 0 Value Decision 5 d = Model A 0,000 5,000 4,000 600 6 d = Model B,000,000,000 600 7 d3 = Model C 6,000 6,000,000 4000 d3 = Model C Probability 0.4 0. 0.4 9 Maximum Expected Value 4000 Expected Value of Perfect Information Frequently information is available which can improve the probability estimates for the states of nature. The expected value of perfect information (EVPI) is the increase in the expected profit that would result if one knew with certainty which state of nature would occur. The EVPI provides an upper bound on the expected value of any sample or survey information. 0 0
Expected Value of Perfect Information EVPI Calculation Step : Determine the optimal return corresponding to each state of nature. Step : Compute the expected value of these optimal returns. Step 3: Subtract the EV of the optimal decision from the amount determined in step (). Expected Value of Perfect Information Calculate the expected value for the optimum payoff for each state of nature and subtract the EV of the optimal decision. EVPI=.4(0,000) +.(,000) +.4(,000) - 4,000 = $,000 Expected Value of Perfect Information Spreadsheet A B C D E F PAYOFF TABLE 3 Decision State of Nature Expected Recommended 4 Alternative s = 0 s = 00 s3 = 0 Value Decision 5 d = Model A 0,000 5,000 4,000 600 6 d = Model B,000,000,000 600 7 d3 = Model C 6,000 6,000,000 4000 d3 = Model C Probability 0.4 0. 0.4 9 Maximum Expected Value 4000 0 Maximum Payoff EVwPI EVPI 0,000,000,000 6000 000
Risk Analysis Risk analysis helps the decision maker recognize the difference between: the expected value of a decision alternative, and the payoff that might actually occur The risk profile for a decision alternative shows the possible payoffs for the decision alternative along with their associated probabilities. Risk Profile Model C Decision Alternative Probability.50.40.30.0.0 5 0 5 0 5 Sensitivity Analysis Sensitivity analysis can be used to determine how changes to the following inputs affect the recommended decision alternative: probabilities for the states of nature values of the payoffs If a small change in the value of one of the inputs causes a change in the recommended decision alternative, extra effort and care should be taken in estimating the input value.
Bayes Theorem and Posterior Probabilities Knowledge of sample (survey) information can be used to revise the probability estimates for the states of nature. Prior to obtaining this information, the probability estimates for the states of nature are called prior probabilities. With knowledge of conditional probabilities for the outcomes or indicators of the sample or survey information, these prior probabilities can be revised by employing Bayes' Theorem. The outcomes of this analysis are called posterior probabilities or branch probabilities for decision trees. Computing Branch Probabilities Branch (Posterior) Probabilities Calculation Step : For each state of nature, multiply the prior probability by its conditional probability for the indicator -- this gives the joint probabilities for the states and indicator. Computing Branch Probabilities Branch (Posterior) Probabilities Calculation Step : Sum these joint probabilities over all states -- this gives the marginal probability for the indicator. Step 3: For each state, divide its joint probability by the marginal probability for the indicator -- this gives the posterior probability distribution. 3 3
Expected Value of Sample Information The expected value of sample information (EVSI) is the additional expected profit possible through knowledge of the sample or survey information. Expected Value of Sample Information EVSI Calculation Step : Determine the optimal decision and its expected return for the possible outcomes of the sample using the posterior probabilities for the states of nature. Step : Compute the expected value of these optimal returns. Step 3: Subtract the EV of the optimal decision obtained without using the sample information from the amount determined in step (). Efficiency of Sample Information Efficiency of sample information is the ratio of EVSI to EVPI. As the EVPI provides an upper bound for the EVSI, efficiency is always a number between 0 and. 4 4
Sample Information Burger Prince must decide whether or not to purchase a marketing survey from Stanton Marketing for $,000. The results of the survey are "favorable" or "unfavorable". The conditional probabilities are: P(favorable 0 customers per hour) =. P(favorable 00 customers per hour) =.5 P(favorable 0 customers per hour) =.9 Should Burger Prince have the survey performed by Stanton Marketing? Influence Diagram Decision Chance Consequence Market Survey Results Avg. Number of Customers Per Hour Market Survey Restaurant Size Profit Posterior Probabilities State Prior Favorable Conditional Joint Posterior 0.4..0.4 00..5.0.5 0.4.9.36.667 Total.54.000 P(favorable) =.54 5 5
Posterior Probabilities State Prior Unfavorable Conditional Joint Posterior 0.4..3.696 00..5.0.7 0.4..04.07 Total.46.000 P(unfavorable) =.46 Formula Spreadsheet Posterior Probabilities A B C D E Market Research Favorable Prior Conditional Joint Posterior 3 State of Nature Probabilities Probabilities Probabilities Probabilities 4 s = 0 0.4 0. =B4*C4 =D4/$D$7 5 s = 00 0. 0.5 =B5*C5 =D5/$D$7 6 s3 = 0 0.4 0.9 =B6*C6 =D6/$D$7 7 P(Favorable) = =SUM(D4:D6) 9 0 Market Research Unfavorable Prior Conditional Joint Posterior State of Nature Probabilities Probabilities Probabilities Probabilities s = 0 0.4 0. =B*C =D/$D$5 3 s = 00 0. 0.5 =B3*C3 =D3/$D$5 4 s3 = 0 0.4 0. =B4*C4 =D4/$D$5 5 P(Unfavorable) = =SUM(D:D4) Solution Spreadsheet Posterior Probabilities A B C D E Market Research Favorable Prior Conditional Joint Posterior 3 State of Nature Probabilities Probabilities Probabilities Probabilities 4 s = 0 0.4 0. 0.0 0.4 5 s = 00 0. 0.5 0.0 0.5 6 s3 = 0 0.4 0.9 0.36 0.667 7 P(Favorable) = 0.54 9 0 Market Research Unfavorable Prior Conditional Joint Posterior State of Nature Probabilities Probabilities Probabilities Probabilities s = 0 0.4 0. 0.3 0.696 3 s = 00 0. 0.5 0.0 0.7 4 s3 = 0 0.4 0. 0.04 0.07 5 P(Favorable) = 0.46 6 6
Decision Tree Top Half I (.54) d d d 3 4 5 6 s (.4) s (.5) s 3 (.667) s (.4) s (.5) s 3 (.667) s (.4) s (.5) s 3 (.667) $0,000 $5,000 $4,000 $,000 $,000 $,000 $6,000 $6,000 $,000 Decision Tree Bottom Half I (.46) d 3 d d 3 7 9 s (.696) s (.696) s (.696) s (.7) s 3 (.07) s (.7) s 3 (.07) s (.7) s 3 (.07) $0,000 $5,000 $4,000 $,000 $,000 $,000 $6,000 $6,000 $,000 Decision Tree I (.54) I (.46) $7,55 3 $,433 d d d 3 d d d 3 4 5 6 7 9 EMV =.4(0,000) +.5(5,000) +.667(4,000) = $3,593 EMV =.4 (,000) +.5(,000) +.667(,000) = $,5 EMV =.4(6,000) +.5(6,000) +.667(,000) = $7,55 EMV =.696(0,000) +.7(5,000) +.07(4,000)= $,433 EMV =.696(,000) +.7(,000) +.07(,000) = $0,554 EMV =.696(6,000) +.7(6,000) +.07(,000) = $9,475 7 7
Expected Value of Sample Information If the outcome of the survey is "favorable, choose Model C. If it is unfavorable, choose Model A. EVSI =.54($7,55) +.46($,433) - $4,000 = $900. Since this is less than the cost of the survey, the survey should not be purchased. Efficiency of Sample Information The efficiency of the survey: EVSI/EVPI = ($900.)/($000) =.4504