2Chapter 4 Decision Analysis Problem FormulationDecision Making without ProbabilitiesDecision Making with ProbabilitiesRisk Analysis and Sensitivity AnalysisDecision Analysis with Sample InformationComputing Branch Probabilities
3Problem FormulationThe first step in the decision analysis process is problem formulation.We begin with a verbal statement of the problem.Then we identify:the decision alternativesthe states of nature (uncertain future events)the payoff (consequences) associated with each specific combination of:decision alternativestate of nature
4Problem Formulation Example: Burger Prince Restaurant Burger Prince Restaurant is consideringopening a new restaurant on Main Street.The company has three differentbuilding designs (A, B, and C), eachwith a different seating capacity.Burger Prince estimates that theaverage number of customers arrivingper hour will be 40, 60, or 80.
5Problem Formulation Decision Alternatives d1 = use building design A d2 = use building design Bd3 = use building design CStates of Natures1 = an average of 40 customers arriving per hours2 = an average of 60 customers arriving per hours3 = an average of 80 customers arriving per hour
6Problem Formulation Payoff Table 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.
7Problem Formulation Payoff Table (Payoffs are Profit Per Week) Average Number ofCustomers Per Hours1 = s2 = s3 = 80Design ADesign BDesign C$10, $15, $14,000$ 8, $18, $12,000$ 6, $16, $21,000
8Problem Formulation Influence Diagram 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.
9Problem Formulation Influence Diagram Profit Average Number of CustomersPer HourStates of Nature40 customers per hour (s1)60 customers per hour (s2)80 customers per hour (s3)DecisionAlternativesDesign A (d1)RestaurantDesignProfitConsequenceDesign B (d2)ProfitDesign C (d3)
10Problem Formulation Decision Tree A decision tree is a chronological representation of the decision problem.A decision tree has two types of nodes:round nodes correspond to chance eventssquare nodes correspond to decisionsBranches leaving a round node represent the different states of nature; branches leaving a square node represent the different decision alternatives.At the end of a limb of the tree is the payoff attained from the series of branches making up the limb.
11Problem Formulation Decision Tree 2 1 3 4 10,000 15,000 14,000 8,000 40 customers per hour (s1)10,000Design A (d1)60 customers per hour (s2)215,00080 customers per hour (s3)14,00040 customers per hour (s1)8,000Design B (d2)60 customers per hour (s2)1318,00080 customers per hour (s3)12,00040 customers per hour (s1)6,000Design C (d3)60 customers per hour (s2)416,00080 customers per hour (s3)21,000
12Decision Making without Probabilities Criteria for Decision MakingThree commonly used criteria for decisionmaking when probability information regarding thelikelihood of the states of nature is unavailable are:the optimistic (maximax) approachthe conservative (maximin) approachthe minimax regret approach.
13Decision Making without Probabilities Optimistic (Maximax) ApproachThe optimistic approach would be used by an optimistic decision maker.The decision with the overall largest payoff is chosen.If the payoff table is in terms of costs, the decision with the overall lowest cost will be chosen (hence, a minimin approach).
14Decision Making without Probabilities Optimistic (Maximax) ApproachThe decision that has the largest single value in the payoff table is chosen.States of NatureDecision (Customers Per Hour)Alternative s s s3Design A d , , ,000Design B d , , ,000Design C d , , ,000MaximaxdecisionMaximaxpayoff
15Decision Making without Probabilities Conservative (Maximin) ApproachThe conservative approach would be used by a conservative decision maker.For each decision the minimum payoff is listed.The decision corresponding to the maximum of these minimum payoffs is selected.If payoffs are in terms of costs, the maximum costs will be determined for each decision and then the decision corresponding to the minimum of these maximum costs will be selected. (Hence, a minimax approach)
16Decision Making without Probabilities Conservative (Maximin) ApproachList the minimum payoff for each decision. Choose the decision with the maximum of these minimum payoffs.MaximindecisionMaximinpayoffDecision MinimumAlternative PayoffDesign A d ,000Design B d ,000Design C d ,000
17Decision Making without Probabilities Minimax Regret ApproachThe 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 corresponding to the minimum of the maximum regrets is chosen.
18Decision Making without Probabilities Minimax Regret ApproachFirst compute a regret table by subtracting each payoff in a column from the largest payoff in that column. The resulting regret table is:States of NatureDecision (Customers Per Hour)Alternative s s s3Design A d , ,000Design B d , ,000Design C d , ,
19Decision Making without Probabilities Minimax Regret ApproachFor each decision list the maximum regret. Choose the decision with the minimum of these values.Decision MaximumAlternative RegretDesign A d ,000Design B d ,000Design C d ,000MinimaxdecisionMinimaxregret
20Decision Making with Probabilities Assigning ProbabilitiesOnce we have defined the decision alternatives and states of nature for the chance events, we focus on determining probabilities for the states of nature.The classical, relative frequency, or subjective method of assigning probabilities may be used.Because only one of the N states of nature can occur, the probabilities must satisfy two conditions:P(sj) > 0 for all states of nature
21Decision Making with Probabilities Expected Value ApproachWe use the expected value approach to identify the best or recommended decision alternative.The expected value of each decision alternative is calculated (explained on the next slide).The decision alternative yielding the best expected value is chosen.
22Expected Value Approach The expected value of a decision alternative is the sum of the weighted payoffs for the decision alternative.The expected value (EV) of decision alternative di is defined aswhere: N = the number of states of natureP(sj ) = the probability of state of nature sjVij = the payoff corresponding to decision alternative di and state of nature sj
23Expected Value Approach Calculate the expected value (EV) for each decision.The decision tree on the next slide can assist in this calculation.Here d1, d2, d3 represent the decision alternatives of Designs A, B, and C.And s1, s2, s3 represent the states of nature of 40, 60, and 80 customers per hour.The decision alternative with the greatest EV is the optimal decision.
25Expected Value Approach EV(d1) = .4(10,000) + .2(15,000)+ .4(14,000) = $12,600Design A d12EV(d2) = .4(8,000) + .2(18,000)+ .4(12,000) = $11,600Design B d213EV(d3) = .4(6,000) + .2(16,000)+ .4(21,000) = $14,000Design C d34Choose the decision alternative with the largest EV:Design C
26Expected 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.
27Expected Value of Perfect Information Expected value of perfect information is defined asEVPI = |EVwPI – EVwoPI|where:EVPI = expected value of perfect informationEVwPI = expected value with perfect informationabout the states of natureEVwoPI = expected value without perfect information about the states of nature
28Expected Value of Perfect Information EVPI CalculationStep 1:Determine the optimal return corresponding to each state of nature.Step 2:Compute the expected value of these optimal returns.Step 3:Subtract the EV of the optimal decision from the amount determined in step (2).
29Expected Value of Perfect Information EVPI CalculationCalculate the expected value for the optimum payoff for each state of nature and subtract the EV of the optimal decision.EVPI= .4(10,000) + .2(18,000) + .4(21,000) - 14,000 = $2,000
30Risk AnalysisRisk analysis helps the decision maker recognize the difference between:the expected value of a decision alternative, andthe payoff that might actually occurThe risk profile for a decision alternative shows the possible payoffs for the decision alternative along with their associated probabilities.
31Risk Analysis Risk Profile for Decision Alternative d3 .50 .40 Probability.30.20.10Profit ($ thousands)
32Sensitivity AnalysisSensitivity analysis can be used to determine how changes to the following inputs affect the recommended decision alternative:probabilities for the states of naturevalues of the payoffsIf 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.
33Sensitivity Analysis Resolving Using Different Values for the Probabilities of the States of NatureP(s1)P(s2)P(s3)EV(d1)EV(d2)EV(d3)Optimal d12,90012,98712,70012,25011,80013,50013,80014,10012,40012,65412,20011,50010,80014,00014,80015,60014,25014,31913,50012,25011,00014,75015,00015,250d3d1 and d3d1d2
34Decision Analysis With Sample Information 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.
35Decision Analysis With Sample Information Example: Burger PrinceBurger Prince must decide whether to purchase amarketing survey from Stanton Marketing for$1,000. The results of the survey are "favorable“or "unfavorable". The branch probabilitiescorresponding to all the chance nodesare listed on the next slide.
36Decision Analysis With Sample Information Branch ProbabilitiesP(Favorable market survey) = .54P(Unfavorable market survey) = .46P(40 customers per hour | favorable) = .148P(60 customers per hour | favorable) = .185P(80 customers per hour | favorable) = .667P(40 customers per hour | unfavorable) = .696P(60 customers per hour | unfavorable) = .217P(80 customers per hour | unfavorable) = .087
37Decision Analysis With Sample Information Influence DiagramDecisionChanceConsequenceMarketSurveyResultsAvg. Numberof CustomersPer HourMarketSurveyRestaurant Design(seating capacity)Profit
40Decision Analysis With Sample Information Decision StrategyA decision strategy is a sequence of decisions and chance outcomes.The sequence of decisions chosen depends on the yet to be determined outcomes of chance events.The optimal decision strategy is based on the Expected Value with Sample Information (EVwSI).The EVwSI is calculated by making a backward pass through the decision tree.
41Decision Analysis With Sample Information Expected Value with Sample Information (EVwSI)Step 1:Determine the optimal decision strategy and its expected returns for the possible outcomes of the sample using the posterior probabilities for the states of nature.Step 2:Compute the expected value of these optimal returns.
43Decision Analysis With Sample Information 4$13,593$17,855d225$12,518I1(.54)d36$17,855EVwSI = .54(17,855)+ .46(11,433)= $14,900.881d1I2(.46)7$11,433d238$10,554$11,433d39$ 9,475
44Decision Analysis With Sample Information Expected Value with Sample Information (EVwSI)EVwSI = .54($17,855) + .46($11,433) = $14,900.88Optimal Decision StrategyIf the outcome of the survey is "favorable”,choose Design C.If the outcome of the survey is “unfavorable”, choose Design A.
45Decision Analysis With Sample Information Risk Profile for Optimal Decision StrategyP(s|I)P(I)PayoffI1 = .54P(s1|I1) = .148P(s2|I1) = .185P(s3|I1) = .667.08.10.36$ 6,000$16,000$21,000Payoffsfor d3I2 =.46P(s1|I2) = .696P(s2|I2) = .217P(s3|I2) = .087.32.10.041.00$10,000$15,000$14,000Payoffsfor d1
46Decision Analysis With Sample Information Risk Profile for Optimal Decision Strategy.22.214.171.124Probability.126.96.36.199.10.08.04Profit ($ thousands)
47Expected 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.EVSI = |EVwSI – EVwoSI|where:EVSI = expected value of sample informationEVwSI = expected value with sample informationabout the states of natureEVwoSI = expected value without sample information about the states of nature
48Expected Value of Sample Information EVSI CalculationSubtract the EVwoSI (the value of the optimal decision obtained without using the sample information) from the EVwSI.EVSI = $14, $14,000 = $900.88ConclusionBecause the EVSI ($900.88) is less than the cost of the survey ($ ), the survey should not be purchased.
49Efficiency of Sample Information The efficiency rating (E) of sample information is the ratio of EVSI to EVPI expressed as a percent.The efficiency rating (E) of the market survey for Burger Prince Restaurant is:
50Computing Branch Probabilities Bayes’ Theorem can be used to compute branch probabilities for decision trees.For the computations we need to know:the initial (prior) probabilities for the states of nature,the conditional probabilities for the outcomes or indicators of the sample information, given each state of nature.A tabular approach is a convenient method for carrying out the computations.
51Computing Branch Probabilities Step 1For each state of nature, multiply the prior probabilityby its conditional probability for the indicator. Thisgives the joint probabilities for the states and indicator.Step 2Sum these joint probabilities over all states. This givesthe marginal probability for the indicator.Step 3For each state, divide its joint probability by themarginal probability for the indicator. This gives theposterior probability distribution.
52Computing Branch Probabilities Example: Burger Prince RestaurantRecall that Burger Prince is considering purchasinga marketing survey from Stanton Marketing. Theresults of the survey are "favorable“ or "unfavorable".The conditional probabilities are:P(favorable |40 customers per hour) = .2P(favorable |60 customers per hour) = .5P(favorable |80 customers per hour) = .9Compute the branch (posterior) probabilitydistribution.