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Chapter 18 Introduction to Decision Analysis

Business Statistics: A Decision-Making Approach 6 th Edition. Chapter 18 Introduction to Decision Analysis. Chapter Goals. After completing this chapter, you should be able to: Describe the decision environments of certainty and uncertainty

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Chapter 18 Introduction to Decision Analysis

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  1. Business Statistics: A Decision-Making Approach 6th Edition Chapter 18Introduction to Decision Analysis

  2. Chapter Goals After completing this chapter, you should be able to: • Describe the decision environments of certainty and uncertainty • Construct a payoff table and an opportunity-loss table • Define and apply the expected value criterion for decision making • Compute the value of perfect information • Develop and use decision trees for decision making

  3. Decision Making Overview Decision Making Decision Environment Decision Criteria Certainty Nonprobabilistic Uncertainty Probabilistic

  4. The Decision Environment Decision Environment Certainty: The results of decision alternatives are known * Certainty Example: Must print 10,000 color brochures Offset press A: $2,000 fixed cost + $.24 per page Offset press B: $3,000 fixed cost + $.12 per page Uncertainty

  5. The Decision Environment (continued) Uncertainty: The outcome that will occur after a choice is unknown Decision Environment Certainty Example: You must decide to buy an item now or wait. If you buy now the price is $2,000. If you wait the price may drop to $1,500 or rise to $2,200. There also may be a new model available later with better features. * Uncertainty

  6. Decision Criteria Decision Criteria Nonprobabilistic Decision Criteria: Decision rules that can be applied if the probabilities of uncertain events are not known. * Nonprobabilistic • maximax criterion • maximin criterion • minimax regret criterion Probabilistic

  7. Decision Criteria (continued) Decision Criteria Probabilistic Decision Criteria: Consider the probabilities of uncertain events and select an alternative to maximize the expected payoff of minimize the expected loss Nonprobabilistic * Probabilistic • maximize expected value • minimize expected opportunity loss

  8. A Payoff Table A payoff table shows alternatives, states of nature, and payoffs

  9. Maximax Solution • The maximax criterion (an optimistic approach): • For each option, find the maximum payoff 1. Maximum Profit 200 120 40

  10. Maximax Solution (continued) • The maximax criterion (an optimistic approach): • For each option, find the maximum payoff • Choose the option with the greatest maximum payoff 2. Greatest maximum is to choose Large factory 1. Maximum Profit 200 120 40

  11. R program for Maximax • act1=c(200,50,-120) #data for action 1 • act2=c(90,120,-30) • act3=c(40,30,20) • nact=3 #number of actions • nj=3 #number of states of nature • payoff=matrix(rbind(act1,act2,act3,act4),nact,nj) • payoff • maxact=rep(0,nact) #initialize • i=1 • while (i<=nact) { • maxact[i]=max(payoff[i,]) • i=i+1} • maxact • maximax=max(maxact) #optimistic criterion maximax • maximax

  12. Maximin Solution • The maximin criterion (a pessimistic approach): • For each option, find the minimum payoff 1. Minimum Profit -120 -30 20

  13. Maximin Solution (continued) • The maximin criterion (a pessimistic approach): • For each option, find the minimum payoff • Choose the option with the greatest minimum payoff 2. Greatest minimum is to choose Small factory 1. Minimum Profit -120 -30 20

  14. R for Maximin (payoff in memory) • minact=rep(0,nact) #initialize • i=1 • while (i<=nact) { • minact[i]=min(payoff[i,]) • i=i+1} • minact #minimum payoffs for each action • maximin=max(minact) #pessimistic criterion maximin • maximin

  15. Opportunity Loss Opportunity loss is the difference between an actual payoff for a decision and the optimal payoff for that state of nature (in each column) Payoff Table The choice “Average factory” has payoff 90 for “Strong Economy”. Given “Strong Economy”, the choice of “Large factory” would have given a payoff of 200, or 110 higher. Opportunity loss = 110 for this cell.

  16. Opportunity Loss (continued) Payoff Table Opportunity Loss Table

  17. R for opportunity loss matrix • opp=payoff #initialize old info in memory of R • j=1 • while (j<=nj) { • opp[1:nact,j]= max(payoff[,j]) • j=j+1} • opp #the opportunity table • opploss=opp-payoff #opportunity loss table • opploss

  18. Minimax Regret Solution • The minimax regret criterion: • For each alternative, find the maximum opportunity loss (or “regret”) Opportunity Loss Table 1. Maximum Op. Loss 140 110 160

  19. R for maximum regret • #assume previous info is in memory of R • maxreg=rep(0,nact) #initialize • i=1 • while (i<=nact) { • maxreg[i]= max(opploss[i,]) • i=i+1} • maxreg #maximum regrets

  20. Minimax Regret Solution (continued) • The minimax regret criterion: • For each alternative, find the maximum opportunity loss (or “regret”) • Choose the option with the smallest maximum loss Opportunity Loss Table 1. Maximum Op. Loss 2. Smallest maximum loss is to choose Average factory 140 110 160

  21. Expected Value Solution • The expected value is the weighted average payoff, given specified probabilities for each state of nature Suppose these probabilities have been assessed for these states of nature

  22. Minimax regret+Expected Value • minimaxregret=min(maxreg) • minimaxregret • prob=c(0.3,0.5,0.2) • length(prob) • # Error if number of probabilities are not equal to number of states of nature • nj • sum(prob) # sum probabilities should be one • evalu=payoff %*%prob • evalu • maxevalu=max(evalu) • maxevalu

  23. Expected Value Solution (continued) Example: EV (Average factory) = 90(.3) + 120(.5) + (-30)(.2) = 81 Maximize expected value by choosing Average factory Expected Values 61 81 31

  24. Expected Opportunity Loss Solution Example: EOL (Large factory) = 0(.3) + 70(.5) + (140)(.2) = 63 Opportunity Loss Table Minimize expected op. loss by choosing Average factory Expected Op. Loss (EOL) 63 43 93

  25. Expected Opportunity Loss in R • expopploss=opploss%*%prob • expopploss • minexpopploss=min(expopploss) • Minexpopploss #min of exp opp loss

  26. Cost of Uncertainty • Cost of Uncertainty (also called Expected Value of Perfect Information, or EVPI) • Cost of Uncertainty = Expected Value Under Certainty (EVUC) – Expected Value without information (EV) so: EVPI = EVUC – EV

  27. Expected Value Under Certainty • Expected Value Under Certainty (EVUC): EVUC = expected value of the best decision, given perfect information 200 120 20 Example: Best decision given “Strong Economy” is “Large factory”

  28. Expected Value Under Certainty (continued) • Now weight these outcomes with their probabilities to find EVUC: 200 120 20 EVUC = 200(.3)+120(.5)+20(.2) = 124

  29. Cost of Uncertainty Solution • Cost of Uncertainty (EVPI) = Expected Value Under Certainty (EVUC) – Expected Value without information (EV) Recall: EVUC = 124 EV is maximized by choosing “Average factory”, where EV = 81 so: EVPI = EVUC – EV = 124 – 81 = 43

  30. Decision Tree Analysis • A Decision tree shows a decision problem, beginning with the initial decision and ending will all possible outcomes and payoffs. Use a square to denote decision nodes Use a circle to denote uncertain events

  31. Sample Decision Tree Strong Economy Large factory Stable Economy Weak Economy Strong Economy Average factory Stable Economy Weak Economy Strong Economy Small factory Stable Economy Weak Economy

  32. Add Probabilities and Payoffs (continued) Strong Economy (.3) 200 Large factory Stable Economy (.5) 50 Weak Economy (.2) -120 (.3) Strong Economy 90 Average factory (.5) Stable Economy 120 (.2) Weak Economy -30 Decision (.3) Strong Economy 40 Small factory (.5) Stable Economy 30 (.2) Weak Economy 20 Uncertain Events (States of Nature) Probabilities Payoffs

  33. Fold Back the Tree Strong Economy (.3) EV=200(.3)+50(.5)+(-120)(.2)=61 200 Large factory Stable Economy (.5) 50 Weak Economy (.2) -120 (.3) Strong Economy EV=90(.3)+120(.5)+(-30)(.2)=81 90 Average factory (.5) Stable Economy 120 (.2) Weak Economy -30 (.3) Strong Economy EV=40(.3)+30(.5)+20(.2)=31 40 Small factory (.5) Stable Economy 30 (.2) Weak Economy 20

  34. Make the Decision Strong Economy (.3) EV=61 200 Large factory Stable Economy (.5) 50 Weak Economy (.2) -120 (.3) Strong Economy EV=81 90 Maximum EV=81 Average factory (.5) Stable Economy 120 (.2) Weak Economy -30 (.3) Strong Economy EV=31 40 Small factory (.5) Stable Economy 30 (.2) Weak Economy 20

  35. Chapter Summary • Examined decision making environments • certainty and uncertainty • Reviewed decision making criteria • nonprobabilistic: maximax, maximin, minimax regret • probabilistic: expected value, expected opp. loss • Computed the Cost of Uncertainty (EVPI) • Developed decision trees and applied them to decision problems

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