UNIT 5 DECISION MAKING

Transcription

1 UNIT 5 DECISION MAKING This unit: UNDER UNCERTAINTY Discusses the techniques to deal with uncertainties 1

2 INTRODUCTION Few decisions in construction industry are made with certainty. Need to look at: The magnitudes of uncertainties How uncertainties might affect decisions How uncertainties influence subsequent outcomes of decisions 2

3 TRADITIONAL STRATEGY 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. 3

4 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. 4

5 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. 5

6 EXPECTED VALUE APPROACH 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 defined as: is 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 6

7 EXPECTED VALUE APPROACH Example lets consider the example of a housing contractor who builds residential houses. The contractor is planning for the number of houses to be built for the coming year. Assume that each house costs $15,000 and sells for $0,000 and that the probability distribution of the market demand in this area for new houses is know as: P 0 = Prob (demand = 00) = 0.2 P 1 = Prob (demand = 10) = 0.4 P 2 = Prob (demand = 20) = 0.3 P 3 = Prob (demand = 30) = 0.1 How many houses should the contractor build for the year? 7

8 EXPECTED VALUE APPROACH What are the states of nature? # of houses sold What are the decision alternatives? # of houses built The payoff table is shown below. Decision (houses to be built) State of nature (Sold houses)

9 EXPECTED VALUE APPROACH The probability distribution of market demand is added to the payoff table to calculate the EV of each decision Decision (houses to be built) State of nature (Sold houses) EV Prob DD The maximum EV is when the contractor builds 20 houses. 9

10 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 is the price that one would be willing to pay in order to gain access to perfect information. The EVPI provides an upper bound on the expected value of any sample or survey information. 10

11 EXPECTED VALUE OF PERFECT INFORMATION EVPI Calculation Step 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). 11

12 EXPECTED VALUE OF PERFECT INFORMATION Decision (houses to be built) State of nature (Sold houses) EV Prob DD Optimal EVPI = =

13 DECISION TREES A decision tree is a chronological representation of the decision problem. They provide a highly effective structure within which one can lay out options and investigate the possible outcomes of choosing those options. They also help you to form a balanced picture of the risks and rewards associated with each possible course of action. 13

14 DECISION TREES A decision tree is read from left to right. Each decision tree has two types of nodes: round nodes correspond to the states of nature, outcome or chance event square nodes correspond to the decision alternatives Lines connecting the nodes show the direction of influence 14

15 DECISION TREES 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. 15

16 DECISION TREES Decision node State of nature nodes Payoffs Decision alternatives: 1. Develop temp sensor 2. Develop pressure sensor 3. Do neither States of nature: 1. Success 2. Failure 16

17 DECISION TREES 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 (Maximax) approach the conservative (Maximin) approach the minimax regret approach 17

18 OPTIMISTIC APPROACH MAXIMAX 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. 18

19 CONSERVATIVE APPROACH MAXIMIN 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.) 19

20 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 20

21 EXAMPLE 1 Consider the following problem with three decision alternatives and three states of nature with the following payoff table representing profits 21

22 EXAMPLE 1 S1 4 S2 4 d1 S3 S1-2 0 d2 S2 3 S3-1 d3 S1 1 S2 5 S3-3 22

23 EXAMPLE 1 - MAXIMAX 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 Maximax payoff 23

24 EXAMPLE 1 - MAXIMAX S1 4 4 S2 4 d1 d2 3 S3 S1 S S3-1 d3 5 S1 S2 1 5 S3-3 24

25 EXAMPLE 1 - MAXIMIN 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 Minimum Payoff Maximin decision Maximin payoff 25

26 EXAMPLE 1 - MAXIMIN S1 4-2 S2 4 d1 d2-1 S3 S1 S S3-1 d3-3 S1 S2 1 5 S3-3 26

27 EXAMPLE 1 MINIMAX REGRET For the minimax regret approach, first compute a regret table by subtracting each payoff in a column from the largest payoff in that column. States of nature Decision S1 S2 S3 d d d Column Max = 4-0 Regret table S1 S2 S3 d d d =-1-(-2) 27

28 EXAMPLE 1 MINIMAX REGRET For each decision list the maximum regret. Choose the decision with the minimum of these values.. Regret table S1 S2 S3 Max regret d d d Minimax regret Minimax regret decision 28

29 EXAMPLE 1 - MINIMAX REGRET S1 4 1 S2 4 d1 d2 4 S3 S1 S S3-1 d3 3 S1 S2 1 5 S3-3 29

30 DECISION TREES WITH PROBABILITIES Decision trees are also used for displaying decision problems with uncertainty. If probabilistic information regarding the states of nature is available, use the expected value (EV) approach. The EV 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 EV is chosen. 30

31 EXAMPLE 2 Company ABC has developed a new line of products. Top management is attempting to decide on the appropriate marketing and production strategy. 3 strategies are being considered: A (Aggressive), B (basic) and C (Cautious). The market conditions under study are denoted by S (Strong) or W (Weak). Management's best estimate of the net profits (in $, mil) in each case is given in the following payoff table. Decision State of nature S W A 30-8 B 20 7 C Management's estimates of the probabilities of a strong or a weak market are 0.45 and 0.55 respectively. Which strategy should be chosen. 31

32 SOLUTION 2 EV=9.1 P=0.45 P= A P= B EV=12.85 P= C EV=12.25 P= P=

33 SENSITIVITY ANALYSIS Other critical points of the studied decision can be concluded from the decision tree analysis. Using the same facts from example 2: EV A = 30*P(S) 8*P(W); since P(S) + P(W) =1, then EV A = 30*P(S) 8*[1-P(S)]; which simplifies into EV A = 38P(S) 8; which is the equation of a straight line 33

34 SENSITIVITY ANALYSIS Similarly solving for EV B & EV C gives: EV B = P(S); and EV C = P(S) All 3 equations can be plotted with EV as the y-axis and P(S) as the x-axis. The resultant graph is shown in next slide 34

35 SENSITIVITY ANALYSIS EV C strategy P(S) < 0.38 B strategy A strategy P(S) > P(S) -10 EVA EVB EVC 35

36 CLASS EXERCISE 1 An investment performance is significantly affected by two variables: the economic environment and whether a competing building is developed. The cash flow under the various scenarios is as follows: 36

37 CLASS EXERCISE 1 Through an assessment of the economic environment, the probability estimates of the likelihood of each of the 3 economic situations under consideration is: The conditional probability of whether the competing building will be built under the 3 economic conditions is estimated as follows: What is the expected value of the investment? 37

38 CLASS EXERCISE 2 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 80, 100, or 120. The payoff table for the three models is: Given that the probability of S 1, S 2 and S 3 is 0.4, 0.2 and 0.4 respectively, calculate the expected value for each decision & the expected value of perfect information. 38

39 CLASS EXERCISE 3 A contractor is concerned about the capacity of the existing dewatering system to be used in a new project to keep the site dry in order to prevent progress delay. Additional pumps will add costs to the budget, and progress delay will cause penalties to the contractor. Three scenarios (S1, S2 and S3) are expected to make the water level in the construction site high. S1 (the rain will be < 6 inch in 12 hour period) has the probability of 0.5, S2 (the rain will reach 6 inch only one time in 12 hour period) has the probability of 0.3, and S3 (the rain will reach 6 inch many times in 12 hour period) has the probability of 0.2. The cost of pumps installation and the penalties for any delay are shown in the following table. 39

40 CLASS EXERCISE 3 Alternative S1 S2 S3 Install 15,000 15,000 65,000 Do nothing 0 20,00 100,000 A. Draw a decision tree to represent this problem B. Which alternative is the best choice? And Why? 40

41 CLASS EXERCISE 4 Two pumping systems A and B are proposed for supplying water to a residential area. The construction costs for system A is $250,000 and for system B is $750,000. if partial failure occurs, it is expected that the damage cost for system A is $80,000 and for system B is $15,000. If complete failure occurs, it is expected that the damage cost for system A is $450,000 and for system B is $400,000. the probabilities of partial and complete failures are 5% and 1% respectively. a. Draw a decision tree to represent this problem b. Which alternative is the best choice? And Why? 41

42 CLASS EXERCISE 5 Adrian is a developer and must build factories for the coming year. Construction for the factories must be in quantities of 20. The cost per factory is $70 if they build 20, $67 if they build 40, $65 if they build 60, and $64 if they build 80. The factory will be sold for $100 each. Any factory left over at the end of the season can be sold (for certain) at $45 each. If Adrian run out of factories during the year, then they will suffer a loss of "goodwill" among their customers. They estimate this goodwill loss to be $5 per customer who was unable to buy a factory. Adrian estimate that the demand for factory this season will be 10, 30, 50, or 70 factory with probabilities of 0.2, 0.4, 0.3, and 0.1 respectively. 42

43 CLASS EXERCISE 5 a. What are the decisions facing Adrian? b. What are the states of nature? c. Construct the payoff table. d. What would be Adrian s decision based on Maximax, Maximin and Minimax Regret criterion? 43

44 THE END Any questions? 44