Decision analysis: part 2 BSAD 30 Dave Novak Source: Anderson et al., 2013 Quantitative Methods for Business 12th edition – some slides are directly from J. Loucks © 2013 Cengage Learning

Overview Risk analysis EVPI calculation EOL calculation Risk profile Sensitivity analysis Changes in states of nature Changes in payoffs EVPI calculation EOL calculation Building and using decision trees

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 Recall that a payoff is the result of a combination of: 1) a decision alternative (you control this), and 2) a state of nature probability (you do not control this)

Risk analysis The risk profile for a decision alternative shows the possible payoffs for the decision alternative along with their associated probabilities We want to identify the probability or likelihood that a particular payoff will occur Basically, we want to list out ALL possible payoffs, and their probabilities of occurrence

Strong Demand Weak Demand PDC example - lecture 15 Payoff table with P(s1) = 0.8 and P(s2) = 0.2 PAYOFF TABLE States of Nature Strong Demand Weak Demand Decision Alternative s1 s2 Small complex, d1 8 7 Medium complex, d2 14 5 Large complex, d3 20 -9

PDC example – lecture 15 Payoffs $8 mil 2 $7 mil d1 d2 $14 mil 1 3 d3 .8 $8 mil 2 s2 .2 d1 $7 mil s1 .8 d2 $14 mil 1 3 s2 .2 d3 $5 mil s1 .8 $20 mil 4 s2 .2 -$9 mil

PDC example - lecture 15 2 small d1 medium d2 1 3 large d3 4

Risk profile d3 (90 unit) decision alternative versus d2 (60 unit) decision alternative

Value of perfect information Calculate EV assuming MOST OPTIMISTIC payoff for both states of nature (does not need to be the same decision) = EVwPI Take the EV associated with your decision (this is the largest EV value across all decisions) = EVwoPI Given imperfect information, this is what we would choose to do

Value of perfect information EVPI = 𝐸𝑉𝑤𝑃𝐼 −𝐸𝑉𝑤𝑜𝑃𝐼 EVPI = $17.4 𝑀 −$14.2 𝑀 EVPI = $3.2 𝑀

Expected opportunity loss Using the regret table from lecture #15, we can calculate “the expected opportunity lost” (EOL) associated with each decision States of Nature Strong Demand Weak Demand Decision Alternative s1 s2 Small complex, d1 12 0 Medium complex, d2 6 2 Large complex, d3 0 16 REGRET TABLE

Expected opportunity loss

Expected opportunity loss The minimum of the EOL values always provides the optimal decision Notice that EVPI = Expected Opportunity Loss (EOL) for decision d3 (90 units) EVPI is ALWAYS equal to the EOL for the optimal decision

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 can be subjective, and therefore subject to change values of the payoffs

Sensitivity analysis If a small change in the value of one of the inputs (state of nature probabilities or payoff values) causes a change in the recommended decision alternative, extra effort and care should be taken in estimating the input value If changes to inputs do not really impact your decision, you can feel more confident about this decision

Sensitivity analysis One way to address the sensitivity question is to select different values for either the state of nature probabilities or the payoff values, and then do some “what if” calculations Say we flip our state of nature probabilities for the PDC problem…

Modified PDC example Payoffs $8 mil 2 $7 mil d1 d2 $14 mil 1 3 d3 .2 $8 mil 2 s2 .8 d1 $7 mil s1 .2 d2 $14 mil 1 3 s2 .8 d3 $5 mil s1 .2 $20 mil 4 s2 .8 -$9 mil

Modified PDC example 2 small d1 medium d2 1 3 large d3 4

Modified PDC example Payoffs $8 mil 2 $7 mil d1 d2 $14 mil 1 3 d3 .5 $8 mil 2 s2 .5 d1 $7 mil s1 .5 d2 $14 mil 1 3 s2 .5 d3 $5 mil s1 .5 $20 mil 4 s2 .5 -$9 mil

Modified PDC example 2 small d1 medium d2 1 3 large d3 4

EV comparison

Graphing for 2 state of nature problem Like a LP with two decision variables, if we only have two states of nature, we can perform sensitivity analysis graphically Generalize relationship for P(s1) and P(s2)

EV calculations for decision variables

Sensitivity graph

Sensitivity graph Solve for each intersection point way we solved for d3 provides highest EV d1 provides highest EV d2 provides highest EV Solve for each intersection point way we solved for internal points in LP Set two linear equations equal to one another and solve for p

Solving for inflection points Intersection of EV(d1) and EV(d2) lines EV(d1) = p + 7 EV(d2) = 9p + 5 So, p + 7 = 9p + 5 2 = 8p p = 2 8 = 0.25

Solving for inflection points Intersection of EV(d2) and EV(d3) lines EV(d2) = 9p + 5 EV(d3) = 2p - 9 So, 9p + 5 = 29p - 9 14 = 20p p = 14 20 = 7 10 = 0.7

Sensitivity graph p = 0.25 p = 0.7 d3 provides highest EV d1 provides

What are managerial implications? If probability of strong demand, P(s1) < 0.25, then choose d1 (30 units) If probability of strong demand, P(s1) = 0.25, then choose either d1 (30 units) or d2 (60 units) If probability of strong demand, 0.25 < P(s1) < 0.7, then choose d2 (60 units) If probability of strong demand, P(s1) = 0.7, then choose either d2 (60 units) or d3 (90 units) If probability of strong demand, P(s1) > 7 then choose d3 (90 units)

What about changes in payoff values? From the PDC problem, we have: EV(d1) = 7.8 EV(d2) = 12.2 EV(d3) = 14.2 So, we conclude that building 90 units is the optimal decision (d3) as long as EV(d3) ≥ 12.2

What about changes in payoff values? Let’s look at changing one of the payoff values for decision alternative d3 (90 units) Hold the state of nature probabilities constant for both s1 (strong demand) and s2 (weak demand) P(s1) = 0.8, and P(s2) = 0.2 Let S = payoff value for d3 assuming s1 W= payoff value for d3 assuming s2

What about changes in payoff values? We can write EV(d3) as: Examine a change in one of the payoff values for a particular decision alternative Here, we will hold payoff for weak demand constant at -9

What about changes in payoff values? Solve for S 0.8S – 1.8 ≥ 12.2 0.8S ≥ 14 S ≥ 17.5 What does this mean? As long as our original state of nature probabilities hold, we should build d3, as long as the payoff under the strong demand scenario is ≥ 17.5 mil

What about changes in payoff values? Examine a change in the other payoff value for decision alternative d3 we will hold payoff for strong demand constant at 20, and investigate changes in weak demand

What about changes in payoff values? Solve for W 16 + 0.2W ≥ 12.2 0.2W ≥ -3.8 W ≥ -19 What does this mean? As long as our original state of nature probabilities hold, we should build d3, as long as the payoff under the weak demand scenario is ≥ -19 mil

What about changes in payoff values? When we hold the state of nature probabilities constant at P(s1) = 0.8, and P(s2) = 0.2, the d3 decision does not seem to be particularly sensitive to variations in the payoff Why? Probability that demand is strong, P(s1), is very high AND expected payoff is very high Using the EV approach, this combination leads us to choose d3

Decision trees Just a graphical representation of the decision-making process Shows a progression over time Squares: decision nodes that we control Circles: chance nodes that we do not control

Decision tree Payoffs $8 mil 2 $7 mil $14 mil 1 3 $5 mil $20 mil 4 For each decision alternative and state of nature pair, we have an expected payoff Given each decision, we are then subject to demand, which we cannot control Payoffs s1 .8 $8 mil 2 d1-build 30 units s2 .2 $7 mil s1 .8 $14 mil 1 d2-build 60 units 3 s2 .2 $5 mil d3-build 90 units s1 .8 $20 mil 4 s2 .2 -$9 mil

Decision trees Hemmingway, Inc., is considering a $5 million research and development (R&D) project. Profit projections appear promising, but Hemmingway's president is concerned because the probability that the R&D project will be successful is only 0.50. Furthermore, the president knows that even if the project is successful, it will require that the company build a new production facility at a cost of $20 million in order to manufacture the product.

Decision trees If the facility is built, uncertainty remains about the demand and thus uncertainty about the profit that will be realized. Another option is that if the R&D project is successful, the company could sell the rights to the product for an estimated $25 million. Under this option, the company would not build the $20 million production facility.

Decision trees Identify the “pieces” or nodes associated with this problem We have to present this information in a time dependent sequence of events

Decision trees Make a decision whether to start the R&D project or not Start R&D project (cost of $5 mil)  proceed Do not start R&D project (cost of zero)  stop

Decision trees

Decision trees If we start R&D, there is a state of nature or chance event that the project will be successful, where: P(R&D success) = 0.5  proceed P(R&D failure) = 0.5  stop and lose $5 mil

Decision trees

Decision trees If R&D is successful, we make a decision whether to: Build the production facility (cost of $20 mil)  proceed Sell the rights to the product for $20 mil  stop

Decision trees

Decision trees If we decide to build the facility, we are subject to state of nature or chance events regarding demand for the product: P(high demand) = 0.5, and the corresponding payoff is $34 mil P(med demand) = 0.3, and the corresponding payoff is $20 mil P(low demand) = 0.2, and the corresponding payoff is $10 mil

Decision tree

Decision trees Populate the decision tree with probabilities, so we can calculate EV for different scenarios We work BACKWARDS to calculate EV for each chance or state of nature node EV (node 4) EV (node 2)

Decision trees

Decision tree

Using the decision tree to guide decision-making Should the company undertake the R&D project? If R&D is successful, what should company do, sell or build?

Using the decision tree to guide decision-making What is EV of your decision strategy? What would selling price need to be for company to consider selling? What about for recovering R&D cost?

Using the decision tree to guide decision-making Developing a risk profile for the optimal strategy Possible Profit Corresponding probability $34 M $20 M $10 M -$5 M

What would the payoff table look like? States of Nature High Demand Med Demand Low Demand PAYOFF TABLE Decision Alternatives Build Sell 34 20 10 20 20 20

What would the regret table look like? States of Nature High Demand Med Demand Low Demand REGRET TABLE Decision Alternatives Build Sell 10 14

Value of perfect information EVwPI = EVwoPI = EVPI =

Expected opportunity loss Using the regret table States of Nature High Demand Med Demand Low Demand REGRET TABLE Decision Alternatives Build Sell 10 14

Summary Risk analysis EVPI calculation EOL calculation Risk profile Sensitivity analysis Changes in states of nature Changes in payoffs EVPI calculation EOL calculation Building and using decision trees