Decision Analysis PDF

Decision Analysis ITIS 1P97: Data Analysis and Business Modelling Instructor: Leila Tahmooresnejad Copyright © 2020 Pearson Education, Inc. Topics Decision analysis Decision table Decision making under uncertainty Decision tree...

Decision Analysis ITIS 1P97: Data Analysis and Business Modelling Instructor: Leila Tahmooresnejad Copyright © 2020 Pearson Education, Inc. Topics Decision analysis Decision table Decision making under uncertainty Decision tree 2 Role of Decision Analysis The purpose of business analytics is to provide decision- makers with information needed to make decisions. Making good decisions requires an assessment of intangible factors and risk attitudes. Decision analysis is the study of how people make decisions, particularly when faced with : Imperfect information Uncertain information A collection of techniques to support decision choices. 3 Formulating Decision Problems Many decisions involve making a choice between a small set of decisions with uncertain consequences. Decision problems involve: 1. Decision alternatives 2. Uncertain events that may occur after a decision is made along with their possible outcomes (which are often called states of nature), one and only one of them will occur. 3. Consequences associated with each decision and outcome, which are usually expressed as payoffs. 4 Formulating Decision Problems The decision maker first selects a decision alternative One of the outcomes of the uncertain event occurs Resulting in the payoff. 5 Example Let's imagine a coffee shop is considering adding a new seasonal drink to their menu for the fall season. The decision alternatives could be: Pumpkin Spice Latte Salted Caramel Hot Chocolate Apple Cinnamon Tea The uncertain events here are the customer preferences and how they might respond to these new additions: High demand for the new seasonal drink. Moderate demand with average sales. Low demand resulting in below-expected sales. 6 Formulating Decision Problems Payoffs are often summarized in a payoff table, a matrix : Rows correspond to decisions Columns correspond to events. 7 Example The payoffs are the net profit from the sale of each drink, considering the cost of ingredients, marketing, and additional training for staff, versus the sales revenue. Here is a simplified payoff table for this scenario: Seasonal High Moderate Low Drinks Demand (A) Demand (B) Demand (C) Pumpkin Spice $1,200 $600 -$100 Latte Salted Caramel $1,000 $500 -$200 Hot Chocolate Apple Cinnamon $800 $400 -$ Tea 8 Decision Strategies Without Outcome Probabilities: Minimize Objective With a minimize objective, the payoffs are costs. Aggressive (Optimistic) Strategy – Choose the decision that minimizes the smallest payoff that can occur among all outcomes for each decision (minimin strategy). Conservative (Pessimistic) Strategy – Choose the decision that minimizes the largest payoff that can occur among all outcomes for each decision (minimax strategy). 9 Decision Strategies Without Outcome Probabilities: Minimize Objective Opportunity Loss Strategy – Choose the decision that minimizes the largest opportunity loss among all outcomes for each decision (minimax regret) 10 The Six Steps in Decision Making 1. Clearly define the problem at hand 2. List the possible alternatives 3. Identify the possible outcomes or states of nature 4. List the payoff of each combination of alternatives and outcomes 5. Select one of the mathematical decision theory models 6. Apply the model and make your decision 11 Example : Selecting a Mortgage Instrument A family is considering purchasing a new home and wants to finance $150,000. Three mortgage options are available and the payoff table for the outcomes is shown below. The payoffs represent total interest paid under three future interest rate situations. – 1-year ARM (Adjusted Rate Mortgage) at a low interest rate (sensitive to interest rate) – 3-year ARM at a slightly higher rate (sensitive to interest rate) – 30-year fixed at a highest rate 12 Example : Selecting a Mortgage Instrument The Potential future change in interest rates represent an uncertain event Total interest cost they might occur represent the payoffs associated with the choice Decision Outcome: Outcome: Outcome: Rates Rise Rates Stable Rates Fall 1-year ARM $61,134 $46,443 $40,161 3-year ARM $56,901 $51,075 $46,721 30-year fixed $54,658 $54,658 $54,658 – The best decision depends on the outcome that may occur. Since you cannot predict the future outcome with certainty, the question is how to choose the best decision, considering risk. 13 Example : Mortgage Decision with the Aggressive Strategy Determine the lowest payoff (interest cost) for each type of mortgage, and then choose the decision with the smallest value (minimin). 14 Example : Mortgage Decision with the Conservative Strategy Determine the largest payoff (interest cost) for each type of mortgage, and then choose the decision with the smallest value (minimax). 15 Understanding Opportunity Loss Opportunity loss represents the “regret” that people often feel after making a nonoptimal decision. In general, the opportunity loss associated with any decision and event is the difference between: the best decision for that particular outcome the payoff for the decision that was chosen. – Opportunity losses can be only nonnegative values. 16 Example : Mortgage Decision with the Opportunity-Loss Strategy Compute the opportunity loss matrix. Step 1:Find the best outcome (minimum cost) in each column. Step Decision Outcome: Rates Rise Outcome: Rates Stable Outcome: Rates Fall 2:Subtract the best 1-year A R M $6,476 $− $− column value 3-year A R M $2,243 $4,632 $6,560 from each 30-year fixed $− $8,215 $14,497 value in the 17 column. 61,134 – 54,658= 6,476 Example : Mortgage Decision with the Opportunity-Loss Strategy Find the “minimax regret” decision Step 3: Determine the maximum opportunity loss for each decision, and then choose the decision with the smallest of these. Using this strategy, we would choose the 1-year ARM. This ensures that, no matter what outcome occurs, we will never be more than $6,476 away from the least cost we could have 18 incurred. Decision Strategies Without Outcome Probabilities: Maximize Objective With a maximize objective, the payoffs are profits. Aggressive (Optimistic) Strategy – Choose the decision that maximizes the largest payoff that can occur among all outcomes for each decision (maximax strategy). Conservative (Pessimistic) Strategy – Choose the decision that maximizes the smallest payoff that can occur among all outcomes for each decision (maximin strategy). 19 Decision Strategies Without Outcome Probabilities: Maximize Objective Opportunity Loss Strategy – Choose the decision that minimizes the maximum opportunity loss among all outcomes for each decision (minimax regret). Note that this is the same as for a minimize objective Calculation of the opportunity losses is different. 20  Summary of Decision Strategies Under Uncertainty – Minimize Objective Objective Strategy Aggressive Conservative Opportunity-Loss Strategy Strategy Strategy Minimize Choose the Find the Find the For each outcome, compute objective decision with smallest largest payoff the the smallest payoff for each for each opportunity loss for each average decision among decision decision as payoff. all Outcomes among all the absolute difference and choose Outcomes between its payoff and the the decision and choose the smallest payoff for that with the decision with outcome. smallest of the smallest of Find the maximum these these opportunity loss for each (minimin). (minimax). decision and choose the decision with the smallest opportunity loss (minimax regret). 21 Summary of Decision Strategies Under Uncertainty - Maximize objective Objective Strategy Aggressive Conservative Opportunity-Loss Strategy Strategy Strategy Maximize Choose the Find the Find the For each outcome, compute objective decision largest smallest the with payoff for payoff for opportunity loss for each the largest each decision each decision decision as the absolute average among all among all difference between its payoff. outcomes outcomes payoff and the largest and choose and choose payoff for that outcome. Find the decision the decision the maximum opportunity with the with the loss for each decision and largest largest of choose the decision with the of these these smallest opportunity loss (maximax). (maximin). (minimax regret). 22 Decision Strategies with Outcome Probabilities In many situations, we might have some assessment of these probabilities, either through some method of forecasting or reliance on expert opinions. If we can assess a probability for each outcome, we can choose the best decision based on the expected value. – The simplest case is to assume that each outcome is equally likely to occur; that is, the probability of each outcome is 1 , N where N is the number of possible outcomes. This is called the average payoff strategy. 23 Example : Mortgage Decision with the Average Payoff Strategy Estimates for the probabilities of each outcome are shown in the table below. For each loan type, compute the expected value of the interest cost and choose the minimum. (61,134 + 46,443 + 40,161) / 3 = 49,246 24 Expected Value Strategy A more general case is when the probabilities of the outcomes are not all the same. This is called the expected value strategy. We may use the expected value calculation: ¥ E [ X ] = å xi f ( xi ) (5.12) i =1 25 Example : Mortgage Decision with the Expected Value Strategy Estimates for the probabilities of each outcome are shown in the table below. For each loan type, compute the expected value of the interest cost and choose the minimum. 0.6 * 61,134 +0.3 * 46,443 + 0.1*40,161 = 54,629.40 26 Decision Trees A decision tree is a graphical model used to structure a decision problem involving uncertainty. – Nodes are points in time at which events take place. – Decision nodes are nodes in which a decision takes place by choosing among several alternatives (typically denoted as squares). – Event nodes are nodes in which an event occurs not controlled by the decision-maker (typically denoted as circles). – Branches are associated with decisions and events. Decision trees model sequences of decisions and outcomes over time. 27 Five Steps of Decision Tree Analysis Define the problem Structure or draw the decision tree Assign probabilities to the states of nature Estimate payoffs for each possible combination of alternatives and states of nature Solve the problem by computing expected monetary values (EMVs) for each state of nature node 28 Example : Creating a Decision Tree Mortgage selection problem To start the decision tree, add a node for selection of the loan type. 29 Example : Creating a Decision Tree Next, for each type of loan, add a node for selection of the uncertain interest rate conditions. 30 Example : Creating a Decision Tree Finally, enter the payoffs of the outcomes associated with each event in the cells immediately below the branches. Sum all payoffs along the paths and place these values next to the terminal nodes. 31 Example : Analyzing a Decision Tree (1 of 2) To find the best decision strategy in a decision tree, “roll back” the tree by computing expected values at event nodes selecting the optimal value of alternative decisions at decision nodes. For one-year ARM, the expected value of the chance events is 0.6* ( -$61,134 ) + 0.3* ( -$46, 443) + 0.1* ( -40,161) = - $54,629.40. 32 Example : Analyzing a Decision Tree At the decision node, choose the minimum from among all decisions; this is -$54,135.20. Best decision branch three-year ARM, which is branch 2. 33 Final Exam The final exam will include all the material covered since the beginning of the course. Final Exam Weight: 40% of the final grade – Multiple Choice – True / False – Problem Solving 34 Probability Probability is the likelihood that an Die Sum 2 Frequency 1 outcome occurs. Probabilities are 3 2 expressed as values between 0 and 1. 4 3 5 4 Probability Rules 6 5 7 6 8 5 9 4 10 3 If events A and B are mutually exclusive, 11 2 12 1 Sum 36 If two events A and B are not mutually exclusive, then 35  Probability Mass Function for Rolling Two Dice values of the random variable X, which represents sum of the rolls of two dice X (Die Sum) Frequency f(X) 2 1 0.03 3 2 0.06 4 3 0.08 5 4 0.11 6 5 0.14 7 6 0.17 8 5 0.14 Probability Distribution for Rolling Two Dice 9 4 0.11 0.2 10 3 0.08 0.15 11 2 0.06 0.1 0.05 12 1 0.03 0 Sum 36 1 2 3 4 5 6 7 8 9 10 11 12  Cumulative Distribution Function for Rolling Two Dice X (Die Sum) Prob. Cumulative Frequency Outcome f(x) Prob. F(X) 2 1 0.0278 0.0278 3 2 0.0556 0.0834 = 0.0278+ 0.0556 4 3 0.0833 0.1667 = 0.0834 + 0.0833 5 4 0.1111 0.2778 6 5 0.1389 0.4167 7 6 0.1667 0.5834 8 5 0.1389 0.7223 9 4 0.1111 0.8334 10 3 0.0833 0.9167 11 2 0.0556 0.9723 12 1 0.0278 1.000 Sum 36 1 Example: Simple linear regression Drawing a Scatterplot Let us draw a scatterplot for the hypothetical example 38  Correlation Coefficient § An expression of the strength of Examples of the the linear relationship Correlation Coefficient Always between +1 and –1 Y Y The correlation coefficient is r (a) Perfect Positive X (b) Positive X Correlation: Correlation: r = +1 0