MNG221- Management Science – Decision Analysis MNG221- Management Science –

Learning objectives Categories of decision situation Components of decision making Decision making without probabilities Decision making with probabilities - Expected value - Decision trees - Expected Opportunity Loss - Expected value of Perfect Information (EVPI) Decision analysis with additional information Utility

Categories of Decision Situation A choice among alternatives Decision Uncertainty (probability assigned to future occurrence) Certainty (no probability assigned to future occurrence) Decision situations can be categorized into two classes: Situations in which probabilities cannot be assigned to future occurrences and, Situations in which probabilities can be assigned.

Components of Decisions Making A decision-making situation includes several components: The Decision to be made The Decision Alternatives The States of Nature The Payoff or Outcome The Probability of an Outcome Occurring

Components of Decision Making The Decision – a choice among several alternatives Payoff tables – is a means of organizing a decision situation given various states of nature. Payoff Table Decision State of Nature Probability a b 1 Payoff 1a Payoff 1b 0.5 2 Payoff 2a Payoff 2b Decision-making situation includes several components, the decisions themselves and the actual events that may occur in the future, known as states of nature. At the time a decision is made, the decision maker is uncertain which states of nature will occur in the future and has no control over them. The Decision table shows the payoff (or outcome) of each decision given different conditions.

Components of Decision Making Alternatives – The possible solutions available to solve decision. States of nature – the actual event that may occur in the future. Payoff Table Decision State of Nature Probability a b 1 Payoff 1a Payoff 1b 0.5 2 Payoff 2a Payoff 2b Decision-making situation includes several components, the decisions themselves and the actual events that may occur in the future, known as states of nature. At the time a decision is made, the decision maker is uncertain which states of nature will occur in the future and has no control over them. The Decision table shows the payoff (or outcome) of each decision given different conditions.

Components of Decision Making Payoff or Outcome – is the result of a combination between an alternative and a state of nature. Probabilities – is the likelihood of an event or state if nature occurring. Payoff Table Decision State of Nature Probability a b 1 Payoff 1a Payoff 1b 0.5 2 Payoff 2a Payoff 2b Decision-making situation includes several components, the decisions themselves and the actual events that may occur in the future, known as states of nature. At the time a decision is made, the decision maker is uncertain which states of nature will occur in the future and has no control over them. The Decision table shows the payoff (or outcome) of each decision given different conditions.

Components of Decisions Making Example: Suppose a distribution company is considering purchasing a computer to increase the number of orders it can process and thus increase its business. If economic conditions remain good, the company will realize a large increase in profit; however, if the economy takes a downturn, the company will lose money. The likelihood of each event occurring has a 50-50 chance.

Decisions Making Analysis A situation in which a decision is to be made may be one of the following: Without Probability or the likelihood of occurrence of an event is not known. With Probability or the likelihood of occurrence of an event is known.

Decision Making Without Probabilities Decision Making Analysis Decision Making Without Probabilities

Decision Making Without Probabilities Payoff Table State of Nature Decision (Purchase) Good Economic Conditions Poor Economic Conditions Apartment Building $50,000 $30,000 Office Building $100,000 ($40,000) Warehouse $10,000 An investor is to purchase one of three types of real estate.

Decision Making Without Probabilities Once the decision situation has been organized into a payoff table, several criteria are available for making the actual decision (Decision Criteria): Maximax Criterion Maximin Criterion Minimax Regret Criterion Hurwicz Criterion Equal likelihood Criterion Once the decision situation has been organized into a payoff table, several criteria are available for making the actual decision. These decision criteria, which will be presented in this section, include maximax, maximin, minimax regret, Hurwicz, and equal likelihood. On occasion these criteria will result in the same decision; however, often they will yield different decisions. The decision maker must select the criterion or combination of criteria that best suits his or her needs.

Decision Making Without Probabilities Maximax Criterion – the decision maker is very optimistic about the future of decision situation and therefore selects the decision that will result in the maximum of the maximum payoffs (Good Situation).

Decision Making Without probabilities Maximax Criterion and Costs It should be noted that the maximax decision rule as presented here deals with profit. However, if the payoff table consisted of costs, the opposite selection would be indicated: the minimum of the minimum costs, or a minimin criterion.

Decision Making Without Probabilities Maximin criterion – the decision maker is very pessimistic and therefore selects the decision that will result in the maximum of the minimum payoff (bad situation) .

Decision Making Without Probabilities Maximin Criterion and Costs If the Payoff Table contained costs instead of profits as the payoffs, the conservative approach would be to select the maximum cost for each decision. Then the decision that resulted in the minimum of these costs would be selected.

Decision Making Without Probabilities Minimax Regret Criterion The decision maker attempts to avoid regret by selecting the decision alternative that minimizes the maximum regret. Regret is the difference between the payoff from the best decision and all other decision payoffs.

Decision Making Without Probabilities Minimax Regret Criterion Example – if the investor chooses to purchase a warehouse and good economic conditions occur, the decision will have a regret of $70,000 ($100,000 - $30000) for not having chosen to purchase and office building.

Decision Making Without Probabilities Minimax Regret criterion

Decision Making Without Probabilities Good Economic Conditions 100,000 - 50,000 = 50,000 100,000 - 100,000 = 0 100,000 - 30,000 = 70,000 Poor Economic Conditions 30,000 - 30,000 = 0 30,000 - (40,000) = 70,000 30,000 - 10,000 = 20,000

Decision Making Without Probabilities According to the minimax regret criterion, the decision should be to purchase the apartment building rather than the office building or the warehouse. The investor will experience the least amount of regret by purchasing the apartment building, since if either the office building or the warehouse, $70,000 worth of regret could result; however, the purchase of the apartment building will result in, at most, $50,000 in regret.

Decision Making Without Probabilities The Hurwicz criterion is a compromise between the maximax and maximin criteria where the decision maker is neither totally optimistic (maximax criterion) nor totally pessimistic (maximin criterion). The decision payoffs are weighted by a coefficient of optimism.

Decision Making Without Probabilities The coefficient of optimism - α, is between zero and one (i.e., 0 ≤α≤ 1.0). If α = 1.0 - decision maker is completely optimistic; If α = 0 - decision maker is completely pessimistic. If α coefficient of optimism, then 1 - α is the coefficient of pessimism

Decision Making Without Probabilities The Hurwicz Criterion It multiplies the best payoff by α (the coefficient of optimism) and the worst payoff by 1 – α for each decision. Example: Assume that α = 0.4, and 1 – α = 0.6. Apartment Building $ 50,000(0.4) + $30,000(0.6) = $38,000 Office Building $100,000(0.4) + -$40,000(0.6) = $16,000 Warehouse $30,000(0.4) + $10,000(0.6) = $18,000

Decision Making without Probabilities The Hurwicz Criterion The Hurwicz criterion multiplies the best payoff by α, the coefficient of optimism, and the worst payoff by 1 - α, for each decision, and the best result is selected.

Decision Making Without Probabilities The Equal Likelihood, or LaPlace, Criterion This assumes that the investor is neutral and that the decision payoff of each state of nature is equally likely to occur and as such are weighted equally. Apartment Building $ 50,000(0.5) + $30,000(0.5) = $40,000 Office Building $100,000(0.5) + -$40,000(0.5) = $30,000 Warehouse $30,000(0.5) + $10,000(0.5) = $20,000

Decision Making Without Probabilities Summary of Criteria Results Criterion Decision Maximax Office building Maximin Apartment building Minimax Regret Apartment building Hurwicz Apartment building Equal likelihood Apartment building A dominant decision is one that has a better payoff than another decision under each state of nature

Decision Making With Probabilities Decision Making Analysis Decision Making With Probabilities

Decision Making With Probabilities It is often possible for the decision maker to know enough about the future states of nature to assign probabilities to their occurrence. Given that probabilities can be assigned, the following are decision criteria available to aid the decision maker: Expected Value and, Expected Opportunity Loss among others

Decision Making With Probabilities Expected Value Computed by multiplying each decision outcome under each state of nature by the probability of its occurrence.

Decision Making With Probabilities EV(apartment) =50,000(.60) + $30,000(.40) = $42,000 EV(office) = $100,000(.60) + $40,000(.40) = $44,000 EV(warehouse) = $30,000(.60) + $10,000(.40)= $22,000 Payoff Table State of Nature Decision (Purchase) Good Economic Conditions 0.6 Poor Economic Conditions 0.4 Apartment Building $50,000 $30,000 Office Building $100,000 ($40,000) Warehouse $10,000

Decision Making With Probabilities Expected Opportunity Loss This is the expected value of the regret for each decision. To use this criterion, we multiply the probabilities by the regret (i.e., opportunity loss) for each decision outcome.

Decision Making With Probabilities EOL(apartment) = $50,000(.60) + $0(.40) = $30,000 EOL(office) = $0(.60) + $70,000(.40) = $28,000 EOL(warehouse) = $70,000(.60) + $20,000(.40) = $50,000 Payoff Table State of Nature Decision (Purchase) Good Economic Conditions 0.6 Poor Economic Conditions 0.4 Apartment Building $50,000 $0 Office Building $70,000 Warehouse $20,000

Decision Making With Probabilities Expected value of Perfect Information (EVPI) The expected value of perfect information is the maximum amount a decision maker would pay for additional information. It is equal to the expected value, with/given perfect information (EVWPI), less the expected value without perfect information (EVWOPI or Maximum EMV)

Decision Making With Probabilities EVWPI – If we had perfect information we would select the best ($100,000 & $30,000) of each outcome (Good & Poor Economic Conditions . Therefore the sum of the best outcome of each state of nature will be multiplied by the probability of each state of nature to find EVWPI.

Decision Making With Probabilities EVWOPI or Maximum EMV – Is the decision alternative that we will choose if we didn’t have perfect information. Therefore it is the maximum expected monetary value calculated without perfect information.

Decision Making With Probabilities Expected value of Perfect Information (EVPI) EV(given perfect information) =$100,000(.60) + $30,000(.40) = $72,000 EV(without perfect information)- OFFICE = $100,000(.60) + -40,000(.40) = $44,000 Payoff Table State of Nature Decision (Purchase) Good Economic Conditions 0.6 Poor Economic Conditions 0.4 Apartment Building $50,000 $30,000 Office Building $100,000 ($40,000) Warehouse $10,000

Decision Making With Probabilities Expected value of Perfect Information (EVPI) EVPI= $72,000 - $44,000 = $28,000 The expected value of perfect information equals the expected opportunity loss for the best decision. Payoff Table State of Nature Decision (Purchase) Good Economic Conditions 0.6 Poor Economic Conditions 0.4 Apartment Building $50,000 $30,000 Office Building $100,000 ($40,000) Warehouse $10,000

Decision Making With Probabilities Decision Trees A Decision Tree is a graphical/pictorial diagram of the decision-making process consisting of square decision nodes, circle probability nodes, and branches representing decision alternatives.

Decision making With probabilities Decision Trees This makes it easier to correctly compute the necessary expected values and to understand the process of making the decision. The decision tree represents the sequence of events in a decision situation.

Decision making With probabilities Determining the best decision by using a decision tree is accomplished by starting with the final outcomes (payoffs) and working backward through the decision tree toward node 1. First the expected value is computed at each probability node. Then branches with the greatest expected value are selected.

Decision Making With Probabilities Decision Trees

Decision Making With Probabilities Decision Trees – expected value is computed at each probability node. EV(node 2) = .60($50,000) +.40($30,000) = $42,000 EV(node 3) = .60($100,000) +.40($40,000) = $44,000 EV(node 4) = .60($30,000) +.40($10,000) = $22,000

Decision Making With Probabilities Sequential Decision Trees – illustrates a situation requiring a series of decisions and where a payoff table is not possible.

Decision Making With Probabilities Sequential Decision Trees – Expected value of all nodal values. First, compute the expected values at nodes 6 and 7: EV(node 6) = .80($3,000,000) +.20($700,000) = $2,540,000 EV(node 7) = .30($2,300,000) +.70($1,000,000) = $1,390,000

Decision Making With Probabilities Sequential Decision Trees – Expected value of all nodal values. Deduct relevant cost at decision node 4 & 5 choose best alternative (Node 4) = 2,540,000 - 800,000 = 1,740,000 (Node 5) = 1,390,000 – 600,000 = 790,000

Decision Making With probabilities Sequential Decision Trees – Expected value of all nodal values. Next, compute EV at nodes 2 and 3 EV(node 2) = .60($2,000,000) +.40($225,000) = $1,290,000 EV(node 3) = .60($1,740,000) +.40($790,000) = $1,360,000

Decision Making With probabilities Sequential Decision Trees – Expected value of all nodal values. Select the decision with the greatest expected value after the cost of each decision is subtracted out: Apartment building:$1,290,000 - 800,000 = $490,000 Land: $1,360,000 - 200,000 = $1,160,000

Decision Analysis With Additional Information Decision Making Analysis Decision Analysis With Additional Information

Decision Analysis with Additional Information It is often possible to gain some amount of additional (imperfect) information that will improve decisions. Bayesian Analysis – additional information is used to alter the marginal probability of the occurrence of an event.

Decision Analysis with Additional Information Conditional Probabilities – is the probability that an event will occur, given that another event has already occurred. Example: p(P|g) = 0.80 p(N|g) = 0.20 p(P|p) = 0.10 p(N|p) = 0.90 P(g) = 0.60 P(p) = 0.40, where g = good economic conditions p = poor economic conditions P = positive economic report N = negative economic report

Decision Analysis with Additional Information Posterior Probabilities – is the altered (revised) marginal probability of an event occuring, based on additional information that can be determined by Bayes' rule. The remaining Posterior Probabilities p(g|N) = 0.250 p(p|P) = 0.077 p(p|N) = 0.750

Decision Trees With Posterior Probabilities

Computing Posterior Probabilities with Tables Tables - may be used where the problem is too complex for Bayes’s rule.

Expected Value With Sample Information Expected Value Of Sample Information – is the difference between the expected value with and without additional information. Efficiency Of Sample Information – is the ratio of the expected value of sample information to the expected value of perfect information.

Utility Some decision makers use a decision criterion other than the expected monetary outcome. This alternative criterion is known as Utility Utility – a measure of satisfaction derived from money. Risk takers – people who are willing to take risks. Risk averters – people who are unwilling to take risks.