MNG221- Management Science –

MNG221- Management Science –

DECISION ANALYSIS

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

Decision

Certainty(no probability assigned

to future occurrence)

Uncertainty(probability assigned to future occurrence)

Decision situations can be categorized into two classes: 1. Situations in which probabilities cannot be assigned to

future occurrences and, 2. Situations in which probabilities can be assigned.

A choice among

alternatives

Components of Decisions Making• A decision-making situation includes

several components:1. The Decision to be made2. The Decision Alternatives 3. The States of Nature4. The Payoff or Outcome5. 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

Probabilitya b

1 Payoff 1a Payoff 1b 0.5


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

Probabilitya b



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

Probabilitya b



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:1. Without Probability or the likelihood

of occurrence of an event is not known.

2. With Probability or the likelihood of occurrence of an event is known.

Decision Making Without ProbabilitiesDecision Making Analysis

Decision Making Without Probabilities

An investor is to purchase one of three types of real estate.

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 $30,000 $10,000


Once the decision situation has been organized into a payoff table, several criteria are available for making the actual decision (Decision Criteria): 1. Maximax Criterion2. Maximin Criterion3. Minimax Regret Criterion4. Hurwicz Criterion5. Equal likelihood Criterion


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.


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) .


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.


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.


Minimax Regret CriterionExample – 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 ProbabilitiesMinimax Regret criterion



100,000 - 50,000 = 50,000100,000 - 100,000 = 0100,000 - 30,000 = 70,000


30,000 - 30,000 = 030,000 - (40,000) = 70,00030,000 - 10,000 = 20,000


• 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.


• 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.

mk:@MSITStore:D:/TJW%20Docs/UG%20Teaching/MNG221/New%20Lecture%20Notes/Introduction%20to%20Management%20Science%209th%20Edition%20-%20Prentice%20Hall.chm::/0131737961/gloss01__gdh.html

mk:@MSITStore:D:/TJW%20Docs/UG%20Teaching/MNG221/New%20Lecture%20Notes/Introduction%20to%20Management%20Science%209th%20Edition%20-%20Prentice%20Hall.chm::/0131737961/gloss01__gdh.html

mk:@MSITStore:D:/TJW%20Docs/UG%20Teaching/MNG221/New%20Lecture%20Notes/Introduction%20to%20Management%20Science%209th%20Edition%20-%20Prentice%20Hall.chm::/0131737961/gloss01__gdc.html


• The coefficient of optimism - α, is between zero and one (i.e., 0 ≤α≤ 1.0).

o If α = 1.0 - decision maker is completely optimistic; o If α = 0 - decision maker is completely pessimistic. o If α coefficient of optimism, theno 1 - α is the coefficient of pessimism


• 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) = $38,000Office 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 CriterionThe 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.


The Equal Likelihood, or LaPlace, CriterionThis 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) = $40,000Office Building $100,000(0.5) + -$40,000(0.5) = $30,000 Warehouse $30,000(0.5) + $10,000(0.5) = $20,000

mk:@MSITStore:D:/TJW%20Docs/UG%20Teaching/MNG221/New%20Lecture%20Notes/Introduction%20to%20Management%20Science%209th%20Edition%20-%20Prentice%20Hall.chm::/0131737961/gloss01__gde.html

mk:@MSITStore:D:/TJW%20Docs/UG%20Teaching/MNG221/New%20Lecture%20Notes/Introduction%20to%20Management%20Science%209th%20Edition%20-%20Prentice%20Hall.chm::/0131737961/gloss01__gdl.html

mk:@MSITStore:D:/TJW%20Docs/UG%20Teaching/MNG221/New%20Lecture%20Notes/Introduction%20to%20Management%20Science%209th%20Edition%20-%20Prentice%20Hall.chm::/0131737961/gloss01__gdl.html


Summary of Criteria ResultsCriterion DecisionMaximax 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 ProbabilitiesDecision Making Analysis

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:1. Expected Value and, 2. Expected Opportunity Loss among others


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


EV(apartment) =50,000(.60) + $30,000(.40) = $42,000EV(office) = $100,000(.60) + $40,000(.40) = $44,000EV(warehouse) = $30,000(.60) + $10,000(.40)= $22,000

Payoff TableState of Nature

Decision(Purchase)


0.6


0.4

Apartment Building $50,000 $30,000Office Building $100,000 ($40,000)Warehouse $30,000 $10,000


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.


EOL(apartment) = $50,000(.60) + $0(.40) = $30,000EOL(office) = $0(.60) + $70,000(.40) = $28,000EOL(warehouse) = $70,000(.60) + $20,000(.40) = $50,000

Payoff TableState of Nature

Decision(Purchase)


0.6


0.4

Apartment Building $50,000 $0Office Building $0 $70,000Warehouse $70,000 $20,000


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)


• 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.


• 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.



EV(given perfect information) =$100,000(.60) + $30,000(.40) = $72,000EV(without perfect information)- OFFICE= $100,000(.60) + -40,000(.40) = $44,000

Payoff Table

State of Nature

Decision(Purchase)


0.6


0.4

Apartment Building

$50,000 $30,000

Office Building $100,000 ($40,000)

Warehouse $30,000 $10,000



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)


0.6


0.4

Apartment Building

$50,000 $30,000

Office Building $100,000 ($40,000)

Warehouse $30,000 $10,000


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 Trees


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


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


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) = $2,540,000EV(node 7) = .30($2,300,000) +.70($1,000,000) = $1,390,000


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 3EV(node 2) = .60($2,000,000) +.40($225,000) = $1,290,000EV(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 = $490,000Land: $1,360,000 - 200,000 = $1,160,000

Decision Analysis With Additional Information

Decision Making Analysis

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.


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) = 0.20p(P|p) = 0.10

p(N|p) = 0.90

P(g) = 0.60P(p) = 0.40, whereg = good economic conditionsp = poor economic conditionsP = positive economic reportN = negative economic report


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 Probabilitiesp(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. State of Nature

Decision NO ACCIDENT .992 ACCIDENT .008

Purchase insurance $500 $ 500

Do not purchase insurance 0 10,000

MNG221- Management Science –

Documents

Transcript of MNG221- Management Science –