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DECISION MAKING USING PROBABILITIES I IEng154 – OPERATIONS RESEARCH 2
Introduction to decision making
Most decisions managers make involve some uncertainty. Specific tasks and knowledge required by managers before quantitative tools can be used effectively include: specification of organizational goals; knowledge about possible actions, possible likelihoods, expected payoffs, and specific criterion (or criteria) on which choice will be based.
Steps in decision making 1. 2.
3.
List all viable alternatives. Identify the future events that may occur. Construct a payoff table.
Example (record and tape company) Viable options:
1. 1.
2. 3.
Expand the plant. Build a new plant. Subcontract out extra production to other record and tape manufacturers.
Future events (states of nature)
2. 1. 2. 3.
4.
High demand Moderate demand Low demand Failure
3.
Payoff table – a table which shows the payoffs (expressed in profits or any other measure of benefit which is appropriate to the situation) which would result from each possible combination of decision alternative and states of nature. Decision maker’s alternatives
States of nature (demand)
Expand
Build
Subcontrac t
High
$500,000
$700,000
$300,000
Moderate
$250,000
$300,00
$150,000
Low
-$250,000
-$400,000
-$10,000
Failure -$450,000 -$800,000 -$100,000 Payoff table for record-tape company expansion decision (payoffs expressed in profits over next 5 years)
Criteria for decision making under uncertainty 1.
Maximax criterion (optimistic criterion) (choose alternative which would maximize his maximum payoff) Decision maker’s alternatives
States of nature (demand)
Expand
Build
Subcontrac t
High
$500,000
$700,000
$300,000
Moderate
$250,000
$300,00
$150,000
Low
-$250,000
-$400,000
-$10,000
Failure
-$450,000
-$800,000
-$100,000
Different decision environments in which decisions are made: 1.
2.
3.
Decision making under conditions of certainty. (only one state in nature exists; that is, complete certainty about the future) Decision making under conditions on uncertainty. (more than one state of nature exists, but the decision maker has no knowledge about the various state, not even sufficient knowledge to permit the assignment of probabilities to the states of nature) Decision making under conditions of risk. (more than one state on nature exists, but now the decision maker has information which will the assignment of probability values to each of the possible states.
Criteria for decision making under uncertainty 2.
Maximin criterion (pessimistic criterion) (choose alternative which would maximize his minimum possible payoff; choose the best of the worst) Decision maker’s alternatives
States of nature (demand)
Expand
Build
Subcontrac t
High
$500,000
$700,000
$300,000
Moderate
$250,000
$300,00
$150,000
Low
-$250,000
-$400,000
-$10,000
Failure
-$450,000
-$800,000
-$100,000
Criteria for decision making under uncertainty 2.
Minimax regret criterion (choose alternative which has the minimum of the maximum regret) Decision maker’s alternatives
States of nature (demand)
Expand
Build
Subcontrac t
High
$500,000
$700,000
$300,000
Moderate
$250,000
$300,00
$150,000
Low
-$250,000
-$400,000
-$10,000
Failure
-$450,000
-$800,000
-$100,000
Criteria for decision making under uncertainty 2.
Criterion of Realism (a middle-ground criterion) Index of optimism, α = between 0 and 1 in value.
Decision maker’s alternatives
States of nature (demand)
Expand
Build
Subcontrac t
High
$500,000
$700,000
$300,000
Moderate
$250,000
$300,00
$150,000
Low
-$250,000
-$400,000
-$10,000
Failure
-$450,000
-$800,000
-$100,000
Decision making under conditions of risk: discrete random variables 1. 2.
3.
Expected value criterion (criterion of Bayes) Criterion of rationality (principle of insufficient reason) Criterion of maximum likelihood
Expected value criterion
This criterion asks the decision maker to calculate the expected value for each decision alternative (sum of the weighted payoffs for that alternative, where the weights are the probability values assigned by the decision maker to the states of nature that can happen).
Example
Beth Perry sells strawberries in a market environment where “tomorrow’s demand for strawberries” is a discrete random variable. Beth purchases strawberries for $3 a case and sells them for $8 a case. The rather high markup reflects the perishability of the item and the great risk of stocking it; the product has no value after the first day it is offered for sale. Beth faces the problem of how many to order today for tomorrow’s business. A 90-day observation of past demand gives the information shown below. Daily demand
No. of days demanded
Probability of each level of demand
10
18
0.20
11
36
0.40
12
27
0.30
13
9
0.10
90
1.00
Solution
Beth uses a three-step procedure: 1.
2.
3.
Determine the conditional profits associated with each possible combination of supply and demand. Find the expected profit resulting from each possible stock level. Choose the stock level with the highest expected profit.
Calculating conditional profit
Conditional profit table Possible stock actions Market size, cases 10 11 12 13
10 cases 11 cases 12 cases 13 cases
Determining expected profits
Weighting each possible value the variable could take by the probability of its taking on that value. Expected profit for stocking 10 cases: $50.00 Expected profit for stocking 11 cases: $53.40 Expected profit for stocking 12 cases: $53.60 Expected profit for stocking 13 cases: $51.40
Expected profit with perfect information
Complete and accurate information about the future, referred to as perfect information, would remove all uncertainty from the problem. Beth would know in advance how many cases were going to be called for each day. She buys and stocks so as to avoid all loses from obsolete stock as well as all opportunity losses which reflect lost profits on unfilled requests for merchandise. The expected profit with perfect information is the maximum profit possible.
Alternative approach: minimizing losses
Choose that course of action which will minimize the expected value of these reductions or losses. Two types of losses: 1. 2.
Obsolescence losses – due to stocking too many. Opportunity losses – due to being out of stock when buyers want to buy.
Calculating conditional loss
Conditional loss table Possible stock actions Market size, cases 10 11 12 13
10 cases 11 cases 12 cases 13 cases
Determining expected loss
Weighting each possible value the variable could take by the probability of its taking on that value.
Expected loss for stocking 10 cases: $6.50 Expected loss for stocking 11 cases: $3.10 Expected loss for stocking 12 cases: $2.90 Expected loss for stocking 13 cases: $5.10
The optimal stock action is the one which will minimize expected losses.
Expected value of perfect information
EVPI (expected value of perfect information)
the difference between the expected profit with perfect information and the best expected daily profit without the predictor. The maximum amount you would be willing to pay, per day, for a perfect predictor because that is the maximum amount by which you can increase your expected daily profit.
Items which have salvage value
Any time the number stocked is more than the demand on the selling day, the computation of conditional profit must take salvage value into consideration.
Example:
Consider the case of fresh blueberries which the retailer orders and receives on the day before the selling day. They cost $5 per case and sell for $8 per case; any remaining unsold at the end of the day can be disposed of at a salvage price of $2 per case. Observation shows that past demand has ranged from 15 to 18 cases per day; there is no reason to believe that level of demand will take on any value in the future.
Conditional profit
Conditional profit for items with salvage value Possible stock actions
15 16
Probabilit y of market size 0.10 0.20
17 18
0.40 0.30
Market size, cases
15 cases
16 cases
17 cases
18 cases
Expected profit table
Expected profit for items with salvage value Possible stock actions
Market size, cases
Probabilit y of
market size
15
0.10
16 17 18
0.20 0.40 0.30
15 case s
Expec ted profit
16 cas es
17 cas es
18 cas es
Seatwork 1.
Benny Weston is famous for his hot sausage sandwiches which he sells each year at the Oklahoma State Fair. For each kilogram of sausage which he stocks and sells, his profit is 65 cents. If he stocks more sausages than he can sell, he can still recover from a complete loss by selling any unsold meat to the Canned Dogfood Company for a net loss of 15 cents per kilogram. He estimates demand in kilogram to be 10% of the total attendance during the week of the fair. This year’sAttendance attendance isProbability estimated to be as follows: 60,000
0.20
80,000
0.60
100,000
0.20
Construct a table of conditional losses. Determine the number of kilograms to stock. What is the expected value of perfect information for fair attendance?
Introduction to decision making
Most decisions managers make involve some uncertainty. Specific tasks and knowledge required by managers before quantitative tools can be used effectively include: specification of organizational goals; knowledge about possible actions, possible likelihoods, expected payoffs, and specific criterion (or criteria) on which choice will be based.
Steps in decision making 1. 2.
3.
List all viable alternatives. Identify the future events that may occur. Construct a payoff table.
Example (record and tape company) Viable options:
1. 1.
2. 3.
Expand the plant. Build a new plant. Subcontract out extra production to other record and tape manufacturers.
Future events (states of nature)
2. 1. 2. 3.
4.
High demand Moderate demand Low demand Failure
3.
Payoff table – a table which shows the payoffs (expressed in profits or any other measure of benefit which is appropriate to the situation) which would result from each possible combination of decision alternative and states of nature. Decision maker’s alternatives
States of nature (demand)
Expand
Build
Subcontrac t
High
$500,000
$700,000
$300,000
Moderate
$250,000
$300,00
$150,000
Low
-$250,000
-$400,000
-$10,000
Failure -$450,000 -$800,000 -$100,000 Payoff table for record-tape company expansion decision (payoffs expressed in profits over next 5 years)
Criteria for decision making under uncertainty 1.
Maximax criterion (optimistic criterion) (choose alternative which would maximize his maximum payoff) Decision maker’s alternatives
States of nature (demand)
Expand
Build
Subcontrac t
High
$500,000
$700,000
$300,000
Moderate
$250,000
$300,00
$150,000
Low
-$250,000
-$400,000
-$10,000
Failure
-$450,000
-$800,000
-$100,000
Different decision environments in which decisions are made: 1.
2.
3.
Decision making under conditions of certainty. (only one state in nature exists; that is, complete certainty about the future) Decision making under conditions on uncertainty. (more than one state of nature exists, but the decision maker has no knowledge about the various state, not even sufficient knowledge to permit the assignment of probabilities to the states of nature) Decision making under conditions of risk. (more than one state on nature exists, but now the decision maker has information which will the assignment of probability values to each of the possible states.
Criteria for decision making under uncertainty 2.
Maximin criterion (pessimistic criterion) (choose alternative which would maximize his minimum possible payoff; choose the best of the worst) Decision maker’s alternatives
States of nature (demand)
Expand
Build
Subcontrac t
High
$500,000
$700,000
$300,000
Moderate
$250,000
$300,00
$150,000
Low
-$250,000
-$400,000
-$10,000
Failure
-$450,000
-$800,000
-$100,000
Criteria for decision making under uncertainty 2.
Minimax regret criterion (choose alternative which has the minimum of the maximum regret) Decision maker’s alternatives
States of nature (demand)
Expand
Build
Subcontrac t
High
$500,000
$700,000
$300,000
Moderate
$250,000
$300,00
$150,000
Low
-$250,000
-$400,000
-$10,000
Failure
-$450,000
-$800,000
-$100,000
Criteria for decision making under uncertainty 2.
Criterion of Realism (a middle-ground criterion) Index of optimism, α = between 0 and 1 in value.
Decision maker’s alternatives
States of nature (demand)
Expand
Build
Subcontrac t
High
$500,000
$700,000
$300,000
Moderate
$250,000
$300,00
$150,000
Low
-$250,000
-$400,000
-$10,000
Failure
-$450,000
-$800,000
-$100,000
Decision making under conditions of risk: discrete random variables 1. 2.
3.
Expected value criterion (criterion of Bayes) Criterion of rationality (principle of insufficient reason) Criterion of maximum likelihood
Expected value criterion
This criterion asks the decision maker to calculate the expected value for each decision alternative (sum of the weighted payoffs for that alternative, where the weights are the probability values assigned by the decision maker to the states of nature that can happen).
Example
Beth Perry sells strawberries in a market environment where “tomorrow’s demand for strawberries” is a discrete random variable. Beth purchases strawberries for $3 a case and sells them for $8 a case. The rather high markup reflects the perishability of the item and the great risk of stocking it; the product has no value after the first day it is offered for sale. Beth faces the problem of how many to order today for tomorrow’s business. A 90-day observation of past demand gives the information shown below. Daily demand
No. of days demanded
Probability of each level of demand
10
18
0.20
11
36
0.40
12
27
0.30
13
9
0.10
90
1.00
Solution
Beth uses a three-step procedure: 1.
2.
3.
Determine the conditional profits associated with each possible combination of supply and demand. Find the expected profit resulting from each possible stock level. Choose the stock level with the highest expected profit.
Calculating conditional profit
Conditional profit table Possible stock actions Market size, cases 10 11 12 13
10 cases 11 cases 12 cases 13 cases
Determining expected profits
Weighting each possible value the variable could take by the probability of its taking on that value. Expected profit for stocking 10 cases: $50.00 Expected profit for stocking 11 cases: $53.40 Expected profit for stocking 12 cases: $53.60 Expected profit for stocking 13 cases: $51.40
Expected profit with perfect information
Complete and accurate information about the future, referred to as perfect information, would remove all uncertainty from the problem. Beth would know in advance how many cases were going to be called for each day. She buys and stocks so as to avoid all loses from obsolete stock as well as all opportunity losses which reflect lost profits on unfilled requests for merchandise. The expected profit with perfect information is the maximum profit possible.
Alternative approach: minimizing losses
Choose that course of action which will minimize the expected value of these reductions or losses. Two types of losses: 1. 2.
Obsolescence losses – due to stocking too many. Opportunity losses – due to being out of stock when buyers want to buy.
Calculating conditional loss
Conditional loss table Possible stock actions Market size, cases 10 11 12 13
10 cases 11 cases 12 cases 13 cases
Determining expected loss
Weighting each possible value the variable could take by the probability of its taking on that value.
Expected loss for stocking 10 cases: $6.50 Expected loss for stocking 11 cases: $3.10 Expected loss for stocking 12 cases: $2.90 Expected loss for stocking 13 cases: $5.10
The optimal stock action is the one which will minimize expected losses.
Expected value of perfect information
EVPI (expected value of perfect information)
the difference between the expected profit with perfect information and the best expected daily profit without the predictor. The maximum amount you would be willing to pay, per day, for a perfect predictor because that is the maximum amount by which you can increase your expected daily profit.
Items which have salvage value
Any time the number stocked is more than the demand on the selling day, the computation of conditional profit must take salvage value into consideration.
Example:
Consider the case of fresh blueberries which the retailer orders and receives on the day before the selling day. They cost $5 per case and sell for $8 per case; any remaining unsold at the end of the day can be disposed of at a salvage price of $2 per case. Observation shows that past demand has ranged from 15 to 18 cases per day; there is no reason to believe that level of demand will take on any value in the future.
Conditional profit
Conditional profit for items with salvage value Possible stock actions
15 16
Probabilit y of market size 0.10 0.20
17 18
0.40 0.30
Market size, cases
15 cases
16 cases
17 cases
18 cases
Expected profit table
Expected profit for items with salvage value Possible stock actions
Market size, cases
Probabilit y of
market size
15
0.10
16 17 18
0.20 0.40 0.30
15 case s
Expec ted profit
16 cas es
17 cas es
18 cas es
Seatwork 1.
Benny Weston is famous for his hot sausage sandwiches which he sells each year at the Oklahoma State Fair. For each kilogram of sausage which he stocks and sells, his profit is 65 cents. If he stocks more sausages than he can sell, he can still recover from a complete loss by selling any unsold meat to the Canned Dogfood Company for a net loss of 15 cents per kilogram. He estimates demand in kilogram to be 10% of the total attendance during the week of the fair. This year’sAttendance attendance isProbability estimated to be as follows: 60,000
0.20
80,000
0.60
100,000
0.20
Construct a table of conditional losses. Determine the number of kilograms to stock. What is the expected value of perfect information for fair attendance?
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