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### Decision Analysis

Chapter 13

Introduction to Decision Analysis

- Used to develop an optimal strategy, when decision maker is faced with several alternatives
- And
- An uncertain/ risk filed pattern of future events
- Examples
- Control equipment for coal fired generating units & uncertain future regarding sulphur content, construction costs etc

Step 1: Problem Formulation

- Verbal statement of the problem
- Identify the decision alternatives,
- the uncertain future events, aka as chance events, and
- The consequences associated with each decision alternative & each chance event outcome

Example 1

- The Pittsburgh Development Corporation (PDC) purchased land that will be the site of a new luxury condominium complex. The location provides a spectacular view of downtown Pittsburgh and the golden triangle where Allegheny and Monogahela rivers meet to for the Ohio river. PDC plans to price the individual units between $300,000 and $1,400,000.PDC commissioned preliminary architectural drawings for three different projects:
- One with 30 condominiums, one with 60 and another with 90 condominium units

- The financial success of the project depends on the size of the size of the complex and the chance event concerning the demand for the condominiums

Objective: select the best size for PDC’s condominium complex

Decision alternatives

- D1 = a small complex with 30 condominiums
- D2= a medium complex with 60 condominiums
- D3 = a large complex with 90 condominiums
Select the best decision - influenced by chance events concerning the demand for the condominiums, known as States of Nature. For PDC these are:

- S1 = Strong demand for the condominiums
- S2 = Weak demand for the condominiums
The Consequence is PDC’s profit

- A graphical device showing the relationships among the decisions, the consequences for a decision problem
Payoff Tables

- A table showing payoffs for all combinations of decision alternatives and states of nature
- Payoffs can be expressed in terms of profit, cost time, distance, or any other measure appropriate for the decision problem being analysed
Decision Trees

- Provides a graphical representation of teh decision making process

Decision Making Without Probabilities

Optimistic Approach

Evaluates each decision in terms of the best payoff

Conservative Approach

Evaluates each decision alternative in terms of the worst payoff that can occur

Minimax Regret Approach

- Neither purely optimistic nor purely conservative
- PDC constructs a small condo complex & demand is actually strong
Profit is $8million. However would have been $20 if gone with large complex, & demand is strong

Thus opportunity lost or regret associated with decision alternative is 20-8= $12million

Rij = | Vj* - Vij|

Where:

Rij = regret associated with decision alternative di & state of nature sj

Vj* = payoff value corresponding to best decision for state of nature sj

Vij = payoff corresponding to decision alternative di & state of nature sj

Note:

Opportunity loss or regret table for the PDC condominium Project ($millions) select the min of the maximum regret values

Expected Value Approach

Remember: trying to quantify the options open to management & help come to a decision, where options have values e.g. profit / contribution etc., as well as probabilities, the concept of expected value (EV) is often used.

Let:

N = the number of state of nature or Outcomes

P(sj)= probability of the state of nature sj or outcome of an event

P(sj) ≥ 0 for all states of nature / outcomes

∑ P(sj) = P(s1) +P(s2) +P(s3) + ………. + P(sN) = 1

EV= expected number of times that this outcome will occur in ‘N’ events = N x p.

or EV (di) = ∑ P(sj) Vij

EV = its probability times the outcome or value of the event over a series of trials. It is a weighted average based on probabilities.

Example 1: calculate daily sales of Product ‘T’

Units Probability EV .

1,000 0.2 200

2,000 0.3 600

3,000 0.4 1200

4,000 0.1 400 .

1.0 2,400 = EV of daily sales

Step 1 Calculate the EV of the project(s), which in the long run should approximate actual average of the event many times over.

In example 1 we do not expect the sales on any one day to be 2,400 units, but in the long run, over a large number of days, the average sales would be equal to 2,400 units per day.

Advantages of expected value:

- Simple to understand and calculated.
- Represents whole distribution by a single figure.
- Arithmetically takes account of the expected variability of all outcomes.
Disadvantages:

- Single figure ==> that the characteristics of the distribution are being ignored.
- Makes the assumption that the decision maker is neutral

Decision Making and Expected Values -- Which project should be chosen?

Rules

- A project with a positive EV should be accepted
- A project with a negative EV should be rejected.
- Choose an option or alternative which has the highest EV of profit (or the lowest EV of cost).
Example 2:

Project AProject B

Probability Profit EVProbability Profit EV___

0.8 5,000 4000 0.1 (2,000) (200)

0.2 6000 1200 0.25000 1000

£5,200 0.6 7000 4200

0.1 8000 800

£6,000

Solution:

Project B has a higher EV of profit. This means that on balance of probabilities it could offer a better return than A and is so arguably a better choice. On the other hand the minimum return from project A would be £5000. In addition with project B there is a small chance of making a loss.

NOTES: Although it appears to be widely used for the purpose , the concept of EV is not particularly well suited to one off decisions. EV can strictly only be interpreted as the value that would be obtained if a large number of similar decisions were taken with the same ranges of outcomes and associated probabilities.

Example 3:

A distributor buys perishable articles for £2 per item and sells them at £5. Demand per day is uncertain and items unsold at the end of the day represent a write off because of perishiability. If he under stocks he loses profit he could have made.

Daily Demand (units) No. of Days p

10 30 0.1

11 60 0.2

12 120 0.4

13 900.3

300 1.0

What stock level should be held from day to day?

Conditional Profit calculation , CP = (10 x £5) - (10 x £2) = £24

(CP = £3 per unit) units demanded units bought

Expected profit, EP = CP x probability of the demand

Sales price = £5 per unitPurchase price = £2 per unit

The optimum stock position given the pattern of demand is: to stock 12 units per day

Expected Value of Perfect Information to stock 12 units per day

IF PDC knew for certain that the state of nature S1 (strong demand) would occur then the best alternative would be d3, with a payoff of $20million

If S1, select d3 and receive a payoff of $20 million

If S2, select d1 and receive a payoff of $8 million

EV to stock 12 units per day with Perfect Information (EVwPI)

- Based on the above information there is a 0.8 probability that the perfect information will indicate a state of nature s1 and the resulting decision d3 will provide a $20 million profit
- Similarly, with a 0.2 probability for the state of nature s2, the optimal decision alternative d1 will provide a $7million profit
- EV of decision strategy that uses perfect information is 0.8 (20) + .02 (7) = $17.4

EV to stock 12 units per day without Perfect Information (EVwoPI)

- Earlier recommended decision using the EV approach was d3, with an EV of $14.2
- (0.8 x 20) + (0.2 x -9) = $14.2

Expected Value of Perfect Information to stock 12 units per day(EVPI)

EVPI = |EVwPI – EVwoPI |

- EVxPI of $17.4 and the EVwoPI is $14.2; therefore the EV of the PI is $3.2 (17.4-14.2)
- In other words $3.2million represents the additional value that can be obtained if perfect information were available about the states of nature
- Real life; PI generally not available
- But for PDC maybe there is some merit in conducting a market survey to establish better the state of demand in the market

Note: to stock 12 units per dayregardless of whether the decision analysis involves maximisation or minimisation, the minimum expected opportunity loss always provides the best decision alternativeIn addition, the minimum expected opportunity loss is always equal to the EVPI

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