NotesAtHKU

Analytics is the science of building models that help us make better decisions.

Descriptive analytics: identify patterns in data.
- Clustering
- PCA
- Hypothesis tests
- Visualizations
Predictive analytics: forecast future trends.

Decision making

Six Steps in Decision Making

Clearly define the problem (Goal to achieve)
List the possible alternatives
Identify the possible outcomes or states of nature
List the payoff of each alternative in each state of nature
Select one of the decision theory models
Apply the model and make your decision

Types of decision making environments

Decisions can be made under different conditions:

Certainty: all information is known.
Risk: probabilities of outcomes are known.
Uncertainty: probabilities of outcomes are not known.

Decision making under risk

Expected monetary value

These quantities are used in decision making under risk:

Metric	Shorthand	Formula
Expected Monetary Value	EMV	$\sum(\text{Payoff}\times\text{Probability})$
Expected Value with Perfect Information	EVwPI	$\sum(\text{Best Payoff}\times\text{Probability})$
Expected Value of Perfect Information	EVPI	$EVwPI - \max\{EMV\}$

Our goal is to maximize the $EMV$ under risk.

Example practice

Consider the following example:

State of Nature	Favorable Market (Profit in $)	Unfavorable Market (Profit in $)	EMV ($)
Construct a large plant	200,000	-180,000	-9,000
Construct a small plant	100,000	-20,000	34,000
Do nothing	0	0	0
Probability	0.45	0.55

We want to maximize the $EMV$ under risk. Hence we choose "Construct a small plant".

Given the opportunity, to decide if we should pay a price for additional information, we have to evaulate $P \leq EVPI$ .

Given the price of $P = 65,000$ :

\begin{align*} EVwPI = 0.45 \times 200,000 &= 90,000\\ EVPI = 90,000 - 34,000 &= 56,000\\ P &\gt EVPI \end{align*}

Therefore, we should not pay for additional information.

Expected opportunity loss

Opportunity loss is the difference of payoff between a decision and the alternative best decision.

Metric	Shorthand	Formula
Expected Opportunity Loss	EOL	$\sum\text{Opp. Loss}\times\text{Probability}$

Our goal is to minimize the $EOL$ under risk.

The choice we take based on $\max\{EMV\}$ is the best decision, which the same decision has the $\min\{EOL\}$ as well.
$\min\{EOL\} = EVPI$

Restoring opportunity loss table

To convert the previous table to an opportunity loss table, we can calculate the opportunity loss for each decision and state of nature.

Large-favorable: $200-200=0k$ as self is best
Small-favorable: $200-100=100k$ as large is best
Small-unfavorable: $0-(-20)=20k$ as do nothing is best
etc...

State of Nature	Favorable Market (Opp. Loss in $)	Unfavorable Market (Opp. Loss in $)	EOL ($)
Construct a large plant	0	180,000	99,000
Construct a small plant	100,000	20,000	56,000
Do nothing	200,000	0	90,000
Probability	0.45	0.55

Restoring profit table

Opp. Loss	State 1	State 2
A	5	1
B	0	3
C	6	0
Prob.	0.3	0.7

We can only restore payoff tables if at least one row of a state is given. Let A-1 = 1, C-2 = 4:

Payoff	State 1	State 2
A	1	3
B	6	1
C	0	4

Example practice

The Café buys donuts each day for $40 per carton of 20 dozen donuts. Any cartons not sold are thrown away at the end of the day. If a carton is sold, the total revenue is $60

Daily Demand (Cartons)	Probability	Cumulative Probability
4	0.05	0.05
5	0.15	0.20
6	0.15	0.35
7	0.20	0.55
8	0.25	0.80
9	0.10	0.90
10	0.10	1.00
Total	1.00

Should we reduce the order size $Q$ from 6 to 5? What is the EMV of $Q=5$ ?

We identify the decision to make is the order size: $Q=6,Q=5$ . Then, we construct the monetary payoff table for different states of nature (demand $D$ ) and calculate the Expected Monetary Value.

The payoff is calculated as $D \times 60 - 40 \times Q$ .

Payoff	D = 4	D = 5	D = 6	D = 7	D = 8	D = 9	D = 10	EMV
Q = 7	-40	20	80	140	140	140	140	104
Q = 6	0	60	120	120	120	120	120	105
Q = 5	40	100	100	100	100	100	100	97
Prob.	0.05	0.15	0.15	0.20	0.25	0.10	0.10	1.00

We want to maximize the $EMV$ under risk. Hence we choose $Q=6$ .

If we can only choose between 6 and 7, what is the EVPI?

$EVwPI = 0 * 0.05 + 60 * 0.15 + 120 * 0.15 + 140 * (1-0.05-0.15*2) = 118$

$EVPI = 118 - 105 = 13$

We can also calculate using the two properties of $EOL$ . We know $\max\{EMV\}$ is the best decision, which is the same decision that has the $\min\{EOL\}$ as well.

$EOL(6) = 0 + 20 * (.20+.25+.1+.1) = 13$

Sensitivity analysis

We can let the probability of a state of nature to be $p$ to work out curves for the $EMV$ . Then, we can deduce the ranges of $p$ in which we should take an action.

Consider the following example:

State of Nature	Favorable Market (Profit in $)	Unfavorable Market (Profit in $)	EMV ($1000)
Construct a large plant	200,000	-180,000	200p - 180(1-p)
Construct a small plant	100,000	-20,000	100p - 20(1-p)
Do nothing	0	0	0
Probability	p	1-p

We can plot the $EMV$ as a function of $p$ to find the range of $p$ in which we should take an action, or use an equality to find the changeover point.

Decision trees

We can use a decision tree to visualize the decision-making process.

The following are parts of the tree:

$\square$ Decision nodes: Where decisions are made.
$\bigcirc$ State-of-Nature nodes: Where probabilities are assigned.
$\triangle$ Payoff nodes: Where payoffs are assigned.
Branches: Represent the possible outcomes.

Finding the best decision

The goal of a decision tree is to find the best choice to make. We can backtrack, start by finding the $EMV$ of each $\bigcirc$ leaves. Travelling upwards, pick the decision at $\square$ that maximizes $EMV$ ,and assign that $EMV$ to the $\square$ . Continue until the root node.

Example

Consider the following example tree:

Starting from the leaves:

$EMV_E = EMV_D = 21600\times.05+16800\times.25+12000\times.4+6000\times.25=11580$

For $\square C$ , we choose "Accept Offer" as $14000>11580$ .

$EMV_B = 0.6\times14000+0.4\times11580=13032$

For $\square A$ , we choose "Reject Offer" as $12000<13032$ .

Hence, best decision is to reject John's offer, and accept Vanessa's offer if presented.

Decision Analysis