8.4 Decision Trees & Expected Value Analysis
Key Takeaways
- The decision tree is the seventh improvement tool listed in Body of Knowledge entry III.C, alongside flowcharts, histograms, Pareto charts, scatter diagrams, check sheets, and control charts.
- Squares represent decision nodes the analyst controls, circles represent chance nodes governed by probability, and triangles represent terminal payoff nodes.
- Expected monetary value at a chance node is the sum of each outcome's probability multiplied by its payoff, and probabilities on branches leaving one chance node must total 1.0.
- Trees are solved by rollback — evaluating from the right-hand payoffs back to the initial decision — choosing the higher expected value at every decision node.
- A decision tree makes assumptions explicit and auditable, which is its main advantage over unstructured judgment even when the probability estimates are uncertain.
8.4 Decision Trees & Expected Value Analysis
The CQIA Body of Knowledge lists decision trees as entry III.C.7, the seventh of the improvement tools, and the entry is written at the Apply level: select, interpret, and apply the basic improvement tools. Candidates study the other six intensively and skip this one, which makes it a reliable place to lose an easy point.
1. What a Decision Tree Does
A decision tree is a diagram that maps a sequence of decisions and uncertain outcomes so that alternatives can be compared quantitatively. It is used when three conditions hold at once:
- There is a genuine choice between two or more courses of action.
- The consequences depend on uncertain events you do not control.
- The outcomes can be assigned a value — cost, profit, time, or units.
Typical quality applications: scrap a nonconforming lot or rework it; sort 100% or ship on a sampling plan; invest in a preventive control or absorb the failure cost; qualify a second supplier or accept single-source risk.
2. Notation
| Symbol | Node type | Meaning |
|---|---|---|
| Square ▢ | Decision node | A choice you control. Branches leaving it are alternatives. |
| Circle ○ | Chance node | An uncertain event you do not control. Branches leaving it are possible outcomes, each with a probability. |
| Triangle ◁ | Terminal node | The end of a path; carries the payoff value. |
Two rules govern branches:
- Branches leaving a chance node must be mutually exclusive and collectively exhaustive, so their probabilities sum to 1.0. A chance node whose branches total 0.8 or 1.15 is malformed — a common exam item.
- Branches leaving a decision node carry no probabilities. You are not guessing which one you will pick; you are choosing.
The tree reads left to right in time and is solved right to left.
3. Expected Monetary Value
The value of a chance node is its expected monetary value (EMV) — the probability-weighted average of its outcomes:
where $P_i$ is the probability of outcome $i$ and $V_i$ is its payoff. EMV is a long-run average, not a prediction of what will happen once. A decision with an EMV of −$4,000 does not cost exactly $4,000; it is the average cost if the same decision were repeated many times under the same conditions.
4. Rollback: A Worked Example
A supplier ships a lot of 5,000 components. Incoming inspection has found defects, and the lot's true defect rate is unknown. Two options:
- Option A — 100% sort. Certain cost of $6,000 in labour, catching every defect.
- Option B — ship as-is. No sorting cost. If the lot is actually good (estimated 70% likely), there is no further cost. If the lot is bad (30%), a field failure occurs costing $40,000 in warranty, containment, and customer penalty.
Step 1 — Evaluate the chance node (Option B):
Step 2 — Value the certain branch (Option A): −$6,000.
Step 3 — Roll back to the decision node. Compare −$6,000 against −$12,000. The sort is better by $6,000 in expected value, so the tree recommends sorting.
Step 4 — Find the indifference point. At what probability of a bad lot would the two options be equal?
Below a 15% chance of a bad lot, shipping as-is has the better expected value; above 15%, sorting does. This sensitivity check is often more valuable than the recommendation itself, because it tells you how wrong your probability estimate can be before the decision flips. When the indifference point sits far from your estimate, the decision is robust; when it sits close, invest in a better estimate before committing.
Reading the Value of Information
The same structure quantifies what better data is worth. If a $500 destructive test would tell you with certainty whether the lot is good or bad, the tree shows that perfect information reduces the expected cost from $6,000 to $1,800 — the 30% of cases where sorting is genuinely needed — so information worth $4,200 is available for $500. This is how a decision tree justifies inspection, testing, or a pilot run.
5. Limitations
| Limitation | Consequence |
|---|---|
| Probabilities are usually estimates | Garbage in, garbage out; always run the sensitivity check |
| EMV is risk-neutral | It ignores that a catastrophic outcome may be unsurvivable even at low probability. A recall that bankrupts the firm is not adequately captured by its expected value. |
| Trees grow quickly | More than three or four sequential decisions becomes unreadable |
| Non-monetary factors are excluded | Safety, regulatory exposure, and reputation resist dollar values |
The risk-neutrality limitation is the important one for a quality professional. Where a branch involves patient harm, regulatory violation, or a safety recall, expected value is the wrong decision rule — that branch must be eliminated regardless of its probability. This is the same logic that makes a severity rating of 9 or 10 in an FMEA require action irrespective of the risk priority number.
6. The Classification-Tree Variant
The term also describes a classification or logic tree used for structured troubleshooting: a branching sequence of yes/no questions that routes a symptom to a cause. Quality applications include defect disposition flows, alarm response guides, and nonconformance classification aids.
Distinguish it from its neighbours, which the exam pairs as distractors:
| Tool | Purpose |
|---|---|
| Decision tree | Compare alternatives under uncertainty using probabilities and payoffs |
| Tree diagram (section 8.1) | Decompose a goal or objective into progressively finer sub-tasks; no probabilities |
| PDPC (section 8.1) | Anticipate what could go wrong in a plan and pre-plan countermeasures |
| Flowchart | Document the sequence of an existing process as it operates |
On a decision tree, what do the square and circle symbols represent, and what rule governs the branches leaving each?
A team must choose between a certain $6,000 sorting cost and shipping a lot as-is, where there is a 25% chance of a $40,000 field failure and a 75% chance of no cost. Which option has the better expected monetary value, and by how much?
A decision tree analysis of a potential product defect shows that continuing to ship has a better expected monetary value than initiating a recall, because the probability of a serious injury is low. Why is expected monetary value the wrong decision rule here?
How does a decision tree differ from the tree diagram covered among the seven management and planning tools?