Prescriptive Analytics and Decision Models

Introduction and context

Descriptive analytics asks what happened, diagnostic analytics asks why it happened, and predictive analytics asks what is likely to happen. Prescriptive analytics goes one step further: it recommends what an organisation should do, given objectives, constraints, available resources and uncertainty. It connects data analysis with managerial action.

A prescriptive recommendation normally contains four components:

1. Decision variables — choices controlled by the decision-maker.
2. Objective — the outcome to maximise or minimise, such as profit, service level, cost or risk.
3. Constraints — limits on money, labour, capacity, time, policy or regulation.
4. Information and uncertainty — forecasts, probabilities, scenarios and assumptions used to evaluate alternatives.

Prescriptive analytics does not replace managerial judgement. It structures choices, makes assumptions visible and enables alternatives to be compared consistently.

Learning objectives

By the end of this chapter, you should be able to:

• distinguish descriptive, predictive and prescriptive analytics;
• construct and interpret decision trees;
• perform what-if and scenario analysis;
• explain the role of simulation in uncertain decision environments;
• formulate and solve a simple linear optimisation model;
• compare alternatives using expected value, risk and constraints;
• communicate recommendations responsibly, including assumptions and limitations.

From prediction to action

A forecast is not itself a decision. For example, a retailer may forecast high demand for a product, but still need to decide how many units to order. Ordering too many creates holding costs and waste; ordering too few creates lost sales. Prescriptive analysis combines the demand forecast with costs, capacity and organisational priorities.

A useful general representation is:

Here, x represents one or more controllable decisions. The objective may be profit maximisation or cost minimisation. A model is useful only when its data, assumptions and decision scope are credible.

Decision environments

Decisions can be made under:

• certainty, when relevant outcomes are known;
• risk, when several outcomes are possible and probabilities can reasonably be assigned;
• uncertainty, when outcomes are possible but reliable probabilities are unavailable.

The same recommendation can change when the environment changes. Consequently, analysts should report not only a preferred option but also the conditions under which that option remains preferable.

Decision trees

A decision tree is a graphical or tabular model of sequential choices and uncertain events. A decision node represents a controllable choice, usually shown as a square in conventional diagrams. A chance node represents an uncertain event, usually shown as a circle. Terminal branches show outcomes such as profit, cost or customer response.

Decision trees are particularly useful when:

• decisions occur in stages;
• later decisions depend on earlier results;
• a small number of scenarios can represent uncertainty;
• probabilities and payoffs can be estimated.

Expected monetary value

When outcomes are represented by monetary payoffs and probabilities, the expected monetary value (EMV) of an alternative is:

where pᵢ is the probability of outcome i, Vᵢ is its monetary value, and ∑ᵢpᵢ = 1. Probabilities must be non-negative and measured over mutually exclusive, collectively exhaustive outcomes. EMV is a long-run average criterion; it does not guarantee the result in any single case.

For a decision node, the analyst selects the branch with the highest EMV when maximising value, or the lowest EMV when the quantities are costs to be minimised. This process is called rollback analysis: calculate values from the terminal outcomes back toward the initial decision.

Worked example: product launch

A firm can launch a product immediately or conduct a market test. A successful launch yields $120,000 and an unsuccessful launch yields −$50,000. The probability of success is 0.60.

For immediate launch:

The market test costs $10,000. Suppose a favourable test leads to a 0.75 probability of success and occurs with probability 0.60. An unfavourable test occurs with probability 0.40 and leads to a 0.25 probability of success. If the firm launches after either test result, its expected operating payoff is:

After subtracting the test cost, the test-and-launch strategy has value $31,000. In this simplified design, immediate launch has the higher EMV ($52,000). The test could still be justified for non-monetary reasons, such as learning, reputation protection or risk reduction, but those benefits would need to be represented explicitly.

A common error is to multiply probabilities incorrectly or to count the test cost more than once. Each complete path probability must be the product of probabilities along that path, and all relevant costs must be included exactly once.

Expected value of sample information

Information can have economic value because it may change a decision. The expected value of perfect information (EVPI) is:

EVPI is non-negative under consistent calculations. It provides an upper limit on what an organisation should pay for perfect information. Real market research is imperfect, so its value cannot exceed EVPI and is usually lower.

What-if analysis

What-if analysis changes one input at a time, or a small group of inputs, to observe the effect on an output. Typical questions include:

• What happens to profit if price falls by 5%?
• How many staff are needed if demand rises by 15%?
• What is the break-even sales volume if fixed cost increases?

For a simple profit model:

The variables must use consistent units: for example, price and costs in dollars per unit, quantity in units and fixed cost in dollars.

Break-even analysis

If P is selling price per unit, v is variable cost per unit and F is fixed cost, the break-even quantity is:

This formula assumes constant unit price and variable cost, a relevant output range in which fixed costs remain fixed, and all produced units can be sold. The denominator is the contribution margin per unit and must be positive. If P ≤ v, selling additional units does not cover variable cost and ordinary break-even analysis is not meaningful.

For example, with F = $30,000, P = $50 and v = $30:

A sensitivity table can show how the answer changes when price, volume or cost changes. One-way analysis is easy to communicate, but it may be misleading when inputs are correlated. For example, a price reduction may increase quantity demanded rather than leave quantity fixed.

Scenario analysis

Scenario analysis evaluates a coherent set of assumptions rather than changing one variable in isolation. A retail organisation might define:

• base case: expected demand and normal operating conditions;
• optimistic case: strong demand and favourable supply conditions;
• adverse case: weak demand, delays or increased costs.

Each scenario should specify assumptions, probabilities if used, outputs and management actions. Scenarios are not necessarily forecasts. They are structured representations of plausible futures.

A scenario matrix can support robust decisions:

Scores must be defined clearly—for example, profit in thousands of dollars or a weighted performance score. A decision-maker may choose the highest expected score, the best adverse-case result, or a diversified combination, depending on risk tolerance.

Useful methods include:

• Best-case criterion: choose the alternative with the strongest favourable outcome;
• Worst-case criterion: choose the alternative with the strongest minimum outcome;
• Minimax regret: minimise the greatest opportunity loss relative to the best alternative in each scenario;
• Expected value: weight outcomes by scenario probabilities.

These criteria are not interchangeable. They reflect different attitudes toward risk and uncertainty.

Simulation

Simulation imitates the operation of a real or proposed system using a model. It is appropriate when relationships are too complex for a simple formula, but the system can be represented by rules and probability distributions.

A basic Monte Carlo procedure is:

1. Define the performance measure, such as annual profit or waiting time.
2. Identify uncertain inputs and their probability distributions.
3. Generate random values for those inputs.
4. Calculate the output for one simulated trial.
5. Repeat many trials.
6. Summarise the output distribution using means, percentiles and risk measures.
7. Validate assumptions and communicate uncertainty.

If simulated outputs are Y₁, Y₂, …, Yₙ, the sample mean is:

where n is the number of trials. The mean has the same units as Y. A percentile, such as the 95th percentile of cost, indicates a value below which approximately 95% of simulated outcomes fall; it is not automatically a 95% probability that the cost equals that value.

Example: inventory uncertainty

Suppose daily demand is simulated from an estimated distribution, and each trial calculates sales, leftover inventory and shortage cost. Repeating the trial 10,000 times may reveal that two ordering policies have similar average profit but very different probabilities of stockout. Management can then select a policy based on both expected profit and service requirements.

Simulation is not optimisation by itself. It evaluates specified policies. A simulation-optimisation approach can test many candidate policies, but the search procedure and performance comparisons must be designed carefully.

Simulation cautions

Results are only as credible as the model. Analysts should distinguish random variation from model uncertainty, use realistic distributions, test extreme inputs and validate outputs against historical behaviour. Random seeds improve reproducibility, but they do not correct biased assumptions. A large number of trials cannot compensate for an incorrectly specified process.

Introductory optimisation

Optimisation identifies the best feasible decision according to a stated objective. A linear programming (LP) model assumes that the objective and constraints are linear functions of decision variables.

A general maximisation model is:

Here, xⱼ are decision quantities; cⱼ are objective contributions per unit; aᵢⱼ are resource requirements; and bᵢ are available resource amounts. Units must be consistent: if xⱼ is products, then aᵢⱼ may be labour hours per product and bᵢ labour hours available.

Worked example: product mix

A workshop makes products A and B. Each A earns $40 contribution and uses 2 labour hours and 1 machine hour. Each B earns $30 contribution and uses 1 labour hour and 2 machine hours. Available resources are 100 labour hours and 80 machine hours.

Let x = units of A and y = units of B. The model is:

At the intersection of the two resource boundaries:

From the first equation, y = 100 − 2x. Substitution gives x + 2(100 − 2x) = 80, so x = 40 and y = 20. Contribution is:

The intercept alternatives produce $2,000 at x = 50, y = 0 and $1,200 at x = 0, y = 40. Therefore, among these corner points, producing 40 units of A and 20 units of B is optimal, assuming divisibility and the stated linear relationships. If products must be whole units, an integer model or checking nearby feasible integer solutions is required.

Interpreting optimisation results

The optimal solution is conditional, not universally correct. Managers should ask:

• Which constraints are binding, meaning fully used at the solution?
• What happens if resource availability changes?
• Are contribution margins reliable?
• Are demand limits, minimum quantities or contractual requirements missing?
• Are fixed costs and implementation costs represented?

A solver can identify a mathematical optimum while overlooking quality, fairness, employee wellbeing or strategic flexibility. Such concerns should be represented as constraints, additional objectives or explicit managerial judgement.

Integrated illustrative case: FreshRoute grocery delivery

Background. FreshRoute is a fictional regional grocery-delivery company. It must allocate next-day delivery capacity between standard and express orders. Standard orders contribute $18 each and use 20 driver-minutes; express orders contribute $25 and use 30 driver-minutes. The depot has 2,400 driver-minutes available. At most 80 express orders can be accepted because of promised time windows. Demand is uncertain.

Problem. Management wants to maximise daily contribution while avoiding a policy that performs poorly during a high-demand day when customer service is important.

Analysis. Let s and e be standard and express orders. The deterministic model is:

The model recommends using available time efficiently while respecting the express limit. However, forecast demand may be below the capacity for standard orders. A demand constraint should therefore be added if the forecast supplies a credible upper bound. The team then performs what-if analysis for driver availability (2,000, 2,400 and 2,800 minutes) and scenario analysis for normal, peak and disruption days.

Simulation uses uncertain demand and driver absence rates to estimate the probability of late deliveries under alternative allocations. Results show that an allocation maximising average contribution creates unacceptable late-delivery risk during disruption days. FreshRoute therefore adopts a policy that reserves driver capacity for express commitments, monitors demand during the day and triggers a contingency courier contract when the simulated risk threshold is exceeded.

Outcome. The final recommendation is not simply the highest-profit mathematical solution. It combines an optimisation baseline, scenario stress tests, simulation evidence and an operational trigger. The decision is documented with assumptions, data owners and review dates.

Learning points. Constraints can express service promises; simulation can expose tail risk; and a robust policy may be preferable to an average-case optimum. The case also illustrates the importance of translating analytical outputs into actions that employees can execute.

Governance, ethics and implementation

Prescriptive systems can cause harm when their objectives or data encode unfairness. Important controls include:

• transparency: document data sources, assumptions, objective functions and constraints;
• accountability: identify who approves, monitors and can override recommendations;
• fairness: test impacts across relevant customer or employee groups;
• privacy and security: collect only necessary data and protect sensitive information;
• human oversight: retain review for high-impact decisions such as credit, employment or healthcare allocation;
• auditability: preserve model versions, inputs, outputs and decision rationales;
• monitoring: detect drift in demand, costs, probabilities and outcomes.

AI tools may assist with forecasting, scenario generation or optimisation interfaces, but generated recommendations require validation. An apparently sophisticated model may be less reliable than a transparent spreadsheet if the data-generating process has changed. Organisations should establish thresholds for automatic action, escalation and periodic reapproval.

Best practices for professional use

1. Define the decision before selecting a technique.
2. Separate controllable variables from forecasts and assumptions.
3. State the objective and all material constraints in plain language.
4. Use consistent units and verify arithmetic independently.
5. Test base, adverse and extreme scenarios.
6. Report ranges, probabilities and sensitivity—not only a single optimum.
7. Validate outputs with subject-matter experts and historical evidence.
8. Consider implementation cost, timing, reversibility and behavioural effects.
9. Document ethical, legal and privacy implications.
10. Communicate a recommendation together with trigger conditions and review dates.

Chapter Summary

Prescriptive analytics converts evidence into recommended action. Decision trees model sequential choices and uncertainty, while expected monetary value provides a probability-weighted comparison when probabilities and payoffs are credible. What-if analysis changes inputs to examine sensitivity; scenario analysis evaluates coherent alternative futures; and simulation estimates distributions of outcomes when systems are complex or uncertain. Linear optimisation allocates scarce resources by maximising or minimising an explicit objective subject to constraints. These methods are complementary: optimisation identifies feasible best choices under stated assumptions, while scenarios and simulation test robustness. Responsible practice requires transparent assumptions, valid data, sensitivity analysis, human oversight and attention to fairness, privacy and implementation risk.

Key Terms

Prescriptive analytics: Analytics that recommends actions under objectives, constraints and uncertainty.

Decision variable: A controllable quantity selected by a decision-maker or model.

Objective function: A mathematical representation of what is maximised or minimised.

Constraint: A condition limiting feasible decisions.

Decision tree: A model showing sequential decisions, chance events and outcomes.

Decision node: A point at which an alternative can be selected.

Chance node: A point representing uncertain outcomes and probabilities.

Expected monetary value: Probability-weighted average monetary payoff.

Rollback analysis: Evaluating a decision tree from terminal outcomes back to the initial choice.

Expected value of perfect information: Maximum justifiable value of knowing the future outcome in advance.

What-if analysis: Testing the effect of changing selected inputs.

Scenario analysis: Evaluating alternatives under coherent sets of assumptions.

Regret: The loss relative to the best decision after the actual scenario is known.

Simulation: Repeated computational imitation of a system to estimate outcome distributions.

Monte Carlo simulation: Simulation using random draws from specified probability distributions.

Optimisation: Selecting the best feasible solution according to an objective.

Linear programming: Optimisation with linear objective and constraint functions.

Binding constraint: A constraint fully used at a solution.

Feasible solution: A decision satisfying every stated constraint.

Robust decision: A decision performing acceptably across several plausible conditions.

Review Questions

Short Answer Questions

1. How does prescriptive analytics differ from predictive analytics?
2. What are the roles of decision nodes and chance nodes in a decision tree?
3. State two assumptions of the basic break-even quantity formula.
4. Why is simulation not, by itself, an optimisation method?
5. What is meant by a binding constraint?

Long Answer Questions

1. Explain how what-if analysis and scenario analysis differ, and describe a situation in which both should be used.
2. Explain expected monetary value, including its assumptions and limitations.
3. Describe the stages of a Monte Carlo simulation study and the main validation issues.
4. Formulate a linear programming model for a two-product organisation with two scarce resources. Define every variable and constraint.
5. Discuss ethical and governance controls needed when prescriptive analytics influences high-impact decisions.

Analytical / Application Questions

1. A firm must choose between two projects with uncertain cash outcomes. Construct a decision tree, calculate EMV for each project and explain whether the EMV criterion is appropriate for a risk-averse manager.
2. A service organisation has fixed costs, unit prices and variable costs that may change under three demand scenarios. Build a scenario table, calculate break-even quantities for each cost structure and recommend a monitoring trigger.
3. A manufacturer has three products, limited labour and machine capacity, and maximum demand for each product. Formulate an LP model, identify likely binding constraints and explain how simulation could test the recommendation when demand and absenteeism are uncertain.

Multiple Choice Questions with Answers

Question 1

Which statement best describes prescriptive analytics?

A. It reports historical sales totals only.

B. It recommends actions using objectives, constraints and uncertainty.

C. It stores operational data in a database.

D. It replaces all managerial judgement.

Correct Answer: B. It recommends actions using objectives, constraints and uncertainty.

Question 2

In a decision tree, which element represents an uncertain event?

A. Decision node

B. Constraint

C. Chance node

D. Objective function

Correct Answer: C. Chance node

Question 3

An outcome has probability 0.6 and value $100,000, while another has probability 0.4 and value −$20,000. What is the EMV?

A. $52,000

B. $60,000

C. $68,000

D. $80,000

Correct Answer: C. $68,000

Question 4

What-if analysis primarily examines:

A. The effect of changing selected inputs on outputs

B. Only legal compliance requirements

C. A database’s storage capacity

D. Historical data without assumptions

Correct Answer: A. The effect of changing selected inputs on outputs

Question 5

Which statement about scenario analysis is correct?

A. It changes exactly one variable at a time.

B. It evaluates coherent sets of assumptions about possible futures.

C. It guarantees the most likely future.

D. It can be used only when probabilities are known.

Correct Answer: B. It evaluates coherent sets of assumptions about possible futures.

Question 6

A simulation output is measured in dollars. What unit does its sample mean have?

A. Trials per dollar

B. Probability squared

C. Dollars

D. Percent only

Correct Answer: C. Dollars

Question 7

Which condition is required for the basic break-even quantity formula Q₀ = F ÷ (P − v)?

A. Variable cost must exceed price.

B. The contribution margin per unit must be positive.

C. Fixed cost must be zero.

D. Quantity must always be negative.

Correct Answer: B. The contribution margin per unit must be positive.

Question 8

In a linear programming model, a feasible solution is one that:

A. Has the largest objective value automatically.

B. Satisfies all stated constraints.

C. Uses every resource completely.

D. Contains only integer variables.

Correct Answer: B. Satisfies all stated constraints.

Question 9

What is a binding constraint?

A. A constraint that is fully used at the solution.

B. A constraint that has been removed.

C. A constraint with no measurable units.

D. A constraint that applies only to forecasts.

Correct Answer: A. A constraint that is fully used at the solution.

Question 10

Why should simulation results be validated?

A. More trials always eliminate model bias.

B. Validation checks whether the model represents the real system adequately.

C. Validation makes probabilities unnecessary.

D. Validation guarantees a mathematical optimum.

Correct Answer: B. Validation checks whether the model represents the real system adequately.