Decision Models and What-If Analysis

Introduction

Business decisions are rarely made with complete information. Managers may need to choose a product launch strategy before demand is known, select a supplier despite uncertain delivery performance, or allocate a limited budget among competing projects. Business intelligence (BI) systems support such decisions by combining data, assumptions, analytical models, and visual communication.

A decision model is a structured representation of a business choice. It identifies the available alternatives, relevant external conditions, consequences, and decision criteria. What-if analysis then examines how the recommended decision changes when assumptions or input values change.

This chapter develops practical methods for decisions under certainty, risk, and uncertainty. It covers payoff tables, decision trees, expected value, sensitivity analysis, goal seeking, simulation, optimization, and multi-criteria decision analysis (MCDA).

Learning Objectives

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

1. Distinguish decisions under certainty, risk, and uncertainty.
2. Construct and interpret payoff tables and decision trees.
3. Calculate expected monetary value and expected opportunity loss.
4. Apply sensitivity analysis and goal seeking to business models.
5. Explain how simulation represents uncertainty in decision models.
6. Formulate basic optimization models and interpret constraints.
7. Apply a transparent MCDA process to competing alternatives.
8. Evaluate model assumptions, limitations, governance, and ethical risks.

1. The Structure of a Decision Model

A business decision model normally contains five elements:

• Decision alternatives: actions controlled by the decision-maker.
• States of nature: external conditions that cannot be controlled, such as high or low demand.
• Payoffs or outcomes: profit, cost, revenue, service level, risk, or another consequence.
• Probabilities: estimates of the likelihood of states of nature, when reliable estimates are available.
• Decision rule: the method used to select an alternative.

A model should separate facts from assumptions. Historical sales may be data; the assumption that next year's demand distribution will resemble last year's may be a modelling judgement.

Decisions under certainty, risk, and uncertainty

A decision is made under certainty when the relevant state of nature is known. It is made under risk when several states are possible and probabilities can reasonably be estimated. It is made under uncertainty when probabilities are unavailable, unreliable, or disputed.

The distinction is important. Expected value is appropriate when probabilities have a defensible interpretation. Using precise probabilities merely to make a spreadsheet appear scientific can create false confidence.

2. Payoff Tables

A payoff table displays the outcome of each alternative under each possible state of nature. For example, a retailer is considering three warehouse strategies. The following figures are estimated annual profits in thousands of dollars.

If demand is known with certainty, the best action is selected separately for the known state: small for low demand, medium for medium demand, and large for high demand.

When probabilities are unavailable, several decision rules are possible:

• Maximax: choose the alternative with the highest possible payoff. This represents an optimistic attitude.
• Maximin: choose the alternative with the best worst-case payoff. This is conservative.
• Minimax regret: choose the alternative that minimizes the largest possible opportunity loss.
• Laplace rule: assume all states are equally likely and choose the highest average payoff, if equal likelihood is a defensible neutral assumption.

For the table, maximax selects the large warehouse because its best payoff is 600. Maximin selects the small warehouse because its worst payoff is 180, higher than 120 and −80.

Regret and opportunity loss

Regret is the payoff forgone because the chosen alternative was not the best alternative for the state that occurred. For a profit table:

Here, i identifies an alternative and j identifies a state of nature. The best payoff is found separately within each column. The regret table is therefore:

Minimax regret selects the medium warehouse because its maximum regret, 180, is the smallest.

3. Decision Trees

A decision tree represents decisions and uncertain events in chronological order. A square usually represents a decision node, a circle a chance node, and terminal branches the resulting payoffs. Decision trees are especially useful when information arrives over time or when an early decision determines later choices.

A tree is evaluated from right to left, a process called rollback. At a chance node, calculate the probability-weighted average of downstream values. At a decision node, select the branch with the highest value for a profit objective or the lowest value for a cost objective.

Figure: A decision tree combines an initial choice with subsequent uncertain outcomes.

A tree can also include a survey, pilot, or market test. The analyst must distinguish the value of information from the cost of obtaining it. Information is useful only if it can change the decision or improve its outcome sufficiently to justify its cost.

4. Expected Value Under Risk

The expected monetary value (EMV) of alternative i is the probability-weighted average of its payoffs:

where p_j is the probability of state j, V_ij is the payoff from alternative i in state j, and ∑p_j = 1. EMV is expressed in the same monetary units as the payoffs. It is appropriate when repeated decisions or long-run average performance is relevant and probabilities are credible.

Using probabilities of low demand 0.30, medium demand 0.45, and high demand 0.25:

Thus, the medium warehouse has the highest EMV of $273,000. This does not mean that it will produce $273,000 in every year. It is a probability-weighted expectation.

Expected opportunity loss

Expected opportunity loss (EOL) applies the same probability calculation to regret:

For the small warehouse:

For the medium warehouse, EOL_medium = 0.30(60) + 0.45(0) + 0.25(180) = 63. For the large warehouse, EOL_large = 0.30(260) + 0.45(40) + 0.25(0) = 96. The medium warehouse again performs best, with the lowest expected opportunity loss.

Risk measures

Two alternatives can have the same EMV but very different risk. A common measure is variance:

The standard deviation is σ_i = √σ_i² and has the same monetary units as the payoff. These measures describe dispersion, not the probability of loss by themselves. Managers should also inspect downside outcomes, probabilities of loss, and cash-flow timing.

5. Sensitivity Analysis

Sensitivity analysis changes one or more inputs to determine whether the decision is robust. Inputs may include demand, price, probability, cost, capacity, or discount rate. The output may be profit, EMV, return on investment, or service level.

One-way and two-way analysis

In one-way analysis, one input changes while others remain constant. A two-way data table changes two inputs simultaneously. A tornado chart ranks variables by their effect on the result, helping managers focus on the most influential assumptions.

For two alternatives, the probability at which their EMVs are equal can be found algebraically. Suppose alternative A has payoffs A_H and A_L under high and low demand, while B has B_H and B_L. Let p be the probability of high demand:

Solving this equation gives the probability threshold. If the estimated probability lies on one side of the threshold, one alternative is preferred; on the other side, the other is preferred. This is useful because it shows how much the forecast would need to change before the decision changes.

Sensitivity analysis does not prove that an assumption is correct. It reveals dependence on assumptions and identifies where better data would have the greatest value.

6. Goal Seeking and Scenario Analysis

Goal seeking works backward from a desired output to find the input required to achieve it. For example, if profit is:

then the quantity required for a target profit T is:

This formula assumes constant unit price and variable cost, a fixed-cost structure, and a positive contribution margin. If price is $50, variable cost is $30, fixed cost is $100,000, and target profit is $50,000:

Spreadsheet goal-seek tools can solve more complex relationships numerically. The analyst should check that the solution is feasible: demand, capacity, cash, and operational constraints may make the calculated input impossible.

Scenario analysis changes a coherent group of assumptions. A base case might use expected demand and current costs; a downside case might combine weak demand, higher costs, and delayed delivery; an upside case might combine strong demand and improved productivity. Scenarios are more realistic than changing unrelated inputs independently, but they can conceal probabilities unless explicitly documented.

7. Simulation

Simulation represents uncertainty by repeatedly sampling possible input values and calculating the resulting output. In a Monte Carlo model, each trial uses randomly sampled values from specified probability distributions.

A simple simulation procedure is:

1. Define the output equation and uncertain inputs.
2. Select distributions based on historical data, expert judgement, or fitting procedures.
3. Generate random values for each trial.
4. Calculate the output for that trial.
5. Repeat many times.
6. Summarize the output distribution using means, percentiles, probabilities, and charts.
7. Validate the model and test sensitivity to distributional assumptions.

For example, annual profit might be calculated for each trial as Profit_t = (Price_t − Cost_t)Quantity_t − FixedCost_t. A simulation can estimate the probability that profit is negative, rather than reporting only an average.

Simulation outputs should include measures such as the mean, median, 5th percentile, 95th percentile, and probability of exceeding a target. More trials reduce random sampling error, but they do not correct a badly specified model. A useful diagnostic is the standard error of a simulated mean, approximately SE ≈ s ÷ √N, where s is the sample standard deviation and N is the number of trials.

Important limitations include dependence between inputs, poor-quality distributions, model bias, and false precision. Demand and price, for example, may be correlated. Treating them as independent can materially distort results.

8. Optimization Models

Optimization identifies the best feasible solution according to an objective function. A linear programming model has the general form:

subject to:

Here, x_j is the decision quantity, c_j is its contribution to the objective, a_ij is the amount of resource i used by decision j, and b_i is the available amount of resource i. Units must be consistent.

For example, a company produces products A and B. A contributes $40 and uses 2 labour hours and 1 machine hour. B contributes $30 and uses 1 labour hour and 2 machine hours. There are 100 labour hours and 80 machine hours:

Solving the intersection of the two binding constraints gives A = 40 and B = 20, with contribution Z = 2,200. A solver should also test corner points and report whether constraints are binding, slack, or infeasible. Real business models may require integer, nonlinear, multi-period, or robust optimization.

Optimization is not a substitute for judgement. The solution is optimal only relative to the objective, data, constraints, and assumptions supplied.

9. Multi-Criteria Decision Analysis

Many decisions involve conflicting criteria. A supplier may be evaluated on cost, quality, delivery reliability, sustainability, and financial stability. MCDA makes these trade-offs explicit.

A basic weighted-sum model is:

where r_ik is the normalized rating of alternative i on criterion k, and w_k is the criterion weight. Usually ∑w_k = 1 and ratings are placed on a common scale, such as 0 to 100.

For benefit criteria, a min–max normalization is:

For cost criteria, reverse the scale:

Normalization is meaningful only when higher values consistently represent better performance after adjustment. Criteria should not be double-counted, and weights should reflect stakeholder priorities rather than the convenience of available data. Sensitivity analysis should test whether small changes in weights reverse the ranking.

MCDA can also use methods such as the analytic hierarchy process, outranking methods, or compromise ranking. The method should match the decision context and be explainable to stakeholders.

10. Illustrative Case: Selecting a Cloud Analytics Platform

Background

The fictional company Northstar Retail wants to select a cloud analytics platform. Three alternatives—Alpha, Beta, and Gamma—meet the basic technical requirements. The decision team identifies annual cost, implementation time, data security, scalability, and user usability as criteria.

Problem

The lowest-cost platform is not necessarily the best. A platform that is difficult to use may reduce adoption, while a platform with weak scalability may create future migration costs.

Analysis and approach

The team assigns weights: cost 0.25, implementation time 0.15, security 0.25, scalability 0.20, and usability 0.15. It scores each platform from 0 to 100 after converting cost and time so that higher scores represent better performance.

Weighted scores are:

Outcome

Gamma ranks first despite its lower usability score. However, the decision team should not accept the ranking automatically. It should test whether Gamma remains first if security receives a lower weight, verify the evidence supporting each score, and negotiate contractual safeguards.

Learning points

The case illustrates that MCDA documents trade-offs rather than eliminating judgement. A transparent model allows stakeholders to challenge weights, ratings, and assumptions. It also supports a balanced decision when financial and non-financial consequences matter.

11. Governance, Ethics, and Professional Practice

Decision models can influence employment, lending, pricing, healthcare access, and public services. Good practice therefore requires:

• documenting data sources, assumptions, formulas, versions, and owners;
• validating calculations independently;
• testing sensitivity and adverse scenarios;
• protecting confidential and personal data;
• checking for discriminatory proxy variables and unequal impacts;
• explaining material decisions in understandable language;
• preserving human accountability for consequential decisions;
• monitoring model performance after implementation.

A model should not be presented as objective merely because it uses mathematics. Historical data may encode past discrimination, and optimization may maximize profit while violating fairness or service obligations. Access controls, audit trails, model approval processes, and periodic review are important controls.

AI-assisted decision tools can accelerate forecasting, simulation, and scenario generation, but generated assumptions and recommendations require human verification. AI systems may introduce opaque reasoning, data leakage, unstable outputs, or automation bias. A responsible organization defines when human review is mandatory and records the rationale for final decisions.

12. Best Practices and Common Errors

A disciplined workflow is:

1. Define the decision, objective, time horizon, and decision-maker.
2. Identify alternatives, states, criteria, constraints, and stakeholders.
3. Collect and clean relevant data.
4. Make assumptions explicit and distinguish estimates from observations.
5. Build the simplest model that answers the question.
6. Validate formulas, units, logic, and extreme cases.
7. Perform what-if, sensitivity, and, when appropriate, simulation analysis.
8. Communicate results with uncertainty and limitations.
9. Implement, monitor, and update the model.

Common errors include confusing revenue with profit, omitting fixed or opportunity costs, using probabilities that do not sum to one, treating correlated variables as independent, double-counting criteria, optimizing an unrealistic objective, and reporting a single forecast without downside information. A technically correct calculation can still support a poor decision if the decision problem was framed incorrectly.

Chapter Summary

Decision models translate business choices into alternatives, uncertain states, outcomes, and decision rules. Payoff tables support transparent comparison under certainty, risk, or uncertainty. Maximax, maximin, minimax regret, and Laplace rules represent different attitudes toward uncertainty. Decision trees show sequential decisions and are evaluated by rollback. Expected monetary value uses credible probabilities to calculate long-run average payoff, while expected opportunity loss evaluates regret.

Sensitivity analysis tests robustness, goal seeking works backward from a target, and scenario analysis changes coherent groups of assumptions. Simulation estimates distributions of outcomes and downside probabilities but depends on credible input distributions and relationships. Optimization identifies the best feasible solution under a specified objective and constraints. MCDA combines weighted criteria after appropriate normalization and makes trade-offs visible.

Effective analysts combine quantitative technique with data quality assessment, stakeholder judgement, ethical governance, explainability, and monitoring. A model is useful not because it produces a precise number, but because it improves the quality, transparency, and resilience of a decision.

Key Terms

Decision model: A structured representation of alternatives, assumptions, uncertainties, outcomes, and decision rules.

Alternative: An action available to the decision-maker.

State of nature: An external condition that affects outcomes but cannot be controlled.

Payoff: The consequence of an alternative under a particular state.

Decision under risk: A decision in which probabilities can reasonably be estimated.

Decision under uncertainty: A decision in which probabilities are unavailable or unreliable.

Maximax: A rule selecting the alternative with the highest possible payoff.

Maximin: A rule selecting the alternative with the best worst-case payoff.

Regret: The payoff lost by not selecting the best alternative for the state that occurs.

Expected monetary value: The probability-weighted average payoff of an alternative.

Expected opportunity loss: The probability-weighted average regret.

Decision tree: A graphical representation of sequential decisions and uncertain events.

Sensitivity analysis: Examination of how outputs change when inputs change.

Goal seeking: Finding an input value that achieves a specified output target.

Scenario analysis: Evaluation of outcomes under coherent sets of assumptions.

Simulation: Repeated calculation using sampled uncertain inputs.

Optimization: Selection of the best feasible solution according to an objective.

Constraint: A restriction on available choices or resources.

MCDA: A method for evaluating alternatives across multiple criteria.

Normalization: Conversion of measures to a common comparison scale.

Review Questions

Short Answer Questions

1. What are the five basic elements of a decision model?
2. Distinguish decisions under risk from decisions under uncertainty.
3. What is the purpose of a decision tree rollback calculation?
4. Why should sensitivity analysis accompany an expected value calculation?
5. State one ethical risk associated with automated decision models.

Long Answer Questions

1. Explain the maximax, maximin, minimax regret, and Laplace decision rules. Discuss the managerial attitude represented by each.
2. Explain expected monetary value and expected opportunity loss, including their assumptions and limitations.
3. Describe the steps in a Monte Carlo simulation and explain why a large number of trials does not guarantee a valid model.
4. Formulate a linear programming model for a business resource-allocation problem of your choice and identify its objective and constraints.
5. Explain how weighted-sum MCDA can support supplier selection and identify at least three risks in using it.

Analytical / Application Questions

1. A firm has two expansion options and uncertain demand. Construct a payoff table, calculate EMV using stated probabilities, and perform a sensitivity analysis to identify the probability threshold at which the preferred option changes.
2. Design a simulation model for estimating the probability that a new product will fail to reach its profit target. Specify uncertain inputs, distributions or evidence sources, outputs, and validation checks.
3. A company must choose a software platform using cost, security, scalability, implementation time, and usability. Develop an MCDA framework, justify weights and normalization, and explain how you would test the robustness and fairness of the ranking.

Multiple Choice Questions with Answers

Question 1

Which decision rule selects the alternative with the best worst-case payoff?

A. Maximax

B. Maximin

C. Laplace

D. Expected monetary value

Correct Answer: B. Maximin

Question 2

What does a regret value represent?

A. The total cost of implementing an alternative

B. The probability that a state of nature occurs

C. The payoff forgone by not selecting the best alternative for the state that occurs

D. The average payoff across all alternatives

Correct Answer: C. The payoff forgone by not selecting the best alternative for the state that occurs

Question 3

An alternative has payoffs of $100 and $40 under two states with probabilities 0.6 and 0.4. What is its EMV?

A. $64

B. $76

C. $84

D. $140

Correct Answer: C. $84

Question 4

In a decision tree, what does a circle normally represent?

A. A decision node

B. A chance node

C. A constraint

D. A terminal payoff

Correct Answer: B. A chance node

Question 5

Which technique works backward from a desired output to find the required input?

A. Goal seeking

B. Minimax regret

C. Normalization

D. Rollback only

Correct Answer: A. Goal seeking

Question 6

What is the main purpose of Monte Carlo simulation in decision analysis?

A. To remove all uncertainty from a model

B. To calculate only the single most likely outcome

C. To estimate a distribution of outcomes by repeatedly sampling uncertain inputs

D. To guarantee an optimal feasible solution

Correct Answer: C. To estimate a distribution of outcomes by repeatedly sampling uncertain inputs

Question 7

In a linear programming model, what does a constraint represent?

A. A restriction on resources or feasible choices

B. The probability of a state of nature

C. The average payoff

D. A stakeholder preference only

Correct Answer: A. A restriction on resources or feasible choices

Question 8

If MCDA weights are used in a weighted-sum model, what should normally be true of the weights?

A. They must all be equal

B. They must sum to 1

C. They must be negative

D. They must be measured in dollars

Correct Answer: B. They must sum to 1

Question 9

Why is sensitivity analysis important?

A. It proves that the base-case assumptions are correct

B. It identifies how changes in assumptions affect the decision or output

C. It eliminates the need for data validation

D. It guarantees that all alternatives have equal risk

Correct Answer: B. It identifies how changes in assumptions affect the decision or output

Question 10

A simulation model treats price and demand as independent even though they are strongly related. What is the likely problem?

A. The model may produce distorted output distributions

B. The model becomes a deterministic optimization model

C. The number of simulation trials becomes irrelevant

D. The payoff table automatically corrects the relationship

Correct Answer: A. The model may produce distorted output distributions