2  Decision Analysis and Data Uncertainty

2.1 Structuring Decision Problems

Decision analysis is a systematic approach to making choices under uncertainty (John S. Hammond et al., 1999). It involves identifying, evaluating, and selecting among alternatives based on available data and the outcomes they may produce. Data uncertainty is a critical factor in this process, as it affects the reliability of the decisions reached. Structuring the problem is the first and perhaps most crucial step in decision analysis, as it lays the groundwork for identifying the best possible course of action.

2.1.1 The Five-Step Structuring Process

Structuring a decision problem involves defining the problem’s context, identifying the decision objectives, recognising the alternatives, understanding the constraints, and considering the risks or uncertainties. This process breaks down complex decisions into more manageable parts, allowing each component to be analysed thoroughly.

1. Define the Problem

The first step is to clearly define the problem that needs to be solved. This includes understanding the background, the context in which the decision is made, and the impact of the decision. A well-defined problem sets the stage for identifying objectives and alternatives.

Example: A company must decide whether to launch a new product. The problem is to determine the most profitable course of action while managing risks associated with market acceptance and production capabilities.

2. Identify Objectives

Objectives are the goals that the decision-maker wishes to achieve. Identifying objectives helps in focusing the decision-making process and in evaluating the potential outcomes of different alternatives.

Example: In the case of the new product launch, objectives could include maximizing profit, achieving a certain market share, or enhancing the company’s reputation.

3. Recognize the Alternatives

Alternatives are the different courses of action available to the decision-maker. Identifying all feasible alternatives is crucial for a comprehensive analysis.

Example: Alternatives for the new product launch could include launching immediately, delaying the launch until further market research is completed, launching in a limited market as a test, or not launching the product at all.

4. Understand the Constraints

Constraints are limitations that affect the decision-making process. These could be financial, operational, legal, or time-related constraints.

Example: Constraints on the new product launch could include budget limits, production capacity, and regulatory compliance requirements.

5. Consider the Risks or Uncertainties

Every decision is subject to some level of uncertainty. Identifying the uncertainties and assessing their impact on the decision is a critical part of the structuring process.

Example: Uncertainties in the product launch decision could include customer acceptance of the new product, competitor reactions, and potential supply chain issues.

Incorporating Data Uncertainty

Data uncertainty can significantly affect decision-making. Incorporating uncertainty into the decision analysis process involves estimating the likelihood of various outcomes and considering how uncertainties in the data might impact the decision.

Example: In the product launch decision, uncertainty about market demand could be addressed by creating several scenarios (e.g., high, moderate, and low demand) and estimating the probability and impact of each.

2.2 Decision Criteria and Utility Theory

In decision analysis, especially under uncertainty, selecting the best course of action often requires a systematic approach to evaluate and compare different alternatives. Decision criteria provide rules or guidelines for making such choices, taking into account the outcomes’ risks, benefits, and trade-offs. Utility theory, a cornerstone of decision criteria, helps quantify an individual’s preferences under uncertainty, offering a way to make choices that align with their risk tolerance and value judgment.

2.2.1 Decision Criteria

Decision criteria are methodologies used to choose among alternatives based on their expected outcomes. These criteria are crucial in scenarios where decisions must be made under uncertainty, and they vary depending on the decision-maker’s attitude towards risk. Some common decision criteria include:

• Expected Value (EV): This criterion suggests selecting the alternative with the highest expected value, calculated by weighting each outcome by its probability of occurrence. It’s a measure of the central tendency of the possible outcomes.

  Example: Choosing an investment option based on the highest expected return.

• Maximin or Minimax Regret: Maximin focuses on maximizing the minimum payoff to avoid the worst-case scenario, suitable for risk-averse individuals. Minimax Regret aims to minimize the maximum regret, which is the difference between the payoff of the chosen alternative and the best possible payoff in hindsight.

  Example: Selecting a supply chain strategy that ensures the least disruption in the worst-case scenario.

• Expected Utility (EU): This criterion takes into account the decision-maker’s risk preferences, using utility functions to convert outcomes into values that reflect the decision-maker’s satisfaction or utility.

2.2.2 Utility Theory

Utility theory is a framework for understanding how individuals make choices under uncertainty (John von Neumann & Oskar Morgenstern, 1944). It posits that every outcome has a utility value attached to it, which represents the individual’s subjective assessment of the outcome’s worth. The theory assumes that individuals choose alternatives that maximize their expected utility, rather than merely maximizing expected monetary value.

Key Concepts of Utility Theory

• Utility Function: A mathematical representation of how utility (satisfaction) varies with changes in wealth or consumption. The shape of the utility function indicates the decision-maker’s risk attitude: concave for risk-averse, convex for risk-seeking, and linear for risk-neutral.

• Risk Aversion: A characteristic of preferring a certain outcome over a gamble with a higher expected value but with risk. A risk-averse individual’s utility function is concave, reflecting diminishing marginal utility for wealth.

• Expected Utility Maximization: The principle that individuals choose the alternative with the highest expected utility, calculated by summing the utilities of all possible outcomes weighted by their probabilities.

Application of Utility Theory

Utility theory is applied in various fields, including economics, finance, and insurance, to model behavior under uncertainty. In decision-making, it helps to:

• Assess investment choices by comparing the expected utility of different financial assets.
• Make insurance decisions, where buying insurance is seen as a risk-averse choice to avoid large losses.
• Guide complex business decisions by evaluating alternatives on their utility, not just their expected monetary outcome.

2.3 Multi-Criteria Decision Analysis (MCDA)

Multi-Criteria Decision Analysis (MCDA) is a framework used to evaluate and prioritise options when multiple, often conflicting, criteria must be considered simultaneously. It supports decision-makers in making choices that best align with their objectives and values, especially in complex situations where trade-offs between different criteria are necessary. MCDA is widely applied in business strategy, environmental management, healthcare, and public policy.

2.3.1 Core Concepts of MCDA

• Criteria: These are the attributes, measures, or aspects that are considered important in the evaluation and decision-making process. Criteria can be quantitative (numerical) or qualitative (descriptive) and reflect the objectives and values of the decision-makers.

• Alternatives: The options or choices available to the decision-maker. MCDA aims to identify the best alternative based on the evaluation against the set criteria.

• Weighting: Since not all criteria are equally important, weights are assigned to express the relative importance of each criterion. Weighting helps in prioritizing the criteria according to the decision-maker’s preferences.

• Scoring: Alternatives are scored based on how well they meet each criterion. Scoring can be based on objective data, subjective assessments, or a combination of both.

• Aggregation: The weighted scores of the alternatives for all criteria are aggregated to determine an overall score for each alternative. This helps in comparing the alternatives comprehensively.

2.3.2 MCDA Methods

Several MCDA methods are available, each with its own approach to dealing with multiple criteria. Some popular methods include:

• Analytic Hierarchy Process (AHP) (Thomas L. Saaty, 1980): A structured technique for organizing and analyzing complex decisions, based on mathematics and psychology. It involves decomposing the decision problem into a hierarchy of more easily comprehended sub-problems, each of which can be analyzed independently.

• Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) (Ching-Lai Hwang & Kwangsun Yoon, 1981): A method based on the concept that the chosen alternative should have the shortest geometric distance from the ideal solution and the farthest from the worst condition.

• Multi-Attribute Utility Theory (MAUT): Focuses on constructing a utility function that captures the decision-maker’s preference over a set of alternatives and criteria.

2.3.3 Application of MCDA

• Business Strategy: MCDA helps companies evaluate different strategic options, considering factors like cost, risk, return on investment, and market potential.

• Environmental Management: Used to balance economic benefits with environmental impacts, such as in assessing the viability of renewable energy projects.

• Healthcare: In public health policies, MCDA can help prioritize healthcare interventions based on effectiveness, cost, accessibility, and equity.

2.3.4 Advantages of MCDA

• Structured Decision-Making: MCDA provides a systematic approach to complex decision problems, making the process transparent and rational.

• Incorporates Multiple Perspectives: By considering various criteria, MCDA allows for a holistic view of the problem, incorporating different stakeholders’ values and objectives.

• Facilitates Communication: The structured nature of MCDA helps in communicating the decision-making process and rationale to all stakeholders.

2.3.5 Challenges in MCDA

• Subjectivity: The selection of criteria, assignment of weights, and scoring can be subjective, potentially biasing the outcome.

• Complexity: The process can become unwieldy with a large number of criteria and alternatives, requiring careful management and analysis.

2.4 Sensitivity Analysis and Risk Assessment

Sensitivity analysis and risk assessment are critical components of decision-making, particularly in fields requiring complex modelling and forecasting — finance, engineering, environmental science, and public policy. These methodologies help decision-makers understand the impact of uncertainty, evaluate model robustness, and manage potential risks effectively.

2.4.1 Sensitivity Analysis

Sensitivity analysis investigates how the variation in the output of a model can be attributed to different variations in its input variables. This analysis is used to identify which inputs are the most critical to the model’s outcomes, helping in prioritizing focus on those variables.

Key Points:

• Objective: To determine how changes in model inputs affect outputs, identifying sensitive inputs that significantly influence the results.
• Process: Involves varying one or more input parameters within a specified range and observing the effect on the model’s output.
• Applications: Used in financial modeling to assess the impact of changes in interest rates or exchange rates, in engineering for design optimization, and in environmental science for assessing the impact of variable factors on ecosystem models.

Benefits:

• Improved Understanding: Helps in understanding the relationship between input and output variables.
• Robustness and Reliability: Assists in evaluating the robustness and reliability of models by identifying the inputs that have the most significant impact on outputs.
• Decision Support: Supports decision-making by highlighting areas where information and control are most needed.

2.4.2 Risk Assessment

Risk assessment involves identifying, analyzing, and evaluating the likelihood and impact of uncertain events on objectives. It’s a systematic process for determining the magnitude of potential threats and the probability of their occurrence, facilitating effective risk management strategies.

Key Points:

• Risk Identification: The process starts with identifying potential risks that could negatively impact the project or investment.
• Risk Analysis: Involves evaluating the likelihood of each risk occurring and its potential impact on project outcomes.
• Risk Evaluation: Prioritizing risks based on their severity and likelihood to focus on managing the most critical threats.

Applications:

• Business and Finance: Assessing risks related to market fluctuations, credit, investments, and operational functions.
• Environmental and Public Health: Identifying potential hazards and their impacts on the environment or public health.
• Project Management: Evaluating risks associated with project timelines, costs, and scopes.

Benefits:

• Proactive Management: Enables proactive identification and mitigation of risks before they manifest.
• Resource Optimization: Helps in allocating resources efficiently to manage risks effectively.
• Enhanced Decision-Making: Improves decision-making by providing a structured approach to assessing and managing uncertainties.

Integrating Sensitivity Analysis with Risk Assessment

Integrating sensitivity analysis with risk assessment provides a comprehensive approach to understanding and managing uncertainty. Sensitivity analysis highlights the critical variables that influence outcomes, while risk assessment evaluates the potential threats and their impacts. Together, they enable decision-makers to develop robust strategies that account for uncertainties, enhancing the resilience and success of projects and investments.

2.5 Sources of Uncertainty and Risk

Uncertainty and risk are inherent in virtually all decision-making, especially in complex and dynamic environments. Knight (Frank H. Knight, 1921) drew the classical distinction: risk refers to situations where outcome probabilities are known or estimable, while uncertainty refers to situations where the underlying probability distribution itself is unknown. Both stem from our inability to predict future events with complete accuracy and from the potential for adverse outcomes. Understanding their sources is essential for developing effective strategies to manage them. These sources are varied and multifaceted, depending on the decision context, the environment, and the specific characteristics of the projects or investments involved.

2.5.1 Types of Uncertainty

1. Parameter Uncertainty: Involves uncertainty about the values of parameters in models or systems, often due to limited data or inherent variability in the data.

2. Model Uncertainty: Arises from the simplifications and assumptions made when developing models to represent complex systems. It questions whether the chosen model accurately reflects reality.

3. Decision Uncertainty: Pertains to the uncertainty in making decisions due to incomplete information about the alternatives or outcomes.

4. Environmental Uncertainty: Relates to unpredictability in the external environment, including economic, political, social, and technological factors.

2.5.2 Sources of Risk

1. Market Risk (Systematic Risk): The risk of losses due to factors that affect the overall performance of the financial markets, such as changes in interest rates, inflation rates, exchange rates, and economic recessions.

2. Credit Risk: The risk that a borrower will default on any type of debt by failing to make required payments.

3. Operational Risk: The risk of loss resulting from inadequate or failed internal processes, people, and systems, or from external events. This includes fraud risks, legal risks, and physical or environmental risks.

4. Liquidity Risk: The risk that an entity will not be able to meet its financial obligations as they come due because it cannot convert assets to cash or cannot obtain cash quickly enough.

5. Strategic Risk: Risks associated with the high-level goals and objectives of an organization, including changes in consumer preferences, technological advances, and competitive pressures.

2.5.3 Managing Uncertainty and Risk

1. Risk Identification and Assessment: The first step in risk management involves identifying potential risks and assessing their likelihood and potential impact.

2. Mitigation Strategies: Developing strategies to reduce the likelihood of risks or to minimize their impact. This could involve diversifying investments, improving internal processes, or implementing safety measures.

3. Contingency Planning: Preparing plans for how to respond if certain risks materialize, ensuring that the organization can continue operations and meet its objectives.

4. Insurance and Hedging: Using financial instruments or insurance policies to transfer or share risk.

5. Continuous Monitoring: Regularly reviewing and updating risk assessments and mitigation strategies to reflect changes in the environment and in the organisation’s objectives and capabilities.

2.6 Probability Distributions and Statistical Inference

Probability distributions and statistical inference are foundational tools for analysing random phenomena and drawing conclusions from data samples. Probability distributions describe how probabilities are assigned to different events or numerical values; statistical inference uses sample data to make predictions or decisions about a population.

2.6.1 Probability Distributions

A probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment. It’s a description of how the probabilities are distributed over the events or numerical values. Probability distributions can be classified into two main categories: discrete and continuous.

Discrete Probability Distributions

• Binomial Distribution: Describes the number of successes in a fixed number of independent Bernoulli trials (yes/no outcomes) with a constant probability of success. It’s characterized by parameters \(n\) (number of trials) and \(p\) (probability of success in each trial).

• Poisson Distribution: Used to model the number of times an event occurs in a fixed interval of time or space, given the average number of times the event occurs over that interval. It’s characterized by the parameter \(\lambda\) (average rate of occurrence).

Continuous Probability Distributions

• Normal (Gaussian) Distribution: Describes a symmetrical, bell-shaped curve defined by the mean (\(\mu\)) and standard deviation (\(\sigma\)), where the bulk of the observations cluster around the mean. It’s used in many natural and social phenomena.

• Uniform Distribution: All outcomes are equally likely within a certain interval. It’s characterized by the minimum and maximum values of the interval.

• Exponential Distribution: Describes the time between events in a Poisson process, representing the likelihood of waiting a certain amount of time until the next event. It’s characterized by the rate parameter \(\lambda\).

2.6.2 Statistical Inference

Statistical inference involves using data from a sample to make generalizations about a population. It includes estimating population parameters, testing hypotheses, and making predictions.

Key Concepts

• Point Estimation: Provides a single value as an estimate of a population parameter, like the sample mean estimating the population mean.

• Confidence Intervals: A range of values used to estimate the true value of a population parameter with a certain level of confidence.

• Hypothesis Testing: A method for testing a hypothesis about a population parameter based on sample data. It involves comparing observed data against the null hypothesis and determining the probability of observing such data if the null hypothesis were true.

• P-Value: The probability of observing the collected data, or something more extreme, if the null hypothesis is true. A low p-value indicates that the observed data are unlikely under the null hypothesis, leading to its rejection.

Applications

Statistical inference is used in various fields to make decisions or predictions when complete certainty is impossible. Applications include:

• Clinical Trials: To determine the efficacy of new drugs.
• Market Research: To understand consumer preferences and behaviours.
• Quality Control: To ensure manufacturing processes meet defined standards.

2.7 Monte Carlo Simulation and Bootstrapping

Monte Carlo Simulation (MCS) is a computational technique used to model the probability of different outcomes in a process that cannot easily be predicted because of random variables (Nicholas Metropolis & Stanislaw Ulam, 1949). It belongs to a class of algorithms that rely on repeated random sampling to obtain numerical results — using randomness to solve problems that may be deterministic in principle. The method is used across finance, engineering, supply chain, and physics for risk assessment, decision-making, and optimisation. Bootstrapping is a closely related resampling approach (described later in this section) that uses the observed sample, rather than an assumed distribution, to estimate uncertainty in a statistic.

2.7.1 Key Features of Monte Carlo Simulation

• Versatility: Applicable to a wide range of problems, including those with a complex mix of variables and relationships.
• Uncertainty Modeling: Allows the incorporation of uncertainty in variables to simulate a wide range of scenarios.
• Outcome Forecasting: Provides a distribution of possible outcomes rather than a single deterministic solution, which helps in understanding the risk and variability.

2.7.2 How Monte Carlo Simulation Works

1. Define a Domain of Possible Inputs: Monte Carlo Simulation relies on a random sampling of inputs from a probability distribution for each variable to simulate different scenarios.

2. Generate Random Inputs: For each variable, generate random values that follow the defined probability distributions.

3. Perform a Deterministic Computation: For each set of random inputs, compute the outputs using a deterministic model.

4. Aggregate the Results: After repeating the process a large number of times, aggregate the results to create a probability distribution of the outcomes.

2.7.3 Case Study: Investment Risk Analysis

Background

Consider an investment portfolio consisting of stocks, bonds, and real estate. The goal is to understand the portfolio’s risk profile over the next year, considering the volatility of the market.

Step 1: Define the Domain of Possible Inputs

• Stocks: Expected return of 8% with a standard deviation of 10%.
• Bonds: Expected return of 4% with a standard deviation of 5%.
• Real Estate: Expected return of 6% with a standard deviation of 8%.

Each asset class has its return modeled as a normal distribution with the given expected returns and standard deviations.

Step 2: Generate Random Inputs

Using a random number generator, sample from the defined distributions to simulate possible returns for each asset class over the next year.

Step 3: Perform Deterministic Computation

For each simulation, calculate the overall portfolio return based on the randomly generated returns for each asset class and their weights in the portfolio.

Step 4: Aggregate the Results

After thousands of simulations, compile the results to create a probability distribution of the portfolio’s expected return.

Results and Analysis

• Probability Distribution: The simulation might show that there is a 95% chance the portfolio’s return will fall between -5% and +20% over the next year.
• Risk Assessment: The wide range indicates significant risk, primarily driven by stock-market volatility.
• Decision Making: An investor seeking lower risk might adjust the portfolio — for instance, by increasing the weight of bonds.

2.7.4 Example problem of Net Present Value

Net Present Value (NPV)

Net Present Value (NPV) is a financial metric used to evaluate the profitability of an investment or project. It represents the difference between the present value of cash inflows and the present value of cash outflows over a period of time. NPV is a core component of corporate finance and investment analysis, helping decision-makers assess the value of undertaking specific investments or projects by considering the time value of money.

The formula for calculating NPV is:

\[ NPV = - \text{Initial Investment} + \sum_{t=1}^{5} \frac{C_t}{(1+r)^t} \]

where:

• \(NPV\) = Net Present Value
• \(C_t\) = Net cash inflow during the period \(t\)
• \(r\) = Discount rate (or the required rate of return)
• \(t\) = Number of time periods
• \(n\) = Total number of periods

The discount rate (\(r\)) is a crucial factor in NPV calculation, reflecting the opportunity cost of capital, or the return that could be earned on an investment with a similar risk profile.

Example

Let’s consider a project with an initial investment of $100,000 (considered a cash outflow, or negative cash flow), and expected to generate an average cash flow of $30,000 annually for 5 years. The company’s required rate of return (discount rate) is 10%.

We’ll calculate the NPV to determine if the project should be undertaken. The formula will be applied as follows, considering the average cash flow remains constant over the period:

Given Data

• Initial Investment: $100,000
• Average Annual Cash Flow: $30,000 for 5 years
• Discount Rate: 10% (0.10)

Calculating NPV

Using the NPV formula:

\[ NPV = - \text{Initial Investment} + \sum_{t=1}^{5} \frac{C_t}{(1+r)^t} \]

where \(C_t = 30,000\) (constant for each year), \(r = 0.10\), and \(t\) is the year.

The calculated NPV of the project is $13,723.60.

Since the NPV is positive, it indicates that the project is expected to generate a return above the required rate of return of 10%. Therefore, according to NPV criteria, the project would be considered a good investment as it adds value to the company.

This example demonstrates how NPV can be a decisive factor in evaluating the feasibility of investments or projects, guiding businesses in making informed decisions by considering the time value of money and the expected profitability.

2.7.5 Monte Carlo Simulation to Estimate Net Present Value

Parameters

• initial_investment = 100,000
• project_lifetime = 5
• average_annual_cash_flow = 30,000
• std_dev_annual_cash_flow = 10,000
• discount_rate = 0.10
• num_simulations = 10,000

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2.8 Worked Example: Monte Carlo NPV — Coffee Shop

Let’s consider a simplified, hypothetical example of a small business considering the purchase of a new piece of machinery.

2.8.1 Scenario:

A local coffee shop wants to purchase a new espresso machine to increase its coffee-making capacity and improve the quality of its coffee.

2.8.2 Cost-Benefit Analysis (CBA):

• Costs:

  • The espresso machine costs $10,000.
  • Installation fees amount to $500.
  • Additional training for staff costs $300.

• Benefits:

  • Increased sales due to higher quality coffee and faster service are estimated to be $3,000 extra per year.
  • The new machine may attract an additional 50 customers per month, with an average spend of $4 per customer, leading to $2,400 annually.

Calculation of Total Costs and Benefits:

• Total Costs (Year 0): $10,800 ($10,000 + $500 + $300)
• Total Benefits (Year 1 onwards): $5,400 per year ($3,000 + $2,400)

Now, let’s move to calculate the NPV of the investment over a 5-year period.

2.8.3 Net Present Value (NPV):

• Discount Rate: Assuming a discount rate of 7%, which reflects the cost of capital and the risk associated with the investment.

We will now calculate the NPV over a 5-year period.

NPV Formula:

\[ NPV = \sum_{t=1}^{n} \frac{R_t}{(1 + d)^t} - C_0 \]

Where: - \(R_t\) = Net cash inflow-outflows during a single period t - \(d\) = Discount rate - \(C_0\) = Initial investment costs - \(n\) = Number of time periods

Given the figures:

• \(R_t\) = $5,400 (for t = 1 to 5)
• \(d\) = 0.07 (7 percent discount rate)
• \(C_0\) = $10,800
• \(n\) = 5 years

Let’s calculate the NPV.

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The Net Present Value (NPV) of the coffee shop’s investment in the new espresso machine over a 5-year period, with a 7% discount rate, is approximately $11,341.07.

This positive NPV suggests that, given the assumptions in our analysis, the investment would add value to the coffee shop and should be considered a financially sound decision. The benefits from increased sales and new customers outweigh the initial costs when considering the time value of money.

2.9 Bootstrapping

Bootstrapping is a non-parametric resampling technique that estimates the sampling distribution of a statistic by drawing many random samples — with replacement — from the observed data. Where Monte Carlo simulation generates samples from an assumed probability distribution, bootstrapping uses the empirical distribution of the data themselves. This makes it particularly valuable when the underlying distribution is unknown or when analytical formulas for standard errors and confidence intervals are intractable.

2.9.1 How Bootstrapping Works

1. Start with the observed sample of size \(n\).
2. Draw a bootstrap resample of size \(n\) by sampling with replacement from the original data. Some observations will appear multiple times; others may not appear at all.
3. Compute the statistic of interest (mean, median, regression coefficient, etc.) on the resample.
4. Repeat steps 2–3 a large number of times (typically \(B = 1{,}000\) to \(10{,}000\)).
5. Build the sampling distribution of the statistic from the \(B\) replicate values, and use it to compute standard errors, confidence intervals, or bias estimates.

2.9.2 Bootstrap vs. Monte Carlo: A Quick Comparison

Aspect	Monte Carlo Simulation	Bootstrapping
Source of samples	Assumed probability distribution	Observed data (resampled with replacement)
Requires distributional assumption	Yes	No
Best for	Forward-looking what-if scenarios	Estimating uncertainty in a statistic from existing data
Typical use	Project NPV, portfolio risk, queueing systems	Confidence intervals, standard errors, hypothesis tests when assumptions fail

2.9.3 Worked Example: Bootstrap Confidence Interval for the Mean

Suppose a coffee-shop manager records daily revenue for 30 days and wants a 95% confidence interval for the true mean daily revenue, without assuming a normal distribution.

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The resulting interval is built from the data alone — no normality assumption, no closed-form variance formula. The same idea generalises to medians, regression coefficients, model accuracy, and any other statistic computable from the sample.

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Summary

Concept	Description
Structuring Decisions
Decision Analysis	Systematic approach for evaluating alternatives and choosing among them when outcomes are uncertain
Five-Step Structuring Process	Define the problem, identify objectives, recognise alternatives, understand constraints, and consider uncertainties
Incorporating Data Uncertainty	Building scenarios with probabilities so the impact of unknowns on the decision is visible up front
Decision Criteria and Utility
Expected Value	Choose the alternative with the highest probability-weighted average outcome
Maximin and Minimax Regret	Risk-averse rule that protects the worst case, or that minimises the gap to the best possible payoff in hindsight
Expected Utility	Choose the alternative with the highest probability-weighted utility, reflecting the decision-maker's preferences
Utility Function and Risk Attitude	Concave for risk-averse, convex for risk-seeking, and linear for risk-neutral preferences over wealth
Multi-Criteria Decision Analysis
MCDA Framework	Method for evaluating alternatives against multiple, often conflicting criteria, weighted by importance
Weighting and Scoring	Assigning relative weights to criteria and scoring each alternative against each criterion
AHP, TOPSIS, and MAUT	Hierarchy-based, distance-based, and utility-based MCDA techniques for structured comparison
Sensitivity Analysis and Risk
Sensitivity Analysis	Studies how variation in inputs drives variation in outputs to identify the variables that most influence results
Risk Assessment Process	Identify potential threats, analyse their likelihood and impact, and prioritise the most severe risks
Integrated Risk Strategy	Combining sensitivity findings with risk assessment to build robust strategies under uncertainty
Sources of Uncertainty and Risk
Parameter and Model Uncertainty	Limited data on parameter values, and assumptions baked into models that may not match reality
Decision and Environmental Uncertainty	Incomplete information about alternatives, and unpredictable shifts in the external environment
Market and Credit Risk	Losses from market-wide moves and from borrower default on debt obligations
Operational, Liquidity, and Strategic Risk	Process failures, inability to convert assets to cash, and exposure to shifts in strategy or competition
Probability and Statistical Inference
Discrete Probability Distributions	Binomial and Poisson distributions for counts of events under defined conditions
Continuous Probability Distributions	Normal, uniform, and exponential distributions for measurements that take any value in a range
Statistical Inference	Point estimates, confidence intervals, hypothesis tests, and p-values for moving from sample to population
Monte Carlo, NPV, and Bootstrapping
Monte Carlo Simulation	Repeated random sampling from an assumed distribution to build a distribution of outcomes for problems that resist closed-form solutions
NPV Coffee Shop Case Study	Discounted-cash-flow valuation of a new espresso machine over five years at a 7 percent discount rate
Bootstrapping	Non-parametric resampling with replacement from observed data to estimate sampling distributions, standard errors, and confidence intervals without distributional assumptions

Hammond, J. S., Keeney, R. L., & Raiffa, H. (1999). Smart choices: A practical guide to making better decisions. Harvard Business School Press.

Hwang, C.-L., & Yoon, K. (1981). Multiple Attribute Decision Making: Methods and Applications. Springer. https://doi.org/10.1007/978-3-642-48318-9

Knight, F. H. (1921). Risk, uncertainty and profit. Houghton Mifflin.

Metropolis, N., & Ulam, S. (1949). The Monte Carlo Method. Journal of the American Statistical Association, 44(247), 335–341. https://doi.org/10.1080/01621459.1949.10483310

Saaty, T. L. (1980). The Analytic Hierarchy Process: Planning, Priority Setting, Resource Allocation. McGraw-Hill.

von Neumann, J., & Morgenstern, O. (1944). Theory of Games and Economic Behavior. Princeton University Press.