📊Probabilistic Decision-Making Unit 11 – Decision Analysis & Trees in Management

Key Concepts and Terminology

• Decision analysis involves structuring and analyzing decisions to identify the best course of action under uncertainty
• Decision trees visually represent the decision-making process, including decision nodes (squares), chance nodes (circles), and terminal nodes (triangles)
• Probability assesses the likelihood of an event occurring, expressed as a value between 0 and 1
  • Joint probability is the likelihood of two or more events occurring simultaneously
  • Conditional probability is the likelihood of an event occurring given that another event has already occurred
• Expected value (EV) is the weighted average of all possible outcomes, calculated by multiplying each outcome's probability by its value and summing the results
• Payoff is the outcome or reward associated with a particular decision path
• Utility represents the decision-maker's preferences and risk attitude, assigning a value to each possible outcome
• Sunk costs are irrelevant past expenditures that should not influence future decisions
• Opportunity cost is the value of the best alternative foregone when making a decision

Decision-Making Process Overview

• Identify the decision problem and objectives, clearly defining the issue at hand and the desired outcomes
• Generate alternatives by brainstorming potential courses of action
• Gather relevant information, including probabilities, costs, and benefits associated with each alternative
• Structure the decision using a decision tree or other appropriate tool
• Evaluate alternatives by calculating expected values and considering risk preferences
• Make a decision based on the analysis and implement the chosen course of action
• Monitor and adapt the decision as new information becomes available or circumstances change
• Conduct a post-decision evaluation to assess the outcome and learn from the experience

Types of Decision Trees

• Single-stage decision trees involve a single decision point followed by chance events and outcomes
• Multi-stage decision trees include multiple sequential decision points, with each decision leading to chance events and subsequent decisions
• Asymmetric decision trees have different numbers of branches emanating from chance nodes, reflecting varying possible outcomes
• Symmetric decision trees have the same number of branches emanating from each chance node
• Deterministic decision trees contain only decision nodes and known outcomes, without any chance events
• Stochastic decision trees incorporate chance nodes and probabilistic outcomes
• Continuous decision trees involve decision variables that can take on a continuous range of values
• Discrete decision trees feature decision variables with a finite set of possible values

Constructing Decision Trees

• Begin with a decision node, represented by a square, indicating the initial decision to be made
• Draw branches from the decision node, each representing a possible course of action or alternative
• For each alternative, consider the possible outcomes and their associated probabilities
  • If the outcome is uncertain, add a chance node (circle) and draw branches for each possible outcome
  • If the outcome is certain, add a terminal node (triangle) and assign the associated payoff value
• Continue adding decision nodes, chance nodes, and terminal nodes until all possible paths have been explored
• Assign probabilities to each branch emanating from a chance node, ensuring they sum to 1
• Assign payoff values to each terminal node, representing the outcome of that particular path

Probability and Expected Value Calculations

• To calculate the expected value (EV) of a chance node, multiply each possible outcome's probability by its associated payoff and sum the results
  • $EV = \sum_{i=1}^{n} (probability_i \times payoff_i)$
• For decision nodes, calculate the EV of each branch and choose the alternative with the highest EV
• Rollback the tree by working backward from the terminal nodes, calculating EVs at each chance node and selecting the best alternative at each decision node
• The overall expected value of the decision tree is the EV at the initial decision node after rollback
• To calculate the expected value of perfect information (EVPI), subtract the EV of the best alternative without additional information from the EV with perfect information
  • $EVPI = EV_{perfect\,information} - EV_{best\,alternative}$

Risk Assessment and Sensitivity Analysis

• Conduct a sensitivity analysis to determine how changes in probabilities or payoffs affect the decision
  • One-way sensitivity analysis varies a single parameter while holding others constant
  • Two-way sensitivity analysis varies two parameters simultaneously
• Use tornado diagrams to visualize the sensitivity of the decision to changes in each parameter
• Assess the decision-maker's risk attitude using utility functions or risk profiles
  • Risk-averse decision-makers prefer certainty and have concave utility functions
  • Risk-neutral decision-makers focus on expected values and have linear utility functions
  • Risk-seeking decision-makers prefer uncertainty and have convex utility functions
• Incorporate the decision-maker's risk attitude by using certainty equivalents or risk-adjusted payoffs in the decision tree

Applications in Management

• Investment decisions, such as evaluating projects or acquisitions based on expected returns and risks
• Resource allocation, determining the optimal distribution of limited resources among competing projects or departments
• Pricing strategies, analyzing the impact of different pricing options on demand, revenue, and profitability
• Capacity planning, deciding on the optimal level of production capacity based on demand forecasts and associated costs
• Marketing campaigns, evaluating the potential success of different marketing strategies and their associated costs and benefits
• Hiring decisions, assessing the value of different candidates based on their expected contributions and the costs of recruitment and training
• Supply chain management, selecting suppliers and transportation options based on costs, reliability, and lead times
• Risk management, identifying and mitigating potential risks in various business functions

Advanced Techniques and Software Tools

• Influence diagrams combine decision trees with Bayesian networks to model complex relationships among variables
• Monte Carlo simulation involves generating random samples from probability distributions to estimate the distribution of possible outcomes
• Markov decision processes (MDPs) model sequential decision-making problems with uncertain outcomes and state transitions
• Dynamic programming is a recursive optimization technique for solving complex decision problems by breaking them down into smaller subproblems
• Decision support systems (DSS) are software tools that assist decision-makers by providing data analysis, visualization, and scenario testing capabilities
  • Examples of DSS include Palisade's @RISK, TreeAge Pro, and Vanguard Studio
• Analytic hierarchy process (AHP) is a structured technique for organizing and analyzing complex decisions based on pairwise comparisons of criteria and alternatives
• Multi-criteria decision analysis (MCDA) methods, such as ELECTRE and PROMETHEE, help decision-makers evaluate alternatives based on multiple, often conflicting, criteria