Glossary

6.4 Prescriptive analytics and optimization

5 min read•july 30, 2024

Prescriptive analytics takes business insights to the next level by recommending optimal actions. It uses mathematical techniques to find the best solutions to complex problems, helping companies make smarter decisions about resources, production, and more.

This powerful tool combines historical data, current conditions, and future predictions to provide actionable recommendations. By defining problems, identifying variables and , and interpreting results, businesses can optimize everything from supply chains to investment portfolios.

Prescriptive Analytics in Business

Concepts and Applications

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  • Prescriptive analytics focuses on finding the best course of action to achieve a specific goal or objective, given a set of constraints and available resources
  • Uses mathematical optimization techniques (, , ) to determine the to a problem
  • Helps businesses make data-driven decisions by providing actionable recommendations based on the analysis of historical data, current conditions, and future predictions
  • Process involves defining the problem, identifying and constraints, formulating the optimization model, solving the model, and interpreting the results

Business Applications

  • optimizes the distribution of limited resources (personnel, equipment, budget) across different projects or activities to maximize efficiency or minimize costs
  • determines the optimal production quantities, inventory levels, and scheduling to meet customer demand while minimizing production costs and maximizing resource utilization
  • improves the flow of goods and services from suppliers to customers by optimizing inventory levels, transportation routes, and distribution networks
  • selects the optimal mix of investments (stocks, bonds, real estate) to maximize returns while minimizing risk based on an investor's preferences and constraints
  • assigns employees to shifts or tasks based on their skills, availability, and labor regulations to minimize labor costs and ensure adequate coverage

Optimization Problem Formulation

Decision Variables and Constraints

  • Decision variables are the unknowns or controllable factors in an optimization problem that can be adjusted to achieve the desired outcome (production quantities, resource allocation, investment amounts)
  • Constraints are the limitations or restrictions that must be satisfied by the decision variables based on resource availability, budget, capacity, or other limiting factors
  • Identifying decision variables and constraints involves translating the real-world problem into mathematical terms using equations or inequalities
  • The is the set of all possible solutions that satisfy the constraints of an optimization problem

Objectives and Problem Formulation

  • Objectives are the goals or criteria that the optimization problem seeks to maximize or minimize (profit, cost, resource utilization)
  • Formulating an optimization problem involves expressing the decision variables, constraints, and objectives in mathematical terms
  • The optimal solution is the point within the feasible region that maximizes or minimizes the
  • Example: A company wants to maximize profit by determining the optimal production quantities of two products (A and B) subject to limited raw material availability and production capacity
    • Decision variables: xAx_A (quantity of product A) and xBx_B (quantity of product B)
    • Constraints: 2xA+3xB≤12002x_A + 3x_B \leq 1200 (raw material constraint), xA+xB≤500x_A + x_B \leq 500 (production capacity constraint), xA,xB≥0x_A, x_B \geq 0 (non-negativity constraints)
    • Objective: Maximize Z=50xA+80xBZ = 50x_A + 80x_B (profit function)

Linear and Integer Programming

Linear Programming (LP)

  • Linear programming is a mathematical optimization technique used to solve problems with linear objective functions and linear constraints
  • LP problems involve continuous decision variables that can take on any real value within the feasible region
  • The is a widely used method for solving LP problems by iteratively improving the solution by moving from one extreme point of the feasible region to another until the optimal solution is found
  • Example: A company produces two products (X and Y) using two resources (A and B). The objective is to maximize profit subject to resource constraints
    • Decision variables: xx (quantity of product X) and yy (quantity of product Y)
    • Constraints: 2x+3y≤1202x + 3y \leq 120 (resource A constraint), 4x+y≤1004x + y \leq 100 (resource B constraint), x,y≥0x, y \geq 0 (non-negativity constraints)
    • Objective: Maximize Z=50x+80yZ = 50x + 80y (profit function)

Integer Programming (IP)

  • Integer programming is an extension of linear programming where some or all of the decision variables are required to be integers
  • IP problems are more complex than LP problems and require specialized algorithms (, ) to solve
  • Branch-and-bound systematically enumerates candidate solutions by solving a series of LP relaxations and discarding suboptimal solutions based on bounding criteria
  • Cutting plane methods (Gomory cuts) tighten the LP relaxations and improve the efficiency of the solution process
  • Software tools (, , ) provide powerful solvers for linear and integer programming problems using algebraic modeling languages

Interpreting Prescriptive Analytics Results

Understanding the Optimal Solution

  • Interpreting the results of prescriptive analytics involves understanding the optimal solution, the values of decision variables, and the associated objective value
  • assesses the impact of changes in the input parameters (constraints, objective coefficients) on the optimal solution and helps identify the robustness and stability of the solution
  • (dual prices) indicate the marginal value of relaxing a constraint by one unit and provide insights into the opportunity cost of resources and resource allocation decisions
  • Example: In the LP example, the optimal solution might be x=30x = 30 and y=20y = 20, resulting in a maximum profit of 3,100.SensitivityanalysismayshowthatincreasingtheavailabilityofresourceAbyoneunitwouldincreasetheprofitby3,100. Sensitivity analysis may show that increasing the availability of resource A by one unit would increase the profit by 10 (shadow price)

Communicating Results and Insights

  • Communicating the results of prescriptive analytics to stakeholders requires translating the mathematical solution into actionable recommendations and business insights
  • Visualization techniques (charts, graphs, dashboards) present the results in a clear and concise manner, highlighting the key findings and their implications for decision-making
  • Limitations and assumptions of the optimization model should be acknowledged, and the results should be used as a decision support tool rather than a definitive answer
  • Example: Presenting the optimal production plan to the management team, emphasizing the expected profit increase and the resource utilization levels, while discussing the model's assumptions and potential risks

Key Terms to Review (25)

Branch-and-bound: Branch-and-bound is a mathematical optimization technique used to solve integer and combinatorial problems by systematically exploring the solution space. It involves dividing the problem into smaller subproblems (branching) and calculating bounds to eliminate subproblems that cannot yield better solutions than the current best (bounding). This method efficiently narrows down the search for optimal solutions in complex optimization tasks.
Constraints: Constraints are limitations or restrictions that affect the options available in decision-making processes. In prescriptive analytics and optimization, constraints define the boundaries within which solutions must be found, ensuring that the recommendations produced are feasible and realistic. They help in establishing what is possible or impossible in a given situation, which is essential for generating effective and actionable insights.
Cutting Plane Methods: Cutting plane methods are optimization techniques used to solve linear programming problems by iteratively refining feasible regions of a solution space. These methods involve generating linear inequalities, known as cutting planes, which help to exclude regions of the search space that do not contain optimal solutions, thus steering the search toward the most promising areas. This approach is particularly useful in large-scale problems where traditional methods may struggle to find optimal solutions efficiently.
Decision variables: Decision variables are the adjustable elements in a mathematical optimization model that represent choices available to decision-makers. These variables are fundamental in prescriptive analytics, as they help quantify the impact of various scenarios on outcomes, allowing organizations to evaluate different strategies and make informed decisions.
Descriptive-prescriptive integration: Descriptive-prescriptive integration refers to the process of combining descriptive analytics, which focuses on understanding past data and trends, with prescriptive analytics, which aims to recommend actions for optimal decision-making. This integration allows organizations to not only comprehend what has happened but also to determine the best course of action moving forward, using data-driven insights and optimization techniques.
Feasible Region: The feasible region is the set of all possible solutions to a linear programming problem that satisfy all constraints. It represents the area where all constraints overlap, indicating the combinations of decision variables that lead to valid solutions for optimization problems.
Goal programming: Goal programming is a mathematical optimization technique that aims to find solutions that satisfy multiple, often conflicting objectives or goals simultaneously. It extends traditional linear programming by allowing decision-makers to prioritize different goals, making it particularly useful in complex decision-making scenarios where trade-offs between goals need to be managed.
Gurobi: Gurobi is a powerful optimization solver used for mathematical programming and combinatorial optimization. It helps businesses and researchers find the best possible solutions to complex problems by allowing them to model their decisions mathematically. With its advanced algorithms and high performance, Gurobi is widely utilized in various industries, from finance to logistics, where making optimal decisions can lead to significant improvements in efficiency and cost savings.
IBM ILOG CPLEX: IBM ILOG CPLEX is a powerful optimization software tool used for solving linear programming, mixed integer programming, and quadratic programming problems. It is widely recognized in the field of prescriptive analytics for its ability to handle large-scale optimization tasks efficiently, making it essential for businesses that need to make data-driven decisions.
Integer Programming: Integer programming is a mathematical optimization technique where some or all of the decision variables are required to be integers. This method is particularly useful in scenarios where decisions are discrete in nature, such as assigning resources, scheduling, or planning. Integer programming combines the elements of linear programming with the requirement of integrality, making it a powerful tool in prescriptive analytics and optimization.
Lindo: Lindo is a modeling language used in the context of optimization and prescriptive analytics. It provides a way to formulate complex optimization problems in a structured manner, allowing users to define variables, constraints, and objective functions easily. This language is particularly useful for businesses to make informed decisions by analyzing various scenarios and determining the best course of action based on available data.
Linear programming: Linear programming is a mathematical method used for optimizing a linear objective function, subject to linear equality and inequality constraints. This technique is widely applied in various fields to find the best possible outcome, such as maximizing profits or minimizing costs, while considering limited resources. The use of linear programming in decision-making helps organizations allocate their resources efficiently and effectively, making it essential in resource allocation and optimization processes.
Objective function: An objective function is a mathematical expression that defines the goal of an optimization problem, usually aimed at maximizing or minimizing a specific quantity. It serves as the central component in prescriptive analytics, guiding decision-making processes by quantifying the outcomes of various choices. The objective function is crucial for determining the best possible solution from a set of feasible options based on constraints and available resources.
Optimal solution: An optimal solution refers to the best possible outcome or decision within a set of constraints in a given scenario, ensuring that objectives are met while minimizing costs or maximizing benefits. It is often derived from mathematical models and algorithms in order to achieve the most efficient use of resources. Finding an optimal solution is crucial in decision-making processes, as it leads to improved performance and effectiveness in achieving goals.
Portfolio optimization: Portfolio optimization is the process of selecting the best mix of assets in an investment portfolio to maximize returns while minimizing risk, given certain constraints. This involves using various quantitative techniques and algorithms to analyze potential investments and their correlations, ensuring that the chosen portfolio aligns with the investor's goals and risk tolerance. The process is crucial for effective investment management and is heavily influenced by advancements in analytics and cognitive technologies.
Predictive-prescriptive synergy: Predictive-prescriptive synergy refers to the combined use of predictive analytics and prescriptive analytics to enhance decision-making processes. Predictive analytics forecasts future outcomes based on historical data, while prescriptive analytics provides recommendations on the best course of action. When these two approaches are integrated, they create a powerful tool that not only anticipates what might happen but also guides decision-makers on how to act effectively in response.
Production planning: Production planning is the process of organizing and coordinating the various elements of production to ensure that goods are produced efficiently, on time, and at the right quality. This involves forecasting demand, scheduling production activities, allocating resources, and managing inventory levels to align with business goals. Effective production planning minimizes waste, optimizes resource usage, and enhances overall productivity, making it a crucial component of operational success.
Resource allocation: Resource allocation refers to the process of distributing available resources, such as time, money, personnel, and technology, to various tasks or projects in order to maximize efficiency and achieve specific goals. This concept is crucial for optimizing operations, as it ensures that resources are used in a way that aligns with strategic objectives while minimizing waste and redundancies.
Robust Optimization: Robust optimization is a mathematical approach used to optimize decision-making under uncertainty, ensuring that solutions remain effective across a range of possible scenarios. This method focuses on minimizing the worst-case scenario impacts, making it particularly useful in situations where data may be incomplete or unreliable. By accounting for variability in inputs, robust optimization provides more reliable and resilient solutions in prescriptive analytics.
Sensitivity Analysis: Sensitivity analysis is a method used to determine how different values of an independent variable impact a particular dependent variable under a given set of assumptions. By varying inputs and observing changes in outcomes, this technique helps identify which variables are most influential, aiding in decision-making and model robustness. It is essential for refining models, assessing risk, and optimizing systems across various applications.
Shadow prices: Shadow prices represent the implicit cost of using a resource in optimization problems, particularly in the context of linear programming. They provide insight into how much the objective function (like profit or cost) would change with a small increase in resource availability. Shadow prices help decision-makers understand the value of resources that are scarce or not directly priced in the market.
Simplex algorithm: The simplex algorithm is a mathematical method used for solving linear programming problems, which are optimization problems where the goal is to maximize or minimize a linear objective function subject to linear equality and inequality constraints. It systematically examines the vertices of the feasible region defined by the constraints to find the optimal solution, making it a fundamental tool in prescriptive analytics and optimization.
Stochastic optimization: Stochastic optimization is a mathematical approach used to find the best solution in situations where uncertainty is present, incorporating random variables and probabilistic constraints. This technique is crucial for decision-making processes, as it allows businesses to optimize their operations while accounting for variability in demand, costs, and other factors. By utilizing stochastic models, organizations can make more informed choices that enhance efficiency and reduce risks associated with uncertain environments.
Supply Chain Optimization: Supply chain optimization refers to the process of enhancing a company's supply chain operations to maximize efficiency, reduce costs, and improve overall performance. This involves analyzing and refining each step in the supply chain, from sourcing raw materials to delivering finished products to customers, ensuring that every component works harmoniously for optimal results.
Workforce scheduling: Workforce scheduling is the process of assigning and managing employee work hours to ensure that the right number of staff is available to meet operational needs while balancing employee availability and preferences. Effective workforce scheduling incorporates various factors, such as demand forecasting, labor laws, and individual employee skills, to optimize performance and productivity within an organization.
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