Mathematical Optimization Lecture

Jul 29, 2024

Lecture Notes: Mathematical Optimization

Introduction

  • Topics Covered:
    • Why Mathematical Optimization?
    • What is Mathematical Optimization?
    • Identifying Applications of Optimization

Importance of Mathematical Optimization

  • Usefulness across Industries:
    • Supply Chain Analytics
    • Manufacturing
    • Airlines
    • Energy
    • Finance
    • Sales and Marketing

Specific Industry Applications

  • Supply Chain Analytics:
    • Location planning for facilities
    • Routing of trucks
    • Inventory placement
  • Manufacturing:
    • Production planning
    • Scheduling
    • Workforce planning
  • Airlines:
    • Crew scheduling
    • Flight and aircraft assignment
  • Energy:
    • Demand-supply ratios
    • Resource planning for emissions targets
  • Finance:
    • Portfolio optimization
  • Sales & Marketing:
    • Campaign optimization
    • Sales territory allocation

Who Uses It?

  • Examples of companies leveraging optimization:
    • Google
    • Microsoft
    • NFL (scheduling complexities)
    • Starbucks
    • HP

Predictive vs. Prescriptive Analytics

  • Predictive Analytics:
    • Answers: "What will happen next?"
    • Limitations in execution and decision-making
  • Prescriptive Analytics:
    • Answers: "What should I do?"
    • Supports actionable strategies using optimization

Key Differences

  • Predictive analytics lacks action-oriented insights; prescriptive analytics focuses on identifying optimal actions.
  • Organizations can start prescriptive analytics without complete predictive analytics maturity.

Understanding Optimization Problems

  • Characteristics of Mathematical Optimization Problems:
    • Finding the best course of action among multiple alternatives
    • Control over certain decision factors
    • Presence of limitations (constraints)
    • Specific objectives (e.g., minimize costs or maximize revenue)

Translating Decision Problems

  • Optimization Model Components:
    • Decision Variables:
      • Continuous, Integer, Binary
    • Constraints:
      • Limitations such as budget, capacity, or specific requirements
    • Objective Function:
      • Formulated to represent what is being optimized

Types of Mathematical Optimization Models

  1. Linear Programming (LP)
  2. Integer Linear Programming (ILP)
  3. Mixed Integer Linear Programming (MIP)
  4. Quadratic Programming (QP)
  5. Stochastic Programming
  6. Non-convex Problems

Interaction with Machine Learning

  • Machine Learning as an optimization problem: minimizing error functions
  • Utilizing machine learning outputs as inputs to optimization models
  • Applications include constrained regression and optimal decision trees

Collaboration between Teams

  • Identify groups familiar with optimization within your organization
  • Share insights and explore how machine learning and optimization can complement each other

Case Study: Widget Production and Distribution

  • Decision Problem:
    • Forecast widget demand and reduce transportation costs
  • Optimization Goals:
    • Minimize costs while respecting production minimums and satisfying distribution demands

Translating to Mathematical Formulation

  • Define model objects in grobi Pi for widgets being produced and shipped
  • Identify production facilities and distribution locations as sets
  • Decision variables represent quantity shipped from production to distribution points
  • Constraints ensure production limits and fulfill demand
  • Objective function aims to minimize transportation costs

Conclusion

  • Next steps involve hands-on exercises to work through model formulations
  • Open relevant notebooks to begin practical application.