Quantitative Methods
for the Management Sciences
45-760
Course Notes

INSTRUCTORS:
GERARD CORNUEJOLS
MICHAEL TRICK
Graduate School of Industrial Administration
Carnegie Mellon University, Pittsburgh, PA 15213 USA

Fall 1998

• Contents
  • Introduction
  • Grading Components
  • Schedule
  • Mathematical Methods in Business
    • History
    • Mathematical Modeling Today
• Basic Linear Algebra
  • Linear Equations
  • Operations on Vectors and Matrices
  • Linear Combinations
  • Inverse of a Square Matrix
  • Determinants
  • Positive Definite Matrices
• Unconstrained Optimization: Functions of One Variable
  • Derivatives
  • Maximum and Minimum
  • Binary Search
  • Convexity
• Unconstrained Optimization: Functions of Several Variables
  • Gradient
  • Maximum and Minimum
  • Global Optima
• Constrained Optimization
  • Equality Constraints (Lagrangians)
    • Geometric Interpretation
    • Economic Interpretation
  • Equality and Inequality Constraints
  • Exercises
• Modeling with Linear Programming
  • Introductory Example
    • Modeling
    • Solving the Model
    • Using Solver
  • Second Example: Market Allocations
    • Modeling
    • Solving the Model
    • Using Solver
  • Common Features
  • Linear Programming Models
    • Diet Problem
      • Problem Definition
      • Problem Modeling
      • Discussion
    • Workforce Planning
      • Problem Definition.
      • Model.
      • Discussion.
    • Financial Portfolio
      • Problem Definition.
      • Model
      • Discussion
    • Production over Time
      • Problem Description
      • Modeling.
      • Discussion.
    • Investing over Time
    • Problems
• Case Study
• The Simplex Method
  • Linear Programs in Standard Form
  • Solution of Linear Programs by the Simplex Method
  • Alternate Optimal Solutions, Degeneracy, Unboudedness, Infeasibility
• Sensitivity Analysis for Linear Programming
  • Tableau Sensitivity Analysis
    • Cost Changes
    • Right Hand Side Changes
    • New Variable
  • Solver Output
    • Tucker Automobiles
    • Carla's Maps
    • Exercises
• Decision Theory
  • Decisions Under Risk
  • Decision Trees
• Game Theory
  • Two-Person Games
    • Pure Strategies
    • Mixed Strategies
  • n-Person Games
• Network Optimization
  • Introduction
  • Terminology
  • Examples
    • Shortest Paths
    • Maximum Flow
    • Transportation Problem
    • Assignment Problem
  • Unifying Model: Minimum Cost Network Flows
  • Generalized Networks
  • Conclusions
• Data Envelopment Analysis
  • Numerical Example
  • Graphical Example
  • Using Linear Programming
  • Applications
  • Strengths and Limitations of DEA
  • References
• About this document ...