Loyola University Chicago

Mathematics and Statistics

MATH 108: Real World Modeling with Mathematics

Course Details
Credit Hours: 3
Prerequisites: None. Fulfills CORE Quantitative Analysis requirement.
Description:  An introduction to mathematical modeling. Topics chosen from linear programming, probability theory, Markov chains, scheduling problems, coding theory, social choice, voting theory, geometric concepts, game theory, graph theory, combinatorics, networks. Emphasis placed upon demonstrating the usefulness of mathematical models in other disciplines, especially the social sciences and business.

Consortium for Mathematics and Its Applications (COMAP), S. Garfunkel, ed., For All Practical Purposes: Mathematical Literacy in Today's World, 10th edition. W. H. Freeman (2015). ISBN-13: 978-1464-12473-0.

Part I - Management Science

Chapter 1: Urban Services [0.5 Weeks] 
Euler Circuits, Finding Euler Circuits, Circuits with Reused Edges

Chapter 2: Business Efficiency [1 Week] 
Hamiltonian Circuits, Fundamental Principle of Counting, Traveling Salesman Problem, Strategies for Solution, Nearest-Neighbor Algorithm, Sorted-Edges Algorithm, Minimum-Cost Spanning Trees, Kruskal's Algorithm

Chapter 3: Planning and Scheduling [1 Week] 
Scheduling Tasks, Assumptions and Goals, List-Processing Algorithm, When is a Schedule Optimal?, Strange Happenings, Critical-Path Schedules, Independent Tasks, Decreasing-Time Lists

Chapter 4: Linear Programming [1 Week] 
Mixture Problems, Mixture Problems Having One Resource, One Product and One Resource: Making Skateboards, Common Features of Mixture Problems, Two Products and One Resource: Skateboards and Dolls, Mixture Charts, Resource Constraints, Graphing the Constraints to Form the Feasible Region, Finding the Optimal Production Policy, General Shape of Feasible Regions, The Role of the Profit Formula: Skateboards and Dolls, Setting Minimum Quantities for Products: Skateboards and Dolls, Drawing a Feasible Region When There are Nonzero Minimum Constraints, Finding Corner Points of a Feasible Region Having Nonzero Minimums, Evaluating the Profit Formula at the Corners of a Feasible Region with Nonzero Minimums, Summary of the Pictorial Method, Mixture Problems Having Two Resources, Two Products and Two Resources: Skateboards and Dolls, The Corner Point Principle, Linear Programming: The Wider Picture, Characteristics of Linear Programming Algorithms, The Simplex Method, An Alternative to the Simplex Method


Part III - Voting and Social Choice

Chapter 9: Social Choice: The Impossible Dream [1.5 Weeks] 
Elections with Only Two Alternatives, Elections with Three or More Alternatives: Procedures and Problems, Plurality Voting and the Condorcet Winner Criterion, The Borda Count and Independence of Irrelevant Alternatives, Sequential Pairwise Voting and the Pareto Condition, the Hare System and Monotonicity, Insurmountable Difficulties: From Paradox to Impossibility, The Voting Paradox of Condorcet, Impossibility, A Better Approach? Approval Voting

Chapter 11: Weighted Voting Systems [2 Weeks] 
How Weighted Voting Works, Notation for Weighted Voting, The Banzhaf Power Index, How to Count Combinations, Equivalent Voting Systems, The Shapley-Shubik Power Index, How to Compute the Shapley-Shubik Power Index, Comparing the Banzhaf and Shapley-Shubik Models


Part IV - Fairness and Game Theory

Chapter 13: Fair Division [1.5 Weeks] 
The Adjusted Winner Procedure, The Knaster Inheritance Procedure, Divide-and-Choose, Cake-Division Procedures: Proportionality, Cake-Division Procedures: The Problem of Envy

Chapter 14: Apportionment [1.5 Weeks] 
The Apportionment Problem, The Hamilton Method, Paradoxes of the Hamilton Method, Divisor Methods, The Jefferson Method, Critical Divisors, The Webster Method, The Hill-Huntington Method, Which Divisor Method is the Best?

Chapter 15: Game Theory: The Mathematics of Competition [as time permits] 
Two-Person Total-Conflict Games: Pure Strategies, Two-Person Total-Conflict Games: Mixed Strategies, A Flawed Approach, A Better Idea, Partial-Conflict Games, Larger Games, Using Game Theory, Solving Games, Practical Applications

Part V - The Digital Revolution

Chapter 16: Identification Numbers [1 Week] 
Check digits, the Zip Code, Bar Codes, Encoding Personal Data

Chapter 17: Transmitting Information [1.5 Weeks] 
Binary Codes, Encoding with Parity-Check Sums, Data Compression, Cryptography


Note: Instructors may vary the topics covered, and length of time devoted to each.