Models
The definition of a model:
Models are simplified versions of the things they
represent.
A useful model accurately represents the
relevant or essential characteristics of the object
or decision being studied. (Like a model
airplane studied in a wind tunnel.)
Models
A model is valuable if you make better decisions when
you use it than when you don’t!
Symbolic World
Analysis
Model Results
Interpretation
Abstraction
Management Decisions
Situation Intuition
Real World
Decision Models
• Force you to be explicit about your objectives
Force you to identify the types of decisions that influence those
objectives
Force you to think carefully about variables to include and their
definitions in terms that are quantifiable
Force you to consider what data are pertinent for quantification
Force you to recognize constraints on values that variables may
assume
Allow communication of your ideas and understanding to facilitate
teamwork
Decision Models
• Inputs
– Decisions which are controllable
– Parameters which are uncontrollable
• Outputs
– Performance variables, or objective functions, that
measure the degree of goal attainment
– Consequence variables that display other consequences
so results can be better interpreted
Structure of Mathematical Models
for Decision Support
Non-Quantitative Models (Qualitative)
Quantitative Models: Mathematically links decision variables,
uncontrollable variables, and result variables
Uncontrollable
Variables
Decision Mathematical Result
Variables Relationships Variables
Intermediate
Variables
Examples - Components of Models
Deterministic –vs- Probabilistic Models
• In deterministic models, all of the relevant data
(parameter values) are assumed to be known with
certainty.
• In probabilistic (stochastic) models, some
parameter input is not known with certainty, thus
causing uncertainty in the other variables.
Optimization- Introduction
• We all face decisions about how to use
limited resources such as:
– Oil in the earth
– Land for dumps
– Time
– Money
– Workers
Mathematical Programming
• MP is a field of management science that
finds the optimal, or most efficient, way of
using limited resources to achieve the
objectives of an individual or a business.
• a.k.a. Optimization
Common Optimization Problems
– A manufacturer wants to develop a production
schedule and an inventory policy that will
satisfy demand in future periods and at the same
time minimize the total production and
inventory costs
– A financial analyst would like to establish an
investment portfolio from a variety of stock and
bond investment alternatives that maximizes the
return on investment
Common Optimization Problems
– A marketing manager wants to determine how
best to allocate a fixed advertising budget
among alternative advertising media such as
web, radio, television, newspaper, and magazine
that maximizes advertising effectiveness
– A company had warehouses in a number of
locations. Given specific customer demands, the
company would like to determine how much
each warehouse should ship to each customer so
that total transportation costs are minimized
Supply Chain
General Network Design Questions (e.g.
Walmart)
• How many Walmart stores are needed?
• Where are locations of those stores?
• What are the capacity allocations?
• What markets does each store serve?
• Which supply sources should be fed into each store?
Optimization Examples
The Harris Corporation
Major electronics company in Melbourne, FL.
Developed a computerized optimization-based
production planning system.
Benefits:
• On-time deliveries increased from 75% to 95%.
• Expanded markets and market share.
• Increased profits by $115 million annually.
Optimization Examples
KeyCorp
One of the largest bank holding companies in the US
($66.8 billion in assets).
Developed a system to measure branch activities,
customer wait times, teller productivity.
Benefits:
• Customer processing time reduced 53%.
• Customer wait time reduced.
• Cost savings of $98 million over 5 years.
Optimization Examples
NYNEX
Major telecommunications provider (16.5 million
customers worldwide).
Developed optimization techniques for network planning.
Benefits:
• Improved quality and reliability of network plans.
• Savings of $33 million.
Optimization Examples
Grantham, May, Van Otterloo & Co., an investment
management firm with $26 billion assets, developed a
mixed integer programming model to design portfolios that
achieve investment objectives while minimizing the
number of stocks and transactions required.
• GE Capital, a $70 billion subsidiary of GE financial
services business, developed an optimization model to
allocate and schedule the rental and debt payments of a
leveraged lease which allowed analysts to target
profitability as well as optimize NPV of rental payments.
Optimization Examples
• TFM Investment Group, which was designated as a market
maker in exchange traded funds (ETFs) in 2001, used
integer programming to minimize the cost of producing
creation units while remaining hedged. A second
optimization technique was used to minimize the beta-
dollar difference between the ETF and the portfolio of
constituent stocks which minimized the tracking error
between the current position in the basket of stocks and the
number of short ETFs in TFM’s portfolio.
Characteristics of Optimization Problems
• Decisions
• Constraints
• Objectives
Characteristics of Optimization Problems
• Optimization problems:
– Can be used to support and improve managerial
decision making
– Maximize or minimize some function, called
the objective function, and have a set of
restrictions known as constraints
– Can be linear or nonlinear
Characteristics of Optimization Problems
Optimization problems have constraints on pursuing the
objective of maximization or minimization.
A feasible solution satisfies all the constraints.
An optimal solution (or optimum) is a feasible solution
that results in the largest possible objective-function
value when maximizing (or smallest when minimizing).
Course Objectives
• To enhance technical and analytical skills of
the participants in optimal decision making
in the field of management and business.
• To lay a solid foundation in problem
formulation and model building in different
business environments and resources
optimization situations.
Course Objectives (continued)
• To provide exposure to various
optimization techniques relevant to industry
for reducing cost and improving
productivity
• To show how modeling and decision
support system techniques can be applied
to real life business problems using
software packages
Linear Optimization Problems
• Linear optimization models are also known as
linear programs
• Linear programming:
– A problem-solving approach developed to help
managers make better decisions
– Numerous applications in today’s competitive
business environment
– For instance, GE Capital uses linear
programming to help determine optimal lease
structuring
Linear Programming
• In a recent survey of Fortune 500 firms, 85% of those
responding said that they used linear programming.
• In this course, we discuss some of the LP models that are
most often applied to real applications. In this course’s
examples, you will discover how to build optimization
models to
– purchase television ads
– schedule postal workers
– create an aggregate labor and production plan at a manufacturing company
– create a blending plan to transform crude oils into end products, etc.
Linear Programming
• In a linear-programming problem, the objective function and
the constraints are linear.
• Functions are linear when each variable appears in a separate
term raised to the first power and is multiplied by a constant
(which could be 0).
– Thus 5x1 + 7x2 is a linear function, but 5x12 +7x1x2 is not.
• Linear constraints (or, standard linear constraints) are linear
functions that are restricted to be "less than or equal to",
"equal to", or "greater than or equal to" a constant.
– Thus 2x1 + 3x2 < 19 is a linear constraint, but
2x1 + 3x2 < 19 and 2x1 + 3x1x2 < 19 are not.
Linear Programming
• Many problems fit into the Linear Programming approach
• These are optimization tasks where both the constraints and
the objective are linear functions
• Given a set of variables we want to assign real values to
them such that they
– Satisfy a set of variable constraints represented by linear
equations and/or linear inequalities
– Maximize/minimize a given linear objective function
Assumptions of Linear Programming
• The decision variables are continuous or divisible,
meaning that 3.333 eggs or 4.266 airplanes is an
acceptable solution
• The parameters are known with certainty
• The objective function and constraints exhibit
constant returns to scale (i.e., linearity)
• There are no interactions between decision
variables
