BUSINESS OPTIMIZATION
AND SIMULATION
Module 1
Introduction
COURSE OBJECTIVES
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Specific goals for this course:
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Identifying, modeling and solving decision-making problems
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Being able to understand and address the most common difficulties that
may appear in both the formulation and the solution of these problems
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Modeling and solving using Excel:
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The most widespread and versatile tool
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Widely used in companies as a decision-making tool
REFERENCES
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Class notes:
Aula Global
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Main reference:
Practical Management Science: Winston - Albright
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Other references:
Spreadsheet Modeling and Decision Analysis: Ragsdale
Applied Management Science: Lawrence, Pasternack
Operations Management: Russell, Taylor
STRUCTURE OF THE COURSE
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Lessons:
1.  Introduction
2.  Optimizing linear models
3.  Optimizing discrete models
4.  Optimizing nonlinear (unconstrained and constrained)
models
5.  Simulation
DECISION SUPPORT SYSTEMS
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DSS in practice
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Decision Support Systems: advanced analytic tools to support decision-making
processes
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DSS use mathematical models to analyze complex situations in finance, industrial or
scientific settings.
Formal disciplines supporting DSS:
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Operations Research: Mathematical and computational algorithms to solve
mathematical models of interest
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Management Science: Use of mathematical modeling, statistics and algorithms to enact
rational management decisions
DECISION SUPPORT SYSTEMS
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The main analytic tools supported in a DSS include:
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Optimization:
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Simulation:
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Find the best possible decision among a very large (uncountable) set of alternatives
Formal approximation of (uncertain) reality (behavior, materials, ideas,...) to save
time and money
Probability and Statistics:
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Help to summarize/analyze information, measure risks, prepare forecasts, etc.
DECISION SUPPORT SYSTEMS
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Using a DSS, Continental Airlines saved 40M$ in 2001 after the 9/11 attacks, by producing optimal
decisions to organize its airplanes
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The Ford company, using DSS, optimized the way they designed and tested their new model
prototypes, saving 250M$
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UPS used DSS to redesign their delivery network, saving 87M$ between 2000 and 2002, and an
additional estimated 189M$ up to 2010
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The NBC network used DSS to improve their strategies to negotiate their advertisement spaces,
increasing their profit in more than 200M$
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AT&T saved more than 100M$ in the late 90's after optimizing the procedures for the recovery of their
network after any severe failure in the system
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British Telecom (BT) uses DSS techniques to optimize the planning of the work to be conducted by the
40000 engineers/programmers in their payroll. It expects to save 250M$ per year
DECISION SUPPORT SYSTEMS
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A large collection of real cases:
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Operations Research: The Science of Better
http://www.scienceofbetter.org
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Maintained by a professional society: INFORMS
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Examples of real-world applications in different areas
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Including an introduction to Operations Research and
Management Science
DECISION-MAKING
METHODOLOGY
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Definition of the problem: Description of the alternative decisions,
identification of an objective, specification of constraints
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Formulation of the model: Translation of these parts into a mathematical or
modeling framework
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Solution of the model: Using modeling languages and optimization algorithms
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Validation of the solution: Is the solution implementable? Is it acceptable? Does
it provide reasonable results?
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Implementation of the solution: translating this solution into operating
instructions
We will emphasize the definition of the problem, the formulation of the model
and its solution
EXAMPLE: TRANSPORTATION
PROBLEM
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Description:
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A company has 2 separate production centers that manufacture a given product.
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This product is then delivered to 3 demand areas (geographically separated markets)
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The delivery of the product to a given area has a cost:
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90 euros per unit and per 100 Km
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A function of the distance between centers and demand
Other relevant information:
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Demands and production capacities
EXAMPLE: TRANSPORTATION
PROBLEM
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Data:
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Distances, demands and capacities:
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The table provides the distances, dij, between the production centers and the
demand areas (in hundreds of Km), the maximum production capacity in each
center, and the estimated demand in each market:
Distances
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M1
M2
M3
Capacity
P1
2.5
1.7
1.8
350
P2
2.5
1.8
1.4
600
Demand
325
300
275
From these data, find the best way to transport the product so that the total delivery
costs are minimized
EXAMPLE: TRANSPORTATION
PROBLEM
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Model:
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Finding a mathematical representation for all relevant parts of our decision
problem
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Decision variables. What we want to compute
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Amount to transport from production center i to market j,
Objective. The criterion to define what is best
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Minimize total delivery costs,
EXAMPLE: TRANSPORTATION
PROBLEM
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Constraints. Limits on what we can or want to do
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Demand in each market,
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Production limits (or demand limits),
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Technical constraints,
Formulate and solve this problem in Excel
ELEMENTS OF AN
OPTIMIZATION PROBLEM
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Elements of a model (components of its formal model):
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Variables (unknowns):
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Objective function to optimize:
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Take-off times for planes, investment amounts, decisions to make (or not to make),
alternatives, etc.
Profits, time, energy, costs, etc.
Constraints: limits on the variables, such as:
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Take-off times for planes limited by safety reasons; amounts to invest by a mutual
fund should be sufficiently diversified
EXAMPLE: TRANSPORTATION
PROBLEM
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A formal model
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Generic data:
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We have i = 1,...,n production centers and j = 1,...,m markets
For all of them we need the values of (parameters):
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ai ≣ maximum production in each center i
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bj ≣ estimated demand in each market j
•  cij
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≣ unit transportation cost from supply center i to market j
Modeling elements:
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Decision variables: Amount to deliver from i to j : xij
EXAMPLE: TRANSPORTATION
PROBLEM
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A formal model
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Objective function: minimize delivery costs
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Constraints: satisfy demand estimates
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Constraints: impose production limits
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Constraints: technical conditions, xij ≥ 0
SOLVING THE PROBLEM
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Once the problem has been formulated, its solution is obtained by applying an optimization
algorithm
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These algorithms are different depending on the properties of the problem to solve:
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Linear and nonlinear problems
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Discrete vs. continuous problems
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Global (vs. local) optimization
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Stochastic optimization
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Multiobjective problems
We will cover linear, discrete (linear) and nonlinear optimization problems
SOLVING THE PROBLEM
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Solving a problem:
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Computing optimal values for the variables
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General solution codes for different classes of optimization problems:
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Linear problems: CPLEX, XPressMP
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Discrete problems: CPLEX, MOSEK
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Nonlinear problems: KNITRO, SNOPT
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Global optimization: BARON
SOLVING THE PROBLEM
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Compute optimal values for the variables:
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General problem-description languages:
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GAMS, AIMMS, AMPL
Web resources: NEOS
http://www.neos-server.org/neos/
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General-purpose solvers:
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Solver for Excel
SOLVING THE PROBLEM
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For a given problem, it is convenient to consider separately the model and the
associated data
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A given model may be used to solve many problems, by using it with different sets of
data
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The approach in the NEOS server
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And in all description language-based systems
Model: mathematical representation of the problem
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Need a language to describe the model
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In Excel: cells with formulas
SOLVING THE PROBLEM
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Solving the problem with Excel
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The Solver tool is available in the menu Data - Solver (verify that this is the case
before introducing the model)
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If it is not available, you should go to File - Options - Complements - Manage, and
you would see the following screen:
Before using the tool, you must introduce the model in the Excel sheet
EXAMPLES: DIET PROBLEM
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A veterinary is helping a poultry farmer to design a diet for the feeding of the animals in
its farm
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The main requirements of this diet are that it should provide a minimum of 3 units of
iron and 4 units of vitamins per week
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These nutrients are obtained from a mix of three different foodstuffs: corn, fishmeal
and synthetic poultry feed.
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Each Kg of corn provides 2.5 units of iron and 1 unit of vitamins, each Kg of fishmeal
has 3 units of iron and 3 units of vitamins, and each Kg of poultry feed has 1 unit of
iron and 3 units of vitamins
The prices per Kg of the corn, fishmeal and feed are 0.3, 0.5 and 0.2 respectively
EXAMPLES: DIET PROBLEM
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Data summary
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Numerical data for the diet problem:
Corn Fishmeal
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Feed
Requirements
Iron
2.5
3
1
3
Vitamins
1
3
2
4
Cost
0.3
0.5
0.2
Your goal:
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The farmer wishes to determine the composition of a diet with
minimum cost that satisfies the nutritional requirements
EXAMPLES: PROD PLANNING
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A production planning problem
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An engineering plant manufactures two alloys, A and B, by combining three elements: iron, lead and tin.
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The following table provides information on the requirements of these three products, their demand and
the unit profits associated to them
Units per Kg
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Resources
Prod A
Prod B Availability
Iron
7
4
56
Lead
3
5
45
Tin
4
3
48
Unit profit
10
8
Write down the mathematical formulation of the model that optimices the total profit for the industry