BUSINESS OPTIMIZATION AND SIMULATION

Module 1Introduction

COURSE OBJECTIVES•  Specific goals for this course:

•  Identifying, modeling and solving decision-making problems

•  Being able to understand and address the most common difficulties that may appear in both the formulation and the solution of these problems

•  Modeling and solving using Excel:

•  The most widespread and versatile tool

•  Widely used in companies as a decision-making tool

REFERENCES•  Class notes:

Aula Global

•  Main reference:

Practical Management Science: Winston - Albright

•  Other references:

Spreadsheet Modeling and Decision Analysis: Ragsdale

Applied Management Science: Lawrence, Pasternack

Operations Management: Russell, Taylor

STRUCTURE OF THE COURSE•  Lessons:

1.  Introduction

2.  Optimizing linear models

3.  Optimizing discrete models

4.  Optimizing nonlinear (unconstrained and constrained) models

5.  Simulation

DECISION SUPPORT SYSTEMS•  DSS in practice

•  Decision Support Systems: advanced analytic tools to support decision-making processes

•  DSS use mathematical models to analyze complex situations in finance, industrial or scientific settings.

•  Formal disciplines supporting DSS:

•  Operations Research: Mathematical and computational algorithms to solve mathematical models of interest

•  Management Science: Use of mathematical modeling, statistics and algorithms to enact rational management decisions

DECISION SUPPORT SYSTEMS•  The main analytic tools supported in a DSS include:

•  Optimization:

•  Find the best possible decision among a very large (uncountable) set of alternatives

•  Simulation:

•  Formal approximation of (uncertain) reality (behavior, materials, ideas,...) to save time and money

•  Probability and Statistics:

•  Help to summarize/analyze information, measure risks, prepare forecasts, etc.

DECISION SUPPORT SYSTEMS•  Using a DSS, Continental Airlines saved 40M$ in 2001 after the 9/11 attacks, by producing optimal

decisions to organize its airplanes

•  The Ford company, using DSS, optimized the way they designed and tested their new model prototypes, saving 250M$

•  UPS used DSS to redesign their delivery network, saving 87M$ between 2000 and 2002, and an additional estimated 189M$ up to 2010

•  The NBC network used DSS to improve their strategies to negotiate their advertisement spaces, increasing their profit in more than 200M$

•  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

•  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•  A large collection of real cases:

•  Operations Research: The Science of Better

http://www.scienceofbetter.org

•  Maintained by a professional society: INFORMS

•  Examples of real-world applications in different areas

•  Including an introduction to Operations Research and Management Science

DECISION-MAKING METHODOLOGY

•  Definition of the problem: Description of the alternative decisions, identification of an objective, specification of constraints

•  Formulation of the model: Translation of these parts into a mathematical or modeling framework

•  Solution of the model: Using modeling languages and optimization algorithms•  Validation of the solution: Is the solution implementable? Is it acceptable? Does

it provide reasonable results?•  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

•  Description:

•  A company has 2 separate production centers that manufacture a given product.

•  This product is then delivered to 3 demand areas (geographically separated markets)

•  The delivery of the product to a given area has a cost:

•  90 euros per unit and per 100 Km

•  A function of the distance between centers and demand

•  Other relevant information:

•  Demands and production capacities

EXAMPLE: TRANSPORTATION PROBLEM

•  Data:•  Distances, demands and capacities:

•  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:

•  From these data, find the best way to transport the product so that the total delivery costs are minimized

DistancesM1 M2 M3 Capacity

P1 2.5 1.7 1.8 350

P2 2.5 1.8 1.4 600Demand 325 300 275

EXAMPLE: TRANSPORTATION PROBLEM

•  Model:

•  Finding a mathematical representation for all relevant parts of our decision problem

•  Decision variables. What we want to compute

•  Amount to transport from production center i to market j,

•  Objective. The criterion to define what is best

•  Minimize total delivery costs,

EXAMPLE: TRANSPORTATION PROBLEM

•  Constraints. Limits on what we can or want to do•  Demand in each market,

•  Production limits (or demand limits),

•  Technical constraints,

•  Formulate and solve this problem in Excel

ELEMENTS OF AN OPTIMIZATION PROBLEM

•  Elements of a model (components of its formal model):

•  Variables (unknowns):

•  Take-off times for planes, investment amounts, decisions to make (or not to make), alternatives, etc.

•  Objective function to optimize:

•  Profits, time, energy, costs, etc.

•  Constraints: limits on the variables, such as:

•  Take-off times for planes limited by safety reasons; amounts to invest by a mutual fund should be sufficiently diversified

EXAMPLE: TRANSPORTATION PROBLEM

•  A formal model

•  Generic data:

•  We have i = 1,...,n production centers and j = 1,...,m markets

•  For all of them we need the values of (parameters):

•  ai ≣ maximum production in each center i

•  bj ≣ estimated demand in each market j

•  cij ≣ unit transportation cost from supply center i to market j

•  Modeling elements:

•  Decision variables: Amount to deliver from i to j : xij

EXAMPLE: TRANSPORTATION PROBLEM

•  A formal model

•  Objective function: minimize delivery costs

•  Constraints: satisfy demand estimates

•  Constraints: impose production limits

•  Constraints: technical conditions, xij ≥ 0

SOLVING THE PROBLEM•  Once the problem has been formulated, its solution is obtained by applying an optimization

algorithm

•  These algorithms are different depending on the properties of the problem to solve:

•  Linear and nonlinear problems

•  Discrete vs. continuous problems

•  Global (vs. local) optimization

•  Stochastic optimization

•  Multiobjective problems

•  We will cover linear, discrete (linear) and nonlinear optimization problems

SOLVING THE PROBLEM•  Solving a problem:

•  Computing optimal values for the variables

•  General solution codes for different classes of optimization problems:

•  Linear problems: CPLEX, XPressMP

•  Discrete problems: CPLEX, MOSEK

•  Nonlinear problems: KNITRO, SNOPT

•  Global optimization: BARON

SOLVING THE PROBLEM•  Compute optimal values for the variables:

•  General problem-description languages:

•  GAMS, AIMMS, AMPL

•  Web resources: NEOS

http://www.neos-server.org/neos/

•  General-purpose solvers:

•  Solver for Excel

SOLVING THE PROBLEM•  For a given problem, it is convenient to consider separately the model and the

associated data

•  A given model may be used to solve many problems, by using it with different sets of data

•  The approach in the NEOS server

•  And in all description language-based systems

•  Model: mathematical representation of the problem

•  Need a language to describe the model

•  In Excel: cells with formulas

SOLVING THE PROBLEM•  Solving the problem with Excel

•  The Solver tool is available in the menu Data - Solver (verify that this is the case before introducing the model)

•  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•  A veterinary is helping a poultry farmer to design a diet for the feeding of the animals in

its farm

•  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

•  These nutrients are obtained from a mix of three different foodstuffs: corn, fishmeal and synthetic poultry feed.

•  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•  Data summary

•  Numerical data for the diet problem:

•  Your goal:

•  The farmer wishes to determine the composition of a diet with minimum cost that satisfies the nutritional requirements

Corn Fishmeal Feed RequirementsIron 2.5 3 1 3

Vitamins 1 3 2 4Cost 0.3 0.5 0.2

EXAMPLES: PROD PLANNING•  A production planning problem

•  An engineering plant manufactures two alloys, A and B, by combining three elements: iron, lead and tin.

•  The following table provides information on the requirements of these three products, their demand and the unit profits associated to them

•  Write down the mathematical formulation of the model that optimices the total profit for the industry

Units per KgResources Prod A Prod B Availability

Iron 7 4 56

Lead 3 5 45

Tin 4 3 48

Unit profit 10 8