Computational Methods for Management and Economics Carla Gomes

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Computational Methods forManagement and
EconomicsCarla Gomes

• Module 3
• OR Modeling Approach

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Overview of OR Modeling Approach
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OR Nature

• Operations research involves research
• approach resembles the scientific method
• OR search for optimality (best, optimal
  solution) is an important theme in OR.
• OR - team approach (mathematics, statistics and
  probability theory, economics, business
  administration, computer science, and engineering
  and other areas relevant to the particular
  organization  
• OR adopts an organizational point of view i.e.,
  it tries to meet the goals of the overall
  organization.

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Overview of OR Modeling Approach

•  Define the problem of interest and gather the
  relevant data
• Formulate a mathematical model to represent the
  problem
• Develop a computer base procedure for deriving
  solutions to the problems from the model
• Test the model and refine it as needed
• Prepare for ongoing application of the model as
  prescribed by the management
• Implement
•  
•  

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•  Define the problem of interest and gather
• the relevant data

Need to develop a well-defined statement of the
problem to be considered.  

• objectives,
• constraints (e.g., what can be done with
  available resources)
• inter-relationships between the area to be
  studied and
• other areas in the organization
• possible alternative course of actions
• time limits for making a decision
• . . .
• Difficult phase - ill-defined nature and its
  difficult to be taught. It depends on the
  particular problem and domain of activity.
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2. Formulate a mathematical model to
represent the problem
Formulate the problem into a way that is
convenient for analysis typically using a
mathematical model. Mathematical models
idealized representations expressed in terms of
mathematical symbols and expressions. Famous
mathematical models F ma (Newtons second law
of motion) and Emc² (Einsteins famous equation
of conservation of energy into mass).  
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• Mathematical model of a business problem 
• Decision variables - they represent quantifiable
  decisions to be made, under our control, say x1,
  x2, , xn. The respective values are to be
  determined.
• Objective function expresses the appropriate
  measure of performance as a mathematical function
  of the decision variables. E.g., P 2 x1 x2
  xn
• Constraints any restriction on the values of
  the variables are also expressed mathematically,
  typically by means of equations (e.g. 2 x1 3
  x2 lt 10)
• Parameters of the model constants (namely
  coefficients and right-end-sides) in the
  constraints and objective function.
• Typical OR model
• Choose the decision variables so as to maximize
  (or minimize) the objective function, subject to
  the specified constraints.

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Mathematical Program

• Optimization problem in which the objective and
  constraints are given as mathematical functions
  and functional relationships.
• Optimize Z f(x1, x2, , xn)
• Subject to
• g1(x1, x2, , xn) , , b1

g2(x1, x2, , xn) , , b2

gm(x1, x2, , xn) , , bm
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Comments
a) Gathering of the relevant data --- frequently
difficult. ? value assigned to parameters are
often rough estimates. ?it is important to
analyze how the solution derived from the model
would change if the value assigned to the
parameters (one at a time) were changed to other
plausible values. This process is referred to as
sensitivity analysis (discussed later).

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b) In general real problems can be modeled using
more than one model. ?The process of testing a
model typically leads to a succession of models
that provide better and better representations of
the problem. ?Even possible that two or more
different LP models completely different types of
models may be developed for the same
problem ?Even possible that two or more
completely different types of models may be
developed to help analyze the same problem.
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3. Develop a computer base procedure for
deriving solutions to the problems from the
model ?This phase can be very easy if we are
using a well known mathematical model such as
LP.
 
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4. Test the model and refine it as needed
model validation   Developing a mathematical
model is like developing a large computer program
in general it has bugs!
Some tips  

• Fresh look at the model to check for obvious
  errors or oversights (including a new person who
  didnt participate in the original process)
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• Retrospective test
• use of historical data to reconstruct the past
  --- how well the model and the resulting
  solution would have performed if they had been
  used. (even though the key issue about using the
  model is to predict the future and things may
  change)
• Careful technical review of the model by
  individuals not involved in the design of the
  model

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5 Prepare for ongoing application of the model
as prescribed by the management   Install a well
documented system for applying the model. The
system includes   Mathematical Model Solution
procedure (including post-optimality
procedure) Operation procedures for
implementation   In general --- computer-based
system often integrating databases and management
information systems. Quite often interactive
system (Decision Support Systems) are used.  

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• 6. Implementation
• Critical phase to make sure that the model
  and recommendations of precious phases are
  properly implemented. The benefits of the study
  are reaped only after this phase.
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• Key Aspects
• Management support
• Involvement of users of system - so that they
  feel also
• ownership of the system and they dont reject
  the system
• Feedback on how well the system is working and if
  the
• assumptions of the system continue to be
  satisfied
• throughout the entire period during which the
  system is used.
• Gradually phase in the system while identifying
  and
• eliminating flaws.
• Adoption of management incentives
• for the effective implementation of the system.
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Advantages of mathematical models

• describe a problem in a very concise way
• provide a bridge to sue very powerful computer
  packages
•  

Pitfalls to avoid when using mathematical
models  

• Make sure that the model is a valid
  representation of the problem
• high correlation between the prediction by the
  model and what would actually happen in the real
  world(much of the model validation work is
  performed during the testing phase).
• We should be able to solve the model
  tractability.