Introduction to Operations Research

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Introduction to Operations Research Prof.
Fernando Augusto Silva Marins www.feg.unesp.br/f
marins fmarins_at_feg.unesp.br
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What Is Management Science
(Operations Research, Operational Research ou
ainda Pesquisa Operacional)?

• Management Science is the discipline that adapts
  the scientific approach for problem solving to
  help managers make informed decisions.

• The goal of management science is to recommend
  the course of action that is expected to yield
  the best outcome with what is available.

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What Is Management Science?

• The basic steps in the management science problem
  solving process involves
• Analyzing business situations (problem
  identification)
• Building mathematical models to describe them
• Solving the mathematical models
• Communicating/implementing recommendations based
  on the models and their solutions (reports)

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The Management Science Process

• The four-step management science process

Problem definition
Mathematical modeling
Solution of the model
Communication/implementationof results
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The Management Science Process

• Management Science is a discipline that adopts
  the scientific method to provide management with
  key information needed in making informed
  decisions.
• The team concept calls for the formation of
  (consulting) teams consisting of members who come
  from various areas of expertise.

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The Management Science Approach

• Logic and common sense are basic components in
  supporting the decision making process.
• The use of techniques such as
• Statistical inference
• Mathematical programming
• Probabilistic models
• Network and computer science
• Simulation

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Using Spreadsheets in Management Science Models

• Spreadsheets have become a powerful tool in
  management science modeling.
• Several reasons for the popularity of
  spreadsheets
• Data are submitted to the modeler in spreadsheets
• Data can be analyzed easily using statistical
  (Data Analysis Statistical Package) and
  mathematical tools (Solver Optimization Package)
  readily available in the spreadsheet.
• Data and information can easily be displayed
  using graphical tools.

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Classification of Mathematical Models

• Classification by the model purpose
• Optimization models
• Prediction models

• Classification by the degree of certainty of the
  data in the model
• Deterministic models (Mathematical Programming)
• Probabilistic (stochastic) models (Simulation)

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Examples of Management Science Applications

• Linear Programming was used by Burger King to
  find how to best blend cuts of meat to minimize
  costs.
• Integer Linear Programming model was used by
  American Air Lines to determine an optimal flight
  schedule.
• The Shortest Route Algorithm was implemented by
  the Sony Corporation to developed an onboard car
  navigation system.

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Examples of Management Science Applications

• Project Scheduling Techniques were used by a
  contractor to rebuild Interstate 10 damaged in
  the 1994 earthquake in the Los Angeles area.
• Decision Analysis approach was the basis for the
  development of a comprehensive framework for
  planning environmental policy in Finland.
• Queuing models are incorporated into the overall
  design plans for Disneyland and Disney World,
  which lead to the development of waiting line
  entertainment in order to improve customer
  satisfaction.

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Is Operations Research really important?
INFORMS 2007
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Sucessos da Pesquisa Operacional em Logística
61 trabalhos 42
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Edelman métodos empregados
Todos finalistas
Somente logística
Simulação estocástica discreta é popular na
indústria...
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FINALISTAS EDELMAN 1984-2007
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FINALISTAS EDELMAN 1984-2007
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Optimization Models

• Many managerial decision situations lend
  themselves to quantitative analyses.
• A Mathematical Model consists of
• Objective function with one or more Control
  /Decision Variables to be optimised.
• Constraints (Functional constraints , ³,
  restrictions that involve expressions with one or
  more Control /Decision Variables)

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The Galaxy Industries Production Problem

• Galaxy manufactures two toy doll models
• Space Ray.
• Zapper.

• Resources are limited to
• 1000 pounds of special plastic.
• 40 hours of production time per week.

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Galaxy Industries Production Problem

• Technological input
• Space Rays uses 2 of plastic and 3 min of labor
• Zappers uses 1 of plastic and 4 min of labor

• Marketing requirement
• Total production cannot exceed 700 dozens.
• Number of dozens of Space Rays cannot exceed
  number of dozens of Zappers by more than 350.

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The Galaxy Industries Production Problem

• The current production plan calls for
• Producing as much as possible of the more
  profitable product, Space Ray (8 profit per
  dozen).
• Use resources left over to produce Zappers (5
  profit
• per dozen), while remaining within the marketing
  guidelines.

• The current production plan consists of
• Space Rays 450 dozen
• Zapper 100 dozen
• Profit 4,100 per week

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• Management is seeking a production schedule that
  will increase the companys profit.

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A Linear Programming model can provide an
insight and an intelligent solution to this
problem.
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Defining Control/Decision Variables

• Ask, Does the decision maker have the authority
  to decide the numerical value (amount) of the
  item?
• If the answer yes it is a control/decision
  variable.
• By very precise in the units (and if appropriate,
  the time frame) of each decision variable.

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The Galaxy Linear Programming Model

• Decisions variables

• X1 Weekly production level of Space Rays
• X2 Weekly production level of Zappers

(in dozens)
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Objective Function

• The objective of all optimization models, is to
  figure out how to do the best you can with what
  youve got.
• The best you can implies maximizing something
  (profit, efficiency...) or minimizing something
  (cost, time...).

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The Galaxy Linear Programming Model

• Objective Function

• Decisions variables
• X1 Weekly production of Space Rays,
• X2 Weekly production of Zappers

Space Ray- 8/dozen Zappers 5/dozen

• Weekly profit, to be maximized

Max 8X1 5X2
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Writing Constraints

• Create a limiting condition in words in the
  following manner(The amount of a resource
  required) (Has some relation to) (The
  availability of the resource)
• Make sure the units on the left side of the
  relation are the same as those on the right
  side.
• Translate the words into mathematical notation
  using known or estimated values for the
  parameters and the previously defined symbols for
  the decision variables.

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Writing Constraints
Decisions variables X1 Space Rays, X2 Zappers

• 2X1 1X2 1000 (Plastic)

Space Rays uses 2 of plastic and 3 min of
labor Zappers uses 1 of plastic and 4 min of labor
3X1 4X2 2400 (Prod Time - Min)
X1 X2 700 (Total production)
There is 1000 of special plastic and 40 hours
(2,400 min) of production time/week. Total
production ? 700, Number Space Rays cannot exceed
number of dozens of Zappers by more than 350,
X1 - X2 350 (Mix)
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Writing Constraints

• Additional constraints

Non negativity constraint - X ³ 0 Lower bound
constraint - X ³ L Upper bound constraint - X
U Integer constraint - X integer Binary
constraint - X 0 or 1
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The Galaxy Linear Programming Model
Non negativity constraint Lower bound constraint
- Upper bound constraint - Integer
constraint Binary constraint

• Max 8X1 5X2 (Weekly profit)
• subject to (the constraints)
• 2X1 1X2 1000 (Plastic)
• 3X1 4X2 2400 (Production Time - Min)
• X1 X2 700 (Total production)
• X1 - X2 350 (Mix)

Is there Additional Constraints?
Xj ? 0, j 1,2 (Nonnegativity)
Integers??
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The Graphical Analysis of Linear Programming
The set of all points that satisfy all the
constraints of the model is called a
FEASIBLE REGION
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• Using a graphical presentation we can
  represent
• All the constraints
• The objective function
• The three types of feasible points.

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Graphical Analysis the Feasible Region
The non-negativity constraints
X2
X1
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Graphical Analysis the Feasible Region
X2
1000
700
Total production constraint X1X2 700
(redundant)
500
Infeasible
Feasible
Production Time 3X14X2 2400
X1
500
700
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Graphical Analysis the Feasible Region
X2
1000
The Plastic constraint 2X1X2 1000
700
Total production constraint X1X2 700
(redundant)
500
Infeasible
Production mix constraint X1-X2 350
Feasible
Production Time 3X14X2 2400
X1
500
700
Interior points.
Boundary points.
Extreme points (5 Vertices).

• There are three types of feasible points

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The search for an optimal solution
Max 8X1 5X2
Start at some arbitrary profit, say profit
2,000...
Then increase the profit, if possible...
X2
...and continue until it becomes infeasible
1000
Optimal Profit 4,360 and optimal solution
700
Space Rays 320 dozen Zappers 360 dozen
600
8X1 5X2 3,000
Current solution Space Rays 450, Zapper
100 and Profit 4,100
8X1 5X2 2,000
400
X1
500
250
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Simulation
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Overview of Simulation

• When do we prefer to develop simulation model
  over an analytic model?
• When not all the underlying assumptions set for
  analytic model are valid.
• When mathematical complexity makes it hard to
  provide useful results.
• When good solutions (not necessarily optimal)
  are satisfactory (In general it is the interest
  of the Enterprises).

- A simulation develops a model to numerically
evaluate a system over some time period.
- By estimating characteristics of the system,
the best alternative from a set of alternatives
under consideration (sceneries) can be selected.
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Overview of Simulation

• Continuous simulation systems monitor the system
  each time a change in its state takes place.

- Discrete simulation systems monitor
changes in a state of a system at discrete
points in time.

• Simulation of most practical problems requires
  the use of a computer program.

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Overview of Simulation

• Approaches to developing a simulation model
• Using add-ins to Excel such as _at_Risk or Crystal
  Ball
• Using general purpose programming languages such
  as FORTRAN, PL/1, Pascal, Basic.
• Using simulation languages such as GPSS, SIMAN,
  SLAM.
• Using a simulator software program (ARENA,
  SIMUL8, PROMODEL).

- Modeling and programming skills, as well as
knowledge of statistics are required when
implementing the simulation approach.
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Monte Carlo Simulation

• Monte Carlo simulation generates random events.
• Random events in a simulation model are needed
  when the input data includes random variables.
• To reflect the relative frequencies of the random
  variables, the random number mapping method is
  used.

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JEWEL VENDING COMPANY an example for the
random mapping technique

• Jewel Vending Company (JVC) installs and stocks
  vending machines.
• Bill, the owner of JVC, considers the
  installation of a certain product (Super Sucker
  jaw breaker) in a vending machine located at a
  new supermarket.

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JEWEL VENDING COMPANY
Bill would like to estimate the expected number
of days it takes for a filled machine to become
half empty.

• Data
• The vending machine holds 80 units of the
  product.
• The machine should be filled when it becomes half
  empty.

• Daily demand distribution is estimated from
  similar vending machine placements.
• P(Daily demand 0 jaw breakers) 0.10
• P(Daily demand 1 jaw breakers) 0.15
• P(Daily demand 2 jaw breakers) 0.20
• P(Daily demand 3 jaw breakers) 0.30
• P(Daily demand 4 jaw breakers) 0.20
• P(Daily demand 5 jaw breakers) 0.05

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Random number mapping The Probability function
Approach
Random number mapping uses the probability
function to generate random demand.
A number between 00 and 99 is selected randomly.
The daily demand is determined by the mapping
demonstrated below.
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34
0.30
34
34
34
0.20
0.20
34
34
0.15
34
34
0.10
0.05
45-74
26-44
10-25
75-94
00-09
95-99
Demand
0
1
2
3
4
5
2
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26-44
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Random number mapping The Cumulative
Distribution Approach
F(X)
1.00
Daily demand X is determined by the random number
Y between0 and 1, such that X is the smallest
value for which F(X) ³ Y.
1.00
0.95
0.75
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0.45
Y 0.34
F(1) .25 lt .34 F(2) .45 gt .34
0.34
0.25
0.10
0.00
45
2
1
2
3
4
5
0
X
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Simulation of the JVC Problem

• A random demand can be generated by hand (for
  small problems) from a table of pseudo random
  numbers.
• Using Excel a random number can be generated by
• The RAND() function
• The random number generation option (ToolsgtData
  Analysis)

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Simulation of the JVC Problem

• An illustration of generating a daily random
  demand.

• Since we have two digit probabilities, we use the
  first two digits of each random number.

0
1
3
4
5
2
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Simulation of the JVC Problem
Simulation is repeated and stops once total
demand reaches 40 or more.
The number of simulated days required for the
total demand to reach 40 or more is recorded.
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Simulation Results and Hypothesis Tests

• The purpose of performing the simulation runs is
  to find the average number of days required to
  sell 40 jaw breakers.
• Each simulation run ends up with (possibly) a
  different number of days.

• Hypothesis test is conducted to test
• whether or not m 16.
• Null hypothesis H0 m 16
• Alternative hypothesis HA m gt 16

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