1 introduction to operations research prof. fernando augusto silva marins fmarins...
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Introduction to Operations Research
Prof. Fernando Augusto Silva Marins
www.feg.unesp.br/[email protected]
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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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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)
What Is Management Science?
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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.
11 INFORMS 2007
Is Operations Research really important?
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61 trabalhos = 42%
Categoria OcorrênciasGoverno 25Transportes 22PCP 21Serviços 16Marketing 11Rede logística 8Finanças 7Adm. Pessoal 7Manufatura 6Compras 5P&D 5Manutenção 4Estoques 3Engenharia 3Armazenagem 2TOTAL 145
Sucessos da Pesquisa Operacional em
Logística
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Edelman: métodos empregados
Todos finalistas
Somente logística
Método OcorrênciasOtimização 61Heurísticas 17Estatística 12Simulação 11Mistos 9DSS 5Contr. estoques 5Análise de risco 4Revenue mngt 4Prog. Dinâmica 4Filas 4Soft systems 2System dynamics 2Expert systems 1Delphi 1Adm. de projetos 1DEA 1N/D 1TOTAL 145
Método Transp PCP Rede Compr. Estoq Armaz. TOTALOtimização 11 10 7 4 1 33Heurísticas 7 5 12Mistos 2 1 1 4Contr. estoques 2 2 4Simulação 1 1 2Revenue mngt 2 2Prog. Dinâmica 1 1 2Expert systems 1 1Análise de risco 1 1TOTAL 22 21 8 5 3 2 61
Simulação estocástica discreta é popular na
indústria...
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Ano Empresa Título do Trabalho1996 South African National Defense Force* "Guns or Butter: Decision Support for Determining the Size and Shape of the
South African National Defense Force (SANDF)"1996 The Finance Ministry of Kuwait "The Use of Linear Programming in Disentangling the Bankruptcies of al-Manakh
Stock Market Crash1996 AT&T Capital "Credit and Collections Decision Automation in AT&T Capital's Small-Ticket
Business"1996 British National Health Service "A New Formula for Distributing Hospital Funds in England"1996 National Car Rental System, Inc. "Revenue Management Program"1996 Procter and Gamble "North American Product Supply Restructuring at Procter & Gamble"1996 Federal Highway Administration/California Department
of Transportation"PONTIS: A System for Maintenance Optimization and Improvement of U.S. Bridge Networks "
1995 Harris Corporation/Semiconductor Sector* "IMPReSS: An Automated Production-Planning and Delivery-Quotation System at Harris Corporation - Semiconductor Sector"
1995 Israeli Air Force "Air Power Multiplier Through Management Excellence"1995 KeyCorp "The Teller Productivity System and Customer Wait Time Model"1995 NYNEX "The Arachne Network Planning System"1995 Sainsbury's "An Information Systems Strategy for Sainsbury’s"1995 SADIA "Integrated Planning for Poultry Production"1994 Tata Iron & Steel Company, Ltd.* "Strategic and Operational Management with Optimization at Tata Steel"1994 Bellcore "SONET Toolkit: A Decision Support System for the Design of Robust and Cost-
Effective Fiber-Optic Networks"1994 Chinese State Planning Commission and the World "Investment Planning for China’s Coal and Electricity Delivery System"1994 Digital Equipment Corp. "Global Supply Chain Management at Digital Equipment Corp."1994 Hanshin Expressway Public Corporation "Traffic Control System on the Hanshin Expressway"1994 U.S. Army "An Analytical Approach to Reshaping the Army"1993 AT&T* "AT&T's Call Processing Simulator (CAPS) Operational Design for Inbound Call
Centers"1993 Frank Russell Company & The Yasuda Fire and Marine
Insurance Co. Ltd."An Asset/Liability Model for a Japanese Insurance Company Using Multistage Stochastic Programming"
1993 North Carolina Department of Public Instruction "Data Envelopment Analysis of Nonhomogeneous Units: Improving Pupil Transportation in North Carolina"
1993 National Aeronautic and Space Administration (NASA) "Management of the Heat Shield of the Space Shuttle Orbiter: Priorities and Recommendations Based on Risk Analysis"
1993 Delta Airlines "COLDSTART: Daily Fleet Assignment Model"1993 Bellcore "An Optimization Approach to Analyzing Price Quotations Under Business Volume
Discounts"FINALISTAS EDELMAN 1984-2007
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Ano Empresa Título do Trabalho1985 Weyerhaeuser Company* Weyerhaeuser Decision Simulator Improves Timber Profits1985 Canadian National Railways "Cost Effective Strategies for Expanding Rail-Line Capacity Using Simulation and
Parametric Analysis"1985 Pacific Gas and Electric Company "PG&E's State-of-the-Art Scheduling Tool for Hydro Systems"1985 New York, NY, Department of Sanitation "Polishing the Big Apple"1985 Eletrobras and CEPEL, Brazil Coordinating the Energy Generation of the Brazilian System1985 United Airlines United Airlines Station Manpower Planning System1984 Blue Bell, Inc.* Blue Bell Trims Its Inventory1984 The Netherlands Rijkswaterstaat and the Rand Planning the Netherlands' Water Resources1984 Austin, Texas, Emergency Medical Services Determining Emergency Medical Service Vehicle Deployment 1984 Pfizer, Inc. "Inventory Management at Pfizer Pharmaceuticals"1984 Monsanto Corporation "Chemical Production Optimization"1984 U.S. Air Force "Improving Utilization of Air Force Cargo Aircraft"
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 “”, “”, “=” 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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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.
Technological input– Space Rays uses 2 of plastic and 3 min of labor– Zappers uses 1 of plastic and 4 min of labor
Galaxy Industries Production Problem
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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 dozenZapper = 100 dozenProfit = $4,100 per week
The Galaxy Industries Production Problem
8(450) + 5(100)
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Management is seeking a production schedule that will
increase the company’s 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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Decisions variables::
The Galaxy Linear Programming Model
–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 you’ve got.
“The best you can” implies maximizing something (profit, efficiency...) or minimizing something (cost, time...).
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Objective Function:
The Galaxy Linear Programming Model
Max 8X1 + 5X2
– Weekly profit, to be maximized
Decisions variables: :
X1 = Weekly production of Space Rays,
X2 = Weekly production of Zappers Space Ray- $8/dozen
Zappers $5/dozen
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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
2X1 + 1X2 1000 (Plastic)
3X1 + 4X2 2400 (Prod Time - Min)
X1 + X2 700 (Total production)
X1 - X2 350 (Mix)
Decisions variables X1 = Space Rays, X2 = Zappers
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,
Space Rays uses 2 of plastic and 3 min of labor
Zappers uses 1 of plastic and 4 min of labor
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Additional constraints
Non negativity constraint - X0
Lower bound constraint - X L
Upper bound constraint - X U
Integer constraint - X = integer
Binary constraint - X = 0 or 1
Writing Constraints
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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)
The Galaxy Linear Programming Model
Xj 0, j = 1,2 (Nonnegativity)
Integers??
Is there Additional Constraints?
Non negativity constraint Lower bound constraint -Upper bound constraint -
Integer constraintBinary constraint
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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 REGIONFEASIBLE 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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The non-negativity constraints
X2
X1
Graphical Analysis – the Feasible Region
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1000
500
Feasible
X2
Infeasible
Production Time3X1+4X2 2400
Total production constraint: X1+X2 700 (redundant)
500
700
The Plastic constraint2X1+X2 1000
X1
700
Graphical Analysis – the Feasible Region
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1000
500
Feasible
X2
Infeasible
Production Time3X1+4X22400
Total production constraint: X1+X2 700 (redundant)
500
700
Production mix constraint:X1-X2 350
The Plastic constraint2X1+X2 1000
X1
700
Graphical Analysis – the Feasible Region
• There are three types of feasible pointsInterior points. Boundary points. Extreme points (5 Vertices).
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The search for an optimal solution
Start at some arbitrary profit, say profit = $2,000...
Then increase the profit, if possible......and continue until it becomes infeasible
Optimal Profit =$4,360 and optimal solution:
600
700
1000
500
X2
X1
8X1 + 5X2 = 2,000
Space Rays = 320 dozen Zappers = 360 dozen
Current solution:Space Rays = 450, Zapper = 100 and Profit = $4,100
Max 8X1 + 5X2
8X1 + 5X2 = 3,000
400
250
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SimulationSimulation
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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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– Continuous simulation systems monitor the system each time a change in its state takes place.
Overview of Simulation
– Simulation of most practical problems requires the use of a computer program.
- Discrete simulation systems monitor changes in a
state of a system at discrete points in time.
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– Approaches to developing a simulation modelUsing add-ins to Excel such as @Risk or Crystal BallUsing 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).
Overview of Simulation
- 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 (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.
JEWEL VENDING COMPANY – an example for the random mapping
technique
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Data– The vending machine holds 80 units of the product.– The machine should be filled when it becomes half empty.
Bill would like to estimate the expected number of days it takesfor a filled machine to become half empty.
Bill would like to estimate the expected number of days it takesfor a filled machine to become half empty.
JEWEL VENDING COMPANY
– Daily demand distribution is estimated from similar vending machine placements.
P(Daily demand = 0 jaw breakers) = 0.10P(Daily demand = 1 jaw breakers) = 0.15P(Daily demand = 2 jaw breakers) = 0.20P(Daily demand = 3 jaw breakers) = 0.30P(Daily demand = 4 jaw breakers) = 0.20P(Daily demand = 5 jaw breakers) = 0.05
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0.100.15
0.20
0.30
0.20
0.05
0 1 2 3 4 5
Random number mapping uses the probability function to generate random demand.
A number between 00 and 99 is selected
randomly.
00-09 10-25 26-44 45-74 75-94 95-99
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The daily demand is determined by the mapping
demonstrated below.
3434343434343434343434343434343434343434
2226-4426-44
Random number mapping – The Probability function Approach
Demand
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1.000.95
0.75
0.45
0.25
0.10
1 2 3 4 50
0.34
1.00
0.00
Random number mapping – The Cumulative Distribution Approach
Daily demand X is determined by the random number Y between0 and 1, such that X is the smallest value for which F(X) Y.
Y = 0.34
2
F(1) = .25 < .34F(2) = .45 > .34
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F(X)
X
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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
(Tools>Data Analysis)
Simulation of the JVC Problem
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Random Two First Total DemandDay Number Digits Demand to Date
1 6506 65 3 32 7761 77 4 73 6170 61 3 104 8800 88 4 145 4211 42 2 166 7452 74 3 19
Random Two First Total DemandDay Number Digits Demand to Date
1 6506 65 3 32 7761 77 4 73 6170 61 3 104 8800 88 4 145 4211 42 2 166 7452 74 3 19
Simulation of the JVC Problem
Since we have two digit probabilities, we use the first two digits of each random number.
00-09 10-25 45-74 75-94 95-9926-44
0 1 3 4 52 3
An illustration of generating a daily random demand.
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Simulation is repeated and stops once total demand reaches 40 or more.
Simulation of the JVC Problem
Random Two First Total DemandDay Number Digits Demand to Date
1 6506 65 3 32 7761 77 4 73 6170 61 3 104 8800 88 4 145 4211 42 2 166 7452 74 3 19
Random Two First Total DemandDay Number Digits Demand to Date
1 6506 65 3 32 7761 77 4 73 6170 61 3 104 8800 88 4 145 4211 42 2 166 7452 74 3 19
The number of “simulated” days required for the total demand to reach 40 or more is recorded.
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– 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.
Simulation Results and Hypothesis Tests
Hypothesis test is conducted to test whether or not m = 16.
Null hypothesis H0 : m = 16
Alternative hypothesis HA : m > 16