15053 February 5 2002

bull1048698 Introduction to Optimization

Handouts Lecture Notes

Overview

bull Course Descriptionbull Course Administration and Logisticsbull What is Management Sciencebull Linear Programming Examples

ndash MSR Marketingndash GTC

bull Handouts Syllabus and General Info Lecture Notes Homework Set 1

Slide 3 contains MIT specific information

Required Materials

bull Coursepack (Course Reader)bull Includes copies of chapters from

ndash Applied Mathematical Programming by BradleyHax amp Magnanti

bull Class website sloanspacemitedubull Other resources

ndash Operations Research by Winstonndash Introduction to Linear Optimization by Bertsimas

amp Tsitsiklisndash Network Flows by Ahuja Magnanti amp Orlin

Grading Policy

bull Homework assignments (27)ndash Weekly ( 10 in total )ndash Nonlinear grading scheme

bull Midterm Exams (25 each)ndash Two in class modterms

bull Final Exam (25)ndash During Finals weekndash Last 13 of subject + modeling

bull Yes it does add up to 102

Course Policy (contrsquod)

bull Webpage

ndash Includesbull class notesbull Spreadsheetsbull Readingsbull assignments (assignment 1 is already there)bull and more

Active Learning

bull Occasionally I will introduce a break in the lecture for you to work on your own or with a partner

bull Please identify your ldquopartnerrdquo nowbull Those on aisle ends may be in a group of siz

e 3

What is Operations ResearchWhat is Management Sciencebull World War II British military leaders asked

scientists and engineers to analyze several military problemsndash Deployment of radarndash Management of convoy bombing antisubmarine

and mining operationsbull The result was called Military Operations

Research later Operations Researchbull MIT was one of the birthplaces of OR

ndash Professor Morse at MIT was a pioneer in the USndash Founded MIT OR Center and helped found ORSA

What is Management Science(Operations Research)bull Today Operations Research and Management

Science mean ldquothe use of mathematical models in providing guidelines to managers for making effective decisions within the state of the current information or in seeking further information if current knowledge is insufficient to reach a proper decisionrdquobull cf Decision science systems analysis

operational research systems dynamicsoperational analysis engineering systemssystems engineering and more

Voices from the past

bull Waste neither time nor money but make the best use of bothndash Benjamin Franklin

bull Obviously the highest type of efficiency is that which can utilize existing material to the best advantagendash Jawaharlal Nehru

bull It is more probable that the average man could with no injury to his health increase his efficiency fifty percentndash Walter Scott

Operations Research Over the Years

bull 1947ndash Project Scoop (Scientific Computation of

Optimum Programs) with George Dantzig andothers Developed the simplex method forlinear programs

bull 1950sndash Lots of excitement mathematical

developments queuing theory mathematicalprogrammingcf AI in the 1960s

bull 1960sndash More excitement more development and grand

plans cf AI in the 1980s

Operations Research Over the Years

bull 1970sndash Disappointment and a settling down NPcompleteness

More realistic expectationsbull 1980s

ndash Widespread availability of personal computersIncreasingly easy access to data Widespreadwillingness of managers to use models

bull 1990sndash Improved use of OR systems

Further inroads of OR technology egoptimization and simulation add-ons to spreadsheets modeling languages large scaleoptimization More intermixing of AI and OR

Operations Research in the 00rsquos

bull LOTS of opportunities for OR as a fieldbull Data data data

ndash E-business data (click stream purchases othertransactional data E-mail and more)

ndash The human genome project and its outgrowthbull Need for more automated decision makingbull Need for increased coordination for efficient

use of resources (Supply chain management)

Optimization

As agelessas time

Optimization in NatureHeron of Alexandria

First Century AD

Angle of

Incidence

Angle of

Incidence

Angle of

Reflection

FERM AT

Angle α1 ofIncidence

Angle α2 ofRefraction

CalculusMaximum

Minimum

Fermat Newton Euler LagrangeG auss and more

Some of the themes of 15053

bull Optimization is everywherebull Models Models Modelsbull The goal of models is ldquoinsightrdquo not

numbersndash paraphrase of Richard Hamming

bull Algorithms Algorithms Algorithms

Optimization is Everywhere

bull The more you know about something the more you see where optimization can be applied

bull Some personal decision makingndash Finding the fastest route home (or to class)ndash Optimal allocation of time for homeworkndash Optimal budgetingndash Selecting a major

Optimization is everywhere

bull Some MIT decision makingndash setting exam times to minimize overlapndash assigning classes to classrooms and time

slots while satisfying constraintsndash figuring out prices for parking and subsidies

for public transportation so as to maximize fairness and permit sufficient access

ndash optimizing in fundraising

Optimization is everywhere

bull Exercise introduce yourself to your neighborand then mention a topic or two with which youare pretty familiar (summer job or major or yourparentrsquos occupation or whatever)

bull Then brainstorm with your neighbor on placeswhere optimization occurs

bull Then choose your favorite 2 or 3 applicationsfrom your list and letrsquos share them

On 15053 and Optimization Tools

bull Optimization is everywhere but optimizationtools are not applied everywhere

bull Goals in 15053 present a variety of toolsfor optimization and illustrate applicationsin manufacturing finance e-businessmarketing and more

bull When you see optimization problems arisein business (and you will) you will know thatthere are tools to help you out

Addressing managerial problems Amanagement science framework

1 Determine the problem to be solved2 Observe the system and gather data3 Formulate a mathematical model of the problem an

d any important subproblems4 Verify the model and use the model for prediction

or analysis5 Select a suitable alternative6 Present the results to the organization7 Implement and evaluate

Linear Programming (our first tooland probably the most important one)bull minimize or maximize a linear objectivebull subject to linear equalities and

inequalities

maximize

subject to

3x + 4y

5x + 8y le 24x y ge 0

Here is the set of feasible solutions

The optimalsolution

Terminology

bull Decision variables eg x and yndash In general there are quantities you can control to

improve your objective which should completely describe the set of decisions to be made

bull Constraints eg 5x + 8y le 24 x ge 0 y ge 0ndash Limitations on the values of the decision variables

bull Objective Function eg 3x + 4yndash Value measure used to rank alternativesndash Seek to maximize or minimize this objectivendash examples maximize NPV minimize cost

MSR Marketing Incadapted from Frontline Systems

bull Need to choose ads to reach at least 15 million people

bull Minimize Costbull Upper bound on number

of ads of each type

TV Radio Mail Newspaper

Audience Size 50000 25000 20000 15000

CostImpression $500 $200 $250 $125

Max of ads 20 15 10 15

Formulating as a math model

Work with your partner

bull What decisions need to be made Expressthese as ldquodecision variablesrdquo

bull What is the objective Express the objectivein terms of the decision variables

bull What are the constraints Express these interms of the decision variables

bull If you have time try to find the best solution

Gemstone Tool Company

bull Privately-held firmbull Consumer and industrial market for construction toolsbull Headquartered in Seattlebull Manufacturing plants in the US Canada and Mexicobull Simplifying assumptions for purposes of illustration

ndash Winnipeg Canada plantndash Wrenches and pliersndash Made from steelndash Injection molding machinendash Assembly machine

Data for the GTC ProblemWrenches Pliers Available

Steel 15 10 15000 tons

Molding Machine 10 10 12000 hrs

Assembly Machine 4 5 5000 hrs

Demand Limit 8000 10000

Contribution($ per item)

$40 $30

We want to determine the number of wrenches and pliersto produce given the available raw materials machinehours and demand

To do with your partner

bull Work with your partner to formulate theGTC problem as a linear program DONOT LOOK AHEAD IN THE NOTES

bull Let P = number of pliers madebull Let W = number of wrenches made

Formulating the GTC Problem

Step 1 Determine Decision Variables P = number of thousands of pliers manufactured W = number of thousands of wrenches manufactured

Step 2 Determine Objective Function Maximize Profit =

The Formulation Continued

Step 3 Determine Constraints

Steel Molding Assembly Plier Demand Wrench Demand

We will show how to solve this in the next lecture

An Algebraic Formulation

bull J = set of items that are manufacturedndash eg S = pliers wrenchesndash pj = unit profit from item jndash dj = maximum demand for item jndash xj = number of units of item j manufactured

bull M = set of manufacturing processesndash eg M = molding and assemblyndash bi = capacity of process indash aij = amount of capacity of process i used in making item j

An Algebraic formulation

bull Maximize

bull subject to

The sameformulation workseven if | J | = 10000and | M | = 100

An Algebraic formulation

bull Maximize

bull subject to

Linear Programs

bull A linear function is a function of the form

bull A mathematical program is a linear program (LP) if the objective is a linear function and the constraints are linear equalities or inequalities

bull Typically an LP has non-negativity constraints

A non-linear program is permitted to havea non-linear objective and constraints

bull maximize f(xy) = xybull subject to x - y22 le 10 3x ndash 4y ge 2 x ge 0 y ge 0

An integer program is a linear programplus constraints that some or all of thevariables are integer valuedbull Maximize 3x1 + 4x2 - 3x3 3x1 + 2x2 - x3 ge 17 3x2 - x3 = 14 x1 ge 0 x2 ge 0 x3 ge 0 and x1 x2 x3 are all integers

An Algebraic formulation withequality constraints

bull Maximize

bull subject to

Linear Programming Assumptions

Proportionality Assumption Contribution from W isproportional to W

Additivity Assumption Contribution to objective function fromP is independent of W

Divisibility Assumption Each variable is allowed to assumefractional values

Certainty Assumption Each linear coefficient of the objectivefunction and constraints is known (and is not a random variable)

Maximize 4W + 3P15W + P le 15

hellip

Some Success Stories

bull Optimal crew scheduling saves American Airlines $20 millionyr

bull Improved shipment routing saves Yellow Freight over $173 millionyr

bull Improved truck dispatching at Reynolds Metals improves on-time delivery and reduces freight cost by $7 millionyr

bull GTE local capacity expansion saves $30 millionyr

Other Success Stories (cont)

bull Optimizing global supply chains saves Digital Equipment over $300 million

bull Restructuring North America Operations Proctor and Gamble reduces plants by 20 saving $200 millionyr

bull Optimal traffic control of Hanshin Expressway in Osaka saves 17 million driver hoursyr

bull Better scheduling of hydro and thermal generating units saves southern company $140 million

Success Stories (cont)

bull Improved production planning at Sadia (Brazil) saves $50 million over three years

bull Production Optimization at Harris Corporation improves on-time deliveries from 75 to 90

bull Tata Steel (India) optimizes response to power shortage contributing $73 million

bull Optimizing police patrol officer scheduling saves police department $11 millionyr

bull Gasoline blending at Texaco results in saving of over $30 millionyr

Summary

bull Answered the question What is Operations Research amp Management Science and provided some historical perspective

bull Introduced the terminology of linear programmingbull Two Examples

1 MSR Marketing2 Gemstone Tool Companyndash Small (2-dimensional) Linear Program non-obvious solution

bull We will discuss this problem in detail in the next lecture

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• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45

Overview

bull Course Descriptionbull Course Administration and Logisticsbull What is Management Sciencebull Linear Programming Examples

ndash MSR Marketingndash GTC

bull Handouts Syllabus and General Info Lecture Notes Homework Set 1

Slide 3 contains MIT specific information

Required Materials

bull Coursepack (Course Reader)bull Includes copies of chapters from

ndash Applied Mathematical Programming by BradleyHax amp Magnanti

bull Class website sloanspacemitedubull Other resources

ndash Operations Research by Winstonndash Introduction to Linear Optimization by Bertsimas

amp Tsitsiklisndash Network Flows by Ahuja Magnanti amp Orlin

Grading Policy

bull Homework assignments (27)ndash Weekly ( 10 in total )ndash Nonlinear grading scheme

bull Midterm Exams (25 each)ndash Two in class modterms

bull Final Exam (25)ndash During Finals weekndash Last 13 of subject + modeling

bull Yes it does add up to 102

Course Policy (contrsquod)

bull Webpage

ndash Includesbull class notesbull Spreadsheetsbull Readingsbull assignments (assignment 1 is already there)bull and more

Active Learning

bull Occasionally I will introduce a break in the lecture for you to work on your own or with a partner

bull Please identify your ldquopartnerrdquo nowbull Those on aisle ends may be in a group of siz

e 3

What is Operations ResearchWhat is Management Sciencebull World War II British military leaders asked

scientists and engineers to analyze several military problemsndash Deployment of radarndash Management of convoy bombing antisubmarine

and mining operationsbull The result was called Military Operations

Research later Operations Researchbull MIT was one of the birthplaces of OR

ndash Professor Morse at MIT was a pioneer in the USndash Founded MIT OR Center and helped found ORSA

What is Management Science(Operations Research)bull Today Operations Research and Management

Science mean ldquothe use of mathematical models in providing guidelines to managers for making effective decisions within the state of the current information or in seeking further information if current knowledge is insufficient to reach a proper decisionrdquobull cf Decision science systems analysis

operational research systems dynamicsoperational analysis engineering systemssystems engineering and more

Voices from the past

bull Waste neither time nor money but make the best use of bothndash Benjamin Franklin

bull Obviously the highest type of efficiency is that which can utilize existing material to the best advantagendash Jawaharlal Nehru

bull It is more probable that the average man could with no injury to his health increase his efficiency fifty percentndash Walter Scott

Operations Research Over the Years

bull 1947ndash Project Scoop (Scientific Computation of

Optimum Programs) with George Dantzig andothers Developed the simplex method forlinear programs

bull 1950sndash Lots of excitement mathematical

developments queuing theory mathematicalprogrammingcf AI in the 1960s

bull 1960sndash More excitement more development and grand

plans cf AI in the 1980s

Operations Research Over the Years

bull 1970sndash Disappointment and a settling down NPcompleteness

More realistic expectationsbull 1980s

ndash Widespread availability of personal computersIncreasingly easy access to data Widespreadwillingness of managers to use models

bull 1990sndash Improved use of OR systems

Further inroads of OR technology egoptimization and simulation add-ons to spreadsheets modeling languages large scaleoptimization More intermixing of AI and OR

Operations Research in the 00rsquos

bull LOTS of opportunities for OR as a fieldbull Data data data

ndash E-business data (click stream purchases othertransactional data E-mail and more)

ndash The human genome project and its outgrowthbull Need for more automated decision makingbull Need for increased coordination for efficient

use of resources (Supply chain management)

Optimization

As agelessas time

Optimization in NatureHeron of Alexandria

First Century AD

Angle of

Incidence

Angle of

Incidence

Angle of

Reflection

FERM AT

Angle α1 ofIncidence

Angle α2 ofRefraction

CalculusMaximum

Minimum

Fermat Newton Euler LagrangeG auss and more

Some of the themes of 15053

bull Optimization is everywherebull Models Models Modelsbull The goal of models is ldquoinsightrdquo not

numbersndash paraphrase of Richard Hamming

bull Algorithms Algorithms Algorithms

Optimization is Everywhere

bull The more you know about something the more you see where optimization can be applied

bull Some personal decision makingndash Finding the fastest route home (or to class)ndash Optimal allocation of time for homeworkndash Optimal budgetingndash Selecting a major

Optimization is everywhere

bull Some MIT decision makingndash setting exam times to minimize overlapndash assigning classes to classrooms and time

slots while satisfying constraintsndash figuring out prices for parking and subsidies

for public transportation so as to maximize fairness and permit sufficient access

ndash optimizing in fundraising

Optimization is everywhere

bull Exercise introduce yourself to your neighborand then mention a topic or two with which youare pretty familiar (summer job or major or yourparentrsquos occupation or whatever)

bull Then brainstorm with your neighbor on placeswhere optimization occurs

bull Then choose your favorite 2 or 3 applicationsfrom your list and letrsquos share them

On 15053 and Optimization Tools

bull Optimization is everywhere but optimizationtools are not applied everywhere

bull Goals in 15053 present a variety of toolsfor optimization and illustrate applicationsin manufacturing finance e-businessmarketing and more

bull When you see optimization problems arisein business (and you will) you will know thatthere are tools to help you out

Addressing managerial problems Amanagement science framework

1 Determine the problem to be solved2 Observe the system and gather data3 Formulate a mathematical model of the problem an

d any important subproblems4 Verify the model and use the model for prediction

or analysis5 Select a suitable alternative6 Present the results to the organization7 Implement and evaluate

Linear Programming (our first tooland probably the most important one)bull minimize or maximize a linear objectivebull subject to linear equalities and

inequalities

maximize

subject to

3x + 4y

5x + 8y le 24x y ge 0

Here is the set of feasible solutions

The optimalsolution

Terminology

bull Decision variables eg x and yndash In general there are quantities you can control to

improve your objective which should completely describe the set of decisions to be made

bull Constraints eg 5x + 8y le 24 x ge 0 y ge 0ndash Limitations on the values of the decision variables

bull Objective Function eg 3x + 4yndash Value measure used to rank alternativesndash Seek to maximize or minimize this objectivendash examples maximize NPV minimize cost

MSR Marketing Incadapted from Frontline Systems

bull Need to choose ads to reach at least 15 million people

bull Minimize Costbull Upper bound on number

of ads of each type

TV Radio Mail Newspaper

Audience Size 50000 25000 20000 15000

CostImpression $500 $200 $250 $125

Max of ads 20 15 10 15

Formulating as a math model

Work with your partner

bull What decisions need to be made Expressthese as ldquodecision variablesrdquo

bull What is the objective Express the objectivein terms of the decision variables

bull What are the constraints Express these interms of the decision variables

bull If you have time try to find the best solution

Gemstone Tool Company

bull Privately-held firmbull Consumer and industrial market for construction toolsbull Headquartered in Seattlebull Manufacturing plants in the US Canada and Mexicobull Simplifying assumptions for purposes of illustration

ndash Winnipeg Canada plantndash Wrenches and pliersndash Made from steelndash Injection molding machinendash Assembly machine

Data for the GTC ProblemWrenches Pliers Available

Steel 15 10 15000 tons

Molding Machine 10 10 12000 hrs

Assembly Machine 4 5 5000 hrs

Demand Limit 8000 10000

Contribution($ per item)

$40 $30

We want to determine the number of wrenches and pliersto produce given the available raw materials machinehours and demand

To do with your partner

bull Work with your partner to formulate theGTC problem as a linear program DONOT LOOK AHEAD IN THE NOTES

bull Let P = number of pliers madebull Let W = number of wrenches made

Formulating the GTC Problem

Step 1 Determine Decision Variables P = number of thousands of pliers manufactured W = number of thousands of wrenches manufactured

Step 2 Determine Objective Function Maximize Profit =

The Formulation Continued

Step 3 Determine Constraints

Steel Molding Assembly Plier Demand Wrench Demand

We will show how to solve this in the next lecture

An Algebraic Formulation

bull J = set of items that are manufacturedndash eg S = pliers wrenchesndash pj = unit profit from item jndash dj = maximum demand for item jndash xj = number of units of item j manufactured

bull M = set of manufacturing processesndash eg M = molding and assemblyndash bi = capacity of process indash aij = amount of capacity of process i used in making item j

An Algebraic formulation

bull Maximize

bull subject to

The sameformulation workseven if | J | = 10000and | M | = 100

An Algebraic formulation

bull Maximize

bull subject to

Linear Programs

bull A linear function is a function of the form

bull A mathematical program is a linear program (LP) if the objective is a linear function and the constraints are linear equalities or inequalities

bull Typically an LP has non-negativity constraints

A non-linear program is permitted to havea non-linear objective and constraints

bull maximize f(xy) = xybull subject to x - y22 le 10 3x ndash 4y ge 2 x ge 0 y ge 0

An integer program is a linear programplus constraints that some or all of thevariables are integer valuedbull Maximize 3x1 + 4x2 - 3x3 3x1 + 2x2 - x3 ge 17 3x2 - x3 = 14 x1 ge 0 x2 ge 0 x3 ge 0 and x1 x2 x3 are all integers

An Algebraic formulation withequality constraints

bull Maximize

bull subject to

Linear Programming Assumptions

Proportionality Assumption Contribution from W isproportional to W

Additivity Assumption Contribution to objective function fromP is independent of W

Divisibility Assumption Each variable is allowed to assumefractional values

Certainty Assumption Each linear coefficient of the objectivefunction and constraints is known (and is not a random variable)

Maximize 4W + 3P15W + P le 15

hellip

Some Success Stories

bull Optimal crew scheduling saves American Airlines $20 millionyr

bull Improved shipment routing saves Yellow Freight over $173 millionyr

bull Improved truck dispatching at Reynolds Metals improves on-time delivery and reduces freight cost by $7 millionyr

bull GTE local capacity expansion saves $30 millionyr

Other Success Stories (cont)

bull Optimizing global supply chains saves Digital Equipment over $300 million

bull Restructuring North America Operations Proctor and Gamble reduces plants by 20 saving $200 millionyr

bull Optimal traffic control of Hanshin Expressway in Osaka saves 17 million driver hoursyr

bull Better scheduling of hydro and thermal generating units saves southern company $140 million

Success Stories (cont)

bull Improved production planning at Sadia (Brazil) saves $50 million over three years

bull Production Optimization at Harris Corporation improves on-time deliveries from 75 to 90

bull Tata Steel (India) optimizes response to power shortage contributing $73 million

bull Optimizing police patrol officer scheduling saves police department $11 millionyr

bull Gasoline blending at Texaco results in saving of over $30 millionyr

Summary

bull Answered the question What is Operations Research amp Management Science and provided some historical perspective

bull Introduced the terminology of linear programmingbull Two Examples

1 MSR Marketing2 Gemstone Tool Companyndash Small (2-dimensional) Linear Program non-obvious solution

bull We will discuss this problem in detail in the next lecture

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45

Slide 3 contains MIT specific information

Required Materials

bull Coursepack (Course Reader)bull Includes copies of chapters from

ndash Applied Mathematical Programming by BradleyHax amp Magnanti

bull Class website sloanspacemitedubull Other resources

ndash Operations Research by Winstonndash Introduction to Linear Optimization by Bertsimas

amp Tsitsiklisndash Network Flows by Ahuja Magnanti amp Orlin

Grading Policy

bull Homework assignments (27)ndash Weekly ( 10 in total )ndash Nonlinear grading scheme

bull Midterm Exams (25 each)ndash Two in class modterms

bull Final Exam (25)ndash During Finals weekndash Last 13 of subject + modeling

bull Yes it does add up to 102

Course Policy (contrsquod)

bull Webpage

ndash Includesbull class notesbull Spreadsheetsbull Readingsbull assignments (assignment 1 is already there)bull and more

Active Learning

bull Occasionally I will introduce a break in the lecture for you to work on your own or with a partner

bull Please identify your ldquopartnerrdquo nowbull Those on aisle ends may be in a group of siz

e 3

What is Operations ResearchWhat is Management Sciencebull World War II British military leaders asked

scientists and engineers to analyze several military problemsndash Deployment of radarndash Management of convoy bombing antisubmarine

and mining operationsbull The result was called Military Operations

Research later Operations Researchbull MIT was one of the birthplaces of OR

ndash Professor Morse at MIT was a pioneer in the USndash Founded MIT OR Center and helped found ORSA

What is Management Science(Operations Research)bull Today Operations Research and Management

Science mean ldquothe use of mathematical models in providing guidelines to managers for making effective decisions within the state of the current information or in seeking further information if current knowledge is insufficient to reach a proper decisionrdquobull cf Decision science systems analysis

operational research systems dynamicsoperational analysis engineering systemssystems engineering and more

Voices from the past

bull Waste neither time nor money but make the best use of bothndash Benjamin Franklin

bull Obviously the highest type of efficiency is that which can utilize existing material to the best advantagendash Jawaharlal Nehru

bull It is more probable that the average man could with no injury to his health increase his efficiency fifty percentndash Walter Scott

Operations Research Over the Years

bull 1947ndash Project Scoop (Scientific Computation of

Optimum Programs) with George Dantzig andothers Developed the simplex method forlinear programs

bull 1950sndash Lots of excitement mathematical

developments queuing theory mathematicalprogrammingcf AI in the 1960s

bull 1960sndash More excitement more development and grand

plans cf AI in the 1980s

Operations Research Over the Years

bull 1970sndash Disappointment and a settling down NPcompleteness

More realistic expectationsbull 1980s

ndash Widespread availability of personal computersIncreasingly easy access to data Widespreadwillingness of managers to use models

bull 1990sndash Improved use of OR systems

Further inroads of OR technology egoptimization and simulation add-ons to spreadsheets modeling languages large scaleoptimization More intermixing of AI and OR

Operations Research in the 00rsquos

bull LOTS of opportunities for OR as a fieldbull Data data data

ndash E-business data (click stream purchases othertransactional data E-mail and more)

ndash The human genome project and its outgrowthbull Need for more automated decision makingbull Need for increased coordination for efficient

use of resources (Supply chain management)

Optimization

As agelessas time

Optimization in NatureHeron of Alexandria

First Century AD

Angle of

Incidence

Angle of

Incidence

Angle of

Reflection

FERM AT

Angle α1 ofIncidence

Angle α2 ofRefraction

CalculusMaximum

Minimum

Fermat Newton Euler LagrangeG auss and more

Some of the themes of 15053

bull Optimization is everywherebull Models Models Modelsbull The goal of models is ldquoinsightrdquo not

numbersndash paraphrase of Richard Hamming

bull Algorithms Algorithms Algorithms

Optimization is Everywhere

bull The more you know about something the more you see where optimization can be applied

bull Some personal decision makingndash Finding the fastest route home (or to class)ndash Optimal allocation of time for homeworkndash Optimal budgetingndash Selecting a major

Optimization is everywhere

bull Some MIT decision makingndash setting exam times to minimize overlapndash assigning classes to classrooms and time

slots while satisfying constraintsndash figuring out prices for parking and subsidies

for public transportation so as to maximize fairness and permit sufficient access

ndash optimizing in fundraising

Optimization is everywhere

bull Exercise introduce yourself to your neighborand then mention a topic or two with which youare pretty familiar (summer job or major or yourparentrsquos occupation or whatever)

bull Then brainstorm with your neighbor on placeswhere optimization occurs

bull Then choose your favorite 2 or 3 applicationsfrom your list and letrsquos share them

On 15053 and Optimization Tools

bull Optimization is everywhere but optimizationtools are not applied everywhere

bull Goals in 15053 present a variety of toolsfor optimization and illustrate applicationsin manufacturing finance e-businessmarketing and more

bull When you see optimization problems arisein business (and you will) you will know thatthere are tools to help you out

Addressing managerial problems Amanagement science framework

1 Determine the problem to be solved2 Observe the system and gather data3 Formulate a mathematical model of the problem an

d any important subproblems4 Verify the model and use the model for prediction

or analysis5 Select a suitable alternative6 Present the results to the organization7 Implement and evaluate

Linear Programming (our first tooland probably the most important one)bull minimize or maximize a linear objectivebull subject to linear equalities and

inequalities

maximize

subject to

3x + 4y

5x + 8y le 24x y ge 0

Here is the set of feasible solutions

The optimalsolution

Terminology

bull Decision variables eg x and yndash In general there are quantities you can control to

improve your objective which should completely describe the set of decisions to be made

bull Constraints eg 5x + 8y le 24 x ge 0 y ge 0ndash Limitations on the values of the decision variables

bull Objective Function eg 3x + 4yndash Value measure used to rank alternativesndash Seek to maximize or minimize this objectivendash examples maximize NPV minimize cost

MSR Marketing Incadapted from Frontline Systems

bull Need to choose ads to reach at least 15 million people

bull Minimize Costbull Upper bound on number

of ads of each type

TV Radio Mail Newspaper

Audience Size 50000 25000 20000 15000

CostImpression $500 $200 $250 $125

Max of ads 20 15 10 15

Formulating as a math model

Work with your partner

bull What decisions need to be made Expressthese as ldquodecision variablesrdquo

bull What is the objective Express the objectivein terms of the decision variables

bull What are the constraints Express these interms of the decision variables

bull If you have time try to find the best solution

Gemstone Tool Company

bull Privately-held firmbull Consumer and industrial market for construction toolsbull Headquartered in Seattlebull Manufacturing plants in the US Canada and Mexicobull Simplifying assumptions for purposes of illustration

ndash Winnipeg Canada plantndash Wrenches and pliersndash Made from steelndash Injection molding machinendash Assembly machine

Data for the GTC ProblemWrenches Pliers Available

Steel 15 10 15000 tons

Molding Machine 10 10 12000 hrs

Assembly Machine 4 5 5000 hrs

Demand Limit 8000 10000

Contribution($ per item)

$40 $30

We want to determine the number of wrenches and pliersto produce given the available raw materials machinehours and demand

To do with your partner

bull Work with your partner to formulate theGTC problem as a linear program DONOT LOOK AHEAD IN THE NOTES

bull Let P = number of pliers madebull Let W = number of wrenches made

Formulating the GTC Problem

Step 1 Determine Decision Variables P = number of thousands of pliers manufactured W = number of thousands of wrenches manufactured

Step 2 Determine Objective Function Maximize Profit =

The Formulation Continued

Step 3 Determine Constraints

Steel Molding Assembly Plier Demand Wrench Demand

We will show how to solve this in the next lecture

An Algebraic Formulation

bull J = set of items that are manufacturedndash eg S = pliers wrenchesndash pj = unit profit from item jndash dj = maximum demand for item jndash xj = number of units of item j manufactured

bull M = set of manufacturing processesndash eg M = molding and assemblyndash bi = capacity of process indash aij = amount of capacity of process i used in making item j

An Algebraic formulation

bull Maximize

bull subject to

The sameformulation workseven if | J | = 10000and | M | = 100

An Algebraic formulation

bull Maximize

bull subject to

Linear Programs

bull A linear function is a function of the form

bull A mathematical program is a linear program (LP) if the objective is a linear function and the constraints are linear equalities or inequalities

bull Typically an LP has non-negativity constraints

A non-linear program is permitted to havea non-linear objective and constraints

bull maximize f(xy) = xybull subject to x - y22 le 10 3x ndash 4y ge 2 x ge 0 y ge 0

An integer program is a linear programplus constraints that some or all of thevariables are integer valuedbull Maximize 3x1 + 4x2 - 3x3 3x1 + 2x2 - x3 ge 17 3x2 - x3 = 14 x1 ge 0 x2 ge 0 x3 ge 0 and x1 x2 x3 are all integers

An Algebraic formulation withequality constraints

bull Maximize

bull subject to

Linear Programming Assumptions

Proportionality Assumption Contribution from W isproportional to W

Additivity Assumption Contribution to objective function fromP is independent of W

Divisibility Assumption Each variable is allowed to assumefractional values

Certainty Assumption Each linear coefficient of the objectivefunction and constraints is known (and is not a random variable)

Maximize 4W + 3P15W + P le 15

hellip

Some Success Stories

bull Optimal crew scheduling saves American Airlines $20 millionyr

bull Improved shipment routing saves Yellow Freight over $173 millionyr

bull Improved truck dispatching at Reynolds Metals improves on-time delivery and reduces freight cost by $7 millionyr

bull GTE local capacity expansion saves $30 millionyr

Other Success Stories (cont)

bull Optimizing global supply chains saves Digital Equipment over $300 million

bull Restructuring North America Operations Proctor and Gamble reduces plants by 20 saving $200 millionyr

bull Optimal traffic control of Hanshin Expressway in Osaka saves 17 million driver hoursyr

bull Better scheduling of hydro and thermal generating units saves southern company $140 million

Success Stories (cont)

bull Improved production planning at Sadia (Brazil) saves $50 million over three years

bull Production Optimization at Harris Corporation improves on-time deliveries from 75 to 90

bull Tata Steel (India) optimizes response to power shortage contributing $73 million

bull Optimizing police patrol officer scheduling saves police department $11 millionyr

bull Gasoline blending at Texaco results in saving of over $30 millionyr

Summary

bull Answered the question What is Operations Research amp Management Science and provided some historical perspective

bull Introduced the terminology of linear programmingbull Two Examples

1 MSR Marketing2 Gemstone Tool Companyndash Small (2-dimensional) Linear Program non-obvious solution

bull We will discuss this problem in detail in the next lecture

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45

Required Materials

bull Coursepack (Course Reader)bull Includes copies of chapters from

ndash Applied Mathematical Programming by BradleyHax amp Magnanti

bull Class website sloanspacemitedubull Other resources

ndash Operations Research by Winstonndash Introduction to Linear Optimization by Bertsimas

amp Tsitsiklisndash Network Flows by Ahuja Magnanti amp Orlin

Grading Policy

bull Homework assignments (27)ndash Weekly ( 10 in total )ndash Nonlinear grading scheme

bull Midterm Exams (25 each)ndash Two in class modterms

bull Final Exam (25)ndash During Finals weekndash Last 13 of subject + modeling

bull Yes it does add up to 102

Course Policy (contrsquod)

bull Webpage

ndash Includesbull class notesbull Spreadsheetsbull Readingsbull assignments (assignment 1 is already there)bull and more

Active Learning

bull Occasionally I will introduce a break in the lecture for you to work on your own or with a partner

bull Please identify your ldquopartnerrdquo nowbull Those on aisle ends may be in a group of siz

e 3

What is Operations ResearchWhat is Management Sciencebull World War II British military leaders asked

scientists and engineers to analyze several military problemsndash Deployment of radarndash Management of convoy bombing antisubmarine

and mining operationsbull The result was called Military Operations

Research later Operations Researchbull MIT was one of the birthplaces of OR

ndash Professor Morse at MIT was a pioneer in the USndash Founded MIT OR Center and helped found ORSA

What is Management Science(Operations Research)bull Today Operations Research and Management

Science mean ldquothe use of mathematical models in providing guidelines to managers for making effective decisions within the state of the current information or in seeking further information if current knowledge is insufficient to reach a proper decisionrdquobull cf Decision science systems analysis

operational research systems dynamicsoperational analysis engineering systemssystems engineering and more

Voices from the past

bull Waste neither time nor money but make the best use of bothndash Benjamin Franklin

bull Obviously the highest type of efficiency is that which can utilize existing material to the best advantagendash Jawaharlal Nehru

bull It is more probable that the average man could with no injury to his health increase his efficiency fifty percentndash Walter Scott

Operations Research Over the Years

bull 1947ndash Project Scoop (Scientific Computation of

Optimum Programs) with George Dantzig andothers Developed the simplex method forlinear programs

bull 1950sndash Lots of excitement mathematical

developments queuing theory mathematicalprogrammingcf AI in the 1960s

bull 1960sndash More excitement more development and grand

plans cf AI in the 1980s

Operations Research Over the Years

bull 1970sndash Disappointment and a settling down NPcompleteness

More realistic expectationsbull 1980s

ndash Widespread availability of personal computersIncreasingly easy access to data Widespreadwillingness of managers to use models

bull 1990sndash Improved use of OR systems

Further inroads of OR technology egoptimization and simulation add-ons to spreadsheets modeling languages large scaleoptimization More intermixing of AI and OR

Operations Research in the 00rsquos

bull LOTS of opportunities for OR as a fieldbull Data data data

ndash E-business data (click stream purchases othertransactional data E-mail and more)

ndash The human genome project and its outgrowthbull Need for more automated decision makingbull Need for increased coordination for efficient

use of resources (Supply chain management)

Optimization

As agelessas time

Optimization in NatureHeron of Alexandria

First Century AD

Angle of

Incidence

Angle of

Incidence

Angle of

Reflection

FERM AT

Angle α1 ofIncidence

Angle α2 ofRefraction

CalculusMaximum

Minimum

Fermat Newton Euler LagrangeG auss and more

Some of the themes of 15053

bull Optimization is everywherebull Models Models Modelsbull The goal of models is ldquoinsightrdquo not

numbersndash paraphrase of Richard Hamming

bull Algorithms Algorithms Algorithms

Optimization is Everywhere

bull The more you know about something the more you see where optimization can be applied

bull Some personal decision makingndash Finding the fastest route home (or to class)ndash Optimal allocation of time for homeworkndash Optimal budgetingndash Selecting a major

Optimization is everywhere

bull Some MIT decision makingndash setting exam times to minimize overlapndash assigning classes to classrooms and time

slots while satisfying constraintsndash figuring out prices for parking and subsidies

for public transportation so as to maximize fairness and permit sufficient access

ndash optimizing in fundraising

Optimization is everywhere

bull Exercise introduce yourself to your neighborand then mention a topic or two with which youare pretty familiar (summer job or major or yourparentrsquos occupation or whatever)

bull Then brainstorm with your neighbor on placeswhere optimization occurs

bull Then choose your favorite 2 or 3 applicationsfrom your list and letrsquos share them

On 15053 and Optimization Tools

bull Optimization is everywhere but optimizationtools are not applied everywhere

bull Goals in 15053 present a variety of toolsfor optimization and illustrate applicationsin manufacturing finance e-businessmarketing and more

bull When you see optimization problems arisein business (and you will) you will know thatthere are tools to help you out

Addressing managerial problems Amanagement science framework

1 Determine the problem to be solved2 Observe the system and gather data3 Formulate a mathematical model of the problem an

d any important subproblems4 Verify the model and use the model for prediction

or analysis5 Select a suitable alternative6 Present the results to the organization7 Implement and evaluate

Linear Programming (our first tooland probably the most important one)bull minimize or maximize a linear objectivebull subject to linear equalities and

inequalities

maximize

subject to

3x + 4y

5x + 8y le 24x y ge 0

Here is the set of feasible solutions

The optimalsolution

Terminology

bull Decision variables eg x and yndash In general there are quantities you can control to

improve your objective which should completely describe the set of decisions to be made

bull Constraints eg 5x + 8y le 24 x ge 0 y ge 0ndash Limitations on the values of the decision variables

bull Objective Function eg 3x + 4yndash Value measure used to rank alternativesndash Seek to maximize or minimize this objectivendash examples maximize NPV minimize cost

MSR Marketing Incadapted from Frontline Systems

bull Need to choose ads to reach at least 15 million people

bull Minimize Costbull Upper bound on number

of ads of each type

TV Radio Mail Newspaper

Audience Size 50000 25000 20000 15000

CostImpression $500 $200 $250 $125

Max of ads 20 15 10 15

Formulating as a math model

Work with your partner

bull What decisions need to be made Expressthese as ldquodecision variablesrdquo

bull What is the objective Express the objectivein terms of the decision variables

bull What are the constraints Express these interms of the decision variables

bull If you have time try to find the best solution

Gemstone Tool Company

bull Privately-held firmbull Consumer and industrial market for construction toolsbull Headquartered in Seattlebull Manufacturing plants in the US Canada and Mexicobull Simplifying assumptions for purposes of illustration

ndash Winnipeg Canada plantndash Wrenches and pliersndash Made from steelndash Injection molding machinendash Assembly machine

Data for the GTC ProblemWrenches Pliers Available

Steel 15 10 15000 tons

Molding Machine 10 10 12000 hrs

Assembly Machine 4 5 5000 hrs

Demand Limit 8000 10000

Contribution($ per item)

$40 $30

We want to determine the number of wrenches and pliersto produce given the available raw materials machinehours and demand

To do with your partner

bull Work with your partner to formulate theGTC problem as a linear program DONOT LOOK AHEAD IN THE NOTES

bull Let P = number of pliers madebull Let W = number of wrenches made

Formulating the GTC Problem

Step 1 Determine Decision Variables P = number of thousands of pliers manufactured W = number of thousands of wrenches manufactured

Step 2 Determine Objective Function Maximize Profit =

The Formulation Continued

Step 3 Determine Constraints

Steel Molding Assembly Plier Demand Wrench Demand

We will show how to solve this in the next lecture

An Algebraic Formulation

bull J = set of items that are manufacturedndash eg S = pliers wrenchesndash pj = unit profit from item jndash dj = maximum demand for item jndash xj = number of units of item j manufactured

bull M = set of manufacturing processesndash eg M = molding and assemblyndash bi = capacity of process indash aij = amount of capacity of process i used in making item j

An Algebraic formulation

bull Maximize

bull subject to

The sameformulation workseven if | J | = 10000and | M | = 100

An Algebraic formulation

bull Maximize

bull subject to

Linear Programs

bull A linear function is a function of the form

bull A mathematical program is a linear program (LP) if the objective is a linear function and the constraints are linear equalities or inequalities

bull Typically an LP has non-negativity constraints

A non-linear program is permitted to havea non-linear objective and constraints

bull maximize f(xy) = xybull subject to x - y22 le 10 3x ndash 4y ge 2 x ge 0 y ge 0

An integer program is a linear programplus constraints that some or all of thevariables are integer valuedbull Maximize 3x1 + 4x2 - 3x3 3x1 + 2x2 - x3 ge 17 3x2 - x3 = 14 x1 ge 0 x2 ge 0 x3 ge 0 and x1 x2 x3 are all integers

An Algebraic formulation withequality constraints

bull Maximize

bull subject to

Linear Programming Assumptions

Proportionality Assumption Contribution from W isproportional to W

Additivity Assumption Contribution to objective function fromP is independent of W

Divisibility Assumption Each variable is allowed to assumefractional values

Certainty Assumption Each linear coefficient of the objectivefunction and constraints is known (and is not a random variable)

Maximize 4W + 3P15W + P le 15

hellip

Some Success Stories

bull Optimal crew scheduling saves American Airlines $20 millionyr

bull Improved shipment routing saves Yellow Freight over $173 millionyr

bull Improved truck dispatching at Reynolds Metals improves on-time delivery and reduces freight cost by $7 millionyr

bull GTE local capacity expansion saves $30 millionyr

Other Success Stories (cont)

bull Optimizing global supply chains saves Digital Equipment over $300 million

bull Restructuring North America Operations Proctor and Gamble reduces plants by 20 saving $200 millionyr

bull Optimal traffic control of Hanshin Expressway in Osaka saves 17 million driver hoursyr

bull Better scheduling of hydro and thermal generating units saves southern company $140 million

Success Stories (cont)

bull Improved production planning at Sadia (Brazil) saves $50 million over three years

bull Production Optimization at Harris Corporation improves on-time deliveries from 75 to 90

bull Tata Steel (India) optimizes response to power shortage contributing $73 million

bull Optimizing police patrol officer scheduling saves police department $11 millionyr

bull Gasoline blending at Texaco results in saving of over $30 millionyr

Summary

bull Answered the question What is Operations Research amp Management Science and provided some historical perspective

bull Introduced the terminology of linear programmingbull Two Examples

1 MSR Marketing2 Gemstone Tool Companyndash Small (2-dimensional) Linear Program non-obvious solution

bull We will discuss this problem in detail in the next lecture

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45

Grading Policy

bull Homework assignments (27)ndash Weekly ( 10 in total )ndash Nonlinear grading scheme

bull Midterm Exams (25 each)ndash Two in class modterms

bull Final Exam (25)ndash During Finals weekndash Last 13 of subject + modeling

bull Yes it does add up to 102

Course Policy (contrsquod)

bull Webpage

ndash Includesbull class notesbull Spreadsheetsbull Readingsbull assignments (assignment 1 is already there)bull and more

Active Learning

bull Occasionally I will introduce a break in the lecture for you to work on your own or with a partner

bull Please identify your ldquopartnerrdquo nowbull Those on aisle ends may be in a group of siz

e 3

What is Operations ResearchWhat is Management Sciencebull World War II British military leaders asked

scientists and engineers to analyze several military problemsndash Deployment of radarndash Management of convoy bombing antisubmarine

and mining operationsbull The result was called Military Operations

Research later Operations Researchbull MIT was one of the birthplaces of OR

ndash Professor Morse at MIT was a pioneer in the USndash Founded MIT OR Center and helped found ORSA

What is Management Science(Operations Research)bull Today Operations Research and Management

Science mean ldquothe use of mathematical models in providing guidelines to managers for making effective decisions within the state of the current information or in seeking further information if current knowledge is insufficient to reach a proper decisionrdquobull cf Decision science systems analysis

operational research systems dynamicsoperational analysis engineering systemssystems engineering and more

Voices from the past

bull Waste neither time nor money but make the best use of bothndash Benjamin Franklin

bull Obviously the highest type of efficiency is that which can utilize existing material to the best advantagendash Jawaharlal Nehru

bull It is more probable that the average man could with no injury to his health increase his efficiency fifty percentndash Walter Scott

Operations Research Over the Years

bull 1947ndash Project Scoop (Scientific Computation of

Optimum Programs) with George Dantzig andothers Developed the simplex method forlinear programs

bull 1950sndash Lots of excitement mathematical

developments queuing theory mathematicalprogrammingcf AI in the 1960s

bull 1960sndash More excitement more development and grand

plans cf AI in the 1980s

Operations Research Over the Years

bull 1970sndash Disappointment and a settling down NPcompleteness

More realistic expectationsbull 1980s

ndash Widespread availability of personal computersIncreasingly easy access to data Widespreadwillingness of managers to use models

bull 1990sndash Improved use of OR systems

Further inroads of OR technology egoptimization and simulation add-ons to spreadsheets modeling languages large scaleoptimization More intermixing of AI and OR

Operations Research in the 00rsquos

bull LOTS of opportunities for OR as a fieldbull Data data data

ndash E-business data (click stream purchases othertransactional data E-mail and more)

ndash The human genome project and its outgrowthbull Need for more automated decision makingbull Need for increased coordination for efficient

use of resources (Supply chain management)

Optimization

As agelessas time

Optimization in NatureHeron of Alexandria

First Century AD

Angle of

Incidence

Angle of

Incidence

Angle of

Reflection

FERM AT

Angle α1 ofIncidence

Angle α2 ofRefraction

CalculusMaximum

Minimum

Fermat Newton Euler LagrangeG auss and more

Some of the themes of 15053

bull Optimization is everywherebull Models Models Modelsbull The goal of models is ldquoinsightrdquo not

numbersndash paraphrase of Richard Hamming

bull Algorithms Algorithms Algorithms

Optimization is Everywhere

bull The more you know about something the more you see where optimization can be applied

bull Some personal decision makingndash Finding the fastest route home (or to class)ndash Optimal allocation of time for homeworkndash Optimal budgetingndash Selecting a major

Optimization is everywhere

bull Some MIT decision makingndash setting exam times to minimize overlapndash assigning classes to classrooms and time

slots while satisfying constraintsndash figuring out prices for parking and subsidies

for public transportation so as to maximize fairness and permit sufficient access

ndash optimizing in fundraising

Optimization is everywhere

bull Exercise introduce yourself to your neighborand then mention a topic or two with which youare pretty familiar (summer job or major or yourparentrsquos occupation or whatever)

bull Then brainstorm with your neighbor on placeswhere optimization occurs

bull Then choose your favorite 2 or 3 applicationsfrom your list and letrsquos share them

On 15053 and Optimization Tools

bull Optimization is everywhere but optimizationtools are not applied everywhere

bull Goals in 15053 present a variety of toolsfor optimization and illustrate applicationsin manufacturing finance e-businessmarketing and more

bull When you see optimization problems arisein business (and you will) you will know thatthere are tools to help you out

Addressing managerial problems Amanagement science framework

1 Determine the problem to be solved2 Observe the system and gather data3 Formulate a mathematical model of the problem an

d any important subproblems4 Verify the model and use the model for prediction

or analysis5 Select a suitable alternative6 Present the results to the organization7 Implement and evaluate

Linear Programming (our first tooland probably the most important one)bull minimize or maximize a linear objectivebull subject to linear equalities and

inequalities

maximize

subject to

3x + 4y

5x + 8y le 24x y ge 0

Here is the set of feasible solutions

The optimalsolution

Terminology

bull Decision variables eg x and yndash In general there are quantities you can control to

improve your objective which should completely describe the set of decisions to be made

bull Constraints eg 5x + 8y le 24 x ge 0 y ge 0ndash Limitations on the values of the decision variables

bull Objective Function eg 3x + 4yndash Value measure used to rank alternativesndash Seek to maximize or minimize this objectivendash examples maximize NPV minimize cost

MSR Marketing Incadapted from Frontline Systems

bull Need to choose ads to reach at least 15 million people

bull Minimize Costbull Upper bound on number

of ads of each type

TV Radio Mail Newspaper

Audience Size 50000 25000 20000 15000

CostImpression $500 $200 $250 $125

Max of ads 20 15 10 15

Formulating as a math model

Work with your partner

bull What decisions need to be made Expressthese as ldquodecision variablesrdquo

bull What is the objective Express the objectivein terms of the decision variables

bull What are the constraints Express these interms of the decision variables

bull If you have time try to find the best solution

Gemstone Tool Company

bull Privately-held firmbull Consumer and industrial market for construction toolsbull Headquartered in Seattlebull Manufacturing plants in the US Canada and Mexicobull Simplifying assumptions for purposes of illustration

ndash Winnipeg Canada plantndash Wrenches and pliersndash Made from steelndash Injection molding machinendash Assembly machine

Data for the GTC ProblemWrenches Pliers Available

Steel 15 10 15000 tons

Molding Machine 10 10 12000 hrs

Assembly Machine 4 5 5000 hrs

Demand Limit 8000 10000

Contribution($ per item)

$40 $30

We want to determine the number of wrenches and pliersto produce given the available raw materials machinehours and demand

To do with your partner

bull Work with your partner to formulate theGTC problem as a linear program DONOT LOOK AHEAD IN THE NOTES

bull Let P = number of pliers madebull Let W = number of wrenches made

Formulating the GTC Problem

Step 1 Determine Decision Variables P = number of thousands of pliers manufactured W = number of thousands of wrenches manufactured

Step 2 Determine Objective Function Maximize Profit =

The Formulation Continued

Step 3 Determine Constraints

Steel Molding Assembly Plier Demand Wrench Demand

We will show how to solve this in the next lecture

An Algebraic Formulation

bull J = set of items that are manufacturedndash eg S = pliers wrenchesndash pj = unit profit from item jndash dj = maximum demand for item jndash xj = number of units of item j manufactured

bull M = set of manufacturing processesndash eg M = molding and assemblyndash bi = capacity of process indash aij = amount of capacity of process i used in making item j

An Algebraic formulation

bull Maximize

bull subject to

The sameformulation workseven if | J | = 10000and | M | = 100

An Algebraic formulation

bull Maximize

bull subject to

Linear Programs

bull A linear function is a function of the form

bull A mathematical program is a linear program (LP) if the objective is a linear function and the constraints are linear equalities or inequalities

bull Typically an LP has non-negativity constraints

A non-linear program is permitted to havea non-linear objective and constraints

bull maximize f(xy) = xybull subject to x - y22 le 10 3x ndash 4y ge 2 x ge 0 y ge 0

An integer program is a linear programplus constraints that some or all of thevariables are integer valuedbull Maximize 3x1 + 4x2 - 3x3 3x1 + 2x2 - x3 ge 17 3x2 - x3 = 14 x1 ge 0 x2 ge 0 x3 ge 0 and x1 x2 x3 are all integers

An Algebraic formulation withequality constraints

bull Maximize

bull subject to

Linear Programming Assumptions

Proportionality Assumption Contribution from W isproportional to W

Additivity Assumption Contribution to objective function fromP is independent of W

Divisibility Assumption Each variable is allowed to assumefractional values

Certainty Assumption Each linear coefficient of the objectivefunction and constraints is known (and is not a random variable)

Maximize 4W + 3P15W + P le 15

hellip

Some Success Stories

bull Optimal crew scheduling saves American Airlines $20 millionyr

bull Improved shipment routing saves Yellow Freight over $173 millionyr

bull Improved truck dispatching at Reynolds Metals improves on-time delivery and reduces freight cost by $7 millionyr

bull GTE local capacity expansion saves $30 millionyr

Other Success Stories (cont)

bull Optimizing global supply chains saves Digital Equipment over $300 million

bull Restructuring North America Operations Proctor and Gamble reduces plants by 20 saving $200 millionyr

bull Optimal traffic control of Hanshin Expressway in Osaka saves 17 million driver hoursyr

bull Better scheduling of hydro and thermal generating units saves southern company $140 million

Success Stories (cont)

bull Improved production planning at Sadia (Brazil) saves $50 million over three years

bull Production Optimization at Harris Corporation improves on-time deliveries from 75 to 90

bull Tata Steel (India) optimizes response to power shortage contributing $73 million

bull Optimizing police patrol officer scheduling saves police department $11 millionyr

bull Gasoline blending at Texaco results in saving of over $30 millionyr

Summary

bull Answered the question What is Operations Research amp Management Science and provided some historical perspective

bull Introduced the terminology of linear programmingbull Two Examples

1 MSR Marketing2 Gemstone Tool Companyndash Small (2-dimensional) Linear Program non-obvious solution

bull We will discuss this problem in detail in the next lecture

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45

Course Policy (contrsquod)

bull Webpage

ndash Includesbull class notesbull Spreadsheetsbull Readingsbull assignments (assignment 1 is already there)bull and more

Active Learning

bull Occasionally I will introduce a break in the lecture for you to work on your own or with a partner

bull Please identify your ldquopartnerrdquo nowbull Those on aisle ends may be in a group of siz

e 3

What is Operations ResearchWhat is Management Sciencebull World War II British military leaders asked

scientists and engineers to analyze several military problemsndash Deployment of radarndash Management of convoy bombing antisubmarine

and mining operationsbull The result was called Military Operations

Research later Operations Researchbull MIT was one of the birthplaces of OR

ndash Professor Morse at MIT was a pioneer in the USndash Founded MIT OR Center and helped found ORSA

What is Management Science(Operations Research)bull Today Operations Research and Management

Science mean ldquothe use of mathematical models in providing guidelines to managers for making effective decisions within the state of the current information or in seeking further information if current knowledge is insufficient to reach a proper decisionrdquobull cf Decision science systems analysis

operational research systems dynamicsoperational analysis engineering systemssystems engineering and more

Voices from the past

bull Waste neither time nor money but make the best use of bothndash Benjamin Franklin

bull Obviously the highest type of efficiency is that which can utilize existing material to the best advantagendash Jawaharlal Nehru

bull It is more probable that the average man could with no injury to his health increase his efficiency fifty percentndash Walter Scott

Operations Research Over the Years

bull 1947ndash Project Scoop (Scientific Computation of

Optimum Programs) with George Dantzig andothers Developed the simplex method forlinear programs

bull 1950sndash Lots of excitement mathematical

developments queuing theory mathematicalprogrammingcf AI in the 1960s

bull 1960sndash More excitement more development and grand

plans cf AI in the 1980s

Operations Research Over the Years

bull 1970sndash Disappointment and a settling down NPcompleteness

More realistic expectationsbull 1980s

ndash Widespread availability of personal computersIncreasingly easy access to data Widespreadwillingness of managers to use models

bull 1990sndash Improved use of OR systems

Further inroads of OR technology egoptimization and simulation add-ons to spreadsheets modeling languages large scaleoptimization More intermixing of AI and OR

Operations Research in the 00rsquos

bull LOTS of opportunities for OR as a fieldbull Data data data

ndash E-business data (click stream purchases othertransactional data E-mail and more)

ndash The human genome project and its outgrowthbull Need for more automated decision makingbull Need for increased coordination for efficient

use of resources (Supply chain management)

Optimization

As agelessas time

Optimization in NatureHeron of Alexandria

First Century AD

Angle of

Incidence

Angle of

Incidence

Angle of

Reflection

FERM AT

Angle α1 ofIncidence

Angle α2 ofRefraction

CalculusMaximum

Minimum

Fermat Newton Euler LagrangeG auss and more

Some of the themes of 15053

bull Optimization is everywherebull Models Models Modelsbull The goal of models is ldquoinsightrdquo not

numbersndash paraphrase of Richard Hamming

bull Algorithms Algorithms Algorithms

Optimization is Everywhere

bull The more you know about something the more you see where optimization can be applied

bull Some personal decision makingndash Finding the fastest route home (or to class)ndash Optimal allocation of time for homeworkndash Optimal budgetingndash Selecting a major

Optimization is everywhere

bull Some MIT decision makingndash setting exam times to minimize overlapndash assigning classes to classrooms and time

slots while satisfying constraintsndash figuring out prices for parking and subsidies

for public transportation so as to maximize fairness and permit sufficient access

ndash optimizing in fundraising

Optimization is everywhere

bull Exercise introduce yourself to your neighborand then mention a topic or two with which youare pretty familiar (summer job or major or yourparentrsquos occupation or whatever)

bull Then brainstorm with your neighbor on placeswhere optimization occurs

bull Then choose your favorite 2 or 3 applicationsfrom your list and letrsquos share them

On 15053 and Optimization Tools

bull Optimization is everywhere but optimizationtools are not applied everywhere

bull Goals in 15053 present a variety of toolsfor optimization and illustrate applicationsin manufacturing finance e-businessmarketing and more

bull When you see optimization problems arisein business (and you will) you will know thatthere are tools to help you out

Addressing managerial problems Amanagement science framework

1 Determine the problem to be solved2 Observe the system and gather data3 Formulate a mathematical model of the problem an

d any important subproblems4 Verify the model and use the model for prediction

or analysis5 Select a suitable alternative6 Present the results to the organization7 Implement and evaluate

Linear Programming (our first tooland probably the most important one)bull minimize or maximize a linear objectivebull subject to linear equalities and

inequalities

maximize

subject to

3x + 4y

5x + 8y le 24x y ge 0

Here is the set of feasible solutions

The optimalsolution

Terminology

bull Decision variables eg x and yndash In general there are quantities you can control to

improve your objective which should completely describe the set of decisions to be made

bull Constraints eg 5x + 8y le 24 x ge 0 y ge 0ndash Limitations on the values of the decision variables

bull Objective Function eg 3x + 4yndash Value measure used to rank alternativesndash Seek to maximize or minimize this objectivendash examples maximize NPV minimize cost

MSR Marketing Incadapted from Frontline Systems

bull Need to choose ads to reach at least 15 million people

bull Minimize Costbull Upper bound on number

of ads of each type

TV Radio Mail Newspaper

Audience Size 50000 25000 20000 15000

CostImpression $500 $200 $250 $125

Max of ads 20 15 10 15

Formulating as a math model

Work with your partner

bull What decisions need to be made Expressthese as ldquodecision variablesrdquo

bull What is the objective Express the objectivein terms of the decision variables

bull What are the constraints Express these interms of the decision variables

bull If you have time try to find the best solution

Gemstone Tool Company

bull Privately-held firmbull Consumer and industrial market for construction toolsbull Headquartered in Seattlebull Manufacturing plants in the US Canada and Mexicobull Simplifying assumptions for purposes of illustration

ndash Winnipeg Canada plantndash Wrenches and pliersndash Made from steelndash Injection molding machinendash Assembly machine

Data for the GTC ProblemWrenches Pliers Available

Steel 15 10 15000 tons

Molding Machine 10 10 12000 hrs

Assembly Machine 4 5 5000 hrs

Demand Limit 8000 10000

Contribution($ per item)

$40 $30

We want to determine the number of wrenches and pliersto produce given the available raw materials machinehours and demand

To do with your partner

bull Work with your partner to formulate theGTC problem as a linear program DONOT LOOK AHEAD IN THE NOTES

bull Let P = number of pliers madebull Let W = number of wrenches made

Formulating the GTC Problem

Step 1 Determine Decision Variables P = number of thousands of pliers manufactured W = number of thousands of wrenches manufactured

Step 2 Determine Objective Function Maximize Profit =

The Formulation Continued

Step 3 Determine Constraints

Steel Molding Assembly Plier Demand Wrench Demand

We will show how to solve this in the next lecture

An Algebraic Formulation

bull J = set of items that are manufacturedndash eg S = pliers wrenchesndash pj = unit profit from item jndash dj = maximum demand for item jndash xj = number of units of item j manufactured

bull M = set of manufacturing processesndash eg M = molding and assemblyndash bi = capacity of process indash aij = amount of capacity of process i used in making item j

An Algebraic formulation

bull Maximize

bull subject to

The sameformulation workseven if | J | = 10000and | M | = 100

An Algebraic formulation

bull Maximize

bull subject to

Linear Programs

bull A linear function is a function of the form

bull A mathematical program is a linear program (LP) if the objective is a linear function and the constraints are linear equalities or inequalities

bull Typically an LP has non-negativity constraints

A non-linear program is permitted to havea non-linear objective and constraints

bull maximize f(xy) = xybull subject to x - y22 le 10 3x ndash 4y ge 2 x ge 0 y ge 0

An integer program is a linear programplus constraints that some or all of thevariables are integer valuedbull Maximize 3x1 + 4x2 - 3x3 3x1 + 2x2 - x3 ge 17 3x2 - x3 = 14 x1 ge 0 x2 ge 0 x3 ge 0 and x1 x2 x3 are all integers

An Algebraic formulation withequality constraints

bull Maximize

bull subject to

Linear Programming Assumptions

Proportionality Assumption Contribution from W isproportional to W

Additivity Assumption Contribution to objective function fromP is independent of W

Divisibility Assumption Each variable is allowed to assumefractional values

Certainty Assumption Each linear coefficient of the objectivefunction and constraints is known (and is not a random variable)

Maximize 4W + 3P15W + P le 15

hellip

Some Success Stories

bull Optimal crew scheduling saves American Airlines $20 millionyr

bull Improved shipment routing saves Yellow Freight over $173 millionyr

bull Improved truck dispatching at Reynolds Metals improves on-time delivery and reduces freight cost by $7 millionyr

bull GTE local capacity expansion saves $30 millionyr

Other Success Stories (cont)

bull Optimizing global supply chains saves Digital Equipment over $300 million

bull Restructuring North America Operations Proctor and Gamble reduces plants by 20 saving $200 millionyr

bull Optimal traffic control of Hanshin Expressway in Osaka saves 17 million driver hoursyr

bull Better scheduling of hydro and thermal generating units saves southern company $140 million

Success Stories (cont)

bull Improved production planning at Sadia (Brazil) saves $50 million over three years

bull Production Optimization at Harris Corporation improves on-time deliveries from 75 to 90

bull Tata Steel (India) optimizes response to power shortage contributing $73 million

bull Optimizing police patrol officer scheduling saves police department $11 millionyr

bull Gasoline blending at Texaco results in saving of over $30 millionyr

Summary

bull Answered the question What is Operations Research amp Management Science and provided some historical perspective

bull Introduced the terminology of linear programmingbull Two Examples

1 MSR Marketing2 Gemstone Tool Companyndash Small (2-dimensional) Linear Program non-obvious solution

bull We will discuss this problem in detail in the next lecture

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
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• Slide 14
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• Slide 44
• Slide 45

Active Learning

bull Occasionally I will introduce a break in the lecture for you to work on your own or with a partner

bull Please identify your ldquopartnerrdquo nowbull Those on aisle ends may be in a group of siz

e 3

What is Operations ResearchWhat is Management Sciencebull World War II British military leaders asked

scientists and engineers to analyze several military problemsndash Deployment of radarndash Management of convoy bombing antisubmarine

and mining operationsbull The result was called Military Operations

Research later Operations Researchbull MIT was one of the birthplaces of OR

ndash Professor Morse at MIT was a pioneer in the USndash Founded MIT OR Center and helped found ORSA

What is Management Science(Operations Research)bull Today Operations Research and Management

Science mean ldquothe use of mathematical models in providing guidelines to managers for making effective decisions within the state of the current information or in seeking further information if current knowledge is insufficient to reach a proper decisionrdquobull cf Decision science systems analysis

operational research systems dynamicsoperational analysis engineering systemssystems engineering and more

Voices from the past

bull Waste neither time nor money but make the best use of bothndash Benjamin Franklin

bull Obviously the highest type of efficiency is that which can utilize existing material to the best advantagendash Jawaharlal Nehru

bull It is more probable that the average man could with no injury to his health increase his efficiency fifty percentndash Walter Scott

Operations Research Over the Years

bull 1947ndash Project Scoop (Scientific Computation of

Optimum Programs) with George Dantzig andothers Developed the simplex method forlinear programs

bull 1950sndash Lots of excitement mathematical

developments queuing theory mathematicalprogrammingcf AI in the 1960s

bull 1960sndash More excitement more development and grand

plans cf AI in the 1980s

Operations Research Over the Years

bull 1970sndash Disappointment and a settling down NPcompleteness

More realistic expectationsbull 1980s

ndash Widespread availability of personal computersIncreasingly easy access to data Widespreadwillingness of managers to use models

bull 1990sndash Improved use of OR systems

Further inroads of OR technology egoptimization and simulation add-ons to spreadsheets modeling languages large scaleoptimization More intermixing of AI and OR

Operations Research in the 00rsquos

bull LOTS of opportunities for OR as a fieldbull Data data data

ndash E-business data (click stream purchases othertransactional data E-mail and more)

ndash The human genome project and its outgrowthbull Need for more automated decision makingbull Need for increased coordination for efficient

use of resources (Supply chain management)

Optimization

As agelessas time

Optimization in NatureHeron of Alexandria

First Century AD

Angle of

Incidence

Angle of

Incidence

Angle of

Reflection

FERM AT

Angle α1 ofIncidence

Angle α2 ofRefraction

CalculusMaximum

Minimum

Fermat Newton Euler LagrangeG auss and more

Some of the themes of 15053

bull Optimization is everywherebull Models Models Modelsbull The goal of models is ldquoinsightrdquo not

numbersndash paraphrase of Richard Hamming

bull Algorithms Algorithms Algorithms

Optimization is Everywhere

bull The more you know about something the more you see where optimization can be applied

bull Some personal decision makingndash Finding the fastest route home (or to class)ndash Optimal allocation of time for homeworkndash Optimal budgetingndash Selecting a major

Optimization is everywhere

bull Some MIT decision makingndash setting exam times to minimize overlapndash assigning classes to classrooms and time

slots while satisfying constraintsndash figuring out prices for parking and subsidies

for public transportation so as to maximize fairness and permit sufficient access

ndash optimizing in fundraising

Optimization is everywhere

bull Exercise introduce yourself to your neighborand then mention a topic or two with which youare pretty familiar (summer job or major or yourparentrsquos occupation or whatever)

bull Then brainstorm with your neighbor on placeswhere optimization occurs

bull Then choose your favorite 2 or 3 applicationsfrom your list and letrsquos share them

On 15053 and Optimization Tools

bull Optimization is everywhere but optimizationtools are not applied everywhere

bull Goals in 15053 present a variety of toolsfor optimization and illustrate applicationsin manufacturing finance e-businessmarketing and more

bull When you see optimization problems arisein business (and you will) you will know thatthere are tools to help you out

Addressing managerial problems Amanagement science framework

1 Determine the problem to be solved2 Observe the system and gather data3 Formulate a mathematical model of the problem an

d any important subproblems4 Verify the model and use the model for prediction

or analysis5 Select a suitable alternative6 Present the results to the organization7 Implement and evaluate

Linear Programming (our first tooland probably the most important one)bull minimize or maximize a linear objectivebull subject to linear equalities and

inequalities

maximize

subject to

3x + 4y

5x + 8y le 24x y ge 0

Here is the set of feasible solutions

The optimalsolution

Terminology

bull Decision variables eg x and yndash In general there are quantities you can control to

improve your objective which should completely describe the set of decisions to be made

bull Constraints eg 5x + 8y le 24 x ge 0 y ge 0ndash Limitations on the values of the decision variables

bull Objective Function eg 3x + 4yndash Value measure used to rank alternativesndash Seek to maximize or minimize this objectivendash examples maximize NPV minimize cost

MSR Marketing Incadapted from Frontline Systems

bull Need to choose ads to reach at least 15 million people

bull Minimize Costbull Upper bound on number

of ads of each type

TV Radio Mail Newspaper

Audience Size 50000 25000 20000 15000

CostImpression $500 $200 $250 $125

Max of ads 20 15 10 15

Formulating as a math model

Work with your partner

bull What decisions need to be made Expressthese as ldquodecision variablesrdquo

bull What is the objective Express the objectivein terms of the decision variables

bull What are the constraints Express these interms of the decision variables

bull If you have time try to find the best solution

Gemstone Tool Company

bull Privately-held firmbull Consumer and industrial market for construction toolsbull Headquartered in Seattlebull Manufacturing plants in the US Canada and Mexicobull Simplifying assumptions for purposes of illustration

ndash Winnipeg Canada plantndash Wrenches and pliersndash Made from steelndash Injection molding machinendash Assembly machine

Data for the GTC ProblemWrenches Pliers Available

Steel 15 10 15000 tons

Molding Machine 10 10 12000 hrs

Assembly Machine 4 5 5000 hrs

Demand Limit 8000 10000

Contribution($ per item)

$40 $30

We want to determine the number of wrenches and pliersto produce given the available raw materials machinehours and demand

To do with your partner

bull Work with your partner to formulate theGTC problem as a linear program DONOT LOOK AHEAD IN THE NOTES

bull Let P = number of pliers madebull Let W = number of wrenches made

Formulating the GTC Problem

Step 1 Determine Decision Variables P = number of thousands of pliers manufactured W = number of thousands of wrenches manufactured

Step 2 Determine Objective Function Maximize Profit =

The Formulation Continued

Step 3 Determine Constraints

Steel Molding Assembly Plier Demand Wrench Demand

We will show how to solve this in the next lecture

An Algebraic Formulation

bull J = set of items that are manufacturedndash eg S = pliers wrenchesndash pj = unit profit from item jndash dj = maximum demand for item jndash xj = number of units of item j manufactured

bull M = set of manufacturing processesndash eg M = molding and assemblyndash bi = capacity of process indash aij = amount of capacity of process i used in making item j

An Algebraic formulation

bull Maximize

bull subject to

The sameformulation workseven if | J | = 10000and | M | = 100

An Algebraic formulation

bull Maximize

bull subject to

Linear Programs

bull A linear function is a function of the form

bull A mathematical program is a linear program (LP) if the objective is a linear function and the constraints are linear equalities or inequalities

bull Typically an LP has non-negativity constraints

A non-linear program is permitted to havea non-linear objective and constraints

bull maximize f(xy) = xybull subject to x - y22 le 10 3x ndash 4y ge 2 x ge 0 y ge 0

An integer program is a linear programplus constraints that some or all of thevariables are integer valuedbull Maximize 3x1 + 4x2 - 3x3 3x1 + 2x2 - x3 ge 17 3x2 - x3 = 14 x1 ge 0 x2 ge 0 x3 ge 0 and x1 x2 x3 are all integers

An Algebraic formulation withequality constraints

bull Maximize

bull subject to

Linear Programming Assumptions

Proportionality Assumption Contribution from W isproportional to W

Additivity Assumption Contribution to objective function fromP is independent of W

Divisibility Assumption Each variable is allowed to assumefractional values

Certainty Assumption Each linear coefficient of the objectivefunction and constraints is known (and is not a random variable)

Maximize 4W + 3P15W + P le 15

hellip

Some Success Stories

bull Optimal crew scheduling saves American Airlines $20 millionyr

bull Improved shipment routing saves Yellow Freight over $173 millionyr

bull Improved truck dispatching at Reynolds Metals improves on-time delivery and reduces freight cost by $7 millionyr

bull GTE local capacity expansion saves $30 millionyr

Other Success Stories (cont)

bull Optimizing global supply chains saves Digital Equipment over $300 million

bull Restructuring North America Operations Proctor and Gamble reduces plants by 20 saving $200 millionyr

bull Optimal traffic control of Hanshin Expressway in Osaka saves 17 million driver hoursyr

bull Better scheduling of hydro and thermal generating units saves southern company $140 million

Success Stories (cont)

bull Improved production planning at Sadia (Brazil) saves $50 million over three years

bull Production Optimization at Harris Corporation improves on-time deliveries from 75 to 90

bull Tata Steel (India) optimizes response to power shortage contributing $73 million

bull Optimizing police patrol officer scheduling saves police department $11 millionyr

bull Gasoline blending at Texaco results in saving of over $30 millionyr

Summary

bull Answered the question What is Operations Research amp Management Science and provided some historical perspective

bull Introduced the terminology of linear programmingbull Two Examples

1 MSR Marketing2 Gemstone Tool Companyndash Small (2-dimensional) Linear Program non-obvious solution

bull We will discuss this problem in detail in the next lecture

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
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• Slide 24
• Slide 25
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• Slide 27
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• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45

What is Operations ResearchWhat is Management Sciencebull World War II British military leaders asked

scientists and engineers to analyze several military problemsndash Deployment of radarndash Management of convoy bombing antisubmarine

and mining operationsbull The result was called Military Operations

Research later Operations Researchbull MIT was one of the birthplaces of OR

ndash Professor Morse at MIT was a pioneer in the USndash Founded MIT OR Center and helped found ORSA

What is Management Science(Operations Research)bull Today Operations Research and Management

Science mean ldquothe use of mathematical models in providing guidelines to managers for making effective decisions within the state of the current information or in seeking further information if current knowledge is insufficient to reach a proper decisionrdquobull cf Decision science systems analysis

operational research systems dynamicsoperational analysis engineering systemssystems engineering and more

Voices from the past

bull Waste neither time nor money but make the best use of bothndash Benjamin Franklin

bull Obviously the highest type of efficiency is that which can utilize existing material to the best advantagendash Jawaharlal Nehru

bull It is more probable that the average man could with no injury to his health increase his efficiency fifty percentndash Walter Scott

Operations Research Over the Years

bull 1947ndash Project Scoop (Scientific Computation of

Optimum Programs) with George Dantzig andothers Developed the simplex method forlinear programs

bull 1950sndash Lots of excitement mathematical

developments queuing theory mathematicalprogrammingcf AI in the 1960s

bull 1960sndash More excitement more development and grand

plans cf AI in the 1980s

Operations Research Over the Years

bull 1970sndash Disappointment and a settling down NPcompleteness

More realistic expectationsbull 1980s

ndash Widespread availability of personal computersIncreasingly easy access to data Widespreadwillingness of managers to use models

bull 1990sndash Improved use of OR systems

Further inroads of OR technology egoptimization and simulation add-ons to spreadsheets modeling languages large scaleoptimization More intermixing of AI and OR

Operations Research in the 00rsquos

bull LOTS of opportunities for OR as a fieldbull Data data data

ndash E-business data (click stream purchases othertransactional data E-mail and more)

ndash The human genome project and its outgrowthbull Need for more automated decision makingbull Need for increased coordination for efficient

use of resources (Supply chain management)

Optimization

As agelessas time

Optimization in NatureHeron of Alexandria

First Century AD

Angle of

Incidence

Angle of

Incidence

Angle of

Reflection

FERM AT

Angle α1 ofIncidence

Angle α2 ofRefraction

CalculusMaximum

Minimum

Fermat Newton Euler LagrangeG auss and more

Some of the themes of 15053

bull Optimization is everywherebull Models Models Modelsbull The goal of models is ldquoinsightrdquo not

numbersndash paraphrase of Richard Hamming

bull Algorithms Algorithms Algorithms

Optimization is Everywhere

bull The more you know about something the more you see where optimization can be applied

bull Some personal decision makingndash Finding the fastest route home (or to class)ndash Optimal allocation of time for homeworkndash Optimal budgetingndash Selecting a major

Optimization is everywhere

bull Some MIT decision makingndash setting exam times to minimize overlapndash assigning classes to classrooms and time

slots while satisfying constraintsndash figuring out prices for parking and subsidies

for public transportation so as to maximize fairness and permit sufficient access

ndash optimizing in fundraising

Optimization is everywhere

bull Exercise introduce yourself to your neighborand then mention a topic or two with which youare pretty familiar (summer job or major or yourparentrsquos occupation or whatever)

bull Then brainstorm with your neighbor on placeswhere optimization occurs

bull Then choose your favorite 2 or 3 applicationsfrom your list and letrsquos share them

On 15053 and Optimization Tools

bull Optimization is everywhere but optimizationtools are not applied everywhere

bull Goals in 15053 present a variety of toolsfor optimization and illustrate applicationsin manufacturing finance e-businessmarketing and more

bull When you see optimization problems arisein business (and you will) you will know thatthere are tools to help you out

Addressing managerial problems Amanagement science framework

1 Determine the problem to be solved2 Observe the system and gather data3 Formulate a mathematical model of the problem an

d any important subproblems4 Verify the model and use the model for prediction

or analysis5 Select a suitable alternative6 Present the results to the organization7 Implement and evaluate

Linear Programming (our first tooland probably the most important one)bull minimize or maximize a linear objectivebull subject to linear equalities and

inequalities

maximize

subject to

3x + 4y

5x + 8y le 24x y ge 0

Here is the set of feasible solutions

The optimalsolution

Terminology

bull Decision variables eg x and yndash In general there are quantities you can control to

improve your objective which should completely describe the set of decisions to be made

bull Constraints eg 5x + 8y le 24 x ge 0 y ge 0ndash Limitations on the values of the decision variables

bull Objective Function eg 3x + 4yndash Value measure used to rank alternativesndash Seek to maximize or minimize this objectivendash examples maximize NPV minimize cost

MSR Marketing Incadapted from Frontline Systems

bull Need to choose ads to reach at least 15 million people

bull Minimize Costbull Upper bound on number

of ads of each type

TV Radio Mail Newspaper

Audience Size 50000 25000 20000 15000

CostImpression $500 $200 $250 $125

Max of ads 20 15 10 15

Formulating as a math model

Work with your partner

bull What decisions need to be made Expressthese as ldquodecision variablesrdquo

bull What is the objective Express the objectivein terms of the decision variables

bull What are the constraints Express these interms of the decision variables

bull If you have time try to find the best solution

Gemstone Tool Company

bull Privately-held firmbull Consumer and industrial market for construction toolsbull Headquartered in Seattlebull Manufacturing plants in the US Canada and Mexicobull Simplifying assumptions for purposes of illustration

ndash Winnipeg Canada plantndash Wrenches and pliersndash Made from steelndash Injection molding machinendash Assembly machine

Data for the GTC ProblemWrenches Pliers Available

Steel 15 10 15000 tons

Molding Machine 10 10 12000 hrs

Assembly Machine 4 5 5000 hrs

Demand Limit 8000 10000

Contribution($ per item)

$40 $30

We want to determine the number of wrenches and pliersto produce given the available raw materials machinehours and demand

To do with your partner

bull Work with your partner to formulate theGTC problem as a linear program DONOT LOOK AHEAD IN THE NOTES

bull Let P = number of pliers madebull Let W = number of wrenches made

Formulating the GTC Problem

Step 1 Determine Decision Variables P = number of thousands of pliers manufactured W = number of thousands of wrenches manufactured

Step 2 Determine Objective Function Maximize Profit =

The Formulation Continued

Step 3 Determine Constraints

Steel Molding Assembly Plier Demand Wrench Demand

We will show how to solve this in the next lecture

An Algebraic Formulation

bull J = set of items that are manufacturedndash eg S = pliers wrenchesndash pj = unit profit from item jndash dj = maximum demand for item jndash xj = number of units of item j manufactured

bull M = set of manufacturing processesndash eg M = molding and assemblyndash bi = capacity of process indash aij = amount of capacity of process i used in making item j

An Algebraic formulation

bull Maximize

bull subject to

The sameformulation workseven if | J | = 10000and | M | = 100

An Algebraic formulation

bull Maximize

bull subject to

Linear Programs

bull A linear function is a function of the form

bull A mathematical program is a linear program (LP) if the objective is a linear function and the constraints are linear equalities or inequalities

bull Typically an LP has non-negativity constraints

A non-linear program is permitted to havea non-linear objective and constraints

bull maximize f(xy) = xybull subject to x - y22 le 10 3x ndash 4y ge 2 x ge 0 y ge 0

An integer program is a linear programplus constraints that some or all of thevariables are integer valuedbull Maximize 3x1 + 4x2 - 3x3 3x1 + 2x2 - x3 ge 17 3x2 - x3 = 14 x1 ge 0 x2 ge 0 x3 ge 0 and x1 x2 x3 are all integers

An Algebraic formulation withequality constraints

bull Maximize

bull subject to

Linear Programming Assumptions

Proportionality Assumption Contribution from W isproportional to W

Additivity Assumption Contribution to objective function fromP is independent of W

Divisibility Assumption Each variable is allowed to assumefractional values

Certainty Assumption Each linear coefficient of the objectivefunction and constraints is known (and is not a random variable)

Maximize 4W + 3P15W + P le 15

hellip

Some Success Stories

bull Optimal crew scheduling saves American Airlines $20 millionyr

bull Improved shipment routing saves Yellow Freight over $173 millionyr

bull Improved truck dispatching at Reynolds Metals improves on-time delivery and reduces freight cost by $7 millionyr

bull GTE local capacity expansion saves $30 millionyr

Other Success Stories (cont)

bull Optimizing global supply chains saves Digital Equipment over $300 million

bull Restructuring North America Operations Proctor and Gamble reduces plants by 20 saving $200 millionyr

bull Optimal traffic control of Hanshin Expressway in Osaka saves 17 million driver hoursyr

bull Better scheduling of hydro and thermal generating units saves southern company $140 million

Success Stories (cont)

bull Improved production planning at Sadia (Brazil) saves $50 million over three years

bull Production Optimization at Harris Corporation improves on-time deliveries from 75 to 90

bull Tata Steel (India) optimizes response to power shortage contributing $73 million

bull Optimizing police patrol officer scheduling saves police department $11 millionyr

bull Gasoline blending at Texaco results in saving of over $30 millionyr

Summary

bull Answered the question What is Operations Research amp Management Science and provided some historical perspective

bull Introduced the terminology of linear programmingbull Two Examples

1 MSR Marketing2 Gemstone Tool Companyndash Small (2-dimensional) Linear Program non-obvious solution

bull We will discuss this problem in detail in the next lecture

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45

What is Management Science(Operations Research)bull Today Operations Research and Management

Science mean ldquothe use of mathematical models in providing guidelines to managers for making effective decisions within the state of the current information or in seeking further information if current knowledge is insufficient to reach a proper decisionrdquobull cf Decision science systems analysis

operational research systems dynamicsoperational analysis engineering systemssystems engineering and more

Voices from the past

bull Waste neither time nor money but make the best use of bothndash Benjamin Franklin

bull Obviously the highest type of efficiency is that which can utilize existing material to the best advantagendash Jawaharlal Nehru

bull It is more probable that the average man could with no injury to his health increase his efficiency fifty percentndash Walter Scott

Operations Research Over the Years

bull 1947ndash Project Scoop (Scientific Computation of

Optimum Programs) with George Dantzig andothers Developed the simplex method forlinear programs

bull 1950sndash Lots of excitement mathematical

developments queuing theory mathematicalprogrammingcf AI in the 1960s

bull 1960sndash More excitement more development and grand

plans cf AI in the 1980s

Operations Research Over the Years

bull 1970sndash Disappointment and a settling down NPcompleteness

More realistic expectationsbull 1980s

ndash Widespread availability of personal computersIncreasingly easy access to data Widespreadwillingness of managers to use models

bull 1990sndash Improved use of OR systems

Further inroads of OR technology egoptimization and simulation add-ons to spreadsheets modeling languages large scaleoptimization More intermixing of AI and OR

Operations Research in the 00rsquos

bull LOTS of opportunities for OR as a fieldbull Data data data

ndash E-business data (click stream purchases othertransactional data E-mail and more)

ndash The human genome project and its outgrowthbull Need for more automated decision makingbull Need for increased coordination for efficient

use of resources (Supply chain management)

Optimization

As agelessas time

Optimization in NatureHeron of Alexandria

First Century AD

Angle of

Incidence

Angle of

Incidence

Angle of

Reflection

FERM AT

Angle α1 ofIncidence

Angle α2 ofRefraction

CalculusMaximum

Minimum

Fermat Newton Euler LagrangeG auss and more

Some of the themes of 15053

bull Optimization is everywherebull Models Models Modelsbull The goal of models is ldquoinsightrdquo not

numbersndash paraphrase of Richard Hamming

bull Algorithms Algorithms Algorithms

Optimization is Everywhere

bull The more you know about something the more you see where optimization can be applied

bull Some personal decision makingndash Finding the fastest route home (or to class)ndash Optimal allocation of time for homeworkndash Optimal budgetingndash Selecting a major

Optimization is everywhere

bull Some MIT decision makingndash setting exam times to minimize overlapndash assigning classes to classrooms and time

slots while satisfying constraintsndash figuring out prices for parking and subsidies

for public transportation so as to maximize fairness and permit sufficient access

ndash optimizing in fundraising

Optimization is everywhere

bull Exercise introduce yourself to your neighborand then mention a topic or two with which youare pretty familiar (summer job or major or yourparentrsquos occupation or whatever)

bull Then brainstorm with your neighbor on placeswhere optimization occurs

bull Then choose your favorite 2 or 3 applicationsfrom your list and letrsquos share them

On 15053 and Optimization Tools

bull Optimization is everywhere but optimizationtools are not applied everywhere

bull Goals in 15053 present a variety of toolsfor optimization and illustrate applicationsin manufacturing finance e-businessmarketing and more

bull When you see optimization problems arisein business (and you will) you will know thatthere are tools to help you out

Addressing managerial problems Amanagement science framework

1 Determine the problem to be solved2 Observe the system and gather data3 Formulate a mathematical model of the problem an

d any important subproblems4 Verify the model and use the model for prediction

or analysis5 Select a suitable alternative6 Present the results to the organization7 Implement and evaluate

Linear Programming (our first tooland probably the most important one)bull minimize or maximize a linear objectivebull subject to linear equalities and

inequalities

maximize

subject to

3x + 4y

5x + 8y le 24x y ge 0

Here is the set of feasible solutions

The optimalsolution

Terminology

bull Decision variables eg x and yndash In general there are quantities you can control to

improve your objective which should completely describe the set of decisions to be made

bull Constraints eg 5x + 8y le 24 x ge 0 y ge 0ndash Limitations on the values of the decision variables

bull Objective Function eg 3x + 4yndash Value measure used to rank alternativesndash Seek to maximize or minimize this objectivendash examples maximize NPV minimize cost

MSR Marketing Incadapted from Frontline Systems

bull Need to choose ads to reach at least 15 million people

bull Minimize Costbull Upper bound on number

of ads of each type

TV Radio Mail Newspaper

Audience Size 50000 25000 20000 15000

CostImpression $500 $200 $250 $125

Max of ads 20 15 10 15

Formulating as a math model

Work with your partner

bull What decisions need to be made Expressthese as ldquodecision variablesrdquo

bull What is the objective Express the objectivein terms of the decision variables

bull What are the constraints Express these interms of the decision variables

bull If you have time try to find the best solution

Gemstone Tool Company

bull Privately-held firmbull Consumer and industrial market for construction toolsbull Headquartered in Seattlebull Manufacturing plants in the US Canada and Mexicobull Simplifying assumptions for purposes of illustration

ndash Winnipeg Canada plantndash Wrenches and pliersndash Made from steelndash Injection molding machinendash Assembly machine

Data for the GTC ProblemWrenches Pliers Available

Steel 15 10 15000 tons

Molding Machine 10 10 12000 hrs

Assembly Machine 4 5 5000 hrs

Demand Limit 8000 10000

Contribution($ per item)

$40 $30

We want to determine the number of wrenches and pliersto produce given the available raw materials machinehours and demand

To do with your partner

bull Work with your partner to formulate theGTC problem as a linear program DONOT LOOK AHEAD IN THE NOTES

bull Let P = number of pliers madebull Let W = number of wrenches made

Formulating the GTC Problem

Step 1 Determine Decision Variables P = number of thousands of pliers manufactured W = number of thousands of wrenches manufactured

Step 2 Determine Objective Function Maximize Profit =

The Formulation Continued

Step 3 Determine Constraints

Steel Molding Assembly Plier Demand Wrench Demand

We will show how to solve this in the next lecture

An Algebraic Formulation

bull J = set of items that are manufacturedndash eg S = pliers wrenchesndash pj = unit profit from item jndash dj = maximum demand for item jndash xj = number of units of item j manufactured

bull M = set of manufacturing processesndash eg M = molding and assemblyndash bi = capacity of process indash aij = amount of capacity of process i used in making item j

An Algebraic formulation

bull Maximize

bull subject to

The sameformulation workseven if | J | = 10000and | M | = 100

An Algebraic formulation

bull Maximize

bull subject to

Linear Programs

bull A linear function is a function of the form

bull A mathematical program is a linear program (LP) if the objective is a linear function and the constraints are linear equalities or inequalities

bull Typically an LP has non-negativity constraints

A non-linear program is permitted to havea non-linear objective and constraints

bull maximize f(xy) = xybull subject to x - y22 le 10 3x ndash 4y ge 2 x ge 0 y ge 0

An integer program is a linear programplus constraints that some or all of thevariables are integer valuedbull Maximize 3x1 + 4x2 - 3x3 3x1 + 2x2 - x3 ge 17 3x2 - x3 = 14 x1 ge 0 x2 ge 0 x3 ge 0 and x1 x2 x3 are all integers

An Algebraic formulation withequality constraints

bull Maximize

bull subject to

Linear Programming Assumptions

Proportionality Assumption Contribution from W isproportional to W

Additivity Assumption Contribution to objective function fromP is independent of W

Divisibility Assumption Each variable is allowed to assumefractional values

Certainty Assumption Each linear coefficient of the objectivefunction and constraints is known (and is not a random variable)

Maximize 4W + 3P15W + P le 15

hellip

Some Success Stories

bull Optimal crew scheduling saves American Airlines $20 millionyr

bull Improved shipment routing saves Yellow Freight over $173 millionyr

bull Improved truck dispatching at Reynolds Metals improves on-time delivery and reduces freight cost by $7 millionyr

bull GTE local capacity expansion saves $30 millionyr

Other Success Stories (cont)

bull Optimizing global supply chains saves Digital Equipment over $300 million

bull Restructuring North America Operations Proctor and Gamble reduces plants by 20 saving $200 millionyr

bull Optimal traffic control of Hanshin Expressway in Osaka saves 17 million driver hoursyr

bull Better scheduling of hydro and thermal generating units saves southern company $140 million

Success Stories (cont)

bull Improved production planning at Sadia (Brazil) saves $50 million over three years

bull Production Optimization at Harris Corporation improves on-time deliveries from 75 to 90

bull Tata Steel (India) optimizes response to power shortage contributing $73 million

bull Optimizing police patrol officer scheduling saves police department $11 millionyr

bull Gasoline blending at Texaco results in saving of over $30 millionyr

Summary

bull Answered the question What is Operations Research amp Management Science and provided some historical perspective

bull Introduced the terminology of linear programmingbull Two Examples

1 MSR Marketing2 Gemstone Tool Companyndash Small (2-dimensional) Linear Program non-obvious solution

bull We will discuss this problem in detail in the next lecture

• Slide 1
• Slide 2
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• Slide 5
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• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45

Voices from the past

bull Waste neither time nor money but make the best use of bothndash Benjamin Franklin

bull Obviously the highest type of efficiency is that which can utilize existing material to the best advantagendash Jawaharlal Nehru

bull It is more probable that the average man could with no injury to his health increase his efficiency fifty percentndash Walter Scott

Operations Research Over the Years

bull 1947ndash Project Scoop (Scientific Computation of

Optimum Programs) with George Dantzig andothers Developed the simplex method forlinear programs

bull 1950sndash Lots of excitement mathematical

developments queuing theory mathematicalprogrammingcf AI in the 1960s

bull 1960sndash More excitement more development and grand

plans cf AI in the 1980s

Operations Research Over the Years

bull 1970sndash Disappointment and a settling down NPcompleteness

More realistic expectationsbull 1980s

ndash Widespread availability of personal computersIncreasingly easy access to data Widespreadwillingness of managers to use models

bull 1990sndash Improved use of OR systems

Further inroads of OR technology egoptimization and simulation add-ons to spreadsheets modeling languages large scaleoptimization More intermixing of AI and OR

Operations Research in the 00rsquos

bull LOTS of opportunities for OR as a fieldbull Data data data

ndash E-business data (click stream purchases othertransactional data E-mail and more)

ndash The human genome project and its outgrowthbull Need for more automated decision makingbull Need for increased coordination for efficient

use of resources (Supply chain management)

Optimization

As agelessas time

Optimization in NatureHeron of Alexandria

First Century AD

Angle of

Incidence

Angle of

Incidence

Angle of

Reflection

FERM AT

Angle α1 ofIncidence

Angle α2 ofRefraction

CalculusMaximum

Minimum

Fermat Newton Euler LagrangeG auss and more

Some of the themes of 15053

bull Optimization is everywherebull Models Models Modelsbull The goal of models is ldquoinsightrdquo not

numbersndash paraphrase of Richard Hamming

bull Algorithms Algorithms Algorithms

Optimization is Everywhere

bull The more you know about something the more you see where optimization can be applied

bull Some personal decision makingndash Finding the fastest route home (or to class)ndash Optimal allocation of time for homeworkndash Optimal budgetingndash Selecting a major

Optimization is everywhere

bull Some MIT decision makingndash setting exam times to minimize overlapndash assigning classes to classrooms and time

slots while satisfying constraintsndash figuring out prices for parking and subsidies

for public transportation so as to maximize fairness and permit sufficient access

ndash optimizing in fundraising

Optimization is everywhere

bull Exercise introduce yourself to your neighborand then mention a topic or two with which youare pretty familiar (summer job or major or yourparentrsquos occupation or whatever)

bull Then brainstorm with your neighbor on placeswhere optimization occurs

bull Then choose your favorite 2 or 3 applicationsfrom your list and letrsquos share them

On 15053 and Optimization Tools

bull Optimization is everywhere but optimizationtools are not applied everywhere

bull Goals in 15053 present a variety of toolsfor optimization and illustrate applicationsin manufacturing finance e-businessmarketing and more

bull When you see optimization problems arisein business (and you will) you will know thatthere are tools to help you out

Addressing managerial problems Amanagement science framework

1 Determine the problem to be solved2 Observe the system and gather data3 Formulate a mathematical model of the problem an

d any important subproblems4 Verify the model and use the model for prediction

or analysis5 Select a suitable alternative6 Present the results to the organization7 Implement and evaluate

Linear Programming (our first tooland probably the most important one)bull minimize or maximize a linear objectivebull subject to linear equalities and

inequalities

maximize

subject to

3x + 4y

5x + 8y le 24x y ge 0

Here is the set of feasible solutions

The optimalsolution

Terminology

bull Decision variables eg x and yndash In general there are quantities you can control to

improve your objective which should completely describe the set of decisions to be made

bull Constraints eg 5x + 8y le 24 x ge 0 y ge 0ndash Limitations on the values of the decision variables

bull Objective Function eg 3x + 4yndash Value measure used to rank alternativesndash Seek to maximize or minimize this objectivendash examples maximize NPV minimize cost

MSR Marketing Incadapted from Frontline Systems

bull Need to choose ads to reach at least 15 million people

bull Minimize Costbull Upper bound on number

of ads of each type

TV Radio Mail Newspaper

Audience Size 50000 25000 20000 15000

CostImpression $500 $200 $250 $125

Max of ads 20 15 10 15

Formulating as a math model

Work with your partner

bull What decisions need to be made Expressthese as ldquodecision variablesrdquo

bull What is the objective Express the objectivein terms of the decision variables

bull What are the constraints Express these interms of the decision variables

bull If you have time try to find the best solution

Gemstone Tool Company

bull Privately-held firmbull Consumer and industrial market for construction toolsbull Headquartered in Seattlebull Manufacturing plants in the US Canada and Mexicobull Simplifying assumptions for purposes of illustration

ndash Winnipeg Canada plantndash Wrenches and pliersndash Made from steelndash Injection molding machinendash Assembly machine

Data for the GTC ProblemWrenches Pliers Available

Steel 15 10 15000 tons

Molding Machine 10 10 12000 hrs

Assembly Machine 4 5 5000 hrs

Demand Limit 8000 10000

Contribution($ per item)

$40 $30

We want to determine the number of wrenches and pliersto produce given the available raw materials machinehours and demand

To do with your partner

bull Work with your partner to formulate theGTC problem as a linear program DONOT LOOK AHEAD IN THE NOTES

bull Let P = number of pliers madebull Let W = number of wrenches made

Formulating the GTC Problem

Step 1 Determine Decision Variables P = number of thousands of pliers manufactured W = number of thousands of wrenches manufactured

Step 2 Determine Objective Function Maximize Profit =

The Formulation Continued

Step 3 Determine Constraints

Steel Molding Assembly Plier Demand Wrench Demand

We will show how to solve this in the next lecture

An Algebraic Formulation

bull J = set of items that are manufacturedndash eg S = pliers wrenchesndash pj = unit profit from item jndash dj = maximum demand for item jndash xj = number of units of item j manufactured

bull M = set of manufacturing processesndash eg M = molding and assemblyndash bi = capacity of process indash aij = amount of capacity of process i used in making item j

An Algebraic formulation

bull Maximize

bull subject to

The sameformulation workseven if | J | = 10000and | M | = 100

An Algebraic formulation

bull Maximize

bull subject to

Linear Programs

bull A linear function is a function of the form

bull A mathematical program is a linear program (LP) if the objective is a linear function and the constraints are linear equalities or inequalities

bull Typically an LP has non-negativity constraints

A non-linear program is permitted to havea non-linear objective and constraints

bull maximize f(xy) = xybull subject to x - y22 le 10 3x ndash 4y ge 2 x ge 0 y ge 0

An integer program is a linear programplus constraints that some or all of thevariables are integer valuedbull Maximize 3x1 + 4x2 - 3x3 3x1 + 2x2 - x3 ge 17 3x2 - x3 = 14 x1 ge 0 x2 ge 0 x3 ge 0 and x1 x2 x3 are all integers

An Algebraic formulation withequality constraints

bull Maximize

bull subject to

Linear Programming Assumptions

Proportionality Assumption Contribution from W isproportional to W

Additivity Assumption Contribution to objective function fromP is independent of W

Divisibility Assumption Each variable is allowed to assumefractional values

Certainty Assumption Each linear coefficient of the objectivefunction and constraints is known (and is not a random variable)

Maximize 4W + 3P15W + P le 15

hellip

Some Success Stories

bull Optimal crew scheduling saves American Airlines $20 millionyr

bull Improved shipment routing saves Yellow Freight over $173 millionyr

bull Improved truck dispatching at Reynolds Metals improves on-time delivery and reduces freight cost by $7 millionyr

bull GTE local capacity expansion saves $30 millionyr

Other Success Stories (cont)

bull Optimizing global supply chains saves Digital Equipment over $300 million

bull Restructuring North America Operations Proctor and Gamble reduces plants by 20 saving $200 millionyr

bull Optimal traffic control of Hanshin Expressway in Osaka saves 17 million driver hoursyr

bull Better scheduling of hydro and thermal generating units saves southern company $140 million

Success Stories (cont)

bull Improved production planning at Sadia (Brazil) saves $50 million over three years

bull Production Optimization at Harris Corporation improves on-time deliveries from 75 to 90

bull Tata Steel (India) optimizes response to power shortage contributing $73 million

bull Optimizing police patrol officer scheduling saves police department $11 millionyr

bull Gasoline blending at Texaco results in saving of over $30 millionyr

Summary

bull Answered the question What is Operations Research amp Management Science and provided some historical perspective

bull Introduced the terminology of linear programmingbull Two Examples

1 MSR Marketing2 Gemstone Tool Companyndash Small (2-dimensional) Linear Program non-obvious solution

bull We will discuss this problem in detail in the next lecture

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
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• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45

Operations Research Over the Years

bull 1947ndash Project Scoop (Scientific Computation of

Optimum Programs) with George Dantzig andothers Developed the simplex method forlinear programs

bull 1950sndash Lots of excitement mathematical

developments queuing theory mathematicalprogrammingcf AI in the 1960s

bull 1960sndash More excitement more development and grand

plans cf AI in the 1980s

Operations Research Over the Years

bull 1970sndash Disappointment and a settling down NPcompleteness

More realistic expectationsbull 1980s

ndash Widespread availability of personal computersIncreasingly easy access to data Widespreadwillingness of managers to use models

bull 1990sndash Improved use of OR systems

Further inroads of OR technology egoptimization and simulation add-ons to spreadsheets modeling languages large scaleoptimization More intermixing of AI and OR

Operations Research in the 00rsquos

bull LOTS of opportunities for OR as a fieldbull Data data data

ndash E-business data (click stream purchases othertransactional data E-mail and more)

ndash The human genome project and its outgrowthbull Need for more automated decision makingbull Need for increased coordination for efficient

use of resources (Supply chain management)

Optimization

As agelessas time

Optimization in NatureHeron of Alexandria

First Century AD

Angle of

Incidence

Angle of

Incidence

Angle of

Reflection

FERM AT

Angle α1 ofIncidence

Angle α2 ofRefraction

CalculusMaximum

Minimum

Fermat Newton Euler LagrangeG auss and more

Some of the themes of 15053

bull Optimization is everywherebull Models Models Modelsbull The goal of models is ldquoinsightrdquo not

numbersndash paraphrase of Richard Hamming

bull Algorithms Algorithms Algorithms

Optimization is Everywhere

bull The more you know about something the more you see where optimization can be applied

bull Some personal decision makingndash Finding the fastest route home (or to class)ndash Optimal allocation of time for homeworkndash Optimal budgetingndash Selecting a major

Optimization is everywhere

bull Some MIT decision makingndash setting exam times to minimize overlapndash assigning classes to classrooms and time

slots while satisfying constraintsndash figuring out prices for parking and subsidies

for public transportation so as to maximize fairness and permit sufficient access

ndash optimizing in fundraising

Optimization is everywhere

bull Exercise introduce yourself to your neighborand then mention a topic or two with which youare pretty familiar (summer job or major or yourparentrsquos occupation or whatever)

bull Then brainstorm with your neighbor on placeswhere optimization occurs

bull Then choose your favorite 2 or 3 applicationsfrom your list and letrsquos share them

On 15053 and Optimization Tools

bull Optimization is everywhere but optimizationtools are not applied everywhere

bull Goals in 15053 present a variety of toolsfor optimization and illustrate applicationsin manufacturing finance e-businessmarketing and more

bull When you see optimization problems arisein business (and you will) you will know thatthere are tools to help you out

Addressing managerial problems Amanagement science framework

1 Determine the problem to be solved2 Observe the system and gather data3 Formulate a mathematical model of the problem an

d any important subproblems4 Verify the model and use the model for prediction

or analysis5 Select a suitable alternative6 Present the results to the organization7 Implement and evaluate

Linear Programming (our first tooland probably the most important one)bull minimize or maximize a linear objectivebull subject to linear equalities and

inequalities

maximize

subject to

3x + 4y

5x + 8y le 24x y ge 0

Here is the set of feasible solutions

The optimalsolution

Terminology

bull Decision variables eg x and yndash In general there are quantities you can control to

improve your objective which should completely describe the set of decisions to be made

bull Constraints eg 5x + 8y le 24 x ge 0 y ge 0ndash Limitations on the values of the decision variables

bull Objective Function eg 3x + 4yndash Value measure used to rank alternativesndash Seek to maximize or minimize this objectivendash examples maximize NPV minimize cost

MSR Marketing Incadapted from Frontline Systems

bull Need to choose ads to reach at least 15 million people

bull Minimize Costbull Upper bound on number

of ads of each type

TV Radio Mail Newspaper

Audience Size 50000 25000 20000 15000

CostImpression $500 $200 $250 $125

Max of ads 20 15 10 15

Formulating as a math model

Work with your partner

bull What decisions need to be made Expressthese as ldquodecision variablesrdquo

bull What is the objective Express the objectivein terms of the decision variables

bull What are the constraints Express these interms of the decision variables

bull If you have time try to find the best solution

Gemstone Tool Company

bull Privately-held firmbull Consumer and industrial market for construction toolsbull Headquartered in Seattlebull Manufacturing plants in the US Canada and Mexicobull Simplifying assumptions for purposes of illustration

ndash Winnipeg Canada plantndash Wrenches and pliersndash Made from steelndash Injection molding machinendash Assembly machine

Data for the GTC ProblemWrenches Pliers Available

Steel 15 10 15000 tons

Molding Machine 10 10 12000 hrs

Assembly Machine 4 5 5000 hrs

Demand Limit 8000 10000

Contribution($ per item)

$40 $30

We want to determine the number of wrenches and pliersto produce given the available raw materials machinehours and demand

To do with your partner

bull Work with your partner to formulate theGTC problem as a linear program DONOT LOOK AHEAD IN THE NOTES

bull Let P = number of pliers madebull Let W = number of wrenches made

Formulating the GTC Problem

Step 1 Determine Decision Variables P = number of thousands of pliers manufactured W = number of thousands of wrenches manufactured

Step 2 Determine Objective Function Maximize Profit =

The Formulation Continued

Step 3 Determine Constraints

Steel Molding Assembly Plier Demand Wrench Demand

We will show how to solve this in the next lecture

An Algebraic Formulation

bull J = set of items that are manufacturedndash eg S = pliers wrenchesndash pj = unit profit from item jndash dj = maximum demand for item jndash xj = number of units of item j manufactured

bull M = set of manufacturing processesndash eg M = molding and assemblyndash bi = capacity of process indash aij = amount of capacity of process i used in making item j

An Algebraic formulation

bull Maximize

bull subject to

The sameformulation workseven if | J | = 10000and | M | = 100

An Algebraic formulation

bull Maximize

bull subject to

Linear Programs

bull A linear function is a function of the form

bull A mathematical program is a linear program (LP) if the objective is a linear function and the constraints are linear equalities or inequalities

bull Typically an LP has non-negativity constraints

A non-linear program is permitted to havea non-linear objective and constraints

bull maximize f(xy) = xybull subject to x - y22 le 10 3x ndash 4y ge 2 x ge 0 y ge 0

An integer program is a linear programplus constraints that some or all of thevariables are integer valuedbull Maximize 3x1 + 4x2 - 3x3 3x1 + 2x2 - x3 ge 17 3x2 - x3 = 14 x1 ge 0 x2 ge 0 x3 ge 0 and x1 x2 x3 are all integers

An Algebraic formulation withequality constraints

bull Maximize

bull subject to

Linear Programming Assumptions

Proportionality Assumption Contribution from W isproportional to W

Additivity Assumption Contribution to objective function fromP is independent of W

Divisibility Assumption Each variable is allowed to assumefractional values

Certainty Assumption Each linear coefficient of the objectivefunction and constraints is known (and is not a random variable)

Maximize 4W + 3P15W + P le 15

hellip

Some Success Stories

bull Optimal crew scheduling saves American Airlines $20 millionyr

bull Improved shipment routing saves Yellow Freight over $173 millionyr

bull Improved truck dispatching at Reynolds Metals improves on-time delivery and reduces freight cost by $7 millionyr

bull GTE local capacity expansion saves $30 millionyr

Other Success Stories (cont)

bull Optimizing global supply chains saves Digital Equipment over $300 million

bull Restructuring North America Operations Proctor and Gamble reduces plants by 20 saving $200 millionyr

bull Optimal traffic control of Hanshin Expressway in Osaka saves 17 million driver hoursyr

bull Better scheduling of hydro and thermal generating units saves southern company $140 million

Success Stories (cont)

bull Improved production planning at Sadia (Brazil) saves $50 million over three years

bull Production Optimization at Harris Corporation improves on-time deliveries from 75 to 90

bull Tata Steel (India) optimizes response to power shortage contributing $73 million

bull Optimizing police patrol officer scheduling saves police department $11 millionyr

bull Gasoline blending at Texaco results in saving of over $30 millionyr

Summary

bull Answered the question What is Operations Research amp Management Science and provided some historical perspective

bull Introduced the terminology of linear programmingbull Two Examples

1 MSR Marketing2 Gemstone Tool Companyndash Small (2-dimensional) Linear Program non-obvious solution

bull We will discuss this problem in detail in the next lecture

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45

Operations Research Over the Years

bull 1970sndash Disappointment and a settling down NPcompleteness

More realistic expectationsbull 1980s

ndash Widespread availability of personal computersIncreasingly easy access to data Widespreadwillingness of managers to use models

bull 1990sndash Improved use of OR systems

Further inroads of OR technology egoptimization and simulation add-ons to spreadsheets modeling languages large scaleoptimization More intermixing of AI and OR

Operations Research in the 00rsquos

bull LOTS of opportunities for OR as a fieldbull Data data data

ndash E-business data (click stream purchases othertransactional data E-mail and more)

ndash The human genome project and its outgrowthbull Need for more automated decision makingbull Need for increased coordination for efficient

use of resources (Supply chain management)

Optimization

As agelessas time

Optimization in NatureHeron of Alexandria

First Century AD

Angle of

Incidence

Angle of

Incidence

Angle of

Reflection

FERM AT

Angle α1 ofIncidence

Angle α2 ofRefraction

CalculusMaximum

Minimum

Fermat Newton Euler LagrangeG auss and more

Some of the themes of 15053

bull Optimization is everywherebull Models Models Modelsbull The goal of models is ldquoinsightrdquo not

numbersndash paraphrase of Richard Hamming

bull Algorithms Algorithms Algorithms

Optimization is Everywhere

bull The more you know about something the more you see where optimization can be applied

bull Some personal decision makingndash Finding the fastest route home (or to class)ndash Optimal allocation of time for homeworkndash Optimal budgetingndash Selecting a major

Optimization is everywhere

bull Some MIT decision makingndash setting exam times to minimize overlapndash assigning classes to classrooms and time

slots while satisfying constraintsndash figuring out prices for parking and subsidies

for public transportation so as to maximize fairness and permit sufficient access

ndash optimizing in fundraising

Optimization is everywhere

bull Exercise introduce yourself to your neighborand then mention a topic or two with which youare pretty familiar (summer job or major or yourparentrsquos occupation or whatever)

bull Then brainstorm with your neighbor on placeswhere optimization occurs

bull Then choose your favorite 2 or 3 applicationsfrom your list and letrsquos share them

On 15053 and Optimization Tools

bull Optimization is everywhere but optimizationtools are not applied everywhere

bull Goals in 15053 present a variety of toolsfor optimization and illustrate applicationsin manufacturing finance e-businessmarketing and more

bull When you see optimization problems arisein business (and you will) you will know thatthere are tools to help you out

Addressing managerial problems Amanagement science framework

1 Determine the problem to be solved2 Observe the system and gather data3 Formulate a mathematical model of the problem an

d any important subproblems4 Verify the model and use the model for prediction

or analysis5 Select a suitable alternative6 Present the results to the organization7 Implement and evaluate

Linear Programming (our first tooland probably the most important one)bull minimize or maximize a linear objectivebull subject to linear equalities and

inequalities

maximize

subject to

3x + 4y

5x + 8y le 24x y ge 0

Here is the set of feasible solutions

The optimalsolution

Terminology

bull Decision variables eg x and yndash In general there are quantities you can control to

improve your objective which should completely describe the set of decisions to be made

bull Constraints eg 5x + 8y le 24 x ge 0 y ge 0ndash Limitations on the values of the decision variables

bull Objective Function eg 3x + 4yndash Value measure used to rank alternativesndash Seek to maximize or minimize this objectivendash examples maximize NPV minimize cost

MSR Marketing Incadapted from Frontline Systems

bull Need to choose ads to reach at least 15 million people

bull Minimize Costbull Upper bound on number

of ads of each type

TV Radio Mail Newspaper

Audience Size 50000 25000 20000 15000

CostImpression $500 $200 $250 $125

Max of ads 20 15 10 15

Formulating as a math model

Work with your partner

bull What decisions need to be made Expressthese as ldquodecision variablesrdquo

bull What is the objective Express the objectivein terms of the decision variables

bull What are the constraints Express these interms of the decision variables

bull If you have time try to find the best solution

Gemstone Tool Company

bull Privately-held firmbull Consumer and industrial market for construction toolsbull Headquartered in Seattlebull Manufacturing plants in the US Canada and Mexicobull Simplifying assumptions for purposes of illustration

ndash Winnipeg Canada plantndash Wrenches and pliersndash Made from steelndash Injection molding machinendash Assembly machine

Data for the GTC ProblemWrenches Pliers Available

Steel 15 10 15000 tons

Molding Machine 10 10 12000 hrs

Assembly Machine 4 5 5000 hrs

Demand Limit 8000 10000

Contribution($ per item)

$40 $30

We want to determine the number of wrenches and pliersto produce given the available raw materials machinehours and demand

To do with your partner

bull Work with your partner to formulate theGTC problem as a linear program DONOT LOOK AHEAD IN THE NOTES

bull Let P = number of pliers madebull Let W = number of wrenches made

Formulating the GTC Problem

Step 1 Determine Decision Variables P = number of thousands of pliers manufactured W = number of thousands of wrenches manufactured

Step 2 Determine Objective Function Maximize Profit =

The Formulation Continued

Step 3 Determine Constraints

Steel Molding Assembly Plier Demand Wrench Demand

We will show how to solve this in the next lecture

An Algebraic Formulation

bull J = set of items that are manufacturedndash eg S = pliers wrenchesndash pj = unit profit from item jndash dj = maximum demand for item jndash xj = number of units of item j manufactured

bull M = set of manufacturing processesndash eg M = molding and assemblyndash bi = capacity of process indash aij = amount of capacity of process i used in making item j

An Algebraic formulation

bull Maximize

bull subject to

The sameformulation workseven if | J | = 10000and | M | = 100

An Algebraic formulation

bull Maximize

bull subject to

Linear Programs

bull A linear function is a function of the form

bull A mathematical program is a linear program (LP) if the objective is a linear function and the constraints are linear equalities or inequalities

bull Typically an LP has non-negativity constraints

A non-linear program is permitted to havea non-linear objective and constraints

bull maximize f(xy) = xybull subject to x - y22 le 10 3x ndash 4y ge 2 x ge 0 y ge 0

An integer program is a linear programplus constraints that some or all of thevariables are integer valuedbull Maximize 3x1 + 4x2 - 3x3 3x1 + 2x2 - x3 ge 17 3x2 - x3 = 14 x1 ge 0 x2 ge 0 x3 ge 0 and x1 x2 x3 are all integers

An Algebraic formulation withequality constraints

bull Maximize

bull subject to

Linear Programming Assumptions

Proportionality Assumption Contribution from W isproportional to W

Additivity Assumption Contribution to objective function fromP is independent of W

Divisibility Assumption Each variable is allowed to assumefractional values

Certainty Assumption Each linear coefficient of the objectivefunction and constraints is known (and is not a random variable)

Maximize 4W + 3P15W + P le 15

hellip

Some Success Stories

bull Optimal crew scheduling saves American Airlines $20 millionyr

bull Improved shipment routing saves Yellow Freight over $173 millionyr

bull Improved truck dispatching at Reynolds Metals improves on-time delivery and reduces freight cost by $7 millionyr

bull GTE local capacity expansion saves $30 millionyr

Other Success Stories (cont)

bull Optimizing global supply chains saves Digital Equipment over $300 million

bull Restructuring North America Operations Proctor and Gamble reduces plants by 20 saving $200 millionyr

bull Optimal traffic control of Hanshin Expressway in Osaka saves 17 million driver hoursyr

bull Better scheduling of hydro and thermal generating units saves southern company $140 million

Success Stories (cont)

bull Improved production planning at Sadia (Brazil) saves $50 million over three years

bull Production Optimization at Harris Corporation improves on-time deliveries from 75 to 90

bull Tata Steel (India) optimizes response to power shortage contributing $73 million

bull Optimizing police patrol officer scheduling saves police department $11 millionyr

bull Gasoline blending at Texaco results in saving of over $30 millionyr

Summary

bull Answered the question What is Operations Research amp Management Science and provided some historical perspective

bull Introduced the terminology of linear programmingbull Two Examples

1 MSR Marketing2 Gemstone Tool Companyndash Small (2-dimensional) Linear Program non-obvious solution

bull We will discuss this problem in detail in the next lecture

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
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• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45

Operations Research in the 00rsquos

bull LOTS of opportunities for OR as a fieldbull Data data data

ndash E-business data (click stream purchases othertransactional data E-mail and more)

ndash The human genome project and its outgrowthbull Need for more automated decision makingbull Need for increased coordination for efficient

use of resources (Supply chain management)

Optimization

As agelessas time

Optimization in NatureHeron of Alexandria

First Century AD

Angle of

Incidence

Angle of

Incidence

Angle of

Reflection

FERM AT

Angle α1 ofIncidence

Angle α2 ofRefraction

CalculusMaximum

Minimum

Fermat Newton Euler LagrangeG auss and more

Some of the themes of 15053

bull Optimization is everywherebull Models Models Modelsbull The goal of models is ldquoinsightrdquo not

numbersndash paraphrase of Richard Hamming

bull Algorithms Algorithms Algorithms

Optimization is Everywhere

bull The more you know about something the more you see where optimization can be applied

bull Some personal decision makingndash Finding the fastest route home (or to class)ndash Optimal allocation of time for homeworkndash Optimal budgetingndash Selecting a major

Optimization is everywhere

bull Some MIT decision makingndash setting exam times to minimize overlapndash assigning classes to classrooms and time

slots while satisfying constraintsndash figuring out prices for parking and subsidies

for public transportation so as to maximize fairness and permit sufficient access

ndash optimizing in fundraising

Optimization is everywhere

bull Exercise introduce yourself to your neighborand then mention a topic or two with which youare pretty familiar (summer job or major or yourparentrsquos occupation or whatever)

bull Then brainstorm with your neighbor on placeswhere optimization occurs

bull Then choose your favorite 2 or 3 applicationsfrom your list and letrsquos share them

On 15053 and Optimization Tools

bull Optimization is everywhere but optimizationtools are not applied everywhere

bull Goals in 15053 present a variety of toolsfor optimization and illustrate applicationsin manufacturing finance e-businessmarketing and more

bull When you see optimization problems arisein business (and you will) you will know thatthere are tools to help you out

Addressing managerial problems Amanagement science framework

1 Determine the problem to be solved2 Observe the system and gather data3 Formulate a mathematical model of the problem an

d any important subproblems4 Verify the model and use the model for prediction

or analysis5 Select a suitable alternative6 Present the results to the organization7 Implement and evaluate

Linear Programming (our first tooland probably the most important one)bull minimize or maximize a linear objectivebull subject to linear equalities and

inequalities

maximize

subject to

3x + 4y

5x + 8y le 24x y ge 0

Here is the set of feasible solutions

The optimalsolution

Terminology

bull Decision variables eg x and yndash In general there are quantities you can control to

improve your objective which should completely describe the set of decisions to be made

bull Constraints eg 5x + 8y le 24 x ge 0 y ge 0ndash Limitations on the values of the decision variables

bull Objective Function eg 3x + 4yndash Value measure used to rank alternativesndash Seek to maximize or minimize this objectivendash examples maximize NPV minimize cost

MSR Marketing Incadapted from Frontline Systems

bull Need to choose ads to reach at least 15 million people

bull Minimize Costbull Upper bound on number

of ads of each type

TV Radio Mail Newspaper

Audience Size 50000 25000 20000 15000

CostImpression $500 $200 $250 $125

Max of ads 20 15 10 15

Formulating as a math model

Work with your partner

bull What decisions need to be made Expressthese as ldquodecision variablesrdquo

bull What is the objective Express the objectivein terms of the decision variables

bull What are the constraints Express these interms of the decision variables

bull If you have time try to find the best solution

Gemstone Tool Company

bull Privately-held firmbull Consumer and industrial market for construction toolsbull Headquartered in Seattlebull Manufacturing plants in the US Canada and Mexicobull Simplifying assumptions for purposes of illustration

ndash Winnipeg Canada plantndash Wrenches and pliersndash Made from steelndash Injection molding machinendash Assembly machine

Data for the GTC ProblemWrenches Pliers Available

Steel 15 10 15000 tons

Molding Machine 10 10 12000 hrs

Assembly Machine 4 5 5000 hrs

Demand Limit 8000 10000

Contribution($ per item)

$40 $30

We want to determine the number of wrenches and pliersto produce given the available raw materials machinehours and demand

To do with your partner

bull Work with your partner to formulate theGTC problem as a linear program DONOT LOOK AHEAD IN THE NOTES

bull Let P = number of pliers madebull Let W = number of wrenches made

Formulating the GTC Problem

Step 1 Determine Decision Variables P = number of thousands of pliers manufactured W = number of thousands of wrenches manufactured

Step 2 Determine Objective Function Maximize Profit =

The Formulation Continued

Step 3 Determine Constraints

Steel Molding Assembly Plier Demand Wrench Demand

We will show how to solve this in the next lecture

An Algebraic Formulation

bull J = set of items that are manufacturedndash eg S = pliers wrenchesndash pj = unit profit from item jndash dj = maximum demand for item jndash xj = number of units of item j manufactured

bull M = set of manufacturing processesndash eg M = molding and assemblyndash bi = capacity of process indash aij = amount of capacity of process i used in making item j

An Algebraic formulation

bull Maximize

bull subject to

The sameformulation workseven if | J | = 10000and | M | = 100

An Algebraic formulation

bull Maximize

bull subject to

Linear Programs

bull A linear function is a function of the form

bull A mathematical program is a linear program (LP) if the objective is a linear function and the constraints are linear equalities or inequalities

bull Typically an LP has non-negativity constraints

A non-linear program is permitted to havea non-linear objective and constraints

bull maximize f(xy) = xybull subject to x - y22 le 10 3x ndash 4y ge 2 x ge 0 y ge 0

An integer program is a linear programplus constraints that some or all of thevariables are integer valuedbull Maximize 3x1 + 4x2 - 3x3 3x1 + 2x2 - x3 ge 17 3x2 - x3 = 14 x1 ge 0 x2 ge 0 x3 ge 0 and x1 x2 x3 are all integers

An Algebraic formulation withequality constraints

bull Maximize

bull subject to

Linear Programming Assumptions

Proportionality Assumption Contribution from W isproportional to W

Additivity Assumption Contribution to objective function fromP is independent of W

Divisibility Assumption Each variable is allowed to assumefractional values

Certainty Assumption Each linear coefficient of the objectivefunction and constraints is known (and is not a random variable)

Maximize 4W + 3P15W + P le 15

hellip

Some Success Stories

bull Optimal crew scheduling saves American Airlines $20 millionyr

bull Improved shipment routing saves Yellow Freight over $173 millionyr

bull Improved truck dispatching at Reynolds Metals improves on-time delivery and reduces freight cost by $7 millionyr

bull GTE local capacity expansion saves $30 millionyr

Other Success Stories (cont)

bull Optimizing global supply chains saves Digital Equipment over $300 million

bull Restructuring North America Operations Proctor and Gamble reduces plants by 20 saving $200 millionyr

bull Optimal traffic control of Hanshin Expressway in Osaka saves 17 million driver hoursyr

bull Better scheduling of hydro and thermal generating units saves southern company $140 million

Success Stories (cont)

bull Improved production planning at Sadia (Brazil) saves $50 million over three years

bull Production Optimization at Harris Corporation improves on-time deliveries from 75 to 90

bull Tata Steel (India) optimizes response to power shortage contributing $73 million

bull Optimizing police patrol officer scheduling saves police department $11 millionyr

bull Gasoline blending at Texaco results in saving of over $30 millionyr

Summary

bull Answered the question What is Operations Research amp Management Science and provided some historical perspective

bull Introduced the terminology of linear programmingbull Two Examples

1 MSR Marketing2 Gemstone Tool Companyndash Small (2-dimensional) Linear Program non-obvious solution

bull We will discuss this problem in detail in the next lecture

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
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• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45

Optimization

As agelessas time

Optimization in NatureHeron of Alexandria

First Century AD

Angle of

Incidence

Angle of

Incidence

Angle of

Reflection

FERM AT

Angle α1 ofIncidence

Angle α2 ofRefraction

CalculusMaximum

Minimum

Fermat Newton Euler LagrangeG auss and more

Some of the themes of 15053

bull Optimization is everywherebull Models Models Modelsbull The goal of models is ldquoinsightrdquo not

numbersndash paraphrase of Richard Hamming

bull Algorithms Algorithms Algorithms

Optimization is Everywhere

bull The more you know about something the more you see where optimization can be applied

bull Some personal decision makingndash Finding the fastest route home (or to class)ndash Optimal allocation of time for homeworkndash Optimal budgetingndash Selecting a major

Optimization is everywhere

bull Some MIT decision makingndash setting exam times to minimize overlapndash assigning classes to classrooms and time

slots while satisfying constraintsndash figuring out prices for parking and subsidies

for public transportation so as to maximize fairness and permit sufficient access

ndash optimizing in fundraising

Optimization is everywhere

bull Exercise introduce yourself to your neighborand then mention a topic or two with which youare pretty familiar (summer job or major or yourparentrsquos occupation or whatever)

bull Then brainstorm with your neighbor on placeswhere optimization occurs

bull Then choose your favorite 2 or 3 applicationsfrom your list and letrsquos share them

On 15053 and Optimization Tools

bull Optimization is everywhere but optimizationtools are not applied everywhere

bull Goals in 15053 present a variety of toolsfor optimization and illustrate applicationsin manufacturing finance e-businessmarketing and more

bull When you see optimization problems arisein business (and you will) you will know thatthere are tools to help you out

Addressing managerial problems Amanagement science framework

1 Determine the problem to be solved2 Observe the system and gather data3 Formulate a mathematical model of the problem an

d any important subproblems4 Verify the model and use the model for prediction

or analysis5 Select a suitable alternative6 Present the results to the organization7 Implement and evaluate

Linear Programming (our first tooland probably the most important one)bull minimize or maximize a linear objectivebull subject to linear equalities and

inequalities

maximize

subject to

3x + 4y

5x + 8y le 24x y ge 0

Here is the set of feasible solutions

The optimalsolution

Terminology

bull Decision variables eg x and yndash In general there are quantities you can control to

improve your objective which should completely describe the set of decisions to be made

bull Constraints eg 5x + 8y le 24 x ge 0 y ge 0ndash Limitations on the values of the decision variables

bull Objective Function eg 3x + 4yndash Value measure used to rank alternativesndash Seek to maximize or minimize this objectivendash examples maximize NPV minimize cost

MSR Marketing Incadapted from Frontline Systems

bull Need to choose ads to reach at least 15 million people

bull Minimize Costbull Upper bound on number

of ads of each type

TV Radio Mail Newspaper

Audience Size 50000 25000 20000 15000

CostImpression $500 $200 $250 $125

Max of ads 20 15 10 15

Formulating as a math model

Work with your partner

bull What decisions need to be made Expressthese as ldquodecision variablesrdquo

bull What is the objective Express the objectivein terms of the decision variables

bull What are the constraints Express these interms of the decision variables

bull If you have time try to find the best solution

Gemstone Tool Company

bull Privately-held firmbull Consumer and industrial market for construction toolsbull Headquartered in Seattlebull Manufacturing plants in the US Canada and Mexicobull Simplifying assumptions for purposes of illustration

ndash Winnipeg Canada plantndash Wrenches and pliersndash Made from steelndash Injection molding machinendash Assembly machine

Data for the GTC ProblemWrenches Pliers Available

Steel 15 10 15000 tons

Molding Machine 10 10 12000 hrs

Assembly Machine 4 5 5000 hrs

Demand Limit 8000 10000

Contribution($ per item)

$40 $30

We want to determine the number of wrenches and pliersto produce given the available raw materials machinehours and demand

To do with your partner

bull Work with your partner to formulate theGTC problem as a linear program DONOT LOOK AHEAD IN THE NOTES

bull Let P = number of pliers madebull Let W = number of wrenches made

Formulating the GTC Problem

Step 1 Determine Decision Variables P = number of thousands of pliers manufactured W = number of thousands of wrenches manufactured

Step 2 Determine Objective Function Maximize Profit =

The Formulation Continued

Step 3 Determine Constraints

Steel Molding Assembly Plier Demand Wrench Demand

We will show how to solve this in the next lecture

An Algebraic Formulation

bull J = set of items that are manufacturedndash eg S = pliers wrenchesndash pj = unit profit from item jndash dj = maximum demand for item jndash xj = number of units of item j manufactured

bull M = set of manufacturing processesndash eg M = molding and assemblyndash bi = capacity of process indash aij = amount of capacity of process i used in making item j

An Algebraic formulation

bull Maximize

bull subject to

The sameformulation workseven if | J | = 10000and | M | = 100

An Algebraic formulation

bull Maximize

bull subject to

Linear Programs

bull A linear function is a function of the form

bull A mathematical program is a linear program (LP) if the objective is a linear function and the constraints are linear equalities or inequalities

bull Typically an LP has non-negativity constraints

A non-linear program is permitted to havea non-linear objective and constraints

bull maximize f(xy) = xybull subject to x - y22 le 10 3x ndash 4y ge 2 x ge 0 y ge 0

An integer program is a linear programplus constraints that some or all of thevariables are integer valuedbull Maximize 3x1 + 4x2 - 3x3 3x1 + 2x2 - x3 ge 17 3x2 - x3 = 14 x1 ge 0 x2 ge 0 x3 ge 0 and x1 x2 x3 are all integers

An Algebraic formulation withequality constraints

bull Maximize

bull subject to

Linear Programming Assumptions

Proportionality Assumption Contribution from W isproportional to W

Additivity Assumption Contribution to objective function fromP is independent of W

Divisibility Assumption Each variable is allowed to assumefractional values

Certainty Assumption Each linear coefficient of the objectivefunction and constraints is known (and is not a random variable)

Maximize 4W + 3P15W + P le 15

hellip

Some Success Stories

bull Optimal crew scheduling saves American Airlines $20 millionyr

bull Improved shipment routing saves Yellow Freight over $173 millionyr

bull Improved truck dispatching at Reynolds Metals improves on-time delivery and reduces freight cost by $7 millionyr

bull GTE local capacity expansion saves $30 millionyr

Other Success Stories (cont)

bull Optimizing global supply chains saves Digital Equipment over $300 million

bull Restructuring North America Operations Proctor and Gamble reduces plants by 20 saving $200 millionyr

bull Optimal traffic control of Hanshin Expressway in Osaka saves 17 million driver hoursyr

bull Better scheduling of hydro and thermal generating units saves southern company $140 million

Success Stories (cont)

bull Improved production planning at Sadia (Brazil) saves $50 million over three years

bull Production Optimization at Harris Corporation improves on-time deliveries from 75 to 90

bull Tata Steel (India) optimizes response to power shortage contributing $73 million

bull Optimizing police patrol officer scheduling saves police department $11 millionyr

bull Gasoline blending at Texaco results in saving of over $30 millionyr

Summary

bull Answered the question What is Operations Research amp Management Science and provided some historical perspective

bull Introduced the terminology of linear programmingbull Two Examples

1 MSR Marketing2 Gemstone Tool Companyndash Small (2-dimensional) Linear Program non-obvious solution

bull We will discuss this problem in detail in the next lecture

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
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• Slide 28
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• Slide 31
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• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45

Optimization in NatureHeron of Alexandria

First Century AD

Angle of

Incidence

Angle of

Incidence

Angle of

Reflection

FERM AT

Angle α1 ofIncidence

Angle α2 ofRefraction

CalculusMaximum

Minimum

Fermat Newton Euler LagrangeG auss and more

Some of the themes of 15053

bull Optimization is everywherebull Models Models Modelsbull The goal of models is ldquoinsightrdquo not

numbersndash paraphrase of Richard Hamming

bull Algorithms Algorithms Algorithms

Optimization is Everywhere

bull The more you know about something the more you see where optimization can be applied

bull Some personal decision makingndash Finding the fastest route home (or to class)ndash Optimal allocation of time for homeworkndash Optimal budgetingndash Selecting a major

Optimization is everywhere

bull Some MIT decision makingndash setting exam times to minimize overlapndash assigning classes to classrooms and time

slots while satisfying constraintsndash figuring out prices for parking and subsidies

for public transportation so as to maximize fairness and permit sufficient access

ndash optimizing in fundraising

Optimization is everywhere

bull Exercise introduce yourself to your neighborand then mention a topic or two with which youare pretty familiar (summer job or major or yourparentrsquos occupation or whatever)

bull Then brainstorm with your neighbor on placeswhere optimization occurs

bull Then choose your favorite 2 or 3 applicationsfrom your list and letrsquos share them

On 15053 and Optimization Tools

bull Optimization is everywhere but optimizationtools are not applied everywhere

bull Goals in 15053 present a variety of toolsfor optimization and illustrate applicationsin manufacturing finance e-businessmarketing and more

bull When you see optimization problems arisein business (and you will) you will know thatthere are tools to help you out

Addressing managerial problems Amanagement science framework

1 Determine the problem to be solved2 Observe the system and gather data3 Formulate a mathematical model of the problem an

d any important subproblems4 Verify the model and use the model for prediction

or analysis5 Select a suitable alternative6 Present the results to the organization7 Implement and evaluate

Linear Programming (our first tooland probably the most important one)bull minimize or maximize a linear objectivebull subject to linear equalities and

inequalities

maximize

subject to

3x + 4y

5x + 8y le 24x y ge 0

Here is the set of feasible solutions

The optimalsolution

Terminology

bull Decision variables eg x and yndash In general there are quantities you can control to

improve your objective which should completely describe the set of decisions to be made

bull Constraints eg 5x + 8y le 24 x ge 0 y ge 0ndash Limitations on the values of the decision variables

bull Objective Function eg 3x + 4yndash Value measure used to rank alternativesndash Seek to maximize or minimize this objectivendash examples maximize NPV minimize cost

MSR Marketing Incadapted from Frontline Systems

bull Need to choose ads to reach at least 15 million people

bull Minimize Costbull Upper bound on number

of ads of each type

TV Radio Mail Newspaper

Audience Size 50000 25000 20000 15000

CostImpression $500 $200 $250 $125

Max of ads 20 15 10 15

Formulating as a math model

Work with your partner

bull What decisions need to be made Expressthese as ldquodecision variablesrdquo

bull What is the objective Express the objectivein terms of the decision variables

bull What are the constraints Express these interms of the decision variables

bull If you have time try to find the best solution

Gemstone Tool Company

bull Privately-held firmbull Consumer and industrial market for construction toolsbull Headquartered in Seattlebull Manufacturing plants in the US Canada and Mexicobull Simplifying assumptions for purposes of illustration

ndash Winnipeg Canada plantndash Wrenches and pliersndash Made from steelndash Injection molding machinendash Assembly machine

Data for the GTC ProblemWrenches Pliers Available

Steel 15 10 15000 tons

Molding Machine 10 10 12000 hrs

Assembly Machine 4 5 5000 hrs

Demand Limit 8000 10000

Contribution($ per item)

$40 $30

We want to determine the number of wrenches and pliersto produce given the available raw materials machinehours and demand

To do with your partner

bull Work with your partner to formulate theGTC problem as a linear program DONOT LOOK AHEAD IN THE NOTES

bull Let P = number of pliers madebull Let W = number of wrenches made

Formulating the GTC Problem

Step 1 Determine Decision Variables P = number of thousands of pliers manufactured W = number of thousands of wrenches manufactured

Step 2 Determine Objective Function Maximize Profit =

The Formulation Continued

Step 3 Determine Constraints

Steel Molding Assembly Plier Demand Wrench Demand

We will show how to solve this in the next lecture

An Algebraic Formulation

bull J = set of items that are manufacturedndash eg S = pliers wrenchesndash pj = unit profit from item jndash dj = maximum demand for item jndash xj = number of units of item j manufactured

bull M = set of manufacturing processesndash eg M = molding and assemblyndash bi = capacity of process indash aij = amount of capacity of process i used in making item j

An Algebraic formulation

bull Maximize

bull subject to

The sameformulation workseven if | J | = 10000and | M | = 100

An Algebraic formulation

bull Maximize

bull subject to

Linear Programs

bull A linear function is a function of the form

bull A mathematical program is a linear program (LP) if the objective is a linear function and the constraints are linear equalities or inequalities

bull Typically an LP has non-negativity constraints

A non-linear program is permitted to havea non-linear objective and constraints

bull maximize f(xy) = xybull subject to x - y22 le 10 3x ndash 4y ge 2 x ge 0 y ge 0

An integer program is a linear programplus constraints that some or all of thevariables are integer valuedbull Maximize 3x1 + 4x2 - 3x3 3x1 + 2x2 - x3 ge 17 3x2 - x3 = 14 x1 ge 0 x2 ge 0 x3 ge 0 and x1 x2 x3 are all integers

An Algebraic formulation withequality constraints

bull Maximize

bull subject to

Linear Programming Assumptions

Proportionality Assumption Contribution from W isproportional to W

Additivity Assumption Contribution to objective function fromP is independent of W

Divisibility Assumption Each variable is allowed to assumefractional values

Certainty Assumption Each linear coefficient of the objectivefunction and constraints is known (and is not a random variable)

Maximize 4W + 3P15W + P le 15

hellip

Some Success Stories

bull Optimal crew scheduling saves American Airlines $20 millionyr

bull Improved shipment routing saves Yellow Freight over $173 millionyr

bull Improved truck dispatching at Reynolds Metals improves on-time delivery and reduces freight cost by $7 millionyr

bull GTE local capacity expansion saves $30 millionyr

Other Success Stories (cont)

bull Optimizing global supply chains saves Digital Equipment over $300 million

bull Restructuring North America Operations Proctor and Gamble reduces plants by 20 saving $200 millionyr

bull Optimal traffic control of Hanshin Expressway in Osaka saves 17 million driver hoursyr

bull Better scheduling of hydro and thermal generating units saves southern company $140 million

Success Stories (cont)

bull Improved production planning at Sadia (Brazil) saves $50 million over three years

bull Production Optimization at Harris Corporation improves on-time deliveries from 75 to 90

bull Tata Steel (India) optimizes response to power shortage contributing $73 million

bull Optimizing police patrol officer scheduling saves police department $11 millionyr

bull Gasoline blending at Texaco results in saving of over $30 millionyr

Summary

bull Answered the question What is Operations Research amp Management Science and provided some historical perspective

bull Introduced the terminology of linear programmingbull Two Examples

1 MSR Marketing2 Gemstone Tool Companyndash Small (2-dimensional) Linear Program non-obvious solution

bull We will discuss this problem in detail in the next lecture

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
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• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45

FERM AT

Angle α1 ofIncidence

Angle α2 ofRefraction

CalculusMaximum

Minimum

Fermat Newton Euler LagrangeG auss and more

Some of the themes of 15053

bull Optimization is everywherebull Models Models Modelsbull The goal of models is ldquoinsightrdquo not

numbersndash paraphrase of Richard Hamming

bull Algorithms Algorithms Algorithms

Optimization is Everywhere

bull The more you know about something the more you see where optimization can be applied

bull Some personal decision makingndash Finding the fastest route home (or to class)ndash Optimal allocation of time for homeworkndash Optimal budgetingndash Selecting a major

Optimization is everywhere

bull Some MIT decision makingndash setting exam times to minimize overlapndash assigning classes to classrooms and time

slots while satisfying constraintsndash figuring out prices for parking and subsidies

for public transportation so as to maximize fairness and permit sufficient access

ndash optimizing in fundraising

Optimization is everywhere

bull Exercise introduce yourself to your neighborand then mention a topic or two with which youare pretty familiar (summer job or major or yourparentrsquos occupation or whatever)

bull Then brainstorm with your neighbor on placeswhere optimization occurs

bull Then choose your favorite 2 or 3 applicationsfrom your list and letrsquos share them

On 15053 and Optimization Tools

bull Optimization is everywhere but optimizationtools are not applied everywhere

bull Goals in 15053 present a variety of toolsfor optimization and illustrate applicationsin manufacturing finance e-businessmarketing and more

bull When you see optimization problems arisein business (and you will) you will know thatthere are tools to help you out

Addressing managerial problems Amanagement science framework

1 Determine the problem to be solved2 Observe the system and gather data3 Formulate a mathematical model of the problem an

d any important subproblems4 Verify the model and use the model for prediction

or analysis5 Select a suitable alternative6 Present the results to the organization7 Implement and evaluate

Linear Programming (our first tooland probably the most important one)bull minimize or maximize a linear objectivebull subject to linear equalities and

inequalities

maximize

subject to

3x + 4y

5x + 8y le 24x y ge 0

Here is the set of feasible solutions

The optimalsolution

Terminology

bull Decision variables eg x and yndash In general there are quantities you can control to

improve your objective which should completely describe the set of decisions to be made

bull Constraints eg 5x + 8y le 24 x ge 0 y ge 0ndash Limitations on the values of the decision variables

bull Objective Function eg 3x + 4yndash Value measure used to rank alternativesndash Seek to maximize or minimize this objectivendash examples maximize NPV minimize cost

MSR Marketing Incadapted from Frontline Systems

bull Need to choose ads to reach at least 15 million people

bull Minimize Costbull Upper bound on number

of ads of each type

TV Radio Mail Newspaper

Audience Size 50000 25000 20000 15000

CostImpression $500 $200 $250 $125

Max of ads 20 15 10 15

Formulating as a math model

Work with your partner

bull What decisions need to be made Expressthese as ldquodecision variablesrdquo

bull What is the objective Express the objectivein terms of the decision variables

bull What are the constraints Express these interms of the decision variables

bull If you have time try to find the best solution

Gemstone Tool Company

bull Privately-held firmbull Consumer and industrial market for construction toolsbull Headquartered in Seattlebull Manufacturing plants in the US Canada and Mexicobull Simplifying assumptions for purposes of illustration

ndash Winnipeg Canada plantndash Wrenches and pliersndash Made from steelndash Injection molding machinendash Assembly machine

Data for the GTC ProblemWrenches Pliers Available

Steel 15 10 15000 tons

Molding Machine 10 10 12000 hrs

Assembly Machine 4 5 5000 hrs

Demand Limit 8000 10000

Contribution($ per item)

$40 $30

We want to determine the number of wrenches and pliersto produce given the available raw materials machinehours and demand

To do with your partner

bull Work with your partner to formulate theGTC problem as a linear program DONOT LOOK AHEAD IN THE NOTES

bull Let P = number of pliers madebull Let W = number of wrenches made

Formulating the GTC Problem

Step 1 Determine Decision Variables P = number of thousands of pliers manufactured W = number of thousands of wrenches manufactured

Step 2 Determine Objective Function Maximize Profit =

The Formulation Continued

Step 3 Determine Constraints

Steel Molding Assembly Plier Demand Wrench Demand

We will show how to solve this in the next lecture

An Algebraic Formulation

bull J = set of items that are manufacturedndash eg S = pliers wrenchesndash pj = unit profit from item jndash dj = maximum demand for item jndash xj = number of units of item j manufactured

bull M = set of manufacturing processesndash eg M = molding and assemblyndash bi = capacity of process indash aij = amount of capacity of process i used in making item j

An Algebraic formulation

bull Maximize

bull subject to

The sameformulation workseven if | J | = 10000and | M | = 100

An Algebraic formulation

bull Maximize

bull subject to

Linear Programs

bull A linear function is a function of the form

bull A mathematical program is a linear program (LP) if the objective is a linear function and the constraints are linear equalities or inequalities

bull Typically an LP has non-negativity constraints

A non-linear program is permitted to havea non-linear objective and constraints

bull maximize f(xy) = xybull subject to x - y22 le 10 3x ndash 4y ge 2 x ge 0 y ge 0

An integer program is a linear programplus constraints that some or all of thevariables are integer valuedbull Maximize 3x1 + 4x2 - 3x3 3x1 + 2x2 - x3 ge 17 3x2 - x3 = 14 x1 ge 0 x2 ge 0 x3 ge 0 and x1 x2 x3 are all integers

An Algebraic formulation withequality constraints

bull Maximize

bull subject to

Linear Programming Assumptions

Proportionality Assumption Contribution from W isproportional to W

Additivity Assumption Contribution to objective function fromP is independent of W

Divisibility Assumption Each variable is allowed to assumefractional values

Certainty Assumption Each linear coefficient of the objectivefunction and constraints is known (and is not a random variable)

Maximize 4W + 3P15W + P le 15

hellip

Some Success Stories

bull Optimal crew scheduling saves American Airlines $20 millionyr

bull Improved shipment routing saves Yellow Freight over $173 millionyr

bull Improved truck dispatching at Reynolds Metals improves on-time delivery and reduces freight cost by $7 millionyr

bull GTE local capacity expansion saves $30 millionyr

Other Success Stories (cont)

bull Optimizing global supply chains saves Digital Equipment over $300 million

bull Restructuring North America Operations Proctor and Gamble reduces plants by 20 saving $200 millionyr

bull Optimal traffic control of Hanshin Expressway in Osaka saves 17 million driver hoursyr

bull Better scheduling of hydro and thermal generating units saves southern company $140 million

Success Stories (cont)

bull Improved production planning at Sadia (Brazil) saves $50 million over three years

bull Production Optimization at Harris Corporation improves on-time deliveries from 75 to 90

bull Tata Steel (India) optimizes response to power shortage contributing $73 million

bull Optimizing police patrol officer scheduling saves police department $11 millionyr

bull Gasoline blending at Texaco results in saving of over $30 millionyr

Summary

bull Answered the question What is Operations Research amp Management Science and provided some historical perspective

bull Introduced the terminology of linear programmingbull Two Examples

1 MSR Marketing2 Gemstone Tool Companyndash Small (2-dimensional) Linear Program non-obvious solution

bull We will discuss this problem in detail in the next lecture

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
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• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45

CalculusMaximum

Minimum

Fermat Newton Euler LagrangeG auss and more

Some of the themes of 15053

bull Optimization is everywherebull Models Models Modelsbull The goal of models is ldquoinsightrdquo not

numbersndash paraphrase of Richard Hamming

bull Algorithms Algorithms Algorithms

Optimization is Everywhere

bull The more you know about something the more you see where optimization can be applied

bull Some personal decision makingndash Finding the fastest route home (or to class)ndash Optimal allocation of time for homeworkndash Optimal budgetingndash Selecting a major

Optimization is everywhere

bull Some MIT decision makingndash setting exam times to minimize overlapndash assigning classes to classrooms and time

slots while satisfying constraintsndash figuring out prices for parking and subsidies

for public transportation so as to maximize fairness and permit sufficient access

ndash optimizing in fundraising

Optimization is everywhere

bull Exercise introduce yourself to your neighborand then mention a topic or two with which youare pretty familiar (summer job or major or yourparentrsquos occupation or whatever)

bull Then brainstorm with your neighbor on placeswhere optimization occurs

bull Then choose your favorite 2 or 3 applicationsfrom your list and letrsquos share them

On 15053 and Optimization Tools

bull Optimization is everywhere but optimizationtools are not applied everywhere

bull Goals in 15053 present a variety of toolsfor optimization and illustrate applicationsin manufacturing finance e-businessmarketing and more

bull When you see optimization problems arisein business (and you will) you will know thatthere are tools to help you out

Addressing managerial problems Amanagement science framework

1 Determine the problem to be solved2 Observe the system and gather data3 Formulate a mathematical model of the problem an

d any important subproblems4 Verify the model and use the model for prediction

or analysis5 Select a suitable alternative6 Present the results to the organization7 Implement and evaluate

Linear Programming (our first tooland probably the most important one)bull minimize or maximize a linear objectivebull subject to linear equalities and

inequalities

maximize

subject to

3x + 4y

5x + 8y le 24x y ge 0

Here is the set of feasible solutions

The optimalsolution

Terminology

bull Decision variables eg x and yndash In general there are quantities you can control to

improve your objective which should completely describe the set of decisions to be made

bull Constraints eg 5x + 8y le 24 x ge 0 y ge 0ndash Limitations on the values of the decision variables

bull Objective Function eg 3x + 4yndash Value measure used to rank alternativesndash Seek to maximize or minimize this objectivendash examples maximize NPV minimize cost

MSR Marketing Incadapted from Frontline Systems

bull Need to choose ads to reach at least 15 million people

bull Minimize Costbull Upper bound on number

of ads of each type

TV Radio Mail Newspaper

Audience Size 50000 25000 20000 15000

CostImpression $500 $200 $250 $125

Max of ads 20 15 10 15

Formulating as a math model

Work with your partner

bull What decisions need to be made Expressthese as ldquodecision variablesrdquo

bull What is the objective Express the objectivein terms of the decision variables

bull What are the constraints Express these interms of the decision variables

bull If you have time try to find the best solution

Gemstone Tool Company

bull Privately-held firmbull Consumer and industrial market for construction toolsbull Headquartered in Seattlebull Manufacturing plants in the US Canada and Mexicobull Simplifying assumptions for purposes of illustration

ndash Winnipeg Canada plantndash Wrenches and pliersndash Made from steelndash Injection molding machinendash Assembly machine

Data for the GTC ProblemWrenches Pliers Available

Steel 15 10 15000 tons

Molding Machine 10 10 12000 hrs

Assembly Machine 4 5 5000 hrs

Demand Limit 8000 10000

Contribution($ per item)

$40 $30

We want to determine the number of wrenches and pliersto produce given the available raw materials machinehours and demand

To do with your partner

bull Work with your partner to formulate theGTC problem as a linear program DONOT LOOK AHEAD IN THE NOTES

bull Let P = number of pliers madebull Let W = number of wrenches made

Formulating the GTC Problem

Step 1 Determine Decision Variables P = number of thousands of pliers manufactured W = number of thousands of wrenches manufactured

Step 2 Determine Objective Function Maximize Profit =

The Formulation Continued

Step 3 Determine Constraints

Steel Molding Assembly Plier Demand Wrench Demand

We will show how to solve this in the next lecture

An Algebraic Formulation

bull J = set of items that are manufacturedndash eg S = pliers wrenchesndash pj = unit profit from item jndash dj = maximum demand for item jndash xj = number of units of item j manufactured

bull M = set of manufacturing processesndash eg M = molding and assemblyndash bi = capacity of process indash aij = amount of capacity of process i used in making item j

An Algebraic formulation

bull Maximize

bull subject to

The sameformulation workseven if | J | = 10000and | M | = 100

An Algebraic formulation

bull Maximize

bull subject to

Linear Programs

bull A linear function is a function of the form

bull A mathematical program is a linear program (LP) if the objective is a linear function and the constraints are linear equalities or inequalities

bull Typically an LP has non-negativity constraints

A non-linear program is permitted to havea non-linear objective and constraints

bull maximize f(xy) = xybull subject to x - y22 le 10 3x ndash 4y ge 2 x ge 0 y ge 0

An integer program is a linear programplus constraints that some or all of thevariables are integer valuedbull Maximize 3x1 + 4x2 - 3x3 3x1 + 2x2 - x3 ge 17 3x2 - x3 = 14 x1 ge 0 x2 ge 0 x3 ge 0 and x1 x2 x3 are all integers

An Algebraic formulation withequality constraints

bull Maximize

bull subject to

Linear Programming Assumptions

Proportionality Assumption Contribution from W isproportional to W

Additivity Assumption Contribution to objective function fromP is independent of W

Divisibility Assumption Each variable is allowed to assumefractional values

Certainty Assumption Each linear coefficient of the objectivefunction and constraints is known (and is not a random variable)

Maximize 4W + 3P15W + P le 15

hellip

Some Success Stories

bull Optimal crew scheduling saves American Airlines $20 millionyr

bull Improved shipment routing saves Yellow Freight over $173 millionyr

bull Improved truck dispatching at Reynolds Metals improves on-time delivery and reduces freight cost by $7 millionyr

bull GTE local capacity expansion saves $30 millionyr

Other Success Stories (cont)

bull Optimizing global supply chains saves Digital Equipment over $300 million

bull Restructuring North America Operations Proctor and Gamble reduces plants by 20 saving $200 millionyr

bull Optimal traffic control of Hanshin Expressway in Osaka saves 17 million driver hoursyr

bull Better scheduling of hydro and thermal generating units saves southern company $140 million

Success Stories (cont)

bull Improved production planning at Sadia (Brazil) saves $50 million over three years

bull Production Optimization at Harris Corporation improves on-time deliveries from 75 to 90

bull Tata Steel (India) optimizes response to power shortage contributing $73 million

bull Optimizing police patrol officer scheduling saves police department $11 millionyr

bull Gasoline blending at Texaco results in saving of over $30 millionyr

Summary

bull Answered the question What is Operations Research amp Management Science and provided some historical perspective

bull Introduced the terminology of linear programmingbull Two Examples

1 MSR Marketing2 Gemstone Tool Companyndash Small (2-dimensional) Linear Program non-obvious solution

bull We will discuss this problem in detail in the next lecture

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
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• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45

Some of the themes of 15053

bull Optimization is everywherebull Models Models Modelsbull The goal of models is ldquoinsightrdquo not

numbersndash paraphrase of Richard Hamming

bull Algorithms Algorithms Algorithms

Optimization is Everywhere

bull The more you know about something the more you see where optimization can be applied

bull Some personal decision makingndash Finding the fastest route home (or to class)ndash Optimal allocation of time for homeworkndash Optimal budgetingndash Selecting a major

Optimization is everywhere

bull Some MIT decision makingndash setting exam times to minimize overlapndash assigning classes to classrooms and time

slots while satisfying constraintsndash figuring out prices for parking and subsidies

for public transportation so as to maximize fairness and permit sufficient access

ndash optimizing in fundraising

Optimization is everywhere

bull Exercise introduce yourself to your neighborand then mention a topic or two with which youare pretty familiar (summer job or major or yourparentrsquos occupation or whatever)

bull Then brainstorm with your neighbor on placeswhere optimization occurs

bull Then choose your favorite 2 or 3 applicationsfrom your list and letrsquos share them

On 15053 and Optimization Tools

bull Optimization is everywhere but optimizationtools are not applied everywhere

bull Goals in 15053 present a variety of toolsfor optimization and illustrate applicationsin manufacturing finance e-businessmarketing and more

bull When you see optimization problems arisein business (and you will) you will know thatthere are tools to help you out

Addressing managerial problems Amanagement science framework

1 Determine the problem to be solved2 Observe the system and gather data3 Formulate a mathematical model of the problem an

d any important subproblems4 Verify the model and use the model for prediction

or analysis5 Select a suitable alternative6 Present the results to the organization7 Implement and evaluate

Linear Programming (our first tooland probably the most important one)bull minimize or maximize a linear objectivebull subject to linear equalities and

inequalities

maximize

subject to

3x + 4y

5x + 8y le 24x y ge 0

Here is the set of feasible solutions

The optimalsolution

Terminology

bull Decision variables eg x and yndash In general there are quantities you can control to

improve your objective which should completely describe the set of decisions to be made

bull Constraints eg 5x + 8y le 24 x ge 0 y ge 0ndash Limitations on the values of the decision variables

bull Objective Function eg 3x + 4yndash Value measure used to rank alternativesndash Seek to maximize or minimize this objectivendash examples maximize NPV minimize cost

MSR Marketing Incadapted from Frontline Systems

bull Need to choose ads to reach at least 15 million people

bull Minimize Costbull Upper bound on number

of ads of each type

TV Radio Mail Newspaper

Audience Size 50000 25000 20000 15000

CostImpression $500 $200 $250 $125

Max of ads 20 15 10 15

Formulating as a math model

Work with your partner

bull What decisions need to be made Expressthese as ldquodecision variablesrdquo

bull What is the objective Express the objectivein terms of the decision variables

bull What are the constraints Express these interms of the decision variables

bull If you have time try to find the best solution

Gemstone Tool Company

bull Privately-held firmbull Consumer and industrial market for construction toolsbull Headquartered in Seattlebull Manufacturing plants in the US Canada and Mexicobull Simplifying assumptions for purposes of illustration

ndash Winnipeg Canada plantndash Wrenches and pliersndash Made from steelndash Injection molding machinendash Assembly machine

Data for the GTC ProblemWrenches Pliers Available

Steel 15 10 15000 tons

Molding Machine 10 10 12000 hrs

Assembly Machine 4 5 5000 hrs

Demand Limit 8000 10000

Contribution($ per item)

$40 $30

We want to determine the number of wrenches and pliersto produce given the available raw materials machinehours and demand

To do with your partner

bull Work with your partner to formulate theGTC problem as a linear program DONOT LOOK AHEAD IN THE NOTES

bull Let P = number of pliers madebull Let W = number of wrenches made

Formulating the GTC Problem

Step 1 Determine Decision Variables P = number of thousands of pliers manufactured W = number of thousands of wrenches manufactured

Step 2 Determine Objective Function Maximize Profit =

The Formulation Continued

Step 3 Determine Constraints

Steel Molding Assembly Plier Demand Wrench Demand

We will show how to solve this in the next lecture

An Algebraic Formulation

bull J = set of items that are manufacturedndash eg S = pliers wrenchesndash pj = unit profit from item jndash dj = maximum demand for item jndash xj = number of units of item j manufactured

bull M = set of manufacturing processesndash eg M = molding and assemblyndash bi = capacity of process indash aij = amount of capacity of process i used in making item j

An Algebraic formulation

bull Maximize

bull subject to

The sameformulation workseven if | J | = 10000and | M | = 100

An Algebraic formulation

bull Maximize

bull subject to

Linear Programs

bull A linear function is a function of the form

bull A mathematical program is a linear program (LP) if the objective is a linear function and the constraints are linear equalities or inequalities

bull Typically an LP has non-negativity constraints

A non-linear program is permitted to havea non-linear objective and constraints

bull maximize f(xy) = xybull subject to x - y22 le 10 3x ndash 4y ge 2 x ge 0 y ge 0

An integer program is a linear programplus constraints that some or all of thevariables are integer valuedbull Maximize 3x1 + 4x2 - 3x3 3x1 + 2x2 - x3 ge 17 3x2 - x3 = 14 x1 ge 0 x2 ge 0 x3 ge 0 and x1 x2 x3 are all integers

An Algebraic formulation withequality constraints

bull Maximize

bull subject to

Linear Programming Assumptions

Proportionality Assumption Contribution from W isproportional to W

Additivity Assumption Contribution to objective function fromP is independent of W

Divisibility Assumption Each variable is allowed to assumefractional values

Certainty Assumption Each linear coefficient of the objectivefunction and constraints is known (and is not a random variable)

Maximize 4W + 3P15W + P le 15

hellip

Some Success Stories

bull Optimal crew scheduling saves American Airlines $20 millionyr

bull Improved shipment routing saves Yellow Freight over $173 millionyr

bull Improved truck dispatching at Reynolds Metals improves on-time delivery and reduces freight cost by $7 millionyr

bull GTE local capacity expansion saves $30 millionyr

Other Success Stories (cont)

bull Optimizing global supply chains saves Digital Equipment over $300 million

bull Restructuring North America Operations Proctor and Gamble reduces plants by 20 saving $200 millionyr

bull Optimal traffic control of Hanshin Expressway in Osaka saves 17 million driver hoursyr

bull Better scheduling of hydro and thermal generating units saves southern company $140 million

Success Stories (cont)

bull Improved production planning at Sadia (Brazil) saves $50 million over three years

bull Production Optimization at Harris Corporation improves on-time deliveries from 75 to 90

bull Tata Steel (India) optimizes response to power shortage contributing $73 million

bull Optimizing police patrol officer scheduling saves police department $11 millionyr

bull Gasoline blending at Texaco results in saving of over $30 millionyr

Summary

bull Answered the question What is Operations Research amp Management Science and provided some historical perspective

bull Introduced the terminology of linear programmingbull Two Examples

1 MSR Marketing2 Gemstone Tool Companyndash Small (2-dimensional) Linear Program non-obvious solution

bull We will discuss this problem in detail in the next lecture

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
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• Slide 18
• Slide 19
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• Slide 34
• Slide 35
• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45

Optimization is Everywhere

bull The more you know about something the more you see where optimization can be applied

bull Some personal decision makingndash Finding the fastest route home (or to class)ndash Optimal allocation of time for homeworkndash Optimal budgetingndash Selecting a major

Optimization is everywhere

bull Some MIT decision makingndash setting exam times to minimize overlapndash assigning classes to classrooms and time

slots while satisfying constraintsndash figuring out prices for parking and subsidies

for public transportation so as to maximize fairness and permit sufficient access

ndash optimizing in fundraising

Optimization is everywhere

bull Exercise introduce yourself to your neighborand then mention a topic or two with which youare pretty familiar (summer job or major or yourparentrsquos occupation or whatever)

bull Then brainstorm with your neighbor on placeswhere optimization occurs

bull Then choose your favorite 2 or 3 applicationsfrom your list and letrsquos share them

On 15053 and Optimization Tools

bull Optimization is everywhere but optimizationtools are not applied everywhere

bull Goals in 15053 present a variety of toolsfor optimization and illustrate applicationsin manufacturing finance e-businessmarketing and more

bull When you see optimization problems arisein business (and you will) you will know thatthere are tools to help you out

Addressing managerial problems Amanagement science framework

1 Determine the problem to be solved2 Observe the system and gather data3 Formulate a mathematical model of the problem an

d any important subproblems4 Verify the model and use the model for prediction

or analysis5 Select a suitable alternative6 Present the results to the organization7 Implement and evaluate

Linear Programming (our first tooland probably the most important one)bull minimize or maximize a linear objectivebull subject to linear equalities and

inequalities

maximize

subject to

3x + 4y

5x + 8y le 24x y ge 0

Here is the set of feasible solutions

The optimalsolution

Terminology

bull Decision variables eg x and yndash In general there are quantities you can control to

improve your objective which should completely describe the set of decisions to be made

bull Constraints eg 5x + 8y le 24 x ge 0 y ge 0ndash Limitations on the values of the decision variables

bull Objective Function eg 3x + 4yndash Value measure used to rank alternativesndash Seek to maximize or minimize this objectivendash examples maximize NPV minimize cost

MSR Marketing Incadapted from Frontline Systems

bull Need to choose ads to reach at least 15 million people

bull Minimize Costbull Upper bound on number

of ads of each type

TV Radio Mail Newspaper

Audience Size 50000 25000 20000 15000

CostImpression $500 $200 $250 $125

Max of ads 20 15 10 15

Formulating as a math model

Work with your partner

bull What decisions need to be made Expressthese as ldquodecision variablesrdquo

bull What is the objective Express the objectivein terms of the decision variables

bull What are the constraints Express these interms of the decision variables

bull If you have time try to find the best solution

Gemstone Tool Company

bull Privately-held firmbull Consumer and industrial market for construction toolsbull Headquartered in Seattlebull Manufacturing plants in the US Canada and Mexicobull Simplifying assumptions for purposes of illustration

ndash Winnipeg Canada plantndash Wrenches and pliersndash Made from steelndash Injection molding machinendash Assembly machine

Data for the GTC ProblemWrenches Pliers Available

Steel 15 10 15000 tons

Molding Machine 10 10 12000 hrs

Assembly Machine 4 5 5000 hrs

Demand Limit 8000 10000

Contribution($ per item)

$40 $30

We want to determine the number of wrenches and pliersto produce given the available raw materials machinehours and demand

To do with your partner

bull Work with your partner to formulate theGTC problem as a linear program DONOT LOOK AHEAD IN THE NOTES

bull Let P = number of pliers madebull Let W = number of wrenches made

Formulating the GTC Problem

Step 1 Determine Decision Variables P = number of thousands of pliers manufactured W = number of thousands of wrenches manufactured

Step 2 Determine Objective Function Maximize Profit =

The Formulation Continued

Step 3 Determine Constraints

Steel Molding Assembly Plier Demand Wrench Demand

We will show how to solve this in the next lecture

An Algebraic Formulation

bull J = set of items that are manufacturedndash eg S = pliers wrenchesndash pj = unit profit from item jndash dj = maximum demand for item jndash xj = number of units of item j manufactured

bull M = set of manufacturing processesndash eg M = molding and assemblyndash bi = capacity of process indash aij = amount of capacity of process i used in making item j

An Algebraic formulation

bull Maximize

bull subject to

The sameformulation workseven if | J | = 10000and | M | = 100

An Algebraic formulation

bull Maximize

bull subject to

Linear Programs

bull A linear function is a function of the form

bull A mathematical program is a linear program (LP) if the objective is a linear function and the constraints are linear equalities or inequalities

bull Typically an LP has non-negativity constraints

A non-linear program is permitted to havea non-linear objective and constraints

bull maximize f(xy) = xybull subject to x - y22 le 10 3x ndash 4y ge 2 x ge 0 y ge 0

An integer program is a linear programplus constraints that some or all of thevariables are integer valuedbull Maximize 3x1 + 4x2 - 3x3 3x1 + 2x2 - x3 ge 17 3x2 - x3 = 14 x1 ge 0 x2 ge 0 x3 ge 0 and x1 x2 x3 are all integers

An Algebraic formulation withequality constraints

bull Maximize

bull subject to

Linear Programming Assumptions

Proportionality Assumption Contribution from W isproportional to W

Additivity Assumption Contribution to objective function fromP is independent of W

Divisibility Assumption Each variable is allowed to assumefractional values

Certainty Assumption Each linear coefficient of the objectivefunction and constraints is known (and is not a random variable)

Maximize 4W + 3P15W + P le 15

hellip

Some Success Stories

bull Optimal crew scheduling saves American Airlines $20 millionyr

bull Improved shipment routing saves Yellow Freight over $173 millionyr

bull Improved truck dispatching at Reynolds Metals improves on-time delivery and reduces freight cost by $7 millionyr

bull GTE local capacity expansion saves $30 millionyr

Other Success Stories (cont)

bull Optimizing global supply chains saves Digital Equipment over $300 million

bull Restructuring North America Operations Proctor and Gamble reduces plants by 20 saving $200 millionyr

bull Optimal traffic control of Hanshin Expressway in Osaka saves 17 million driver hoursyr

bull Better scheduling of hydro and thermal generating units saves southern company $140 million

Success Stories (cont)

bull Improved production planning at Sadia (Brazil) saves $50 million over three years

bull Production Optimization at Harris Corporation improves on-time deliveries from 75 to 90

bull Tata Steel (India) optimizes response to power shortage contributing $73 million

bull Optimizing police patrol officer scheduling saves police department $11 millionyr

bull Gasoline blending at Texaco results in saving of over $30 millionyr

Summary

bull Answered the question What is Operations Research amp Management Science and provided some historical perspective

bull Introduced the terminology of linear programmingbull Two Examples

1 MSR Marketing2 Gemstone Tool Companyndash Small (2-dimensional) Linear Program non-obvious solution

bull We will discuss this problem in detail in the next lecture

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
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• Slide 44
• Slide 45

Optimization is everywhere

bull Some MIT decision makingndash setting exam times to minimize overlapndash assigning classes to classrooms and time

slots while satisfying constraintsndash figuring out prices for parking and subsidies

for public transportation so as to maximize fairness and permit sufficient access

ndash optimizing in fundraising

Optimization is everywhere

bull Exercise introduce yourself to your neighborand then mention a topic or two with which youare pretty familiar (summer job or major or yourparentrsquos occupation or whatever)

bull Then brainstorm with your neighbor on placeswhere optimization occurs

bull Then choose your favorite 2 or 3 applicationsfrom your list and letrsquos share them

On 15053 and Optimization Tools

bull Optimization is everywhere but optimizationtools are not applied everywhere

bull Goals in 15053 present a variety of toolsfor optimization and illustrate applicationsin manufacturing finance e-businessmarketing and more

bull When you see optimization problems arisein business (and you will) you will know thatthere are tools to help you out

Addressing managerial problems Amanagement science framework

1 Determine the problem to be solved2 Observe the system and gather data3 Formulate a mathematical model of the problem an

d any important subproblems4 Verify the model and use the model for prediction

or analysis5 Select a suitable alternative6 Present the results to the organization7 Implement and evaluate

Linear Programming (our first tooland probably the most important one)bull minimize or maximize a linear objectivebull subject to linear equalities and

inequalities

maximize

subject to

3x + 4y

5x + 8y le 24x y ge 0

Here is the set of feasible solutions

The optimalsolution

Terminology

bull Decision variables eg x and yndash In general there are quantities you can control to

improve your objective which should completely describe the set of decisions to be made

bull Constraints eg 5x + 8y le 24 x ge 0 y ge 0ndash Limitations on the values of the decision variables

bull Objective Function eg 3x + 4yndash Value measure used to rank alternativesndash Seek to maximize or minimize this objectivendash examples maximize NPV minimize cost

MSR Marketing Incadapted from Frontline Systems

bull Need to choose ads to reach at least 15 million people

bull Minimize Costbull Upper bound on number

of ads of each type

TV Radio Mail Newspaper

Audience Size 50000 25000 20000 15000

CostImpression $500 $200 $250 $125

Max of ads 20 15 10 15

Formulating as a math model

Work with your partner

bull What decisions need to be made Expressthese as ldquodecision variablesrdquo

bull What is the objective Express the objectivein terms of the decision variables

bull What are the constraints Express these interms of the decision variables

bull If you have time try to find the best solution

Gemstone Tool Company

bull Privately-held firmbull Consumer and industrial market for construction toolsbull Headquartered in Seattlebull Manufacturing plants in the US Canada and Mexicobull Simplifying assumptions for purposes of illustration

ndash Winnipeg Canada plantndash Wrenches and pliersndash Made from steelndash Injection molding machinendash Assembly machine

Data for the GTC ProblemWrenches Pliers Available

Steel 15 10 15000 tons

Molding Machine 10 10 12000 hrs

Assembly Machine 4 5 5000 hrs

Demand Limit 8000 10000

Contribution($ per item)

$40 $30

We want to determine the number of wrenches and pliersto produce given the available raw materials machinehours and demand

To do with your partner

bull Work with your partner to formulate theGTC problem as a linear program DONOT LOOK AHEAD IN THE NOTES

bull Let P = number of pliers madebull Let W = number of wrenches made

Formulating the GTC Problem

Step 1 Determine Decision Variables P = number of thousands of pliers manufactured W = number of thousands of wrenches manufactured

Step 2 Determine Objective Function Maximize Profit =

The Formulation Continued

Step 3 Determine Constraints

Steel Molding Assembly Plier Demand Wrench Demand

We will show how to solve this in the next lecture

An Algebraic Formulation

bull J = set of items that are manufacturedndash eg S = pliers wrenchesndash pj = unit profit from item jndash dj = maximum demand for item jndash xj = number of units of item j manufactured

bull M = set of manufacturing processesndash eg M = molding and assemblyndash bi = capacity of process indash aij = amount of capacity of process i used in making item j

An Algebraic formulation

bull Maximize

bull subject to

The sameformulation workseven if | J | = 10000and | M | = 100

An Algebraic formulation

bull Maximize

bull subject to

Linear Programs

bull A linear function is a function of the form

bull A mathematical program is a linear program (LP) if the objective is a linear function and the constraints are linear equalities or inequalities

bull Typically an LP has non-negativity constraints

A non-linear program is permitted to havea non-linear objective and constraints

bull maximize f(xy) = xybull subject to x - y22 le 10 3x ndash 4y ge 2 x ge 0 y ge 0

An integer program is a linear programplus constraints that some or all of thevariables are integer valuedbull Maximize 3x1 + 4x2 - 3x3 3x1 + 2x2 - x3 ge 17 3x2 - x3 = 14 x1 ge 0 x2 ge 0 x3 ge 0 and x1 x2 x3 are all integers

An Algebraic formulation withequality constraints

bull Maximize

bull subject to

Linear Programming Assumptions

Proportionality Assumption Contribution from W isproportional to W

Additivity Assumption Contribution to objective function fromP is independent of W

Divisibility Assumption Each variable is allowed to assumefractional values

Certainty Assumption Each linear coefficient of the objectivefunction and constraints is known (and is not a random variable)

Maximize 4W + 3P15W + P le 15

hellip

Some Success Stories

bull Optimal crew scheduling saves American Airlines $20 millionyr

bull Improved shipment routing saves Yellow Freight over $173 millionyr

bull Improved truck dispatching at Reynolds Metals improves on-time delivery and reduces freight cost by $7 millionyr

bull GTE local capacity expansion saves $30 millionyr

Other Success Stories (cont)

bull Optimizing global supply chains saves Digital Equipment over $300 million

bull Restructuring North America Operations Proctor and Gamble reduces plants by 20 saving $200 millionyr

bull Optimal traffic control of Hanshin Expressway in Osaka saves 17 million driver hoursyr

bull Better scheduling of hydro and thermal generating units saves southern company $140 million

Success Stories (cont)

bull Improved production planning at Sadia (Brazil) saves $50 million over three years

bull Production Optimization at Harris Corporation improves on-time deliveries from 75 to 90

bull Tata Steel (India) optimizes response to power shortage contributing $73 million

bull Optimizing police patrol officer scheduling saves police department $11 millionyr

bull Gasoline blending at Texaco results in saving of over $30 millionyr

Summary

bull Answered the question What is Operations Research amp Management Science and provided some historical perspective

bull Introduced the terminology of linear programmingbull Two Examples

1 MSR Marketing2 Gemstone Tool Companyndash Small (2-dimensional) Linear Program non-obvious solution

bull We will discuss this problem in detail in the next lecture

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
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• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45

Optimization is everywhere

bull Exercise introduce yourself to your neighborand then mention a topic or two with which youare pretty familiar (summer job or major or yourparentrsquos occupation or whatever)

bull Then brainstorm with your neighbor on placeswhere optimization occurs

bull Then choose your favorite 2 or 3 applicationsfrom your list and letrsquos share them

On 15053 and Optimization Tools

bull Optimization is everywhere but optimizationtools are not applied everywhere

bull Goals in 15053 present a variety of toolsfor optimization and illustrate applicationsin manufacturing finance e-businessmarketing and more

bull When you see optimization problems arisein business (and you will) you will know thatthere are tools to help you out

Addressing managerial problems Amanagement science framework

1 Determine the problem to be solved2 Observe the system and gather data3 Formulate a mathematical model of the problem an

d any important subproblems4 Verify the model and use the model for prediction

or analysis5 Select a suitable alternative6 Present the results to the organization7 Implement and evaluate

Linear Programming (our first tooland probably the most important one)bull minimize or maximize a linear objectivebull subject to linear equalities and

inequalities

maximize

subject to

3x + 4y

5x + 8y le 24x y ge 0

Here is the set of feasible solutions

The optimalsolution

Terminology

bull Decision variables eg x and yndash In general there are quantities you can control to

improve your objective which should completely describe the set of decisions to be made

bull Constraints eg 5x + 8y le 24 x ge 0 y ge 0ndash Limitations on the values of the decision variables

bull Objective Function eg 3x + 4yndash Value measure used to rank alternativesndash Seek to maximize or minimize this objectivendash examples maximize NPV minimize cost

MSR Marketing Incadapted from Frontline Systems

bull Need to choose ads to reach at least 15 million people

bull Minimize Costbull Upper bound on number

of ads of each type

TV Radio Mail Newspaper

Audience Size 50000 25000 20000 15000

CostImpression $500 $200 $250 $125

Max of ads 20 15 10 15

Formulating as a math model

Work with your partner

bull What decisions need to be made Expressthese as ldquodecision variablesrdquo

bull What is the objective Express the objectivein terms of the decision variables

bull What are the constraints Express these interms of the decision variables

bull If you have time try to find the best solution

Gemstone Tool Company

bull Privately-held firmbull Consumer and industrial market for construction toolsbull Headquartered in Seattlebull Manufacturing plants in the US Canada and Mexicobull Simplifying assumptions for purposes of illustration

ndash Winnipeg Canada plantndash Wrenches and pliersndash Made from steelndash Injection molding machinendash Assembly machine

Data for the GTC ProblemWrenches Pliers Available

Steel 15 10 15000 tons

Molding Machine 10 10 12000 hrs

Assembly Machine 4 5 5000 hrs

Demand Limit 8000 10000

Contribution($ per item)

$40 $30

We want to determine the number of wrenches and pliersto produce given the available raw materials machinehours and demand

To do with your partner

bull Work with your partner to formulate theGTC problem as a linear program DONOT LOOK AHEAD IN THE NOTES

bull Let P = number of pliers madebull Let W = number of wrenches made

Formulating the GTC Problem

Step 1 Determine Decision Variables P = number of thousands of pliers manufactured W = number of thousands of wrenches manufactured

Step 2 Determine Objective Function Maximize Profit =

The Formulation Continued

Step 3 Determine Constraints

Steel Molding Assembly Plier Demand Wrench Demand

We will show how to solve this in the next lecture

An Algebraic Formulation

bull J = set of items that are manufacturedndash eg S = pliers wrenchesndash pj = unit profit from item jndash dj = maximum demand for item jndash xj = number of units of item j manufactured

bull M = set of manufacturing processesndash eg M = molding and assemblyndash bi = capacity of process indash aij = amount of capacity of process i used in making item j

An Algebraic formulation

bull Maximize

bull subject to

The sameformulation workseven if | J | = 10000and | M | = 100

An Algebraic formulation

bull Maximize

bull subject to

Linear Programs

bull A linear function is a function of the form

bull A mathematical program is a linear program (LP) if the objective is a linear function and the constraints are linear equalities or inequalities

bull Typically an LP has non-negativity constraints

A non-linear program is permitted to havea non-linear objective and constraints

bull maximize f(xy) = xybull subject to x - y22 le 10 3x ndash 4y ge 2 x ge 0 y ge 0

An integer program is a linear programplus constraints that some or all of thevariables are integer valuedbull Maximize 3x1 + 4x2 - 3x3 3x1 + 2x2 - x3 ge 17 3x2 - x3 = 14 x1 ge 0 x2 ge 0 x3 ge 0 and x1 x2 x3 are all integers

An Algebraic formulation withequality constraints

bull Maximize

bull subject to

Linear Programming Assumptions

Proportionality Assumption Contribution from W isproportional to W

Additivity Assumption Contribution to objective function fromP is independent of W

Divisibility Assumption Each variable is allowed to assumefractional values

Certainty Assumption Each linear coefficient of the objectivefunction and constraints is known (and is not a random variable)

Maximize 4W + 3P15W + P le 15

hellip

Some Success Stories

bull Optimal crew scheduling saves American Airlines $20 millionyr

bull Improved shipment routing saves Yellow Freight over $173 millionyr

bull Improved truck dispatching at Reynolds Metals improves on-time delivery and reduces freight cost by $7 millionyr

bull GTE local capacity expansion saves $30 millionyr

Other Success Stories (cont)

bull Optimizing global supply chains saves Digital Equipment over $300 million

bull Restructuring North America Operations Proctor and Gamble reduces plants by 20 saving $200 millionyr

bull Optimal traffic control of Hanshin Expressway in Osaka saves 17 million driver hoursyr

bull Better scheduling of hydro and thermal generating units saves southern company $140 million

Success Stories (cont)

bull Improved production planning at Sadia (Brazil) saves $50 million over three years

bull Production Optimization at Harris Corporation improves on-time deliveries from 75 to 90

bull Tata Steel (India) optimizes response to power shortage contributing $73 million

bull Optimizing police patrol officer scheduling saves police department $11 millionyr

bull Gasoline blending at Texaco results in saving of over $30 millionyr

Summary

bull Answered the question What is Operations Research amp Management Science and provided some historical perspective

bull Introduced the terminology of linear programmingbull Two Examples

1 MSR Marketing2 Gemstone Tool Companyndash Small (2-dimensional) Linear Program non-obvious solution

bull We will discuss this problem in detail in the next lecture

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
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• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45

On 15053 and Optimization Tools

bull Optimization is everywhere but optimizationtools are not applied everywhere

bull Goals in 15053 present a variety of toolsfor optimization and illustrate applicationsin manufacturing finance e-businessmarketing and more

bull When you see optimization problems arisein business (and you will) you will know thatthere are tools to help you out

Addressing managerial problems Amanagement science framework

1 Determine the problem to be solved2 Observe the system and gather data3 Formulate a mathematical model of the problem an

d any important subproblems4 Verify the model and use the model for prediction

or analysis5 Select a suitable alternative6 Present the results to the organization7 Implement and evaluate

Linear Programming (our first tooland probably the most important one)bull minimize or maximize a linear objectivebull subject to linear equalities and

inequalities

maximize

subject to

3x + 4y

5x + 8y le 24x y ge 0

Here is the set of feasible solutions

The optimalsolution

Terminology

bull Decision variables eg x and yndash In general there are quantities you can control to

improve your objective which should completely describe the set of decisions to be made

bull Constraints eg 5x + 8y le 24 x ge 0 y ge 0ndash Limitations on the values of the decision variables

bull Objective Function eg 3x + 4yndash Value measure used to rank alternativesndash Seek to maximize or minimize this objectivendash examples maximize NPV minimize cost

MSR Marketing Incadapted from Frontline Systems

bull Need to choose ads to reach at least 15 million people

bull Minimize Costbull Upper bound on number

of ads of each type

TV Radio Mail Newspaper

Audience Size 50000 25000 20000 15000

CostImpression $500 $200 $250 $125

Max of ads 20 15 10 15

Formulating as a math model

Work with your partner

bull What decisions need to be made Expressthese as ldquodecision variablesrdquo

bull What is the objective Express the objectivein terms of the decision variables

bull What are the constraints Express these interms of the decision variables

bull If you have time try to find the best solution

Gemstone Tool Company

bull Privately-held firmbull Consumer and industrial market for construction toolsbull Headquartered in Seattlebull Manufacturing plants in the US Canada and Mexicobull Simplifying assumptions for purposes of illustration

ndash Winnipeg Canada plantndash Wrenches and pliersndash Made from steelndash Injection molding machinendash Assembly machine

Data for the GTC ProblemWrenches Pliers Available

Steel 15 10 15000 tons

Molding Machine 10 10 12000 hrs

Assembly Machine 4 5 5000 hrs

Demand Limit 8000 10000

Contribution($ per item)

$40 $30

We want to determine the number of wrenches and pliersto produce given the available raw materials machinehours and demand

To do with your partner

bull Work with your partner to formulate theGTC problem as a linear program DONOT LOOK AHEAD IN THE NOTES

bull Let P = number of pliers madebull Let W = number of wrenches made

Formulating the GTC Problem

Step 1 Determine Decision Variables P = number of thousands of pliers manufactured W = number of thousands of wrenches manufactured

Step 2 Determine Objective Function Maximize Profit =

The Formulation Continued

Step 3 Determine Constraints

Steel Molding Assembly Plier Demand Wrench Demand

We will show how to solve this in the next lecture

An Algebraic Formulation

bull J = set of items that are manufacturedndash eg S = pliers wrenchesndash pj = unit profit from item jndash dj = maximum demand for item jndash xj = number of units of item j manufactured

bull M = set of manufacturing processesndash eg M = molding and assemblyndash bi = capacity of process indash aij = amount of capacity of process i used in making item j

An Algebraic formulation

bull Maximize

bull subject to

The sameformulation workseven if | J | = 10000and | M | = 100

An Algebraic formulation

bull Maximize

bull subject to

Linear Programs

bull A linear function is a function of the form

bull A mathematical program is a linear program (LP) if the objective is a linear function and the constraints are linear equalities or inequalities

bull Typically an LP has non-negativity constraints

A non-linear program is permitted to havea non-linear objective and constraints

bull maximize f(xy) = xybull subject to x - y22 le 10 3x ndash 4y ge 2 x ge 0 y ge 0

An integer program is a linear programplus constraints that some or all of thevariables are integer valuedbull Maximize 3x1 + 4x2 - 3x3 3x1 + 2x2 - x3 ge 17 3x2 - x3 = 14 x1 ge 0 x2 ge 0 x3 ge 0 and x1 x2 x3 are all integers

An Algebraic formulation withequality constraints

bull Maximize

bull subject to

Linear Programming Assumptions

Proportionality Assumption Contribution from W isproportional to W

Additivity Assumption Contribution to objective function fromP is independent of W

Divisibility Assumption Each variable is allowed to assumefractional values

Certainty Assumption Each linear coefficient of the objectivefunction and constraints is known (and is not a random variable)

Maximize 4W + 3P15W + P le 15

hellip

Some Success Stories

bull Optimal crew scheduling saves American Airlines $20 millionyr

bull Improved shipment routing saves Yellow Freight over $173 millionyr

bull Improved truck dispatching at Reynolds Metals improves on-time delivery and reduces freight cost by $7 millionyr

bull GTE local capacity expansion saves $30 millionyr

Other Success Stories (cont)

bull Optimizing global supply chains saves Digital Equipment over $300 million

bull Restructuring North America Operations Proctor and Gamble reduces plants by 20 saving $200 millionyr

bull Optimal traffic control of Hanshin Expressway in Osaka saves 17 million driver hoursyr

bull Better scheduling of hydro and thermal generating units saves southern company $140 million

Success Stories (cont)

bull Improved production planning at Sadia (Brazil) saves $50 million over three years

bull Production Optimization at Harris Corporation improves on-time deliveries from 75 to 90

bull Tata Steel (India) optimizes response to power shortage contributing $73 million

bull Optimizing police patrol officer scheduling saves police department $11 millionyr

bull Gasoline blending at Texaco results in saving of over $30 millionyr

Summary

bull Answered the question What is Operations Research amp Management Science and provided some historical perspective

bull Introduced the terminology of linear programmingbull Two Examples

1 MSR Marketing2 Gemstone Tool Companyndash Small (2-dimensional) Linear Program non-obvious solution

bull We will discuss this problem in detail in the next lecture

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
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• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45

Addressing managerial problems Amanagement science framework

1 Determine the problem to be solved2 Observe the system and gather data3 Formulate a mathematical model of the problem an

d any important subproblems4 Verify the model and use the model for prediction

or analysis5 Select a suitable alternative6 Present the results to the organization7 Implement and evaluate

Linear Programming (our first tooland probably the most important one)bull minimize or maximize a linear objectivebull subject to linear equalities and

inequalities

maximize

subject to

3x + 4y

5x + 8y le 24x y ge 0

Here is the set of feasible solutions

The optimalsolution

Terminology

bull Decision variables eg x and yndash In general there are quantities you can control to

improve your objective which should completely describe the set of decisions to be made

bull Constraints eg 5x + 8y le 24 x ge 0 y ge 0ndash Limitations on the values of the decision variables

bull Objective Function eg 3x + 4yndash Value measure used to rank alternativesndash Seek to maximize or minimize this objectivendash examples maximize NPV minimize cost

MSR Marketing Incadapted from Frontline Systems

bull Need to choose ads to reach at least 15 million people

bull Minimize Costbull Upper bound on number

of ads of each type

TV Radio Mail Newspaper

Audience Size 50000 25000 20000 15000

CostImpression $500 $200 $250 $125

Max of ads 20 15 10 15

Formulating as a math model

Work with your partner

bull What decisions need to be made Expressthese as ldquodecision variablesrdquo

bull What is the objective Express the objectivein terms of the decision variables

bull What are the constraints Express these interms of the decision variables

bull If you have time try to find the best solution

Gemstone Tool Company

bull Privately-held firmbull Consumer and industrial market for construction toolsbull Headquartered in Seattlebull Manufacturing plants in the US Canada and Mexicobull Simplifying assumptions for purposes of illustration

ndash Winnipeg Canada plantndash Wrenches and pliersndash Made from steelndash Injection molding machinendash Assembly machine

Data for the GTC ProblemWrenches Pliers Available

Steel 15 10 15000 tons

Molding Machine 10 10 12000 hrs

Assembly Machine 4 5 5000 hrs

Demand Limit 8000 10000

Contribution($ per item)

$40 $30

We want to determine the number of wrenches and pliersto produce given the available raw materials machinehours and demand

To do with your partner

bull Work with your partner to formulate theGTC problem as a linear program DONOT LOOK AHEAD IN THE NOTES

bull Let P = number of pliers madebull Let W = number of wrenches made

Formulating the GTC Problem

Step 1 Determine Decision Variables P = number of thousands of pliers manufactured W = number of thousands of wrenches manufactured

Step 2 Determine Objective Function Maximize Profit =

The Formulation Continued

Step 3 Determine Constraints

Steel Molding Assembly Plier Demand Wrench Demand

We will show how to solve this in the next lecture

An Algebraic Formulation

bull J = set of items that are manufacturedndash eg S = pliers wrenchesndash pj = unit profit from item jndash dj = maximum demand for item jndash xj = number of units of item j manufactured

bull M = set of manufacturing processesndash eg M = molding and assemblyndash bi = capacity of process indash aij = amount of capacity of process i used in making item j

An Algebraic formulation

bull Maximize

bull subject to

The sameformulation workseven if | J | = 10000and | M | = 100

An Algebraic formulation

bull Maximize

bull subject to

Linear Programs

bull A linear function is a function of the form

bull A mathematical program is a linear program (LP) if the objective is a linear function and the constraints are linear equalities or inequalities

bull Typically an LP has non-negativity constraints

A non-linear program is permitted to havea non-linear objective and constraints

bull maximize f(xy) = xybull subject to x - y22 le 10 3x ndash 4y ge 2 x ge 0 y ge 0

An integer program is a linear programplus constraints that some or all of thevariables are integer valuedbull Maximize 3x1 + 4x2 - 3x3 3x1 + 2x2 - x3 ge 17 3x2 - x3 = 14 x1 ge 0 x2 ge 0 x3 ge 0 and x1 x2 x3 are all integers

An Algebraic formulation withequality constraints

bull Maximize

bull subject to

Linear Programming Assumptions

Proportionality Assumption Contribution from W isproportional to W

Additivity Assumption Contribution to objective function fromP is independent of W

Divisibility Assumption Each variable is allowed to assumefractional values

Certainty Assumption Each linear coefficient of the objectivefunction and constraints is known (and is not a random variable)

Maximize 4W + 3P15W + P le 15

hellip

Some Success Stories

bull Optimal crew scheduling saves American Airlines $20 millionyr

bull Improved shipment routing saves Yellow Freight over $173 millionyr

bull Improved truck dispatching at Reynolds Metals improves on-time delivery and reduces freight cost by $7 millionyr

bull GTE local capacity expansion saves $30 millionyr

Other Success Stories (cont)

bull Optimizing global supply chains saves Digital Equipment over $300 million

bull Restructuring North America Operations Proctor and Gamble reduces plants by 20 saving $200 millionyr

bull Optimal traffic control of Hanshin Expressway in Osaka saves 17 million driver hoursyr

bull Better scheduling of hydro and thermal generating units saves southern company $140 million

Success Stories (cont)

bull Improved production planning at Sadia (Brazil) saves $50 million over three years

bull Production Optimization at Harris Corporation improves on-time deliveries from 75 to 90

bull Tata Steel (India) optimizes response to power shortage contributing $73 million

bull Optimizing police patrol officer scheduling saves police department $11 millionyr

bull Gasoline blending at Texaco results in saving of over $30 millionyr

Summary

bull Answered the question What is Operations Research amp Management Science and provided some historical perspective

bull Introduced the terminology of linear programmingbull Two Examples

1 MSR Marketing2 Gemstone Tool Companyndash Small (2-dimensional) Linear Program non-obvious solution

bull We will discuss this problem in detail in the next lecture

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
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• Slide 36
• Slide 37
• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45

Linear Programming (our first tooland probably the most important one)bull minimize or maximize a linear objectivebull subject to linear equalities and

inequalities

maximize

subject to

3x + 4y

5x + 8y le 24x y ge 0

Here is the set of feasible solutions

The optimalsolution

Terminology

bull Decision variables eg x and yndash In general there are quantities you can control to

improve your objective which should completely describe the set of decisions to be made

bull Constraints eg 5x + 8y le 24 x ge 0 y ge 0ndash Limitations on the values of the decision variables

bull Objective Function eg 3x + 4yndash Value measure used to rank alternativesndash Seek to maximize or minimize this objectivendash examples maximize NPV minimize cost

MSR Marketing Incadapted from Frontline Systems

bull Need to choose ads to reach at least 15 million people

bull Minimize Costbull Upper bound on number

of ads of each type

TV Radio Mail Newspaper

Audience Size 50000 25000 20000 15000

CostImpression $500 $200 $250 $125

Max of ads 20 15 10 15

Formulating as a math model

Work with your partner

bull What decisions need to be made Expressthese as ldquodecision variablesrdquo

bull What is the objective Express the objectivein terms of the decision variables

bull What are the constraints Express these interms of the decision variables

bull If you have time try to find the best solution

Gemstone Tool Company

bull Privately-held firmbull Consumer and industrial market for construction toolsbull Headquartered in Seattlebull Manufacturing plants in the US Canada and Mexicobull Simplifying assumptions for purposes of illustration

ndash Winnipeg Canada plantndash Wrenches and pliersndash Made from steelndash Injection molding machinendash Assembly machine

Data for the GTC ProblemWrenches Pliers Available

Steel 15 10 15000 tons

Molding Machine 10 10 12000 hrs

Assembly Machine 4 5 5000 hrs

Demand Limit 8000 10000

Contribution($ per item)

$40 $30

We want to determine the number of wrenches and pliersto produce given the available raw materials machinehours and demand

To do with your partner

bull Work with your partner to formulate theGTC problem as a linear program DONOT LOOK AHEAD IN THE NOTES

bull Let P = number of pliers madebull Let W = number of wrenches made

Formulating the GTC Problem

Step 1 Determine Decision Variables P = number of thousands of pliers manufactured W = number of thousands of wrenches manufactured

Step 2 Determine Objective Function Maximize Profit =

The Formulation Continued

Step 3 Determine Constraints

Steel Molding Assembly Plier Demand Wrench Demand

We will show how to solve this in the next lecture

An Algebraic Formulation

bull J = set of items that are manufacturedndash eg S = pliers wrenchesndash pj = unit profit from item jndash dj = maximum demand for item jndash xj = number of units of item j manufactured

bull M = set of manufacturing processesndash eg M = molding and assemblyndash bi = capacity of process indash aij = amount of capacity of process i used in making item j

An Algebraic formulation

bull Maximize

bull subject to

The sameformulation workseven if | J | = 10000and | M | = 100

An Algebraic formulation

bull Maximize

bull subject to

Linear Programs

bull A linear function is a function of the form

bull A mathematical program is a linear program (LP) if the objective is a linear function and the constraints are linear equalities or inequalities

bull Typically an LP has non-negativity constraints

A non-linear program is permitted to havea non-linear objective and constraints

bull maximize f(xy) = xybull subject to x - y22 le 10 3x ndash 4y ge 2 x ge 0 y ge 0

An integer program is a linear programplus constraints that some or all of thevariables are integer valuedbull Maximize 3x1 + 4x2 - 3x3 3x1 + 2x2 - x3 ge 17 3x2 - x3 = 14 x1 ge 0 x2 ge 0 x3 ge 0 and x1 x2 x3 are all integers

An Algebraic formulation withequality constraints

bull Maximize

bull subject to

Linear Programming Assumptions

Proportionality Assumption Contribution from W isproportional to W

Additivity Assumption Contribution to objective function fromP is independent of W

Divisibility Assumption Each variable is allowed to assumefractional values

Certainty Assumption Each linear coefficient of the objectivefunction and constraints is known (and is not a random variable)

Maximize 4W + 3P15W + P le 15

hellip

Some Success Stories

bull Optimal crew scheduling saves American Airlines $20 millionyr

bull Improved shipment routing saves Yellow Freight over $173 millionyr

bull Improved truck dispatching at Reynolds Metals improves on-time delivery and reduces freight cost by $7 millionyr

bull GTE local capacity expansion saves $30 millionyr

Other Success Stories (cont)

bull Optimizing global supply chains saves Digital Equipment over $300 million

bull Restructuring North America Operations Proctor and Gamble reduces plants by 20 saving $200 millionyr

bull Optimal traffic control of Hanshin Expressway in Osaka saves 17 million driver hoursyr

bull Better scheduling of hydro and thermal generating units saves southern company $140 million

Success Stories (cont)

bull Improved production planning at Sadia (Brazil) saves $50 million over three years

bull Production Optimization at Harris Corporation improves on-time deliveries from 75 to 90

bull Tata Steel (India) optimizes response to power shortage contributing $73 million

bull Optimizing police patrol officer scheduling saves police department $11 millionyr

bull Gasoline blending at Texaco results in saving of over $30 millionyr

Summary

bull Answered the question What is Operations Research amp Management Science and provided some historical perspective

bull Introduced the terminology of linear programmingbull Two Examples

1 MSR Marketing2 Gemstone Tool Companyndash Small (2-dimensional) Linear Program non-obvious solution

bull We will discuss this problem in detail in the next lecture

• Slide 1
• Slide 2
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• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45

Here is the set of feasible solutions

The optimalsolution

Terminology

bull Decision variables eg x and yndash In general there are quantities you can control to

improve your objective which should completely describe the set of decisions to be made

bull Constraints eg 5x + 8y le 24 x ge 0 y ge 0ndash Limitations on the values of the decision variables

bull Objective Function eg 3x + 4yndash Value measure used to rank alternativesndash Seek to maximize or minimize this objectivendash examples maximize NPV minimize cost

MSR Marketing Incadapted from Frontline Systems

bull Need to choose ads to reach at least 15 million people

bull Minimize Costbull Upper bound on number

of ads of each type

TV Radio Mail Newspaper

Audience Size 50000 25000 20000 15000

CostImpression $500 $200 $250 $125

Max of ads 20 15 10 15

Formulating as a math model

Work with your partner

bull What decisions need to be made Expressthese as ldquodecision variablesrdquo

bull What is the objective Express the objectivein terms of the decision variables

bull What are the constraints Express these interms of the decision variables

bull If you have time try to find the best solution

Gemstone Tool Company

bull Privately-held firmbull Consumer and industrial market for construction toolsbull Headquartered in Seattlebull Manufacturing plants in the US Canada and Mexicobull Simplifying assumptions for purposes of illustration

ndash Winnipeg Canada plantndash Wrenches and pliersndash Made from steelndash Injection molding machinendash Assembly machine

Data for the GTC ProblemWrenches Pliers Available

Steel 15 10 15000 tons

Molding Machine 10 10 12000 hrs

Assembly Machine 4 5 5000 hrs

Demand Limit 8000 10000

Contribution($ per item)

$40 $30

We want to determine the number of wrenches and pliersto produce given the available raw materials machinehours and demand

To do with your partner

bull Work with your partner to formulate theGTC problem as a linear program DONOT LOOK AHEAD IN THE NOTES

bull Let P = number of pliers madebull Let W = number of wrenches made

Formulating the GTC Problem

Step 1 Determine Decision Variables P = number of thousands of pliers manufactured W = number of thousands of wrenches manufactured

Step 2 Determine Objective Function Maximize Profit =

The Formulation Continued

Step 3 Determine Constraints

Steel Molding Assembly Plier Demand Wrench Demand

We will show how to solve this in the next lecture

An Algebraic Formulation

bull J = set of items that are manufacturedndash eg S = pliers wrenchesndash pj = unit profit from item jndash dj = maximum demand for item jndash xj = number of units of item j manufactured

bull M = set of manufacturing processesndash eg M = molding and assemblyndash bi = capacity of process indash aij = amount of capacity of process i used in making item j

An Algebraic formulation

bull Maximize

bull subject to

The sameformulation workseven if | J | = 10000and | M | = 100

An Algebraic formulation

bull Maximize

bull subject to

Linear Programs

bull A linear function is a function of the form

bull A mathematical program is a linear program (LP) if the objective is a linear function and the constraints are linear equalities or inequalities

bull Typically an LP has non-negativity constraints

A non-linear program is permitted to havea non-linear objective and constraints

bull maximize f(xy) = xybull subject to x - y22 le 10 3x ndash 4y ge 2 x ge 0 y ge 0

An integer program is a linear programplus constraints that some or all of thevariables are integer valuedbull Maximize 3x1 + 4x2 - 3x3 3x1 + 2x2 - x3 ge 17 3x2 - x3 = 14 x1 ge 0 x2 ge 0 x3 ge 0 and x1 x2 x3 are all integers

An Algebraic formulation withequality constraints

bull Maximize

bull subject to

Linear Programming Assumptions

Proportionality Assumption Contribution from W isproportional to W

Additivity Assumption Contribution to objective function fromP is independent of W

Divisibility Assumption Each variable is allowed to assumefractional values

Certainty Assumption Each linear coefficient of the objectivefunction and constraints is known (and is not a random variable)

Maximize 4W + 3P15W + P le 15

hellip

Some Success Stories

bull Optimal crew scheduling saves American Airlines $20 millionyr

bull Improved shipment routing saves Yellow Freight over $173 millionyr

bull Improved truck dispatching at Reynolds Metals improves on-time delivery and reduces freight cost by $7 millionyr

bull GTE local capacity expansion saves $30 millionyr

Other Success Stories (cont)

bull Optimizing global supply chains saves Digital Equipment over $300 million

bull Restructuring North America Operations Proctor and Gamble reduces plants by 20 saving $200 millionyr

bull Optimal traffic control of Hanshin Expressway in Osaka saves 17 million driver hoursyr

bull Better scheduling of hydro and thermal generating units saves southern company $140 million

Success Stories (cont)

bull Improved production planning at Sadia (Brazil) saves $50 million over three years

bull Production Optimization at Harris Corporation improves on-time deliveries from 75 to 90

bull Tata Steel (India) optimizes response to power shortage contributing $73 million

bull Optimizing police patrol officer scheduling saves police department $11 millionyr

bull Gasoline blending at Texaco results in saving of over $30 millionyr

Summary

bull Answered the question What is Operations Research amp Management Science and provided some historical perspective

bull Introduced the terminology of linear programmingbull Two Examples

1 MSR Marketing2 Gemstone Tool Companyndash Small (2-dimensional) Linear Program non-obvious solution

bull We will discuss this problem in detail in the next lecture

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
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• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45

Terminology

bull Decision variables eg x and yndash In general there are quantities you can control to

improve your objective which should completely describe the set of decisions to be made

bull Constraints eg 5x + 8y le 24 x ge 0 y ge 0ndash Limitations on the values of the decision variables

bull Objective Function eg 3x + 4yndash Value measure used to rank alternativesndash Seek to maximize or minimize this objectivendash examples maximize NPV minimize cost

MSR Marketing Incadapted from Frontline Systems

bull Need to choose ads to reach at least 15 million people

bull Minimize Costbull Upper bound on number

of ads of each type

TV Radio Mail Newspaper

Audience Size 50000 25000 20000 15000

CostImpression $500 $200 $250 $125

Max of ads 20 15 10 15

Formulating as a math model

Work with your partner

bull What decisions need to be made Expressthese as ldquodecision variablesrdquo

bull What is the objective Express the objectivein terms of the decision variables

bull What are the constraints Express these interms of the decision variables

bull If you have time try to find the best solution

Gemstone Tool Company

bull Privately-held firmbull Consumer and industrial market for construction toolsbull Headquartered in Seattlebull Manufacturing plants in the US Canada and Mexicobull Simplifying assumptions for purposes of illustration

ndash Winnipeg Canada plantndash Wrenches and pliersndash Made from steelndash Injection molding machinendash Assembly machine

Data for the GTC ProblemWrenches Pliers Available

Steel 15 10 15000 tons

Molding Machine 10 10 12000 hrs

Assembly Machine 4 5 5000 hrs

Demand Limit 8000 10000

Contribution($ per item)

$40 $30

We want to determine the number of wrenches and pliersto produce given the available raw materials machinehours and demand

To do with your partner

bull Work with your partner to formulate theGTC problem as a linear program DONOT LOOK AHEAD IN THE NOTES

bull Let P = number of pliers madebull Let W = number of wrenches made

Formulating the GTC Problem

Step 1 Determine Decision Variables P = number of thousands of pliers manufactured W = number of thousands of wrenches manufactured

Step 2 Determine Objective Function Maximize Profit =

The Formulation Continued

Step 3 Determine Constraints

Steel Molding Assembly Plier Demand Wrench Demand

We will show how to solve this in the next lecture

An Algebraic Formulation

bull J = set of items that are manufacturedndash eg S = pliers wrenchesndash pj = unit profit from item jndash dj = maximum demand for item jndash xj = number of units of item j manufactured

bull M = set of manufacturing processesndash eg M = molding and assemblyndash bi = capacity of process indash aij = amount of capacity of process i used in making item j

An Algebraic formulation

bull Maximize

bull subject to

The sameformulation workseven if | J | = 10000and | M | = 100

An Algebraic formulation

bull Maximize

bull subject to

Linear Programs

bull A linear function is a function of the form

bull A mathematical program is a linear program (LP) if the objective is a linear function and the constraints are linear equalities or inequalities

bull Typically an LP has non-negativity constraints

A non-linear program is permitted to havea non-linear objective and constraints

bull maximize f(xy) = xybull subject to x - y22 le 10 3x ndash 4y ge 2 x ge 0 y ge 0

An integer program is a linear programplus constraints that some or all of thevariables are integer valuedbull Maximize 3x1 + 4x2 - 3x3 3x1 + 2x2 - x3 ge 17 3x2 - x3 = 14 x1 ge 0 x2 ge 0 x3 ge 0 and x1 x2 x3 are all integers

An Algebraic formulation withequality constraints

bull Maximize

bull subject to

Linear Programming Assumptions

Proportionality Assumption Contribution from W isproportional to W

Additivity Assumption Contribution to objective function fromP is independent of W

Divisibility Assumption Each variable is allowed to assumefractional values

Certainty Assumption Each linear coefficient of the objectivefunction and constraints is known (and is not a random variable)

Maximize 4W + 3P15W + P le 15

hellip

Some Success Stories

bull Optimal crew scheduling saves American Airlines $20 millionyr

bull Improved shipment routing saves Yellow Freight over $173 millionyr

bull Improved truck dispatching at Reynolds Metals improves on-time delivery and reduces freight cost by $7 millionyr

bull GTE local capacity expansion saves $30 millionyr

Other Success Stories (cont)

bull Optimizing global supply chains saves Digital Equipment over $300 million

bull Restructuring North America Operations Proctor and Gamble reduces plants by 20 saving $200 millionyr

bull Optimal traffic control of Hanshin Expressway in Osaka saves 17 million driver hoursyr

bull Better scheduling of hydro and thermal generating units saves southern company $140 million

Success Stories (cont)

bull Improved production planning at Sadia (Brazil) saves $50 million over three years

bull Production Optimization at Harris Corporation improves on-time deliveries from 75 to 90

bull Tata Steel (India) optimizes response to power shortage contributing $73 million

bull Optimizing police patrol officer scheduling saves police department $11 millionyr

bull Gasoline blending at Texaco results in saving of over $30 millionyr

Summary

bull Answered the question What is Operations Research amp Management Science and provided some historical perspective

bull Introduced the terminology of linear programmingbull Two Examples

1 MSR Marketing2 Gemstone Tool Companyndash Small (2-dimensional) Linear Program non-obvious solution

bull We will discuss this problem in detail in the next lecture

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
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• Slide 42
• Slide 43
• Slide 44
• Slide 45

MSR Marketing Incadapted from Frontline Systems

bull Need to choose ads to reach at least 15 million people

bull Minimize Costbull Upper bound on number

of ads of each type

TV Radio Mail Newspaper

Audience Size 50000 25000 20000 15000

CostImpression $500 $200 $250 $125

Max of ads 20 15 10 15

Formulating as a math model

Work with your partner

bull What decisions need to be made Expressthese as ldquodecision variablesrdquo

bull What is the objective Express the objectivein terms of the decision variables

bull What are the constraints Express these interms of the decision variables

bull If you have time try to find the best solution

Gemstone Tool Company

bull Privately-held firmbull Consumer and industrial market for construction toolsbull Headquartered in Seattlebull Manufacturing plants in the US Canada and Mexicobull Simplifying assumptions for purposes of illustration

ndash Winnipeg Canada plantndash Wrenches and pliersndash Made from steelndash Injection molding machinendash Assembly machine

Data for the GTC ProblemWrenches Pliers Available

Steel 15 10 15000 tons

Molding Machine 10 10 12000 hrs

Assembly Machine 4 5 5000 hrs

Demand Limit 8000 10000

Contribution($ per item)

$40 $30

We want to determine the number of wrenches and pliersto produce given the available raw materials machinehours and demand

To do with your partner

bull Work with your partner to formulate theGTC problem as a linear program DONOT LOOK AHEAD IN THE NOTES

bull Let P = number of pliers madebull Let W = number of wrenches made

Formulating the GTC Problem

Step 1 Determine Decision Variables P = number of thousands of pliers manufactured W = number of thousands of wrenches manufactured

Step 2 Determine Objective Function Maximize Profit =

The Formulation Continued

Step 3 Determine Constraints

Steel Molding Assembly Plier Demand Wrench Demand

We will show how to solve this in the next lecture

An Algebraic Formulation

bull J = set of items that are manufacturedndash eg S = pliers wrenchesndash pj = unit profit from item jndash dj = maximum demand for item jndash xj = number of units of item j manufactured

bull M = set of manufacturing processesndash eg M = molding and assemblyndash bi = capacity of process indash aij = amount of capacity of process i used in making item j

An Algebraic formulation

bull Maximize

bull subject to

The sameformulation workseven if | J | = 10000and | M | = 100

An Algebraic formulation

bull Maximize

bull subject to

Linear Programs

bull A linear function is a function of the form

bull A mathematical program is a linear program (LP) if the objective is a linear function and the constraints are linear equalities or inequalities

bull Typically an LP has non-negativity constraints

A non-linear program is permitted to havea non-linear objective and constraints

bull maximize f(xy) = xybull subject to x - y22 le 10 3x ndash 4y ge 2 x ge 0 y ge 0

An integer program is a linear programplus constraints that some or all of thevariables are integer valuedbull Maximize 3x1 + 4x2 - 3x3 3x1 + 2x2 - x3 ge 17 3x2 - x3 = 14 x1 ge 0 x2 ge 0 x3 ge 0 and x1 x2 x3 are all integers

An Algebraic formulation withequality constraints

bull Maximize

bull subject to

Linear Programming Assumptions

Proportionality Assumption Contribution from W isproportional to W

Additivity Assumption Contribution to objective function fromP is independent of W

Divisibility Assumption Each variable is allowed to assumefractional values

Certainty Assumption Each linear coefficient of the objectivefunction and constraints is known (and is not a random variable)

Maximize 4W + 3P15W + P le 15

hellip

Some Success Stories

bull Optimal crew scheduling saves American Airlines $20 millionyr

bull Improved shipment routing saves Yellow Freight over $173 millionyr

bull Improved truck dispatching at Reynolds Metals improves on-time delivery and reduces freight cost by $7 millionyr

bull GTE local capacity expansion saves $30 millionyr

Other Success Stories (cont)

bull Optimizing global supply chains saves Digital Equipment over $300 million

bull Restructuring North America Operations Proctor and Gamble reduces plants by 20 saving $200 millionyr

bull Optimal traffic control of Hanshin Expressway in Osaka saves 17 million driver hoursyr

bull Better scheduling of hydro and thermal generating units saves southern company $140 million

Success Stories (cont)

bull Improved production planning at Sadia (Brazil) saves $50 million over three years

bull Production Optimization at Harris Corporation improves on-time deliveries from 75 to 90

bull Tata Steel (India) optimizes response to power shortage contributing $73 million

bull Optimizing police patrol officer scheduling saves police department $11 millionyr

bull Gasoline blending at Texaco results in saving of over $30 millionyr

Summary

bull Answered the question What is Operations Research amp Management Science and provided some historical perspective

bull Introduced the terminology of linear programmingbull Two Examples

1 MSR Marketing2 Gemstone Tool Companyndash Small (2-dimensional) Linear Program non-obvious solution

bull We will discuss this problem in detail in the next lecture

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
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• Slide 38
• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45

Formulating as a math model

Work with your partner

bull What decisions need to be made Expressthese as ldquodecision variablesrdquo

bull What is the objective Express the objectivein terms of the decision variables

bull What are the constraints Express these interms of the decision variables

bull If you have time try to find the best solution

Gemstone Tool Company

bull Privately-held firmbull Consumer and industrial market for construction toolsbull Headquartered in Seattlebull Manufacturing plants in the US Canada and Mexicobull Simplifying assumptions for purposes of illustration

ndash Winnipeg Canada plantndash Wrenches and pliersndash Made from steelndash Injection molding machinendash Assembly machine

Data for the GTC ProblemWrenches Pliers Available

Steel 15 10 15000 tons

Molding Machine 10 10 12000 hrs

Assembly Machine 4 5 5000 hrs

Demand Limit 8000 10000

Contribution($ per item)

$40 $30

We want to determine the number of wrenches and pliersto produce given the available raw materials machinehours and demand

To do with your partner

bull Work with your partner to formulate theGTC problem as a linear program DONOT LOOK AHEAD IN THE NOTES

bull Let P = number of pliers madebull Let W = number of wrenches made

Formulating the GTC Problem

Step 1 Determine Decision Variables P = number of thousands of pliers manufactured W = number of thousands of wrenches manufactured

Step 2 Determine Objective Function Maximize Profit =

The Formulation Continued

Step 3 Determine Constraints

Steel Molding Assembly Plier Demand Wrench Demand

We will show how to solve this in the next lecture

An Algebraic Formulation

bull J = set of items that are manufacturedndash eg S = pliers wrenchesndash pj = unit profit from item jndash dj = maximum demand for item jndash xj = number of units of item j manufactured

bull M = set of manufacturing processesndash eg M = molding and assemblyndash bi = capacity of process indash aij = amount of capacity of process i used in making item j

An Algebraic formulation

bull Maximize

bull subject to

The sameformulation workseven if | J | = 10000and | M | = 100

An Algebraic formulation

bull Maximize

bull subject to

Linear Programs

bull A linear function is a function of the form

bull A mathematical program is a linear program (LP) if the objective is a linear function and the constraints are linear equalities or inequalities

bull Typically an LP has non-negativity constraints

A non-linear program is permitted to havea non-linear objective and constraints

bull maximize f(xy) = xybull subject to x - y22 le 10 3x ndash 4y ge 2 x ge 0 y ge 0

An integer program is a linear programplus constraints that some or all of thevariables are integer valuedbull Maximize 3x1 + 4x2 - 3x3 3x1 + 2x2 - x3 ge 17 3x2 - x3 = 14 x1 ge 0 x2 ge 0 x3 ge 0 and x1 x2 x3 are all integers

An Algebraic formulation withequality constraints

bull Maximize

bull subject to

Linear Programming Assumptions

Proportionality Assumption Contribution from W isproportional to W

Additivity Assumption Contribution to objective function fromP is independent of W

Divisibility Assumption Each variable is allowed to assumefractional values

Certainty Assumption Each linear coefficient of the objectivefunction and constraints is known (and is not a random variable)

Maximize 4W + 3P15W + P le 15

hellip

Some Success Stories

bull Optimal crew scheduling saves American Airlines $20 millionyr

bull Improved shipment routing saves Yellow Freight over $173 millionyr

bull Improved truck dispatching at Reynolds Metals improves on-time delivery and reduces freight cost by $7 millionyr

bull GTE local capacity expansion saves $30 millionyr

Other Success Stories (cont)

bull Optimizing global supply chains saves Digital Equipment over $300 million

bull Restructuring North America Operations Proctor and Gamble reduces plants by 20 saving $200 millionyr

bull Optimal traffic control of Hanshin Expressway in Osaka saves 17 million driver hoursyr

bull Better scheduling of hydro and thermal generating units saves southern company $140 million

Success Stories (cont)

bull Improved production planning at Sadia (Brazil) saves $50 million over three years

bull Production Optimization at Harris Corporation improves on-time deliveries from 75 to 90

bull Tata Steel (India) optimizes response to power shortage contributing $73 million

bull Optimizing police patrol officer scheduling saves police department $11 millionyr

bull Gasoline blending at Texaco results in saving of over $30 millionyr

Summary

bull Answered the question What is Operations Research amp Management Science and provided some historical perspective

bull Introduced the terminology of linear programmingbull Two Examples

1 MSR Marketing2 Gemstone Tool Companyndash Small (2-dimensional) Linear Program non-obvious solution

bull We will discuss this problem in detail in the next lecture

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
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• Slide 39
• Slide 40
• Slide 41
• Slide 42
• Slide 43
• Slide 44
• Slide 45

Gemstone Tool Company

bull Privately-held firmbull Consumer and industrial market for construction toolsbull Headquartered in Seattlebull Manufacturing plants in the US Canada and Mexicobull Simplifying assumptions for purposes of illustration

ndash Winnipeg Canada plantndash Wrenches and pliersndash Made from steelndash Injection molding machinendash Assembly machine

Data for the GTC ProblemWrenches Pliers Available

Steel 15 10 15000 tons

Molding Machine 10 10 12000 hrs

Assembly Machine 4 5 5000 hrs

Demand Limit 8000 10000

Contribution($ per item)

$40 $30

We want to determine the number of wrenches and pliersto produce given the available raw materials machinehours and demand

To do with your partner

bull Work with your partner to formulate theGTC problem as a linear program DONOT LOOK AHEAD IN THE NOTES

bull Let P = number of pliers madebull Let W = number of wrenches made

Formulating the GTC Problem

Step 1 Determine Decision Variables P = number of thousands of pliers manufactured W = number of thousands of wrenches manufactured

Step 2 Determine Objective Function Maximize Profit =

The Formulation Continued

Step 3 Determine Constraints

Steel Molding Assembly Plier Demand Wrench Demand

We will show how to solve this in the next lecture

An Algebraic Formulation

bull J = set of items that are manufacturedndash eg S = pliers wrenchesndash pj = unit profit from item jndash dj = maximum demand for item jndash xj = number of units of item j manufactured

bull M = set of manufacturing processesndash eg M = molding and assemblyndash bi = capacity of process indash aij = amount of capacity of process i used in making item j

An Algebraic formulation

bull Maximize

bull subject to

The sameformulation workseven if | J | = 10000and | M | = 100

An Algebraic formulation

bull Maximize

bull subject to

Linear Programs

bull A linear function is a function of the form

bull A mathematical program is a linear program (LP) if the objective is a linear function and the constraints are linear equalities or inequalities

bull Typically an LP has non-negativity constraints

A non-linear program is permitted to havea non-linear objective and constraints

bull maximize f(xy) = xybull subject to x - y22 le 10 3x ndash 4y ge 2 x ge 0 y ge 0

An integer program is a linear programplus constraints that some or all of thevariables are integer valuedbull Maximize 3x1 + 4x2 - 3x3 3x1 + 2x2 - x3 ge 17 3x2 - x3 = 14 x1 ge 0 x2 ge 0 x3 ge 0 and x1 x2 x3 are all integers

An Algebraic formulation withequality constraints

bull Maximize

bull subject to

Linear Programming Assumptions

Proportionality Assumption Contribution from W isproportional to W

Additivity Assumption Contribution to objective function fromP is independent of W

Divisibility Assumption Each variable is allowed to assumefractional values

Certainty Assumption Each linear coefficient of the objectivefunction and constraints is known (and is not a random variable)

Maximize 4W + 3P15W + P le 15

hellip

Some Success Stories

bull Optimal crew scheduling saves American Airlines $20 millionyr

bull Improved shipment routing saves Yellow Freight over $173 millionyr

bull Improved truck dispatching at Reynolds Metals improves on-time delivery and reduces freight cost by $7 millionyr

bull GTE local capacity expansion saves $30 millionyr

Other Success Stories (cont)

bull Optimizing global supply chains saves Digital Equipment over $300 million

bull Restructuring North America Operations Proctor and Gamble reduces plants by 20 saving $200 millionyr

bull Optimal traffic control of Hanshin Expressway in Osaka saves 17 million driver hoursyr

bull Better scheduling of hydro and thermal generating units saves southern company $140 million

Success Stories (cont)

bull Improved production planning at Sadia (Brazil) saves $50 million over three years

bull Production Optimization at Harris Corporation improves on-time deliveries from 75 to 90

bull Tata Steel (India) optimizes response to power shortage contributing $73 million

bull Optimizing police patrol officer scheduling saves police department $11 millionyr

bull Gasoline blending at Texaco results in saving of over $30 millionyr

Summary

bull Answered the question What is Operations Research amp Management Science and provided some historical perspective

bull Introduced the terminology of linear programmingbull Two Examples

1 MSR Marketing2 Gemstone Tool Companyndash Small (2-dimensional) Linear Program non-obvious solution

bull We will discuss this problem in detail in the next lecture

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Data for the GTC ProblemWrenches Pliers Available

Steel 15 10 15000 tons

Molding Machine 10 10 12000 hrs

Assembly Machine 4 5 5000 hrs

Demand Limit 8000 10000

Contribution($ per item)

$40 $30

We want to determine the number of wrenches and pliersto produce given the available raw materials machinehours and demand

To do with your partner

bull Work with your partner to formulate theGTC problem as a linear program DONOT LOOK AHEAD IN THE NOTES

bull Let P = number of pliers madebull Let W = number of wrenches made

Formulating the GTC Problem

Step 1 Determine Decision Variables P = number of thousands of pliers manufactured W = number of thousands of wrenches manufactured

Step 2 Determine Objective Function Maximize Profit =

The Formulation Continued

Step 3 Determine Constraints

Steel Molding Assembly Plier Demand Wrench Demand

We will show how to solve this in the next lecture

An Algebraic Formulation

bull J = set of items that are manufacturedndash eg S = pliers wrenchesndash pj = unit profit from item jndash dj = maximum demand for item jndash xj = number of units of item j manufactured

bull M = set of manufacturing processesndash eg M = molding and assemblyndash bi = capacity of process indash aij = amount of capacity of process i used in making item j

An Algebraic formulation

bull Maximize

bull subject to

The sameformulation workseven if | J | = 10000and | M | = 100

An Algebraic formulation

bull Maximize

bull subject to

Linear Programs

bull A linear function is a function of the form

bull A mathematical program is a linear program (LP) if the objective is a linear function and the constraints are linear equalities or inequalities

bull Typically an LP has non-negativity constraints

A non-linear program is permitted to havea non-linear objective and constraints

bull maximize f(xy) = xybull subject to x - y22 le 10 3x ndash 4y ge 2 x ge 0 y ge 0

An integer program is a linear programplus constraints that some or all of thevariables are integer valuedbull Maximize 3x1 + 4x2 - 3x3 3x1 + 2x2 - x3 ge 17 3x2 - x3 = 14 x1 ge 0 x2 ge 0 x3 ge 0 and x1 x2 x3 are all integers

An Algebraic formulation withequality constraints

bull Maximize

bull subject to

Linear Programming Assumptions

Proportionality Assumption Contribution from W isproportional to W

Additivity Assumption Contribution to objective function fromP is independent of W

Divisibility Assumption Each variable is allowed to assumefractional values

Certainty Assumption Each linear coefficient of the objectivefunction and constraints is known (and is not a random variable)

Maximize 4W + 3P15W + P le 15

hellip

Some Success Stories

bull Optimal crew scheduling saves American Airlines $20 millionyr

bull Improved shipment routing saves Yellow Freight over $173 millionyr

bull Improved truck dispatching at Reynolds Metals improves on-time delivery and reduces freight cost by $7 millionyr

bull GTE local capacity expansion saves $30 millionyr

Other Success Stories (cont)

bull Optimizing global supply chains saves Digital Equipment over $300 million

bull Restructuring North America Operations Proctor and Gamble reduces plants by 20 saving $200 millionyr

bull Optimal traffic control of Hanshin Expressway in Osaka saves 17 million driver hoursyr

bull Better scheduling of hydro and thermal generating units saves southern company $140 million

Success Stories (cont)

bull Improved production planning at Sadia (Brazil) saves $50 million over three years

bull Production Optimization at Harris Corporation improves on-time deliveries from 75 to 90

bull Tata Steel (India) optimizes response to power shortage contributing $73 million

bull Optimizing police patrol officer scheduling saves police department $11 millionyr

bull Gasoline blending at Texaco results in saving of over $30 millionyr

Summary

bull Answered the question What is Operations Research amp Management Science and provided some historical perspective

bull Introduced the terminology of linear programmingbull Two Examples

1 MSR Marketing2 Gemstone Tool Companyndash Small (2-dimensional) Linear Program non-obvious solution

bull We will discuss this problem in detail in the next lecture

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To do with your partner

bull Work with your partner to formulate theGTC problem as a linear program DONOT LOOK AHEAD IN THE NOTES

bull Let P = number of pliers madebull Let W = number of wrenches made

Formulating the GTC Problem

Step 1 Determine Decision Variables P = number of thousands of pliers manufactured W = number of thousands of wrenches manufactured

Step 2 Determine Objective Function Maximize Profit =

The Formulation Continued

Step 3 Determine Constraints

Steel Molding Assembly Plier Demand Wrench Demand

We will show how to solve this in the next lecture

An Algebraic Formulation

bull J = set of items that are manufacturedndash eg S = pliers wrenchesndash pj = unit profit from item jndash dj = maximum demand for item jndash xj = number of units of item j manufactured

bull M = set of manufacturing processesndash eg M = molding and assemblyndash bi = capacity of process indash aij = amount of capacity of process i used in making item j

An Algebraic formulation

bull Maximize

bull subject to

The sameformulation workseven if | J | = 10000and | M | = 100

An Algebraic formulation

bull Maximize

bull subject to

Linear Programs

bull A linear function is a function of the form

bull A mathematical program is a linear program (LP) if the objective is a linear function and the constraints are linear equalities or inequalities

bull Typically an LP has non-negativity constraints

A non-linear program is permitted to havea non-linear objective and constraints

bull maximize f(xy) = xybull subject to x - y22 le 10 3x ndash 4y ge 2 x ge 0 y ge 0

An integer program is a linear programplus constraints that some or all of thevariables are integer valuedbull Maximize 3x1 + 4x2 - 3x3 3x1 + 2x2 - x3 ge 17 3x2 - x3 = 14 x1 ge 0 x2 ge 0 x3 ge 0 and x1 x2 x3 are all integers

An Algebraic formulation withequality constraints

bull Maximize

bull subject to

Linear Programming Assumptions

Proportionality Assumption Contribution from W isproportional to W

Additivity Assumption Contribution to objective function fromP is independent of W

Divisibility Assumption Each variable is allowed to assumefractional values

Certainty Assumption Each linear coefficient of the objectivefunction and constraints is known (and is not a random variable)

Maximize 4W + 3P15W + P le 15

hellip

Some Success Stories

bull Optimal crew scheduling saves American Airlines $20 millionyr

bull Improved shipment routing saves Yellow Freight over $173 millionyr

bull Improved truck dispatching at Reynolds Metals improves on-time delivery and reduces freight cost by $7 millionyr

bull GTE local capacity expansion saves $30 millionyr

Other Success Stories (cont)

bull Optimizing global supply chains saves Digital Equipment over $300 million

bull Restructuring North America Operations Proctor and Gamble reduces plants by 20 saving $200 millionyr

bull Optimal traffic control of Hanshin Expressway in Osaka saves 17 million driver hoursyr

bull Better scheduling of hydro and thermal generating units saves southern company $140 million

Success Stories (cont)

bull Improved production planning at Sadia (Brazil) saves $50 million over three years

bull Production Optimization at Harris Corporation improves on-time deliveries from 75 to 90

bull Tata Steel (India) optimizes response to power shortage contributing $73 million

bull Optimizing police patrol officer scheduling saves police department $11 millionyr

bull Gasoline blending at Texaco results in saving of over $30 millionyr

Summary

bull Answered the question What is Operations Research amp Management Science and provided some historical perspective

bull Introduced the terminology of linear programmingbull Two Examples

1 MSR Marketing2 Gemstone Tool Companyndash Small (2-dimensional) Linear Program non-obvious solution

bull We will discuss this problem in detail in the next lecture

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Formulating the GTC Problem

Step 1 Determine Decision Variables P = number of thousands of pliers manufactured W = number of thousands of wrenches manufactured

Step 2 Determine Objective Function Maximize Profit =

The Formulation Continued

Step 3 Determine Constraints

Steel Molding Assembly Plier Demand Wrench Demand

We will show how to solve this in the next lecture

An Algebraic Formulation

bull J = set of items that are manufacturedndash eg S = pliers wrenchesndash pj = unit profit from item jndash dj = maximum demand for item jndash xj = number of units of item j manufactured

bull M = set of manufacturing processesndash eg M = molding and assemblyndash bi = capacity of process indash aij = amount of capacity of process i used in making item j

An Algebraic formulation

bull Maximize

bull subject to

The sameformulation workseven if | J | = 10000and | M | = 100

An Algebraic formulation

bull Maximize

bull subject to

Linear Programs

bull A linear function is a function of the form

bull A mathematical program is a linear program (LP) if the objective is a linear function and the constraints are linear equalities or inequalities

bull Typically an LP has non-negativity constraints

A non-linear program is permitted to havea non-linear objective and constraints

bull maximize f(xy) = xybull subject to x - y22 le 10 3x ndash 4y ge 2 x ge 0 y ge 0

An integer program is a linear programplus constraints that some or all of thevariables are integer valuedbull Maximize 3x1 + 4x2 - 3x3 3x1 + 2x2 - x3 ge 17 3x2 - x3 = 14 x1 ge 0 x2 ge 0 x3 ge 0 and x1 x2 x3 are all integers

An Algebraic formulation withequality constraints

bull Maximize

bull subject to

Linear Programming Assumptions

Proportionality Assumption Contribution from W isproportional to W

Additivity Assumption Contribution to objective function fromP is independent of W

Divisibility Assumption Each variable is allowed to assumefractional values

Certainty Assumption Each linear coefficient of the objectivefunction and constraints is known (and is not a random variable)

Maximize 4W + 3P15W + P le 15

hellip

Some Success Stories

bull Optimal crew scheduling saves American Airlines $20 millionyr

bull Improved shipment routing saves Yellow Freight over $173 millionyr

bull Improved truck dispatching at Reynolds Metals improves on-time delivery and reduces freight cost by $7 millionyr

bull GTE local capacity expansion saves $30 millionyr

Other Success Stories (cont)

bull Optimizing global supply chains saves Digital Equipment over $300 million

bull Restructuring North America Operations Proctor and Gamble reduces plants by 20 saving $200 millionyr

bull Optimal traffic control of Hanshin Expressway in Osaka saves 17 million driver hoursyr

bull Better scheduling of hydro and thermal generating units saves southern company $140 million

Success Stories (cont)

bull Improved production planning at Sadia (Brazil) saves $50 million over three years

bull Production Optimization at Harris Corporation improves on-time deliveries from 75 to 90

bull Tata Steel (India) optimizes response to power shortage contributing $73 million

bull Optimizing police patrol officer scheduling saves police department $11 millionyr

bull Gasoline blending at Texaco results in saving of over $30 millionyr

Summary

bull Answered the question What is Operations Research amp Management Science and provided some historical perspective

bull Introduced the terminology of linear programmingbull Two Examples

1 MSR Marketing2 Gemstone Tool Companyndash Small (2-dimensional) Linear Program non-obvious solution

bull We will discuss this problem in detail in the next lecture

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The Formulation Continued

Step 3 Determine Constraints

Steel Molding Assembly Plier Demand Wrench Demand

We will show how to solve this in the next lecture

An Algebraic Formulation

bull J = set of items that are manufacturedndash eg S = pliers wrenchesndash pj = unit profit from item jndash dj = maximum demand for item jndash xj = number of units of item j manufactured

bull M = set of manufacturing processesndash eg M = molding and assemblyndash bi = capacity of process indash aij = amount of capacity of process i used in making item j

An Algebraic formulation

bull Maximize

bull subject to

The sameformulation workseven if | J | = 10000and | M | = 100

An Algebraic formulation

bull Maximize

bull subject to

Linear Programs

bull A linear function is a function of the form

bull A mathematical program is a linear program (LP) if the objective is a linear function and the constraints are linear equalities or inequalities

bull Typically an LP has non-negativity constraints

A non-linear program is permitted to havea non-linear objective and constraints

bull maximize f(xy) = xybull subject to x - y22 le 10 3x ndash 4y ge 2 x ge 0 y ge 0

An integer program is a linear programplus constraints that some or all of thevariables are integer valuedbull Maximize 3x1 + 4x2 - 3x3 3x1 + 2x2 - x3 ge 17 3x2 - x3 = 14 x1 ge 0 x2 ge 0 x3 ge 0 and x1 x2 x3 are all integers

An Algebraic formulation withequality constraints

bull Maximize

bull subject to

Linear Programming Assumptions

Proportionality Assumption Contribution from W isproportional to W

Additivity Assumption Contribution to objective function fromP is independent of W

Divisibility Assumption Each variable is allowed to assumefractional values

Certainty Assumption Each linear coefficient of the objectivefunction and constraints is known (and is not a random variable)

Maximize 4W + 3P15W + P le 15

hellip

Some Success Stories

bull Optimal crew scheduling saves American Airlines $20 millionyr

bull Improved shipment routing saves Yellow Freight over $173 millionyr

bull Improved truck dispatching at Reynolds Metals improves on-time delivery and reduces freight cost by $7 millionyr

bull GTE local capacity expansion saves $30 millionyr

Other Success Stories (cont)

bull Optimizing global supply chains saves Digital Equipment over $300 million

bull Restructuring North America Operations Proctor and Gamble reduces plants by 20 saving $200 millionyr

bull Optimal traffic control of Hanshin Expressway in Osaka saves 17 million driver hoursyr

bull Better scheduling of hydro and thermal generating units saves southern company $140 million

Success Stories (cont)

bull Improved production planning at Sadia (Brazil) saves $50 million over three years

bull Production Optimization at Harris Corporation improves on-time deliveries from 75 to 90

bull Tata Steel (India) optimizes response to power shortage contributing $73 million

bull Optimizing police patrol officer scheduling saves police department $11 millionyr

bull Gasoline blending at Texaco results in saving of over $30 millionyr

Summary

bull Answered the question What is Operations Research amp Management Science and provided some historical perspective

bull Introduced the terminology of linear programmingbull Two Examples

1 MSR Marketing2 Gemstone Tool Companyndash Small (2-dimensional) Linear Program non-obvious solution

bull We will discuss this problem in detail in the next lecture

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An Algebraic Formulation

bull J = set of items that are manufacturedndash eg S = pliers wrenchesndash pj = unit profit from item jndash dj = maximum demand for item jndash xj = number of units of item j manufactured

bull M = set of manufacturing processesndash eg M = molding and assemblyndash bi = capacity of process indash aij = amount of capacity of process i used in making item j

An Algebraic formulation

bull Maximize

bull subject to

The sameformulation workseven if | J | = 10000and | M | = 100

An Algebraic formulation

bull Maximize

bull subject to

Linear Programs

bull A linear function is a function of the form

bull A mathematical program is a linear program (LP) if the objective is a linear function and the constraints are linear equalities or inequalities

bull Typically an LP has non-negativity constraints

A non-linear program is permitted to havea non-linear objective and constraints

bull maximize f(xy) = xybull subject to x - y22 le 10 3x ndash 4y ge 2 x ge 0 y ge 0

An integer program is a linear programplus constraints that some or all of thevariables are integer valuedbull Maximize 3x1 + 4x2 - 3x3 3x1 + 2x2 - x3 ge 17 3x2 - x3 = 14 x1 ge 0 x2 ge 0 x3 ge 0 and x1 x2 x3 are all integers

An Algebraic formulation withequality constraints

bull Maximize

bull subject to

Linear Programming Assumptions

Proportionality Assumption Contribution from W isproportional to W

Additivity Assumption Contribution to objective function fromP is independent of W

Divisibility Assumption Each variable is allowed to assumefractional values

Certainty Assumption Each linear coefficient of the objectivefunction and constraints is known (and is not a random variable)

Maximize 4W + 3P15W + P le 15

hellip

Some Success Stories

bull Optimal crew scheduling saves American Airlines $20 millionyr

bull Improved shipment routing saves Yellow Freight over $173 millionyr

bull Improved truck dispatching at Reynolds Metals improves on-time delivery and reduces freight cost by $7 millionyr

bull GTE local capacity expansion saves $30 millionyr

Other Success Stories (cont)

bull Optimizing global supply chains saves Digital Equipment over $300 million

bull Restructuring North America Operations Proctor and Gamble reduces plants by 20 saving $200 millionyr

bull Optimal traffic control of Hanshin Expressway in Osaka saves 17 million driver hoursyr

bull Better scheduling of hydro and thermal generating units saves southern company $140 million

Success Stories (cont)

bull Improved production planning at Sadia (Brazil) saves $50 million over three years

bull Production Optimization at Harris Corporation improves on-time deliveries from 75 to 90

bull Tata Steel (India) optimizes response to power shortage contributing $73 million

bull Optimizing police patrol officer scheduling saves police department $11 millionyr

bull Gasoline blending at Texaco results in saving of over $30 millionyr

Summary

bull Answered the question What is Operations Research amp Management Science and provided some historical perspective

bull Introduced the terminology of linear programmingbull Two Examples

1 MSR Marketing2 Gemstone Tool Companyndash Small (2-dimensional) Linear Program non-obvious solution

bull We will discuss this problem in detail in the next lecture

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An Algebraic formulation

bull Maximize

bull subject to

The sameformulation workseven if | J | = 10000and | M | = 100

An Algebraic formulation

bull Maximize

bull subject to

Linear Programs

bull A linear function is a function of the form

bull A mathematical program is a linear program (LP) if the objective is a linear function and the constraints are linear equalities or inequalities

bull Typically an LP has non-negativity constraints

A non-linear program is permitted to havea non-linear objective and constraints

bull maximize f(xy) = xybull subject to x - y22 le 10 3x ndash 4y ge 2 x ge 0 y ge 0

An integer program is a linear programplus constraints that some or all of thevariables are integer valuedbull Maximize 3x1 + 4x2 - 3x3 3x1 + 2x2 - x3 ge 17 3x2 - x3 = 14 x1 ge 0 x2 ge 0 x3 ge 0 and x1 x2 x3 are all integers

An Algebraic formulation withequality constraints

bull Maximize

bull subject to

Linear Programming Assumptions

Proportionality Assumption Contribution from W isproportional to W

Additivity Assumption Contribution to objective function fromP is independent of W

Divisibility Assumption Each variable is allowed to assumefractional values

Certainty Assumption Each linear coefficient of the objectivefunction and constraints is known (and is not a random variable)

Maximize 4W + 3P15W + P le 15

hellip

Some Success Stories

bull Optimal crew scheduling saves American Airlines $20 millionyr

bull Improved shipment routing saves Yellow Freight over $173 millionyr

bull Improved truck dispatching at Reynolds Metals improves on-time delivery and reduces freight cost by $7 millionyr

bull GTE local capacity expansion saves $30 millionyr

Other Success Stories (cont)

bull Optimizing global supply chains saves Digital Equipment over $300 million

bull Restructuring North America Operations Proctor and Gamble reduces plants by 20 saving $200 millionyr

bull Optimal traffic control of Hanshin Expressway in Osaka saves 17 million driver hoursyr

bull Better scheduling of hydro and thermal generating units saves southern company $140 million

Success Stories (cont)

bull Improved production planning at Sadia (Brazil) saves $50 million over three years

bull Production Optimization at Harris Corporation improves on-time deliveries from 75 to 90

bull Tata Steel (India) optimizes response to power shortage contributing $73 million

bull Optimizing police patrol officer scheduling saves police department $11 millionyr

bull Gasoline blending at Texaco results in saving of over $30 millionyr

Summary

bull Answered the question What is Operations Research amp Management Science and provided some historical perspective

bull Introduced the terminology of linear programmingbull Two Examples

1 MSR Marketing2 Gemstone Tool Companyndash Small (2-dimensional) Linear Program non-obvious solution

bull We will discuss this problem in detail in the next lecture

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An Algebraic formulation

bull Maximize

bull subject to

Linear Programs

bull A linear function is a function of the form

bull A mathematical program is a linear program (LP) if the objective is a linear function and the constraints are linear equalities or inequalities

bull Typically an LP has non-negativity constraints

A non-linear program is permitted to havea non-linear objective and constraints

bull maximize f(xy) = xybull subject to x - y22 le 10 3x ndash 4y ge 2 x ge 0 y ge 0

An integer program is a linear programplus constraints that some or all of thevariables are integer valuedbull Maximize 3x1 + 4x2 - 3x3 3x1 + 2x2 - x3 ge 17 3x2 - x3 = 14 x1 ge 0 x2 ge 0 x3 ge 0 and x1 x2 x3 are all integers

An Algebraic formulation withequality constraints

bull Maximize

bull subject to

Linear Programming Assumptions

Proportionality Assumption Contribution from W isproportional to W

Additivity Assumption Contribution to objective function fromP is independent of W

Divisibility Assumption Each variable is allowed to assumefractional values

Certainty Assumption Each linear coefficient of the objectivefunction and constraints is known (and is not a random variable)

Maximize 4W + 3P15W + P le 15

hellip

Some Success Stories

bull Optimal crew scheduling saves American Airlines $20 millionyr

bull Improved shipment routing saves Yellow Freight over $173 millionyr

bull Improved truck dispatching at Reynolds Metals improves on-time delivery and reduces freight cost by $7 millionyr

bull GTE local capacity expansion saves $30 millionyr

Other Success Stories (cont)

bull Optimizing global supply chains saves Digital Equipment over $300 million

bull Restructuring North America Operations Proctor and Gamble reduces plants by 20 saving $200 millionyr

bull Optimal traffic control of Hanshin Expressway in Osaka saves 17 million driver hoursyr

bull Better scheduling of hydro and thermal generating units saves southern company $140 million

Success Stories (cont)

bull Improved production planning at Sadia (Brazil) saves $50 million over three years

bull Production Optimization at Harris Corporation improves on-time deliveries from 75 to 90

bull Tata Steel (India) optimizes response to power shortage contributing $73 million

bull Optimizing police patrol officer scheduling saves police department $11 millionyr

bull Gasoline blending at Texaco results in saving of over $30 millionyr

Summary

bull Answered the question What is Operations Research amp Management Science and provided some historical perspective

bull Introduced the terminology of linear programmingbull Two Examples

1 MSR Marketing2 Gemstone Tool Companyndash Small (2-dimensional) Linear Program non-obvious solution

bull We will discuss this problem in detail in the next lecture

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Linear Programs

bull A linear function is a function of the form

bull A mathematical program is a linear program (LP) if the objective is a linear function and the constraints are linear equalities or inequalities

bull Typically an LP has non-negativity constraints

A non-linear program is permitted to havea non-linear objective and constraints

bull maximize f(xy) = xybull subject to x - y22 le 10 3x ndash 4y ge 2 x ge 0 y ge 0

An integer program is a linear programplus constraints that some or all of thevariables are integer valuedbull Maximize 3x1 + 4x2 - 3x3 3x1 + 2x2 - x3 ge 17 3x2 - x3 = 14 x1 ge 0 x2 ge 0 x3 ge 0 and x1 x2 x3 are all integers

An Algebraic formulation withequality constraints

bull Maximize

bull subject to

Linear Programming Assumptions

Proportionality Assumption Contribution from W isproportional to W

Additivity Assumption Contribution to objective function fromP is independent of W

Divisibility Assumption Each variable is allowed to assumefractional values

Certainty Assumption Each linear coefficient of the objectivefunction and constraints is known (and is not a random variable)

Maximize 4W + 3P15W + P le 15

hellip

Some Success Stories

bull Optimal crew scheduling saves American Airlines $20 millionyr

bull Improved shipment routing saves Yellow Freight over $173 millionyr

bull Improved truck dispatching at Reynolds Metals improves on-time delivery and reduces freight cost by $7 millionyr

bull GTE local capacity expansion saves $30 millionyr

Other Success Stories (cont)

bull Optimizing global supply chains saves Digital Equipment over $300 million

bull Restructuring North America Operations Proctor and Gamble reduces plants by 20 saving $200 millionyr

bull Optimal traffic control of Hanshin Expressway in Osaka saves 17 million driver hoursyr

bull Better scheduling of hydro and thermal generating units saves southern company $140 million

Success Stories (cont)

bull Improved production planning at Sadia (Brazil) saves $50 million over three years

bull Production Optimization at Harris Corporation improves on-time deliveries from 75 to 90

bull Tata Steel (India) optimizes response to power shortage contributing $73 million

bull Optimizing police patrol officer scheduling saves police department $11 millionyr

bull Gasoline blending at Texaco results in saving of over $30 millionyr

Summary

bull Answered the question What is Operations Research amp Management Science and provided some historical perspective

bull Introduced the terminology of linear programmingbull Two Examples

1 MSR Marketing2 Gemstone Tool Companyndash Small (2-dimensional) Linear Program non-obvious solution

bull We will discuss this problem in detail in the next lecture

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A non-linear program is permitted to havea non-linear objective and constraints

bull maximize f(xy) = xybull subject to x - y22 le 10 3x ndash 4y ge 2 x ge 0 y ge 0

An integer program is a linear programplus constraints that some or all of thevariables are integer valuedbull Maximize 3x1 + 4x2 - 3x3 3x1 + 2x2 - x3 ge 17 3x2 - x3 = 14 x1 ge 0 x2 ge 0 x3 ge 0 and x1 x2 x3 are all integers

An Algebraic formulation withequality constraints

bull Maximize

bull subject to

Linear Programming Assumptions

Proportionality Assumption Contribution from W isproportional to W

Additivity Assumption Contribution to objective function fromP is independent of W

Divisibility Assumption Each variable is allowed to assumefractional values

Certainty Assumption Each linear coefficient of the objectivefunction and constraints is known (and is not a random variable)

Maximize 4W + 3P15W + P le 15

hellip

Some Success Stories

bull Optimal crew scheduling saves American Airlines $20 millionyr

bull Improved shipment routing saves Yellow Freight over $173 millionyr

bull Improved truck dispatching at Reynolds Metals improves on-time delivery and reduces freight cost by $7 millionyr

bull GTE local capacity expansion saves $30 millionyr

Other Success Stories (cont)

bull Optimizing global supply chains saves Digital Equipment over $300 million

bull Restructuring North America Operations Proctor and Gamble reduces plants by 20 saving $200 millionyr

bull Optimal traffic control of Hanshin Expressway in Osaka saves 17 million driver hoursyr

bull Better scheduling of hydro and thermal generating units saves southern company $140 million

Success Stories (cont)

bull Improved production planning at Sadia (Brazil) saves $50 million over three years

bull Production Optimization at Harris Corporation improves on-time deliveries from 75 to 90

bull Tata Steel (India) optimizes response to power shortage contributing $73 million

bull Optimizing police patrol officer scheduling saves police department $11 millionyr

bull Gasoline blending at Texaco results in saving of over $30 millionyr

Summary

bull Answered the question What is Operations Research amp Management Science and provided some historical perspective

bull Introduced the terminology of linear programmingbull Two Examples

1 MSR Marketing2 Gemstone Tool Companyndash Small (2-dimensional) Linear Program non-obvious solution

bull We will discuss this problem in detail in the next lecture

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An integer program is a linear programplus constraints that some or all of thevariables are integer valuedbull Maximize 3x1 + 4x2 - 3x3 3x1 + 2x2 - x3 ge 17 3x2 - x3 = 14 x1 ge 0 x2 ge 0 x3 ge 0 and x1 x2 x3 are all integers

An Algebraic formulation withequality constraints

bull Maximize

bull subject to

Linear Programming Assumptions

Proportionality Assumption Contribution from W isproportional to W

Additivity Assumption Contribution to objective function fromP is independent of W

Divisibility Assumption Each variable is allowed to assumefractional values

Certainty Assumption Each linear coefficient of the objectivefunction and constraints is known (and is not a random variable)

Maximize 4W + 3P15W + P le 15

hellip

Some Success Stories

bull Optimal crew scheduling saves American Airlines $20 millionyr

bull Improved shipment routing saves Yellow Freight over $173 millionyr

bull Improved truck dispatching at Reynolds Metals improves on-time delivery and reduces freight cost by $7 millionyr

bull GTE local capacity expansion saves $30 millionyr

Other Success Stories (cont)

bull Optimizing global supply chains saves Digital Equipment over $300 million

bull Restructuring North America Operations Proctor and Gamble reduces plants by 20 saving $200 millionyr

bull Optimal traffic control of Hanshin Expressway in Osaka saves 17 million driver hoursyr

bull Better scheduling of hydro and thermal generating units saves southern company $140 million

Success Stories (cont)

bull Improved production planning at Sadia (Brazil) saves $50 million over three years

bull Production Optimization at Harris Corporation improves on-time deliveries from 75 to 90

bull Tata Steel (India) optimizes response to power shortage contributing $73 million

bull Optimizing police patrol officer scheduling saves police department $11 millionyr

bull Gasoline blending at Texaco results in saving of over $30 millionyr

Summary

bull Answered the question What is Operations Research amp Management Science and provided some historical perspective

bull Introduced the terminology of linear programmingbull Two Examples

1 MSR Marketing2 Gemstone Tool Companyndash Small (2-dimensional) Linear Program non-obvious solution

bull We will discuss this problem in detail in the next lecture

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An Algebraic formulation withequality constraints

bull Maximize

bull subject to

Linear Programming Assumptions

Proportionality Assumption Contribution from W isproportional to W

Additivity Assumption Contribution to objective function fromP is independent of W

Divisibility Assumption Each variable is allowed to assumefractional values

Certainty Assumption Each linear coefficient of the objectivefunction and constraints is known (and is not a random variable)

Maximize 4W + 3P15W + P le 15

hellip

Some Success Stories

bull Optimal crew scheduling saves American Airlines $20 millionyr

bull Improved shipment routing saves Yellow Freight over $173 millionyr

bull Improved truck dispatching at Reynolds Metals improves on-time delivery and reduces freight cost by $7 millionyr

bull GTE local capacity expansion saves $30 millionyr

Other Success Stories (cont)

bull Optimizing global supply chains saves Digital Equipment over $300 million

bull Restructuring North America Operations Proctor and Gamble reduces plants by 20 saving $200 millionyr

bull Optimal traffic control of Hanshin Expressway in Osaka saves 17 million driver hoursyr

bull Better scheduling of hydro and thermal generating units saves southern company $140 million

Success Stories (cont)

bull Improved production planning at Sadia (Brazil) saves $50 million over three years

bull Production Optimization at Harris Corporation improves on-time deliveries from 75 to 90

bull Tata Steel (India) optimizes response to power shortage contributing $73 million

bull Optimizing police patrol officer scheduling saves police department $11 millionyr

bull Gasoline blending at Texaco results in saving of over $30 millionyr

Summary

bull Answered the question What is Operations Research amp Management Science and provided some historical perspective

bull Introduced the terminology of linear programmingbull Two Examples

1 MSR Marketing2 Gemstone Tool Companyndash Small (2-dimensional) Linear Program non-obvious solution

bull We will discuss this problem in detail in the next lecture

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Linear Programming Assumptions

Proportionality Assumption Contribution from W isproportional to W

Additivity Assumption Contribution to objective function fromP is independent of W

Divisibility Assumption Each variable is allowed to assumefractional values

Certainty Assumption Each linear coefficient of the objectivefunction and constraints is known (and is not a random variable)

Maximize 4W + 3P15W + P le 15

hellip

Some Success Stories

bull Optimal crew scheduling saves American Airlines $20 millionyr

bull Improved shipment routing saves Yellow Freight over $173 millionyr

bull Improved truck dispatching at Reynolds Metals improves on-time delivery and reduces freight cost by $7 millionyr

bull GTE local capacity expansion saves $30 millionyr

Other Success Stories (cont)

bull Optimizing global supply chains saves Digital Equipment over $300 million

bull Restructuring North America Operations Proctor and Gamble reduces plants by 20 saving $200 millionyr

bull Optimal traffic control of Hanshin Expressway in Osaka saves 17 million driver hoursyr

bull Better scheduling of hydro and thermal generating units saves southern company $140 million

Success Stories (cont)

bull Improved production planning at Sadia (Brazil) saves $50 million over three years

bull Production Optimization at Harris Corporation improves on-time deliveries from 75 to 90

bull Tata Steel (India) optimizes response to power shortage contributing $73 million

bull Optimizing police patrol officer scheduling saves police department $11 millionyr

bull Gasoline blending at Texaco results in saving of over $30 millionyr

Summary

bull Answered the question What is Operations Research amp Management Science and provided some historical perspective

bull Introduced the terminology of linear programmingbull Two Examples

1 MSR Marketing2 Gemstone Tool Companyndash Small (2-dimensional) Linear Program non-obvious solution

bull We will discuss this problem in detail in the next lecture

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Some Success Stories

bull Optimal crew scheduling saves American Airlines $20 millionyr

bull Improved shipment routing saves Yellow Freight over $173 millionyr

bull Improved truck dispatching at Reynolds Metals improves on-time delivery and reduces freight cost by $7 millionyr

bull GTE local capacity expansion saves $30 millionyr

Other Success Stories (cont)

bull Optimizing global supply chains saves Digital Equipment over $300 million

bull Restructuring North America Operations Proctor and Gamble reduces plants by 20 saving $200 millionyr

bull Optimal traffic control of Hanshin Expressway in Osaka saves 17 million driver hoursyr

bull Better scheduling of hydro and thermal generating units saves southern company $140 million

Success Stories (cont)

bull Improved production planning at Sadia (Brazil) saves $50 million over three years

bull Production Optimization at Harris Corporation improves on-time deliveries from 75 to 90

bull Tata Steel (India) optimizes response to power shortage contributing $73 million

bull Optimizing police patrol officer scheduling saves police department $11 millionyr

bull Gasoline blending at Texaco results in saving of over $30 millionyr

Summary

bull Answered the question What is Operations Research amp Management Science and provided some historical perspective

bull Introduced the terminology of linear programmingbull Two Examples

1 MSR Marketing2 Gemstone Tool Companyndash Small (2-dimensional) Linear Program non-obvious solution

bull We will discuss this problem in detail in the next lecture

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Other Success Stories (cont)

bull Optimizing global supply chains saves Digital Equipment over $300 million

bull Restructuring North America Operations Proctor and Gamble reduces plants by 20 saving $200 millionyr

bull Optimal traffic control of Hanshin Expressway in Osaka saves 17 million driver hoursyr

bull Better scheduling of hydro and thermal generating units saves southern company $140 million

Success Stories (cont)

bull Improved production planning at Sadia (Brazil) saves $50 million over three years

bull Production Optimization at Harris Corporation improves on-time deliveries from 75 to 90

bull Tata Steel (India) optimizes response to power shortage contributing $73 million

bull Optimizing police patrol officer scheduling saves police department $11 millionyr

bull Gasoline blending at Texaco results in saving of over $30 millionyr

Summary

bull Answered the question What is Operations Research amp Management Science and provided some historical perspective

bull Introduced the terminology of linear programmingbull Two Examples

1 MSR Marketing2 Gemstone Tool Companyndash Small (2-dimensional) Linear Program non-obvious solution

bull We will discuss this problem in detail in the next lecture

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Success Stories (cont)

bull Improved production planning at Sadia (Brazil) saves $50 million over three years

bull Production Optimization at Harris Corporation improves on-time deliveries from 75 to 90

bull Tata Steel (India) optimizes response to power shortage contributing $73 million

bull Optimizing police patrol officer scheduling saves police department $11 millionyr

bull Gasoline blending at Texaco results in saving of over $30 millionyr

Summary

bull Answered the question What is Operations Research amp Management Science and provided some historical perspective

bull Introduced the terminology of linear programmingbull Two Examples

1 MSR Marketing2 Gemstone Tool Companyndash Small (2-dimensional) Linear Program non-obvious solution

bull We will discuss this problem in detail in the next lecture

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Summary

bull Answered the question What is Operations Research amp Management Science and provided some historical perspective

bull Introduced the terminology of linear programmingbull Two Examples

1 MSR Marketing2 Gemstone Tool Companyndash Small (2-dimensional) Linear Program non-obvious solution

bull We will discuss this problem in detail in the next lecture

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