1

15.053 February 1, 2005

Introduction to Optimization

Handouts: – Lecture Notes– Info Sheet– 1-page subject description

Note: no laptops in class.Note: 18.06 is not a prerequisite for 15.053 this semester.

However, we will assume that you are comfortable in working with matrices and in solving systems of equations (the level of matrix operations assumed in 18.02)

2

Introductions

James Orlin– Professor of Operations Research at Sloan– Co-director of the MIT Operations Research Center

3

Goals for this LectureCourse AdministrationBackground on Operations Research (Management Science) and Optimization Course Themes and Goals

Linear Programming Examples– MSR Marketing– DTC

Some time to choose a homework partner and fill in a info sheet

4

Required Materials

Introduction to Operations Research (4th edition)by Wayne Winston

Class Website:

Assignment 1 will be due next Thursday

It contains subject information (which is accessible from web.mit.edu/jorlin/www/)

-

-

5

Grading Policy

Homework assignments (25%)

Midterm Exams (40%: 20% each)– March 10th, 7:30 to 9:30 PM– April 21st, 7:30 to 9:30 PM

Final Exam (20%)– During Finals week

Recitation quizzes (16%)

Extra credit mini-project (5%)

6

“Active Learning”

Occasionally I will introduce a break in the lecture for you to work on your own or with an in-class partner. “Cognitive balancing.”

Please identify your “partner” now.

Those on aisle ends may be in a group of size 3.

7

What is Operations Research? What is Management Science?

World War II : British military leaders asked scientists and engineers to analyze several military problems– Deployment of radar – Management of convoy, bombing, antisubmarine, and

mining operations.The result was called Operations Research

MIT was one of the birthplaces of OR– Professor Morse at MIT was a pioneer in

the US– Founded MIT OR Center and helped to

found ORSA

8

What is Management Science (Operations Research)?

Operations Research (O.R.) is the discipline of applying advanced analytical methods to help make better decisions.

It is the “Science of Better”

9

Voices from the pastWaste neither time nor money, but Waste neither time nor money, but make the best use of both.make the best use of both.

---- Benjamin FranklinBenjamin Franklin

Obviously, the highest type of efficiency is that which can utilize existing material to the best advantage.

-- Jawaharlal Nehru

It is more probable that the average man could, with no injury to his health, increase his efficiency fifty percent.

-- Walter Scott

10

Operations Research Over the Years

1947 – Project Scoop (Scientific Computation of Optimum

Programs) with George Dantzig and others. Developed the simplex method for linear programs.

1950's– Lots of excitement, mathematical developments,

queuing theory, mathematical programming.

1960's– More excitement, more development and grand

plans.

11

Operations Research Over the Years1970's– Disappointment, and a settling down. NP-

completeness. More realistic expectations. 1980's– Widespread availability of personal computers.

Increasingly easy access to data. Widespread willingness of managers to use models.

1990's– Improved use of O.R. systems.

Further inroads of O.R. technology, e.g., optimization and simulation add-ins to spreadsheets, modeling languages, large scale optimization. More intermixing of A.I. and O.R.

12

Operations Research in the 2000’sLOTS of opportunities for OR as a field

Data, data, data

– E-business data (click stream, purchases, other transactional data, E-mail and more)

– Sensor data

– The human genome project and its outgrowth

Need for more automated decision making

Need for increased coordination for efficient use of resources (Supply chain management)

13

Some Skills for Operations Researchers

Modeling Skills– Take a real world situation, and model it using

mathematics

Methodological Toolkit– Optimization– Probabilistic Models

As ageless as time

Optimization

Thanks to Tom Magnanti for these slides

Optimization in NatureHeron of Alexandria

bb

Angle of Angle of

ReflectionReflection

aa

Angle of Angle of

IncidenceIncidence

b’b’Angle of Angle of

IncidenceIncidence

First Century A.DFirst Century A.D

FERMATFERMAT

Angle Angle αα11 of of

IncidenceIncidenceAngle Angle αα22 of of

RefractionRefraction

16281628--2929

17

Calculus

Maximum

Minimum

Fermat, Newton, Euler, Lagrange, Gauss, and more

18

Some of the themes of 15.053

Optimization is everywhereModels, Models, ModelsThe goal of models is “insight” not numbers– paraphrase of Richard Hamming

Algorithms, Algorithms, Algorithms

19

Optimization is EverywhereIt is embedded in language, and part of the way we think.– firms want to maximize value to shareholders– people want to make the best choices– We want the highest quality at the lowest price– When playing games, we want the best strategy– When we have too much to do, we want to optimize the use of

our time– etc.

Take 3 minutes with your partner to brainstorm on where optimization might be used. (business, or sports, or personal uses, or politics, or …)“Optimize the ….” or “optimal ….”

American AirlinesOptimal crew scheduling saves $20 million/yearOptimal yield management contributes $500 million per year

Revenue management saves National Car rental $56 million in one yearBetter truck dispatching at Reynolds Metals improves on-time delivery and reduces annual freight costs by $7 millionImproved shipment routing saves Yellow freight over $17.3 million/year

GTE local access capacity expansion saves $30 million per yearOptimizing its global supply chains saves Digital Equipment over $300 millionRestructuring North America operations, Proctor and Gamble reduces plants by 20%, saving $200 million/yearTaha Steel (India) maximizes production in response to power shortages contributing $73 millionImproved production planning at Sadia (Brazil) saves $50 million over three yearsProduction optimization at Harris Corporation improves on-time deliveries from 75% to 90%Optimal traffic control of the Hanshin Expressway in Osaka saves 17 million driver hours with benefits of $320 millionBetter scheduling of hydro and thermal generating units saves Southern Company $140 million

20

From Google Search

Search phrase number of hits“optimize the supply chain” 12,000“optimize (the) return” 16,490 “optimal experience” 39,800“optimal investment” 57,000“optimal system” 63,800“optimal decision” 66,000“optimize your PC” 81,000“optimal choice” 163,000“optimal design” 285,000“optimal health” 382,000

21

On 15.053 and Optimization Tools

Some goals in 15.053: – Present a variety of tools for optimization

– Illustrate applications in manufacturing, finance, e-business, marketing and more.

– Prepare students to recognize opportunities for mathematical optimization as they arise

22

Linear Programming

minimize or maximize a linear objectivesubject to linear equalities and inequalities

Example. Max is in a pie eating contest that lasts 1 hour. Each torte that he eats takes 2 minutes. Each apple pie that he eats takes 3 minutes. He receives 4 points for each torte and 5 points for each pie. What should Max eat so as to get the most points?

Step 1. Determine the decision variablesLet x be the number of tortes eaten by Max.Let y be the number of pies eaten by Max.

23

Max’s linear program

Step 2. Determine the objective functionStep 3. Determine the constraints

Maximize z = 4x + 5y (objective function)

subject to 2x + 3y ≤ 60 (constraint)

x ≥ 0 ; y ≥ 0 (non-negativity constraints)

A feasible solution satisfies all of the constraints.x = 10, y = 10 is feasible; x = 10, y = 15 is infeasible.An optimal solution is the best feasible solution.The optimal solution is x = 30, y = 0.

24

TerminologyDecision variables: e.g., x and y. – In general, these are quantities you can control to improve

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

Constraints: e.g., 2x + 3y ≤ 24 , x ≥ 0 , y ≥ 0

– Limitations on the values of the decision variables.

Objective Function. e.g., 4x + 5y– Value measure used to rank alternatives– Seek to maximize or minimize this objective– examples: maximize NPV, minimize cost

25

MSR Marketing Inc.adapted from Frontline Systems

•Need to choose ads to reach at least 1.5 million people

•Minimize Cost

•Upper bound on number of ads of each type

TV Radio Mail Newspaper

Audience Size 50,000 25,000 20,000 15,000

Cost/Impression $500 $200 $250 $125

Max # of ads 20 15 10 15

26

Formulating as a math modelWork with your partner

1. The decisions are how many ads of each type to choose. Let x1 be the number of TV ads selected. Let x2, x3, x4 denote the number of radio, mail, and newspaper ads. These are the “decision variables.”

2. What is the objective? Express the objective in terms of the decision variables.

3. What are the constraints? Express these in terms of the decision variables.

4. If you have time, try to find the best solution.2 minuteswrite on the board the decision variablesthe objective function and the constraints

27

David’s Tool Corporation (DTC)

Motto: “We may be no Goliath, but we think big.”

Manufacturer of slingshots kits and stone shields.

28

Data for the DTC ProblemSlingshot

KitsStone Shields

Resources

Stone Gathering

time

2 hours 3 hours 100 hours

Stone Smoothing

1 hour 2 hours 60 hours

Delivery time

1 hour 1 hour 50 hours

Demand 40 30

Profit 3 shekels 5 shekels

29

To do with your partner

Take two minutes to work with your partner to formulate the DTC problem as a linear program.

Let K = number of slingshot kits madeLet S = number of stone shields made

30

Formulating the DTC Problem as an LP

Step 1: Determine Decision VariablesK = number of slingshot kits manufacturedS = number of stone shields manufactured

Step 2: Write the Objective Function as a linear function of the decision variablesMaximize Profit =

Step 3: Write the constraints as linear functions of the decision variables

subject to

31

The Formulation Continued

Step 3: Determine Constraints

Stone gathering:

Smoothing:

Delivery:

Slingshot demand:

Shield demand:

We will show how to solve this in Lecture 3.

2K + 3S <= 100K + 2S <= 60 K + S <= 50K <= 40S <= 50K >= 0 and S >= 0We ignore integrality constraints.

32

Addressing managerial problems: A management science framework

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

and 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

33

How Problems get large, and what to do.

Suppose that there are 10,000 products and 100 raw materials and processes that lead to constraints.

New technique used: write an “algebraic version of the model”

34

An Algebraic Formulation

n = number of items that are manufactured– e.g., in the previous example, n = 2;

m = number of resource constraints– e.g., m = 3, {gathering, smoothing, and

delivery.}

xj = number of units of item j manufactured

pj = unit profit from item j, e.g., p1 = 3;

dj = maximum demand for item j; e.g., bi = amount of resource i availableaij = amount of resource i used in making item j

Decision Variables

Data for objective function

Data for constraints

35

An Algebraic formulation

1=∑

n

j jj

p x

11for to

n

ij j ij

a x b i m=

≤ =∑1for toj jx d j n≤ =

0 for = 1 to jx j n≥

Maximize

subject to

36

Linear ProgramsA linear function is a function of the form:

f(x1, x2, . . . , xn) = c1x1 + c2x2 + . . . + cnxn

= ∑i=1 to n cixi

e.g., 3x1 + 4x2 - 3x4.

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

e.g., 3x1 + 4x2 - 3x4 ≥ 7x1 - 2x5 = 7

Typically, an LP has non-negativity constraints.

37

A non-linear program is permitted to have a non-linear objective and constraints.

maximize f(x,y) = xysubject to x - y2/2 ≤ 10

3x – 4y ≥ 2x ≥ 0, y ≥ 0

38

An integer program is a linear program plus constraints that some or all of the variables

are integer valued.

Maximize 3x1 + 4x2 - 3x33x1 + 2x2 - x3 ≥ 17

3x2 - x3 = 14x1 ≥ 0, x2 ≥ 0, x3 ≥ 0 and x1 , x2, x3 are all integers

39

Linear Programming Assumptions

Maximize 4 K + 3 S

2 K + 3 S ≤ 10….

Proportionality Assumption Contribution from K is proportional to K

Additivity Assumption Contribution to objective function from K is independent of S.

Divisibility Assumption Each variable is allowed to assume fractional values.

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

40

Some Success StoriesOptimal crew scheduling saves American Airlines $20 million/yr.

Improved shipment routing saves Yellow Freight over $17.3 million/yr.

Improved truck dispatching at Reynolds Metals improves on-time delivery and reduces freight cost by $7 million/yr.

GTE local capacity expansion saves $30 million/yr.

Mention Cindy Barnhart’s success at United Airlines: her fleet routing models are estimated to save United $100,000,000 per year.Our solutions have an estimated savings of $25 million per year.Mention our work with CSXMention Interfaces.

41

Other Success Stories (cont.)

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

Restructuring North America Operations, Proctor and Gamble reduces plants by 20%, saving $200 million/yr.

Optimal traffic control of Hanshin Expressway in Osaka saves 17 million driver hours/yr.

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

42

Success Stories (cont.)

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

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

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

Optimizing police patrol officer scheduling saves police department $11 million/yr.

Gasoline blending at Texaco results in saving of over $30 million/yr.

43

Summary

Answered the question: What is Operations Research & Management Science? and provided some historical perspective.

Introduced the terminology of linear programming

Two Examples:1. MSR Marketing2. David’s Tool Company

“Lecture check” problems. Section 3.1 Problems 1-4.