Friday 17:00-19:45

405

BUS 557

Mathematical Programming

What will this class be about?• Modeling of Optimization Problems

– Linear Programming– Transportation Problems– Network Models– CPM-PERT– Integer Programming

• Mathematical Structure of Linear Models

– Geometric– Algebraic

• Techniques for Solution and Analysis • Modeling Languages and Solvers

What are the goals for the course?

After this course, you should be able to:

• Given an optimization problem, formulate an appropriate linear model.

• Use a modeling language and/or commercial solver to solve the model.

• Understand the basic mathematical structure of the model.

• Understand the techniques used to solve the model.

• Analysis the model.

Course Requirements

• Attendance

• Participation

• Reading and Presentation

• Homework

• Exams

Homework and Presentation

• There will be approximately 7 problem sets. It will be solved as a hardcopy.

• Homework is due at the beginning of Friday’s class each week.

• You will be given an essay in order to presentation.

Grading• Your grade will correspond to your learning and understanding of the course material.

• Some areas to keep in mind

– Good proof technique– Accurate self-assessment– Class participation

• Weighting

– 30% Midterm Exam– 20% Homework, Participation, Presentation– 50% Final Exam

Textbook

Render, B., Stair, M.R., Hanna, E.M.(2009), Quantitative Analysis for Management, 10th Edition, Prentice-Hall, Inc.

Anderson, R.D., Sweeney, J.D., Williams, A.T., Martin, K.(2008), Quantitative Methods for Business, Thomson Higher Education.

Taha, H. A. (2007), Operations Research: An Introduction, 8th Edition, Prentice-Hall, Inc.

Essays

RememberEquations

Inequalities

Rectangular Coordinate Systems

Matrix(Gaussian Elimination Method)

Equations

An equation states the equality of two algebraic expressions. The algebraic expressions may be stated in terms of one or more variables.

The solution of an equation consist of those numbers which, when substituted for the variables, make the equation true. The numbers, or values of the variables, which make the equation true are referred to as the roots of the equations.

First Degree Equations in One Variable

First Degree Equations in two Variable

- The Elimination Method- Substitution Method

IntervalA subset of the real line is called an interval if it

contains at least two numbers and also contains all real numbers between any two of its element.

x<3 is an interval2<x<5 is an intervalx=5 is not an interval

If a and b are real numbers and a<b,

1- The open interval from a to b, denoted by (a,b), consisting of all real numbers x satisfying a<x<b.

2- The closed interval from a to b, denoted by [a,b], consisting of all real numbers x satisfying a≤x≤ b.

3- The half open interval from a to b, denoted by [a,b), consisting of all real numbers x satisfying a≤ x< b.

4- The half open interval from a to b, denoted by (a,b], consisting of all real numbers x satisfying a<x≤ b.

this intervals are illustrated as follows;….

InequalitiesThe order properties of the real numbers are

summarized in the following rules for inequalities;

If a, b and c are real numbers, then;

1- a<b a+c<b+c

2- a<b a-c<b-c

3- a<b, c>0 a.c<b.c

4- a<b, c<0 a.c>b.c

5- a>0

6- 0<a<b or a<b<0

10

a

1 1

a b

Solve the following inequalities

1-

2-

3-

4-

5-

6-

2 3 3x x

2 12

xx

25

1x

5 2 1 11x

3 1 5 3 2 15x x x

30

5

x

x

If x is a real number, then;

a)

b)

If x,y are real numbers, ,solve the following,

a) x+y b) x-y c) 2x+3y

d) x.y e)

23 7 .... ....x x

23 7 .... ....x x

-3<x<7 and 2<y<5

2 2x y

Solve the inequalities,

2

2

2

2

2

1) x <9

2) 1<x <9

3) x-3 <16

4) x -3x 0

5) x -2x-3<0

RECTANGULAR COORDİNATE SYSTEMS

The axes divide the coordinate plane into four

quadrants.

GAUSSIAN ELIMINATION METHOD

The Gaussian elimination method begins with the original system of equations and transforms it, using row operations, into an equivalent system from which the solution may be read directly.

Gaussian elimination transformation for 2x2 systems.

1 1 1 1 1

2 2 2 2 2

a x b x c

a x b x c

1 1 1

2 2 2

1 0

0 1

x x v

x x v

1 1

2 2

x v

x v

Original system Transformed system is the solution set 1 2,v v

Basic Row Operations

1- Both sides of an equation may be multiplied by a nonzero constant.2- Nonzero multiples of one equation may be added to another equation.3- The order of equations may be interchanged.

Example:

Solve the following system of equations by the Gaussian elimination method.

5 20 25

4 7 26

x y

x y

Example-2

0563

1342

9211

0563

17720

9211

third theto rowfirst the times3- add

second theto rowfirst the times2- add

271130

17720

9211

271130

10

9211

217

27

third theto row second the

times3- add

2

1by row

second emultily th

23

21

217

27

00

10

9211

3100

10

9211

217

27

first theto row second

the times1- Add

2-by row thirdeMultily th

3100

10

01

217

27

235

211

3100

2010

1001

second thetorow third thetimes andfirst theto

row thirdthe times- Add

27

211

The solution x=1,y=2,z=3 is now evident.

Examples1-

2-

3-

3 2 6

15 10 30

x y

x y

6 12 24

1.5 3 9

x y

x y

2 3 7

4

x y

x y

n-Variable Systems, n≥3

Graphical analysis for three-variable systems. Gaussian elimination procedure for 3x3 systems.

Example:

1 2 3

1 2 3

1 2 3

6

2 3 4

4 5 10 13

x x x

x x x

x x x

Selected Applications

Product Mix problem

A company produces three products, each of which must be processed through three different departments. The following the table summarizes the hours required per unit of each product in each department. In addition, the weekly capacities are stated for each department in terms of work-hours available. What is desired is to determine whether there are any combinations of the three products which would exhaust the weekly capacities of the three departments.

Department

ProductHours

Available per week1 2 3

A 2 3,5 3 1.200

B 3 2,5 2 1.150

C 4 3 2 1.400