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Fri day 17:00-19:45 405

BUS 557 Mathematical Programming. Fri day 17:00-19:45 405. 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

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Fri day 17:00-19:45 405

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  1. BUS 557 Mathematical Programming Friday 17:00-19:45405

  2. 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

  3. 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.

  4. Course Requirements • Attendance • Participation • Reading and Presentation • Homework • Exams

  5. 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.

  6. 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

  7. 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

  8. Remember • Equations • Inequalities • Rectangular Coordinate Systems • Matrix(GaussianEliminationMethod)

  9. 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.

  10. First Degree Equations in One Variable • First Degree Equations in two Variable - The Elimination Method - Substitution Method

  11. Interval • A 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 interval • 2<x<5 is an interval • x=5 is not an interval

  12. 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;….

  13. Inequalities • The 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 

  14. Solve the following inequalities 1- 2- 3- 4- 5- 6-

  15. 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)

  16. Solve the inequalities,

  17. RECTANGULAR COORDİNATE SYSTEMS Theaxesdividethecoordinateplaneintofourquadrants.

  18. 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. GAUSSIAN ELIMINATION METHOD Original system Transformed system is the solution set

  19. 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.

  20. Example-2

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

  22. 1- 2- 3- Examples

  23. Graphical analysis for three-variable systems. Gaussian elimination procedure for 3x3 systems. Example: n-Variable Systems, n≥3

  24. 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. Selected Applications

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