IE521 Advanced Optimization Example Set 2
Fall 2011 1. Consider a linear programming problem in the standard form, described in terms of the following initial tableau: The entries , , , , in the tableau are unknown parameters. 0 2 3 0 0 0 1 0 1 0 0 0 0 1
IE521 Advanced Optimization Lecture 9
Dr. Zeliha Akca
December 2011
1 / 31
Sensitivity Analysis
Sensitivity analysis investigates how the optimal solution is sensitive to the parameters of the linear program. Parameters are objective function coefficient
IE521 Advanced Optimization AMPL Exercises 1
Fall 2011 1. A group of young entrepreneurs earns a (temporarily) steady living by acquiring inadequately supervised items from electronics stores and re-selling them. Each item has a street value, a weight, an
IE521 Advanced Optimization Example Questions
Fall 2011 1. Consider a road divided into n segments that is illuminated by m lamps. Let pj be the power of the j th lamp. The illumination Ii of the ith segment is assumed to be m aij pj j=1 where aij are kno
IE521 Advanced Optimization Lecture 10
Dr. Zeliha Akca
December 2011
1 / 27
Integer Programming
An integer programming model is a mathematical programming model in which some decision variables are integers. Note that we are interested in integer program
IE521 Advanced Optimization Lecture 3
Dr. Zeliha Akca
October 2011
1 / 14
Reading
Bertsimas 1.5 Bertsimas 2.1 - 2.4
2 / 14
Problem of the Week
A coal production company produces coal for 4 customers using 3 mines. For each mine, the unit production cost,
IE521 Advanced Optimization Lecture 2
Dr. Zeliha Akca
October 2011
1 / 48
Previous Lecture: Recall
System? Modeling? Model types? Mathematical programming? min veya max s.t. f (x1 , x2 , ., xn ) gi (x1 , x2 , ., xn ) bi (x1 , x2 , ., xn ) X
Decision Var
IE521 Advanced Optimization Lecture 4
Dr. Zeliha Akca
October 2011
1 / 23
Reading
Bertsimas 2.3 - 2.7
2 / 23
Problem of the Week
RADIATION TREATMENT
Radiation therapy is a technique used in the treatment of most cancer. Radiation is depositted to the par
IE521 Advanced Optimization Lecture 5
Dr. Zeliha Akca
October 2011
1 / 22
Reading
Bertsimas 3.1-3.3.
2 / 22
Recall
There is a one-to-one correspondence between the extreme points of a polyhedron and the basic feasible solutions. Constructing a basic solu
IE521 Advanced Optimization Lecture 6
Dr. Zeliha Akca
November 2011
1 / 12
Reading
Bertsimas 3.3, 3.5.
2 / 12
Recall
Feasible and Improving Direction: d Rn is feasible direction if ^ + d P for some R+ . x d Rn is an improving direction if c d < 0. Constr
IE521 Advanced Optimization Lecture 8
Dr. Zeliha Akca
December 2011
1 / 23
Hw3 and Midterm 1 Grades
Hw3 is posted. Midterm 1 was graded. Max grade = 86 Average = 44.3
2 / 23
Recall Previous Lectures
Example: For the following LP, x1 and x2 are basic vari
IE 521 Advanced Optimization Lecture 1
Dr. Zeliha Akca
September 2011
1 / 31
Reading
Lecture covers Bertsimas Sections 1.1, 1.2, 1.4 and some other resources. P. E. Gill, W. Murray, M. Saunders, J. Tomlin, M. Wright, "George B. Dantzig and Systems Optimi
Advanced Optimization AmplPart1
Zeliha Akca
What is Ampl?
A modeling language that helps us to develop and apply mathematical programming models. Can solve linear, nonlinear, integer, mixed integer models. Can write any piece of code that you can write i
IE521 Advanced Optimization Lecture 11
Dr. Zeliha Akca
December 2011
1 / 28
Integer Programming Models
The optimal solution is a solution that gives the best objective function value. If minimization problem, a feasible but not optimal solution is an upp
Optimization Techniques Ampl2
Sets, Writing Models and Data Files Using Sets
Zeliha Akca
Product Mix Example:
3 products: TV sets, stereos and speakers. 5 different parts: Chassis, picture tubes, speaker cones, power supplies and electronics units. Pro
IE521 Advanced Optimization Lecture 12
Dr. Zeliha Akca
January 2012
1 / 23
Large scale Linear Programming
Linear programs occurring in practice can be extremely large. They may have a large number of variables or constraint. For large LPs, the constraint
Microsoft Excel 14.0 Answer Report Worksheet: [Workbook1]Sheet1 Report Created: 12/10/2011 5:28:00 PM Result: Solver found a solution. All constraints and optimality conditions are satisfied. Solver Engine Engine: Simplex LP Solution Time: 0.636929 Second
1. First, multiply each constraint with -1 in order to get a positive 1 coefficient for x3 and x4. With this we can use x3 and x4 as basic variables. Initial table can be constructed as follows. All reduced costs are postivie, therefore dual feasible. How
IE521 Advanced Optimization Example Set 3
Fall 2011
1. Consider the tableau given below for a maximization problem. Give conditions on the unknowns a1 , a2 , a3 , b, c that make the following statements true. 10 4 1 b -c -1 a2 a3 2 a1 -4 3 0 1 0 0 0 0 1 0
IE521 Advanced Optimization HW2
Due November 16 Fall 2011
1. In a school trip there are students with age 7 to 17. The average age of the students is 14. Give an LP formulation in order to find the minimum number of students who are older than 14 years-ol
IE521 Advanced Optimization Fall 2011, HW3
Due December 14
1. (20 points) Write the g(p) function for the following linear programs (function g(p) is the optimal solution to the LP in which the constraints are relaxed by paying a price p). Simplify the fu