20
Solving integer programs
We now have a systematic understanding of how to solve linear programs. However, we have so far developed no ideas about how to solve the kinds of integer programs that we encountered earlier. The only exception is the s

5
The tableau
Solving simple linear programs graphically suggests the importance of extreme points of the feasible region. In the last section, we related the geometric idea of an extreme point to the algebraic idea of a basic feasible solution. Ou

4
Basic feasible solutions
To summarize the main idea from the last section, a basis for an m-by-n matrix A is a list of numbers chosen from {1, 2, , . . . , n} such that the matrix AB with columns indexed by this list is invertible. The correspond

3
Linear algebra and basic solutions
To study linear programs, we make crucial use of some basic ideas from linear algebra. Before proceeding further, we quickly review some of these ideas. As usual, we use Rn to denote the vector space of column v

2
Linear programs
In this section, we discuss the ingredients of linear programs more carefully. As an example for our discussion, here is a simple linear program:
maximize
(2.1)
x1 + x2 subject to x1 2 x1 + 2x2 4 x1 , x2 0.
Any linea

22
The dual simplex method
We have seen how, when a linear program in standard equality form has a nondegenerate basic optimal solution, the optimal solution of the dual problem provides sensitivity information with respect to small changes to the

21
Scheduling problems
In this section we consider a very important class of problems involving integer variables, arising from scheduling. We suppose we have n jobs, where job j takes time pj 0 to process. We cannot process two or more jobs simul

1
Introduction
An example To illustrate the idea of linear programming, we begin with an example. Consider the simple distribution problem illustrated below.
Imagine we want to transport a total of ten pianos, from their current locations at three

Random variables A quantity whose value depends on the result of a random experiment A function that assigns a numerical value to every outcome in the sample space of a random experiment A random variable X is called discrete if its possible values f

Counting Number of ways of picking n objects from a collection if N objects is N C n = Example: In poker, what is the probability of getting (a) A full-house (b) Three-of-a-kind (c) Two pairs a. b. c. = = = = = = =0.0475
The sample space S of an exp

OR 320/520 Optimization I 10/18/07. Prof. Bland Parts of this handout are adapted from notes of Prof. A.S. Lewis.
The Transportation Problem
Read: Chapter 3 of the AMPL Book.
Recall that at the beginning of the term we discussed examples of the Ass

OR&IE 320/520 Prof. Bland
10/11/07
Balancing Return and Risk: A Simplified Example of Parametric Programmimg This problem is adapted from Bradley, Hax, and Magnanti. An investor has $5,000,000 and two potential investments. Let xj for j = 1, 2 deno

6
The simplex method
We next formalize the method we developed in the previous section. We again consider a general linear program in standard equality form: maximize cT x subject to Ax = b x 0. As before, we introduce a new variable z to keep tra

7
Finding an initial tableau
To begin the simplex method for solving a linear program in standard equality form, we need to nd an initial feasible tableau. Sometimes, this is easy. For example, the rst linear program we studied was the problem
ma

19
Complementary slackness
Over the last few sections we have seen how we can use duality to verify the optimality of a feasible solution for a linear program. If we are able to nd a feasible solution for the dual problem with dual objective value

18
Interpreting the dual
In the previous two sections we showed how to associate with any maximization linear program a dual problem, derived by considering upper bounds on the problems objective value. Duality is a powerful theoretical tool: optim

17
Duality for general linear programs
In the previous section, we introduced the powerful idea of a certicate of optimality for a feasible solution x of a linear program in standard equality form. A certicate of optimality is simply a feasible sol

16
Duality
We have now seen in some detail how we can use the simplex method to solve linear programs, and how, at termination, the simplex method provides a proof of optimality. This proof, as we have seen, consists of a tableau equivalent to the

15
Multicommodity tranportation
Linear programs in practice are often very large. Typically these massive models arise from interfacing smaller models. In this section we describe a typical example. In Section 10 we introduced a model called the tr

14
The revised simplex method
maximize cT x = z subject to Ax = b x 0,
Consider once again the standard equality-form linear program
Corresponding to any basis B is a tableau that is unique except for the order in which we write the equations. Th

13
Fundamental theorems
Now that we have made the simplex method a reliable nite algorithm, we can use it to deduce some striking properties of general linear programs. We start by asking when a linear program in standard equality form, maximize

12
Termination
In order for the simplex method to be a reliable algorithm, we must be sure that, starting from a feasible tableau, it terminates after a nite number of iterations. In this section we shall see that the smallest subscript rule ensure

10
The transportation problem
In this section we study a classical and very useful linear programming model: the transportation problem. We begin with a simple example of a transportation problem, taken from the seminal book on linear programming [

9
Degeneracy
In Section 5, we introduced the idea of a degenerate tableau, by which we mean at least one of the numbers i on the right-hand side of the body of the b tableau is zero. Thus a tableau and the corresponding basis are degenerate when on

8
Phase 1 of the simplex method
Let us summarize the method we sketched in the previous section, for nding a feasible basis for the constraint system
n
(8.1)
aij xj = bi (i = 1, 2, . . . , m)
j=1
xj 0
(j = 1, 2, . . . , n).
We can assume

OR 320/520 Optimization I Professor Bland
11/013/07
Sparsity and Packed Storage
Assume we have an m n constraint matrix A, where m and n are large, but (the fraction of entries in A that are nonzero) is very small. Then rather than store A explic

OR 320/520 Optimization I Prof. Bland
Fall 2007
A manufacturer has 80 tons of natural beef, 120 tons of natural pork, and 210 tons of recycled magazines on hand for the production of "Grandma's All Natural Sausages" during the current period. There

OR 320/520 Optimization I Professor Bland
11/29/07
Integer Linear Programming
(see BHM Ch. 9, Sections 1, 3, 4, 5, 8)
n
Minimize
j=1 n
cj x j aij xj = bi (i = 1, . . . , m)
j=1
s.t. (ILP)
xj 0 and integer (j = 1, . . . , n).
(May maximize ra