ORIE 3300/5300 - Fall 2010
Solutions for Recitation 1
1. When we resolve transportation.mod with the right hand side of the rst constraint as 3 instead of 2, AMPL returns: MINOS 5.5: infeasible problem. 1 iterations This is because by changing the rst con

Copyright 2003 by Robert Fourer, David M. Gay and Brian W. Kernighan
15
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Network Linear Programs
Models of networks have appeared in several chapters, notably in the transportation
problems in Chapter 3. We now return to the formulation of these models

6 You have decided to enter the candy business. You are considering
producing two types of candies: Slugger Candy and Easy Out Candy,
both of which consist solely of sugar, nuts, and Chocolate. At present,
you have in stock 100 oz of sugar, 20 oz of nuts,

Copyright 2003 by Robert Fourer, David M. Gay and Brian W. Kernighan
1
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Production Models:
Maximizing Profits
As we stated in the Introduction, mathematical programming is a technique for solving certain kinds of problems notably maximizing profits and

The Further Discussion of the
Simplex Algorithm
How to solve the following problem?
min z 2 x 1 3x 2
0.5 x 1 0.25 x 2 4
x 1 3x 2 20
s.t.
x1
x 2 10
x 1 , x 2 0
(1) The Big M
Method
(2) The Two-Phase
Simplex Method
How to spot an infeasible LP?
The Furt

The Simplex Method for UpperBounded Variables
1 Definitions
Upper-Bound Constraint
- A constraint of the form xiui (where ui is a constant), providing
a upper bound on xi.
Upper-Bound Substitution
- For upper-bound constraint xiui, the relationship i
x
wh

LINEAR PROGRAMMING
ORIE 3300/5300 Optimization I
A.S. Lewis
D. Shmoys
October 2, 2014
Planning, scheduling, and design problems in large organizations or engineering systems can often be modeled mathematically using variables satisfying linear equations a

Exercises
1. Suppose that in solving a max LP, we obtain the tableau in the
following table, in which there is no artificial variable and a1, a2,
a3, d, c1, c2 are unknown constants. Determine the values of the
constants in order that the following conclu

The Application of LP
Problem 1. A Work-Scheduling Problem
A post office requires different numbers of full-time employees on different
days of the week. The number of full-time employees required on each
day is given in Table 1. Union rules state that ea

The Computer Package of LP
1 LINDO
- To Solve LP, IP and QP
2 LINGO
- To Solve LP and NLP
3 MATLAB - LINPROG
- To Solve LP, IP, QP, Multi-Goal Programming and NLP
The Computer Package of LP-LINDO
Using LINDO to Solve LP
Input Given Data
Solution
Press But

The Simplex Algorithm
The Standard Form of LP
1. Definition
Standard Form - An LP is said to be in standard form if its all
constraint are equations and all variables are nonnegative
max z c1 x 1 c 2 x 2 c n x n
a 11 x 1 a 12 x 2 a 1n x n b1
a 21 x 1 a

Duality and Sensitivity Analysis
Matrices Description of the Simplex Algorithm
Revised Simplex Algorithm
Finding the Dual of an LP
Economic Interpretation of the Dual Problem
The Duality Theorem
Shadow Prices
The Dual Simplex Method
Sensitivity Analysis
M

Finding the Dual of an LP
Normal Max (Min) Problem
A max (min) problem in which all variables are required
to be nonnegative and all constraint are ( ) constraints.
Primal Problem
max Z CX
AX b
s.t.
X 0
Dual Problem
Dual Problem
min w Yb
YA C
s.t.
Y

The Revised Simplex Algorithm
1. The Product Form of the Inverse
Suppose that we are solving an LP with m constraints. Assume that we
have found that xk should enter the basis, in row r.
Let the column for xk in the current simplex tableau be
a1k
a2 k

The Dual Simplex Method
1. Definitions
Max Problem
Simplex Method
- A method that maintains a primal feasibility (because each
constraint in the initial tableau has a nonnegative right-hand
side), and obtains an optimal solution when dual feasibility (a
n

Sensitivity Analysis
1 Principle of Sensitivity Analysis
Sensitivity analysis is concerned with how changes in an
LPs parameters affect the LPs optimal solution.
BV
XBV
XBV
I
z
0
XNBV
B-1N
CBV B-1N- CNBV
BV (for a max problem)
is optimal
rhs
B-1b
CBV B-1b

Assignment problem
1 The Mathematical Model
Example 1 The personal department want to assign 4 different tasks to 4
persons,
each person can perform only one task. The assessment score (100 scale) of the 4
persons is given in the Table 1 below , how to as

LINEAR PROGRAMMING
ORIE 3300/5300 Optimization I
A.S. Lewis
D. Shmoys
October 2, 2014
Planning, scheduling, and design problems in large organizations or engineering systems can often be modeled mathematically using variables satisfying linear equations a

LINEAR PROGRAMMING
ORIE 3300/5300 Optimization I
A.S. Lewis
D. Shmoys
August 26, 2014
Planning, scheduling, and design problems in large organizations or engineering systems can often be modeled mathematically using variables satisfying linear equations a

LINEAR PROGRAMMING
ORIE 3300/5300 Optimization I
A.S. Lewis
D. Shmoys
August 26, 2014
Planning, scheduling, and design problems in large organizations or engineering systems can often be modeled mathematically using variables satisfying linear equations a

LINEAR PROGRAMMING
ORIE 3300/5300 Optimization I
A.S. Lewis
D. Shmoys
September 2, 2014
Planning, scheduling, and design problems in large organizations or engineering systems can often be modeled mathematically using variables satisfying linear equations

LINEAR PROGRAMMING
ORIE 3300/5300 Optimization I
A.S. Lewis
D. Shmoys
September 8, 2014
Planning, scheduling, and design problems in large organizations or engineering systems can often be modeled mathematically using variables satisfying linear equations

LINEAR PROGRAMMING
ORIE 3300/5300 Optimization I
A.S. Lewis
D. Shmoys
September 8, 2014
Planning, scheduling, and design problems in large organizations or engineering systems can often be modeled mathematically using variables satisfying linear equations

LINEAR PROGRAMMING
ORIE 3300/5300 Optimization I
A.S. Lewis
D. Shmoys
September 4, 2014
Planning, scheduling, and design problems in large organizations or engineering systems can often be modeled mathematically using variables satisfying linear equations

LINEAR PROGRAMMING
ORIE 3300/5300 Optimization I
A.S. Lewis
D. Shmoys
September 17, 2014
Planning, scheduling, and design problems in large organizations or engineering systems can often be modeled mathematically using variables satisfying linear equation

LINEAR PROGRAMMING
ORIE 3300/5300 Optimization I
A.S. Lewis
D. Shmoys
September 8, 2014
Planning, scheduling, and design problems in large organizations or engineering systems can often be modeled mathematically using variables satisfying linear equations

Fall 2014 ORIE 3300/5300
OPTIMIZATION I (Linear programming)
Instructor: Adrian Lewis
Course outline
Planning, scheduling, and design problems in large organizations or engineering systems
can often be modeled mathematically using variables satisfying lin

1
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5 11 8
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8
This is a puzzle from a flight
magazine. We want to fill in
this 7 by 7 array so that
each entry is an integer from
1 to 7. The entry for (row 6,
column 3) must be 5, and

Consider the following LP:
max
x1
s.t.
2x1
x1
+ x2
+ x2
, x2
2
0
There are only two variables, so visualizing the geometry is easy.
Lets draw the feasible region on the board.
1
Now consider
max x1
s.t.
x1
x1
x1
+
x2
+ 2x2
,
x2
2
4
0
2
Here, again, is the

In the previous lecture,
we recalled how to compute the inverse by applying elementary row operations.
2
matrix
to be
inverted
1
4 2
1
2
1
4 0
0
2
1
2
2
5
0
3
3
8
identity
matrix
1
0
0
3
0
1
0
0
0 5
1
Add -2 times first row to second,
and -1 times second