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 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

1 Fill in the blanks.
(1) A linear programming consists of
,
and
.
(2) The linear programming obtained by omitting all integers or 0-1
constraints on variables is called
.
(3) If constraint i of an LP is a constraint, we convert it to an
equality constrai

Integer Programming
Introduction
to
IP
Problems
The Branch-and-Bound Method for Solving IP
Implicit Enumeration
The Cutting Plane Algorithm
The Application of IP
Introduction to Integer Programming
1 Examples
Example 1 Items ( piece ) 1
Loading
Problem

Network Optimization
Basic Definitions
Minimum Spanning Tree Problems
Shortest Path Problems
Maximum Flow Problems
Basic Definitions
The Seven Bridges of Konigsberg
L.Euler (1707-1783)
Inaugurator
Theory
of
Graph
Basic Definitions
Example 1. Suppose t

3
Find all the basic feasible solutions to the following LP.
max z 2 x 1 3x 2 4 x 3 7 x 4
2 x 1 3x 2 x 3 4 x 4 8
s .t . x 1 2 x 2 6 x 3 7x 4 3
x 1 , x 2 , x 3 , x 4 0

2 Convert the following LP into standard form.
min z 3x 1 4 x 2 2 x 3 5 x 4
4 x 1 x 2 2 x 3 x 4 2
x 1 x 2 3x 3 x 4 14
s .t .
2 x 1 3x 2 x 3 2 x 4 2
x 1 , x 2 , x 3 0, x 4urs

7 The following Table lists the initial simplex tableau and the optimal
simplex for solving an max LP. Find the values of the unknown
constants in the Table.
cj 3 2
2 0
0 0
XBV x1 x2 x3 x4 x5 x6 rhs
x4 1
1 1 1 0
0
B
x5 A 1 2 0 1
0 15
x6 2
C 1 0 0
1 20
z -

10 Consider the following LP
max z 2 x 1 x 2 5 x 3 6 x 4
x 3 x 4 8
2x 1
s .t . 2 x 1 2 x 2 x 3 2 x 4 12
x 1 , x 2 , x 3 , x 4 0
*
Given the optimal dual solution is y 1* 4, y 2 1 , then use the
complementary slackness to solve the primal problem.

9 Find the dual of following LP
max z x 1 2 x 2 3x 3 4 x 4
x 1 x 2 x 3 3 x 4 5
6 x 7x 2 3x 3 5 x 4 8
s .t . 1
12 x 1 9 x 2 9 x 3 9 x 4 20
x 1 , x 2 0, x 3 0, x 4urs

4 Use the simplex algorithm to solve the following LP, indicate that
which extreme piont in the feasible region the bfs in each simplex tableau
corresponds to, and identify that which case the LP falls into.
max z 2 x 1 5 x 2
4
x1
2 x 2 12
s .t .
3x 2 x

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

1
1
4
5
6
7
3
4
5
6
3
14
2
3
2
6
9
11 4
11
7
9
5 11 8
7
8
9
5
3
10
7
11 4
9
9
6
9
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

LINEAR PROGRAMMING
ORIE 3300/5300 Optimization I
A.S. Lewis
D. Shmoys
September 29, 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 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 24, 2014
Planning, scheduling, and design problems in large organizations or engineering systems can often be modeled mathematically using variables satisfying linear equation

The Graphical Solution of Two-Variable LP
Feasible Solution
- The point satisfying the LPs constraints and all LPs sign
restrictions
Feasible Region
- The set of all feasible solution
Optimal Solution
- A point in the feasible region with the largest (s

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

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

set Profs;
set Offices;
param c cfw_Profs,Offices;
var x cfw_Profs,Offices >= 0;
param cc cfw_p in Profs, o in Offices :=
if c[p,o] >= 5 then 99 else c[p,o];
minimize unhappiness:
sum cfw_p in Profs, o in Offices cc[p,o]*x[p,o];
subject to each_prof_gets_

set Types;
param demand cfw_Types;
set Machines;
param prod_cost cfw_Machines, Types;
param days_reqd cfw_Machines, Types;
param days_avail cfw_Machines;
var x cfw_Machines, Types >=0;
minimize
total_cost:
# amount produced
sum cfw_m in Machines, t in Typ

set Profs;
set Offices;
param c cfw_Profs,Offices;
var x cfw_Profs,Offices >= 0;
minimize unhappiness:
sum cfw_p in Profs, o in Offices ?
subject to
?
each_prof_gets_an_office cfw_p in Profs:
subject to
each_office_gets_a_prof ?