ORIE 3300/5300 - Fall 2010
Solutions for Recitation 3
Problem 1-4 in AMPL book (a) First, we have to know what products (types of cars) we are going to produce. We will declare this set as set P; Then, for each car type we need: time it takes to produce t

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

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
1
_
_
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

Copyright 2003 by Robert Fourer, David M. Gay and Brian W. Kernighan
15
_
_
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,

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

5 Use the Big M method and the two-phase simplex method to solve the
following LP respectively.
max z 10 x 1 15 x 2 12 x 3
5 x 1 3x 2 x 3 9
5 x 1 6 x 2 15 x 3 15
s .t .
2 x x 2 x 3 5
1
x 1 , x 2 , x 3 0

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

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

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

12 Use the simplex algorithm to solve the following LP
max z 5 x 1 5 x 2 13 x 3
x 1 x 2 3x 3 20
s .t . 12 x 1 4 x 2 10 x 3 90
x 1 , x 2 , x 3 0
And then using the resulted optimal tableau answer the following
questions
(1) If the right-hand side of the

14 Use the implicit enumeration to solve the following 0-1 IP
min z 4 x 1 3x 2 2 x 3
2 x 1 5 x 2 3x 3 4
4 x x 2 3x 3 3
s .t . 1
x 2 x 3 1
x 1 , x 2 , x 3 0 or 1.

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

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
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

Conclusion: invest in the constraint giving the highest profit rise.
Incremental change does not change my optimal solution set:
Each resource has a shadow price yi .
The optimal profit is:
z() = z(0) + yT
4

Weak duality theorem: y T b = y T Ax cT x
y b cT x
Weak duality: valmin valmax
Strong duality: If either value is finite, then valmin = valmax
5.
Game Theory
Row-column player game in a m x n matrix.
Row player has m strategies, column player has n numb

3.
Tableu Method
ck
yT
z
Ak
A1
J
b
figure out the blocking variable
? y T = cTJ A1
J
? ck = ck y T Ak
determine min(ratio b/Ak )
? watch out! Ak = A1
J Ak
? if min occurs at multiple variables, choose any
pivot Ak [watch out! not Ak ] into identity and

Optimal value in terms of y:
T 1
z = cT x = cTJ b = cTJ (A1
T b = z
J b) = (cJ AJ )b = y
Set is optimal when all costs are non-negative:
cT = cT yT A 0
generealizing this to any vector y
it becomes the dual constraint: cT y T A
ck = ck yT Ak 0 for all k
m

1.
Preliminaries
AJ - set, b - right-hand vector, x - solution
Which variables are basic?
? xj where for each index j J
Which variables are non-basic?
? xj = 0 where for each index j
/J
Is the set basic?
? Check if invertible = det(AJ ) 6= 0
Is the