Chapter 2 - Introduction to Optimization & Linear Programming : S-1
Chapter 2
Introduction to Optimization & Linear Programming
1.
If an LP model has more than one optimal solution it has an infinite number of alternate optimal
solutions. In Figure 2.8, t

A Contract Award Problem
B&G Construction has 4 building projects and
can purchase cement from 3 companies for the
following costs:
Co.1
Co.2
Co.3
Needs
(tons)
Max.
CostperDeliveredTonofCement
Project1 Project2 Project3 Project4 Supply
$120
$115
$130
$12

Sensitivity Analysis in a
Spreadsheet
Chapter 4
Introduction
When solving an LP problem we assume
that values of all model coefficients are
known with certainty.
Sensitivity analysis helps answer questions
about how sensitive the optimal solution is to
ch

Chapter 7
Goal Programming and
Multiple Objective
Optimization
1
Introduction
Most of the optimization problems considered
to this point have had a single objective.
Two other modeling techniques:
Goal Programming (GP): no one specific objective
function

The Shortest Path Problem (Sec. 5.2)
Many decision problems boil down to
determining the shortest (or least costly) route
or path through a network.
Ex. Emergency Vehicle Routing
This is a special case of a transshipment
problem where:
There is one su

Sensitivity Analysis (III) and
the Simplex Method
1
The Simplex Method
To use the simplex method, we first convert all
inequalities to equalities by adding slack
variables to <= constraints and subtracting
slack variables from >= constraints.
For example

Lecture 19 Difference Equations (II)
The time path of the solution to a difference equation
The general solution consists of the sum of two components:
yt 1 ayt h t
First-order case:
Solution: yt = complementary function + particular solution
yt yc y p
y

Sensitivity Analysis (II)
1
Sensitivity Report
Report Created: 2012/2/24 ? ? 12:45:04
Engine: Standard LP/Quadratic
Objective Cell (Max)
Cell
Name
$D$6 Unit Profits Total Profit
Final Value
66100
Decision Variable Cells
Final
Cell
Name
Value
$B$5 Number t

Chapter4
Sensitivity Analysis (I)
1
Introduction
When solving an LP problem we assume
that values of all model coefficients are
known with certainty.
Such certainty rarely exists.
Sensitivity analysis helps answer
questions about how sensitive the opti

QMDS300
Quantitative Decision
Analysis
1
Chapter1(Ragsdale)
Introduction to Modeling
and Decision Analysis
2
Introduction
We face numerous decisions in life &
business.
We can use computers to analyze the
potential outcomes of decision
alternatives.
Sprea

Chapter3,Part1
Modeling and Solving LP
Problems in a Spreadsheet
Introduction
Solving LP problems graphically is only
possible when there are no more than two
decision variables
Few real-world LP have fewer than two
decision variables
We can now use sp

Managerial Decision
Modeling
6th edition
Cliff T. Ragsdale
Chapter7
Goal Programming and
Multiple Objective
Optimization
Introduction
Most of the optimization problems considered to
this point have had a single objective.
Often, more than one objective

Chapter6
Integer Linear Programming
1
Introduction
When one or more variables in an LP problem
must assume an integer value we have an
Integer Linear Programming (ILP) problem.
ILPs occur frequently
Scheduling workers
Manufacturing airplanes
Integer

Chapter2
Introduction to Optimization
and Linear Programming
1
Introduction
We all face decision about how to use
limited resources such as:
Oil in the earth
Land for dumps
Time
Money
Workers
2
Mathematical Programming.
MP is a field of management

Chapter3,Part3
More Examples on Modeling
and Solving LP Problems in a
Spreadsheet
1
Benchmarking schools
Outputs
n
1
2
3
4
School
A
B
C
D
1
86
82
81
81
2
75
72
79
73
I nputs
3
71
67
80
69
1
0.06
0.05
0.08
0.06
2
260
320
340
460
3
11.3
10.5
12
13.1
2
Steak

Lecture 7
Solving nonlinear equations
Solving nonlinear equations
Given a cubic equation
3 42 4 17 0 , one way to determine the
3
2
real roots of the above equation is to plot f ( ) 4 4 17
against :
As shown in the graph below, the curve crosses the hor

Lecture 5 - 6
Elementary Linear Algebra (III)
Quadratic Forms
For functions in quadratic form, the terms are squares of variables or products of
two variables. Here are some examples:
f ( x, y ) 2 x 2 y 2 3xy
g ( x, y ) x 2 y 2 2 xy
f1 ( x1 , x2 , x3 ) 3

Lecture 3 - 4
Elementary Linear Algebra (II)
Eigenvectors and Eigenvalues
Definition
Given a square matrix A , if there exist a vector x and scalar , such that
Ax x, then x is said to be an eigenvector of A and is the corresponding
eigenvalue of A.
Exampl

Lecture 15-17
Optimization with equality constraints
Supplementary Example 1: (Two-variable case)
Optimize the utility function u f x, y xy 2 x subject to the budget constraint
4 x 2 y 60 .
Step (1):
rewrite the objective function as the Lagrangian functi

Lecture 2
Elementary Linear Algebra (I)
Linear Combination of Vectors
Generally, a vector W R n is said to be a linear combination of the vectors
V1 , V2 , , Vm if
W 1V1 2V2 mVm , where i are scalars and not all of them 0.
Example 1
1 4 5
2 2 4
5

Lecture 1
A Review of Matrix Algebra
Matrix
A system of equations being written in an abbreviated form as a rectangular array of numbers enclosed
by brackets is called a matrix (pl. matrices). Each number in the array is called an element or entry of
the

Lecture 8-9
Numerical
approximation:
Taylor
series
and
Maclaurin series; L' H o pital ' s Rule
Functional Approximation using Polynomials
Taylor Polynomial
For infinitely differentiable functions such as polynomial, rational,
trigonometric, logarithmic an

Lecture 10
Review on partial differentiation
Partial Differentiation
Given a function of two variables z f ( x , y ) , we can perform differentiation on f
by keeping one of the two independent variables constant. This process is known
as partial different

The Maximal Flow Problem
In some network problems, the objective is to
determine the maximum amount of flow that can
occur through a network.
The arcs in these problems have upper and
lower flow limits.
Examples
How much water can flow through a netwo

Chapter3,Part2
More Examples on Modeling
and Solving LP Problems in a
Spreadsheet
1
A Blending Problem:
The Agri-Pro Company (Sec.3.12)
Agri-Pro has received an order for 8,000 pounds of
chicken feed to be mixed from the following feeds.
Percent of Nutri

Jack Potts
Jack Potts
Investment
A
B
C
D
E
M1
M2
M3
Amount
Invested
$500,000
$275,685
$0
$500,000
$429,921
$70,079
$50,000
$500,000
Totals:
Min
$0
$0
$0
$0
$0
$50,000
$50,000
$50,000
Max
$500,000
$500,000
$500,000
$500,000
$500,000
$500,000
$500,000
$500,

Chapter 6
Integer Linear Programming
1
Introduction
When one or more variables in an LP problem
must assume an integer value we have an
Integer Linear Programming (ILP) problem.
ILPs occur frequently
Scheduling workers
Manufacturing airplanes
Integer