3. T2 - 3
Optimization is derived from the Latin
word “optimus”, the best.
Optimization characterizes the activities
involved to find “the best”.
People have been “optimizing” forever,
but the roots for modern day optimization
can be traced to the Second World War.
Introduction
4. T2 - 4
Operational Research
Operational Research originated from the
activities performed by multidisciplinary
teams formed in the British armed forces
involved in solving complex strategic and
tactical problems in World War II.
Waddington describes the main objectives of
the Operational Research Section in the
British armed forces as
“The prediction of the effects of
new weapons and tactics.”
5. T2 - 5
Motivation for Operational Research
Many problems associated with the
Allied military effort were simply too
complicated to expect adequate
solutions from a single individual, or
even a single discipline.
Due to the diversity of the membership,
one of the earliest groups in Britain
became known as “Blacket's circus”.
They were very successful. Any idea
why?
6. T2 - 6
OR
All persons selected were talented men + wartime
pressure + synergism generated from the
interactions of different disciplines
Due to their success, other allied nations adopted
the same approach.
Because the work assigned to these groups were
in the nature of military operations their work
became known as operational research in the
United Kingdom and as operations research in
the United States.
The abbreviation OR is commonly used for both
operational research and operations research.
Wartime examples: radar deployment, anti-
aircraft fire control, fleet convoy sizing, submarine
detection.
7. T2 - 7
Operational Research
Operational Research was defined by the Operational
Research Society of Great Britain as follows:
"Operational research is the application of the methods of
science to complex problems arising in the direction and
management of large systems of men, machines, materials
and money in industry, business, government, and defense.
The distinctive approach is to develop a scientific model of
the system, incorporating measurements of factors such as
chance and risk, with which to predict and compare the
outcomes of alternative decisions, strategies or controls.
The purpose is to help management determine its policy
and actions scientifically."
8. T2 - 8
Current
Definition
“Operational Research ("OR"), also
known as Operations Research or
Management Science ("OR/MS")
looks at an organisation's operations
and uses mathematical or computer
models, or other analytical
approaches, to find better ways of
doing them.”
9. T2 - 9
Operations Research
• Although the US effort started later than the British, it produced more
fundamental advances in the mathematical techniques for analyzing
military problems.
The Operations Research Society of America
had the following definition:
“Operations research is concerned with
scientifically deciding how to best design
and operate man-machine systems,
usually under conditions requiring the
allocation of scarce resources”.
10. T2 - 10
IFORS
“Operational Research can be described as a
scientific approach to the solution of problems in the
management of complex systems.
In a rapidly changing environment an understanding
is sought which will facilitate the choice of more
effective solutions which, typically, may involve
complicated interaction among people, materials, and
money.”
IFORS - International Federation of Operational
Research Societies (www.ifors.org):
11. T2 - 11
After the War...
Many of the scientists in the OR groups turned their
activities to applying their approach to civilian problems.
Some returned to universities to develop a sound
foundation for the hastily developed techniques, others
concentrated on developing new techniques.
First civilian organizations interested were large profit
making corporations. For example, petroleum
companies were the first to use linear programming on a
large scale for production planning.
First only big business could afford it.
Applications in the service industries did not start until
the mid 1960s.
One concurrent technological development has been
critical for OR...
12. T2 - 12
Electronic Computers
It is generally accepted that without computers, OR and
optimization would not be what they are today.
Earlier mathematical models (such as calculus, Lagrange
multipliers) relied on sophistication of technique to solve
the problem classes for which they were suited.
Methods of mathematical optimization (e.g., Linear
Programming) rely far less on mathematical sophistication
than they do on an unusual “adaptibility to the mode of
solution inherent in the modern digital computer”.
Particularly striking is the simplicity of these methods of
mathematics coupled with their iterative processes (i.e.,
the repeated performance of a relatively simple set of
operations)
13. T2 - 13
The Computer and Linear Programming
Consider the case of Linear Programming (more in later
lectures):
The first large scale computer became a practical reality in
1946 at the University of Pennsylvania.
This was just one year before the development of simplex.
The simplex method for linear programming consists only of a
few steps and these steps require only the most basic
mathematical operations which a computer is well suited to
handle.
However, these steps must be repeated over and over before
one finally obtains an answer.
The first successful computer solution of a LP problem was in
January 1952 on the National Bureau of Standards SEAC
computer.
14. T2 - 14
By the way...
You will often hear the phrase “programming” as in:
mathematical programming,
linear programming,
nonlinear programming,
mixed integer programming, etc.
15. T2 - 15
By the way ... (cont.)
This has (in principle) nothing to do with modern
day computer programming.
In the early days, a set of values
which represented a solution to a
problem was referred to as a
“program”.
However, nowadays you program (software) to
find a program!
16. T2 - 16
Types of Optimization
Do not forget:
Optimization methods fall in the category of
“decision support systems/methods”
Note: There are MANY different
optimization methods/algorithms
However, they are can be grouped by
fundamental principles of:
model, or
solution method/algorithm
17. T2 - 17
Learning Objectives
Explain linear programming
Formulate linear programming
problems
Solve linear programming problems
using graphical methods
Corner point
Iso-profit line
18. T2 - 18
What Is
Linear Programming?
Mathematical technique
Not computer programming
Allocates scarce resources to
achieve an objective
Pioneered by George Dantzig in
World War 2
Developed workable solution in 1947
Called Simplex Method
19. T2 - 19
Linear Programming
Applications
Find product mix given machine &
labor hours to maximize profit
Schedule production given demand
to minimize costs
Allocate police given limited patrol
cars to minimize response time
Plan menus given minimum daily diet
requirements to minimize cost
20. T2 - 20
Linear Programming
Requirements
One objective
Maximize or minimize
Constraints (e.g., limited resources)
Alternative courses of action
(decision variables)
Divisible (fractional values)
Non-negative
Linear relationships
27. T2 - 27
Formulation Solution
Summary
Max Z = 4X1 + 5X2
subject to:
1X1 + 2X2 40
4X1 + 3X2 120
X1, X2 0
28. T2 - 28
Graphical Solution
Method Steps
Draw graph with vertical & horizontal
axes (1st quadrant only)
Plot constraints as lines, then as planes
Use (X1,0), (0,X2) for line
Find feasible region
Find optimal solution
Corner point method
Iso-profit line method
33. T2 - 33
0
10
20
30
40
0 10 20 30 40
X1
X2
Graphical Solution
Inequality is a plane.
To find which side of
line inequality applies,
test any 2 points in
1X1 + 2X2 40
39. T2 - 39
Graphical Solution
Inequality is a plane.
To find which side of
line inequality applies,
test any 2 points in
4X1 + 3X2 120
0
10
20
30
40
0 10 20 30 40
X1
X2
42. T2 - 42
Graphical Solution
Inequality is satisfied
by all points BELOW
the line.
0
10
20
30
40
0 10 20 30 40
X1
X2
43. T2 - 43
Graphical Solution
Feasible region is
intersection of all
planes. It satisfies
all constraints.
0
10
20
30
40
0 10 20 30 40
X1
X2
44. T2 - 44
Graphical Solution
Feasible region is
intersection of all
planes. It satisfies
all constraints.
Feasible region
0
10
20
30
40
0 10 20 30 40
X1
X2
45. T2 - 45
Optimal Solution:
Corner Point Method
A
B
C
D
Optimal solution
is at corner point
of feasible region.
Feasible region
0
10
20
30
40
0 10 20 30 40
X1
X2
46. T2 - 46
Optimal Solution:
Corner Point Method
A (0, 0) (4)(0) + (5)(0) = 0
B (0, 20) (4)(0) + (5)(20) = 100
C (24, 8) (4)(24) + (5)(8) = 136
D (30, 0) (4)(30) + (5)(0) = 120
Point C has highest profit. Produce 24 baseball
hats (X1) & 8 western hats (X2).
Point Coordinates Profit: Z = 4X1 + 5X2
47. T2 - 47
Finding Point C
Coordinates
Point C is the intersection of 2 lines. To solve
simultaneously, multiply 1st equation by - 4:
Add to 2nd equation:
Substitute into 2nd equation to get :
4 1 2 40 4 8 160
4 3 120
4 8 160
8
24
1 2 1 2
1 2
1 2
2
1
1
X X X X
X X
X X
X
X
X
a f
62. T2 - 64
Problem Setup*
Hours Required to
Produce 1 Unit
Dept. VCR
X1
TV
X2
Available
Hrs./Week
Electronic 4 3 240
Assembly 2 1 100
Profit/unit $7 $5
63. T2 - 65
Formulation Solution*
Max Z = 7X1 + 5X2
subject to:
4X1 + 3X2 240
2X1 + 1X2 100
X1, X2 0
69. T2 - 71
Optimal Solution:
Corner Point Method*
Feasible
Region
0
20
40
60
80
100
120
0 10 20 30 40 50 60
Number VCR’s
A
B
C
D
No.
TV's
Electronics
Assembly
70 80
70. T2 - 72
Optimal Solution:
Corner Point Method*
A (0, 80) (7)(0) + (5)(80) = 400
B (30, 40) (7)(30) + (5)(40) = 410
C (50, 0) (7)(50) + (5)(0) = 350
D (0, 0) (7)(0) + (5)(0) = 0
Point B is the optimum solution.
Point Coordinates Profit: Z = 4X1 + 5X2
71. T2 - 73
Point B Coordinates*
To solve simultaneously,
multiple 2nd equation by - 2:
Add to 1st Equation:
Substitute into 1st equation to get :
2 2 1 100 4 2 200
4 3 240
4 2 200
40
30
1 2 1 2
1 2
1 2
2
1
1
X X X X
X X
X X
X
X
X
a f
77. T2 - 79
Graphical Solution
0
20
40
60
80
0
Tons,
Color
Chemical
20 40 60 80
Tons, BW Chemical
Draw axes.
- 40 - 20
78. T2 - 80
Graphical Solution
Feasible
Region
0
20
40
60
80
0
Tons,
Color
Chemical
20 40 60 80
Tons, BW Chemical
- 40 - 20
Everything is possible
without constraints.
79. T2 - 81
Graphical Solution
Feasible
Region
0
20
40
60
80
0
Tons,
Color
Chemical
20 40 60 80
Tons, BW Chemical
- 40 - 20
Find values
satisfying X1 0.
80. T2 - 82
Graphical Solution
Feasible
Region
20
40
60
80
0
Tons,
Color
Chemical
20 40 60 80
Tons, BW Chemical
0
Find values
satisfying X1 0.
81. T2 - 83
Graphical Solution
Feasible
Region
20
40
60
80
0
Tons,
Color
Chemical
20 40 60 80
Tons, BW Chemical
0
Find values
satisfying X2 0.
82. T2 - 84
Graphical Solution
Feasible
Region
0
20
40
60
80
0
Tons,
Color
Chemical
20 40 60 80
Tons, BW Chemical
Find values
satisfying X2 0.
83. T2 - 85
Graphical Solution
Feasible
Region
0
20
40
60
80
0
Tons,
Color
Chemical
20 40 60 80
Tons, BW Chemical
BW Draw X1 30 as
equality: X1 = 30.
84. T2 - 86
Graphical Solution
Feasible
Region
0
20
40
60
80
0
Tons,
Color
Chemical
20 40 60 80
Tons, BW Chemical
BW Find values
satisfying X1 30.
85. T2 - 87
Graphical Solution
Feasible
Region
0
20
40
60
80
0
Tons,
Color
Chemical
20 40 60 80
Tons, BW Chemical
BW Find values
satisfying X1 30.
86. T2 - 88
Graphical Solution
Feasible
Region
0
20
40
60
80
0
Tons,
Color
Chemical
20 40 60 80
Tons, BW Chemical
BW
Color
Draw X2 20 as
equality: X2 = 20.
87. T2 - 89
Graphical Solution
Feasible
Region
0
20
40
60
80
0
Tons,
Color
Chemical
20 40 60 80
Tons, BW Chemical
BW
Color
Find values
for X2 20.
88. T2 - 90
Graphical Solution
Feasible
Region
0
20
40
60
80
0
Tons,
Color
Chemical
20 40 60 80
Tons, BW Chemical
BW
Color
Find values
for X2 20.
89. T2 - 91
Graphical Solution
Feasible
Region
0
20
40
60
80
0
Tons,
Color
Chemical
20 40 60 80
Tons, BW Chemical
BW
Color
Total
Draw X1 + X2 60
as equality:
X1 + X2 = 60.
90. T2 - 92
Graphical Solution
Feasible
Region
0
20
40
60
80
0
Tons,
Color
Chemical
20 40 60 80
Tons, BW Chemical
BW
Color
Total
Find values for
X1 + X2 60.
91. T2 - 93
Graphical Solution
Feasible
Region
0
20
40
60
80
0
Tons,
Color
Chemical
20 40 60 80
Tons, BW Chemical
BW
Color
Total
Find values for
X1 + X2 60.
92. T2 - 94
Optimal Solution:
Corner Point Method
Feasible
Region
0
20
40
60
80
0
Tons,
Color
Chemical
20 40 60 80
Tons, BW Chemical
BW
Color
Total
Find corner
points.
93. T2 - 95
Optimal Solution:
Corner Point Method
Feasible
Region
0
20
40
60
80
0
Tons,
Color
Chemical
20 40 60 80
Tons, BW Chemical
BW
Color
Total
Find corner
points.
A
B
94. T2 - 96
Optimal Solution:
Corner Point Method
A (40, 20) (2500)(40) + (3000)(20) = 160,000
B (30, 30) (2500)(30) + (3000)(30) = 165,000
Point A has lowest cost. Produce 40 tons BW
chemicals (X1) & 20 tons color chemicals (X2).
Point Coord. Z = 2500X1 + 3000X2
96. T2 - 98
Optimal Solution:
Iso-Profit Line Method
Feasible
Region
0
20
40
60
80
0
Tons,
Color
Chemical
20 40 60 80
Tons, BW Chemical
BW
Color
Total
97. T2 - 99
Optimal Solution:
Iso-Profit Line Method
0
20
40
60
80
0
Tons,
Color
Chemical
20 40 60 80
Tons, BW Chemical
BW
Color
Total
Lower
Cost
98. T2 - 100
Optimal Solution:
Iso-Profit Line Method
0
20
40
60
80
0
Tons,
Color
Chemical
20 40 60 80
Tons, BW Chemical
BW
Color
Total
Lower
Cost
99. T2 - 101
Optimal Solution:
Iso-Profit Line Method
0
20
40
60
80
0
Tons,
Color
Chemical
20 40 60 80
Tons, BW Chemical
BW
Color
Total
Lower
Cost
100. T2 - 102
Optimal Solution:
Iso-Profit Line Method
0
20
40
60
80
0
Tons,
Color
Chemical
20 40 60 80
Tons, BW Chemical
BW
Color
Total
Lower
Cost
101. T2 - 103
Optimal Solution:
Iso-Profit Line Method
0
20
40
60
80
0
Tons,
Color
Chemical
20 40 60 80
Tons, BW Chemical
BW
Color
Total
Lower
Cost
102. T2 - 104
Conclusion
Explained linear programming
Formulated linear programming
problems
Solved linear programming problems
using graphical methods
Corner point
Iso-profit line
103. T2 - 105
This Class...
What was the most important thing
you learned in class today?
What do you still have questions
about?
How can today’s class be improved?
Please take a moment to answer the
following questions in writing:
