Intro_to_OR.pdf

Chapter 1
Chapter 1
Introduction
Introduction

Body of Knowledge
Body of Knowledge

Problem Solving and Decision Making

Quantitative Analysis and Decision Making

Quantitative Analysis

Models of Cost, Revenue, and Profit
Models of Cost, Revenue, and Profit
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© 2008 Thomson South
© 2008 Thomson South-
-Western. All Rights Reserved
Western. All Rights Reserved

Management Science Techniques

Body of Knowledge
Body of Knowledge

The
The body
body of
of knowledge
knowledge involving
involving quantitative
quantitative
approaches
approaches to
to decision
decision making
making is
is referred
referred to
to as
as
•
•Management
Management Science
Science
•
•Operations
Operations Research
Research
•
•Decision
Decision Science
Science

It
It had
had its
its early
early roots
roots in
in World
World War
War II
II and
and is
is flourishing
flourishing
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It
It had
had its
its early
early roots
roots in
in World
World War
War II
II and
and is
is flourishing
flourishing
in
in business
business and
and industry
industry due,
due, in
in part,
part, to
to:
:
•
•numerous
numerous methodological
methodological developments
developments (e
(e.
.g
g.
.
simplex
simplex method
method for
for solving
solving linear
linear programming
programming
problems)
problems)
•
•a
a virtual
virtual explosion
explosion in
in computing
computing power
power

7
7 Steps
Steps of
of Problem
Problem Solving
Solving
(First
(First 5
5 steps
steps are
are the
the process
process of
of decision
decision making
making)
)
1
1.
. Identify
Identify and
and define
define the
the problem
problem.
.
2
2.
. Determine
Determine the
the set
set of
of alternative
alternative solutions
solutions.
.
3
3.
. Determine
Determine the
the criteria
criteria for
for evaluating
evaluating alternatives
alternatives.
.
4
4.
. Evaluate
Evaluate the
the alternatives
alternatives.
.
3
3
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4
4.
. Evaluate
Evaluate the
the alternatives
alternatives.
.
5
5.
. Choose
Choose an
an alternative
alternative (make
(make a
a decision)
decision).
.
---------------------------------------------------------------------
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6
6.
. Implement
Implement the
the selected
selected alternative
alternative.
.
7
7.
. Evaluate
Evaluate the
the results
results.
.

Nature of OR
Nature of OR

OR
OR is
is being
being regarded
regarded as
as the
the application
application of
of scientific
scientific
methods
methods to
to decision
decision making
making.
.

OR
OR attempts
attempts to
to provide
provide a
a systematic
systematic and
and rational
rational
approach
approach to
to the
the fundamental
fundamental problems
problems involved
involved in
in
the
the control
control of
of systems
systems by
by making
making decisions
decisions which,
which, in
in a
a
sense,
sense, achieve
achieve the
the best
best results
results considering
considering all
all the
the
information
information that
that can
can be
be profitably
profitably used
used.
.
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information
information that
that can
can be
be profitably
profitably used
used.
.

As
As per
per Churchman,
Churchman, OR
OR can
can be
be defined
defined as
as the
the
application
application of
of scientific
scientific methods,
methods, techniques
techniques and
and tools
tools to
to
problems
problems involving
involving the
the operations
operations of
of systems
systems so
so as
as to
to
provide
provide those
those in
in control
control of
of operations
operations with
with optimum
optimum
solutions
solutions to
to the
the problems
problems.
.

Significant
Significant features
features of
of OR
OR are
are as
as follows
follows:
:
•
• Decision
Decision Making
Making
•
• It
It helps
helps in
in decision
decision making
making.
. Steps
Steps in
in decision
decision
making
making are
are:
:
– Define the problem
– Establish the criterion (maximization of profit, utility and
minimization of cost etc)
– Select the alternative courses of action.
– Determine the model to be used and the values of the
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– Determine the model to be used and the values of the
parameters of the process
– Evaluate the alternative and choose the one that is optimal.
• Scientific Approach
• OR employs scientific methods for the purpose of
solving problems.
• It is formalized process of reasoning.

• Objective
• OR attempts to locate the best or optimal
solution to the problem under consideration.
• Inter-disciplinary team approach
• OR is inter-disciplinary in nature requires a
team approach to the problem.
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• Digital Computer
– Use of digital computer has become an integral part of the
operations research approach to decision making.
– Computers are required due to complexity of the model,
volume of data required or the computations to be made.

Define
Define
the
the
Problem
Problem
Identify
Identify
the
the
Alternatives
Alternatives
Determine
Determine
the
the
Criteria
Criteria
Identify
Identify
the
the
Alternatives
Alternatives
Choose
Choose
an
an
Alternative
Alternative
Structuring the Problem
Structuring the Problem Analyzing the Problem
Analyzing the Problem

Decision
Decision-
-Making Process
Making Process
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Analysis
Analysis Phase
Phase of
of Decision
Decision-
-Making
Making Process
Process

Qualitative
Qualitative Analysis
Analysis
•
• based
based largely
largely on
on the
the manager’s
manager’s judgment
judgment and
and
experience
experience
•
• includes
includes the
the manager’s
manager’s intuitive
intuitive “feel”
“feel” for
for the
the
problem
problem
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problem
problem
•
• is
is more
more of
of an
an art
art than
than a
a science
science

Analysis
Analysis Phase
Phase of
of Decision
Decision-
-Making
Making Process
Process

Quantitative
Analysis
•
• analyst
analyst will
will concentrate
concentrate on
on the
the quantitative
quantitative facts
facts
or
or data
data associated
associated with
with the
the problem
problem
•
• analyst
analyst will
will develop
develop mathematical
mathematical expressions
expressions
that
that describe
describe the
the objectives,
objectives, constraints,
constraints, and
and
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that
that describe
describe the
the objectives,
objectives, constraints,
constraints, and
and
other
other relationships
relationships that
that exist
exist in
in the
the problem
problem
•
• analyst
analyst will
will use
use one
one or
or more
more quantitative
quantitative
methods
methods to
to make
make a
a recommendation
recommendation


Potential
Potential Reasons
Reasons for
for a
a Quantitative
Analysis
Approach
Approach to
to Decision
Decision Making
Making
•
•The
The problem
problem is
is complex
complex.
.
•
•The
The problem
problem is
is very
very important
important.
.
•
•The
The problem
problem is
is new
new.
.
•
•The
The problem
problem is
is repetitive
repetitive.
.
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•
•The
The problem
problem is
is repetitive
repetitive.
.


Quantitative Analysis Process
Quantitative Analysis Process
•
•Model Development
Model Development
•
•Data Preparation
Data Preparation
•
•Model Solution
Model Solution
•
•Report Generation
Report Generation
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Model Development
Model Development

Models
Models are
are representations
representations of
of real
real objects
objects or
or situations
situations

Three
Three forms
forms of
of models
models are
are:
:
•
•Iconic
Iconic models
models -
- physical
physical replicas
replicas (scalar
(scalar
representations)
representations) of
of real
real objects
objects
•
•Analog
Analog models
models -
- physical
physical in
in form,
form, but
but do
do not
not
physically
physically resemble
resemble the
the object
object being
being modeled
modeled
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physically
physically resemble
resemble the
the object
object being
being modeled
modeled
•
•Mathematical
Mathematical models
models -
- represent
represent real
real world
world
problems
problems through
through a
a system
system of
of mathematical
mathematical
formulas
formulas and
and expressions
expressions based
based on
on key
key
assumptions,
assumptions, estimates,
estimates, or
or statistical
statistical analyses
analyses

Advantages of Models
Advantages of Models

Generally,
Generally, experimenting
experimenting with
with models
models (compared
(compared to
to
experimenting
experimenting with
with the
the real
real situation)
situation):
:
•
•requires
requires less
less time
time
•
•is
is less
less expensive
expensive
•
•involves
involves less
less risk
risk

The
The more
more closely
closely the
the model
model represents
represents the
the real
real
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The
The more
more closely
closely the
the model
model represents
represents the
the real
real
situation,
situation, the
the accurate
accurate the
the conclusions
conclusions and
and predictions
predictions
will
will be
be.
.

Mathematical Models
Mathematical Models

Objective
Objective Function
Function –
– a
a mathematical
mathematical expression
expression that
that
describes
describes the
the problem’s
problem’s objective,
objective, such
such as
as maximizing
maximizing
profit
profit or
or minimizing
minimizing cost
cost

Constraints
Constraints –
– a
a set
set of
of restrictions
restrictions or
or limitations,
limitations, such
such as
as
production
production capacities
capacities

Uncontrollable
Uncontrollable Inputs
Inputs –
– environmental
environmental factors
factors that
that are
are
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Uncontrollable
Inputs –
– environmental
environmental factors
factors that
that are
are
not
not under
under the
the control
control of
of the
the decision
decision maker
maker

Decision
Decision Variables
Variables –
– controllable
controllable inputs
inputs;
; decision
decision
alternatives
alternatives specified
specified by
by the
the decision
decision maker,
maker, such
such as
as the
the
number
number of
of units
units of
of Product
Product X
X to
to produce
produce

Mathematical Models
Mathematical Models

Deterministic
Deterministic Model
Model –
– if
if all
all uncontrollable
uncontrollable inputs
inputs to
to the
the
model
model are
are known
known and
and cannot
cannot vary
vary

Stochastic
Stochastic (or
(or Probabilistic)
Probabilistic) Model
Model –
– if
if any
any uncontrollable
uncontrollable
inputs
inputs are
are uncertain
uncertain and
and subject
subject to
to variation
variation

Stochastic
Stochastic models
models are
are often
often more
more difficult
difficult to
to analyze
analyze.
.
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Mathematical Models
Mathematical Models

Cost/benefit
Cost/benefit considerations
considerations must
must be
be made
made in
in selecting
selecting
an
an appropriate
appropriate mathematical
mathematical model
model.
.

Frequently
Frequently a
a less
less complicated
complicated (and
(and perhaps
perhaps less
less
precise)
precise) model
model is
is more
more appropriate
appropriate than
than a
a more
more
complex
complex and
and accurate
accurate one
one due
due to
to cost
cost and
and ease
ease of
of
solution
solution considerations
considerations.
.
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Transforming Model Inputs into Output
Transforming Model Inputs into Output
(Environmental Factors)
(Environmental Factors)
Controllable
Controllable
Output
Output
17
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Controllable
Controllable
Inputs
Inputs
(Decision
(Decision
Variables)
Variables)
Output
Output
(Projected
(Projected
Results)
Results)
Mathematical
Mathematical
Model
Model

Example:
Example: 2.1, Maximization case (NDV)
2.1, Maximization case (NDV)
A
A firm
firm is
is engaged
engaged in
in producing
producing two
two products,
products, A
A and
and B
B.
.
Each
Each unit
unit of
of product
product A
A requires
requires 2
2 Kg
Kg of
of raw
raw material
material and
and 4
4
labor
labor hours
hours for
for processing,
processing, whereas
whereas each
each unit
unit of
of product
product B
B
requires
requires 3
3 Kg
Kg of
of raw
raw material
material and
and 3
3 labor
labor hours
hours.
. Every
Every
week,
week, the
the firm
firm has
has an
an availability
availability of
of 60
60 Kg
Kg of
of r
raw
aw material
material
and
and 96
96 labor
labor hours
hours.
. One
One unit
unit of
of product
product A
A sold
sold yields
yields Rs
Rs.
. 40
40
and
and one
one unit
unit of
of product
product B
B sold
sold gives
gives Rs
Rs.
. 35
35 as
as profit
profit.
.
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and
and one
one unit
unit of
of product
product B
B sold
sold gives
gives Rs
Rs.
. 35
35 as
as profit
profit.
.
Formulate
Formulate this
this problem
problem of
of linear
linear programming
programming problem
problem to
to
determine
determine as
as to
to how
how many
many units
units of
of each
each of
of the
the products
products
should
should be
be produced
produced per
per week
week so
so that
that the
the firm
firm can
can earn
earn the
the
maximum
maximum profit
profit.
. Assume
Assume that
that there
there is
is no
no mktg
mktg.
. constraint
constraint
so
so that
that all
all that
that is
is produced
produced can
can be
be sold
sold.
.

Solution of Example 2.1
Solution of Example 2.1
•
•Define the objective function
Define the objective function
•
•Define constraints
Define constraints
•
•Non
Non-
-negativity constraints
negativity constraints
•
•Formulate LPP.
Formulate LPP.
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Example:
Example: Example 2.2, Minimization case
Example 2.2, Minimization case
(NDV)
(NDV)

The
The agricultural
agricultural research
research institute
institute suggested
suggested to
to a
a
farmer
farmer to
to spread
spread out
out at
at least
least 4800
4800 kg
kg of
of a
a special
special
phosphate
phosphate fertilizer
fertilizer and
and not
not less
less than
than 7200
7200 kg
kg of
of a
a
special
special nitrogen
nitrogen fertilizer
fertilizer to
to raise
raise productivity
productivity of
of crops
crops
in
in his
his fields
fields.
. There
There are
are two
two sources
sources for
for obtaining
obtaining
these
these---
---mixtures
mixtures A
A
B
B.
. Both
Both of
of these
these are
are available
available in
in
bags
bags weighing
weighing 100
100 kg
kg each
each and
and they
they cost
cost Rs
Rs.
. 40
40 and
and
20
20
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bags
bags weighing
weighing 100
100 kg
kg each
each and
and they
they cost
cost Rs
Rs.
. 40
40 and
and
Rs
Rs.
. 24
24 resp
resp.
. Mixture
Mixture A
A contains
contains phosphate
phosphate and
and
nitrogen
nitrogen equivalent
equivalent of
of 20
20 kg
kg and
and 80
80 kg
kg resp
resp.
. while
while
mixture
mixture B
B contains
contains these
these ingredients
ingredients equivalent
equivalent of
of 50
50
kg
kg each
each.
.

Write
Write this
this as
as a
a LPP
LPP to
to determine
determine how
how many
many bags
bags of
of
each
each type
type the
the farmer
farmer should
should buy
buy in
in order
order to
to obtain
obtain the
the
required
required fertilizer
fertilizer at
at minimum
minimum cost
cost.
.

Model Solution
Model Solution

The
The analyst
analyst attempts
attempts to
to identify
identify the
the alternative
alternative (the
(the
set
set of
of decision
decision variable
variable values)
values) that
that provides
provides the
the
“best”
“best” output
output for
for the
the model
model.
.

The
The “best”
“best” output
output is
is the
the optimal
optimal solution
solution.
.

If
If the
the alternative
alternative does
does not
not satisfy
satisfy all
all of
of the
the model
model
constraints,
constraints, it
it is
is rejected
rejected as
as being
being infeasible
infeasible,
, regardless
regardless
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constraints,
constraints, it
it is
is rejected
rejected as
as being
being infeasible
infeasible,
, regardless
regardless
of
of the
the objective
objective function
function value
value.
.

If
If the
the alternative
alternative satisfies
satisfies all
all of
of the
the model
model constraints,
constraints,
it
it is
is feasible
feasible and
and a
a candidate
candidate for
for the
the “best”
“best” solution
solution.
.

Model Solution
Model Solution

One
One solution
solution approach
approach is
is trial
trial-
-and
and-
-error
error.
.
•
•Might
Might not
not provide
provide the
the best
best solution
solution
•
•Inefficient
Inefficient (numerous
(numerous calculations
calculations required)
required)

Special
Special solution
solution procedures
procedures have
have been
been developed
developed for
for
specific
specific mathematical
mathematical models
models.
.
•
•Some
Some small
small models/problems
models/problems can
can be
be solved
solved by
by
22
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•
•Some
Some small
small models/problems
models/problems can
can be
be solved
solved by
by
hand
hand calculations
calculations
•
•Most
Most practical
practical applications
applications require
require using
using a
a
computer
computer

Model Solution
Model Solution

A
A variety
variety of
of software
software packages
packages are
are available
available for
for
solving
solving mathematical
mathematical models
models.
.
•
•Microsoft
Microsoft Excel
Excel
•
•The
The Management
Management Scientist
Scientist
•
•LINGO
LINGO
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Model Testing and Validation
Model Testing and Validation

Often,
Often, goodness/accuracy
goodness/accuracy of
of a
a model
model cannot
cannot be
be
assessed
assessed until
until solutions
solutions are
are generated
generated.
.

Small
Small test
test problems
problems having
having known,
known, or
or at
at least
least
expected,
expected, solutions
solutions can
can be
be used
used for
for model
model testing
testing and
and
validation
validation.
.

If
If the
the model
model generates
generates expected
expected solutions,
solutions, use
use the
the
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If
If the
the model
model generates
generates expected
expected solutions,
solutions, use
use the
the
model
model on
on the
the full
full-
-scale
scale problem
problem.
.

If
If inaccuracies
inaccuracies or
or potential
potential shortcomings
shortcomings inherent
inherent in
in
the
the model
model are
are identified,
identified, take
take corrective
corrective action
action such
such
as
as:
:
•
•Collection
Collection of
of more
more-
-accurate
accurate input
input data
data
•
•Modification
Modification of
of the
the model
model

Report Generation
Report Generation

A
A managerial
managerial report,
report, based
based on
on the
the results
results of
of the
the
model,
model, should
should be
be prepared
prepared.
.

The
The report
report should
should be
be easily
easily understood
understood by
by the
the
decision
decision maker
maker.
.

The
The report
report should
should include
include:
:
•
•the
the recommended
recommended decision
decision
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•
•the
the recommended
recommended decision
decision
•
•other
other pertinent
pertinent information
information about
about the
the results
results (for
(for
example,
example, how
how sensitive
sensitive the
the model
model solution
solution is
is to
to the
the
assumptions
assumptions and
and data
data used
used in
in the
the model)
model)

Implementation and Follow
Implementation and Follow-
-Up
Up

Successful
Successful implementation
implementation of
of model
model results
results is
is of
of
critical
critical importance
importance.
.

Secure
Secure as
as much
much user
user involvement
involvement as
as possible
possible
throughout
throughout the
the modeling
modeling process
process.
.

Continue
Continue to
to monitor
monitor the
the contribution
contribution of
of the
the model
model.
.

It
It might
might be
be necessary
necessary to
to refine
refine or
or expand
expand the
the model
model.
.
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It
It might
might be
be necessary
necessary to
to refine
refine or
or expand
expand the
the model
model.
.

Example: Austin Auto Auction
An
An auctioneer
auctioneer has
has developed
developed a
a simple
simple mathematical
mathematical
model
model for
for deciding
deciding the
the starting
starting bid
bid he
he will
will require
require
when
when auctioning
auctioning a
a used
used automobile
automobile.
.
Essentially
Essentially,
, he
he sets
sets the
the starting
starting bid
bid at
at seventy
seventy percent
percent
of
of what
what he
he predicts
predicts the
the final
final winning
winning bid
bid will
will (or
(or
should)
should) be
be.
. He
He predicts
predicts the
the winning
winning bid
bid by
by starting
starting
with
with the
the car's
car's original
original selling
selling price
price and
and making
making two
two
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with
with the
the car's
car's original
original selling
selling price
price and
and making
making two
two
deductions,
deductions, one
one based
based on
on the
the car's
car's age
age and
and the
the other
other
based
based on
on the
the car's
car's mileage
mileage.
.
The
The age
age deduction
deduction is
is $
$800
800 per
per year
year and
and the
the mileage
mileage
deduction
deduction is
is $
$.
.025
025 per
per mile
mile.
.


Question
Question:
:
Develop
Develop the
the mathematical
mathematical model
model that
that will
will give
give the
the
starting
starting bid
bid (
(B
B )
) for
for a
a car
car in
in terms
terms of
of the
the car's
car's original
original
price
price (
(P
P ),
), current
current age
age (
(A
A)
) and
and mileage
mileage (
(M
M )
).
.
28
28
Slide
Slide

Answer:
Answer:
The expected winning bid can be expressed as:
The expected winning bid can be expressed as:
P
P -
- 800(
800(A
A)
) -
- .025(
.025(M
M )
)
The entire model is:
The entire model is:
B
B = .7(expected winning bid)
= .7(expected winning bid)
29
29
Slide
Slide
B
B = .7(expected winning bid)
= .7(expected winning bid)
B
B = .7(
= .7(P
P -
- 800(
800(A
A)
) -
- .025(
.025(M
M ))
))
B
B = .7(
= .7(P
P )
)-
- 560(
560(A
A)
) -
- .0175(
.0175(M
M )
)


Question
Question:
:
Suppose
Suppose a
a four
four-
-year
year old
old car
car with
with 60
60,
,000
000 miles
miles on
on the
the
odometer
odometer is
is being
being auctioned
auctioned.
. If
If its
its original
original price
price was
was
$
$12
12,
,500
500,
, what
what starting
starting bid
bid should
should the
the auctioneer
auctioneer
require?
require?

Answer
Answer:
:
30
30
Slide
Slide

Answer
Answer:
:
B
B =
= .
.7
7(
(12
12,
,500
500)
) -
- 560
560(
(4
4)
) -
- .
.0175
0175(
(60
60,
,000
000)
) =
= $
$5
5,
,460
460


Question
Question:
:
The
The model
model is
is based
based on
on what
what assumptions?
assumptions?

Answer
Answer:
:
The
The model
model assumes
assumes that
that the
the only
only factors
factors
influencing
influencing the
the value
value of
of a
a used
used car
car are
are the
the original
original
price,
price, age,
age, and
and mileage
mileage (not
(not condition,
condition, rarity,
rarity, or
or other
other
31
31
Slide
Slide
price,
price, age,
age, and
and mileage
mileage (not
(not condition,
condition, rarity,
rarity, or
or other
other
factors)
factors).
.
Also,
Also, it
it is
is assumed
assumed that
that age
age and
and mileage
mileage devalue
devalue a
a
car
car in
in a
a linear
linear manner
manner and
and without
without limit
limit.
. (Note,
(Note, the
the
starting
starting bid
bid for
for a
a very
very old
old car
car might
might be
be negative!)
negative!)

Example: Iron Works, Inc.
Iron Works, Inc. manufactures two
Iron Works, Inc. manufactures two
products made from steel and just received
products made from steel and just received
this month's allocation of
this month's allocation of b
b pounds of steel.
pounds of steel.
It takes
It takes a
a1
1 pounds of steel to make a unit of product 1
pounds of steel to make a unit of product 1
and
and a
a2
2 pounds of steel to make a unit of product 2.
pounds of steel to make a unit of product 2.
Let
Let x
x and
and x
x denote this month's production level of
denote this month's production level of
32
32
Slide
Slide
Let
Let x
x1
1 and
and x
x2
2 denote this month's production level of
denote this month's production level of
product 1 and product 2, respectively. Denote by
product 1 and product 2, respectively. Denote by p
p1
1 and
and
p
p2
2 the unit profits for products 1 and 2, respectively.
the unit profits for products 1 and 2, respectively.
Iron Works has a contract calling for at least
Iron Works has a contract calling for at least m
m units of
units of
product 1 this month. The firm's facilities are such that at
product 1 this month. The firm's facilities are such that at
most
most u
u units of product 2 may be produced monthly.
units of product 2 may be produced monthly.


Mathematical Model
Mathematical Model
•
•The total monthly profit =
The total monthly profit =
(profit per unit of product 1)
(profit per unit of product 1)
x (monthly production of product 1)
+ (profit per unit of product 2)
+ (profit per unit of product 2)
33
33
Slide
Slide
=
= p
p1
1x
x1
1 +
+ p
p2
2x
x2
2
We want to maximize total monthly profit:
We want to maximize total monthly profit:
Max
Max p
p1
1x
x1
1 +
+ p
p2
2x
x2
2


Mathematical Model (continued)
•
•The total amount of steel used during monthly
The total amount of steel used during monthly
production equals:
production equals:
(steel required per unit of product 1)
(steel required per unit of product 1)
+ (steel required per unit of product 2)
34
34
Slide
Slide
=
= a
a1
1x
x1
1 + a
+ a2
2x
x2
2
This quantity must be less than or equal to the
This quantity must be less than or equal to the
allocated
allocated b
b pounds of steel:
pounds of steel:
a
a1
1x
x1
1 +
+ a
a2
2x
x2
2
b
b


•
•The monthly production level of product 1 must
The monthly production level of product 1 must
be greater than or equal to
be greater than or equal to m
m :
:
x
x1
1
m
m
•
•The monthly production level of product 2 must
The monthly production level of product 2 must
be less than or equal to
be less than or equal to u
u :
:
35
35
Slide
Slide
be less than or equal to
be less than or equal to u
u :
:
x
x2
2
u
u
•
•However, the production level for product 2
However, the production level for product 2
cannot be negative:
cannot be negative:
x
x2
2
0
0


Mathematical Model Summary
Mathematical Model Summary
Max
Max p
p1
1x
x1
1 +
+ p
p2
2x
x2
2
s.t.
s.t. a
a1
1x
x1
1 +
+ a
a2
2x
x2
2
b
b
x
x1
1
m
m
x
x2
2
u
u
Objectiv
Objectiv
e
e
Function
Function
Constraint
Constraint
s
s
36
36
Slide
Slide
x
x2
2
u
u
x
x2
2
0
0
Function
Function
“Subject to”
“Subject to”


Question:
Question:
Suppose
Suppose b
b = 2000,
= 2000, a
a1
1 = 2,
= 2, a
a2
2 = 3,
= 3, m
m = 60,
= 60, u
u = 720,
= 720,
p
p1
1 = 100,
= 100, p
p2
2 = 200. Rewrite the model with these
= 200. Rewrite the model with these
specific values for the uncontrollable inputs.
specific values for the uncontrollable inputs.
37
37
Slide
Slide


Answer:
Answer:
Substituting, the model is:
Substituting, the model is:
Max 100
Max 100x
x1
1 + 200
+ 200x
x2
2
s.t. 2
s.t. 2x
x1
1 + 3
+ 3x
x2
2
2000
2000
x
x1
1
60
60
38
38
Slide
Slide
x
x1
1
60
60
x
x2
2
720
720
x
x2
2
0
0


Linear Programming
Linear Programming

Integer Linear
Integer Linear
Programming
Programming

PERT/CPM
PERT/CPM

Inventory Models
Inventory Models

Decision Analysis
Decision Analysis

Goal Programming
Goal Programming

Analytic Hierarchy
Analytic Hierarchy
Process
Process

Forecasting
Forecasting
39
39
Slide
Slide
Inventory Models
Inventory Models

Waiting Line Models
Waiting Line Models

Simulation
Simulation
Forecasting
Forecasting

Markov
Markov-
-Process Models
Process Models

Dynamic Programming
Dynamic Programming

The Management Scientist
The Management Scientist Software
Software

12 Modules
12 Modules
40
40
Slide
Slide

General Statement of Linear Programming
General Statement of Linear Programming

In
In general
general terms,
terms, a
a linear
linear programming
programming problem
problem can
can be
be written
written as
as:
:
Maximize
Maximize Z=c
Z=c1
1x
x1
1 +
+ c
c2
2x
x2
2 +
+ ………
………+
+ c
cn
nx
xn
n (Objective
(Objective Function)
Function)
Subject
Subject to
to
a
a11
11x
x1
1 +
+ a
a12
12x
x2
2 +
+ ……………
……………..
.. +
+ a
a1
1n
nx
xn
n ≤
≤ b
b1
1
a
a21
21x
x1
1 +
+ a
a22
22x
x2
2 +
+ ……………
……………..
.. +
+ a
a2
2n
nx
xn
n ≤
≤ b
b2
2
.
.
.
.
.
.
41
41
Slide
Slide
.
.
a
am
m1
1x
x1
1 +
+ a
am
m2
2x
x2
2 +
+ …………
…………..
.. +
+ a
amn
mnx
xn
n ≤
≤ b
bm
m
x
x1
1,
, x
x2
2,
, ……………
……………x
xn
n ≥
≥ 0
0
Where,
Where, c
cj
j,
, a
aij
ij,
, b
bi
i (
(i
i=
=1
1,
, 2
2,
,…
…..
..,
, m
m;
; j=
j=1
1,
, 2
2,
,…
…..
..,
, n)
n) are
are known
known as
as constants
constants and
and x
xj
j’s
’s are
are
decisions
decisions variables
variables.
.
The
The c
cj
j’s
’s are
are termed
termed as
as profit
profit coefficients,
coefficients, a
aij
ij’s
’s the
the technological
technological coefficients
coefficients and
and b
bi
i’s
’s the
the
resource
resource values
values.
.

Solution to linear programming problems
Solution to linear programming problems

LPPs can be solved by
LPPs can be solved by
•
•Graphic method
Graphic method
•
•Applying algebraic method (Simplex Method)
Applying algebraic method (Simplex Method)
42
42
Slide
Slide

Intro_to_OR.pdf

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