Download Markov Analysis

MARKOV ANALYSIS
 Andrei Markov, a Russian Mathematician
developed the technique to describe the
movement of gas in a closed container in
1940
 In 1950s, management scientists began to
recognize that the technique could be
adapted to decision situations which fit the
pattern Markov described
MARKOV ANALYSIS




In the early 1960s, the procedure
has been used to describe
Marketing Strategies
Plant maintenance decisions
Stock market movements
Account Receivable Projections
MARKOV ANALYSIS
Markov analysis is used to analyze the
current state and movement of a
variable to predict the future
occurrences and movement of this
variable by the use of presently
known probabilities.
TERMS USED
Market share – The fraction of a population
that shops at a particular store or market.
When expressed as a fraction, market
shares can be used in place of state
probabilities.
TERMS USED
State probability – The probability of an
event occurring at a point in time, say, the
probability that a person will purchase a
product at a given store during a given
month.
TERMS USED
Transition probability – The conditional
probability that we will be in a future state
given a current state.
Characteristics of Markov
Process
The following are the characteristics of Markov
process as applied in business.
1. The time structure. It is the period of time
when a set of outcomes occurs
2. The state variables. These are variables that
describe the condition, or state, of the
system at each trial or period of the process.
3. At a specific point, often at the beginning
or end of a time period in the process, the
condition being monitored may changed
from one state to another.
Characteristics of Markov
Process
4. The decision maker is often uncertain about
whether the state variable will change or
remain the same; and this uncertainty is
reflected in a set of transition probabilities.
5. At a given point in the structure of the
process, the set of transition probabilities
representing the uncertainty of moving to
any future state depend only on the current
state.
Assumptions of Markov
Analysis
The following are the assumptions of Markov
analysis as applied in business.
1. The transition probabilities for a given
beginning state of the system equals one.
2. The probabilities apply to all system
participants.
3. The transition probabilities are constant
over time.
4. The states are independent over time.
Matrix of Transition Probabilities

It is the likelihood that the system
in a given current state will remain in
the same state or move to another
state in the next period.

It is a proportion of customers
which a company retains or loses from
period to period.
Matrix of Transition Probabilities
P=
P
11
P
21

P
m1
P
12
P
22

P
m2




P
1n
P
2n

P
mn
Matrix of Transition Probabilities
Mn = Mn-1(P)
where:
Mn = matrix of market share at period
n
Mn-1= matrix of market share at period
n-1
P
= matrix of transition probabilities
STEADY STATE (Equilibrium)
 One of the major properties of
Markov chains is that, in the long
run, the process usually stabilizes.
 A stabilized system is said to
approach
steady
state
or
equilibrium when the system’s
state probabilities have become
independent of time.
STEADY STATE (Equilibrium)
 One of the major properties of
Markov chains is that, in the long
run, the process usually stabilizes.
 A stabilized system is said to
approach
steady
state
or
equilibrium when the system’s
state probabilities have become
independent of time.
Matrix of Transition Probabilities
Example: Find the steady state
probabilities,
[Pp
Sp] = [Pp Sp]
 0.8 0.2 
0.15 0.85


Market Share Analysis
A community has two gasoline service stations,
Petron and Shell. The residents of the
community purchase gasoline at the two
stations on a monthly basis. The marketing
department of the Petron Company surveyed a
number of residents and found that the
customers were not totally loyal to either brand
of gasoline. Customers were willing to change
service stations as a result of advertising ,
service and other factors.
Market Share Analysis
The marketing department found that if a
customer bought their gasoline from Petron in
any given month, there was only a 0.60
probability that the customers would buy from
Petron the next month and a .40 probability
that the customer would buy their gas from
Shell the next month. Likewise, if a customer
traded with Shell in a given month, there was
an .80 probability that the customer would
purchase gasoline from Shell in the next month
and a .20 probability that the customer would
purchase gasoline from Petron.
Market Share Analysis
Required:
1. Set-up the transition probability matrix for
this problem.
2. Determine the probability that the customer
will trade in a) Petron and b) Shell?
3. Determine the steady-state probabilities and
4. If there are 3,000 customers in the
community
who
purchase
gasoline,
determine the number of customers that
each company can anticipate in the long run
Predicting Future Market
Shares
Suppose in the canteen of a certain university which
offers three types of combo meals, A, B and C, on a given
day, 35% of the customers choose A, 30% choose B, and
35% choose C. Based on the canteen staff observations,
those ordering A on one day, 40% will reorder A the next
day, while 25% will order B and 35% will order C. Of
those who order B on a day, 30% will reorder B the next
day, 40% will order A and 30% will order C. Of those who
order C on a day, 50% will reorder C the next day, 25%
will order A and 25% will order B. The transition
probabilities of the problem can be illustrated using a tree
diagram.
Plant Maintenance Schedule
A particular production machine could be
assigned the states “operating” and
“breakdown”. The transition probabilities
could then reflect the probability of a
machine’s either breaking down or
operating in the next time period. (i.e.,
month, day, or year)
Plant Maintenance Schedule
As an example, consider a machine
having the following daily transition
matrix.
Day 1
T= Operate
Breakdown
Day 2
Operate
Breakdown
.90
.10
.70
.30
Plant Maintenance Schedule
The steady-state probabilities for this example
are
.88 = steady-state probabilities for this example
are
.12 = steady-state probabilities of the machine’s
breaking down
If management decides that the long –run
probability of .12 for a breakdown is excessive, it
might
consider
increasing
preventive
maintenance, which would change the transition
matrix for this example.
Market Share Analysis
The marketing department found that if a
customer bought their gasoline from Petron in
any given month, there was only a 0.60
probability that the customers would buy from
Petron the next month and a .40 probability
that the customer would buy their gas from
Shell the next month. Likewise, if a customer
traded with Shell in a given month, there was
an .80 probability that the customer would
purchase gasoline from Shell in the next month
and a .20 probability that the customer would
purchase gasoline from Petron.
Market Share Analysis
Required:
1. Set-up the transition probability matrix for
this problem.
2. Determine the probability that the customer
will trade in a) Petron and b) Shell?
3. Determine the steady-state probabilities and
indicate the number of customers that each
company can anticipate in the long run.
EXAMPLE #1
For the past twelve months, a computer
printer functioned 80% of the time
correctly during the current month if it
had functioned correctly in the preceding
month. This means that 20% of the time
the printer did not function correctly for
a given month when it was functioning
correctly during the preceding month.
EXAMPLE #1
Moreover, it was observed that 85% of
the time the printer remained
incorrectly adjusted for any given
month if it was incorrectly adjusted the
preceding month and 15% of the time
the printer operated correctly in a given
month when it did not operate correctly
during the preceding month.
EXAMPLE #1
a. What is the probability that the
printer will be functioning correctly one
month from now?
b. What is the probability that the
printer will be functioning correctly two
months
from now?
APPLICATIONS
The mathematical function that describes this
distribution, also called the density function, is given by
1
f ( ) 
e
 2
( X  )2
2 2
where:
X – is the value of the random variable
- standard deviation of the distribution
- mean of the distribution
= 2.14
e = 2.718