OR-Notes are a series of introductory notes on topics that fall under the broad heading of the field of operations research (OR). They were originally used by me in an introductory OR course I give at Imperial College. They are now available for use by any students and teachers interested in OR subject to the following conditions.

A full list of the topics available in OR-Notes can be found here.

Let:

state 1 = % of fleet buyers having a target % of 100%

state 2 = % of fleet buyers having a target % of 70%

state 3 = % of fleet buyers having a target % of 50%

state 4 = % of fleet buyers having a target % of 20%

then s_{1} (the state of the system in year one) is given by

s_{1}= [0.05, 0.30, 0.45, 0.20]

and the transition matrix P is given by

P = | 0.6 0.3 0.1 - | | - 0.7 0.3 - | | - 0.4 0.4 0.2 | | - 0.2 0.5 0.3 |

The state-transition diagram is shown below.

We are asked for the % of fleet buyers having a target % for Ford cars of 50%

- in 2 years time; and
- in the long-run.

The state of the system after one year s_{2}=s_{1}P

= [0.05, 0.30, 0.45, 0.20] | 0.6 0.3 0.1 - | | - 0.7 0.3 - | | - 0.4 0.4 0.2 | | - 0.2 0.5 0.3 |

= [0.03, 0.445, 0.375, 0.15]

Note here that as a numerical check all the elements of s_{2}
add to one.

Hence the state of the system in 2 years time s_{3} = s_{2}P

= [0.03, 0.445, 0.375, 0.15] | 0.6 0.3 0.1 - | | - 0.7 0.3 - | | - 0.4 0.4 0.2 | | - 0.2 0.5 0.3 |

= [0.018, 0.5005, 0.3615, 0.12]

Note again that as a numerical check all the elements of s_{3}
add to one.

Hence the % of fleet buyers having a target % of 50% for Ford cars in 2 years time is 36.15%.

To work out the corresponding % in the long-run we *assume* that
the system reaches equilibrium in the sense that s_{t} = s_{t-1}
(=[x_{1},x_{2},x_{3},x_{4} say).

Hence from s_{t} = s_{t-1}P we have that

[x_{1},x_{2},x_{3},x_{4}] =[x_{1},x_{2},x_{3},x_{4}] | 0.6 0.3 0.1 - | | - 0.7 0.3 - | | - 0.4 0.4 0.2 | | - 0.2 0.5 0.3 |

(and note also that x_{1}+x_{2}+x_{3}+x_{4}=1).

Hence we have the five equations

x_{1}= 0.6x_{1 }x_{2}= 0.3x_{1}+ 0.7x_{2}+ 0.4x_{3}+ 0.2x_{4 }x_{3}= 0.1x_{1}+ 0.3x_{2}+ 0.4x_{3}+ 0.5x_{4 }x_{4}= 0.2x_{3}+ 0.3x_{4 }x_{1}+ x_{2}+ x_{3}+ x_{4}= 1

Now from the first equation above 0.4x_{1}=0 i.e. x_{1}=0
so substituting this into the other equations above and rearranging we
get

0.3x_{2}= 0.4x_{3}+ 0.2x_{4}(1) 0.6x_{3}= 0.3x_{2}+ 0.5x_{4}(2) 0.7x_{4}= 0.2x_{3}(3) x_{2}+ x_{3}+ x_{4}= 1 (4)

From equation (3)

x_{4}= (0.2/0.7)x_{3}(5)

so substituting equation (5) into equation (1) we get

0.3x_{2}= 0.4x_{3}+ 0.2(0.2/0.7)x_{3}i.e. x_{2}= [(0.4 + 0.2(0.2/0.7))/0.3]x_{3 }(6)

Substituting equations (5) and (6) into equation (4) we get

x_{3}[[(0.4 + 0.2(0.2/0.7))/0.3] + 1 + (0.2/0.7)] = 1 i.e. x_{3}= 0.3559

Hence from equation (5) we have

x_{4}= (0.2/0.7)(0.3559) i.e. x_{4}= 0.1017

and from equation (6) we have

x_{2}= [(0.4 + 0.2(0.2/0.7))/0.3](0.3559) i.e. x_{2}= 0.5423

Hence we have

x_{1}= 0 x_{2}= 0.5423 x_{3}= 0.3559 x_{4}= 0.1017

**As a numerical check we can substitute these values back into our
original equations (equations (1)-(5) above). If we do this we will see
that the values for x _{1}, x_{2}, x_{3} and x_{4}
given above are consistent with these equations (to within rounding errors).
**

Hence we have that the % of fleet buyers having a target % for Ford cars of 50% in the long-run is 35.59%.

In general it takes a long time for a system to reach the long-run state and hence, in practice, we would not expect this state ever to be reached (due to changing circumstances rendering the transition matrix invalid).

However note that in this particular example the state of the system
after just two years (s_{3} = [0.018, 0.5005, 0.3615, 0.12]) is
close to the long-run system state (= [0, 0.5423, 0.3559, 0.1017]) and
so it is likely that we will (effectively) reach the long-run state in
just a few years.

For example, the percentage of fleet buyers having a target percentage for Ford cars of 50% the figure after two years is 36.15% and in the long-run is 35.59% - so these figures are very close to each other.

The advantages and disadvantages of using Markov theory to forecast fleet buyers' behaviour include:

- Markov theory simple to apply and understand
- Sensitivity calculations (i.e. "what-if" questions) are easily carried out
- Markov theory gives us an insight into changes in the system over time
- P may be dependent upon the current state of the system (e.g. the transition
matrix probabilities may depend upon the % of fleet buyers in a particular
system state - for example if a high % of fleet buyers currently have a
target percentage of 70% for Ford cars this may influence others to aim
for that target percentage). If P is dependent upon both time and the current
state of the system i.e. P a function of both t and s
_{t}then the basic Markov equation becomes s_{t}=s_{t-1}P(t-1,s_{t-1}). - How do we cope with the effect of promotional advertising campaigns?
- Markov theory is only a simplified model of a complex decision-making process.
- What this analysis has neglected is the size of the fleet controlled by each buyer - the behaviour of a buyer with a large fleet is more important than the behaviour of a buyer with a small fleet.