A coach operator needs to plan and setup an optimal number of coaches from London

to Nottingham. The intermediate stops that need to be included are: first Milton

Keynes, then Birmingham and$/$or Leicester. Figure $3$ demonstrates the routes and the

capacity $($maximum number of coaches allowed$)$ of each route. Decide the maximum

number of coaches running from London to Nottingham to exploit demand of travellers.

London

$($L$)$

Milton

Keynes $($M$)$

Birmingham

$($B$)$

Leicester $($E$)$

Nottingham

$($N$)$

$7$

$3$

$4$

$6$

$4$

$3$

Note: Capital letter within the bracket is the abbreviation of the

corresponding city name.

Figure $3.$ Graph of coach routes from London to Nottingham.

a$)$ Which type of network model does the problem described above exemplify?

$[5\%]$

b$)$ Formulate this problem as a linear programming model. Clearly explain the decision

variables, the objective function and the constraints.