**Algebraic Formulation**

Two algebraic formulations are discussed. The first, is for the Christchurch City Council (CCC) roading network, and uses roughness (IRI) as the sole indicator of condition. The second incorporates the roads structure as well as roughness to indicate condition, yet data and solving capacity are not currently available.

The formulation shown is for the structure-inclusive model.

**Indices:**

*f,g* : the structural number of a road.
*i,j* : the IRI value of a road.
*m,n *: the type of maintenance undertaken. *m,n* = 0 when
there is no maintenance performed.
*t* : the year, from 1 to *T*.
*v *: the average daily volume of traffic on a road (over all
lanes).
*z *: the composition of vehicles on a road (% cars, heavy vehicles
and so on).

**Parameters:**

*a _{z,v,i,f} *: the initial area of road in m

**Decision Variables:**

*A _{z,v,m,i,f,t} *: the total area in m

**Model:**

Minimise

**(1)**

Subject to

for all *z, v, j , g, t* ¹ T.
**(2)**

for all *t*. **(3)**

for
all *z, v, i*. **(4)**

for all *z, v, f*.**(5)**

for
all *z, v, i*. **(6)**

**(7)**

**Objective Function (1)**

The objective of this model is to minimise the cost of roads to the New Zealand public. The costs of roads to the New Zealand public comprise both road user costs and maintenance costs, as the public fund maintenance through taxation paid to Central Government. The objective function in this formulation reflects this objective and contains four separate terms. The first term from the objective function computes the total discounted cost of road maintenance over the planning horizon. The second term calculates the discounted road user costs incurred by motorists when driving along road sections. The third term computes the discounted increase in road user costs that occur when maintenance is being performed. The final term in the objective function is the perpetuities of user costs, costs during maintenance and maintenance costs calculated in constraint seven. The inclusion of these perpetuities is discussed in further detail in Section 2.9.7.

With this objective in mind, the solution to this formulation may indicate to an appropriate level of annual maintenance funding. The level of funding may presently be too low, or alternatively too high. Whichever case holds, a likely outcome is the improvement of road quality at the present budget level due to the formulation optimising road maintenance.

**Maintenance State Change Constraint set (2)**

This set of constraints works by tracking the structural number and
IRI value of road sections with similar characteristics over the duration
of the planning horizon. The IRI of a road section can change in two ways.
Firstly, if no maintenance is performed during a year, then the IRI on
a road section increases by the parameter *p*. Secondly, if a maintenance
is performed on a road section, the IRI gets reduced to the IRI reset value
of that specific maintenance type. The structural number can also change
in two ways. Firstly, if no maintenance is carried out, then the structural
number of a road section decreases by the parameter *d*. Secondly,
when maintenance is performed, the structural number gets reset to the
structural number reset associated with that maintenance type.

**Budget constraint set (3)**

The budget constraints ensure that the funds spent on maintenance during a year cannot exceed the funding available. As funds that are not required for use in a year can be placed in an account at the start of that year, interest is earned and the inflated amount will accrue to the next year when it can be used. Thus, the amount of funding available for road maintenance in an individual year is the annual budget plus any funds unused in previous years. These are equality constraints to ensure that when the funding available in a year exceeds the actual amount spent on maintenance, the remainder becomes unused funds which can then be utilised in following years.

**Setting of Initial Area constraint set (4)**

This constraint set ensures that the area of all road sections of the same road type, IRI value and structural number at the start of the planning horizon equates to the summation of the equivalent ‘Area of Road’ decision variables in the first year regardless of their maintenance type.

**IRI Target constraint set (5)**

For each road type with an IRI greater than a set limit in the final
year of the planning horizon these constraints force the area of road,
to be less than some fraction of the initial area of that road type. The
target level for each road type’s IRI is taken as the 80^{th} percentile
of initial roughness values; that is, the IRI value lower than which 80%
of the initial area of roads in a road type are. These values are different
for all road types, ranging from just 2.8 for the high volume, high user
cost Tertiary roads, to 6.4 for low volume Secondary roads. The target
fraction that has been used in the base run is 20%. Thus, the constraint
ensures that at the end of the planning horizon less than 20% of the area
of a given road type can have an IRI equal to or exceeding the initial
80^{th} percentile IRI value. This set of constraints have been
included to prevent the dramatic reduction in road quality towards the
end of the planning horizon that can be evident in solutions generated
by Chong & Thompson with dTIMS.

**Structural Number Target constraint set (6)**

This set of constraints acts in the same manner as the target constraints for IRI. They ensure that road sections within a road type with structural numbers equal to or less than the target have a combined area that is less than a certain percentage of the total area. Again, this is by no means a complete method. The problems that may have occurred with the IRI targets could again provide setbacks.

Due to the lack of structural data and the solver capacity (without decomposition) a full results set could not be attained.

Back to the main project page... Back to Management Science Honours Project page...

Page updated: 1 November 2000.