Features of the Deterministic Approach to Road Fund Allocation 65

The features of the deterministic approach to road fund allocation were defined as in the following paragraphs.

1. It was based on only quantifiable indicators for purposes of objectivity and certainty of outcomes.

2. It included economic efficiency and equity indicators.

3. It included engineering attributes and the application of a pavement

management system.

4. It considered the time horizon of impacts.

5. It was modelled from a decision maker’s perspective.

6. It was set on two stage structure with the following components.

▪ The first component applied the value function model (VFM) by Keeney and Raiffa (1976) and von Winterfeldt and Edwards (1986) for the

estimation of the input parameters for an initial allocation of the road fund by

the three road types in Ghana.

▪ The second component involved the use of the concept of efficiency frontier to determine the input parameters for the internal division of the

proportion of the road fund allocated to each type by economic efficiency and

equity components.

3.8.1.1 Description of the Value Function Model (VFM)

The VFM defines a score for selected attributes which are used to evaluate an

alternative element for an investment option. It estimates separate ratio scales

described as value scores for the attributes by a dimensionless scale using a defined

value form. The set of decision alternatives ( )_a

i indexed by are ranked

on the set of attributes indexed as . The ratio scale is derived as a

value score for each attribute is expressed as;

a

a

a

i= 1... n

c

j

c

j=

c

1...

c

n

( )_x

v

j j Equation 3.1 Where;

v

j = a value function scaled from 0 to 1 per attribute

x

j = is the measure of effectiveness on an attribute spacex.  It is designed to satisfy the functional form;

( ) ( )

( )

{_v

_x

_v

_x

_v

_n

_x

_n

}

imise , ... max 2 2 1 1         Equation 3.2 Where;

Rj

v

j∈ i is an objective function of n-dimensional attributes with feasible decision

solutions. The overall optimisation function for each alternative is expressed in an

aggregated form;

( ) ( )

( )

{_v

_x

_v

_x

_v

_x

}

a

i= 1 1 + 2 2 +... n n Equation 3.3 n i=1,2,...

This can be expressed either in a multiplicative form or an addtive form. It is

expressed in a multiplicative form where there is weak difference independence

between the attributes. An attribute is weak-different independent of the other

attributes if the order for preference consequences involving only changes in

pairs of levels does not depend on the levels at which are fixed. The

multiplicative form is expressed as;

x

1

x

x

2.... n

x

1

x

2....

x

n

(_x

_x

_x

)

n

[

_kk

_j

_v

_j

( )_x

_j j n kv = + +

Π

= 1 ... 1 1 2 1

]

Equation 3.4 Where;

k

i= is an assigned weight on

v

i

( )x

i and 1, 1 =

∑

− n i i

k

0

_k

, 1,i 1,2,...n. i p = p

k

= is an additional scaling constant that characterizes the interaction effect of different measures on preference.

It is expressed in an additive form where there is preferential independence on the

attributes. A pair of attributes

{_x

_x}

2

1, is preferentially independent of the other

attributes

{_x

_x}

n ,

3 if the preference order for consequences involving only changes

in and does not depend on the levels at which are fixed. The additive

form is expressed as:

x

1

x

2

x

3....

x

n

(_x

_x

_x

)

n

_k

_v(_x

_j j j j n v

∑

= = 1 2 1 ....

)

Equation 3.5

1. Axioms on the Value Function Model: The VFM is set on the axioms of

transtivity, continuity and completeness.

(i) Transitivity: This indicates that if an option A is preferred over B and B is

preferred over C then A is preferred over C.

(ii) Continuity: This implies that if option A is preferred over B and B is preferred

over C. There should be some probability (P) that A will happen and some

probability (1-P) that C will happen, so that the agent is indifferent about

accepting this probability or being sure of getting B.

(iii) Completeness: If the agent is indifferent between result A and B then it should

be able to replace one with the other.

2. Reasons for VFM Application in this Research: The VFM was applied in this

research for the following reasons.

(i) It assumes that outcomes are known with certainty and applies quantitative

values for reliability, objectivity and transparency.

(ii) It defines a separate weighted value score for each attribute.

(iii) It transforms attributes into a dimensionless scale.

(iv) It offers different dimensions of the value forms which could be linear,

exponential or user defined as indicated in Figure 3.6. The exponential

function allows for the inclusion of time dimensions in the analysis.

Linear Function Exponential Function

x 1 0 1 0 x 1 0 x

User Defined Function Figure 3.6: Types of Value Forms

(v) The global importance of attributes reflects the importance of an attribute as a

stable characteristic that does not depend on a specific stimulus set. The local

importance of an attribute reflects the importance in judgment and depends on

the stimuli set under consideration.

(vi) It provides a logical structure

3. Limitations of the VFM: The major limitation is that it does not allow for

intransitivity. (Luce and Raiffa's, 1957) but this is not required in the context

of application in this study. It is also that argued a single super-value cannot

encompass all the different dimensions of the plurality of values (Rosenberger,

2001). Since each attribute is defined on a dimensionless scale this was not

considered to be a problem.

3.8.1.2. Description of the Concept of Efficiency Frontier:

The concept of Efficiency Frontier is based on the combination of two variables in

possible proportions to determine an optimal decision point for an expected return.

The variables are combined in different forms to sum up to a fixed total. The

combined option that produces the greatest value closest to an expected return is

defined as an efficiency lotus. The mathematical expression of the concept is

expressed as; Equation 3.6

m

e

i i i

∑

= 2 1 max Subject to

_em

g and i=1,2,3....n i i i =

∑

= 2 1 Where;

= is the worth of variable eat a set proportion.

e

i

= is the worth of variable mat a set proportion.

m

i

= the fixed total to which the different combinations of

_e

and

_m

must add up to.

g _i

i

1. Reasons for the Application of the Concept of Efficiency Frontier: The

application of the concept of efficiency frontier in this study was to

determine an optimal level of combined proportions of economic and

equity factors for road fund allocation. The selected indicator for assessing

economic efficiency was to maximise Net Present Value (NPV). The equity

indicator was based on affordability factor derived from VOC and income per

capita.

(i) The Net Present Value (NPV): It is defined as the difference between

discounted benefits and costs and estimated as;

∑

₊− = _t r Ct Bt NPV ) 1 ( . Equation 3.7

The NPV was adopted as an indicator for economic assessment on the basis of the

following reasons.

▪ It is an objectively quantified indicator and allows for comparison of alternatives.

▪ It allows investment alternatives to be ranked in order of their contribution to economic growth parameter;

▪ It maximises the economic worth of a project subject to budget constraints;

▪ It focuses on the total welfare gain of a project over the whole life; ▪ It presents a common unit to all the agencies and it is easy to

understand.

The properties of the NPV as compared to other decision indices is summarised in

Table 3.6.

Table 3.6: Economic Decision Criteria

NPV IRR NPV/Capital FYRR

Project Economic Validity Very Good Very Good Very Good Poor Mutually Exclusive Projects Very Good Poor Good Poor

Project Timing Fair Poor Poor Good

Project Screening Poor Very Good Poor Under Budget Constraint Fair Poor Very Good Poor Source: HDM-4 Version 2

(ii) The affordability Factor was adopted as an egalitarian equity measure to

provide leverage for road users with different levels of per capita income. The

rational was to compensate those who spend a higher proportion of their

income on transport costs by allocating higher proportions of the road

maintenance funds to such roads. It was estimated as income per km of travel

minus VOC/km. VOC was adopted as a proxy for transport costs for the

following reasons.

▪ Roads in poor condition have higher VOC’s and there is a high elasticity between VOCs and transport costs (Pratt, 2003).

▪ Transport cost is estimated as the sum of VOC and Profit and VOC’s constitutes a significant proportion of transport costs. For example, in

Ghana VOC is about 83 percent of transport cost. Table 3.7 provides

the details.

Table 3.7: Vehicle Operation Cost Components in Ghana

Item Weight ( Percentage)

Fuel 64.04 Cost of Vehicle 10.75 Comprehensive Insurance 3.11 Tyres 2.65 Spare Parts 12.60 Driver’s salaries 3.15

Driver’s Mate salaries 0.11

Lubricant 3.59

Total 100 Source:National Transport Co-ordinating Council, Ghana, 2004.

▪ VOC is policy sensitive and a major dynamic driving force that lead to changes in transportation costs (Nijkamp and Blass, 1996);

▪ VOC presents the single most objective common metric of

measurement for all maintainable road projects;

▪ It responds to long term trends.

▪ The study is on maintainable roads which are already open to traffic.

(iii) Income was used to adjust the VOC such that those with low income levels

who pay higher transport fares due to the high VOC resulting from poor road

condition will have higher preference in road fund allocation than others.

Income was selected as a strategic variable for the development of the

affordability factor because of the following reasons.

▪ It is a measurable indicator and the information is easily obtained. ▪ It is highlighted as important in determining social and distributional

impacts of transport by the DfT's recent rapid evidence assessment,

(DFT, 2005).

Estimated values of the efficiency and equity indicators at constrained budget levels

were combined such that for example an efficiency indicator generated at a 90 percent

constrained budget level was combined with an equity indicator generated at 10

percent constrained budget level to add up to 100. The process was repeated for all the

possible combinations of the corresponding decile proportions at which the budget

was constrained to generate the values on each indicator. The efficiency lotus was

defined as the combined proportions of the efficiency and equity indicators at a

constrained budget level which was closest to the combined proportions of efficiency

and equity indicators at unconstrained budget level.

The efficiency variable was defined as the stimuli of the NPV indicator within an I x I

impact matrix in the order of where; is a specific road section,

(

is the NPV/Cap estimated for the road section,

)

e

s

ei s

( )

i is the decile proportion at which budget was constrained to generate the corresponding NPV value. The equity indicator was

defined as the stimuli of the affordability factor which was determined within a J x I

impact matrix in the order of

_s

_mj where; is a specific road section, s

(

m

)

is income 72

per capita on a road type minus VOC/km for the particular road section and

(

is the decile proportion at which budget was constrained to generate the corresponding

VOC value. Table 3.8 presents the form of matrix representation from which the

values the efficiency and equity indicators were generated.

)

j

Table 3.8: Impact Matrix on Selected Variables

Road Section

Values of Variables at Decile Budget Proportions (

s

and )

ei

s

mj

s

e10

s

e20

s

e30

s

e40

s

e50

s

e60

s

e70

s

e80

s

e90

1

s

m10

s

m20

s

m30

s

m40

s

m50

s

m60

s

m70

s

m80

s

m90

s

e10

s

e20

s

e30

s

e40

s

e50

s

e60

s

e70

s

e80

s

e90

2

s

m10

s

m20

s

m30

s

m40

s

m50

s

m60

s

m70

s

m80

s

m90

s

en

s

en

s

en

s

en

s

en

s

en

s

en

s

en

s

en

n

s

mn

s

mn

s

mn

s

mn

s

mn

s

mn

s

mn

s

mn

s

mn

On the basis of Equation 3.6 the combined proportions of the efficiency and equity

indicators were estimated as;

_s

_s

Subject to

mj i ei

∑

=

2 1

max

∑

_i2₌₁

_s

_ei

_s

_mj=g and the

efficiency lotus was defined as illustrated in Figure 3.7. From Figure 3.7, if the

different combinations of and at different decile proportions of unconstrained

budget levels are identified as A, B, C, and D and is E is determined as the expected

return at an unconstrained budget level, then D is defined as the efficiency lotus since

it gives the closest value to E.

s

ei

s

mj

B C A E D (Maximum Fund Allocation Level) Efficiency Lotus Co mb in ed Variab les Budget Proportions

Figure 3.7: Efficiency Lotus for Fund Allocation on Efficiency and Equity Basis

In document Development of models for optimal road maintenance fund allocation : A case of Ghana (Page 82-91)