Systematic budget allocation for transportation construction projects: a case in Taiwan

Abstract

It has always been a daunting task for any government to allocate the budgets for public infrastructure effectively so as to maximize the benefits of the limited resources. The most important contribution this paper makes is to conduct an empirical study of the budget allocations of the six region-based road system construction plans in northern Taiwan using the transportation budget allocation model devised in this paper. After reviewing the results, the experts consulted while preparing this paper believe that these results are both objective and able to meet the actual requirements. The competent authority also believes that they are better than those obtained using their original approach. In particular, the construction of this model takes into consideration the highest-level supervisors. To a certain degree, it caters for the practices in budget allocation so that the managers are able to assume the authority corresponding to their responsibility concerning management efficiency and final outcomes.

Appendices

Appendix A: Fuzzy analytic hierarchy process method

This paper applies the Fuzzy Analytic Hierarchy Process (FAHP) developed by Buckley (1985) to derive such weightings. The paper makes fuzzy the paired comparison values of Saaty’s AHP method (1980) and obtains fuzzy weightings based on geometric means. Below is a simple description of the theory and the method.

Let A = [a _ij ] be a positive reciprocal matrix. If the geometric mean of the n number of the assessment criteria is $r_{i} = \left( {\prod\nolimits_{j = 1}^{n} {a_{ij} } } \right)^{1/n}$, Saaty (1980) defines λ _max as the largest eigenvalue of Matrix A. Therefore, the weighting w _i is the item of the normalized eigenvector of corresponding λ _max, i.e., the respective criterion weighting $w_{i} = {{r_{i} } \mathord{\left/ {\vphantom {{r_{i} } {(r_{1} + \cdot \cdot \cdot + r_{n} )}}} \right. \kern-\nulldelimiterspace} {(r_{1} + \cdot \cdot \cdot + r_{n} )}}$. Buckley (1985) assumes a fuzzy positive reciprocal matrix $\tilde{A} = [\tilde{a}_{ij} ]$. The extended geometric mean method is used to define the fuzzy geometric mean and the weightings of individual criteria as Eq. A-1:

$$ \tilde{r}_{i} = (\tilde{a}_{i1} \otimes \tilde{a}_{i2} \otimes \cdots \otimes \tilde{a}_{in} )^{1/n} ,\tilde{w}_{i} = \tilde{r}_{i} \otimes (\tilde{r}_{1} \oplus \cdots \oplus \tilde{r}_{n} )^{ - 1} . $$

(A-1)

If there are 7 judges, and they all integrate the opinions from many participating judges (scholars, experts, control government controllers and supervisors, transportation authorities and organizers), the geometric mean method proposed by Buckley (1985) can be used to compute an average weighting matrix, as shown by Eq. A-2:

$$ \tilde{e}_{ij} = (\tilde{e}_{ij}^{1} \otimes \tilde{e}_{ij}^{2} \otimes \cdots \otimes \tilde{e}_{ij}^{T} )^{1/T} ,\tilde{e}^{l} = \left[ {\tilde{e}_{ij} } \right]^{l} ,{\text{ for each aspect}} . $$

(A-2)

Let $\tilde{e}_{ij}^{T}$ be the judgment criterion i of the Tth judge in relation to the fuzzy comparison value of the criterion j. Therefore, the paired comparison matrix between in the l aspect should be $\tilde{e}^{l}$.

Appendix B: Fuzzy multi-criteria grade classification model

The Fuzzy multi-criteria grade classification model (FMGCM; Teng et al. 2007) mentioned in this paper deals precisely with the problems of priority in implementing transportation construction projects. The transportation construction projects at the highest priority level will be given top priority in budget allocation while the implementation of those with the lowest priority level will hinge on whether any money remains. The following are the theoretical methods and procedures of Fuzzy MCDM applied in this paper.

Procedure 1

List all transportation construction projects to be evaluated.

List s _i transportation construction projects: $A = \left\{ {A_{1} , \ldots ,A_{i} , \ldots ,A_{{s_{i} }} } \right\}(s_{i} \ge 1).$

Procedure 2

Formulate evaluation criteria.

Formulate n evaluation criteria: $C = \left\{ {C_{1} , \ldots ,C_{j} , \ldots ,C_{n} } \right\}(n \ge 1).$

Procedure 3

Distinguish Need Grades.

Use each evaluation criterion to size up the performance of the suggested transportation construction projects; distinguish P Need Grades $V_{j} = \left\{ {V_{j1} , \ldots ,V_{jk} , \ldots ,V_{jp} } \right\}(P \ge 2)$ in accordance with the performance—the grade priority tends to decrease, namely, V _j1 indicates the Most Needed while V _jp the A Least Needed.

Procedure 4

Determine the weights for evaluation criteria.

Determine the importance of n evaluation criteria; the experts in the related fields can be invited to assign respective weights: $\tilde{w} = \left\{ {\tilde{w}_{1} , \ldots ,\tilde{w}_{j} , \ldots ,\tilde{w}_{n} } \right\}$.

Procedure 5

Establish evaluation matrix.

In light of the judgments and weightings made by the members on the evaluation panel, the fuzzy evaluation matrix $\tilde{R}_{{s_{i} }}$for each transportation construction project A _i under the evaluation criteria C _j can be formulated as follows:

$$ \tilde{R}_{{s_{i} }} = \left[ {\tilde{r}_{ijk} } \right]_{n \times P} ,\quad i = 1,2, \ldots ,n. $$

(B-1)

In the formula $\tilde{r}_{ijk}$indicates the $A_{{s_{i} }}$fuzzy performance value of the V _jk Grade in the evaluation criteria C _j .

Due to the qualitative nature of the evaluation criteria for transportation construction projects, the related experts, optimally 5–9 people, can be organized to form an evaluation panel. The job of each member is to make his/her judgment on the grade of each criterion. Each member is required to tick one grade out of P grades in each criterion.

Suppose there are T members on the evaluation panel. The fuzzy performance value $\tilde{r}_{ijk}$of each grade in each criterion for each transportation construction project can be obtained through the following formula.

$$ \tilde{r}_{ijk} = {\raise0.7ex\hbox{${T_{ijk} }$} \!\mathord{\left/ {\vphantom {{T_{ijk} } T}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$T$}}, \quad \forall i,j,k. $$

(B-2)

$$ T_{ijk} = \sum\limits_{f = 1}^{T} {T_{ijk}^{f} }, \quad \forall i,j,k $$

(B-3)

$$ T_{ijk}^{f} = \left\{ {\begin{array}{*{20}c} {1,} & {{\text{Indicates the member's judgement that }}A_{{s_{i} }} {\text{ belongs to Grade }}k{\text{ in the criterion }}C_{j} .} \\ {0,} & {\text{others}} \\ \end{array} } \right. $$

(B-4)

$$ \sum\limits_{f = 1}^{T} {\sum\limits_{k = 1}^{P} {T_{ijk} } } = T $$

(B-5)

$$ 0 \le \tilde{r}_{ijk} \le 1. $$

(B-6)

Procedure 6

Calculate the fuzzy evaluation vector after the weighting.

When considering n evaluation criteria simultaneously, we have the following $\tilde{E}_{i}$—the fuzzy evaluation vector after the weighting.

$$ \tilde{E}_{i} = \left( {\tilde{E}_{i1} , \ldots ,\tilde{E}_{ik} , \ldots ,\tilde{E}_{ip} } \right) = \tilde{w} \otimes \tilde{R}_{{s_{i} }} $$

(B-7)

In the formula

$$ \tilde{E}{}_{ik} = \sum\limits_{j = 1}^{n} {\tilde{w}_{j} \tilde{r}_{ijk} , \quad k = 1,2, \ldots ,P} $$

(B-8)

$$ \sum\limits_{k = 1}^{P} {\tilde{E}_{ik} } = 1, \quad i = 1,2, \ldots ,n. $$

(B-9)

Procedure 7

Grade classification.

Since $\tilde{E}_{ik}$ refers to the $A_{{s_{i} }}$ degree of membership of the Grade k in n evaluation criteria, the priority sequence of transportation construction projects is the result of the grade classification of s _i transportation construction projects—$A_{1} ,A_{2} , \ldots ,A_{{s_{i} }}$ when $\tilde{E}_{i1} ,\tilde{E}_{i2} , \ldots ,\tilde{E}_{iP}$ are known information.

(1) If there are only two grades classified

• a. If there are two grades (namely, P = 2, V = {X, Y}) classified in the grade classification, and the degree of membership of Grade X meets the value of M (which is decided by the evaluation panel, for instance, θ = 0.5 or θ = 0.7), then $A_{{s_{i} }}$ belongs to X; if the degree of membership of Grade X fails to meets the value of θ, then $A_{{s_{i} }}$ belongs to Y.

• b. Let α _x and α _y be the degree of membership of $A_{{s_{i} }} \in X$ and $A_{{s_{i} }} \in Y$ respectively and α _x  + α _y  = 1, then there are three possibilities:

  • (a) ${\text{If}}\,\alpha_{x} > \theta ,\quad {\text{then}}\;A_{{s_{i} }} \in X;$

  • (b) ${\text{If}}\;\alpha_{x} = \theta ,\quad {\text{then}}\;A_{{s_{i} }} \in X\;{\text{or}}\;A_{{s_{i} }} \in Y$

  • (c) ${\text{If}}\;\alpha_{y} > \theta ,\quad {\text{then}}\;A_{{s_{i} }} \in Y.$

(2) If there are three grades classified

If there are three grades classified ((P = 3), V = {X, Y, Z}),

• a. In classifying the grade of $A_{{s_{i} }}$, use V ₁ = {X, Y or Z} and V ₂ = {Y, Z} to conduct evaluation to obtain their respective degrees of membership $\left( {\alpha_{1} ,\bar{\alpha }_{1} } \right)$ and $\left( {\alpha_{2} ,\bar{\alpha }_{2} } \right)$, $\left( {\alpha_{1} + \bar{\alpha }_{1} = 1} \right)$ and $\left( {\alpha_{2} + \bar{\alpha }_{2} = 1} \right)$.

• b. Use the values of α ₁ and α ₂ to make the grade classification:

  • (a) First make the grade classification based on the value of α ₁

    • (i) ${\text{If}}\;\alpha_{1} \ge \theta ,\quad {\text{then}}\;A_{{s_{i} }} \in X$

    • (ii) ${\text{If}}\;\overline{{\alpha_{1} }} \ge \theta ,\quad {\text{then}}\;A_{{s_{i} }} \in X\;{\text{or}}\;A_{{s_{i} }} \in Z;$

  • (b) Then make the grade classification based on the value of α ₂

    • (i) ${\text{If}}\;\alpha_{2} \ge \theta ,\quad {\text{then}}\;A_{{s_{i} }} \in Y;$

    • (ii) ${\text{If}}\;\overline{{\alpha_{2} }} \ge \theta ,\quad {\text{then}}\;A_{{s_{i} }} \in Z;$

(3) If there are P grades classified,

• a. For P evaluation grades $V = \left\{ {V_{1} ,V_{2} , \ldots ,V_{P} } \right\}\left( {P \ge 2} \right),P - 1$ evaluation sets composed of every two immediate grades can be construed:

  $$ \begin{array}{*{20}c} {V_{1}^{'} = \left\{ {V_{1} ,V_{2} \;{\text{or}}\;V_{3} \;{\text{or}}\; \ldots ,V_{P} } \right\}} \\ {V_{2}^{'} = \left\{ {V_{1} ,V_{2} \;{\text{or}}\;V_{3} \;{\text{or}}\; \ldots ,V_{P} } \right\}} \\ \vdots \\ {V_{P - 1}^{'} = \left\{ {V_{P - 1} ,V_{P} } \right\}} \\ \end{array} $$

• b. Evaluate the same project $A_{{s_{i} }}$ in accordance with $V_{1}^{'} ,V_{2}^{'} , \ldots ,V_{P - 1}^{'}$ and obtain the corresponding degrees of membership—$\alpha_{1} ,\alpha_{2} , \ldots ,\alpha_{P - 1}.$

• c. Make a grade classification according to the rules, as shown below

  • (1) ${\text{If}}\;\alpha_{1} \ge \theta ,\quad {\text{then}}\;A_{{s_{i} }} \in V_{1} ;\;{\text{otherwise}},$

  • (2) $\begin{array}{c}{\text{If}}\;\alpha_{2} \ge \theta ,\quad {\text{then}}\;A_{{s_{i} }} \in V_{2} ;\;{\text{otherwise}} \\ \vdots \\ {{\text{For}}\;(P - 1),\;{\text{if}}\;\alpha_{p - 1} \ge \theta ,\quad {\text{then}}\;A_{{s_{i} }} \in V_{P - 1} ;\;{\text{otherwise}},\;A_{{s_{i} }} \in V_{P} .}\end{array}$

• d. It can be seen from the above classification logic and inferring process that the problem of grade classification will be easily resolved if the degrees of membership of $\alpha_{1} ,\alpha_{2} , \ldots ,\alpha_{P - 1}$ are calculated.

Procedure 8

Calculate the degree of membership for the evaluation value of each grade.

(1) In this paper, the inference by Hsinng and Tsaur (1991) is adopted to calculate the degree of membership for each evaluation value of each grade. They are as follows:

$$ \begin{array}{*{20}c} {\alpha_{1} = E_{i1} = \sum\limits_{k = 1}^{1} {E_{ik} } } \\ {\alpha_{2} = \alpha_{1} + E_{i2} = \sum\limits_{k = 1}^{2} {E_{ik} } } \\{\alpha_{3} = \alpha_{2} + E_{i3} = \sum\limits_{k = 1}^{3} {E_{ik} } } \\\vdots \\ {\alpha_{P - 1} = \alpha_{P - 2} + E_{{i\left( {P - 1} \right)}} = \sum\limits_{k = 1}^{P - 1} {E_{ik} } } \\ {\alpha_{P} = \sum\limits_{k = 1}^{\text{P}} {E_{ik} } } \\ \end{array} $$

(B-10)

(2) $\alpha_{1} ,\alpha_{2} , \ldots ,\alpha_{P}$ are the actual degrees of membership of the transportation construction project $A_{{s_{i} }}$ when $A_{{s_{i} }}$ is evaluated at the grades $1,2, \ldots ,P$.

$$ \begin{aligned} E_{i\Upsigma } = & \left( {\alpha_{1} ,\alpha_{2} , \ldots ,\alpha_{P} } \right) \\ = & \left( {\sum\limits_{k = 1}^{1} {E_{ik} } ,\sum\limits_{k = 1}^{2} {E_{ik} } , \ldots, \sum\limits_{k = 1}^{P} {E_{ik} } } \right) \\ \end{aligned} $$

(B-11)

Procedure 9

Grade Classification for all transportation construction projects.

All transportation construction projects can be classified into different grades in accordance with the degree of membership for each grade evaluation set as well as the value of θ determined by the evaluation panel.

$$ {\text{If }}\alpha_{id} \ge \theta ,\;{\text{then}}\;A_{{s_{i} }} \in V_{d} ,\quad \forall i,k $$

(B-12)

$$ {\text{And}}, \, d = \min \left\{ {\left. k \right|\alpha_{ik} \ge \theta } \right\}. $$

(B-13)

Appendix C: Fuzzy multi-criteria project ranking model

When evaluating and ranking the projects at the same grade, we suggest combining scoring methods with the multi-criteria property to construct a fuzzy multi-criteria project ranking model.

Suppose there are n _k projects at grade k, and then under n evaluation criteria C _j  (j = 1, 2, …, n), T evaluation members carry out the individual scoring using a fuzzy scale and linguistic expressions respectively for the n criteria; the scope of scoring is 1, 2, …, Q, and a higher score indicates more urgent need.

Suppose the fth evaluation member gives the project the score $\tilde{A}_{{s_{i} }} \left( {s_{i} = 1,2, \ldots ,n_{k} } \right)$ at grade k under criterion C _j , and the result is expressed as $\tilde{E}_{ij}^{f}$, then $\tilde{A}_{{s_{ij} }}$—the average score given to $A_{{s_{i} }}$under criterion C _j —is

$$ \tilde{A}_{{s_{ij} }} = {{\sum\limits_{f = 1}^{T} {\tilde{E}_{ij}^{f} } } \mathord{\left/ {\vphantom {{\sum\limits_{f = 1}^{T} {\tilde{E}_{ij}^{f} } } T}} \right. \kern-\nulldelimiterspace} T},\quad \forall i,j $$

(C-1)

If the weights of n evaluation criteria are taken into account, the average weighting of the score of $A_{{s_{i} }}$ at grade k is

$$ \tilde{R}_{{s_{i} }} = \sum\limits_{j = 1}^{n} {{\tilde{\text{w}}}_{j} \circ } \tilde{A}_{{s_{ij} }} ,\quad \forall i,j $$

(C-2)

The sign “$\circ$” indicates the calculation of the fuzzy numbers, including fuzzy addition and fuzzy multiplication. The fuzzy numbers in the equation are fuzzy subsets of real numbers. It is an extension of the concept of confidence intervals. According to the definition by Dubois and Prade (1978), fuzzy number $\tilde{A}_{{}}$ refers to a fuzzy set. Its membership function $\mu_{{\tilde{A}}} (x)$: R → [0, 1], and x refers to the score for the assessed item. It carries the following characteristics:

• (1) $\mu_{{\tilde{A}}} (x)$: Continuous mapping of domain R [0, 1];

• (2) $\mu_{{\tilde{A}}} (x)$: a Convex Fuzzy subset;

• (3) $\mu_{{\tilde{A}}} (x)$: a Normalization Fuzzy subset, with a real number x₀, to make $\mu_{{\tilde{A}}} (x_{0} ) = 1$.

The numbers that meet the above criteria are fuzzy numbers. Below are the details of the characteristics of the triangular fuzzy number $\tilde{A} = \left( {L,M,U} \right)$ and calculations (Buckley 1985), as shown in Eqs. C-3–C-10 and Fig. 8.

$$ \mu_{{\tilde{A}}} (x) = \left[ \begin{gathered} \begin{array}{*{20}c} {{{\left( {x - L} \right)} \mathord{\left/ {\vphantom {{\left( {x - L} \right)} {\left( {M - L} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {M - L} \right)}}} & {L \le x \le M} \\ \end{array} \hfill \\ \begin{array}{*{20}c} {{{\left( {U - x} \right)} \mathord{\left/ {\vphantom {{\left( {U - x} \right)} {\left( {U - M} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {U - M} \right)}}} & {M \le x \le U} \\ \end{array} \hfill \\ \begin{array}{*{20}c} {\begin{array}{*{20}c} {} & {} & {} \\ \end{array} 0} & {\begin{array}{*{20}c} {} & {} \\ \end{array} } & {\text{otherwise}} \\ \end{array} \hfill \\ \end{gathered} \right. $$

(C-3)

Fig. 8

Triangular fuzzy number and α-level

• (1) Addition of fuzzy numbers

$$ (L_{1} ,M_{1} ,U_{1} ) \oplus (L_{2} ,M_{2} ,U_{2} ) = (L_{1} \oplus L_{2} ,M_{1} \oplus M_{2} ,U_{1} \oplus U_{2} ) $$

(C-4)

• (2) Multiplication of fuzzy numbers

$$ (L_{1} ,M_{1} ,U_{1} ) \otimes (L_{2} ,M_{2} ,U_{2} ) = (L_{1} \otimes L_{2} ,M_{1} \otimes M_{2} ,U_{1} \otimes U_{2} ),\quad {\text{for}}\;L_{1} \ge 0 $$

(C-5)

Any real number $k \ge 0$

$$ k \otimes \tilde{A} = (k,k,k) \otimes (L,M,U) = (kL,kM,kU) $$

(C-6)

• (3) Subtraction of fuzzy numbers

$$ (L_{1} ,M_{1} ,U_{1} ) - (L_{2} ,M_{2} ,U_{2} ) = (L_{1} - U_{2} ,M_{1} - M_{2} ,U_{1} - L_{2} ) $$

(C-7)

• (4) Division of fuzzy numbers

  $$ (L_{1} ,M_{1} ,U_{1} ) \div (L_{2} ,M_{2} ,U_{2} ) = (L_{1} /U_{2} ,M_{1} /M_{2} ,U_{1} /L_{2} ),\quad {\text{for}}\;L_{1} \ge 0,L_{2} \ge 0 $$

  (C-8)

• (5) Inverse of fuzzy numbers

$$ (L_{1} ,M_{1} ,U_{1} )^{ - 1} = (1/U_{1} ,1/M_{1} ,1/L_{1} ),\quad {\text{for}}\;L_{1} \ge 0 $$

(C-9)

• (6) α-level:

  $$ \begin{aligned} \forall {{\upalpha}} \in & [0,1 ],\tilde{A}\;{\text{of}}\;{{\upalpha}} - {\text{cut}}\;{\text{shows}}\;{}^{{{\upalpha}}}\tilde{A} ,\;{\text{and}} \\ {}^{\alpha }\tilde{A} = & [ (M - L ) {{\upalpha}} + L, - (U - M )\alpha + U ]= [{}^{{{\upalpha}}}L ,{}^{{{\upalpha}}}U ]\\ \end{aligned} $$

The average weighting score $\tilde{R}_{{s_{i} }}$ is defuzzed, and this value can be used as the reference to rank the n _k transportation construction projects at grade k according to need priority. The bigger the average value of weighting, the higher the need priority in terms of the comprehensive consideration of n evaluation criteria; therefore, the project has a greater advantage in the competition for funding.