To evaluate the sustainability of public sector investment in transportation infrastructure and service, this study proposes a mathematical programming model to solve multi-objectives optimization problems. A non-linear multi-objectives model, in which transport diversity and the gaps between sustainable target and present situation are simultaneously considered as the objectives, for resource allocation is thus established.
3.3.1 Fuzzy multi-objectives programming
Most real-world optimization problems are multi-objectives in nature. This implies that several objectives should be simultaneously considered. However, the complexity of multi-objectives optimization problems comes from the fact that there is no single optimal solution for these problems but rather a set of trade-offs called efficient solutions or Pareto-optimal solutions subject to resource constraints. The most crucial challenge is to find efficient feasible solutions in many multi-objectives planning issues. Consequently, methodologies to solve resource allocation problems, ones of the most discussed issues in combinatorial optimization theory, should address the following questions: Given a set of limited resource, what is the best allocation for a given task according to distinct targets? The best or potential efficient solutions should be determined considering a set of diverse and conflicting criteria (Belfares et al., 2007). Therefore, some researchers utilized heuristic algorithms to solve multi-objectives optimization problems (Altiparmak and Karaoglan, 2008;
de Cea et al., 2008). However, an aggregative approach is employed to reduce the multi-objectives problem to a single objective optimization problem in most cases. A convex solution set is the necessary condition that aggregative approaches generate proper Pareto optimal solutions (Das and Dennis, 1997).
This study aims to construct a prototype model to analyze the allocation of investment for urban public transit system. The previous literature exploring operations for public transportation has identified single objective, such as to minimize total costs or to maximize the consumer surplus, and constraints comprise service capacity and user trip demand (Ceder and Wilson, 1986; Martins and Vaz Pato, 1998). However, meeting the multiple objectives of sustainable transportation implies making trade-offs in considering the benefits and costs to different stakeholders. Lee and Moore (1973) argued that single objective models have neglected conflicting targets influencing decision processes related to transportation issues.
Some scholars solved the resource allocation problems for transportation operation by analytical models (van Nes and Bovy, 2000; Aldaihani et al., 2004; Ceselli et al., 2008), simulation analysis (Alterkawi, 2006) and heuristic algorithms (Gao et al., 2004; Jha et al., 2007).
During recent decades, fuzzy multi-objectives programming, which is a good method for identifying compromised solutions for optimization problem, has been applied to solve multi-objective linear, as well as nonlinear, programming problems (Bit et al., 1992; Lee and Li, 1993; Chang, 2007). Fuzzy multi-objectives programming combines fuzzy set theory and multi-criteria decision-making problems. The objective functions are represented via a fuzzy set, and the decision rule is used to select the solution with the highest membership of the decision sets. Zimmermann (1978) developed a fuzzy linear program, identical to the max-min program, and applied the fuzzy set theory concept with suitable membership functions. Solutions obtained by fuzzy multi-objectives programming are always efficient and optimize the comprised solution.
Moreover, fuzzy multi-objectives programming has been used in many fields. Li and Lee (1990) proposed a two-phase approach to get non-dominated solution and adapt it to de Novo programming with fuzzy parameters. Bhattacharya et al. (1992) utilized fuzzy multi-objectives programming to solve a multi-objective facility location problem. The genetic algorithm approach has been proved to be able to solve fuzzy multi-objectives programming with fuzzy nonlinear function goals and nonlinear constraints (Sasaki et al.
1995). Liang (2008) utilized fuzzy linear programming to assist in interactive multi-objectives transportation planning decisions. Furthermore, some studies concluded that using fuzzy multi-objectives programming for large problems, a compromise solution can easily be found, and is applicable to all types of multi-objective transportation problem (Bit et al., 1993; Islam and Roy, 2006). All the achievements of past studies increase the practicability of fuzzy multi-objectives programming.
Accordingly, fuzzy multi-objectives programming is utilized in this allocation model. In the compromise programming, the weights indicate the importance of the relative deviation of the objectives from the ideal, but in the fuzzy multi-objectives programming they express the importance of the deviations from the anti-ideal (Martinson, 1993). Following the procedures of the fuzzy multi-objectives programming algorithm, the ideal solution set
I
* ={W
s*} and the anti-ideal solution setI
# ={W
s#}should first be determined for the basic model, where*
W denotes the independently optimal performance for each indicator
s s whileW
s# represents the worst performance for each indicator s due to the optimization of the objective indicators non-s. For example, the model considers two objectives including transport diversity and the gap between sustainable goal and present value, e.g.s
=1,2.W
1* shows the optimal solution when transport diversity is identified as the objective function.Conversely,
W illustrates the worst value among the performance for transport diversity in
1#the optimization for minimizing the gap between sustainable target and present situation.
Furthermore, both the ideal and anti-ideal solution set are employed as a reference point to define the membership function,
DS
s(W
s), indicating the satisfaction degree of each objectiveW . The membership functions are represented as Eqn. 3-18 for minimization
s problems.Moreover, a compromise-grade λ , referring to overall satisfaction of the optimization model, is expressed as Eqn. 3-19.
)}
Through maximizing λ , the multi-objective problem can be transformed into the following problem and the compromised solutions, including the values of decision variables
x , compromise-grade
i λ and compromised objectivesW with each degree of satisfaction
sDS , are thus obtained.
s3.3.2 Resource Allocation Model
Aggregate indicators representing stakeholder needs for optimal public investment to support transport diversity are identified. From a research perspective, this study integrates research and draws on techniques from literatures in economics, sociology and environmentalism to achieve recommendations for resource allocation. This study considers numerous complicating factors affecting public investment allocation including externality, safety, accessibility, mobility, reliability, affordability, resource over-utilization, operator profit and level of universal design. From a policy perspective, this work characterizes improved resource allocation in conflict stakeholder needs.
Capturing all of the characteristics of urban public transit system in a single model is a challenging task. This study first lays the indicators referring to urban public transit stakeholders including users and operators of MRT and bus along with non-users for a precise mathematical description of problem in this section. Then objectives and constraints of the proposed resource allocation model are formulated.
According to the mentioned indicators in Section 3.2, this study selects 10 indicators representing urban public transit stakeholder needs (
y ), such as accessibility, affordability
and operator profit in both MRT and bus system, reliability and mobility for bus operation, as well as emission, safety and energy over-consumption for non-users (as shown in Eqn. 3-20 ~ Eqn. 3-29). The existent value of transportation infrastructure and service are marked with the suffix 0 for each variable and calculated as a constant in following analyses.0
bus private
Emi
T T
V
=α
4× +α
5× (3-29)To calculate the probability in Eqn. 3-27, this study assumes that the average travel time during the interval of bus stops follows the normal distribution. The bus reliability is assessed by the cumulative distribution function of bus headway indicated in Eqn. 3-30.
∫
−∞ −Additionally, Eqn. 3-31 reveals the travel time of bus affected by the length of bus exclusive lane when the average travel speed of system is assumed as external factor. The operational cost, as shown in Eqn. 3-32, relates to the policy variable namely headway. The monthly trips of MRT and bus, indicated as Eqn. 3-33 and Eqn. 3-34, respectively, are influenced through the satisfaction of each need in terms of urban public transit stakeholders.
The functions of adjustment factors are established by expert consensus building meeting.
Moreover, this study explores the connections among the metadata from GIS through regression model. The analytical results reveal that a non-linear regression model, particularly a logarithm regression model (shown as Eqn. 3-14), has a better goodness of fit and the coefficients are statistic significant at level 0.05. A detailed description of regression formulations are attached in appendix B.
)
The Taipei metropolitan area, the largest in Taiwan, provides the empirical study to discuss the managerial implications of the model. Except the unavailable data including driver behavior and patterns of conflict, the average speed assumed as constant here and the tiny variation of trips can not express the accident rate well. Besides, it is difficult in quantitatively determining the impact of level of universal design on system behavior. Therefore, accident rate and level of universal design are excluded from this study. Furthermore, it is assumed that the impacts of subsystems pedestrians, bicycles, private vehicles, as well as the land use patterns are given. Diverse transport stakeholders have different needs for urban transport
infrastructure and services. The main issue in transport diversity thus becomes how to more equitably satisfy diverse stakeholder needs. Transport diversity is defined as different levels of satisfaction within stakeholder needs, expressed as appropriate indicators and measured using the variations in achievement among indicators.
Additionally, minimizing the indicator gaps, the remainder of the needs achievement, between the expected goals and present values (as shown in Eqn. 3-35) is a key objective in urban transportation planning and thus the first objective in proposed model. The normalized value prevents indicator gaps resulting from differences in unit scale.
∑
⎟⎟where Oygoal and Othresholdy represent the expected goal and minimum threshold of indicator
y , respectively, and set via collaborative planning, specifically through consensus
building, based on stakeholder and public opinions, along with feedback from experts.V is
y the present value of indicatory . The value of the normalized gap exceeds 0 and the degree
of need satisfaction increases as the gap approaches 0. Meanwhile,n denotes the positive
i remainder of the gap of indicators, namely the achievement indicated by Eqn. 3-36. The non-negative achievement avoids misleading evaluation in transport diversity.⎟⎟
Moreover, the second objective is to maximize transport diversity in the form of Entropy (as shown in Eqn. 3-37) for equitably achieving the various conflicting needs of urban public transit stakeholders. Transport diversity calculated with Eqn. 3-37 comprises two components:
richness, measured by the number of stakeholder groups, which determines the number of terms in the summation, and equability, measured by the evenness of needs distribution across groups.
Besides, this study considers constraints. Budget constraint indicated as Eqn. 3-38 expresses the limited resource which should be allocated efficiently and equitably. Eqn. 3-39 denotes transportation capacity for public transit system. Moreover, the total trips in the Taipei metropolitan area are constant (as shown in Eqn. 3-40) due to the deficient consideration of trip generation. Because MRT operator makes a fixed positive profit, Eqn. 3-41 prevent bus operators from money-losing. The domain of each policy variable x is identified from Eqn.
3-42 to Eqn. 3-44, respectively. The upper boundaries of policy variables, x3, x5 and x6, are employed to keep the unreasonable negative travel costs and headway off.
7
C
HAPTER4 E
MPIRICALS
TUDYThe objective of this chapter is to demonstrate the methodologies presented in Chapter 3.
Prior to these demonstrations, a preliminary spatiotemporal analysis adopting the Taipei metropolitan area data is introduced in Section 4.1. Subsequently, the empirical study of the approach for exploring the causality and behavior of urban transportation systems, as well as of the approach for examining the impact of resource allocation on transport diversity are shown in Section 4.2 and 4.3, respectively.