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A smoothing approach for solving transportation problem with road toll pricing and capacity expansions
 Robert Ebihart Msigwa^{1}Email authorView ORCID ID profile,
 Yue Lu^{2},
 Yingen Ge^{3, 4} and
 Liwei Zhang^{5}
https://doi.org/10.1186/s1366001507594
© Msigwa et al. 2015
 Received: 28 April 2015
 Accepted: 11 July 2015
 Published: 30 July 2015
Abstract
In this paper, we establish a bilevel optimization model for the equilibrium transportation problem concerning both capacity expansion and road toll pricing under the user equilibrium conditions. The bilevel optimization problem is reformulated as a mathematical programming problem with complementarity constraints (MPCC), which fails to satisfy the MangasarianFromovitz constraint qualification (MFCQ). We adopt a smoothing approach to overcome the lack of constraint qualifications in the MPCC problem. Under mild conditions, it has been proven that the sequence of the global optimal solutions generated by solving corresponding smoothing subproblems converges to one optimal solution of the original MPCC problem. Numerical experiments show that the proposed method is practical in solving user equilibrium transportation problems with capacity expansion combining road toll pricing.
Keywords
 bilevel programming
 perturbation approach
 FischerBurmeister function
 road toll pricing
 capacity expansion
1 Introduction
During the past decade, bilevel programming problems have received remarkable considerations in the decentralized planning problems concerning the decision progress with a hierarchical structure, which differ from the classical optimization problems as required to solve two levels of optimization tasks, e.g. the upper level (the leader problem) and the lower level (the follower problem). Two roots have dominated the study of bilevel programming. The first root dates back to the Stackelberg game [1], which is a famous problem in game theory. The second one stems from mathematical programming problems, in which the optimization problem becomes a constraint in another optimization problem [2]. Besides the paper above, a variety of literature has contributed to this topic from theoretical aspects to the computational point of view; see [3–6] and the references therein.
A lot of practical applications can be reformulated as bilevel programming problems, for instance [7–9]. Despite the extensive use in real life, bilevel programming problems are still difficult to solve in the optimization field because they require more computation time even for a small problems [5, 10]. If the lowerlevel problem is convex, people always substitute it by the corresponding KarushKuhnTucker (KKT) optimality conditions, so the whole model becomes a mathematical program with complementarity constraints (MPCC). Due to the hardness of the structure, such a problem does not satisfy the Mangasarian Fromovitz constraints qualification (MFCQ) and leads to the nonexistence of the Lagrange multipliers in the KKT optimality conditions [11]. As a result, many stateoftheart algorithms for solving nonlinear programming problems cannot be applied to this MPCC problem directly. Instead, various methods have been proposed in different branches of numerical optimization to solve bilevel programming problems, such as penalty type approaches [12], the branchbound approach [13], the Taylor approach [14], and the neural network approach [15]. A comprehensive review of using standard nonlinear programming (NLP) methods to solve MPCCs can be found in Fletcher and Leyffer [16].
In this paper, we consider the combined road toll pricing and capacity expansion problem, which is reformulated as a bilevel programming problem [17]. The upper level regarded as a leader aims to minimize the total system travel time/cost, and the lower level considered as a follower determines the individual user travel time subject to user equilibrium conditions. Because our lowerlevel problem is convex, the whole model can be transformed into an MPCC problem and further converted to a nonlinear program (NLP) problem. When the idea of smoothing methods is combined with the technique of variational analysis, a perturbationbased approach is proposed to relax the difficulty of solving the MPCC problem. Also, the objective function and constraints in the corresponding perturbed NLP problem are differentiable; a linear independent constraint qualification holds at every feasible point. Consequently, a sequential quadratic programming (SQP) solver in MATLAB is adopted to solve the smoothing subproblems. The numerical results show that the smoothing approach is efficient to solve the combined road toll pricing and capacity expansion problems compared with other solvers.
This paper is organized as follows. Section 2 establishes a bilevel optimization model for the capacity expansion problem with road toll pricing strategy under the user equilibrium conditions. Section 3 uses the proposed method to solve the combined pricing and capacity expansion problem. In Section 4, we report numerical results by the proposed method for different scale capacity expansion problems. Some concluding remarks are presented in the last section.
2 Bilevel programming problems
A bilevel programming involves two competing decisionmaking parties acting at different levels: one is the upperlevel decision makers (leader); the other is a lowerlevel decision maker (follower). Although the two levels interact with each other, yet each set has their decision variables and objectives and attempts to optimize their goals in sequence. The leader can adjust the performance of the overall system by setting some parameters to influence the decisions of the road users.
Recently, many problems found in the transportation literature have been reformulated as a bilevel programming, particularly in discrete network design problems [17, 18]. Gao et al. [19] introduced a traditional bilevel programming model for the discrete network design problem and new solution algorithm for analyzing the existing relationship between the improved flows and the new addition links in the existing urban network. Numerical results for proposed algorithm produced a better solution and performed efficiently in practice. Marcotte [20] conducted an extensive study of a continuous and nonlinear design problem where the problem was reformulated as a bilevel programming problem. The findings showed that heuristics can produce near optimal solutions. Suh and Kim [21] presented specific issues associated with solving a bilevel transportation planning model in which there is a publicprivate interaction. Also the study discussed issues on solving a large bilevel programming problem, which contribute to building a normative theory necessary for resources allocation in a mixed economy system.
In addition, the application of the bilevel programming to the network design problem reformulated as a nonlinear problem was studied by Friesz et al. [22], where the lowerlevel problem substituted with equivalent variational inequality problem. LeBlanc and Boyce [23] investigated a nonlinear bilevel network design problem while utilizing the user equilibrium route choice problem as the lowerlevel problem. Apart from the mentioned references, many researchers have reformulated second best toll pricing as a bilevel programming problem or mathematical program with equilibrium constraints (MPEC) [9, 24]. In these references, the upperlevel models are the leaders/managers responsible for planning where to add a new link and timing signals, and how much to charge road users. The lower level minimizes the individual route choice under user equilibrium conditions [25, 26] corresponding to these controls. Although a bilevel model provides a flexible platform for both the upperlevel and the lowerlevel problems and achieves the optimal solution simultaneously, these problems are difficult to solve because most of these problems are nonlinear and entail a nonconvex programming problem.
One advantage of dealing with the convex bilevel programming problem is that under mild constraint qualification, the lowerlevel problem can be replaced by its KarushKuhnTucker (KKT) optimality conditions to obtain an equivalence single level mathematical programming problem. Although bilevel programming has been used in various applications, one of the essential conditions for applying bilevel programming to solve designed problems is the availability of the efficient algorithms. In transportation road networks various algorithms have been proposed for solving the bilevel programming problems, such as the simulated annealing [27], the genetic algorithm [28], the ant colony algorithm [29]. These algorithms have also succeeded to solve other branches of network design problems [30–32] and [33]. However, in traffic assignment problems even when the upperlevel and the lowerlevel problems are convex, the resulting bilevel program itself may be nonconvex [34]. For that reason, up to now most of the proposed approaches are inapplicable when the size of the problem becomes big. Therefore, it is important to find advanced theoretical and methodological methods for handling such as bilevel problems efficiently. In this study, we adopt the smoothing based on the FB function to solve a combined road toll pricing and capacity expansion problem.
2.1 Mathematical formulation
Consider a road network \(G= (N, A) \) connected by sets of links and nodes denoted by A and N, respectively. Let r and s be the origin and destination on a given network, respectively. The set of origin and destination denoted by r and s is represented by w. Each origindestination (OD) pair w is connected by a set of paths (routes) represented by \(K_{w}\). Let \(q_{w}\) and \(u_{w}\) be the demand and the minimum travel time/cost between an OD pair w, respectively. The flow and travel time/cost on link a are given by \(x_{a}\) and \(t_{a}\), respectively. While \(f^{w}_{k}\) and \(c^{w}_{k}\) are the flow and travel time/cost experienced by travelers along the path \(k\in K_{w}\); \(\delta^{w}_{a, k} = 1\) if link a is part of path k connecting OD pair w and 0 otherwise.
We formulate the combined road toll pricing and capacity expansion problem as a bilevel programming problem under budget constraints.

\(q_{w}\) is the travel demand between OD pair w,

\(D_{w}(u_{w})\) is the demand function between OD pair w,

\(D^{1}_{w}(q_{w})\) is the inverse of the demand function, where \(D^{1}_{w}(q_{w})=h_{w}(q_{w})\), \(\forall{w\in{W}}\),

\(t_{a}(x_{a},y_{a})\) is the unit cost of travel on link a, where the t denote the vector of \(t_{a}(x_{a},y_{a})\), \(\forall{a\in{A}}\),

\(c_{a}\) is the capacity of each link, \(\forall{a\in{A}}\),

\({\bar{y}_{a}}\) is the upper bound for the link capacity expansion, \(\forall{a\in{A}}\),

\(g_{a}(y_{a})\) is the cost of improving link a,

\(g_{a}\) is a twice continuously differentiable and nondecreasing function,

F is the upperlevel objective function,

f is the lowerlevel objective function,

\(y_{a}\) is the link capacity, \(\forall{a\in{A}}\),

\(\tau_{a}\) is the link parameter for road pricing, \(\forall{a\in{A}}\),

θ is the conversion coefficient converting investment cost to travel cost,

\(\delta_{\overline{K}}(\overline{E}(z, u,v))\) is the indicator function,

\(\mathbf{B}_{\delta}(\bar{z}, \bar{u},\bar{v})\) is the open ball with center \((\bar{z},\bar{u},\bar{v})\) and radius δ.
The bilevel model consists of two problems: the leader problem and the follower problem (lowerlevel problem), which can be written as follows.
The leader problem
The follower problem
3 A perturbation approach for MPCC
Lemma 3.1
Proof
Theorem 3.1
Assume that \(\bar{f}\) is levelbounded. Then the function \(\kappa(\mu)\) is continuous at 0 with respect to \(\mathbb{R}_{+}\) and the setvalued mapping \(S(\mu)\) is outer semicontinuous at 0 with respect to \(\mathbb{R}_{+}\).
Proof
Corollary 3.2
As the linear independence constraint qualification holds at any feasible solution of (\(\mathrm{P}_{\mu}\)), we see that if there exists \(\bar{\lambda}\) such that the above KKT condition holds, then \(\bar {\lambda}\) is unique. The following proposition gives the secondorder sufficient conditions at a KKT point of (\(\mathrm{P}_{\mu}\)).
Proposition 3.3
Proof
4 Numerical examples
Example 1
(A 5link network)
Parameters for the 5link network
Link  \(\boldsymbol{A_{a}}\)  \(\boldsymbol{B_{a}}\)  \(\boldsymbol{C_{a}}\)  \(\boldsymbol{\tau_{a}}\) 

12  1  10  2  5 
13  2  5  3  3 
23  4  10  4  2 
24  5  15  2  3 
34  1  7  1  4 
\(q_{14}=15\)  \(\alpha_{14}= 2\)  \(\varphi_{14}=4\)  θ = 1.0 
Results for the 5link network for different link tolls
Link flow  1  3  5  \(\boldsymbol{\tau_{1\cdots{5}}}\) 

\(x_{1}\)  1.514  1.381  1.274  1.282 
\(x_{2}\)  2.875  2.282  1.169  2.108 
\(x_{3}\)  0.00  0.00  0.00  0.00 
\(x_{4}\)  1.514  1.381  1.274  1.282 
\(x_{5}\)  2.875  2.875  1.169  2.108 
\(y_{1}\)  0.000  0.000  0.000  0.000 
\(y_{5}\)  1.875  1.282  0.169  1.108 
\(q_{14}\)  4.390  3.662  2.442  3.390 
\({F_{\mathrm{max}}}\)  178.303  144.662  110.568  140.699 
Net benefit  1,250.7  1,151.8  1,099.7  1,153.9 
Example 2
(A 16link network)
Parameters for a 16link network
Link  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16 

\(\tau_{a}\)  1  5  2  4  3  2.5  1  2  6  1  4  3  1  2  3.5  1.5 
\({\varphi_{w}}\)  4  4  4  4  4  4  4  4  4  4  4  4  4  4  4  4 
Results for a 16link network for different link tolls
Link flow  1  3  5  \(\boldsymbol{\tau_{1\cdots16}}\) 

\(x_{2}\)  3.375  4.00  2.00  1.844 
\(x_{3}\)  1.561  0.00  0.00  0.00 
\(x_{6}\)  1.625  0.522  0.00  1.189 
\(x_{8}\)  1.813  0.00  2.00  1.844 
\(x_{9}\)  1.561  0.00  0.00  0.00 
\(x_{12}\)  1.625  0.522  0.00  1.189 
\(x_{14}\)  1.813  4.00  2.00  1.844 
\(x_{15}\)  1.561  0.00  2.00  0.00 
\(x_{16}\)  1.625  0.522  0.00  1.189 
Net benefit  9,566  8,973  8,105  9,494 
\(y_{2}\)  0.322  1.846  0.000  0.000 
\(y_{3}\)  0.000  0.000  0.000  0.000 
\(y_{6}\)  0.000  0.000  0.000  0.000 
\(y_{8}\)  0.000  1.000  0.000  0.000 
\(y_{9}\)  0.000  0.000  0.000  0.000 
\(y_{12}\)  0.000  0.000  0.000  1.433 
\(y_{14}\)  0.482  2.964  2.187  0.000 
\(y_{15}\)  0.367  2.168  1.828  1.639 
\(q_{16}\)  5.00  4.522  2.000  3.033 
\({F_{\mathrm{max}}}\)  364.833  188.231  55.891  181.561 
Table 4 shows that charging a single link and expanding others links provides a better result in the system optimum resulting in an increased network performance. It can be seen form this result that it is not always advantageous to consider a large number of road toll or capacity links when improving networks. This result indicates that road tolls have the ability to discourage the trips of road users resulting in a reduction of traffic congestion and the investments can be seen as a wastage. The type of road toll pricing associated with other factors such as quality of public transport services with induced demand depending on the time measured may significantly affect the capacity expansion strategy when usage is underpriced.
Example 3
(A 17link network)
Parameters for a 17link network
Link  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17 

\(\tau_{a}\)  1  5  2  4  3  2.5  1  2  6  1  4  3  1  2  3.5  1.5  4 
\({\varphi_{w}}\)  4  4  4  4  4  4  4  4  4  4  4  4  4  4  4  4  4 
Comparison results for a 17link network with tolls
Link flow  1  3  5  \(\boldsymbol{\tau_{1\cdots17}}\) 

\(x_{1}\)  2.135  0.938  0.386  2.037 
\(x_{2}\)  2.135  0.938  0.386  1.202 
\(x_{3}\)  1.000  0.938  0.386  1.202 
\(x_{4}\)  2.000  1.363  0.00  0.775 
\(x_{5}\)  0.000  0.000  0.000  0.835 
\(x_{6}\)  1.135  0.000  0.000  0.000 
\(x_{7}\)  1.000  0.938  0.386  1.202 
\(x_{8}\)  2.000  1.363  0.000  0.775 
\(x_{9}\)  2.000  1.363  0.00  1.610 
\(x_{10}\)  3.057  1.363  0.000  0.775 
\(x_{13}\)  0.078  0.000  0.000  0.835 
\(x_{14}\)  2.057  2.301  0.386  1.977 
\(x_{17}\)  0.078  0.000  0.000  0.835 
Net benefit  7,823  8,233  11,494  12,629 
\(y_{1}\)  0.328  0.000  0.000  0.000 
\(y_{2}\)  1.135  0.000  0.000  0.202 
\(y_{3}\)  0.000  0.000  0.000  0.000 
\(y_{4}\)  2.130  1.377  0.000  0.001 
\(y_{5}\)  0.000  0.000  0.000  0.000 
\(y_{6}\)  0.000  0.000  0.000  0.000 
\(y_{7}\)  0.000  0.369  0.000  0.413 
\(y_{8}\)  0.000  0.000  0.000  0.000 
\(y_{9}\)  0.000  0.000  0.000  2.431 
\(y_{10}\)  0.057  0.000  0.000  0.420 
\(y_{13}\)  1.009  0.000  0.000  0.263 
\(y_{14}\)  3.551  0.000  0.000  0.000 
\(q_{17}\)  4.135  2.301  0.386  2.812 
\({F_{\mathrm{max}}}\)  622.9811  398.818  82.2225  480.6378 

The capacity expansion relieves congestion and lower congestion charges has a negative effects on the price.

Capacity expansion improves transportation service; particularly traffic congestion would lead to a higher willingnesstopay by road users, which has a positive effect on the price.
5 Conclusions
In this paper, we have formulated the capacity expansion with the combined road pricing problem as a bilevel program, where the upper level optimizes the link capacity expansion vector and maximizes the social welfare, while the lower level determines the demand and the flow satisfying the Wardrop principles. Then the bilevel program is transformed to the MPCC model. The smoothing approach is proposed to solve the MPCC problem and this approach overcomes the lack of a suitable set of constraint qualifications. Under the mild conditions, the convergence property studied in this paper shows that the global optimal solution of the perturbed problem converges to the original solutions of the MPCC problem.
The perturbationbased approach and the established model were tested on 5link, 16link, and 17link road networks, widely used to analyze transportation networks. The numerical experiments indicate that the proposed model can be applied to solve various user equilibrium transportation problems efficiently. The proposed model can be employed to analyze the multimodal transportation networks to improve the environmental pollution caused by transport emissions.
The proposed model with the findings can be used by the planner to allocate the links for pricing and expansion under budget constraints. Although the proposed model may be computationally timedemanding and it may take time to find the optimal solution for largesized network design, yet it can easily be converted to a smaller dimensional problem and solved. The numerical examples show that the proposed model can produce a better solution of the combined road toll pricing and capacity expansion problem after solving the model several times with different values of the parameters.
Declarations
Acknowledgements
This research was supported by the National Natural Science Foundation of P.R. China (No. 11071029, No. 91330206 and No. 91130007). The authors would like to thank Dr. Xiantao Xiao in Dalian University of Technology for his helpful comments and suggestions on the subject of this paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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