Abstract

This paper proposes a model to address an urban transport planning problem involving combined network design and signal setting in a saturated network. Conventional transport planning models usually deal with the network design problem and signal setting problem separately. However, the fact that network capacity design and capacity allocation determined by network signal setting combine to govern the transport network performance requires the optimal transport planning to consider the two problems simultaneously. In this study, a combined network capacity expansion and signal setting model with consideration of vehicle queuing on approaching legs of intersection is developed to consider their mutual interactions so that best transport network performance can be guaranteed. We formulate the model as a bilevel program and design an approximated global optimization solution method based on mixed-integer linearization approach to solve the problem, which is inherently nnonlinear and nonconvex. Numerical experiments are conducted to demonstrate the model application and the efficiency of solution algorithm.

1. Introduction

The growth of urban population, social progress, and economic prosperity lead to an increasing travel demand. Therefore, it is imperative for the planners to understand how to determine the optimal transportation planning to ensure best transportation system performance so that the fast growing travel demand can be addressed in the planned transport network. One important issue in transportation planning is to find out the optimal decision on road capacity design, that is, the optimal road capacity expansions and optimal planning on new links addition. Another intrinsic issue to be addressed in urban transportation planning is the equilibrium network signal setting, which seeks to optimize the road capacity allocation in the network by devising the signal control strategy.

In the literature, network design problem (NDP) has received extensive research attention for decades [1–5]. Historically, the NDP has emerged to cater for the faster growing travel demand via the optimal decision on the expansion of urban road network, subject to the limited resources available for system improvement. The objective of the NDP is to optimize a specific performance index of the transportation system (e.g., minimize the total monetary travel cost), accounting for drivers’ routing behavior at the same time. There are mainly three types of the NDP: one that is dealing with the continuous capacity expansion of links, referred to as continuous NDP (CNDP) [1, 2, 6], another one that determines the optimal decision of the new links addition, referred to as discrete NDP (DNDP) [7–9], and a combination of the former two types, referred to as mixed NDP (MNDP) [10]. Recently, some researchers also considered travel demands variation over planning horizon in NDP, which is categorized into a time-dependent NDP [11, 12]. The NDP is often formulated as a bilevel problem with upper-level describing the expansion decision-making of transport planners to minimize the investment and travel cost, while lower-level is representing the traffic assignment on the network for given expansion plan. For comprehensive review on models and algorithms of the NDP, one can refer to Yang and Bell [13] and Farahani et al. [14]. The equilibrium network signal setting (ENSS) problem involves the optimal signal control of the transport system to improve the network performance (e.g., minimize delay, fuel consumption, or travel time). Studies on ENSS include two categories: local optimization of signal setting for individual intersection and global signal setting for the network. Local methods are widely studied in previous studies [15–18], which assume fixed and exogenously given network flows. On the other hand, global methods [19–22] determine signal setting parameters and endogenous equilibrium network flows simultaneously.

The solution of both the CNDP and ENSS problems is a challenging issue because the model formulations are intrinsically nonlinear and nonconvex, which makes it difficult to even find solution algorithm for local optimum [23, 24]. Sheffi and Powell [20] proposed a gradient projection method on solving the ENSS problem and it is subsequently extended in the work of Cascetta et al. [25], Lee and Machemehl [26], and Cipriani and Fusco [27]. Chiou [28] exploited another descent approach implementing four variants of gradient-based methods to solve the general CNDP. In addition, based on the results of Tobin and Friesz [29], a number of sensitivity analysis based algorithms were introduced to solve bilevel transport planning and management problems: Friesz et al. [4], Sumalee et al. [30], and Connors et al. [31] applied this method to the general CNDP and relevant problems; Yang and Yagar [22] developed an algorithm to solve the ENSS with capacity constraints; Ying et al. [32] further developed a sensitivity analysis based algorithm on ENSS, which significantly improved computing efficiency by avoiding the path-enumerating process. Moreover, Meng et al. [33] imposed an augmented Lagrangian method to solve an equivalent single-level problem of CNDP with nonconvex and continuously differentiable properties, which was transformed via a marginal function tool from the bilevel program. However, for the nonconvex nonlinear CNDP and the ENSS problems, most of the gradient-based algorithms can only obtain local optimal solutions and the solutions are heavily dependent on the selection of initial points. Besides, getting gradient information will be much more difficult with the increasing size of the network. The other type of solution method is metaheuristic algorithm. Abdulaal and LeBlanc [1] suggested Hooke-Jeeves algorithm to solve the CNDP by transferring the CNDP into an unconstrained nonlinear program. Then a simple metaheuristic algorithm called equilibrium decomposed optimization method was developed by Suwansirikul et al. [6] for practical use. It increased the computing efficiency through decomposing the CNDP into interacting subproblems and calculating user equilibrium at each iteration. Besides, genetic algorithm [24, 26, 34, 35], simulated annealing [36, 37], and tabu search [18, 38, 39] are also applied to the CNDP and ENSS problem. The advantage of the metaheuristic algorithms is that they are simple and can be easily implemented on the nonconvex nonlinear problems with many local optimal solutions. What is more, as derivative information is not needed, they avoid the troublesome gradient calculation process [35]. However, generally long computational time is needed in the stochastic global search, and the global optimal solution cannot be guaranteed. Recently, Wang and Lo [40] proposed a method to obtain the global optimal solution of linearized CNDP. This method is soon generalized in the work of Luathep et al. [10] for solving the mixed NDP. Many other research works developed global optimization solution algorithms to solve the network design problems [41–43]. Szeto et al. [44] applied artificial bee colony algorithm to solve network design problem.

Basically, most of the previous research works investigated the two planning problems, road capacity design in NDP and equilibrium network signal setting (ENSS), separately. However, in practice, NDP determines the link capacity, while ENSS determines the capacity allocation; and the combination of the road capacity design and the capacity allocation design is needed that fully determines the resultant equilibrium travel pattern and the transportation network performance. Most of previous studies addressed the two problems separately, regarding the other problem as exogenously given, which ignores the mutual interaction between them in determining the transport network performance. Ziyou and Yifan [45] developed an integrated model to combine the concept of reserve capacity and CNDP, which aimed to maximize the reserve capacity of the network. A heuristic algorithm based on sensitivity analysis was applied to solve their problem. Cantarella et al. [18] used LOSS (local optimization of signal setting) method to study the discrete network design problem (lane layout) combined with signal setting and proposed several heuristic methods to solve it. Nevertheless, the presented solution algorithms cannot guarantee the global optimization solutions of the two problems, both of which did not notice the queuing phenomenon on an approaching leg of intersection. In this paper, we would present a combined network design and signal setting problem (CNDSSP) with consideration of queuing vehicles storage constraint on approaching legs of intersections to minimize the total transportation system cost. The problem is formulated as a nonlinear and nonconvex model. Moreover, we design an approximated global optimization solution method to solve the formulated model, which if solved by a local optimal algorithm may result in urban transportation plan that is deviated from the true optimal solution. Following the approximated global optimization solution idea introduced in the work of Wang and Lo [40], we transform the originally formulated bilevel program into a single-level mixed-integer linear program by applying a mixed-integer linearization approach. Noting that the solution efficiency of this method is highly restricted by the number of introduced integer variables, we apply a new mixed-integer linearization model, which significantly reduces the number of integer variables needed and thus improves the solution efficiency. Specifically, the number of integer variables introduced for the linearization follows logarithmic relationship with the size of the partition scheme, which requires much smaller number of integer variables even if high linearization accuracy and refined partition scheme are needed. In summary, this study contributes to literature in two aspects: firstly, we propose a combined network design and signal setting transport planning model with consideration of queuing vehicle storage restriction on approaching legs of intersections to take into account their interaction and ensure the true optimal transportation planning. Secondly, an efficient global optimization solution method is developed to seek the approximated globally optimal design of the transportation planning problem. Numerical studies are conducted to demonstrate the solution quality by comparing the proposed model and solution algorithm with the other existing models and solution methods.

The remainder of this paper is organized as follows. In Section 2, the bilevel model of combined network design and signal setting is stated and then reformulated as a single-level mixed-integer program (MIP). Section 3 demonstrates the single-level model linearization process. A numerical example is given in Section 4 to illustrate the proposed model and solution method and to compare the computational results with other models and algorithms. Some final remarks are presented in Section 5.

2. Model Formulation

The following notation is used in the formulation.

Parameters  : the set of links in the network,  : the set of links with signal control, ,  : the set of links without signal control, ,  : the set of approaching links for intersection in the network,  : the set of signal-controlled intersections in the network,  : the set of origin-destination (OD) pairs,  : the set of paths between OD pair ,  : the fixed OD travel demand, ,  : the link-path incidence matrix, , , , , where if link belongs to path between OD pair , and otherwise,  : the conversion factor which turns travel time into monetary cost.

Variables  : the traffic flow of link ,  : the green split of link ; if ,  : the queuing delay on link ; if ,  : the capacity of link after expansion,  : path flow of path between OD pair ,  : travel time of path between OD pair ,  : minimum path travel cost between OD pair .

Functions  : travel time function of link , which is the function of link flow , green split , and link capacity ,  : the link expansion cost function, , which is assumed to be the function of ,  : the link storage capacity function, , which is assumed to be the function of .

2.1. Bilevel Model for CNDSSP

Given the OD demand pattern, which is supposed to be relatively stable within the planning horizon and current network layout, the combined continuous network design and signal control problem in this study aims to determine which links should be expanded, how much the link capacity should be improved, and what the green split of each signal-controlled link is. The objective is to minimize the total travel time cost and the investment on link capacity enhancement subject to the feasible range of signal settings and constraints of maximum link capacity expansions.

The combined network design and signal setting problem can be modeled into a bilevel program stated as follows:while the equilibrium traffic flow is determined as follows:where and are the upper and lower bound of green split of link , is the current capacity of link , and is the maximum link capacity after expansion due to the restriction of urban land use planning.

In the above formulation, the upper-level of the model is to seek the optimal green splits and road capacity enhancement to minimize the total travel time cost and investment on link capacity expansion. Here, total travel time cost includes all flow-dependent running time, signal delay, and queuing delay at link exits due to road saturation in the network. As was done in the work of Yang and Yagar [22], the running time and signal delay are captured by travel time function and the queuing delay is endogenously solved and denoted as . Constraint (2) states the given green split restriction and entails the conservation equation depending on the specific signal phase structure of an intersection. Equation (3) describes the boundary constraints of link capacity expansion. As a reasonable signal control should consider the queuing length on the approaching leg of an intersection to avoid blocking the upstream section, (4) enforces that number of queuing vehicles on an approaching leg should be restricted to the link storage capacity. and can be endogenously obtained by solving the lower-level program, which determines the equilibrium travel pattern following the deterministic user equilibrium (UE) principle. Constraint (6) ensures that the total travel demand and link flow conservation are fulfilled. Equation (7) denotes link exit capacity constraints. The Lagrange multipliers associated with the capacity constraints are exactly the corresponding queuing delay due to limited link capacity. For any given signal settings and link capacity, drivers’ route choice behavior and the resultant equilibrium travel pattern can be predicted through the lower-level program with the assumption that all drivers perfectly know all the travel costs and queuing delay information of the whole network. Note that the queuing delay is an implicit variable in the bilevel model.

2.2. Single-Level MIP for CNDSSP

In this section, we transform the bilevel problem into a single-level program, which is then linearized into a mixed-integer linear program (MILP), whose global optimality can be guaranteed.

The UE condition assumes that each driver has perfect information of travel time on each possible path and chooses the route with minimum travel time; that is, the travel times on all the routes used are equal and less than those experienced on any unused routes [46]. It implies that the following relationship should be satisfied:wherein , which is the total travel time cost for individual traveler, including both the queuing delay and the running time and signal delay.

The above UE condition can be expressed into the form of complementary condition:

Next, a set of mixed-integer constraints can be introduced to substitute (9). Referring to Lo [47, 48] and Wang and Lo [40], the equivalent set of mixed-integer constraints are stated as follows:where is a large enough negative number; is a large enough positive number; is a very small positive number; and is a 0-1 variable. If , then ; on the contrary, if , equals zero. The detailed verification of the equivalence between the set of constraints (10) and UE condition can be referred to the work of Lo [47, 48] and Wang and Lo [40]. Note that constraints (10) are now linear in , , , and .

In the bilevel formulation, queuing delay is implicit as mentioned above. To transform the bilevel program into single-level problem, should be converted into an explicit variable. The following process shows how is converted and how to formulate the mixed-integer constraints associated with (7).

Since queues may only occur on saturated roads, the following relationship should be enforced [22]:which means if a link is unsaturated, that is, , then the queuing delay on this link is zero; otherwise, if a link is oversaturated, the queuing delay is greater than zero. Constraint (11) is similar to that in (8). Therefore, constraint (11) can also be rewritten into the form of complementary problem in

Then following the transformation procedure as imposed on (9), the constraints in terms of mixed-integer linear constraints in (13) can be developed to replace (12):where is a binary variable. For , we have

On the other hand, if , since , we have

Thus, the set of constraints (13) is equivalent to the “if-then” conditions in (11). Note that is an explicit variable in constraints (13), which are linear in , , and but remain nonlinear in and .

In summary, the single-level MIP for CNDSSP can be formulated as follows:subject to the following:(i)Signal variable constraints:(ii)Capacity constraints and demand conservation constraints:(iii)UE constraints:(iv)Queuing delay constraints:(v)Definitional constraints:

In the above single-level mixed-integer model for CNDSSP, the UE conditions and queuing delay constraints are represented by a set of mixed-integer constraints (19)-(20), rather than in a lower-level program formulation. One can note that the nonlinearity stems from the link travel time function, the objective function, and capacity and queuing delay constraints; that is, this single-level model is a mixed-integer nonlinear program (MINLP). In the next section, we will employ linearization technique to transform the MINLP into MILP, through which global optimal solution can be obtained.

3. Linearization of the Single-Level Model

Mixed-integer models would be used to formulate the nonlinear terms into piecewise-linear approximations. Indeed, many mixed-integer models have been studied to capture piecewise-linear functions [49–52]. However, different formulations have different properties and their computational efficiencies vary from each other. Recently, several new and existing modeling efforts of piecewise-linear functions are reviewed in the work of Vielma et al. [53]. In this study, we apply the logarithmic formulation Log model (logarithmic branching convex combination model) to linearize the nonlinear terms, which significantly reduces the number of variables and constraints needed by introducing a binary in through an injective function to identify an active polytope in the linearization process. Specifically, the logarithmic relationship between the number of binary variables and the number of polytopes and constraints makes sure that the size of the linearization model will not be prohibitively large even when high approximation accuracy and refined partition scheme are required.

This section includes three parts. The details of linearizing the two-dimensional functions and MILP formulation are expressed in Section 3.1. The objective function and the storage capacity function are then transformed in Section 3.2. The MILP model for the CNDSSP is summarized in Section 3.3. Finally, we adopt the MILP formulation for the case of discrete link capacity improvement.

3.1. Linearization of Two-Dimensional Functions

In the single-level MIP proposed above, there are several nonlinear terms that exist (i.e., , , and ) and all of them are multivariate functions with multiple variables. In this study, the nonlinear terms will be approximated by piecewise-linear functions. In this way, the MINLP can be transformed into the MILP.

For the last nonlinear term , it is used to cover both running time and signal delay at link exit. The signal delay stems from the signal interruption of traffic and could be determined by approximated formula developed from local traffic conditions, such as Webster’s two-term delay formula for signalized intersections [54] and the polynomial cost function [17]. However, as was pointed out in Yang and Yagar [22], both of the two formulas have drawbacks in describing the signal delay, especially for the saturated links. Therefore, following Yang and Yagar [22], we use the link performance function to depict both the running time on the link and signal delay at link exit. The formulation is given as follows: For signal-controlled link : For non-signal-controlled link :where is free-flow travel time of link and given as known constants. Note that (22) is a function with three variables and it includes the term in its formulation. If the term is linearized first and substituted by a new variable, (22) will have the same form as (23) and both of them are two-dimensional. Since the other two nonlinear terms, that is, and , are both two-dimensional, all the nonlinear terms in the single-level model can be regarded as two-dimensional and be linearized through the same piecewise-linear approximation method. Now is taken as the example to demonstrate the detailed piecewise-linearization process by applying Log model.

Let be represented by a new variable ; that is, . The bounded domain of would be portioned into a finite family of polytopes , which is usually taken to be a triangulation of the field [51]. For Log model, the family of polytopes needs to be topologically compatible or equivalent with a triangulation known as or “Union Jack” [55]. The Log model needs only logarithmic number of binary variables and thus a branching scheme with a logarithmic number of dichotomies, which is introduced in Vielma and Nemhauser [56]. In this paper, the domain of is to be divided into subsets as shown in Figure 1 when . The feasible region of (i.e., ) is partitioned into intervals in dimension; that is, , , and (i.e., ) is partitioned into intervals in dimension; that is, , . Note that the lengths of these intervals are not necessarily the same.

Figure 1 

An equivalent triangulation of the feasible region.

Apparently, with the increasing number of and , the piecewise-linear approximation will be more accurate. However, not only does the approximation accuracy depend on the total number of polytopes; the detailed partition scheme also impacts the results remarkably. For example, in Figure 2, although the piecewise-linear function only has two pieces, the minimum of is equal to the actual minimum of original function. As compared to , the minimum of , which has three segments, is larger than the actual minimum. Hence, advanced partition method, which better describes the profile characteristics of the graph of nonlinear functions, can be studied in the future. In this paper, we just apply the commonly used evenly partitioned scheme.

Figure 2 

Different piecewise-linear approximations for .

Following Log method in Vielma et al. [53], the function for can be substituted by a set of MILP constraints as follows:

In the above formulation, is the family of partitioned triangles and denotes corner-point, which may belong to more than one triangle in the partitioned domain; represents the coordinates of corner-point ; is the weighted factor of convex combination related to corner-point , and, hence, . Since must fall into a single triangle of the domain, that is, the active triangle, can be approximated in the form of a weighted function of value of vertices of this triangle; can also be calculated from a weighted function of the coordinates of the vertices of the active triangle, as is shown in (24). Equation (25) entails that the sum of weighted factors of all vertices in the domain is equal to 1. Indeed, only of the active triangle is larger than zero while the weighted factors of the other inactive triangles are all equal to zero.

As is not associated with any specific polytope, constraints (24)-(25) only require the nonnegativity of . However, constraints (26) ensure that all nonzero are related to the corner-points of one polytope in . In (26), and the set of dichotomies is referred to a branching scheme for the family of polytopes that is equivalent to . For each particular triangle , the vertices of can be expressed by , where or for each . The index set is divided into two subsets and . The first is given by , where Dim refers to dimension number of the domain and it is equal to two in this case because our nonlinear function is two-dimensional; is the number of intervals divided in dimension and note that is even because of triangulation. Assume that each interval index can be represented by a fixed binary vector . The selection of is arbitrary on condition that the two successive vectors and differ in only one bit for each . The set of binary vectors with this property is known as Gray code or reflected binary code [57]. Then, for each , and , where . The second set is denoted by . For each , and . The size of (26) is equal to and only dependent on when Dim is fixed. A simple example of this branching scheme of an equivalent triangulation is given in detail in the appendix.

Similarly, a new variable is introduced to denote the term of in the single-level problem. Then, we apply Log model for piecewise-linear approximation, and the function for can be written into a set of MILP constraints as follows:where is the family of polytopes, which are partitioned triangles of the domain of ; that is, . For link flow , the upper bound can be set as the sum of travel demands of the OD pairs, at least one viable path of which includes link , or simply the sum of all OD demands. Since OD demands are given as constants in our model, the upper bound can be easily defined. As for queuing delay , there is no explicit upper bound in the constraints. However, considering the realistic condition of queuing delay at an intersection, should not be unbounded. can be set as a positive large enough number which fully covers the feasible region of , for example, four times signal cycle or free-flow travel time of the whole link . Both and are set equal to zero.

Now consider the linearization of link travel time function. For signal-controlled link, is substituted by . Thus, the link travel time function can be regarded as a function with respect to both and , that is, , , the piecewise-linear approximation of which is represented as follows:where is the family of polytopes which is the triangulation of . The range of is defined as , where and . For non-signal-controlled link , since is originally two-dimensional, the piecewise-linear functions can still be expressed as (28) by changing to .

3.2. Linearization of Objective Function and Storage Capacity Function

The objective function of the single-level optimization model is the combination of actual total system travel cost and associated link expansion cost. The first term of the objective, that is, , is nonlinear and nonconvex. But, through the UE principle, it is easy to rewrite the first part into a linear form. When UE condition is satisfied, all used paths of the same OD pair must be equal to the minimum travel time , which includes running time, signal delay, and queuing delay in this model. Then, the total system travel cost can be rewritten into the equivalent form of , where all OD demands are given constants and is the factor to convert travel time to monetary travel cost, which is also a constant. Thus, the first term of the objective function is linear. The second term of the objective function , which determines the link expansion expense, is a function of . In practical use, due to various topographic and geological conditions of land, the link investment cost function can be quite different. If is linear, the whole objective function is linear, and no transformation is needed. Otherwise, the linearization method Log applied in the last section can be used to transform the nonlinear functions into piecewise-linear approximations. As is only a single variable function, the linearization process is much easier than multivariate functions, like . The domain of is one-dimensional and partitioned into line segments rather than triangles. In conclusion, the investment function can also be expressed in a linear form and thus the whole objective function is linear.

For the link storage capacity function , which is also a single variable function with respect to , it can be treated in the identical way as in the link expansion cost function. That is, we can always have a linear or piecewise-linearized link storage capacity function . In this paper, for illustration purpose, we simply assume linear functions of and .

3.3. The Reformulated MILP Model

In summary, the reformulated MILP model of the original single-level optimization program is summarized as follows:subject to the following:(i)Green split constraints for links with signal control are as follows:(ii)Capacity constraints and demand conservation constraints are as follows:(iii)UE condition constraints are as follows:(iv)Queuing delay constraint is as follows:(v)Linearization of function for is as follows:(vi)Linearization of function for is as follows:(vii)For links with signal control , linearization of the link travel time function is as follows, while, for links without signal control , in the following constraints is substituted by : (viii)Demand conservation and definitional constraints are as follows: