A robust urban traffic network design with signal settings

Information Sciences 334–335 (2016) 144–160

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Information Sciences journal homepage: www.elsevier.com/locate/ins

A robust urban traffic network design with signal settings Suh-Wen Chiou∗ Department of Information Management, National Dong Hwa University, Dahsueh Rd., Shou-Feng, Hualien, 97401, Taiwan

a r t i c l e

i n f o

Article history: Received 5 August 2014 Revised 27 September 2015 Accepted 13 November 2015 Available online 21 November 2015 Keywords: Uncertainty Urban traffic network Bi-level programming Cutting plane method Robust optimization

a b s t r a c t An urban traffic network (UTN) accounting for signal setting and link capacity expansion under uncertain travel demand is considered. In order to mitigate vulnerability of UTN system, a robust bi-level model is firstly presented. A trust-region cutting plane projection (TCPP) is proposed in this paper to solve the proposed robust bi-level model. Numerical computations for proposed bi-level model were performed using various test road networks. Computational comparisons for proposed approach were also made with other heuristics. It indicates that the proposed approach can substantially enhance greater system performance of UTN system as compared to other alternatives while incurring less CPU time. In particular, in comparison with a recently proposed heuristic, the proposed approach improved robustness of UTN system with significance whilst maintaining optimality of nominal solutions. © 2015 Elsevier Inc. All rights reserved.

1. Introduction For most urban traffic networks, severe travel delays would be incurred by all road users as a result of inappropriate design of signal settings and insufficient provision of link capacity particularly under uncertain travel demand. To deal with growing travel demand and alleviating traffic congestion with success for most urban traffic networks (UTNs), a bi-level programming (BLP) model has been employed [6,8,11–13,19,21,22,31,40,46–48]. For example, [11] proposed a general bi-level programming approach for continuous transportation network design problem. [21] proposed a non-linear constrained bi-level program for urban network design where both directions of existing roads and signal settings at junctions were taken into account. A scatter search algorithm based on a random descent method was proposed to solve the proposed bi-level program. Numerical tests were conducted on a real dimension network. The local optimal solution can be found by reasonable computation times. Recently, [19] presented a comprehensive review for urban transportation network design problem which includes a variety of definitions, classifications, objectives, constraints, and solution methods of road network design and public transit network design problems. [46] also proposed a BLP for a continuous network design problem with tradable credit scheme and equity constraints. At the upper level, the government chooses optimal capacity enhancement for some existing links to minimize the total system costs under a budget constraint. The lower level chooses the optimal route based on the generalized cost. A relaxation algorithm was proposed to solve the proposed BLP model. To deal with compromises among interacting decision entities distributed throughout a multi-level hierarchical system which appears quite often in many real-world applications, [23] presented a tri-level decision-making model with multiple followers. The Stackelberg equilibrium in a three-level vertical structure and the Nash equilibrium among multiple followers at the same horizontal level can be achieved using a fuzzy programming approach. To exploit maximum reserve capacity of UTN system, [12] proposed a min–max bi-level programming problem for urban city traffic signal design. A new hybrid strategy using ∗

Corresponding author. Tel.: +886 3 8633108; fax: +886 3 8633100. E-mail address: [email protected]

http://dx.doi.org/10.1016/j.ins.2015.11.022 0020-0255/© 2015 Elsevier Inc. All rights reserved.

S.-W. Chiou / Information Sciences 334–335 (2016) 144–160

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delay-minimizing signal settings was proposed to mitigate travel delay to all road users. The merits of proposed strategy have been numerically demonstrated with two moderate road networks when compared to other alternatives. More recently, [6] proposed a bi-level programming model to solve multi-modal bus lane design in transportation road networks. The upper level problem aims to minimize average travel time of road users and the difference of passengers’ comfort among all bus lines is minimized via optimal design of bus frequencies. The lower level problem is a multi-modal transportation network equilibrium model for the joint modal split and traffic assignment problem. A simple numerical test has been conducted using a column generation algorithm combined with the branch-and-bound method. Good results have been reported with regard to the objective of effectively reducing travel times for all users and balancing transit service level among all bus lines. To address optimal road capacity expansion of existing links under uncertainty, [40] proposed a reliability-based bi-level traffic network design model with advanced traveler information system. The objective at the upper level is to maximize the reliability of the total travel time whilst drivers’ travel time uncertainty at lower level is being effectively reduced via traffic information provided by advanced traveler information systems. The proposed reliability-based bi-level network design model was solved by the particle swarm optimization and primary results were reported using the Sioux Falls real data network. As it has been widely recognized from literature [2,18,29,32,35,38], the BLP is generally a non-convex problem due to the implicit form in constraints. Ways of characterizing the implicit function form in constraints for BLP can be achieved by a gap function as investigated in [28,31,45]. Employing a certain form of gap function, Meng et al. [31] gave an equivalent continuously differentiable model for BLP. An augmented Lagrangian method was proposed to solve the problem. Since in general the gap function is non-convex, optimal solutions for the BLP turn out to be fairly difficult to obtain in practice. In this regard, [28] proposed a viable global optimization method to solve the BLP model effectively. Based on selected gap functions, the proposed BLP model in [28] with continuous variables can be conveniently transferred into a sequence of single-level concave sub-problems. A combination of multi-cutting plane approach and penalty method was proposed to solve the BLP. Due to intensive computational efforts in solving sub-problems, however, the approach presented in [28] may simply solve very modest networks only. Instead of solving the BLP directly, [45] on the other hand presented a mixed transportation network design model using a global optimization approach with equilibrium constraints. The user equilibrium condition in [45] was characterized by a path-based complementarity problem. A BLP model was transferred into a single-level optimization problem with a set of mixed-integer constraints. Because it is generally challenging to calculate all path flows for a moderately large network, the techniques developed by [45] were relatively not deemed practicable when performed on a road network of realistic size. The difficulty inherent to the BLP makes attractive the use of heuristics. One meta-heuristic algorithm such as Genetic Algorithm (GA) is often preferred as an alternative to solve the BLP [9,21,42,44]. In order to solve a class of nonlinear BLPs, [44] presented a novel evolutionary algorithm. In [44], the lower level problem is a convex programming problem for each given upper level decision variable. A novel algorithm was proposed to address the nonlinearity of the BLPs with non-differentiable upper level objective function and constraints without requiring the differentiability of the objective function. For urban traffic networks, [42] presented a GA-based heuristic to optimize traffic signals with promising results from empirical studies. [9] also employed a hybrid harmony search and hill-climbing heuristic to improve system performance of UTN with stochastic flows. The effectiveness of proposed GA-based heuristics has been demonstrated numerically with example test networks. Recently, [50] presented a bi-objective optimization model for coordinated traffic signals. A signal timing optimization model was presented to simultaneously optimize the cycle length, offsets, green splits and phase sequences to minimize traffic delay and the risk associated with human exposure to traffic emissions. The Pareto optimal solutions for bi-objective optimization model were solved by GA with a numerical example. Robust optimization is often employed as an effective tool to hedge against uncertainty [3–5,14,24,43]. In order to mitigate vulnerability of UTN system, for instance, [14] proposed a robust optimization approach [3–5] for traffic network design problem ([14,15,24,33,34] and references therein). A linear programming model was presented by [14] to solve a UTN system design together with numerical test results at a small toy network. Considering a UTN system under uncertain travel demand, [43] proposed a robust system optimal signal control and conducted numerical results. More recently, [24] proposed a robust optimization approach for dynamic traffic signal control together with emission considerations. A mixed integer linear program was proposed to explicitly capture vehicle spillback, and minimizes travel delay together with reduced emissions. In order to perform against the worst-case scenario without loss optimality of nominal solution, [49] presented a set-based model to determine robust optimal signal timings. A cutting plane solution algorithm was employed to minimize travel delays in the presence of the worst-case scenario of travel demands. Computational results were conducted at simple small isolated signal-controlled junctions. Moreover, [15] considered a BLP dynamic congestion pricing problem under travel uncertainty. A BLP cellular particle swarm optimization approach was proposed to find optimal congestion pricing when traffic flows correspond to dynamic user equilibrium on the network. Numerical comparisons for BLP dynamic congestion pricing problem were made with two other robust dynamic solution approaches. As it was shown from numerical results, the proposed approach considerably outperformed other alternatives both in terms of solution quality and computational efficiency. More recently, [37] proposed an integrated model to tackle BLP with signal-settings under uncertain travel demand. A bi-objective BLP model was presented to determine the Pareto solutions. Although the proposed signal settings are less sensitive to fluctuations of traffic flow, the network effect of route choice from travelers has not yet been taken into account and therefore solutions obtained are only sub-optimal. For a general BLP, because of the non-linearity of equilibrium constraint with respect to design variables, most solution heuristics mentioned above [14,15,24,43,49] can simply solve the BLPs only locally. Considering a UTN system under uncertain travel demand, at the best of author’s knowledge, there is very limited work in the literature to contribute in this area except a more recent preliminary study in [13]. The work given in [13] was, however,

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only limited to a small-scale signal control problem while taking no account of network design parameters. No numerical comparison for proposed heuristic was either made with other alternative. Based on a crude bundle type gradient approach, the Gradient Based Projection (GBP) delivered in [13] could fail at worst-case scenario due to kinky structure of the bi-level problem. In this paper, we intend to fill this gap both theoretically and empirically. For a UTN system under uncertain travel demand, a robust BLP model is firstly proposed to account for signal settings at junctions and capacity expansion at links simultaneously. The Stackelberg solution of the BLP is characterized by robust signal-setting and link capacity expansion against a worst-case scenario that might be taken by future travel demand growth. In this regard, the worst-case performance measure of UTN serves as an upper bound estimate. Due to the non-convexity of the BLP, a trust region based cutting plane projection (TCPP) is firstly proposed with global convergence. Numerical computations and comparisons are made in turn with other well-known heuristics. In order to investigate the price of robustness of proposed heuristic, two new indices are proposed to measure the price of robustness of solution heuristics for BLP at worst-case scenarios. The originality and contributions made from this paper can be briefly summarized as follows. A robust BLP model is firstly established to simultaneously determine optimal signal settings and link capacity expansion under uncertain travel demand for UTN systems. The performance measure (PM) of UTN system, whilst maximized with respect to a travel demand growth factor, is minimized with respect to signal settings and link capacity expansion. Considering the non-convexity of BLP, a trust region cutting plane projection (TCPP) with global convergence is firstly proposed. Numerical computations are performed using various initial data sets at example road networks of realistic size. Two new indices are firstly introduced to measure the price of robustness at worst-case scenarios. The rest of the paper is organized as follows. Section 2 introduces a robust bi-level optimization model for UTN system under uncertain travel demand. A trust region based cutting plane approach is proposed in Section 3. Numerical computations and comparisons are made in Section 4 with recently proposed heuristics. The price and gain of heuristics solving a robust BLP are analyzed. Conclusions for this paper and extensions of the proposed approach to topics of interest are briefly summarized in Section 5. 2. A robust bi-level model In the presence of uncertain travel demand, a robust bi-level model is proposed. A UTN system design under uncertain travel demand is regarded as a worst-case Stackelberg game. The decision maker with the leader at the upper level and road users with the followers at the lower level are trying to realize a best solution on their own with respect to some certain but generally different objectives. The decision maker at the upper level has the leadership in playing the game. Under a worst-case realization of uncertain travel demand, the leader determines a set of signal settings and link capacity expansion simultaneously in anticipation of road users’ action. The route choice chosen by users at the lower level strongly relies on signal settings and link capacity expansion which are determined at the upper level. The constraints for equilibrium flow at the lower level are defined in part by a variational inequality. Notation used for a UTN system design under uncertain travel demand is introduced now. 2.1. Notation G(N, L ) W Rw  = (ζ , θ , ϕ )

λa λmin τ jlm m ( j, l )

sa Da Sa WD WS MD MS q

μ f h y u

δ

c (y,  , f ) π

a road network with node set N and link set L. a set of origin-destination (OD) pairs. a set of routes between OD pair w, ∀w ∈ W . set of signal setting variables respectively for the reciprocal of common cycle time, start and duration of greens. duration of effective green for link a. minimum green. clearance time between the end of green for group j and the start of green for incompatible group l at junction m. collection of numbers 0 and 1 for each pair of incompatible signal groups at junction m; where collection equals 0 if the start of green for signal group j proceeds that of l, and collection equals 1, otherwise. saturation flow on link a. rate of delay on link a. number of stops per unit time on link a. weighting factor for rate of delay on link a. weighting factor for number of stops on link a. monetary factor associated with Da . monetary factor associated with Sa . a matrix of travel demand for OD pairs. OD demand growth factor. a vector of link flow. a vector of route flow between points of entry to exit from network. a vector of link capacity expansion. a vector of link capacity expansion upper bound. a link-route incidence matrix. a OD-route incidence matrix. a vector of link travel cost. a vector of minimum travel cost between OD pair.

S.-W. Chiou / Information Sciences 334–335 (2016) 144–160

C V (y ) B

147

a vector of route travel cost. a vector of link capacity expansion investment cost. budget constraint for investment cost.

2.2. A lower level problem under uncertain travel demand Let K (μ ) denote a feasible equilibrium flow with respect to some realization taken by unknown demand growth factor μ, i.e.

K (μ ) = { f : f = δ h, h = μq, h ≥ 0}

(1)

For a UTN system with a set of link capacity expansion y and signal settings  in the presence of a realization taken by unknown demand growth factor μ, the traffic flow can be characterized as follows. To find a f ∈ K (μ ) such that

c (y,  , f )( f¯ − f ) ≥ 0

(2)

for every f¯ ∈ K (μ ). 2.3. An upper level problem under uncertain travel demand In the presence of uncertain travel demand, the UTN system can be expressed as a following robust bi-level problem:

Min Max

y, , f

μ

P (y,  , f (μ, y,  ))

Sub ject to 0 ≤ ya ≤ ua ,

(3)

∀a ∈ L

(4)

ζ ≤ζ ≤ζ

(5)

λmin ζ ≤ ϕ jm ≤ 1, ∀ j, m

(6)

fa (μ, y,  ) ≤ sa λa + ya ,

∀a ∈ L

θ jm + ϕ jm + τ jlm ζ ≤ θlm + m ( j, l ), 

(7)

j = l, ∀ j, l, m

(8)

Va (ya ) ≤ B

(9)

a

According to [10], the performance measure (PM) of UTN system can be evaluated in terms of traffic delays and queues constrained by investment cost, i.e.

P (y,  , f ) =



Da (y,  , f )WD MD + Sa (y,  , f )WS MS +



a∈L

Va (ya )

(10)

a

In Constraints (3)–(9): Constraint (3) is a bound for link capacity expansion. (5) is a bound for common cycle time. The Constraints (6)–(8) are respectively for green phase, link capacity and clearance time for signal setting variables, and (9) is for investment budget constraint. The performance measure P (y,  , f ) in (10), is defined as a sum of a weighted linear combination of the rate of delay and number of stops per unit time for all traffic streams constrained by link capacity expansion investment cost V (y ). A flow f ∗ under uncertain travel demand is solved by a parametric variational inequality (2). A triplet of saddle points (μ∗ , y∗ ,  ∗ ) exists with network flow f ∗ if the following condition holds:

P (y∗ ,  ∗ , f (μ, y∗ ,  ∗ )) ≤ P (y∗ ,  ∗ , f (μ∗ , y∗ ,  ∗ )) ≤ P (y,  , f (μ∗ , y,  ))

(11)

or

P (y∗ ,  ∗ , f (μ, y∗ ,  ∗ )) ≤ P (y∗ ,  ∗ , f ∗ ) ≤ P (y,  , f (μ∗ , y,  ))

(12)

μ∗

is maximized under link In (12), at the first inequality a worst-case realization taken by unknown demand growth factor capacity expansion y∗ and signal-setting  ∗ . Analogously, at the second inequality of (12) a link capacity expansion y∗ and signal settings  ∗ are simultaneously minimized under a worst-case scenario of uncertain demand with a growth factor μ∗ . 2.4. Directional derivatives for UTN system under uncertain demand For a worst-case scenario of uncertain demand with growth factor μ∗ , the first-order directional derivatives for traffic flow can be evaluated in this section. Let ∇ f denote the perturbation of responding flow f ∗ with respect to change y,  and μ. Introduce

K ( μ ) = {∇ f : ∇ f = δ ( h ), ( h ) = μ, ∃ h ∈ K0 }

(13)

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with



(i ) h p f ree, K0 = h : (ii ) h p ≥ 0, i f (iii ) h p = 0, i f

i f h∗k > 0, = 0, Cp = πw = 0, Cp > πw

h∗p h∗p

 ∀ p ∈ Rw , ∀w ∈ W

(14)

For every f ∈ K ( μ ), according to [36], the directional derivative ∇ f ∈ K ( μ ) along a direction y and  can be determined.

(∇y c(y∗ ,  ∗ , f ∗ ) y + ∇ c(y∗ ,  ∗ , f ∗ )  + ∇ f c(y∗ ,  ∗ , f ∗ )∇ f )( f − ∇ f ) ≥ 0

(15)

∇ and ∇ f in (15) are evaluated at when perturbations The gradients ∇y in y,  and μ are specified. According to [36], the directional derivatives ∇y f , ∇ f and ∇μ f are piecewise linear and differentiable almost everywhere along direction y,  and μ. The generalized gradients for responding flow f ∗ with respect to y,  and μ, can be characterized in the following manner: let co denote a convex hull, it implies c (y∗ ,  ∗ ,

f ∗ ),

c (y∗ ,  ∗ ,

f ∗)

c (y∗ ,  ∗ ,

f ∗)

( y∗ ,  ∗ ,

f ∗)

  ∂y  f (μ∗ , y∗ ,  ∗ ) = co n→∞ lim ∇y f (μ(n) , y(n) ,  (n) ) : (μ(n) , y(n) ,  (n) ) → (μ∗ , y∗ ,  ∗ ),∇y f (μ(n) , y(n) ,  (n) )exist

(16)



∂  f (μ∗ , y∗ ,  ∗ ) = co n→∞ lim ∇ f (μ(n) , y(n) ,  (n) ) : (μ(n) , y(n) ,  (n) ) → (μ∗ , y∗ ,  ∗ ),∇ f (μ(n) , y(n) ,  (n) )exist



(17)

  ∂μ  f (μ∗ , y∗ ,  ∗ ) = co n→∞ lim ∇μ f (μ(n) , y(n) ,  (n) ) : (μ(n) , y(n) ,  (n) ) → (μ∗ , y∗ ,  ∗ ),∇μ f (μ(n) , y(n) ,  (n) )exist (18)

2.5. A single-level problem under uncertain demand According to results given in (16)–(18), the generalized gradients for PM in (10) can be calculated as follows:

∇y P = (∇y D + ∇ f D(y∗ ,  ∗ , f ∗ )∇y f )WD MD + (∇y S + ∇ f S(y∗ ,  ∗ , f ∗ )∇y f )WS MS + ∇V (y∗ )

(19)

∇ P = (∇ D + ∇ f D(y∗ ,  ∗ , f ∗ )∇ f )WD MD + (∇ S + ∇ f S(y∗ ,  ∗ , f ∗ )∇ f )WS MS

(20)

The PM in (10) can be re-expressed as

P˜1 (y,  ) =



Da (y,  )WD MD +Sa (y,  )WS MS +



Va (ya )

(21)

a

a∈L

And the generalized gradients can be expressed as

∇y P˜1 = ∇y P

(22)

∇ P˜1 = ∇ P

(23)

and

Now, a robust bi-level model for (3)–(9) can be reduced into a robust single-level optimization in the following way.

Min Max P1 (μ, y,  ) = y,

μ



Da (y,  )WD MD + Sa (y,  )WS MS +

Va (ya )

(24)

a

a∈L

Sub ject to 0 ≤ ya ≤ ua ,



∀a ∈ L

ζ ≤ζ ≤ζ λmin ζ ≤ ϕ jm ≤ 1, ∀ j, m fa (μ, y,  ) + y∇y fa +  ∇ fa + μ∇μ fa ≤ sa λa + ya , ∀a ∈ L θ jm + ϕ jm + τ jlm ζ ≤ θlm + m ( j, l ), j = l, ∀ j, l, m 

Va (ya ) ≤ B

a

Analogous to (12), a triplet of saddle points (μ∗ , y∗ ,  ∗ ) exists for (24) if the following condition holds:

P1 (μ, y∗ ,  ∗ ) ≤ P1 (μ∗ , y∗ ,  ∗ ) ≤ P1 (μ∗ , y,  )

(25)

S.-W. Chiou / Information Sciences 334–335 (2016) 144–160

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At the first inequality of (25), a worst-case realization taken by unknown demand growth factor μ∗ is determined such that the PM is maximized. At the second inequality of (25), the link capacity expansion y∗ and signal settings  ∗ are simultaneously determined such that PM in (25) is minimized. Let

P1 (μ ) = P1 (μ, y∗ ,  ∗ ) = Min P1 (μ, y,  )

(26)

y,

where (y∗ ,  ∗ ) = Arg min P1 (μ, y,  ), and y,

P1 (y,  ) = P1 (μ∗ , y,  ) = Max P1 (μ, y,  )

(27)

μ

where μ∗ = Arg max P1 (μ, y,  ). The inequalities in (25) can be re-expressed. μ

P1 (μ ) ≤ P1 (μ∗ , y∗ ,  ∗ ) ≤ P1 (y,  )

(28)

(μ∗ , y∗ ,  ∗ ) can be optimally determined by alternately solving two following

A triplet of the worst-case Stackelberg solution sub-problems (29) and (30). First, provided with demand factor μ, a perturbation μ can be determined such that μ∗ = μ + μ; i.e.

Max μ

P1 (μ ) =



Da (y∗ ,  ∗ )WD MD + Sa (y∗ ,  ∗ )WS MS +

Va (y∗a )

(29)

a

a∈L

fa∗ + μ∇μ fa ≤ sa λ∗a + y∗a ,

Sub ject to



∀a ∈ L

For a linear programming problem (29), it can be solved by a SIMPLEX method where the generalized gradient ∇μ f is characterized in (18) and determined in (15). Next, provided with link capacity y and signal-setting  , a pair of ( y,  )can be determined simultaneously such that y∗ = y + y and  ∗ =  +  ; i.e.

Min

( y,  )

P¯1 (y,  ) =



Da (y,  )WD MD + Sa (y,  )WS MS +



Va (ya )

(30)

a

a∈L

∀a ∈ L ζ ≤ζ ≤ζ λmin ζ ≤ ϕ jm ≤ 1, ∀ j, m fa∗ + y∇y fa +  ∇ fa ≤ sa λa + ya , ∀a ∈ L θ jm + ϕ jm + τ jlm ζ ≤ θlm + m ( j, l ), j = l, ∀ j, l, m Sub ject to 0 ≤ ya ≤ ua



Va (ya ) ≤ B

a

In (30) the generalized gradients ∇y f and ∇ f are characterized in (16) and (17), and solved by inequality (15). Regarding solutions of Problem (30), it will be discussed next section. Definition 1 (A worst-case Stackelberg solution). We say that (μ∗ , y∗ ,  ∗ ) is a triplet of worst-case Stackelberg solutions for (24) if and only if there exists a simultaneous link capacity expansion y∗ and signal settings  ∗ under a worst-case realization of a demand growth factor μ∗ such that the equalities in (28) hold, i.e.

P1 (μ∗ ) = P1 (μ∗ , y∗ ,  ∗ ) = P1 (y∗ ,  ∗ )

(31)

In (31) we have jointly

μ∗ = Arg max P1 (μ )

(32)

(y∗ ,  ∗ ) = Arg min P1 (y,  )

(33)

and

According to (26) and (27), we have a worst-case Stackelberg solution such that

P1 (μ∗ , y∗ ,  ∗ ) = Max Min P1 (μ, y,  ) = Min Max P1 (μ, y,  ) μ

y,

y,

μ

(34)

Or, by means of (11), we have

P (y∗ ,  ∗ , f ∗ ) = Max Min P (y,  , f (μ, y,  )) = Min Max P (y,  , f (μ, y,  )) μ

y,

y,

μ

(35)

3. A computationally tractable solution scheme In the presence of a realization taken by unknown demand growth factor that is most unfavorable, perturbations of parametric flow can be evaluated using piecewise linear directional derivatives as shown in (16)–(18), respectively for decision variables of interest. Due to kinky structure of the BLP, the Model (30) is non-convex theoretically and computationally intractable. As compared to a recent proposed heuristic in [13], in this section, a trust region model is firstly proposed to enhance system performance of (30) and good solutions can be found with less computational efforts.

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3.1. A trust region method A trust region method is a kind of important and efficient methods in the area of constrained optimization. This class of methods iteratively solves a restricted sub-problem based on a quadratic function model and gets a robust solution. Trust region methods are also efficient for non-smooth optimization problems [7,16,17]. Consider a following optimization problem

Min x

(x )

(36)

Let k denote a trust region radius, we introduce Hk an approximate Hessian matrix for sub-problem (k ) at iterate k, and gk a sub-gradient for (k ) at iterate k. The trust region sub-problem (k ) at iterate k for (36) is to

Min d

1 2

(k) (d ) = gk d + dHk d

d ≤ k

sub ject to

(37)

3.2. A cutting plane model for problem (30) For a non-smooth optimization problem, the cutting plane method has been widely used in the literature [20,25–27,30,39]. To make a lower approximation of P1 (·, · ) at (y(k ) ,  (k ) ), a cutting plane model along a perturbed direction ( y(i ) ,  (i ) ) is (i )

constructed using a bundle of past sub-gradients {∇ P1 , 1 ≤ i ≤ k}. The generalized gradient of an upper bound estimate P1 (·, · ) with respect to the signal settings and link capacity expansion can be determined by means of (19) and (20). A convex hull of all points of the form lim ∇ P1 k→∞

(k )

where the subsequence {y(k ) ,  (k ) } converges to a limit value (y∗ ,  ∗ ) can

be established as follows for (30).

  ∂ P1 (y∗ ,  ∗ ) = co lim ∇ P1 (y(k) ,  (k) ) : (y(k) ,  (k) ) → (y∗ ,  ∗ ), ∇ P1 (y(k) ,  (k) )exists

(38)

k→∞

Let P1

P1

(k )

(k )

denote a lower approximation of P1 (y,  ) close to (y(k ) ,  (k ) ) for (30) such that

 

y − y (i ) (i ) (i ) (i ) (i ) ≈ Max ∇ P1 (y ,  ) + P1 (y ,  ) ≤ P¯1  −  (i ) 1≤i≤k

Supposing that P1

(k )

(k )

is convex, let



(k )



(i )

(i )

(i )

εi,k = P1 (y ,  ) − P1 (y ,  ) + ∇ P1 (y , 

(i )

y (k ) − y (i ) )  (k ) −  (i )

(39)

(40)

denote an error bound for a linear approximation of P1 (y(k ) ,  (k ) ) and εi,k ≥ 0 due to local convexity of P1 (y ). Let ( y(k ) ,  (k ) ) denote a search direction from current iterate (y(k ) ,  (k ) ) to the next one such that



 y(k) ,  (k) = y − y(k) ,  −  (k)

A linear approximation of P1

(k )

(41)

in (39) can be expressed in the following way:

 

(k ) y ( k ) (k ) P1 ≈ Max ∇ P1 (y(i) ,  (i) ) − ε ,  (k ) ) i,k + P1 (y  (k) 1≤i≤k

(42)

Since P1 (y(k ) ,  (k ) ) is constant in (42), for convenience let us skip this, a cutting plane model for a linear approximation of P1 (y(k ) ,  (k ) )along a perturbed direction ( y(k ) ,  (k ) ) can be constructed as follows. (k )

P1 cp (y



(k )

,

(k )

; y

(k )

, 

(k )





y ( k ) ) = Max ∇ P1 (y ,  ) − εi,k  (k) 1≤i≤k (i )

(i )

(43)

The general concept of cutting plane model for (30) is based on a piecewise linear lower approximation of P1 (y(k ) ,  (k ) ) in the neighborhood of iterate (y(k ) ,  (k ) ). Such approximations are built by employing the supporting hyper-plane to the graph (k )

(k )

corresponds to one of minimization problem P1 , received at the iterates {(y(i ) ,  (i ) )}i≤k . However, the minimization of P1 step of cutting plane method which may give us very slow convergence. The cutting plane Model (43) usually becomes more and (k )

more crude an approximation of P1 the farther away from (y(k ) ,  (k ) ). To improve slow convergence in (43) for a minimization problem (30), a trust region based cutting plane is proposed instead. 3.3. A trust region cutting plane projection (TCPP) Due to the kinky structure of P1 in (30), the cutting plane model (43) may be such a poor approximation of P1 that d (k ) =

( y(k) ,  (k) ) is not a descent direction for P1 . To solve a minimization Problem (30) effectively, the bundle sub-gradients

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151

enriched with information around the actual iterate (y(k ) ,  (k ) ) are projected onto null space of active constraints in search for a better local optimum constrained by a proposed trust region based Model (37). By consecutively implementing successive projections of past computed sub-gradients constrained by (37), it will help us to dilate the direction provided by the negativity of sub-gradient at current iterate, which may effectively improve the local solutions obtained for (30). A trust region based cutting plane projection (TCPP) with global convergence is now established. Theorem 1 (A trust region cutting plane projection (TCPP)). following a quadratic model given in (37), let d (k ) = ( y(k ) ,  (k ) ) denote a solution for problem (30).

1 2

ν + d ( k ) Bk d ( k )

Min d (k )

(44)

∇ P1 (y(i) ,  (i) )d (k) − εi,k ≤ ν, 1 ≤ i ≤ k

(k )

d ≤ k sub ject to

Let β (k ) = (y(k ) ,  (k ) ) and Pr (β ) denote the projection of β on a domain set  for a minimization problem (30) such that

β − Pr (β ) = inf β − z z∈

(45)

A sequence of iterates {β (k ) } for a minimization problem (30) can be determined in accordance with

β (k+1) = Pr(β (k) + ld (k) ), k = 1, 2, . . . 

(46)

where l ∈ (0, 2 ) is the step length which minimizes P¯1(k ) in (30) and d (k ) solves problem (44). Let β ∗ be a minimum point, then

β (k) − β ∗ is monotonically decreasing and β (k) − β ∗ ≤ β (1) − β ∗ , i.e. the sequence of points {β (k) } generated by TCPP is bounded. Proof. For any x and y in the domain set  for (30), by definition of the projection, we have

Pr (x ) − Pr (y ) ≤ x − y

(47)

thus for β (k+1 ) we have

(k+1)

2

2

β − β ∗ = Pr (β (k) + ld (k) ) − β ∗

2 ≤ β (k) + ld (k) − β ∗

2

2 = β (k) − β ∗ + l 2 d (k) + 2l (β (k) − β ∗ )d (k)

let d (k ) =

β∗

(48)

− β (k ) , thus in (48)

(k+1)

2

2

2

β − β ∗ ≤ β (k) − β ∗ + l 2 d (k) − 2l (β (k) − β ∗ )(β (k) − β ∗ )

2

2 = β (k) − β ∗ + l 2 (β ∗ − β (k) ) − 2l (β (k) − β ∗ )(β (k) − β ∗ )

2

2 = β (k) − β ∗ + l (l − 2 ) (β ∗ − β (k) )

2

2 = β (k) − β ∗ + l (l − 2 ) d (k)

(49)

since 0 < l < 2, we have

(k+1)

2

2

β − β ∗ < β ( k ) − β ∗

for k = 1, 2, 3 . . . It implies β (k ) − β ∗ is monotonically decreasing and β (k ) − β ∗ ≤ β (1 ) − β ∗ .

(50) 

Theorem 2 (Convergence of TCPP). If the solution set ∗ for a minimization problem (30) is nonempty then the proposed TCPP is globally convergent, i.e.

lim

k→∞

(k )

β − β ∗ 2 = 0 ∀β ∗ ∈ ∗

(51)

Proof. Since β ∗ ∈ ∗ , following the results in Theorem 1, it is easy to check that every sequence {β (k ) } generated by the proposed TCPP is bounded. We prove this theorem by contradiction. Suppose

lim

k→∞

then



(k )

β − β ∗ 2 = η > 0

 

2  β (k) ⊂ ϒ = β ∈  : η ≤ β − β ∗ 2 , β − β ∗ 2 ≤ β (1) − β ∗ , ∀β ∗ ∈ ∗

(52)

(53)

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and ϒ is a closed bounded set. From (49), we have

(k+1)

2

2

2

β − β ∗ ≤ β ( k ) − β ∗ − l ( 2 − l ) d ( k )

(54)

where 0 < l < 2. Because ϒ ∩ ∗ = ϕ , then on ϒ we let



2

X (β (k) ) = l (2 − l ) d (k)

(55)

Since X (β (k ) ) is Lipschitz continuous on ϒ , there exists a ε ≥ 0 such that

inf X (β (k) ) = ε

(56)

β (k) ∈ϒ

From (52), there is a k0 > 0 such that for all k > k0 ,

(k )

β − β ∗ 2 < η + ε

(57)

2

On the other hand, from (54) and (56),

(k+1)

2

2 ε

β − β ∗ ≤ β ( k ) − β ∗ < η −

(58)

2



This contradicts (52) and completes this proof. 3.4. A computationally tractable solution scheme

According to Definition 1, the Stackelberg solution for robust BLP can be characterized. The optimal link capacity expansion and signal settings are simultaneously determined by solving two sub-problems (29) and (30) alternately. The corresponding PM in (30) can be thus continuously improved from iteration to iteration. The corresponding link capacity expansion and signal settings are jointly updated in the following way until a search direction y(k ) and  (k ) defined in (41) vanishes.











y(k+1) ,  (k+1) = y(k) ,  (k) + y(k) ,  (k)



(59)

A computationally tractable scheme for two sub-problems (29) and (30) is given below. Step 1. Start with initial link capacity expansion y(k ) together with signal settings  (k ) and demand growth factor μ(k ) . Set index k = 0 and set stopping threshold ε . Step 2. Solve a maximization problem (29) to obtain demand growth factor μ(k+1 ) in the presence of link capacity expansion y(k ) and signal settings  (k ) . Update the upper bound estimate P¯1 = P1 (μ(k+1 ) , y(k ) ,  (k ) ) in (29) with worst-case realization taken by demand growth factor μ(k+1 ) . Details about setting upper bound estimate about P¯1 with respect to μ are described below. Step 2–1: Characterize the responding flow f (μ(k ) , y(k ) ,  (k ) ) by (2) subject to link capacity expansion y(k ) and signal settings  (k ) under the worst-case realization taken by a demand growth factor μ(k ) . Step 2–2: Perform the first-order sensitivity analysis for f (μ(k ) , y(k ) ,  (k ) ). Characterize the directional derivative ∇μ f (μ(k) , y(k) ,  (k) ) along direction of change μ(k) by (15). Step 2–3. Determine optimal direction of change μ(k ) by (29) such that a new demand growth factor μ(k+1 ) can be determined as μ(k+1 ) = μ(k ) + μ(k ) . Step. 2–4. Update the upper bound estimate P¯1 = P1 (μ(k+1 ) , y(k ) ,  (k ) ) for new demand growth factor μ(k+1 ) . Step 3. Solve a minimization problem (30) to obtain a link capacity expansion y(k+1 ) and signal settings  (k+1 ) under the worstcase of realization of demand growth factor μ(k+1 ) . Update the lower bound estimate P1 = P1 (μ(k+1 ) , y(k+1 ) ,  (k+1 ) )in (30). Details about setting lower bound estimate about P1 with respect to link capacity expansion y and signal settings  are described below. Step 3–1: Determine the responding flow f (μ(k+1 ) , y(k ) ,  (k ) ) by (2) in the presence of the worst-case with demand growth factor μ(k+1 ) . Step 3–2: Perform the first-order sensitivity analysis and characterize directional derivatives ∇y f (μ(k+1 ) , y(k ) ,  (k ) ) and ∇ f (μ(k+1) , y(k) ,  (k) ) along direction of change y(k) and  (k) by (15). Step 3–3. Perform the first-order sensitivity analysis for the upper bound estimate P1 (y,  ) in (30) with respect to perturbation y(k ) and  (k ) . Characterize generalized gradient ∇y P˜1 and ∇ P˜1 via (22) and (23). Step 3–4. Determine optimal directions of change y and  by constructing a cutting plane model for a linear approximation of lower bound estimate P1 (y(k ) ,  (k ) ) via (43). Step 3–5. Conduct a TCPP approach given in Theorem 1. Determine link capacity expansion y(k+1 ) and signal settings  (k+1 ) along a determined direction y and  such that (y(k+1 ) ,  (k+1 ) ) = (y(k ) ,  (k ) ) + ( y(k ) ,  (k ) ). Step. 3–6. Update the lower bound estimate P1 = P1 (μ(k+1 ) , y(k+1 ) ,  (k+1 ) ) for new link capacity expansion y(k+1 ) and signal

settings  (k+1 ) in the presence of the worst-case with demand growth factor μ(k+1 ) . Step 4. If the distance of PM between upper bound estimate P1 and lower bound estimate P1 is within a threshold ε , i.e. |P1 − P1 | ≤ ε then stop. According to (31), we have a worst-case Stackelberg equilibrium solution for (24) with optimal link capacity expansion and signal settings, i.e. (y∗ ,  ∗ ) = (y(k+1 ) ,  (k+1 ) ). Otherwise, increase index k by 1 and go to Step 2.

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1

153

2 candidate link for capacity expansion 100

3

12

4

11

signalized junction

5

6

9

8

7

10

16

18

17

13

14

15

23

22

24

21

19

20

Fig. 1. Sioux Falls real-data test network [2].

4. Numerical computations and comparisons In this section, numerical computations were performed at a benchmark problem of aggregated Sioux Falls city network [1] and a grid-size road network as respectively shown in Figs. 1 and 2. The performance measure (PM) used in problem (3)–(9) for a robust BLP was calculated on the basis of results given in [10]. The travel time and link investment cost functions for a UTN system with link capacity expansion in problem (3)–(9) were adopted from [41]. This numerical test includes 11 signalcontrolled junctions and 4 candidate links for capacity expansion. In order to illustrate effectiveness of the proposed TCPP, three sets of initial data with travel demand growth factors were given in Table 1. The performance measure (PM) calculated was also given for various sets of initial settings. Three solution alternatives for BLP: the augmented Lagrangian algorithm (ALA) [31], the GA approach employed in [42] and recently proposed gradient-based projection (GBP) in [13] were chosen for comparison with TCPP at the worst-case scenarios. Implementations for carrying out the following computations were made on Intel Xeon 2.5 GHz

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3

signal-controlled junction

candidate link for capacity expansion

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

Fig. 2. 25-node grid network.

processor with 32 GB RAM under Window 7 using C++ compiler. The stopping criterion is set when the relative difference in the PM value is less than 0.15%. 4.1. Computational results for PM Computational results for TCPP and three other alternatives were summarized in Tables 2–4 for three sets of initial data. As it is shown in Tables 2–4, the improvement rates in the value of PM of heuristics ranged from 14% to 42% over three initial data sets. As it is also seen from Tables 2–4, TCPP achieved the greatest improvement rate in the value of PM of all heuristics at three sets of initial data. Also, TCPP took the least CPU time to solve a robust BLP as compared to other alternatives. As for other heuristics, GA took the greatest CPU time to solve a robust BLP. ALA proposed in [31] achieved the least improvement rate of all heuristics. As it is shown in Tables 2–4, the recently proposed GBP achieved a comparable improvement rate in the value of PM as that did the TCPP. However, as it is observed in Tables 2–4, the GBP took more CPU time to find a Stackelberg solution for a BLP at worst-case scenarios than that did TCPP. As it is mentioned in the literature, a BLP is typically a non-convex problem. The results from Tables 2–4 for heuristics also indicate this consequence of a non-convex problem (24) where various local solutions are obtained for various sets of initial settings. Due to the kinky structure of the BLP, the recent proposed GBP heuristic in [12] was not reliably performed at some worst-case scenarios as robustly as that did TCPP. In order to understand the robustness of solution heuristics at worst-case scenarios, we continue to discuss these issues for BLPs. 4.2. Price of robustness According to the solution procedure given in Section 3.4, the resulting PM value is updated by alternately solving two Problems (29) and (30) until the difference in PM is within a predetermined threshold. A Stackelberg solution for (24) at worst-case scenario was achieved with very modest CPU time. A robust solution is often regarded as a solution for a worst-case scenario where the feasibility of solution is pursued at the expense of solution optimality against parameter ambiguity. For a robust BLP it becomes a trade-off between optimizing system performance and mitigation against the worst-case scenarios. To investigate the efficiency of heuristics x solving a BLP at worst-case scenarios, two new computational indices are introduced: the price

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Table 1 Initial data for Sioux Falls city network.

ϕ1,2 /ζ ϕ2,2 /ζ ϕ1,6 /ζ ϕ2,6 /ζ ϕ1,8 /ζ ϕ2,8 /ζ ϕ1,16 /ζ ϕ2,16 /ζ ϕ1,17 /ζ ϕ2,17 /ζ ϕ1,19 /ζ ϕ2,19 /ζ ϕ1,20 /ζ ϕ2,20 /ζ ϕ1,10 /ζ ϕ2,10 /ζ ϕ1,11 /ζ ϕ2,11 /ζ ϕ1,12 /ζ ϕ2,12 /ζ ϕ1,18 /ζ ϕ2,18 /ζ y(9,10) y(15,10) y(7,8) y(9,8) 1/ζ

μ

PM (in $)

1st set data

2nd set data

3rd set data

25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 0.0 0.0 0.0 0.0 60 0.91 1426

50 40 50 40 50 40 50 40 50 40 50 40 50 40 50 40 50 40 50 40 50 40 2.0 2.0 2.0 2.0 100 0.90 1615

55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 5.0 5.0 5.0 5.0 120 0.92 1509

where 1/ζ and ϕ jm /ζ respectively denote the common cycle time and green durations for signal group j at junction m measured in seconds and y(a,b) denotes link (a, b) Capacity expansion and upper bound for link (a, b) is 25.0.

of non-robustness (PoNR) with respect to heuristic x at worst-case scenario, and the price of robustness (PoR) with respect to heuristic x at nominal condition. For the PoNR of heuristic x at a worst-case scenario, we compare the PM value of a BLP model with that of a robust BLP model. For the PoR of heuristic x solving a robust BLP model at nominal condition, we compare the PM value of a robust BLP model with that of a BLP model. By doing so, the efficiency of various heuristics solving the BLP at worst-case scenarios can be evaluated and compared. The larger value of PoNR with respect to heuristicx, the better the heuristic x solved a BLP model at a worst-case scenario since more infeasibility gain would be obtained at a worst-case scenario. Moreover, the smaller value of PoR of heuristic x, the better the heuristic x solved a robust BLP model at nominal condition because less optimality loss would be incurred at nominal condition. The PoNR can be defined as follows. First, let the PoNR of a heuristic x solving a BLP at worst-case scenario denoted by ρx+ . Let ˜ Px denote the PM value for a BLP model using heuristic x at a worst-case scenario. Let Px∗ denote the PM value for a robust BLP model using heuristic x at a worst-case scenario. The PoNR of heuristic x can be thus represented as a percent ratio of the PM value for a BLP model over that for a robust BLP model at a worst-case scenario, i.e.



ρ = 100% × + x

P˜x Px∗



(60)

Similarly, the PoR, ρx− , of heuristic x solving a BLP at nominal condition can be defined as follows. Let Pˆx denote the PM value for a robust BLP model at nominal condition using heuristic x, and let Px denote the PM value for a BLP using heuristic x with nominal travel demand. The PoR of heuristic x can be represented as a percent ratio of the PM value for a robust BLP model over that for a BLP model at nominal condition, i.e.



ρx− = 100% ×

Pˆx P x



(61)

4.3. Comparison of heuristics Numerical comparisons for PoNR and PoR of solution heuristics: ALA, GA, GBP and TCPP at three initial data sets were made and results were plotted at Figs. 3–4, respectively. As it is shown in Fig. 3, all solution heuristics achieved various PoNR values at three distinct initial data sets. Among them the proposed TCPP achieved the largest PoNR value of all heuristics. The prices of

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S.-W. Chiou / Information Sciences 334–335 (2016) 144–160 Table 2 Computational results for Sioux Falls city network at 1st data set.

ϕ1,2 /ζ ϕ2,2 /ζ ϕ1,6 /ζ ϕ2,6 /ζ ϕ1,8 /ζ ϕ2,8 /ζ ϕ1,16 /ζ ϕ2,16 /ζ ϕ1,17 /ζ ϕ2,17 /ζ ϕ1,19 /ζ ϕ2,19 /ζ ϕ1,20 /ζ ϕ2,20 /ζ ϕ1,10 /ζ ϕ2,10 /ζ ϕ1,11 /ζ ϕ2,11 /ζ ϕ1,12 /ζ ϕ2,12 /ζ ϕ1,18 /ζ ϕ2,18 /ζ y(9,10) y(15,10) y(7,8) y(9,8) 1/ζ

μ

PM (in $) CPU time (in seconds) Improvement rate (%)

ALA

GA

GBP

TCPP

40 40 42 38 37 43 41 39 35 45 45 35 44 36 41 39 38 42 41 39 55 35 5.6 10.2 9.8 8.7 90 1.02 1123 107 21.2

55 55 54 56 49 61 60 50 52 58 62 48 49 61 60 50 55 55 54 56 50 60 6.2 11.5 12.5 10.5 120 1.03 1029 551 27.8

44 46 45 45 50 40 42 48 51 39 40 50 51 39 42 48 46 44 45 45 50 40 4.5 10.5 12.3 11.6 100 1.05 987 23 30.8

62 60 59 63 58 64 65 57 61 61 56 66 65 57 56 66 65 57 58 64 57 65 5.2 9.9 11.5 7.4 132 1.09 944 19 33.8

160 140 120

TCPP

100

GBP

80

GA

60

ALA

40 20 0

1

2

3

Fig. 3. PoNR for heuristics at three initial data sets.

non-robustness of TCPP at all initial data sets showed the superiority of TCPP over all other heuristics when solving BLP at worstcase scenarios. Of all heuristics, TCPP achieved the largest values of PoNR at three initial data sets. Moreover TCPP enjoyed the greatest infeasibility gain against potential risk of uncertain demand for UTN system. On the other hand, the recently proposed GBP has a less advantage on the PoNR as compared to TCPP when solving BLP at worst-case scenarios. Also, the ALA achieved the least value of PoNR of all heuristics and showed marginal benefit of a robust BLP model at a worst-case scenario. Similarly, as it is shown in Fig. 4, all solution heuristics achieved various PoR values at three distinct initial data sets. Among them the TCPP, again, achieved the least one of all heuristics in all cases. The prices of robustness of TCPP showed the greatest advantage over all other heuristics at all initial data sets due to the least optimality loss incurred. The GBP has a less advantage

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Table 3 Computational results for Sioux Falls city network at 2nd data set.

ϕ1,2 /ζ ϕ2,2 /ζ ϕ1,6 /ζ ϕ2,6 /ζ ϕ1,8 /ζ ϕ2,8 /ζ ϕ1,16 /ζ ϕ2,16 /ζ ϕ1,17 /ζ ϕ2,17 /ζ ϕ1,19 /ζ ϕ2,19 /ζ ϕ1,20 /ζ ϕ2,20 /ζ ϕ1,10 /ζ ϕ2,10 /ζ ϕ1,11 /ζ ϕ2,11 /ζ ϕ1,12 /ζ ϕ2,12 /ζ ϕ1,18 /ζ ϕ2,18 /ζ y(9,10) y(15,10) y(7,8) y(9,8) 1/ζ

μ

PM (in $) CPU time (in seconds) Improvement rate (%)

ALA

GA

GBP

TCPP

45 45 42 48 39 51 51 39 45 45 45 45 44 46 45 45 38 52 41 49 54 36 9.6 9.2 11.2 12.7 100 1.01 1271 112 21.3

54 56 55 55 50 60 60 50 52 58 62 48 49 61 59 51 55 55 54 56 50 60 7.2 10.5 8.5 8.5 120 1.02 1109 576 31.3

40 40 41 39 38 42 40 40 35 45 45 35 45 35 41 39 38 42 40 40 39 41 10.1 10.5 8.6 11.2 90 1.05 975 25 39.6

54 58 57 55 56 56 57 55 52 60 61 51 59 53 57 55 54 58 56 56 51 61 11.5 9.8 9.2 7.5 122 1.10 939 17 41.9

140 120 100

TCPP

80

GBP

60

GA

40

ALA

20 0

1

2

3

Fig. 4. PoR for heuristics at three initial data sets.

on the PoR as compared to TCPP when solving a BLP at nominal condition. ALA achieved the greatest value of PoR of all heuristics and therefore was incurred the worst loss of optimality at nominal conditions. To investigate the competitive advantage of heuristics at worst-case scenarios, the relative percent differences of PoNR and PoR of heuristics are in turn considered. The relative percent differences of PoNR and PoR of heuristic x can be defined as follows. The relative difference of PoNR using heuristic x (RPoNR) at a worst-case scenario can be represented as follows:

 rx+

=

P˜x − Px∗ Px∗

× 100%

(62)

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S.-W. Chiou / Information Sciences 334–335 (2016) 144–160 Table 4 Computational results for Sioux Falls city network at 3rd data set.

ϕ1,2 /ζ ϕ2,2 /ζ ϕ1,6 /ζ ϕ2,6 /ζ ϕ1,8 /ζ ϕ2,8 /ζ ϕ1,16 /ζ ϕ2,16 /ζ ϕ1,17 /ζ ϕ2,17 /ζ ϕ1,19 /ζ ϕ2,19 /ζ ϕ1,20 /ζ ϕ2,20 /ζ ϕ1,10 /ζ ϕ2,10 /ζ ϕ1,11 /ζ ϕ2,11 /ζ ϕ1,12 /ζ ϕ2,12 /ζ ϕ1,18 /ζ ϕ2,18 /ζ y(9,10) y(15,10) y(7,8) y(9,8) 1/ζ

μ

PM (in $) CPU time (in seconds) Improvement rate (%)

ALA

GA

GBP

TCPP

45 45 45 45 49 41 42 48 51 39 40 50 51 39 42 48 46 44 45 45 53 37 13.1 10.6 9.2 8.5 100 1.02 1298 111 14.0

42 40 40 42 39 43 45 37 40 42 36 46 50 32 36 46 45 37 38 44 48 34 10.5 8.4 9.6 5.0 92 1.04 1123 559 25.6

38 42 40 40 35 45 40 40 36 44 46 34 45 35 41 39 38 42 39 41 38 42 10.2 9.2 10.5 5.5 90 1.05 967 26 35.9

50 50 51 49 50 50 51 49 46 54 54 46 53 47 51 49 48 52 48 52 46 54 9.8 9.1 8.1 5.7 110 1.09 932 18 38.2

Table 5 RPoNR (in %) for 25-node grid network. Data set

ALA

GBP

GA

TCPP

1 2 3 4 5 6 7 8 9 10

9.9 8.17 10.53 21.33 15.53 17.10 14.98 15.07 15.66 16.02

12.23 17.80 17.26 22.49 25.93 23.55 25.42 24.98 26.08 25.55

8.21 13.07 11.59 20.75 20.81 22.19 19.91 21.55 20.19 21.88

15.3 25.7 26.92 31.28 32.22 35.07 37.11 34.89 36.77 36.09

The relative difference of PoR using heuristic x (RPoR) at nominal condition can be represented as follows:

 rx− =

Pˆx − P x P x

× 100%

(63)

Numerical computations using a 25 node grid-size road network as shown in Fig. 2 were performed and comparisons were made extensively for heuristics: ALA, GA, GBP and TCPP at 10 initial data sets. Numerical comparisons for RPoNR and RPoR of heuristics: ALA, GA, GBP and TCPP were also made. The results were summarized in Tables 5 and 6. As it is observed in Tables 5 and 6, TCPP, again, successfully solved the BLP model at worst-case scenarios with the largest RPoNR gain in all cases. Moreover, TCPP enjoyed the least RPoR loss of all. Due to the non-convexity of BLP, GA and GBP failed to perform robustly at worst-case scenarios and accordingly incurred with rather larger RPoR loss. ALA showed the least RPoNR gain at worst-case scenario in all cases and incurred the worst RPoR loss at nominal condition of all heuristics.

S.-W. Chiou / Information Sciences 334–335 (2016) 144–160

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Table 6 RPoR (in %) for 25-node grid network. Data set

ALA

GBP

GA

TCPP

1 2 3 4 5 6 7 8 9 10

27.91 28.17 28.53 27.33 29.53 27.10 28.98 29.07 28.66 26.02

9.23 7.80 7.26 9.49 8.11 9.15 9.42 8.98 8.08 7.95

13.21 15.07 12.59 18.75 19.21 12.19 17.91 15.55 16.19 17.88

4.32 3.78 3.92 4.28 4.22 3.07 3.11 3.89 3.77 4.09

5. Conclusions and further issues In this paper, we presented a robust bi-level model for a UTN system under uncertain travel demand. A new solution scheme was proposed to reduce overall travel delay to all road users in the presence of uncertain travel demand. Numerical computations were performed using test road networks of realistic size. As compared to recent proposed heuristic, a trust region based cutting plane projection (TCPP) with global convergence was presented to improve the solutions at worst-case scenarios. Two new robustness measures were introduced in order to evaluate the effectiveness and robustness of TCPP. In comparison with other solution heuristics, it obviously indicated that TCPP outperformed all other heuristics at worst-case scenarios while incurring a relatively slight loss of optimality at nominal conditions. The superiority of TCPP has again been demonstrated by extensive numerical comparisons using a grid-size road network. Considering a general road network, most of travel time delay strongly depends on correct and continuous operations of effective signal control at junctions. The growing complexity of signal-controlled road junctions has increased a formidable risk of sudden loss of road network capacity due to signal setting failure at some of junctions. The vulnerability of road networks as a result of capacity reduction at links downstream is of growing concern [19]. The method investigated in this paper can be applied to a general urban road network design with signal-controlled junctions in the presence of road capacity uncertainty. The present work on UTN system design indicated that the proposed TCPP is attractive for BLP at worst-case scenarios. We will continue to apply TCPP on various topics of network design and discuss these issues of interest in subsequent papers. Acknowledgments The author appreciates a lot to Editor-in-Chief: Prof. Witold Pedrycz and a couple of reviewers for their helpful and constructive comments on earlier versions of this manuscript. The work reported in this paper has been supported by grants NSC/MOST 98-2410-H-259-009-MY3, 101-2628-H-259-001-MY2 and 104-2221-E-259-029-MY3 from Taiwan National Science Council. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]

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