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Transportation Research Part B journal homepage: www.elsevier.com/locate/trb

Optimal network-level traﬃc signal control: A benders decomposition-based solution algorithm Rasool Mohebifard, Ali Hajbabaie∗ Department of Civil and Environmental Engineering, Washington State University, Pullman, Washington, United States

a r t i c l e

i n f o

Article history: Received 18 August 2018 Revised 22 January 2019 Accepted 26 January 2019

Keywords: Traﬃc signal control Cell transmission model Benders decomposition Flow holding-back problem

a b s t r a c t This paper formulates the network-level traﬃc signal timing optimization problem as a Mixed-Integer Non-Linear Program (MINLP) and presents a customized methodology to solve it with a tight optimality gap. The MINLP is based on the Cell Transmission Model (CTM) network loading concept and captures the fundamental ﬂow-density diagram of the CTM explicitly by considering closed-form constraints in the model and thus, eliminates the ﬂow holding-back problem. The proposed solution algorithm is based on the Benders decomposition technique and decomposes the original MINLP to an equivalent Integer Program (IP) (Master problem), and a new MINLP (Primal problem). We will show that the new MINLP has only one optimal non-holding-back solution that can be found by a CTM simulation run. We will prove that the proposed solution technique guarantees convergence to optimal solutions with a ﬁnite number of iterations. Furthermore, we propose a dual estimation algorithm for the new MINLP (the Primal problem), which utilizes a simulation-based approach to generate Benders cuts instead of solving a complex optimization program. We applied the proposed solution technique to a simulated network of 20 intersections under various demand patterns and observed an optimality gap of at most 2% under all tested conditions. We compared the solutions of the proposed algorithm with two benchmark algorithms and found reductions in total travel time ranging from 7.0% to 35.7%. © 2019 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license. (http://creativecommons.org/licenses/by-nc-nd/4.0/)

1. Introduction Network-level traﬃc signal timing optimization often involves a large number of mixed-integer decision variables, nonlinearity, and an excessive number of constraints. In fact, signal timing optimization in an arterial street is shown to be an NP-complete problem (Wünsch, 2008). Therefore, the existing classical optimization techniques cannot solve the problem when either the network is large, or the study period is long. Researchers have used heuristic and metaheuristic approaches to ﬁnd solutions to the signal timing optimization problem. For instance, genetic algorithms (Abu-Lebdeh and Benekohal, 1997; Ceylan and Bell, 2004; Park et al., 1999; Sun et al., 2003; Tan et al., 2017), neural networks (Castro et al., 2014; Dai et al., 2011; Saito and Fan, 20 0 0; Srinivasan et al., 2006), feedback control (Zhao and Gao, 2005; Zlatkovic and Zhou, 2015), or fuzzy logic (Araghi et al., 2014; Chiu and Chand, 1993; ∗

Corresponding author. E-mail address: [email protected] (A. Hajbabaie).

https://doi.org/10.1016/j.trb.2019.01.012 0191-2615/© 2019 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license. (http://creativecommons.org/licenses/by-nc-nd/4.0/)

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Wei et al., 2001; Wen et al., 2015) are shown to be effective algorithms to ﬁnd high quality solutions to the problem. However, evaluating the solution quality or ﬁnding optimality bounds for these approaches is not a trivial task (Lin and Wang, 2004). On the other hand, exact algorithms are mostly centralized and thus become computationally expensive as spatial or temporal scales of a network grow. The available exact algorithms in the traﬃc signal control domain are still limited in the application scale and the inherent assumptions. The assumptions vary from the linearization or simpliﬁcation of the network loading models to the relaxation of integer or binary variables that result in ﬁnding sub-optimal solutions. Developing a formulation and an eﬃcient solution technique not only facilitates the implementation of exact signal timing algorithms but also, can serve as a benchmark for other heuristic or approximated signal timing approaches. This paper presents the development of a network-level signal timing optimization program and an eﬃcient solution technique to ﬁnd globally optimal solutions. We formulate the problem using the Cell Transmission Model (CTM) (Daganzo, 1995, 1994) network loading concept. We follow the signal timing optimization formulation developed by (Lo, 2001) but, represent the ﬂow-density diagram of the CTM for different types of cells by several constraints in the optimization program. Therefore, the proposed optimization program captures the fundamental ﬂow-density relationship of the CTM explicitly and eliminates the ﬂow holding-back problem. The ﬂow holding-back problem happens when vehicles are artiﬁcially “held back” regardless of the available downstream capacity (Doan and Ukkusuri, 2012; Lo, 2001; Mohebifard and Hajbabaie, 2018a; Nie, 2011; Pavlis and Recker, 2009; Zheng and Chiu, 2011). Because the ﬂow-density diagram of the CTM is nonlinear, the proposed MINLP is very hard to solve. We will develop a customized methodology based on Benders decomposition technique (Benders, 1962; Geoffrion, 1972) to solve the problem eﬃciently. The proposed solution technique converts the MINLP to an IP and several CTM simulation runs and signiﬁcantly reduces the problem complexity. We will prove that the solution technique converges to the optimal solutions of the original problem with a ﬁnite number of iterations. Moreover, at any intermediate iteration of the algorithm, the solutions are feasible to the original problem. This property prevents the complexities of ﬁnding feasibility cuts for infeasible solutions, which is a common issue in Benders decomposition technique. We also present a dual estimation algorithm for the MINLP, to generate Benders cuts, that utilizes CTM simulation and prevents the complexities of converting network loading models into standard mathematical constraints. The dual estimation algorithm has a parallel structure that can be utilized to reduce the run times. In the rest of this paper, we will present a review of relevant literature on signal control. Then, we will detail the problem formulation and the proposed solution technique. We will continue the discussion by presenting the numerical results, concluding remarks, and trends for future research. 2. Literature review Different formulations and solution techniques have been proposed for the optimal network-level signal control problem. Lo (1999) proposed a CTM-based signal timing program to minimize the delay of vehicles by optimizing the green time of each intersection approach. He linearized the non-linear ﬂow-density diagram of the CTM to reduce the computational complexity. Although the formulation was applied to an isolated intersection without any turning movements, the study provided promising insights into the application of CTM to a transportation network. Lo (2001) stated that the previous formulation suffered from the ﬂow holding-back issue and converted the ﬂow-density diagram into several constraints with binary variables. However, the formulation did not consider turning movements and was applied to only two intersections with one-way streets and a total of 15 cells. Lin and Wang (2004) extended Lo’s (1999) linearized program by modifying the objective function to minimize the total delay and the number of stops. They considered a penalty term in the objective function to capture the lost time associated with changes in the phasing plan. However, the linearization caused ﬂow holding-back problem. Beard and Ziliaskopoulos (2006) integrated Lo’s (1999) formulation with the system optimal traﬃc assignment and stated that the optimization program resulted in a set of ﬂows and densities that do not fall “on” the fundamental diagram (ﬂow holding-back problem), and thus the ﬂow of vehicles is not realistic. Similarly, Guilliard et al. (2016) developed an MILP for optimal traﬃc signal timing. They used a linear queue transmission model to simplify the CTM and be able to use nonhomogeneous time steps in the formulation with binary signal decision variables. However, the formulation suffered from the holding-back issue. Although the ﬂow-density linearization simpliﬁed the signal timing problem signiﬁcantly, this approximation did not allow capturing the ﬂow of vehicles realistically and caused ﬂow holding-back problem. Wada et al. (2017) developed an MILP for a coordinated signal control in an arterial street under both deterministic and stochastic demands based on the variational theory (VT) (Daganzo, 2005a, 2005b). They showed that the MILP had a network structure that allowed ﬁnding its solutions by solving two problems over two equivalent node-link networks (i.e., VT and signal-constraint networks). The node-link networks have a special structure that should be constructed according to the geometry of an arterial street, shape of the ﬂow-density diagram, lost time, and signal timing constraints (e.g., cycle lengths, minimum and maximum green times). Although the solution technique was applied effectively to an arterial street, its extension to a transportation network with turning movements changes the structure of the node-link networks from acyclic (the case for an arterial street) to cyclic. This change requires a new solution technique with a different node-link representation of the problem that needs to be further studied. Wong and Wong (2003) and Wong and Heydecker (2011) proposed a lane-based optimization program for signal timing of an isolated intersection. The formulation optimized the signal settings and lane markings (i.e., deﬁning permissible movements for each lane) at the same time. This integrated framework allowed designing a more eﬃcient movement allocation

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strategy at the entry approaches of an intersection. They used intersection capacity maximization and cycle length minimization as two possible objective functions. Although their numerical analysis showed that the lane marking optimization can increase the capacity of an intersection, it needed an excessive number of decision variables to ensure a feasible and safe movement (without conﬂicting) of vehicles. Lee et al. (2017b, 2017a) considered a hierarchical approach for the signal timing of isolated intersections. In this approach, an upper level optimization program optimized the signal settings for three consecutive cycles ahead of the current cycle. Moreover, they employed the max-pressure control algorithm (Varaiya, 2013) to adjust the signal settings as the most up-to-date information of traﬃc ﬂow becomes available. The proactive nature of the solution technique prevented it from ﬁnding low quality solutions for each individual intersection, but the solutions were sub-optimal for a network as the interaction of neighboring intersections was not considered. Yu et al. (2018a) argued that recent advances in connected and autonomous vehicles created an opportunity for better utilization of intersection capacity. Thus, they proposed an integrated approach to optimize trajectory of vehicles approaching an intersection and its signal timing parameters to minimize the delay of vehicles. They optimized the trajectory of leading vehicles in a platoon and used Newel’s car-following model (Newell, 2002) for the following vehicles. Their analysis results showed that the integrated optimization outperformed an actuated control strategy. The reviewed studies for isolated intersections provide invaluable information on local signal control strategies. However, network-level signal timing techniques can provide additional operational beneﬁts in congested urban-street networks where queue spillbacks and gridlocks reduce the network capacity. In such conditions, the interactions among various intersections cannot be ignored. Researchers have used heuristic and distributed solution techniques to overcome the computational complexities of the network-level signal control problem. Abu-Lebdeh and Benekohal (20 0 0), Ceylan and Bell (2004), Lo and Chow (2004), and Stevanovic et al. (2007) developed genetic algorithms for the network-level signal timing problem. Hajbabaie et al. (2011) used genetic algorithms and approximate dynamic programming for signal timing optimization in oversaturated ﬂow networks. They compared the performance of these methods and used them to test the effects of different traﬃc management strategies on network performance (Hajbabaie et al., 2010; Hajbabaie and Benekohal, 2011). Putha et al. (2012) developed an ant colony algorithm based on the formulation proposed by Girianna and Benekohal (2002) to coordinate signals in oversaturated networks and compared the results to genetic algorithms. Zhang et al. (2010) proposed a robust CTM-based approach to optimize traﬃc signals in the presence of day-to-day demand variation for an arterial. Later, the proposed signal timing formulation was incorporated in a bi-level model, where the upper level minimized the delay of vehicles and the lower layer minimized the human exposure to vehicular emissions (Zhang et al., 2013). The proposed signal timing formulation by Zhang et al. (2010, 2013) considered the non-linear ﬂowdensity diagram of the CTM explicitly by converting the ﬂow-feasibility constraints into several mixed-integer constraints using a customized Big-M approach (Pavlis and Recker, 2009). This formulation has many binary variables, in addition to the binary variables of the NEMA phasing conﬁguration. Thus, they used a GA-based solution technique to solve the program due to its complexity that was not capable of providing optimality gaps. Lertworawanich et al. (2011) also considered the non-linear ﬂow-density diagram in their signal timing formulation with three objective functions. The objective functions were spillover minimization, delay minimization, and throughput maximization. They used a genetic algorithm-based goal programming technique to solve the problem. Although considering the non-linear ﬂow-density diagram of the CTM eliminated the ﬂow holding-back issue, it made the signal timing problem very complex so that it could not be solved with exact solution techniques. Ukkusuri et al. (2013) adopted the linearized CTM-based formulation of Ziliaskopoulos (20 0 0) in a bilevel signal timing and dynamic traﬃc assignment problem. They used a heuristic procedure to solve the bi-level program, but the solution technique did not address the holding-back problem in the formulation. Hajbabaie and Benekohal (2015) developed a program and a genetic algorithm-based solution technique for simultaneous signal timing optimization and traﬃc assignment for oversaturated transportation networks. However, their approach did not provide optimality gaps for the obtained solutions. Yu et al. (2018b) formulated the optimal signal control with dynamic user equilibrium traﬃc assignment problem. They used the double queue traﬃc ﬂow model (Ma et al., 2014; Osorio et al., 2011) for network loading. They stated that binary signal control variables made the problem intractable, and thus they approximated them with continuous variables. Moreover, they developed a genetic algorithm-based solution technique to solve the problem. Han et al. (2014) showed that the error of approximating binary variables with continuous variables grows with time in congested networks with queue spillbacks. Hence, Han and Gayah (2015) proposed a generalized approximation method and used a particle swarm technique (Kennedy, 2011) to optimize signal settings in an arterial street with bounded errors. However, the errors of the approximation in their numerical analysis, although within a theoretical bound, grew with time in the presence of queue spillbacks. Hence, approximating binary signal variables with continuous variables resulted in computational errors especially in oversaturated ﬂow conditions. The reviewed heuristic/metaheuristic-based solution techniques had promising performances and provided invaluable knowledge on traﬃc signal control; however, they lacked the proof of convergence and did not guarantee ﬁnding optimal signal timing parameters. Therefore, evaluating the quality of their solutions is not a straightforward task. Wongpiromsarn et al. (2012) adopted the backpressure routing algorithm from communication and power networks (Tassiulas and Ephremides, 1992) and proposed a distributed approach for traﬃc signal timing. The proposed algorithm determined the movements that needed to receive the green time from a set of predeﬁned movements at each time step based on the queue lengths on each intersection approach. They showed that the approach resulted in stable ﬂow conditions if queue lengths did not exceed the link capacities (no queue spillovers). The approach did not coordinate the decisions of

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intersections as such, could not ensure ﬁnding globally optimal solutions. Timotheou et al. (2013) proposed a distributed cooperative approach that relied on dividing a network into several subnetworks (spatial decomposition) and solving the problem for each subnetwork for a limited number of time steps (temporal decomposition). The optimization program in this study was a mixed integer program based on the CTM. However, this approach required each intersection to either receive vehicles from or send vehicles to the adjacent intersections which limited its application to networks with two-way arterials. Timotheou et al. (2015) further improved the distributed signal timing solution technique by incorporating the ADMM approach (Boyd et al., 2011) into the distributed optimization component. They used a two-phase solution technique: The ﬁrst phase solved the signal timing problem by relaxing the integrality constraints of the signal timing decision variables, and the second phase rounded the real-valued decision variables to binary values for the signal indications. Despite its real-time solutions on the evaluated case studies, the convergence rate of the algorithm might vary across different cases and might not result in real-time solutions. Mehrabipour and Hajbabaie (2017) proposed a distributed-coordinated approach to optimize network-level signal timing parameters in real-time. They distributed the central problem to several intersection-level problems to reduce the computation complexity. They created effective coordination between the intersection-level problems to push the solutions towards optimality. They numerically showed that the algorithm found near-optimal solutions in their test cases, where an optimal solution could be found by classical optimization techniques. However, ﬁnding nearoptimal solutions was not mathematically proven. Islam and Hajbabaie (2017) developed a link-level formulation for signal timing optimization and a distributed solution approach that was incorporated into Vissim (PTV Group, 2013) with 100% connected vehicle market penetration rate. The objective function was to maximize the network throughput and minimize the number of vehicles at each intersection approach. Although they compared the solutions with a state-of-the-practice signal optimizer, they did not show the optimality gaps and the quality of the solutions. The discussed distributed signal timing approaches are scalable and computationally eﬃcient, but they do not guarantee ﬁnding optimal solutions. 2.1. Literature review summary and contributions of the paper The review of literature shows that many network-level signal timing studies employed CTM for network loading. These studies can be divided into two groups based on their approach to reducing the problem complexity. The ﬁrst group replaced the non-linear CTM constraints that represented the ﬂow-density relationship with a set of linear inequality constraints. This simpliﬁcation reduced computational complexity and allowed the application of exact algorithms such as branch and bound to solve the problem. However, the approximation led to the ﬂow holding-back problem. As such, the optimized signals are based on unrealistic ﬂow-density relationships, and thus do not represent the real-world traﬃc ﬂow accurately. The second group captured the nonlinear ﬂow-density diagram of the CTM by converting the diagram into several mixedinteger constraints but could not solve the problem with exact algorithms due to its computational complexity and used heuristic/metaheuristic algorithms. These approaches do not guarantee the optimality of the solutions and do not provide an optimality gap. This study ﬁlls this knowledge gap by: 1. Formulating a CTM-based signal timing optimization program with explicit representation of the CTM ﬂow-density diagram to avoid the ﬂow holding-back problem. 2. Developing a customized solution technique based on the Benders decomposition technique to solve the problem eﬃciently by converting the original MINLP to an IP and several CTM simulation runs. 3. Proving the optimality, convergence, and feasibility properties of the solution technique. 4. Developing a dual estimation algorithm for the MINLP to generate Benders cuts with a simulation-based approach. 3. Problem formulation The problem formulation is based on the CTM (Daganzo, 1995, 1994). The CTM is one of the numerical solution techniques for solving the hydrodynamic traﬃc ﬂow model (LWR) (Lighthill and Whitham, 1955; Richards, 1956) based on the discretization of both time and space. In the CTM, each network link is divided into homogenous segments called cells. We ﬁrst deﬁne the temporal and spatial distribution of the problem. We let T denote the set of all time steps and C, Co , Cr , Cs , Ci , Cd , and Cm represent the set of all network, ordinary, resource, sink, intersection, diverge, and merge cells, respectively. We let I denote the set of all intersections and deﬁne P(i) and S(i) as the set of predecessor and successor cells of each cell i ∈ C, respectively. We let E(k) denote the set of all intersection cells of intersection k ∈ I with through and left-turning movements and O(i) represent the set of all cells of intersection k ∈ I with conﬂicting through and left turning movements with the movement i ∈ E(k). Finally, R represents the set of all concurrent through and right turn movement with adjacent movements. We categorize the cells into ﬁve groups: ordinary, resource, sink, merge, and diverge cells. Ordinary cells receive vehicles from and send vehicles to only one adjacent cell. Resource cells represent the entry points to a network where vehicles enter the network according to a predeﬁned demand proﬁle, while sink cells absorb vehicles leaving the network. Merge cells receive vehicles from multiple adjacent upstream cells. Diverge cells send vehicles to multiple adjacent downstream cells. We have also deﬁned intersection cells, which are like ordinary cells but, have a variable saturation ﬂow rate to represent the effect of traﬃc signals. The notations that are used in this paper are presented in Table 1. The decision variables are the indications of traﬃc signal gti at intersection cell i ∈ Ci at time step t ∈ T. We deﬁne gti equal to one when the signal indication is green and zero otherwise. The objective function of the problem is to maximize

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R. Mohebifard and A. Hajbabaie / Transportation Research Part B 121 (2019) 252–274 Table 1 Deﬁnition of sets, decision variables, and parameters. Sets T set of all time steps C set of all network cells A set of all links between cells set of all ordinary cells Co set of all resource cells Cr set of all sink cells Cs set of all intersection cells Ci set of all diverge cells Cd set of all merge cells Cm I set of all intersections S(i) set of all successor cells of cell i ∈ C P(i) set of all predecessor cells of cell i ∈ C E(k) set of all intersection cells of intersection k ∈ I with through and left-turning movements O(i) set of all cells of intersection k ∈ I with conﬂicting through and left turning movements with the movement i ∈ E(k) R set of all concurrent through and right turn movement with adjacent movements Decision variables gti signal indication; 0 for red, and 1 for green signal for cell i ∈ Ci at time step t ∈ T number of vehicles advancing from cell i ∈ C to cell j ∈ S(i) at time step t ∈ T yti j Variables state variable; number of vehicles in cell i ∈ C at time step t ∈ T variable saturation ﬂow rate in intersection cell i ∈ Ci at time step t ∈ T dual value of constraint i at time step t ∈ T λ L Benders cut Z objective function of the original and the Primal problem μ objective function of the Master problem Parameters xti qti

t i

Dti Qit Ni

βit δ ij τ VL

ρ k

ε

demand of resource cell i ∈ Cr at time step t ∈ T saturation ﬂow rate in cell i ∈ C at time step t ∈ T capacity of cell i ∈ C in terms of the number vehicles it can hold portion of ﬂow entering intersection cell i ∈ Ci at time step t ∈ T from the total ﬂow leaving its upstream diverge cell Kronecker’s delta (δ ij = 1 when i = j; otherwise δ ij = 0) duration of each time step, seconds speed limit, mi/hr ratio of backward shockwave speed to the free ﬂow speed iteration number optimality gap; the difference between the objective function value of the Primal and Master problems small perturbation

the cumulative number of vehicles in sink cells, see (1). This objective function maximizes network throughput while encouraging vehicles to leave the network as soon as possible (Tajalli and Hajbabaie, 2018a, 2018b). Note that the presented formulation in this paper can be modiﬁed to consider different linear objective functions such as minimization of total travel t time, τ ∀i∈C\Cs ∀t∈T xti , or total delay, τ ∀i∈C\Cs ∀t∈T (xti − j∈S(i ) yi j ).

(P1 )

max Z =

∀i∈Cs

t

x ∀t ∈T i

(1)

Constraint (2) ensures the ﬂow conservation for all cells utilizing the Kronecker’s delta concept (δ io = 1 when i = o; otherwise δ io = 0). The constraint shows that the difference between the number of vehicles xt+1 − xti at consecutive time steps i t + 1 ∈ T and t ∈ T in ordinary cell i ∈ C\Cs ∪Cr (δ io = 1, δ is = 0, δ ir = 0) is equal to the number of vehicles k∈P (i ) ytki coming to t the cell minus the number of vehicles j∈S(i ) yi j that leave it at time step t ∈ T. The number of vehicles coming to resource t cell i ∈ Cr (δ io = 0, δ is = 0, δ ir = 1) at time step t ∈ T is Dti and the number of vehicles leaving it is j∈S(i ) yi j . The number of vehicles coming to the sink cell i ∈ Cs (δ io = 0, δ is = 1, δ ir = 0 ) is k∈P (i ) ytki and the number vehicles leaving it is zero. Moreover, sink cell occupancies at time step t ∈ T show the total number of completed trips from the beginning of the analysis period up to time step t.

xt+1 − xti = (δio + δis ) i

ytki + δir Dti − (δio + δir )

k∈P ( i )

∀i ∈ C, o ∈ C \Cs ∪ Cr , r ∈ Cr , s ∈ Cs , t ∈ T

yti j

j∈S (i )

(2)

Constraints (3)–(5) guarantee ﬁnding a set of dynamic conﬂict-free signal plans, where all movements at intersections are protected. Fig. 1 shows the deﬁned signal-related sets in these constraints. Constraint (3) ensures that at most two movements from the set of through and left-turning movements E(k) of intersection k ∈ I receive a green signal indication at time step t ∈ T. Furthermore, only one of the conﬂicting movements i ∈ E(k) and j ∈ O(i) can have the green indication at each time step t ∈ T, see Constraint (4). Constraint (5) sets the signal

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Fig. 1. Illustration of signal-related sets in Constraints (3)–(5).

indication of the concurrent through and right turning movements (i, j) ∈ R equal. Hence, the variables of Constraints (3)–(5) are signal indications gti for each intersection cell i ∈ Ci and time step t ∈ T. Note that the constraints do not enforce a ﬁxed cycle length to ensure as much ﬂexibility as possible in the signal plans.

gtj ≤ 2

∀k ∈ I, t ∈ T

(3)

j∈E (k )

gti + gtj ≤ 1

∀k ∈ I, i ∈ E (k ), j ∈ O(i ), t ∈ T

∀(i, j ) ∈ R, t ∈ T

gti = gtj

(4) (5)

Constraint (6) adjusts the saturation ﬂow rate in intersection cell i ∈ Ci at time step t ∈ T. The saturation ﬂow rate at a cell is zero when the signal is red (gti = 0) and equal to Qit otherwise. This constraint captures the effect of signal timing parameters (gti ) on the ﬂow of vehicles inside the network. Note that qti and gti are the variables in this constraint.

∀i ∈ Ci , t ∈ T

qti = gti Qit

(6)

Constraints (7) and (8) specify a maximum and minimum green time for through and left-turning movement in cell i ∈ ζ E(k) at intersection k ∈ I, respectively (Beard and Ziliaskopoulos, 2006). The variables in these constraints are gti (or gi ). t+G max +1

ζ =t

t+Gmin

ζ =t+1

ζ

gi ≤ Gmax

ζ

∀k ∈ I, i ∈ E (k ), t ∈ T , t ≤ |T | − Gmax

gi ≥ gt+1 − gti Gmin i

∀k ∈ I, i ∈ E (k ), t ∈ T , t ≤ |T | − Gmin

(7)

(8)

Constraint (9) ensures the ﬂow-feasibility in ordinary, resource, and sink cells. The ﬂow of vehicles advancing from cell i ∈ Co ∪Cr ∪Cs to its immediate downstream cell j ∈ S(i) is the minimum of the number of vehicles xti that are present in the cell, saturation ﬂow rate Qit of the sending cell, saturation ﬂow rate Q tj of the receiving cell, and available capacity ρ (N j − xtj ) of the receiving cell, at time step t ∈ T (Daganzo, 1994). For intersection cell i ∈ Ci , the same requirements are applied with variable saturation ﬂow rate qti , see Constraint (10). In Constraints (9) and (10), yti j , xti , xtj , and qti are the variables.

ρ N j −xtj

∀i∈Co∪Cr ∪Cs , j∈S(i ),t ∈T

ρ N j −xtj

∀i∈Ci , j∈S(i ),t ∈T

yti j = min xti ,Qit ,Q tj , yti j = min xti ,qti ,Q tj ,

(9) (10)

Constraint (11) shows the ﬂow-feasibility in diverge cells when they are followed by three downstream cells j, k, and m (Daganzo, 1994). The ﬂow of vehicles is proportional to predeﬁned turning percentages, β tj at each time step t ∈ T. In

this constraint, when one of the receiving cells is full (ρ (N j − xtj ) = 0), ﬂow to other immediate downstream cells of the diverge cell will be zero too, while the other cells might still have an available capacity for receiving vehicles. This condition represents the ﬁrst-in-ﬁrst-out assumption at intersections and captures the effect of queue spillovers on decreasing the capacity of intersections. The variables in this constraint are represented by yti j , xti , xtj , xtk , and xtm . Note that the expressions

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that include index m should be removed from the constraint for diverge cells with only two downstream cells. Please see Appendix A for the illustration of this case.

yti j

= β min t j

xti ,

Qit ,

ρ N j −xtj Qkt ρ Nk −xtk Qmt ρ Nm −xtm , , t, , t , β tj β tj βk βkt βm βmt

Q tj

(11)

∀i∈Cd , j∈S(i ), k∈S(i ), m∈S(i ), j=k=m, t ∈T We have adopted the feasibility constraints proposed in (Ukkusuri et al., 2012) for merge cells. Constraint (12) ensures that the number of vehicles that merge cell i ∈ Cm can receive from each of its immediate upstream cells is proportional to the number of vehicles that the upstream cells can send to the merge cell. Variables of this constraint are yti j , xti , xtj , and xtk . Please see Appendix A for the illustration of this constraint. Constraint (12) does not consider any priorities for merging movements. More details on including priorities are available in (Daganzo, 1994).

t

yti j = min xti ,Qi

min Q tj ,ρ N j −xtj t t min 1, k∈P ( j ) min xk ,Qk

∀ j∈Cm , i∈P ( j ),t ∈T

(12)

The min(.) functions in Constraints (9)–(12) guarantee that traﬃc ﬂow follows the fundamental ﬂow-density relationship of the CTM by setting the ﬂow of vehicles between neighboring cells exactly equal to one of the terms included in the min(.) function. The min(.) operator can be linearized with the big-M technique (Lo, 2001; Mohebifard and Hajbabaie, 2018b). This technique requires adding many auxiliary binary variables to the problem and hence increases the complexity of the problem. As an alternative, a constraint with the min(.) operator can be replaced with several linear constraints with lessthan-or-equal-to operator without adding auxiliary binary variables. These constraints will ensure the ﬂow conservation; however, do not guarantee that ﬂow of vehicles between adjacent cells takes the minimum of the right-hand-side values. Therefore, the ﬂow can be less than the minimum value and the holding-back issue can happen. Although this approach reduces complexity of the problem, ﬂow of vehicles will not be “on” the ﬂow-density fundamental diagram of CTM, and thus ﬂow of vehicles is not realistic. Furthermore, the min(.) operators in Constraints (9)–(12) result in a more complex optimization program compared to the formulations developed by Lo (2001) and Lin and Wang (2004), to name a few. However, we will show later in Section 4 that this complexity can be utilized in development of a more eﬃcient solution technique. Finally, Constraint (13) ensures non-negativity of cell occupancies and ﬂows, and Constraint (14) ensures that signal timing variables are binary.

xti ≥0,yti j ≥0 gti ∈{0, 1}

∀i∈C , j∈S(i ),t ∈T ∀i∈Ci ,t ∈T

(13) (14)

4. Solution technique The signal timing decision variables are binary and additional binary variables will be introduced if Constraints (9)–(12) are to be converted to standard inequality constraints using the big-M technique. Note that the signal timing optimization is a computationally complex problem even without the min(.) operators in the constraints and cannot be solved even for a small-size network in a reasonable amount of time. However, the developed program has a speciﬁc structure that can be utilized to solve it eﬃciently: if signal timing decision variables are given, there is only one set of non-holding-back ﬂow decision variables that are feasible. These ﬂows can be found by a CTM simulation. We utilize this idea using the generalized Benders decomposition technique (Arslan and Karas¸ an, 2016; Benders, 1962; Fontaine and Minner, 2014; Geoffrion, 1972; Nourinejad et al., 2018) to develop an eﬃcient solution technique for the MINLP model. In this section, we show that the proposed methodology will always ﬁnd a feasible solution if one exists and improves it towards global optimality through iterations. 4.1. Generalized benders decomposition technique Benders (1962) proposed an algorithm to solve complex MILPs with complicating variables. The idea is to ﬁx the complicating variables temporarily to ﬁnd other decision variables, temporarily ﬁx them to ﬁnd the complicating variables, and go through this process iteratively. Later, Geoffrion (1972) generalized this algorithm to a broader class of nonlinear programs. The generalized Benders decomposition algorithm (Geoffrion, 1972) decomposes the original problem into two problems. The Primal problem is the original problem, whose complicating variables are temporarily ﬁxed to some initial values. Therefore, the constraints that only include these variables are removed from the problem. The Primal problem ﬁnds the optimal values of non-complicating variables, objective function value, and dual multipliers associated with constraints including both non-complicating and ﬁxed complicating variables. These values will be used in a Master problem. The Master problem is constructed by relaxing the dual of the original problem to optimize the values of the complicating variables whose values were temporarily ﬁxed in the Primal problem. The constraints are those of the original problem that

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include the complicating variables (the constraints that were relaxed in the Primal problem) and Benders cuts. Note that the values of non-complicating decision variables are ﬁxed in the Master problem. The Benders cuts are constraints that will be added to the Master problem iteratively. Solutions of the Master problem are new values for the complicating variables. Once the new values are found, the Primal problem can be solved again, and a new cut will be added to the Master problem. The feasible region of the Master problem becomes smaller by adding new cuts, and hence the algorithm converges to the optimal solutions. The objective function value of the Primal problem provides a lower bound for the objective function of the original problem (for a maximization problem) because the Primal problem is the original problem with ﬁxed values of complicating variables. The Master problem, on the other hand, is a relaxed problem and provides an upper bound for the objective function of the original problem. The algorithm iterates between the Primal and Master problems until the difference between the upper and lower bounds reaches an acceptable gap, where the algorithm can be terminated. We applied the discussed algorithm to the developed MINLP for signal timing optimization. The Primal problem solves the original MINLP problem with predetermined signal timing decision variables (gti = gˆti : ∀i ∈ Ci , t ∈ T ) and relaxes all constraints that include the signal timing decision variables (i.e., Constraints (3)–(5), (7), (8), and (14)). The Master problem is constructed by including only the constraints that contain signal timing decision variables and Benders cuts. Therefore, if we temporarily ﬁx the signal timing decision variables (gˆti : ∀i ∈ Ci , t ∈ T ), the Primal problem (P2) is as follows: Primal Problem:

(P2 ) max Z =

(1 )

xti

∀i∈Cs ∀t∈T

s.t.

(2 ), (9 )-(13 ) and the following constraints : ∀i ∈ Ci , t ∈ T

qti = gˆti Qit

(15)

The optimal solutions to the Primal problem are used to construct the Benders cuts and the Master problem. The cut is constructed by adding objective function (1) and Constraint (15) which is multiplied by its dual values (Geoffrion, 1972), see t (16). In the cut, λ∗ i : ∀i ∈ Ci , t ∈ T are the dual values of Constraint (15). Eq. (16) represents a Benders cut, and the values of t the variables with asterisks are the optimal solutions to the Primal problem. Note that the estimation procedure of λ∗ i is discussed later in Section 4.3.

L∗ gti =

∀i∈Cs ∀t∈T

x∗ti +

λ∗ ti q∗ti − gti Qit

(16)

∀i∈Ci ∀t∈T

Finally, the Benders cut along with Constraints (3)–(5), (7), and (8) construct the Master problem, (P3). The objective function of the Master problem is to maximize μ such that the constraints that include the signal timing decision variables are satisﬁed. At iteration k of the algorithm, a new cut L∗k is added to the Master problem. Therefore, the number of cuts in the Master problem increases in each iteration, resulting in a smaller feasible region and convergence of the algorithm (the formal proofs are in the following discussions). Master Problem:

(P3 )

max μ s.t.

(17)

(3 )-(5 ), (7 ), (8 ), (14 ), and the following constraints : L∗k gti ≥ μ k = 1, . . . , K

(18)

As is shown, the original MINLP is converted to the Primal problem, which is an MINLP, and the Master problem, which is an IP. Although the Primal problem is still an MINLP, it can be solved very eﬃciently because its optimal solutions can be found with a simple CTM simulation run. 4.2. Proof of optimality In this section, several properties of the developed solution technique are discussed. One of the main features of the solution technique is its convergence to the optimal solutions of the original problem within a ﬁnite number of iterations (Proposition 3). Before discussing this property, we show that the Primal problem has only one feasible non-holding back solution that can be found by a CTM simulation run (Proposition 1) and the solutions to the Master problem are always feasible to the original problem (Proposition 2). Proposition 1. The Primal problem has only one feasible non-holding back solution that can be found by a CTM simulation run. Proof. If we separate the constraints of (P2) over t ∈ T, starting from t = 0, y0i j for all i ∈ C and j ∈ S(i) can be found from Constraints (9)–(12). Then, the values of x1i for all i ∈ C can be derived from Constraint (2). With known x1i , the ﬂow of vehicles between cells for the next time step (t = 1) can be found similarly. By repeating this procedure, the values

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of all other variables can be found which is unique because of the equality constraints. This procedure is indeed a CTM simulation. According to Proposition 1, the optimal solutions of the Primal problem can be found by a simple CTM simulation rather than solving a complex optimization program. Moreover, with the proposed decomposition technique, the Primal problem only includes the constraints representing network loading, which are separated from the rest of the optimization program. Remark 1. Based on Proposition 1, the Primal problem has only one feasible solution, and hence its feasible area is convex by deﬁnition: a feasible region is convex if any linear combination of any two points (that are not required to be distinctive) is still in the feasible region (Boyd and Vandenberghe, 2004; Gallier, 2011). Moreover, the solution of the Primal problem is bounded because under any signal timing indication, according to Constraint (9), at most Qit vehicles can leave the network (getting to the sink cells), and hence the maximum number of vehicles in the sink cells, |T | 1 1 2 1 2 t i∈Cs [Qi + (Qi + Qi ) + . . . + (Qi + Qi + . . . + Qi )], is bounded by [|T | (|T | + 1 ) i∈Cs maxt∈T {Qi }]/2, where |.| is the cardi |T | 1 1 2 1 2 nality of a set. It is trivial that + (Qi + Qi + . . . + Qi )] ≤ i∈Cs [maxt∈T {Qit }(1 + 2 + . . . + |T |)], i∈Cs [Qi + (Qi + Qi ) + . . . and the right-hand side expression is equal to [|T |(|T | + 1) i∈Cs maxt∈T {Qit }]/2. Proposition 2. Solutions to the Master problem are always feasible to the original problem (P1). Proof. We will show that solutions to the Master problem do not violate any constraints of (P1). Among Constraints (2)–(14) of (P1), the Master problem satisﬁes Constraints (3)–(5), (7), (8), and (14) that are on the signal timing decision variables because these constraints are included in the constraints of the Master problem, see (P3). Moreover, solutions to the Master problem are binary signal indications that set the variable saturation ﬂow rate qti in Constraint (6) to either zero or Qit at time step t ∈ T for intersection cell i ∈ Ci . Furthermore, 0 ≤ qti ≤ Qit does not violate Constraints (2), and (9)–(13), which are all the remaining constraints. Consequently, the solutions to the Master problem are feasible to (P1). Proposition 2 states an important property of the solution technique. Because solutions to the Master problem are always feasible to (P1), the solution technique can be terminated at any intermediate iterations with a feasible solution available. Moreover, one of the drawbacks and complexities of the Benders decomposition technique is that solutions to the master problem are not necessarily feasible to the original problem. This issue requires additional efforts for ﬁnding feasible solutions and valid cuts, which adds to the computational complexity of the Benders based solution algorithm. The proposed solution technique does not have this issue. Remark 2. Program (P1) has |T||Ci | integer signal variables, |T||C| real-valued variables for the cell occupancies, |T||A| realvalued variables for the ﬂow of vehicles between cells, |T||Ci | real-valued variables for the variable saturation ﬂow rates (|.| is the cardinality of a set), and more than 4|T||A| integer variables for converting the min(.) operator. Note that four binary variables are required to represent a min(.) operator that includes four expressions for each link and time step, similar to the case of Constraint (9) with the big-M technique, please see (Mohebifard and Hajbabaie, 2018b) for more details). Thus, (P1) is a large optimization program even for a small network. Moreover, conventional solution techniques such as branch and bound require ﬁnding optimal solutions for |T|(|Ci | + 4|A|) integer and |T|(|C| + |A| + |Ci |) real-valued variables. However, the proposed solution technique reduces the original problem into a problem with only |T||Ci | integer variables that should be solved iteratively. Proposition 3. The solution technique converges to optimal solutions with a ﬁnite number of iterations with the optimality gap ε ≥ 0. Proof. The proof directly follows Theorem 2.4 in (Geoffrion, 1972) by considering Propositions 1 and 2, and Remark 1 that states that the Primal problem is convex for each set of feasible signal timing decision variables with only one feasible solution.

4.3. Dual value estimation algorithm Notice that the Benders cut in (16) requires the dual values of Constraint (15) in (P2). However, (P2) is an MINLP, for which the conventional duality theories do not hold. Although several generalizations of the duality theory have been proposed (Hooker, 1994; Hooker and Ottosson, 2003), the available approaches are computationally complex. Thus, we utilize the special structure of (P2) whose solutions can be found by a simple simulation run to determine dual values. We ﬁrst introduce a linearized version of (P2) and prove in Proposition 4 that it is equivalent to (P2) and then use the conventional duality theory to show that the dual values of Constraint (15) can be estimated by several simulations (Proposition 5). Proposition 4. The Primal problem is equivalent to the following linear program, where yˆti j : ∀i ∈ C, j ∈ S(i ), t ∈ T are the results of (P1) that are found by a CTM simulation.

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Linearized Primal Problem:

max Z =

(P4)

(1 )

xti

∀i∈Cs ∀t∈T

s.t.

(2 ), (13 ), (15 ), and the following constraints : ρ N j − xtj ∀i ∈ C \ Cd ∪ Cm , j ∈ S(i ), t ∈ T

ytij ≤ xti , Qit , Q tj ,

∀i ∈ Ci , j ∈ S(i ), t ∈ T

yti j ≤ qti

yti j

≤β

t j

(19)

xti

, Qit

(20)

ρ N j − xtj Qkt ρ Nk − xtk Qmt ρ Nm − xtm , t , , t , , t , βj β tj βk βkt βm βmt Q tj

(21)

∀i ∈ Cd , j ∈ S(i ), k ∈ S(i ), m ∈ S(i ), j = k = m, t ∈ T yti j ≤ yˆti j

∀ j ∈ Cm , i ∈ P ( j ), t ∈ T

(22)

yti j ≥ yˆti j

∀i ∈ C, j ∈ S(i ), t ∈ T

(23)

Proof. Please see Appendix B for the proof. Note that Constraints (19) and (21) are a compact representation of several linear constraints. For instance, Constraints (19) represents four constraints of yti j ≤ xti , yti j ≤ Qit , yti j ≤ Q tj , and yti j ≤ ρ (N j − xtj ) for i ∈ C\Cd ∪Cm , j ∈ S(i), t ∈ T. Moreover, prior to solving (P4), a CTM simulation should be solved (see Proposition 1 and Appendix B) to ﬁnd yˆti j : ∀i ∈ C, j ∈ S(i ), t ∈ T for Constraint (23). The linearized Primal problem (P4) can be solved instead of the original MINLP in the Primal problem. Notice that the two problems are equivalent, and Proposition 1 and Remark 1 are still valid for (P4). Moreover, because (P4) is linear, the conventional duality theory Bradley et al., 1977) holds for it, and thus dual values of Constraint (15) for the Benders cut in (16) can be found by solving (P4). We show in the next proposition that the linearization of the Primal problem can be further utilized to eliminate the need to solve an optimization program for the Primal problem and to develop a more general solution technique. Proposition 5. Dual values of Constraint (15) can be found by measuring the changes in the objective function by changing the signal indications. Proof. Dual value of a constraint shows the changes in the objective function per unit increase in the right-hand side of the t constraint (Bradley et al., 1977). Mathematically, dual value of Constraint (15), λ∗ i , can be written as follows:

λ∗ ti = Zgˆti Qit + − Zgˆti Qit /

∀i ∈ Ci , t ∈ T

(24)

where Zgˆt Q t is the objective function value when the right-hand side of Constraint (15) is gˆti Qit , and Zgˆt Q t + is the objeci

i

i

i

tive function value when the right-hand side of the constraint is increased by , gˆti Qit + . If we take = Qit in the latter equation, the dual values of Constraint (15) can be estimated as follows:

λ∗ ti = Z(gˆti +1 )Qit − Zgˆti Qit /Qit

∀i ∈ Ci , t ∈ T

(25)

In Eq. (25), Z(gˆt +1 )Q t , Zgˆt Q t , and λ∗ i are variables. Moreover, Eq. (25) has a concrete interpretation: The dual value of i i i i Constraint (15) is equal to the changes in the objective function over Qit when each signal indication changes individually. This deﬁnition means that changing the signal indication from red (gˆti = 0) to green (gˆti = 1) in intersection cell i ∈ Ci at time t

step t ∈ T gives its corresponding dual value λ∗ i . On the other hand, when the signal is already green, the dual value is zero because each intersection cell can process at most Qit vehicles per time step (Constraint (19)) and increasing the right-hand side of the constraint does not improve the objective function. Note that objective function Z might not change linearly with gti Qit , but gti Qit can only take values of zero (when gti = 0) or Qit (when gti = 1) in the right-hand side of Constraint (15). Consequently, only selecting = Qit will result in the saturation ﬂow rate of Qit to evaluate the effect of changing the signal from red to green. Since the Primal (or linearized) problem with ﬁxed signal indications has only one feasible solution (see Proposition 1), the dual values can be found by several CTM simulation runs. Fig. 2 shows the proposed algorithm for ﬁnding the dual values for the Benders cut (16). The dual estimation algorithm in Fig. 2 requires several simple simulation runs that might be time-consuming. However, the simulation runs are independent, and thus they can be performed in parallel. Parallel computation signiﬁcantly reduces the runtime of the dual estimation algorithm. t

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Fig. 2. The proposed procedure for ﬁnding dual values of Constraint (15).

Remark 3. The proposed dual value estimation algorithm in Fig. 2 eliminates the need for solving any optimization program for the Primal problem, and thus captures the complexities of the network loading representation by a simulation run without the need to represent the network loading model by explicit mathematical constraints. Accordingly, different network loading models such as link transmission (Yperman, 2007) and double queue (Ma et al., 2014; Osorio et al., 2011) models can be used instead of CTM without altering the solution technique. 4.4. Steps of the developed solution technique In summary, the steps of the developed solution technique are as follows: Step 0 (Initialization): Set k ← 0, the optimality gap ε , and ﬁnd an initial feasible solution gˆti for each i ∈ Ci ,t ∈ T. Solving the Master problem without Constraint (18) gives an initial feasible solution. Step 1 (Solution to the Primal problem): Set k ← k + 1 and run a CTM simulation using gˆti : ∀i ∈ Ci , t ∈ T and ﬁnd Z ∗ , x∗ti : ∀i ∈ C, t ∈ T and q∗tj: ∀ j ∈ Ci , t ∈ T . Step 2 (Dual estimation): For each intersection cell i ∈ Ci and each time step t, ﬁnd the dual values, λ∗ i , according to Fig. 2. Step 3 (Benders cut): Construct L∗ k (gti ) using Eq. (16). t

Step 4 (Solving Master problem): Add L∗ k (gti ) to the Master problem and solve (P3) to ﬁnd μ∗ and a new set of gˆti for all i ∈ Ci ,t ∈ T. Step 5 (Stopping criterion): If the stopping criterion (μ∗ − Z∗ )/μ∗ ≤ ε is met, gˆti are the optimal solutions. Otherwise, go to Step 1. Remark 4. Different linear objective functions (e.g., travel time or delay minimizations) can be used instead of maximizing the cumulative number of vehicles in the sink cells in the solution technique. This modiﬁcation can be considered by using the desired objective instead of ∀i∈Cs ∀t∈T xti in the problem formulation or solution technique. Note that the objective function should keep the problem representation as a “maximization” problem (e.g., negative of delay maximization); otherwise, the Master problem and dual value estimation algorithm should be modiﬁed for a minimization problem. 5. Case study Fig. 3 shows a portion of downtown Springﬁeld network in Illinois that is used as the case study network. This network has 20 intersections with a combination of one-way and two-way streets, different numbers of lanes, and various turning

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Fig. 3. Modiﬁed downtown Springﬁeld, Illinois network.

movements. The case study network is analyzed for 300 time steps (30 min). Moreover, we used CPLEX (2009) for solving the Master problem. The case study network is analyzed with the following demand proﬁles: - Symmetric undersaturated (Demand Proﬁle 1): 400 veh/hr/lane in all directions - Symmetric oversaturated (Demand Proﬁle 2): 900 veh/hr/lane in all directions - Asymmetric undersaturated (Demand Proﬁle 3): 100 veh/hr/lane in the North-South and 400 veh/hr/lane in the EastWest direction - Asymmetric near-saturation (Demand Proﬁle 4): 400 veh/hr/lane in the North-South and 900 veh/hr/lane in the EastWest direction Moreover, the network is loaded with deﬁned demand proﬁles for the entire study period, 300 time steps. We also set the capacity of resource cells to inﬁnity so that all vehicles can be considered in the analysis even if queue of vehicles at the boundary intersections reaches the resource cells and prevents other vehicles entering the network. Table 2 shows other characteristics of the case study network. Note that the case study network has 316 cells (see Table 2), 42,600 signal timing integer decision variables, 597,600 integer variables to represent the min(.) operator, and 253,800 real variables that represent the cell occupancies, ﬂows between cells and variable saturation ﬂow rates of the intersection cells. Moreover, the Master problem has 150,240 constraints at the start of the algorithm and one constraint, the Benders cut, will be added to the problem at each iteration. The Master problem is solved relatively fast as it does not include any network loading constraint. As such, it only needs to ﬁnd a feasible signal timing that results in the highest value of the Benders cuts.

264

R. Mohebifard and A. Hajbabaie / Transportation Research Part B 121 (2019) 252–274 Table 2 Characteristics of the case study network. Parameter

Value

Parameter

Value

Free ﬂow speed (mi/hr) Saturation ﬂow rate of links (veh/hr/lane) Time interval duration (sec) Total number of cells Length of cells (ft) Saturation ﬂow rate cells (veh/time step/lane)

25 1800 6 316 225 3

Jam density of cells (veh/cell/lane) Turning percentages (%) Minimum green time for through movements (sec) Minimum green time for left-turning movements (sec) Maximum green time for through movements (sec) Maximum green time for left-turning movements (sec)

12 10 18 6 60 24

6. Results This section presents the results of applying the developed solution technique to the case study network. Fig. 4 shows the upper and lower bounds on the objective function of the problem (the cumulative number of vehicles in the sink cells) for the four demand proﬁles at each iteration. The termination criterion for the algorithm was set to 2% gap between the bounds. The ﬁgure shows that in the beginning, the gap between the lower bound (feasible solution) and the upper bound was on average 49.8%. However, after the ﬁrst 150 iterations, the gap was reduced to 5% for all four demand proﬁles. The algorithm satisﬁed the termination criterion in at most 500 iterations (occurred for Demand Proﬁle 3 in Fig. 4.c). Similar trends were observed in our other tests indicating that the proposed solution technique numerically converges to the optimal solutions as well.

5.00

Upper and Lower Bounds (×106 veh)

Upper and Lower Bounds (×106 veh)

1.00 0.90 0.80

0.70 0.60

4.00 3.00

2.00 1.00

0

100

200

300

400

500

0

100

200

Iteration Number Lower Bound

Lower Bound

Upper Bound

(a) Demand Profile 1

400

500

Upper BOund

(b) Demand Profile 2

0.53

1.70

Upper and Lower Bounds (×106 veh)

Upper and Lower Bounds (×106 veh)

300

Iteration Number

0.46 0.39 0.32 0.25

1.45 1.20 0.95 0.70

0

100

200

300

400

Iteration Number Lower Bound

Upper Bound

(c) Demand Profile 3

500

0

100

200

300

400

Iteration Number Lower Bound

Upper Bound

(d) Demand Profile 4

Fig. 4. Convergence rate of the developed solution technique.

500

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Table 3 Network performance measures in the case study network.

Performance measure Demand proﬁle 1

Number of completed trips Total travel time (hr) Total delay (hr) Average speed (mph)

Demand proﬁle 2

BDT

Vistro

% Diff. BDT vs. Vistro

GA

% Diff. BDT vs. GA

BDT

Vistro

% Diff. BDT vs. Vistro

GA

% Diff. BDT vs. GA

4585

4539

1.0

4371

4.9

9477

6634

42.9

7991

18.6

179.26

207.47

−13.6

223.76

−19.9

606.94

943.55

−35.7

883.45

−31.3

49.37

75.75

−34.8

95.93

−48.5

337.07

738.58

−54.4

642.10

−47.5

18.09

15.84

14.2

14.23

27.1

11.05

5.27

109.6

6.72

64.5

GA

% Diff. BDT vs. GA

BDT

Vistro

% Diff. BDT vs. Vistro

GA

% Diff. BDT vs. GA

Performance measure Demand proﬁle 3

Number of completed trips Total travel time (hr) Total delay (hr) Average speed (mph)

Demand proﬁle 4

BDT

Vistro

% Diff. BDT vs. Vistro

2244

2240

0.2

2122

5.8

6349

6329

0.3

5944

6.8

88.57

95.28

−7.0

110.06

−19.5

270.09

299.15

−9.7

353.39

−23.6

22.54

27.96

−19.4

45.40

−50.3

85.89

110.55

−22.3

174.58

−50.8

18.61

17.64

5.5

14.64

27.2

17.02

15.73

8.2

12.59

35.2

BDT: The customized Benders decomposition technique, GA: Genetic algorithm.

We calculated several network performance measures for the signal timing parameters found by the following scenarios: (a) The customized Benders decomposition technique (BDT) that is developed in the present paper. (b) A state of practice signal timing optimizer, Vistro (PTV, 2014). We modeled the case study network in Vistro with the same minimum and maximum green times deﬁned in Table 2 and optimized the cycle lengths, green splits, and offsets at each intersection. Moreover, we used the coordination option of Vistro to coordinate all intersections inside the network. Vistro uses genetic algorithms and we used the maximum allowed number of iterations (10,0 0 0) and population size (100) in Vistro. The minimum and maximum cycle length values were 36 and 168 s, respectively, with a 6 s increment. Then, we used the optimized signals in a CTM simulation and reported the network performance measures. (c) A Genetic Algorithm-based technique (GA). For this approach, we adopted a similar algorithm that is developed in (Hajbabaie, 2012; Hajbabaie and Benekohal, 2013) for ﬁnding dynamic signal timings. The elitist GA algorithm with a roulette wheel selection approach was used. A blend crossover operator with a 50% probability and an alternate random mutation operator with a 0.5% probability was used. Moreover, we used cost function scaling factor of one for ﬁnding the probability of selecting candidate solutions for crossover or mutation in the roulette wheel selection component. These parameters were found through a ﬁne-tuning process using the case study network with Demand Proﬁle 2 and 30 min analysis period. Furthermore, the CTM was employed as the network loading model to evaluate the generated solutions. The maximum number of 10 0 0 iterations were used to optimize the signals for each demand proﬁle. Note that this approach found dynamic signal timing parameters without any ﬁxed cycle length, similar to the proposed approach. Table 3 shows the number of completed trips, total travel time, total delay, and average speeds in the discussed scenar ios. Total travel time is τ ∀i∈C\Cs ∀t∈T xti , total delay is τ ∀t∈T ∀i∈C\Cs (xti − ∀ j∈S(i ) yti j ), and weighted average speed of vehicles is VL [ ∀t∈T ∀i∈C\Cs xti ( ∀ j∈S(i ) yti j /xti )]/ ∀t∈T ∀i∈C\Cs xti . Table 3 shows that in Demand Proﬁles 1 and 3 (undersaturated ﬂow conditions), the number of completed trips in BDT is 1.0% and 0.2% more than Vistro, and 4.9% and 5.8% more than GA, respectively. The BDT signal timings reduced the total travel time by 13.6% and 7.0% compared to Vistro and 19.9% and 19.5% compared to the solutions of GA for Demand Proﬁles 1 and 3, respectively. In the oversaturated condition (Demand Proﬁle 2), BDT solutions reduced the total travel time and delay by at least 31.3% and increased the average speed of vehicles by at least 64.5%. Thus, BDT resulted in more noticeable improvements in all performance measures. Furthermore, Table 3 shows a similar trend of improvements in BDT compared to Vistro and GA for Demand Proﬁle 4. We observed a similar trend in our other tests indicating that the near-optimal solutions of the proposed methodology can lead to signiﬁcant improvements in network performance.

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R. Mohebifard and A. Hajbabaie / Transportation Research Part B 121 (2019) 252–274 Table 4 The minimum, average, and maximum green time (second) of through and left-turning movements in the 5th St. and Washington St. Intersection number

Movement

Demand proﬁle 1 min

avg

max

Demand proﬁle 2

Demand proﬁle 3

Demand proﬁle 4

min

avg

max

min

avg

max

min

avg

max

26.2 25.3 12.8 21.2 12.2 21.0 14.3

54.0 48.0 24.0 36.0 24.0 30.0 24.0

18.0 18.0 6.0 18.0 6.0 18.0 6.0

21.8 23.1 11.5 21.2 10.3 24.9 8.9

36.0 42.0 24.0 42.0 24.0 48.0 24.0

18.0 18.0 6.0 18.0 6.0 18.0 6.0

20.4 19.8 12.1 18.2 12.2 19.6 13.6

48.0 30.0 24.0 24.0 18.0 30.0 24.0

36.0 42.0 24.0 36.0 36.0 24.0 36.0 24.0

18.0 6.0 6.0 18.0 18.0 6.0 18.0 6.0

23.5 22.6 10.5 21.5 21.2 12.3 21.6 11.6

36.0 48.0 24.0 36.0 36.0 24.0 36.0 24.0

18.0 18.0 6.0 18.0 18.0 6.0 18.0 6.0

27.8 24.6 12.5 23.3 21.3 12.7 22.2 13.4

48.0 60.0 24.0 48.0 42.0 24.0 36.0 24.0

5th St. 1 2 3 4

T T L T L T L

18.0 18.0 6.0 18.0 6.0 18.0 6.0

19.5 22.2 11.9 18.2 11.0 18.8 10.7

30.0 60.0 24.0 24.0 18.0 36.0 18.0

T T L T T L T L

18.0 6.0 6.0 18.0 18.0 6.0 18.0 6.0

21.7 22.6 10.5 22.5 20.1 12.5 22.2 11.9

36.0 36.0 24.0 42.0 36.0 24.0 42.0 24.0

18.0 18.0 6.0 18.0 6.0 18.0 6.0

Washington St. 2 6 10 14 18

18.0 6.0 6.0 18.0 18.0 6.0 18.0 6.0

20.4 23.5 12.1 21.1 20.4 15.4 21.0 14.4

T: Through movement, L: Left-turning movement.

Table 5 The change in network performance measures by relaxing the minimum and maximum green time constraints (%). Criteria

Demand proﬁle 1

Demand proﬁle 2

Demand proﬁle 3

Demand proﬁle 4

Network throughput (veh) Total travel time (hr) Total delay (hr) Average speed (mph)

1.2 −11.4 −43.2 13.7

1.7 −9.0 −17.8 12.3

1.0 −8.4 −35.0 10.0

1.8 −11.1 −37.4 13.9

The minimum, average, and maximum green times for through and left-turning movements at all intersections of 5th St. and Washington St. (see Fig. 3) are shown in Table 4. The table shows that the minimum green times are 18 and 6 s respectively for the through and left-turning movements, which happens to be equal to the deﬁned minimum green durations in Table 2. This trend indicates that Constraint (8) for the minimum green times was binding in the test cases. On the other hand, most of the maximum green times for the through movements did not meet the deﬁned maximum green time of 60 s in test cases, except for two intersections. Fig. 5 shows the green time duration of through movements for 5th St. and Washington St. The green times change dynamically over time to ensure the best network performance. Moreover, the green durations are within the deﬁned minimum and maximum green times, 18 and 60 s respectively for the minimum and maximum green time of through movements in Table 2. The green distributions and their corresponding bandwidths for Demand Proﬁle 2 for the same two arterial streets are shown in Fig. 6. The ﬁgure shows that although no explicit constraints on the offsets and signal coordination are included in the model, the maximum value of the objective function occurred when the signals were coordinated. The effects of relaxing the minimum and maximum green time constraints (Constraints (7) and (8)) are investigated, and the results are summarized in Table 5. Although it is obvious that relaxing any constraint results in solutions that are not worse than the original solutions, it is important to evaluate the beneﬁts of relaxing such constraints with the future presence of connected and autonomous vehicles (Mirheli et al., 2018; Mirhelli et al., 2019; Yu et al., 2018). Such vehicles can respond more eﬃciently to traﬃc signals with frequent changes in the signal indications as they offer signiﬁcantly shorter reaction times. Table 5 shows that relaxing these constraints increased average vehicle speed by 10.0% to 13.9% and reduced the travel time and delay of the vehicles by 8.4% to 43.2% while increasing the network throughput by at most 1.8%. Hence, relaxing the constraints have a signiﬁcant positive effect on the performance of the network. Fig. 7 shows the runtimes of the solution technique at each iteration. A computer with 8 cores and 24 gigabytes of memory was used. The total runtimes for convergence were 10.5, 5.2, 12.7, and 11.8 h for Demand Proﬁles 1 to 4, respectively. Moreover, the average runtime for each iteration was 107 s. Fig. 7(a) shows that the Primal problem (including the dual estimation algorithm) was solved in at most 16 s. The runtime is quite consistent over iterations, which indicates that the computational complexity of the Primal problem does not change. The Master problem’s runtime ﬂuctuates in the ﬁrst 100 iterations. However, the runtimes show a linear increase with the number of iterations beyond the initial iterations. The increasing trend is due to an increase in the size of the problem as a result of adding more Benders cuts.

Green Time Duration (sec)

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267

60 50 40 30 20 10 0

3

6

9

12

15

18

21

24

27

30

Cycle Nnumber Intersection 1

Intersection 2

Intersection 3

Intersection 4

GreenTime Duration (sec)

(a) 5th St. (north-south direction) 60 50 40 30 20 10 0

3

6

9

12

15

18

21

24

27

30

Cycle Nnumber Intersection 2

Intersection 6

Intersection 10

Intersection 14

Intersection 18

(b) Washington St. (east-west direction) Fig. 5. Green duration of through signal indications of two arterial streets for Demand Proﬁle 2.

Finally, we compared the solutions of the proposed Benders decomposition technique (BDT) with CPLEX at three levels. At the ﬁrst level, CPLEX is used to solve the original unrelaxed problem. At the second level, CPLEX is used to solve the original problem with relaxed min(.) functions. Note that the problem still includes integer signal timing variables. The third level introduces another relaxation, where in addition to the min(.) functions, the integrality constraints are relaxed. In other words, this problem does not include any integer variable. We set the maximum CPLEX runtime equal to 15 h in all cases and reported the best solution it found within this timeframe. Note that these solutions may not be the optimal solutions. Table 6 shows the cumulative number of vehicles in the sink cells (the objective of P1) that were found by all approaches. The objective values of BDT are 653,350, 1344,914, 319,632, and 900,747 vehicles for Demand Proﬁles 1 to 4, respectively. These solutions were found in at most 13 h. CPLEX could not ﬁnd a feasible integer solution to the original problem with no relaxation in 15 h. On the other hand, CPLEX could ﬁnd a feasible solution to the problem with relaxed min(.)operators (i.e., Constraints (9)–(12)). These operators were each replaced with several constraints with a less-than-or-equal-to operators similar to the formulations proposed in (Lin and Wang, 2004; Lo, 1999). This relaxation reduced the number of integer variables signiﬁcantly, hence CPLEX found feasible integer solutions for the signal indications in 15 h. However, Table 6 shows that the objective values of CPLEX solutions are 11.7%, 7.7%, 8.8%, and 9.8% lower than the objective values found by BDT for Demand Proﬁles 1 to 4, respectively. Note that the reported CPLEX solutions are those that could be found within 15 h runtime and not the optimal solutions. Table 6 also shows the results of relaxing both min(.) functions and integrality constraints (He et al., 2010). In this scenario, all integrality constraints of (P1) are relaxed, and the original problem is converted into a linear program (LP) whose objective value is a “theoretical” upper bound for (P1). Accordingly, solutions of the linear program do not provide integer signal indications and allow vehicles in conﬂicting approaches pass intersections at the same time. The results show that the objective values of BDT are 2.0%, 8.6%, 1.7%, and 3.2% lower than their theoretical upper bounds respectively for Demand Proﬁles 1 to 4. Note that the gap of Demand Proﬁle 2 is more than the other demand proﬁles because it has the highest demand level. These results show the capability of BDT in ﬁnding integer solutions, eliminating ﬂow-holding-back, ensuring the signal feasibility constraints, and still being less than 8.6% different from the theoretical upper bounds in the evaluated scenarios.

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Fig. 6. Green time distribution in two arterials for Demand Proﬁle 2.

Fig. 7. Runtimes of solving (a) Primal problem (including dual estimation) and (b) Master problem.

7. Conclusion We developed an MINLP that optimizes the green time duration of traﬃc signals based on the CTM network loading concept. The formulation eliminates the ﬂow holding-back problem by considering the fundamental diagram of the CTM explicitly as hard constraints in the program. The proposed program is computationally expensive, and thus we proposed a Benders decomposition technique to decompose the MINLP to an IP and several CTM simulation runs. The solution technique signiﬁcantly reduced the complexity of the original MINLP, for which exact solutions could not be found eﬃciently. Furthermore, we showed that the solution technique converges to optimal solutions within a ﬁnite number of iterations, and all the solutions in the intermediate iterations are feasible to the original problem. This property eliminates the need for ﬁnding feasible solutions and feasibility cuts when solutions to the master problem are not feasible to the original problem. We also proposed a simulation-based dual value estimation algorithm to generate Benders cuts without solving a complex optimization program.

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269

Table 6 Comparison of objective function values (cumulative number of vehicles in sink cells) of the proposed solution technique with the solutions of CPLEX. Demand proﬁle 1

Demand proﬁle 2

Solution technique

Problem

Objective value (veh)

Runtime (hr)

Objective value (veh)

Runtime (hr)

Benders decomposition technique (BDT) CPLEX∗

Central problem (P1)

653,350

10.5

1344,914

5.2

Central problem (P1) Central problem with integer signal variables and relaxed min(.) operators∗∗ Central problem with relaxed integer signal variables and relaxed min(.) operators

NFI 584,842

15.0 15.0

NFI 1248,408

15.0 15.0

666,722

0.02

1472,138

0.04

Demand proﬁle 3

Demand proﬁle 4

Solution technique

Problem

Objective value (veh)

Runtime (hr)

Objective value (veh)

Runtime (hr)

Benders decomposition technique (BDT) CPLEX∗

Central problem (P1)

319,632

12.7

900,747

11.8

Central problem (P1) Central problem with integer signal variables and relaxed min(.) operators∗∗ Central problem with relaxed integer signal variables and relaxed min(.) operators

NFI 293,782

15.0 15.0

NFI 820,488

15.0 15.0

325,061

0.02

930,762

0.02

NFI: No feasible integer solution was found. ∗ The maximum runtime of CPLEX was set to 15 h. ∗∗ CPLEX could not ﬁnd the optimal solutions in this scenario, and the reported solutions are the best solutions that could be found within 15 h runtime.

The solution technique was applied to a case study network with four different demand proﬁles. The results show that the solution technique found the optimal solutions with at most a 2% optimality gap and outperformed the signal plans derived from Vistro and a GA-based algorithm. The signal plans of the developed program decreased the total travel time and delay of vehicles by 7.0% to 54.4% compared to the benchmark solutions. The proposed solution technique provides insights to optimizing traﬃc control problems when the complexity is mostly due to the network loading representation. The network loading principles might be very easy to handle in a simulation script by a series of “if-then” and “for-loop” commands. In the optimization formulation, the network loading principles should be converted into standard constraints that requires adding many additional auxiliary variables and constraints to the problem that makes the problem intractable. However, we showed that the developed solution technique does not require expressing the network loading equations with explicit standard constraints. Instead, we can use simulation that takes a set of ﬁxed decision variables as input and gives the desired performance measure, i.e., the objective function value. Then, by updating the values of the decision variables and iterating the procedure, the optimal solutions can be found. This solution technique is a promising framework for integrating mathematical programming and traﬃc simulation. Furthermore, the proposed algorithm can serve as a benchmark for distributed or decentralized signal timing techniques to evaluate their solution quality. The proposed solution technique can theoretically ﬁnd the optimal signal timing parameters within a predeﬁned optimality gap; however, it follows a centralized architecture that might not allow ﬁnding solutions in the desired time span. We used CPLEX to solve the IP in the Master problem with a standard branch-andcut algorithm. The run-time of solving the Master problem can be reduced further by developing customized algorithms. Note that the proposed formulation and solution technique do not enforce a common cycle. Thus, customizing the formulation and solution technique so that they can consider a common cycle length will be an interesting topic for further research. Appendix A This appendix illustrates Constraints (11) and (12) for diverge and merge cells, respectively. Consider the following diverge cells with two and three downstream cells. For each case, Constraint (11) is as follows:

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Constraint (12) for the case of two and three merging movements is as follows:

Appendix B The proof of Proposition 4 is discussed in this appendix. We presented an equivalent linear program (P4) for the Primal problem (P2). Constraints (2), (13), and (15) in the Primal problem are linear and thus we used them without any changes in the linear program (P4). However, ﬂow-feasibility Constraints (9)–(12) should be linearized. These constraints are shown below:

ρ N j − xtj

∀i ∈ Co ∪ Cr ∪ Cs , j ∈ S(i ), t ∈ T

ρ N j − xtj

∀i ∈ Ci , j ∈ S(i ), t ∈ T

ytij = min xti , Qit , Q tj , yti j = min xti , qti , Q tj ,

Qt

ρ (N j −xt ) Q t

ρ (N −xt ) Q t

k j k yti j = β tj min xti , Qit , βjt , , βkt , , βmt , β tj βkt m j k ∀i ∈ Cd , j ∈ S(i ), k ∈ S(i ), m ∈ S(i ), j = k = m, t ∈ T

yti j

= min

xti

, Qit

ρ (Nm −xtm ) βmt

(9) (10)

(11)

min Q tj , ρ N j − xtj t t min 1, k∈P ( j ) min xk , Qk

∀ j ∈ Cm , i ∈ P ( j ), t ∈ T

(12)

We linearized each of the above constraints separately, starting from Constraint (9). This constraint is an “equality” constraint that can be equivalently written as two “inequality” constraints as follows (Bradley et al., 1977):

ρ N j − xtj

∀i ∈ Co ∪ Cr ∪ Cs , j ∈ S(i ), t ∈ T

(9-1)

ρ N j − xtj

∀i ∈ Co ∪ Cr ∪ Cs , j ∈ S(i ), t ∈ T

(9-2)

ytij ≤ min xti , Qit , Q tj , ytij ≥ min xti , Qit , Q tj ,

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Moreover, it is trivial that when yti j is smaller than minimum of several variables (Constraint (9-1)) it will be smaller than each variable individually. Thus, Constraint (9-1) is equivalent to constraints (9-1-1) to (9-1-4). In other words, Constraints (9-1-1) to (9-1-4) forces yti j to be less than or equal to the minimum of xti , Qit , Q tj , and ρ (N j − xtj ).

ytij ≤ xti

∀i ∈ Co ∪ Cr ∪ Cs , j ∈ S(i ), t ∈ T

(9-1-1)

ytij ≤ Qit

∀i ∈ Co ∪ Cr ∪ Cs , j ∈ S(i ), t ∈ T

(9-1-2)

∀i ∈ Co ∪ Cr ∪ Cs , j ∈ S(i ), t ∈ T ≤ ρ N j − xtj ∀i ∈ Co ∪ Cr ∪ Cs , j ∈ S(i ), t ∈ T

(9-1-3)

ytij ≤ Q tj ytij

(9-1-4)

On the other hand, value of yti j : ∀i ∈ Co ∪ Cr ∪ Cs , j ∈ S(i ), t ∈ T is equal to the solutions of a CTM simulation according

to Proposition 1. Suppose that yˆtij: ∀i ∈ Co ∪ Cr ∪ Cs , j ∈ S(i ), t ∈ T is the solution of a CTM simulation. Thus, we can write

ytij = yˆtij = min{xti , Qit , Q tj , ρ (N j − xtj )}: ∀i ∈ Co ∪ Cr ∪ Cs , j ∈ S(i ), t ∈ T . Accordingly, we can substitute the right-hand side of Constraint (9–2) with yˆti j and rewrite it as Constraint (26):

∀i ∈ Co ∪ Cr ∪ Cs , j ∈ S(i ), t ∈ T

ytij ≥ yˆtij

(26)

We can write Constraint (9) as several equivalent linear Constraints (26) and (27) with the discussed modiﬁcations. Note that Constraint (27) is the compact representation of linear constraints (9-1-1) to (9-1-4).

ytij ≤ xti , Qit , Q tj , ytij

≥

ρ N j − xtj

∀i ∈ Co ∪ Cr ∪ Cs , j ∈ S(i ), t ∈ T

∀i ∈ Co ∪ Cr ∪ Cs , j ∈ S(i ), t ∈ T

yˆtij

(27) (26)

Moreover, we can linearize other nonlinear Constraints (10)–(12) with the same approach. Thus, Constraint (10) is represented by linear Constraints (28) and (29) while Constraint (11) is converted to Constraints (30) and (31). We followed the same linearization approach for Constraints (10) and (11) as Constraint (9). The compact representation of these constraints are as follows. Constraints (28) and (29) are the linearized form of Constraints (10) and Constraints (30) and (31) are the linearized form of Constraint (11). Moreover, Constraint (12) required one further step for linearization because of the min(.) operators that are multiplied on the right-hand side of this constraint. Thus, we replaced nonlinear expression min{xti , Qit } min{1, (32) and (33).

yti j ≤ xti , qti , Q tj ,

min{Q tj ,ρ (N j −xtj )} t t } k∈P ( j ) min{x ,Q } k

ρ N j − xtj

k

by its equal value yˆti j and wrote it with two linear inequality Constraints

∀i ∈ Ci , j ∈ S(i ), t ∈ T

∀i ∈ Ci , j ∈ S(i ), t ∈ T Q tj ρ N j − xtj Qkt ρ Nk − xtk Qmt ρ Nm − xtm t t t t yi j ≤ β j xi , Qi , t , , t , , t , βj β tj βk βkt βm βmt ∀i ∈ Cd , j ∈ S(i ), k ∈ S(i ), m ∈ S(i ), j = k = m, t ∈ T yti j ≥ yˆti j

(28) (29)

(30)

yti j ≥ yˆti j

∀i ∈ Cd , j ∈ S(i ), k ∈ S(i ), m ∈ S(i ), j = k = m, t ∈ T

(31)

yti j ≤ yˆti j

∀ j ∈ Cm , i ∈ P ( j ), t ∈ T

(32)

yti j ≥ yˆti j

∀ j ∈ Cm , i ∈ P ( j ), t ∈ T

(33)

Finally, we combined Constraints (26)–(33) and represented the linearized Primal problem as (P4). Linearized Primal Problem:

(P4 )

max Z =

xti

(1 )

∀i∈Cs ∀t∈T

s.t.

(2 ), (13 ), (15 ), and the following constraints: ytij ≤ xti , Qit , Q tj , ρ N j − xtj ∀i ∈ C \ Cd ∪ Cm , j ∈ S(i ), t ∈ T

(19)

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yti j ≤ qti

∀i ∈ Ci , j ∈ S(i ), t ∈ T Qt

yti j ≤ β tj xti , Qit , βjt , j

ρ (N j −xtj ) Qkt ρ (Nk −xtk ) Qmt ρ (Nm −xtm ) , βt , , βt , β tj βkt βmt m k

(20)

(21)

∀i ∈ Cd , j ∈ S(i ), k ∈ S(i ), m ∈ S(i ), j = k = m, t ∈ T yti j ≤ yˆti j

∀ j ∈ Cm , i ∈ P ( j ), t ∈ T

(22)

yti j ≥ yˆti j

∀i ∈ C, j ∈ S(i ), t ∈ T

(23)

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