Progression Optimization in Priority Arterial Signal Networks

Copyright ~ IFAC Control in Transportation Systems, Braunschweig, Germany, 2000 PROGRESSION OPTIMIZATION IN PRIORITY ARTERIAL SIGNAL NETWORKS Chronis Stamatiadis, Nathan H. Gartner Dept. ofCivil and Environmental Engineering, University ofMassachusetts, Lowell, MA 01854, U.S.A. Abstract: Arterial progression methods, adapted for application to grid networks, use mixed integer linear programming models for the maximization of the bandwidths. Due to the number of discrete variables involved, these methods are computationally demanding and inefficient when applied to large networks. This paper develops a fast heuristic procedure, which can be applied to both uniform and variable bandwidth optimization. The procedure uses the traffic characteristics of the network and involves an iterative decomposition into priority arterial sub-networks. This facilitates the determination of the optimal values for the integer variables and enables the application of bandwidth optimization methods to large-scale networks. Copyright ID 2000 IFAC Keywords: Traffic Control; Bandwidth Optimization; Progressions; Mixed Integer Linear Programming; Heuristics. 1. INTRODUCTION arterial. The MULTIBAND model is also formulated as a mixed-integer linear problem. A network version of the variable bandwidth formulation was also introduced (Stamatiadis and Gartner, 1996). The multi-band design has been shown to provide significant benefits in terms of common performance measures such as delays, number of stops and fuel consumption over conventional uniform bandwidth models. However, the multi-band formulation increases the size of the continuous variable set and of the constraint set by 20-60010 compared with the uniform band formulation; while, the size of the integer variables set remains the same. This makes the model more computationally demanding. Arterial progression methods, which use the width of green bands along the arterial as the objective of optimization, are often used for traffic signal control. Numerous computer programs were developed for finding optimal signal timing plans based on bandwidth maximization. There are two categories of such programs: one uses uniform bandwidth optimization, while the other variable bandwidth optimization. Two of the most versatile and advanced programs in the first category are PASSER11 and MAXBAND, which provide offset, split, cycle length and left turn phase sequence optimization on single arterial streets. The MAXBAND formulation (Little et al., 1981) includes several decision variables that are integers and, therefore. mixedinteger linear programming (MILP) is used for solving the problem. The model is applicable to single arterials as well as to networks (Chang et al., 1988). The primary difficulty in solving these models is due to the multidimensionality of the discrete variables of the problems: the large number of discrete variables, as well as their wide feasible ranges, make the solution of the program computationally demanding. This paper presents a network decomposition procedure for more efficient solution of the variable bandwidth network progression problem. The procedure is based on the traffic characteristics of the network. The network is decomposed into priority sub-networks which facilitates the determination of the optimal values for the integer variables. This improves dramatically the efficiency of the solution and thus enables to handle large-scale networks. similar to the ones found in many metropolitan areas. The solution that is obtained is equal or nearly equal A more elaborate approach was developed by Gartner et al. (1990, 1991) in the MULTIBAND model, which is an extension of MAXBAND. By incorporating into the model a traffic-dependent criterion, MULTIBAND calculates individual bandwidths for each directional link of the arterial while still maintaining main street platoon progression. The individual bands depend on the actual traffic volumes that each link carries and the resulting signal-timing plan is tailored to the varying traffic flows along the 555 Outbound Inbound • WI+IJ , ·,• ~ Outbound '""iiI!i!i5i!i!i2i!i\ Inbound - ar:oIi~_ I!.,+IJ ' , ~ +Oll.(/+IJ) . rt·t/J : .. i. .. +(1)).(1+ IJ) : ..; 'I+IJ . , ,~ , ' .,-:,, :~ . ~. fy .. 'u Si} . Figure 1: Time-space diagram for unifonn bandwidth optimization 4>(lj}.(lcI) to the globally optimal solution obtained by solving the original model. The procedure can be applied to both unifonn and variable bandwidth optimization models. The speed and the quality of the iterative decomposition procedure may allow on-line implementation of the method in the future. 2. )Jk(, ji ~ I:!.ij = 'ij; DESCRIPTION OF THE MODEL queue arterial); outbound (inbound) red time at Sij; w,,(wu )= interference time time for speed on link Lij (ft/sec). In the case of unifonn bands, the objective function has the following fonn: The geometric relations for the unifonn bandwidth model are shown in Figure 1. Consider a network with m arterials and each arterial having nj signalized intersections. Let SI] denote the i'h signal on the Jh arterial of the network and LI] denote the i'h link (between signals i and i+ 1) of the Jh arterial, with) = 1 ,..., m and i = 1 ,... , n]. All time variables are defined in units of the cycle time. The following variables are defined: cycle time (sec); C= outbound (inbound) bandwidth on bj ( b; ) = 'y(ru )= clearance advancement of outbound (inbound) bandwidth at Sij to clear turning-in traffic before arrival of main-street platoon; outbound (inbound) progression The multi-band optimization model consists of several blocks of constraints dealing with the individual arterials of the network as well as a set of network constraints that ensures that continuous bands are being produced on all the intersecting arterials. The basic unifonn bandwidth optimization fonnulation is presented first. variables, ) = internode offsets, time from the center of the outbound (inbound) red at SI] to the center of the outbound (inbound) red at SkI; directional node phase shift, time from center of Tu to nearest center of Maximize f.(b) +k) .~) (1) ;=1 where k) is the target ratio of inbound to outbound bandwidth for arterial). The directional interference constraints ensure that the progression bands use only the green time and they do not cross through the red time. Only one such constraint is needed for each signal Sij and each directional band: (2.a) from (2.b) right (left) side of red at S,) to left (right) side of outbound (inbound) green band; travel time on link i of arterial) in the The arterial loop constraints result from the fact that all signals must be synchronized, i.e., that they operate with a common cycle time. In Figure 1 it can be seen that for each link L ,) the summation of the outbound (inbound) direction; 556 internode offsets and directional node phase shifts is an integer multiple of the cycle time as follows: cI>(ij),(I+I,i) + ~(lj),i+ + ~i+I'J - ~lj =Kij (6) (3) An additional important decision capability that can be added through the Mll..P formulation of the problem is in identifying the optimal left-turn phase sequence with respect to the through green at any signal Sij. Up to four additional integer variables may be added at each node to identify all possible left-turn phase sequences. where 1C;j is an integer variable. The same principle of signal synchronization applies to closed loops of the network consisting of more than 2 links, resulting in the network loop constraints. For simplicity, we drop the arterial index in the notation of nodes and internode offsets and we define the intranode offset OJijk as the time from the center of the red at S; for traffic moving from S, to Sf' to the center of red in the crossing direction at the same node for traffic moving The network loop from Sj to Sk (Figure 2). constraints specify that the summation of internode and intranode offsets around a loop of intersecting arterials must be an integer multiple of the cycle time: cl>ij +roljk +cI>jk +ro jkl +cI>kI +ro kll +cI>/i +ro/ij = Il n In the MULTlBAND model the width of the directional bands may differ from link to link. The bandwidth can be individually weighted with respect to its contribution to the overall objective function. The link specific bands generated by MULTlBAND are symmetric about the centerline of the arterial progression band. The geometry of MULTlBAND is shown in Figure 3. The bandwidths and the interference variables are redefined as follows: (4) where Pn is the integer variable of the nth network loop. The number of network loop constraints and the choice of a fundamental set of loops are given by Gartner (2). hi} ( ~ Wij )jiT~l( ) = outbound (inbound) bandwidth of link i on arterial j; there is now one specific band for each link Lij; = the time from right (left) side of red at Sij to the centerline of the outbound (inbound) green band; the reference point at each signal has been moved from the edges to the centerline of the band. The objective function now has the form: .. 1 n·-I _ Maximize " -L ~ (<x lj . hlj -Ki... h) L. n- -1 lj lj j=1 where Fig. 2: Closed loop of intersecting arterials <XI) J aij and (8) 1=1 are the link specific weighting coefficients for the outbound and inbound directions respectively. The following coefficients are used: The cycle time C (sec) and the link specific progression speeds V'j and vij are treated as decision variables as well. This formulation introduces considerable flexibility in the calculation of the best progression scheme. Each of these variables must be constrained by upper and lower bounds as follows: Cl' Cl = lower and upper bounds on cycle length; (eij , j,), (elj' IIj) = lower and upper bounds on outbound (inbound) speed Vi} ( 'I+JJ tl2?mazwZl2?i2: T, ezzzzmzzzzmnt vlj ); hy» = lower and upper bounds on change in outbound (inbound) speed (gl)' hij)' (gij' vij (vij). p2??2unaZZd The corresponding constraints will be: Cl $ C $ C2 (5) Fig. 3: Geometric relations for the variable bandwidth optimization model 557 where. Vy (~ Sy ( • use more economical mathematical programming codes; (9) and • to optimize larger-scale networks; )= outbound (inbound) directional flow • obtain more reliable convergence to optimal solutions; rate on link L,;; sij ) = saturation flow rate outbound (inbound) • analyze a larger number of alternatives at a much reduced cost; and directional volume on link 11;;; an integer exponential power; the values 0, 1,2 and 4 were used P= • enable the use of codes in real time, e.g., in a multilevel RT-TRACS system (Gartner et al., 1995). In the case of variable bandwidths the band must be constrained from both sides so that neither edge of the band crosses through the red time. For each signal Sy and for each link specific directional band there are two interference constraints, as follows: wij+bij/25,1-rij and 0~2/jib+ w (10.a) wij +bij/25,1-r;j and wij 0~2/j,b+ Branch-and-bound is essentially a strategy of "divide and conquer". The idea is to partition the feasible region into more manageable subdivisions and, eventually, to fathom the entire tree of integer solutions. It is advantageous to decompose mixedinteger programming problems into smaller subproblems, in order to reduce the number of integer variables that have to be considered in each subproblem, and to restrict as much as possible the allowable range of each variable. Several authors have proposed solutions for the uniform bandwidth problem along these lines, e.g., Mireault (1991), Chaudhary et al. (1991) and Pillai et al. (1994). (lO.b) The same relationship must be valid at both ends of the band, i.e., at signals Sy and S;+lJ' The arterial loop constraints, and the network loop constraints are not affected by the variable bandwidth extension of the procedure and remain as in expressions (3) and (4). The cycle time and the progression speed, as well as, the left-turn phase in pattern constraints remain unchanged MULTffiAND. The new procedure that is described in this paper does not merely exploit the mathematical structure of the mixed-integer problem, as most other approaches do, but is primarily based on the traffic characteristics of the network. As such, it does not require any modification of the optimization code as would be required by some of the other approaches. A priority arterial sub-network consisting of an arterial tree is selected from the original traffic network based on its geometry and on the volume of traffic that each link is carrying. The priority sub-network is optimized first and the results are then used for the solution of the entire network. Alternative sub-networks can be selected in a heuristic procedure if further improvements are desired. Most importantly, the new decomposition process does not require relaxation of any integer variables unlike previous approaches. The intermediate solution obtained are each a feasible optimal solution for the priority sub-network considered. This feature, in combination with the fact that the priority sub-network contains the bulk of the traffic volumes in the network, enables the achievement of faster and better solutions. Results indicate that an optimal solution is obtained in the majority of cases during the first iteration. The decomposition procedure is described below. It has been shown (Gartner et al., 1990, 1991, Stamatiadis and Gartner, 1996, 1997) that the multiband approach offers substantial flexibility for the design of progressions that result in significant It improvements in all performance measures. produces optimal variable progression schemes tailored to the traffic demand and capacity of each individual road section along each arterial street, while simultaneously it optimizes progressions on the crossing arterials as well. 3. THE NETWORK DECOMPOSITION APPROACH The MULTffiAND model uses mixed-integer linear programming for determining the optimal solution. The principal difficulty in solving mixed-integer problems is the number of integer variables and their range, i.e., the size of the integer feasible set. Virtually all MILP codes use branch-and-bound strategies for calculating the optimal values. More efficient computational procedures will result in the ability to obtain improved solutions and, ultimately, will lead to improved performance of the traffic network. Computational procedures are just as important as are more accurate traffic models, especially in an era of increasing real time applications. In the case of progression optimization in networks this enables the following: Let P J be a progression optimization problem obtained by considering only a sub-network Ni of the original network N, and P2 be another progression optimization problem obtained by setting a selected subset of the arterial loop and the network loop integer variables in the network problem to a specific Then the heuristic network set of values. 'i'iR The performance of the heuristic solution was tested on two networks: the first in downtown Ann-Arbor, Michigan (3 x 5 grid), shown in Figure Sa, and the second in downtown Memphis, Tennessee (4 x 4 The grids are not grid), shown in figure 5b. complete, as there are not signals at each of the nodes. The shaded arterials indicate the priority arterial sub-networks utilized in this analysis. The results are shown in Table 2. It must be noted that the values of the objective function of MAXBAND and MULTIBAND are not directly comparable. All the runs were executed on a 200MHz Pentium computer. The procedure was applied to both the uniform and the variable bandwidth models. For the uniform bandwidth case the heuristic calculated an optimal solution - there are multiple optimal solutions - for both sample networks. In the case of the multi-band model, the heuristic located the optimal solution only for one of the test networks. However, the execution times were significantly reduced by as much as 1:263 compared to the original times. set of values. Then the heuristic network decomposition method, shown in Figure 4, is as follows: ~ Step 1: Identify a new priority sub-network N, c N y Step 2: Optimize P1 for N" and save the resulting ~ values of the arterial loop and network loop integer variables ICy', f.Jn' for all the links and network loops ofN,. Step 3: Optimize P2 by setting the integer variables calculated in step 2 (IC;; = ICy'. f.Jn =f.Jn' : 1<:,1' ~ ~ f.Jn E P1). Step 4: Calculate the value of the objective function. If it is better than the previous solution, save it. Step 5: Stop if all priority sub-networks have been considered; otherwise go back to step 1. A priority sub-network is chosen so that it contains only a "tree" of arterials, eliminating any network loops. The arterials contained in the "tree" should include the principal arterials of the network and can be chosen based on the following criteria: 1. Choose the principal arterial of the network to be the trunk of the tree and include only crossing arterials in the sub-network; 2. The tree consists of the maximum number of arterial two-way links without forming any network loops. The solution of both P 1 and P2 can be obtained very quickly due to the reduced number of integer variables. Table 1 shows the number of integer variables in the original problem and in the two subproblems P 1 and P2 of the network decomposition approach for an m x n closed grid network (m x n intersections and m+ n arterials) . (b) Select Priority Tree Sub-Network N, :N, c N Optimiz. N, and save K.j: (ij) Optimiz.]I; by setting K.j: (ij) E E N, N, No Reoort optimal Fig. 5: The (a) Ann-Arbor,Michigan and the (b) Memphis, Tennessee networks. FigA The network decomposition procedure 559 Table 1: Size of MILP problem for an m grid network. Var. MAXBAND b,b 2(m+l) MULTIBAND 2(m(n-I)+ +n(m-!)) z x 5. REFERENCES Little, lD.C., Kelson, M.D., Gartner, N.H. (1981). MAXBAND: A program for setting signals on arteries and triangular networks. Transportation Research Record 795, 40-46. n closed Heuriltk -Step1 HeuristkStep2 2(mn-l) 2(2mn-m-n) Chang E.,C-P, Cohen, S.L, Liu, C, Chaudhary, N.A., Messer, C. (1988). MAXBAND-86: A program for optimizing left-turn phase sequence in multiarterial closed networks. Transportation Research Record 1181,61-67. Gartner, N.H., Assmann, S.F., Lasaga, F., Hou, D.L. (1990). MULTffiAND - A variable-bandwidth arterial progression scheme. Transportation Research Record 1287,212-222. I w,w 2(2mn-m-n) 2(2mn-m-n) 2(mn-l) 2(2mn-m-n) K (2mn-m-n) (2mn-m-n) mn-I mn-m-n+1 Jl (m-l)(n-I) (m-I)(n-l) 0 (m-I)(n-I) Total int. 3mn-2m-2n+1 3mn-2m-2n+ I mn-I 2(mn-m-n) Example Networks / No. of integers 4x6 53 53 23 30 3x7 44 44 20 24 Gartner, N.H., Assmann, S.F., Lasaga, F., Hou, D.L., (1991). A multi-band approach to arterial traffic signal optimization. Transportation Research Vol. 25B(I), 55-74. Stamatiadis, C., Gartner, N.H., (1996). MULTIBAND-96: A program for variable bandwidth progression optimization of multi-arterial traffic networks. Transportation Research Record 1554, 9-17. Table 2: Objective Function Values and Execution Times of the original and heuristic solutions. Network Memphis, TN (8/17) Model MAXB. Obj. Value 3.4682 Enc. MULl. 7.9381 Ann Arbor, MI (8/14) MAXB. MULl. 2.9381 4.8930 6,735sec 15,012sec 4,235sec 9,995sec Obj. Value 3.4682 (100"10) 7.9381 (100%) 2.9381 (100%) 37680 (77%) EIec. 50sec (11135) 57sec (1/263) 44sec (1/%) 51sec (1/1%) Time Time 4. Stamatiadis, C., Gartner, N.H., (1997), Concurrent Progression Schemes in Multi-arterial Traffic Networks.. IFACIIFIPIIFORS Symp., Chania, Greece. Gartner, N.H., Stamatiadis, c., Tarnoff, P.l (1995). Development of advanced traffic signal control strategies for IVHS: A multi-level design. Transportation Research Record 1494,98-105. Mireault, P., (1991). A branch-and-bound algorithm for the traffic signal synchronization problem with variable speed. TRSTAN I, Montreal., Canada. Chaudhary, N.A., Pinnoi, A., Messer, c., (1991). Proposed enhancements to MAXBAND-86 program. Transportation Research Record 1324, 98-104. Pillai, R.S., Rathi, A.K., Cohen, S., (1994). A restricted branch-and-bound approach for setting the left turn phase sequence in signalized networks. Transportation Research, Vol. 32B(5). CONCLUSIONS This paper describes a new network decomposition procedure for the solution of the mixed integer linear programming formulation of the multi-band network progression problem. It is shown that the procedure can yield considerable reductions in computational effort without or with little compromise in the quality of the results. Reductions range from 1:100 to 1:300 compared with the original formulation with the most dramatic reductions occurring in the case of the more of variable bandwidth complex problem optimization. By achieving these results one can obtain more easily optimal solutions for large-scale networks, analyze a larger number of alternatives, as well as implement this strategy in an on-line system. Overall, more efficient computational procedures result in the ability to obtain improved solutions and, ultimately, lead to improved performance of the traffic network. They are just as important as are more accurate modeling techniques. 560