Little, John D. C.

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ALFRED P. SLOAN SCHOOL OF MANAGEMENTA Mixed Integer Linear Program for SynchronizingTraffic Signals for Maximal Bandwidth109-65John D. C. LittleMASSACHUSETTSINSTITUTE OF TECHNOLOGY50 MEMORIAL DRIVECAMBRIDGE, MASSACHUSETTS 02139A Mixed Integer Linear Program for SynchronizingTraffic Signals for Maximal Bandwidth109-65John D. C. LittleJanuary 13, 1965Working PaperWork reported was supported in part by U. S. Army ResearchOffice (Durham) under contract Da-31-124-ARO-D-209.tOI-LfAbstractTraffic signals can be synchronized so that a car, starting at one endof the street and traveling at preassigned speeds, can go to the other endwithout stopping for a red light. The portion of a signal cycle for whichthis is possible is called the bandwidth for that direction. Ordinarilythe bandwidth in each direction is single, i.e., is not split into two ormore intervals within a cycle. For this case we formulate the followingproblem as a mixed integer linear program: Given an arbitrary number ofsignals, red times for each signal, upper and lower limits on signal period,upper and lower limits on speed each way between adjacent signals, limitson change in speed and a fixed constant of proportionality between the twobandwidths, find a common signal period, speeds between signals, and therelative phasing of the signals so as to maximize the sum of the bandwidths.1 . IntroductionIn an earlier paper J. Morgan and the author have developed analgorithm for solving the following two problems in synchronizing trafficsignals along an arterial street:1. Given an arbitrary number of signals, a common signal period, thegreen and red times for each signal, and specified travel times between ad-jacent signals, synchronize the signals to produce bandwidths that are equalin each direction and as large as possible.2. Adjust the synchronization to increase one bandwidth to somespecified, feasible value and maintain the other as large as is then poss-ible.Subsequently, R. Oliver pointed out to the author that a two-signal versionof problem 1 could be set up as a linear program. We here expand this ideainto a rather general formulation of the maximal bandwidth problem. Theformulation comes out as a fairly sizable mixed integer linear program.For problems 1 and 2 this offers no advantages and many disadvantages.However, the linear programming format opens up the possibility of solvingmore general problems involving the introduction of new decision variables.For example, maximal bandwidth calculations as usually performed havea disconcerting feature. On a long street the critical signals that con-struct bandwidth may turn out to be very far apart. Then a small change inone of the speeds along the street is likely, upon re-solving the problem,to result in a different synchronization and a different bandwidth. Yet,drivers do not hold their speeds particularly constant and, indeed, as iawell known, tend to adjust their speeds to the signals. Thus, it makes agood deal of sense to let speed between signals be a variable. At the same- 2 -time, it is a fairly simple matter in a linear program to put limits on thespeeds that can be assumed.Another variable that can be introduced explicitly in a linear programis the signal period. In Problem 1, period is a constant and, although itis not too difficult to use our earlier methods to examine a considerablenumber of values in an organized way, the continuous variation of the linearprogram formulation seems preferable.2. DefinitionsConsider a two-way street having n traffic signals. Directions on thestreet will be identified as outbound and inbound . The signals will be de-noted S., S ,...,S with the subscript increasing in the outbound direction,i 2 nUnder some circumstances, a car can start at one end of the street and,by traveling at preassigned speeds, go to the other end without stopping.The portion of a signal cycle during which this is possible will be calledthe bandwidth in this direction. (Ordinarily the bandwidth in each cycleis single, i.e., is not split into two or more intervals within a cycle. Werestrict ourselves to this case.)Figure 1 shows a space-time diagram for travel on a street. Heavyhorizontal lines indicate when the signals are red. The zig-zag lines re-present trajectories of cars passing unimpeded along the street in thedirections shown. Changes in slope correspond to changes in speed. The setof possible unimpeded trajectories in a given direction forms a green bandwhose horizontal width is the bandwidth for that direction. Although drawnbut once, it is clear that the green bands appear once per cycle in parallelbands across the diagram.Certain signals, usually two or three of the n, form the ultimatelimitation or bandwidth and will be called critical signals . A signal S.is said to be a critical signal if one side of S.'s red touches the greenCOUJ_JUJ PcCOIcCO CO COtCOozZ>omh-Z>O3OCOE«00EI<uu«9O.CO0)Uo0030NV1SIQ- J -Land in one direction and the other side touches the green band in the otherdirection. Thus, in Figure 1 signals S, and S are cirtical, but no others1 iare.3. Basic Maximal Bandwidth LPFirst we set up the basic Problem 1 above as a mixed integer linearprogram.3.1 NotationLetr = red time of S on street under study, (cycles)b (b) = outbound (inbound) bandwidth, (cycles)t (t ) = travel time from S to S in outbound direction (traveltime from S. to S in inbound direction), (cycles)0. (0.) = distance from center of red at S to the center of aparticular red at S • See Figure 2. The two reds arechosen so that each is immediately to the left (right) ofthe same outbound (inbound) green band. 0. (0.) is posi-tive if S. 's center of red lies to the right (left) ofS 's. (cycles)w (w.) = distance from the right (left) side of S 's red to thegreen band. See Figure 2. (cycles)m. = 0. + 0.J J .1Note that a quantity having dimensions of time can always be expressed incycles by dividing by the period.3. 2 FormulationFrom Figure 2 can be read the identities*-s.w.Figure 2. Geometry of the green bands. Notice that9+9. must be an integral number of cycles- 4 -(1/2) r1+ »1+ t±- wi- (l/2)r4=£(1/2) rx+ wt+ tt- w±- (1/2)rt=t.The values of and 0, have the very important constraint that their sumis an integer, but otherwise they are unrestricted. Therefore, the abovetwo constraints can be replaced by(w +vx) - (tv^) - \ « < rrV - ( t i+s. )m. = integer .iFrom Figure 2 we also see thatw + b < 1 - rand that a similar expression holds for the inbound direction. Consequently,Problem 1 can be represented by the linear program:LP1. Find b, b, w^ w^ n^ toMax (b + b)Subject to:b = b (LP1.1)WjL + b < 1 - r / (LP1.2a)_ _ V i = l,...,nwt+ b < 1 - r (LP1.2b)(w.+w ) - (w.-t-w.) - m. = (r.-r ) - (t.+t.) )i = 2. ...,n (LP1.3)llii i l 1 ill ' ' vmi = integer ( (LP1.4)b, b, v., w >LP1 has 3n equations and 3n+l unknowns, not counting slack or artificial variables4. More Decision VariablesNext we let period and speed be variables. Each will be constrainedby upper and lower limits. In addition changes in speed from one street- 5 -segment to the next will be limited. Finally, instead of requiring equalbandwidths in each direction, we generalize to let one bandwidth be a fixedproportion of the other.4.1 NotationLetk = constant of proportionality between b and b.T = signal period, (seconds)d = distance from S. , to S., i = 2, .... n . (meters)i l-l iv. (v.) = speed between S. and S outbound (S and S. inbound),i = 2, ..., n . (meters/second)We wish to put limits on speed, say,:e < v. < f i = 2, . . .,n ,i - i — ior, equivalently,111.i i iWe also wish to constrain change of speed. This will be done by constrainingchange in reciprocal speed. Although the two are not quite the same, con-straining change in the reciprocal surely satisfies the basic intention,which is to have a means of preventing large abrupt speed changes. For nota-tion, we use.1_ < ±_ - jL < -L- i = 3, . .., n .hi" Vi Vi-l- g iLimits for T can be denotedT. < T < T,i — <Next, lety. (y,) = time to travel from S. , to S J outbound (S. to S. , inbound),3xJx l-l i i l-li = 2, . ..,n. (cycles)Thenz = 1/T = signal frequency (cycles/second)d.x_(di/fi)z <Y±< (di/ei)z(di/hi)z < y±- y < (dj./gi)zt = £ y1 k=2k2, . . ., nSimilar expressions can be written involving y. and t.straint relating b and b isb = k bFinally, the con-4„ 2 FormulationThe above can be assembled intoLP2o Find b, b, w . w, y. y., m. toMax (b + b)Subject to:b = kb(1/T2 ) < z < (1/T1)w. + b < 1-ri — iw. + b < i- r.l — i<W " (Vwi ) - mi + k?2 (W = (rr r i>m. = integer(d./f.) z < y. < (d./e.) zl l i i(dj/fj) z < y±< (dj^/e ) zi=l, . . . ,ni=2,...,n(LP2.1)(LP2.2)(LP2.3a)(LP 2. 3b)(LP 2. 4a)(LP2.4b)(LP2.5a)(LP2.5b)- 7(d./h.) z < y. - y. < (d./g.) z ) (LP2.6a)11 — 1 l-l li /S i=3, ...n(d /h.) z <y±- y± _ l< (d./g.) z J (LP2.6b)b, b, w , w^ y^ y ± , >LP2 involves (lln-10) constraints and 4n variables, not counting slackand artificial variables5. The SynchronizationThe linear programming variables determine the synchronization of thesignals. Let9 = relative phase of S and S , measured as the time fromJ 1 Jthe center of a red of S, to the next center of red ofS . (cycles) By convention < < 1. See figure 1.A set of Q., j = 2, . ..., n ; will be called a synchronization.From Figure 2, we see0. = mani lwhere "man" stands for "mantissa of" (e.g., man (5.2) = .2, man (-1.3) = .7)Here= l/2(r -r.) + w, + t. - w. .i 1 l liiThis can easily be computed from the linear program results.ReferencesJ. T. Morgan and J: D. C. Little, "Synchronizing Traffic Signals forMaximal Bandwidth," Operations Research, 12, 896-912, (Nov-Dec. 1964)AUG 2 7 7J

A mixed integer linear program for synchronizing traffic signals for maximal bandwidth.

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A mixed integer linear program for synchronizing traffic signals for maximal bandwidth.

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