Memirs f the Schl f Engineering, Okayama University Vl. 20, N.2, February 1986 Inflw Cntrl n Expressway Cnsidering Traffic Equilibria Hirshi INOUYE* (Received February 14, 1986) SYNOPSIS When expressway and rads cexist, it is necessary t establish a reasnable traffic share between them. It may be practiced by the regulatin f tll-rate f expressway. But at an ccasinal traffic cngestin, the reasnable share is disturbed, s that sme traffic cntrl means shuld be taken. In this paper, we deal hw t cntrl inf lws n expressway, frm a viewpint f the ptimal share between expressway and rads. The minimizatin f ttal travel cst in a system is aimed under traffic equilibrium cnditins. The prblem is frmed as a tw-stage prgramming mdel, and a simple example slving the prblem is shwed. 1. INTRODUCTION S far, the traffic cntrl n expressway mainly aims hw t disslve speedily the traffic cngestin caused by an accident. But nwadays, natural traffic cngestins freequently ccur n expressway with the increase f traffic demand. Accrdingly, it becmes t be necessary t cntrl inflws n expressway. When inflws are limited at interchanges, the traffic flw n a parallel rad increaes, and traffic cngestin n it becmes intensified. Fr this reasn, the limitatin f inflws n express- * Department f Civil Engineering 47
48 Hirshi INOUYE way may nt be allwed, if traffic cngestin gets wrse. It may be allwed nly if the scial benefit is imprved thrugh the cntl. We aim the minimizatin f ttal travel cst in a netwrk instead f the prmtin f scial benefit. By the way, the direct cntrl f inflws is nt practical under present facilities f expressway and peratins f traffic in ur natin. A pssible way is t cntrl inflws indirectly, thrugh the cntrl f the number f pen bths. On the ther hand, expressway users may decide their behavirs n infrmatins ffered by the traffic peratr f expressway. In this case, users will act such that their utility is maximized r their lss is minimized. S that, the traffic peratr can decide the ptimal cntrl strategy, taking int accunt user's behavir. This can be cnsidered as a tw-persn game, in which the traffic peratr have the initiative and can decide the ptimum strategy ahead f users. 2. FORMULATION Nw we cnsider a netwrk which cnsists f an expressway, such rads that are substitutive fr expressway and crss cnnectins with expressway. As traffic demands, we take int accunt nly thse cncerned with expressway. Other demands are treated as fixed values. In rder t simplify, a kind f vehicles is assumed. rad :... inflw link Fig.l An Example f Expressway and Rad Netwrk Let dente sij:demand frm i t j, Xijp:flw n rute p frm i t j, rijpk=l:if link k is cntained in rute p frm i t j O:therwise, Cijp:tll rate n rute p frm i t j.
Inflw Cntrl n Expressway Cnsidering Traffic Equilibria 49 We devide the set f 1 ink int tw sets. One is the set f inf lw links t expressway and the ther is the set f ther rdinary links. The frmer is represented by L i and the latter is by La. It is assumed that the travel time n a rdinary link k is a mntne increasing functin f link flw, dented by Tk=fk(X k ). On the ther hand, we assume that the travel time n inflw link k is a functin f bth capcity Yk and flw X k, dented by Tk=gk(Yk'X k ). Inf lw 1ink capacity depends n the number f pen bths, but we assume it as a cntinuus variable fr cnvenience. The travel time n a inflw link represents the delay caused by queues at tll-gate. It shuld be mntne increasing fr flw and mntne decreasing fr capacity. The flws which enter frm interchanges ut f cntrl area are trated as fixed values. The fixed flws n link k is dented by XkO Nw we assume any user acts such that his travel cst is minimized. As travel cst, we take the sum f the value f travel time and tllrate n the rute. Then the distributin f flws, n a netwrk under given inflw link capacities, is btained thrugh traffic equilibrium cnditins. That is the slutin f the fllwing ptimizatin prblem(l), (2). minimize fr x, subject t Where v J(y,x) X k = L Xijp'rijpk + X ko ijp LXijp = Sij P Xijp f, 0 is the average time value. On the ther hand, X k + v L J gk(yk,x)dx + kel 0 Itk Itij Itij,p L XijpCijp ijp ( 1 ) we assume that the traffic peratr cntrls the number f pen bths such that user's ttal travel cst is minimized, taking int accunt user's rute chice behavirs. As travel cst, we take the ttal value f time lss and tll-rate n a netwrk. user's ttal travel cst is represented by F(y,x) (2) (3) (4) (5) Then, where, Itk (6)
50 Hirshi INOUYE Inflw link capacity Yk shuld be under less than the supremum 9k' which is equivalent t the capacity when all bths are pened, that Yk ~ 9k The traffic peratr links, taking int kel i (7) can decide n the ptimum capacitiy f inflw accunt user's rute chice behavir. On the ther hand, user's can nt knw the strategy f traffic peratr. by the peratr. They merely decide their behavirs n infrmatins prvided this tw-persn game, S that the traffic peratr has an init.iative in and can take the ptimum strategy ahead f users. Whence, this prblem can be frmed as the fllwing tw-level ptimizatin prblem (3). Where, F(y*,x(y*)) min F(y,x(y)) (8) y sub. t ~ Yk ~ 9k J(y,x(y))=min J(y,x) x sub. t Xk = L: xijp'rijpk+ XkO ijp L:xijp = Sij P Xijp ~ 0 \fk \fij \fij,p y* is the ptimal slutin fr inflw link capacities, and x(y) is the parametric ptimal slutin f the inferir prblem crrespnding t given inflw link capacities y. Clearly x(y) shws ne-t-ne. crrespndence, s that if the inferir prblem is feasible, an equilibrium slutin, Le. the stackelberg slutin exists. This tw-level ptimizatin prblem may be slved by the fllwing algrithm. (1) Select an feasible initial value y(l), and set n=l. (2) Slve the inferir prblem fr x under given capacities yin). (3) Cmpute the gradient f bjective functin f the superir prblem, using the ptimality cnditins f the inferir prblem. (4) Slve the superir prblem, using the gradient. Write yin) fr the ptimal slutin. If Ily(n)-y(n-1) 11< E:, then stp. Otherwise, set n=n+1, and g t step (2). (9) (10) (11) (12) (13) s
1I Inflw Cntrl n Expressway Cnsidering Traffic Equilibria 51 3. EXAMPLE 3 1 8 2 10 I X 122-- t ~ ~ L_l ~. J7 4 3 6 Fig.2 Netwrk fr Example Nw, we cnsider a simple netwrk illustrated in Fig.2. Link 1 and 2 are inflw links and the ther are rdinary links. We take int cnsideratin traffic demands nly frm nde 3 t 4 and frm nde 1 t 2. Flws with respect t ther 00 pairs are fixed. Fr each 00 pairs, tw rutes by expressway and by rad are cnsidered respectively. If these rutes are balanced in travel cst, the equilibrium cnditins fr the inferir prblem are represented by x121+x122 = 812 x341+x 342 = 8 34 V(T2+TS+T7+T10)+c121 V(T1+T3+T4+TS+Tg)+c341 These equatins may be cnsidered as simultaneus equatins f rute flws, x 121 x122 x341 s that implicit functins h 1 (Y1'Y2) h 2 (Y1'Y2) h 3 (Y1'Y2) x342 h 4 (Y1'Y2) (18) exist. The partial differentials f them are expressed as fllws. (14) (15) (16) (17) 1 1-1 0 1 T' 5 1 (19 ) T '+T'+T'+T'+T' lx 3 4 5 9 -T ' ly
52 Hirshi INOUYE dx 121-1 l 1 1 0 0 0 dy2 dx122 --- 0 0 1 1 0 dy 2 (20) dx --- 341 T '+T'+T'+T ' -T T' 0 -T ' dy2 2x 5 7 10 S 5 2y dx342 --- T' 0 T'+T'+T'+T' -T 0 dy 5 3 4 5 9 2 6 Where, dt dt dt k k k Tk=dX (kel ), T '=-- T '=-- kx dx (keli) k k ky dyk On the ther hand, the ttal travel cst is represented by F = v[(xio+x341)t1+(x20+x121)t2+(x30+x341)t3 +(X40+x341)T4+(XSO+x121+x341)TS+(X60+x342)T6 +(X70+x121)T7+(XSO+x122)TS+(X90+x341)T9 + (X100+x121)T10 (21) If implicit functins (1S) are substituted int (21t, ttal travel time F becmes a functin f Y1 and Y2' dented by F (Y1'Y2) _ The gradient f F is given by dx 121 XT '+A 1 ly (22) X T '+ A 2 2y ~ dx122 ay;- dx 341 (23) dy2 dx 342 dy 2 where, A (24) Then, we can cmpute the gradient f the evaluatin functin F, frm (19), (20), (22 ) and ( 23) _ The ptimality cnditins fr the superir prblem are as fllws_
Inflw Cntrl n Expressway Cnsidering Traffic Equilibria 53 ali' ayk < 0 (if Yk=Y k ) 0 (if O<Yk<Y k ) (k=1,2) ~ 0 (if Yk=0) (25) If Yk < Yk and af/ayk < 0, then ttal travel cst may be decreased by expand f the inflw link capacity. On the ther hand, if Yk > 0, and af/ayk > 0, then we can nt reduce ttal travel cst by increase f the inflw link capacity. In this case, we may take inflw restrictin strategy at that interchange, thrugh the restrictin f the number f pen bths. 4. CONCLUDING REMARKS An inflw cntrl mdel n expressway was dealt n ptimal traffic share between expressway and rads. It was assumed that any user acts reasnably, s as t minimize his travel cst under given cnditins. This is valid, if infrmatins with respect t traffic cngestins and travel times are ffered t users quickly and accurately. The mdel prpsed is still cnceptinal. There are sme prblems t use practically. One f them is hw t calculate in case f large netwrk. The ther is hw much des the cntrl prduce effects n ttal travel cst. These are the present pending questins. REFERENCES (1) J.G.Wardrp: Prc. Inst. Civil Engrs., 1 (1952), 325 (2) M.J.Beckmann, C.B.McGuire & C.B.Winsten: "Studies in the Ecnmics f Transprtatin", Yale University Press (1956) (3) H.Vn Stackelberg: "The Thery f Market Ecnmy", Oxfrd University Press (1952)