CHOICE OF THE CONTROL VARIABLES OF AN ISOLATED INTERSECTION BY GRAPH COLOURING

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1 Yugoslav Journal of Operatons Research 25 (25), Number, 7-3 DOI:.2298/YJOR38345B CHOICE OF THE CONTROL VARIABLES OF AN ISOLATED INTERSECTION BY GRAPH COLOURING Vladan BATANOVIĆ Mhalo Pupn Insttute, Volgna 5, Belgrade, Serba Slobodan GUBERINIĆ Mhalo Pupn Insttute, Volgna 5, Belgrade, Serba Radvo PETROVIĆ Mhalo Pupn Insttute, Volgna 5, Belgrade, Serba Receved: Аugust 23 / Accepted: October 23 Abstract: Ths paper deals wth the problem of groupng traffc streams nto sgnal groups on a sgnalzed ntersecton. Determnaton of the complete sets of sgnal groups,.e. the groups of traffc streams on one ntersecton, controlled by one control varable s defned n ths paper as a graph-colorng problem. The complete sets of sgnal groups are obtaned by colorng the complement of the graph of dentcal ndcatons. It s shown that the mnmal number of sgnal groups n the complete set of sgnal groups s equal to the chromatc number of the complement of the graph wth dentcal ndcatons. The problem of fndng all complete sets of sgnal groups wth mnmal cardnalty s formulated as a lnear programmng problem where the values of varables belong to a set {,}. Keywords: Traffc control, Sgnalzed ntersecton, Sgnal group, Graph colorng, Optmzaton. MSC: 9C35.

2 8 V. Batanovć, S. Gubernć, R. Petrovć / Choce of the Control Varables. INTRODUCTION Vehcles approachng an ntersecton are ready to perform certan "maneuver",.e. to drve straght through, turn left, or turn rght at the ntersecton. The vehcles whch perform the same maneuver and form the same queue on an approach, n one or several lanes, represent a flow component that can be consdered separately from other flow components that perform other maneuvers [], [2]. Such an arrval flow component s termed as a traffc stream. In fact, ths s the smallest flow component that can be controlled by a separate traffc sgnal,.e. by a sequence of sgnal ndcatons dfferent from the sequences on other sgnals. Traffc streams on an ntersecton are elements of the set of traffc streams S.e. S =,,,,, }, () where { 2 I J, and J s the set of traffc stream ndces: J = {,2,,,, I } = {,2,,,, I,, I}. Indces =,2,, I are assgned to vehcle traffc streams, and ndces = I +,, I to a pedestran and other traffc streams. Elements of set S are components of vector = (, 2,, I), whch descrbes the uncontrolled system nput and represent passenger vehcle flows, pedestran flows, flows of publc transport vehcles, etc. For an exact statement and soluton of traffc control problems, t s necessary to study the relatons n the set of traffc streams S. The most mportant relatons are: conflctness, non-conflctness and compatblty... Conflctness of Traffc Streams Some pars of traffc streams use along the part of ther traectores, through the ntersecton, the same space, so-called the conflct area. Traectores of these streams cross or merge. Between such streams, there exsts a conflct. The set of all pars of traffc streams where elements of a par are n conflct represents the conflctness relaton. Thus, the conflctness relaton C can be defned n the followng way: C S S (2) C = {(, ) traectores of and cross or merge,, J } The graph of conflctness G k s defned by the set S and the relaton C : = S, C ). G k ( (3)

3 V. Batanovć, S. Gubernć, R. Petrovć / Choce of the Control Varables 9 Snce there s a conflct between any two streams whose traectores cross or merge, t s obvous that the conflctness relaton s symmetrcal: (, ) C C,, J. (4) (, ) Relaton C s not reflexve (a stream cannot be n conflct by tself). Therefore, (, ) C, ( J )..2. Non-conflctness of Traffc Streams The non-conflctness relaton of traffc streams represents a set of ordered pars of traffc streams, where the traectores of the elements of the pars nether cross nor merge. Thus, ths relaton s the set of all pars of traffc streams that are not mutually n conflct: C ( C (5) 2 = C = S S ) \ The graph of non-conflctness s defned by the set S and the relaton C 2, as G k = ( S, C2 ). Traectores traversed by dfferent traffc streams through the ntersecton have to be known n order to determne whether a par of traffc streams can smultaneously gan the rght-of-way,.e. whether the streams are compatble..3. Compatblty of Traffc Streams Snce the man obectve of the traffc control by traffc lghts s to gve the rght-ofway to some traffc streams, and to stop others n the set of traffc streams of an ntersecton, t s necessary to fnd the traffc streams whch can smultaneously get the rght-of-way. Therefore, the traffc stream compatblty relaton s ntroduced. It s defned by a set of traffc streams pars, such that elements of a par can smultaneously get the rght-of-way. The traffc stream compatblty relaton plays an mportant role n solvng traffc control problems related to: Decdng whether a traffc control by traffc lghts should be ntroduced at an ntersecton, Assgnng control varables to traffc streams or to subsets of traffc streams, The traffc control process on an ntersecton. The factors to be consdered when defnng the compatblty relaton are: The ntersecton geometry, Factors related to the traffc safety process, for whch traffc engneers expert estmatons are needed. The analyss of the ntersecton geometry consders mutual relatons of traectores of traffc streams. Obvously, when traectores of two traffc streams do not cross, these streams can smultaneously get the rght-of-way,.e. they are compatble. On the other hand, when traectores of two traffc streams do cross, the streams are n a conflct and ther smultaneous movement through the ntersecton should not be permtted. However f volumes are not hgh, a "flterng" of one stream through another stream can be

4 2 V. Batanovć, S. Gubernć, R. Petrovć / Choce of the Control Varables permtted n some cases. When determnng the compatblty relaton, some specal requrements should be taken nto account, e.g., t s requred sometmes that some streams have to pass through the ntersecton wthout any dsturbance although, flterng could be permtted f only ther volumes are consdered. These requrements are usually acheved by so called drectonal sgnals. When only geometrcal factors are consdered, the relaton of conflctness and the relaton of non-conflctness can be defned. It means that when determnng the compatblty relaton of traffc streams, besdes data on geometrcal features of traffc stream traectores, t s necessary to consder some other factors,.e. t s necessary to lst: Pars of conflctng traffc streams that can smultaneously get the rght-ofway, The traffc streams requred to pass through the ntersecton wthout any dsturbance (the streams to whch the rght-of-way s gven by drectonal sgnals). Some pars of conflctng traffc streams can be, at the same tme, the pars of compatble streams (although the streams are conflctng). Therefore, t s necessary to dvde the conflcts nto allowed and forbdden [3]. Forbdden conflcts can be regulated only by traffc lghts, whle allowed conflcts are solved by traffc partcpants themselves, respectng prorty rules prescrbed by traffc regulatons. Wthout traffc lghts, conflcts are solved by "flterng" one stream through another. Obvously, the possblty of flterng depends on vehcle spacng nterval, whch depends on volume of traffc streams. Snce the volumes change durng a day, and there are perods wth very hgh volume dfferences, such as mornng peak, afternoon peak, off-peak and nght perods, stuatons may arse that two conflctng traffc streams may smultaneously have the rght-of-way n one perod but not n some other. The set of traffc streams pars, whch comprse condtonally compatble streams,.e. conflctng streams allowed to pass smultaneously through an ntersecton, can be thus defned as follows: C2 = {(, ) (, ) C,, J, streams and can smulta neously (6) get the rght of way} The problem of ntroducng traffc sgnals for the traffc control on an ntersecton s actually a problem of the same knd. It s necessary to determne when traffc lghts have to be ntroduced n order to remove conflcts,.e. to determne the values of traffc stream volumes when flterng s not possble any more. Before the traffc sgnals were ntroduced, traffc partcpants themselves, usng flterng and respectng prorty rules, were solvng all the conflcts. When volumes of conflctng traffc streams reach a level where flterng becomes dffcult, the ntroducton of traffc lghts becomes unavodable because traffc partcpants themselves cannot solve the conflcts. The values of traffc stream volumes that ustfy an ntroducton of the sgnalzaton of an ntersecton are gven n tables n traffc-engneerng handbooks. Not ntroducng traffc lghts when these levels are reached can lead to many negatve effects, such as an enormous number of stops and delays, ncrease n the number of traffc accdents, etc. Therefore, conflcts at all conflct

5 V. Batanovć, S. Gubernć, R. Petrovć / Choce of the Control Varables 2 ponts on a not sgnalzed ntersecton are prevented by traffc partcpants, respectng prorty rules, whle at a sgnalzed ntersecton traffc lghts are used n order to avod conflcts at most of the conflct ponts, wth a possblty of conflcts n some conflct ponts stll left for "self-regulaton" by traffc partcpants. The compatblty relaton of traffc stream pars whose elements can smultaneously get the rght-of-way s: C = (7) 2 C2 C2 In some cases, t may be necessary to control the traffc n such a way that certan streams can pass through an ntersecton wthout condtonal conflcts. Then they cannot gan the rght-of-way smultaneously wth any other conflctng streams although, t would be ustfed f only volumes were consdered. For controllng these streams, the drectonal sgnals are used. If the set of streams that have to pass through the ntersecton wthout any conflct s denoted by S, where S S, then the set of pars of traffc streams that can smultaneously get the rght-of-way s defned by the followng expresson: C 3 = C2 \ {(, ) (, ) C2, ( or S )} (8) Assumng that each traffc stream s compatble wth tself then, n order to defne the set of pars that determne the compatblty relaton, set of pars C 3 should be extended by the dagonal Δ S n set S. Therefore, the compatblty relaton can be defned as: C = C Δ 3 S, (9) where Δ = {(, ), J } () S Relaton C s symmetrc and reflexve. Compatblty graph of traffc streams s defned by the set of traffc streams S and the compatblty relaton C: G c = ( S, C). () Snce the set S s fnte, and the relaton C s symmetrc and reflexve, graph s a fnte, non-orented graph, wth a loop at each node. The ncdence matrx of ths graph s B = [ b ] I I, where I = card S. Elements of the adacency matrx are defned as b, =, (, ) C (, ) C G c, (, J ) (2) A compatblty graph does not have to be a connected graph.

6 22 V. Batanovć, S. Gubernć, R. Petrovć / Choce of the Control Varables 2. CONTROL VARIABLE Introducton of a traffc control system on an ntersecton means to nstall sgnals that wll control traffc streams by dfferent lght ndcatons. The basc ntenton of traffc sgnals ntroducton s to prevent smultaneous movement of ncompatble traffc streams. The traffc control at an ntersecton comprses thus, gvng and cancelng the rghtof-way to partcular traffc streams. Gvng and cancelng the rght-of-way s performed by dfferent sgnal ndcatons. The ndcatons get the meanngs by conventon. Green ndcaton for vehcles means allowed passage, whle red means forbdden passage. Amber ndcaton appearng after green ndcaton, as well as after red/red-amber nforms drvers that the rght-of-way wll be changed. The duraton of amber and red-amber ntervals n some countres are determned by traffc regulatons and most frequently, t s specfed as 3s for amber and 2s for red-amber ndcaton. Sgnals that control pedestran streams usually have only two ndcatons: red ("stop") and green ("walk"). The most frequently used sequence of sgnal ndcatons for vehcles and pedestrans s presented n Fgure. However, n some countres there are other sequences, such as flashng amber before a steady amber ndcaton, or drect swtchng from red to green, etc. a) Sgnal sequence for vehcles b) Sgnal sequence for pedestrans Legend: red ndcaton green ndcaton amber ndcaton red-amber ndcaton Fgure : The sequences of sgnal ndcatons for vehcles and pedestrans The control of traffc lghts,.e. formng and mplementng of specfed sgnal sequences s performed by an electronc devce a traffc controller. The controller changes sgnal ndcatons by usng sequence of pulses. Changes of sgnal ndcatons are descrbed by a mathematcal varable, so-called control varable. Control varable can be assgned to every traffc stream. However, as compatble traffc streams can smultaneously gan and loose the rght of way, t s possble that a subset of traffc streams, comprsng several compatble streams, can be controlled by a sngle control varable []. Therefore, among the frst problems to be solved when ntroducng traffc lghts control at an ntersecton s the problem of establshng a correspondence between traffc streams and traffc sgnal sequences,.e. to determne the control varables whch control the traffc streams. The smplest way to assgn control varables to traffc streams s to

7 V. Batanovć, S. Gubernć, R. Petrovć / Choce of the Control Varables 23 assgn one control varable to one traffc stream. However, there are practcal reasons why ths assgnment s not always used. Techncal and economc consderatons cause a tendency to mnmze the number of control varables. Namely, the traffc controller should be smpler, wth a smaller number of modules that form control varables and thus, t would gve a cheaper soluton. Modern traffc controllers can mplement more complex control algorthms than those used before ther ntroducton. By ncreasng the number of control varables, the combnatoral nature of traffc control problems s emphaszed, whch gves way to mprove the performances of the control system. 3. SIGNAL GROUP Varous ntersecton performance ndces depend on the choce of the traffc control system for an ntersecton. Among these performance ndces are: total delay or total number of vehcle stops n a defned nterval, total flow through the ntersecton (for saturated ntersectons), capacty factor, lnear combnaton of delays and number of stops, etc. Values of these performance ndces depend on the assgnment of control varables to traffc streams. The best results are, obvously, obtaned f each traffc stream s controlled by one control varable. If the number of control varables s smaller than the number of traffc streams, certan constrants have to be ntroduced, expressng the requrement that several traffc streams smultaneously get and loose the rght-of-way. The consequence of ntroducng such constrants s the "corrupton" of optmum values of performance ndces, compared wth the case when each traffc stream s controlled by ts own control varable. Reducton n the number of control varables results n smplfcaton of traffc control problems, and also n a possblty to use cheaper and smpler traffc controllers. In real-tme traffc control systems, n whch data on current values of traffc stream parameters are used for determne values of control varables, a partcular attenton has to be pad on choosng the approprate set of control varables and assgnng them to traffc streams. Determnaton of the set of control varables s very complex due to all mentoned reasons. Ths problem, n fact, s the problem of parttonng the set of traffc streams S nto subsets of traffc streams so that control of each subset can be performed by a sngle control varable. A subset of traffc streams that can smultaneously gan and loose the rght-of-way,.e. whch can be controlled by a sngle control varable, s called a sgnal group. A sgnal group can also be defned as: A sgnal group s a set of traffc streams controlled by dentcal traffc sgnal ndcatons. Some authors defne a sgnal group as the set of sgnals on varous traffc lghts that always show the same ndcaton [4]. For traffc equpment manufacturers, a sgnal group s a controller module, whch always produces one sequence of traffc sgnal ndcatons. It s obvous that the traffc streams belongng to the same sgnal group have to be mutually compatble. However, ths condton s not suffcent. Namely, sgnals used for control of traffc streams of varous types - vehcle, pedestran, tram, etc., cannot always have the same ndcatons, whch s necessary f they are to belong to the same sgnal group. Vehcle streams are, for example, controlled by sgnal sequences wth four

8 24 V. Batanovć, S. Gubernć, R. Petrovć / Choce of the Control Varables ndcatons, whle for pedestran streams only two ndcatons are used. Therefore, sgnal groups are formed so to contan only the same types of traffc streams and the set of traffc streams S has to be parttoned n several subsets: the subset of vehcle traffc streams, the subset of pedestran traffc streams, etc. Accordng to the sgnal group defnton, for the ntersecton presented n Fgure 2 together wth ts compatblty graph, the sgnal groups are the followng subsets: D =,, }, D =, }, D = }, etc. { { 3 3 { 6 2 G c : Fgure 2: Intersecton and ts compatblty graph A sgnal group D p represents a subset of the set of traffc streams S and can be presented as follows: Dp =,,,,, } (3) { p p2 pe pe( p) where pe S, e E p, and E p s the set of traffc stream ndces n sgnal group D p,.e. E = {,2,, e,, E( p)}. p 3.. The Relaton of Identcal Sgnal Indcatons (Identty Relaton) In order to form sgnal groups, t s necessary to determne for each par of compatble traffc streams whether they can be controlled by traffc lghts whch always have dentcal ndcatons. The set of such traffc streams pars represents a relaton n the set of traffc streams S. Snce ths relaton determnes whether dentcal traffc lght ndcatons can be used for controllng traffc the streams pars, t s called the relaton of dentcal sgnal ndcatons, or the dentty relaton. The dentty relaton C s defned as: α C α = {(, ) traffc streams, can be controlled by a songle control vrable, J } (4)

9 V. Batanovć, S. Gubernć, R. Petrovć / Choce of the Control Varables 25 Relaton C α can be presented as: C α = C \ C 4,, J where C 4 = {(, ) ( (, ) C) ( S f, S l, f, l F, f l),, J } (5) 2 The subsets S, S,, S f,, S represent subsets of the same type (vehcles, pedestrans, trams, etc.) of traffc streams. Traffc streams of one type are controlled by the sgnals whch have the same sequences of ndcatons. For vehcle traffc streams, for example, ths sequence s: green, amber, red, red-amber. The set F s the ndex set of traffc stream types,.e. sgnal types: F F = {,2,, f, F} (6) The collecton S 2 f F = { S, S,, S,, S } (7) represents a partton of set S. Hence, we have: F f = f S = S (8) f l S S =, ( f F, l F, f l) (9) The relaton of dentcal traffc sgnal ndcatonsc α s: а) Reflexve,.e. (, ) C, ( J ) (2) α b) Symmetrc,.e. (, ) C (, ) C, (, J ) (2) α α The dentty relaton corresponds to an dentty graph: G = S, C ) = ( S, Γ ), (22) α ( α α where Γ α s Γ α : S P( S ). The dentty graph gven n Fgure 3 refers to the ntersecton wth 6 traffc streams presented n Fgure 2 together wth ts dentty graph. There are two traffc stream types.

10 26 V. Batanovć, S. Gubernć, R. Petrovć / Choce of the Control Varables Vehcle traffc streams belong to subset S = {, 2,..., 5} and pedestran traffc 2 stream belong to type two,.e. S = { 6 }. If traffc streams of varous types pass through an ntersecton (F>), the dentty graph G α s a non-connected graph. The number of connected components s equal to or greater than the number of stream types F. Graph G α s a non-orented graph wth a loop n each node. 2 G α : Fgure 3: The ntersecton wth 6 traffc streams and ts dentty graph Snce graphs = ( S, C) and G S, C ) have the same set of nodes, and G c α = ( α Cα C then, the dentty graph G α s a spannng subgraph of the compatblty graph G. c 3.2 The Complete Set of Sgnal Groups The dentty relaton C α gven by (4) defnes the set of traffc streams pars that can be controlled by dentcal sgnal ndcatons, whle the dentty graph G α enables determnaton of all subsets of set S that represent sgnal groups. A set of nodes of any subgraph of dentty graph G α, such that the subgraph s a complete graph, represents, n fact, a sgnal group. Snce a complete subgraph of a graph represents a clque, a sgnal group can be also defned n the followng way: A sgnal group s a clque (n Berges sense [5]) of the graph of dentcal sgnal ndcatons G α. Therefore, for traffc control at an ntersecton, t s necessary to determne a set of sgnal groups such that each element of set S belongs to one and only one sgnal group,.e. to a clque of graph G α. Such a set of sgnal groups s called the complete set of sgnal groups, and t represents a partton of set S. For one graph of dentcal sgnal ndcatons, there exst several complete sets of sgnal groups. Ths means that one ntersecton can be controlled n several ways, based on the choce of the complete set of sgnal groups. Introducng an approprate measure for comparson of complete sets of sgnal groups, the choce of the complete set can be formulated as an optmzaton problem: Fnd a complete set of sgnal groups such that the

11 V. Batanovć, S. Gubernć, R. Petrovć / Choce of the Control Varables 27 value of the chosen performance ndex s optmal. The set of feasble solutons for ths problem s the collecton of all complete sets of sgnal groups. The performance ndex that s often optmzed s the cost of the traffc controller. Havng n mnd that one control varable s assgned to each sgnal group and that each control varable s realzed by a separate module of control equpment, t s obvous that the equpment cost depends on the number of sgnal groups n a chosen complete set of sgnal groups. 4. PROBLEM STATEMENT In ths paper, we propose the method for solvng the followng problem: Fnd all complete sets of sgnal groups when the graph of dentcal ndcatons s gven. The soluton of ths problem ncludes the soluton of the followng problem mportant for the practce: Fnd a complete set of sgnal groups whch contans the mnmal number of sgnal groups. The problem of fndng a complete set of sgnal groups s the problem of parttonng a set S. Snce a sgnal group s a clque of graph G α, t s necessary to determne all clques of graph G α, and to choose a set of clques such that each element of the set S belongs to one and only one clque n the chosen set of clques of graph G α. The ntroducton of a graph G α, whch s the complement of a graph G α, enables a transformaton of the problem: Fnd all complete sets of sgnal groups f the graph of dentcal ndcatons s gven, as a problem of colorng graph G α. It s enabled by the fact that for each clque (sgnal group) n graph G α, there s a correspondent stable set n graph G α. The stable or the ndependent set s a set of vertces n a graph where no two vertces are adacent. It s obvous that vertces of each stable set,.e. the sgnal group, can be colored by one color. It means that complete set of sgnal groups can be obtaned by colorng vertces of each stable set of the graph G α by the same color. Then, the stable sets are called the color classes. An assgnment of the colors to color classes s a vertex colorng of a graph G α. More formally, a vertex colorng wth k colors means assgnng k colors to the vertces of the graph G α = ( S,C α ). It s defned by the next functon: c: S {,2,,k}, such that cu ( ) cv ( ) for each edge ( u, v ) G α. The Chromatc number χ(g α ) of the graph G α s the mnmal value of k such that there exsts a vertex colorng of the graph G α wth k colors [6]. The maxmal number of colors for the vertces colorng of graph G α s equal to the number of traffc streams I,.e. k max = S = I, and the mnmal value of k s equal to the chromatc number χ(g α ),.e. k mn = χ(g α ).

12 28 V. Batanovć, S. Gubernć, R. Petrovć / Choce of the Control Varables It means that all complete sets of sgnal groups can be determned by the vertex colorng of the graph G α, wth k colors for all nteger values of k between k mn and k max. e. for χ k. ( ) I G a 5. METHOD FOR PROBLEM SOLUTION Method for fndng all complete sets of sgnal groups conssts of the followng steps:. Determnaton of all color classes of graph G α ; 2. Fndng the chromatc number of graph G α ; 3. Colorng of graph G α for all ntegers values of k belongng to the nterval [χ(g α, I]. All color classes can be found by usng the program CLIQUE [2]. It means that the result of the applcaton of the program CLIQUE s a collecton of all color classes: Q = {Q, Q 2,,Q l,,q L }. It means that l s the element of the ndex set L = {,2, l,,l}. One partton of set S s obtaned by k colorng of graph G α. But, colorng of graph G α by usng k colors s not a unque process. It means that there can be more possbltes for colorng the graph G α by k colors. The color classes used for colorng one partton of the set S are elements of the collecton Q. The choce of the color classes can be exactly descrbed by the ntroducton of the selecton vector x: x=[x, x 2,, x l,,x L ], x l {,}. The assgnment of the separate values to the varable x l has the next meanng: x l =,, f the color class Q s ncluded n the chosen partton of the set S, f t s not l Snce k colorng s a partton of the set S, every element of the set S has to be present n only one color class ncluded n that partton. The problem of determnng the chromatc number and the correspondng partton sets of set S, contanng the mnmal number of elements, has now the followng exact formulaton: Fnd all selecton vectors that enable the achevement of the mnmal value of the functon: P = a T x = L l= x subect to constrants Hx = b, l, where a T = [a, a 2,, a l,, a L ] =[,,,,,],

13 V. Batanovć, S. Gubernć, R. Petrovć / Choce of the Control Varables 29 {,} x l, l {,2,..., L} [ h l ] I L H h l =, where =,, f traffc stream s an element of the color class Q f t s not [ b,b,...,b ] T [ ] T b = =. 2 I,,..., The program COMP [2] can be used, f necessary, besdes the partton sets wth mnmal cardnalty, to fnd all complete sets of sgnal groups. The use of the proposed method s presented by the next example. 6. EXAMPLE For the ntersecton and ts dentty graph and the complement of that graph, presented n Fgure 4, fnd the chromatc number χ(g α ) of graph G α,.e. of the complement of the dentty graph G α, and fnd also a vertex colorng of graph G α wth the cardnalty χ(g α ). l 2 G α : G α Fgure 4: The ntersecton wth 6 traffc streams ts dentty graph and complement graph of dentty graph All color classes of graph G α are obtaned by usng the program CLIQUE. Q = {Q, Q 2,,Q l,,q 2 } = {{ },{ 2 },{ 3 },{ 4 },{ 5 },{ 6 },{, 2 },{, 3 }, {, 5 },{{ 2, 5 },{ 4, 5 },{{, 2, 5 }}.

14 3 V. Batanovć, S. Gubernć, R. Petrovć / Choce of the Control Varables [ a,a,...,a ] T [ ] T a = =. 2 2,,..., H = [ b,b,...,b ] T [ ] T b = =. 2 6,,..., The formulated problem s solved by colorng graph G α, usng the program MINA []. Chromatc number of graph G α s P ( G ) 4 = χ. mn a = The optmal collectons of the color classes (complete sets of sgnal groups contanng mnmal number of elements) are: * D = { 3}{, 6}{,, 2}{, 4, 5 } * D 2 = { 4}{, 6}{,, 3}{, 2, 5 } * D 3 = { 2}{, 6}{,, 3}{, 4, 5 } * D 4 = { 3}{, 4}{, 6}{,, 2, 5 } * The complete set of sgnal groups = { }, { }, {, }, { } presented n Fgure 5. D s , 5

15 V. Batanovć, S. Gubernć, R. Petrovć / Choce of the Control Varables 3 G α : Fgure 5: The complement graph of the dentty graph colored by the mnmal number of colors 7. CONCLUSION The exact defnton of the sgnal group, based on the relatons and the correspondng graphs, defned over the set of traffc streams s ntroduced. The graph of dentcal ndcatons and ts complement graph G α = ( S,C α ) are used to show that k-colorng of the complement of the graph of dentcal ndcatons can be used for fndng the complete sets of sgnal groups on an sgnalzed ntersecton. The mnmal number of sgnal groups n one complete set of sgnal groups s equal to the chromatc number of that graph. The color classes are equvalent to sgnal groups. The program CLIQUE [2] s used to fnd all color classes,.e. all subsets of the set S colored by the same color. The program COMP [2] s developed to fnd all complete sets of sgnal groups. The problem of fndng all complete sets of sgnal groups wth the mnmal cardnalty, whch s equal to the chromatc number χ(g α ), s formulated as a lnear programmng problem where the values of varables belong to set {,}. The program MINA [] s developed to solve ths problem. REFERENCES [] Gubernć, S., Šenborn, G., Lazć, B., Optmal Traffc Control: Urban Intersecton, CRC Press, Boca Raton, 27. [2] Gubernć S., Mtrovć Mnć S., Sgnal Group: Defntons and Algorthms, Yugoslav Journal of Operatonal Research, 3 (993) [3] Forchhammer N., Poulsen L., Traffc Streams n Relaton to Traffc Lghts the Operatve Conflct Matrx, L.M. Ercsson Techncal Note, Unpublshed, 968. [4] Pavel, G., Planen von Sgnalanlagen für den Strassenverkehr, Krschbaum Verlag, Bonn-Bad Godesberg, 974. [5] Berge, C., Théore des graphes et ses applcaton, Dunod, Pars, 958. [6] Cvetkovć D., Kovačevć-Vučć V., Kombnatorna optmzaca, DOPIS, Beograd, 996.