Trac Mo deling and Variational Inequalities using GAMS y z Steven P Dirkse Michael C Ferris April Abstract We describ e how several trac assignment and design problems can b e formulated within the GAMS mo deling language using newly developed mo deling and interface to ols The fundamental problem is user equilibrium where multiple drivers comp ete nonco op eratively for the resources of the trac network A description of how these mo dels can b e written as complementarity problems variational inequalities mathematical programs with equilibrium constraints or sto chastic lin ear programs is given At least one general purp ose solution technique for each mo del format is briey outlined Some observations relating to particular mo del solutions are drawn Introduction Mo dels that p ostulate ways to assign trac within a transp ortation network for a given demand have b een used in planning and analysis for many years see and references therein A p opular technique for such assignment is to use the shortest path b etween the origin and destination p oints of a given journey Of course such a path dep ends not only on the physical distance b etween these two p oints but also on the mo de of transp ort and the congestion exp erienced during the trip This material is based on research supp orted by National Science Foundation Grant CCR y GAMS Development Corp oration Potomac Street NW Washington DC stevegamscom z Computer Sciences Department University of Wisconsin Madison West Day ton Street Madison Wisconsin ferriscswiscedu To account for congestion trac assignment mo dels use the notion of user equilibrium or Wardropian equilibrium In this context the travel time used to dene the distance b etween the origin and destination is a function of the length and capacity of each arc of the path and the total ow on that path Thus the fact that many users can travel along an arc will aect the time it takes any particular user to traverse the arc User equilib rium o ccurs when all users travel along their shortest path where distance is measured using the ab ove denition for time Underlying the mo del is the notion of nonco op erative b ehavior everyone is out for themselves This should b e contrasted with the notion of system equilibrium when a trac controller assigns every vehicle to a particular path to minimize the total distance traveled Note that these kinds of o dels are typically used to predict the steadystate volume of trac on a network not to lo ok at the dynamic b ehavior Due to the absence of an overall ob jective in the user equilibrium this problem is more easily cast in the context of a variational inequality In this pap er we describ e how to formulate solve and extend mo dels for trac assignment using the notion of a mixed complementarity problem a sp e cialization of the variational inequality Many pap ers have discussed the formulation and solution of user equilibriu m problems using complementar ity and variational inequality mo dels as well as the applications of these problems to urban planning see The rst two sections of the pap er show how the user equilibrium prob lem is cast as a mixed complementarity problem MCP formulated within the GAMS mo deling language and solved using the PATH algorithm Some examples of problems formulated using these to ols are describ ed in There are many cases when a mo deler wishes to optimize an ob jec tive function sub ject to the system b eing in equilibrium These problems are commonly called Mathematical Programs with Equilibrium Constraints MPECs The recent monograph describ es the current state of opti mality theory and algorithms related to such problems In Section we describ e the basic structure of an MPEC and give some examples of trac design problems that can b e formulated as such Section describ es extensions to the GAMS mo deling language that allow MPECs to b e formulated within the language Furthermore an out line of new to ols for large scale implementation is given These to ols allow algorithm developers direct access to relevant function and derivative val ues via subroutine library calls It is intended that this suite of routines MPECLIB will foster the development of new applications and test prob lems in the MPEC format In the ensuing section we show how these to ols are used in an analysis of a tolling problem over a network representing SiouxFalls The problem is cast as an MPEC and two algorithms based on an implicit programming approach namely DFO and the bundletrust region algorithm are used to demonstrate the ability of these to ols and the p ower of the mo deling format The nal section of the pap er treats some mo deling issues related to trac assignment and path choice in networks sub ject to failure Here again we show how recently developed mo deling to ols are eective for investigating complex issues in transp ortation design and analysis MCP mo dels user equilibri um The mixed complementarity problem MCP is dened in terms of some n n lower and upp er b ounds R and u R satisfying u n and a nonlinear function F B R Here B represents the b ox B n n u fz R z u g The variables z R solve MCP F u if i i i n n for some variables w R and v R w v z u i n i i i i i F z w v i n i i i and w z and v u z i n i i i i i i Note that any solution z of MCP trivially satises the following implications z z u v F z i i i i i i z u z w F z i i i i i i and z u w v F z i i i i i i Several interesting sp ecial cases of MCP exist for particular choices of and u When and u it follows from that w v and n is satised for any z R Hence the problem b ecomes the classical square system of nonlinear equations F z Many techniques used to solve MCP are inspired by techniques used for such systems Another sp ecial case is when and u whereup on it can b e easily seen that the problem b ecomes z F z Here we have introduced the notation that signies the two adjacent quantities are orthogonal that is in addition to the explicit inequalities z and F z we have T z F z In eect this enables us to rewrite and more succinctly as z w u z v Problem is commonly called the nonlinear complementarity problem NCP and has b een the sub ject of much research over the past three decades A plethora of applications can b e found in this pap er is concerned with applications arising from trac and transp ortation man agement The general variational inequality VIF C is dened using an arbitrary n convex set C R as the following system of inequalities z C hF z y z i y C It can b e reformulated as an MCP using a transformation involving multi pliers We consider two cases separately If the feasible set C is a b ox then it is elementary to show that and the MCP dened by F and C are completely equivalent as their solution sets are identical When C is p olyhedral rather than rectangular can b e reduced to an MCP by explicitly including the dual variables to the constraints dening C Thus mn given a b ox B and a set X fz Az bg where A R it can b e T m shown that with C B X is equivalent to VIH B R where T F z A u H z u Az b When equality constraints are used to dene X the asso ciated dual variables u are free Two advantages to using the MCP formulation as opp osed to the NCP are the explicit treatment of simple b ounds on the variables z and the availability of free variables which enable the explicit representation of equality constraints This is more ecient than introducing extra variables and equations to deal with b ounds and equality constraints We now pro ceed to show how to mo del the user equilibrium problem as an MCP In all mo dels used for the analysis of trac congestion there is a transp ortation network given by a set of no des N and a set of arcs A In the equilibrium setting it is usually assumed that drivers comp ete nonco op eratively for the resources of the network in an attempt to minimize their costs where the cost of traveling along a given arc a A is a nonlinear function c f of the total ow vector f with comp onents f b A Let a b cf denote the vector with comp onents c f a A There are two subsets a of N that represent the set of origin no des O and destination no des D resp ectively The set of origindestination OD pairs is a given subset W of O D asso ciated with each such pair is a travel demand that represents the required ow from the origin no de to the destination no de There are at least two equilibriu m techniques used for generating mo dels of trac congestion on such a network The rst mo del is based on consid ering all the paths b etween the origindestination pairs and the second uses a multicommodity formulation representing each origin or destination no de as a dierent commo dity Both of these formulations use the Wardropian characterizations of equilibria a sp ecial case of a Nash equilibriu m see A path based formulation The given path based formulation follows For each w W let P w represent the set of paths connecting the OD pair w and P represent the set of all paths joining all OD pairs of the network Let denote the ow p on path p P let b e the cost of ow on this path which is a function p of the path ow vector Let b e the arcpath incidence matrix with entries if path p P traverses arc a A ap otherwise It is clear that f and are related by f When the path cost on each path p is assumed to b e the sum of the p arc costs on