Solving the toll design problem with multiple user groups

Solving the toll design problem with multiple user groupsMei Chena,*,David H.Bernstein b,1aDepartment of Civil Engineering,University of Kentucky,Lexington,KY 40506-0281,USA b Department of Computer Science,James Madison University,Harrisonburg,VA 22807,USAReceived 19March 2002;received in revised form 31October 2002;accepted 8November 2002AbstractCongestion pricing has been regarded as an efficient method to reduce network-wide travel cost.In this study,a methodology for toll design is developed to provide policy-makers with suggestions on both where to charge tolls and how much the tolls should be.As opposed to the traditional approach of marginal social cost pricing,this methodology is capable of dealing with the more realistic case,in which only a small number of links can be tolled.Furthermore,this methodology can accommodate multiple user groups.Specifically,we make several simplifying assumptions which enable us to convert a traditional bilevel formulation of the toll design problem into a single level,standard nonlinear optimization problem.Ó2003Elsevier Ltd.All rights reserved.Keywords:Toll design problem;Route choice;Congestion1.IntroductionCongestion pricing has been widely considered as an effective method to alleviate urban traffic congestion,through changing road users Õroute choice (see Beckmann et al.,1956;Dafermos and Sparrow,1971;Smith,1979;Ferrari,1995;Gabriel and Bernstein,1997;Bergendorffet al.,1997;Hearn and Yildirim,2001;Labb e et al.,1998;Yang and Lam,1996;Bellei et al.,2002;Verhoef,2002)and departure time (see Vickrey,1967;Arnott et al.,1990;Bernstein,1993;Chen and Bernstein,1995;Yang and Meng,1998).In addition,the issue of finding the appropriate toll on the network has been discussed in many studies recently,such as Ferrari (1995),Bergendorffet al.(1997),Dial (1999a,b),Wie and Tobin (1998),Yang and Lam (1996),Labb e et al.(1998),Yang and Meng (1998),Yang (1999),and Verhoef (2002).However,many of these are theoretically*Corresponding author.Tel.:+1-859-257-9262;fax:+1-859-257-4404.E-mail addresses:mchen@ (M.Chen),bernstdh@ (D.H.Bernstein).1Tel.:+1-540-568-1671;fax:+1-540-568-2745.Transportation Research Part B 38(2004)61–79/locate/trb0191-2615/$-see front matter Ó2003Elsevier Ltd.All rights reserved.doi:10.1016/S0191-2615(03)00002-X62M.Chen,D.H.Bernstein/Transportation Research Part B38(2004)61–79optimal tolls in that they commonly assume that it is possible to place tolls on any link of the network.In order to describe the reality where policymakers are generally restricted to place tolls on only a very limited roadway links such as bridges and tunnels,Chen et al.(1999)presented a simplified formulation of the toll design problem with a predetermined set of links that can be bb e et al.(1998)looked at a similar situation while assuming that congestion is not an influential factor when making route choice decisions.Yang and Lam(1996)obtained the de-rivatives of equilibriumflows with respect to tolls and used them in the heuristic in solving the road pricing problem when placing tolls on certain links of the network.Verhoef(2002)suggested a heuristic algorithm forfinding the second-best optimal tolls based on linear cost functions. Most previous studies assume that all road users have identical characteristics.We relax that assumption.We begin by formulating the toll design problem as a bilevel program.We then make certain simplifying assumption and convert it into a single level problem.This simplified problem is a standard nonlinear programming problem,which can be solved relatively easily,compared to its original version.In addition,the single level toll design problem can be further simplified by using a linear cost function instead of the conventional nonlinear cost function.In many transportation engineering and planning practices particularly road pricing,it is necessary to consider characteristics of various user groups.For example,trucks are usually charged a higher toll than cars because their impact on travel time of other vehicles in the traffic is more significant.In terms of route choice,it is common belief that travelers respond differently to congestion tolls.Specifically,low-income travelers tend to switch to toll-free alternatives or change their travel schedule(if possible)to avoid the tolls,while high-income travelers may be more willing to pay for a less congested trip.The effect of congestion tolls on route choice be-havior can be evaluated through grouping road users according to their income levels.The methodology is crucial to the evaluation of the impact of congestion tolls on various income groups,and to the design of a convincing revenue distribution package that could ease the ar-guments on the social equity consequences of the congestion pricing plan.The toll design problem with multiple user groups is defined as:given the characteristics of a network,the set of links that can be tolled,the travel demand for each user group between each O–D pair,and the link performance functions,find the optimal link toll pattern of the network that would induce the lowest total travel cost(exclusive of tolls).Similar to that with single user group discussed by Chen et al.(1999),the toll design problem with multiple user groups has equilibrium constraints.In this case,a user chooses his/her route based on the route choices of other users in the same group as well as users of different groups.The equilibrium with multiple user groups can be defined as follows:at equilibrium,no user from any group will be able to further decrease his/her travel cost(inclusive of tolls)by unilaterally changing routes.It is generally assumed in the toll design problem that travelers have multiple route choices available to them and they have perfect information on travel times and costs on all alternative routes.Particularly,road users always choose the routes that minimize their individual cost of traveling(inclusive of tolls).2.Bilevel toll design problem with multiple user groupsThe toll design problem with multiple user groups can be formulated as a bilevel programming problem,similar to the toll design model for single user group presented by Chen et al.(1999).Specifically,the toll design problem is formulated as a mathematical program with another op-timization problem embedded in it as a constraint.Consider a transportation network comprised of a finite set of links,A (with cardinality j A j )and nodes,N (with cardinality j N j ).Let W (with cardinality j W j )denote the set of O–D pairs,P w denote the set of routes that connecting O–D pair w 2W .Then P ¼S w 2W P w denotes the complete set of routes in the network.Let U denote the set of user groups on the network.Then D uw denote the travel demand of type u users between O–D pair w 2W .The relationship between routes and links is described by an indicator function,d ap .That is,d ap ¼1if link a is in route p and d ap ¼0otherwise.Further,let f denote a vector of link flows,thus f ¼ðf ua :u 2U ;a 2A Þ,in which f ua denotes the flow of type u user on link A .Let F denote a vector of route flows,thus F ¼ðF up :u 2U ;p 2P Þ,in which F up represents the flow of type u user on route p ,and let F u ¼ðF up :p 2P Þdenote the vector of route flows of users from group u 2U .Hence the set of feasible route flow patterns,X ,can be given byX ¼F 2R þ:X p 2P wF up (¼D uw ;u 2U ;w 2W )ð1Þand the set of feasible link flow patterns,x ,can be given byx ¼f f 2R þ:f ¼D T F ;F 2X g ð2ÞAssociated with each link and route is a travel time function and a travel cost function for each user group.The vector of link travel times is denoted by t ¼ðt ua :u 2U ;a 2A Þ,the vector of link travel costs (exclusive of tolls )is denoted by c ¼ðc ua :u 2U ;a 2A Þ,and the vector of link tolls is denoted by s ¼ðs ua :u 2U ;a 2A Þ.Similarly,the vector of route travel times is denoted by T ¼ðT up :u 2U ;p 2P Þ,the vector of route travel costs (exclusive of tolls )is denoted by C ¼ðC up :u 2U ;p 2P Þ,the vector of route travel cost (exclusive of tolls )of users from group u 2U is denoted by C u ¼ðC up :p 2P Þ,the vector of route tolls is denoted by W ¼ðW up :u 2U ;p 2P Þ,and the vector of route tolls on users from group u 2U is denoted byW u ¼ðW up :p 2P Þ.It is assumed that all route travel times and costs are additive,i.e.,T up ¼P a 2p t ua ,C up ¼P a 2p c ua ,and W up ¼P a 2p s ua for all u 2U .Then the set of equilibrium route flows is given byE s ¼ÈF Ã2X :½C u ðF ÃÞþW u T ðF u ÀF Ãu ÞP 0;F 2XÉð3Þwhich is simply the set of solutions to the variational inequality formulation of the equilibrium problem.So,the set of the equilibrium link flows can be given bye s ¼f f Ã2x :f üD T F ;F 2E s gð4ÞHence,a route-based bilevel toll design problem can be formulated as follows:min s ;F X u 2U X w 2W ;p 2P w C up ðF ÞT ÁF up !ð5a ÞM.Chen,D.H.Bernstein /Transportation Research Part B 38(2004)61–7963subject toF2E sð5bÞs ua P0u2U;a2Ið5cÞs ua¼0u2U;a2Ið5dÞwhere I A denotes the set of links that can be tolled.In this problem,there are basically two classes of decision-makers;the transportation planner chooses the toll that minimizes the total travel cost(exclusive of tolls),while each individual driver chooses the route that minimizes his or her travel cost(inclusive of tolls),given the behavior of other drivers.The upper level of this bilevel toll design problem can be viewed as the‘‘trans-portation plannerÕs problem’’,in which the transportation planner tries tofind a toll pattern that minimizes the total travel cost.It has a nonlinear objective function,which is the total travel cost. The lower level of the toll design problem can be viewed as the‘‘road usersÕproblem’’,in which each road user tries to minimize his or her travel cost under the influence of congestion tolls. Solving this toll design problem with multiple user groups is very difficult because of the nondifferentiability and nonconvexity of the equilibrium constraint.Moreover,the solution procedure of the lower level problem is further complicated by the asymmetric interactions among multiple user groups.In the following section,this bilevel problem will be converted into a single level problem,which is much easier to solve.3.Single level toll design problem with multiple user groupsAssume that,based on the pre-toll equilibrium,a set of‘‘reasonable’’routes that will be used after the tolls are put in place can be found.Then,the equilibrium constraint in the toll design problem can be dramatically simplified,as discussed by Chen et al.(1999).Let R w denote the set of reasonable routes connecting O–D pair w2W.Then similarly,for the case of multiple user groups,a routeflow pattern,F,is now an equilibrium if and only ifC upðFÞþW up¼C uqðFÞþW uq u2U;w2W;p;q2R wð6ÞThat is,F is an equilibrium if and only if the costs on all routes connecting a particular O–D pair are equal.Hence,the toll design problem can be written as follows:min s;F Xu2UXw2W;p2R wC upðFÞTÁF up!ð7aÞsubject toC upðFÞþW up¼C uqðFÞþW uq u2U;w2W;p;q2R wð7bÞX p2R w F up¼D uw u2U;w2Wð7cÞ64M.Chen,D.H.Bernstein/Transportation Research Part B38(2004)61–79F P0ð7dÞs ua P0u2U;a2Ið7eÞs ua¼0u2U;a2Ið7fÞNote that this is a single level problem.In fact,when C is linear,this is a convex quadratic minimization problem with linear equality and nonnegativity constraints.When the BPR function is used to represent the link travel cost,this problem is a nonlinear optimization problem with nonlinear and linear equality constraints,as well as nonnegativity constraints.Particularly,both the objective function and the constraints are differentiable.In such cases,this problem can be solved using a variety of algorithms for solving standard nonlinear programming problems. However,constraint(7b)is likely nonconvex because it is the difference between the two convex functions,C upðFÞþW up and C uqðFÞþW uq(e.g.,when the BPR function is used to represent the link travel cost).Therefore,the single level toll design problem is usually nonconvex,which cannot guarantee a global optimal solution.Nevertheless,if we modify the formulation(7b)somewhat,as shown in(8),the solution of this problem can be facilitated.Specifically,let H uw denotes the vector of total route cost(inclusive of tolls),i.e.,H uw¼C upðFÞþW up u2U;w2W;p2R wð8Þthen the toll design problem(7)can be written asmin s;F;H Xu2UXw2W;p2R wðH uwÀW upÞTÁF up!ð9aÞsubject toC upðFÞþW up¼H uw u2U;w2W;p2R wð9bÞXp2R wF up¼D uw u2U;w2Wð9cÞF P0ð9dÞs ua P0u2U;a2Ið9eÞs ua¼0u2U;a2Ið9fÞH P0ð9gÞNote that H uw with cardinality j UÂW j is added into the problem as a decision variable,which is independent of all other variables.Since R w is defined as the set of used routes between O–D pair w2W,H uw turns out to be the equilibrium route cost for each O–D pair,which is the minimum possible route cost under the equilibriumflow.The objective function(9a)remains convex since H and W are independent of each other. Constraint(9b)is still a nonlinear equality.However,if we relax this constraint toC upðFÞþW up6H uw u2U;w2W;p2R wð10ÞM.Chen,D.H.Bernstein/Transportation Research Part B38(2004)61–7965then the toll design problem will look like the following:min s;F;H Xu2UXw2W;p2R wðH uwÀW upÞTÁF up!ð11aÞsubject toC upðFÞþW up6H uw u2U;w2W;p2R wð11bÞXp2R wF up¼D uw u2U;w2Wð11cÞF up P0u2U;w2W;p2R wð11dÞs ua P0u2U;a2Ið11eÞs ua¼0u2U;a2Ið11fÞH P0ð11gÞEven though they are both standard nonlinear programming problems,(11)may be easier to solve than(9),since most algorithms for nonlinear programming problems require a feasible solution to start with,and it is rather easy tofind a feasible point that satisfies inequality con-straint(11b)instead of equality constraint(9b).Furthermore,many algorithms require that this starting point lies strictly inside rather than on the boundary of the feasible region.Here,we prove that problem(11)and problem(9)have the same solution set.Note that besides the aforemen-tioned two constraints,all other constraints are the same for the two problems.Therefore,it is just necessary to compare constraints(11b)and(9b).Theorem.Problem(11)and problem(9)have the same solution set.Proof.Any solutionðFÃ;sÃ;HÃÞof problem(9)that satisfies constraint(9b)also satisfies con-straint(11b)since(9b)is a special case of(11b).Thus,any solution to problem(9)must be a solution of(11).On the other hand,if there exists an optimal solutionðFÃ;sÃ;HÃÞto the problem(11),which only satisfiesC upðFÃÞþWÃup <HÃuwu2U;w2W;p2R wð12Þand at the same time satisfies all other constraints of(11),thenHÃuw ÀWÃup>C upðFÃÞu2U;w2W;p2R wð13Þand thenX u2UXw2W;p2R wðHÃuwÀWÃupÞÁFÃup!>Xu2UXw2W;p2R wC upðFÃÞÁFÃup!ð14ÞThe right-hand side of(14)is equivalent to the original objective of the toll design problem(7). Hence,(14)shows that the solutionðFÃ;sÃ;HÃÞresults in a larger objective value than the original minimization formulation(7).Thus the objective function can be further minimized,which 66M.Chen,D.H.Bernstein/Transportation Research Part B38(2004)61–79contradicts the assumption that ðF Ã;s Ã;H ÃÞis an optimal solution to the problem (11).Therefore,such a solution does not exist.In other words,all optimal solutions of problem (11)have to satisfy (9b),which means that they are also solutions to the problem (9).Hence we can conclude that problems (11)and (9)have the same solution set,and we can solve(11)instead of (9).ÃSince (9)is equivalent to (7)and (11),it follows that (11)and (7)are also equivalent.However,the numbers of decision variables in these two problems are different.For (7)the total number of decision variables is j U ÂS w 2W R w j þj U ÂA j ,while for (11)this number is j U ÂS w 2W R w j þj U ÂA j þj U ÂW j because of the addition of decision variable H .4.Equivalence of the two toll design problemsFirst,look at the objective functions in problems (5)and (7).Since R w is the set of used routes,then,F up ¼0;u 2U ;w 2W ;p 2P w ÀR w .Therefore,Eq.(5a)is equivalent to (7a),and it is suf-ficient to compare the feasible region of problems (5)and (7)only.The feasible region of (5)consists of the solutions to the variational inequality problem (3),which can be alternatively formulated as a complementarity problem as follows:F up ½C up ðF ÞþW up ÀH uw ¼0u 2U ;w 2W ;p 2P wð15a ÞC up ðF ÞþW up ÀH uw P 0u 2U ;w 2W ;p 2P w ð15b ÞX p 2P w F up ¼D uw u 2U ;w 2Wð15c ÞIn addition,all nonnegativity constraints should be satisfied.Next,we will use this to prove the equivalency between the two toll design problems.Lemma 1.For any route flow vector F ¼ðF up :u 2U ;w 2W ;p 2P w Þthat belongs to the feasible region of (5),it belongs to the feasible region of (7).Proof.Let R w denote the set of used routes connecting O–D pair w 2W .Then for any F ¼ðF up :u 2U ;w 2W ;p 2P w Þin the feasible region of (5),the complementarity condition (15)should be satisfied.According to the definition of the deterministic equilibrium,the complementarity con-dition (15)can be expanded as system (16)as follows:C up ðF ÞþW up ÀH uw ¼0u 2U ;w 2W ;p 2R w ð16a ÞC up ðF ÞþW up ÀH uw >0u 2U ;w 2W ;p 2P w ÀR w ð16b ÞF up >0u 2U ;w 2W ;p 2R w ð16c ÞF up ¼0u 2U ;w 2W ;p 2P w ÀR w ð16d ÞX p 2P wF up ¼D uw u 2U ;w 2Wð16e ÞEq.(16a)is equivalent to (7b)if variable H is disregarded,and Eqs.(16d)and (16e)imply (7c).Eq.(16c)is contained in (7d).Since (7b)–(7d)constitute the feasible region for flows of problem M.Chen,D.H.Bernstein /Transportation Research Part B 38(2004)61–796768M.Chen,D.H.Bernstein/Transportation Research Part B38(2004)61–79(7),we can conclude that for any F that belongs to the feasible region of(5),it also belongs to the feasible region of(7).ÃLemma2.For any route flow vector F¼ðF up:u2U;w2W;p2P wÞthat belongs to the feasible region of(7),it belongs to the feasible region of(5).Proof.For any F¼ðF up:u2U;w2W;p2P wÞin the feasible region of(7),the following equations hold:F up P0u2U;w2W;p2R wð17ÞF up¼0u2U;w2W;p2P wÀR wð18ÞC upðFÞþW up¼C uqðFÞþW uq u2U;w2W;p;q2R wð19ÞXF up¼D uw u2U;w2Wð20Þp2R wLetC upðFÞþW up¼H uw u2U;w2W;p2R wð21ÞThen Eq.(16a)as well as the following hold:F up½C upðFÞþW upÀH uw ¼0u2U;w2W;p2R wð22ÞEq.(18)also guaranteesF up½C upðFÞþW upÀH uw ¼0u2U;w2W;p2P wÀR wð23ÞNow we compare the C upðFÞþW up and H uw when w2W;p2P wÀR w.IfC upðFÞþW up<H uw u2U;w2W;p2P wÀR wð24Þthen route p would have a lower cost than those used routes.Based on the equilibrium principles, the following holds:F p>0u2U;w2W;p2P wÀR wð25Þwhich is contradictory to Eq.(18).Thus,(24)is false and Eq.(26)holds:C upðFÞþW up P H uw u2U;w2W;p2P wÀR wð26ÞEqs.(16a)and(26)can be combined into(15b),Eqs.(22)and(23)can be combined into(15a), and Eqs.(18)and(20)imply(15c).At the same time,all nonnegativity constraints are satisfied. Since equation system(15)is the complementary formulation of the equilibrium constraint of(5), any F which is from the feasible region of(7)should be also in the feasible region of(5).ÃBased on Lemmas1and2,we can conclude that the single level toll design problem(7)is equivalent to the bilevel toll design problem(5).5.AlgorithmsMany algorithms have been developed for solving constrained nonlinear optimization prob-lems.In general,these algorithms can be categorized into types such as feasible direction methods and interior point algorithms.There have been various software packages developed using these algorithms,which are capable of solving the constrained nonlinear programming problems.Such products include MINOS,a projected Lagrangian method combined with a reduced-space ap-proach,LANCELOT,a trust region method performed on an augmented Lagrangian,SNOPT,a sequential quadratic programming method,and LOQO,an interior point method for nonconvex nonlinear programs.Among these packages,MINOS,LANCELOT,and SNOPT use algorithms in the category of feasible direction method,and LOQO uses an interior point method.In this study,we found that LOQO has the best performance in providing solutions.Observing that the nonconvexity of the toll design problem arises from its nonlinear con-straints,we will make an attempt to linearize these nonlinear functions using the iterative linea-rization method introduced by Rosen (1966).By solving a series of linearly constrained optimization problems,such effort could greatly reduce the complexity of the solution procedure.We begin by introducing the procedures necessary to solve the toll design problem.5.1.Procedures for solving toll design problemsAs discussed previously,the single level deterministic toll design problem (11)is much easier to solve than the bilevel problem (5),however,in order to convert the bilevel problem into the single level one,the set of routes that will be used after tolls are in place has to be identified.It is common knowledge that at equilibrium,there may exist multiple route flow patterns that would generate the same link flow pattern on the network.For example,consider the network shown in Fig.1,in which there are four links and four routes.Route 1consists of links 1and 3,route 2consists of links 1and 4,route 3consists of links 2and 3,and route 4consists of links 2and 4.Let F i ,i ¼1;2;3;4,denote the flow on route i ,and f j ,j ¼1;2;3;4,denote the flow on link j .Thenf 1¼F 1þF 2f 2¼F 3þF 4f 3¼F 1þF 3f 4¼F 2þF4M.Chen,D.H.Bernstein /Transportation Research Part B 38(2004)61–7969If the equilibrium linkflow pattern is f1¼f2¼f3¼f4¼2,then one possible routeflow pat-tern is F1¼F2¼F3¼F4¼2,another possible routeflow pattern is F1¼2,F2¼F3¼0,and F4¼2.So,the set of used routes is not unique.In order to identify this set,we use the method introduced by Larsson et al.(1998),which can uniquelyfind the most likely routeflows given the equilibrium linkflows.In this method,the equilibrium linkflow solution is defined as a macrostate,which arises as the result of the route choices of the individual travelers.The travelersÕroute choices between minimum and equal cost routes of a particular O–D pair define a set of microstates.All microstates consistent with the macrostate are argued to be equally probable,since the travelers are indifferent to which route they use,as long as it is the minimum cost route(Smith,1978,1983).Based on well-known principles from information theory,it is argued that the most likely routeflow solution is the one which corresponds to the largest number of microstates among those which are consistent with the observed macrostate.Even though this method is designed for the single user group case,it also applies to the sit-uation with multiple user groups.Let RÃw denote the set of equilibrium routes between O–D pairw2W,that is,RÃw¼f p2P w:C up¼H uw;u2U gð27ÞThen,after applying StirlingÕs approximation,the most likely routeflows can be found by solving the maximum entropy problem(28):maxF ÀXu2UXw2W;p2RÃwF up ln F up!ð28aÞsubject toX p2RÃw F up¼D uw u2U;w2Wð28bÞXw2W;p2RÃwd ap F up¼f ua u2U;a2Að28cÞF up P0u2U;w2W;p2RÃwð28dÞThis is a standard nonlinear programming problem with linear constraints.A variety of al-gorithms could be applied to this problem.However,it should be mentioned that because of theentropy formulation in the objective function,F up>0holds for all u2U,w2W,and p2RÃw .Infact,it is rather difficult to get the exact solution to this entropy problem,since the objective function becomes extremelyflat when approaching the optimum.However,for the purpose of identifying the most likely routes that carryflow,a near-optimum solution is good enough.In this near-optimum solution,although none of the routeflows would be zero exactly,some of them could be very close to zero.Theoretically,these routes are used,while from an engineering point of view,these routes certainly should not be included in the set of used routes.A tolerance e>0 will be specified to exclude these routes,i.e.,the set of‘‘reasonable’’routes between O–D pair w2W can be defined asR w¼f p2RÃw :F up P e;u2U gð29aÞ70M.Chen,D.H.Bernstein/Transportation Research Part B38(2004)61–79and then[R¼R wð29bÞw2WIt is possible that some routes that were not in the‘‘no-toll’’route set will be attractive after the tolls are in place(i.e.,if the tolls are placed on links that are not in that route).Such routes should, of course,be included.While it is difficult to identify such routes a priori,one can discover them ex post.Hence,one could apply an iterative procedure in which the initial route set is based on the ‘‘no-toll’’equilibrium and,after the‘‘with-toll’’equilibrium is found for this route set,new routes could be generated by solving a sequence of shortest route problems.Then the toll design problem could be re-solved.As it turns out,this issue did not arise in the examples in this study.The procedure for solving the toll design problem is shown in Fig.2.5.2.Interior point algorithmThe basic idea of an interior point algorithm is to approximate constrained optimization problems by unconstrained problems.By using this type of method,the constraints are placed into the objective function via either a penalty or barrier parameter,to either penalize the vio-lation of constraints by imposing a high cost for it,or prevent the intermediate solutions from leaving the feasible region.Vanderbei and Shanno(1998)proposed an interior point algorithm for solving nonconvex nonlinear programming problems.This is an extension of the interior point algorithm for qua-dratic programming problems(Vanderbei,1994).The basic idea of this algorithm is to use NewtonÕs method tofind the feasible search direction.In the case of toll design problems,the nonlinear constraints are all equalities,and the only inequalities are the nonnegativity constraints. (If one adds the constraint that the toll on trucks should be higher than the toll on cars,it can be converted into an equality constraint by adding slack variables.)The inequalities can be elimi-nated if they are added to the objective function as barrier terms.A Lagrangian is then formulated to incorporate those equality constraints.Let g denote the system of all constraints in the toll design problem except those with simple bounds on variables.That is,g represents equation system of(11b),(11c),(11f).Note that these constraints should already be converted to equality constraints by adding slack variables.Let h represent the system of simple bounds as well as bounds on those added slack variables.Let l denote the barrier parameter,k denote the vector of Lagrangian multiplier,and Y denote the vector with all components being1.Then the Lagrangian L can be written as L¼Xu2UC uðFÞT F uÀlðlog½hðF;sÞ ÞT YÀk T gðF;sÞð30ÞBy solving the Newton system of thefirst-order KKT condition of the Lagrangian,one canfind the descent direction from the current solution,provided that a feasible starting point is used. In order tofind the step size,merit function(31)is used,in which q denotes the penalty pa-rameter.Specifically,this function can prevent the selection of a step length that will lead to the infeasibility of the subsequent solution point.u q;l¼Xu2U C uðFÞT F uÀl½log hðF;sÞ T Yþq2gðF;sÞk k22ð31ÞTests on those examples presented in this study show that this algorithm gives at least the same solution,if not better,to toll design problems,compared with LANCELOT and SNOPT.5.3.Iterative linearization methodThe iterative linearization method,proposed by Rosen(1966),essentially consists of NewtonÕs method with a convex or linear programming subproblem solved at each iteration.In this method, the nonlinear constraints are linearized using afirst-order approximation.For the deterministic based toll design problem,the only nonlinear constraint is(11b).Let C upðFÞdenote thefirst-order approximation of the cost function.For the toll design problem(11),the following subproblem is solved:。